The three rods depicted in the figure have identical geometrical dimensions. In arrangement (a), heat flows from the hot end at a rate of 40 W . The thermal conductivities of aluminium and copper are 200 W m ⁻¹ ^ C ⁻¹ and 400 W m ⁻¹ ^ C ⁻¹ respectively. Determine the rate of heat flow when the rods are joined as in arrangement (a).

Step-by-step solution

Let the thermal resistance of the aluminium rod be R_ Al = L K_ Al A = R . Since K_ Cu = 2K_ Al , the thermal resistance of the copper rod is R_ Cu = L K_ Cu A = R 2 . For arrangement (a), the equivalent thermal resistance is: R_a = R_ Al + R_ Cu + R_ Al = R + R 2 + R = 5R 2 The rate of heat flow in arrangement (a) is: H_a = T R_a = 100 5R 2 = 40 R Given H_a = 40 W , we have 40 R = 40 R = 1 ^ C W ⁻¹ . For arrangement (b), the equivalent thermal resistance is: R_b = R_ Al + R_ Cu R_ Al R_ Cu + R_ Al = R + R 2 R R 2