At body temperature the latent heat of vaporization of water is Lv = 2.26e3 kJ/kg.
(a) A naked person with a surface area A = 1.55 m2 and a skin temperature = 37.3 degrees is in a sauna room at T = 82.0 degrees. The person's skin has an emissivity e = 0.910. She evaporates sweat from her body to balance the rate of heat absorption by radiation.
(i) Neglecting any heat production due to metabolism, what is the net rate of heat absorption by the person due to radiation? in W
(ii) How much sweat must be evaporated per hour to balance a net rate of heat absorption of H = 0.850e3 W? in kg/hr
To find the net rate of heat absorption by the person due to radiation, we can use the Stefan-Boltzmann Law, which states that the power radiated by an object is proportional to the fourth power of its temperature.
(i) The net rate of heat absorption by the person due to radiation can be calculated using the following formula:
H_absorption = ε * σ * A * (T^4 - Ts^4)
where:
H_absorption is the net rate of heat absorption by radiation,
ε is the emissivity of the person's skin (given as 0.910),
σ is the Stefan-Boltzmann constant (σ = 5.67e-8 W/(m^2K^4)),
A is the surface area of the person (given as 1.55 m^2),
T is the temperature of the sauna room (given as 82.0 degrees Celsius),
Ts is the skin temperature (given as 37.3 degrees Celsius).
Let's calculate the net rate of heat absorption by the person due to radiation:
First, convert the temperatures to Kelvin:
T = 82.0 + 273.15 = 355.15 K
Ts = 37.3 + 273.15 = 310.45 K
Now, substitute these values into the formula:
H_absorption = 0.910 * 5.67e-8 * 1.55 * (355.15^4 - 310.45^4)
H_absorption ≈ 63.14 W
So, the net rate of heat absorption by the person due to radiation is approximately 63.14 W.
(ii) To calculate how much sweat must be evaporated per hour to balance a net rate of heat absorption of 0.85e3 W, we'll need to use the latent heat of vaporization of water.
The rate at which heat is absorbed due to evaporation can be calculated using the formula:
H_evaporation = m * Lv
where:
H_evaporation is the rate of heat absorbed by evaporation,
m is the mass of water evaporated per second,
Lv is the latent heat of vaporization of water (given as 2.26e3 kJ/kg).
To convert the Lv from kJ/kg to W/(kg/s), divide it by 1000:
Lv = 2.26e3 / 1000 = 2.26e0 kJ/(kg/s)
To find the mass of water evaporated per second, we divide the net rate of heat absorption by the latent heat of vaporization:
m = H_evaporation / Lv
Convert H_evaporation to W:
H_evaporation = 0.85e3 W
Now, substitute these values into the formula:
m = 0.85e3 / 2.26e0 = 0.3754 kg/s
To find the mass evaporated per hour, multiply the mass per second by 3600 (the number of seconds in an hour):
m_hour = m * 3600 = 0.3754 * 3600 = 1351.44 kg/hr
So, approximately 1351.44 kg/hr of sweat must be evaporated to balance a net rate of heat absorption of 0.85e3 W.
To answer these questions, we need to use the formula for the net rate of heat absorption due to radiation, as well as the formula for the rate of water evaporation.
(i) The net rate of heat absorption by radiation can be calculated using the Stefan-Boltzmann Law:
Q = A * e * σ * (T^4 - Ts^4)
Where:
- Q is the net rate of heat absorption in Watts
- A is the surface area in square meters
- e is the emissivity
- σ is the Stefan-Boltzmann constant (5.67e-8 W/m^2K^4)
- T is the temperature of the surroundings (in Kelvin)
- Ts is the temperature of the person's skin (in Kelvin)
First, convert the skin temperature to Kelvin:
Ts = 37.3 + 273.15 = 310.45 K
Now, substitute the values into the equation to calculate the net rate of heat absorption:
Q = 1.55 * 0.910 * 5.67e-8 * (82.0^4 - 310.45^4)
Q ≈ 171.71 W
Therefore, the net rate of heat absorption by radiation is approximately 171.71 Watts.
(ii) The rate of water evaporation can be calculated using the formula:
E = H / Lv
Where:
- E is the rate of water evaporation in kg/s
- H is the net rate of heat absorption in Watts
- Lv is the latent heat of vaporization of water in kJ/kg (which needs to be converted to J/kg)
First, convert the latent heat of vaporization from kJ/kg to J/kg:
Lv = 2.26e3 * 1000 = 2.26e6 J/kg
Now, convert the net rate of heat absorption from Watts to J/s:
H = 0.850e3 = 0.850e3 * 1 = 850 J/s
Substitute the values into the equation to find the rate of water evaporation:
E = 850 / 2.26e6
E ≈ 0.375 x 10^-3 kg/s
To convert this to kg/hr, multiply by the number of seconds in an hour:
0.375 x 10^-3 * 3600 = 1.35 kg/hr
Therefore, approximately 1.35 kg/hr of sweat must be evaporated to balance a net rate of heat absorption of 0.850e3 W.