Question

A cylinder of radius r and of thermal conductivity K1 is

surrounded by a cylindrical shell of inner radius r and outer radius 2r made of a material of thermal conductivity K2. The effective thermal conductivity of the system is??

Answer 1

Answer on Question #50847-Physics-Molecular-Physics-Thermodynamics

A cylinder of radius $r$ and of thermal conductivity $K_{1}$ is surrounded by a cylindrical shell of inner radius $r$ and outer radius $2r$ made of a material of thermal conductivity $K_{2}$ . The effective thermal conductivity of the system is?

Solution

Since two cylinders are in parallel, the equivalent termal resistance $R$ of the combination is given by

or

$$\frac {1}{R} = \frac {1}{R _ {1}} + \frac {1}{R _ {2}}$$ $$\frac {K A}{x} = \frac {K _ {1} A _ {1}}{x _ {1}} + \frac {K _ {2} A _ {2}}{x _ {2}}.$$

Here $A = \pi (2r)^{2} = 4\pi r^{2}$ , $A_{1} = \pi r^{2}$ and $A = \pi ((2r)^{2} - r^{2}) = 3\pi r^{2}$ and $x = x_{1} = x_{2}$ .

Substituting these values in our equation, we get

$$4 K = K _ {1} + 3 K _ {2}, o r K = \frac {K _ {1} + 3 K _ {2}}{4}.$$

Answer: $\frac{K_1 + 3K_2}{4}$ .

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