A cylinder of radius R is surrounded by a cylindrical shell of inner radius R and outer radius 2R. The thermal conductivity of the material of the inner cylinder is K1 and that of the outer cylinder is K2. Assuming no loss of heat, the effective thermal conductivity of the system for heat flowing along the length of the cylinder is:

Question

A cylinder of radius R is surrounded by a cylindrical shell of inner radius R and outer radius 2R. The thermal conductivity of the material of the inner cylinder is K1 and that of the outer cylinder is K2. Assuming no loss of heat, the effective thermal conductivity of the system for heat flowing along the length of the cylinder is: (A) K1 + K2 (B) (K1 + K2/2) (C) (2K1 + 3K2/5) (D) (K1 + 3K2/4)

Tardigrade Admin · Accepted Answer

Keq = (K1A1+K2A2/A1+A2) =(K1(π R2)+K2(3π R2)/4π R2) = (K1 + 3K2/4)

K₁ + K₂

$\frac{K _{1} + K _{2}}{2}$

$\frac{2 K _{1} + 3 K _{2}}{5}$

$\frac{K _{1} + 3 K _{2}}{4}$

$K_{e q} = \frac{K _{1} A _{1} + K _{2} A _{2}}{A _{1} + A _{2}}$
$= \frac{K _{1} ( π R ^{2} ) + K _{2} ( 3 π R ^{2} )}{4 π R ^{2}}$
$= \frac{K _{1} + 3 K _{2}}{4}$

K1 + K2

2K1​+K2​​

52K1​+3K2​​

4K1​+3K2​​

Keq​=A1​+A2​K1​A1​+K2​A2​​ =4πR2K1​(πR2)+K2​(3πR2)​ =4K1​+3K2​​

K₁ + K₂

$\frac{K _{1} + K _{2}}{2}$

$\frac{2 K _{1} + 3 K _{2}}{5}$

$\frac{K _{1} + 3 K _{2}}{4}$

$K_{e q} = \frac{K _{1} A _{1} + K _{2} A _{2}}{A _{1} + A _{2}}$
$= \frac{K _{1} ( π R ^{2} ) + K _{2} ( 3 π R ^{2} )}{4 π R ^{2}}$
$= \frac{K _{1} + 3 K _{2}}{4}$