Saturday, April 17, 2010

Heat balance of nudist colonies

One of the three mechanisms for heat transfer is Radiation.  Just about everything outside of a lab that has a temperature radiates light in some visible or invisible form.  The hotter the temperature, the more visible the light (which is how our incandescent lights work). Radiation at the temperatures we live in (70 F) are not as small as I originally thought-- ~400 Watts per square meter of surface, and it's almost entirely in the non-visible infrared light spectrum.  Fortunately, everything around is throwing ~400 Watts/m^2 back and forth, so if the environment itself is enclosed, with no heat loss outward, then everything is just throwing out 400 Watts/m^2 while simultaneously absorbing 400 W/m^2 back.  

--In a sort of gross approximation, imagine two of those heat dishes facing each other, radiating to each other, keeping each other mutually warm.  awww... Loving...

Our radiant environment and the human body:
Us mortals are homeostatically around 98.6 deg F (actually, proven to be more in the 99 deg range, but 98.6 is exactly 37 deg C, so the math is easier).
The Radiant heat transfer given ideally by the Stefan-Boltzmann Law, which equals (5.67*10^-8)*([Temperature of one surface]^4  -   [Temperature of other surface]^4)
If we think of one surface as the human body, and the other surface as everything else around us (let's say, at 71 deg F, or 22 C), the net radiant heat transfer between the body and the room is roughly:
5.67*10^-8 * (37^4 - 22^4)= 84 Watts/m^2.
If we multiply by 1 square meter (a guess at the area of the human body exposed to the environment), then we get ~80 Watts net heat loss to the environment.
If we add in conductance, that equation looks like this, assuming a thermal conductance of the skin to the surrounding air of R-2:  (98.6 F - 71 F)/(R-2)*3.412[Btu/hr-Watt]=  255 Watts of heat loss to the environment.  YIKES!  brrr....
And, if you were wondering about that R-2 of the convection resistance next to the skin, I am indeed assuming for all of this that we're totally naked.  You'll see soon why we invented clothes.

[Caveats here for those who really can see what's going on and want to rightfully criticize my spherical cow approach here-- this will be a rather conservative estimate, as we'll see later.  The skin of the human body is not usually 98.6 deg F, but a bit lower, resulting in lower radiation as well as convective heat losses]

So, to keep track, we've got the following heat losses:
84 Watts of Radiation  +
255 Watts of Conduction/Convection
~330 Watts lost to the environment

Ok, now what about heat gains?  us mortals are also little power plants, converting food into thought, and waste heat (with about 20% efficiency overall).  Given our diets and average activites, we eat about 2000-3000 kilocalories of food a day.  Do some conversions, and we get a total of about 150 Watts of heat production  from just being living breathing things.

So, the net heat balance:
330 Watts loss-150 Watts of heat gain= 170 Watts of net cooling off.

Again... brrr.  We invented clothes for a reason.  It's because 70 degrees is a little chilly of a radiant environment to be naked, or so it seems.  Due to the aforementioned conservativeness of the calculation, it not actually as bad as 170 Watts of heat loss, and it's also a dynamic system, so our bodies are pretty good at reducing heat loss in these situations by lowering skin temperature.

But while continuing on the theme--
how warm would it have to be to be comfortable nude, standing, in a radiantly heated room?
The answer, per all my assumptions here is:
87 degrees Fahrenheit!  exhibitionists take note!  or put on some clothes already.

Ok, now on to some real work.

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