che_february-2025 - 22

Geometry of sonic
choking
Both Refs. 1 and 2 discuss
in depth the three
geometric situations
where sonic choking
occurs. Because the
examples from Ref. 2
to be explored here are
designed to examine
all the
FIGURE 1. Each of these pipe configurations can
result in sonic choking
Quick review of sonic choking
Sonic choking occurs when the gas
bulk velocity reaches the gas sonic
velocity somewhere in a pipe system.
This is another way of saying
that the Mach number reaches 1.
When this happens in a sequential
run of pipes, then lowering the
downstream pressure will not produce
any additional flow. Figures 3
and 4 from Ref. 1 and the associated
discussion there do a good job
of
elaborating.
Further discussion
on sonic choking can also be found
in Ref. 2.
When the first article was published
in 2000, the author was under
the impression that a sonic choking
point involved a shock wave. After
discussions of this concept with several
other knowledgeable engineers
over the years since the publication,
he is now unsure of this, and has
stopped calling it a shock wave. Regardless,
a sonic choking point is (at
minimum) similar to a shock wave in
that it involves a discontinuity in the
flow field. Therefore, whether or not
there is a shock wave is somewhat
irrelevant. A discontinuity exists, and
that is what is important.
these three geometries,
including cases where
two, or all three, geometries
are happening in
the same system, the
three situations are discussed
here.
NOMENCLATURE
A
cp, cv
D
f
L
m∙
possibilities of
M
P, Po
R
T, To
v
Z
1. Endpoint choking
(Figure 1, top diagram).
Endpoint choking occurs
when the gas pressure
in the pipe cannot
drop down to the discharge pressure
without accelerating to sonic
velocity thereby resulting in a choke
point (and pressure discontinuity) at
the end of the pipe.
2. Expansion choking (Figure 1,
middle diagram). Expansion choking
occurs when there is an increase in
pipe area, such as an expander, diverging
tee, or discharge into a large
header pipe. Here, the gas cannot
navigate to the pipe discharge
downstream without accelerating to
sonic velocity at the point where the
area increases. A choke point and
pressure discontinuity thus occurs.
3. Restriction choking (Figure 1,
bottom diagram). Restriction choking
occurs when there is restriction
(resultting from an orifice or valve,
for example) where the gas cannot
navigate through the restriction without
accelerating to sonic velocity. A
choke point and pressure discontinuity
thus occurs at the restriction.
γ
FIGURE 2. The diagram here shows types of solvable
boundary condition combinations for a pipe or
pipe sequence (flow from left to right)
22
Solvable boundary conditions
In a sequence of pipes, there are
three types of solvable boundary
conditions [2], as depicted in Figure
2. Repeateed reference to Figure 2,
diagrams A, B and C will be made
throughout the following seven examples
in order to point out which
part of the choked pipe system is
solved with which combination of
boundary conditions.
Cross-sectional area (ft2 / m2)
Specific heat at constant pressure and
volume (Btu/lbm · R, kJ/kg · K)
Inner diameter (ft / m)
Friction factor, Darcy-Weisbach (dimensionless
quantity)
Length (ft / m)
Mass flowrate (lbm/s / kg/s)
Mach number (dimensionless quantity)
Static and stagnation pressure (psi / kPa)
Molar gas constant (Btu/lbm-R or J/kg-K)
Static and stagnation temperature (°F / °C)
Specific volume (ft3/lbm, m3/kg, inverse of
density)
Compressibility factor, correction for non-ideal
gas (dimensionless quantity)
Isentropic expansion coefficient, also known
as k (dimensionless quantity)
Important equations
A form of the mass-conservation
equation is highly useful for gas-flow
calculations in pipes, in general, and
for sonic-choking calculations specifically
(Equation (1) [1, 2]).
(1)
Because choked flow occurs when
the Mach number equals 1, Equation
(1) becomes Equation (2).
(2)
A closed-form solution of adiabatic
flow in constant-diameter pipes can
be found in most compressible-flow
textbooks where the pipe is
horizontal
with a constant friction factor
(Equation (3) [1, 2]):
(3)
Note that Equation (3) is sometimes
presented without all the parenthetical
groupings, which can be
misleading. For example, the author
did this in Equation 11 from Ref. 1.
The Equation (3) form here is preCHEMICAL
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