Question

An important concern in the study of heat transfer is to determine the steady-state temperature distribution of a plate when 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 A, B, C, and D 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. For instance, 10+25B+ D , or 4A-B- D 35 25 25 10 #30 10 #30 35 35 Write a system of tour equations whose solution gives estimates tor the temperatures A. B. C, and D E! Solve the system

Answer 1

Going as per the instructions, the equTion for the temperature at each nodes will be

1025 + B+ D

25 +30 +A+C B=

$C = \frac{35 + 30 + B + D}{4}$

35 + 10 A+C

On simplifying the eqn becomes

4A - B- D-35

4B-A-C=55

4C-B-D = 65

4D-A-C 45

Here we have 4 sets of equations and 4 unknown variables.

This can be solved using the matrix method,

4 -1 0 -1 A = 35

-1 4 -1 0 B = 55

0 -1 4 -1 C = 65

-1 0 -1 4 D = 45

This is of the form

AX = B

So to find X we need inverse of A

Were A is

4 1 0 1 1 41 0 0 1 4-1 1 0 4

So that

Here the inverse of the matrix is

7/24 1/12 1/24 1/12 1/12 7/24 1/12 1/24 1/24 1/12 7/24 1/12 1/12 1/24 1/12 | 7/24

To get the matrix X we multiply the inverse of A with matrix B

So we get, the matrix X as

85/4

105/4

115/4

95/4

So the temperature of four points are

A = 21.25

B = 26.25

C = 28.75

D = 23.75