# -*- coding: utf-8 -*-
'''Chemical Engineering Design Library (ChEDL). Utilities for process modeling.
Copyright (C) 2016, Caleb Bell <Caleb.Andrew.Bell@gmail.com>
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.'''
from __future__ import division
from math import log10
from scipy.constants import R
__all__ = ['size_control_valve_l', 'size_control_valve_g', 'cavitation_index',
'FF_critical_pressure_ratio_l', 'is_choked_turbulent_l',
'is_choked_turbulent_g', 'Reynolds_valve',
'loss_coefficient_piping', 'Reynolds_factor']
N1 = 0.1 # m^3/hr, kPa
N2 = 1.6E-3 # mm
N4 = 7.07E-2 # m^3/hr, m^2/s
N5 = 1.8E-3 # mm
N6 = 3.16 # kg/hr, kPa, kg/m^3
N7 = 4.82 # m^3/hr kPa K
N8 = 1.10 # kPa kg/hr K
#N9 = 2.60E1 # m^3/hr kPa K at 15 deg C
N9 = 2.46E1 # m^3/hr kPa K at 0 deg C
N18 = 8.65E-1 # mm
N19 = 2.5 # mm
#N22 = 1.84E1 # m^3/hr kPa K at 15 deg C
N27 = 7.75E-1 # kg/hr kPa K at 0 deg C
N32 = 1.4E2 # mm
rho0 = 999.10329075702327 # Water at 288.15 K
[docs]def cavitation_index(P1, P2, Psat):
r'''Calculates the cavitation index of a valve with upstream and downstream
absolute pressures `P1` and `P2` for a fluid with a vapor pressure `Psat`.
.. math::
\sigma = \frac{P_1 - P_{sat}}{P_1 - P_2}
Parameters
----------
P1 : float
Absolute pressure upstream of the valve [Pa]
P2 : float
Absolute pressure downstream of the valve [Pa]
Psat : float
Saturation pressure of the liquid at inlet temperature [Pa]
Returns
-------
sigma : float
Cavitation index of the valve [-]
Notes
-----
Larger values are safer. Models for adjusting cavitation indexes provided
by the manufacturer to the user's conditions are available, making use
of scaling the pressure differences and size differences.
Values can be calculated for incipient cavitation, constant cavitation,
maximum vibration cavitation, incipient damage, and choking cavitation.
Has also been defined as:
.. math::
\sigma = \frac{P_2 - P_{sat}}{P_1 - P_2}
Another definition and notation series is:
.. math::
K = xF = \frac{1}{\sigma} = \frac{P_1 - P_2}{P_1 - P_{sat}}
Examples
--------
>>> cavitation_index(1E6, 8E5, 2E5)
4.0
References
----------
.. [1] ISA. "RP75.23 Considerations for Evaluating Control Valve
Cavitation." 1995.
'''
return (P1 - Psat)/(P1 - P2)
[docs]def FF_critical_pressure_ratio_l(Psat, Pc):
r'''Calculates FF, the liquid critical pressure ratio factor,
for use in IEC 60534 liquid valve sizing calculations.
.. math::
F_F = 0.96 - 0.28\sqrt{\frac{P_{sat}}{P_c}}
Parameters
----------
Psat : float
Saturation pressure of the liquid at inlet temperature [Pa]
Pc : float
Critical pressure of the liquid [Pa]
Returns
-------
FF : float
Liquid critical pressure ratio factor [-]
Examples
--------
From [1]_, matching example.
>>> FF_critical_pressure_ratio_l(70100.0, 22120000.0)
0.9442375225233299
References
----------
.. [1] IEC 60534-2-1 / ISA-75.01.01-2007
'''
return 0.96 - 0.28*(Psat/Pc)**0.5
[docs]def is_choked_turbulent_l(dP, P1, Psat, FF, FL=None, FLP=None, FP=None):
r'''Calculates if a liquid flow in IEC 60534 calculations is critical or
not, for use in IEC 60534 liquid valve sizing calculations.
Either FL may be provided or FLP and FP, depending on the calculation
process.
.. math::
\Delta P > F_L^2(P_1 - F_F P_{sat})
\Delta P >= \left(\frac{F_{LP}}{F_P}\right)^2(P_1 - F_F P_{sat})
Parameters
----------
dP : float
Differential pressure across the valve, with reducer/expanders [Pa]
P1 : float
Pressure of the fluid before the valve and reducers/expanders [Pa]
Psat : float
Saturation pressure of the fluid at inlet temperature [Pa]
FF : float
Liquid critical pressure ratio factor [-]
FL : float, optional
Liquid pressure recovery factor of a control valve without attached fittings [-]
FLP : float, optional
Combined liquid pressure recovery factor with piping geometry factor,
for a control valve with attached fittings [-]
FP : float, optional
Piping geometry factor [-]
Returns
-------
choked : bool
Whether or not the flow is choked [-]
Examples
--------
>>> is_choked_turbulent_l(460.0, 680.0, 70.1, 0.94, 0.9)
False
>>> is_choked_turbulent_l(460.0, 680.0, 70.1, 0.94, 0.6)
True
References
----------
.. [1] IEC 60534-2-1 / ISA-75.01.01-2007
'''
if FLP and FP:
return dP >= (FLP/FP)**2*(P1-FF*Psat)
elif FL:
return dP >= FL**2*(P1-FF*Psat)
else:
raise Exception('Either (FLP and FP) or FL is needed')
[docs]def is_choked_turbulent_g(x, Fgamma, xT=None, xTP=None):
r'''Calculates if a gas flow in IEC 60534 calculations is critical or
not, for use in IEC 60534 gas valve sizing calculations.
Either xT or xTP must be provided, depending on the calculation process.
.. math::
x \ge F_\gamma x_T
x \ge F_\gamma x_{TP}
Parameters
----------
x : float
Differential pressure over inlet pressure, [-]
Fgamma : float
Specific heat ratio factor [-]
xT : float, optional
Pressure difference ratio factor of a valve without fittings at choked
flow [-]
xTP : float
Pressure difference ratio factor of a valve with fittings at choked
flow [-]
Returns
-------
choked : bool
Whether or not the flow is choked [-]
Examples
--------
Example 3, compressible flow, non-choked with attached fittings:
>>> is_choked_turbulent_g(0.544, 0.929, 0.6)
False
>>> is_choked_turbulent_g(0.544, 0.929, xTP=0.625)
False
References
----------
.. [1] IEC 60534-2-1 / ISA-75.01.01-2007
'''
if xT:
return x >= Fgamma*xT
elif xTP:
return x >= Fgamma*xTP
else:
raise Exception('Either xT or xTP is needed')
[docs]def Reynolds_valve(nu, Q, D1, FL, Fd, C):
r'''Calculates Reynolds number of a control valve for a liquid or gas
flowing through it at a specified Q, for a specified D1, FL, Fd, C, and
with kinematic viscosity `nu` according to IEC 60534 calculations.
.. math::
Re_v = \frac{N_4 F_d Q}{\nu \sqrt{C F_L}}\left(\frac{F_L^2 C^2}
{N_2D^4} +1\right)^{1/4}
Parameters
----------
nu : float
Kinematic viscosity, [m^2/s]
Q : float
Volumetric flow rate of the fluid [m^3/s]
D1 : float
Diameter of the pipe before the valve [m]
FL : float, optional
Liquid pressure recovery factor of a control valve without attached
fittings []
Fd : float
Valve style modifier [-]
C : float
Metric Kv valve flow coefficient (flow rate of water at a pressure drop
of 1 bar) [m^3/hr]
Returns
-------
Rev : float
Valve reynolds number [-]
Examples
--------
>>> Reynolds_valve(3.26e-07, 360, 150.0, 0.9, 0.46, 165)
2966984.7525455453
References
----------
.. [1] IEC 60534-2-1 / ISA-75.01.01-2007
'''
return N4*Fd*Q/nu/(C*FL)**0.5*(FL**2*C**2/(N2*D1**4) + 1)**0.25
[docs]def loss_coefficient_piping(d, D1=None, D2=None):
r'''Calculates the sum of loss coefficients from possible
inlet/outlet reducers/expanders around a control valve according to
IEC 60534 calculations.
.. math::
\Sigma \xi = \xi_1 + \xi_2 + \xi_{B1} - \xi_{B2}
\xi_1 = 0.5\left[1 -\left(\frac{d}{D_1}\right)^2\right]^2
\xi_2 = 1.0\left[1 -\left(\frac{d}{D_2}\right)^2\right]^2
\xi_{B1} = 1 - \left(\frac{d}{D_1}\right)^4
\xi_{B2} = 1 - \left(\frac{d}{D_2}\right)^4
Parameters
----------
d : float
Diameter of the valve [m]
D1 : float
Diameter of the pipe before the valve [m]
D2 : float
Diameter of the pipe after the valve [m]
Returns
-------
loss : float
Sum of the four loss coefficients [-]
Examples
--------
In example 3, non-choked compressible flow with fittings:
>>> loss_coefficient_piping(0.05, 0.08, 0.1)
0.6580810546875
References
----------
.. [1] IEC 60534-2-1 / ISA-75.01.01-2007
'''
loss = 0.
if D1:
loss += 1. - (d/D1)**4 # Inlet flow energy
loss += 0.5*(1. - (d/D1)**2)**2 # Inlet reducer
if D2:
loss += 1.0*(1. - (d/D2)**2)**2 # Outlet reducer (expander)
loss -= 1. - (d/D2)**4 # Outlet flow energy
return loss
[docs]def Reynolds_factor(FL, C, d, Rev, full_trim=True):
r'''Calculates the Reynolds number factor `FR` for a valve with a Reynolds
number `Rev`, diameter `d`, flow coefficient `C`, liquid pressure recovery
factor `FL`, and with either full or reduced trim, all according to
IEC 60534 calculations.
If full trim:
.. math::
F_{R,1a} = 1 + \left(\frac{0.33F_L^{0.5}}{n_1^{0.25}}\right)\log_{10}
\left(\frac{Re_v}{10000}\right)
F_{R,2} = \min(\frac{0.026}{F_L}\sqrt{n_1 Re_v},\; 1)
n_1 = \frac{N_2}{\left(\frac{C}{d^2}\right)^2}
F_R = F_{R,2} \text{ if Rev < 10 else } \min(F_{R,1a}, F_{R,2})
Otherwise :
.. math::
F_{R,3a} = 1 + \left(\frac{0.33F_L^{0.5}}{n_2^{0.25}}\right)\log_{10}
\left(\frac{Re_v}{10000}\right)
F_{R,4} = \frac{0.026}{F_L}\sqrt{n_2 Re_v}
n_2 = 1 + N_{32}\left(\frac{C}{d}\right)^{2/3}
F_R = F_{R,4} \text{ if Rev < 10 else } \min(F_{R,3a}, F_{R,4})
Parameters
----------
FL : float
Liquid pressure recovery factor of a control valve without attached
fittings []
C : float
Metric Kv valve flow coefficient (flow rate of water at a pressure drop
of 1 bar) [m^3/hr]
d : float
Diameter of the valve [m]
Rev : float
Valve reynolds number [-]
full_trim : bool
Whether or not the valve has full trim
Returns
-------
FR : float
Reynolds number factor for laminar or transitional flow []
Examples
--------
In Example 4, compressible flow with small flow trim sized for gas flow
(Cv in the problem was converted to Kv here to make FR match with N32, N2):
>>> Reynolds_factor(FL=0.98, C=0.015483, d=15., Rev=1202., full_trim=False)
0.7148753122302025
References
----------
.. [1] IEC 60534-2-1 / ISA-75.01.01-2007
'''
if full_trim:
n1 = N2/(min(C/d**2, 0.04))**2 # C/d**2 must not exceed 0.04
FR_1a = 1 + (0.33*FL**0.5)/n1**0.25*log10(Rev/10000.)
FR_2 = 0.026/FL*(n1*Rev)**0.5
if Rev < 10:
FR = FR_2
else:
FR = min(FR_2, FR_1a)
else:
n2 = 1 + N32*(C/d**2)**(2/3.)
FR_3a = 1 + (0.33*FL**0.5)/n2**0.25*log10(Rev/10000.)
FR_4 = min(0.026/FL*(n2*Rev)**0.5, 1)
if Rev < 10:
FR = FR_4
else:
FR = min(FR_3a, FR_4)
return FR
[docs]def size_control_valve_l(rho, Psat, Pc, mu, P1, P2, Q, D1, D2, d, FL, Fd):
r'''Calculates flow coefficient of a control valve passing a liquid
according to IEC 60534. Uses a large number of inputs in SI units. Note the
return value is not standard SI. All parameters are required.
This sizing model does not officially apply to liquid mixtures, slurries,
non-Newtonian fluids, or liquid-solid conveyance systems. For details
of the calculations, consult [1]_.
Parameters
----------
rho : float
Density of the liquid at the inlet [kg/m^3]
Psat : float
Saturation pressure of the fluid at inlet temperature [Pa]
Pc : float
Critical pressure of the fluid [Pa]
mu : float
Viscosity of the fluid [Pa*s]
P1 : float
Inlet pressure of the fluid before valves and reducers [Pa]
P2 : float
Outlet pressure of the fluid after valves and reducers [Pa]
Q : float
Volumetric flow rate of the fluid [m^3/s]
D1 : float
Diameter of the pipe before the valve [m]
D2 : float
Diameter of the pipe after the valve [m]
d : float
Diameter of the valve [m]
FL : float
Liquid pressure recovery factor of a control valve without attached
fittings []
Fd : float
Valve style modifier []
Returns
-------
C : float
Metric Kv valve flow coefficient (flow rate of water at a pressure drop
of 1 bar) [m^3/hr]
Examples
--------
From [1]_, matching example 1 for a globe, parabolic plug,
flow-to-open valve.
>>> size_control_valve_l(rho=965.4, Psat=70.1E3, Pc=22120E3, mu=3.1472E-4,
... P1=680E3, P2=220E3, Q=0.1, D1=0.15, D2=0.15, d=0.15,
... FL=0.9, Fd=0.46)
164.9954763704956
From [1]_, matching example 2 for a ball, segmented ball,
flow-to-open valve.
>>> size_control_valve_l(rho=965.4, Psat=70.1E3, Pc=22120E3, mu=3.1472E-4,
... P1=680E3, P2=220E3, Q=0.1, D1=0.1, D2=0.1, d=0.1,
... FL=0.6, Fd=0.98)
238.05817216710483
References
----------
.. [1] IEC 60534-2-1 / ISA-75.01.01-2007
'''
# Pa to kPa, according to constants in standard
P1, P2, Psat, Pc = P1/1000., P2/1000., Psat/1000., Pc/1000.
# m to mm, according to constants in standard
D1, D2, d = D1*1000., D2*1000., d*1000.
Q = Q*3600. # m^3/s to m^3/hr, according to constants in standard
nu = mu/rho # kinematic viscosity used in standard
dP = P1 - P2
FF = FF_critical_pressure_ratio_l(Psat=Psat, Pc=Pc)
choked = is_choked_turbulent_l(dP=dP, P1=P1, Psat=Psat, FF=FF, FL=FL)
if choked:
# Choked flow, equation 3
C = Q/N1/FL*(rho/rho0/(P1-FF*Psat))**0.5
else:
# non-choked flow, eq 1
C = Q/N1*(rho/rho0/dP)**0.5
Rev = Reynolds_valve(nu=nu, Q=Q, D1=D1, FL=FL, Fd=Fd, C=C)
if Rev > 10000 and (D1 != d or D2 != d):
# liquid, using Fp and FLP
FP = 1
Ci = C
def iterate_piping_turbulent(Ci):
loss = loss_coefficient_piping(d, D1, D2)
FP = (1 + loss/N2*(Ci/d**2)**2)**-0.5
loss_upstream = loss_coefficient_piping(d, D1)
FLP = FL*(1 + FL**2/N2*loss_upstream*(Ci/d**2)**2)**-0.5
choked = is_choked_turbulent_l(dP, P1, Psat, FF, FLP=FLP, FP=FP)
if choked:
# Choked flow with piping, equation 4
C = Q/N1/FLP*(rho/rho0/(P1-FF*Psat))**0.5
else:
# Non-Choked flow with piping, equation 4
C = Q/N1/FP*(rho/rho0/dP)**0.5
if Ci/C < 0.99:
C = iterate_piping_turbulent(C)
return C
C = iterate_piping_turbulent(Ci)
elif Rev <= 10000:
# Laminar
def iterate_piping_laminar(C):
Ci = 1.3*C
Rev = Reynolds_valve(nu=nu, Q=Q, D1=D1, FL=FL, Fd=Fd, C=Ci)
if Ci/d**2 > 0.016*N18:
FR = Reynolds_factor(FL=FL, C=Ci, d=d, Rev=Rev, full_trim=False)
else:
FR = Reynolds_factor(FL=FL, C=Ci, d=d, Rev=Rev, full_trim=True)
if C/FR >= Ci:
Ci = iterate_piping_laminar(Ci) # pragma: no cover
return Ci
C = iterate_piping_laminar(C)
return C
[docs]def size_control_valve_g(T, MW, mu, gamma, Z, P1, P2, Q, D1, D2, d, FL, Fd, xT):
r'''Calculates flow coefficient of a control valve passing a gas
according to IEC 60534. Uses a large number of inputs in SI units. Note the
return value is not standard SI. All parameters are required. For details
of the calculations, consult [1]_. Note the inlet gas flow conditions.
Parameters
----------
T : float
Temperature of the gas at the inlet [K]
MW : float
Molecular weight of the gas [g/mol]
mu : float
Viscosity of the fluid at inlet conditions [Pa*s]
gamma : float
Specific heat capacity ratio [-]
Z : float
Compressibility factor at inlet conditions, [-]
P1 : float
Inlet pressure of the gas before valves and reducers [Pa]
P2 : float
Outlet pressure of the gas after valves and reducers [Pa]
Q : float
Volumetric flow rate of the gas at *273.15 K* and 1 atm specifically
[m^3/s]
D1 : float
Diameter of the pipe before the valve [m]
D2 : float
Diameter of the pipe after the valve [m]
d : float
Diameter of the valve [m]
FL : float
Liquid pressure recovery factor of a control valve without attached
fittings [-]
Fd : float
Valve style modifier [-]
xT : float
Pressure difference ratio factor of a valve without fittings at choked
flow [-]
Returns
-------
C : float
Metric Kv valve flow coefficient (flow rate of water at a pressure drop
of 1 bar) [m^3/hr]
Examples
--------
From [1]_, matching example 3 for non-choked gas flow with attached
fittings and a rotary, eccentric plug, flow-to-open control valve:
>>> size_control_valve_g(T=433., MW=44.01, mu=1.4665E-4, gamma=1.30,
... Z=0.988, P1=680E3, P2=310E3, Q=38/36., D1=0.08, D2=0.1, d=0.05,
... FL=0.85, Fd=0.42, xT=0.60)
72.58664545391052
From [1]_, roughly matching example 4 for a small flow trim sized tapered
needle plug valve. Difference is 3% and explained by the difference in
algorithms used.
>>> size_control_valve_g(T=320., MW=39.95, mu=5.625E-5, gamma=1.67, Z=1.0,
... P1=2.8E5, P2=1.3E5, Q=0.46/3600., D1=0.015, D2=0.015, d=0.015, FL=0.98,
... Fd=0.07, xT=0.8)
0.016498765335995726
References
----------
.. [1] IEC 60534-2-1 / ISA-75.01.01-2007
'''
# Pa to kPa, according to constants in standard
P1, P2 = P1/1000., P2/1000.
# m to mm, according to constants in standard
D1, D2, d = D1*1000., D2*1000., d*1000.
Q = Q*3600. # m^3/s to m^3/hr, according to constants in standard
# Convert dynamic viscosity to kinematic viscosity
Vm = Z*R*T/(P1*1000)
rho = (Vm)**-1*MW/1000.
nu = mu/rho # kinematic viscosity used in standard
dP = P1 - P2
Fgamma = gamma/1.40
x = dP/P1
Y = max(1 - x/(3*Fgamma*xT), 2/3.)
choked = is_choked_turbulent_g(x, Fgamma, xT)
if choked:
# Choked, and flow coefficient from eq 14a
C = Q/(N9*P1*Y)*(MW*T*Z/xT/Fgamma)**0.5
else:
# Non-choked, and flow coefficient from eq 8a
C = Q/(N9*P1*Y)*(MW*T*Z/x)**0.5
Rev = Reynolds_valve(nu=nu, Q=Q, D1=D1, FL=FL, Fd=Fd, C=C)
if Rev > 10000 and (D1 != d or D2 != d):
# gas, using xTP and FLP
FP = 1.
def iterate_piping_coef(Ci):
loss = loss_coefficient_piping(d, D1, D2)
FP = (1. + loss/N2*(Ci/d**2)**2)**-0.5
loss_upstream = loss_coefficient_piping(d, D1)
xTP = xT/FP**2/(1 + xT*loss_upstream/N5*(Ci/d**2)**2)
choked = is_choked_turbulent_g(x, Fgamma, xTP=xTP)
if choked:
# Choked flow with piping, equation 17a
C = Q/(N9*FP*P1*Y)*(MW*T*Z/xTP/Fgamma)**0.5
else:
# Non-choked flow with piping, equation 11a
C = Q/(N9*FP*P1*Y)*(MW*T*Z/x)**0.5
if Ci/C < 0.99:
C = iterate_piping_coef(C)
return C
C = iterate_piping_coef(C)
elif Rev <= 10000:
# Laminar;
def iterate_piping_laminar(C):
Ci = 1.3*C
Rev = Reynolds_valve(nu=nu, Q=Q, D1=D1, FL=FL, Fd=Fd, C=Ci)
if Ci/d**2 > 0.016*N18:
FR = Reynolds_factor(FL=FL, C=Ci, d=d, Rev=Rev, full_trim=False)
else:
FR = Reynolds_factor(FL=FL, C=Ci, d=d, Rev=Rev, full_trim=True)
if C/FR >= Ci:
Ci = iterate_piping_laminar(Ci)
return Ci
C = iterate_piping_laminar(C)
return C