A long, uniform rod of length L = 1.0 m and cross-sectional area A = 1.0 × 1 0 − 4 m 2 has its lateral surface perfectly insulated. One end of the rod is maintained at a temperature T 1 = 100 K and

Question

A long, uniform rod of length L = 1.0 m and cross-sectional area A = 1.0 × 1 0 − 4 m 2 has its lateral surface perfectly insulated. One end of the rod is maintained at a temperature T 1 ​ = 100 K and the other end at T 2 ​ = 400 K. The thermal conductivity of the rod material varies with temperature as k ( T ) = k 0 ​ ( 1 + α T ) , where k 0 ​ = 10 W/(m K) and α = 0.01 K − 1 . Calculate the steady-state heat current (in Watts) flowing through the rod.

TestHub · Accepted Answer

In steady state, the heat current H is constant along the rod. From Fourier's law, H = − k ( T ) A d x d T ​ . Integrating this expression from x = 0 to x = L and from T = T 1 ​ to T = T 2 ​ : H ∫ 0 L ​ d x = − A ∫ T 1 ​ T 2 ​ ​ k 0 ​ ( 1 + α T ) d T H L = − A k 0 ​ [ T + 2 α T 2 ​ ] T 1 ​ T 2 ​ ​ H = − L A k 0 ​ ​ [ ( T 2 ​ − T 1 ​ ) + 2 α ​ ( T 2 2 ​ − T 1 2 ​ ) ] Substituting the given values: H = − 1.0 ( 1.0 × 1 0 − 4 ) × 10 ​ [ ( 400 − 100 ) + 2 0.01 ​ ( 40 0 2 − 10 0 2 ) ] H = − 1 0 − 3 [ 300 + 0.005 ( 160000 − 10000 ) ] H = − 1 0 − 3 [ 300 + 0.005 × 150000 ] = − 1 0 − 3 [ 300 + 750 ] = − 1.05 W The magnitude of the heat current is 1.05 W.

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