Question

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 \(25-\mathrm{cm}\)-diameter, \(1.7-\mathrm{m}\)-long cylinder, determine the fractions of heat lost from each person by perspiration.

Short answer

Answer: To find the fraction of heat each person loses through perspiration, we need to follow the steps outlined in the solution above: 1. Calculate the total heat generated by each person 2. Define the dimensions and clothing variables 3. Find the surface area of the exposed body considering the clothing coverage ratio 4. Calculate the heat transfer for each person due to convection 5. Calculate the fraction of heat lost through perspiration. After going through these steps, we can find the fractions for each person (Fraction1 and Fraction2) representing the heat lost through perspiration.

Step-by-step solution

Calculate the total heat generated by each person

As given, each person generates 60 W of metabolic heat steadily. Therefore, the total heat generated by each person, \(Q_{total}\), is simply 60 W.

Define the dimensions and clothing variables

We are given the dimensions of the person as a cylinder with a diameter of \(25 ~\mathrm{cm}\) and length of \(1.7~\mathrm{m}\). We are also given the thickness and thermal conductivity of the clothing materials. Let's define the variables: - \(D = 0.25 ~\mathrm{m}\) (diameter of the cylinder) - \(L = 1.7 ~\mathrm{m}\) (length of the cylinder) - \(k_{leather} = 0.159 ~\mathrm{W} / \mathrm{m} \cdot \mathrm{K}\) (thermal conductivity of leather) - \(k_{synthetic} = 0.13 ~\mathrm{W} / \mathrm{m} \cdot \mathrm{K}\) (thermal conductivity of synthetic fabric)

Find the surface area of the exposed body considering the clothing coverage ratio

To calculate the surface area of the exposed body for each person, we need to consider the coverage ratio of the clothes. Person 1 is wearing clothes covering half of the body, and Person 2's clothes are covering the body completely. The total surface area of the cylindrical body, \(A_{total}\), can be calculated as follows: \(A_{total} = 2 \pi r L + 2 \pi r^2\) \(A_{total} = 2 \pi(D/2) L + 2 \pi(D/2)^2\) Person 1: \(A_{exposed1} = \frac{1}{2}A_{total}\) Person 2: \(A_{exposed2}= 0\)

Calculate the heat transfer for each person due to convection

The heat transfer due to convection for each person can be obtained using the formula: \(Q_{conv} = h * A_s * (\Delta T)\) The heat transfer coefficient, \(h\), and the temperature difference, \(\Delta T\), are given: \(h = 15~\mathrm{W} / \mathrm{m}^2 \cdot \mathrm{K}\) \(\Delta T = T_{in} - T_{out} = 32^{\circ} \mathrm{C} - 30^{\circ} \mathrm{C} = 2 \mathrm{K}\) Therefore, the heat transfer due to convection for each person: Person 1: \(Q_{conv1} = h * A_{exposed1} * (\Delta T)\) Person 2: \(Q_{conv2} = h * A_{exposed2} * (\Delta T)\) (Note that this equals 0 since \(A_{exposed2}=0\))

Calculate the fraction of heat lost through perspiration

Now, we can calculate the heat lost through perspiration by subtracting the heat transfer due to convection from the total heat transfer for each person. Then, divide the heat lost through perspiration by the total heat transfer to find the fraction of heat loss: Person 1: \(Q_{pers1} = Q_{total} - Q_{conv1}\) \(Fraction1 = \frac{Q_{pers1}}{Q_{total}}\) Person 2: \(Q_{pers2} = Q_{total} - Q_{conv2}\) \(Fraction2 = \frac{Q_{pers2}}{Q_{total}}\) These fractions represent the fractions of heat lost through perspiration for each person.