# Convergent-Divergent / De Laval Nozzle Dimensions

I'm creating a super sonic ping pong ball cannon like the one recently shown on Mythbusters. I plan on using pressure rated steel pipe so it's as safe as realistically possible and showing it off at science and technology exhibits for kids. I think it will be a big hit and a great way to get kids interested in physics and science.

The device is pretty simple and well explained in this video here: https://www.youtube.com/watch?v=YYNCGZCul1Q

The key to the whole thing is ~100 psi on one side of a burst disk with a de Laval nozzle attached to a barrel on the other side. The barrel and nozzle is all in a pretty strong vacuum. The pressure chamber is using 4" pipe and the barrel is 1.5" diameter.

Since the most common usage for de Laval nozzles is in rocket motors, I ended up here. Reading and research has led me to believe that a convergent angle of 30 degrees is common in rockets. I've also read that the divergent angle of 15 degrees works but less works as well. What I can't find anywhere is the throat size.

With the goal to get the highest velocity possible using 100 PSI and 4" diameter on the convergent side and 1.5" diameter on the divergent side which is the best convergent angle, divergent angle, and most importantly, the throat diameter.

I've found equations that I think would provide this information but they are well beyond my ability to solve so any help would be greatly appreciated!

Thank you very much for the response! It was fascinating! I did actually build the entire thing and it worked great. The trick in the end was to use a large burst disk and pressurizing the chamber until it failed (generally 600-700kPa - that took a LOT of experimentation to get right BTW). It made reloading a bit of a pain but I improved that with some pneumatic bits to help with opening and closing the chamber. The nozzle ended up being a very simple one that I machined to some rules of thumb I found somewhere - not optimal but worked well enough in the end.

I plan on rebuilding the entire thing some day as a demonstration project to show off as it's actually pretty impressive to see in action. I'll absolutely use your information to improve it. The cannon makes quite a "crack" when the ping pong ball breaks the sound barrier. I used a rifle chronometer to measure speed and while it was quite variable (due to the burst disk inconsistencies) I would routinely get speeds in the 400-500+ m/s range. That's fast enough to actually "scorch" the faux fur on stuffed animals - in addition to blowing holes clear through them - with a ping pong ball in case anyone is curious :)

Thank you again for taking the time to respond!

• That's awesome. – 0xDBFB7 Apr 26 at 18:16
• This post landed in the "low quality" review queue, likely because "Thank you very much for the response" triggered some keywords (SE does not work like a forum, and is strictly Question-Answer). This, however, contains what the actual practical solution to the original problem ended up like, and the results. That's impressive, 6 years after the original post! All answers should answer the question, and this certainly does. – SE - stop firing the good guys Apr 26 at 20:10

This is really an underconstrained problem. A lot of it also belongs on physics.stackexchange, or the like, or is down to specifics of your design constraints. However, I shall summarize the basics that are relevant to rocketry and space travel here.

The ratio of the final area to the throat area is the "expansion ratio" of the nozzle, and is the main factor influencing how fast your propellant leaves. At the throat, the flow is mach 1, which is a speed that will vary by propellant composition and temperature.

From there, the nozzle expands, and the increased area causes the exhaust to accelerate due the weird way in which supersonic flows work. The final velocity is given by $$v_e$$ in the following: $$v_e = \sqrt{ \left( \frac{2 \kappa}{\kappa - 1} \right) \left( \frac{R_u}{M} T_c \right) \left( 1 - \left(\frac{p_e}{p_c}\right)^{(\kappa-1)/\kappa} \right) }$$ $$R_u$$ is the universal gas constant. AFAICT, dry air at $$T_c=293.15\text{ K}$$ (i.e. $$20^\circ\text{ C}$$) has an average molar mass $$M \approx 28.9647\text{ g/mol}$$ and a heat capacity ratio $$\kappa \approx 1.400$$. The chamber pressure is $$p_c=100\text{ PSI}\approx 689\text{ kPa}$$ (use metric please!), and you're expanding it by some ratio to get a new pressure $$p_e$$. If the ratio is infinity, the nozzle is "ideally expanded", and $$p_e=0\text{ kPa}$$.

As it happens, I wrote a handy web calculator which can be applied to this case. Expanding to vacuum (i.e., "perfectly") using the above parameters will give you an exit velocity of $$v_e \approx 767.5\text{ m/s}$$. In practice, you'll get lower than this because you can't make an infinitely big rocket nozzle (and if you did, it wouldn't push the ball).

Of course, exit velocity is only half the story; you also need to consider mass flow (which is calculated as the density of the air at that pressure, times the hole's area, times mach 1). If you make an teensy hole and expand the flow to almost that speed, very little mass of air will escape, and this will not be strong enough to push the ball to a high speed before the end of the pipe. Conversely, if the hole is too large, plenty of mass will flow, but the flow will not accelerate very much above mach 1 (for the above, about $$343\text{ m/s}$$).

One needs to work out how much the gas force from the exhaust accelerates the projectile, and the length of pipe required. Indeed, there are lots of ways to extend this analysis in various ways and to various degrees of accuracy, but they get more and more into the realm of general Physics, which I won't go into here because it's off-topic. At least now you have a way to calculate the velocity and mass flow, though.

TL;DR, though: my guess is that a reasonably sized hole will work fine, even if you don't hit upon the exactly optimal size (which would be very difficult to calculate, given all the variables, anyway). Good luck with your project, even six years later! Be sure to heed the safety warnings in the very video you linked; this sort of engineering can be dangerous.