When throttling an engine, the main limiting factor is injector stiffness, or the ratio of injector pressure drop to chamber pressure. A rule of thumb is that injector stiffness should be above 15% to avoid combustion instabilities. Injector stiffness is controlled by two equations: mdot = p_c * A_t / cstar, and mdot = Cd * A_inj * sqrt(2 * rho *dP). The first equation tells us that if mdot decreases by a factor of 2, p_c also decreases by a factor of two. The second equation tells us that if mdot decreases by a factor of two, dP decreases by a factor of four. This means that as mass flow decreases, the injector stiffness will decrease until it drops below the critical limit.
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. In 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, 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.
Our plan to successfully throttle Hephaestus hinges on a significant amount of cold-flow testing to obtain a flow curve that we will use to calibrate our throttle valves. However, cold-flow testing at nominal pressures will result in inaccurate data because there is no chamber pressure, which would result in a much larger dP across the injector than there would be during hotfire. To accurately characterize flow response, we need to enforce the same flow through the system during a cold flow test as there will be during a hotfire.
Currently, we are thinking of two ways to do this. The first is to make "mock injectors" that have smaller/less orifices to account for the increase in dP. So, we would run fluid through the system at nominal pressures, but with a smaller orifice area, or simply less orifices. This can be designed to perfectly offset the greater dP. For nitrous, this approach is feasible, as we can just decrease the number of slots on the pintle. However, for the IPA, making the annulus smaller than it already is (it is nominally 10 thou) will be a machining headache. Therefore, for the IPA, we will use another method. This other method consists of water flowing at off-nominal conditions, i.e. with manifold pressures equal to injector dP during hotfire + atmospheric pressure so that the dP across the injector for this cold flow is the same as the hotfire.
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.
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.