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Fanno Flow

Flow with friction

Consider a flow through constant area pipe as shown in Fig. 1. A subsonic or supersonic flow enters in the pipe at section 1 and leaves at section 2. Thermodynamic properties along with the velocity of the flow change from their initial value at station 1 to the station 2 in the presence of friction force. This 1D flow with friction is called as Fanno flow. Analysis of this flow would lead to prediction of properties of the flow at the exit for known inlet conditions and pipe configuration.

Here we will be considering the effect of friction between pipe wall and fluid. However this assumption will be used only in momentum equation. Hence total temperature can be considered to be constant in the flow process. The

1D governing equations for this flow are ,

ρ₁u₁ = ρ₂u₂ (mass conservation)

And

The main change takes place in momentum equation. Therefore consider the integral form of momentum equation for 1D flow.

Lets assume the flow to be steady through the pipe. Hence,

For integration of pressure and momentum terms, area is the cross sectional area while for the shear stress term area s the circumferential area of the pipe. Therefore the wetted area for shear stress is πDl.Here D is diameter and l is length of the pipe. Negative sign should be associated with the shear stress term since shear acts in the direction opposite to the flow. Hence the momentum equation is,

p₁ + ρu₁₂ = p₁ + ρu₁₂ + F

where term F corresponds the frictional force and can be expressed in terms of pipe dimensions and friction coefficient.

Let’s try to express the change in static and total properties of the flow from station 1 and 2 of the pipe due to consideration of wall shear or friction.

Since total temperature is constant, we can express the static temperature ratio as

Hence,

From, mass conservation equation, we can get,

ρ₁u₁ = ρ₂u₂

But,

Substituting this expression in mass conservation equation, we get,

We can use ideal gas equation to calculate the density ratio from pressure and temperature ratio.

The earlier equations (1 to 3) are for static property ratios. For total property ratios between two stations we have,

These expression provide the ratios of thermodynamic properties for the known Mach number at station 1 and 2.

Reference conditions for Fanno flow

If the inlet flow, either subsonic or supersonic, attains Mach number equal to 1 or sonic condition, at the station 2, then such a condition is taken as reference for calculations of Fanno flow. The corresponding length of the pipe is terms as critical length of the pipe. We can use the reference conditions for frictional pipe flow analysis. The expressions for property ratios are then given as,

Since M₂ is equal to 1, the Mach number at station 1 (M₁) is the free stream Mach number (M_∞).