First slide

Heat transfer

Question

A cylinder of radius R made of material of conductivity $K_{1}$ is surrounded by a cylindrical shell of internal radius R and outer radius 2R made of material of conductivity $K_{2}$ , the two ends of the system are kept at two different temperatures. If there is no loss of heat from the surface, the effective conductivity of the system is

Moderate

A

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

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

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

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

Solution

Both are parallel
$t h e e f f e c t i v e c o e f f i c i e n t o f t h e r m a l c o n d u c t i v i t y f o r a p a r a l l e l c o m b i n a t i o n o f c o n d u c t o r s i s K_{Parallel} = \frac{K_{1} A_{1} {+K}_{2} A_{2}}{A_{1} {+A}_{2}} \begin{array}{l} {base area of inner cylinder isA}_{1} {=πR}^{2} \\ {area of outer cylinder is A}_{2} {=π(2R)}^{2} {-πR}^{2} \\ A_{2} {=3πR}^{2} \end{array} K_{P} = \frac{K_{1} {.πR}^{2} {+K}_{2} {.3πR}^{2}}{{πR}^{2} {+3πR}^{2}} K_{e f f e c t i v e} {=K}_{Parallel} = \frac{K_{1} {+3K}_{2}}{4}$

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