Climber's Energy Budget
© 2007 L.L. Williams
I have long noticed a rule of thumb for how quickly it takes me to hike or climb through elevation gains in the mountains: 1000 vertical feet per hour. On steeper terrain, the distance is shorter but it takes longer to move up the steeper slope. On more moderate terrain, you make better time but the distance is greater. It always averages out to about 1000 vertical feet per hour.
Is there something intrinsic about 1000 feet per hour? I think so.
Let's say you and your pack weigh 200 pounds. Then, moving through 1000 feet in the earth's gravitational field requires 2x105 foot-pounds of energy (in english units).
Another unit of energy measure is the calorie. This is not the widely used unit of food energy, however. It turns out that the food unit known as the calorie is 1000 of the english energy units known as calories.
Converting the power required to raise 200 pounds up 1000 feet against earth's gravity in one hour into units of food calories per day, we find 2x105 foot-pounds per hour = 1500 food calories per day. This 1500 calories of food energy per day is very near to what an athlete may be able to eat in a single day. So it appears that the rule of thumb for how fast a climber can climb in the gravitational field is indeed reflecting the fundamental power scale derived from how much food energy a climber can ingest. That is, the power output of a human being reflects the rate of energy input as food.
Another numerical fact of interest is that 2x105 foot-pounds per hour works out to about 75 watts. The power output of a human being working against gravity is roughly the same as a 75 watt lightbulb.
If a human body was 100% efficient, all food energy could go into work against gravity. However, the human engine, like most engines, does generate waste heat. The waste heat can be estimated from the temperature of a human body, about 100o F.
There is a basic law of physics that describes the amount of power radiated by an object at a given temperature, the so-called blackbody formula. A blackbody is an idealized body which, at thermal equilibrium, emits precisely the amount of energy it absorbs at every frequency. For typical cases on earth's surface, these frequencies are in the infrared. It turns out that human flesh is well approximated as a blackbody in the infrared.
The radiative energy released is 5.7x10-8 watts per square meter per degree kelvin4. Kelvin (K) is a unit of temperature, and the power radiated by a blackbody goes as the fourth power of the temperature. This means something twice as hot emits 16 times the energy. It turns out 100o F is about 310o K. So a human emits about 530 watts per square meter.
However, if the climber is in a room-temperature environment of around 300o K, the climber will absorb from the environment around 460 watts per square meter, for a net loss of around 70 watts per square meter. If the climber is in freezing temperatures of 273o K, the absorption from the environment is only around 320 watts per square meter, for a net loss of 210 watts per square meter.
A typical human adult has something less than 2 meters of surface area, so our heat loss can range from 140 watts at room temperature to 420 watts at freezing temperatures. Of course, these numbers are true only for exposed flesh; insulating clothing will reduce the loss. Yet it is clear that maintaining body temperature can dominate a climber's energy budget in cold temperatures and the energy required to climb can be a minor fraction of the energy required to stay warm.
Notice that both the energy required to climb against gravity and the energy required to stay warm increase with the size of the person. The energy required to climb goes as the third power of the size of the person (volume) while the energy to stay warm goes as the second power (area). Note that many top endurance athletes are small in stature; a 300 pound linebacker will have difficulty summiting Everest no matter how strong he is. In Scott's fatal expedition to the South Pole, Scott made a last minute change to choose a larger man to join the final push to the pole. Given the fine line between life and death, this choice may have mortally tipped the scales for the entire party since the larger man required more food to stay warm and move himself over the ice.