Computer Application in Mechanical Power Engineering - III

by

Dr. Hossam S.S. AbdelMeguid

2013-2014

Mansoura University

Faculty of Engineering

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Dr. H

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Mansoura University

Faculty of Engineering

2012-2013

Computer Application In Mechanical Power Engineering - III

Dr. Hossam S.S. AbdelMeguid

ii

ISBN: 978-977-6005-89-1

DEPOSIT NUMBER: 23598/2012

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Table of Contents PREFACE .......................................................................................................... IX

PART 1: FUNDAMENTALS ................................................................................ 1

1 FUNDAMENTALS ......................................................................................... 1

1.1 Introduction ..................................................................................... 1

1.2 Incentives for process modelling ....................................................... 1

1.3 Systems ............................................................................................ 3 1.3.1 Classification based on thermodynamic principles .......................................................... 3 1.3.2 Classification based on number of phases ...................................................................... 4

1.4 Classification of Models.................................................................... 5

1.5 State variables and state equations.................................................. 6

1.6 Classification of theoretical models .................................................. 6 1.6.1 Steady state vs. unsteady state ...................................................................................... 6 1.6.2 Lumped vs. distributed parameters................................................................................ 7 1.6.3 Linear vs. non-linear ...................................................................................................... 7 1.6.4 Continuous vs. discrete.................................................................................................. 8 1.6.5 Deterministic vs. probabilistic ........................................................................................ 8

1.7 Building steps for a mathematical model ......................................... 8

1.8 Conservation Laws ......................................................................... 10 1.8.1 Total mass balance ...................................................................................................... 11 1.8.2 Component balance .................................................................................................... 12 1.8.3 Momentum balance .................................................................................................... 12 1.8.4 Energy balance ............................................................................................................ 13

1.9 Microscopic balance ....................................................................... 13

1.10 Macroscopic balance .................................................................. 14

1.11 Transport rates ........................................................................... 15 1.11.1 Mass Transport ....................................................................................................... 15 1.11.2 Momentum transport ............................................................................................. 16 1.11.3 Energy transport ..................................................................................................... 18

.1.1 Thermodynamic relations ........................................................... 20

1.13 Phase Equilibrium ....................................................................... 21

1.14 Chemical kinetics ........................................................................ 24

1.15 Control Laws ............................................................................... 25

1.16 Degrees of Freedom .................................................................... 26

1.17 Model solution ............................................................................ 27

.1.1 Model validation ......................................................................... 28

1.19 Problems ..................................................................................... 29

2 MATHEMATICAL MODELS FOR MECHANICAL AND CHEMICAL PROCESSES ................ 30

2.1 Lumped Parameter Systems ........................................................... 31 2.1.1 Liquid Storage Tank ..................................................................................................... 31 2.1.2 Stirred Tank Heater ..................................................................................................... 34 2.1.3 Gas-Phase Pressurized CSTR ........................................................................................ 39

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2.1.4 Mixing Process ............................................................................................................ 41 2.1.5 Heat Exchanger ........................................................................................................... 45 2.1.6 Heat Exchanger with Steam ......................................................................................... 49

2.2 Examples of Distributed Parameter Systems................................... 51 2.2.1 Liquid Flow in a Pipe .................................................................................................... 51 2.2.2 Velocity profile inside a pipe ........................................................................................ 53 2.2.3 Temperature profile in a heated cylindrical Rod ........................................................... 55 2.2.4 Heat Exchanger: Distributed parameter model ............................................................. 58

2.3 Problems ........................................................................................ 60

PART 2: SIMULTANEOUS LINEAR EQUATIONS ............................................... 63

3 INTRODUCTION ......................................................................................... 63

3.1 Special types of matrices? .............................................................. 63

4 SYSTEM OF EQUATIONS .............................................................................. 66

4.1 Solving systems of equations using matrix algebra ......................... 66

4.2 A consistent / inconsistent system of equations.............................. 68

4.3 Distinguish between a consistent and inconsistent system of equations ................................................................................................. 69

4.4 A unique solution of a system of equation ...................................... 75

5 GAUSS-SEIDEL METHOD ............................................................................. 79

5.1 Another method to solve a set of simultaneous linear equations .... 79

5.2 Pitfall of most iterative methods .................................................... 84

5.3 Matlab code for Gauss Seidel ......................................................... 89

6 CASE STUDY (STEADY-STATE ANALYSIS OF A SYSTEM OF REACTORS) ........................ 91

7 PROBLEMS ............................................................................................... 95

PART 3: ORDINARY DIFFERENTIAL EQUATIONS ............................................. 99

8 INTRODUCTION ......................................................................................... 99

8.1 Initial Value Problems .................................................................. 101 8.1.1 Taylor Series Expansion ............................................................................................. 102 8.1.2 Truncation Errors ...................................................................................................... 103

9 EULER'S METHOD.................................................................................... 104

9.1 Algorithm of Euler Method ........................................................... 105

10 MODIFIED EULER'S (HEUN'S) METHOD ...................................................... 108

10.1 Algorithm of Heun’s Method ..................................................... 108

10.2 Matlab code for Euler method .................................................. 111

11 RUNGE-KUTTA METHOD........................................................................ 112

11.1 Runge-Kutta Method of second order ....................................... 112 11.1.1 Algorithm of Second Order Runge-Kutta Method .................................................. 114

11.2 Fourth Order Runge-Kutta ......................................................... 115 11.2.1 Algorithm of Fourth Order Runge-Kutta Method ................................................... 117

11.3 Runge-Kutta with adaptive step size ......................................... 118

11.4 Matlab code for 4th order Runge-Kutta method ........................ 119

12 SYSTEM OF COUPLED ODE'S ............................................................ 121

12.1 Euler's method for system of coupled ODE's .............................. 122

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12.2 Algorithm of Euler method to system of ODEs ........................... 122

12.3 Matlab code or a coupled system of ODE using Euler method ... 124

12.4 Fourth order Runge-Kutta method for system of coupled ODE's 125 12.4.1 Algorithm of Coupled 4th Order Runge-Kutta .......................................................... 125

12.5 Matlab code or a coupled system of ODE using 4th order Runge-Kutta 128

12.6 Stability of the integration methods .......................................... 129

12.7 Stiff differential equations......................................................... 131

12.8 Other solution techniques ......................................................... 133

13 BOUNDARY VALUE PROBLEM .................................................................. 135

13.1 Introduction .............................................................................. 135

13.2 Numerical methods for the solution of BVP problems................ 138

13.3 Shooting methods ..................................................................... 139 13.3.1 algorithm of the shooting method ......................................................................... 143

13.4 Matlabe code for 4th order Runge-Kutta –shooting method for a linear BVP .............................................................................................. 147

13.5 Finite-Difference Methods ......................................................... 148

13.6 Finite-Difference Methods for Linear Problems ......................... 152

13.7 Handling different types boundary conditions ........................... 154

13.8 Other solution techniques ......................................................... 156

14 CASE STUDY (USING ODES TO ANALYZE THE TRANSIENT RESPONSE OF A REACTOR) . 157

15 PROBLEMS ......................................................................................... 161

PART 4: PARTIAL DIFFERENTIAL EQUATIONS .............................................. 163

16 INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS .................................. 163

16.1 Introduction .............................................................................. 163

16.2 Basics ........................................................................................ 164 16.2.1 Conservation Principles ......................................................................................... 164 16.2.2 Partial Differential Equation (PDE) ......................................................................... 164 16.2.3 From Ordinary to a Partial Differential Equation..................................................... 165 16.2.4 Types and Classification of Partial Differential Equations ........................................ 168

16.3 Discretization Methods ............................................................. 172

16.4 The Nature of Numerical Methods ............................................ 173 16.4.1 The Task ............................................................................................................... 173 16.4.2 The Discretization Concept .................................................................................... 174 16.4.3 The Structure of the Discretization Equation .......................................................... 175

16.5 Methods of Deriving the Discretization Equations ..................... 176 16.5.1 Taylor-Series Formulation...................................................................................... 176 16.5.2 Control-Volume Formulation ................................................................................. 177 16.5.3 An Illustrative Example for Control volume method ............................................... 179

16.6 Basic Rules ................................................................................ 182

16.7 Summery .................................................................................. 183

17 FINITE DIFFERENCE METHODS ................................................................. 184

17.1 Introduction .............................................................................. 184

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17.2 Forming Finite Differences ........................................................ 184 17.2.1 Taylor Series ......................................................................................................... 185

18 FINITE DIFFERENCE FORMULATION FOR ELLIPTIC EQUATION ........................... 188

18.1 Laplace's Equation (Steady Conduction) .................................... 188

18.2 The Direct Method of Solving Laplace's Equation ...................... 192 18.2.1 Solving the System of Algebraic Equations ............................................................. 194

18.3 A Numerical Example for Elliptic (Laplace's) Equation ............... 195

18.4 Matlab Code and Script for Elliptic (Laplace's) Equation (Heat Conduction) ........................................................................................... 201

18.5 Poisson's Equation (Steady Heat conduction with Generation) .. 206

18.6 A Numerical Example for Poisson's Equation ............................. 207

18.7 Matlab Code and Script for Poisson’s Equation ......................... 210

19 PARABOLIC EQUATION (UNSTEADY HEAT CONDUCTION) ............................... 219

19.1 Unsteady Conduction ................................................................ 219

19.2 Simple Explicit Scheme .............................................................. 220

19.3 Implicit Schemes ....................................................................... 223

19.4 Numerical example for unsteady 1D heat conduction ............... 224 19.4.1 Example and Matlab code for unsteady 1D heat conduction .................................. 226

19.5 Unsteady Conduction in Two and Three Dimensions ................. 234 19.5.1 Example and Matlab code for 1D and 2D Unsteady Conduction .............................. 235

19.6 Unsteady Conduction in 2D and 3D with Heat generation ......... 243

20 CONVECTION AND DIFFUSION (CONDUCTION) ............................................. 247

20.1 Steady 1-D Convection and Diffusion (conduction) .................... 248 20.1.1 A Preliminary Derivation........................................................................................ 249 20.1.2 The Upwind Scheme.............................................................................................. 250

20.2 Unsteady Convection Diffusion with Heat Generation in 1D ...... 252 20.2.1 Example and Matlab codes .................................................................................... 253

20.3 Unsteady Convection Diffusion with Heat Generation in 2D ...... 258 20.3.1 Example and Matlab codes .................................................................................... 259

21 HYPERBOLIC EQUATION ......................................................................... 265

22 ALTERNATIVE BOUNDARY CONDITIONS ...................................................... 266

PART5: PROGRAMMING WITH MATLAB ..................................................... 267

23 MATLAB BASICS ................................................................................... 267

23.1 The MATLAB Working Environment .......................................... 267

23.2 The m-file .................................................................................. 270

23.3 Inline function ........................................................................... 273

23.4 Control Flow ............................................................................. 274 23.4.1 The for loop .......................................................................................................... 274 23.4.2 The while loops ..................................................................................................... 276 23.4.3 The if-else-end constructions................................................................................. 277 23.4.4 The switch-case constructions ............................................................................... 278

23.5 Relations and logical operations ............................................... 279

23.6 Rounding to integers ................................................................. 280

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23.7 Matlab Graphics ....................................................................... 282 23.7.1 2-D graphics .......................................................................................................... 283 23.7.2 3-D graphs ............................................................................................................ 286

23.8 Animation ................................................................................. 287

REFERENCES ............................................................................................... 289

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Preface

This book evolved from lecture notes developed last year of teaching this

material, mostly in computer application in mechanical power engineering and

process modelling and simulation at Mansoura University and NDETI. The

course is taken by graduate students, along with post graduate students.

Exercises and assignments are an important aspect of any such course and

many have been developed in conjunction with this book. Rather than

lengthening the text, they are available on the book's webpage:

Https://sites.google.com/site/hssaleh/

The webpage also contains Matlab m-files that illustrate how to implement

different numerical methods for soling ordinary and partial deferential

equations, and that may serve as a starting point for further study of the

methods. A number of the exercises require programming on the part of the

student, or making changes to the Matlab programs provided. Some of these

exercises are fairly simple, designed to enable students to observe first hand

the behaviour of numerical methods described in the text. Others are more

open-ended and could form the basis for a many project.

The book is organized into five main parts. Part 1 deals with fundamentals of

modelling and simulation. Part 2 deals with solving linear system of equation.

Part 3 concerns time-dependent problems, starting with the initial value

problem for ODEs and moving on to initial-boundary value problems. Part 4

concerns the finite difference solution of energy equation in 1, 2 and 3

dimensions, with an emphasis on systems arising from finite difference

approximations. Part 5 gives a brief introduction about the programming with

matlab and explores its main features that will be used in this book.

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The emphasis is on building an understanding of the essential ideas that

underlie the development, analysis, and practical use of numerical methods

for solving ODE and PDE. Stability theory necessarily plays a large role, and I

have attempted to explain several key concepts, their relation to one another,

and their practical implications. I include some proofs of convergence in order

to motivate the various definitions of "stability" and to show how they relate

to error estimates, but have not attempted to rigorously prove all results in

complete generality. I have also tried to give an indication of some of the more

practical aspects of the algorithms without getting too far into implementation

details. My goal is to form a foundation from which students can approach the

vast literature on more advanced topics and further explore the theory and/or

use of numerical solution methods for ODE and PDE according to their

interests and needs.

I have also been influenced by other books covering these same topics, and

many excellent ones exist at all levels. Advanced books go into more detail on

countless subjects only briefly discussed here, including for example [1-4].

There are also a number of general introductory books that may be useful as a

complement to the presentation found here, including for example [5-9]

Dr. Hossam S.S. AbdelMeguid,

September, 2012