Answer 08/98

Answer to the Question 08/98

BODY HEAT

The question was:

How much heat is produced by a human body (in Watts)?

(27/7/98) Itzhak Shapir (e-mail microwav@rafael.co.il) solved the problem by considering the food intake by a man, and reached the conclusion that it is 58Watts.

Solution:

1. An average human consumes about 1200 food calories (1,200,000 calories) which are about 5,000,000 Joules per day (24 hours).

2. Most of that energy is processed in the body and results in heat. In that case, the heat power is 5,000,000/(24*3600) J/sec ~ 58W

(3/99) Tal Kuzniz (e-mail kuzniz@eng.tau.ac.il) from Tel Aviv University (Israel) and Mark Davison (e-mail davi-ph0@paisley.ac.uk) from University of Paisley (Scotland) suggested to look at thermal radiation balance between the body and the surrounding. The solutions used different estimates of the total area of the body, as well as different estimates of the normal temperature difference between the body and the air. (We need such temperature difference that a (naked) person does not feel hot (i.e. does not sweat) or does not feel cold.) A little opinion poll which was conducted converged on (or rather averaged at) 25-26C as ideal temperature.

Solution:

Body area A=1.5m²
Body temperature: T=310K
Air temperature: t=299K
Stefan-Boltzmann constant: S=5.67*10^-8W/(m²K⁴)
Skin emissivity close to unity.
Power=S*A*(T⁴-t⁴).
This expression leads to estimate of 105W, which looks like a nice estimate.

Y. Kantor: Here is my personal "solution" of the problem, which is based on heat conductance:

Solution:

Person (or rather, naked person) feels well at temperature of about 25C. At temperatures below that one needs clothing to reduce the heat loss while at higher temperatures one sweats to increase heat loss. So, we may assume that when the temperature difference between the body and the environment is 12C the heat flow by thermal conductivity is equal to heat production. If we assume that the human body is a sphere (of course) then the temperature field around it will be T=const + A/r, and the total heat current will be 4{pi}Ak, where k=2.6J/m*sec*deg is the heat conductivity of the air. (The latter expression is simply 4{pi}r² multiplied by conductivity k and multiplied by the temperature gradient A/r². The prefactor A in the expression for temperature is determined by assuming that the temperature difference is 12C when r is the radius of the sphere R. Thus A=12*R, and the final result for heat loss is 4{pi} k *12*R. Assuming that the "equivalent sphere" representing the human being has radius somewhere between 0.2m and 0.6m, we find that the heat production is between 80W and 240W.

For those who are not happy representing a human being as a sphere, let us assume that it is a prolate ellipsoid with larger semi-axis a=0.8m and the minor semi-axis b=0.2m. In this case the coefficient A in the above derivation should be the temperature difference (12C) multiplied by the capacitance of the ellipsoid (see Landau and Lifshitz, Electrodynamics of Continuous Medium). That capacitance is sqrt(a^2-b^2)/Arch(a/b)=0.4m. Substituting this number we get heat current of 160W.

Y. Kantor: We note that radiation and heat conduction together remove more heat than is produced by human body. Maybe our estimates are wrong by a factor of 2 or more... We are still waiting for comments and more quantitative estimates.

(12/2000) Lawrence Weinstein (e-mail weinstei@physics.odu.edu) from the Physics Department of the Old Dominion University, Virginia, wrote us the following e-mail:

(1) 1200 calories per day is much too few. It is typically more like 2000-2500 (at least for well fed Americans). This will double the estimated heat available from 58 W to between 100 and 120 W. (The rule of thumb is that one person = 100 W light bulb.)

(2) Skin temperature is not 37 C; core temperature is 37 C. This is especially true for the extremities. The average skin temperature will vary significantly to maintain the heat balance. This temperature is controled by varying the blood flow to the surface and to the extremities. Presumably, 25 C is the lowest temperature at which people are comfortable. This implies that skin and extremity temperatures will be much lower than 37 C at this point. A better reference would be the highest comfortable temperature in a high humidity environment (to minimize the cooling effects of sweating). I would guess this to be about 30-31 C. (This implies a comfort zone (without sweating) of about 5 C.)

If we use 33 C for the body's surface temperature (may be too low for the torso but too high for the extremities) and 25 C for the ambient temperature, then we get 65 W for the radiation output and about 110 W for conductive output (using the prolate elipsoid) for a total of 175 W.
If we use 37 C for the body's surface temperature at an ambient temperature of 31 C, then we get 60 W for the radiation output and 80 W for the prolate elipsoid for a total of 140 W.
These numbers are reasonable because your body is less metabolically active and cools at night, emitting significantly less heat for at least 8 hours.

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