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The way we plan on controlling our engine's mass flow rate is by placing ball valves upstream of the combustion chamber that will regulate the mass flow into the combustion chamber. But wait, you may say – isn't the mass flow rate set by the injector? Well, yes, but the mass flow rate that the injector outputs is dependent on the pressure drop across the injector. An upstream valve will create a pressure drop across itself, which will decrease the injector manifold pressure. This will decrease the dP, which will decrease the mass flow of the injector. When the mass flow decreases, the chamber pressure will decrease proportionally. Thus, the main challenge with throttling is relating ball valve pulse width (if using a servo) to mass flow reductionIn the case of the nitrous, a reduction in valve opening area will decrease the mass flow across the valve, which will cause the chamber pressure to decrease. The decreased chamber pressure will then induce a greater dP across the valve, which will cause more nitrous crossing the valve to flash boil. This means that the nitrous entering the injector manifold will be at a lower density, since more of it will be a gas. And since mdot = Cd*A_inj * sqrt(2 * rho * dP), a decreased rho will cause a greater dP per unit of massflow. Less massflow but more dP per unit of massflow means that the nitrous injector pressure drop will remain somewhat constant over a wide range of throttle levels. In the case of the IPA, a reduction in valve opening area will also decrease the mass flow across the valve, which will then decrease the chamber pressure. However, although this reduced chamber pressure will cause a higher dP across the valve, the IPA will not boil, which means its density will not change. So, the dP across the IPA injector will decrease. This means that the limiting propellant (in regards to stiffness) is the IPA.
For regenerative cooling, a another limiting factor is cooling efficiency, which is what we aim to get valuable data on for our research. When an engine throttles down, there is less fuel (and ox) massflow, which means less fuel in the regenerative channels cooling the engine walls. We aim to characterize the effect of less fuel flow on the cooling efficiency of our engine.
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All of this, however, is just to obtain the Cd of our flow at nominal flow rates. When throttling down, our flow rates will change. Since our fuel is incompressible, its Cd will remain constant when we throttle down, so we can use the same Cd for all throttle levels for the fuel. However, for the nitrous, this is not the case. This is because we are modeling the nitrous as an incompressible fluid (which it is not) and wrapping all of its flash-boiling into a very low Cd. However, the amount of flash-boiling is governed by the dP across the injector. If the dP is high, more flash-boiling will occur; if the dP is low, less flash-boiling will occur. This means that as we throttle down, our Cd will increase because the injector dP decreases, which makes the nitrous behave more like an incompressible fluid. Currently, we are thinking of obtaining Cd at a bunch of different injector dP's, and then fitting that data to a curve. If we know how Cd changes with mass flow, we can characterize our throttle valvesthe effective valve opening area (Cd*A) will likely be a nonlinear function of servo pulse width. However, obtaining this nonlinear function isn't impossible. To calibrate our valves, we are thinking to have a fluid circuit that ends at the throttle valve, with a differential pressure sensor reading the pressure before and after the valve. This will allow us to calculate the Cd*A of the valve using the SPI equation: mdot = Cd*A_inj*sqrt(2*rho*dP). This is assuming we know how much nitrous/CO2 we're putting into our tanks, which will allow us to integrate the SPI equation to solve for Cd*A. Although the curve that we get will most likely be nonlinear, some papers have observed a linear portion of the curve over a wide range of throttle.
Finally, yet another difficult part about throttling a rocket engine is performing initial open loop testing. From cold-flows, a relationship between pulse width and effective flow area can be obtained. However, to close the loop on your control system, you need to empirically obtain data from a hotfire, i.e. the response time (tau) of valve actuation, which depends on how far/how fast your propellant travels through your plumbing and injector atomization/mixing. Open loop characterization is challenging because you are setting the valve to a few opening areas – the system is not controlling itself yet. The problem with this is that the IPA tank pressure decreases way faster than the nitrous pressure (the IPA tank is pressurized via blowdown). This will gradually make the mixture ratio more ox-rich over time, which will increase the risk of engine melting. Based on our RPA sims, our engine could survive a mixture ratio of even 4.5, but we will limit our maximum mixture ratio to 3.5. We found that two hot fire tests, each one testing a couple different throttle levels, should be enough to characterize our system to close the loop while keeping the mixture ratio at the end of the burn less than ~3.5.