Preface xiii
About the Companion Website xv
1 Problem Formulation, Models, and Solution Strategies 1
1.1 Introduction 1
1.1.1 Rationale for Modeling and Some Unwanted Complications 5
1.2 Algebraic Equations: Force Resolution and Vapor–Liquid Equilibria (VLE) 6
1.3 Macroscopic Balances—Lumped-Parameter Models 8
1.3.1 Recognizing When a Model Suggests Potential Hazards 10
1.4 Force Balances—Newton’s Second Law of Motion 10
1.5 Distributed Parameter Models—Microscopic Balances 11
1.5.1 Fluid Flow and Heat Transfer Combined 14
1.6 Using the Equations of Change Directly 14
1.7 Discretizing a Problem that is Continuous in Time or Space 16
1.8 A Contrast: Deterministic Models and Stochastic Processes 19
1.9 Problems with Integer-Valued Variables 20
1.10 Empiricisms and Data Interpretation 25
1.10.1 Energy Output of Candle Flame 25
1.10.2 Correlations Developed from Experimental Data 26
1.10.3 Frictional Resistance for Transitional Flow 27
1.10.4 Speech Recognition 28
1.11 Conclusion 29
Problems 30
References 35
2 Algebraic Equations 37
2.1 Introduction 37
2.1.1 An Interpolation Example 38
2.2 Elementary Methods 39
2.2.1 Newton–Raphson (Newton’s Method of Tangents) 39
2.2.2 Regula Falsi (False Position Method) 42
2.2.3 Dichotomous Search 44
2.2.4 Golden Section Search 46
2.3 Simultaneous Linear Algebraic Equations 47
2.3.1 Crout’s (or Cholesky’s) Method 48
2.3.2 Matrix Inversion 50
2.3.3 Iterative Methods of Solution 52
2.4 Simultaneous Nonlinear Algebraic Equations 54
2.4.1 Successive Substitution 54
2.4.2 Newton’s Method 55
2.4.3 An Example Problem with Simultaneous Nonlinear Algebraic Equations 56
2.4.4 Pattern Search for Solution of Nonlinear Algebraic Equations, Sequential Simplex, and the Rosenbrock Method 57
2.4.5 An Example of a Pattern Search Application: Ship Hull Design 59
2.5 Algebraic Equations with Constraints 61
2.5.1 Further Practice with the Lagrange Multiplier 62
2.6 Conclusion 62
Problems 63
References 76
3 Vectors and Tensors 77
3.1 Introduction 77
3.2 Elementary Operations 78
3.2.1 An Illustration 80
3.3 Review of Some Basic Mechanics 81
3.3.1 Force Equilibrium 81
3.3.2 Frictional Force 82
3.3.3 Equating Moments 83
3.3.4 Calculation of Centroids 84
3.3.5 Projectile Motion 85
3.4 Other Important Vector Operations 86
3.4.1 Dot and Cross-Products 86
3.4.2 Coriolis Effect 87
3.4.3 Differentiation of Vectors 87
3.4.4 Gradient, Divergence, and Curl 88
3.5 Green’s Theorem 91
3.5.1 The Divergence Theorem of Gauss 94
3.6 Stokes’ Theorem 96
3.7 Conclusion 97
Problems 98
References 103
4 Numerical Quadrature 105
4.1 Introduction 105
4.2 Trapezoid Rule 105
4.3 Simpson’s Rule 107
4.4 Newton–Cotes Formulae 109
4.5 Roundoff and Truncation Errors 109
4.6 Romberg Integration 111
4.7 Adaptive Integration Schemes 112
4.7.1 Simpson’s Rule 113
4.8 Gaussian Quadrature and the Gauss–Kronrod Procedure 114
4.9 Integrating Discrete Data 118
4.10 Multiple Integrals (Cubature) 121
4.11 Monte Carlo Methods 123
4.12 Conclusion 126
Problems 128
References 136
5 Analytic Solution of Ordinary Differential Equations 137
5.1 Some Introductory Examples 137
5.1.1 The RC Circuit 137
5.1.2 Cooling by Natural Convection 138
5.1.3 Heat Loss from an Insulated Steam Pipe 139
5.1.4 Dissolution of an Inorganic Metal Salt 140
5.2 First-Order Ordinary Differential Equations 140
5.2.1 An Example Exercise: The Streeter–Phelps Model for Dissolved Oxygen Sag 142
5.3 Nonlinear First-Order Ordinary Differential Equations 143
5.3.1 Riccati Equations 144
5.3.2 Graphical Interpretation 146
5.3.3 Solutions with Elliptic Integrals and Elliptic Functions 146
5.4 Higher-Order Linear ODEs with Constant Coefficients 149
5.4.1 The LRC Circuit 151
5.4.2 Use of the Laplace Transform for Solution of ODEs 152
5.4.3 Finding Time-Domain Response by Inversion of Laplace Transform 155
5.5 Higher-Order Equations with Variable Coefficients 156
5.6 Bessel’s Equation and Bessel Functions 158
5.6.1 Bessel’s Equation in Extended Surface Heat Transfer 162
5.6.2 An Example from Lubrication Theory 163
5.7 Power Series Solutions of Ordinary Differential Equations 164
5.7.1 Power Series Example for a First-Order ODE 165
5.7.2 A Complete Power Series Example for a Second-Order ODE with Numerical Confirmation 166
5.7.3 Guided Exercise: Power Series Solution of a Second-Order ODE 168
5.8 Regular Perturbation 169
5.9 Linearization 171
5.9.1 Beam Deflection Under Load 172
5.10 Frequency Response for Model Development 175
5.11 Conclusion 179
Problems 180
References 192
6 Numerical Solution of Ordinary Differential Equations 193
6.1 An Illustrative Example 193
6.2 The Euler Method 194
6.2.1 Modified Euler Method 196
6.2.2 A Recommended Exercise 197
6.3 Runge–Kutta Methods 197
6.4 Simultaneous Ordinary Differential Equations 201
6.4.1 Some Potential Difficulties Illustrated 201
6.5 Limitations of Fixed Step-Size Algorithms 203
6.6 Richardson Extrapolation 206
6.7 Multistep Methods 207
6.8 Split Boundary Conditions 208
6.9 Finite-Difference Methods 211
6.10 Stiff Differential Equations 212
6.10.1 Gear’s method 213
6.11 BDF (Backward Differentiation Formula) Methods 214
6.12 Bulirsch–Stoer Method 216
6.13 Phase Space 217
6.13.1 Identifying a Strange Attractor 217
6.13.2 The Binet Equation 217
6.14 Summary 222
Problems 224
References 236
7 Analytic Solution of Partial Differential Equations 237
7.1 Introduction 237
7.2 Classification of Partial Differential Equations and Boundary Conditions 237
7.3 Fourier Series 238
7.3.1 A Preview of the Utility of Fourier Series 241
7.4 The Product Method (Separation of Variables) 244
7.5 Parabolic Equations 245
7.5.1 Implementing Different Boundary Conditions 245
7.5.2 Diffusion in a Plane Sheet 249
7.5.3 Cylindrical Coordinates 250
7.5.4 The Annulus or Hollow Cylinder 255
7.5.5 Spherical Coordinates 257
7.5.6 Multiple Spatial Variables 259
7.6 Elliptic Equations 261
7.6.1 Rectangular Coordinates 261
7.6.2 Elliptic Equations in Cylindrical Coordinates 269
7.6.3 Elliptic Equations in Spherical Coordinates 274
7.6.4 Neutron Diffusion 274
7.7 Application to Hyperbolic Equations 276
7.7.1 The Vibrating String Problem 276
7.7.2 Membranes, Drums, and Chains 277
7.7.3 The Schrodinger Equation 280
7.7.4 Telegrapher’s Equations 284
7.8 Applications of the Laplace Transform 286
7.9 Approximate Solution Techniques 288
7.9.1 Galerkin MWR Applied to a PDE 290
7.9.2 The Rayleigh–Ritz Method 291
7.9.3 Collocation 293
7.9.4 Orthogonal Collocation for Partial Differential Equations 296
7.10 The Cauchy–Riemann Equations, Conformal Mapping, and Solutions for the Laplace Equation 297
7.11 Conclusion 300
Problems 301
References 312
8 Numerical Solution of Partial Differential Equations 315
8.1 Introduction 315
8.2 Finite Difference Approximations for Derivatives 316
8.3 Boundary Conditions 317
8.4 Elliptic Partial Differential Equations 318
8.4.1 An Iterative Numerical Procedure: Gauss–Seidel 320
8.4.2 Improving the Rate of Convergence with SOR 321
8.5 Parabolic Partial Differential Equations 326
8.5.1 An Elementary, Explicit Numerical Procedure 326
8.5.2 Du Fort–Frankel Scheme 331
8.5.3 Von Neumann Stability Analysis 332
8.5.4 The Crank–Nicolson Method 333
8.5.5 Alternating-Direction Implicit (ADI) Method 335
8.5.6 Three Spatial Dimensions 339
8.6 Hyperbolic Partial Differential Equations 340
8.6.1 The Method of Characteristics 343
8.6.2 The Leapfrog Method 344
8.6.3 Lax–Wendroff Method 345
8.7 Problems with Moving Boundaries 348
8.8 Elementary Problems with Convective Transport 350
8.9 A Numerical Procedure for Two-Dimensional Flow and Transport Problems 354
8.9.1 Vorticity Transport at Low Reynolds Numbers 356
8.9.2 The Deep Cavity at Large Reynolds Numbers 359
8.9.3 Adding Heat and Mass Transfer to Vorticity Transport Models 360
8.9.4 Adding Buoyancy to Vorticity Transport 361
8.9.5 Vorticity Transport and the Rayleigh–Benard Scenario 363
8.9.6 Vorticity Transport in More Difficult Geometries 365
8.9.7 Flow in the Entrance of a Duct Formed by Parallel Planes 366
8.9.8 Free Convection from Horizontal Ducts and the Effect of Pr 368
8.10 MacCormack’s Method 371
8.11 Adaptive Grids 373
8.11.1 Von Mises Transformation 374
8.11.2 Elliptic Grid Generation 376
8.12 Conclusion 378
Problems 381
References 402
9 Integro-Differential Equations 405
9.1 Introduction 405
9.2 An Example of Three-Mode Control 408
9.3 Population Problems with Hereditary Influences 409
9.4 An Elementary Solution Strategy 412
9.4.1 A Practice Exercise with an Elementary IDE 413
9.4.2 Extending the Solution Strategy to Higher-Order Equations 414
9.5 VIM: The Variational Iteration Method 416
9.6 Integro-Differential Equations and the Spread of Infectious Disease 421
9.7 Examples Drawn from Population Balances 423
9.7.1 Particle Size in Coagulating Systems 429
9.7.2 Application of the Population Balance to a Continuous Crystallizer 430
9.8 Conclusion 432
Problems 433
References 442
10 Time-Series Data and the Fourier Transform 443
10.1 Introduction 443
10.2 A Nineteenth-Century Idea 446
10.3 The Autocorrelation Coefficient 447
10.3.1 Example of Periodogram Construction 449
10.4 A Fourier Transform Pair 450
10.5 The Fast Fourier Transform 451
10.5.1 Discrete Fourier Transform (DFT) Example 452
10.5.1.1 A Recommended Exercise 453
10.6 Aliasing and Leakage 456
10.6.1 Proximate Signals with Significant Amplitude Disparity 461
10.7 Smoothing Data by Filtering 463
10.8 Modulation (Beats) 466
10.9 Some Important Examples 468
10.9.1 Using a Smartphone for Collection of Time-series Data 468
10.9.2 Bridges and Structural Integrity 470
10.9.3 Two Signals Received Simultaneously 471
10.9.3.1 Suggested Exercise: Computing the Cross Spectrum for Two Signals, x(t) and y(t) 475
10.9.4 Applications of Coherence 476
10.9.5 Seismometry and Time-series Data 476
10.9.6 Decaying Turbulence in a Box 479
10.9.7 Bubbles and the Gas–Liquid Interface 480
10.9.8 Shock and Vibration Events in Transportation 482
10.10 Conclusion and Some Final Thoughts 483
Problems 484
References 498
11 An Introduction to the Calculus of Variations and the Finite Element Method 499
11.1 Some Preliminaries 499
11.1.1 Elementary Principles of Extrema 499
11.1.2 Principle of Least Action 500
11.2 Notation for the Calculus of Variations 503
11.3 Brachistochrone Problem 503
11.4 Other Examples 505
11.4.1 Minimum Surface Area 505
11.4.2 Systems of Particles 507
11.4.3 Vibrating String 508
11.4.4 Laplace’s Equation 508
11.4.5 Boundary Value Problems 509
11.5 The Rayleigh–Ritz Method and Sturm–Liouville Problems 511
11.6 Contemporary COV Analyses of Old Structural Problems 515
11.6.1 Flexing of a Rod of Small Cross Section 515
11.6.2 The Optimal Column Shape 516
11.7 Systems with Surface Tension 517
11.8 Less Familiar COV Applications 519
11.9 The Connection Between COV and the Finite Element Method 522
11.10 Conclusion 527
Problems 528
References 531
Index 533
