Rocket Fluid Dynamics (10/13/21)

Rocket Engine Fluid Dynamics

Fluids is a critical subject in the field of propulsion engineering and I got by in the past by keeping a few formulas handy and consulting my textbooks a lot, but I’ve decided to go back and properly teach myself fluid dynamics. I now am confident enough to attack any science subject now that I’ve completed calculus BC. 

The basic formulas in fluid dynamics are Bernoulli’s equation, Torricelli's equation, the definition of pressure and behavior of laminar flow. Turbulent flow is what we’ll be dealing with on the rocket, but for now it’s best to treat it as just more chaotic laminar flow. Bernoulli’s equation is effectively the sum energy within a moving fluid, which includes the pressure, the gravitational potential and the kinetic energy of the fluid. In some Bernoulli derived equations we add something called the discharge coefficient, which is a unitless coefficient that corrects for the imperfection of surfaces and orifices as surfaces are not perfect. Typically cd values range around 0.4-0.98. Cd is always empirically measured. Toricelli’s equation is a simplified version of Bernoulli’s equation that is used to calculate the speed of a fluid exiting a tank approaching osmosis. Other relevant concepts are pressure which is just force over a given area and is constant throughout a fluid. These formulas predict incompressible fluid behavior and assume no viscosity or capillary action, which are typically also combined with the discharge coefficient. 

These concepts are relevant to my rocket in many ways, the first being the pressure fed design of the feed system, which operates on a conservative ideal gas law due to uncharacterized thermal behaviors. It’s essentially just a big tank of helium pressurizing the 2 propellant tanks, driving the propellant towards the combustion chamber as the fluids seek a lower pressure. Our mass flow drops off due to the dropping pressure in the tanks, which is because the gas is expanding to fill the place of the burned propellant, thereby slowing the flow of fluid out of the tanks. It’ll still all get burned but not as fast as the initial burn. In our plumbing design we have to be wary of the pressure drop incurred with certain design choices, so 90 degree fittings have a pressure drop with liquids but gases don’t care. We just have to be wary of the area delta which will incur a pressure drop. Things that incur p-drops include: material change, change in area, threaded surfaces and fitting interiors. Ideally the best rocket system would be a smooth straight tube from each tank outlet right to the propellant valve, smooth without any fittings which can be achieved with some extra machine work or by welding up the assemblies together. The propellants leave the tanks to then approach our propellant valve which opens letting them into the LOx dome and fuel inlet which lead to the injector which is an impinging design that operates entirely on shrinking the area to increase the velocity of the fluids (still subsonic) and shoot them at each other. One important thing to note is that when the area shrinks the pressure drops and the velocity goes up exponentially. This can sometimes set the plumbing on fire but we’ve picked large enough propellant lines so that this is not an issue. The injector shoots the fluids at each other at about a 60 degree angle, which through empirical testing has been determined to be one of the better impingement angles. The injector we went with was loosely based on the lr-101 injector bust scaled up by 50% and we added film cooling. Film cooling is essentially just shooting jets of fuel around the perimeter of the chamber to cool the combustion around the edges, throwing off the mixture ratio on the perimeter and lowering the hot side temperature of our ablative liner. During the injection process it is important that the propellants atomize so that they mix properly and this is achieved in our design with the use of many small elements. This drives our c* (combustion efficiency, pronounced CEE-STAR) to somewhere in the 94-98% range. C* goes up with higher combustion pressures which can be achieved in this case by higher tank pressures or in a pump fed design by picking a more powerful pump. The best of rocket engines can reach 99.7-99.9% c*. Once the propellants are injected, and burned they are accelerated by the use of a Delaval nozzle, which basically converts the pressure head into velocity energy. And that in short and with skipping some steps, is my rocket engine :)

Ismail

Sources: 

Physics (2005). Physics. For high school students. GIANCOLI