Iterative Methods for Solving Sets of Equations

2.1 The Gauss-Seidel Method

The Gauss-Seidel method may be used to solve a set of linear or nonlinear algebraic equations. We will illustrate the method by solving a heat transfer problem. For steady state, no heat generation, and constant k, the heat conduction equation is simplified to Laplace equation

(2T = 0

For 2-dimensional heat transfer in Cartesian coordinate

[pic] + [pic] = 0

The above equation can be put in the finite difference form. We divide the medium of interest into a number of small regions and apply the heat equation to these regions. Each sub-region is assigned a reference point called a node or a nodal point. The average temperature of a nodal point is then calculated by solving the resulting equations from the energy balance. Accurate solutions can be obtained by choosing a fine mesh with a large number of nodes. We will discuss an example from Incropera’s1 text to illustrate the method.

Example 2.1-1

A long column with thermal conductivity k = 1 W/m(oK is maintained at 500oK on three surfaces while the remaining surface is exposed to a convective environment with h = 10 W/m2(oK and fluid temperature T(. The cross sectional area of the column is 1 m by 1 m. Using a grid spacing (x = (y = 0.25 m, determine the steady-state temperature distribution in the column and the heat flow to the fluid per unit length of the column.

Solution

The cross sectional area of the column is divided into many sub-areas called a grid or nodal network with 25 nodes as shown in Figure 2.1-1. There are 12 nodal points with unknown temperature, however only 8 unknowns need to be solved due to symmetry so that the nodes to the left of the centerline are the same as those to the right.
[pic]
Figure 2.1-1 The grid for the column cross sectional area.

The energy balance is now applied to