Question

A cylindrical rod of length 50 cm and cross sectional area 1 cm² is fitted between a large ice chamber at 0°C and an evacuated chamber maintained at 27°C as shown in figure (29-E12). Only small portions of the rod are inside the chamber and the rest is thermally insulated from the surrounding. The cross section going into the evacuated chamber is blackened so that it completely absorbs any radiation falling on it. The temperature of the blackened end is 17°C when steady state is reached. Stefan constant σ = 6 × 10⁻⁸ W m⁻² K⁻⁴. Find the thermal conductivity of the material of the rod.

Answer 1

Length, l = 50 cm
Cross sectional area, A =1 cm²
Stefan's constant, σ = 6 × 10⁻⁸ W m⁻² K⁻⁴
Temperature of the blackened end = 17°C
Temperature of the chamber = 27°C
Temperature of one end of the rod = 0°C

According to Stefan's law,

$$\begin{gathered} \frac{\Delta Q}{\mathrm{\Delta }t}=\epsilon \mathit{·}A\mathit{}\sigma \left(T_2^4-T_1^4\right) \\ \frac{\Delta Q}{\mathrm{\Delta }t\mathit{·}A}=\epsilon \mathit{}\sigma \left(T_2^4-T_1^4\right) \\ \frac{\Delta Q}{\mathrm{\Delta }t\mathit{·}A}=1\times 6\times {10}^{-8}\left({\left(300\right)}^4-{\left(290\right)}^4\right) \\ =6\times 10.3...\left(1\right) \end{gathered}$$

$$\begin{gathered} \mathrm{Also},\frac{\Delta Q}{\mathrm{\Delta }t}=\frac{\mathrm{K}\mathrm{A}\left(T_1-T_2\right)}{l} \\ \frac{\Delta Q}{\mathrm{\Delta }t\mathit{·}A}=\frac{\mathrm{K}\times 17}{0.5}...\left(2\right) \\ \mathrm{From}\left(1\right)\mathrm{and}\left(2\right), \\ 6\times 10.3=\frac{K\times 17}{0.5} \\ K=1.8\mathrm{W}/°\mathrm{C} \end{gathered}$$