The other day I ran up a really big hill - Mount Carmel in Israel, in fact - wearing my Garmin Forerunner 305.
It keeps track of your speed and heart rate and is supposed to tell you how many calories your burned (it knows my weight, age and gender). It's pretty good on the flats, but completely wrong in the climb!
Here are the pertinent numbers (just the first few from a longer climb):
1........5.7 mph......55 ft....162
2........4.4 mph....325 ft....128
3........3.8 mph....450 ft....115
4........4.3 mph....337 ft....133
The actual effort expended in miles 2 - 4 was very high: with a 10-15% grade you're really working (and it was a hot July day) yet the calories were very low (especially for a 200 pound guy).
I'm a physicist and I decided to do a bit of "back of the envelope math".
You recall from high school that an object with mass m being raised by a distance h increases its potential energy by m x g x h
Using round numbers (g=10 m/s^2), a 100 kg person climbing 100 meters increases his potential energy by 100 kJ.
The body is roughly 25% efficient in converting calories to "work done", which conveniently means that 1 kJ of "net work" costs about 1 kcal of "food work". (1 cal is approx 4.2 J).
This suggests a simple modification to the conventional "calorie calculators" you find online. To get a more accurate estimate of your work done you can add the following term:
Calories due to climbing = (body mass in kg) x (height climbed in meters) / 100
You add this correction to whatever your conventional "running on the flat" calculator comes up with.
After I had come up with this equation I found a blog
which talked about the same problem and which included a reference to a scientific paper from the Journal of Applied Physiology
which describes measurements done on athletes to come up with accurate estimates of the calories burned as they run on a treadmill. They found pretty much the same relationship (although you have to wade through the descriptions and the graphs to get to the essence) - that when you climb, you expend about 40 J / kg / m (see figure 3), with only a weak dependence on the slope (once it's a "real" slope - more than a few % grade).
I look forward to hearing people's thoughts on this - and perhaps one day Garmin will include a better equation in their running GPS.