A rod of length L with sides fully insulated is made of a material whose thermal conductivity K varies with temperature as $K = \frac{α}{T},$ where $α$ is constant. The ends of rod are at temperature $T_{1}$ and $T_{2} (T_{2} > T_{1}) .$ Find the rate of heat flow per unit area of rod.

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$q = - \frac{L}{α A} \ln (\frac{T_{1}}{T_{2}})$

$q = \frac{α A}{L} \ln (\frac{T_{2}}{T_{1}})$

$q = \frac{L}{α A} \ln (\frac{T_{1}}{T_{2}})$

$q = \frac{α A}{L} \ln (\frac{T_{1}}{T_{2}})$

answer is C.

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Detailed Solution

$\frac{d Q}{d t} = q = - K A \frac{d T}{d x}$

Heat transfer across each differential element is same.

$q = - [\frac{α}{T}] A \frac{d T}{d x}$

$\int_{0}^{L} q d x = - α A \int_{T_{1}}^{T_{2}} \frac{d T}{T} \Rightarrow q = \frac{- α A}{L} \ln (\frac{T_{2}}{T_{1}})$

$q = \frac{α A}{L} \ln (\frac{T_{1}}{T_{2}})$

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