Numerical Methods with Chemical Engineering Applications

Contents
List of Illustrations
List of Tables
List of Programs
Preface
Acknowledgments
1 Mathematical Modeling and Structured Programming
	1.1 Elements of a Mathematical Model
	1.2 Some Analytically Solvable Chemical Engineering Models
		1.2.1 Vapor–Liquid Equilibrium
		1.2.2 Batch Reactor with a Single Reactant
		1.2.3 Batch Reactor with Three Species
		1.2.4 Continuous Stirred Tank Reactor
		1.2.5 Reaction–Diffusion in a Slab
		1.2.6 Unsteady Diffusion
	1.3 Computer Math
		1.3.1 Three Key Operations
		1.3.2 Words and Round-Off Error
	1.4 Structured Programming
		1.4.1 Learning to Program
		1.4.2 The for Loop
		1.4.3 The while Loop
		1.4.4 If-else Statements
		1.4.5 Modular Programming
	1.5 Further Reading
	Problems
	Computer Problems
2 Linear Algebraic Equations
	2.1 Definition of a Linear System of Algebraic Equations
	2.2 Does a Solution Exist?
	2.3 Conditioning
		2.3.1 Vector Spaces
		2.3.2 Vector Norms
		2.3.3 Matrix Norms
		2.3.4 Condition Number
	2.4 Cramer’s Rule
	2.5 Gauss Elimination
		2.5.1 Algorithm
		2.5.2 Computational Cost
	2.6 Pivoting
		2.6.1 A Simple Example
		2.6.2 Algorithm
	2.7 Case Study: Multi-component Flash
		2.7.1 Problem Statement
		2.7.2 Solution
	2.8 Calculation of the Determinant by Gauss Elimination
	2.9 Banded Matrices
	2.10 Simultaneous Gauss Elimination
	2.11 LU Decomposition
		2.11.1 Doolittle’s Method
		2.11.2 Implementation
	2.12 Case Study: Linear Process Flow Sheet
		2.12.1 Problem Statement
		2.12.2 Solution
	2.13 Iterative Methods
		2.13.1 Jacobi’s Method
		2.13.2 Gauss–Seidel Method
		2.13.3 Successive Relaxation Method
		2.13.4 Convergence
	2.14 Application: Polynomial Regression
	2.15 Implementation in MATLAB
	2.16 Further Reading
	Problems
	Computer Problems
3 Nonlinear Algebraic Equations
	3.1 Introduction
		3.1.1 Superposition
		3.1.2 Differences Between Linear and Nonlinear Equations
	3.2 Solution Approaches
		3.2.1 Bracketing Methods
		3.2.2 Fixed-Point (Open) Methods
	3.3 Fixed-Point Methods for Single Nonlinear Algebraic Equations
		3.3.1 Picard’s Method
		3.3.2 Newton’s Method
	3.4 Convergence of General Fixed-Point Iteration
		3.4.1 Will We Find a Solution?
		3.4.2 Rate of Convergence
	3.5 Case Study: Eigenvalue Calculation in Unsteady Diffusion
		3.5.1 Problem Statement
		3.5.2 Solution
	3.6 Case Study: Extension of a Tethered Polymer by a Force
		3.6.1 Problem Statement
		3.6.2 Solution
	3.7 Systems of Nonlinear Algebraic Equations
		3.7.1 Picard’s Method
		3.7.2 Newton–Raphson
	3.8 Case Study: Nonlinear Flash
		3.8.1 Problem Statement
		3.8.2 Solution
		3.8.3 A Variation on the Problem
	3.9 Continuation Methods
		3.9.1 Zero-Order Continuation
		3.9.2 First-Order Continuation
	3.10 Case Study: Liquid–Liquid Phase Diagram from the van Laar Model
		3.10.1 Problem Statement
		3.10.2 Solution
	3.11 Implementation in MATLAB
	3.12 Further Reading
	Problems
	Computer Problems
4 Initial Value Problems
	4.1 Introduction
		4.1.1 Classification of ODE Problems
		4.1.2 General Solution Approach
	4.2 First-Order ODE IVPs
		4.2.1 Explicit (Forward) Euler
		4.2.2 Implicit (Backward) Euler
		4.2.3 Predictor–Corrector
		4.2.4 Runge–Kutta Methods
	4.3 Numerical Stability
		4.3.1 Linear IVPs
		4.3.2 Nonlinear IVPs
	4.4 Case Study: CSTR with a Fluctuating Inlet
		4.4.1 Problem Statement
		4.4.2 Solution
	4.5 Case Study: Liquid Level Dynamics and Control
		4.5.1 Problem Statement
		4.5.2 Solution
	4.6 Systems of ODE IVPs
		4.6.1 Explicit Methods
		4.6.2 Implicit Euler
	4.7 Higher-Order ODE IVPs
	4.8 Numerical Stability for Systems of ODEs
		4.8.1 Analysis by Matrix Diagonalization
		4.8.2 Stiff Systems
	4.9 Case Study: Non-Isothermal PFR
		4.9.1 Problem Statement
		4.9.2 Solution
	4.10 Implementation in MATLAB
	4.11 Further Reading
	Problems
	Computer Problems
5 Dynamical Systems
	5.1 Introduction
	5.2 Equilibrium Points and Their Stability
	5.3 Dynamical Systems on the Plane
		5.3.1 Linear Systems on the Plane – Classification of Steady States
		5.3.2 Nonlinear Systems on the Plane
	5.4 Nonlinear Phenomena
		5.4.1 Periodic Solutions
		5.4.2 Limit Cycles
	5.5 Bifurcations
		5.5.1 One-Dimensional Systems
		5.5.2 Two-Dimensional Systems
	5.6 Deterministic Chaos
	5.7 Case Study: Non-Isothermal CSTR
		5.7.1 Problem Statement
		5.7.2 Solution
	5.8 Implementation in MATLAB
	5.9 Further Reading
	Problems
	Computer Problems
6 Boundary Value Problems
	6.1 ODE Boundary Value Problems
	6.2 Finite Difference Approximations
		6.2.1 Discretization in Space
		6.2.2 Derivation of Finite Difference Equations
	6.3 Solution Procedure
		6.3.1 Dirichlet Boundary Conditions
		6.3.2 Neumann and Robin Boundary Conditions
		6.3.3 Position-Dependent Terms
		6.3.4 Mesh Refinement
	6.4 Case Study: Reaction–Diffusion in a Packed Bed
		6.4.1 Problem Statement
		6.4.2 Solution
	6.5 Coupled BVPs
		6.5.1 Approach #1: Appending the Unknown Vectors
		6.5.2 Approach #2: Interlacing the Variables
	6.6 Case Study: Coupled Heat and Mass Transfer
		6.6.1 Problem Statement
		6.6.2 Solution
	6.7 Implementation in MATLAB
	6.8 Further Reading
	Problems
	Computer Problems
7 Partial Differential Equations
	7.1 Introduction
	7.2 Solution of IBVPs – Method of Lines
		7.2.1 Solution Method
		7.2.2 Numerical Stability
	7.3 Case Study: Unsteady Diffusion Through a Membrane
		7.3.1 Problem Statement
		7.3.2 Solution
	7.4 Two-Dimensional BVPs
	7.5 Case Study: 2D Reaction–Diffusion
		7.5.1 Problem Statement
		7.5.2 Solution
	7.6 Implementation in MATLAB
	7.7 Further Reading
	Problems
	Computer Problems
8 Interpolation and Integration
	8.1 Introduction
	8.2 Polynomial Interpolation
		8.2.1 Linear Interpolation
		8.2.2 Quadratic Interpolation
		8.2.3 Newton’s Divided Differences and Polynomial Interpolation
		8.2.4 Equally Spaced Data
		8.2.5 Error of Newton’s Interpolating Polynomials
	8.3 Newton–Coates Integration
		8.3.1 Trapezoidal Rule (n = 1)
		8.3.2 Simpson’s 1/3 Rule (n = 2)
		8.3.3 Higher-Order Formulas
	8.4 Applications of Trapezoidal Rule
		8.4.1 Multiple Trapezoidal Rule
		8.4.2 Richardson Extrapolation
		8.4.3 Romberg Integration
	8.5 Gauss Quadrature
	8.6 Case Study: Concentrated Differential Distillation
		8.6.1 Problem Statement
		8.6.2 Solution
	8.7 Implementation in MATLAB
	8.8 Further Reading
	Problems
	Computer Problems
Appendix A MATLAB “Tutorial”
	A.1 Data Structures
	A.2 Matrix and Vector Manipulation
	A.3 Order of Operations
	A.4 Displaying Data on the Screen
	A.5 Logical Flow
	A.6 Loops
	A.7 Plotting
	A.8 Functions
Appendix B Determinant of a Matrix
Index