"""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.
This module contains functions for calculating two-phase pressure drop. It also
contains correlations for flow regime.
For reporting bugs, adding feature requests, or submitting pull requests,
please use the `GitHub issue tracker <https://github.com/CalebBell/fluids/>`_
or contact the author at Caleb.Andrew.Bell@gmail.com.
.. contents:: :local:
Interfaces
----------
.. autofunction:: two_phase_dP
.. autofunction:: two_phase_dP_methods
.. autofunction:: two_phase_dP_acceleration
.. autofunction:: two_phase_dP_gravitational
.. autofunction:: two_phase_dP_dz_acceleration
.. autofunction:: two_phase_dP_dz_gravitational
Two Phase Pressure Drop Correlations
------------------------------------
.. autofunction:: Beggs_Brill
.. autofunction:: Lockhart_Martinelli
.. autofunction:: Friedel
.. autofunction:: Chisholm
.. autofunction:: Kim_Mudawar
.. autofunction:: Baroczy_Chisholm
.. autofunction:: Theissing
.. autofunction:: Muller_Steinhagen_Heck
.. autofunction:: Gronnerud
.. autofunction:: Lombardi_Pedrocchi
.. autofunction:: Jung_Radermacher
.. autofunction:: Tran
.. autofunction:: Chen_Friedel
.. autofunction:: Zhang_Webb
.. autofunction:: Xu_Fang
.. autofunction:: Yu_France
.. autofunction:: Wang_Chiang_Lu
.. autofunction:: Hwang_Kim
.. autofunction:: Zhang_Hibiki_Mishima
.. autofunction:: Mishima_Hibiki
.. autofunction:: Bankoff
Two Phase Flow Regime Correlations
----------------------------------
.. autofunction:: Mandhane_Gregory_Aziz_regime
.. autofunction:: Taitel_Dukler_regime
"""
__all__ = ['two_phase_dP', 'two_phase_dP_methods', 'two_phase_dP_acceleration',
'two_phase_dP_dz_acceleration', 'two_phase_dP_gravitational',
'two_phase_dP_dz_gravitational',
'Beggs_Brill', 'Lockhart_Martinelli', 'Friedel', 'Chisholm',
'Kim_Mudawar', 'Baroczy_Chisholm', 'Theissing',
'Muller_Steinhagen_Heck', 'Gronnerud', 'Lombardi_Pedrocchi',
'Jung_Radermacher', 'Tran', 'Chen_Friedel', 'Zhang_Webb', 'Xu_Fang',
'Yu_France', 'Wang_Chiang_Lu', 'Hwang_Kim', 'Zhang_Hibiki_Mishima',
'Mishima_Hibiki', 'Bankoff',
'Mandhane_Gregory_Aziz_regime', 'Taitel_Dukler_regime']
from math import cos, exp, log, log10, pi, radians, sin, sqrt
from fluids.constants import deg2rad, g
from fluids.core import Bond, Confinement, Froude, Reynolds, Suratman, Weber
from fluids.friction import friction_factor
from fluids.numerics import implementation_optimize_tck, splev
from fluids.two_phase_voidage import Lockhart_Martinelli_Xtt, homogeneous
Beggs_Brill_dat = {'segregated': (0.98, 0.4846, 0.0868),
'intermittent': (0.845, 0.5351, 0.0173),
'distributed': (1.065, 0.5824, 0.0609)}
def _Beggs_Brill_holdup(regime, lambda_L, Fr, angle, LV):
if regime == 0:
a, b, c = 0.98, 0.4846, 0.0868
elif regime == 2:
a, b, c = 0.845, 0.5351, 0.0173
elif regime == 3:
a, b, c = 1.065, 0.5824, 0.0609
HL0 = a*lambda_L**b*Fr**-c
if HL0 < lambda_L:
HL0 = lambda_L
if angle > 0.0: # uphill
# h used instead of g to avoid conflict with gravitational constant
if regime == 0:
d, e, f, h = 0.011, -3.768, 3.539, -1.614
elif regime == 2:
d, e, f, h = 2.96, 0.305, -0.4473, 0.0978
elif regime == 3:
# Dummy values for distributed - > psi = 1.
d, e, f, h = 2.96, 0.305, -0.4473, 0.0978
elif angle <= 0: # downhill
d, e, f, h = 4.70, -0.3692, 0.1244, -0.5056
C = (1.0 - lambda_L)*log(d*lambda_L**e*LV**f*Fr**h)
if C < 0.0:
C = 0.0
# Correction factor for inclination angle
Psi = 1.0 + C*(sin(1.8*angle) - 1.0/3.0*sin(1.8*angle)**3)
if (angle > 0 and regime == 3) or angle == 0:
Psi = 1.0
Hl = HL0*Psi
return Hl
[docs]def Beggs_Brill(m, x, rhol, rhog, mul, mug, sigma, P, D, angle, roughness=0.0,
L=1.0, g=g, acceleration=True):
r'''Calculates the two-phase pressure drop according to the Beggs-Brill
correlation ([1]_, [2]_, [3]_).
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Mass quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
sigma : float
Surface tension, [N/m]
P : float
Pressure of fluid (used only if `acceleration=True`), [Pa]
D : float
Diameter of pipe, [m]
angle : float
The angle of the pipe with respect to the horizontal, [degrees]
roughness : float, optional
Roughness of pipe for use in calculating friction factor, [m]
L : float, optional
Length of pipe, [m]
g : float, optional
Acceleration due to gravity, [m/s^2]
acceleration : bool
Whether or not to include the original acceleration component, [-]
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Notes
-----
The original acceleration formula is fairly primitive and normally
neglected. The model was developed assuming smooth pipe, so leaving
`roughness` to zero may be wise.
Note this is a "mechanistic" pressure drop model - the gravitational
pressure drop cannot be separated from the frictional pressure drop.
Examples
--------
>>> Beggs_Brill(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
... sigma=0.0487, P=1E7, D=0.05, angle=0, roughness=0.0, L=1.0)
686.9724506803469
References
----------
.. [1] Beggs, D.H., and J.P. Brill. "A Study of Two-Phase Flow in Inclined
Pipes." Journal of Petroleum Technology 25, no. 05 (May 1, 1973):
607-17. https://doi.org/10.2118/4007-PA.
.. [2] Brill, James P., and Howard Dale Beggs. Two-Phase Flow in Pipes,
1994.
.. [3] Shoham, Ovadia. Mechanistic Modeling of Gas-Liquid Two-Phase Flow in
Pipes. Pap/Cdr edition. Richardson, TX: Society of Petroleum Engineers,
2006.
'''
# 0 - segregated; 1 - transition; 2 - intermittent; 3 - distributed
qg = x*m/rhog
ql = (1.0 - x)*m/rhol
A = 0.25*pi*D*D
Vsg = qg/A
Vsl = ql/A
Vm = Vsg + Vsl
Fr = Vm*Vm/(g*D)
lambda_L = Vsl/Vm # no slip liquid holdup
L1 = 316.0*lambda_L**0.302
L2 = 0.0009252*lambda_L**-2.4684
L3 = 0.1*lambda_L**-1.4516
L4 = 0.5*lambda_L**-6.738
if (lambda_L < 0.01 and Fr < L1) or (lambda_L >= 0.01 and Fr < L2):
regime = 0
elif (lambda_L >= 0.01 and L2 <= Fr <= L3):
regime = 1
elif (0.01 <= lambda_L < 0.4 and L3 < Fr <= L1) or (lambda_L >= 0.4 and L3 < Fr <= L4):
regime = 2
elif (lambda_L < 0.4 and Fr >= L1) or (lambda_L >= 0.4 and Fr > L4):
regime = 3
else:
raise ValueError('Outside regime ranges')
LV = Vsl*sqrt(sqrt(rhol/(g*sigma)))
if angle is None:
angle = 0.0
angle = deg2rad*angle
if regime != 1:
Hl = _Beggs_Brill_holdup(regime, lambda_L, Fr, angle, LV)
else:
A = (L3 - Fr)/(L3 - L2)
Hl = (A*_Beggs_Brill_holdup(0, lambda_L, Fr, angle, LV)
+ (1.0 - A)*_Beggs_Brill_holdup(2, lambda_L, Fr, angle, LV))
rhos = rhol*Hl + rhog*(1.0 - Hl)
mum = mul*lambda_L + mug*(1.0 - lambda_L)
rhom = rhol*lambda_L + rhog*(1.0 - lambda_L)
Rem = rhom*D/mum*Vm
fn = friction_factor(Re=Rem, eD=roughness/D)
x = lambda_L/(Hl*Hl)
if 1.0 < x < 1.2:
S = log(2.2*x - 1.2)
else:
logx = log(x)
# from horner(-0.0523 + 3.182*log(x) - 0.8725*log(x)**2 + 0.01853*log(x)**4, x)
S = logx/(logx*(logx*(0.01853*logx*logx - 0.8725) + 3.182) - 0.0523)
if S > 7.0:
S = 7.0 # Truncate S to avoid exp(S) overflowing
ftp = fn*exp(S)
dP_ele = g*sin(angle)*rhos*L
dP_fric = ftp*L/D*0.5*rhom*Vm*Vm
# rhos here is pretty clearly rhos according to Shoham
if P is None:
P = 101325.0
if not acceleration:
dP = dP_ele + dP_fric
else:
Ek = Vsg*Vm*rhos/P # Confirmed this expression is dimensionless
dP = (dP_ele + dP_fric)/(1.0 - Ek)
return dP
[docs]def Friedel(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0.0, L=1.0):
r'''Calculates two-phase pressure drop with the Friedel correlation.
.. math::
\Delta P_{friction} = \Delta P_{lo} \phi_{lo}^2
.. math::
\phi_{lo}^2 = E + \frac{3.24FH}{Fr^{0.0454} We^{0.035}}
.. math::
H = \left(\frac{\rho_l}{\rho_g}\right)^{0.91}\left(\frac{\mu_g}{\mu_l}
\right)^{0.19}\left(1 - \frac{\mu_g}{\mu_l}\right)^{0.7}
.. math::
F = x^{0.78}(1 - x)^{0.224}
.. math::
E = (1-x)^2 + x^2\left(\frac{\rho_l f_{d,go}}{\rho_g f_{d,lo}}\right)
.. math::
Fr = \frac{G_{tp}^2}{gD\rho_H^2}
.. math::
We = \frac{G_{tp}^2 D}{\sigma \rho_H}
.. math::
\rho_H = \left(\frac{x}{\rho_g} + \frac{1-x}{\rho_l}\right)^{-1}
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
sigma : float
Surface tension, [N/m]
D : float
Diameter of pipe, [m]
roughness : float, optional
Roughness of pipe for use in calculating friction factor, [m]
L : float, optional
Length of pipe, [m]
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Notes
-----
Applicable to vertical upflow and horizontal flow. Known to work poorly
when mul/mug > 1000. Gives mean errors on the order of 40%. Tested on data
with diameters as small as 4 mm.
The power of 0.0454 is given as 0.045 in [2]_, [3]_, [4]_, and [5]_; [6]_
and [2]_ give 0.0454 and [2]_ also gives a similar correlation said to be
presented in [1]_, so it is believed this 0.0454 was the original power.
[6]_ also gives an expression for friction factor claimed to be presented
in [1]_; it is not used here.
Examples
--------
Example 4 in [6]_:
>>> Friedel(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
... sigma=0.0487, D=0.05, roughness=0.0, L=1.0)
738.6500525002241
References
----------
.. [1] Friedel, L. "Improved Friction Pressure Drop Correlations for
Horizontal and Vertical Two-Phase Pipe Flow." , in: Proceedings,
European Two Phase Flow Group Meeting, Ispra, Italy, 1979: 485-481.
.. [2] Whalley, P. B. Boiling, Condensation, and Gas-Liquid Flow. Oxford:
Oxford University Press, 1987.
.. [3] Triplett, K. A., S. M. Ghiaasiaan, S. I. Abdel-Khalik, A. LeMouel,
and B. N. McCord. "Gas-liquid Two-Phase Flow in Microchannels: Part II:
Void Fraction and Pressure Drop.” International Journal of Multiphase
Flow 25, no. 3 (April 1999): 395-410. doi:10.1016/S0301-9322(98)00055-X.
.. [4] Mekisso, Henock Mateos. "Comparison of Frictional Pressure Drop
Correlations for Isothermal Two-Phase Horizontal Flow." Thesis, Oklahoma
State University, 2013. https://shareok.org/handle/11244/11109.
.. [5] Thome, John R. "Engineering Data Book III." Wolverine Tube Inc
(2004). http://www.wlv.com/heat-transfer-databook/
.. [6] Ghiaasiaan, S. Mostafa. Two-Phase Flow, Boiling, and Condensation:
In Conventional and Miniature Systems. Cambridge University Press, 2007.
'''
# Liquid-only properties, for calculation of E, dP_lo
v_lo = m/rhol/(pi/4*D**2)
Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D)
fd_lo = friction_factor(Re=Re_lo, eD=roughness/D)
dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2)
# Gas-only properties, for calculation of E
v_go = m/rhog/(pi/4*D**2)
Re_go = Reynolds(V=v_go, rho=rhog, mu=mug, D=D)
fd_go = friction_factor(Re=Re_go, eD=roughness/D)
F = x**0.78*(1-x)**0.224
H = (rhol/rhog)**0.91*(mug/mul)**0.19*(1 - mug/mul)**0.7
E = (1-x)**2 + x**2*(rhol*fd_go/(rhog*fd_lo))
# Homogeneous properties, for Froude/Weber numbers
voidage_h = homogeneous(x, rhol, rhog)
rho_h = rhol*(1-voidage_h) + rhog*voidage_h
Q_h = m/rho_h
v_h = Q_h/(pi/4*D**2)
Fr = Froude(V=v_h, L=D, squared=True) # checked with (m/(pi/4*D**2))**2/g/D/rho_h**2
We = Weber(V=v_h, L=D, rho=rho_h, sigma=sigma) # checked with (m/(pi/4*D**2))**2*D/sigma/rho_h
phi_lo2 = E + 3.24*F*H/(Fr**0.0454*We**0.035)
return phi_lo2*dP_lo
[docs]def Gronnerud(m, x, rhol, rhog, mul, mug, D, roughness=0.0, L=1.0):
r'''Calculates two-phase pressure drop with the Gronnerud correlation as
presented in [2]_, [3]_, and [4]_.
.. math::
\Delta P_{friction} = \Delta P_{gd} \phi_{lo}^2
.. math::
\phi_{gd} = 1 + \left(\frac{dP}{dL}\right)_{Fr}\left[
\frac{\frac{\rho_l}{\rho_g}}{\left(\frac{\mu_l}{\mu_g}\right)^{0.25}}
-1\right]
.. math::
\left(\frac{dP}{dL}\right)_{Fr} = f_{Fr}\left[x+4(x^{1.8}-x^{10}
f_{Fr}^{0.5})\right]
.. math::
f_{Fr} = Fr_l^{0.3} + 0.0055\left(\ln \frac{1}{Fr_l}\right)^2
.. math::
Fr_l = \frac{G_{tp}^2}{gD\rho_l^2}
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
D : float
Diameter of pipe, [m]
roughness : float, optional
Roughness of pipe for use in calculating friction factor, [m]
L : float, optional
Length of pipe, [m]
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Notes
-----
Developed for evaporators. Applicable from 0 < x < 1.
In the model, if `Fr_l` is more than 1, `f_Fr` is set to 1.
Examples
--------
>>> Gronnerud(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
... D=0.05, roughness=0.0, L=1.0)
384.12541144474085
References
----------
.. [1] Gronnerud, R. "Investigation of Liquid Hold-Up, Flow Resistance and
Heat Transfer in Circulation Type Evaporators. 4. Two-Phase Flow
Resistance in Boiling Refrigerants." Proc. Freudenstadt Meet., IIR/C.
R. Réun. Freudenstadt, IIF. 1972-1: 127-138. 1972.
.. [2] ASHRAE Handbook: Fundamentals. American Society of Heating,
Refrigerating and Air-Conditioning Engineers, Incorporated, 2013.
.. [3] Mekisso, Henock Mateos. "Comparison of Frictional Pressure Drop
Correlations for Isothermal Two-Phase Horizontal Flow." Thesis, Oklahoma
State University, 2013. https://shareok.org/handle/11244/11109.
.. [4] Thome, John R. "Engineering Data Book III." Wolverine Tube Inc
(2004). http://www.wlv.com/heat-transfer-databook/
'''
G = m/(pi/4*D**2)
V = G/rhol
Frl = Froude(V=V, L=D, squared=True)
if Frl >= 1:
f_Fr = 1
else:
f_Fr = Frl**0.3 + 0.0055*(log(1./Frl))**2
dP_dL_Fr = f_Fr*(x + 4*(x**1.8 - x**10*sqrt(f_Fr)))
phi_gd = 1 + dP_dL_Fr*((rhol/rhog)/sqrt(sqrt(mul/mug)) - 1)
# Liquid-only properties, for calculation of E, dP_lo
v_lo = m/rhol/(pi/4*D**2)
Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D)
fd_lo = friction_factor(Re=Re_lo, eD=roughness/D)
dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2)
return phi_gd*dP_lo
[docs]def Chisholm(m, x, rhol, rhog, mul, mug, D, roughness=0.0, L=1.0,
rough_correction=False):
r'''Calculates two-phase pressure drop with the Chisholm (1973) correlation
from [1]_, also in [2]_ and [3]_.
.. math::
\frac{\Delta P_{tp}}{\Delta P_{lo}} = \phi_{ch}^2
.. math::
\phi_{ch}^2 = 1 + (\Gamma^2 -1)\left\{B x^{(2-n)/2} (1-x)^{(2-n)/2}
+ x^{2-n} \right\}
.. math::
\Gamma ^2 = \frac{\left(\frac{\Delta P}{L}\right)_{go}}{\left(\frac{
\Delta P}{L}\right)_{lo}}
For Gamma < 9.5:
.. math::
B = \frac{55}{G_{tp}^{0.5}} \text{ for } G_{tp} > 1900
.. math::
B = \frac{2400}{G_{tp}} \text{ for } 500 < G_{tp} < 1900
.. math::
B = 4.8 \text{ for } G_{tp} < 500
For 9.5 < Gamma < 28:
.. math::
B = \frac{520}{\Gamma G_{tp}^{0.5}} \text{ for } G_{tp} < 600
.. math::
B = \frac{21}{\Gamma} \text{ for } G_{tp} > 600
For Gamma > 28:
.. math::
B = \frac{15000}{\Gamma^2 G_{tp}^{0.5}}
If `rough_correction` is True, the following correction to B is applied:
.. math::
\frac{B_{rough}}{B_{smooth}} = \left[0.5\left\{1+ \left(\frac{\mu_g}
{\mu_l}\right)^2 + 10^{-600\epsilon/D}\right\}\right]^{\frac{0.25-n}
{0.25}}
.. math::
n = \frac{\ln \frac{f_{d,lo}}{f_{d,go}}}{\ln \frac{Re_{go}}{Re_{lo}}}
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
D : float
Diameter of pipe, [m]
roughness : float, optional
Roughness of pipe for use in calculating friction factor, [m]
L : float, optional
Length of pipe, [m]
rough_correction : bool, optional
Whether or not to use the roughness correction proposed in the 1968
version of the correlation
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Notes
-----
Applicable for 0 < x < 1. n = 0.25, the exponent in the Blassius equation.
Originally developed for smooth pipes, a roughness correction is included
as well from the Chisholm's 1968 work [4]_. Neither [2]_ nor [3]_ have any
mention of the correction however.
Examples
--------
>>> Chisholm(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6,
... mug=14E-6, D=0.05, roughness=0.0, L=1.0)
1084.1489922923738
References
----------
.. [1] Chisholm, D. "Pressure Gradients due to Friction during the Flow of
Evaporating Two-Phase Mixtures in Smooth Tubes and Channels."
International Journal of Heat and Mass Transfer 16, no. 2 (February
1973): 347-58. doi:10.1016/0017-9310(73)90063-X.
.. [2] Mekisso, Henock Mateos. "Comparison of Frictional Pressure Drop
Correlations for Isothermal Two-Phase Horizontal Flow." Thesis, Oklahoma
State University, 2013. https://shareok.org/handle/11244/11109.
.. [3] Thome, John R. "Engineering Data Book III." Wolverine Tube Inc
(2004). http://www.wlv.com/heat-transfer-databook/
.. [4] Chisholm, D. "Research Note: Influence of Pipe Surface Roughness on
Friction Pressure Gradient during Two-Phase Flow." Journal of Mechanical
Engineering Science 20, no. 6 (December 1, 1978): 353-354.
doi:10.1243/JMES_JOUR_1978_020_061_02.
'''
G_tp = m/(pi/4*D**2)
n = 0.25 # Blasius friction factor exponent
# Liquid-only properties, for calculation of dP_lo
v_lo = m/rhol/(pi/4*D**2)
Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D)
fd_lo = friction_factor(Re=Re_lo, eD=roughness/D)
dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2)
# Gas-only properties, for calculation of dP_go
v_go = m/rhog/(pi/4*D**2)
Re_go = Reynolds(V=v_go, rho=rhog, mu=mug, D=D)
fd_go = friction_factor(Re=Re_go, eD=roughness/D)
dP_go = fd_go*L/D*(0.5*rhog*v_go**2)
Gamma = sqrt(dP_go/dP_lo)
if Gamma <= 9.5:
if G_tp <= 500:
B = 4.8
elif G_tp < 1900:
B = 2400./G_tp
else:
B = 55.0/sqrt(G_tp)
elif Gamma <= 28:
if G_tp <= 600:
B = 520./sqrt(G_tp)/Gamma
else:
B = 21./Gamma
else:
B = 15000./sqrt(G_tp)/Gamma**2
if rough_correction:
n = log(fd_lo/fd_go)/log(Re_go/Re_lo)
B_ratio = (0.5*(1 + (mug/mul)**2 + 10**(-600*roughness/D)))**((0.25-n)/0.25)
B = B*B_ratio
phi2_ch = 1 + (Gamma**2-1)*(B*x**((2-n)/2.)*(1-x)**((2-n)/2.) + x**(2-n))
return phi2_ch*dP_lo
[docs]def Baroczy_Chisholm(m, x, rhol, rhog, mul, mug, D, roughness=0.0, L=1.0):
r'''Calculates two-phase pressure drop with the Baroczy (1966) model.
It was presented in graphical form originally; Chisholm (1973) made the
correlation non-graphical. The model is also shown in [3]_.
.. math::
\frac{\Delta P_{tp}}{\Delta P_{lo}} = \phi_{ch}^2
.. math::
\phi_{ch}^2 = 1 + (\Gamma^2 -1)\left\{B x^{(2-n)/2} (1-x)^{(2-n)/2}
+ x^{2-n} \right\}
.. math::
\Gamma ^2 = \frac{\left(\frac{\Delta P}{L}\right)_{go}}{\left(\frac{
\Delta P}{L}\right)_{lo}}
For Gamma < 9.5:
.. math::
B = \frac{55}{G_{tp}^{0.5}}
For 9.5 < Gamma < 28:
.. math::
B = \frac{520}{\Gamma G_{tp}^{0.5}}
For Gamma > 28:
.. math::
B = \frac{15000}{\Gamma^2 G_{tp}^{0.5}}
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
D : float
Diameter of pipe, [m]
roughness : float, optional
Roughness of pipe for use in calculating friction factor, [m]
L : float, optional
Length of pipe, [m]
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Notes
-----
Applicable for 0 < x < 1. n = 0.25, the exponent in the Blassius equation.
The `Chisholm_1973` function should be used in preference to this.
Examples
--------
>>> Baroczy_Chisholm(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6,
... mug=14E-6, D=0.05, roughness=0.0, L=1.0)
1084.1489922923738
References
----------
.. [1] Baroczy, C. J. "A systematic correlation for two-phase pressure
drop." In Chem. Eng. Progr., Symp. Ser., 62: No. 64, 232-49 (1966).
.. [2] Chisholm, D. "Pressure Gradients due to Friction during the Flow of
Evaporating Two-Phase Mixtures in Smooth Tubes and Channels."
International Journal of Heat and Mass Transfer 16, no. 2 (February
1973): 347-58. doi:10.1016/0017-9310(73)90063-X.
.. [3] Mekisso, Henock Mateos. "Comparison of Frictional Pressure Drop
Correlations for Isothermal Two-Phase Horizontal Flow." Thesis, Oklahoma
State University, 2013. https://shareok.org/handle/11244/11109.
'''
G_tp = m/(pi/4*D**2)
n = 0.25 # Blasius friction factor exponent
# Liquid-only properties, for calculation of dP_lo
v_lo = m/rhol/(pi/4*D**2)
Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D)
fd_lo = friction_factor(Re=Re_lo, eD=roughness/D)
dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2)
# Gas-only properties, for calculation of dP_go
v_go = m/rhog/(pi/4*D**2)
Re_go = Reynolds(V=v_go, rho=rhog, mu=mug, D=D)
fd_go = friction_factor(Re=Re_go, eD=roughness/D)
dP_go = fd_go*L/D*(0.5*rhog*v_go**2)
Gamma = sqrt(dP_go/dP_lo)
if Gamma <= 9.5:
B = 55.0/sqrt(G_tp)
elif Gamma <= 28:
B = 520./sqrt(G_tp)/Gamma
else:
B = 15000./sqrt(G_tp)/Gamma**2
phi2_ch = 1 + (Gamma**2-1)*(B*x**((2-n)/2.)*(1-x)**((2-n)/2.) + x**(2-n))
return phi2_ch*dP_lo
[docs]def Muller_Steinhagen_Heck(m, x, rhol, rhog, mul, mug, D, roughness=0.0, L=1.0):
r'''Calculates two-phase pressure drop with the Muller-Steinhagen and Heck
(1986) correlation from [1]_, also in [2]_ and [3]_.
.. math::
\Delta P_{tp} = G_{MSH}(1-x)^{1/3} + \Delta P_{go}x^3
.. math::
G_{MSH} = \Delta P_{lo} + 2\left[\Delta P_{go} - \Delta P_{lo}\right]x
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
D : float
Diameter of pipe, [m]
roughness : float, optional
Roughness of pipe for use in calculating friction factor, [m]
L : float, optional
Length of pipe, [m]
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Notes
-----
Applicable for 0 < x < 1. Developed to be easily integrated. The
contribution of each term to the overall pressure drop can be
understood in this model.
Examples
--------
>>> Muller_Steinhagen_Heck(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6,
... mug=14E-6, D=0.05, roughness=0.0, L=1.0)
793.4465457435081
References
----------
.. [1] Müller-Steinhagen, H, and K Heck. "A Simple Friction Pressure Drop
Correlation for Two-Phase Flow in Pipes." Chemical Engineering and
Processing: Process Intensification 20, no. 6 (November 1, 1986):
297-308. doi:10.1016/0255-2701(86)80008-3.
.. [2] Mekisso, Henock Mateos. "Comparison of Frictional Pressure Drop
Correlations for Isothermal Two-Phase Horizontal Flow." Thesis, Oklahoma
State University, 2013. https://shareok.org/handle/11244/11109.
.. [3] Thome, John R. "Engineering Data Book III." Wolverine Tube Inc
(2004). http://www.wlv.com/heat-transfer-databook/
'''
# Liquid-only properties, for calculation of dP_lo
v_lo = m/rhol/(pi/4*D**2)
Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D)
fd_lo = friction_factor(Re=Re_lo, eD=roughness/D)
dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2)
# Gas-only properties, for calculation of dP_go
v_go = m/rhog/(pi/4*D**2)
Re_go = Reynolds(V=v_go, rho=rhog, mu=mug, D=D)
fd_go = friction_factor(Re=Re_go, eD=roughness/D)
dP_go = fd_go*L/D*(0.5*rhog*v_go**2)
G_MSH = dP_lo + 2*(dP_go - dP_lo)*x
return G_MSH*(1-x)**(1/3.) + dP_go*x**3
[docs]def Lombardi_Pedrocchi(m, x, rhol, rhog, sigma, D, L=1.0):
r'''Calculates two-phase pressure drop with the Lombardi-Pedrocchi (1972)
correlation from [1]_ as shown in [2]_ and [3]_.
.. math::
\Delta P_{tp} = \frac{0.83 G_{tp}^{1.4} \sigma^{0.4} L}{D^{1.2}
\rho_{h}^{0.866}}
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
sigma : float
Surface tension, [N/m]
D : float
Diameter of pipe, [m]
L : float, optional
Length of pipe, [m]
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Notes
-----
This is a purely empirical method. [3]_ presents a review of this and other
correlations. It did not perform best, but there were also correlations
worse than it.
Examples
--------
>>> Lombardi_Pedrocchi(m=0.6, x=0.1, rhol=915., rhog=2.67, sigma=0.045,
... D=0.05, L=1.0)
1567.328374498781
References
----------
.. [1] Lombardi, C., and E. Pedrocchi. "Pressure Drop Correlation in Two-
Phase Flow." Energ. Nucl. (Milan) 19: No. 2, 91-99, January 1, 1972.
.. [2] Mekisso, Henock Mateos. "Comparison of Frictional Pressure Drop
Correlations for Isothermal Two-Phase Horizontal Flow." Thesis, Oklahoma
State University, 2013. https://shareok.org/handle/11244/11109.
.. [3] Turgut, Oğuz Emrah, Mustafa Turhan Çoban, and Mustafa Asker.
"Comparison of Flow Boiling Pressure Drop Correlations for Smooth
Macrotubes." Heat Transfer Engineering 37, no. 6 (April 12, 2016):
487-506. doi:10.1080/01457632.2015.1060733.
'''
voidage_h = homogeneous(x, rhol, rhog)
rho_h = rhol*(1-voidage_h) + rhog*voidage_h
G_tp = m/(pi/4*D**2)
return 0.83*G_tp**1.4*sigma**0.4*L/(D**1.2*rho_h**0.866)
[docs]def Theissing(m, x, rhol, rhog, mul, mug, D, roughness=0.0, L=1.0):
r'''Calculates two-phase pressure drop with the Theissing (1980)
correlation as shown in [2]_ and [3]_.
.. math::
\Delta P_{{tp}} = \left[ {\Delta P_{{lo}}^{{1/{n\epsilon}}} \left({1 -
x} \right)^{{1/\epsilon}} + \Delta P_{{go}}^{{1/
{(n\epsilon)}}} x^{{1/\epsilon}}} \right]^{n\epsilon}
.. math::
\epsilon = 3 - 2\left({\frac{{2\sqrt {{{\rho_{{l}}}/
{\rho_{{g}}}}}}}{{1 + {{\rho_{{l}}}/{\rho_{{g}}}}}}}
\right)^{{{0.7}/n}}
.. math::
n = \frac{{n_1 + n_2 \left({{{\Delta P_{{g}}}/{\Delta
P_{{l}}}}} \right)^{0.1}}}{{1 + \left({{{\Delta P_{{g}}} /
{\Delta P_{{l}}}}} \right)^{0.1}}}
.. math::
n_1 = \frac{{\ln \left({{{\Delta P_{{l}}}/
{\Delta P_{{lo}}}}} \right)}}{{\ln \left({1 - x} \right)}}
.. math::
n_2 = \frac{\ln \left({\Delta P_{{g}} / \Delta P_{{go}}}
\right)}{{\ln x}}
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
D : float
Diameter of pipe, [m]
roughness : float, optional
Roughness of pipe for use in calculating friction factor, [m]
L : float, optional
Length of pipe, [m]
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Notes
-----
Applicable for 0 < x < 1. Notable, as it can be used for two-phase liquid-
liquid flow as well as liquid-gas flow.
Examples
--------
>>> Theissing(m=0.6, x=.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
... D=0.05, roughness=0.0, L=1.0)
497.6156370699538
References
----------
.. [1] Theissing, Peter. "Eine Allgemeingültige Methode Zur Berechnung Des
Reibungsdruckverlustes Der Mehrphasenströmung (A Generally Valid Method
for Calculating Frictional Pressure Drop on Multiphase Flow)." Chemie
Ingenieur Technik 52, no. 4 (January 1, 1980): 344-345.
doi:10.1002/cite.330520414.
.. [2] Mekisso, Henock Mateos. "Comparison of Frictional Pressure Drop
Correlations for Isothermal Two-Phase Horizontal Flow." Thesis, Oklahoma
State University, 2013. https://shareok.org/handle/11244/11109.
.. [3] Greco, A., and G. P. Vanoli. "Experimental Two-Phase Pressure
Gradients during Evaporation of Pure and Mixed Refrigerants in a Smooth
Horizontal Tube. Comparison with Correlations." Heat and Mass Transfer
42, no. 8 (April 6, 2006): 709-725. doi:10.1007/s00231-005-0020-7.
'''
# Liquid-only flow
v_lo = m/rhol/(pi/4*D**2)
Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D)
fd_lo = friction_factor(Re=Re_lo, eD=roughness/D)
dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2)
# Gas-only flow
v_go = m/rhog/(pi/4*D**2)
Re_go = Reynolds(V=v_go, rho=rhog, mu=mug, D=D)
fd_go = friction_factor(Re=Re_go, eD=roughness/D)
dP_go = fd_go*L/D*(0.5*rhog*v_go**2)
# Handle x = 0, x=1:
if x == 0:
return dP_lo
elif x == 1:
return dP_go
# Actual Liquid flow
v_l = m*(1-x)/rhol/(pi/4*D**2)
Re_l = Reynolds(V=v_l, rho=rhol, mu=mul, D=D)
fd_l = friction_factor(Re=Re_l, eD=roughness/D)
dP_l = fd_l*L/D*(0.5*rhol*v_l**2)
# Actual gas flow
v_g = m*x/rhog/(pi/4*D**2)
Re_g = Reynolds(V=v_g, rho=rhog, mu=mug, D=D)
fd_g = friction_factor(Re=Re_g, eD=roughness/D)
dP_g = fd_g*L/D*(0.5*rhog*v_g**2)
# The model
n1 = log(dP_l/dP_lo)/log(1.-x)
n2 = log(dP_g/dP_go)/log(x)
n = (n1 + n2*(dP_g/dP_l)**0.1)/(1 + (dP_g/dP_l)**0.1)
epsilon = 3 - 2*(2*sqrt(rhol/rhog)/(1.+rhol/rhog))**(0.7/n)
dP = (dP_lo**(1./(n*epsilon))*(1-x)**(1./epsilon)
+ dP_go**(1./(n*epsilon))*x**(1./epsilon))**(n*epsilon)
return dP
[docs]def Jung_Radermacher(m, x, rhol, rhog, mul, mug, D, roughness=0.0, L=1.0):
r'''Calculates two-phase pressure drop with the Jung-Radermacher (1989)
correlation, also shown in [2]_ and [3]_.
.. math::
\frac{\Delta P_{tp}}{\Delta P_{lo}} = \phi_{tp}^2
.. math::
\phi_{tp}^2 = 12.82X_{tt}^{-1.47}(1-x)^{1.8}
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
D : float
Diameter of pipe, [m]
roughness : float, optional
Roughness of pipe for use in calculating friction factor, [m]
L : float, optional
Length of pipe, [m]
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Notes
-----
Applicable for 0 < x < 1. Developed for the annular flow regime in
turbulent-turbulent flow.
Examples
--------
>>> Jung_Radermacher(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6,
... mug=14E-6, D=0.05, roughness=0.0, L=1.0)
552.0686123725568
References
----------
.. [1] Jung, D. S., and R. Radermacher. "Prediction of Pressure Drop during
Horizontal Annular Flow Boiling of Pure and Mixed Refrigerants."
International Journal of Heat and Mass Transfer 32, no. 12 (December 1,
1989): 2435-46. doi:10.1016/0017-9310(89)90203-2.
.. [2] Kim, Sung-Min, and Issam Mudawar. "Universal Approach to Predicting
Two-Phase Frictional Pressure Drop for Adiabatic and Condensing Mini/
Micro-Channel Flows." International Journal of Heat and Mass Transfer
55, no. 11-12 (May 2012): 3246-61.
doi:10.1016/j.ijheatmasstransfer.2012.02.047.
.. [3] Filip, Alina, Florin Băltăreţu, and Radu-Mircea Damian. "Comparison
of Two-Phase Pressure Drop Models for Condensing Flows in Horizontal
Tubes." Mathematical Modelling in Civil Engineering 10, no. 4 (2015):
19-27. doi:10.2478/mmce-2014-0019.
'''
v_lo = m/rhol/(pi/4*D**2)
Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D)
fd_lo = friction_factor(Re=Re_lo, eD=roughness/D)
dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2)
Xtt = Lockhart_Martinelli_Xtt(x, rhol, rhog, mul, mug)
phi_tp2 = 12.82*Xtt**-1.47*(1.-x)**1.8
return phi_tp2*dP_lo
[docs]def Tran(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0.0, L=1.0):
r'''Calculates two-phase pressure drop with the Tran (2000) correlation,
also shown in [2]_ and [3]_.
.. math::
\Delta P = dP_{lo} \phi_{lo}^2
.. math::
\phi_{lo}^2 = 1 + (4.3\Gamma^2-1)[\text{Co} \cdot x^{0.875}
(1-x)^{0.875}+x^{1.75}]
.. math::
\Gamma ^2 = \frac{\left(\frac{\Delta P}{L}\right)_{go}}{\left(\frac
{\Delta P}{L}\right)_{lo}}
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
sigma : float
Surface tension, [N/m]
D : float
Diameter of pipe, [m]
roughness : float, optional
Roughness of pipe for use in calculating friction factor, [m]
L : float, optional
Length of pipe, [m]
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Notes
-----
Developed for boiling refrigerants in channels with hydraulic diameters of
2.4 mm to 2.92 mm.
Examples
--------
>>> Tran(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
... sigma=0.0487, D=0.05, roughness=0.0, L=1.0)
423.2563312951232
References
----------
.. [1] Tran, T. N, M. -C Chyu, M. W Wambsganss, and D. M France. "Two-Phase
Pressure Drop of Refrigerants during Flow Boiling in Small Channels: An
Experimental Investigation and Correlation Development." International
Journal of Multiphase Flow 26, no. 11 (November 1, 2000): 1739-54.
doi:10.1016/S0301-9322(99)00119-6.
.. [2] Kim, Sung-Min, and Issam Mudawar. "Universal Approach to Predicting
Two-Phase Frictional Pressure Drop for Adiabatic and Condensing Mini/
Micro-Channel Flows." International Journal of Heat and Mass Transfer
55, no. 11-12 (May 2012): 3246-61.
doi:10.1016/j.ijheatmasstransfer.2012.02.047.
.. [3] Choi, Kwang-Il, A. S. Pamitran, Chun-Young Oh, and Jong-Taek Oh.
"Two-Phase Pressure Drop of R-410A in Horizontal Smooth Minichannels."
International Journal of Refrigeration 31, no. 1 (January 2008): 119-29.
doi:10.1016/j.ijrefrig.2007.06.006.
'''
# Liquid-only properties, for calculation of dP_lo
v_lo = m/rhol/(pi/4*D**2)
Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D)
fd_lo = friction_factor(Re=Re_lo, eD=roughness/D)
dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2)
# Gas-only properties, for calculation of dP_go
v_go = m/rhog/(pi/4*D**2)
Re_go = Reynolds(V=v_go, rho=rhog, mu=mug, D=D)
fd_go = friction_factor(Re=Re_go, eD=roughness/D)
dP_go = fd_go*L/D*(0.5*rhog*v_go**2)
Gamma2 = dP_go/dP_lo
Co = Confinement(D=D, rhol=rhol, rhog=rhog, sigma=sigma)
phi_lo2 = 1 + (4.3*Gamma2 -1)*(Co*x**0.875*(1-x)**0.875 + x**1.75)
return dP_lo*phi_lo2
[docs]def Chen_Friedel(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0.0, L=1.0):
r'''Calculates two-phase pressure drop with the Chen modification of the
Friedel correlation, as given in [1]_ and also shown in [2]_ and [3]_.
.. math::
\Delta P = \Delta P_{Friedel}\Omega
For Bo < 2.5:
.. math::
\Omega = \frac{0.0333Re_{lo}^{0.45}}{Re_g^{0.09}(1 + 0.4\exp(-Bo))}
For Bo >= 2.5:
.. math::
\Omega = \frac{We^{0.2}}{2.5 + 0.06Bo}
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
sigma : float
Surface tension, [N/m]
D : float
Diameter of pipe, [m]
roughness : float, optional
Roughness of pipe for use in calculating friction factor, [m]
L : float, optional
Length of pipe, [m]
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Notes
-----
Applicable ONLY to mini/microchannels; yields drastically too low
pressure drops for larger channels. For more details, see the `Friedel`
correlation.
It is not explicitly stated in [1]_ how to calculate the liquid mixture
density for use in calculation of Weber number; the homogeneous model is
assumed as it is used in the Friedel model.
The bond number used here is 1/4 the normal value, i.e.:
.. math::
Bo = \frac{g(\rho_l-\rho_g)D^2}{4\sigma}
Examples
--------
>>> Chen_Friedel(m=.0005, x=0.9, rhol=950., rhog=1.4, mul=1E-3, mug=1E-5,
... sigma=0.02, D=0.003, roughness=0.0, L=1.0)
6249.247540588871
References
----------
.. [1] Chen, Ing Youn, Kai-Shing Yang, Yu-Juei Chang, and Chi-Chung Wang.
"Two-Phase Pressure Drop of Air-water and R-410A in Small Horizontal
Tubes." International Journal of Multiphase Flow 27, no. 7 (July 2001):
1293-99. doi:10.1016/S0301-9322(01)00004-0.
.. [2] Kim, Sung-Min, and Issam Mudawar. "Universal Approach to Predicting
Two-Phase Frictional Pressure Drop for Adiabatic and Condensing Mini/
Micro-Channel Flows." International Journal of Heat and Mass Transfer
55, no. 11-12 (May 2012): 3246-61.
doi:10.1016/j.ijheatmasstransfer.2012.02.047.
.. [3] Choi, Kwang-Il, A. S. Pamitran, Chun-Young Oh, and Jong-Taek Oh.
"Two-Phase Pressure Drop of R-410A in Horizontal Smooth Minichannels."
International Journal of Refrigeration 31, no. 1 (January 2008): 119-29.
doi:10.1016/j.ijrefrig.2007.06.006.
'''
# Liquid-only properties, for calculation of E, dP_lo
v_lo = m/rhol/(pi/4*D**2)
Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D)
fd_lo = friction_factor(Re=Re_lo, eD=roughness/D)
dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2)
# Gas-only properties, for calculation of E
v_go = m/rhog/(pi/4*D**2)
Re_go = Reynolds(V=v_go, rho=rhog, mu=mug, D=D)
fd_go = friction_factor(Re=Re_go, eD=roughness/D)
F = x**0.78*(1-x)**0.224
H = (rhol/rhog)**0.91*(mug/mul)**0.19*(1 - mug/mul)**0.7
E = (1-x)**2 + x**2*(rhol*fd_go/(rhog*fd_lo))
# Homogeneous properties, for Froude/Weber numbers
rho_h = 1./(x/rhog + (1-x)/rhol)
Q_h = m/rho_h
v_h = Q_h/(pi/4*D**2)
Fr = Froude(V=v_h, L=D, squared=True) # checked with (m/(pi/4*D**2))**2/g/D/rho_h**2
We = Weber(V=v_h, L=D, rho=rho_h, sigma=sigma) # checked with (m/(pi/4*D**2))**2*D/sigma/rho_h
phi_lo2 = E + 3.24*F*H/(Fr**0.0454*We**0.035)
dP = phi_lo2*dP_lo
# Chen modification; Weber number is the same as above
# Weber is same
Bo = Bond(rhol=rhol, rhog=rhog, sigma=sigma, L=D)/4 # Custom definition
if Bo < 2.5:
# Actual gas flow, needed for this case only.
v_g = m*x/rhog/(pi/4*D**2)
Re_g = Reynolds(V=v_g, rho=rhog, mu=mug, D=D)
Omega = 0.0333*Re_lo**0.45/(Re_g**0.09*(1 + 0.5*exp(-Bo)))
else:
Omega = We**0.2/(2.5 + 0.06*Bo)
return dP*Omega
[docs]def Zhang_Webb(m, x, rhol, mul, P, Pc, D, roughness=0.0, L=1.0):
r'''Calculates two-phase pressure drop with the Zhang-Webb (2001)
correlation as shown in [1]_ and also given in [2]_.
.. math::
\phi_{lo}^2 = (1-x)^2 + 2.87x^2\left(\frac{P}{P_c}\right)^{-1}
+ 1.68x^{0.8}(1-x)^{0.25}\left(\frac{P}{P_c}\right)^{-1.64}
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
P : float
Pressure of fluid, [Pa]
Pc : float
Critical pressure of fluid, [Pa]
D : float
Diameter of pipe, [m]
roughness : float, optional
Roughness of pipe for use in calculating friction factor, [m]
L : float, optional
Length of pipe, [m]
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Notes
-----
Applicable for 0 < x < 1. Corresponding-states method developed with
R-134A, R-22 and R-404A in tubes of hydraulic diameters of 2.13 mm,
6.25 mm, and 3.25 mm. For the author's 119 data points, the mean deviation
was 11.5%. Recommended for reduced pressures larger than 0.2 and tubes of
diameter 1-7 mm.
Does not require known properties for the gas phase.
Examples
--------
>>> Zhang_Webb(m=0.6, x=0.1, rhol=915., mul=180E-6, P=2E5, Pc=4055000,
... D=0.05, roughness=0.0, L=1.0)
712.0999804205617
References
----------
.. [1] Zhang, Ming, and Ralph L. Webb. "Correlation of Two-Phase Friction
for Refrigerants in Small-Diameter Tubes." Experimental Thermal and
Fluid Science 25, no. 3-4 (October 2001): 131-39.
doi:10.1016/S0894-1777(01)00066-8.
.. [2] Choi, Kwang-Il, A. S. Pamitran, Chun-Young Oh, and Jong-Taek Oh.
"Two-Phase Pressure Drop of R-410A in Horizontal Smooth Minichannels."
International Journal of Refrigeration 31, no. 1 (January 2008): 119-29.
doi:10.1016/j.ijrefrig.2007.06.006.
'''
# Liquid-only properties, for calculation of dP_lo
v_lo = m/rhol/(pi/4*D**2)
Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D)
fd_lo = friction_factor(Re=Re_lo, eD=roughness/D)
dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2)
Pr = 0.5 if (Pc is None or P is None) else P/Pc
phi_lo2 = (1-x)**2 + 2.87*x**2/Pr + 1.68*x**0.8*sqrt(sqrt(1-x))*Pr**-1.64
return dP_lo*phi_lo2
[docs]def Bankoff(m, x, rhol, rhog, mul, mug, D, roughness=0.0, L=1.0):
r'''Calculates two-phase pressure drop with the Bankoff (1960) correlation,
as shown in [2]_, [3]_, and [4]_.
.. math::
\Delta P_{tp} = \phi_{l}^{7/4} \Delta P_{l}
.. math::
\phi_l = \frac{1}{1-x}\left[1 - \gamma\left(1 - \frac{\rho_g}{\rho_l}
\right)\right]^{3/7}\left[1 + x\left(\frac{\rho_l}{\rho_g} - 1\right)
\right]
.. math::
\gamma = \frac{0.71 + 2.35\left(\frac{\rho_g}{\rho_l}\right)}
{1 + \frac{1-x}{x} \cdot \frac{\rho_g}{\rho_l}}
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
D : float
Diameter of pipe, [m]
roughness : float, optional
Roughness of pipe for use in calculating friction factor, [m]
L : float, optional
Length of pipe, [m]
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Notes
-----
This correlation is not actually shown in [1]_. Its origin is unknown.
The author recommends against using this.
Examples
--------
>>> Bankoff(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
... D=0.05, roughness=0.0, L=1.0)
4746.0594424533965
References
----------
.. [1] Bankoff, S. G. "A Variable Density Single-Fluid Model for Two-Phase
Flow With Particular Reference to Steam-Water Flow." Journal of Heat
Transfer 82, no. 4 (November 1, 1960): 265-72. doi:10.1115/1.3679930.
.. [2] Thome, John R. "Engineering Data Book III." Wolverine Tube Inc
(2004). http://www.wlv.com/heat-transfer-databook/
.. [3] Moreno Quibén, Jesús. "Experimental and Analytical Study of Two-
Phase Pressure Drops during Evaporation in Horizontal Tubes," 2005.
doi:10.5075/epfl-thesis-3337.
.. [4] Mekisso, Henock Mateos. "Comparison of Frictional Pressure Drop
Correlations for Isothermal Two-Phase Horizontal Flow." Thesis, Oklahoma
State University, 2013. https://shareok.org/handle/11244/11109.
'''
# Liquid-only properties, for calculation of dP_lo
v_lo = m/rhol/(pi/4*D**2)
Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D)
fd_lo = friction_factor(Re=Re_lo, eD=roughness/D)
dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2)
gamma = (0.71 + 2.35*rhog/rhol)/(1. + (1.-x)/x*rhog/rhol)
phi_Bf = 1./(1.-x)*(1 - gamma*(1 - rhog/rhol))**(3/7.)*(1. + x*(rhol/rhog -1.))
return dP_lo*phi_Bf**(7/4.)
[docs]def Xu_Fang(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0.0, L=1.0):
r'''Calculates two-phase pressure drop with the Xu and Fang (2013)
correlation. Developed after a comprehensive review of available
correlations, likely meaning it is quite accurate.
.. math::
\Delta P = \Delta P_{lo} \phi_{lo}^2
.. math::
\phi_{lo}^2 = Y^2x^3 + (1-x^{2.59})^{0.632}[1 + 2x^{1.17}(Y^2-1)
+ 0.00775x^{-0.475} Fr_{tp}^{0.535} We_{tp}^{0.188}]
.. math::
Y^2 = \frac{\Delta P_{go}}{\Delta P_{lo}}
.. math::
Fr_{tp} = \frac{G_{tp}^2}{gD\rho_{tp}^2}
.. math::
We_{tp} = \frac{G_{tp}^2 D}{\sigma \rho_{tp}}
.. math::
\frac{1}{\rho_{tp}} = \frac{1-x}{\rho_l} + \frac{x}{\rho_g}
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
sigma : float
Surface tension, [N/m]
D : float
Diameter of pipe, [m]
roughness : float, optional
Roughness of pipe for use in calculating friction factor, [m]
L : float, optional
Length of pipe, [m]
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Notes
-----
Examples
--------
>>> Xu_Fang(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
... sigma=0.0487, D=0.05, roughness=0.0, L=1.0)
604.0595632116267
References
----------
.. [1] Xu, Yu, and Xiande Fang. "A New Correlation of Two-Phase Frictional
Pressure Drop for Condensing Flow in Pipes." Nuclear Engineering and
Design 263 (October 2013): 87-96. doi:10.1016/j.nucengdes.2013.04.017.
'''
A = pi/4*D*D
# Liquid-only properties, for calculation of E, dP_lo
v_lo = m/rhol/A
Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D)
fd_lo = friction_factor(Re=Re_lo, eD=roughness/D)
dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2)
# Gas-only properties, for calculation of E
v_go = m/rhog/A
Re_go = Reynolds(V=v_go, rho=rhog, mu=mug, D=D)
fd_go = friction_factor(Re=Re_go, eD=roughness/D)
dP_go = fd_go*L/D*(0.5*rhog*v_go**2)
# Homogeneous properties, for Froude/Weber numbers
voidage_h = homogeneous(x, rhol, rhog)
rho_h = rhol*(1-voidage_h) + rhog*voidage_h
Q_h = m/rho_h
v_h = Q_h/A
Fr = Froude(V=v_h, L=D, squared=True)
We = Weber(V=v_h, L=D, rho=rho_h, sigma=sigma)
Y2 = dP_go/dP_lo
phi_lo2 = Y2*x**3 + (1-x**2.59)**0.632*(1 + 2*x**1.17*(Y2-1)
+ 0.00775*x**-0.475*Fr**0.535*We**0.188)
return phi_lo2*dP_lo
[docs]def Yu_France(m, x, rhol, rhog, mul, mug, D, roughness=0.0, L=1.0):
r'''Calculates two-phase pressure drop with the Yu, France, Wambsganss,
and Hull (2002) correlation given in [1]_ and reviewed in [2]_ and [3]_.
.. math::
\Delta P = \Delta P_{l} \phi_{l}^2
.. math::
\phi_l^2 = X^{-1.9}
.. math::
X = 18.65\left(\frac{\rho_g}{\rho_l}\right)^{0.5}\left(\frac{1-x}{x}
\right)\frac{Re_{g}^{0.1}}{Re_l^{0.5}}
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
D : float
Diameter of pipe, [m]
roughness : float, optional
Roughness of pipe for use in calculating friction factor, [m]
L : float, optional
Length of pipe, [m]
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Notes
-----
Examples
--------
>>> Yu_France(m=0.6, x=.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
... D=0.05, roughness=0.0, L=1.0)
1146.9833225539571
References
----------
.. [1] Yu, W., D. M. France, M. W. Wambsganss, and J. R. Hull. "Two-Phase
Pressure Drop, Boiling Heat Transfer, and Critical Heat Flux to Water in
a Small-Diameter Horizontal Tube." International Journal of Multiphase
Flow 28, no. 6 (June 2002): 927-41. doi:10.1016/S0301-9322(02)00019-8.
.. [2] Kim, Sung-Min, and Issam Mudawar. "Universal Approach to Predicting
Two-Phase Frictional Pressure Drop for Adiabatic and Condensing Mini/
Micro-Channel Flows." International Journal of Heat and Mass Transfer
55, no. 11-12 (May 2012): 3246-61.
doi:10.1016/j.ijheatmasstransfer.2012.02.047.
.. [3] Xu, Yu, Xiande Fang, Xianghui Su, Zhanru Zhou, and Weiwei Chen.
"Evaluation of Frictional Pressure Drop Correlations for Two-Phase Flow
in Pipes." Nuclear Engineering and Design, SI : CFD4NRS-3, 253 (December
2012): 86-97. doi:10.1016/j.nucengdes.2012.08.007.
'''
# Actual Liquid flow
v_l = m*(1-x)/rhol/(pi/4*D**2)
Re_l = Reynolds(V=v_l, rho=rhol, mu=mul, D=D)
fd_l = friction_factor(Re=Re_l, eD=roughness/D)
dP_l = fd_l*L/D*(0.5*rhol*v_l**2)
# Actual gas flow
v_g = m*x/rhog/(pi/4*D**2)
Re_g = Reynolds(V=v_g, rho=rhog, mu=mug, D=D)
X = 18.65*sqrt(rhog/rhol)*(1-x)/x*Re_g**0.1/sqrt(Re_l)
phi_l2 = X**-1.9
return phi_l2*dP_l
[docs]def Wang_Chiang_Lu(m, x, rhol, rhog, mul, mug, D, roughness=0.0, L=1.0):
r'''Calculates two-phase pressure drop with the Wang, Chiang, and Lu (1997)
correlation given in [1]_ and reviewed in [2]_ and [3]_.
.. math::
\Delta P = \Delta P_{g} \phi_g^2
.. math::
\phi_g^2 = 1 + 9.397X^{0.62} + 0.564X^{2.45} \text{ for } G >= 200 kg/m^2/s
.. math::
\phi_g^2 = 1 + CX + X^2 \text{ for lower mass fluxes}
.. math::
C = 0.000004566X^{0.128}Re_{lo}^{0.938}\left(\frac{\rho_l}{\rho_g}
\right)^{-2.15}\left(\frac{\mu_l}{\mu_g}\right)^{5.1}
.. math::
X^2 = \frac{\Delta P_l}{\Delta P_g}
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
D : float
Diameter of pipe, [m]
roughness : float, optional
Roughness of pipe for use in calculating friction factor, [m]
L : float, optional
Length of pipe, [m]
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Notes
-----
Examples
--------
>>> Wang_Chiang_Lu(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6,
... mug=14E-6, D=0.05, roughness=0.0, L=1.0)
448.29981978639137
References
----------
.. [1] Wang, Chi-Chuan, Ching-Shan Chiang, and Ding-Chong Lu. "Visual
Observation of Two-Phase Flow Pattern of R-22, R-134a, and R-407C in a
6.5-Mm Smooth Tube." Experimental Thermal and Fluid Science 15, no. 4
(November 1, 1997): 395-405. doi:10.1016/S0894-1777(97)00007-1.
.. [2] Kim, Sung-Min, and Issam Mudawar. "Universal Approach to Predicting
Two-Phase Frictional Pressure Drop for Adiabatic and Condensing Mini/
Micro-Channel Flows." International Journal of Heat and Mass Transfer
55, no. 11-12 (May 2012): 3246-61.
doi:10.1016/j.ijheatmasstransfer.2012.02.047.
.. [3] Xu, Yu, Xiande Fang, Xianghui Su, Zhanru Zhou, and Weiwei Chen.
"Evaluation of Frictional Pressure Drop Correlations for Two-Phase Flow
in Pipes." Nuclear Engineering and Design, SI : CFD4NRS-3, 253 (December
2012): 86-97. doi:10.1016/j.nucengdes.2012.08.007.
'''
G_tp = m/(pi/4*D**2)
# Actual Liquid flow
v_l = m*(1-x)/rhol/(pi/4*D**2)
Re_l = Reynolds(V=v_l, rho=rhol, mu=mul, D=D)
fd_l = friction_factor(Re=Re_l, eD=roughness/D)
dP_l = fd_l*L/D*(0.5*rhol*v_l**2)
# Actual gas flow
v_g = m*x/rhog/(pi/4*D**2)
Re_g = Reynolds(V=v_g, rho=rhog, mu=mug, D=D)
fd_g = friction_factor(Re=Re_g, eD=roughness/D)
dP_g = fd_g*L/D*(0.5*rhog*v_g**2)
X = sqrt(dP_l/dP_g)
if G_tp >= 200:
phi_g2 = 1 + 9.397*X**0.62 + 0.564*X**2.45
else:
# Liquid-only flow; Re_lo is oddly needed
v_lo = m/rhol/(pi/4*D**2)
Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D)
C = 0.000004566*X**0.128*Re_lo**0.938*(rhol/rhog)**-2.15*(mul/mug)**5.1
phi_g2 = 1 + C*X + X**2
return dP_g*phi_g2
[docs]def Hwang_Kim(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0.0, L=1.0):
r'''Calculates two-phase pressure drop with the Hwang and Kim (2006)
correlation as in [1]_, also presented in [2]_ and [3]_.
.. math::
\Delta P = \Delta P_{l} \phi_{l}^2
.. math::
C = 0.227 Re_{lo}^{0.452} X^{-0.32} Co^{-0.82}
.. math::
\phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2}
.. math::
X^2 = \frac{\Delta P_l}{\Delta P_g}
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
sigma : float
Surface tension, [N/m]
D : float
Diameter of pipe, [m]
roughness : float, optional
Roughness of pipe for use in calculating friction factor, [m]
L : float, optional
Length of pipe, [m]
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Notes
-----
Developed with data for microtubes of diameter 0.244 mm and 0.792 mm only.
Not likely to be suitable to larger diameters.
Examples
--------
>>> Hwang_Kim(m=0.0005, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
... sigma=0.0487, D=0.003, roughness=0.0, L=1.0)
798.302774184557
References
----------
.. [1] Hwang, Yun Wook, and Min Soo Kim. "The Pressure Drop in Microtubes
and the Correlation Development." International Journal of Heat and
Mass Transfer 49, no. 11-12 (June 2006): 1804-12.
doi:10.1016/j.ijheatmasstransfer.2005.10.040.
.. [2] Kim, Sung-Min, and Issam Mudawar. "Universal Approach to Predicting
Two-Phase Frictional Pressure Drop for Adiabatic and Condensing Mini/
Micro-Channel Flows." International Journal of Heat and Mass Transfer
55, no. 11-12 (May 2012): 3246-61.
doi:10.1016/j.ijheatmasstransfer.2012.02.047.
.. [3] Xu, Yu, Xiande Fang, Xianghui Su, Zhanru Zhou, and Weiwei Chen.
"Evaluation of Frictional Pressure Drop Correlations for Two-Phase Flow
in Pipes." Nuclear Engineering and Design, SI : CFD4NRS-3, 253 (December
2012): 86-97. doi:10.1016/j.nucengdes.2012.08.007.
'''
# Liquid-only flow
v_lo = m/rhol/(pi/4*D**2)
Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D)
# Actual Liquid flow
v_l = m*(1-x)/rhol/(pi/4*D**2)
Re_l = Reynolds(V=v_l, rho=rhol, mu=mul, D=D)
fd_l = friction_factor(Re=Re_l, eD=roughness/D)
dP_l = fd_l*L/D*(0.5*rhol*v_l**2)
# Actual gas flow
v_g = m*x/rhog/(pi/4*D**2)
Re_g = Reynolds(V=v_g, rho=rhog, mu=mug, D=D)
fd_g = friction_factor(Re=Re_g, eD=roughness/D)
dP_g = fd_g*L/D*(0.5*rhog*v_g**2)
# Actual model
X = sqrt(dP_l/dP_g)
Co = Confinement(D=D, rhol=rhol, rhog=rhog, sigma=sigma)
C = 0.227*Re_lo**0.452*X**-0.320*Co**-0.820
phi_l2 = 1 + C/X + 1./X**2
return dP_l*phi_l2
[docs]def Zhang_Hibiki_Mishima(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0.0,
L=1.0, flowtype='adiabatic vapor'):
r'''Calculates two-phase pressure drop with the Zhang, Hibiki, Mishima and
(2010) correlation as in [1]_, also presented in [2]_ and [3]_.
.. math::
\Delta P = \Delta P_{l} \phi_{l}^2
.. math::
\phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2}
.. math::
X^2 = \frac{\Delta P_l}{\Delta P_g}
For adiabatic liquid-vapor two-phase flow:
.. math::
C = 21[1 - \exp(-0.142/Co)]
For adiabatic liquid-gas two-phase flow:
.. math::
C = 21[1 - \exp(-0.674/Co)]
For flow boiling:
.. math::
C = 21[1 - \exp(-0.358/Co)]
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
sigma : float
Surface tension, [N/m]
D : float
Diameter of pipe, [m]
roughness : float, optional
Roughness of pipe for use in calculating friction factor, [m]
L : float, optional
Length of pipe, [m]
flowtype : str
One of 'adiabatic vapor', 'adiabatic gas', or 'flow boiling'
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Notes
-----
Seems fairly reliable.
Examples
--------
>>> Zhang_Hibiki_Mishima(m=0.0005, x=0.1, rhol=915., rhog=2.67, mul=180E-6,
... mug=14E-6, sigma=0.0487, D=0.003, roughness=0.0, L=1.0)
444.9718476894804
References
----------
.. [1] Zhang, W., T. Hibiki, and K. Mishima. "Correlations of Two-Phase
Frictional Pressure Drop and Void Fraction in Mini-Channel."
International Journal of Heat and Mass Transfer 53, no. 1-3 (January 15,
2010): 453-65. doi:10.1016/j.ijheatmasstransfer.2009.09.011.
.. [2] Kim, Sung-Min, and Issam Mudawar. "Universal Approach to Predicting
Two-Phase Frictional Pressure Drop for Adiabatic and Condensing Mini/
Micro-Channel Flows." International Journal of Heat and Mass Transfer
55, no. 11-12 (May 2012): 3246-61.
doi:10.1016/j.ijheatmasstransfer.2012.02.047.
.. [3] Xu, Yu, Xiande Fang, Xianghui Su, Zhanru Zhou, and Weiwei Chen.
"Evaluation of Frictional Pressure Drop Correlations for Two-Phase Flow
in Pipes." Nuclear Engineering and Design, SI : CFD4NRS-3, 253 (December
2012): 86-97. doi:10.1016/j.nucengdes.2012.08.007.
'''
# Actual Liquid flow
v_l = m*(1-x)/rhol/(pi/4*D**2)
Re_l = Reynolds(V=v_l, rho=rhol, mu=mul, D=D)
fd_l = friction_factor(Re=Re_l, eD=roughness/D)
dP_l = fd_l*L/D*(0.5*rhol*v_l**2)
# Actual gas flow
v_g = m*x/rhog/(pi/4*D**2)
Re_g = Reynolds(V=v_g, rho=rhog, mu=mug, D=D)
fd_g = friction_factor(Re=Re_g, eD=roughness/D)
dP_g = fd_g*L/D*(0.5*rhog*v_g**2)
# Actual model
X = sqrt(dP_l/dP_g)
Co = Confinement(D=D, rhol=rhol, rhog=rhog, sigma=sigma)
if flowtype == 'adiabatic vapor':
C = 21*(1 - exp(-0.142/Co))
elif flowtype == 'adiabatic gas':
C = 21*(1 - exp(-0.674/Co))
elif flowtype == 'flow boiling':
C = 21*(1 - exp(-0.358/Co))
else:
raise ValueError("Only flow types 'adiabatic vapor', 'adiabatic gas, \
and 'flow boiling' are recognized.")
phi_l2 = 1 + C/X + 1./X**2
return dP_l*phi_l2
[docs]def Mishima_Hibiki(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0.0, L=1.0):
r'''Calculates two-phase pressure drop with the Mishima and Hibiki (1996)
correlation as in [1]_, also presented in [2]_ and [3]_.
.. math::
\Delta P = \Delta P_{l} \phi_{l}^2
.. math::
C = 21[1 - \exp(-319D)]
.. math::
\phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2}
.. math::
X^2 = \frac{\Delta P_l}{\Delta P_g}
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
sigma : float
Surface tension, [N/m]
D : float
Diameter of pipe, [m]
roughness : float, optional
Roughness of pipe for use in calculating friction factor, [m]
L : float, optional
Length of pipe, [m]
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Notes
-----
Examples
--------
>>> Mishima_Hibiki(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6,
... mug=14E-6, sigma=0.0487, D=0.05, roughness=0.0, L=1.0)
732.4268200606265
References
----------
.. [1] Mishima, K., and T. Hibiki. "Some Characteristics of Air-Water Two-
Phase Flow in Small Diameter Vertical Tubes." International Journal of
Multiphase Flow 22, no. 4 (August 1, 1996): 703-12.
doi:10.1016/0301-9322(96)00010-9.
.. [2] Kim, Sung-Min, and Issam Mudawar. "Universal Approach to Predicting
Two-Phase Frictional Pressure Drop for Adiabatic and Condensing Mini/
Micro-Channel Flows." International Journal of Heat and Mass Transfer
55, no. 11-12 (May 2012): 3246-61.
doi:10.1016/j.ijheatmasstransfer.2012.02.047.
.. [3] Xu, Yu, Xiande Fang, Xianghui Su, Zhanru Zhou, and Weiwei Chen.
"Evaluation of Frictional Pressure Drop Correlations for Two-Phase Flow
in Pipes." Nuclear Engineering and Design, SI : CFD4NRS-3, 253 (December
2012): 86-97. doi:10.1016/j.nucengdes.2012.08.007.
'''
# Actual Liquid flow
v_l = m*(1-x)/rhol/(pi/4*D**2)
Re_l = Reynolds(V=v_l, rho=rhol, mu=mul, D=D)
fd_l = friction_factor(Re=Re_l, eD=roughness/D)
dP_l = fd_l*L/D*(0.5*rhol*v_l**2)
# Actual gas flow
v_g = m*x/rhog/(pi/4*D**2)
Re_g = Reynolds(V=v_g, rho=rhog, mu=mug, D=D)
fd_g = friction_factor(Re=Re_g, eD=roughness/D)
dP_g = fd_g*L/D*(0.5*rhog*v_g**2)
# Actual model
X = sqrt(dP_l/dP_g)
C = 21*(1 - exp(-0.319E3*D))
phi_l2 = 1 + C/X + 1./X**2
return dP_l*phi_l2
def friction_factor_Kim_Mudawar(Re):
if Re < 2000:
return 64./Re
elif Re < 20000:
return 0.316/sqrt(sqrt(Re))
else:
return 0.184*Re**-0.2
[docs]def Kim_Mudawar(m, x, rhol, rhog, mul, mug, sigma, D, L=1.0):
r'''Calculates two-phase pressure drop with the Kim and Mudawar (2012)
correlation as in [1]_, also presented in [2]_.
.. math::
\Delta P = \Delta P_{l} \phi_{l}^2
.. math::
\phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2}
.. math::
X^2 = \frac{\Delta P_l}{\Delta P_g}
For turbulent liquid, turbulent gas:
.. math::
C = 0.39Re_{lo}^{0.03} Su_{go}^{0.10}\left(\frac{\rho_l}{\rho_g}
\right)^{0.35}
For turbulent liquid, laminar gas:
.. math::
C = 8.7\times 10^{-4} Re_{lo}^{0.17} Su_{go}^{0.50}\left(\frac{\rho_l}
{\rho_g}\right)^{0.14}
For laminar liquid, turbulent gas:
.. math::
C = 0.0015 Re_{lo}^{0.59} Su_{go}^{0.19}\left(\frac{\rho_l}{\rho_g}
\right)^{0.36}
For laminar liquid, laminar gas:
.. math::
C = 3.5\times 10^{-5} Re_{lo}^{0.44} Su_{go}^{0.50}\left(\frac{\rho_l}
{\rho_g}\right)^{0.48}
This model has its own friction factor calculations, to be consistent with
its Reynolds number transition. As their model was regressed with these
equations, more error is obtained when using any other friction factor
calculation. The laminar equation 64/Re is used up to Re=2000, then the
Blasius equation with a coefficient of 0.316, and above Re = 20000,
.. math::
f_d = \frac{0.184}{Re^{0.2}}
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
sigma : float
Surface tension, [N/m]
D : float
Diameter of pipe, [m]
L : float, optional
Length of pipe, [m]
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Notes
-----
The critical Reynolds number in this model is 2000, with a Reynolds number
definition using actual liquid and gas flows. This model also requires
liquid-only Reynolds number to be calculated.
No attempt to incorporate roughness into the model was made in [1]_.
The model was developed with hydraulic diameter from 0.0695 to 6.22 mm,
mass velocities 4 to 8528 kg/m^2/s, flow qualities from 0 to 1, reduced
pressures from 0.0052 to 0.91, superficial liquid Reynolds numbers up to
79202, superficial gas Reynolds numbers up to 253810, liquid-only Reynolds
numbers up to 89798, 7115 data points from 36 sources and working fluids
air, CO2, N2, water, ethanol, R12, R22, R134a, R236ea, R245fa, R404A, R407C,
propane, methane, and ammonia.
Examples
--------
>>> Kim_Mudawar(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
... sigma=0.0487, D=0.05, L=1.0)
840.4137796786074
References
----------
.. [1] Kim, Sung-Min, and Issam Mudawar. "Universal Approach to Predicting
Two-Phase Frictional Pressure Drop for Adiabatic and Condensing Mini/
Micro-Channel Flows." International Journal of Heat and Mass Transfer
55, no. 11-12 (May 2012): 3246-61.
doi:10.1016/j.ijheatmasstransfer.2012.02.047.
.. [2] Kim, Sung-Min, and Issam Mudawar. "Review of Databases and
Predictive Methods for Pressure Drop in Adiabatic, Condensing and
Boiling Mini/Micro-Channel Flows." International Journal of Heat and
Mass Transfer 77 (October 2014): 74-97.
doi:10.1016/j.ijheatmasstransfer.2014.04.035.
'''
# Actual Liquid flow
v_l = m*(1-x)/rhol/(pi/4*D**2)
Re_l = Reynolds(V=v_l, rho=rhol, mu=mul, D=D)
fd_l = friction_factor_Kim_Mudawar(Re=Re_l)
dP_l = fd_l*L/D*(0.5*rhol*v_l**2)
# Actual gas flow
v_g = m*x/rhog/(pi/4*D**2)
Re_g = Reynolds(V=v_g, rho=rhog, mu=mug, D=D)
fd_g = friction_factor_Kim_Mudawar(Re=Re_g)
dP_g = fd_g*L/D*(0.5*rhog*v_g**2)
# Liquid-only flow
v_lo = m/rhol/(pi/4*D**2)
Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D)
Su = Suratman(L=D, rho=rhog, mu=mug, sigma=sigma)
X = sqrt(dP_l/dP_g)
Re_c = 2000 # Transition Reynolds number
if Re_l < Re_c and Re_g < Re_c:
C = 3.5E-5*Re_lo**0.44*sqrt(Su)*(rhol/rhog)**0.48
elif Re_l < Re_c and Re_g >= Re_c:
C = 0.0015*Re_lo**0.59*Su**0.19*(rhol/rhog)**0.36
elif Re_l >= Re_c and Re_g < Re_c:
C = 8.7E-4*Re_lo**0.17*sqrt(Su)*(rhol/rhog)**0.14
else: # Turbulent case
C = 0.39*Re_lo**0.03*Su**0.10*(rhol/rhog)**0.35
phi_l2 = 1 + C/X + 1./X**2
return dP_l*phi_l2
[docs]def Lockhart_Martinelli(m, x, rhol, rhog, mul, mug, D, L=1.0, Re_c=2000.0):
r'''Calculates two-phase pressure drop with the Lockhart and Martinelli
(1949) correlation as presented in non-graphical form by Chisholm (1967).
.. math::
\Delta P = \Delta P_{l} \phi_{l}^2
.. math::
\phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2}
.. math::
X^2 = \frac{\Delta P_l}{\Delta P_g}
+---------+---------+--+
|Liquid |Gas |C |
+=========+=========+==+
|Turbulent|Turbulent|20|
+---------+---------+--+
|Laminar |Turbulent|12|
+---------+---------+--+
|Turbulent|Laminar |10|
+---------+---------+--+
|Laminar |Laminar |5 |
+---------+---------+--+
This model has its own friction factor calculations, to be consistent with
its Reynolds number transition and the procedure specified in the original
work. The equation 64/Re is used up to Re_c, and above it the Blasius
equation is used as follows:
.. math::
f_d = \frac{0.184}{Re^{0.2}}
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
D : float
Diameter of pipe, [m]
L : float, optional
Length of pipe, [m]
Re_c : float, optional
Transition Reynolds number, used to decide which friction factor
equation to use and which C value to use from the table above.
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Notes
-----
Developed for horizontal flow. Very popular. Many implementations of this
model assume turbulent-turbulent flow.
The original model proposed that the transition Reynolds number was 1000
for laminar flow, and 2000 for turbulent flow; it proposed no model
for Re_l < 1000 and Re_g between 1000 and 2000 and also Re_g < 1000 and
Re_l between 1000 and 2000.
No correction is available in this model for rough pipe.
[3]_ examined the original data in [1]_ again, and fit more curves to the
data, separating them into different flow regimes. There were 229 datum
in the turbulent-turbulent regime, 9 in the turbulent-laminar regime, 339
in the laminar-turbulent regime, and 42 in the laminar-laminar regime.
Errors from [3]_'s curves were 13.4%, 3.5%, 14.3%, and 12.0% for the above
regimes, respectively. [2]_'s fits provide further error.
Examples
--------
>>> Lockhart_Martinelli(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6,
... mug=14E-6, D=0.05, L=1.0)
716.4695654888484
References
----------
.. [1] Lockhart, R. W. & Martinelli, R. C. (1949), "Proposed correlation of
data for isothermal two-phase, two-component flow in pipes", Chemical
Engineering Progress 45 (1), 39-48.
.. [2] Chisholm, D."A Theoretical Basis for the Lockhart-Martinelli
Correlation for Two-Phase Flow." International Journal of Heat and Mass
Transfer 10, no. 12 (December 1967): 1767-78.
doi:10.1016/0017-9310(67)90047-6.
.. [3] Cui, Xiaozhou, and John J. J. Chen."A Re-Examination of the Data of
Lockhart-Martinelli." International Journal of Multiphase Flow 36, no.
10 (October 2010): 836-46. doi:10.1016/j.ijmultiphaseflow.2010.06.001.
.. [4] Kim, Sung-Min, and Issam Mudawar. "Universal Approach to Predicting
Two-Phase Frictional Pressure Drop for Adiabatic and Condensing Mini/
Micro-Channel Flows." International Journal of Heat and Mass Transfer
55, no. 11-12 (May 2012): 3246-61.
doi:10.1016/j.ijheatmasstransfer.2012.02.047.
'''
v_l = m*(1-x)/rhol/(pi/4*D**2)
Re_l = Reynolds(V=v_l, rho=rhol, mu=mul, D=D)
v_g = m*x/rhog/(pi/4*D**2)
Re_g = Reynolds(V=v_g, rho=rhog, mu=mug, D=D)
if Re_l < Re_c and Re_g < Re_c:
C = 5.0
elif Re_l < Re_c and Re_g >= Re_c:
# Liquid laminar, gas turbulent
C = 12.0
elif Re_l >= Re_c and Re_g < Re_c:
# Liquid turbulent, gas laminar
C = 10.0
else: # Turbulent case
C = 20.0
# Frictoin factor as in the original model
fd_l = 64./Re_l if Re_l < Re_c else 0.184*Re_l**-0.2
dP_l = fd_l*L/D*(0.5*rhol*v_l**2)
fd_g = 64./Re_g if Re_g < Re_c else 0.184*Re_g**-0.2
dP_g = fd_g*L/D*(0.5*rhog*v_g**2)
X = sqrt(dP_l/dP_g)
phi_l2 = 1 + C/X + 1./X**2
return dP_l*phi_l2
two_phase_correlations = {
# 0 index, args are: m, x, rhol, mul, P, Pc, D, roughness=0.0, L=1
'Zhang_Webb': (Zhang_Webb, 0),
# 1 index, args are: m, x, rhol, rhog, mul, mug, D, L=1
'Lockhart_Martinelli': (Lockhart_Martinelli, 1),
# 2 index, args are: m, x, rhol, rhog, mul, mug, D, roughness=0.0, L=1
'Bankoff': (Bankoff, 2),
'Baroczy_Chisholm': (Baroczy_Chisholm, 2),
'Chisholm': (Chisholm, 2),
'Gronnerud': (Gronnerud, 2),
'Jung_Radermacher': (Jung_Radermacher, 2),
'Muller_Steinhagen_Heck': (Muller_Steinhagen_Heck, 2),
'Theissing': (Theissing, 2),
'Wang_Chiang_Lu': (Wang_Chiang_Lu, 2),
'Yu_France': (Yu_France, 2),
# 3 index, args are: m, x, rhol, rhog, mul, mug, sigma, D, L=1
'Kim_Mudawar': (Kim_Mudawar, 3),
# 4 index, args are: m, x, rhol, rhog, mul, mug, sigma, D, roughness=0.0, L=1
'Friedel': (Friedel, 4),
'Hwang_Kim': (Hwang_Kim, 4),
'Mishima_Hibiki': (Mishima_Hibiki, 4),
'Tran': (Tran, 4),
'Xu_Fang': (Xu_Fang, 4),
'Zhang_Hibiki_Mishima': (Zhang_Hibiki_Mishima, 4),
'Chen_Friedel': (Chen_Friedel, 4),
# 5 index: args are m, x, rhol, rhog, sigma, D, L=1
'Lombardi_Pedrocchi': (Lombardi_Pedrocchi, 5),
# Misc indexes:
'Chisholm rough': (Chisholm, 101),
'Zhang_Hibiki_Mishima adiabatic gas': (Zhang_Hibiki_Mishima, 102),
'Zhang_Hibiki_Mishima flow boiling': (Zhang_Hibiki_Mishima, 103),
'Beggs-Brill': (Beggs_Brill, 104)
}
_unknown_msg_two_phase = "Unknown method; available methods are %s" %(list(two_phase_correlations.keys()))
[docs]def two_phase_dP_methods(m, x, rhol, D, L=1.0, rhog=None, mul=None, mug=None,
sigma=None, P=None, Pc=None, roughness=0.0, angle=0,
check_ranges=False):
r'''This function returns a list of names of correlations for two-phase
liquid-gas pressure drop for flow inside channels.
24 calculation methods are available, with varying input requirements.
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
D : float
Diameter of pipe, [m]
L : float, optional
Length of pipe, [m]
rhog : float, optional
Gas density, [kg/m^3]
mul : float, optional
Viscosity of liquid, [Pa*s]
mug : float, optional
Viscosity of gas, [Pa*s]
sigma : float, optional
Surface tension, [N/m]
P : float, optional
Pressure of fluid, [Pa]
Pc : float, optional
Critical pressure of fluid, [Pa]
roughness : float, optional
Roughness of pipe for use in calculating friction factor, [m]
angle : float, optional
The angle of the pipe with respect to the horizontal, [degrees]
check_ranges : bool, optional
Added for Future use only
Returns
-------
methods : list
List of methods which can be used to calculate two-phase pressure drop
with the given inputs.
Examples
--------
>>> len(two_phase_dP_methods(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6, sigma=0.0487, D=0.05, L=1.0, angle=30.0, roughness=1e-4, P=1e5, Pc=1e6))
24
'''
usable_indices = []
if rhog is not None and sigma is not None:
usable_indices.append(5)
if rhog is not None and sigma is not None and mul is not None and mug is not None:
usable_indices.extend([4, 3, 102, 103]) # Differs only in the addition of roughness
if rhog is not None and mul is not None and mug is not None:
usable_indices.extend([1,2, 101]) # Differs only in the addition of roughness
if mul is not None and P is not None and Pc is not None:
usable_indices.append(0)
if (rhog is not None and mul is not None and mug is not None
and sigma is not None and P is not None and angle is not None):
usable_indices.append(104)
return [key for key, value in two_phase_correlations.items() if value[1] in usable_indices]
[docs]def two_phase_dP(m, x, rhol, D, L=1.0, rhog=None, mul=None, mug=None, sigma=None,
P=None, Pc=None, roughness=0.0, angle=None, Method=None):
r'''This function handles calculation of two-phase liquid-gas pressure drop
for flow inside channels. 23 calculation methods are available, with
varying input requirements. A correlation will be automatically selected if
none is specified. The full list of correlation can be obtained with the
`AvailableMethods` flag.
If no correlation is selected, the following rules are used, with the
earlier options attempted first:
* If rhog, mul, mug, and sigma are specified, use the Kim_Mudawar model
* If rhog, mul, and mug are specified, use the Chisholm model
* If mul, P, and Pc are specified, use the Zhang_Webb model
* If rhog and sigma are specified, use the Lombardi_Pedrocchi model
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
D : float
Diameter of pipe, [m]
L : float, optional
Length of pipe, [m]
rhog : float, optional
Gas density, [kg/m^3]
mul : float, optional
Viscosity of liquid, [Pa*s]
mug : float, optional
Viscosity of gas, [Pa*s]
sigma : float, optional
Surface tension, [N/m]
P : float, optional
Pressure of fluid, [Pa]
Pc : float, optional
Critical pressure of fluid, [Pa]
roughness : float, optional
Roughness of pipe for use in calculating friction factor, [m]
angle : float, optional
The angle of the pipe with respect to the horizontal, [degrees]
Returns
-------
dP : float
Pressure drop of the two-phase flow, [Pa]
Other Parameters
----------------
Method : string, optional
A string of the function name to use, as in the dictionary
two_phase_correlations.
Notes
-----
These functions may be integrated over, with properties recalculated as
the fluid's quality changes.
This model considers only the frictional pressure drop, not that due to
gravity or acceleration.
Examples
--------
>>> two_phase_dP(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
... sigma=0.0487, D=0.05, L=1.0)
840.4137796786074
'''
if Method is None:
if rhog is not None and mul is not None and mug is not None and sigma is not None:
Method2 = 'Kim_Mudawar' # Kim_Mudawar preferred
elif rhog is not None and mul is not None and mug is not None:
Method2 = 'Chisholm' # Second choice, indexes 1 or 2
elif mul is not None and P is not None and Pc is not None:
Method2 = 'Zhang_Webb' # Not a good choice
elif rhog is not None and sigma is not None:
Method2 = 'Lombardi_Pedrocchi' # Last try
else:
raise ValueError('All possible methods require more information \
than provided; provide more inputs!')
else:
Method2 = Method
if Method2 == "Zhang_Webb":
return Zhang_Webb(m=m, x=x, rhol=rhol, mul=mul, P=P, Pc=Pc, D=D, roughness=roughness, L=L)
elif Method2 == "Lockhart_Martinelli":
return Lockhart_Martinelli(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, L=L)
elif Method2 == "Bankoff":
return Bankoff(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, roughness=roughness, L=L)
elif Method2 == "Baroczy_Chisholm":
return Baroczy_Chisholm(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, roughness=roughness, L=L)
elif Method2 == "Chisholm":
return Chisholm(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, roughness=roughness, L=L)
elif Method2 == "Gronnerud":
return Gronnerud(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, roughness=roughness, L=L)
elif Method2 == "Jung_Radermacher":
return Jung_Radermacher(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, roughness=roughness, L=L)
elif Method2 == "Muller_Steinhagen_Heck":
return Muller_Steinhagen_Heck(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, roughness=roughness, L=L)
elif Method2 == "Theissing":
return Theissing(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, roughness=roughness, L=L)
elif Method2 == "Wang_Chiang_Lu":
return Wang_Chiang_Lu(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, roughness=roughness, L=L)
elif Method2 == "Yu_France":
return Yu_France(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, roughness=roughness, L=L)
elif Method2 == "Kim_Mudawar":
return Kim_Mudawar(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, sigma=sigma, D=D, L=L)
elif Method2 == "Friedel":
return Friedel(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, sigma=sigma, D=D, roughness=roughness, L=L)
elif Method2 == "Hwang_Kim":
return Hwang_Kim(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, sigma=sigma, D=D, roughness=roughness, L=L)
elif Method2 == "Mishima_Hibiki":
return Mishima_Hibiki(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, sigma=sigma, D=D, roughness=roughness, L=L)
elif Method2 == "Tran":
return Tran(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, sigma=sigma, D=D, roughness=roughness, L=L)
elif Method2 == "Xu_Fang":
return Xu_Fang(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, sigma=sigma, D=D, roughness=roughness, L=L)
elif Method2 == "Zhang_Hibiki_Mishima":
return Zhang_Hibiki_Mishima(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, sigma=sigma, D=D, roughness=roughness, L=L)
elif Method2 == "Chen_Friedel":
return Chen_Friedel(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, sigma=sigma, D=D, roughness=roughness, L=L)
elif Method2 == "Lombardi_Pedrocchi":
return Lombardi_Pedrocchi(m=m, x=x, rhol=rhol, rhog=rhog, sigma=sigma, D=D, L=L)
elif Method2 == "Chisholm rough":
return Chisholm(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D,
L=L, roughness=roughness, rough_correction=True)
elif Method2 == "Zhang_Hibiki_Mishima adiabatic gas":
return Zhang_Hibiki_Mishima(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug,
sigma=sigma, D=D, L=L, roughness=roughness,
flowtype='adiabatic gas')
elif Method2 == "Zhang_Hibiki_Mishima flow boiling":
return Zhang_Hibiki_Mishima(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug,
sigma=sigma, D=D, L=L, roughness=roughness,
flowtype='flow boiling')
elif Method2 == "Beggs-Brill":
return Beggs_Brill(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug,
sigma=sigma, P=P, D=D, angle=angle, L=L,
roughness=roughness, acceleration=False, g=g)
else:
raise ValueError(_unknown_msg_two_phase)
[docs]def two_phase_dP_acceleration(m, D, xi, xo, alpha_i, alpha_o, rho_li, rho_gi,
rho_lo=None, rho_go=None):
r'''This function handles calculation of two-phase liquid-gas pressure drop
due to acceleration for flow inside channels. This is a discrete
calculation for a segment with a known difference in quality (and ideally
known inlet and outlet pressures so density dependence can be included).
.. math::
\Delta P_{acc} = G^2\left\{\left[\frac{(1-x_o)^2}{\rho_{l,o}
(1-\alpha_o)} + \frac{x_o^2}{\rho_{g,o}\alpha_o} \right]
- \left[\frac{(1-x_i)^2}{\rho_{l,i}(1-\alpha_i)}
+ \frac{x_i^2}{\rho_{g,i}\alpha_i} \right]\right\}
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
D : float
Diameter of pipe, [m]
xi : float
Quality of fluid at inlet, [-]
xo : float
Quality of fluid at outlet, [-]
alpha_i : float
Void fraction at inlet (area of gas / total area of channel), [-]
alpha_o : float
Void fraction at outlet (area of gas / total area of channel), [-]
rho_li : float
Liquid phase density at inlet, [kg/m^3]
rho_gi : float
Gas phase density at inlet, [kg/m^3]
rho_lo : float, optional
Liquid phase density at outlet, [kg/m^3]
rho_go : float, optional
Gas phase density at outlet, [kg/m^3]
Returns
-------
dP : float
Acceleration component of pressure drop for two-phase flow, [Pa]
Notes
-----
The use of different gas and liquid phase densities at the inlet and outlet
is optional; the outlet densities conditions will be assumed to be those of
the inlet if they are not specified.
There is a continuous variant of this method which can be integrated over,
at the expense of a speed. The differential form of this is as follows
([1]_, [3]_):
.. math::
- \left(\frac{d P}{dz}\right)_{acc} = G^2 \frac{d}{dz} \left[\frac{
(1-x)^2}{\rho_l(1-\alpha)} + \frac{x^2}{\rho_g\alpha}\right]
Examples
--------
>>> two_phase_dP_acceleration(m=1, D=0.1, xi=0.372, xo=0.557, rho_li=827.1,
... rho_gi=3.919, alpha_i=0.992, alpha_o=0.996)
706.8560377214725
References
----------
.. [1] Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat
Transfer, 3E. New York: McGraw-Hill, 1998.
.. [2] Awad, M. M., and Y. S. Muzychka. "Effective Property Models for
Homogeneous Two-Phase Flows." Experimental Thermal and Fluid Science 33,
no. 1 (October 1, 2008): 106-13.
doi:10.1016/j.expthermflusci.2008.07.006.
.. [3] Kim, Sung-Min, and Issam Mudawar. "Review of Databases and
Predictive Methods for Pressure Drop in Adiabatic, Condensing and
Boiling Mini/Micro-Channel Flows." International Journal of Heat and
Mass Transfer 77 (October 2014): 74-97.
doi:10.1016/j.ijheatmasstransfer.2014.04.035.
'''
G = 4*m/(pi*D*D)
if rho_lo is None:
rho_lo = rho_li
if rho_go is None:
rho_go = rho_gi
in_term = (1.-xi)**2/(rho_li*(1.-alpha_i)) + xi*xi/(rho_gi*alpha_i)
out_term = (1.-xo)**2/(rho_lo*(1.-alpha_o)) + xo*xo/(rho_go*alpha_o)
return G*G*(out_term - in_term)
[docs]def two_phase_dP_dz_acceleration(m, D, x, rhol, rhog, dv_dP_l, dv_dP_g, dx_dP,
dP_dL, dA_dL):
r'''This function handles calculation of two-phase liquid-gas pressure drop
due to acceleration for flow inside channels. This is a continuous
calculation, providing the differential in pressure per unit length and
should be called as part of an integration routine ([1]_, [2]_, [3]_).
.. math::
-\left(\frac{\partial P}{\partial L}\right)_{A} = G^2
\left(\left(\frac{1}{\rho_g} - \frac{1}{\rho_l}\right)\frac{\partial P}
{\partial L}\frac{\partial x}{\partial P} +
\frac{\partial P}{\partial L}\left[x \frac{\partial (1/\rho_g)}
{\partial P} + (1-x) \frac{\partial (1/\rho_l)}{\partial P}
\right] \right) - \frac{G^2}{\rho_{hom}}\frac{1}{A}\frac{\partial A}
{\partial L}
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
D : float
Diameter of pipe, [m]
x : float
Quality of fluid [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
dv_dP_l : float
Derivative of mass specific volume of the liquid phase with respect to
pressure, [m^3/(kg*Pa)]
dv_dP_g : float
Derivative of mass specific volume of the gas phase with respect to
pressure, [m^3/(kg*Pa)]
dx_dP : float
Derivative of mass quality of the two-phase fluid with respect to
pressure (numerical derivatives may be convenient for this), [1/Pa]
dP_dL : float
Pressure drop per unit length of pipe, [Pa/m]
dA_dL : float
Change in area of pipe per unit length of pipe, [m^2/m]
Returns
-------
dP_dz : float
Acceleration component of pressure drop for two-phase flow, [Pa/m]
Notes
-----
This calculation has the `homogeneous` model built in to it as its
derivation is shown in [1]_. The discrete calculation is more flexible as
different void fractions may be used.
Examples
--------
>>> two_phase_dP_dz_acceleration(m=1, D=0.1, x=0.372, rhol=827.1,
... rhog=3.919, dv_dP_l=-5e-12, dv_dP_g=-4e-7, dx_dP=-2e-7, dP_dL=120.0,
... dA_dL=0.0001)
20.137876617489034
References
----------
.. [1] Shoham, Ovadia. Mechanistic Modeling of Gas-Liquid Two-Phase Flow in
Pipes. Pap/Cdr edition. Richardson, TX: Society of Petroleum Engineers,
2006.
.. [2] Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat
Transfer, 3E. New York: McGraw-Hill, 1998.
.. [3] Kim, Sung-Min, and Issam Mudawar. "Review of Databases and
Predictive Methods for Pressure Drop in Adiabatic, Condensing and
Boiling Mini/Micro-Channel Flows." International Journal of Heat and
Mass Transfer 77 (October 2014): 74-97.
doi:10.1016/j.ijheatmasstransfer.2014.04.035.
'''
A = 0.25*pi*D*D
G = m/A
t1 = (1.0/rhog - 1.0/rhol)*dP_dL*dx_dP + dP_dL*(x*dv_dP_g + (1.0 - x)*dv_dP_l)
voidage_h = homogeneous(x, rhol, rhog)
rho_h = rhol*(1.0 - voidage_h) + rhog*voidage_h
return -G*G*(t1 - dA_dL/(rho_h*A))
[docs]def two_phase_dP_gravitational(angle, z, alpha_i, rho_li, rho_gi,
alpha_o=None, rho_lo=None, rho_go=None, g=g):
r'''This function handles calculation of two-phase liquid-gas pressure drop
due to gravitation for flow inside channels. This is a discrete
calculation for a segment with a known difference in elevation (and ideally
known inlet and outlet pressures so density dependence can be included).
.. math::
- \Delta P_{grav} = g \sin \theta z \left\{\frac{ [\alpha_o\rho_{g,o}
+ (1-\alpha_o)\rho_{l,o}] + [\alpha_i\rho_{g,i} + (1-\alpha_i)\rho_{l,i}]}
{2}\right\}
Parameters
----------
angle : float
The angle of the pipe with respect to the horizontal, [degrees]
z : float
The total length of the pipe, [m]
alpha_i : float
Void fraction at inlet (area of gas / total area of channel), [-]
rho_li : float
Liquid phase density at inlet, [kg/m^3]
rho_gi : float
Gas phase density at inlet, [kg/m^3]
alpha_o : float, optional
Void fraction at outlet (area of gas / total area of channel), [-]
rho_lo : float, optional
Liquid phase density at outlet, [kg/m^3]
rho_go : float, optional
Gas phase density at outlet, [kg/m^3]
g : float, optional
Acceleration due to gravity, [m/s^2]
Returns
-------
dP : float
Gravitational component of pressure drop for two-phase flow, [Pa]
Notes
-----
The use of different gas and liquid phase densities and void fraction
at the inlet and outlet is optional; the outlet densities and void fraction
will be assumed to be those of the inlet if they are not specified. This
does not add much accuracy.
There is a continuous variant of this method which can be integrated over,
at the expense of a speed. The differential form of this is as follows
([1]_, [2]_):
.. math::
-\left(\frac{dP}{dz} \right)_{grav} = [\alpha\rho_g + (1-\alpha)
\rho_l]g \sin \theta
Examples
--------
Example calculation, page 13-2 from [3]_:
>>> two_phase_dP_gravitational(angle=90, z=2, alpha_i=0.9685, rho_li=1518.,
... rho_gi=2.6)
987.237416829999
The same calculation, but using average inlet and outlet conditions:
>>> two_phase_dP_gravitational(angle=90, z=2, alpha_i=0.9685, rho_li=1518.,
... rho_gi=2.6, alpha_o=0.968, rho_lo=1517.9, rho_go=2.59)
994.5416058829999
References
----------
.. [1] Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat
Transfer, 3E. New York: McGraw-Hill, 1998.
.. [2] Kim, Sung-Min, and Issam Mudawar. "Review of Databases and
Predictive Methods for Pressure Drop in Adiabatic, Condensing and
Boiling Mini/Micro-Channel Flows." International Journal of Heat and
Mass Transfer 77 (October 2014): 74-97.
doi:10.1016/j.ijheatmasstransfer.2014.04.035.
.. [3] Thome, John R. "Engineering Data Book III." Wolverine Tube Inc
(2004). http://www.wlv.com/heat-transfer-databook/
'''
if rho_lo is None:
rho_lo = rho_li
if rho_go is None:
rho_go = rho_gi
if alpha_o is None:
alpha_o = alpha_i
angle = radians(angle)
in_term = alpha_i*rho_gi + (1. - alpha_i)*rho_li
out_term = alpha_o*rho_go + (1. - alpha_o)*rho_lo
return g*z*sin(angle)*(out_term + in_term)/2.
[docs]def two_phase_dP_dz_gravitational(angle, alpha, rhol, rhog, g=g):
r'''This function handles calculation of two-phase liquid-gas pressure drop
due to gravitation for flow inside channels. This is a differential
calculation for a segment with an infinitesimal difference in elevation for
use in performing integration over a pipe as shown in [1]_ and [2]_.
.. math::
-\left(\frac{dP}{dz} \right)_{grav} = [\alpha\rho_g + (1-\alpha)
\rho_l]g \sin \theta
Parameters
----------
angle : float
The angle of the pipe with respect to the horizontal, [degrees]
alpha : float
Void fraction (area of gas / total area of channel), [-]
rhol : float
Liquid phase density, [kg/m^3]
rhog : float
Gas phase density, [kg/m^3]
g : float, optional
Acceleration due to gravity, [m/s^2]
Returns
-------
dP_dz : float
Gravitational component of pressure drop for two-phase flow, [Pa/m]
Notes
-----
Examples
--------
>>> two_phase_dP_dz_gravitational(angle=90, alpha=0.9685, rhol=1518,
... rhog=2.6)
493.6187084149995
References
----------
.. [1] Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat
Transfer, 3E. New York: McGraw-Hill, 1998.
.. [2] Kim, Sung-Min, and Issam Mudawar. "Review of Databases and
Predictive Methods for Pressure Drop in Adiabatic, Condensing and
Boiling Mini/Micro-Channel Flows." International Journal of Heat and
Mass Transfer 77 (October 2014): 74-97.
doi:10.1016/j.ijheatmasstransfer.2014.04.035.
'''
angle = radians(angle)
return g*sin(angle)*(alpha*rhog + (1. - alpha)*rhol)
Dukler_XA_tck = implementation_optimize_tck([[-2.4791105294648372, -2.4791105294648372, -2.4791105294648372,
-2.4791105294648372, 0.14360803483759585, 1.7199938263676038,
1.7199938263676038, 1.7199938263676038, 1.7199938263676038],
[0.21299880246561081, 0.16299733301915248, -0.042340970712679615,
-1.9967836909384598, -2.9917366639619414, 0.0, 0.0, 0.0, 0.0],
3])
Dukler_XC_tck = implementation_optimize_tck([[-1.8323873272724698, -1.8323873272724698, -1.8323873272724698,
-1.8323873272724698, -0.15428198203334137, 1.7031193462360779,
1.7031193462360779, 1.7031193462360779, 1.7031193462360779],
[0.2827776229531682, 0.6207113329042158, 1.0609541626742232,
0.44917638072891825, 0.014664597632360495, 0.0, 0.0, 0.0, 0.0],
3])
Dukler_XD_tck = implementation_optimize_tck([[0.2532652936901574, 0.2532652936901574, 0.2532652936901574,
0.2532652936901574, 3.5567847823070253, 3.5567847823070253,
3.5567847823070253, 3.5567847823070253],
[0.09054274779541564, -0.05102629221303253, -0.23907463153703945,
-0.7757156285450911, 0.0, 0.0, 0.0, 0.0],
3])
XA_interp_obj = lambda x: 10**float(splev(log10(x), Dukler_XA_tck))
XC_interp_obj = lambda x: 10**float(splev(log10(x), Dukler_XC_tck))
XD_interp_obj = lambda x: 10**float(splev(log10(x), Dukler_XD_tck))
[docs]def Taitel_Dukler_regime(m, x, rhol, rhog, mul, mug, D, angle, roughness=0.0,
g=g):
r'''Classifies the regime of a two-phase flow according to Taitel and
Dukler (1976) ([1]_, [2]_).
The flow regimes in this method are 'annular', 'bubbly', 'intermittent',
'stratified wavy', and 'stratified smooth'.
The four dimensionless parameters used are 'X', 'T', 'F', and 'K'.
.. math::
X = \left[\frac{(dP/dL)_{l,s,f}}{(dP/dL)_{g,s,f}}\right]^{0.5}
.. math::
T = \left[\frac{(dP/dL)_{l,s,f}}{(\rho_l-\rho_g)g\cos\theta}\right]^{0.5}
.. math::
F = \sqrt{\frac{\rho_g}{(\rho_l-\rho_g)}} \frac{v_{g,s}}{\sqrt{D g \cos\theta}}
.. math::
K = F\left[\frac{D v_{l,s}}{\nu_l} \right]^{0.5} = F \sqrt{Re_{l,s}}
Note that 'l' refers to liquid, 'g' gas, 'f' friction-only, and 's'
superficial (i.e. if only the mass flow of that phase were flowing in the
pipe).
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Mass quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
D : float
Diameter of pipe, [m]
angle : float
The angle of the pipe with respect to the horizontal, [degrees]
roughness : float, optional
Roughness of pipe for use in calculating friction factor, [m]
g : float, optional
Acceleration due to gravity, [m/s^2]
Returns
-------
regime : str
One of 'annular', 'bubbly', 'intermittent', 'stratified wavy',
'stratified smooth', [-]
X : float
`X` dimensionless group used in the calculation, [-]
T : float
`T` dimensionless group used in the calculation, [-]
F : float
`F` dimensionless group used in the calculation, [-]
K : float
`K` dimensionless group used in the calculation, [-]
Notes
-----
The original friction factor used in this model is that of Blasius.
Examples
--------
>>> Taitel_Dukler_regime(m=0.6, x=0.112, rhol=915.12, rhog=2.67,
... mul=180E-6, mug=14E-6, D=0.05, roughness=0.0, angle=0)[0]
'annular'
References
----------
.. [1] Taitel, Yemada, and A. E. Dukler. "A Model for Predicting Flow
Regime Transitions in Horizontal and near Horizontal Gas-Liquid Flow."
AIChE Journal 22, no. 1 (January 1, 1976): 47-55.
doi:10.1002/aic.690220105.
.. [2] Brill, James P., and Howard Dale Beggs. Two-Phase Flow in Pipes,
1994.
.. [3] Shoham, Ovadia. Mechanistic Modeling of Gas-Liquid Two-Phase Flow in
Pipes. Pap/Cdr edition. Richardson, TX: Society of Petroleum Engineers,
2006.
'''
angle = radians(angle)
A = 0.25*pi*D*D
# Liquid-superficial properties, for calculation of dP_ls, dP_ls
# Paper and Brill Beggs 1991 confirms not v_lo but v_sg
v_ls = m*(1.0 - x)/(rhol*A)
Re_ls = Reynolds(V=v_ls, rho=rhol, mu=mul, D=D)
fd_ls = friction_factor(Re=Re_ls, eD=roughness/D)
dP_ls = fd_ls/D*(0.5*rhol*v_ls*v_ls)
# Gas-superficial properties, for calculation of dP_gs
v_gs = m*x/(rhog*A)
Re_gs = Reynolds(V=v_gs, rho=rhog, mu=mug, D=D)
fd_gs = friction_factor(Re=Re_gs, eD=roughness/D)
dP_gs = fd_gs/D*(0.5*rhog*v_gs*v_gs)
X = sqrt(dP_ls/dP_gs)
F = sqrt(rhog/(rhol-rhog))*v_gs/sqrt(D*g*cos(angle))
# Paper only uses kinematic viscosity
nul = mul/rhol
T = sqrt(dP_ls/((rhol-rhog)*g*cos(angle)))
K = sqrt(rhog*v_gs*v_gs*v_ls/((rhol-rhog)*g*nul*cos(angle)))
F_A_at_X = XA_interp_obj(X)
X_B_transition = 1.7917 # Roughly
if F >= F_A_at_X and X <= X_B_transition:
regime = 'annular'
elif F >= F_A_at_X:
T_D_at_X = XD_interp_obj(X)
if T >= T_D_at_X:
regime = 'bubbly'
else:
regime = 'intermittent'
else:
K_C_at_X = XC_interp_obj(X)
if K >= K_C_at_X:
regime = 'stratified wavy'
else:
regime = 'stratified smooth'
return regime, X, T, F, K
[docs]def Mandhane_Gregory_Aziz_regime(m, x, rhol, rhog, mul, mug, sigma, D):
r'''Classifies the regime of a two-phase flow according to Mandhane,
Gregory, and Azis (1974) flow map.
The flow regimes in this method are 'elongated bubble', 'stratified',
'annular mist', 'slug', 'dispersed bubble', and 'wave'.
The parameters used are just the superficial liquid and gas velocity (i.e.
if only the mass flow of that phase were flowing in the pipe).
Parameters
----------
m : float
Mass flow rate of fluid, [kg/s]
x : float
Mass quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
sigma : float
Surface tension, [N/m]
D : float
Diameter of pipe, [m]
Returns
-------
regime : str
One of 'elongated bubble', 'stratified', 'annular mist', 'slug',
'dispersed bubble', or 'wave', [-]
v_gs : float
The superficial gas velocity in the pipe (x axis coordinate), [ft/s]
v_ls : float
The superficial liquid velocity in the pipe (x axis coordinate), [ft/s]
Notes
-----
[1]_ contains a Fortran implementation of this model, which this has been
validated against. This is a very fast flow map as all transitions were
spelled out with clean transitions.
Examples
--------
>>> Mandhane_Gregory_Aziz_regime(m=0.6, x=0.112, rhol=915.12, rhog=2.67,
... mul=180E-6, mug=14E-6, sigma=0.065, D=0.05)
('slug', 0.9728397701853173, 42.05456634236875)
References
----------
.. [1] Mandhane, J. M., G. A. Gregory, and K. Aziz. "A Flow Pattern Map for
Gas-liquid Flow in Horizontal Pipes." International Journal of
Multiphase Flow 1, no. 4 (October 30, 1974): 537-53.
doi:10.1016/0301-9322(74)90006-8.
'''
A = 0.25*pi*D*D
Vsl = m*(1.0 - x)/(rhol*A)
Vsg = m*x/(rhog*A)
# Convert to imperial units
Vsl, Vsg = Vsl/0.3048, Vsg/0.3048
# X1 = (rhog/0.0808)**0.333 * (rhol*72.4/62.4/sigma)**0.25 * (mug/0.018)**0.2
# Y1 = (rhol*72.4/62.4/sigma)**0.25 * (mul/1.)**0.2
X1 = (rhog/1.294292)**0.333 * sqrt(sqrt(rhol*0.0724/(999.552*sigma))) * (mug*1.8E5)**0.2
Y1 = sqrt(sqrt(rhol*0.0724/999.552/sigma)) * (mul*1E3)**0.2
if Vsl < 14.0*Y1:
if Vsl <= 0.1:
Y1345 = 14.0*(Vsl/0.1)**-0.368
elif Vsl <= 0.2:
Y1345 = 14.0*(Vsl/0.1)**-0.415
elif Vsl <= 1.15:
Y1345 = 10.5*(Vsl/0.2)**-0.816
elif Vsl <= 4.8:
Y1345 = 2.5
else:
Y1345 = 2.5*(Vsl/4.8)**0.248
if Vsl <= 0.1:
Y456 = 70.0*(Vsl/0.01)**-0.0675
elif Vsl <= 0.3:
Y456 = 60.0*(Vsl/0.1)**-0.415
elif Vsl <= 0.56:
Y456 = 38.0*(Vsl/0.3)**0.0813
elif Vsl <= 1.0:
Y456 = 40.0*(Vsl/0.56)**0.385
elif Vsl <= 2.5:
Y456 = 50.0*(Vsl/1.)**0.756
else:
Y456 = 100.0*(Vsl/2.5)**0.463
Y45 = 0.3*Y1
Y31 = 0.5/Y1
Y1345 = Y1345*X1
Y456 = Y456*X1
if Vsg <= Y1345 and Vsl >= Y31:
regime = 'elongated bubble'
elif Vsg <= Y1345 and Vsl <= Y31:
regime = 'stratified'
elif Vsg >= Y1345 and Vsg <= Y456 and Vsl > Y45:
regime = 'slug'
elif Vsg >= Y1345 and Vsg <= Y456 and Vsl <= Y45:
regime = 'wave'
else:
regime = 'annular mist'
elif Vsg <= (230.*(Vsl/14.)**0.206)*X1:
regime = 'dispersed bubble'
else:
regime = 'annular mist'
return regime, Vsl, Vsg
Mandhane_Gregory_Aziz_regimes = {'elongated bubble': 1, 'stratified': 2,
'slug':3, 'wave': 4,
'annular mist': 5, 'dispersed bubble': 6}