Could a Stirling engine be used to pump propellants into engines?

A stirling engine is a type of engine which converts thermal energy into mechanical energy. An example of a rather large one is shown here

I know this is pretty much the same principle that an expander cycle engine uses, but could a stirling engine possibly make a rocket engine more efficient, and more reliable? The nozzle of the engine could be attached to the heating part of the stirling engine and the rotational energy could be used to drive the pumps for the engine. Also would a stirling driven engine be confined to the same limited amount of thrust an expander engine has?

• Turbopumps have a lot in common with jet engines as used in aircraft and helicopters. In such applications, power density (both mass and volume per unit of power), reliability, and efficiency are major criteria influencing design, with some compromise in thermodynamic efficiency accepted for the sake of other factors. We don't see Stirling engines in aircraft, presumably because turbines do best on the factors which matter most. Rocketry would place similar emphasis, so would favor a similar solution. – Anthony X Jun 5 '17 at 1:50
• What would you attach the radiator part to? Vacuum is a very lousy heat conductor. – SF. Jun 8 '17 at 1:02
• To start the engine it is necessary to pump the propellants into the combustion chamber before the nozzle is hot and the stirling engine would run. Without very efficient cooling, the combustion chamber and nozzle walls would be destroyed very fast. – Uwe Jun 9 '17 at 19:58

The Merlin engine used by SpaceX has a chamber temperature of around 2500 Kelvin, and liquid oxygen has a temperature of around 50 Kelvin. The Carnot efficiency for this is $$\eta = 1 - \dfrac{T_c}{T_h} = 98\%$$ and stirling engines have an efficiency of 50%, so we have an efficiency of 49%, which is the ratio we have between the hot temperature and the power produced. This means to reach 1900kW of power, our sterling engine would have to move almost 3800kW of power. Depending on the temperature gradient between the thrust chamber and the hot side of the sterling engine, this would require several square meters of surface area.