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Gas flows from the tank, into the pipe, and discharges:
Units: atm=atmosphere,
cm=centimeter, cP=centiPoise, ft=foot, g=gram, hr=hour, k=kilo (1000), kPa=kiloPascal,
kg=kilogram, km=kilometer, lb=pound, m=meter, min=minute, mm=millimeter, M=Mega (million,
10^{6}) or Thousand (10^{3}) depending on context, MM=Million,
MPa=MegaPascal, Mscfh=thousand std cubic foot per hour, MMscfd=million std cubic foot per
day, mol=mole, N=Newton, N/m^{2}=Newton per square meter (same as Pascal),
Normal=std conditions, Pa=Pascal, s=second, psf=pound per square foot (lb/ft^{2}), psi=pound
per square inch, psia=psi (absolute), psig=psi (gage), scfd=standard cubic foot per day,
scfh=standard cubic foot per hour, scfm=standard cubic foot per minute.
Standard conditions are 15^{ o}C (i.e. 288.15 K, 59^{ o}F,
518.67^{ o}R) and 1 atm (i.e. 101,325 N/m^{2}, 14.696 psia). If gage
pressure units are selected, the calculation assumes atmospheric pressure is 1 atm. The
program uses 1 atm to convert between gage and absolute pressures.
Introduction
The choked flow calculation computes the mass flow rate through a pipe based on tank
pressure and temperature, pipe length and diameter, minor losses, discharge pressure, and
gas properties. Temperatures, pressures, densities, velocities, and Mach numbers are
computed at all transition points (in the tank, at the pipe entrance, in the pipe at the
exit, and in the surroundings at the discharge). As the gas flows through the pipe, the
pressure drops. Hence, density drops and velocity increases. If choking occurs, it will
only occur at the pipe exit (Munson et al., 1998) for flow through a constant diameter
pipe.
In the calculation, gas flowing steadily from a tank into a pipe and discharging to the atmosphere (or another
tank) is modeled using Fanno flow. Fanno flow assumes that the pipe is adiabatic.
Adiabatic means that there is no heat transfer into or out of the pipe. This is physically
accomplished by insulating the pipe. Even if the pipe is not insulated, the adiabatic
assumption is probably more realistic than an isothermal assumption for short lengths of
pipe. In longer pipelines that are isothermal (constant temperature) and subsonic, the Weymouth calculation may be more suitable for computing
flow rates and pressure drops.
Equations
For a gas flowing steadily from a tank to a pipe under Fanno flow conditions (adiabatic),
the procedure follows that of Fanno flow in Gerhart et al. (1992) and Munson et al.
(1998). An example in Perry (1984) uses figures which represent Fanno flow for choked
flow. The following equations are solved simultaneously to compute mass flow rate,
temperatures, pressures, velocities, and Mach numbers.
Gas specific gravity and pipe crosssectional area:
Gas densities using ideal gas law:
The calculation checks to see if choked flow occurs. If flow is choked, then choking
occurs at location 2. Choked flow occurs if P_{3}<P_{2}^{*}.
P_{2}^{*} is the static pressure at location 2 if flow is choked.
If choked flow occurs, then the * is dropped from the superscripts at location 2 for
simplicity. ΣK_{m} represents losses due to elbows and the pipe entrance. M_{1}
is computed from:
Conditions are stagnant in the tank. Assuming the tank exit behaves isentropically:
P_{t} = P_{o,1}
T_{t} = T_{o,1}
ρ_{t} = ρ_{o,1 }
and:
Since choked flow occurs at location 2, M_{2}=1.0 and:
Velocities are:
Since pipe diameter and mass flow rate are constant along the pipe, the Reynolds number
is the same at locations 1 and 2:
Moody friction factor is computed from the following equations:
If laminar flow (Re < 4000 and any ε/D), then
If turbulent flow (4000 < Re < 10^{8} and ε/D <
0.05), then Colebrook equation (Munson et al., 1998, p. 494):
If fully turbulent flow (Re > 10^{8} and 0 < ε/D < 0.05), then
Streeter et al. (1998, p. 289):
Mass flow rate, density at standard conditions, and flow rate at standard conditions
are:
Variables
The units shown for the variables are SI (International System of Units); however, the
equations above are valid for any consistent set of units. Our calculation allows a
variety of units; all unit conversions are accomplished internally.
A = Pipeline crosssectional area, m^{2}.
D = Pipe inside diameter, m.
f = Moody friction factor. (Note that Moody friction factor is 4 times the Fanning
friction factor. Fanning friction factor is often used by chemical engineers.)
k = Gas specific heat ratio. Specific heat at constant pressure divided by specific heat
at constant volume, C_{p}/C_{v}. Default values at 15^{o}C or 20^{o}C
from Munson et al. (1998). k actually varies with temperature and can range from, using
methane as an example, 1.32 at 50^{o}F to 1.28 at 250^{o}F (GPSA, 1998, p.
136).
K_{m} = Minor loss coefficient for pipe entrance, bends, etc. Since flow is choked
(if the calculation proceeds), do not include an exit loss coefficient.
ln = Natural logarithm (base e, where e=2.71828...).
log = Common logarithm (base 10).
L = Pipe length, m.
M = Mach number.
M_{w} = Molecular weight of gas, kg/mole.
P = Absolute pressure, N/m^{2} absolute.
Q = Flow rate at standard conditions, Normal m^{3}/s.
Re = Reynolds number.
R_{u} = Universal gas constant, 8.3144126 Nm/molK (CRC, 1983, p. F208).
S = Specific gravity of gas, relative to air (S_{air}=1).
T = Absolute temperature, Kelvin.
V = Velocity, m/s.
W = Mass flow rate, kg/s.
ε = Pipe roughness, m. Default values from Munson et al. (1998).
μ = Gas dynamic viscosity, kg/ms. The program assumes this is constant even though
temperature (thus viscosity) varies along the pipe.
π = 3.14159....
ρ = Gas density, kg/m^{3}.
Σ = Summation.
Subscripts:
t = Tank.
1 = Pipe entrance.
2 = Pipe exit.
3 = Surroundings.
o = Stagnation property.
s = std = Standard (or "Normal") conditions. The word "standard" is
used for English units. The term "Normal" is used with SI units, as in Nm^{3}/s
which means "Normal m^{3}/s". Standard and Normal conditions for our
choked flow calculation are 15^{ o}C (i.e. 288.15 K, 59^{ o}F, 518.67^{
o}R) and 1 atm (i.e. 101,325 N/m^{2}, 14.696 psia) from Perry (1984, p.
3167). Some industries use different temperature and pressure for standard (or Normal)
conditions. To convert the mass flow rate computed by our choked flow calculation to
volumetric flow at a different T_{s} and P_{s}, use our gas flow conversions page.
If gage pressure units are selected, the calculation assumes atmospheric pressure is P_{s}.
The program uses P_{s} to convert between gage and absolute pressures.
Minor Loss Coefficients (K_{m}).
From Munson et al. (1998).
Fitting 
K_{m} 
Fitting 
K_{m} 
Pipe Entrance (Tank to Pipe): 

Elbows: 

Square Connection 
0.5 
Regular 90^{o}, flanged 
0.3 
Rounded Connection 
0.2 
Regular 90^{o}, threaded 
1.5 


Long radius 90^{o}, flanged 
0.2 


Long radius 90^{o}, threaded 
0.7 
180^{o} return bends: 

Long radius 45^{o}, threaded 
0.2 
Flanged 
0.2 
Regular 45^{o}, threaded 
0.4 
Threaded 
1.5 


Error Messages given by calculation
Input checks:
"Need S > 1e8", "Need Viscosity > 1e20 Ns/m^{2}
", "Need k > 1.0000001", "Need D > 1e9
m", "Need Pipe Roughness > 0", "Need T_{t} >
1e8 Kelvin", "Need P_{t} > 1e8 N/m^{2}
absolute", "Need P_{3} > 1e8 N/m^{2} absolute",
"Need P_{t} > P_{3}". "Need K_{m}+ f L / D
≥ 1e8". Specific gravity, viscosity, specific heat ratio, pipe diameter,
surface roughness, tank temperature, tank pressure, discharge pressure must be acceptable
values.
"Need L > 0". Pipe length can be 0.0 since the software can
model an orifice without actually having a pipe.
Runtime messages:
"Flow is choked". This message will appear during normal running of the
program if flow is choked.
"Flow is subsonic". If the conditions are such (e.g. low P_{t},
high P_{3}, high L, high K_{m}, low D), then flow will not be choked and
the calculation will stop.
"Mach at 1 not found". The program is unable to compute the Mach number
at the pipe inlet.
"Re or ε/D out of range". Reynolds number or roughnesstodiameter
ratio is out of range.
"Moody f not found". The program is unable to determine the friction
factor. Reynolds number or roughnesstodiameter ratio may be out of range.
References
Chemical Rubber Company (CRC). 1983. CRC Handbook of Chemistry and
Physics. Weast, Robert C., editor. 63rd edition. CRC Press, Inc. Boca Raton, Florida.
Gerhart, P. M., R. J. Gross, and J. I. Hochstein. 1992. Fundamentals of Fluid
Mechanics. AddisonWesley Pub. Co. 2ed.
Gas Processors Suppliers Association (GPSA). 1998. Engineering Data Book (fps
[footpoundsecond] version). Gas Processors Association. 11ed.
Perry, R. H. and D. W. Green, editors. 1984. Perry's Chemical Engineers' Handbook.
McGrawHill, Inc. 6ed.
Munson, B.R., D. F. Young, and T. H. Okiishi. 1998. Fundamentals of Fluid Mechanics.
John Wiley and Sons, Inc. 3ed.
Streeter, V. L., E. B. Wylie, and K. W. Bedford. 1998. Fluid Mechanics. McGrawHill.
9ed.
© 20102020 LMNO Engineering, Research, and
Software, Ltd. All rights reserved.
Please contact us for consulting or questions about choked gas flow in pipes.
LMNO Engineering, Research, and Software, Ltd.
7860 Angel Ridge Rd. Athens, Ohio 45701 USA Phone: (740) 7072614
LMNO@LMNOeng.com https://www.LMNOeng.com

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