Sunday, April 4, 2010

running power, part 3 (climbing)

Last time, I described how it's not so simple to come up with a model for the power required in running without a detailed analysis of human body kinetics. Bicycles are so wonderful because they act as an interface between human kinetics and propulsion. Assuming the bike fits the rider, the details of human kinetics cease to be important, and the simple power-speed relationship of the machine versus the road are simply analyzed.

Ed Coyle's model as described by Ed Burke is simple: 1 kcal per kg per km traveled (assuming standard Earth gravity). But obviously that's simplistic. Running up the World Trade Center stairs, for example, obviously takes more energy than running a similar distance on level ground.

The record in the Empire State Building Run-Up is held by Australian Paul Crake, in 9:33 in the 2003 race. That corresponds to a VAM of 2010 meters/hr for 9 min 33 sec. That's 0.558 meters/second.

Then this VAM corresponds to running 1.675 m/sec = 1.675 W/kg. Lifting COM to that altitude is 5.47 W/kg. If clothing is 2% of total weight, then that's 5.58 W/kg.

Again, Coyle's model for running energy used is 1 kcal/kg/km. Assuming a metabolic efficency of 23%, that's 1 kJ/kg/km. However, this number includes some assumption for wind resistance, which I prefer to model explicitly. Assuming A = 0.68 meters² (40 cm by 1.7 meters) and CD = 0.8, with air density running at 250 meters/minute, that yields 0.04 kJ/km/kg due to wind resistance. So I could reduce Coyle's number to 0.96 kJ/km/kg, a correction which is smaller than the precision of the obviously crude estimate.

So now I simply need to estimate the distance traveled by the runners. The stairs appear quite steep:

Empire State Building Run-Up

Assume for example the runners go 2 meters for every meter uphill, including landings. Then this VAM corresponds to running 1.12 m/sec = 1.07 W/kg. Adding in those 5.58 W/kg brings the total to 6.65 W/kg.

Wind resistance is another factor, of course. Again assuming CD = 0.8, A = 0.68 meters², M = 70 kg, and air density ρ = 1.2 kg/m³, that's 0.013 W/kg, bringing the total to 6.66 W/kg.

Curiously, that's close to the 6.7 W/kg claimed by Daniel Coyle (related to Ed?) as the lactate threshold needed to win the Tour de France. This race is a lot shorter than the one hour or so one can sustain efforts at near lactate threshold. On the other hand, the Empire State Building Run-Up doesn't have the prestige of the Tour de France, either, and the participants don't have the luxury of the preparation or training available to Tour contenders.

The issue with simply adding potential energy to Ed Coyle's 1 kJ/kg/km, however, is that it predicts, as is the case with a bike coasting downhill, that downhill running might be accomplished with zero power. Of course this is far from the truth. So something more is needed.

I'll conjecture on that next time.

I'm doubting Coyle's model applies to climbing. I suspect energy efficiency is better uphill: less bouncing. Hard to believe running would allow for more power output than cycling.


Andrew Coggan said...

Hi Dan,

1) 1 kCal/kg/km isn't really Ed's model, but a common rule-of-thumb (based on research) that is used in exercise physiology.

2) AFAIK, Ed and Dan Coyle aren't related.

3) There are various models of the power requirement of running published in the scientific literature...running isn't really my "thing" so I can't point you to them directly, but I believe that Phil Skiba has cited them in the development of his GOVSS metric, so you might check his site (

Andy Coggan

rusty@feepish said...

trivia related to this post:

pretty sure george hincapie won the empire state building stairclimb at least once in the 90s.

source was probably velonews. google says the internet knows nothing about it.