Analytical and Approximate Methods in Transport Phenomena

Marcio L. de Souza-Santos

• Publisher: CRC Press
• Year: 2007
• ISBN:
  9781000612066
• Edition: 1
• 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 ...