Compressible Pipe Flow (Fanno and Rayleigh) Calculator

Fanno flow describes adiabatic flow with wall friction in a constant-area duct. No heat is added or removed, only friction acts on the flow, driving the Mach number toward 1 (the sonic reference state) regardless of whether the flow starts subsonic or supersonic. The friction ratios T/T*, p/p*, p0/p0*, and V/V*, plus the dimensionless friction length 4f-bar L*/D, describe how far along the duct the flow travels before reaching M=1.

Rayleigh flow describes frictionless flow with heat addition or rejection in a constant-area duct. Wall friction is neglected, only heat transfer changes the flow properties, driving the Mach number toward 1 (the sonic reference state) for heat addition regardless of whether the flow starts subsonic or supersonic. The ratios T0/T0*, T/T*, p/p*, p0/p0*, and rho/rho* describe how the flow properties change relative to that sonic reference state.

Fanno flow has friction but no heat transfer (adiabatic), while Rayleigh flow has heat transfer but no friction (frictionless). Both apply to a constant-area duct and both use M=1 as their reference state, but the physical mechanism driving the flow toward that state is different: mechanical friction in Fanno flow, thermal energy addition or rejection in Rayleigh flow.

M=1 is chosen as the reference (starred) state for both flows by definition, so every ratio X/X* equals 1 when M=1, since X and X* are the same value at that point. The dimensionless friction length 4f-bar L*/D is the one exception in Fanno flow, it equals 0 at M=1 because no further duct length is needed once the flow has already reached the sonic state.

4f-bar L*/D is the dimensionless maximum duct length (measured in diameters) that a Fanno flow at Mach number M could travel before friction brings it exactly to M=1. f-bar is the mean Darcy friction factor over that length, L* is the physical duct length to reach the sonic state, and D is the duct diameter. A duct longer than this value cannot pass the same mass flow rate at the same upstream Mach number, the flow chokes.

Yes. Both sets of relations are valid for any Mach number greater than 0, subsonic or supersonic. In Fanno flow, friction drives a subsonic flow up toward M=1 and drives a supersonic flow down toward M=1, approaching the sonic state from opposite directions. Rayleigh flow behaves the same way with heat addition, while heat rejection drives the Mach number away from 1 in both regimes.

Isentropic flow, normal shock, Fanno flow, and Rayleigh flow are the four classical building blocks of one-dimensional compressible-flow analysis. Isentropic flow relations describe area change with no friction, heat transfer, or shocks. The normal shock describes an abrupt, irreversible jump. Fanno flow adds friction alone in a constant-area duct, and Rayleigh flow adds heat transfer alone in a constant-area duct. Real duct problems often combine more than one of these effects in sequence.

As friction acts along a constant-area duct, the Mach number is driven monotonically toward 1, regardless of starting subsonic or supersonic. Once M reaches 1, no further increase in duct length can push the flow past the sonic point for a fixed upstream mass flow rate and stagnation conditions, the flow is choked, and adding more duct length instead forces the upstream conditions (such as mass flow rate) to adjust.

Yes. Heat addition always drives the Mach number toward 1 in Rayleigh flow, whether the flow starts subsonic (M increases toward 1) or supersonic (M decreases toward 1). Heat rejection does the opposite, driving the Mach number away from 1 in both regimes. This is why Rayleigh flow is used to model combustor thermal choking.

Any calorically perfect gas works if you set the correct gamma (ratio of specific heats): air and diatomic gases use gamma of about 1.4, combustion products typically use gamma of about 1.30 to 1.33, and monatomic gases like helium or argon use gamma of about 1.67. The formulas are identical, only the exponents and coefficients change.

In Fanno flow, stagnation pressure always decreases toward the sonic state (p0/p0* falls as M moves away from 1 in either direction, meaning p0/p0* is greater than 1 for both subsonic and supersonic M), reflecting the irreversible loss caused by friction. In Rayleigh flow, stagnation pressure decreases with heat addition (moving M toward 1) and increases with heat rejection, since heat addition to a flow is also an irreversible, entropy-generating process even without friction.