Compact Heat Exchangers

3Fundamentals of Finite Element and Finite Volume Methods

3.1 Introduction

There are three distinct methods of numerical solution techniques: finite difference, finite volume and finite element methods. The purpose in each is to convert the differential equations into algebraic equations. The main differences between the three methods are associated with the way the differential equations are converted to algebraic equations.

Finite difference methods (FDM) describe the unknown variable of the flow problem by means of point samples at the node points of a grid of coordinate lines.

In the finite element method (FEM), the continuous variable in the domain is replaced by piecewise continuous functions defined in sub‐regions (elements). The approximating function is substituted in differential equations resulting in residuals which are minimized by weighted residuals (Lewis et al. [1]). This process determines the unknown coefficients in the assumed profile.

In the finite volume method (FVM), discretization of the domain is carried out into number of cells. Here the flux entering the face and leaving is used by integrating the differential equations in the cell. This produces algebraic equations. Incidentally, the sub‐domain methods (with weighting function of 1) gives the same algebraic equations as the finite volume method.

3.2 Finite Element Method

The finite element method is a numerical tool for determining approximate solutions to a large class of engineering problems. ...