Wednesday, January 29, 2014

Gee whiz, it's cold!

A friend posted on Facebook something to the effect that the temperature outside had risen from 2 degrees all the way up to 4. Being a mathematician and a smart ass, my natural response was to say that it is twice as warm. You can only imagine the intense ripples of laughter as this comment echoed through the facebookiverse!

For all the silliness of my statement, there is something of a serious question buried in there. What does it mean to be twice as hot?

Let's review the original question, but in Celsius. 2 degrees F is -16.67 degrees C, and 4 degrees F is -15.56 degrees C. So... If I take the ratio in Celsius, I get 0.93. If I do the computation in Celsius, then 4 degrees F is 0.93 times as hot as 2 degrees F. That's just silly.

The problem is that both the Fahrenheit and Celsius scales are offset. Zero is not zero, so to speak. More precisely, zero thermal energy does not occur at 0 degrees F, or at 0 degrees C for that matter. Zero thermal energy occurs at absolute zero, which is -459.67 degrees F or -273.15 degrees C. At that point, there is no thermal energy.

So, let's repeat the computation. 2 degrees F is 461.67 degrees above absolute zero. 4 degrees F is 463.67 degrees above absolute zero. The ratio is now 1.004. Thus 4 degrees F is 0.4% hotter than 2 degrees F. It has 0.4% more thermal energy.

But then, I got thinking about it some more. The difference between 2 and 4 degrees F pretty minimal. I might not even notice it. Being from Wisconsin, I would wear a tee shirt, shorts and sandals at either temperature. While drinking a beer named after a cow.
A real Wisconsin drink for when it gets above 2 degrees F

But there is a big difference between 2 degrees and 32 degrees. At 32 degrees F, I would typically start sweating profusely. But according to the previous formula, I would divide 491.67 (32 degrees F) by 461.67, to get 1.065. 32 degrees F has only 6.5% more heat than 2 degrees F. Now that just doesn't seem right!

Then I remembered something that my buddy Jean-Baptiste Fourier told me. The rate of flow of heat is proportional to the difference in temperature. Like, if I set my thermostat to 20 degrees above the outdoor temperature, my gas bill will be twice as high than if I set it to 10 degrees above the outdoor temperature.
Winner of the 1819 Green Bay Flyfishing contest
and author of "Théorie analytique de la chaleur"
Let's say that my thermostat is set to a greedy 72 degrees F. When the outside temp is 2 degrees F, the temperature differential is 70 degrees. When it is a tropical 32 degrees F, the differential is only 40 degrees F. The house will loose heat 70 / 40 times as fast at the lower temperature. In other words, for a house at 72 degrees F, 2 degrees is 1.75 times as cold as 32 degrees F. That is more or less the difference I would expect to see in heating bills.

And if I compare 4 degrees F to 2 degrees F?  If I set the thermostat to 68 degrees F, then the ratio (68 - 2) / (68 - 4) is the answer. In terms of the utility bill, 4 degrees F is 3% warmer than 2 degrees F.

I have three answers so far:

1. If I just do dumb arithmetic, then 4 degrees F is twice as warm as 2 degrees F .
2. If we look at the total amount of heat energy, then 4 degrees F is 0.3% hotter than 2 degrees F.
3. My utility bill says that 4 degrees F is 3% warmer than 2 degrees F.

Of course, there is one other way to look at it. How cold does it feel?

I would be tempted to start with the temperature differential thing, but I am not sure what to use for the temperature of skin. My hands are pretty much always warmer than my wife's hands. Now think this through carefully... When we are outside making snowmen, the temperature differential between my skin and the cold air is thus bigger than her own temperature differential. So, it follows that it is colder out for me than it is for her.

I really don't know why she is always the one complaining that we need to move to California.