# -*- coding: utf-8 -*- """Chemical Engineering Design Library (ChEDL). Utilities for process modeling. Copyright (C) 2018, 2019, 2020 Caleb Bell Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. This module contains correlations, standards, and solvers for orifice plates and other flow metering devices. Both permanent and measured pressure drop is included, and models work for both liquids and gases. A number of non-standard devices are included, as well as limited two-phase functionality. For reporting bugs, adding feature requests, or submitting pull requests, please use the `GitHub issue tracker `_ or contact the author at Caleb.Andrew.Bell@gmail.com. .. contents:: :local: Flow Meter Solvers ------------------ .. autofunction:: differential_pressure_meter_solver Flow Meter Interfaces --------------------- .. autofunction:: differential_pressure_meter_dP .. autofunction:: differential_pressure_meter_C_epsilon .. autofunction:: differential_pressure_meter_beta .. autofunction:: dP_orifice Orifice Plate Correlations -------------------------- .. autofunction:: C_Reader_Harris_Gallagher .. autofunction:: C_eccentric_orifice_ISO_15377_1998 .. autofunction:: C_quarter_circle_orifice_ISO_15377_1998 .. autofunction:: C_Miller_1996 .. autofunction:: orifice_expansibility .. autofunction:: orifice_expansibility_1989 .. autodata:: ISO_15377_CONICAL_ORIFICE_C Nozzle Flow Meters ------------------ .. autofunction:: C_long_radius_nozzle .. autofunction:: C_ISA_1932_nozzle .. autofunction:: C_venturi_nozzle .. autofunction:: nozzle_expansibility Venturi Tube Meters ------------------- .. autodata:: ROUGH_WELDED_CONVERGENT_VENTURI_TUBE_C .. autodata:: MACHINED_CONVERGENT_VENTURI_TUBE_C .. autodata:: AS_CAST_VENTURI_TUBE_C .. autofunction:: dP_venturi_tube .. autofunction:: C_Reader_Harris_Gallagher_wet_venturi_tube .. autofunction:: dP_Reader_Harris_Gallagher_wet_venturi_tube Cone Meters ----------- .. autodata:: CONE_METER_C .. autofunction:: diameter_ratio_cone_meter .. autofunction:: cone_meter_expansibility_Stewart .. autofunction:: dP_cone_meter Wedge Meters ------------ .. autofunction:: C_wedge_meter_ISO_5167_6_2017 .. autofunction:: C_wedge_meter_Miller .. autofunction:: diameter_ratio_wedge_meter .. autofunction:: dP_wedge_meter Flow Meter Utilities -------------------- .. autofunction:: discharge_coefficient_to_K .. autofunction:: K_to_discharge_coefficient .. autofunction:: velocity_of_approach_factor .. autofunction:: flow_coefficient .. autofunction:: flow_meter_discharge .. autodata:: all_meters """ from __future__ import division from math import sqrt, cos, sin, tan, atan, pi, radians, exp, acos, log10, log from fluids.friction import friction_factor from fluids.core import Froude_densimetric from fluids.numerics import interp, secant, brenth, NotBoundedError, implementation_optimize_tck, bisplev from fluids.constants import g, inch, inch_inv, pi_inv, root_two __all__ = ['C_Reader_Harris_Gallagher', 'differential_pressure_meter_solver', 'differential_pressure_meter_dP', 'flow_meter_discharge', 'orifice_expansibility', 'discharge_coefficient_to_K', 'K_to_discharge_coefficient', 'dP_orifice', 'velocity_of_approach_factor', 'flow_coefficient', 'nozzle_expansibility', 'C_long_radius_nozzle', 'C_ISA_1932_nozzle', 'C_venturi_nozzle', 'orifice_expansibility_1989', 'dP_venturi_tube', 'diameter_ratio_cone_meter', 'diameter_ratio_wedge_meter', 'cone_meter_expansibility_Stewart', 'dP_cone_meter', 'C_wedge_meter_Miller', 'C_wedge_meter_ISO_5167_6_2017', 'dP_wedge_meter', 'C_Reader_Harris_Gallagher_wet_venturi_tube', 'dP_Reader_Harris_Gallagher_wet_venturi_tube', 'differential_pressure_meter_C_epsilon', 'differential_pressure_meter_beta', 'C_eccentric_orifice_ISO_15377_1998', 'C_quarter_circle_orifice_ISO_15377_1998', 'C_Miller_1996', 'all_meters', ] CONCENTRIC_ORIFICE = 'orifice' # normal ECCENTRIC_ORIFICE = 'eccentric orifice' CONICAL_ORIFICE = 'conical orifice' SEGMENTAL_ORIFICE = 'segmental orifice' QUARTER_CIRCLE_ORIFICE = 'quarter circle orifice' CONDITIONING_4_HOLE_ORIFICE = 'Rosemount 4 hole self conditioing' ORIFICE_HOLE_TYPES = [CONCENTRIC_ORIFICE, ECCENTRIC_ORIFICE, CONICAL_ORIFICE, SEGMENTAL_ORIFICE, QUARTER_CIRCLE_ORIFICE] ORIFICE_CORNER_TAPS = 'corner' ORIFICE_FLANGE_TAPS = 'flange' ORIFICE_D_AND_D_2_TAPS = 'D and D/2' ORIFICE_PIPE_TAPS = 'pipe' # Not in ISO 5167 ORIFICE_VENA_CONTRACTA_TAPS = 'vena contracta' # Not in ISO 5167, normally segmental or eccentric orifices # Used by miller; modifier on taps TAPS_OPPOSITE = '180 degree' TAPS_SIDE = '90 degree' ISO_5167_ORIFICE = 'ISO 5167 orifice' ISO_15377_ECCENTRIC_ORIFICE = 'ISO 15377 eccentric orifice' ISO_15377_QUARTER_CIRCLE_ORIFICE = 'ISO 15377 quarter-circle orifice' ISO_15377_CONICAL_ORIFICE = 'ISO 15377 conical orifice' MILLER_ORIFICE = 'Miller orifice' MILLER_ECCENTRIC_ORIFICE = 'Miller eccentric orifice' MILLER_SEGMENTAL_ORIFICE = 'Miller segmental orifice' MILLER_CONICAL_ORIFICE = 'Miller conical orifice' MILLER_QUARTER_CIRCLE_ORIFICE = 'Miller quarter circle orifice' UNSPECIFIED_METER = 'unspecified meter' LONG_RADIUS_NOZZLE = 'long radius nozzle' ISA_1932_NOZZLE = 'ISA 1932 nozzle' VENTURI_NOZZLE = 'venuri nozzle' AS_CAST_VENTURI_TUBE = 'as cast convergent venturi tube' MACHINED_CONVERGENT_VENTURI_TUBE = 'machined convergent venturi tube' ROUGH_WELDED_CONVERGENT_VENTURI_TUBE = 'rough welded convergent venturi tube' HOLLINGSHEAD_ORIFICE = 'Hollingshead orifice' HOLLINGSHEAD_VENTURI_SMOOTH = 'Hollingshead venturi smooth' HOLLINGSHEAD_VENTURI_SHARP = 'Hollingshead venturi sharp' HOLLINGSHEAD_CONE = 'Hollingshead v cone' HOLLINGSHEAD_WEDGE = 'Hollingshead wedge' CONE_METER = 'cone meter' WEDGE_METER = 'wedge meter' __all__.extend(['ISO_5167_ORIFICE','ISO_15377_ECCENTRIC_ORIFICE', 'MILLER_ORIFICE', 'MILLER_ECCENTRIC_ORIFICE', 'MILLER_SEGMENTAL_ORIFICE', 'LONG_RADIUS_NOZZLE', 'ISA_1932_NOZZLE', 'VENTURI_NOZZLE', 'AS_CAST_VENTURI_TUBE', 'MACHINED_CONVERGENT_VENTURI_TUBE', 'ROUGH_WELDED_CONVERGENT_VENTURI_TUBE', 'CONE_METER', 'WEDGE_METER', 'ISO_15377_CONICAL_ORIFICE', 'MILLER_CONICAL_ORIFICE', 'MILLER_QUARTER_CIRCLE_ORIFICE', 'ISO_15377_QUARTER_CIRCLE_ORIFICE', 'UNSPECIFIED_METER', 'HOLLINGSHEAD_ORIFICE', 'HOLLINGSHEAD_CONE', 'HOLLINGSHEAD_WEDGE', 'HOLLINGSHEAD_VENTURI_SMOOTH', 'HOLLINGSHEAD_VENTURI_SHARP']) __all__.extend(['ORIFICE_CORNER_TAPS', 'ORIFICE_FLANGE_TAPS', 'ORIFICE_D_AND_D_2_TAPS', 'ORIFICE_PIPE_TAPS', 'ORIFICE_VENA_CONTRACTA_TAPS', 'TAPS_OPPOSITE', 'TAPS_SIDE']) __all__.extend(['CONCENTRIC_ORIFICE', 'ECCENTRIC_ORIFICE', 'CONICAL_ORIFICE', 'SEGMENTAL_ORIFICE', 'QUARTER_CIRCLE_ORIFICE']) def flow_meter_discharge(D, Do, P1, P2, rho, C, expansibility=1.0): r'''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. .. math:: m = \left(\frac{\pi D_o^2}{4}\right) C \frac{\sqrt{2\Delta P \rho_1}} {\sqrt{1 - \beta^4}}\cdot \epsilon Parameters ---------- D : float Upstream internal pipe diameter, [m] Do : float Diameter of orifice at flow conditions, [m] P1 : float Static pressure of fluid upstream of orifice at the cross-section of the pressure tap, [Pa] P2 : float Static pressure of fluid downstream of orifice at the cross-section of the pressure tap, [Pa] rho : float Density of fluid at `P1`, [kg/m^3] C : float Coefficient of discharge of the orifice, [-] expansibility : float, optional Expansibility factor (1 for incompressible fluids, less than 1 for real fluids), [-] Returns ------- m : float Mass flow rate of fluid, [kg/s] Notes ----- This is formula 1-12 in [1]_ and also [2]_. 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 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. ''' beta = Do/D beta2 = beta*beta return (0.25*pi*Do*Do)*C*expansibility*sqrt((2.0*rho*(P1 - P2))/(1.0 - beta2*beta2)) def orifice_expansibility(D, Do, P1, P2, k): r'''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. .. math:: \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 ---------- D : float Upstream internal pipe diameter, [m] Do : float Diameter of orifice at flow conditions, [m] P1 : float Static pressure of fluid upstream of orifice at the cross-section of the pressure tap, [Pa] P2 : float Static pressure of fluid downstream of orifice at the cross-section of the pressure tap, [Pa] k : float Isentropic exponent of fluid, [-] Returns ------- expansibility : float, 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. It is said in [1]_ that for liquids this should not be used. The result can be forced by setting `k` to a really high number like 1E20. Examples -------- >>> orifice_expansibility(D=0.0739, Do=0.0222, P1=1E5, P2=9.9E4, k=1.4) 0.9974739057343425 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. ''' beta = Do/D beta2 = beta*beta beta4 = beta2*beta2 return (1.0 - (0.351 + beta4*(0.93*beta4 + 0.256))*( 1.0 - (P2/P1)**(1./k))) def orifice_expansibility_1989(D, Do, P1, P2, k): r'''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. .. math:: \epsilon = 1- (0.41 + 0.35\beta^4)\Delta P/\kappa/P_1 Parameters ---------- D : float Upstream internal pipe diameter, [m] Do : float Diameter of orifice at flow conditions, [m] P1 : float Static pressure of fluid upstream of orifice at the cross-section of the pressure tap, [Pa] P2 : float Static pressure of fluid downstream of orifice at the cross-section of the pressure tap, [Pa] k : float Isentropic exponent of fluid, [-] Returns ------- expansibility : float 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: .. math:: \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. Examples -------- >>> orifice_expansibility_1989(D=0.0739, Do=0.0222, P1=1E5, P2=9.9E4, k=1.4) 0.9970510687411718 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. ''' return 1.0 - (0.41 + 0.35*(Do/D)**4)*(P1 - P2)/(k*P1) def C_Reader_Harris_Gallagher(D, Do, rho, mu, m, taps='corner'): r'''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. .. math:: 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} .. math:: M_2' = \frac{2L_2'}{1-\beta} .. math:: A = \left(\frac{19000\beta}{Re_{D}}\right)^{0.8} .. math:: 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): .. math:: C += 0.11(0.75-\beta)\left(2.8-\frac{D}{0.0254}\right) If the orifice has corner taps: .. math:: L_1 = L_2' = 0 If the orifice has D and D/2 taps: .. math:: L_1 = 1 .. math:: L_2' = 0.47 If the orifice has Flange taps: .. math:: L_1 = L_2' = \frac{0.0254}{D} Parameters ---------- D : float Upstream internal pipe diameter, [m] Do : float Diameter of orifice at flow conditions, [m] rho : float Density of fluid at `P1`, [kg/m^3] mu : float Viscosity of fluid at `P1`, [Pa*s] m : float Mass flow rate of fluid through the orifice, [kg/s] taps : str The orientation of the taps; one of 'corner', 'flange', 'D', or 'D/2', [-] Returns ------- C : float 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 :math:`0.10 \le \beta \le 0.56`) or for :math:`\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 :math:`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. 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 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. .. [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. ''' A_pipe = 0.25*pi*D*D v = m/(A_pipe*rho) Re_D = rho*v*D/mu Re_D_inv = 1.0/Re_D beta = Do/D if taps == 'corner': L1, L2_prime = 0.0, 0.0 elif taps == 'flange': L1 = L2_prime = 0.0254/D elif taps == 'D' or taps == 'D/2' or taps == ORIFICE_D_AND_D_2_TAPS: L1 = 1.0 L2_prime = 0.47 else: raise ValueError('Unsupported tap location') beta2 = beta*beta beta4 = beta2*beta2 beta8 = beta4*beta4 A = 2648.5177066967326*(beta*Re_D_inv)**0.8 # 19000.0^0.8 = 2648.51.... M2_prime = 2.0*L2_prime/(1.0 - beta) # These two exps expnL1 = exp(-L1) expnL2 = expnL1*expnL1 expnL3 = expnL1*expnL2 delta_C_upstream = ((0.043 + expnL3*expnL2*expnL2*(0.080*expnL3 - 0.123)) *(1.0 - 0.11*A)*beta4/(1.0 - beta4)) # The max part is not in the ISO standard t1 = log10(3700.*Re_D_inv) if t1 < 0.0: t1 = 0.0 delta_C_downstream = (-0.031*(M2_prime - 0.8*M2_prime**1.1)*beta**1.3 *(1.0 + 8.0*t1)) # C_inf is discharge coefficient with corner taps for infinite Re # Cs, slope term, provides increase in discharge coefficient for lower # Reynolds numbers. x1 = 63.095734448019314*(Re_D_inv)**0.3 # 63.095... = (1e6)**0.3 x2 = 22.7 - 0.0047*Re_D t2 = x1 if x1 > x2 else x2 # max term is not in the ISO standard C_inf_C_s = (0.5961 + 0.0261*beta2 - 0.216*beta8 + 0.000521*(1E6*beta*Re_D_inv)**0.7 + (0.0188 + 0.0063*A)*beta2*beta*sqrt(beta)*( t2)) C = (C_inf_C_s + delta_C_upstream + delta_C_downstream) if D < 0.07112: # Limit is 2.8 inches, .1 inches smaller than the internal diameter of # a sched. 80 pipe. # Suggested to be required not becausue of any effect of small # diameters themselves, but because of edge radius differences. # max term is given in [4]_ Reader-Harris, Michael book # There is a check for t3 being negative and setting it to zero if so # in some sources but that only occurs when t3 is exactly the limit # (0.07112) so it is not needed t3 = (2.8 - D*inch_inv) delta_C_diameter = 0.011*(0.75 - beta)*t3 C += delta_C_diameter return C _Miller_1996_unsupported_type = "Supported orifice types are %s" %str( (CONCENTRIC_ORIFICE, SEGMENTAL_ORIFICE, ECCENTRIC_ORIFICE, CONICAL_ORIFICE, QUARTER_CIRCLE_ORIFICE)) _Miller_1996_unsupported_tap_concentric = "Supported taps for subtype '%s' are %s" %( CONCENTRIC_ORIFICE, (ORIFICE_CORNER_TAPS, ORIFICE_FLANGE_TAPS, ORIFICE_D_AND_D_2_TAPS, ORIFICE_PIPE_TAPS)) _Miller_1996_unsupported_tap_pos_eccentric = "Supported tap positions for subtype '%s' are %s" %( ECCENTRIC_ORIFICE, (TAPS_OPPOSITE, TAPS_SIDE)) _Miller_1996_unsupported_tap_eccentric = "Supported taps for subtype '%s' are %s" %( ECCENTRIC_ORIFICE, (ORIFICE_FLANGE_TAPS, ORIFICE_VENA_CONTRACTA_TAPS)) _Miller_1996_unsupported_tap_segmental = "Supported taps for subtype '%s' are %s" %( SEGMENTAL_ORIFICE, (ORIFICE_FLANGE_TAPS, ORIFICE_VENA_CONTRACTA_TAPS)) def C_Miller_1996(D, Do, rho, mu, m, subtype='orifice', taps=ORIFICE_CORNER_TAPS, tap_position=TAPS_OPPOSITE): r'''Calculates the coefficient of discharge of any of the orifice types supported by the Miller (1996) [1]_ correlation set. These correlations cover a wide range of industrial applications and sizes. Most of them are functions of `beta` ratio and Reynolds number. Unlike the ISO standards, these correlations do not come with well defined ranges of validity, so caution should be applied using there correlations. The base equation is as follows, and each orifice type and range has different values or correlations for :math:`C_{\infty}`, `b`, and `n`. .. math:: C = C_{\infty} + \frac{b}{{Re}_D^n} Parameters ---------- D : float Upstream internal pipe diameter, [m] Do : float Diameter of orifice at flow conditions, [m] rho : float Density of fluid at `P1`, [kg/m^3] mu : float Viscosity of fluid at `P1`, [Pa*s] m : float Mass flow rate of fluid through the orifice, [kg/s] subtype : str, optional One of 'orifice', 'eccentric orifice', 'segmental orifice', 'conical orifice', or 'quarter circle orifice', [-] taps : str, optional The orientation of the taps; one of 'corner', 'flange', 'D and D/2', 'pipe', or 'vena contracta'; not all orifice subtypes support the all tap types [-] tap_position : str, optional The rotation of the taps, used **only for the eccentric orifice case** where the pressure profile is are not symmetric; '180 degree' for the normal case where the taps are opposite the orifice bore, and '90 degree' for the case where, normally for operational reasons, the taps are near the bore [-] Returns ------- C : float Coefficient of discharge of the orifice, [-] Notes ----- Many of the correlations transition at a pipe diameter of 100 mm to different equations, which will lead to discontinuous behavior. It should also be noted the author of these correlations developed a commercial flow meter rating software package, at [2]_. He passed away in 2014, but contributed massively to the field of flow measurement. The numerous equations for the different cases are as follows: For all **regular (concentric) orifices**, the `b` equation is as follows and n = 0.75: .. math:: b = 91.706\beta^{2.5} Regular (concentric) orifice, corner taps: .. math:: C_{\infty} = 0.5959 + 0.0312\beta^2.1 - 0.184\beta^8 Regular (concentric) orifice, flange taps, D > 58.4 mm: .. math:: C_{\infty} = 0.5959 + 0.0312\beta^{2.1} - 0.184\beta^8 + \frac{2.286\beta^4}{(D_{mm}(1.0 - \beta^4))} - \frac{0.856\beta^3}{D_{mm}} Regular (concentric) orifice, flange taps, D < 58.4 mm: .. math:: C_{\infty} = 0.5959 + 0.0312\beta^{2.1} - 0.184\beta^8 + \frac{0.039\beta^4}{(1.0 - \beta^4)} - \frac{0.856\beta^3}{D_{mm}} Regular (concentric) orifice, 'D and D/2' taps: .. math:: C_{\infty} = 0.5959 + 0.0312\beta^{2.1} - 0.184\beta^8 + \frac{0.039\beta^4}{(1.0 - \beta^4)} - 0.01584 Regular (concentric) orifice, 'pipe' taps: .. math:: C_{\infty} = 0.5959 + 0.461\beta^{2.1} + 0.48\beta^8 + \frac{0.039\beta^4}{(1.0 - \beta^4)} For the case of a **conical orifice**, there is no tap dependence and one equation (`b` = 0, `n` = 0): .. math:: C_{\infty} = 0.734 \text{ if } 250\beta \le Re \le 500\beta \text{ else } 0.730 For the case of a **quarter circle orifice**, corner and flange taps have the same dependence (`b` = 0, `n` = 0): .. math:: C_{\infty} = (0.7746 - 0.1334\beta^{2.1} + 1.4098\beta^8 + \frac{0.0675\beta^4}{(1 - \beta^4)} + 0.3865\beta^3) For all **segmental orifice** types, `b` = 0 and `n` = 0 Segmental orifice, 'flange' taps, D < 10 cm: .. math:: C_{\infty} = 0.6284 + 0.1462\beta^{2.1} - 0.8464\beta^8 + \frac{0.2603\beta^4}{(1-\beta^4)} - 0.2886\beta^3 Segmental orifice, 'flange' taps, D > 10 cm: .. math:: C_{\infty} = 0.6276 + 0.0828\beta^{2.1} + 0.2739\beta^8 - \frac{0.0934\beta^4}{(1-\beta^4)} - 0.1132\beta^3 Segmental orifice, 'vena contracta' taps, D < 10 cm: .. math:: C_{\infty} = 0.6261 + 0.1851\beta^{2.1} - 0.2879\beta^8 + \frac{0.1170\beta^4}{(1-\beta^4)} - 0.2845\beta^3 Segmental orifice, 'vena contracta' taps, D > 10 cm: .. math:: C_{\infty} = 0.6276 + 0.0828\beta^{2.1} + 0.2739\beta^8 - \frac{0.0934\beta^4}{(1-\beta^4)} - 0.1132\beta^3 For all **eccentric orifice** types, `n` = 0.75 and `b` is fit to a polynomial of `beta`. Eccentric orifice, 'flange' taps, 180 degree opposite taps, D < 10 cm: .. math:: C_{\infty} = 0.5917 + 0.3061\beta^{2.1} + .3406\beta^8 -\frac{.1019\beta^4}{(1-\beta^4)} - 0.2715\beta^3 .. math:: b = 7.3 - 15.7\beta + 170.8\beta^2 - 399.7\beta^3 + 332.2\beta^4 Eccentric orifice, 'flange' taps, 180 degree opposite taps, D > 10 cm: .. math:: C_{\infty} = 0.6016 + 0.3312\beta^{2.1} -1.5581\beta^8 + \frac{0.6510\beta^4}{(1-\beta^4)} - 0.7308\beta^3 .. math:: b = -139.7 + 1328.8\beta - 4228.2\beta^2 + 5691.9\beta^3 - 2710.4\beta^4 Eccentric orifice, 'flange' taps, 90 degree side taps, D < 10 cm: .. math:: C_{\infty} = 0.5866 + 0.3917\beta^{2.1} + .7586\beta^8 - \frac{.2273\beta^4}{(1-\beta^4)} - .3343\beta^3 .. math:: b = 69.1 - 469.4\beta + 1245.6\beta^2 -1287.5\beta^3 + 486.2\beta^4 Eccentric orifice, 'flange' taps, 90 degree side taps, D > 10 cm: .. math:: C_{\infty} = 0.6037 + 0.1598\beta^{2.1} -.2918\beta^8 + \frac{0.0244\beta^4}{(1-\beta^4)} - 0.0790\beta^3 .. math:: b = -103.2 + 898.3\beta - 2557.3\beta^2 + 2977.0\beta^3 - 1131.3\beta^4 Eccentric orifice, 'vena contracta' taps, 180 degree opposite taps, D < 10 cm: .. math:: C_{\infty} = 0.5925 + 0.3380\beta^{2.1} + 0.4016\beta^8 - \frac{.1046\beta^4}{(1-\beta^4)} - 0.3212\beta^3 .. math:: b = 23.3 -207.0\beta + 821.5\beta^2 -1388.6\beta^3 + 900.3\beta^4 Eccentric orifice, 'vena contracta' taps, 180 degree opposite taps, D > 10 cm: .. math:: C_{\infty} = 0.5922 + 0.3932\beta^{2.1} + .3412\beta^8 - \frac{.0569\beta^4}{(1-\beta^4)} - 0.4628\beta^3 .. math:: b = 55.7 - 471.4\beta + 1721.8\beta^2 - 2722.6\beta^3 + 1569.4\beta^4 Eccentric orifice, 'vena contracta' taps, 90 degree side taps, D < 10 cm: .. math:: C_{\infty} = 0.5875 + 0.3813\beta^{2.1} + 0.6898\beta^8 - \frac{0.1963\beta^4}{(1-\beta^4)} - 0.3366\beta^3 .. math:: b = -69.3 + 556.9\beta - 1332.2\beta^2 + 1303.7\beta^3 - 394.8\beta^4 Eccentric orifice, 'vena contracta' taps, 90 degree side taps, D > 10 cm: .. math:: C_{\infty} = 0.5949 + 0.4078\beta^{2.1} + 0.0547\beta^8 + \frac{0.0955\beta^4}{(1-\beta^4)} - 0.5608\beta^3 .. math:: b = 52.8 - 434.2\beta + 1571.2\beta^2 - 2460.9\beta^3 + 1420.2\beta^4 Examples -------- >>> C_Miller_1996(D=0.07391, Do=0.0222, rho=1.165, mu=1.85E-5, m=0.12, taps='flange', subtype='orifice') 0.599065557156788 References ---------- .. [1] Miller, Richard W. Flow Measurement Engineering Handbook. McGraw-Hill Education, 1996. .. [2] "RW Miller & Associates." Accessed April 13, 2020. http://rwmillerassociates.com/. ''' A_pipe = 0.25*pi*D*D v = m/(A_pipe*rho) Re = rho*v*D/mu D_mm = D*1000.0 beta = Do/D beta3 = beta*beta*beta beta4 = beta*beta3 beta8 = beta4*beta4 beta21 = beta**2.1 if subtype == MILLER_ORIFICE or subtype == CONCENTRIC_ORIFICE: b = 91.706*beta**2.5 n = 0.75 if taps == ORIFICE_CORNER_TAPS: C_inf = 0.5959 + 0.0312*beta21 - 0.184*beta8 elif taps == ORIFICE_FLANGE_TAPS: if D_mm >= 58.4: C_inf = 0.5959 + 0.0312*beta21 - 0.184*beta8 + 2.286*beta4/(D_mm*(1.0 - beta4)) - 0.856*beta3/D_mm else: C_inf = 0.5959 + 0.0312*beta21 - 0.184*beta8 + 0.039*beta4/(1.0 - beta4) - 0.856*beta3/D_mm elif taps == ORIFICE_D_AND_D_2_TAPS: C_inf = 0.5959 + 0.0312*beta21 - 0.184*beta8 + 0.039*beta4/(1.0 - beta4) - 0.01584 elif taps == ORIFICE_PIPE_TAPS: C_inf = 0.5959 + 0.461*beta21 + 0.48*beta8 + 0.039*beta4/(1.0 - beta4) else: raise ValueError(_Miller_1996_unsupported_tap_concentric) elif subtype == MILLER_ECCENTRIC_ORIFICE or subtype == ECCENTRIC_ORIFICE: if tap_position != TAPS_OPPOSITE and tap_position != TAPS_SIDE: raise ValueError(_Miller_1996_unsupported_tap_pos_eccentric) n = 0.75 if taps == ORIFICE_FLANGE_TAPS: if tap_position == TAPS_OPPOSITE: if D < 0.1: b = 7.3 - 15.7*beta + 170.8*beta**2 - 399.7*beta3 + 332.2*beta4 C_inf = 0.5917 + 0.3061*beta21 + .3406*beta8 -.1019*beta4/(1-beta4) - 0.2715*beta3 else: b = -139.7 + 1328.8*beta - 4228.2*beta**2 + 5691.9*beta3 - 2710.4*beta4 C_inf = 0.6016 + 0.3312*beta21 - 1.5581*beta8 + 0.6510*beta4/(1-beta4) - 0.7308*beta3 elif tap_position == TAPS_SIDE: if D < 0.1: b = 69.1 - 469.4*beta + 1245.6*beta**2 -1287.5*beta3 + 486.2*beta4 C_inf = 0.5866 + 0.3917*beta21 + 0.7586*beta8 -.2273*beta4/(1-beta4) - .3343*beta3 else: b = -103.2 + 898.3*beta - 2557.3*beta**2 + 2977.0*beta3 - 1131.3*beta4 C_inf = 0.6037 + 0.1598*beta21 - 0.2918*beta8 + 0.0244*beta4/(1-beta4) - 0.0790*beta3 elif taps == ORIFICE_VENA_CONTRACTA_TAPS: if tap_position == TAPS_OPPOSITE: if D < 0.1: b = 23.3 -207.0*beta + 821.5*beta**2 -1388.6*beta3 + 900.3*beta4 C_inf = 0.5925 + 0.3380*beta21 + 0.4016*beta8 -.1046*beta4/(1-beta4) - 0.3212*beta3 else: b = 55.7 - 471.4*beta + 1721.8*beta**2 - 2722.6*beta3 + 1569.4*beta4 C_inf = 0.5922 + 0.3932*beta21 + .3412*beta8 -.0569*beta4/(1-beta4) - 0.4628*beta3 elif tap_position == TAPS_SIDE: if D < 0.1: b = -69.3 + 556.9*beta - 1332.2*beta**2 + 1303.7*beta3 - 394.8*beta4 C_inf = 0.5875 + 0.3813*beta21 + 0.6898*beta8 -0.1963*beta4/(1-beta4) - 0.3366*beta3 else: b = 52.8 - 434.2*beta + 1571.2*beta**2 - 2460.9*beta3 + 1420.2*beta4 C_inf = 0.5949 + 0.4078*beta21 + 0.0547*beta8 +0.0955*beta4/(1-beta4) - 0.5608*beta3 else: raise ValueError(_Miller_1996_unsupported_tap_eccentric) elif subtype == MILLER_SEGMENTAL_ORIFICE or subtype == SEGMENTAL_ORIFICE: n = b = 0.0 if taps == ORIFICE_FLANGE_TAPS: if D < 0.1: C_inf = 0.6284 + 0.1462*beta21 - 0.8464*beta8 + 0.2603*beta4/(1-beta4) - 0.2886*beta3 else: C_inf = 0.6276 + 0.0828*beta21 + 0.2739*beta8 - 0.0934*beta4/(1-beta4) - 0.1132*beta3 elif taps == ORIFICE_VENA_CONTRACTA_TAPS: if D < 0.1: C_inf = 0.6261 + 0.1851*beta21 - 0.2879*beta8 + 0.1170*beta4/(1-beta4) - 0.2845*beta3 else: # Yes these are supposed to be the same as the flange, large set C_inf = 0.6276 + 0.0828*beta21 + 0.2739*beta8 - 0.0934*beta4/(1-beta4) - 0.1132*beta3 else: raise ValueError(_Miller_1996_unsupported_tap_segmental) elif subtype == MILLER_CONICAL_ORIFICE or subtype == CONICAL_ORIFICE: n = b = 0.0 if 250.0*beta <= Re <= 500.0*beta: C_inf = 0.734 else: C_inf = 0.730 elif subtype == MILLER_QUARTER_CIRCLE_ORIFICE or subtype == QUARTER_CIRCLE_ORIFICE: n = b = 0.0 C_inf = (0.7746 - 0.1334*beta21 + 1.4098*beta8 + 0.0675*beta4/(1.0 - beta4) + 0.3865*beta3) else: raise ValueError(_Miller_1996_unsupported_type) C = C_inf + b*Re**-n return C # Data from: Discharge Coefficient Performance of Venturi, Standard Concentric Orifice Plate, V-Cone, and Wedge Flow Meters at Small Reynolds Numbers orifice_std_Res_Hollingshead = [1.0, 5.0, 10.0, 20.0, 30.0, 40.0, 60.0, 80.0, 100.0, 200.0, 300.0, 500.0, 1000.0, 2000.0, 3000.0, 5000.0, 10000.0, 100000.0, 1000000.0, 10000000.0, 50000000.0 ] orifice_std_logRes_Hollingshead = [0.0, 1.6094379124341003, 2.302585092994046, 2.995732273553991, 3.4011973816621555, 3.6888794541139363, 4.0943445622221, 4.382026634673881, 4.605170185988092, 5.298317366548036, 5.703782474656201, 6.214608098422191, 6.907755278982137, 7.600902459542082, 8.006367567650246, 8.517193191416238, 9.210340371976184, 11.512925464970229, 13.815510557964274, 16.11809565095832, 17.72753356339242 ] orifice_std_betas_Hollingshead = [0.5, 0.6, 0.65, 0.7] orifice_std_beta_5_Hollingshead_Cs = [0.233, 0.478, 0.585, 0.654, 0.677, 0.688, 0.697, 0.700, 0.702, 0.699, 0.693, 0.684, 0.67, 0.648, 0.639, 0.632, 0.629, 0.619, 0.615, 0.614, 0.614 ] orifice_std_beta_6_Hollingshead_Cs = [0.212, 0.448, 0.568, 0.657, 0.689, 0.707, 0.721, 0.725, 0.727, 0.725, 0.719, 0.707, 0.688, 0.658, 0.642, 0.633, 0.624, 0.61, 0.605, 0.602, 0.595 ] orifice_std_beta_65_Hollingshead_Cs = [0.202, 0.425, 0.546, 0.648, 0.692, 0.715, 0.738, 0.748, 0.754, 0.764, 0.763, 0.755, 0.736, 0.685, 0.666, 0.656, 0.641, 0.622, 0.612, 0.61, 0.607 ] orifice_std_beta_7_Hollingshead_Cs = [0.191, 0.407, 0.532, 0.644, 0.696, 0.726, 0.756, 0.772, 0.781, 0.795, 0.796, 0.788, 0.765, 0.7, 0.67, 0.659, 0.646, 0.623, 0.616, 0.607, 0.604 ] orifice_std_Hollingshead_Cs = [orifice_std_beta_5_Hollingshead_Cs, orifice_std_beta_6_Hollingshead_Cs, orifice_std_beta_65_Hollingshead_Cs, orifice_std_beta_7_Hollingshead_Cs ] orifice_std_Hollingshead_tck = implementation_optimize_tck([ [0.5, 0.5, 0.5, 0.5, 0.7, 0.7, 0.7, 0.7], [0.0, 0.0, 0.0, 0.0, 2.302585092994046, 2.995732273553991, 3.4011973816621555, 3.6888794541139363, 4.0943445622221, 4.382026634673881, 4.605170185988092, 5.298317366548036, 5.703782474656201, 6.214608098422191, 6.907755278982137, 7.600902459542082, 8.006367567650246, 8.517193191416238, 9.210340371976184, 11.512925464970229, 13.815510557964274, 17.72753356339242, 17.72753356339242, 17.72753356339242, 17.72753356339242 ], [0.23300000000000026, 0.3040793845022822, 0.5397693379388018, 0.6509414325648643, 0.6761419937262648, 0.6901697401156808, 0.6972240707909276, 0.6996759572505151, 0.7040223363705952, 0.7008741587711967, 0.692665226515394, 0.6826387818678974, 0.6727930643166521, 0.6490542161859936, 0.6378780959698012, 0.6302027504736312, 0.6284904523610422, 0.616773266650063, 0.6144108030024114, 0.6137270770149181, 0.6140000000000004, 0.21722222222222212, 0.26754856063815036, 0.547178981607613, 0.6825835849471493, 0.6848255120880751, 0.712775784969247, 0.7066842545008245, 0.7020345744268808, 0.6931476737316041, 0.6710886785478944, 0.6501218695989138, 0.6257164975579488, 0.5888463567232898, 0.6237505336392806, 0.578149766754485, 0.5761890160080455, 0.5922303103985014, 0.5657790974864929, 0.6013376373672517, 0.5693593555949975, 0.5528888888888888, 0.206777777777778, 0.2644342350096853, 0.4630985572034346, 0.6306849522311501, 0.6899260188747366, 0.7092703879134302, 0.7331416654072416, 0.7403866219900521, 0.7531493636395633, 0.7685019053395048, 0.771007019842085, 0.7649533772965396, 0.7707020081746302, 0.6897832472092346, 0.6910618341373851, 0.6805763529796045, 0.6291884772151493, 0.6470904244660671, 0.5962879899497537, 0.6353096798316025, 0.6277777777777779, 0.19100000000000003, 0.23712276889270198, 0.44482842661392175, 0.6337225464930397, 0.6926462978136392, 0.7316874888663132, 0.7542057211530093, 0.77172737538752, 0.7876049778429112, 0.795143180926116, 0.7977570986094262, 0.7861445043222344, 0.777182818678971, 0.7057345800650827, 0.6626698628526632, 0.6600690433654985, 0.6323396431072075, 0.6212684034830293, 0.616281323630018, 0.603728515722033, 0.6040000000000001 ], 3, 3 ]) def C_eccentric_orifice_ISO_15377_1998(D, Do): r'''Calculates the coefficient of discharge of an eccentric orifice based on the geometry of the plate according to ISO 15377, first introduced in 1998 and also presented in the second 2007 edition. It also appears in BS 1042-1.2: 1989. .. math:: C = 0.9355 - 1.6889\beta + 3.0428\beta^2 - 1.7989\beta^3 This type of plate is normally used to avoid obstructing entrained gas, liquid, or sediment. Parameters ---------- D : float Upstream internal pipe diameter, [m] Do : float Diameter of orifice at flow conditions, [m] Returns ------- C : float Coefficient of discharge of the eccentric orifice, [-] Notes ----- No correction for where the orifice bore is located is included. The following limits apply to the orifice plate standard [1]_: * Bore diameter above 50 mm. * Pipe diameter between 10 cm and 1 m. * Beta ratio between 0.46 and 0.84 * :math:`2\times 10^5 \beta^2 \le Re_D \le 10^6 \beta` The uncertainty of this equation for `C` is said to be 1% if `beta` is under 0.75, otherwise 2%. The `orifice_expansibility` function should be used with this method as well. Additional specifications are: * The thickness of the orifice should be between 0.005`D` and 0.02`D`. * Corner tappings should be used, with hole diameter between 3 and 10 mm. The angular orientation of the tappings matters because the flow meter is not symmetrical. The angle should ideally be at the top or bottom of the plate, opposite which side the bore is on - but this can cause issues with deposition if the taps are on the bottom or gas bubbles if the taps are on the taps. The taps are often placed 30 degrees away from the ideal position to counteract this effect, with under an extra 2% error. Some comparisons with CFD results can be found in [2]_. Examples -------- >>> C_eccentric_orifice_ISO_15377_1998(.2, .075) 0.6351923828125 References ---------- .. [1] TC 30/SC 2, ISO. ISO/TR 15377:1998, Measurement of Fluid Flow by Means of Pressure-Differential Devices - Guide for the Specification of Nozzles and Orifice Plates beyond the Scope of ISO 5167-1. .. [2] Yashvanth, S., Varadarajan Seshadri, and J. YogeshKumarK. "CFD Analysis of Flow through Single and Multi Stage Eccentric Orifice Plate Assemblies," 2017. ''' beta = Do/D C = beta*(beta*(3.0428 - 1.7989*beta) - 1.6889) + 0.9355 return C def C_quarter_circle_orifice_ISO_15377_1998(D, Do): r'''Calculates the coefficient of discharge of a quarter circle orifice based on the geometry of the plate according to ISO 15377, first introduced in 1998 and also presented in the second 2007 edition. It also appears in BS 1042-1.2: 1989. .. math:: C = 0.73823 + 0.3309\beta - 1.1615\beta^2 + 1.5084\beta^3 Parameters ---------- D : float Upstream internal pipe diameter, [m] Do : float Diameter of orifice at flow conditions, [m] Returns ------- C : float Coefficient of discharge of the quarter circle orifice, [-] Notes ----- The discharge coefficient of this type of orifice plate remains constant down to a lower than normal `Re`, as occurs in highly viscous applications. The following limits apply to the orifice plate standard [1]_: * Bore diameter >= 1.5 cm * Pipe diameter <= 50 cm * Beta ratio between 0.245 and 0.6 * :math:`Re_d \le 10^5 \beta` There is also a table in [1]_ which lists increased minimum upstream pipe diameters for pipes of different roughnesses; the higher the roughness, the larger the pipe diameter required, and the table goes up to 20 cm for rusty cast iron. Corner taps should be used up to pipe diameters of 40 mm; for larger pipes, corner or flange taps can be used. No impact on the flow coefficient is included in the correlation. The recommended expansibility method for this type of orifice is :obj:`orifice_expansibility`. Examples -------- >>> C_quarter_circle_orifice_ISO_15377_1998(.2, .075) 0.77851484375000 References ---------- .. [1] TC 30/SC 2, ISO. ISO/TR 15377:1998, Measurement of Fluid Flow by Means of Pressure-Differential Devices - Guide for the Specification of Nozzles and Orifice Plates beyond the Scope of ISO 5167-1. ''' beta = Do/D C = beta*(beta*(1.5084*beta - 1.16158) + 0.3309) + 0.73823 return C def discharge_coefficient_to_K(D, Do, C): r'''Converts a discharge coefficient to a standard loss coefficient, for use in computation of the actual pressure drop of an orifice or other device. .. math:: K = \left[\frac{\sqrt{1-\beta^4(1-C^2)}}{C\beta^2} - 1\right]^2 Parameters ---------- D : float Upstream internal pipe diameter, [m] Do : float Diameter of orifice at flow conditions, [m] C : float Coefficient of discharge of the orifice, [-] Returns ------- K : float 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. Examples -------- >>> discharge_coefficient_to_K(D=0.07366, Do=0.05, C=0.61512) 5.2314291729754 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. ''' beta = Do/D beta2 = beta*beta beta4 = beta2*beta2 root_K = (sqrt(1.0 - beta4*(1.0 - C*C))/(C*beta2) - 1.0) return root_K*root_K def K_to_discharge_coefficient(D, Do, K): r'''Converts a standard loss coefficient to a discharge coefficient. .. math:: C = \sqrt{\frac{1}{2 \sqrt{K} \beta^{4} + K \beta^{4}} - \frac{\beta^{4}}{2 \sqrt{K} \beta^{4} + K \beta^{4}} } Parameters ---------- D : float Upstream internal pipe diameter, [m] Do : float Diameter of orifice at flow conditions, [m] K : float Loss coefficient with respect to the velocity and density of the fluid just upstream of the orifice, [-] Returns ------- C : float 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. Examples -------- >>> K_to_discharge_coefficient(D=0.07366, Do=0.05, K=5.2314291729754) 0.6151200000000001 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. ''' beta = Do/D beta2 = beta*beta beta4 = beta2*beta2 root_K = sqrt(K) return sqrt((1.0 - beta4)/((2.0*root_K + K)*beta4)) def dP_orifice(D, Do, P1, P2, C): r'''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. .. math:: \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 ---------- D : float Upstream internal pipe diameter, [m] Do : float Diameter of orifice at flow conditions, [m] P1 : float Static pressure of fluid upstream of orifice at the cross-section of the pressure tap, [Pa] P2 : float Static pressure of fluid downstream of orifice at the cross-section of the pressure tap, [Pa] C : float Coefficient of discharge of the orifice, [-] Returns ------- dP : float Non-recoverable pressure drop of the orifice plate, [Pa] Notes ----- This formula can be well approximated by: .. math:: \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. Examples -------- >>> dP_orifice(D=0.07366, Do=0.05, P1=200000.0, P2=183000.0, C=0.61512) 9069.474705745388 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. ''' beta = Do/D beta2 = beta*beta beta4 = beta2*beta2 dP = P1 - P2 delta_w = (sqrt(1.0 - beta4*(1.0 - C*C)) - C*beta2)/( sqrt(1.0 - beta4*(1.0 - C*C)) + C*beta2)*dP return delta_w def velocity_of_approach_factor(D, Do): r'''Calculates a factor for orifice plate design called the `velocity of approach`. .. math:: \text{Velocity of approach} = \frac{1}{\sqrt{1 - \beta^4}} Parameters ---------- D : float Upstream internal pipe diameter, [m] Do : float Diameter of orifice at flow conditions, [m] Returns ------- velocity_of_approach : float Coefficient of discharge of the orifice, [-] Notes ----- Examples -------- >>> velocity_of_approach_factor(D=0.0739, Do=0.0222) 1.0040970074165514 References ---------- .. [1] American Society of Mechanical Engineers. Mfc-3M-2004 Measurement Of Fluid Flow In Pipes Using Orifice, Nozzle, And Venturi. ASME, 2001. ''' return 1.0/sqrt(1.0 - (Do/D)**4) def flow_coefficient(D, Do, C): r'''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. .. math:: \text{Flow coefficient} = \frac{C}{\sqrt{1 - \beta^4}} Parameters ---------- D : float Upstream internal pipe diameter, [m] Do : float Diameter of flow meter characteristic dimension at flow conditions, [m] C : float Coefficient of discharge of the flow meter, [-] Returns ------- flow_coefficient : float 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]_. Examples -------- >>> flow_coefficient(D=0.0739, Do=0.0222, C=0.6) 0.6024582044499308 References ---------- .. [1] American Society of Mechanical Engineers. Mfc-3M-2004 Measurement Of Fluid Flow In Pipes Using Orifice, Nozzle, And Venturi. ASME, 2001. .. [2] Miller, Richard W. Flow Measurement Engineering Handbook. 3rd edition. New York: McGraw-Hill Education, 1996. ''' return C*1.0/sqrt(1.0 - (Do/D)**4) def nozzle_expansibility(D, Do, P1, P2, k, beta=None): r'''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. .. math:: \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 ---------- D : float Upstream internal pipe diameter, [m] Do : float Diameter of orifice of the venturi or nozzle, [m] P1 : float Static pressure of fluid upstream of orifice at the cross-section of the pressure tap, [Pa] P2 : float Static pressure of fluid downstream of orifice at the cross-section of the pressure tap, [Pa] k : float Isentropic exponent of fluid, [-] beta : float, optional Optional `beta` ratio, which is useful to specify for wedge meters or flow meters which have a different beta ratio calculation, [-] Returns ------- expansibility : float 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. Mathematically the equation cannot be evaluated at `k` = 1, but if the limit of the equation is taken the following equation is obtained and is implemented: .. math:: \epsilon = \sqrt{\frac{- D^{4} P_{1} P_{2}^{2} \log{\left(\frac{P_{2}} {P_{1}} \right)} + Do^{4} P_{1} P_{2}^{2} \log{\left(\frac{P_{2}}{P_{1}} \right)}}{D^{4} P_{1}^{3} - D^{4} P_{1}^{2} P_{2} - Do^{4} P_{1} P_{2}^{2} + Do^{4} P_{2}^{3}}} Note also there is a small amount of floating-point error around the range of `k` ~1+1e-5 to ~1-1e-5, starting with 1e-7 and increasing to the point of giving values larger than 1 or zero in the `k` ~1+1e-12 to ~1-1e-12 range. Examples -------- >>> nozzle_expansibility(D=0.0739, Do=0.0222, P1=1E5, P2=9.9E4, k=1.4) 0.994570234456 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. ''' if beta is None: beta = Do/D beta2 = beta*beta beta4 = beta2*beta2 tau = P2/P1 if k == 1.0: '''Avoid a zero division error: from sympy import * D, Do, P1, P2, k = symbols('D, Do, P1, P2, k') beta = Do/D tau = P2/P1 term1 = k*tau**(2/k )/(k - 1) term2 = (1 - beta**4)/(1 - beta**4*tau**(2/k)) term3 = (1 - tau**((k - 1)/k))/(1 - tau) val= sqrt(term1*term2*term3) print(simplify(limit((term1*term2*term3), k, 1))) ''' limit_val = (P1*P2**2*(-D**4 + Do**4)*log(P2/P1)/(D**4*P1**3 - D**4*P1**2*P2 - Do**4*P1*P2**2 + Do**4*P2**3)) return sqrt(limit_val) term1 = k*tau**(2.0/k)/(k - 1.0) term2 = (1.0 - beta4)/(1.0 - beta4*tau**(2.0/k)) if tau == 1.0: '''Avoid a zero division error. Obtained with: from sympy import * tau, k = symbols('tau, k') expr = (1 - tau**((k - 1)/k))/(1 - tau) limit(expr, tau, 1) ''' term3 = (k - 1.0)/k else: # This form of the equation is mathematically equivalent but # does not have issues where k = `. term3 = (P1 - P2*(tau)**(-1.0/k))/(P1 - P2) # term3 = (1.0 - tau**((k - 1.0)/k))/(1.0 - tau) return sqrt(term1*term2*term3) def C_long_radius_nozzle(D, Do, rho, mu, m): r'''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. .. math:: C = 0.9965 - 0.00653\beta^{0.5} \left(\frac{10^6}{Re_D}\right)^{0.5} Parameters ---------- D : float Upstream internal pipe diameter, [m] Do : float Diameter of long radius nozzle orifice at flow conditions, [m] rho : float Density of fluid at `P1`, [kg/m^3] mu : float Viscosity of fluid at `P1`, [Pa*s] m : float Mass flow rate of fluid through the nozzle, [kg/s] Returns ------- C : float Coefficient of discharge of the long radius nozzle orifice, [-] Notes ----- Examples -------- >>> C_long_radius_nozzle(D=0.07391, Do=0.0422, rho=1.2, mu=1.8E-5, m=0.1) 0.9805503704679863 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. ''' A_pipe = pi/4.*D*D v = m/(A_pipe*rho) Re_D = rho*v*D/mu beta = Do/D return 0.9965 - 0.00653*sqrt(beta)*sqrt(1E6/Re_D) def C_ISA_1932_nozzle(D, Do, rho, mu, m): r'''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. .. math:: 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 ---------- D : float Upstream internal pipe diameter, [m] Do : float Diameter of nozzle orifice at flow conditions, [m] rho : float Density of fluid at `P1`, [kg/m^3] mu : float Viscosity of fluid at `P1`, [Pa*s] m : float Mass flow rate of fluid through the nozzle, [kg/s] Returns ------- C : float Coefficient of discharge of the nozzle orifice, [-] Notes ----- Examples -------- >>> C_ISA_1932_nozzle(D=0.07391, Do=0.0422, rho=1.2, mu=1.8E-5, m=0.1) 0.9635849973250495 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. ''' A_pipe = pi/4.*D*D v = m/(A_pipe*rho) Re_D = rho*v*D/mu beta = Do/D C = (0.9900 - 0.2262*beta**4.1 - (0.00175*beta**2 - 0.0033*beta**4.15)*(1E6/Re_D)**1.15) return C def C_venturi_nozzle(D, Do): r'''Calculates the coefficient of discharge of an Venturi style nozzle used for measuring flow rate of fluid, based on the geometry of the nozzle. .. math:: C = 0.9858 - 0.196\beta^{4.5} Parameters ---------- D : float Upstream internal pipe diameter, [m] Do : float Diameter of nozzle orifice at flow conditions, [m] Returns ------- C : float Coefficient of discharge of the nozzle orifice, [-] Notes ----- Examples -------- >>> C_venturi_nozzle(D=0.07391, Do=0.0422) 0.9698996454169576 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. ''' beta = Do/D return 0.9858 - 0.198*beta**4.5 # Relative pressure loss as a function of beta reatio for venturi nozzles # Venturi nozzles should be between 65 mm and 500 mm; there are high and low # loss ratios , with the high losses corresponding to small diameters, # low high losses corresponding to large diameters # Interpolation can be performed. venturi_tube_betas = [0.299160, 0.299470, 0.312390, 0.319010, 0.326580, 0.337290, 0.342020, 0.347060, 0.359030, 0.365960, 0.372580, 0.384870, 0.385810, 0.401250, 0.405350, 0.415740, 0.424250, 0.434010, 0.447880, 0.452590, 0.471810, 0.473090, 0.493540, 0.499240, 0.516530, 0.523800, 0.537630, 0.548060, 0.556840, 0.573890, 0.582350, 0.597820, 0.601560, 0.622650, 0.626490, 0.649480, 0.650990, 0.668700, 0.675870, 0.688550, 0.693180, 0.706180, 0.713330, 0.723510, 0.749540, 0.749650] venturi_tube_dP_high = [0.164534, 0.164504, 0.163591, 0.163508, 0.163439, 0.162652, 0.162224, 0.161866, 0.161238, 0.160786, 0.160295, 0.159280, 0.159193, 0.157776, 0.157467, 0.156517, 0.155323, 0.153835, 0.151862, 0.151154, 0.147840, 0.147613, 0.144052, 0.143050, 0.140107, 0.138981, 0.136794, 0.134737, 0.132847, 0.129303, 0.127637, 0.124758, 0.124006, 0.119269, 0.118449, 0.113605, 0.113269, 0.108995, 0.107109, 0.103688, 0.102529, 0.099567, 0.097791, 0.095055, 0.087681, 0.087648] venturi_tube_dP_low = [0.089232, 0.089218, 0.088671, 0.088435, 0.088206, 0.087853, 0.087655, 0.087404, 0.086693, 0.086241, 0.085813, 0.085142, 0.085102, 0.084446, 0.084202, 0.083301, 0.082470, 0.081650, 0.080582, 0.080213, 0.078509, 0.078378, 0.075989, 0.075226, 0.072700, 0.071598, 0.069562, 0.068128, 0.066986, 0.064658, 0.063298, 0.060872, 0.060378, 0.057879, 0.057403, 0.054091, 0.053879, 0.051726, 0.050931, 0.049362, 0.048675, 0.046522, 0.045381, 0.043840, 0.039913, 0.039896] #ratios_average = 0.5*(ratios_high + ratios_low) D_bound_venturi_tube = [0.065, 0.5] def dP_venturi_tube(D, Do, P1, P2): r'''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. .. math:: \epsilon = \frac{\Delta\bar w }{\Delta P} The :math:`\epsilon` value is looked up in a table of values as a function of beta ratio and upstream pipe diameter (roughness impact). Parameters ---------- D : float Upstream internal pipe diameter, [m] Do : float Diameter of venturi tube at flow conditions, [m] P1 : float Static pressure of fluid upstream of venturi tube at the cross-section of the pressure tap, [Pa] P2 : float Static pressure of fluid downstream of venturi tube at the cross-section of the pressure tap, [Pa] Returns ------- dP : float 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. Examples -------- >>> dP_venturi_tube(D=0.07366, Do=0.05, P1=200000.0, P2=183000.0) 1788.5717754177406 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. ''' # Effect of Re is not currently included beta = Do/D epsilon_D65 = interp(beta, venturi_tube_betas, venturi_tube_dP_high) epsilon_D500 = interp(beta, venturi_tube_betas, venturi_tube_dP_low) epsilon = interp(D, D_bound_venturi_tube, [epsilon_D65, epsilon_D500]) return epsilon*(P1 - P2) def diameter_ratio_cone_meter(D, Dc): r'''Calculates the diameter ratio `beta` used to characterize a cone flow meter. .. math:: \beta = \sqrt{1 - \frac{d_c^2}{D^2}} Parameters ---------- D : float Upstream internal pipe diameter, [m] Dc : float Diameter of the largest end of the cone meter, [m] Returns ------- beta : float Cone meter diameter ratio, [-] Notes ----- A mathematically equivalent formula often written is: .. math:: \beta = \frac{\sqrt{D^2 - d_c^2}}{D} Examples -------- >>> diameter_ratio_cone_meter(D=0.2575, Dc=0.184) 0.6995709873957624 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. ''' D_ratio = Dc/D return sqrt(1.0 - D_ratio*D_ratio) def cone_meter_expansibility_Stewart(D, Dc, P1, P2, k): r'''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]_. .. math:: \epsilon = 1 - (0.649 + 0.696\beta^4) \frac{\Delta P}{\kappa P_1} Parameters ---------- D : float Upstream internal pipe diameter, [m] Dc : float Diameter of the largest end of the cone meter, [m] P1 : float Static pressure of fluid upstream of cone meter at the cross-section of the pressure tap, [Pa] P2 : float Static pressure of fluid at the end of the center of the cone pressure tap, [Pa] k : float Isentropic exponent of fluid, [-] Returns ------- expansibility : float 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. Examples -------- >>> cone_meter_expansibility_Stewart(D=1, Dc=0.9, P1=1E6, P2=8.5E5, k=1.2) 0.9157343 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. ''' dP = P1 - P2 beta = diameter_ratio_cone_meter(D, Dc) beta *= beta beta *= beta return 1.0 - (0.649 + 0.696*beta)*dP/(k*P1) def dP_cone_meter(D, Dc, P1, P2): r'''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]_. .. math:: \Delta \bar \omega = (1.09 - 0.813\beta)\Delta P Parameters ---------- D : float Upstream internal pipe diameter, [m] Dc : float Diameter of the largest end of the cone meter, [m] P1 : float Static pressure of fluid upstream of cone meter at the cross-section of the pressure tap, [Pa] P2 : float Static pressure of fluid at the end of the center of the cone pressure tap, [Pa] Returns ------- dP : float 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. Examples -------- >>> dP_cone_meter(1, .7, 1E6, 9.5E5) 25470.093437973323 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. ''' dP = P1 - P2 beta = diameter_ratio_cone_meter(D, Dc) return (1.09 - 0.813*beta)*dP def diameter_ratio_wedge_meter(D, H): r'''Calculates the diameter ratio `beta` used to characterize a wedge flow meter as given in [1]_ and [2]_. .. math:: \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 ---------- D : float Upstream internal pipe diameter, [m] H : float 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 ------- beta : float Wedge meter diameter ratio, [-] Notes ----- Examples -------- >>> diameter_ratio_wedge_meter(D=0.2027, H=0.0608) 0.5022531424646643 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 ''' H_D = H/D t0 = 1.0 - 2.0*H_D t1 = acos(t0) t2 = t0 + t0 t3 = sqrt(H_D - H_D*H_D) t4 = t1 - t2*t3 return sqrt(pi_inv*t4) def C_wedge_meter_Miller(D, H): r'''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: .. math:: C = 0.7883 + 0.107(1 - \beta^2) For 1 to 1.5 inch lines: .. math:: C = 0.6143 + 0.718(1 - \beta^2) For 1.5 to 24 inch lines: .. math:: C = 0.5433 + 0.2453(1 - \beta^2) Parameters ---------- D : float Upstream internal pipe diameter, [m] H : float 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 ------- C : float 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. Examples -------- >>> C_wedge_meter_Miller(D=0.1524, H=0.3*0.1524) 0.7267069372687651 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. ''' beta = diameter_ratio_wedge_meter(D, H) beta *= beta if D <= 0.7*inch: # suggested limit 0.5 inch for this equation C = 0.7883 + 0.107*(1.0 - beta) elif D <= 1.4*inch: # Suggested limit is under 1.5 inches C = 0.6143 + 0.718*(1.0 - beta) else: C = 0.5433 + 0.2453*(1.0 - beta) return C def C_wedge_meter_ISO_5167_6_2017(D, H): r'''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). .. math:: C = 0.77 - 0.09\beta Parameters ---------- D : float Upstream internal pipe diameter, [m] H : float 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 ------- C : float 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. Examples -------- >>> C_wedge_meter_ISO_5167_6_2017(D=0.1524, H=0.3*0.1524) 0.724792059539853 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. ''' beta = diameter_ratio_wedge_meter(D, H) return 0.77 - 0.09*beta def dP_wedge_meter(D, H, P1, P2): r'''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]_. .. math:: \Delta \bar \omega = (1.09 - 0.79\beta)\Delta P Parameters ---------- D : float Upstream internal pipe diameter, [m] H : float 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] P1 : float Static pressure of fluid upstream of wedge meter at the cross-section of the pressure tap, [Pa] P2 : float Static pressure of fluid at the end of the wedge meter pressure tap, [ Pa] Returns ------- dP : float 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. Examples -------- >>> dP_wedge_meter(1, .7, 1E6, 9.5E5) 20344.849697483587 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. ''' dP = P1 - P2 beta = diameter_ratio_wedge_meter(D, H) return (1.09 - 0.79*beta)*dP def C_Reader_Harris_Gallagher_wet_venturi_tube(mg, ml, rhog, rhol, D, Do, H=1): r'''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`. .. math:: C = 1 - 0.0463\exp(-0.05Fr_{gas, th}) \cdot \min\left(1, \sqrt{\frac{X}{0.016}}\right) .. math:: Fr_{gas, th} = \frac{Fr_{\text{gas, densionetric }}}{\beta^{2.5}} .. math:: \phi = \sqrt{1 + C_{Ch} X + X^2} .. math:: C_{Ch} = \left(\frac{\rho_l}{\rho_{1,g}}\right)^n + \left(\frac{\rho_{1, g}}{\rho_{l}}\right)^n .. math:: 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] .. math:: X = \left(\frac{m_l}{m_g}\right) \sqrt{\frac{\rho_{1,g}}{\rho_l}} .. math:: {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 ---------- mg : float Mass flow rate of gas through the venturi tube, [kg/s] ml : float Mass flow rate of liquid through the venturi tube, [kg/s] rhog : float Density of gas at `P1`, [kg/m^3] rhol : float Density of liquid at `P1`, [kg/m^3] D : float Upstream internal pipe diameter, [m] Do : float Diameter of venturi tube at flow conditions, [m] H : float, 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 ------- C : float 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: .. math:: 0.4 \le \beta \le 0.75 .. math:: 0 < X \le 0.3 .. math:: Fr_{gas, th} > 3 .. math:: \frac{\rho_g}{\rho_l} > 0.02 .. math:: D \ge 50 \text{ mm} 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 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. ''' V = 4.0*mg/(rhog*pi*D*D) Frg = Froude_densimetric(V, L=D, rho1=rhol, rho2=rhog, heavy=False) beta = Do/D beta2 = beta*beta Fr_gas_th = Frg/(beta2*sqrt(beta)) n = max(0.583 - 0.18*beta2 - 0.578*exp(-0.8*Frg/H), 0.392 - 0.18*beta2) t0 = rhog/rhol t1 = (t0)**n C_Ch = t1 + 1.0/t1 X = ml/mg*sqrt(t0) # OF = sqrt(1.0 + X*(C_Ch + X)) C = 1.0 - 0.0463*exp(-0.05*Fr_gas_th)*min(1.0, sqrt(X/0.016)) return C def dP_Reader_Harris_Gallagher_wet_venturi_tube(D, Do, P1, P2, ml, mg, rhol, rhog, H=1.0): r'''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`. .. math:: Y = \frac{\Delta \bar \omega}{\Delta P} - 0.0896 - 0.48\beta^9 .. math:: Y_{max} = 0.61\exp\left[-11\frac{\rho_{1,g}}{\rho_l} - 0.045 \frac{Fr_{gas}}{H}\right] .. math:: \frac{Y}{Y_{max}} = 1 - \exp\left[-35 X^{0.75} \exp \left( \frac{-0.28Fr_{gas}}{H}\right)\right] .. math:: X = \left(\frac{m_l}{m_g}\right) \sqrt{\frac{\rho_{1,g}}{\rho_l}} .. math:: {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 ---------- D : float Upstream internal pipe diameter, [m] Do : float Diameter of venturi tube at flow conditions, [m] P1 : float Static pressure of fluid upstream of venturi tube at the cross-section of the pressure tap, [Pa] P2 : float Static pressure of fluid downstream of venturi tube at the cross- section of the pressure tap, [Pa] ml : float Mass flow rate of liquid through the venturi tube, [kg/s] mg : float Mass flow rate of gas through the venturi tube, [kg/s] rhol : float Density of liquid at `P1`, [kg/m^3] rhog : float Density of gas at `P1`, [kg/m^3] H : float, 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 ------- C : float 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. 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 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. ''' dP = P1 - P2 beta = Do/D X = ml/mg*sqrt(rhog/rhol) V = 4*mg/(rhog*pi*D*D) Frg = Froude_densimetric(V, L=D, rho1=rhol, rho2=rhog, heavy=False) Y_ratio = 1.0 - exp(-35.0*X**0.75*exp(-0.28*Frg/H)) Y_max = 0.61*exp(-11.0*rhog/rhol - 0.045*Frg/H) Y = Y_max*Y_ratio rhs = -0.0896 - 0.48*beta**9 dw = dP*(Y - rhs) return dw # Venturi tube loss coefficients as a function of Re as_cast_convergent_venturi_Res = [4E5, 6E4, 1E5, 1.5E5] as_cast_convergent_venturi_Cs = [0.957, 0.966, 0.976, 0.982] machined_convergent_venturi_Res = [5E4, 1E5, 2E5, 3E5, 7.5E5, # 5E5 to 1E6 1.5E6, # 1E6 to 2E6 5E6] # 2E6 to 1E8 machined_convergent_venturi_Cs = [0.970, 0.977, 0.992, 0.998, 0.995, 1.000, 1.010] rough_welded_convergent_venturi_Res = [4E4, 6E4, 1E5] rough_welded_convergent_venturi_Cs = [0.96, 0.97, 0.98] as_cast_convergent_entrance_machined_venturi_Res = [1E4, 6E4, 1E5, 1.5E5, 3.5E5, # 2E5 to 5E5 3.2E6] # 5E5 to 3.2E6 as_cast_convergent_entrance_machined_venturi_Cs = [0.963, 0.978, 0.98, 0.987, 0.992, 0.995] venturi_Res_Hollingshead = [1.0, 5.0, 10.0, 20.0, 30.0, 40.0, 60.0, 80.0, 100.0, 200.0, 300.0, 500.0, 1000.0, 2000.0, 3000.0, 5000.0, 10000.0, 30000.0, 50000.0, 75000.0, 100000.0, 1000000.0, 10000000.0, 50000000.0] venturi_logRes_Hollingshead = [0.0, 1.6094379124341003, 2.302585092994046, 2.995732273553991, 3.4011973816621555, 3.6888794541139363, 4.0943445622221, 4.382026634673881, 4.605170185988092, 5.298317366548036, 5.703782474656201, 6.214608098422191, 6.907755278982137, 7.600902459542082, 8.006367567650246, 8.517193191416238, 9.210340371976184, 10.308952660644293, 10.819778284410283, 11.225243392518447, 11.512925464970229, 13.815510557964274, 16.11809565095832, 17.72753356339242] venturi_smooth_Cs_Hollingshead = [0.163, 0.336, 0.432, 0.515, 0.586, 0.625, 0.679, 0.705, 0.727, 0.803, 0.841, 0.881, 0.921, 0.937, 0.944, 0.954, 0.961, 0.967, 0.967, 0.97, 0.971, 0.973, 0.974, 0.975] venturi_sharp_Cs_Hollingshead = [0.146, 0.3, 0.401, 0.498, 0.554, 0.596, 0.65, 0.688, 0.715, 0.801, 0.841, 0.884, 0.914, 0.94, 0.947, 0.944, 0.952, 0.959, 0.962, 0.963, 0.965, 0.967, 0.967, 0.967] CONE_METER_C = 0.82 '''Constant loss coefficient for flow cone meters''' ROUGH_WELDED_CONVERGENT_VENTURI_TUBE_C = 0.985 '''Constant loss coefficient for rough-welded convergent venturi tubes''' MACHINED_CONVERGENT_VENTURI_TUBE_C = 0.995 '''Constant loss coefficient for machined convergent venturi tubes''' AS_CAST_VENTURI_TUBE_C = 0.984 '''Constant loss coefficient for as-cast venturi tubes''' ISO_15377_CONICAL_ORIFICE_C = 0.734 '''Constant loss coefficient for conical orifice plates according to ISO 15377''' cone_Res_Hollingshead = [1.0, 5.0, 10.0, 20.0, 30.0, 40.0, 60.0, 80.0, 100.0, 150.0, 200.0, 300.0, 500.0, 1000.0, 2000.0, 3000.0, 4000.0, 5000.0, 7500.0, 10000.0, 20000.0, 30000.0, 100000.0, 1000000.0, 10000000.0, 50000000.0 ] cone_logRes_Hollingshead = [0.0, 1.6094379124341003, 2.302585092994046, 2.995732273553991, 3.4011973816621555, 3.6888794541139363, 4.0943445622221, 4.382026634673881, 4.605170185988092, 5.0106352940962555, 5.298317366548036, 5.703782474656201, 6.214608098422191, 6.907755278982137, 7.600902459542082, 8.006367567650246, 8.294049640102028, 8.517193191416238, 8.922658299524402, 9.210340371976184, 9.903487552536127, 10.308952660644293, 11.512925464970229, 13.815510557964274, 16.11809565095832, 17.72753356339242 ] cone_betas_Hollingshead = [0.6611, 0.6995, 0.8203] cone_beta_6611_Hollingshead_Cs = [0.066, 0.147, 0.207, 0.289, 0.349, 0.396, 0.462, 0.506, 0.537, 0.588, 0.622, 0.661, 0.7, 0.727, 0.75, 0.759, 0.763, 0.765, 0.767, 0.773, 0.778, 0.789, 0.804, 0.803, 0.805, 0.802 ] cone_beta_6995_Hollingshead_Cs = [0.067, 0.15, 0.21, 0.292, 0.35, 0.394, 0.458, 0.502, 0.533, 0.584, 0.615, 0.645, 0.682, 0.721, 0.742, 0.75, 0.755, 0.757, 0.763, 0.766, 0.774, 0.781, 0.792, 0.792, 0.79, 0.787 ] cone_beta_8203_Hollingshead_Cs = [0.057, 0.128, 0.182, 0.253, 0.303, 0.343, 0.4, 0.44, 0.472, 0.526, 0.557, 0.605, 0.644, 0.685, 0.705, 0.714, 0.721, 0.722, 0.724, 0.723, 0.725, 0.731, 0.73, 0.73, 0.741, 0.734 ] cone_Hollingshead_Cs = [cone_beta_6611_Hollingshead_Cs, cone_beta_6995_Hollingshead_Cs, cone_beta_8203_Hollingshead_Cs ] cone_Hollingshead_tck = implementation_optimize_tck([ [0.6611, 0.6611, 0.6611, 0.8203, 0.8203, 0.8203], [0.0, 0.0, 0.0, 0.0, 2.302585092994046, 2.995732273553991, 3.4011973816621555, 3.6888794541139363, 4.0943445622221, 4.382026634673881, 4.605170185988092, 5.0106352940962555, 5.298317366548036, 5.703782474656201, 6.214608098422191, 6.907755278982137, 7.600902459542082, 8.006367567650246, 8.294049640102028, 8.517193191416238, 8.922658299524402, 9.210340371976184, 9.903487552536127, 10.308952660644293, 11.512925464970229, 13.815510557964274, 17.72753356339242, 17.72753356339242, 17.72753356339242, 17.72753356339242 ], [0.06600000000000003, 0.09181180887944293, 0.1406341453010674, 0.27319769866300025, 0.34177839953532274, 0.4025880076725502, 0.4563149328810349, 0.5035445307357295, 0.5458473693359689, 0.583175639128474, 0.628052124545805, 0.6647198135005781, 0.7091524396786245, 0.7254729823419331, 0.7487816963926843, 0.7588145502817809, 0.7628692532631826, 0.7660482147214834, 0.7644188319583379, 0.7782644144006241, 0.7721508139116487, 0.7994728794028244, 0.8076742194714519, 0.7986221420822799, 0.8086240532850298, 0.802, 0.07016232064017663, 0.1059162635703894, 0.1489681838592814, 0.28830815748629207, 0.35405213706957395, 0.40339795504063664, 0.4544570323055189, 0.5034637712201067, 0.5448190156693709, 0.5840164245031125, 0.6211559598098063, 0.6218648844980823, 0.6621745760710729, 0.7282379546292953, 0.7340030734801267, 0.7396324865779599, 0.7489736798953754, 0.7480726412914717, 0.7671564751169978, 0.756853660688892, 0.7787029642272745, 0.7742381131312691, 0.7887584162443445, 0.7857610450218329, 0.7697076645551957, 0.7718300910596032, 0.05700000000000002, 0.07612544859943549, 0.12401733415778271, 0.24037452209595875, 0.29662463502593156, 0.34859536586855205, 0.39480085719322505, 0.43661601622480606, 0.48091259102454764, 0.5240691286186233, 0.5590609288020619, 0.6144556048716696, 0.6471713640567137, 0.6904158809061184, 0.7032590252050219, 0.712177974557301, 0.7221845303680273, 0.721505707129694, 0.7249822376264551, 0.7218890085289907, 0.7221848475768714, 0.7371751354515526, 0.7252385062304629, 0.7278943803933404, 0.7496546607029086, 0.7340000000000001 ], 2, 3 ]) wedge_Res_Hollingshead = [1.0, 5.0, 10.0, 20.0, 30.0, 40.0, 60.0, 80.0, 100.0, 200.0, 300.0, 400.0, 500.0, 5000.0, 1.00E+04, 1.00E+05, 1.00E+06, 5.00E+07] wedge_logRes_Hollingshead = [0.0, 1.6094379124341003, 2.302585092994046, 2.995732273553991, 3.4011973816621555, 3.6888794541139363, 4.0943445622221, 4.382026634673881, 4.605170185988092, 5.298317366548036, 5.703782474656201, 5.991464547107982, 6.214608098422191, 8.517193191416238, 9.210340371976184, 11.512925464970229, 13.815510557964274, 17.72753356339242 ] wedge_beta_5023_Hollingshead = [0.145, 0.318, 0.432, 0.551, 0.61, 0.641, 0.674, 0.69, 0.699, 0.716, 0.721, 0.725, 0.73, 0.729, 0.732, 0.732, 0.731, 0.733] wedge_beta_611_Hollingshead = [0.127, 0.28, 0.384, 0.503, 0.567, 0.606, 0.645, 0.663, 0.672, 0.688, 0.694, 0.7, 0.705, 0.7, 0.702, 0.695, 0.699, 0.705] wedge_betas_Hollingshead = [.5023, .611] wedge_Hollingshead_Cs = [wedge_beta_5023_Hollingshead, wedge_beta_611_Hollingshead] wedge_Hollingshead_tck = implementation_optimize_tck([ [0.5023, 0.5023, 0.611, 0.611], [0.0, 0.0, 0.0, 0.0, 2.302585092994046, 2.995732273553991, 3.4011973816621555, 3.6888794541139363, 4.0943445622221, 4.382026634673881, 4.605170185988092, 5.298317366548036, 5.703782474656201, 5.991464547107982, 6.214608098422191, 8.517193191416238, 9.210340371976184, 11.512925464970229, 17.72753356339242, 17.72753356339242, 17.72753356339242, 17.72753356339242 ], [0.14500000000000005, 0.18231832425722, 0.3339917130006919, 0.5379467710226973, 0.6077700659940896, 0.6459542943925077, 0.6729757007770231, 0.6896405007576225, 0.7054863114589583, 0.7155740600632635, 0.7205446407610863, 0.7239576816068966, 0.7483627568160166, 0.7232963355919931, 0.7366325320490953, 0.7264222143567053, 0.7339605394126009, 0.7330000000000001, 0.1270000000000001, 0.16939873865132285, 0.2828494933525669, 0.4889107009077842, 0.5623120043524101, 0.6133092379676948, 0.6437092394687915, 0.6629923366662017, 0.6782934366011034, 0.687302374134782, 0.6927470053128909, 0.6993992364234898, 0.7221204483546849, 0.6947577293284015, 0.7063701306810815, 0.6781614534359871, 0.7185326811948407, 0.7050000000000001 ], 1, 3 ]) beta_simple_meters = frozenset([ISO_5167_ORIFICE, ISO_15377_ECCENTRIC_ORIFICE, ISO_15377_CONICAL_ORIFICE, ISO_15377_QUARTER_CIRCLE_ORIFICE, MILLER_ORIFICE, MILLER_ECCENTRIC_ORIFICE, MILLER_SEGMENTAL_ORIFICE, MILLER_CONICAL_ORIFICE, MILLER_QUARTER_CIRCLE_ORIFICE, CONCENTRIC_ORIFICE, ECCENTRIC_ORIFICE, CONICAL_ORIFICE, SEGMENTAL_ORIFICE, QUARTER_CIRCLE_ORIFICE, UNSPECIFIED_METER, HOLLINGSHEAD_VENTURI_SHARP, HOLLINGSHEAD_VENTURI_SMOOTH, HOLLINGSHEAD_ORIFICE, LONG_RADIUS_NOZZLE, ISA_1932_NOZZLE, VENTURI_NOZZLE, AS_CAST_VENTURI_TUBE, MACHINED_CONVERGENT_VENTURI_TUBE, ROUGH_WELDED_CONVERGENT_VENTURI_TUBE]) all_meters = frozenset(list(beta_simple_meters) + [CONE_METER, WEDGE_METER, HOLLINGSHEAD_CONE, HOLLINGSHEAD_WEDGE]) '''Set of string inputs representing all of the different supported flow meters and their correlations. ''' _unsupported_meter_msg = "Supported meter types are %s" % all_meters def differential_pressure_meter_beta(D, D2, meter_type): r'''Calculates the beta ratio of a differential pressure meter. Parameters ---------- D : float Upstream internal pipe diameter, [m] D2 : float Diameter of orifice, or venturi meter orifice, or flow tube orifice, or cone meter end diameter, or wedge meter fluid flow height, [m] meter_type : str One of {'conical orifice', 'orifice', 'machined convergent venturi tube', 'ISO 5167 orifice', 'Miller quarter circle orifice', 'Hollingshead venturi sharp', 'segmental orifice', 'Miller conical orifice', 'Miller segmental orifice', 'quarter circle orifice', 'Hollingshead v cone', 'wedge meter', 'eccentric orifice', 'venuri nozzle', 'rough welded convergent venturi tube', 'ISA 1932 nozzle', 'ISO 15377 quarter-circle orifice', 'Hollingshead venturi smooth', 'Hollingshead orifice', 'cone meter', 'Hollingshead wedge', 'Miller orifice', 'long radius nozzle', 'ISO 15377 conical orifice', 'unspecified meter', 'as cast convergent venturi tube', 'Miller eccentric orifice', 'ISO 15377 eccentric orifice'}, [-] Returns ------- beta : float Differential pressure meter diameter ratio, [-] Notes ----- Examples -------- >>> differential_pressure_meter_beta(D=0.2575, D2=0.184, ... meter_type='cone meter') 0.6995709873957624 ''' if meter_type in beta_simple_meters: beta = D2/D elif meter_type == CONE_METER or meter_type == HOLLINGSHEAD_CONE: beta = diameter_ratio_cone_meter(D=D, Dc=D2) elif meter_type == WEDGE_METER or meter_type == HOLLINGSHEAD_WEDGE: beta = diameter_ratio_wedge_meter(D=D, H=D2) else: raise ValueError(_unsupported_meter_msg) return beta _meter_type_to_corr_default = { CONCENTRIC_ORIFICE: ISO_5167_ORIFICE, ECCENTRIC_ORIFICE: ISO_15377_ECCENTRIC_ORIFICE, CONICAL_ORIFICE: ISO_15377_CONICAL_ORIFICE, QUARTER_CIRCLE_ORIFICE: ISO_15377_QUARTER_CIRCLE_ORIFICE, SEGMENTAL_ORIFICE: MILLER_SEGMENTAL_ORIFICE, } def differential_pressure_meter_C_epsilon(D, D2, m, P1, P2, rho, mu, k, meter_type, taps=None, tap_position=None, C_specified=None, epsilon_specified=None): r'''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 ---------- D : float Upstream internal pipe diameter, [m] D2 : float Diameter of orifice, or venturi meter orifice, or flow tube orifice, or cone meter end diameter, or wedge meter fluid flow height, [m] m : float Mass flow rate of fluid through the flow meter, [kg/s] P1 : float Static pressure of fluid upstream of differential pressure meter at the cross-section of the pressure tap, [Pa] P2 : float Static pressure of fluid downstream of differential pressure meter or at the prescribed location (varies by type of meter) [Pa] rho : float Density of fluid at `P1`, [kg/m^3] mu : float Viscosity of fluid at `P1`, [Pa*s] k : float Isentropic exponent of fluid, [-] meter_type : str One of {'conical orifice', 'orifice', 'machined convergent venturi tube', 'ISO 5167 orifice', 'Miller quarter circle orifice', 'Hollingshead venturi sharp', 'segmental orifice', 'Miller conical orifice', 'Miller segmental orifice', 'quarter circle orifice', 'Hollingshead v cone', 'wedge meter', 'eccentric orifice', 'venuri nozzle', 'rough welded convergent venturi tube', 'ISA 1932 nozzle', 'ISO 15377 quarter-circle orifice', 'Hollingshead venturi smooth', 'Hollingshead orifice', 'cone meter', 'Hollingshead wedge', 'Miller orifice', 'long radius nozzle', 'ISO 15377 conical orifice', 'unspecified meter', 'as cast convergent venturi tube', 'Miller eccentric orifice', 'ISO 15377 eccentric orifice'}, [-] taps : str, optional The orientation of the taps; one of 'corner', 'flange', 'D', or 'D/2'; applies for orifice meters only, [-] tap_position : str, optional The rotation of the taps, used **only for the eccentric orifice case** where the pressure profile is are not symmetric; '180 degree' for the normal case where the taps are opposite the orifice bore, and '90 degree' for the case where, normally for operational reasons, the taps are near the bore [-] C_specified : float, optional If specified, the correlation for the meter type is not used - this value is returned for `C` epsilon_specified : float, optional If specified, the correlation for the fluid expansibility is not used - this value is returned for :math:`\epsilon`, [-] Returns ------- C : float Coefficient of discharge of the specified flow meter type at the specified conditions, [-] expansibility : float 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. The latest ISO formulations for `expansibility` are used with the Miller correlations. 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) ''' # # Translate default meter type to implementation specific correlation if meter_type == CONCENTRIC_ORIFICE: meter_type = ISO_5167_ORIFICE elif meter_type == ECCENTRIC_ORIFICE: meter_type = ISO_15377_ECCENTRIC_ORIFICE elif meter_type == CONICAL_ORIFICE: meter_type = ISO_15377_CONICAL_ORIFICE elif meter_type == QUARTER_CIRCLE_ORIFICE: meter_type = ISO_15377_QUARTER_CIRCLE_ORIFICE elif meter_type == SEGMENTAL_ORIFICE: meter_type = MILLER_SEGMENTAL_ORIFICE if meter_type == ISO_5167_ORIFICE: C = C_Reader_Harris_Gallagher(D, D2, rho, mu, m, taps) epsilon = orifice_expansibility(D, D2, P1, P2, k) elif meter_type == ISO_15377_ECCENTRIC_ORIFICE: C = C_eccentric_orifice_ISO_15377_1998(D, D2) epsilon = orifice_expansibility(D, D2, P1, P2, k) elif meter_type == ISO_15377_QUARTER_CIRCLE_ORIFICE: C = C_quarter_circle_orifice_ISO_15377_1998(D, D2) epsilon = orifice_expansibility(D, D2, P1, P2, k) elif meter_type == ISO_15377_CONICAL_ORIFICE: C = ISO_15377_CONICAL_ORIFICE_C # Average of concentric square edge orifice and ISA 1932 nozzles epsilon = 0.5*(orifice_expansibility(D, D2, P1, P2, k) + nozzle_expansibility(D=D, Do=D2, P1=P1, P2=P2, k=k)) elif meter_type in (MILLER_ORIFICE, MILLER_ECCENTRIC_ORIFICE, MILLER_SEGMENTAL_ORIFICE, MILLER_QUARTER_CIRCLE_ORIFICE): C = C_Miller_1996(D, D2, rho, mu, m, subtype=meter_type, taps=taps, tap_position=tap_position) epsilon = orifice_expansibility(D, D2, P1, P2, k) elif meter_type == MILLER_CONICAL_ORIFICE: C = C_Miller_1996(D, D2, rho, mu, m, subtype=meter_type, taps=taps, tap_position=tap_position) epsilon = 0.5*(orifice_expansibility(D, D2, P1, P2, k) + nozzle_expansibility(D=D, Do=D2, P1=P1, P2=P2, k=k)) elif meter_type == LONG_RADIUS_NOZZLE: epsilon = nozzle_expansibility(D=D, Do=D2, P1=P1, P2=P2, k=k) C = C_long_radius_nozzle(D=D, Do=D2, rho=rho, mu=mu, m=m) elif meter_type == ISA_1932_NOZZLE: epsilon = nozzle_expansibility(D=D, Do=D2, P1=P1, P2=P2, k=k) C = C_ISA_1932_nozzle(D=D, Do=D2, rho=rho, mu=mu, m=m) elif meter_type == VENTURI_NOZZLE: epsilon = nozzle_expansibility(D=D, Do=D2, P1=P1, P2=P2, k=k) C = C_venturi_nozzle(D=D, Do=D2) elif meter_type == AS_CAST_VENTURI_TUBE: epsilon = nozzle_expansibility(D=D, Do=D2, P1=P1, P2=P2, k=k) C = AS_CAST_VENTURI_TUBE_C elif meter_type == MACHINED_CONVERGENT_VENTURI_TUBE: epsilon = nozzle_expansibility(D=D, Do=D2, P1=P1, P2=P2, k=k) C = MACHINED_CONVERGENT_VENTURI_TUBE_C elif meter_type == ROUGH_WELDED_CONVERGENT_VENTURI_TUBE: epsilon = nozzle_expansibility(D=D, Do=D2, P1=P1, P2=P2, k=k) C = ROUGH_WELDED_CONVERGENT_VENTURI_TUBE_C elif meter_type == CONE_METER: epsilon = cone_meter_expansibility_Stewart(D=D, Dc=D2, P1=P1, P2=P2, k=k) C = CONE_METER_C elif meter_type == WEDGE_METER: beta = diameter_ratio_wedge_meter(D=D, H=D2) epsilon = nozzle_expansibility(D=D, Do=D2, P1=P1, P2=P1, k=k, beta=beta) C = C_wedge_meter_ISO_5167_6_2017(D=D, H=D2) elif meter_type == HOLLINGSHEAD_ORIFICE: v = m/((0.25*pi*D*D)*rho) Re_D = rho*v*D/mu C = float(bisplev(D2/D, log(Re_D), orifice_std_Hollingshead_tck)) epsilon = orifice_expansibility(D, D2, P1, P2, k) elif meter_type == HOLLINGSHEAD_VENTURI_SMOOTH: v = m/((0.25*pi*D*D)*rho) Re_D = rho*v*D/mu C = interp(log(Re_D), venturi_logRes_Hollingshead, venturi_smooth_Cs_Hollingshead, extrapolate=True) epsilon = nozzle_expansibility(D=D, Do=D2, P1=P1, P2=P2, k=k) elif meter_type == HOLLINGSHEAD_VENTURI_SHARP: v = m/((0.25*pi*D*D)*rho) Re_D = rho*v*D/mu C = interp(log(Re_D), venturi_logRes_Hollingshead, venturi_sharp_Cs_Hollingshead, extrapolate=True) epsilon = nozzle_expansibility(D=D, Do=D2, P1=P1, P2=P2, k=k) elif meter_type == HOLLINGSHEAD_CONE: v = m/((0.25*pi*D*D)*rho) Re_D = rho*v*D/mu beta = diameter_ratio_cone_meter(D, D2) C = float(bisplev(beta, log(Re_D), cone_Hollingshead_tck)) epsilon = cone_meter_expansibility_Stewart(D=D, Dc=D2, P1=P1, P2=P2, k=k) elif meter_type == HOLLINGSHEAD_WEDGE: v = m/((0.25*pi*D*D)*rho) Re_D = rho*v*D/mu beta = diameter_ratio_wedge_meter(D=D, H=D2) C = float(bisplev(beta, log(Re_D), wedge_Hollingshead_tck)) epsilon = nozzle_expansibility(D=D, Do=D2, P1=P1, P2=P1, k=k, beta=beta) elif meter_type == UNSPECIFIED_METER: epsilon = orifice_expansibility(D, D2, P1, P2, k) # Default to orifice type expansibility if C_specified is None: raise ValueError("For unspecified meter type, C_specified is required") else: raise ValueError(_unsupported_meter_msg) if C_specified is not None: C = C_specified if epsilon_specified is not None: epsilon = epsilon_specified return C, epsilon def err_dp_meter_solver_m(m_D, D, D2, P1, P2, rho, mu, k, meter_type, taps, tap_position, C_specified, epsilon_specified): m = m_D*D C, epsilon = differential_pressure_meter_C_epsilon(D, D2, m, P1, P2, rho, mu, k, meter_type, taps=taps, tap_position=tap_position, C_specified=C_specified, epsilon_specified=epsilon_specified) m_calc = flow_meter_discharge(D=D, Do=D2, P1=P1, P2=P2, rho=rho, C=C, expansibility=epsilon) err = m - m_calc return err def err_dp_meter_solver_P2(P2, D, D2, m, P1, rho, mu, k, meter_type, taps, tap_position, C_specified, epsilon_specified): C, epsilon = differential_pressure_meter_C_epsilon(D, D2, m, P1, P2, rho, mu, k, meter_type, taps=taps, tap_position=tap_position, C_specified=C_specified, epsilon_specified=epsilon_specified) m_calc = flow_meter_discharge(D=D, Do=D2, P1=P1, P2=P2, rho=rho, C=C, expansibility=epsilon) return m - m_calc def err_dp_meter_solver_D2(D2, D, m, P1, P2, rho, mu, k, meter_type, taps, tap_position, C_specified, epsilon_specified): C, epsilon = differential_pressure_meter_C_epsilon(D, D2, m, P1, P2, rho, mu, k, meter_type, taps=taps, tap_position=tap_position, C_specified=C_specified, epsilon_specified=epsilon_specified) m_calc = flow_meter_discharge(D=D, Do=D2, P1=P1, P2=P2, rho=rho, C=C, expansibility=epsilon) return m - m_calc def err_dp_meter_solver_P1(P1, D, D2, m, P2, rho, mu, k, meter_type, taps, tap_position, C_specified, epsilon_specified): C, epsilon = differential_pressure_meter_C_epsilon(D, D2, m, P1, P2, rho, mu, k, meter_type, taps=taps, tap_position=tap_position, C_specified=C_specified, epsilon_specified=epsilon_specified) m_calc = flow_meter_discharge(D=D, Do=D2, P1=P1, P2=P2, rho=rho, C=C, expansibility=epsilon) return m - m_calc def differential_pressure_meter_solver(D, rho, mu, k=None, D2=None, P1=None, P2=None, m=None, meter_type=ISO_5167_ORIFICE, taps=None, tap_position=None, C_specified=None, epsilon_specified=None): r'''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 ---------- D : float Upstream internal pipe diameter, [m] rho : float Density of fluid at `P1`, [kg/m^3] mu : float Viscosity of fluid at `P1`, [Pa*s] k : float, optional Isentropic exponent of fluid; required unless `epsilon_specified` is specified , [-] D2 : float, optional Diameter of orifice, or venturi meter orifice, or flow tube orifice, or cone meter end diameter, or wedge meter fluid flow height, [m] P1 : float, optional Static pressure of fluid upstream of differential pressure meter at the cross-section of the pressure tap, [Pa] P2 : float, optional Static pressure of fluid downstream of differential pressure meter or at the prescribed location (varies by type of meter) [Pa] m : float, optional Mass flow rate of fluid through the flow meter, [kg/s] meter_type : str One of {'conical orifice', 'orifice', 'machined convergent venturi tube', 'ISO 5167 orifice', 'Miller quarter circle orifice', 'Hollingshead venturi sharp', 'segmental orifice', 'Miller conical orifice', 'Miller segmental orifice', 'quarter circle orifice', 'Hollingshead v cone', 'wedge meter', 'eccentric orifice', 'venuri nozzle', 'rough welded convergent venturi tube', 'ISA 1932 nozzle', 'ISO 15377 quarter-circle orifice', 'Hollingshead venturi smooth', 'Hollingshead orifice', 'cone meter', 'Hollingshead wedge', 'Miller orifice', 'long radius nozzle', 'ISO 15377 conical orifice', 'unspecified meter', 'as cast convergent venturi tube', 'Miller eccentric orifice', 'ISO 15377 eccentric orifice'}, [-] taps : str, optional The orientation of the taps; one of 'corner', 'flange', 'D', or 'D/2'; applies for orifice meters only, [-] tap_position : str, optional The rotation of the taps, used **only for the eccentric orifice case** where the pressure profile is are not symmetric; '180 degree' for the normal case where the taps are opposite the orifice bore, and '90 degree' for the case where, normally for operational reasons, the taps are near the bore [-] C_specified : float, optional If specified, the correlation for the meter type is not used - this value is used for `C` epsilon_specified : float, optional If specified, the correlation for the fluid expansibility is not used - this value is used for :math:`\epsilon`. Many publications recommend this be set to 1 for incompressible fluids [-] Returns ------- ans : float 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. If a meter has already been calibrated to have a known `C`, this may be provided and it will be used in place of calculating one. 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 ''' if k is None and epsilon_specified is not None: k = 1.4 if m is None and D is not None and D2 is not None and P1 is not None and P2 is not None: # Initialize via analytical formulas C_guess = 0.7 D4 = D*D D4 *= D4 D24 = D2*D2 D24 *= D24 m_guess = root_two*pi*C_guess*D2*D2*sqrt(D4*rho*(P1 - P2)/(D4 - D24))*0.25 m_D_guess = m_guess/D # Diameter to mass flow ratio # m_D_guess = 40 # if rho < 100.0: # m_D_guess *= 1e-2 return secant(err_dp_meter_solver_m, m_D_guess, args=(D, D2, P1, P2, rho, mu, k, meter_type, taps, tap_position, C_specified, epsilon_specified), low=1e-40)*D elif D2 is None and D is not None and m is not None and P1 is not None and P2 is not None: args = (D, m, P1, P2, rho, mu, k, meter_type, taps, tap_position, C_specified, epsilon_specified) try: return brenth(err_dp_meter_solver_D2, D*(1-1E-9), D*5E-3, args=args) except: try: return secant(err_dp_meter_solver_D2, D*.3, args=args, high=D, low=D*1e-10) except: return secant(err_dp_meter_solver_D2, D*.75, args=args, high=D, low=D*1e-10) elif P2 is None and D is not None and D2 is not None and m is not None and P1 is not None: args = (D, D2, m, P1, rho, mu, k, meter_type, taps, tap_position, C_specified, epsilon_specified) try: return brenth(err_dp_meter_solver_P2, P1*(1-1E-9), P1*0.5, args=args) except: return secant(err_dp_meter_solver_P2, P1*0.5, low=P1*1e-10, args=args, high=P1, bisection=True) elif P1 is None and D is not None and D2 is not None and m is not None and P2 is not None: args = (D, D2, m, P2, rho, mu, k, meter_type, taps, tap_position, C_specified, epsilon_specified) try: return brenth(err_dp_meter_solver_P1, P2*(1+1E-9), P2*1.4, args=args) except: return secant(err_dp_meter_solver_P1, P2*1.5, args=args, low=P2, bisection=True) else: raise ValueError('Solver is capable of solving for one of P1, P2, D2, or m only.') # Set of orifice types that get their dP calculated with `dP_orifice`. _dP_orifice_set = set([ISO_5167_ORIFICE, ISO_15377_ECCENTRIC_ORIFICE, ISO_15377_CONICAL_ORIFICE, ISO_15377_QUARTER_CIRCLE_ORIFICE, MILLER_ORIFICE, MILLER_ECCENTRIC_ORIFICE, MILLER_SEGMENTAL_ORIFICE, MILLER_CONICAL_ORIFICE, MILLER_QUARTER_CIRCLE_ORIFICE, HOLLINGSHEAD_ORIFICE, CONCENTRIC_ORIFICE, ECCENTRIC_ORIFICE, CONICAL_ORIFICE, SEGMENTAL_ORIFICE, QUARTER_CIRCLE_ORIFICE]) _missing_C_msg = "Parameter C is required for this orifice type" def differential_pressure_meter_dP(D, D2, P1, P2, C=None, meter_type=ISO_5167_ORIFICE): r'''Calculates 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 ---------- D : float Upstream internal pipe diameter, [m] D2 : float Diameter of orifice, or venturi meter orifice, or flow tube orifice, or cone meter end diameter, or wedge meter fluid flow height, [m] P1 : float Static pressure of fluid upstream of differential pressure meter at the cross-section of the pressure tap, [Pa] P2 : float Static pressure of fluid downstream of differential pressure meter or at the prescribed location (varies by type of meter) [Pa] C : float, optional Coefficient of discharge (used only in orifice plates, and venturi nozzles), [-] meter_type : str One of {'conical orifice', 'orifice', 'machined convergent venturi tube', 'ISO 5167 orifice', 'Miller quarter circle orifice', 'Hollingshead venturi sharp', 'segmental orifice', 'Miller conical orifice', 'Miller segmental orifice', 'quarter circle orifice', 'Hollingshead v cone', 'wedge meter', 'eccentric orifice', 'venuri nozzle', 'rough welded convergent venturi tube', 'ISA 1932 nozzle', 'ISO 15377 quarter-circle orifice', 'Hollingshead venturi smooth', 'Hollingshead orifice', 'cone meter', 'Hollingshead wedge', 'Miller orifice', 'long radius nozzle', 'ISO 15377 conical orifice', 'unspecified meter', 'as cast convergent venturi tube', 'Miller eccentric orifice', 'ISO 15377 eccentric orifice'}, [-] Returns ------- dP : float 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 ''' if meter_type in _dP_orifice_set: if C is None: raise ValueError(_missing_C_msg) dP = dP_orifice(D=D, Do=D2, P1=P1, P2=P2, C=C) elif meter_type == LONG_RADIUS_NOZZLE: if C is None: raise ValueError(_missing_C_msg) dP = dP_orifice(D=D, Do=D2, P1=P1, P2=P2, C=C) elif meter_type == ISA_1932_NOZZLE: if C is None: raise ValueError(_missing_C_msg) dP = dP_orifice(D=D, Do=D2, P1=P1, P2=P2, C=C) elif meter_type == VENTURI_NOZZLE: raise NotImplementedError("Venturi meter does not have an implemented pressure drop correlation") elif (meter_type == AS_CAST_VENTURI_TUBE or meter_type == MACHINED_CONVERGENT_VENTURI_TUBE or meter_type == ROUGH_WELDED_CONVERGENT_VENTURI_TUBE or meter_type == HOLLINGSHEAD_VENTURI_SMOOTH or meter_type == HOLLINGSHEAD_VENTURI_SHARP): dP = dP_venturi_tube(D=D, Do=D2, P1=P1, P2=P2) elif meter_type == CONE_METER or meter_type == HOLLINGSHEAD_CONE: dP = dP_cone_meter(D=D, Dc=D2, P1=P1, P2=P2) elif meter_type == WEDGE_METER or meter_type == HOLLINGSHEAD_WEDGE: dP = dP_wedge_meter(D=D, H=D2, P1=P1, P2=P2) else: raise ValueError(_unsupported_meter_msg) return dP