First slide

Conduction

Question

A cylinder of radius R made of a material of thermal conductivity & is surrounded by a cylindrical shell of inner radius .E and outer radius 2 .R made of a material of thermal conductivity K₂. The two ends of the combined system are maintained at two different temperatures. There is no loss of heat across the cylindrical surface and the systems in steady state. The effective thermal conductivity of the system is

Difficult

A

$K_{1} + K_{2}$
B

$\frac{K_{1} K_{2}}{K_{1} + K_{2}}$
C

$\frac{K_{1} + 3 K_{2}}{4}$
D

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

Solution

Heat flowing per second through cylinder of radius R
$Q_{1} = K_{1} ({πR}^{2}) (\frac{T_{1} - T_{2}}{l})$
Heat flowing per second through outer shell of radius 2 R
$Q_{2} = K_{2} [π (2 R)^{2} - π (R)^{2}] (\frac{T_{1} - T_{2}}{l})$
Total $Q = Q_{1} + Q_{2}$
$= (K_{1} + 3 K_{2}) {πR}^{2} (\frac{T_{1} - T_{2}}{l}) . . . . (1)$
Let K be the equivalent thermal conductivity of the system. Then
$Q = Kπ (2 R)^{2} (\frac{T_{1} - T_{2}}{l}) . . . . (2)$
From eqs. (1) and (2), we have
$(K_{1} + 3 K_{2}) = 4 K$
$K = \frac{K_{1} + 3 K_{2}}{4}$

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