Practical Numerical Methods for Chemical Engineers Using Excel with VBA

Transcription

1 Practical Numerical Methods for Chemical Engineers Using Excel with VBA By Richard A. Davis Copyright Richard A. Davis. All rights reserved. No part of this book may be reproduced in any form or by any means, without written permission from the publisher. Excel is a registered trademark of Microsoft Corp. ( Other company and product names mentioned herein are the trademarks of their respective owners. Disclaimer Use the examples, programs, and algorithms in this book or associated web site at your own risk! No warranties are expressed or implied, that the examples, programs, and algorithms in this book or associated web site are free of error, or will meet your requirements for any particular application. The publisher disclaims all liability for direct or consequential damages resulting from any use of the examples, programs or algorithms in this book or associated web site, including, but not limited to, Excel worksheets, VBA programs, and macros ISBN-13: ISBN-10: Contact the publisher (rdavis@d.umn.edu) to inquire about options for customizing any part of the content of this book for your unique courses or particular needs.

2 Contents CHAPTER 1 DO YOU HAVE A CAREER IN MODELING? INTRODUCTION TO NUMERICAL METHODS Linear versus Nonlinear Equations Common Features of Numerical Methods Computer Round-off Error MATHEMATICAL MODEL BUILDING Research versus Design Laws of Conservation Steady-state versus Equilibrium Generation Lumped vs. Distributed Systems Modeling Assumptions Dimensionless Equations Degrees of Freedom What Do We Mean by Order? Model Validation Stochastic Modeling EXPERT PROBLEM SOLVERS APPENDIX: PRIMER ON REACTION ENGINEERING CHAPTER 2 REAL ENGINEERS USE EXCEL GETTING STARTED WITH EXCEL Calculators vs. Spreadsheets Moving Around in a Worksheet (Things you may not have tried ) AUTO FILL ARRAY OPERATIONS DOCUMENT YOUR CALCULATIONS Cell Referencing and Named Cells Significant Figures and Scientific Notation Use Symbols and Combine Text with Numbers Troubleshoot with Formula Auditing Tools Text Wrapping and Breaking Text Drawing Tools Charts and Graphs FUNCTION LIBRARY Unit Conversions Logical, Lookup and Reference Functions WHAT-IF ANALYSIS USING GOAL SEEK AND DATA TABLE EXCEL ADD-INS Analysis ToolPak Solver DATA VALIDATION TOOLS CHAPTER 3 LEARN TO SPEAK VBA MACRO ENABLED WORKBOOKS VISUAL BASIC EDITOR (VBE) Properties Window PROCEDURES: USER FUNCTIONS AND SUBROUTINES Text, Breaks and Multiple Statements per Line Descriptive Comments VARIABLES AND CONSTANTS STRUCTURED PROGRAM FLOW CONTROL Loop Program Flow Control If-Then-Else Program Flow Control Case Program Flow Control... 89

3 iv PRACTICAL NUMERICAL METHODS 3.6 ARITHMETIC AND WORKSHEET FUNCTIONS IN VBA Speed up VBA Calculations Pseudo Random Number Generator MACRO CONTROLS ON THE WORKSHEET ARRAYS Range Variables and Worksheet References for Communication between VBA and Excel Sorting Arrays COMMUNICATION WITH THE USER: MSGBOX AND INPUTBOX MsgBox InputBox User Forms Read/Write Data To/From Files Status Bar FORMAT Numbers for Output DEBUGGING YOUR CODE PNM SUITE OF MACROS CHAPTER 4 LINEAR EQUATIONS LINEAR FUNCTIONS SMALL LINEAR ALGEBRAIC SYSTEMS GAUSSIAN ELIMINATION Crout Reduction Thomas Algorithm for Tridiagonal Matrix Inversion Cholesky Decomposition of a Symmetric Matrix MATRIX ALGEBRA USING EXCEL WORKSHEET ARRAY FUNCTIONS Matrix Solution Improvement SUCCESSIVE SUBSTITUTION (ITERATION) Iteration on Circular References in Excel Relaxation EPILOGUE ON LINEAR EQUATIONS CHAPTER 5 TAYLOR SERIES ANALYSIS DERIVATIVE APPROXIMATIONS Finite-difference Derivative Approximations Derivatives Approximations Using Unevenly Spaced Data Richardson s Extrapolation SENSITIVITY ANALYSIS Maximum Uncertainty CHAPTER 6 NONLINEAR EQUATIONS GRAPHICAL SOLUTION FINDING ROOTS OF A SINGLE EQUATION Bisection Regula Falsi Excel s Goal Seek Successive Substitution (Ordinary Iteration) Wegstein s Acceleration of Fixed-point Iteration Newton s Method Secant Method ROOTS FOR SIMULTANEOUS EQUATIONS Ordinary Iteration Aitken s 2 and Steffensen Methods of Series Convergence Acceleration Equation Sequence and Arrangement Newton s Method for Simultaneous Equations Bairstow s Method of Polynomial Factorization Excel s Solver for Simultaneous Equations Newton s Method with Continuation for Root Finding TRANSFORMATION AND SCALING INITIALIZING ITERATIVE METHODS EPILOGUE ON NONLINEAR EQUATION SOLUTION METHODS

4 CONTENTS v CHAPTER 7 OPTIMIZATION DEFINE THE OPTIMIZATION PROBLEM CALCULUS MAXIMUM VS. MINIMUM DIRECT SEARCH METHODS Graphical Direct Search Methods Bracketing with Quadratic Interpolation Golden Sections Powell s Method for Multiple Dimensions Luus-Jaakola Global Optimization INDIRECT SEARCH METHODS Response Surfaces Newton s Method Excel s Solver CONSTRAINTS ON THE SEARCH REGION Penalty Functions for Inequality Constraints Slack Variables Variable Transformation Lagrange Multipliers Redundancy and Separation Desirability Functions for Multiobjective Optimization VARIABLE AND PARAMETER SCALING SENSITIVITY ANALYSIS EPILOGUE ON OPTIMIZATION CHAPTER 8 UNCERTAINTY ANALYSIS MODELS OF MEASUREMENT UNCERTAINTY Precision versus Accuracy Standard Uncertainty from Random Errors Confidence Intervals for Expanded Uncertainty Fixed Uncertainty from Systematic Errors Uncertainty in Quoted Values How (Not) to Report Uncertainty UNCERTAINTY PROPAGATION The Law of Propagation of Uncertainty How (Not) to Handle Unit Conversions in Uncertainty Analysis Correlation Between Variables Expanded Uncertainty from Degrees of Freedom in Combined Uncertainty using the Welch- Satterthwaite Formula Uncertainty in Replicated Experiments Jitter Macro MONTE CARLO SIMULATION OF UNCERTAINTY PROPAGATION Latin Hypercube Sampling Correlation in Monte Carlo Analysis LOG NORMAL DISTRIBUTION EPILOGUE ON UNCERTAINTY ANALYSIS CHAPTER 9 LEAST-SQUARES REGRESSION AND DATA MODELING LEAST-SQUARES REGRESSION LINEAR REGRESSION Excel s Chart Tools and Trend line Slope and Intercept Functions MULTIVARIABLE LINEAR REGRESSION Excel s LINEST Array Function NONLINEAR REGRESSION Select Good Initial Guesses for Nonlinear Model Parameters MODEL VALIDATION: ASSESSMENT AND UNCERTAINTY ANALYSIS Linearization Average Absolute and Relative Deviation Coefficient of Determination: R 2 and Adjusted R Standard Uncertainty in Regression with Parameter Correlation...364

5 vi PRACTICAL NUMERICAL METHODS Centering Data to Eliminate Correlation in Linear Model Parameters P-values and t-statistics for Hypothesis Testing in Linear Models Residual Plots and Analysis Normal Residual Plots Weighted Least-Squares Regression Jackknife Method of Regression Parameter Uncertainty Analysis Gauss-Newton with Uncertainty Analysis Levenberg-Marquardt Least Squares Regression Monte Carlo Analysis of Uncertainty in Regression Parameters Plot Confidence Intervals and Error Bars CHOOSING AN EMPIRICAL MODEL CHAPTER 10 INTERPOLATION LINEAR INTERPOLATION LAGRANGE POLYNOMIAL INTERPOLATION RATIONAL INTERPOLATING FUNCTIONS CUBIC SPLINES Cubic Spline Derivative Approximation Integrating with Cubic Splines Constrained Cubic Splines EPILOGUE ON INTERPOLATION CHAPTER 11 INTEGRATION GRAPHICAL INTEGRATION (RECTANGLE RULE) TRAPEZOIDAL RULE Improper Integrals and the Midpoint Rule Integration Accuracy and Precision Romberg Integration with Error Control Data Integration by the Trapezoidal Rule with Uncertainty Propagation SIMPSON S RULES GAUSS QUADRATURE WITH ADAPTIVE ERROR CONTROL MULTIPLE INTEGRATION Double Integration by Simpson s Rules Multiple Integration by Gauss-Kronrod Quadrature Monte Carlo Integration EPILOGUE ON INTEGRATION CHAPTER 12 INITIAL-VALUE PROBLEMS EULER S METHOD Stability and Stiffness Simultaneous First-order Differential Equations TRAPEZOIDAL RULES FOR IVPS RUNGE-KUTTA METHODS Integration Accuracy versus Precision Adaptive Runge-Kutta Methods MULTI-STEP METHODS DIFFERENTIAL-ALGEBRAIC SYSTEMS EPILOGUE ON IVP CHAPTER 13 BOUNDARY-VALUE PROBLEMS AND PDES SHOOTING METHOD FINITE-DIFFERENCE Symmetric Boundary Conditions Finite-differences for Elliptic PDEs ORTHOGONAL COLLOCATION Orthogonal Collocation on Finite Elements PARABOLIC PDES Method of Lines Crank-Nicholson Implicit Method CHAPTER 14 REVIEW OF PRACTICAL NUMERICAL METHODS

6 CONTENTS vii 14.1 APPENDIX A: EXCEL WORKSHEET FUNCTIONS FOR NUMERICAL METHODS APPENDIX B: VBA FUNCTIONS AND OBJECTS FOR NUMERICAL METHODS APPENDIX C: VBA USER FUNCTIONS IN THE EXCEL WORKBOOK PNMSUITE APPENDIX D: VBA MACROS IN THE EXCEL WORKBOOK PNMSUITE REFERENCES INDEX Book Type Conventions for Excel and VBA menus and formulas / sub menu tree in the Visual Basic Editor > sub menu tree from Excel ribbon tabs and groups Calibri Excel menu syntax, Excel worksheet text Courier Visual Basic of Applications syntax Italics keyboard operation (or emphasis)