Problem Solving with Mathematical Software Packages 1

Transcription

1 C H A P T E R 1 Problem Solvig with Mathematical Software Packages EFFICIENT PROBLEM SOLVING THE OBJECTIVE OF THIS BOOK As a egieerig studet or professioal, you are almost always ivolved i umerical problem solvig o a persoal computer. The objective of this book is to eable you to solve umerical problems that you may ecouter i your studet or professioal career i a most effective ad efficiet maer. The tools that are typically used for egieerig or techical problem solvig are mathematical software packages that execute o persoal computers o the desktop. I order to solve your problems most efficietly ad accurately, you must be able to select the appropriate software package for the particular problem at had. The you must also be proficiet i usig your selected software tool. I order to help you achieve these objectives, this book provides a wide variety of problems from differet areas of chemical, biochemical, ad related egieerig ad scietific disciplies. For some of these problems, the complete solutio process is demostrated. For some problems, partial solutios or hits for the solutio are provided. Other problems are left as exercises for you to solve. Most of the chapters of the book are orgaized by chemical ad biochemical egieerig subject areas. The various chapters cotai betwee five ad twety-eight problems that represet may of the problem types that require a computer solutio i a particular subject area. All problems preseted i the book have the same geeral format for your coveiece. The cocise problem topic is first followed by a listig of the egieerig or scietific cocepts demostrated by the problem. The the umerical methods utilized i the solutio are idicated just before the detailed problem statemet is preseted. Typically a particular problem presets all of the detailed equatios that are ecessary for solutio, icludig the appropriate uits i a variety of systems, with Système Iteratioal d Uités (SI) beig the most commoly used. Physical properties are either give directly i the problem or i the appedices. Complete ad partial solutios are provided to may of the problems. These solutios will help you lear to formulate ad the to solve the usolved problems i the book as well as the problems that you will face i your studet ad/or professioal career. 1

2 2 CHAPTER 1 PROBLEM SOLVING WITH MATHEMATICAL SOFTWARE PACKAGES Three widely used mathematical software packages are used i this book for solvig the various problems: POLYMATH, * Excel, ad MATLAB. Each of these packages has specific advatages that make it the most appropriate for solvig a particular problem. I some cases, a combied use of several packages is most desirable. These mathematical software packages that solve the problems utilize what are called umerical methods. This book presets the fudametal ad practical approaches to settig up problems that ca the be solved by mathematical software that utilizes umerical methods. It also gives much practical iformatio for problem solvig. The details of the umerical methods are beyod the scope of this book, ad referece ca be made to textbook by Costatiides ad Mostoufi. 1 More advaced ad extesive treatmet of umerical methods ca be foud i the book of Press et al. 2 The first step i solvig a problem usig mathematical software is to prepare a mathematical model of the problem. It is assumed that you have already leared (or you will lear) how to prepare the model of a problem from i particular subject area (such as thermodyamics, fluid mechaics, or biochemical egieerig). A geeral approach advocated i this book is to start with a very simple model ad the to make the model more complex as ecessary to describe the problem. Egieerig ad scietific fudametals are importat i model buildig. The first step i the solutio process is to characterize the problem accordig to the type of the mathematical model that is formulated: a system of algebraic equatios or a system of ordiary differetial equatios, for example. Whe the problem is characterized i these terms, the software package that efficietly solves this type of problems ca be utilized. Most of the later part of this chapter is devoted to learig how to characterize a problem i such mathematical terms. I order to put the use of mathematical software packages for problem solvig ito proper perspective, it is importat ad iterestig to review the history i which maual problem solvig has bee replaced by umerical problem solvig. 1.2 FROM MANUAL PROBLEM SOLVING TO USE OF MATHEMATICAL SOFTWARE The problem solvig tools o the desktop that were used by egieers prior to the itroductio of the hadheld calculators (i.e., before 1970) are show i Figure 1 1. Most calculatios were carried out usig the slide rule. This required carryig out each arithmetic operatio separately ad writig dow the results of such operatios. The highest precisio of such calculatios was to three decimal digits at most. If a calculatioal error was detected, the all the slide rule ad arithmetic calculatios had to be repeated from the poit where the error occurred. The results of the calculatios were typically typed, ad had-draw * POLYMATH is a product of Polymath Software ( Excel is a trademark of Microsoft Corporatio ( MATLAB is trademark of The Math Works, Ic. (

3 1.2 FROM MANUAL PROBLEM SOLVING TO USE OF MATHEMATICAL SOFTWARE 3 Figure 1 1 The Egieer s Problem Solvig Tools Prior to 1970 graphs were ofte prepared. Temperature- ad/or compositio-depedet thermodyamic ad physical properties that were eeded for problem solvig were represeted by graphs ad omographs. The values were read from a straight lie passed by a ruler betwee two poits. The highest precisio of the values obtaied usig this techique was oly two decimal digits. All i all, maual problem solvig was a tedious, time-cosumig, ad error-proe process. Durig the slide rule era, several techiques were developed that eabled solvig realistic problems usig the tools that were available at that time. Aalytical (closed form) solutios to the problems were preferred over umerical solutios. However, i most cases, it was difficult or eve impossible to fid aalytical solutios. I such cases, cosiderable effort was ivested to maipulate the model equatios of the problem to brig them ito a solvable form. Ofte model simplificatios were employed by eglectig terms of the equatios which were cosidered less importat. Short-cut solutio techiques for some types of problems were also developed where a complex problem was replaced by a simple oe that could be solved. Graphical solutio techiques, such as the McCabe- Thiele ad Pocho-Savarit methods for distillatio colum desig, were widely used. After digital computers became available i the early 1960 s, it became apparet that computers could be used for solvig complex egieerig problems. Oe of the first textbooks that addressed the subject of umerical solutio of problems i chemical egieerig was that by Lapidus. 3 The textbook by Caraha, Luther ad Wilkes 4 o umerical methods ad the textbook by Heley ad Rose 5 o material ad eergy balaces cotai may example problems for umerical solutio ad associated maiframe computer programs (writte i the FORTRAN programmig laguage). Solutio of a egieerig problem usig digital computers i this era icluded the followig stages: (1) derive the model equatios for the problem at had, (2) fid the appropriate umerical method

4 4 CHAPTER 1 PROBLEM SOLVING WITH MATHEMATICAL SOFTWARE PACKAGES (algorithm) to solve the model, (3) write ad debug a computer laguage program (typically FORTRAN) to solve the problem usig the selected algorithm, (4) validate the results ad prepare documetatio. Problem solvig usig umerical methods with the early digital computers was a very tedious ad time-cosumig process. It required expertise i umerical methods ad programmig i order to carry out the 2d ad 3rd stages of the problem-solvig process. Thus the computer use was justified for solvig oly large-scale problems from the 1960 s through the mid 1980 s. Mathematical software packages started to appear i the 1980 s after the itroductio of the Apple ad IBM persoal computers. POLYMATH versio 1.0, the software package which is extesively used i this book, was first published i 1984 for the IBM persoal computer. Itroductio of mathematical software packages o maiframe ad ow persoal computers has cosiderably chaged the approach to problem solvig. Figure 1 2 shows a flow diagram of the problem-solvig process usig such a package. The user is resposible for the preparatio of the mathematical model (a complete set of equatios) of the problem. I may cases the user will also eed to provide data or correlatios of physical properties of the compouds ivolved. The complete model ad data set must be fed ito the mathematical software package. It is also the user s resposibility to categorize the problem type. The problem category will determie the type of umerical algorithm to be used for the solutio. This issue will be discussed i detail i the ext sectio. The mathematical software package will the solve the problem usig the selected umerical techique. The results obtaied together with the model defiitio ca serve as partial or complete documetatio of the problem ad its solutio. Figure 1 2 Problem Solvig with Mathematical Software Packages

5 1.3 CATEGORIZING PROBLEMS ACCORDING TO THE SOLUTION TECHNIQUE USED CATEGORIZING PROBLEMS ACCORDING TO THE SOLUTION TECHNIQUE USED Mathematical software packages cotai various tools for problem solvig. I order to match the tool to the problem i had, you should be able to categorize the problem accordig to the umerical method that should be used for its solutio. The discussio i this sectio details the various categories for which represetative examples are icluded i the book. Note that the study of the followig categories (a) through (e) is highly recommeded prior to usig Chapters 7 through 14 of this book that are associated with particular subject areas. Categories (f) through () are advaced topics that should be reviewed prior to advaced problem solvig. (a) Cosecutive Calculatios These calculatios do ot require the use of a special umerical techique. The model equatios ca be writte oe after aother. O the left-had side a variable ame appears (the output variable), ad the right-had side cotais a costat or a expressio that may iclude costats ad previously defied variables. Such equatios are usually called explicit equatios. A typical example for such a problem is the calculatio of the pressure usig the va der Waals equatio of state. R = T c = P c = 72.9 T = 350 V = 0.6 (1-1) a ( 24 64) R 2 2 = (( T c ) P c ) b = ( RT c ) ( 8P c ) P = ( RT) ( V b) a V 2 The various aspects associated with the solutio of this type of problem are described i detail i Problems 4.1 ad 5.1 (Molar Volume ad Compressibility from Redlich-Kwog Equatio). I those completely solved problems, the advatages of the differet software packages (POLYMATH, Excel, ad MATLAB) i the various stages of the solutio process are also demostrated. To gai the most beefit from usig this book, you should proceed ow to study Problems 4.1 ad 5.1 ad the retur to this poit.

6 6 CHAPTER 1 PROBLEM SOLVING WITH MATHEMATICAL SOFTWARE PACKAGES (b) System of Liear Algebraic Equatios A system of liear algebraic equatios ca be represeted by the equatio: Ax = b (1-2) where A is a matrix of coefficiets, x is a 1 vector of ukows ad b a 1 vector of costats. Note that the umber of equatios is equal to the umber of the ukows. A detailed descriptio of the various aspects of the solutio of systems of liear equatios is provided i Problem 2.4 (Steady-State Material Balaces o a Separatio Trai). (c) Oe Noliear (Implicit) Algebraic Equatio A sigle oliear equatio ca be writte i the form fx ( ) = 0 (1-3) where f is a fuctio ad x is the ukow. Additioal explicit equatios, such as those show i Sectio (a), may also be icluded. Solved problems associated with the solutio of oe oliear equatio are preseted i Problems 2.1 (Molar Volume ad Compressibility Factor from Va Der Waals Equatio), 2.9 (Gas Volume Calculatios usig Various Equatios of State), 2.10 (Bubble Poit Calculatio for a Ideal Biary Mixture), ad 2.13 (Adiabatic Flame Temperature i Combustio). These problems should be reviewed before proceedig further. The use of the various software packages for solvig sigle oliear equatios is demostrated i solved Problems 4.2 ad 5.2 (Calculatio of the Flow Rate i a Pipelie). Please study those solved problems as well. (d) Multiple Liear ad Polyomial Regressios Give a set of data of measured (or observed) values of a depedet variable: y i versus idepedet variables x 1i, x 2i, x i, multiple liear regressio attempts to fid the best values of the parameters a 0, a 1, a for the equatio ŷ i = a 0 + a 1 x 1, i + a 2 x 2, i + + a x, i (1-4) where ŷ i is the calculated value of the depedet variable at poit i. The best parameters have values that miimize the squares of the errors N S = ( y i yˆ i ) 2 i = 1 (1-5) where N is the umber of available data poits. I polyomial regressio, there is oly oe idepedet variable x, ad Equatio (1-4) becomes 2 ŷ i = a 0 + a 1 x i + a 2 x i + + a x i (1-6)

7 1.3 CATEGORIZING PROBLEMS ACCORDING TO THE SOLUTION TECHNIQUE USED 7 Multiple liear ad polyomial regressios usig POLYMATH are demostrated i detail i solved Problems 3.3 (Correlatio of Thermodyamic ad Physical Properties of -Propae) ad 3.5 (Heat Trasfer Correlatios from Dimesioal Aalysis). The use of Excel ad MATLAB for the same purpose is demostrated respectively i Problems 4.4 ad 5.4 (Correlatio of the Physical Properties of Ethae). These examples should be studied before proceedig further. (e) Systems of First-Order Ordiary Differetial Equatios (ODEs) Iitial Value Problems A system of simultaeous first-order ordiary differetial equatios ca be writte i the followig (caoical) form dy1 = f1 dx dy2 = f2 dx M dy = f dx ( y, y, Ky, x) 1 ( y, y, K y, x) 1 ( y, y, K y, x) (1-7) where x is the idepedet variable ad y 1, y 2, y are depedet variables. To obtai a uique solutio of simultaeous first-order ODEs, it is ecessary to specify values of the depedet variables (or their derivatives) at specific values of the idepedet variable. If those values are specified at a commo poit, say x 0, y ( x ) = y y ( x ) = 1,0 y ( x ) = y M 2,0 0 y,0 (1-8) the the problem is categorized as a iitial value problem. The solutio of systems of first-order ODE iitial value problems is demostrated i Problems 2.14 (Usteady-state Mixig i a Tak) ad 2.16 (Heat Exchage i a Series of Taks) where POLYMATH is used to obtai the solutio. The use of Excel ad MATLAB for systems of first-order ODEs is demostrated respectively i Problems 4.3 ad 5.3 (Adiabatic Operatio of a Tubular Reactor for Crackig of Acetoe). (f) System of Noliear Algebraic Equatios (NLEs) A system of oliear algebraic equatios is defied by f(x) = 0 (1-9) where f is a vector of fuctios, ad x is a vector of ukows. Note that

8 8 CHAPTER 1 PROBLEM SOLVING WITH MATHEMATICAL SOFTWARE PACKAGES the umber of equatios is equal to the umber of the ukows. Solved problems i the category of NLEs are Problems 8.11 (Flow Distributio i a Pipelie Network) ad 6.6 (Expeditig the Solutio of Systems of Noliear Algebraic Equatios). More advaced treatmet of systems of oliear equatios (obtaied whe solvig a costraied miimizatio problem), is demostrated alog with the use of various software packages i Problems 4.5 ad 5.5 (Complex Chemical Equilibrium by Gibbs Eergy Miimizatio). (g) Higher Order ODEs Cosider the -th order ordiary differetial equatio d z dx dz G z dx d z dx K d z, x dx 2 1 =,,, 2 1 (1-10) This equatio ca be trasformed by a series of substitutio to a system of first-order equatios (Equatio (1-7)). Such a trasformatio is demostrated i Problems 6.5 (Shootig Method for Solvig Two-Poit Boudary Value Problems) ad 8.18 (Boudary Layer Flow of a Newtoia Fluid o a Flat Plate). (h) Systems of First-Order ODEs Boudary Value Problems ODEs with boudary coditios specified at two (or more) poits of the idepedet variable are classified as boudary value problems. Examples of such problems ad demostratio of the solutio techiques used ca be foud i Problems 6.4 (Iterative Solutio of ODE Boudary Value Problem) ad 6.5 (Shootig Method for Solvig Two-Poit Boudary Value Problems). (i) Stiff Systems of First-Order ODEs Systems of ODEs where the depedet variables chage o various time (idepedet variable) scales which differ by may orders of magitude are called Stiff systems. The characterizatio of stiff systems ad the special techiques that are used for solvig such systems are described i detail i Problem 6.2 (Solutio of Stiff Ordiary Differetial Equatios). (j) Differetial-Algebraic System of Equatios (DAEs) The system defied by the equatios: dy = f dx g( y, z) = 0 ( y, z, x) (1-11) with the iitial coditios y(x 0 ) = y 0 is called a system of differetial-algebraic equatios. Demostratio of oe particular techique for solvig DAEs ca be foud i Problem 6.7 (Solvig Differetial Algebraic Equatios DAEs).

9 1.3 CATEGORIZING PROBLEMS ACCORDING TO THE SOLUTION TECHNIQUE USED 9 (k) Partial Differetial Equatios (PDEs) Partial differetial equatios where there are several idepedet variables have a typical geeral form: 2 2 T T T = α t x y (1-12) A problem ivolvig PDEs requires specificatio of iitial values ad boudary coditios. The use of the Method of Lies for solvig PDEs is demostrated i Problems 6.8 (Method of Lies for Partial Differetial Equatios) ad 9.14 (Usteady-State Coductio i Two Dimesios). (l) Noliear Regressio I oliear regressio, a oliear fuctio g K K yˆ i = g( a0, a1 a, x1, i, x2, i, x, i ) (1-13) is used to model the data by fidig the values of the parameters a 0, a 1 a that miimize the squares of the errors show i Equatio (1-5). Detailed descriptio of the oliear regressio problem ad the method of solutio usig POLYMATH ca be foud i Problem 3.1 (Estimatio of Atoie Equatio Parameters usig Noliear Regressio). The use of Excel ad MATLAB for oliear regressio is demostrated i respective Problems 4.4 ad 5.4 (Correlatio of the Physical Properties of Ethae). (m) Parameter Estimatio i Dyamic Systems This problem is similar to the oliear regressio problem except that there is o closed form expressio for ŷ i, but the squares of the errors fuctio to be miimized must be calculated by solvig the system of first order ODEs dy ˆ = f dt 0 1K 1 2 K, ( a, a a, x, x, x t) (1-14) Problem 6.9 (Estimatig Model Parameters Ivolvig ODEs usig Fermetatio Data) describes i detail a parameter idetificatios problem ad demostrates its solutio usig POLYMATH ad MATLAB. () Noliear Programmig (Optimizatio) with Equity Costraits The oliear programmig problem with equity costrais is defied by: Miimize f ( x) Subject to h(x) = 0 (1-15) where f is a fuctio, x is a -vector of variables ad h is a m-vector (m<) of fuctios.

10 10 CHAPTER 1 PROBLEM SOLVING WITH MATHEMATICAL SOFTWARE PACKAGES Problems 4.5 ad 5.5 (Complex Chemical Equilibrium by Gibbs Eergy Miimizatio) demostrate several techiques for solvig oliear optimizatio problems with POLYMATH, Excel, ad MATLAB. 1.4 EFFECTIVE USE OF THIS BOOK Readers who wish to begi solvig realistic problems with mathematical software packages are recommeded to iitially study the items listed i categories (a) through (e) i the previous sectio. I additio, differet subject areas will require completio of some of the advaced topics listed i categories (f) through () of the previous sectio. Table 1 1 shows the advaced topics associated with the various subjects covered i Chapters 7 to 14 of this book. For example, you will be able to achieve effective solutios for the problems associated with Chapter 8 (Fluid Mechaics) if you iitially study the categories (f) System of Noliear Algebraic Equatios (NLEs), (g) Higher Order ODEs, (h) Systems of First- Order ODEs Boudary Value Problems, ad (k) Partial Differetial Equatios (PDEs). It is recommeded that the advaced topics listed i Table 1 1 be studied before you begi to work o a chapter related to a particular subject area. Table 1 1 Advaced Topic Prerequisites for Chapters 7 through 14 Chapter No. Subject Area Advaced Topics Required 7 Thermodyamics (f) 8 Fluid Mechaics (f), (g), (h), ad (k) 9 Heat Trasfer (g), (h), ad (k) 10 Mass Trasfer (g), (h), ad (k) 11 Chemical Reactio Egieerig (f), (g), (h), (l), ad (m) 12 Phase Equilibria ad Distillatio (f) ad (j) 13 Process Dyamics ad Cotrol (f), (g), (h), ad (i) 14 Biochemical Egieerig (f), (g), (h), (i), (l), ad () Several of the advaced topics are typically cosidered i the advaced subject areas of Numerical Methods, Advaced Mathematics, ad Optimizatio. Usig the problems pertiet to a particular topic i these advaced subject areas ca be very beeficial i the learig process. The problems associated with the various topics are show i Table 1 2, where solved or partially solved problem umbers are show i bold umerals Some problems may require special solutio techiques. Solved problems that demostrate some special techiques are listed i Table 1-3. If a problem matches oe or more of the categories i this table, the a examiatio of the similar solved problem ca be very helpful.

11 1.4 EFFECTIVE USE OF THIS BOOK 11 Table 1 2 List of Problems Associated with Advaced Topics a Topic Pertiet Problem Numbers* (f) System of Noliear Algebraic Equatios (NLEs) 4.5, 5.5, 6.6, 7.13, 7.14, 8.9, 8.10, 8.12, 8.13, 10.2,12.1, 12.2, 12.3, 12.4, 12.5, 12.8, 12.9, 14.8, 14.11, (g) Higher Order ODEs , 8.18, 9.2, 9.5, 10.11, 13.1, 13.2, 13.5, 13.7, 13.12, 14.5 (h) Systems of First-Order ODEs Boudary Value Problems 6.4,6.5,8.1,8.2,8.3,8.4,8.18, 9.1, 9.2, 9.6, 9.7, 10.1, 10.3, 10.5, 10.6,10.7, 10.8, 10.9, 10.10, 10.12, 14.5 (i) Stiff Systems of First-Order ODEs 6.1, 6.2 (j) Differetial-Algebraic System of Equatios 6.7, 12.10, (DAEs) (k) Partial Differetial Equatios (PDEs) 6.8, 8.17, 9.12, 9.13, 9.14, 10.13, 10.14, (l) Noliear Regressio 3.1, 3.2, 3.3, 3.4, 4.4, 5.4, 11.7, 13.4, 13.8, 14.2, 14.7, 14.8, 14.13, 14.14, (m) Parameter Estimatio i Dyamic Systems 6.9 () Noliear Programmig (Optimizatio) 4.5, 5.5, 6.9, 14.4, 14.6, 14.11, a Solved ad partially solved problems are idicated i bold. Table 1 3 List of Problems Associated with Special Problem Solvig Techiques Pertiet Problem Differetial ad Algebraic Equatios Number(s) Plottig Solutio Trajectory for a Algebraic Equatio Usig the ODE 7.1, 7.5 Solver Usig the l Hôpital s Rule for Udefied Fuctios at the Begiig or 7.11, 7.12 Ed Poit of Itegratio Iterval Usig If Statemet to Avoid Divisio by Zero 8.1 Switchig Variables O ad Off durig Itegratio 8.16 Retaiig a Value whe a Coditio Is Satisfied 8.16 Geeratio of Error Fuctio 8.17 Fuctios Udefied at the Iitial Poit 9.2, 9.5 Usig If Statemet to Match Differet Equatios to the Same Variable 9.2, 10.4, 8.6 Ill-Coditioed Systems 6.3 Coversio of a System of Noliear Equatios ito a Sigle Equatio 6.3 Selectio of Iitial Estimates for Noliear Equatios 6.6 Modificatio of Strogly Noliear Equatios for Easier Solutio 6.6 Coversio of a Noliear Algebraic Equatio to a Differetial 7.1, 7.5 Equatio Data Modelig ad Aalysis Usig Residual Plot for Data Aalysis 2.5, 3.1, 3.3, 3.5, 3.8, 3.14 Usig Cofidece Itervals for Checkig Sigificace of Parameters 2.5, 3.1, 3.3, 3.8 Trasformatio of Noliear Models for Liear Represetatios 2.5, 3.3, 3.5, 3.8 Checkig Liear Depedecy amog Idepedet Variables 3.11

12 12 CHAPTER 1 PROBLEM SOLVING WITH MATHEMATICAL SOFTWARE PACKAGES 1.5 SOFTWARE USAGE WITH THIS BOOK The problems preseted withi this text ca be solved with a variety of mathematical software packages. The problems alog with the appedices provide all the ecessary equatios ad parameters for obtaiig problem solutios. However, POLYMATH is extesively utilized to demostrate problem solutios throughout the book because it is extremely easy to use, ad because the equatios are etered i basically the same mathematical form as they are writte. Recet ehacemets to POLYMATH have eabled the cotet of this book to be expaded to more directly support problem solvig with Excel ad MAT- LAB. This is due to the ability of POLYMATH to export a problem solutio to a workig spreadsheet i Excel or the ecessary m-file of the problem for MAT- LAB. The export process to Excel results i a complete workig spreadsheet with the same otatio ad logic as utilized i the POLYMATH problem solutio code. I additio, all the itrisic fuctios (such as log, exp, si, etc.) are automatically coverted. POLYMATH also provides a Excel Add-I which allows the solutio of systems of ordiary differetial equatios withi Excel. This is separate software which operates i Excel ad eables the same solutio algorithms available i POLYMATH to be utilized withi Excel. Complete details ad a itroductio to problem solvig i Excel are give i Chapter 4. For MATLAB, the m-file for a particular POLYMATH program solutio is automatically geerated with traslatio of the program logic ad the eeded itrisic fuctios. Also, the geerated statemets i the m-file are automatically ordered as required by MATLAB. This use of MATLAB ad the use of the geeratio of m-files from POLYMATH are discussed i Chapter 5. Withi this book, the problems are discussed with referece to importat ad ecessary equatios. The solutios are preseted usig the POLYMATH codig as this provides a clear represetatio of the problem formulatio alog with the mathematical equatios ad logic ecessary for problem solutio. The Excel ad MATLAB solutios ca easily be achieved through the use of the POLYMATH program via the automated export capability. All of the completely worked or partially worked problems i the book are available i files for all three packages POLYMATH, Excel, ad MATLAB. Access to these files is provided from the book web site as discussed i the ext sectio. You ca obtai a iexpesive educatioal versio of the latest POLYMATH software from the book s web site, Thus you have a choice of the software with which you wish to geerate problem solutios while usig this book. POLYMATH is highly recommeded as it is used throughout the book ad is widely accepted as the most coveiet mathematical software for a studet or a ovice to lear ad use. You are ecouraged to lear to solve problems with POLYMATH, Excel, ad MATLAB as this book will highlight ad utilize some of the particular capabilities that each package eables.

13 1.6 WEB-BASED RESOURCES FOR THIS BOOK WEB-BASED RESOURCES FOR THIS BOOK A special web site is dedicated to the cotiuig support of this book ad this is idetified throughout the book with the ico o the left. This special web site provides readers with the followig materials: www 1. Solutio files for the worked ad partially worked problems for POLYMATH, Excel, ad MATLAB 2. Access to the latest Educatioal Versio of POLYMATH at a special reduced price that is available oly to book owers for educatioal use 3. MATLAB templates for geeral problem solvig 4. Additioal problems as they become available 5. Correctios to the prited book 6. Special resources for studets 7. Special resources for istructors who are usig this book Note: Istructors should obtai more details from oe of the authors via from their home istitutio to michael.cutlip@uco.edu shacham@bgu.ac.il GENERAL REFERENCES 1. Costatiides, A., ad Mostoufi, N., Numerical Methods for Chemical Egieers with MATLAB Applicatios, 1st ed., Upper Saddle River, NJ: Pretice Hall, Press, W. H., Teukolsky, S, A., Vetterlig, W. T., ad Flaery, B. P., Numerical Recipes i C++: The Art of Scietific Computig, 2d ed., Cambridge, Eglad: Cambridge Uiversity Press, Lapidus, L., Digital Computatio for Chemical Egieers, New York: McGraw-Hill, Caraha, B., Luther, H. A. ad Wilkes, J. O., Applied Numerical Methods, New York: Wiley, Heley, E. J., ad E. M. Rose, Material ad Eergy Balace Computatio, New York: Wiley, 1969.