Applied Mathematics and Modeling for Chemical Engineers

Richard G. Rice, PhD is Emeritus Professor in the Department of Chemical Engineering at Louisiana State University, Baton Rouge, LA, USA. Duong D. Do, PhD is Emeritus Professor in the School of Chemical Engineering at the University of Queensland, Australia. James E. Maneval, PhD is Professor in the Department of Chemical Engineering at Bucknell University, Lewisburg, PA, USA.

• Preface to the Third Edition xvPart I 11 Formulation of Physicochemical Problems 31.1 Introduction 31.2 Illustration of the Formulation Process (Cooling of Fluids) 31.2.1 Model I: Plug Flow 31.2.2 Model II: Parabolic Velocity 61.3 Combining Rate and Equilibrium Concepts (Packed-Bed Adsorber) 71.4 Boundary Conditions and Sign Conventions 81.5 Summary of the Model Building Process 91.6 Model Hierarchy and its Importance in Analysis 101.6.1 Level 1 101.6.2 Level 2 111.6.3 Level 3 131.6.4 Level 4 13Problems 15References 202 Modeling with Linear Algebra and Matrices 212.1 Introduction 212.2 Basic Concepts of Systems of Linear Equations 212.3 Matrix Notation 222.3.1 Matrices 222.3.2 Vectors 222.3.3 Scalars 222.3.4 Matrices and Vectors with Special Structure 222.4 Matrix Algebra and Calculus Operations 242.4.1 Equality 242.4.2 Addition and Subtraction 242.4.3 Multiplication 242.4.4 Division 262.4.5 Further Algebraic Properties of Matrices 272.4.6 Basic Differential and Integral Relations for Matrices 282.5 Problem 1: Solution of N Equations in N Unknowns 292.5.1 Analytical Results 292.5.2 Computational Approach: Gauss Elimination 302.6 Problem 2: The Matrix Eigenvalue Problem 322.6.1 Problem Statement and Formal Solution 322.6.2 Computing Eigensystems: Basic Procedure 332.7 Singular Systems 342.7.1 Consistent and Inconsistent Systems 342.7.2 Solution Structure for Consistent Systems 352.7.3 Formulation and Characteristics of Non-Square Problems 362.7.4 Over-Determined Systems: Least-Squares Solution 372.7.5 Under-Determined Systems 382.8 Computational Linear Algebra 402.8.1 The LU Factorization 402.8.2 The QR Factorization 402.8.3 The SVD Factorization 402.8.4 Large-Scale Problems and Iterative Methods 41Problems 42References 473 Solution Techniques for Models Yielding Ordinary Differential Equations 493.1 Geometric Basis and Functionality 493.2 Classification of ODE 503.3 First-Order Equations 503.3.1 Exact Solutions 513.3.2 Equations Composed of Homogeneous Functions 523.3.3 Bernoulli’s Equation 523.3.4 Riccati’s Equation 523.3.5 Linear Coefficients 543.3.6 First-Order Equations of Second Degree 543.4 Solution Methods for Second-Order Nonlinear Equations 553.4.1 Derivative Substitution Method 553.4.2 Homogeneous Function Method 583.5 Linear Equations of Higher Order 593.5.1 Second-Order Unforced Equations: Complementary Solutions 603.5.2 Particular Solution Methods for Forced Equations 643.5.3 Summary of Particular Solution Methods 703.6 Coupled Simultaneous ODE 713.7 Eigenproblems 743.8 Coupled Linear Differential Equations 743.9 Summary of Solution Methods for ODE 75Problems 75References 874 Series Solution Methods and Special Functions 894.1 Introduction to Series Methods 894.2 Properties of Infinite Series 904.3 Method of Frobenius 914.3.1 Indicial Equation and Recurrence Relation 914.4 Summary of the Frobenius Method 984.5 Special Functions 984.5.1 Bessel’s Equation 994.5.2 Modified Bessel’s Equation 1004.5.3 Generalized Bessel’s Equation 1004.5.4 Properties of Bessel Functions 1024.5.5 Differential Integral and Recurrence Relations 103Problems 105References 1075 Integral Functions 1095.1 Introduction 1095.2 The Error Function 1095.2.1 Properties of Error Function 1105.3 The Gamma and Beta Functions 1105.3.1 The Gamma Function 1105.3.2 The Beta Function 1115.4 The Elliptic Integrals 1115.5 The Exponential and Trigonometric Integrals 113Problems 113References 1166 Staged-Process Models: The Calculus of Finite Differences 1176.1 Introduction 1176.1.1 Modeling Multiple Stages 1176.2 Solution Methods for Linear Finite Difference Equations 1186.2.1 Complementary Solutions 1186.3 Particular Solution Methods 1216.3.1 Method of Undetermined Coefficients 1216.3.2 Inverse Operator Method 1226.4 Nonlinear Equations (Riccati Equation) 122Problems 124References 1267 Probability and Statistical Modeling 1277.1 Concepts and Results From Probability Theory 1277.1.1 Experiments and Random Variables 1277.1.2 Probabilities and Distribution Functions 1287.1.3 Characteristics of Distributions Functions 1317.1.4 The Cumulative Distribution Function 1327.2 Concepts and Results From Mathematical Statistics 1347.2.1 Populations Samples and Sampling 1347.2.2 Sample Statistics and Sampling Distributions 1347.3 Statistical Analysis and Modeling 1377.3.1 Confidence Interval for the Mean of a Population 1377.3.2 Hypothesis Tests for the Population Mean 1387.3.3 Hypothesis Tests: Comparing Multiple Means 1407.3.4 Linear Models and Linear Regression 143Problems 150References 1548 Approximate Solution Methods for ODE: Perturbation Methods 1558.1 Perturbation Methods 1558.1.1 Introduction 1558.2 The Basic Concepts 1578.2.1 Gauge Functions 1578.2.2 Order Symbols 1588.2.3 Asymptotic Expansions and Sequences 1588.2.4 Sources of Nonuniformity 1598.3 The Method of Matched Asymptotic Expansion 1608.3.1 Outer Solutions 1608.3.2 Inner Solutions 1608.3.3 Matching 1618.3.4 Composite Solutions 1618.3.5 General Matching Principle 1628.3.6 Composite Solution of Higher Order 1628.4 Matched Asymptotic Expansions for Coupled Equations 1638.4.1 Outer Expansion 1638.4.2 Inner Expansion 1648.4.3 Matching 164Problems 165References 173Part II 1759 Numerical Solution Methods (Initial Value Problems) 1779.1 Introduction 1779.2 Type of Method 1799.3 Stability 1809.4 Stiffness 1859.5 Interpolation and Quadrature 1869.6 Explicit Integration Methods 1879.7 Implicit Integration Methods 1889.8 Predictor–Corrector Methods and Runge–Kutta Methods 1899.8.1 Predictor–Corrector Methods 1899.9 Runge–Kutta Methods 1899.10 Extrapolation 1919.11 Step Size Control 1929.12 Higher-Order Integration Methods 192Problems 192References 19510 Approximate Methods for Boundary Value Problems: Weighted Residuals 19710.1 The Method of Weighted Residuals 19710.1.1 Variations on a Theme of Weighted Residuals 19810.2 Jacobi Polynomials 20510.2.1 Rodrigues Formula 20510.2.2 Orthogonality Conditions 20510.3 Lagrange Interpolation Polynomials 20610.4 Orthogonal Collocation Method 20610.4.1 Differentiation of a Lagrange Interpolation Polynomial 20610.4.2 Gauss–Jacobi Quadrature 20710.4.3 Radau and Lobatto Quadrature 20810.5 Linear Boundary Value Problem: Dirichlet Boundary Condition 20910.6 Linear Boundary Value Problem: Robin Boundary Condition 21110.7 Nonlinear Boundary Value Problem: Dirichlet Boundary Condition 21310.8 One-Point Collocation 21510.9 Summary of Collocation Methods 21510.10 Concluding Remarks 216Problems 217References 22511 Introduction to Complex Variables and Laplace Transforms 22711.1 Introduction 22711.2 Elements of Complex Variables 22711.3 Elementary Functions of Complex Variables 22811.4 Multivalued Functions 22911.5 Continuity Properties for Complex Variables: Analyticity 23011.5.1 Exploiting Singularities 23111.6 Integration: Cauchy’s Theorem 23211.7 Cauchy’s Theory of Residues 23311.7.1 Practical Evaluation of Residues 23411.7.2 Residues at Multiple Poles 23511.8 Inversion of Laplace Transforms by Contour Integration 23511.8.1 Summary of Inversion Theorem for Pole Singularities 23711.9 Laplace Transformations: Building Blocks 23711.9.1 Taking the Transform 23711.9.2 Transforms of Derivatives and Integrals 23811.9.3 The Shifting Theorem 24011.9.4 Transform of Distribution Functions 24011.10 Practical Inversion Methods 24211.10.1 Partial Fractions 24211.10.2 Convolution Theorem 24311.11 Applications of Laplace Transforms for Solutions of ODE 24311.12 Inversion Theory for Multivalued Functions: The Second Bromwich Path 24811.12.1 Inversion When Poles and Branch Points Exist 25011.13 Numerical Inversion Techniques 25011.13.1 The Zakian Method 25011.13.2 The Fourier Series Approximation 252Problems 253References 25712 Solution Techniques for Models Producing PDEs 25912.1 Introduction 25912.1.1 Classification and Characteristics of Linear Equations 26112.2 Particular Solutions for PDEs 26312.2.1 Boundary and Initial Conditions 26312.3 Combination of Variables Method 26412.4 Separation of Variables Method 26912.4.1 Coated Wall Reactor 26912.5 Orthogonal Functions and Sturm–Liouville Conditions 27212.5.1 The Sturm–Liouville Equation 27212.6 Inhomogeneous Equations 27512.7 Applications of Laplace Transforms for Solutions of PDEs 279Problems 285References 30213 Transform Methods for Linear PDEs 30513.1 Introduction 30513.2 Transforms in Finite Domain: Sturm–Liouville Transforms 30513.2.1 Development of Integral Transform Pairs 30613.2.2 The Eigenvalue Problem and the Orthogonality Condition 30913.2.3 Inhomogeneous Boundary Conditions 31313.2.4 Inhomogeneous Equations 31613.2.5 Time-Dependent Boundary Conditions 31713.2.6 Elliptic Partial Differential Equations 31713.3 Generalized Sturm–Liouville Integral Transform 32013.3.1 Introduction 32013.3.2 The Batch Adsorber Problem 320Problems 327References 33114 Approximate and Numerical Solution Methods for PDEs 33314.1 Polynomial Approximation 33314.2 Singular Perturbation 33814.3 Finite Difference 34314.3.1 Notations 34314.3.2 Essence of the Method 34414.3.3 Tridiagonal Matrix and the Thomas Algorithm 34514.3.4 Linear Parabolic Partial Differential Equations 34514.3.5 Nonlinear Parabolic Partial Differential Equations 34914.4 Orthogonal Collocation for Solving PDEs 35014.4.1 Elliptic PDE 35014.4.2 Parabolic PDE: Example 1 35314.4.3 Coupled Parabolic PDE: Example 2 354Problems 355References 362Appendix A: Review of Methods for Nonlinear Algebraic Equations 363A.1 The Bisection Algorithm 363A.2 The Successive Substitution Method 364A.3 The Newton–Raphson Method 366A.4 Rate of Convergence 367A.4.1 Definition of Speed of Convergence 367A.5 Multiplicity 368A.5.1 Multiplicity 368A.6 Accelerating Convergence 369References 369Appendix B: Derivation of the Fourier–Mellin Inversion Theorem 371References 374Appendix C: Table of Laplace Transforms 375Appendix D: Numerical Integration 381D.1 Basic Idea of Numerical Integration 381D.2 Newton Forward Difference Polynomial 381D.3 Basic Integration Procedure 382D.3.1 Trapezoid Rule 382D.3.2 Simpson’s Rule 383D.4 Error Control and Extrapolation 384D.5 Gaussian Quadrature 384D.6 Radau Quadrature 386D.7 Lobatto Quadrature 388D.8 Concluding Remarks 389References 389Appendix E: Nomenclature 391Appendix F: Statistical Tables 395Postface 399Index 401