Ideal gas is assumed to be compressed and expanded isentropically inside a cylinder. I want to drive a differential equation which relate the piston position to its pressure , i,e some ode in the form of
dp/dt = f (p,gamma,x,t)
where: Gamma is the specific heat ratio.
Edit: there was missing a factor -gamma in the expression for dp/dt.
Consider a cylinder of volume V=A(L-x), where x is the position of the piston.
For an isentropic process in an ideal gas, we have pV^gamma=const, and using the expression for the volume above, we can express the pressure as p=p0 * (1 - x/L)^(-gamma), where p0 is the pressure at x=0.
When differentiating with respect to time, we get
dp/dt = u * (p0*gamma/L) * (1 - x/L)^(-gamma-1), where u=dx/dt is the piston displacement velocity.
I would suggest Lagrange dynamics approach. The piston in the cylinder may be considered a simple system with a single degree of freedom. This degree of freedom will be the coordinate (location) of the piston (rather than the pressure. Kinetic energy of the moving piston and potential energy of the gas in the cylinder, along with the dissipative function (heat energy losses) and external force applied to the piston should be taken into account. The pressure will depend on coordinate of the piston, but if the processes are not slow, it may also depend on time (i.e., on the "history", change of temperature, etc.). The pressure will be related to the piston's coordinate, but this relationship alone does not suffice to describe the dynamics of the system.
@Sverre,
I did the same to drive the pressure equation. In fact, you can drive two different forms i.e. the one you drive which consists the piston position (x) and speed (u). And the second one is the using, let say V = V0 + Ax, where V0 = AL the initial volume. The we can write the equation as pV^gamma = p0V0^gamma. Substituting for V and putting p to the RHS of the equation. Then you get an ode which does not contain the position of the piston. Anyhow, I agree with your approach.
Now a new problem arise that I wanted to control the velocity of the piston using e.g. a linear motor. This means that the force should be considered. As @Igor suggests, by considering the piston as a single degree of freedom system (i.e. mass – damper system). The damper may be considered as a friction b/n the piston and the cylinder. However, to make the problem simple I would like to consider no HEAT dissipation during the process.
@ Igor,
You said at on your comment " but this relationship alone does not suffice to describe the dynamics of the system. "
which relationship do you mean? The one that you describe or Sverre's derivation?
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