A rod of length L with sides fully insulated is of a material whose thermal conductivity varies with temperature as K = T α , where α is a constant. The ends of the rod are kept at temperature T 1

Question

A rod of length L with sides fully insulated is of a material whose thermal conductivity varies with temperature as K = T α ​ , where α is a constant. The ends of the rod are kept at temperature T 1 ​ and T 2 ​ . The temperature T at x, where x is the distance from the end whose temperature is T 1 ​ is

TestHub · Accepted Answer

Rod diagram: T1 at left, T2 at right, differential dx and position x

At steady state the heat current is constant.
$C = k A \frac{d T}{d x} = \frac{α A}{T} \frac{d T}{d x}$
$C \int_{0}^{x} d x = α A \int_{T_{1}}^{T} \frac{d T}{T}$
$C x = α A ln (\frac{T}{T _{1}})$
$C L = α A ln (\frac{T _{2}}{T _{1}})$
$\frac{x}{L} = \frac{l n ( T / T _{1} )}{l n ( T _{2} / T _{1} )}$
$T = T_{1} (\frac{T _{2}}{T _{1}})^{x / L}$
Answer A.

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