For a conducting cylinder with variable thermal conductivity along the axis the magnitude of rate of heat flow if temperature difference across ends is ( T 1 − T 2 ) , Solid cylinder, heat flows a

Question

For a conducting cylinder with variable thermal conductivity along the axis the magnitude of rate of heat flow if temperature difference across ends is $(T_{1} - T_{2})$ ,

Solid cylinder, heat flows along the axis; thermal conductivity varies as $k = k_{0} (1 + \frac{x}{3 R})$ , where x is the distance from the left end.

TestHub · Accepted Answer

In steady state, the same heat current crosses every axial slice.
$A = π R^{2}, L = 3 R$
$R = \int_{0}^{3 R} \frac{d x}{k ( x ) A}$
$R = \frac{1}{k _{0} π R ^{2}} \int_{0}^{3 R} \frac{d x}{1 + x / ( 3 R )}$
$R = \frac{3 R l n 2}{k _{0} π R ^{2}} = \frac{3 l n 2}{π k _{0} R}$
$\dot{Q} = \frac{T _{1} - T _{2}}{R} = \frac{π k _{0} R}{3 l n 2} (T_{1} - T_{2})$
Answer B.

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