The key temperature would be the peak water temperature in the systems. That would control the vapor pressure.
Hi Machinbird,
The peak temperature would indeed define the local max pressure. But if the evaporated volume of water is small compared to the overall volume, the overall presssure throughout the system would be much lower, more a function of the average temperature in the system.
Regarding the Flash Boiler:
Yes indeed, it won't generate indefinite pressure, but in order to ensure the rapid expansion I would expect the temperature (and thus the local /instantaneous pressure) to be much higher than the average required for expansion and moving of the catapult. So if the catapult won't move the pressure should rise to some extent until the temperature of the whole steam in the system is identical to the boiler temperature. Then there would be equilibrium.
I don't think these catapults work at equilibrium of the whole system as in that case the force exerted on the catapult would decrease wgeb expanding/moving