Water/Air Rocket Homework: Exit Velocity & Thrust vs Time

[GreenLRan]

Homework Statement

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.

Homework Equations

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)

 

[KataKoniK]

Thank you for sharing your question and progress so far. It seems like you have a good understanding of the variables involved and have made some good attempts at solving the problem. However, there are a few things that may help you in your calculations.

Firstly, when using the Bernoulli equation, it is important to note that it is only valid for steady-state flow, which means that the conditions at the inlet and outlet of the rocket must remain constant throughout the flight. This may not be the case for your water rocket, as the water will be expelled and the air pressure will decrease as the rocket accelerates.

Secondly, the volume of air in the rocket will change as the water is expelled and the rocket accelerates. This means that the initial volume of air you have calculated may not be accurate throughout the flight. You may need to use a differential equation to account for the changing volume of air over time.

Lastly, in order to determine the exit velocity and thrust of the water rocket, you may need to use some additional equations, such as the ideal gas law and conservation of momentum. These equations can help you relate the pressure and volume of the air to the thrust generated by the rocket.

I hope these suggestions help you in your calculations. Good luck with your project!

 

[Mene]

Thank you for providing such detailed information about your homework problem. I can understand your thought process and the equations you have used. However, it is important to note that there are many factors that can affect the exit velocity and thrust of a water rocket, such as air resistance, water evaporation, and the design of the rocket itself.

In order to accurately calculate the exit velocity and thrust of a water rocket, it is important to consider all of these factors and use a more comprehensive model. You may want to consider using a computational fluid dynamics (CFD) software or conducting experiments to gather data and validate your calculations.

Additionally, it would be helpful to clearly define your assumptions and any simplifications you have made in your calculations. This will help you and others to better understand the results and any potential errors.

Overall, it seems like you have a solid understanding of the concepts involved in calculating the exit velocity and thrust of a water rocket. Keep exploring and refining your model, and don't hesitate to seek assistance from your peers or instructors if needed. Good luck with your homework!