Thursday, August 11, 2011

20 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.

*  *  *

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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  1. Nitpick: K = C + 273.15, *not* F + 273.15.

  2. allow me to correct your calculations

    To convert Fahrenheit degrees to Kelvins you must subtract 32, multiply by 5/9 and then add 273.

    So your temperatures in Kelvins are:
    T0 = 294.26 K
    Tf = 310.93 K

    Your final pressure is then Pf = P0*Tf/T0 = 965266.02*310.93/294.26 = 1019948.9 Pag = 147.93 psig

    The compressibility factor for air at these conditions is .999 so ideal gas is a good assumption, but your friend is really only gaining about 8 psi, not 12.

  3. Actually, you have to convert F to C (celsius) before adding 273 to obtain K. e.g. 70F ~ 21C ~ 294K

  4. yeah thats what the subtracting 32 and multiplying by 5/9 is for

  5. Not only does tire pressure change with temperature, it changes with altitude. Where I live in Sacramento at 22 feet above sea level, in your 70 degree house, when you pump up your tires to their max, drive up the the mountains at 5000+ feet, and do a climb up to 6500/7000ft elevation, and ride in 80 degree temps, you have overinflation. Add to that most clincher rims are labeled "max inflation 130" and you're really asking for trouble.
    There is a laboratory in Finland;, the only independent bicycle tire lab in the world, where rolling resistance has been measured to be LESS at lower pressures!

  6. Interesting thought, and I'd expect explosive decompression as the failure mode. I wouldn't want to bike next to him, that's for sure!

  7. Somehow, I'd be skeptical of your theory. I usually run my tires towards the high end and have had no flats during record heat. I also got no pinch flats last winter when it was cold.

    Still, perhaps your obsessive friend would welcome another correction factor!

  8. Steve: The Ideal Gas Law is not Ron's pet theory. Its not really up for debate as its been proven over and over again since the mid 1800's.

  9. Thank you for this, living in Phoenix where the temperature delta between pumping my tires up indoors and then riding on the summer asphalt can easily reach >50F, I've often wondered about this specific question. In summer I pump 7 to 10 lbs less than the recommended pressure, and hope that manufacturers account for some thermal expansion when they recommend pressure. Of course, there's also thermal contraction to consider in colder places in wintertime--I wonder what people who ride in Duluth experience in December, when the delta between indoors and out might be 80F.

  10. The correction to the temperature conversion has been made. Thanks!

  11. You're back! Nice to read the blog again Ron!

  12. Anonymous2:24 PM

    "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"

    Correct me if I'm wrong, but a change from 70deg F to 100deg F isn't a 43% change in temperature as the zero point (0deg F) isn't really zero it's just a random point where the Fahrenheit scale happens to come to zero (there's still heat energy there at zero degrees Fahrenheit). To get the real percentage change in temperature you'd have have to convert to degrees Kelvin first.
    70F = 294K
    100F = 311K
    So the actual percentage change is about 6% which, of course, is the same as the change in pressure in the tires.

  13. The theory to which I referred was the one that heat caused the flats, not the laws involving compressible gases. He could probably pump those puppies 10psi over the posted max and have no problem all summer. He might complain about the harsh ride. Have any of his flats been what you'd expect from overinflation?

  14. While I haven't been too scientifically rigorous in investigating the cause of his bike tires (factors such as age of wheel, type of terrain etc), I think over-inflation and the simple math behind it is representative of flats in hot weather. I seriously doubt inner tubes can maintain their structural integrity with over-inflation and pressure rise on top of that.

  15. Another take on this that most readers might appreciate is what happens in the winter. Pump your MTB tires up to 35psi in your warm garage, then go out and ride in 30 degree weather....pinch flat city on your favorite trail.

  16. Anonymous1:41 PM

    Pumping the tyres (I am English) up in the garage at 70F would not fill the tyre with 70F air. To compress to the 140 psi you have to put a lot of work into the air (just feel the bottom of your pump). I presume that the tires weren't pumped up so slowly, or thru' a cooler, to reduce this effect. Thus, the air in the tyre would be a greater than 70 psi to start.

    Great Blog!

  17. Anonymous6:41 AM

    I came across this post somehow, and started to laugh. Good to see there are sharper engineers out there who caught some of the errors and misconceptions in the original post and sent in corrections!
    Structural integrity of an inner tube…? Geez, I could comment all day.

    So, air is now an ideal gas?… ;)
    Ron- if you profess to be proficient in understanding these things, you really do need to at least have a grasp of your fundamentals.

  18. @Anonymous above - To highlight this concept, taking air as ideal gas at normal pressure and temperature is a valid approach for first order approximation. Feel free to enlighten me with a more rigorous mathematical model in your research paper. Ofcourse, easier said than done for anonymous posters!

  19. Richard, AZ11:37 PM

    Great and fresh site Ron! Just wished you'd blog more frequently.

  20. Tire pressure is very important. Correct tyre pressure can help to extend the life of your tyre, improve vehicle safety and maintain fuel efficiency. It is also advisable to always check your tire and you can also wash your tire daily.


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