Question

A cylinder of radius $R$ made of material of thermal conductivity $K_1$ is surrounded by a cylindrical shell of inner radius $R$ and outer radius $3R$ made of a material of thermal conductivity $K_2$. The two ends of a combined systems are maintained at two different temperatures. What is the thermal conductivity of the system?

• A. $K_1+K_2$
• B. $\frac{K_1+8K_2}{9}$
• C. $\frac{K_1{K}_2}{K_1+K_2}$
• D. $\frac{8K_1+K_2}{9}$

Answer 1

The correct option is B $\frac{K_1+8K_2}{9}$

Area of inner cylinder $A_1=\pi R^2$
Area of outer cylinder $A_2=\pi \left(\right(3R{)}^2-R^2)$
Let $K$ be thermal conductivity of the equivalent system,
Effective area of equivalent system is $A=\pi (3R{)}^2$

Two cylinders are in parallel, therefore equivalent thermal resistance R is given by
$\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}$
$\frac{KA}{l}=\frac{K_1{A}_1}{l}+\frac{K_2{A}_2}{l}$
$K\pi (3R{)}^2=K_1\pi R^2+K_2\pi \left(\right(3R{)}^2-R^2)$
$9K=K_1+8K_2$
$K=\frac{K_1+8K_2}{9}$