Polaris Chamber & Nozzle

Updated 11/17:

Design Process

To design the chamber and nozzle, the specifications we already know include the chamber pressure, OF mixture ratio, chamber temperature, and mass flow, as these are design choices based on the tank and overall constraints, as well as the properties of the propellants. To find T0, R, and gamma, we used NASA CEA. 

For other values, such as contraction angle, expansion angle, charactaristic length, and contraction ratio, constraints were chosen based on literature or previous projects. For instance, 15 degrees for an expansion angle in a conical nozzle is standard and known to be close to the ideal value. For our design, a conical shaped nozzle was the best choice, as it provides almost as good performance as the ideal bell nozzle, but it is much easier to machine. The contraction angle is less specific in what is ideal, but 30-45 degrees is normal. Since our chamber is fairly wide compared to the throat (large contraction area ratio), we chose a more gradual angle to promote better gas flow, but this choice is not critical. The charactaristic length is the total volume of the chamber (injector to throat) divided by the area of the throat, which is basically a metric for the overall size of the chamber. This value needs to be high enough to allow enough space for the gas to mix and fully combust in the chamber. If this value is too small, we have non combusted propellant being expelled through the nozzle, resulting in lost potential energy and less efficient combustion. If the chamber is too large, than the fully combusted gases spend more time in the chamber, increasing the heat stress on the chamber components. The ideal value is usually found emperically, so there are ranges of accepted values for different propellant combinations. We observed other nitrous-ethanol projects with values between 20" and 300," which is a broad range to choose from. At this point, we basically chose a chamber length that would give roughly a 2 to 1 length to diameter aspect ratio in the chamber. This made the chamber roughly 6 inches long from the injector to the start of contraction. This gave a charactaristic length of 57.8". Furhermore, the values closer to 300" were on engines that did not go on a rocket, so it is unreasonable to assume that the amount of mass involved in such a large chamber is reasonable for a flight application. Lastly, the contraction ratio is the ratio of the cross sectional area of the chaber to the area of the throat. If this ratio is too large, than the faceplate of the injector has a large area exposed to the hot gases of the chamber, resulting in large thermal stresses. Also, this can create gas flow problems in the chamber, and the stagnant gases remaining in the sides of the chamber further cause thermal problems. If the contraction area ratio is too small, we run out of room fitting the injector we want, especially in our design with impinging triplets. Additionally, there is a point where you can no longer assume that the velocity in the chamber is small (near M=0) compared to the throat. From the chart below, you can see that typical values are as low as 2 for large engines and up to 10 for very small engines. For Polaris, initially we had a very large contraction ratio to fit in the injector we wanted, and we determined that getting the value down to 10 was along the limits of what was possible. Essentially, the smallest chamber diameter we could accomplish was 3.25," and then we slightly increased the mass flow to attain a throat diameter that made our contraction ratio less than 10. This gave us a mass flow rate of 1.18 kg/s. 

References:

1. https://ocw-mit-edu.ezproxyberklee.flo.org/courses/16-512-rocket-propulsion-fall-2005/resources/lecture_3/ (Exit pressure)

2. https://www.eucass.eu/doi/EUCASS2017-474.pdf (L*, contraction ratio)

3. https://ntrs.nasa.gov/api/citations/19710019929/downloads/19710019929.pdf (angles, general)

4. https://yang.gatech.edu/publications/Journal/JPP%20(2008,%20Thakre).pdf(graphite)