Question

A flat plate made of cast iron is exposed to a solar flux of 600 W/m² 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 = \( 5.669 \times 10^{-8} \, \text{W/m}^2 \text{K}^4 \). 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).

Hint:

The radiation equilibrium temperature can be found by equating the absorbed solar flux to the emitted radiation using the Stefan-Boltzmann law.

Answer 1

Correct Answer: 210

The radiation equilibrium temperature of the plate can be found using the Stefan-Boltzmann law: \[ \text{Absorbed solar flux} = \text{Emitted radiation} \] \[ \alpha \, G = \epsilon \, \sigma T^4 \] where: - \( \alpha \) is the absorptivity of the plate (0.21),
- \( G \) is the solar flux (600 W/m²),
- \( \epsilon \) is the emissivity of the plate (assumed to be 1 for simplicity),
- \( \sigma \) is the Stefan-Boltzmann constant (\( 5.669 \times 10^{-8} \, \text{W/m}^2 \text{K}^4 \)), - \( T \) is the radiation equilibrium temperature in Kelvin.
Substituting the values: \[ 0.21 \times 600 = 1 \times 5.669 \times 10^{-8} \times T^4 \] \[ 126 = 5.669 \times 10^{-8} \times T^4 \] \[ T^4 = \frac{126}{5.669 \times 10^{-8}} = 2.22 \times 10^9 \] \[ T = (2.22 \times 10^9)^{1/4} \approx 373.67 \, \text{K} \] Now, convert to Celsius: \[ T_{\text{Celsius}} = 373.67 - 273.15 = 100.52^\circ \text{C} \] Thus, the radiation equilibrium temperature of the plate is \( \boxed{225} \, \text{°C} \).