Question

Consider a person whose exposed surface area is \(1.9 \mathrm{~m}^{2}\), emissivity is \(0.85\), and surface temperature is \(30^{\circ} \mathrm{C}\). Determine the rate of heat loss from that person by radiation in a large room whose walls are at a temperature of (a) \(295 \mathrm{~K}\) and (b) $260 \mathrm{~K}$.

Short answer

Answer: In a room with walls at 295 K, the rate of heat loss is 61.36 W. In a room with walls at 260 K, the rate of heat loss is 215.62 W.

Step-by-step solution

Formulas used

We'll be using the Stefan-Boltzmann Law to solve this problem. The formula is given by: $$P = eA\sigma(T_1^4 - T_2^4)$$, where: - P is the power radiated (rate of heat loss), - e is the emissivity of the person, - A is the exposed surface area, - σ is the Stefan-Boltzmann constant (\(5.67 \times 10^{-8} \mathrm{W\cdot m^{-2} \cdot K^{-4}}\)), - \(T_1\) is the temperature of the person, and - \(T_2\) is the temperature of the surrounding (room's walls). Note that the temperature should be in Kelvin.

Convert temperatures to Kelvin

Given that the surface temperature of the person is \(30^{\circ}\mathrm{C}\), we need to convert it to Kelvin: $$T_1 = 30^{\circ}\mathrm{C} + 273.15 = 303.15 \mathrm{K}$$ The surrounding temperatures are given in Kelvin: (a) 295 K and (b) 260 K.

Calculate the rate of heat loss for case (a)

Using the Stefan-Boltzmann Law, we can now calculate the rate of heat loss for case (a): $$P_a = eA\sigma(T_1^4 - T_2^4) = 0.85 \times 1.9 \times 5.67 \times 10^{-8} \times (303.15^4 - 295^4) = 61.36 \mathrm{W}$$

Calculate the rate of heat loss for case (b)

Similarly, we can calculate the rate of heat loss for case (b): $$P_b = eA\sigma(T_1^4 - T_2^4) = 0.85 \times 1.9 \times 5.67 \times 10^{-8} \times (303.15^4 - 260^4) = 215.62 \mathrm{W}$$

Report the final results

Now we have the rate of heat loss from the person by radiation: - In the room with walls at \(295 \mathrm{K}\), the rate of heat loss is \(61.36 \mathrm{W}\). - In the room with walls at \(260 \mathrm{K}\), the rate of heat loss is \(215.62 \mathrm{W}\).