Introduction

For the ME140 Rocket Project, each team was tasked with designing a hybrid rocket to maximize either total impulse or specific impulse. Our team of four decided to compete for specific impulse. In the end our team places second in the competition with an Isp max of 166.0. This details our engineering process.

Fuel Grain Iteration 

From our project proposal, our team selected a fuel grain design known as the finocyl, which looks like a star or asterisk. An advantage of the finocyl over other fuel grain designs is cleaner burn and a larger area under the thrust curve. We discovered a key disadvantage of this design early on when trying to manufacture it. The  6-point design required that we fixture the fuel grain three different times, which was slow and imprecise. In addition to being inefficient, this fixturing process––combined with the milling operation––significantly changed the outer diameter and shape of the fuel grain. This led us to be out of specification for a number of fires. 

From here, we decided to simplify the design but preserve the finocyl concept. We moved from a 6-point design to a 4-point design, allowing us to fixture the part only once compared to the three times of the previous design. 

We used this 4-point design for our first fire and conducted it with the TA nozzle. We held off on manufacturing our nozzle until we had a good idea of the mixture ratios we could expect. From our first fire, we learned an approximate mixture ratio but found that the surface area of our design was too low to allow for adequate fuel burn. You can see fire data presented in Table 1 below and the fuel grain designs in Table 2. 

In our next fire, we added holes to this design to increase the surface area significantly but without dramatically changing the cross-sectional area. This was important as we wanted to maintain a high chamber pressure (~8 bar). We added 15/64” holes to the off 45˚ angles of the fuel grain which dramatically increased performance. A happy accident then occurred when inserting the fuel grains: the bottom section was misaligned by 45 ˚ from the top section, which inadvertently created a mixing chamber between the two halves. Adding a mixing chamber slows down the flow of the oxygen and promotes burning. From here, we looked to improve our specific impulse by increasing the holes’ size to increase the surface area while slightly increasing in the cross sectional area. 

In our third fire, we incorporated a nozzle. There were a couple of caveats: although the internal geometry of this nozzle was designed to our initial plan (described below), due to time constraints, we were unable to precisely machine the exit geometry such that the expansion ratio closely matched the data. The nozzle exit area was thus 1.705 in2 for an expansion ratio of 6.158.  This created an overexpanded nozzle which limited our maximum specific impulse. 

This meant that in our final fire we could precisely tune the exit area to the expansion ratio target given by our previous fire’s data. This was extremely helpful. For our final fire, our nozzle had an expansion ratio of 2.11.

Nozzle Design

Our nozzle was designed based on research done about atmospheric nozzles in Modern Compressible Flow [1]. Here, Anderson recommends geometry for the converging section was a 45˚ angle. We then set the nozzle inlet area to be the max allowed by the rocket test stand. To calculate the throat area we use Eqn. 1 below. 

From here we add additional length, 0.2 inches, to the throat to promote laminar flow along the transition from the converging to diverging section. Based on Solidworks flow simulation results, this proves to be beneficial. Because of the relatively sharp transition from the converging section to throat section, the flow continues to be compressed and speed up beyond the sharp edge. Adding this section not only makes the flow settle at sonic conditions before being accelerated in the diverging section, but it also allows for some error in our throat area. If the area is not exactly small enough to be choked, we ensure that it gets to choked conditions in this middle section. 

Finally, in designing our diverging section, we chose an angle of 15˚ as a good mix between different performance metrics [1]. The length of this section was determined via the ideal expansion ratio mentioned above in II.A. If we were to design this nozzle again, we would experiment with added curvature to the diverging section. Perhaps rapidly expanding early on and then a more gradual ~10˚ section as it approaches the ends of the nozzle. This more closely replicates the polynomial designs of some of SpaceX’s most recent rocket engines [2].

Final Results

Unfortunately on our fourth and final fire, the lab equipment was not behaving as expected. The lab data showed a high degree of erraticity. Nonetheless, we got a reading that marked our specific impulse at 165.14 N · s/kg, the best of all four of our fires. 

Another key aspect of our investigation concerned regression rate. The regression rate refers to the rate at which the fuel comes off the chamber’s sides and therefore how it mixes with the oxidizer. With this information, we might estimate the required wall thickness of our fuel grain to approximate how the fuel will burn and maintain an optimal mixture ratio during our fire. We used the regression rate to estimate a theoretical fuel mass flow rate of the fuel (Eqn 3), which we compared to the average mass flow rate from our fire (Table 1), excluding the fourth fire due to the inaccurate sensor readings. We calculated this based on a formula found in Boardman [3] which is shown below in Eqn 2. 

Here, G is the oxidizer mass velocity,  a and n are constants found in literature [4], ṁo is the oxidizer mass flow rate, ṁf is the fuel mass flow rate, Ac is the cross-sectional area, and As is the internal surface area of the combustion port. 

Another key data point was the c* efficiency. This is defined as the experimental c* at a particular mixture ratio compared to the c* for the ideal scenario. The data is presented in Table 1 above. We see values around 88% for our first three fires (excluding the anomaly of the last fire). The theoretical c* values represents an adiabatic process, with 100% combustion efficiency, and isentropic flow to the nozzle. Our fires are inherently not perfect, however, values close to 90% are typical of a well designed rocket.

Key Takeaways

In the process of this project, we learned many interesting things about rockets. For one we discovered the importance of machining process. When machining our fuel grains we had a lot of issues with shrinkage and asymmetry. This resulted from us clamping the fuel grain in the vice on the mill and in the chucks of the lathe. Additionally, when milling the internal design, the heat of the drill bit changed the internal geometry enough to go out of spec. We got around this mostly by machining the outer diameter in two stages. The first stage got it to approximately 15 thousandths of an inch over specification, after which we would mill the internal design. Lastly, we would turn the outer diameter precisely to spec. This new process worked much better for our team.

Another key takeaway concerned the fuel grain design. In speaking with other teams, a key trend emerged concerning the design of the central fuel grain hole. It seems as though the central hole area most directly affects the chamber pressure. The oxygen flow appears to be radial in nature with the preponderance of the flow in the middle. Implementing a smaller hole in the middle will thus dramatically increase the pressure in the system. This proved true for our designs. As we moved from a large central plus sign to a smaller plus sign and smaller holes, our performance increased. Teams with small hole sizes, particularly in the middle had chamber pressures at or exceeding the max allowable. Potentially implementing a smaller diameter for the plus sign slots could achieve this, although balancing this objective with manufacturability could be difficult.

References 

[1] Modern Compressible Flow, 2nd Edition, 1990, by John D. Anderson Jr.

[2] Merlin Engines, Space Exploration Technology Corporation, 2017. https://www.spacex.com/news/2013/03/26/merlin-engines 

[3] Boardman, Terry A., Alan Holzman, andGeorge P. Sutton. Hybrid Propellant Rockets, Rocket Propulsion Elements Chapter 15, 6th Edition.
[4] Zilliac, Greg and Karabeyoglu, M. Arif, Hybrid Rocket Fuel Regression Rate Data and Modeling, NASA Ames Research Center, Moffett Field, 12-July 2006. 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit