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Two walls of thickness $l_{1}$ and $l_{2}$ and thermal conductivities $K_{1}$ and $K_{2}$ are in contact. In the steady state, if the temperature at the outer faces are $θ_{1}$ and $θ_{2}$ find the temperature at the common wall

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$\frac{K_{1} θ_{1} l_{2} + K_{2} θ_{2} l_{1}}{K_{1} l_{2} + K_{2} l_{1}}$

$\frac{K_{1} θ_{1} l_{2} + K_{2} θ_{2} l_{2}}{K_{1} l_{1} + K_{2} l_{2}}$

$\frac{K_{1} θ_{2} l_{1} + K_{2} θ_{1} l_{2}}{K_{1} l_{2} + K_{2} l_{1}}$

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answer is A.

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Detailed Solution

In steady state ${(\frac{Q}{t})}_{1} = {(\frac{Q}{t})}_{2}$

$\frac{K_{1} A (θ_{1} - θ)}{l_{1}} = \frac{K_{2} A (θ - θ_{2})}{l_{2}}$ $θ = \frac{K_{1} θ_{1} l_{2} + K_{2} θ_{2} l_{1}}{K_{1} l_{2} + K_{2} l_{1}}$

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