Applied Mathematics And Modeling For Chemical Engineers – Richard G. Rice, Duong D. Do – 1st Edition

Description

The revolution created in 1960 by the publication and widespread adoption of the textbook Transport by Bird et al. ushered in a new era for engineering. This book has nurtured several generations on the importance of problem formulation by elementary balances.

Modeling (or idealization) of processes has now become standard operating procedure; but; unfortunately; the sophistication of the modeling exercise has not been matched by textbooks on the of such models in quantitative mathematical terms. Moreover; the widespread availability of computer software packages has weakened the generational skills in classical analysis.

The purpose of this book is to attempt to bridge the gap between classical analysis and modern applications. Thus; emphasis is directed in Chapter 1 to the proper representation of a physicochemical situation into correct mathematical language. It is important to recognize that if a problem is incorrectly posed in the first instance; then any will do.

The thought process of “idealizing;” or approximating an actual situation; is now commonly called “modeling.” Such models of natural and man-made processes can only be fully accepted if they fit the reality of experiment. We try to give emphasis to this well-known truth by selecting literature examples; which sustain experimental verification.

Following the model stage; we introduce classical methods in Chapters 2 and 3 for solving ordinary equations (ODE); adding new material in Chapter 6 on solution methods; which include perturbation techniques and elementary numerical solutions.

This seems altogether appropriate; since most models are in the first instance. Finally; because of the propensity of staged processing in engineering; we introduce analytical methods to deal with important classes of finite-difference equations in Chapter 5.

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  • Part I. 1
    1. Formulation of Physicochemical Problems 3
    1.1 Introduction 3
    1.2 Illustration of the Formulation Process (Cooling of Fluids) 3
    1.3 Combining Rate and Equilibrium Concepts (Packed Bed Adsorber) 7
    1.4 Boundary Conditions and Sign Conventions 8
    1.5 Models with Many Variables: Vectors and Matrices 10
    1.6 Matrix Definition 10
    1.7 Types of Matrices 11
    1.8 Matrix Algebra 12
    1.9 Useful Row Operations 13
    1.10 Direct Elimination Methods 14
    1.11 Iterative Methods 18
    1.12 Summary of the Model Building Process 19
    1.13 Model Hierarchy and its Importance in Analysis 19
    Problems 25

    2. Solution Techniques for Models Yielding Ordinary Differential Equations 31
    2.1 Geometric Basis and Functionality 31
    2.2 Classification of ODE 32
    2.3 First-Order Equations 32
    2.4 Solution Methods for Second-Order Nonlinear Equations 37
    2.5 Linear Equations of Higher Order 42
    2.6 Coupled Simultaneous ODE 55
    2.7 Eigenproblems 59
    2.8 Coupled Linear Differential Equations 59
    2.9 Summary of Solution Methods for ODE 60
    Problems 60
    References 73

    3. Series Solution Methods and Special Functions 75
    3.1 Introduction to Series Methods 75
    3.2 Properties of Infinite Series 76
    3.3 Method of Frobenius 77
    3.4 Summary of the Frobenius Method 85
    3.5 Special Functions 86
    Problems 93
    References 95

    4. Integral Functions 97
    4.1 Introduction 97
    4.2 The Error Function 97
    4.3 The Gamma and Beta Functions 98
    4.4 The Elliptic Integrals 99
    4.5 The Exponential and Trigonometric Integrals 101
    Problems 102
    References 104

    5. Staged-Process Models: The Calculus of Finite Differences 105
    5.1 Introduction 105
    5.2 Solution Methods for Linear Finite Difference Equations 106
    5.3 Particular Solution Methods 109
    5.4 Nonlinear Equations (Riccati Equations) 111
    Problems 112
    References 115

    6. Approximate Solution Methods for ODE: Perturbation Methods 117
    6.1 Perturbation Methods 117
    6.2 The Basic Concepts 120
    6.3 The Method of Matched Asymptotic Expansion 122
    6.4 Matched Asymptotic Expansions for Coupled Equations 125
    Problems 128
    References 136

    Part II. 137
    7. Numerical Solution Methods (Initial Value Problems) 139
    7.1 Introduction 139
    7.2 Type of Method 142
    7.3 Stability 142
    7.4 Stiffness 147
    7.5 Interpolation and Quadrature 149
    7.6 Explicit Integration Methods 150
    7.7 Implicit Integration Methods 152
    7.8 Predictor-Corrector Methods and Runge-Kutta Methods 152
    7.9 Runge-Kutta Methods 153
    7.10 Extrapolation 155
    7.11 Step Size Control 155
    7.12 Higher Order Integration Methods 156
    Problems 156
    References 159

    8. Approximate Methods for Boundary Value Problems: Weighted Residuals 161
    8.1 The Method of Weighted Residuals 161
    8.2 Jacobi Polynomials 179
    8.3 Lagrange Interpolation Polynomials 172
    8.4 Orthogonal Collocation Method 172
    8.5 Linear Boundary Value Problem: Dirichlet Boundary Condition 175
    8.6 Linear Boundary Value Problem: Robin Boundary Condition 177
    8.7 Nonlinear Boundary Value Problem: Dirichlet Boundary Condition 179
    8.8 One-Point Collocation 181
    8.9 Summary of Collocation Methods 182
    8.10 Concluding Remarks 183
    Problem 184
    References 192
    9. Introduction to Complex Variables and Laplace Transforms 193
    9.1 Introducton 193
    9.2 Elements of Complex Variables 193
    9.3 Elementary Functions of Complex Variables 194
    9.4 Multivalued Functions 195
    9.5 Continuity Properties for Complex Variables: Analyticity 196
    9.6 Integration: Cauchy’s Theorem 198
    9.7 Cauchy’s Theory of Residues 201
    9.8 Inversion of Laplace Transforms by Contour Integration 202
    9.9 Laplace Transformations: Building Blocks 204
    9.10 Practical Inversion Methods 209
    9.11 Applications of Laplace Transforms for Solutions of ODE 211
    9.12 Inversion Theory for Multivalued Functions: the Second Bromwich Path 215
    9.13 Numerical Inversion Techniques 218
    Problems 221
    References 225

    10. Solution Techniques for Models Producing PDEs 227
    10.1 Introduction 227
    10.2 Particular Solutions for PDES 231
    10.3 Combination of Variables Method 233
    10.4 Separation of Variables Method 238
    10.5 Orthogonal Functions and Sturm-Liouville Conditions 241
    10.6 Inhomogeneous Equations 245
    10.7 Applications of Laplace Transforms for Solutions of PDES 248
    Problems 254
    References 271

    11. Transform Methods for Linear PDEs 273
    11.1 Introduction 273
    11.2 Transforms in Finite Domain: Sturm-Liouville Transforms 273
    11.3 Generalized Sturm-Liouville Integral Transforms 289
    Problems 297
    References 301

    12. Approximate and Numerical Solution Methods for PDEs 303
    12.1 Polynomial Approximation 303
    12.2 Singular Perturbation 310
    12.3 Finite Difference 315
    12.4 Orthogonal Collocation for Solving PDEs 324
    12.5 Orthogonal Collocation on Finite Elements 330
  • Citation
    • Full Title: Applied Mathematics And Modeling For Chemical Engineers
    • Author/s: / Duong D. Do
    • ISBN-10: 0471303771
    • Edition: 1st Edition
    • Topic: Math
    • Subtopic: Mathematics
    • File Type: eBook
    • Idioma: English

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