Applied Mathematics And Modeling For Chemical Engineers E-Book

Richard G. Rice

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 827

Veröffentlichungsjahr: 2012

Beschreibung

Enables chemical engineers to use mathematics to solve common on-the-job problems With its clear explanations, examples, and problem sets, Applied Mathematics and Modeling for Chemical Engineers has enabled thousands of chemical engineers to apply mathematical principles to successfully solve practical problems. The book introduces traditional techniques to solve ordinary differential equations as well as analytical methods to deal with important classes of finite-difference equations. It then explores techniques for solving partial differential equations from classical methods to finite-transforms, culminating with??numerical methods??including orthogonal collocation. This Second Edition demonstrates how classical mathematics solves a broad range of new applications that have arisen since the publication of the acclaimed first edition. Readers will find new materials and problems dealing with such topics as: * Brain implant drug delivery * Carbon dioxide storage * Chemical reactions in nanotubes * Dissolution of pills and pharmaceutical capsules * Honeycomb reactors used in catalytic converters * New models of physical phenomena such as bubble coalescence Like the first edition, this Second Edition provides plenty of worked examples that explain each step on the way to finding a problem's solution. Homework problems at the end of each chapter are designed to encourage readers to more deeply examine the underlying logic of the mathematical techniques used to arrive at the answers. Readers can refer to the references, also at the end of each chapter, to explore individual topics in greater depth. Finally, the text's appendices provide additional information on numerical methods for solving algebraic equations as well as a detailed explanation of numerical integration algorithms. Applied Mathematics and Modeling for Chemical Engineers is recommended for all students in chemical engineering as well as professional chemical engineers who want to improve their ability to use mathematics to solve common on-the-job problems.

Leseprobe

Contents

Cover

Title Page

Copyright

Preface to the Second Edition

Part I

Chapter 1: Formulation of Physicochemical Problems

1.1 Introduction

1.2 Illustration of the Formulation Process (Cooling of Fluids)

1.3 Combining Rate and Equilibrium Concepts (Packed Bed Adsorber)

1.4 Boundary Conditions and Sign Conventions

1.5 Models with Many Variables: Vectors and Matrices

1.6 Matrix Definition

1.7 Types of Matrices

1.8 Matrix Algebra

1.9 Useful Row Operations

1.10 Direct Elimination Methods

1.11 Iterative Methods

1.12 Summary of the Model Building Process

1.13 Model Hierarchy and its Importance in Analysis

References

Chapter 2: Solution Techniques for Models Yielding Ordinary Differential Equations

2.1 Geometric Basis and Functionality

2.2 Classification of ODE

2.3 First-Order Equations

2.4 Solution Methods for Second-Order Nonlinear Equations

2.5 Linear Equations of Higher Order

2.6 Coupled Simultaneous ODE

2.7 Eigenproblems

2.8 Coupled Linear Differential Equations

2.9 Summary of Solution Methods for ODE

References

Chapter 3: Series Solution Methods and Special Functions

3.1 Introduction to Series Methods

3.2 Properties of Infinite Series

3.3 Method of Frobenius

3.4 Summary of The Frobenius Method

3.5 Special Functions

References

Chapter 4: Integral Functions

4.1 Introduction

4.2 The Error Function

4.3 The Gamma and Beta Functions

4.4 The Elliptic Integrals

4.5 The Exponential and Trigonometric Integrals

References

Chapter 5: Staged-Process Models: The Calculus of Finite Differences

5.1 Introduction

5.2 Solution Methods for Linear Finite Difference Equations

5.3 Particular Solution Methods

5.4 Nonlinear Equations (Riccati Equation)

References

Chapter 6: Approximate Solution Methods for ODE: Perturbation Methods

6.1 Perturbation Methods

6.2 The Basic Concepts

6.3 The Method of Matched Asymptotic Expansion

6.4 Matched Asymptotic Expansions for Coupled Equations

References

Part II

Chapter 7: Numerical Solution Methods (Initial Value Problems)

7.1 Introduction

7.2 Type of Method

7.3 Stability

7.4 Stiffness

7.5 Interpolation and Quadrature

7.6 Explicit Integration Methods

7.7 Implicit Integration Methods

7.8 Predictor–Corrector Methods and Runge–Kutta Methods

7.9 Runge–Kutta Methods

7.10 Extrapolation

7.11 Step Size Control

7.12 Higher Order Integration Methods

References

Chapter 8: Approximate Methods for Boundary Value Problems: Weighted Residuals

8.1 The Method of Weighted Residuals

8.2 Jacobi Polynomials

8.3 Lagrange Interpolation Polynomials

8.4 Orthogonal Collocation Method

8.5 Linear Boundary Value Problem: Dirichlet Boundary Condition

8.6 Linear Boundary Value Problem: Robin Boundary Condition

8.7 Nonlinear Boundary Value Problem: Dirichlet Boundary Condition

8.8 One-Point Collocation

8.9 Summary of Collocation Methods

8.10 Concluding Remarks

References

Chapter 9: Introduction to Complex Variables and Laplace Transforms

9.1 Introduction

9.2 Elements of Complex Variables

9.3 Elementary Functions of Complex Variables

9.4 Multivalued Functions

9.5 Continuity Properties for Complex Variables: Analyticity

9.6 Integration: Cauchy's Theorem

9.7 Cauchy's Theory of Residues

9.8 Inversion of Laplace Transforms by Contour Integration

9.9 Laplace Transformations: Building Blocks

9.10 Practical Inversion Methods

9.11 Applications of Laplace Transforms for Solutions of ODE

9.12 Inversion Theory for Multivalued Functions: the Second Bromwich Path

9.13 Numerical Inversion Techniques

References

Chapter 10: Solution Techniques for Models Producing PDEs

10.1 Introduction

10.2 Particular Solutions for PDEs

10.3 Combination of Variables Method

10.4 Separation of Variables Method

10.5 Orthogonal Functions and Sturm–Liouville Conditions

10.6 Inhomogeneous Equations

10.7 Applications of Laplace Transforms for Solutions of PDEs

References

Chapter 11: Transform Methods for Linear PDEs

11.1 Introduction

11.2 Transforms in Finite Domain: Sturm–Liouville Transforms

11.3 Generalized Sturm–Liouville Integral Transform

References

Chapter 12: Approximate and Numerical Solution Methods for PDEs

12.1 Polynomial Approximation

12.2 Singular Perturbation

12.3 Finite Difference

12.4 Orthogonal Collocation for Solving PDEs

12.5 Orthogonal Collocation on Finite Elements

References

Appendix A: Review of Methods for Nonlinear Algebraic Equations

A.1 The Bisection Algorithm

A.2 The Successive Substitution Method

A.3 The Newton–Raphson Method

A.4 Rate of Convergence

A.5 Multiplicity

A.6 Accelerating Convergence

References

Appendix B: Derivation of The Fourier–Mellin Inversion Theorem

References

Appendix C: Table of Laplace Transforms

Appendix D: Numerical Integration

D.1 Basic Idea of Numerical Integration

D.2 Newton Forward Difference Polynomial

D.3 Basic Integration Procedure

D.4 Error Control and Extrapolation

D.5 Gaussian Quadrature

D.6 Radau Quadrature

D.7 Lobatto Quadrature

D.8 Concluding Remarks

References

Appendix E: Nomenclature

Postface

Index

Cover illustration: Roberta Fox

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Library of Congress Cataloging-in-Publication Data:

Rice, Richard G.

Applied mathematics and modeling for chemical engineers / Richard G.

Rice, Duong D. Do — 2nd ed.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-118-02472-0

Preface to the Second Edition

While classical mathematics has hardly changed over the years, new applications arise regularly. In the second edition, we attempt to teach applicable mathematics to a new generation of readers involved in self-study and traditional university coursework. As in the first edition, we lead the reader through homework problems, providing answers while coaxing students to think more deeply about how the answer was uncovered. New material has been added, especially homework problems in the biochemical area, including diffusion in skin and brain implant drug delivery, and modern topics such as carbon dioxide storage, chemical reactions in nanotubes, dissolution of pills and pharmaceutical capsules, honeycomb reactors used in catalytic converters, and new models of important physical phenomenon such as bubble coalescence.

The presentation of linear algebra using vectors and matrices has been moved from an Appendix and interspersed between Chapters 1 and 2, and used in a number of places in the book, notably Chapters 11 and 12, where MATLAB referenced solutions are provided.

The model building stage in Chapter 1 has thus been augmented to include models with many variables using vectors and matrices. Chapter 6 begins teaching applied mathematics for solving ordinary differential equations, both linear and nonlinear. The chapter culminates with teaching how to solve arrays of coupled linear equations using matrix methods and the analysis of the eigenproblem, leading to eigenvalues and eigenvectors. Classical methods for solving second-order linear equations with nonconstant coefficients are treated using series solutions via the method of Frobenius in Chapter 3. Special functions are also inspected, especially Bessel's functions, which arise frequently in chemical engineering owing to our propensity toward cylindrical geometry. Integral functions, often ignored in other textbooks, are given a careful review in Chapter 4, with special attention to the widely used error function. In Chapter 5, we study the mathematics of staged processing, common in chemical engineering unit operations, and develop the calculus of finite difference equations, showing how to obtain analytical solutions for both linear and nonlinear systems. This chapter adds a new homework problem dealing with economics and finite difference equations used by central banks to forecast personal consumption. These five chapters would provide a suitable undergraduate course for third or fourth year students. To guide the teacher, we again have used a scheme to indicate homework problem challenges: subscripts 1 denotes mainly computational or setup problems, while subscripts 2 and 3 require more synthesis and analysis. Problems with an asterisk are the most difficult and are more suited for graduate students.

The approximate technique for solving equations, especially nonlinear types, is treated in Chapter 6 using perturbation methods. It culminates with teaching the method of matched asymptotic expansion. Following this, other approximate methods suitable for computer implementation are treated in Chapter 7 as numerical solution by finite differences, for initial value problems. In Chapter 8, computer-oriented boundary value problems are addressed using weighted residuals and the methods of orthogonal collocation. Complex variables and Laplace transforms are given a combined treatment in Chapter 9, illustrating the intimate connection to the Fourier–Mellon complex integral, which is the basis for the Laplace transform.

Treatment of partial differential equations (PDEs) begins in Chapter 10, where classical methods are taught, including the combination of variables approach, the separation of variables method and the important orthogonality conditions arising from the Sturm–Liouville equation, and finally, solutions using Laplace transforms and the method of residues. After these classical methods, we introduce finite transform methods in Chapter 11, and exploit the othogonality condition to introduce a universal transform called the Sturm–Liouville transform. This method produces as subsets the famous Hankel and Fourier transforms. The concept of Hilbert space is explained and the solution of coupled partial differential equations is illustrated using the famous batch adsorber problem. The last chapter (12) of the book deals with approximate and numerical solution methods for PDE, treating polynomial approximation, singular perturbation, and finite difference methods. Orthogonal collocation methods applied to PDEs are given an extensive treatment. Appendices provide useful information on numerical methods to solve algebraic equations and a careful explanation of numerical integration algorithms.

After 17 years in print, we would be remiss by not mentioning the many contributions and suggestions by users and teachers from around the world. We especially wish to thank Professor Dean O. Harper of the University of Louisville and his students who forwarded errata that proved most helpful in revision for the second edition. Thanks also to Professor Morton M. Denn of CCNY for suggesting the transfer from the Appendix of vectors and matrix methods to the first two chapters. Reviewers who have used the book also gave many good ideas for revision in the second edition, and we wish to thank colleagues from Bucknell University, notably Professors James E. Maneval and William E. King. We continue to be grateful to students and colleagues who point out errors, typos, and suggestions for additional material; we particularly appreciate the help of Professor Benjamin J. McCoy of UC Davis and Ralph E. White of the University of South Carolina. In the final analysis, any remaining errors, and the selection of material to include, are the responsibility of the authors, and we apologize for not including all the changes and additions suggested by others.

Richard G. Rice

Tazewell, TN, USA

Duong D. Do

University of Queensland, Australia

Part I

Two roads diverged in a yellow wood,And sorry I could not travel bothAnd be one traveler, long I stoodAnd looked down one as far as I couldTo where it bent in the undergrowth;

Then took the other, as just as fair,And having perhaps the better claim,Because it was grassy and wanted wear;Though as for that the passing thereHad worn them really about the same,

And both that morning equally layIn leaves no step had trodden black.Oh, I kept the first for another day!Yet knowing how way leads on to way,I doubted if I should ever come back.

I shall be telling this with a sighSomewhere ages and ages hence:Two roads diverged in a wood, and I-I took the one less traveled by,And that has made all the difference.

—Robert Frost (“The Road Not Taken”)

Chapter 1

Formulation of Physicochemical Problems

1.1 Introduction

Modern science and engineering require high levels of qualitative logic before the act of precise problem formulation can occur. Thus, much is known about a physicochemical problem beforehand, derived from experience or experiment (i.e., empiricism). Most often, a theory evolves only after detailed observation of an event. This first step usually involves drawing a picture of the system to be studied.

The second step is the bringing together of all applicable physical and chemical information, conservation laws, and rate expressions. At this point, the engineer must make a series of critical decisions about the conversion of mental images to symbols, and at the same time, how detailed the model of a system must be. Here, one must classify the real purposes of the modeling effort. Is the model to be used only for explaining trends in the operation of an existing piece of equipment? Is the model to be used for predictive or design purposes? Do we want steady-state or transient response? The scope and depth of these early decisions will determine the ultimate complexity of the final mathematical description.

The third step requires the setting down of finite or differential volume elements, followed by writing the conservation laws. In the limit, as the differential elements shrink, then differential equations arise naturally. Next, the problem of boundary conditions must be addressed, and this aspect must be treated with considerable circumspection.

When the problem is fully posed in quantitative terms, an appropriate mathematical solution method is sought out, which finally relates dependent (responding) variables to one or more independent (changing) variables. The final result may be an elementary mathematical formula or a numerical solution portrayed as an array of numbers.

1.2 Illustration of the Formulation Process (Cooling of Fluids)

We illustrate the principles outlined above and the hierarchy of model building by way of a concrete example: the cooling of a fluid flowing in a circular pipe. We start with the simplest possible model, adding complexity as the demands for precision increase. Often, the simple model will suffice for rough, qualitative purposes. However, certain economic constraints weigh heavily against overdesign, so predictions and designs based on the model may need be more precise. This section also illustrates the “need to know” principle, which acts as a catalyst to stimulate the garnering together of mathematical techniques. The problem posed in this section will appear repeatedly throughout the book, as more sophisticated techniques are applied to its complete solution.

1.2.1 Model I: Plug Flow

As suggested in the beginning, we first formulate a mental picture and then draw a sketch of the system. We bring together our thoughts for a simple plug flow model in Fig. 1.1a. One of the key assumptions here is plug flow, which means that the fluid velocity profile is plug shaped, in other words, uniform at all radial positions. This almost always implies turbulent fluid flow conditions, so that fluid elements are well mixed in the radial direction, hence the fluid temperature is fairly uniform in a plane normal to the flow field (i.e., the radial direction).

Figure 1.1 (a) Sketch of plug flow model formulation. (b) Elemental or control volume for plug flow model. (c) Control volume for Model II.

If the tube is not too long or the temperature difference is not too severe, then the physical properties of the fluid will not change much, so our second step is to express this and other assumptions as a list:

1. A steady-state solution is desired.

2. The physical properties (ρ, density; Cp, specific heat; k, thermal conductivity, etc.) of the fluid remain constant.

3. The wall temperature is constant and uniform (i.e., does not change in the z or r direction) at a value Tw.

4. The inlet temperature is constant and uniform (does not vary in r direction) at a value T0, where T0 > Tw.

5. The velocity profile is plug shaped or flat, hence it is uniform with respect to z or r.

6. The fluid is well mixed (highly turbulent), so the temperature is uniform in the radial direction.

7. Thermal conduction of heat along the axis is small relative to convection.

The third step is to sketch, and act upon, a differential volume element of the system (in this case, the flowing fluid) to be modeled. We illustrate this elemental volume in Fig. 1.1b, which is sometimes called the “control volume.”

We act upon this elemental volume, which spans the whole of the tube cross section, by writing the general conservation law

(1.1)

Since steady state is stipulated, the accumulation of heat is zero. Moreover, there are no chemical, nuclear, or electrical sources specified within the volume element, so heat generation is absent. The only way heat can be exchanged is through the perimeter of the element by way of the temperature difference between wall and fluid. The incremental rate of heat removal can be expressed as a positive quantity using Newton's law of cooling, that is,

(1.2)

As a convention, we shall express all such rate laws as positive quantities, invoking positive or negative signs as required when such expressions are introduced into the conservation law (Eq. 1.1). The contact area in this simple model is simply the perimeter of the element times its length.

The constant heat transfer coefficient is denoted by h. We have placed a bar over T to represent the average between T(z) and T(z + Δ z)

(1.3)

In the limit, as , we see

(1.4)

Now, along the axis, heat can enter and leave the element only by convection (flow), so we can write the elemental form of Eq. 1.1 as

(1.5)

The first two terms are simply mass flow rate times local enthalpy, where the reference temperature for enthalpy is taken as zero. Had we used ( − ) for enthalpy, the term would be cancelled in the elemental balance. The last step is to invoke the fundamental lemma of calculus, which defines the act of differentiation