The concept of thermal resistance described in the previous problem can be used to find the temperature distribution in the flat square plate shown in Figure P43(a). The plate’s edges are insulated so that no heat can escape, except at two points where the edge temperature is heated to Ta

and T_b, respectively. The temperature varies through the plate, so no single point can describe the plate’s temperature. One way to estimate the temperature distribution is to imagine that the plate consists of four sub squares and to compute the temperature in each sub square. Let R be the thermal resistance of the material between the centers of adjacent sub squares. Then we can think of the problem as a network of electrical resistors, as shown in part (b) of the figure. Let q_ij be the heat flow rate between the points whose temperatures are T_i and T_j. If T_a and T_b remain constant for some time, then the heat energy stored in each sub square is constant also, and the heat flow rate between each sub square is constant. Under these conditions, conservation of energy says that the heat flow into a sub square equals the heat flow out. Applying this principle to each sub square gives the following equations.

Substituting q = (T_i =- T_j )/R, we find that R can be canceled out of every equation, and they can be rearranged as follows:

These equations tell us that the temperature of each sub square is the average of the temperatures in the adjacent sub squares!

Solve these equations for the case where T_a = 150ºC and T_b = 20ºC.