For a rocket engine that runs on (deeply) cryogenic fuel like LH2, e.g. SSME, is the fuel kept liquid throughout the entire length of the regenerative cooling circuit in the nozzle? Or is it allowed to boil or go supercritical at some point in the circuit? I imagine it would be a really bad idea to do so because generally, gases are worse heat conductors than true liquids.
At least for the SSME, the hydrogen exiting the nozzle cooling circuit was a supercritical fluid.
Data I used when working on a simulation of the SSME shows at 104% throttle setting the hydrogen was at 5911 psi (40.7 MPa) and 445 deg R (247 K).
This slide doesn't show the nozzle cooling circuit exit properties but it does show the mixer outlet properties (purple arrows), 5336 psi (36.7 MPa) and -183 F (153 K). This slide is also from a newer generation of the SSME than I worked with and tends to run cooler and lower pressure.
Here is a phase diagram of hydrogen from here.
Reference for the properties slide (it's slightly different from the copy I have and shows 5310/-193 for the mixer outlet properties, but I can't be bothered to make a new screenshot and do the unit conversions again).