Question 64: Thermodynamics & KTG

Question

A cylinder of radius $$r$$ and 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 effective thermal conductivity of the system is

Accepted Answer

Both the cylinders are in parallel, for the heat flow from one end as shown
So, $$K_{\mathrm{eq}}=\frac{K_1 A_1+K_2 A_2}{A_1+A_2}$$
where, $$A_1=$$ area of cross-section of inner cylinder $$=\pi R^2$$
and $$\quad A_2=$$ area of cross-section of cylindrical shell
$$\begin{aligned}
& =\pi\left\{(2 R)^2-(R)^2ight\}=3 \pi R^2 \
\Rightarrow \quad K_{\mathrm{eq}} & =\frac{K_1\left(\pi R^2ight)+K_2\left(3 \pi R^2ight)}{\pi R^2+3 \pi R^2}=\frac{1}{4}\left(K_1+3 K_2ight)
\end{aligned}$$

Answer

$$\frac{1}{3}\left(K_1+2 K_2\right)$$

Answer

$$\frac{1}{2}\left(2 K_1+3 K_2\right)$$

Answer

$$\frac{1}{3}\left(3 K_2+2 K_1\right)$$

Answer

$$\frac{1}{4}\left(K_1+3 K_2\right)$$

Thermodynamics & KTG

A cylinder of radius $r and of thermal conductivity K_1 is surrounded by a cylindrical shell of inner radius r and outer radius 2 r made of a material of thermal conductivity K_2$. The effective thermal conductivity of the system is

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