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HEAT TRANSFER IN PROCESS ENGINEERING

ABOUT THE AUTHOR DR. EDUARDO CAO is a professor of heat transfer unit operation in the chemical engineering departments of Buenos Aires University and ITBA (Instituto Tecnologico de Buenos Aires). He also works as a consultant in the process department of Techint, one of the largest engineering groups in the world.

HEAT TRANSFER IN PROCESS ENGINEERING Eduardo Cao

New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto

Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. ISBN: 978-0-07-162613-2 MHID: 0-07-162613-1 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-162408-4, MHID: 0-07-162408-2. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. To contact a representative please e-mail us at [email protected] Information contained in this work has been obtained by The McGraw-Hill Companies, Inc. (“McGraw-Hill”) from sources believed to be reliable. However, neither McGraw-Hill nor its authors guarantee the accuracy or completeness of any information published herein, and neither McGraw-Hill nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information. This work is published with the understanding that McGraw-Hill and its authors are supplying information but are not attempting to render engineering or other professional services. If such services are required, the assistance of an appropriate professional should be sought. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and its licensors reserve all rights in and to the work. Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill’s prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms. THE WORK IS PROVIDED “AS IS.” McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise.

CONTENTS

Chapter 1. Introduction

1

Chapter 2. Heat Transfer Fundamentals

3

2-1 Concept of Unit Operations / 3 2-2 Heat and the First Law of Thermodynamics / 3 2-3 Temperature and Internal Energy / 5 2-4 Continuous and Steady-State Systems / 5 2-5 Formulation of the First Law for Continuous Systems 2-6 Thermodynamic Variables / 8 2-7 Heat Transfer Mechanisms / 9 Glossary / 13

/

6

Chapter 3. Heat Conduction in Solids

15

3-1 General Equation of Conduction in Solids / 15 3-2 Heat Conduction Through a Wall / 16 3-3 Steady-State Conduction Heat Transfer in Multidimensional Systems 3-4 Thermal Insulation / 28 Glossary / 34 References / 34

/

26

Chapter 4. Convection 4-1 4-2 4-3 4-4

Forced Convection Over a Flat Plate / 35 Forced Convection Inside Tubes / 44 Forced Convection Around Submerged Objects Natural Convection / 66 Glossary / 76 References / 77

35

/

60

Chapter 5. Fundamentals of Heat Transfer Between Fluids

79

5-1 The Enthalpy Balance / 79 5-2 Heat Transfer Area and Heat Transfer Coefficient / 81 5-3 Expression of the Overall Heat Transfer Coefficient as a Function of the Film Coefficients / 82 5-4 Mean Temperature Difference Between Two Fluids / 86 5-5 Calculation of the Overall Heat Transfer Coefficient and Pressure Drop for Double-Tube Heat Exchangers / 94 5-6 Heat Exchanger Process Specifications / 98 5-7 Design Procedure for a Double-Tube Heat Exchanger / 102 Glossary / 111 References / 113

v

vi

CONTENTS

Chapter 6. Shell-and-Tube Heat Exchangers

115

6-1 Design and Construction Standards / 115 6-2 Principal Components of a Shell-and-Tube Heat Exchanger / 116 6-3 Baffles / 117 6-4 Tubes and Tube Distribution / 122 6-5 Tubes to Tubesheets Joint / 124 6-6 Multipass Heat Exchangers / 127 6-7 Heat Exchangers with Multiple Shell Passes / 129 6-8 Front Heads (Inlet Heads) / 132 6-9 Fixed-Tubesheet and Removable-Bundle Heat Exchangers / 133 6-10 Tube Vibration / 141 6-11 Specification Sheet / 145 6-12 Design and Construction of Heat Exchangers / 145 References / 146

Chapter 7. Thermal Design of Shell-and-Tube Heat Exchangers 7-1 7-2 7-3 7-4 7-5 7-6 7-7

147

Basic Principles / 147 LMTD Correction Factors for Other Flow Configurations / 151 Basic Design Parameters of a Heat Exchanger / 157 Convection Film Coefficient and Pressure Drop in the Tube Side / 160 Heat Transfer Coefficient and Pressure Drop at the Shell Side / 164 Design and Rating of Heat Exchangers / 193 Heat Exchanger Effectiveness / 205 Glossary / 213 Glossary of the Heat Exchanger Geometry / 214 References / 215

Chapter 8. Finned Tubes

217

8-1 Double-Tube Heat Exchangers with Longitudinal Fins 8-2 Air Coolers / 231 Glossary / 251 References / 253

/

217

Chapter 9. Plate Heat Exchangers

255

9-1 Operating Principles and General Description / 255 9-2 Series-Parallel Combination / 258 9-3 Components of a Gasketed Plate Heat Exchanger / 260 9-4 Use and Limitations of Gasketed Plate Heat Exchangers / 264 9-5 Welded and Semiwelded Plate Heat Exchangers / 265 9-6 Plate Heat Exchangers versus Shell-and-Tube Heat Exchangers / 265 9-7 Typical Heat Transfer Coefficients / 266 9-8 Heat Transfer and Pressure-Drop Correlations for Plate Heat Exchangers 9-9 LMTD Correction Factor / 269 9-10 Fouling Resistances / 269 9-11 Number of Transfer Units and Specific Pressure Drop / 272 Glossary / 273 References / 274

Chapter 10. Condensation of Vapors 10-1 10-2

Mechanisms of Condensation / 275 Condensation of Single-Component Vapors

/

267

275 /

277

CONTENTS

10-3 Single-Component Vapors Condensers / 292 10-4 Desuperheater Condensers / 303 10-5 Condensation of Vapor Mixtures / 309 10-6 The Use of Steam as Process Heating Medium / Glossary / 327 References / 329

318

Chapter 11. Boiling 11-1 11-2 11-3 11-4

331

Mechanisms of Heat Transfer to Boiling Liquids Pool Boiling / 345 Flow Boiling in Tubes / 352 Reboilers / 367 Glossary / 392 References / 394

/

331

Chapter 12. Thermal Radiation 12-1 12-2

vii

395

Heat Transfer by Radiation Through Transparent Media / 395 Radiant Heat Transmission in Absorbing Media / 426 Glossary / 438 References / 438

Chapter 13. Process Fired Heaters

439

13-1 Simplified Schematic / 439 13-2 Combustion / 439 13-3 Applications of Fired Heaters / 446 13-4 Heater Types / 448 13-5 Heater Components / 452 13-6 Elements for the Design and Verification of the Capacity of a Fired Heater Glossary / 482 References / 483

/

458

Appendix A. Distillation Control Scheme / 487 Total and Partial Condenser / 488 Simulation of Distillation Columns / 488 Phase Equilibrium in a Multicomponent System Mass and Energy Balances / 490 Condensation and Boiling Curves / 492

485

/

489

Appendix B. LMTD Correction Factors for E-Shell Heat Exchangers

495

Appendix C. LMTD Correction Factors for Air Coolers

501

Appendix D. Tube Count Tables

505

viii

CONTENTS

Appendix E. Tube Layout

509

Appendix F. Fouling Factors

523

Appendix G. Typical Heat Transfer Coefficients

527

Appendix H. Dimensions of Tubes According to BWG Standard

529

Appendix I. Physical Properties of Pure Substances

531

Appendix J. Heat Exchanger Data Sheet

547

Appendix K. Unit Conversions

551

Index

555

CHAPTER 1

INTRODUCTION

This book has been written with the double purpose of being used as a textbook for university courses in unit operations and as a consulting book in the professional field of process engineering. Thus every attempt has been made to maintain a difficult equilibrium in the weights assigned to theory and practice. It is assumed that the reader has taken courses in thermodynamics and is familiar with energy balances and the calculation of physical properties in simple systems with and without change of phase. Thus only a few basic concepts considered convenient for the fluidity of the redaction were included. It is also assumed that the reader have some basic knowledge of fluid mechanics and can write mechanical energy balances and calculate pressure drops in piping systems. A difficult decision was to choose the approach to Chap. 4 on convection. In the chemical engineering curricula in many universities, unit operations courses are preceded by a course in transport phenomena. In other universities, they are not. Some knowledge of the theory of transport phenomena helps in an understanding of the principles of convective heat transfer and allows a more elegant treatment of this subject, but a treatment based on transport equations also increases the mathematical complexity and is not particularly useful in daily engineering practice. Therefore, after rewriting Chap. 4 a few times, it was finally decided to avoid presenting a theoretical approach based on transport equations so as not to drive away readers who are not friends of mathematics. Other chapters present subjects associated with the thermal design of different kinds of heat transfer equipment. Nowadays, in the professional field, this task is performed almost exclusively by commercial software. There are many programs on the market, some of them developed by important companies that have well-known heat transfer researchers working for them. Additionally, since these programs are used worldwide, many users supply important feedback that allows error corrections and fine-tuning of the correlations. Thus it is difficult to conceive of heat transfer equipment design without the help of these programs. However, most of these programs are presented as a “black box,” and the suppliers offer very little information about their content and the correlations used. This makes their use difficult. It is necessary for users to know how the program will use its input data to evaluate the importance of each input data field. For example, to calculate the boiling heat transfer coefficient, some programs use correlations based on critical properties. Other programs do not use the critical properties and calculate boiling heat transfer coefficients based on properties that are difficult to predict, such as surface tension of the boiling liquid. This means that some variables that are important in some programs are not important in others. Boiling heat transfer is probably the most complicated subject. There are limits to the maximum heatflux density that sometimes define equipment design. Prediction of these limits is ambiguous and varies according to the approach of different authors. Frequently, when comparing designs created with different software packages, one discovers important differences in the calculated heat transfer areas due to different criteria in the adoption of these limits. In some cases, the limits are incorporated as simple “rules of thumb” without any theoretical background and can be changed by the user if desired, which obviously changes the program results. Thus it is important that users have the necessary knowledge to be capable of investigating the calculation path on which the design is performed In heat transfer equipment design, many independent variables must be adopted. To simplify the use of this software by nonexperienced users, the programs usually have default values for many of the variables. There is a natural tendency to accept the first results offered by the program without investigating the effect of modifications in some of these variables. Sometimes, however, the modification of the default values allows a significant improvement in the design.

1

2

CHAPTER ONE

For example, a change in the number of tube rows or the air face velocity in an air cooler can be translated in a substantial decrease in the required heat transfer area. This is why it is very important that users know the theoretical background behind the programs to make an efficient use of them. On the other hand, the complexity and diversity of situations that may exist in the design of heat transfer equipment make it impossible to cover all the possible situations with general and simple correlations such as those presented in this book. This book includes design methods that, based on the author’s experience, allow readers to obtain reasonable results in most cases. However, it is not possible to guarantee that in certain specific situations the use of these general methods do not result in appreciable deviations in the heat transfer coefficients. Therefore, the recommendation is to use the commercial software, but doing so with enough knowledge of the subject to be able to evaluate and analyze the results. To that end, having a grasp of the simple tools presented in this book will allow process engineers to perform their own calculations and to detect the critical aspects of the design.

CHAPTER 2

HEAT TRANSFER FUNDAMENTALS

2-1 CONCEPT OF UNIT OPERATIONS A complex chemical process can be divided in elemental stages involving physical or chemical transformations. Stages where materials suffer physical changes are called unit operations. If transformations of a chemical nature are involved, they are usually called unit processes. Unit operations can modify the momentum, energy, or composition of the different phases of a system. Let us consider the system in Fig. 2-1. Tank A contains a two-component liquid mixture. Pump B circulates the liquid through heat exchanger C, where it is heated with steam. The heated liquid goes into flash vessel D, where the pressure is reduced, and part of the liquid vaporizes. The vapor, richer in the more volatile component than the liquid, is condensed in watercooled condenser E, whereas the liquid fraction exits the separator as a bottom-product stream In this simple sketch, several elemental units can be identified. They are 1. 2. 3. 4.

The pumping of the fluid by means of pump B The heat exchange in heat exchanger C The phase separation in separator D The condensation of vapor in condenser E

Each of these units is a unit operation. Unit operations normally are classified in three groups, which generally are studied separately. They are • Momentum-transfer operations • Heat transfer operations • Mass-transfer operations The first group includes all operations in which mechanical transformations are involved. They are associated with variations in system pressure, momentum, and kinetic energy due to the action of pumps or compressors or mechanical separations such as filtration, centrifugation, etc. The second group, which is the object of this book, corresponds to systems in which thermal energy is transfer to fluids, with or without changes of phase, such as heat exchange between two fluids, condensation, vaporization, etc. The third group refers to systems in which one or more components of a multicomponent system are transferred between two phases, thus achieving a change in the composition of both phases. Examples of this type of operation include absorption, distillation, solvent extraction, etc. Many times, classification into one of these three groups is not very clear, and some operations must be studied using the principles of both heat and mass transfer simultaneously, such as the condensation of multicomponent mixtures.

2-2 HEAT AND THE FIRST LAW OF THERMODYNAMICS Heat can be defined as energy that is transferred because of the existence of a temperature difference between two systems or between two parts of a system. When we speak about heat, we always refer to energy in transit. It cannot be said that a system accumulates heat. Heat is transferred to a system, and once 3

4

CHAPTER TWO

Steam E

A C

Cooling water

D

B

FIGURE 2-1

Different unit operations in a process.

it enters the system, this energy is transformed in other types of energy, such as kinetic or internal energy or mechanical work. The relationship among these types of energies is known as the first law of thermodynamics or the energy conservation law. The mathematical expression for a closed system is as follows: Q = W + ΔU + ΔEc where

(2-2-1)

Q = heat added to the system ΔU = change in internal energy ΔEc = change in kinetic energy W = work done by the system

We shall clarify a few aspects of this expression: 1. The sign convention adopted is the usual in thermodynamics, which is to consider positive the heat added to the system and the work performed by the system. 2. The action of electric or magnetic fields is not considered. However, in such cases it is possible to consider their effect by introducing the mechanical or thermal equivalent of their energy. 3. It must be noted that we didn’t include the potential energy, since we consider the weight of the object as an external force, so its action is treated as mechanical work performed by an external agent. For example, let’s consider an object in free fall. Since in this case there is no internal energy change or heat added to the system, Eq. (2-2-1) reduces to W + ΔEc = 0

(2-2-2)

W is the work performed by the force m · g (mass × acceleration due to gravity); this means that W = m · g · Δh The expression of the first law of thermodynamics for this system therefore is m · g · Δh + ΔEc = 0

(2-2-3)

The first term is what sometimes is called potential energy. 4. Usually, in process engineering, when heat terms are present, the kinetic energy terms are of lower order of magnitude and can be disregarded. The expression of the first law in this case is Q = W + ΔU

(2-2-4)

5. It must be noted that Eq. (2-2-1) may be considered a definition of the internal energy if it is written as ΔU = Q – W – ΔEc

(2-2-5)

We can see that a duality of concepts exists because we cannot define the internal energy without accepting the validity of the first law, but at the same time, to express the first law, we must postulate the existence of this type of energy. Thus the internal energy and the first law of thermodynamics constitute a single concept because they cannot be made independent of each other.

HEAT TRANSFER FUNDAMENTALS

5

2-3 TEMPERATURE AND INTERNAL ENERGY The internal energy, since it is a state function, is related to the thermodynamic parameters of the system— pressure, temperature, and composition—and can be calculated from them. When the composition of the system remains unchanged, the internal energy is a function of pressure and temperature. Usually, the changes in internal energy with pressure are not significant as long as these pressure changes do not produce changes of phase in the system, so the internal energy sometimes may be considered a function of temperature exclusively. This is why the term thermal energy is used sometimes. This expression, although not having a precise thermodynamic meaning, is both graphic and intuitive. Internal energy is stored in a system at the molecular or atomic level. This means that atoms or molecules can exist in different energy levels, which correspond, for example, to different velocities or frequencies of their movements or oscillations. Temperature is the macroscopic observable that is more directly related to these energy levels. Thus molecules or atoms with more energy are observed as more “hot.” For example, in the case of monatomic gases, the kinetic energy of the molecule is related to temperature by the expression 1 3 m ⋅u2 = k ⋅T 2 2

(2-3-1)

where m = molecular mass u = molecular velocity k = Boltzman constant T = absolute temperature Equation (2-3-1) can be explained by saying that the internal energy of the gas is stored as kinetic energy and that it can be measured through its temperature, both magnitudes being related by that expression. The energy units employed in different units systems are International system: joule (J) Technical metric system: kilocalorie (kcal) English system: British thermal unit (Btu) And their equivalence is 1 kcal = 3.968 Btu = 4183 J

2-4 CONTINUOUS AND STEADY-STATE SYSTEMS Most process plants nowadays operate in continuous mode. This means that material circulates through the system, and a steady state is achieved, in which all the operating parameters (i.e., pressure, temperature, composition, and flow rate) at any point in the system remain constant over time. For example, in the system in Fig. 2-1, from the piping inlet section up to the condenser outlet, a steady state exists, which means that at any point in the system the operating parameters do not change with time. In continuous systems, mass and energy balances are expressed per unit time, so we speak about kilograms per hour of flow rate or British thermal units per hour or joules per second (watts) of energy added to the system. The equivalences then become 1 kcal/h = 3.968 Btu/h = 1.162 W From now on, and unless specifically stated, we shall use the symbol Q to designate heat flux, this is, the heat delivered to a system per unit time.

6

CHAPTER TWO

2-5 FORMULATION OF THE FIRST LAW FOR CONTINUOUS SYSTEMS To understand how the first law of thermodynamics can be applied to a continuous system, let us consider the system presented in Fig. 2-2. This represents a portion of a system in steady state in which a fluid circulates at a flow rate W(kg/s). Within the boundaries we are considering, a continuous heat flux Q (J/s) is added, and a mechanical work per unit time Ws (J/s) is performed. In section 1 of the system, the fluid parameters—pressure, temperature, density, velocity, internal energy, and geometric height—are p1, t1, ρ1, v1, u1, and z1. In section 2 of the system, the fluid parameters will be different, and the corresponding increments are indicated in the figure. Since the fluid is in motion, if we consider the mass that in a certain instant is enclosed within the boundaries of the control volume (sections 1 and 2) after a certain time increment Δt, this mass will occupy the volume contained between sections 1' and 2'. Let Δx1 and Δx2 be the displacements of the system boundaries. Since in all the intermediate points the thermodynamic parameters remained constant, we can imagine this evolution as if m kg of the fluid, originally contained between sections 1 and 1', were moved to the volume contained between sections 2 and 2', changing its pressure, temperature, internal energy, density, kinetic energy, and height. We shall apply the first law to this evolution. The change in the internal energy of the displaced mass is m · Δu

(2-5-1)

⎛ v2 ⎞ m⋅Δ ⎜ ⎟ ⎝ 2⎠

(2-5-2)

The kinetic energy change is

ρ1 + Δ ρ p1 + Δ p z1 + Δ z u1 + Δ u v1 + Δ v

Ws = Work per unit time

2′

2

A2

Δ x2

Q = Heat per unit time 1′ 1 u1 v 1 ρ1 ρ1 z 1

Δ x1 A1

FIGURE 2-2 Application of the first law to a continuous system.

HEAT TRANSFER FUNDAMENTALS

7

To calculate the work received or performed by the system, it must be considered that in addition to the external work, there is work performed by the pressure forces acting on the boundaries of the system (inlet and outlet sections) and work performed by the gravitational force. We shall first consider the pressure forces. In the inlet section (section 1), this force equals the product of the pressure times the area over which the force is exerted, which is the area of the cross section A1. Since the boundary of the system moved from section 1 to section 1', it can be considered that this force performed an amount of work given by p1 · A1 · Δx1

(2-5-3)

The product A1 · Δx1 is the volume that the mass m occupied at the thermodynamic conditions of the inlet section. The work then can be expressed as m

p1 ρ1

(2-5-4)

This work is done on the system. In the outlet section (section 2), the system performs work against the force exerted by the pressure p1 + Δp. This work can be expressed as −m

( p1 + Δp) (ρ + Δρ)

(2-5-5)

The difference between both terms can be written as p m ⋅ Δ ⎛⎜ ⎞⎟ ⎝ ρ⎠

(2-5-6)

Regarding the work performed by the gravitational forces, again, it can be considered that the evolution suffered by the system is the translation of a mass m from position z1 to position z1 + Δz. The work is m · g · Δz

(2-5-7)

If Q is the amount of heat received or given per unit time, the heat per unit mass will be the quotient of Q and the circulating mass flow rate. And the heat received or given by the mass m will be Q m W

(2-5-8)

Similarly, the circulation work performed or received by mass m will be Ws m W

(2-5-9)

Then the expression of the first law can be written as m

Q Ws = m ⎡⎢ + gΔz + Δu + Δ ( v 2 /2) + Δ ( p/ρ) ⎤⎥ W ⎣W ⎦

(2-5-10)

or Q = Ws + W ⎡⎣ gΔz + Δu + Δ ( v 2 /2) + Δ ( p/ρ) ⎤⎦

(2-5-11)

The function u + p/ρ is the specific enthalpy i, so we get the final expression of the first law for a circulating system as Q = Ws + W ⎡⎣ gΔz + Δi + Δ( v 2 /2) ⎤⎦

(2-5-12)

8

CHAPTER TWO

In systems where heat transfer operations are performed, the terms Q and W · Δi are usually of a higher order of magnitude than the rest of the terms. For example, in the system shown in Fig. 2-1, the power delivered by the pump or the changes in hydrostatic height or fluid velocity have little influence on the temperature change experienced by the fluid. Then the energy balance can be expressed as Q = W ⋅ Δi

(2-5-13)

This is the equation we shall use in this course for the energy balance in heat transfer equipment.

2-6 THERMODYNAMIC VARIABLES The thermodynamic state of a system is defined when its state variables—pressure, temperature, and composition—are known. If these variables are defined, it is possible to calculate all the other intensive properties, such as u = specific internal energy (J/kg) i = specific enthalpy (J/kg) v = 1/ρ = specific volume (m3/kg) as well as all other properties that thermodynamics define for mathematical simplicity (e.g., entropy, free energy, etc.). All these thermodynamic properties are related by a set of relationships in partial derivatives called Maxwell equations. If the relationship between any of these intensive thermodynamic variables and the state variables (pressure, temperature, and composition) is known, then it is possible to calculate the rest of them. But this first relationship must be obtained experimentally. The thermodynamic property that is the easiest to measure is volume. If we have a function of the type v = v ( p, T , composition)

(2-6-1)

it will be possible to calculate entropies, enthalpies, and any other property. Functions of the type of Eq. (2-6-1) are called equations of state. There are some systems for which it is possible to find an equation of state that adequately represents the system volume in both the liquid and vapor phases. From this equation, a thermodynamic-mathematical package can be developed that allows the calculation of all other properties. These systems normally consist of nonpolar hydrocarbon mixtures. In other cases, when it is not possible to find a suitable equation of state, it is necessary to use other types of experimental information. Modern process-simulation programs calculate all thermodynamic properties from an equation of state or other experimental information using these mathematical relationships. However, it is usual to define some other auxiliary variables that allow manual calculations to be simplified. For example, for a system that performs an evolution without a change of state, the specific heat at constant pressure is defined as ⎛ ∂i ⎞ cP = ⎜ ⎟ ⎝ ∂T ⎠ P

(2-6-2)

or the specific heat at constant volume is defined as ⎛ ∂u ⎞ cV = ⎜ ⎟ ⎝ ∂T ⎠ V

(2-6-3)

In many constant-composition systems without change of phase, enthalpy can be considered a function of temperature exclusively (the variation with pressure is weak). It is then possible to calculate the enthalpy change between two states at different temperatures as Δi = cp ⋅ Δt

(2-6-4)

HEAT TRANSFER FUNDAMENTALS

9

(In this book we will only use the specific heat at constant pressure, so we will drop the subscript and will designate it as c.) The specific heats may be considered approximately constant within certain temperature ranges. It is then possible to put them into tables or graphs that are useful tools for hand calculations. Specific heat values for several substances can be found in the tables of App. I. Another useful auxiliary variable is the latent heat of vaporization of a pure substance λ. This is defined as the enthalpy difference between the saturated vapor and the liquid at its boiling temperature and same pressure. Latent heats of vaporization for different substances are also included in App. I.

2-7

HEAT TRANSFER MECHANISMS It is normally accepted that there are three basic heat transfer mechanisms—conduction, convection, and radiation. In what follows we shall make a brief presentation of them, and they will be examined in more detail in following chapters

2-7-1

Conduction Description of the Mechanism. When there is a temperature gradient within a substance, heat is transferred through it by molecular contact. In the case of solids, it is normally accepted that the molecular energy is associated with some form of vibration. Each molecule vibrates in a fixed position and can interchange energy with its neighbors. If heat is supplied to one part of the solid, the molecules vibrate faster. As they vibrate more, the bonds between molecules are shaken more. This passes vibration on to the next molecule and so on. Thus heat spreads through the solid in the direction of decreasing temperature. It has been observed that the heat flux is, within certain limits, proportional to the temperature variation per unit length and to the area through which the heat flows. That is, Q = − kA

dT dx

(2-7-1)

where x is the axis corresponding to the direction of the temperature variation, Q is the heat flux, A is the area perpendicular to the heat flux, and k is a proportionality constant. In this expression, the negative sign means that the direction of the heat flow is the direction in which temperature decreases. C In the rest of the book we will work frequently with heat-flux denTA sities, meaning heat transferred per unit time and per unit area. The TB heat flux density will be designated as q. Thus it will be q = −k

dT dx

(2-7-2) A

In the case of gases, the model that explains conduction heat transmission is quite different from that for solids. Gas molecules move with a velocity related to their temperature [see Eq. (2-3-1)]. Let’s assume that we have a stagnant gas enclosed within two walls, A and B, at different temperatures, with TA > TB (Fig. 2-3). Let C be an imaginary planar surface located between both walls. This planar surface is crossed in both directions by gas molecules. If the system pressure is constant, the same number of molecules crosses the surface in one direction as the other. Then, at this surface, the average velocity of all the molecules is zero. This means that the gas is stagnant even though the molecules move individually.

Equimolar flows in both directions

B

Net heat flow FIGURE 2-3 Thermal conductivity in gases can be explained by molecular transport of energy.

10

CHAPTER TWO

However, since the molecules crossing the surface from left to right come from a higher-temperature region, they have more energy than those crossing the surface from right to left. The result is a net energy flux through the surface. The higher the thermal gradient, the higher will be the heat flux. Therefore, again, q = −k

dT dx

By means of the kinetic theory, it is possible to calculate a value for the proportionality constant, which is a function of the temperature and the nature of the gas. Note Actually, it is not possible to have a completely stagnant fluid in a heat transfer situation. The temperature differences always create density differences that provoke fluid motion. The movement of a mass of fluid between regions of different temperature provides an additional mechanism for energy transport. The “pure conduction” model we have just described for the gas is an ideal situation for a case in which the distance between the walls is very small. In the case of liquids, the model is more complex, and we shall not try to explain it. But it can be seen that although the phenomenologic descriptions of these mechanisms look very different, the final expression that allows the calculation of the heat flux is the same. Thus, even though the mechanisms are different for solids, gases, and liquids, the fact that proportionality between the heat flux and the thermal gradient can be established makes possible that all these mechanisms can be grouped together under the common designation of conduction. The mathematical expression of Eq. (2-7-2) then is the only possible definition of this mechanism. Thus we can say that conduction is a heat transfer mechanism whose characteristic is a proportionality between the heat flux and the thermal gradient and which, at the microscopic level, can be explained with different models depending on whether the conduction takes place in a gas, a liquid, or a solid. The proportionality coefficient in Eq. (2-7-2) is called thermal conductivity. As it will be explained later, in the case of fibrous or porous materials, the thermal conductivity must include many effects, such as conduction through the solid fibers, convective effects in the air separating the fibers, and radiation heat transfer between the fibers. What we really do is transfer to the proportionality constant our inability to explain these phenomena with more rigor. Maybe someday we will have a better knowledge of the nature of matter, and what today is generically called “conduction heat transfer” will be studied with more specific models in each case. The thermal conductivity units in the different systems are 1 kcal/(h · m · °C) = 0.671 Btu/(h · ft · °F) = 1.162 W/(m · K) Some Characteristics of the Mechanism of Conduction 1. For conduction to occur, the presence of matter is necessary. We shall see that there is another mechanism, radiation that allows the propagation of heat through a vacuum. 2. This mechanism is restricted to heat propagation through stagnant matter. In the case of fluids in motion, the heat transfer mechanism is convection. 3. Conduction is the only possible heat transfer mechanism in opaque solids. 2-7-2

Convection Description of the Mechanism. Convection is a heat transfer mechanism present in fluids in motion. Let’s consider a solid wall at temperature Tw in contact with a moving fluid at temperature Tf (Fig. 2-4). We can see that even though we say that the fluid temperature is Tf , there is a small thickness layer of fluid in which the temperature varies continuously from Tw to Tf . This is known as thermal boundary layer. Owing to the existence of this temperature gradient, heat will flow from the solid wall into the fluid. Let’s assume that the fluid is in motion, with velocity in the direction parallel to the solid wall. It is possible to observe experimentally that a velocity profile, as shown in the right side of the figure, appears

HEAT TRANSFER FUNDAMENTALS

Tf Fluid

Velocity profile V = V(y)

Temperature profile T = T(y)

11

y

A

Tw FIGURE 2-4

Solid

x

Boundary layer and velocity distribution in a fluid flowing over a flat plate.

in the system. This graph shows the fluid velocity at any point as a function of the distance to the wall. The fluid velocity is zero at the solid-fluid interface and reaches a constant value beyond a certain distance from the solid. This region of variable fluid velocity is called the velocity boundary layer. In this model, we are assuming that the fluid velocity has only a component in the direction parallel to the solid surface and that the streamlines are straight lines. This is known as laminar movement, and it takes place only at low fluid velocities. If the fluid velocity increases, eddies or turbulences appear in the fluid, provoking erratic fluid movements in other directions. Since the velocity in a region immediately adjacent to the solid wall is nil, the fluid in this region behaves like a solid, and the heat flows into it by conduction. Then, ⎛ dT ⎞ q = −k ⎜ ⎟ ⎝ dy ⎠ y = 0

(2-7-3)

Let’s consider an imaginary volume element located at a certain distance from the wall, at position A in Fig. 2-4. This situation is shown enlarged in Fig. 2-5. Heat enters the volume element through the lower side by conduction and exits through the upper side. If conduction were the only heat transfer mechanism, both heat flows would be equal. However, the fluid that is entering into this volume absorbs part of the energy and leaves at a higher temperature. Thus the heat flux through the lower and upper sides of the volume must be different.

Conduction heat outlet

Cold fluid inlet

Hot fluid outlet

Conduction heat inlet FIGURE 2- 5

Heat balance on a volume element within the boundary layer.

12

CHAPTER TWO

Therefore, the dT/dy values also will be different. This means that the fluid movement distorts the temperature profile. Then, even though Eq. (2-7-3) is absolutely general, the value of the derivative is affected by the hydrodynamics of the system. For simple geometries and laminar flow, it is possible to obtain an expression for the temperature distribution function analytically, and then the heat flux can be calculated by means of Eq. (2-7-3). However, laminar flow is seldom found in process engineering, and the most common situation is that dT/dy cannot be calculated. It can be observed experimentally that within a certain range of system parameters, the heat flow is proportional to the temperature difference between the fluid and the wall. Thus q = h(Tw − T f )

(2-7-4)

where h is a constant of proportionality called the convection film coefficient. This coefficient is a function of system geometry, the properties of the fluid, and fluid velocity. The relationship between h and the system parameters is usually found experimentally. As a general rule, the higher the fluid velocity, the higher is the convection film coefficient. This is due to the fact that the fluid streams are an additional mechanism for heat removal, the magnitude of which increases with the increase in the velocity of fluid renewal over the surface. The units of h in the several units systems are 1 kcal/(h · m2 · °C) = 0.205 Btu/(h · ft2 · °F) = 1.162 w/(m2 · K) It is important to emphasize that the heat delivered by a solid to a fluid enters the fluid initially by conduction, and then the fluid streams absorb this energy and transport it into and through the fluid. It does not make sense trying to distinguish between heat transferred by conduction and heat transferred by convection. It is the same energy that is transported by the combined action of fluid movement and thermal conductivity of the fluid. The global phenomenon is expressed by Eq. (2-7-4), which allows calculation of the total heat flux through the interface. Natural and Forced Convection. Fluid movement may be caused by the application of external pressure gradients, for example, by the action of pumps or compressors. It is said that in this case the convection is forced, meaning that the velocity profile is imposed externally on the system independent of the heat transfer. However, in the absence of fluid movers, if heat is applied to a fluid, the temperature gradients provoke density differences that cause the fluid to move. This movement is the result of heat flux and is called natural convection. This means that every time heat is transferred to or from a fluid, the mechanism will be convection, either natural or forced.

2-7-3 Thermal Radiation Explanation of the Mechanism. Radiation is probably the primary heat transfer mechanism between molecules. In contrast to the other mechanisms, the heat source does not need a physical medium to transfer its energy. Energy can be transmitted from an emitting to a receiving body through a vacuum. This is why it is possible to receive solar energy on the earth. Radiation is propagated by electromagnetic waves. The emission of electromagnetic energy is not continuous. It is emitted as a multiple of a minimum amount of energy called a quantum or photon. This means that the body emits a certain number of quantums. The energy of the quantum is related to the wavelength of the emitted radiation. To emit energy, a body must have been excited previously from the exterior. Owing to the natural tendency of things to return to equilibrium, the body emits energy so as to return to its original state. This decrease in the energy of the body corresponds to the radiation emitted. The emission takes place by the same mechanisms by which the energy was absorbed. The change in the energy level of the molecules may be related to an increase in the rotation or vibrational movements of its atoms or electrons as they move to higher-energy orbits or even to changes in the energy of the atomic nucleus. These different mechanisms

HEAT TRANSFER FUNDAMENTALS

13

produce radiation of very different wavelengths. The electromagnetic spectrum ranges from wavelengths of 10–11 cm (cosmic rays) to 105 cm ( low frequency radio waves). There is a small portion of this spectrum in which the emission and absorption of energy produce changes in temperature of the related bodies. This is the region between 0.5- and 50-μm wavelengths. This is called thermal radiation. Any body at an absolute temperature T emits an amount of energy within this wavelength range that is proportional to the fourth power of its absolute temperature. This is q = σ ⋅ε ⋅T 4

(2-7-5)

where q = energy emitted by the body per unit time and per unit area σ = Boltzman’s constant = 4.88 × 10−8 kcal/(m2 · h · K4) = 5.672 × 10−8 W/(m2 · K4) ε = emissivity of the body (This is a parameter that is always less than 1 and depends on the type and conditions of the surface.) Later in this text we shall see that when a body at temperature TA is in an infinite ambient at temperature TB, it exchanges heat with the surroundings at a rate that can be expressed as Q = σ ⋅ ε ⋅ A(TA4 − TB4 )

(2-7-6)

Some Characteristics of the Radiation Mechanism 1. The mechanism of heat transfer by radiation is superimposed on that of heat transfer by convection. Both mechanisms can be considered to be in parallel. Thus the energy exchanged between a body and the ambient that surrounds it can be expressed as Q = σ ⋅ ε ⋅ A(TA4 − TB4 ) + h ⋅ A(TA − TB )

(2-7-7)

where TA is the temperature of the body and TB is the temperature of the surroundings. 2. We see that as a difference between conduction and convection, the heat exchanged does not depend on the temperature difference but on the fourth power of the temperatures. This mechanism thus is particularly important in systems operating at high temperatures, such as process furnaces or steam generators. On the other hand, in low-temperature systems, sometimes it can be neglected, unless the convection film coefficient is also particularly low, as in natural convection systems in the air.

2-7-4 Other Heat Transfer Mechanisms Even though it is commonly accepted that conduction, convection, and radiation are the primary heat transfer mechanisms, there are some more complex situations that must be studied separately and that could be considered in the same category as these three. These are cases where a fluid experiences a change of state, such as boiling and condensation. In these cases, the heat transfer rate is associated with the velocity by which the change of state takes place, which frequently is related to the kinetics of a mass-transfer process or requires more complex models to be predicted. These mechanisms will be studied in specific later chapters.

GLOSSARY Ec = kinetic energy (J) g = acceleration of gravity (m/s2) h = convection film coefficient [W/(m2 · K)]

14

CHAPTER TWO

h = height (m) i = specific enthalpy (J/kg) k = thermal conductivity [W/(m · K)] k = Boltzman’s constant m = mass (kg) Q = heat (J) or heat flow (W) q = heat flux density (W/m2) T = temperature (K) u = molecular velocity (m/s) u = specific internal energy (J/kg) U = internal energy (J) v = velocity (m/s) v = specific volume (m3/kg) W = mass flow (kg/s) Ws = work per unit mass (J/kg) x, y, z = coordinates—distance (m) σ = Boltzman’s constant ε = emissivity Subscripts w = wall f = fluid

CHAPTER 3

HEAT CONDUCTION IN SOLIDS

Conduction is a heat transfer mechanism resulting from the energy exchange between neighbor molecules as a consequence of molecular contact. In conduction heat transfer, atoms or molecules interact by elastic and inelastic collisions to propagate the energy from regions of higher temperature to regions of lower temperature. Conduction is the only possible heat transfer mechanism in opaque solids. In liquids and gases, pure conduction cannot exist. The fluid motion always offers a vehicle for energy transport, and the transfer of heat is a combination of both effects generically designated as convection. This is why, in the present chapter, we shall study only heat conduction within solids. Only steady-state systems will be studied. For conduction of heat in nonstationary systems, see ref. 2.

3-1 GENERAL EQUATION OF CONDUCTION IN SOLIDS In Chap. 2, we presented the equation of heat conduction in one direction as dT (3-1-1) dx This is a simplified expression. In a general sense, the conduction heat-flux density can be represented by a vector q, which direction is the principal direction of the propagation of heat in the solid medium. This direction coincides with the direction of the temperature-gradient vector ∇T . The temperature gradient, in turn, is a vector whose direction is that of the greatest temperature variation per unit length and whose module is the value of this derivative. In a Cartesian coordinates system, the components of the gradient of a scalar field T in the coordinate axes are the partial derivatives of T with respect to each coordinate. This is qx = − k

∂T ∇T x = ∂x

∂T ∇T y = ∂y

∂T ∇T x = ∂z

(3-1-2)

And the most general conduction equation is q = − k ∇T

(3-1-3)

This vector equation can be decomposed in the following three scalar equations: qx = − k

∂T ∂x

qy = − k

∂T ∂y

qz = − k

∂T ∂z

(3-1-4)

However, in most cases it is possible to choose the coordinates axes so that the direction of one of them coincides with the direction of the gradient vector. For example, in the case of an infinite planar wall whose faces are kept at different temperatures (as in Fig. 3-2), it is evident that the gradient direction is perpendicular to the wall. Then, if we choose the z-axis coincident with that direction, it will be ∂T ∂T = =0 ∂x ∂y

and

qz = − k

dT dz

(3-1-5)

15

16

CHAPTER THREE

z A(x, y, z) or A(r, θ, z)

y

FIGURE 3-1

r

θ

x

Rectangular and cylindrical coordinates.

In a cylindrical system, the coordinates of any point are r, θ, and z. In Fig. 3-1, the coordinates of a point A are shown in both systems. The components of the gradient vector in cylindrical coordinates are ∇Tr =

∂T ∂r

∇Tθ =

1 ∂T r ∂θ

∇Tz =

∂T ∂z

(3-1-6)

The most usual case in process heat transfer involving cylindrical geometry is heat conduction through the wall of a cylindrical pipe. In this case, the direction of the vector q is radial. Then the partial derivatives with respect to θ and z are zero, and it is qr = −k

3-2

dT dr

(3-1-7)

HEAT CONDUCTION THROUGH A WALL

3-2-1 Case of Planar Geometry Let’s consider a planar wall of thickness L and infinite dimensions in the other two directions (Fig. 3-2). On each face of the wall, a constant temperature is maintained. Let TA be the temperature of the left face and TB be the temperature of the right face. If we make the z axis coincide with the direction normal to the wall, the temperatures will be T = TA T = TB

at z = 0 at z = L

The direction of the vector q will be perpendicular to the wall, so it will only have a component in the z axis.

HEAT CONDUCTION IN SOLIDS

17

Temperature profile T = T(z)

T = TA

T = TB

z=0

z=L

FIGURE 3-2 Temperature profile in a homogeneous planar wall in steady state.

Since we are dealing with a steady state, the amount of heat per unit time crossing any plane normal to z is the same. Then qz is constant with changes in z. Thus we have dqz =0 dz Then

d 2T =0 dz 2

and

(3-2-1) dT = C1 dz

Therefore, finally, T = C1z + C2

(3-2-2)

The integration constants are evaluated with the boundary conditions, and this results in C1 = (TB − TA )/L C2 = TA The temperature profile is a straight line, as shown in Fig. 3-2. Then q = k (TB − TA )/L

(3-2-3)

Composite Walls. In the case of a solid wall composed of two layers of different materials I and II, as shown in Fig. 3-3, the equation deduced for the simple wall can be applied to each material. Then, if Ti is the temperature of the plane separating both layers, qI = kI (T1 − Ti ) / Δz I = constant = CI

(3-2-4)

qII = kII (Ti − T2 ) /Δz II = constant = CII

(3-2-5)

Since in the plane of union q is the same for both materials, CI = CII = q, and then q

ΔzI = T1 − Ti kI

q

Δ z II = Ti − T2 kII

(3-2-6)

18

CHAPTER THREE

T

T = T1

T = Ti

T = T2

Δz I

Δz II

FIGURE 3-3 Temperature profile in a composite planar wall in steady-state heat transfer.

Adding both equations ⎛ Δz Δz ⎞ q ⎜ I + II ⎟ = T1 − T2 kII ⎠ ⎝ kI

(3-2-7)

q = U (T1 − T2 )

(3-2-8)

1 Δz I Δz II = + U kI kII

(3-2-9)

this can be written

where

If the wall is composed of more than two materials, the general expression is Δz 1 =∑ i U ki

(3-2-10)

U is called the overall heat transfer coefficient, and its utility lies on the fact that it allows a direct calculation of the heat flux without calculating the temperatures at the intermediate planes. Heat Transfer Through a Wall Separating Two Fluids. In the preceding case we assumed that the temperatures of both faces of the wall were known. However, the usual case is that these temperatures are unknown. Let’s consider the case of a composite solid wall separating two fluids at different temperatures. For example, it could be the wall of a vessel containing a warm liquid. The wall may be composed of a steel plate with a mineral wool thermal insulation layer. The temperature of the contained liquid is Ti , whereas the temperature of the exterior air is To. Heat will flow from the liquid to the exterior through the composite solid wall (Fig. 3-4). The heat-flux density qz will be constant through the wall. Heat is transferred from the liquid to the internal wall by convection. For heat transfer to exist, a temperature difference between the bulk of the liquid and the solid wall is necessary. We will call T1 the interior wall temperature, which, as we can see in the figure, is lower than Ti . Then q = hi (Ti − T1 ) where hi is the convection heat transfer coefficient.

(3-2-11)

HEAT CONDUCTION IN SOLIDS

19

Ti

T1

T2

Liquid

T3

Air

Δz II

Δz I

To

Steel

Mineral wool

FIGURE 3-4 Temperature profile in a wall separating two fluids in steadystate heat transfer.

Since heat must flow by conduction through the steel and the mineral wool, a temperature gradient must exist in both materials. Let T2 be the temperature of the separation plane between the steel and the insulation, and let T3 be the temperature of the insulation external face. We then can write q=

kI (T1 − T2 ) Δz1

(3-2-12)

q=

kII (T2 − T3 ) Δz2

(3-2-13)

From the external face of the insulation to the ambient air, heat is transferred by a combined convectionradiation mechanism, so q = ho (T3 − To )

(3-2-14)

where ho is the combined convection-radiation coefficient. Since the heat-flux density is constant through the wall, we can isolate the temperature differences in Eqs. (3-2-11) through (3-2-14), and this results in Ti − T1 =

q hi

(3-2-15a)

T1 − T2 = q

Δz I kI

(3-2-15b)

T2 − T3 = q

Δz II kII

(3-2-15c)

T3 − To =

q ho

(3-2-15d)

20

CHAPTER THREE

Adding member to member these equations, the intermediate temperatures cancel, and the result is ⎛ 1 1 Δz Δz ⎞ Ti − To = q ⎜ + + 1 + 2 ⎟ k II ⎠ ⎝ hi ho k I

(3-2-16)

The term within parentheses is the reciprocal of the overall heat transfer coefficient, so generalizing, q = U (Ti − To ) ⎛ Δz 1 1⎞ U = ⎜∑ 1 + + ⎟ kI hi ho ⎠ ⎝

and

(3-2-17) −1

(3-2-18)

Controlling Resistance. It is useful to analyze Eqs. (3-2-17) and (3-2-18) making an analogy with an electric circuit in which the heat flux is analogous to the electric current, the temperature difference acts as potential difference, and U is the reciprocal of the resistance. Then each term within the parentheses can be considered as a partial resistance representing any of the solid-fluid interfaces or any material layer. All these resistance are in series, and the total resistance is the sum of the partial resistances. Frequently, one of these resistances is much smaller than the others and can be neglected without causing an appreciable error in the calculations. For example, the resistance of the metallic wall of a vessel is usually very small because the thickness is small and the thermal conductivity is very high (metals are good heat conductors). When one of the partial resistances is much larger than the others, the total resistance is similar to it, which is then called the controlling resistance. For example, when a liquid container is insulated to avoid heat losses, low-thermal-conductivity materials are employed, and most of the total resistance and most of the temperature drop take place in the insulation layer. Example 3-1 A vessel contains a liquid at 160°C and is exposed to air at 20°C. The thickness of the steel wall is 10 mm. Calculate the heat loss and the temperature profile through the system if the vessel is insulated with mineral wool [thermal conductivity 0.072 W/(m · K)] with the following thicknesses: a. 25 mm b. 38 mm Assume that the convection heat transfer coefficient between the liquid and the metallic wall is 500 W/(m 2 · K) and that the combined convection-radiation coefficient between the insulation and the ambient air is 5 W/(m 2 · K). The thermal conductivity of steel is 43 W/(m · K). First of all, we must clarify that the convection heat transfer coefficient between the insulated wall and the ambient air is a function of the external temperature of the wall (as we shall see in Chap. 4). To assume this value as constant is a simplification that in this case is acceptable because between both cases this temperature will not change significantly. Therefore, referring to Fig. 3-4,

Solution

a. For 25-mm thickness, U=

1 1 = 1.82 W/(m 2 ⋅ K) = Δzi 1 1 0.01 0.025 1 1 ∑ k + h + h 43 + 0.072 + 500 + 5 i i o

q = U ΔT = 1.82(160 − 20) = 254.8 W/m 2 The temperature drop between the liquid and the internal wall (see figure) is Ti – T = q/hi = 254.8/500 = 0.51°C

HEAT CONDUCTION IN SOLIDS

21

T1 = 160 – 0.51 = 159.5°C

Then

The temperature drop through the metal is T1 – T2 = q/(k1/Δz1) = 254.8/(43/0.01) = 0.059°C T2 = T1 – 0.059 = 159.4°C

Then

The temperature drop in the insulation is T2 – T3 = q/(k2/Δz2) = 254.8/(0.072/0.025) = 88.47°C T3 = T2 – 88.47= 70.92°C

Then

Finally, the temperature drop between the insulation and the external air is 70.92 – 20 = 50.92°C (which must coincide with q/ho = 254.8/5 within rounding errors). It must be noted that practically there is no temperature drop between the liquid and the external face of the steel wall because the two involved resistances are very small. The highest temperature drop, as may be expected, is in the insulation layer; even so, an important temperature drop also takes place between the external face of the insulation and the air. The temperature of the external face of the insulation is 70.92°C. This is too high and may be objectionable from a safety point of view because it may provoke burns if a person touches the external wall. (Normally, temperatures higher than 60°C are not accepted in surfaces that may be in contact with plant operators.) One way to reduce this temperature is to increase the insulation thickness, as we shall see. b. For 38-mm thickness, U=

1 1 = = 1.37 W/(m 2 ⋅ K) Δzi 1 1 0.01 0.038 1 1 ∑ k + h + h 43 + 0.072 + 500 + 5 i i o

q = U ΔT = 1.37(160 − 20) = 191.8 W/m 2 The temperature drop between the liquid and the internal wall is Ti − T1 = q/hi = 191.8/500 = 0.38°C Then

T1 = 160 − 0.38 = 159.6°C

The temperature drop in the metal is T1 − T2 = q/(k1/Δz1) = 191.8/(43/0.01) = 0.044°C Then

T2 = T1 − 0.044 = 159.55°C

The temperature drop in the insulation layer is T2 − T3 = q/(k2/Δz2) = 191.8/(0.072/0.038) = 101.2°C

22

CHAPTER THREE

T3 = T2 − 101.2 = 58.3°C

Then

It can be noted that there is now a larger temperature drop in the insulation owing to the higher thickness.

3-2-2

Cylindrical Geometries The other important case in process engineering is that of cylindrical geometry. For example, let’s consider the case of a cylindrical tube with a warm internal fluid flowing inside (Fig. 3-5). The tube is exposed to air at a lower temperature, so there is a loss of heat through the tube wall. The internal wall of the tube is at a temperature Ti, and the external wall is at a temperature To. Let Ri and Ro be the internal and external radii. In steady state, the amount of heat flowing through any concentric surface must be the same. This means that Q = 2π rqr = constant

(3-2-19)

rqr = C1

(3-2-20)

C1 r

(3-2-21)

Then

or qr = Since dT dr

(3-2-22)

dT C1 = dr r

(3-2-23)

qr = − k we get −k

Integrating this differential equation with the following boundary conditions, For r = Ri → T = Ti and for r = Ro → T = To, the result is − k (Ti − To ) = C1 ln

Ri Ro

To Ro Ri

FIGURE 3-5

Ti

Heat conduction through a cylindrical wall.

(3-2-24)

HEAT CONDUCTION IN SOLIDS

Thus

C1 =

Then

qr =

k (Ti − To ) R ln o Ri

23

(3-2-25)

k (Ti − To ) r ln Ro Ri

(3-2-26)

where qr is the heat-flux density through a concentric surface radius r. The product of this heat-flux density times the flow area is the total heat flow through that surface. This is Q = 2π Lrqr

(3-2-27)

And from Eq. (3-2-26), we get Q = 2π L k

Ti − To R ln o Ri

(3-2-28)

This equation allows calculating the heat flow through the tube wall. Composite Cylindrical Walls. As in the case of the composite planar wall, we can assume that we have a cylindrical wall composed of two different materials, for example, a metallic tube with an insulating layer (Fig. 3-6). Here again, we can consider that instead of the internal and external wall temperatures, we know the bulk temperatures of the interior fluid Tf and the external air To. Air

T3 T2 Liquid Tf

T1

Steel Insulation R1 R2 R3 FIGURE 3-6

Composite cylindrical wall.

24

CHAPTER THREE

Let’s call T1 the internal wall temperature, T2 the temperature of the metal-insulation separation surface, and T3 the temperature of the external insulation face. The convection heat transfer from the internal fluid to the tube can be expressed as Q = 2 πLR1hi (T f − T1 )

(3-2-29)

where hi is the convection coefficient. For both solid media, Eq. (3-2-28) is valid, so we can write Q = 2 πLk1

Q = 2 πLk2

T1 − T2 R ln 2 R1 T2 − T3 R ln 3 R2

(3-2-30)

(3-2-31)

The same heat flow transfers from the external insulation face to the air by a combined convection-radiation mechanism. This can be expressed as Q = 2 πR3 Lho (T3 − To )

(3-2-32)

In Eqs. (3-2-29) through (3-2-32), we can isolate the temperature differences and add the equations so that the intermediate temperatures are canceled and we get 1 ln( R2 /R1 ) ln( R3 /R2 ) ⎤ Q ⎡ 1 + + + ⎢ ⎥ = (T f − To ) 2 πL ⎣ hi R1 ho R3 k1 k2 ⎦

(3-2-33)

If we multiply and divide by R3, we get 1 Q = (T f − To ) 2 πLR3 U o

(3-2-34)

where 1 ⎡ 1 1 R ln( R2 /R1 ) R3 ln( R3 /R2 ) ⎤ =⎢ + + 3 + ⎥ U o ⎣ hi ( R1 /R3 ) ho k2 k1 ⎦

(3-2-35)

where Uo is the overall heat transfer coefficient based on the external surface. Example 3-2 A 3-in Schedule 40 pipe (Di = 77.9 mm, Do = 88.9 mm) conducting gas oil at 190°C is insulated with 25 mm of mineral wool [conductivity 0.05 W/(m · K)]. The combined radiationconvection heat transfer coefficient for the external wall is estimated as 10 W/(m2 · K). Calculate the heat loss per meter of pipe and the temperatures of the external face of the insulation and of the metal wall, assuming that the film convection coefficient for the internal side is 200 W/(m2 · K) and the ambient temperature is 20°C. The metal conductivity is 80 W/(m · K). Solution

With reference to Fig. 3-6,

R1 = 38.95 mm R2 = 44.45 mm R3 = 69.45 mm hi = 200 W/(m2 · K)

ho = 10 W/(m2 · K) k1 = 80 W/(m · K) k2 = 0.05 W/(m · K)

HEAT CONDUCTION IN SOLIDS

25

We shall calculate the individual resistances 1 1 = = 0.0089 m 2 ⋅ K/W hi ( R1 /R3 ) 200 x (38.95 / 69.45) 1 1 = = 0.1 m 2 ⋅ K /W ho 10 R3 ln( R2 /R1 ) 0.06945 ln(0.04445 / 0.03895) = = 1.14 × 10 −4 m 2 ⋅ K/W k1 80 R3 ln( R3 /R2 ) 0.06945 ln(0.06945 / 0.04445) = 0.61 m 2 ⋅ K/W = k2 0.05 From an analysis of the results, we can see that 1. The resistance corresponding to the metal wall is clearly negligible in comparison with the other terms, so it is not necessary to include it in the calculations. This means that there will be no temperature drop through the metal, and we can consider T1 = T2. 2. The term corresponding to the internal liquid-solid resistance is also two orders of magnitude lower than the insulation resistance. It must be considered that a value of hi = 200 W/(m2 · K) is quite low for liquids flowing inside pipes, and it could correspond to a very viscous fluid or a fluid circulating at a very low velocity. Then, in the majority of actual cases, the relative importance of this term will be even lower. This explains the usual practice of also neglecting this term when dealing with insulated pipes and to consider that the metal temperature coincides with the inner fluid temperature. 3. The terms corresponding to the insulation resistance and to the exterior convection-radiation resistance are of the same order of magnitude, and both should be considered. The overall heat transfer resistance then is 1 = 0.0089 + 0.1 + 0.61 = 0.7189 m 2 ⋅ K/W U And the heat loss per meter of tube is Q = 2 πR3U (T f − Ta ) =

2 π ⋅ 0.06945(190 − 20) = 103 W/m 0.7189

The temperature drops can be obtained by multiplying the heat flow times the corresponding resistances, that is, ΔTi , i +1 =

103 Q Ri , i +1 = Ri , i +1 = 236 Ri , i +1 2πLR3 2π ⋅ 0.06945

T f − T1 = 236 × 0.0089 = 2.1 K T1 − T3 = 236 × 0.61 = 143.9 K T3 − Ta = 236 × 0.1 = 23.6 K Then T3 = 43.6°C and T1 = 187.9°C.

26

CHAPTER THREE

3-3 STEADY-STATE CONDUCTION HEAT TRANSFER IN MULTIDIMENSIONAL SYSTEMS Up to now, we have considered cases in which the direction of the vector representing the heat flux is known beforehand, and it is possible to make this direction coincide with one of the coordinate axes. In these cases, the vector equation Eq. (3-1-3) reduces to a simple differential equation that can be integrated as in the analyzed cases. However, there are situations in which this procedure is not possible. For example, let’s consider a buried pipeline with a hot fluid circulating into it (Fig. 3-7). We want to calculate its heat loss. If we write the conduction equation in cylindrical coordinates for the medium in which the pipeline is buried, we get 1 ∂(rqr ) 1 ∂qθ ∂qz + =0 + r ∂r r ∂θ ∂z

(3-3-1)

Assuming that there is no heat flow in the axial direction, we get 1 ∂(rqr ) 1 ∂qθ =0 + r ∂r r ∂θ

(3-3-2)

The boundary conditions for the integration of this differential equation can be T = To on the pipeline surface T = TA on the surface of the ground It can be appreciated in Fig. 3-7 that it is not possible to make the direction of heat flow coincide with the direction of any of the coordinate axes, and Eq. (3-3-2) must be integrated as a differential equation in partial derivatives. The analytical treatment is quite complex, so this type of problem normally is handled with numerical or graphic methods. However, in bidimensional systems, where only two limit temperatures are involved, it is possible to define a conduction shape factor S as Q = S (T1 − T2 )

(3-3-3)

Surface at T = TA

Surface at T = To

Direction of heat flow FIGURE 3-7

Heat flow from a buried pipeline.

HEAT CONDUCTION IN SOLIDS T2

T2 d r

T1

r

S = 4.π.k.r 1– r 2d

T1

S = 2.π.r.k

Hemispherical cavity

Sphere buried in a solid

Insulation

T2

T2

d T1

S = 4.r.k if d = 0 S = 8.r.k if d >>r

T1

L

S=

2r

T2

2.π.L.k

S= r T 1

2.π.L.k ln 2L r

2r Vertical cylinder in a semi-infinite solid

Disc buried in a solid

d

27

d ln ( r +

T1

( d )2 – 1) r

T2

r1

(

e Pipe buried in a solid

2.π.L.k r2 S = 2–r 2–r 2 e 1 2 cosh–1 2.r 1.r2

)

Conduction heat transfer between two parallel pipes buried in a solid T2 T1

d

a

S= b

6 2a ) L.k 3.5 d ln ( 1/4 3/4 ) b a

(5.7 +

Rectangular duct buried in a solid

FIGURE 3-8

Shape factor S for usual geometries.

S values for the usual geometries are included in Fig. 3-8 with the corresponding values of T1 and T2. For other geometries, ref. 1 can be consulted. Example 3-3 A 6-in underground pipeline (Dext = 0.168 m) transports crude oil at 45°C. The centerline of the pipe is 1.5 m below grade. The conductivity of the soil is 0.9 W/(m · K). Assuming that the ground surface is at 0°C, calculate the heat loss per kilometer. If the mass flow is 120000 kg/h (33.33 kg/s) and its specific heat is 2500 J/(kg · K), what will be the temperature drop of the oil? The thermal conductivity of a soil depends on many factors, with the humidity being one of the most important. The following table indicates some typical values:

Solution

Typical Soil Conductivity [W/(m · K)] Average soil Dry sand Sand with 8% water Humid clay

0.9 0.3 0.6 2.5

28

CHAPTER THREE

Let’s assume that the convection heat resistance between the oil and the pipe wall is negligible, so it is possible to assume that the external temperature of the pipe is 45°C. If kg is the soil conductivity, Tc and Tg are the pipe and ground temperatures, and d is the depth at which the pipeline is buried, the heat loss will be S=

2 π Lkg ⎡d d 2 ⎤ ln ⎢ + ⎛ ⎞ − 1 ⎥ ⎝r⎠ ⎥⎦ ⎢⎣ r

=

2 π L × 0.9 = 1.58 L 1.5 ⎡ + 17.852 − 1 ⎤⎥ ln ⎢ ⎣ 0.084 ⎦

Q = 1.58(T1 − T2 ) = 1.58(45 − 0) = 71.1 W/m = 71.1 kW/km L If W is the oil mass flow and c is its specific heat, the temperature change of the oil per kilometre will be ΔT =

Q 71,100 = = 0.85ºC Wc 33.33 × 2500

In the case of crude oils, it is necessary to avoid too high a temperature drop because the increase in viscosity may require an uneconomical pumping power. If, for example, it is defined that the minimum pumping temperature is 30°C, it would be necessary to install a crude heater every 17 km to raise the oil temperature again to 45°C.

3-4 THERMAL INSULATION When a piece of equipment or a pipe operates at temperatures sensibly different from ambient temperature, it is usually necessary to thermally insulate it. Thermal insulation may be necessary for the following reasons: 1. For energy savings. The energy lost to the ambient needs to be supplied at the expense of a higher fuel or electricity consumption. In these cases, the cost of a higher insulation thickness must be economically balanced again the higher fuel consumption. 2. To reduce the external temperature of equipment and piping to avoid burns to the operating personnel. This is called personal-protection thermal insulation, and the insulation thickness is calculated so as to have external surface temperatures lower than 55 or 60°C. 3. For process reasons. In some cases it is necessary to keep process fluids at certain minimum temperatures to avoid undesirable reactions, solidification, precipitation, polymerization, etc. In this case, insulation thicknesses result from process calculations. 4. In low temperature processes to avoid condensation of ambient humidity over the surface of piping and equipment. This can lead to ice formation, with water weeping, turning the soil muddy or slippery.

3-4-1 Characteristics of a Good Thermal Insulation System A good thermal insulation system must meet the following conditions: 1. The materials employed must be compatible with the process and must be inert when the process fluids come in contact with them. 2. The insulation system must combine a low thermal conductivity with adequate mechanical resistance. Frequently, ladders are leaned against insulated pipes, or plant operators step on insulated surfaces, which ideally must not suffer dents or damage in these situations.

HEAT CONDUCTION IN SOLIDS

29

3. It must have a barrier to avoid humidity penetration. The presence of water within the insulation increases the thermal conductivity dramatically. 4. In plants where combustible materials are processed, it is necessary to evaluate the behavior of the insulation in case of fire. Insulation material must present low combustibility and smoke production. Plastic foams such as polyurethane usually present this problem. Thermal insulation manufacturers are now offering special foams with low combustibility indexes.

3-4-2 Thermal Insulation Materials Materials used in thermal insulation applications generally are grouped according to the temperature range in which they are employed. The temperatures that limit these ranges are somewhat arbitrary. When the temperature reaches an upper limit, materials may be damaged or become uneconomical because their thermal conductivity increases. A lower limit usually means that the material is not competitive because there are cheaper materials that can perform satisfactorily. Within each temperature range, the selection is made taking into consideration other properties and cost. Cryogenic Range (–260 to –100°C). Within this range, there are two types of insulation: vacuum and massive, the latter consisting of one or more solid phases distributed with a gas, such as dry air, to produce a very low thermal conductivity. Vacuum insulation systems, consisting typically of highly polished metal supporting walls with a vacuum space between them (sometimes with multiple metal reflective foils or specified powders inside), are usually custom-designed and installed by the insulation vendors. The theory behind vacuum cryogenic insulation systems is that of the Dewar flask, similar to those used for cold or hot beverage conservation. It relies in part on vacuum between the walls and in part on reflection of radiant heat. Without gas inside, heat transfer will be mainly by radiation. Coating the inside hot surface facing the evacuated area reduces heat transfer to a level proportional to the emissivity of the coating (e.g., 0.01 for silver). To reduce the heat transfer between the two walls even more, radiation shields are installed. This technique is based on the fact that radiation heat transfer often can be cut in half by installing a radiation shield between the hot and cold surfaces. Powders such as expanded perlite, diatomaceous earth, or fibers may be used for this purpose. The multilayer type, a series of reflective foil shields of aluminum separated by low-conductivity fillers such as fiberglass, comes in blanket form. Less costly than the evacuated forms just described are foam types. Foamed polyurethane and polystyrene, either foamed into flexible sheets, foamed in situ, or foamed in rigid insulation sections may be used within this range, but they are not completely resistant to water vapor permeation. Foamed glass exhibits a somewhat better performance. Low-Temperature Range (–100 to 0°C). Some of the evacuated types of insulation are used in the lower end of this range, but foams are employed more. In low temperatures, the main problem is moisture permeation. Table 3-1 shows water permeability values for four generic insulation materials. It can be seen that the permeability of fibrous materials is much higher than the permeability of foams. The microstructure of a foam consists of closed cells, and moisture cannot migrate through the insulation, as in the case of fibers. TABLE 3-1 Water Permeability Values for Four Generic Insulation Materials Material

Permeability (perm-in)

Foamed glass Urethane foam Polyestirene foam Fiberglass

0.00 0.3–0.6 1–4 100–200

30

CHAPTER THREE

Penetration by water or water vapor can increase the conductivity of insulation dramatically, so it is always necessary to protect the external side of the insulation with a moisture barrier. Water vapor permeation is not a serious problem until the moisture condenses. In high-temperature insulation, the problem is not serious because the vapor pressure of water is higher at the temperature of the internal face of the insulation than at the temperature of the external face. This tends to expel the humidity. In such cases, only protection against rain or water splashing is necessary. When the temperature of the internal face is lower than the dew-point temperature of the ambient air, however, as in cold-temperature insulation, water vapor may penetrate and condense into the insulation owing to a driving force that is proportional to the temperatures difference between both faces of the insulation. Even when a moisture barrier is installed, it is possible to have water vapor penetration through unsealed joints or sites where the barrier may be damaged. Localized freezing of this water vapor can cause a minor damage. During plant shutdowns, however, the ice melts, and water may travel through the insulation, thus spreading the problem. To avoid water penetration, moisture barriers are used. There are three different types: • Structural barriers. These are rigid sheets of reinforced plastic, aluminum, or stainless steel or galvanized iron plate, flat or corrugated. • Membrane barriers. These are metal foils, laminated foils, treated papers, plastic films, and coated felt or paper. They are either part of the insulation or may be supplied separately. • Coating barriers. These are in fluid form, usually as a paint. The material can be asphaltic, resinous, or polymeric. These provide a seamless coating but require time to dry. If, for any incident, water gets into the insulation anyway, the material must be heated and dried. This can be problematic in the case of plastic foams because they usually do not resist high temperatures well. Intermediate Range (0 to 520°C). This is the usual range in the process industry. The commonly used materials include the following: Calcium Silicate. This is a mixture of lime and silica, usually reinforced with organic or inorganic fibers and molded in the desired shape, either as blocks or panels, half-round segments for pipes, or special forms for the insulation of valves and fittings. The reason why it is usually preferred over other materials is its compressive strength. It can be used at up to 900°C. Mineral Wool. This is obtained by melting rocks and slag in furnaces at very high temperatures. The fibers usually are bonded together with a heat-resistant binder. The maximum temperature of use is almost as high as that for calcium silicate, but its compressive strength is much lower. It is available in both rigid molded form for piping or vessels and as flexible blankets for irregular surfaces. Fiberglass (Up to 400°C). Molten glass can be extruded in fibers to manufacture blankets or mattresses without binders, or they can be agglomerated, forming rigid or semirigid molded elements. Its maximum use temperature is lower than that of mineral wool. Additionally, glass fibers are irritating to the skin, which makes installation work uncomfortable. Cellular Glass or Cellular Foam (Up to 400°C). The same material used in low-temperatures applications also can be used in this range. High-Temperature Range (More than 550°C). Usually in this temperatures range, not only high temperatures but also other process conditions may be severe as well. Insulation may be required to resist abrasion and erosion by molten materials, direct flame impingement, corrosive atmospheres, and severe thermal shocks. The application of these types of insulation usually requires special installation procedures and specialized personnel. The materials usually employed are Miner fiber or calcium silicate up to 900°C Ceramic fibers (Al2O3-SiO2) up to 1400°C Molded ceramic refractories up to 1650°C Metallic oxides fibers such as Al2O3 or ZrO2 up to 1650°C Carbon fibers up to 2000°C

HEAT CONDUCTION IN SOLIDS

3-4-3

31

Factors Affecting Thermal Conductivity Insulation materials usually present a fibrous, cellular, or porous structure in which a solid matrix enclose spaces where air is present. Within this structure, heat transmission is due to a combination of the three basic mechanisms: conduction, convection, and radiation. Conduction takes place through the solid matrix that forms the insulation, for example, the solid fibers in a mineral wool blanket. It is clear, then, that to reduce conduction heat transfer, it is necessary that the base material be of low conductivity and that the number of fibers be as low as possible. This means a lowdensity material. In the free space, air is enclosed. The temperature differences between fibers create interior natural convection streams that originate the second heat transfer mechanism. The size, shape, and orientation of these hollow spaces affect the convection heat transmission through the insulation. It is convenient to create small cells to limit air movement. This means that the higher the material density, the lower will be the convection contribution to heat transfer (the opposite of conduction). As explained previously, another way to reduce convection is vacuum insulation, but this is not practicable in most cases. The third mechanism is radiation heat transfer between fibers through the insulation thickness. This mechanism depends on the fourth power of the absolute temperatures, so it is very important in hightemperature insulation. The radiation heat transfer is reduced with radiation shields. This means the installation in the radiation path of solid elements that absorb and re-emit energy, thus reducing its thermal level. Thus, to create a good insulation material, it is necessary to reduce the radiation transmission with an adequate number of absorbers, to create small cells to minimize air convective movement, and to install as little mass as possible to reduce conduction but at the same time maintain good mechanical properties. All the above-mentioned mechanisms work together in the insulation, forming a complex system. To simplify examination of the heat transfer mechanisms and their combined effects, the familiar term thermal conductivity, or simply k, is used. As we see, there are too many factors for k to be a description of an intrinsic property of a homogeneous material, as it purports to be. However, it is the only practical way to define the characteristics of an insulation, so it is used widely. The relative importance of the different heat transfer mechanisms changes when the insulation temperature changes. At high temperatures, radiation between fibers is the dominant mechanism, so it is advantageous to use insulation materials with rather high density. On the other hand, at low temperatures, when the importance of radiation decreases, a high density favors conduction and is harmful to the resulting conductivity. This means that a specific material may perform better than another at a certain temperature, but when the temperature changes, the situation may invert. Comparisons must be done with care, and ideally, thermal conductivity values have to be obtained at the temperature at which the material will be used. Table 3-2 shows typical ranges in which the thermal conductivity of insulation materials may vary. As can be appreciated, the ranges are ample, and it is always necessary to consult the vendor for more specific information. TABLE 3-2 Temperatures, Conductivities, and Typical Densities of Insulation Materials Material Urethane foam Fiberglass blankets Fiberglass panels Calcium silicate Molded mineral fiber Cellular glass Mineral fiber blankets

3-4-4

Temperature (°C)

k [W/(m · K)]

–150 to 100 –150 to 200 0 to 450 25 to 650 up to1000 –200 to 250 up to 750

0.015–0.020 0.024–0.085 0.033–0.08 0.031–0.12 0.051–0.13 0.028–0.10 0.04–0.05

Density (kg/m3) 25–45 9–45 35–90 90–150 9–45 100–140 110–170

Installation Piping. Piping is usually insulated with flexible blankets of mineral wool or fiberglass or with rigid half-round premolded segments. Blankets are installed by wrapping the pipe and tying them with galvanized wire. Sometimes the blankets are manufactured with hexagonal wire netting in one of their faces

32

CHAPTER THREE

Insulation with blankets Parker screws Metal jacket

Insulation with rigid sections Parker screws Metal jacket Tie wire every 0.25 m

Blanket

Tie wire every 0.25 m Hexagonal wire netting FIGURE 3-9

Staggered joints Rigid section

Pipe insulation systems.

to facilitate installation. Sometimes they are supplied with vapor barrier incorporated, which may consist on asphaltic felt or paper in one of their faces. Once the insulation is secured, a vapor barrier (e.g., felt embedded in asphaltic paint) can be applied, and then it may be covered with an aluminum, stainless steel, or galvanized iron jacket. The jacket is installed by wrapping the insulated pipe with metal sheets with a certain overlap and fixing the sheets with Parker screws or rivets. Valves and flanges usually are covered by insulation-filled metallic boxes with fast clamps that can be removed and reinstalled easily. Premolded sections are rigid half segments that are supplied in different sizes to match the pipe diameters. The end joints of both half segments are staggered by beginning a length of pipe with half- and fulllength sections of insulation. The segments then are fastened with wire or metal strips and covered with the metallic jacket (Fig. 3-9). Premolded sections are manufactured in a variety of materials, such as calcium silicate, different fibers with organic or inorganic binders, rigid polyurethane foam, etc. When long vertical pipe lengths are insulated (e.g., piping connected to distillation columns), support rings are welded to the pipe to avoid insulation displacement owing to excessive weight. Tanks and Vessels. Tanks and vessels may be insulated with either flexible blankets or rigid panels. Usually the vessel manufacturer installs pins, clips, or studs where the material is fixed. The vessel or tank is insulated in successive rings that, when completed, are fastened with wire or metal bands. Insulating cement may be applied to seal the joints. Metal jackets or another type of finishing is applied to protect the insulation from rain or moisture. Vertical vessels or tanks must be provided with support rings approximately every 2 m to avoid excessive weight. To apply insulation on hemispherical or semielliptical vessel heads, usually support rings are welded to the vessel. These rings allow building a wire net to hold the insulation in position. Joints are sealed with insulation cement, and the metal cover is site-manufactured in radial segments. Metal jacket installation is a craftsman work that must be performed with care because it greatly influences the aesthetic of the plant.

3-4-5

Economical Thickness When the reason for the use of thermal insulation is to reduce energy expenses, optimal thickness results from an economic calculation that considers insulation and energy costs. This calculation is greatly influenced by many parameters such as ambient temperatures, process conditions, maintenance and installation cost, amortization period, etc. Companies may perform these calculations with computer programs and, with their results, construct tables of recommended thicknesses as a function of the process temperatures and pipe diameters for a given set of the remaining parameters. These tables then are adopted as standards. For example, Table 3-3 shows recommended insulation thicknesses for calcium silicate insulation in a hightemperature application for mild weathers adopted by an oil company. When the reason for thermal insulation is personal protection, calculations are performed to have a temperature in the outer insulation face not higher than 55°C. In this case, different thicknesses are obtained, as shown in Table 3-4.

HEAT CONDUCTION IN SOLIDS

33

TABLE 3-3 Economical Thickness in Millimeters for Calcium Silicate Insulation (Hot Surfaces) Diameter (in)

Temperature (°C) 50

75

100

125

150

175

200

250

300

350

400

450

500

550

600

½

25

25

25

25

38

38

38

38

50

75

75

75

89

89

89

¾

25

25

25

38

38

38

38

38

50

75

75

75

89

89

89

1

25

25

25

38

38

38

38

50

75

75

75

89

89

89

100

1½

25

25

38

38

38

38

50

50

75

75

75

89

89

100

100

2

25

25

38

38

38

50

50

75

75

89

89

89

100

100

100

3

25

38

38

38

50

50

50

75

89

89

89

100

100

140

140

4

25

38

38

38

50

50

50

75

89

89

100

100

100

140

140

6

25

38

38

38

50

50

75

75

89

89

100

100

100

140

140

8

25

38

38

38

50

50

75

75

89

89

100

100

140

140

140

10

25

38

38

38

50

50

75

75

89

89

100

100

140

140

140

12

25

38

38

38

50

50

75

75

89

89

100

100

140

140

140

14

25

38

38

38

50

50

75

75

89

89

100

100

140

140

140

16

25

38

38

38

50

50

75

75

89

89

100

100

140

140

140

18

25

38

38

38

50

50

75

75

89

89

100

100

140

140

140

20

25

38

38

38

50

50

75

75

89

89

100

100

140

140

140

Plane

25

38

50

50

50

63

89

100

100

114

114

140

153

165

165

TABLE 3-4 Calcium Silicate Insulation Thickness for Personal Protection Diameter (in)

Temperature (°C) 50

75

100

125

150

175

200

250

300

350

400

450

500

550

600

½

0

25

25

25

25

25

25

25

25

25

38

38

38

50

50

¾

0

25

25

25

25

25

25

25

25

25

38

38

50

50

63

1

0

25

25

25

25

25

25

25

25

38

38

38

50

50

63

1½

0

25

25

25

25

25

25

25

25

38

38

50

50

63

75

2

0

25

25

25

25

25

25

25

38

38

50

50

63

63

75

3

0

25

25

25

25

25

25

25

38

38

50

63

75

75

89

4

0

25

25

25

25

25

25

38

38

50

50

63

75

89

100

6

0

25

25

25

25

25

25

38

50

50

63

75

89

100

125

8

0

25

25

25

25

25

25

38

50

63

75

75

89

100

125

10

0

25

25

25

25

25

25

38

50

63

75

89

100

114

125

12

0

25

25

25

25

25

38

38

50

63

75

89

114

125

140

14

0

25

25

25

25

25

38

50

63

75

89

100

114

125

140

16

0

25

25

25

25

25

38

50

63

75

89

100

114

140

150

18

0

25

25

25

25

25

38

50

63

75

89

100

125

140

150

20

0

25

25

25

25

25

38

50

63

75

89

114

125

140

165

Plane

0

25

25

38

38

50

63

75

100

125

150

189

216

250

300

34

CHAPTER THREE

GLOSSARY A = area (m2) c = specific heat [J/(kg · K)] h = convection film coefficient [W/(m2 · K)] k = thermal conductivity [W/(m · K)] L = length (m) m = dimensionless parameter = k/hrm q = heat flux (W/m2) Q = heat flow (W) r = radial coordinate or radius (m) rm = thickness or radius (m) R = radius (m) S = parameter in Eq. (3-2-40) (W/K) T = temperature (K) u = internal energy (J/kg) U = overall heat transfer coefficient [W/(m2 · K)] W = power per unit area (W/m2) W = mass flow (kg/s) x, y, z = cartesian coordinates

REFERENCES 1. 2. 3. 4.

Rohsenow W. M., Hartnett J. P.: Handbook of Heat Transfer. New York: McGraw-Hill, 1973. Kreith F.: Principles of Heat Transfer. New York: International Textbook Company, 1973. Wong H. Y.: Handbook of Essential Formulae and Data on Heat Transfer. London: Longman Group, 1977. McAdams W.: Heat Transmision. New York: McGraw-Hill, 1954.

CHAPTER 4

CONVECTION

Convection is a heat transfer mechanism that takes place at a solid-fluid interface. The fluid acts as a vehicle for energy transport, and the heat-flux density depends on the velocity with which the fluid is renewed over the surface. Two different flow patterns can exist: laminar and turbulent. In laminar flow, the fluid streamlines are stable, and then, once a steady state is reached, the velocity at every point is constant. In turbulent flow, even though we also speak about steady state, meaning that an average constant velocity exists at every location, there are random fluctuations around this average value. These random fluid movements are called eddies, and they cannot be predicted and provoke an internal agitation that greatly influences heat transfer. In laminar-flow systems, it is possible to write differential equations for the momentum and energy balances that can be solved to find the velocity and temperature profiles in the fluid. By knowing the temperature distribution, it is possible to calculate the heat-flux density at the interface. In turbulent-flow systems, since the temperature and velocity distributions are not continuous functions, this approach is not possible, and it is necessary to use experimental correlations. This will be explained in more detail later in this chapter.

4-1 FORCED CONVECTION OVER A FLAT PLATE 4-1-1

Boundary Layer Let’s consider the flat plate shown in Fig. 4-1 over which a fluid circulates. The fluid approaches the plate with a velocity v∞. The plate exerts a perturbation on the flow. This perturbation takes place in a region that extends up to a certain distance from the plate, so beyond this distance the fluid velocity is also v∞, which is called the free-stream velocity. The region of the fluid where the solid surface exerts its influence on the flow is called the boundary layer. Since the velocity increases asymptotically to v∞, it is usual to define the boundary layer as the fluid region where the velocity is below 99 percent of the free-stream velocity. We shall analyze how the solid perturbs the flow and how is the velocity field within the boundary layer.

4-1-2 Shear Stress in a Moving Fluid Refering again to Fig 4-1,when the fluid comes in contact with the plate, shear stresses develop in the fluid that tend to slow down the fluid layers closer to the solid surface. We can think on this with the simple logic of the action and reaction principle. The fluid exerts a drag force over the plate, so the plate must exert an opposite force over the fluid. If the plate is fixed and cannot move, the effect is that the fluid begins to slow down. The effect appears initially in the layers of fluid closer to the solid. The layer immediately adjacent to the solid stops completely, and this layer, in turn, begins to slow down the adjacent layers, and a velocity profile, as shown in Fig. 4-2, develops in the fluid. This effect is transmitted from one layer to another by shear stresses. Let’s consider a control differential volume belonging to the fluid located at a certain distance y1 from the solid (Fig. 4-3). We shall call this differential volume I.

35

36

CHAPTER FOUR

y

v∞

v∞

x Aproximation velocity FIGURE 4-1

(Free-stream velocity)

Region of perturbated flow (boundary layer) Solid wall

Fluid flowing over a flat plate.

y

v = v (y)

Fluid velocity as a function of the distance to the wall

FIGURE 4-2 Velocity profile in a fluid moving over a plate.

||

Δy | y

Δz

y1

z

Δx

|τyx |y = y 1 + Δy

(Action exerted by volume element I on volume element II)

|τyx |y = y 1 + Δy

(Action exerted by volume element II on volume element I)

|τyx |y = y 1

(Action exerted on volume element I by the fluid below)

Solid wall x

FIGURE 4-3

Shear forces in the boundary layer.

As mentioned previously, the fluid that is below this volume exerts a slowdown effect over the volume. This effect is explained by the action of a force that appears on the lower horizontal face of the control volume I. We shall call τ (shear stress) the force exerted per unit area. Shear stresses are normally designated by two subscripts. The first one indicates the plane in which the force acts, designated by its normal. In this case, the shear stress appears in a plane whose normal is y. The second subscript indicates the direction of the force. In this case, the direction is that of the x axis. This shear stress thus will be designated as τyx.

CONVECTION

37

If y1 is the vertical coordinate of the lower face of the control volume, the shear stress acting on this face will be |τyx|y = y1, and the force will be |τyx|y = y1ΔxΔz. This differential volume, in turn, exerts a similar slowdown effect on the fluid that is above it If volume II is a differential volume immediately adjacent to volume I (even though it was represented separately for drawing clarity), we see that a shear stress appears on the lower face of volume II whose value is |τyx|y = y1+Δy . It is then obvious that differential volume II exerts over differential volume I a force of the same magnitude and opposite direction as shown in the figure. Any differential volume such as volume I, then, is subject to the action of two forces of opposite directions. For an infinite fluid moving over a flat plate, these forces decrease in intensity as one moves away from the plate. This means that the force acting on the lower face of volume I is higher than that acting on the upper face, and the result is a net force opposing to the fluid movement The shear stress at every fluid location is related to the velocity gradient by τ yx = μ

∂v x ∂y

(4-1-1)

This means that as the shear stress decreases, the slope of the velocity profile also decreases, and the fluid velocity remains constant at a certain distance from the plate. This distance is the boundary-layer thickness. This means that beyond the boundary layer, there are no more shear stresses on the fluid. This region is called the undisturbed free stream or potential flow regime.

4-1-3

Boundary-Layer Development Let’s refer to Fig. 4-4. The x coordinate indicates the horizontal distance to the plate leading edge, and the y coordinate is the vertical distance to the plate. When the fluid comes in contact with the plate, the fluid layers closer to the plate slow down, and the boundary layer begins to develop. As long as the fluid progresses along the x axis, the perturbation extends to the fluid layers farther from the solid, and the thickness of the boundary layer increases with x. The figure shows the development of the boundary layer and the velocity profiles at different values of the x coordinate.

v∞

v∞

y

v∞

Boundary limit contour

v∞

δ = δ (x) x FIGURE 4-4 Velocities distribution in the boundary layer.

4-1-4 Laminar and Turbulent Boundary Layers The boundary layer can be laminar or turbulent. However, even in a turbulent regime, there is always a region near the leading edge of the plate where the boundary layer is completely laminar (Fig. 4-5). This is so because in this region the velocity gradients are high, so the shear stresses are also high. These shear stresses exert a laminating action over the fluid movement. However, as the fluid progresses over the plate, the velocity gradients decrease, and eddy streams can penetrate into the boundary layer, which becomes turbulent. We shall call xc (critical distance) the value of x when the boundary layer becomes turbulent. Even in a turbulent boundary layer, the region near the solid wall, where fluid velocities are very low, is in laminar

38

CHAPTER FOUR

Potential flow region (V = V∞) V∞

V∞

Turbulent boundary layer

Laminar sublayer

xc FIGURE 4-5

Laminar and turbulent regions in the boundary layer.

flow. This region is called the laminar sublayer. The value of xc depends on the approximation velocity and on the roughness of the surface. It is normally accepted that for regular-roughness surfaces, xc =

5 × 10 5 μ ρv∞

(4-1-2)

or xc ρv∞ = 5 × 10 5 μ

(4-1-3)

The force that the fluid exerts over the plate or the plate exerts over the fluid per unit area is τ 0 = τ yx

y=0

=μ

∂v x ∂y

(4-1-4) y=0

For a laminar boundary layer, it is possible to develop theoretical expressions for this derivative, and it is then possible to calculate the force analytically. Theoretical solutions to this case were developed by Blasius,1 and his results coincide remarkably well with experimental data14 (Fig. 4-6). However, in a turbulent regime, the momentum and energy-balance equations cannot be integrated, and the derivative cannot be calculated. The usual approach then is to define a coefficient called the drag coefficient or friction factor Cf as τ 0 = Cf

1 2 ρv∞ 2

(4-1-5)

And the friction factor is obtained empirically. Usually, experimental data are correlated as Cf = Cf (ρ, v, μ, L) and a mean friction factor for the entire plate is defined. The correlations normally are expressed as ⎛ ρv L ⎞ Cf = Cf ⎜ ∞ ⎟ ⎝ μ ⎠

(4-1-6)

CONVECTION

39

1

Slope = 0.332

v 0.5 v∞

x = 1.0 cm x = 2.0 cm x = 2.5 cm x = 4.0 cm x = 5.0 cm x = 7.5 cm x = 10.0 cm x = 12.5 cm x = 15.0 cm x = 17.0 cm

v∞ = 8 m/s Thin plate (No.1) Blasius

0

0

1

2

3

4

5

6

7

ρ v∞ x μ

FIGURE 4-6 Velocity profile in a laminar boundary layer according to Blasius with experimental data of Hansen.14 (National Advisory Committee for Aeronautics NACA TM 585.)

where ρv∞L/μ is the Reynolds number for the entire plate (ReL). For example, for Reynolds numbers between 5 × 105 and 1 × 107 and excluding the region of boundarylayer development, it was found2 that C f = 0.072Re −0.2

(4-1-7)

4-1-5 Heat Transfer from a Flat Plate to a Fluid in Motion Let’s consider now that the plate is at a temperature Tw , whereas the fluid approximation temperature is T∞. When the fluid comes in contact with the plate, its temperature begins to increase owing to the heat flux. For low x values, the perturbation extends only to the region of the fluid close to the interface, but as long as the fluid moves farther, the perturbation progresses, and a temperature profile develops. For any x at enough distance from the plate, the temperature is again T∞. The region where the temperature changes from Tw to T∞ is called the thermal boundary layer. Figure 4-7 shows the evolution of the thermal boundary layer and the temperature profiles in the system. Since the velocity of the fluid that is in contact with the plate is nil, in that region the fluid behaves like a solid, and there will be a conduction heat flow from the plate to the fluid with a heat-flux density given by ⎛ ∂T ⎞ q = −k ⎜ ⎟ ⎝ ∂y ⎠ y =0

(4-1-8)

Let’s consider a differential volume of fluid within the boundary layer (Fig. 4-8). There is a fluid flow entering the volume on the left side and leaving on the right side. Since the flowing fluid receives heat by conduction from the plate, it suffers a temperature increase, which means that the fluid is removing heat from the control volume.

40

CHAPTER FOUR

T∞ T∞

T∞

T∞

Fluid velocity Thermal boundary layer

Tw

Tw

Tw

FIGURE 4-7 Temperatures distribution in the boundary layer of a fluid flowing over a flat plate.

q = –k

dT dy y = y1 + Δy

T + ΔT

T

y1

FIGURE 4-8

T = T (y)

dT q = –k dy y = y 1

Heat balance on a volume element in the boundary layer.

Since part of the heat entering the volume by conduction is removed by the fluid, the heat-flux density on the upper side is smaller than that on the lower side. Since the conduction heat-flux density is related to the thermal gradient, it is clear that ∂T/∂y decreases as we move away from the plate. The shape of the temperature profile depends on the hydrodynamic conditions and on the properties of the fluid—density, specific heat, and viscosity. In laminar-flow systems, it is possible to develop theoretical models that allow one to calculate the partial derivative of Eq. (4-1-8) and solve the problem analytically. Pohlhausen10 used the velocity profiles calculated previously by Blasius to obtain the solution of the heat transfer problem and obtain the temperature profiles in the system. At low Reynolds numbers, the analytical solution coincides with experimental data (Fig. 4-9). However, in the majority of systems in process engineering, this approach is not possible, and an empirical coefficient called the convection film coefficient is defined. This coefficient plays the same role as the friction factor in fluid mechanics.

4-1-6 The Convection Film Coefficient Since the analytical treatment has the limitations explained earlier, the usual approach to the problem consists of using empirical correlations. These correlations generally are expressed in terms of a coefficient called the convection film coefficient, defined as the quotient between the heat flux per unit area and the temperature difference between the wall and the bulk of the fluid.

CONVECTION

41

1.0 0.8 Pr = 50 15

7

3

1 0.8

1.2

1.6

0.6

T – Tw 0.6 T∞ – Tw 0.4 0.2 0

0

0.4

0.8

y x

2.0

2.4

2.8

3.2

3.6

4.0

x v∞ ρ μ

FIGURE 4-9 Dimensionless temperature distributions in a fluid flowing over a heated plate in laminar flow.

This coefficient may be defined either for the entire surface or for an area element. In the first case, a mean coefficient hL , valid for the entire length of the plate, is defined as Q / A = hL (Tw − T∞)

(4-1-9)

where Q is the total heat delivered by the plate to the fluid and A is the total area of the plate. In the second case, a local coefficient hx is defined. This coefficient is a function of the position x, and the definition equation is dQ = hx (Tw − T∞ ) dA

(4-1-10)

where dQ is the heat delivered to the fluid per unit time by a differential area dA located at a distance x from the leading edge of the plate. The total heat delivered to the fluid in the entire length of the plate can be calculated as Q=

L

∫A hx (Tw − T∞ )dA = ∫0 hx (Tw − T∞ )b dx

(4-1-11)

where b is the plate width. Then, comparing with Eq. (4-1-9), we get L

hL

∫ hx dx = 0 L

(4-1-12)

It is quite evident that the film coefficients depend on the velocity distribution of the fluid, on its thermal conductivity, and on the properties that affect its capacity to transport energy (i.e., specific heat and density). As a general expression, it can be written hL = hL ( v∞ , L , ρ, μ, c, k )

(4-1-13)

42

CHAPTER FOUR

or hx = hx ( x , ρ, c p , μ, k , v∞ )

(4-1-14)

and these relations are found experimentally.

4-1-7 Effect of Fluid Velocity on the Film Coefficient h It was explained that within the boundary layer there is always a sublayer close to the solid surface that is free of turbulences. The turbulence existing in the remaining portion of the boundary layer exerts a mixing effect and homogenizes the fluid properties. Then, in the turbulent boundary layer region, the velocity and temperature gradients are much less steep than in the laminar sublayer. This is shown in Fig. 4-10.

Temperature profile

Turbulent boundary layer (small gradient)

δ

Laminar sublayer (steep gradient)

δ1

FIGURE 4-10 Temperature profile and laminar sublayer thickness.

As an approximation, we can assume that the temperature profile can be replaced by the two tangents drawn at y = 0 and y = δ, as shown in Fig. 4-10. The intersection of both tangents defines a distance δ1 that approximately coincides with the laminar sublayer thickness. We have explained that the heat-flux density always can be calculated as q=

dQ dT = −k dA dy

(4-1-15) y =0

And then q=k

Tw − T∞ δ1

(4-1-16)

But remembering the definition of the convection film coefficient, that is, q=

dQ = h(Tw − T∞ ) dA

(4-1-17)

and comparing Eq. (4-1-16) with Eq. (4-1-17), we get h=

k δ1

(4-1-18)

CONVECTION

43

When the fluid velocity is high, the turbulent eddies penetrate deeper within the boundary layer, and the laminar sublayer thickness is reduced, thus increasing h. This means that as a general rule, the higher the fluid velocity, the higher the film coefficient will be.

4-1-8 Correlations for Heat Flow from a Planar Surface to a Moving Fluid With the help of dimensional analysis, it is possible to demonstrate that the functions represented by Eqs. (4-1-13) and (4-1-14) will be of the type ⎛ ρxv∞ c pμ ⎞ hx x = f⎜ , ⎝ μ k k ⎟⎠

(4-1-19)

⎛ ρLv∞ c pμ ⎞ hL L = f⎜ , ⎝ μ k k ⎟⎠

(4-1-20)

or for the mean coefficient

The left side of Eqs. (4-1-19) and (4-1-20) are called the local and mean Nusselt numbers, respectively (Nux and NuL). Note Dimensional analysis is a technique that allows combining the physical magnitudes that express the solution of a problem in a smaller number of dimensional groups, thus reducing the number of variables to correlate. This subject can be examined more closely in ref. 2. The correlations usually accepted for the case of a fluid circulating over a flat plate are the following: 1. For laminar flow ( ReX < 5.105), Nu x =

xhx = 0.33Re x1/ 2 Pr1/ 3 k

(4-1-21)

or k hx = 0.33 Re x1/ 2 Pr1/ 3 x

(4-1-22)

According to Eq. (4-1-12), the mean Nusselt number for the entire plate, if the laminar regime exists over the complete length of the plate, can be obtained by integrating the previous expression from x = 0 to x = L and dividing by L, that is, L

hL

∫ hx .dx = 0.33 = 0 L

= 0.66

L

k ⎛ Lv∞ρ ⎞ L ⎜⎝ μ ⎟⎠

L

∫0

1/ 2

k ⎛ ρv∞⎞ 1/ 2 ⎜ μ ⎟ ⎠ x ⎝

⎛ c pμ ⎞ ⎜ k ⎟ ⎝ ⎠

1/ 2

⎛ cμ ⎞ ⎜⎝ k ⎟⎠

1/ 3

dx (4-1-23)

1/ 3

Nu L = 0.66Re L1/2 Pr1/3

(4-1-24)

44

CHAPTER FOUR

2. For turbulent flow (Rex > 5.105),2 Nu x =

xhx = 0.0288 Pr1/ 3 Re x 0.8 k

(4-1-25)

and the mean coefficient for a plate of length L can be calculated with Eq. (4-1-12), and the following expression for NuL is obtained: NuL =

hL = 0.036 Pr1/ 3 Re L 0.8 k

(4-1-26)

It is necessary to note that to obtain the preceding expression, we have neglected the existence of the laminar zone between x = 0 and x = xc . The expression then is only valid for L >> xc. The laminar region can be included in the analysis if Eq. (4-1-21) is used between x = 0 and x = xc and Eq. (4-1-25) between x = xc and x = L. Considering ReC = 5.105, this leads to the following expression:2 Nu L = 0.036 Pr1/ 3 (Re 0L.8 − 23, 200)

(4-1-27)

4-2 FORCED CONVECTION INSIDE TUBES 4-2-1 Flow of Fluids Inside Tubes—Velocity Distribution Let’s consider a fluid entering into a tube with velocity vo. At the section immediately after the entrance, almost all the fluid maintains the inlet velocity except in the region very close to the tube wall, where the fluid is slowed down owing to the shear-stress effect. A boundary layer then starts developing. As long as the fluid progresses into the tube, the boundary layer extends toward the center of the tube, and successive fluid layers begin to slow down. Since the total mass flow is constant from the inlet to the outlet, if the velocity of the fluid close to the tube wall decreases, the velocity near the center of the tube must increase. Then a velocity profile as shown in Fig. 4-11 begins to develop. At a certain distance from the tube inlet, the boundary layer has reached the center of the tube, and from then on, the velocity profile remains constant. We then speak of a fully developed velocity profile. The distance extending from the inlet section to the section where the velocity profile is fully developed is called the entrance region Le. The boundary layer can be in either laminar or turbulent flow. Whatever the

Entrance region L e

Nonperturbated region Boundary layer

At the entrance v = v0 = constant

FIGURE 4-11

The more external layers begin to slow down

The perturbation progresses. The central velocity increases

The boundary layer reaches the center of the pipe. Fully developed velocity profile

Development of the velocity profile in a fluid flowing into a pipe.

CONVECTION

45

velocity with which the fluid enters the tube, there is always a region near the entrance where the velocity gradients are very high and the boundary layer is laminar. The central core can be in turbulent flow if the inlet velocity is high (Fig. 4-12a). As long as the influence of the wall extends to the bulk of the fluid, the shear stresses decrease. It is then possible that eddies penetrate into the boundary layer, which becomes turbulent (Fig. 4-12b). Finally, at a certain point downstream, the boundary layer reaches the center of the tube (Fig. 4-12c). There is always a region close to the tube wall where the boundary layer is in laminar flow. This is the laminar sublayer.

Laminar sublayer Turbulent boundary layer Nonperturbated region

Fully developed turbulent layer

Turbulent nucleous

Turbulent nucleous

(a)

FIGURE 4-12

Laminar sublayer

Laminar sublayer Turbulent B.L

Laminar boundary layer

(b)

(c)

In-tube flow boundary layer regions.

It was found that the necessary condition for the boundary layer to remain in laminar flow is Re =

Dvρ < 2100 μ

(4-2-1)

where v is a mean velocity defined as v=

W ρat

(4-2-2)

where W = mass flow (kg/s) ρ = density (kg/m3) at = flow area (= cross section of the tube)(m2) In cases when the boundary layer remains in laminar flow, it is possible to find the mathematical function of the velocity profile theoretically, and for the fully developed profile region, the well-known parabolic distribution function is obtained, that is, vz = vz ,max ⎡⎣1 − (r /R)2 ⎤⎦

(4-2-3)

In this equation vz, max is the velocity at the center of the tube. The total mass flow circulating through the pipe is equal to the integral of the mass flow density in the cross section of the tube. Let’s consider an area element with the shape of an annulus, as shown in

46

CHAPTER FOUR

vz (r ) R r r + dr

dA = 2πr dr FIGURE 4-13

Parabolic velocity distribution in laminar flow.

Fig. 4-13. Since vz is a function of the radial coordinate, in all points of dA the velocity is the same. The mass flow through dA then will be dW = vz ρ dA

(4-2-4)

W = ∫ dW = ∫ vz ρ dA

(4-2-5)

Integrating in the entire section,

and comparing with Eq. (4-2-2), we get v=

∫

vz ρ dA ρat

∫ vz dA = ∫ 0 vz ,max ⎡⎣1 − (r /R)2 ⎤⎦ 2πr dr = vz ,max R

=

at

πR 2

(4-2-6)

2

(the integration details are left to the reader). This means that in laminar flow, the mean velocity in any section is half the velocity at the tube center. In turbulent flow, the velocity profile is more planar. The velocity varies from 0 to about 99 percent of the maximum value in a very short distance, and a turbulent core exists where the velocity is practically uniform (Fig. 4-14). Then a velocity measurement with a Pitot tube at the center of the pipe can be considered, for all practical purposes, the mean velocity in the pipe section.

vz = vz (r )

FIGURE 4-14 Velocity distribution in turbulent flow.

CONVECTION

4-2-2

47

Pressure Drop Owing to the friction at the tube wall, the fluid will suffer a pressure drop when circulating through the tube. This pressure drop can be related to the shear stress at the tube wall. Let’s consider a portion of the fluid contained between two pipe sections which are separated by a length dz. The forces acting over this portion of the fluid are shown in Fig. 4-15. Then, making a balance of forces, we get 2 πR dz τ |r = R

(p – Δp) πR 2

p πR 2

dz FIGURE 4-15

Forces acting on a segment of fluid.

dpπR 2 = τr = R 2 π R dz dp = τr = R

2 dz R

(4-2-7) (4-2-8)

And the shear stress at the tube wall is ⎛ dv ⎞ τr = R = −μ ⎜ z ⎟ ⎝ dr ⎠ r = R

(4-2-9)

(the minus sign is necessary because the radial coordinate has its origin at the center of the pipe, so the derivative is negative). In laminar flow, the partial derivative in Eq. (4-2-9) can be calculated from the velocity distribution of Eq. (4-2-3). Then, combining with Eq. (4-2-6), we get τ r = R = μvz ,max

2 4v =μ R R

(4-2-10)

And replacing in Eq. (4-2-8), we get dp = 8μ

v v dz = 32μ 2 dz R2 D

(4-2-11)

This expression relates the pressure drop in a pipe with the mean fluid velocity in laminar flow. When the flow regime is turbulent, the partial derivative of Eq. (4-2-9) cannot be calculated, and an analytical solution is not possible. The usual approach is to define a friction factor as τr = R =

1 f ρv 2 2

(4-2-12)

48

CHAPTER FOUR

and the correlation of the friction factor as a function of the rest of the system variables is obtained experimentally. Combining Eq. (4-2-8) and Eq. (4-2-12), we get dp = 4 f

dz 1 2 ρv D2

(4-2-13)

The friction factor is a function of the Reynolds number and the roughness of the tube surface. A wellknown graph representing the friction factor as a function of the Reynolds number (Fig. 4-16) is due to Moody.9 The curve’s parameter is the relative roughness, defined as the quotient between the tube surface roughness and the pipe diameter. For commercial steel pipes in process installations, it is normally accepted that a standard roughness of 45 μm can be used in the calculations. This graph has different regions.

FIGURE 4-16 Fanning friction factor versus Reynolds number based on Moody.9 (Reproduced with permission of the American Society of Mechanical Engineers.)

The region where 2100 < Re corresponds to the laminar flow. In that region, the mathematical expression for the friction factor can be obtained combining Eq. (4-2-11) and Eq. (4-2-13), which gives f =

16 Re

(4-2-14)

At high Reynolds numbers, the dependence of the friction factor on the Reynolds number is very weak, and it mainly depends on the surface roughness of the tube. The intermediate region corresponds to the transition regime, in which the friction factor depends on both the Reynolds number and the surface roughness. This graph allows calculation of the frictional pressure drop in a pipe. Knowing the mass flow W circulating in the pipe, if D is the internal diameter, we calculate Re =

GD μ

(4-2-15)

CONVECTION

49

where G = mass flow density or mass velocity = 4W/πD2. With the Reynolds number it is possible to obtain the friction factor, and finally. we can calculate the pressure drop by integrating Eq. (4-2-13), and we get, Δp=4f

L ρv 2 D 2

(4-2-16)

4-2-3 Nonisothermal Flow Inside Tubes Let’s consider the system shown in Fig. 4-17, where a fluid at temperature Tb1 contained in a tank is pumped through a pipe to a second tank. If part of the pipe is heated, the fluid temperature will increase, and the fluid will arrive at the second tank at a higher temperature.

Tb 1

FIGURE 4-17

Heated length

Tb 2

Fluid heating into a pipe.

Let’s assume that the second tank is perfectly mixed, and we shall call Tb2 the temperature of the fluid contained in it. If the mass transferred from one tank to the other per unit time is W (kg/s), the heat flow (heat delivered per unit time) will be that necessary to heat up W kg/s from a temperature Tb1 to temperature Tb2. This can be expressed as Q = W (ib 2 − ib1 )

(4-2-17)

where ib1 and ib2 are the specific enthalpies of the fluid at temperatures Tb1 and Tb2. Assuming that the enthalpies are only a function of temperature (i.e., the case of uncompressible fluids), Eq. (4-2-17) can be expressed as Q = Wc(Tb 2 − Tb1 )

(4-2-18)

where c is the specific heat or mass heat capacity [J/(kg · K)]. Let’s analyze now what happens in the heated length. The fluid at temperature Tb1 enters the heated pipe. We shall assume that the internal surface of the tube is at a uniform temperature Tw , where Tw > Tb1. Owing to this temperature difference, heat begins to flow from the tube surface to the fluid, and the regions of the fluid closer to the tube wall begin to increase their temperature. In the same way that in fluid dynamics it is assumed that the velocity of the fluid in contact with the solid wall is zero, in heat transfer, the hypothesis normally assumed is that at the interfaces there is no discontinuity in the temperature profile, and then the temperature of the fluid in contact with the tube wall is also Tw. As long as the fluid progresses through the pipe, the perturbation extends toward the interior of the conduit, and finally, at a certain length, all the fluid has increased its temperature above the initial Tb1 (Fig. 4-18). Thus a temperature distribution T = f(r, z) develops into the pipe, where z is the axial coordinate. If the pipe were of an infinite length, at the outlet end, all the fluid would be heated up to the wall temperature, and a planar temperature profile again would exist. If, on the other hand, we consider that the pipe has a certain finite length, at the outlet section, the fluid will exit with a certain temperature distribution T = f(r) like the profile indicated with (c) in Fig. 4-18.

50

CHAPTER FOUR Tw

Tw

(a) Constant fluid temperature = Tb

1

FIGURE 4-18

(b) Temperature profile starts developing. Regions near the tube wall increase the temperature

Tw

(c) The perturbation extends to the entire section. The mean temperature continues increasing

(d) In an infinite length tube all the fluid finally reaches temperature Tw

Development of the temperature profile in a pipe.

The fluid leaving the heated tube enters the second tank, where it is perfectly mixed, resulting a uniform temperature Tb2. It is said that Tb2 is the mixing-cup temperature at the outlet section. At any section of the pipe, it is possible to define the mixing-cup temperature as the temperature that a fluid passing the cross-sectional area of the conduit during a given time interval would assume if the fluid were collected and mixed in a cup. The mixing-cup temperature also can be called the average bulk temperature in the section under consideration. The mixing-cup temperature corresponding to any section of the pipe can be calculated if the velocity and temperature profiles of the fluids in that section are known. If we apply the expression of the first law of thermodynamics for a circulating system to the heated pipe we are considering, we get Q = outlet enthalpy per unit time – inlet enthalpy per unit time = i2 − i1

(4-2-19)

The inlet enthalpy to the system is easy to calculate. Since the fluid enters the system at a uniform temperature Tb1, and selecting the reference state of the enthalpy function such that i = cT

(enthalpy per unit mass)

(4-2-20)

the inlet enthalpy then will be i1 = WcTb1 (J/s)

(4-2-21)

To calculate the outlet enthalpy, it is necessary to calculate the flow integral across the outlet section. Let’s consider an annulus-shaped surface element belonging to the outlet section (Fig. 4-19) with the area dA = 2πrdr

vz (r)

T (r )

(4-2-22)

R r r + dr

dA = 2πrdr FIGURE 4-19 Temperature and velocity profiles in a tube section.

CONVECTION

51

Since the fluid temperature and velocity in any pipe section are a function of the radial coordinate, they can be considered constant in that surface element. The mass flow through the surface element then will be dW = 2πrdr ρv

(4-2-23)

di2 = 2 πrdr ρ vc T

(4-2-24)

And the enthalpy flow is

The enthalpy flow through the entire outlet section then will be i2 =

R

∫0 2πrdr ρvcT

(4-2-25)

By combining Eqs. (4-2-19), (4-2-21), and (4-2-25), we get Q=

R

∫0 2πrdr ρvcT − WcTb

(4-2-26)

1

But comparing with Eq. (4-2-18) results in Wc(Tb 2 − Tb1 ) =

R

∫0 2πrdr ρvcT − WcTb

(4-2-27)

1

The mixing-cup or average bulk fluid temperature Tb2 then will be

∫ 0 ρcvT (2πrdr ) = ∫ 0 ρcvT (2πrdr ) = ∫ 0 vTrdr R R Wc ∫ 0 ρcv(2πrdr ) ∫ 0 vrdr R

Tb 2 =

R

R

(4-2-28)

Equation (4-2-28) allows calculation of the average bulk temperature at any section if you know the velocity and temperature profiles corresponding to that section. This is the temperature we can measure with a thermometer inserted into the pipe downstream of the heater at a distance long enough that the fluid can reach a uniform temperature. In practice, this distance is only a few diameters. Additionally, in the majority of cases, we shall deal with turbulent flow, and the temperature profiles are almost planar owing to the mixing effect produced by the turbulent eddies. In this case, the average bulk fluid temperature practically coincides with the temperature at the center of the pipe. Then, by inserting a thermometer into the pipe as shown in Fig. 4-20, it is possible to measure with a good approximation the mixing-cup or average bulk temperature of the fluid, and it is always a good practice to install the instrument a few diameters downstream of the outlet section of the heater. If the total heat flow transferred to the fluid is known, the mixture temperature at the outlet section can be obtained from Tb 2 = Tb1 +

Q Wc

(4-2-29)

From now on, every time we shall speak about the temperature of a fluid at a pipe section without any other explanation, we shall be referring to its mixing-cup or average bulk temperature.

52

CHAPTER FOUR

Temperature Indicator

FIGURE 4-20 Average bulk temperature measurement.

4-2-4 Development of the Temperature Profile We have explained how the temperature profile develops into the fluid as it circulates into the pipe (see Fig. 4-18). Let’s define a dimensionless temperature as T *(r , z ) =

Tw − T(r , z ) Tw − Tb ( z )

(4-2-30)

where Tb(z) is the average bulk temperature at the section under consideration. It can be observed that the temperature distribution T*(r, z) becomes practically independent from z after a short length following the inlet section. This means that when the temperature distribution in the system is represented in terms of this dimensionless temperature, we get a profile as shown in Fig. 4-21. The length of the inlet region after which the temperature profile is completely developed in laminar flow approximately equals 0.05D(Re · Pr). In turbulent flow, this thermal inlet region is much shorter, and the temperature profile develops very quickly.

T ∗ = f (r )

Inlet Section T∗ = 1

FIGURE 4-21

Region of temperature profile development

Fully developed temperature profile

Development of the dimensionless temperature profile.

CONVECTION

53

4-2-5 Heat Transfer Rate from the Wall to the Fluid—Local Film Heat Transfer Coefficient Let’s consider that at a certain section of the heated pipe the average bulk temperature of the fluid is T (from now on, we shall drop the subscript b to indicate bulk temperatures). If Tw is higher than T, heat will flow from the surface of the pipe to the fluid. (we have to point out that the analysis can be done assuming that the heat transfer is in the opposite direction, and all the conclusion we shall obtain will continue to be valid.) The magnitude of the heat flux that is established depends on the hydrodynamics of the system. Since the velocity of the fluid in contact with the solid wall is nil, the fluid in this region behaves like a solid, and the only heat transfer mechanism is conduction. Then we can always say that qr

r=R

=k

dT dr

(4-2-31) r=R

(We didn’t include the minus sign because we want q to be positive when flowing from the wall to the fluid, in the negative direction of the radial coordinate.) This means that should we know the temperature distribution in the system, it would be possible to calculate the heat flux to the fluid with Eq. (4-2-31) As in the case of a flat plate, in certain simple cases and in laminar flow, it is possible to obtain an analytical solution to the problem. However, the analytical procedure is limited to laminar-flow systems, which normally have little practical interest because most process equipment work in a turbulent regime. The practical approach, then, is to define a local film convective heat transfer coefficient hloc as dQ = hloc dA(Tw − T )

(4-2-32)

dq = hloc (Tw − T )

(4-2-33)

or

where dA is the differential lateral area of the pipe (=2πRdz), and dQ is the heat flow transmitted through the wall. The problem to be solved is to find an empirical correlation to calculate hloc as a function of the system parameters. Before addressing this subject, we shall analyze some characteristics of the convective film heat transfer coefficients we have just defined.

4-2-6 Physical Interpretation of the Coefficient hloc We have seen that the heat flux at the tube wall can be expressed as dq r = R = k

dT dr

(4-2-34) r=R

Combining this expression with Eq. (4-2-33), we get k

dT dr

r =R

= hloc (Tw − T )

(4-2-35)

If we write this equation as a function of the dimensionless temperature T* defined by Eq. (4-2-30) and of a dimensionless radial coordinate r* = r/D, we can write −k

dT * = hloc D dr * r = R

(4-2-36)

54

CHAPTER FOUR

or hloc D dT * =− k dr * r = R

(4-2-37)

The left side of Eq. (4-2-37) is called the local Nusselt number (Nuloc) and is a function of the coordinate z because hloc also is. Thus we see that the Nusselt number, as well as the film convection coefficient, can be interpreted as a local temperature gradient evaluated at the tube wall. But we have seen in Sec. 4-2-3 that the dimensionless temperature profile remains constant after a certain inlet length. This means that after this inlet section, the local coefficient hloc is constant and independent of the axial position.

4-2-7 Mean Temperature Difference for the Entire Tube Length Continuing with the analysis of the system we were considering, we see that the fluid that enters the heated pipe with a temperature T1 leaves with a temperature T2. This means that while the fluid circulates through the pipe, its temperature increases. Figure 4-22 represents the evolution of the fluid temperature as a function of the axial coordinate z.

Temperature

Tw

T2

T1 Position z=0

z=L

FIGURE 4-22 Temperature variation in the longitudinal direction.

We see that the temperature difference between the wall and the fluid varies from (Tw − T1) at the inlet end to (Tw − T2 ) at the opposite end. We want to define a mean temperature difference between the tube wall and the fluid. If we admit that the film coefficient hloc is constant in the entire tube length, we get dQ = WcdT = hloc πDo dx (Tw − T ) ∴

h πD dx dT = loc o Tw − T Wc

(4-2-38)

(4-2-39)

Integrating, Wc =

hloc πDo L T −T ln w 1 Tw − T2

(4-2-40)

CONVECTION

55

Multiplying both members by (T2 − T1) = (Tw − T1) − (Tw − T2) Wc(T2 − T1 ) = Q = hloc A

(Tw − T1 ) − (Tw − T2 ) = hloc AΔTln T −T ln w 1 Tw − T2

(4-2-41) (4-2-42)

In this equation, ΔTln is the logarithmic mean temperature difference. This is a mean value between the temperature differences at both ends of the tube. Since Eq. (4-2-42) allows calculation of the total heat flow, the local film coefficient hloc will be called, in what follows, merely h on the understanding that it must be used with the logarithmic mean temperature difference ΔTln. 4-2-8 Experimental Determination of the Convection Film Coefficients—Dimensionless Numbers We shall analyze what are the system variables influencing the film heat transfer coefficients. First, the film coefficient h will be a function of the hydrodynamic conditions and flow regime, which are characterized by the Reynolds number. Then h will be a function of all the variables included in the Reynolds number. It also will depend on the fluid properties related to the energy transport. These are the thermal conductivity k and the heat capacity c. Finally, h also will be a function of the tube length L. We then can write h = f (D, v, ρ, μ, c, k, L)

(4-2-43)

Using the techniques of dimensional analysis, we come to the conclusion that the mathematical solution of the problem can be expressed by a relationship of the type ⎛ Dvρ cμ L ⎞ hD = f⎜ , , k k D ⎟⎠ ⎝ μ

(4-2-44)

We have assumed that the fluid properties are constant. In the case of fluids whose properties change considerably with temperature along the system under consideration, the usual approach is to evaluate the physical properties at the arithmetic mean fluid temperature between the inlet and the outlet. The dimensionless groups included in Eq. (4-2-44) are called Nusselt (hD/k), Reynolds (Dvρ/μ), and Prandtl (cμ/k) numbers. The function indicated in Eq. (4-2-44) must be obtained experimentally. 4-2-9

Influence of Wall Temperature As was the case for a flat plate, in the proximity of the tube wall there is always a sublayer in laminar flow. This laminar sublayer represents an important resistance to the heat transfer. The thickness of this sublayer depends mainly on the fluid viscosity in the region of the fluid near the tube wall. For example, let’s consider the following two cases: Case 1: Water circulates through a tube of diameter D with a mass flow rate W kg/s. At a certain section of the tube, the average bulk temperature of the fluid is T = 323 K. The tube wall temperature is 373 K. This means that the fluid is being heated. Case 2: Through the same tube circulates an identical water flow rate. The average bulk temperature at a considered section is again 323 K, but the tube wall is at 273 K. This means that the fluid is being cooled. We see that both cases have the same bulk temperatures, so the Reynolds and Prandtl numbers will be the same. However, in case 1, since the fluid near the tube wall is at high temperature, the thickness of the laminar sublayer will be smaller than in case 2. (This is due to the fact that the viscosity of liquids decreases with increasing temperatures.) The film coefficient h then will be higher in case 1 than in case 2.

56

CHAPTER FOUR

This means that we have to add another variable to the list defined by Eq. (4-2-43), which is the viscosity at the tube wall temperature. The mathematical function expressed by Eq. (4-2-44) then is modified as Nu =

⎛ Dvρ c pμ L μ w ⎞ hD = f⎜ , , , ⎟ k D μ ⎠ k ⎝ μ

(4-2-45)

(It must be noted that in the case of gases, the effect is the opposite because the viscosity of gases increases with temperature.)

4-2-10

Empirical Correlations The form of Eq. (4-2-45) is determined experimentally, and the following empirical correlations are found: 1. Laminar flow (Re < 2100). One of the best known correlations is due to Sieder and Tate:3 hD 0.33 ⎛ μ ⎞ = 1.86 ⎡⎣ Re ⋅ Pr( D /L ) ⎤⎦ ⎜ ⎟ k ⎝ μw ⎠

0.14

(4-2-46)

This equation is valid for heat transfer in laminar flow and with constant tube wall temperature. All the physical properties except μw must be evaluated at the average bulk temperature at the section under consideration or at a mean temperature between the average bulk temperatures of the inlet and outlet sections. Equation (4-2-46) has been reported to have a maximum error of 20 percent for Re · Pr(D/L) > 10. For smaller mass velocities, the values of h obtained are too small. 2. Turbulent flow (Re > 10,000). In the turbulent region, the following correlation, also due to Sieder and Tate,3 is used: Nu =

⎛ μ ⎞ hD = 0.023 Re 0.8 ⋅ Pr 0.33 ⎜ ⎟ k ⎝ μw ⎠

0.14

(4-2-47)

It can be seen, analyzing Eq. (4-2-47), that in turbulent regime the film coefficient h is independent of the ratio D/L. This is due to the fast development of the temperature profiles in turbulent flow. 3. Transition regime (2100 < Re < 10,000). In this region, the flow pattern can be unstable, and thus it is not possible to predict the Nusselt number. In certain cases, as we shall see later, a graphic interpolation between laminar and turbulent curves is performed. However, it is advisable not to design heat transfer equipment operating in this region because of the uncertainties of the results. An empirical expression for cases where the transition zone cannot be avoided is11 ⎛ Re 0.66 − 125 ⎞ ⎡ ⎛ D ⎞ 0.66 ⎤ −0.66 ⎛ μ ⎞ h = 0.116 ⎜ ⎥ Pr ⎟⎠ ⎢1 + ⎜⎝ L ⎟⎠ ⎜⎝ μ ⎟⎠ Re cρv ⎝ w ⎣ ⎦

0.14

(4-2-48)

4-2-11 Other Dimensionless Groups The heat transfer correlations are sometimes expressed as a function of other dimensionless groups that are combinations of the ones we have introduced. A dimensionless group that frequently appears in the literature is the Stanton number, which is defined as St =

Nu h = Re ⋅ Pr ρcv

(4-2-49)

CONVECTION

57

Another is the dimensionless group jH, which is defined7 as ⎛μ ⎞ jH = St ⋅ Pr 2 / 3 ⎜ w ⎟ ⎝ μ ⎠

0.14

=

h 2/3 ⎛ μ w ⎞ Pr ⎜ ⎟ cG ⎝ μ ⎠

0.14

(4-2-50)

where G is the mass velocity or mass flow density = ρv [kg/(s · m2)]. The dimensionless number jH then results exclusively in a function of the Reynolds number and the ratio D/L, which according to Eqs. (4-2-46) and (4-2-47) can be expressed as follows: For laminar flow, jH = 1.86 Re −2 / 3 ( D /L )1/ 3

(4-2-51)

For turbulent flow, jH = 0.023 Re −0.2

(4-2-52)

A graph showing jH as a function of the Reynolds number is presented in Fig. 4-23. μ –0.14 h.D Re–1Pr–1/3 μw k

0.01 0.009 0.008 0.007 0.006 0.005

( )

=

μ –0.14 h Pr2/3 μw c.G

( )

L/D = 60

0.004 (T2 – T1) D μ = Pr 2/3 μ –0.14 0.003 (Tw – T )ln 4.L w

120

( )

180 240

0.002

103

104

Re

105

FIGURE 4-23 Graph of jH versus Reynolds number. (From E. Sieder and G. Tate, Ind Eng Chem 28:1429–1435, 1936. Reproduced with permission from the American Chemical Society.)

In the transition region, the curves were drawn by graphic interpolation between the laminar and turbulent region curves. However, as was explained earlier, it is advisable to avoid this region. Other useful expressions for jH can be obtained by combining the definition of Eq. (4-2-50) with Eq. (4-2-42) and an enthalpy balance for the system. We can write Q=

h(Tw − T1 ) − (Tw − T2 ) πDL = Wc(T2 − T1 ) T −T ln w 1 Tw − T2 h=

(T − T ) Wc ln w 1 πDL (Tw − T2 )

(4-2-53)

(4-2-54)

and replacing in Eq. (4-2-50), we get jH =

D Tw − T1 2 / 3 ⎛ μ ⎞ ln Pr ⎜ ⎟ 4 L Tw − T2 ⎝ μw ⎠

−0.14

(4-2-55)

This expression allows direct calculation of the outlet temperature without explicit calculation of the film coefficient h.

58

CHAPTER FOUR

D. Kern8 presented another definition of jH as ⎛ μ ⎞0.14 hD ⎛ cμ ⎞−1/ 3 ⎛ μ w ⎞0.14 = jH (Kern) = Nu ⋅ Pr −1/ 3 ⎜ w ⎟ ⎜ ⎟ ⎜ ⎟ k ⎝k⎠ ⎝μ ⎠ ⎝μ ⎠

(4-2-56)

Please note the difference with the definition of Eq. (4-2-50). The coefficient jH defined in Eq. ( 4-2-50) was multiplied by the Reynolds number. The advantage of this definition is that the curves of jH (Kern), as a function of the Reynolds number, always slopes upward, and interpolation in the transition region is easier. Example 4-1 A natural gas stream at a pressure of 30 bar and at a temperature of 20°C must be expanded to approximately atmospheric pressure. Since the expansion of a gas through a valve produces a temperature decrease owing to Joule Thompson effect, it is necessary to preheat the gas stream to avoid too low temperatures that may produce ambient humidity condensation and icing on the exterior surfaces of the tubes. It was calculated that the gas must be heated up to a temperature of 40°C before the expansion. The operation will be performed in an indirect heater. This is a unit consisting of a coil, into which the gas circulates, submerged in a hot-water bath (the water is maintained at high temperature with gas burners). For a first estimation of the coil size, we assume that the water temperature can be regulated so that the tube wall temperature is maintained at 60°C. The coil will be built with 2-in Schedule 40 pipe. The gas mass flow is 2,235 kg/h, and its properties are 30

60

Viscosity (cP)

Temp. (°C)

0.0119

0.0134

Thermal conductivity [W/(m · K)]

0.0355

Specific heat [J/(kg · K)] Density (kg/m3)

2,381 23.367

It is desired to know: a. What tube length will be necessary, and what will be the pressure drop b. What will be the gas outlet temperature if the gas mass flow is doubled and if it is reduced to half the design value a. The gas inlet temperature to the coil is 20°C, and the outlet temperature is 40°C. Then all the physical properties will be evaluated at the mean temperature of 30°C. The amount of heat to be delivered is

Solution

Q = Wc(t2 − t1) = (2,235/3,600) × 2,381 × (40 − 20) = 29,564 W The inlet diameter of a 2-in Schedule 40 pipe is 0.0525 m. Thus Flow area = at = π(0.0525)2/4 = 2.164 × 10–3 G = ρv = W/at = 2,235/(3,600 × 2.164 × 10–3) = 287 kg/(s · m2) v = 12.2 m/s Re = DiG/μ = 0.0525 × 287/(0.0119 × 10–3) = 1.266 × 106 Pr = cμ/k = 2,381 × (0.0119 × 10–3)/0.0355 = 0.798 (μ/μw)0.14 = (0.0119/0.0134)0.14 = 0.983

CONVECTION

59

In turbulent flow, hi = 0.023(k/Di)Re0.8Pr0.33(μ/μw)0.14 = 1,081 W/(m2 · K) If the entire wall temperature is maintained at 60°C, the logarithmic mean temperature difference is (Tw − t2 ) − (Tw − t1 ) (60 − 40) − (60 − 20) = 28.8°C = T −t ln(20 / 40) ln w 2 Tw − t1 The necessary area then will be Ai = Q/(hi. LMTD) = 29,564/(1,081 × 28.8) = 0.95 m2, which corresponds to a tube length L = Ai /πDi = 0.95/3.14 × 0.0525 = 5.76 m. The roughness of a commercial steel pipe is 0.045 mm. The relative roughness, then, is 9 × 10–4. The friction factor obtained from Fig. 4-6 is 0.005. The frictional pressure drop in the coil will be ΔP = 4f(L/Di)ρv2/2 = 4 × 0.005 × (5.76/0.0525) × 23.367 × 12.22/2 = 3,815 Pa b. For a mass flow of 4,470 kg/h, the outlet temperature will be different. We do not know the mean temperature of the gas within the coil. As the physical properties of the gas do not change sensibly within the operating temperatures range, we can use the same physical properties as in the preceding case. Since the film coefficient varies with the 0.8 power of the velocity, in this case it will be hi = 1,081 × 20.8 = 1,882 W/(m2 · K) To calculate the outlet temperature, an iterative procedure will be followed: 1. 2. 3. 4.

Assume t2. Calculate logarithmic mean temperature difference (LMTD). Calculate Q = hiAiLMTD. Verify t2 = t1 + Q/Wc.

For an outlet temperature of 38°C, we get LMTD = 30.1. Thus, Q = 1,882 × 0.95 × 30.1 = 53,815 W t2 = 20 + 53,815/(1.241 × 2,381) = 38.2°C We see that the amount of heat transferred is sensibly higher than in the preceding case, and the outlet gas temperature decreases only 2°C. The pressure drop varies with the velocity squared (the friction factor is roughly constant), and it will then be ΔP = 15,260 Pa In the same way, for a gas mass flow of 1,117 kg/h, with the same procedure we obtain t2 = 42°C. 4-2-12

Equivalent Diameter The heat transfer correlations used for fluids flowing within tubes can be extended for non-circular-section conduits. In such cases, it is necessary to replace the internal diameter in Reynolds and Nusselt numbers by an equivalent diameter of the section under consideration. The equivalent diameter is defined as Deq = 4RH

(4-2-57)

where RH is the hydraulic radius, which is RH =

flow area heat transfer perimeter

(4-2-58)

60

CHAPTER FOUR

Example 4-2 A double-pipe heat exchanger is formed by two concentric tubes. One fluid circulates through the inner tube, and the other one through the annular space. Calculate the heat transfer film coefficient for a gasoline stream flowing through the annular space of an exchanger of this type in the following conditions: External diameter of the internal tube = 25.4 mm. Internal diameter of the external tube = 39 mm. Flow rate = 5.2 m3/h Properties: ρ = 710 kg/m3; c = 2,442 J/(kg · K); μ = 0.4 cP; k = 0.18 W/(m · K) Neglect the correction for tube wall temperature. Solution Flow area = π(0.039 − 0.0254 )/4 = 0.000687 m 2

2

2

Heat transfer perimeter = π × 0.0254 = 0.0797 m 2 /m Deq = 4 × 0.000687/0.0797 = 0.0344 m v=

5.2 = 2.1 m/s 3,600 × 0.000687

Pr =

Re =

Devρ 0.0344 × 2.1 × 710 = 128, 200 = μ 0.4 × 10 −3

2,442 × 0.4 × 10 −3 = 5.4 0.18

h = 0.023

k 0.18 Re 0.8 × Pr 0.33 = 0.023 128,200 0.8 5.4 0.33 = 2,565 W/(m 2 ⋅ K) Deq 0.0344

4-3 FORCED CONVECTION AROUND SUBMERGED OBJECTS 4-3-1 Fluids Flow over Submerged Bodies Up to now, in all cases we analyzed, the fluid flow was parallel to the solid surface. However, there are important cases in process equipment where the fluid is incident over the solid surface perpendicularly or at angles to the axes of these bodies. The most important case is that of a fluid flowing at right angles to cylinders. This situation is frequent in heat exchangers, where a fluid circulates inside the tubes of a bundle and another fluid circulates externally in a cross-flow arrangement. Refer to Fig. 4-24. The fluid approaching the cylinder with a velocity v∞ reduces its velocity while it moves toward the body surface. While the fluid slows, the pressure increases. At point A (called the stagnant point), fluid velocity is zero, and the pressure at this point rises approximately one velocity head (that is, ρv2/2) above the pressure in the incoming free stream. From that point, the fluid accelerates again while it flows along the contour of the cylinder. The maximum velocity is reached at an angle θ of approximately 90 degrees, and the pressure is at a minimum in this zone. From this point, the velocity decreases again with another increase in pressure, and a second stagnant point is reached for θ = 180 degrees with maximum pressure. This idealized behavior corresponds to fluids in laminar flow at very low fluid velocities. The pressure distribution is symmetric with respect to the plane corresponding to θ = 90 degrees, which means that the pressure forces do not exert a net force over the body. All the drag force is due to tangential viscous stress. Characterization of the flow regime is done employing a Reynolds number (Dv∞ρ/μ), and the condition for the flow pattern described corresponds to Reynolds numbers below 10.

CONVECTION

61

v∞

θ A

FIGURE 4-24

Fluid flow past a cylinder at low velocities.

A

FIGURE 4-25

Boundary-layer separation at high fluid velocities.

For higher fluid velocities, the viscous forces increase, and part of the kinetic energy of the fluid is irreversibly dissipated as friction. The pressure decrease that takes place in the front half of the cylinder cannot be recovered completely in the rear half. Thus the streamlines separate from the solid surface, adopting the shape indicated in Fig. 4-25. A turbulence region appears in the rear part of the cylinder, and the average pressure in the rear half is smaller than in the front half. The angle at which the streamlines separate from the surface also depends on the Reynolds number. Since the pressure distribution is not symmetric, the integral of the pressure force over the solid surface is not nil and results in a net force acting on the cylinder. The total force thus is the sum of a viscous and a pressure effect. Since the pressure and velocity distributions can only be obtained analytically in simple geometries and laminar flow, in the more general case it is necessary to find another approach to calculate the net force acting on the body. The experimental results normally are expressed by means of a drag coefficient cD defined as 1 F = cD ρv∞2 Ap 2 In this equation, Ap is the body area projected in a direction normal to the flow. The drag coefficients are correlated as a function of the Reynolds number. Figure 4-26 correlates the drag coefficients for the case of cylinders and spheres.

62

CHAPTER FOUR 100 80 60 40 20 10 8 6 cD

4 2 Cylinders 1 0.8 0.6

Spheres

0.4 0.2 0.1 0.1 0.2 0.5 1

2

5 10

102

103

104

105

106

ReD

FIGURE 4-26 Drag coefficient versus Reynolds number for long circular cylinders and spheres in cross-flow. (From C.E. Lapple and C.B. Shepherd, “Calculation of particle trajectories,” IEC 32:606, (1940) Reproduced with permission from the American Chemical Society.)

4-3-2 Convection Heat Transfer Coefficient If a body exposed to a fluid stream exchanges heat with the fluid, a heat transfer coefficient hm can be defined as Q = hm A(Tw − T∞)

(4-3-1)

where Tw = temperature of the body surface T∞ = temperature of the fluid far from the surface (bulk temperature) A = body area hm = average heat transfer coefficient for the entire surface It can be easily understood that since the hydrodynamic conditions are not constant over the body surface, the heat-flux density will not be either. This means that it is also possible to define a local heat transfer coefficient as dQ = hloc dA(Tw − T∞)

(4-3-2)

Here, hloc will depend on the position of dA over the surface. Figure 4-27 is a plot of the local Nusselt number (hlocD/k) versus the angular coordinate θ at given parameters of the Reynolds number for air flowing in cross-flow pattern past a cylinder. It can be appreciated that the local Nusselt number is highly dependent on the position, and the location of the zone with higher heat transfer coefficients depending on the flow regime.

CONVECTION

63

800

700

600 Re 219,000 500

186,000

Nu(θ)

170,000 400 140,000 101,300 300

70,800

200

100

80

40

120

160

Degrees from the stagnant point FIGURE 4-27 Unit heat transfer coefficient around a cylinder normal to an air stream. (From W. H. Giedt, “Investigation of Variation of Point Unit Heat Transfer Coefficient around a Cylinder Normal to an Air Stream”Trans ASME 71: 375-381 (1949). Used with permission from the American Society of Mechanical Engineers.)

However, for practical purposes, calculation of the mean heat transfer coefficient hm defined by Eq. (4-3-1) is of much higher interest. The usual expressions are of the type Nu = f(Pr, Re) where

Nu = average Nusselt number = Re = Reynolds number = Pr = Prandtl number =

hm D k

Dvρ μ

cμ k

Unless specifically stated, all the physical properties to be used in these correlations must be evaluated at the film temperature Tf , defined as T f = (Tw + T∞)/2

64

CHAPTER FOUR

In what follows, the best-known correlations for simple geometries will be presented Correlations for Single Cylinders in Cross-Flow. For gases flowing past cylinders, the correlations are of the type Nu = BRe n

(4-3-3)

where B and n are obtained as a function of the Reynolds number according to Table 4-1.12 In the case of liquids, the values resulting from Eq. (4-3-3) must be multiplied by 1.1Pr1/3. TABLE 4-1 Values of the Constants to Be Used in Eq. (4-3-3) Re

B

n

0.4–4 4–40 40–4,000 4,000–40,000 40,000–400,000

0.891 0.821 0.615 0.174 0.0239

0.33 0.385 0.466 0.615 0.805

Correlations for Spheres. The following correlations were suggested:4 For liquids with 1 < Re < 70,000, Nu = 2 + 0.6 Re1/ 2 ⋅ Pr1/ 3

(4-3-4)

Nu = 0.33Re 0.6

(4-3-5)

Nu = 2.2Pr + 0.48Re1/ 2 ⋅ Pr

(4-3-6)

Nu = 0.37Re 0.6 ⋅ Pr1/3

(4-3-7)

For air with 20 < Re < 150,000,

For other gases with 1 < Re < 25,

And for 25 < Re < 150,000,

Banks of Tubes. This is the case of highest interest in process engineering because it is the geometry found in most heat exchangers. This subject was studied widely by Bergelin, Brown, and Doberstein as part of a research program on heat exchangers at the University of Delaware.5 The experiments were performed with an ideal bank of tubes, as shown in Fig. 4-28. The half tubes located at both sides simulated the continuity of the bank. An ample range of Reynolds numbers was covered, and different geometric arrays were investigated. The definition of the Reynolds number employed to correlate the results is Re =

DGm μ

(4-3-8)

where D = external diameter of the tubes Gm = mass velocity = W/am W = mass flow (kg/s) am = minimum flow area in the system. (This is the free area at the central plane of a tube’s row.)

CONVECTION

65

c

Flow direction FIGURE 4-28

Ideal bank of tubes.

If c = free distance between tubes (see figure), b = width of the tubes bank in the direction normal to the plane of the drawing, and Nt = the number of tubes per row, then am = cbN t (m 2 ) Square and triangular tube patterns with different tube spacing were tested. In all cases, the heat transfer coefficient and the pressure drop of the fluid across the bank were measured. The convection heat transfer coefficient can be correlated with an expression of the type ⎛ h ⎞ ⎛ cμ ⎞ ⎛ μ ⎞ j=⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ cGm ⎠ ⎝ k ⎠ ⎝ μ w ⎠

0.14

⎛ DGm ⎞ = f⎜ ⎝ μ ⎟⎠

(4-3-9)

And the pressure drop is expressed in terms of a friction factor, that is, Δ p= 2f

⎛ μ ⎞ Gm2 Nf ⎜ ⎟ ρ ⎝ μw ⎠

0.14

(4-3-10)

where Nf is the number of tube rows in the bank. The experimental correlations are shown in Fig. 7-27 (in Chap. 7). Analysis of the curves shows the existence of different flow regimes. It is generally accepted that the region of Reynolds numbers below 100 corresponds to a laminar-flow regime. At a Reynolds number around 100, a transition region begins, with the appearance of random eddies. It has been observed that for square arrays, eddies appear simultaneously and suddenly in all the tube rows. In staggered arrays, on the other hand, the eddies appear first in the most downstream rows, and if

66

CHAPTER FOUR

the Reynolds number is increased, the phenomenon propagates upstream. This makes the transition from one regime to the other more gradual in staggered arrays (triangles or rotated squares) than in nonstaggered arrays (squares). (See the shape of the curves in Fig. 7-27.) The results shown in the figure correspond to a bank with 10 tube rows. These results also can be used for a higher number of rows, but the results must be corrected if the number of rows is less than 10.

4-4

NATURAL CONVECTION

4-4-1 Heat Transfer Mechanism in Natural Convection Let’s consider a body at initial temperature To that at a certain time is submerged into a fluid at temperature T∞, where To > T∞. Owing to the temperature difference, heat begins to flow from the body to the fluid. The heat transfer provokes an increase in the temperature of the fluid. Thus the density of the fluid closer to the body decreases below the density of the bulk of the fluid. This density difference creates an ascendant movement of the fluid closer to the body, which is replaced by fresh fluid coming from the colder zones The shapes of the streamlines for several geometries are shown in Fig. 4-29. This fluid movement, in turn, helps to increase the heat transfer by convection because it allows the renewal of the fluid in the vicinity of the body surface.

Thot

Thot

Sphere

Hot vertical plane

Thot

Tcold

Hot horizontal plane

Cold horizontal plane

Tcold

Thot

Enclosed space between walls at different temperatures FIGURE 4-29

Streamlines in natural convection.

CONVECTION

67

The difference with respect to the other cases we studied previously is that now the heat transfer is what provokes the movement of the fluid. The fluid velocity is not imposed externally. It is not a predefined variable. The velocity profile depends on the thermal characteristics of the system. The heat transfer coefficient in natural convection depends, as always, on the fluid properties c, ρ, μ, and k. The cause of the fluid movement is the change in density created by the applied temperature difference. The physical property that indicates how the density varies with temperature is the thermal expansion coefficient β, which is defined as 1 ⎛ ∂ρ ⎞ β=− ⎜ ⎟ ρ ⎝ ∂T ⎠ p=cte

(4-4-1)

For the case of an ideal gas, ρ=

Mp RT

(4-4-2)

Then 1 ⎛ ∂ρ ⎞ 1 β=− ⎜ ⎟ = ρ ⎝ ∂T ⎠ p T

(4-4-3)

We see that for an ideal gas the thermal expansion coefficient is the reciprocal of the absolute temperature. For other fluids, β must be calculated from tabulated data of densities at different temperatures. The convection heat transfer coefficient obviously will be a function of the temperature difference between solid and fluid as well because this temperature difference is the cause of the fluid movement. We then can say h = h [c, μ, ρ, β, (To − T∞ ), k , L ]

(4-4-4)

where L is a characteristic length of the geometry. By applying dimensional analysis techniques, we can find a set of dimensionless groups to correlate the phenomenon. One possible solution is ⎡ L3ρ2 gβ(To − T∞ ) cμ ⎤ hL = f⎢ , ⎥ k ⎦ k μ2 ⎣

(4-4-5)

The first group within brackets is called the Grashoff number (Gr), that is, Gr =

L3ρ2 gβ(To − T∞ ) μ2

(4-4-6)

The other dimensionless groups in Eq. (4-4-12) are the Nusselt and Prandtl numbers. It is observed experimentally that the correlations are usually of the type Nu = a(Gr ⋅ Pr)b

(4-4-7)

This means that the product Gr · Pr behaves as a single dimensionless group in the correlations. The physical properties included in the dimensionless groups are evaluated at the film temperature, which is the arithmetic mean value between the surface temperature and the temperature of the bulk of the fluid.

68

CHAPTER FOUR

4-4-2 Correlations for Different Geometries The correlations for the cases of higher practical interest are presented below. Vertical Plates and Vertical Cylinders. According to Eckert and Jackson,6 Nu L =

hL = 0.555(GrL ⋅ Pr)1/ 4 k

for GrL ⋅ Pr < 10 9

(4-4-8)

Nu L =

hL = 0.021(GrL ⋅ Pr)2 / 5 k

for GrL ⋅ Pr > 10 9

(4-4-9)

In these expressions, L is the height of the plate or cylinder, and h is the average coefficient for the entire plate. Horizontal Cylinders. According to McAdams,4 Nu =

hD = 0.53(GrD ⋅ Pr)1/ 4 k

for 10 4 < GrD ⋅ Pr < 10 9

(4-4-10)

Nu =

hD = 0.114(GrD ⋅ Pr)1/ 3 k

for 10 9 < GrD ⋅ Pr < 1012

(4-4-11)

Horizontal Planar Surfaces. According to McAdams,4 for hot horizontal planes facing upward or cold horizontal planes facing downward, Nu L = 0.54(GrL ⋅ Pr)1/ 4 Nu L = 0.14(GrL ⋅ Pr)1/ 3

for 10 5 < GrL ⋅ Pr < 2 × 10 7 for 2 × 10 7 < GrL ⋅ Pr < 3 × 1010

(4-4-12) (4-4-13)

For hot horizontal plates facing downward and cold horizontal plates facing upward in the range of 3 × 105 < Gr · Pr < 1010, the recommended correlation is Nu L = 0.27(GrL ⋅ Pr)1/ 4

(4-4-14)

In these expressions, the characteristic length is the length of one side if the surface is a square or the mean value of both sides if it is a rectangle or 0.9 times the diameter if it is a circle. Spheres. According to Ranz and Marshall,13 for single spheres in an infinite fluid, Nu D = 2 + 0.60 GrD1/4 ⋅ Pr1/ 3

(4-4-15)

Example 4-3 Calculate what must be the temperature of the bath in Example 4-1 to reach the required 60°C at the tube wall. Solution The heat flux transferred from the external fluid (hot water) to the tube wall must be the same

as that transferred from the tube wall to the process fluid. Thus Q = 29,564 W = ho Ao(T − 60) where T is the temperature of the water, and Ao is the external area of the coil. An iterative procedure must be employed because ho is also a function of the water temperature. The steps are as follows:

CONVECTION

69

1. Assume T. 2. Calculate ho. 3. Verify Q. Let’s assume the T = 90°C. The film temperature of the water is (90 + 60)/2 = 75°C. At this temperature, the physical properties of the water are Density = 974.8 kg/m3 Viscosity = 0.377 cP. Thermal conductivity = 0.666 W/(m · K) Specific heat = 4,190 J/(kg · K) To calculate the thermal expansion coefficient β, we need to know the density at other temperatures. At 80°C, the density of water is 971.8; thus, β=

1 Δρ 1 974.8 − 971.8 = = 6.15 × 10 −4 5 ρ ΔT 973.3

Then ⎛ D 3ρ2 g βΔT cμ ⎞ ⎛ 0.0604 3 × 974.82 × 9.8 × 6.15 × 10 −4 × 30 4,190 × 0.377 × 10 −3 ⎞ Gr.Pr = ⎜ o 2 ⎟ = ⎜ ⎟ = 6.3 × 108 0.666 k ⎠ ⎝ μ (0.377 × 10 −3 )2 ⎠ ⎝ Nu = 0.53(Gr · Pr)0.25 = 84 ho = Nu · k/Do = 84(0.666/0.0604) = 926 W/(m2 · K) Q = hoAo(T − Tw) = 926(πDoL)(90 − 60) = 926(π0.0604 × 5.76) × 30 = 30,347 W This is approximately equal to 29,567 W. It is important to point out that this calculation assumes that the heat transfer surfaces are clean and free from deposits or fouling. When the heat transfer surfaces get dirty, additional resistances to heat transfer must be considered in the calculations. This subject will be explained in more detail in Chap. 5. Simplified Correlations for Air at Atmospheric Pressure. McAdams4 presented simplified correlations that are valid for natural convection in air. These expressions can be obtained by substituting the physical properties of air in the Grashoff and Prandtl numbers so that the coefficient h is merely a function of the temperature difference and the characteristic length. These simplified correlations are shown in Table 4-2. Since they are dimensional equations, the units indicated in the table must be used for the correlations to be valid. In Table 4-3, the physical properties of air at several temperatures are included to allow calculation of the dimensionless groups. 4-4-3 Combined Coefficient of Convection and Radiation When a body is immersed in an infinite ambient, superimposed on the convective mechanism there exists radiation heat transfer, so the heat transfer rate is the sum of the contributions of both mechanisms, as was explained in Chap. 2. That is, q = h(Tw − To ) + σε (Tw4 − To4 )

(4-4-23)

In this expression, σ is 5.67 × 10–8 W/(m2 · K4), and ε is the surface emissivity, which depends on the type and condition of the surface. The thermal insulation of process equipment is usually finished with an aluminum or galvanized iron jacket. Table 4-4 indicates the emissivity of commonly used materials.

70

CHAPTER FOUR

TABLE 4-2

Simplified Equations for Natural Convection in Air at Atmospheric Pressure and Moderate Temperatures

Geometry

Characteristic Dimension L

Flow Regime

Heat Transfer Coefficient ha [W/(m2 · K)]

Range of Gr · Pr

Vertical plates and cylinders

Height

Laminar Turbulent

104–109 109–1013

ha = 1.42(ΔT/L)1/4 (4-4-16) ha = 1.31ΔT1/3 (4-4-17)

Horizontal cylinders

External diameter

Laminar Turbulent

104–109 109–1013

ha = 1.32(ΔT/L)1/4 (4-4-18) ha = 1.24ΔT1/3 (4-4-19)

Horizontal plates, upper surface cold or lower surface hot

As defined in the text

Laminar Turbulent

105–2 × 107 2 × 107–3 × 1010

ha = 1.32(ΔT/L)1/4 (4-4-20) ha = 1.52ΔT1/3 (4-4-21)

Horizontal plates, upper surface hot or lower surface cold

As defined in the text

Laminar

3 × 105–3 × 1010

ha = 0.59(ΔT/L)1/4 (4-4-22)

Note: L in meters; ΔT in °C. These correlations can be extended to higher or lower pressures with respect to atmospheric pressure by multiplying by the following factors: (p/1.033)1/2 for laminar flow and (p/1.032)2/3 for turbulent flow, with p in absolute atmospheres.

TABLE 4-3

Properties of Air Kinematic Viscosity μ/ρ (m2/s)

Temperature (°C)

Thermal Conductivity k [W/(m · K)]

10

0.0250

120.3 × 106

14.4 × 10–6

0.711

20

0.0257

102.9 × 106

15.3 × 10–6

0.709

30

0.0264

87.4 × 106

16.2 × 10–6

0.707

40

0.0272

75.8 × 106

17.2 × 10–6

0.705

50

0.028

65.7 × 10

18.2 × 10

–6

0.704

60

0.0287

57.0 × 106

19.2 × 10–6

0.702

20.2 × 10

–6

0.701

ρ2 gβ μ

2

[1/(m3 · K)]

6

Prandtl Number

70

0.0295

49.4 × 10

80

0.0303

43.1 × 106

21.3 × 10–6

0.699

90

0.0310

38.1 × 106

22.4 × 10–6

0.697

0.0318

33.7 × 10

23.5 × 10

0.695

100

TABLE 4-4

6

6

–6

Emissivities ε

Material Aluminum plate Aluminum-based paint Steel plate White paint Asphaltic mastic

0.1–0.2 0.3–0.7 0.94–0.97 0.84–0.92 0.93

A radiation heat transfer coefficient is sometimes defined as hR =

σε(Tw4 − To4 ) Tw − To

(4-4-24)

Equation (4-4-23) then can be written as Q = (h + hR ) A(Tw − To )

(4-4-25)

CONVECTION

71

4-4-4 Heat Loss through an Insulated Wall in Natural Convection In Chap. 3 we studied the equations to calculate the heat loss through an insulated wall. Let’s consider a vessel containing a hot liquid whose walls are insulated with mineral wool. The heat loss through the wall can be calculated as q = U (Ti − To )

(4-4-26)

1 1 δ 1 = + + U hi k ho

(4-4-27)

with

In Eq. (4-4-27), hi is the natural convection coefficient of the internal side (liquid-wall), and ho is the wallair coefficient; δ is the insulation thickness, and k is its thermal conductivity (the resistance of the metallic wall was neglected). The calculation is not straightforward because both hi and ho depend on the temperature of the walls in contact with the fluids, and these are not known. It is necessary, therefore, to follow an iterative procedure to solve the problem. The proposed methodology is 1. 2. 3. 4.

Make a first guess for U using approximate values for the h’s. Calculate q. Estimate the internal and external temperatures of the wall with ΔT = q/h. With these temperatures, calculate the film temperatures and the convection coefficients with the respective correlations. 5. Recalculate U and q with the new coefficients. 6. Recalculate the wall temperatures. 7. Compare with the previous ones and go back to step 4 if necessary. For a first guess of the heat transfer coefficients, the following values are suggested: For water and aqueous solutions For light organic liquids For heavy organic liquids For gases (combined convection radiation coefficient) For steam as the heating medium

1,000 W/(m2 · K) 350 W/(m2 · K) 75 W/(m2 · K) 10 W/(m2 · K) 8,500 W/(m2 · K)

Example 4.4 Calculate the heat loss of a vertical cylinder tank with 20-m diameter and 10-m height insulated with 11/2-in mineral wool in the vertical walls. The tank roof is not insulated. The tank contains a product whose properties are given in the following table and has to be maintained at a minimum temperature of 22°C when the external temperature is 0°C. It can be considered that the tank is full with liquid to near the roof height. The roof is in contact with the internal vapor atmosphere. The volatility of the product is low, so this internal atmosphere is mainly air. The thermal conductivity of the insulation is 0.048 W/(m · K). Liquid Properties Temperature (°C) ρ (kg/m3) μ (cP) c [J/(kg · K)] k [W/(m ·K)]

22

10

802.4 18 2,260 0.148

809.7 27 2,115 0.148

72

CHAPTER FOUR

Solution

We shall first calculate the thermal expansion coefficient β using density data: β=−

1 Δρ 1 809.7 − 802.4 = = 7.5 × 10 −4 K −1 ρ ΔT 806 12

(to be assumed as constant)

We shall calculate the heat loss by natural convection and radiation from the vertical wall and roof next: Vertical wall Calling

T1: liquid temperature (22°C) T2: metallic wall temperature T3: external temperature of the insulation T4: air temperature (0°C) hi: internal heat transfer coefficient ho: external convection-radiation combined coefficient

We shall first calculate an overall heat transfer coefficient using the estimated h’s mentioned in the preceding section. This will be 1 1 1 Δx 1 1 0.038 = + + = + + = 0.905 (m m 2 ⋅ K)/W U hi ho kins 75 10 0.048 With this coefficient, the heat loss through the wall would be q = U (T1 − T4 ) = (22 − 0) / 0.905 = 24.3 W/m 2 And temperatures T2 and T3 may be calculated as T2 = T1 −

q 24.3 = 22 − = 21.6 hi 75

T3 = T4 +

q 24.3 = 0+ = 2.4 ho 10

The film temperatures for calculation of the physical properties are For air: (2.4 + 0)/2 = 1.2 For the liquid: (22 + 21.6)/2 = 21.8 And the properties are Liquid

Air

Temp (°C)

21.6

1.2

ρ (kg/m3)

802.4

1.33

μ (cP)

18

0.018

c [J/(kg · K)]

2,260

965

k [W/(m · K)]

0.148

0.024

Pr

274

0.71 2.4

ΔT

0.4

L

10

10

Gr

5.84E+9

4.69E11

Gr · Pr

1.6E12

3.34E11

h conv = 0.021k/L(Gr · Pr)0.4

23.6

2.08

CONVECTION

73

The radiation coefficient for the external wall is hR = 5.67 × 10 −8 ε(T34 − T44 ) / (T3 − T4 ) = 5.67 × 10 −8 × 0.2(275.4 4 − 2734 ) / 2.4 = 0.93 W/(m 2 ⋅ K) The external combined coefficient then is 2.08 + 0.93 = 3.01 W/(m2 · K). Calculating U again, we get 1 1 1 0.038 = + + = 1.15 (m 2 ⋅ K)/W U 3.01 23.6 0.048 Then q = U ΔT =

22 = 19.1 W/m 2 1.15

And recalculating the wall temperatures, we get T2 = T1 −

q 19.1 = 22 − = 22.6 hi 28

T3 = T4 +

q 19.1 = 6.3 = 0+ ho 3.01

Since these temperatures do not differ too much from the previous estimation, the physical properties of the fluids will be the same. Recalculating the coefficients: Liquid

Air

Temp (°C)

21.6

1.2

ρ (kg/m3)

802.4

1.33

μ (cP)

18

0.018

c [J/(kg · K)]

2,260

965

k [W/(m · K)]

0.148

0.024

Pr

274

0.71

ΔT

0.4

6.3

L

10

10

Gr

5.84E+9

1.19E12

Gr · Pr

1.6E12

8.47E11

23.6

3.03

β

0.00364

h conv = 0.021k/L (Gr · Pr)0.4

The radiation coefficient for the external wall results: hR = 5.67 × 10 −8 ε(T34 − T44 ) / (T3 − T4 ) = 5.67 × 10 −8 × 0.2(279.34 − 2734 ) / 6.3 = 0.95 W/(m 2 ⋅ K) The external combined coefficient then is 3.03 + 0.95 = 3.98 W/(m2 · K). Recalculating U, we get 1 1 1 0.038 = + + = 1.08 (m 2 ⋅ K)/W U 3.98 23.6 0.048

74

CHAPTER FOUR

Then q = U ΔT =

22 = 20.3 W/m 2 1.08

A new iteration is not necessary. The heat loss through the wall will be 20.3 × π × D × L = 20.3 × π × 20 × 10 = 12,750 W. Heat Loss through the Roof If T1= internal temperature = 22°C Tw = wall temperature (equal at both sides because there is no insulation) Ta = air temperature hi = internal coefficient ho = external coefficient then it must be hi(Ti − Tw) = ho(Tw − To). Each one of the coefficients must be calculated by iteration with the corresponding ΔT. As a first guess, we can assume that the wall temperature is the mean value between the internal temperature and the air temperature; this is 11°C. The film temperatures for the internal and external air are, respectively, 16.5 and 5.5°C. In both cases we shall use the correlations for a hot horizontal flat surface facing upward or a cold surface facing downward. Internal Air

External Air

Temp (°C)

16.5

5.5

ρ (kg/m )

1.262

1.31

μ (cP)

0.019

0.018

c [J/(kg · K)]

950

957

k [W/(m · K)]

0.025

0.025

Pr

0.71

0.71

ΔT

11

11

L

20

20

Gr

1.31E13

1.59E13

Gr · Pr

9.32E12

1.13E13

3

β

0.00345

0.00359

h conv = 0.14(Gr · Pr)0.33

3.39

3.5

It must be noted that even the product Gr · Pr exceeds the recommended value for the correlation we use it because we do not have a more specific correlation available. The radiation heat transfer coefficients are calculated as hRi = 5.67 × 10 −8 ε(Ti4 − Tw4 ) / (Ti − Tw ) = 5.67 × 10 −8 × 0.9(2954 − 284 4 ) /11 = 4.95 W/(m 2 ⋅ K) hRo = 5.67 × 10 −8 ε(Tw4 − To4 ) / (Tw − To ) = 5.67 × 10 −8 × 0.9(284 4 − 2734 ) /11 = 4.41 W/(m 2 ⋅ K) The combined radiation-convection coefficients then will be hi = 3.39 + 4.95 = 8.34 W/(m2 · K) ho = 3.5 + 4.41 = 7.91 W/(m2 · K)

CONVECTION

75

And the overall coefficient will be 1 1 1 = + = 0.246 ∴U = 4.05 W/(m 2 ⋅ K) U 8.34 7.91 The heat-flux density through the roof then will be q = U ΔT = 4.05 × 22 = 89 W/(m 2 ⋅ K) Since the internal and external coefficients are approximately equal, the wall temperature will be very close to the assumed value that was estimated as the arithmetic mean temperature between the internal and external temperatures. Recalculation thus is not necessary. Heat loss through the roof = πD 2 / 4 × 89 = 3.14 × 20 2 / 4 × 89 = 27, 946W Heat Loss through the Floor For a plate at temperature Tw lying on a semi-infinite solid, the heat loss can be calculated as Q = 2 Dks (Tw − Ts ) where D is the plate diameter, Ks is the thermal conductivity of the solid, and Ts is the temperature of the solid far from the plate (see Fig. 3-8). If the soil conductivity is 1.5 W/(m · K), this will be Q = 2 × 20 × 1.5(Tw − 0) = 60(Tw − 0) W To calculate hi , we must use the expression corresponding to a cold plate facing upward, that is, Nu = 0.27(Gr · Pr)0.25 After some trials, we shall assume a wall temperature of 21°C. Since this value is sensibly equal to the bulk liquid temperature, we shall use the physical properties at this temperature ⎛ D 3ρ2 gβΔT cμ ⎞ ⎡ (0.9 × 20)3 × 802.4 2 × 9.8 × 7.5 × 10 −4 × 1 2,115 × 18 × 10 −3 ⎤ 14 Gr ⋅ .Pr = ⎜ ⎟=⎢ ⎥ = 0.23 × 10 0.148 k ⎠ ⎣ μ2 (18 × 10 −3 )2 ⎦ ⎝ The product Gr · Pr is again out of the range of the available correlation, but we shall use it anyway. We get Nu = 591. hi = Nu.k/0.9D = 591 × 0.148/18 = 4.85 W/(m2 · K) Verification of the floor temperature: hi(20 − Tw)πD2/4 = 60(Tw − 0) 4.85(22 − 21) × 314 = 60(21 − 0) And 1,520 = 1,260 reasonably well. Then Q = (1,520 + 1,260)/2 = 1,390 W The total heat loss of the tank thus is Q = 12,750 + 27,946 + 1,390 = 42,086 W

76

CHAPTER FOUR

GLOSSARY A = area (m2) Ap = projected area (m2) a = flow area (m2) c = heat capacity at constant pressure [J/(kg · K)] cD = drag coefficient D = diameter (m) F = force (N) G = mass flow density [kg/(m · s)] h = film coefficient [W/(m2 · K)] i = enthalpy per unit mass (J/kg) i = enthalpy flow (W) jH = dimensionless factor k = thermal conductivity [W/(m · K)] L = length (m) M = molecular weight Nu = Nusselt number p = pressure Pr = Prandtl number Q = heat flux (W) q = heat-flux density (W/m2) R = ideal gas constant R = radius (m) r = radial coordinate (m) Re = Reynolds number T = temperature (K) U = overall heat transfer coefficient [W/(m2 · K)] v = velocity (m/s) W = mass flow (kg/s) x, y, z = rectangular coordinates β = thermal expansion coefficient (1/K) δ = thickness (m) ρ = density (kg/m3) μ = viscosity [kg/(m · s · cP)] τ = shear stress Subscripts 1 = inlet 2 = outlet b = bulk

CONVECTION

f i loc m o t w

77

= friction factor = internal = local = mean = external = tube = wall

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Blasius H.: “Grenzschleten in Flussig Keiten mit Kleiner Reibung,” M: Z. Math u Phys 56(1), 1908. Kreith F., Bohn M.: Principles of Heat Transfer, 6th ed., Brooks Cole, 1998. Sieder E., Tate G.: “Heat Transfer and Pressure Drop of Liquids in Tubes,” Ind. Eng. Chem. 28:1429–1435, 1936. McAdams W. H.: Heat Transmission. New York: McGraw-Hill, 1954. Bergelin O. P., Brown G. A., Doberstein S. C.: “Heat Transfer and Fluid Friction During Flow across Tube Banks,” Trans ASME, 74:953–960, 1952. Eckert E. R. G., Jackson W.: NACA Report 1015, July 1950. Colburn A. P.: Trans AICh E, 29:174, 1933. Kern D.: Process Heat Transfer. New York: McGraw-Hill, 1950. Moody L.: “Friction Factor for Pipe Flow,” Trans ASME, 66:671–684, 1944. Pohlausen E.: “Der Warmeaustausch zwischen festen Korpern und Flüssigkeiten mit kleiner Reibung und kleiner Wärmeleitung,” Z Angew Math Mech, 1:115, 1921. Hansen M.: Z Ver Deut Ing Beih Verfarenstech, 4:91, 1934. Hilpert R.: Forsch Gebiete Ingenieurw, 4:215–224, 1933. Ranz W.E., Marshall W. R.: Chem. Eng. Progr., 48:141–146, 173–180, 1952. Hansen M.: “Velocity Distribution in a Boundary Layer of a Submerged Plate,” NACA Report 582, 1930.

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CHAPTER 5

FUNDAMENTALS OF HEAT TRANSFER BETWEEN FLUIDS

In almost all process industries, it is necessary to perform heat transfer unit operations over fluid streams. The purpose of these operations may be either to heat up or cool down a fluid stream or to produce a change of phase either by vaporization or condensation. For a fluid stream to deliver heat, it is necessary to have another stream capable of receiving it. Then both streams can interchange heat, and this operation is performed in a piece of equipment called a heat exchanger. The streams handled in a process plant can be divided into two groups: 1. Process streams 2. Utility streams Process streams are the streams that participate in the mass balance of the plant. They can be feedstocks, reaction products, fractionation products, effluents, intermediate streams, etc. Utility streams are the streams that do not participate in the mass balance of the plant because they do not mix with the process streams, but they can affect the energy balances. The most typical examples are the steam used as heating media and the cooling water of the plant refrigeration circuits. Usually in process industries, steam and cooling water installations are centralized, and they serve several process units. The streams in a heat exchanger can be process or utility streams. If there is a process stream that needs to be heated up and another stream that needs to be cooled down, it may be possible to put them in contact as a way to save energy. A typical example can be found in distillation operations. Usually the feed stream to a distillation column has to be heated up before being introduced into the column. The column bottom product, in turn, must be cooled down before being sent to storage. Since the temperature of the column bottoms is always higher that the column feed temperature, it is possible to exchange heat between them, as shown in Fig. 5-1 (unit A). If it is not possible to combine process streams this way, it will be necessary to perform the heating or cooling operations with utility streams. For example, in Fig. 5-1 we can see a reboiler that is used to vaporize the liquid column bottoms, delivering the heat required by the process. This unit (B) receives steam as heating media. (Note: Reboiler is the term specifically used to designate heat exchangers used to vaporize distillation column bottoms.) As another example, at the top of the column we can see a condenser (C). This piece of equipment is used to condense the light vapors coming from the column. In this case, the heat is withdrawn with cooling water. Even though there is no universally accepted criterion, we shall normally use the term heat exchanger to designate units in which sensible heat is transferred between streams. When there is a change of phase of the process stream, the units will be designated as condensers, vaporizers, reboilers, etc. In what follows, we shall study the principles of heat transfer between fluids.

5-1 THE ENTHALPY BALANCE Most process plants operate in a continuous mode, and we normally refer to the heat transferred per unit time. We shall call W the mass flow rate (kg/s) of a stream. The subscript h will make reference to the hot stream, and subscript c will designate the cold stream. In all cases, subscript 1 will denote heat exchanger inlet conditions, whereas subscript 2 will refer to outlet conditions of any stream. 79

80

CHAPTER FIVE

C

Cooling water

Light product A

Feed

Heavy product

Steam

B FIGURE 5-1

Heat transfer units in a distillation process

To designate the fluid temperatures, we shall use T for the hot-stream temperatures and t for the coldstream temperatures. Q will be the heat flow or heat exchanged per unit time (J/s) If a hot stream delivers heat to a cold medium, it will suffer an enthalpy decrease. Then Q = Wh (ih1 − ih 2 )

(5-1-1)

where i is the specific enthalpy (J/kg). If the fluid cools down without a change of phase, the enthalpy difference can be expressed as ih1 − ih 2 = ch (T1 − T2 )

(5-1-2)

where c is the heat capacity or specific heat [J/(kg · K)]. It then can be said that Q = Whch (T1 − T2 )

(5-1-3)

If the hot fluid is a vapor of a pure substance, saturated at a certain pressure, when delivering heat, it will suffer an isothermal condensation, and ih1 − ih 2 = λ h

(latent heat of condensation)

(5-1-4)

Then, in this case, Q = Wh λ h

(5-1-5)

If the fluid is a mixture of vapors, the condensation will not be isothermal, and Eq. (5-1-1) must be used, calculating ih1 as the specific enthalpy of the vapor mixture at its initial state and ih2 as the enthalpy of the condensed liquid at its final state. It is also possible that the hot fluid is a superheated vapor, which initially cools down to its saturation temperature and then undergoes an isothermal condensation. Additionally, this condensation can be followed by further cooling of the condensed liquid. In this case, Eq. (5-1-1) also must be applied, calculating the specific enthalpies of the initial and final state.

FUNDAMENTALS OF HEAT TRANSFER BETWEEN FLUIDS

81

The heat delivered by the hot stream is received by the cold stream. This fluid will suffer an evolution inverse to the one just described. If the fluid heats up without a change of phase, then Q = Wccc (t2 − t1 )

(5-1-6)

And if it undergoes an isothermal vaporization, then Q = Wc λ c

(5-1-7)

Q = Wc (ic 2 − ic1 )

(5-1-8)

where λ c is the latent heat of vaporization. Generically, for any situation,

Obviously, the heat delivered by the hot fluid is the same heat absorbed by the cold fluid. Thus Q = Wc (ic 2 − ic1 ) = Wh (ih1 − ih 2 )

(5-1-9)

This expression may assume the following particular forms: 1. Heat exchange between two fluids without change of phase: Whch (T1 − T2 ) = Wccc (t2 − t1 )

(5-1-10)

2. Condensation of a pure saturated vapor with a refrigerant medium that exchanges sensible heat: Wh λ h = Wccc (t2 − t1 )

(5-1-11)

3. Evaporation of a pure saturated liquid with a heating medium that exchanges sensible heat: Whch (T1 − T2 ) = Wc λ c

(5-1-12)

4. Evaporation of a pure saturated liquid heated with steam or another condensing pure vapor: Wh λ h = Wc λ c

5-2

(5-1-13)

HEAT TRANSFER AREA AND HEAT TRANSFER COEFFICIENT To perform a heat transfer operation between two fluids, the following are required: 1. A temperature difference between both streams. The higher this difference, the higher will be the heat transfer rate. 2. Both fluids must be separated by a surface through which the heat can be transferred. This is called the heat transfer area. For example, if one of the fluids circulates through a tube, and the other is outside the tube, the heat transfer area is the surface of the tube. It is easy to imagine that the larger the contact area between both fluids, the higher will be the amount of heat that can be exchanged per unit time. The two preceding statements can be summarized in the following expression: Q = UAΔT

(5-2-1)

where the proportionality constant U is called the overall heat transfer coefficient and ΔT is the temperature difference between both streams. We see that in problems dealing with heat transfer between fluids, there are two types of equations. Equations (5-1-1) to (5-1-13) represent the heat balances. They allow calculation of the amount of heat that

82

CHAPTER FIVE

must be transferred to achieve a certain process condition in the streams that participate in the operation. These equations are associated with the thermodynamics of the process and are completely independent of the design of the equipment in which the process is performed. On the other hand, Eq. (5-2-1) represents the kinetics of the heat transfer process. It allows calculation of the area of the heat transfer device needed to achieve a heat flux Q between two streams whose temperature difference is ΔT. This area depends on the heat transfer coefficient U, which can be modified by changing the characteristics of the apparatus. The basic objective of the design will be to reach the highest possible value for U compatible with all process restrictions. In any equipment design problem, both types of equations are always present. This means that it will be necessary to combine a balance equation with a kinetic equation that will allow evaluating whether the heat transfer area is large enough to achieve the desired objective. The third type of equation includes those necessary to calculate the heat transfer coefficient U. All the art and science of heat transfer are centered on this point. The heat transfer coefficient U is calculated as a function of the film coefficients, as we shall see in what follows.

5-3 EXPRESSION OF THE OVERALL HEAT TRANSFER COEFFICIENT AS A FUNCTION OF THE FILM COEFFICIENTS 5-3-1

Individual Heat Transfer Resistances Let’s assume that we have a tube with a hot fluid at temperature T circulating through it. The tube is submerged into a cold fluid at temperature t (Fig. 5-2). Let Di and Do be the diameters of the internal and external surfaces of the tube, respectively. In order for heat to flow by conduction through the tube wall, from the internal to the external surfaces, there must be a temperature difference between them. Let us call Twi the temperature of the internal surface and Two the temperature of the external surface of the tube wall. We then can express the heat transfer rate between the internal fluid and the wall as Q = hi Ai (T − Twi )

(5-3-1)

where hi is the convection film coefficient for the internal fluid. Through the tube wall, heat is transmitted by conduction, and we can write Q=

kAm (Twi − Two ) 1/ 2( Do − Di )

(5-3-2)

where k = thermal conductivity of the tube material Fluid at temperature t Two Twi Do

Di

FIGURE 5-2

Fluid at temperature T

Heat transfer through a tube wall.

FUNDAMENTALS OF HEAT TRANSFER BETWEEN FLUIDS

83

Am = logarithmic mean area Am =

πL ( D0 − Di ) D ln 0 Di

(5-3-3)

The heat transfer from the external wall to the cold fluid can be expressed as Q = ho Ao (Two − t )

(5-3-4)

Equations (5-3-1), (5-3-2), and (5-3-4) can be written as T − Twi = Q

Twi − Two = Q Two − t = Q

1 hi Ai

(5-3-5)

1/ 2( Do − Di ) kAm

(5-3-6)

1 ho Ao

(5-3-7)

By summing the previous expressions, we get ⎡ 1 1/ 2( Do − Di ) ⎤ 1 T −t = Q⎢ + + ⎥ h A h A kAm o o ⎣ i i ⎦

(5-3-8)

Comparing with Eq. (5-2-1), we see that 1/ 2( Do − Di ) 1 1 1 = + + UA hi Ai ho Ao kAm

(5-3-9)

This expression is not completely defined because we have not yet decided which is the area A. In this case, we have an internal area Ai and an external area Ao. Either of them can be used to define a heat transfer coefficient U. Obviously, the value of this coefficient will depend on which area we choose. If we refer to the external area, the expression defines a coefficient Uo , and if we refer to the internal area, we shall be defining a coefficient Ui , and it will be 1/ 2( Do − Di ) 1 1 1 1 = = + + U o Ao Ui Ai hi Ai ho Ao kAm

(5-3-10)

1 1 1 1/ 2( Do − Di ) = + + U o h1 ( Ai /Ao ) ho k ( Am /Ao )

(5-3-11)

1/ 2( Do − Di ) 1 1 1 = + + Ui hi ho ( Ao /Ai ) k ( Am /Ai )

(5-3-12)

or

Either of these two coefficients can be used. However, since normally the heat exchanger tubes are standardized by their external diameter, the current practice is to use the coefficient Uo defined by Eq. (5-3-11). From now on, we shall always make reference to it, unless specifically otherwise indicated. For simplicity, we shall drop the subscript and use the symbol U. For the same reason, the external area simply shall be called A.

84

CHAPTER FIVE

Since Eq. (5-2-1) can be written as Q=

ΔT (1/UA)

(5-3-13)

if we remember Ohm’s law of the electric circuits, that is, i=

ΔV R

(5-3-14)

where i = current intensity ΔV = electrical potential difference R = resistance then, looking at Eqs. (5-3-13) and (5-3-14), we see that an analogy between both phenomena can be established, and we can consider the term 1/(UA) as a resistance to the heat transmission and ΔT as the driving force that provokes the heat flux. The resistance, in turn, is formed by three resistances in series, represented by the three terms of Eq. (5-3-12). Normally, the heat exchanger tubes are made from metal, and owing to their high thermal conductivity, the third term of Eq. (5-3-12) can be neglected, resulting in 1 1 1 = + U hi ( Ai /Ao ) ho

(5-3-15)

It is possible to define an internal film coefficient referred to the external area hio as hio = hi(Ai /Ao) and then Eq. (5-3-15) can be written as 1 1 1 = + U hio ho 5-3-2

(5-3-16)

Fouling Resistance When heat transfer equipment has been operating for a certain time, scale, algae, suspended solids, insoluble salts, or other fouling agents can deposit over the internal or external surfaces of the tubes. This adds two additional resistances to those previously considered in the calculation of U because the heat has to flow by conduction through the two scale layers, as shown in Fig. 5-3. We shall call the additional internal and external fouling resistances Rfi and Rf0, respectively. Their units are [(m2 · K · s)/J]. Then the overall resistance to the heat transmission, once these fouling layers are formed, is given by 1 1 1 = + + Rfi + Rfo U hio ho

(5-3-17)

The combined fouling resistance is the sum Rf = Rfi + Rf0

(5-3-18)

As long as the tubes foul and the resistance to heat transfer grows, the heat transfer capacity of the unit is reduced. For this reason, heat transfer equipment designers must make allowances to anticipate these deposits. It is necessary to perform an estimation of the value that this resistance can reach in the service time between two exchanger cleanings.

FUNDAMENTALS OF HEAT TRANSFER BETWEEN FLUIDS

Rfo

FIGURE 5-3

85

Rfi

Fouling resistances.

This resistance can be estimated on the basis of designer knowledge and previous experience with the fluids or with the help of tables that give typical values for different applications. A fouling resistances table is included in App. F. Once the fouling resistances corresponding to each fluid have been selected, the overall heat transfer coefficient can be calculated with Eq. (5-3-17). The heat transfer area then is obtained as A=

Q U ΔT

(5-3-19)

When the heat exchanger is put into service, during the initial plant startup or after a thorough cleaning operation, the fouling layer is not present, and the overall heat transfer coefficient is given by Eq. (5-3-16). We shall call Uc (clean coefficient) the term defined by 1 1 1 = + Uc ho hio

(5-3-20)

Uc then will be higher than the design U. This means that at the beginning of operations, the exchanger will be capable to transfer a heat flux Q' given by Q' = Uc AΔT

(5-3-21)

with Q' > Q. Usually it is always possible to adjust the performance of a unit when it is oversized, for example, by reducing the fluid flow rates (see Examples 5-6 and 5-7), so this oversizing normally is not a problem. As long as the unit fouls, the heat transfer resistance increases up to the design value. At that time, the exchanger must be cleaned because otherwise it will not be capable of transferring the required heat flow. Example 5-1 It is necessary to exchange 500,000 W between two fluids. Their mean temperature difference is 50 K. The film heat transfer coefficients ho and hio were calculated as 1,500 and 3,000 J/(s · m2 · K), respectively. Previous experience allows us to assume that fouling resistances of 0.0002 and 0.0003 (m2 · K)/W will be deposited annually on the internal and external surfaces of the tubes, respectively. Calculate the required heat transfer area of the exchanger that will allow continuous operation for 1 year between cleanings. Solution

The clean heat transfer coefficient will be Uc = (1/hio + 1/ho)−1 = (1/1,500 + 1/3,000)−1 = 1,000 J/(s · m2 · K)

86

CHAPTER FIVE

The accumulated resistance after 1 year will be Rf = Rfi + R fo = 0.0003 + 0.0002 = 0.0005 (m2 · s · K)/J And the overall dirty heat transfer coefficient will be U = (1/Uc + R f)−1 = (1/1,000 + 0.0005)−1 = 666 J/(s · m2 · K) The required area is A = Q/(UΔT) = 500,000/(666 × 50) = 15 m2 If fouling were not considered, the required area would have been A = Q/(Uc ΔT ) = (500,000/1,000 × 50) = 10 m2 This means that fouling considerations make a 50 percent increase in the heat transfer area necessary. 5-3-3

Controlling Resistance When calculating the overall heat transfer coefficient, it may happen that one of the three terms of the equation 1 1 1 = + + Rf U hio ho

(5-3-22)

is of a higher order of magnitude than the others. In this case, it is said that this is a controlling resistance, and the overall coefficient U practically coincides with the corresponding film coefficient. For example, if ho Wc cc

T2 Larger dQ/dx or dt /dx t1

T2 Smaller dQ/dx t1

Longitudinal coordinate FIGURE 5-11

91

Longitudinal coordinate

T-x curves in countercurrent configuration.

T1

T1 T2

t2

T2

t2 t1

A

B

C

t1

D

Heat exchanged

Heat exchanged

(a) Parallel current

(b) Countercurrent

FIGURE 5-12

T-Q curves in parallel-current and countercurrent configurations.

and B is the same as that transferred between sections C and D, the heat transfer area existing between sections A and B is smaller than that between sections C and D because a higher temperature difference exists in the first case.

5-4-3

Logarithmic Mean Temperature Difference Figure 5-13 represents a countercurrent heat exchanger. At the section where the axial coordinate is x, the temperature of the hot fluid is T, and the temperature of the cold fluid is t. At the section corresponding to coordinate x + dx, these temperatures are T + dT and t + dt. In this case, both increments are positive because both temperatures increase with coordinate x. A heat balance is dQ = WhchdT = Wcccdt

(5-4-10)

dQ = UdA(T − t) = Uπ Dodx(T − t)

(5-4-11)

dQ = dT Whch

(5-4-12)

dQ = dt Wccc

(5-4-13)

but From Eq. (5-4-10), we obtain

92

CHAPTER FIVE

T2

t1

t

t + dt

T

T + dT

t2

T1 T1

t2 T2

t1 x

x + dx

FIGURE 5-13 Temperatures evolution in a countercurrent heat exchanger.

Substracting, we get ⎛ 1 1 ⎞ dQ ⎜ − = d (T − t ) ⎝ Whch Wccc ⎟⎠

(5-4-14)

Additionally, considering Eq. (5-4-11), dQ = (T − t ) U πDo dx

(5-4-15)

Dividing Eq. (5-4-14) by Eq. (5-4-15), we get ⎛ 1 1 ⎞ d (T − t ) = πDo dxU ⎜ − T −t ⎝ Whch Wccc ⎟⎠

(5-4-16)

This differential equation can be integrated with the following limits: For x = 0, T – t = T2 – t1. For x = L, T – t = T1 – t2. And we get ⎛ 1 1 ⎞ T −t πDo L.U ⎜ − = ln 1 2 ⎟ T2 − t1 ⎝ Whch Wccc ⎠

(5-4-17)

FUNDAMENTALS OF HEAT TRANSFER BETWEEN FLUIDS

93

The total heat exchanged in the unit can be expressed as Q = Whch(T1 − T2)

(5-4-18)

Q = Wccc(t2 − t1)

(5-4-19)

T −T 1 = 1 2 Whch Q

(5-4-20)

t −t 1 = 2 1 Wccc Q

(5-4-21)

Or

Then

Substituting Eqs. (5-4-20) and (5-4-21) into Eq. (5-4-17) and reordering gives ⎤ ⎡ ⎢ (T1 − t2 ) − (T2 − t1 ) ⎥ Q = (πDo L )U ⎢ ⎥ T −t ⎢ ⎥ ln 1 2 T2 − t1 ⎢⎣ ⎥⎦

(5-4-22)

The term within brackets is the logarithmic mean temperature difference between both fluids. This means the logarithmic mean between the ΔT values at both ends of the exchanger. By comparing with Eq. (5-4-5), we see that Eq. (5-4-22) tells us that the mean temperature difference that must be used to establish a relationship between the total area of the unit and the total heat transferred is this logarithmic mean temperature difference (LMTD). Equation (5-4-22) has been demonstrated for a countercurrent heat exchanger but is also valid for a co-current configuration. In this case, the expression for LMTD is LMTD =

(T1 − t1 ) − (T2 − t2 ) (T − t ) ln 1 1 (T2 − t2 )

(5-4-23)

We must remark that LMTD can be used as the mean temperature difference for the unit as long as all the hypotheses made in the deduction are fulfilled. These are • The overall heat transfer coefficient is constant. • The fluid heat capacities are constant. These hypotheses are not always satisfied, particularly in cases in which in addition to sensible heat, the fluids also exchange latent heat. In such cases, it is necessary to develop special expressions or to divide the unit into area elements and perform a numerical integration. Example 5-2 It is desired to cool down 0.34 kg/s of an aqueous solution with a heat capacity similar to water from 60 to 50°C. The cooling fluid will be 0.3 kg/s of water at 25°C. A double-pipe heat exchanger will be used. The external diameter of the interior tube is 0.025 m. The overall heat transfer coefficient is estimated as 1,600 W/(m2 · K). Calculate the necessary tube length if a. A countercurrent arrangement is used. b. A co-current arrangement is used.

94

CHAPTER FIVE

Solution

Heat balance: Q = Whch(T1 − T2) = 0.34 × 4,180 × (60 − 50) = 14,210 J/s

Cooling-water outlet temperature: t2 = t1 + Q/Wccc = 25 + 14,210/(0.30 × 4,180) = 36.3°C Calculation of LMTD: a. Countercurrent: DMLT =

(T1 − t2 ) − (T2 − t1 ) (60 − 36..5) − (50 − 25) = = 24.3°C (T − t ) (60 − 36.5) ln ln 1 2 (50 − 25) (T2 − t1 )

b. Co-current: DMLT =

(T1 − t1 ) − (T2 − t2 ) (60 − 25)) − (50 − 36.3) = = 22.7°C (T − t ) (60 − 25) ln ln 1 1 (50 − 36.3) (T2 − t2 )

The required area in each case is A=

Q U ⋅ DMLT

This means that 14,210/(1,600 × 24.3) = 0.36 m2

for countercurrent

14,210/(1,600 × 22.7) = 0.391 m2

for co-current

and

The length of the unit in each case is L=

A πDo

This means that L = 0.365/(π × 0.025) = 4.64 m

for countercurrent

L = 0.391/(π × 0.025) = 4.97m

for co-current

and

5-5 CALCULATION OF THE OVERALL HEAT TRANSFER COEFFICIENT AND PRESSURE DROP FOR DOUBLE-TUBE HEAT EXCHANGERS The overall heat transfer coefficient U can be calculated from the film coefficients according to Eq. (5-3-22). To calculate these coefficients, the usual correlations for heat transfer within tubes must be employed.

FUNDAMENTALS OF HEAT TRANSFER BETWEEN FLUIDS

5-5-1

95

Internal Coefficient hi hi is calculated as a function of the Reynolds number, defined as Re =

Di ρv Di G = μ μ

(5-5-1)

where G = mass flow density = ρv. The correlations are 1. For a laminar regime (Re < 2,100),2 Nu = 1.86[Re · Pr(Di /L)]0.33(μ/μw)0.14

(5-5-2)

where Nu = Nusselt number = hi Di /k Pr = Prandtl number = cμ/k μ/μw = ratio between viscosity at the mean fluid temperature and viscosity at the mean tube wall temperature In Eq. (5-5-2), L is the length of the fluid path before a mixing or homogenization takes place. We can think that for a double-tube heat exchanger such as that shown in Fig. 5-14, the temperature of the fluid in the annulus becomes uniform when the fluid passes from one tube to the following through the

Gland

Gland

See detail Gland

Packing (a) Single hairpin T1

t2

t1 T2 (b) Hairpins in series FIGURE 5-14

Hairpin heat exchanger.

96

CHAPTER FIVE

union tees. Then L should be taken as the length of one pass. For the fluid circulating inside the internal tube, Kern1 suggests adopting the total length for L. This means the sum of the lengths of all passes. All the physical properties of Eq. (5-5-2), with the exception of μw, must be taken at the mean fluid temperature in the tube. 2. For a turbulent regime (Re > 10,000),2 Nu = 0.023Re0.8 · Pr0.33(μ/μw)0.14

(5-5-3)

where the physical properties also should be evaluated at the mean fluid temperature, with the exception of μw, which is evaluated at the mean wall temperature. 3. The transition regime (2,100 < Re < 10,000) is a highly unstable zone, and all the correlations that have been suggested present important deviations. Some authors1 perform graphic interpolations between the correlations corresponding to the other two regimes, but the best recommendation is to avoid this zone in the design. If this is not possible, Eq (4-2-48) should be used. 4. For water,3 the following dimensional correlation has been suggested: hi = 1,423(1 + 0.0146t)v0.8/Di0.2

(5-5-4)

where hi = J/(m · s · K) t = mean temperature of the water (°C) v = velocity (m/s) Di = internal diameter (m) 2

Kern1 presents a graph whose values have good agreement with this correlation, and he extends the application range up to water velocities of 0.3 to 3 m/s, diameters between 0.01 and 0.05 m, and temperatures between 5 and 95°C.

5-5-2 Calculation for the Annulus Fluid The same correlations used for the tube fluid are still valid, but the internal diameter must be substituted by the annulus equivalent hydraulic diameter. This is defined as Deq = 4 ×

flow area wetted perimeter

(5-5-5)

For heat transfer, the wetted perimeter is that corresponding to the internal tube. This is πDo. Then Deq = 4 ×

(

π Ds2 − Do2 4 πDo

)

(5-5-6)

where Ds is the internal diameter of the external tube. 5-5-3 Pressure Drop in Double-Tube Heat Exchangers Each time a heat exchanger must be designed or rated, it is necessary to predict the pressure drop of the fluids through the heat exchanger. This can be calculated with the usual expressions for fluid flow using a friction factor that is obtained as a function of the Reynolds number. Calculations for the Internal-Tube Fluid. The usual expression to calculate pressure drop through a pipe is Δp = 4 f

L v2 ρ Di 2

(5-5-7)

FUNDAMENTALS OF HEAT TRANSFER BETWEEN FLUIDS

97

This equation is valid for isothermal fluids. It is usually corrected for heating or cooling by multiplying by a factor (μ/μw)a, where a = −0.14 for turbulent flow and −0.25 for laminar flow. The friction factor that must be used in Eq. (5-5-7) is the Fanning factor. The graphs of the friction factor as a function of the Reynolds number can be obtained from Chap. 4, or the following expressions can be used: 1. For the zone of Re < 2,100, the friction factor can be calculated with the following expression derived from the Hagen Pouiseuille equation: f = 16/Re

(5-5-8)

2. In the turbulent region (Re > 2,100), the friction factor depends on the roughness of the tube material. However some simplified correlations that are valid for particular situations have been suggested. For 3/4- or 1-in smooth tubes, a recommended expression4 is f = 0.0014 + 0.125/Re0.32

(5-5-9)

And for commercial steel heat exchanger tubes,5 f = 0.0035 + 0.264/Re0.42

(5-5-10)

Calculations for the Annulus Fluid. In this case, the same expressions are valid, but again, the internal diameter must be substituted by the equivalent hydraulic diameter. The equivalent hydraulic diameter is also defined by Eq. (5-5-5), but the wetted perimeter is, in this case, the friction perimeter, which is the sum ' of the perimeters of both tubes. This equivalent hydraulic diameter will be called Deq and is ' Deq = 4×

(

)

π Ds2 − Do2 / 4 π( Do + Ds )

(5-5-11)

5-5-4 Double-Tube Heat Exchanger Applications The double-tube heat exchanger is one of the simplest pieces of equipment in which perform heat exchange in a continuous mode between two fluids. Figure 5-14 represents a hairpin of a small home-made heat exchanger. A hairpin is formed by two sets of concentric tubes with the corresponding connection pieces. In cases where a higher heat transfer area is required, it is possible to add several hairpins, one after the other, in a series configuration (see Fig. 5-14b). It can be seen that all parts of the unit are standard pipe fittings, so the assembly of this type of equipment does not required highly skilled labor and can be performed in a modest shop by a pipe fitter. It also can be appreciated in the figure that in order to avoid fluid leaks, it is necessary to install packing elements at both ends with their corresponding glands. This is necessary because the unit must be disassembled for cleaning, so welded unions are not allowed. In packed unions, it is usual to have some leakage, so the glands must be adjusted periodically. Additionally, disassembling the unit is complicated and time-consuming. These are the reasons why this type of unit is not used very much in industry. Additionally, the maximum tube length that is usually employed is 6 m. Longer tubes can present too high a deflection and distortion of the annular space, which can cause poor flow distribution. The maximum heat transfer area of a hairpin thus is quite small, and it would be necessary to use a great number of hairpins for most industrial applications. The unit thus is not compact, and maintenance labor is high. The tubes employed in the construction of these heat exchangers are usually any of the combinations indicated in Table 5-1. Even though an increase in tube diameter increases the cross sections and makes it possible to handle higher flow rates, it also results in an unfavorable heat transfer-area/flow-area ratio. For example, a 3-cm-diameter tube has a flow area equal to 0.032π/4 = 0.0007 m2. The lateral area per meter of tube is π × D = π × 0.03 = 0.094 m2/m. If, instead, two 2-cm-diameter tubes were employed, the flow area would

98

CHAPTER FIVE

TABLE 5-1

Standard Combinations for Double-Tube Heat Exchangers Dimensions in Inches

φ External Tube 2 2½ 3 4

φ Internal Tube ¾ ¾ ¾ ¾

1 1 1 1

1¼ 1¼ 1¼ 1¼

1½ 1½

2 2

3

Tubesheet

Split ring

Header flange FIGURE 5-15

Shell flange Backup flange Multitube hairpin heat exchanger.

be 2 × π × 0.022/4 = 0.00063 m2 (a little smaller than before). The lateral heat transfer area, however, is 2πD = 2 × π × 0.02 = 0.125 m2/m. This means that by using two 2-cm-diameter tubes, we get a somewhat higher fluid velocity (which means better heat transfer coefficients) and higher heat transfer area than with a single 3-cm-diameter tube. Thus, from the heat transfer point of view, the first option is more convenient. This is the reason why the diameter of the tubes cannot be increased indefinitely. If the flow rates of the fluids require higher flow areas than those indicated in Table 5-1, additional units in parallel should be used. It is also possible, when it is necessary to increase the flow area for the internal-tube fluid, to manufacture heat exchangers with more than one internal tube inside a single external tube. In this case, the construction is more complex because it is always necessary to be able to disassemble the unit for cleaning. These units are specifically called hairpin heat exchangers. Figure 5-15 shows a sectional view of one hairpin heat exchanger model. The tubes are inserted and welded into a tubesheet. The tubesheet has a slot where a split metallic ring is inserted. This ring is split into two halves so that it can be inserted and withdrawn from the slot. The backup flange compresses the split ring against the body flange, and the unit thus is closed. To disassemble the unit, both end heads are withdrawn, and after removing the split ring, the tubes and tubesheet assembly can be extracted to the right. These units are usually constructed by specialized companies, and there are several patent-protected closure systems. They can be a cost-effective solution in cases where the flow rates are small and it is necessary to keep a countercurrent configuration. They are limited to a few tubes because for bigger sizes this type of construction becomes difficult. They are not a competitive solution against the shell and tube heat exchangers that will be studied in later chapters. We can then say that hairpin heat exchangers are limited to applications where the required heat transfer area is less than 10 or 15 m2.

5-6 HEAT EXCHANGER PROCESS SPECIFICATIONS The thermal design of a heat exchanger consists of the definition of its geometric configuration to perform a certain process service. This service is determined by the process conditions that must be achieved, and the designer must look for a unit capable of satisfying them at a minimum cost.

FUNDAMENTALS OF HEAT TRANSFER BETWEEN FLUIDS

99

The process data consist of • • • •

Specifications related to thermal performance Specifications related to pressure drop of the streams Fouling factors Geometric restrictions

We shall see how these specifications must be considered in the design.

5-6-1 Specifications Related to Thermal Performance The process variables that define the performance of a heat exchanger are related by the thermal balance equations: Q = Whch (T1 − T2 )

(5-6-1)

Q = Wccc (t2 − t1 )

(5-6-2)

In this system of equations, there are seven process variables: Wh, Wc, T1, T2, t1, t2, and Q. For the system to be defined, it is necessary to specify five, and the two remaining can be calculated with Eqs. (5-6-1) and (5-6-2). For example, a usual situation is to know the process flow rates, both inlet temperatures, and the outlet temperature of one of the streams. Then the other outlet temperature and the heat duty Q can be calculated from Eqs. (5-6-1) and (5-6-2). In other cases, the process conditions require to cool-down a hot stream with flow rate Wh from a certain inlet temperature T1 to an outlet temperature T2 using cooling water available at a certain t1. The design conditions for industrial cooling systems usually define an upper limit for the cooling-water return temperature. In this case, t2 is also part of the process data, and with Eqs. (5-6-1) and (5-6-2), Wc and Q can be calculated. When the required five variables are not defined in the process specifications, they must be adopted. For example, it is possible that neither the cooling-water flow rate nor the return temperature were defined to the designer. In this case, the designer has to adopt them. To adopt these values, it must be considered that the greater the water-mass flow to remove a certain heat flow, the smaller is the temperature increase t2 – t1 that this water will suffer. This is illustrated in Fig. 5-16. Curve h represents the evolution of the hot fluid. Curve a shows the evolution of the cooling water if a mass flow Wca is employed. If a cooling-water mass flow Wcb is used,

T1 h T2 a b

t1

FIGURE 5-16

Effect of the cooling-water flow rate on the temperature difference.

100

CHAPTER FIVE

the evolution of the cooling-water temperature is represented by curve b. In this case, Wcb > Wca. Thus we can see that the mean temperature difference between the hot fluid (curve h) and the cold fluid (curve a or curve b) is higher if a higher amount of cooling water is used (curve b). This higher ΔT means that a smaller heat exchanger area will be necessary to transfer the same Q. The problem therefore is to economically balance a smaller heat exchanger against an increase in the operational cost owing to higher water consumption (pumping power, water treatment, etc.). If the mean temperature difference between both streams is considerably higher than the temperature change of each stream, the area reduction that can be achieved by increasing the flow rate may not be significant, as illustrated in the following examples. Example 5-3 Here, 20 kg/s of benzene [c = 1,650 J/(kg · K)] must be cooled down from 35 to 30°C. It is intended to use 4 kg/s of cooling water, available at 20°C. What would be the reduction in the heat exchanger area if the water flow rate were doubled? Solution

For the original case, with a cooling-water flow of 4 kg/s, it will be Q = Whch(T1 − T2) = 20 × 1,650 × (35 − 30) = 165,000 J/s

The cooling-water outlet temperature will be t2 = t1 + Q/Wccc = 20 + 165,000/(4 × 4,180) = 29.86°C For a countercurrent configuration, the LMTD is LMTD =

(35 − 29.86) − (30 − 20) = 7.3°C (35 − 29.86) ln (30 − 20)

The required area will be A1 = Q/(U · LMTD) = Q/(7.3U) If the water-mass flow is doubled, for the same duty, the new outlet temperature must be t'2 = 20 + 165,000/(8 × 4,180) = 24.93°C and then DMLT =

(35 − 24.93) − (30 − 20) = 10.03°C (35 − 24.93) ln (30 − 20)

The required area thus will be A2 = Q/(U · DMLT) = Q/(10.03U) We see that even if the heat transfer coefficients are the same, the higher water flow allows a reduction of about 30 percent in area. This difference can be even higher if we consider that with a higher mass flow, the velocity will be higher and a better heat transfer coefficient probably can be achieved. Example 5-4 to 60°C.

Repeat Example 5-3 considering that the benzene now must be cooled down from 65

The heat duty Q is the same as in the preceding case. Then, if 4 kg/s of water is employed, the outlet temperature again will be 29.86°C. Then

Solution

FUNDAMENTALS OF HEAT TRANSFER BETWEEN FLUIDS

DMLT =

101

(65 − 29.86) − (60 − 20) = 37.51°C (65 − 29.86) ln (60 − 20)

The required area thus will be A1 = Q/(37.51U) If the water flow rate is doubled, the outlet temperature will be 24.93°C, and then DMLT =

(65 − 24.93) − (60 − 20) = 40.03°C (65 − 24.93) ln (60 − 20)

And the new area will be A2 = Q/(40.03U) We see that the area reduction achieved in this case is only 6 percent. This is so because with the higher water flow rate, even though its outlet temperature is 5°C lower, this has little effect on the LMTD value. 5-6-2

Specifications Concerning Pressure Drop To improve the heat transfer coefficients, it is necessary to increase the fluid velocities. It is better, from the heat transfer point of view, to use a small-diameter, high-length tube rather than a shorter one with a higher diameter, both having the same heat transfer area. However, the increase in velocity also means higher pressure drops. If the heat exchanger has to be installed in an existing process, the designer must adhere to the maximum allowable pressure drop. If the heat exchanger is for a new process, it may happen that the designer defines the heat exchanger pressure drop, and then the required pumps can be specified to overcome this Δ p. In these cases, the problem consists of balancing a higher heat exchanger cost against a higher pumping power, and the most cost-effective solution must be adopted.

5-6-3

Fouling Factors Adoption of the fouling factor should not be the designer’s responsibility. The fouling factor is a datum that belongs to the process technology, and it must be supplied to the design engineers just like the physical properties of the fluids handled. In cases where the thermal design is performed by the equipment vendor, it is essential that the purchaser include the Rf value in the technical specifications because in this way the proposals received from different vendors will be homogeneous and comparable. In certain cases, the designer can suggest, based on his or her expertise, fouling factors that can be accepted or not by the purchaser, but in any case, both parties must agree as to what the design fouling factor will be. Tables in App. F can be used as a guide in the absence of more specific information.

5-6-4

Geometric Constraints Usually there exist certain design constraints based on mechanical and layout considerations. They may be a maximum tube length, position (horizontal or vertical), relative nozzle orientation, etc. It is convenient for the designer to know the characteristics of the place where the heat exchanger has to be installed. It must be noted that a design problem admits more than one solution, and the designer must choose the most suitable for the particular case.

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5-7 DESIGN PROCEDURE FOR A DOUBLE-TUBE HEAT EXCHANGER We shall describe the necessary steps to design a double-tube heat exchanger. We assume that all the process conditions previously explained are completely defined. This means that we know the flow rates, inlet and outlet temperatures, allowable pressure drops in both streams, and fouling resistances. We also assume that a maximum tube length Lt also has been defined. The heat exchanger thus will be designed using this tube length, and we shall calculate how many hairpins in series are necessary to achieve the required heat duty. The unit configuration will be similar to that shown in Fig. 5-14b. The design procedure consists of the following steps: 1. Selection of Tube Diameter. The usual combinations between internal and external tubes diameters that can be adopted are shown in Table 5-1. The smaller the flow area for both fluids, the higher are the velocity and heat transfer coefficients, but this also results in a higher pressure drop. As a first approach, it is suggested to select the tube diameter considering the fluid velocities. For example, in case of lowviscosity liquids, it is advisable to work with velocities of about 1–2 m/s. For highly viscous fluids, this value must be lower. The flow area for each fluid then will be a=

W ρv

(5-7-1)

Once the flow area for the annulus and tube have been calculated, it is possible to select the required tube diameters. If the flow rates are high, it may be necessary to add units in parallel. At the end of the calculations, when the final length is known, the pressure drop for each fluid can be calculated, and it is possible that this may require a change in the selected diameters. This velocity criterion thus must be considered only as a first approximation to design. 2. Calculation of the Film Coefficients. Once the fluid velocities are known, it is possible to calculate the Reynolds numbers for the tube and annulus. Then it is possible to obtain the film heat transfer coefficients with the correlations presented in Sec. 5-5. The physical properties of both fluids must be obtained at the average temperature between inlet and outlet. The viscosities at the wall temperature cannot be calculated from the beginning because this temperature is unknown. Then, as a first approximation, it is assumed that the factors (μ/μw)0.14 are unity. With this simplification, a first value of the film coefficients ho and hio is obtained. Then, by equating the heat transfer rates at both sides of the tube wall, we get ho(T − Tw) = hio(Tw − t)

(5-7-2)

hio(T − Tw) = ho(Tw − t)

(5-7-3)

if the annulus fluid is the hot fluid or

if the tube fluid is the hot fluid. In Eqs. (5-7-2) and (5-7-3), T and t are the average temperatures of the hot and cold fluids, respectively, whereas Tw is a mean temperature of the wall, not necessarily coincident with the average of the wall temperatures at both ends of the exchanger. From Eqs. (5-7-2) and (5-7-3), Tw can be obtained. With Tw , it is possible to calculate the correction factors (μ/μw)0.14, and the previous values for ho and hio can be corrected. An iterative procedure could be proposed, but usually the first correction is good enough. 3. Calculation of the Overall Heat Transfer Coefficient. With the individual film coefficients and the fouling resistance available, the overall heat transfer coefficient can be calculated: 1 1 1 = + + Rf U ho hio

(5-7-4)

FUNDAMENTALS OF HEAT TRANSFER BETWEEN FLUIDS

103

4. Calculation of the LMTD. With the inlet and outlet temperatures available, the LMTD can be calculated by Eqs. (5-4-22) or (5-4-23) depending on the flow configuration. 5. Calculation of the Heat Transfer Area. The heat transfer area can be calculated as A=

Q U ⋅ DMLT

(5-7-5)

6. Calculation of the Total Tube Length and Number of Tubes. The required tube length then will be A πDo

(5-7-6)

nt = L /Lt

(5-7-7)

L= And the number of tubes in series is obtained as

7. Calculation of the Pressure Drop. The friction factors for both streams can be calculated with the correlations presented in Sec. 5-5, and the pressure drop for each fluid will be Δ pt = 4 f

L v2 ⎛ μ ⎞ ρ D 2 ⎜⎝ μ w ⎟⎠

a

(5-7-8)

where a = –0.14 for Re > 2,100 and a = –0.25 for Re < 2,100. In Eq. (5-7-8), D must be replaced by Di or ' as required. Deq For heat exchangers with more than one tube, the annulus fluid suffers an additional pressure drop when passing from one tube to the next through the connecting tees. This pressure drop can be calculated as Δ pr =

nt ρ v 2 2 2

(5-7-9)

Then, for the annulus fluid, the total pressure drop is Δ p = Δ pt + Δ pr

(5-7-10)

8. Changes to the Original Design. If the calculated pressure drop is excessive, it will be necessary to increase the flow area, either by increasing the tube diameters or installing more branches in parallel. If the calculated pressure drop is smaller than allowable, then a reduction in the flow area can be attempted. In both cases, the design procedure must be reinitiated. Since it is not possible to modify the flow area of the internal tube without affecting the annulus, the design of this type of equipment is difficult to optimize. Many times it is necessary to accept a poor utilization of the allowable pressure drop in one stream in order to satisfy the requirement in the other. In any case, as was explained, this type of heat exchanger is limited to low-area and low-cost applications. Later on in this text we shall study other types of heat exchangers where, owing to the existence of a greater number of design variables, it is possible to optimize the allowable pressure drops in both streams. Once the thermal design of a heat exchanger is completed, the mechanical design must be performed. This means verifying tubes thickness, selecting materials, rating nozzles, choosing gaskets, etc. and elaborating the mechanical drawings as well. These subjects are under the scope of the mechanical engineering and will not be treated in this text. Example 5-5 Here, 0.8 kg/s of a solvent whose properties are indicated below must be cooled down from 40 to 30°C. A stream of ethylene glycol at 5°C will be used as coolant. The outlet temperature of ethylene glycol is limited to 25°C. Design a suitable heat exchanger for this service. The pressure

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CHAPTER FIVE

drop for both streams must not be higher than 110,000 N/m2. The combined fouling resistance must be 5 × l0–4 (s · m2 · K)/J. The properties of the solvent at 35°C are ρ = 790 kg/m3 c = 1,922 J/(kg · K) μ = 0.95 cP = 0.95 × 10–3 kg/ms k = 0.187 J/(s · m · K) Solution

The properties of ethylene glycol are

ρ = 1,010 kg/m3 c = 2,340 J/(kg · K) k = 0.264 J/(s · m · K) Viscosity (kg/ms): T, °C μ

10 0.028

20 0.020

30 0.014

35 0.012

Ethylene glycol is used as intermediate cooling medium in refrigeration applications owing to its low freezing point. These substances are called brines. In this case, the operating condition of the refrigeration cycle establishes a maximum return temperature for the hot brine, and the brine mass flow must be selected not to surpass this limit. If we select the outlet temperature as the allowable maximum, the mass flow will be at the minimum. The heat duty is Q = WhCh(T1 − T2) = 0.8 × 1,922 × (40 − 30) = 15,376 J/s and the brine mass flow is Wc = Q/[cc(t2 − t1)] = 15,376/[2,340(25 − 5)] = 0.32 kg/s 1. Selection of Diameters

For a preliminary selection of diameters, we shall adopt a solvent velocity of 1 m/s. The required flow area thus will be at = Wh /ρv = 0.80/(790 × 1) = 1.01 × 10−3 m2 If the solvent circulates through the internal tube, this section corresponds to a diameter of Di =

4 at = 0.036 m π

This is roughly the diameter of a 1¼-in Schedule 40 pipe with dimensions Di = 0.035 m Do = 0.0421 m at = πDi2/4 = 9.62 × 10−4 m2 If the exchanger is built with this pipe, the solvent velocity will be v = W/(ρ at) = 0.80/(790 × 9.62 × 10−4) = 1.05 m/s

FUNDAMENTALS OF HEAT TRANSFER BETWEEN FLUIDS

105

It can be seen in Table 5-1 that a possible standard combination consists in using a 2-in external pipe. Let’s select a 2-in Schedule 40 pipe. Its internal diameter is Ds = 0.0525 m The annular flow area thus is aS = π

0.05252 − 0.04212 Ds2 − Do2 = π = 7.75 × 10 −4 m 2 4 4

The brine velocity into the annulus thus is v = Wc /(as ρ) = 0.32/(7.75 × 10−4 × 1,010) = 0.408 m/s This is acceptable for a viscous fluid. If too low a velocity were obtained in the preceding step, using a higher brine mass flow may be considered to get better heat transfer coefficients. 2. Calculation of the Heat Transfer Coefficients

Calculations for the solvent: Re t =

Pr =

Di vρ 0.035 × 1.05 × 790 = = 30, 560 μ 0.95 × 10 −3 cμ 1, 922 × 0.95 × 10 −3 = = 9.76 k 0.187

⎛ μ ⎞ hi = 0.023 Re t0.8 Pr 0.33 ⎜ ⎝ μ w ⎟⎠ ⎛ μ ⎞ = 1011⎜ ⎝ μ w ⎟⎠

0.14

⎛ μ ⎞ k = 0.023 × 30, 560 0.8 × 9.760.33 × ⎜ Di ⎝ μ w ⎟⎠

0.14

×

0.187 0.035

0.14

As a first approximation, we neglect the viscosity correction factors. Then hio = hi

Di 0.035 = 1, 011 = 839 J/(s ⋅ m 2 ⋅ K) Do 0.0421

Calculations for the brine: Equivalent diameter = 4RH RH =

flow area 7.75 × 10 −4 = = 5.85 × 10 −3 π × 0.0421 internal tube perimeter ∴ Deq = 4 × 5.85 × 10−3 = 0.023 m

The mean temperature of ethylene glycol is 15°C. At this temperature, the viscosity is 0.024 kg/(m · s). The Reynolds number for heat transfer is Re s =

Deq vρ μ

=

0.023 × 0.408 × 1, 010 = 394 0.024

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CHAPTER FIVE

The heat transfer coefficient can be calculated from Deq ⎞ ⎛ = 1.86 ⎜ Re s Pr ⎝ L ⎟⎠

ho Deq k

0.33

⎛ μ ⎞ ⎜⎝ μ ⎟⎠ W

0.14

Let’s assume that the exchanger shall be built with 6-m pipes. Considering L as the length of each tube and neglecting as a first approximation the viscosity correction, we get ho = 1.86

k Deq

Deq ⎞ ⎛ ⎜⎝ Re s Pr L ⎟⎠

0.33

= 1.86

2340 × 0.024 ⎞ 0.023 ⎤ 0.264 ⎡ 394 × ⎛⎜ ⎟⎠ × ⎝ 0.264 6 ⎥⎦ 0.023 ⎢⎣

0.333

= 144 J/(s ⋅ m 2 ⋅ K)

Calculation of the wall temperature: ho(Tw − t) = hio(T − Tw) where T = mean temperature of the hot fluid = 35°C t = mean temperature of the cold fluid = 15°C Tw = mean wall temperature ∴ 144(Tw − 15) = 839(35 − Tw) from where Tw = 32°C At this temperature, the viscosity of ethylene glycol is 0.013 kg/(m · s). The correction factor for ethylene glycol is ⎛ μ ⎞ ⎜⎝ μ ⎟⎠ w

0.14

0.024 ⎞ 0.14 = ⎛⎜ = 1.08 ⎝ 0.013 ⎟⎠

Then ho = 144 × 1.08 = 156 The correction factor for the solvent is negligible because the wall temperature is very close to the mean fluid temperature. 3. Calculation of the Overall Heat Coefficient

⎛ 1 ⎞ 1 U =⎜ + + Rf ⎟ ⎝ ho hio ⎠

−1

−1 1 1 = ⎛⎜ + + 5 × 10 −4 ⎟⎞ = 124 J/(s ⋅ m 2 ⋅ K) ⎝ 155 389 ⎠

4. Calculation of the LMTD

LMTD =

(T1 − t2 ) − (T2 − t1 ) (40 − 25)) − (30 − 5) = = 19.5°C (T − t ) (40 − 25) ln ln 1 2 (30 − 5) (T2 − t1 )

5. Calculation of the Heat Transfer Area

The required area thus will be A=

Q 15, 376 = = 6.36 m 2 U (LMTD) 124 × 19.5

The area of each tube is πDo L = π × 0.0421 × 6 = 0.794 m2

FUNDAMENTALS OF HEAT TRANSFER BETWEEN FLUIDS

107

Then the number of tubes in series will be 6.36/0.794 = 8 tubes Then the proposed unit consists in four hairpins (eight tubes) 6 m in length made with 1¼-in internal tubes and 2-in external tubes connected in series. Now it is necessary to check pressure drops. 6. Calculation of Pressure Drops

Calculations for ethylene glycol: The equivalent diameter of the annulus for pressure-drop calculation is D'eq = Ds − Do = 0.0525 − 0.0421 = 0.0104 m The Reynolds number is Re 's =

' vρ Deq μ

=

0.0104 × 0.408 × 1010 = 178 0.024

For laminar flow, it will be f = 16/Re's = 0.090 Δp = 4 f

L ρv 2 ⎛ μ ⎞ ' 2 ⎜⎝ μ w ⎟⎠ Deq

0.25

= 4 × 0.090 ×

8×6 0.4082 ⎛ 0.013 ⎞ 0.25 × 1, 010 × × = 119,0000 N/m 2 ⎝ 0.024 ⎠ 0.0104 2

The pressure-drop terms, corresponding to the connections between hairpins, (4 × ρv2 /2) are in this case negligible. We see that the brine pressure drop slightly exceeds the allowable limit, and we can assume that it will be tolerated by the process. If it were desired to reduce this Δ p, an option would be to slightly reduce the brine mass flow. Even though this option would result in a brine outlet temperature higher than the allowable 25°C, the excess also would be very small. If neither of these options is acceptable, the only possibility is to increase the tube diameter to have a higher flow area. Calculations for the solvent: The friction factor is f = 0.0035 ×

0.264 0.264 = 0.0035 × = 6.9 × 10 −3 Re t0.42 30, 560 0.42

and it results in Δp = 4 f

L ρv 2 6×8 1.052 = 4 × 6.9 × 10 −3 × × 790 × = 16,900 N/m 2 Di 2 0.035 2

This is much smaller than the allowable maximum. This means that the velocity of the solvent could be increased, which would improve hi . However, since the controlling resistance is the ethylene glycol film, this would have little effect on the overall heat transfer coefficient U. Additionally, if the internal tube diameter is reduced, the annulus fluid velocity and ho also would be reduced . It would not be possible to compensate for this effect with a reduction in the external tube diameter because the annulus fluid pressure drop is very close to the maximum. Thus this design can be considered definitive. Example 5-6 Calculate the necessary tube length for the conditions of Example 5-6 if, instead of ethylene glycol, water at the same inlet temperature (5°C) and with the same pressure-drop limitations is employed.

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CHAPTER FIVE

Since a lower-viscosity fluid is employed, a higher velocity could be adopted. If we choose 2 m/s for the water, its mass flow would be

Solution

W = ρvas = 1,000 × 2 × 7.75 × 10−4 = 1.55 kg/s The temperature change for the water then will be t2 − t1 =

Q 15, 376 = = 2.4°C Wc cc 1.55 × 4,180

Then t2 = 5 + 2.4 = 7.4°C The LMTD then will be (40 − 7.4) − (30 − 5) = 28.6°C (40 − 7.4) ln (30 − 5) To calculate the film heat transfer coefficient, we shall use Eq. (5-5-4) with the annulus equivalent diameter LMTD =

ho = 1, 423(1 + 0.0146t )

⎛ 20.8 ⎞ v 0.8 = 5,750 J/(s ⋅ m 2 ⋅ K) 0.2 = 1, 423 (1 + 0.0146 × 6.2) ⎜ Deq ⎝ 0.0230.2 ⎟⎠

If we neglect the change in the physical properties of the solvent with temperature, the film heat transfer coefficient for the internal tube would be the same as before. Then 1 ⎛ 1 ⎞ U =⎜ + + 5 × 10 −4 ⎟ ⎝ 5, 750 839 ⎠

−1

= 535 J/(s ⋅ m 2 )

Thus A=

Q 15, 376 = = 1 m2 U (LMTD) 535 × 28.6

The required length is 1.0 A = = 7.6 m πDo π × 0.042 A hairpin with two tubes 4 m in length can be used (area = 1.05 m2). Calculation of the Water Pressure Drop

The viscosity of water at the mean temperature of 6°C is 1.5 × 10–3 kg/(m · s). Then Re s =

' Deq vρ

μ

=

0.0104 × 2 × 1, 000 = 13, 866 1.5 × 10 −3

For turbulent flow, f = 0.0035 + Δp= 4f

0.264 0.264 = 0.0035 + = 8.3 × 10 −3 Re 0.42 13, 8660.42

L ρv 2 8 22 = 4 × 8.3 × 10 −3 × 1, 000 × = 51,076 N/m 2 Deq 2 0.0104 2

FUNDAMENTALS OF HEAT TRANSFER BETWEEN FLUIDS

109

Again, the pressure drop for the connecting tees can be neglected. We see that it is possible to cool down the solvent with a heat exchanger whose area is six times smaller than before and with a lower pressure drop. This is due to (1) a much higher heat transfer coefficient because a lower-viscosity fluid is employed and (2) a higher LMTD because a higher coolant flow rate is employed. It can be noticed that the controlling resistance is now in the solvent film. This means that an important reduction in the heat transfer area could be obtained by increasing its velocity. It then would be possible to improve the design by adopting a smaller tube diameter. Also notice that the reduction in the tube length results in a reduction in the pressure drop, so there is a higher margin to increase the velocities. Example 5-7 Calculate what will be the solvent outlet temperature in the exchanger of Example 5-6 at the beginning of operation when the fouling resistance is nil. Solution

If the water flow rate is maintained, the overall heat transfer coefficient now will be Uc =

1 1 + = 732 J/(s ⋅ m 2 ⋅ K) 5, 750 839

With this coefficient, a higher heat flux will be achieved. This will result in a lower solvent outlet temperature and a higher water outlet temperature. In turn, this means a lower LMTD than in the preceding example. The equilibrium point is reached for the values of t2, T2, and Q that satisfy the following set of equations: Q = Wc cc (t2 − t1 )

(1)

Wc cc (t2 − t1 ) = Wh ch (T1 − T2 )

(2)

Q = UA

(T1 − t2 ) − (T2 − t1 ) (T − t ) ln 1 2 (T2 − t1 )

(3)

This system can be solved by a trial-and-error procedure. We shall assume t2 and calculate Q with Eq. (1). Then, with the same t2, we calculate T2 with Eq. (2), and by substituting these values in Eq. (3), we calculate a new Q. The t2 value for which the same Q is obtained by both calculations is the solution of the system (see Fig. 5-17). Calculation Progress Iteration t2 Q T2 Q

1 10 32, 300 18.9 16, 081

2 9 25, 916 23.14 18, 451

3 8 19, 437 27.36 20, 660

4 8, 5 20, 084 26.93 20, 440

A water outlet temperature of 8.1°C is obtained. This corresponds to a solvent outlet temperature of 27°C, which means that it is possible to cool down the solvent to a temperature 8°C lower than with a dirty unit. Example 5-8 What should be the cooling-water mass flow for the unit in Example 5-7 (clean conditions) to have the design (30°C) outlet temperature in the solvent? If the water flow rate is reduced, the water will heat up to a higher temperature. This would reduce the LMTD, and the solvent outlet temperature also would increase. Additionally, the reduction in the water flow will result in a lower heat transfer coefficient ho.

Solution

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CHAPTER FIVE

Start Assume t2

Q = Wc cc (t2 – t1) = 1.55 × 4,180 × (t 2 – t1)

T2 = T1 –

Q Q = 40 – Wh ch 0.8 × 1,922

Q = UA (LMTD) = 732 × 1.05 ×

(40 – t 2) – (T2 – 5) ln

40 – t2 T2 – 5

Same Q ? Yes End FIGURE 5-17

Flow diagram for Example 5-7.

The problem can be solved by assuming different Wc values. For each Wc , the water outlet temperature will be calculated assuming that the heat transferred is the same as in Example 5-6 (15, 376 J/s). We then shall calculate the LMTD. The required heat transfer coefficient to transfer 15,376 W with that temperature difference will be U=

Q A(LMTD)

This value is compared with the heat transfer coefficient that can be attained in the exchangers with the assumed flow rates, which is ⎛ 1 1⎞ U =⎜ + ⎟ ⎝ ho hio ⎠

−1

The water flow rate that makes both values equal is the one that allows the desired heat duty. The calculation flow diagram is shown in Fig. 5-18, and the results are listed in the table below. Trial Wc t2 LMTD U(1) t ho U(2)

1

2

3

4

0.5 12.35 26.3 556 8.6 2350 620

0.4 14.19 25.4 576 9.6 2000 590

0.35 15.5 24.26 603 10.2 1830 570

0.38 14.68 25.15 582 9.8 1950 586

FUNDAMENTALS OF HEAT TRANSFER BETWEEN FLUIDS

111

Start

Assume Wc t2 = t1 +

LMTD =

15,376 Q =5+ 4,180 Wc Wc c c

(T2 – t1) – (T1 – t2) (T2 – t1)

ln

U=

=

(40 – t2) – (30 – 5) ln

(T1 – t2)

(40 – t2) (30 – 5)

15,376 Q = A (LMTD) 1.05 (LMTD)

(1)

t = Mean temperature of water = 1/2 (5 + t2)

Water velocity =

ho =

U=(

Wc as ρ

1,423 (1+ 0.0146 t ) v 0.8 0.023 0.2

1 1 + ho hio

–1

)

=(

1 1 –1 + ) (2) ho 839

Same U ?

End FIGURE 5-18

Flow diagram for Example 5-8.

We see that it is possible to reach the desired temperature by reducing the flow rate to one-third the design value.

GLOSSARY a = flow area (m2) A = heat transfer area (m2) Ai = internal tube area (m2) Ao = external tube area (m2)

112

CHAPTER FIVE

Am = mean tube area (m2) c = specific heat [J/(kg · K)] D = diameter (m) Di = internal diameter of the internal tube (m) Do = external diameter of the internal tube (m) Ds = internal diameter of the external tube (m) Deq = equivalent diameter for h calculations (m) D'eq = equivalent diameter for Δ p calculations (m) f = friction factor G = mass velocity (kg/s) = ρv h = film coefficient [J/(s · m2 · K)] ho = external film coefficient [J/(s · m2 · K)] hi = internal film coefficient [J/(s · m2 · K)] hio = internal film coefficient referred to the external area [J/(s · m2 · K)] i = specific enthalpy (J/kg) k = thermal conductivity [J/(s · m · K)] L = heat transfer length (m) Lt = tube length (m) P = pressure (N/m2) Pr = Prandtl number Q = heat exchanged (J/s) Rf = fouling resistance Re = Reynolds number T = hot-fluid temperature (K or °C) t = cold-fluid temperature (K or °C) v = velocity (m/s) W = mass flow (kg/s) x = length coordinate (m) λ = heat of condensation (J/kg) μ = viscosity [kg/(m · s)] ρ = density (kg/m3) Subscripts 1 = inlet 2 = outlet c = cold h = hot i = internal o = external w = wall

FUNDAMENTALS OF HEAT TRANSFER BETWEEN FLUIDS

113

REFERENCES 1. Kern D.: Process Heat Transfer. New York: McGraw-Hill, 1954. 2. Sieder E.N., Tate G.E.: “Heat Transfer and Pressure drop of Liquids in Tubes” Ind Eng Chem 28:1429–1436, 1930. 3. Perry R., Chilton C.: Chemical Engineers Handbook, 5th ed., New York: McGraw-Hill, 1973. 4. Drew T. B., Koo E. C., McAdams W.H.: Trans AlCh E 28:56–72, 1932. 5. Wilson R. E., McAdams W. H., Seltzer H.: Ind Eng Chem 14:105–119, 1922.

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CHAPTER 6

SHELL-AND-TUBE HEAT EXCHANGERS 6-1 DESIGN AND CONSTRUCTION STANDARDS The quality of a heat exchanger or any other piece of process equipment depends on a great number of constructive details ranging from the quality of the construction materials to the way in which it is packed for shipment, passing through the mechanical design of its components, constructive techniques, tolerances, tests, and many others. It would be impossible for someone requesting a quotation for the supply of process equipment or any other industrial material to specify all the necessary details to ensure the quality of the supply. However, the specification of all these details is necessary to be sure that the proposals coming from different vendors will be comparable and will accommodate the quality level intended. To solve this problem, engineers make use of standards prepared by standards development organizations. These are governmental or private associations of companies related to a certain type of activity, either users or vendors that develop, publish, and maintain constructive standards and recommendations for almost any conceivable product. These standards are prepared by specialist in the particular subject, taking into consideration the interests of purchasers and vendors, and they establish all the requirements to which the construction of the product must adhere in its different stages. The standards then are made public for use. Their principal objective is to provide a common reference point for purchasers and vendors, thus avoiding the necessity of specifying all the aspects previously mentioned by just specifying that the provision has to comply with the prescriptions of a certain standard. Usually adherence to these standards is not mandatory by law. However, the use of standards is beneficial for vendors and purchasers because standards make it possible to clearly define similar and uniform conditions for all the vendors participating in a bid process and to prevent further discussions regarding the quality of the provision. A good standard must specify all the details that may result in conflicts between vendors and purchasers, thus avoiding ambiguous terms such as good engineering practice, usual practice, etc. Frequently, different standards development organizations produce sector-based standards that respond to the particular requirements of a specific type of industry. For example, the American Petroleum Institute (API) is an association that normally develops standards for the severe service conditions of the oil-processing industry. Often the quality requirements for this industry are not necessary in other applications, and other standards development organizations such as the Tubular Exchanger Manufacturers Association (TEMA) or the American National Standards Institute (ANSI) may have different standards for the same type of product. The engineer has to choose the standard that best adapts to the particular requirements of the project. The adoption of an unnecessarily stringent standard will increase the cost of the project without any benefit. Heat Exchanger Manufacturing Standards. In what follows, the main features and characteristics of shell-and-tube heat exchangers will be explained. This is by far the most common heat exchanger type employed in the process industry. The design and construction of these units, as with any other pressure vessel, normally follow the prescriptions of the American Society of Mechanical Engineers (ASME) Boiler and Pressure Vessel Code, Section VII, Division 1. In U.S.A this code is mandatory in many States, and because it is a recognized code, it is adopted in many projects around the world. TEMA issues the broadly known TEMA Standards.1 They are prepared and updated by a technical committee of association members. These standards complement the ASME Code in the particular subjects related to heat exchangers. The ASME stamp is required for all the exchangers manufactured and designed according to TEMA Standards.

115

116

CHAPTER SIX

Three classes of mechanical standards are defined: • Class R: designates heat exchangers for the severe requirements of petroleum and other related processing applications. • Class C: indicates generally moderate requirements of commercial and general process applications. • Class B: specifies design and application for chemical process service. Heat exchanger Class C and B designs are more compact and economical than Class R designs. The users must select the design/fabrication code designation for their individual application. The standards do not include recommendations for the selection of a particular class. Many chemical plants select the most severe class, Class R, because they prefer a more robust construction. In design or construction aspects that are general for any pressure vessel, TEMA Standards refer to the ASME code (e.g., construction materials, hydraulic tests, tensile strengths, etc.), but the TEMA Standards include specific details for heat exchangers, such as tubesheet design, tube-to-tubesheet joints, tolerances, etc. There are other heat exchangers design and construction codes than the TEMA Standards. For example, API Standard 6607 establishes additional requirements for the severe service of the petroleum industry. However, since the TEMA Standards are the best-known heat exchanger standards, We shall make reference to them in the following topics. It must be understood that adherence to these standards is voluntary, and another standard or no standard can be adopted based on the particular characteristics of the project.

6-2 PRINCIPAL COMPONENTS OF A SHELL-AND-TUBE HEAT EXCHANGER The double-tube heat exchanger described in Chap. 5 is limited to applications in which the heat flux Q and the fluids flow rates are small. If higher flow rates have to be processed, it would be necessary to install several units in parallel, and the resulting design would not be compact, and maintenance would be difficult. To avoid these inconveniences, the shell-and-tube heat exchanger is normally used. The basic idea consists of installing several internal tubes into another tube of much bigger diameter, designated as the shell. This construction is illustrated in Fig. 6-1, and Table 6-1 provides the nomenclature for the figures. The shell is closed at both ends by the tubesheets. These normally have a considerable thickness and are penetrated by the tubes. The tube-to-tubesheet joints must be hermetically sealed to avoid fluid leakage from one side of the tubesheet to the other. One possible joint method is the welded construction indicated in the detail of Fig. 6-1. The tubesheets are bolted to the heads, which act as collectors and distributors of the fluid circulating inside the tubes. This fluid enters the exchanger through a head nozzle and goes into and through the tubes up to the opposite head, where it exits. The other fluid goes into the shell through an inlet nozzle, fills all the

TABLE 6-1 1 2 3 4 5 6 7 8 9 10 11 12 13

Component Nomenclature

Channel cover Channel Channel nozzle Bonnet Bonnet nozzle Shell flange, stationary head side Bolting Gasket Stationary tubesheet Fixed tubesheet Shell nozzle Shell flange rear end side Impingement plate

14 15 16 17 18 19 20 21 22 23 24 25 26

Tubes Shell Expansion joint Transversal baffle Fixed tubesheet flange Pass partition plate Tie rod and spacers Vent connection Split backing flange Floating tubesheet Floating head Lantern ring Packing

27 28 29 30 31 32 33 34 35 36 37 38 39

Note: Reference numbers correspond to Figs. 6-1, 6-13, 6-22, 6-26, 6-28, 6-29, and 6-31. Source: Used with permission from the TEMA Standards.

Stuffing box Gland Floating tubesheet and skirt Loose flange Split ring Reversing bonnet Floating head coverplate Shell cover Lifting lug Drain plug Shell cover flange Floating head cover flange Return bonnet flange

SHELL-AND-TUBE HEAT EXCHANGERS

18 4

10

117

14 20

7

17

16 10

21

11

5 See detail A 8

Tubesheet

Tubes

Fixed tubesheet heat exchanger

Detail A

TEMA TYPE BEM For parts nomenclature see Table 6-1

Welded unions Shell

FIGURE 6-1

Fixed tubesheet heat exchanger.

space surrounding the tubes, and moves toward the outlet nozzle, where it exits. So both fluids are separated by the tubes wall, which constitute the heat transfer area . If N is the number of tubes of the heat exchanger, L is the length of the tubes, and Do is the external diameter of the tubes, the heat transfer area will be A = πDo NL

6-3

BAFFLES We know that the heat transfer coefficients increase when fluids velocity and turbulence increase. The shellside fluid velocity can be modified by installing a set of baffles that force the fluid movement in a direction perpendicular to the axis of the tubes. By changing the separation between these baffles, it is possible to change fluid velocity. The most common type of baffle is the segmental baffle. Segmental baffles are circular plates, about the same diameter as the shell, to which a horizontal or vertical cut has been made. The baffles are installed into the shell in such a way that the cuts of two consecutive baffles are rotated 180 degrees. The baffles obviously must be perforated to allow the tubes to pass through them. This type of exchanger is illustrated in Fig. 6-2. The shell fluid must follow the illustrated path to reach the outlet nozzle. It then can be seen that the shell fluid velocity has a component in the direction perpendicular to the tubes and a component in the direction parallel to the axis of the tubes. The velocity component in the direction perpendicular to the tubes is usually the most important for heat transfer. The fluid velocity

118

CHAPTER SIX

2 Consecutive baffles

Longitudinal section FIGURE 6-2

Segmental baffles.

in the direction perpendicular to the tubes depends on the separation between the baffles. The closer the baffles are to each other, the smaller is the flow area, and the higher is the fluid velocity. This has a direct influence on the heat transfer coefficient, which increases with velocity. Of course, this increment in the velocity represents an increase in the frictional pressure drop. The exchanger designer has to choose the baffle separation to attain the maximum possible value of the heat transfer coefficient without exceeding the allowable pressure drop of the shell-side fluid. We shall later study the available correlations to calculate the heat transfer coefficients and friction factor for a heat exchanger shell3. The most common baffle cut is about 25 percent of the diameter. This means that the height of the baffle window is 25 percent of the shell diameter. Figure 6-3 shows the effects in the shell fluid streamlines corresponding to different baffle cuts. For low baffle cuts (Fig. 6-3a), the velocity of the fluid at the window is high, with further reconversion in pressure with high turbulence and eddy formation. A large amount of the fluid energy is spent in the window area, where there are few tubes, resulting in an inefficient conversion of pressure drop in heat transfer. If, on the other hand, the baffle cut is high, there may be short circuits between the baffle tips, as shown in Fig. 6-3c, and there will be important zones of the fluid with low and erratic velocities and low heat transfer coefficients.6 The optimum seems to be in an intermediate situation such as that illustrated in Fig. 6-3b. Other types of baffles, not as common as segmental baffle, are shown in Fig. 6-4. These are called disk and doughnut baffles.

SHELL-AND-TUBE HEAT EXCHANGERS

Baffles

Main flow

(a) Small baffle cut or large baffle spacing

Eddies

(c) Large cut or small spacing FIGURE 6-3

Effect of the baffle cut.

2 Consecutives baffles

Longitudinal section FIGURE 6-4

Disk and doughnut baffles.

Shell diameter

(b) Normal cut and spacing

119

120

CHAPTER SIX

Drain Holes. Precautions must be taken to allow complete drainage of the shell side when the heat exchanger is taken out of operation. For this reason, drain holes are provided in baffles with horizontal cuts to avoid trapping liquid between consecutive baffles, as shown in Fig. 6-5. Baffle Spacing. As was explained previously, an increase in the heat transfer coefficient is achieved by reducing the baffle spacing. However, there is a lower limit for the baffle spacing, set by the TEMA Standards, of one-fifth the shell diameter and never less than 2 in (51 mm). Spacing higher than the shell diameter is not common. If higher spacing is employed, the effect may be similar to that seen in Fig. 6-3c Drain hole with poor heat transfer coefficients. The TEMA Standards also define a maximum spacing for baffles, FIGURE 6-5 Drain holes. which is defined by mechanical considerations. This is so because the purpose of baffles is not only to improve the heat transfer but also to support the tubes. If a heat exchanger tube is not supported adequately, it can vibrate owing to the effect of the externally circulating fluid. In many cases this has contributed to tube destruction. This topic will be explained later in more detail . The TEMA Standards specify the maximum unsupported span that heat exchanger tubes may have. Some tubes are supported at every baffle hole, with the unsupported span equal to the baffle spacing. But the tubes passing through the window are only supported every second baffle, so the unsupported span is equal to twice the baffle spacing. The maximum unsupported spans allowed by the standards for different tube materials and diameters are shown in Table 6-2.

TABLE 6-2

Maximum Unsupported Span (m) Tube Material and Temperature Limit, °C

External Tube Diameter, (mm) 3/4 (19) 1 (25) 11/4 (32) 11/2 (38) 2 (50)

Carbon Steel and Low-Alloy Steels (399) High-Alloy Steels (454) Nickel-Copper (315) Nickel (454) Nickel-Chromium-Iron (537) 1.52 1.88 2,23 2.54 3.17

Aluminum and Aluminum Alloys Copper and Copper Alloys, Titanium at the Maximum Temperature Allowed by the ASME Code 1.32 1.62 1.93 2.21 2.79

Source: Extracted with permission from the TEMA Standards.

The baffles must be maintained firmly in their position because any vibration with respect to the tubes may wear and eventually destroy the tube at the baffle location. This is achieved with tie rods (see Fig. 6-1). Tie rods may be welded to the baffles or installed as shown on Fig. 6-6. In this construction, the tie rods pass through the baffle holes, and concentric spacers with higher diameters than the baffle holes are inserted in the tie rods to maintain the baffles in position.4 Provision must be made in the tubesheet layout for these rods, which is usually accomplished by omitting a tube at the selected location. This must be taken into consideration in defining the number of tubes that may be installed in a shell. Tie rods usually are threaded into the back of only one tubesheet, being free at the other end, and terminate with the last baffle by means of a lock washer.

SHELL-AND-TUBE HEAT EXCHANGERS

121

Tubesheet Spacer Baffle Tie rod Washer Nut

FIGURE 6-6 Tie rod and baffle spacers.

The minimum number of tie rods that may be installed is also defined in the TEMA Standards as follows: Shell Diameter (m) Up to 0.4 0.4–0.83 0.83–1.22 More

Number of Tie Rods 4 6 8 10

Tolerances. The tube holes must be drilled in the baffles with a certain tolerance over the tube outside diameter. When the maximum unsupported length is 914 mm or less, this tolerance is 1/32 in (0.8 mm) in diameter. For larger unsupported lengths, it must be 1/64 in (0.4 mm). The maximum diametral clearance between the baffle and the interior shell diameter is defined by TEMA as follows: Nominal Shell Diameter, m 0.15–0.43 0.45–0.99 1.016–1.37 1.39–1.75 1.78–2.1 2.15–2.54

Shell Internal Diameter Minus Baffle Diameter, mm 3.2 4.8 6.4 7.9 9.5 11.1

6-4 TUBES AND TUBE DISTRIBUTION The dimensions of the tubes used in the construction of heat exchangers usually respond to Birmingham Wire Gage (BWG) Standards. The tubes are designated by their outside diameters and a code that is related to the wall thickness. The table included in App. H shows the dimensional characteristics of tubes according to BWG Standards.

122

CHAPTER SIX

Tube diameters ranging from 1/4 in (6.35 mm) to 1.5 in (38 mm) may be used. However, almost all heat exchangers are constructed with 3/4-in (19-mm) or 1-in (25.4-mm) tubes. The length of the tubes is defined by the designer. It is not advisable to use tube lengths higher than 6 m because the construction is more difficult. Cleaning operations are also more difficult with long tubes. However, there are certain instances in which it is imperative to maintain the countercurrent configuration, and large heat transfer areas are necessary (e.g., in natural gas conditioning plants to exchange heat between the inlet and outlet gas). In such cases, tubes up to 20 m long can be used.

6-4-1

Materials A number of materials can be used in heat exchanger tubes. The material is selected based on the characteristics of the fluids from the corrosion point of view. The material of the tubes is in contact with the tube-side and shell-side fluids and must be resistant to both. Wall thickness is selected based on a mechanical calculation to resist the design pressures. The most commonly used materials are carbon steel, alloy steel, stainless steel, bronze, nickel, and their alloys (e.g., Monel, Inconel, and Hastelloy). For very corrosive fluids, more exotic materials such as titanium or even glass can be used.

6-4-2

Impingement Protection If the fluids are erosive, or when inlet velocities are high, it is necessary to install shield plates under the inlet nozzle to avoid the direct impact of the fluid on the surfaces of the tubes. These plates are called impingement protection plates, and their location can be appreciated in Fig. 6-29. The installation of impingement plates is mandatory if the inlet fluid is a gas with abrasive particles and in heat exchangers that receive liquid-vapor mixtures or vapor with entrained liquid, where the erosive effects are intense. For single-phase fluids, TEMA Standards define when the installation of impingement protection is required, depending on the kinetic energy of the fluid entering the shell. For liquids with abrasive particles, impingement protection is required if the inlet-line ρv2 exceeds 740 kg/(m · s2). For clean fluids, impingement protection is required if the inlet-line ρv2 exceeds 2,232 kg/(m · s2). TEMA defines two other important areas that also limit the inlet or outlet velocities—the shell entrance or exit area and the bundle entrance or exit area (Fig. 6-7). Shell Entrance or Exit Area. When an impingement plate is provided, this area is the unrestricted radial flow area between the inside diameter of the shell at the nozzle and the face of the impingement plate. If no impingement plate is provided, the shell entrance or exit area also includes the flow area between the tubes within the projection of the nozzle. Bundle Entrance or Exit Area. This is the flow area between the tubes within the compartments between first baffle and tubesheet. The area of the impingement plate, if any, must be substracted. The TEMA Standards establish that in no case should the shell or bundle entrance or exit areas produce a ρv2 in excess of 5953 kg/(m · s2).

6-4-3 Tube Patterns The tube pattern is the geometric disposition of tubes in the tubesheet. Figure 6-8 shows the most common tube pattern types. The tube pitch Pt is the minimum separation between tube axes. A tube pattern is defined by the type (e.g., square, rotated square, triangle, or rotated triangle), by the tube diameter, and by the tube pitch. Usually, triangular patterns have higher heat transfer coefficients and pressure drops than square arrays. For removable bundles, the square patterns have the advantage of allowing mechanical cleaning of the

SHELL-AND-TUBE HEAT EXCHANGERS

Shell entrance area

With impingement protection

Without impingement protection

First baffle

Tubesheet

Bundle entrance area

First tube row FIGURE 6-7

Shell entrance area and bundle entrance area.

Pt c c

Flow direction

Pt

Triangular (also called 30° triangular)

Flow direction Rotated triangle (also called 60° triangular)

c Pt Pt c Square

Flow direction

FIGURE 6-8 Tube patterns.

Rotated square

Flow direction

123

124

CHAPTER SIX

TABLE 6.3 Square Arrays

Triangular Arrays

Di , Tube Diameter 3/4

in (19 mm) 1 in (25 mm) 11/4 in (32 mm) 11/2 in (39 mm)

Pt , Pitch 1 in (25 mm) 11/4 in (32 mm) 19/16 (40 mm) 17/8 in (48 mm)

Di , Tube Diameter 3/4

in (19 mm) in (19 mm) 1 in (25 m) 11/4 in (32 mm) 11/2 in (39 mm) 3/4

Pt , Pitch 15/16 in (24 mm) 1 in (25 mm) 11/4 in (32 mm) 19/16 in (40 mm) 17/8 (48 mm)

outside surfaces of the tubes. When 3/4- or 1-in tubes are used, the clearance between tubes c (see Fig. 6-8) is about 5 or 6 mm. The most common tube patterns are indicated in Table 6-3.

6-5 TUBES TO TUBESHEETS JOINT There are basically two tube-to-tubesheet joining methods: 1. Expanded joint 2. Welded joint

6-5-1

Expanded Joints When a tube is submitted to internal pressure, it suffers an expansion and increases its diameter. When the stress to which the material is subjected is lower than the yield strength, the material will recover its original shape when the applied pressure disappears. However, if the stress exceeds the yield strength, the recovery is only partial, and a permanent deformation remains. The stress distribution as a function of the radial coordinate in the wall of a tube submitted to internal pressure is shown in Fig. 6-9. It can be seen that the stress is inversely proportional to the radius. This means that the material near the internal wall is subject to a higher stress than the more external material. When the heat exchanger is assembled, the tube is introduced in the tubesheet hole with a clearance between them. Then internal pressure is applied, and the tube bulges out. As the pressure is increased, the material of the tube can reach its elastic limit. We can imagine the tube and tubesheet assembly as a large plate with a hole in its center and pressure acting on the hole (Fig. 6-10). The stress distribution is continuous in both materials, as shown in the figure. With increasing pressure, enlargement continues, and the plastic zone spreads outward, including part of the tubesheet material. When the pressure is released, the tubesheet material that had not reached the elastic limit tries to recover its original size, but the material of the tube, which had reached the elastic limit, does not spring back and is compressed by the recovery of the tubesheet. This creates a tight bond between tube and tubesheet. The resulting joint has good mechanical resistance and is the most common joint method employed in heat exchanger construction. There are several techniques to achieve this effect. The most usual is roller expanding. A tube roller is a special tool that is introduced into the tube and can be expanded against the tube wall by the action of a mandrel. Rotation of the tool does the job by squeezing the tube wall. Other techniques are based on the application of hydraulic pressure directly to the tube end or even detonation of explosive charges consisting of Primacord contained in a polyethylene cylinder inserted into the tube. The strength and tightness of an expanded joint improve if annular grooves are provided in the tubesheet holes. Then the tube material expands into the grooves, as shown in Fig. 6-11a. The TEMA Standards define the number and dimensions of these grooves for the different heat exchanger classes.

SHELL-AND-TUBE HEAT EXCHANGERS

τ

τ = τ(r ) RO p Ri

r

FIGURE 6-9 Distribution of stresses in the wall of a tube submitted to internal pressure.

τ τ = τ(r )

r

p

Tubesheet Tubesheet FIGURE 6-10

Stress distribution in the tube wall and tubesheet under internal pressure.

(a) Grooved

(b) Grooved and flared

FIGURE 6-11 Tube-to-tubesheet joint.

125

126

CHAPTER SIX

When the shell-side pressure is higher than the pressure into the exchanger heads (which means that there is an outward force acting on the tubesheet), the strength of the joint can be improved by flaring or beading the end of the tubes, as shown on Fig. 6-11b.

6-5-2 Welded Joints (a)

(b)

There are several techniques to weld the tubes to the tubesheet.5 Some of them are represented in Fig. 6-12. A welded joint is tighter than an expanded joint, but the possibility of using this technique depends on the tube and tubesheet materials. When materials of different thicknesses are welded, the material with the higher mass can dissipate heat more easily than the thinner material, and it may be difficult to bring both pieces to the welding temperature simultaneously. For this reason, sometimes the alternative shown in Fig. 6-12d is adopted. This allows one (c) (d) to weld materials of similar thicknesses. Welding tubes to the tubesheet is a difficult operation and requires that the weld proceFIGURE 6-12 Welded joints dures be carefully prepared and tested before being applied. Another technique that may be used is brazing. In brazing, coalescence is produced by heating the tube-tubesheet assembly to a temperature above 430°C. This melts a nonferrous filler metal that flows into the space between the tube and hole by capillary action. The base metals have higher melting points than the filler and do not melt. This joining method does not produce joints of high strength and normally is used in combination with expansion.

6-6 MULTIPASS HEAT EXCHANGERS The heat exchanger of Fig. 6-1 can be modified as indicated in Fig. 6-13 by changing both heads. With this configuration, the tube-side fluid, which enters into the exchanger through the left head (front head), circulates through half the tubes up to the opposite head (rear head). Then the fluid inverts its direction and circulates back through the other half of the tubes up to the front head, where it exits. Since the fluid circulates twice through the exchanger, this is called a two-pass heat exchanger. Please note that the front (left) head must be partitioned to avoid a fluid short circuit from inlet to outlet nozzles. This is achieved with installation of a pass-partition plate, as indicated in the figure. If we compare the heat exchangers in Figs. 6-1 and 6-13, if both have the same shell diameter and length, the heat transfer area will be the same. However, in the case of Fig. 6-13, because the incoming fluid is distributed into a smaller number of tubes, the tube-side flow area will be half that of Fig. 6-1. Thus, if both units have to process the same flow rate, the fluid velocity in the two-pass heat exchanger will be double that in the single-pass exchanger. The tube-side-fluid heat transfer coefficient thus will be higher in the two-pass unit. We could have reached the same result with a heat exchanger with half the number of tubes but twice the length. However, the length of a heat exchanger usually is limited by the available plot area, and additionally, as was mentioned earlier, for construction and maintenance reasons, tube lengths higher than 6 m are seldom used. Thus the alternative of increasing the number of tube passes leads to more compact designs. By partitioning the heat exchanger heads adequately, it is possible to increase the number of tube passes, and heat exchangers with two, four, six, eight, or more passes can be constructed, as shown in Fig. 6-14. Usually an even number of passes is adopted. It is possible to use an odd number of passes, but if an even number of passes is employed, the inlet and outlet nozzles are at the same end, which usually makes the piping layout and cleaning operations more simple. Figure 6-15 shows possible pass-partition configurations for different numbers of passes. Since in each pass the temperature of the fluid changes, different sectors in each head will contain fluid at different

SHELL-AND-TUBE HEAT EXCHANGERS

4

18

10

8

11

21

10

8

18

19

7

7

36

Double pass fixed tubesheet heat exchanger TEMA TYPE BEM For parts nomenclature see Table 6-1

FIGURE 6-13 Two-pass heat exchanger.

Four passes

Eight passes

Six passes FIGURE 6-14

Multipass heat exchangers.

127

21

128

CHAPTER SIX

2 Passes

4 Passes

6 Passes

8 Passes

FIGURE 6-15

Pass partitions.

temperature. Since high temperature gradients can cause undesirable thermal stress on the tubesheet, it is recommended that the pass-partition configuration be such that the fluid temperature difference between adjacent sectors is less than 28°C (50°F).6 The pass-partition plates reduce the tubesheet area available for tubes allocation, as can be appreciated in Fig. 6-13. This normally makes it necessary to eliminate some tube rows. The tables in App. D indicate the number of tubes that can be allocated on a tubesheet for different shell diameters and tube passes. It can be seen that for the same shell diameter, the higher the number of tube passes, the smaller will be the number of tubes that may be allocated. Also, in App. E, several tube layouts for the most common tube patterns and shell diameters are included It is left to the discretion of each manufacturer to establish a system of standard shell diameters within TEMA Standards to achieve the economies peculiar to his individual design and manufacturing facilities. However, the diameters indicated in the graphs and tables in the appendices are quite typical for all manufacturers.

6-7 HEAT EXCHANGERS WITH MULTIPLE SHELL PASSES Let us consider a heat exchanger with four tube passes, as shown in Fig. 6-16. Let’s assume that the tubeside fluid is the cold fluid. This fluid enters the exchanger at temperature t1. In each pass, its temperature increases. The temperature evolution is represented in the lower part of the figure. Since in every pass the temperature of the cold fluid gets closer to that of the hot fluid, the ΔT available for heat transfer is reduced, and the amount of heat transferred in each successive pass is lower. This means that the increase in temperature of the cold fluid is higher in the first passes than in the later ones. It can be seen in the figure that the inlet temperature at the last tube pass is t’’’. This stream must receive heat from the shell-side fluid, which at this section of the exchanger is at its lowest temperature T2. In order for heat transfer to the cold fluid in the fourth tube pass to be possible, it is necessary that T2 > t’’’. We have explained that the temperature change experienced by the cold fluid in the last tube pass is small, so the cold

SHELL-AND-TUBE HEAT EXCHANGERS

t2

129

T1

t ″′

t″

t′

T2

t1 T1

t2 t″

T2 t ″′ t′

t2 FIGURE 6-16 Temperatures evolution in a four-pass heat exchanger.

stream outlet temperature t2 cannot be much higher than T2 either. This means that with this type of heat exchanger, it is not possible to achieve an important temperature cross between both fluids. This limitation does not exist in true countercurrent units. However, the length of a countercurrent heat exchanger would be four times that of the four-pass exchanger with the same fluid velocity and heat transfer area. Then the tubes would be excessively long. One way to improve the heat transfer characteristics of multipass heat exchangers is to install longitudinal baffles. These are plates whose width is similar to the shell diameter and whose length is somewhat smaller than the tube length so as to allow the shell-side fluid to cross from the upper to the lower part or vice versa at one of the exchanger ends. Above and below the longitudinal baffle, transversal baffles are installed. The resulting configuration is illustrated in the upper part of Fig. 6-17. If the tube-side fluid performs four passes, the schematic of the unit and the evolution of the streams temperatures are those shown in Fig. 6-18. In the lower diagram, the dashed curve corresponds to the evolution of the shell-side fluid. It can be appreciated that in this case it is simpler to achieve the temperature cross with T2 < t2. This is so because the hot fluid that exits the unit at T2 is in contact only with the two coldest passes of the tubeside fluid. We can see that in this case t’’’ > T2, which could not have been possible without the longitudinal baffle. This means that the longitudinal baffle improves the heat transfer characteristics in units operating with difficult temperature programs (high-temperature cross between both streams). The heat exchanger shown in Fig. 6-18 has two shell passes. It is also possible to build units with three or more shell passes. In theses cases, it is necessary to install additional longitudinal baffles, as shown in Fig. 6-19. The higher the number of shell passes, the higher is the temperature cross between both streams that can be achieved. We shall explain later how to decide if a particular temperature program makes it necessary to install longitudinal baffles. A heat exchanger configuration is designated with two digits. The first digit indicates the number of shell passes, and the second is the number of tube passes. Thus, for example, the heat exchanger in Fig. 6-17 is a 1-2 configuration, whereas that in Fig. 6-18 is a 2-4 configuration because it has two shell passes and four tube passes

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FIGURE 6-17

Heat exchanger with longitudinal baffle.

T1

t2

t″′

t″

t′

T2

t1

T1 t2 T2

t″′

t″ t′ t1 FIGURE 6-18 Temperatures evolution in a 2-4 heat exchanger.

SHELL-AND-TUBE HEAT EXCHANGERS

FIGURE 6-19

131

Heat exchanger with three shell passes.

Main stream Bypass streams FIGURE 6-20

Leakage between shell and longitudinal baffle.

In units constructed with longitudinal baffles, it is important to achieve a tight seal in the baffle-shell joint. Otherwise, the shell fluid may short circuit the tube bundle and reach the outlet nozzle without performing a good heat exchange with the other fluid. This is shown in Fig. 6-20. To achieve a tight seal, it is necessary to use specially designed gaskets that can be adjusted from the outside before proceeding to the final closure of the unit. In large-diameter heat exchangers, the longitudinal baffle can be welded to the shell internally. However, this is not possible in the removable-bundle units that we shall describe later. This is why it is usually preferred to avoid the installation of longitudinal baffles. This is possible because, as we shall see, a 2-4 configuration can be achieved by adequately combining two 1-2 units. It also must be noted that a heat exchanger with two shell passes and two tube passes is equivalent to a countercurrent heat exchanger with half the tube numbers and double the tube lengths.

6-8 FRONT HEADS (INLET HEADS) There are several types of front-head designs, as illustrated in Fig. 6-21. Two of the major considerations in the selection of heads are accessibility to the tubes and piping convenience. If a process will foul the exchanger repeatedly so that cleaning is a major factor, a head or coverplate that can be removed easily

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(a) Bonnet single-pass axial connection

(b) Bonnet double-pass radial connections

(c) Channel double-pass radial connections

(d) Channel integrated with shell single-pass radial connection

3 2

1

(e) Special design for high pressure (1) Closure (2) Backing ring (3) Pass partition

FIGURE 6-21 Front head types. (Used with permission from Standards of the Tubular Exchangers Manufacturers Association.)

is an obvious choice. One step in this direction is to locate connections on the sides, not on the ends, of removable heads. The radial flange connections shown in Fig. 6-21b permit easy removal of the head with minimum disturbance to the piping of the whole installation. The next step is to use a channel, as in Fig. 6-21c and d. Open-end heads that are fabricated from cylindrical sections are called channels. These are fitted with easily removable coverplates so that tubes can be cleaned without disturbing piping. They are used widely where fouling conditions are encountered or where frequent access for inspection is desired. The heads shown in Fig. 6-21a and b are called bonnets. They can be either cast or fabricated. Sometimes the channel is integral with the shell, as in Fig. 6-21d. In this case, only the coverplate can be removed. The design shown in Fig. 6-21e is used in very-high-pressure applications. The head and tubesheet are built in a single piece. The internal pressure compresses the coverplate against the seat, and longitudinal efforts are absorbed by the backup ring, which is stressed only in shear. Pass-partition plates (3), if required, are built with rather thin plate because the differential pressure to which they are submitted is only given by the pressure drop of the fluid through the tubes. All these designs can be used as front or rear heads. However, for removable-bundle designs, special types of rear heads must be employed, as we shall consider later.

SHELL-AND-TUBE HEAT EXCHANGERS

133

6-9 FIXED-TUBESHEET AND REMOVABLE-BUNDLE HEAT EXCHANGERS 6-9-1

Fixed-Tubesheet Heat Exchangers The heat exchanger shown in Fig. 6-1 has the tubesheets welded to the shell, as can be appreciated in the detail included in the figure. This means that once the unit is built, it is not possible to have access to the shell side for cleaning or inspection. This type of heat exchanger is called a fixed-tubesheet exchanger. The construction sequence is as follows: 1. 2. 3. 4. 5. 6. 7.

Tubesheet and baffle holes are drilled. The tubes are installed in one of the tubesheets. The baffles and spacers are introduced through the tubes and tie rods. The shell is placed in position. The second tubesheet is installed. The shell is welded to the tubesheets. The tubes are rolled into the tubesheets. The two main disadvantages of this type of unit are

1. It cannot be disassembled for cleaning or inspection. 2. If the temperature difference between the fluids is high or the linear thermal expansion coefficients of the tube and shell materials are very different, when the exchanger comes into operation, differential expansion between shell and tubes creates forces acting on the tube-to-tubesheet joints that can damage the unit. The first of these problems cannot be solved with this type of heat exchanger. This means that this design is not suitable for cases in which both fluids have a fouling tendency (if only one fluid is fouling, it can be allocated to the tube side because the tube interiors can be cleaned mechanically). In some cases even though a mechanical cleaning of the tube exteriors is not possible, they can be cleaned chemically by circulating a solvent or detergent. However, this alternative is not feasible in many applications where the fouling characteristics of the fluid require mechanical removal procedures. The second problem, differential expansion between the shell and tube bundle, can be solved by installing a shell expansion joint such as that shown in Fig. 6-1. Such expansion joints act as elastic bellows absorbing the differential expansion without transmitting forces to the tubesheets. During mechanical design of the heat exchanger, it is determined if the installation of an expansion joint is necessary. However, the most common approach to solving the two previously mentioned limitations of the fixedtubesheet design is to adopt a removable-bundle construction, as will be explained below.

6-9-2 Removable-Bundle Heat Exchangers In these units, the tube bundle can be removed, and this allows access to the tube exteriors to perform mechanical cleaning. There are several types of removable-bundle constructions that will be analyzed below. Pull-Through Floating Head. In the design shown in Fig. 6-22, the right head is not fixed to the shell but is free to move longitudinally into the shell as the differential expansion requires. This is called a floating head. The opposite tubesheet is bolted to the shell and to the left head, which is called a fixed head. This tubesheet is called a stationary tubesheet because it does not have relative movement with respect to the shell.

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15

14

20

17

23

38

37

35

19

34 4 11 8

7

8

5

Pull through floating head heat exchanger TEMA TYPE BET For parts nomenclature see Table 6-1

FIGURE 6-22

Pull-through floating head.

The assembly constituted by this tubesheet, tubes, baffles, and floating head is the removable bundle. To remove the bundle, the procedure is 1. The flanged joint of the fixed head is disassembled. 2. The removable bundle is withdrawn from the left side. During this operation, the floating head passes through the shell. This is why it is called a pull-through floating head. Access to the exterior of the tubes then is possible.

b a

FIGURE 6-23 Floating head in a singlepass heat exchanger.

If the tube layout pattern is square, then it is possible to perform mechanical cleaning of the exterior of the tubes. In a triangular pattern, this cleaning is not possible even if the bundle is removed. This design does not present the problem of differential expansion stress because the bundle may expand without creating any stress. The main disadvantages of this design are 1. This design is not very suitable for single-pass heat exchangers. In these cases, it is necessary to install special flanges through the shell cover. Figure 6-23 illustrates one possible design. The flange a is screwed to the outlet nozzle and can be removed. Then the packing gland is loosened, and the bundle can be pulled through to the left. This can add considerably to the cost and adds a potential source of leaking. Also, the advantage of free expansion is lost. 2. The location of the floating-head flanges requires that the tube circle diameter d1 be considerably smaller than the shell

SHELL-AND-TUBE HEAT EXCHANGERS

diameter Ds. This can be easily appreciated in Fig. 6-24. This leaves an empty peripheral lane where the shell-side fluid may canalize, as shown in Fig. 6-25. This allows part of the fluid to circulate between baffle windows without penetrating into the bundle. This stream is called a bypass stream. Bypass streams can be eliminated by installing sealing strips, as will be discussed below. The installation of sealing strips does not eliminate the need to increase the shell diameter with respect to the tube circle, which increases the cost of the unit. 3. A third problem is that the floating-head flange gasket is a hidden gasket. Any damage to this gasket will not be detected and may cause the higher-pressure fluid to contaminate the other fluid. If this cannot be tolerated because of chemical compatibility, toxicity, etc., other designs sometimes are preferred.

Ds

135

d1

FIGURE 6-24 In a TEMA T heat exchanger, the tube circle is considerably smaller than the shell diameter.

Split-Ring Floating-Head Heat Exchanger. This design aims to solve the second of the problems pointed out for the pullthrough floating-head heat exchanger: the existence of a bypass region between the shell and the tube circle. As was explained, this diametral clearance is necessary for allocation of the removable-head flange. In the split-ring floating-head design, the term (Ds – d1), although larger than for the fixed-tubesheet design, is a good deal smaller than for the pull-through floating-head design. The construction is shown in Fig. 6-26. It can be seen that the floating head is not bolted to the tubesheet. It is bolted to a backup ring, and the tubesheet is tightened between the ring and the head flange. This construction makes it possible for the tubesheet diameter to be smaller than the bolt circle of the flanged joint. The backup ring is not a single piece. It is split into two halves. This makes it possible for both parts to be removed from the unit (once unbolted from the head flange) before pulling the bundle through the shell.

By pass streams

FIGURE 6-25

Bypass streams.

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9 18

8

18

11

6

15

20

17

14

12

37

35 21

1 34

35

24

38

22 7

3

8

7

23 22

34

Split ring floating head heat exchanger TEMA TYPE AES For parts nomenclature see Table 6-1

FIGURE 6-26

Split-ring floating-head heat exchanger.

The disassembly procedure is as follows: 1. 2. 3. 4. 5.

The shell cover (34) is removed. The flanged joint is unbolted. The ring is split, and both halves are removed. The stationary head in the opposite end is removed. The tube bundle is pulled out, passing the tubesheet through the shell.

Figure 6-27 is a comparison between a pull-through design and a split-ring design. It can be seen that the second alternative allows a smaller shell diameter. U-Tube Heat Exchanger Another way to solve the problem of the differential expansion is with the U-tube construction shown in Fig. 6-28. This design has the advantage of a lower cost because it eliminates one head, but its principal limitations are 1. It is not possible to clean the interior of the tubes because it is not possible to pass a cleaning rod through them. 2. This construction cannot be used for single-pass exchangers. 3. With the exception of the outermost tubes, individual tubes cannot be replaced. Any leaking tube must be plugged. 4. In very large diameters, support of the tubes is difficult. (The U-tube bundle becomes susceptible to vibration hazards.)

SHELL-AND-TUBE HEAT EXCHANGERS

Ds

d1

Ds

137

d1

FIGURE 6-27 Comparison between TEMA T and TEMA S floating heads for the same tube-circle diameter. It can be appreciated that a smaller shell diameter is possible with the TEMA S type.

3

18

9

11

6

15

14 20 17

21

4 19

8

7

36 U-tube heat exchanger TEMA TYPE BEU For parts nomenclature see Table 6-1

FIGURE 6-28

U-tube heat exchanger.

As a counterpart, the number of gaskets is minimal. This makes this design particularly attractive in high-pressure service. Outside-Packed Floating-Head Design. Figure 6-29 shows this design. The floating head can move axially, and the shell side is sealed by a packing (26) that is compressed by means of a packing gland (28). The floating tubesheet skirt (29) diameter is smaller than that of the shell. It thus can be removed to the left, passing through the shell, when the unit is disassembled. The slip-on backing flange (30) is a loose flange that can be removed to the right after removal of the split shear ring (31). Any leakage at the floating-head joint is easily detectable, but this type of packed joint should not be used when lethal, toxic, or flammable fluids are to be contained. Leakage through a stuffing box is more likely than is failure of a gasketed joint.

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8

1

3

9

18

6

11

13

20

14

17

27

26

30

28

33

19

29 7

18

2

8

15

27

26 28

29

31

8

30

Outside-packed floating-head heat exchanger TEMA TYPE AEP For parts nomenclature see Table 6-1

FIGURE 6-29

Outside-packed floating-head heat exchanger.

This unit is very limited in its design pressure and temperature ranges. Design pressure must be lower than 40 bar, and design temperature must be lower than 590 K. API Standard 660, which is used in the petroleum industry, does not allow this type of construction. Figure 6-30 shows a double-tubesheet design. This design can be used where a contamination of one fluid with the other because of tube-to-tubesheet joint leakage cannot be tolerated. This double-tubesheet

FIGURE 6-30

Double-tubesheet design.

SHELL-AND-TUBE HEAT EXCHANGERS

5

4

18

9

11

13

15

20

17

14

23

12

139

25 39

19

26 8

7

36

32

36 25

26

Externally sealed floating-tubesheet Heat exchanger

TEMA TYPE BEW For parts nomenclature see Table 6-1

FIGURE 6-31

Externally sealed floating-tubesheet heat exchanger.

construction also can be used in other types of heat exchangers, such as the fixed-tubesheet or U-tube designs, but it is not feasible in the other floating-head designs. Externally Sealed Floating-Tubesheet or Floating-Head Outside-Packed Lantern Ring. In this construction (Fig. 6-31), an outside-packed lantern ring provides the seal between the shell-side and tube-side fluids as the tubesheet moves back and forth with expansion and contraction of tubes. Weep holes in the ring vent any seepage to the atmosphere. The construction is relatively simple, and there is no bypass area. Maintenance is also relatively simple. However, it is even more limited than the outside-packed floating-head heat exchanger. The TEMA Standards allow this type of construction only for water, steam, air, lubricating oil, or similar services in conditions less severe than 2106 N/m2 and 423 K. 6-9-3 Comparison among the Different Heat Exchangers Types Table 6-4 lists the advantages and disadvantages of the different types of heat exchangers we have analyzed. 6-9-4 Standard Types and TEMA Designation The several types of rear heads (described previously) can be combined with different types of front heads and shells, giving origin to a number of possible combinations. The TEMA Standards include a code to describe the heat exchanger type using three letters. The first letter designates the inlet (front) head. The

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TABLE 6-4

Features of Principal Heat Exchangers Designs1, 2

U-Tube (Type U)

Fixed Tubesheet (Types L, M, and N)

Pull-Through Floating Head (Type T)

Floating-Head Outside-Packed Lantern Ring (Type W)

Split-BackingRing FloatingHead (Type S)

Floating-Head Outside-Packed Stuffing Box (Type P)

Relative cost increases from A (least expensive) through E (most expensive)

A

B

C

C

D

E

Provision for differential expansion

Individual tubes free to expand

Expansion joint in shell

Floating head

Floating head

Floating head

Floating head

Removable bundle Yes

No

Yes

Yes

Yes

Yes

Individual tubes replaceable

Only those in outside rows

Yes

Yes

Yes

Yes

Yes

Tube interiors cleanable

Difficult to do mechanically; can do chemically

Yes, mechanically or chemically

Yes, mechanically or chemically

Yes, mechanically or chemically

Yes, mechanically or chemically

Yes, mechanically or chemically

Tube exteriors with triangular pitch cleanable

Chemically only

Chemically only

Chemically only

Chemically only

Chemically only

Chemically only

Tubes exterior with sqare pitch cleanable

Yes, mechanically or chemically

Chemically only

Yes, mechanically or chemically

Yes, mechanically or chemically

Yes, mechanically or chemically

Yes, mechanically or chemically

Number of tube passes

Any practical even number possible

No practical limitations

No practical limitations (single-pass floating-head requires packing joint)

Limited to single- or double-pass

No practical limitations (single-pass floating-head requires packing joint)

No practical limitations

Internal gaskets eliminated

Yes

Yes

No

Yes

No

Yes

Type of Design

second letter refers to the shell type. The third letter designates the rear (outlet or return) head. For example, a heat exchanger of type AEL is a heat exchanger with front-head type A, shell type E, and rear-head type L. The meaning of these codes can be seen in Fig. 6-32. Shell type K is used as a evaporator or reboiler. The rest of the elements in the figure already have been analyzed. Figures 6-1, 6-13, 6-22, 6-26, 6-28, 6-29, and 6-31 indicate the corresponding code for each type of exchanger represented.

6-10 TUBE VIBRATION 6-10-1 Vibration Damage The tubes of a heat exchanger are supported at both ends by the tubesheets and in intermediate positions by baffles. Each tube section between two consecutive supports is a flexible element and can vibrate like the strings of a guitar. Like the strings of a musical instrument, each tube section has a natural frequency

SHELL-AND-TUBE HEAT EXCHANGERS

FIGURE 6-32 TEMA designations. (From Standards of the TEMA with permission.)

141

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of vibration that depends on its mass (both that of the tube and that of the fluid into it), on its moment of inertia, on the way it is supported, and more important, on the distance between supports. For example, a tube section included between two consecutive baffles has a different frequency of vibration than another one of the same length included between the tubesheet and a baffle or the end of a U-tube bend. The natural frequency of vibration also depends on the density of the shell-side fluid and the tensile strength at which the tube is submitted (the same way as tightening the strings of a guitar changes the frequency of vibration). The flow of the shell-side fluid past the tubes may produce certain periodic unbalanced forces over the tubes. When the frequency of these forces coincides with the natural frequency of the tubes, resonance occurs, and the tubes may start vibrating with large amplitude. Under certain conditions, the vibration amplitude may be large enough to produce severe damage to the tubes. Damage can result from any of the following independent conditions or combination thereof: 1. Collision damage. Impact of the tubes against each other or against the shell can result in failure. The impacted area of the tube develops the characteristic flattened, boat-shaped spot, generally at the midspan of the unsupported length. The tube wall eventually wears thin, causing failure. 2. Baffle damage. Baffle tube holes require a manufacturing clearance over the tube outer diameter to facilitate fabrication. When large fluid forces are present, the tube can impact the baffle hole, causing thinning of the tube wall in a uneven circumferential manner, usually the width of the baffle thickness. Continuous thinning over a period of time results in tube failure. 3. Tubesheet clamping effect. Tubes are normally expanded into the tubesheet to minimize the crevice between the outer tube wall and the tubesheet hole. The natural frequency of the tube span adjacent to the tubesheet is increased by the clamping effect. However, the stresses owing to any lateral deflection of the tube are also maximal at the location where the tube emerges from the tubesheet, contributing to possible tube breakage. 4. Material defect propagation. There are some vibration frequencies that cannot produce damage to a homogeneous material. However, if a material contains flaws oriented with respect to the stress field, they can propagate, and the tube may fail. 5. Acoustic vibration. This phenomenon may appear in heat exchangers working with gas in the shell side. Acoustic resonance is due to gas-column oscillation. This oscillation creates an acoustic vibration and generates a sound wave. This is not usually associated with tube damage unless the acoustic resonance frequency approaches the tube natural frequency, but the heat exchanger and even the associated piping may vibrate with loud noise. 6-10-2 Vibration Mechanisms There are four basic flow-induced vibration mechanisms that can occur in a tube bundle: 1. Fluidelastic instability. This is the most damaging in that it results in extremely large amplitudes of vibration. This mechanism is associated with high fluid velocities. A tube exposed to a fluid circulating at high velocity in a direction perpendicular to the tube starts vibrating, describing oval orbits around its axis. 2. Vortex shedding. This is also known as Von Karman turbulence. A tube exposed to an incident cross-flow provokes instability in the flow and the simultaneous shedding of discrete vorticities alternately from the sides of the tube. This phenomenon is referred to as vortex shedding. Alternate shedding of the vorticities produces harmonically varying lift and drag forces that may cause movement of the tube. When the tube oscillation frequency approaches the tube natural frequency within about 20 percent, the tube starts vibrating at its natural frequency. 3. Turbulent buffeting. This refers to unsteady forces developed on a tube exposed to a highly turbulent flow. The turbulence has a wide spectrum of frequencies around a dominating frequency that increases as the cross-flow increases. The effect is similar to the wind shaking or buffeting a canvas tent. This effect is observed only in heat exchangers handling high-velocity gases. 4. Acoustic vibration. As explained previously, this phenomenon consists on the vibration of a certain mass of gas, which can be excited by any of the preceding mechanism and provokes the appearance of sound waves.

143

SHELL-AND-TUBE HEAT EXCHANGERS

Region of high velocity

Region of high velocity

(a) FIGURE 6-33

6-10-3

Region of high velocity

(b)

(c)

Restricted entrance areas.

Failure Regions

a

Tube failures have been reported in nearly all locations within a heat exchanger. But those of primary concern are 1. Nozzle entrance and exit area. In these regions, impingement plates b (Fig. 6-33a), small nozzle diameter (Fig. 6-33b), or large outer-tube limits (Fig. 6-33c) can contribute to restricted entrance or exit areas. These restricted areas usually create high local velocities that can result in damaging flow-induced vibration. FIGURE 6-34 U-tubes bends. 2. U-tubes bends. Outer rows of U-bends (like tube a in Fig. 6-34) have a lower natural frequency of vibration and therefore are more susceptible to flow-induced vibration failures than the inner rows. 3. Tubesheet region. Unsupported tube spans adjacent to the tubesheets are frequently longer than those in the baffled region of a heat exchanger owing to the space required to allocate inlet and outlet nozzles (Fig. 6-35). The higher unsupported span results in lower natural vibration frequencies of the tubes in this region. The possible high local velocities, in conjunction with the lower natural frequency, make this a region of primary concern in preventing damaging vibrations. 4. Baffle window region. Tubes located in baffle windows have unsupported spans equal to multiples of the baffle spacing. Long, unsupported tube spans result in a reduced natural frequency of vibration, and tubes have a greater tendency to vibrate and suffer damage. 6-10-4 Treatment of the Problem The evaluation of potential flow-induced vibration problems in heat exchanger design is not part of the heat transfer field, so it will not be presented here in detail. However, in many cases, to solve vibration problems, it is necessary to make changes in the exchanger geometry, and this affects the thermal design.

FIGURE 6-35

Unsupported span is higher at exchanger ends.

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The tube vibration phenomenon is complex. The TEMA Standards provide a methodology to predict vibration damage, and the calculation procedure may be consulted there. However, it is made clear that the TEMA guarantee does not cover vibration damage. Commercial heat exchanger design programs usually include a vibration-evaluation module based on this methodology, that alerts the designer when vibration may be expected with a certain design. The method proposed by TEMA consists of calculating the natural frequency of vibration for each tube section within the exchanger. This frequency, as was explained, depends on the unsupported span, type of support, tensile or compression stress, and physical properties of tube material and fluids. The method includes criteria to avoid tube vibration caused by all the mechanisms explained previously. Vibration owing to fluidelastic instability is avoided if the fluid velocity at each section of the heat exchanger is below a certain critical velocity. This critical velocity is a function of the tube geometry, its natural frequency of vibration, and the viscous damping effect of the shell-side fluid. If the fluid velocity at any location is higher than the critical velocity calculated for that location, potential tube damage can be expected. It is important to note that this velocity must be calculated point by point at every location in the shell. This requires considering internal bypass streams, leakage streams between tubes and baffles and between baffles and the shell, the effect of impingement plates, pass-partition lanes, sealing strips, etc. The calculation equations are included in the standards. Regarding vortex shedding vibration, the TEMA Standards include equations and graphs that allow calculation of the vibration frequency resulting from this mechanism. If the tube natural frequency is less than double the vortex shedding frequency, there is a potential risk of vibration damage. In this case, it is necessary to calculate the amplitude of the tube vibration, and this must not exceed 2 percent of the tube diameter. The standards also include the necessary equations to calculate the amplitude of the vibrations associated with the turbulent buffeting mechanism, and it also must be checked that this amplitude is smaller than 2 percent of the tube diameter. Regarding acoustic vibration, it is necessary first to calculate the acoustic vibration frequency of the shell. This depends on the physical properties of the circulating fluids. The TEMA Standards then define criteria to compare this frequency with the vortex shedding and turbulent buffeting frequencies and decide if the acoustic resonance effect will be present. This would be a noise source. 6-10-5

Corrective Measures When the conclusion of a vibration analysis is that there exists a potential risk of tube damage, the design needs to be changed. The design normally is corrected by 1. Reducing the unsupported tube span. This can be achieved by using baffles with no tubes in the window area (NTIW type = no tube in window baffles). 2. Reducing the fluid velocity in the critical regions. For example, by changing the tube pattern, by increasing the baffle spacing (although the counterpart is that the unsupported span increases), by increasing the nozzle diameters, or by eliminating some tubes. The problem of the acoustic resonance may be corrected by adding a deresonating baffle parallel to the cross-flow direction to increase the shell acoustic frequency.

6-11

SPECIFICATION SHEET The TEMA Standards include a template form to specify the characteristics of a heat exchanger. This is called a specification sheet or data sheet, and it is used as a basic engineering document in the design, purchasing, and construction stages of a heat exchanger. This document is shown in App. J. The terminology included in the specification sheet was explained earlier, with the exception of some terms that will be defined next: Operating pressure This is the pressure to which the unit is submitted in normal operation.

SHELL-AND-TUBE HEAT EXCHANGERS

145

Design pressure This is the maximum pressure at which the unit still needs to continue operating, resulting from unusual conditions, for example, during special operating procedures or as consequence of process excursions that must be tolerated. This is the pressure used for the mechanical designer to calculate thicknesses of plates, tubes, flanges, etc. Hydraulic test pressure Any pressure vessel designed per ASME code can withstand at ambient temperature, in the absence of dynamic loads and for a limited time, pressures higher than the design pressure. Thus, if a hydraulic test is specified, the test pressure will be 1.3 times the design pressure. Design temperature This is the maximum temperature that may coexist with the design pressure.

6-12 DESIGN AND CONSTRUCTION OF HEAT EXCHANGERS The stages in the project of a heat exchanger are 1. 2. 3. 4.

Process specification Thermal design Mechanical design Construction

We shall discuss which heat exchanger aspects must be defined in each of these stages and which are the engineering documents resulting from them. 1. Process specification. As we discussed for double-tube heat exchangers, in this stage the process conditions and design constrains are defined. The following data must be specified: flow rates, inlet and outlet temperatures, physical properties, fouling factors, allowable pressure drops, and design and operating pressures and temperatures. Also, this stage includes the geometric specifications that must be adopted during design, such as maximum tube length, shell and tubes fluid allocation, TEMA type and class, and sometimes construction materials. In this stage, the process engineer will fill in all these fields in the specification sheet, thus defining the design basis. 2. Thermal design. The methods employed to perform the thermal design will be studied in Chap. 7. The result of this thermal design is the definition of the heat exchanger geometry, including tube diameter and length, tube pattern and pitch, number of tubes and shell passes, shell diameter, number and type of baffles, and nozzle size. This is normally a process engineering activity, and further design activities correspond to the field of mechanical engineering. At the end of the thermal design, the specification sheet will be almost complete. It will be possible to sketch an outline drawing of the exchanger, indicating location of nozzles, main dimensions, types of supports, or any other information the process engineer wants to define. 3. Mechanical design. In this stage, all the heat exchanger components are mechanically designed, the materials specification is completed, mechanical tests are specified, and detailed drawings are prepared. 4. Construction. Constructive and shop drawings usually are left to the manufacturer. In this stage, all the necessary details for the construction and welding procedures and methods are defined. Technical Specification. All the previously mentioned stages in a heat exchanger project can be carried on by different companies or different departments of the same company. Sometimes, the exchanger manufacturer performs the mechanical design or even the thermal design, depending on the type of contract and the engineering capacity of the purchaser. Usually, and especially if the heat exchanger is part of a complete project of a process unit, an engineering contractor performs the exchanger design and prepares the request for tender documentation. The advantage of this approach is that all vendors will quote on the basis of the same design, which will make proposal evaluation easier.

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Usually, the purchaser of process equipment or its engineering contractor wants to supervise and monitor the different stages of the engineering and construction according to a quality plan. A document called the material requisition defines the scope of the supply, the drawing approval procedure, and the shop inspection and test procedures to be followed. It also specifies the technical documentation that must be delivered, such as detailed drawings, materials quality certificates, welding procedures, etc. The material requisition also includes a technical specification with specific details and standard drawings of the different components (e.g., nameplates, saddles, lifting lugs, insulation, painting, etc.) adopted by the purchaser. All these documents are included with the request for quotation.

REFERENCES 1. 2. 3. 4. 5. 6. 7.

TEMA: Standards of the Tubular Exchangers Manufacturers Association. Perry R. H., Chilton C.H.: Chemical Engineer’s Handbook. New York: McGraw-Hill, 1973. Ludwing: Applied Process Design for Chemical and Petrochemical Plants. Gulf Publishing, 1964. Kern D.: Process Heat Transfer. New York: McGraw-Hill, 1954. Yorkell: “Heat Exchanger Tube to Tubesheet Connections,” Chem. Eng., Feb 8:78–94, 1982. Gardner K.: in Heat Exchanger Design and Theory Sourcebook. New York: McGraw-Hill, 1974. ANSI/API Standard 660- Shell and Tube Heat Exchangers, 7th Ed., American Petroleum Institute, 2003.

CHAPTER 7

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

7-1

BASIC PRINCIPLES

7-1-1 Heat Exchangers Design Techniques The thermal design of a heat exchanger can be performed by the supplier, the purchaser, or an engineering company. Nowadays, there are many commercial design programs developed by research institutes or software suppliers that can be purchased or licensed usually on an annual basis. The task of heat exchanger design consists of the definition of a large number of geometric parameters, such as length, diameter, and thickness of the tubes; tube pattern and pitch; baffle spacing; shell diameter and number of tubes; and number of passes in tube-side and shell-side. Some of these parameters may have been defined in advance by project specifications or designer preferences (typically tube type and pattern). Regarding tube length, usually the cost of a heat exchanger decreases if longer tubes are used, but sometimes, tube length is limited by layout restrictions. Once these parameters are defined, the remaining geometric characteristics (e.g., number of tubes, number of passes, baffle spacing, etc.) are defined by the thermal design program, following an optimization path that results in the minimum cost or minimum area. In some cases, the thermal design program interacts with mechanical design modules and computeraided design (CAD) software to prepare the engineering drawings of the heat exchanger. Obviously, access to these tools depends on the size of the company, the degree of specialization, and the volume of heat exchanger design jobs usually performed and is normally limited to engineering companies or process equipment suppliers. In small companies that have their own design department, heat exchanger design can be performed using simpler programs or with the help of manual calculations. Before starting a heat exchanger design, it is important to analyze the amount of effort that should be devoted to the task. This decision must take into consideration the following factors: 1. Cost of the engineering work 2. Heat exchanger cost (The design of a large and high-cost unit obviously will deserve more attention than that for a small unit.) 3. The effect that a malfunction of the unit may have on the entire process (Equipment that is critical for the process justifies more effort in the design stage.) The accuracy of the calculated results is another point to be considered. The two most frequent sources of uncertainty are the physical properties of the fluids and fouling factors, and quite often, because of these uncertainties, it is not worth using highly sophisticated design tools for the prediction of convection film coefficients. Another important aspect to be considered is which part of the exchanger deserves more attention. The concept of controlling resistance always must be kept in mind. For example, if fouling is the controlling resistance in a heat exchanger, it is more important to have a good prediction of the fouling factor than to have a very accurate method to calculate the film coefficients.

147

148

CHAPTER SEVEN

Fouling deserves a special paragraph. Caution needs to be exerted in selecting fouling factors. An excessively large fouling factor will result in an oversized heat exchanger. A large fouling factor is sometimes adopted as a safety margin to cover uncertainties in fluid properties and even in process knowledge. This is not an advisable procedure for many reasons. First, the effect of fouling resistance on heat exchanger design depends on the relative magnitude of fouling with respect to the fluid film resistances. It is thus not possible to know in advance which will be the resulting oversizing. If fouling is the controlling resistance, too large a fouling factor may result in a unit with two or three times more area than really necessary. Additionally, a numerical value for the fouling factor written in a specification sheet usually has the effect of setting a precedent for future designs that may result in unnecessary penalization. Unfortunately, the experimental determination and validation of fouling factors is not a usual practice in process industries. This is due to the fact that to get reliable results, long- and medium-term programs must be established, and dedicated instrumentation installation may be necessary Thus, in the absence of more specific information, the fouling resistances included in App. F can be adopted This chapter will examine the methods of Kern1 and Bell9 for heat exchanger design. Both approaches require only a small calculation capacity and can be performed either manually or with simple spreadsheets. Whenever possible, the correlations are presented numerically and not graphically so that they can be easily programmed. When the original sources did not include numerical expressions, regressions of the graphic correlations were performed. This chapter deals only with heat exchangers in which there is no phase change of the streams. Other types of units, such as condensers and reboilers, will be treated in later chapters.

7-1-2 Cross Mixing Hypothesis and Mean Temperature Difference When two circulating fluids exchange heat in a co-current or countercurrent pattern, as in the double-pipe heat exchanger shown in Fig. 7-1, the heat exchanged can be calculated as Q = UA(LMTD)

(7-1-1)

LMTD is the logarithmic mean temperature difference, which can be calculated with the expressions included in Fig. 7-1 corresponding to each one of the two cases.

Wc t1

t2

Wh T1

T2 t2 Co-current configuration

Wh T1

T2 Wc t1 Countercurrent configuration T1

T1 T2 t2

t2

t1

t1

DMLT =

FIGURE 7-1

T2

(T1 – t1) – (T2 – t2) (T – t ) ln 1 1 (T2 – t2)

(T1 – t2) – (T2 – t1) DMLT = (T1 – t2) ln (T2 – t1)

LMTD in co-current and countercurrent configurations.

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THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

The overall heat transfer coefficient can be calculated from the film coefficients of both fluids as 1 1 = + Rf U Uc

(7-1-2)

1 1 1 = + U c ho hio

(7-1-3)

hio = hi

Di Do

(7-1-4)

where U = overall heat transfer coefficient Uc = clean overall heat transfer coefficient Rf = combined fouling resistance hio = convection heat coefficient of the tube-side fluid referred to the external surface ho = annulus fluid convection film coefficient Di = internal diameter of the inner tube Do = external diameter of the inner tube In shell and tubes heat exchangers with only one tube pass, the inlet and outlet nozzles can be arranged in a countercurrent or co-current configuration, as shown in Fig. 7-2. However, in this type of heat exchanger, the shell fluid follows the path indicated in Fig. 7-3 owing to the presence of the baffles. We see that there is a component of the flow perpendicular to the tubes (cross-flow), and at the same time, there is a velocity component in the longitudinal direction, mainly at baffle windows. This means that the fluid circulation is not truly countercurrent or co-current because of this cross-flow component. However, in the design of this type of heat exchanger, it is usually assumed that at any cross section of the heat exchanger, the shell fluid is completely mixed, so a uniform shell fluid temperature can be considered in the entire cross section.

Parallel-current circulation

Countercurrent circulation

FIGURE 7-2

Nozzle arrangements in a shell-and-tube heat exchanger.

FIGURE 7-3

Shell-fluid path in a baffled heat exchanger.

150

CHAPTER SEVEN

Then, if it is admitted that the film coefficients for all the tubes are the same, the tube-side fluid also will heat up or cool down uniformly in all the tubes because they will be exposed to the same external temperatures. It is then possible to consider that at each cross section the temperature of each fluid is uniform, and the temperature evolution can be represented by diagrams such as those in Fig. 7-1, and it is valid to use the logarithmic mean temperature difference to calculate the heat transferred. Equations (7-1-1) through (7-1-4) thus are valid. In these equations, A is the total area of the unit, which can be calculated by means of the expressions A = Na1L a1 = πDo

(7-1-5) (7-1-6)

where A = heat exchanger area N = number of tubes a1 = tube surface area per unit length = πDo L = heat exchanger tube length Do = external diameter of the tubes The overall heat transfer coefficient can be calculated by means of Eqs. (7-1-2) through (7-1-4). In those equations, ho is the film convection coefficient for the fluid circulating through the shell of the heat exchanger. The calculation of this coefficient will be explained later. Example 7-1 Calculation of the LMTD It is desired to cool down 14 kg/s of an oil having a heat capacity of 1,964 J/(kg · °C) from an inlet temperature of 107°C to 40°C. The operation will be performed using 50 kg/s of cold water available at 20°C. Calculate the LMTD using a. A countercurrent heat exchanger b. A parallel-flow heat exchanger Solution

The heat to be exchanged is Q = Whch(T1 − T2) = 14 × 1,964 × (107 − 40) = 1.842 × 106 J/s

The cold water outlet temperature will be t2 = t1 + Q/Wccc = 20 + 1.842 × 106/50 × 4,180 = 28.8°C This value is independent on the heat exchanger configuration. For a countercurrent unit, LMTD =

(T1 − t2 ) − (T2 − t1 ) (107 − 28.8) − (40 − 20) = = 42.6°C T −t 107 − 28.8 ln ln 1 2 40 − 20 T2 − t1

For a parallel-current unit, LMTD =

(T1 − t1 ) − (T2 − t2 ) (107 − 20) − (40 − 28.8) = = 36.9°C T −t 107 − 20 ln ln 1 1 40 − 28.8 T2 − t 2

We see that the LMTD in the countercurrent configuration is higher than that in a co-current configuration. This means that if a co-current heat exchanger is used, a larger heat transfer area will be necessary. The ratio of the required areas for both cases is Apc Acc

=

LMTDcc 42.6 = = 1.15 LMTD pc 36.9

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

7-2

151

LMTD CORRECTION FACTORS FOR OTHER FLOW CONFIGURATIONS In a heat exchanger having more than one tube pass, the situation is different. It is no longer possible to consider a single temperature for each fluid at every cross section of the heat exchanger. Let’s consider a heat exchanger such as that shown schematically in Fig. 7-4. In a section such as AA' we can accept that the shell fluid is uniformly mixed. However, the fluid circulating through the tubes increases its temperature from t1 to t ' in the first pass and then from t ' to t2 in the second pass, so it crosses section AA' twice, with different temperatures each time. In this way, the shell fluid is in co-current flow with the first tube pass and in countercurrent flow with the second. We have seen in Example 7-1 that a countercurrent heat exchanger requires less heat transfer area than a co-current unit with the same overall heat transfer coefficient. In a 1-2 heat exchanger, the heat transfer area will be higher than in a countercurrent unit and lower than in a co-current unit. This problem is normally handled by keeping the form of the equation Q = UAΔT

(7-2-1)

where ΔT is an effective mean temperature difference between both streams, which must be higher than the LMTD corresponding to a parallel-flow arrangement but lower than the LMTD of a countercurrent configuration. If LMTDcc is the logarithmic mean temperature difference for a countercurrent configuration, it is possible to define a factor Ft , called the LMTD correction factor, as ΔT = LMTD cc Ft

(7-2-2) T2

t2 A

t′

A′ T1

t1

T1 T2 t2 t′ t1

FIGURE 7-4 Temperature diagram in a 1-2 heat exchanger.

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CHAPTER SEVEN

Ft is always lower than 1 because ΔT < DMLTcc. Equation (7-2-1) then can be written as Q = UA ⋅ LMTD cc Ft

(7-2-3)

Then the mean temperature difference for any heat exchanger is calculated as if the configuration were countercurrent and then multiplying by the correction factor, as shown in Eq. (7-2-3). For this reason, the ΔT defined by Eq. (7-2-2) is called the effective mean temperature difference to distinguished it from the LMTDcc, which is not the real temperature difference of the unit. From now on, the logarithmic mean temperature difference for a countercurrent configuration, LMTDcc will be designated simply as LMTD, dropping the subscripts. 7-2-1 Calculation of Ft Factors Ft factors can be calculated theoretically.2,13 The mathematical deduction of the final expressions is tedious and will not be presented here. The value of this factor depends on the type of unit (number of shell and tubes passes) and the inlet and outlet temperatures. A heat exchanger with one shell pass and two tube passes (1-2) has approximately the same Ft as a 1-4, 1-6, or 1-8 unit with the same inlet and outlet temperatures. This means that Ft is only a weak function of the number of tube passes (as long as it is even). Thus only the Ft values for 1-2, 2-4, 3-6, 4-8, etc. configurations are usually shown. For any other case, the value corresponding to the same number of shell passes must be used independently of the number of tube passes. Ft values are calculated as a function of two dimensionless parameters defined as R=

T1 − T2 t 2 − t1

and

S=

t 2 − t1 T1 − t1

Appendix B provides curves to calculate Ft factors for several flow arrangements. The analytical expressions corresponding to these curves, which can be used in computer programs, are Ft =

R2 + 1 R −1

ln [ (1 − Px )/(1 − RPx ) ]

⎡ (2 / P ) − 1 − R + R 2 + 1 ⎤ x ⎥ ln ⎢ ⎢⎣ (2 / Px ) − 1 − R − R 2 + 1 ⎥⎦

(7-2-4)

where RS − 1 ⎞ 1/ N s 1− ⎛ ⎝ S −1 ⎠ Px = RS − 1 ⎞ 1/ N s R−⎛ ⎝ S −1 ⎠ Ns is the number of simple shells or number of shell passes. If R = 1, the preceding expression is undefined, and the following equation must be applied: Ft =

( Px R2 + 1)/(1 − Px ) ⎡ (2 / P ) − 1 − R + R 2 + 1 ⎤ x ⎥ ln ⎢ ⎢⎣ (2 / Px ) − 1− − R − R 2 + 1 ⎥⎦

where Px =

S ( N s − N s S + S)

(7-2-5)

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

153

T1 t2

Temperature cross T2 t1

FIGURE 7-5 Temperature cross in a countercurrent heat exchanger.

Note: A downloadable interactive spreadsheet to perform these calculations is available at http://www. mhprofessional.com/product.php?isbn=0071624082

7-2-2 Ft Factor as Configuration-Selection Criterion In a co-current heat exchanger, the outlet temperature of the hot fluid can never be lower than the outlet temperature of the cold fluid. In the limit case, where both temperatures are equal, the LMTDpc becomes zero, and an infinite area would be required. In a countercurrent heat exchanger, on the other hand, it is possible to have a temperature cross as shown in Fig. 7-5, obtaining a t2 > T2. The countercurrent heat exchanger is, among all heat exchangers, the one which allows the higher temperature-cross, because, theoretically, with an infinite area, it would be possible to withdraw one of the fluids at the inlet temperature of the other, resulting in any of the situations shown in Fig. 7-6. If the product of the mass flow rate times the specific heat of the hot fluid (Whch) is higher than that of the cold fluid (Wccc), the situation is that of Fig. 7-6a, and in the opposite case, we would have the situation of Fig. 7-6b. In the case of multipass heat exchangers, it is possible to have some temperature cross, but there is a maximum temperature cross that is possible for each configuration and inlet temperatures. This fixes an upper limit to the amount of heat that may be transferred between both streams for each configuration. In this situation, the Ft factor is 0, so an infinite area would be necessary to reach this limit. Even though a heat exchanger with Ft > 0 theoretically could do the job, if we observe the shape of the Ft curves, we can appreciate that for Ft values lower than 0.7 or 0.8, the slope of the curves is almost vertical, and Ft decreases drastically with a minor change in the parameters. So it is not safe to work in these zones, and it is recommended never to design with an Ft lower than 0.75. It must be kept in mind that the Ft curves have been developed theoretically, and in their deduction, certain hypotheses corresponding to the ideality of the heat exchange configuration were assumed. These hypotheses may not be true in a real heat exchanger, so it is not recommended to approach the steepest region of the curves in which the true Ft may be quite different from the theoretical one. By examining the Ft curves, we see that for the same process conditions (e.g., the same inlet and outlet temperatures), when the number of shell passes increases, the Ft factor also increases. This means that if a

t2 = T1

T1 T2

t2

t1 Case a. Wh ch > Wc cc

T2 = t1 Case b. Wh ch < Wc cc

FIGURE 7-6 Temperature diagrams in a countercurrent heat exchanger with infinite area.

154

CHAPTER SEVEN

1-2 heat exchanger has an Ft factor lower than 0.75 for the required process conditions, it may be possible to perform the service adopting a configuration 2-4, 3-6, 4-8, etc. Example 7-2 It is required to cool down 20 kg/s of an aqueous solution whose specific heat is 4,180 J/(kg · K) from an inlet temperature of 67°C to 37°C. The operation will be performed with cooling water available at 17°C. Determine the number of shell passes of the heat exchanger using a water mass flow of (a) 40 kg/s and (b) 20 kg/s. Solution

a. With a water flow rate of 40 kg/s, the water outlet temperature can be calculated with the following heat balance: 4,180 × 20 × (67 − 37) = 4,180 × 40 × (t2 − 17) From which we get t2 = 32°C. Calculating the parameters we have R=

67 − 37 =2 32 − 17

and

S=

32 − 17 = 0.3 67 − 17

For a 1-2 heat exchanger with these parameters, we get Ft = 0.85. Thus it is possible to use this configuration for the required service. b. If a water mass flow of 20 kg/s is employed, the water outlet temperature calculated with a mass balance is 47°C and then R=

67 − 37 =1 47 − 17

and

S=

47 − 17 = 0.6 67 − 17

For a 1-2 configuration, a too low Ft factor is obtained with these parameters. This means that a 1-2 heat exchanger is not possible for these process conditions. With R = 1 and S = 0.6, if we move to a 2-4 configuration, we get Ft = 0.9, and it is thus the minimum configuration with this water flow rate. If the water mass flow is reduced further, it may not be possible to use a 2-4 configuration, and a 3-6 unit may be necessary. In all cases, a countercurrent configuration is always possible because then Ft = 1. The reason for using multipass configurations is to increase the in-tube fluid velocity, improving the heat transfer coefficient, because in order to have good fluid velocity with only one tube pass, an excessively long heat exchanger may be necessary in some cases. This will be better explained in Example 7-3. Additionally, in services requiring floating heads, the countercurrent configuration presents construction issues and is normally avoided. Obviously, U-tubes cannot be used in a one-pass configuration. Example 7-3 It is known that to achieve a reasonable value of the heat transfer coefficient when cooling a solution with the characteristics of that in Example 7-2, a minimum fluid velocity of 1 m/s is necessary. In this case, the overall heat transfer coefficient will be about 1,300 W/(m2 · K). For cleaning considerations, the solution must circulate into the tubes. It is desired to design a heat exchanger to cool 20 kg/s of solution using 20 kg/s of cooling water. The process conditions are those of case (b) in Example 7-2. The heat exchanger must be constructed with 3/4-in BWG 16 tubes that have an internal diameter of 0.0157 m and an external diameter of 0.019 m. Suggest a preliminary design. Let’s consider as a first choice the use of a pure countercurrent configuration for which Ft = 1. The heat duty is

Solution

Q = Whch(T1 − T2) = 20 × 4,180 × (67 − 37) = 2.508 × 106 J/s The flow area of each tube is π × 0.01572/4 = 1.94 × 10−4 m2

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

155

Assuming a 1,000 kg/m3 density for the solution, if we want to get a velocity of 1m/s, the number of tubes in parallel will be Nt =

0.020 m 3 /s 102 tubes. 1 m/s × 1.94 × 10 −4 m 2/tube

This is the maximum number of tubes in parallel that can be used because otherwise the fluid velocity would be too low. For a countercurrent unit, the effective mean temperature difference coincides with the LMTD. However, since T1 − t2 = T2 − t1 = 20°C, the LMTD is undetermined. Then ΔT = T1 − t2 = T2 − t1 = 20°C. To transfer 2,508, kW with a coefficient equal to 1,300 W/(m2 · K) and a temperature difference of 20°C, the required heat transfer area is A=

Q 2.508 × 10 6 = = 96.5 m 2 U ⋅ DMLT 1, 300 × 20

The heat transfer area of the 102 tubes per meter of exchanger length is π × 0.019 × 102 = 6 m2/m of exchanger length The required heat exchanger length thus would be L=

96.5 = 16 m 6

This is an excessive length for a single-shell heat exchanger. To achieve this length, it would be necessary to use two or three shells connected in series, as shown in Fig. 7-7, to keep the countercurrent configuration. If a more compact design is desired, a multipass heat exchanger must be used. The alternative of a 1-2 configuration must be precluded according to the results of Example 7-2. We shall consider the possibility of using a single-shell 2-4 heat exchanger with longitudinal baffle (TEMA type F). As analyzed in Example 7-2, for a 2-4 configuration, Ft = 0.9; then ΔT = 20 × 0.9 = 18°C.

T1

t2

t1

T2 FIGURE 7-7

Countercurrent arrangement with three shells.

156

CHAPTER SEVEN

To keep the velocity of 1 m/s, the unit must have no more than 102 tubes in each pass. This means 408 tubes total. The required heat transfer area is A = Q/UΔT = 2.508 × 106/(1,300 × 18) = 107 m2 The heat transfer area of the 408 tubes per meter length is 408 × π × 0.019 = 24 m2/m of heat exchanger The total length will be L = 107/24 = 4.5 m An acceptable decision thus will be to use a heat exchanger with 408 tubes arranged in four passes with 4.5 m length with two shell passes. The designer now must decide which is the required shell diameter to contain the 408 tubes and choose the baffle spacing to have a good heat transfer coefficient in the shell side. Of course, this is a preliminary design that must be confirmed with the design methods explained later on in this chapter. 7-2-3 Combination of Heat Exchangers Ft factors are characteristic of a heat exchange configuration rather than of a particular type of heat exchanger. For example, with two 1-2 heat exchangers connected as shown in Fig. 7-8, we obtain the same heat exchange configuration as using a 2-4 exchanger. Both shells constitute a unit. In this case, if both shells are identical, a single overall heat transfer coefficient can be used for the whole unit, and the Ft correction factor can be calculated from temperatures T1, T2, t1, and t2 as if it were a single 2-4 heat exchanger. The heat exchanged thus is Q = UA × LMTD × Ft

(7-2-6)

where LMTD =

(T1 − t2 ) − (T2 − t1 ) T −t ln 1 2 T2 − t1

In this case, A is the total area of the two shells. T2

T1

t1

t2

FIGURE 7-8 A 2-4 configuration using two shells.

(7-2-7)

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

t2

T2

t2

T1

t1

t1

T2

T1 T1

T1

t2

T2

t1

t2 T2 t1

(a) FIGURE 7-9

157

(b)

Different nozzle arrangements in a 1-2 heat exchanger.

7-2-4 Inlet Nozzle Orientation A multipass heat exchanger may have the inlet nozzles of both fluids at the same end of the unit or at opposite ends. This is shown in Fig. 7-9 for the case of a 1-2 heat exchanger. It can be demonstrated2 that the Ft values for both cases are identical. It must be noted in Fig 7-9b that close to the cold fluid outlet, the direction of the heat flow in the second tube pass is inverted because the temperature of the hot fluid has decreased below that of the cold fluid owing to the effect of the first pass of the tube-side fluid.

7-3 BASIC DESIGN PARAMETERS OF A HEAT EXCHANGER To design a heat exchanger means to determine the principal geometric characteristics of the unit. These are 1. 2. 3. 4. 5. 6. 7.

Heat exchanger type (e.g., fixed tubesheet, U tubes, floating heads, etc.) Diameter and array of tubes (e.g., square, triangular, rotated square and tube pitch) Fluid allocation (which will be the tube fluid and which will be the shell fluid) Number of shells in series, number of shell passes Number of tubes, number of tube passes and shell diameter Tubes length Baffles type and baffles spacing

The thermal design of a heat exchanger is always a compromise between higher heat transfer coefficients and lower pressure drop in the fluids. We know that in order to improve the heat transfer coefficient, it is necessary to increase the fluid velocities. This always produces an increase in frictional pressure drops. Sometimes this can be solved by increasing the pumping power. Thus the goal is to find the best solution considering the investment cost (i.e., heat exchanger area) and operative cost (i.e., pumping power). In other cases, the allowable pressure drops are defined by process considerations, and the designer must accept these limits at any resulting cost. He then must find the design parameter combination that results in the highest heat transfer coefficient within the pressure drop limitations. We shall briefly analyze what factors must be considered to adopt the design parameters.

158

CHAPTER SEVEN

Heat Exchanger Type. The advantages and disadvantages of each type of heat exchanger were analyzed in Chap. 6 and will not be repeated here. The selection takes into consideration the cleaning facility, relative cost, thermal expansion, fluids toxicity, etc. Tube Diameter and Tube Pattern. The tubes used in heat exchanger manufacture are usually standardized by their external diameter. The most used standard is BWG. The tube thickness must be selected according to the internal and external pressures. Despite the fact that the thickness selection corresponds to the mechanical design, this is information that affects the thermal design and must be selected from the beginning. Appendix H shows the tube dimensions based on BWG standards. It can be seen that the higher the BWG designation, the lower is the tube thickness for the same internal diameter. The most popular diameters are 1 or 3/4 in. Smaller tubes are difficult to clean, whereas larger diameters result in an unfavorable flow-area:transfer-area ratio. Square tubes pattern allow mechanical cleaning of the external tubes surfaces, which is not possible with triangular arrays. On the other hand, triangular arrays allow more tubes for the same shell diameter and tube pitch, and heat transfer coefficients are somewhat higher. At any rate, these variables do not affect the design significantly, and the tube diameter, as well as the array and tube pitch, is adopted beforehand based on design preferences, and they are seldom changed during the design process. Fluid Allocation. It is usually preferred to allocate the most fouling fluid to the tube side. The reason is that it is easier to clean the tube interiors than the exteriors. If one of the fluids is more corrosive, it also may be convenient to send it through the tube side because the shell then can be built with a lower-quality and cheaper material. The tube material must be resistant to both fluids. If the flow rates of both streams are very different, it may be that the decision as to which will be the tube-side and shell-side fluids will be based on the possibility of reaching a good velocity in both fluids. However, there are other ways to act on the fluid velocities, and normally, the fluid allocation is based on the other considerations mentioned previously. Number of Shells and/or Shell Passes. Use of the Ft factor as a criterion to decide the number of shell passes was explained and analyzed in Examples 7-2 and 7-3. The use of shells with longitudinal baffles (TEMA type F) results in construction complications because the design must guarantee a good closure between the shell and the longitudinal baffle. Thus it is normally preferred to avoid this type of construction, and the multipass configuration is obtained combining adequately connected shells (see Fig. 7-8). Number of Tubes and Number of Tube Passes. To obtain a specific heat transfer area, a combination of number and length of tubes must be selected. The higher the number of tubes, the smaller will be the length required. The cost of a heat exchanger with long tubes is lower than that of a unit with shorter tubes and the same area because the labor cost increases with the number of tubes to be welded. In addition, the cost of tubesheets and heads increases significantly with shell diameter. This is why the usual practice is to adopt the greatest length allowed by the equipment layout. The higher the number of tubes, the higher is the flow area, and the smaller is the velocity of the tubeside fluid. If a high number of tubes is adopted, it may be necessary to increase the number of tube passes to get a reasonable velocity and good heat transfer coefficient. Let’s consider a countercurrent heat exchanger such as that shown in Fig. 7-10a. This heat exchanger can be transformed in a two-pass heat exchanger with about the same number of tubes by changing the heads as shown in Fig. 7-10b. If both exchangers are put in operation with the same flow rates, the tube-side fluid velocity in case b will be twice as high as that in case a. The film heat transfer coefficient for turbulent flow within the tubes changes with the power 0.8 of the velocity. Then, with the change performed, the coefficient hi increased by 20.8 = 1.74. If we assume that the friction factor does not change too much with the velocity increase in turbulent flow, the pressure drop can be considered to be proportional to the square of the velocity and to the length

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

(a) FIGURE 7-10

159

(b)

(a) Countercurrent and (b) double-pass heat exchangers.

of the fluid path. Since the path length in case b is twice that of case a, the pressure drop increases eight times with the modification performed. The effect of this modification on the overall U depends on the relative importance of the tube-side coefficient with respect to other resistances. It must be kept in mind that the overall heat transfer coefficient is given by Eqs. (7-1-2) and (7-1-3), and it may happen, especially if the tube-side film is not the controlling resistance, that the improvement achieved in the overall heat transfer coefficient U may be much lower than a 1.74 factor, making the modification unworthy. Obviously, the higher the number of tubes in the heat exchanger, the larger must be the shell diameter. Once the number of tubes, tube passes, and tube array have been selected, it is necessary to determine the necessary shell diameter. This can be done with the help of App. D and with the tube distribution diagrams in App. E. The shell diameter is related to the separation of the baffles because an increase in the shell diameter must be compensated during design with a reduction in baffle spacing to keep a high shell-side fluid velocity. Tube Length. The selection of the tube length was analyzed earlier in connection with the other design parameters. We should add that sometimes it may be convenient to consult the tube suppliers to know what are the commercial lengths available on the market. If the heat exchanger is designed with a tube length that is not a submultiple of a commercial size, a waste of material may result. Baffle Type and Spacing. The most popular type of baffle is the segmental baffle, and the height of the circular segment corresponding to the baffle window is usually about 25 percent of the shell diameter. Baffle spacing determines the shell-side fluid velocity. The smaller the spacing, the higher is the fluid velocity, the higher is the film coefficient, and the higher is the pressure drop. Baffle spacing should be adopted to get the highest convection film coefficient within the pressure-drop limitations. The minimum baffle spacing allowed by the TEMA Standard is 1/5 of the shell diameter, but not less than 2 in. The other function of baffles is to provide mechanical support to the tubes, mitigating bending and vibration. This is why TEMA Standards define maximum values for unsupported tube length. Since the tubes crossing the baffle window are supported every other baffle, the unsupported span is twice the baffle spacing, as shown in Fig. 7-11. Sometimes baffles with no tubes in the window area must be employed (baffle type NTIW).

Maximum unsupported span FIGURE 7-11

Maximum unsupported tube span.

160

CHAPTER SEVEN

7-4 CONVECTION FILM COEFFICIENT AND PRESSURE DROP IN THE TUBE SIDE 7-4-1 Convection Film Coefficient The convection film coefficient for the fluid circulating inside the tubes of a heat exchanger can be calculated with the usual correlations for in-tube convection. A Reynolds number is defined as, Re =

Di Gt μ

(7-4-1)

where Gt = mass velocity = W/at at = tube side flow area = ( N πDi2 ) / 4 n N = number of tubes n = number of tube passes Di = tube internal diameter

(7-4-2)

And the results are correlated using this Reynolds number as follows: 1. For laminar flow (Re < 2,100), according to Sieder and Tate,3 hi Di D 0.33 ⎛ μ ⎞ = 1.86 ⎛ Re ⋅ Pr i ⎞ ⎜⎝ μ ⎟⎠ ⎝ k L⎠ w

0.14

4Wnc ⎞ = 1.86 ⎛ ⎝ πkLN ⎠

0.33

⎛ μ ⎞ ⎜⎝ μ ⎟⎠ w

0.14

(7-4-3)

In a multipass heat exchanger, L is still the length of one tube, and not the length of the total path L · n because in the exchanger heads, the temperature of the fluid becomes uniform, and the temperature profile must start developing again. 2. For the turbulent region (Re > 10,000), the correlation4 is ⎛ μ ⎞ hi Di = 0.023 Re 0.8 ⋅ Pr 0.33 ⎜ ⎟ k ⎝ μw ⎠

0.14

(7-4-4)

The average deviation of this correlation was reported as +15 and −10 percent for Reynolds numbers higher than 10,000. For the particular case of water at moderates temperature and in turbulent regime, the following dimensional correlation is suggested: hi = 1, 423(1 + 0.0146t )

v 0.8 Di0.2

(7-4-5)

where hi = W/(m2 · K) t = °C (mean temperature of the water) v = m/s Di = m 3. Transition zone (2,100 < Re < 10,000). This is a highly nonstable region, and it is not possible to find a suitable correlation representing the experimental results. The following correlation was suggested,18 but the best recommendation is to avoid this region during design: ⎛ Re 0.66 − 125 ⎞ ⎡ ⎛ Di ⎞ 0.66 ⎤ −0.66 ⎛ μ ⎞ hi = 0.116 ⎜ ⎜⎝ μ ⎟⎠ ⎟⎠ ⎢1 + ⎝ L ⎠ ⎥ Pr Re cGt ⎝ w ⎣ ⎦

0.14

(7-4-6)

161

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

7-4-2 Frictional Pressure Drop The frictional pressure drop for fluids circulating in the tube side of a heat exchanger can be considered as the sum of two effects: 1. The pressure drop along the tubes 2. The pressure drop owing to the change in direction in the exchanger heads The pressure drop along the tubes can be calculated with the Fanning equation: Δp = 4 fn

L Gt2 ⎛ μ ⎞ ⎜ ⎟ Di 2ρ ⎝ μ w ⎠

a

(7-4-7)

where the exponent a is −0.14 for turbulent flow and −0.25 in laminar flow. In this equation, nL is the total fluid path length corresponding to n passes. The friction factor in laminar flow is f =

16 Re

(7-4-8)

For heat exchanger tubes, the friction factor in the turbulent region can be determined with the following equation owing to Drew, Koo, and McAdams:6 f = 0.0014 + 0.125 Re −0.32

(7-4-9)

It is normally accepted that Eq. (7-4-8) can be used for 2,100 < Re, whereas Eq. (7-4-9) must be used for Re > 2,100. The friction factors obtained with Eq. (7-4-9) are for smooth tubes. Some authors suggest to increase them by 20 percent for commercial steel heat exchanger tubes. The pressure drop corresponding to the change in direction at the exchanger heads in multipass heat exchangers can be calculated as Δpr = 4 n

Gt2 2ρ

(7-4-10)

where n is the number of tube passes. The total pressure drop thus will be Δ pT = Δ pt + Δ pr

(7-4-11)

Example 7-4 A heat exchanger has 300 3/4-in BWG 16 tubes, 2 m length, arranged in two passes. It is used to heat up 58 kg/s of an oil whose properties are ρ = 790 kg/m3 c = 2,100 J/(kg · K) k = 0.133 W/(m · K) Viscosity as a Function of Temperature [kg/(m · s)] T (°C) μ [kg/(m · s)]

57 3.6 × 10

67 −3

3.02 × 10

77 −3

147

2.5 × 10

−3

7.4 × 10

177 −4

5.2 × 10−4

162

CHAPTER SEVEN

The oil inlet temperature is 57°C, and the desired outlet temperature is 77°C. The heating medium is saturated steam at 927 kPa (condensation temperature = 177°C). Decide if the unit is appropriated for the service and, in that case, what fouling factor is allowed by the heat exchanger. Assume that the oil circulates into the tubes. We must analyze some particular characteristics of heat exchangers in which steam is the heating medium.

Solution

1. If we look at App. G, we notice that the heat transfer coefficients for condensing steam at pressures higher than atmospheric are extremely high. Then, in the expression ⎛ 1 ⎞ 1 U =⎜ + + Rf ⎟ ⎝ hio ho ⎠

−1

the term 1/ho is much smaller than the other two terms. Thus, even if we have a high error in the prediction of ho , this error has little influence in the calculated U. It is therefore usual, in steamheated exchangers, to assume for the condensing steam heat transfer film coefficient a value of 8,500 W/(m2 · K), which is not necessary to verify. Thus, to calculate the overall heat transfer coefficient, it is only necessary to calculate the cold-side film coefficient, and it will be ⎛ 1 ⎞ 1 U =⎜ + + Rf ⎟ ⎝ hio 8, 500 ⎠

−1

2. In these types of units, one fluid (the condensing steam) suffers a change of state. Since it is a purecomponent fluid, the condensation is isothermal, and Q = Wh λ = Wc cc (t 2 − t1 ) where λ is the latent heat of condensation. It can be seen in the Ft graphs that if one fluid is isothermal, Ft equals 1 independently of the number of shell passes. Thus Q = UA × LMTD = UA

(T − t1 ) − (T − t2 ) t −t = UA 2 1 T − t1 T − t1 ln ln T − t2 T − t2

The mean temperature of the cold fluid is 1/2(57 + 77) = 67°C. At this temperature, the viscosity is 3.02 × 10−3 kg/ms. Then Pr =

cμ 0.00302 × 2,100 = = 47.68 k 0.133

2 at = tube-side flow area = 1 πDi2 N = π × 0.0157 × 300 = 0.029 m 2 n 4 4 × 2

Gt = W/at = 58/0.029 = 1,997 kg/(m2 · s) Re = DiGt /μ = 0.0157 × 1,997/3.02 × 10−3 = 10,381 hi = 0.023 Re 0.8 ⋅ Pr 0.33 ×

0.133 ⎛ μ ⎞ w ⎟⎠ 0.0157 ⎜⎝ μw

0.14

k ⎛ μ ⎞ Di ⎜⎝ μ w ⎟⎠

0.14

⎛ μ ⎞ = 1,151⎜ ⎝ μ w ⎟⎠

= 0.023 × 10, 3810.8 × 47.680.333 0.14

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

As a first approximation, we assume that ⎛ μ ⎞ ⎜⎝ μ ⎟⎠ w

163

0.14

= 1, to be checked later. Then

hi = 1,151W/(m2 · K) hio = 1,151

Di 0.0157 = 1,151 = 951 W/(m 2 ⋅ K) Do 0.019

We shall calculate the mean tube wall temperature. The heat transferred from the steam to the tube walls per unit area can be expressed as Q/A = ho(T − Tw) And the heat transferred from the tube walls to the oil is Q/A = hio(Tw − t) If we take for t the mean temperature of the oil, which is 67°C, and we equate both expressions, we get 8,500 × (177 − Tw) = 951 × (Tw − 67) from which we get Tw = 165°C We can obtain the viscosity at the wall temperature. Adopting μw = 5.2 × 10−4 kg/(m · s), then ⎛ μ ⎞ ⎜⎝ μ ⎠⎟ w

0.14

0.00302 ⎞ 0.14 = ⎛⎜ = 1.27 ⎝ 0.00052 ⎟⎠

By applying this correction to the hio value, we get hio = 951 × 1.27 = 1,207 W/(m2 · K) It would be possible to perform a new iteration and recalculate Tw , but this is usually not necessary. The clean overall coefficient (not including fouling) thus will be ⎛ 1 1 ⎞ Uc = ⎜ + ⎟ ⎝ ho hio ⎠

−1

1 ⎞ ⎛ 1 =⎜ + ⎝ 8, 500 1, 207 ⎟⎠

−1

= 1,,056 W/(m 2 ⋅ K)

The heat duty to be transferred in the unit is Q = Wccc(t2 − t1) = 58 × 2,100 × (77 − 57) = 2.436 × 106W The logarithmic mean temperature difference is t2 − t1 77 − 57 = = 109.7°°C 177 − 57 T − t1 ln ln 177 − 77 T − t2 The heat exchanger area is A = πDoLN = π × 0.019 × 2 × 300 = 35.8 m2

164

CHAPTER SEVEN

With a temperature driving force of 109.7°C, the required heat transfer coefficient is U=

Q 2.436 × 10 6 = = 620 W/(m 2 ⋅ K) A ⋅ DMLT 35.8 × 109.7

Since this value is smaller than the calculated Uc , the unit is suitable for the required service. The fouling resistance allowed by this heat exchanger is ⎛1 1 ⎞ 1 1 Rf = ⎜ − ⎟ = − = 6.6 × 10 −4 (m 2 ⋅ K)/W U U , 620 1 056 ⎝ c⎠ Let’s now calculate the pressure drop of the oil: f = 0.0014 + 0.125 Re−0.32 = 0.0014 + 0.125 × 10,381−0.32 = 7.88 × 10−3 Δpt = 4 fn

LGt2 4 × 7.88 × 10 −3 × 2 × 2 × 1, 9972 = = 20, 269 N/m 2 Di 2ρ 0.0157 × 2 × 790 Δpr =

4 nGt2 4 × 2 × 1, 9972 = = 20,192 2ρ 2 × 790

Then ΔpT = Δpt + Δpr = 20,269 + 20,192 = 40,461 N/m2

7-5 HEAT TRANSFER COEFFICIENT AND PRESSURE DROP AT THE SHELL SIDE 7-5-1

Kern Method Calculation of the shell-side heat transfer coefficient and pressure drop is considerably more complex than that of the tube side. The first calculation methods were developed in the 1930s and 1940s. The method that has been the most popular was proposed by Kern.1 This method was used extensively in heat exchanger design. However, the error can be high in some cases. At present, other methods are reported to be more accurate, and we can recommend the Kern method for preliminary estimation, but not as a final design tool. Shell-Side Reynolds Number. In the region between two consecutive baffles of a heat exchanger, the main flow direction is normal to the tubes, as shown in Fig. 7-12. We see that the fluid velocity is subject to continuous fluctuations owing to the reduction in flow area when the fluid crosses a tube row in comparison with the flow area in the space between two consecutive rows. Additionally, the width of the shell cross section changes from zero at the bottom and top of the shell to a maximum in the central plane. Hence it is not possible to define a single value for the cross-flow area. This means that in order to define a fluid velocity, the definition of the flow area must be arbitrary. Kern considers a flow area that corresponds with the hypothetical tube row at the shell central plane. We called it hypothetical because it may happen that there is no tube row at the central plane, but rather two tube rows shifted a certain distance up and down the central plane. This hypothetical flow area is shown in Fig. 7-13. We shall call Pt, or tube pitch, the separation between tube axes, and c is the clearance or free distance between two adjacent tubes in the same row (see Fig. 7-13). The number of clearances existing in the central row can be obtained approximately by dividing the shell diameter by Pt. If B is baffle spacing, the area of any of these clearances is cB, and it results that the shell flow area is aS =

DS cB Pt

(7-5-1)

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

Baffle tips

FIGURE 7-12

Pt

Cross-flow in the region between baffles.

c

B

FIGURE 7-13

Shell-side flow area.

165

166

CHAPTER SEVEN

where Ds is the shell internal diameter. It is then possible to define a mass velocity for the shell fluid as Gs =

W as

(7-5-2)

It must be noted that the free distance between tubes that correspond with the central plane of a tubes row coincides with the tube clearance c in triangular or square arrays. In the case of rotated triangular or square patterns (see Fig. 6-8), these magnitudes are different, and Kern does not indicate how to calculate as in these cases. However, square and triangular arrays are the most popular tubes patterns. Equivalent Diameter. To complete the definition of the Reynolds number, Kern uses an equivalent diameter for the shell. The usual definition of an equivalent diameter is De = 4 × hydraulic radius = 4 ×

flow area wetted periimeter

(7-5-3)

Even though the main component of the fluid velocity is in the direction perpendicular to the tubes axes, Kern defines the hydraulic radius as if the fluid were flowing in a direction parallel to the tubes. The hydraulic radius in Eq. (7-5-3) thus is defined by the tube distribution pattern in the tubesheet. The ratio of flow area to wetted perimeter for square and triangular arrays can be obtained from Fig. 7-14, where the shaded area represents the flow area in Eq. (7-5-3). The result for a square pattern is De =

4 × ( Pt2 − πDo2 / 4) πDo

(7-5-4)

For the triangular array shown in Fig. 7-14, the wetted perimeter for the portion of the tubesheet represented corresponds to half of a tube. Then De =

4 × (0.5Pt × 0.86 Pt − 0.5πDo2 / 4) 0.5πDo

(7-5-5)

Table 7-1 shows the equivalent diameters calculated with Eqs. (7-5-4) and (7-5-5) for the most common arrays. Then the shell Reynolds number is defined as Re s =

DeGs μ

(7-5-6)

Pt c Pt

c

Square array FIGURE 7-14 Tube patterns.

Triangular array

167

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

TABLE 7-1

Equivalent Diameters Calculated for the Most Common Arrays

Tube Diameter 3/4 in (0.019 m)

1 in (0.0254 m) 11/4 in (0.0317 m) 3/4 in (0.019 m) 3/4 in (0.019 m) 1 in (0.0254 m) 11/4 in (0.0317 m)

Pattern

Pitch (see Fig. 6-8)

Equivalent Diameter

Square Square Square Triangle Triangle Triangle Triangle

1 in (0.0254 m) 11/4 in (0.0317 m) 19/16 in (0.0397 m) 15/16 in (0.0238 m) 1 in (0.0254 m) 11/4 in (0.0317 m) 19/16 in (0.0397 m)

0.95 in (0.0241 m) 0.99 in (0.0251 m) 1.23 in (0.0312 m) 0.55 in (0.0139 m) 0.73 in ( 0.0185 m) 0.72 in (0.0183 m) 0.91 in (0.0231 m)

And the heat transfer and friction-factor data are correlated as a function of this Reynolds number. Correlation for the Heat transfer Coefficient ho . The correlation suggested by Kern for a shell with 25 percent cut segmental baffles is ⎛ μ ⎞ ho De = 0.36 Re s0.55 Pr 0.33 ⎜ k ⎝ μ w ⎟⎠

0.14

(7-5-7)

Graphical correlations for other baffle cuts can be found in ref. 7. Pressure Drop. According to Kern, the pressure drop in the shell fluid is proportional to the number of times the fluid crosses the tube bundle. If NB is the number of baffles, the shell fluid crosses the bundle NB + 1 times. The pressure drop is also proportional to the length of the path at every bundle cross, which can be represented by the shell diameter. Thus ΔpS = f

( N B + 1) DS GS2 ⎛ μ w ⎞ De 2ρ ⎜⎝ μ ⎟⎠

0.14

(7-5-8)

The friction factor can be correlated as a function of the shell Reynolds number. The correlations are presented graphically in the original source. A numerical regression of the curves leads to the following expressions: 1. For Re5 < 500,

{

f = exp 5.1858 − 1.7645 ln(Re S ) + 0.13357[ln(Re S )]2

}

(7-5-9)

2. For Re5 > 500, f = 1.728 Re S−0.188

(7-5-10)

Heat Exchangers with 2-4 Configuration. In the case of heat exchangers with two shell passes in a single shell with a longitudinal baffle (TEMA type F), such as that shown in Fig. 7-15, it must be considered that the flow area for the mass velocity calculation is half that of a heat exchanger with the same shell diameter without a longitudinal baffle. The number of bundle crosses to calculate the pressure drop will be twice the number in a heat exchanger with the same baffle spacing and no longitudinal baffle.

7-5-2 Factors Affecting the Performance Not Considered in the Kern Method The Kern method has been criticized8 for a number of reasons. Among them, we can mention the following:

168

CHAPTER SEVEN

FIGURE 7-15

Shell with longitudinal baffle.

Bypass and Leakage Streams. The method assumes that the shell fluid flow is perpendicular to the tubes. This is not true for the following reasons:14 1. At the baffles window, the flow is parallel to the tubes. 2. In the region between baffle tips, part of the fluid diverts toward the periphery of the bundle where frictional resistance is lower, as shown in Fig. 7-16. This fraction of the flow is called the bypass stream. The magnitude of this stream is higher in heat exchangers having a big gap between the tube bundle and

FIGURE 7-16

Bypass streams.

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

169

Sealing strip

FIGURE 7-17

Location of sealing strips.

the shell diameter, as in TEMA type T floating-head heat exchangers. Obviously, this flow is ineffective for the heat transfer because it by passes the tube bundle. One way to solve this problem is to install sealing strips, which are flat bars inserted in the baffles, as shown in Fig. 7-17. These sealing strips redirect the flow toward the center of the bundle. A similar problem appears in multipass heat exchangers, where the separation between tubes must be increased to allocate the pass-partition plates into the exchanger headers. This creates additional bypass lanes, and part of the shell fluid flows through them without contacting the tubes. This fraction of the flow is called pass-partition flow (Fig. 7-18). Sometimes this effect is reduced by installing false tubes (closed at both ends) called dummy tubes5 that act as deflectors (Fig. 7-19). 3. Since the baffle holes crossed by the tubes are drilled with a diameter slightly larger than the tube diameter, there is a clearance between the tube and the baffle, and part of the shell fluid leaks across this clearance. There is also leakage through the free area between the baffle and the shell. These baffle-shell

Pass-partition flow

FIGURE 7-18

Bypass streams through the pass-partition lanes.

170

CHAPTER SEVEN

Tubesheet False tubes (dummy tubes)

FIGURE 7-19

Heat exchanger tubes

Installation of dummy tubes to block bypass lanes.

leaks are completely inefficient from the point of view of the heat exchange. This is not the case with the leaks between baffle and tube, which takes place with a certain amount of heat exchange with the tube fluid (see Fig. 7-20). Thus the total fluid flow in the shell of a heat exchanger is the sum of the following fractions: • • • •

A cross-flow fraction, which is the fraction that really circulates across the bundle A bypass fraction Leaks between the tubes and baffle Leaks between the baffle and shell These flow fractions are represented in Figs. 7-20 and 7-21.

Leakage baffle shell

Cross-flow fraction Leakage tube baffle FIGURE 7-20

Leakage flows.

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

171

Cross-flow fraction

Bypass fraction

FIGURE 7-21

Bypass flows.

The relative magnitude of the several flow fractions is shown in Table 7-2 as a percentage of the total shell fluid flow. The experimental data were obtained in heat exchangers with the following range of parameters: Shell diameter between 3 and 44 in Tube diameters from 1/4 to 3/4 in Baffle cut from 9.6 to 50 percent Baffle-spacing:shell-diameter ratio from 0.1 to 0.8 Number of sealing strips from 0 to 3 for every 5 tube rows Reynolds number between 2 and 100,000 Prandtl number between 0.7 and 9,000 It is easy to appreciate the serious deviation we may have when calculating a heat transfer coefficient with the assumption that 100 percent of the fluid circulates in cross-flow, as in Kern’s method. Tube Pattern. As we explained, Kern defines an equivalent diameter for the shell-side flow. The definition of this equivalent diameter depends on the tube pattern. For square and triangular tubes, it is calculated with Eqs. (7-5-4) and (7-5-5). This approach was objected to by other authors for the following reasons: 1. The equivalent diameter is defined for a flow direction parallel to the tubes, whereas the shell-side flow is mainly normal to the tubes. 2. It is not possible to have a correlation that is valid for every geometry if geometric similitude does not exist. TABLE 7-2

Relative Magnitude of the Several Flow Fractions

Cross-flow Bypass Baffle-shell leakage Baffle-tube leakage

Turbulent Flow

Laminar Flow

10–45% 15–35% 6–21% 9–23%

10–50% 30–80% 6–48% 9–10%

172

CHAPTER SEVEN

T1 t2

T2

t1

FIGURE 7-22 Theoretical temperature diagram.

This means that there should be a different correlation for each geometry. When geometric similitude exists, any characteristic length can be used to define the Reynolds number. Thus there is no need to define the equivalent diameter. This is why the concept of equivalent diameter was abandoned by most authors and the present approach is to use the tube diameter for the Reynolds number and obtain a different correlation for each tube pattern. Effect of Temperature-Profile Distortion. The effect of leakage and bypass flows is not only to reduce the heat transfer coefficients but also to produce a distortion in the temperature profiles that may seriously affect the behavior of the unit. For example, let’s assume that it is desired to heat up a certain fluid from an inlet temperature t1 to temperature t2 using a hot stream available at T1. If we know the mass flow rates of both streams, it is possible, with a heat balance, to calculate the outlet temperature T2, resulting in a temperature diagram such as that in Fig. 7-22. Let’s assume that the cold fluid is the shell-side stream. Since the total flow decomposes in different fractions, and each one is heated up with different efficiency, their temperature evolutions also will be different. For example, let’s consider the total flow divided in two fractions Wb and Wa that exchange heat with different efficiencies, as shown in Fig. 7-23. Curve b shows the evolution of the most efficient flow fraction. The fraction corresponding to the less efficient streams (e.g., leakage between baffle and shell) can follow an evolution such as that of curve a. The resultant from both is curve c, which represents the heat balance. It can be seen that the true temperature difference between the hot fluid and the stream Wb (which is mainly responsible of the heat transfer because Wa corresponds to the less efficient fractions) is lower than

T1 tb t2

T2 c

b a

t1

FIGURE 7-23 Temperature evolution of the different flow fractions.

ta

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

173

T1 = tb

T2

t2 b c a

ta

t1

FIGURE 7-24 Limit case where flow of the “inefficient” fraction is large.

that resulting from a simple heat balance (curve c). Thus the required heat transfer area will be higher in comparison with an exchanger with no leaks. The outlet temperature t2 results from t2 =

Wat a + Wbtb Wa + Wb

If the “inefficient” flow fraction is very high, to get the resulting outlet temperature t2, it will be necessary that the “efficient” fraction b reaches a much higher temperature to compensate inefficiencies of stream a As a limit case, we may have a situation in which curve b reaches the hot fluid temperature at T1, as shown in Fig. 7-24. In this case, an infinite heat transfer area would be necessary. Thus special precautions must be taken when leakage flows are important and the temperature difference between hot and cold streams is small. In these cases, the use of global methods can lead to considerable errors because of the reduction in the true temperature difference with respect to the theoretical value calculated without considering the relative efficiency of the different flow fractions.

7-5-3

Bell Method In 1950 at Delaware University, a research program in shell-and-tube heat exchangers was developed with the sponsorship of the Tubular Exchanger Manufacturers Association (TEMA) and the American Society of Mechanical Engineers (ASME). Many researchers worked in this program and published their conclusions in several reports over that decade. The final synthesis of the study was published as a heat exchanger design method by Kenneth Bell9 in 1963. This is now known as Bell or Delaware method. Fundamentals of the Method. The method is based on heat transfer and pressure-drop experimental data corresponding to an ideal tube bank (a tube bank with infinite width). The ideal tube bank is materialized with a rectangular heat exchanger in pure cross-flow. Half-tubes are installed at the borders to simulate continuity (Fig. 7-25). The data corresponding to the ideal bank17 are later corrected with coefficients that take into account the characteristics of a particular exchanger (such as leakage areas, baffle cut, bypass streams, etc.). Ideal-Tube-Bank Data.

Data are correlated as a function of a Reynolds number defined as Re m =

WDo Sm μ

(7-5-11)

174

CHAPTER SEVEN

Geometry

Rows

Tube diameter

Pitch ratio

10

3/8"

10

Rows

Tube diameter

Pitch ratio

1.25

10

3/8"

1.5

3/8"

1.25

10

3/4"

1.25

14

3/8"

1.25

14

3/4"

1.25

10

3/8"

1.5

Geometry

Flow direction Flow direction FIGURE 7-25

Flow direction

Experimental ideal tube banks in Delaware research program.

where W = shell-fluid mass flow Do = external diameter of tubes μ = viscosity Sm = cross-flow area Sm is the minimum cross-sectional flow area in the tube bank. In square or triangular arrays, this is the free area available for normal flow calculated at the central plane of a tube row (Fig. 7-26). The mass velocity or mass flow density is defined as Gm =

W Sm

(7-5-12)

And the experimental data are expressed as a function of a friction factor f and a Colburn coefficient j defined as f=

2 Δpb ρ ⎛ μ ⎞ 4Gm2 N c ⎜⎝ μ w ⎟⎠

−0.14

(7-5-13)

Cross-flow area Sm

Flow direction FIGURE 7-26

Cross-flow area.

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

175

where Δ pb = tube-bank pressure drop Nc = number of tube rows perpendicular to the flow and j=

⎛μ ⎞ hB Pr 2 / 3 ⎜ w ⎟ cGm ⎝ μ ⎠

0.14

(7-5-14)

where hB is the heat transfer coefficient corresponding to the ideal tube bank. (In the definition of hB, LMTD is used to express the temperature difference. In a heat exchanger, Nc will be the number of tube rows included between the edges of two consecutive baffles.) The experimental data are plotted as a function of the Reynolds number for different tube patterns and different pitch:diameter ratio. This graph is shown in Fig. 7-27a. The data in the figure correspond to an ideal tube bank with 10 tube rows. A numerical regression of these data is presented in Fig. 7-27b. It can be seen that the friction-factor curves corresponding to square arrays present a minimum, which corresponds to the regime transition from laminar to turbulent.15 In the staggered arrays (i.e., triangular or rotated square), the transition is not clear. The difficulty in appreciating the regime transition is due to the fact that this phenomenon is evidenced by a sudden change in the pressure drop. Since the flow area is not constant, the fluid experiences accelerations and deaccelerations, then the inertial terms of the NavierStokes equation are not nil and the turbulence effects are masked by them. The effect is more pronounced in staggered arrays. However, it is normally accepted that the laminar-flow region extends up to a Reynolds number on the order of 100. In this region, the inertial effects are small compared with the viscous forces, and f is approximately proportional to the reciprocal of the Reynolds number.

1

1.25 1.25 1.50 1.50

1.25

0.14

f=

2.Δpb.ρ 4.Gm2 .Nc

μw μ

0.14

10

f facto 1.25 1.50 1.25 1.50

0.01

j fac

tor

j=

h c.Gm

c.μ k

2/3

μw μ

r

0.1

0.001

1

10

102

103

104

105

106

Do .Gm μ FIGURE 7-27 (a) j and f factors for the ideal tube bank. (From O. P. Bergelin, G. A. Brown, and S. C. Doberstein,”Heat Transfer and Fluid Friction During Flow Across Banks of Tubes,” Trans ASME 4:958, 1952. With permission from the American Society of Mechanical Engineers.) (b) Numerical regression of the curves in part a.

176 CORRELATION FOR SHELL SIDE j FACTOR Mathematical regression of curves in Figure 7-27 j = (h/c Gm) Pr 2/3 (μw /μ)0.14 (See variables definition in the text) INTERVAL Geometry and PT/Do ratio Triangle 1.25 Square 1.25 Rotated sq 1.25 Triangle 1.5 Square 1.5

Rem < 100

100 < Rem< 3000

Rem > 3000

j = a.Remb

ln j = a + b ln Rem + c (ln Rem )2

j = a.Remb

a

b

a

b

c

a

b

1.81 0.97 1.81 1.34 0.88

–0.72 –0.62 –0.72 –0.68 –0.64

1.70 3.29 0.32 –0.4548 1.3132

–1.25 –1.90 –0.76 –0.5413 –1.2937

0.065 0.121 0.025 0.00718 0.0743

0.275 0.275 0.275 0.275 0.275

–0.38 –0.38 –0.38 –0.38 –0.38

CORRELATION FOR SHELL SIDE f FACTOR Mathematical regression of curves in Figure 7-27 f = (2.ρ.Δp/4 Nc Gm2) (μw /μ)m (See variables definition in the text) ln f = a + b ln Rem + c (ln Rem)2 + d (ln Rem)3 INTERVAL

Rem < 100

Geometry and PT/Do ratio Triangle 1.25 Square 1.25 Rotated sq 1.25 Triangle 1.5 Square 1.5

FIGURE 7-27

100 < Rem < 1300

a

b

c

d

a

b

4.376 3.923 3.923 3.196 3.04

–1.025 –0.984–0.984 –1.176 –1.13

0 0 0 0.0557 0.031

0 0 0 0 0

5.293 3.871 6.30 3.472 –8.015

–1.864 –0.498 –2.403 –1.547 5.152

(Continued)

1300 < Rem < 3000 c

d

a

0.1584 –0.00472 5.293 –0.2052 0.0221 3.871 0.226 –0.00721 6.3 0.1425 –0.00454 3.472 –1.166 0.0771 –8.015

Rem > 3000

b

c

d

a

b

c

d

–1.864 –0.498 –2.403 –1.547 5.152

0.1584 –0.2052 0.226 0.1425 –1.166

–0.00472 0.00221 –0.0072 –0.00454 0.0771

5.293 –7.907 6.30 3.472 –6.666

–1.864 1.774 –2.403 –1.547 1.152

0.1584 –0.16 0.226 0.1425 –0.0981

–0.00472 0.00407 –0.0072 –0.00454 0.00237

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

177

The region between Reynolds numbers 100 and 4,000 corresponds to a transition regime. The occasional appearance of turbulent eddies can be observed in this range. The friction factor is higher than could be expected if calculated with the laminar-flow correlation. The difference corresponds to the energy wasted in eddy formation. In staggered arrays (i.e., triangular and rotated-square arrays), eddy appearance starts at the outermost downstream tube row, and the phenomenon propagates upstream as the Reynolds number is increased. In square arrays, eddies appear simultaneously in the entire tube bundle, provoking a sudden increase in the friction factor. The higher the distance between tube rows, the lower is the friction-factor increase. Fully turbulent flow corresponds to Rem > 4,000. In this region, the friction factor is proportional to a power of Rem ranging from −0.2 to −0.4. Influence of the Number of Tube Rows. It is known that in laminar flow the heat transfer coefficient decreases with increasing distance from the start of heating. [For example, see Eq. (7-4-3) for in-tube convection.]This is due to the fact that with increasing distance to the tube inlet, the temperature gradient at the tube wall decreases and it also decreases the heat transfer coefficient. This phenomenon also exists during flow across tube banks.. For large heat exchangers in deep laminar flow, it can result in a decrease in the average heat transfer coefficient by a factor of 2 or more compared with what would have been predicted based on flow across a 10-row tube bank such as that used in the Delaware research work. Bell proposed to introduce a correction factor that depends on the total number of tube rows in the fluid path across the exchanger, which is Nc' = ( N c + N w )( N B + 1)

(7-5-15)

where Nc = number of tube rows in each cross-flow section (number of tube rows between baffle tips) Nw = effective number of tube rows in the baffle window [defined in Eq. (7-5-31)] The correction factor is 1. For Rem < 20, ⎛ N' ⎞ X= ⎜ c ⎟ ⎝ 10 ⎠

−0.18

(7-5-16a)

2. For Rem >100, X=1

(7-5-16b)

⎧ ⎡ ⎛ N ' ⎞ −0.18 ⎤ ⎫⎪ ⎪ ⎥⎬ X = 1 − ⎨(1.217 − 0.0121 Re m ) ⎢1 − ⎜ c ⎟ ⎢⎣ ⎝ 10 ⎠ ⎥⎦ ⎪ ⎪⎩ ⎭

(7-5-16c)

3. For 20 ≤ Rem ≤ 100,

Thus, for a real exchanger, the shell-side heat transfer coefficient is related to the 10-row ideal tube bank by hexchanger = hideal bank ⋅ X

(7-5-17)

The number of tube rows does not have any effect on the friction factor. Other Correction Factors for Tubular Heat Exchangers. As we have mentioned, the data corresponding to an ideal tube bank must be corrected to take into account the effects of leakage, bypass streams, and baffle windows when used in real heat exchanger design.

178

CHAPTER SEVEN

TABLE 7-3

Influence of By-pass Area in Shellside Film Coefficient and Shellside Pressure Drop Laminar Flow

Turbulent Flow

Percent Bypass Area with Respect to Cross-Flow Area

Δ p with Bypass Δp without Bypass

h with Bypass h without Bypass

Δp with Bypass Δp without Bypass

h with Bypass h without Bypass

Ideal bank 17.8% bypass without sealing devices 27% bypass without sealing device 17.8% bypass with one sealing device 27% bypass with one sealing device

100% 42%

100% 76–79%

100% 43%

100% 77–80%

22%

60–65%

31%

69–73%

68%

90–94%

67%

90–94%

63%

85–88%

Bypass Effects. Some studies were performed in nonideal tube banks where part of the fluid had the potential to circulate through the peripheral lane between the tube bundle and the shell.19 It was observed that even in cases where the bypass area was not very important, a big fraction of the total flow deviates toward this region. The fraction of the total flow bypassing the bundle could be as high as 75 percent in laminar flow and 50 percent in turbulent flow, with a consequent reduction in heat transfer coefficient and pressure drop (the bypass lane has a lower frictional resistance than the tube bundle). The results are shown in Table 7-3 (these data correspond to a particular exchanger; they are not general). It can be observed that the influence of the sealing devices is very important The effect of the sealing devices is illustrated in Fig. 7-28. All the experiments reported in the table were performed comparing a tube bank with bypass with an ideal tube bank at the same total flow. These sealing effects are introduced in Bell’s method by means of a correction factor for h, namely, ξh =

hBP hB

(7-5-18)

where hBP = coefficient of the bank with bypass area hB = coefficient of an ideal bank And for pressure drop, it is defined as ξ Δp =

Δ pBP Δ pB

(7-5-19)

where Δ pBP = pressure drop of the bank with bypass area Δ pB = pressure drop of an ideal bank

Sealing device

Bypass streams

FIGURE 7-28

Effect of sealing strips.

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

179

TABLE 7-4 Values of α

For ξΔp For ξh

Laminar

Turbulent

5 1.5

4 1.35

The correction factors are calculated as ⎡ ⎛ 2NS ⎞ ⎤ ξ h = ξ Δp = exp ⎢ − αFBP ⎜ 1 − 3 ⎥ N C ⎟⎠ ⎥⎦ ⎝ ⎢⎣ where FBP = bypass fraction =

SBP bypass area = Sm crosss-flow area

(7-5-20)

(7-5-21)

NS = number of pairs of sealing devices in the bank NC = number of tube rows in a cross-flow section (number of tube rows included between edges of two consecutive baffles) α = coefficient depending on the flow regime obtained from Table 7-4. As it can be seen, the values of α are also different for heat transfer and pressure drop calculations If NS > 1/2 NC, take ξ h = ξ ΔP = 1

(7-5-22)

The cross-flow and bypass areas for a heat exchanger can be calculated as Sm = cross-flow area = ( DS − N CL Do ) B

(7-5-23)

where DS = shell diameter NCL = number of tubes in the central row B = baffle spacing SBP = [ DS − ( N CL − 1) Pt − Do ] B

(7-5-24)

where Pt is the separation between tube axes (tube pitch). Note: Equation (7-5-24) can be used for triangular and square arrays. In the case of rotated arrays, Pt must be substituted by the separation between tube axes in a direction perpendicular to the flow in the central row. It is also assumed that pass-partition lanes in the direction parallel to the flow are adequately sealed with dummy tubes. Effect of the Baffle Window. be analyzed separately.

The effect of the baffle window on heat transfer and pressure drop will

Effect of the baffle window on pressure drop. Up to now, we have analyzed the pressure drop corresponding to a cross-flow section, which is the portion of the tube bank limited by the edges of two consecutive baffles, as shown in Fig. 7-29. We shall now consider the pressure drop in the baffle window. It was observed that in order to correlate the experimental data properly, it was necessary to use a velocity vz defined as the geometric mean between the cross-flow velocity and the window velocity. Thus vm = cross-flow fluid velocity = vw = fluid velocity through window =

Gm ρ

Gw W = ρ Sw ρ

(7-5-25) (7-5-26)

180

CHAPTER SEVEN

Cross-flow section FIGURE 7-29

Cross-flow section.

where Sw is the free-flow area through the window, that is, 2

π A ⎛ D ⎞⎛ D ⎞ ⎛D ⎞ A ⎛ Sw = π ⎜ S ⎟ − sin ⎞⎟ ⎜ S ⎟ ⎜ S − BC ⎟ − NWT Do2 ⎠ ⎝ 2 ⎠ 360 ⎜⎝ 4 2⎠⎝ 2 ⎠⎝ 2

(7-5-27)

where NWT is the number of tubes in the window, BC is the baffle cut, and A is the baffle central angle (Fig. 7-30). Then vZ = vm vW

(7-5-28)

(The need to define this velocity arises from the observation that by increasing the baffle spacing, the pressure drop through the window changes despite the fact that vw remains constant.) Pressure Drop Through a Baffle Window in Laminar Flow (Re < 100). The fluid velocity is not uniform throughout the window, and it has components in the directions parallel and normal to the tubes. The total pressure drop through the window will be the sum of three contributions. The first is due to the flow parallel to the tubes, another is due to the normal flow, and the third one is due to direction changes. Data are correlated as a function of the parameter

vZ μ c

Baffle tip

BC = baffle cut

A A = central angle

FIGURE 7-30

Baffle central angle and baffle cut.

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

181

Pt

c FIGURE 7-31 Tube pitch and clearance.

where c is the clearance between tubes which is c = Pt − Do (Fig. 7-31). The pressure drop through the window in laminar flow thus is ⎛ v μ ⎞ ⎛ B ⎞ ⎛ 2ρv Z2 ⎞ ⎛ v μ⎞ Δpwl = 28 ⎜ z ⎟ NW + 26 ⎜ z ⎟ ⎜ + ⎝ c ⎠ ⎝ Dv ⎠ ⎝ Dv ⎟⎠ ⎜⎝ 2 ⎟⎠

(7-5-29)

where Dv is the hydraulic diameter of the window which is Dv =

4 Sw πNWT Do + πDS

(7-5-30)

A 360

NW is a parameter called the effective number of tubes in the window and is defined as NW =

0.8 BC σ

(7-5-31)

where σ is the distance between two tube rows measured in the flow direction (Fig. 7-32). On the right side of Eq. (7-5-29), the first term is the contribution to pressure drop owing to the flow normal to the tubes; the second is the parallel- flow contribution and the last one is due to direction change.

σ

Flow direction FIGURE 7-32

Definition of σ.

182

CHAPTER SEVEN

Pressure Drop Through the Baffle Window in Turbulent Flow (Re >100). The correlation for pressure

drop through the window in turbulent flow is Δpwt = (2 + 0.6 N w )

ρvz2 2

(7-5-32)

Effect of the baffle window on the heat transfer coefficient. The influence that the tubes located at the window exert on the heat transfer was studied by plugging the cross-flow tubes so that the heat transfer was solely due to the window tubes. It was observed that the experimental data can be correlated adequately by multiplying the ideal-tube-bank coefficient by a correction factor φ that includes all the effects exerted by the window. Thus hNL = φhB

(7-5-33)

where hB = ideal-tube-bank coefficient hNL = heat transfer coefficient for a heat exchanger with baffles but no leakages (since the effect of leakage has not been considered yet) And it was found that φ can be correlated as ⎛S ⎞ φ = 1 − r + 0.524r 0.32 ⎜ m ⎟ ⎝ Sw ⎠

0.03

(7-5-34)

where r=

number of tubes in window N wt = total number of tubes N

Thus the shell-side heat transfer coefficient for a heat exchanger with no leakage can be calculated as 0.14 ⎤ ⎡ ⎛μ ⎞ ⎥ φξ h X hNL = ⎢ jcGm Pr 2 / 3 ⎜ w ⎟ μ ⎝ ⎠ ⎥⎦ ⎢⎣

(7-5-35)

The term within the brackets corresponds to hB , and the others are the correction factors. Effect of Leakage. Effect of leakage on pressure drop. Owing to the leakage existing between baffles and tubes and between baffles and shell, both pressure drop and heat transfer coefficients differ from the values of an ideal bank. This subject was studied by Sullivan and Bergelin.10,12 The mathematical treatment was simplified by Bell, who assumed that the ratio between leakage flow rate and cross-flow flow rate is independent of the flow regime and depends only on the ratio between leakage area and cross-flow area. This is a simplification of the problem. Some other methods allow calculation of the individual flow rates of the different flow fractions as a function of the total flow and geometric parameters. Then, according to Bell, the ratio between ΔpL (pressure drop for a heat exchanger with leakage) and ΔpNL (pressure drop for a similar heat exchanger with the same total flow but without leakage) can be represented by a curve of the type shown in Fig. 7-33. The upper curve corresponds to a heat exchanger where the leakage occurs between tubes and baffles exclusively, and the lower curve corresponds to a heat exchanger with leakage only between the shell and baffles. It can be appreciated that [1 − (ΔpL/ΔpNL)] for tube-baffle leakage is approximately half the value of [1 − (ΔpL/ΔpNL)] corresponding to shell-baffle leakage. Thus the mathematical treatment can be simplified using a single curve, which is that of tube-baffle leakage (Fig. 7-34).

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

183

1 Δp

tube-baffle (1– ΔpNLL ) For leakage

0.8 ΔpL ΔpNL

SSB =0 SL

0.6 Δp

shell-baffle ( 1– ΔpNLL ) For leakage

0.4 0.2 0

0.05

0.1

0.15

0.2

SSB =1 SL

0.3

0.25

SL Sm FIGURE 7-33

Baffle leakage effect on pressure drop. (Courtesy of Professor K. Bell )

0.7 0.6

ΔpL ΔpNL

0.4

1–

0.5

0.3 0.2 0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

SL Sm FIGURE 7-34

Correlation for the leakage effect.

The ordinate of the graph represents the value (1 − ΔpL/ΔpNL) for a heat exchanger with leakage only between tubes and baffles. The mathematical expression of the curve is11 ⎡ ⎛ ⎛ Δ pL ⎞ SL SL ⎞ ⎤ ⎜⎝ 1 − Δ p ⎟⎠ = 0.57 S + 0.27 ⎢1 − exp ⎜⎝ −20 S ⎟⎠ ⎥ NL o m m ⎦ ⎣

(7-5-36)

⎛ ⎛ Δ pL ⎞ Δ pL ⎞ STB + 2 SSB = ⎜1 − ⎜⎝ 1 − Δ p ⎟⎠ Δ pNL ⎟⎠ o SL ⎝ NL exchanger

(7-5-37)

and it is

184

CHAPTER SEVEN

where STB = leakage area between tubes and baffle SSB = leakage area between baffle and shell and STB + SSB = SL

(7-5-38)

STB and SSB can be calculated as follows: STB = N BT

π 2 DBT − Do2 4

(

)

(7-5-39)

where NBT = number of tubes passing across the baffle DBT = diameter of the baffle holes and 360 − A ⎞ π 2 SSB = ⎛ D − DB2 ⎝ 360 ⎠ 4 S

(

)

(7-5-40)

where DS = shell diameter DB = baffle diameter Effect of leakage on heat transfer. With similar simplifications to those assumed in pressure drop calculations, it is possible to draw curves showing the ratio hL/hNL (where hL = heat transfer coefficient for a heat exchanger with leakage, and hNL = heat transfer coefficient for a heat exchangers with no leakage and operating with the same flow rate) as a function of SL/Sm. Here again, it can be observed that the value of (1 − hL/hNL) for tube-baffle leakage is roughly half the value corresponding to shell-baffle leakage. Then it is also possible to use a single curve, which can be correlated as11 ⎡ ⎛ ⎛ hL ⎞ SL SL ⎞ ⎤ ⎜⎝ 1 − h ⎟⎠ = 0.45 S + 0.1 ⎢1 − exp ⎜⎝ −30 S ⎟⎠ ⎥ NL o m m ⎦ ⎣

(7-5-41)

⎛ ⎛ hL ⎞ h ⎞ S + 2SSB = ⎜ 1 − L ⎟ TB ⎜⎝ 1 − h ⎟⎠ SL ⎝ hNL ⎠ o NL exchanger

(7-5-42)

And then

With these equations, knowing (1 − hL/hNL) and (1 − ΔpL/ΔpNL), it is possible to calculate hL/hNL and ΔpL/ΔpNL, which are the correction factors to be applied to correct the values of h and Δ p corresponding to an exchanger without leakage. Conclusions. Heat Transfer. The shell-side heat transfer coefficient of a heat exchanger can be obtained as 0.14 ⎤⎛ ⎡ ⎛ μ ⎞ h ⎞ ⎥ ⎜ φξ h X L ⎟ h = ⎢ jcGm Pr −2 / 3 ⎜ ⎟ hNL ⎠ ⎝ μw ⎠ ⎥⎦ ⎝ ⎢⎣

(7-5-43)

which equals Ideal-tube-bank coefficient × window correction × bypass correction × number-of-tube-rows correction × leakage correction

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

185

Pressure Drop. The total shell-side pressure drop must be calculated by adding the pressure drops corresponding to all cross-flow sections and all baffle windows. The pressure drop in a cross-flow section without leakage correction is

ΔpBP =

4 fN cGm2 ⎛ μ w ⎞ 2ρ ⎝⎜ μ ⎠⎟

0.14

ξ Δp

(7-5-44)

= Δp ideal tube bank × bypass correction The pressure drop through each window can be obtained as follows: 1. Laminar flow: Δpw = 28

vZ μ v μBC 2ρv Z2 N c + 26 Z + Dv Dv 2 ( Pt − Do )

(7-5-45)

2. Turbulent flow: Δpw = (2 + 0.6 N w )

ρv Z2 2

(7-5-46)

The total pressure drop is obtained by adding all cross-flow sections and all windows as ⎛ ⎛ Δ pL ⎞ N ⎞ Δ p total = 2 Δ pBP ⎜ 1 + W ⎟ + [ ( N B − 1) ΔpBP + N B Δ pW ] ⎜ NC ⎠ ⎝ ⎝ Δ pNL ⎟⎠

(7-5-47)

In Eq. (7-5-47), the first term on the right side represents the pressure drop through the inlet and outlet sections, the first term within the square brackets represents the remaining cross-flow sections, and the second term in the square brackets corresponds to the pressure drops through the windows. For example, let’s consider the heat exchanger in Fig. 7-35, where NB is the number of baffles (=4), Sections I and V are the inlet and outlet sections. These are treated as no-leakage sections. Since it can be observed that the number of tube rows crossed by the fluid in these two sections is Nw + Nc instead of Nw, the correction factor (NW + NC)/NC is applied. The pressure drop through these two sections is ⎛ N ⎞ 2 ΔpBP ⎜ 1 + w ⎟ Nc ⎠ ⎝

I

II

III

FIGURE 7-35 A heat exchanger with four baffles.

IV

V

186

CHAPTER SEVEN

The pressure drop through the cross-flow sections identified as sections II, III, and IV is ⎛ Δ pL ⎞ ⎛ Δ pL ⎞ ( N B − 1) Δ pBP ⎜ = 3Δ pBP ⎜ ⎟ ⎝ Δ pNL ⎠ ⎝ Δ pNL ⎟⎠ And the pressure drop through the four windows is ⎛ Δ pL ⎞ ⎛ Δ pL ⎞ N B Δ pW ⎜ = 4 Δ pW ⎜ ⎟ ⎝ Δ pNL ⎠ ⎝ Δ pNL ⎟⎠ Computational Techniques Nowadays, heat exchanger design is performed almost exclusively using commercial software, as are many other activities. The use of commercial software makes the designer’s task faster and easier, and in addition, the designer responsibility is backed by the prestige of the software supplier. These programs are developed by companies that have specialists in heat transfer and sponsor research programs at universities or have their own research facilities. Normally, their design programs are more elaborated than the simple methods presented in this chapter. For example, Heat Transfer Research, Inc. (HTRI) uses a calculation method that subdivides the heat exchanger into a series of three-dimensional cells (Fig. 7-36). First, the heat exchanger is divided into increments in the axial direction. Then it is divided into rows in the vertical direction and, finally, into sections in the horizontal direction. The number of rows and sections depends on the geometry of the heat exchanger (i.e., type of baffle, baffle-cut orientation, number of tube passes, etc.). The total number of cells necessary is such that in each one there is a predominant type of shell flow (axial in window or cross-flow) and only one pass in tubes. In each cell, the program calculates the heat transfer and pressure drop using HTRI proprietary correlations and local values of temperature, pressure, and physical properties. The model is solved with finiteelements analysis, and the heat exchanged is calculated as Q = ΣΔQi = ΣΔAiUi ( ΔTml )i

Increments

Rows

7-5-4

Sections FIGURE 7-36

Heat exchanger cells.

(7-5-48)

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

187

This rigorous technique allows one to predict the behavior of heat exchangers with complex geometries. It must be noted that this model uses neither the concept of logarithmic mean temperature difference nor the Ft correction factor. 7-5-5 Accuracy of Correlations As we have explained previously, the heat exchanger design task is presently performed almost exclusively with computational techniques. However, we must realize that for decades, all heat exchangers (most of them still in operation) were designed using Kern’s and Bell’s method. The information regarding accuracy of the correlations must be analyzed with care. As we have seen, there are certain geometries for which we know that these methods present problems. For example, we know that Kern’s method cannot be applied to a TEMA type T floating-head heat exchanger without sealing strips or with unsealed pass-partition lanes. Whitley18 presents a study of the errors found in heat transfer coefficient and pressure-drop predictions obtained with the Kern and Bell methods, among others. In some cases, Kern’s correlations are reported to present errors as high as 850 percent. However, it is not informed what are the types of heat exchangers presenting so high deviations. Probably if those “pathologic” cases were not considered, the errors may have been considerably lower. According to Palen and Taborek,14 the Bell-Delaware method allows one to predict shell-side film coefficients that range from 50 percent lower to 100 percent higher than the real values, whereas the error range for pressure-drop prediction varies from 50 percent lower to 200 percent higher than the real values. For the entire set of data analyzed, the mean error for heat transfer was 15 percent below the real values (safe side) for all Reynolds numbers, whereas the mean error for pressure-drop prediction was about 100 percent above the real values for Reynolds numbers lower than 10. Commercial program suppliers offer very little information about the correlations used, and comparative studies among their results and accuracy have not been not published. Example 7-5 It is desired to heat up 38.88 kg/s of a liquid A from 40 to 61°C using process hot stream B, available at 104°C with a flow rate of 33.33 kg/s. The intention is to use an existing heat exchanger whose characteristics are indicated below. A 0.0009 (m2 · K)/W fouling resistance is anticipated. Verify if the unit is suitable for the service, and calculate the fluid pressure drops. Note: A downloadable spreadsheet with the calculations of this example is available at http://www. mhprofessional.com/product.php?isbn=0071624082 Fluids Data

W (kg/s) ρ (kg/m3) c [J/(kg · K)] μ [kg/(m · s)] k [W/(m · K)]

Fluid A (Tube Side)

Fluid B (Shell Side)

38.88 716 2140 0.62 × 10−3 0.129

33.33 578 2640 0.16 × 10−3 0.0917

Heat Exchanger Data N = number of tubes = 414 two-tube passes, type AES Tubes 3/4 in BWG 16 (Do = 0.019 m, Di = 0.0157 m) Pattern Δ, Pt = 1 in (0.0254 m) L = tube length = 4.267 m Central baffle spacing B = 0.234 m Shell diameter Ds = 0.609 m

188

CHAPTER SEVEN

Tubes layout and other geometric dimensions according to Fig. 7-37. 1. Tube-side film coefficient calculation: a 't = tube flow area =

πDi2 π × 0.01572 = = 1.936 × 10 −4 m 2 4 4

at = total tube-side flow area = Gt = mass velocity =

a t' N 1.935 × 10 −4 × 414 = = 0.0401 m 2 n 2 Wt 38.88 = = 970 kg/(m 2 ⋅ s) at 0.0401

DS DB

σ BC

NCL Nc

SSB Detail STB

N Ntw NCL NBT Nc σ Ds DB BC A FIGURE 7-37

= Total number of tubes = 414 Do = Number of tubes in window = 75 DBT = Number of tubes in the central row = 21 = Number of tubes crossing the baffle = 339 = Number of tubes rows between baflles tips = 13 = 0.022 m D = 0.019 m o = 0.609 m D BT = 0.0198 m = 0.6043 m P = 0.0254 m t = 0.1522 c = 0.064 m = 120º

Figure for Example 7-5.

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

Re t = Reynolds number = Pr = Prandtl number = hi = 0.023

⎛ μ ⎞ k × Re 0.8 × Pr 0.33 ⎜ Di ⎝ μ w ⎟⎠

189

DiGt 0.0157 × 970 = 24,568 = μ 0.62 × 10 −3

cμ 2,140 × 0.62 × 10 −3 = = 10.29 k 0.129

0.14

= 0.023 ×

0.129 × 24, 5680.8 × 10.290.33 = 1,327 W/(m m 2 ⋅ K) 0.0157

All fluid properties have been considered constant, so the correction factor μ/μW is unity. The same criteria will be adopted for the shell-side fluid. Di 0.0157 = 1, 327 = 1,096 W/(m 2 ⋅ K) Do 0.019

hio = hi 2. Tube-side pressure drop:

f = friction factor = 1.2 × (0.0014 + 0.125 Re −0.32 ) = 7.58 × 10 −3 Δpt = pressure drop in tube = 4 fn

L Gt2 4.267 × 970 2 = 4 × 7.58 × 10 −3 × 2 × = 10, 843 N/m 2 Di 2ρ 0.0157 × 2 × 716 Δpr =

4.n.Gt2 = 5, 256 N / m 2 2ρ

Δ pT = Δ pt + Δ pr = 16,099 N/m 2 3. Shell-side heat transfer coefficient: Both Kern’s and Bell’s method will be used. 3-a. Kern’s method: aS = shell-side flow area =

DS cB 0.609 × (6.35 × 10 −3 ) × 0.234 = = 0.0356 m 2 Pt 0.0254

GS = mass velocity =

W 33.33 = = 936.2 kg/(m 2 ⋅ s) aS 0.0356

De = equivalent diameter = 0.0182 m Re S = Reynolds number =

ho = shell coefficient = 0.36

De GS 0.0182 × 936.2 = 106,492 = μ 0.16 × 10 −3

⎛ 2640 × 0.16 × 10 −3 ⎞ k 0.0917 Re 0S.55 Pr1/ 3 = 0.36 106, 4920.55 ⎜ ⎟ De 0.0182 0.0917 ⎝ ⎠

= 1,756 W/(m 2 ⋅ K) 3-b. Bell’s method: Sm = cross-flow area = ( DS − N CL Do ) B = (0.609 − 21 × 0.019) × 0.234 = 0.0491 m 2 Gm = mass velocity =

W 33.33 = = 678.3 kg/(m 2 ⋅ s) Sm 0.0491

1/ 3

190

CHAPTER SEVEN

Re m = Reynolds number =

Do Gm 0.019 × 678.3 = 80,544 = μ 0.16 × 100 −3

j = Colburn coefficient = 0.00376 (In this case, Pt /Do = 1.33. The curves corresponding to Pt /Do = 1.25 were used.) 3-b-1. Bypass correction: FBP = bypass fraction =

[ DS − ( NCL − 1)Pt − Do ] B = [0..609 − (21 − 1)0.0254 − 0.019] × 0.234 = 0.390 0.0491

Sm

∴ α = 1.35 If two pairs of sealing strips are used,

{

}

ξ h = exp −1.35FBP ⎡⎣1 − (2 × N s / N c )0.333 ⎤⎦ = exp{−1.35 × 0.39 × [1 − (2 × 2/13)0.333 ]} = 0.842 Please note that if no sealing strips were used (Ns = 0), the bypass correction factor would be 0.59 and this reduces the heat transfer coefficient to about 30 percent. Observe that Kern’s method remains insensitive to these important effects 3-b-2. Window correction: r= SW = flow area at window = =

N wt 75 = = 0.181 N 414

πDS2 A ⎛ A D D2 − sin ⎞ ⎛ S ⎞ ( 0.5 DS − BC ) − N wt π o 4 360 ⎝ 2⎠⎝ 2 ⎠ 4

π × 0.6092 120 π × 75 × 0.0192 = 0.0357 m 2 × − (sin 60° ) × 0.3045 × (0.3045 − 0.152) − 4 4 360

⎛S ⎞ φ = correction factor = 1 − r + 0.524 r 0.32 ⎜ m ⎟ ⎝ SW ⎠

0.03

0.0491 ⎞ 0.03 = 1 − 0.181 + 0.524 × 0.1810.32 ⎛ = 1.125 ⎝ 0.0357 ⎠

3-b-3. Leakages correction: STB = tube-baffle leakage area = N BT SSB = baffle-shell leakage area =

π 2 π DBT − D02 = 339 0.01982 − 0.0192 = 0.00826 m 2 4 4

(

)

(

)

360 − A π 2 360 − 120 π DS − DB2 = 0.6092 − 0.60432 = 0.003 m 2 360 4 360 4

(

)

(

)

S L = total leakage area = STB + SSB = 0.003 + 0.0082 = 0.0112 m 2 S L 0.0112 = = 0.228 Sm 0.0491 ⎡ ⎛ ⎛ hL ⎞ SL SL ⎞ ⎤ −3 ⎜⎝ 1 − h ⎟⎠ = 0.45 S + 0.1 ⎢1 − exp ⎜⎝ −30 S ⎟⎠ ⎥ = 0.45 × 0.228 + 0.1(1 − 1.07 × 10 ) = 0.203 NL o m m ⎣ ⎦

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

1− ∴

hL ⎛ h ⎞ STB + 2 SSB 0.00826 + 2 × 0.003 = 1− L = 0.203 = 0.257 hNL ⎜⎝ hNL ⎟⎠ o SL 0.0112

hL = 1 − 0.257 = 0.743 hNL

3-b-4. The correction for number of tube rows is negligible in turbulent flow. 3-b-5. Shell-side heat transfer coefficient: −0.14 ⎡ ⎤ ⎛ h ⎞ ⎛μ ⎞ h = ⎢ jcGm Pr −2 / 3 ⎜ w ⎟ ⎥ φξ h ⎜ L ⎟ ⎝ μ ⎠ ⎝ hNL ⎠ ⎢⎣ ⎥⎦

= (0.00376 × 2,640 × 678.3 × 4.606−0.66 × 1) × 1.1251 × 0.8426 × 0.743 = 1,714 W/(m 2 ⋅ K) Both methods in this particular situation give similar results, but this is not normally the case. 4. Shell-side pressure drop 4-a. Kern’s method: f = friction factor = 1.728 Re −S0.188 = 1.728(106, 492)−0.188 = 0.196 Δp = f

GS2 DS ( N B + 1) 0.196 × 936.22 × 0.609 × 18 = 89,500 N/m 2 = 2ρDe 2 × 578 × 0.0182

4-b. Bell’s method: 4-b-1. Friction factor: f = 0.095 (with Re m = 80,544; see Fig. 7-27) 4-b-2. Bypass correction: ⎧⎪ ⎡ ⎛ 2 N ⎞ 1/ 3 ⎤ ⎪⎫ ⎡ ⎛ 2 × 2 ⎞ 1/ 3 ⎤ ⎪⎫ ⎪⎧ s ⎥ exp . = − × × ξ Δp = exp ⎨− αFBP ⎢1 − ⎜ 4 0 39 ⎨ ⎬ ⎟ ⎥ ⎬ = 0.602 ⎢1 − ⎜⎝ ⎟ 13 ⎠ ⎦ ⎭⎪ ⎢⎣ ⎝ N c ⎠ ⎥⎦ ⎪⎭ ⎣ ⎪⎩ ⎩⎪ 4-b-3. Pressure drop in a cross-flow section without leakage: ΔpBP =

4 fNC Gm2 (μ w / μ)0.14 ξ Δp 2ρ

=

4 × 0.095 × 13 × 678.32 × 0.602 = 1183.6 N/m 2 2 × 578

4-b-4. Pressure drop through window: VZ = mean velocity =

W = ρ Sm SW 578

33.33 = 1.38 m/s 0.0491 × 0.0357

ΔpW = pressure drop through window = (2 + 0.6 NW ) Thus, since NW = 0.8

ρvZ2 2

BC 0.8 × 0.15225 = = 5.54 σ Pt sin 60°

∴ Δ pW = (2 + 0..6 × 5.54) × 578 ×

1.382 = 2,930 N/m 2 2

191

192

CHAPTER SEVEN

4-b-5 Leakage correction: ⎡ ⎛ ⎛ −20 S L ⎞ ⎤ Δ pL ⎞ SL −4.56 ) = 0.397 ⎜⎝ 1 − Δ p ⎟⎠ = 0.57 S + 0.27 ⎢1 − exp ⎜⎝ S ⎟ ⎥ = 0.57 × 0.228 + 0.27 × (1 − e NL o m m ⎠⎦ ⎣ 1− ∴

Δ pL ⎛ Δ pL ⎞ STB + 2 S SB = ⎜1 − = 0.397 × 1.27 = 0.504 Δ pNL ⎝ Δ pNL ⎟⎠ o SL

Δ pL = 1 − 0.504 = 0.496 Δ pNL

4-b-6 Pressure drop through the entire unit: ⎛ N ⎞ Δ pL ΔpS = 2 Δ pBP ⎜ 1 + W ⎟ + [ ( N B − 1) Δ pBP + N B Δ pW ] Δ pNL N ⎝ c ⎠ 5.54 ⎞ 2 = 2 × 1,183.6 × ⎛⎜ 1 + ⎟ + (16 × 1,183.6 + 17 × 2930) × 0.496 = 37,445 N/m ⎝ 13 ⎠ Note the difference between the two methods. 5. Verification of the heat transfer area: ⎛ 1 ⎞ 1 U =⎜ + + Rf ⎟ ⎝ ho hio ⎠

−1

1 ⎞ ⎛ 1 =⎜ + + 9 × 10 −4 ⎟ ⎝ 1, 714 1, 096 ⎠

−1

= 417 W/(m 2 ⋅ K)

Q = Wc cc (t2 − t1 ) = 38.88 × 2,140 × (60 − 40) = 1.664 × 10 6 W T2 = T1 − R=

Q 1.664 × 10 6 = 104 − = 85°C Wh ch 2, 640 × 33.33

104 − 85 = 0.95 60 − 40

and

S=

60 − 40 = 0.312 104 − 40

For a 1-2 configuration, we get Ft = 0.967. Then DMLT =

(T1 − t2 ) − (T2 − t1 ) (104 − 60) − (85 − 40) = = 44.5°C T −t 104 − 60 ln ln 1 2 85 − 40 T2 − t1

A (calculated) =

Q 1, 664, 064 = 92.7 m 2 = U ΔT 417 × (44.5 × 0.967)

A (real) = πDo NL = π × 0.019 × 414 × 4.267 = 105.4 m 2 Excess area =

105.4 − 92.7 × 100 = 14% 92.7

It is possible now to estimate the wall temperature by equating the heat transfer rate on both sides of the tube wall, as in Example 5-5. A more rigorous expression considering the individual fouling resistances would be ⎛ 1 ⎞ ⎜⎝ h + R fi ⎟⎠ io

−1

⎛ 1 ⎞ (Tw − t ) = ⎜ + R fo ⎟ ⎝ ho ⎠

−1

(T − Tw )

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

193

From which Tw and the viscosity correction factors can be calculated, feeding back the procedure if necessary.

7-6 DESIGN AND RATING OF HEAT EXCHANGERS Two different types of problems can exist in relation with heat exchanger design or evaluation: 1. To verify adequacy of an existing or proposed unit (rating) 2. To design a new unit to perform a certain service In what follows we shall see how to proceed in either of these cases. 7-6-1

Rating Existing Units This task consists of deciding whether a unit will allow us to reach a proposed process objective. For example, let’s assume that we want to cool down a certain process stream with known mass flow from a temperature T1 to a final T2. It was decided to use cooling water with a certain inlet temperature and with a proposed flow rate. There is a heat exchanger available whose geometric dimensions are known, and it is desired to know if the objective will be reached with this unit. To do this, we assume that the desired outlet temperatures will be obtained, so it is possible to calculate the LMTD. Since the exchanger configuration is known, we can calculate the Ft correction factor and the effective temperature difference. Since the flow rates and flow areas are known, it is possible to calculate the heat transfer coefficients ho and hio. By adding the fouling resistance based on Eq. (7-1-2), we obtain the overall heat transfer coefficient U. It is then possible to calculate a heat transfer area by means of A (calculated) =

Q U ΔT

(7-6-1)

This is the surface area required by a heat exchanger whose overall heat transfer coefficient is U to exchange a heat-duty Q with a “driving force” ΔT. This value can be compared with the real area of the unit we are verifying. If Acalc > Areal, the heat exchanger will not be able to transfer the required heat. If the exchanger is put into operation anyway, a heat-duty Q' will be transferred, with Q' < Q. The result is that the outlet temperature of the hot fluid will be higher than desired, and the outlet temperature of the cooling water will be lower than proposed. In case it happens that Acalc < Areal, the unit will transfer a higher amount of heat than originally proposed. In any case, if calculation of the real outlet temperature of the fluids is desired, it would be necessary to solve the following set of equations: Q = Wh ch (T1 − T2 )

(7-6-2)

Q = Wc cc (t 2 − t1 )

(7-6-3)

Q = UAreal ⋅ DMLT ⋅ Ft

(7-6-4)

Ft = f (T1 , T2 , t1 , t2 )

(7-6-5)

In this system, the unknowns are Q, Ft, T2, and t2. The resolution of this equation system, as it is written, presents the difficulty that it cannot be solved algebraically, and iterative procedures are necessary. In Sec. 7-7 we shall explain a technique that allows a straightforward solution and avoids this iterative procedure. However in many cases, it is not necessary to know the exact outlet temperatures, but it is enough to know if the unit is capable of transferring, at a minimum, the required duty. Continuing with our example, we should consider that if it were really necessary to establish and maintain an exact value for

194

CHAPTER SEVEN

the process-fluid outlet temperature with an oversized heat exchanger, it is always possible to adjust it by reducing the cooling-water flow. The other thing that always must be checked is that the fluid pressure drops are lower than the allowable values. In what follows we shall develop a detailed procedure with all the steps for verification of an existing heat exchanger. The flow rates and inlet temperatures will be known, as well as the outlet temperature of one fluid. In this case, with a heat balance, we can calculate the outlet temperature of the other fluid. It is also possible that the outlet temperatures of both fluids have been defined, in which case we shall use the heat balance to calculate one of the flow rates. Whatever the situation, the steps are as follows: 1. Write the heat balance, that is, Q = Wh ch (T1 − T2 ) = Wc cc (t 2 − t1 )

(7-6-6)

and then calculate Q and the remaining unknown, flow rate or temperature. 2. With the four temperatures known, calculate LMTD as LMTD =

(T1 − t 2 ) − (T2 − t1 ) T −t ln 1 2 T2 − t1

(7-6-7)

3. Calculate R and S parameters: R=

T1 − T2 t 2 − t1

(7-6-8)

S=

t 2 − t1 T1 − t1

(7-6-9)

4. For the exchanger configuration (i.e., 1-2, 2-4, etc.), obtain Ft from graphs or analytical expressions included in appendices (graphs in App. B). 5. Obtain the physical properties of the fluids at the mean temperatures. These will be (T1 + T2)/2 and (t1 + t2)/2 for the hot and cold fluids, respectively These mean temperatures will be called T and t. In order to calculate the heat transfer coefficients, it will be necessary to calculate (μ/μW) for each fluid, where μ is the viscosity at the mean temperature, and μW is the viscosity at the tube wall temperature. The tube wall temperature is not known beforehand. Thus it is necessary to assume a tube wall temperature that will be verified later. To make a first guess, it must be considered that the tube wall temperature will have an intermediate value between T and t and will be closer to the temperature of the fluid with higher film coefficient h. Anyway, the exponent 0.14 means that the importance of this factor is not very high, and normally, the first guess is a “good enough” value. 6. The flow area for the fluid flowing into the tubes is calculated as at =

Na t' n

(7-6-10)

The mass velocity of the tube-side fluid can be calculated as Gt = ρvt = Wt / at

(7-6-11)

7. Calculate the Reynolds number for the tube-side fluid: Re t =

Di Gt μ

(7-6-12)

195

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

8. Calculate hi with Eqs. (7-4-3) through (7-4-5) based on the Reynold number. 9. Correct for the external diameter: hio = hi

Di Do

(7-6-13)

10. Calculate the shell-side heat transfer coefficient based on the procedures explained in Sec. 7-5. 11. With the calculated values of hio and ho, the tube wall temperature assumed in step 5 can be checked. To do this, it is necesssary to equate the heat transfer rates at both sides of the wall: hio (Tw − t ) = ho (T − Tw )

(7-6-14)

if the shell-side fluid is the hot fluid, or hio (T − Tw ) = ho (Tw − t )

(7-6-15)

if the tube-side fluid is the hot fluid. And then we can solve Eq. (7-6-15) for Tw . 12. The clean heat transfer coefficient can be calculated as ⎛ 1 1⎞ Uc = ⎜ + ⎟ ⎝ hio ho ⎠

−1

(7-6-16)

13. The total heat transfer coefficient can be obtained as ⎛ 1 ⎞ U =⎜ + Rf ⎟ ⎝ Uc ⎠

−1

(7-6-17)

14. The calculated area is Acalc =

Q U ΔT

(7-6-18)

15. The real area of the heat exchanger is calculated with its geometric dimensions: Areal = N πDo L

(7-6-19)

16. If Areal > Acalc, it will be possible to use the equipment from the heat transfer point of view. The excess area will be Percent excess =

Areal − Acalc × 100 Acalc

(7-6-20)

Tube-side pressure-drop calculation: With Ret calculated in step 7, obtain the friction factor with Eq. (7-4-8) or Eq. (7-4-9), and calculate the straight-tube pressure drop with Eq. (7-4-7). Then calculate the pressure drops in the headers with Eq. (7-4-10). The total pressure drop will be the sum of both effects, as indicated in Eq. (7-4-11). 17. Shell-side pressure-drop calculation: It can be calculated with the methods explained in Sec. 7-5. 18. The heat exchanger will be suitable for the required service if the two following conditions are satisfied: a. Areal > Acalc, as explained in step 16 b. Both the tube-side and shell-side calculated Δp values must be lower than allowable values.

196

CHAPTER SEVEN

7-6-2 Design of a Heat Exchanger If a totally new heat exchanger must be designed to perform a certain service, all its geometric characteristics must be defined by the designer. In general terms, the procedure is to propose the heat exchanger geometry and then apply the rating methodology detailed in Sec. 7-6-1. In this case, the goal is to minimize the difference between the assumed and calculated areas because any excess area results in an unnecessary cost. Additionally, the heat transfer coefficients must be as high as possible because then the heat transfer area will be at a minimum. The obvious limitation to the increase in heat transfer coefficients is the allowable pressure drop for both fluids. As a general rule, we can say that the design will be optimal when the Δ p values of both fluids are close to the maximum allowable values (because then the heat transfer coefficients also will be close to the maximum) and the heat transfer area is enough, but with little excess, to transfer the required heat duty. If any of these conditions are not satisfied, it will be necessary to change the geometry of the unit. We have explained in Sec. 7-3 how the several geometric parameters affect the performance of the unit, and these criteria must be considered to improve the design. To initially adopt the heat exchanger geometry, the following procedure is suggested: 1. With a heat balance, the unknown process variable (flow rate or temperature) of one of the streams can be found. Then calculate LMTD. 2. Select the number of shell passes or shells configuration with the criteria explained in Sec. 7-2 and with the procedure explained in Example 7-3. If there is no limitation owing to Ft considerations, a heat exchanger with one shell pass will be selected. It must be remembered that the countercurrent configuration allows the highest ΔT because Ft = 1, and this possibility must be investigated. Many times, however, a pure countercurrent configuration is avoided because it makes the removable-bundle construction difficult or because it makes a very high tube length or installation of shells in series necessary. So usually exploration of alternatives starts with one shell pass and 2n' tubes passes, with n' being any integer number, calculate Ft . 3. Considering the type of fluids to be handled, a first guess of the overall heat transfer coefficient can be obtained using the table in App. G. 4. With the assumed U, an approximate value of the heat transfer area can be calculated as A' =

Q U ⋅ DMLT ⋅ Ft

(7-6-21)

5. With the criteria explained in Sec. 7-3, it is possible to choose the tube diameter, pattern, and pitch and decide the allocation of fluids on the shell and tube sides. 6. Select the number of tubes per pass. This is normally selected so as to have a reasonable tube-side fluid velocity. For example, if we adopt a fluid velocity of 1 m/s, the number of tubes in parallel will be np =

Wt ρa 't (1 m/s)

7. Select the number of tubes, tube length, and number of tube passes. Here, it is necessary to find a combination of number of tubes and tube length to satisfy the value of A' calculated in step 3, which is A' = NπDoL

(7-6-22)

At the same time, the number of tube passes must be selected in such a way that the quotient between N and np is an integer. This means that one must adopt the number of tube passes n as n = N/np

(7-6-23)

adjusting np to get an integer. At the same time, the heat exchanger length must be reasonable, and the unit must be proportionate (see step 8).

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

197

8. Selection of shell diameter. The tables in App. D show the number of tubes that can be allocated in a certain shell diameter for different exchanger types. Once the number of tubes has been selected with the help of these tables, it is possible to find the necessary shell diameter. Usually the number of tubes obtained from the tables will not match exactly with the number of tubes calculated in step 7. Anyway, at this stage we are only looking for a first approximation to the final design, to be verified later. 9. With the number of tubes determined in step 8, calculate the corrected heat transfer area as A = NπDo L

(7-6-24)

This is usually somewhat different from the area calculated in step 7 because we may have slightly changed the number of tubes. 10. Finally, the baffle separation must be selected. As a first guess. we shall try to obtain a Reynolds number that gives a reasonably high shell-side heat transfer coefficient. This may require a few trial calculations with Eq. (7-5-7). Once the heat exchanger is completely defined, we can proceed to the rating procedure explained in Sec. 7-6-1. Example 7-6 It is desired to cool down 33.33 kg/s of methanol from 65°C to 30°C using cooling water at 25°C. The cooling water outlet temperature must not be higher than 40°C because of cooling system restrictions. Pressure drops of 1.50 × 105 N/m2 for the process stream and 1 × 105 N/m2 for the cooling water are allowed. The total fouling resistance must be 5 × 10−4 (s · m2 · K)/J. The physical properties of methanol are k = 0.21 J/(m · s · K) ρ = 800 kg/m3 c = 2,508 J/(kg · K) Viscosity [kg/(m · s)]: T (ªC) μ

74

60

50

40

30

0.00030

0.00036

0.00040

0.00047

0.00054

Note: A downloadable spreadsheet with the calculations for this example is available at http://www. mhprofessional.com/product.php?isbn=0071624082 Solution

a. Propose a tentative geometry: 1. Calculation of water-mass flow: Q = Wh ch (T1 − T2 ) = 33.33 × 2, 508 × (65 − 30) = 2.925 × 10 6 W ∴ Wc =

Q 2.925 × 10 6 = = 46.66 kg/s cc (t2 − t1 ) 4,180(40 − 25)

2. Select a tentative overall heat transfer coefficient: With the help of the table in App. G, we choose as a first guess U = 450 W/(m2 · K) 3. Exchanger configuration:

198

CHAPTER SEVEN

We shall calculate the Ft factor for different configurations: R=

65 − 30 = 2.33 40 − 25

and

S=

40 − 25 = 0.375 65 − 25

With these values, for a 1-2 configuration (one shell pass), Ft < 0.75. For two shell passes, with these parameters, we get Ft = 0.82. It thus will be necessary to use a configuration with two shell passes and 2n tubes passes, where n is an integer. We can obtain this configuration by connecting two shells in series (one pass each), as show in Fig. 7-8. (except that in this case the cold fluid will be the tubeside fluid).The number of tube passes will be decided later. 4. Calculation of LMTD: LMTD =

Then ΔT

(T1 − t2 ) − (T2 − t1 ) (65 − 40) − (300 − 25) = = 12.4°C T −t 65 − 40 ln ln 1 2 30 − 25 T2 − t1

= LMTD · Ft = 12.4 × 0.82 = 10.17.

5. Estimation of the heat transfer area: With the assumed U, we can estimate an area A' =

Q 2.925 × 10 6 = = 639.4 m 2 U ΔT 450 × 10.17

Since this area will be split into two shells, they will have approximately 320 m2 each. 6. Selection of tube diameters and pattern. Let’s adopt 3/4-in BWG 14 tubes. According to the table in App. H, the internal diameter is 0.0147 m. We shall adopt a triangular pattern with 0.0254-m pitch. 7. Selection of tube length, number of tubes, and number of tube passes. The cooling water will circulate inside the tubes. As a first guess, we shall adopt a water velocity of 2 m/s. The flow area of each tube is a t' =

πDi2 π × 0.01472 = = 1.69 × 10 −4 m 2 4 4

To have a velocity of 2 m/s, the number of tubes in parallel must be np =

W 46.66 = = 138 tubes ρat' v 1, 000 × 1.69 × 10 −4 × 2

If we assume, for the first trial, a 6-m tube length, the total number of tubes must be N=

A 320 = = 893 tubes in each shelll πDo L π × 0.019 × 6

Thus the number of tube passes in each shell will be n=

N 893 = = 6.47 n p 138

We shall adopt six tube passes in each shell. Then the proposed configuration consists in two 1-6 shells connected in series, making an overall 2-12 configuration.

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

199

We can see in the tables of App. D that with a triangular pattern with 3/4-in tubes separated by 1 in and six tube passes, the tube number most close to 892 corresponds to 834 tubes in a 33-in shell diameter (0.838 m). The area of each shell is A = πDo LN = π × 0.019 × 6 × 834 = 298.6 m 2 This means that the total area of the unit is 597 m2. Thus we shall propose an arrangement consisting in two shells in series with 834 tubes each and with a tube length of 6 m arranged in six passes per shell. As a first trial, we shall adopt a baffle spacing of 0.3 m. Thus each shell shall have 19 baffles. b. Verification of the proposed design: 1. Calculation of the tube-side film coefficient: Flow area at = a t' Water velocity =

N 834 = 1.69 × 10 −4 = 0.0236 m 2 n 6 Wc 46.66 = = 1.98 m/s ρat 1, 000 × 0.0235

The mean temperature of the cooling water is 32.5°C. hi = 1, 423(1 + 0.0146t )

v 0.8 Di0.2

= 1, 423(1 + 0.0146 × 32.5)

1.980.8 0.01470.2

= 8,420 W/(m 2 ⋅ K)

Then hio = hi

Di 0.0147 = 8, 420 = 6,515 W/(m 2 ⋅ K) Do 0.019

2. Calculation of the shell-side film coefficient: Ds cB 0.838 × 6.35 × 10 −3 × 0.3 = = 0.0629 m 2 Pt 0.0254 W 33.33 GS = mass velocity = = = 530 kg/(m 2 ⋅ s) as 0.0629

as = flow area =

For a 30-degree triangular tube pattern with 19-mm tubes separated by 25.4 mm, the equivalent shell-side diameter is 18.5 mm (0.0185 m). Re s = ho = 0.36 Re 0S.55 Pr 0.33

De Gs 0.0185 × 530 = = 24,526 μ 0.0004

k 0.21 = 0.36 × 24, 5260.55 × 4.770.33 × = 1, 777 W/(m 2 ⋅ K) De 0.0185

We have assumed that the viscosity correction factor is unity. We can now verify this assumption. It will be ho (T − Tw ) = hio (T w − t ) where T and t are the mean temperatures of methanol and water, respectively. Then 1, 777(47 − Tw ) = 6, 514(Tw − 32.5)

200

CHAPTER SEVEN

From which we get Tw = 35.5°C. This temperature practically coincides with the mean temperature of water, so we shall not perform any correction to the water film coefficient. For methanol, the viscosity at 35.5°C is 0.0005 kg/ms; thus ⎛ μ ⎞ ⎜μ ⎟ ⎝ w⎠

0.14

⎛ 0.0004 ⎞ =⎜ ⎝ 0.0005 ⎟⎠

0.14

= 0.96

Then ho = 1, 777 × 0.96 = 1,706 W/(m 2 ⋅ K) 3. Calculation of the overall heat coefficient: ⎛ 1 ⎞ 1 U =⎜ + + Rf ⎟ h h ⎝ io ⎠ o

−1

1 ⎛ 1 ⎞ =⎜ + + 5 × 10 −4 ⎟ ⎝ 1, 706 6, 514 ⎠

−1

= 807 W/(m 2 ⋅ K)

4. Calculation of the heat transfer area. A=

Q 2, 925, 700 = = 357 (total for both h shells) U ΔT 807 × 10.16

This value is sensibly smaller than the 597 m2 we had proposed. The design thus must be modified because this excess area cannot be accepted. Nevertheless, we shall calculate the pressure drops for both fluids to know how the proposed design must be modified 5. Calculation of the cooling-water pressure drop. For the water flowing inside tubes, Re t =

Di vρ 0.0147 × 1.98 × 1, 000 = = 35, 495 μ 8.2 × 10 −4

The friction factor is f = 0.0014 + 0.125 Re −0.32 = 5.78 × 10 −3 The pressure drop in the straight tubes in each shell is Δpt = 4 f

Ln ρv 2 6 × 6 1, 000 × 1.982 = 4 × 5.77 × 10 −3 × × = 1.1079 × 10 5 N/m 2 Di 2 0.0147 2

And the pressure drop in the return headers is Δpr = 4 n

1.982 ρv 2 = 4 × 6 × 1, 000 × = 4.704 × 10 4 N/m 2 2 2

So the pressure drop for each shell is Δ pT = Δ pr + Δ pt = 1.578 × 10 5 N/m 2 And for the complete unit, it will be 3.156 × 105 N/m2, which largely exceeds the maximum allowable. 6. Calculation of the methanol pressure drop: f = friction factor = 1.728 Re −0.188 = 0.258

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

Δps = f

GS2 DS ( N B + 1) ⎛ μ w ⎞ ⎜⎝ μ ⎟⎠ 2 Deρ

0.14

=

201

0.258 × 530 2 × 0.838 × 20 1 = 4.280 × 10 4 N/m 2 2 × 0.0185 × 800 0.96

And for both shells, Δ p = 8.56 × 104 N/m2. c. Modification of the proposed design: The design we have rated has an excessive pressure drop in the water side and an excess in heat transfer surface. In a next design step, we can reduce the water velocity by decreasing the number of tube passes. This will affect the heat transfer coefficient, but since the controlling resistances are in the methanol side and fouling, the net effect on the overall U will not be important. We also must reduce the heat transfer area. An obvious solution is to decrease the tube length, which also will help to reduce the pressure drop. However, to lower the cost of the unit, it is usually more effective to reduce the number of tubes. Since with the help of a spreadsheet it is very easy to explore alternatives, we shall analyze the option of maintaining the 6-m tube length as well as the option of using 4-m tube length. In both cases we shall reduce the number of tubes and the number of tube passes with the objective of obtaining the smaller area requested for the heat transfer without exceeding the fluid pressure drops. In the shell side, we see that the pressure drop calculated in the first trial is lower than acceptable. This means that we still have some margin to reduce the baffle spacing and increase the shell-side film coefficient. However, since we shall reduce the shell diameter, and this also will increase the velocity, we shall, for the moment, keep the 0.3-m spacing. Second trial: We shall try the following parameters: Ds

n

N

L

B

A

736 mm

4

646

6m

0.3 m

462 m2

And the calculated result is Ret 30,500

Res

hio

ho

U

Δ ps

Δ pt

A Required

27,925

5,778

1,832

820

95,200

159,000

350 m2

We see that we still have a 24 percent excess area, and the water pressure drop is still excessive. We can move forward in the same direction with a smaller number of tubes and a smaller number of tube passes. Third trial: We shall try the following parameters: Ds

n

N

L

B

A

685 mm

2

578

6

0.3

414 m2

And we get the following results: Ret 17,100

Res

hio

ho

U

Δ ps

Δ pt

A Required

29,970

3,627

1,905

768

100,870

27718

374 m2

We still have a 9 percent excess area. We shall further reduce the number of tubes. It is interesting to note that the decrease in the water velocity does not influence the overall U because of the relatively low shell-side film coefficient. Fourth trial: We shall try the following parameters: Ds

n

N

L

B

A

635 mm

2

492

6

0.30

352 m2

202

CHAPTER SEVEN

And the results are Res

hio

ho

U

Δ ps

Δ pt

A Required

32,367

4,126

1,987

802

107,377

37,122

358 m2

Ret 20,000

In this case, the required area is larger than the actual area. The defect is −1.6 percent. However, since the shell-side pressure drop is still lower than allowable, we can get some improvement in U reducing the baffle spacing. Fifth trial: We shall try with the following parameters: Ds

n

N

L

B

A

635

2

492

6

0.27

352 m2

With the following results: Ret

Res

hio

ho

U

Δ ps

Δ pt

A Required

20,000

35,900

4,126

2,106

821

144,000

37,122

352 m2

And this design can be considered acceptable. We shall now see what design results using 4-m tubes. Sixth trial: The following geometry is proposed: Ds

n

N

L

B

A

0.787 m

4

750

4

0.22

358 m2

And the results are Ret 12,700

Res

hio

ho

U

Δps

Δpt

A Required

35,600

5,127

1,766

932

143,800

92,575

308 m2

This design has 14 percent excess area, so from the heat transfer point of view it is more attractive. However, even though the heat transfer areas of the last two designs are very similar, the cost of the unit in trial 5 probably will be considerably lower owing to the economies in the tubesheet and manufacturing labor. Note that the tube-side coefficient is considerably higher in trial 6 than in trial 5, but this difference has little importance in the overall coefficient.

7-6-3 Design Programs Heat exchanger design is usually performed with the help of computer programs. The structure of these design programs is usually complex owing to the great number of parameters that must be adopted (e.g., TEMA type, number of shell and tubes passes, diameter and number of tubes, clearances, sealing strips, tube pattern and pitch, tube length, type of baffles and baffle cut, baffles spacing, and shell diameter). Many of these variables depend on user preference, project specifications, or layout restrictions and are defined by the user in the program input. Thus the optimization path followed by the program depends on which are the design parameters defined by the user. For example, we include the flow diagram in Fig. 7-38, which shows the design strategy of a program for a specific situation.

1 Initialize U = 1/Rf

3

Select shell determine actual number of tubes

Ncs = 1 n=2 Calculate Ft Ft < 0.75? no L = Lmax

Adopt the maximum value for B n=n+2

Calculate velocity and tube side Δp Ncs = Ncs + 1 yes

v = vmax or Δp > Δpmax?

n=n+2

Was n reduced in the previous step?

no

ATOTAL = Q/U.ΔT

n = 2?

no

yes

no

2

Calculate Δps

no yes

no Is the first Δp? Decrease L

n = 12? ATOTAL A= Ncp.Ncs N=

A π.Do.L

N > Nmax?

yes Ncp = Ncp + 1 yes

no 1 FIGURE 7-38

yes

Structure of a heat exchanger design program.

B > Bmin?

no

decrease B

Δps > Δpmax? yes Was B reduced in the last step? yes Use the previous value

no Ncp = 1

yes

yes 2

Calculate ho Calculate Ucalc

Calculate hi

Ucalc > U?

3

end

no

U = Ucalc

2

203

204

CHAPTER SEVEN

The program starts by assuming an initial value of the overall heat transfer coefficient. We know that this coefficient is given by 1 1 1 = + + Rf U ho hio

(7-6-25)

The maximum possible value that U can have is 1/Rf , a situation that is reached when the film coefficients are very high. Thus the program starts by adopting for U the value given by Umax = 1/Rf. The area calculated with this first approach therefore will be the minimum, and it will increase with the evolution of the calculations. Then the heat exchanger configuration is adopted. In this case, the use of countercurrent heat exchangers was removed from consideration owing to cleaning difficulties. The simplest configuration thus will be a single shell with one shell pass and 2n' tube passes, with n' being any integer. More complex configurations will be obtained by adding shells in series. For the simplest configuration, the Ft factor is calculated. If it is lower than 0.75, the program moves to a 2-4 configuration, and so on, finally obtaining the simplest structure that allows operating with a satisfactory Ft. At the beginning, it is assumed that each shell has two tube passes (remember that the number of tube passes may be changed without affecting Ft). The program also assumes as an initial guess for the tube length the maximum allowable value that has been defined by design restrictions. Then the total heat transfer area is calculated as Atotal = Q/UΔT where Atotal represents the area of all the individual shells that are necessary. If there is also a restriction on the maximum shell diameter or maximum number of tubes per shell, it may be necessary to use shells in parallel. At the beginning of the calculation, it is assumed that no parallel trains are necessary. The area of each shell will be A=

Atotal N cs N cp

where Ncs is the number of shells in series and Ncp is the number of parallel branches. Then the number of tubes per shell is determined. If this number is higher than the maximum allowable per shell, parallel branches will be added. The shell diameter then can be selected, and the real number of tubes per shell is determined by means of a computerized version of the tables in App. D. Then the tube-side fluid velocity and pressure drop are calculated. If in the first calculation these values are higher than the maximum allowed, the program continues with the calculation of heat transfer coefficients because, since the initially assumed U is very high, in successive calculation steps the number of tubes will increase and ΔpT will decrease. If ΔpT is lower than allowable, the program always will try to increase the velocity by increasing the number of tube passes (up to a maximum of 12) unless any recent trial has shown that this would result in an unacceptable pressure drop. Then the tube-side film coefficient is calculated, and the program moves to shell-side calculations. The maximum baffle spacing allowed by the TEMA Standards is assumed (which is defined by tube support considerations), and in successive calculation steps it will be reduced if the shell-side pressure drop allows. If, even with the highest spacing, the allowable Δp is exceeded, the tube length is reduced, which results in an increase of the number of tubes in parallel. Finally, the shell-side heat transfer coefficient is calculated, which allows calculation of the overall coefficient U with Eq. (7-6-25). This value is compared with the previously assumed value. If the calculated U is smaller than the previous U, the calculated value is adopted, restarting the calculations.

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

205

7-7 HEAT EXCHANGER EFFECTIVENESS 7-7-1 Calculation of Heat Exchanger Outlet Temperatures In Sec. 7-6-1 we mentioned that it is sometimes necessary to calculate the heat exchanger outlet temperatures of the fluids when the inlet temperatures are known. This may be necessary in the following cases: • When one wants to know the behavior of the heat exchanger in conditions different from those of the design • In process-simulation programs (In this case, every piece of equipment is represented by a module or transfer function that calculates the outlet streams for any inlet condition.) • When one wants to adapt an existing heat exchanger to a new process condition In these types of problems, we know all the geometric characteristics of the heat exchanger and the inlet streams. If we assume that the physical properties are constant, we can calculate the film heat transfer coefficients h and then the overall U. As explained in Sec. 7-6-1, the equation system that must be solved is Q = Wh ch (T1 − T2 )

(7-7-1)

Q = Wc cc (t 2 − t1 )

(7-7-2)

(T1 − t 2 ) − (T2 − t1 ) Ft T −t ln 1 2 T2 − t1 Ft = f (T1 , T2 , t1 , t2 )

Q = UA

(7-7-3) (7-7-4)

In this system, the unknowns are Q, T2, t2, and Ft . The system can be solved mathematically by an iterative procedure (Fig. 7-39). When using this algorithm in computer programs, it is necessary to restrict the size of the iteration steps to achieve convergence in the calculations and to avoid indeterminate solutions in the Ft calculations. A more direct method for solving this equation system was proposed by W.Kays and A.M.London.16 This method is particularly useful for cases where the outlet temperatures are not known. They define a new parameter called effectiveness of the heat exchanger, which will be designated with the symbol ε and is defined as ε=

Q (Wc)min (T1 − t1 )

(7-7-5)

where (Wc)min = Wc cc

if Wc cc < Wh ch

(Wc)min = Wh ch

if Wh ch < Wc cc

Another parameter called number of transfer units (NTU) is defined as NTU =

UA (Wc)min

(7-7-6)

And a parameter called heat capacity ratio R' is defined as R' =

(Wc)min (Wc)max

(7-7-7)

206

CHAPTER SEVEN

Assume Q

Calculate T2 [Equation (7-7-1)] Calculate t2 [Equation (7-7-2)] Assume new Q

Calculate Ft [Equation (7-7-4)] Calculate Q [Equation (7-2-3)]

no

Q calculated = Q assumed? Yes End

FIGURE 7-39 Flow diagram for solving the equation system (7-7-1) through (7-7-4).

where (Wc)max is the Wc product that was not defined as (Wc)min. The general form of the correlations is ε = f ( R ', NTU) And the function f depends on the heat exchanger configuration. The curves representing these functions are shown in Fig. 7-40. As can be seen, there are effectiveness curves for any kind of configuration, such as countercurrent, co-current, 1-2, 2-4, 3-6, etc. Curves for other configurations may be found in ref. 16. These graphs can be constructed through mathematical solutions to Eqs.(7-7-1) through (7-7-4). Analytical expressions of the effectiveness functions also can be developed theoretically, and they are included in Table 7-5. Use of the effectiveness concept will be better understood with completion of the following examples. Example 7-7 Calculate the outlet temperature of the oil in the heat exchanger in Example 7-4 when the heat exchanger is initially put into service and the fouling factor is 0. Note: A downloadable spreadsheet to perform these calculations is available at http://www. mhprofessional.com/product.php?isbn=0071624082 Solution This is a specific case in which one of the fluids exchanges latent heat isothermally.

A fluid that is capable of yielding or receiving heat without changing its temperature can be considered a fluid with an infinite heat capacity. (The heat capacity is the ratio between the heat received or yielded and the temperature change that the fluid undergoes.). Thus (Wc)min = Wc cc

and R ' = 0

since (Wc)max = ∞

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

207

The area of the heat exchanger is A = 35.8 m2. When the unit is put into service, Rf will equal 0, and then U = Uc = 1056 W/(m2 · K). Then NTU =

35.8 × 1, 056 UA = = 0.310 (Wc)min 58 × 2,100

Parallel-current configuration

Countercurrent configuration

80

1.00

60 40 20 0

0

Effectiveness ε %

R=0 0.25

80

0.50 0.75 1.00

60 40 20 0

0

R=0

80

0.25 0.50 0.75 1.00

60 40 20 0

1 2 3 4 5 Number of transfer units NTU = U.A/(W.c)min Configuration 1-2

100

100 Effectiveness ε %

R=0 0.25 0.50 0.75

0

1 2 3 4 5 Number of transfer units NTU = U.A /(W.c)min Configuration 2-4

100 Effectiveness ε %

Effectiveness ε %

100

60 40 20 0

1 2 3 4 5 Number of transfer units NTU = U.A/(W.c)min

R=0 0.25 0.50 0.75 1.00

80

0

1 2 3 4 5 Number of transfer units NTU = U.A /(W.c)min

Configuration 3-6 R=0 0.25 0.50 0.75 1.00

80 60 40 20 0

Configuration 4-8 100

0

1 2 3 4 5 Number of transfer units NTU = U.A/(W.c)min

Effectiveness ε %

Effectiveness ε %

100

R=0 0.25 0.50 0.75 1.00

80 60 40 20 0

0

1

2

3

4

5

Number of transfer units NTU = U.A/(W.c)min

FIGURE 7-40 Heat exchanger effectiveness graphs. (From W. Kays and A.M London, Compact Heat Exchangers. New York: McGraw-Hill, 1952- Courtesy of Professor W. Kays.)

208

CHAPTER SEVEN

TABLE 7-5

Heat Exchanger Effectiveness Configuration

Effectiveness Expression 1 − exp [ (− NTU(1 + R ')] 1+ R' 1 − exp [ − NTU(1 − R ')] ε= 1 − R ' exp [ − NTU(1 − R ')]

Parallel flow

ε=

Countercurrent flow

ε= 1-2 exchanger (one shell pass; even number of tube passes)

NTU 1 + NTU

for R ' = 1 2

ε= (1 + R ') + R '2 + 1

n-2n exchanger (n shell passes; 2n, 2n + 2, 2n + 4, etc. tube passes)

for R ' ≠ 1

1 + exp(− NTU R '2 + 1) 1 − exp(− NTU R '2 + 1)

ε=

(1 − εa R ') − (1 − εa )n (1 − εa R ')n − R ' (1 − εa )n

for R ' ≠ 1

ε=

nεa 1 + (n − 1)εa

for R ' = 1

n

where εa is calcculated as in 1-2 exchanger with NTU = NTU/nn

With NTU = 0.310 and R' = 0, from the graph corresponding to the 1-2 configuration we obtain ε = 0.267. Thus ε = 0.267 =

t − 57 Q W c (t − t ) = c c 2 1 = 2 (Wc)min (T1 − t1 ) Wc cc (T1 − t1 ) 177 − 57

Isolating t2, we get t2 = 89°C. Example 7-8 Four 1-2 heat exchangers are connected as shown in Fig. 7-41, thus making a 4-8 configuration. The cold-fluid mass flow is 18 kg/s, and its specific heat is 3,000 J/(kg · K). The hot-fluid mass flow is 25 kg/s, and its specific heat is 2,889 J/(kg · K). The inlet temperatures are T1 = 170°C and t1 = 87°C. The heat transfer area of each shell is 40 m2, and the overall heat transfer coefficient is 1,012 W/(m2 · K). Calculate the outlet temperatures. Solution

We can consider the whole assembly as a single unit with a 4-8 configuration. The NTU

will be NTU =

UA 1, 012 × 160 = =3 (Wc)min 1.8 × 3, 000

Since (Wc)min is in this case Wccc, R' =

(Wc)min 18 × 3, 000 = = 0.75 (Wc)max 25 × 2, 880

Entering the graph corresponding to a 4-8 configuration with NTU = 3 and R' = 0.75, we obtain ε = 0.8. Then ε=

Wc cc (t 2 − t1 ) t − 87 = 2 = 0.8 Wc cc (T1 − t1 ) 170 − 87

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

209

T2 t1

t′

T ″′

t″

T″

t ″′

T′

t2

T1

FIGURE 7-41

Example 7-8 heat exchangers.

Isolating t2, we get t2 = 153.4. The heat exchanged is then Q = Wc cc (t2 − t1 ) = 54, 000(153.4 − 87) = 3.585 × 10 6 W The outlet temperature of the hot fluid can be obtained from T2 = T1 −

Q 3.585 × 10 6 = 170 − = 120.2°C Wh ch 25 × 2, 880

If you want to calculate the intermediate temperatures, each one of the four shells can be considered individually. The heat transfer area of each shell is 40 m2, and the number of transfer units is NTU =

40 × 1, 012.5 UA = = 0.75 (Wc)min 18 × 3, 000

With NTU = 0.75 and R' = 0.75, from the graph corresponding to a 1-2 configuration, we find ε = 0.433. If we consider the lowest shell, ε= t ''' =

W c (t − t ''') t2 − t ''' Q = c c 2 = Wc cc (T1 − t ''') Wc cc (T1 − t ''') T1 − t '''

t2 − εT1 153.4 − 0.433 × 170 = = 140.722°C 1− ε 1 − 0.433

Temperature T ' can be obtained from a heat balance: Wc cc (t 2 − t ''') = Wh ch (T1 − T ')

210

CHAPTER SEVEN

∴ T ' = T1 −

Wc cc (t2 − t ''') = T1 − R '(t2 − t ''') = 160..49°C Wh ch

Following the same procedure with the other units, we have t '' =

t '''− εT ' 140.72 − 0.433 × 160.49 = = 125.62°C 1− ε 1 − 0.433

T '' = T '− R '(t '''− t '') = 160.49 − 0.75(140.72 − 125.62) = 149.16°C t' =

t ''− εT '' 125.62 − 0.433 × 149.16 = = 1007.63°C 1− ε 1 − 0.433

T ''' = 149.16 − 0.75(125.62 − 107.63) = 135.66°C

7-7-2 Physical Interpretation of the Effectiveness We have explained in Sec. 7-2 that the maximum heat exchange that is possible thermodynamically between two streams is achieved with a countercurrent heat exchanger with infinite area. If Wccc = (Wc)min, the temperatures diagram is as shown in Fig. 7-42a. In this case, the outlet temperature of the cold fluid reaches the inlet temperature of the hot fluid. The heat exchanged is Q = Wc cc (t 2 − t1 ) = (Wc)min (T1 − t1 )

(7-7-8)

If, on the other hand, Whch = (Wc)min, the temperature diagram is that of Fig. 7-42b. In this case, the hot fluid leaves the exchanger at a temperature equal to the cold-fluid inlet temperature. The heat exchanged is Q = Wh ch (T1 − T2 ) = (Wc)min (T1 − t1 )

(7-7-9)

We see that in both cases the heat exchanged can be expressed as Q = (Wc)min (T1 − t1 )

(7-7-10)

We can then assign a physical interpretation to the definition of the effectiveness given by Eq. (7-7-5), saying that the effectiveness of a heat exchanger is the ratio between the heat exchanged in the unit and the maximum amount of heat that is possible thermodynamically to exchange between the two streams. This

Wc.cc = (W.c)min

Wh.ch = (W.c)min

T1 = t2

T1 T2

t2 T2 = t1

t1

(a)

(b)

FIGURE 7-42 Temperature diagrams for an infinite-area countercurrent heat exchanger.

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

211

thermodynamic maximum could be achieved by putting the streams in contact in a countercurrent heat exchanger with infinite area. It is important to understand that the effectiveness is not a concept related to a good or bad design of a heat exchanger. If it is necessary to design a heat exchanger to perform a certain process service, defined by the flow rates of the fluids and by the desired inlet and outlet temperatures, the effectiveness of the unit is already defined, and its value is ε=

t 2 − t1 T1 − t1

if Wc cc = (Wc)min

(7-7-11)

if Wh ch = (Wc)min

(7-7-12)

or ε=

T1 − T2 T1 − t1

This value is independent on the exchanger design, so it must not be thought that a unit with a low effectiveness is related to poor design. Physical Interpretation of the Number of Transfer Units The number of transfer units has been defined as NTU =

UA (Wc)min

(7-7-13)

Despite the fact that the definition includes U and A, which are values obtained after the thermal design, the product of these two magnitudes is constant and independent of the exchanger design once the pass configuration has been adopted. This can be better understood if we write UA =

Q Ft ⋅ DMLT

(7-7-14)

We see that all the magnitudes on the right side of the equation are defined by the process conditions once the configuration is adopted. This means that NTU can be calculated before performing the thermal design as NTU =

Q (Wc)min Ft ⋅ DMLT

(7-7-15)

The concept of transfer unit was used much more extensively in the theory of mass-transfer operations than in heat transfer. The basic idea is that the length of a mass transfer or heat transfer device can be expressed as the product of two magnitudes, that is, L = HTU · NTU

(7-7-16)

where HTU is the height of a transfer unit. This is a magnitude that does depend on the characteristics of the device; this means the area per unit length and the heat transfer coefficient. For example, in the case of a single-pass countercurrent heat exchanger, the height of the transfer unit can be calculated as HTU =

(Wc)min U πDo N

(7-7-17)

Thus the definitions of NTU and HTU satisfy Eq. (7-7-16). This means that once the process is defined, the number of transfer units is also defined. Then, depending on the characteristics of the heat exchanger that will be used (i.e., number and diameters of tubes, etc.), we shall have different values of U and then different heights of the transfer unit. The length of the heat exchanger then can be calculated as the product of HTU and NTU.

212

CHAPTER SEVEN

7-7-3 Analysis of the Effectiveness Graphs An analysis of the effectiveness graphs, especially for countercurrent and parallel-current configurations, may help to clarify some concepts. Looking at the countercurrent-configuration graph, we see that effectiveness tends toward 1 when the number of transfer units tends toward infinity, and this is true for any R'. This conclusion is in agreement with the interpretation of the effectiveness definition explained previously. In the case of a parallel-flow configuration, if the heat transfer area tends toward infinity, the outlet temperatures of both streams would be equal. The effectiveness is lower than 1 because the heat exchanged is lower than in the countercurrent case. For the particular case in which the heat capacities W × c of both fluids are identical, the limit value of the effectiveness is 0.5. A temperatures diagram for a parallel-flow infinite-area heat exchanger is shown in Fig. 7-43. When the Wc values of both streams are equal, the outlet temperature T2 will be T2 = t2 = 1/2(T1 + t1) and Q = Wc[T1 − 1/2(T1 + t1)] = 1/2Wc(T1 − t1) Then, from the definition of effectiveness, we get ε = 1/2. We also can analyze the case when R' = 0. Since R' =

(Wc)min (Wc)max

R' will be 0 when the heat capacity of one of the fluids is infinite. This situation, as explained previously, exists when there is a change of phase of a single-component fluid. In this case, that stream can receive or yield energy without changing its temperature. In this case. the thermal diagram is as shown in Fig. 7-44. We can see that the diagram is the same for countercurrent or parallel-current configurations. In particular, if the area is infinite, it may happen that the outlet temperature of the cold fluid equals that of the hot fluid even for the parallel-current case. This means that the effectiveness versus NTU curves coincide for any configuration in which R' = 0. (The same happens with the Ft curves. This means that when one of the fluids is isothermal, it does not matter what the configuration is.) For all other cases different from R' = 0, at the same R ' and NTU, the countercurrent exchanger has a higher effectiveness than the parallel-current exchanger, which means that it has a higher heat transfer capacity. This can be appreciated in the effectiveness graphs by comparing ordinates for the same abscissa.

T1

T2 = t2

t1 Q FIGURE 7-43 Thermal diagram in a parallel-flow heat exchanger with infinite area.

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

T

T

t2

t1

Parallel streams

213

t2

t1

Countercurrent

FIGURE 7-44 Thermal diagrams for parallel flow or countercurrent flow are identical when one fluid is isothermal.

We see that for low NTU values, the differences in effectiveness for both types are not very large. The parallel-current heat exchanger loses effectiveness at high NTU values because at the entrance end the temperature difference may be high (which means a high driving force), but it greatly reduces toward the outlet end, and the performance of the unit deteriorates. The difference between both types is important with high NTU values. On the other hand, for small NTU values, both units behave in a similar way.

GLOSSARY A = heat transfer area (m2) c = specific heat (J/kg . K) Ft = LMTD correction factor (dimensionless) f = friction factor (dimensionless) G = mass velocity = W/flow area = ρv(kg/s) h = film coefficient (W/m2 . K) hi = internal h (W/m2 . K) ho = external h (W/m2 . K) hio = internal h referred to the external area = hiDi /Do (W/m2 . K) hB = ideal tube-bank h (W/m2 . K) hBP = film coefficient for a tube bank with bypass (W/m2 . K) hNL = h of a tubes bank without leakage (W/m2 . K) hL = h of a tube bank with leakage (W/m2 . K) j = Colburn coefficient (dimensionless) k = thermal conductivity (W/m . K) Q = heat exchanged per unit time (W) R = parameter for Ft calculation (dimensionless) Rf = fouling ressistance (m2 . K/W) S = parameter for Ft calculation (dimensionless) T = hot-fluid temperature, mean or generic value (K or °C) t = cold-fluid temperature, mean or generic value (K or °C) U = overall heat transfer coefficient (W/m2 . K) Uc = clean overall heat transfer coefficient (W/m2 . K)

214

CHAPTER SEVEN

v = velocity (m/s) W = mass flow (kg/s) X = correction factor for number of tube rows (dimensionless) Δp = pressure drop (N/m2) Δ pt = Δp in straight tube length (N/m2) Δ pr = Δp in return headers (N/m2) Δ pT = total Δp for the tube-side fluid (N/m2) Δ ps = Shell side fluid Δp (N/m2) Δ pB = ideal tube bank Δp (N/m2) Δ pBP = Δp of a tube bank with bypass (N/m2) pL = Δp for an exchanger section considering leakage (N/m2) ΔpNL = Δp for an exchanger section with no leakage (N/m2) ξh = bypass correction coefficient for h calculation (dimensionless) ξΔp = bypass correction coefficient for Δ p calculation (dimensionless) α = constant in Eq. (7-5-20) (dimensionless) ε = heat exchanger effectiveness (dimensionless) φ = window correction factor (dimensionless) μ = viscosity (kg/m.s) or (cP) ρ = density (kg/m3) Subscripts 1 = inlet 2 = outlet c = cold h = hot i = internal m = cross-flow o = external s = shell t = tubes w = wall or window Glossary of the Heat Exchanger Geometry A = heat transfer area (m2) A = baffle central angle (°) a1 = external area per unit length of tube = πDo (m2/m) a 't = flow area of a tube = πDi2/4 (m2) at = total flow area for the tubeside fluid = Na't/n (m2) B = baffle spacing (m)

THERMAL DESIGN OF SHELL-AND-TUBE HEAT EXCHANGERS

215

BC = baffle cut (m) c = clearance between tubes (m) DB = baffle diameter (m) DBT = diameter of the baffle hole (m) De = shell-equivalent diameter according to Kern (m) Di = internal diameter of a tube (m) Do = external diameter of a tube (m) DS = internal shell diameter (m) FBP = bypass fraction (dimensionless) L = tube length (m) N = number of tubes (dimensionless) n = number of tube passes (dimensionless) NB = Number of baffles (dimensionless) NBT = number of tubes crossing a baffle (dimensionless) Nc = number of tube rows between baffles borders (dimensionless) NCL = number of tubes in the central row (dimensionless) NS = number of pairs of sealing strips (dimensionless) NW = effective number of tube rows in a window (dimensionless) Nwt = number of tubes in window (dimensionless) Np = number of tubes in parallel (number of tubes per pass) Pt = distance between tube axes (tube pitch) (m) SL = total leakage area (m2) Sm = cross-flow area (m2) SSB = leakage area between shell and baffle (m2) STB = lakage area between tubes and baffle (m2) SBP = bypass area (m2) SW = flow area at window (m2)

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

Kern D.: Process Heat Transfer. New York: McGraw-Hill, 1954. Underwood A. J.: J Inst Petroleum Technol 20:145–158, 1934. Sieder E. N., Tate G. E.: “Heat Transfer and Pressure Drop of Liquids in Tubes,” Ind Eng Chem 28:1429–1436, 1936. Dittus F. W., Boelter L. M.: University of California Publ Eng 2:443, 1930. Lord R. C., Minton P. E., Slusser R. P.: “Design of Heat Exchangers,” Chem Eng January 26:98–115, 1970. Drew T. B., Koo E. C., McAdams W. H.: Trans AICHE 28:56–72, 1932. Ludwig E.: Applied Process Design for Chemical and Petrochemical Plants. Gulf Publishing Company, Houston, 1964. Taborek J.: “Survey of Shell Side Flow Correlations,” in Heat Exchangers Design Handbook: Hemisphere Publishing Co., 1981. 9. Bell K.: Bulletin No. 5 “Final Report of the Cooperative Research Program on Shell andTube Heat Exchangers,” University of Delaware, Newark, Delaware, 1963.

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10. Bergelin O. P., Bell K., Leighton M. D.: “Heat Transfer and Fluid Friction during Flow across Banks of Tubes VI—The Effect of Internal Leakages within Segmentally Baffled Exchangers,” Trans ASME 80:53–60, January 1958. 11. Rohsenow, H.: Handbook of Heat Transfer. New York: McGraw-Hill, 1973. 12. Bergelin O. P., Brown G. A., Colburn A. P.: “Heat Transfer and Fluid Friction during Flow across Banks of Tubes V. A. Study of a Cylindrical Baffled Heat Exchanger without Internal Leakages,” Trans ASME 74:841–850, 1952. 13. Bowman R. A., Mudller A. C., Tagle W. M.: Trans ASME 62:283–294, 1940. 14. Palen J., Taborek J.: “Solution of Shell Side Pressure Drop and Heat Transfer by Stream Analysis Method,” Chem Eng Progr Symp Series 65(92):53–63, 1969. 15. Bergelin O. P., Brown G. A., Doberstein S. C.: “Heat Transfer and Fluid Friction during Flow across Banks of Tubes-IV—A Study of the Transition Zone Between Viscous and Turbulent Flow,” Trans ASME 74:953–960, 1952. 16. Kays W. M., London A. L.: Compact Heat Exchangers. New York: McGraw-Hill, 1951. 17. Sullivan F. W., Bergelin O. P.: “Heat Transfer and Fluid Friction in a Shell and Tube Heat Exchanger with a Single Baffle,” Chem Eng Progr Symp. Series 18(52):85–94, 1956. 18. Whitley D. L.: “Calculating Heat Exchangers Shell Side Pressure Drop,” Chem Eng Progr (9):57–65 1961. 19. Bergelin O. P., Bell K. J., Leighton M. D.: “Heat Transfer and Fluid Friction during Flow across Banks of Tubes VI: By Passing between Tube Bundle and Shell,” Chem Eng Progr 29(55):48–58, 1959.

CHAPTER 8

FINNED TUBES

One way to increase the heat transfer capacity of heat exchanger tubes consists of attaching to them metal pieces, called fins, that extend the heat transfer surface. This makes it possible to have a larger external surface for the same internal surface. In cases where the heat transfer controlling coefficient is the external film, this may result in a significant increase in the heat-flux density per unit internal area. Fins can be transverse to the tube axis, as in Fig. 8-5, or longitudinal, as in Fig. 8-1. This chapter will examine two applications of finned tubes in the process industry: hairpin exchangers with longitudinal fins (either double-tube or multitube) and air coolers, usually provided with transverse fins.

8-1 DOUBLE-TUBE HEAT EXCHANGERS WITH LONGITUDINAL FINS 8-1-1 Use of Fins in Double-Tube Heat Exchangers Let’s consider a double-tube heat exchanger such as those studied in Chap. 5 with longitudinal fins welded to the internal tube, as shown in Fig. 8-1. We shall assume that the hot fluid circulates through the annulus and the cold fluid through the internal tube. The fins are in contact with the hot fluid, receiving heat by convection, and they transfer this heat by conduction to the base of the fin and to the internal tube. We shall analyze the situation at a generic cross section of the heat exchanger. In the considered section, the annulus fluid temperature is T, and the cold fluid temperature is t. Since the fins are welded to the tube, the temperature at the base of the fins Tb is practically equal to the tube-wall temperature. In the rest of the fin surface, however, the temperature will be higher because it is in contact with the hot fluid. Figure 8-2 is a plot of the temperature at the fin surface Tf as a function of the inverse radial coordinate x (the origin of the radial coordinate is at the fin tip). Thus Tf = Tf (x) It can be seen that there is a temperature gradient dTf /dx such that there is a conduction heat flow toward the base of the fin. With relation to Fig. 8-3, if we consider a slice of fin with height dx and length dz, the hot fluid delivers heat to the fin through a heat transfer area dA = 2 dz dx

(8-1-1)

The heat flow through this surface, assuming that the surface is clean, will be dQ f = (2dz dx )h f [T − T f ( x )]

(8-1-2)

where hf is the convection heat transfer coefficient between the hot fluid and the metal. In the presence of a fouling resistance on the fin surface, it is convenient to incorporate it into the convection coefficient, defining a coefficient corrected by fouling as ⎛ 1 ⎞ h 'f = ⎜ + R fo ⎟ h ⎝ f ⎠

−1

(8-1-3)

217

218

CHAPTER EIGHT

Section of a double-tube heat exchanger

Tube with longitudinal fins FIGURE 8-1

Longitudinal fins.

Temperature

Hot fluid temperature

Tb x

Tf (x) = Temperature at the fin surface

FIGURE 8-2 Temperature profile in a longitudinal fin.

in which case dQ f = (2dz dx )h f' [T − T f ( x )]

(8-1-4)

In what follows, we shall express everything per unit tube length. We shall call Nf = number of fins per tube Ao = area of tube in correspondence with the external diameter per unit length (plain tube area), that is, Ao = πDo (m2/m of tube)

(8-1-5)

Af = fin surface per unit tube length (consider both sides of the fins). If H is the fins height, it will be Af = 2HNf

(m2/m of tube) (neglecting the area of the fin shoulder)

(8-1-6)

219

FINNED TUBES

dQf

Qc

Qc + dQc

dz

b dx H FIGURE 8-3

Heat balance in a longitudinal fin.

AD = exposed external area of tube (not covered by fins), that is, AD = Ao − Nf b

(8-1-7)

where b is the fin thickness. All the heat penetrating into the fin is transmitted by conduction toward the base of the fin owing to the temperature gradient. In the control volume considered in Fig. 8-3, it can be seen that the conduction heat flow Qc increases toward the base of the fin as a consequence of the heat incorporated through the lateral surface, so in steady state, it must be dQc = dQf. The conduction heat flow, which enters the internal tube through the base of the fin, equals the convection heat transferred to the entire surface of the fin. Then, if Qb (Watt) is the heat flow entering the internal tube through the base of the fin (which corresponds to the value of Qc at x = H), and if dQb/dz is the flow per unit tube length (W/m), we can write dQb = dz

∫ Af h 'f [T − T f ( x )] dA = h 'f (T − T f )m A f

(8-1-8)

where the term in parentheses is a mean temperature difference between the fluid and the fin surface. If Tb is the temperature at the base of the fin, we can write (T − T f )m = Ω(T − Tb )

(8-1-9)

where Ω is a coefficient whose value is between 0 and 1 because at any point on the fin surface, the temperature difference between the annulus fluid and the metal is smaller than that existing at the base of the fin. Then dQb (W/m) = A f Ωh 'f (T − Tb ) dz

(8-1-10)

The heat flow penetrating into the internal tube is the sum of that penetrating through the base of the fin (Qb) plus the convection heat transferred from the annulus fluid to the bare portion of the tube (area AD).

220

CHAPTER EIGHT

Calling QD the heat flow through AD , the heat flow per unit length through the bare area is dQD (W/m) = h 'f AD (T − Tb ) dz

(8-1-11)

And the total heat flow per unit length through the internal tube wall is dQ = AD h 'f (T − Tb ) + A f h 'f Ω(T − Tb ) = h f' (T − Tb )(ΩA f + AD ) dz

(8-1-12)

So the calculation can be performed as if the entire metal were at a uniform temperature (that of the tube wall) but affecting the area of the fin by an efficiency coefficient that indicates that this area is not as effective as the bare tube for heat transfer. This is why Ω is called fin efficiency. The efficiency of a fin depends on its geometry, on the thermal conductivity of its material, and also on the convection heat transfer coefficient of the external side. In certain cases, it is possible to deduce analytical expressions to calculate the fin efficiency as a function of these parameters. For example, in the next section we shall deduce the expressions to calculate the efficiency of a fin with a relatively simple geometry such as that shown in Fig. 8-1. Since the usual practice is to refer the heat transfer coefficients to the external plain area of the tube Ao , we shall define a heat transfer coefficient to be used with this area as h 'fo =

h 'f ( AD + ΩA f ) Ao

(8-1-13)

And the heat transfer between the external fluid and the tube can be expressed as dQ = h fo ' Ao (T − Tb ) dz

(8-1-14)

From now on, the procedure is the same as for the case of plain tubes. From the tube at Tb , heat is transmitted to the internal fluid through the resistances corresponding to the internal film coefficient and internal fouling; that is, ⎞ dQ ⎛ 1 =⎜ + R fi ⎟ dz ⎝ hio ⎠

−1

Ao (Tb − t )

(8-1-15)

Isolating the temperatures differences in Eqs. (8-1-14) and (8-1-15) and summing both equations we get ⎞ dQ ⎛ 1 1 =⎜ + + R fi ⎟ dz ⎝ hio h 'fi ⎠

−1

Ao (T − t )

(8-1-16)

Defining ⎛ 1 ⎞ 1 U =⎜ + + R fi ⎟ ⎝ hio h 'fi ⎠

−1

(8-1-17)

it is dQ = U πDo (T − t ) dz

(8-1-18)

Equations (8-1-14) and (8-1-15) allows calculation of the heat-flux density (watts per unit tube length) in a section of a heat exchanger where the fluid temperatures are T and t.

221

FINNED TUBES

As was demonstrated in Chap. 5, this expression can be integrated into all the exchanger length, and for a countercurrent or co-current configuration, we get Q = UAo (LMTD)

(8-1-19)

where Ao is now the total area of the exchanger (not per unit length).

8-1-2 Derivation of the Fin Efficiency for the Case of Longitudinal Fins with Constant Section The efficiencies of different types of fins can be calculated theoretically with certain assumptions. We shall see how the analytical expression for fin efficiency in the case of rectangular fins can be obtained. The reader not interested in this type of mathematical derivation can move to the next section. We shall assume that the fin is surrounded by a hot fluid at temperature T, so the transfer of heat takes place from the exterior into the tube. However, the final expressions are valid if the heat transfer is in the opposite direction. We shall assume that the hot fluid temperature does not vary with the radial coordinate x. It is also assumed that the fin thickness b is small enough to neglect the temperature gradients across its width. We also shall assume that the heat transferred through the top of the fin (fin shoulder) is negligible owing to the small thickness. If we consider the volume element shown in Fig. 8-3, with thickness b, length dz, and height dx, we can see that a conduction heat flow Qc is entering the control volume through the right face (bdz) and that a conduction heat flow Qc + dQc is leaving through the left face. Additionally, a convection heat flow dQf enters through the lateral area 2dxdz. In steady state, the sum of these terms is zero, so it must be dQc = dQ f

(8-1-20)

Qc is a conduction heat flow and can be expressed as Qc = − k

dT f ( x ) dx

(bdz )

(8-1-21)

where Tf (x) is the metal temperature at a certain x, which is assumed constant through the fin thickness. The convection heat entering the fin can be expressed as a function of the temperature difference between the external fluid and the metal. This is dQ f = h 'f [T − T f ( x )](2dx dz )

(8-1-22)

Then dQ f

= h 'f 2dz[T − T f ( x )]

(8-1-23)

d 2T f ( x ) dQc = −k (bdz ) dx dx 2

(8-1-24)

dx Differentiating Eq. (8-1-21), we get

Combining Eqs. (8-1-20), (8-1-23), and (8-1-24), we get − kb

d 2T f ( x ) dx 2

− 2h 'f [T − T f ( x )] = 0

(8-1-25)

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CHAPTER EIGHT

Since T is a constant for the integration in x, we can also write kb

d 2 [T − T f ( x )] dx 2

− 2h 'f [T − T f ( x )] = 0

(8-1-26)

And defining a variable θ = [T − Tf (x)], d 2θ − 2h 'f θ = 0 dx 2

(8-1-27)

d 2θ 2h 'f θ − =0 kb dx 2

(8-1-28)

kb Rearranging, we get

The general solution to this differential equation is θ = C1e mx + C2 e− mx

(8-1-29)

where ⎛ 2h 'f ⎞ m=⎜ ⎝ kb ⎟⎠

1/ 2

(8-1-30)

Now we must evaluate the integration constants. We shall call θ0 the value of θ at x = 0 (at the fin shoulder), so from Eq 8-1-29 θ0 = C1 + C2

(8-1-31)

Since we assumed that there is no heat flow through the fin shoulder, for x = 0, the temperature gradient must also be zero ⎛ dθ ⎞ =0 ⎜⎝ dx ⎟⎠ x=0

(8-1-32)

Since dθ/dx = m(C1emx − C2e–mx), this condition requires that C1 − C2 = 0. Then C1 = C2 =

θ0 2

(8-1-33)

Thus Eq. (8-1-29) results in θ = θ0

e mx + e − mx = θ0 cosh mx 2

(8-1-34)

If we call θb the value of θ for x = H ( at the base of the fin), it results, θb = θ0 cosh mH

(8-1-35)

The conduction heat flow for different values of x can be obtained as follows: Combining Eqs. (8-1-24) and (8-1-25), we get dQc = 2h 'f dz[T − T f ( x )] = 2h f dzθ dx

(8-1-36)

FINNED TUBES

223

Differentiating with respect to x, we get d 2Qc dθ = 2h 'f dz dx dx 2

(8-1-37)

d 2Qc 2h 'f Qc = 0 − kb dx 2

(8-1-38)

Combining with Eq. (8-1-21) gives

As before, the solution to this differential equation is Qc = C 1' e mx + C '2 e − mx

(8-1-39)

C '1 = −C '2

(8-1-40)

Since for x = 0, Qc = 0, the result is

Differentiating Eq. (8-1-39) and calculating the value of the derivative at x = 0, we get dQc dx

(8-1-41)

= mC '1 − mC '2 x= 0

And according to Eq. (8-1-36), dQc = 2h 'f dzθ0 = m C 1' − m C '2 dx

(8-1-42)

And combining with Eq. (8-1-40), we get C1 =

h 'f dzθ0 m

and

C2 = −

h 'f dzθ0

(8-1-43)

m

Thus Qc =

Qb = Qc

x=H

h 'f dzθ0 m =

(emx − e− mx ) =

h 'f dzθ0 m

2h 'f dzθ0

(emH − e− mH ) =

m

sinh mx

2h 'f dzθ0 m

sinh mH

(8-1-44)

(8-1-45)

Dividing Eqs. (8-1-45) and (8-1-35), we get Qb 2h 'f dz = tanh mH m θb

(8-1-46)

which can be written as Qb = h 'f (2 Hdz )

tanh mH (T − Tb ) mH

(8-1-47)

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CHAPTER EIGHT

Comparing with Eq. (8-1-10), the result is Ω=

tanh mH mH

(8-1-48)

Interpretation of the Results. The film heat transfer coefficient for the fluid external to the tube is given by Eq (8-1-13): h 'fo =

h 'f ( AD + ΩA f ) Ao

It is evident from this expression that by increasing the fin area, we improve the value of h 'fo , which also improves the overall heat transfer coefficient. However, the effect of increasing Af is weighted by the fin efficiency. If the efficiency Ω is small, an increase in the fin area will not improve the overall heat transfer coefficient significantly. The following table shows the values of Ω calculated with Eq. (8-1-48) for different values of mH: mH Ω

0.0 1

0.1 0.997

0.2 0.987

0.3 0.971

0.4 0.949

0.5 0.924

0.6 0.895

0.7 0.863

0.8 0.830

mH Ω

0.9 0.795

1.0 0.762

1.1 0.728

1.2 0.695

1.3 0.663

1.4 0.632

2.0 0.482

3.0 0.332

4.0 0.25

It can be observed in this table that for high values of mH, the efficiency of the fin is small. Thus, for example, if mH = 4, Ω = 0.25, which means that every 4 m2 of fin area is equivalent to 1 m2 of bare tube surface. In these cases, the installation of fins may not be cost-effective, and it is more convenient to increase the number or length of tubes. Since 1/ 2 ⎛ 2h f' ⎞ mH = ⎜ H ⎝ kb ⎟⎠

we can analyze the different factors affecting the value of the efficiency as follows: 1. Heat transfer coefficient hf . The higher hf is, the higher is the product mH, and the smaller is the fin efficiency. This means that the installation of fins is advantageous when the convection film coefficients are small, such as in cases where the fluids external to the tubes are gases or viscous fluids. On the other hand, with fluids that have intrinsic high values of hf , the percentage improvement obtained with fins is small. 2. Fin height. The greater the fin height, the smaller is the efficiency. Obviously, if the fin height is increased, Af also increases, so h 'f o always will increase regardless the efficiency reduction. But it must be interpreted that every square meter of additional surface obtained by increasing the fin height is less effective every time. A situation is reached where the marginal benefit is so low that the results are uneconomical. 3. Fin thickness. The greater the thickness, the higher is the fin efficiency, but the lower is the number of fins that can be installed in the tube. Usually it is better to use small-thickness fins. BWG 20-gauge (0.9-mm-thick) plate is typical. 4. Thermal conductivity of the fin material. The higher the thermal conductivity, the higher is the fin efficiency.

FINNED TUBES

225

8-1-3 Film Coefficients and Frictional Pressure Drop for Longitudinal Fins To use the preceding expressions, it is necessary to calculate the heat transfer coefficients between the annulus fluid and the metal (fins and bare tube surface). As in the case of double-tube heat exchangers with plain tubes, the correlations are based on Reynolds and Nusselt numbers defined with an equivalent diameter. The equivalent diameter is four times the hydraulic radius. The hydraulic radius is the quotient between the flow area and the transference perimeter. For heat transfer, this perimeter is that of the fins and the bare portion of the internal tube. To calculate the friction factor, the perimeter of the external tube also must be considered. The expressions are De (heat ) =

4( πDio2 / 4 − πDo2 / 4 − N f bH ) πDo − N f b + 2 HN f

(8-1-49)

and D 'e (friction) =

4( πDio2 / 4 − πDo2 / 4 − N f bH ) πDo − N f b + 2HN + πDio

(8-1-50)

In these expressions, Dio is the internal diameter of the external tube, and Nf is the number of fins per tube. With these equivalent diameters, Reynolds numbers for heat transfer and friction can be defined, considering that the flow area for velocity calculation is (8-1-51)

as = πDio2 / 4 − πDo2 / 4 − N f bH

According to Kern,1 the correlations to calculate the heat transfer coefficients in the case of finned tubes are different from those for plain tubes, and he proposes a graph whose values are correlated adequately with the following expressions: Friction factor. For Re < 2,100, 16 Re

(8-1-52)

f = 0.109 Re −0.255

(8-1-53)

f= For Re > 2,100,

Heat transfer coefficient. For Re < 2,100, ⎛ μ ⎞ Nu = 0.359 Re 0.324 Pr 0.33 ⎜ ⎝ μ w ⎟⎠

0.14

(8-1-54)

For Re > 10,000, ⎛ μ ⎞ Nu = 0.01783 Re 0.835 Pr 0.33 ⎜ ⎝ μ w ⎟⎠

0.14

(8-1-55)

226

CHAPTER EIGHT

The main difference is that the dimensionless group L/D is not included in the laminar-flow correlation. The author does not mention if the correlation is valid in a particular range of L/D. The expression for turbulent flow is quite similar to that used for plain tubes, but the coefficients are about 30 percent lower. This is due to the fact that longitudinal fins reduce turbulence. As usual, there is a transition regime for Reynolds number values between 2,100 and 10,000, where the correlations are not accurate. A linear interpolation can be done in this region, and the results are as follows: For 2,100 < Re < 6,000, ⎛ μ ⎞ Nu = ⎡⎣ 4.28 + 4.8 × 10 −3 (Re − 2,100) ⎤⎦ Pr 0.33 ⎜ ⎟ ⎝ μw ⎠

0.14

(8-1-56)

For 6,000 < Re < 10,000, ⎛ μ ⎞ Nu = ⎡⎣ 23 + 3.75 × 10 −3 (Re − 6, 000) ⎤⎦ Pr 0.33 ⎜ ⎟ ⎝ μw ⎠

0.14

(8-1-57)

Calculation of Wall Temperatures. To evaluate the quotients (μ/μw), it is necessary to evaluate the wall temperatures at the annulus side and at the internal-tube side. Since an important temperature drop takes place along the fin, both temperatures will be different. Calling T and t the mean temperatures of the hot and cold fluids, the mean wall temperature at the annulus side is t +

Q h f ( AD + A f ) LN

T−

Q h f ( AD + A f ) LN

if the annulus fluid is the cold fluid or

if the annulus fluid is the hot fluid. And the wall temperature corresponding to the internal side is t +

Q hi Ai LN

T−

Q hi Ai LN

if the cold fluid is in the internal tube or

if the tube-side fluid is the hot fluid.

8-1-4 Multipass Heat Exchangers with Longitudinal Flow In Chap. 5 we explained some constructive aspects of hairpin heat exchangers (see Fig. 5-15). When the fluid circulating through the annular space has a low heat transfer coefficient, these units can be built using tubes with longitudinal fins. In these cases, the calculation methods presented earlier can be used, but it is

227

FINNED TUBES

necessary to have the precaution of modifying the equations according to the geometry. That is, Flow area as = πDio2 / 4 − N πDo2 / 4 − NN f bH De (heat ) =

4(πDio2 / 4 − N πDo2 / 4 − NN f bH )

D 'e (friction) =

N πDo − NN f b + 2 HNN f 4( πDio2 / 4 − N πDo2 / 4 − NN f bH ) N πDo − NN f b + 2 HNN f + πDio

(8-1-58)

(8-1-59)

(8-1-60)

In this case, N is the number of tubes. It is also possible to build shell-and-tube heat exchangers with longitudinal fins. In this case, transverse baffles will not be installed, and the shell-side fluid circulates with longitudinal flow, so we can use the equations presented earlier to calculate the heat transfer coefficients. This construction allows us to have two or more tube passes with one shell pass. Example 8-1 When a natural gas stream suffers an isoenthalpic expansion, reducing its pressure, a temperature decrease takes place. This is due to the Joule-Thompson effect (because in real gases enthalpy is not only a function of temperature but also of pressure). If the gas contains some water vapor, this temperature decrease can lead to the formation of solid components called hydrates. These are icelike substances, but they form at temperatures about 15°C and can cause obstructions in piping systems. It is then necessary to heat the gas before the expansion. A stream consisting in 75,330 kg/h of natural gas, molecular weight 16.88, at 20°C and 4,000 kPa, must be heated up to 40°C before its expansion. A hot-oil stream, available at 200°C, will be used for this service. The hot-oil return temperature should be 150°C. The physical properties of the streams are as follows: Hot Oil Mean temperature,°C Specific heat, J/(kg · K) Viscosity, cP Density, kg/m3 Thermal conductivity, W/(m · K)

175 2800 0.46 790 0.112

Gas 30 2510 0.0119 30.71 (= Mp/zRT) 0.0358

Available pressure drops are 0.5 bar for the hot oil and 0.1 bar for the gas. Fouling resistances should be 0.00053 (m2 · K)/W for each fluid. A suitable heat exchanger must be designed. The selected tubes are 25.4 mm OD, 19.86 mm ID, with 20 fins per tube. Fins will be 12.5 mm high and 0.889 mm (0.035 in) thick. Fin thermal conductivity is 43.25 W/(m · K). Note: Downloadable interactive spreadsheet with the calculations of this example is available at http://www.mhprofessional.com/product.php?isbn=0071624082 Solution

Q = Wccc (t2 − t1) = (75,330/3,600) × 2,510 × 20 = 1.05 × 106 W Wh = Q/ch(T1 − T2) = 1.05 × 106/2,800 × 50 = 7.5 kg/s = 27,000 kg/h After some preliminary trials, a shell-and-tube heat exchanger with the following configuration is proposed:

228

CHAPTER EIGHT

Number of tubes: 70 in four tube passes. From a geometric analysis, we find that a shell diameter of 489 mm is necessary for this number of tubes. We shall calculate the required tube length. Hot fluid inside tubes: Flow area = at =

N πDi2 70 π × 0.01986 2 = = 0.00542 m 2 4 4 n 4

Gt = mass flow density =

Di × Gt 0.01986 × 1, 383 = 59, 730 = μ 0.46 × 10 −3

Re t =

Pr = hi = 0.023

Wh 7.5 = = 1, 383 kg/(m 2 ⋅ s) 0.00542 at

2, 800 × 0.46 × 10 −3 = 11.7 0.112

k 0.112 Re 0.8 Pr 0.33 = 0.023 × × 59, 730 0.8 × 11.70.33 = 1,900 W/(m 2 ⋅ K) Di 0.01986 hio = hi

Di 19.86 = 1, 900 = 1,485 W/(m 2 ⋅ K) Do 25.4

External coefficient Flow area ⎛ πD 2 N πD 2 ⎞ π × 0.4892 70 × π × 0.0254 2 o as = ⎜ i − − NN f bH ⎟ = − 4 4 4 ⎝ 4 ⎠ − 70 × 20 × 0.889 × 10 −3 × 0.0125 = 0.1367 m 2 Velocity =

75, 330 W = = 4.98 m/s aS ρ 3, 600 × 0.1367 × 30.7

Equivalent diameter (friction) 4( πDio2 / 4 − N πDo2 / 4 − NN f bH ) = N πDo − NN f b + 2 HNN f + πDio =

4( π 0.4892 / 4 − 70 π 0.0254 2 / 4 − 70 × 20 × 0.889 × 10 −3 × 0.0125) 70 × π × 0.0254 − 70 × 20 × 0.889 × 10 −3 + 2 × 70 × 20 × 0.0125 + π × 0.489

= 0.01338 m

Equivalent diameter (heat) 4(πDio2 / 4 − N πDo2 / 4 − NN f bH ) = N πDo − NN f b + 2 HNN f =

4(π 0.4892 / 4 − 70 π 0.0254 2 / 4 − 70 × 20 × 0.889 × 10 −3 × 0.0125) = 0.01391 m 70 × π × 0.0254 − 70 × 20 × 0.889 × 10 −3 + 2 × 70 × 20 × 0.0125 Re (friction) =

D ' evρ 0.01338 × 4.98 × 30.7 = = 171, 900 μ 0.0119 × 10 −3

FINNED TUBES

Re (heat ) =

Devρ 0.01391 × 4.98 × 30.7 = = 178, 700 μ 0.0119 × 10 −3

For the first trial, we shall neglect the viscosity correction factor. Thus Pr =

2, 510 × 0.0119 × 10 −3 3.58 × 10 −2

= 0.834

Nu = 0.01783 Re 0.835 Pr 0.33 = 0.01783 × 178, 700 0.835 × 0.834 0.33 = 407 h f = 407

k 3.58 × 10 −2 = 407 × = 1,050 W/(m 2 ⋅ K) De 0.01391

⎛ 1 ⎞ Fouling corrected coefficient h 'f = ⎜ + R fo ⎟ ⎝ hf ⎠

−1

−1

⎛ 1 ⎞ =⎜ + 5.3 × 10 −4 ⎟ = 674 ⎝ 1, 050 ⎠

Calculation of fin efficiency: 1/ 2 ⎛ ⎞ 2 × 674 ⎛ 2h f' ⎞ m=⎜ =⎜ ⎝ kb ⎟⎠ ⎝ 43.25 × 0.889 × 10 −3 ⎟⎠

1/ 2

= 187.3

mH = 187.3 × 0.0125 = 2.34 Ω=

tan h(mH ) = 0.419 mH

AD = πDo − N f b = 0.0254 π − 20 × 0.889 × 10 −3 = 0.0620 m 2/m A f = N f 2 H = 20 × 2 × 0.0125 = 0.5 m 2/m Ao = πDo = π × 0.0254 = 0.0798 m 2/m h 'fo =

h 'f ( AD + ΩA f ) Ao

=

674(0.0620 + 0.419 × 0.5) = 2,297 W/(m 2 ⋅ K) 0.07798

Internal fouling resistance referred to the external diameter is R fio = R f i ×

Do 0.0254 = 5.3 × 10 −4 × = 6.8 × 10 −4 (m 2 ⋅ K)/W Di 0.0198 −1

−1 ⎛ 1 1 ⎞ 1 1 ⎞ ⎛ −4 U = ⎜ R fio + + = 560 W/(m 2 ⋅ K) + ⎟ = ⎜⎝ 6.8 × 10 + ⎟ hio h 'fo ⎠ 1, 485 2, 297 ⎠ ⎝

LMTD =

(200 − 40 ) − (150 − 20 ) = 144.48°C 200 − 40 ln 150 − 20

229

230

CHAPTER EIGHT

LMTD correction factor: R=

20 − 40 = 0.4 150 − 200 A=

and

S=

200 − 150 = 0.277 Ft ≅ 1 200 − 20

Q 1, 050, 400 = = 12.98 m 2 U (DMLT) 560 × 144.48

L=

A 12.98 = = 2.31 m πDo N π × 0.0254 × 70

Viscosity correction factor: The hot and cold fluid mean temperatures are T = 175°C Wall temperature at the external side = t +

and

t = 30°C

Q 1, 050, 400 = 41°C = 30 + h f ( AD + A f ) LN 1, 050(0.0617 + 0..5) × 2.31 × 70

Wall temperature at the internal side = T −

Q 1, 050, 400 = 120.8°C = 175 − hi Ai NL 70 × π × 0.0199 × 2.31 × 1, 930

The gas viscosity at 41°C is sensibly equal to the viscosity at 30°C, so the factor (μ/μw)0.14 is unity. We also shall neglect this correction for the hot oil. Pressure-drop calculations: Shell side f = 0.109 Re −0.255 = 0.109 × 171, 900 − 0.255 = 5.04 × 10 −3 Δp = 4 f

L v2 2.31 4.982 ρ = 4 × 5.04 × 10 −3 × × 30.07 × = 1,332 N/m 2 = 0.013 bar D 'e 2 0.0133 2

Tube side f = 1.2(0.0014 + 0.125 / Re 0.32 ) = 6.11 × 10 −3 Δpt = 4 f

Ln v 2 2.3 × 4 790 × 1.762 ρ = 4 × 6.11 × 10 −3 × × = 13,800 N/m 2 Di 2 0.0199 2 Δpr = 4 nρ

v2 790 × 1.762 =4×4× = 19,382 N/m 2 2 2

Δ p = Δ pt + Δ pr = 33,187 N/m 2 It can be observed that the fins transform the shell-side coefficient from 674 to 2,297 W/(m2 · K). However, the relatively high value of the fouling resistance, in addition to a hot-oil coefficient of 1,539 W/(m2 · K), dampens the effect of the fins on the overall heat transfer coefficient. Since the hot oil is close to its allowable Δ p, it is not possible to improve hi . So it may be necessary to evaluate whether the higher cost from the use of finned tubes still can be an attractive option against the use of a conventional heat exchanger with a larger area.

FINNED TUBES

231

8-2 AIR COOLERS Air coolers are heat exchangers in which heat is removed from a process stream in a cooling or condensing operation using air as a refrigerant medium. The obvious advantage over a water-cooled exchanger is that a cooling water circuit is not necessary, thus avoiding the cooling tower, circulation pumps, water-treating systems, and piping. As a counterpart, air coolers are bigger units and more difficult to allocate in a plant. Air coolers usually are installed on piperacks or at the highest level of process structures to reduce the required ground area. The cost of these units is usually higher than that of water-cooled shell-and-tube heat exchangers. They are the only option in locations where cooling water is not available, as is usually the situation in oil and gas fields, or in process plants where the cooling-water systems are operating at their maximum capacity and it is necessary to install additional process coolers. A typical application is as distillation column overhead condensers as an alternative to shell-and-tubes condenser, which will be studied in Chap. 10. They are also used to cool down final products before sending them to storage tanks, as refrigeration-cycle condensers, as interstage coolers in compressor circuits, etc. Even though they can be constructed with plain tubes, it is usual to employ finned tubes with transverse fins, and this is the reason why they are included in this chapter. The minimum temperature at which the process stream can be cooled down is about 5°C above the ambient air temperature. The air temperature varies during the day and during different seasons of the year, so it is necessary to have meteorologic information to decide which air temperature will be used for design. The usual practice is to choose an air temperature that is not exceeded during a certain percentage of time over the year (e.g., 95 percent of the time). This value is obtained from statistics covering a certain number of years. In very critical applications, it is possible to choose the absolute historical maximum of the site, but it is important to evaluate this subject to avoid unnecessary oversizing. 8-2-1 Components of an Air Cooler An air cooler is composed of 3,7 1. 2. 3. 4.

One or more tubes bundles Fans to circulate air through the external side of the bundle (air circulation is usually in cross-flow.) A plenum between the fans and the bundle A supporting structure that must be high enough to allow the entrance of air below the bundle with a reasonably low velocity 5. Platforms and ladders for maintenance 6. Optionally, louvers to regulate the airflow or systems to modify the angle of the fan blades to control the process temperature, thus saving electric power (see Fig. 8-4) Tube Bundle. The tube bundle includes a set of tubes, headers, lateral frame, and tube supports. The tube length is usually defined by layout considerations. As a general rule, the longer the tubes, the lower will be the exchanger cost per unit area, but in very long exchangers with few tubes, it may be difficult to locate the fans to cover the entire surface of the bundle. Since the film heat transfer coefficients on the air side are usually low, finned tubes are used to increase the heat transfer area on the external side. There are several types of fins. The most common are6 1. Tension-wrapped. This is the most common type because of economics. It is a rectangular-section aluminum ribbon tension-wrapped around the tube. The contact between the fin and the tube is through the ribbon edge. Since this contact is not perfect, fin efficiency is affected. 2. Embedded. These fins are built by helically wrapping a strip of aluminum to embed it in a precut helical groove and then peening back the edges of the groove against the base of the fin to tightly secure it. 3. Extruded. These fins are extruded from the wall of an aluminum tube that is integrally bonded to the base tube for the full length.

232

CHAPTER EIGHT

1 2 9 3 4 5 4 6 7 8 Induced draft 6 4 5 4 3 10

1 2 9 7 8

Forced draft 1. Fan 2. Fan ring 3. Plenum

4. Nozzle 5. Header 6. Tube bundle

7. Drive assembly 8. Column support 9. Inlet bell 10. Outlet bell

FIGURE 8-4 Air cooler configurations. Courtesy of Hudson Products Corporation.

4. Footed. These fins are built by wrapping an aluminum strip that is footed at the base as it is wrapped on the tube. The tube is completely covered by the fin feet. 5. Others. These are tubes at whose exterior surface fins are fixed by other means, such as brazing, welding, or hot immersion (see Fig. 8-5). Finned tubes can be built in diameters ranging from 5/8 to 6 in. For air coolers, the most popular diameter is 1 in. Fin heights ranges from 1/2 to 1 in. (The most popular sizes are 1/2 and 5/8 in.) Finned tubes are manufactured with 275–433 fins per meter (7–11 fins per inch). The ratio between the fin area and the plain-tube external area (without fins) varies from 7 to 25. The larger the tubes, the lower is the unit cost for the same heat transfer area. The tube bundle section is rectangular and typically consists of 2 to 10 tube rows in a triangular pattern. The separation between tubes axes is typically 21/2 tube diameters. The free area for airflow allowed by the tubes is roughly 50 percent of the projected area.

FINNED TUBES

233

Temperature

Wrapped

Embedded FIGURE 8-5

Extruded

Footed

Different fin types.

The tubes are roller-expanded or welded to tubesheets that are part of the headers. The headers are rectangular boxes. One of their faces is the tubesheet, and the opposite face is the cover. The cover can be bolted to allow removal or welded to the header. When the cover is welded, threaded holes are drilled just opposite each tube. The holes are provided with screwed plugs that can be removed for internal tube cleaning (Fig. 8-6). The headers can have pass-partition plates, as in the case of shell-and-tube heat exchangers. This allows one to increase the process fluid velocity and/or obtain a more countercurrent circulation configuration. The tube bundles are usually installed in a horizontal position, with the air always entering at the lower side and discharging vertically upward. The pass partition can be done horizontally or vertically. Horizontal partitions allow approximating the configuration of a countercurrent pattern, with the process stream entering at the upper pass. Fans. Fans are of the axial-flow type. The air cooler is a forced-draft cooler when the fan pushes air through the bundle. If the bundle is located at the suction side of the fans, it is called an induced-draft cooler. Fans normally have 2–20 paddles. The fan diameter is related to the bundle width. Usually, the fan diameter is limited to 4 or 5 m, but fans with up to 20-m diameters have been manufactured. Fan paddles can be manufactured in aluminum, fiberglass, plastic, or steel and can be straight or contoured. The paddles pitch can be adjusted either manually or automatically to regulate the airflow. The automatic control is done with pneumatic devices installed in the fan central hub (that rotate with the fan) and whose air supply is through a rotating coupling. Each bundle is usually air fed by at least two fans. This is a safety measure to avoid complete air loss in case of a fan failure, as well as a means to achieve certain control of the airflow (e.g., switching off one fan in winter). Fan coverage is defined as the ratio between the projected area of the fans and the projected area of the bundles they serve. This ratio must be 0.4 at a minimum to achieve good air distribution. Fans are driven by electric motors, and velocity reduction is achieved with reducing gears or V belts. The tangential tip speed always must be lower than 60 m/s for mechanical reasons.

234

CHAPTER EIGHT 16 9 14

10 3

13

1

4

5

2 8

11

7

6

12 3

18 17

16

15

Plug header

16

9 3 1

10 13 5

18

14 11

17

6

12 3

4

15

Cover plate header 1. Tube sheet 2. Plug sheet 3. Top and bottom plates 4. End plate 5. Tube 6. Pass partition

7. Stiffener 8. Plug 9. Nozzle 10. Side frame 11. Tube spacer 12. Tube support cross-member

13. Tube keeper 14. Vent 15. Drain 16. Instrument connection 17. Cover plate 18. Gasket

Typical construction of tube bundles with plug and cover plate headers FIGURE 8-6 Header types. (Courtesy of Hudson Products Corporation.)

The fan power requirements depend on the airflow rate and air pressure drop, which is related to the number of tubes in the flow direction. Sometimes it is preferred to split the service into a higher number of fans to avoid high-power motors. Most air cooler fan power is lower than 35 kW. Plenums. The air plenum is a totally enclosed space that allows accommodation of the airflow between the fan and the tube bundles to produce a uniform air distribution. Plenums can be designed as pyramidal or straight-section. Structure. The structure is formed by columns and beams to support the unit at enough height over ground level to allow the air to enter below the bundle without excessive velocity. In process plants, it is

FINNED TUBES

Bay width

235

Unit width

Tubes length

One bay with 3 tubes bundles Bay width

One bay with 2 fans and 2 tubes bundles

2 single fan bays with 4 tubes bundles Unit width

2 bays with 2 fans each and 6 tubes bundles

FIGURE 8-7 Air cooler configurations.

common for air coolers to be installed on the piperacks to avoid occupying ground area. In these cases, the piperack structure supports the air cooler. Configuration. The assembly of one or more tube bundles, served by one ore more fans, complete with a plenum, structure, and auxiliary devices, is called a bay. When large flow rates are handled, it is common for the unit to be divided in several bays to facilitate transport and construction. Figure 8-7 shows some air cooler configurations. Forced Draft versus Induced Draft. Advantages of induced-draft coolers are 1. A better distribution of air on the entire section of the bundle 2. Less possibility of hot-air recirculation to the intake section. (This is so because in these units the hot air is discharged upward at a velocity that is about two or three times the intake velocity, owing to the ratio between the fan area and the cross section of the bundle.) 3. Higher capacity in case of fan failure owing to a natural draft effect. (This effect is much greater in induced-draft units than in forced-draft units.) 4. Less affected by rain, snow, or climatic effects because most of the top face of the bundle is covered

236

CHAPTER EIGHT

Nm = number of fins per meter of tube Do Db W b

Air flow

L

nf = number of rows

ST

ntf = number of tubes per row

SF SL

FIGURE 8-8 Air cooler nomenclature.

Disadvantages are 1. Higher power consumption, especially if the air temperature increase is important 2. Air outlet temperature limited to about 90°C to avoid potential damage to the fan components owing to high temperature 3. Fan maintenance more difficult and must be done in the hot air generated by the natural convection effect 8-2-2 Heat Transfer in Air Coolers Heat transfer in air coolers can be described by the general equation Q = UAΔT

(8-2-1)

But it is necessary to clarify the meaning of the different terms of the equation owing to the particular issues resulting from the presence of fins and a cross-flow configuration. The nomenclature we shall use is explained in Fig. 8-8. Heat Transfer Area. We shall call Ao = plain tube external area per unit length of tube = πDo . AD = exposed area of tube (not covered by fins) per unit tube length = Ao(1 − bNm). Af = fin area per unit tube length = π2Nm(Db2 − Do2)/4. Usually the heat transfer area in Eq. (8-2-1) is taken as the plain-tube area. This is A = Ao Ln f ntf

(8-2-2)

Overall Heat Transfer Coefficient U. It was already explained in Sec. 8-1 that the fin surface behaves differently from the exposed tube surface, and in order to define a film heat transfer coefficient for the external side, a fin efficiency Ω must be considered, so h 'fo = h 'f ( AD + Ω A f ) /Ao

(8-2-3)

FINNED TUBES

237

The meaning of the fin efficiency was explained in Sec. 8-1. It depends on the geometry and on the parameter (bh'f /k)1/2, where h'f is the film coefficient for the external surface of the fins, and k is the thermal conductivity of the fin. The higher the thermal conductivity of the material, the higher is the fin efficiency. Additionally, the lower the external film coefficient h'f , the higher is the fin efficiency. This is so because if h'f is intrinsically high; a significant increase in the heat transfer rate is not achieved with the use of fins. On the other hand, for low heat transfer coefficients, as is the case with air coolers, the improvement is considerable. The value of h'f includes the effect of fouling resistance on the external side. Usually this resistance is neglected because it may be considered that air is a clean fluid, but if it is desired to include this effect, it must be calculated as 1 1 = + R fo h 'f hf

(8-2-4)

where hf is the clean coefficient. To calculate the overall heat transfer resistance, the internal film coefficient and internal fouling resistance are included, and we get 1 1 1 = + + R fio U h 'fo hio

(8-2-5)

where hio is the internal coefficient referred to the external tube area, hio = hi × Di / Do, and Rfio is the internal fouling resistance, also referred to the external diameter (Rfio = Rfi × Do /Di). This correction is usually neglected. Fin Efficiency. In Sec. 8-1 we performed the analytical derivation to calculate the efficiency of a longitudinal fin. Air coolers have transverse fins, and the mathematical deduction of fin efficiency is more complex. The analytical expressions for different fin geometries can be found in ref. 4. The graph in Fig. 8-9 allows one to obtain the efficiency of constant-thickness circular fins, which are the most common type. The nomenclature for the geometric parameters can be found in Fig. 8-8. Figure 8-10 includes the mathematical expressions of these curves.4 To obtain the fin efficiency, it is necessary first to calculate the film coefficient hf. This calculation will be explained later. Temperatures Difference. In an air cooler, air circulates in a perpendicular direction to the tube-side fluid. The tube-side fluid, in turn, flows trough several tube rows (usually between two and eight) and can have several tube passes. Let’s consider an air cooler such as that shown in Fig. 8-11, where the tube-side fluid performs only one pass, so all the tubes are in parallel. We also can assume that the air velocity only has a component in the direction perpendicular to the tubes, so there is no effect of longitudinal mixing. This can be interpreted as if fictitious baffles (as indicated by the vertical lines in the figure) are present. The inlet temperatures of the hot fluid and air are T1 and t1, respectively. At point A, at the first horizontal tube row, and in correspondence with plane 1–1', the temperature difference between the two streams is T1 − t1. If we move to plane 2–2' , in position A', there will be a smaller temperature difference between the hot fluid and the inlet air because the process fluid temperature decreases. Similarly, at plane 3–3', in position A'', the temperature difference will be even smaller, and it continues to decrease as we move along the tubes in the first row. Let’s now analyze the second tube row. The air temperature leaving the first row varies along the tube because the heat transferred from the first row varies from point to point as the ΔT for heat transfer decreases. At the second row, the temperature differences also vary, and the temperature changes experienced by the air are different from those in the preceding row. This means that in every tube segment, defined by its longitudinal position and by the tube row to which it belongs, the rate of heat transfer is different, and the temperature change of each fluid is also different. The mean outlet temperature of the process stream will be the mixing temperature of the streams coming from all tube rows.

238

CHAPTER EIGHT

1.0 0.9 0.8 0.7 0.6 Db /Do = 1

Ω 0.5

1.4 1.6 1.8

0.4

2.0

0.3 3.0 4.0

0.2 0.1 0

3.0

2.0

1.0

(Db – Do) FIGURE 8-9

4.0

5.0

hf′ 2·kf ·b

Efficiency of annular fins of constant section.

m=

2 ⋅ h 'f k f ⋅b

H=

D b − Do 2

b ⎛ D ⎞ Y = ⎛⎜ H + ⎞⎟ ⎜ 1 + 0.35 ln b ⎟ ⎝ ⎠ 2 ⎝ Do ⎠ Ω=

tan h(mY ) mY

FIGURE 8-10 Mathematical expressions for the curves in Fig. 8-9.

To calculate the total heat transferred in the unit, it is possible to develop mathematical models that predict the temperatures evolution with finite-increments techniques, analytical solutions, or iterative methods. The most common approach is to adopt the same methodology used in shell-and-tube heat exchangers and calculate the heat transfer rate by defining a factor Ft as Q = UA(LMTD)Ft

(8-2-6)

where LMTD is the logarithmic mean temperature difference assuming a countercurrent configuration. The correction factor Ft can be calculated with the methods mentioned earlier. Another possible model we may have used is to consider that the air suffers axial mixing after each tube row and adopts a uniform temperature before passing to the next row. In this case, the air temperature is a function of the position of the row under consideration but not of the longitudinal coordinate. With this assumption, the mathematical derivations of the Ft relations are simpler and can be obtained analytically. However, in long-tube air coolers, this hypothesis is far from reality, and the nonmixed model must be used.

FINNED TUBES

239

t2

T1

1 D

2 D′

3 D″

4 D ″′

5 D ″″

C

C′

C″

C ″′

C ″″

B

B′

B″

B ″′

B″″

A

A′

A″

A″′

A″″

1′

2′

3′

4′

5′

T2

t1 FIGURE 8-11

Single-pass air cooler with no longitudinal mixing of the air stream.

In the configuration in Fig. 8-11, the tube-side fluid performs only one pass through the tubes. But usually, multipass units such as that represented in Fig. 8-12 are employed. In this case, the mathematical model is even more complicated, and the Ft value is different from that of a single-pass unit. A mathematical model to calculate the true temperature difference in this type of equipment, with different numbers of tube passes, for mixed and unmixed airflow was proposed by Cordero and Pignotti.5 The graphs in App. C allow calculation of the Ft factors using the same parameters as those employed for shell-and-tube heat exchangers: R=

T1 − T2 t 2 − t1

(8-2-7)

t2

1 D

2 D′

3 D″

4 D ″′

5 D ″″

C

C′

C″

C ″′

C ″″

B

B′

B″

B ″′

B ″″

A

A′

A″

A″′

A″″

1′

2′

3′

4′

5′

T1

T2

t1 FIGURE 8-12

Double-pass air cooler with no longitudinal mixing of the air stream.

240

CHAPTER EIGHT

S=

t 2 − t1 T1 − t1

(8-2-8)

These graphs are presented for configurations with one, two, or three tube passes. If the unit has more than three passes, Ft is practically 1. The passes performed by the hot fluid always must have opposite direction to the airflow. This means that the hot-fluid inlet is always in the upper pass, as shown in Fig. 8-12. Calculation of the Heat Transfer Coefficient. A method to calculate the film heat transfer coefficients and the pressure drop for a finned tube bundle was proposed by Kern1 and shall be described below. The method is valid for a tube bank with transverse fins of constant thickness in which the tubes are arranged in a triangular pattern, as shown in Fig. 8-8. This is the geometry usually employed in air coolers. The following nomenclature for the heat transfer coefficients will be used: hf = film coefficient for the air side, for the tube and fin surface, without including either fouling effects or the different efficiency of the fin surface with respect to the bare-tube portion. h'f = coefficient hf corrected for external fouling. It is related to hf by the expression 1 1 = + R fo h 'f hf

(8-2-9)

where Rfo is the external fouling resistance. It is frequently assumed that air is a clean fluid, and then, hf = h'f . h'fo = film coefficient corrected for fin efficiency, and considering as reference area the external surface of the plain tube, h 'fo = (ΩA f + AD )

h 'f Ao

(8-2-10)

hi = internal heat transfer coefficient (process side). This is calculated with the correlations included in Chaps. 4 and 10 for forced convection and condensation depending on the case. h'i = internal coefficient corrected for fouling Rfi = the internal fouling resistance. Then 1 1 = + R fi h 'i hi

(8-2-11)

h'io = internal heat transfer coefficient corrected for fouling, referred to the exterior area of the tube, h'i(Di /Do). U = overall heat transfer coefficient based on the external area of the tubes. Thus 1 1 1 = + U h 'io h 'fo

(8-2-12)

The film coefficient hf is obtained with the following correlation: j=

h f De k

Pr −1/ 3 = 0.0959(Re S )0.718

(8-2-13)

FINNED TUBES

241

In this expression, De is an equivalent diameter for heat transfer defined as De =

2( A f + AD ) π (projected perimeter )

(8-2-14)

Projected perimeter is the sum of all external dimensions in the plant view of a finned tube per unit length. That is, Projected perimeter = 2( Db − Do ) N m + 2(1 − bN m )

(8-2-15)

The Reynolds number for heat transfer is defined as Re S =

De GS DeWc = μ aS μ

(8-2-16)

where Wc is the airflow (air is always the cold stream) and aS is the flow area, which is calculated by subtracting the projected area obstructed by the tubes from the cross section of the bay aF (aF = WL). aS = WL − ntf L[ Do + N m ( Db − Do )b ]

(8-2-17)

ntf is the number of tubes in a row. If even and odd tubes rows have different number of tubes, a mean value is used. Fin Efficiency. The fin efficiency is calculated with the graph of Fig. 8-9 or the analytical expressions of Fig. 8-10. For other geometries, ref. 1 can be used. Calculation of Air Pressure Drop. The pressure drop of the air through the bundle can be calculated as Δp =

fGS2 L p ⎛ D 'e ⎞ 2ρD 'e ⎜⎝ ST ⎟⎠

0.4

⎛ SL ⎞ ⎜S ⎟ ⎝ T⎠

0.6

(8-2-18)

where f = friction factor GS = mass flow density, the same as defined for heat transfer coefficient calculation (GS = Wc /aS) Lp = length of path = nf SF SF , SL, ST see Fig. 8-8 D'e = equivalent diameter for friction D'e is different from the equivalent diameter used for heat transfer coefficient calculations and is defined as D 'e =

4 × net free volume Lntf ( A f + AD )

(8-2-19)

The net free volume is the volume between the central planes of two consecutive rows minus the portion of that volume occupied by the tubes and fins. That is, Net free volume = WLSF − ntf π

⎛ D 2 − Do2 ⎞ Do2 L − N m ntf Lπ ⎜ b ⎟⎠ b 4 4 ⎝

(8-2-20)

The friction factor f is obtained as a function of a Reynolds number Re'S, defined as Re'S =

D 'e GS D 'e Wc = μ aS μ

(8-2-21)

242

CHAPTER EIGHT

And the correlation is f = 1.276 Re'S −0.14

(8-2-22)

Fan Power Consumption. The power consumption of each fan can be calculated as Power =

volumetric flow of each fan × pressure developed fan efficiency × driver efficiency

(8-2-23)

The volumetric flow of the fan is the air-mass flow (kg/s) divided by the air density at the fan ρfan (kg/m3): Fan volumetric flow = (Wc /number of fans) = (Wc /number of fans) RTfan ρfan MAp

(8-2-24)

In induced-draft fans, the air temperature at the fan is higher than the ambient air temperature; thus its density is lower, and the volumetric flow is higher than in the case of a forced draft. This means that the power ratio between an induced-draft fan and a forced-draft fan is approximately equal to the ratio of the outlet and inlet absolute air temperatures. The pressure developed by the fan is the sum of the velocity head (ρfanvfan2/2) and the frictional pressure drop through the bundle, calculated as indicated earlier. The air velocity at the fan depends on the adopted fan diameter. The smaller this diameter, the higher will be the velocity for the same volumetric flow. The efficiency of the fans employed in air coolers is usually about 75 percent, whereas the efficiency of the electric motor and transmission system is about 95 percent. 8-2-3

Detailed Design Procedure Preliminary Design. As usual in heat transfer equipment, the design procedure of an air cooler consists of proposing a preliminary design and verifying its heat transfer capacity by calculating the heat transfer coefficient and the corrected mean temperature different, as was explained in the preceding section. Depending on the results of the verification, the proposed geometry or the airflow will be modified in an attempt to obtain a solution with an acceptable combination of heat transfer area, fan power, and process-side pressure drop. It thus is necessary to have guidelines to set up a preliminary design that can be used as starting point. The design input data usually consist of the following: 1. Hot-fluid process conditions (Wh , T1, and T2) 2. Inlet air temperature t1 3. Some restriction or preference regarding the footprint. It is possible that the width or length of the unit will be subject to some restriction or be defined for layout considerations. When the size of the unit is large, the more economical solution is obtained with the longest acceptable tubes. If the unit will be mounted on a pipe rack, the length is fixed and must coincide with the width of the pipe rack. 4. The characteristics of the finned tubes (i.e., tube diameter, fin diameter and thickness, number of fins per unit tube length, and fin material) usually are adopted from the beginning and are seldom modified during the design process. They are usually defined by the project specifications or by the manufacturing workshop facilities. The most popular finned tube is 1 in with aluminum fins 5/8-in height with 11 fins per inch (433 fins per meter), and the tubes are arranged in a triangular equilateral pattern separated by 21/2 in between axes. 5. Sometimes there are also preferences regarding the maximum power per fan, which in some cases can affect unit design, requiring a higher number of fans. To come up with a preliminary design, a first guess of the heat transfer coefficient is assumed, and an air face velocity and number of tube rows are adopted based on typical values for different applications.

243

FINNED TUBES

TABLE 8-1 Typical Values of the Overall Heat Transfert Coefficients in Air Coolers [W/(m2 · K)], Using 5/8-in Aluminum Fins in 1-in OD Tubes with 393 Fins per Meter Condensation

Gas Cooling

Amine regenerator Ammonia

600–700

Refrigerant 12

420–500

Heavy naphtha

400–500

Light gasoline Light hydrocarbons Light naphtha

540 540–600

Reactor effluent, re-forming, platforming Steam

570–670

450–550 450–550

Air or combustion gas at 50 psig (Δ P = 1 psi) Air or combustion gas at 100 psig (Δ p = 2 psi) Air or combustion gas at 100 psig (Δ p = 5 psi) Hydrocarbon gas at 15–50 psig (Δ p = 1 psi) Ammonia reactor stream Gas hydrocarbon at 15–50 psig (Δ p = 1 psi) Gas hydrocarbon at 50–250 psig (Δ p = 3 psi) Gas hydrocarbon at 250–1500 psig (Δ p = 5 psi)

Liquid Cooling 60

Machine cooling water

740–900

112

Fuel oil Re-forming or platforming liquids

115–170 480

170–220

Light gas oil

450–550

500–600 170–220

Light hydrocarbons Light naphtha

510–680 510

280–340

Process water

680–820

400–500

Distillation residue Tar

170

60–120 30–60

800–1200

(Courtesy of Hudson Products Corporation.)

Table 8-1 shows some typical values of the overall heat transfer coefficient for different process services that can be used as an initial estimation. The number of tube rows is related to the temperatures approximation between the air and the process flow. A parameter Z is defined as Z=

T1 − T2 T1 − t1

(8-2-25)

This parameter can be calculated from the process conditions. The higher the value of Z, the deeper must be the tube bundle within practical and economic considerations. Since an isothermal condenser has Z = 0, it should have a low depth, typically no more than four rows. A process cooler with a high temperature range (e.g., Z > 0.8), on the other hand, should be as deep as possible, say, 8–10 tube rows. The air face velocity VF, defined as VF =

Wc ρAWL

(8-2-26)

is also related to the number of rows because if the number of rows is high, the velocity should be reduced to avoid excessive air pressure drop. Table 8-2 can be used for the preliminary design. The procedure is 1. 2. 3. 4. 5.

Calculate the heat duty Q. Adopt a heat transfer coefficient with the help of Table 8-1. Calculate Z . From Table 8-2, obtain the number of tube rows nf and air face velocity vF . Assume an air outlet temperature. This temperature must be adopted considering the process temperatures. In multipass heat exchangers, some temperature cross is possible. In single-pass units, the air outlet temperature must be lower than the process outlet temperature. 6. Calculate the air mass flow as Wc =

Q cc (t 2 − t1 )

(8-2-27)

244

CHAPTER EIGHT

TABLE 8-2

Design Recommendations as a Function of Z Parameter

Z

nf = Number of Rows

VF (m/s)

0.4 0.5 0.6 0.8–1

4 5 6 8–10

3.3 3 2.8 2–2.4

(Courtesy of Hudson Products Corporation.)

7. Calculate the face area (aF = WL) as aF =

Wc ρ A vF

(8-2-28)

8. Select W or L according to layout convenience, and calculate the other one. The resulting unit should be proportionate. If the unit is excessively long and narrow, it will be necessary to install more fans to adequately cover the surface of the bundle. The more usual solution is to install two fans per bay, so it may be convenient to have an L / W ratio approximately equal to 2.5. 9. Calculate the LMTD. As a first approximation, the correction factor can be neglected, unless an air outlet temperature considerably higher than the process outlet temperature was adopted. 10. Calculate the plain tube area required for heat transfer as A=

Q U (MLDT)

(8-2-29)

A πDo L

(8-2-30)

11. Calculate the required number of tubes as N= 12. Calculate the number of tubes per row ntf as ntf =

N nf

(8-2-31)

13. With the separation between tubes axes ST (see Fig. 8-8), verify that W = ST ntf

(8-2-32)

In cases of agreement, proceed with the detailed design. Otherwise, modify the geometry within the design restrictions. Detailed Design. The following steps should be followed. We assume that the tube length was adopted, which is the case when the unit will be mounted on a pipe rack. 1. With the finally proposed geometry, recalculate the number of tubes N = A/πDo. 2. ntf = number of tubes per row = N/nf . 3. W = unit width = ST × ntf . If W is larger than the maximum bay width, which is usually limited for transport and erection considerations, it may be necessary to divide the unit in two or more bays. Each bay will handle part of the job. The reader can adapt the procedure for this case. 4. Airflow = Wc = vFρAWL . 5. Air outlet temperature = t2 = t1 + Q/Wccc.

FINNED TUBES

6. 7. 8. 9. 10.

11. 12. 13.

14.

245

Adopt the number of hot-fluid passes n. Calculate Ft and LMTD. Calculate the hot-fluid velocity into the tubes. With suitable correlations, calculate the heat transfer coefficient hio and pressure drop Δ pt . In case Δ pt > Δ p allowable, if the number of tube passes can be reduced without seriously affecting hio , or it is considered that hio will not be controlling, reduce n and go back to step 7. If it is not possible to reduce the number of passes, it will be necessary to increase the number of tubes (thus increasing the area) and go back to step 1. Calculate the external-side coefficient ho and the overall coefficient U. Verify A = Q/UΔT. If the calculated area is larger than the proposed area, investigate if it is possible to improve hio and U by increasing the number of tube passes without compromising the tube-side pressure drop. If it is not possible to increase the number of passes, it will be necessary to increase the heat transfer area A and go back to step 1. If the calculated area is considerably lower than the proposed area, the design must be changed by reducing the number of tubes. Calculate the air-side pressure drop and fan power. Distribute the total service in a suitable number of fans with diameters that allow covering at least 40 percent of the bay face section.

In this procedure we have assumed the tube length, number of rows, and air face velocity are constant. We obviously assume that these values have been selected with good criteria. If the final design results are unsatisfactory (e.g., an excessively long and narrow unit), the designer should think about modifying these parameters as well. The effect of airflow on the design also must be investigated. If the temperature approach between air and process is small, an increase in the airflow (or vF) may have a considerable impact on the LMTD, and considerable savings in area can be obtained. The graphs shown in Figs. 8-13 and 8-14 offer a tool to perform a fast design. They represent the value of h'fo and pressure drop per tube row as a function of the face velocity vF , calculated with the correlations presented earlier, for two of the most popular tube pattern (1-in tubes with aluminum fins 15.87 mm high and 0.381 mm thick with 394 or 433 fins per meter in a triangular pattern with 60.32 mm of pitch). Example 8-2 Design an air cooler to cool down 180,000 kg/h (50 kg/s) of heavy-gas oil from 76°C to 66°C. The maximum allowable tube length is 7 m. The tubes will be 25.4 mm OD and 19.3 mm ID. They will be provided with aluminum fins 15.875 mm in height (5/8 in). The fin thickness will be 0.381 mm, and the tubes will have 394 fins per meter. The tubes will be arranged in a triangular pattern with 60.32 mm of pitch. The air design temperature is 30°C. A pressure drop of 50 kPa is allowed for the process side. An internal fouling resistance of 0.0003 (m2 · K)/W must be used. Fouling is considered negligible on the air side. The following physical properties will be used:

Density, kg/m3 Viscosity, cP Thermal conductivity, W/(m · K) Specific heat, J/(kg · K)

Product

Air

770 5.6 0.14 2452

0.0209 0.0267 1011

The unit will be installed at sea level (atmospheric pressure 101.3 kPa). Note: A downloadable interactive spreadsheet with the calculations of this example is available at http://www.mhprofessional.com/product.php?isbn=0071624082

CHAPTER EIGHT

Pressure drop per tubes row 5/8" fins in 1" tubes. Triangular pattern 2 1/2"

60

Δp (Pa/row)

394 fins per meter 433 fins per meter

40

20

0.5

FIGURE 8-13

1.0

1.5

2.0 2.5 vF (m/s)

3.0

3.5

Pressure drop per tube row.

Heat Transfer Coefficient h′fo Aluminum 5/8" fins in 1" tubes. Triangular array 2 1/2"

1200

1000 h ′fo (W/m2K)

246

800

394 fins/meter 433 fins/meter

600

400 0.5

FIGURE 8-14

1.0

1.5

Heat transfer coefficient.

2.5 2.0 vF (m/s)

3.0

3.5

FINNED TUBES

Solution

247

Preliminary design: We shall follow the steps indicated earlier in the text.

1. Q = Whch(T1 − T2) = 50 × 2452 × (76 − 66) = 1.226 × 106 W 2. This particular application is not specifically mentioned in Table 8-1, but we can assume that the value of the overall heat transfer coefficient will be higher than 170 W/(m2 · K) (corresponding to fuel oil) and lower than 450 W/(m2 · K) (corresponding to light-gas oil). We shall adopt 250 W/(m2 · K). 3. Z = (T1 − T2)/(T1 − t1) = 10/(76 − 30) = 0.22 4. We shall adopt four tube rows and a face velocity of 3.3 m/s (see Table 8-2). 5. We shall tentatively assume an air outlet temperature of 45°C. 6. The airflow will be Wc = Q/cc(t2 − t1) = 1.226 × 106/(1,011 × 15) = 80.84 kg/s 7. The inlet air density is ρ = Mp/RT = 29 × 1/0.082 × 303 = 1.16 kg/m3. The face area to get a face velocity of 3.3 m/s is aF = Wc / vFρ = 80.84/(3.3 × 1.16) = 21.1 m2 8. Using 7-m tubes, the width of the unit is W = aF / L = 21.1/7 = 3.02 m. The geometry looks reasonable. 9. LMTD = [(T1 − t2) − (T2 − t1)] / ln(T1 − t2)/(T2 − t1) = [(76 − 45) − (66 − 30)]/ln(31/36) = 33.4°C 10. A = Q/(U · LMTD) = 1.226 × 106/(250 × 33.4) = 146.8 m2 11. N = A/(πDoL) = 146.8/(3.14 × 0.0254 × 7) = 262 12. ntf = 262/4 = 65.5 13. For the proposed geometry, SL = ST = 0.0603 m. Then W = ST × ntf = 65.5 × 0.0603 = 3.94 m. This is larger than required for a face velocity of 3.3 m/s. For a first trial, we shall keep this geometry, and we shall correct the airflow, that is, Wc = vF × ρ × W × L = 3.3 × 1.16 × 3.94 × 7 = 105.5 kg/s The air outlet temperature then will be t2 = t1 + Q/Wccc = 30 + 1.226 × 106/(105.5 × 1011) = 41.5 LMTD = [(76 − 41.5) − (66 − 30)]/ln(34.5/36) = 35.2 Detailed design: The values adopted for the first trial are 1. 2. 3. 4. 5. 6. 7.

N = 262 ntf = 65.5 (two rows with 65 and two rows with 66) W = 3.94 m Wc = 105.5 kg/s t2 = 41.5 We shall assume two passes for the hot fluid. LMTD = 35.2 R = (T1 − T2)/(t2 − t1) = 10/11.5 = 0.87 S = (t2 − t1)/(T1 − t1) = 11.5/46 = 0.25 Ft = ~1. Then ΔT = 35.2°C.

248

CHAPTER EIGHT

Calculations for the hot fluid: The flow area for the tube-side fluid is at = N(πDi2/4)/n = 262(3.14 × 0.01932/4)/2 = 0.0380 m2 The velocity of the fluid into the tubes is vt = Wh /(ρ × at) = 50/(770 × 0.038) = 1.7 m/s 9. Ret = Divtρ/μ = 0.0193 × 1.7 × 770/5.6 × 10–3 = 4,511 f = 1.2(0.0014 + 0.125/Re0.32) = 0.0118 Δ pt = 4fn(L/Di)ρvt2/2 = 4 × 0.0118 × 2 × (7/0.0193) × 770 × 1.72/2 = 38,095 Pa Δ pr = 4nρvt2/2 = 4 × 2 × 770 × 1.72/2 = 8,900 Δ p = 46,996 Pa 10. We see that the flow regime corresponds to the transition zone, which is not recommended. However, to achieve a Reynolds number of 10,000, it would be necessary to double the velocity, which would make the pressure drop higher than allowable. In this case, the pressure drop is close to the allowable, so it is not possible to improve the internal film coefficient. For the transition regime, the correlation is ⎛ Re 0.66 − 125 ⎞ 0.66 −0.66 hi = 0.116cρv ⎜ ⎟ [1 + Di /L ) ]Pr Re ⎝ ⎠ −3 ⎞ ⎛ ⎛ 4, 5110.66 − 125 ⎞ 0.66 ⎤ 2,452 × 5.6e ⎡ 1 (0.0193/7) = 0.116 × 2,452 × 770 × 1.7 ⎜ + ⎟ ⎟⎣ ⎦ ⎜⎝ 0.14 4,511 ⎠ ⎝ ⎠

−0.66

= 534 W/(m 2 ⋅ K) hio = hi × Di /Do = 534 × 0.0193/0.0254 = 405 W/(m2 · K) Calculations for the air side: The coefficient h'fo can be obtained directly from Fig. 8-14 (because the tube pattern coincides with the parameters of the graph) with a face velocity of 3.3 m/s. However, in order to illustrate how to proceed in a general case, we shall use the complete procedure. Ao = plain tube area per meter of tube = πDo= 3.14 × 0.0254 = 0.0798 m2/m AD = bare tube surface per meter of tube = Ao(1 − bNm) = 0.0798(1 − 0.000381 × 394) = 0.0678 m2/m Af = fin surface = π.2Nm(Db2 − Do2)/4 = 3.14 × 2 × 394 × (0.057152 − 0.02542)/4 = 1.621 m2/m Projected perimeter = 2(Db − Do)Nm + 2(1 − bNm) = 2(0.05715 − 0.0254) × 394 + 2(1 − 0.000381 × 394) = 26.718 m Equivalent diameter = 2(Af + AD)/π(projected perimeter) = 2 × (1.621 + 0.0678)/(π × 26.718) = 0.0403 m aS = WL − ntf L[Do + Nm(Db − Do)b] = 3.94 × 7 − 65.5 × 7[0.0254 + 394 × (0.05715 − 0.0254) × 0.000381] = 13.748 m2 ReS (heat) = DeWc / aSμ = 0.0403 × 105.5/(13.748 × 0.0209 × 10−3) = 14,784 j = 0.0959ReS0.718 = 0.0959 × 147840.718 = 94.57 Pr = ccμ/k = 1,011 × 0.0209 × 10−3/0.0267 = 0.79

FINNED TUBES

249

Since a fouling factor on the air side is not considered, hf = h'f = j(k/De)Pr1/3 = 94.57 × (0.0267/0.0403) × 0.791/3 = 58.1 W/(m2 · K) Calculation of the fin efficiency: m = (Db − Do)[h'f /(2kf b)]1/2 kf = aluminum thermal conductivity = 200 W/(m · K) m = (0.05715 − 0.0254)[58.1/(2 × 200 × 0.000381)]1/2 = 0.62 The fin efficiency can be calculated with Fig. 8-10 or with the analytical expressions of Fig. 8-11. Thus m= H=

2 × 58.1 = 39.05 200 × 0.000381

0.05715 − 0.0254 = 0.01587 2

0.000381 ⎞ ⎛ 0.05715 ⎞ ⎛ Y = ⎜ 0.01587 + ⎟ = 0.0206 ⎟⎠ ⎜⎝ 1 + 0.35 ln ⎝ 2 0.0254 ⎠ Ω=

tanh mY = 0.83 mY

External coefficient: h'fo = h'f(AD + ΩAf )/Ao = 58.1(0.0678 + 0.83 × 1.621)/0.0798 = 1,028 W/(m2 · K) Overall coefficient: U = (1/h'fo + 1/hio + Rfi)–1 The internal fouling resistance is corrected for the internal/external diameters ratio. Then Rfi = 0.0003 × (0.0254/0.0193) = 0.00039 (m2 · K)/W Then U = (1/405 + 1/1,028 + 0.00039)–1 = 261 W/(m2 · K) This reasonably agrees with the assumed 250 W/(m2 · K), indicating a slight area excess. It is important to note that the controlling resistance is that of the internal fluid, and it was calculated using the correlation for transition regime. Considering a safety margin in the heat transfer area may be a good idea in these cases. Calculation of the fan power: We must calculate the pressure drop of the air when flowing past the tubes. To calculate the equivalent diameter for friction, we must first calculate the net free volume of the bundle with Eq. (8-2-20). For this geometry, SF = 52.23 mm. Thus Net free volume = WLSF − ntf πLDo2/4 − Nm ntf Lπb(Db2 − Do2)/4 = 3.93 × 7 × 0.05223 − 65.5 × 3.14 × 7 × 0.02542/4 − 394 × 65.5 × 7 × 3.14 × 0.000381(0.057152 − 0.02542)/4 = 1.066 m3

250

CHAPTER EIGHT

Equivalent diameter = D'e = 4 × net free volume/[Lntf (Af + Ao)] = 4 × 1.066/[7 × 65.5(1.621 + 0.0798)] = 0.0055 m Re'S = D'eWc/aSμ = 0.0055 × 105.5/(13.748 × 0.0209 × 10–3) = 2022 f = 1.276Re'S0.14 = 0.440 Pressure drop through the bundle: Mean air density in the tube bundle (at an average temperature of 35.7°C) = 1.146 kg/m3. Thus Δp =

fGS2 n f SF ⎛ D 'e ⎞ 2ρD 'e ⎜⎝ ST ⎟⎠

= 0.44 ×

0.4

⎛ SL ⎞ ⎜⎝ S ⎟⎠

0.6

=

T

(105.5 /13.748)2 × 4 × 0.05223 ⎛ 0.0055 ⎞ 0.4 ⎛ 0.06032 ⎞ 0.6 = 164.7 Pa 2 × 1.146 × 0.0055

⎜⎝ ⎟ 0.06032 ⎠

⎜⎝ ⎟ 0.06032 ⎠

This is 164.7/4 = 41 Pa per row. We could have obtained this value from Fig. 8-13 for a face velocity of 3.3 m/s. The pressure that must be developed by the fans is the sum of frictional pressure drop and velocity pressure: Pressure developed = Δ p + 1/2ρfanvfan2 If forced draft is used, the density of the air at the fan corresponds to the inlet temperature: ρfan = Mp/RT = 29 × 101,300/8,306 × 303 = 1.167 kg/m3 If two fans of 2.5 m diameter are used, the air velocity at the fan is vfan = Wc /2ρfan(1/4π × 2.52) = 105.5/(2 × 1.167 × 4.906) = 9.21 m/s The pressure developed by the fans thus will be 164.7 + 1/2 × 1.167 × 9.212 = 214.2 Pa The volumetric flow in the fans will be Wc /(2ρfan) = 105.5/(2 × 1.167) = 45.2 m3/s. The driver power will be Volumetric flow × pressure developed/(fan efficiency × driver efficiency) = 45.2 × 214.2/(0.75 × 0.95) = 13,600 W = 13.6 kW The ratio between the projected area of the fan and the face area of the bay should be at a minimum 40 percent to have a uniform distribution. Projected area of the fans: 2 × 4.906 = 9.81 m2. Face area = 21.1 m2. 9.81/21.1 = 0.46 > 0.40

OK

Temperature Control of the Process Side. If the temperature difference between the process side and air is small, the changes in air temperature may have a pronounced effect on the unit capacity. Usually, the air design temperature is taken as the maximum possible temperature for the meteorologic conditions of the site. This means that on cold days, the capacity of the unit will be higher than design. In most cases, this is not an issue because the higher the heat removed, the better it is for the process. However, in air coolers

FINNED TUBES

251

operating as distillation columns over condensers, temperature fluctuations may lead to undesirable pressure fluctuations. In other cases it may be desired to keep the temperature within a narrow range for other process reasons. The more common systems to control the process temperatures include2 1. Adjust the blade pitch. This changes the characteristic curve of the fan and makes it possible to change the airflow without changing the fan speed. The required blade angle decreases as the air temperature drops, and this conserves fan power. Different systems can be used to modify the blade angle. The simpler systems require stopping the fan and making the change manually. This system is used for seasonal changes (winter/summer) There exist hydraulic systems that make it possible to change the blade angle with the fan in operation, and these can be used for automatic temperature control. 2. Variable-speed drivers. Usually electronic speed controllers that allow changing the motor rpms are used. 3. Adjustable louvers. Placing these at the air inlet permits control of airflow. These again can be automatic or manual. Even though they can control the airflow, louvers do not reduce fan power requirements. 4. A simple control in units with more than one fan can be obtained by switching off one or more of the fans in winter.

GLOSSARY A = heat transfer area (m2) Ao = plain tube area per unit length (=πDo) (m2/m) AD = bare area of the tube per unit length (not covered by fins) (m2/m) Af = area of fins per unit length of tube (m2/m) aS = airflow area (m2) aF = face area (m2) Ai = internal area of the tube per unit length (m2/m) b = fin thickness (m or mm) Di = internal diameter of the tube (m) Do = external diameter of the tube (m) Db = fin diameter (m) De = equivalent diameter for heat transfer (m) D'e = equivalent diameter for friction (m) Dio = internal diameter of the external tube f = friction factor Ft = LMTD correction factor G = mass flow density [kg/(s · m2)] GS = mass flow density of air based on area aS [kg/(s · m2)] h = convection film coefficient [W/(m2 · K)] hf = convection film coefficient for tube and fins [W/(K · m2) of fin and tube area] h'f = convection film coefficient for tube and fins corrected by fouling [W/(K · m2) of fin and tube area] h'fo = convection film coefficient for tubes and fins corrected by fouling and by fin efficiency, referred to the plain area of tube [W/(K · m2) of plain tube surface] hi = internal film coefficient [W/(m2 · K)]

252

CHAPTER EIGHT

hio = internal coefficient referred to the external area [W/(m2 · K)] H = height of longitudinal fin j = dimensionless number defined by Eq. (8-2-13) k = thermal conductivity [W/(m · K)] kf = thermal conductivity of the fin [W/(m · K)] L = tube length (m) Lp = nf × SF (m) M = molecular weight M = parameter defined in Fig. 8-10 m = parameter defined in Eq. (8-1-29) N = number of tubes or number of tubes per shell n = number of tube passes Nm = number of transverse fins per meter of tube nf = number of tube rows ntf = number of tubes per row Nf = number of longitudinal fins of the tube Q = heat flow (W) Qf = heat flow on the fin surface (W) Qc = conduction heat flow through the section of the fin (W) R = universal gas constant [8,306 (Pa · m3)/(K · kmol)] R = dimensionless number to obtain Ft Re = Reynolds number ReS = air Reynolds number for heat transfer Re'S = air Reynolds number for friction Rfi = iternal fouling resistance [(m2 · K)/W] Rfo = external fouling resistance [(m2 · K)/W] S = dimensionless number to obtain Ft SL = geometric parameter (see Fig. 8-8) ST = geometric parameter (see Fig. 8-8) SF = geometric parameter (see Fig. 8-8) T = hot-fluid temperature t = cold-fluid temperature Tb = temperature at the fin base (K) Tf = fin temperature U = overall heat transfer coefficient [W/(m2 · K)] v = velocity (m/s) vF = face velocity (m/s) vfan = air velocity at the fan (m/s) W = mass flow (kg/s) Z = dimensionless parameter (Z = RS) ρ = density (kg/m3)

FINNED TUBES

253

ρfan = air density at fan (kg/m3) θ = temperature difference between the external fluid and the fin surface (K) μ = viscosity [kg/(m · s)] Ω = fin efficiency Subscripts A = air 1 = inlet 2 = outlet c = cold h = hot

REFERENCES 1. Kern D.: Process Heat Transfer. New York: McGraw-Hill, 1954. 2. Ludwig E.: Applied Process Design for Chemical and Petrochemical Plants. Gulf Publishing Company, Houston, Texas. 3. Engineering Data Book, 11th ed. Gas Processors Suppliers Association, Tulsa Oklahoma, 1998. 4. Weierman C.: “Correlations Ease the Selection of Finned Tubes”. Oil and Gas Journal 74, 36, pp 97–100, 1976. 5. Pignotti A., Cordero G.: “Mean Temperature Difference in Multipass Crossflow” and “Mean Temperature Difference Charts for Air Coolers” Journal of Heat Transfer 105: pp 584–597, 1983. 6. Brown R., Ganapathy V.: “Design of Air Cooled Exchangers” Chem. Eng March 27, 1978, pp 107–124. 7. A.P.I Standard 661- Air Cooled Heat Exchangers for General Refinery Service, 6th ed. Feb 2006.

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CHAPTER 9

PLATE HEAT EXCHANGERS

9-1 OPERATING PRINCIPLES AND GENERAL DESCRIPTION A thin planar metallic plate separating two circulating fluids at different temperatures can act as a heat transfer surface, as indicated in Fig. 9-1. To complete the device, two more plates must be added at both sides to form the channels where the fluids circulate. The plates are kept separated by means of elastomeric gaskets, as shown in Fig. 9-2. The assembly is clamped together in a frame by clamping bolts that must exert the necessary pressure to maintain the plates in position. A schematic drawing of a plate heat exchanger is shown in Fig. 9-3. The gaskets are such that the separation between plates is only a few millimeters to achieve high fluid velocities and hence high heat transfer coefficients. This assembly is the basis of an elemental plate heat exchanger. The main objection to this type of device is that it has important limitations regarding the maximum flow rates that it can handle, and it would not be practical to indefinitely increase the size of the plates when higher flow rates and heat duty are required. In this case it is necessary to increase the number of plates, making both fluids circulate through alternate channels separated by the heat transfer plates, as shown in Fig. 9-4. This is basically the arrangement of a plate heat exchanger. Basically, a plate heat exchanger consists of a pack of corrugated metal plates with portholes for the passage of the two fluids. Heat transfer takes place through the plates. The plate pack is assembled between a fixed frame plate and a movable pressure plate and is compressed by tightening bolts (Fig. 9-5). The plates are fitted with gaskets that seal the interplate channels. The number of plates is defined by the heat transfer requirements. The plates and pressure plate are suspended by an upper carrying bar and located by a lower guiding bar, both of which are fixed to a support column. The plates are corrugated. The corrugations provide reinforcement as well as a large number of plateto-plate contact points. These contact points provide full plate support and make it possible to have high operating pressures with very thin-gauge plates. The usual gap between consecutive plates is about 5 mm. The plates form the channels where the fluids circulate. The fluids enter and exit these channels through portholes located at the corners of the plates. These portholes may be perforated or not, as required by the desired circulation scheme. The portholes form the distribution headers, which distribute the fluids into the circulation channels. The fluids enter the unit through inlet nozzles perforated in the frames and pass into the distribution headers. From the headers, the fluids distribute into the circulation channels following a specific flow pattern defined during the design stage. This is achieved by the positioning of elastomeric gaskets to blank off certain flowpaths and by blinding specific portholes of individual heat exchange plates. A suitable design of gaskets and portholes allows different flow configurations. Some models of heat exchange plates are shown in Fig. 9-14. Coming back to Fig. 9-4, we see that this unit has nine plates. Figure 9-6 is an expanded view showing the configuration of plates. Plates 1 and 9 are closure plates (they do not act as heat transfer surfaces), and plates 2 to 8 are the heat transfer plates. It can be seen that each branch in which the cold stream is divided exchanges heat through the plates forming its channel with two branches of the hot stream, and vice versa, except for the outermost branches, which only exchange through one plate. In this case, the plates are arranged for diagonal flow because the inlet and outlet ports to each channel are located at diagonally opposed corners. In some models, the flow pattern can be parallel, as shown in Fig. 9-13a, c, and d and in Fig. 9-15.

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Planar plate Cold fluid

Hot fluid

Closure plates Heat exchange plate Hot fluid

FIGURE 9-1

Heat transfer through a plate.

Cold fluid

Gaskets FIGURE 9-2

Principle of the plate heat exchanger.

Heat exchange plate

Compression bolts

Closure plates FIGURE 9-3

Compression frame

Elemental plate heat exchanger.

9

8

7

6

5

4

3

2

1

Cold fluid

Hot fluid FIGURE 9-4

Fluids circulation in a plate heat exchanger.

PLATE HEAT EXCHANGERS

2

257

4

4

6

1 6

5

2 5 1 7

3

3

1-Fixed frame plate 3-Tightening bolts 5-Lower guiding bar 7-Plates FIGURE 9-5

7

2-Movable pressure plate 4-Upper carrying bar 6-Support column

Component parts of a plate heat exchanger.

Headers

Cold fluid

9

8

7

6

5

4

3

2

1

Hot fluid Channels FIGURE 9-6

Expanded view of the heat exchanger of Fig. 9-4.

The gasket design of the different plates makes the fluid circulate according the desired flow configuration. For example, the hot fluid enters the exchanger through the lower-right porthole of plate 1. The gasket design of plate 2 prevents the fluid from entering the channel formed by plates 1 and 2, so it must cross plate 2. The cold fluid, which enters the exchanger through the upper-right porthole of plate 9 (always looking at the exchanger from plate 1), when it reaches and crosses plate 2, can get into the channel formed by plates 1 and 2 and circulates through this channel, exiting through the lower-left porthole of plate 2 (Fig. 9-7). The complete diagram of plates for this exchanger is shown in Fig. 9-8. Figure 9-9 is a schematic representation of the exchanger configuration. The exchanger must always be looked at from plate 1. When the fluid streams cross a plate, it is because the porthole at the corresponding position is perforated. For example, in the heat exchanger in Fig. 9-9, we see that plate 1 is only crossed by the hot fluid at the lower-right and upper-left positions, so the plate is perforated in those corners. This type of plate heat exchanger is referred to by some vendors as a gasketed plate heat exchanger because there are other types of plate heat exchanger that are all-welded (without gaskets) as we shall see later on in the text.

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Plate 2

The gasket prevents the hot fluid from entering the channel formed by plates 1 and 2. FIGURE 9-7 circulation.

9

8

FIGURE 9-8

9

7

The gasket design defines the flow

6

5

4

3

2

1

Gasket configuration.

8

7

6

5

4

3

2

1 Upper right Upper left Observation direction Lower left Lower right

FIGURE 9-9

9-2

Schematic representation.

SERIES-PARALLEL COMBINATION In the exchanger we have just described, all the channels where fluid circulates are in parallel. However, sometimes it is necessary, in order to achieve higher velocities, to arrange plate heat exchangers with more than one pass, the same as in shell-and-tube heat exchangers. For example, let’s assume that the hot fluid must perform two passes, whereas the cold fluid requires only one. The configuration of plates can be as indicated in Fig. 9-10. The hot fluid performs two passes, and each pass is made up of two channels in parallel. The cold fluid path, in turn, consists of one pass with four channels in parallel. The plates of this exchanger can be built easily

PLATE HEAT EXCHANGERS

9

8

7

6

5

4

3

2

259

1 Upper left Upper right Observation direction Lower right Lower left

FIGURE 9-10

Schematic representation of a heat exchanger with two passes in one of the streams.

from the circulation diagram in Fig. 9-10,where you can see in which position each plate is crossed by a fluid stream. Then, looking at Fig. 9-10, we can deduce the perforation diagram of the plates. It is as follows: Plate 1: Only one perforation at the lower left Plate 2: Both lower perforations and one perforation at the upper left Plate 3: Perforated at the four positions Plate 4: Same as plate 3 Plate 5: One perforation at the lower right and both upper perforations Plate 6: Same as plate 5 Plate 7: Must be perforated at all four positions Plate 8: Same as plate 7 Plate 9: Both lower perforations and one perforation at the upper left The gaskets, diagram can be deduced as follows: The channel formed by plates 1 and 2 must allow the communication of the lower-right and upper-left headers. This effect is achieved by the design of the gasket of plate 2, which must have these perforations communicated. The channel formed by plates 2 and 3 must allow communication of the lower-left and upper-right headers, and at the same time, it must prevent the fluid from the other headers from entering into this channel. This effect is achieved with the design of the gasket of plate 3. The complete diagram of plates and gaskets is shown in Fig. 9-11. Another representation of the fluid circulation in this exchanger can be seen in Fig. 9-12. In this way, it is possible to arrange the plates so as to achieve any desired circulation pattern and modify the fluid

Perforation diagram

9

8

7

6

5

4

3

2

1

3

2

1

Gasket diagram

9 FIGURE 9-11

8

7

6

5

4

Diagram of the plates and gaskets of the heat exchanger of Fig. 9-10.

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9 8 7 6 5 4 3 2 1

FIGURE 9-12

Expanded representation of the heat exchanger of Fig. 9-10.

velocities as dictated by the heat transfer requirements. Obviously, the lower the number of channels in parallel for each fluid, the higher is its velocity, and the higher is the heat transfer coefficient, a counterpart being an increase in the pressure drop.

9-3 COMPONENTS OF A GASKETED PLATE HEAT EXCHANGER 9-3-1 Heat Exchanger Plates Heat exchanger plates are corrugated. The shape of the corrugations is a characteristic of each plate model and is carefully studied by the manufacturers. The purpose of the corrugations is to provide turbulence in order to increase the heat transfer coefficients and, at the same time, to increase the structural strength of the assembly. Each plate must be supported mechanically to resist the pressure difference of the fluids on both sides of the plate. The most severe case is when one of the fluids is at its maximum pressure and the other one is at zero pressure. This situation may occur during startup or when the unit is being emptied for cleaning. It is necessary to avoid plate deformations owing to these pressure effects. The plate’s corrugations provide many contact points between plates, thus preventing deformations. There are many models of corrugated plates. In the “washboard” type, corrugations are parallel among themselves and perpendicular to the flow direction in the channel. This type of corrugation is shown in Figs. 9-13a and 9-14a. The most common corrugation pattern is the herringbone pattern, which is shown in Figs. 9-13b and 9-14b. In this case, the corrugations of consecutive plates are rotated 180 degrees, so between two plates there are a large number of contact points where the corrugations cross. This allows very rigid plate support and a high degree of turbulence in the fluids. Other models, not used very often, are shown in Fig. 9-13c and d. Each manufacturer has several types of plate for different applications. Selection of the plates for any application depends on the process requirements in terms of heat transfer coefficients and allowable heat exchanger pressure drop. The models that allow higher heat transfer coefficients for a given flow rate also produce higher pressure drops. Thus, for any application, the designer must choose the plate type offering the best balance of both effects.

PLATE HEAT EXCHANGERS

(a) FIGURE 9-13

(b)

(c)

261

(d)

Models of plates.

(a)

(b) FIGURE 9-14 Different corrugations: (a) plates with parallel corrugations, (b) plates with transversal corrugations.

The plates are manufactured by cold forming in hydraulic presses. Each manufacturer owns the matrixes corresponding to its standard models and usually has a certain stock of nonperforated plates. When a heat exchanger is purchased, the manufacturer designs the fluids circulation pattern, and then the plates are perforated in the positions required by the design. Thus plate heat exchangers must be considered to be semistandard units, and since the perforation and assembly are a rather fast operation, the delivery time is shorter than with other types of heat exchangers. The effective heat transfer area of a plate can be as small as 0.033 m2 in laboratory models up to 3 m2 in larger models. The maximum number of plates that can be assembled in a unit is limited by the compression strength that can be achieved with the bolts and maldistribution of flow among the flow channels, but a maximum of 500 plates is possible in certain cases. Construction Materials. Plates can be made from different materials, such as stainless steel, copper and its alloys, aluminum, titanium, nickel or nickel-molybdenum alloys, and others. Plate thickness is between 0.5 and 1.2 mm. Since the plates are so thin, it is necessary to use corrosion-resistant materials. This is why carbon steel is seldom employed in plate manufacturing, and stainless steel is used as a minimum quality. The plate material must be suitable for cold forming. Certain chrome, zirconium, and titanium alloys are difficult to cold form and are not used in plate heat exchanger manufacture. Table 9-1 indicates how easily the plates can be cold formed using different materials.

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TABLE 9-1

Cold-Forming Characteristics of Various Materials

Material

Type

Common steel Stainless steel (Mo < 4%) Stainless Steel (Mo > 4%) Titanium Copper Copper-nickel (70/30 or 90/10) Bronze Monel Incolloy 825 Nickel Inconel Hastelloy B Hastelloy C Aluminum Zirconium Tantalium

a a b b b aob a b b a b c b b d a

Key: a = all types can be cold formed; b = some types can be cold formed; c = very difficult to cold form; d = impossible to cold form.

9-3-2

Gaskets For plate gaskets, elastomeric materials are employed, such as natural rubber, styrene butadiene rubber (SBR), nitrile rubber (acrylonitrile butadiene), butyl rubber (a copolymer of isobutylene and a small amount of isoprene), silicones, or other elastomers such as neoprene, hypalon, or viton.7 Sometimes, compressed fibers are used for high-temperature applications. Plastic materials of the polytetrafluoroethylene (PTFE) type (e.g., Teflon or Fluon) are not suitable for gasket manufacture because they have poor elastic recovery behavior. The selection of a suitable gasket material is very important in plate heat exchanger design because the gaskets are the limiting factor for the maximum operating temperature of the exchanger. Table 9-2 is a guide showing the maximum temperatures at which the different materials can be used. For higher temperatures the vendors must be consulted.

TABLE 9-2

Maximum Service Temperature for Gaskets Material

Max. Temperature

Natural rubber, styrene, neoprene Nitrile rubber, viton Butyl rubber Silicone

70°C 100°C 120°C 140°C

Regarding chemical resistance, SBR is suitable for general applications in aqueous systems at low temperatures. Nitrile rubber combines the general-purpose characteristics of SBR with a good resistance to fats and aliphatic hydrocarbons. Butyl rubber offers excellent chemical resistance to many chemical agents, such as acids, alkalys, and some ketones and amines, but is not very resistant to fats. Silicone is used in a limited number of applications but is the most suitable material for sodium hypochlorite and some

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263

low-temperature applications. Fluoroelastomers such as viton are more expensive but are the most suitable for high-temperature oils (140°C). With a proper formulation, they can be used for 98% sulfuric acid at temperatures up to 100°C. The cross sections of gaskets depend on the plate design. They can be rectangular, trapezoidal, or oval. The width normally ranges from 5–15 mm. Gasket height before compression is 15–50 percent higher than the compressed height. Gaskets are installed into grooves formed in each plate and glued with special cements. The cement must provide a good union during service and opening of the heat exchanger but must be removed easily when the gasket has to be replaced.

9-3-3 Frame The frame is formed by two robust plates (frame plates) at both ends of the unit. These plates must absorb the force resulting from the fluid pressure. For example, a 5-bar pressure acting on a 1-m2 surface exerts a force of 500,000 N (~50 tons). The force that must be applied on the frame plates to keep the whole assembly tight is also important. This force is absorbed by the tightening bolts. Usually, one of the frame plates contains the inlet and outlet nozzles and is fixed to the system piping. The other frame plate, also called the movable pressure plate, can be slid back for inspection and cleaning when the unit is unbolted (see Fig. 9-17). With this configuration, it is not necessary to remove the piping components for maintenance. However, this construction is possible only when both fluids perform a single pass through the exchanger, as in Fig. 9-15. In a multipass exchanger, it is unavoidable to have piping connections in both frame plates. Frames are usually free-standing or, for smaller units, attached to structural steel work. Figure 9-16 shows some frame models.

FIGURE 9-15 When both fluids make a single pass, all inlet and outlet nozzles are in the fixed frame plate.

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FIGURE 9-16 Types of frames.

9-4 USE AND LIMITATIONS OF GASKETED PLATE HEAT EXCHANGERS 9-4-1 General Characteristics The gasketed plate heat exchanger is suitable for services where the process conditions are moderate. It was conceived initially in the 1930s to meet the hygienic demands of the dairy industry. The exchanger can be completely disassembled for cleaning, as shown in Fig. 9-17, which is a primary requirement in the food industry. Another characteristic is that the heat transfer area can be enlarged, if necessary, by adding more plates. The only requirement is to oversize the tightening bolts in length and diameter initially so that they can contain a higher number of plates and supply the required closure force. Maintenance is very simple, and cleaning is effected by brushing or with a pressure water hose. Chemical cleaning is also very effective owing to the high turbulence that can be achieved in the circulating medium. Gasket replacement is also very simple, and if a plate is corroded, it can be changed easily. Another advantage, mentioned previously, is that delivery time is shorter than for other types of heat exchangers because they are semistandard units. The heat transfer coefficients of plate heat exchangers are usually much higher than those of shell-andtube heat exchangers, and the heat transfer area is smaller. Additionally, since they are compact in nature, they require a smaller footprint than shell-and-tube units, especially considering the additional space required for bundle removal. Owing to their compactness, these heat exchangers found an important application in the marine industry in the refrigeration circuits of ship diesel motors, where seawater is used to cool down the primary water, which, in turn, cools the motor cylinders.

9-4-2

Pressure and Temperature Limitations The maximum operating temperatures are related to the resistance of the gaskets. We have already treated this subject. The maximum operating pressures are limited by the ability to achieve a tight closure without gasket leakage. This depends on the size of the plates, the frame type, and the number of plates, so it is not possible to give general rules. In some cases, it is possible to operate at pressures as high as 30 bar. However, this type of equipment is seldom used for pressures higher than 10 bar.

PLATE HEAT EXCHANGERS

FIGURE 9-17

265

Cleaning of a plate heat exchanger.

Regarding capacity, some vendors offer plates capable of operating with flow rates up to 3,600 m3/h. The maximum flow rates are limited by the size of the plate portholes that form the distribution headers.

9-5 WELDED AND SEMIWELDED PLATE HEAT EXCHANGERS Plate heat exchangers also can be supplied in a semiwelded construction. The plates are welded in pairs, and each pair is supplied with gaskets that allow coupling with other similar pairs. The resulting assembly allows maintenance and cleaning of the channels of one fluid, whereas the channels of the other fluid are hermetically welded. This type of construction is employed in cases of clean fluids where leakage cannot be tolerated (e.g., ammonia or caustic soda), whereas the other fluid is innocuous and requires cleaning (e.g., cooling water). It is also possible to construct plate heat exchangers that are 100 percent welded. In this case, cleaning is not possible in any of the sides, but a very economical gasketless design results. Welding is performed by brazing, and this type of heat exchangers is called a brazed plate heat exchanger.

9-6 PLATE HEAT EXCHANGERS VERSUS SHELL-AND-TUBE HEAT EXCHANGERS The main objection to the gasketed plate heat exchanger is the possibility of leakage. This usually disqualifies this type of equipment in critical process services and is the reason why it is limited in acceptance in the refinery and petrochemical industry.

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Plate heat exchangers are used for moderate pressures and temperatures and innocuous fluids. These are the conditions normally found in the food and beverage industry, and the additional cleaning facility explains the wide acceptance of the plate heat exchanger in such applications. In the food industry, it is also frequent that heating operations must be carried out without submitting the products to high temperatures or excessive heating time to avoid decomposition or alterations of flavor. The plate heat exchanger is very suitable in these cases because the high heat transfer coefficients allow working with lower wall temperatures and lower residence time. When the decision is a matter of cost, there are several factors that may favor one or the other type of exchanger. One the most important factors are the materials of construction. Plate heat exchangers are manufactured with stainless steel as the minimum-quality material. Therefore, when corrosion is not an issue, the shell-and-tube heat exchanger may have some advantage because it can be built with cheaper materials. On the other hand, if high-cost corrosion-resistant materials are necessary, the plate heat exchanger is usually favored because it is a lighter unit.5

9-7 TYPICAL HEAT TRANSFER COEFFICIENTS Some typical values of heat transfer coefficients for plate heat exchangers are listed in Table 9-3. Values of specific pressure drop are included as well.4 This concept will be explained later. TABLE 9-3 Typical Overall Heat Transfer Coefficients Overall Heat Transfer Coefficient

Physical Properties of Product η = Kinematic Viscosity (m2/s)

Transversel Corrugation Plate

Herringbone Corrugation Plate

ρc = Heat Capacity per Unit Volume [J/(m3 · K)] Service Water to water or steam

Viscous aqueous solution to water or steam Mineral oil to water or steam Mineral oil to mineral oil Mineral oil to water or steam Organic solvent to water or steam Vegetable oil to water or steam

K = Thermal Conductivity [J/(s · m · K)] −6

η = 0.6 × 10 ρc = 4.18 × 106 K = 0.62 η = 50 × 10−6 ρc = 3.97 × 106 K = 0.39 η = 50 × 10−6 ρc = 2.09 × 106 K = 0.11 η = 50 × 10−6 ρc = 2.09 × 106 K = 0.11 η = 100 × 10−6 ρc = 2.09 × 106 K = 0.11 η = 1 × 10−6 ρc = 2.09 × 106 K = 0.19 η = 100 × 10−6 ρc = 2.09 × 106 K = 0.15

Specific Pressure Drop

W/(m · K)

W/(m · K)

m of Liquid Column per Transfer Unit

3,100–3,900

3,000–3,700

1.5–3

1,000–1,200

700–800

11–20

450–580

300–350

15–30

210–270

120–190

18–40

330–400

190–290

22–51

1,850–2,100

1,500–1,950

870–1,000

810–930

2.5–3

7–10

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267

9-8 HEAT TRANSFER AND PRESSURE-DROP CORRELATIONS FOR PLATE HEAT EXCHANGERS Both heat transfer rate and pressure drop are highly influenced by the plate geometry and the corrugation design. Since each plate model is exclusive from a specific manufacturer, the thermal design of this type of equipment is always performed by the vendor, which has the correlations applicable to his plates. So these heat exchangers are usually purchased on the basis of a process specification, where flow rates, inlet and outlet temperatures, fouling, and allowable pressure drops are specified by the purchaser, and the vendor selects a unit capable to satisfy these operating conditions. Nevertheless, we shall present some general correlations that may be useful to analyze changes in the operating conditions and to perform preliminary sizing. Reynolds Number. An equivalent diameter can be defined for the channel as De =

4 × flow area 4 be 2e = ≅ wetted perimeter 2(e + bφa ) φa

(9-8-1)

where b = channel width e = channel depth The channel depth e equals the thickness of the corrugated plate minus the thickness of the metal sheet. Since the plates are in touch with each other, the thickness of the corrugated plates s can be obtained by dividing the length of the plate pack L (which is always shown in the vendor drawings) by the number of plates (see Figure 9-18). φa is a multiplication factor representing the enhancement of the heat transfer area owing to the corrugations. Is a quotient between the actual area of a plate and its planar projected area. It is usually between 1.1 and 1.25. This factor should be supplied by the vendor (see Example 9-1). t e

s

e = channel depth s = thickness of the corrugated plate t = thickness of the metal sheet L s= number of plates FIGURE 9-18

L

Plate nomenclature.

A Reynolds number then can be defined as Re =

De vρ 2evρ = μ μφa

(9-8-2)

W ρn p be

(9-8-3)

The velocity v is defined as v= where W = mass flow (kg/s) ρ = density (kg/m3) np = number of channels in parallel

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Typical velocities in plate heat exchangers for aqueous fluids range from 0.3–0.9 m/s, but local velocities can be considerably higher owing to the effect of corrugations. Film Coefficients. For a plate heat exchanger, the usual dimensionless numbers are defined: hDe k 0.14 Nu ⎛ μ ⎞ Colburn factor = J H = Re ⋅ Pr 0.33 ⎜⎝ μw ⎟⎠ Nusselt number = Nu =

(9-8-4) (9-8-5)

The experimental results are usually expressed with correlations of the type JH = f (Re), where the correlation parameters depend on the plate geometry. An important factor in the correlations is the corrugation angle. Figure 9-19 shows a graph of JH versus Re for herringbone corrugations at given parameters of the corrugation angle. This is a general graph that may change depending on the plate manufacturer. 1 β = 30º β = 45º β β = 60º

0.1 JH

268

0.01 0.001 1

10

100

1000

10000

100000

Re FIGURE 9-19

Colburn JH factor for plate heat exchangers with herringbone corrugations.

Pressure Drop. The pressure drop through the channels of a plate heat exchanger can be expressed as a function of a friction factor in the usual way: Δ pplates =

4 fLe ns ρv 2 2 De

(9-8-6)

where ns is the number of channels in series and Le is the effective length of the channel, which is the diagonal distance between inlet and outlet portholes. The friction factor is correlated as a function of the Reynolds number. Figure 9-20 is a graph of f versus Re for herringbone corrugations at given parameters of the corrugations angle. This is a general graph, and specific plate models may present deviations with respect to it. The pressure drop in the unit is the sum of the pressure drop in the channels and the pressure drop in the distribution headers. The pressure drop in the distribution headers depends on the flow configuration.3 By knowing the pass configuration, it is possible to estimate this additional pressure drop. As an estimation, we can consider 1.5 velocity heads for each pass, that is, Δ pheaders = where Gc =

1.5ns Gc2 2ρ

4W πDc2

(9-8-7)

(9-8-8)

Here, Dc is the diameter of the distribution header. Then Δ pexchanger = Δ pplates + Δ pheaders

(9-8-9)

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269

100 β = 30º β = 45º β = 60º

f

10 1

β

0.1 0.01

1

10

100

1000

10000

100000

Re FIGURE 9-20 Friction factor for a plate heat exchanger with herringbone corrugations.

If the pressure drop in the distribution header is significant, it may result in an uneven flow distribution among the channels of the same pass. This uneven distribution may affect the thermal behavior. The most advanced computation programs take this effect in consideration and calculate the flow distribution among the channels.

9-9

LMTD CORRECTION FACTOR The heat transfer area is calculated as in other types of exchangers by means of Q = UAΔT

(9-9-1)

1 1 1 = + + Rf U hc hh

(9-9-2)

where

and ΔT = MLDT ⋅ Ft

(9-9-3) The LMTD correction factor Ft takes into consideration the efficiency reduction that results as a consequence of the partial co-current flow when the flow configurations of both streams are different. Experimental results of Ft factors for some flow configurations can be found in ref. 2. Marriot1 presents a graph of Ft as a function of the number of transfer units for different flow configurations (Fig. 9-21). It should be noted that even configurations with the same number of passes in both streams present an Ft factor smaller than 1, which could be explained as the result of border effects and stagnant zones in the channel.

9-10

FOULING RESISTANCES Fouling resistances in plate heat exchangers are usually much smaller than in shell-and-tube heat exchangers. The reasons are pointed out by Marriot:1 1. The high degree of turbulence keeps solids in suspension. 2. Heat transfer surfaces are smooth. For some types, a mirror finish may be available. 3. There are no dead spaces where fluids can stagnate and solids deposit, such as, for example, in the vicinity of baffles in shell-and-tube heat exchangers.

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1.0 LMTD correction factor Ft

270

1/1

Passes hot fluid/cold fluid

0.9

4/4

2/2

0.8

3/3

2/1

0.7

3/1

0.6 0.5

4/1 0

1

2

3

4

5

6

7

8

9

10

11

Number of transfer units NTU FIGURE 9-21 LMTD correction factor for plate heat exchangers. From J. Marriot, Chem. Eng., April 5 1971. Reproduced with permission.)

4. Since the plates are built with higher-quality materials, there are no corrosion products to which fouling may adhere. 5. High film coefficients tend to lead to lower surface temperatures for the cold fluid (the cold fluid is usually the culprit as far as fouling is concerned). 6. Since cleaning is a very simple operation, the interval between cleanings is usually smaller. A good estimation of the fouling resistance is very important in plate heat exchangers. This is due to the usually high heat transfer film coefficients in the circulating fluids, which frequently make fouling the controlling resistance with a direct incidence in the cost of the unit.8 It can be an expensive mistake to specify for a plate heat exchanger the same fouling factors used for shell-and-tube heat exchangers. Table 9-4 indicates some typical fouling resistances for plate heat TABLE 9-4

Fouling Resistances for Plate Heat Exchangers7

Fresh pure water, condensate, etc. Lake, river, or sea water Sewer or brackish water Acetic acid Acetone Benzene Carbon tetrachloride Difenile (Dowtherm, etc.) Ethyl acetate Ethyl eter Ethanol Etylene glycol, 25% weight n-Heptane Methanol Methyl ethyl ketone n-Octane Toluene ClH, 96% weight Sulfuric acid, 30% weight Magnesium chloride brine Beer, milk Wine Sugar solution, 17°Bx

0.2–0.5 × 10−4 0.5–1 × 10−4 1–2 × 10−4 0.5–1 × 10−4 0.5–1 × 10−4 0.5–1 × 10−4 0.5–1 × 10−4 0.5–1 × 10−4 0.5–1 × 10−4 0.5–1 × 10−4 0.5–1 × 10−4 0.5–1 × 10−4 0.5–1 × 10–4 0.5–1 × 10−4 0.5–1 × 10−4 0.5–1 × 10−4 0.5–1 × 10−4 0.5–1 × 10−4 Depends on material Depends on material 0.1–0.3 × 10−4 0.5–1 × 10−4 1.5–2 × 10−4

Note: Values are given in (h · m2)/kcal. To convert to SI, they must be multiplied by 0.86.

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271

exchangers. The table assumes operation at an economical pressure drop (~30 kPa/NTU). The total resistance must be obtained adding the individual resistances corresponding to both sides. Some plate heat exchanger manufacturers prefer to specify a percentage of excess area above that required for clean conditions instead of using fouling resistances. Example 9-1 To cool down 45,000 kg/h of alcohol from 55.4°C to 43°C, a heat exchanger with the configuration indicated in Fig. 9-22 is available. Cooling water at 25°C will be used. The water outlet temperature should not be higher than 40°C. A fouling resistance of 0.0001 (m2 · K)/W will be adopted for each fluid. Verify if the unit is suitable, and calculate the pressure drops for each fluid. Physical properties:

Thermal conductivity, W/(m · K) Viscosity, kg/ms Specific heat, J/(kg · K) Density, kg/m3 Prandtl Number (Pr)

100 mm

Alcohol

Water

0.148 0.65 × 10−3 3400 765 15.4

0.62 0.75 × 10−3 4,200 1,000 5.16

HO

CI

946 mm

HI

CO

360 mm Characteristics of the plate Effective area per plate: 0.319 m2 Channel depth: e = 0.005 m Area increment factor φ = 1.14 Channel width b = 0.36 m Corrugations angle = 45 degrees FIGURE 9-22

Number of plates: 61 (59 heat transfer plates and 2 closure plates) Passes in series per fluid: 3 Number of channels per pass: 10 Area of the unit: 18.87 m2

Illustration for Example 9-1.

Solution

Heat exchanged = Whch(T1 − T2) = 45,000/3,600 × 3,400 × (55.4 − 43) = 5.27 × 105 W Water flow rate = Q/[cc(t2 − t1)] = 5.27 × 105/[4,200(40 − 25)] = 8.36 kg/s = 30,096 kg/h as = flow area per channel = eb = 0.005 × 036 = 0.0018 m2 Equivalent diameter De = 2e/φa = 2 × 0.005/1.14 = 0.00877 m

272

CHAPTER NINE

W1 = mass flow per channel = W/10 (kg/h) Velocity = W1/(ρas × 3,600) Re = Devρ/μ JH (graph) f h = JH(k/De)Re · Pr 0.33

Δ pplates =

4 fns ρv 2 ⎛ Le ⎞ ⎜⎝ D ⎟⎠ (For simplicity, we took Le as 946 mm) 2 e

Flow area of the distribution headers = πDc2/4 m2 Mass velocity in the distribution headers Gc [kg/(s · m2)]

Δ pheaders = Δ ptotal

1.5nsGc2 2ρ

Alcohol

Water

4,500 0.9 9,370 0.012 0.2 4,640

3,009 0.46 5,400 0.0145 0.22 9,600

80,208 N/m2

30,128 N/m2

0.00785 1,592

0.00785 1,061

7,454

2,532

87,662

32,660

The nozzle arrangement of the figure corresponds to a countercurrent configuration. MLDT =

(T1 − t2 ) − (T2 − t1 ) (55.4 − 40) − (43 − 25) = = 16.7°C (T − t ) 15.4 ln ln 1 2 18 (T2 − t1 )

⎛ 1 1 ⎞ U = ⎜ + + Rf1 + Rf2 ⎟ ⎝ hc hh ⎠

−1

1 ⎛ 1 ⎞ =⎜ + + 2 × 0.0001⎟ ⎝ 4, 640 9, 600 ⎠

−1

= 1,900

Q = UA ⋅ MLDT = 1, 900 × 18.87 × 16.7 = 599,000 W The unit is approximately 13 percent oversized.

9-11 NUMBER OF TRANSFER UNITS AND SPECIFIC PRESSURE DROP Both the friction factor and the heat transfer coefficient are functions of the Reynolds number. When different plate types are compared for the same process service, it may happen that one of them has better JH values for the same Reynolds number, which may make it look more attractive. But it may present too high a pressure drop, so it may be necessary to work with lower velocities. This means that the comparison cannot be effected at the same Reynolds number. It is more useful to use other parameters, which are the number of transfer units (NTUs) and the specific pressure drop ε. The number of transfer units was already defined in relation to shell-and-tube heat exchangers. For a plate heat exchanger, the same definition applies:4,6 NTU =

UA (Wc)min

(9-11-1)

The specific pressure drop for each stream is defined as the quotient between pressure drop and the number of transfer units. This is εh =

Δ ph NTU

εc =

Δ pc NTU

where Δph and Δpc are the pressure drops for the hot and cold streams, respectively.

(9-11-2)

PLATE HEAT EXCHANGERS

273

To compare the performance of different plates, the analysis always must be performed with the same fluids. Then, if the physical properties of the fluids are constant, the heat transfer coefficients will be exclusively a function of the velocities and geometry. If the heat exchanger configuration is adopted such that the velocities of both fluids are approximately the same (which is usual in plate heat exchanger design), then, for every geometry, the overall U will be a function of this velocity. This means that U = U(v) where the subscript only denotes the functional dependence. Since the area of the unit can be expressed as A = Apnpns

(9-11-3)

W = ρvnpbe

(9-11-4)

U( v ) Ap ns

(9-11-5)

where Ap is the area of the plate, and

NTU will be NTU =

(ρvc)min be

And then NTU = ϕ( v ) ns

(9-11-6)

for each model of plate. And ε=

4 f (1/ 2)ρv 2 Lns Δp = = ε( v ) NTU ϕ ( v ) ns

(9-11-7)

This means that ε is also a function of the velocity for each model of plate. So it is possible to use ε as a valid parameter to correlate the data instead of using the velocity. Plate heat exchanger vendors can have curves of ε versus v for the different types of plates. These curves can be used to make a preliminary selection of the plate type for a certain service. Since the specific pressure drop can be calculated for each service with only the process conditions [because NTU = Q/(Ft · LMTD)], with these curves, the manufacturer can select, for each type of plate, the fluid velocity necessary to achieve the desired specific pressure drop. This allows defining the number of plates in parallel in each pass by calculating U and selecting the number of passes in series with Eq. (9-11-6).

GLOSSARY A = total heat transfer area (m2) Ap = heat transfer area of one plate (m2) b = channel width (m) c = specific heat [J/(kg · K)] Dc = distribution header diameter (m) De = channel equivalent diameter (m) e = channel depth (m) (see Fig. 9-18)

274

CHAPTER NINE

f = friction factor (dimensionless) Ft = LMTD correction factor (dimensionless) h = film coefficient [J/(s · m2 · K)] JH = Colburn factor (dimensionless) k = thermal conductivity [J/(s · m · K)] Le = effective plate length (m) np = number of parallel channels per pass ns = number of passes s = thickness of the corrugated plate (m) (see Fig. 9-18) t = metal sheet thickness (m) v = velocity (m/s) W = mass flow (kg/s) ρ = density (kg/m3) μ = viscosity (kg/ms) β = corrugations angle (degrees) ε = specific pressure drop (N/m2) φa = area correction factor owing to corrugations (dimensionless) ϕ = number of transfer units per pass

REFERENCES 1. Marriot J.: “Where and How to Use Plate Heat Exchangers,” Chem Eng, May 4, 1971, p. 127. 2. Buonopane R., Troupe R. A., Morgan J. C.: “Heat Transfer Design Method for Plate Heat Exchangers,” Chem Eng Progr 59(7):57–61, 1963. 3. Watson E. L., McKillop A. A., Dunkley W. L., Perry R. L.: “Plate Heat Exchangers: Flow Characteristics,” Ind Eng Chem 52(9):733–739, 1960. 4. Thermal Handbook. Sweden: Alfa Laval. 5. Heat Transfer Handbook: Design and Application of Paraflow Heat Exchangers. New York: APV Company. 6. Usher J.: “Evaluating Plate Heat Exchangers,” in Process Heat Exchange. New York: McGraw-Hill, 1979. 7. Cowan C. T.: “Choosing Materials of Construction for Plate Heat Exchangers,” in Process Heat Exchange. New York: McGraw-Hill, 1979. 8. Cross P. H.: “Preventing Fouling in Plate Heat Exchangers,” in Process Heat Exchange. New York: McGraw-Hill, 1979.

CHAPTER 10

CONDENSATION OF VAPORS

10-1 MECHANISMS OF CONDENSATION Condensation is the process by which a vapor transforms into a liquid. For condensation to take place, it is necessary to remove heat from the condensing fluid by means of a cooling medium. If the vapor is a pure substance, as long as the pressure remains constant, condensation takes place isothermally. The temperature of the process is the saturation temperature of the vapor at the prevailing pressure. In this case, the amount of heat that must be removed per unit mass to achieve condensation of a saturated vapor is called the latent heat of condensation λ. The process can be performed in a continuous fashion in a heat exchanger, which receives the specific name of condenser, as represented in Fig. 10-1. The heat balance can be expressed as Q = Wh (i2 − i1 )

(10-1-1)

For a pure vapor, the enthalpy difference is the latent heat of condensation λ; then Q = Wh λ h = Wc cc (t2 − t1 )

(10-1-2)

We have assumed that the cooling medium is a fluid that exchanges sensible heat. It is also possible to use as refrigerant a fluid that performs the opposite change of phase, removing heat from the condensing fluid and suffering a vaporization process (this is the case with ammonia or propane vaporization in refrigeration cycles). In these cases, the enthalpy balance can be written as Q = Wh λ h = WR λ R

(10-1-3)

where the subscript R denotes the refrigerant. In the foregoing discussion we have assumed that the condensing fluid is a pure substance that comes into the condenser at its saturation temperature. If, instead, the vapor is superheated, it will be necessary to extract from it an additional amount of heat to bring it to the saturation temperature, and the enthalpy balance will be Q = Wh (i2 − i1 ) = Wh ch v (T1 − Ts ) + Wh λ h

(10-1-4)

where chv is the specific heat of the hot fluid at vapor state, Ts is the saturation temperature, and λh is the latent heat of condensation at the saturation temperature. The first term on the right side of Eq. (10-1-4) is called sensible heat (it is associated with a temperature change), whereas the second term is called latent heat. If the vapor to be condensed is not a pure substance but a mixture of different components, the condensation will not be isothermal because there is a temperature range in which vapor and liquid may coexist. This temperature range extends between the dew-point temperature (where the first drop of condensate forms) and the bubble-point temperature (where complete condensation of the mixture is reached). Bubble-point and dew-point temperatures of a mixture depend on its composition and pressure. Figure 10-2 is a condensation temperature versus composition diagram for a binary mixture at a certain pressure. In this diagram, the evolution of a vapor that is cooled down from an initial temperature T1 is represented. When the dew-point temperature TR is reached, condensation starts. From then on, the composition 275

CHAPTER TEN

Saturated vapor Wh T1

t2

Cooling medium Wc t1 FIGURE 10-1

T2 Condensate

Shell-and-tube condenser.

T1

Vapor C

A C′

TR

A′ A″

C″

Temperature

276

TB

C ″′

Liquid

1

0 Mole fraction of the more volatile component FIGURE 10-2

Condensation of a binary mixture.

of the vapor phase, while the temperature decreases, passes through points A, A', and A", whereas the composition of the condensate liquid evolves through points C, C ', C ", etc. When the bubble-point temperature TB is reached, the condensation is complete, and the liquid composition C '" coincides with the original composition of the vapor. We then can see that there is a temperatures range for the condensation of the mixture that extends between the dew-point and bubble-point temperatures. The heat balance of the process is always expressed with Eq. (10-1-1), where i1 is the inlet enthalpy and i2 is the outlet enthalpy (which may correspond to the bubble-point or even lower temperature if the condensate is subcooled). Both enthalpies are evaluated at the temperatures corresponding to the fluid evolution. Dropwise and Filmwise Condensation. To condense a vapor, it is necessary to extract heat from it. This heat extraction is performed with a refrigerant fluid that must be at a lower temperature than the fluid to be condensed.

CONDENSATION OF VAPORS

277

Both streams, vapor and refrigerant, are separated by a solid wall that will be at an intermediate temperature between the fluid temperatures. For condensation to occur, it is necessary for the wall temperature to be below the dewpoint temperature of the vapor at the prevailing pressure. Vapor The condensation of a vapor over a cold surface may take place by two different mechanisms: dropwise or Cold wall filmwise condensation. The first one takes place when the condensate exhibits little surface affinity for the wall. The vapor condenses in the form of small droplets that grow on the surface. These droplets act as nucleation centers for the condensation of additional vapor, increasing their size. When the weight of the droplets overcomes the surface attraction forces, the droplet falls down, leaving bare metal on which successive droplets of condensate may form. The second mechanism is filmwise condensation. Let’s consider a vertical-plane surface in contact with a vapor Descending (Fig. 10-3). In film condensation, the vapor condenses film on the cold surface, forming a continuous film. This film FIGURE 10-3 Filmwise condensation on a descends owing to the action of gravity, and the liquid flows vertical surface. downward. While the film drains, more liquid is incorporated into the film owing to vapor condensation. The film thickness increases downward because the amount of liquid in the film also increases. All the condensation heat must be removed by the refrigerant fluid, which is on the other side of the solid wall. So all the condensation heat, transferred by the condensing vapor to the vapor-liquid interface, must go through the condensate film to pass into the refrigerant. This means that the condensate film constitutes an additional resistance to heat transfer. The values of the heat transfer coefficients for filmwise condensation thus are lower than those of dropwise condensation. However, most fluids form condensate films. Additionally, dropwise condensation is an unstable process that turns into film condensation in a rather unpredictable way. Thus, since it is not possible to guarantee dropwise condensation, the usual assumption is to design on the hypothesis of film condensation, which implies a conservative approach.

10-2 CONDENSATION OF SINGLE-COMPONENT VAPORS 10-2-1

Filmwise Condensation: Nusselt Theory Let’s analyze the case of a vertical plate at temperature Tw in contact with a pure-component vapor saturated at temperature Tv . The vapor will condense over the solid surface, forming a condensate film, as shown in Fig. 10-4. The condensate flows downward owing to gravity. Since more fluid incorporates into the film owing to vapor condensation as the condensate flows down, the liquid flow rate in the film, at a distance x + dx from the upper edge of the plate is higher than the flow rate at distance x. This difference corresponds to the amount of vapor condensed on the surface Bdx per unit time. If the mass of vapor condensing on the surface Bdx per unit time is designated as dW ′, we can define a vapor mass velocity toward the interface as J=

dW ' Bdx

[kg/(s · m2)]

(10-2-1)

If W is the mass flow of condensate in the film (kg/s), a mass balance gives W

x + dx

−W

x

= dW ' = JBdx

(10-2-2)

278

CHAPTER TEN

B

x W (flow rate in the condensate film)

dx

dW ′

W + dW

FIGURE 10-4

Liquid-film layer on a vertical surface.

So dW = dW ' = JBdx

(10-2-3)

The model we shall develop in what follows is due to Nusselt1 and is based on the following hypotheses: 1. The vapor is saturated, and it only delivers latent heat. 2. The drainage of the condensate film from the surface is by laminar flow only. It is assumed that the vapor velocity is low, so the vapor does not exert any drag force on the film. In other words, the shear stress at the liquid-vapor interface is nil. 3. A thermodynamic equilibrium exists at the interface. This means that the liquid temperature at the interface is also Tv , so there is no heat transfer resistance in the vapor phase. 4. The vapor condenses on the interface and delivers its latent heat of condensation to the liquid film. This amount of heat must be transmitted to the cold wall (and to the refrigerant that is on the other side) through the liquid film. It is assumed that the heat is transferred through the film by conduction. Since the temperatures at both sides of the film are Tv (at the vapor-liquid interface) and Tw (at the solid surface), the temperature gradient for the heat transmission is (Tv − Tw)/δ(x), where δ(x) is the film thickness, which is a function of position x. Condensation Heat Transfer Film Coefficient. When a mass flow dW ' (kg/s) condenses on the surface of the film, it delivers to the film an amount of heat per unit time given by dQ = dW 'λ

(10-2-4)

279

CONDENSATION OF VAPORS

This heat is delivered to a film surface area dA = Bdx

(10-2-5)

and must be transmitted by conduction through the liquid film. Applying Fourier’s conduction law, we have dQ = k L

(Tv − Tw ) dA δ( x )

(10-2-6)

And the heat-flux density will be q=

k dQ dW ' =λ = λJ = L (Tv − Tw ) dA dA δ( x )

(10-2-7)

We see that since the film thickness is a function of the position, the heat-flux density also will be. We can define the local heat transfer coefficient as q( x ) = h( x ) (Tv − Tw )

(10-2-8)

And comparing Eqs. (10-2-7) and (10-2-8), we get h( x ) =

kL δ( x )

(10-2-9)

This means that if we know the film thickness at a certain position, we can calculate the local heat transfer coefficient by the simple Eq. (10-2-9). In what follows, we shall try to obtain a mathematical expression to calculate the film thickness at any location in the film. Velocity Profile in a Descending Film. The condensate layer drains owing to the force of gravity. At each point into the film, there is a fluid velocity that depends on the position. Figure 10-5 shows the velocity profile at a certain section of the condensate layer. In contact with the solid wall, the velocity is zero. According to the Nusselt’s hypothesis, the velocity of the film at the vapor-liquid interface is not affected by the vapor. This means that there is no momentum transfer between the vapor and the liquid. Since there is no shear stress, the slope of the velocity profile at the interface must be 0. Let’s consider a volume element belonging to the liquid layer, as shown in Fig. 10-5. Owing to the existence of a velocity profile, there are shear forces acting at the faces of the volume element, as indicated in the figure. The volume element must be in an equilibrium, resulting from the action of these forces and its own weight. If we analyze the forces acting on the volume element, we have

Velocity distribution in the condensate film vx = f (y)

∂τyx τyx + ∂y

τyx Δy

FIGURE 10-5 Velocity distribution in the liquid film.

1. On the right side, near the cold vertical surface, there is a tangential force acting upward and tending to support the volume. This force is −τ yx Δ x Δ z

280

CHAPTER TEN

2. On the opposite side there is a tangential force acting downward owing to the more rapid movement of the liquid downward as the distance from the surface is increased. This force is ∂τ y x ⎞ ⎛ ⎜⎝ τ y x + ∂y Δy⎟⎠ Δx Δz 3. The weight of the volume element acting down, that is, ρg Δx ΔyΔz Equating these forces, ρg Δx ΔyΔz +

∴

∂τ y x

Δx ΔyΔz = 0

∂y

∂τ y x

(10-2-10)

= − ρg

(10-2-11)

τ y x = −ρgy + C1

(10-2-12)

∴ τ y x = ρg(δ ( x ) − y)

(10-2-13)

∂y

Integrating,

Since for y = δ(x), it is τyx = 0

(Note that even though δ(x) is a function of x, it is a constant for the integration over y.) Since dv x dy

τ yx = μ

(10-2-14)

we get dv x =

ρg(δ ( x ) − y) μ

dy

(10-2-15)

Integrating again, we get vx =

ρgyδ ( x ) μ

−

ρgy 2 + C2 2μ

(10-2-16)

ρgy 2 2μ

(10-2-17)

For y = 0, v = 0 ∴ C2 = 0, then vx =

ρgyδ ( x ) μ

−

This is the expression of the velocity profile in a cross section of the film in which the film thickness is δ(x).

CONDENSATION OF VAPORS

281

The mean velocity in that section is v=

1 δ (x )

δ( x )

∫0

v x dy =

1 δ (x )

δ ( x ) ⎛ ρgδ (x ) y

∫0

⎜⎝

μ

−

ρgy 2 ⎞ dy 2μ ⎟⎠

ρgδ (x ) ⎞ 1 ρgδ 2(x ) 1 ⎛ ρgδ (x ) = = − ⎜ δ (x ) ⎝ 2μ 6μ ⎟⎠ 3 μ 3

Condensate-Film Thickness.

3

(10-2-18)

The condensate mass flow at any film section is W = ρvBδ ( x )

(10-2-19)

The change in the mass flow between a section at x and a section at x + dx is ⎛ ρ2 gδ (3x ) B ⎞ ρ2 gδ (2x ) B d δ( x ) dW = d (ρv δ ( x ) B) = d ⎜ ⎟= μ ⎝ 3μ ⎠

(10-2-20)

But, according to Eq. (10-2-3), dW = dW ' = JBdx; then Jdx =

ρ2 gδ (2x ) μ

d δ( x )

(10-2-21)

but q k (Tv − Tw ) = λ δ( x ) λ

(10-2-22)

ρ2 gδ 2(x ) d δ (x ) k (Tv − Tw ) dx = δ (x ) λ μ

(10-2-23)

ρ2 gδ 3(x ) d δ (x ) k (Tv − Tw ) dx = λ μ

(10-2-24)

J=

∴

∴

Equation (10-2-24) is a differential equation that can be integrated to find δ(x) as a function of the position x. The integration is done with the following boundary conditions: At x = 0, δ(x) = 0. At x = x (generic), δ(x) = δ(x). And we get ⎡ 4 k μ(Tv − Tw ) x ⎤ δ( x ) = ⎢ ⎥ ρ2 gλ ⎣ ⎦ This equation gives the film thickness at position x.

1/ 4

(10-2-25)

282

CHAPTER TEN

Condensation Heat Transfer Coefficient. cient is

According to Eq. (10-2-9), the local heat transfer coeffi-

h( x ) =

k δ( x )

⎛ k 3ρ2 gλ ⎞ =⎜ ⎝ 4μΔTx ⎟⎠

1/ 4

(10-2-26)

In practice, what interests us most is not the local heat transfer coefficient, but the mean value for the whole surface. Thus, if we consider a vertical plate of height L, we can define the mean coefficient as Q = hA(Tv − Tw )

(10-2-27)

where Q is the heat flow transferred to the whole plate. For a surface element dA = Bdx, it will be dQ = h(x ) (Tv − Tw )Bdx

(10-2-28)

Then Q=

∫ A dQ = ∫ A h(x ) (Tv − Tw )B dx

⎛ k 3ρ2 gλΔT 3 ⎞ =⎜ ⎟⎠ 4μ ⎝

1/4

(10-2-29)

dx ∫ A x1/ 4 B

Remembering that dx

∫ x1/ 4

=

4 3/ 4 x 3

(10-2-30)

the heat flow transferred will be Q=

4 3/ 4 ⎛ k 3ρ2 λΔT 3 L3 ⎞ ⎟⎠ 3 ⎜⎝ μ

1/ 4

(10-2-31)

B

Then, according to the definition given in Eq. (10-2-27), h=

Q Q 4 3/ 4 ⎛ k 3ρ2 λ g ⎞ = = AΔT LBΔT 3 ⎜⎝ μΔTL ⎟⎠

1/ 4

(10-2-32)

This is ⎛ k 3ρ2 λ g ⎞ h = 0.943 ⎜ ⎝ μ L ΔT ⎟⎠

1/ 4

(10-2-33)

Equation (10-2-33) is the final expresion that allows calculating the mean heat transfer coefficient for the condensation of a saturated vapor at temperature Tv over a vertical surface at Tw (where ΔT = Tv − Tw) as a function of the physical properties of the condensate and the height of the plate. The expression also can be applied to condensation over vertical tubes.

CONDENSATION OF VAPORS

283

Condensate film

α

FIGURE 10-6

Film condensate on an inclined surface.

It can be demonstrated that if condensation occurs over an inclined surface whose normal forms a certain angle α with the vertical (see Fig. 10-6), the heat transfer coefficient is ⎛ k 3ρ2 λ g sin α ⎞ h = 0.943 ⎜ ⎟⎠ μ L ΔT ⎝

1/ 4

(10-2-34)

This expression can be obtained easily if in the deduction of the velocity profile it is considered that the component of the gravity acceleration in the direction of movement is g sin α instead of g.

Condensate film

Condensation Over Horizontal Tubes. When a vapor condenses over the exterior surface of a horizontal tube, as in Fig. 10-7, it also forms a condensate film on the tube, and it is possible to develop a similar model to that for vertical surfaces. The deduction is more complex since the system geometry also is, and will not be developed here (see ref. 2). The final expression is ⎛ k 3ρ2 λ g ⎞ h = 0.725 ⎜ ⎝ μ Do ΔT ⎟⎠

FIGURE 10-7 tube.

Film condensate on a horizontal

1/ 4

(10-2-35)

Colburn-Type Expressions. Equations (10-2-34) and (10-2-35) are called Nusselt equations. They can be written in terms of dimensionless numbers to obtain the so-called Colburn equations. These equations are expressed as a function of a condensate-film Reynolds number. We shall analyze what the expressions of the film Reynolds number are for the most usual geometries. Condensation Over a Vertical Plate. Let’s consider a cross section at the lower end of the liquid layer, as shown in Fig. 10-8. The flow area for the liquid flow at this section is Bδf , where δf is the film thickness at the bottom end. The total amount of vapor condensing per unit time in the whole film is Wf (kg/s). The amount of heat transferred per unit time then will be Q = Wf λ (Note that Wf is the mass flow at the bottom end of the film.)

(10-2-36)

284

CHAPTER TEN

B

Condensate film

δf

FIGURE 10-8

Section plane

Cross section at the bottom end of the liquid film.

It is possible to define an equivalent diameter for the bottom section of the condensate film as Deq =

4δ f B 4 × flow area = = 4δ f wetted perimeter B

(10-2-37)

And we can define a mass velocity G as G=

mass flow W f = flow area Bδ f

(10-2-38)

We also can define a Reynolds number as Re f =

Deq G μ

=

4δ f W δ fμB

=

4W f Bμ

(10-2-39)

Calling G' = Wf /B

(10-2-40)

4G ' μ

(10-2-41)

The Reynolds number thus can be expressed as Re f =

where G' must be interpreted as a mass flow per unit film width [(kg)/(m · s)]. Condensation Over a Vertical Tube. It is also possible to define a Reynolds number for condensation over a vertical tube. A cross section at the base of the tube showing the film layer is shown in Fig. 10-9. If p is the tube perimeter (= πD), the equivalent diameter will be Deq =

4 pδ f p

= 4δ f

(10-2-42)

CONDENSATION OF VAPORS

285

δf

Flow area = perimeter × δf

FIGURE 10-9 Condensate layer cross section at the base of a vertical tube.

and G = mass velocity =

Wf pδ f

(10-2-43)

then Re f =

W f 4δ f

=

pδ f μ

4W f pμ

(10-2-44)

It is possible to define a mass flow per unit tube perimeter as G' =

Wf p

(10-2-45)

And then again, Re f =

4G ' μ

(10-2-46)

Condensation Over a Vertical Tube Bundle. When the condensation takes place over a bundle of N vertical tubes, if W is the total condensate mass flow, the flow over each tube is Mass flow = W/N

(10-2-47)

And the Reynolds number is Re f =

4G ' μ

where G' =

W pN

(10-2-48)

The heat transfer coefficients can be expressed as a function of the film Reynolds number, as we shall explain in what follows.

286

CHAPTER TEN

According to Eq. (10-2-36), Q=Wλ

(10-2-49)

(From now on, we shall use W to express the total mass flow rate, dropping the subscript f, which was used to emphasize that it is the condensate mass flow at the bottom of the film.) At the same time, Q = hAΔT

(10-2-50)

where A = condensation area = pL for condensation over a tube = pLN for condensation on a tube bundle = BL for a plane surface Combining Eqs. (10-2-49) and (10-2-50), we get λW AΔT

h=

(10-2-51)

which can be expressed as h=

λW λG ' = pL ΔT L ΔT

h=

λW λG ' = pLN ΔT L ΔT

h=

λW λG ' = BL ΔT L ΔT

for condensation on a tube

(10-2-52)

for condensation on a tube bundle

(10-2-53)

for condensation on a vertical plane

(10-2-54)

Thus, in all cases, the result is ΔT =

λG ' hL

(10-2-55)

Substituting Eq. (10-2-55) in Eq. (10-2-33), we get ⎛ μ2 ⎞ h⎜ 3 2 ⎟ ⎝ k ρ g⎠

1/ 3

⎛ 4G ' ⎞ = 1.47 ⎜ ⎝ μ ⎟⎠

−1/ 3

(10-2-56)

Equation (10-2-56) is called a Colburn-type expression for condensation on vertical surfaces. Horizontal Tubes. When a vapor condenses over the surface of a horizontal tube, it also may form a condensate film. The film covers the tube surface and drains by gravity action, increasing its thickness from top to bottom, as shown in Fig. 10-10. It is also possible to define a mass flow per unit film length. We shall define G" = W/LN where N is the number of tubes and L is the tube length. Dimensions of G" are also kg/(m · s).

CONDENSATION OF VAPORS

287

A

L

A

FIGURE 10-10

Section AA

Condensate film on a horizontal tube.

Operating similarly to the case of vertical surfaces, substituting in Eq. (10-2-35), we come to ⎛ μ2 ⎞ h⎜ 3 2 ⎟ ⎝ k ρ g⎠

1/ 3

⎛ 4G " ⎞ = 1.51 ⎜ ⎝ μ ⎟⎠

−1/ 3

(10-2-57)

Example 10-1 We must condense 1,400 kg/h of methanol vapors coming from a distillation column at 120 kPa. The condensation temperature at this pressure is 68.8°C. An existing condenser with 100 tubes, 19 mm external diameter, and 3 m in length will be used. Calculate which should be the wall temperature of the tubes if the condenser is installed in vertical and horizontal positions. Solution

Physical properties of liquid methanol at 68.8°C are

ρ = 738 kg/m3

k = 0.161 W/(m · K)

μ = 0.32 cP

c = 3,773 J/(kg · K)

λ = 1,090 kJ/kg W = 1,400 kg/h = 0.389 kg/s Condenser area = πDo LN = 3.14 × 0.019 × 3 × 100 = 17.9 m2 The heat flow yielded during condensation will be Q = W λ = 0.389 × 1,090 = 423.8 kW For a vertical condenser, G ' = W/πDoN = 0.389/(3.14 × 0.019 × 100) = 6.52 × 10–2 kg/(m · s) ⎛ μ2 ⎞ h⎜ 3 2 ⎟ ⎝ k ρ g⎠

1/ 3

⎛ 4G ' ⎞ = 1.47 ⎜ ⎝ μ ⎟⎠

⎛ μ2 ⎞ h = 0.161 ⎜ 3 2 ⎟ ⎝ k ρ g⎠

−1/ 3

−1/ 3

⎛ 4 × 6.52 × 10 −2 ⎞ = 1.47 ⎜ ⎝ 0.32 × 10 −3 ⎟⎠

⎡ (0.32 × 10 −3 )2 ⎤ = 0.161 ⎢ ⎥ 2 3 ⎣ 0.161 × 738 × 9.8 ⎦

−0.33

= 0.161

−0.33

= 887

288

CHAPTER TEN

Thus heat flow must be Q = h(T − Tw ) A ∴ Tw = T −

Q 423, 800 = 68.8 − = 42.1°C Ah 887 × 17.9

In the case of a horizontal condenser, according to the definition of G", we get G" = W/NL = 0.389/3 × 100 = 1.29 × 10–3 kg/(m · s) But it must be considered that in a horizontal tube bundle, part of the condensate formed on a tube drains over the tubes below, thus increasing their condensate flow rate. Nusselt equations, then, would not be valid. Kern2 proposes to correct the correlations using in this case a G" value calculated as G" =

W N 2/3 L

where N is the total number of tubes in the bundle. Then G" = 0.389/(3 × 1000.66) = 6.2 × 10−3 kg/(m · s) ⎛ μ2 ⎞ h⎜ 3 2 ⎟ ⎝ k ρ g⎠

1/ 3

⎛ 4G ′′ ⎞ = 1.51 ⎜ ⎝ μ ⎟⎠

⎛ μ2 ⎞ h = 0.35 ⎜ 3 2 ⎟ ⎝ k ρ g⎠

−1/ 3

−1/ 3

⎛ 4 × 6.2 × 10 −3 ⎞ = 1.51 ⎜ ⎝ 0.32 × 10 −3 ⎟⎠

⎡ (0.32 × 10 −3 )2 ⎤ = 0.35 ⎢ ⎥ 3 2 ⎣ 0.161 × 738 × 9.8 ⎦

−0.33

= 0.35

−0.33

= 1, 927 W/(m 2 ⋅ K)

And the necessary wall temperature is Tw = T −

Q 423, 800 = 68.8 − = 56.7°C Ah 1, 967 × 17.9

We can see that a lower wall temperature (and hence a lower refrigerant temperature) is required for the vertical condenser because of the smaller heat transfer coefficient.

10-2-2 Film Condensation in Turbulent Flow One of the hypotheses assumed by Nusselt was that of laminar flow in the condensate film. This is true for film Reynolds numbers [Eqs. (10-2-41) and (10-2-46)] lower than 1,600 (some authors extend the laminar Reynolds number range up to 2,100). Dukler3 developed a semitheoretical model for turbulent-flow condensate films. In those cases where the vapor velocity is still low (Nusselt hypothesis about nonexistence of shear stress at the vapor liquid interface continues being valid), the Dukler solution can be represented by the graph in Fig. 10-11. In this graph, the abscissa corresponds to the film Reynolds number, and the parameter is the Prandtl number of the condensate. For low Reynolds numbers, all curves are close to Nusselt’s prediction (dashed line in Fig. 10-11). However, for high Reynolds numbers, the coefficients calculated with Dukler’s model are considerably higher than those obtained with Nusselt’s equations (except for very low Prandtl numbers, which correspond to liquid metals). This is explained by the presence of important convective effects in the condensate film when flow becomes turbulent.

CONDENSATION OF VAPORS

h

k3

μ2 ρ2 g

289

Prandtl number 1/3

10 5 2 1 0.5

0.1

200

1,000

10,000

100,000

Terminal Reynolds Number = 4 G ′/μ FIGURE 10-11 Dukler’s plot of the condensate film coefficient. (From Chem. Eng. Progr. 55(10): 62–67, 1959. Reproduced with permission from the American Institute of Chemical Engineers.)

10-2-3

Effect of Vapor Velocity Up to now, we have assumed that the shear stress is nil at the vapor-liquid interface. This hypothesis may be far from reality in cases where the vapor velocity is considerable, situations found more frequently when condensation takes place inside tubes. We shall examine some models that take into account this effect. Condensation Inside Horizontal Tubes. In the case of condensation into horizontal tubes, two different flow regimes may exist: stratified and annular. These are represented in Fig. 10-12. ( There are other twophase intermediate flow patterns, but these are the two most distincts ones.) The former of these regimes corresponds to the case where the vapor velocity is low. In this case, gravitational forces prevail, and film

Stratified FIGURE 10-12

Annular

Flow patterns in condensation within horizontal tubes.

290

CHAPTER TEN

coefficients can be obtained with a model similar to Nusselt’s (the film regime is always laminar in horizontal tubes). Kern2 proposed a correction to take into account the flooded portion of the tube and suggested the following correlation for the stratified flow pattern: ⎡ k 3ρ (ρ − ρV ) g ⎤ h = 0.761 ⎢ L L L ⎥ G '' μ L ⎣ ⎦

1/ 3

⎡ k 3ρ (ρ − ρV ) gλ ⎤ = 0.815 ⎢ L L L ⎥ μ L Di ΔT ⎣ ⎦

1/ 4

(10-2-58)

When the vapor velocity is high, vapor displaces the liquid against the tube wall, and an annular flow patterns results. Several correlations were proposed for this annular flow regime. The Boyko-Krushilin correlation5 allows calculation of a mean condensation coefficient for a stream whose inlet liquid mass fraction is xi and whose outlet mass fraction is xo (x = mass of liquid present/total mass). The expresion is ⎛DG ⎞ hDi = 0.024 ⎜ i T ⎟ kL ⎝ μL ⎠

0.8

⎡ (ρ /ρ ) 0.5 + (ρm /ρv )o 0.5 ⎤ PrL 0.43 ⎢ m v i ⎥ 2 ⎣ ⎦

(10-2-59)

where GT is the mass flow density calculated with the total flow (vapor + liquid). Then (ρm /ρv )i = 1 + (ρm /ρv )o = 1 +

ρL − ρV xi ρV

(10-2-60)

ρL − ρV xo ρV

(10-2-61)

Another correlation is that of Carpenter and Colburn:6 2 hμ L 1/2 ⎛ fGVm ⎞ 1/2 = 0.065.PrL ⎜ k L ρL ⎝ 2ρV ⎟⎠

1/2

(10-2-62)

where ⎛ Gv2 + GVi GVo + Gv2o ⎞ GVm = ⎜ i ⎟ 3 ⎝ ⎠

1/2

(10-2-63)

In Eq 10-2-62 f is the friction factor evaluated for a single-phase system at a Reynolds number of ReVm =

Di GVm μV

(10-2-64)

What we still must decide is when we should use the stratified or annular model. A simple recipe consists in calculating the coefficient with both models and adopting the higher. Condensation Inside Vertical Tubes. The effect of the circulating vapor will be different based on whether it circulates upward or downward. In the first case the effect of vapor is to slow down the liquid drainage, whereas in the second case the effect is to accelerate it. Let’s consider first the case of vapor circulating downward. If the vapor velocity is low, it is possible to use the same model as for condensation outside tubes and employ Dukler’s graph. If, on the other hand, vapor velocities are high, the annular model described for horizontal tubes may be used. Again, the recommendation is to try both models and adopt the higher coefficient or an average of both.7

CONDENSATION OF VAPORS

291

For vapor flowing upward, the design must be done keeping the vapor velocity low enough to avoid flooding, which may happen when vapor prevents film drainage. One criterion suggested by Wallis8 establishes that flooding will be avoided if it is vG∗1/ 2 + v ∗L1/ 2 < 0.6 where vG∗ = ⎡⎣ρG vG2 /(ρL − ρG )gDi ⎤⎦

1/2

(10-2-65)

and v ∗L = ⎡⎣ρL vG2 /(ρL − ρG )gDi ⎤⎦

1/2

(10-2-66)

If it is checked that flooding will not be an issue, the shear stress at the vapor-liquid interface will be low enough to have no effect on the liquid-film drainage, and h can be calculated with the methods employed for low vapor velocity. Condensation Outside Tubes with High Vapor Velocities. In distillation-column condensers, where water is the cooling medium, condensation is usually in the shell side, so the interiors of the tubes can be cleaned easily (the condensing vapor is a clean fluid). In this case, baffled exchangers are used, and baffle spacing determines the velocity of the vapor. There is little information in the open literature about design methods for these cases. A model proposed by Mueller7 consists of adding a convective effect depending on vapor velocity to the coefficient calculated with Dukler’s correlation, The model is based on the following equations: Nu =

h 'f D kL

(

= X 4 ⋅ Re 2LG + Nu 4f

)1/ 4

(10-2-67)

where Re LG =

DGρL μ L ρG

(10-2-68)

vapor mass flow shell flow area

(10-2-69)

1 ⎞ 1/ 3 X = 0.9 ⎛ 1 + ⎝ RH ⎠

(10-2-70)

G= and

h'f is the coefficient corrected by vapor velocity. The parameters included in the correlations are defined as R= H=

ρL μ L ρG μ G

cL (Tsat − Tw ) PrL λ

(10-2-71)

(10-2-72)

292

CHAPTER TEN

Nu f =

hf D

(10-2-73)

kL

hf is the coefficient obtained with Dukler’s model without considering the effect of vapor velocity. It must be noted that if vapor velocity tends to be 0, in Eq. (10-2-67), Nu = Nuf , and the coefficient coincides with Dukler’s model. This means that this model is more general, and it is convenient to employ it in all cases. In the following section we shall see some application examples of these models for condensers design, but before that, we shall explain the use of this type of equipment in distillation installations. For readers who are not familiar with the distillation operation, we include App. A, which illustrates some basic concepts about this subject.

10-3 SINGLE-COMPONENT VAPORS CONDENSERS A condenser is a piece if equipment in which a vapor is condensed by heat extraction with a cooling medium. In process industry, the most usual applications are the condensers of distillation columns (see Fig. 10-13). The vapors exiting the top of the distillation column come into the condenser. Heat is removed by the cooling medium, and vapor condenses. Vent

ρV Cooling water H

ρL

Reflux R Product D

FIGURE 10-13

Horizontal gravity-flow condenser.

The condensate stream is split in two other streams. One of them is returned to the column as reflux R to wash the vapors ascending through the column, thus extracting the heavy components, and the other is withdrawn as distillate product D. The cooling medium is usually water. If no cooling water is available, air-cooled condensers may be employed. Sometimes a process stream that must be heated is used as coolant. The more common type is the shell-and-tube condenser. This is a heat exchanger with some modifications with respect to those used in single-phase applications. Figure 10-14 shows two possible ways to install the condenser. The figure at the left corresponds to an installation with forced reflux return and is the more usual. The condensate discharges from the condenser into a reflux accumulator. The reflux pumps take the condensate from the accumulator, pumping it back to the column and extracting the distillate. The condenser is usually installed above the accumulator, and the accumulator must be at enough height above the pumps to provide the required net positive suction head (the condensate is a liquid at its bubble-point temperature, so the levels difference must be higher than the required NPSH). If it is possible to install the condenser at a level above the top of the column, a gravityreturn reflux system can be used. This system is shown in the figure at the right. The obvious advantage is

CONDENSATION OF VAPORS

293

Vent

Vent

R

R

D

D FIGURE 10-14

Forced reflux and gravity reflux condensers.

simplicity because pumps are not required. The liquid seal in the return line, shown in the right side figure, avoids vapor reverse flow through this line. The vapors coming from the column suffer a pressure drop when they circulate through the condenser. In the case of gravity reflux, for the fluid to circulate, it is necessary that the hydrostatic pressure difference between the vapor and liquid legs be high enough to overcome the frictional pressure drop. Referring to Fig. 10-13, this can be expressed as Hg(ρL − ρV ) ≥ Δpfriction

(10-3-1)

where Δpfriction = Δpcondenser + Δppiping + Δpvalve

(10-3-2)

During steady-state operation, the liquid level in the return pipe will stabilize in a position H that satisfies the equality in Eq. (10-3-1). Usually the vapor density is much lower than that of liquid, and it can be written approximately as H=

Δpfriction ρL g

(10-3-3)

Equation (10-3-3) allows calculating the height difference between the condenser bottom and the column that is required for the operation. This condition must be checked during the design of the system because if the height is not enough to allow the condensate discharge, the liquid level will reach the condenser, and as the condenser floods, the heat transfer surface available for condensation will be reduced. The value of Δpfriction depends on the vapor mass flow into the condenser. If the column is operated above its design capacity, the increase in Δpfriction may cause condenser flooding, and this becomes a system bottleneck. The gravity-return reflux system, even it presents the advantage of elimination of the reflux pump, makes it necessary to install the condenser at a high level, which in most cases is uneconomical because it requires expensive structures. Forced-return reflux systems are more stable than gravity-return systems because the hydraulics of the system are independent of the operating conditions of the column. Condensers are usually provided with a vent line, as shown in Fig. 10-14. This is used to purge the condenser, eliminating the inert noncondensable gases that may be present in the system. These gases may consist of air or light components that enter the system dissolved in the feed stream and cannot be condensed. Air also can get into the system during plant shutdowns. If these gases are not purged, they reach the condenser, and because they cannot condense, they accumulate there, so their partial pressures increase with time. This has two harmful effects on the condenser operation. Since the total pressure at which the condenser operates is the sum of the partial pressures of vapor and noncondensable gas, the higher the partial pressure

294

CHAPTER TEN

Purge

Vapor inlet

Bulk velocity Difussion direction of noncondensibles

Condensate Inerts concentration

Position FIGURE 10-15

Inerts concentration profile in a condenser.

of the last one, the lower will be the partial pressure of the condensing vapors to maintain the total pressure. But the partial pressure of the condensing vapors is related to the condensing temperature. To condense the vapors at a lower partial pressure, a lower temperature is required. This would make a higher refrigerant flow rate or a lower refrigerant temperature necessary. If this is not possible, the capacity of the system is affected. The second effect of the accumulation of inert components is a very pronounced reduction in the condensing heat transfer coefficient. Othmer9 analyzed the case of condensation of water vapor at low pressure. With a temperature difference between the vapor and cooling media of 11°C, the condensing coefficient is reduced from 11,000 to 6,000 W/(m2 · K) when the inert concentration is 1 percent and to 4,000 W/(m2 · K) for a 2 percent concentration. This is due to the fact that inert gases accumulate over the heat transfer surface, thus forming a cold film through which the condensing vapor must diffuse to reach the cold surface. This creates an additional heat transfer resistance in the process. To avoid inert gas buildup, part of the vapor-inert mixture is vented, either in a continuous or discontinuous way. This maintains a low inert concentration in the condenser. Purging is usually performed at the opposite end from the vapor inlet nozzle because this is the zone of the condenser where inert concentrations are higher. To favor the inert purging, it is convenient to increase the vapor velocity into the condenser. This effect is explained as follows: As long as vapors condense, the mole fraction of inert gas in the mixture increases. This means that there is an inert concentration gradient across the condenser, as shown in Fig. 10-15. As a consequence of this gradient, inerts would tend to diffuse toward the inlet end. Since it is desired to keep a high inert concentration at the outlet end to minimize the loss of vapors with the purge, it is necessary to increase the vapor velocity near the outlet end to counteract diffusion. This is attained with a proper baffle spacing.

10-3-1 Different Types of Condensers Condensers can be horizontal or vertical. Also, condensation can take place inside or outside the tubes. We shall briefly analyze the characteristics of each type. Vertical Condenser with Condensation on the Shell Side. The installation of this type is represented in Fig. 10-16. The condensate formed on the tubes surface drains downward, forming a film. This film is interrupted by the presence of baffles (Fig. 10-17). Since the baffle holes are drilled with diameters only

CONDENSATION OF VAPORS

295

CW

FIGURE 10-16

Vertical condenser installation.

0.8 mm higher than the tube diameter, the film thickness cannot be greater than this, so part of the condensate is retained by the baffle and is reentrained by the vapor. Usually, in this type of unit, the condensation coefficient depends more on the vapor velocity than on the film thickness, and a conservative assumption is to calculate the resistance of the liquid film as if the baffles were not present. Horizontal Condenser with Condensation on the Shell Side. This is a horizontal heat exchanger. The cooling medium, usually water, circulates inside the tubes. The vapors condense on the exterior surface of the tubes. A condensate layer is formed over the tubes. The condensate flows downward around the tube perimeter and drops to the bottom of the unit. FIGURE 10-17 Effect of If vapor velocities are low, the condensation coefficient could be prebaffles in vertical shell-side dicted with the Nusselt model. It was explained on Example 10-1 that a condensation. correction to the Nusselt model is necessary to take into account that the liquid draining from a tube may drop on the tubes below, thus increasing the condensate load on the lower tubes. When baffles are installed in a horizontal condenser, they should have drain areas at the bottom so that condensate can flow toward the outlet. Otherwise, condensate would be trapped between baffles, obstructing the vapor circulation, as is the case in Fig. 10-18. Another solution, shown in Fig. 10-19, is to rotate the baffles 90 degrees. Vertical Condenser with In-Tube Condensation. Figure 10-20 shows different ways to install a vertical condenser with in-tube condensation. A condensate film forms on the interior surface of the tubes. The film drains downward into the lower head. Vapor and liquid flow in the same direction. This favors the condensation heat transfer coefficient because it decreases the film thickness. Horizontal Condenser with In-Tube Condensation. When cooling water is used as refrigerant, it is usually preferred to allocate the water in the tube side for cleaning facility. However, in air-cooled condensers,

296

CHAPTER TEN

FIGURE 10-18

Wrong installation of baffles.

Baffle with drain area FIGURE 10-19

Rotated baffles

Baffles for horizontal condensers.

Vent CW

CW Vent

Distillation column FIGURE 10-20

Condensate to pump

Distillation column

Vertical condenser installation.

or when a refrigeration cycle using boiling propane, ammonia, or other fluids is employed, condensation is usually on the tube side If the condenser has only one tube pass, or in a U-tube condenser, each tube receives the same amount of vapor and produces the same amount of condensate. If a multipass heat exchanger with return heads is employed, however, vapor-liquid separation may take place into the heads. Then, in the following pass, the lower tubes will receive proportionately more liquid than the upper tubes. Since a mathematical model taking into account this phase separation would be complicated, the effect is usually not considered, and it is assumed that there is a uniform behavior within all tubes. Figure 10-21 shows a condenser refrigerated by a propane cycle.

CONDENSATION OF VAPORS

297

Vaporized propane Vapor to condense

Condensate Liquid propane FIGURE 10-21

Propane-refrigerated condenser.

Partial Condensers. Very frequently, the top product of a distillation column is not obtained as a liquid but as a vapor, for example, in the case of hydrocarbon stabilizer units. In these cases, the condenser only condenses the required reflux mass flow. All the installation diagrams previously considered can be easily adapted for partial condensation. In this case, we shall not have the liquid product line, and the vapor outlet line (which in total condensers is used for inerts purging) must have the necessary diameter to evacuate the noncondensed vapors (see Fig. 10-22).

10-3-2 Design of Pure-Fluid Condensers Process condenser design is performed with a similar methodology to that used for single-phase heat exchangers. The main difference lies in the fact that in order to calculate the film coefficients corresponding to the condensing vapor side, the correlations studied in previous sections of this chapter must be employed.

CW Light uncondensible fraction

Liquid reflux FIGURE 10-22

Partial condenser.

CHAPTER TEN

As usual, the types of problems that can exist are as follows: 1. To verify whether a unit with known geometry will accomplish a certain process service (rating) 2. To find the proper geometry of a unit to accomplish a certain process service (design) Problems of the second type are solved by proposing a preliminary geometry based on typical heat transfer coefficients obtained from tables and rating the unit for the desired process conditions, so the task is transformed in the first type of problem. We shall study first the case of single-component total condensers with no condensate subcooling and receiving vapors at the dew-point condition. These hypotheses, even though they look very restrictive, correspond to conditions usually found in binary distillation units in which the top product of the distillation is almost a single-component fluid. Heat Balance of the Condenser and Mean Temperature Difference. In the simple case of a pure-fluid condenser with cooling water as the refrigerant medium, the heat balance is Q = Wh λ = Wc cc (t2 − t1 )

(10-3-4)

To calculate the heat transfer area of any heat exchanger device, the general expression is Q=

A

∫0 U (T − t ) dA

(10-3-5)

where T − t is the local temperature difference existing at any location in the unit, which can be obtained from a diagram such as that in Fig. 10-23 (which represents the particular case of a countercurrent condenser). Usually, a mean temperature difference is defined as Q = UAΔTmean

(10-3-6)

If U can be considered constant, then A

ΔTmean

∫ (T − t ) dA = 0

(10-3-7)

A

T Temperature

298

t2

t1

Position FIGURE 10-23 Temperatures diagram in the condenser of a pure substance.

299

CONDENSATION OF VAPORS

When for each fluid there exists a proportionality constant between the amount of heat that the fluid yields or receives and the temperature variation it experiences, it can be demonstrated that for parallel or countercurrent configurations, ΔTmean is the logarithmic mean temperature difference. Thus, for this case, ΔTmean =

(T − t2 ) − (T − t1 ) (T − t2 ) ln (T − t1 )

(10-3-8)

The case of pure-component condensers can be analyzed as a particular case of a heat exchanger where the proportionality constant between the heat yielded by the condensing steam and its temperature variation is infinite. (A fluid with an infinite heat capacity can exchange heat without changing its temperature.) For this particular case, it is then valid to use the LMTD. It can easily be seen that for an isothermal condenser, the LMTD for a countercurrent configuration coincides with that of a parallel-current configuration. If the condenser has two or more tube or shell passes, since these geometries correspond to different combinations, parallel-countercurrent, the value of ΔTmean also will be the same. This means that the Ft factor, which corrects the temperature difference in multipass units, is always unity when one of the fluids is isothermal. In a design problem, to transfer a certain amount of heat Q, the higher the temperature difference, the smaller will be the heat transfer area required. Usually the inlet temperature of the cooling medium t1 is an input for the designer, for example, the available cooling water temperature. Thus, in order to have a high value of ΔTmean, it is necessary to have a low refrigerant outlet temperature. According to Eq. (10-3-4), though, to transfer a certain amount of heat, the lower the refrigerant outlet temperature, the higher mass flow Wc will be necessary. Thus the equipment designer must adopt t2 as a compromise between a bigger exchanger and a higher refrigerant consumption. Condensation Heat transfer Coefficient. The overall heat transfer coefficient can be expressed as 1 1 1 = + + Rf U hc hR

(10-3-9)

In Eq. (10-3-9), hR is the film coefficient corresponding to the cooling medium. This is calculated with the correlations usually employed for single-phase heat exchangers. The fouling resistance Rf is obtained from tables or from the design basis of the project. To calculate the condensation heat transfer coefficient, it is necessary to choose the proper correlation depending on the type of condenser. The following situations are possible: Shell-Side Condensation: Vertical Condenser. Use the model corresponding to Eqs. (10-2-67) through (10-2-73). The value of hf is calculated with Dukler’s graph. Shell-Side Condensation: Horizontal Condenser. Again, the model represented by Eqs. (10-2-67) through (10-2-73) is used. In this case, to calculate hf , it must be considered that the condensate formed on the tube surface drops on the tubes below, increasing the condensate load and film thickness in the lower tubes. As explained previously, to take into account this effect, Kern 2 proposed to calculate hf with Eq. (10-2-57) but using for G" the following expression: G" =

Wh L N 2/3

(10-3-10)

where N is the total number of tubes of the condenser and Wh is the total condensate mass flow. Condensation Inside Horizontal Tubes. Use Eq. (10-2-58) for stratified flow and Eq. (10-2-59) or Eq. (10-2-62) for annular flow, and choose the higher calculated coefficient. Condensation Inside Vertical Tubes. For downward vapor flow, use Dukler’s model (low vapor velocity) and annular flow [Eqs. (10-2-59) to (10-2-62), high vapor velocities]. Choose the higher calculated coefficient.

300

CHAPTER TEN

Example 10-2 It is desired to condense 8,000 kg/h of acetone at a pressure of 120 kPa using cooling water at 35°C. At this pressure, the condensation temperature of acetone is 61°C. A cooling water flow rate of 100 m3/h will be used. After some preliminary trials, the following design was proposed: N = number of tubes: 192 L = tube length: 4 m Tube diameters: Do: 19.05 mm; Di: 14.4 mm; triangular array, 30 degrees; tube pitch: 23.81 mm Ds = shell diameter = 438 mm n = number of tube passes: 2 B = baffles spacing: 450 mm Physical properties are as follows:

3

Density, kg/m Viscosity, cP Thermal conductivity, W/(m · K) Specific heat, J/(kg · K) Heat of condensation, kJ/kg

Vapor

Liquid

2.54 0.0072 0.0147 1,381 497.5

738 0.215 0.142 2,255

Verify whether this unit is suitable to be installed in horizontal and vertical positions, and calculate the pressure drops for acetone and the cooling water. Use a total fouling factor of 0.0005 (m2 · K)/W. Solution

Water will flow inside the tubes. Heat balance Q = Whλ = (8,000/3,600) × 497.5 = 1,105 kW Water mass flow Wc = 100,000/3,600 = 27.77 kg/s Water flow area at = (πDi2/4)N/n = (π × 0.01442/4) × 192/2 = 0.0156 m2 Water velocity = Wc /ρat = 27.7/(1,004 × 0.0156) = 1.76 m/s

Water outlet temperature = 35 + Q/Wccc = 35 + 1.105 × 106/(27.7 × 4,180) = 44.5°C Cooling-water film coefficient: hi = 1,423(1 + 0.0146t)V 0.8/Di0.2 = 1,423(1 + 0.0146 × 39.8) × 1.760.8/0.01440.2 = 8,250 hio = hiDi /Do = 8,250 × 0.0144/0.019 = 6,250 W/(m2 · K) Shell-side heat transfer coefficient: Free distance between tubes c = Pt − Do = 23.8 − 19.05 = 4.8 mm Shell-side flow area as = (DscB)/Pt = (0.438 × 0.0048 × 0.45)/0.0238 = 0.0398 m2 The inlet vapor flow is 8,000 kg/h. At the outlet, the vapor flow is zero because all the vapor condenses. We shall consider a mean vapor flow Ws = 8,000/2 = 4,000 kg/h. Mass velocity Gs = Ws /as = 4,000/(3,600 × 0.0398) = 27.89 kg/(s · m2)

CONDENSATION OF VAPORS

301

According to Eq. (10-2-68), ReLG = (DoGsρL)/(ρGμL) = (0.019 × 27.89 × 738)/(0.215 × 10−3 × 2.54) = 715,800 According to Eq. (10-2-71), R = ρLμL/ρGμG = (738 × 0.215 × 10–3)/(2.54 × 0.0072 × 10–3) = 8,676 Liquid Prandtl number PrL = cLμL/kL= 2,255 × 0.215 × 10–3/0.142 = 3.414 According to Eq. (10-2-72), H = cL(T − Tw)/PrLλ Let’s assume a wall temperature of 50°C (to be checked later). Then H = 2,255(61 − 50)/(3.414 × 497,500) = 0.0146 According to Eq. (10-2-70), X = 0.9(1 + 1/RH)0.33 = 0.9[1 + 1/(8,676 × 0.0146)]0.33 = 0.902 To calculate hf , we must consider horizontal and vertical installation separately. Vertical condenser: G' = W/(NπDo) = [8,000/(3,600 × 3.14 × 192 × 0.019)] = 0.194 kg/(m · s) Ref = 4G'/μL = 4 × 0.194/0.215 × 10–3 = 3,609 From Dukler’s graph, with Ref = 3,609 and PrL = 3.414, we get an ordinate value = 0.28. This means that hf (μL2/ρL2gkL3)0.33 = 0.28 Then hf = 0.28(ρL2gkL3/μ2)0.33 = 0.28[7382 × 9.8 × 0.1423/(0.215 × 10−3)2]0.33 = 1,772 W/(m2 · K) Nu f = hf Do /kL = 1,772 × 0.019/0.142 = 237 From Eq. (10-2-67), Nu = hDo /kL = (X4 · ReLG2 + Nuf4)1/4 = (0.9024 × 715,8002 + 2374)1/4 = 764 W/(m2 · K) Then h = Nu · kL/Do = 764 × 0.142/0.019 = 5,711 W/(m2 · K) Horizontal condenser: The value of hf is calculated from hf (μ2/k3ρ2g)1/3 = 1.5(4G"/μ)–1/3 where G" = W/LN2/3 = 8,000/(3,600 × 4 × 1922/3) = 0.0172 kg/ms

302

CHAPTER TEN

Then hf [(0.215 × 10–3)2/(0.1423 × 7,382 × 9.8)]0.33 = 1.5(4 × 0.0172/0.215 × 10–3)–0.33 from which hf = 1,413 W/(m2 · K). Then Nu f = hf Do /kL = 1,413 × 0.019/0.142 = 189 X and ReLG are the same as for the vertical condenser. Then Nu = hDo/kL = (X4 · ReLG2 + Nuf4)1/4 = (0.9024 × 7,158,002 + 1894)1/4 = 764 h = 764kL/Do = 568 × 0.142/0.019 = 5,711 W/(m2 · K) It produces the same coefficient from both horizontal and vertical condensers. Verification of the wall temperature: hio(Tw − t) = ho(T − Tw) 6,250(Tw − 39.75) = 5,711(61 − Tw) Solving for Tw gives Tw = 49.9°C, almost equal to the assumed 50°C. Overall heat transfer coefficient: ⎛ 1 ⎞ 1 U =⎜ + + Rf ⎟ ⎝ hio ho ⎠

−1

1 ⎞ ⎛ 1 =⎜ + + 0.0005⎟ ⎝ 6, 250 5, 711 ⎠ DMLT =

−1

= 1,197 W/(m 2 ⋅ K)

44.5 − 35 = 20.9 61 − 35 ln 61 − 44.5

Required area = Q/U · DMLT = 1,105 ×106/1,197 × 20.9 = 44.16 m2 Condenser area = πLNDo = π × 4 × 192 × 0.019 = 45.8 m2 > 44.16 (okay) Shell-side pressure drop: For simplicity, we shall use Kern’s method. Shell side equivalent diameter = 0.0139 m Vapor Reynolds number at inlet: Re =

WDeq asμ

=

2.222 × 0.0139 = 107,699 0.0072 × 10 −3 × 0.0398

The friction factor is 0.195 ΔpS for the inlet mass flow = f

0.14 ( N B + 1) DS GS2 ⎛ μ w ⎞ 0.195 × 9 × 0.438 55.822 = = 344,000 N/m 2 ⎜ ⎟ 0.0139 2 × 2.5 Deq 2ρ ⎝ μ ⎠

Since at the outlet the vapor velocity is 0, Δps calculated with the outlet mass flow also will be 0. The mean value then is 17,000 N/m2.

CONDENSATION OF VAPORS

10-4

303

DESUPERHEATER CONDENSERS The vapor coming from the top of a distillation column is always in thermodynamic equilibrium with the liquid leaving the first tray of the column, so it is always at its dew-point temperature and comes into the condenser in this condition. However, there are other types of processes where it may be necessary to condense a vapor that comes into the condenser at a temperature higher than its dew-point temperature—this means in a superheated condition. Let’s consider the case of a propane refrigeration cycle such as that represented in Fig. 10-24. The chiller is a heat exchanger used to cool down a process stream by means of a liquid-propane stream that absorbs heat and vaporizes. The liquid-propane stream is indicated as (3) in the figure, and the vaporized propane is the stream at (4). The vaporized propane is compressed by the compressor up to a pressure high enough to allow propane condensation using cooling water or atmospheric air as the cooling medium. So this pressure is basically defined by the available air or cooling-water temperature. The condensed propane is later expanded in a valve down to the chiller pressure. This is an isoenthalpic evolution that produces a partial vaporization at the expense of a temperature drop, thus achieving a cold stream that closes the cycle. Figure 10-24 includes a representation of the cycle in a T-S diagram. It must be noted that in evolution 4–1, which takes place in the compressor, the temperature of the fluid increases up to a value T1 that is higher than the condensation temperature T2, which corresponds to pressure p1. The condenser then will have two zones in series. In the first zone, there is heat transfer in single phase. Vapors cool down from the inlet temperature to the dew-point temperature corresponding to the condenser operating pressure. The second zone corresponds to the vapor condensation, approximately isothermal. In a countercurrent unit, the temperature diagram is that shown in Fig. 10-25. The heat balance can be expressed by the following equations: Q = Qs + Qλ

(10-4-1)

Qs = sensible heat = Wh ch v (T1 − T2 )

(10-4-2)

Compressor 1 4 Condenser

Chiller Process stream to be cooled down

2 Receiver 3 Expansion valve T

1

T1 T2

P1

2 3

4 S

FIGURE 10-24

Propane refrigeration cycle.

Cooling water

304

CHAPTER TEN

T1

T2 t2

t3 t1

FIGURE 10-25

Temperature diagram of the compressor postcondenser.

Qλ = latent heat = Wh λ

(10-4-3)

Q = Wc cc (t2 − t1 )

(10-4-4)

The temperature of the cooling water at the section corresponding to the transition between the two zones is t3 = t 2 −

Qs Wc cc

(10-4-5)

This configuration is equivalent to having two heat exchangers in series, and the heat transfer area of each zone can be calculated using the heat transfer coefficient and LMTD corresponding to that zone. For the desuperheating zone (sensible heat), the result will be (T2 − t3 ) − (T1 − t2 ) (T − t ) ln 2 3 (T1 − t2 )

ΔTS = MLDTS =

(10-4-6)

For the condensing zone (latent heat), the result will be (T2 − t3 ) − (T2 − t1 ) (T − t ) ln 2 3 (T2 − t1 )

(10-4-7)

Qs U s ΔTs

(10-4-8)

Qλ U λ ΔTλ

(10-4-9)

ΔTλ = MLDTλ =

And the area of each zone can be calculated as As =

Aλ =

The overall heat transfer coefficients for both zones are usually very different because the heat transfer mechanisms also are very different.

CONDENSATION OF VAPORS

305

In the desuperheating zone, heat transfer takes place by a mechanism called reflashing. If the temperature of the tube wall is lower than the vapor dew-point temperature, condensate will form on the tube wall despite the fact that the global fluid temperature corresponds to a superheated condition. When this condensate drains from the tube into the bulk of the hot vapor phase, it revaporizes (reflashes), extracting heat from the vapor. This is a new and very effective heat transfer mechanism that contributes to vapor cooling and improves the heat transfer coefficient. However, most design methods do not take this effect into account and calculate Us for the desuperheating zone with a vapor film coefficient obtained via the single-phase methods of Chap. 7. This is a conservative approach. In many cases, though, since the sensible-heat zone is considerably smaller than the latent-heat zone, the effect of this simplification on the total area of the condenser is not significant. The overall heat transfer coefficient for the condensing zone Uλ is calculated with the methods of this chapter. The cooling-water flow rate must be selected high enough to avoid temperature crosses between both streams. For example, let’s consider the situation shown in Fig. 10-26. Despite the fact that the water outlet temperature is lower than the hot-vapor inlet temperature, the temperature evolutions shown on this diagram are thermodynamically impossible because somewhere in the condenser, the cooling water should be heated above the temperature of the fluid from which it receives heat. The recommendation to avoid such situations is to choose the water flow rate so that its outlet temperature is always below the vapor condensation temperature T2. In the foregoing discussion, we assumed that the condenser configuration was countercurrent. If the condenser has two or more tube passes, the situation will be that of Fig. 10-27, where in each zone the hot

T1 t2 T2

t1

FIGURE 10-26

An impossible thermodynamic evolution.

T1 T2 t2

t1

FIGURE 10-27 Temperature diagram in a condenser with two passes in the cooling side.

306

CHAPTER TEN

fluid is in contact with two (or more) passes of the cold fluid. In this case, it would not be valid to use the LMTDs defined by Eqs. (10-4-6) and (10-4-7). According to Kern, provided that the outlet temperature of the cold fluid is below the vapor dew-point temperature, which means that the heat transfer mechanism in the desuperheating zone is that of reflashing, it is acceptable to use the temperatures differences defined by Eqs. (10-4-6) and (10-4-7) as if the configuration were pure countercurrent. The justification is that the error resulting from this simplification is somewhat compensated for by the error of not considering the reflashing mechanism in the calculation of the heat transfer coefficient. This means that, independent of the pass configuration, a countercurrent diagram can be assumed, and a temperature t3 (which in this case does not have any physical interpretation) is calculated with Eq. (10-4-5), and then the LMTDs for each zone are calculated. Certain design programs calculate an Ft correction factor using the four extreme temperatures and then apply this correction factor to both LMTDs. This procedure does not look to have any other justification than to introduce an additional safety factor in the design. Example 10-3 It is desired to condense 5,144 kg/h of propane coming from the discharge of a refrigeration compressor at 1,300 kPa and 60°C. Cooling water at 30°C will be used. At 1,300 kPa, the condensing temperature of propane is 37.2°C. A temperature rise of 4°C for the cooling water is adopted. After some trials, the following design is proposed: N = number of tubes: 672 L = tube length: 4 m Tube diameters: Do: 19.05 mm; Di: 14.4 mm; square array; tube separation: 23.81 mm Ds = shell diameter: 762 mm n = number of tube passes: 6 B = baffle spacing: 150 mm Physical properties:

Density, kg/m3 (at Tsat) Viscosity, cP Thermal conductivity, W/(m · K) Specific heat, J/(kg · K) Heat of condensation, kJ/kg

Vapor

Liquid

28.1 0.009 0.020 2,030 314

476 0.09 0.088 3,020

Verify if this unit is suitable if installed in a horizontal position. Calculate the pressure drops for the propane and for the cooling water. Assume fouling factors of 0.0003 (m2 · K)/W for cooling water and 0.0001 (m2 · K)/W for propane. Water will flow in the tube-side Heat balance:

Solution

Qλ = Whλ = (5,144/3,600) × 314 = 449 kW Qs = Whch(T1 − T2) = (5,144/3,600) × 2,030 × (60 − 37.2) = 66.14 kW Q = Qs + Qλ = 515.2 kW Water mass flow: Wc = Q/cc(t2 − t1) = 5.152 × 106/(4,180 × 4) = 30.81 kg/s = 111,000 kg/h

CONDENSATION OF VAPORS

307

T1 = 60 T2 = 37.2 t2 = 34

t3 = 33.5

t1 = 30

FIGURE 10-28

Illustration for Example 10-3.

The unit has six tube passes. We shall use the simplification of assuming a countercurrent configuration, and we shall correct the LMTD for both zones with a correction factor Ft calculated with the four extremes temperatures. See Fig. 10-28. Temperature t3, corresponding to the cooling water at the fictitious transition point, will be t3 = t2 − Qs/Wccc = 34 − 66,140/(30.81 × 4,180) = 33.5°C The logarithmic mean temperature differences for each zone will be MLDTs = [(60 − 34) − (37.2 − 33.5)]/ln(26/3.7) = 11.4 MLDTλ = [(37.2 − 33.5) − (37.2 − 30)]/ln(3.7/7.2) = 5.2 With the four extreme temperatures, we calculate Ft . The parameters required for the calculation are R = (T1 − T2)/(t2 − t1) = 5.7

and

S = (t2 − t1)/(T1 − t1) = 0.133

Thus we get Ft = 0.917. The corrected temperature differences are ΔTs = MLDTs · Ft = 11.4 × 0.917 = 10.4°C ΔTλ = MLDTλ · Ft = 5.2 × 0.917 = 4.8°C Heat transfer coefficient for the cooling water: Flow area at = (πDi2/4)N/n = (π × 0.01442/4) × 672/6 = 0.0182 m2 Water velocity = Wc /ρat = 30.89/(1,000 × 0.0182) = 1.7 m/s Film coefficient, tube side: hi = 1,423(1 + 0.0146t)v0.8/Di0.2 = 1,423(1 + 0.0146 × 32)1.70.8/0.01440.2 = 7,453 W/(m2 · K) hio = hi/(Di/Do) = 7,453 × (14.4/19.05) = 5,633 W/(m2 · K)

308

CHAPTER TEN

Sensible-heat zone: We shall first calculate the heat transfer coefficient for the desuperheating zone. To simplify calculations, we shall use Kern’s method. Clearance between tubes c = Pt − Do = 23.8 − 19.05 = 4.8 mm Shell-side flow area as = (DscB)/Pt = (0.762 × 0.0048 × 0.15)/0.0238 = 0.0231 m2 The equivalent diameter for this array is Deq = 4 × (Pt2 − πDo2/4)/πDo = 4 × (0.02382 − 3.14 × 0.0192/4)/(3.14 × 0.019) = 0.019 m The Reynolds number is ReK = DeqW/(asμG) = 0.019 × 1.429/(0.0231 × 0.009 × 10–3) = 130,600 (We use the subscript K to differentiate it from other Reynolds numbers that will be used in the calculations.) ho = 0.36(kG/Deq)ReK0.55 · PrG0.33 = 0.36 × (0.02/0.019) × 130,6000.55 × 0.9280.33 = 240 W/(m2 · K) Then Us = (1/hio + 1/ho + Rf) − 1 = (1/5,633 + 1/240 + 0.0004)–1 = 210 W/(m2 · K) The required area for the desuperheating zone is As = Qs/UsΔTs = 66,140/(210 × 10.4) = 30.3 m2 Latent-heat zone: The vapor flow at the inlet is 5,144 kg/h. At the condenser outlet, the vapor flow is zero because it condenses totally. We consider an average mass flow Ws = 5,144/2 = 2,572 kg/h. Mass velocity Gs = Ws/as = 2,572/(3,600 × 0.0231) = 30.92 kg/(s · m2) By Eq. (10-2-68), ReLG = (DoGsρL)/(ρGμL) = (0.019 × 30.92 × 476)/(0.09 × 10–3 × 28.1) = 110,600 By Eq. (10-2-71), R = ρLμL/ρGμG = (476 × 0.09 × 10–3)/(28.1 × 0.009 × 10–3) = 169 Liquid Prandtl number PrL = cLμL/kL = 3,020 × 0.09 × 10–3/0.088 = 3.08 By Eq. (10-2-72), H = cL(T − Tw)/PrLλ We shall assume a wall temperature of 33°C, which can be verified later. Then H = 3,020(37.2 − 33)/(3.08 × 314,000) = 0.013 According to Eq. (10-3-70), X = 0.9(1 + 1/RH)0.33 = 0.9[1 + 1/(169 × 0.013)]0.33 = 1.01

CONDENSATION OF VAPORS

309

For a horizontal condenser, hf is calculated from the expression hf (μ2/k3ρ2g)1/3 = 1.5(4G"/μ)–1/3 where G" = W/LN2/3 = 5,144/(3,600 × 4 × 672 2/3) = 4.6 × 10–3 kg/ms Then h f [(0.09 × 10–3)2/(0.0883 × 4,762 × 9.8)]0.33 = 1.5(4 × 4.6 × 10–3/0.09 × 10–3)–0.33 from which we get hf = 1,339 W/(m2 · K) Nuf = hf Do/kL = 1,339 × 0.019/0.088 = 289 Nu = h'f Do/kL = (X4 · ReLG2 + Nuf4)1/4 = (1.014 × 110,6002 + 2894)1/4 = 374 h'f = ho = 374kL/Do = 547 × 0.088/0.019 = 1,732 W/(m2 · K) Then, in the condensation zone, the overall heat coefficient will be Uλ = (1/hio + 1/ho + Rf)–1 = (1/1,732 + 1/5,633 + 0.0004)–1 = 862 W/(m2 · K) Verification of the wall temperature: Tw = T2 − Uλ(T2 − t)/ho = 37.2 − 862 × (37.2 − 31.7)/1,732 = 34.4 It is not considered necessary to correct H. Thus the area required for condensation is Aλ = Qλ/ΔTλUλ = 449,000/(4.8 × 862) = 108 m2 Total area: The total area required then will be 108 + 30.3 = 138.3 m2. The condenser area is A = πDoLN = 3.14 × 0.019 × 4 × 672 = 160 m2 Therefore, the unit has about 15 percent excess area.

10-5 CONDENSATION OF VAPOR MIXTURES When the vapor to be condensed is not a pure substance, condensation is not isothermal. This is what happens in the condenser of a distillation column separating hydrocarbon mixtures. The vapors coming from the top of the distillation column are a mixture of components with different condensation temperatures. When the mixture condenses, its temperature decreases while condensation progresses. Initially, those components with higher condensation temperatures (called heavier components) condense in a higher proportion. As long as the condensation progresses, the temperature decreases, and the lighter components condense. If the mixture contains an important proportion of heavy components and a few lights, most of the condensation heat corresponds to the components with higher condensation temperatures. Representing the temperature evolution as a function of the heat removed, a diagram such as that in Fig. 10-29a results. These curves are called condensation curves.

310

CHAPTER TEN

Temp

Temp

Heat delivered

Heat delivered

(a)

(b)

FIGURE 10-29

Different types of condensation curves.

On the other hand, for a mixture containing a higher proportion of light components, the condensation curve will be of the type shown in Fig. 10-29b. In this case, with the initial removal of a small amount of heat, the heavy components condense, and the condensation temperature decreases rapidly, whereas at the end of the condensation the temperature remains more constant. Let’s consider a countercurrent condenser and represent the T-Q diagram for the condensing fluid and the cooling water in both cases. The temperature difference between both streams changes along the condenser; then, to calculate the heat transfer area, we should integrate A=

dQ

∫ U ΔT

(10-5-1)

We have demonstrated that in cases where it a linear relationship exists between the heat received or yielded by each fluid and the temperature change that the fluid experiences, the preceding expression may be integrated analytically, resulting in A=

Q U (LMTD)

(10-5-2)

This situation is represented by the dashed lines in Fig. 10-30, which show an ideal evolution of the condensing fluid, assuming the existence of a linear relationship between heat and temperature, in which case it would be valid the use of the LMTD. We see that in the case of Fig. 10-30a, the actual temperature difference ΔT is higher than that corresponding to the ideal situation, whereas in the case of Fig. 10-30b, it is lower. This means that in the first case, the use of LMTD would result in an error by excess in the calculation of the area, whereas in the second case we should have an error by deficit. Actual evolution Temp Actual evolution

Temp T1

Ideal evolution

t2 ΔT

T2

T1

Ideal evolution

t2 ΔT

t1

t1 Heat delivered (a) FIGURE 10-30

T2

Heat delivered (b)

True temperature difference can be higher or smaller than the LMTD.

CONDENSATION OF VAPORS

311

It is thus necessary to know the condensation curve in order to design the condenser. To obtain this curve, it is necessary to solve, for each temperature within the condensation interval, the equations representing the thermodynamic equilibrium between vapor and liquid, which requires us to know the liquidvapor equilibrium relationship. In very ideal systems, it is possible to use simplified models that assumed that the equilibrium constants are not a function of the phase composition and only depend on the temperature (such as Raoult’s law). This is explained in more detail in App. A. But nowadays, these calculations invariably are made with simulation programs that use complex thermodynamic packages, making possible an exact prediction of phase equilibria. The condensation curve then is usually supplied to the exchanger designer as part of the process information.

10-5-1 Diffusional Processes in the Condensation of Multicomponent Mixtures When we studied Nusselt’s condensation theory, we described a model that considered that the vapor is in contact with a condensate film and the temperature in the bulk of the vapor phase is the same as the vaporliquid interface temperature. The liquid film then was the only resistance to heat transfer. In the case of multicomponent mixtures, the situation is different. When the vapor condenses at the interface, because the molecules that preferably condense are those of the heavier components, the mixture near the interface becomes richer in light components. This results in a concentration gradient in the direction normal to the interface. In the case of a binary mixture, a graphic representation such as that in Fig. 10-31 is possible. The graph shows the mole fraction of the heavier component yA as a function of the distance to the interface. We see that at the interface, the concentration is yi, whereas in the bulk of the vapor phase it is y, where yi < y. This means that the proportion of A molecules that can reach the interface and condense is lower than that which would result if the concentration at the interface were equal to the bulk concentration. The A molecules must diffuse through a layer with a higher proportion of B molecules to reach the interface. This adds an additional resistance to heat transfer. Let’s consider a condenser such as that represented in Fig. 10-32, in which the vapor temperature varies from inlet to outlet. The diagram also shows the cooling-water temperature evolution in a countercurrent arrangement. Through a differential heat transfer area dA, the amount of heat exchanged per unit time is dQ. Considering the condensing fluid side, dQ must include the sensible heat required to reduce in dT the temperature of the vapor-liquid stream circulating through the condenser, plus the contribution of the latent heat that corresponds to the condensation of dW (kg/s) of vapor.

Vapor

yA y

Diffussion of the condensing component

Cooling medium t yi

Condensate film FIGURE 10-31 Concentration profile of the heavy component of a binary mixture in the vicinity of the condensation surface.

312

CHAPTER TEN

A

dA

A

T T1

dT t2 T2

dt

t1 Q dQ FIGURE 10-32 Temperature evolutions in a condenser.

Tc

Vapor T

dQs

Cooling medium t dA

dW

Condensate film FIGURE 10-33 condensation.

Condensate film in multicomponent

Schematically, the process can be represented by Fig. 10-33. Here, T, Tc , and t are the temperatures of the vapor, the interface, and the cooling media at a cross section of the condenser such as AA in Fig. 10-32. If we consider a heat transfer differential area dA in the vicinity of this section, we can explain the condensation mechanism as follows: 1. The vapor at temperature T delivers heat to the interface at Tc . This transfer is by convection, and it can be expresed as dQS = hG (T − Tc )dA

(10-5-3)

where dQs is the convection heat flow transferred from the gas to the liquid film per unit time, and hG is the gas-phase convection heat transfer coefficient.

CONDENSATION OF VAPORS

313

2. In addition to the aforementioned heat flow, there exists a condensation of a mass vapor dW (kg/s) on dA. When this vapor condenses, it delivers to the condensate film its latent heat of condensation. Thus the total amount of heat that incorporates to the film on dA will be the sum of both terms dQ = dQS + λdW

(10-5-4)

This heat must be transmitted through the film to be removed by the cooling medium. The resistance to heat transfer consists of the liquid film, fouling, and the cooling-medium boundary layer. These resistances are in series, and it is possible to define a combined heat transfer coefficient as ⎛ 1 ⎞ −1 1 U ' = ⎜⎜ + + R f ⎟⎟ ⎝ h f' hR ⎠

(10-5-5)

The first term within the parentheses represents the resistance of the condensate film, which can be calculated with the models we studied for pure fluids; the second term is the resistance of the refrigerant; and the third term is the fouling resistance. The result thus will be dQ = U '(Tc − t )dA

(10-5-6)

In order to obtain an expression for the heat transfer rate, let’s consider a control volume, as indicated by the dashed lines in Fig. 10-34. This control volume includes the vapor phase contained in the portion of condenser corresponding to dA and extends up to the vapor-liquid interface without including it. A vapor flow Wv at temperature T enters the control volume. A fraction dW of this vapor condenses over the surface dA. This mass leaves the control volume at the vapor state (because the interface is outside the control volume) and at a temperature Tc. The enthalpy change of this mass of vapor between its inlet and outlet conditions to the control volume is dWcv (T − Tc ) where cV is the vapor specific heat. Additionally, there is a convection heat transfer from the bulk of the vapor phase to the interface given by dQs = hG dA(T − Tc )

(10-5-7) Heat transfer surface

T – dT

Wv – dW Vapor phase

Tc T

dQs

dQ Refrigerant t

dW

Wv

Condensate film

FIGURE 10-34 Control volume for energy balances in a multicomponent condenser.

314

CHAPTER TEN

The vapor flow WV that enters the control volume at temperature T leaves at temperature T − dT. Then its enthalpy change is WV cV dT (neglecting the mass flow reduction dW). The enthalpy balance for the control volume then can be expressed as WV cV dT + dWcV (T − Tc ) − dQs = 0

(10-5-8)

If it is accept that the second term is small in comparison with the first one (which is usually the situation), then WV cV dT = hG (T − Tc)dA

(10-5-9)

Combining Eqs. (10-5-6) and (10-5-7), we get dQ = (Tc − t ) U ′dA

(10-5-10)

dQS = (T − Tc ) hG dA

(10-5-11)

Adding both equations and defining γ = dQS /dQ = WV .cV .dT/dQ, we obtain ⎛ 1 γ ⎞ dQ ⎜ + = T −t ⎝ U ' dA hG dA ⎟⎠

(10-5-12)

The overall heat transfer coefficient corresponds to the sum of all the resistances, so γ γ 1 1 1 1 = + = + + + Rf U U ' hG hR h ' f hG

(10-5-13)

To be consistent with the methodology we always use, we can separate the resistances at both sides of the tube wall, defining a heat transfer coefficient for the hot fluid as γ 1 1 = + hh h f' hG

(10-5-14)

1 1 1 = + + Rf U hh hR

(10-5-15)

So we get the usual expression

The heat transfer area then can be calculated as dA =

dQ U (T − t )

(10-5-16)

and A=

dQ

∫ U (T − t )

(10-5-17)

To use this expression, it is necessary to know the condensation curve, which is sectioned in intervals of dQ. Superimposing this curve with the T-Q curve for the refrigerant, we can obtain for each interval the

CONDENSATION OF VAPORS

315

value of (T − t) and the ratio of sensible heat to total heat γ. Since the film coefficients hG , hR , and h'f can be calculated for the conditions of each interval, it is possible to calculate the integral numerically. 10-5-2 General Method for Condenser Design The design method for a multicomponent condenser with condensation range includes the case of condensation of a pure vapor (making γ = 0).We thus shall develop a calculation method for the rating of a condenser with a condensation range whose results are completely general and applicable to any condensation case. This procedure only covers the two-phase region. If the condenser also includes a desuperheating zone, this must be rated separately. To simplify the treatment, we shall use Kern’s correlations for calculation of single-phase heat transfer coefficients. As was explained in Chap. 7, this method can result in important errors, specially in floatingheads type condensers. However, the procedure can be easily adapted to be used with more elaborated methods for the prediction of shell-side heat transfer coefficients. The input data to use this model include the condensation curve. This curve is obtained from the process simulation using programs such as Hysis, Aspen, ProII, etc. This curve represents the thermodynamic evolution of the hot stream while its temperature is reduced from T1 to T2. For each intermediate temperature, it is necessary to know the enthalpy of the stream, the vapor mass fraction (kilograms of vapor per total kilograms), and the molecular weight of the vapor. This subject is explained with a little more detail in App. A. The number of intermediate points depends on the number of intervals in which it is desired to divide the calculations. Usually, five or six intervals are enough. Two different types of condensation curves can be calculated: differential and integral. Integral condensation assumes that vapor and condensate remain in contact and in thermodynamic equilibrium with each other along their path through the condenser. Vertical condensers with tube-side condensation are the best example of integral condensation. Other cases approximating integral condensation correspond to condensation within horizontal tubes, condensation outside vertical tubes, and condensation outside horizontal tubes in cross-flow (TEMA type X). In differential condensation, the liquid condensate is separated from the vapor stream. This modifies the equilibrium and reduces the dew point of the remaining vapor. A typical example is the reflux countercurrent condenser, in which vapor flows upward inside tubes, while the condensate film formed on the tube surfaces drains in countercurrent. Horizontal condensers with shell types E or J are an intermediate case. To be conservative, in these cases one should use the differential condensation curve. However, the usual practice in industrial design has been to use the integral condensation curves in all cases. The number of points in the condensation curve define the number of heat transfer area calculation intervals: n points of the condensation curve define n − 1 calculation intervals. Then, for the interval enclosed by points i and i + 1, it is possible to calculate ΔQ = heat to be removed in the interval = Wh(ii+1 − ii), where i = specific enthalpy ΔQs = vapor sensible heat = Wvcv(Ti+1 − Ti) γ = ΔQs/ΔQ Knowing the mass flow of the cold stream, its physical properties, and the inlet temperature, it is possible to calculate the temperature change that this stream will suffer in each interval. This is done by assuming a countercurrent configuration, and calculating ti − ti+1 = ΔQ/Wccc . The T − Q curve for the refrigerant then is superimposed on the hot-stream temperature curve. With the four extreme temperatures for the whole process, an LMTD correction factor Ft is calculated. Then, for each interval, it is possible to calculate the LMTD as LMTD = [(Ti +1 − ti +1) − (Ti − ti)]/ln[(Ti +1 − ti +1)/(Ti − ti)] And then apply the correction factor Ft to each interval. For a better understanding of the procedure, the following example is included.

316

CHAPTER TEN

Example 10-4 It is desired to condense 0.277 kg/s of a hydrocarbon mixture coming from the top of a distillation column at 49.2°C and 4.5 at(a). Condensation takes place in the range of 49.2–27.4°C. The refrigerant is cooling water, available at 20°C, with a mass flow of 3.031 kg/s. Vapor specific heat is 1,800 J/(kg · K). The condensation curve corresponds to the first five rows of the following table, and the calculations for each interval are shown on the next rows. The LMTD correction factor was calculated with the four extreme temperatures (T1 = 49.2, T2 = 27.4, t1 = 20, t2 = 28.2) for a configuration with one shell pass and n tube passes resulting in Ft = 0.783. Vapor temperature, °C

49.2

44.3

39.0

33.4

27.4

Vapor mass fraction, kg vap/kg total Enthalpy of mixture, kJ/kg Vapor mole weight, kg/kmol Vapor mass flow, kg/s Interval ΔQ, kW ΔQs, kW γ Water temperature, °C LMTD, °C LMTD · Ft

1 375.4 58 0.277

0.728 281.6 56 0.201

0.476 187.2 54 0.132

0.233 93.8 52.5 0.064

0 0 51 0

I 26 2.42 0.093 28.2

II 26 1.91 0.073 26.1

19.5 15.29

III 26 1.33 0.051 24.1

16.5 12.89

IV 26 0.69 0.026 22.0

13.0 10.21

20 9.2 7.23

The following properties for vapor and condensate may be considered constant in all the condensation range: Liquid

Vapor

Density = 561 kg/m3 Viscosity = 0.15 cP = 0.15 × 10–3 kg/(m · s) Thermal conductivity = 0.09 W/(m · K) Specific heat = 2,500 J/(kg · K) Prandtl number = 4.16

Compressibility factor = 0.9 Viscosity = 0.81 × 10–5 kg/ms Thermal conductivity = 0.018 W/(m · K) Specific heat = 1,800 J/(kg · K) Prandtl number = 0.81

Note: A downloadable spreadsheet with the calculations of this example is available at http://www.mhprofessional. com/product.php?isbn=0071624082

A horizontal heat exchanger with the following geometry was proposed: N = number of tubes: 42 n = number of tube passes: 2 L = tube length: 6.09 m Tube diameters Do = 0.019; Di = 0.0144 Ds = shell diameter = 0.254 m Pt = tube pitch = 0.0238 m Triangular array 30 degrees c = tube clearance = 0.0048 m Segmental baffles with spacing B = 0.063 m Fouling resistances of 0.0001 (m2 · K)/W for the condensing vapor and 0.0003 (m2 · K)/W for the cooling water must be considered.

CONDENSATION OF VAPORS

Solution

317

We shall calculate first the heat transfer coefficient for the cooling water. at = tube-side flow area = (πDi2/4)N/n = 0.00341 m2 v = velocity = Wc /(1,000at) = 3.031/1,000 × 0.00341 = 0.886 m/s t = mean temperature of water = 24°C hio = 1,423(1 + 0.0146t)v0.8/Di0.2 × (Di /Do) = 3,092 (m2 · K)/W

To calculate the vapor-side coefficient, we must first calculate the coefficient corresponding to the condensate film resistance h'f . This requires initially calculating h f (without considering the vapor drag effect over the condensate film) and correcting it later for vapor velocity. We shall calculate a single hf for the whole unit using Nusselt’s condensation model: ⎛ μ 2 ⎞ h⎜ 3 L 2 ⎟ ⎝ k L ρL g ⎠

1/ 3

⎛ 4G " ⎞ = 1.51 ⎜ ⎝ μ L ⎟⎠

−1/ 3

where G" is calculated as G" = W condensate/LN 0.66 = 0.277/(420.66 × 6.09) = 0.00386 kg/(m · s). Substituting the liquid physical properties, the result is hf = 1,386 W/(m2 · K). Thus Nuf = hf Do /kL = 1,386 × 0.019/0.09 = 292.7 For each calculation interval, this coefficient must be corrected to include the effect of the vapor drag forces. The corrected condensate film coefficient is obtained with the following equations: Nu'f = G=

h'f D kL

(

= X 4 ⋅ Re 2LG + Nu 4f

)1/ 4

where Re LG =

DGρL μ L ρG

1 ⎞ 1/ 3 vapor mass flow X = 0.9 ⎛ 1 + ⎝ RH ⎠ shell-siide flow area cL (Tsat − Tw ) ρ Lμ L R= H= ρG μ G PrL ⋅ λ

(The heat of condensatio λ can be estimated as the quotient between “latent heat” and condensate mass flow.) Then λ=

Q − Wv cv (T1 − T2 ) 104, 000 − 0.277 × 1, 800 × (49.2 − 7.4) = = 336,000 J/kg Wv 0.277

This model makes it necessary to perform an iterative calculation to find Tw . For each interval, we shall assume a value of Tw that will be verified by means of Tw =

ho T − U (T − t ) ho

Once h'f is calculated, we can obtain ho (total heat coefficient of the vapor side) as ⎛ 1 γ ⎞ ho = ⎜ ' + ⎟ ⎝ h f hG ⎠

−1

318

CHAPTER TEN

This requires calculation of hG (single-phase vapor heat transfer coefficient). To simplify its calculation, we shall use Kern’s method. A vapor mass velocity Gv is calculated as the quotient between the vapor mass flow of each interval and the shell-side flow area. this flow area is obtained as as = DscB/Pt = 0.254 × 0.0048 × 0.063/0.023 = 0.00333 m2 The Kern Reynolds number for this geometry is calculated as ReK = DeqGv/μv . The equivalent diameter for this geometry is 0.0137 m, and then hG = 0.36 × ReK0.55 × Prv0.33 × kv/Deq For each interval, the overall U is calculated as U = (1/ho + 1/hio + Rfi + Rfo)–1 And finally, the required area for the interval will be A = ΔQ/UΔT. The results are shown in the following table: ρ vapor

10.97

10.76

10.55

10.44

10.35

W vapor Re R Tw H X Nu'f h'f ReK hG ho U Ainterval

0.274 555,658 946 40 0.0187 0.919 690 3,271 144,756 305 1,629 748

0.201 412,595 965 36 0.0161 0.920 599 2,841 105,383 256 1,560 733

0.132 275,092 984 32 0.0133 0.924 499 2,367 68,904 203 1,483 715

.0646 136,018 994 28 0.0101 0.930 381 1,807 33,728 137 1,335 679

0 0 1,003 24 0.00662 0.947 293 1,386 0 0 1,386 692

2.3

2.78

3.65

5.25

The total required area will be the summation of the intervals areas, which is 13.98 m2. The area of the proposed unit is 15.2 m2. This means an 8 percent excess area.

10-6 THE USE OF STEAM AS PROCESS HEATING MEDIUM In preceding sections we considered the condensation of a process vapor using a refrigerant medium, which normally consists on a utility stream. However, the most common case of vapor condensation is when steam is used as a process heating medium. In this case, the condensing steam is the utility stream, whereas the process stream is the cold fluid receiving the heat. Because of the wide application of this process, we shall study some particular characteristics of steam installations. 10-6-1 Process Heating Systems To deliver heat to a process using an external energy source, the following possibilities can be considered: • Direct heating with a flame • Heating with a fluid that delivers sensible heat • Heating with a fluid that delivers latent heat

CONDENSATION OF VAPORS

319

Direct heating with open flames in all process equipment would require the installation of expensive safety systems and better-quality materials to resist higher temperatures. The usual solution thus is to use some fluid as a vehicle to transport thermal energy where it is required. This means that heat is delivered to a circulating fluid via burning a solid, liquid, or gas fuel in a central piece of equipment, which may be a furnace or steam generator, and this fluid transports the energy to all consumers, usually in a closed cycle. So there is only one piece of equipment working with open flame, and it is built under strict safety specifications by specialized companies, and its design is optimized to reach high thermal efficiencies, whereas in the rest of the equipment heating is performed safely and economically by the circulating fluid. The circulating fluid used as heating medium may be a fluid that undergoes a change of phase, delivering its latent heat, or may be a single-phase fluid that performs its function by receiving and delivering sensible heat. For example, 1 kg of water in the vapor state condenses delivering an amount of energy equal to 2,000 J. If it is desired to deliver the same amount of heat using liquid water, which cools down from 100 to 50°C, 10 kg of water should be employed. This means that by using a fluid that exchanges latent heat, it is necessary to employ a lower mass flow to transport the same amount of energy. The fluid most commonly used as a heating fluid is steam. Its main advantages are • It has a high value of latent heat of condensation and vaporization. • The raw material to produce it (softened or demineralized water) is relatively inexpensive. • The film coefficients for steam condensation are extremely high, which means that the size of the heat transfer equipment will be relatively small. In certain cases, it is necessary to deliver heat to a process at high temperature (200°C or more). In these cases, if steam were used, the vapor pressure of this steam would be too high. Then the cost of process equipment increases because a higher pressure makes it necessary to use higher plates thicknesses, and safety specifications are stricter. In such situations, it is sometimes preferred to use as the heating medium any of the so-called thermal fluids. They are mineral oils or synthetic oils that have low vapor pressures so that they continue being liquids at high temperatures. The process equipment working with them then have low design pressures even for high operating temperatures. Steam System Installations. Figure 10-35 shows schematically a steam system installation. We can distinguish the following elements: Water-Treatment Unit. Dissolved solids may precipitate and produce scaling in the boiler and heat transfer equipment. The required quality of the water in the circuit depends on the boiler’s operating pressure, and the dissolved solids level is maintained with periodic and continuous purges. To reduce purges, the makeup water is treated to eliminate hardness and other dissolved salts. The treatment unit may include clarifiers, filters, softeners, and ion-exchange or reverse-osmosis equipment depending on the quality of both the raw and treated water. Deaereator and Condensate Tank. Since the steam condensate is free from ions, it is recovered to be recycled to the boiler. Thus the treating unit only needs to treat the makeup water that replaces purges and system losses. The deaereator eliminates dissolved air that can cause corrosion. Boiler or Steam Generator. Here, the steam is generated by the combustion of any fuel. The steam pressure is controlled by regulating the fuel supply. Sometimes it is necessary to have two or more steam qualities, for example, a high-pressure steam for turbines operation or high-temperature process heating and a low-pressure steam for lower thermal level applications. In these cases, low-pressure steam is obtained by reducing the pressure in a control valve. Industrial Process. The steam consumption in the different process equipment is represented schematically in Fig. 10-36 as a stirred jacketed vessel. As can be seen in the figure, the steam flow is regulated with an inlet valve, and the condensate is removed through a trap that avoids loss of uncondensed steam. We shall analyze in more detail the operation of these two elements for a better understanding of the steam condensation mechanism.

320

CHAPTER TEN

High-pressure header 7

PC

6a

Low-pressure header

PC

6b

5 Fuel

10 8 9

3 1

2 4

1- Raw feed water 2- Water treatment 3-Deaereator and treated water storage 4-Boiler feed pump 5-Steam generator (boiler) 6a- High-pressure consumptions 6b- Low-pressure consumptions 7- Pressure-reducing valve 8-Process equipment 9-Steam trap 10- Control valve

FIGURE 10-35 Components of a steam system.

Wc cold fluid

t1

Wh steam

t Wc

Wh condensate FIGURE 10-36

Steam-heated jacketed vessel.

Let’s consider the steam-heated jacketed vessel shown in this figure. Since the liquid is agitated, we may consider that its temperature is uniform. Let t be the temperature of the process liquid in the vessel. The steam into the jacket is at its saturation temperature and condenses, delivering heat to the process fluid. The process fluid enters the vessel at a temperature t1, and it immediately mixes with the vessel contents, increasing its temperature to t.

321

CONDENSATION OF VAPORS

If the process fluid mass flow is W, the heat that must be delivered is Q = Wc cc (t − t1 )

(10-6-1)

This heat is delivered by the steam. Assuming that the inlet steam is saturated, the heat is Q = Wh λ

(10-6-2)

because the steam condenses, delivering its heat of condensation. Now, how can we know that only condensate exits through the jacket-bottom nozzle? If this connection were an open pipe or a manual valve at a fixed opening position, it is probable that an important fraction of the inlet steam would escape through the outlet nozzle without yielding its heat of condensation. Thus the steam consumption would be much higher than that resulting from Eq. (10-6-2). Then it is necessary to install some device that only allows the passage of the condensate that has already delivered its latent heat, retaining the uncondensed steam. This function is performed by devices called steam traps. There are many different kinds of steam traps, and they will be described below. First, however, we must explain another problem that can exist in steam installations, which is the presence of air. Effect of Air in Steam Systems. Air can get into steam systems during shutdowns, when the installation cools down. When steam is again introduced into the system, a steam-air mixture will result. The presence of air in a steam system is harmful for the following reasons: 1. Steam will condense at a temperature that depends on its partial pressure. Since air is present in the mixture, the partial pressure of steam is reduced, and the condensation temperature will be lower, thus decreasing the temperature-driving force for heat transfer. 2. When steam condenses over cold surfaces, because air does not condense, the air-steam proportion in the vicinity of the heat transfer surfaces can be considerably higher than in the bulk of the mixture. This creates diffusional resistances because steam molecules must diffuse through an air-enriched layer to reach the heat transfer surfaces. This was already explained in Sec. 10-4. It was reported that only 1 percent of air in a steam system can result in a 50 percent decrease of the heat transfer coefficients. 3. Additionally, the presence of air increases corrosion rates. The usual practice to deal with this problem is to purge the installation during startup. Manual purge valves are installed in cold and dead-end points, and they are left open for a certain amount of time, purging a steam-air mixture so that most of the air is eliminated. This obviously results in a loss of steam. But after startup, air still can get into the system, dissolved in the makeup water, so this purging should be done periodically. To reduce this procedure, steam traps also must perform the air purging, so they should allow the discharge of air and condensate but retain the steam.

C

Different Types of Steam Traps. There are many types of steam traps, each one with its advantages and limitations. We shall describe the most common types. Thermostatic Bellows Trap (Fig. 10-37). The principal element is a thermostatic bellows A filled with a fluid (usually an alcohol mixture) that can vaporize or condense depending on the temperature of the trap. The boiling point of this mixture is selected based on the temperatures range in which the trap has to operate.

FIGURE 10-37

A

D B

Thermostatic steam trap.

322

CHAPTER TEN

A

D

C

B E

FIGURE 10-38

Float-type steam trap.

Thus these traps operate on the difference in temperature between steam and cooled condensate and air. Steam increases the temperature inside the thermostatic element, which causes the fluid to vaporize and expand, thus closing the discharge at C. As condensate and noncondensable gases back up in upstream piping (called the cooling leg), the temperature begins to drop, the thermostatic element contracts, and the valve opens. The amount of condensate backed up ahead of the trap depends on the load conditions, steam pressure, and size of the piping. The discharge of the trap is intermittent. Thermostatic traps also can be used for venting air from a steam system. When air collects, the temperature drops, and the thermostatic air vent automatically discharges the air at slightly below-steam temperature throughout the entire operating pressure. These traps are small in size and have high discharge capacity, but they offer little resistance to water hammer and may be affected by corrosion because the bellows is made of a very thin metal sheet. Float-Type Trap (Fig. 10-38). The float-type trap is a mechanical trap that operates on the density principle. A lever C connects the valve float B to the valve and seat D. Once condensate reaches a certain level in the trap, the float rises, opening the orifice and draining condensate. A water seal formed by the condensate prevents live-steam loss. Since the discharge valve is under water, it is not capable of venting air and noncondensables. This is why these traps are usually provided with an auxiliary thermostatic vent, as shown in Fig. 10-39. When the accumulation of air and noncondensables causes a significant temperature drop, a thermostatic air vent H in the top of the trap discharges it. This type of trap has a continuous condensate discharge because the float adopts an equilibrium position that allows evacuation of all the incoming condensate. These traps easily adapt to variable condensate flow and allow a stable operation without the process fluctuations caused by discontinuousdischarge traps. Inverted-Bucket Steam Trap (Fig. 10-40). The principal element is an inverted bucket B that can move vertically inside a casing. The bucket acts through a lever opening and closing the discharge valve. When the bucket is at the lower position, the valve opens, and when the bucket rises, the valve closes. The inlet port to the trap is located below the bucket. When the trap is full of condensate, the weight of the bucket makes it sink, opening the valve, and the condensate can discharge (left side of Fig. 10-40). When steam comes into the trap, it displaces the condensate and gets trapped below the bucket. This makes the bucket rise and closes the valve (right side of Fig. 10-40). When the trap cools down, the steam condenses, and condensate gets below the bucket, which sinks again, reopening the discharge. A small orifice is placed at the top of the bucket to vent air. Otherwise, air would accumulate below the bucket, and because it does not condense, the trap would remain closed. The air escapes through this orifice and accumulates at the top of the trap. When the trap opens, the air is eliminated by the condensate

CONDENSATION OF VAPORS

323

H

B

FIGURE 10-39

Float-type steam trap with auxiliary vent.

C

C

A

A B B

Trap in open position FIGURE 10-40

Trap in closed position

Inverted-bucket steam trap.

pressure. Obviously, some steam can escape through the orifice too, so it must be of a small diameter to reduce such steam loss. This limits the air-purging capacity of this type of trap. However, the trap is robust and has good mechanical resistance. Thermodynamic Trap (Fig. 10-41). This type of trap has a body A with inlet and outlet ports, a cover B, and a control disk C that is free to move vertically. The body has two concentric ring-shaped seats. The inner seat D surrounds the inlet orifice E and the exterior seat F close to the cover. Between these round seats is the discharge orifice G. Both seats are carefully ground, as well as the disk C, so that the disk closes against the two concentric faces at the same time, separating inlet from outlet. The interior part of the cover has a central stop H that limits the lift of the disk, so when the disk is in the upper position, there is still some room between the upper face of the disk and the lower face of the cover. This space is called the control chamber. When the disk closes against the seats, the control chamber remains isolated from the inlet and outlet ports. We shall see how this trap works (see Fig. 10-42). Let’s assume that the trap is cold initially. When air and cold condensate come into the trap, they pass through the inlet orifice E, raise the disk, and flow radially from the center to the space between both seats D and F and discharge through the orifice G (Fig. 10-42a). When steam and hot condensate come into the trap, since their specific volume is higher, the velocity below the disk increases. According to the Bernoulli principle, this velocity increase provokes a pressure reduction at the lower face of the disk, and it seats, closing the trap (Fig. 10-42b).

324

CHAPTER TEN

D H

F

C B E

G

A F

D E

G

Plant view of the seat FIGURE 10-41

Thermodynamic trap.

B

(a) FIGURA 10-42

B

(b)

D

F

(c)

Operation of a thermodynamic trap.

But part of the steam could escape into the control chamber and be trapped there, where its velocity is converted completely into pressure. The steam trapped in the control chamber exerts a downward force on the disk equal to the product of the pressure times the area of the upper face of the disk, whereas the steam below exerts an upward force equal to the pressure times the area of the orifice E, which is sensibly smaller (Fig. 10-42c). Then, while the trap is hot and there is steam into the control chamber, the trap remains closed. The control chamber losses heat to the ambient but maintains the temperature because it receives heat from the steam below the disk. When cold condensate and air come into the trap, it cools down. This makes the steam in the control chamber condense, so its pressure decreases. The condensate pressure then can raise the disk, and the cycle restarts. This type of trap is robust, small, and economic. It has only one moving part, the disk, which is made from hardened stainless steel, so its service time is high. These traps cannot be used with pressures lower than 1 bar(g), and their capacity is small. Some steam is always lost when the trap opens, so their efficiency is not very high.

10-6-2 Regulation of the Steam Flow Going back to Fig. 10-36, we shall now analyze the action of the inlet valve that regulates the steam flow to the jacket. We mentioned that the heat flow delivered to the unit is that necessary to heat the process stream from the inlet temperature t1 up to t. If the process flow Wc can fluctuate, it is necessary to adjust the steam flow to maintain the temperature of the process. Let’s suppose that somebody is operating the valve so as to keep constant the temperature t. The heat flow is Q = Wc cc (t − t1 )

(10-6-3)

CONDENSATION OF VAPORS

325

The steam mass flow required to deliver this heat is Wh =

Q λ

(10-6-4)

(In what follows, as a simplification, we shall assume that λ is independent of pressure, and we shall neglect the sensible-heat terms in the steam enthalpy.) Combining the last two equations, we get Wh =

Wc cc (t − t1 ) λ

(10-6-5)

It is obvious that the higher the process stream flow Wc, the higher is the amount of steam that must be admitted into the jacket. It’s intuitive, and we know from daily experience that in order to increase the steam flow, we must open the steam inlet valve to the jacket. However, we also know that once the steady state is reached, it is not possible to admit more steam into the jacket than it can condense. So the inlet steam flow, which equals the amount of condensate exiting through the trap, depends on the heat transfer capacity of the jacket. In other words, if U is the overall heat transfer coefficient and ΔT is the temperature difference between the steam in the jacket and the process liquid, the heat transferred will be Q = UAΔT

(10-6-6)

And then Wh =

UAΔT λ

(10-6-7)

This expression tells us that the steam flow entering the jacket is determined by the heat transfer capacity of the vessel (UA) and by the temperature difference. We can consider that U is independent of the process flow rate Wc within a certain range. But then we see that the valve opening, which regulates the steam flow, is not included directly in Eq. (10-6-7). Therefore, why can the action of the valve modify the steam flow? The explanation lies in the term ΔT of Eq. (10-6-7). ΔT is the temperature difference between the fluids at both sides of the jacket wall. This is ΔT = T − t

(10-6-8)

where T is the temperature at which steam condenses and t is the temperature of the liquid which is in the vessel. T is the condensation temperature at the pressure existing into the jacket and depends on this pressure. But this pressure equals the pressure in the steam header minus the pressure drop in the valve. When the valve is opened, its Δp decreases, and the steam pressure into the jacket increases. This makes the steam condensation temperature increase and also increases ΔT in Eq. (10-6-7). This will be better understood with the following example. Example 10-5 A jacketed vessel such as that in Fig. 10-36 has a heat transfer area of 10 m2, and the overall heat transfer coefficient U is 350 W/(m2 · K). The pressure in the steam header is 3 bar(abs). The specific heat of the process fluid is 2,500 J/(kg · K), and the inlet temperature is 20°C. The fluid must be heated in the vessel up to 70°C. Calculate the temperature in the steam jacket when the mass flow of the process fluid is a. 1 kg/s b. 1.5 kg/s

326

CHAPTER TEN

c. 1.78 kg/s d. 0.8 kg/s Solution

The amount of heat to be delivered to the process fluid per unit mass is Δic = cc (t − t1 ) = 2, 500 × (70 − 20) = 125 kJ/kg

Thus, to process 1 kg/s, the heat to be delivered is Q = 1 × 125 = 125 kW The steam flow that must be fed to the jacket is Wh =

Q λ

And taking λ = 2.115×106 J/kg (which shall be considered constant in the entire problem), the result will be Wh =

125 kJ/s = 0.059 kg/s 2,115 kJ/kg

The temperature difference required to transfer this heat is ΔT =

Q 125 x10 3 = = 35.7°C UA 350 × 10

Then the temperature inside the jacket must be T = 35.7 + 70 = 105.7 The steam pressure corresponding to this temperature is 123 kPa. Then, since the pressure in the distribution header is 3 bar (300 kPa), to heat up 1 kg/s of process liquid, the steam flow must be throttled in the valve so as to have a pressure drop of 300 − 123 = 177 kPa. The following table shows the calculation for the other process conditions requested in the example. Case

b

c

d

Wc Q(W) = 125,000 × Wc Wh (kg/s) = Q/2.115 × 106 ΔT = Q/(350 × 10) T = 70 + ΔT Psteam

1.5 187,500 0.088 53.5 123.5 220

1.78 222,500 0.105 63.5 133.5 300

0.8 100, 000 0.047 29 99 98

We shall analyze the results of this table. We see that in order to move from condition (a) to condition (b) with a higher process liquid flow rate, it was necessary to open the steam valve to increase the jacket pressure. If we want to move to condition (c) with a higher process flow, it can be seen that a jacket pressure of 3 bar (300 kPa) is necessary. But this pressure equals the header pressure. This means that the valve pressure drop must be zero. This is a limit condition that ideally happens when the valve is completely open. It is obviously the maximum pressure that can be established into the jacket, so process condition (c) determines the maximum capacity of the system.

CONDENSATION OF VAPORS

327

If it is desired to increase the operating capacity further, it will be necessary to increase the heat transfer area of the vessel. According to Eq. (10-6-7), this allows us to increase the heat flow with a lower ΔT. If a steam-heated vessel has its admission valve completely open and cannot reach the desired process condition, the steam pressure downstream of the control valve must be measured. If this pressure does not differ from the header pressure, there is some problem in vessel design (the heat transfer area is too small, the heat transfer coefficients are different from expected, or fouling is too high). On the other hand, if the valve downstream pressure is lower than the header pressure, it means that the valve is too small because even at its maximum opening its pressure drop for the required steam flow is too high. Let’s now see what happens if we attempt to reduce the process flow to 0.8 kg/s (condition d). We see that to reduce the capacity so much, it would be necessary to have inside the jacket a very low temperature fluid that corresponds to a subatmospheric pressure. In this condition, the condensate cannot discharge from the trap. Thus it builds up into the jacket, reducing the heat transfer surface exposed to the steam. This reduction in A allows us to decrease the heat transfer rate without making a subatmospheric pressure into the jacket necessary. But this type of operation can lead to fluctuating process conditions. Possible solutions to this problem may be 1. To install a condensate receiver below the vessel level with a barometric leg that provides the necessary height to overcome the negative Δp. 2. To install an inverted check valve to allow admission of air into the jacket when pressure falls below atmospheric. In this way, the steam partial pressure is reduced, and it is possible to have a lower condensation temperature with a total pressure equal to atmospheric.

GLOSSARY A = area (m2) as = shell-side flow area (m2) B = condensate layer width (m) B = baffle spacing (m) c = specific heat [J/(kg · K)] c = clearance between tubes (m) D = diameter (m) Deq = equivalent diameter (m) Ds = shell diameter (m) f = friction factor g = gravity acceleration (m/s2) G = mass flow density [kg/(s · m2)] G' = flow per unit length at the bottom of a vertical condensate layer [kg/(m · s)] G" = flow per unit length on a horizontal condensate layer [kg/(m · s)] H = parameter of Eq. (10-3-72) h = height (m) h = heat transfer coefficient (generic) [W/(m2 · K)] hf = heat transfer coefficient for the condensation of a vapor on a film without considering the effect of vapor velocity [W/(m2 · K)] h'f = heat transfer coefficient for the condensation of a vapor on a film corrected for the effect of vapor velocity

328

CHAPTER TEN

hG = single-phase heat transfer coefficient [W/(m2 · K)] i = specific enthalpy (J/kg) J = mass flow density [kg/(s · m2)] k = thermal conductivity [W/(m · K)] L = tube length (m) M = mole weight N = number of tubes n = number of passes p = pressure (N/m2) Pt = separation between tube axes in a row (m) Pr = Prandtl number Q = heat flow (J/s) q = heat-flow density [J/(s · m2)] R = parameter in Eq. (10-3-71) Re = Reynolds number Rf = fouling resistance [(m2 · K)/W] T = hot-fluid temperature (°C) t = cold-fluid temperature (°C) U = overall heat transfer coefficient [W/(m2 · K)] U' = combined heat transfer resistance including the condensate layer, fouling, and refrigerant film coefficient [W/(m2 · K)] v = velocity (m/s) W = mass flow (kg/s) Ws = shell-side mass flow (kg/s) W ' = mass flow toward the film surface (kg/s) Wf = mass flow at the bottom of the condensate layer (kg/s) x = longitudinal coordinate (m) y = longitudinal coordinate (m) z = longitudinal coordinate (m) λ = heat of condensation (J/kg) δ = film thickness (m) δf = film thickness at the film bottom (m) μ = viscosity [kg/(m · s)] ρ = density (kg/m3) τ = shear stress (N/m2) Subscripts 1 = inlet 2 = outlet c = cold h = hot

CONDENSATION OF VAPORS

329

i = internal K = Kern's method L = liquid m = mean o = external R = refrigerant T = total (liquid + vapor) V = vapor w = wall x = position

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

Nusselt W.: “Surface Condensation of Water Vapor,” Zeitschr Ver Deutsch Ing 60(27):541–546, 1916. Kern D.: Process Heat Exchange. New York: McGraw-Hill, 1950. Dukler A. E.: “Dynamics of vertical falling film systems,”: Chem. Eng. Progr. 55(10):62–67, 1959. Collier J.: Convective Boiling and Condensation. London: McGraw-Hill, 1967. Boyko I. D., Krushilin G. N.: “Heat Transfer and Hydraulic Resistance during Condensation of Steam in a Horizontal Tube and a Bundle of Tubes,” Int. J. Heat Mass Transfer 10:361–373, 1967. Carpenter E. F., Colburn A. P.: “The Effect of Vapor Velocity on Condensation Inside tubes,” Inst. Mech. Engrs Proc. General Discussion on Heat Transfer. New York: ASME, 1951, pp. 20–26. Mueller A. C.: in Heat Exchanger Design Handbook. New York: Hemisphere Publishing, 1983. Wallis G. B.: “Flooding Velocities for Air and water in Vertical Tubes,” U.K.A.E.A Rep. AEEW-R-123, 1961. Othmer D. F.: Ind Eng Chem 21:576, 1929.

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CHAPTER 11

BOILING

11-1 MECHANISMS OF HEAT TRANSFER TO BOILING LIQUIDS The change of phase from liquid to vapor can take place following two different mechanisms: evaporation and boiling. Let’s assume that we have a subcooled liquid (at a temperature below the equilibrium temperature at the existing pressure). If this liquid comes in contact with a heating surface at a higher temperature, heat will flow from the heating surface to the liquid. The heat flux can be expressed as q = h(TW − t F) where tF is the liquid temperature (away from the surface), TW is the temperature of the surface, and h is the convection heat transfer coefficient corresponding to the hydrodynamic conditions of the system, which, in natural convection, is a function of the temperature difference (TW − tF). For low heat transfer rates (or low wall temperatures), the only effect of the convection heat flux is to progressively increase the liquid temperature. When heat flux exists, there is a temperature profile, as shown in Fig. 11-1. As long as the liquid is being heated and the temperature at the free surface of the liquid increases, the vapor pressure of the liquid at the liquid-vapor interface also increases, and liquid molecules abandon the liquid phase. This phenomenon is called evaporation, and its characteristic is that the transition from liquid to vapor state occurs only at the free surface of the liquid. Evaporation takes place at any liquid temperature, and the driving force for this mechanism is the partial pressure difference of the volatile component between the liquid surface and the vapor phase. Let’s assume that now it has been decided to increase the heat transfer rate in the system of Fig 11-1. To achieve this, it will be necessary to increase the temperature of the heating surface. It is then possible that the solid-surface temperature TW will be increased above the liquid-saturation temperature TSAT . Since the liquid temperature varies continuously from TW to tF , part of the liquid also will be above the saturation temperature TSAT . In this case it is possible that vapor bubbles begin to appear near the wall. This process is called boiling. While the bulk temperature of the liquid tF (away from the solid wall) is still below the saturation temperature, once the bubbles formed at the heating surface depart from the surface and contact the colder liquid, collapse, and yield their latent heat of condensation to the liquid. This phenomenon is called subcooled boiling and may be described as formation of vapor bubbles at the heating surface with further condensation of them when they depart from the surface. If the heat transfer to the liquid continues, its temperature will increase, and finally, all the liquid will reach the saturation temperature. In this case, the bubbles do not collapse any more and can reach the liquid surface. This is called saturated boiling. The three mechanisms are shown in Fig. 11-2.

11-1-1 Formation of Vapor Bubbles in Boiling Liquids: Equilibrium Condition of a Bubble For a better understanding of the different boiling heat transfer mechanisms, it is necessary to review some topics dealing with the formation of bubbles over a solid-liquid interface. We shall first consider the equilibrium condition of a vapor bubble submerged into a liquid and, for the moment, far from any solid surface.

331

332

CHAPTER ELEVEN

tF

Gas-liquid interface

Subcooled liquid

TW FIGURE 11-1

Heating surface

Temperature profile in a heated liquid.

tF

tF

tF

TW

TW

TW

(a) Small TW – tF tF < TSAT single-phase convection

(b) Higher TW – tF tF < TSAT subcooled boiling

(c) tF = TSAT saturated boiling

FIGURE 11-2

Liquid vaporization mechanisms.

The study of the equilibrium condition of bubbles is a classic in physics books, and it can be easily demonstrated that the necessary condition for the existence of a bubble is that the pressure inside the bubble be higher than the exterior pressure by an amount given by 2σ/r, where σ is the surface tension and r is the bubble radius. We shall apply this concept to the case of a vapor bubble in thermodynamic equilibrium with the surrounding liquid. Let’s assume that we have a system at a constant temperature TG consisting of a single-component liquid pool and a spherical vapor nucleus with radius r. Since we assume that the system is in thermodynamic equilibrium, the pressure above the liquid-vapor interface, which is also the pressure inside the bubble, shall be the equilibrium vapor pressure at temperature TG . We shall call this pressure pG . Note: In fact, the vapor pressure over a curve interface is slightly different from the value obtained from the saturation curve. This is a simplification of the problem, and for a more rigorous treatment, ref. 1 may be consulted. As stated previously, there must be a pressure difference between the interior and the exterior of the bubble, so if pF is the exterior pressure, then pG − pF =

2σ r

(11-1-1)

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333

Let TSAT be the saturation temperature corresponding to pF. This means that the pairs ( pG , TG) and ( pF , TSAT) determine two points belonging to the vapor-pressure curve of the liquid. Since pG > pF , it is also true that TG > TSAT . This means that the equilibrium condition of the bubble requires that the system temperature TG be higher than the saturation temperature corresponding to the imposed liquid pressure. In other words, the liquid must be superheated, where TG − TSAT is the value of superheat required by the bubble to exist in equilibrium. If the temperature of the system is increased above TG , the bubble grows, whereas if the temperature decreases below TG , the bubble collapses. The slope of the saturation curve of a pure liquid can be expressed by the Clausius Clapeyron equation: λ dp = dT T ( vV − v L )

(11-1-2)

where λ is the heat of vaporization and vV and vL are the specific volumes of vapor and liquid, respectively. T is the absolute temperature (K). Since vV >> vL, and assuming that the ideal gas equation of state can be applied to the vapor phase, MpvV = RT

(11-1-3)

dp λM = dT p RT 2

(11-1-4)

Equation (11-1-2) then can be rewritten as

Remembering that the points ( pF , TSAT) and (pG , TG) belong to the saturation curve, Eq. (11-1-4) can be integrated between these points, resulting in ln

pG λM ⎛ 1 1 ⎞ = − pF R ⎜⎝ TSAT TG ⎟⎠

(11-1-5)

It also must be considered that the pressure difference between the interior and exterior of the bubble is small in comparison with the absolute value of the pressure pF , and since ln(1 + x) ≅ x for x 0, the necessary value of TG − TSAT for the existence of a nucleus in equilibrium is smaller than that calculated with Eq. (11-1-8) because the correction factor is smaller than 1. When a liquid does not wet the surface at all, then θ = 180 degrees. In this case, Φ equals 0, which means that no superheat is required. The equilibrium condition is that the liquid temperature coincides with the saturation temperature. This is the situation on a planar interface.

Liquid Gas bubble θ Solid FIGURE 11-3

Bubble contact angle θ.

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335

Φ 1

0

0

FIGURE 11-4

90

180

270

θ(o)

Superheat correction factor.

FIGURE 11-5

Case of θ = 0.

One may think that it is not possible to have contact angles over an interface that are higher than 180 degrees. However, normally, all surfaces have pits or cavities, and a vapor nucleus can exist in them, as shown in Fig. 11-6. In a cavity of this type, with an irregular shape, the curvature direction of the interface can be inverted. In this case, the contact angle can be higher than 180 degrees, so the correction factor Φ is negative. This means that the superheat is also negative, and a vapor nucleus can exist even at temperatures lower than the saturation temperature. This explains the reason why the pores that normally are present on the solid surface can act as nucleation centers because even at low temperatures it is possible for a vapor embryo to be present in them and this vapor embryo can give origin to a bubble when the temperature is increased. The way in which a bubble grows is shown in Fig. 11-7. Let’s imagine a vapor nucleus in the interior of a pit, represented by the situation indicated as 1 in the figure. We have already explained that this nucleus can exist even when the temperature of the system is lower than the saturation temperature. If now the surface temperature is raised above the saturation temperature, the bubble will grow and will move to position 2 with a positive value of Φ because now θ < l80 degrees. If the bubble continues growing and moves to position 3, the curvature radius of the bubble decreases with respect to that of situation 2. Since the superheat required for the equilibrium condition of the bubble is inversely proportional to the radius, the result is that situation 3 requires a higher superheat than situation 2. This means that to achieve the bubble growth from situation 2 to situation 3, it is necessary to increase the temperature. The same can be said to pass from situation 3 to situation 4. However, once the bubble radius equals that of the cavity mouth (situation 4), to attain further growth and to pass into situation 5, the curvature radius of the bubble must start increasing, and the necessary overheating for the equilibrium of the system decreases.

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CHAPTER ELEVEN

Liquid

Solid θ

Vapor

FIGURE 11-6

Vapor embryo into a cavity.

5 4 3 2

1

FIGURE 11-7

Bubble growth.

Since the degree of superheat that the system has reached up to that moment is higher than the bubble equilibrium temperature, the bubble starts growing spontaneously and finally departs from the surface. This means that in the process of bubble growth, there is a critical instant, represented by situation 4 in Fig. 11-7, after which bubble growth is spontaneous and continues up to bubble departure. If the temperature of the system reaches or surpasses the superheat required by this critical instant, the bubble will be released. This critical instant is reached when the bubble has a radius equal to that of the cavity mouth. It must be noted that when moving from situation 3 to situation 4, the contact angle θ also decreases. Since the reduction in θ produces an opposite effect to a reduction in r, the combination of both effects causes the critical bubble radius to be somewhat different than the mouth-cavity radius. However, this difference is not taken into account in the majority of the models.

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337

Nucleation in a Temperature Gradient. The preceding discussion of vapor nucleation considered only uniform temperature in all the system. However, this situation is not real when the solid surface over which the bubbles are formed acts as a heating surface. In this case, if the surface temperature is TW1, for heat transfer to exist, it is required that this temperature be higher than the liquid temperature. This means that a temperature gradient exists in the liquid phase. Figure 11-8 represents a heating surface with an active nucleation site. Well away from the wall, the liquid temperature is tF . In this case, tF has been represented below TSAT . This corresponds to a situation where the bulk of the fluid is subcooled, but the conclusions are also valid for saturated boiling with tF = TSAT . The change in the liquid temperature from TW1 to tF takes place in a zone near the solid, which is called the boundary layer. A usual simplification is to assume a linear temperature profile through the boundary layer. Let’s assume that this profile is represented by line I in Fig. 11-8.

When the wall temperature is TW 3, those cavities with radii between r1 and r 2 will produce bubbles

Temperature axis r2 r1

TW 3 III II Bubble equilibrium curve

I

TW 2 = TWONB TW 1 TSAT

δ δ = boundary layer thickness for a wall temperature = TW 1

tF rC

Distance to the wall

FIGURE 11-8

Temperature profiles and bubble equilibrium condition.

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CHAPTER ELEVEN

If h is the heat transfer coefficient, the heat-flux density at the interface is k

TW 1 − t F =q δ

(11-1-9)

where δ is the distance in which the temperature varies from TW1 to tF. Since q = h(TW 1 − t F)

(11-1-10)

it follows that δ=

k h

(11-1-11)

Typically, for the case of stagnant water, the boundary-layer thickness is about 0.1 mm. The dashed line in Fig. 11-8 also represents Eq. (11-1-8): TG = TSAT +

2 RTSAT 2σ λMpF r

(11-1-12)

This equation represents the equilibrium temperature for a bubble of radius r. The plotted curve is the function TG = f (r) given by Eq. (11-1-12). It was explained previously that during bubble growth, there is a critical instant when the size of the bubble reaches that of the cavity mouth. The following theory was postulated by Hsu2 and establishes that the bubble will grow and release if the temperature of the liquid surrounding the top of the bubble, when this critical instant is reached, exceeds the temperature required for the nucleus to remain in equilibrium given by Eq. (11-1-12). Let’s consider the behavior of the cavity represented in Fig. 11-8. Figure 11-9 represents the situation in more detail. The temperature at the top of the bubble can be obtained from the line representing the temperature profile at a distance r = r1 from the wall (r1 = cavity radius). This temperature is Tb . According to Hsu2, the temperature required for bubble growth is given by the equilibrium curve. This value is Ta. In the situation represented in Fig. 11-9, since Ta is higher than Tb, the bubble cannot progress, and this situation is the same for any cavity radius because for any value of r, the temperature profile is below the equilibrium temperature. Then none of the cavities, whatever their radii, have reached the necessary temperature to produce bubbles. Coming back to Fig. 11-8, let’s assume that the surface temperature is increased up to TW2. In this case, the temperature profile changes, and it becomes tangent to the curve that represents the equilibrium condition at a point corresponding to a radius rc. In this situation, the cavities whose radii are equal to rc would have reached the condition required for nucleation, and they will start producing bubbles. TW 2 is the wall temperature at which the production of bubbles starts. It is called the onset of nucleation boiling temperature (TWONB) Let’s assume that the wall temperature is increased further up to a value TW 3 so that the temperature profile in the system corresponds to line III in Fig 11-8 (see also Fig. 11-10). It can be seen that now there is a range of r values for which the system temperature is higher than that required for bubble growth. This range corresponds to r values between r1 and r2. It means that when the wall temperature is TW 3 , all the cavities whose sizes are between r1 and r2 will be active, producing bubbles. Thus, as the wall temperature is being raised, the number of active sites increases and the boiling rate also increases. This model, then, allows us to calculate the wall temperature at which nucleate boiling starts (TWONB), which corresponds to TW 2 in Fig. 11-8. This temperature can be obtained by finding the TW value that makes the temperature profile tangent to the bubble equilibrium curve. It can be demonstrated geometrically that the abscissa of the tangency point is half the abscissa at which the tangent intersects the asymptote of the equilibrium curve given by T = TSAT .

BOILING

Temperature axis Temperature profile in the liquid Bubble equilibrium curve

Ta = temperature required for the bubble to progress Tb = temperature at top of the bubble r1

Distance to the wall

FIGURE 11-9

Wall superheat required for bubble growth.

Temperature axis TW 3 Temperature profile in the liquid Bubble equilibrium curve

Distance to the wall r1 FIGURE 11-10

r2

Range of active cavity sizes at a certain wall superheat.

339

340

CHAPTER ELEVEN

Temperature axis

Temperature profile in the liquid

TWONB

Bubble equilibrium curve

TTAN

TSAT

rC

Distance to the wall FIGURE 11-11

In saturated boiling TWONB − TTAN = 1/2 (TTAN − TSAT).

With relation to Fig. 11-11, if the temperature corresponding to the tangency point is TTAN, the result will be 1 TWONB − TTAN = TTAN − TSAT = (TWONB − TSAT) 2

(11-1-13)

According to Eq. (11-1-8), at the tangency point, TTAN − TSAT =

2 2σRTSAT λMpF rC

(11-1-14)

While the equation of the line representing the temperature profile can be written TWONB − TTAN =

qrC k

(11-1-15)

Multiplying Eqs. (11-1-14) and (11-1-15) together and considering Eq. (11-1-13), we get 2 q 2σRTSAT 1 (TWONB − TSAT)2 = λMkpF 4

(11-1-16)

Since pF and TSAT define a point on the saturation curve, the specific volume corresponding to this state will be called vVSAT . And applying the ideal gas state equation, we get vVSAT =

RTSAT MpF

(11-1-17)

And Eq. (11-1-16) can be rewritten as q=

k (T − T )2 4 B WONB SAT

(11-1-18)

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341

where B=

2σTSAT vVSAT λ

(11-1-19)

A more rigorous deduction gives the following expression4 for B: B=

2σTSAT ( vV − v L )SAT λ

(11-1-20)

Equation (11-1-18) gives a relationship between the heat-flux density and the wall temperature at the onset of nucleate boiling. Since up to that moment the heat transfer regime was single-phase convection, defined by q = h(TW − t F )

(11-1-21)

if a correlation for h is available (the usual correlations for convection in liquids can be used), Eqs. (11-1-18) and (11-1-21) can be solved, and it is possible to calculate the values of q and TWONB for the onset of nucleate boiling. Some authors consider that the presence of a bubble produces a distortion of the temperature profile so that the shape of the isotherm is as shown in Fig. 11-12. In this case, if rc is the radius of the first bubble reaching the equilibrium condition, the isotherm that passes near its top is, away from the bubble, at a distance nrc from the solid surface, where n is a factor greater than 1.

Temperature axis Evolution of the temperature profile when changing TW TWONB q Bubble equilibrium curve n rC Distance to the wall

Isotherm at the bubble temperature TG rC

FIGURE 11-12

Distortion of the isotherms in the vicinity of a bubble.

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CHAPTER ELEVEN

Some authors suggest a value between 1 and 2 for n. Apparently, the value n = 1 reproduces with good agreement experimental results for the boiling of water. Han and Griffith3 suggested to assume n = 1.5 based on theoretical considerations. With this model, Eq. (11-1-18) can be modified as q=

k ⎛ TWONB − TSAT ⎞ ⎜ ⎟⎠ 4B ⎝ n

2

(11-1-22)

One deficiency of the previously exposed theory is that it assumes a continuous distribution of cavity sizes from r = 0 to r = ∞. However, for a cavity to act as a nucleation site, it is necessary that its size is not too big. In such cases, the liquid completely fills the cavity, and if a vapor embryo is not present, the cavity cannot generate a bubble. This will be better understood with the following example.4 Example 11-1 Calculate the critical radius and wall temperature required for the onset of nucleation in the case of water at atmospheric pressure and at its boiling temperature (saturated boiling). 1. Obtaining the temperature profile in the liquid. Up to the start of boiling, the natural convection film coefficient for a planar surface in contact with a liquid is given by

Solution

h = 0.14

k L ⎛ β L g ΔT ρ L2 D 3 cL μ L ⎞ ⎜ ⎟ D⎝ kL ⎠ μ L2

1/ 3

For water at 100°C, βL = 7.2 × 10–4 K–1 ρL = 960 kg/m3 PrL = 1.9

μL = 2.97 × 10–4 kg/(m · s] kL = 0.68 W/(m · K) g = 9.8 m/s2

⎡ 7.2 × 10 −4 × 9.8 × 960 2 × 1.9 ⎤ h = 0.14 × 0.68 ⎢ ⎥ (2.97 × 10 −4 )2 ⎣ ⎦

0.33

ΔT 0.33 = 453ΔT 0.33 = 453(TW − TSAT )0.33

Since in this case tF = TSAT . The boundary-layer thickness is δ=

k 0.68 = = 1.5 × 10 −3 (TW − TSAT )−0.33 h 453(TW − TSAT )0.33

This means that the boundary-layer thickness is a function of TW and for each TW, the temperature profile is given by a line with a slope (TW – TSAT)/δ. Figure 11-13 shows the temperature profile for three different values of the wall temperature. 2. Bubble equilibrium curve. The bubble equilibrium curve is given by TG − TSAT = For water at 100°C, σ = 0.058 N/m M = 18 TSAT = 373 K λ = 2.257 × 106 J/kg pF = 1.01 × 105N/m2

2 RTSAT 2σ B = r λMpF r

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343

Temperature axis TW 2 = TWONB δ2 = δONB = boundary layer thickness at the onset of nucleation

TW 3 TW 2 III

Bubble equilibrium curve TG = f (r )

II

I

TW 1 T SAT

Distance to the wall δ1 δ2 FIGURE 11-13

δ3

Boundary-layer thickness at different wall superheats.

The ideal gas constant is R = 8,307 J/(kg · mol · K). With these values, we get B = 3.26 × 10–5. The bubble equilibrium curve then can be written as TG − TSAT =

3.26 × 10 −5 r

This is the equation of a hyperbole with asymptotes T = TSAT and r = 0. 3. Determination of the wall temperature for the start of boiling. We must calculate the wall temperature at which the temperature profile is tangent to the bubble equilibrium curve. It has been demonstrated that in that condition, Eqs. (11-1-18) and (11-1-21) must be satisfied simultaneously. Then q=

k 0.68 (T − T )2 = (TWONB − 373)2 4 B WONB SAT 4 × 3.26 × 10 −5

and q = h(TWONB − TSAT ) = 453(TWONB − TSAT )4 / 3 This is a two-equations system with two unknowns. Solving, we get TWONB = 373.02 K and q = 2.2 W/m2. The value of rc then can be found as δ ONB 1.5 × 10 −3 = (TWONB − TSAT)−0.33 2 2 = 0.75 × 10 −3 (373.02 − 373)−0.33 = 2.76 × 10 −3 m = 2.76 mm

rC =

This means that according to the theoretical prediction, with a wall temperature only 0.02 K above the saturation temperature, boiling already starts. However, it can be found experimentally that a few degrees of superheat are always necessary for the start of boiling. The discrepancy lies in the fact that not all the cavities or pores of the heating surface can act as nucleation sites. If the size of a cavity is large, once the first vapor bubble detaches, the liquid completely fills the cavity, and it cannot continue acting as a nucleation site.

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Note that according to these calculations, the critical radius is 2.76 mm, which hardly can be accepted as a pore. It is normally accepted that only cavities whose radii are 5 μm or less (5 × l0–6 m) can act as nucleation sites. Some authors reduce this figure to 1 μm or even less. If we accept 5 μm as the minimum pore radius capable of generating bubbles, this means that the wall temperature has to be raised up to the value that makes the liquid temperature at 5 μm from the wall reach the required superheat predicted by Eq. (11-1-8) for a cavity r = 5 μm. This is illustrated in Fig. 11-14. Since in this case TG ≅ TW , then TG ≅ TW = TSAT +

3.26 × 10 −5 3.26 × 10 −5 = 373 + = 378.9 K r 5 × 10 −6

If we accept that the minimum active-site radius is 1 μm, then the required wall temperature for the start of boiling is 405.6 K.

11-1-3 Bubble Diameter at Detachment As explained previously, once a bubble has reached its critical radius, it continues growing spontaneously. For a certain amount of time, owing to surface-tension forces, the bubble remains adhered to the solid surface, but as it continues growing, the buoyancy forces, which depend on the third power of the radius, overcome the surface-tension forces, and the bubble detaches from the solid. Several models were proposed to predict the growth velocity of bubbles and the detachment diameter. Detachment diameter is obtained by a balance between surface-tension and buoyancy forces. The following expression is obtained31 1/ 2

⎡ ⎤ σ Detachment diameter = Dd = 0.0208θ ⎢ ⎥ ⎣ g(ρ L − ρV ) ⎦

(11-1-23)

where θ is the contact angle of the bubble at detachment (in degrees).

Temperature axis

TW TG

Bubble equilibrium curve

Distance to the wall 5 μm FIGURE 11-14

Wall superheat for a 5-μm cavity.

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11-2

345

POOL BOILING To study the boiling mechanism, it is necessary to distinguish between the boiling of stagnant liquids (pool boiling) and that of moving liquids, which is called forced convective boiling. We shall first study the case of pool boiling.

11-2-1

Pool-Boiling Curve We shall consider the case of a liquid at its saturation temperature TSAT contained in a vessel. This liquid is being heated by means of a heating surface at a temperature TW , so TW > TSAT . The temperatures difference TW – TSAT is called ΔTSAT . The results of investigations about heat transfer rates in pool boiling are usually plotted on a graph of surface heat flux against heater-wall surface temperature or, more frequently, against the wall superheat ΔTSAT rather than the wall temperature itself. The resulting curve—the boiling curve—is like that shown in Fig. 11-15. (When the relationship between q and ΔT is not linear, the definition of a heat transfer coefficient h is not particularly useful, and this is why it is preferred to present the results directly as a plot of q against ΔT.) It can be observed on the curve in the figure the existence of several regions. In the region corresponding to low wall temperatures F (curve AB), heat transfer takes place without bubble formaD q tion. Heat is removed by natural convection because the D′ temperature required for the onset of nucleate boiling, TWONB, C has not been reached. In this region, all the correlations for natural convection are valid. B″ When point B is reached, the wall superheat becomes sufficient to cause vapor bubble formation at the heating surface. This was analyzed in detail in the preceding section, and we E B B′ also explained how to predict the value of this temperature. When bubble production starts, the heat transfer coefficient suddenly increases as a consequence of the liquid agitaA tion produced by the bubbles. This explains the discontinuity presented by the curve at this point. ΔTSAT The boiling curve in Fig. 11-15 can be constructed in two FIGURE 11-15 Typical pool-boiling curve. ways: 1. Increasing the wall temperature slowly and continuously and measuring the resulting heat flux 2. Increasing the heat flux slowly and continuously and measuring the resulting wall temperature (This is the procedure when heat is supplied by an electric resistance. The electric power delivered to the resistance is increased progressively, and this provokes an increase in the wall temperature.) If the boiling curve is constructed by increasing the wall temperature, it is observed that when point B is reached, a slight increase in TW provokes a jump in heat flux to point B''. If, on the other hand, the curve is constructed by increasing the heat flux, when point B is reached, a slight increase in q causes a decrease in the wall temperature to point B' because, owing to the increase in the heat transfer coefficient, a smaller ΔT is required to maintain the heat flux. Region B'CD corresponds to the nucleate boiling regime; this means with bubble production. It can be observed now that the slope of the boiling curve is higher than that of region AB, indicating an increase in the film boiling coefficient. The agitation effect produced by the detachment of the bubbles explains the high values of the heat transfer coefficients in this region. As long as the difference TW – TSAT is being increased, the heat flux also increases, but a moment is reached when a change begins in the hydrodynamic regime of the system. The bubble production is so high that they begin to agglomerate before detaching, and a vapor blanket forms over the surface. This vapor blanket prevents the liquid from flowing onto the surface.

346

CHAPTER ELEVEN

As long as ΔT is increased further, more and more heating surface is being covered by this blanket, and the heat transfer coefficient falls. This corresponds to region DE of the boiling curve. In this region, a quite unique phenomenon in heat transfer takes place, consisting of a reduction in heat flux with an increase in the driving force. Point D is called the burnout point. This is so because when the experimental construction of the boiling curve is performed, increasing the heat flux as the independent variable and measuring the resulting wall temperature, once point D is reached, a further increase in the heat flux moves the system to situation D', with a sudden increase in the temperature of the heating element (normally an electric resistance). In this case, the resistance may burn out, and that is the reason for the expression. In region EF, the heating surface is covered by a stable vapor film, and the heat has to flow to the liquid through it. In this region, the wall temperature is high enough for the radiation effects to become significant, and heat flux begins again to increase with the increase in ΔT. This mechanism is known as film boiling. This effect sometimes can be observed if a drop of water falls onto a very hot iron plate. In this case, sometimes it can be seen that the drop takes a certain amount of time to evaporate and remains, for a while, moving rapidly over the surface up to its final consumption. This is due to the formation of a vapor film that insulates the drop from the surface, and the heat transfer is through this vapor film.

11-2-2

Boiling-Curve Correlations Several correlations have been proposed for each region of the boiling curve. They represent the experimental results with more or less success. It was not possible to find general correlations that adjust the results of different authors. The explanation may be in the fact that boiling is a very complex phenomenon and depends on a large number of variables. In the following sections, we shall present the best-known correlations. Single-Phase Convection (Zone AB). In this zone, the usual single-phase natural convection correlations may be used. For example, for natural convection in a turbulent regime from a horizontal plane, a wellknown correlation is ⎛ βg ΔTD 3ρ 2 c μ ⎞ hD L L L = 0.14 ⎜ ⎟ kL kL ⎠ μ 2L ⎝

1/ 3

(11-2-1)

where D is the diameter of the surface. Onset of Nucleation Boiling (Point B). The model based on Hsu’s theory to predict the onset of nucleate boiling was already explained (see Sec. 11-1). Nucleate Boiling Zone (Zone B'D). This region of the boiling curve is the most important for design because most heat transfer equipment normally operates in it. Many researchers tried to correlate the heat transfer coefficients, and many expressions have been proposed. Most of these correlations look completely different and are based on different variables. This is confirmation that a good understanding of this phenomenon is still lacking, and a large amount of research is still to be done. In what follows we shall present the best-known correlations. Even the usual form of correlation is to express the heat transfer coefficient as h = aΔT b. Substituting h with q/ΔT or ΔT with q/h, it is possible to present the correlations as h = f (q) or q = f (ΔT). We shall use all these forms. (a) Rohsenow Correlation. Rohsenow7 proposed a correlation of the type Nu = a Re b PrLc

(11-2-2)

For definitions of Nusselt and Reynolds numbers, the bubble detachment diameter predicted by Eq. (11-1-23) is used.

347

BOILING

The velocity in Reynolds number is taken as the superficial velocity of liquid toward the heating surface. If q is the heat-flux density and λ is the latent heat of vaporization, the vaporization rate on the surface (kg/s) is given by q/λ, and the velocity of the liquid toward the surface to replace the vaporized mass will be q/λρL. Hence the Reynolds number will be Re =

1/ 2

⎡ ⎤ σ ⎢ ⎥ g ( − ) ρ ρ L V ⎦ ⎣

q λρL

ρL μL

(11-2-3)

It must be noted that the contact angle θ of the Eq. (11-1-23) has not been included. This is due to the fact that this variable is very difficult to measure, and it was decided not to include it in the correlation, The Nusselt number then will be 1/ 2

Nu =

⎤ hDd h ⎡ σ = ⎢ ⎥ kL k L ⎣ g(ρL − ρV ) ⎦

(11-2-4)

Re (1 − n) Pr − m

(11-2-5)

The correlation proposed by Rohsenow is Nu =

1 CSF

which can be arranged to give n

1/ 2 m +1 ⎧⎪ q ⎡ ⎤ ⎪⎫ ⎛ cL μ L ⎞ cL ΔTSAT σ = CSF ⎨ ⎢ ⎥ ⎬ ⎜ λ μ λ g(ρL − ρV ) ⎦ ⎪ ⎝ k L ⎟⎠ ⎭ ⎩⎪ L ⎣

(11-2-6)

The original equation had n = 0.33 and m = 0.7. Afterwards, Rohsenow recommended that m be changed to 0 for the case of water. Values of CSF for several solid-liquid combinations are included in Table 11-1. A more detailed study of CSF values was performed by Vachon and colleagues.6 The author’s recommendation, in case a specific value for the particular fluid surface combination is not available, is to assume as a first approximation CSF = 0.013. It is not indicated how to later improve this first approximation. (b) McNelly Equation. McNelly13 proposed the following equation: ⎛ qc ⎞ h = 0.225 ⎜ L ⎟ ⎝ λ ⎠

0.69

0.31 ⎛ ρL ⎞ ⎛ pk L ⎞ − 1⎟ ⎜⎝ ⎟⎠ ⎜ σ ⎝ ρV ⎠

0.33

(11-2-7)

where p is the system pressure. (c) Forster and Zuber Equation. These authors8 proposed a correlation of the type Nu =

hrb = 0.0015 Re b0.62 PrL0.33 kL

(11-2-8)

TABLE 11-1 Values of the CSF Constant in Rhosenhow’s Equation Fluid-Surface Combination

CSF

Fluid-Surface Combination

Water-nikel Water–stainless steel Water-copper Carbon tetrachloride–copper

0.006 0.015 0.013 0.013

n-Butylic alcohol–copper CO3K2 50%–copper CO3K2 35%–copper Isopropylic alcohol–copper

CSF 0.003 0.0027 0.0054 0.0022

348

CHAPTER ELEVEN

where rb =

ΔT ⎛ 2 πk L ρ L cL σ ⎞ ⎛ ρ L ⎞ λρV ⎜⎝ Δ pSAT ⎟⎠ ⎜⎝ Δ pSAT ⎟⎠ Re b =

πk L cL μL

2 ⎛ ρL ΔTSAT ⎞ ⎜ ρ λ ⎟ ⎝ ⎠ V

0.225

(11-2-9)

2

(11-2-10)

Substituting, we get ⎛ k 0.79 c 0.45ρ0.49 ⎞ 1.24 0.75 q = 0.00122 ⎜ 0.5L 0.29L 0.24L 0.24 ⎟ ΔTSAT Δ pSAT ⎝ σ μ L λ ρV ⎠

(11-2-11)

where ΔpSAT is the difference between the vapor pressures at the heating surface and liquid temperatures. (d) Mostinsky Equation. Mostinsky12 proposed the following dimensional correlation: ⎡ ⎛ p⎞ h = 0.00417 pc0.69 q 0.7 ⎢1.8 ⎜ ⎟ ⎢⎣ ⎝ pc ⎠

0.17

⎛ p⎞ + 4⎜ ⎟ ⎝ pc ⎠

1.2

10 ⎛ p⎞ ⎤ + 10 ⎜ ⎟ ⎥ ⎝ pc ⎠ ⎥⎦

(11-2-12)

where pc (critical pressure) must be expressed in kPa and q in W/m2 to get h in W/(m2 · K). If pc is expressed in N/m2, the value of the constant is 3.7 × 10–5. (e) Stephan and Abdelsalam Correlations. These researchers14 made a regression of experimental results obtained from different authors. Using dimensional analysis techniques, they obtained the following dimensionless groups to express the solution of the problem: ⎛ qDd ⎞ X1 = ⎜ ⎟ ⎝ k L TSAT ⎠

⎛ α 2ρ ⎞ L X2 = ⎜ ⎟ D σ ⎝ d ⎠

⎛c T D 2⎞ X 3 = ⎜ L SAT2 d ⎟ α ⎝ ⎠

⎛ λD 2 ⎞ X 4 = ⎜ 2d ⎟ ⎝ α ⎠

⎛ρ ⎞ X5 = ⎜ V ⎟ ⎝ ρL ⎠

⎛c μ ⎞ X6 = ⎜ L L ⎟ ⎝ kL ⎠

⎛ρ c k ⎞ X7 = ⎜ W W W ⎟ ⎝ ρ L cL k L ⎠

⎛ ρ − ρV ⎞ X8 = ⎜ L ⎟ ⎝ ρL ⎠

(11-2-13)

It is interesting to note that the dimensionless group X 7 is a ratio of physical properties of the heatingsurface material and liquid properties. In these expressions, α is the liquid thermal diffusivity (k/ρc), and Dd is the bubble detachment diameter given by Eq. (11-1-23). TSAT is the absolute saturation temperature. The authors correlated the Nusselt number (hDd /kL) as a function of these dimensionless groups. It was not possible to find a single correlation to adjust the experimental data for all types of compounds. The authors, then, divided the data in four groups corresponding to the following four types of substances: 1. 2. 3. 4.

Water Hydrocarbons Cryogenic fluids Refrigerants

Since in the experimental information analyzed by these authors there were no measurements of the contact angle θ included in the definition of Dd , they correlated the data using typical values of θ for each of the four groups of substances. The values of θ, as well as the correlations they obtained, are shown in Table 11-2. Boiling of Mixtures. All the previous correlations are recommended by their authors for boiling of singlecomponent fluids. In the case of mixtures, the heat transfer rate is considerably reduced by the existence of diffusive resistances. In these cases, the film coefficients must be corrected as indicated in Sec. 11-4-2.

BOILING

TABLE 11-2

Stephan and Abdelsalam Correlations

Group Water Hydrocarbons Cryogenics Refrigerants

349

Range (p/pc)

θ, degrees

Equation

10 < p/pc < 0.88 5 × 10–3 < p/pc < 0.9 4 × 10–3 < p/pc < 0.97 3 × 10–3 < p/pc < 0.78

45 35 1 35

(11-2-14) (11-2-15) (11-2-16) (11-2-17)

Correlation Nu = 0.24 × 10 X1 X4 X3 X8 Nu = 0.054X50.34X10.67X8–4.33X40.248 Nu = 4.82X10.624X70.117X30.374X4–0.329X50.257 Nu = 207X10.745X50.581X6 0.533 7

0.67

–1.58

1.26

5.22

–4

Maximum Heat Flux. For the maximum heat-flux density (point D of the boiling curve), the Zuber18,19 equation is normally accepted: 1/ 4

⎡ (ρ − ρ )σg ⎤ qmax = 0.18ρV λ ⎢ L 2V ⎥ ρV ⎣ ⎦

(11-2-14)

This equation is valid for planar surfaces facing upward. It can be extended to single tubes immersed in a big mass of liquid. However, it cannot be used for tube bundles without correction. This will be explained later. Another correlation owing to Mostinsky, also valid for a single tube immersed in a boiling liquid, is ⎛ p⎞ qmax = 367 pc ⎜ ⎟ ⎝ pc ⎠

0.35

⎛ p⎞ ⎜⎝ 1 − p ⎟⎠ c

0.9

(11-2-15)

where pressures are expressed in kPa and qmax in W/m2. Transition Flux Zone (Zone DE). This is a very unstable zone, and no correlations for it are available at present. Point of Minimum Heat Flux (Point E). applied:

The theoretical correlation developed by Zuber19 can be 1/ 4

⎤ ⎥ ⎥⎦

⎡ (ρ − ρ )σg qmin = 0.125ρV λ ⎢ L 2 V ρL ⎢⎣

(11-2-16)

Afterwards, Kutateladze20 proposed to change the coefficient to 0.09. Stable Film Boiling Zone (Zone EF).

The Bromley correlation9 can be used: 1/ 4

⎡ k 3 (ρ − ρV )ρV λg ⎤ h = 0.62 ⎢ V L ⎥ μV Do ΔT ⎣ ⎦

(11-2-17)

where Do is the tube diameter.

11-2-3 Comparison among the Several Nucleate Boiling Correlations The nucleate boiling region is that of highest interest because most process equipment is designed to operate in this zone. We can see that different physical variables are used for different authors to correlate the same phenomena. It is quite obvious, then, that different results will be obtained depending on the correlation employed. The differences can be important, as will be seen in the following examples.

350

CHAPTER ELEVEN

Example 11-2 Calculate the values of the nucleate boiling heat transfer coefficients for water, ammonia, and benzene at atmospheric pressure corresponding to the following values of ΔTSAT (=TW – TSAT): 4.3°C, 9.1°C, and 13°C using Rohsenow, Forster and Zuber, Mostinsky, McNelly, and Stephan correlations. (We have chosen these values of ΔT because we had at hand experimental data for water.) The following table includes the physical properties necessary for the calculations: Water TSAT , K cL, J/(kg · K) λ, J/kg ρL , kg/m3 ρV , kg/m3 μL , kg/(m · s) kL , W/(m · K) σ, N/m pc, N/m2 θ, ° Δ pSAT/ΔTSAT

NH3

373 4,180 2.25 × 106 958 0.597 0.275 × 10–3 0.688 0.0588 221 × 105 45 3,906

Benzene

244.2 4,472 1.37 × 106 689 0.843 0.24 × 10–3 0.502 0.0325 112 × 105 35 5,479

353 1,985 3.97 × 105 820 2.82 0.3 × 10–3 0.129 0.021 48.9 × 105 35 3,306

The last row of the table is the slope of the vapor pressure curve, necessary for Forster and Zuber correlations. Substituting the physical properties into the correlations, it is possible to obtain expressions of the type q = aΔTSATb And we come to Correlation Rohsenow Mostinsky McNelly Stephan Forster and Zuber

Water

Ammonia

3

154.2ΔTSAT 46.45ΔTSAT3.33 21ΔTSAT3.22 64.2ΔTSAT3.05 1.68ΔTSAT1.24ΔpSAT0.75 = 830ΔTSAT1.99

3

Benzene

50.59ΔTSAT 14.12ΔTSAT3.33 46.2ΔTSAT3.22 0.044ΔTSAT3.92 1.66ΔTSAT1.24ΔpSAT0.75 = 1057ΔTSAT1.99

1.6ΔTSAT3 3.4ΔTSAT3.33 16.3ΔTSAT3.22 8ΔTSAT3 0.508ΔTSAT1.24ΔpSAT0.75 = 221ΔTSAT1.99

And calculating the values of q for the selected temperature differences, we get 1. Water at atmospheric pressure (values of q in W/m2): ΔTSAT

Rohsenow

Mostinsky

McNelly

Stephan

Forster & Zuber

Experimental

4.3 9.1 13

11,200 1.16 × 105 3.38 × 105

5,900 7.25 × 104 2.37 × 105

2,300 2.57 × 104 8.11 × 105

5,490 5.4 × 104 1.60 × 105

15,100 6.72 × 104 1.36 × 105

55,000 2.36 × 105 5.17 × 105

2. Ammonia at atmospheric pressure (values of q in W/m2): ΔTSAT

Rohsenow

Mostinsky

McNelly

Stephan

Forster & Zuber

4.3 9.1 13

4,022 3.8 × 104 1.11 × 105

1,816 2.2 × 104 7.23 × 104

5,060 5.66 × 104 1.78 × 105

13.3 252 1,023

19,200 8.56 × 104 1.74 × 105

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BOILING

3. Benzene at atmospheric pressure (values of q in W/m2): ΔTSAT

Rohsenow

Mostinsky

McNelly

Stephan

Forster & Zuber

4.3 9.1 13

130 1,200 3,500

440 5,300 17,400

1,780 19,900 62,900

640 6,000 17,500

4,030 17,900 36,400

Note the important dispersion in the calculated results. Expression for h as a Function of the Heat-Flux Density.

Nucleate boiling correlations are of the type

q = aΔT b

(11-2-18)

where ΔT = TW – t (we use the small t for cold-fluid temperature). Combining this expresion with q = hcΔT

(11-2-19)

(we added the subscript c to identify the cold fluid, which is the boiling fluid), we get hc = cq d

(11-2-20)

where d = (b – 1)/b and c = a1/b. As we shall see, when the correlations are expressed this way, the deviations in the calculated h at same q are much smaller than those compared at the same ΔT. This will be shown in the following example. Example 11-3 For the same correlations and fluids of Example 11-2, calculate the boiling film coefficient for similar q values. We shall first write all the correlations using the transformation equations explained above

Solution

Correlation

Water

Rohsenow Mostinsky McNelly Stephan Forster and Zuber

h = 5.35q h = 3.16q0.7 h = 2.56q0.69 h = 3.05q0.66 h = 29.3q0.5 0.66

Ammonia

Benzene

h = 3.69q h = 2.21q0.7 h = 3.27q0.69 h = 0.45q0.75 h = 33q0.5

h =1.16q0.66 h = 1.44q0.7 h = 2.37q0.69 h = 1.99q0.66 h = 15q0.5

0.66

For each fluid of Example 11-2, we shall calculate hc for different q values: Water 2

Ammonia

Benzene

q (W/m )

1,000

10,000

50,000

1,000

10,000

50,000

1,000

10,000

50,000

Rohsenow Mostinsky McNelly Stephan Forster and Zuber

532 397 300 389 926

2,460 2,000 1,470 1,800 2,930

7,200 6,150 4,470 4,930 6,550

350 279 384 80 1,040

1,610 1,400 1,880 450 3,300

4,660 4,300 5,700 1,500 7,400

110 181 280 190 470

510 910 1,360 500 1,500

1,460 2,800 4,140 2,500 3,350

It can be seen that except for some particular cases (e.g., the big dispersion showed by the Stephan correlation for the case of ammonia), when the results are presented this way, the differences among the correlations look smaller.

352

CHAPTER ELEVEN

Effect of the Controlling Resistances. The heat-flux density q depends not only on the boiling side coefficient but also on the fouling resistance and heating medium coefficient. This is q = U(T – t)

(11-2-21)

U = (1/hh + 1/hc + Rf)–1

(11-2-22)

where

The boiling heat transfer coefficients are usually very high, so frequently the controlling resistance for heat transfer lies in the combined effect of fouling and the hot-fluid coefficient. This means that independent of the differences that the boiling correlations may present, their influence in the calculation of q is normally dampened. Thus, when calculation of the boiling h is performed with expressions of the type h = cqd, the deviations are flattened, and the results obtained with different correlations are not very dissimilar. This is not an indication of the quality of the correlations but rather of the low influence that the selection of a good correlation has in the final equipment design. The concept is illustrated in the following example. Example 11-4 Calculate, using the previously studied correlations, the area of a coil to evaporate 500 kg/h of benzene (0.138 kg/s) at atmospheric pressure (boiling temperature = 80°C) using 140°C steam as the heating medium. Assume that the combined resistance (1/hh + Rf) corresponding to the heating steam and fouling is 0.0005 (m2 · K)/W. We shall assume that the benzene is fed at its boiling temperature, so only the latent heat of vaporization has to be delivered. The following procedure is proposed: 1. 2. 3. 4.

Assume a value for q as a first trial. With the boiling correlations (hc = cqd ), calculate hc . Calculate 1/U = 1/hc + (1/hh + Rf). Calculate q = U(T – t) = U(140 –80), and compare with the assumed q. With the calculated q, go back to step 1 up to convergence. 5. The required area then will be A = W λ /q = (0.138 × 3.97 × 105)/q. The final calculations are shown in the following table: Correlation Rohsenow Mostinsky McNelly Stephan Forster and Zuber

Q (W/m2) 5 × 10 8 × 104 9 × 104 7.5 × 104 8.1 × 104 4

hc 1,470 3,895 6,211 3,280 4,270

U

U(140–80)

A (m2)

847 1,321 1,512 1,242 1,360

5.08 × 10 7.92 × 104 9.07 × 104 7.45 × 104 8.17 × 104

1.1 0.69 0.61 0.73 0.67

4

The damping effect of the other resistances can be observed. The results obtained look more presentable.

11-3 FLOW BOILING IN TUBES A case of great interest in process engineering is the flow boiling of liquids in a tube. To study this case, it is necessary to analyze some aspects of fluid mechanics dealing with two-phase flow.

11-3-1 Two-Phase Flow Patterns When a liquid-vapor mixture flows in a tube, the flow pattern can assume different configurations depending on the relative amount of each phase, their physical properties, and the flow rates. The flow patterns

BOILING

353

that can exist are different for vertical and horizontal flow. We shall analyze the case of two-phase vertical upward flow because it is the case that most interests us. A description of the different two-phase flow patterns in horizontal tubes can be found in ref. 4. The different flow patterns that will be described next follow an order of increasing vapor quality x (x = vapor mass flow/total mass flow). Bubbly Flow. This pattern appears when the vapor-phase mass flow is small. The vapor phase is distributed as discrete bubbles in a continuous liquid phase (Fig. 11-16a). Slug Flow. In slug flow, the vapor bubbles are approximately the diameter of the pipe. The length of these bubbles can vary considerably, and they are separated from the pipe wall by a slowly moving liquid film. The liquid flow is contained in liquid slugs that separate successive gas bubbles (Fig. 11-16b). Churn or Semiannular Flow. Churn flow is formed by the breakdown of large vapor bubbles in slug flow. The vapor flows in a somewhat chaotic manner through the liquid, which is mainly displaced to the tube wall. The flow has an oscillatory or time-varying character, hence the descriptive name churn flow. This region is also called semiannular flow (Fig. 11-16c). Wispy Annular Flow. The flow takes the form of a relatively thick liquid film on the walls of the pipe carrying small vapor bubbles, together with a considerable amount of liquid entrained in a central vapor core. This entrained liquid phase appears as large droplets that have agglomerated into long, irregular filaments or wisps (Fig. 11-16d). Annular Flow. A liquid film forms at the pipe wall with a continuous central gas or vapor core. Largeamplitude waves usually are present on the surface of the film, and the continued breakup of these waves forms a source for droplet entrainment that occurs in the central core. As distinct from the wispy annular pattern, the droplets are separated rather than agglomerated (Fig. 11-16e). Mist Flow. The liquid is only present in the form of small droplets entrained in the vapor phase that occupy the entire pipe (Fig. 11-16f ). Frequently, the distinction among the different flow patterns is not so clear. Some authors describe other intermediate patterns, and no uniform criterion exists in the definition of one pattern or the other.

(a)

(b)

(c)

(d)

(e)

(f)

Bubble

Slug

Churn

Wispy annular

Annular

Mist

FIGURE 11-16

Two-phase flow patterns.

354

CHAPTER ELEVEN

When a fluid flows in a heated tube, as long as the fluid receives heat, it experiences a boiling process, and the vapor quality x increases along its path. This means that the two-phase flow pattern changes from one type to the following as in the system of Fig. 11-18.

11-3-2

Flow-Pattern Characterization The type of flow pattern existing in a two-phase system depends on the fluid mass flow and the relative amounts of vapor and liquid present. The vapor quality of the circulating fluid x, can be defined as x=

WV WV + WL

(11-3-1)

where WV = vapor mass flow rate (kg/s) WL = liquid mass flow rate (kg/s) WV + WL = W = total mass flow rate (kg/s) The mass velocity G is the mass flow rate divided by the flow area: G=

W at

(11-3-2)

For characterization of the different flow patterns that may be present in a section of the tube, some graphs known as flow-pattern maps have been developed. The Hewit and Roberts32 flow-pattern map for vertical flow is shown in Fig. 11-17a. This map is developed as a function of two variables, which are the vapor and liquid mass velocities, defined as GL = G(1 – x)

GV = Gx

and

Another flow-pattern map for vertical upflow was presented by Fair16 and is shown in Fig. 11-17b. The author restricts the validity of this map to the sizing of vertical thermosiphon reboilers. Liquid and Vapor Volume Fractions. Other parameters used frequently in the analysis of two-phase systems are the liquid or vapor volume fractions (RL or RV). These are the fraction of volume or cross section of tube occupied by the liquid or vapor, respectively. These parameters cannot be calculated directly from the values of x and the liquid and vapor densities due to the fact that both phases normally flow with different velocities. The first authors to study two-phase flow systems and correlate the different flow parameters were Lockhardt and Martinelli.15 Different authors later proposed modifications to their original correlations. Fair16 presents a correlation to calculate RL as a function of a parameter Xtt defined as 1 − x ⎞ 0.9 ⎛ ρV ⎞ X tt = ⎛⎜ ⎝ x ⎟⎠ ⎝⎜ ρL ⎠⎟

0.5

⎛ μL ⎞ ⎝⎜ μ ⎠⎟

0.1

(11-3-3)

V

This parameter is known as the Lockhardt-Martinelli parameter for turbulent flow in both phases. The correlation proposed by Fair is ⎛ ⎞ 1 RL = ⎜ 2⎟ ⎝ 1 + 21X ' + X ' ⎠ where X' = 1/Xtt .

0.5

(11-3-4)

BOILING

355

105 104 Wispy annular

Annular GV [kg/(m2·s)]

103 102 Churn Bubble

10 Slug

1 10–1

10

102

103

GL

104

105

106

[kg/(m2·s)]

(a)

GT [kg/(m2·s)]

1000

Bubbly flow

Annular flow

Slug flow

Mist flow

100

10 0.1

1

10 1/Xtt (b)

FIGURE 11-17 (a) Hewit and Roberts map for two-phase vertical upflow. (b) Fair map for two-phase vertical upflow.

11-3-3 Frictional Pressure Drop The frictional pressure drop for a two-phase system circulating in a pipe can be calculated as Δ pTP = φ 2 ΔpL where ΔpTP = frictional pressure drop of the two-phase system ΔpL = frictional pressure drop if only the liquid phase circulates in the pipe

(11-3-5)

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φ2 is a correction factor (the fact that φ2 rather than φ is used has no special significance and is only due to historic reasons). φ2 was originally correlated by Lockhardt and Martinelli. The original correlation was later modified by other authors. Fair16 proposed the following: φ2 = 1 + 21X '+ X '2

(11-3-6)

11-3-4 Heat Transfer Mechanisms for Boiling Inside Vertical Tubes Let’s consider a vertical tube fed at its base with subcooled liquid with enthalpy i1 and temperature t1. Let W be the mass flow rate. The tube is externally heated, so there is a heat flux through the wall to the circulating fluid. Depending on the boundary condition for heat transfer, we can have different situations. For example, it is usual in laboratory conditions to work with electrically heated tubes. In this case, if a constant electrical power per unit length is imposed, the boundary condition is constant heat flux for the whole tube surface. Then the tube wall temperature will vary from point to point to deliver this heat flux according to q = constant = hi (TW ( z ) − t F ( z ) )

(11-3-7)

where hi is the heat transfer coefficient and TW(z) and tF(z) are the local temperatures of the wall and circulating fluid corresponding to a tube section at coordinate z. The heat transfer coefficient hi , as well as tF(z) and TW(z) are a function of z. A more usual situation in industrial processes is to have the tube heated by an external fluid. For example, let’s assume that the heating medium at the external side of the tube is condensing steam at temperature T. Let ho be its condensation heat transfer coefficient. The wall temperature of the tube at every section must be such that ho (T − TW ( z ) ) = hi (TW ( z ) − t F ( z ) )

(11-3-8)

or TW ( z ) =

ho T + hi t F ( z ) hi + ho

(11-3-9)

Figure 11-18 shows the variation of fluid temperature tF(z) and wall temperature TW(z) for a heated tube along the z axis. The origin of z coordinate is at the tube entrance. During the first stage of its evolution, the liquid temperature increases owing to the sensible heat transfer received (zone A in Fig. 11-18). As the fluid temperature increases, the wall temperature also increases according to Eq. (11-3-9). Somewhere a condition is reached in which the wall and fluid temperatures are such that vapor bubbles begin to form at the tube wall. This point corresponds to the onset of nucleate boiling. Note that even though the wall temperature at this location must be higher than the saturation temperature, the bulk of the fluid still has not reached this saturation temperature. This means that the liquid is still subcooled. When the vapor bubbles produced at the tube wall depart and come in contact with the colder fluid, they recondense and deliver their heat of condensation to the liquid. From the point where nucleate boiling begins, the flow pattern starts being two-phase flow. To the right of the tube in Fig. 11-18, the different flow patterns are indicated, and farther to the right are the different heat transfer regions. Region B corresponds to subcooled boiling. At the point where nucleate boiling starts, the microconvective effect created by bubble detachment produces an increase in the heat transfer coefficient h, the consequence of which is a sudden decrease in the wall temperature according to Eq. (11-3-9).

BOILING Wall and fluid temperature

Heat transfer mechanism

Flow regime Single-phase vapor

G

357

Convective heat transfer to vapor Liquid fraction = 0

Liquid-deficient region

Mist flow

F

Dry wall condition

Forced convection through liquid ring

Annular flow

E

Ceases bubbles production at the tube wall D

Fluid temp. Wall temp. Temperature axis

Saturated nucleate boiling

Slug flow

C

Bubbly flow Subcooled boiling

B

Bulk temperature of liquid reaches saturation temp Onset of nucleate boiling

Single-phase liquid

A

Convective heat transfer to liquid

TSAT

FIGURE 11-18 Temperature profile, flow patterns, and heat transfer regions in a heated tube.

As the temperature of the liquid continues increasing, at a certain location it reaches the saturation temperature. This happens at a distance from the inlet section zSAT given by W (iSAT − i1 ) =

zSAT

∫0

qdA

(11-3-10)

where iSAT is the saturation enthalpy of the liquid stream per unit mass. Note that iSAT is the average enthalpy of the stream at section zSAT . However, since there is a radial temperature profile, and since in the region near the tube wall the fluid temperature is higher than the saturation temperature, there must exist at the center of the tube a region with fluid temperatures below tSAT . Nevertheless, it is usual to accept Eq. (11-3-10) as a valid criterion to define the start of saturated boiling (start of region C in Fig. 11-18). After the point where saturated boiling starts, the most characteristic variable to represent the condition of flow is the vapor quality or vapor mass fraction, defined as z

x=

∫z

qdA

SAT

Wλ

=

iz − izSAT λ

(11-3-11)

This vapor quality or vapor mass fraction represents the ratio between the vapor mass flow and the total mass flow as expressed by Eq. (11-3-1). As long as x increases, the flow pattern experiences further transformations and may turn into slug flow or annular flow, as shown in Fig. 11-18. If the circulating fluid is a pure single component, its temperature remains almost constant during the entire saturated boiling region (except for the change in boiling

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temperature as the pressure changes along the tube). The heat transfer coefficient hi increases as long as the vaporization proceeds owing to the increase in velocity caused by the reduction in the fluid density. Hence the wall temperature decreases slowly along the z axis (even though this is difficult to appreciate in Fig. 11-18). After the annular flow pattern starts, there is a change in the heat transfer mechanism consisting of suppression of the nucleate boiling mechanism. This means that bubble production at the tube wall stops, and heat must flow by forced convection through the liquid film to the liquid-vapor interface, where evaporation takes place. In this region we cannot speak about boiling any more. It is called the two-phase force convection region (region E in Fig. 11-18). If heat transfer continues, the liquid film may disappear completely, and the flow pattern changes from annular to mist flow. This transition is known as tube-wall dryout. In this region the heat transfer is directly from the tube wall to the vapor. The effect is a sudden reduction in the heat transfer coefficient and, consequently, an increase in the wall temperature, as shown in Fig. 11-18. In this region, indicated as F in the figure, the liquid droplets evaporate, and finally, a single vapor phase will be present. If heat transfer continues further, there is an increase in the fluid temperature, and according to Eq. (11-3-9), the wall temperature also increases (zone G).

11-3-5 Correlations for Boiling Heat Transfer Inside Tubes We shall consider the correlations that can be used to calculate the heat transfer coefficient in each region. Single-Phase Liquid Convection Region. In this region the convective heat transfer coefficient can be calculated with the usual single-phase convection correlations. For example, the Dittus-Boelter correlation can be used for turbulent flow: ⎛ k ⎞⎛ μ ⎞ hi = 0.023 Re 0.8 Pr 0.33 ⎜ ⎟ ⎜ ⎝ Di ⎠ ⎝ μW ⎟⎠

0.14

(11-3-12)

Onset of Subcooled Nucleate Boiling. This point can be predicted by Hsu’s theory, which was already explained for pool boiling. Figure 11-19 represents the radial temperature profiles in the liquid t = f(r) corresponding to different tube sections and superimposed on the same diagram the bubble equilibrium curve given by Eq. (11-1-8) is represented. At a certain tube section, before the start of boiling, the temperature profile is represented by line AB. As long as the fluid and wall temperatures increase in upper sections, the temperature profile moves upward until it becomes tangent to the bubble equilibrium curve. This situation corresponds to the tube section where nucleation starts. In this section, Eq. (11-1-18) or Eq. (11-1-22) applies, and it is possible to calculate the wall temperature combining Eqs. (11-1-21), (11-1-22), (11-3-9), and (11-3-12). There must be taken into consideration the restrictions explained in Example 11-1 regarding the maximum size of cavities that can act as active nucleation sites. Subcooled Nucleate Boiling Region. It is generally accepted that in boiling heat transfer, the heat flux can be calculated as the addition of two mechanisms: a macroscopic two-phase forced convection heat transfer and a microconvective effect caused by the agitation produced by bubble detachment at the solid wall. This approach has been proposed by several authors: q = qc + qn

(11-3-13)

The microconvective effect can be explained by a relationship of the type qn = ϕ(TW − TSAT )n

(11-3-14)

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Temperature axis

Temperature profile at ONB

TWONB TW A

Equation (11-1-8)

Temperature profile when wall temperature = TW TSAT TFONB

B

Boundary layer thickness

TFA

Distance to the wall FIGURE 11-19

Start of boiling in a heated tube.

This is a correlation of the same type as those studied for saturated nucleate boiling. But it must be noted that in the expression of ΔT, the saturation temperature, rather than the actual fluid temperature, is used, even though the liquid is subcooled. The exponent n ranges from 2 to 4. The second mechanism, forced-convection heat transfer, is calculated the same way as single-phase liquid heat transfer: qc = hL (TW − t F ( z ) )

(11-3-15)

where hL is calculated with Eq. (11-3-12). A conservative simplification normally assumed in reboiler design is to ignore the microconvective effect and consider this region and the single-phase liquid region as a single region. Start of Saturated Boiling. The transition from subcooled boiling to saturated boiling is defined from a thermodynamic viewpoint. It is the point at which the liquid reaches the saturation temperature found on the basis of simple heat-balance calculations. In this region the bubbles formed at the tube wall, once they detach from the surface, remain in the fluid without recondensation. The wall temperature can be calculated using Eq. (11-3-9) with tF (z) = TSAT . If, as a simplification, it is assumed that the pressure may be considered constant along the tube, the saturation temperature also will be constant. If, on the other hand, the pressure drop due to friction and hydrostatic head effects cannot be neglected, the saturation temperature will change along tube length z. For example, consider the tube in Fig. 11-20. At the inlet section, the pressure is p1. If a pressure gradient dp/dz exists in the system, we shall have a curve like curve I, representing the pressure along the tube length. Curve II represents the saturation temperature at the pressure corresponding to each z value. This means that points as A and A' indicate conditions p and t corresponding to the saturation curve. Curve III represents the evolution of liquid temperature along the tube length. The inlet temperature is t1. From t1, the temperature increases as long as the liquid is heated up (subcooled region). The evolution is represented by segment BC of curve III. At point C, the fluid temperature reaches the saturation temperature, and saturated boiling starts. From then on, the evolution of the fluid temperature follows curve CD corresponding to saturation. Please note

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z

z D

Curve I p vs z

Saturated region

C

Subcooled region

Curve III t vs z

A′

A

B

z=0 Pressure FIGURE 11-20

Curve II t sat vs z

p1

pA

Temp

tA

t1

Pressure and temperature variation along the tube.

that in Fig. 11-18, on the contrary, it has been supposed that the pressure drop is negligible, so the saturated boiling region is represented by an essentially constant fluid temperature. Heat Transfer in the Saturated Boiling Region. From the section where saturated boiling starts, the vapor bubbles formed at the tube wall remain in the system without recondensing, and the vapor mass fraction begins to increase along tube length z. Since the total mass flow is constant, the average fluid velocity increases when the fluid density decreases as a consequence of this vaporization. This also produces an increase in the heat transfer coefficient with the progress of vaporization. As in the case of the subcooled region, the heat transfer mechanism can be considered as the superimposition of two effects: the macroconvective effect and a second effect originated by bubble detachment from the heat transfer surface. The difference is that in the subcooled case, the ΔT values acting as driving forces are different for both mechanisms. They are TW – TSAT for the nucleation mechanism and TW – tF(z) for the macroconvection. In the case of saturated boiling, since the saturation temperature and fluid temperature coincide along the whole region, both mechanisms occur with the same driving force. Thus q = qC + qN = (hN + hTP )(TW − t F )

(11-3-16)

where hN is the coefficient corresponding to the microconvective effect of bubble detachment and hTP is the two-phase macroconvective coefficient. As a difference with the subcooled boiling region, where the macroconvective coefficient was calculated with Eq. (11-3-12) assuming only liquid flow, in the saturated boiling region, where the vapor generated remains in the system, the velocity in the Reynolds number will be a function of the mass vapor fraction. It is usually accepted that hTP is related to the single-phase convection coefficient by hTP = f( X ) hL n

(11-3-17)

where hL is the coefficient for the total flow assumed to be liquid and Xtt is the Lockhardt-Martinelli parameter. Many correlations of the type in Eq. (11-3-17) were proposed. One of the best known is due to Dengler and Addoms:11 hTP = 3.5( X ')0.5 hL

(11-3-18)

where X' is the reciprocal of Xtt (X' = 1/Xtt). Calculation of hN to complete Eq. (11-3-16) can be performed with the usual expressions for nucleate boiling presented in Sec. 11-3. Chen17 has developed a correlation that has been recommended for the saturated nucleate boiling region, as well as for the two-phase force convection region. This will be studied later .

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361

Suppression of Saturated Nucleate Boiling. As was explained earlier, when the heat transfer coefficient hTP increases because of the increase in the vapor mass fraction, a condition called suppression of saturated nucleate boiling can be reached. In this condition, the mechanism of bubble production disappears. This transition normally takes place in the annular flow region, but it can occur in any of the flow patterns described in Sec. 11-3. The suppression of nucleate boiling can be explained using the same model employed to predict the onset of nucleate boiling. In the entire saturated boiling region, if no pressure gradient exists, the fluid temperature tF(z) remains constant. Hence the heat-flux density increases with the vapor mass fraction as a consequence of the increase in hTP . Figure 11-21 represents the evolution of the temperature profile in the liquid phase while the heat flux increases. It can be seen that with the increase in q, the boundary layer thickness decreases because the coefficient hTP is given by hTP =

kL δ

(11-3-19)

The curve representing the bubble equilibrium condition given by Eq. (11-1-8) is also shown in the figure. Curve A corresponds to a condition where nucleate boiling exists because there are cavities with superheat higher than necessary to act as nucleation sites. However, when the temperature profile moves from B to C, the nucleate boiling disappears. This is so because at a very short distance from the wall, less than a bubble radius, the temperature has decreased to a value that is lower than that required for the bubble to grow and detach. Then curve B corresponds to the limit of nucleate boiling. Accepting that coefficient hTP is given by Eq. (11-3-18), it can be demonstrated that the heat-flux density at the point of nucleate boiling suppression is qS =

49 BhL 2 k L X tt

(11-3-20)

where B is defined by Eq. (11-1-20).

Temperature axis

TW q

A

B

C TSAT

δ1

δ2

δ3

Distance to the wall FIGURE 11-21

Suppression of nucleate boiling.

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Two-Phase Force-Convection Region. This region can be associated to any of the two-phase flow patterns, but if fluid velocities are not extremely high, the usual situation is that this region is associated with the annular flow. Heat is transferred by conduction and convection through the liquid film, and vapor is being generated continuously at the interface. It is normally considered that the vapor is at its saturation temperature corresponding to the local pressure. However, the existence of a heat flux from the interface requires the existence of, at least, a small temperature drop. This is normally neglected, but it can be important in certain cases, particularly at low reduced pressures ( pr < 0.001). It is then assumed that the total temperature drop (TW – TSAT) takes place in the liquid film, and the smaller the liquid film thickness and the higher its turbulence, the smaller this temperature drop will be. The heat transfer coefficient can reach extremely high values in this region. Values for water as high as 200,000 W/(m2 · K) have been reported. Following Martinelli recommendations, several authors have correlated their two-phase convection heat transfer data in the form of Eq.(11-3-17). These correlations can be found in the ref. 21. In what follows, we shall consider in more detail one of the most frequently used correlations, which can be used not only for the two-phase convective region but also can be extended for the nucleate boiling region, either saturated or subcooled. Chen Correlation.17 Chen performed research work comparing about 600 experimental points with the results predicted by several authors' correlations. None of these correlations could be considered satisfactory in the whole range. Chen therefore proposed a new correlation, and it proved very successful in correlating all the available data. The proposed correlation covers both the saturated nucleate boiling region and the two-phase forcedconvection region for pure liquids, either water or organics. Some authors suggest extending it to the subcooled nucleate boiling region following a procedure that will be explained later. As was mentioned earlier, the basic assumption is that the saturated convective boiling coefficient results from the superposition of two additive mechanisms: the two-phase convection plus a microconvective effect associated with nucleation and the detachment of bubbles. Thus h = hN + hTP

(11-3-21)

However, there is an interaction between both mechanisms that causes the individual effects to be modified. The equations describing these mechanisms thus also must be modified as follows. The convective component can be represented by a Dittus-Boelter type of equation: 0.8 0.4 hTP = 0.023 Re TP PrTP

kTP D

(11-3-22)

where the thermal conductivity and the Reynolds and Prandtl numbers are effective values associated with the two-phase fluid. However, since heat is transferred to a liquid film in annular and dispersed flow, Chen argued that it is reasonable to use the liquid thermal conductivity in Eq. (11-3-22). Similarly, the values of the Prandtl modulus for liquid and vapor are normally of the same magnitude, and it may be expected that the two-phase value will be close to this value. Regarding Re, a parameter F is defined such that ⎛ Re ⎞0.8 ⎡ ⎤ 0.8 Re TP F = ⎜ TP ⎟ = ⎢ ⎥ ⎣ G (1 − x ) D /μ L ⎦ ⎝ Re L ⎠

(11-3-23)

Equation (11-3-22) now may be rewritten as ⎛k ⎞ hTP = 0.023 Re 0L.8 PrL0.4 ⎜ L ⎟ F ⎝D⎠

(11-3-24)

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363

As can be appreciated in Eq. (11-3-23), ReL is the Reynolds number that would exist if only the liquid phase circulates in the system. Regarding the calculation of F, since this ratio is a flow parameter only, it may be expected that it can be expressed as a function of the Lockhardt-Martinelli factor Xtt . This was found to be the case. Figure 11-22 is a representation of F versus Xtt . The analysis of Forster and Zuber, developed for pool boiling, was taken as a basis for evaluation of the nucleate boiling component, and this can be expressed by Eqs. (11-2-8) through (11-2-11). These equations are based on the premise that the Reynolds number for the nucleate boiling mechanism is governed by the bubble growth rate. The value of ΔTSAT included in the expression of the Reynolds number [Eq. (11-2-10)] is the difference TW – TSAT . This difference appears in the Forster and Zuber model as the bubble superheat because it is assumed that the bubble forms in a region where the temperatures is approximately equal to TW (close to the tube wall). However, in some cases, the actual superheat of a bubble can be quite different from ΔTSAT . Figure 11-23 represents the actual superheat of a bubble ΔTe, which is the difference between the temperature in the region where the bubble grows and the saturation temperature. This difference depends on the shape of the temperature profile existing in the boundary layer. As can be appreciated in the figure, the actual superheat of the fluid in which the bubble grows is lower than the wall superheat ΔTSAT (ΔTSAT = TW – TSAT). The difference between this lower actual superheat ΔTe and the wall superheat is small for the case of pool boiling and was neglected by Forster and Zuber, but it is significant in the forced-convection case because the boundary layer is much thinner and the temperature gradients much steeper, so the bubble temperature may be quite lower than the wall temperature. Equation (11-2-11) then must be written in terms of effective ΔT and Δp: ⎛ k 0.79 c 0.45ρ 0.49 ⎞ hN = 0.00122 ⎜ 0.5L 0.29L 0.2L4 0.24 ⎟ ΔTe0.24 Δpe0.75 ⎝ σ μ L ρV λ ⎠

(11-3-25)

102

F = (ReTP / ReL )0.8

F 10

Approximate region of data 1

10–1 10–1

1

10

102

1/Xtt FIGURE 11-22 F factor versus 1/Xtt. (Reprinted with permission from Chen JC: “Correlation for Boiling Heat Transfer to Saturated Fluids in Convective Flow,” I&EC Process Design and Developments 5(3):326, 1966. Copyright by the American Chemical Society.)

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Temperature axis

Temperature axis

TW

Temperature profile

TW Temperature profile

ΔTe

ΔTe

TSAT Distance to the wall

Pool boiling FIGURE 11-23

TSAT

Distance to the wall

Convective boiling

Temperature profiles in pool and convective boiling.

Chen defines a suppression factor S as ⎛ ΔTe ⎞ S=⎜ ⎝ ΔTSAT ⎟⎠

0.99

(11-3-26)

The exponent 0.99 allows S to appear to the first power in the final equation. Using the Clausius-Clapeyron equation for the vapor pressure, the result is ⎛ ΔTe ⎞ S=⎜ ⎝ ΔTSAT ⎟⎠

0.24

⎛ Δpe ⎞ ⎜⎝ Δp ⎟⎠ SAT

0.75

(11-3-27)

with which we finally get ⎛ ⎞ k 0.79 c 0.45ρ0.49 0.24 0.75 hN = 0.00122 ⎜ 0.5 L 0.29L 0.24L 0.24 ⎟ ΔTSAT S ΔpSAT ⎝ σ μ L λ ρV ⎠

(11-3-28)

It might be expected that S would approach unity at low flows and zero at high flows. Chen suggested that S can be represented as a function of the local two-phase Reynolds number ReTP , which was confirmed experimentally. The function S versus ReTP is shown in Fig. 11-24. An analysis of Figs. 11-22 and 11-24 may contribute to a better comprehension of this phenomenon. F increases with increasing values of 1/Xtt , thus indicating that the turbulence associated with the two-phase flow increases with higher vapor mass fractions. At low mass flows and low vapor mass fractions, S approaches unity, indicating that the microconvective mechanism plays a more important role, whereas at high mass velocities and vapor mass fractions, the macroconvective mechanism becomes preponderant. Thus it is logical that S approaches 0. Chen used the Forster and Zuber equation as the basis for his analysis. However, it is possible to use any other of the previously mentioned correlations to calculate the nucleate boiling contribution. At present,

S = (ΔTe /ΔT )0.99

BOILING

1 0.9 0.8

365

S

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 104

Approximate region of data

105 ReTP = ReL.F 1.25

106

FIGURE 11-24 S factor versus ReTP . (Reprinted with permission from Chen JC: “Correlation for Boiling Heat Transfer to Saturated Fluids in Convective Flow,” I&EC Process Design and Developments 5(3):326, 1966. Copyright by the American Chemical Society.)

some authors prefer the Mostinsky equation [Eq. (11-2-12)] owing to its simplicity. The expression for hN then could be written as 1.2 ⎡ ⎛ p ⎞ 0.17 ⎛ p⎞ ⎤ hN = 0.00417 pc0.69 q 0.7 ⎢1.8 ⎜ ⎟ + 4⎜ ⎟ ⎥ S ⎝ pc ⎠ ⎥⎦ ⎢⎣ ⎝ pc ⎠

(11-3-29)

with pressures in kPa and q in W/m2 to get h in W/(m2 · K). (The third term of the polynomial has been neglected.) Equations (11-3-28) and (11-3-29) can only be applied to the boiling of single-component fluids or fluids with a small boiling range. In the case of mixtures, diffusion resistances appear at the vapor-liquid interface, and hN has to be corrected with a factor Fc (see Eq. 11-4-17). This effect is similar to what we explained in condensation of vapor mixtures. In the case of boiling, since the lighter components preferably vaporize, the composition of the liquid near the interface is richer in heavy components, and the lighter components in the liquid mixture must diffuse through this film to reach the interface. It also was found that in the case of mixtures, hTP also must be corrected for diffusion effects, but the correction factors are difficult to predict. We shall not include them, but in high-boiling-range mixtures this may have an effect on the calculated results. The Chen correlation can be extended for use in the subcooled boiling region by applying the principle of mechanisms superposition: q = hN (TW − TSAT ) + hTP (TW − TF ( z ) )

(11-3-30)

In this case, hTP is obtained from Eq. (11-3-24) with F = 1 (this means that it coincides with hL), and hN is evaluated from Eq. (11-3-29), with the value of S obtained from the single-phase-liquid Reynolds number. This extension of the Chen correlation has been tested against experimental information with satisfactory results. Limits to the Convective Boiling Correlations. Mist Flow Region. As was explained earlier, in cases where the vapor mass fraction at the outlet region of the tube is very high, it is possible to experience a transition from annular to mist flow. In heat transfer equipment design, this is an undesirable circumstance because there is a substantial deterioration in the performance of the unit. When in a thermosiphon reboiler mist flow exists near the tube outlet, the local wall temperature approaches the heating medium temperature, indicating a very low value of the heat transfer coefficient.

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This condition may be predicted as a function of the total mass velocity and the Lockhardt-Martinelli parameter calculated for the outlet conditions. Fair proposed to calculate a critical mass velocity for mist flow as G (mist flow) = 2441Xtt

(11-3-31)

where G is in kg/(m2 · s). If the mass velocity at the tube outlet is higher than that calculated with Eq. (11-3-31), it is possible to have mist flow at the tube outlet, and the Chen correlation is not valid. Upper Limit to the Heat-Flux Density. The curve representing the heat-flux density versus temperature difference for flow boiling is similar to that for pool boiling and also presents a maximum. Several correlations have been proposed to predict the maximum heat-flux density. A correlation21 that has proven to be successful for a wide variety of fluids, including water, alcohols, and hydrocarbons, is ⎛ D2 ⎞ qmax = 23, 600 ⎜ i ⎟ ⎝ L ⎠

0.35

⎛ p⎞ pc0.61 ⎜ ⎟ ⎝ pc ⎠

0.25

⎛ p⎞ ⎜⎝ 1− p ⎟⎠ c

(11-3-32)

where qmax = maximum heat-flux density (W/m2) pc = critical pressure (kPa) p = absolute pressure (kPa) Di = internal tube diameter (m) L = tube length (m) And the previously studied correlations (such as that of Chen) cannot be used when they predict heat-flux densities higher than that calculated with Eq. (11-3-32). In-Tube Film Boiling. For in-tube boiling, it is also possible, if temperature differences are high, to have a stable film-boiling regime in which heat transfer to the liquid takes place through a vapor film. The ΔT value must be high enough to move away from the transition region (where q decreases with an increase in ΔT). An approximate value for the minimum ΔT required for film boiling can be calculated28 as ⎡ ⎛ ⎛ p ⎞−2 ⎤ p⎞ ΔTfb = 0.555⎢52 ⎜1 − ⎟ + 0.04 ⎜ ⎟ ⎥ ⎢⎣ ⎝ pc ⎠ ⎝ pc ⎠ ⎥⎦

(11-3-33)

where ΔTfb is expressed in °C. The film boiling heat transfer coefficient can be estimated21,29 by means of the following equation: 0.22 ⎛ p ⎞0.38 ⎛ p⎞ hfb = 105 pc0.5 ΔT −0.33 ⎜ ⎟ ⎜1 − ⎟ ⎝ pc ⎠ ⎝ pc ⎠

(11-3-34)

with pc in kPa and hfb in W/(m2 · K). This equation must be considered only as a conservative estimation because the available information is scarce. The Critical Pressure. Many of the correlations presented in this chapter are based on the use of critical pressure. The critical pressure of a single-component fluid is the maximum pressure at which two phases in equilibrium can exist. At this pressure, liquid and vapor densities became equal. Any fluid at a pressure higher than this presents a unique phase, called supercritical, whose density decreases continuously with an increase in temperature without presenting a two-phase state. For pure substances, the critical pressures of fluids are tabulated27 and can be found easily. In the case of mixtures, the situation is more complex. The critical pressure of a mixture can be obtained by calculating with a process simulator the bubble and dew-point temperatures at different pressures. The

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367

pressure at which both temperatures coincide corresponds to the critical state. Some process simulators include the calculation of critical properties of all the streams routinely. For a multicomponent mixture with known composition, an approximation consists of calculating a pseudo-critical pressure, defined as psc = ∑ pci xi

(11-3-35)

where xi is the mole fraction of component i and pci is its critical pressure. For petroleum fractions, the method proposed by the American Petroleum Institute (API)25 and reproduced in Gas Processors Suppliers Association (GPSA) manuals26 allows the calculation of pseudocritical properties as a function of the average boiling temperature and specific gravity.

11-4 11-4-1

REBOILERS Different Types The purpose of a reboiler is to vaporize the bottoms of a distillation column, thus producing the necessary amount of vapor required for the operation of the column, which is determined by the reflux ratio adopted by the designer. For a brief description of the distillation process, App. A can be consulted. A reboiler consists of a heat exchanger, heated by a hot fluid, which can be steam, hot oil, or any available process stream. The hot stream transfers heat to the column bottoms liquid, which vaporizes into the reboiler. There are several types of reboilers, and they will be described below. Forced-Circulation Reboiler. The reboiler shown in Fig. 11-25 is called a forced-circulation reboiler because the liquid is fed into it by means of a pump. The liquid coming from the lowest tray gets into the bottom end of the column, which acts as an accumulator. From there, it is taken up by the pump and is forced through the reboiler (stream W), whereas a fraction B is withdrawn as bottom distillation product. The device indicated as LC is a level controller that opens or closes the product outlet valve, thus preventing the column bottom from getting flooded or dried out. The liquid flow rate fed into the reboiler must be higher than the required vapor production, so the reboiler outlet stream is a liquid-vapor mixture. The vapor V passes to the lowest tray, and the liquid L comes back to the column bottoms from where it is recirculated. For stable system operation, without pressure fluctuations or pulsating flows, it is recommended to keep a circulation ratio W:V of at least 3:1. The higher this ratio, the more stable will be the hydraulic behavior of the system. Another reason to work with high circulation ratios is that in this way the fouling problems on the tube surfaces are reduced. The bottom part of a distillation column is where high-boiling-point impurities accumulate. If the reboiler feed were completely vaporized, nonvolatile compounds (salts or dissolved substances) potentially could precipitate or deposit on the tube walls. This reboiler type has the particularly attractive feature of allowing any desired circulation rate because the rate is defined by the capacity of the pump and is practically independent of reboiler design. It is a very convenient alternative in cases where the column bottom product is very viscous. The principal disadvantage is that it requires installation of a pumping system in a location that is problematic for the following reasons: 1. The column bottom liquid is at its boiling temperature. The pump is then prone to cavitation, and it is necessary to raise the column bottom above the pump foundation level by a value dictated by the required net positive suction head (NPSH). This usually means 3 or 4 m. 2. Temperatures at column bottoms are high, and the pump and mechanical seals are normally subject to severe service conditions that may result in frequent maintenance. For these reasons, natural-circulation units, as described below, are normally preferred.

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CHAPTER ELEVEN

V

L

LC

Heating medium

B

W FIGURE 11-25

Forced-circulation reboiler.

Vertical Thermosiphon Reboiler. The thermosiphon reboiler is the most common type in the process industry. The basic idea consists of eliminating the circulation pump and using the difference between liquid and vapor densities as the driving force for fluid circulation. A schematic diagram of a vertical thermosiphon reboiler can be seen in Fig. 11-26. The unit is a vertical heat exchanger in which the liquid coming from the column bottoms flows into and through the tubes, where it is vaporized. In the vaporization process, a vapor-liquid mixture is generated into the tubes. The average density of this mixture is lower than that of the liquid. With reference to Fig. 11-26, between points A and B, the hydrostatic pressure difference is given by ρLgz1. Between points C and B, the hydrostatic pressure difference will be the product of an average density corresponding to the vapor-liquid mixture and gz2. Since this density is lower than that of the liquid, it results that the hydrostatic pressure difference is higher for segment AB than for segment BC. Then a fluid circulation is established between the column and the reboiler. As the liquid in the reboiler partially vaporizes, a vapor-liquid mixtures exits the unit from the top outlet nozzle and flows back into the column, where both phases separate as in a forced-circulation reboiler. The circulation rate that results in the system will be such that the frictional pressure drop balances the hydrostatic pressure difference. Since normally z1 ≅ z2 = z, this hydrostatic pressure difference will be given by gz(ρL – ρaverage liquid-vapor) This means that the higher z (= reboiler length) is, the higher will be the circulation rate. This has an effect on the heat transfer coefficient that improves with the increase in fluid velocity. In well-designed thermosiphon reboilers, it is possible to achieve very high circulation rates and hence good values of the heat transfer coefficient. The piping configuration also is very simple, which simplifies plant layout. Cleaning of this type of reboiler is also a simple task because the process fluid is inside

BOILING

369

V C L LC

A

Heating medium z1

z2

B

FIGURE 11-26

Vertical thermosiphon reboiler.

the tubes. The shell-side heating fluid is normally a clean fluid, so low-cost fixed tubesheet units can be employed. The main disadvantage is that the column has to be raised to achieve the necessary height for fluid circulation. The distillation column is itself a tall piece of equipment, and an additional height increase may have a significant impact on the mechanical design and cost. An important aspect that must be considered in design is the temperature increase required by the fluid to achieve the desired vaporization fraction. For example, with reference to Fig. 11-27, let TA be the temperature of the bottom tray liquid. This liquid falls down through the downcomer to the column bottom, where it mixes with the liquid fraction coming back from the reboiler, resulting in a mixture temperature TC . If the liquid consists of a mixture of several components with different boiling points, because the lighter components vaporize preferably, the reboiler outlet temperature TB will be higher than the inlet temperature TC. If the circulation ratio (total inlet flow/vaporized flow) is high, the vaporization process does not produce a significant change in the liquid-phase composition; thus the outlet bubble-point temperature TB will not be very different from the inlet bubble-point temperature TC. If, on the other hand, the circulation ratio is low, this temperature change can rise up by several degrees and has to be considered for the following reasons: 1. For temperature-sensitive fluids, undesirable transformations may occur. 2. The temperature increase in the boiling fluid reduces the available ΔT between the hot and cold fluids. In certain cases, when the temperature of the heating fluid is very close to that of the column bottom product, this may have a significant impact on the required heat transfer area. In other cases, this can be beneficial because it moves the operating condition away from the critical heat flow. Another aspect that may be considered a disadvantage of the two reboiler types considered up to now (Figs. 11-25 and 11-27) is related to their efficiency from the mass-transfer point of view. The objective of a distillation is always to obtain the bottoms product as heavy as possible (this means rich in high-boilingpoint components).

370

CHAPTER ELEVEN

TA V TB

TB L

LC

TC

TC

TC FIGURE 11-27

Temperatures in a reboiler system.

If we consider the unit shown in Fig. 11-27, we can see that the stream richest in heavy components corresponds to the liquid fraction exiting the reboiler. But this fraction is mixed in the column bottom with the liquid coming down from the lowest tray, which is lighter. Hence the distillation bottom product, which is a mixture of both streams, does not correspond to the heaviest composition. The quality of the product has been degraded with this mixture. Sometimes the configuration shown in Fig. 11-28 is used. In this case, the liquid from the bottom tray is extracted completely and send to the reboiler. The outlet vapor-liquid mixture is separated into the column, so the liquid fraction is withdrawn as distillation bottoms without being recirculated. This is called once-through circulation. It can be observed that the vapor generated in the reboiler is in thermodynamic equilibrium with the bottom product of the distillation. This means that the reboiler behaves as an additional column tray, and from the mass-transfer point of view, this is more efficient than the recirculation configuration. This configuration is also particularly attractive in services subject to polymerization or product decomposition caused by high temperatures because it minimizes the residence time in the reboiler and avoids repeated contact of the fluid with the heating surface as it is the case in recirculation units. If the distillation column is a packed-type column instead of a tray-type, this configuration becomes more complex because it is necessary to add a liquid drawoff tray. The other disadvantage of this design is that both the liquid bottom product and the reboiler vapor product are always external data for the reboiler designers, so they have no chance to handle the recirculation rate as a design variable to improve the heat transfer coefficients Additionally, it was explained earlier that it is always convenient to work with a vaporization ratio lower than 30 percent to avoid fouling problems. This type of reboiler, then, should not be selected when the vapor generation is higher than 3/7 of the bottom-product flow rate.

BOILING

371

V L LC z2

FIGURE 11-28

z1

Once through the vertical reboiler.

The maximum temperature in this system is the reboiler outlet temperature, which coincides with the bottom-product temperature. This means that with this type of unit it is not necessary to reach temperatures higher than the bubble point of the bottom product, as was the situation with recirculation reboilers. Figure 11-29 shows a configuration pretending to combine the advantages of the recirculation and at the same time reduce the efficiency loss originating in the mixture of the liquid fraction coming back from the reboiler with the bottom-tray liquid. In this case, the liquid fed to the reboiler has approximately the composition of the bottom tray, and the bottom product of the distillation has a similar composition to that of the reboiler outlet liquid fraction. Horizontal Thermosiphon Reboiler. In this type of reboiler, the vaporization takes place in the shell side while the heating fluid circulates through the tubes. Since normally the process fluid presents more fouling issues than the heating medium, in certain cases this design may not be very attractive because of the difficult cleaning. This type of reboiler requires more plot area than a vertical unit, and recirculation ratios are not so high. To reduce the frictional pressure drop, the double-split configuration (TEMA H shell) shown in Fig. 11-30 is frequently adopted. As in the case of a vertical thermosiphon, it is possible to connect this unit in a oncethrough circulation diagram. Kettle Reboilers. The kettle reboiler is shown in Fig. 11-31, and the way it is connected to the distillation column is shown in Fig. 11-32. It is a TEMA BKT or BKU heat exchanger. The liquid feed gets into the unit through a nozzle (3) and fills the shell side up to a level determined by the height of an overflow weir (4). The liquid pool contained into the shell receives heat from the tubes and vaporizes. The amount of vapor produced obviously depends on the heat transfer area and the temperature of the heating medium. The fraction of the feed that is not vaporized flows over the weir and exits through a nozzle (5) as distillation bottom product.

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CHAPTER ELEVEN

V

L

LC

FIGURE 11-29 An improved column bottom arrangement.

LC

FIGURE 11-30

Horizontal thermosiphon reboiler.

Heating medium

BOILING

373

Vapor outlet 6 1 Heating medium inlet

Liquid level

3 Feed inlet

4 Weir

5 Bottoms product outlet

2 Heating medium outlet FIGURE 11-31

Kettle reboiler.

V

H LC

W B FIGURE 11-32

Kettle reboiler arrangement.

The purpose of the vapor space above the liquid level is to allow separation of the vapor and entrained liquid droplets so that only vapor exits through the outlet nozzle (6). The tube bundle is a simple floatinghead type or U tubes, and the heating fluid circulating in the tubes may be steam or any other fluid. The difference between this unit and the thermosiphon or forced-circulation reboiler is that there is no recirculation or liquid coming back from the reboiler to the column. It can be observed that the vapors generated into the unit are in thermodynamic equilibrium with the distillation bottom product extracted after the weir through the nozzle (5). This means that in this configuration the reboiler behaves as an additional equilibrium stage from the mass-transfer point of view. The fluid velocities in this type of reboiler are very low, and frictional pressure drops are negligible. The only pressure drops in the system are due to friction in the inlet and outlet piping. These must be overcome by the difference in liquid levels between the column bottom and the reboiler (indicated as H in Fig. 11-32). This difference is very small, and the reboiler and the column bottom behave as communicating vessels. It is necessary neither to raise the column to allow liquid circulation, as is the case of thermosiphon units, nor to install a circulation pump.

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CHAPTER ELEVEN

As a counterpart, the reboiler is more expensive, especially in cases where the operating pressure is high because the big shell diameters make it necessary to use greater plate thickness in its construction. These units also require more plot area than vertical types. However, they have the advantage, in comparison with thermosiphon units, that the thermal design is not conditioned by the hydraulics of the system. This gives the designer more freedom to modify the relative location between the column and the reboiler. This is particularly important in the case of big units. When the amount of distillation bottom product is small in comparison with the amount of vapor, this type of reboiler has a tendency to act as an impurities concentrator. Since these units do not have the high circulation rates of thermosiphons, it is convenient to limit the vaporization to not more than 80 percent of the feed rate. This means that the 20 percent constituting the bottoms product acts as a purge, thus eliminating the concentrated impurities. Internal Reboiler. In certain less frequent cases, when the amount of heat to be transferred in the reboiler is small, the construction shown in Fig. 11-33 may be used. In this case, a tube bundle is installed directly in the bottom of the column. The cost of the shell is avoided, and there is no connecting piping. However, the column must be provided with the internal supports for the bundle and a big flange. The tube bundle is comparatively expensive because it is necessary to employ more tubes with smaller lengths. Other Applications of Boiling Heat Transfer Equipment. Among all the process equipment in which boiling heat transfer takes place, the most common is the distillation-column reboiler. This is why in the following topics we shall only make reference to this application. But there are other process equipment apart from reboilers in which fluids are vaporized. However, the calculation methods that can be employed in their design are the same as we shall study for reboilers. This means that even though we shall always make reference to the distillation process, the same treatment can be extended to other applications as well. For example, let us consider the case of Fig. 11-34. It represents a waste-heat-recovery boiler of a chemical plant. In this plant, an exothermic vapor-phase reaction takes place, and the high-temperature reaction products must be cooled down before passing to the fractionation section. The waste heat of the gases is recovered in the equipment shown in the figure to generate steam that is later used in other sectors of the plant. The gases circulate through the shell side of the exchanger while water is vaporized within the tubes. A steam-water mixture exits the outlet nozzle into the separator S. There, both phases are separated, and the steam is withdrawn from the upper nozzle while the liquid is mixed with the fresh feed (which replaces the vaporized mass) and recirculates through the tubes. This recirculation is caused by the difference in

V LC

FIGURE 11-33

Internal reboiler.

BOILING

375

Steam

S

LC

Hot gases from reactor

Makeup water

FIGURE 11-34

Thermosiphon steam generator.

densities of the water-steam mixture in the tubes and that of the liquid in the return leg. It is evident that this unit can be assimilated to a thermosiphon reboiler and can be designed with the same techniques that will be studied for this case. Let us analyze the case of the chiller represented in Fig. 10-24. In this case, a process stream is cooled down using as cooling medium a refrigerant fluid such as ammonia, propane, or a halogenated hydrocarbon and follows the refrigerating cycle shown in the figure. In the chiller, the refrigerant receives heat from the process stream and vaporizes. This type of unit operates as a kettle reboiler and must be designed as such.

11-4-2 General Aspects of the Thermal Design of Reboilers It has already been mentioned that the prediction of boiling heat transfer coefficients presents many difficulties, and research work has led to the development of a variety of correlations that, when compared, reveal considerable differences. It is unavoidable, then, that these uncertainties extend to the design of boiling heat transfer equipment, especially to the case of reboilers. Several authors have published design methods that proved to be useful for a particular application or fluid but had little success when extended to other situations. In recent years, with the appearance of commercial software, it became more difficult to find systematic design methods for this type of equipment in the open literature. This section is based on the information quoted most frequently in the references, but it is recommended that one verifies manual calculations with commercial programs and vice versa to be sure that the results are interpreted adequately. The following general considerations must be taken into account: 1. There are critical phenomena that impose limits to the heat transfer rate (e.g., the transition from nucleate to film boiling or the mist flow in convective boiling) that must be checked. High temperature differences between hot and cold fluids can lead to these effects and generally are not convenient. 2. Except for these critical effects, the heat transfer coefficients for boiling liquids are usually very high, and frequently, the controlling resistance is fouling. However, when the unit is clean, the wall temperature

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CHAPTER ELEVEN

in contact with the boiling liquid is higher than that assumed for design, and a critical condition may be reached. This condition should be checked if it is not possible to adjust the temperature of the heating medium.

11-4-3 Kettle Reboiler Design Heat Balance of a Kettle Reboiler. With reference to Fig. 11-32, the liquid from the column bottom is at the bubble-point temperature corresponding to the column operating pressure. Let this temperature be tF . When this liquid goes into the reboiler, it receives a heat flow Q, increasing its enthalpy and suffering a partial vaporization. Two phases are then formed, V and B, that are, respectively, the vapor going back to the column and the distillation bottom product. Since it is considered that the reboiler is an ideal stage, these two streams will be at thermodynamic equilibrium at temperature t, which is the temperature inside the reboiler. If the liquid coming down from the bottom plate is a pure component (which roughly may be the situation in binary distillations), since the column and reboiler pressures are practically the same, the temperature t coincides with the temperature of stream W. This means that the operation of the unit is practically isothermal. Only latent heat is transferred, and the enthalpy balance is reduced to Q = Vλ

(11-4-1)

If, instead, the column bottom product is a multicomponent mixture, the vaporization process will generate two phases with different composition as well as different from the composition of W. Since B and W are two liquid streams at their respective bubble points but with different compositions, their temperatures also will be different. This means that the liquid W receives sensible and latent heat in the reboiler and it increases its temperature and is partially vaporized. It is generally accepted that when the liquid feed enters the reboiler, its temperature is instantaneously increased as the result of the mixture with a big mass of fluid at temperature t, so the boiling fluid temperature in all the reboiler interiors may be considered constant and equal to the outlet temperature. This means that the heating medium must deliver all the heat (sensible and latent) to a fluid at temperature t. If the heating medium is, in turn, isothermal (e.g., in the case of condensing steam) at temperature T, the temperature difference for heat transfer will be ΔT = T − t

(11-4-2)

The value of the heat flux Q required for the distillation normally is obtained as a result of the process simulation when the heat balance of the column is solved and the enthalpies of streams W, V, and B are calculated. This balance can be expressed as Q = ViV + BiB − WiW

(11-4-3)

Even though the calculation of enthalpies is performed directly by thermodynamic models employed by the simulator, some reboiler design methods require separation of the heat flux Q into a fraction of sensible heat and a fraction of latent heat. By making use of a mean liquid specific heat cL, we can define QS = WcL (t − t F )

(sensible heat)

(11-4-4)

QL = Q − QS

(latent heat)

(11-4-5)

Calculation of the Reboiler Area. The design of this type of reboiler follows a method originally published by Palen and Small.22 The method assumes that the boiling heat transfer coefficient can be calculated by the addition of two contributions. One of them is the macroscopic convection as consequence of the two-phase fluid circulation through the tube bundle. This is a convective mechanism in which the fluid

BOILING

377

moving across the bundle is liquid and vapor. The convection heat transfer coefficient that allows the calculation of this contribution will be called hTP . Superimposed to this mechanism, there exists a microagitation effect caused by bubble nucleation and detachment at the heat transfer surface. The coefficient associated with this mechanism will be called hNB . Then ho = hNB + hTP

(11-4-6)

In a kettle reboiler, the liquid velocities are small, and the effect of hNB is much more important than that of hTP . High precision in the calculation of hTP thus is not required, and normally, it is assumed to have a value in the range of 250–500 W/(m2 · K) without further verification. hNB , in turn, is calculated with the methods and correlations mentioned in Sec. 11-2. However, it must be considered that those correlations are valid only for simple geometries such as, for example, a single tube submerged into a boiling liquid. In the case of tube bundles, some corrections are necessary because the vapor bubbles produced on the surface of a tube affect the tubes above it. The general expression is hNB = hNB1 Fb Fc

(11-4-7)

where hNB1 = nucleate boiling coefficient for a single tube submerged into a fluid with similar properties to that in the reboiler and submitted to the same heat-flux density Fc = boiling-range correction factor Fb = bundle-geometry correction factor In what follows, we shall analyze how to calculate these terms. Calculation of hNB1 . In Sec. 11-2 we studied some expressions for calculation of the boiling heat transfer coefficient for a single tube submerged into a fluid. Some of these equations were expressed in the form q = a ΔT b

(11-4-8)

q = hNB1ΔT

(11-4-9)

hNB1 = aΔT b−1

(11-4-10)

hNB1 = cq d

(11-4-11)

If we consider that

then it can be written

or

where c = a1/b

and

d = (b – 1)/b

When the only mechanism present is the nucleate boiling, all these expressions are equivalent and any one of these forms is valid. However, if the heat-flux density results from the addition of two effects, Eq. (11-4-9) is no longer valid, and the transformation from one type of equation into another cannot be done. This subject is not very clear in the references, and some confusion may arise. Thus it is very important to remark that hNB1 must be evaluated at the same heat-flux density as that of the reboiler. This means that an equation like Eq. (11-4-11) must be used, where q has to be taken as the total heat-flux density considering all the mechanisms present.

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CHAPTER ELEVEN

It was explained in Sec. 11-2 that important deviations may be found among the results of the different nucleate boiling correlations available. Among all the possible correlations, most authors prefer the Mostinsky correlation [Eq. (11-2-12)], very probably because of its simplicity. This is expressed as hNB1 = 0.00417 pc0.69 q 0.7 Fp

(11-4-12)

where Fp is a pressure correction factor, expressed as a function of p/pc (pressures in kPa), that is, 1.2 10 ⎡ ⎛ p ⎞ 0.17 ⎛ p⎞ ⎛ p⎞ ⎤ ⎢ Fp = 1.8 ⎜ ⎟ + 4 ⎜ ⎟ + 10 ⎜ ⎟ ⎥ ⎢ ⎝ pc ⎠ ⎝ pc ⎠ ⎝ pc ⎠ ⎥ ⎦ ⎣

(11-4-13)

For reboilers design, Palen21 suggested considering only the first term of the polynomial. Calculation of Fb , the Correction Factor for Tube Bundle Geometry. The correlations studied for a single tube cannot be employed for a tube bundle because the bubbles produced on the surface of a tube affect the neighboring tubes. In one of his first papers, Palen22 considered that the boiling heat transfer coefficient of a tube bundle is smaller than that of a single tube (about 30 percent). Some years later, in another paper, the same author explained that the bundle heat transfer coefficient is about two or three times higher than the coefficient of a single tube. It is remarkable that between the first and second values there is 10 times difference that remained hidden for years. This gives one an idea on how difficult is boiling heat transfer coffecients experimental measurement. Afterwards, in successive papers,21 Palen proposed different correlations for the calculation of Fb. One of them is Fb = 1 + 0.1 ⎡⎣(1 − ψ ) /ψ ⎤⎦

0.75

(11-4-14)

where ψ = ( πDb L /A)

(11-4-15)

Db is the tube bundle diameter (diameter of the minimum circle completely containing all the tubes). To evaluate it, it is necessary to perform the tubes distribution. Another correlation proposed by Taborek24 is ⎤ ⎡ 0.785 Db Fb = 1 + 0.1 ⎢ − 1⎥ 2 ⎥⎦ ⎢⎣ C1 ( Pt / Do ) Do

0.75

(11-4-16)

where C1 is unity for 90- and 45-degree tube arrays and 0.866 for triangle arrays. All these equations require one to know the geometry of the tube bundle. In early design stages, when this information is not available, it is suggested to adopt Fb = 2 for kettle reboilers. For the case of horizontal thermosiphons (whose calculation method is similar), it is suggested to adopt Fb = 2.3 for preliminary calculations. Calculation of Fc , Boiling-Range Correction Factor. This factor is important when the liquid presents a boiling range. This means that the reboiler feed temperature is different from the fluid temperature into the reboiler. All the correlations studied previously for nucleate boiling are valid for single-component fluids. In the case of mixtures, the heat transfer coefficients can be considerably smaller owing to diffusion effects that create additional resistances to heat transfer, as was the case in multicomponent condensers. Several correlations were proposed to correct the heat transfer coefficient to include diffusion effects. The following was proposed by Palen and Young,21 and we adopt it because of its simplicity: Fc = 1/(1 + 0.023q 0.15 BR 0.75 )

(11-4-17)

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379

where BR is the boiling range of the mixture in °C (difference between dew point and bubble point of the reboiler feed at the operating pressure). Please note that since the vaporization in the reboiler is not total, BR normally is higher than the temperature change that the fluid actually experiences. In Eq. (11-4-17), q is in W/m2. Maximum Heat-Flux Density and Film Boiling. The previously studied correlations are valid only for low ΔT between the tube wall and boiling fluid. Otherwise, it is possible to reach the transition to film boiling regime with a consequent reduction of the heat transfer coefficient. It is then necessary to have a method to establish the validity limit of the correlations. Normally, this is handled through calculation of qmax (maximum heat-flux density for the tube bundle). Mostinsky12 proposed a correlation to calculate the maximum heat-flux density for a single tube: 0.9 ⎛ p ⎞0.35 ⎛ p⎞ q1, max = 367 pc ⎜ ⎟ ⎜1 − ⎟ ⎝ pc ⎠ ⎝ pc ⎠

(11-4-18)

pc is expressed in kPa and q1, max in W/m2. In a tube bundle, the maximum heat-flux density is lower than for a single tube because the vapor generated in the lower tubes increases the amount of vapor in the upper tubes, thus making access of liquid to the tube wall more difficult. Palen and Small22 suggested correcting the maximum heat-flux density for a single tube by means of qb , max = q1, max φ b

(11-4-19)

where φb is a correction factor related to the geometry of the tube bundle. Several expressions were proposed for the calculation of φb. The simplest21 is φb = 3.1ψ

(11-4-20)

with the restriction φb ≤ 1

(11-4-21)

where ψ is the same parameter defined in Eq. (11-4-15). It must be noted that for a single tube, ψ = 1. The equation tells us that essentially it must be considered that φb = 1 for all cases with ψ higher than 1/3. In these cases, which correspond to small tube bundles, the behavior is similar to that of a single tube. The bigger the tube bundle and the more compact its design, the smaller are the values of ψ and φb . This means that blockage of the heat transfer surface by the vapor occurs at lower heat-flux densities. A certain safety margin with respect to these values must be adopted. It can be considered that when the heat-flux density at which the reboiler operates is higher than 0.5qb, max, there is a potential risk of vapor interference. In these cases it is recommended to modify the tube bundle geometry, increasing ψ (e.g., increasing the separation between the tubes). A heat-flux density higher than 0.7qb, max never should be used for design. Kern’s Recommendations for Maximum Heat Flux. In the 1950s, Donald Kern suggested upper limits to the maximum heat-flux density that should be adopted for reboiler design. These values were 94,600 W/m2 [30,000 Btu/(h · ft2)] for aqueous substances and 37,800 W/m2 [12,000 Btu/(h · ft2) for organics. These recommendations were not supported by any theory but were obtained by industrial experience. The engineering practice around the world followed Kern's recommendations for decades. Since these heat-flux densities are rather low, they normally were the limiting factor in design, and usually, the heat transfer area simply was adopted as the quotient between Q and this heat-flux density. At present, the theories that we have presented predict maximum heat-flux densities generally higher than Kern’s values. However, owing to the weakness of the theoretical models and the complexity of the problem, even today, Kern’s limits are used widely, and some commercial design programs still maintain them as a user option.

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CHAPTER ELEVEN

A1

Db Ds FIGURE 11-35

Kettle reboiler section.

Shell Sizing. The shell diameter must be larger than the tube bundle diameter Db to allow vapor-liquid separation. Usually the shell diameter is approximately twice the tube bundle diameter (Fig. 11-35). It is recommended to calculate the shell cross-sectional area above the level of the weir (A1 in the figure) using a critical vapor velocity defined as vc = k1

ρL − ρV ρV

(11-4-22)

with k1 = 0.03 to 0.09 m/s. If VV = WV /ρV = m3/s of vapor at the outlet, then it must be A1 ≥

VV vc N N

(11-4-23)

where NN is the number of vapor outlet nozzles (if the tube length is higher than 5 m, it is usual to have two vapor outlet nozzles). Example 11-5 Reboiler with boiling range. A kettle reboiler for a distillation column operating at 400 kPa pressure must be designed. The column bottom product entering the reboiler has a flow rate of 10,000 kg/h and a temperature of 104.5°C (essentially equal to its bubble-point temperature). Fifty percent of the feed is vaporized into the reboiler, and the other 50 percent is extracted as liquid product. The outlet temperature (temperature inside the reboiler) is 113.3°C. The dew point of the feed at 400 kPa is 119°C. The required heat flux obtained from the process simulation is 470 kW. The heating media is steam. The available steam is at 717 kPa. To perform this service, it is proposed to use a reboiler with the following characteristics: TEMA type AKU N = number of tubes: 44 n = number of tube passes: 2 Tube diameters: Do = 0.019 mm; Di = 0.0144 mm Pt = tube pitch: 0.0238-mm triangle Tube length: 3.3 m Fouling resistances of 0.0001 for the tube side and 0.0003 for the shell side must be considered. Verify the suitability of this unit. Physical properties: Cold fluid (properties at the outlet temperature 113.3°C):

BOILING

Vapor cV = 2,100 J/(kg · K) μV = 0.0083 cP ρV = 11.7 kg/m3 kV = 0.0213 W/(m · K) pc = 3214 kPa (critical pressure)

381

Liquid cL = 2,700 J/(kg · K) μL = 0.142 cP ρL = 564 kg/m3 kL = 0.087 W/(m · K)

We shall use the correlations for the prediction of qmax, even though we know that this may not be the most conservative approach. The available steam pressure is 717 kPa. If a pressure drop for the steam control valve of 100 kPa is assumed, and considering a 50-kPa pressure drop in the reboiler, the inlet and outlet steam pressures will be 617 and 567 kPa, respectively. At these pressures, the corresponding condensation temperatures are 160 and 155.5°C. Thus we shall assume a mean temperature for the condensing steam of 157.7°C. The temperature difference for heat transfer thus will be ΔT = 157.7 – 113.3 = 44.4°C (remember that for the boiling fluid, the outlet temperature must be used). For the steam side, we shall assume hio = 8500 W/(m2 · K). The heat transfer area is Solution

A = Nπ DoL = 44 × π × 0.019 × 3.3 = 8.66 m2 The heat-flux density thus will be q = Q/A = 470,000/8.66 = 54,272 W/m2 We shall verify this value using Mostinsky correlations [Eqs. (11 -4-12) to (11-4-17)]: hNB1 = 0.00417(3214)0.69 × 54,272 0.7Fp = 2260Fp Fp =1.8(p/pc)0.17 + 4(p/pc)1.2 + 10(p/pc)10 = 1.8(400/3214)0.17 + 4(400/3214)1.2 + 10(400/3214)10 = 1.59 hNB1 = 2260 × 1.59 = 3593 W/(m2 · K) Correction factor for tube bundle geometry: For calculation of this parameter, it is necessary to analyze the tube-sheet layout. A TEMA AKU shell with two tube passes is considered. 44 tubes, 19 mm OD, with 23.8-mm pitch in a 30-degree triangular distribution corresponds to a tube bundle diameter Db = 240 mm. With Db = 0.240, Pt = 0.0238, Do = 0.019, C1 = 0.866, from Eq. (11-4-16), we obtain Fb = 1.4. Correction factor for boiling-range Fc: The boiling range of the mixture is BR = 119 – 104.5 = 14.5°C. Then Fc =1/(1 + 0.023q0.15BR0.75) = 1/(1 + 0.023 × 54,2720.15 × 14.50.75) = 0.532 hNB = hNB1FbFc = 3,593 × 1.4 × 0.532 = 2,670 W/(m2 · K) The shell-side coefficient then will be ho = hNB + hTP = 2,670 + 500 = 3,170 W/(m2 · K) The total heat transfer coefficient will be U = (1/ho + 1/hio + Rf)–1 = (1/3,170 + 1/8,500 + 0.0004)–1 = 1,200 W/(m2 · K) Verification

q = UΔT = 1,200 × 44.4 = 53,300 W/(m2 · K) This value is in good agreement with the 54,272 W/m2 adopted for calculation of the boiling h. Otherwise, ho should be recalculated. The area required by the unit will be A = Q/q = 470,000/53,300 = 8.8 m2

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The unit is slightly undersized. Since the difference is very small, a new calculation is not considered necessary. Verification of the maximum heat-flux density: The preceding calculations are only valid if the maximum nucleate boiling heat-flux density is not exceeded. Calculating q1, max with Eq. (11-4-18) with p = 400 kPa and pc = 3214 kPa, we get q1, max = 504,000 W/m2 The geometric correction factor is φb = 3.1ψ. Thus ψ = πDb L/A = π × 0.24 × 3.3/8.66 = 0.29 φb = 0.89 Then qb, max= 0.89 × 504,000 = 448,500 W/m2 Since this value is much higher than the design heat-flux density, the geometry of the unit is satisfactory. Example 11-6 Design a reboiler for the same process conditions as Example 11-5 but with a vapor production of 85,100 kg/h. Analyze the convenience of using a heating steam of higher temperature. The heat duty will be proportional to the preceding case, that is,

Solution

Q = (85,100/5,000) × 470 = 8,000 kW In the preceding example, we noticed that the maximum heat-flux density was much higher than that used in the design. One could think that it is possible to work with a higher-temperature steam, which would allow a higher heat-flux density. However, since it is a bigger unit, the geometric factor φb and hence the maximum heat-flux density will be smaller. Let’s maintain the design with 3/4-in (19-mm) tubes with a 15/16-in (23.8-mm) triangular pitch and a tube length of 3.3 m. To study the impact of the scale factor in the design, the following procedure will be used: 1. 2. 3. 4. 5. 6. 7.

A value for q is assumed. Calculate A = Q/q. Calculate the number of tubes N = A/πDoL. Draw the tube distribution, and estimate the tube bundle diameter Db . Calculate ψ and φb . Calculate qb, max = q1, maxφb . Assuming a safety factor of 0.7, if q > 0.7qb, max, a smaller value for the heat-flux density must be assumed. Otherwise, the assumed heat-flux density is acceptable. 8. Calculate the film coefficients and the total U. Then, with the heat-flux density resulting from the preceding steps, calculate the required ΔT as ΔT = q/U. 9. The required steam temperature will be Th = Tc + ΔT = 113 + ΔT.

The results of the final iterations corresponding to steps 1 to 7 are shown in the following table: q

A

N

Db, mm

ψ

φb

qb, max

0.7qb, max

100,000 65,000

80 123

405 624

598 685

0.0774 0.057

0.24 0.176

121,100 89,056

84,700 62,300 OK

This means that the design must be based on a maximum heat-flux density of about 62,300 W/m2, and a 624-tube reboiler is necessary.

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Calculation of the heat transfer coefficients: Recalculating hNB1 and Fc with q = 62,300 W/m2, the result is hNB1 = 3,957 W/(m2 · K), and Fc = 0.527. The geometric factor Fb is calculated with Db = 0.685 using Eq. (11-4-16), and the result is Fb = 1.94. hNB = hNB1FbFc = 3,957 × 1.94 × 0.527 = 4,045 W/(m2 · K) ho = hNB + hTP = 4,045 + 500 = 4,545 W/(m2 · K) U = (1/hio + 1/ho + Rf)–1 = 1,355 W/(m2 · K) Then ΔT = q/U = 62,300/1,355 = 47°C This means that the maximum steam temperature that can be used is 113.3 + 47 = 160.3°C.

11-4-4

Horizontal Thermosiphon Reboiler The heat transfer mechanism in a horizontal thermosiphon reboiler is essentially the same as that in a kettle reboiler, and Eq. (11-4-6) is applied. However, in this case it may be expected that the convective contribution (hTP) is incremented as a consequence of the higher liquid circulation velocity. Several models with different degrees of complexity were proposed to take into account the convective effects. One of the best known is that of Fair and Klip.23 These models require a significant number of calculations and the available information to support their use is scarce, so a conservative recommendation is to design them as kettle reboilers.

11-4-5 Axial-Flow Reboilers This type includes forced-circulation and vertical thermosiphon reboilers. In these units, boiling takes place into the tubes, and both the boiling heat transfer coefficient and the frictional pressure drops depend on the two-phase Reynolds number ReTP = GF1.25(1 – x)Di/μL (see Eq. 11-3-23). In a forced-circulation reboiler, the flow rate is imposed by the circulation pump, but in a thermosiphon unit, as explained in Sec. 11-4, the flow rate results from the hydraulic balance in the circuit, where the circulation driving force is the density difference between the reboiler liquid feed and the vapor liquid mixture in the tubes. But the density of the vapor-liquid mixture depends, in turn, on the heat transfer capacity of the reboiler. This means that the thermal and hydraulic calculations are coupled and cannot be solved independently. The usual approach is to assume a total flow rate and calculate the required heat transfer area to reach the desired vaporization for this flow rate. The hydraulic balance then is verified, and based on the results, the assumed flow rate is modified, repeating the calculations up to final agreement. It must be noted that the vapor mass fraction x included in the Reynolds number ReTP changes as the vaporization progresses. Hence both ReTP and the frictional pressure drop per unit length, as well as the density of the vapor-liquid mixture, change along the tube length. The calculation method requires that the reboiler be divided into intervals small enough to assume that the fluid conditions are uniform, and for each interval, the heat transfer surface and frictional pressure drops are calculated. The total value results from summation of the individual values for all the intervals. The detailed procedure is explained in the following section. Vertical Thermosiphon Reboiler Design. Assume that it is necessary to design a thermosiphon reboiler to vaporize WV (kg/h) of a liquid mixture at a distillation column bottom. The column bottom conditions correspond to a pressure pA. At this pressure, the liquid is at its bubble-point temperature TA. The liquid-feed mass flow to the reboiler W will be higher than the vaporization rate Wv owing to the recirculated fraction. It is usual to design with circulation ratios W/Wv of at least 4:1. We assume a

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CHAPTER ELEVEN

mass flow rate W. As already explained, it must be verified that the reboiler design allows this circulation rate. The vapor mass fraction at the reboiler outlet thus will be xE =

Wv W

(11-4-24)

If the liquid is a multicomponent mixture, the boiling temperature increases as long as vaporization progresses. It is then necessary to have a vaporization curve showing the amount of heat to be delivered to the circulating fluid for different vapor mass fractions between 0 and xE to achieve that vapor fraction, as well as the temperatures corresponding to each point. This curve can be calculated with a process simulator and can be presented in a table such as Table 11-3. TABLE 11-3 The Boiling-Curve Data x

T

ΔQ

0 . . . xE

tA . . . tE

0 . . . Q = ΣΔQ

We must remark that the pressure also changes along the tube length. However, for the calculation of this table, it is usual to neglect the effects of pressure changes, and the table is calculated at a constant pressure equal to the column bottom pressure In certain cases, if the circulation ratio is high or the boiling range of the product is small, the table may not be necessary, and it can be assumed that the fluid vaporizes at constant temperature. In this case, the boiling curve will be a linear function of x and can be expressed as ΔQ = λW Δx

(11-4-25)

where λ is a mean heat of vaporization. As in other design cases, it is necessary to assume a geometric configuration and then verify that it allows the required heat flux. We shall then assume that the following variables were adopted: tube diameter, number of tubes, shell diameter, heating-fluid flow rate, and temperatures. With this information, the shell-side heat transfer coefficient can be calculated. To simplify the explanation, it will be assumed that the heating medium is isothermal condensing steam at temperature T. Liquid Single-Phase Region. We refer to Fig. 11-36. The liquid at the distillation column bottom is at pA and TA, where TA is the bubble-point temperature corresponding to pA. The liquid pressure at the lower tube sheet level will be pB, and neglecting the frictional pressure drop in the reboiler inlet piping, this is pB = p A + ρL gz1

(11-4-26)

Since pB is higher than pA, the bubble-point temperature at pB is also higher than TA, meaning that the reboiler liquid feed is subcooled. The first reboiler region thus will be liquid single-phase heat transfer. As was explained in Sec. 11-3 (see Fig. 11-18), at a certain reboiler section, the bubble production at the tube wall starts, and the heat transfer mechanism turns into a subcooled nucleate boiling regime. From then on, the liquid continues increasing its temperature, whereas the pressure decreases, up to the section in

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V E L LC

A

z1

z2 C Single-phase region B

FIGURE 11-36

Thermosiphon reboiler with single-phase region.

which the mean liquid temperature reaches the bubble point at the prevailing pressure. There the saturated boiling regime starts. Coming back to Fig. 11-36, let’s assume that section C corresponds to the start of saturated boiling. In this section, pressure pc and temperature Tc correspond to a point on the vapor-pressure curve. This was already explained in Sec. 11-3 (see Fig. 11-20). The pressure-temperature conditions corresponding to point C can be obtained as the intersection between the saturation curve of the fluid (vapor pressure versus temperature) and the curve representing the pressure-temperature evolution of the fluid as it flows up through the tube. If t(z) and p(z) represent the temperature and pressure condition of the fluid corresponding to a coordinate z, this evolution is represented by the equation dt( z ) t( z ) − t B = dz ( p( z ) − pB) dp( z )

(11-4-27)

dz And the saturation curve is ⎛ dt ⎞ tSAT( z ) − t A = ⎜ ⎟ ( p − p A) ⎝ dp ⎠ S

(11-4-28)

where the derivative is the slope of the saturation curve (bubble-point temperature versus pressure). This also can be calculated with the help of a process simulator. In the equation, p is the saturation pressure at

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CHAPTER ELEVEN

tSAT(z). At point C, the last two equations must be satisfied, and considering that tA equals tB because there is no heating between A and B, combining both expressions, we have

pB − pC = pB − p A

⎛ dt ⎞ ⎜⎝ dp ⎟⎠ S −

dt /dz ⎛ dt ⎞ + dp /dz ⎜⎝ dp ⎟⎠ S

(11-4-29)

The derivatives of temperature and pressure with respect to z can be calculated easily because it is a liquid single-phase region. (Even though part of this region corresponds to subcooled boiling, it is conservative to calculate it as a liquid single-phase region.) Then, with the total mass flow rate W, it is possible to calculate the total heat transfer coefficient USP for this single-phase region, and it is WcL dt = ( πDi Ndz )USP (T − t A ) ∴

dt πDi NUSP (T − t A ) = dz WcL

(11-4-30) (11-4-31)

and the pressure drop is dp = − ρL gdz − dF − vdv

(11-4-32)

The last two terms represent the frictional and acceleration pressure drops. For this single-phase region, these terms can be neglected, resulting in dp = −ρL g dz

(11-4-33)

Then, with the preceding equations, it is possible to calculate the pressure and temperature corresponding to point C where saturated boiling starts. The heat flux corresponding to the single-phase region is QSP = WcL (TC − TA )

(11-4-34)

And the heat transfer area corresponding to this region may be calculated as QSP USP (T − t A )

(11-4-35)

ΔzSP = ASP / ( πDo N )

(11-4-36)

ASP = The length of this single-phase region will be

We must remark that this length is usually not significant with respect to the total area of the reboiler. Thus it is often neglected, and calculations can be started directly with the saturated boiling region assuming that TC = TA and pC = pB. Saturated Boiling Region. As was explained in Sec. 11-3, the boiling heat transfer coefficient in this region is calculated with the Chen correlation. For this purpose, we shall divide the unit into intervals, as indicated in Table 11-3, and for each interval with the corresponding value of x, we shall calculate ReTP , and with it we shall calculate Xtt and S, which are the necessary parameters for the calculation of hN and hTP . Then, for each one of the intervals, we can calculate

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hi = hN + hTP

(11-4-37)

Di Do

(11-4-38)

hio = hi

⎛ 1 ⎞ 1 U =⎜ + + Rf ⎟ ⎝ hio ho ⎠

−1

(11-4-39)

It is then possible to calculate the area and length of each interval as ΔA =

ΔQ U (T − t( z ) )

Δz =

(11-4-40)

ΔA πDo N

(11-4-41)

Knowing ΔZ, it is possible to calculate the frictional pressure drop corresponding to the interval. It can be obtained with Eqs. (11-3-5) and (11-3-6). For each interval, the Reynolds number for liquid flowing alone [ReL = W(1 – x)Di /at μL] is calculated, and with it, the Fanning friction factor fL can be obtained. Then the frictional pressure drop for the interval can be calculated as Δz 1 ⎡ W (1 − x ) ⎤ Δ p f = φ Δ pL = 4 φ f L ⎢ ⎥ Di 2ρ L ⎣ at ⎦ 2

2

2

(11-4-42)

The two-phase mixture density in each interval can be calculated as ρTP = ρL RL + ρV (1 − RL )

(11-4-43)

where RL is obtained with Eqs. (11-3-3) and (11-3-4). Verification of the Recirculation Rate. With reference to Fig. 11-36, the pressure pB is given by Eq. (11-4-26) (neglecting the frictional pressure drop in the inlet piping). The mass flow W circulating through the circuit must be such that pB is the sum of 1. 2. 3. 4.

Frictional pressure drop in the boiler Hydrostatic pressure owing to the liquid-vapor column inside the tubes Acceleration pressure drop Outlet pressure

We shall see how to calculate these terms. Frictional Pressure Drop. The frictional pressure drops correspond to the sum of 1. Pressure drop in the single-phase region BC. This can be calculated as 4f

Δz SP W 2 Di 2ρL at2

(11-4-44)

where f is the friction factor for liquid flow in the single-phase region. This pressure drop is not significant and often is neglected.

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2. Frictional pressure drop in the double-phase region. This is calculated as the summation ΣΔ pf corresponding to all the intervals. 3. Outlet frictional pressure drop. This can be calculated as W2 (1 − x E )2 φ2E aE2 ρL

(11-4-45)

where aE is the area of the outlet nozzle and φE2 is calculated with Eq. (11-3-6) for the outlet condition. Hydrostatic Pressure Corresponding to the Liquid-Vapor Column. This will be the sum of that corresponding to the liquid single-phase region and all the double-phase intervals. This is ρLgΔzSP + ΣρTPgΔz. Acceleration Pressure Drop. Since the velocity of the vapor-liquid mixture exiting the reboiler is much higher than the liquid velocity at the reboiler inlet, there is an acceleration term that must be considered (this is analogous to the ΔρV 2 term in the Bernouilli equation). This term can be expressed as 2 (1 − x E )2 1 ⎤ W 2 ⎡ xE + − ⎥ ⎢ ρ L RLE ρ L ⎥⎦ at2 ⎢⎣ ρV RVE

(11-4-46)

where subscript E refers to the exit condition. Outlet Pressure. This pressure is pA. Then the hydraulic verification requires the following equation to be satisfied: ρ L gz1 = 4 f

ΔzSP W 2 + ∑ ρTP gΔz + ∑ Δp f + Di 2ρ L at2

(1 − x E )2 1 ⎤ W 2 W 2 ⎡ x2 − ⎥ + 2 (1 − x E )2 φ 2E + 2 ⎢ E + ρ L RLE ρ L ⎦ aE ρ L at ⎣ ρV RVE

(11-4-47)

Limitations to the Heat-Flux Density. As was pointed out in Sec. 11-3, in flow boiling it is also necessary to establish limits to the validity of the correlations. These limits correspond to the beginning of the transition boiling regime. A simple correlation to predict the maximum heat-flux density is expressed by Eq. (11-3-32). Thus, if the maximum heat-flux density resulting from the calculation is higher than that obtained with Eq. (11-3-32), it will be necessary to decrease the temperature of the heating medium. Again, in this case the values suggested by Kern (94,600 W/m2 for aqueous fluids and 37,800 W/m2 for organics) can be adopted as the upper limit of the heat-flux density. However, at present, these values are considered excessively conservative and may lead to oversized designs. When working at low pressures, if ΔT is increased above a certain value, another phenomenon may appear, consisting of a hydraulic instability that provokes a pulsating flow in the reboiler circuit. This effect was studied by Blumenkrantz,24 and the conclusion is that this limitation can be stabilized by increasing the frictional resistance in the reboiler inlet piping either by the addition of a valve or by using a smaller diameter. However, at moderate pressures, the critical temperature corresponding to the start of transition boiling is reached before the occurrence of this hydraulic instability, so the limit to ΔT is given by Eq. (11-3-32). If either the heat-flux density or the mass vapor fraction at the reboiler outlet is too low, it is also possible to have an unsatisfactory operation. This is so because in these cases the thermosiphon effect can be too small to have a predictable circulation. Thus it is not recommended to design with heat-flux densities lower than 5,000 W/m2. If this condition cannot be achieved because of a very low available ΔT, it is preferable to use a forced-circulation or kettle reboiler. The other limit that must be verified is that the outlet vapor fraction is not so large as to have mist flow. A correlation that was presented to predict the mist-flow condition is given by Eq. (11-3-31). The recommendation, then, is to calculate the mass flow density, and if it is higher than the value predicted by

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Eq. (1-3-31) for the outlet value of Xtt , the design must be adjusted so as to work with higher recirculation rates. This is another reason why it is not recommended to design with circulation rates lower than 3:1 for hydrocarbons or 10:1 for water. Example 11-7 It is desired to design a vertical thermosiphon reboiler to vaporize 4,138 kg/h of the bottom liquids of a distillation column. The column bottom temperature is 142.1°C, and the pressure is 1,261 kPa. The following data corresponding to the boiling curve was obtained with a process simulator at 1,261 kPa, assuming a liquid inlet flow to the reboiler of 12,143 kg/h (which roughly corresponds to a recirculation ratio about 3:1). Q (kW) 0 82 164 246 328 410

Temperature (°C)

Vapor Mass Fraction x (kg/kg)

142 146.2 148.1 151.28 153.2 156.4

0 0.0682 0.1364 0.2046 0.2728 0.34

The following physical properties of the vapor and liquid may be assumed constant in the whole operating range: Vapor Specific heat, J/(kg · K) Viscosity, cP Thermal conductivity, W/(m · K) Density, kg/m3 Critical pressure, kPa Molecular weight Latent heat, J/kg

2,400 0.01 0.027

70 234

Líquid 3,010 0.1 0.055 510 3,167 70

The slope of the vapor-pressure curve of the column bottom liquids (bubble-point curve) is 0.039°C/kPa BR = boiling range of the mixture (difference between dew point and bubble point) = 18°C With the molecular weight, the vapor density may be calculated: ρV = Mp/zRT = 25.635 kg/m3 The heating media will be steam. A condensing temperature of 187°C will be assumed. We shall use 19-mm OD and 14.4-mm ID tubes. A fouling factor of 0.00027 (m2 · K)/W will be adopted. Design the unit and calculate the necessary liquid height to achieve the assumed recirculation rate. Note: A downloadable spreadsheet with the calculations of this example is available at http://www. mhprofessional.com/product.php?isbn=0071624082 Solution

We shall adopt 42 tubes, and we shall calculate their necessary length. The mass flow density

will be G = W/(NπDi2/4) = 12,143/[3,600 × 42 × 0.01442 × (3.14/4)] = 493 kg/(s · m2) As a first approach, we can neglect the liquid single-phase region. This implies assuming that the liquid enters into the reboiler at its bubble-point temperature, and from the beginning, a saturated boiling regime exists. We shall adjust the results later. To calculate the heat transfer coefficient hN using Eq. (11-3-29), it would be necessary to know in advance the value of q. Since this value is not known, an iterative procedure is necessary. We shall first assume a value for q, and with it, we shall calculate hN . At the end of the calculation, this can be verified and the calculations repeated if necessary. We shall start with q = 45,000 W/m2. Also with

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CHAPTER ELEVEN

q = 45,000 W/m2, the correction factor for boiling range Fc = 0.499 [(Eq. (11-4-17)] is calculated. The following table contains the calculation results for all the intervals: Interval ΔQ, kW t ΔT = (T – t = 187 – t) x Xtt [Eq. (11-3-3)] F (Fig. 11-22) ReL = [G(1 – x)Di/μL] ReTP = (ReLF1.25) hTP , W/(m2 · K) [Eq. (11-3-24)] S (Fig. 11-24) hN, W/(m2 · K) [Eq. (11-3-29) with q = 45,000 W/m2] hNFc hi, W/(m2 · K) [Eq. (11-4-37)] hio, W/(m2 · K) [Eq. (11-4-38)] U [Eq. (11-4-39)] ΔA, m2[Eq. (11-4-40)]

1

2

3

4

5

82 144.1 42.8 0.0341 5.722 1.170 68,623 83,516 1,348 0.408 1,236

82 147.14 39.85 0.1023 1.993 1.835 63,778 136,237 1,995 0.28 848

82 149.7 37.5 0.1705 1.722 2.463 58,932 181,867 2,514 0.217 658

82 152.2 34.75 0.2387 0.8016 3.105 54,087 222,981 2,959 0.179 543

82 154.8 32.2 0.3064 0.588 3.875 49,278 260,207 3,348 0.154 467

617 1,966 1,490 945 2.02

423 2,419 1,833 1,072 1.92

328 2,842 2,154 1,174 1.87

271 3,231 2,448 1,256 1.88

233 3,582 2,715 1,323 1.93

The total area will be A = ΣΔA = 9.62 m2. The tube length should be L = A/(πNDo) = 9.62/(3.14 × 42 × 0.019) = 3.84 m The average heat-flux density is 410/9.62 = 42,618 W/m2. It is noteworthy that the importance of hN in the calculation of hi is not significant, so it is not necessary to feed back the calculated value in place of the assumed 45,000 W/m2. Verification of the maximum heat-flux density: The maximum heat-flux density will be verified with Eq. (11-3-32): qmax = 23,600(0.01442/3.84)0.35 × 3,1670.61(1,261/3,167)0.25(1 – 1,261/3,167) OK = 49,123W/m2 > 42,618 Verification of mist flow at reboiler outlet: We shall verify that mist flow does not exist at the unit outlet using Eq. (11-3-31): G = 2,441XttE = 2,441 × 0.512 = 1,251 > 493

OK

Calculation of the height of the liquid leg required to achieve the desired circulation rate: We shall first calculate the frictional pressure drop and the hydrostatic pressure of the liquid vapor column: Interval Δz [Eq. (11-4-41)] Xtt [Eq. (11-3-3)] X' = 1/Xtt RL [Eq. (11-3-4)] ρTP [Eq. (11-4-43)] ρTP.gΔz φ2 [Eq. (11-3-6)] f [Eq. (7-4-9)] Δpf [Eq. (11-4-42)]

1 0.808 5.722 0.174 0.461 249 1,971 4.7 0.0059 1,055

2 0.766 1.993 0.501 0.291 166 1,252 11.78 0.0060 2,206

3

4

5

0.747 1.722 0.580 0.225 135 988 19.64 0.0061 3,117

0.750 0.8016 1.247 0.186 116 852 28.73 0.0063 3,934

0.768 0.588 1.700 0.159 103 773 39.55 0.0064 4,705

The total frictional pressure drop will be ΣΔpfi = 15,016 Pa. The hydrostatic pressure will be ΣρTPgΔz =

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5,836 Pa. The friction loss at the outlet nozzle can be calculated with Eq. (11-4-45) where aE is the area of the reboiler outlet nozzle. Assuming a 0.2-m (8-in) diameter, aE = 0.0314 m2. With Eq. (11-4-45), we get 451 Pa. The acceleration pressure drop is calculated with Eq. (11-4-46), where subscript E refers to the outlet conditions (corresponding to a mass vapor fraction = 0.34). The calculated value is 2,217 Pa. Then Hydrostatic pressure of the liquid vapor column Friction loss at the outlet Acceleration loss Friction loss in tubes Total

5,836 Pa 451 Pa 2,217 Pa 15,016 Pa 23,521 Pa

Then, according to Eq. (11-4-47), the required height is z1 = 23,521/gρL = 23,521/9.8 × 510) = 4.71 m. If the reboiler were built with a 3.84-m tube length as was calculated, to achieve the desired circulation rate, it would be necessary to operate with a liquid level into the column 0.9 m above the upper tubesheet level. In this situation, the liquid would be obstructing the reboiler outlet nozzle. Thus it is necessary to modify the design. With the help of a spreadsheet, it is easy to modify the design by changing the number of tubes to 50. The following table contains the calculation results: Interval

1

2

3

F (Fig. 11-22) ReL = [G(1 – x)Di/μL] ReTP = (ReLF1.25) hTP , W/(m2 · K) [Eq. (11-3-24)] S (Fig. 11-24) hN Fc, W/(m2 · K) [Eq. (11-3-29) with q = 45,000 W/m2] hi, W/(m2 · K) [Eq. (11-4-37)] hio, W/(m2 · K) [Eq. (11-4-38)] U [Eq. (11-4-39)] ΔA, m2 [Eq. (11-4-40)]

1.170 57,643 70,153 1,173 0.458 644

1.835 53,573 114,439 1,735 0.322 454

2.463 49,503 152,768 2,186 0.253 357

3.105 45,433 187,304 2,574 0.211 297

4

3.875 41,393 218,573 2,912 0.182 257

5

1,817 1,377 898 2.13

2,189 1,659 1,010 2.04

2,543 1,928 1,103 1.99

2,871 2,176 1,180 2.00

3,169 2,402 1,244 2.05

The required area is then A = ΣΔA = 10.12 m2. The tube length necessary for heat transfer is 10.12/(50 × π × 0.019) = 3.42 m. The hydraulic calculation results are Hydrostatic pressure liquid + vapor Friction loss at outlet Acceleration loss Friction loss in tubes Total

5,194 451 1,564 9,854 17,063

Required liquid height for circulation is 17,063/(9.8 × 512) = 3.41 m, which is acceptable. Example 11-8 Calculate the length of the liquid single-phase region for Example 11-7 (with the 50 tubes). Solution

Using Eq. (11-4-29) and the nomenclature of Fig. 11-36, we get pB − pC ΔzSP = pB − p A z1 ΔzSP = 3.14 m

0.039 (dt /dp)s = dt /dz dt /dz − + (dt /dp)s − + 0.039 dp /dz dp /dz

392

CHAPTER ELEVEN

In the single-phase region, hi is calculated with Eq. (11-3-24) with F = 1. We get hi = 1,002 W/(m2 · K). hio = 759 W/(m2 · K) U = (1/759 + 1/8,500 + 0.00027)–1 = 586 W/(m2 · K) dt/dz = πDoNU(T – t)/WcL = π × 0.019 × 50 × 586 × (187 – 142)/[(12,143/3,600) × 3,010] = 7.74°C/m dp/dz = – ρLg = – 510 × 9.8 = – 4,998 Pa/m = – 4.99 kPa/m ΔzSP = 3.14(0.039)/[(7.74/4.99) + 0.039] = 0.077 m The temperature at which boiling starts will be tC = tB + dt/dz. ΔzSP = 142.6°C. We see that the length of the single-phase region is not significant with respect to the total tube length, which confirms our previous assumption. Otherwise, it would be necessary to add this length to the value calculated for the convective boiling region. In a rigorous way, it would be necessary to recalculate the convective boiling region later because the temperature at the section where boiling starts is different from the reboiler inlet temperature. However, the differences are very small, and the effort is not necessary. In units working at very low pressures (vacuum service), the single-phase region may be important and must be considered.

GLOSSARY at = flow area of a tube (m) B = constant defined in Eq. (11-1-19) (K · m) CSF = coefficient for Eq. (11-2-6) (dimensionless) D = diameter (m) Dd = bubble detachment diameter (m) F = correction factor in Chen’s Eq. (11-3-23) f = friction factor G = mass flow density or mass velocity (kg/s) g = gravity acceleration (m/s2) i = enthalpy per unit mass (J/kg) h = film coefficient [W/(m2 · K)] hL = single-phase film coefficient calculated assuming all the fluid at liquid state [W/(m2 · K)] hN = hNB = nucleation contribution to the film coefficient [W/(m2 · K)] hNB1 = nucleation coefficient for a single tube hTP = two-phase convection contribution to the film coefficient [W/(m2 · K)] k = thermal conductivity [W/(m · K)] M = molecular weight (kg/kmol) m = exponent in Eq. (11-2-6) (dimensionless) n = exponent in Eq. (11-2-6) (dimensionless) Nu = Nusselt number (dimensionless) p = pressure (N/m2) pc = critical pressure (N/m2) (in Mostinsky dimensional equations, kPa) pG = equilibrium saturation pressure at temperature TG (N/m2) pF = system pressure described in Sec. 11-1 (N/m2) Pr = Prandtl number (dimensionless) q = heat-flux density at the interface (W/m2)

BOILING

qN qC r rc rb R Re ReL ReTP RL RV S T TG tF TSAT TWONB vV vL vVSAT x Xtt X' ΔTSAT ΔpSAT σ λ δ θ φ2 Φ ρ μ β

393

= contribution to q owing to nucleation effect (W/m2) = contribution to q owing to convection effect (W/m2) = bubble radius (m) = critical radius of the cavity (m) = characteristic radius defined by Eq (11-3-8) (m) = Ideal gases constant [(N · m)/(K · mol)] = Reynolds number (dimensionless) = Reynolds number for the liquid flowing alone (dimensionless) = Reynolds number for the two-phase flow (dimensionless) = volumetric fraction of the tube occupied by the liquid (dimensionless) = volumetric fraction of the tube occupied by the vapor (dimensionless) = Chen’s supression factor (dimensionless) = temperature (generic or of the hot fluid medium) (K) = system temperature defined in Sec. 11-1 (K) = temperature in the bulk of the fluid (K) = saturation temperature corresponding to the system pressure (K) = wall temperature at which boiling starts (K) = vapor specific volume (m3/kg) = liquid specific volume (m3/kg) = saturated vapor specific volume (m3/kg) = vapor quality (vapor mass flow/total mass flow) = Lokhardt-Martinelli parameter [Eq. (11-3-3)] (dimensionless) = reciprocal of Xtt (dimensionless) = difference between wall temperature and saturation temperature (K) = difference between the vapor pressure at wall temperature and the vapor pressure at saturation temperature (K) = surface tension (N/m) = heat of vaporization (J/kg) = boundary layer thickness (m) = bubble contact angle (degrees or radians) = correction factor defined by Eq. (11-3-5) = correction factor for bubble contact angle = density (kg/m3) = viscosity (kg/ms) = expansion coefficient (1/K)

Subscripts L = liquid V = vapor TP = two phase i = internal o = external

394

CHAPTER ELEVEN

REFERENCES 1. Thomson W.: Phil Mag 42(4):448, 1871. 2. Hsu Y. Y.: “On the size of range of active nucleation cavities on a heating surface,” Trans ASME J Heat Transfer 84:207, 1962. 3. Han C. Y., Griffith P.: “The mechanism of heat transfer in nucleate pool boiling. Pt 1: Bubble initiation growth an departure,” Int J Heat Mass Transfer 8(6):887–904, 1965. 4. Collier J.: Convective Boiling and Condensation. London: McGraw-Hill, 1972. 5. Fritz W.: “Berechnung des Maximal Volume von Dampfblasen,” Phys Z 36:379, 1935. 6. Vachon R.: Paper presented at U.S. National Heat Transfer Conference, Seattle, WA, paper 67-HT-34, 1967. 7. Rohsenow W.: “A method of correlating heat transfer data for surface boiling of liquids,” Trans ASME 74:969, 1952. 8. Forster H., Zuber N.: “Bubble dynamics and boiling heat transfer,” AIChE J 1:532, 1955. 9. Bromley L. A.: “Heat transfer in stable film boiling,” Chem Eng Progr 46:221–227, 1950. 10. Alves G.: Chem Process Eng 50(9):449–456, 1964. 11. Dengler C. E., Addoms J. N.: “Heat transfer mechanism for vaporization of water in a vertical tube,” Chem Eng Progr Symp Series 57(18):95–103, 1956. 12. Mostinsky I. L.:“ Application of the rule of corresponding states for the calculation of heat transfer and critical heat flux,” Teploenergetika 4:66, 1963 English Abstr. Br Chem Eng 8(8):586, 1963. 13. McNelly M. J.: “A correlation of the rates of heat transfer to nucleate boiling liquids,” J. Imp Coll. Chem Eng Soc 7:18, 1953. 14. Stephan K., Abdelsalam M.: “Heat transfer correlations for natural convection boiling,” Int J Heat Mass Transfer 25:73–85, 1979. 15. Lokhardt R. W., Martinelli R.: “Proposed correlation of data for isothermal two phase two components flow in pipes,” Chem Eng Progr 45(1):39–48, 1949. 16. Fair J. R.: “What you need to design thermosiphon reboilers,” Petroleum Refiner 39(2):105–123, 1960. 17. Chen J.: “Correlation for boiling heat transfer to saturated fluids in convective flow,” I&E.C Process Design and Development 5(3):322–329, July 1966. 18. Zuber N.: “On the stability of boiling heat transfer,” Trans ASME 80:711, 1958. 19. Zuber N., Tribus M., Westwater J. W.: “The hydrodinamic crisis in pool boiling of saturated and subcooled liquids,” Int Dev Heat Transfer 2:230, 1961. 20. Kutateladze S. S.: “A hydrodinamic theory of changes in the boiling process under free convection,” Izv Akad Nauk SSSR Otd Tekh Nauk 4:529, 1951. 21. Palen J. W., in Heat Exchangers Design Handbook. New York: Hemisphere Publishing, 1983. 22. Palen J. W., Small W. M.: “A new way to design kettle and internal reboilers,” Hydrocarbon Proc 43(11):199, 1964. 23. Fair J. R., Klip A.: “Thermal design of horizontal reboilers,” Chem Eng Progr March 1983: 83–96. 24. Blumenkrantz A., Taborek J.: “Application of stability analysis for design of natural circulation boiling systems and comparison with experimental data,” AIChE Symp Series 68(118) 1971. 25. Procedures 4D4.1 and 4D3.1. Technical Data Book: Petroleum Refining. Washington: American Petroleum Institute, 1980. 26. Engineering Data Book, 11th ed: Gas Processors Suppliers Association, Tulsa, Oklahoma, 1998. 27. Perry: Chemical Engineers Handbook. New York: McGraw-Hill. 28. Cichelli M. T., Bonilla C. F.: “Heat Transfer to Boiling Liquids under Pressure,” Trans AIChe 41:755, 1945. 29. Frederking, Clark: “Natural Convection Film Boiling on a Sphere.” Adv Cryogen Eng 8:501, 1963. 30 Bankoff S. G.: “Ebullition from solid surfaces in the absence of a pre existing gaseous phase,” Trans ASME 79:735, 1957. 31 Fritz W.: “Berechnung des Maximal Volume von Dampfblasen,” Phys. Z 36:379, 1935. 32 Hewitt G. F., Roberts D. N.: “Studies of two phase flow patterns by simultaneous X rays and flash photography," A.E.R.E.-M 2159 HMSO, 1969.

CHAPTER 12

THERMAL RADIATION

12-1 HEAT TRANSFER BY RADIATION THROUGH TRANSPARENT MEDIA 12-1-1 Characteristics of the Radiant Energy Radiation is a heat transfer mechanism that, unlike conduction and convection, does not require a physical medium to propagate. Even more, if the propagation medium contains certain gases as water vapor or CO2, part of the energy will be absorbed by these gases, and a smaller amount of energy will reach the receptor. Radiant energy propagates better in a vacuum than through any other medium, either gas, liquid, or solid. Radiant energy is sometimes envisioned as transported by electromagnetic waves and at other times as transported by photons, whose energy depends on the associated wavelength. It is also known that radiation travels with the speed of light. Radiant energy can be emitted by any body, solid, liquid, or gas. For a body to emit radiant energy, its atoms or molecules need to be previously excited by the absorption of any form of energy. Then, when the body emits radiation, the energy of its atoms or molecules decreases to the previous level. These changes in the energy level may consist of a modification of the vibration or rotation modes of the molecules, electrons jumping from one orbital to another with a lower energy level, or a decrease in the energy of the atomic nucleus. Owing to this variety of mechanisms, radiation may consist of photons having very different energy; thus radiant energy encompasses a wide range of wavelengths, and the body can emit a continuous spectrum. The electromagnetic spectrum does not extend uniformly from λ = 0 to λ = infinite because the radiant energy is concentrated in a certain region. The wavelengths of the complete electromagnetic spectrum extend from λ = 10–13 m (cosmic rays) to values of λ as high as 1,000 m (long-wavelength radio waves). There is a relatively narrow zone of this spectrum which is associated with the thermal energy of the emitting body. (This is the internal energy related to the macroscopic observable parameter that is temperature.) This wavelength interval ranges from 0.1–100 μm. All bodies, as a consequence of their temperature, emit radiation within this wavelength range. The higher the temperature of the body, the higher is the amount of energy emitted within this wavelength range. (Also, the distribution of energy within the wavelength spectrum changes with temperature.) The portion of the radiant energy emitted by a body as a consequence of its thermal level is called thermal radiation. Within this wavelength interval, there is a very narrow band (from 0.4 to 0.7 μm) corresponding to radiation that can affect the human optic nerve. This is the visible radiation. When a certain body, as a consequence of its temperature, emits an important amount of radiation within the visible region, it appears luminous (e.g., the filament of an incandescent light bulb, a flame, a very hot iron, etc.). Cold bodies, on the other hand, emit little energy within the visible range, and they lack luminosity. They can, however, reflect the light emitted by other hot sources. When we observe an object illuminated by a light source (e.g., solar light), the radiation we see is not that emitted by the object, but the radiation from the source reflected by the object. The inverse process to emission takes place when a radiant energy beam is incident on the surface of a body. In this case, it may result in an increase in the energy level of atoms or molecules of the body by absorption of part of the incident radiation. In the case of solid surfaces, this absorption takes place within a very thin layer under the surface. In the case of liquids, the radiant energy can travel deeper into the medium, and this distance can be even much higher with gases.

395

396

CHAPTER TWELVE

Not all gases absorb thermal radiation. Elemental gases (N2, H2, and O2) do not absorb thermal radiation to a significant degree. Some other gases, such as water vapor and CO2, are capable of absorbing in very definite bands of the spectrum. Solids and liquids, on the other hand, can absorb energy of all wavelengths. In the same way that the effect of radiant energy emission is to decrease the temperature of an emitting body (unless it is simultaneously receiving energy from another source), the effect of radiant energy absorption is to increase the temperature of the absorber. If a body is at a certain temperature in a steady state, this is so because it has reached a thermal level such that the energy it is emitting equals the energy it is absorbing from other sources. If two bodies are facing each other, there will be a net heat transmission from the warmer body to the colder one. If the two bodies reach the thermal equilibrium, the energy emitted by each equals the energy each absorbs from the other. There is no net heat transmission between them. This condition is reached when the temperatures of both bodies are equal. Thus it is evident that bodies capable of emitting a higher amount of energy at a certain temperature also must be capable of absorbing energy in a higher proportion because this is the only way in which they can reach thermal equilibrium with other bodies that are not as good emitters. A body can reach a steady state within a system when it absorbs the same amount of energy than it emits. For example, a body exposed to solar radiation interchanges energy with the sun, which acts as a source, and also with the rest of the universe that surrounds it. It is possible that the body reaches a temperature that is higher than the surroundings (e.g., a piece of iron remaining under the sun in summer). In this system, the sun acts as the energy source (it emits more energy than it absorbs), the body reaches a steady state (emitting the same amount that it absorbs), and the rest of the world acts as a final receptor of energy. There are some other wavelengts of the electromagnetic spectrum capable of producing a temperature increase in an absorbing body. For example, microwaves used for cooking correspond to wavelengths of about 10–1 m. However, this type of radiation is not produced by a hot source but rather by electronic devices. The waves do not belong to the field of thermal radiation, respond to different mechanisms, and will not be studied here. This chapter will deal only with radiation resulting directly from thermal excitation, and we shall refer to it hereafter merely as radiation.

12-1-2 Absorption, Reflection, and Transmission Coefficients When radiation is incident on a surface, a portion of the total energy is absorbed in the material, a portion is reflected from the surface, and the remainder is transmitted through the body. The fraction absorbed is called the absorptivity or absorption coefficient α, the fraction reflected is the reflectivity or reflection coefficient ρ, and the fraction transmitted is the transmissivity or transmission coefficient τ. These magnitudes are related by α + ρ+ τ =1

(12-1-1)

The relative values of these coefficients depend on the material of the body and the state of its surface. In an opaque body, an incident beam can penetrate only a very short distance into the body. This distance is on the order of a micron for metals. The absorbed energy increases the temperature of the zone immediately adjacent to the surface, and then heat penetrates into the body by conduction. We may think, then, that all the process takes place at the body surface, and then, for an opaque solid, it can be considered that α+ρ=1

(12-1-2)

A body with α = ρ = 0 is called a white body (then τ = 1). A material that behaves roughly as a white body is glass. A body with ρ = 1 is a mirror. Glass is not a perfect white body. The fact that it heats up when exposed to solar radiation demonstrates that its absorption coefficient is not nil. A black body is one that absorbs all the incident radiation. Then, for a black body, α = 1. For all real bodies, α < 1. This indicates that the body reflects part of the incident radiation. The absorptivity or absorption coefficient depends on the nature of the body, its temperature, and the nature of the incident radiation. This means that any body will present different values of the absorption coefficient when changes the source from which it is absorbing radiant energy.

THERMAL RADIATION

12-1-3

397

Kirchoff’s Law Let’s consider a big cavity into which there is a small body a whose external area is Aa (Fig. 12-1). This body is in thermal equilibrium with the cavity walls, which are assumed to be insulated, and it is also assumed that there is nothing else in the cavity other than body a. Let t be the temperature of the system in equilibrium. Both body a and the internal walls of the cavity emit radiant energy. The energy emitted by the body per unit time and per unit area is what we shall call Wa (emissive power of body a).

I Aa

a Wa

FIGURE 12-1

Body in an infinite cavity.

Let I be the irradiation over the surface of the body (which is the incident energy per unit area and per unit time). Then the surface of the body emits an amount of energy AaWa and absorbs an amount of energy AaαaI, where αa is the absorption coefficient of the body for the radiation coming from the cavity. Since we have assumed that the system is in thermal equilibrium, Aa Wa = Aa α a I

(12-1-3)

This means that I=

Wa αa

(12-1-4)

If now we replace body a with another body c, whose area is Ac, at the same temperature as the cavity (we reserve the letter b for the blackbody), we have I=

Wc αc

(12-1-5)

Since we have assumed that the dimensions of the cavity are very large in comparison with the bodies, we can assume that the presence of these bodies does not affect the behavior of the cavity. Then irradiation I remains constant when we change the bodies. Then Wa Wc Wi = = =I α a αc αi

(12-1-6)

398

CHAPTER TWELVE

This relationship is valid as long as the bodies a, c, and i are at the same temperature as the cavity. In other words, the bodies and the cavity are in thermal equilibrium. This expression is known as Kirchhoff’s law, and it says that the quotient between the emissive power of a body and the coefficient of absorption exhibited by that body for the radiation coming from a big cavity at the same temperature is a constant. This means that a body that is capable of absorbing a higher amount of radiation also must be the best emitter because it must get rid of all the absorbed energy to reach equilibrium with other bodies. In particular, if we place into the cavity a body capable of absorbing all the incident energy (blackbody), we can write I=

Wa Wc Wb = = α a αc 1

(12-1-7)

Since for any body other than a blackbody α is always lower than 1, Eq. (12-1-7) shows that the blackbody is the maximum emitter at the considered temperature. The emissivity ε of a body is the quotient between its emissive power and the emissive power of the blackbody at the same temperature. For a generic body a, this is Wa Wb

(12-1-8)

εa = α a

(12-1-9)

εa = By comparing Eqs. (12-1-7) and (12-1-8), we get

This is another expression of Kirchhoff’s law, and it establishes that the emissivity of a body is equal to the absorption coefficient of that body for the radiation coming from a big cavity in thermal equilibrium with it. The emissivity of a body is a function of its temperature, but the absorption coefficient depends not only on its temperature but also on the characteristics of the incident radiation. If the temperature or the nature of the source with which the body is exchanging energy is modified, the value of the absorption coefficient will change. Equation (12-1-9) is only valid for the case of a body in thermal equilibrium with a big cavity, and it cannot, in principle, be generalized for other situations. Equation (12-1-7) indicates that the irradiation I in a cavity of large dimensions is equal to the emissive power of the blackbody at the same temperature.

12-1-4 The Cavity as Blackbody The blackbody is a theoretical concept. No real body absorbs all the energy it receives. For experimental purposes, though, a blackbody can be simulated by a cavity such as a hollow sphere maintained at a uniform temperature with a small hole in its wall (Fig. 12-2). Any radiation entering through this hole has little probability of reaching the orifice again and escaping because it repeatedly strikes and reflects on the interior surface, and part of the energy is absorbed in each reflection. When the original radiation beam finally reaches the hole again and escapes, it has been so weakened by repeated reflection that the energy leaving the cavity is negligible. Thus, for any external observer, the orifice behaves as a blackbody because it absorbs all the radiation that is incident on it. We have explained that the irradiation within the cavity equals the emissive power of the blackbody. A small hole in the cavity wall will not modify the situation, so the energy coming out of the hole equals the emissive power of the blackbody at the temperature of the cavity. Thus a small hole placed in the wall of a big enclosure behaves as a blackbody from the point of view of either absorption or emission of radiant energy, and it is a way to approximate the characteristics of an element of an area belonging to a blackbody. A big cavity also acts as a blackbody surface for any body placed into it because all the energy emitted by the body is being absorbed in successive reflections at the cavity walls with little probability to reach back the surface of the emitter.

THERMAL RADIATION

FIGURE 12-2 completely.

399

Radiation coming into a big cavity is absorbed

12-1-5 Monochromatic Emissive Power Any body emits energy within a range of wavelengths. The monochromatic emissive power at a certain wavelength is the amount of energy emitted per unit time, per unit area, and per unit wavelength interval. For example, if W50 is the energy emitted in the interval from 0 to 50 μm and W51 is the energy emitted in the interval from 0 to 51 μm, always per unit time and per unit area, then the monochromatic emissive power at the considered wavelength (in this case 50 μm) is Wλ =

W51 − W50 1 μm

(12-1-10)

Any body shows an emission spectrum that is characteristic of the nature of the body and its temperature. The emission spectrum is the distribution W λ = f(λ) (Fig. 12-3). The total emissive power of the body is then ∞

∫0 Wλ d λ

Monochromatic emissive power (Wλ )

W=

Wavelength (λ) FIGURE 12-3

Emission spectrum.

(12-1-11)

400

CHAPTER TWELVE

The unit for W (total emissive power) is W/m2, whereas the unit for Wλ (monochromatic emissive power) is W/(m2 · μm). 12-1-6 Spectral Radiant Energy Distribution for a Blackbody A relationship showing how the emissive power of a blackbody is distributed among the different wavelengths was derived by Max Plank3 in 1,900 through his quantum theory. According to this theory, Wbλ =

2πC1λ −5 eC2 / λT − 1

(12-1-12)

where C1 = hc 2 C2 = hc /k where h and k are the Planck and Boltzman constants, respectively, and c is the speed of light in a vacuum = 2.9979 × 108 m/s. Figure 12-4 is a plot of Eq. (12-1-12) showing Wbλ as a function of λ at given parameters of the body temperature. From the graph it is possible to extract some conclusions:

108

Blackbody temperature 106 [Wbλ(λ,t) (W/m 2·μm)]

Monochromatic emissive power of the blackbody

107

6560 K 2780 K

105

1670 K 104 833 K 103

556 K

102 Locus of the maximum values of Wbλ 10 Violet

Red

4 6 8 Wavelength λ (μm)

Visible region 0.4–0.7 μm FIGURE 12-4

Emission spectrum of the blackbody.

10

12

THERMAL RADIATION

401

1. The total energy emitted per unit time and per unit area over the entire wavelength spectrum, which is the emissive power of the blackbody, increases with the temperature of the body. This value can be obtained as the total area under a curve corresponding to any temperature. It can be demonstrated that Wb =

∞ 2πC1λ

∫0

e

−5

C2 / λT

dλ = σT 4 −1

(12-1-13)

where σ is known as the Stephan Boltzman constant, and its value can be obtained by calculating the integral in Eq. (12-1-13), resulting in σ = 5.672 × 10 −8 W/(m 2 ⋅ K 4 ) 2. When the temperature increases, the wavelength corresponding to the peak of emission moves toward the smaller wavelength’s zone. The relationship between the wavelength λmax at which Wbλ is a maximum and the absolute temperature is called Wien’s displacement law. Thus λ maxT = 2897.6 μm ⋅ K

(12-1-14)

where λmax is the wavelength for which the monochromatic emissive power is a maximum at temperature T. The dashed line in Fig. 12-4 is the locus of all points represented by Eq. (12-1-14). The shaded portion of the graph shows the region of the spectrum corresponding to visible light. It can be noticed that at temperatures about 1,000 K, a certain part of the emitted energy falls within this range, and the human eye begins to detect radiation. At this temperature, the object glows with a dull-red color (within the visible spectrum, the higher wavelengths correspond to red). As the temperature is increased, the peak of emission moves toward smaller wavelengths, and the color changes becoming nearly white at about 1,500 K. Equation (12-1-14) made it possible to calculate the temperature of the sun. Solar radiation was resolved, and it was observed that the maximum emissive power corresponds to a wavelength of 0.25 μm. Then, by application of Eq. (12-1-14), it can be concluded that the temperature of the sun is 6,366 K.1

12-1-7 Emission from Real Surfaces Any real surface (not black) emits energy with a spectral distribution different from that of the blackbody. With reference to Fig. 12-5, if at a certain temperature curve b represents the emissive power of the blackbody and curve a represents that of any other body at the same temperature, we can define the monochromatic emissivity of that body at the wavelength and temperature considered as ελ =

Wλ = f(λ,T ) Wbλ

(12-1-15)

The last part of the equation merely means that the monochromatic emissivity is a function of the wavelength and temperature. While the total emissivity of the body at the considered temperature is ∞

W ε= = Wb

∞

∫0 Wλ d λ = ∫0 Wbλ ελ d λ = f (T ) ∞ Wb ∫0 Wbλ d λ

(12-1-16)

since usually ελ is not a constant for all wavelengths, when dealing with real bodies, not only is the amount of energy emitted at a certain temperature smaller to that of the blackbody, but also the spectral distribution is different from that of the blackbody.

CHAPTER TWELVE

Monochromatic emissive power [kWh/(m2·μm)]

402

350 300 Wbλ 250 Blackbody b

200 150 100

Real body a 50

0 FIGURE 12-5

12-1-8

1

2

4 3 Wavelength (μm)

5

6

Emission spectrum of real surfaces.

Gray Bodies Gray bodies are bodies whose surfaces present a constant emissivity for any wavelength and any temperature. In this case, the curve representing the emissive power is similar to that of the blackbody, with the difference being that all the ordinates are reduced by a constant factor, which is the emissivity of the gray surface (Fig. 12-6). For a gray surface, the total emissivity becomes ∞

∫ Wbλ ελ d λ = ε = constant ε= 0∞ λ ∫0 Wbλ d λ Real surfaces do not behave as gray surfaces, but the concept of gray body greatly simplifies the problems of radiant heat transfer and is an approximation used frequently in technical calculations.

12-1-9

Monochromatic Absorption Coefficient When a surface absorbs radiant energy, it preferably exhibits a higher capacity to absorb radiation from certain wavelengths. We shall define the monochromatic absorption coefficient for a specific wavelength as the fraction of the total incident monochromatic energy that is absorbed by the surface. This coefficient is denoted as αλ. Then, if the monochromatic energy of a certain wavelength that is incident on a surface per unit area and per unit time is Iλ, we get αλ =

energy absorbed = f (λ , T ) Iλ

This monochromatic absorption coefficient is a function of the temperature of the surface and the wavelength under consideration.

THERMAL RADIATION

403

Emissive monochromatic power [kW/(m2·μm)]

350 W λ1 W λ2 = constant = W bλ1 W b λ2

300 250

ελ = ε = 1 (blackbody)

Wb λ1 200

ελ = ε = 0.6 (gray body) 150 W λ1

W bλ2

100 W λ2 50

1 FIGURE 12-6

3 4 2 Wavelength (μm)

5

6

Emission spectrum of gray surfaces.

The total absorption coefficient defined in Sec. 12-1-2 is the fraction of the energy that is absorbed with respect to the total incident energy of all wavelengths. Then ∞

∫ αλ Iλ d λ α= 0 ∞ ∫0 I λ d λ

(12-1-17)

This coefficient is a function of the temperature of the body but also depends on the distribution of the incident radiation I λ = f(λ), as can be seen in Eq. (12-1-17). According to Kirchhoff’s law, when a body absorbs energy from a cavity in equilibrium with it, ε = α . Since a big cavity behaves as a blackbody thermally, we can say that Kirchhoff’s law expresses the equality between the emissivity at a certain temperature and the absorption coefficient for the radiation emitted by a blackbody at the same temperature as that of the considered surface. It then follows that the equality between emissivity and absorption coefficient requires that the incident radiation have the same spectral distribution as that of the blackbody. In these conditions, the absorption coefficient is ∞

∫ α λ I bλ d λ α= 0 ∞ ∫0 I bλ d λ

(12-1-18)

where the energy distribution Ibλ corresponds to the black body. If the source with which the body exchanges energy is a gray body, the absorption coefficient wil be the same, since ∞

∫ α λ (εI bλ ) d λ α= 0 ∞ ∫0 (εI bλ ) d λ

(12-1-19)

404

CHAPTER TWELVE

where ε (= constant) is the emissivity of the source. Then the absorption coefficient of a real surface is the same, whether the surface absorbs energy from a black source or from a gray source at equal temperatures.

12-1-10 Generalization of Kirchhoff’s Law for Gray Surfaces We shall demonstrate that the absorption coefficient of a gray body is constant and independent of the nature and temperature of the source from which it is absorbing energy. Let’s consider a gray body absorbing energy coming from a black source in equilibrium with it. Under these conditions, α = ε. This is true whatever the temperature is. However, since ε remains constant when the temperature changes (definition of gray surface), it follows that α also must be constant. Thus a gray body absorbing energy from a black source exhibits the same absorption coefficient for any temperature of the source. However, since changing the temperature of the source also changes the spectral distribution of the incident energy, if the gray body remains insensible to this change, we can conclude that the gray surface has a constant monochromatic absorption coefficient. If αλ is constant, the total absorption coefficient of a gray body for the radiation coming from any source, defined by Eq. (12-1-17), also will be constant and independent of the spectral distribution of the incident energy or, which is the same, independent of the temperature and nature of the source. We can see that the gray-body concept results in a simplification much more powerful than that suggested by the initial definition. A gray body thus is a body whose emissivity is constant and equal to its absorption coefficient whatever the temperatures of the body and the source are and whether or not thermal equilibrium exists between them.

12-1-11 Intensity of Radiation The intensity of radiation or intensity of emission is a magnitude used to indicate the total amount of radiation leaving a surface that propagates in a certain direction. Figure 12-7 represents an element of area dA belonging to an energy-emitting body surrounded by a hemisphere with radius r constructed around the radiating surface. All the energy emitted by dA will reach the hemisphere, whose area is 2πr2. Each

θ

dθ dω = dS r2

r sinθ

r dθ dS

r sinθ d φ

r

dA FIGURE 12-7

φ

dφ

Geometry for the definition of the intensity of radiation.

THERMAL RADIATION

405

element of area belonging to the hemisphere subtends a solid angle dω with center at dA. The unit of the solid angle is the steradian. In the same way as the ratio between the length of an elemental arc belonging to a circumference and the radius of the circumference is a measure of the subtended angle in radians; the measure of a solid angle in steradians is the ratio between the differential area belonging to the hemisphere that subtends the angle and the square of the radius. In the hemisphere surrounding dA, there are therefore 2π steradians. We can see in the figure that if θ is the zenith angle and φ is the azimuthal angle, the area of the surface element dS is dS = (r sin θ d φ)(rd θ)

(12-1-20)

d ω = sin θ d φ d θ

(12-1-21)

and

The intensity of radiation emitted by the element of area dA in a certain direction is the energy emitted in that direction per unit time, per unit solid angle, and per unit area of emitting surface projected onto a plane normal to the direction of emission. Thus i=

dQ d ω (dA cos θ)

(12-1-22)

where dQ = energy emitted per unit time within the solid angle dω dω = element of solid angle dA = element of emitting surface θ = angle formed by the direction of emission and the normal to dA (Fig. 12-8)

n

θ dω dA cos θ

dA

FIGURE 12-8

Nomenclature for Eq. (12-1-22).

The reason to include cos θ is because it results in simpler final expressions. This is due to the fact that for the case of a blackbody, the intensity of radiation thus defined is independent of the direction of emission, as will be explained below. For the case of real surfaces, the intensity of radiation will be a function of θ.

406

CHAPTER TWELVE

The total energy emitted by the surface per unit time and per unit area is the emissive power, which, according to Eq. (12-1-22), is W=

1 dQ = ∫ i cos θ d ω = dA ∫

φ= 2 π θ= π / 2

∫

φ= 0

∫

i cos θ sin θ d φ d θ

(12-1-23)

θ= 0

12-1-12 Monochromatic Intensity of Radiation It is also possible to define the monochromatic intensity of radiation as the energy emitted per unit time, per unit projected area, per unit solid angle, and per unit wavelength interval. The relationship between the total intensity of radiation defined in the preceding section and the monochromatic intensity of radiation is the same as that which exists between the total emissive power and the monochromatic emissive power. This means that it can be interpreted that the total intensity of radiation corresponds to the integration of the monochromatic intensity of radiation over the entire range of wavelengths. The monochromatic intensity of radiation depends on the characteristics of the emitting surface, on its temperature, on the direction of emission, and on the wavelength. This is iλ = f( λ ,θ,φ,T )

(12-1-24)

whereas the total intensity of radiation is i=

∞

∫0 iλ d λ = f(T ,θ,φ)

(12-1-25)

We shall explain in what follows that for the case of a blackbody, the intensity of radiation depends only on the temperature and not on the direction of emission.

12-1-13 Spacial Distribution of a Blackbody Emission Let’s assume that we have two elements of area dA1 and dA2, both being black surfaces and in thermal equilibrium. Let dA2 belong to a spherical surface whose center is occupied by dA1 (Fig. 12-9). If both

dω1 dA 2 θ

r dA1 cosθ dω2

dA1

FIGURE 12-9 Heat exchange between two elements of area belonging to blackbodies. dA1 is located at the center of a sphere to which dA2 belongs.

THERMAL RADIATION

407

elements are in thermal equilibrium, the amount of energy that each element absorbs from the other must be the same. However, if both surfaces are black, the energy that each element absorbs is the total that is incident on it. The energy emitted by dA1 that strikes dA2 and is absorbed; thus can be expressed as dQ1→2 = iθ dA1 cos θ d ω1

(12-1-26)

where iθ is the intensity of radiation emitted by dA1 in the direction θ. The solid angle is dA2 r2

(12-1-27)

iθ dA1dA2 cos θ r2

(12-1-28)

d ω1 = Then dQ1→2 =

In turn, the energy emitted by dA2 and reaching dA1 is dQ2→1 = in dA2 d ω 2

(12-1-29)

where in is the intensity of radiation emitted by dA2 in the direction connecting both elements of area, which is the direction normal to dA2 [for the same reason, the cosine does not appear in Eq. (12-1-29)]. But dω2 can be obtained as dω2 =

dA1 cos θ r2

(12-1-30)

(It must be remembered that to obtain the value of the solid angle, we must imagine a hemispherical surface with its center at dA2, intersect it with the solid angle, and divide the intersection area by the square of the radius.) Then dQ2→1 =

in dA1dA2 cosθ r2

(12-1-31)

Since thermal equilibrium exists, dQ1→2 = dQ2→1 , and it can be deduced from Eqs. (12-1-28) and (12-1-31) that iθ = in

(12-1-32)

This means that the intensity of radiation emitted by a blackbody in the normal direction n is the same as that emitted in an arbitrary direction θ. In other words, the intensity of radiation from a blackbody depends only on its temperature and not on the direction of emission.

12-1-14 Directional Emissive Power: Cosine Law—Real Surface Emission Distribution It is possible to define a directional emissive power as the amount of energy emitted per unit time, per unit solid angle, and per unit area of emitting surface (not projected). The directional emissive power, denoted as E θ , is related to the intensity of radiation by Eθ =

dQ = iθ cos θ dAd ω

(12-1-33)

408

CHAPTER TWELVE

For a blackbody, since iθ is independent of θ, we get Ebθ = ib cos θ

(12-1-34)

where ib depends only on the temperature. This expression is known as Lambert’s law or the cosine law, and the surfaces that follow it are called lambertians.Usually, real surfaces do not follow this law, and the intensity of emitted radiation varies with the direction. Figure 12-10 is a plot in polar coordinates showing the characteristics of directional emission for several surfaces.

60°

20°

40°

0°

20°

40°

60°

1 2

4

3

5

0.14 0.12

0.1

0.08

0.06

0.04 0.02

0

0.02

0.04

0.06 0.08

iθ

0.1

1 = Chromium 2 = Manganese 3 = Aluminum 4 = Nickel polished 5 = Nickel dull

0.12 0.14

iθ 0° 20°

20° 40°

40°

8 60°

60° 9 10

7

6 = Ice 7 = Glass 8 = Copper oxide 9 = Wood 10 = Paper

6 80°

1.0

80°

0.8

0.6

0.4

0.2

0

iθ

0.2

0.4

0.6

0.8

1.0

iθ

FIGURE 12-10 Directional emission from real surfaces.

Usually, for electric conductors, the intensity of emission is relatively constant for small values of θ (angle between the normal to the surface and the direction of the radiant beam) and then increase when viewed from wider angles.7 For electric nonconductors, the intensity shows a maximum in the direction normal to the surface, and this value is practically constant up to θ = π /4. For higher values, the intensity decreases, and it is nil for θ = π /2.

12-1-15 Intensity of Radiation and Emissive Power of a Blackbody Since in the case of a blackbody the intensity of emission does not depend on θ, Eq. (12-1-23) can be easily integrated to find the total emissive power:

THERMAL RADIATION

Wb =

φ = 2π

θ = π/2

∫φ = 0 ∫θ = 0

ib cos θ sin θ d θd φ = 2πib ∫

π/2 0

409

sin θ cos θ d θ (12-1-35)

1

= 2πib ∫ 1/ 2d (sin 2 θ) 0

∴ Wb = πib

This expression relates the emissive power to the intensity of emission for a blackbody.

12-1-16 Radiant Heat Transfer between Blackbodies: View Factor Let’s consider two black surfaces such that part of the radiation emitted by each one can reach the other (Fig. 12-11). Let these surfaces be A1 and A2. On each surface we can consider an element of area. Let them be dA1 and dA2, respectively. The radiant beams that leave each element of area and are intercepted by the other form angles θ1 and θ2 with the respective normals.

n1

dA 2 A2

θ1

r θ2

n2

dA1 A1 FIGURE 12-11

Heat exchange between two area elements of blackbodies.

The heat flow leaving dA1 and reaching dA2 is dQ1→2 = ib 1 dA1 cos θ1

dA2 cos θ2 r2

(12-1-36)

We wrote directly ib1 without any other subscript because in the case of blackbodies it is not a function of the angle. Subscript 1 denotes the temperature T1. By integrating this expression over both surfaces, we obtain the total heat flow leaving A1 and striking A2. Since A2 is a blackbody, this energy is absorbed completely. Then Q1→2 =

∫A ∫A 1

2

Wb1 cos θ1 cos θ2 πr 2

dA1dA2

(12-1-37)

The total energy that leaves A1 and expands through all the half-space is Q1→∞ = Wb1 A1

(12-1-38)

410

CHAPTER TWELVE

It is possible to define a magnitude F12 as Q1→2 = Wb1 A1 F12

(12-1-39)

where F12 is called the view factor between A1 and A2 evaluated on the basis of area A1. It represents the fraction of the energy leaving A1 and striking A2 with respect to the total energy leaving A1. (Some authors prefer the expression shape factor rather than view factor.) In this particular case, since both bodies are black, the energy that reaches A2 is absorbed completely. Now we can perform the same analysis for the heat flow from A2 to A1: dA1 cos θ1 r2 Integrating and defining a view factor for the heat flow from A2 to A1, we get dQ2→1 = ib2 dA2 cos θ2

Q2→1 = Wb2 ∫

A1

∫A

2

cos θ1 cos θ2 dA1dA2 = Wb2 A2 F21 πr 2

(12-1-40)

(12-1-41)

The total heat flux leaving A2 is Q2→∞ = Wb2 A2

(12-1-42)

Equation (12-1-41) defines the view factor F21 as the fraction of the energy emitted by A2 that reaches A1. It is called the view factor between A1 and A2 evaluated on the basis of area A2. Comparing Eqs. (12-1-37), (12-1-39), and (12-1-41), we see that A1 F12 = A2 F21 =

∫A ∫A 1

2

dA1dA2 cos θ1 cos θ2 πr 2

(12-1-43)

Then the net rate of radiant heat transfer between A1 and A2 is Q12 = Wb1 A1 F12 − Wb2 A2 F21

(12-1-44)

Q12 = A1F12 (Wb1 − Wb2 ) = A1F12 σ (T14 − T24 )

(12-1-45)

or

It can be noted that if both surfaces are at the same temperature, the net rate of heat transfer is obviously nil. The integrals defining the view factors, of the type of Eq. (12-1-43), have been calculated for the most common geometries, and they can be found in the references in the form of graphs or tables.6 Figures 12-12, 12-13, and 12-14 are graphs of the view factors between parallel planes, between a differential element of area and a parallel plane, and between perpendicular planes. Figure 12-15 is a graph of the view factor between a plane and a bank of tubes parallel to it. Since the view factor F12 represents the fraction of the energy emitted by surface A1 that strikes surface A2, it is evident that the sum of the vision factors of a body with respect to all other bodies in the universe is unity. Example 12-1 Calculate the net rate of radiant heat transfer between two black surfaces consisting of two parallel 1 × 2 m rectangles separated by 1 m and at 800 and 300 K, respectively. Solution

(

Q12 = A1F12 (Wb1 − Wb2 ) = A1F12 σ T14 − T24

)

F12 is obtained from the graph in Fig. 12-12. In curve 3, we get F12 = 0.3. Q12 = 2 × 0.3 × 5.672 × 10 −8 × (800 4 − 300 4 ) = 13, 663 W

THERMAL RADIATION

411

1

View factor F or F

0.8 7 6 5 4

8 0.6

3 0.4

2 1

0.2

0

0

1

2 Ratio

3 4 5 smaller side or diameter distance between planes

6

7

Radiation between parallel planes directly opposed 1, 2, 3 5, 6, 7, 8 1, 5 3, 7 2, 6 4, 8

Direct radiation between the planes F Planes connected by nonconducting but reradiating walls F Discs 2:1 Rectangles Squares Long narrow rectangles

FIGURE 12-12 View factors for parallel squares, rectangles, and disks. (From Hottel HC: in McAdams WH (ed): Heat Transmission. New York: McGraw-Hill, 1954. Reproduced with permission of the editors.)

3.5 FdA1

A2 =

0.02

L2

0.03

3.0

L1

0.04

2.5 Ratio D/L2

0.05

D

0.06

2.0

dA

0.08

1.5

0.10 0.12 0.14

1.0

L1 and L2 are the sides of the rectangle. D is the distance from dA to the rectangle

0.16 0.18 0.20 0.22 0.24

0.5

0

0.5

1.0

1.5

2.0

2.5 3.0 Ratio D/L1

3.5

4.0

4.5

5.0

5.5

FIGURE 12-13 View factor for a differential area located below one corner of a finite rectangular plane FdA1→A2. (From Hottel HC: in McAdams WH (ed): Heat Transmission. New York: McGraw-Hill, 1954. Reproduced with permission of the editors.)

0.50 Z = 0.1

View factor F12

0.40

0.30

0.2 0.3 0.4

x

0.6

A1

A2

0.8 1.0 0.20

0

0

1.0

2.0

z

A1 = area in which is based the heat transfer equation

1.5 2.0 3.0 4.0 6.0 8.0 Scale changes here

0.10

y

3.0

Z = z/x

4.0

6.0

8.0

Y = y/x

10

Y FIGURE 12-14 View factor for perpendicular planes with a common edge. (From Hottel HC: in McAdams WH (ed): Heat Transmission. New York: McGraw-Hill, 1954. Reproduced with permission of the editors.)

Radiation from a plane to one or two rows above and parallel to the plane refractory

1.0 0.9 A

First row Second row

0.8 0.7

radiating plane A = Total to two rows when both are present B = Total to one row when only one is present C = Total to the first row when both are present D = Direct to the first row E = Total to the second row when two are present F = Direct to the second row

B

0.6 D 0.5

C

0.4

E

0.3 F 0.2 0.1

Tubes in equilateral triangular pattern. Refractories reradiating walls behind the tubes

0 1

2 3 4 5 6 7 8 distance between tube axis in a row Ratio tube diameter

FIGURE 12-15 View factor between a plane and one or two tube rows parallel to it, with and without a refractory wall. (From Hottel HC: in McAdams WH (ed): Heat Transmission. New York: McGraw-Hill, 1954. Reproduced with permission of the editors.)

412

THERMAL RADIATION

413

12-1-17 View-Factor Algebra It is sometimes possible to calculate the view factor for a particular geometric arrangement by substracting or adding other known view factors corresponding to simpler arrangements. This procedure is known as view-factor algebra. It is illustrated in the following examples. Example 12-2 A 0.01-m2 radiant heater at 1,000 K, oriented as shown in Fig. 12-16, is used to heat a floor 6 m wide and 8.5 m long. Assuming that the surface of the floor behaves as a blackbody, calculate the heat flux that it receives from the source. dA

3m

3

3m

2 6m 4

5.5 m

5

8.5 m FIGURE 12-16

Figure for Example 12-2.

Let 1 designate the heater. Owing to its small size, it can be considered as a differential surface element when compared with the size of the floor. Let the floor be divided into surfaces 2, 3, 4, and 5, as shown in the figure. Then

Solution

Q1( 2+3 + 4+5) = Q1 2 + Q13 + Q14 + Q15 Q12 = A1 F12Wb1 Q13 = A1 F13Wb1 Q14 = A1 F14 Wb1 Q15 = A1 F15Wb1 Q1floor = A1Wb1 ( F12 + F13 + F14 + F15 ) Observe that F1 floor = ∑ F1 i

414

CHAPTER TWELVE

From Fig. 12-13, we get F12 = F14 = 0.14 and F13 = F15 = 0.164. Thus Q1floor = 0.01 × 5.672 × 10 −8 × 1, 000 4 × (0.14 + 0.164 + 0.14 + 0.164) = 345 W It was assumed that the heating element is also a blackbody. Example 12-3

Calculate the view factors between surfaces A1 and A3 in Fig. 12-17.

A1

A2 A3

FIGURE 12-17

Solution

Figure for Example 12-3.

We can write F1(2, 3) = F12 + F13

Factors F1(2, 3) and F12 can be obtained from Fig. 12-14, and then F13 can be easily calculated by difference. 12-1-18

Electric Analogy Equation (12-1-45) can be interpreted as the equation describing an electric circuit where there are two opposite electromotive forces (Wb1 and Wb2) and a resistance 1/A1F12. The electric current in the circuit represents the heat flux Q1↔2 (Fig. 12-18). In this simple case, this electric analogy does not look particularly useful, but its advantage will be appreciated when we deal with more complex systems. Wb1 = σ T14

i = Q1W2

Wb 2 = σT24

(A1F12)–1 = R

FIGURE 12-18 Electric analogy for the heat exchange between two blackbodies.

12-1-19 Radiant Heat Exchange among Three Blackbodies Let us consider a system consisting of three blackbodies exchanging heat among themselves (Fig. 12-19). The energy exchanged between 1 and 2 can be expressed as Q12 = A1 F12 (Wb1 − Wb2 )

(12-1-46)

THERMAL RADIATION

415

A3 A1

Q13

Q23 Q12

A2 FIGURE 12-19

Heat exchange among three blackbodies.

Wb1

Wb2

(A1F12)–1

(A 1F13)–1

(A 2F23)–1

Wb3

FIGURE 12-20 bodies.

Electric analogy for the heat exchange among three black-

The energy exchanged between 2 and 3 is Q23 = A2 F23 (Wb2 − Wb3 )

(12-1-47)

And the energy exchanged between 1 and 3 is Q13 = A1 F13 (Wb1 − Wb3 )

(12-1-48)

Each one of these equations represents a loop of the electric network of Fig. 12-20. The current through each resistor represents one of the binary heat fluxes of Eqs. (12-1-46) through (12-1-48). The amount of energy that body 1 is exchanging with the rest of the system is the algebraic sum of Q1↔2 and Q1↔3. This value is represented in the circuit by the current circulating through the electromotive force Wb1. Depending on the direction of this current, body 1 would be delivering or absorbing energy to or from the system. The same can be said about surfaces A2 and A3. 12-1-20 Insulating Walls or Refractories Let us consider the system represented in Fig. 12-21, where surface A1 is a heat source, A2 is a heat sink, and the A3’s are walls insulated from the exterior. For example, the figure can be a representation of an

416

CHAPTER TWELVE

A1

Q indirect

A3

A3

Q direct

A2 FIGURE 12-21 Heat exchange between two bodies in the presence of refractory reradiating walls.

electric oven where A1 is the surface of the heating electrical resistor, A2 is the object that is being heated, and the A3’s are the oven walls. Since the walls A3 are insulated, it is not possible for energy to be delivered to or withdrawn from the system through them. This is normally the case with refractory walls of ovens and furnaces, so we shall refer to this type of surface either as insulating walls or refractories. When the system reaches its steady state, the energy emitted by A1 must be totally absorbed by A2. (Note that for this system to reach a steady state it is necessary that energy be withdrawn continuously through A2. For example, A2 can be the surface of a coil internally refrigerated by a circulating fluid.) In a system such as this, with a hot source and a cold energy receptor, the insulating walls necessarily must reach an intermediate temperature between the source and the receptor. Thus there must be an energy exchange between the walls and surfaces A1 and A2, and owing to the adiabatic character of the walls, the energy absorbed by them must be equal to the energy they emit. From the total energy that is incident on A3, a portion is reflected and a portion is absorbed. (A3 is not necessarily a black surface.) But all the energy absorbed is emitted again toward the rest of the system. This behavior is similar to that of a blackbody at a temperature such that its emissive power is equal to the energy that is incident on it. We then can represent this system by an electric network such as that in Fig. 12-20, to which we shall impose the additional condition that Q1↔3 = Q3↔2. In this case, there is no current circulation through the branch containing Wb3, so it can be eliminated from the network without modification of the current distribution in the rest of the circuit. The resulting network is shown in Fig. 12-22. The interpretation of the circuit is as follows: From the total energy leaving A1, a portion strikes A2 directly. This fraction is the current circulating through the upper branch and is given by (A1F12)–1

Wb1

Wb2

Q12 (A1F13)–1 Q13

(A 2F23)–1 Q23

FIGURE 12-22 Electric circuit representing the heat exchange between two blackbodies in the presence of refractory reradiating walls.

THERMAL RADIATION

Q12 = A1 F12 (Wb1 − Wb2 )

417

(12-1-49)

where F12 is the view factor representing the direct vision between the two surfaces. Another fraction of the energy leaving A1 reaches the refractory wall, where it is absorbed and reemitted again to finally reach A2. This energy is represented by the current in the lower branch. We see that the refractory offers an alternative path for the heat transmission between A1 and A2. The triangle of resistances in Fig. 12-22 can be solved easily to find the total resistance (first adding the two resistances in series and then solving the parallel with the upper branch). The total resistance will −1 be called ( A1 F12 ) . The equivalent circuit is that of Fig. 12-23, where the expression for the total resistance is A1 F12 = A1 F12 +

(A1F12)–1

Wb1

FIGURE 12-23

A1 F13 A2 F23 A1 F13 + A2 F23

(12-1-50)

Wb2

Equivalent resistance of the circuit in Fig. 12-22.

This means that the effective view factors F can be calculated from the binary view factors between the several surfaces of the system. The graph in Fig. 12-15 shows curves representing the view factors F12 for the case of radiant heat transfer from an emitting planar surface to a bank of tubes, including, in some of the curves, the presence of refractory walls behind the tubes. Also, curves 5, 6, 7, and 8 in Fig. 12-12 represent the view factors F12 for parallel planes connected by reradiant refractory walls. The main difference that exists between the circuits in Figs. 12-20 and 12-22 is that in the latter case, the refractory wall has an emissive power (electric potential) that “floats” between the emissive powers Wb1 and Wb2 at a level that depends on the magnitudes of Wb1 and Wb2 and on the thermal resistances participating in the circuit. Usually a refractory wall is a real surface (not black). However, we explained that with the purpose of calculating the heat flux between A1 and A2, it is possible to replace the refractory wall with a blackbody having a fictitious temperature such that its emissive power is equal to the incident irradiation. In the particular case where the refractory is a blackbody, this is exactly the situation, and the temperature, determined by T3 =

4

Wb3 σ

(12-1-51)

coincides with the real temperature of the refractory wall. However, in a more general situation, the temperature determined by Eq. (12-1-51) will be the ficitious temperature we mentioned, and it will not coincide with the real temperature of the refractories. 12-1-21 Case of Several Insulating Walls For Eq. (12-1-50) to be valid, it is necessary that all the insulating walls can be represented by a single node in the circuit and have the same electrical potential. This requires the hypothesis that all the refractory walls

418

CHAPTER TWELVE

are at the same temperature. If the geometry of the systems is such that we presume there will be areas at different temperatures, they must be treated as different surfaces. For example, let’s consider the case of heat transfer between two surfaces A1 and A2 connected by refractory walls AS and AR, as shown in Fig. 12-24. In this case, the equivalent electric circuit is that shown in the right side of the figure which also can be solved to find the effective view factor F12 . T1

Wb1

Q1R

Wb2

Q1S QRS

TR

(A1F12)–1

TS

R

Q12

(A2F2S )–1

(A1F1S )–1

QS2

QR 2

(A 2 F2R)–1

(A1F1R)–1

T2 (AR FRS )–1

S FIGURE 12-24

Heat transfer between two blackbodies with two reradiating refractory walls.

As a general case, where a higher number of refractory surfaces are involved, the system can be solved as a quotient of determinants of the following form (see ref. 2):

A1 F 12 =

A1F12

A2 F2 R

A2 F2 S

...

A1F1R

AR FRR − AR

AR FRS

...

A1F1S

AR FRS

AS FSS − AS

...

...

...

...

...

AR FRR − AR

AR FRS

....

AR FRS

AS FSS − AS

...

...

...

...

(12-1-52)

FRR is the view factor of the wall R with respect to itself (concave surfaces can intercept part of the radiation emitted by themselves). All the view factors of a body are related by F11 + F12 + F1i = 1

(12-1-53)

If we apply the general solution to the more simple case of a single refractory wall, we get

A1 F 12

A2 F2 R ⎞ ⎛ A1 F12 ⎜⎝ A F AR FRR − AR ⎟⎠ AAF F = 1 1R = A1 F12 − 1 2 1R 2 R AR FRR − AR AR FRR − AR

(12-1-54)

Additionally, the summation of FRi values equals 1; thus (FR1 + FR 2 + FRR ) AR = AR

(12-1-55)

THERMAL RADIATION

419

Thus AR FRR − AR = −( AR FR1 + AR FR 2 )

(12-1-56)

Substituting in the solution of the determinant A1 F 12 = A1 F12 +

A1 F1R A2 F2 R A1 F1R + A2 F2 R

(12-1-57)

which is the same expression we get with the equivalent electric circuit. 12-1-22 Enclosures with Gray Surfaces In all the cases considered up to now, all the energy striking a surface was absorbed completely because we considered only black surfaces. In the case of real surfaces, not all the incident energy is absorbed because it is partly reflected. The reflected energy can come back to the emitter or to other bodies in the universe according to the view factors of the reflecting surface. This energy, in turn, is partly reflected by those other bodies. Thus the steady state is reached through a process of successive reflections and absorptions in which the energy is being transmitted among the different bodies participating in the system. The solution of this type of system is extremely complex, but it is possible to simplify the mathematical treatment by assuming that the bodies behave as gray surfaces. To solve the problem, we must perform the following analysis: Let’s consider a gray body that is exchanging energy with the ambient. Since it is not a blackbody, it will reflect part of the incident energy. Then the radiation leaving its surface per unit time and per unit area, which is called the radiosity J, is the sum of the radiation emitted and reflected (Fig. 12-25). If I is the irradiation to which this surface is submitted (density of incident energy), the absorbed fraction of it is α. The reflected energy is I (1− α ) = I ρ Then J = W + Iρ

(12-1-58)

W = Wb ε

(12-1-59)

J −W ρ

(12-1-60)

where W is the emissive power, given by

From Eq. (12-1-58), we get I=

J I

ρI

W

I = incident energy ρI = reflected energy αI = absorbed energy W = emitted by the body according to its thermal level J = radiosity

αI FIGURE 12-25

Behavior of a gray surface.

420

CHAPTER TWELVE

The net energy flux at the body surface (heat exchanged with the exterior) is Q1ext = A1 ( J − I )

(12-1-61)

Substituting I, the result is ⎛ J −W ⎞ Q1ext = A1 ⎜ J − ρ ⎟⎠ ⎝

(12-1-62)

Considering Eq. (12-1-59), and since (1 – ρ) = ε because the body is gray and opaque, the result is Q1ext =

A1ε (Wb − J ) 1− ε

(12-1-63)

If the surface were a blackbody, the energy flux emitted per unit area would be Wb . The energy flux per unit area coming from the gray surface is J, which is related to Wb by Wb − J = Q1ext Wb1 = σ T14

(1 – ε1) A1ε1 J

FIGURE 12-26 Representation of a gray surface.

1− ε A1ε

(12-1-64)

We see that the higher is the energy exchange between the surface and the rest of the system, the higher is the difference between Wb and J. If we analyze this from the point of view of the electric analogy, we could think that the gray body acts as an electromotive force Wb, but with an internal resistance that provokes a voltage drop, so the electric potential for an external observer is J (see Fig. 12-26). We shall call this resistance gray-to-black transformation resistance. Let’s consider the case of two gray surfaces with areas A1 and A2 at temperatures T1 and T2 and with emissivities ε1 and ε2. Let’s transform these bodies from gray to black. The blackbody emissive powers at these temperatures are

Wb1 = σT14

and

Wb2 = σT24

(12-1-65)

We then add the gray-to-black transformation resistances. From the side of body A1 we have 1 − ε1 A1ε1

(12-1-66)

1 − ε2 A2 ε 2

(12-1-67)

And from the side of body A2 we have

With the addition of these resistances, the blackbody emissive powers are transformed into the radiosities J1 and J2 (Fig. 12-27). From here on, we have the same circuits that we analyzed earlier for the case of black surfaces. We shall consider some examples: 1. Heat exchange between gray bodies in a vacuum or surrounded by nonabsorbing gases (Fig. 12-28), where F12 is the view factor, as before 2. Heat exchange between gray bodies surrounded by refractory walls at a uniform temperature (Fig. 12-29) 3. Heat exchange between two gray bodies in the presence of refractory walls at different temperatures (in this case, two walls) (Fig. 12-30)

Wb1 = σT14

(1 – ε1) A1ε1 J1

FIGURE 12-27

Wb1

Wb2 = σ T24

(1 – ε2) A2ε2

Wb 2 = σ T24

J2

Radiosities of two gray surfaces.

Wb1 = σ T14

FIGURE 12-28

(1 – ε2) A 2ε2

(1 – ε1) A 1ε 1

1 (A1F12)

Electric circuit representing heat exchange between two gray bodies.

(1 – ε1) A1ε1

(A1F12

(1 – ε2) A 2ε2

)–1

Wb2

(A 2F23)–1

(A1F13)–1

FIGURE 12-29 Electric circuit representing heat exchange between two gray bodies in the presence of reradiating refractory walls at uniform temperature.

Wb1

(1 – ε1) A1ε1

(1 – ε2) A2ε2

(A1F12)–1

Wb 2

(A2F2R )–1

(A1F1R)–1 R (A1F1S

(A2F2S)–1

)–1

(ARFRS )–1

S FIGURE 12-30 Electric circuit representing heat exchange between two gray bodies in the presence of reradiating refractory walls at two different temperatures.

421

422

CHAPTER TWELVE

In all these cases, the network is solved by means of a determinant, and finally, it reduces to the case shown in Fig. 12-31, where F12 designes the view factor including the effect of the refractory walls. The total resistance of the circuit shown in Fig. 12-31 will be called 1/A1F12, and the equivalent circuit is that shown in Fig. 12-32. Here, we made reference to A1, but A2 also can be used, and the resistance also may be called 1/A2F21.

Wb1

(1 – ε1) A1ε1

1 A1F12

(1 – ε2) A2 ε2

Wb 2

FIGURE 12-31 General case for two gray surfaces where the view factor includes the presence of refractory walls.

Wb1

1 A1F12

Wb2

FIGURE 12-32 Total view factor including the presence of refractory walls and the effect of surface emissivities.

12-1-23 Heat Transfer between a Gray Body and a Body That Completely Surrounds It Let’s consider the case of a gray body at temperature T1, completely surrounded by a second body at temperature T2. The situation is that shown in Fig. 12-33. If the view factor of body 1 with respect to itself is nil, then F12 = 1, and with reference to Fig. 12-28, the total resistance of the circuit is 1 1 − ε1 1 1 − ε2 = + + A1F12 A1ε1 A1 A2 ε 2

(12-1-68)

1 1 − ε1 1 − ε 2 A1 = + 1+ F12 ε1 ε 2 A2

(12-1-69)

This means that

And if the quotient A1/A2 is small (e.g., in the case of a small body in an infinite ambient), Eq. (12-1-69) reduces to F12 = ε1

(12-1-70)

THERMAL RADIATION

Wa3

W1: Energy emitted by A1. This energy strikes on A2 and is partially absorbed (Wa1). The nonabsorbed fraction is reflected (Wr1). If area A2 is much larger than A1, the reflected fraction strikes again on other region of A2, where another fraction of the energy is absorbed. Since there is a very low probability that this energy comes back to A1, it is finaly completely absorbed in a multiple reflections process. A2 behaves as a blackbody since it absorbs all the energy it receives

Wa2

Body at T2 Wr 2 T1 Wr3 Wa4

FIGURE 12-33

423

Wr1 W1

Wa1

Heat transfer between a gray body and another body that completely surrounds it.

Then the heat lost by body A1 can be calculated as

(

Q = A1ε1 (Wb1 − Wb2 ) = A1ε1σ T14 − T24

)

(12-1-71)

This expression was already used in previous chapters to calculate the radiant heat loss from a body to an infinite ambient. We were in debt for its justification. It is interesting to note that the emissivity ε2 does not appear in Eq. (12-1-71). This is so because since we assume that A2 has infinite dimensions, it behaves like a blackbody because all the energy received from A1 is being absorbed in successive reflections until it is completely exhausted, with very little probability that part of this energy comes back to body A1 (see Fig. 12-33). Example 12-4 Two parallel plates 1 m wide and 1 m in length are separated by a distance of 0.5 m between them. One plate is at T1 = 1,000 K, and the other is at T2 = 500 K. Both plates are immersed in an ambient at 27°C (300 K) that can be considered infinite with respect to the dimensions of the plates (Fig. 12-34). Calculate the heat exchanged by each plate and by the ambient. Consider that both surfaces are gray, with emissivities ε1 = 0.5 and ε2 = 0.3. The ambient can be treated as a third body, and the equivalent electric network is represented in the Fig. 12-34. The view factor F12 is obtained from Fig. 12-12 (curve 2). We get F12 = F21 = 0.4. Since each plate only has a view of the other plate and the ambient, the result is

Solution

F13 = 1 – F12 = 0.6 F23 = 1 – F21 = 0.6 And since A1 and A2 are unity, the resistances are ( A1F12 )−1 = 1/ 0.4 = 2.5 1 − ε1 1 − 0.5 = =1 0.5 ε1 A1

( A1F13 )−1 = ( A2 F23 )−1 = 1/ 0.6 = 1.66 1 − ε 2 1 − 0.3 = = 2.33 ε 2 A2 0.3

Wb1 = 5.672 × 10 −8 × 1, 000 4 = 56, 720 W/m 2 Wb2 = 5.672 × 10 −8 × 500 4 = 3, 570 W/m 2 Wb3 = 5.672 × 10 −8 × 300 4 = 459 W/m 2

424

CHAPTER TWELVE

T1 = 1,000 K Two finite plates in an ambient at lower temperature

T2 = 500 K Ambient 300 K

Wb1

(1 – ε1) A1ε1

(1 – ε2) A2ε2 Wb2

(A1F12)–1

(A2F23)–1

(A1F13)–1

Wb3

Electric circuit

56.72 kW

1 Q1

2.5

J1

J2

1.66

1.66

Q3

J3 0.46 kW

2.33

3.57 kW

Q2

Numerical values FIGURE 12-34

Figure for Example 12-4.

The electric network can be solved by any method used for electric circuit resolution. A simple way to solve it is to consider that the sum of all the currents (heat fluxes) coming to a node must be zero. Then 56.72 − J1 J −J J − 0.46 = 1 2 + 1 1 2.5 1.66 0.46 − J2 J2 − 3.57 J1 − J2 = + 2.5 1.66 2.33 We have two equations with two unknowns. Solving, we get J1 = 30.45

and

J2 = 9.796

By knowing the electric potentials of these points, we can find the currents in the network. We get Q1 = 26.27 kW

Q2 = 2.67 kW

Q3 = 23.6 kW

Q1-2 = 8.264 kW

Q1-3 = 18.0 kW Q2-3 = 5.6 kW

This means that most of the radiant energy emitted by both plates is transferred to the ambient. Only 2.67 kW from a total of 26.26 kW emmitted by plate 1 is absorbed by plate 2.

THERMAL RADIATION

12-1-24

425

Radiant Shields One possible way to reduce the radiant heat transfer between two surfaces is to interpose a screen or shield between them. We know from our personal experience that if we do not want to feel the effects of solar radiation, we must stay under a shadow or that if we interpose a screen between a quartz infrared heater and ourselves, we do not feel its radiation. Let’s consider the case of two infinite parallel planes at temperatures T1 and T2, one acting as a source and the other acting as a sink. Here again, the view factor F12 = 1, and then Eq. (12-1-68) is also valid. Let’s consider now that we interpose between both planes a screen A3. Since this screen can exchange energy only with the planes A1 and A2, it will finally reach steady state when the energy received from A1 equals that transmitted to A2. The energy exchange between plane A1 and plane A3 can be represented by the left loop in Fig. 12-35, whereas the energy exchange between plane A3 and plane A2 corresponds to the right loop. Since Q(1↔2) = Q(2↔3) (plane 3 does not supply or withdraw energy to or from the system), both branches can be joined and we get the equivalent circuit of Fig. 12-36.

4

(1 – ε1)

Wb1 = σT1

FIGURE 12-35

A.ε1

(1 – ε3) Wb3 = σ T34 A.ε3

1 A.

Wb3 =

σ T34

(1 – ε3) A.ε3

1 A

(1 – ε2) A.ε2

Wb2 = σT24

Heat exchange between two parallel planes with a radiant screen separating them.

Wb1 = σ T14

FIGURE 12-36

(1 – ε1) A.ε1

(1 – ε3) A.ε3

1 A.

(1 – ε3) A.ε3

1 A

(1 – ε2) A.ε2 Wb2 = σT24

Simplification of the circuit in Fig. 12-35.

Example 12-5 Calculate the radiant heat-flux density exchanged between two black infinite planes at 900 and 400 K, respectively, and in what proportion the heat flux is reduced if a radiant screen with emissivity 0.4 is interposed between them. The view factors for infinite parallel planes are unity. In the case of the two planes facing each other without a screen, the heat exchanged is

Solution

(

)

Q = σ T14 − T24 = 5.672 × 10 −8 (900 4 − 400 4 ) = 35, 760 W/m 2 A When the screen is added, the system can be represented by Fig. 12-36. The total resistance of the circuit is 1 1 − ε3 1 − ε3 1 2 ⎛ 1 − 0.4 ⎞ 5 + + + = ⎜1 + ⎟= ε3 A A A ⎝ A ε3 A 0.4 ⎠ A Then

(

)

4 4 5.672 × 10 −8 (900 4 − 400 4 ) Q σ T1 − T2 = = 7,152 W/m 2 = 5 5 A

426

CHAPTER TWELVE

We see that the net energy exchanged is reduced to one-fifth by the simple addition of the screen. This is greatly due to its relatively low emissivity, which makes the screen reflect most of its incident energy. We shall see what would be the situation if we consider that the screen is a blackbody. In this case, the total resistance of the network is 2/A; then

(

)

4 4 5.672 × 10 −8 (900 4 − 400 4 ) Q σ T1 − T2 m2 = = 17, 880 W/m = 2 2 A

This means that in this case the heat exchanged is reduced to 50 percent from the original. It can be demonstrated easily that if additional screens are added in the radiation path, the net heat exchanged is

(

4 4 Q σ T1 − T2 = 2(n + 1) A

)

(12-1-72)

where n is the number of screens.

12-2 RADIANT HEAT TRANSMISSION IN ABSORBING MEDIA 12-2-1 Radiation Absorbing Gases and Inert Gases Up to now, we considered the transmission of radiant energy, assuming that this energy propagates in a vacuum or in the presence of a gas completely transparent to radiation. Certain gases, however, are capable of absorbing and emitting energy. Up to a temperature about 2,800 K, the energy emitted by a gas is almost exclusively due to changes in the vibration or rotation of molecules. Those gases whose molecules are formed by atoms of the same chemical element are not capable of absorbing or emitting radiant thermal energy. These gases (N2, O2, and H2) are transparent to thermal energy. On the other hand, gases with asymmetric molecules, such as CO2, CO, H2O, etc., can absorb and emit energy in the range of wavelengths corresponding to thermal radiation. 12-2-2 Description of the Gas Energy-Absorption Mechanism: Beer’s Law Let dA be an element of a surface belonging to a blackbody surrounded by a gas capable of absorbing energy. Let iboλ be the monochromatic intensity of radiation of a certain wavelength emitted by the surface in the direction θ (Fig. 12-37). x = distance from the plate ibλ = monochromatic intensity of radiation in the beam at a certain distance from the plate ibλ is smaller than the radiation emitted due to the absorption of energy by the gas.

θ

o

dA FIGURE 12-37

ibλ = intensity of emitted monochromatic radiation

Radiation from an element of black surface to a gas that surrounds it.

THERMAL RADIATION

427

The absorption of energy by the gas is a volume-dependent process. In a length dx, the intensity of radiation decreases in a value that, according to the Beer’s law, is given by − dibλ = kλ ibλ dx

(12-2-1)

where kλ is a function of the concentration of absorbent molecules of the gas (in other words, a function of the partial pressure of the components of the gas capable of absorbing radiant energy). The boundary conditions for the integration of Eq. (12-2-1) are For x = 0, ibλ = ibλo For x = x , ibλ = ibλ ( x ) Then ibλ ( x )

∫i

o bλ

dibλ = ibλ

x

∫0 − k dx

(12-2-2)

As a result of the integration, we obtain the following expression for the intensity of radiation as a function of the distance to the surface element: ibλ ( x ) = iboλ e − kλ x

(12-2-3)

This relationship indicates that the intensity of radiation reduces exponentially with the distance through the gas. Let’s assume that x tends to 0; then ibλ ( x ) = iboλ

(12-2-4)

This means that if the width is small, the gas behaves as transparent. If dQλ is the monochromatic radiation emitted by the element of area per unit time within the solid angle dω, we have dQλ , emitted = iboλ dA cos θ d ω

(12-2-5)

At a certain distance x from the surface, the intensity of the beam has been weakened according to Eq. (12-2-3), and then the amount of energy remaining in the beam is iboλ dA cos θ d ωe − kλ x

(12-2-6)

dQλ , absorbed = iboλ dA cos θ d ω (1 − e − kλ x )

(12-2-7)

and the absorbed energy is given by

We can define the monochromatic absorption factor of the gas up to a certain distance from the emitting surface as the fraction of the energy that is absorbed by the gas mass contained in the solid angle up to the considered distance x. This fraction is taken with respect to the total energy emitted by the surface in that direction: αλ =

iboλ dA cos θ d ω (1 − e − kλ x ) = (1 − e − kλ x ) iboλ dA cos θ d ω

(12-2-8)

By analyzing Eq. (12-2-8), we see that 1. The monochromatic absorption factor is independent from the direction because kλ is not a function of θ. 2. The monochromatic absorption factor is a function of the distance to the emitting surface. This means that it depends on the gas mass crossed by the beam (it is not exclusively a function of the thermodynamic

428

CHAPTER TWELVE

parameters of the gas). If the distance is very high, α tends to 1, meaning that at a big distance to the surface the gas has absorbed all the monochromatic energy emitted by the surface. 3. The monochromatic absorption factor depends on the wavelength. This dependence is very strong. The emission and absorption spectra of gases are in general discontinuous. This means that there are wavelength bands in which the gas absorbs and some others in which it is completely transparent.

12-2-3

Hemispheric Monochromatic Absorption Factor Let’s assume again that we have an element of black surface surrounded by a hemisphere of absorbing gas with radius L (Fig. 12-38).

n

dω

dA

FIGURE 12-38 Geometry for the definition of gas radiation parameters.

The monochromatic energy emitted by the elemental surface within the differential solid angle dω is given by Eq. (12-2-5). To calculate the total monochromatic energy emitted by dA, we must integrate Eq. (12-2-5) in all the half-space. We get

∫ ibλ cos θ dAd ω = πibλ dA o

o

(12-2-9)

(For details about this integration, see Sec. 12-1-14.) But all the energy absorbed within the solid angle and up to a radius L is, according to Eq. (12-2-7), dQλ , absorbed = iboλ dA cos θ d ω (1 − e − kλ L )

(12-2-10)

The term within parentheses is the monochromatic absorption factor corresponding to the radius L. Since this is independent of θ, the integration of Eq. (12-2-10) is performed in the same way as that of Eq. (12-2-9). It then results that the total amount of energy absorbed in the entire hemisphere of radius L is dQλ ,1→G = ∫ iboλ cos θ dAd ωα λ = ∫

φ= 2π

φ= 0

=

π iboλ α λ dA

θ= π / 2 o ibλ α λ

∫θ= 0

cos θ sin θ dAd ϕd θ

(12-2-11)

THERMAL RADIATION

429

Since πiboλ = Wbλ1 (monochromatic emissive power of the blackbody at the surface temperature T1) : dQλ ,1→G = α λ Wbλ1dA

(12-2-12)

The fraction of the absorbed energy with respect to the energy emitted by the surface can be calculated by dividing Eqs. (12-2-11) and (12-2-9), giving Energy absorbed π iboλ α λ dA = = αλ Energy emitted π iboλ dA

(12-2-13)

For this reason, the absorption factor defined by Eq. (12-2-8) is also called the hemispheric monochromatic absorption factor because its definition can be expressed by Eq. (12-2-13). The hemispheric monochromatic absorption factor represents the fraction of the monochromatic energy of the considered wavelength that is absorbed by the gas contained in a hemisphere of radius L with respect to the total monochromatic energy emitted by a surface element located at its center. Since α λ = 1 − e − kλ L, it is clear that this factor depends on L. This means that it depends on the volume of gas and also, through kλ, on the nature of the gas, its pressure, temperature, and wavelength considered. 12-2-4

Monochromatic Hemispheric Emissivity In the same way that gases absorb energy, they are capable of emitting it. As with absorption, the emission of energy by a gas is a volumetric phenomenon taking place in the entire volume of the gas. To study the radiant energy emission by a gas, the same geometry used for definition of the absorption coefficient will be used: a hemispherical gas volume with a blackbody element of area located at its center. As a consequence of its temperature, the gas emits radiation. Part of the emitted radiation reaches the element of area. If the monochromatic energy reaching the element of area per unit time is called dQλG→1, it is possible to define monochromatic hemispheric emissivity as ελ =

dQλG→1 dA WbλG

(12-2-14)

where WbλG is the monochromatic emissive power of the blackbody at the gas temperature. It is important to note that the gas emissivity is defined with reference to the receptor area and not the emitter area (because no area can be associated with the gas). Since dA is a black surface, all the incident energy is absorbed, and then the net rate of energy exchange between the gas and the element of area is dQλ ,1G = dQλ ,1→G − dQλ , G→1

(12-2-15)

and combining Eqs. (12-2-12) and (12-2-14), we get dQλ ,1G = dA(ε λWbλG − α λWb λ1 )

(12-2-16)

It’s important to realize that both ελ and αλ are a function of the gas volume. The higher this volume, the higher will be ελ and αλ, and the higher will be the heat flux in both directions. 12-2-5 Total Emissivity and Total Absorption Factor We have explained that the monochromatic absorption factor and the monochromatic gas emissivity are almost noncontinuous functions of the wavelength. Unlike the case of solid bodies, the gas absorption and emission mechanisms exist only in narrow bands of the electromagnetic spectrum. In each wavelength interval where the gas is absorbent, both the absorption factor and the emissivity are functions like α λ = 1 − e − kλ L where kλ depends on the wavelength, as well as on system pressure and temperature.

(12-2-17)

430

CHAPTER TWELVE

However, in technical calculations, we need to have total absorption factors and emissivities for the entire range of thermal radiation. These total coefficients and their dependence on the gas parameters were defined and correlated by Hottel4,5 for the cases of CO2 and water vapor. The definitions of the total absorption factor and total emissivity are based on the same geometry used for the hemispheric monochromatic factors. We must consider an element of area located at the center of a gas hemisphere of radius L containing the gas. If dQG→1 is the amount of energy (corresponding to all the wavelengths) reaching the area element per unit time, the total emissivity of the gas is defined as dQG→1 = ε WbG dA

(12-2-18)

where WbG is the emissive power of the blackbody at the gas temperature. This means that WbG = σTG4

(12-2-19)

In turn, if dQ1→G is the energy emitted by dA and absorbed by the gas, the total absorption factor of the gas can be defined as dQ1→G = α Wb1 dA

(12-2-20)

where Wb1 is the emissive power of the blackbody at the temperature of the surface. This is Wb1 = σT14

(12-2-21)

The net heat exchange between the gas and the surface element then can be calculated as

(

dQ1G = dA σ εTG4 − α T14

)

(12-2-22)

12-2-6 Graphs for Gas Emissivity and Absorption Factor Total gas emissivities and absorption factors were correlated for the cases of CO2 and water vapor, which are the gases present in the products of all combustion. The dependence of these total coefficients on the parameters of the system differs completely from Eq. (12-2-17), which is only valid for the monochromatic radiation. Calculation of the emissivity and absorption factor of a gas at temperature TG contained in a hemisphere of radius L exchanging energy with an element of black surface located at its center can be calculated with graphs, as explained in what follows. For CO2, if pc is the partial pressure of CO2 in the gas mixture, the emissivity can be obtained from Fig. 12-39 as a function of the gas temperature at given parameters of the product pcL. This graph is only valid for a total pressure of 1 atm (101.3 kPa). For other conditions, a correction factor Cc must be obtained from Fig. 12-40 as a function of the total pressure and the product pcL. The emissivity then is calculated as ε CO2 = eCc

(12-2-23)

where e is the value obtained from Fig. 12-39. For water vapor, the procedure is quite similar. If pw is the partial pressure of water vapor in the gas mixture, it is possible to obtain from Fig. 12-41 an emissivity e as a function of TG and pwL.This graph is valid for a total pressure of 101.3 kPa and for pw tending to 0. For other conditions, a correction factor obtained from Fig. 12-42 must be used. It must be observed that the abscissa is in this case (p + pw)2. The emissivity of water vapor is then the product. ε H 2O = eCw

(12-2-24)

431

THERMAL RADIATION

0.30 0.20 pc L = 105 Pa·m

0.15 0.10

6 × 104

0.08

3 × 104

Emissivity (ε)

0.06 0.05 0.04

104 6,000

0.03 3,000 0.02

1,000

0.015 600

100 0.01 0.008 60

0.006 0.005 0.004 0.003 500

300

30 1,000

1,500 2,000 Temperature (K)

2,500

FIGURE 12-39 Emissivity of CO2 at 1 atm (abs) total pressure. (From H.C Hottel: in W.H McAdams: Heat Transmission. New York: McGraw-Hill, 1954 Reproduced with permission of the editors.)

2 1 2 3 4 5 6 7

1.5

Cc

1 0.8 0.6 0.5 0.4 0.3 0.05

Curve 1 2 3 4 5 6 7

7 6 5 4 3 2 1

0.08 0.1

0.2

0.3

0.5

1.0

2.0

pc L (Pa·m) 600 1,500 4,000 8,000 15,000 30,000 80,000

3.0

5.0

Total pressure (atm) FIGURE 12-40 Correction factor for CO2 emissivity for pressures different from 1 atm (abs). (From H.C Hottel: in W.H McAdams: Heat Transmission. New York: McGraw-Hill, 1954 Reproduced with permission of the editors.)

CHAPTER TWELVE

0.7 0.6 0.5 0.4

pwL = 6 × 105 Pa·m

0.3 3 × 105 0.2 105 0.15

2 × 104

6 × 104

104

3 × 104

0.12 Emissivity (ε)

432

0.10 0.08 0.07 0.06

6,000 4,000

0.05

3,000

0.04

2,000 0.03 600

1,500

400

1,000

0.02 0.015 300 0.012 0.01 0.009 0.008 0.007 500

200

1,000

1,500 Temperature (K)

2,000

2,500

FIGURE 12-41 Emissivity of water vapor at 1 atm (abs) total pressure and water vapor partial pressure tending to 0. (From H.C Hottel: in W.H McAdams: Heat Transmission. New York: McGraw-Hill 1954 Reproduced with permission of the editors.)

For a mixture containing CO2 and water vapor together, an additional correction is necessary. Since the emission bands of water vapor and CO2 present a certain overlap, when both gases are present, part of the energy emitted by one of them is absorbed by the other. Then the mixture emissivity, which in case of no overlap would be the sum of the emissivities of both components, is reduced. The mixture emissivity then can be calculated as ε = ε CO2 + ε H 2O − Δε

(12-2-25)

where Δε is obtained from Fig. 12-43. The absorption factor can be obtained in a similar way. The absorption factor of a mixture containing CO2 and water vapor is α = α CO2 + α H 2O − Δα

(12-2-26)

THERMAL RADIATION

1.8

1 2 3 4 5 6 7

1.6 1.4 1.2 Cw

433

Curve pw L (Pa·m)

1.0

1 2 3 4 5 6 7

0.8 76 5 4 32 1

0.6 0.4 0.2 0

0.2

0.4 0.6 0.8 (p + pw)/2 (atm)

1.5 × 103 8.0 × 103 1.5 × 104 3.0 × 104 8.0 × 104 1.5 × 105 3.0 × 105

1 atm = 101300 Pa 1.2

1.0

FIGURE 12-42 Correction factor for water vapor emissivity. (From H.C Hottel: in W.H McAdams : Heat Transmission. New York: McGraw-Hill, 1954 Reproduced with permission of the editors.)

0.7

0.7 400 K

0.6

0.5 1 2

0.3 0.2

0

0

0.2

0.4

4 5 6 0.6 0.8

pw pc + pw

0.5

0.4

0.4

0.3 0.2

3

0.1

0.5

Δε

Δε

0.4

0.1 1.0

0

1 2

0.3

3 4

0.2

0.2

0.4 0.6 pw pc + pw

0.8

1.0

0

Curve 1 2 3 4 5 6

1 2 3 4

pc L + p w L (Pa·m) 1.5 × 105 1.0 × 105 5 × 104 3 × 104 1.5 × 104 1.0 × 104

5

0.1

5 6 0

1,200 K and more

0.6

Δε

0.6

0.7 800 K

1 atm = 101,300 Pa 6 0

0.2

0.4

0.6 pw pc + pw

0.8 1.0

FIGURE 12-43 Emissivity correction for mixtures of CO2 and water vapor. (From H.C Hottel: in W.H McAdams: Heat Transmission. New York: McGraw-Hill, 1954 Reproduced with permission of the editors.)

where ⎛T ⎞ α CO2 = e'Cc' ⎜ G ⎟ ⎝ T1 ⎠

0.65

(12-2-27)

Where e' is obtained from Fig. 12-39 but entering the graph with the surface temperature T1 as abscissa instead of gas temperature and calculating the parameter as pcL(T1/TG) instead of pcL. For water vapor, ⎛T ⎞ α H 2 O = e'Cw' ⎜ G ⎟ ⎝ T1 ⎠

0.45

(12-2-28)

434

CHAPTER TWELVE

where e' and C'w are obtained from Figs. 12-41 and 12-42. To find e’, we enter the graph in Fig. 12-41 with T1 as abscissa and with pwL(T1/TG) as the parameter. The correction coefficients are obtained in the same way as for emissivity calculation. A detailed calculation is explained in Example 12-6. 12-2-7 Radiant Heat Exchange between a Gas and a Finite Surface Up to now, we only considered the case of heat exchange between a gas hemisphere and an element of area located at its center. For other geometries, the problem is solved by defining an equivalent mean hemispherical beam length. The equivalent mean hemispherical beam length is a fictitious length characteristic of the considered geometry that allows one to obtain, using the graphs in Figs. 12-39 through 12-43, emissivities and absorption factors for the gas such that the net heat exchanged between the gas and the surface can be expressed as QG 1 = A(εWbG − αWb1 )

(12-2-29)

(

(12-2-30)

or QG 1 = Aσ εTG4 − αT14

)

The more usual case is that of a gas exchanging energy with the walls of the enclosure in which it is contained. In these cases, Table 12-1 allows one to obtain the equivalent mean hemispherical beam length for simple geometries. TABLE 12-1

Equivalent Mean Hemispherical Beam Length6 Shape

L

Infinite cylinder Circular cylinder with height = diameter Clearance between infinite parallel planes Cube

Diameter 2/ × diameter 3 2 × distance between planes 2/ × edge 3

For geometries not included in Table 12-1, an approximate value can be calculated as L = 3.4

volume surface area

12-2-8 Equivalent Gray Gas Emissivity Making an extension of the gray-body concept, we can define a gray gas as that whose emissivity equals the absorption factor. In this case, Eq. (12-2-29) reduces to QG 1 = Aε(WbG − Wb1 )

(12-2-31)

As we have seen in the preceding section, gases such as CO2 and water vapor are very far from being gray. However, we can keep the form of Eq. (12-2-31) if we define an equivalent gray gas emissivity as εg =

εWbG − αWb1 WbG − Wb1

(12-2-32)

In this way, the net heat exchange between the gas and the surface can be expressed as QG 1 = Aε g (WbG − Wb1 )

(12-2-33)

THERMAL RADIATION

435

And the equation becomes simpler because instead of ε and α , it requires a single parameter εg. Of course, in order to use this expression, both ε and α have to be calculated anyway. So this approach does not look to offer any particular advantage. However, by using a single parameter, the equation can be represented by the equivalent electric network of Fig. 12-44a, and this will be useful when dealing with more complex cases, as we shall later see. If the solid surface is supposed to have an emissivity ε1, the gray-to-black transformation resistance must be included, as shown in Fig. 12-44b.

Wb1 = σT14

1 A1εg

WbG = σTG 4

(a) Heat exchange between a gas and a blackbody FIGURE 12-44

Wb1 = σT14

(1 – ε1) A1ε1

1 A1εg

WbG = σTG 4

(b) Heat exchange between a gas and a gray body

Heat exchange between a gas and a solid surface.

Example 12-6 The exhaust gases of an internal combustion motor are discharged into the atmosphere through an exhaust pipe with 0.1-m internal diameter. The pressure within the pipe is 125 kPa. The components mole fractions are H2O: 0.096 CO2: 0.085 N2: 0.747 O2: 0.072 (Note: This composition corresponds to the combustion gases when a fuel with a molecular composition CH1.8 is burned with 20 percent excess air. In Chap. 13, the stoichiometry of combustion will be explained in detail, and this will allow you to calculate the composition of the products of any combustion.) If the wall temperature is assumed to be at 700 K, and the emissivity of the walls is 0.8, calculate the net radiant heat exchanged between the gas and the wall. Solution

The partial pressures of the absorbing gases (CO2 and H2O) are

For CO2: For H2O:

pc = 0.085 × 125 = 10.62 kPa pw = 0.096 × 125 = 12 kPa

The equivalent mean hemispherical beam length, assuming that the pipe is an infinite cylinder, is equal to the diameter according to Table 12-1. This is 0.1 m. Then pc × L = 10.62 × 0.1 = 1.062 kPa · m pw × L = 12 × 0.1 = 1.2 kPa · m CO2 emissivity: From Fig. 12-39, with TG = 1,500 K and pcL = 1,062 Pa · m (0.0105 atm · m), we get e = 0.037. The correction factor from Fig. 12-40 with p = 125 kPa (1.23 atm) and pcL = 1,062 Pa · m is Cc = 1.05. Thus εCO2 = 0.037 × 1.05 = 0.0388

436

CHAPTER TWELVE

Water vapor emissivity: With pwL = 1,200 Pa · m (0.0118 atm · m) and TG = 1,500 K, from Fig. 12-41, we get e = 0.015. The correction factor from Fig. 12-42 is obtained by entering with an abscissa of 1/2(1.23 + 0.118) = 0.673 atm and 1,200 Pa · m as the parameter. We obtain Cw = 1.27. Thus εH2O = 0.015 × 1.27 = 0.019 Mixture emissivity: The correction factor from Fig. 12-43 is negligible. Thus the emissivity of the mixture will be the sum ε = εCO2 + εH2O = 0.0388 + 0.019 = 0.0578 CO2 absorption factor: Entering Fig. 12-39 with abscissa T1 = 700 K, and calculating the value of the parameter as PcL(T1/TG) = 1,062(700/1,500) = 495 Pa · m (0.00488 atm · m) we get e = 0.035. The correction factor C'c = Cc = 1.05. Then αCO2 = 0.035 × 1.05 × (1,500/700)0.65 = 0.060 Water vapor absorption factor: Entering Fig. 12-41 with an abscissa of 700 K and a parameter value of pwL(T1/TG) = 1,200 (700/1,500) = 560 Pa · m (0.0052 atm · m) we get e = 0.024. The correction factor is C'w = Cw = 1.27. Then αH2O = 0.024 × 1.27 × (1,500/700)0.45 = 0.043 Mixture absorption factor: Again, neglecting the mixture correction, the absorption factor will be the sum of αCO2 + αH2O = 0.060 + 0.043 = 0.102 The equivalent gray gas emissivity thus will be εg =

εWbG − αWb1 0.0578 × 1, 500 4 − 0.102 × 700 4 = = 0.055 WbG − Wb1 1, 500 4 − 700 4

The area of the pipe per meter length is A1 = π × D = 0.314 m2/m. Thus the total resistance of the electric circuit represented in Fig. 12-44b is 1 − ε1 1 1 1 − 0. 8 = 58.7 + = + A1εg A1ε1 0.314 × 0.055 0.314 × 0.8 and the net heat exchanged is QG1 =

WbG − Wb1 = 58.7

(

σ TG4 − T14 58.7

) = 5.678 × 10 −8 (1, 500 4 − 700 4 ) = 4, 664 W 58.7

12-2-9 Radiant Heat Transfer between Surfaces in the Presence of Absorbing Gases Let’s first consider the case of two planar parallel plates at temperatures T1 and T2 exchanging energy when a vacuum or a transparent gas exists between them. The heat exchange rate can be expressed as Q12 = A1 F12 (Wb1 − Wb 2 )

(12-2-34)

THERMAL RADIATION

437

This expression has its electric analogy in a circuit where the resistance is (A1F12)–1. If the surfaces are gray, instead of F12, we use F12. If now we introduce a nontransparent gas between the plates, the heat exchanged will not be the same. The gas will absorb energy according to its coefficient α, and it will only allow the passage of a fraction (1 – α) = τG12. We could think that the resistance to the heat transmission between plates 1 and 2 has increased in a factor 1/τG12 owing to the presence of the gas. The gas, in turn, can exchange energy with both surfaces. The radiant energy exchanged between the gas and plate 1 is given by A1εg1(WbG – Wb1), and the energy exchanged between the gas and plate 2 is A2εg2(WbG – Wb2). The electric circuit in Fig. 12-45 represents this system. 1 A1F12(1 – αg )

Wb1

1 A1εg1

Wb2

1 A2εg2 WbG

FIGURE 12-45

System with two solid surfaces and a gas.

If the gas is enclosed in the space between the plates without delivering or withdrawing energy to or from the system, it will reach a steady-state temperature in between those of plates 1 and 2. This steady state is reached when the gas temperature is such that all the energy received from the hot source is yielded to the cold one. Then we can eliminate in the electric circuit the branch containing WbG without modifying the current distribution in the rest of the network. The network represented in Fig. 12-46 is obtained. We see that in this case the final result is a net heat transmission from plate 1 to plate 2. The energy flow from plate 1 to plate 2 is the sum of two fractions. The first one comes directly to plate 2 through the resistance of the upper branch. This is the energy flow allowed by the gas according to its transparency coefficient. The rest of the energy emitted by plate 1 is absorbed by the gas. But the gas reemits it, and it finally reaches A2. This is the current circulating through the lower resistances. The total energy exchanged is the sum of both currents. If temperatures T1 and T2 are known, the network can be solved and the current distribution found. It is then possible to calculate the potential of point G (WbG), and the gas temperature can be obtained from TG =

WbG σ

(12-2-35)

1 A1F12(1 – αg )

Wb1

1 A1εg1

FIGURE 12-46

4

1 A2εg 2

Gas enclosed between plates.

Wb 2

438

CHAPTER TWELVE

GLOSSARY A = area (m2) F = direct view factor F = view factor including refractories F = view factor between gray surfaces I = intensity of radiation (W/m2) Q = heat flux (W) r = radial coordinate (m) T = temperature (K) W = emissive power (W/m2) Wλ = monochromatic emissive power [W/(m2 · m)] α = absorption coefficient ε = emissivity λ = wavelength (m) ω = solid angle ρ = reflection coefficient σ = Stephan Boltzman constant τ = transmissivity θ, φ = angular coordinates

Subscripts b = blackbody n = normal direction θ = direction θ λ = wavelength λ

REFERENCES 1. 2. 3. 4. 5. 6. 7.

Kreith F.: Principles of Heat Transfer. New York: Internacional Textbook Company, 1968. Hottel H. C.: in McAdams W.H : Heat Transmission. New York: McGraw-Hill, 1954. Planck M.: The theory of Heat Radiation. New York: Dover, 1959. Hottel H. C.: “Radiant Heat Transmission,” Mech Eng 52:699–704, 1930. Hottel H. C., Egbert: “Radiant Heat Transmission from Water Vapor,” AIChE Trans 38:531–565, 1942. Rohsenow M., Hartnett J. P.: Handbook of Heat Transfer. New York: McGraw-Hill, 1973. Schmidt E., Eckert E.: “Über die Richtunsverteilun der Warmestrahlung von Oberflachen,” Forsch Geb Ingenieurwes 6:175–183, 1935.

CHAPTER 13

PROCESS FIRED HEATERS

13-1

SIMPLIFIED SCHEMATIC A process furnace or fired heater consists of a combustion chamber or firebox with one or more burners and containing a tubular coil where a process fluid circulates. The chamber walls normally are metallic, interiorly covered by a refractory lining. A gaseous or liquid fuel is injected in the burners with the necessary combustion air. The combustion reaction that takes place in the burners generates hot flue gases that are the product of the combustion. Part of the energy liberated in the combustion process is used to heat up the product circulating into the coil. Another part is lost through the stack (because the flue gas is still hot when it leaves the system), and there also is a conduction heat loss through the walls of the chamber. A schematic is shown on Fig. 13-1. Before proceeding with a description of the several elements constituting the furnace, we shall review some aspects related to the process of combustion.

13-2 13-2-1

COMBUSTION Stoichiometry Combustion is an exothermic chemical reaction between oxygen and a fuel. Usually the fuel is a hydrocarbon and the oxygen is that contained in air. If the amount of oxygen is enough to convert all the fuel to CO2 and H2O (the principal products of combustion), it is said that the combustion is complete. If the amount of oxygen in the burner is not enough for complete burning of the fuel, the combustion gases will contain unburned hydrocarbon and CO. The combustion reactions of the most important hydrocarbons are Methane: CH4 + 2O2 = CO2 + 2H2O 7 Ethane: C2H6 + O2 = 2CO2 + 3H2O 2 Propane: C3H8 + 5O2 = 3CO2 + 4H2O Ethylene: C2H4 + 3O2 = 2CO2 + 2H2O Any sulfur present in the fuel will react according to S + O2 = SO2 One mole of oxygen is required for each carbon atom, 0.25 mole of oxygen for every hydrogen atom, and 1 mole for every sulfur atom. Thus, for a fuel containing n carbon atoms, m hydrogen atoms, and s sulfur atoms, the reaction can be written CnHmSs + (n + 0.25m + s)O2 = nCO2 + 0.5mH2O + sSO2 In industrial burners, the amount of oxygen employed is not stoichiometric because an oxygen excess is always necessary to achieve complete burning of the fuel.

439

440

CHAPTER THIRTEEN

Flue gas WH

Stack

Coil

i2 Product Wp

Refractories i1 Firebox Burner

Air WA

FIGURE 13-1

Fuel WF Atomizing steam (optional) WV

Schematic of a process fired heater.

When gaseous fuels are employed (natural gas or refinery fuel gas), 10–15 percent excess air normally is used. Liquid-fuel burners normally require a higher air excess because it is more difficult to obtain a good mixture between the air and the fuel, especially if high-viscosity fuel oil is used. In these cases, the excess air can be 25 percent or even more. Most liquid-fuel burners require the addition of steam, which is injected into the burner and mixed with the fuel oil to atomize it at the burner outlet nozzle. The fuel exits the nozzle at high velocity, divided into small droplets that can burn easily. If e is the oxygen excess (e = excess air/stoichiometric air), and considering that the atmospheric nitrogen:oxygen ratio is 3.762:1, the combustion reaction for a molecule with a chemical formula CnHmSs can be written CnHmSs + (1 + e)(n + 0.25m + s)O2 + 3.762(1 + e)(n + 0.25m + s)N2 = nCO2 + 0.5mH2O + sSO2 + e(n + 0.25m + s)O2 + 3.762(1 + e)(n + 0.25m + s)N2 To this equation we should add the atomizing steam, if used. Even though the atomizing steam does not take part in the combustion reaction, it is important to consider it among the combustion products because it increases the partial pressure of the water vapor, and this is important for the calculation of gas emissivities and the enthalpy balance. It also should be considered that atmospheric air always contains some water vapor. However, we shall neglect this contribution in the calculations. Fuel gas also can contain nitrogen and CO2. These components will incorporate directly into the combustion gases. If, additionally, some