Question

Two metal rods, one silver and the other gold, are attached to each other. The free end of the silver rod is connected to a steam chamber, with a temperature of 100°C, and the free end of the gold rod to an ice water bath, with a temperature of 0°C. The rods are 5.0 cm long and have a square cross-section, 2.0 cm on a side. How much heat flows through the two rods in 60 s? The thermal conductivity of silver is 417 W/(m·K), and that of gold is 291 W/(m·K). No heat is exchanged between the rods and the surroundings, except at the ends.

Answer 1

There is an electrical analogy with conduction heat transfer that can be exploited in problem solving. The analog of is current, and the analog of the temperature difference, , is voltage difference. From this perspective the slab is a pure resistance to heat transfer and we can define,

where , the thermal resistance. The thermal resistance increases as increases, as decreases, and as decreases.

The concept of a thermal resistance circuit allows ready analysis of problems such as a composite slab (composite planar heat transfer surface). In the composite slab shown in Figure, the heat flux is constant with . The resistances are in series and sum to . If is the temperature at the left, and is the temperature at the right, the heat transfer rate is given by

Here, slab1 = silver, slab2 = gold, T_L = 100^o C and T_R = 0^o C, L = L1 = L2 = 0.05 m, A = 0.0004 m²

The amount of heat flown through the rod in 60 s is,