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. 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 reduction.
For regenerative cooling, a 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. Hopefully we can obtain this same relationship!