che_february-2025 - 23

(top) shows the input for Example
1, endpoint choking, and Figure 3
(bottom) shows the output. Here,
the Figure 2a boundary condition
combination is used. From Figure 3
(bottom), any discharge pressure at
or below 95.3 psia (657.1 kPa) will
result in choking.
FIGURE 3. Endpoint choking input data (top) and
output data (bottom) are shown for Example 1
ferred because it makes clear the
otherwise ambiguous parenthetical
groups. Equations (1) to (3) are applicable
for real gases.
Preface to all examples
In order to provide closed-form solutions
in Microsoft Excel, simplifying
assumptions are required. All
examples assume the flowing gas is
air, follows the ideal gas law, is calorically
perfect, with an isentropic expansion
coefficient of 1.4. All pipes
are assumed to be adiabatic, constant-diameter
and horizontal, which
allows Equation (3) to be used. Entrance
losses are ignored.
Including real-world behavior, such
as heat transfer, real gas behavior,
sloped pipes and varying friction
factors, is important and will be discussed
in a subsequent section.
Space limitations do not allow a
thorough discussion of all examples,
so brief information will be given
about each one. Refer to Ref. 2 for
more details.
Single-choke-point examples
Example 1: Endpoint choking,
single choking point. Figure 3
Example 2: Expansion choking,
single choking point. Figure 4
(top) shows the input for Example
2 (expansion choking), and Figure
4 (bottom) shows the output. The
input data for Pipe 1 is the same
as for Example 1. Here, the Figure
2a boundary condition combination
is used for Pipe 1. How is Pipe 3
solved? Once the choked flowrate
is determined for Pipe 1, that becomes
a known boundary condition
to use for the inlet of Pipe 3. Hence,
Pipe 3 is solved using the Figure 2c
boundary-condition combination.
From the Law of Conservation of
Mass, the mass flowrate in Pipe 3 is
the same as that in Pipe 1. Further,
from the energy conservation law,
the stagnation enthalpy at the inlet
of Pipe 3 must match the outlet of
Pipe 1 [2] regardless of whether the
pipes are adiabatic or not.
Example 3: Restriction choking,
single choking point. Figure
5 (top) shows the input for Example
3 (restriction choking) and Figure
5 (bottom) shows the output. The
input data for Pipe 1 is the same
as Example 1. Here, the Figure 2a
boundary condition combination is
used for Pipe 1. Once the choked
flowrate at point J2 is determined,
that becomes a known boundary
condition to use for the inlet of Pipe
2. Hence, Pipe 2 is solved using
the Figure 2c boundary condition
combination. Similar to Example
2, mass and energy conservation
across J2 will determine the mass
flowrate and stagnation enthalpy at
the inlet of Pipe 2.
Multiple choke points in series
Example 4: Two choking points
in series with expansion choking
first. Example 4 is similar to Example
2 (Figure 4, top), except for one
change. The downstream pressure
is reduced from 100.6 psia (693.5
kPa) to 50 psia (344.7 kPa). Figure
6 (top) shows the input. This results
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in endpoint choking occurring at
point J4. Hence there are two sonic
choking points. Figure 6 (bottom)
shows the output. Similar to Example
2, the Figure 2a boundary
condition combination is used for
Pipe 1 and the Figure 2c boundary
condition combination is used for
Pipe 3. Mass and energy conservation
across J3 determine the conditions
at the inlet of Pipe 3, similar to
Example 2.
Example 5: Two choking points
in series with restriction choking
first. Example 5 is the same
as Example 3 (with Figure 5, top),
except with one change. The downstream
pressure is reduced from
100.6 psia (693.5 kPa) to 50 psia
(344.7 kPa). Figure 7 (top) shows
the input. This results in endpoint
choking occurring at J4. Hence,
there are two sonic choking points.
Figure 7 (bottom) shows the output.
Similar to Example 3, the Figure 2a
boundary condition combination is
used for Pipe 1 and the Figure 2c
boundary condition combination is
used for Pipe 2. Mass and energy
conservation across J2 determine
FIGURE 4. Expansion choking input data (top) and
output data (bottom) are shown for Example 2
23
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