Numerical Methods for Chemical Engineering: Applications in MATLAB

Kenneth J Beers,Ebooks Corporation.9780511256509, 0511256507

Suitable for a first year graduate course, this textbook unites the applications of numerical mathematics and scientific computing to the practice of chemical engineering. Written in a pedagogic style, the book describes basic linear and nonlinear algebric systems all the way through to stochastic methods, Bayesian statistics and parameter estimation. These subjects are developed at a level of mathematics suitable for graduate engineering study without the exhaustive level of the theoretical mathematical detail. The implementation of numerical methods in MATLAB is integrated within each chapter and numerous examples in chemical engineering are provided, with a library of corresponding MATLAB programs. This book will provide the graduate student with essential tools required by industry and research alike. Supplementary material includes solutions to homework problems set in the text, MATLAB programs and tutorial, lecture slides, and complicated derivations for the more advanced reader

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Table of contents :
Cover……Page 1
Half-title……Page 3
Title……Page 5
Copyright……Page 6
Contents……Page 7
Preface……Page 11
Linear systems of algebraic equations……Page 15
Scalars, real and complex……Page 17
Vector notation and operations……Page 18
Multiplication of a square N × N matrix A with an N-dimensional vector v……Page 22
Matrix transposition……Page 23
Elimination methods for solving linear systems……Page 24
Elementary row operations……Page 25
Gaussian elimination to place Ax = b in upper triangular form……Page 27
Basic algorithm for solving Ax = b by Gaussian elimination……Page 31
Partial pivoting……Page 33
Interpreting Ax = b as a linear transformation……Page 37
Multiplication of matrices……Page 39
Vector spaces and basis sets……Page 40
Gram–Schmidt orthogonalization……Page 42
The null space and the existence/uniqueness of solutions……Page 43
The determinant……Page 46
General properties of the determinant function……Page 48
Computing the determinant value……Page 49
Matrix inversion……Page 50
LU decomposition……Page 52
Cholesky decomposition……Page 56
Submatrices and matrix partitions……Page 58
Example. Modeling a separation system……Page 59
Sparse and banded matrices……Page 60
Example. Solving a boundary value problem from fluid mechanics……Page 61
Banded and sparse matrices……Page 64
Treatment of sparse, banded matrices in MATLAB……Page 66
Fill-in (why Gaussian elimination is sometimes impractical)……Page 68
MATLAB summary……Page 70
Problems……Page 71
Existence and uniqueness of solutions to a nonlinear algebraic equation……Page 75
Iterative methods and the use of Taylor series……Page 76
Newton’s method for a single equation……Page 77
Performance of Newton’s method for a single equation……Page 78
Formal convergence properties of Newton’s method for a single equation……Page 81
The secant method……Page 83
Finding complex solutions……Page 84
Systems of multiple nonlinear algebraic equations……Page 85
Newton’s method for multiple nonlinear equations……Page 86
Performance of Newton’s method for an example system of two equations……Page 88
Estimating the Jacobian and quasi-Newton methods……Page 91
Robust reduced-step Newton method……Page 93
The backtracking weak line search method……Page 94
The trust-region Newton method……Page 95
Solving nonlinear algebraic systems in MATLAB……Page 97
Example. 1-D laminar flow of a shear-thinning polymer melt……Page 99
Homotopy……Page 102
Example. Steady-state modeling of a condensation polymerization reactor……Page 103
Rate equations for polycondensation……Page 104
Steady-state model of a stirred-tank polycondensation reactor……Page 106
Effect of Da and mass transfer upon polymer chain length……Page 107
Bifurcation analysis……Page 108
Example. Bifurcation points of a simple quadratic equation……Page 110
Numerical calculation of bifurcation points……Page 111
MATLAB summary……Page 112
Problems……Page 113
Orthogonal matrices……Page 118
A specific example of an orthogonal matrix……Page 119
Eigenvalues and eigenvectors defined……Page 120
A is triangular……Page 121
Analytical computation of eigenvectors……Page 122
Multiplicity and formulas for the trace and determinant……Page 123
Eigenvalues and the existence/uniqueness properties of linear systems……Page 124
Estimating eigenvalues; Gershgorin’s theorem……Page 125
Matrix norm, spectral radius, and condition number……Page 127
Applying Gershgorin’s theorem to study the convergence of iterative linear solvers……Page 128
Irreducible matrices……Page 130
Eigenvector properties of a general N × N complex matrix……Page 131
Special eigenvector properties of normal matrices……Page 135
Computing all eigenvalues and eigenvectors with eig……Page 137
Computing extremal eigenvalues and their eigenvectors with eigs……Page 138
Computing extremal eigenvalues……Page 140
Finding the next largest eigenvalues of a positive-semidefinite matrix……Page 142
QR decomposition of a real matrix……Page 143
Iterative QR method for computing all eigenvalues……Page 145
Improving the efficiency of the QR method……Page 146
Example. QR method for a real 4 × 4 matrix……Page 147
Normal mode analysis……Page 148
The generalized eigenvalue problem……Page 150
Eigenvalue problems in quantum mechanics……Page 151
Numerical solution of a differential equation eigenvalue problem……Page 152
Singular value decomposition (SVD)……Page 155
SVD analysis and the existence/uniqueness properties of linear systems……Page 157
Least-squares approximate solutions……Page 159
SVD in MATLAB……Page 160
Computing the roots of a polynomial……Page 162
Problems……Page 163
4 Initial value problems……Page 168
Initial value problems of ordinary differential equations (ODE-IVPs)……Page 169
Polynomial interpolation……Page 170
Newton interpolation……Page 171
Hermite interpolation……Page 174
Other types of interpolation……Page 175
Newton–Cotes integration……Page 176
Gaussian quadrature……Page 177
Preliminary definitions……Page 178
Gaussian quadrature……Page 179
Gaussian quadrature with w(x) = 1 and the use of quadl……Page 180
Multidimensional integrals……Page 181
Monte Carlo integration……Page 182
Linear ODE systems and dynamic stability……Page 183
Stability of the steady state of a linear system……Page 184
Example. Stability of steady states for nonlinear ODE systems……Page 186
Explicit single-step methods……Page 190
Implicit multistep methods……Page 192
Stiffness and the choice of integration method……Page 194
ODE solvers in MATLAB……Page 195
Example. Stiffness and the QSSA in chemical kinetics……Page 197
Accuracy and stability of single-step methods……Page 199
Numerical accuracy and the order of an integration method……Page 200
Absolute stability of an integration method……Page 201
Error rejection……Page 204
Stiff systems from discretized PDEs……Page 205
Stiff stability of BDF methods……Page 206
Symplectic methods for classical mechanics……Page 208
BDF method for DAE systems of index one……Page 209
Example. Heterogeneous catalysis in a packed bed reactor……Page 213
Parametric continuation……Page 217
Example. Multiple steady states in a nonisothermal CSTR……Page 218
MATLAB summary……Page 221
Problems……Page 222
Local methods for unconstrained optimization problems……Page 226
Gradient methods……Page 227
Strong and weak line searches……Page 230
Choosing the search direction……Page 231
A gradient minimizer routine……Page 232
Conjugate gradient method applied to quadratic cost functions……Page 234
Newton line search methods……Page 237
The dogleg method……Page 239
Newton methods for large problems……Page 241
Example. A simple cost function……Page 242
Example. Fitting a kinetic rate law to time-dependent data……Page 244
Lagrangian methods for constrained optimization……Page 245
Optimization with equality constraints……Page 246
Treatment of inequality constraints……Page 249
Sequential quadratic programming (SQP)……Page 254
Example. Finding the closest points on two ellipses……Page 256
Example. Optimal steady-state design of a CSTR……Page 258
Optimal control……Page 259
An open-loop optimal control routine……Page 261
Dynamic programming……Page 262
Example. A simple 1-D optimal control problem……Page 264
Problems……Page 266
BVPs from conservation principles……Page 272
The finite difference method applied to a 2-D BVP……Page 274
Finite difference approximations……Page 276
Finite difference solution of a Poisson BVP……Page 277
Extending the finite difference method……Page 278
Chemical reaction and diffusion in a spherical catalyst pellet……Page 279
Dimensionless formulation……Page 280
Treatment of Dirichlet and von Neumann boundary conditions……Page 281
Numerical solution in MATLAB……Page 283
Finite differences for a convection/diffusion equation……Page 284
Central difference scheme (CDS)……Page 285
Why does upwind differencing work?……Page 287
Characteristics and types of PDEs……Page 289
Modeling a tubular chemical reactor with dispersion; treating multiple fields……Page 293
Solution by upwind finite differences……Page 294
Numerical issues for discretized PDEs with more than two spatial dimensions……Page 296
The Jacobi, Gauss–Seidel, and successive over-relaxation (SOR) methods……Page 299
The conjugate gradient method for positive-definite matrices……Page 300
The generalized minimum residual (GMRES) Krylov subspace method……Page 301
The use of preconditioners……Page 302
Example. 3-D heat transfer in a stove top element……Page 306
Finite differences in complex geometries……Page 308
The finite volume method……Page 311
The finite element method (FEM)……Page 313
Automatic mesh generation……Page 314
Weighted-residual methods and the Galerkin formulation of FEM……Page 318
Solving Poisson’s equation in two dimensions with the FEM……Page 319
FEM in MATLAB……Page 323
Numerical solution of a 2-D BVP using the MATLAB PDE toolkit……Page 324
MATLAB summary……Page 325
Problems……Page 326
Condensation polymers……Page 331
Chain length distribution in linear condensation polymers; joint and conditional probabilities……Page 332
Gelation of multifunctional monomers (more on conditional probabilities and mathematical expectation)……Page 336
Important probability distributions……Page 339
Bernoulli trials……Page 341
The random walk problem……Page 342
The binomial distribution……Page 343
The Gaussian (normal) distribution……Page 345
The Gaussian distribution with nonzero mean……Page 347
The Poisson distribution……Page 348
Random vectors and multivariate distributions……Page 350
The Boltzmann and Maxwell distributions……Page 351
Brownian dynamics and stochastic differential equations (SDEs)……Page 352
The Langevin equation……Page 354
The Wiener process……Page 355
Stochastic Differential Equations (SDEs)……Page 356
Itos stochastic calculus……Page 357
Example. Stochastic calculus in quantitative finance……Page 360
The Fokker–Planck equation……Page 361
The Einstein relation……Page 365
General formulation of SDEs; Brownian motion in multiple dimensions……Page 366
Markov chains……Page 367
Monte Carlo simulation in statistical mechanics……Page 368
Example. Monte Carlo simulation of a 2-D Ising lattice……Page 370
Field theory and stochastic PDEs……Page 372
Monte Carlo integration……Page 374
Simulated annealing……Page 375
Genetic programming……Page 376
MATLAB summary……Page 378
Problems……Page 379
General problem formulation……Page 386
Example. Fitting kinetic parameters of a chemical reaction……Page 387
Fitting the rate law to initial rate measurements in a batch reactor……Page 388
Transforming the batch reactor data to obtain a linear regression problem……Page 389
Fitting the rate law from multiple sources……Page 390
Single-response linear regression……Page 391
Linear least-squares regression……Page 392
Example. Least-squares .tting of rate law parameters to transformed batch data……Page 393
Example. Comparing protein expression data for two bacterial strains……Page 394
The Bayesian view of statistical inference……Page 395
Bayes’ theorem……Page 396
Bayesian view of single-response regression……Page 397
Some general considerations about the selection of a prior……Page 400
Numerical treatment of nonlinear least-squares problems……Page 402
Selecting a prior for single-response data……Page 403
Noninformative prior for theta……Page 404
Non informative prior for sigma……Page 407
The posterior density for single-response data……Page 408
Confidence interval for the mean of a population and the t-distribution……Page 409
Confidence intervals for model parameters……Page 411
Confidence intervals on the model predictions……Page 412
Least-squares fitting and confidence interval generation in MATLAB……Page 413
Nonlinear least-squares calculations in MATLAB……Page 414
Example. Fitting the rate constant from the full curve of concentration of C vs. time for a batch reactor experiment……Page 416
MCMC techniques in Bayesian analysis……Page 417
MCMC computation of posterior predictions……Page 418
Example. Protein expression data for bacterial strains……Page 420
MCMC computation of marginal posterior densities……Page 421
Computing highest probability density (HPD) regions from marginal posterior distributions……Page 423
Example. Batch reactor chemical reaction data……Page 425
Applying eigenvalue analysis to experimental design……Page 426
Bayesian multiresponse regression……Page 428
Reduction of the multiresponse posterior density to the previous result for single-response data……Page 430
Numerically computing the parameter estimate……Page 431
Example. Fitting the rate constant from multiresponse dynamic batch reactor data……Page 432
MCMC simulation with the multiresponse marginal posterior density……Page 433
Analysis of composite data sets……Page 435
Example. Numerical analysis of composite data sets, applied to the problem of estimating the rate constant of a reaction from multiple reactor data sets……Page 436
Bayesian testing and model criticism……Page 440
Null hypothesis testing……Page 441
Schwartz’s Bayesian information criterion (BIC)……Page 442
MATLAB summary……Page 445
Problems……Page 446
Fourier series and transforms in one dimension……Page 450
Gibbs oscillations……Page 451
Exponential form of the Fourier series……Page 452
The Fourier transform……Page 453
The discrete Fourier transform……Page 454
The fast Fourier transform (FFT)……Page 457
Special properties of the Fourier transform of a real function and the power spectrum……Page 458
1-D Fourier transforms in MATLAB……Page 459
Aliasing……Page 460
Convolution of two signals……Page 461
Correlation of two signals……Page 463
The discrete d-dimensional Fourier transform……Page 464
FFT in multiple dimensions in MATLAB……Page 465
Scattering theory……Page 466
Applying Fourier analysis……Page 470
Scattering peaks from samples with periodic structure……Page 471
Problems……Page 473
References……Page 475
Index……Page 478