## Friday, July 22, 2011

### The Heat Index and Jenson's Inequality

I've mentioned Jensen's Inequality before, which deals with the average output of a nonlinear function. If the function increases nonlinearly with increases in the input, then the average output will be greater than the output of the average input. How about a real world example? In chemical reactions, the reaction rate increases in a nonlinear fashion with temperature. So if you run a reaction for a time period at T1 and then later for the same time at T2 (so that the avereage temperature for the reaction is Tavg = (T1 + T2)/2), the total extent of the reaction will actually be greater than if the reaction had been run strictly at Tavg. The reaction ran much faster at the higher temperature, more than enough to compensate for the slowness at the lower temperature.

I realized another use for Jensen's inequality while sweltering through the recent heat and humidity wave that started in the Midwest and is now blanketing the East coast. This was actually the result of a lively discussion on the Minnesota Forecaster's blog on the impact of a wind on the heat index.

The heat index is based on a remarkably long set of factors. You can read the original research article (open access) if you like - it's a pretty straight forward engineering analysis. I (and you) can certainly suggest other assumptions to be made, but realize that the calculations are all laid out for you, so feel free to recalculate out your own results. The biggest issue I had with the original report was that it was assumed that there was a constant wind speed of 5.6 mph.

Heat transfer from forced convection (wind) is nonlinear (I think you can see now where I am going with this, huh), in this case varying with the 0.3 power - a decreasing exponent. In this case then, Jensen's inequality is reversed. If the wind blows for a fixed time at speed v1 and the later at v2, the cooling effect will be worse than if it had just blown at vavg. I've never felt a constant wind speed so this is actually an important result. The fact that the wind is assumed at a non-zero constant actually understates the heat index, especially in the lack of wind. The slightest breeze is more important than any incremental increase, something we all appreciated on some of those near windless days.