Compressible flow and compressor sizing (fluids.compressible)¶

fluids.compressible.
Panhandle_A
(SG, Tavg, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7, Ps=101325.0, Zavg=1, E=0.92)[source]¶ Calculation function for dealing with flow of a compressible gas in a pipeline with the Panhandle A formula. Can calculate any of the following, given all other inputs:
 Flow rate
 Upstream pressure
 Downstream pressure
 Diameter of pipe
 Length of pipe
A variety of different constants and expressions have been presented for the Panhandle A equation. Here, a new form is developed with all units in base SI, based on the work of [R2171].
\[Q = 158.02053 E \left(\frac{T_s}{P_s}\right)^{1.0788}\left[\frac{P_1^2 P_2^2}{L \cdot {SG}^{0.8539} T_{avg}Z_{avg}}\right]^{0.5394}D^{2.6182}\]Parameters: SG : float
Specific gravity of fluid with respect to air at the reference temperature and pressure Ts and Ps, []
Tavg : float
Average temperature of the fluid in the pipeline, [K]
L : float, optional
Length of pipe, [m]
D : float, optional
Diameter of pipe, [m]
P1 : float, optional
Inlet pressure to pipe, [Pa]
P2 : float, optional
Outlet pressure from pipe, [Pa]
Q : float, optional
Flow rate of gas through pipe, [m^3/s]
Ts : float, optional
Reference temperature for the specific gravity of the gas, [K]
Ps : float, optional
Reference pressure for the specific gravity of the gas, [Pa]
Zavg : float, optional
Average compressibility factor for gas, []
E : float, optional
Pipeline efficiency, a correction factor between 0 and 1
Returns: Q, P1, P2, D, or L : float
The missing input which was solved for [base SI]
Notes
[R2171]‘s original constant was 4.5965E3, and it has units of km (length), kPa, mm (diameter), and flowrate in m^3/day.
The form in [R2271] has the same exponents as used here, units of mm (diameter), kPa, km (length), and flow in m^3/hour; its leading constant is 1.9152E4.
The GPSA [R2371] has a leading constant of 0.191, a bracketed power of 0.5392, a specific gravity power of 0.853, and otherwise the same constants. It is in units of mm (diameter) and kPa and m^3/day; length is stated to be in km, but according to the errata is in m.
[R2471] has a leading constant of 1.198E7, a specific gravity of power of 0.8541, and a power of diameter which is under the root of 4.854 and is otherwise the same. It has units of kPa and m^3/day, but is otherwise in base SI units.
[R2571] has a leading constant of 99.5211, but its reference correction has no exponent; other exponents are the same as here. It is entirely in base SI units.
[R2671] has pressures in psi, diameter in inches, length in miles, Q in ft^3/day, T in degrees Rankine, and a constant of 435.87. Its reference condition power is 1.07881, and it has a specific gravity correction outside any other term with a power of 0.4604.
References
[R2171] (1, 2, 3) Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton, FL: CRC Press, 2005. [R2271] (1, 2) Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe. Crane, 2009. [R2371] (1, 2) GPSA. GPSA Engineering Data Book. 13th edition. Gas Processors Suppliers Association, Tulsa, OK, 2012. [R2471] (1, 2) Campbell, John M. Gas Conditioning and Processing, Vol. 2: The Equipment Modules. 7th edition. Campbell Petroleum Series, 1992. [R2571] (1, 2) Coelho, Paulo M., and Carlos Pinho. “Considerations about Equations for Steady State Flow in Natural Gas Pipelines.” Journal of the Brazilian Society of Mechanical Sciences and Engineering 29, no. 3 (September 2007): 26273. doi:10.1590/S167858782007000300005. [R2671] (1, 2) Ikoku, Chi U. Natural Gas Production Engineering. Malabar, Fla: Krieger Pub Co, 1991. Examples
>>> Panhandle_A(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, Tavg=277.15) 42.56082051195928

fluids.compressible.
Panhandle_B
(SG, Tavg, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7, Ps=101325.0, Zavg=1, E=0.92)[source]¶ Calculation function for dealing with flow of a compressible gas in a pipeline with the Panhandle B formula. Can calculate any of the following, given all other inputs:
 Flow rate
 Upstream pressure
 Downstream pressure
 Diameter of pipe
 Length of pipe
A variety of different constants and expressions have been presented for the Panhandle B equation. Here, a new form is developed with all units in base SI, based on the work of [R2777].
\[Q = 152.88116 E \left(\frac{T_s}{P_s}\right)^{1.02}\left[\frac{P_1^2 P_2^2}{L \cdot {SG}^{0.961} T_{avg}Z_{avg}}\right]^{0.51}D^{2.53}\]Parameters: SG : float
Specific gravity of fluid with respect to air at the reference temperature and pressure Ts and Ps, []
Tavg : float
Average temperature of the fluid in the pipeline, [K]
L : float, optional
Length of pipe, [m]
D : float, optional
Diameter of pipe, [m]
P1 : float, optional
Inlet pressure to pipe, [Pa]
P2 : float, optional
Outlet pressure from pipe, [Pa]
Q : float, optional
Flow rate of gas through pipe, [m^3/s]
Ts : float, optional
Reference temperature for the specific gravity of the gas, [K]
Ps : float, optional
Reference pressure for the specific gravity of the gas, [Pa]
Zavg : float, optional
Average compressibility factor for gas, []
E : float, optional
Pipeline efficiency, a correction factor between 0 and 1
Returns: Q, P1, P2, D, or L : float
The missing input which was solved for [base SI]
Notes
[R2777]‘s original constant was 1.002E2, and it has units of km (length), kPa, mm (diameter), and flowrate in m^3/day.
The form in [R2877] has the same exponents as used here, units of mm (diameter), kPa, km (length), and flow in m^3/hour; its leading constant is 4.1749E4.
The GPSA [R2977] has a leading constant of 0.339, and otherwise the same constants. It is in units of mm (diameter) and kPa and m^3/day; length is stated to be in km, but according to the errata is in m.
[R3077] has a leading constant of 1.264E7, a diameter power of 4.961 which is also under the 0.51 power, and is otherwise the same. It has units of kPa and m^3/day, but is otherwise in base SI units.
[R3177] has a leading constant of 135.8699, but its reference correction has no exponent and its specific gravity has a power of 0.9608; the other exponents are the same as here. It is entirely in base SI units.
[R3277] has pressures in psi, diameter in inches, length in miles, Q in ft^3/day, T in degrees Rankine, and a constant of 737 with the exponents the same as here.
References
[R2777] (1, 2, 3) Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton, FL: CRC Press, 2005. [R2877] (1, 2) Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe. Crane, 2009. [R2977] (1, 2) GPSA. GPSA Engineering Data Book. 13th edition. Gas Processors Suppliers Association, Tulsa, OK, 2012. [R3077] (1, 2) Campbell, John M. Gas Conditioning and Processing, Vol. 2: The Equipment Modules. 7th edition. Campbell Petroleum Series, 1992. [R3177] (1, 2) Coelho, Paulo M., and Carlos Pinho. “Considerations about Equations for Steady State Flow in Natural Gas Pipelines.” Journal of the Brazilian Society of Mechanical Sciences and Engineering 29, no. 3 (September 2007): 26273. doi:10.1590/S167858782007000300005. [R3277] (1, 2) Ikoku, Chi U. Natural Gas Production Engineering. Malabar, Fla: Krieger Pub Co, 1991. Examples
>>> Panhandle_B(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, Tavg=277.15) 42.35366178004172

fluids.compressible.
Weymouth
(SG, Tavg, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7, Ps=101325.0, Zavg=1, E=0.92)[source]¶ Calculation function for dealing with flow of a compressible gas in a pipeline with the Weymouth formula. Can calculate any of the following, given all other inputs:
 Flow rate
 Upstream pressure
 Downstream pressure
 Diameter of pipe
 Length of pipe
A variety of different constants and expressions have been presented for the Weymouth equation. Here, a new form is developed with all units in base SI, based on the work of [R3383].
\[Q = 137.32958 E \frac{T_s}{P_s}\left[\frac{P_1^2 P_2^2}{L \cdot {SG} \cdot T_{avg}Z_{avg}}\right]^{0.5}D^{2.667}\]Parameters: SG : float
Specific gravity of fluid with respect to air at the reference temperature and pressure Ts and Ps, []
Tavg : float
Average temperature of the fluid in the pipeline, [K]
L : float, optional
Length of pipe, [m]
D : float, optional
Diameter of pipe, [m]
P1 : float, optional
Inlet pressure to pipe, [Pa]
P2 : float, optional
Outlet pressure from pipe, [Pa]
Q : float, optional
Flow rate of gas through pipe, [m^3/s]
Ts : float, optional
Reference temperature for the specific gravity of the gas, [K]
Ps : float, optional
Reference pressure for the specific gravity of the gas, [Pa]
Zavg : float, optional
Average compressibility factor for gas, []
E : float, optional
Pipeline efficiency, a correction factor between 0 and 1
Returns: Q, P1, P2, D, or L : float
The missing input which was solved for [base SI]
Notes
[R3383]‘s original constant was 3.7435E3, and it has units of km (length), kPa, mm (diameter), and flowrate in m^3/day.
The form in [R3483] has the same exponents as used here, units of mm (diameter), kPa, km (length), and flow in m^3/hour; its leading constant is 1.5598E4.
The GPSA [R3583] has a leading constant of 0.1182, and otherwise the same constants. It is in units of mm (diameter) and kPa and m^3/day; length is stated to be in km, but according to the errata is in m.
[R3683] has a leading constant of 1.162E7, a diameter power of 5.333 which is also under the 0.50 power, and is otherwise the same. It has units of kPa and m^3/day, but is otherwise in base SI units.
[R3783] has a leading constant of 137.2364; the other exponents are the same as here. It is entirely in base SI units.
[R3883] has pressures in psi, diameter in inches, length in miles, Q in ft^3/hour, T in degrees Rankine, and a constant of 18.062 with the exponents the same as here.
References
[R3383] (1, 2, 3) Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton, FL: CRC Press, 2005. [R3483] (1, 2) Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe. Crane, 2009. [R3583] (1, 2) GPSA. GPSA Engineering Data Book. 13th edition. Gas Processors Suppliers Association, Tulsa, OK, 2012. [R3683] (1, 2) Campbell, John M. Gas Conditioning and Processing, Vol. 2: The Equipment Modules. 7th edition. Campbell Petroleum Series, 1992. [R3783] (1, 2) Coelho, Paulo M., and Carlos Pinho. “Considerations about Equations for Steady State Flow in Natural Gas Pipelines.” Journal of the Brazilian Society of Mechanical Sciences and Engineering 29, no. 3 (September 2007): 26273. doi:10.1590/S167858782007000300005. [R3883] (1, 2) Ikoku, Chi U. Natural Gas Production Engineering. Malabar, Fla: Krieger Pub Co, 1991. Examples
>>> Weymouth(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, Tavg=277.15) 32.07729055913029

fluids.compressible.
Spitzglass_high
(SG, Tavg, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7, Ps=101325.0, Zavg=1, E=1.0)[source]¶ Calculation function for dealing with flow of a compressible gas in a pipeline with the Spitzglass (high pressure drop) formula. Can calculate any of the following, given all other inputs:
 Flow rate
 Upstream pressure
 Downstream pressure
 Diameter of pipe (numerical solution)
 Length of pipe
A variety of different constants and expressions have been presented for the Spitzglass (high pressure drop) formula. Here, the form as in [R3989] is used but with a more precise metric conversion from inches to m.
\[Q = 125.1060 E \left(\frac{T_s}{P_s}\right)\left[\frac{P_1^2 P_2^2}{L \cdot {SG} T_{avg}Z_{avg} (1 + 0.09144/D + \frac{150}{127}D)} \right]^{0.5}D^{2.5}\]Parameters: SG : float
Specific gravity of fluid with respect to air at the reference temperature and pressure Ts and Ps, []
Tavg : float
Average temperature of the fluid in the pipeline, [K]
L : float, optional
Length of pipe, [m]
D : float, optional
Diameter of pipe, [m]
P1 : float, optional
Inlet pressure to pipe, [Pa]
P2 : float, optional
Outlet pressure from pipe, [Pa]
Q : float, optional
Flow rate of gas through pipe, [m^3/s]
Ts : float, optional
Reference temperature for the specific gravity of the gas, [K]
Ps : float, optional
Reference pressure for the specific gravity of the gas, [Pa]
Zavg : float, optional
Average compressibility factor for gas, []
E : float, optional
Pipeline efficiency, a correction factor between 0 and 1
Returns: Q, P1, P2, D, or L : float
The missing input which was solved for [base SI]
Notes
This equation is often presented without any correction for reference conditions for specific gravity.
This model is also presented in [R4089] with a leading constant of 1.0815E2, the same exponents as used here, units of mm (diameter), kPa, km (length), and flow in m^3/hour.
References
[R3989] (1, 2) Coelho, Paulo M., and Carlos Pinho. “Considerations about Equations for Steady State Flow in Natural Gas Pipelines.” Journal of the Brazilian Society of Mechanical Sciences and Engineering 29, no. 3 (September 2007): 26273. doi:10.1590/S167858782007000300005. [R4089] (1, 2) Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton, FL: CRC Press, 2005. Examples
>>> Spitzglass_high(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, Tavg=277.15) 29.42670246281681

fluids.compressible.
Spitzglass_low
(SG, Tavg, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7, Ps=101325.0, Zavg=1, E=1.0)[source]¶ Calculation function for dealing with flow of a compressible gas in a pipeline with the Spitzglass (low pressure drop) formula. Can calculate any of the following, given all other inputs:
 Flow rate
 Upstream pressure
 Downstream pressure
 Diameter of pipe (numerical solution)
 Length of pipe
A variety of different constants and expressions have been presented for the Spitzglass (low pressure drop) formula. Here, the form as in [R4191] is used but with a more precise metric conversion from inches to m.
\[Q = 125.1060 E \left(\frac{T_s}{P_s}\right)\left[\frac{2(P_1 P_2)(P_s+1210)}{L \cdot {SG} \cdot T_{avg}Z_{avg} (1 + 0.09144/D + \frac{150}{127}D)}\right]^{0.5}D^{2.5}\]Parameters: SG : float
Specific gravity of fluid with respect to air at the reference temperature and pressure Ts and Ps, []
Tavg : float
Average temperature of the fluid in the pipeline, [K]
L : float, optional
Length of pipe, [m]
D : float, optional
Diameter of pipe, [m]
P1 : float, optional
Inlet pressure to pipe, [Pa]
P2 : float, optional
Outlet pressure from pipe, [Pa]
Q : float, optional
Flow rate of gas through pipe, [m^3/s]
Ts : float, optional
Reference temperature for the specific gravity of the gas, [K]
Ps : float, optional
Reference pressure for the specific gravity of the gas, [Pa]
Zavg : float, optional
Average compressibility factor for gas, []
E : float, optional
Pipeline efficiency, a correction factor between 0 and 1
Returns: Q, P1, P2, D, or L : float
The missing input which was solved for [base SI]
Notes
This equation is often presented without any correction for reference conditions for specific gravity.
This model is also presented in [R4291] with a leading constant of 5.69E2, the same exponents as used here, units of mm (diameter), kPa, km (length), and flow in m^3/hour. However, it is believed to contain a typo, and gives results <1/3 of the correct values. It is also present in [R4291] in imperial form; this is believed correct, but makes a slight assumption not done in [R4191].
This model is present in [R4391] without reference corrections. The 1210 constant in [R4191] is an approximation necessary for the reference correction to function without a square of the pressure difference. The GPSA version is as follows, and matches this formulation very closely:
\[Q = 0.821 \left[\frac{(P_1P_2)D^5}{L \cdot {SG} (1 + 91.44/D + 0.0018D)}\right]^{0.5}\]The model is also shown in [R4491], with diameter in inches, length in feet, flow in MMSCFD, pressure drop in inH2O, and a rounded leading constant of 0.09; this makes its predictions several percent higher than the model here.
References
[R4191] (1, 2, 3, 4) Coelho, Paulo M., and Carlos Pinho. “Considerations about Equations for Steady State Flow in Natural Gas Pipelines.” Journal of the Brazilian Society of Mechanical Sciences and Engineering 29, no. 3 (September 2007): 26273. doi:10.1590/S167858782007000300005. [R4291] (1, 2, 3) Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton, FL: CRC Press, 2005. [R4391] (1, 2) GPSA. GPSA Engineering Data Book. 13th edition. Gas Processors Suppliers Association, Tulsa, OK, 2012. [R4491] (1, 2) PetroWiki. “Pressure Drop Evaluation along Pipelines” Accessed September 11, 2016. http://petrowiki.org/Pressure_drop_evaluation_along_pipelines#Spitzglass_equation_2. Examples
>>> Spitzglass_low(D=0.154051, P1=6720.3199, P2=0, L=54.864, SG=0.6, Tavg=288.7) 0.9488775242530617

fluids.compressible.
Oliphant
(SG, Tavg, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7, Ps=101325.0, Zavg=1, E=0.92)[source]¶ Calculation function for dealing with flow of a compressible gas in a pipeline with the Oliphant formula. Can calculate any of the following, given all other inputs:
 Flow rate
 Upstream pressure
 Downstream pressure
 Diameter of pipe (numerical solution)
 Length of pipe
This model is a more complete conversion to metric of the Imperial version presented in [R4595].
\[Q = 84.5872\left(D^{2.5} + 0.20915D^3\right)\frac{T_s}{P_s}\left(\frac {P_1^2  P_2^2}{L\cdot {SG} \cdot T_{avg}}\right)^{0.5}\]Parameters: SG : float
Specific gravity of fluid with respect to air at the reference temperature and pressure Ts and Ps, []
Tavg : float
Average temperature of the fluid in the pipeline, [K]
L : float, optional
Length of pipe, [m]
D : float, optional
Diameter of pipe, [m]
P1 : float, optional
Inlet pressure to pipe, [Pa]
P2 : float, optional
Outlet pressure from pipe, [Pa]
Q : float, optional
Flow rate of gas through pipe, [m^3/s]
Ts : float, optional
Reference temperature for the specific gravity of the gas, [K]
Ps : float, optional
Reference pressure for the specific gravity of the gas, [Pa]
Zavg : float, optional
Average compressibility factor for gas, []
E : float, optional
Pipeline efficiency, a correction factor between 0 and 1
Returns: Q, P1, P2, D, or L : float
The missing input which was solved for [base SI]
Notes
Recommended in [R4595] for use between vacuum and 100 psi.
The model is simplified by grouping constants here; however, it is presented in the imperial unit set inches (diameter), miles (length), psi, Rankine, and MMSCFD in [R4595]:
\[Q = 42(24)\left(D^{2.5} + \frac{D^3}{30}\right)\left(\frac{14.4}{P_s} \right)\left(\frac{T_s}{520}\right)\left[\left(\frac{0.6}{SG}\right) \left(\frac{520}{T_{avg}}\right)\left(\frac{P_1^2  P_2^2}{L}\right) \right]^{0.5}\]References
[R4595] (1, 2, 3, 4) GPSA. GPSA Engineering Data Book. 13th edition. Gas Processors Suppliers Association, Tulsa, OK, 2012. [R4695]  Oliphant, “Production of Natural Gas,” Report. USGS, 1902.
Examples
>>> Oliphant(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, Tavg=277.15) 28.851535408143057

fluids.compressible.
Fritzsche
(SG, Tavg, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7, Ps=101325.0, Zavg=1, E=1)[source]¶ Calculation function for dealing with flow of a compressible gas in a pipeline with the Fritzsche formula. Can calculate any of the following, given all other inputs:
 Flow rate
 Upstream pressure
 Downstream pressure
 Diameter of pipe
 Length of pipe
A variety of different constants and expressions have been presented for the Fritzsche formula. Here, the form as in [R4797] is used but with all inputs in base SI units.
\[Q = 93.500 \frac{T_s}{P_s}\left(\frac{P_1^2  P_2^2} {L\cdot {SG}^{0.8587} \cdot T_{avg}}\right)^{0.538}D^{2.69}\]Parameters: SG : float
Specific gravity of fluid with respect to air at the reference temperature and pressure Ts and Ps, []
Tavg : float
Average temperature of the fluid in the pipeline, [K]
L : float, optional
Length of pipe, [m]
D : float, optional
Diameter of pipe, [m]
P1 : float, optional
Inlet pressure to pipe, [Pa]
P2 : float, optional
Outlet pressure from pipe, [Pa]
Q : float, optional
Flow rate of gas through pipe, [m^3/s]
Ts : float, optional
Reference temperature for the specific gravity of the gas, [K]
Ps : float, optional
Reference pressure for the specific gravity of the gas, [Pa]
Zavg : float, optional
Average compressibility factor for gas, []
E : float, optional
Pipeline efficiency, a correction factor between 0 and 1
Returns: Q, P1, P2, D, or L : float
The missing input which was solved for [base SI]
Notes
This model is also presented in [R4797] with a leading constant of 2.827, the same exponents as used here, units of mm (diameter), kPa, km (length), and flow in m^3/hour.
This model is shown in base SI units in [R4897], and with a leading constant of 94.2565, a diameter power of 2.6911, main group power of 0.5382 and a specific gravity power of 0.858. The difference is very small.
References
[R4797] (1, 2, 3) Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton, FL: CRC Press, 2005. [R4897] (1, 2) Coelho, Paulo M., and Carlos Pinho. “Considerations about Equations for Steady State Flow in Natural Gas Pipelines.” Journal of the Brazilian Society of Mechanical Sciences and Engineering 29, no. 3 (September 2007): 26273. doi:10.1590/S167858782007000300005. Examples
>>> Fritzsche(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, Tavg=277.15) 39.421535157535565

fluids.compressible.
Muller
(SG, Tavg, mu, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7, Ps=101325.0, Zavg=1, E=1)[source]¶ Calculation function for dealing with flow of a compressible gas in a pipeline with the Muller formula. Can calculate any of the following, given all other inputs:
 Flow rate
 Upstream pressure
 Downstream pressure
 Diameter of pipe
 Length of pipe
A variety of different constants and expressions have been presented for the Muller formula. Here, the form as in [R4999] is used but with all inputs in base SI units.
\[Q = 15.7743\frac{T_s}{P_s}E\left(\frac{P_1^2  P_2^2}{L \cdot Z_{avg} \cdot T_{avg}}\right)^{0.575} \left(\frac{D^{2.725}}{\mu^{0.15} SG^{0.425}}\right)\]Parameters: SG : float
Specific gravity of fluid with respect to air at the reference temperature and pressure Ts and Ps, []
Tavg : float
Average temperature of the fluid in the pipeline, [K]
mu : float
Average viscosity of the fluid in the pipeline, [Pa*s]
L : float, optional
Length of pipe, [m]
D : float, optional
Diameter of pipe, [m]
P1 : float, optional
Inlet pressure to pipe, [Pa]
P2 : float, optional
Outlet pressure from pipe, [Pa]
Q : float, optional
Flow rate of gas through pipe, [m^3/s]
Ts : float, optional
Reference temperature for the specific gravity of the gas, [K]
Ps : float, optional
Reference pressure for the specific gravity of the gas, [Pa]
Zavg : float, optional
Average compressibility factor for gas, []
E : float, optional
Pipeline efficiency, a correction factor between 0 and 1
Returns: Q, P1, P2, D, or L : float
The missing input which was solved for [base SI]
Notes
This model is presented in [R4999] with a leading constant of 0.4937, the same exponents as used here, units of inches (diameter), psi, feet (length), Rankine, pound/(foot*second) for viscosity, and 1000 ft^3/hour.
This model is also presented in [R5099] in both SI and imperial form. The SI form was incorrectly converted and yields much higher flow rates. The imperial version has a leading constant of 85.7368, the same powers as used here except with rounded values of powers of viscosity (0.2609) and specific gravity (0.7391) rearranged to be inside the bracketed group; its units are inches (diameter), psi, miles (length), Rankine, pound/(foot*second) for viscosity, and ft^3/day.
This model is shown in base SI units in [R5199], and with a leading constant of 15.7650, a diameter power of 2.724, main group power of 0.5747, a specific gravity power of 0.74, and a viscosity power of 0.1494.
References
[R4999] (1, 2, 3) Mohitpour, Mo, Golshan, and Allan Murray. Pipeline Design and Construction: A Practical Approach. 3rd edition. New York: Amer Soc Mechanical Engineers, 2006. [R5099] (1, 2) Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton, FL: CRC Press, 2005. [R5199] (1, 2) Coelho, Paulo M., and Carlos Pinho. “Considerations about Equations for Steady State Flow in Natural Gas Pipelines.” Journal of the Brazilian Society of Mechanical Sciences and Engineering 29, no. 3 (September 2007): 26273. doi:10.1590/S167858782007000300005. Examples
>>> Muller(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, mu=1E5, ... Tavg=277.15) 60.45796698148659

fluids.compressible.
IGT
(SG, Tavg, mu, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7, Ps=101325.0, Zavg=1, E=1)[source]¶ Calculation function for dealing with flow of a compressible gas in a pipeline with the IGT formula. Can calculate any of the following, given all other inputs:
 Flow rate
 Upstream pressure
 Downstream pressure
 Diameter of pipe
 Length of pipe
A variety of different constants and expressions have been presented for the IGT formula. Here, the form as in [R52102] is used but with all inputs in base SI units.
\[Q = 24.6241\frac{T_s}{P_s}E\left(\frac{P_1^2  P_2^2}{L \cdot Z_{avg} \cdot T_{avg}}\right)^{5/9} \left(\frac{D^{8/3}}{\mu^{1/9} SG^{4/9}}\right)\]Parameters: SG : float
Specific gravity of fluid with respect to air at the reference temperature and pressure Ts and Ps, []
Tavg : float
Average temperature of the fluid in the pipeline, [K]
mu : float
Average viscosity of the fluid in the pipeline, [Pa*s]
L : float, optional
Length of pipe, [m]
D : float, optional
Diameter of pipe, [m]
P1 : float, optional
Inlet pressure to pipe, [Pa]
P2 : float, optional
Outlet pressure from pipe, [Pa]
Q : float, optional
Flow rate of gas through pipe, [m^3/s]
Ts : float, optional
Reference temperature for the specific gravity of the gas, [K]
Ps : float, optional
Reference pressure for the specific gravity of the gas, [Pa]
Zavg : float, optional
Average compressibility factor for gas, []
E : float, optional
Pipeline efficiency, a correction factor between 0 and 1
Returns: Q, P1, P2, D, or L : float
The missing input which was solved for [base SI]
Notes
This model is presented in [R52102] with a leading constant of 0.6643, the same exponents as used here, units of inches (diameter), psi, feet (length), Rankine, pound/(foot*second) for viscosity, and 1000 ft^3/hour.
This model is also presented in [R53102] in both SI and imperial form. Both forms are correct. The imperial version has a leading constant of 136.9, the same powers as used here except with rounded values of powers of viscosity (0.2) and specific gravity (0.8) rearranged to be inside the bracketed group; its units are inches (diameter), psi, miles (length), Rankine, pound/(foot*second) for viscosity, and ft^3/day.
This model is shown in base SI units in [R54102], and with a leading constant of 24.6145, and the same powers as used here.
References
[R52102] (1, 2, 3) Mohitpour, Mo, Golshan, and Allan Murray. Pipeline Design and Construction: A Practical Approach. 3rd edition. New York: Amer Soc Mechanical Engineers, 2006. [R53102] (1, 2) Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton, FL: CRC Press, 2005. [R54102] (1, 2) Coelho, Paulo M., and Carlos Pinho. “Considerations about Equations for Steady State Flow in Natural Gas Pipelines.” Journal of the Brazilian Society of Mechanical Sciences and Engineering 29, no. 3 (September 2007): 26273. doi:10.1590/S167858782007000300005. Examples
>>> IGT(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, mu=1E5, Tavg=277.15) 48.92351786788815

fluids.compressible.
isothermal_gas
(rho, fd, P1=None, P2=None, L=None, D=None, m=None)[source]¶ Calculation function for dealing with flow of a compressible gas in a pipeline for the complete isothermal flow equation. Can calculate any of the following, given all other inputs:
 Mass flow rate
 Upstream pressure (numerical)
 Downstream pressure (analytical or numerical if an overflow occurs)
 Diameter of pipe (numerical)
 Length of pipe
A variety of forms of this equation have been presented, differing in their use of the ideal gas law and choice of gas constant. The form here uses density explicitly, allowing for nonideal values to be used.
\[\dot m^2 = \frac{\left(\frac{\pi D^2}{4}\right)^2 \rho_{avg} \left(P_1^2P_2^2\right)}{P_1\left(f_d\frac{L}{D} + 2\ln\frac{P_1}{P_2} \right)}\]Parameters: rho : float
Average density of gas in pipe, [kg/m^3]
fd : float
Darcy friction factor for flow in pipe []
P1 : float, optional
Inlet pressure to pipe, [Pa]
P2 : float, optional
Outlet pressure from pipe, [Pa]
L : float, optional
Length of pipe, [m]
D : float, optional
Diameter of pipe, [m]
m : float, optional
Mass flow rate of gas through pipe, [kg/s]
Returns: m, P1, P2, D, or L : float
The missing input which was solved for [base SI]
Notes
The solution for P2 has the following closed form, derived using Maple:
\[ \begin{align}\begin{aligned}P_2={P_1 \left( {{ e}^{0.5\cdot{\frac {1}{{m}^{2}} \left( C{m}^{2} +\text{ lambertW} \left({\frac {BP_1}{{m}^{2}}{{ e}^{{\frac {C{m}^{ 2}+BP_1}{{m}^{2}}}}}}\right){}{m}^{2}+BP_1 \right) }}} \right) ^{1}}\\B = \frac{\pi^2 D^4}{4^2} \rho_{avg}\\C = f_d \frac{L}{D}\end{aligned}\end{align} \]A wide range of conditions are impossible due to choked flow. See P_isothermal_critical_flow for details. An exception is raised when they occur.
The 2 multiplied by the logarithm is often shown as a power of the pressure ratio; this is only the case when the pressure ratio is raised to the power of 2 before its logarithm is taken.
A number of limitations exist for this model:
 Density dependence is that of an ideal gas.
 If calculating the pressure drop, the average gas density cannot be known immediately; iteration must be used to correct this.
 The friction factor depends on both the gas density and velocity, so it should be solved for iteratively as well. It changes throughout the pipe as the gas expands and velocity increases.
 The model is not easily adapted to include elevation effects due to the acceleration term included in it.
 As the gas expands, it will change temperature slightly, further altering the density and friction factor.
There are many commercial packages which perform the actual direct integration of the flow, such as OLGA Dynamic Multiphase Flow Simulator, or ASPEN Hydraulics.
References
[R55105] Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe. Crane, 2009. [R56105] Kim, J. and Singh, N. “A Novel Equation for Isothermal Pipe Flow.”. Chemical Engineering, June 2012, http://www.chemengonline.com/anovelequationforisothermalpipeflow/?printmode=1 [R57105] Wilkes, James O. Fluid Mechanics for Chemical Engineers with Microfluidics and CFD. 2 edition. Upper Saddle River, NJ: Prentice Hall, 2005. Examples
>>> isothermal_gas(rho=11.3, fd=0.00185, P1=1E6, P2=9E5, L=1000, D=0.5) 145.4847572636031

fluids.compressible.
isothermal_work_compression
(P1, P2, T, Z=1)[source]¶ Calculates the work of compression or expansion of a gas going through an isothermal process.
\[W = zRT\ln\left(\frac{P_2}{P_1}\right)\]Parameters: P1 : float
Inlet pressure, [Pa]
P2 : float
Outlet pressure, [Pa]
T : float
Temperature of the gas going through an isothermal process, [K]
Z : float
Constant compressibility factor of the gas, []
Returns: W : float
Work performed per mole of gas compressed/expanded [J/mol]
Notes
The full derivation with all forms is as follows:
\[ \begin{align}\begin{aligned}W = \int_{P_1}^{P_2} V dP = zRT\int_{P_1}^{P_2} \frac{1}{P} dP\\W = zRT\ln\left(\frac{P_2}{P_1}\right) = P_1 V_1 \ln\left(\frac{P_2} {P_1}\right) = P_2 V_2 \ln\left(\frac{P_2}{P_1}\right)\end{aligned}\end{align} \]The substitutions are according to the ideal gas law with compressibility:
The work of compression/expansion is the change in enthalpy of the gas. Returns negative values for expansion and positive values for compression.
An average compressibility factor can be used where Z changes. For further accuracy, this expression can be used repeatedly with small changes in pressure and the work from each step summed.
This is the best possible case for compression; all actual compresssors require more work to do the compression.
By making the compression take a large number of stages and cooling the gas between stages, this can be approached reasonable closely. Integrally geared compressors are often used for this purpose.
References
[R58108] Couper, James R., W. Roy Penney, and James R. Fair. Chemical Process Equipment: Selection and Design. 2nd ed. Amsterdam ; Boston: Gulf Professional Publishing, 2009. Examples
>>> isothermal_work_compression(1E5, 1E6, 300) 5743.425357533477

fluids.compressible.
polytropic_exponent
(k, n=None, eta_p=None)[source]¶ Calculates one of:
 Polytropic exponent from polytropic efficiency
 Polytropic efficiency from the polytropic exponent
\[ \begin{align}\begin{aligned}n = \frac{k\eta_p}{1  k(1\eta_p)}\\\eta_p = \frac{\left(\frac{n}{n1}\right)}{\left(\frac{k}{k1} \right)} = \frac{n(k1)}{k(n1)}\end{aligned}\end{align} \]Parameters: k : float
Isentropic exponent of the gas (Cp/Cv) []
n : float, optional
Polytropic exponent of the process []
eta_p : float, optional
Polytropic efficiency of the process, []
Returns: n or eta_p : float
Polytropic exponent or polytropic efficiency, depending on input, []
References
[R59109] Couper, James R., W. Roy Penney, and James R. Fair. Chemical Process Equipment: Selection and Design. 2nd ed. Amsterdam ; Boston: Gulf Professional Publishing, 2009. Examples
>>> polytropic_exponent(1.4, eta_p=0.78) 1.5780346820809246

fluids.compressible.
isentropic_work_compression
(T1, k, Z=1, P1=None, P2=None, W=None, eta=None)[source]¶ Calculation function for dealing with compressing or expanding a gas going through an isentropic, adiabatic process assuming constant Cp and Cv. The polytropic model is the same equation; just provide n instead of k and use a polytropic efficiency for eta instead of a isentropic efficiency. Can calculate any of the following, given all the other inputs:
 W, Work of compression
 P2, Pressure after compression
 P1, Pressure before compression
 eta, isentropic efficiency of compression
\[W = \left(\frac{k}{k1}\right)ZRT_1\left[\left(\frac{P_2}{P_1} \right)^{(k1)/k}1\right]/\eta_{isentropic}\]Parameters: T1 : float
Initial temperature of the gas, [K]
k : float
Isentropic exponent of the gas (Cp/Cv) or polytropic exponent n to use this as a polytropic model instead []
Z : float, optional
Constant compressibility factor of the gas, []
P1 : float, optional
Inlet pressure, [Pa]
P2 : float, optional
Outlet pressure, [Pa]
W : float, optional
Work performed per mole of gas compressed/expanded [J/mol]
eta : float, optional
Isentropic efficiency of the process or polytropic efficiency of the process to use this as a polytropic model instead []
Returns: W, P1, P2, or eta : float
The missing input which was solved for [base SI]
Notes
For the same compression ratio, this is always of larger magnitude than the isothermal case.
The full derivation is as follows:
For constantheat capacity “isentropic” fluid,
\[ \begin{align}\begin{aligned}V = \frac{P_1^{1/k}V_1}{P^{1/k}}\\W = \int_{P_1}^{P_2} V dP = \int_{P_1}^{P_2}\frac{P_1^{1/k}V_1} {P^{1/k}}dP\\W = \frac{P_1^{1/k} V_1}{1  \frac{1}{k}}\left[P_2^{11/k}  P_1^{11/k}\right]\end{aligned}\end{align} \]After performing the integration and substantial mathematical manipulation we can obtain:
\[W = \left(\frac{k}{k1}\right) P_1 V_1 \left[\left(\frac{P_2}{P_1} \right)^{(k1)/k}1\right]\]Using PV = ZRT:
\[W = \left(\frac{k}{k1}\right)ZRT_1\left[\left(\frac{P_2}{P_1} \right)^{(k1)/k}1\right]\]The work of compression/expansion is the change in enthalpy of the gas. Returns negative values for expansion and positive values for compression.
An average compressibility factor should be used as Z changes. For further accuracy, this expression can be used repeatedly with small changes in pressure and new values of isentropic exponent, and the work from each step summed.
For the polytropic case this is not necessary, as eta corrects for the simplification.
References
[R60110] Couper, James R., W. Roy Penney, and James R. Fair. Chemical Process Equipment: Selection and Design. 2nd ed. Amsterdam ; Boston: Gulf Professional Publishing, 2009. Examples
>>> isentropic_work_compression(P1=1E5, P2=1E6, T1=300, k=1.4, eta=0.78) 10416.873455626454

fluids.compressible.
isentropic_efficiency
(P1, P2, k, eta_s=None, eta_p=None)[source]¶ Calculates either isentropic or polytropic efficiency from the other type of efficiency.
\[ \begin{align}\begin{aligned}\eta_s = \frac{(P_2/P_1)^{(k1)/k}1} {(P_2/P_1)^{\frac{k1}{k\eta_p}}1}\\\eta_p = \frac{\left(k  1\right) \log{\left (\frac{P_{2}}{P_{1}} \right )}}{k \log{\left (\frac{1}{\eta_{s}} \left(\eta_{s} + \left(\frac{P_{2}}{P_{1}}\right)^{\frac{1}{k} \left(k  1\right)}  1\right) \right )}}\end{aligned}\end{align} \]Parameters: P1 : float
Initial pressure of gas [Pa]
P2 : float
Final pressure of gas [Pa]
k : float
Isentropic exponent of the gas (Cp/Cv) []
eta_s : float, optional
Isentropic efficiency of the process, []
eta_p : float, optional
Polytropic efficiency of the process, []
Returns: eta_s or eta_p : float
Isentropic or polytropic efficiency, depending on input, []
Notes
The form for obtained eta_p from eta_s was derived with SymPy.
References
[R61111] Couper, James R., W. Roy Penney, and James R. Fair. Chemical Process Equipment: Selection and Design. 2nd ed. Amsterdam ; Boston: Gulf Professional Publishing, 2009. Examples
>>> isentropic_efficiency(1E5, 1E6, 1.4, eta_p=0.78) 0.7027614191263858

fluids.compressible.
isentropic_T_rise_compression
(T1, P1, P2, k, eta=1)[source]¶ Calculates the increase in temperature of a fluid which is compressed or expanded under isentropic, adiabatic conditions assuming constant Cp and Cv. The polytropic model is the same equation; just provide n instead of k and use a polytropic efficienty for eta instead of a isentropic efficiency.
\[T_2 = T_1 + \frac{\Delta T_s}{\eta_s} = T_1 \left\{1 + \frac{1} {\eta_s}\left[\left(\frac{P_2}{P_1}\right)^{(k1)/k}1\right]\right\}\]Parameters: T1 : float
Initial temperature of gas [K]
P1 : float
Initial pressure of gas [Pa]
P2 : float
Final pressure of gas [Pa]
k : float
Isentropic exponent of the gas (Cp/Cv) or polytropic exponent n to use this as a polytropic model instead []
eta : float
Isentropic efficiency of the process or polytropic efficiency of the process to use this as a polytropic model instead []
Returns: T2 : float
Final temperature of gas [K]
Notes
For the ideal case (`eta`=1), the model simplifies to:
\[\frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{(k1)/k}\]References
[R62112] Couper, James R., W. Roy Penney, and James R. Fair. Chemical Process Equipment: Selection and Design. 2nd ed. Amsterdam ; Boston: Gulf Professional Publishing, 2009. [R63112] GPSA. GPSA Engineering Data Book. 13th edition. Gas Processors Suppliers Association, Tulsa, OK, 2012. Examples
>>> isentropic_T_rise_compression(286.8, 54050, 432400, 1.4) 519.5230938217768

fluids.compressible.
T_critical_flow
(T, k)[source]¶ Calculates critical flow temperature Tcf for a fluid with the given isentropic coefficient. Tcf is in a flow (with Ma=1) whose stagnation conditions are known. Normally used with converging/diverging nozzles.
\[\frac{T^*}{T_0} = \frac{2}{k+1}\]Parameters: T : float
Stagnation temperature of a fluid with Ma=1 [K]
k : float
Isentropic coefficient []
Returns: Tcf : float
Critical flow temperature at Ma=1 [K]
Notes
Assumes isentropic flow.
References
[R64114] (1, 2) Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006. Examples
Example 12.4 in [R64114]:
>>> T_critical_flow(473, 1.289) 413.2809086937528

fluids.compressible.
P_critical_flow
(P, k)[source]¶ Calculates critical flow pressure Pcf for a fluid with the given isentropic coefficient. Pcf is in a flow (with Ma=1) whose stagnation conditions are known. Normally used with converging/diverging nozzles.
\[\frac{P^*}{P_0} = \left(\frac{2}{k+1}\right)^{k/(k1)}\]Parameters: P : float
Stagnation pressure of a fluid with Ma=1 [Pa]
k : float
Isentropic coefficient []
Returns: Pcf : float
Critical flow pressure at Ma=1 [Pa]
Notes
Assumes isentropic flow.
References
[R65115] (1, 2) Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006. Examples
Example 12.4 in [R65115]:
>>> P_critical_flow(1400000, 1.289) 766812.9022792266

fluids.compressible.
P_isothermal_critical_flow
(P, fd, D, L)[source]¶ Calculates critical flow pressure Pcf for a fluid flowing isothermally and suffering pressure drop caused by a pipe’s friction factor.
\[P_2 = P_{1} e^{\frac{1}{2 D} \left(D \left(\operatorname{LambertW} {\left ( e^{\frac{1}{D} \left( D  L f_d\right)} \right )} + 1\right) + L f_d\right)}\]Parameters: P : float
Inlet pressure [Pa]
fd : float
Darcy friction factor for flow in pipe []
D : float
Diameter of pipe, [m]
L : float
Length of pipe, [m]
Returns: Pcf : float
Critical flow pressure of a compressible gas flowing from P1 to Pcf in a tube of length L and friction factor fd [Pa]
Notes
Assumes isothermal flow. Developed based on the isothermal_gas model, using SymPy.
The isothermal gas model is solved for maximum mass flow rate; any pressure drop under it is impossible due to the formation of a shock wave.
References
[R66116] Wilkes, James O. Fluid Mechanics for Chemical Engineers with Microfluidics and CFD. 2 edition. Upper Saddle River, NJ: Prentice Hall, 2005. Examples
>>> P_isothermal_critical_flow(P=1E6, fd=0.00185, L=1000., D=0.5) 389699.7317645518

fluids.compressible.
is_critical_flow
(P1, P2, k)[source]¶ Determines if a flow of a fluid driven by pressure gradient P1  P2 is critical, for a fluid with the given isentropic coefficient. This function calculates critical flow pressure, and checks if this is larger than P2. If so, the flow is critical and choked.
Parameters: P1 : float
Higher, source pressure [Pa]
P2 : float
Lower, downstream pressure [Pa]
k : float
Isentropic coefficient []
Returns: flowtype : bool
True if the flow is choked; otherwise False
Notes
Assumes isentropic flow. Uses P_critical_flow function.
References
[R67117] API. 2014. API 520  Part 1 Sizing, Selection, and Installation of Pressurerelieving Devices, Part I  Sizing and Selection, 9E. Examples
Examples 12 from API 520.
>>> is_critical_flow(670E3, 532E3, 1.11) False >>> is_critical_flow(670E3, 101E3, 1.11) True

fluids.compressible.
stagnation_energy
(V)[source]¶ Calculates the increase in enthalpy dH which is provided by a fluid’s velocity V.
\[\Delta H = \frac{V^2}{2}\]Parameters: V : float
Velocity [m/s]
Returns: dH : float
Incease in enthalpy [J/kg]
Notes
The units work out. This term is pretty small, but not trivial.
References
[R68118] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006. Examples
>>> stagnation_energy(125) 7812.5

fluids.compressible.
P_stagnation
(P, T, Tst, k)[source]¶ Calculates stagnation flow pressure Pst for a fluid with the given isentropic coefficient and specified stagnation temperature and normal temperature. Normally used with converging/diverging nozzles.
\[\frac{P_0}{P}=\left(\frac{T_0}{T}\right)^{\frac{k}{k1}}\]Parameters: P : float
Normal pressure of a fluid [Pa]
T : float
Normal temperature of a fluid [K]
Tst : float
Stagnation temperature of a fluid moving at a certain velocity [K]
k : float
Isentropic coefficient []
Returns: Pst : float
Stagnation pressure of a fluid moving at a certain velocity [Pa]
Notes
Assumes isentropic flow.
References
[R69119] (1, 2) Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006. Examples
Example 121 in [R69119].
>>> P_stagnation(54050., 255.7, 286.8, 1.4) 80772.80495900588

fluids.compressible.
T_stagnation
(T, P, Pst, k)[source]¶ Calculates stagnation flow temperature Tst for a fluid with the given isentropic coefficient and specified stagnation pressure and normal pressure. Normally used with converging/diverging nozzles.
\[T=T_0\left(\frac{P}{P_0}\right)^{\frac{k1}{k}}\]Parameters: T : float
Normal temperature of a fluid [K]
P : float
Normal pressure of a fluid [Pa]
Pst : float
Stagnation pressure of a fluid moving at a certain velocity [Pa]
k : float
Isentropic coefficient []
Returns: Tst : float
Stagnation temperature of a fluid moving at a certain velocity [K]
Notes
Assumes isentropic flow.
References
[R70120] (1, 2) Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006. Examples
Example 121 in [R70120].
>>> T_stagnation(286.8, 54050, 54050*8, 1.4) 519.5230938217768

fluids.compressible.
T_stagnation_ideal
(T, V, Cp)[source]¶ Calculates the ideal stagnation temperature Tst calculated assuming the fluid has a constant heat capacity Cp and with a specified velocity V and temperature T.
\[T^* = T + \frac{V^2}{2C_p}\]Parameters: T : float
Tempearture [K]
V : float
Velocity [m/s]
Cp : float
Ideal heat capacity [J/kg/K]
Returns: Tst : float
Stagnation temperature [J/kg]
References
[R71121] (1, 2) Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006. Examples
Example 121 in [R71121].
>>> T_stagnation_ideal(255.7, 250, 1005.) 286.79452736318405