Question

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

Answer 1

The correct option is A $\frac{K_1+K_2}{2}$

Since the rods are maintained at the same temperature difference (between their ends), they can be assumed to be connected in parallel.
Thus, the thermal current can be written as:
$H=H_1+H_2=\frac{K_{1}A(T_1-T_2)}{d}+\frac{K_{2}A(T_1-T_2)}{d}$
Let $K_{eq}$ be the equivalent thermal conductivity of the composite rod.
$\Rightarrow \frac{K_{eq}\times 2A(T_1-T_2)}{d}=\frac{A(T_1-T_2)}{d}\times [K_1+K_2]$
$\Rightarrow K_{eq}=\left[\frac{K_1+K_2}{2}\right]$
Hence, option (a) is the correct answer.