Calculating characteristic length for heat transfer along nozzle injector

[YungPlantEng]

We have an injection nozzle supply water to a quench tower that adiabatically cools incoming gas from 2400 to 200 F. I am trying to calculate external nozzle temperature (not accounting for ash accumulation/oxidation of the surface) and am having trouble understanding how to confirm the characteristic length for the external gas stream against the nozzle. Am I just able to make the annulus assumption? The flow is very turbulent approaching .3-.5 the speed of the sound.