Home made 3d printed rocket nozzle

One of the most important parts in a rocket engine is the nozzle. A well designed nozzle will provide the optimum expansion of the exhaust gases and maximize thrust. I am working on a home made rocket, and decided to do a few tests with a real nozzle just to get a real feel of how these devices work.

So for starters, how does a rocket nozzle work and what is it for? The nozzle is basically a tube that redirects the combustion gases into one direction so to push the rocket. While doing so, it also accelerates the gases and lowers their temperature. And all of that without a single moving part. Rocket nozzles achieve this because they belong to the category of Convergent-Divergent Nozzles, or de Laval Nozzles, which take advantage of the properties of gases at supersonic speed. I won't get into much detail about the thermodynamics of ConDi Nozzles, as there are many resources that explain them perfectly (https://en.wikipedia.org/wiki/De_Laval_nozzle).
The important point here is how to design the nozzle, and for that there are a few things we need to know:
- The properties of  the gas that will flow through the nozzle (i.e. air), or gamma-
- The amount of air we can provide per second (mass flow).
- Our working pressures.
About air, all we need to know is the heat capacity ratio, which happens to be just about 1.4.
The amount of air is a bit more tricky. In my case it is limited by the air compressor I use to power the system. It's not so important to know the exact amount of air that will flow as it is to be sure that our nozzle becomes the limiting factor. In order for the air to go supersonic (locally), it needs to choke the nozzle. That means that you must be able to supply air enough to saturate it, or conversely, that the nozzle must be small enough to saturate with the air you can provide. For this reason, the simplest thing you can do is to design the nozzle with a throat smaller than the smallest area of your air feed system. Just measure the ducts of your feed system and choose a smaller size for the throat. My throat is about 4 mm diameter because the smallest duct of my air compressor has a diameter of 6mm. It could actually be even a bit bigger than the duct and still choke (for a bigger area, a smaller pressure will be able to choke the same mass flow) due to the pressure losses, but this way we have some margin.
Finally, the working pressures (along with gamma) will define the expansion ration of the nozzle (the only thing we are left to know to fix its geometry). Expansion ratio, the ratio between the area at the end of the nozzle, and the area at the throat, follows the following formula:

Taking into account that Pe, the expanded pressure will be equal to ambient pressure (1.013 bars) and that my air compressor can give up to 8 bars, that results in an expansion ratio of about 4, so the expansion radius will be about twice that of the throat. I will build my nozzle with a contraction angle of about 30º and an expansion angle of 15º.

As you can see in the pictures, the nozzle is a simple revolution solid.

Notice I let a pretty big input hole. That is because I use 3/8" plumbing to feed the system. That is conventional plumbing that you can buy in any hardware store and is pretty easy to work with and to seal. I also printed a small cap to transform a PVC tube into an adapter to hold the nozzle in place. This way I can put it on top of a weight scale to measure thrust. The picture below shows my poor man's engine test bench, where I can measure pressure vs force.

I didn't expect much of this at first, but the results impressed me. The whole "engine" (the chamber plus nozzle) weight less than 10 grams, and it delivers more than 200 grams of thrust. A force to weight ratio of more than 20. Not bad for 10 grams of plastic.