Page 194 - 48Fundamentals of Compressible Fluid Mechanics

156 CHAPTER 9. FANNO FLOW

a

all supersonic b c

flow

mixed supersonic

with subsonic

flow with a shock
the nozzle
between
is still
choked !

Fig. 9.8: The Mach numbers at entrance and exit of tube and mass flow rate for Fanno Flow
as a function of the

14
Should the mathematical ) .
derivations be inserted to flow. In this range, the flow rate decreases since ( N b
demonstrate it? To summarize the above discussion the figures 9.8 exhibits the developed
. Somewhat different then the sub-
sonic branch the the mass flow rate is constant even the flow in the tube is com-
b
, N mass flow rate as a function of WYX(Z [
of N
pletely subsonic. This situation is because the “double” choked condition in the
nozzle. The exit Mach N is a continuous monotonic function that decreases with
is a non continuous function with a jump at point when
shock occurs at the entrance “moves” into the nozzle.
b
WYX(Z [
. The entrance Mach N
as a function of N . The figure was calculated
for N and subtracting
b
Figure 9.9 exhibits the N
.

by utilizing the data from figure 9.2 by obtaining the WYX(Z [
In the figure 9.10
b
the given W`XaZ [
and finding the corresponding N
The figure 9.10 exhibits the entrance Mach number as a function of the
N . Obviously there can be two extreme possibilities for the subsonic exit branch.
Subsonic velocity occurs for supersonic entrance velocity, one, when the shock
wave occurs at the tube exit and, two, at the tube entrance . In the figure 9.10
shown
with only shock at the exit only. Obviously, and as can be observed, the larger
only for WYX(Z [
H QS#"
H QS V two extremes are shown. For W`XaZ [
and W`XaZ [
creates larger differences between exit Mach number for the different shock
H QS_I
must occurs even for shock at the entrance.
WYX(Z [
b
14 Note that $&% increases with decreases of '% but this effect is less significant.
location. The larger W`XaZ [
larger N