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100 g Payload – Lunar Descent – appendix A Section 5.4.2, Page X Propulsion System Design Propellant and Propulsion System Selection Selection of the Lunar Lander Propulsion system began with a preliminary propellant/propulsion system type vs. required propellant mass study to determine the relative masses between the available choices. Using the ideal rocket equation as a starting point, and assuming a vehicle inert mass fraction, finert, along with a mission ∆V and payload mass, mpay it is possible to derive an equation that computes the required propellant mass for a given propulsion systems specific impulse, Isp.Humble (5.4.2-1) Assuming a payload mass of 85 kg, minimum and maximum finert values of 0.1 and 0.25 based off of historical data and a ∆V of 1950 m/s, Eq. (5.4.2-1) was used to compute the two dashed lines of propellant mass vs. Isp for the two inert mass fractions shown in Figure (5.4.2-1). 240 220 200 180 160 Mono-Prop Propellant mass [kg] 140 Hybrid 120 100 80 60 Bi-Prop 40 20 150 200 250 300 350 400 450 500 550 600 Isp [s] Fig. (5.4.2-1) Propellant mass required vs. propulsion system Isp (Thaddaeus Halsmer) Author: Thaddaeus Halsmer 100 g Payload – Lunar Descent – appendix A Section 5.4.2, Page X Because the inert mass fractions were chosen as historical limits, when data points were plotted for various propulsion systems empirical Isp values, they fall within this envelope. The result of propellant mass vs. Isp study shows that an exponential propellant mass penalty is paid for choosing a propulsion system with a lower Isp. Assuming that lunar descent should conclude with a fully controlled soft landing leads to the requirement of an engine with variable thrust. Varying the thrust of a rocket engine is done by changing the mass flow rate of propellants as seen in the vacuum thrust equation where Isp is assumed constant, is the propellant mass flow rate and go is the Earth gravity acceleration constant. Mono-propellant, bi-propellant and hybrid propulsion systems were all initially considered because they fulfill this requirement by varying the propellant mass flow rates using flow control devises in the propellant feed systems. Both mono-propellant and hybrid propulsions systems have the added advantage of being throttled by controlling only a single fluid mass flow rate where in a bi-propellant engine both oxidizer and fuel flow rates must be controlled simultaneously to maintain the correct oxidizer to fuel ratio. An additional constraint on the propulsion system selection was that there are currently no commercially available rocket engines sold that meet the variable thrust requirement and scale needed for this mission. This made the cost of development of the chosen propulsion system another significant factor. Because of the complexity of bi-propellant rocket engines and their associated feed systems it is not realistic to suggest that such a system could be developed with sufficient reliability within the time and cost constraints in this project, therefore the bi-propellant option was eliminated as a possibility on this basis. Based on expert advice it was determined that an H2O2 / Polyethylene Radial Flow Hybrid propulsion system of appropriate scale could be developed within one year and Author: Thaddaeus Halsmer 100 g Payload – Lunar Descent – appendix A Section 5.4.2, Page X Heister for less than $250000. Using CEAGordon we determined that this system could easily achieve a vacuum Isp >300 seconds. Using this information and the associated propellant masses shown in Fig. (5.4.2-1) the H2O2 / Polyethylene hybrid propulsion system was selected over the mono-propellant system because of its significant propellant mass savings and low cost of development. Radial Flow Hybrid Rocket Engine Overview Traditional hybrid rocket engines contain a solid fuel grain in the combustion chamber and liquid oxidizer is injected into axial ports in the fuel grain, however, this configuration leads to engines that have a high Length/Diameter and are not well suited to our compact landing vehicle. Because of the dimension constraints an experimentally proven radial flow configuration was chosen that allows the combustion chamber to have a much larger diameter and shorter length.Heister Fig. (5.4.2-2) Radial Flow Hybrid engine configuration (Thaddaeus Halsmer) Figure (5.4.2-2) presents the architecture of a radial flow hybrid rocket engine. It also shows how the oxidizer is injected around the combustion chamber perimeter between the two Polyethylene fuel plates, with the combustion gasses exiting through the hole in the lower fuel grain plate and into the nozzle. Chamber Pressure & Feed System Selection Space engines enjoy the benefit of being able to achieve high efficiency at relatively low chamber pressures and one of the benefits of this is that the entire oxidizer feed system mass can be reduced because of the low operating pressure. CEA was initially used to do a preliminary Isp sensitivity study to reductions in chamber pressure and it was found that Author: Thaddaeus Halsmer 100 g Payload – Lunar Descent – appendix A Section 5.4.2, Page X benefits of reducing the chamber pressure far outweighed the minimal change in Isp. A key assumption that was made in determining the chamber pressure operating range was that the engine would be designed for a 10:1 throttle ratio which has been experimentally proven to be possible without extraordinary difficulty.Sutton Because the chamber pressure of the engine changes by almost exactly the same factor that the thrust is throttled, we assumed that the change in chamber pressure would be 10:1 for an engine with 10:1 throttle ratio. Based on this the maximum combustion chamber pressure, at maximum thrust, will be 10 times the minimum possible chamber pressure at which steady combustion can be maintained. While the exact minimum chamber pressure will need to be experimentally determined for this particular design hybrid engines using the same propellants have demonstrated steady operation at chamber pressures as low as 0.21 MPa therefore that will be the assumed minimum chamber pressure for this design making the maximum steady state chamber pressure 2.1 MPa. Based on these pressures the optimum feed system architecture is a gas-pressurized propellant feed system. This is because of its ability to supply the required oxidizer injection pressures without the added complexity of pumps. Mean Isp and Combustion Chamber Sizing Sizing of the combustion chamber for the Radial Flow Hybrid is based on the dimensions of the fuel grains shown in Fig. (5.4.2-2) and these are dictated by the fuel grain regression rate, the desired oxidizer to fuel ratio and thrust. Hybrid engine fuel regression rates can be modeled using Equation (5.4.2-2) (5.4.2-2) where r is the fuel regression rate in m/s, Go is the total oxidizer mass flux in kg/m2-s, and a and n are empirically derived constants. By taking Eq. (5.4.2-2) and multiplying it by the fuel grain density ρf and the burn area Ab, the equation for the fuel mass flow rate is given as Equation (5.4.2-3). (5.4.2-3) Author: Thaddaeus Halsmer 100 g Payload – Lunar Descent – appendix A Section 5.4.2, Page X Fig. (5.4.2-3) Fuel grain dimension definitions (Thaddaeus Halsmer) Using the variable definitions in Fig. (5.4.2-3) along with the definition of Go and substituting them into Eq. (5.4.2-3) yields an expression for the fuel mass flow rate as a function of the fuel grain/combustion chamber geometry (5.4.2-4) Assuming that all terms in (5.4.2-4) are constant except h, which is changing at a rate proportional to the fuel regression rate r, taking its inverse, lumping all constant terms in to a single term k, and multiplying by the oxidizer mass flow rate gives Eq. (5.4.2-5) (5.4.2-5) which is an expression for how the engines oxidizer to fuel ratio changes as the fuel grains regress. In Eq. (5.4.2-5), n is a constant that has been empirically proven to be approximately 0.8. From Eq. (5.4.2-5) we can see that the O/F ratio will change as the engine burns and this directly affects the engines efficiency or Isp. Because h is an increasing function of time the O/F ratio will be increasing and because maximum Isp occurs near the stoichiometric O/F ratio it is necessary to size the fuel plates so that initially the engine is fuel rich and as the oxidizer fraction increases through the burn time the mean Isp is maximized. Author: Thaddaeus Halsmer 100 g Payload – Lunar Descent – appendix A Section 5.4.2, Page X Using CEA integrated into a Matlab script along with an empirical fuel regression rate of Caravella 0.000559 m/s and Eq. (5.4.2-5) it was possible to iteratively find an initial O/F ratio that maximized the mean Isp over the burn time. Figure (5.4.2-4) is an example of the program output, over a 200 s burn time, and it demonstrates the significant O/F shift along with the corresponding changes in Isp. 340 330 320 310 Isp [s] 300 290 280 270 260 0 2 4 6 8 10 12 14 16 18 20 O/F ratio Fig. (5.4.2-4) Isp vs. O/F ratio for 200 s burn time (Thaddaeus Halsmer) Through the previously discussed analysis an initial O/F ratio was found. Using Eq. (5.4.2-3) along with the initial O/F ratio, Isp, and thrust the initial dimensions of the fuel grains are determined and from these the dimensions of the combustion chamber computed. Nozzle Sizing Analysis For rocket engines operating in space, as the nozzle area ratio Aexit/Athroat is increased, their Isp also increases. However, the cost of increasing the Isp with a increase in nozzle area ratio is the added mass and size of the physically larger nozzle. Because an increase in Isp reduces the required propellant mass there exists theoretically a optimum nozzle mass and corresponding area ratio. To make a preliminary examination of tradeoff between the nozzle area ratio and the propellant mass required for lunar descent a Matlab script was written that used CEA to compute the Isp of the engine for a given area ratio along with the corresponding nozzle mass and propellant mass. Thrust and burn time were kept constant during this analysis and only the nozzle area ratio was varied. The Author: Thaddaeus Halsmer 100 g Payload – Lunar Descent – appendix A Section 5.4.2, Page X mass of the nozzle was determined using an empirically derived equation that estimates the mass of a carbon phenolic hybrid engine nozzle.Humble Inputs for this equation are the nozzle area ratio , and total burned propellant mass mprop. Nozzle + Prop mass [kg] 128 Pc = 1.72 MPa, 2000N thrust 127.5 127 126.5 126 50 100 150 200 nozzle area ratio Fig. (5.4.2-5) Mass penalty vs. nozzle area ratio trade (Thaddaeus Halsmer) Figure (5.4.2-5) is an example of the output generated by the script. It shows that the minimum combined mass of the nozzle and propellant occurs at an area ratio of approximately 150. Based on this analysis an area ratio of 100 was chosen because there is only a approximately 0.5 kg mass increase by using the smaller nozzle area ratio and its dimensions were near the maximum that could be integrated into the vehicle. The throat area of the nozzle is computed as a function of the engine thrust F, coefficient of thrust Cf, and the chamber pressure Po using Equation (5.4.2-6) (5.4.2-6) Author: Thaddaeus Halsmer 100 g Payload – Lunar Descent – appendix A Section 5.4.2, Page X where Cf is computed using CEA and the thrust correction factory was assumed to be 0.96. Sutton Once the nozzle area ratio is known and along with the throat diameter the exit diameter is easily computed and the length of the nozzle estimated using Eq. (5.4.2- 7) which was derived from empirical data for 80% bell contour nozzles (5.4.2-7) where Dt is the throat diameter of the nozzle derived from Equation (5.4.2-6).Sutton Author: Thaddaeus Halsmer

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