Three rods, each of length 20 cm and cross-sectional area 1 cm ^2 , are joined together to form a triangle ABC . The thermal conductivities of these rods are K_ AB = 50 J s ⁻¹ m ⁻¹ ^ C ⁻¹ , K_ BC = 200 J s ⁻¹ m ⁻¹ ^ C ⁻¹ , and K_ AC = 400 J s ⁻¹ m ⁻¹ ^ C ⁻¹ . The junctions A , B , and C are maintained at temperatures of 40^ C , 80^ C , and 80^ C respectively. Determine the rate of heat flowing through the rods AB , A

Step-by-step solution

The rate of heat flow through a rod is given by H = KA T L . Given: Length of each rod, L = 20 cm = 0.2 m Cross-sectional area, A = 1 cm ^2 = 10⁻⁴ m ^2 Temperatures at junctions: T_A = 40^ C , T_B = 80^ C , T_C = 80^ C For rod AB : T_ AB = T_B - T_A = 80^ C - 40^ C = 40^ C H_ AB = K_ AB A T_ AB L = 50 10⁻⁴ 40 0.2 = 2000 10⁻⁴ 0.2 = 1 W For rod AC : T_ AC = T_C - T_A = 80^ C - 40^ C = 40^ C H_ AC = K_ AC A T_ AC L = 400 10⁻⁴ 40 0.2 = 16000 10⁻⁴ 0.2 = 8 W For rod BC : T_ BC = T_B - T_C = 80^ C - 80^ C = 0^ C H_ BC = K