First slide

Conduction

Question

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

Moderate

A

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

$\frac{K_{1} T_{1} d_{2} + K_{2} T_{2} d_{1}}{K_{1} d_{2} + K_{2} d_{1}}$
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 temperature be T then $\frac{K_{1} A (T_{1} - T)}{d_{1}} = \frac{K_{2} A (T - T_{2})}{d_{2}} \Rightarrow T = \frac{K_{1} T_{1} d_{2} + K_{2} T_{2} d_{1}}{K_{1} d_{2} + K_{2} d_{1}}$

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