Boyle's law needs revision

[Ben]

I was reading a pneumatics book when I noticed in an equation for an air spring they assumed the air spring was isentropic. Isentropic means no change in entropy, which means no heat transfer and no irreversibilities like friction.

Technically this is more correct because Boyle's law assumes the temperature is constant (isothermal), which means there is some heat transfer to keep it constant. While in real life the temperature difference is small if existent, assuming isentropic pressure chambers would be closer to reality because if the pressure chambers were isothermal then you would lose some energy. For the sake of analysis I'd rather keep that energy since it will be leaving in the form of heat very slowly anyway.

What this means is that we need to change the equation we use. The form for a polytropic process is

P1*V1^n = P2*V2^n

where n is an exponent that describes the system.

n = 1 describes an isothermal system because it reduces to Boyle's law.

n = k, where k is the ratio of specific heats for the gas, describes an isentropic system. For air k equals about 1.4. This means that different gases would have different properties, but only slightly.

Assuming either isothermal or isentropic pressure chambers both should be reasonable approximations of reality and the assumption would be acceptable to me as long as the assumption is made clear. It would be more realistic to assume an exponent between 1 and k because there are some losses to friction if a piston is used, though they should be negligibly small. In the future I will assume the process is isentropic rather than isothermal and use the appropriate relationships.

[cantab]

Don't you just use the ideal gas law? PV = nRT

Whether a pressure chamber is isothermal or isentropic will depend on timescales. If the pressure is charged and release rapidly, it will likely be isentropic. If, however, it is left pressurised for long enough to equilibrate thermally with the surroundings, then it's isothermal (assuming ambient temperature is constant).

In any case, I doubt the difference is significant.

[Ben]

You can use the ideal gas law yes, but if the temperature changes it doesn't work alone any longer.

For us the different will not be very significant as I said earlier, but to be as technically correct as possible I'll assume an isentropic pressure relationship. The heat transfer will be nearly non-existent, which supports either argument, but Boyle's law loses energy so I'll use the isentropic version. Using either is fine as long as the assumption is made clear.

If it makes the math easier, keep using Boyle's law. The difference between the two is large only at higher temperatures from my understanding. Right now out of boredom I'm working on a model of an air gun and to prevent the math relatively simple I'm going to use Boyle's law.

Perhaps I shouldn't have posted this because it might just end up confusing people.