I am trying to determine the exit velocity and thrust of a water rocket as a function of time. The total rocket volume is .002m^3 (2 liter coke bottle) and 1kg of water and air initially at .35MPa. Therefore the initial volume of water is .001m^3 and the volume of air is .001m^3 as well.
T = 298K
Vol_tot = Vol_water + Vol_air (only unknown is M_air where Vol_air = M_air*R_air*T/(P_air))
therefore Vol_water = .001m^3 = Vol_air
I have calculated the initial velocity and exit area to be 22.36m/s and 2.38E-5 m^2 respectively.
The Attempt at a Solution
I am assuming isentropic expansion of air.
I started by using the Bernouilli equation Po = Pe + 1/2 * rho_water * Ue^2 (where Po is the total air pressure, Pe is assumed to be equal to the ambient air pressure (constant .1MPa)). The Po will change as a function of time, as will the exit velocity (Ue).
I started by trying Po(t) = Mair*Rair*T/(Vol_air(t)) = Pe + 1/2 * rho_water*Ue^2
where Vol_air(t) = Vol_tot - Vol_water(t)
and Vol_water(t) = Vol_water_initial - dVol_water(t)
and dVol_water(t) = dm(t)/rho_water = mdot*dt/rho_water = Ue*Ae*dt
Po(t) = Mair*Rair*T/(Vol_tot - (Vol_water_initial + (U_final*t_final - V_initial*t_initial)*Ae))
= Pe + 1/2*rho_water *V_initial^2
I solved this equation for V_final and used matlab to get a Velocity vs Time graph and I can tell it's wrong. It seems that there should be an exponential term in there somewhere.
Any help is greatly appreciated. Thanks!
( I was thinking about doing dPo/dt = d (const/Vol)/dt, however I'm unsure about the volume/time relationship)