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, Rx, 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 Rx 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.
