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. When you throttle down, mass flow through the engine decreases. This decreases the engine chamber pressure linearly (since p = mdot*cstar/A_t) but decreases the injector pressure drop nonlinearly (dP = (mdot/(Cd*A_inj))^2 / (2*rho)) for an incompressible flow). This means that as mass flow decreases, the injector stiffness will decrease until it drops below the critical limit. 

For regenerative cooling, however, 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 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 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. The orifice area can be sized to offset the greater dP, such that a smaller total injector area perfectly offsets the larger dP. The other method we are thinking of is to cold-flow test 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. We prefer the second option much more than the first, as we wouldn't need to make mock injectors with even smaller orifices (since our engine is small, it might not even be possible to make our fuel annulus smaller). But we haven't seen people do it this way, so we need to look deeper into whether there's anything wrong with this method before we decide to do it. I don't see anything wrong with the strategy at the moment though – as long flowrates in a cold flow are equal to the flowrates during a hotfire, I would expect valve calibration for that coldflow to also work similarly for a hotfire.

We could also combine these two methods. The main reason why we don't like the "mock injector" method is because it would require us to make our annular gap even smaller than it currently is (6 thou) which is borderline impossible. We have talked with some engineers and they said the main concern with method #2 is cavitation. We still don't understand how cavitation would be different for method #2, as our current understanding is that it only depends on injector pressure drop. If we find that different amounts of cavitation are a concern, we can use method #2 to test our annular flow (since that's incompressible fuel) and method #1 to test our nitrous mass flow. For the nitrous, we will just make an aluminum cylinder with less radial holes to offset the greater pressure drop. This would be really easy to test – another testament to why pintle injectors are so awesome.