"""Chemical Engineering Design Library (ChEDL). Utilities for process modeling.
Copyright (C) 2016, 2017, 2018, 2020 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 void fraction/holdup in
two-phase flow. This is an important parameter for predicting pressure drop.
Also included are empirical "two phase viscosity" definitions which do not
have a physical meaning but are often used in pressure drop correlations.
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:: liquid_gas_voidage
.. autofunction:: liquid_gas_voidage_methods
.. autofunction:: density_two_phase
.. autofunction:: gas_liquid_viscosity
.. autofunction:: gas_liquid_viscosity_methods
Void Fraction/Holdup Correlations
---------------------------------
.. autofunction:: Thom
.. autofunction:: Zivi
.. autofunction:: Smith
.. autofunction:: Fauske
.. autofunction:: Chisholm_voidage
.. autofunction:: Turner_Wallis
.. autofunction:: homogeneous
.. autofunction:: Chisholm_Armand
.. autofunction:: Armand
.. autofunction:: Nishino_Yamazaki
.. autofunction:: Guzhov
.. autofunction:: Kawahara
.. autofunction:: Baroczy
.. autofunction:: Tandon_Varma_Gupta
.. autofunction:: Harms
.. autofunction:: Domanski_Didion
.. autofunction:: Graham
.. autofunction:: Yashar
.. autofunction:: Huq_Loth
.. autofunction:: Kopte_Newell_Chato
.. autofunction:: Steiner
.. autofunction:: Rouhani_1
.. autofunction:: Rouhani_2
.. autofunction:: Nicklin_Wilkes_Davidson
.. autofunction:: Gregory_Scott
.. autofunction:: Dix
.. autofunction:: Sun_Duffey_Peng
.. autofunction:: Xu_Fang_voidage
.. autofunction:: Woldesemayat_Ghajar
Utilities
---------
.. autofunction:: Lockhart_Martinelli_Xtt
.. autofunction:: two_phase_voidage_experimental
Gas/Liquid Viscosity
--------------------
.. autofunction:: Beattie_Whalley
.. autofunction:: McAdams
.. autofunction:: Cicchitti
.. autofunction:: Lin_Kwok
.. autofunction:: Fourar_Bories
.. autofunction:: Duckler
"""
from math import cos, exp, log, pi, radians, sin, sqrt
from fluids.constants import g
from fluids.core import Froude
__all__ = ['Thom', 'Zivi', 'Smith', 'Fauske', 'Chisholm_voidage', 'Turner_Wallis',
'homogeneous', 'Chisholm_Armand', 'Armand', 'Nishino_Yamazaki',
'Guzhov', 'Kawahara', 'Baroczy', 'Tandon_Varma_Gupta', 'Harms',
'Domanski_Didion', 'Graham', 'Yashar', 'Huq_Loth',
'Kopte_Newell_Chato', 'Steiner', 'Rouhani_1', 'Rouhani_2',
'Nicklin_Wilkes_Davidson', 'Gregory_Scott', 'Dix',
'Sun_Duffey_Peng', 'Xu_Fang_voidage', 'Woldesemayat_Ghajar',
'Lockhart_Martinelli_Xtt', 'two_phase_voidage_experimental',
'density_two_phase', 'Beattie_Whalley', 'McAdams', 'Cicchitti',
'Lin_Kwok', 'Fourar_Bories','Duckler', 'liquid_gas_voidage',
'liquid_gas_voidage_methods', 'gas_liquid_viscosity',
'gas_liquid_viscosity_methods',
'two_phase_voidage_correlations', 'liquid_gas_viscosity_correlations']
### Models based on slip ratio
[docs]def Thom(x, rhol, rhog, mul, mug):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_ as given in [2]_.
.. math::
\alpha = \left[1 + \left(\frac{1-x}{x}\right)\left(\frac{\rho_g}
{\rho_l}\right)^{0.89}\left(\frac{\mu_l}{\mu_g}\right)^{0.18}
\right]^{-1}
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
mul : float
Viscosity of liquid [Pa*s]
mug : float
Viscosity of gas [Pa*s]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
Based on experimental data for boiling of water. [3]_ presents a slightly
different model. However, its results are almost identical. A comparison can
be found in the unit tests. Neither expression was found in [1]_ in a brief
review.
Examples
--------
>>> Thom(.4, 800, 2.5, 1E-3, 1E-5)
0.9801482164042417
References
----------
.. [1] Thom, J. R. S. "Prediction of Pressure Drop during Forced
Circulation Boiling of Water." International Journal of Heat and Mass
Transfer 7, no. 7 (July 1, 1964): 709-24.
doi:10.1016/0017-9310(64)90002-X.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Dalkilic, A. S., S. Laohalertdecha, and S. Wongwises. "Effect of
Void Fraction Models on the Two-Phase Friction Factor of R134a during
Condensation in Vertical Downward Flow in a Smooth Tube." International
Communications in Heat and Mass Transfer 35, no. 8 (October 2008):
921-27. doi:10.1016/j.icheatmasstransfer.2008.04.001.
'''
return (1 + (1-x)/x * (rhog/rhol)**0.89 * (mul/mug)**0.18)**-1
# return x*((mug/mul)**(111/1000)*(rhol/rhog)**(111/200))**1.6/(x*(((mug/mul)**(111/1000)*(rhol/rhog)**(111/200))**1.6 - 1) + 1)
[docs]def Zivi(x, rhol, rhog):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_ as given in [2]_ and [3]_.
.. math::
\alpha = \left[1 + \left(\frac{1-x}{x}\right)
\left(\frac{\rho_g}{\rho_l}\right)^{2/3}\right]^{-1}
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
Based on experimental data for boiling of water.
More complicated variants of this are also in [1]_.
Examples
--------
>>> Zivi(.4, 800, 2.5)
0.9689339909056356
References
----------
.. [1] Zivi, S. M. "Estimation of Steady-State Steam Void-Fraction by Means
of the Principle of Minimum Entropy Production." Journal of Heat
Transfer 86, no. 2 (May 1, 1964): 247-51. doi:10.1115/1.3687113.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Dalkilic, A. S., S. Laohalertdecha, and S. Wongwises. "Effect of
Void Fraction Models on the Two-Phase Friction Factor of R134a during
Condensation in Vertical Downward Flow in a Smooth Tube." International
Communications in Heat and Mass Transfer 35, no. 8 (October 2008):
921-27. doi:10.1016/j.icheatmasstransfer.2008.04.001.
'''
return (1 + (1-x)/x * (rhog/rhol)**(2/3.))**-1
[docs]def Smith(x, rhol, rhog):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_, also given in [2]_ and [3]_.
.. math::
\alpha = \left\{1 + \left(\frac{1-x}{x}\right)
\left(\frac{\rho_g}{\rho_l}\right)\left[K+(1-K)
\sqrt{\frac{\frac{\rho_l}{\rho_g} + K\left(\frac{1-x}{x}\right)}
{1 + K\left(\frac{1-x}{x}\right)}}\right] \right\}^{-1}
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
[1]_ is an easy to read paper and has been reviewed.
The form of the expression here is rearranged somewhat differently
than in [1]_ but has been verified to be numerically equivalent. The form
of this in [3]_ is missing a square root on a bracketed term; this appears
in multiple papers by the authors.
Examples
--------
>>> Smith(.4, 800, 2.5)
0.959981235534199
References
----------
.. [1] Smith, S. L. "Void Fractions in Two-Phase Flow: A Correlation Based
upon an Equal Velocity Head Model." Proceedings of the Institution of
Mechanical Engineers 184, no. 1 (June 1, 1969): 647-64.
doi:10.1243/PIME_PROC_1969_184_051_02.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Dalkilic, A. S., S. Laohalertdecha, and S. Wongwises. "Effect of
Void Fraction Models on the Two-Phase Friction Factor of R134a during
Condensation in Vertical Downward Flow in a Smooth Tube." International
Communications in Heat and Mass Transfer 35, no. 8 (October 2008):
921-27. doi:10.1016/j.icheatmasstransfer.2008.04.001.
'''
K = 0.4
x_ratio = (1-x)/x
root = sqrt((rhol/rhog + K*x_ratio) / (1 + K*x_ratio))
alpha = (1 + (x_ratio) * (rhog/rhol) * (K + (1-K)*root))**-1
return alpha
[docs]def Fauske(x, rhol, rhog):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_, as given in [2]_ and [3]_.
.. math::
\alpha = \left[1 + \left(\frac{1-x}{x}\right)
\left(\frac{\rho_g}{\rho_l}\right)^{0.5}\right]^{-1}
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
[1]_ has not been reviewed. However, both [2]_ and [3]_ present it the
same way.
Examples
--------
>>> Fauske(.4, 800, 2.5)
0.9226347262627932
References
----------
.. [1] Fauske, H., Critical two-phase, steam-water flows, in: Heat Transfer
and Fluid Mechanics Institute 1961: Proceedings. Stanford University
Press, 1961, p. 79-89.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Dalkilic, A. S., S. Laohalertdecha, and S. Wongwises. "Effect of
Void Fraction Models on the Two-Phase Friction Factor of R134a during
Condensation in Vertical Downward Flow in a Smooth Tube." International
Communications in Heat and Mass Transfer 35, no. 8 (October 2008):
921-27. doi:10.1016/j.icheatmasstransfer.2008.04.001.
'''
return (1 + (1-x)/x*sqrt(rhog/rhol))**-1
[docs]def Chisholm_voidage(x, rhol, rhog):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_, as given in [2]_ and [3]_.
.. math::
\alpha = \left[1 + \left(\frac{1-x}{x}\right)\left(\frac{\rho_g}
{\rho_l}\right)\sqrt{1 - x\left(1-\frac{\rho_l}{\rho_g}\right)}
\right]^{-1}
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
[1]_ has not been reviewed. However, both [2]_ and [3]_ present it the
same way.
Examples
--------
>>> Chisholm_voidage(.4, 800, 2.5)
0.949525900374774
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 1,
1973): 347-58. doi:10.1016/0017-9310(73)90063-X.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Dalkilic, A. S., S. Laohalertdecha, and S. Wongwises. "Effect of
Void Fraction Models on the Two-Phase Friction Factor of R134a during
Condensation in Vertical Downward Flow in a Smooth Tube." International
Communications in Heat and Mass Transfer 35, no. 8 (October 2008):
921-27. doi:10.1016/j.icheatmasstransfer.2008.04.001.
'''
S = sqrt(1 - x*(1-rhol/rhog))
alpha = (1 + (1-x)/x*rhog/rhol*S)**-1
return alpha
[docs]def Turner_Wallis(x, rhol, rhog, mul, mug):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_, as given in [2]_ and [3]_.
.. math::
\alpha = \left[1 + \left(\frac{1-x}{x}\right)^{0.72}\left(\frac{\rho_g}
{\rho_l}\right)^{0.4}\left(\frac{\mu_l}{\mu_g}\right)^{0.08}
\right]^{-1}
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
mul : float
Viscosity of liquid [Pa*s]
mug : float
Viscosity of gas [Pa*s]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
[1]_ has not been reviewed. However, both [2]_ and [3]_ present it the
same way, if slightly differently rearranged.
Examples
--------
>>> Turner_Wallis(.4, 800, 2.5, 1E-3, 1E-5)
0.8384824581634625
References
----------
.. [1] J.M. Turner, G.B. Wallis, The Separate-cylinders Model of Two-phase
Flow, NYO-3114-6, Thayer's School Eng., Dartmouth College, Hanover, New
Hampshire, USA, 1965.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Dalkilic, A. S., S. Laohalertdecha, and S. Wongwises. "Effect of
Void Fraction Models on the Two-Phase Friction Factor of R134a during
Condensation in Vertical Downward Flow in a Smooth Tube." International
Communications in Heat and Mass Transfer 35, no. 8 (October 2008):
921-27. doi:10.1016/j.icheatmasstransfer.2008.04.001.
'''
return (1 + ((1-x)/x)**0.72 * (rhog/rhol)**0.4 * (mul/mug)**0.08)**-1
### Models using the Homogeneous flow model
[docs]def homogeneous(x, rhol, rhog):
r'''Calculates void fraction in two-phase flow according to the homogeneous
flow model, reviewed in [1]_, [2]_, and [3]_.
.. math::
\alpha = \frac{1}{1 + \left(\frac{1-x}{x}\right)\frac{\rho_g}{\rho_l}}
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
Examples
--------
>>> homogeneous(.4, 800, 2.5)
0.995334370139969
References
----------
.. [1] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [2] Dalkilic, A. S., S. Laohalertdecha, and S. Wongwises. "Effect of
Void Fraction Models on the Two-Phase Friction Factor of R134a during
Condensation in Vertical Downward Flow in a Smooth Tube." International
Communications in Heat and Mass Transfer 35, no. 8 (October 2008):
921-27. doi:10.1016/j.icheatmasstransfer.2008.04.001.
.. [3] Woldesemayat, Melkamu A., and Afshin J. Ghajar. "Comparison of Void
Fraction Correlations for Different Flow Patterns in Horizontal and
Upward Inclined Pipes." International Journal of Multiphase Flow 33,
no. 4 (April 2007): 347-370. doi:10.1016/j.ijmultiphaseflow.2006.09.004.
'''
if x == 1.0:
return 1.0
elif x == 0.0:
return 0.0
return 1.0/(1.0 + (1.0 - x)/x*(rhog/rhol))
[docs]def Chisholm_Armand(x, rhol, rhog):
r'''Calculates void fraction in two-phase flow according to the model
presented in [1]_ based on that of [2]_ as shown in [3]_, [4]_, and [5]_.
.. math::
\alpha = \frac{\alpha_h}{\alpha_h + (1-\alpha_h)^{0.5}}
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
Examples
--------
>>> Chisholm_Armand(.4, 800, 2.5)
0.9357814394262114
References
----------
.. [1] Chisholm, Duncan. Two-Phase Flow in Pipelines and Heat Exchangers.
Institution of Chemical Engineers, 1983.
.. [2] Armand, Aleksandr Aleksandrovich. The Resistance During the Movement
of a Two-Phase System in Horizontal Pipes. Atomic Energy Research
Establishment, 1959.
.. [3] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [4] Dalkilic, A. S., S. Laohalertdecha, and S. Wongwises. "Effect of
Void Fraction Models on the Two-Phase Friction Factor of R134a during
Condensation in Vertical Downward Flow in a Smooth Tube." International
Communications in Heat and Mass Transfer 35, no. 8 (October 2008):
921-27. doi:10.1016/j.icheatmasstransfer.2008.04.001.
.. [5] Woldesemayat, Melkamu A., and Afshin J. Ghajar. "Comparison of Void
Fraction Correlations for Different Flow Patterns in Horizontal and
Upward Inclined Pipes." International Journal of Multiphase Flow 33,
no. 4 (April 2007): 347-370. doi:10.1016/j.ijmultiphaseflow.2006.09.004.
'''
alpha_h = homogeneous(x, rhol, rhog)
return alpha_h/(alpha_h + sqrt(1-alpha_h))
[docs]def Armand(x, rhol, rhog):
r'''Calculates void fraction in two-phase flow according to the model
presented in [1]_ as shown in [2]_, [3]_, and [4]_.
.. math::
\alpha = 0.833\alpha_h
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
Examples
--------
>>> Armand(.4, 800, 2.5)
0.8291135303265941
References
----------
.. [1] Armand, Aleksandr Aleksandrovich. The Resistance During the Movement
of a Two-Phase System in Horizontal Pipes. Atomic Energy Research
Establishment, 1959.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Dalkilic, A. S., S. Laohalertdecha, and S. Wongwises. "Effect of
Void Fraction Models on the Two-Phase Friction Factor of R134a during
Condensation in Vertical Downward Flow in a Smooth Tube." International
Communications in Heat and Mass Transfer 35, no. 8 (October 2008):
921-27. doi:10.1016/j.icheatmasstransfer.2008.04.001.
.. [4] Woldesemayat, Melkamu A., and Afshin J. Ghajar. "Comparison of Void
Fraction Correlations for Different Flow Patterns in Horizontal and
Upward Inclined Pipes." International Journal of Multiphase Flow 33,
no. 4 (April 2007): 347-370. doi:10.1016/j.ijmultiphaseflow.2006.09.004.
'''
return 0.833*homogeneous(x, rhol, rhog)
[docs]def Nishino_Yamazaki(x, rhol, rhog):
r'''Calculates void fraction in two-phase flow according to the model
presented in [1]_ as shown in [2]_.
.. math::
\alpha = 1 - \left(\frac{1-x}{x}\frac{\rho_g}{\rho_l}\right)^{0.5}
\alpha_h^{0.5}
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
[1]_ is in Japanese.
[3]_ either shows this model as iterative in terms of voidage, or forgot
to add a H subscript to its second voidage term; the second is believed
more likely.
Examples
--------
>>> Nishino_Yamazaki(.4, 800, 2.5)
0.931694583962682
References
----------
.. [1] Nishino, Haruo, and Yasaburo Yamazaki. "A New Method of Evaluating
Steam Volume Fractions in Boiling Systems." Journal of the Atomic Energy
Society of Japan / Atomic Energy Society of Japan 5, no. 1 (1963):
39-46. doi:10.3327/jaesj.5.39.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Woldesemayat, Melkamu A., and Afshin J. Ghajar. "Comparison of Void
Fraction Correlations for Different Flow Patterns in Horizontal and
Upward Inclined Pipes." International Journal of Multiphase Flow 33,
no. 4 (April 2007): 347-370. doi:10.1016/j.ijmultiphaseflow.2006.09.004.
'''
alpha_h = homogeneous(x, rhol, rhog)
return 1 - sqrt((1-x)*rhog/x/rhol)*sqrt(alpha_h)
[docs]def Guzhov(x, rhol, rhog, m, D):
r'''Calculates void fraction in two-phase flow according to the model
in [1]_ as shown in [2]_ and [3]_.
.. math::
\alpha = 0.81[1 - \exp(-2.2\sqrt{Fr_{tp}})]\alpha_h
Fr_{tp} = \frac{G_{tp}^2}{gD\rho_{tp}^2}
\rho_{tp} = \left(\frac{1-x}{\rho_l} + \frac{x}{\rho_g}\right)^{-1}
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
m : float
Mass flow rate of both phases, [kg/s]
D : float
Diameter of the channel, [m]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
Examples
--------
>>> Guzhov(.4, 800, 2.5, 1, .3)
0.7626030108534588
References
----------
.. [1] Guzhov, A. I, Vasiliĭ Andreevich Mamaev, and G. E Odisharii︠a︡. A
Study of Transportation in Gas-Liquid Systems. Une Étude Sur Le
Transport Des Systèmes Gaz-Liquides. Bruxelles: International Gas Union,
1967.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
921-27. doi:10.1016/j.icheatmasstransfer.2008.04.001.
.. [3] Woldesemayat, Melkamu A., and Afshin J. Ghajar. "Comparison of Void
Fraction Correlations for Different Flow Patterns in Horizontal and
Upward Inclined Pipes." International Journal of Multiphase Flow 33,
no. 4 (April 2007): 347-370. doi:10.1016/j.ijmultiphaseflow.2006.09.004.
'''
rho_tp = ((1-x)/rhol + x/rhog)**-1
G = m/(pi/4*D**2)
V_tp = G/rho_tp
Fr = Froude(V=V_tp, L=D, squared=True) # squaring in undone later; Fr**0.5
alpha_h = homogeneous(x, rhol, rhog)
return 0.81*(1 - exp(-2.2*sqrt(Fr)))*alpha_h
[docs]def Kawahara(x, rhol, rhog, D):
r'''Calculates void fraction in two-phase flow according to the model
presented in [1]_, also reviewed in [2]_ and [3]_. This expression is for
microchannels.
.. math::
\alpha = \frac{C_1 \alpha_h^{0.5}}{1 - C_2\alpha_h^{0.5}}
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
D : float
Diameter of the channel, [m]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
C1 and C2 were constants for different diameters. Only diameters of
100 and 50 mircometers were studied in [1]_. Here, the coefficients are
distributed for three ranges, > 250 micrometers, 250-75 micrometers, and
< 75 micrometers.
The `Armand` model is used for the first, C1 and C2 are 0.03 and
0.97 for the second, and C1 and C2 are 0.02 and 0.98 for the third.
Examples
--------
>>> Kawahara(.4, 800, 2.5, 100E-6)
0.9276148194410238
References
----------
.. [1] Kawahara, A., M. Sadatomi, K. Okayama, M. Kawaji, and P. M.-Y.
Chung. "Effects of Channel Diameter and Liquid Properties on Void
Fraction in Adiabatic Two-Phase Flow Through Microchannels." Heat
Transfer Engineering 26, no. 3 (February 16, 2005): 13-19.
doi:10.1080/01457630590907158.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Dalkilic, A. S., S. Laohalertdecha, and S. Wongwises. "Effect of
Void Fraction Models on the Two-Phase Friction Factor of R134a during
Condensation in Vertical Downward Flow in a Smooth Tube." International
Communications in Heat and Mass Transfer 35, no. 8 (October 2008):
921-27. doi:10.1016/j.icheatmasstransfer.2008.04.001.
'''
if D > 250E-6:
return Armand(x, rhol, rhog)
elif D > 75E-6:
C1, C2 = 0.03, 0.97
else:
C1, C2 = 0.02, 0.98
alpha_h = homogeneous(x, rhol, rhog)
return C1*sqrt(alpha_h)/(1. - C2*sqrt(alpha_h))
### Miscellaneous correlations
[docs]def Lockhart_Martinelli_Xtt(x, rhol, rhog, mul, mug, pow_x=0.9, pow_rho=0.5,
pow_mu=0.1, n=None):
r'''Calculates the Lockhart-Martinelli Xtt two-phase flow parameter in a
general way according to [2]_. [1]_ is said to describe this. However,
very different definitions of this parameter have been used elsewhere.
Accordingly, the powers of each of the terms can be set. Alternatively, if
the parameter `n` is provided, the powers for viscosity and phase fraction
will be calculated from it as shown below.
.. math::
X_{tt} = \left(\frac{1-x}{x}\right)^{0.9} \left(\frac{\rho_g}{\rho_l}
\right)^{0.5}\left(\frac{\mu_l}{\mu_g}\right)^{0.1}
.. math::
X_{tt} = \left(\frac{1-x}{x}\right)^{(2-n)/2} \left(\frac{\rho_g}
{\rho_l}\right)^{0.5}\left(\frac{\mu_l}{\mu_g}\right)^{n/2}
Parameters
----------
x : float
Quality at the specific tube interval [-]
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
mul : float
Viscosity of liquid [Pa*s]
mug : float
Viscosity of gas [Pa*s]
pow_x : float, optional
Power for the phase ratio (1-x)/x, [-]
pow_rho : float, optional
Power for the density ratio rhog/rhol, [-]
pow_mu : float, optional
Power for the viscosity ratio mul/mug, [-]
n : float, optional
Number to be used for calculating pow_x and pow_mu if provided, [-]
Returns
-------
Xtt : float
Xtt Lockhart-Martinelli two-phase flow parameter [-]
Notes
-----
Xtt is best regarded as an empirical parameter.
If used, n is often 0.2 or 0.25.
Examples
--------
>>> Lockhart_Martinelli_Xtt(0.4, 800, 2.5, 1E-3, 1E-5)
0.12761659240532292
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] Woldesemayat, Melkamu A., and Afshin J. Ghajar. "Comparison of Void
Fraction Correlations for Different Flow Patterns in Horizontal and
Upward Inclined Pipes." International Journal of Multiphase Flow 33,
no. 4 (April 2007): 347-370. doi:10.1016/j.ijmultiphaseflow.2006.09.004.
'''
if n is not None:
pow_x = (2-n)/2.
pow_mu = n/2.
return ((1-x)/x)**pow_x * (rhog/rhol)**pow_rho * (mul/mug)**pow_mu
[docs]def Baroczy(x, rhol, rhog, mul, mug):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_ as given in [2]_, [3]_, and [4]_.
.. math::
\alpha = \left[1 + \left(\frac{1-x}{x}\right)^{0.74}\left(\frac{\rho_g}
{\rho_l}\right)^{0.65}\left(\frac{\mu_l}{\mu_g}\right)^{0.13}
\right]^{-1}
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
mul : float
Viscosity of liquid [Pa*s]
mug : float
Viscosity of gas [Pa*s]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
Examples
--------
>>> Baroczy(.4, 800, 2.5, 1E-3, 1E-5)
0.9453544598460807
References
----------
.. [1] Baroczy, C. Correlation of liquid fraction in two-phase flow with
applications to liquid metals, Chem. Eng. Prog. Symp. Ser. 61 (1965)
179-191.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Dalkilic, A. S., S. Laohalertdecha, and S. Wongwises. "Effect of
Void Fraction Models on the Two-Phase Friction Factor of R134a during
Condensation in Vertical Downward Flow in a Smooth Tube." International
Communications in Heat and Mass Transfer 35, no. 8 (October 2008):
921-27. doi:10.1016/j.icheatmasstransfer.2008.04.001.
.. [4] Woldesemayat, Melkamu A., and Afshin J. Ghajar. "Comparison of Void
Fraction Correlations for Different Flow Patterns in Horizontal and
Upward Inclined Pipes." International Journal of Multiphase Flow 33,
no. 4 (April 2007): 347-370. doi:10.1016/j.ijmultiphaseflow.2006.09.004.
'''
Xtt = Lockhart_Martinelli_Xtt(x, rhol, rhog, mul, mug,
pow_x=0.74, pow_rho=0.65, pow_mu=0.13)
return (1 + Xtt)**-1
[docs]def Tandon_Varma_Gupta(x, rhol, rhog, mul, mug, m, D):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_ also given in [2]_, [3]_, and [4]_.
For 50 < Rel < 1125:
.. math::
\alpha = 1- 1.928Re_l^{-0.315}[F(X_{tt})]^{-1} + 0.9293Re_l^{-0.63}
[F(X_{tt})]^{-2}
For Rel > 1125:
.. math::
\alpha = 1- 0.38 Re_l^{-0.088}[F(X_{tt})]^{-1} + 0.0361 Re_l^{-0.176}
[F(X_{tt})]^{-2}
.. math::
F(X_{tt}) = 0.15[X_{tt}^{-1} + 2.85X_{tt}^{-0.476}]
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
mul : float
Viscosity of liquid [Pa*s]
mug : float
Viscosity of gas [Pa*s]
m : float
Mass flow rate of both phases, [kg/s]
D : float
Diameter of the channel, [m]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
[1]_ does not specify how it defines the liquid Reynolds number.
[2]_ disagrees with [3]_ and [4]_; the later variant was selected, with:
.. math::
Re_l = \frac{G_{tp}D}{\mu_l}
The lower limit on Reynolds number is not enforced.
Examples
--------
>>> Tandon_Varma_Gupta(.4, 800, 2.5, 1E-3, 1E-5, m=1, D=0.3)
0.9228265670341428
References
----------
.. [1] Tandon, T. N., H. K. Varma, and C. P. Gupta. "A Void Fraction Model
for Annular Two-Phase Flow." International Journal of Heat and Mass
Transfer 28, no. 1 (January 1, 1985): 191-198.
doi:10.1016/0017-9310(85)90021-3.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Dalkilic, A. S., S. Laohalertdecha, and S. Wongwises. "Effect of
Void Fraction Models on the Two-Phase Friction Factor of R134a during
Condensation in Vertical Downward Flow in a Smooth Tube." International
Communications in Heat and Mass Transfer 35, no. 8 (October 2008):
921-27. doi:10.1016/j.icheatmasstransfer.2008.04.001.
.. [4] Woldesemayat, Melkamu A., and Afshin J. Ghajar. "Comparison of Void
Fraction Correlations for Different Flow Patterns in Horizontal and
Upward Inclined Pipes." International Journal of Multiphase Flow 33,
no. 4 (April 2007): 347-370. doi:10.1016/j.ijmultiphaseflow.2006.09.004.
'''
G = m/(pi/4*D**2)
Rel = G*D/mul
Xtt = Lockhart_Martinelli_Xtt(x, rhol, rhog, mul, mug)
Fxtt = 0.15*(Xtt**-1 + 2.85*Xtt**-0.476)
if Rel < 1125:
alpha = 1 - 1.928*Rel**-0.315/Fxtt + 0.9293*Rel**-0.63/Fxtt**2
else:
alpha = 1 - 0.38*Rel**-0.088/Fxtt + 0.0361*Rel**-0.176/Fxtt**2
return alpha
[docs]def Harms(x, rhol, rhog, mul, mug, m, D):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_ also given in [2]_ and [3]_.
.. math::
\alpha = \left[1 - 10.06Re_l^{-0.875}(1.74 + 0.104Re_l^{0.5})^2
\left(1.376 + \frac{7.242}{X_{tt}^{1.655}}\right)^{-0.5}\right]^2
.. math::
Re_l = \frac{G_{tp}(1-x)D}{\mu_l}
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
mul : float
Viscosity of liquid [Pa*s]
mug : float
Viscosity of gas [Pa*s]
m : float
Mass flow rate of both phases, [kg/s]
D : float
Diameter of the channel, [m]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
[1]_ has been reviewed.
Examples
--------
>>> Harms(.4, 800, 2.5, 1E-3, 1E-5, m=1, D=0.3)
0.9653289762907554
References
----------
.. [1] Tandon, T. N., H. K. Varma, and C. P. Gupta. "A Void Fraction Model
for Annular Two-Phase Flow." International Journal of Heat and Mass
Transfer 28, no. 1 (January 1, 1985): 191-198.
doi:10.1016/0017-9310(85)90021-3.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Dalkilic, A. S., S. Laohalertdecha, and S. Wongwises. "Effect of
Void Fraction Models on the Two-Phase Friction Factor of R134a during
Condensation in Vertical Downward Flow in a Smooth Tube." International
Communications in Heat and Mass Transfer 35, no. 8 (October 2008):
921-27. doi:10.1016/j.icheatmasstransfer.2008.04.001.
'''
G = m/(pi/4*D**2)
Rel = G*D*(1-x)/mul
Xtt = Lockhart_Martinelli_Xtt(x, rhol, rhog, mul, mug)
return (1 - 10.06*Rel**-0.875*(1.74 + 0.104*sqrt(Rel))**2
*1.0/sqrt(1.376 + 7.242/Xtt**1.655))
[docs]def Domanski_Didion(x, rhol, rhog, mul, mug):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_ also given in [2]_ and [3]_.
if Xtt < 10:
.. math::
\alpha = (1 + X_{tt}^{0.8})^{-0.378}
Otherwise:
.. math::
\alpha = 0.823- 0.157\ln(X_{tt})
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
mul : float
Viscosity of liquid [Pa*s]
mug : float
Viscosity of gas [Pa*s]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
[1]_ has been reviewed. [2]_ gives an exponent of -0.38 instead of -0.378
as is in [1]_. [3]_ describes only the novel half of the correlation.
The portion for Xtt > 10 is novel; the other is said to be from their 31st
reference, Wallis.
There is a discontinuity at Xtt = 10.
Examples
--------
>>> Domanski_Didion(.4, 800, 2.5, 1E-3, 1E-5)
0.9355795597059169
References
----------
.. [1] Domanski, Piotr, and David A. Didion. "Computer Modeling of the
Vapor Compression Cycle with Constant Flow Area Expansion Device."
Report. UNT Digital Library, May 1983.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Dalkilic, A. S., S. Laohalertdecha, and S. Wongwises. "Effect of
Void Fraction Models on the Two-Phase Friction Factor of R134a during
Condensation in Vertical Downward Flow in a Smooth Tube." International
Communications in Heat and Mass Transfer 35, no. 8 (October 2008):
921-27. doi:10.1016/j.icheatmasstransfer.2008.04.001.
'''
Xtt = Lockhart_Martinelli_Xtt(x, rhol, rhog, mul, mug)
if Xtt < 10:
return (1 + Xtt**0.8)**-0.378
else:
return 0.823 - 0.157*log(Xtt)
[docs]def Graham(x, rhol, rhog, mul, mug, m, D, g=g):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_ also given in [2]_ and [3]_.
.. math::
\alpha = 1 - \exp\{-1 - 0.3\ln(Ft) - 0.0328[\ln(Ft)]^2\}
.. math::
Ft = \left[\frac{G_{tp}^2 x^3}{(1-x)\rho_g^2gD}\right]^{0.5}
.. math::
\alpha = 0 \text{ for } F_t \le 0.01032
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
mul : float
Viscosity of liquid [Pa*s]
mug : float
Viscosity of gas [Pa*s]
m : float
Mass flow rate of both phases, [kg/s]
D : float
Diameter of the channel, [m]
g : float, optional
Acceleration due to gravity, [m/s^2]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
[1]_ has been reviewed. [2]_ does not list that the expression is not
real below a certain value of Ft.
Examples
--------
>>> Graham(.4, 800, 2.5, 1E-3, 1E-5, m=1, D=0.3)
0.6403336287530644
References
----------
.. [1] Graham, D. M. "Experimental Investigation of Void Fraction During
Refrigerant Condensation." ACRC Technical Report 135. Air Conditioning
and Refrigeration Center. College of Engineering. University of Illinois
at Urbana-Champaign., December 1997.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Dalkilic, A. S., S. Laohalertdecha, and S. Wongwises. "Effect of
Void Fraction Models on the Two-Phase Friction Factor of R134a during
Condensation in Vertical Downward Flow in a Smooth Tube." International
Communications in Heat and Mass Transfer 35, no. 8 (October 2008):
921-27. doi:10.1016/j.icheatmasstransfer.2008.04.001.
'''
G = m/(pi/4*D**2)
Ft = sqrt(G**2*x**3/((1-x)*rhog**2*g*D))
if Ft < 0.01032:
return 0
else:
return 1 - exp(-1 - 0.3*log(Ft) - 0.0328*log(Ft)**2)
[docs]def Yashar(x, rhol, rhog, mul, mug, m, D, g=g):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_ also given in [2]_ and [3]_.
.. math::
\alpha = \left[1 + \frac{1}{Ft} + X_{tt}\right]^{-0.321}
.. math::
Ft = \left[\frac{G_{tp}^2 x^3}{(1-x)\rho_g^2gD}\right]^{0.5}
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
mul : float
Viscosity of liquid [Pa*s]
mug : float
Viscosity of gas [Pa*s]
m : float
Mass flow rate of both phases, [kg/s]
D : float
Diameter of the channel, [m]
g : float, optional
Acceleration due to gravity, [m/s^2]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
[1]_ has been reviewed; both [2]_ and [3]_ give it correctly.
Examples
--------
>>> Yashar(.4, 800, 2.5, 1E-3, 1E-5, m=1, D=0.3)
0.7934893185789146
References
----------
.. [1] Yashar, D. A., M. J. Wilson, H. R. Kopke, D. M. Graham, J. C. Chato,
and T. A. Newell. "An Investigation of Refrigerant Void Fraction in
Horizontal, Microfin Tubes." HVAC&R Research 7, no. 1 (January 1, 2001):
67-82. doi:10.1080/10789669.2001.10391430.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Dalkilic, A. S., S. Laohalertdecha, and S. Wongwises. "Effect of
Void Fraction Models on the Two-Phase Friction Factor of R134a during
Condensation in Vertical Downward Flow in a Smooth Tube." International
Communications in Heat and Mass Transfer 35, no. 8 (October 2008):
921-27. doi:10.1016/j.icheatmasstransfer.2008.04.001.
'''
G = m/(pi/4*D**2)
Ft = sqrt(G**2*x**3/((1-x)*rhog**2*g*D))
Xtt = Lockhart_Martinelli_Xtt(x, rhol, rhog, mul, mug)
return (1 + 1./Ft + Xtt)**-0.321
[docs]def Huq_Loth(x, rhol, rhog):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_, also given in [2]_, [3]_, and [4]_.
.. math::
\alpha = 1 - \frac{2(1-x)^2}{1 - 2x + \left[1 + 4x(1-x)\left(\frac
{\rho_l}{\rho_g}-1\right)\right]^{0.5}}
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
[1]_ has been reviewed, and matches the expressions given in the reviews
[2]_, [3]_, and [4]_; the form of the expression is rearranged somewhat
differently.
Examples
--------
>>> Huq_Loth(.4, 800, 2.5)
0.9593868838476147
References
----------
.. [1] Huq, Reazul, and John L. Loth. "Analytical Two-Phase Flow Void
Prediction Method." Journal of Thermophysics and Heat Transfer 6, no. 1
(January 1, 1992): 139-44. doi:10.2514/3.329.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Dalkilic, A. S., S. Laohalertdecha, and S. Wongwises. "Effect of
Void Fraction Models on the Two-Phase Friction Factor of R134a during
Condensation in Vertical Downward Flow in a Smooth Tube." International
Communications in Heat and Mass Transfer 35, no. 8 (October 2008):
921-27. doi:10.1016/j.icheatmasstransfer.2008.04.001.
.. [4] Woldesemayat, Melkamu A., and Afshin J. Ghajar. "Comparison of Void
Fraction Correlations for Different Flow Patterns in Horizontal and
Upward Inclined Pipes." International Journal of Multiphase Flow 33,
no. 4 (April 2007): 347-370. doi:10.1016/j.ijmultiphaseflow.2006.09.004.
'''
B = 2*x*(1-x)
D = sqrt(1 + 2*B*(rhol/rhog -1))
return 1 - 2*(1-x)**2/(1 - 2*x + D)
[docs]def Kopte_Newell_Chato(x, rhol, rhog, mul, mug, m, D, g=g):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_ also given in [2]_.
.. math::
\alpha = 1.045 - \exp\{-1 - 0.342\ln(Ft) - 0.0268[\ln(Ft)]^2
+ 0.00597[\ln(Ft)]^3\}
.. math::
Ft = \left[\frac{G_{tp}^2 x^3}{(1-x)\rho_g^2gD}\right]^{0.5}
.. math::
\alpha = \alpha_h \text{ for } F_t \le 0.044
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
mul : float
Viscosity of liquid [Pa*s]
mug : float
Viscosity of gas [Pa*s]
m : float
Mass flow rate of both phases, [kg/s]
D : float
Diameter of the channel, [m]
g : float, optional
Acceleration due to gravity, [m/s^2]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
[1]_ has been reviewed. If is recommended this expression not be used above
Ft values of 454.
Examples
--------
>>> Kopte_Newell_Chato(.4, 800, 2.5, 1E-3, 1E-5, m=1, D=0.3)
0.6864466770087425
References
----------
.. [1] Kopke, H. R. "Experimental Investigation of Void Fraction During
Refrigerant Condensation in Horizontal Tubes." ACRC Technical Report
142. Air Conditioning and Refrigeration Center. College of Engineering.
University of Illinois at Urbana-Champaign., August 1998.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
'''
G = m/(pi/4*D**2)
Ft = sqrt(G**2*x**3/((1-x)*rhog**2*g*D))
if Ft < 0.044:
return homogeneous(x, rhol, rhog)
else:
return 1.045 - exp(-1 - 0.342*log(Ft) - 0.0268*log(Ft)**2 + 0.00597*log(Ft)**3)
### Drift flux models
[docs]def Steiner(x, rhol, rhog, sigma, m, D, g=g):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_ also given in [2]_ and [3]_.
.. math::
\alpha = \frac{x}{\rho_g}\left[C_0\left(\frac{x}{\rho_g} + \frac{1-x}
{\rho_l}\right) +\frac{v_{gm}}{G} \right]^{-1}
.. math::
v_{gm} = \frac{1.18(1-x)}{\rho_l^{0.5}}[g\sigma(\rho_l-\rho_g)]^{0.25}
.. math::
C_0 = 1 + 0.12(1-x)
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
sigma : float
Surface tension of liquid [N/m]
m : float
Mass flow rate of both phases, [kg/s]
D : float
Diameter of the channel, [m]
g : float, optional
Acceleration due to gravity, [m/s^2]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
[1]_ has been reviewed.
Examples
--------
>>> Steiner(0.4, 800., 2.5, sigma=0.02, m=1, D=0.3)
0.895950181381335
References
----------
.. [1] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition.
Berlin; New York:: Springer, 2010.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Dalkilic, A. S., S. Laohalertdecha, and S. Wongwises. "Effect of
Void Fraction Models on the Two-Phase Friction Factor of R134a during
Condensation in Vertical Downward Flow in a Smooth Tube." International
Communications in Heat and Mass Transfer 35, no. 8 (October 2008):
921-27. doi:10.1016/j.icheatmasstransfer.2008.04.001.
'''
G = m/(pi/4*D**2)
C0 = 1 + 0.12*(1-x)
vgm = 1.18*(1-x)/sqrt(rhol)*sqrt(sqrt(g*sigma*(rhol-rhog)))
return x/rhog*(C0*(x/rhog + (1-x)/rhol) + vgm/G)**-1
[docs]def Rouhani_1(x, rhol, rhog, sigma, m, D, g=g):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_ as given in [2]_ and [3]_.
.. math::
\alpha = \frac{x}{\rho_g}\left[C_0\left(\frac{x}{\rho_g} + \frac{1-x}
{\rho_l}\right) +\frac{v_{gm}}{G} \right]^{-1}
.. math::
v_{gm} = \frac{1.18(1-x)}{\rho_l^{0.5}}[g\sigma(\rho_l-\rho_g)]^{0.25}
.. math::
C_0 = 1 + 0.2(1-x)
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
sigma : float
Surface tension of liquid [N/m]
m : float
Mass flow rate of both phases, [kg/s]
D : float
Diameter of the channel, [m]
g : float, optional
Acceleration due to gravity, [m/s^2]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
The expression as quoted in [2]_ and [3]_ could not be found in [1]_.
Examples
--------
>>> Rouhani_1(0.4, 800., 2.5, sigma=0.02, m=1, D=0.3)
0.8588420244136714
References
----------
.. [1] Rouhani, S. Z, and E Axelsson. "Calculation of Void Volume Fraction
in the Subcooled and Quality Boiling Regions." International Journal of
Heat and Mass Transfer 13, no. 2 (February 1, 1970): 383-93.
doi:10.1016/0017-9310(70)90114-6.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Woldesemayat, Melkamu A., and Afshin J. Ghajar. "Comparison of Void
Fraction Correlations for Different Flow Patterns in Horizontal and
Upward Inclined Pipes." International Journal of Multiphase Flow 33,
no. 4 (April 2007): 347-370. doi:10.1016/j.ijmultiphaseflow.2006.09.004.
'''
G = m/(pi/4*D**2)
C0 = 1 + 0.2*(1-x)
vgm = 1.18*(1-x)/sqrt(rhol)*sqrt(sqrt(g*sigma*(rhol-rhog)))
return x/rhog*(C0*(x/rhog + (1-x)/rhol) + vgm/G)**-1
[docs]def Rouhani_2(x, rhol, rhog, sigma, m, D, g=g):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_ as given in [2]_ and [3]_.
.. math::
\alpha = \frac{x}{\rho_g}\left[C_0\left(\frac{x}{\rho_g} + \frac{1-x}
{\rho_l}\right) +\frac{v_{gm}}{G} \right]^{-1}
.. math::
v_{gm} = \frac{1.18(1-x)}{\rho_l^{0.5}}[g\sigma(\rho_l-\rho_g)]^{0.25}
.. math::
C_0 = 1 + 0.2(1-x)(gD)^{0.25}\left(\frac{\rho_l}{G_{tp}}\right)^{0.5}
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
sigma : float
Surface tension of liquid [N/m]
m : float
Mass flow rate of both phases, [kg/s]
D : float
Diameter of the channel, [m]
g : float, optional
Acceleration due to gravity, [m/s^2]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
The expression as quoted in [2]_ and [3]_ could not be found in [1]_.
Examples
--------
>>> Rouhani_2(0.4, 800., 2.5, sigma=0.02, m=1, D=0.3)
0.44819733138968865
References
----------
.. [1] Rouhani, S. Z, and E Axelsson. "Calculation of Void Volume Fraction
in the Subcooled and Quality Boiling Regions." International Journal of
Heat and Mass Transfer 13, no. 2 (February 1, 1970): 383-93.
doi:10.1016/0017-9310(70)90114-6.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Woldesemayat, Melkamu A., and Afshin J. Ghajar. "Comparison of Void
Fraction Correlations for Different Flow Patterns in Horizontal and
Upward Inclined Pipes." International Journal of Multiphase Flow 33,
no. 4 (April 2007): 347-370. doi:10.1016/j.ijmultiphaseflow.2006.09.004.
'''
G = m/(pi/4*D**2)
C0 = 1 + 0.2*(1-x)*sqrt(sqrt(g*D))*sqrt(rhol/G)
vgm = 1.18*(1-x)/sqrt(rhol)*sqrt(sqrt(g*sigma*(rhol-rhog)))
return x/rhog*(C0*(x/rhog + (1-x)/rhol) + vgm/G)**-1
[docs]def Nicklin_Wilkes_Davidson(x, rhol, rhog, m, D, g=g):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_ as given in [2]_ and [3]_.
.. math::
\alpha = \frac{x}{\rho_g}\left[C_0\left(\frac{x}{\rho_g} + \frac{1-x}
{\rho_l}\right) +\frac{v_{gm}}{G} \right]^{-1}
.. math::
v_{gm} = 0.35\sqrt{gD}
.. math::
C_0 = 1.2
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
m : float
Mass flow rate of both phases, [kg/s]
D : float
Diameter of the channel, [m]
g : float, optional
Acceleration due to gravity, [m/s^2]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
Examples
--------
>>> Nicklin_Wilkes_Davidson(0.4, 800., 2.5, m=1, D=0.3)
0.6798826626721431
References
----------
.. [1] D. Nicklin, J. Wilkes, J. Davidson, "Two-phase flow in vertical
tubes", Trans. Inst. Chem. Eng. 40 (1962) 61-68.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Woldesemayat, Melkamu A., and Afshin J. Ghajar. "Comparison of Void
Fraction Correlations for Different Flow Patterns in Horizontal and
Upward Inclined Pipes." International Journal of Multiphase Flow 33,
no. 4 (April 2007): 347-370. doi:10.1016/j.ijmultiphaseflow.2006.09.004.
'''
G = m/(pi/4*D**2)
C0 = 1.2
vgm = 0.35*sqrt(g*D)
return x/rhog*(C0*(x/rhog + (1-x)/rhol) + vgm/G)**-1
[docs]def Gregory_Scott(x, rhol, rhog):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_ as given in [2]_ and [3]_.
.. math::
\alpha = \frac{x}{\rho_g}\left[C_0\left(\frac{x}{\rho_g} + \frac{1-x}
{\rho_l}\right) +\frac{v_{gm}}{G} \right]^{-1}
.. math::
v_{gm} = 0
.. math::
C_0 = 1.19
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
Examples
--------
>>> Gregory_Scott(0.4, 800., 2.5)
0.8364154370924108
References
----------
.. [1] Gregory, G. A., and D. S. Scott. "Correlation of Liquid Slug
Velocity and Frequency in Horizontal Cocurrent Gas-Liquid Slug Flow."
AIChE Journal 15, no. 6 (November 1, 1969): 933-35.
doi:10.1002/aic.690150623.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Woldesemayat, Melkamu A., and Afshin J. Ghajar. "Comparison of Void
Fraction Correlations for Different Flow Patterns in Horizontal and
Upward Inclined Pipes." International Journal of Multiphase Flow 33,
no. 4 (April 2007): 347-370. doi:10.1016/j.ijmultiphaseflow.2006.09.004.
'''
C0 = 1.19
return x/rhog*(C0*(x/rhog + (1-x)/rhol))**-1
[docs]def Dix(x, rhol, rhog, sigma, m, D, g=g):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_ as given in [2]_ and [3]_.
.. math::
\alpha = \frac{x}{\rho_g}\left[C_0\left(\frac{x}{\rho_g} + \frac{1-x}
{\rho_l}\right) +\frac{v_{gm}}{G} \right]^{-1}
.. math::
v_{gm} = 2.9\left(g\sigma\frac{\rho_l-\rho_g}{\rho_l^2}\right)^{0.25}
.. math::
C_0 = \frac{v_{sg}}{v_m}\left[1 + \left(\frac{v_{sl}}{v_{sg}}\right)
^{\left(\left(\frac{\rho_g}{\rho_l}\right)^{0.1}\right)}\right]
.. math::
v_{gs} = \frac{mx}{\rho_g \frac{\pi}{4}D^2}
.. math::
v_{ls} = \frac{m(1-x)}{\rho_l \frac{\pi}{4}D^2}
.. math::
v_m = v_{gs} + v_{ls}
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
sigma : float
Surface tension of liquid [N/m]
m : float
Mass flow rate of both phases, [kg/s]
D : float
Diameter of the channel, [m]
g : float, optional
Acceleration due to gravity, [m/s^2]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
Has formed the basis for several other correlations.
Examples
--------
>>> Dix(0.4, 800., 2.5, sigma=0.02, m=1, D=0.3)
0.8268737961156514
References
----------
.. [1] Gary Errol. Dix. "Vapor Void Fractions for Forced Convection with
Subcooled Boiling at Low Flow Rates." Thesis. University of California,
Berkeley, 1971.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Woldesemayat, Melkamu A., and Afshin J. Ghajar. "Comparison of Void
Fraction Correlations for Different Flow Patterns in Horizontal and
Upward Inclined Pipes." International Journal of Multiphase Flow 33,
no. 4 (April 2007): 347-370. doi:10.1016/j.ijmultiphaseflow.2006.09.004.
'''
vgs = m*x/(rhog*pi/4*D**2)
vls = m*(1-x)/(rhol*pi/4*D**2)
G = m/(pi/4*D**2)
C0 = vgs/(vls+vgs)*(1 + (vls/vgs)**((rhog/rhol)**0.1))
vgm = 2.9*sqrt(sqrt(g*sigma*(rhol-rhog)/rhol**2))
return x/rhog*(C0*(x/rhog + (1-x)/rhol) + vgm/G)**-1
[docs]def Sun_Duffey_Peng(x, rhol, rhog, sigma, m, D, P, Pc, g=g):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_ as given in [2]_ and [3]_.
.. math::
\alpha = \frac{x}{\rho_g}\left[C_0\left(\frac{x}{\rho_g} + \frac{1-x}
{\rho_l}\right) +\frac{v_{gm}}{G} \right]^{-1}
.. math::
v_{gm} = 1.41\left[\frac{g\sigma(\rho_l-\rho_g)}{\rho_l^2}\right]^{0.25}
.. math::
C_0 = \left(0.82 + 0.18\frac{P}{P_c}\right)^{-1}
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
sigma : float
Surface tension of liquid [N/m]
m : float
Mass flow rate of both phases, [kg/s]
D : float
Diameter of the channel, [m]
P : float
Pressure of the fluid, [Pa]
Pc : float
Critical pressure of the fluid, [Pa]
g : float, optional
Acceleration due to gravity, [m/s^2]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
Examples
--------
>>> Sun_Duffey_Peng(0.4, 800., 2.5, sigma=0.02, m=1, D=0.3, P=1E5, Pc=7E6)
0.7696546506515833
References
----------
.. [1] K.H. Sun, R.B. Duffey, C.M. Peng, A thermal-hydraulic analysis of
core uncover, in: Proceedings of the 19th National Heat Transfer
Conference, Experimental and Analytical Modeling of LWR Safety
Experiments, 1980, pp. 1-10. Orlando, Florida, USA.
.. [2] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
.. [3] Woldesemayat, Melkamu A., and Afshin J. Ghajar. "Comparison of Void
Fraction Correlations for Different Flow Patterns in Horizontal and
Upward Inclined Pipes." International Journal of Multiphase Flow 33,
no. 4 (April 2007): 347-370. doi:10.1016/j.ijmultiphaseflow.2006.09.004.
'''
G = m/(pi/4*D**2)
Pr = P/Pc if Pc is not None else 0.5
C0 = (0.82 + 0.18*Pr)**-1
vgm = 1.41*sqrt(sqrt(g*sigma*(rhol-rhog)/rhol**2))
return x/rhog*(C0*(x/rhog + (1-x)/rhol) + vgm/G)**-1
# Correlations developed in reviews
[docs]def Xu_Fang_voidage(x, rhol, rhog, m, D, g=g):
r'''Calculates void fraction in two-phase flow according to the model
developed in the review of [1]_.
.. math::
\alpha = \left[1 + \left(1 + 2Fr_{lo}^{-0.2}\alpha_h^{3.5}\right)\left(
\frac{1-x}{x}\right)\left(\frac{\rho_g}{\rho_l}\right)\right]^{-1}
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
m : float
Mass flow rate of both phases, [kg/s]
D : float
Diameter of the channel, [m]
g : float, optional
Acceleration due to gravity, [m/s^2]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
Claims an AARD of 5.0%, and suitability for any flow regime,
mini and micro channels, adiabatic, evaporating, or condensing flow,
and for Frlo from 0.02 to 145, rhog/rhol from 0.004-0.153, and x from 0 to
1.
Examples
--------
>>> Xu_Fang_voidage(0.4, 800., 2.5, m=1, D=0.3)
0.9414660089942093
References
----------
.. [1] Xu, Yu, and Xiande Fang. "Correlations of Void Fraction for Two-
Phase Refrigerant Flow in Pipes." Applied Thermal Engineering 64, no.
1-2 (March 2014): 242-51. doi:10.1016/j.applthermaleng.2013.12.032.
'''
G = m/(pi/4*D**2)
alpha_h = homogeneous(x, rhol, rhog)
Frlo = G**2/(g*D*rhol**2)
return (1 + (1 + 2*Frlo**-0.2*alpha_h**3.5)*((1-x)/x)*(rhog/rhol))**-1
[docs]def Woldesemayat_Ghajar(x, rhol, rhog, sigma, m, D, P, angle=0, g=g):
r'''Calculates void fraction in two-phase flow according to the model of
[1]_.
.. math::
\alpha = \frac{v_{gs}}{v_{gs}\left(1 + \left(\frac{v_{ls}}{v_{gs}}
\right)^{\left(\frac{\rho_g}{\rho_l}\right)^{0.1}}\right)
+ 2.9\left[\frac{gD\sigma(1+\cos\theta)(\rho_l-\rho_g)}
{\rho_l^2}\right]^{0.25}(1.22 + 1.22\sin\theta)^{\frac{P}{P_{atm}}}}
.. math::
v_{gs} = \frac{mx}{\rho_g \frac{\pi}{4}D^2}
.. math::
v_{ls} = \frac{m(1-x)}{\rho_l \frac{\pi}{4}D^2}
Parameters
----------
x : float
Quality at the specific tube interval []
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
sigma : float
Surface tension of liquid [N/m]
m : float
Mass flow rate of both phases, [kg/s]
D : float
Diameter of the channel, [m]
P : float
Pressure of the fluid, [Pa]
angle : float
Angle of the channel with respect to the horizontal (vertical = 90),
[degrees]
g : float, optional
Acceleration due to gravity, [m/s^2]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
Strongly recommended.
Examples
--------
>>> Woldesemayat_Ghajar(0.4, 800., 2.5, sigma=0.2, m=1, D=0.3, P=1E6, angle=45)
0.7640815513429202
References
----------
.. [1] Woldesemayat, Melkamu A., and Afshin J. Ghajar. "Comparison of Void
Fraction Correlations for Different Flow Patterns in Horizontal and
Upward Inclined Pipes." International Journal of Multiphase Flow 33,
no. 4 (April 2007): 347-370. doi:10.1016/j.ijmultiphaseflow.2006.09.004.
'''
vgs = m*x/(rhog*pi/4*D**2)
vls = m*(1-x)/(rhol*pi/4*D**2)
first = vgs*(1 + (vls/vgs)**((rhog/rhol)**0.1))
second = 2.9*sqrt(sqrt((g*D*sigma*(1 + cos(radians(angle)))*(rhol-rhog))/rhol**2))
if P is None:
P = 101325.0
third = (1.22 + 1.22*sin(radians(angle)))**(101325./P)
return vgs/(first + second*third)
# x, rhol, rhog 2ill be the minimum inputs
two_phase_voidage_correlations = {'Thom' : (Thom, ('x', 'rhol', 'rhog', 'mul', 'mug')),
'Zivi' : (Zivi, ('x', 'rhol', 'rhog')),
'Smith' : (Smith, ('x', 'rhol', 'rhog')),
'Fauske' : (Fauske, ('x', 'rhol', 'rhog')),
'Chisholm_voidage' : (Chisholm_voidage, ('x', 'rhol', 'rhog')),
'Turner Wallis' : (Turner_Wallis, ('x', 'rhol', 'rhog', 'mul', 'mug')),
'homogeneous' : (homogeneous, ('x', 'rhol', 'rhog')),
'Chisholm Armand' : (Chisholm_Armand, ('x', 'rhol', 'rhog')),
'Armand' : (Armand, ('x', 'rhol', 'rhog')),
'Nishino Yamazaki' : (Nishino_Yamazaki, ('x', 'rhol', 'rhog')),
'Guzhov' : (Guzhov, ('x', 'rhol', 'rhog', 'm', 'D')),
'Kawahara' : (Kawahara, ('x', 'rhol', 'rhog', 'D')),
'Baroczy' : (Baroczy, ('x', 'rhol', 'rhog', 'mul', 'mug')),
'Tandon Varma Gupta' : (Tandon_Varma_Gupta, ('x', 'rhol', 'rhog', 'mul', 'mug', 'm', 'D')),
'Harms' : (Harms, ('x', 'rhol', 'rhog', 'mul', 'mug', 'm', 'D')),
'Domanski Didion' : (Domanski_Didion, ('x', 'rhol', 'rhog', 'mul', 'mug')),
'Graham' : (Graham, ('x', 'rhol', 'rhog', 'mul', 'mug', 'm', 'D', 'g')),
'Yashar' : (Yashar, ('x', 'rhol', 'rhog', 'mul', 'mug', 'm', 'D', 'g')),
'Huq_Loth' : (Huq_Loth, ('x', 'rhol', 'rhog')),
'Kopte_Newell_Chato' : (Kopte_Newell_Chato, ('x', 'rhol', 'rhog', 'mul', 'mug', 'm', 'D', 'g')),
'Steiner' : (Steiner, ('x', 'rhol', 'rhog', 'sigma', 'm', 'D', 'g')),
'Rouhani 1' : (Rouhani_1, ('x', 'rhol', 'rhog', 'sigma', 'm', 'D', 'g')),
'Rouhani 2' : (Rouhani_2, ('x', 'rhol', 'rhog', 'sigma', 'm', 'D', 'g')),
'Nicklin Wilkes Davidson' : (Nicklin_Wilkes_Davidson, ('x', 'rhol', 'rhog', 'm', 'D', 'g')),
'Gregory_Scott' : (Gregory_Scott, ('x', 'rhol', 'rhog')),
'Dix' : (Dix, ('x', 'rhol', 'rhog', 'sigma', 'm', 'D', 'g')),
'Sun Duffey Peng' : (Sun_Duffey_Peng, ('x', 'rhol', 'rhog', 'sigma', 'm', 'D', 'P', 'Pc', 'g')),
'Xu Fang voidage' : (Xu_Fang_voidage, ('x', 'rhol', 'rhog', 'm', 'D', 'g')),
'Woldesemayat Ghajar' : (Woldesemayat_Ghajar, ('x', 'rhol', 'rhog', 'sigma', 'm', 'D', 'P', 'angle', 'g'))}
_unknown_two_phase_voidage_corr = 'Method not recognized; available methods are %s' %list(two_phase_voidage_correlations.keys())
# All the available arguments are:
#{'rhol', 'angle=0', 'x', 'P', 'mug', 'rhog', 'D', 'g', 'Pc', 'sigma', 'mul', 'm'}
[docs]def liquid_gas_voidage_methods(x, rhol, rhog, D=None, m=None, mul=None, mug=None,
sigma=None, P=None, Pc=None, angle=0.0, g=g,
check_ranges=False):
r'''This function returns a list of liquid-gas voidage correlation names
which can perform the calculation with the provided inputs. The holdup is
for two-phase liquid-gas flow inside channels. 29 calculation methods are
available, with varying input requirements.
Parameters
----------
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
D : float, optional
Diameter of pipe, [m]
m : float, optional
Mass flow rate of fluid, [kg/s]
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]
angle : float, optional
Angle of the channel with respect to the horizontal (vertical = 90),
[degrees]
g : float, optional
Acceleration due to gravity, [m/s^2]
check_ranges : bool, optional
Added for future use only
Returns
-------
methods : list, only returned if AvailableMethods == True
List of methods which can be used to calculate two-phase liquid-gas
voidage with the given inputs.
Examples
--------
>>> len(liquid_gas_voidage_methods(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6, sigma=0.0487, D=0.05))
27
'''
vals = {'x': x, 'rhol': rhol, 'rhog': rhog, 'D': D, 'm': m, 'mul': mul,
'mug': mug, 'sigma': sigma, 'P': P, 'Pc': Pc, 'angle': angle,
'g': g, 'check_ranges': check_ranges}
usable_methods = []
for method, value in two_phase_voidage_correlations.items():
f, args = value
if all(vals[i] is not None for i in args):
usable_methods.append(method)
return usable_methods
[docs]def liquid_gas_voidage(x, rhol, rhog, D=None, m=None, mul=None, mug=None,
sigma=None, P=None, Pc=None, angle=0, g=g, Method=None):
r'''This function handles calculation of two-phase liquid-gas voidage
for flow inside channels. 29 calculation methods are available, with
varying input requirements. A correlation will be automatically selected if
none is specified.
This function is used to calculate the (liquid) holdup as well, as:
.. math::
\text{holdup} = 1 - \text{voidage}
If no correlation is selected, the following rules are used, with the
earlier options attempted first:
* TODO: defaults
Parameters
----------
x : float
Quality of fluid, [-]
rhol : float
Liquid density, [kg/m^3]
rhog : float
Gas density, [kg/m^3]
D : float, optional
Diameter of pipe, [m]
m : float, optional
Mass flow rate of fluid, [kg/s]
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]
angle : float, optional
Angle of the channel with respect to the horizontal (vertical = 90),
[degrees]
g : float, optional
Acceleration due to gravity, [m/s^2]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Other Parameters
----------------
Method : string, optional
A string of the function name to use, as in the dictionary
two_phase_voidage_correlations.
Notes
-----
Examples
--------
>>> liquid_gas_voidage(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
... sigma=0.0487, D=0.05)
0.9744097632663492
'''
if Method is None:
Method2 = 'homogeneous'
else:
Method2 = Method
if Method2 == "Thom":
return Thom(x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug)
elif Method2 == "Zivi":
return Zivi(x=x, rhol=rhol, rhog=rhog)
elif Method2 == "Smith":
return Smith(x=x, rhol=rhol, rhog=rhog)
elif Method2 == "Fauske":
return Fauske(x=x, rhol=rhol, rhog=rhog)
elif Method2 == "Chisholm_voidage":
return Chisholm_voidage(x=x, rhol=rhol, rhog=rhog)
elif Method2 == "Turner Wallis":
return Turner_Wallis(x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug)
elif Method2 == "homogeneous":
return homogeneous(x=x, rhol=rhol, rhog=rhog)
elif Method2 == "Chisholm Armand":
return Chisholm_Armand(x=x, rhol=rhol, rhog=rhog)
elif Method2 == "Armand":
return Armand(x=x, rhol=rhol, rhog=rhog)
elif Method2 == "Nishino Yamazaki":
return Nishino_Yamazaki(x=x, rhol=rhol, rhog=rhog)
elif Method2 == "Guzhov":
return Guzhov(x=x, rhol=rhol, rhog=rhog, m=m, D=D)
elif Method2 == "Kawahara":
return Kawahara(x=x, rhol=rhol, rhog=rhog, D=D)
elif Method2 == "Baroczy":
return Baroczy(x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug)
elif Method2 == "Tandon Varma Gupta":
return Tandon_Varma_Gupta(x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, m=m, D=D)
elif Method2 == "Harms":
return Harms(x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, m=m, D=D)
elif Method2 == "Domanski Didion":
return Domanski_Didion(x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug)
elif Method2 == "Graham":
return Graham(x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, m=m, D=D, g=g)
elif Method2 == "Yashar":
return Yashar(x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, m=m, D=D, g=g)
elif Method2 == "Huq_Loth":
return Huq_Loth(x=x, rhol=rhol, rhog=rhog)
elif Method2 == "Kopte_Newell_Chato":
return Kopte_Newell_Chato(x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, m=m, D=D, g=g)
elif Method2 == "Steiner":
return Steiner(x=x, rhol=rhol, rhog=rhog, sigma=sigma, m=m, D=D, g=g)
elif Method2 == "Rouhani 1":
return Rouhani_1(x=x, rhol=rhol, rhog=rhog, sigma=sigma, m=m, D=D, g=g)
elif Method2 == "Rouhani 2":
return Rouhani_2(x=x, rhol=rhol, rhog=rhog, sigma=sigma, m=m, D=D, g=g)
elif Method2 == "Nicklin Wilkes Davidson":
return Nicklin_Wilkes_Davidson(x=x, rhol=rhol, rhog=rhog, m=m, D=D, g=g)
elif Method2 == "Gregory_Scott":
return Gregory_Scott(x=x, rhol=rhol, rhog=rhog)
elif Method2 == "Dix":
return Dix(x=x, rhol=rhol, rhog=rhog, sigma=sigma, m=m, D=D, g=g)
elif Method2 == "Sun Duffey Peng":
return Sun_Duffey_Peng(x=x, rhol=rhol, rhog=rhog, sigma=sigma, m=m, D=D, P=P, Pc=Pc, g=g)
elif Method2 == "Xu Fang voidage":
return Xu_Fang_voidage(x=x, rhol=rhol, rhog=rhog, m=m, D=D, g=g)
elif Method2 == "Woldesemayat Ghajar":
return Woldesemayat_Ghajar(x=x, rhol=rhol, rhog=rhog, sigma=sigma, m=m, D=D, P=P, angle=angle, g=g)
else:
raise ValueError(_unknown_two_phase_voidage_corr)
[docs]def density_two_phase(alpha, rhol, rhog):
r'''Calculates the "effective" density of fluid in a liquid-gas flow. If
the weight of fluid in a pipe pipe could be measured and the volume of
the pipe were known, an effective density of the two-phase mixture could be
calculated. This is directly relatable to the void fraction of the pipe,
a parameter used to predict the pressure drop. This function converts
void fraction to effective two-phase density.
.. math::
\rho_m = \alpha \rho_g + (1-\alpha)\rho_l
Parameters
----------
alpha : float
Void fraction (area of gas / total area of channel), [-]
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
Returns
-------
rho_lg : float
Two-phase effective density [kg/m^3]
Notes
-----
**THERE IS NO THERMODYNAMIC DEFINITION FOR THIS QUANTITY. DO NOT USE THIS
VALUE IN SINGLE-PHASE CORRELATIONS.**
Examples
--------
>>> density_two_phase(.4, 800, 2.5)
481.0
References
----------
.. [1] 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.
'''
return alpha*rhog + (1. - alpha)*rhol
[docs]def two_phase_voidage_experimental(rho_lg, rhol, rhog):
r'''Calculates the void fraction for two-phase liquid-gas pipeflow. If
the weight of fluid in a pipe pipe could be measured and the volume of
the pipe were known, an effective density of the two-phase mixture could be
calculated. This is directly relatable to the void fraction of the pipe,
a parameter used to predict the pressure drop. This function converts
that measured effective two-phase density to void fraction for use in
developing correlations.
.. math::
\alpha = \frac{\rho_m - \rho_l}{\rho_g - \rho_l}
Parameters
----------
rho_lg : float
Two-phase effective density [kg/m^3]
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
Returns
-------
alpha : float
Void fraction (area of gas / total area of channel), [-]
Notes
-----
Examples
--------
>>> two_phase_voidage_experimental(481.0, 800, 2.5)
0.4
References
----------
.. [1] 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.
'''
return (rho_lg - rhol)/(rhog - rhol)
### two-phase viscosity models
[docs]def Beattie_Whalley(x, mul, mug, rhol, rhog):
r'''Calculates a suggested definition for liquid-gas two-phase flow
viscosity in internal pipe flow according to the form in [1]_ and shown
in [2]_ and [3]_.
.. math::
\mu_m = \mu_l(1-\alpha_m)(1 + 2.5\alpha_m) + \mu_g\alpha_m
.. math::
\alpha_m = \frac{1}{1 + \left(\frac{1-x}{x}\right)\frac{\rho_g}{\rho_l}}
\text{(homogeneous model)}
Parameters
----------
x : float
Quality of the gas-liquid flow, [-]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
Returns
-------
mu_lg : float
Liquid-gas viscosity (**a suggested definition, potentially useful
for empirical work only!**) [Pa*s]
Notes
-----
This model converges to the liquid or gas viscosity as the quality
approaches either limits.
Examples
--------
>>> Beattie_Whalley(x=0.4, mul=1E-3, mug=1E-5, rhol=850, rhog=1.2)
1.7363806909512365e-05
References
----------
.. [1] Beattie, D. R. H., and P. B. Whalley. "A Simple Two-Phase Frictional
Pressure Drop Calculation Method." International Journal of Multiphase
Flow 8, no. 1 (February 1, 1982): 83-87.
doi:10.1016/0301-9322(82)90009-X.
.. [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.
.. [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.
'''
alpha = homogeneous(x, rhol, rhog)
return mul*(1. - alpha)*(1. + 2.5*alpha) + mug*alpha
[docs]def McAdams(x, mul, mug):
r'''Calculates a suggested definition for liquid-gas two-phase flow
viscosity in internal pipe flow according to the form in [1]_ and shown
in [2]_ and [3]_.
.. math::
\mu_m = \left(\frac{x}{\mu_g} + \frac{1-x}{\mu_l}\right)^{-1}
Parameters
----------
x : float
Quality of the gas-liquid flow, [-]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
Returns
-------
mu_lg : float
Liquid-gas viscosity (**a suggested definition, potentially useful
for empirical work only!**) [Pa*s]
Notes
-----
This model converges to the liquid or gas viscosity as the quality
approaches either limits.
[3]_ states this is the most common definition of two-phase liquid-gas
viscosity.
Examples
--------
>>> McAdams(x=0.4, mul=1E-3, mug=1E-5)
2.4630541871921184e-05
References
----------
.. [1] McAdams, W. H. "Vaporization inside Horizontal Tubes-II Benzene-Oil
Mixtures." Trans. ASME 39 (1949): 39-48.
.. [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.
.. [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.
'''
return 1./(x/mug + (1. - x)/mul)
[docs]def Cicchitti(x, mul, mug):
r'''Calculates a suggested definition for liquid-gas two-phase flow
viscosity in internal pipe flow according to the form in [1]_ and shown
in [2]_ and [3]_.
.. math::
\mu_m = x\mu_g + (1-x)\mu_l
Parameters
----------
x : float
Quality of the gas-liquid flow, [-]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
Returns
-------
mu_lg : float
Liquid-gas viscosity (**a suggested definition, potentially useful
for empirical work only!**) [Pa*s]
Notes
-----
This model converges to the liquid or gas viscosity as the quality
approaches either limits.
Examples
--------
>>> Cicchitti(x=0.4, mul=1E-3, mug=1E-5)
0.0006039999999999999
References
----------
.. [1] Cicchitti, A., C. Lombardi, M. Silvestri, G. Soldaini, and R.
Zavattarelli. "Two-Phase Cooling Experiments: Pressure Drop, Heat
Transfer and Burnout Measurements." Centro Informazioni Studi
Esperienze, Milan, January 1, 1959.
.. [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.
.. [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.
'''
return x*mug + (1. - x)*mul
[docs]def Lin_Kwok(x, mul, mug):
r'''Calculates a suggested definition for liquid-gas two-phase flow
viscosity in internal pipe flow according to the form in [1]_ and shown
in [2]_.
.. math::
\mu_m = \frac{\mu_l \mu_g}{\mu_g + x^{1.4}(\mu_l - \mu_g)}
Parameters
----------
x : float
Quality of the gas-liquid flow, [-]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
Returns
-------
mu_lg : float
Liquid-gas viscosity (**a suggested definition, potentially useful
for empirical work only!**) [Pa*s]
Notes
-----
This model converges to the liquid or gas viscosity as the quality
approaches either limits.
Examples
--------
>>> Lin_Kwok(x=0.4, mul=1E-3, mug=1E-5)
3.515119398126066e-05
References
----------
.. [1] Lin, S., C. C. K. Kwok, R. -Y. Li, Z. -H. Chen, and Z. -Y. Chen.
"Local Frictional Pressure Drop during Vaporization of R-12 through
Capillary Tubes." International Journal of Multiphase Flow 17, no. 1
(January 1, 1991): 95-102. doi:10.1016/0301-9322(91)90072-B.
.. [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.
'''
return mul*mug/(mug + x**1.4*(mul - mug))
[docs]def Fourar_Bories(x, mul, mug, rhol, rhog):
r'''Calculates a suggested definition for liquid-gas two-phase flow
viscosity in internal pipe flow according to the form in [1]_ and shown
in [2]_ and [3]_.
.. math::
\mu_m = \rho_m\left(\sqrt{x\nu_g} + \sqrt{(1-x)\nu_l}\right)^2
Parameters
----------
x : float
Quality of the gas-liquid flow, [-]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
rhol : float
Density of the liquid, [kg/m^3]
rhog : float
Density of the gas, [kg/m^3]
Returns
-------
mu_lg : float
Liquid-gas viscosity (**a suggested definition, potentially useful
for empirical work only!**) [Pa*s]
Notes
-----
This model converges to the liquid or gas viscosity as the quality
approaches either limits.
This was first expressed in the equalivalent form as follows:
.. math::
\mu_m = \rho_m\left(x\nu_g + (1-x)\nu_l + 2\sqrt{x(1-x)\nu_g\nu_l}
\right)
Examples
--------
>>> Fourar_Bories(x=0.4, mul=1E-3, mug=1E-5, rhol=850, rhog=1.2)
2.127617150298565e-05
References
----------
.. [1] Fourar, M., and S. Bories. "Experimental Study of Air-Water
Two-Phase Flow through a Fracture (Narrow Channel)." International
Journal of Multiphase Flow 21, no. 4 (August 1, 1995): 621-37.
doi:10.1016/0301-9322(95)00005-I.
.. [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.
.. [3] Aung, NZ, and T. Yuwono. "Evaluation of Mixture Viscosity Models in
the Prediction of Two-Phase Flow Pressure Drops." ASEAN Journal on
Science and Technology for Development 29, no. 2 (2012).
'''
rhom = 1./(x/rhog + (1. - x)/rhol)
nul = mul/rhol # = nu_mu_converter(rho=rhol, mu=mul)
nug = mug/rhog # = nu_mu_converter(rho=rhog, mu=mug)
return rhom*(sqrt(x*nug) + sqrt((1. - x)*nul))**2
[docs]def Duckler(x, mul, mug, rhol, rhog):
r'''Calculates a suggested definition for liquid-gas two-phase flow
viscosity in internal pipe flow according to the form in [1]_ and shown
in [2]_, [3]_, and [4]_.
.. math::
\mu_m = \frac{\frac{x\mu_g}{\rho_g} + \frac{(1-x)\mu_l}{\rho_l} }
{\frac{x}{\rho_g} + \frac{(1-x)}{\rho_l} }
Parameters
----------
x : float
Quality of the gas-liquid flow, [-]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
rhol : float
Density of the liquid, [kg/m^3]
rhog : float
Density of the gas, [kg/m^3]
Returns
-------
mu_lg : float
Liquid-gas viscosity (**a suggested definition, potentially useful
for empirical work only!**) [Pa*s]
Notes
-----
This model converges to the liquid or gas viscosity as the quality
approaches either limits.
This has also been expressed in the following form:
.. math::
\mu_m = \rho_m \left[x\left(\frac{\mu_g}{\rho_g}\right)
+ (1 - x)\left(\frac{\mu_l}{\rho_l}\right)\right]
According to the homogeneous definition of two-phase density.
Examples
--------
>>> Duckler(x=0.4, mul=1E-3, mug=1E-5, rhol=850, rhog=1.2)
1.2092040385066917e-05
References
----------
.. [1] Fourar, M., and S. Bories. "Experimental Study of Air-Water
Two-Phase Flow through a Fracture (Narrow Channel)." International
Journal of Multiphase Flow 21, no. 4 (August 1, 1995): 621-37.
doi:10.1016/0301-9322(95)00005-I.
.. [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.
.. [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.
.. [4] Aung, NZ, and T. Yuwono. "Evaluation of Mixture Viscosity Models in
the Prediction of Two-Phase Flow Pressure Drops." ASEAN Journal on
Science and Technology for Development 29, no. 2 (2012).
'''
return (x*mug/rhog + (1. - x)*mul/rhol)/(x/rhog + (1. - x)/rhol)
liquid_gas_viscosity_correlations = {'Beattie Whalley': (Beattie_Whalley, 1),
'Fourar Bories': (Fourar_Bories, 1),
'Duckler': (Duckler, 1),
'McAdams': (McAdams, 0),
'Cicchitti': (Cicchitti, 0),
'Lin Kwok': (Lin_Kwok, 0)}
liquid_gas_viscosity_correlations_list = ['Beattie Whalley', 'Fourar Bories', 'Duckler', 'McAdams', 'Cicchitti', 'Lin Kwok']
[docs]def gas_liquid_viscosity_methods(rhol=None, rhog=None, check_ranges=False):
r'''This function returns a list of methods which can be used for calculating
two-phase liquid-gas viscosity.
Six calculation methods are available; three of them require only `x`,
`mul`, and `mug`; the other three require `rhol` and `rhog` as well.
Parameters
----------
rhol : float, optional
Liquid density, [kg/m^3]
rhog : float, optional
Gas density, [kg/m^3]
check_ranges : bool, optional
Added for compatibility only, never used
Returns
-------
methods : list
List of methods which can be used to calculate two-phase liquid-gas
viscosity with the given inputs.
Examples
--------
>>> gas_liquid_viscosity_methods()
['McAdams', 'Cicchitti', 'Lin Kwok']
>>> gas_liquid_viscosity_methods(rhol=1000, rhog=2)
['Beattie Whalley', 'Fourar Bories', 'Duckler', 'McAdams', 'Cicchitti', 'Lin Kwok']
'''
methods = ['McAdams', 'Cicchitti', 'Lin Kwok']
if rhol is not None and rhog is not None:
methods = liquid_gas_viscosity_correlations_list
return methods
_gas_liquid_viscosity_method_unknown = 'Method not recognized; available methods are %s' %list(liquid_gas_viscosity_correlations.keys())
[docs]def gas_liquid_viscosity(x, mul, mug, rhol=None, rhog=None, Method=None):
r'''This function handles the calculation of two-phase liquid-gas viscosity.
Six calculation methods are available; three of them require only `x`,
`mul`, and `mug`; the other three require `rhol` and `rhog` as well.
The 'McAdams' method will be used if no method is specified.
The full list of correlation can be obtained with the `AvailableMethods`
flag.
**ALL OF THESE METHODS ARE ONLY SUGGESTED DEFINITIONS, POTENTIALLY
USEFUL FOR EMPIRICAL WORK ONLY!**
Parameters
----------
x : float
Quality of fluid, [-]
mul : float
Viscosity of liquid, [Pa*s]
mug : float
Viscosity of gas, [Pa*s]
rhol : float, optional
Liquid density, [kg/m^3]
rhog : float, optional
Gas density, [kg/m^3]
Returns
-------
mu_lg : float
Liquid-gas viscosity (**a suggested definition, potentially useful
for empirical work only!**) [Pa*s]
Other Parameters
----------------
Method : string, optional
A string of the function name to use, as in the dictionary
liquid_gas_viscosity_correlations.
Notes
-----
All of these models converge to the liquid or gas viscosity as the quality
approaches either limits. Other definitions have been proposed, such as
using only liquid viscosity.
These values cannot just be plugged into single phase correlations!
Examples
--------
>>> gas_liquid_viscosity(x=0.4, mul=1E-3, mug=1E-5, rhol=850, rhog=1.2, Method='Duckler')
1.2092040385066917e-05
>>> gas_liquid_viscosity(x=0.4, mul=1E-3, mug=1E-5)
2.4630541871921184e-05
'''
if Method is None:
Method = 'McAdams'
if Method == 'Beattie Whalley':
return Beattie_Whalley(x, mul, mug, rhol=rhol, rhog=rhog)
elif Method == 'Fourar Bories':
return Fourar_Bories(x, mul, mug, rhol=rhol, rhog=rhog)
elif Method == 'Duckler':
return Duckler(x, mul, mug, rhol=rhol, rhog=rhog)
elif Method == 'McAdams':
return McAdams(x, mul, mug)
elif Method == 'Cicchitti':
return Cicchitti(x, mul, mug)
elif Method == 'Lin Kwok':
return Lin_Kwok(x, mul, mug)
else:
raise ValueError(_gas_liquid_viscosity_method_unknown)