Question

Consider a person whose exposed surface area is \(1.7 \mathrm{~m}^{2}\), emissivity is \(0.5\), and surface temperature is \(32^{\circ} \mathrm{C}\). Determine the rate of heat loss from that person by radiation in a large room having walls at a temperature of (a) \(300 \mathrm{~K}\) and (b) $280 \mathrm{~K}$.

Short answer

Answer: The rate of heat loss from the person by radiation in the room with walls at a temperature of 300 K is approximately 29.19 W, and in the room with walls at a temperature of 280 K, it is approximately 63.37 W.

Step-by-step solution

Convert temperature to Kelvin

First, we need to convert the person's surface temperature from Celsius to Kelvin. To do this, we add 273.15 to the Celsius temperature: $$ T_{person} = 32^{\circ} \mathrm{C} + 273.15 = 305.15 \mathrm{~K} $$

Recall the Stefan-Boltzmann Law

The Stefan-Boltzmann Law states that the power radiated by a black body is: $$ P = e \cdot A \cdot \sigma \cdot (T_{body}^4 - T_{environment}^4) $$ where \(e\) is the emissivity, \(A\) is the surface area, \(\sigma\) is the Stefan-Boltzmann constant (\(5.67 \times 10^{-8} \mathrm{Wm}^{-2}\mathrm{K}^{-4}\)), and \(T_{body}\) and \(T_{environment}\) are the body and environment temperatures in Kelvin, respectively.

Calculate the rate of heat loss for the 300 K room

For the first scenario, plug the given values and the wall temperature of 300 K into the Stefan-Boltzmann Law: $$ P_{300} = 0.5 \cdot 1.7 \mathrm{~m}^{2} \cdot 5.67 \times 10^{-8} \mathrm{Wm}^{-2}\mathrm{K}^{-4} \cdot (305.15^4 - 300^4) \mathrm{~W} $$ Calculate the result: $$ P_{300} \approx 29.19 \mathrm{~W} $$

Calculate the rate of heat loss for the 280 K room

For the second scenario, plug the given values and the wall temperature of 280 K into the Stefan-Boltzmann Law: $$ P_{280} = 0.5 \cdot 1.7 \mathrm{~m}^{2} \cdot 5.67 \times 10^{-8} \mathrm{Wm}^{-2}\mathrm{K}^{-4} \cdot (305.15^4 - 280^4) \mathrm{~W} $$ Calculate the result: $$ P_{280} \approx 63.37 \mathrm{~W} $$

Present the results

The rate of heat loss from the person by radiation in the room with walls at a temperature of 300 K is approximately 29.19 W, and in the room with walls at a temperature of 280 K, it is approximately 63.37 W.