For the Moment of Inertia rocket I’m going to be using a Cesaroni Vmax motor. In particular, the 29mm 3-grain H410. 410 Newtons is a lot for this size motor: 42 kgf, or 92 lbf. It makes you wonder, how do they get that much thrust out of such a small motor? It has to have either a high chamber pressure, a large throat, or both.
With a high chamber pressure you need a big expansion ratio to be properly expanded. With lower chamber pressure and a large throat the expansion ratio would be lower, but multiplied by the large throat size the nozzle exit area still comes out large. The thing is, the nozzle exit diameter is limited by the form factor of the rear closure, which I’m guessing was designed before they started making Vmax motors.
I’m betting that the nozzle design is size constrained such that Vmax motors are underexpanded. To know for sure, I need to know the chamber pressure, throat area, and exit area. I ordered one of the motors so I can measure the throat and exit areas. For the chamber pressure I’m going to have to estimate.
I fiddled around in RPA until I got an engine model that matched pretty well the numbers I do know. The thrust curve for the motor isn’t flat so the chamber pressure changes over the run, but I think using average thrust to get an average chamber pressure is a good strategy here.
Chamber pressure came out to 870 psia, and sure enough, the nozzle is properly expanded for one and a third atmospheres.
What can I do with this information? Well, my plan is to make a graphite nozzle extension built into my motor retainer and get extra thrust. This modification will make the motor be experimental so I’ll have to wait until I get level 2 certification to fly it.
The place where I launch is at 5000 ft elevation, about 0.83 atmospheres ambient pressure. Proper expansion would be a nozzle area ratio of 8.8 instead of 6.25. That would mean extending the exit diameter from 0.75″ to 0.89″. It should get 1.6% more thrust, or 417 instead of 410 Newtons.
I’ve been thinking about this a little more. The amount of additional expansion in the nozzle extension is pretty small. Basically, the existing nozzle expands from 59 atmospheres to 1.33, and the nozzle extension only goes from 1.33 to 0.83. So the benefit of the nozzle extension is correspondingly small. This raises a few questions.
How far off does my estimate of chamber pressure have to be to make the existing nozzle properly expanded in the first place? In my RPA model I had to drop the chamber pressure to 650 psi to make it properly expanded for sea level. I can’t see any way that the motor would make that much thrust at 650 psi. I’ve measured the throat diameter at 0.3″ so the thrust coefficient would have to be 2, and the RPA model says it should only be 1.46. I could be wrong about the chamber pressure and it’s not as underexpanded as I think, but I bet it’s not that low.
Is it worth doing the nozzle extension? Well, maybe, maybe not. If you are looking from an objective performance standpoint it depends on how much you care about pushing for the last little bit. You also have to balance the thrust increase against any mass increase to add the nozzle extension. I’m planning on making the extension an integral part of the motor retaining ring that I need anyway so it shouldn’t add much weight, but it might add a little if it has to be bigger than it would otherwise need to be. However, I’m really doing this because it’s something cool to do, and the nozzle extension is definitely a cool part of the project that I want to do.
How am I going to know if it’s working? I would have to do static tests with and without the nozzle extension. The difference is small enough that I’d have to test a number of motors to get a statistically significant result. In certification test data that I’ve seen they generally test at least three and come up with a standard deviation around 1% or under. I suppose if I come up with two groups at least one standard deviation apart I can conclude that it’s helping, but I’d probably need three standard deviations to be able to say how much it’s helping. With each reload costing $20 it’s something that I might do some day, but It’s not on my to-do list.
I also realized that I don’t need to make the nozzle extension out of graphite. The exhaust has cooled somewhat by the time it gets to the end of the nozzle, and the duration that it will be exposed to the hot exhaust is so short that I’m sure anodized aluminum will do fine.