原创 A simple measurement puzzle

About once a month, I check my car tyres, since correct pressure is necessary for good car handling, a smooth ride, and good gas mileage. When I checked my front tyres recently with my Topeak digital gauge (image below), one was at the correct pressure (30 psi), but the other was higher, at about 33 psi. What puzzled me was that they had both been at 30 psi the previous time. I know tyres can lose pressure, but I had never heard of a case where the pressure increased on its own.

This inexpensive digital-pressure gauge is a pleasure to use: It reads up to 160 psi/11 bar (useful for bike tyres and suspensions) to three significant figures; can be switched among psi, Bar, or kg/cm² readings; and handles Presta and Schrader valves -- a big improvement over the old "pencil-type" mechanical tyre-pressure gauges.

 

I gave this some thought and saw only two possibilities at first: Someone was playing a practical joke on me (very unlikely); or my previous reading for just that one tyre alone was in error (also unlikely, as all tyres were measured twice, and at the same time).

Then I looked at the car and saw that the tyre that read high was in full sunlight, while the other was in the shadow, and a black tyre certainly does heat up from solar radiation. Mystery solved -- or maybe not. I pulled out my custom-made "back of the envelope" pad and did a quick calculation using the ideal gas law:

...where P is pressure, V is volume, T is the absolute temperature, and K is a constant, which depends on the amount and type of gas (here, the value is irrelevant).

My hand-made "back of the envelope" pad reminds me that doing quick, rough estimates is often a good first step to understanding the parameters of a problem.

 

Thus if the pressure I measured was about 10% higher than the original, and the volume was constant, then the temperature of the air in the tyre also should have gone up about 10%. The "cold" ambient temperature was about 77°F (25°C) or about 300K (remember, this is a rough estimate we're doing), so the delta rise would be 30K (30°C), or about 55°F.

Then I worried that perhaps a change in the tyre's volume would affect my estimate, but I realized it was a non-issue for two reasons. First, a car tyre is not an easily expandable balloon; it is a rubber enclosure restrained by steel-wire belts. Therefore, its volume stays fairly constant, especially for modest variations around a nominal value (this is a type of assumption we often use in many simplified models).

Second, even if the tyre did expand slightly due to the increase in internal pressure, that would actually cause a decrease in the resultant pressure -- again, the gas law. (I recall seeing a complex differential equation embodying the relationship among a tyre's construction, pressure, and volume, for more advanced modeling.)

Was solar heating the answer to my mystery? I don’t know, as I have no way of measuring the internal air temperature. I suppose I could do some thermal modeling, or even use an application such as COMSOL Multiphysics for a thermal/mechanical simulation, but it's not worth the effort.

So the question of sunlight heating the tyre and raising the pressure remains a slightly open mystery. My "gut" tells me that a 30°C/55°F rise for a black-rubber tyre in full sunlight is possible, but that's where I have to stop.

Do you think it was solar-heating effect? Can you think of any other causes? Have you ever had a similar "simple" measurement mystery, where you are not sure of the actual cause of the observed effect?