Boyle's law needs revision
Posted: Sun Dec 07, 2008 1:30 pm
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.
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.