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 (b).

Step-by-step solution

Let the thermal resistance of the aluminium rod be R_ Al = R . Since the thermal resistance is R = L KA and K_ Cu = 2K_ Al , the thermal resistance of the copper rod is R_ Cu = R 2 . In arrangement (a), the three rods are in series. The equivalent thermal resistance is: R_a = R_ Al + R_ Cu + R_ Al = R + R 2 + R = 5R 2 The rate of heat flow is H_a = T R_a = 100 5R 2 = 40 R . Given H_a = 40 W , we have 40 R = 40 R = 1 K W ⁻¹ . In arrangement (b), one aluminium rod is in series with a parallel combination of a copper