Understanding the mathematical modeling of chemical processes is fundamental to the successful career of a researcher in chemical engineering. This book reviews, introduces, and develops the mathematics that is most frequently encountered in sophisticated chemical engineering models.

The result of a collaboration between a chemical engineer and a mathematician, both of whom have taught classes on modeling and applied mathematics, the book provides a rigorous and in-depth coverage of chemical engineering model formulation and analysis as well as a text which can serve as an excellent introduction to linear mathematics for engineering students. There is a clear focus in the choice of material, worked examples, and exercises that make it unusually accessible to the target audience. The book places a heavy emphasis on applications to motivate the theory, but simultaneously maintains a high standard of rigor to add mathematical depth and understanding.


Contents:
Model Formulation
Some Ordinary Differential Equations
Finite Dimensional Vector Spaces
Tensors
Linear Difference Equations
Linear Differential Equations
Hilbert Spaces
Partial Differential Equations
Problems


Readership: Graduate students, academics and researchers in chemical engineering.

LINEAR MATHEMATICAL MODELS IN CHEMICAL ENGINEERING
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Contents
Preface v
1. Model Formulation 1
1.1 Classical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Macroscopic balances . . . . . . . . . . . . . . . . . . . . . 2
1.1.1.1 Mass and energy balances . . . . . . . . . . . . . 2
1.1.1.2 Balances involving chemical kinetics . . . . . . . 18
1.1.2 The quasi steady state assumption . . . . . . . . . . . . . . 27
1.1.3 Di®erential balances . . . . . . . . . . . . . . . . . . . . . . 32
1.1.3.1 Coordinate systems . . . . . . . . . . . . . . . . . 32
1.1.3.2 Constitutive equations . . . . . . . . . . . . . . . 36
1.1.3.3 Operator notation . . . . . . . . . . . . . . . . . 38
1.1.3.4 Mass and energy balances . . . . . . . . . . . . . 42
1.1.3.5 Problems in °uid mechanics . . . . . . . . . . . . 59
1.1.3.6 Summary of common boundary conditions . . . . 64
1.1.3.7 Symmetry . . . . . . . . . . . . . . . . . . . . . . 66
1.2 Abstract control volumes . . . . . . . . . . . . . . . . . . . . . . . 69
2. Some Ordinary Di®erential Equations 79
2.1 First order equations . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.1.1 Separable equations . . . . . . . . . . . . . . . . . . . . . . 79
2.1.2 Linear, ¯rst order equations . . . . . . . . . . . . . . . . . 80
2.1.3 Exact equations . . . . . . . . . . . . . . . . . . . . . . . . 82
2.1.4 Homogeneous equations . . . . . . . . . . . . . . . . . . . . 83
2.1.5 Bernoulli equation . . . . . . . . . . . . . . . . . . . . . . . 85
2.1.6 Clairaut’s equation . . . . . . . . . . . . . . . . . . . . . . 86
2.1.7 Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . 87
2.2 Second order equations . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.2.1 Dependent variable does not occur explicitly . . . . . . . . 89
2.2.2 Free variable does not occur explicitly . . . . . . . . . . . . 90
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2.2.3 Homogeneous equations . . . . . . . . . . . . . . . . . . . . 91
2.3 Higher order equations . . . . . . . . . . . . . . . . . . . . . . . . . 92
2.4 Variable transformations . . . . . . . . . . . . . . . . . . . . . . . . 92
2.5 The importance of being Lipschitz . . . . . . . . . . . . . . . . . . 94
3. Finite Dimensional Vector Spaces 97
3.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.3 Span, linear independence, and basis . . . . . . . . . . . . . . . . . 101
3.3.1 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.4 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.4.1 Isomorphisms of vector spaces . . . . . . . . . . . . . . . . 107
3.4.2 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.4.3 Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.4.4 Representation of subspaces . . . . . . . . . . . . . . . . . 113
3.5 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.5.1 Matrix algebra . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.5.2 Gauss elimination . . . . . . . . . . . . . . . . . . . . . . . 119
3.5.3 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.5.3.1 Basic properties of determinants . . . . . . . . . 124
3.5.3.2 Calculation of determinants . . . . . . . . . . . . 127
3.5.3.3 The derivative of a determinant . . . . . . . . . . 129
3.5.4 The classical adjoint matrix . . . . . . . . . . . . . . . . . 129
3.6 Systems of linear algebraic equations . . . . . . . . . . . . . . . . . 130
3.6.1 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.6.2 Applications of rank . . . . . . . . . . . . . . . . . . . . . . 135
3.6.3 Solution structure . . . . . . . . . . . . . . . . . . . . . . . 141
3.6.4 The null and range space of a matrix . . . . . . . . . . . . 148
3.6.5 Overdetermined systems . . . . . . . . . . . . . . . . . . . 151
3.7 The algebraic eigenvalue problem . . . . . . . . . . . . . . . . . . . 152
3.7.1 Finding eigenvalues and eigenvectors . . . . . . . . . . . . 153
3.7.2 Multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . 160
3.7.3 Similar matrices . . . . . . . . . . . . . . . . . . . . . . . . 161
3.7.3.1 Equivalence relations . . . . . . . . . . . . . . . . 163
3.7.4 Eigenspaces and eigenbases . . . . . . . . . . . . . . . . . . 164
3.7.4.1 Diagonalization of simple and semi-simple
matrices . . . . . . . . . . . . . . . . . . . . . . . 166
3.7.5 Generalized eigenspaces . . . . . . . . . . . . . . . . . . . . 167
3.7.5.1 Generalized eigenbases . . . . . . . . . . . . . . . 171
3.7.6 Jordan canonical form . . . . . . . . . . . . . . . . . . . . . 175
3.7.7 Jordan form of real matrices with complex eigenvalues . . 179
3.7.8 Powers and exponentials of matrices . . . . . . . . . . . . . 183
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3.7.9 Location of eigenvalues . . . . . . . . . . . . . . . . . . . . 186
3.8 Geometry of vector spaces . . . . . . . . . . . . . . . . . . . . . . . 188
3.8.1 Vector products . . . . . . . . . . . . . . . . . . . . . . . . 188
3.8.1.1 Inner product . . . . . . . . . . . . . . . . . . . . 188
3.8.1.2 Cross product . . . . . . . . . . . . . . . . . . . . 190
3.8.1.3 Triple scalar product . . . . . . . . . . . . . . . . 191
3.8.1.4 Dyad or outer product . . . . . . . . . . . . . . . 193
3.8.2 Gram-Schmidt orthogonalization . . . . . . . . . . . . . . . 194
3.8.3 Eigenrows . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
3.8.4 Real, symmetric matrices . . . . . . . . . . . . . . . . . . . 198
4. Tensors 201
4.1 De¯nitions and basic concepts . . . . . . . . . . . . . . . . . . . . . 202
4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
4.2.1 Matrices as operators . . . . . . . . . . . . . . . . . . . . . 204
4.2.2 Equivalence transformations . . . . . . . . . . . . . . . . . 208
4.3 The adjoint operator . . . . . . . . . . . . . . . . . . . . . . . . . . 210
4.4 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
4.4.1 Transformation rules . . . . . . . . . . . . . . . . . . . . . 214
4.4.2 Invariants of tensors . . . . . . . . . . . . . . . . . . . . . . 218
4.5 Some tensors from physics and engineering . . . . . . . . . . . . . 220
4.5.1 Fourier’s law . . . . . . . . . . . . . . . . . . . . . . . . . . 221
4.5.2 The stress tensor . . . . . . . . . . . . . . . . . . . . . . . 227
4.6 Vectors and tensors in curvilinear coordinates . . . . . . . . . . . . 235
4.6.1 Proper transformations . . . . . . . . . . . . . . . . . . . . 237
4.6.2 Vectors and transformations at a point . . . . . . . . . . . 238
4.6.3 Covariance and contravariance . . . . . . . . . . . . . . . . 240
4.6.4 The physical components . . . . . . . . . . . . . . . . . . . 246
5. Linear Di®erence Equations 249
5.1 Linear equations with constant coe±cients . . . . . . . . . . . . . . 259
5.1.1 Homogeneous solutions . . . . . . . . . . . . . . . . . . . . 260
5.1.2 Particular solutions . . . . . . . . . . . . . . . . . . . . . . 263
5.2 Single, ¯rst order equations . . . . . . . . . . . . . . . . . . . . . . 266
5.3 Single, higher order equations . . . . . . . . . . . . . . . . . . . . . 267
5.3.1 Solution by variable transformation . . . . . . . . . . . . . 268
5.3.1.1 Euler’s equation . . . . . . . . . . . . . . . . . . . 268
5.3.2 Reduction of order . . . . . . . . . . . . . . . . . . . . . . . 268
5.3.3 Particular solution by variation of parameters . . . . . . . 270
5.4 Systems of linear di®erence equations . . . . . . . . . . . . . . . . 272
5.4.1 Basic theorems . . . . . . . . . . . . . . . . . . . . . . . . . 273
5.4.2 Particular solution by variation of parameters . . . . . . . 275
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5.4.3 Equations with constant coe±cients . . . . . . . . . . . . . 277
5.4.3.1 Homogeneous solutions . . . . . . . . . . . . . . . 277
5.4.3.2 Particular solutions for constant inhomogeneous
term . . . . . . . . . . . . . . . . . . . . . . . . . 282
5.5 Non linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . 285
5.5.1 Riccati’s equation . . . . . . . . . . . . . . . . . . . . . . . 286
6. Linear Di®erential Equations 287
6.1 Linear equations with constant coe±cients . . . . . . . . . . . . . . 288
6.1.1 Homogeneous solutions . . . . . . . . . . . . . . . . . . . . 288
6.1.2 Particular solutions . . . . . . . . . . . . . . . . . . . . . . 290
6.2 Single, higher order equations . . . . . . . . . . . . . . . . . . . . . 293
6.2.1 Solution by variable transformation . . . . . . . . . . . . . 294
6.2.1.1 Euler’s equation . . . . . . . . . . . . . . . . . . . 294
6.2.2 Reduction of order . . . . . . . . . . . . . . . . . . . . . . . 295
6.2.3 Particular solution by variation of parameters . . . . . . . 296
6.3 Systems of linear di®erential equations . . . . . . . . . . . . . . . . 299
6.3.1 Basic theorems . . . . . . . . . . . . . . . . . . . . . . . . . 300
6.3.2 Particular solution by variation of parameters . . . . . . . 301
6.3.3 Equations with constant coe±cients . . . . . . . . . . . . . 302
6.3.3.1 Homogeneous solutions . . . . . . . . . . . . . . . 303
6.3.3.2 Particular solutions for constant inhomogeneous
term . . . . . . . . . . . . . . . . . . . . . . . . . 307
6.3.3.3 Dealing with complex eigenvalues . . . . . . . . . 310
6.3.3.4 Classi¯cation of steady states . . . . . . . . . . . 311
6.3.3.5 Stability of nonlinear ODEs . . . . . . . . . . . . 319
6.4 Series solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
6.5 Some common functions de¯ned by ODEs . . . . . . . . . . . . . . 333
6.5.1 Exponential and trigonometric functions . . . . . . . . . . 333
6.5.2 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . 334
6.5.3 Legendre functions . . . . . . . . . . . . . . . . . . . . . . 340
7. Hilbert Spaces 345
7.1 In¯nite dimensional vector spaces . . . . . . . . . . . . . . . . . . . 346
7.1.1 Countable and uncountable in¯nities . . . . . . . . . . . . 346
7.1.2 Normed spaces . . . . . . . . . . . . . . . . . . . . . . . . . 348
7.1.3 Bases in in¯nite dimensional spaces . . . . . . . . . . . . . 351
7.1.4 The function spaces Lp[0; 1] . . . . . . . . . . . . . . . . . 353
7.2 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
7.2.1 Inner products . . . . . . . . . . . . . . . . . . . . . . . . . 354
7.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
7.2.3 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . 357
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7.2.4 Orthogonal projections . . . . . . . . . . . . . . . . . . . . 361
7.2.5 Orthogonal complements . . . . . . . . . . . . . . . . . . . 363
7.3 Linear operators in Hilbert spaces . . . . . . . . . . . . . . . . . . 363
7.3.1 The adjoint operator . . . . . . . . . . . . . . . . . . . . . 364
7.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
7.3.3 Sturm-Liouville operators . . . . . . . . . . . . . . . . . . . 367
7.4 Eigenvalue problems . . . . . . . . . . . . . . . . . . . . . . . . . . 368
7.4.1 Sturm-Liouville Problems . . . . . . . . . . . . . . . . . . . 372
7.4.2 Conversion of linear equations to SLP . . . . . . . . . . . . 376
7.5 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
7.5.1 Fourier sine series . . . . . . . . . . . . . . . . . . . . . . . 377
7.5.2 Fourier cosine series . . . . . . . . . . . . . . . . . . . . . . 378
7.5.3 Complete Fourier series . . . . . . . . . . . . . . . . . . . . 378
7.5.4 Gibb’s phenomena . . . . . . . . . . . . . . . . . . . . . . . 381
7.5.5 Generalized Fourier series . . . . . . . . . . . . . . . . . . . 383
8. Partial Di®erential Equations 389
8.1 Fourier series methods . . . . . . . . . . . . . . . . . . . . . . . . . 389
8.1.1 Classi¯cation of second order PDEs . . . . . . . . . . . . . 390
8.1.2 Inner product method . . . . . . . . . . . . . . . . . . . . . 392
8.1.3 PDEs with Sturm-Liouville operators . . . . . . . . . . . . 396
8.1.3.1 Homogeneous problem . . . . . . . . . . . . . . . 397
8.1.3.2 Homogeneous problem with transcendental
equation for eigenvalues . . . . . . . . . . . . . . 399
8.1.3.3 Inhomogeneous PDE . . . . . . . . . . . . . . . . 409
8.1.3.4 Inhomogeneous, time varying boundary
conditions . . . . . . . . . . . . . . . . . . . . . . 411
8.1.4 Other self-adjoint PDEs . . . . . . . . . . . . . . . . . . . . 415
8.2 Finite Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . 425
8.3 First order PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
8.4 First order PDE and Cauchy’s method . . . . . . . . . . . . . . . . 432
8.4.1 Cauchy’s method for linear equations . . . . . . . . . . . . 434
8.5 Similarity transformation . . . . . . . . . . . . . . . . . . . . . . . 448
9. Problems 455
Appendix 497
A.1 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
Index 501