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Figure 9.3:
Schematic of Example (9.1)

^{9.4}
Solution
For isentropic, the flow to the pipe inlet, the
temperature and the total pressure at the pipe inlet are
the same as the those in the reservoir.
Thus, finding the total pressure and temperature at the pipe inlet
is the solution.
With the Mach number and temperature known at the exit,
the total temperature at the entrance can be obtained by knowing the
.
For given Mach number (
) the following is obtained.
Fanno Flow 
Input: M 
k = 1.4 
M 
fld 
P/P* 
P0/P0* 
ρ/ρ* 
U/U* 
T/T* 
0.9 
0.0145124 
1.12913 
1.00886 
1.09338 
0.914598 
1.0327 
So, the total temperature at the exit is
To ``move'' to the other side of the tube the
is added as
The rest of the parameters can be obtained with the new
either
from the table by interpolations or utilizing attached program.
Fanno Flow 
Input: fld 
k = 1.4 
M 
fld 
P/P* 
P0/P0* 
ρ/ρ* 
U/U* 
T/T* 
0.358856 
3.21 
3.01404 
1.74047 
2.57639 
0.388141 
1.16987 
Note that the subsonic branch is chosen.
the stagnation ratios has to be added for
Fanno Flow 
Input: M 
k = 1.4 
M 
fld 
P/P* 
P0/P0* 
ρ/ρ* 
U/U* 
T/T* 
0.35886 
3.20989 
3.014 
1.74046 
2.57636 
0.388145 
1.16987 
The total pressure
can be found from the combination of the
ratios as follows:

Figure 9.4:
The schematic of Example (9.2)

Another academic question/example:
Solution
 (a)
Assuming that the pressure vessel very much larger than the pipe
therefore, the velocity in the vessel can be assumed
small enough so it can be neglected.
Thus, the stagnation conditions can be approximated
as the condition in the tank.
It further assumed that the flow
through the nozzle can be approximated as isentropic.
Hence,
and
The mass flow rate through the system is constant and for simplicity
reason point 1 is chosen in which,
The density and speed of sound are unknowns and needed to be computed.
With the isentropic relationship the Mach number at point
one is known the following can be found either from the
Table or PottoGDC
Isentropic Flow 
Input: M 
k = 1.4 
M 
T/T0 
ρ/ρ0 
A/A* 
P/P0 
PAR 
F/F* 
3 
0.357143 
0.0762263 
4.23457 
0.0272237 
0.115281 
0.653256 
The temperature is
With the temperature the speed of sound can be calculated as
The pressure at point 1 can be calculated as
The density as a function of other properties at point 1 is
The mass flow rate can be evaluated from equation (9.2)
 (b)
 First, a check whether the flow is shockless
by comparing the flow resistance and the maximum possible resistance.
From the table or by using PottoGDC,
to obtain the flowing







3.0000 
0.52216 
0.21822 
4.2346 
0.50918 
1.9640 
0.42857 
and the conditions of the tube are
Since
the flow is chocked and with a shock wave.
The exit pressure determines the location of the shock, if a shock
exists, by comparing ``possible''
to
.
Two possibilities needed to be checked; one, the shock at the entrance
of the tube, and two, shock at the exit and comparing the pressure ratios.
First, the possibility that the shock wave occurs immediately
at the entrance for which the ratio for
are (shock wave table)






3.0000 
0.47519 
2.6790 
3.8571 
10.3333 
0.32834 
After shock wave the flow is subsonic with ``
''
.
(fanno flow table)







0.47519 
1.2919 
2.2549 
1.3904 
1.9640 
0.50917 
1.1481 
The stagnation values for
are







0.47519 
0.95679 
0.89545 
1.3904 
0.85676 
1.1912 
0.65326 
The ratio of exit pressure to the chamber total pressure is
The actual pressure ratio
is smaller
than the case in which shock occurs at the entrance.
Thus, the shock is somewhere downstream.
One possible way to find the exit temperature,
is by finding
the location of the shock.
To find the location of the shock ratio of the pressure ratio,
is needed.
With the location of shock, ``claiming'' up stream from the exit
through shock to the entrance.
For example, calculating the parameters for shock location with
known
in the ``y'' side.
Then either utilizing shock table or the program to obtained
the upstream Mach number.
The procedure of the calculations:
 1)
 Calculated the entrance Mach number assuming the shock occurs at the
exit:
 a)
 set
assume the flow in the entire tube is supersonic:
 b)
 calculated
Note this Mach number is the high Value.
 2)
 Calculated the entrance Mach assuming shock at the entrance.
 a)
 set
 b)
 add
and calculated
' for subsonic branch
 c)
 calculated
for
'
Note this Mach number is the low Value.
 3)
 according your root finding algorithm^{9.5}calculated or
guess the shock location and then compute as above the new
.
 a)
 set
 b)
 for the new
and compute the new
' as on
the subsonic branch
 c)
 calculated
' for the
'
 d)
 Add the leftover of
and calculated the
 4)

guess new location for the shock according to your finding root
procedure
and according the result repeat previous stage until the solution is obtained.






3.0000 
1.0000 
0.22019 
0.57981 
1.9899 
0.57910 
 (c)
The way that numerical procedure of solving this problem is by
finding
that will produce
.
In the process
and
must be calculated (see
the chapter on the program with its algorithms.).

Next: Supersonic Branch
Up: Fanno Flow
Previous: The working equations
Index
Created by:Genick BarMeir, Ph.D.
On:
20071121
