1.4.1 Introduction

The analytical solutions examined in the previous sections are convenient since they provide accurate results for arbitrary inputs with minimal computational effort. However, many problems of practical interest are too complicated to allow an analytical solution. In such cases, numerical solutions are required. Analytical solutions remain useful as a way to test the validity of numerical solutions under limiting conditions.

Numerical solutions are generally more computationally complex and are not unconditionally accurate. Numerical solutions are only approximations to a real solution, albeit approximations that are extremely accurate when done correctly. It is relatively straightforward to solve even complicated problems using numerical techniques.

The steps required to set up a numerical solution using the finite difference approach remain the same even as the problems become more complex. The result of a numerical model is not a functional relationship between temperature and position but rather a prediction of the temperatures at many discrete positions. The first step is to define small control volumes that are distributed through the computational domain and to specify the locations where the numerical model will compute the temperatures (i.e., the nodes). The control volumes used in the numerical model are small but finite, as opposed to the infinitesimally small (differential) control volume that is defined in order to derive an analytical solution. It is necessary to perform an energy balance on each control volume. This requirement may seem daunting given that many control volumes will be...