A cylinder of radius r and of thermal conductivity K1 is surrounded by a cylindrical shell of inner radius r and outer radius 2 r made of a material of thermal conductivity K2 . The effective thermal conductivity of the system is

Question

A cylinder of radius r and of thermal conductivity K1 is surrounded by a cylindrical shell of inner radius r and outer radius 2 r made of a material of thermal conductivity K2 . The effective thermal conductivity of the system is (A)  (1/3)(K1+2K2)  (B)  (1/2)(2K1+3K2)  (C)  (1/4)(3K2+2K1)  (D)  (1/4)(K1+3K2)

Tardigrade Admin · Accepted Answer

Both the cylinders are in parallel, for the heat flow from one end as shown

Hence,

K_{e q} = \frac{K _{1} A _{1} + K _{2} A _{2}}{A _{1} + A _{2}}

where

A_{1} =

area of cross-section of inner cylinder

= π R^{2}

and

A_{2} =

area of cross-section of cylindrical shell

= π {(2 R)^{2} - (R)^{2}} = 3 π R^{2}

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

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

Q. A cylinder of radius $r$ and of thermal conductivity $K_{1}$ is surrounded by a cylindrical shell of inner radius $r$ and outer radius $2 r$ made of a material of thermal conductivity $K_{2} .$ The effective thermal conductivity of the system is

$\frac{1}{3} (K_{1} + 2 K_{2})$

$\frac{1}{2} (2 K_{1} + 3 K_{2})$

$\frac{1}{4} (3 K_{2} + 2 K_{1})$

$\frac{1}{4} (K_{1} + 3 K_{2})$

Q. A cylinder of radius r and of thermal conductivity K1​ is surrounded by a cylindrical shell of inner radius r and outer radius 2r made of a material of thermal conductivity K2​. The effective thermal conductivity of the system is

31​(K1​+2K2​)

21​(2K1​+3K2​)

41​(3K2​+2K1​)

41​(K1​+3K2​)

Q. A cylinder of radius $r$ and of thermal conductivity $K_{1}$ is surrounded by a cylindrical shell of inner radius $r$ and outer radius $2 r$ made of a material of thermal conductivity $K_{2} .$ The effective thermal conductivity of the system is

$\frac{1}{3} (K_{1} + 2 K_{2})$

$\frac{1}{2} (2 K_{1} + 3 K_{2})$

$\frac{1}{4} (3 K_{2} + 2 K_{1})$

$\frac{1}{4} (K_{1} + 3 K_{2})$