Preface xi
About the Companion Website xiii
1 Introduction to Modeling 1
1.1 Numerical Methods 3
1.1.1 Linear vs. Nonlinear Equations 5
1.1.2 Standard Features of Numerical Methods 6
1.1.3 Significant Figures and Computer Round-Off Error 7
1.2 Mathematical Model Building 12
1.2.1 Laws of Conservation 14
1.2.2 Lumped vs. Distributed Systems 16
1.2.3 Degrees of Freedom 23
1.2.4 Dimensionless Equations 23
1.2.5 Stochastic vs. Deterministic Modeling 24
1.2.6 Model Verification and Validation 24
1.3 Expert Problem-Solvers 25
2 MATLAB Skill Preparation 27
2.1 Introduction 27
2.1.1 MATLAB User Interface 27
2.1.2 MATLAB File Extensions 29
2.2 Some Useful MATLAB Commands and Functions 33
2.2.1 MATLAB Commands 33
2.2.2 MATLAB Functions 33
2.3 Vector and Matrix Operations 39
2.3.1 Vector Operations 39
2.3.2 Matrix Operations 40
2.4 Loops & Conditional Statements 42
2.5 Solving Equations 43
2.5.1 Linear and Nonlinear Equations and Systems 44
2.6 Optimization Toolbox 46
2.6.1 Minimization vs. Maximization Problems 46
2.6.2 Optimization Solvers 46
Problems 49
3 Model Fitting and Spline Functions 51
3.1 Introduction 51
3.1.1 Linear Equations 52
3.1.1.1 Linear Models 52
3.1.1.2 Model Linearization 52
3.1.2 Algebraic Nonlinear Models 53
3.1.2.1 Nonlinear Models 53
3.1.2.2 Solution Methods for Nonlinear Problems 53
3.1.3 Model Fitting 55
3.1.3.1 Least Squares Procedure 55
3.1.3.2 Regression Analysis 55
3.1.4 Spline Functions 57
3.1.4.1 Linear Interpolation 57
3.1.4.2 Polynomial Interpolation 57
3.1.4.3 Cubic Spline Interpolation 58
3.2 Curve Fitting Toolbox in MATLAB 58
3.2.1 Two Ways to Curve Fitting 58
3.2.2 An Example of Using Curve Fitting App 59
3.2.2.1 Open the Curve Fitting App in Command Window 59
3.2.2.2 Selecting Data and the Default Linear Model 60
3.2.2.3 Selecting Better Models 60
3.3 Model Fitting via Optimization Algorithms 65
3.4 Spline Functions for Complex Models 65
3.4.1 Spline Interpolation in MATLAB – interp1 66
3.4.2 Spline Interpolation in MATLAB – interp2 68
3.5 Application in Chemical Engineering 69
3.5.1 Using Linear Equations 69
3.5.2 Using Nonlinear Equations 72
3.6 Summary 75
Problems 76
4 Optimization 79
4.1 Optimization Overview and Solver Options 79
4.1.1 What Is Optimization? 79
4.1.2 Built-In Function and MATLAB’s Toolbox for Optimization 80
4.2 Extreme Value Problem 82
4.2.1 Unconstrained Functions 82
4.2.2 Constrained Functions 92
4.3 Linear Programming 93
4.3.1 Standard Procedure of Linear Programming 94
4.3.2 Solving Linear Programming in MATLAB 95
4.4 Quadratic Programming 96
4.4.1 Standard Procedure of Quadratic Programming 96
4.4.2 Solving Quadratic Programming in MATLAB 97
4.5 Nonlinear Programming 98
4.6 Application in Chemical Engineering 100
4.6.1 Reaction Kinetics Optimization 100
4.6.2 Reaction Productivity Optimization 101
4.6.3 Refinery Economics 103
4.6.4 Chemical Production Optimization 105
4.6.5 Productivity Profit Optimization 107
4.6.6 Catalyst System Design Optimization 108
4.6.7 Reaction Kinetic 109
4.7 Summary 110
Problems 110
5 Enhanced Optimization 115
5.1 Introduction 115
5.1.1 Differences Among Optimization Solvers in MATLAB 115
5.1.2 The Differences in Algorithms 116
5.2 Enhanced Optimization Through Treatment of Constraints on the Search Region 119
5.2.1 Lagrange Method 119
5.2.1.1 Method of Lagrange Multiplier 119
5.2.1.2 Method of Optimizing S Function 124
5.2.2 Convert Inequality Constraints to Equality Constraints 126
5.2.3 Penalty Function Method 128
5.2.4 Enhanced Constraints Method 133
5.3 Application in Chemical Engineering 135
5.3.1 Optimization of a Three-Stage Compressor 135
5.3.2 Productivity Profit Optimization 138
5.3.3 Optimization for the Operation of a Continuous Stirred Tank Reactor (cstr) 140
5.4 Summary 145
Problems 145
6 Global Optimization 147
6.1 Introduction 147
6.1.1 Local vs. Global Optima 147
6.1.2 Four Standard Heuristic Techniques (Often Incorporate Randomization) 148
6.2 Global Optimization Toolbox in MATLAB 151
6.2.1 Global Optimization Solvers in MATLAB 151
6.2.2 The Differences in the Syntax of the Solvers 152
6.3 The Selection of Global Optimization Methods 156
6.3.1 Based on Constraints 156
6.3.2 Based on the Number of Variables 159
6.3.3 Based on Computational Time 162
6.4 Application in Chemical Engineering 166
6.4.1 Optimization of a Three-Stage Compressor 166
6.4.2 Productivity Profit Optimization 172
6.4.3 Optimization for the Operation of a Continuous Stirred Tank Reactor (cstr) 179
6.5 Summary 186
Problems 186
7 Optimal Experimental Design 189
7.1 Introduction 189
7.1.1 Experimental Design 189
7.1.2 Experimental Error 190
7.1.2.1 Random Error 190
7.1.2.2 Systematic Error 192
7.2 Design of Experiments (DOE) 192
7.2.1 Statistical Design of Experiments 192
7.2.2 MATLAB Functions for the Design of Experiments 193
7.2.3 Orthogonal Experimental Design 196
7.3 Model Building with Experimental Data 200
7.3.1 The Relationships Among M Values vs. Three Factors 201
7.3.2 The Power Relationships and Multiplication used for Model Building 203
7.3.3 The Model Fitting to Determine the Unknown Parameters 203
7.4 Application in Chemical Engineering 208
7.4.1 Experimental Design for Nanoparticle-Reinforced Silica Gels 208
7.4.2 Experimental Design for Turbulence Model Development of a Flotation Cell 211
7.4.2.1 The Relationships Among M Values vs. Four Factors 212
7.4.2.2 The Power Relationships and Multiplication Used for Model Building 216
7.4.2.3 The Model Fitting to Determine the Unknown Parameters 217
7.5 Summary 220
Problems 220
8 Data Statistics 223
8.1 Introduction 223
8.1.1 Data Types 223
8.1.2 Physical Units 226
8.1.3 Dimensional Analysis 228
8.1.4 Some Dimensionless Groups 228
8.1.5 Buckingham Pi Method 231
8.1.6 Model Analysis 232
8.2 Data Statistics Toolbox in MATLAB 234
8.2.1 Statistics and Machine Learning Toolbox 234
8.2.2 Statistics Visualization 234
8.2.2.1 Scatter Plots 234
8.2.2.2 Box Plots 235
8.2.2.3 Distribution Plots 236
8.2.3 Analysis of Variance 238
8.2.3.1 One-Way ANOVA 238
8.2.3.2 ANOVA with Multiple Responses 238
8.2.4 Regression 241
8.2.5 Principal Component Analysis 248
8.3 Application in Chemical Engineering 250
8.3.1 Application of Buckingham Pi (π) Method for a Flotation Rate Model 250
8.3.2 Statistics Visualization for Particle Size Distributions 251
8.4 Summary 261
Problems 262
9 Complex Equation Systems 265
9.1 Introduction 265
9.1.1 Dynamic Systems 265
9.1.2 Lump Parameter System and Distributed Parameter System 266
9.2 Ordinary Differential Equations (ODEs) 267
9.2.1 ODEs for Dynamic Systems 267
9.2.2 Higher-Order ODEs 267
9.2.3 Numerical Integration and Differentiation 268
9.2.4 Numerical Method to Solve ODEs 271
9.2.4.1 Euler’s Explicit Method 271
9.2.4.2 Euler’s Implicit Method 274
9.2.4.3 Runge–Kutta Methods 276
9.2.4.4 Finite-Difference Method 277
9.2.5 MATLAB ODE Solvers 279
9.3 Partial Differential Equation Toolbox in MATLAB (PDE) 283
9.3.1 Forms of Equations Can Be Solved 283
9.3.2 MATLAB PDE Solver pdepe 284
9.4 Application in Chemical Engineering 288
9.4.1 Complex Chemical Reactions 288
9.4.2 Liquid Level Response 289
9.4.3 Plane Poiseuille Flow 292
9.5 Summary 295
Problems 295
10 Process Integration and Optimization 299
10.1 Introduction 299
10.1.1 Process Simulators 299
10.1.2 Flowsheet Simulation and Design 300
10.2 Process Integration and Optimization 300
10.2.1 Process Models 300
10.2.2 Simulating Novel Processes and Equipment 300
10.2.3 Physical and Chemical Property Estimation 303
10.2.4 Process Optimization 305
10.3 Application in Chemical Engineering 305
10.3.1 Introduction to Solvent Extraction 305
10.3.2 Ternary Diagram (Partially Soluble Ternary Systems) 306
10.3.3 Ternary Diagram – One Equilibrium Stage 306
10.3.4 Ternary Diagram – Cross-Current Two Stages Solvent Extraction 310
10.3.5 Ternary Diagram – Counter-Current Two Stages Solvent Extraction 310
10.3.6 Ternary Diagram – Counter-Current N Stages Solvent Extraction 313
10.3.7 Case Study – Cross-Current Multistages Solvent Extraction 314
10.4 Summary 331
Problems 331
References 333
Index 335
