Applied Mathematics and Modeling for Chemical Engineers
Description
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 is ideal 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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Persons
Richard G. Rice, PhD is Emeritus Professor in the Department of Chemical Engineering at Louisiana State University, Baton Rouge, LA, USA.
Duong D. Do, PhD is Emeritus Professor in the School of Chemical Engineering at the University of Queensland, Australia.
James E. Maneval, PhD is Professor in the Department of Chemical Engineering at Bucknell University, Lewisburg, PA, USA.
Content
Preface to the Third Edition xv
Part I 1
1 Formulation of Physicochemical Problems 3
1.1 Introduction 3
1.2 Illustration of the Formulation Process (Cooling of Fluids) 3
1.2.1 Model I: Plug Flow 3
1.2.2 Model II: Parabolic Velocity 6
1.3 Combining Rate and Equilibrium Concepts (Packed-Bed Adsorber) 7
1.4 Boundary Conditions and Sign Conventions 8
1.5 Summary of the Model Building Process 9
1.6 Model Hierarchy and its Importance in Analysis 10
1.6.1 Level 1 10
1.6.2 Level 2 11
1.6.3 Level 3 13
1.6.4 Level 4 13
Problems 15
References 20
2 Modeling with Linear Algebra and Matrices 21
2.1 Introduction 21
2.2 Basic Concepts of Systems of Linear Equations 21
2.3 Matrix Notation 22
2.3.1 Matrices 22
2.3.2 Vectors 22
2.3.3 Scalars 22
2.3.4 Matrices and Vectors with Special Structure 22
2.4 Matrix Algebra and Calculus Operations 24
2.4.1 Equality 24
2.4.2 Addition and Subtraction 24
2.4.3 Multiplication 24
2.4.4 Division 26
2.4.5 Further Algebraic Properties of Matrices 27
2.4.6 Basic Differential and Integral Relations for Matrices 28
2.5 Problem 1: Solution of N Equations in N Unknowns 29
2.5.1 Analytical Results 29
2.5.2 Computational Approach: Gauss Elimination 30
2.6 Problem 2: The Matrix Eigenvalue Problem 32
2.6.1 Problem Statement and Formal Solution 32
2.6.2 Computing Eigensystems: Basic Procedure 33
2.7 Singular Systems 34
2.7.1 Consistent and Inconsistent Systems 34
2.7.2 Solution Structure for Consistent Systems 35
2.7.3 Formulation and Characteristics of Non-Square Problems 36
2.7.4 Over-Determined Systems: Least-Squares Solution 37
2.7.5 Under-Determined Systems 38
2.8 Computational Linear Algebra 40
2.8.1 The LU Factorization 40
2.8.2 The QR Factorization 40
2.8.3 The SVD Factorization 40
2.8.4 Large-Scale Problems and Iterative Methods 41
Problems 42
References 47
3 Solution Techniques for Models Yielding Ordinary Differential Equations 49
3.1 Geometric Basis and Functionality 49
3.2 Classification of ODE 50
3.3 First-Order Equations 50
3.3.1 Exact Solutions 51
3.3.2 Equations Composed of Homogeneous Functions 52
3.3.3 Bernoulli's Equation 52
3.3.4 Riccati's Equation 52
3.3.5 Linear Coefficients 54
3.3.6 First-Order Equations of Second Degree 54
3.4 Solution Methods for Second-Order Nonlinear Equations 55
3.4.1 Derivative Substitution Method 55
3.4.2 Homogeneous Function Method 58
3.5 Linear Equations of Higher Order 59
3.5.1 Second-Order Unforced Equations: Complementary Solutions 60
3.5.2 Particular Solution Methods for Forced Equations 64
3.5.3 Summary of Particular Solution Methods 70
3.6 Coupled Simultaneous ODE 71
3.7 Eigenproblems 74
3.8 Coupled Linear Differential Equations 74
3.9 Summary of Solution Methods for ODE 75
Problems 75
References 87
4 Series Solution Methods and Special Functions 89
4.1 Introduction to Series Methods 89
4.2 Properties of Infinite Series 90
4.3 Method of Frobenius 91
4.3.1 Indicial Equation and Recurrence Relation 91
4.4 Summary of the Frobenius Method 98
4.5 Special Functions 98
4.5.1 Bessel's Equation 99
4.5.2 Modified Bessel's Equation 100
4.5.3 Generalized Bessel's Equation 100
4.5.4 Properties of Bessel Functions 102
4.5.5 Differential Integral and Recurrence Relations 103
Problems 105
References 107
5 Integral Functions 109
5.1 Introduction 109
5.2 The Error Function 109
5.2.1 Properties of Error Function 110
5.3 The Gamma and Beta Functions 110
5.3.1 The Gamma Function 110
5.3.2 The Beta Function 111
5.4 The Elliptic Integrals 111
5.5 The Exponential and Trigonometric Integrals 113
Problems 113
References 116
6 Staged-Process Models: The Calculus of Finite Differences 117
6.1 Introduction 117
6.1.1 Modeling Multiple Stages 117
6.2 Solution Methods for Linear Finite Difference Equations 118
6.2.1 Complementary Solutions 118
6.3 Particular Solution Methods 121
6.3.1 Method of Undetermined Coefficients 121
6.3.2 Inverse Operator Method 122
6.4 Nonlinear Equations (Riccati Equation) 122
Problems 124
References 126
7 Probability and Statistical Modeling 127
7.1 Concepts and Results From Probability Theory 127
7.1.1 Experiments and Random Variables 127
7.1.2 Probabilities and Distribution Functions 128
7.1.3 Characteristics of Distributions Functions 131
7.1.4 The Cumulative Distribution Function 132
7.2 Concepts and Results From Mathematical Statistics 134
7.2.1 Populations Samples and Sampling 134
7.2.2 Sample Statistics and Sampling Distributions 134
7.3 Statistical Analysis and Modeling 137
7.3.1 Confidence Interval for the Mean of a Population 137
7.3.2 Hypothesis Tests for the Population Mean 138
7.3.3 Hypothesis Tests: Comparing Multiple Means 140
7.3.4 Linear Models and Linear Regression 143
Problems 150
References 154
8 Approximate Solution Methods for ODE: Perturbation Methods 155
8.1 Perturbation Methods 155
8.1.1 Introduction 155
8.2 The Basic Concepts 157
8.2.1 Gauge Functions 157
8.2.2 Order Symbols 158
8.2.3 Asymptotic Expansions and Sequences 158
8.2.4 Sources of Nonuniformity 159
8.3 The Method of Matched Asymptotic Expansion 160
8.3.1 Outer Solutions 160
8.3.2 Inner Solutions 160
8.3.3 Matching 161
8.3.4 Composite Solutions 161
8.3.5 General Matching Principle 162
8.3.6 Composite Solution of Higher Order 162
8.4 Matched Asymptotic Expansions for Coupled Equations 163
8.4.1 Outer Expansion 163
8.4.2 Inner Expansion 164
8.4.3 Matching 164
Problems 165
References 173
Part II 175
9 Numerical Solution Methods (Initial Value Problems) 177
9.1 Introduction 177
9.2 Type of Method 179
9.3 Stability 180
9.4 Stiffness 185
9.5 Interpolation and Quadrature 186
9.6 Explicit Integration Methods 187
9.7 Implicit Integration Methods 188
9.8 Predictor-Corrector Methods and Runge-Kutta Methods 189
9.8.1 Predictor-Corrector Methods 189
9.9 Runge-Kutta Methods 189
9.10 Extrapolation 191
9.11 Step Size Control 192
9.12 Higher-Order Integration Methods 192
Problems 192
References 195
10 Approximate Methods for Boundary Value Problems: Weighted Residuals 197
10.1 The Method of Weighted Residuals 197
10.1.1 Variations on a Theme of Weighted Residuals 198
10.2 Jacobi Polynomials 205
10.2.1 Rodrigues Formula 205
10.2.2 Orthogonality Conditions 205
10.3 Lagrange Interpolation Polynomials 206
10.4 Orthogonal Collocation Method 206
10.4.1 Differentiation of a Lagrange Interpolation Polynomial 206
10.4.2 Gauss-Jacobi Quadrature 207
10.4.3 Radau and Lobatto Quadrature 208
10.5 Linear Boundary Value Problem: Dirichlet Boundary Condition 209
10.6 Linear Boundary Value Problem: Robin Boundary Condition 211
10.7 Nonlinear Boundary Value Problem: Dirichlet Boundary Condition 213
10.8 One-Point Collocation 215
10.9 Summary of Collocation Methods 215
10.10 Concluding Remarks 216
Problems 217
References 225
11 Introduction to Complex Variables and Laplace Transforms 227
11.1 Introduction 227
11.2 Elements of Complex Variables 227
11.3 Elementary Functions of Complex Variables 228
11.4 Multivalued Functions 229
11.5 Continuity Properties for Complex Variables: Analyticity 230
11.5.1 Exploiting Singularities 231
11.6 Integration: Cauchy's Theorem 232
11.7 Cauchy's Theory of Residues 233
11.7.1 Practical Evaluation of Residues 234
11.7.2 Residues at Multiple Poles 235
11.8 Inversion of Laplace Transforms by Contour Integration 235
11.8.1 Summary of Inversion Theorem for Pole Singularities 237
11.9 Laplace Transformations: Building Blocks 237
11.9.1 Taking the Transform 237
11.9.2 Transforms of Derivatives and Integrals 238
11.9.3 The Shifting Theorem 240
11.9.4 Transform of Distribution Functions 240
11.10 Practical Inversion Methods 242
11.10.1 Partial Fractions 242
11.10.2 Convolution Theorem 243
11.11 Applications of Laplace Transforms for Solutions of ODE 243
11.12 Inversion Theory for Multivalued Functions: The Second Bromwich Path 248
11.12.1 Inversion When Poles and Branch Points Exist 250
11.13 Numerical Inversion Techniques 250
11.13.1 The Zakian Method 250
11.13.2 The Fourier Series Approximation 252
Problems 253
References 257
12 Solution Techniques for Models Producing PDEs 259
12.1 Introduction 259
12.1.1 Classification and Characteristics of Linear Equations 261
12.2 Particular Solutions for PDEs 263
12.2.1 Boundary and Initial Conditions 263
12.3 Combination of Variables Method 264
12.4 Separation of Variables Method 269
12.4.1 Coated Wall Reactor 269
12.5 Orthogonal Functions and Sturm-Liouville Conditions 272
12.5.1 The Sturm-Liouville Equation 272
12.6 Inhomogeneous Equations 275
12.7 Applications of Laplace Transforms for Solutions of PDEs 279
Problems 285
References 302
13 Transform Methods for Linear PDEs 305
13.1 Introduction 305
13.2 Transforms in Finite Domain: Sturm-Liouville Transforms 305
13.2.1 Development of Integral Transform Pairs 306
13.2.2 The Eigenvalue Problem and the Orthogonality Condition 309
13.2.3 Inhomogeneous Boundary Conditions 313
13.2.4 Inhomogeneous Equations 316
13.2.5 Time-Dependent Boundary Conditions 317
13.2.6 Elliptic Partial Differential Equations 317
13.3 Generalized Sturm-Liouville Integral Transform 320
13.3.1 Introduction 320
13.3.2 The Batch Adsorber Problem 320
Problems 327
References 331
14 Approximate and Numerical Solution Methods for PDEs 333
14.1 Polynomial Approximation 333
14.2 Singular Perturbation 338
14.3 Finite Difference 343
14.3.1 Notations 343
14.3.2 Essence of the Method 344
14.3.3 Tridiagonal Matrix and the Thomas Algorithm 345
14.3.4 Linear Parabolic Partial Differential Equations 345
14.3.5 Nonlinear Parabolic Partial Differential Equations 349
14.4 Orthogonal Collocation for Solving PDEs 350
14.4.1 Elliptic PDE 350
14.4.2 Parabolic PDE: Example 1 353
14.4.3 Coupled Parabolic PDE: Example 2 354
Problems 355
References 362
Appendix A: Review of Methods for Nonlinear Algebraic Equations 363
A.1 The Bisection Algorithm 363
A.2 The Successive Substitution Method 364
A.3 The Newton-Raphson Method 366
A.4 Rate of Convergence 367
A.4.1 Definition of Speed of Convergence 367
A.5 Multiplicity 368
A.5.1 Multiplicity 368
A.6 Accelerating Convergence 369
References 369
Appendix B: Derivation of the Fourier-Mellin Inversion Theorem 371
References 374
Appendix C: Table of Laplace Transforms 375
Appendix D: Numerical Integration 381
D.1 Basic Idea of Numerical Integration 381
D.2 Newton Forward Difference Polynomial 381
D.3 Basic Integration Procedure 382
D.3.1 Trapezoid Rule 382
D.3.2 Simpson's Rule 383
D.4 Error Control and Extrapolation 384
D.5 Gaussian Quadrature 384
D.6 Radau Quadrature 386
D.7 Lobatto Quadrature 388
D.8 Concluding Remarks 389
References 389
Appendix E: Nomenclature 391
Appendix F: Statistical Tables 395
Postface 399
Index 401
1
FORMULATION 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 (?, density; Cp, 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 Tw.
- The inlet temperature is constant and uniform (does not vary in r direction) at a value T0, where T0 > Tw.
- The velocity profile is plug-shaped or flat and constant at a value of ?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 (?z), 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
(1.1)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,
(1.2)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 + ?z)
(1.3)In the limit, as ?z 0, we see
(1.4)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
(1.5)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
(1.6)We rearrange the conservation law into the form required for taking limits and then divide by ?z:
(1.7)Taking limits, one at a time, then yields the sought after differential equation
(1.8)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:
(1.9)where
It is clear that ? 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
(1.10)This suggests we define a new dependent variable, namely,
(1.11)hence Eq. 1.9 now reads simply
(1.12)This can be integrated directly by separation of variables, so we rearrange to get
(1.13)Integrating term by term yields
(1.14)where ln K is any (arbitrary) constant of integration. Using logarithm properties, we can solve directly for ?
(1.15)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
(1.16)Thus, when z = 0, ? (0) must take a value T0 - Tw, so K must also take this value.
Our final result for computational purposes:
(1.17)We note that all arguments of mathematical functions must be dimensionless, so the above result yields a dimensionless temperature
(1.18)and a dimensionless length scale
(1.19)Thus, a problem with six parameters, two external conditions...
