process heat transfer principles and applications

1.2 Fourier’s Law of Heat Conduction 21.3 The Heat Conduction Equation 6 1.5 The Conduction Shape Factor 19 1.6 Unsteady-State Conduction 24 1.7 Mechanisms of Heat Conduction 31 2 Convec

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This book is dedicated to C.C.S.

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Process Heat Transfer

Principles and Applications

R.W Serth

Department of Chemical and Natural Gas Engineering,

Texas A&M University-Kingsville,

Kingsville, Texas, USA

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British Librar y Cataloguing in Publication Data

Serth, R W.

Process heat transfer : principles and applications

1 Heat - Transmission 2 Heat exchangers 3 Heat exchangers - Design

4 Heat - Transmission - Computer programs

I Title

621.4 ′ 022

Librar y of Congress Catalog number: 2006940583

ISBN: 978-0-12-373588-1

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Typeset by Charon Tec Ltd (A Macmillan Company), Chennai, India

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06 07 08 09 10 11 10 9 8 7 6 5 4 3 2 1

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1.2 Fourier’s Law of Heat Conduction 2

1.3 The Heat Conduction Equation 6

1.5 The Conduction Shape Factor 19

1.6 Unsteady-State Conduction 24

1.7 Mechanisms of Heat Conduction 31

2 Convective Heat Transfer 43

2.2 Combined Conduction and Convection 44

2.4 Forced Convection in Pipes and Ducts 53

2.5 Forced Convection in External Flow 62

3.4 The Overall Heat-Transfer Coefficient 93

3.6 Analysis of Double-Pipe Exchangers 102

3.7 Preliminary Design of Shell-and-Tube Exchangers 106

3.8 Rating a Shell-and-Tube Exchanger 109

3.9 Heat-Exchanger Effectiveness 114

4 Design of Double-Pipe Heat Exchangers 127

4.2 Heat-Transfer Coefficients for Exchangers without Fins 128

4.3 Hydraulic Calculations for Exchangers without Fins 128

4.4 Series/Parallel Configurations of Hairpins 131

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5 Design of Shell-and-Tube Heat Exchangers 187

6.2 Ideal Tube Bank Correlations 246

6.3 Shell-Side Heat-Transfer Coefficient 248

6.4 Shell-Side Pressure Drop 250

6.6 Correlations for the Correction Factors 259

6.7 Estimation of Clearances 260

7 The Stream Analysis Method 277

7.2 The Equivalent Hydraulic Network 278

7.3 The Hydraulic Equations 279

7.4 Shell-Side Pressure Drop 281

7.5 Shell-Side Heat-Transfer Coefficient 281

7.6 Temperature Profile Distortion 282

8.7 Significance of the Pinch 335

8.8 Threshold Problems and Utility Pinches 337

8.9 Feasibility Criteria at the Pinch 337

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9 Boiling Heat Transfer 385

10.3 Design of Kettle Reboilers 449

10.4 Design of Horizontal Thermosyphon Reboilers 467

10.5 Design of Vertical Thermosyphon Reboilers 473

10.6 Computer Software 488

11.1 Introduction 540

11.2 Types of Condensers 540

11.3 Condensation on a Vertical Surface: Nusselt Theory 545

11.4 Condensation on Horizontal Tubes 549

11.5 Modifications of Nusselt Theory 552

11.6 Condensation Inside Horizontal Tubes 562

11.7 Condensation on Finned Tubes 568

12.3 Air-Side Heat-Transfer Coefficient 637

12.4 Air-Side Pressure Drop 638

12.5 Overall Heat-Transfer Coefficient 640

12.6 Fan and Motor Sizing 640

12.7 Mean Temperature Difference 643

12.8 Design Guidelines 643

12.9 Design Strategy 644

12.10 Computer Software 653

Appendix A Thermophysical Properties of Materials 682

Appendix B Dimensions of Pipe and Tubing 717

Appendix C Tube-Count Tables 729

Appendix D Equivalent Lengths of Pipe Fittings 737

Appendix E Properties of Petroleum Streams 740

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This book is based on a course in process heat transfer that I have taught for many years The coursehas been taken by seniors and first-year graduate students who have completed an introductorycourse in engineering heat transfer Although this background is assumed, nearly all students needsome review before proceeding to more advanced material For this reason, and also to make thebook self-contained, the first three chapters provide a review of essential material normally covered

in an introductory heat transfer course Furthermore, the book is intended for use by practicingengineers as well as university students, and it has been written with the aim of facilitating self-study.Unlike some books in this field, no attempt is made herein to cover the entire panoply of heat trans-fer equipment Instead, the book focuses on the types of equipment most widely used in the chemicalprocess industries, namely, shell-and-tube heat exchangers (including condensers and reboilers),air-cooled heat exchangers and double-pipe (hairpin) heat exchangers Within the confines of a sin-gle volume, this approach allows an in-depth treatment of the material that is most relevant from anindustrial perspective, and provides students with the detailed knowledge needed for engineeringpractice This approach is also consistent with the time available in a one-semester course.Design of double-pipe exchangers is presented in Chapter 4 Chapters 5–7 comprise a unit dealingwith shell-and-tube exchangers in operations involving single-phase fluids Design of shell-and-tubeexchangers is covered in Chapter 5 using the Simplified Delaware method for shell-side calcula-tions For pedagogical reasons, more sophisticated methods for performing shell-side heat-transferand pressure-drop calculations are presented separately in Chapter 6 (full Delaware method) andChapter 7 (Stream Analysis method) Heat exchanger networks are covered in Chapter 8 I nor-mally present this topic at this point in the course to provide a change of pace However, Chapter

8 is essentially self-contained and can, therefore, be covered at any time Phase-change operationsare covered in Chapters 9–11 Chapter 9 presents the basics of boiling heat transfer and two-phaseflow The latter is encountered in both Chapter 10, which deals with the design of reboilers, andChapter 11, which covers condensation and condenser design Design of air-cooled heat exchang-ers is presented in Chapter 12 The material in this chapter is essentially self-contained and, hence,

it can be covered at any time

Since the primary goal of both the book and the course is to provide students with the edge and skills needed for modern industrial practice, computer applications play an integral role,and the book is intended for use with one or more commercial software packages HEXTRAN(SimSci-Esscor), HTRI Xchanger Suite (Heat Transfer Research, Inc.) and the HTFS Suite (AspenTechnology, Inc.) are used in the book, along with HX-Net (Aspen Technology, Inc.) for pinchcalculations HEXTRAN affords the most complete coverage of topics, as it handles all types of heatexchangers and also performs pinch calculations for design of heat exchanger networks It doesnot perform mechanical design calculations for shell-and-tube exchangers, however, nor does itgenerate detailed tube layouts or setting plans Furthermore, the methodology used by HEXTRAN

knowl-is based on publicly available technology and knowl-is generally less refined than that of the other softwarepackages The HTRI and HTFS packages use proprietary methods developed by their respectiveresearch organizations, and are similar in their level of refinement HTFS Suite handles all types

of heat exchangers; it also performs mechanical design calculations and develops detailed tubelayouts and setting plans for shell-and-tube exchangers HTRI Xchanger Suite lacks a mechanicaldesign feature, and the module for hairpin exchangers is not included with an academic license.Neither HTRI nor HTFS has the capability to perform pinch calculations

As of this writing, Aspen Technology is not providing the TASC and ACOL modules of the HTFSSuite under its university program Instead, it is offering the HTFS-plus design package Thispackage basically consists of the TASC and ACOL computational engines combined with slightlymodified GUI’s from the corresponding BJAC programs (HETRAN and AEROTRAN), and packagedwith the BJAC TEAMS mechanical design program This package differs greatly in appearance and

to some extent in available features from HTFS Suite However, most of the results presented in thetext using TASC and ACOL can be generated using the HTFS-plus package

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Software companies are continually modifying their products, making differences between thetext and current versions of the software packages unavoidable However, many modificationsinvolve only superficial changes in format that have little, if any, effect on results More substantivechanges occur less frequently, and even then the effects tend to be relatively minor Nevertheless,readers should expect some divergence of the software from the versions used herein, and theyshould not be unduly concerned if their results differ somewhat from those presented in the text.Indeed, even the same version of a code, when run on different machines, can produce slightlydifferent results due to differences in round-off errors With these caveats, it is hoped that thedetailed computer examples will prove helpful in learning to use the software packages, as well as

in understanding their idiosyncrasies and limitations

I have made a concerted effort to introduce the complexities of the subject matter graduallythroughout the book in order to avoid overwhelming the reader with a massive amount of detail

at any one time As a result, information on shell-and-tube exchangers is spread over a number ofchapters, and some of the finer details are introduced in the context of example problems, includingcomputer examples Although there is an obvious downside to this strategy, I nevertheless believethat it represents good pedagogy

Both English units, which are still widely used by American industry, and SI units are used in thisbook Students in the United States need to be proficient in both sets of units, and the same is true

of students in countries that do a large amount of business with U.S firms In order to minimizethe need for unit conversion, however, working equations are either given in dimensionless form

or, when this is not practical, they are given in both sets of units

I would like to take this opportunity to thank the many students who have contributed to thiseffort over the years, both directly and indirectly through their participation in my course I wouldalso like to express my deep appreciation to my colleagues in the Department of Chemical andNatural Gas Engineering at TAMUK, Dr Ali Pilehvari and Mrs Wanda Pounds Without their help,encouragement and friendship, this book would not have been written

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= 1895.6 Btu/h ·◦F

Heat transfer coefficient 1 W/m2· K = 0.17612 Btu/h · ft2·◦F

Kinematic viscosity and thermal 1 m2/s = 3.875 × 104ft2/h

= 0.068523 lbf/ft

= (5/9)(◦F + 459.67) = (5/9)(◦R)

= 1.5850 × 104gal/min (gpm)lbf: pound force and lbm: pound mass

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Physical Constants

0.08314 bar · m3/kmol · K

8314 J/kmol · K1.986 cal/mol · K1.986 Btu/lb mole ·◦R10.73 psia · ft3/lb mole ·◦R

1545 ft · lbf/lb mole ·◦RStandard gravitational acceleration g 9.8067 m/s2

32.174 ft/s24.1698 × 108ft/h2Stefan-Boltzman constant σSB 5.670 × 10−8W/m2· K4

1.714 × 10−9Btu/h · ft2·◦R4

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Figure 3.1 Reprinted, with permission, from Extended Surface Heat Transfer by D Q Kern and

Table 3.1 Reprinted, with permission, from Perry’s Chemical Engineers’ Handbook, 7th edn.,

Table 3.2 Reproduced, with permission, from J W Palen and J Taborek, Solution of shell side flow

pressure drop and heat transfer by stream analysis method, Chem Eng Prog Symposium

Table 3.5 Reprinted, with permission, from Perry’s Chemical Engineers’ Handbook, 7th edn.,

and H Liu Reproduced by permission of Taylor & Francis, a division of Informa plc

Figure 5.3 Reproduced, with permission, from R Mukherjee, Effectively design shell-and-tube heat

Editor-in-Chief Reproduced by permission of Taylor & Francis, a division of Informa plc

Figure 7.1 Reproduced, with permission, from J W Palen and J Taborek, Solution of shell side flow

pressure drop and heat transfer by stream analysis method, Chem Eng Prog Symposium

Table, p 283 Reproduced, with permission, from R Mukherjee, Effectively design shell-and-tube heat

Figure 8.20 Reprinted from Computers and Chemical Engineering, Vol 26, X X Zhu and X R Nie,

Pressure Drop Considerations for Heat Exchanger Network Grassroots Design, pp 1661–

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Item Special Credit Line

L S Tong and Y S Tang Reproduced by permission of Taylor & Francis, a division

of Informa plc

Figure 10.6 Reproduced, with permission, from A W Sloley, Properly design thermosyphon

Appendix 10.A Reprinted, with permission, from Chemical Engineers’ Handbook, 5th edn.,

S Kakac and H Liu Reproduced by permission of Taylor & Francis, a division ofInforma plc

1990 by The McGraw-Hill Companies, Inc

Figure 11.11 Reprinted, with permission, from G Breber, J W Palen and J Taborek, Prediction

of tubeside condensation of pure components using flow regime criteria, J Heat

Transfer,102, 471–476, 1980 Originally published by ASME.

S Kakac and H Liu Reproduced by permission of Taylor & Francis, a division ofInforma plc

Reproduced by permission of Taylor & Francis, a division of Informa plc

K Raznjeviˇc Reproduced by permission of Taylor & Francis, a division ofInforma plc

Table A.3 Reprinted, with permission, from Heat Transfer, 7th edn., by J P Holman Copyright

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Item Special Credit Line

Table A.8 Reprinted, with permission, from ASME Steam Tables, American Society of Mechanical

Engineers, New York, 1967 Originally published by ASME

Table A.9 Reprinted, with permission, from Flow of Fluids Through Valves, Fittings and Pipe, Technical

N B Vargaftik Reproduced by permission of Taylor & Francis, a division of Informa plc

Reproduced by permission of Taylor & Francis, a division of Informa plc

Table A.15 Reprinted, with permission, from Chemical Engineers’ Handbook, 5th edn., R H Perry and

Figure A.1 Reprinted, with permission, from Chemical Engineers’ Handbook, 5th edn., R H Perry and

Figure A.2 Reprinted, with permission, from Chemical Engineers’ Handbook, 5th edn., R H Perry and

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1 HEAT

CONDUCTION

Contents

1.1 Introduction 21.2 Fourier’s Law of Heat Conduction 21.3 The Heat Conduction Equation 61.4 Thermal Resistance 15

1.5 The Conduction Shape Factor 191.6 Unsteady-State Conduction 241.7 Mechanisms of Heat Conduction 31

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1.1 Introduction

Heat conduction is one of the three basic modes of thermal energy transport (convection andradiation being the other two) and is involved in virtually all process heat-transfer operations Incommercial heat exchange equipment, for example, heat is conducted through a solid wall (often

a tube wall) that separates two fluids having different temperatures Furthermore, the concept ofthermal resistance, which follows from the fundamental equations of heat conduction, is widely used

in the analysis of problems arising in the design and operation of industrial equipment In addition,many routine process engineering problems can be solved with acceptable accuracy using simplesolutions of the heat conduction equation for rectangular, cylindrical, and spherical geometries.This chapter provides an introduction to the macroscopic theory of heat conduction and its engi-neering applications The key concept of thermal resistance, used throughout the text, is developedhere, and its utility in analyzing and solving problems of practical interest is illustrated

1.2 Fourier’s Law of Heat Conduction

The mathematical theory of heat conduction was developed early in the nineteenth century byJoseph Fourier [1] The theory was based on the results of experiments similar to that illustrated

in Figure 1.1 in which one side of a rectangular solid is held at temperature T1, while the opposite

side is held at a lower temperature, T2 The other four sides are insulated so that heat can flow

only in the x-direction For a given material, it is found that the rate, q x, at which heat (thermalenergy) is transferred from the hot side to the cold side is proportional to the cross-sectional area,

A, across which the heat flows; the temperature difference, T1− T2; and inversely proportional to

the thickness, B, of the material That is:

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The constant of proportionality, k, is called the thermal conductivity Equation (1.1) is also applicable

to heat conduction in liquids and gases However, when temperature differences exist in fluids, vection currents tend to be set up, so that heat is generally not transferred solely by the mechanism

acceptable to treat k as a constant, particularly if the temperature difference is moderate When the

temperature dependence must be taken into account, a linear function is often adequate, particularlyfor solids In this case,

where a and b are constants.

Thermal conductivities of a number of materials are given in Appendices 1.A–1.E Many othervalues may be found in various handbooks and compendiums of physical property data Processsimulation software is also an excellent source of physical property data Methods for estimatingthermal conductivities of fluids when data are unavailable can be found in the authoritative book

Equation (1.4) is not subject to the restriction of constant k Furthermore, when k is constant, it can

be integrated to yield Equation (1.1) Hence, Equation (1.4) is the general one-dimensional form of

Fourier’s law The negative sign is necessary because heat flows in the positive x-direction when the temperature decreases in the x-direction Thus, according to the standard sign convention that

q x is positive when the heat flow is in the positive x-direction, q x must be positive when dT /dx is

negative

It is often convenient to divide Equation (1.4) by the area to give:

ˆq x ≡ q x /A = −k dT

where ˆq x is the heat flux It has units of J/s · m2

= W/m2or Btu/h · ft2 Thus, the units of k are

W/m · K or Btu/h · ft ·◦F

Equations (1.1), (1.4), and (1.5) are restricted to the situation in which heat flows in the x-direction

only In the general case in which heat flows in all three coordinate directions, the total heat flux is

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obtained by adding vectorially the fluxes in the coordinate directions Thus,

where→ˆq is the heat flux vector and→i ,→j ,→k are unit vectors in the x-, y-, z-directions, respectively.

Each of the component fluxes is given by a one-dimensional Fourier expression as follows:

ˆq x = −k ∂T ∂x ˆq y = −k ∂T ∂y ˆq z = −k ∂T ∂z (1.7)Partial derivatives are used here since the temperature now varies in all three directions Substitutingthe above expressions for the fluxes into Equation (1.6) gives:

Example 1.1

The block of 304 stainless steel shown below is well insulated on the front and back surfaces, and

the temperature in the block varies linearly in both the x- and y-directions, find:

(a) The heat fluxes and heat flows in the x- and y-directions.

(b) The magnitude and direction of the heat flux vector

5°C

10°C

x y

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y

q

45°

1.3 The Heat Conduction Equation

The solution of problems involving heat conduction in solids can, in principle, be reduced to thesolution of a single differential equation, the heat conduction equation The equation can be derived

by making a thermal energy balance on a differential volume element in the solid For the case of

conduction only in the x-direction, such a volume element is illustrated in Figure 1.2 The balance

equation for the volume element is:

{rate of thermal energy in} − {rate of thermal energy out} + {net rate of thermal

energy generation} = {rate of accumulation of thermal energy} (1.10)The generation term appears in the equation because the balance is made on thermal energy, nottotal energy For example, thermal energy may be generated within a solid by an electric current

or by decay of a radioactive material

The rate at which thermal energy enters the volume element across the face at x is given by the product of the heat flux and the cross-sectional area, ˆq x

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of strength ˙q per unit volume, the net rate of generation is ˙qAx Finally, the rate of accumulation

is given by the time derivative of the thermal energy content of the volume element, which is

ρc(T − T ref )Ax, where T ref is an arbitrary reference temperature Thus, the balance equationbecomes:

∂y

k∂T

∂y

+ ∂

When k is constant, it can be taken outside the derivatives and Equation (1.11) can be

written as:

ρc k

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Table 1.1 The Heat Conduction Equation

k ∂T

∂y

+∂z∂

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(The partial derivative is replaced by a total derivative because x is the only independent variable

in the equation.) Integrating on both sides of the equation gives:

dT

dx = C1

A second integration gives:

T = C1x + C2Thus, it is seen that the temperature varies linearly across the solid The constants of integrationcan be found by applying the boundary conditions:

(1) At x = 0 T = T1

(2) At x = B T = T2

The first boundary condition gives T1= C2and the second then gives:

T2= C1B + T1

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Solving for C1we find:

It is seen that in this case of variable k, the temperature profile is not linear across the solid.

The constants of integration can be evaluated by applying the same boundary conditions as in theprevious example, although the algebra is a little more tedious The results are:

C2= aT1+bT

2 12

C1= a (T2− T1)

B +2B b (T22− T12)

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As before, the heat flow is found using Fourier’s law:

This equation is exactly the same as the one obtained above by solving the conduction equation

Hence, using Equation (1.1) with an average value of k gives the correct result This is a consequence

of the assumed linear relationship between k and T

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As shown in the sketch, the solid is in the form of a hollow cylinder and the outer and inner surfaces

are maintained at temperatures T1and T2, respectively The ends of the cylinder are insulated sothat heat can flow only in the radial direction There is no heat flow in the angular (φ) directionbecause the temperature is the same all the way around the circumference of the cylinder Thefollowing conditions apply:

(1) No heat flow in z-direction ⇒ ∂T

∂z = 0(2) Uniform temperature in φ-direction ⇒ ∂T

d dr

r dT dr

= 0Integrating once gives:

r dT

dr = C1Separating variables and integrating again gives:

T = C1ln r + C2

It is seen that, even with constant k, the temperature profile in curvilinear systems is nonlinear.

The boundary conditions for this case are:

ˆq r = −k dT

dr = −k C1

r = k(T1− T2)

r ln(R2/R1)The area across which the heat flows is:

A r = 2πrL where L is the length of the cylinder Thus,

q r = ˆq r A r = 2πkL(T1− T2)

ln(R2/R1)

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Notice that the heat-transfer rate is independent of radial position The heat flux, however, depends

on r because the cross-sectional area changes with radial position.

Example 1.5

The block shown in the diagram below is insulated on the top, bottom, front, back, and the side at

x = B The side at x = 0 is maintained at a fixed temperature, T1 Heat is generated within the block

at a rate per unit volume given by:

• Conduction only in x-direction

• Constant thermal conductivity

The appropriate form of the heat conduction equation is then:

dT

dx = Ŵe kγ −γx + C1

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A second integration yields:

T = − Ŵe −γx

kγ2 + C1x + C2The boundary conditions are:

(1) At x = 0 T = T1

(2) At x = B dT

dx = 0

The second boundary condition results from assuming zero heat flow through the insulated

boundary (perfect insulation) Thus, at x = L:

q x = −kA dT dx = 0 ⇒ dT dx = 0

This condition is applied using the equation for dT/dx resulting from the first integration:

0 = Ŵe kγ −γB + C1Hence,

C1= −Ŵe −γB

kγ Applying the first boundary condition to the equation for T :

T1= −Ŵe

(0)

kγ2 + C1(0) + C2Hence,

Now at steady state, all the heat generated in the block must flow out through the un-insulated side

at x = 0 Hence, the maximum temperature must occur at the insulated boundary, i.e., at x = B (This intuitive result can be confirmed by setting the first derivative of T equal to zero and solving for x.) Thus, setting x = B in the last equation gives:

T max = T1+ Ŵ

kγ2(1 − e −γB) −ŴBLe kγ −γBFinally, the solution is obtained by substituting the numerical values of the parameters:

T max= 20 + 10

0.5(0.1)2(1 − e−0.1) −10 × 1.0 e−0.1

0.5 × 0.1

T max∼= 29.4◦C

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The procedure illustrated in the above examples can be summarized as follows:

(1) Write down the conduction equation in the appropriate coordinate system

(2) Impose any restrictions dictated by the physical situation to eliminate terms that are zero ornegligible

(3) Integrate the resulting differential equation to obtain the temperature profile

(4) Use the boundary conditions to evaluate the constants of integration

(5) Use the appropriate form of Fourier’s law to obtain the heat flux

(6) Multiply the heat flux by the cross-sectional area to obtain the rate of heat transfer

In each of the above examples there is only one independent variable so that an ordinary differentialequation results In unsteady-state problems and problems in which heat flows in more than onedirection, a partial differential equation must be solved Analytical solutions are often possible ifthe geometry is sufficiently simple Otherwise, numerical solutions are obtained with the aid of acomputer

1.4 Thermal Resistance

The concept of thermal resistance is based on the observation that many diverse physicalphenomena can be described by a general rate equation that may be stated as follows:

Ohm’s Law of Electricity is one example:

In this case, the quantity that flows is electric charge, the driving force is the electrical potential

difference, E, and the resistance is the electrical resistance, R, of the conductor.

In the case of heat transfer, the quantity that flows is heat (thermal energy) and the driving force

is the temperature difference The resistance to heat transfer is termed the thermal resistance, and

is denoted by R th Thus, the general rate equation may be written as:

q = T R

th

(1.16)

In this equation, all quantities take on positive values only, so that q and T represent the absolute

values of the heat-transfer rate and temperature difference, respectively

An expression for the thermal resistance in a rectangular system can be obtained by comparingEquations (1.1) and (1.16):

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Table 1.2 Expressions for Thermal Resistance

A p= prime surface area

A f= fin surface area

ηf= fin efficiency

R th= ln(R2/R1)

These results, along with a number of others that will be considered subsequently, are summarized

in Table 1.2 When k cannot be assumed constant, the average thermal conductivity, as defined in

the previous section, should be used in the expressions for thermal resistance

The thermal resistance concept permits some relatively complex heat-transfer problems to besolved in a very simple manner The reason is that thermal resistances can be combined in thesame way as electrical resistances Thus, for resistances in series, the total resistance is the sum ofthe individual resistances:

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Figure 1.3 Heat transfer through a composite material.

In general, when thermal resistances occur in parallel, heat will flow in more than one direction

In Figure 1.3, for example, heat will tend to flow between materials B and C, and this flow will be normal to the primary direction of heat transfer In this case, the one-dimensional calculation of q

using Equations (1.16) and (1.22) represents an approximation, albeit one that is generally quiteacceptable for process engineering purposes

Example 1.6

A 5-cm (2-in.) schedule 40 steel pipe carries a heat-transfer fluid and is covered with a 2-cm layer of

calcium silicate insulation (k = 0.06 W/m · K) to reduce the heat loss The inside and outside pipe

diameters are 5.25 cm and 6.03 cm, respectively If the inner pipe surface is at 150◦C and the exteriorsurface of the insulation is at 25◦C, calculate:

(a) The rate of heat loss per unit length of pipe

(b) The temperature of the outer pipe surface

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ln

5.0153.015

2π × 0.06 = 0.000513 + 1.349723

= 1.350236 K/W

q r=1.350236125 ∼= 92.6 W/m of pipe(b) Writing Equation (1.16) for the pipe wall only:

q r=150 − T0

R pipe

92.6 = 150 − T00.000513

T0= 150 − 0.0475 ∼= 149.95◦C

Clearly, the resistance of the pipe wall is negligible compared with that of the insulation, andthe temperature difference across the pipe wall is a correspondingly small fraction of the totaltemperature difference in the system

It should be pointed out that the calculation in Example 1.6 tends to overestimate the rate ofheat transfer because it assumes that the insulation is in perfect thermal contact with the pipe wall.Since solid surfaces are not perfectly smooth, there will generally be air gaps between two adjacentsolid materials Since air is a very poor conductor of heat, even a thin layer of air can result in asubstantial thermal resistance This additional resistance at the interface between two materials

is called the contact resistance Thus, the thermal resistance in Example 1.5 should really bewritten as:

R th = R pipe + R insulation + R contact (1.25)The effect of the additional resistance is to decrease the rate of heat transfer according toEquation (1.16) Since the contact resistance is difficult to determine, it is often neglected

or a rough approximation is used For example, a value equivalent to an additional 5 mm ofmaterial thickness is sometimes used for the contact resistance between two pieces of thesame material [3] A more rigorous method for estimating contact resistance can be found inRef [4]

A slightly modified form of the thermal resistance, the R-value, is commonly used for insulations and other building materials The R-value is defined as:

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where B is the thickness of the material and k is its thermal conductivity Comparison with Equation (1.18) shows that the R-value is the thermal resistance, in English units, of a slab of material having

a cross-sectional area of 1 ft2 Since the R-value is always given for a specified thickness, the thermal conductivity of a material can be obtained from its R-value using Equation (1.26) Also, since R-values

are essentially thermal resistances, they are additive for materials arranged in series

Example 1.7

Triple-glazed windows like the one shown in the sketch below are often used in very cold

cli-mates Calculate the R-value for the window shown The thermal conductivity of air at normal room

temperature is approximately 0.015 Btu/h · ft ·◦F

0.08 in thick glass panes

0.25 in air gaps

The R-value for the window is obtained using the additive property for materials in series:

R-value = 3R glass + 2R air

= 3 × 0.0148 + 2 × 1.3889

R-value ∼= 2.8

1.5 The Conduction Shape Factor

The conduction shape factor is a device whereby analytical solutions to multi-dimensional heat duction problems are cast into the form of one-dimensional solutions Although quite restricted

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con-Table 1.3 Conduction Shape Factors (Source: Ref [5])

T2

T1 D1 D2

W

Case 5

Horizontal circular cylinder

of length L midway between

∞ ∞

∞

Case 6

Circular cylinder of length L

T1

T2

z D

d

⫹

(Continued)

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L D

Case 9

difference T1− T2across

the walls

L

L L

Case 10

semi-finite medium of thermal

conductivity k and T2

T2

T1D

0.930 ln(W /w) − 0.050

W w

T1

T2L

in scope, the shape factor method permits rapid and easy solution of multi-dimensional

heat-transfer problems when it is applicable The conduction shape factor, S, is defined by the

relation:

where T is a specified temperature difference Notice that S has the dimension of length Shape

factors for a number of geometrical configurations are given in Table 1.3 The solution of a problem

involving one of these configurations is thus reduced to the calculation of S by the appropriate

formula listed in the table

The thermal resistance corresponding to the shape factor can be found by comparing Equation(1.16) with Equation (1.27) The result is:

This is one of the thermal resistance formulas listed in Table 1.2 Since shape-factor problemsare inherently multi-dimensional, however, use of the thermal resistance concept in such cases

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will, in general, yield only approximate solutions Nevertheless, these solutions are usually entirelyadequate for process engineering calculations.

From Table 1.3, the shape factor for a buried horizontal cylinder is:

Suppose the pipeline of the previous example is covered with 1 in of magnesia insulation

(k = 0.07 W/m · K) What is the rate of heat loss per foot of pipe?

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The resistance of the earth is obtained by means of the shape factor for a buried horizontal cylinder.

In this case, however, the diameter of the cylinder is the diameter of the exterior surface of theinsulation Thus,

z = 2 ft = 24 in.

D = 12 + 2 = 14 in.

2z/D = 4814 =3.4286Therefore,

k ins= 0.07 × 0.57782 = 0.0404 Btu/h · ft ·◦F

Hence,

R insulation= ln(R 2πk2/R1)

ins L = ln(7/6)2π × 0.0404 × 1

= 0.6073 h ·◦F/Btu

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We consider first the case of a semi-infinite solid illustrated in Figure 1.4 The rectangular solid

occupies the region from x = 0 to x = ∞ The solid is initially at a uniform temperature, T0 At time

t = 0, the temperature of the surface at x = 0 is changed to T sand held at that value The temperature

within the solid is assumed to be uniform in the y- and z-directions at all times, so that heat flows only

in the x-direction This condition can be achieved mathematically by allowing the solid to extend

to infinity in the ±y- and ±z-directions If T s is greater that T0, heat will begin to penetrate into thesolid, so that the temperature at any point within the solid will gradually increase with time That

is, T = T (x, t), and the problem is to determine the temperature as a function of position and time.

Assuming no internal heat generation and constant thermal conductivity, the conduction equationfor this situation is:

1α

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The error function, erf, is defined by:

This function, which occurs in many diverse applications in engineering and applied science, can

be evaluated by numerical integration Values are listed in Table 1.4

Table 1.4 The Error Function

Tiêu đề	Process Heat Transfer Principles and Applications
Tác giả	R.W. Serth
Trường học	Department of Chemical and Natural Gas Engineering, Texas A&M University-Kingsville
Chuyên ngành	Process Heat Transfer Principles and Applications
Thể loại	Book
Năm xuất bản	2007
Thành phố	Kingsville

Định dạng
Số trang	770
Dung lượng	7,18 MB