Basics of Fluid Mechanics, 2014a

452 CHAPTER 11. COMPRESSIBLE FLOW ONE DIMENSIONAL
11.7.6.1 Variations of The Tube Length ( 4fL
D ) Effects
In the analysis of this effect, it should be assumed that back pressure is constant and/or
low as possible as needed to maintain a choked flow. First, the treatment of the two
branches are separated.
Fanno Flow Subsonic branch
For converging nozzle feeding, increasing
the tube length results in increasing the
exit Mach number (normally denoted herein as
M 2 ). Once the Mach number reaches maximum
(M =1), no further increase of the exit
Mach number can be achieved with same pressure
ratio mass flow rate. For increase in the
pipe length results in mass flow rate decreases.
It is worth noting that entrance Mach number
is reduced (as some might explain it to reduce
the flow rate). The entrance temperature increases
as can be seen from Figure (11.24).
T 0
T
constant pressure
lines
Fanno lines
1’’
1’
1
s
2
2’
2’’
Fig. -11.24. The effects of the increase of
on the Fanno Line.
4 fL
D
The velocity therefore must decrease
because the loss of the enthalpy (stagnation temperature) is “used.” The density decrease
because ρ =
P
RT
and when pressure is remains almost constant the density
decreases. Thus, the mass flow rate must decrease. These results are applicable to the
converging nozzle.
In the case of the converging–diverging feeding nozzle, increase of the dimensionless
friction, 4fL
D
, results in a similar flow pattern as in the converging nozzle. Once
the flow becomes choked a different flow pattern emerges.
11.7.6.2 Fanno Flow Supersonic Branch
There are several transitional points that
change the pattern of the flow. Point a
is the choking point (for the supersonic
branch) in which the exit Mach number
reaches to one. Point b is the maximum
possible flow for supersonic flow and is not
dependent on the nozzle. The next point,
referred here as the critical point c, is the
point in which no supersonic flow is possible
in the tube i.e. the shock reaches
to the nozzle. There is another point d,
in which no supersonic flow is possible in
the entire nozzle–tube system. Between
these transitional points the effect parameters
such as mass flow rate, entrance and
exit Mach number are discussed.
Å
Ñ
Å ¾
all supersonic
flow
Ñ ÓÒ×Ø
Å ½
a
Å ½
b
mixed supersonic
with subsonic
flow with a shock
the nozzle
between
is still
choked
Ä
Fig. -11.25. The Mach numbers at entrance
and exit of tube and mass flow rate for Fanno
Flow as a function of the 4fL . D
c
Å ½