Air expansion in the water rocket

We study the thermodynamics of the water rocket in the thrust phase, taking into account the expansion of the air with water vapor, vapor condensation and the energy taken from the environment. We set up a simple experimental device with a stationary bottle and verified that the gas expansion in the bottle is well approximated by a polytropic process $PV^\beta$= constant, where the parameter $\beta$ depends on the initial conditions. We find an analytical expression for $\beta$ that only depends on the thermodynamic initial conditions and is in good agreement with the experimental results.