A person is standing outdoors in the shade where the temperature is 22 °C. (a) What is the radiant energy absorbed per second by his head when it is covered with hair? The surface area of the hair (assumed to be flat) is 180 cm2 and its emissivity is 0.84. (b) What would be the radiant energy absorbed per second by the same person if he were bald and the emissivity of his head were 0.63?
T=273+22=295 K
radiant emittance R
radiant emittance of the black body R*
Stefan-Boltzmann law
R*=σT⁴
Stefan-Boltzmann constant σ =5.67•10⁻⁸ W•m⁻²•K⁻⁴
R= =kR*=kσT⁴
E=RAt =kσT⁴At
E₁=R₁At =k₁σT⁴At
=0.84•5.67•10⁻⁸•180•10⁻⁶•295⁴•1=0.065 J
E₂=R₂At =k₂σT⁴At
To calculate the radiant energy absorbed per second by the person's head, we can use the Stefan-Boltzmann Law and the equation for radiant energy.
(a) When the person's head is covered with hair:
The Stefan-Boltzmann Law states that the radiant energy emitted or absorbed by a surface is proportional to the fourth power of its temperature. The equation for radiant energy (E) is given by:
E = ε * σ * A * T^4
Where:
E = Radiant energy absorbed per second
ε = Emissivity of the surface
σ = Stefan-Boltzmann constant (approximately 5.67 x 10^-8 W/m^2K^4)
A = Surface area
T = Temperature in Kelvin (K)
We need to convert the temperature from Celsius (°C) to Kelvin (K):
T = 22 + 273 = 295 K
Now we can calculate the radiant energy absorbed per second:
E = 0.84 * 5.67 x 10^-8 * 0.018 m^2 * (295 K)^4
Note: The given hair surface area is in cm^2, so we need to convert it to m^2 since the Stefan-Boltzmann constant is in SI units.
E = 0.84 * 5.67 x 10^-8 * (0.018 / 100) m^2 * (295 K)^4
Simplifying the equation:
E = 0.84 * 5.67 x 10^-8 * 0.00018 * (295^4) W
Now, calculate the result to find the radiant energy absorbed per second by the person's head when it is covered with hair.
(b) When the person is bald:
Follow the same steps as above, but this time use an emissivity value of 0.63:
E = 0.63 * 5.67 x 10^-8 * 0.018 * (295 K)^4 W
Simplify the equation further:
E = 0.63 * 5.67 x 10^-8 * 0.00018 * (295^4) W
Now, calculate the result to find the radiant energy absorbed per second by the person's head when they are bald.