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Computer
Simulation
I divided the water rocket simulation into four phases. The first phase is from launch to water "burn out". This is the time that water is coming out of the nozzle. During this phase the thrust is calculated with incompressible fluid dynamics theory using the Bernoulli Equation. (I do assume steady state flow during the entire duration of phase one. This is not quite accurate for the first 0.0007 seconds of thrust while the water establishes steady state flow. The difference in maximum altitude is measured in hundredths of an inch.) The mass of the rocket is changing as the water is expelled. Even though the water is slowing down during phase one (due to the air pressure decreasing), the acceleration is increasing because the rocket mass is decreasing. (Got all that?) Phase two begins as the last of the water is expelled. There is still air pressure in the bottle that forces air through the throat. If the air pressure is above the critical pressure ratio, (approximately 2.1 for air) there is a sonic shock wave established in the throat of the bottle. This is called sonic flow (or choked flow) in the world of compressible fluid dynamics. The mass flow rate can be computed knowing only the air pressure; it is completely independent of surrounding (ambient) pressure. Phase three begins just as the air pressure reached the critical pressure where sonic flow at the throat can not be maintained. The compressible fluid dynamic equations during this phase are different. This is "unchoked flow". Here, the ambient air pressure will have an effect on the mass flow rate. (The pressure difference due to altitude is not that great so it is neglected in my model. Again, no significant change in the performance prediction.) Phase three ends when there is no more air pressure (differential) in the bottle. Phase four is the coast up to maximum altitude (apogee). The mass of the bottle doesn't change and only gravity and aerodynamic drag slow the bottle down. (I do assume that the bottle continues to fly straight. If you don't put fins on the bottle or it is not spin stabilized, the bottle will tumble in phase three.) Some of the other programs I have seen don't take into account phases two or three. They just let the bottle coast (phase four) as soon as the water is gone. Other programs approximate phases two and three by giving the bottle a velocity increase based upon the remaining pressure and thus remaining impulse in the bottle. That is a better technique. My program takes as much into account as is reasonable (I've pointed out a few of the simplifying assumptions). The above discussion lets you know why I think my program is best. My program is not yet available to the public. I am still working on a graphical subroutine for plotting the key parameters. I have a summary table but no fancy graphics. (Who would want just the answers without the pictures anyway?) When I have completed the program and tested it, I am thinking about putting it up on this web page for public download. Leave me feedback to let me know what you think. |
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