First slide

Heat transfer

Question

A cylinder of radius r and thermal conductivity $K_{1}$ is surrounded by a cylinder shell of internal radius r and outer radius 3r whose thermal conductivity is $K_{2}$ . There is no loss of heat from the surface of the cylinder when the ends of the combined system are maintained at temperatures $T_{1}$ and $T_{2}$ . The effective thermal conductivity of the system under steady state is

Moderate

A

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

$K_{1} {+K}_{2}$
C

$\frac{K_{1} {+8K}_{2}}{9}$
D

$\frac{{8K}_{1} {+K}_{2}}{9}$

Solution

the cylinders are In parallel
$K_{Parallel} = \frac{K_{1} A_{1} {+K}_{2} A_{2}}{A_{1} {+A}_{2}} ---(1), here K is coefficient of thermal conductivity, A is area of cross section \begin{array}{l} {area of cross section inner cylinder is A}_{1} {=πr}^{2}, here r = radius \\ area of cross section o u t e r cylinder is A_{2} {=9πr}^{2} {-πr}^{2} \end{array} A_{2} {=8πr}^{2} s u b s t i t u t e g i v e n v a l u e s i n e q n (1) K_{P} = \frac{K_{1} {πr}^{2} {+K}_{2} {.8πr}^{2}}{{πr}^{2} {+8πr}^{2}} K_{Parallel} = \frac{K_{1} {+8K}_{2}}{9}$

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