In a study of heat transfer in a grid of wires the temperatu

In a study of heat transfer in a grid of wires, the temperature at an exterior node is maintained at a constant value (In Degree F) as shown in the figure. When the grid is in thermal equilibrium, the temperature at an interior node is the average of the temperatures at the four adjacent nodes. For instance T_1 = + 300 T_2/4, or 4T_1 - T_2 = 300. Find the temperatures T_1, T_2, T_3 when the grid is in thermal equilibrium. T_2 = 180 + 200 + T_3/4 4T_2 = 380 + T_1 + T_3 T_3 = 108 + 0 + 0 + T_2/4 4T_3 = 108 + T_2

Solution

For all the temperature T1 , T2, T3 we can construct similar equations to find the solution

T1 = 0+0+300+T2 / 4 ; 4T1 -T2 = 300

T3 = 0 + 0 +108 + T2 / 4 ; 4T3 -T2 = 108

T2 = T1 +T3 +180 +200 /4 ; 4T2 - T1 -T3 = 380

Hence Solving the equation we get

T1 = 109.5

T2 = 137.71

T3 = 61.42