Orifice plates, flow nozzles, Venturi tubes, cone and wedge meters (fluids.flow_meter)

fluids.flow_meter.C_ISA_1932_nozzle(D, Do, rho, mu, m)[source]

Calculates the coefficient of discharge of an ISA 1932 style nozzle used for measuring flow rate of fluid, based on the geometry of the nozzle, mass flow rate through the nozzle, and the density and viscosity of the fluid.

\[C = 0.9900 - 0.2262\beta^{4.1} - (0.00175\beta^2 - 0.0033\beta^{4.15}) \left(\frac{10^6}{Re_D}\right)^{1.15}\]
Parameters
Dfloat

Upstream internal pipe diameter, [m]

Dofloat

Diameter of nozzle orifice at flow conditions, [m]

rhofloat

Density of fluid at P1, [kg/m^3]

mufloat

Viscosity of fluid at P1, [Pa*s]

mfloat

Mass flow rate of fluid through the nozzle, [kg/s]

Returns
Cfloat

Coefficient of discharge of the nozzle orifice, [-]

References

1

American Society of Mechanical Engineers. Mfc-3M-2004 Measurement Of Fluid Flow In Pipes Using Orifice, Nozzle, And Venturi. ASME, 2001.

2

ISO 5167-3:2003 - Measurement of Fluid Flow by Means of Pressure Differential Devices Inserted in Circular Cross-Section Conduits Running Full – Part 3: Nozzles and Venturi Nozzles.

Examples

>>> C_ISA_1932_nozzle(D=0.07391, Do=0.0422, rho=1.2, mu=1.8E-5, m=0.1)
0.9635849973250495
fluids.flow_meter.C_Reader_Harris_Gallagher(D, Do, rho, mu, m, taps='corner')[source]

Calculates the coefficient of discharge of the orifice based on the geometry of the plate, measured pressures of the orifice, mass flow rate through the orifice, and the density and viscosity of the fluid.

\[\begin{split}C = 0.5961 + 0.0261\beta^2 - 0.216\beta^8 + 0.000521\left(\frac{ 10^6\beta}{Re_D}\right)^{0.7}\\ + (0.0188 + 0.0063A)\beta^{3.5} \left(\frac{10^6}{Re_D}\right)^{0.3} \\ +(0.043 + 0.080\exp(-10L_1) -0.123\exp(-7L_1))(1-0.11A)\frac{\beta^4} {1-\beta^4} \\ - 0.031(M_2' - 0.8M_2'^{1.1})\beta^{1.3}\end{split}\]
\[M_2' = \frac{2L_2'}{1-\beta}\]
\[A = \left(\frac{19000\beta}{Re_{D}}\right)^{0.8}\]
\[Re_D = \frac{\rho v D}{\mu}\]

If D < 71.12 mm (2.8 in.) (Note this is a continuous addition; there is no discontinuity):

\[C += 0.11(0.75-\beta)\left(2.8-\frac{D}{0.0254}\right)\]

If the orifice has corner taps:

\[L_1 = L_2' = 0\]

If the orifice has D and D/2 taps:

\[L_1 = 1\]
\[L_2' = 0.47\]

If the orifice has Flange taps:

\[L_1 = L_2' = \frac{0.0254}{D}\]
Parameters
Dfloat

Upstream internal pipe diameter, [m]

Dofloat

Diameter of orifice at flow conditions, [m]

rhofloat

Density of fluid at P1, [kg/m^3]

mufloat

Viscosity of fluid at P1, [Pa*s]

mfloat

Mass flow rate of fluid through the orifice, [kg/s]

tapsstr

The orientation of the taps; one of ‘corner’, ‘flange’, ‘D’, or ‘D/2’, [-]

Returns
Cfloat

Coefficient of discharge of the orifice, [-]

Notes

The following limits apply to the orifice plate standard [1]:

The measured pressure difference for the orifice plate should be under 250 kPa.

There are roughness limits as well; the roughness should be under 6 micrometers, although there are many more conditions to that given in [1].

For orifice plates with D and D/2 or corner pressure taps:

  • Orifice bore diameter muse be larger than 12.5 mm (0.5 inches)

  • Pipe diameter between 50 mm and 1 m (2 to 40 inches)

  • Beta between 0.1 and 0.75 inclusive

  • Reynolds number larger than 5000 (for \(0.10 \le \beta \le 0.56\)) or for \(\beta \ge 0.56, Re_D \ge 16000\beta^2\)

For orifice plates with flange pressure taps:

  • Orifice bore diameter muse be larger than 12.5 mm (0.5 inches)

  • Pipe diameter between 50 mm and 1 m (2 to 40 inches)

  • Beta between 0.1 and 0.75 inclusive

  • Reynolds number larger than 5000 and also larger than \(170000\beta^2 D\).

This is also presented in Crane’s TP410 (2009) publication, whereas the 1999 and 1982 editions showed only a graph for discharge coefficients.

References

1(1,2)

American Society of Mechanical Engineers. Mfc-3M-2004 Measurement Of Fluid Flow In Pipes Using Orifice, Nozzle, And Venturi. ASME, 2001.

2

ISO 5167-2:2003 - Measurement of Fluid Flow by Means of Pressure Differential Devices Inserted in Circular Cross-Section Conduits Running Full – Part 2: Orifice Plates.

3

Reader-Harris, M. J., “The Equation for the Expansibility Factor for Orifice Plates,” Proceedings of FLOMEKO 1998, Lund, Sweden, 1998: 209-214.

4

Reader-Harris, Michael. Orifice Plates and Venturi Tubes. Springer, 2015.

Examples

>>> C_Reader_Harris_Gallagher(D=0.07391, Do=0.0222, rho=1.165, mu=1.85E-5, 
... m=0.12, taps='flange')
0.5990326277163659
fluids.flow_meter.C_Reader_Harris_Gallagher_wet_venturi_tube(mg, ml, rhog, rhol, D, Do, H=1)[source]

Calculates the coefficient of discharge of the wet gas venturi tube based on the geometry of the tube, mass flow rates of liquid and vapor through the tube, the density of the liquid and gas phases, and an adjustable coefficient H.

\[C = 1 - 0.0463\exp(-0.05Fr_{gas, th}) \cdot \min\left(1, \sqrt{\frac{X}{0.016}}\right)\]
\[Fr_{gas, th} = \frac{Fr_{\text{gas, densionetric }}}{\beta^{2.5}}\]
\[\phi = \sqrt{1 + C_{Ch} X + X^2}\]
\[C_{Ch} = \left(\frac{\rho_l}{\rho_{1,g}}\right)^n + \left(\frac{\rho_{1, g}}{\rho_{l}}\right)^n\]
\[n = \max\left[0.583 - 0.18\beta^2 - 0.578\exp\left(\frac{-0.8 Fr_{\text{gas, densiometric}}}{H}\right),0.392 - 0.18\beta^2 \right]\]
\[X = \left(\frac{m_l}{m_g}\right) \sqrt{\frac{\rho_{1,g}}{\rho_l}}\]
\[{Fr_{\text{gas, densiometric}}} = \frac{v_{gas}}{\sqrt{gD}} \sqrt{\frac{\rho_{1,g}}{\rho_l - \rho_{1,g}}} = \frac{4m_g}{\rho_{1,g} \pi D^2 \sqrt{gD}} \sqrt{\frac{\rho_{1,g}}{\rho_l - \rho_{1,g}}}\]
Parameters
mgfloat

Mass flow rate of gas through the venturi tube, [kg/s]

mlfloat

Mass flow rate of liquid through the venturi tube, [kg/s]

rhogfloat

Density of gas at P1, [kg/m^3]

rholfloat

Density of liquid at P1, [kg/m^3]

Dfloat

Upstream internal pipe diameter, [m]

Dofloat

Diameter of venturi tube at flow conditions, [m]

Hfloat, optional

A surface-tension effect coefficient used to adjust for different fluids, (1 for a hydrocarbon liquid, 1.35 for water, 0.79 for water in steam) [-]

Returns
Cfloat

Coefficient of discharge of the wet gas venturi tube flow meter (includes flow rate of gas ONLY), [-]

Notes

This model has more error than single phase differential pressure meters. The model was first published in [1], and became ISO 11583 later.

The limits of this correlation according to [2] are as follows:

\[0.4 \le \beta \le 0.75\]
\[0 < X \le 0.3\]
\[Fr_{gas, th} > 3\]
\[\frac{\rho_g}{\rho_l} > 0.02\]
\[D \ge 50 \text{ mm}\]

References

1

Reader-harris, Michael, and Tuv Nel. An Improved Model for Venturi-Tube Over-Reading in Wet Gas, 2009.

2

ISO/TR 11583:2012 Measurement of Wet Gas Flow by Means of Pressure Differential Devices Inserted in Circular Cross-Section Conduits.

Examples

>>> C_Reader_Harris_Gallagher_wet_venturi_tube(mg=5.31926, ml=5.31926/2, 
... rhog=50.0, rhol=800., D=.1, Do=.06, H=1)
0.9754210845876333
fluids.flow_meter.C_long_radius_nozzle(D, Do, rho, mu, m)[source]

Calculates the coefficient of discharge of a long radius nozzle used for measuring flow rate of fluid, based on the geometry of the nozzle, mass flow rate through the nozzle, and the density and viscosity of the fluid.

\[C = 0.9965 - 0.00653\beta^{0.5} \left(\frac{10^6}{Re_D}\right)^{0.5}\]
Parameters
Dfloat

Upstream internal pipe diameter, [m]

Dofloat

Diameter of long radius nozzle orifice at flow conditions, [m]

rhofloat

Density of fluid at P1, [kg/m^3]

mufloat

Viscosity of fluid at P1, [Pa*s]

mfloat

Mass flow rate of fluid through the nozzle, [kg/s]

Returns
Cfloat

Coefficient of discharge of the long radius nozzle orifice, [-]

References

1

American Society of Mechanical Engineers. Mfc-3M-2004 Measurement Of Fluid Flow In Pipes Using Orifice, Nozzle, And Venturi. ASME, 2001.

2

ISO 5167-3:2003 - Measurement of Fluid Flow by Means of Pressure Differential Devices Inserted in Circular Cross-Section Conduits Running Full – Part 3: Nozzles and Venturi Nozzles.

Examples

>>> C_long_radius_nozzle(D=0.07391, Do=0.0422, rho=1.2, mu=1.8E-5, m=0.1)
0.9805503704679863
fluids.flow_meter.C_venturi_nozzle(D, Do)[source]

Calculates the coefficient of discharge of an Venturi style nozzle used for measuring flow rate of fluid, based on the geometry of the nozzle.

\[C = 0.9858 - 0.196\beta^{4.5}\]
Parameters
Dfloat

Upstream internal pipe diameter, [m]

Dofloat

Diameter of nozzle orifice at flow conditions, [m]

Returns
Cfloat

Coefficient of discharge of the nozzle orifice, [-]

References

1

American Society of Mechanical Engineers. Mfc-3M-2004 Measurement Of Fluid Flow In Pipes Using Orifice, Nozzle, And Venturi. ASME, 2001.

2

ISO 5167-3:2003 - Measurement of Fluid Flow by Means of Pressure Differential Devices Inserted in Circular Cross-Section Conduits Running Full – Part 3: Nozzles and Venturi Nozzles.

Examples

>>> C_venturi_nozzle(D=0.07391, Do=0.0422)
0.9698996454169576
fluids.flow_meter.C_wedge_meter_ISO_5167_6_2017(D, H)[source]

Calculates the coefficient of discharge of an wedge flow meter used for measuring flow rate of fluid, based on the geometry of the differential pressure flow meter according to the ISO 5167-6 standard (draft 2017).

\[C = 0.77 - 0.09\beta \]
Parameters
Dfloat

Upstream internal pipe diameter, [m]

Hfloat

Portion of the diameter of the clear segment of the pipe up to the wedge blocking flow; the height of the pipe up to the wedge, [m]

Returns
Cfloat

Coefficient of discharge of the wedge flow meter, [-]

Notes

This standard applies for wedge meters in line sizes between 50 and 600 mm; and height ratios between 0.2 and 0.6. The range of allowable Reynolds numbers is large; between 1E4 and 9E6. The uncertainty of the flow coefficient is approximately 4%. Usually a 10:1 span of flow can be measured accurately. The discharge and entry length of the meters must be at least half a pipe diameter. The wedge angle must be 90 degrees, plus or minus two degrees.

The orientation of the wedge meter does not change the accuracy of this model.

There should be a straight run of 10 pipe diameters before the wedge meter inlet, and two of the same pipe diameters after it.

References

1

ISO/DIS 5167-6 - Measurement of Fluid Flow by Means of Pressure Differential Devices Inserted in Circular Cross-Section Conduits Running Full – Part 6: Wedge Meters.

Examples

>>> C_wedge_meter_ISO_5167_6_2017(D=0.1524, H=0.3*0.1524)
0.724792059539853
fluids.flow_meter.C_wedge_meter_Miller(D, H)[source]

Calculates the coefficient of discharge of an wedge flow meter used for measuring flow rate of fluid, based on the geometry of the differential pressure flow meter.

For half-inch lines:

\[C = 0.7883 + 0.107(1 - \beta^2)\]

For 1 to 1.5 inch lines:

\[C = 0.6143 + 0.718(1 - \beta^2)\]

For 1.5 to 24 inch lines:

\[C = 0.5433 + 0.2453(1 - \beta^2)\]
Parameters
Dfloat

Upstream internal pipe diameter, [m]

Hfloat

Portion of the diameter of the clear segment of the pipe up to the wedge blocking flow; the height of the pipe up to the wedge, [m]

Returns
Cfloat

Coefficient of discharge of the wedge flow meter, [-]

Notes

There is an ISO standard being developed to cover wedge meters as of 2018.

Wedge meters can have varying angles; 60 and 90 degree wedge meters have been reported. Tap locations 1 or 2 diameters (upstream and downstream), and 2D upstream/1D downstream have been used. Some wedges are sharp; some are smooth. [2] gives some experimental values.

References

1

Miller, Richard W. Flow Measurement Engineering Handbook. 3rd edition. New York: McGraw-Hill Education, 1996.

2

Seshadri, V., S. N. Singh, and S. Bhargava. “Effect of Wedge Shape and Pressure Tap Locations on the Characteristics of a Wedge Flowmeter.” IJEMS Vol.01(5), October 1994.

Examples

>>> C_wedge_meter_Miller(D=0.1524, H=0.3*0.1524)
0.7267069372687651
fluids.flow_meter.K_to_discharge_coefficient(D, Do, K)[source]

Converts a standard loss coefficient to a discharge coefficient.

\[C = \sqrt{\frac{1}{2 \sqrt{K} \beta^{4} + K \beta^{4}} - \frac{\beta^{4}}{2 \sqrt{K} \beta^{4} + K \beta^{4}} }\]
Parameters
Dfloat

Upstream internal pipe diameter, [m]

Dofloat

Diameter of orifice at flow conditions, [m]

Kfloat

Loss coefficient with respect to the velocity and density of the fluid just upstream of the orifice, [-]

Returns
Cfloat

Coefficient of discharge of the orifice, [-]

Notes

If expansibility is used in the orifice calculation, the result will not match with the specified pressure drop formula in [1]; it can almost be matched by dividing the calculated mass flow by the expansibility factor and using that mass flow with the loss coefficient.

This expression was derived with SymPy, and checked numerically. There were three other, incorrect roots.

References

1

American Society of Mechanical Engineers. Mfc-3M-2004 Measurement Of Fluid Flow In Pipes Using Orifice, Nozzle, And Venturi. ASME, 2001.

2

ISO 5167-2:2003 - Measurement of Fluid Flow by Means of Pressure Differential Devices Inserted in Circular Cross-Section Conduits Running Full – Part 2: Orifice Plates.

Examples

>>> K_to_discharge_coefficient(D=0.07366, Do=0.05, K=5.2314291729754)
0.6151200000000001
fluids.flow_meter.cone_meter_expansibility_Stewart(D, Dc, P1, P2, k)[source]

Calculates the expansibility factor for a cone flow meter, based on the geometry of the cone meter, measured pressures of the orifice, and the isentropic exponent of the fluid. Developed in [1], also shown in [2].

\[\epsilon = 1 - (0.649 + 0.696\beta^4) \frac{\Delta P}{\kappa P_1}\]
Parameters
Dfloat

Upstream internal pipe diameter, [m]

Dcfloat

Diameter of the largest end of the cone meter, [m]

P1float

Static pressure of fluid upstream of cone meter at the cross-section of the pressure tap, [Pa]

P2float

Static pressure of fluid at the end of the center of the cone pressure tap, [Pa]

kfloat

Isentropic exponent of fluid, [-]

Returns
expansibilityfloat

Expansibility factor (1 for incompressible fluids, less than 1 for real fluids), [-]

Notes

This formula was determined for the range of P2/P1 >= 0.75; the only gas used to determine the formula is air.

References

1

Stewart, D. G., M. Reader-Harris, and NEL Dr RJW Peters. “Derivation of an Expansibility Factor for the V-Cone Meter.” In Flow Measurement International Conference, Peebles, Scotland, UK, 2001.

2

ISO 5167-5:2016 - Measurement of Fluid Flow by Means of Pressure Differential Devices Inserted in Circular Cross-Section Conduits Running Full – Part 5: Cone meters.

Examples

>>> cone_meter_expansibility_Stewart(D=1, Dc=0.9, P1=1E6, P2=8.5E5, k=1.2)
0.9157343
fluids.flow_meter.dP_Reader_Harris_Gallagher_wet_venturi_tube(D, Do, P1, P2, ml, mg, rhol, rhog, H=1)[source]

Calculates the non-recoverable pressure drop of a wet gas venturi nozzle based on the pressure drop and the geometry of the venturi nozzle, the mass flow rates of liquid and gas through it, the densities of the vapor and liquid phase, and an adjustable coefficient H.

\[Y = \frac{\Delta \bar \omega}{\Delta P} - 0.0896 - 0.48\beta^9\]
\[Y_{max} = 0.61\exp\left[-11\frac{\rho_{1,g}}{\rho_l} - 0.045 \frac{Fr_{gas}}{H}\right]\]
\[\frac{Y}{Y_{max}} = 1 - \exp\left[-35 X^{0.75} \exp \left( \frac{-0.28Fr_{gas}}{H}\right)\right]\]
\[X = \left(\frac{m_l}{m_g}\right) \sqrt{\frac{\rho_{1,g}}{\rho_l}}\]
\[{Fr_{\text{gas, densiometric}}} = \frac{v_{gas}}{\sqrt{gD}} \sqrt{\frac{\rho_{1,g}}{\rho_l - \rho_{1,g}}} = \frac{4m_g}{\rho_{1,g} \pi D^2 \sqrt{gD}} \sqrt{\frac{\rho_{1,g}}{\rho_l - \rho_{1,g}}}\]
Parameters
Dfloat

Upstream internal pipe diameter, [m]

Dofloat

Diameter of venturi tube at flow conditions, [m]

P1float

Static pressure of fluid upstream of venturi tube at the cross-section of the pressure tap, [Pa]

P2float

Static pressure of fluid downstream of venturi tube at the cross- section of the pressure tap, [Pa]

mlfloat

Mass flow rate of liquid through the venturi tube, [kg/s]

mgfloat

Mass flow rate of gas through the venturi tube, [kg/s]

rholfloat

Density of liquid at P1, [kg/m^3]

rhogfloat

Density of gas at P1, [kg/m^3]

Hfloat, optional

A surface-tension effect coefficient used to adjust for different fluids, (1 for a hydrocarbon liquid, 1.35 for water, 0.79 for water in steam) [-]

Returns
Cfloat

Coefficient of discharge of the wet gas venturi tube flow meter (includes flow rate of gas ONLY), [-]

Notes

The model was first published in [1], and became ISO 11583 later.

References

1

Reader-harris, Michael, and Tuv Nel. An Improved Model for Venturi-Tube Over-Reading in Wet Gas, 2009.

2

ISO/TR 11583:2012 Measurement of Wet Gas Flow by Means of Pressure Differential Devices Inserted in Circular Cross-Section Conduits.

Examples

>>> dP_Reader_Harris_Gallagher_wet_venturi_tube(D=.1, Do=.06, H=1, 
... P1=6E6, P2=6E6-5E4, ml=5.31926/2, mg=5.31926, rhog=50.0, rhol=800.,)
16957.43843129572
fluids.flow_meter.dP_cone_meter(D, Dc, P1, P2)[source]

Calculates the non-recoverable pressure drop of a cone meter based on the measured pressures before and at the cone end, and the geometry of the cone meter according to [1].

\[\Delta \bar \omega = (1.09 - 0.813\beta)\Delta P\]
Parameters
Dfloat

Upstream internal pipe diameter, [m]

Dcfloat

Diameter of the largest end of the cone meter, [m]

P1float

Static pressure of fluid upstream of cone meter at the cross-section of the pressure tap, [Pa]

P2float

Static pressure of fluid at the end of the center of the cone pressure tap, [Pa]

Returns
dPfloat

Non-recoverable pressure drop of the orifice plate, [Pa]

Notes

The recoverable pressure drop should be recovered by 6 pipe diameters downstream of the cone meter.

References

1

ISO 5167-5:2016 - Measurement of Fluid Flow by Means of Pressure Differential Devices Inserted in Circular Cross-Section Conduits Running Full – Part 5: Cone meters.

Examples

>>> dP_cone_meter(1, .7, 1E6, 9.5E5)
25470.093437973323
fluids.flow_meter.dP_orifice(D, Do, P1, P2, C)[source]

Calculates the non-recoverable pressure drop of an orifice plate based on the pressure drop and the geometry of the plate and the discharge coefficient.

\[\Delta\bar w = \frac{\sqrt{1-\beta^4(1-C^2)}-C\beta^2} {\sqrt{1-\beta^4(1-C^2)}+C\beta^2} (P_1 - P_2)\]
Parameters
Dfloat

Upstream internal pipe diameter, [m]

Dofloat

Diameter of orifice at flow conditions, [m]

P1float

Static pressure of fluid upstream of orifice at the cross-section of the pressure tap, [Pa]

P2float

Static pressure of fluid downstream of orifice at the cross-section of the pressure tap, [Pa]

Cfloat

Coefficient of discharge of the orifice, [-]

Returns
dPfloat

Non-recoverable pressure drop of the orifice plate, [Pa]

Notes

This formula can be well approximated by:

\[\Delta\bar w = \left(1 - \beta^{1.9}\right)(P_1 - P_2)\]

The recoverable pressure drop should be recovered by 6 pipe diameters downstream of the orifice plate.

References

1

American Society of Mechanical Engineers. Mfc-3M-2004 Measurement Of Fluid Flow In Pipes Using Orifice, Nozzle, And Venturi. ASME, 2001.

2

ISO 5167-2:2003 - Measurement of Fluid Flow by Means of Pressure Differential Devices Inserted in Circular Cross-Section Conduits Running Full – Part 2: Orifice Plates.

Examples

>>> dP_orifice(D=0.07366, Do=0.05, P1=200000.0, P2=183000.0, C=0.61512)
9069.474705745388
fluids.flow_meter.dP_venturi_tube(D, Do, P1, P2)[source]

Calculates the non-recoverable pressure drop of a venturi tube differential pressure meter based on the pressure drop and the geometry of the venturi meter.

\[\epsilon = \frac{\Delta\bar w }{\Delta P}\]

The \(\epsilon\) value is looked up in a table of values as a function of beta ratio and upstream pipe diameter (roughness impact).

Parameters
Dfloat

Upstream internal pipe diameter, [m]

Dofloat

Diameter of venturi tube at flow conditions, [m]

P1float

Static pressure of fluid upstream of venturi tube at the cross-section of the pressure tap, [Pa]

P2float
Static pressure of fluid downstream of venturi tube at the

cross-section of the pressure tap, [Pa]

Returns
dPfloat

Non-recoverable pressure drop of the venturi tube, [Pa]

Notes

The recoverable pressure drop should be recovered by 6 pipe diameters downstream of the venturi tube.

Note there is some information on the effect of Reynolds number as well in [1] and [2], with a curve showing an increased pressure drop from 1E5-6E5 to with a decreasing multiplier from 1.75 to 1; the multiplier is 1 for higher Reynolds numbers. This is not currently included in this implementation.

References

1

American Society of Mechanical Engineers. Mfc-3M-2004 Measurement Of Fluid Flow In Pipes Using Orifice, Nozzle, And Venturi. ASME, 2001.

2

ISO 5167-4:2003 - Measurement of Fluid Flow by Means of Pressure Differential Devices Inserted in Circular Cross-Section Conduits Running Full – Part 4: Venturi Tubes.

Examples

>>> dP_venturi_tube(D=0.07366, Do=0.05, P1=200000.0, P2=183000.0)
1788.5717754177406
fluids.flow_meter.dP_wedge_meter(D, H, P1, P2)[source]

Calculates the non-recoverable pressure drop of a wedge meter based on the measured pressures before and at the wedge meter, and the geometry of the wedge meter according to [1].

\[\Delta \bar \omega = (1.09 - 0.79\beta)\Delta P\]
Parameters
Dfloat

Upstream internal pipe diameter, [m]

Hfloat

Portion of the diameter of the clear segment of the pipe up to the wedge blocking flow; the height of the pipe up to the wedge, [m]

P1float

Static pressure of fluid upstream of wedge meter at the cross-section of the pressure tap, [Pa]

P2float

Static pressure of fluid at the end of the wedge meter pressure tap, [ Pa]

Returns
dPfloat

Non-recoverable pressure drop of the wedge meter, [Pa]

Notes

The recoverable pressure drop should be recovered by 5 pipe diameters downstream of the wedge meter.

References

1

ISO/DIS 5167-6 - Measurement of Fluid Flow by Means of Pressure Differential Devices Inserted in Circular Cross-Section Conduits Running Full – Part 6: Wedge Meters.

Examples

>>> dP_wedge_meter(1, .7, 1E6, 9.5E5)
20344.849697483587
fluids.flow_meter.diameter_ratio_cone_meter(D, Dc)[source]

Calculates the diameter ratio beta used to characterize a cone flow meter.

\[\beta = \sqrt{1 - \frac{d_c^2}{D^2}}\]
Parameters
Dfloat

Upstream internal pipe diameter, [m]

Dcfloat

Diameter of the largest end of the cone meter, [m]

Returns
betafloat

Cone meter diameter ratio, [-]

References

1

Hollingshead, Colter. “Discharge Coefficient Performance of Venturi, Standard Concentric Orifice Plate, V-Cone, and Wedge Flow Meters at Small Reynolds Numbers.” May 1, 2011. https://digitalcommons.usu.edu/etd/869.

Examples

>>> diameter_ratio_cone_meter(D=0.2575, Dc=0.184)
0.6995709873957624
fluids.flow_meter.diameter_ratio_wedge_meter(D, H)[source]

Calculates the diameter ratio beta used to characterize a wedge flow meter as given in [1] and [2].

\[\beta = \left(\frac{1}{\pi}\left\{\arccos\left[1 - \frac{2H}{D} \right] - 2 \left[1 - \frac{2H}{D} \right]\left(\frac{H}{D} - \left[\frac{H}{D}\right]^2 \right)^{0.5}\right\}\right)^{0.5}\]
Parameters
Dfloat

Upstream internal pipe diameter, [m]

Hfloat

Portion of the diameter of the clear segment of the pipe up to the wedge blocking flow; the height of the pipe up to the wedge, [m]

Returns
betafloat

Wedge meter diameter ratio, [-]

References

1

Hollingshead, Colter. “Discharge Coefficient Performance of Venturi, Standard Concentric Orifice Plate, V-Cone, and Wedge Flow Meters at Small Reynolds Numbers.” May 1, 2011. https://digitalcommons.usu.edu/etd/869.

2

IntraWedge WEDGE FLOW METER Type: IWM. January 2011. http://www.intra-automation.com/download.php?file=pdf/products/technical_information/en/ti_iwm_en.pdf

Examples

>>> diameter_ratio_wedge_meter(D=0.2027, H=0.0608)
0.5022531424646643
fluids.flow_meter.differential_pressure_meter_C_epsilon(D, D2, m, P1, P2, rho, mu, k, meter_type, taps=None)[source]

Calculates the discharge coefficient and expansibility of a flow meter given the mass flow rate, the upstream pressure, the second pressure value, and the orifice diameter for a differential pressure flow meter based on the geometry of the meter, measured pressures of the meter, and the density, viscosity, and isentropic exponent of the fluid.

Parameters
Dfloat

Upstream internal pipe diameter, [m]

D2float

Diameter of orifice, or venturi meter orifice, or flow tube orifice, or cone meter end diameter, or wedge meter fluid flow height, [m]

mfloat

Mass flow rate of fluid through the flow meter, [kg/s]

P1float

Static pressure of fluid upstream of differential pressure meter at the cross-section of the pressure tap, [Pa]

P2float

Static pressure of fluid downstream of differential pressure meter or at the prescribed location (varies by type of meter) [Pa]

rhofloat

Density of fluid at P1, [kg/m^3]

mufloat

Viscosity of fluid at P1, [Pa*s]

kfloat

Isentropic exponent of fluid, [-]

meter_typestr

One of (‘ISO 5167 orifice’, ‘long radius nozzle’, ‘ISA 1932 nozzle’, ‘venuri nozzle’, ‘as cast convergent venturi tube’, ‘machined convergent venturi tube’, ‘rough welded convergent venturi tube’, ‘cone meter’, ‘wedge meter’), [-]

tapsstr, optional

The orientation of the taps; one of ‘corner’, ‘flange’, ‘D’, or ‘D/2’; applies for orifice meters only, [-]

Returns
Cfloat

Coefficient of discharge of the specified flow meter type at the specified conditions, [-]

expansibilityfloat

Expansibility factor (1 for incompressible fluids, less than 1 for real fluids), [-]

Notes

This function should be called by an outer loop when solving for a variable.

Examples

>>> differential_pressure_meter_C_epsilon(D=0.07366, D2=0.05, P1=200000.0, 
... P2=183000.0, rho=999.1, mu=0.0011, k=1.33, m=7.702338035732168,
... meter_type='ISO 5167 orifice', taps='D')
(0.6151252900244296, 0.9711026966676307)
fluids.flow_meter.differential_pressure_meter_beta(D, D2, meter_type)[source]

Calculates the beta ratio of a differential pressure meter.

Parameters
Dfloat

Upstream internal pipe diameter, [m]

D2float

Diameter of orifice, or venturi meter orifice, or flow tube orifice, or cone meter end diameter, or wedge meter fluid flow height, [m]

meter_typestr

One of (‘ISO 5167 orifice’, ‘long radius nozzle’, ‘ISA 1932 nozzle’, ‘venuri nozzle’, ‘as cast convergent venturi tube’, ‘machined convergent venturi tube’, ‘rough welded convergent venturi tube’, ‘cone meter’, ‘wedge meter’), [-]

Returns
betafloat

Differential pressure meter diameter ratio, [-]

Examples

>>> differential_pressure_meter_beta(D=0.2575, D2=0.184, 
... meter_type='cone meter')
0.6995709873957624
fluids.flow_meter.differential_pressure_meter_dP(D, D2, P1, P2, C=None, meter_type='ISO 5167 orifice')[source]

Calculates either the non-recoverable pressure drop of a differential pressure flow meter based on the geometry of the meter, measured pressures of the meter, and for most models the meter discharge coefficient.

Parameters
Dfloat

Upstream internal pipe diameter, [m]

D2float

Diameter of orifice, or venturi meter orifice, or flow tube orifice, or cone meter end diameter, or wedge meter fluid flow height, [m]

P1float

Static pressure of fluid upstream of differential pressure meter at the cross-section of the pressure tap, [Pa]

P2float

Static pressure of fluid downstream of differential pressure meter or at the prescribed location (varies by type of meter) [Pa]

Cfloat, optional

Coefficient of discharge (used only in orifice plates, and venturi nozzles), [-]

meter_typestr, optional

One of (‘ISO 5167 orifice’, ‘long radius nozzle’, ‘ISA 1932 nozzle’, ‘as cast convergent venturi tube’, ‘machined convergent venturi tube’, ‘rough welded convergent venturi tube’, ‘cone meter’, ‘cone meter’), [-]

Returns
dPfloat

Non-recoverable pressure drop of the differential pressure flow meter, [Pa]

Notes

See the appropriate functions for the documentation for the formulas and references used in each method.

Wedge meters, and venturi nozzles do not have standard formulas available for pressure drop computation.

Examples

>>> differential_pressure_meter_dP(D=0.07366, D2=0.05, P1=200000.0, 
... P2=183000.0, meter_type='as cast convergent venturi tube')
1788.5717754177406
fluids.flow_meter.differential_pressure_meter_solver(D, rho, mu, k, D2=None, P1=None, P2=None, m=None, meter_type='ISO 5167 orifice', taps=None)[source]

Calculates either the mass flow rate, the upstream pressure, the second pressure value, or the orifice diameter for a differential pressure flow meter based on the geometry of the meter, measured pressures of the meter, and the density, viscosity, and isentropic exponent of the fluid. This solves an equation iteratively to obtain the correct flow rate.

Parameters
Dfloat

Upstream internal pipe diameter, [m]

rhofloat

Density of fluid at P1, [kg/m^3]

mufloat

Viscosity of fluid at P1, [Pa*s]

kfloat

Isentropic exponent of fluid, [-]

D2float, optional

Diameter of orifice, or venturi meter orifice, or flow tube orifice, or cone meter end diameter, or wedge meter fluid flow height, [m]

P1float, optional

Static pressure of fluid upstream of differential pressure meter at the cross-section of the pressure tap, [Pa]

P2float, optional

Static pressure of fluid downstream of differential pressure meter or at the prescribed location (varies by type of meter) [Pa]

mfloat, optional

Mass flow rate of fluid through the flow meter, [kg/s]

meter_typestr, optional

One of (‘ISO 5167 orifice’, ‘long radius nozzle’, ‘ISA 1932 nozzle’, ‘venuri nozzle’, ‘as cast convergent venturi tube’, ‘machined convergent venturi tube’, ‘rough welded convergent venturi tube’, ‘cone meter’, ‘wedge meter’), [-]

tapsstr, optional

The orientation of the taps; one of ‘corner’, ‘flange’, ‘D’, or ‘D/2’; applies for orifice meters only, [-]

Returns
ansfloat

One of m, the mass flow rate of the fluid; P1, the pressure upstream of the flow meter; P2, the second pressure tap’s value; and D2, the diameter of the measuring device; units of respectively, kg/s, Pa, Pa, or m

Notes

See the appropriate functions for the documentation for the formulas and references used in each method.

The solvers make some assumptions about the range of values answers may be in.

Note that the solver for the upstream pressure uses the provided values of density, viscosity and isentropic exponent; whereas these values all depend on pressure (albeit to a small extent). An outer loop should be added with pressure-dependent values calculated in it for maximum accuracy.

It would be possible to solve for the upstream pipe diameter, but there is no use for that functionality.

Examples

>>> differential_pressure_meter_solver(D=0.07366, D2=0.05, P1=200000.0, 
... P2=183000.0, rho=999.1, mu=0.0011, k=1.33, 
... meter_type='ISO 5167 orifice', taps='D')
7.702338035732167
>>> differential_pressure_meter_solver(D=0.07366, m=7.702338, P1=200000.0, 
... P2=183000.0, rho=999.1, mu=0.0011, k=1.33, 
... meter_type='ISO 5167 orifice', taps='D')
0.04999999990831885
fluids.flow_meter.discharge_coefficient_to_K(D, Do, C)[source]

Converts a discharge coefficient to a standard loss coefficient, for use in computation of the actual pressure drop of an orifice or other device.

\[K = \left[\frac{\sqrt{1-\beta^4(1-C^2)}}{C\beta^2} - 1\right]^2\]
Parameters
Dfloat

Upstream internal pipe diameter, [m]

Dofloat

Diameter of orifice at flow conditions, [m]

Cfloat

Coefficient of discharge of the orifice, [-]

Returns
Kfloat

Loss coefficient with respect to the velocity and density of the fluid just upstream of the orifice, [-]

Notes

If expansibility is used in the orifice calculation, the result will not match with the specified pressure drop formula in [1]; it can almost be matched by dividing the calculated mass flow by the expansibility factor and using that mass flow with the loss coefficient.

References

1

American Society of Mechanical Engineers. Mfc-3M-2004 Measurement Of Fluid Flow In Pipes Using Orifice, Nozzle, And Venturi. ASME, 2001.

2

ISO 5167-2:2003 - Measurement of Fluid Flow by Means of Pressure Differential Devices Inserted in Circular Cross-Section Conduits Running Full – Part 2: Orifice Plates.

Examples

>>> discharge_coefficient_to_K(D=0.07366, Do=0.05, C=0.61512)
5.2314291729754
fluids.flow_meter.flow_coefficient(D, Do, C)[source]

Calculates a factor for differential pressure flow meter design called the flow coefficient. This should not be confused with the flow coefficient often used when discussing valves.

\[\text{Flow coefficient} = \frac{C}{\sqrt{1 - \beta^4}}\]
Parameters
Dfloat

Upstream internal pipe diameter, [m]

Dofloat

Diameter of flow meter characteristic dimension at flow conditions, [m]

Cfloat

Coefficient of discharge of the flow meter, [-]

Returns
flow_coefficientfloat

Differential pressure flow meter flow coefficient, [-]

Notes

This measure is used not just for orifices but for other differential pressure flow meters [2].

It is sometimes given the symbol K. It is also equal to the product of the diacharge coefficient and the velocity of approach factor [2].

References

1

American Society of Mechanical Engineers. Mfc-3M-2004 Measurement Of Fluid Flow In Pipes Using Orifice, Nozzle, And Venturi. ASME, 2001.

2(1,2)

Miller, Richard W. Flow Measurement Engineering Handbook. 3rd edition. New York: McGraw-Hill Education, 1996.

Examples

>>> flow_coefficient(D=0.0739, Do=0.0222, C=0.6)
0.6024582044499308
fluids.flow_meter.flow_meter_discharge(D, Do, P1, P2, rho, C, expansibility=1.0)[source]

Calculates the flow rate of an orifice plate based on the geometry of the plate, measured pressures of the orifice, and the density of the fluid.

\[m = \left(\frac{\pi D_o^2}{4}\right) C \frac{\sqrt{2\Delta P \rho_1}} {\sqrt{1 - \beta^4}}\cdot \epsilon\]
Parameters
Dfloat

Upstream internal pipe diameter, [m]

Dofloat

Diameter of orifice at flow conditions, [m]

P1float

Static pressure of fluid upstream of orifice at the cross-section of the pressure tap, [Pa]

P2float

Static pressure of fluid downstream of orifice at the cross-section of the pressure tap, [Pa]

rhofloat

Density of fluid at P1, [kg/m^3]

Cfloat

Coefficient of discharge of the orifice, [-]

expansibilityfloat, optional

Expansibility factor (1 for incompressible fluids, less than 1 for real fluids), [-]

Returns
mfloat

Mass flow rate of fluid, [kg/s]

Notes

This is formula 1-12 in [1] and also [2].

References

1

American Society of Mechanical Engineers. Mfc-3M-2004 Measurement Of Fluid Flow In Pipes Using Orifice, Nozzle, And Venturi. ASME, 2001.

2

ISO 5167-2:2003 - Measurement of Fluid Flow by Means of Pressure Differential Devices Inserted in Circular Cross-Section Conduits Running Full – Part 2: Orifice Plates.

Examples

>>> flow_meter_discharge(D=0.0739, Do=0.0222, P1=1E5, P2=9.9E4, rho=1.1646, 
... C=0.5988, expansibility=0.9975)
0.01120390943807026
fluids.flow_meter.nozzle_expansibility(D, Do, P1, P2, k, beta=None)[source]

Calculates the expansibility factor for a nozzle or venturi nozzle, based on the geometry of the plate, measured pressures of the orifice, and the isentropic exponent of the fluid.

\[\epsilon = \left\{\left(\frac{\kappa \tau^{2/\kappa}}{\kappa-1}\right) \left(\frac{1 - \beta^4}{1 - \beta^4 \tau^{2/\kappa}}\right) \left[\frac{1 - \tau^{(\kappa-1)/\kappa}}{1 - \tau} \right] \right\}^{0.5}\]
Parameters
Dfloat

Upstream internal pipe diameter, [m]

Dofloat

Diameter of orifice of the venturi or nozzle, [m]

P1float

Static pressure of fluid upstream of orifice at the cross-section of the pressure tap, [Pa]

P2float

Static pressure of fluid downstream of orifice at the cross-section of the pressure tap, [Pa]

kfloat

Isentropic exponent of fluid, [-]

betafloat, optional

Optional beta ratio, which is useful to specify for wedge meters or flow meters which have a different beta ratio calculation, [-]

Returns
expansibilityfloat

Expansibility factor (1 for incompressible fluids, less than 1 for real fluids), [-]

Notes

This formula was determined for the range of P2/P1 >= 0.75.

References

1

American Society of Mechanical Engineers. Mfc-3M-2004 Measurement Of Fluid Flow In Pipes Using Orifice, Nozzle, And Venturi. ASME, 2001.

2

ISO 5167-3:2003 - Measurement of Fluid Flow by Means of Pressure Differential Devices Inserted in Circular Cross-Section Conduits Running Full – Part 3: Nozzles and Venturi Nozzles.

Examples

>>> nozzle_expansibility(D=0.0739, Do=0.0222, P1=1E5, P2=9.9E4, k=1.4)
0.9945702344566746
fluids.flow_meter.orifice_expansibility(D, Do, P1, P2, k)[source]

Calculates the expansibility factor for orifice plate calculations based on the geometry of the plate, measured pressures of the orifice, and the isentropic exponent of the fluid.

\[\epsilon = 1 - (0.351 + 0.256\beta^4 + 0.93\beta^8) \left[1-\left(\frac{P_2}{P_1}\right)^{1/\kappa}\right]\]
Parameters
Dfloat

Upstream internal pipe diameter, [m]

Dofloat

Diameter of orifice at flow conditions, [m]

P1float

Static pressure of fluid upstream of orifice at the cross-section of the pressure tap, [Pa]

P2float

Static pressure of fluid downstream of orifice at the cross-section of the pressure tap, [Pa]

kfloat

Isentropic exponent of fluid, [-]

Returns
expansibilityfloat, optional

Expansibility factor (1 for incompressible fluids, less than 1 for real fluids), [-]

Notes

This formula was determined for the range of P2/P1 >= 0.80, and for fluids of air, steam, and natural gas. However, there is no objection to using it for other fluids.

References

1

American Society of Mechanical Engineers. Mfc-3M-2004 Measurement Of Fluid Flow In Pipes Using Orifice, Nozzle, And Venturi. ASME, 2001.

2

ISO 5167-2:2003 - Measurement of Fluid Flow by Means of Pressure Differential Devices Inserted in Circular Cross-Section Conduits Running Full – Part 2: Orifice Plates.

Examples

>>> orifice_expansibility(D=0.0739, Do=0.0222, P1=1E5, P2=9.9E4, k=1.4)
0.9974739057343425
fluids.flow_meter.orifice_expansibility_1989(D, Do, P1, P2, k)[source]

Calculates the expansibility factor for orifice plate calculations based on the geometry of the plate, measured pressures of the orifice, and the isentropic exponent of the fluid.

\[\epsilon = 1- (0.41 + 0.35\beta^4)\Delta P/\kappa/P_1\]
Parameters
Dfloat

Upstream internal pipe diameter, [m]

Dofloat

Diameter of orifice at flow conditions, [m]

P1float

Static pressure of fluid upstream of orifice at the cross-section of the pressure tap, [Pa]

P2float

Static pressure of fluid downstream of orifice at the cross-section of the pressure tap, [Pa]

kfloat

Isentropic exponent of fluid, [-]

Returns
expansibilityfloat

Expansibility factor (1 for incompressible fluids, less than 1 for real fluids), [-]

Notes

This formula was determined for the range of P2/P1 >= 0.75, and for fluids of air, steam, and natural gas. However, there is no objection to using it for other fluids.

This is an older formula used to calculate expansibility factors for orifice plates.

In this standard, an expansibility factor formula transformation in terms of the pressure after the orifice is presented as well. This is the more standard formulation in terms of the upstream conditions. The other formula is below for reference only:

\[\epsilon_2 = \sqrt{1 + \frac{\Delta P}{P_2}} - (0.41 + 0.35\beta^4) \frac{\Delta P}{\kappa P_2 \sqrt{1 + \frac{\Delta P}{P_2}}}\]

[2] recommends this formulation for wedge meters as well.

References

1

American Society of Mechanical Engineers. MFC-3M-1989 Measurement Of Fluid Flow In Pipes Using Orifice, Nozzle, And Venturi. ASME, 2005.

2

Miller, Richard W. Flow Measurement Engineering Handbook. 3rd edition. New York: McGraw-Hill Education, 1996.

Examples

>>> orifice_expansibility_1989(D=0.0739, Do=0.0222, P1=1E5, P2=9.9E4, k=1.4)
0.9970510687411718
fluids.flow_meter.velocity_of_approach_factor(D, Do)[source]

Calculates a factor for orifice plate design called the velocity of approach.

\[\text{Velocity of approach} = \frac{1}{\sqrt{1 - \beta^4}}\]
Parameters
Dfloat

Upstream internal pipe diameter, [m]

Dofloat

Diameter of orifice at flow conditions, [m]

Returns
velocity_of_approachfloat

Coefficient of discharge of the orifice, [-]

References

1

American Society of Mechanical Engineers. Mfc-3M-2004 Measurement Of Fluid Flow In Pipes Using Orifice, Nozzle, And Venturi. ASME, 2001.

Examples

>>> velocity_of_approach_factor(D=0.0739, Do=0.0222)
1.0040970074165514