An important concern in the study of heat transfer is to det

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 Ti, 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. For instance or 4T 30 20° 200 2 100 400 4 3 10 400 30 30 Write a system of four equations whose solution gives esti- mates for the temperatures T1, Solve the system of equations from Exercise 33. [Hint: To speed up the calculations, interchange rows 1 and 4 before starting \"replace\" operations.]

Solution

here according to the question

the temperature at a node is approximately equal to the average of 4 nearest node that is left,right,above and below node

so a system of 4 equation whose solution gives estimates for the temperature T₁, T₂, T₃, T₄ is given by

T₁=(10+20+T₂+T₄)/4

4T₁-T₂-T₄=30 .............................................(1)

T₂=(20+40+T₁+T₃)/4

4T₂-T₁-T₃=60 .............................................(2)

T₃=(30+40+T₂+T₄)/4

4T₃-T₂-T₄=70 ..............................................(3)

AND

T₄=(10+30+T₁+T₃)/4

4T₄-T₁-T₃=40 .............................................(4)

SO THE 4 EQUATION ARE

4T₁-T₂-T₄=30

4T₂-T₁-T₃=60

4T₃-T₂-T₄=70

4T₄-T₁-T₃=40

NOW ON SOLVING THESE 4 EQUATION YOU WILL GET THE TEMPERATURE OF EACH NODE