Applied Mathematics and Modeling for Chemical Engineers

Front Matter......Page 1
Preface......Page 4
Table of Contents......Page 0
Table of Contents......Page 6
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1.1 Introduction......Page 16
1.2 Illustration of the Formulation Process (Cooling of Fluids)......Page 17
1.3 Combining Rate and Equilibrium Concepts (Packed Bed Adsorber)......Page 23
1.4 Boundary Conditions and Sign Conventions......Page 26
1.5 Summary of the Model Building Process......Page 29
1.6 Model Hierarchy and Its Importance in Analysis......Page 30
1.8 Problems......Page 41
2.1 Geometric Basis and Functionality......Page 50
2.3 First Order Equations......Page 52
2.3.1 Exact Solutions......Page 54
2.3.2 Equations Composed of Homogeneous Functions......Page 56
2.3.4 Riccati\'s Equation......Page 58
2.3.5 Linear Coefficients......Page 62
2.3.6 First Order Equations of Second Degree......Page 63
2.4 Solution Methods for Second Order Nonlinear Equations......Page 64
2.4.1 Derivative Substitution Method......Page 65
2.4.2 Homogeneous Function Method......Page 71
2.5 Linear Equations of Higher Order......Page 74
2.5.1 Second Order Unforced Equations: Complementary Solutions......Page 76
2.5.2 Particular Solution Methods for Forced Equations......Page 85
2.5.3 Summary of Particular Solution Methods......Page 101
2.6 Coupled Simultaneous ODE......Page 102
2.7 Summary of Solution Methods for ODE......Page 109
2.9 Problems......Page 110
3.1 Introduction to Series Methods......Page 117
3.2 Properties of Infinite Series......Page 119
3.3 Method of Frobenius......Page 121
3.3.1 Indicial Equation and Recurrence Relation......Page 122
3.4 Summary of the Frobenius Method......Page 139
3.5 Special Functions......Page 140
3.5.1 Bessel\'s Equation......Page 141
3.5.2 Modified Bessel\'s Equation......Page 143
3.5.3 Generalized Bessel Equation......Page 144
3.5.4 Properties of Bessel Functions......Page 148
3.5.5 Differential, Integral and Recurrence Relations......Page 150
3.6 References......Page 154
3.7 Problems......Page 155
4.2 The Error Function......Page 161
4.2.1 Properties of Error Function......Page 162
4.3.1 The Gamma Function......Page 163
4.4 The Elliptic Integrals......Page 165
4.5 The Exponential and Trigonometric Integrals......Page 169
4.7 Problems......Page 171
5.1 Introduction......Page 177
5.1.1 Modeling Multiple Stages......Page 178
5.2 Solution Methods for Linear Finite Difference Equations......Page 179
5.2.1 Complementary Solutions......Page 180
5.3.1 Method of Undetermined Coefficients......Page 185
5.3.2 Inverse Operator Method......Page 187
5.4 Nonlinear Equations (Riccati Equation)......Page 189
5.6 Problems......Page 192
6.1.1 Introduction......Page 197
6.2.1 Gauge Functions......Page 202
6.2.2 Order Symbols......Page 203
6.2.3 Asymptotic Expansions and Sequences......Page 204
6.2.4 Sources of Nonuniformity......Page 206
6.3 The Method of Matched Asymptotic Expansion......Page 208
6.3.1 Matched Asymptotic Expansions for Coupled Equations......Page 215
6.4 References......Page 220
6.5 Problems......Page 221
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7.1 Introduction......Page 237
7.2 Type of Method......Page 242
7.3 Stability......Page 244
7.4 Stiffness......Page 255
7.5 Interpolation and Quadrature......Page 258
7.6 Explicit Integration Methods......Page 261
7.7 Implicit Integration Methods......Page 264
7.8.1 Predictor-Corrector Methods......Page 265
7.8.2 Runge-Kutta Methods......Page 266
7.10 Step Size Control......Page 270
7.12 References......Page 272
7.13 Problems......Page 273
8.1 The Method of Weighted Residuals......Page 280
8.1.1 Variations on a Theme of Weighted Residuals......Page 283
8.2.1 Rodrigues Formula......Page 297
8.2.2 Orthogonality Conditions......Page 298
8.3 Lagrange Interpolation Polynomials......Page 301
8.4 Orthogonal Collocation Method......Page 302
8.4.1 Differentiation of a Lagrange Interpolation Polynomial......Page 303
8.4.2 Gauss-Jacobi Quadrature......Page 305
8.4.3 Radau and Lobatto Quadrature......Page 307
8.5 Linear Boundary Value Problem - Dirichlet Boundary Condition......Page 308
8.6 Linear Boundary Value Problem - Robin Boundary Condition......Page 313
8.7 Nonlinear Boundary Value Problem - Dirichlet Boundary Condition......Page 316
8.8 One-Point Collocation......Page 321
8.9 Summary of Collocation Methods......Page 323
8.11 References......Page 325
8.12 Problems......Page 326
9.1 Introduction......Page 343
9.2 Elements of Complex Variables......Page 344
9.3 Elementary Functions of Complex Variables......Page 346
9.4 Multivalued Functions......Page 347
9.5 Continuity Properties for Complex Variables: Analyticity......Page 349
9.6 Integration: Cauchy\'s Theorem......Page 353
9.7 Cauchy\'s Theory of Residues......Page 357
9.7.1 Practical Evaluation of Residues......Page 359
9.7.2 Residues at Multiple Poles......Page 361
9.8 Inversion of Laplace Transforms by Contour Integration......Page 362
9.8.1 Summary of Inversion Theorem for Pole Singularities......Page 365
9.9.1 Taking the Transform......Page 366
9.9.2 Transforms of Derivatives and Integrals......Page 369
9.9.3 The Shifting Theorem......Page 372
9.9.4 Transform of Distribution Functions......Page 373
9.10.1 Partial Fractions......Page 375
9.10.2 Convolution Theorem......Page 378
9.11 Applications of Laplace Transforms for Solutions of ODE......Page 380
9.12 Inversion Theory for Multivalued Functions: The Second Bromwich Path......Page 390
9.12.1 Inversion when Poles and Branch Points Exist......Page 394
9.13.1 The Zakian Method......Page 395
9.13.2 The Fourier Series Approximation......Page 400
9.15 Problems......Page 402
10.1 Introduction......Page 409
10.1.1 Classification and Characteristics of Linear Equations......Page 414
10.2 Particular Solutions for PDEs......Page 417
10.2.1 Boundary and Initial Conditions......Page 418
10.3 Combination of Variables Method......Page 421
10.4 Separation of Variables Method......Page 432
10.4.1 Coated Wall Reactor......Page 433
10.5.1 The Sturm-Liouville Equation......Page 438
10.6 Inhomogeneous Equations......Page 446
10.7 Applications of Laplace Transforms for Solutions of PDEs......Page 456
10.8 References......Page 467
10.9 Problems......Page 468
11.1 Introduction......Page 499
11.2.1 Development of Integral Transform Pairs......Page 500
11.2.2 The Eigenvalue Problem and the Orthogonality Condition......Page 507
11.2.3 Inhomogeneous Boundary Conditions......Page 517
11.2.4 Inhomogeneous Equations......Page 524
11.2.5 Time-Dependent Boundary Conditions......Page 526
11.2.6 Elliptic Partial Differential Equations......Page 529
11.3.2 The Batch Adsorber Problem......Page 534
11.4 References......Page 550
11.5 Problems......Page 551
12.1 Polynomial Approximation......Page 559
12.2 Singular Perturbation......Page 575
12.3 Finite Difference......Page 585
12.3.1 Notations......Page 586
12.3.2 Essence of the Method......Page 587
12.3.3 Tridiagonal Matrix and the Thomas Algorithm......Page 589
12.3.4 Linear Parabolic Partial Differential Equations......Page 591
12.3.5 Nonlinear Parabolic Partial Differential Equations......Page 599
12.3.6 Elliptic Equations......Page 601
12.4.1 Elliptic PDE......Page 606
12.4.2 Parabolic PDE: Example 1......Page 611
12.4.3 Coupled Parabolic PDE: Example 2......Page 613
12.5 Orthogonal Collocation on Finite Elements......Page 616
12.6 References......Page 628
12.7 Problems......Page 629
A.1 The Bisection Algorithm......Page 643
A.2 The Successive Substitution Method......Page 645
A.3 The Newton-Raphson Method......Page 648
A.4 Rate of Convergence......Page 652
A.5 Multiplicity......Page 654
A.6 Accelerating Convergence......Page 655
A.7 References......Page 656
B.1 Matrix Definition......Page 657
B.2 Types of Matrices......Page 659
B.3 Matrix Algebra......Page 660
B.4 Useful Row Operations......Page 662
B.5.1 Basic Procedure......Page 664
B.5.2 Augmented Matrix......Page 665
B.5.3 Pivoting......Page 667
B.5.4 Scaling......Page 668
B.5.6 Gauss-Jordan Elimination......Page 669
B.5.7 LU Decomposition......Page 671
B.6.1 Jacobi Method......Page 672
B.7 Eigenproblems......Page 673
B.8 Coupled Linear Differential Equations......Page 674
B.9 References......Page 675
Appendix C: Derivation of the Fourier-Mellin Inversion Theorem......Page 676
Appendix D: Table of Laplace Transforms......Page 684
E.1 Basic Idea of Numerical Integration......Page 689
E.2 Newton Forward Difference Polynomial......Page 690
E.3.1 Trapezoid Rule......Page 691
E.3.2 Simpson\'s Rule......Page 693
E.4 Error Control and Extrapolation......Page 695
E.5 Gaussian Quadrature......Page 696
E.6 Radau Quadrature......Page 700
E.7 Lobatto Quadrature......Page 703
E.9 References......Page 706
Appendix F: Nomenclature......Page 707
Postface......Page 711
A......Page 713
B......Page 714
C......Page 715
D......Page 717
E......Page 718
F......Page 719
G......Page 720
I......Page 721
L......Page 722
M......Page 724
N......Page 725
P......Page 726
Q......Page 727
R......Page 728
S......Page 729
T......Page 730
Z......Page 731