Temperature variation in transitional flow

Hej everybody,
i am modelling rarefied gasflow (transitional regime) trough a long tube (vacuum at the end). I would like to include some temperature derivations in the model. I know that the transitional flow interface does not give the correct prandtl number in this case even if I consider the correct viscosity.
I thought some coupling between different transitional flow interfaces and some iterative calculations. As an example this means dividing the tube in two parts. For the first calculation simulate the whole tube with Temperature T1 and then take the second part of the tube for an other transitional flow interface with Temperature T2 and take the middle pressure of the first simulation as inlet dor the second interface. (see attached model) Do I need some additional weak constraints (for the correct velocity) and does this generally work?
And if this works how can I account for the flow comming (backscattered) from the second part of the tube (with T2) to the first part (with T1)?

Thanks
Laura