Fanno Line Flows

• If we consider a problem of frictional adiabatic flow through a duct, the governing Eqs (41.1), (41.3), (38.8), (41.5) and (41.6) are valid between any two points "1" and "2".
  
  
• Equation (41.2a) requires to be modified in order to take into account the frictional force, R_x, of the duct wall on the flow and we obtain

So, for a frictional flow, we have the situation of six equations and seven unknowns.

• If all the conditions of "1" are known, the no. of possible states for "2" is 2. With an infinite number of possible states "2" for a given state "1", what do we observe if all possible states "2" are plotted on a T - s diagram, The locus of all possible states "2" reachable from state "1" is a continuous curve passing through state "1". The question is how to determine this curve? The simplest way is to assume different values of T₂. For an assumed value of T₂, the corresponding values of all other properties at " 2 " and R_x can be determined.

Fig 41.3 Fanno line representation of constant area adiabatic flow

• The locus of all possible downstream states is called Fanno line and is shown in Fig. 41.3. Point " b " corresponds to maximum entropy where the flow is sonic. This point splits the Fanno line into subsonic (upper) and supersonic (lower) portions.
  
  
• If the inlet flow is supersonic and corresponds to point 1 in Fig. 41.3, then friction causes the downstream flow to move closer to point "b" with a consequent decrease of Mach number towards unity.
  
  
• Note that each point on the curve between point 1 and "b" corresponds to a certain duct length L. As L is made larger, the conditions at the exit move closer to point "b". Finally, for a certain value of L, the flow becomes sonic.
  Any further increase in L is not possible without a drastic revision of the inlet conditions.
  
  
• Consider the alternative case where the inlet flow is subsonic, say, given the point 1' in Fig. 41.3. As L increases, the exit conditions move closer to point "b". If L is increased to a sufficiently large value, then point "b" is reached and the flow at the exit becomes sonic. The flow is again choked and any further increase in L is not possible without an adjustment of the inlet conditions.