PROBLEM 34E

An important concern in the study of heat transfer is to determine the steady-state temperature distribution of a thin plate when the temperature around the boundary is known. Assume the plate shown in the figure represents a cross section of a metal beam, with negligible heat flow in the direction perpendicular to the plate. Let T1,…,T4 denote the temperatures at the four interior nodes of the mesh in the figure. The temperature at a node is approximately equal to the average of the four nearest nodes—to the left, above, to the right, and below.3 For instance,

T 1 = (10 + 20 T2 + T4)/4, or 4T1 – T2 – T4 = 30

Solve the system of equations from Exercise 33. [Hint: To speed up the calculations, interchange rows 1 and 4 before starting “replace” operations.]

Exercise 33:

Write a system of four equations whose solution gives estimates for the temperatures T1,…,T4.

Solution:

Step 1

In this problem we need to find the temperature at the nodes 1, 2, 3 and 4 given in the figure.