Question

Two rods $A$ and $B$ of different materials are welded together as shown in figure. Their thermal conductivities are $K_1$ and $K_2$. The thermal conductivity of the composite rod will be :

• A. $\frac{3(K_1 + K_2)}{2}$
• B. $K_1 + K_2$
• C. $2( K_1 + K_2)$
• D. $\frac{K_1 + K_2}{2}$

Answer 1

Answer: (D)

$\frac{K_1 + K_2}{2}$

Answer 2

$R _{1}=\frac{ d }{ k _{1} A }$
$R _{2}= d K _{2} A$
$\operatorname{Req}=\frac{ R _{1} R _{2}}{ R _{1}+ R _{2}}$
$\frac{ d }{ keq (2 A )}=\frac{\left(\frac{ d }{ k _{1} A }\right)\left(\frac{ d }{ k _{2} A }\right)}{ A }\left(\frac{1}{ k _{1}}+\frac{1}{ k _{2}}\right)$
$\frac{1}{2 keq }=\frac{\frac{1}{ k _{1} k _{2}}}{\frac{ k _{1}+ k _{2}}{ k _{1} k _{2}}}=\frac{1}{ k _{1}+ k _{2}}$
$keq =\frac{ k _{1}+ k _{2}}{2}$