# -*- coding: utf-8 -*- '''Chemical Engineering Design Library (ChEDL). Utilities for process modeling. Copyright (C) 2016, 2017, 2018 Caleb Bell Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.''' from __future__ import division from math import exp, log, pi, sin, cos, radians from scipy.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', 'liquid_gas_voidage', 'gas_liquid_viscosity', 'two_phase_voidage_correlations', 'liquid_gas_viscosity_correlations'] ### Models based on slip ratio 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) 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 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 = ((rhol/rhog + K*x_ratio) / (1 + K*x_ratio))**0.5 alpha = (1 + (x_ratio) * (rhog/rhol) * (K + (1-K)*root))**-1 return alpha 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*(rhog/rhol)**0.5)**-1 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 = (1 - x*(1-rhol/rhog))**0.5 alpha = (1 + (1-x)/x*rhog/rhol*S)**-1 return alpha 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 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. ''' return 1.0/(1.0 + (1.0 - x)/x*(rhog/rhol)) 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 + (1-alpha_h)**0.5) 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) 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 - ((1-x)*rhog/x/rhol)**0.5*alpha_h**0.5 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*Fr**0.5))*alpha_h 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*alpha_h**0.5/(1. - C2*alpha_h**0.5) ### Miscellaneous correlations 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: pow_x = (2-n)/2. pow_mu = n/2. return ((1-x)/x)**pow_x * (rhog/rhol)**pow_rho * (mul/mug)**pow_mu 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 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 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*Rel**0.5)**2 *(1.376 + 7.242/Xtt**1.655)**-0.5) 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) 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 = (G**2*x**3/((1-x)*rhog**2*g*D))**0.5 if Ft < 0.01032: return 0 else: return 1 - exp(-1 - 0.3*log(Ft) - 0.0328*log(Ft)**2) 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 = (G**2*x**3/((1-x)*rhog**2*g*D))**0.5 Xtt = Lockhart_Martinelli_Xtt(x, rhol, rhog, mul, mug) return (1 + 1./Ft + Xtt)**-0.321 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 = (1 + 2*B*(rhol/rhog -1))**0.5 return 1 - 2*(1-x)**2/(1 - 2*x + D) 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 = (G**2*x**3/((1-x)*rhog**2*g*D))**0.5 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 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)/rhol**0.5*(g*sigma*(rhol-rhog))**0.25 return x/rhog*(C0*(x/rhog + (1-x)/rhol) + vgm/G)**-1 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)/rhol**0.5*(g*sigma*(rhol-rhog))**0.25 return x/rhog*(C0*(x/rhog + (1-x)/rhol) + vgm/G)**-1 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)*(g*D)**0.25*(rhol/G)**0.5 vgm = 1.18*(1-x)/rhol**0.5*(g*sigma*(rhol-rhog))**0.25 return x/rhog*(C0*(x/rhog + (1-x)/rhol) + vgm/G)**-1 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*(g*D)**0.5 return x/rhog*(C0*(x/rhog + (1-x)/rhol) + vgm/G)**-1 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 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*(g*sigma*(rhol-rhog)/rhol**2)**0.25 return x/rhog*(C0*(x/rhog + (1-x)/rhol) + vgm/G)**-1 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) C0 = (0.82 + 0.18*P/Pc)**-1 vgm = 1.41*(g*sigma*(rhol-rhog)/rhol**2)**0.25 return x/rhog*(C0*(x/rhog + (1-x)/rhol) + vgm/G)**-1 # Correlations developed in reviews 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 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*((g*D*sigma*(1 + cos(radians(angle)))*(rhol-rhog))/rhol**2)**0.25 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'))} # All the available arguments are: #{'rhol', 'angle=0', 'x', 'P', 'mug', 'rhog', 'D', 'g', 'Pc', 'sigma', 'mul', 'm'} 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, AvailableMethods=False): 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. The full list of correlation can be obtained with the `AvailableMethods` flag. 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), [-] 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. Other Parameters ---------------- Method : string, optional A string of the function name to use, as in the dictionary two_phase_voidage_correlations. AvailableMethods : bool, optional If True, function will consider which methods which can be used to calculate two-phase liquid-gas voidage with the given inputs and return them as a list instead of performing a calculation. 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 ''' def list_methods(myargs): usable_methods = [] for method, value in two_phase_voidage_correlations.items(): f, args = value if all(myargs[i] is not None for i in args): usable_methods.append(method) return usable_methods if AvailableMethods: return list_methods(locals()) if not Method: Method = 'homogeneous' if Method in two_phase_voidage_correlations: f, args = two_phase_voidage_correlations[Method] kwargs = {} for arg in args: kwargs[arg] = locals()[arg] return f(**kwargs) else: raise Exception('Method not recognized; available methods are %s' %list(two_phase_voidage_correlations.keys())) 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 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 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 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) 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 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]_ and [3]_. .. 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)) 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*((x*nug)**0.5 + ((1. - x)*nul)**0.5)**2 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)} def gas_liquid_viscosity(x, mul, mug, rhol=None, rhog=None, Method=None, AvailableMethods=False): 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] methods : list, only returned if AvailableMethods == True List of methods which can be used to calculate two-phase liquid-gas viscosity with the given inputs. Other Parameters ---------------- Method : string, optional A string of the function name to use, as in the dictionary liquid_gas_viscosity_correlations. AvailableMethods : bool, optional If True, function will consider which methods which can be used to calculate two-phase liquid-gas viscosity with the given inputs and return them as a list instead of performing a calculation. 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 ''' def list_methods(): methods = ['McAdams', 'Cicchitti', 'Lin Kwok'] if rhol is not None and rhog is not None: methods = list(liquid_gas_viscosity_correlations.keys()) return methods if AvailableMethods: return list_methods() if not Method: Method = 'McAdams' if Method in liquid_gas_viscosity_correlations: f, i = liquid_gas_viscosity_correlations[Method] if i == 0: return f(x, mul, mug) elif i == 1: return f(x, mul, mug, rhol=rhol, rhog=rhog) else: raise Exception('Method not recognized; available methods are %s' %list(liquid_gas_viscosity_correlations.keys()))