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I want to do a simple (physically plausible but not physically accurate) simulation of a gas in a cylinder as it works on or is worked on by a piston. Wikipedia gives a good example of an adiabatic process which closely fits my situation, but I want to account for heat exchange between the cylinder and the gas inside (not adiabatic).

I need a way to express my simulation with differential equations (will be solved numerically) with respect to time, where I know the volume at any given time and the change of volume with respect to time at any given time. I cannot fully express the equation for temperature with respect to time, but I know, for any given moment in time, the temperatures of the gas and cylinder and physical parameters like thermal conductance which will allow me to express at least the change in temperature with respect to time due to heat loss. A complication to this is that I plan to vary the thermal conductance by displacement of the cylinder, as more or less of the cylinder's inner surface is exposed to the gas. Since volume of the gas and displacement of the cylinder are proportional to each other, this isn't really so bad.

Can I simply express the change in pressure due to time as the sum of change in pressure due to change in volume (adiabatic equation) plus the change in pressure due to change in temperature (by differentiating ideal gas law)? How would you go about solving this? Am I just keeping track of too many variables?

To clarify, I know:

  • Volume and change in volume with respect to time at any given time
  • Change in temperature with respect to time at any given time
  • Initial conditions for all variables

And I want to find:

  • Pressure
  • Temperature
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  • $\begingroup$ At any moment the ideal gas law should hold. It sounds like you always have everything except. Pressure so you can solve for it - although you say you want temperature, above you say you know it...? Or rather you know the change in temperature so you just integrate with respect to time. But maybe you didn't explain yourself very well? $\endgroup$
    – Floris
    Sep 15, 2014 at 2:02
  • $\begingroup$ Differentiating the Ideal Gas Law should be valid so long as you consider the IGL applicable to your simplified model. How are you modelling the thermal boundary conditions? Constant temperature, heat-flux, or does the wall have a finite thermal mass? $\endgroup$
    – Bryson S.
    Sep 15, 2014 at 2:02
  • $\begingroup$ Finite thermal mass. From Floris's confusion, I am thinking I should do the math and show my solution, then see if people think it makes sense. I'll be back online in a few hours to do that. $\endgroup$
    – Void Star
    Sep 15, 2014 at 14:59
  • $\begingroup$ Or maybe in a few more hours... sigh... life... $\endgroup$
    – Void Star
    Sep 16, 2014 at 7:22

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