The TV($\gamma$-1) = constant formula appears to be unbounded, which begs the question how high can we go?
Let's say you have a theoretical fire piston, and want to increase the adiabatic temperature as high as possible. What are the maximum temperatures achievable in such a process? How would you do it?
If we play around with some parameters of the thought experiment:
- Single-use situation where the equipment is destroyed in the process
- Multiple-use situation - perhaps with tungsten or a refractory cylinder?
- How would you increase the temperature of the compressed gas?
- How would you increase the longevity of the system? Some kind of magnetic confinement system?
- What practical factors limit the maximum temperature one can reach?
Let's set aside cost for now in this thought experiment.
I guess what I'd do to increase the temperature are:
Increase the size of the energy reservoir to do the work to compress the gas. If it is chemical energy (say based on a one-off gun barrel driving the fire piston) then increase the quantity of propellant. If it is a flywheel, increase the rotational kinetic energy.
Increase the efficiency of energy transfer between the energy reservoir and the piston. For example it seems mechanically coupling a flywheel to the piston would be easier than one based on chemical batteries. But still, how though? How would you suddenly connect a flywheel spinning at thousands of RPM to a linear piston without the shock of engagement destroying the apparatus?
Have a piston with lots of inertia - so a heavy tungsten piston head which is also advantageous for its high melting point, with perhaps an osmium body. For a one off system, a material with a high vaporization energy may be preferable to one with a high melting point.
Use an inert gas that does not disassociate, such as helium, neon, argon or xenon. Would a heavy atomic mass noble gas be better?
Presumably materials can be exposed to temperatures exceeding their melting or boiling points for very short times without substantial degradation, if their thermal conductivity and mass is high relative to the thermal impulse. Is this correct and is there a way of estimating how high temperatures a tungsten cylinder can be exposed to before it degrades within one cycle?
Geometry of the piston head. The change in volume with stroke distance is linear for a cylinder in a tube. If the piston head had a taper, and the tube's end had a countersink, would this cause a sharper compression event at the end, and therefore higher temperatures?
In terms of practical factors which limit the maximum achievable temperature, I think things like the material melting points, gas disassociation or even ionization energies, reflectivity of the interior (at very high temperatures radiative heat transfer to the cylinder is also an issue), size of energy reservoir, efficiency of energy transfer to the piston, inertia of the piston would be factors.
And the last question, of course, is how would you estimate the temperature-time history of the adiabatically compressed gas?