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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
Understand the fundamentals of applied mathematics with this up-to-date introduction
Applied mathematics is the use of mathematical concepts and methods in various applied or practical areas, including engineering, computer science, and more. As engineering science expands, the ability to work from mathematical principles to solve and understand equations has become an ever more critical component of engineering fields. New engineering processes and materials place ever-increasing mathematical demands on new generations of engineers, who are looking more and more to applied mathematics for an expanded toolkit.
Applied Mathematics and Modeling for Chemical Engineers provides this toolkit in a comprehensive and easy-to-understand introduction. Combining classical analysis of modern mathematics with more modern applications, it offers everything required to assess and solve mathematical problems in chemical engineering. Now updated to reflect contemporary best practices and novel applications, this guide promises to situate readers in a 21st century chemical engineering field in which direct knowledge of mathematics is essential.
Readers of the third edition of Applied Mathematics and Modeling for Chemical Engineers will also find:
- Detailed treatment of ordinary differential equations (ODEs) and partial differential equations (PDEs) and their solutions
- New material concerning approximate solution methods like perturbation techniques and elementary numerical solutions
- Two new chapters dealing with Linear Algebra and Applied Statistics
Applied Mathematics and Modeling for Chemical Engineers isideal for graduate and advanced undergraduate students in chemical engineering and related fields, as well as instructors and researchers seeking a handy reference.
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Table of Contents
COVER
TITLE PAGE
COPYRIGHT
DEDICATION
PREFACE TO THE THIRD EDITION
PART I
1 FORMULATION OF PHYSICOCHEMICAL PROBLEMS
1.1 INTRODUCTION
1.2 ILLUSTRATION OF THE FORMULATION PROCESS (COOLING OF FLUIDS)
1.3 COMBINING RATE AND EQUILIBRIUM CONCEPTS (PACKED‐BED ADSORBER)
1.4 BOUNDARY CONDITIONS AND SIGN CONVENTIONS
1.5 SUMMARY OF THE MODEL BUILDING PROCESS
1.6 MODEL HIERARCHY AND ITS IMPORTANCE IN ANALYSIS
PROBLEMS
REFERENCES
2 MODELING WITH LINEAR ALGEBRA AND MATRICES
2.1 INTRODUCTION
2.2 BASIC CONCEPTS OF SYSTEMS OF LINEAR EQUATIONS
2.3 MATRIX NOTATION
2.4 MATRIX ALGEBRA AND CALCULUS OPERATIONS
2.5 PROBLEM 1: SOLUTION OF
N
EQUATIONS IN
N
UNKNOWNS
2.6 PROBLEM 2: THE MATRIX EIGENVALUE PROBLEM
2.7 SINGULAR SYSTEMS
2.8 COMPUTATIONAL LINEAR ALGEBRA
PROBLEMS
REFERENCES
3 SOLUTION TECHNIQUES FOR MODELS YIELDING ORDINARY DIFFERENTIAL EQUATIONS
3.1 GEOMETRIC BASIS AND FUNCTIONALITY
3.2 CLASSIFICATION OF ODE
3.3 FIRST‐ORDER EQUATIONS
3.4 SOLUTION METHODS FOR SECOND‐ORDER NONLINEAR EQUATIONS
3.5 LINEAR EQUATIONS OF HIGHER ORDER
3.6 COUPLED SIMULTANEOUS ODE
3.7 EIGENPROBLEMS
3.8 COUPLED LINEAR DIFFERENTIAL EQUATIONS
3.9 SUMMARY OF SOLUTION METHODS FOR ODE
PROBLEMS
REFERENCES
4 SERIES SOLUTION METHODS AND SPECIAL FUNCTIONS
4.1 INTRODUCTION TO SERIES METHODS
4.2 PROPERTIES OF INFINITE SERIES
4.3 METHOD OF FROBENIUS
4.4 SUMMARY OF THE FROBENIUS METHOD
4.5 SPECIAL FUNCTIONS
PROBLEMS
REFERENCES
5 INTEGRAL FUNCTIONS
5.1 INTRODUCTION
5.2 THE ERROR FUNCTION
5.3 THE GAMMA AND BETA FUNCTIONS
5.4 THE ELLIPTIC INTEGRALS
5.5 THE EXPONENTIAL AND TRIGONOMETRIC INTEGRALS
PROBLEMS
REFERENCES
6 STAGED‐PROCESS MODELS: THE CALCULUS OF FINITE DIFFERENCES
6.1 INTRODUCTION
6.2 SOLUTION METHODS FOR LINEAR FINITE DIFFERENCE EQUATIONS
6.3 PARTICULAR SOLUTION METHODS
6.4 NONLINEAR EQUATIONS (RICCATI EQUATION)
PROBLEMS
REFERENCES
7 PROBABILITY AND STATISTICAL MODELING
7.1 CONCEPTS AND RESULTS FROM PROBABILITY THEORY
7.2 CONCEPTS AND RESULTS FROM MATHEMATICAL STATISTICS
7.3 STATISTICAL ANALYSIS AND MODELING
PROBLEMS
REFERENCES
8 APPROXIMATE SOLUTION METHODS FOR ODE: PERTURBATION METHODS
8.1 PERTURBATION METHODS
8.2 THE BASIC CONCEPTS
8.3 THE METHOD OF MATCHED ASYMPTOTIC EXPANSION
8.4 MATCHED ASYMPTOTIC EXPANSIONS FOR COUPLED EQUATIONS
PROBLEMS
REFERENCES
NOTE
PART II
9 NUMERICAL SOLUTION METHODS (INITIAL VALUE PROBLEMS)
9.1 INTRODUCTION
9.2 TYPE OF METHOD
9.3 STABILITY
9.4 STIFFNESS
9.5 INTERPOLATION AND QUADRATURE
9.6 EXPLICIT INTEGRATION METHODS
9.7 IMPLICIT INTEGRATION METHODS
9.8 PREDICTOR–CORRECTOR METHODS AND RUNGE–KUTTA METHODS
9.9 RUNGE–KUTTA METHODS
9.10 EXTRAPOLATION
9.11 STEP SIZE CONTROL
9.12 HIGHER‐ORDER INTEGRATION METHODS
PROBLEMS
REFERENCES
10 APPROXIMATE METHODS FOR BOUNDARY VALUE PROBLEMS: WEIGHTED RESIDUALS
10.1 THE METHOD OF WEIGHTED RESIDUALS
10.2 JACOBI POLYNOMIALS
10.3 LAGRANGE INTERPOLATION POLYNOMIALS
10.4 ORTHOGONAL COLLOCATION METHOD
10.5 LINEAR BOUNDARY VALUE PROBLEM: DIRICHLET BOUNDARY CONDITION
10.6 LINEAR BOUNDARY VALUE PROBLEM: ROBIN BOUNDARY CONDITION
10.7 NONLINEAR BOUNDARY VALUE PROBLEM: DIRICHLET BOUNDARY CONDITION
10.8 ONE‐POINT COLLOCATION
10.9 SUMMARY OF COLLOCATION METHODS
10.10 CONCLUDING REMARKS
PROBLEMS
REFERENCES
NOTES
11 INTRODUCTION TO COMPLEX VARIABLES AND LAPLACE TRANSFORMS
11.1 INTRODUCTION
11.2 ELEMENTS OF COMPLEX VARIABLES
11.3 ELEMENTARY FUNCTIONS OF COMPLEX VARIABLES
11.4 MULTIVALUED FUNCTIONS
11.5 CONTINUITY PROPERTIES FOR COMPLEX VARIABLES: ANALYTICITY
11.6 INTEGRATION: CAUCHY'S THEOREM
11.7 CAUCHY'S THEORY OF RESIDUES
11.8 INVERSION OF LAPLACE TRANSFORMS BY CONTOUR INTEGRATION
11.9 LAPLACE TRANSFORMATIONS: BUILDING BLOCKS
11.10 PRACTICAL INVERSION METHODS
11.11 APPLICATIONS OF LAPLACE TRANSFORMS FOR SOLUTIONS OF ODE
11.12 INVERSION THEORY FOR MULTIVALUED FUNCTIONS: THE SECOND BROMWICH PATH
11.13 NUMERICAL INVERSION TECHNIQUES
PROBLEMS
REFERENCES
12 SOLUTION TECHNIQUES FOR MODELS PRODUCING PDES
12.1 INTRODUCTION
12.2 PARTICULAR SOLUTIONS FOR PDES
12.3 COMBINATION OF VARIABLES METHOD
12.4 SEPARATION OF VARIABLES METHOD
12.5 ORTHOGONAL FUNCTIONS AND STURM–LIOUVILLE CONDITIONS
12.6 INHOMOGENEOUS EQUATIONS
12.7 APPLICATIONS OF LAPLACE TRANSFORMS FOR SOLUTIONS OF PDES
PROBLEMS
REFERENCES
NOTES
13 TRANSFORM METHODS FOR LINEAR PDEs
13.1 INTRODUCTION
13.2 TRANSFORMS IN FINITE DOMAIN: STURM–LIOUVILLE TRANSFORMS
13.3 GENERALIZED STURM–LIOUVILLE INTEGRAL TRANSFORM
PROBLEMS
REFERENCES
14 APPROXIMATE AND NUMERICAL SOLUTION METHODS FOR PDES
14.1 POLYNOMIAL APPROXIMATION
14.2 SINGULAR PERTURBATION
14.3 FINITE DIFFERENCE
14.4 ORTHOGONAL COLLOCATION FOR SOLVING PDES
PROBLEMS
REFERENCES
NOTE
APPENDIX A: REVIEW OF METHODS FOR NONLINEAR ALGEBRAIC EQUATIONS
A.1. THE BISECTION ALGORITHM
A.2. THE SUCCESSIVE SUBSTITUTION METHOD
A.3. THE NEWTON–RAPHSON METHOD
A.4. RATE OF CONVERGENCE
A.5. MULTIPLICITY
A.6. ACCELERATING CONVERGENCE
REFERENCES
APPENDIX B: DERIVATION OF THE FOURIER–MELLIN INVERSION THEOREM
REFERENCES
NOTE
APPENDIX C: TABLE OF LAPLACE TRANSFORMS*
NOTE
APPENDIX D: NUMERICAL INTEGRATION
D.1. BASIC IDEA OF NUMERICAL INTEGRATION
D.2. NEWTON FORWARD DIFFERENCE POLYNOMIAL
D.3. BASIC INTEGRATION PROCEDURE
D.4. ERROR CONTROL AND EXTRAPOLATION
D.5. GAUSSIAN QUADRATURE
D.6. RADAU QUADRATURE
D.7. LOBATTO QUADRATURE
D.8. CONCLUDING REMARKS
REFERENCES
APPENDIX E: NOMENCLATURE
APPENDIX F: STATISTICAL TABLES
POSTFACE
INDEX
END USER LICENSE AGREEMENT
List of Tables
Chapter 4
TABLE 4.1 Selected Values for Bessel Functions
TABLE 4.2 Zeros for
J
n
(
x
); Values of
x
to Produce
J
n
(
x
) = 0...
TABLE 4.3 Values of
x
to Satisfy
x J
1
(
x
) =
N J
0
(
x
)
Chapter 5
TABLE 5.1 Selected Values for Error Function
TABLE 5.2 Selected Values for Gamma Function
TABLE 5.3 Selected Values for Complete Elliptic Integrals k = sinϕ...
Chapter 7
TABLE 7.1 Probability Mass Function for the Number, x of subsystems out of ...
TABLE 7.2 Data Organization and Basic Sample Statistics Used for the Hypoth...
TABLE 7.3 Analysis of Variance (ANOVA) Table for Summarizing the Computatio...
TABLE 7.4 Analysis of Variance (ANOVA) Table for Summarizing the Computatio...
Chapter 9
TABLE 9.1 Comparison of Stability Behavior
Chapter 10
TABLE 10.1 Comparison of Accuracy for Approximate Solution
TABLE 10.2 Computations Using Orthogonal Collocation: Diffusion in Catalyst...
TABLE 10.3 Computations Using Orthogonal Collocation: Effect of Boundary Re...
TABLE 10.4 Computations Using Orthogonal Collocation: Nonlinear Reaction Ki...
Chapter 11
TABLE 11.1 Transforms of Differentials and Products
TABLE 11.2 Set of Five Constants for
α
and
K
for the Zakian Method
TABLE 11.3 Comparison Between the Numerical Inverse Obtained by the Zakian M...
TABLE 11.4 Comparison Between the Numerical Inverse Obtained by the Zakian M...
TABLE 11.5 Comparison Between the Numerical Inverse Obtained by the Zakian M...
TABLE 11.6 Comparison Between the Numerical Inverse Obtained by the Fourier ...
TABLE 11.7 Comparison Between the Numerical Inverse Obtained by the Fourier ...
TABLE 11.8 Comparison Between the Numerical Inverse Obtained by the Fourier ...
Chapter 12
TABLE 12.1 Eigenvalues for .
Chapter 14
TABLE 14.1 Comparison Between the Approximate and the Exact Solutions
TABLE 14.2 Collocation Formulas
Appendix D
TABLE D.1 Quadrature Points and Weights for Gaussian Quadrature.
TABLE D.2 Quadrature Points and Weights for Laguerre Quadrature for N = 3 an...
TABLE D.3 Quadrature Points and Weights for Hermite Quadrature for N = 5 and...
TABLE D.4 Quadrature Points and Weights for Radau Quadrature with the Last E...
TABLE D.5 Quadrature Points and Weights for Radau Quadrature with the First ...
TABLE D.6 Quadrature Points and Weights for Radau Quadrature with the Last E...
TABLE D.7 Quadrature Points and Weights for Radau Quadrature with the First ...
TABLE D.8 Quadrature Points and Weights for Lobatto Quadrature: Weighting Fu...
TABLE D.9 Quadrature Points and Weights for Lobatto Quadrature: Weighting Fu...
Appendix F
TABLE F.1 Cumulative Distribution Function for the Standard Normal Distribut...
TABLE F.2 Cumulative Distribution Function Values for the t‐distribution for...
TABLE F.3 Upper‐Tail Probability Values of the f‐distribution for Given Prob...
TABLE F.3a
p
= 0.10
TABLE F.3b
p
= 0.05
TABLE 0
p
= 0.01
List of Illustrations
Chapter 1
FIGURE 1.1 (a) Sketch of plug flow model formulation. (b) Elemental or contr...
FIGURE 1.2 Packed‐bed adsorber.
FIGURE 1.3 Expected temperature profile for cooling fluids in a pipe at an a...
FIGURE 1.4 Schematic diagram of heat removal from a solvent bath.
FIGURE 1.5 Shell element and the system coordinate.
FIGURE 1.6 Plot of the effectiveness factor versus
mL
2
.
FIGURE 1.7 Schematic diagram of shell for heat balance.
Chapter 2
FIGURE 2.1 Classification of non‐square systems of linear equations. (a) Ove...
FIGURE 2.2 Upper‐triangular forms of the augmented matrix
[A|B]
for non‐squa...
Chapter 3
FIGURE 3.1 Expected behavior of
C
A
(
z
).
FIGURE 3.2 Composition profiles in catalytic reactor.
FIGURE 3.3 Double‐pipe heat exchanger.
Chapter 4
FIGURE 4.1 Plots of
J
0
(
x
),
J
1
(
x
),
Y
0
(
x
),
I
0
(
x
), and
K
0
(
x
).
FIGURE 4.2 (a) Pin promoters attached to heat exchange surface. (b) Geometry...
Chapter 5
FIGURE 5.1 Pendulum problem.
FIGURE 5.2 (a) Plot of Ei(
x
) versus
x
. (b) Plot of Si(
x
) and Ci(
x
) versus
x
....
Chapter 6
FIGURE 6.1 Continuous countercurrent extraction cascade.
FIGURE 6.2 Graphical stage‐to‐stage calculations.
Chapter 7
FIGURE 7.1 The normal probability distribution function (Eq. 7.10) for diffe...
FIGURE 7.2 Selected
t
‐distributions for various degrees of freedom (
a
).
FIGURE 7.3 A selection of
f
‐distributions for various combinations of numera...
FIGURE 7.4 Data and least‐squares line fit to the data showing the effect of...
Chapter 8
FIGURE 8.1 Plots of the exact solution and the zero‐order outer solution (Eq...
FIGURE 8.2 Plots of the exact solution and the zero‐order composite solution...
FIGURE 8.3 Plots of the exact solution and the first‐order composite solutio...
FIGURE 8.4 Plots of the numerically exact solution (continuous line) and the...
Chapter 9
FIGURE 9.1 Schematic diagram of the plug flow reactor.
FIGURE 9.2 Plots of numerical solutions using the explicit Euler method.
FIGURE 9.3 Plots of the numerical solutions for the backward Euler method.
FIGURE 9.4 Computations of the numerical solutions for the trapezoidal metho...
FIGURE 9.5 Relative errors between the numerical solutions and the exact sol...
FIGURE 9.6 Plots of
y
1
,
y
2
(Eqs. 9.65a, b) versus time.
FIGURE 9.7 Graphical representation of Euler method and local and global tru...
Chapter 10
FIGURE 10.1 Diffusion and reaction in a slab catalyst.
FIGURE 10.2 Comparison of concentration profiles for weighted residual metho...
FIGURE 10.3 Plots of three Jacobi polynomials.
FIGURE 10.4 Typical plots of the Lagrangian interpolation polynomials.
FIGURE 10.5 Concentration profiles for
φ
= 10, illustrating advantages ...
FIGURE 10.6 Concentration profiles for
φ
= 100, with number of collocat...
FIGURE 10.7 Plots of concentration profiles for
φ
= 10.
FIGURE 10.8 Approximate concentration profiles for second‐order chemical kin...
Chapter 11
FIGURE 11.1 Representation of
s
in the complex plane.
FIGURE 11.2 Curve in complex plane.
FIGURE 11.3 Closed contour,
s
1
→
s
2
.
FIGURE 11.4 First Bromwich path for pole singularities.
FIGURE 11.5 Unit step function at
t
= 0.
FIGURE 11.6 Delayed unit step function.
FIGURE 11.7 Delayed ramp function.
FIGURE 11.8 Simulation of
δ
(
t
).
FIGURE 11.9 Inlet composition disturbance for Example 11.13.
FIGURE 11.10 Response of
c
A
(
t
) to a step change in .
FIGURE 11.11 Impulse response of CSTR.
FIGURE 11.12 Eliminating multivalued behavior by restricting
θ
to 0 ≤
θ
...
FIGURE 11.13 Contour
C
2
(second Bromwich path), which does not enclose branc...
FIGURE 11.14 Alternative Br
2
path.
Chapter 12
FIGURE 12.1 Plot of (
T
−
T
s
)/(
T
0
−
T
s
) versus .
FIGURE 12.2 Schematic diagram of CVD reactor.
FIGURE 12.3 Schematic diagram of a coated wall reactor.
FIGURE 12.4 Plot of Eq. 12.213 versus
ζ
.
FIGURE 12.5 Temperature deviation at pellet center.
FIGURE 12.6 Response curves at exit of packed bed adsorber,
L
= 100 cm,
ε
...
FIGURE 12.7 Dimensionless temperature profiles for plug flow heat exchanger....
FIGURE 12.8 Solution to Example 12.8, with and .
FIGURE 12.9 Dimensionless center temperature
ψ
=
(T(0, τ) − Tf)/(Ts − T
...
FIGURE 12.10 Loshmilt diffusion cell response curve as plots of the average ...
FIGURE 12.11 Exit composition from kidney dialysis module.
FIGURE 12.12 Start‐up flow in tube.
FIGURE 12.13 Sand molds for smelters.
FIGURE 12.14 Membrane solubility experiment.
FIGURE 12.15 Pressure response for membrane gas solubilty.
FIGURE 12.16 Dissolution of spheres in finite volume.
FIGURE 12.17 Wetted wall tower.
FIGURE 12.18 Oil extraction from seeds.
Chapter 13
FIGURE 13.1 Schematic diagram of the transform pair.
FIGURE 13.2 Temperature profile in a slab object.
FIGURE 13.3 Mapping diagram.
FIGURE 13.4 Temperature profiles in a cylinder.
FIGURE 13.5 Plot of the mean concentration (Eq. 13.134) versus
τ
.
FIGURE 13.6 Plot of
Q
4
/
Q
1
versus Δ with Biot as parameter.
FIGURE 13.7 Batch adsorber.
FIGURE 13.8 Shell element of adsorbent particle.
FIGURE 13.9 Rule 1 of the generalized integral transform.
FIGURE 13.10 Rule 2 of the generalized integral transform.
FIGURE 13.11 Rule 3 of the generalized integral transform.
FIGURE 13.12 Plots of LHS and RHS of the transcendental Eq. 13.198 versus ....
FIGURE 13.13 Plots of
A
b
versus nondimensional
τ
with
B
as the varying ...
Chapter 14
FIGURE 14.1 Plots of exact and approximate average concentrations.
FIGURE 14.2 Exact and approximate concentration profiles.
FIGURE 14.3 Exact concentration profiles at
τ
= 0.005, 0.01, 0.015, 0.0...
FIGURE 14.4 Plots of the mean concentration versus
τ
.
FIGURE 14.5 Plots of the average mass transfer rate versus .
FIGURE 14.6 Schematic diagram of the two subdomains.
FIGURE 14.7 Behavior of the inner solutions.
FIGURE 14.8 Plots of uptake versus time.
FIGURE 14.9 Plots of
y
1
=
y
(
x
= 0.2,
t
) versus time for (
a
) forward differen...
FIGURE 14.10 Plots of
I
versus
τ
for
N
= 2, 3, 5, and 10.
FIGURE 14.11 Locations of the discrete points.
Appendix A
FIGURE A.1 Graphical representation of the bisection technique.
FIGURE A.2 Graphical representation of one‐dimensional contraction mapping....
FIGURE A.3 Graphical representation of the Newton–Raphson method.
FIGURE A.4 Graphical representation of the secant method.
Appendix D
FIGURE D.1 Graphical representation of the trapezoid rule integration.
FIGURE D.2 Selection of subintervals for trapezoid rule.
Guide
Cover
Table of Contents
Title Page
Copyright
Dedication
Preface to the Third Edition
Begin Reading
Appendix A: Review of Methods for Nonlinear Algebraic Equations
Appendix B: Derivation of the Fourier–Mellin Inversion Theorem
Appendix C: Table of Laplace Transforms
Appendix D: Numerical Integration
Appendix E: Nomenclature
Appendix F: Statistical Tables
Postface
Index
End User License Agreement
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APPLIED MATHEMATICS AND MODELING FOR CHEMICAL ENGINEERS
Third Edition
RICHARD G. RICEEmeritus Professor (LSU)Punta Gorda, Florida, USA
DUONG D. DOEmeritus Professor (University of Queensland)Middle Park, Queensland, Australia
JAMES E. MANEVALProfessor, Department of Chemical EngineeringBucknell UniversityLewisburg, Pennsylvania, USA
This edition first published 2023.© 2023 John Wiley & Sons, Inc.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.
The right of Richard G. Rice, Duong D. Do and James E. Maneval to be identified as the authors of this work has been asserted in accordance with law.
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Library of Congress Cataloging‐in‐Publication Data Applied forHB ISBN: 9781119833857
Cover Design: WileyCover Image: Courtesy P.J Jones
The third edition is dedicated to the many talented students I taught at the University of Queensland, Australia. Three of these are particularly noteworthy, first, to my coauthor Professor Duong D. Do, who as a student, and later as a teacher, was the best there ever was. Next, to Dr. Peter J. Jones, a University Gold Medalist who rose through the ranks to become a V.P. of Shell International and in retirement found a new career as an artist, as evidenced by the cover of this book, which he painted as a theme for the poem by the famous Australian poet Banjo Patterson, “Clancy of the Overflow,” used to start Part One of this book. Finally, to Andrew N. Liveris, who became a powerful captain of industry as CEO of Dow Chemical and who never forgot where he came from by endowing the University with a new Chemical Engineering Building, now named after him.
Good on you, lads, and so long it's been good to know you.
RGR
PREFACE TO THE THIRD EDITION
Textbooks must be continuously updated to meet modern demands by teachers, students, and professionals who use the book. To that end, we have revised and added new material for the Third Edition to teach a new Chapter 2: Modeling with Linear Algebra and Matrices and a new Chapter 7: Probability and Statistical Modeling. A number of teachers wanted a more compact treatment of Linear Algebra, and there was also a need to teach how to handle Big Data for modern AI applications. We have revised most of the chapters in Part II in an effort to condense the advanced material. Part I now comprises Chapters 1 through 6, which can be used for a one or two semester course for junior or senior students in chemical engineering. Part II comprises Chapters 9–14 and was mainly intended for first‐year graduate students, but many teachers will find Chapter 7 on Probability and Statistical Modeling and Chapter 8 on Perturbation Methods attractive and suitable for both graduate and undergraduate students. New example problems have been added and solved to stimulate students to engage in the many homework problems provided. Most problems have a final solution given at the end of the problem. Complete analysis and formal solutions can be found in the Solution Manual for the Third Edition, which will be published and sold at the same time as the book. As in the previous editions, challenges in homework problems are given a subscript, rating the degree of difficulty: subscript 1 denotes mainly computational and setup problems, while subscripts 2 and 3 require more derivation, analysis, and background. Problems with an asterisk are the most challenging and are more suitable for graduate students.
We continue to be grateful to colleagues and students for pointing out typos, errors, and suggestions for new topics, so we look forward to your communications.
RICHARD G. RICE, [email protected] Gorda, FL, USA
DUONG D. DO, [email protected] Park, Queensland, Australia
JAMES E. MANEVAL, [email protected], PA, USA
PART I
I had written him a letter which I had, for want of better
Knowledge, sent to where I met him down the Lachlan years ago;
He was shearing when I knew him, so I sent the letter to him,
Just on spec, addressed as follows, “Clancy of The Overflow.”
And an answer came directed in a writing unexpected
(And I think the same was written with a thumb‐nail dipped in tar)
‘Twas his shearing mate who wrote it, and verbatim I will quote it:
“Clancy’s gone to Queensland droving, and we don't know where he are.”
In my wild erratic fancy visions came to me of Clancy
Gone a‐droving “down the Cooper” where the Western drovers go;
As the stock are slowly stringing, Clancy rides behind them singing,
For the drover's life has pleasures that the townfolk never know.
And the bush has friends to meet him, and their kindly voices greet him
In the murmur of the breezes and the river on its bars,
And he sees the vision splendid of the sunlit plains extended,
And at night the wondrous glory of the everlasting stars.
I am sitting in my dingy little office, where a stingy
Ray of sunlight struggles feebly down between the houses tall,
And the foetid air and gritty of the dusty, dirty city,
Through the open window floating, spreads its foulness over all.
And in place of lowing cattle, I can hear the fiendish rattle
Of the tramways and the buses making hurry down the street;
And the language uninviting of the gutter children fighting
Comes fitfully and faintly through the ceaseless tramp of feet.
And the hurrying people daunt me, and their pallid faces haunt me
As they shoulder one another in their rush and nervous haste,
With their eager eyes and greedy, and their stunted forms and weedy,
For townfolk have no time to grow, they have no time to waste.
And I somehow rather fancy that I'd like to change with Clancy,
Like to take a turn at droving where the seasons come and go,
While he faced the round eternal of the cash‐book and the journal—
But I doubt he'd suit the office, Clancy, of The Overflow.
—Banjo Patterson (“Clancy of the Overflow”)
1FORMULATION OF PHYSICOCHEMICAL PROBLEMS
1.1 INTRODUCTION
Modern science and engineering require high levels of qualitative logic before the act of precise problem formulation can occur. Thus, much is known about a physicochemical problem beforehand, derived from experience or experiment (i.e., empiricism). Most often, a theory evolves only after detailed observation of an event. This first step usually involves drawing a picture of the system to be studied.
The second step is the bringing together of all applicable physical and chemical information, conservation laws, and rate expressions. At this point, the engineer must make a series of critical decisions about the conversion of mental images to symbols and, at the same time, how detailed the model of a system must be. Here, one must classify the real purposes of the modeling effort. Is the model to be used only for explaining trends in the operation of an existing piece of equipment? Is the model to be used for predictive or design purposes? Do we want steady‐state or transient response? The scope and depth of these early decisions will determine the ultimate complexity of the final mathematical description.
The third step requires the setting down of finite or differential volume elements, followed by writing the conservation laws. In the limit, as the differential elements shrink, then differential equations arise naturally. Next, the problem of boundary and initial conditions must be addressed, and this aspect must be treated with considerable circumspection.
When the problem is fully posed in quantitative terms, an appropriate mathematical solution method is sought out, which finally relates dependent (responding) variables to one or more independent (changing) variables. The final result may be an elementary mathematical formula or a numerical solution portrayed as an array of numbers.
1.2 ILLUSTRATION OF THE FORMULATION PROCESS (COOLING OF FLUIDS)
We illustrate the principles outlined above and the hierarchy of model building by way of a concrete example: the cooling of a fluid flowing in a circular pipe. We start with the simplest possible model, adding complexity as the demands for precision increase. Often, the simple model will suffice for rough, qualitative purposes. However, certain economic constraints weigh heavily against overdesign, so predictions and designs based on the model may need be more precise. This section also illustrates the “need to know” principle, which acts as a catalyst to stimulate the garnering together of mathematical techniques. The problem posed in this section will appear repeatedly throughout the book, as more sophisticated techniques are applied to its complete solution.
1.2.1 Model I: Plug Flow
As suggested in the beginning, we first formulate a mental picture and then draw a sketch of the system. We bring together our thoughts for a simple plug flow model in Figure 1.1a. One of the key assumptions here is plug flow, which means that the fluid velocity profile is plug‐shaped, in other words, uniform at all radial positions. This almost always implies turbulent fluid flow conditions, so that fluid elements are well mixed in the radial direction; hence, the fluid temperature is fairly uniform in a plane normal to the flow field (i.e., the radial direction).
FIGURE 1.1 (a) Sketch of plug flow model formulation. (b) Elemental or control volume for plug flow model. (c) Control volume for Model II.
If the tube is not too long or the temperature difference is not too severe, then the physical properties of the fluid will not change much, so our second step is to express this and other assumptions as a list:
A steady‐state solution is desired.
The physical properties (
ρ
, density;
C
p
, specific heat;
k
, thermal conductivity, etc.) of the fluid remain constant.
The wall temperature is constant and uniform (i.e., does not change in the
z
or
r
direction) at a value
T
w
.
The inlet temperature is constant and uniform (does not vary in
r
direction) at a value
T
0
, where
T
0
>
T
w
.
The velocity profile is plug‐shaped or flat and constant at a value of
υ
0
; hence, it is uniform
with respect to
(
wrt
)
z
or
r
.
The fluid is well mixed (highly turbulent), so the temperature is uniform in the radial direction.
Thermal conduction of heat along the axis is small relative to convection.
The heat transfer coefficient
h
at the wall is taken to be constant.
The third step is to sketch, and act upon, a differential volume element of the system (in this case, the flowing fluid) to be modeled. We illustrate this elemental volume in Figure 1.1b, which is sometimes called the “control volume,” which has a volume A (Δz), where A is tube cross‐sectional area.
We act upon this elemental volume, which spans the whole of the tube cross section, by writing the general conservation law
Since steady state is stipulated, the accumulation of heat is zero. Moreover, there are no chemical, nuclear, or electrical sources specified within the volume element, so heat generation is absent. The only way heat can be exchanged is through the perimeter of the element by way of the temperature difference between wall and fluid. The incremental rate of heat removal can be expressed as a positive quantity using Newton's law of cooling, that is,
As a convention, we shall express all such rate laws as positive quantities, invoking positive or negative signs as required when such expressions are introduced into the conservation law (Eq. 1.1). The contact area in this simple model is simply the perimeter of the element times its length.
The constant heat transfer coefficient is denoted by h. We have placed a bar over T to represent the average between T(z) and T(z + Δz)
In the limit, as Δz → 0, we see
Now, along the axis, heat can enter and leave the element only by convection (flow), so we can write the elemental form of Eq. 1.1 as
The first two terms are simply mass flow rate times local enthalpy, where the reference temperature for enthalpy is taken as zero. Had we used Cp (T − Tref) for enthalpy, the term Tref would be canceled in the elemental balance. The last step is to invoke the fundamental lemma of calculus, which defines the act of differentiation
We rearrange the conservation law into the form required for taking limits and then divide by Δz:
Taking limits, one at a time, then yields the sought after differential equation
where we have canceled the negative signs.
Before solving this equation, it is good practice to group parameters into a single term (lumping parameters). For such elementary problems, it is convenient to lump parameters with the lowest‐order term as follows:
where
It is clear that λ must take units of reciprocal length.
As it stands, the above equation is classified as a linear, inhomogeneous equation of first order, which in general must be solved using the so‐called integrating factor method, as we discuss later in Section 3.3.
Nonetheless, a little common sense will allow us to obtain a final solution without any new techniques. To do this, we remind ourselves that Tw is everywhere constant and that differentiation of a constant is always zero, so we can write
This suggests we define a new dependent variable, namely,
hence Eq. 1.9 now reads simply
This can be integrated directly by separation of variables, so we rearrange to get
Integrating term by term yields
where ln K is any (arbitrary) constant of integration. Using logarithm properties, we can solve directly for θ
It now becomes clear why we selected the form ln K as the arbitrary constant in Eq. 1.14.
All that remains is to find a suitable value for K. To do this, we recall the boundary condition denoted as T0 in Figure 1.1a, which in mathematical terms has the meaning
Thus, when z = 0, θ (0) must take a value T0 − Tw, so K must also take this value.
Our final result for computational purposes:
We note that all arguments of mathematical functions must be dimensionless, so the above result yields a dimensionless temperature
and a dimensionless length scale
Thus, a problem with six parameters, two external conditions (T0, Tw), and one each dependent and independent variable has been reduced to only two elementary (dimensionless) variables, connected as follows:
1.2.2 Model II: Parabolic Velocity
In the development of Model I (plug flow), we took careful note that the assumptions used in this first model building exercise implied “turbulent flow” conditions, such a state being defined by the magnitude of the Reynolds number (υ0d/v), where d is tube diameter and ν = μ/ρ, this number must always exceed 2100 for this model to be applicable. For slower flows, the velocity is no longer plug‐shaped, and in fact when Re < 2100, the shape is parabolic as follows:
where υ0 now denotes the average velocity, and υz denotes the locally varying value (Bird et al. 1960). Under such conditions, our earlier assumptions must be carefully reassessed; specifically, we will need to modify items 5–7 in the previous list:
The
z
‐directed velocity profile is parabolic and depends on the position
r
.
The fluid is not well mixed in the radial direction, so account must be taken of radial heat conduction.
Because convection is smaller, axial heat conduction may also be important.
These new physical characteristics cause us to redraw the elemental volume as shown in Figure 1.1c. The control volume now takes the shape of a ring of thickness Δr and length Δz. Heat now crosses two surfaces, the annular area normal to fluid flow, and the area along the perimeter of the ring. We shall need to designate additional (vector) quantities to represent heat flux (rate per unit normal area) by molecular conduction:
The net rate of heat gain (or loss) by conduction is simply the flux times the appropriate area normal to the flux direction. The conservation law (Eq. 1.1) can now be written for the element shown in Figure 1.1c.
The new notation is necessary, since we must deal with products of terms, either or both of which may be changing.
We rearrange this to a form appropriate for the fundamental lemma of calculus. However, since two position coordinates are now allowed to change, we must define the process of partial differentiation, for example,
which, of course, implies holding r constant as denoted by subscript (we shall delete this notation henceforth). Thus, we divide Eq. 1.24 by 2πΔzΔr and rearrange to get
Taking limits, one at a time, then yields
The derivative wrt z implies holding r constant, so r can be placed outside this term; thus, dividing by r and rearranging shows
At this point, the equation is insoluble since we have one equation and three unknowns (T, qz, qr). We need to know some additional rate law to connect fluxes q to temperature T. Therefore, it is now necessary to introduce the famous Fourier's law of heat conduction, the vector form of which states that heat flux is proportional to the gradient in temperature
and the two components of interest here are
Inserting these two new equations into Eq. 1.28, along with the definition of υz, yields finally a single equation, with one unknown T(r,z)
The complexity of Model II has now exceeded our poor powers of solution, since we have much that we need to know before attempting such second‐order partial differential equations (PDEs). We shall return to this problem occasionally as we learn new methods to effect a solution and as new approximations become evident.
1.3 COMBINING RATE AND EQUILIBRIUM CONCEPTS (PACKED‐BED ADSORBER)
The occurrence of a rate process and a thermodynamic equilibrium state is common in chemical engineering models. Thus, certain parts of a whole system may respond so quickly that, for practical purposes, local equilibrium may be assumed. Such an assumption is an integral (but often unstated) part of the qualitative modeling exercise.
To illustrate the combination of rate and equilibrium principles, we next consider a widely used separation method, which is inherently unsteady, packed‐bed adsorption. We imagine a packed bed of finely granulated (porous) solid (e.g., charcoal) contacting a binary mixture, one component of which selectively adsorbs (physisorption) onto and within the solid material. The physical process of adsorption is so fast, relative to other slow steps (diffusion within the solid particle), that in and near the solid particles, local equilibrium exists
where q denotes the average composition of the solid phases, expressed as moles solute adsorbed per unit volume solid particle, and C* denotes the solute composition (moles solute per unit volume fluid), which would exist at equilibrium. We suppose that a single film mass transport coefficient controls the transfer rate between flowing and immobile (solid) phase.
It is also possible to use the same model even when intraparticle diffusion is important (Rice 1982) by simply replacing the film coefficient with an “effective” coefficient. Thus, the model we derive can be made to have wide generality.
We illustrate a sketch of the physical system in Figure 1.2. It is clear in the sketch that we shall again use the plug flow concept, so the fluid velocity profile is flat. If the stream to be processed is dilute in the adsorbable species (adsorbate), then heat effects are usually ignored, so isothermal conditions will be taken. Finally, if the particles of solid are small, the axial diffusion effects, which are Fickian‐like, can be ignored, and the main mode of transport in the mobile fluid phase is by convection.
FIGURE 1.2 Packed‐bed adsorber.
Interphase transport from the flowing fluid to immobile particles obeys a rate law, which is based on departure from the thermodynamic equilibrium state. Because the total interfacial area is not known precisely, it is common practice to define a volumetric transfer coefficient, which is the product kca, where a is the total interfacial area per unit volume of packed column. The incremental rate expression (moles/time) is then obtained by multiplying the volumetric transfer coefficient (kca) by the composition linear driving force and this times the incremental volume of the column (AΔz)
We apply the conservation law (Eq. 1.1) to the adsorbable solute contained in both phases as follows:
where υ0 denotes superficial fluid velocity (velocity that would exist in an empty tube), ε denotes the fraction void (open) volume, hence (1 − ε) denotes the fractional volume taken up by the solid phase. Thus, ε is volume fraction between particles and is often called interstitial void volume; it is the volume fraction through which fluid is convected. The rate of accumulation has two possible sinks: accumulation in the fluid phase (C) and in the solid phase (q).
By dividing by AΔz, taking limits as before, we deduce that the overall balance for solute obeys
Similarly, we may make a solute balance on the immobile phase alone, using the rate law, Eq. 1.33, noting adsorption removes material from the flowing phase and adds it to the solid phase. Now, since the solid phase loses no material and generates none (assuming chemical reaction is absent), then the solid phase balance is
which simply states that rate of accumulation equals rate of transfer to the solid. Dividing by the elementary volume, AΔz, yields
We note that as equilibrium is approached (as C → C*)
Such conditions correspond to “saturation”; hence, no further molar exchange occurs. When this happens to the whole bed, the bed must be “regenerated,” for example, by passing a hot, inert fluid through the bed, thereby desorbing solute.
The model of the system is now composed of Eqs. 1.32, : There are three equations and three unknowns (C, C*, q).
To make the system model more compact, we attempt to eliminate q. Now, since q = KC*
The solution to this set of PDEs can be effected by suitable transform methods (e.g., the Laplace transform) for certain types of boundary conditions and initial conditions (BCs and ICs). For the adsorption step, these are
The condition on q implies (cf. Eq. 1.32)
Finally, if the bed was indeed initially clean, as stated above, then it must also be true
We thus have three independent conditions (note, we could use either Eq. 1.40 or Eq. 1.42, since they are linearly dependent) corresponding to three derivatives:
As we demonstrate later, in Chapter 12, linear systems of equations can be solved exactly only when there exists one BC or IC for each order of a derivative. The above system is now properly posed and will be solved as an example in Chapter 12 using Laplace transform.
1.4 BOUNDARY CONDITIONS AND SIGN CONVENTIONS
As we have seen in the previous sections, when time is an independent variable, the boundary condition is usually an initial condition, meaning we must specify the state of the dependent variable at some time t0 (usually t0 = 0). For the steady state, we have seen that integrations of the applicable equations always produce arbitrary constants of integration. These integration constants must be evaluated, using stipulated boundary conditions to complete the model's solution.
For the physicochemical problems occurring in chemical engineering, most boundary or initial conditions are (or can be made to be) of the homogeneous type; a condition or equation is taken to be homogeneous if, for example, it is satisfied by y(x), and is also satisfied by λy(x), where λ is an arbitrary constant. The three classical types for such homogeneous boundary conditions at a point, say x0, are the following:
Most often, the boundary values for a derived model are not homogeneous, but can be made to be so. For example, Model II in Section 1.2 portrays cooling of a flowing fluid in a tube. Something must be said about the fluid temperature at the solid wall boundary, which was specified to take a constant value Tw. This means all along the tube length, we can require
As it stands, this does not match the condition for homogeneity. However, if we define a new variable θ
then it is clear that the wall condition will become homogeneous, of type (i)
When redefining variables in this way, one must be sure that the original defining equation is unchanged. Thus, since the derivative of a constant (Tw) is always zero, then Eq. 1.31 for the new dependent variable θ is easily seen to be unchanged
It often occurs that the heat (or mass) flux at a boundary is controlled by a heat (or mass) transfer coefficient, so for a circular tube, the conduction flux is proportional to a temperature difference, as follows:
Care must be taken to ensure that sign conventions are obeyed. In our cooling problem (Model II, Section 1.2 ), it is clear that
So, h (T − Tc) must be positive, which it is, since the coolant temperature Tc < T(R, z).
This boundary condition also does not identify exactly with the type (iii) homogeneous condition given earlier. However, if we redefine the dependent variable to be θ = T − Tc, then we have
which is identical in form with the type (iii) homogeneous boundary condition when we note the equivalence: θ = y, h/k = β, r = x, and R = x0. It is also easy to see that the original convective diffusion Eq. 1.31 is unchanged when we replace T with θ. This is a useful property of linear equations.
Finally, we consider the type (ii) homogeneous boundary condition in physical terms. For the pipe flow problem, if we had stipulated that the tube wall was well insulated, then the heat flux at the wall becomes nil, so
This condition is of the homogeneous type (ii) without further modification.
Thus, we see that models for a fluid flowing in a circular pipe can sustain any one of the three possible homogeneous boundary conditions.
Sign conventions can be troublesome to students, especially when they encounter type (iii) boundary conditions. It is always wise to double‐check to ensure that the sign of the left‐hand side is the same as that of the right‐hand side. Otherwise, negative transport coefficients will be produced, which is thermodynamically impossible. To guard against such inadvertent errors, it is useful to produce a sketch showing the qualitative shape of the expected profiles.
In Figure 1.3, we sketch the expected shape of temperature profile for a fluid being cooled in a pipe. The slope of temperature profile is such that ∂T/∂r ≤ 0. If we exclude the centerline (r = 0), where exactly ∂T/∂r = 0 (the symmetry condition), then we always have ∂T/∂r < 0. Now, since fluxes (which are vector quantities) are always positive when they move in the positive direction of the coordinate system, then it is clear why the negative sign appears in Fourier's law
FIGURE 1.3 Expected temperature profile for cooling fluids in a pipe at an arbitrary position z1.
Thus, since ∂T/∂r < 0, we see −k∂T/∂r > 0, so that flux qr > 0. This convention thus ensures that heat moves down a temperature gradient, so transfer is always from hot to cold regions. For a heated tube, flux is always in the anti‐r direction; hence, it must be a negative quantity. Similar arguments hold for mass transfer where Fick's law is applicable, so that the radial component of flux in cylindrical coordinates would be
1.5 SUMMARY OF THE MODEL BUILDING PROCESS
These introductory examples are meant to illustrate the essential qualitative nature of the early part of the model building stage, which is followed by more precise quantitative detail as the final image of the desired model is made clearer. It is a property of the human condition that minds change as new information becomes available. Experience is an important factor in model formulation, and there have been recent attempts to simulate the thinking of experienced engineers through a format called Expert Systems. The following step‐by‐step procedure may be useful for beginners:
Draw a sketch of the system to be modeled, and label/define the various geometric, physical, and chemical quantities.
Carefully select the important dependent (response) variables.
Select the possible independent variables (e.g.,
z
,
t
), changes in which must necessarily affect the dependent variables.
List the parameters (physical constants, physical size, and shape) that are expected to be important; also note the possibility of nonconstant parameters (e.g., viscosity changing with temperature,
μ
(
T
)).
Draw a sketch of the expected behavior of the dependent variable(s), such as the “expected” temperature profile we used for illustrative purposes in
Figure 1.3
.
Establish a “control volume” for a differential or finite element (e.g.,
Constant Stirred Tank Reactor
[
CSTR
]) of the system to be modeled; sketch the element, and indicate all inflow–outflow paths.
Write the “conservation law” for the volume element: Write flux and reaction rate terms using general symbols, which are taken as positive quantities, so that signs are introduced only as terms are inserted according to the rules of the conservation law,
Eq. 1.1
.
After rearrangement into the proper differential format, invoke the fundamental lemma of calculus to produce a differential equation.
Introduce specific forms of flux (e.g., J
r
= −
D∂C
/
∂r
) and rate (R
A
=
kC
A
); note the opposite of generation is depletion, so when a species is depleted, its loss rate must be entered with the appropriate sign in the conservation law (i.e., replace “+ generation” with “− depletion” in
Eq. 1.1
).
Write down all possibilities for boundary values of the dependent variables; the choice among these will be made in conjunction with the solution method selected for the defining (differential) equation.
Search out solution methods, and consider possible approximations for (i) the defining equation, (ii) the boundary conditions, and (iii) an acceptable final solution.
It is clear that the modeling and solution effort should go hand in hand, tempered, of course, by available experimental and operational evidence.
1.6 MODEL HIERARCHY AND ITS IMPORTANCE IN ANALYSIS
As pointed out in Section 1.1 regarding the real purposes of the modeling effort, the scope and depth of these decisions will determine the complexity of the mathematical description of a process. If we take the scope and depth as the barometer for generating models, we will obtain a hierarchy of models where the lowest level may be regarded as a black box, and the highest is where all possible transport processes known to humans in addition to all other concepts (such as thermodynamics) are taken into account. Models, therefore, do not appear in isolation, but rather they belong to a family where the hierarchy is dictated by the number of rules (transport principles, thermodynamics). It is this family that provides engineers with capabilities to predict and understand the phenomena around us. The example of cooling of a fluid flowing in a tube (Models I and II) in Section 1.2 illustrated two members of this family. As the level of sophistication increases, the mathematical complexity increases. If one is interested in exactly how heat is conducted through the metal casing and is disposed of to the atmosphere, then the complexity of the problem must be increased by writing down a heat balance relation for the metal casing (taking it to be constant at a value Tw is, of course, a model, albeit the simplest one). Furthermore, if one is interested in how the heat is transported near the entrance section, one must write down heat balance equations before the start of the tube, in addition to Eq. 1.31 for the active, cooling part of the tube. In addition, the nature of the boundary conditions must be carefully scrutinized before and after the entrance zone in order to properly describe the boundary conditions.
To further demonstrate the concept of model hierarchy and its importance in analysis, let us consider a problem of heat removal from a bath of hot solvent by immersing steel rods into the bath and allowing the heat to dissipate from the hot solvent bath through the rod and thence to the atmosphere (Figure 1.4).
FIGURE 1.4 Schematic diagram of heat removal from a solvent bath.
For this elementary problem, it is wise to start with the simplest model first to get some feel about the system response.
1.6.1 Level 1
In this level, let us assume that
(a) the rod temperature is uniform, that is, from the bath to the atmosphere;
(b) ignore heat transfer at the two flat ends of the rod;
(c) overall heat transfer coefficients are known and constant;
(d) very little solvent evaporates from the solvent air interface.
The many assumptions listed above are necessary to simplify the analysis (i.e., to make the model tractable).
Let T0 and T1 be the atmosphere and solvent temperatures, respectively. The steady‐state heat balance (i.e., no accumulation of heat by the rod) shows a balance between heat collected in the bath and that dissipated by the upper part of the rod to atmosphere
where T is the temperature of the rod, and L1 and L2 are lengths of rod exposed to solvent and to atmosphere, respectively. Here, hG and hL represent local heat transfer coefficients. Obviously, the volume elements are finite (not differential), being composed of the volume above the liquid of length L2 and the volume below of length L1.
Solving for T from Eq. 1.52 yields
where
Equation 1.53 gives us a very quick estimate of the rod temperature and how it varies with exposure length. For example, if α is much greater than unity (i.e., long L1 section and high liquid heat transfer coefficient compared with gas coefficient), the rod temperature is then very near T1. Taking the rod temperature to be represented by Eq. 1.53, the rate of heat transfer is readily calculated from Eq. 1.52 by replacing T:
When α = hLL1/hGL2 is very large, the rate of heat transfer becomes simply
Thus, the heat transfer is controlled by the segment of the rod exposed to the atmosphere. It is interesting to note that when the heat transfer coefficient contacting the solvent is very high (i.e., α ≫ 1), it does not really matter how much of the rod is immersed in the solvent.
Thus, for a given temperature difference and a constant rod diameter, the rate of heat transfer can be enhanced by either increasing the exposure length L2 or increasing the heat transfer rate by blowing air over the exposed rod. However, these conclusions are tied to the assumption of constant rod temperature, which becomes tenuous as atmospheric exposure is increased.
To account for effects of temperature gradients in the rod, we must move to the next level in the model hierarchy, which is to say that a differential volume must be considered.
1.6.2 Level 2
Let us relax part of the assumption (a) of the first model by assuming only that the rod temperature below the solvent liquid surface is uniform at a value T1. This is a reasonable proposition, since the liquid has a much higher thermal conductivity than air. The remaining three assumptions of the level 1 model are retained.
Next, choose an upward pointing coordinate x with the origin at the solvent–air surface. Figure 1.5 shows the coordinate system and the elementary control volume.
FIGURE 1.5 Shell element and the system coordinate.
Applying a heat balance around a thin shell segment with thickness Δx gives
where the first and the second terms represent heat conducted into and out of the element, and the last term represents heat loss to atmosphere. We have concluded that temperature gradients are likely to exist in the part of the rod exposed to air but are unlikely to exist in the submerged part.
Dividing Eq. 1.56 by πR2Δx and taking the limit as Δx → 0 yield the following first‐order differential equation for the heat flux, q:
Assuming the rod is homogeneous, that is, the thermal conductivity is uniform, so flux along the axis is related to the temperature according to Fourier's law of heat conduction (Eq. 1.29). Substitution of Eq. 1.29 into Eq. 1.57 yields
Equation 1.58 is a second‐order ordinary differential equation, and to solve this, two conditions must be imposed. One condition was stipulated earlier:
The second condition (heat flux) can also be specified at x = 0 or at the other end of the rod, that is, x = L2. Heat flux is the sought‐after quantity, so it cannot be specified a priori. One must then provide a condition at x = L2. At the end of the rod, one can assume Newton's law of cooling prevails, but since the rod length is usually longer than the diameter, most of the heat loss occurs at the rod's lateral surface, and the flux from the top surface is small, so write approximately:
Equation 1.58 is subjected to the two boundary conditions (Eq. 1.59a,b), and its solution is
where
We will discuss the method of solution of such second‐order equations in Chapter 3.
Once we know the temperature distribution of the rod above the solvent–air interface, then the rate of heat loss can be calculated either of two ways. In the first, we know that the heat flow through area πR2 at x = 0 must be equal to the heat released into the atmosphere, that is,
Applying Eq. 1.60 to Eq. 1.62 gives
Here, the important group called effectiveness factor has been introduced:
This dimensionless group represents the ratio of actual heat loss to the (maximum) loss rate when gradients are absent.
Figure 1.6 shows the log–log plot of η versus the dimensionless group mL2. We note that the effectiveness factor approaches unity when mL2 is much less than unity, and it behaves like 1/mL2 as mL2 becomes very large.
FIGURE 1.6 Plot of the effectiveness factor versus mL2.
In the limit for small mL2, we can write
This is the most effective heat transfer condition. This is physically achieved when
