What is the Math behind bottle rocketry?

[Astrofox]

I have a good question for the forums - what are the mathematics behind water rocketry?
Well, more specifically, how do I calculate the best water levels without having to guess and check with simulators?

It's just something I want to know more about. Also, since this is related to this topic, how do you calculate the force of drag without the drag coefficient? I need to know this so that I can build better, more aerodynamic rockets using simple materials.

Thanks!

[anoymous]

Well, you can't calculate the drag force without the coefficient. But you could try to measure it indirect:
If you know the initial speed (after the thrust phase), and you know the height of apogee, you can calculate the deceleration of your rocket. When comparing this to g, I think you can calculate the drag (and maybe the drag coëfficient?

[Astrofox]

So, I've come up with some more questions involving the Math behind bottle rocketry - How does one determine the Specific Impulse of a bottle rocket by using the volume of the pressure vessel, the pressure of the air inside the pressure chamber, and the volume water takes up?

[david2665]

Out of my own personal interest, I began looking into this.

Searching the web, brought up a few, quite insightful resources, mostly from educational institutions:

https://www.ohio.edu/mechanical/program ... ysis1.html which contains a link to an interesting .pdf file (nielson equations)
http://www.et.byu.edu/~wheeler/benchtop ... t_eqns.pdf
http://stemstates.org/assets/files/430_ ... %20430.pdf

and the most useful I found….
http://research.che.tamu.edu/groups/Sem ... CTIONS.doc page 21 onwards.


I plugged the equations into excel, iterating line by line over time, to come up with a mathematical simulator:


However, whilst the output looks good, I checked it against the Clifford Heath simulator, and the results were not the same….

Dave

[anachronist]

I just found this thread. Earlier this year I spent a few months developing my own simulator, which accounts for just about everything that can possibly be accounted for. I documented the calculations in 5 posts of my blog starting here: http://www.nablu.com/2017/07/the-water-rocket.html

To answer Astrofox's original question: For a given initial pressure (like 100 psi), there's an optimal combination of empty weight and total fill volume that achieves the maximum altitude, for a rocket with a given drag coefficient. Because all the mathematical relationships are nonlinear, the only realistic way to calculate this is by numerical analysis. So yes, you need a simulator.

-Alex