GATE 2022 ME (ME1) – Question 35
A flat plate made of cast iron is exposed to a solar flux of 600 W/m$^2$ at an ambient temperature of 25 °C. Assume that the entire solar flux is absorbed by the plate. Cast iron has a low temperature absorptivity of 0.21. Use Stefan-Boltzmann constant = $5.669 \times 10^{-8}$ W/m$^2$-K$^4$. Neglect all other modes of heat transfer except radiation.
Under the aforementioned conditions, the radiation equilibrium temperature of the plate is __________ °C (round off to the nearest integer).
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Correct answer: 223
Explanation
The plate emits long-wave radiation with the emissivity equal to its low-temperature absorptivity, 0.21, and it also receives radiation from the surroundings at 25 °C (298 K). The balance is $600 = \varepsilon\sigma(T^4 - T_a^4)$. So $T^4 = \frac{600}{0.21 \times 5.669 \times 10^{-8}} + 298.15^4 = 5.040 \times 10^{10} + 7.90 \times 10^{9} = 5.830 \times 10^{10}$ and $T = 495.9$ K $= 222.7$ °C, which is 223 °C.