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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 the 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. Hopefully we can obtain this same relationship!
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.