I'm designing a single open cycle expander rocket engine using propane. However, I've been unable to find out whether or not the fuel is split into the turbine and combustion chamber before or after it circulates through the cooling loop.
I would assume that splitting the fuel before the phase change would make it more efficient as it wouldn't necessitate the use of valves to achieve the correct ration of fuel going through the turbine, as well as the fact that because of the smaller amount of fuel (in the RL-10 I believe it's around 3%, however I believe this would vary in fuels like cryogenic propane) in the cooling loop, it would concentrate the thermal energy where it is most useful, in the working fluid.
However, I'm not sure if this is even possible, since there is so little fuel traveling through the cooling loop that it could prove to be insufficient in cooling down the combustion chamber walls, as it could be evaporated to quickly and significantly reduce its cooling ability, as it has become less conductive (despite propane's incredible range as a liquid of -188 to -42 degrees Celsius).