A.7.2.1.1 Propulsion Assumptions1
A.7.2.1.1 Propulsion Assumptions
Due to the short time span and the complexity of ourdesign task, many assumptions were made to ease our calculations. Our assumptions, while not entirely accurate, still lead us to believe our final numbers are reasonable.
During calculation of the propellant, we assumed for the first stage for a ground launch, sea level specific impulse would be used. Specific impulse increases as the launch vehicle rises so the sizing calculations create more fuel for the first stage than what is needed. For the second and third stage, we assume that the launch vehicle was out of the Earth’s atmosphere. Therefore, vacuum specific impulse is used to calculate the amount of propellant needed for those stages. Historically, many launch vehicles are within one percent of the Earth’s atmosphere at first stage burn out.1
For a balloon launch, all stages are calculated at vacuum specific impulse due to the height of the balloon at launch. During launch, the balloon is within one percent of the atmospheric pressure and therefore the assumption of vacuum conditions is reasonable.
Another way we simplify our calculations iswe assume that thrust is constant. For calculation of the mass flow rate, the specific impulse we use to calculate mass is used in calculating exhaust velocity, and therefore mass flow rate is also assumed constant. For a first stage ground launch, the specific impulse is constantly changing throughout flight, but would require iterations with trajectory that are not feasible at this time. For a balloon flight, constant thrust and mass flow rate is reasonable since specific impulse is constant in vacuum.
During calculation of specific impulses, we use the mean of frozen and equilibrium flow. Historically, the actual specific impulse will lie somewhere between the two.1 The main difference between frozen and equilibrium flow is what happens after the throat of the nozzle. During frozen flow, the constituent products we assumeremains in product form throughout the nozzle.2 Constituent products consist of the molecules created during the process of combustion. During equilibrium flow, the constituent products will reform to find a balance between reactants (fuel and oxidizer) and combusted products.2 Neither of these methods allows for an accurate measure of specific impulse because one is too high and one too low, but a mean value is usually used.
To size the nozzle for each stage of our launch vehicle, we find a pressure ratio to prevent separation and avoid adding mass to the structure without sacrificing specific impulse.3 We assume constant chamber pressure and exit pressure equal to ambient pressure to creating fully expanded flow in the nozzle. Full expansion flow creates optimum thrust. Using the pressure ratio, the expansion ratio is found which specifies the size of the nozzle.
We also assume a nozzle shape. Because our launch vehicle is small, we need a small nozzle like the conical nozzle. Conical nozzles are simple in design and we assume it has a circular cross section. According to Humble, conical nozzles typically have half-angles that range from 12 to 18 degrees.1 In our analysis, we assume our conical nozzles to have a half-angle of 12 degrees. We pick the lower half angle because larger half-angles mean a greater thrust loss due to the inability of the flow to expand so quickly. By choosing the half-angle, it is reasonable for us to assume our nozzle has one hundred percent expansion efficiency.
References
1 Humble, R. W., Henry, G. N., Larson, W. J., Space Propulsion Analysis and Design, McGraw-Hill, New York, NY, 1995.
2 Heister, Stephen D., AAE 539 class., PurdueUniversity, West Lafayette, IN, 1/14/08.
3 Tsohas, John, AAE 450 class., PurdueUniversity, West Lafayette, IN, 2/18/08.
Authors:Dana Lattibeaudiere, Nicole Wilcox