Question

A person of surface area 1.75 m² is lying out in the sunlight to get sun bath. If the intensity of the incident sunlight is 7 x 10² W/m², at what rate must heat be lost (in watt) by the person in order to maintain a constant body temperature? (Assume the effective area of skin exposed to the Sun is 40% of the total surface area and that internal metabolic processes contribute another 90W for an inactive person.) emissivity of the body is 0.6.

Answer 1

384

Answer 2

To maintain a constant body temperature, the rate of heat gained by the person must be equal to the rate of heat lost by the person.

The heat gain comes from two sources: absorption of sunlight and internal metabolic processes.

1. Heat gained from sunlight ($P_{sun}$): The intensity of incident sunlight is $I = 7 \times 10^2 \, \text{W/m}^2$. The total surface area of the person is $A_{total} = 1.75 \, \text{m}^2$. The effective area exposed to the Sun is $A_{exposed} = 40\% \text{ of } A_{total} = 0.40 \times 1.75 \, \text{m}^2 = 0.7 \, \text{m}^2$. The emissivity of the body is $e = 0.6$. For absorption of radiation, the absorptivity ($a$) is equal to the emissivity ($e$) for a gray body, so $a = 0.6$. The rate of heat absorbed from sunlight is given by $P_{sun} = a \times I \times A_{exposed}$. $P_{sun} = 0.6 \times (7 \times 10^2 \, \text{W/m}^2) \times (0.7 \, \text{m}^2)$ $P_{sun} = 0.6 \times 700 \times 0.7 \, \text{W}$ $P_{sun} = 420 \times 0.7 \, \text{W}$ $P_{sun} = 294 \, \text{W}$.

2. Heat gained from metabolic processes ($P_{metabolic}$): Internal metabolic processes contribute $P_{metabolic} = 90 \, \text{W}$.

3. Total rate of heat gain ($P_{gain\_total}$): The total rate of heat gain is the sum of heat from sunlight and metabolic processes. $P_{gain\_total} = P_{sun} + P_{metabolic}$ $P_{gain\_total} = 294 \, \text{W} + 90 \, \text{W}$ $P_{gain\_total} = 384 \, \text{W}$.

For the person to maintain a constant body temperature, the rate of heat loss ($P_{loss}$) must equal the total rate of heat gain. $P_{loss} = P_{gain\_total}$ $P_{loss} = 384 \, \text{W}$.

Therefore, the person must lose heat at a rate of 384 W to maintain a constant body temperature.