Black Body Net Heat Absorption Problem

[MattMark'90]

Homework Statement

A naked person, whose skin area is 1.7 m², sits in a sauna that has a wall
temperature of 61^oC. If the person’s skin temperature is 37^oC, find the net rate at
which the person absorbs heat by radiative transfer (assume an emissivity e of 1).
How much liquid must the person consume after 30 minutes to replace the sweat
evaporated as a result of this heat absorption. (assume the latent heat of vaporisation
of sweat is 2427 kJ/kg at 37^oC.

Homework Equations

Stefan-Boltzmann Law: P = e_$$\sigma$$AT⁴

The Attempt at a Solution

Plugging in the numbers and using the the temperature of the sauna walls and human body termperature to give the NET rate of absorption:

P = 1 x 5.67x10^-8 x 1.7 x (61⁴ - 37⁴)
= 1.15 W

Multiplying this power by the 30 minute time period will give the heat energy, Q, absorbed by the person:

P x t = 1.15 x 30 x 60 = 2077J

We are given the latent heat of vaporisation of sweat so dividing Q by this value will give the mass of water lost:

M = 2077 / 2427x10³ = 0.9 g

This mass of water seems too small to be reasonable. Although this calculation will only give an approximate value I would nevertheless expect it to be above 10g at the very least. Having checked my calculation I cannot find any obvious mistakes, my only real doubt is my calculation of the net rate of absorption but I cannot see how else I would calculate this value.

Thanks for the help.

 

[Redbelly98]

Welcome to PF.

Homework Equations

Stefan-Boltzmann Law: P = e_$$\sigma$$AT⁴

The Attempt at a Solution

Plugging in the numbers and using the the temperature of the sauna walls and human body termperature to give the NET rate of absorption:

P = 1 x 5.67x10^-8 x 1.7 x (61⁴ - 37⁴)
= 1.15 W

You need to use absolute temperatures here, not the °C values. The rest of your method looks pretty good, though note that water's heat of vaporization is around 2300 J/g, not 2300x10³J/g.

 

[MattMark'90]

Thanks for the reply. I made the absolute temperature correction and got 229g/229mL which seems much more reasonable.

Thanks again!

 

[Redbelly98]

You're welcome! Yup, 8 ounces, or a small glass of water, definitely reasonable.