The only one I know of is the LP LOX pump on RD-0120. A multi-spool design is already a default for jet engines, which are basically turbine-driven air pumps. A multi-spool pump can save at least the weight of the duct connecting the LP and HP pump if nothing else, which develops a boundary layer in the liquid pumped, not to mention the additional boundary layer if this duct has one or more flexible joints in it. If the oxygen HP pump is too difficult to use a multi-spool design, then at least the HP fuel pump can use this design to save some weight, ideally combining the LP and HP pumps and the preburner in a single assembly just like a small jet engine. (But of course, no jet engine has a backend pressure of 200 bar or more.)
1$\begingroup$ I'm very interested in this question. Can you provide a link or edit your question to show that the RD-0120 uses this design? I can't find a dual spool pump on the only schematic I can find: qph.fs.quoracdn.net/main-qimg-ce9b18d79f9652c443c989c5695c5541 One reference claims the NK-33 had a dual spool LO2 pump but I can't confirm that either. link.springer.com/chapter/10.1007/978-3-319-27748-6_18 $\endgroup$– Organic MarbleMay 22, 2019 at 15:12
1$\begingroup$ @OrganicMarble I got it wrong. It's the LOX boost pump, not the LH2 boost pump that's twin spool. Two sources mentioned this fact. The first one is History of Liquid Propellant Rocket Engine in the USSR CADB section and the second one is this lpre.de/resources/articles/AIAA-1995-2540.pdf. $\endgroup$– Meatball PrincessMay 22, 2019 at 17:10
2$\begingroup$ Thanks! I'll be watching this one. $\endgroup$– Organic MarbleMay 22, 2019 at 17:13
The reason for multiple shafts/spools is to allow the inner/higher pressure stages to spin at a different (greater) rate than the outer lower pressure stages.
In aviation jet engines this solves two quite specific problems. Namely:
Efficiently creating higher pressure/pressure-ratio combustion, for thermodynamic efficiency (across a range of flow-rates).
Having a wide range of throttle settings/current engine speed/altitude that don't cause a surge condition.
Cryogenic turbo-pumps don't have these problems, or at least not generally.
The ingress-ed fluid is already as dense as it's going to be as pressure/density are no longer as tightly coupled.
The low speed conditions for surge in jets doesn't happen as much in rocketry.
There are also reasons not to have compressor stages in rockets. In particular the limiting factor of spool speed is often cavitaion (no relevant for jets). This means a lot of turbo-pump assemblies couldn't make use of a second spool without risking cavitation. 'Boost' compressors can be used to increase the pressure in the system to prevent this but this is added weight and complexity etc.
There are advantages specific to pump assemblies too, but they are quite specific.
For example the second shaft isolates some of the rise in pressure due to the pre-burner. For single shaft staged combustion engines (i.e.: RD-0120) this is important the seals used to prevent leaking along the shaft are complex/expensive/consume helium. This isolation reduces the pressure gradient those seals have to operate at which can only be a good thing.
Balancing all this up is a bit complex and I wouldn't be able to predict whether or not that make a multi-spool design viable. However I hope that gives some insight at least to why it isn't as obvious a choice as in jets.
The combustion that powers turbo-pumps (analogous to the combustion chamber in a jet) operates at (and requires) a relatively high pressure.
In a jet engine the spool powers a fan at the front. In a turbo-pump assembly this goes on to drive the compressor of the 'other' propellant, the one that isn't flowing through the turbo-pump itself. In both cases if the shaft isn't perfectly sealed, some of the high pressure fluid from the middle can leak along the shaft to what ever the shaft is driving. In the case of a jet this isn't much of a problem. If a tiny amount of the combustion products leak out of the front, its doesn't matter too much (everything is oxidiser rich and under low pressure). In a turbo-pump any such leakage would be very bad. As one of the fluids is fuel rich, the other is oxidiser rich. And its in a confined space. AKA no space today for anything in close proximity.
To avoid this a really complex set of seals are used, and in the middle an inert gas (helium) is injected under huge pressure. This keeps things-that-go-boom-when-together separated. But its not an ideal arrangement. Its heavy and consumes helium which needs to be stored at high pressure. Which means high pressure tanks. Which means more weight more things to go wrong, more cost to develop and manufacture etc.
The twin spool design helps here. In a 2 spool design it's the low pressure spool (inner axle/outer stages) that does the driving. Hence its only the low pressure stages that needs the aforementioned elaborate sealing. If the inner stage fluid leaks into the lower pressure stages, its not super-bad as you are still in the same type of propellant, and it wont cause any extra combustion.
This means you only have to seal against the lower pressure stage, which makes everything easier (lighter).
$\begingroup$ @MeatballPrincess: I have clearly skimmed over a few things for brevity. I am happy to answer follow up questions. $\endgroup$ Jun 19, 2019 at 11:13
$\begingroup$ can you elaborate more on this statement "For example the second shaft isolates some of the rise in pressure due to the pre-burner. For single shaft staged combustion engines (i.e.: RD-0120) this is important the seals used to prevent leaking along the shaft are complex/expensive/consume helium. This isolation reduces the pressure gradient those seals have to operate at which can only be a good thing." $\endgroup$ Jun 19, 2019 at 19:30
$\begingroup$ @MeatballPrincess, a bit long for a comment so i've edited my answer. Hope that helps. $\endgroup$ Jun 20, 2019 at 9:17
$\begingroup$ twin spool means there're two shafts in a single pump pumping a single propellant e.g. LH2, and without the booster LP pump, not 2 pumps pumping 2 propellant, that's already part of my assumption of the original question. In a twin spool LH2 pump, the LH2 goes LPC->HPC->Fuel Rich PB->HPT->LPT->Main Combustion Chamber, no seal needed at all, only controlled leakage. $\endgroup$ Jun 20, 2019 at 15:03
1$\begingroup$ @MeatballPrincess the RD-0120 uses a fuel-rich staged combustion cycle and a single shaft to drive both the fuel and oxidizer turbopumps. Leakage along this shaft is critical and seals are very much needed. The problem is the LP turbine or compressor needs to be connected to the LOx pump via a mechanical connection. Also if this advantage is not present, even less reason to use a twin spool design. $\endgroup$ Jun 20, 2019 at 15:40