One end of a rod of length 20cm is inserted in a furnace at 800K … - Vidyadip

Question

One end of a rod of length 20cm is inserted in a furnace at 800K The sides of the rod are covered with an insulating material and the other end emits radiation like a blackbody. The temperature of this end is 750K in the steady state. The temperature of the surrounding air is 300K. Assuming radiation to be the only important mode of energy transfer between the surrounding and the open end of the rod, find the thermal conductivity of the rod. Stefan constant $\sigma=6.0\times10^{-8}\text{Wm}^{-2}\text{K}^{-4}.$

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Answer

$\sigma=6\times10^{-8}\text{w/m}^2\text{-k}^4$
$\text{L}=20\text{cm}=0.2\text{m},\ \text{K}=?$
$\Rightarrow\text{E}=\frac{\text{KA}(\theta_1-\theta_2)}{\text{d}}=\text{A}\sigma\big(\text{T}_1^4-\text{T}_2^4\big)$
$\Rightarrow\text{K}=\frac{\text{s}(\text{T}_1-\text{T}_2)\times\text{d}}{\theta_1-\theta_2}=\frac{6\times10^{-8}\times(750^4-300^4)\times2\times10^{-1}}{50}$
$\Rightarrow\text{K}=73.993\approx74.$

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