Modified Bernoulli equation

[tim jones]

Supercannon II is an interesting test case, since pressure and everything else is non-constant. have you tried using an iterative (spreadsheet) method? It may provide a decent reality check.

What kind of Mach numbers are you seeing at the nozzle?

[Ben]

I've done some analysis with spreadsheets, but it's not ideal because the pressure was dropping and not all of it makes perfect sense. I "smoothed out" the data for the graphs I posted in the most recent Supercannon II thread.

This summer I would like to attach Supercannon II directly to the air compressor for regulated tests.

I could calculate some Mach numbers, but I'm a little busy with work at the moment. You're free to look at the slightly compensated plots I just referred to and figure something out for yourself though: http://forums.sscentral.org/37487-post5.html

The nozzle is 1/2 inch in diameter.

Edit: I just did the math. At 100 PSI it's about 21.84 m/s (at the nozzle) which is 0.0634 Ma based on the data we took that day.

[Ben]

I took a few minutes to redo the equation with the unsteady Bernoulli equation combined with minor loss coefficients. The differential equation is relatively simple, but still fairly complicated. While I have no idea if solving it would be difficult, solving it by hand could introduce some errors because I'm not perfect, so I'm going to solve it with MATLAB tonight and then see how feasible it is. If it is indeed realistic, that's perfect. If it just needs another coefficient on top of it, that's fine too. As long as it makes sense for the most part then it's perfectly fine.

[Ben]

I've partially solved this equation by hand. It still needs some work, but I've made good progress. I'm going to check my work against MATLAB later, but I shouldn't have made any mistakes so far (or so I hope). Once this is done I can make a spreadsheet that should make predicting the performance of an air pressure water gun trivial if you have all the important data.

[Ben]

I have a new respect for math. One of the simplified solutions to this equation not counting initial data is over 1200 characters long and that's one of the short solutions with all of the large parts that are constants as new variables...

After examining this, while it's obviously possible to solve, an implicit solution might be easiest. I don't know how to do that in spreadsheets, but I'll investigate.

[Ben]

The plan at the moment is to figure out how to do implicit solutions in a spreadsheet. I'm not having luck figuring that out, so I might post some code for a variety of computer math programs. I'll probably use Scilab.