Analytical and Approximate Methods in Transport Phenomena

Marcio L. de Souza-Santos

• Publisher: CRC Press
• Year: 2007
• Print ISBN: 9780849334085
• eBook ISBN: 9781000612066
• Topic: Mathematics
• Subtopic: Analytic Geometry
• Index: Mathematics

On the job or in the field, when facing a problem with differential equations and boundary conditions, most likely you don't have time to read through several publications in search of a method that may or may not solve your problem. Organized for quick and easy access to practical solutions, Analytical and Approximate Methods in Transport Pheno

1 Problems 111;One Variable, 1st Order, 1st Kind Boundary Condition

1.1 INTRODUCTION

This chapter presents methods to solve problems with one independent variable involving first-order differential equation and first-kind boundary condition. Mathematically, this class of problems can be summarized as $f\left(\varphi ,\omega ,\frac{\text{d}\varphi }{\text{d}\omega }\right)$, firstkind boundary condition.

1.1.1 MODEL AND REALITY

As shown throughout the text, even for apparently simple problems presented in this chapter, the solutions are possible only after a series of assumptions. Therefore, every mathematical model just approximately reproduces the relationships among the involved variables during real processes or phenomena. On the other hand, in science and engineering, there is always the need of relying on models to design any equipment or system or to predict the behavior of processes.

Thus, modeling starts with assumptions made about the behavior of real processes. There are several levels of assumptions, which in turn reflect the level of the model complexity. Of course, a model may or may not be a reasonable representation of reality. What is considered reasonable is a matter of conventions or criteria, but all those are based on comparisons between model and reality. Such comparisons are possible by measuring the variables involved in the real phenomena against the respective predicted values computed by the theoretical solution. Usually in engineering applications, deviations below 5% between the measured and predicted values are acceptable. This may not be the case for several other more critical applications.

Let us take, for instance, the case presented in the following section and illustrated by Figure 1.1.

A body exchanges heat with surrounding environment and the temperature inside that body has to be determined against time. Therefore, the temperature in the body would constitute the main variable of the problem. In any case, the simple question of what and how comparisons would take place requires examination. For instance, during the process of heat exchange between the body and environment, temperature will vary along positions inside the body. Consequently, the physical and chemical properties of the material of the body and the immediate environment would change ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• Acknowledgments
• List of Symbols
• Chapter 1 Problems 111; One Variable, 1st Order, 1st Kind Boundary Condition
• Chapter 2 Problems 112; One Variable, 1st Order, 2nd Kind Boundary Condition
• Chapter 3 Problems 113; One Variable, 1st Order, 3rd Kind Boundary Condition
• Chapter 4 Problems 121; One Variable, 2nd Order, 1st Kind Boundary Condition
• Chapter 5 Problems 122; One Variable, 2nd Order, 2nd Kind Boundary Condition
• Chapter 6 Problems 123; One Variable, 2nd Order, 3rd Kind Boundary Condition
• Chapter 7 Problems 211; Two Variables, 1st Order, 1st Kind Boundary Condition
• Chapter 8 Problems 212; Two Variables, 1st Order, 2nd Kind Boundary Condition
• Chapter 9 Problems 213; Two Variables, 1st Order, 3rd Kind Boundary Condition
• Chapter 10 Problems 221; Two Variables, 2nd Order, 1st Kind Boundary Condition
• Chapter 11 Problems 222; Two Variables, 2nd Order, 2nd Kind Boundary Condition
• Chapter 12 Problems 223; Two Variables, 2nd Order, 3rd Kind Boundary Condition
• Chapter 13 Problems 311; Three Variables, 1st Order, 1st Kind Boundary Condition
• Chapter 14 Problems 312; Three Variables, 1st Order, 2nd Kind Boundary Condition
• Chapter 15 Problems 313; Three Variables, 1st Order, 3rd Kind Boundary Condition
• Chapter 16 Problems 321; Three Variables, 2nd Order, 1st Kind Boundary Condition
• Appendix A Fundamental Equations of Transport Phenomena
• Appendix B Fundamental Aspects of Ordinary Differential Equations
• Appendix C Method of Power Series and Special Functions
• Appendix D Laplace Transform
• Appendix E Method of Weighted Residuals
• Appendix F Method of Similarity
• Appendix G Fourier Series and Method of Separation of Variables
• Appendix H Fourier Transforms
• Appendix I Generalized Representation by Series
• Index