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Fanno Flow Curve Analysis

Analysis of Fanno Flow

We can use the Eq. for critical analysis of Fanno flow. For this purpose lets replace dp/p and du/u of this equation.

Lets consider again and replace the term du/u, we get,

Modified differential form of ideal gas equation can be re-written using equation 1 as

Now let’s consider the differential form of momentum and replace the term du/u and dp/p using (2), we get,

We can clearly observe that dM/dx > 0 if M<1 and dM/dx < 0 for M>1.

Hence from Eq. (3), we can clearly state that, for subsonic flow Mach number increases in the presence of friction, while Mach number for supersonic flow in the presence of friction. It becomes clear that, for increases Mach number, velocity of the subsonic flow increases in the poor subsonic flow while that of supersonic flow decreases due to decrease in Mach number. From Eq. 1, it becomes clear that temperature of the subsonic flow decreases due to increase in Mach number in the presence of friction while temperature increases for supersonic flow with friction. We can prove from Eq. 2, that pressure decreases for subsonic flow while pressure increases for supersonic flow with friction. For entropy change we can arrive at the expression as,

For supersonic flow dM is negative however 1 - M₂ is also negative, therefore ds is positive for supersonic flow. For subsonic flow dM is positive however 1 - M₂ is also positive, therefore ds is positive for supersonic flow. This expression clearly proves that entropy increases for both subsonic and supersonic flows while sonic flow is isentropic.

Fanno line or curve

Typical Fanno line for a particular mass flow rate is shown in Fig. 1. Previously proved facts are clearly evident in this figure. For a subsonic flow through a frictional pipe, enthalpy decreases and entropy increases. With increase in length of pipe, more expansion of the flow takes place for the subsonic flow due to increase in Mach number, decrease in pressure and increase in velocity. For a particular length of pipe flow, inlet subsonic flow attains sonic state at the exit. Corresponding length of the pipe is called as critical length. Further increase in length of pipe doesn’t change state at the exit however inlet conditions change and subsonic flow becomes lower subsonic. Hence the flow condition for the critical pipe length is called as choked flow. For supersonic flow at the entry to a constant area pipe, deceleration of flow takes place and flow attains lower supersonic conditions. With increase in length of the pipe, Mach number at the exit decreases due to deceleration and at a particular pipe length flow becomes sonic at the exit. Further increase length of the pipe doesn’t change exit conditions while inlet conditions become subsonic. Therefore choked conditions can be said to be attained for critical length of pipe for which entry is supersonic. This curve shows that there is only one critical point for flow with friction. This critical point corresponds to maximum entropy and hence the sonic state.

Like Rayleigh curve, h-s diagram for Fanno flow shown in Fig. 1 corresponds to a particular value of mass flow rate. Fanno curve for various mass flow rat conditions is given in Fig. 2. Explanation for Fanno curve for various mass flow rates can be obtained on similar line as it has been obtained for Rayleigh flow for various mass flow rates where instead of changing the applied heat flux we have to change the pipe length. Fanno curve for increased mass flow rate is also seen to be shrunk like Rayleigh curve.