Thursday, August 11, 2011

Heat Wave, Gas Laws & Your Bike Tire

Staying cool out there?

Most of us have gone through or are still experiencing the 'heat wave' here in the United States. Temperatures in some places have taken on record proportions. I remember sweating absolute buckets in mid-July here in Western New York. The same route that I have biked for the past 2 years made me more uncomfortable than I ever remember in memory. Some other friends reported sweating Gatorade colored perspiration. I wonder , gee hows that for perspective?

Interestingly, in the midst of the debt crisis, a report from NYT probably slipped by quietly. It wrote that this past July was the 4th warmest on record in the United States according to NOAA studies. That should come as a surprise only to those who still wish to have their heads in the sand about climate change. I mean, the IPCC reports on the global warming phenomenon don't cost a squat and still out there for anyone to read. 20-30 years from now, I wonder whether the idea of a long bicycle ride will bear new meaning as riders struggle to stay cool.

Anyway having said this, there's a certain friend of mine, (who is a bit naive when it comes to bike technicalities), who pumps his tires to their absolute limits before his rides. It is a religious act for him. It does not satisfy him if its 139 psi. He needs all 140 in his pocket! Its as if his bike wouldn't move an inch if he hasn't dialed exactly that number into his tires.

I do keep wondering from time to time whether this has anything to do with the obscenely high number of flat tires he has obtained particularly during this summer. He's told me that he's not had this many in a long time and he's getting frustrated! Well, could one of his problems be that laser focused air pumping addiction?

When you pump air into your tire and go out for a ride, things change inside that tire that you normally would not think of. If I actually believed that he would actually be even remotely interested in some basic math, I would tell him about two beautiful thermodynamic relationships discovered by a bunch of cool people in the 17th and 18th centuries.

In the early 1600's, Robert Boyle sad that the pressure of a gas is inversely proportional to its volume, if temperature is kept constant.

A century later, Joseph Gay-Lussac asserted that pressure of a gas is also directly proportional to its temperature, if volume is kept constant.

The former shows a hyperbolic relationship between pressure and volume, the latter a linear relation between pressure and temperature.

Mathematically, these relationships can be expressed thus :

If you'd put them both together and assumed that your tire volume remains the same while riding, it can be said :

where P1, T1 are pressure and temperature at one instance in time and P2 and T2 are the states at another.

If my friend religiously pumped up his pressure to 140 psi (=P1) in the 70 degree F (=T1) comforts of his home, and then went out to ride in a muggy 100 degree F temperature (=T2,  a 43% change from his house), we can solve for the pressure in his tire, P2.

Thermodynamicists like to stick with absolute temperatures like kelvin, instead of empirical ones like degree F. To convert F to K, you add 273 to the Fahrenheit temperature. Then T1 = 343 K and T2 = 373 K. 

Since kelvin is an SI unit, you can't do math with apples and oranges and so pressures would need to be in Pascals. 1 psi = 6895 Pa. Converting, P1 = 965266.02 Pa.

Following our intentions to then solve for P2, 

Converting this pascal value back to psi, we get the modified pressure = 152.24 psi.

So a 43% temperature increase has just shot the pressure up by 9% ! This basic math doesn't consider the other heat additions through braking and side wall deflections. 

Ofcourse, I won't tell my buddy about all this. There's some amusement in seeing how many flat tires he'll be getting in the coming days through that nasty pumping addiction.

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Correction (Aug 11) : The heat wave has apparently fried a chunk of my brain too. The comments from some readers were right. The conversion factor of "273" I used to convert F to K was actually to convert C to K. Correcting this, 1 deg F = 256 K, and so T1 = 294 K and T2 = 311 K. The correct math then is :

which, as it turns out, is 8 psi over-inflated (6% increase, not 9%).

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