A system consist of a cylinder surrounded by a cylindrical shell. A...

A system consist of a cylinder surrounded by a cylindrical shell. A cylinder is a radius

R

and is made of material of thermal conductivity

K

, whereas a cylindrical shell has inner radius

R

and outer radius

2 R

and is made of material of thermal conductivity twice as that of cylinder. Assuming the system in steady state and negligible heat loss across the cylindrical surface, find the effective thermal conductivity of the system, if the two ends of the combined system are maintained at two different temperatures.

Options

A $3 K$
B $\frac{2}{3} K$
C $\frac{7 K}{4}$
D $\frac{5 K}{4}$

Correct answer

C. $\frac{7 K}{4}$

Step-by-step solution

\therefore \quad R_1=\frac{L}{K_1 \cdot \pi R^2}

and

\begin{aligned} R_2 & =\frac{L}{K_2\left(4 \pi R^2-\pi R^2\right)}=\frac{L}{3 K_2 \pi R^2} \\ R_{\text{eq }} & =\frac{L}{K_{\text{eq }} \cdot 4 \pi R^2}\end{aligned}

R_1 and R_2

are in parallel.

\therefore \quad \frac{1}{R_{\text{eq }}}=\frac{1}{R_1}+\frac{1}{R_2}

\begin{aligned} \Rightarrow & & \frac{4 \pi R^2 K_{\text{eq }}}{L} & =\frac{\pi R^2 K_1}{L}+\frac{3 \pi R^2 K_2}{L} \\ & \Rightarrow & 4 K_{\text{eq }} & =K_1+3 K_2\end{aligned}

On putting values of

K_1 and K_2

, we get

\begin{aligned} 4 K_{\text{eq }} & =K+3(2 K) \quad\left[\because K_1=K, K_2=2 K\right] \\ K_{\text{eq }} & =\frac{K+6 K}{4}=\frac{7 K}{4}\end{aligned}

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