Question 20: Thermodynamics & KTG

Question

A cylinder of radius $R$ made of a material of thermal conductivity $k_1$ is surrounded by a cylindrical shell of inner radius $R$ and outer radius $2 R$ made of a material of thermal conductivity $k_2$. The two ends of a combined system are maintained at two different temperature. There is no loss of heat across the cylindrical surface and the system is in steady state. The effective thermal conductivity of the system is

Accepted Answer

$$ \begin{aligned} &	ext { Since, two cylinders are in parallel combination }\ &\begin{array}{ll} 	herefore & \frac{1}{R_e}=\frac{1}{R_1}+\frac{1}{R_2} \ \Rightarrow & \frac{k_e A_e}{l}=\frac{k_1 A_1}{l}+\frac{k_2 A_2}{l} \ \Rightarrow & k_e \pi(2 R)^2=k_1 \pi(R)^2+k_2 \pi\left[(2 R)^2-(R)^2ight] \ \Rightarrow & 4 k_e=k_1+3 k_2 \ 	herefore & k_e=\frac{k_1+3 k_2}{4} \end{array} \end{aligned} $$

Answer

$k_1+k_2$

Answer

$\frac{k_1+3 k_2}{4}$

Answer

$\frac{k_1 k_2}{k_1+k_2}$

Answer

$\frac{3 k_1+k_2}{4}$

Thermodynamics & KTG

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