It's time to reconsider that model...
First, Minetti shows for every grade tested, walking is more efficient than running. So then why not always walk? Well, obviously human kinetics limit the speed at which walking retains its efficiency, so beyond a certain speed, running is the preferred choice.
Really, I don't care about the cost of running a 45% grade. In a race, I'd never run a 45% grade: at whatever speed I can run that, I can certainly walk with less energy cost. So to estimate at what grade walking becomes preferred, I'll make some crude estimates. First, I assume I can gain altitude, best case, at around 1000 meters / hour. Then I'll assume I can walk up to 8 km / hr (around 5 mph). The grade, then, at which I can walk at my maximum speed is then 1 / 8 = 12.5%. I'll therefore assume I run if the climb is less steep than 12%, walk if it's steeper than 12%. I'll combine Minetti's data (walking > 12%, running < 12%) into a single data set, and refit.
So back to the fitting equation. I had a transcendental equation last time I thought was really slick and which I really wanted to work. Problem is: it didn't work. The fitting parameters in the model weren't matching what was expected from the underlying physics (most notably, the intercept was way too negative). So I decided that maybe Minetti was onto something with his polynomial. The differences in my approach:
- I fitted the polynomial to my g', the sine of the angle, rather than to grade, since g' is a more natural unit relating climbing rate to speed on the ground, and
- I made sure the highest order of the polynomial was even, as opposed to odd, to prevent it from exploding in the negative direction for sufficiently steep descents.
The result is shown in this plot:
Also shown in the plot is an "ideal" curve in which I assume all useful work goes into raising altitude. I set the metabolic efficiency to a relatively high 25.6% to prevent this ideal curve from moving above the fitted curve: walking can't be more efficient than "ideal". You can see the data, here in the walking regime, approaches the ideal curve but then rises above it. At some point it's faster to walk further up a more gradual grade than to take a short-cut up a steeper grade. This is consistent with everyone's experience, I'm sure.
Following a suggestion by Gary Gellin, I then I took the ideal curve and assuming the same metabolic efficiency applies to cycling, I added in additional power for drivetrain losses, added weight of the bike (assuming a bike weighing only 10% of total body + clothing weight), and a coefficient of rolling resistance of 0.5% (high-pressure road tires on a slightly rough road). Interestingly, this curve crosses the running curve at around 15.7%. This suggests if an athlete is similarly adapted to running and cycling, then it's faster to run (rather walk) a hill with a grade over 15.7% than it is to ride it, even assuming the bike has sufficiently low gears to allow for an efficient cadence.
Cadence is an interesting issue. For example, this study shows that cycling can have a very high metabolic efficiency (30%) at 50 rpm, but it drops at higher cadences, although riders had increased endurance at higher cadence. So focusing too strongly on efficiency alone is a mistake.
In this run-bike comparison, I neglect wind resistance, which I assume will be similar for the runner and the cyclist. This assumption is likely favorable to the runner, who is more upright, but also lacks a bike, which is typically around 1/3 of total wind resistance.
Next time, I'll compare running on the flats to cycling up Old La Honda Road, following Gary's observation that world class 5 km times may be a good predictor of what a world-class cyclist would do up Old La Honda (5.4 km @ 7.3%). Then I'll look at the benefit of drafting in running.