Computational Techniques for Process Simulation and Analysis Using MATLAB 1st Edition by Niket Kaisare ISBN‎ B077BC7BNF ‎ 978-1498762120

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ISBN 10:‎ B077BC7BNF
ISBN 13: ‎ 978-1498762120
Author: Niket Kaisare

MATLAB® has become one of the prominent languages used in research and industry and often described as “the language of technical computing”. The focus of this book will be to highlight the use of MATLAB® in technical computing; or more specifically, in solving problems in Process Simulations. This book aims to bring a practical approach to expounding theories: both numerical aspects of stability and convergence, as well as linear and nonlinear analysis of systems.

The book is divided into three parts which are laid out with a “Process Analysis” viewpoint. First part covers system dynamics followed by solution of linear and nonlinear equations, including Differential Algebraic Equations (DAE) while the last part covers function approximation and optimization. Intended to be an advanced level textbook for numerical methods, simulation and analysis of process systems and computational programming lab, it covers following key points

• Comprehensive coverage of numerical analyses based on MATLAB for chemical process examples.

• Includes analysis of transient behavior of chemical processes.

• Discusses coding hygiene, process animation and GUI exclusively.

• Treatment of process dynamics, linear stability, nonlinear analysis and function approximation through contemporary examples.

• Focus on simulation using MATLAB to solve ODEs and PDEs that are frequently encountered in process systems.

Computational Techniques for Process Simulation and Analysis Using MATLAB® 1st Table of contents:

Chapter 1 Introduction

1.1 OVERVIEW

1.1.1 A General Model

1.1.2 A Process Example

1.1.3 Analysis of Dynamical Systems

1.2 STRUCTURE OF A MATLABR CODE

1.2.1 Writing Our First MATLABR Script

1.2.2 MATLABR Functions

1.2.3 Using Array Operations in MATLABR

1.2.4 Loops and Execution Control

1.2.5 Section Recap

1.3 APPROXIMATIONS AND ERRORS IN NUMERICAL METHODS

1.3.1 Machine Precision

1.3.2 Round-Off Error

1.3.3 Taylor’s Series and Truncation Error

1.3.4 Trade-Off between Truncation and Round-Off Errors

1.4 ERROR ANALYSIS

1.4.1 Convergence and Stability

1.4.2 Global Truncation Error

1.5 OUTLOOK

Section I Dynamic Simulations and Linear Analysis

Chapter 2 Linear Algebra

2.1 INTRODUCTION

2.1.1 Solving a System of Linear Equations

2.1.2 Overview

2.2 VECTOR SPACES

2.2.1 Definition and Properties

2.2.2 Span, Linear Independence, and Subspaces

2.2.3 Basis and Coordinate Transformation

2.2.3.1 Change of Basis

2.2.4 Null (Kernel) and Image Spaces of a Matrix

2.2.4.1 Matrix as Linear Operator

2.2.4.2 Null and Image Spaces in MATLAB®

2.3 SINGULAR VALUE DECOMPOSITION

2.3.1 Orthonormal Vectors

2.3.2 Singular Value Decomposition

2.3.3 Condition Number

2.3.3.1 Singular Values, Rank, and Condition Number

2.3.3.2 Sensitivity of Solutions to Linear Equations

2.3.4 Directionality

2.4 EIGENVALUES AND EIGENVECTORS

2.4.1 Orientation for This Section

2.4.2 Brief Recap of Definitions

2.4.3 Eigenvalue Decomposition

2.4.4 Applications

2.4.4.1 Similarity Transform

2.4.4.2 Linear Differential Equations

2.4.4.3 Linear Difference Equations

2.5 EPILOGUE

EXERCISES

Chapter 3 Ordinary Differential Equations: Explicit Methods

3.1 GENERAL SETUP

3.1.1 Some Examples

3.1.2 Geometric Interpretation

3.1.3 Euler’s Explicit Method

3.1.4 Euler’s Implicit Method

3.1.5 Stability and Step-Size

3.1.5.1 Stability of Euler’s Explicit Method

3.1.5.2 Error and Stability of Euler’s Implicit Method

3.1.6 Multivariable ODE

3.1.6.1 Nonlinear Case

3.2 SECOND-ORDER METHODS: A JOURNEY THROUGH THE WOODS

3.2.1 Some History

3.2.2 Runge-Kutta (RK-2) Methods

3.2.2.1 Derivation for RK-2 Methods

3.2.2.2 Heun’s Method

3.2.2.3 Other RK-2 Methods

3.2.3 Step-Size Halving: Error Estimate for RK-2

3.2.4 Richardson’s Extrapolation

3.2.5 Other Second-Order Methods (*)

3.2.5.1 Trapezoidal Rule: An Implicit Second-Order Method

3.2.5.2 Second-Order Adams-Bashforth Methods

3.2.5.3 Predictor-Corrector Methods

3.2.5.4 Backward Differentiation Formulae

3.3 HIGHER-ORDER RUNGE-KUTTA METHODS

3.3.1 Explicit Runge-Kutta Methods: Generalization

3.3.2 Error Estimation and Embedded RK Methods

3.3.2.1 MATLAB® Solver ode23

3.3.3 The Workhorse: Fourth-Order Runge-Kutta

3.3.3.1 Classical RK-4 Method(s)

3.3.3.2 Kutta’s 3/8th Rule RK-4 Method

3.4 MATLAB® ODE45 SOLVER: OPTIONS AND PARAMETERIZATION

3.5 CASE STUDIES AND EXAMPLES

3.5.1 An Ideal PFR

3.5.1.1 Simulation of PFR as ODE-IVP

3.5.1.2 Numerical Integration for PFR Design

3.5.1.3 Comparison of ODE-IVP with Integration

3.5.2 Multiple Steady States: Nonisothermal CSTR

3.5.2.1 Model and Problem Setup

3.5.2.2 Simulation of Transient CSTR

3.5.2.3 Step Change in Inlet Temperature

3.5.3 Hybrid System: Two-Tank with Heater

3.5.4 Chemostat: Preview into “Stiff” System

3.6 EPILOGUE

EXERCISES

Chapter 4 Partial Differential Equations in Time

4.1 GENERAL SETUP

4.1.1 Classification of PDEs

4.1.2 Brief History of Second-Order PDEs

4.1.3 Classification of Second-Order PDEs and Practical Implications

4.1.3.1 Elliptic PDE

4.1.3.2 Hyperbolic PDE

4.1.3.3 First-Order Hyperbolic PDEs

4.1.3.4 Parabolic PDE

4.1.4 Initial and Boundary Conditions

4.2 A BRIEF OVERVIEW OF NUMERICAL METHODS

4.2.1 Finite Difference

4.2.2 Method of Lines

4.2.3 Finite Volume Methods

4.2.4 Finite Element Methods

4.3 HYPERBOLIC PDE: CONVECTIVE SYSTEMS

4.3.1 Finite Differences in Space and Time

4.3.1.1 Upwind Difference in Space

4.3.1.2 Forward in Time Central in Space (FTCS) Differencing

4.3.1.3 Lax-Friedrichs Scheme

4.3.1.4 Higher-Order Methods

4.3.2 Crank-Nicolson: Second-Order Implicit Method

4.3.2.1 Preview of Numerical Solution

4.3.3 Solution Using Method of Lines

4.3.3.1 MoL with Central Difference in Space

4.3.3.2 MoL with Upwind Difference in Space

4.3.4 Numerical Diffusion

4.4 PARABOLIC PDE: DIFFUSIVE SYSTEMS

4.4.1 Finite Difference in Space and Time

4.4.2 Crank-Nicolson Method

4.4.3 Method of Lines Using MATLAB® ODE Solvers

4.4.3.1 MoL with Central Difference in Space

4.4.4 Methods to Improve Stability

4.5 CASE STUDIES AND EXAMPLES

4.5.1 Nonisothermal Plug Flow Reactor

4.5.2 Packed Bed Reactor with Multiple Reactions

4.5.3 Steady Graetz Problem: Parabolic PDE in Two Spatial Dimensions

4.5.3.1 Heat Transfer in Fluid Flowing through a Tube

4.5.3.2 Effect of Velocity Profile

4.5.3.3 Calculation of Nusselt Number

4.6 EPILOGUE

EXERCISES

Chapter 5 Section Wrap-Up: Simulation and Analysis

5.1 BINARY DISTILLATION COLUMN: STAGED ODE MODEL

5.1.1 Model Description

5.1.2 Model Equations and Simulation

5.1.3 Effect of Parameters: Reflux Ratio and Relative Volatility

5.2 STABILITY ANALYSIS FOR LINEAR SYSTEMS

5.2.1 Motivation: Linear Stability Analysis of a Chemostat

5.2.1.1 Phase Portrait at the Steady State

5.2.1.2 Trivial Steady State and Analysis

5.2.2 Eigenvalues, Stability, and Dynamics

5.2.2.1 Dynamics When Eigenvalues Are Real and Distinct

5.2.2.2 An Example

5.2.2.3 Summary

5.2.3 Transient Growth in Stable Linear Systems

5.2.3.1 Defining Normal and Nonnormal Matrices

5.2.3.2 Analysis of Nonnormal Systems

5.3 COMBINED PARABOLIC PDE WITH ODE-IVP: POLYMER CURING

5.4 TIME-VARYING INLET CONDITIONS AND PROCESS DISTURBANCES

5.4.1 Chemostat with Time-Varying Inlet Flowrate

5.4.2 Zero-Order Hold Reconstruction in Digital Control

5.5 SIMULATING SYSTEM WITH BOUNDARY CONSTRAINTS

5.5.1 PFR with Temperature Profile Specified

5.6 WRAP-UP

EXERCISES

Section II Linear and Nonlinear Equations and Bifurcation

Chapter 6 Nonlinear Algebraic Equations

6.1 GENERAL SETUP

6.1.1 A Motivating Example: Equation of State

6.2 EQUATIONS IN SINGLE VARIABLE

6.2.1 Bisection Method

6.2.2 Secant and Related Methods

6.2.2.1 Regula-Falsi: Method of False Position

6.2.2.2 Brent’s Method

6.2.3 Fixed Point Iteration

6.2.4 Newton-Raphson in Single Variable

6.2.5 Comparison of Numerical Methods

6.3 NEWTON-RAPHSON: EXTENSIONS AND MULTIVARIATE

6.3.1 Multivariate Newton-Raphson

6.3.2 Modified Secant Method

6.3.3 Line Search and Other Methods

6.4 MATLAB® SOLVERS

6.4.1 Single Variable Solver: fzero

6.4.2 Multiple Variable Solver: fsolve

6.5 CASE STUDIES AND EXAMPLES

6.5.1 Recap: Equation of State

6.5.2 Two-Phase Vapor-Liquid Equilibrium

6.5.2.1 Bubble Temperature Calculation

6.5.2.2 Dew Temperature Calculation

6.5.2.3 Generating the T–x–y Diagram

6.5.3 Steady State Multiplicity in CSTR

6.5.4 Recap: Chemostat

6.5.5 Integral Equations: Conversion from a PFR

6.5.5.1 First-Order Kinetics

6.5.5.2 Complex Kinetics

6.6 EPILOGUE

EXERCISES

Chapter 7 Special Methods for Linear and Nonlinear Equations

7.1 GENERAL SETUP

7.1.1 Ordinary Differential Equation–Boundary Value Problems

7.1.2 Elliptic PDEs

7.1.3 Outlook of This Chapter

7.2 TRIDIAGONAL AND BANDED SYSTEMS

7.2.1 What Is a Banded System?

7.2.1.1 Tridiagonal Matrix

7.2.2 Thomas Algorithm a.k.a TDMA

7.2.2.1 Heat Conduction Problem

7.2.2.2 Thomas Algorithm

7.2.3 ODE-BVP with Flux Specified at Boundary

7.2.4 Extension to Banded Systems

7.2.5 Elliptic PDEs in Two Dimensions

7.3 ITERATIVE METHODS

7.3.1 Gauss-Siedel Method

7.3.2 Iterative Method with Under-Relaxation

7.4 NONLINEAR BANDED SYSTEMS

7.4.1 Nonlinear ODE-BVP Example

7.4.1.1 Heat Conduction with Radiative Heat Loss

7.4.2 Modified Successive Linearization–Based Approach

7.4.3 Gauss-Siedel with Linearization of Source Term

7.4.4 Using fsolve with Sparse Systems

7.5 EXAMPLES

7.5.1 Heat Conduction with Convective or Radiative Losses

7.5.2 Diffusion and Reaction in a Catalyst Pellet

7.5.2.1 Linear System and Thiele Modulus

7.5.2.2 Langmuir-Hinshelwood Kinetics in a Pellet

7.6 EPILOGUE

EXERCISES

Chapter 8 Implicit Methods: Differential and Differential Algebraic Systems

8.1 GENERAL SETUP

8.1.1 Stiff System of Equation

8.1.1.1 Stiff ODE in Single Variable

8.1.2 Implicit Methods for Distributed Parameter Systems

8.1.3 Differential Algebraic Equations

8.2 MULTISTEP METHODS FOR DIFFERENTIAL EQUATIONS

8.2.1 Implicit Adams-Moulton Methods

8.2.2 Higher-Order Adams-Moulton Method

8.2.3 Explicit Adams-Bashforth Method

8.2.4 Backward Difference Formula

8.2.5 Stability and MATLAB® Solvers

8.2.5.1 Explicit Adams-Bashforth Methods

8.2.5.2 Implicit Euler and Trapezoidal Methods

8.2.5.3 Implicit Adams-Moulton Methods of Higher Order

8.2.5.4 BDF/NDF Methods

8.2.5.5 MATLAB® Nonstiff Solvers

8.2.5.6 MATLAB® Stiff Solvers

8.3 IMPLICIT SOLUTIONS FOR DIFFERENTIAL EQUATIONS

8.3.1 Trapezoidal Method for Stiff ODE

8.3.1.1 Adaptive Step-Sizing

8.3.1.2 Multivariable Example

8.3.2 Crank-Nicolson Method for Hyperbolic PDEs

8.3.2.1 Exploiting Sparse Structure for Efficient Simulation

8.4 DIFFERENTIAL ALGEBRAIC EQUATIONS

8.4.1 An Introductory Example

8.4.1.1 Direct Substitution

8.4.1.2 Formulating and Solving a DAE

8.4.2 Index of a DAE and More Examples

8.4.2.1 Example 2: Pendulum in Cartesian Coordinate System

8.4.2.2 Example 3: Heterogeneous Catalytic Reactor

8.4.3 Solution Methodology: Overview

8.4.3.1 Solving Algebraic Equation within ODE

8.4.3.2 Combined Approach

8.4.4 Solving Semiexplicit DAEs Using ode15s in MATLAB®

8.5 CASE STUDIES AND EXAMPLES

8.5.1 Heterogeneous Catalytic Reactor: Single Complex Reaction

8.5.2 Flash Separation/Batch Distillation

8.6 EPILOGUE

EXERCISES

Chapter 9 Section Wrap-Up: Nonlinear Analysis

9.1 NONLINEAR ANALYSIS OF CHEMOSTAT: “TRANSCRITICAL” BIFURCATION

9.1.1 Steady State Multiplicity and Stability

9.1.2 Phase-Plane Analysis

9.1.3 Bifurcation with Variation in Dilution Rate

9.1.4 Transcritical Bifurcation

9.2 NONISOTHERMAL CSTR: “TURNING-POINT” BIFURCATION

9.2.1 Steady States: Graphical Approach

9.2.2 Stability Analysis at Steady States

9.2.3 Phase-Plane Analysis

9.2.4 Turning-Point Bifurcation

9.3 LIMIT CYCLE OSCILLATIONS

9.3.1 Oscillations in Linear Systems

9.3.2 Limit Cycles: van der Pol Oscillator

9.3.2.1 Relaxation vs. Harmonic Oscillations

9.3.3 Oscillating Chemical Reactions

9.4 SIMULATION OF METHANOL SYNTHESIS IN TUBULAR REACTOR

9.4.1 Steady State PFR with Pressure Drop

9.4.1.1 Reaction Kinetics

9.4.1.2 Input Parameters and Initial Processing

9.4.1.3 Steady State PFR Model

9.4.2 Transient Model

9.5 TRAJECTORY OF A CRICKET BALL

9.5.1 Solving the ODE for Trajectory

9.5.2 Location Where the Ball Hits the Ground

9.5.3 Animation

9.6 WRAP-UP

EXERCISES

Section III Modeling of Data

Chapter 10 Regression and Parameter Estimation

10.1 GENERAL SETUP

10.1.1 Orientation

10.1.2 Some Statistics

10.1.3 Some Other Considerations in Regression

10.2 LINEAR LEAST SQUARES REGRESSION

10.2.1 Fitting a Straight Line

10.2.2 General Matrix Approach

10.2.3 Goodness of Fit

10.2.3.1 Maximum Likelihood Solution

10.2.3.2 Error and Coefficient of Determination

10.3 REGRESSION IN MULTIPLE VARIABLES

10.3.1 General Multilinear Regression

10.3.2 Polynomial Regression

10.3.3 Singularity and SVD

10.4 NONLINEAR ESTIMATION

10.4.1 Functional Regression by Linearization

10.4.2 MATLAB® Solver: Linear Regression

10.4.3 Nonlinear Regression Using Optimization Toolbox

10.5 CASE STUDIES AND EXAMPLES

10.5.1 Specific Heat: Revisited

10.5.2 Antoine’s Equation for Vapor Pressure

10.5.2.1 Linear Regression for Benzene

10.5.2.2 Nonlinear Regression for Ethylbenzene

10.5.3 Complex Langmuir-Hinshelwood Kinetic Model

10.5.3.1 Case 1: Experiments Performed at Single Concentration of B

10.5.3.2 Case 2: Experiments Performed at Different Initial Concentrations of B

10.5.4 Reaction Rate: Differential Approach

10.6 EPILOGUE

10.6.1 Summary

10.6.2 Data Tables

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