An important concern in the study of heat transfer is to determine the steady-st

Question

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 that the plate in the figure represents a cross section of a metal beam, with negligible

heat flow in the direction perpendicular to the plate. Let T1; T2; T3; T4 denote the temper-

ature at the four interior nodes of the mesh in the figure. The temperature at a node is equal to the average of the four nearest nodes- the left, right, above and below. For instance,

T1 = (10 + 20 + T2 + T4)=4.

(a) Find a system of four equations whose solution will give T1; T2; T3; T4.

(b) Solve the above system.

Explanation / Answer

T1 = (10 + 20 + T2 + T4)/4... 4T1 - T2 - T4=30 ,,,,,4* T2=60+T1+T3 .......4T2-T1-T3=60........ 4*T4 -T1- T3= 40 .......4T3-T2-T4=70......HENCE THE EQUATIONS ARE ............................... 4T1 - T2 - T4=30................. .-T1+4T2-T3=60............... -T1- T3+4T4= 40........... -T2+4T3-T4=70.............. solving the equations we get T1=20 T2= 27.5 T3=30 T4=22.5

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