Question

Two walls of thickness $d_{1}$ and $d_{2}$, thermal conductivities $K_{1}$ and $K_{2}$ are in contact. In the steady state if the temperatures at the outer surfaces are $T_{1}$ and $T_{2}$, the temperature at the common wall will be

• 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. $\frac{\left(K_{1} \, d_{1}+K_{2} \, d_{2}\right) T_{1} \, T_{2}}{T_{1}+T_{2}}$
• D. $\frac{K_{1}\, d_{1} T_{1}+K_{2} d_{2}\, T_{2}}{K_{1} \, d_{1}+K_{2} \, d_{2}}$

Answer 1

Answer: (B)

Heat flow across both the walls will be same.
$\frac{K_{1} A}{d_{1}}\left(T-T_{1}\right)=\frac{K_{2} A}{d_{2}}\left(T_{2}-T\right) $
$K_{1} \,T \,d_{2}-K_{1} \,T_{1} \,d_{2}$
$=K_{2} \,T_{2}\, d_{1}-K_{2} \,T\, d_{1} $
$T=\frac{K_{1} \,T_{1} \,d_{2}+K_{2} \,T_{2} \,d_{1}}{K_{1}\, d_{2}+K_{2}\, d_{1}}$