Fanno Flow - Applications

Applications

The Fanno flow model is often used in the design and analysis of nozzles. In a nozzle, the converging or diverging area is modeled with isentropic flow, while the constant area section afterwards is modeled with Fanno flow. 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. Point 2 labels the nozzle throat, where M = 1 if the flow is choked. Point 3 labels the end of the nozzle where the flow transitions from isentropic to Fanno. With a high enough initial pressure, supersonic flow can be maintained through the constant area duct, similar to the desired performance of a blowdown-type supersonic wind tunnel. However, these figures show the shock wave before it has moved entirely through the duct. If a shock wave is present, the flow transitions from the supersonic portion of the Fanno line to the subsonic portion before continuing towards M = 1. The movement in Figure 4 is always from the left to the right in order to satisfy the second law of thermodynamics.

The Fanno flow model is also used extensively with the Rayleigh flow model. These two models intersect at points on the enthalpy-entropy and Mach number-entropy diagrams, which is meaningful for many applications. However, the entropy values for each model are not equal at the sonic state. The change in entropy is 0 at M = 1 for each model, but the previous statement means the change in entropy from the same arbitrary point to the sonic point is different for the Fanno and Rayleigh flow models. If initial values of si and Mi are defined, a new equation for dimensionless entropy versus Mach number can be defined for each model. These equations are shown below for Fanno and Rayleigh flow, respectively.

begin{align}
Delta S_F &= frac{s - s_i}{c_p} = lnleft \
Delta S_R &= frac{s - s_i}{c_p} = lnleft
end{align}

Figure 5 shows the Fanno and Rayleigh lines intersecting with each other for initial conditions of si = 0 and Mi = 3. The intersection points are calculated by equating the new dimensionless entropy equations with each other, resulting in the relation below.

Interestingly, the intersection points occur at the given initial Mach number and its post-normal shock value. For Figure 5, these values are M = 3 and 0.4752, which can be found the normal shock tables listed in most compressible flow textbooks. A given flow with a constant duct area can switch between the Fanno and Rayleigh models at these points.

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