Is there anything fundamentally wrong with 'autogenous expander cycle`?

Question

This question is a spinoff from here.

To set the context, here is my understanding of the expander cycle:

1. Closed expander cycle is very efficient, perhaps more efficient than staged combustion cycle. It also has very little mechanical complexity as compared to other cycles (look at the simplicity of BE-7) which makes it very reliable (look at the longevity of RL-10). But closed expander cycle doesn't scale well, and engines with thrust over 150 kN are not practical.
2. Open expander cycle does not have the same scale limitations (BE-3U has 710 kN of thrust, and LE-9 is expected to have close to 1,500 kN of thrust). But it is less efficient because a small portion of the propellant is ejected unburnt. This puts it closer to gas-generator cycle engines in terms of efficiency.

So, why not route the exhaust of the turbine of the open expander cycle back to the tanks to pressurize them autogenously (see diagram and description below)? This should combine the best of both worlds: efficiency of the closed expander cycle with power of the open expander cycle. But, as far as I know, this hasn't been done yet - so, maybe I'm missing something?

Brief description

The cycle works pretty much in the same way as the open dual expander cycle - but instead of discarding the output of the turbopumps, the propellant is returned to its respective tanks.

As far as I understand, only about 2% of the propellant needs to be diverted to run the pumps. But I have only a single source for this (page 5 from here). So, if anyone has better numbers on how much of the propellant is usually used to run the pumps in an open expander cycle - would really appreciate the info.

As the propellant returns back to the tanks, most of it condenses back into a liquid form as it comes in contact with the subcoold propellant remaining in the tanks (I understand that this is what happens in plain autogenous pressurization as well). Different methods of injecting the propellant back into the tanks can be used to control the rate of condensation.

The temperature of the propellant returned to the tanks is around 400K (this also comes from the same page 5 from here - so, any validation or invalidation of this would be helpful). And since only about 2% of the propellant is returned, it's not enough to significantly heat up the rest of the propellant in the tanks.

The last point is true only while there is a lot of propellant still left in the tanks - but won't be true once the tanks are almost empty. At this point, the exhaust of the turbopumps would need to be largely discarded - as in the regular open expander cycle.

Answer 1

Based on the discussion in the comments and on some additional research, I'll try to answer my own question:

The design, at least as is, does not seem to be workable.

First, while it takes only 2% of propellant mass to run the pumps when the working fluid is hydrogen, it will take much more than 2% for other fluids. Based on my rough calculations, it would take 10% - 12% of methane, and 8% - 10% of oxygen to run their respective pumps. One other potential fuel to use might be propane, but I got conflicting numbers trying to calculate how much propane would be needed to run the pumps.

Second, pumping around 10% of propellants back to the tanks will affect the temperature of the tanks significantly. In fact, in case of methane or hydrogen, it will actually vaporize the propellant in the tanks. For oxygen, it will get it very close to the boiling point (though, as long as you pump less than 12% back, oxygen shouldn't vaporize). For propane, it shouldn't be a problem since propane has a very high boiling point (231K) - but again, not sure how much propane needs to be pumped back for the scheme to work.

Lastly, and perhaps more importantly, in this design most of the heat energy extracted from the nozzle does no useful work. About 90% of the heated up propellant is burnt immediately. Thus, only about 10% of the energy is used to run the pumps. So, unless there is a lot of extra heat available, this design will not work. And if there is a lot of extra heat available, a closed expander cycle would probably work better.

Another potential issue brought up in the comments was that cooling the propellant once it returns to the tanks will be difficult. I don't believe that's a fundamental problem and can be addresses in a number of ways (e.g. releasing the gas close to the bottom of the tanks and let it cool as it bubbles up) - but I don't have a definitive proof for this.

To sum up: this design will not work with fuels like hydrogen and methane. It might work with propane but even then, probably would not be the most efficient way to use heat energy. This is probably why it hasn't been used anywhere.

Answer 2

Maybe you should look at a split dual expander cycle. In this case the propellants are passed through a pump in two stages/two consecutive pumps.

After the first stage, some of the propellent is sent to the combustion chamber and the remainder goes through the high pressure pump and then is sent to cool the chamber/nozzle and the heated propellent changes phase and drives the corresponding turbine. Maybe (part of) the propellent exiting the turbine can be used for autogeneous pressurization (after heat exchange with the fresh propellent, if necessary).