Axel
I think your objections are quite valid. They would be the basis of any reasonable investigation of the concept.

Eric
Your third suggestion is the one I see. The venturi section was sort of supposed to represent sonic flow. Below the venturi, the flow slows down for pressure recovery. The slowing as the area widens is also the injection plane since there are no formal injectors. The flow through the lower chamber will be higher velocity than normal because of the high speed hot gas-gas injection. The lower chamber restriction will be neccesary to create the second sonic throat before expansion.

I appreciate the patience with my incomplete (and sometimes wrong) descriptions. I see the flow as subsonic in the solid chamber. Sonic in the venturi with the pressure drop to the lower chamber. Subsonic in the expanded lower chamber. Sonic again through the true throat. Supersonic through the expansion nozzle.

If the solid exhaust is subsonic as it moves past the liquid fuel inlets, then the pressure in the exhaust gasses at that point must be less than that in the tanks/fuel lines. So, high pressure in the solid combustion chamber must be reduced before it encounters the fuel injectors. If the flow is constricted, like say through a venturi, then it will accelerate thus reducing the local pressure in accordance with Bernoulli’s principle. The liquid fuel then enters the exhaust stream and becomes entrained in the flow, though hopefully not combusting until they are carried downstream to the secondary chamber. Unfortunately, I cannot currently see anyway to take advantage of the secondary combustion event with this configuration. Any combustion will add energy to the fluid, which will increase the temperature and pressure of the flow as a result. Since the flow is subsonic, the pressure wave will be able to find its way back upstream to the fuel injectors, and possibly as far back as the feed lines and tanks. The gasses will always prefer to flow from higher pressure to lower, so the geometry of the secondary chamber will have to be designed to immediately relieve this pressure. In my mind this implies that the secondary chamber will have to be an expansion nozzle. Thus, leading to my previous comment about thrust augmented solids.

Now, there are a couple of possibilities for getting around this limitation without resorting to pumps. The simplest conceptually, but possibly problematic in implementation, is to use the solid exhaust gases to drive a piston which increases the pressure in the liquid tanks. This could be a direct pneumatic arrangement, which I’m not sure could be made compact enough to be practical for a rocket. Or, the heat of the solid exhaust could be transferred through a heat exchanger into a secondary fluid, which then expands (possibly by vaporizing) to provide additional pressurization to the liquid fuel tanks.

The other possibility is to use a nozzle on the solid exhaust to choke the flow and cause it to go sonic in the throat. In this arrangement, I’m imagining the liquid fuel inlets are like backward facing steps downstream of the sonic point in the nozzle. As the Mach 1+ fluid passes the step, the flow stream will attempt to turn the corner, but if the maximum turn angle, as given by the Prandtl-Meyer function can be kept below 90 degrees, then there should be a small vortex generated in the corner of the step which could be used to entrain the liquid fuel into the passing exhaust gasses. (See also Prandtl-Meyer expansion fan.)

I am still not sure that the secondary combustion event can be sufficiently isolated from even this kind of injector design. Perhaps I’ll get my flow solver running on my laptop sometime in the near future and try to run a few tests.

http://en.wikipedia.org/wiki/Ejector_pump
http://en.wikipedia.org/wiki/Venturi_effect

I’m not really familiar with the physics here, and won’t dig into it. Efficiencies, pressures, ISP, cooling, etc. … figuring this out and make it all work seems to be a major development effort to me. So you save on developing a low pressure liquid rocket engine, but at the price of having to develop (or at least integrate and test) both liquid and solid plus inventing a rocket exhaust driven ejector pump.

In some ways you combine the disadvantages of solid and liquid. Operationally you have to support both solid and liquid infrastructure. Throttling will be very limited, especially if you use the liquid fuel to cool the injector/ejector. Controlled turning off of the engine (e.g. for stage separation) is very hard, especially if you would like to have a reusable system. Restarting the engine in flight (e.g. for fly back or vertical landing) is even harder.

You all probably know about this man but just in case you don’t:
http://www.nakka-rocketry.net/
(Warning: one can easily spend a lot of time at that site).

I don’t know him and he seems like a very busy person but maybe he’ll have time to look at the idea and/or possibly recommend someone else for it?

If this can be made to work, I see solid pressure at 3-4 times that of the main combustion chamber and perhaps 10 times that of the liquid tanks. Hopefully that scales. I would see a progression from shop air to sugar rocket to higher amateur sizes before any realistic space craft engine development. If it is all workable, I would expect a steady progression in sizes used IF it is a trully cheap development process.

The inefficiency is a given. The results might not be. If it takes 10 times as much enthalphy to pump the fluids as a normal pump, that just means that solid mass times temperature will have to be 10 times that of a rational pump. The extra mass and heat is used for reaction in the main thrust chamber and through the nozzle and is therefore not wasted. The inefficiency shows up as higher dead mass for the solid because it has to be larger and probably higher pressure to make up for the pump performance losses. Since I am suggesting pumping flow on the order of 20% of the total MDOT, it seems possible that the pumping ineficiencies will be overcome with the pure brute force of the solid component.

I don’t see how the lower chamber pressure could be higher than the higher chamber. That would prevent flow in the right direction. I think you need the nozzle after the liquid burn to bring it back up to supersonic.

It could be thought of as an afterburner.

It is possible that the whole process may be self-regulating. If the chamber pressure grows enough to slow the incoming exhaust gas/fuel/oxidizer stream, then less fuel/oxidizer will be entrained by the exhaust gasses. This will reduce the amount of downstream combustion, which in turn, reduces the chamber pressure. If the system behaves in this manner, then it is likely that some form of dynamic equilibrium could be established.

As David Summers points out, if you can keep the flow supersonic past the fuel injectors, then there is no way for them to be affected by the combustion events down stream. If you use a converging-diverging nozzle to force the primary exhaust gasses to go supersonic, then the rise in the secondary chamber pressure should not exceed 2-3 times the pressure of the gas as it leaves the nozzle1. Otherwise, the flow through the throat of the nozzle will not be able to attain supersonic flow speeds.

Here’s a question; if the flow remains supersonic through the secondary combustion, then is there really a need for a second nozzle? If not, then would this look more like a solid rocket with a thrust augmented nozzle?