First slide

Conduction

Question

Two walls of thickness d₁ and d₂ and thermal conductivities K₁ and K₂ are in contact. In the steady state, if the temperatures at the outer surfaces are T₁ and T₂ the temperatures at the common wall is

Moderate

A

$\frac{K_{1} T_{1} d_{2} + K_{2} T_{2} d_{1}}{K_{1} d_{2} + K_{2} d_{1}}$
B

$\frac{K_{1} T_{1} + K_{2} T_{2}}{d_{1} + d_{2}}$
C

$(\frac{K_{1} d_{1} + K_{2} d_{2}}{T_{1} + T_{2}}) T_{1} T_{2}$
D

$\frac{K_{1} T_{1} d_{1} + K_{2} T_{2} d_{2}}{K_{1} d_{1} + K_{2} d_{2}}$

Solution

Let the temperature of common wall be T. Then
$\begin{aligned} \frac{K_{1} A (T_{1} - T) t}{d_{1}} = \frac{K_{1} A (T - T_{1}) t}{d_{2}} \\ \frac{K_{1} (T_{1} - T)}{d_{1}} = \frac{K_{2} (T - T_{2})}{d_{2}} \end{aligned}$
$\begin{matrix} d_{2} [K_{1} (T_{1} - T)] = d_{1} [K_{2} (T - T_{2})] \\ K_{1} T_{1} d_{2} - K_{1} d_{2} T = d_{1} K_{2} T - d_{1} K_{2} T_{2} \\ K_{1} T_{1} d_{2} + K_{2} T_{2} d_{1} = T [K_{1} d_{2} + d_{1} K_{2}] \\ T = [\frac{K_{1} T_{1} d_{2} + K_{2} T_{2} d_{1}}{K_{1} d_{2} + K_{2} d_{1}}] \end{matrix}$

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