Question

An average person generates heat at a rate of \(240 \mathrm{Btu} / \mathrm{h}\) while resting in a room at \(70^{\circ} \mathrm{F}\). Assuming one-quarter of this heat is lost from the head and taking the emissivity of the skin to be \(0.9\), determine the average surface temperature of the head when it is not covered. The head can be approximated as a 12 -in-diameter sphere, and the interior surfaces of the room can be assumed to be at the room temperature.

Short answer

Based on the given information and calculations using the Stefan-Boltzmann law, the average surface temperature of the head when it is not covered is approximately 91.22°F. Note that this is an approximation, and individual variations may exist due to factors such as head size, skin emissivity, and environmental conditions.

Step-by-step solution

Determine heat loss from the head

We are given that one-quarter of the total heat generated is lost from the person's head. Since the heat generation rate is \(240 \mathrm{Btu} / \mathrm{h}\), the heat loss from the head will be: heat_loss_head = \(\frac{1}{4} * 240 \mathrm{Btu} / \mathrm{h} = 60 \mathrm{Btu} / \mathrm{h}\)

Find the head's surface area

The head is approximated as a 12-inch diameter sphere. To find its surface area, we first calculate the radius and then use the formula for the surface area of a sphere. radius = \(\frac{12 \text{ inches}}{2} = 6 \text{ inches} = 0.5 \text{ ft}\), since there are \(12 \text{ inches}\) per \(1 \text{ ft}\) surface_area = \(4 \pi \cdot radius^2 = 4 \pi (0.5)^2 \approx 3.14 \mathrm{ft^2}\)

Convert heat loss to Standard International (SI) units

Since we will work with the Stefan-Boltzmann law, it's best to use SI units. We need to convert the heat_loss_head from Btu/h to Watts heat_loss_head_Si = \(60 \mathrm{Btu} / \mathrm{h} \times \frac{1055.06 \mathrm{J}}{1 \mathrm{Btu}} \times \frac{1 \mathrm{h}}{3600 \mathrm{s}} \approx 17.584 \mathrm{W}\)

Convert room temperature to Kelvin

The equation we use requires the temperature to be in Kelvin. Therefore, we need to convert the given room temperature: room_temp_kelvin = \((70^{\circ} \mathrm{F} - 32) \times \frac{5}{9} + 273.15 \approx 294.26 \mathrm{K}\)

Apply the Stefan-Boltzmann Law

We use the Stefan-Boltzmann law to find the surface temperature of the head. The equation is: \(Q = \epsilon \sigma A (T^4 - T_0^4)\) Here, \(Q\) is the heat loss (in Watts), \(\epsilon\) is the emissivity of the skin, \(\sigma\) is Stefan-Boltzmann constant \((5.67 \times 10^{-8} \mathrm{W/m^2K^4})\), \(A\) is the surface area (in \(m^2\)), \(T\) is the surface temperature (in K), and \(T_0\) is the room temperature (in K).

Rearrange the Stefan-Boltzmann Law to solve for T

We need to solve for \(T\) in the equation. The equation becomes: \(T = \sqrt[4]{\frac{Q}{\epsilon \sigma A} + T_0^4}\)

Substitute the known values and calculate T

Now, plug in all the known values: \(\epsilon = 0.9\) \(\sigma = 5.67 \times 10^{-8} \mathrm{W/m^2K^4}\) \(A = 3.14 \mathrm{ft^2} \times \frac{1 \mathrm{m^2}}{10.764 \mathrm{ft^2}} \approx 0.2917 \mathrm{m^2}\) \(T_0 = 294.26 \mathrm{K}\) Then, we can calculate the temperature using the modified equation: \(T = \sqrt[4]{\frac{17.584 \mathrm{W}}{0.9 \times 5.67 \times 10^{-8} \mathrm{W/m^2K^4} \times 0.2917 \mathrm{m^2}} + (294.26 \mathrm{K})^4} \approx 306.75 \mathrm{K}\)

Convert T back to Fahrenheit

Finally, we can convert the temperature back to Fahrenheit: \(T_{F} = (T - 273.15) \times \frac{9}{5} + 32 \approx 91.22^{\circ} \mathrm{F}\) So, the average surface temperature of the head when it is not covered is approximately \(91.22^{\circ} \mathrm{F}\).