Exact Closed-Form Solution for Fanno Flow of a Calorically Perfect Ideal Gas

Abstract

This paper presents an explicit Lambert-W inversion of the classical Fanno relation for steady, adiabatic, quasi-one-dimensional flow of a calorically perfect ideal gas in a constant-area duct with wall friction. The formulation isolates the downstream Mach number as a function of the upstream Mach number, the specific-heat ratio, and the prescribed friction-length parameter 4fL/D. The appropriate real branch of the Lambert-W function is selected from the inlet sonic regime: the W-1 branch for subsonic inlet flow and the W0 branch for supersonic inlet flow. The resulting expression eliminates the conventional iterative inversion of the implicit Fanno equation for admissible, unchoked cases. Numerical comparisons with the classical implicit Fanno relation show agreement to numerical precision for representative subsonic and supersonic cases. The expression may be useful for instruction, code verification, preliminary sizing and rapid evaluation of idealized compressible duct systems, and reduced-order analysis of microchannels where wall friction is a dominant effect.