Fanno flow

The frictional effect is modeled as a shear stress at the wall acting on the fluid with uniform properties over any cross section of the duct.

The Fanno flow model begins with a differential equation that relates the change in Mach number with respect to the length of the duct, dM/dx.

[2][3] One must keep in mind, however, that the value of the Fanning friction factor can be difficult to determine for supersonic and especially hypersonic flow velocities.

The resulting relation is shown below where L* is the required duct length to choke the flow assuming the upstream Mach number is supersonic.

Equally important to the Fanno flow model is the dimensionless ratio of the change in entropy over the heat capacity at constant pressure, cp.

The above equation can be rewritten in terms of a static to stagnation temperature ratio, which, for a calorically perfect gas, is equal to the dimensionless enthalpy ratio, H: The equation above can be used to plot the Fanno line, which represents a locus of states for given Fanno flow conditions on an H-ΔS diagram.

Each point on the Fanno line corresponds with a different Mach number, and the movement to choked flow is shown in the diagram.

The Fanno line defines the possible states for a gas when the mass flow rate and total enthalpy are held constant, but the momentum varies.

Differential equations can also be developed and solved to describe Fanno flow property ratios with respect to the values at the choking location.

For given upstream conditions at point 1 as shown in Figures 3 and 4, calculations can be made to determine the nozzle exit Mach number and the location of a normal shock in the constant area duct.

Figure 1 A Fanno Line is plotted on the dimensionless H-ΔS axis.
Figure 2 Common thermodynamic property ratios plotted as a function of Mach number using the Fanno flow model.
Figure 3 A supersonic nozzle leading into a constant area duct is depicted. The initial conditions exist at point 1. Point 2 exists at the nozzle throat, where M = 1. Point 3 labels the transition from isentropic to Fanno flow. Points 4 and 5 give the pre- and post-shock wave conditions, and point E is the exit from the duct.
Figure 4 The H - S diagram is depicted for the conditions of Figure 3. Entropy is constant for isentropic flow, so the conditions at point 1 move down vertically to point 3. Next, the flow follows the Fanno line until a shock changes the flow from supersonic to subsonic. The flow then follows the Fanno line again, almost reaching a choked condition before exiting the duct.
Figure 5 Fanno and Rayleigh Line Intersection Chart.