SATHEE: Heat and Thermodynamics 3 Question 3

Heat and Thermodynamics 3 Question 3

3. A cylinder of radius $R$ is surrounded by a cylindrical shell of inner radius $R$ and outer radius $2 R$ . The thermal conductivity of the material of the inner cylinder is $K_{1}$ and that of the outer cylinder is $K_{2}$ . Assuming no loss of heat, the effective

thermal conductivity of the system for heat flowing along the length of the cylinder is

(a) $\frac{K_{1} + K_{2}}{2}$

(b) $\frac{K_{1} + 3 K_{2}}{4}$

(d) $K_{1} + K_{2}$

(2019 Main, 12 Jan I)

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Solution:

Both the given cylinders are in parallel as heat flow is given along length. In parallel, equivalent thermal conductivity of system is

$K_{e q} = \frac{K_{1} A_{1} + K_{2} A_{2}}{A_{1} + A_{2}}$

So, in given system

$K_{eq} = \frac{K_{1} (π R^{2}) + K_{2} [π (2 R)^{2} - π R^{2}]}{(π R^{2}) + (4 π R^{2} - π R^{2})}$ or $K_{eq} = \frac{K_{1} + 3 K_{2}}{4}$

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Heat and Thermodynamics 3 Question 3

3. A cylinder of radius R is surrounded by a cylindrical shell of inner radius R and outer radius 2R. The thermal conductivity of the material of the inner cylinder is K1 and that of the outer cylinder is K2. Assuming no loss of heat, the effective

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3. A cylinder of radius $R$ is surrounded by a cylindrical shell of inner radius $R$ and outer radius $2 R$ . The thermal conductivity of the material of the inner cylinder is $K_{1}$ and that of the outer cylinder is $K_{2}$ . Assuming no loss of heat, the effective