A car mechanic is working in a shop whose interior space is not heated. Comfort for the mechanic is provided by two radiant heaters that radiate heat at a total rate of \(4 \mathrm{~kJ} / \mathrm{s}\). About 5 percent of this heat strikes the mechanic directly. The shop and its surfaces can be assumed to be at the ambient temperature, and the emissivity and absorptivity of the mechanic can be taken to be \(0.95\) and the surface area to be \(1.8 \mathrm{~m}^{2}\). The mechanic is generating heat at a rate of \(350 \mathrm{~W}\), half of which is latent, and is wearing medium clothing with a thermal resistance of \(0.7 \mathrm{clo}\). Determine the lowest ambient temperature in which the mechanic can work comfortably.

To determine the heat rate received by the mechanic directly from the radiant heaters, we can multiply the total heat rate of the heaters by the percentage of the heat that strikes the mechanic directly. Heat rate received from the radiant heaters: \(Q_{r} = 0.05 \times 4000~\mathrm{J/s} = 200~\mathrm{W}\)

The mechanic loses heat through the following ways: 1. By radiation (assuming the surroundings are at ambient temperature). 2. Through clothing (assuming the surroundings are at ambient temperature). 3. Half of the generated heat is latent and not used by the mechanic. First, we calculate the heat loss through radiation using the equation: \(Q_{rad} = \sigma \times \epsilon \times A \times (T_{m}^{4} - T_{a}^{4})\) Where \(\sigma \approx 5.67 \times 10^{-8}~\mathrm{W/m^2\cdot K^4}\) is the Stefan-Boltzmann constant, \(\epsilon = 0.95\) is the emissivity, \(A = 1.8~\mathrm{m^2}\) is the surface area of the mechanic, \(T_{m}\) is the mechanic's effective temperature, and \(T_{a}\) is the ambient temperature. Second, we calculate the heat loss through clothing using the equation: \(Q_{clo} = \frac{T_{m} - T_{a}}{R}\) Where \(R = 0.155\times0.7~\mathrm{K\cdot m^2/W}\) is the thermal resistance of the clothing. The total heat loss \(Q_{loss} = Q_{rad} + Q_{clo} + 350/2~\mathrm{W}\)

To find the lowest ambient temperature at which the mechanic can work comfortably, we need to balance the heat rates: \(Q_{r} = Q_{loss}\) \(200~\mathrm{W} = 5.67 \times 10^{-8}\times 0.95 \times 1.8 \times (T_{m}^4 - T_{a}^4) + \frac{T_{m} - T_{a}}{0.155\times0.7} + 175~\mathrm{W}\) Assuming that the comfortable work environment is when the mechanic's effective temperature is at \(37^\circ \mathrm{C}\), i.e. the normal body temperature, we have \(T_{m}= 310~\mathrm{K}\). Using this assumption, we can solve for the ambient temperature: \(200 - 175 = 5.67 \times 10^{-8}\times 0.95 \times 1.8 \times (310^4 - T_{a}^4) + \frac{310 - T_{a}}{0.155\times0.7}\) Solving for \(T_{a}\), we find: \(T_{a} \approx 273~\mathrm{K}\) Thus, the lowest ambient temperature in which the mechanic can work comfortably is approximately \(273~\mathrm{K}\), or \(0^\circ \mathrm{C}\).