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# -*- coding: utf-8 -*- 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.'''
'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']
### Models based on slip ratio
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 quite similar, as may be compared as follows. Neither expression was found in [1]_ in a brief review.
>>> from sympy import * >>> x, rhol, rhog, mug, mul = symbols('x, rhol, rhog, mug, mul') >>> Z = (rhol/rhog)**Rational(555,1000)*(mug/mul)**Rational(111,1000) >>> gamma = Z**1.6 >>> alpha = (gamma*x/(1 + x*(gamma-1))) >>> alpha x*((mug/mul)**(111/1000)*(rhol/rhog)**(111/200))**1.6/(x*(((mug/mul)**(111/1000)*(rhol/rhog)**(111/200))**1.6 - 1) + 1) >>> alpha.subs([(x, .4), (rhol, 800), (rhog, 2.5), (mul, 1E-3), (mug, 1E-5)]) 0.980138792146901
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 x*((mug/mul)**(111/1000)*(rhol/rhog)**(111/200))**1.6/(x*(((mug/mul)**(111/1000)*(rhol/rhog)**(111/200))**1.6 - 1) + 1)
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. '''
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 rearanged 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. '''
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. '''
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. '''
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]
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. '''
### Models using the Homogeneous flow model
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. '''
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. '''
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. '''
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. '''
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. '''
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. ''' else:
### Miscellaneous correlations
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}
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 emperical 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. '''
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. ''' pow_x=0.74, pow_rho=0.65, pow_mu=0.13)
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. ''' else:
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
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. ''' *(1.376 + 7.242/Xtt**1.655)**-0.5)
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. ''' else:
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\}
Ft = \left[\frac{G_{tp}^2 x^3}{(1-x)\rho_g^2gD}\right]^{0.5}
\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. ''' else:
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}
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. '''
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 rearanged 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. '''
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\}
Ft = \left[\frac{G_{tp}^2 x^3}{(1-x)\rho_g^2gD}\right]^{0.5}
\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. ''' else:
### Drift flux models
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}
v_{gm} = \frac{1.18(1-x)}{\rho_l^{0.5}}[g\sigma(\rho_l-\rho_g)]^{0.25}
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. '''
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}
v_{gm} = \frac{1.18(1-x)}{\rho_l^{0.5}}[g\sigma(\rho_l-\rho_g)]^{0.25}
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. '''
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}
v_{gm} = \frac{1.18(1-x)}{\rho_l^{0.5}}[g\sigma(\rho_l-\rho_g)]^{0.25}
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. '''
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}
v_{gm} = 0.35\sqrt{gD}
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. '''
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}
v_{gm} = 0
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. '''
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}
v_{gm} = 2.9\left(g\sigma\frac{\rho_l-\rho_g}{\rho_l^2}\right)^{0.25}
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]
v_{gs} = \frac{mx}{\rho_g \frac{\pi}{4}D^2}
v_{ls} = \frac{m(1-x)}{\rho_l \frac{\pi}{4}D^2}
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. '''
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}
v_{gm} = 1.41\left[\frac{g\sigma(\rho_l-\rho_g)}{\rho_l^2}\right]^{0.25}
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. '''
# Correlations developed in reviews
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. '''
r'''Calculates void fraction in two-phase flow according to the model of [1]_ as given in [2]_ and [3]_.
.. 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}}}}
v_{gs} = \frac{mx}{\rho_g \frac{\pi}{4}D^2}
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. '''
#print([Woldesemayat_Ghajar(0.4, 800., 2.5, sigma=0.2, m=1, D=0.3, P=1E6, angle=45)])
#print(Sun_Duffey_Peng(0.4, 800., 2.5, sigma=0.02, m=1, D=0.3, P=1E5, Pc=7E6)) #print(Rouhani_2(0.4, 800., 2.5, sigma=0.02, m=1, D=0.3)) #print(Steiner(0.4, 800., 2.5, sigma=0.02, m=1, D=0.3))
#print([Kopte_Newell_Chato(.4, 800, 2.5, 1E-3, 1E-5, m=10000001, D=0.3)]) #print(Huq_Loth(.4, 800, 2.5)) #print([Yashar(.4, 800, 2.5, 1E-3, 1E-5, m=1, D=0.3)]) #print([Graham(.4, 800, 2.5, 1E-3, 1E-5, m=1, D=0.3)]) #print(Graham(.4, 800, 2.5, 1E-3, 1E-5, m=1, D=0.3))
#print([Tandon_Varma_Gupta(.4, 800, 2.5, 1E-3, 1E-5, m, 0.3) for m in [1, .1]]) #print([Tandon_Varma_Gupta(.4, 800, 2.5, 1E-3, 1E-5, .1, 0.3)]) |