Question

A cylinder of radius $R$ is surrounded by a cylinder shell of inner radius $R$ and outer radius $2R$. The thermal conductivity of the material of the cylinder is $K_1$ and that of the outer cylinder is $K_2$. Assuming no loss of heat, the effective thermal conductivity of the system for heat flowing along the length of the cylinder is:

• A. $\frac{K_1+3K_2}{4}$
• B. $\frac{K_1+K_2}{2}$
• C. $\frac{3K_1+K_2}{4}$
• D. $K_1+K_2$

Answer 1

The correct option is A $\frac{K_1+3K_2}{4}$

Let $\Delta T=(T_2-T_1)$ be the temperature difference between the two ends and $x$ be the length of the cylinder.


The cross-sectional area of the inner cylinder is $A_1=\pi R^2$, having conductivity $K_1$.

Similarly, cross-section area of the outer cylinder is $A_2=\pi (2R{)}^2-\pi R^2=3\pi R^2$ and having conductivity $K_2$.

Since temperature at the ends of both cylinder is equal, so they are in parallel combination.

In parallel combination, effective Thermal Resistance is given as

$\frac{1}{R_{eff}}=\frac{1}{R_1}+\frac{1}{R_2}$

As, $R=\frac{l}{KA}$

$\frac{1}{R_{eff}}=\frac{K_1{A}_1}{x}+\frac{K_2{A}_2}{x}$

$\Rightarrow \frac{1}{R_{eff}}=\frac{K_1\left(\pi R^2\right)}{x}+\frac{K_2\left(3\pi R^2\right)}{x}$

$\Rightarrow \frac{1}{R_{eff}}=\frac{\pi R^2}{x}(K_1+3K_2)......\left(1\right)$

Two cylinders can be replaced by one cylinder having resistance $R_{eff}$, length$l=\text{x}$, Area $A=\pi (2R{)}^2=4\pi R^2$ and having same temperature difference.

Let its conductivity be $K_{eff}$

$\Rightarrow R_{eff}=\frac{x}{K_{eff}\left(4\pi R^2\right)}$

Substituting this value in equation $\left(1\right)$

$\Rightarrow \frac{K_{eff}\left(4\pi R^2\right)}{x}=\frac{\pi R^2}{x}(K_1+3K_2)$

$\Rightarrow K_{eff}=\frac{K_1+3K_2}{4}$

Hence, option $\left(\text{A}\right)$ is correct.