Question

Two materials A and B of different materials are welded together. Both the materials are having the same cross-sectional area and the same length. Their thermal conductivities are $K_1$ and $K_2$. What is the thermal conductivity of the composite material?

Answer 1

Heat transfer due to conduction

1. Conduction is the process of heat transfer, in which the transfer of heat takes place due to the collision of neighboring molecules.
2. Conduction generally takes place in solids, since the molecules of the solid materials are closely packed.
3. The equation for the heat transfer due to the thermal conduction is:

$H=\frac{KA∆T}{L}$

Here, $H$ is the heat transfer, $K$ is the thermal conductivity, $A$ is the cross-sectional area, $L$ is the length of the material, and $∆T$ is the change in temperature.

Formula used

The equation for the heat transfer in the first material is given as follows;

$H_1=\frac{K_{1}A∆T}{L}$

The equation for the heat transfer in the second material is given as follows:

$H_2=\frac{K_{2}A∆T}{L}$

Calculation for the heat transfer in the composite material

1. The total heat transfer in the composite material is the addition of the heat transfers in the two materials.
2. Since the two materials are welded together the cross-sectional area of the composite material will be twice the area of any material.

$$\begin{gathered} H=H_1+H_2 \\ \frac{K\left(2A\right)∆T}{L}=\frac{K_{1}A∆T}{L}+\frac{K_{2}A∆T}{L} \\ 2K=K_1+K_2 \\ K=\frac{K_1+K_2}{2} \end{gathered}$$

Hence, the thermal conductivity of the combined material is$\frac{K_1+K_2}{2}$.