Question

A black wood stove has a surface area of \(1.20 \mathrm{m}^{2}\) and a surface temperature of \(175^{\circ} \mathrm{C} .\) What is the net rate at which heat is radiated into the room? The room temperature is \(20^{\circ} \mathrm{C}\).

Short answer

Answer: The net rate at which heat is radiated into the room is approximately 920.83 W.

Step-by-step solution

Convert temperatures to Kelvin

To use the Stefan-Boltzmann Law, we need to convert the given temperatures to Kelvin. The conversion formula is K = C + 273.15. Stove temperature in Kelvin: \(T_s = 175 + 273.15 = 448.15 \mathrm{K}\) Room temperature in Kelvin: \(T_r = 20 + 273.15 = 293.15 \mathrm{K}\)

Write the Stefan-Boltzmann Law for power emitted and absorbed

The Stefan-Boltzmann Law states that the power radiated per unit area by a black body is directly proportional to the fourth power of its absolute temperature. \(P = \sigma A T^4\), where \(P\) = radiated power, \(\sigma\) = Stefan-Boltzmann constant \((5.67 \times 10^{-8}\) W m\(^{-2}\) K\(^{-4})\), \(A\) = surface area, and \(T\) = absolute temperature.

Calculate the power emitted by the stove

Using the Stefan-Boltzmann Law, we can find the power emitted by the stove. \(P_s = \sigma A_s {T_s}^4\) \(P_s = (5.67 \times 10^{-8}\) W m\(^{-2}\) K\(^{-4})\) × \(1.20 \mathrm{m^2}\) × \((448.15 \mathrm{K})^4\) \(P_s \approx 1241.99 \mathrm{W}\)

Calculate the power absorbed by the room

Similarly, we can find the power absorbed by the room. \(P_r = \sigma A_s {T_r}^4\) \(P_r = (5.67 \times 10^{-8}\) W m\(^{-2}\) K\(^{-4})\) × \(1.20 \mathrm{m^2}\) × \((293.15 \mathrm{K})^4\) \(P_r \approx 321.16 \mathrm{W}\)

Find the net rate of heat radiation

The net rate of heat radiation is the difference between the power emitted by the stove and the power absorbed by the room. Net rate of heat radiation = \(P_s - P_r\) Net rate of heat radiation = \(1241.99 \mathrm{W} - 321.16 \mathrm{W}\) Net rate of heat radiation \(\approx 920.83 \mathrm{W}\) Thus, the net rate at which heat is radiated into the room is approximately \(920.83 \mathrm{W}\).