A few months ago I posted a piece on Usain Bolt. It's interesting to consider his physical limits. Not considering his metabolic efficiency, only about eight percent of the energy he uses goes into moving his body down the track. The the remainder is used to overcome air resistance. That's his speed limit. 100 and 200 meter sprints largely use anaerobic metabolism which can produce enormous amounts of power (over three horsepower). The problem is it only lasts for a short amount of time. Its "fuel" runs out and his core body temperature soars. Trying to run 25 or 30 seconds at that rate might be lethal. Fortunately humans have ways to regulate core body temperature.

Bicycle racers are at or near the top of human power output endurance. A world class rider can generate about 6 watts per kilogram of body weight. A seventy kilogram racer could generate about 420 watts of power on average for periods over an hour.^{1} The spinning motion of riding a bike is one of the most efficient mechanisms of turning energy from food into motion. It varies with people and training, but 25% efficiency is close to average. So for every watt delivered to the pedals, three are lost as waste heat.. that heat has to go somewhere. Our bike racer would need to get rid of 1260 watts - not far from the 1500 watts the burner on an electric range pumps out. Usain Bolt had to get rid of much more heat, but only for a short period of time.^{2}

In humans one of the most important mechanisms for getting rid of heat is sweating. It takes a lot of energy to evaporate water .. just think of the extra heat as waste energy. For fun consider a three cases: a normal person at room temperature, a normal person on a hot day in Arizona, and a Tour de France racer attacking an uphill segment in the French Alps.

Evaporation of sweat, radiation, convection and conduction are the four mechanisms we use to transfer heat. To simplify matters imagine our people are naked so we don't have to worry about the impact of clothing. Conduction and convection are usually lumped together in these calculations and aren't major factors in these three cases. (We'd have to study them if you were in the water or holding a block of ice against yourself.)

Room Temperature

Sitting around in a room at 23°C (73°F) a 70 kg person generates about 100 watts. The details will be in the footnotes, but here's the gist of it. Radiation transfers about 130 watts to the surrounding environment at 23°Ç.^{3} This is why our most comfortable temperature range if we're not exerting ourselves is in the low 20°C s. We don't crack a sweat unless we're exerting ourselves. That changes dramatically with rising air temperature and/or physical exertion.

Arizona in the Summer

Consider a 45°C day - not uncommon in the Phoenix area. Now the body is absorbing heat from the surrounding environment. Now the body receives about 150 watts from the air adding to the 100 watts it generates just sitting around. You start sweating when the air temperature nears body temperature. You need to sweat about 400 grams of water an hour to keep your core body temperature stable.^{4} Move around and that soars and don't even think of getting out of the shade. Note if you wipe the sweat off, you need to sweat even more as it must evaporate on the skin to be effective.

Tour de France mountain stage leader.

Sweating dominates heat removal. The racer is generating over 1300 watts of waste heat corresponding to the need to sweat out about two liters of water an hour. A normal adult in average physical shape can sweat about 1.5 liters per hour. Endurance athletes can increase this to over three liters per hour.

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^{1} Aside: this is insane. If you have an exercise machine calibrated to show watts delivered, see what you can do. Figures over 5.5 watts/kg make one suspect doping. A few people seem to have bodies that can do it naturally, but 7 watts/kg would almost certainly indicate doping.

^{2} There are estimates of around 10% efficient .. so he had to get rid of well over ten thousand watts of waste heat .. his body couldn't sweat it out in real time, so his core temperature soared.

^{3} To calculation the radiative power transfer you need to convert to degrees Kelvin and use the heat transfer equation:

P = eσA(T^{4}_{hot }- T^{4}_{cold})

where σ is the Stefan-Boltzmann constant, A is your skin area .. about 2 square meters for an adult male, and e is the emissivity of skin - in the infrared human skin is nearly a perfect radiator and 0.97 is commonly used. We'll go with 1.0 for a quick back of the envelope. Your skin temperature is about 34°C or 307°K and 23°C is 296°K. Plugging this in gives about 133 watts .. say about 130 for a simple number.

If you want to estimate the surface area of your skin the Du Bois formula is generally considered to be the best approximation for most people.

A ~ 0.007184 * W^{0.425} * H^{0.725}

where W is body mass in kilograms and H is height in centimeters and A is in square meters. My guess is very few people know the surface area of their skin, but it's an important number when you're trying to work out heat flow.

^{4} For the sweating component note a 70 kg person sweats that the rate of 25 grams an hour at rest. It takes 2428 Joules to evaporate a gram of water at 34°C (note it's almost 10% higher than the amount needed at 100°C).. so the cooling power is approximately 25g/h * 2428 J/g * 1/3600 s/h ~ 17 watts. You can use the same equation to calculate how much sweat per hour is needed to transfer a given amount of heat away from the body.

This is in dry air. It's much more difficult to evaporate sweat in humid air. Tokyo in late July through August is hot and humid - it promises to be a real challenge for many athletes and it seems unlikely records will fall in endurance sports.

Note sweating chills blood just under the skin by transferring heat to water on the skin causing it to evaporate. The chilled blood then moves to the body's core where it picks up heat and drops the core temperature a bit. It's just like a radiator in a car.

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