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 travelling 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 it's cooling ability as it has become less conductive (despite propane's incredible range as a liquid of -188 to -42 Celcius).
The paper Cycles for Earth to Orbit Propulsion shows that for open (bleed) expansion cycles, the flow paths diverge at the pump outlet. Two designs are shown, the second has a split pump.
The wikipedia article says the right thing
only a small portion of the propellant is heated and used to drive the turbine and is then bled off
but its accompanying schematic shows all the flow being heated.