Consider two identical people each generating \(60 \mathrm{~W}\) of metabolic heat steadily while doing sedentary work, and dissipating it by convection and perspiration. The first person is wearing clothes made of 1 -mm-thick leather \((k=\) \(0.159 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) ) that covers half of the body while the second one is wearing clothes made of 1 -mm-thick synthetic fabric \((k=0.13 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) that covers the body completely. The ambient air is at \(30^{\circ} \mathrm{C}\), the heat transfer coefficient at the outer surface is \(15 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and the inner surface temperature of the clothes can be taken to be \(32^{\circ} \mathrm{C}\). Treating the body of each person as a 25 -cm-diameter, \(1.7-\mathrm{m}\)-long cylinder, determine the fractions of heat lost from each person by perspiration.

We can use the formula for heat conduction through a cylindrical wall: \(q = \dfrac{2 \pi \mathrm{L} k \Delta T}{\ln (r_{2}/r_{1})}\) where: - \(q\) - heat transfer by convection (W) - \(L\) - length of the cylinder (1.7 m) - \(k\) - thermal conductivity of the material (W/m⋅K) - \(\Delta T\) - temperature difference between the inner and outer surfaces (K) - \(\ln (r_{2}/r_{1})\) - logarithmic mean radius of the cylindrical wall The temperature difference can be calculated as: \(\Delta T = T_{\text{inner}} - T_{\text{outer}} = 32 - 30 = 2 \mathrm{~K}\) Now we can calculate the heat dissipation by convection for each person using their respective material properties. Person 1 (Leather): Using the given thickness and thermal conductivity for leather: \(k_{1} = 0.159 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) \(r_{1} = 0.125 \mathrm{~m}\) (radius of the body) \(r_{2} = r_{1} + 0.1/2 = 0.125 + 0.001 = 0.126 \mathrm{~m}\) (radius of the body plus leather) We can now calculate the heat transfer \(q_{1}\) by substituting these values into the formula: \(q_{1} = \dfrac{2 \pi (1.7) (0.159)(2)}{\ln (0.126/0.125)}\) Person 2 (Synthetic Fabric): Using the given thickness and thermal conductivity for synthetic fabric: \(k_{2} = 0.13 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) \(r_{1} = 0.125 \mathrm{~m}\) (radius of the body) \(r_{2} = r_{1} + 0.1/2 = 0.125 + 0.001 = 0.126 \mathrm{~m}\) (radius of the body plus synthetic fabric) We can now calculate the heat transfer \(q_{2}\) by substituting these values into the formula: \(q_{2} = \dfrac{2 \pi (1.7) (0.13)(2)}{\ln (0.126/0.125)}\) Don't forget, Person 1 is only covered by the clothing around half of their body, so the heat loss from convection needs to be multiplied by 0.5. Finally, calculate \(q_{1}\) and \(q_{2}\).

We will now calculate the fraction of heat lost due to perspiration for each person by comparing the total metabolic heat generated (60 W) to the amount dissipated by convection (\(q_{1}\), \(q_{2}\)). Fraction of heat loss due to perspiration = \(\dfrac{\text{Metabolic Heat - Heat Dissipated by Convection}}{\text{Metabolic Heat}}\) Person 1 (Leather) Fraction of heat loss due to perspiration: \(F_{1} = \dfrac{60 - 0.5q_{1}}{60}\) Person 2 (Synthetic Fabric) Fraction of heat loss due to perspiration: \(F_{2} = \dfrac{60 - q_{2}}{60}\) Calculate \(F_{1}\) and \(F_{2}\) and interpret them as percentages to find the fractions of heat loss due to perspiration for each person.