Wednesday, March 13, 2013

Climbing Rate, Bike vs Running

Inspired by this article:
What’s the Steepest Gradient for a Road Bike? (byRhett Allain in WIRED),

I decided to use Mr Allain's calculations and ask a slightly different question:  At what gradient of hill climbing (assuming a good paved road) is it easier to leave the bike behind and just run or walk?

My presumption, based on experience, is that at some point of gradient, the extra weight of the bike is more of a disadvantage than the added efficiency of the machine.

Mr. Allain is asking at what point can you not ride a road bike up a steep grade, my question is slightly different.

I wanted to make a graph of climbing rates relative to gradient and see if there is a crossover point that makes sense.

I used these numbers for the upper limits of what is possible on steep grades.  Then I looked up the one hour records for running and bicycling to determine limits of output at the low end of the gradient range.  The one hour endurance records are at zero gradient, but for my graphing idea, I needed climbing rates, so I invented a strategy of assuming that if the grade for the one hour record attempts were changed to 1 percent, then the times would be shortened by 1 percent.  This is not a justifiable value, but it gives a start to make a graph.

This is what I came up with:

I used the information from Mr. Allain's article to estimate that the bike climb rate would go to zero at 45 percent incline.  The net result of this initial estimate is that at about 33% grade, the bike looses it's advantage and should be left behind.  This also makes sense because in the associated article, the pro bike riders choose to push the bike on a grade of 27 percent.  Another caveat, this graph has very few data points so the projected curves are questionable.

This graph gives a starting point, but more investigation would be good.  If you sort out the effort of both the rider and the runner into energy expended going forward and energy expended going upward, the climbing rate upward for the bike has a higher energy demand based on the larger combined mass of bike and rider.  The forward speed efficiency advantage of the bike will change with grade.  In Mr. Allain's chart he has only considered the effort for going up based on E = mgh.  But, with a consideration of forward speed, the relative forward speed of the two travelers becomes important.  the energy input is fixed, then as the gradient gets steeper there is going to be less energy left for forward progress because more energy goes into climbing.  At some point the heavier bike and rider no longer has energy left over from the climb requirement to make forward progress.   At that point the lighter runner still has some energy left to make forward progress.

Using BikeCalculator, I put in the numbers 300 watts and 40% grade, then for a 154 pound rider on a 20 pound bike, the calculated forward speed is 2 mph.  This is more than I would have guessed.  If you take out the weight of the bike in the same calculator, the forward progress is 2.33 mph; not as much of a disparity as I was originally guessing. 

Based on the fact that pro bike riders start pushing the bike uphill at 27% grade, and assuming they could go even faster up that 27% grade if they did not have to push the bike, something is missing in my calculation.   The program BikeCalculator is probably missing some of the energy needed for the case of the steeper gradients.

I have done a similar calculation for energy needed riding a unicycle.  On the unicycle a significant energy is expended essentially rocking up and down with the motion of balancing and peddling.  This is quite different than the case of a road bike on low or flat grades, there is almost no wasted motion of the rider's mass going up and down.  But, as the hill gets steep, the torque of peddling will force much more dramatic motion, side to side and accelerating/decelerating, and then a significant part of the energy of the rider gets used up just moving the center of mass up and down.  This energy expenditure is what is missing using a typical bicycle energy calculation.  Also, on the steep hill, the efficiency of a still body moving in only the desired direction starts to weigh in favor of the runner.

Here are the basic numbers for the chart:

empire state building climb = 1050 ft in 9 min 33 sec = 9.55 min = .15917 hr : = 6596 ft /hr, grade 50%

mt washington climb = .912 mi = 4815 ft, 49min 24 sec = .8233 hr :  = 5448 ft / hr, avg 12%grade 

One hour bike ride record converted to climb rate estimate at 1% grade: 1831 ft/ hr

One hour run record converted to climb rate estimate a 1% grade: 691 ft/hr

To be continued

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