che_february-2025 - 25

determine conditions downstream of
choke points. In most real systems,
Equation (3) will not get you far. Realworld
effects, such as heat transfer
between ambient surroundings and
the gas will need to be accounted
for. Other effects include real gas
behavior, sloped or vertical pipes,
variable friction factors and complex
pipe networks. A capable software
tool that accounts for all these effects
is essential to obtain accurate
predictions in such situations [4].
Do you know what γ (or k) is?
The isentropic expansion coefficient
is commonly called γ by gas dynamicists
(or
k by thermodynamicists).
In engineering school, the author
learned the following relationship,
shown in Equation (4):
(4)
FIGURE 7. Restriction choking input data (top) and
output data (bottom) are shown for Example 5 for
two choking points in series
rate knowledge of the stagnation
pressure and temperature, Po, and
To, must exist at each local choke
point. When conditions are ideal,
like the examples presented here,
it is more straightforward to determine
these conditions using equations
such as Equation (3). However,
when real-world conditions
must be accounted for, it is quite
complicated to determine Po and
To at each choke point. Indeed,
determining these values depends
on knowing the mass flowrate -
which is the variable for which we
are solving. In such cases, heavy
numerical iteration is needed using
the governing conservation equations.
The original article [1] discusses
these in more detail.
Simplified to complicated
Simplifications such as Equation (3)
and the examples presented here
are useful to gain an appreciation
of how to navigate single or multiple
sonic choking points - and how to
However, a little-known fact [5] is
that Equation (4) is actually an approximation,
and is the simplified
form of the full definition of γ, as
shown in Equation (5):
(5)
It is often the case that the " correction
term " on the righthand side
of Equation (5) is close to unity, thus
making Equation (4) approximately
correct. On the other hand, the author
has found that the complete
Equation (5) is commonly required in
gas-pipe analysis in order to obtain
accurate results.
Part of the reason for discussing
this here is that after Ref. 1 was published
in 2000, there was a respectful,
but critical, letter to the editor in a subsequent
issue specifically about the
assumptions in the 2000 article. An
exchange of letters occurred, where
Equation (5) was offered as being preferred
to Equation (4), which satisfied
the critic that the assumptions used in
Ref. 1 were, in fact, sound.
Sonic choking analysis software
In order to accurately predict sonic
choking in pipe systems, software
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solutions should be able to handle
all types of sonic choking, including
multiple choking points in series
and parallel. Further, a proper energy
balance should be performed on the
pipes and choke points. Features
such as heat transfer, sloped pipes
and real-gas modeling is recommended.
For a complete list, see
Section 9 in Ref. 2.
Corrections to Ref. 1
The revisiting of the 2000 sonic
choking article allows us to correct
some typographical errors in Ref. 1
that I found after publication. All errors
were the fault of the author.
Equation (14c) in Ref. 1 should
have a negative sign, as shown in
Equation (6).
(6)
Equation (14e) in Ref. 1 should
read, as follows in Equation (7).
25
FIGURE 8. Input data (top) and output data (bottom)
are shown for Example 6, for three choking
points in series
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