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Solution
The stagnation conditions at the reservoir will be maintained
through out tube because the process is isentropic.
Hence the stagnation temperature can be written
and
and both of them are known (the
condition at the reservoir).
For the point where the static pressure is known, the Mach
number can be calculated utilizing
the pressure ratio.
With known Mach number, the temperature, and velocity can be
calculated.
Finally, the cross section can be calculated with all these
information.
In the point where the static pressure is known
From Table (4.2) or from Figure
(4.3) or by utilizing the enclosed program,
PottoGDC, or simply by using the equations that
Isentropic Flow 
Input: Pbar 
k = 1.4 
M 
T/T0 
ρ/ρ0 
A/A* 
P/P0 
PAR 
F/F* 
0.886393 
0.864201 
0.694283 
1.01155 
0.6 
0.606928 
0.531054 
With these values the static temperature and the density can be
calculated.
The velocity at that point is
The tube area can be obtained from the mass conservation as
For a circular tube the diameter is about 1[cm].

^{4.3} ^{4.4}
Solution
With known Mach number at point A all the ratios of the static
properties to total (stagnation) properties can be calculated.
Therefore, the stagnation pressure at point A is known
and stagnation temperature can be calculated.
At
(supersonic flow) the ratios are
Isentropic Flow 
Input: M 
k = 1.4 
M 
T/T0 
ρ/ρ0 
A/A* 
P/P0 
PAR 
F/F* 
2 
0.555556 
0.230048 
1.6875 
0.127805 
0.21567 
0.593093 
With this information the pressure at Point B expressed
The corresponding Mach number for this pressure ratio is
1.8137788 and
.
The stagnation temperature can be ``bypassed'' to calculated the
temperature at point

Solution
To obtain the Mach number at point B by finding the ratio of
the area to the critical area.
This relationship can be obtained by
With the value of
from the Table
(4.2) or from PottoGDC
two solutions can be obtained.
The two possible solutions:
the first supersonic M = 1.6265306 and second subsonic
M = 0.53884934.
Both solution are possible and acceptable.
The supersonic branch solution is possible only if there where
a transition at throat where M=1.
Isentropic Flow 
Input: A/A* 
k = 1.4 
M 
T/T0 
ρ/ρ0 
A/A* 
P/P0 
PAR 
F/F* 
0.538865 
0.945112 
0.868378 
1.27211 
0.820715 
1.04404 
0.611863 
1.62655 
0.653965 
0.345848 
1.27211 
0.226172 
0.287717 
0.563918 

Next: Mass Flow Rate (Number)
Up: Isentropic ConvergingDiverging Flow in
Previous: Relationship Between the Mach
Index
Created by:Genick BarMeir, Ph.D.
On:
20071121
