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Figure:
Unchoked flow calculations
|
This pair of parameters is the most natural to examine
because, in most cases, this information is the only information that
is provided.
For a given pipe

, neither the entrance Mach number
nor the exit Mach number are given (sometimes the entrance
Mach number is give see the next section).
There is no exact analytical solution.
There are two possible approaches to solve this problem:
one, by building a representative function and find a
root (or roots) of this representative function.
Two, the problem can be solved by an iterative procedure.
The first approach require using root finding method and either method
of spline method or the half method found to be good.
However, this author experience show that these methods
in this case were found to be relatively slow.
The Newton-Rapson method is much faster but not
were found to be unstable (at lease in the way that was implemented by
this author).
The iterative method used to solve constructed on the properties
of several physical quantities must be in a certain range..
The first fact is that the pressure ratio

is always between
0 and 1 (see Figure (
9.18)).
In the figure, a theoretical extra tube is added in such a length that
cause the flow to choke (if it really was there).
This length is always positive (at minimum is zero).
The procedure for the calculations is as the following:
- Calculate the entrance Mach number,
assuming the
(chocked flow);
- Calculate the minimum pressure ratio
for
(look at table (9.1))
- Check if the flow is choked:
There are two possibilities to check it.
- a)
- Check if the given
is smaller than
obtained from
the given
, or
- b)
- check if the
is larger than
,
- Calculate the
based on the
,
- calculate
based on
,
- calculate the new
, based on the new
,
(remember that
),
- calculate the corresponding
and
,
- calculate the new and ``improve'' the
by
Note, when the pressure ratios are matching also the
will also match.
- Calculate the ``improved/new''
based on the improve
fanno flow!entrance Mach number calculations
- calculate the improved
as
- calculate the improved
based on the improved
.
- Compare the abs (
)
and if not satisfied
returned to stage (6)
until the solution is obtained.
To demonstrate how this procedure is working consider a typical example of
and
.
Using the above algorithm the results are exhibited in the following
figure.
Figure 9.19:
The results of the algorithm showing the conversion rate for unchoked
Fanno flow model with a given
and pressure ratio.
|
Figure (
9.19) demonstrates that the conversion
occur at about 7-8 iterations. With better first guess this conversion
procedure will converts much faster (under construction).
Next: Subsonic Fanno Flow for
Up: The Practical Questions and
Previous: The Practical Questions and
Index
Created by:Genick Bar-Meir, Ph.D.
On:
2007-11-21
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