  Analytical/quasi-analytical solutions are proposed for a steady,
compressible, single-phase flow in a rectilinear duct subjected to heating
followed by cooling. The flow is driven by the pressure ratio between an
upstream tank and a variable outlet pressure. The article proposes a
methodology to determine the full flow behavior, as a function of pressure
ratio and heat-flux distribution. Following an analogy done with the study of
compressible flows in nozzles, a behavioral classification of non-adiabatic
compressible flows is proposed through the definition of critical pressure
ratios. It is demonstrated that a critical pressure ratio distinguishes
subsonic and supersonic outlet regimes and that there cannot be a steady
shock-wave in such configuration. The behavior of this critical pressure ratio
is studied for limit cases of heat-flux, delineating physical boundaries. An
abacus is also proposed for a given couple of heating and cooling powers, as
both values are needed to characterize the flow. Results are studied for
parameters such as pressure ratio and outlet heat power. A short validation of
a numerical simulation tool is provided, yielding excellent results and very
small relative errors.
