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 © World Scientific Publishing Co. Pte. Ltd. http://www.worldscibooks.com/engineering/6774.html 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 xi LINEAR MATHEMATICAL MODELS IN CHEMICAL ENGINEERING © World Scientific Publishing Co. Pte. Ltd. http://www.worldscibooks.com/engineering/6774.html xii Linear Mathematical Models in Chemical Engineering 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 LINEAR MATHEMATICAL MODELS IN CHEMICAL ENGINEERING © World Scientific Publishing Co. Pte. Ltd. http://www.worldscibooks.com/engineering/6774.html Contents xiii 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 LINEAR MATHEMATICAL MODELS IN CHEMICAL ENGINEERING © World Scientific Publishing Co. Pte. Ltd. http://www.worldscibooks.com/engineering/6774.html xiv Linear Mathematical Models in Chemical Engineering 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 LINEAR MATHEMATICAL MODELS IN CHEMICAL ENGINEERING © World Scientific Publishing Co. Pte. Ltd. http://www.worldscibooks.com/engineering/6774.html Contents xv 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