Preface viii

1 Problem Formulation and Model Development 1

Introduction 1

Algebraic Equations from Vapor–Liquid Equilibria (VLE) 3

Macroscopic Balances: Lumped-Parameter Models 4

Force Balances: Newton’s Second Law of Motion 6

Distributed Parameter Models: Microscopic Balances 6

Using the Equations of Change Directly 8

A Contrast: Deterministic Models and Stochastic Processes 10

Empiricisms and Data Interpretation 10

Conclusion 12

Problems 13

References 14

2 Algebraic Equations 15

Introduction 15

Elementary Methods 16

Newton–Raphson (Newton’s Method of Tangents) 16

Regula Falsi (False Position Method) 18

Dichotomous Search 19

Golden Section Search 20

Simultaneous Linear Algebraic Equations 20

Crout’s (or Cholesky’s) Method 21

Matrix Inversion 23

Iterative Methods of Solution 23

Simultaneous Nonlinear Algebraic Equations 24

Pattern Search for Solution of Nonlinear Algebraic Equations 26

Sequential Simplex and the Rosenbrock Method 26

An Example of a Pattern Search Application 28

Algebraic Equations with Constraints 28

Conclusion 29

Problems 30

References 32

3 Vectors and Tensors 34

Introduction 34

Manipulation of Vectors 35

Force Equilibrium 37

Equating Moments 37

Projectile Motion 38

Dot and Cross Products 39

Differentiation of Vectors 40

Gradient Divergence and Curl 40

Green’s Theorem 42

Stokes’ Theorem 43

Conclusion 44

Problems 44

References 46

4 Numerical Quadrature 47

Introduction 47

Trapezoid Rule 47

Simpson’s Rule 48

Newton–Cotes Formulae 49

Roundoff and Truncation Errors 50

Romberg Integration 51

Adaptive Integration Schemes 52

Simpson’s Rule 52

Gaussian Quadrature and the Gauss–Kronrod Procedure 53

Integrating Discrete Data 55

Multiple Integrals (Cubature) 57

Monte Carlo Methods 59

Conclusion 60

Problems 62

References 64

5 Analytic Solution of Ordinary Differential Equations 65

An Introductory Example 65

First-Order Ordinary Differential Equations 66

Nonlinear First-Order Ordinary Differential Equations 67

Solutions with Elliptic Integrals and Elliptic Functions 69

Higher-Order Linear ODEs with Constant Coefficients 71

Use of the Laplace Transform for Solution of ODEs 73

Higher-Order Equations with Variable Coefficients 75

Bessel’s Equation and Bessel Functions 76

Power Series Solutions of Ordinary Differential Equations 78

Regular Perturbation 80

Linearization 81

Conclusion 83

Problems 84

References 88

6 Numerical Solution of Ordinary Differential Equations 89

An Illustrative Example 89

The Euler Method 90

Modified Euler Method 91

Runge–Kutta Methods 91

Simultaneous Ordinary Differential Equations 94

Some Potential Difficulties Illustrated 94

Limitations of Fixed Step-Size Algorithms 95

Richardson Extrapolation 97

Multistep Methods 98

Split Boundary Conditions 98

Finite-Difference Methods 100

Stiff Differential Equations 100

Backward Differentiation Formula (BDF) Methods 101

Bulirsch–Stoer Method 102

Phase Space 103

Summary 105

Problems 106

References 109

7 Analytic Solution of Partial Differential Equations 111

Introduction 111

Classification of Partial Differential Equations and Boundary Conditions 111

Fourier Series 112

A Preview of the Utility of Fourier Series 114

The Product Method (Separation of Variables) 116

Parabolic Equations 116

Elliptic Equations 122

Application to Hyperbolic Equations 127

The Schrödinger Equation 128

Applications of the Laplace Transform 131

Approximate Solution Techniques 133

Galerkin MWR Applied to a PDE 134

The Rayleigh–Ritz Method 135

Collocation 137

Orthogonal Collocation for Partial Differential Equations 138

The Cauchy–Riemann Equations Conformal Mapping and Solutions for the Laplace Equation 139

Conclusion 142

Problems 143

References 146

8 Numerical Solution of Partial Differential Equations 147

Introduction 147

Finite-Difference Approximations for Derivatives 148

Boundaries with Specified Flux 149

Elliptic Partial Differential Equations 149

An Iterative Numerical Procedure: Gauss–Seidel 151

Improving the Rate of Convergence with Successive Over-Relaxation (SOR) 152

Parabolic Partial Differential Equations 154

An Elementary Explicit Numerical Procedure 154

The Crank–Nicolson Method 155

Alternating-Direction Implicit (ADI) Method 157

Three Spatial Dimensions 158

Hyperbolic Partial Differential Equations 158

The Method of Characteristics 160

The Leapfrog Method 161

Elementary Problems with Convective Transport 162

A Numerical Procedure for Two-Dimensional Viscous Flow Problems 165

MacCormack’s Method 170

Adaptive Grids 171

Conclusion 173

Problems 176

References 183

9 Integro-Differential Equations 184

Introduction 184

An Example of Three-Mode Control 185

Population Problems with Hereditary Infl uences 186

An Elementary Solution Strategy 187

VIM: The Variational Iteration Method 188

Integro-Differential Equations and the Spread of Infectious Disease 192

Examples Drawn from Population Balances 194

Particle Size in Coagulating Systems 198

Application of the Population Balance to a Continuous Crystallizer 199

Conclusion 201

Problems 201

References 204

10 Time-Series Data and the Fourier Transform 206

Introduction 206

A Nineteenth-Century Idea 207

The Autocorrelation Coeffi cient 208

A Fourier Transform Pair 209

The Fast Fourier Transform 210

Aliasing and Leakage 213

Smoothing Data by Filtering 216

Modulation (Beats) 218

Some Familiar Examples 219

Turbulent Flow in a Deflected Air Jet 219

Bubbles and the Gas–Liquid Interface 220

Shock and Vibration Events in Transportation 222

Conclusion and Some Final Thoughts 223

Problems 224

References 227

11 An Introduction to the Calculus of Variations and the Finite-Element Method 229

Some Preliminaries 229

Notation for the Calculus of Variations 230

Brachistochrone Problem 231

Other Examples 232

Minimum Surface Area 232

Systems of Particles 232

Vibrating String 233

Laplace’s Equation 234

Boundary-Value Problems 234

A Contemporary COV Analysis of an Old Structural Problem 236

Flexing of a Rod of Small Cross Section 236

The Optimal Column Shape 237

Systems with Surface Tension 238

The Connection between COV and the Finite-Element Method (FEM) 238

Conclusion 241

Problems 242

References 243

Index 245

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