A pipe containing static fluid at ambient temperature is located in an air plenum (edit: duct) with air flowing along the pipe at an elevated temperature.
Assuming the pipe is full of water and static and the pipe is in the centre of the duct with a certain air velocity within the plenum. Is there a way to calculate the temperature of the inner and outer surfaces of the pipe with respect to time?
I have tried a few steady state calculations using a resistance network and assuming flat plate forced convection on the outer of the pipe, then using the specific heat capacity of the pipe and a time step to calculate an increase in temperature of the pipe, however I believe this is the completely wrong approach for multiple reasons one of which is that it does not take into account that there is fluid within the pipe.
Edit: I am assuming the air flow around the pipe is uniform and constant temperature along the pipe. Also, as the water in the pipe is static, that the water temperature does not change with length along the pipe. Illustration of problem: