Question

a cylinder of radius and length l is made up of substance, whose thermal conductivity K varies with the distance x
from the axis as K = K1x + K2 Determine the efficiency thermal conductivity between the flat faces of the cylinder​

Answer 1

AnSwer :

Let us subdivide the entire cylinder into a number of coaxial cylindrical shells of infinitesimally small thickness dx.

Cross sectional area of the shell is 2πx(dx).

using the expression for the effective thermal conductivity.

${ : {\implies {\sf K_{eff} = \dfrac{ \sum A_iK_i}{ \sum A_i} }}}$

${ : {\implies {\sf {{K_{eff} = \dfrac{1}{ \sum A_i} } \int \limits_{ 0}^{ R}K(2\pi xdx)}}}}$

${ :{\implies{ \sf{K_{eff} = \dfrac{1}{\pi R^{2}}\int\limits_{0}^{R} (K_1x + K_2) 2\pi x(dx)}}}}$

${ : {\implies { \sf{K_{eff} = \dfrac{2}{ R^{2}}\int\limits_{0}^{R} (K_1 {x}^{2} + K_2x)dx}}}}$

${ : {\implies {\boxed { \sf{K_{eff} = \dfrac{1}{3}(2K_1R +3 K_2) }}}}}$

$\:$

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