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Q.

A cylinder is made up of two coaxial layers, one of radius R and the other of inner radius R and outer radius 2R. The inner and the outer portions are respectively made up of substances of thermal conductivities K and 2K respectively. If the effective thermal conductivity between the flat ends of the cylinder is $\frac{x k}{4}$ , then the value of x is ________

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answer is 7.

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Detailed Solution

Here $A_{1} = π R^{2}; K_{1} = K$
And $A_{2} = 4 π R^{2} - π R^{2} = 3 π R^{2}; K_{2} = 2 K$
Using, the expression for the effective thermal conductivity $K_{e f f},$ for slabs in parallel,
. $K_{e f f} = \frac{A_{1} K_{1} + A_{2} K_{2}}{A_{1} + A_{2}} = \frac{π R^{2} (K) + 3 π R^{2} (2 K)}{π R^{2} + 3 π R^{2}} = \frac{7 K}{4}$

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