Tuesday, 23 July 2013

Cost and time of stair climbing

I came across a short negative opinion piece by Hamilton Nolan complaining about mayor Bloomberg's attempt to encourage people climb more stairs. The original Times pieces reads
Mr. Bloomberg said on Wednesday that he had issued an executive order requiring city agencies to promote the use of stairways and use smart design strategies for all new construction and major renovations.
Nolan replies, "Will this get you in good shape? No, it will not."

Alex Hutchinson in turn replies to Nolan's claim that although it is extremely unlikely that stair climbing will "cure" the world of obesity, every little bit helps. And more importantly there are side benefits like (surprisingly?) taking less time to get places, making for more aesthetically pleasing spaces, and maybe even safer buildings (if everyone is used to taking stairs, it sure helps when the elevator is out of order).

Such initiatives are only intended as small steps in fighting obesity (pun intended). It may help, it may not. Nolan's article claims climbing a few stairs will not get you fit. Alex sees otherwise, and has the citations to back up his view.

I for one support the expansion of stairs. If you think of the movies, stairs make for grand entrances and make a statement, whereas elevators are utilitarian; they only get the characters moving to the next scene.

But enough about design. How many calories does one actually burn going up a flight of stairs? In Nolan's piece he cites a Livestrong factoid that reads
A 150-lb person must climb stairs for 6 hours and 30 minutes to burn the amount of calories in 1 lb of body fat.
Nolan cites this as evidence that stair climbing is too inefficient to be a good workout.

Opinions and health research aside, I was interested to find out how that number was obtained, so I started ab initio to see if I could replicate the numbers. 

First of all, 1lb of (human) fat contains of about 3500 Calories of energy. In the metric world this is 14,600 kJ of energy. In terms of pure vertical energy terms, to raise an object h meters into the air costs energy of the order
E = mgh
where m is mass (kg), and g is the gravitational acceleration 9.81 m/s2. But not all energy goes in the vertical direction; some is lost to horizontal motion and other efficiency considerations. After some unexciting calculations (which I decided to omit), one may say, alternatively, that for a 150 lb person stair climbing can be said to cost 0.15 Calories per step at 15 cm climb per step.

Thus to burn 3500 Calories you must climb 3500/0.15 = 23,333 steps, or 3.5 km.

How fast does it take to climb that high? The fastest run up the 320m ascent of the Empire State Building is usually around 10 minutes (at 1576 steps). That's about 0.53 m/s. At that pace one could ascend 3.5 km in just under two hours (110 min). Most people will race the Empire building at a pace about half of that (similar to how the average marathon finishing pace is around half the winning time). This would extrapolate to middle of the pack runner taking about 4 hours to reach 3.5 km and burn 1lb of body fat.

Using some novel analysis in an 2010 article, Minetti et al found for vertical ascents higher than 500m one the maximum sustainable energy output is constant after about 400-500m. A person's rate of climb (measured as power output, W) is therefore surprisingly similar for all greater heights. Factoring in some breaks that would be taken and an slower pace if not racing, we see how 6+ hours is a realistic estimate.