Two-phase flow voidage (fluids.two_phase_voidage)

This module contains functions for calculating void fraction/holdup in two-phase flow. This is an important parameter for predicting pressure drop. Also included are empirical “two phase viscosity” definitions which do not have a physical meaning but are often used in pressure drop correlations.

For reporting bugs, adding feature requests, or submitting pull requests, please use the GitHub issue tracker or contact the author at Caleb.Andrew.Bell@gmail.com.

Interfaces

fluids.two_phase_voidage.liquid_gas_voidage(x, rhol, rhog, D=None, m=None, mul=None, mug=None, sigma=None, P=None, Pc=None, angle=0, g=9.80665, Method=None)

This function handles calculation of two-phase liquid-gas voidage for flow inside channels. 29 calculation methods are available, with varying input requirements. A correlation will be automatically selected if none is specified.

This function is used to calculate the (liquid) holdup as well, as:

$$\text{holdup} = 1 - \text{voidage}$$

If no correlation is selected, the following rules are used, with the earlier options attempted first:

• TODO: defaults

Parameters

xfloat

Quality of fluid, [-]

rholfloat

Liquid density, [kg/m^3]

rhogfloat

Gas density, [kg/m^3]

Dfloat, optional

Diameter of pipe, [m]

mfloat, optional

Mass flow rate of fluid, [kg/s]

mulfloat, optional

Viscosity of liquid, [Pa*s]

mugfloat, optional

Viscosity of gas, [Pa*s]

sigmafloat, optional

Surface tension, [N/m]

Pfloat, optional

Pressure of fluid, [Pa]

Pcfloat, optional

Critical pressure of fluid, [Pa]

anglefloat, optional

Angle of the channel with respect to the horizontal (vertical = 90), [degrees]

gfloat, optional

Acceleration due to gravity, [m/s^2]

Returns

alphafloat

Void fraction (area of gas / total area of channel), [-]

Other Parameters

Methodstr, optional

A string of the function name to use, as in the dictionary two_phase_voidage_correlations.

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

fluids.two_phase_voidage.liquid_gas_voidage_methods(x, rhol, rhog, D=None, m=None, mul=None, mug=None, sigma=None, P=None, Pc=None, angle=0.0, g=9.80665, check_ranges=False)

This function returns a list of liquid-gas voidage correlation names which can perform the calculation with the provided inputs. The holdup is for two-phase liquid-gas flow inside channels. 29 calculation methods are available, with varying input requirements.

Parameters

xfloat

Quality of fluid, [-]

rholfloat

Liquid density, [kg/m^3]

rhogfloat

Gas density, [kg/m^3]

Dfloat, optional

Diameter of pipe, [m]

mfloat, optional

Mass flow rate of fluid, [kg/s]

mulfloat, optional

Viscosity of liquid, [Pa*s]

mugfloat, optional

Viscosity of gas, [Pa*s]

sigmafloat, optional

Surface tension, [N/m]

Pfloat, optional

Pressure of fluid, [Pa]

Pcfloat, optional

Critical pressure of fluid, [Pa]

anglefloat, optional

Angle of the channel with respect to the horizontal (vertical = 90), [degrees]

gfloat, optional

Acceleration due to gravity, [m/s^2]

check_rangesbool, optional

Added for future use only

Returns

methodslist, only returned if AvailableMethods == True

List of methods which can be used to calculate two-phase liquid-gas voidage with the given inputs.

Examples

>>> len(liquid_gas_voidage_methods(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6, sigma=0.0487, D=0.05))
27

fluids.two_phase_voidage.density_two_phase(alpha, rhol, rhog)

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.

$$\rho_m = \alpha \rho_g + (1-\alpha)\rho_l$$

Parameters

alphafloat

Void fraction (area of gas / total area of channel), [-]

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

Returns

rho_lgfloat

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.

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.

Examples

>>> density_two_phase(.4, 800, 2.5)
481.0

fluids.two_phase_voidage.gas_liquid_viscosity(x, mul, mug, rhol=None, rhog=None, Method=None)

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

xfloat

Quality of fluid, [-]

mulfloat

Viscosity of liquid, [Pa*s]

mugfloat

Viscosity of gas, [Pa*s]

rholfloat, optional

Liquid density, [kg/m^3]

rhogfloat, optional

Gas density, [kg/m^3]

Returns

mu_lgfloat

Liquid-gas viscosity (a suggested definition, potentially useful for empirical work only!) [Pa*s]

Other Parameters

Methodstr, optional

A string of the function name to use, as in the dictionary liquid_gas_viscosity_correlations.

Notes

All of these models converge to the liquid or gas viscosity as the quality approaches either limits. Other definitions have been proposed, such as using only liquid viscosity.

These values cannot just be plugged into single phase correlations!

Examples

>>> gas_liquid_viscosity(x=0.4, mul=1E-3, mug=1E-5, rhol=850, rhog=1.2, Method='Duckler')
1.2092040385066917e-05
>>> gas_liquid_viscosity(x=0.4, mul=1E-3, mug=1E-5)
2.4630541871921184e-05

fluids.two_phase_voidage.gas_liquid_viscosity_methods(rhol=None, rhog=None, check_ranges=False)

This function returns a list of methods which can be used for calculating two-phase liquid-gas viscosity. Six calculation methods are available; three of them require only x, mul, and mug; the other three require rhol and rhog as well.

Parameters

rholfloat, optional

Liquid density, [kg/m^3]

rhogfloat, optional

Gas density, [kg/m^3]

check_rangesbool, optional

Added for compatibility only, never used

Returns

methodslist

List of methods which can be used to calculate two-phase liquid-gas viscosity with the given inputs.

Examples

>>> gas_liquid_viscosity_methods()
['McAdams', 'Cicchitti', 'Lin Kwok']
>>> gas_liquid_viscosity_methods(rhol=1000, rhog=2)
['Beattie Whalley', 'Fourar Bories', 'Duckler', 'McAdams', 'Cicchitti', 'Lin Kwok']

Void Fraction/Holdup Correlations

fluids.two_phase_voidage.Thom(x, rhol, rhog, mul, mug)

Calculates void fraction in two-phase flow according to the model of [1] as given in [2].

$$\alpha = \left[1 + \left(\frac{1-x}{x}\right)\left(\frac{\rho_g} {\rho_l}\right)^{0.89}\left(\frac{\mu_l}{\mu_g}\right)^{0.18} \right]^{-1}$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

mulfloat

Viscosity of liquid [Pa*s]

mugfloat

Viscosity of gas [Pa*s]

Returns

alphafloat

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.

References

1(1,2)

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.

Examples

>>> Thom(.4, 800, 2.5, 1E-3, 1E-5)
0.9801482164042417

fluids.two_phase_voidage.Zivi(x, rhol, rhog)

Calculates void fraction in two-phase flow according to the model of [1] as given in [2] and [3].

$$\alpha = \left[1 + \left(\frac{1-x}{x}\right) \left(\frac{\rho_g}{\rho_l}\right)^{2/3}\right]^{-1}$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

Returns

alphafloat

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].

References

1(1,2)

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.

Examples

>>> Zivi(.4, 800, 2.5)
0.9689339909056356

fluids.two_phase_voidage.Smith(x, rhol, rhog)

Calculates void fraction in two-phase flow according to the model of [1], also given in [2] and [3].

$$\alpha = \left\{1 + \left(\frac{1-x}{x}\right) \left(\frac{\rho_g}{\rho_l}\right)\left[K+(1-K) \sqrt{\frac{\frac{\rho_l}{\rho_g} + K\left(\frac{1-x}{x}\right)} {1 + K\left(\frac{1-x}{x}\right)}}\right] \right\}^{-1}$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

Returns

alphafloat

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.

References

1(1,2,3)

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(1,2)

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.

Examples

>>> Smith(.4, 800, 2.5)
0.959981235534199

fluids.two_phase_voidage.Fauske(x, rhol, rhog)

Calculates void fraction in two-phase flow according to the model of [1], as given in [2] and [3].

$$\alpha = \left[1 + \left(\frac{1-x}{x}\right) \left(\frac{\rho_g}{\rho_l}\right)^{0.5}\right]^{-1}$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

Returns

alphafloat

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.

References

1(1,2)

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(1,2)

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(1,2)

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.

Examples

>>> Fauske(.4, 800, 2.5)
0.9226347262627932

fluids.two_phase_voidage.Chisholm_voidage(x, rhol, rhog)

Calculates void fraction in two-phase flow according to the model of [1], as given in [2] and [3].

$$\alpha = \left[1 + \left(\frac{1-x}{x}\right)\left(\frac{\rho_g} {\rho_l}\right)\sqrt{1 - x\left(1-\frac{\rho_l}{\rho_g}\right)} \right]^{-1}$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

Returns

alphafloat

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.

References

1(1,2)

Chisholm, D. “Pressure Gradients due to Friction during the Flow of Evaporating Two-Phase Mixtures in Smooth Tubes and Channels.” International Journal of Heat and Mass Transfer 16, no. 2 (February 1, 1973): 347-58. doi:10.1016/0017-9310(73)90063-X.

2(1,2)

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(1,2)

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.

Examples

>>> Chisholm_voidage(.4, 800, 2.5)
0.949525900374774

fluids.two_phase_voidage.Turner_Wallis(x, rhol, rhog, mul, mug)

Calculates void fraction in two-phase flow according to the model of [1], as given in [2] and [3].

$$\alpha = \left[1 + \left(\frac{1-x}{x}\right)^{0.72}\left(\frac{\rho_g} {\rho_l}\right)^{0.4}\left(\frac{\mu_l}{\mu_g}\right)^{0.08} \right]^{-1}$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

mulfloat

Viscosity of liquid [Pa*s]

mugfloat

Viscosity of gas [Pa*s]

Returns

alphafloat

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.

References

1(1,2)

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(1,2)

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(1,2)

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.

Examples

>>> Turner_Wallis(.4, 800, 2.5, 1E-3, 1E-5)
0.8384824581634625

fluids.two_phase_voidage.homogeneous(x, rhol, rhog)

Calculates void fraction in two-phase flow according to the homogeneous flow model, reviewed in [1], [2], and [3].

$$\alpha = \frac{1}{1 + \left(\frac{1-x}{x}\right)\frac{\rho_g}{\rho_l}}$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

Returns

alphafloat

Void fraction (area of gas / total area of channel), [-]

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.

Examples

>>> homogeneous(.4, 800, 2.5)
0.995334370139969

fluids.two_phase_voidage.Chisholm_Armand(x, rhol, rhog)

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].

$$\alpha = \frac{\alpha_h}{\alpha_h + (1-\alpha_h)^{0.5}}$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

Returns

alphafloat

Void fraction (area of gas / total area of channel), [-]

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.

Examples

>>> Chisholm_Armand(.4, 800, 2.5)
0.9357814394262114

fluids.two_phase_voidage.Armand(x, rhol, rhog)

Calculates void fraction in two-phase flow according to the model presented in [1] as shown in [2], [3], and [4].

$$\alpha = 0.833\alpha_h$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

Returns

alphafloat

Void fraction (area of gas / total area of channel), [-]

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.

Examples

>>> Armand(.4, 800, 2.5)
0.8291135303265941

fluids.two_phase_voidage.Nishino_Yamazaki(x, rhol, rhog)

Calculates void fraction in two-phase flow according to the model presented in [1] as shown in [2].

$$\alpha = 1 - \left(\frac{1-x}{x}\frac{\rho_g}{\rho_l}\right)^{0.5} \alpha_h^{0.5}$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

Returns

alphafloat

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.

References

1(1,2)

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.

Examples

>>> Nishino_Yamazaki(.4, 800, 2.5)
0.931694583962682

fluids.two_phase_voidage.Guzhov(x, rhol, rhog, m, D)

Calculates void fraction in two-phase flow according to the model in [1] as shown in [2] and [3].

$$\alpha = 0.81[1 - \exp(-2.2\sqrt{Fr_{tp}})]\alpha_h Fr_{tp} = \frac{G_{tp}^2}{gD\rho_{tp}^2} \rho_{tp} = \left(\frac{1-x}{\rho_l} + \frac{x}{\rho_g}\right)^{-1}$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

mfloat

Mass flow rate of both phases, [kg/s]

Dfloat

Diameter of the channel, [m]

Returns

alphafloat

Void fraction (area of gas / total area of channel), [-]

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.

Examples

>>> Guzhov(.4, 800, 2.5, 1, .3)
0.7626030108534588

fluids.two_phase_voidage.Kawahara(x, rhol, rhog, D)

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.

$$\alpha = \frac{C_1 \alpha_h^{0.5}}{1 - C_2\alpha_h^{0.5}}$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

Dfloat

Diameter of the channel, [m]

Returns

alphafloat

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.

References

1(1,2)

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.

Examples

>>> Kawahara(.4, 800, 2.5, 100E-6)
0.9276148194410238

fluids.two_phase_voidage.Baroczy(x, rhol, rhog, mul, mug)

Calculates void fraction in two-phase flow according to the model of [1] as given in [2], [3], and [4].

$$\alpha = \left[1 + \left(\frac{1-x}{x}\right)^{0.74}\left(\frac{\rho_g} {\rho_l}\right)^{0.65}\left(\frac{\mu_l}{\mu_g}\right)^{0.13} \right]^{-1}$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

mulfloat

Viscosity of liquid [Pa*s]

mugfloat

Viscosity of gas [Pa*s]

Returns

alphafloat

Void fraction (area of gas / total area of channel), [-]

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.

Examples

>>> Baroczy(.4, 800, 2.5, 1E-3, 1E-5)
0.9453544598460807

fluids.two_phase_voidage.Tandon_Varma_Gupta(x, rhol, rhog, mul, mug, m, D)

Calculates void fraction in two-phase flow according to the model of [1] also given in [2], [3], and [4].

For 50 < Rel < 1125:

$$\alpha = 1- 1.928Re_l^{-0.315}[F(X_{tt})]^{-1} + 0.9293Re_l^{-0.63} [F(X_{tt})]^{-2}$$

For Rel > 1125:

$$\alpha = 1- 0.38 Re_l^{-0.088}[F(X_{tt})]^{-1} + 0.0361 Re_l^{-0.176} [F(X_{tt})]^{-2}$$

$$F(X_{tt}) = 0.15[X_{tt}^{-1} + 2.85X_{tt}^{-0.476}]$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

mulfloat

Viscosity of liquid [Pa*s]

mugfloat

Viscosity of gas [Pa*s]

mfloat

Mass flow rate of both phases, [kg/s]

Dfloat

Diameter of the channel, [m]

Returns

alphafloat

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:

$$Re_l = \frac{G_{tp}D}{\mu_l}$$

The lower limit on Reynolds number is not enforced.

References

1(1,2)

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(1,2)

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(1,2)

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.

4(1,2)

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.

Examples

>>> Tandon_Varma_Gupta(.4, 800, 2.5, 1E-3, 1E-5, m=1, D=0.3)
0.9228265670341428

fluids.two_phase_voidage.Harms(x, rhol, rhog, mul, mug, m, D)

Calculates void fraction in two-phase flow according to the model of [1] also given in [2] and [3].

$$\alpha = \left[1 - 10.06Re_l^{-0.875}(1.74 + 0.104Re_l^{0.5})^2 \left(1.376 + \frac{7.242}{X_{tt}^{1.655}}\right)^{-0.5}\right]^2$$

$$Re_l = \frac{G_{tp}(1-x)D}{\mu_l}$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

mulfloat

Viscosity of liquid [Pa*s]

mugfloat

Viscosity of gas [Pa*s]

mfloat

Mass flow rate of both phases, [kg/s]

Dfloat

Diameter of the channel, [m]

Returns

alphafloat

Void fraction (area of gas / total area of channel), [-]

Notes

[1] has been reviewed.

References

1(1,2)

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.

Examples

>>> Harms(.4, 800, 2.5, 1E-3, 1E-5, m=1, D=0.3)
0.9653289762907554

fluids.two_phase_voidage.Domanski_Didion(x, rhol, rhog, mul, mug)

Calculates void fraction in two-phase flow according to the model of [1] also given in [2] and [3].

if Xtt < 10:

$$\alpha = (1 + X_{tt}^{0.8})^{-0.378}$$

Otherwise:

$$\alpha = 0.823- 0.157\ln(X_{tt})$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

mulfloat

Viscosity of liquid [Pa*s]

mugfloat

Viscosity of gas [Pa*s]

Returns

alphafloat

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.

References

1(1,2,3)

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(1,2)

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(1,2)

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.

Examples

>>> Domanski_Didion(.4, 800, 2.5, 1E-3, 1E-5)
0.9355795597059169

fluids.two_phase_voidage.Graham(x, rhol, rhog, mul, mug, m, D, g=9.80665)

Calculates void fraction in two-phase flow according to the model of [1] also given in [2] and [3].

$$\alpha = 1 - \exp\{-1 - 0.3\ln(Ft) - 0.0328[\ln(Ft)]^2\}$$

$$Ft = \left[\frac{G_{tp}^2 x^3}{(1-x)\rho_g^2gD}\right]^{0.5}$$

$$\alpha = 0 \text{ for } F_t \le 0.01032$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

mulfloat

Viscosity of liquid [Pa*s]

mugfloat

Viscosity of gas [Pa*s]

mfloat

Mass flow rate of both phases, [kg/s]

Dfloat

Diameter of the channel, [m]

gfloat, optional

Acceleration due to gravity, [m/s^2]

Returns

alphafloat

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.

References

1(1,2)

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(1,2)

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.

Examples

>>> Graham(.4, 800, 2.5, 1E-3, 1E-5, m=1, D=0.3)
0.6403336287530644

fluids.two_phase_voidage.Yashar(x, rhol, rhog, mul, mug, m, D, g=9.80665)

Calculates void fraction in two-phase flow according to the model of [1] also given in [2] and [3].

$$\alpha = \left[1 + \frac{1}{Ft} + X_{tt}\right]^{-0.321}$$

$$Ft = \left[\frac{G_{tp}^2 x^3}{(1-x)\rho_g^2gD}\right]^{0.5}$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

mulfloat

Viscosity of liquid [Pa*s]

mugfloat

Viscosity of gas [Pa*s]

mfloat

Mass flow rate of both phases, [kg/s]

Dfloat

Diameter of the channel, [m]

gfloat, optional

Acceleration due to gravity, [m/s^2]

Returns

alphafloat

Void fraction (area of gas / total area of channel), [-]

Notes

[1] has been reviewed; both [2] and [3] give it correctly.

References

1(1,2)

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(1,2)

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(1,2)

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.

Examples

>>> Yashar(.4, 800, 2.5, 1E-3, 1E-5, m=1, D=0.3)
0.7934893185789146

fluids.two_phase_voidage.Huq_Loth(x, rhol, rhog)

Calculates void fraction in two-phase flow according to the model of [1], also given in [2], [3], and [4].

$$\alpha = 1 - \frac{2(1-x)^2}{1 - 2x + \left[1 + 4x(1-x)\left(\frac {\rho_l}{\rho_g}-1\right)\right]^{0.5}}$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

Returns

alphafloat

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.

References

1(1,2)

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(1,2)

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(1,2)

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.

4(1,2)

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.

Examples

>>> Huq_Loth(.4, 800, 2.5)
0.9593868838476147

fluids.two_phase_voidage.Kopte_Newell_Chato(x, rhol, rhog, mul, mug, m, D, g=9.80665)

Calculates void fraction in two-phase flow according to the model of [1] also given in [2].

$$\alpha = 1.045 - \exp\{-1 - 0.342\ln(Ft) - 0.0268[\ln(Ft)]^2 + 0.00597[\ln(Ft)]^3\}$$

$$Ft = \left[\frac{G_{tp}^2 x^3}{(1-x)\rho_g^2gD}\right]^{0.5}$$

$$\alpha = \alpha_h \text{ for } F_t \le 0.044$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

mulfloat

Viscosity of liquid [Pa*s]

mugfloat

Viscosity of gas [Pa*s]

mfloat

Mass flow rate of both phases, [kg/s]

Dfloat

Diameter of the channel, [m]

gfloat, optional

Acceleration due to gravity, [m/s^2]

Returns

alphafloat

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.

References

1(1,2)

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.

Examples

>>> Kopte_Newell_Chato(.4, 800, 2.5, 1E-3, 1E-5, m=1, D=0.3)
0.6864466770087425

fluids.two_phase_voidage.Steiner(x, rhol, rhog, sigma, m, D, g=9.80665)

Calculates void fraction in two-phase flow according to the model of [1] also given in [2] and [3].

$$\alpha = \frac{x}{\rho_g}\left[C_0\left(\frac{x}{\rho_g} + \frac{1-x} {\rho_l}\right) +\frac{v_{gm}}{G} \right]^{-1}$$

$$v_{gm} = \frac{1.18(1-x)}{\rho_l^{0.5}}[g\sigma(\rho_l-\rho_g)]^{0.25}$$

$$C_0 = 1 + 0.12(1-x)$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

sigmafloat

Surface tension of liquid [N/m]

mfloat

Mass flow rate of both phases, [kg/s]

Dfloat

Diameter of the channel, [m]

gfloat, optional

Acceleration due to gravity, [m/s^2]

Returns

alphafloat

Void fraction (area of gas / total area of channel), [-]

Notes

[1] has been reviewed.

References

1(1,2)

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.

Examples

>>> Steiner(0.4, 800., 2.5, sigma=0.02, m=1, D=0.3)
0.895950181381335

fluids.two_phase_voidage.Rouhani_1(x, rhol, rhog, sigma, m, D, g=9.80665)

Calculates void fraction in two-phase flow according to the model of [1] as given in [2] and [3].

$$\alpha = \frac{x}{\rho_g}\left[C_0\left(\frac{x}{\rho_g} + \frac{1-x} {\rho_l}\right) +\frac{v_{gm}}{G} \right]^{-1}$$

$$v_{gm} = \frac{1.18(1-x)}{\rho_l^{0.5}}[g\sigma(\rho_l-\rho_g)]^{0.25}$$

$$C_0 = 1 + 0.2(1-x)$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

sigmafloat

Surface tension of liquid [N/m]

mfloat

Mass flow rate of both phases, [kg/s]

Dfloat

Diameter of the channel, [m]

gfloat, optional

Acceleration due to gravity, [m/s^2]

Returns

alphafloat

Void fraction (area of gas / total area of channel), [-]

Notes

The expression as quoted in [2] and [3] could not be found in [1].

References

1(1,2)

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(1,2)

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(1,2)

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.

Examples

>>> Rouhani_1(0.4, 800., 2.5, sigma=0.02, m=1, D=0.3)
0.8588420244136714

fluids.two_phase_voidage.Rouhani_2(x, rhol, rhog, sigma, m, D, g=9.80665)

Calculates void fraction in two-phase flow according to the model of [1] as given in [2] and [3].

$$\alpha = \frac{x}{\rho_g}\left[C_0\left(\frac{x}{\rho_g} + \frac{1-x} {\rho_l}\right) +\frac{v_{gm}}{G} \right]^{-1}$$

$$v_{gm} = \frac{1.18(1-x)}{\rho_l^{0.5}}[g\sigma(\rho_l-\rho_g)]^{0.25}$$

$$C_0 = 1 + 0.2(1-x)(gD)^{0.25}\left(\frac{\rho_l}{G_{tp}}\right)^{0.5}$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

sigmafloat

Surface tension of liquid [N/m]

mfloat

Mass flow rate of both phases, [kg/s]

Dfloat

Diameter of the channel, [m]

gfloat, optional

Acceleration due to gravity, [m/s^2]

Returns

alphafloat

Void fraction (area of gas / total area of channel), [-]

Notes

The expression as quoted in [2] and [3] could not be found in [1].

References

1(1,2)

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(1,2)

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(1,2)

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.

Examples

>>> Rouhani_2(0.4, 800., 2.5, sigma=0.02, m=1, D=0.3)
0.44819733138968865

fluids.two_phase_voidage.Nicklin_Wilkes_Davidson(x, rhol, rhog, m, D, g=9.80665)

Calculates void fraction in two-phase flow according to the model of [1] as given in [2] and [3].

$$\alpha = \frac{x}{\rho_g}\left[C_0\left(\frac{x}{\rho_g} + \frac{1-x} {\rho_l}\right) +\frac{v_{gm}}{G} \right]^{-1}$$

$$v_{gm} = 0.35\sqrt{gD}$$

$$C_0 = 1.2$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

mfloat

Mass flow rate of both phases, [kg/s]

Dfloat

Diameter of the channel, [m]

gfloat, optional

Acceleration due to gravity, [m/s^2]

Returns

alphafloat

Void fraction (area of gas / total area of channel), [-]

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.

Examples

>>> Nicklin_Wilkes_Davidson(0.4, 800., 2.5, m=1, D=0.3)
0.6798826626721431

fluids.two_phase_voidage.Gregory_Scott(x, rhol, rhog)

Calculates void fraction in two-phase flow according to the model of [1] as given in [2] and [3].

$$\alpha = \frac{x}{\rho_g}\left[C_0\left(\frac{x}{\rho_g} + \frac{1-x} {\rho_l}\right) +\frac{v_{gm}}{G} \right]^{-1}$$

$$v_{gm} = 0$$

$$C_0 = 1.19$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

Returns

alphafloat

Void fraction (area of gas / total area of channel), [-]

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.

Examples

>>> Gregory_Scott(0.4, 800., 2.5)
0.8364154370924108

fluids.two_phase_voidage.Dix(x, rhol, rhog, sigma, m, D, g=9.80665)

Calculates void fraction in two-phase flow according to the model of [1] as given in [2] and [3].

$$\alpha = \frac{x}{\rho_g}\left[C_0\left(\frac{x}{\rho_g} + \frac{1-x} {\rho_l}\right) +\frac{v_{gm}}{G} \right]^{-1}$$

$$v_{gm} = 2.9\left(g\sigma\frac{\rho_l-\rho_g}{\rho_l^2}\right)^{0.25}$$

$$C_0 = \frac{v_{sg}}{v_m}\left[1 + \left(\frac{v_{sl}}{v_{sg}}\right) ^{\left(\left(\frac{\rho_g}{\rho_l}\right)^{0.1}\right)}\right]$$

$$v_{gs} = \frac{mx}{\rho_g \frac{\pi}{4}D^2}$$

$$v_{ls} = \frac{m(1-x)}{\rho_l \frac{\pi}{4}D^2}$$

$$v_m = v_{gs} + v_{ls}$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

sigmafloat

Surface tension of liquid [N/m]

mfloat

Mass flow rate of both phases, [kg/s]

Dfloat

Diameter of the channel, [m]

gfloat, optional

Acceleration due to gravity, [m/s^2]

Returns

alphafloat

Void fraction (area of gas / total area of channel), [-]

Notes

Has formed the basis for several other correlations.

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.

Examples

>>> Dix(0.4, 800., 2.5, sigma=0.02, m=1, D=0.3)
0.8268737961156514

fluids.two_phase_voidage.Sun_Duffey_Peng(x, rhol, rhog, sigma, m, D, P, Pc, g=9.80665)

Calculates void fraction in two-phase flow according to the model of [1] as given in [2] and [3].

$$\alpha = \frac{x}{\rho_g}\left[C_0\left(\frac{x}{\rho_g} + \frac{1-x} {\rho_l}\right) +\frac{v_{gm}}{G} \right]^{-1}$$

$$v_{gm} = 1.41\left[\frac{g\sigma(\rho_l-\rho_g)}{\rho_l^2}\right]^{0.25}$$

$$C_0 = \left(0.82 + 0.18\frac{P}{P_c}\right)^{-1}$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

sigmafloat

Surface tension of liquid [N/m]

mfloat

Mass flow rate of both phases, [kg/s]

Dfloat

Diameter of the channel, [m]

Pfloat

Pressure of the fluid, [Pa]

Pcfloat

Critical pressure of the fluid, [Pa]

gfloat, optional

Acceleration due to gravity, [m/s^2]

Returns

alphafloat

Void fraction (area of gas / total area of channel), [-]

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.

Examples

>>> Sun_Duffey_Peng(0.4, 800., 2.5, sigma=0.02, m=1, D=0.3, P=1E5, Pc=7E6)
0.7696546506515833

fluids.two_phase_voidage.Xu_Fang_voidage(x, rhol, rhog, m, D, g=9.80665)

Calculates void fraction in two-phase flow according to the model developed in the review of [1].

$$\alpha = \left[1 + \left(1 + 2Fr_{lo}^{-0.2}\alpha_h^{3.5}\right)\left( \frac{1-x}{x}\right)\left(\frac{\rho_g}{\rho_l}\right)\right]^{-1}$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

mfloat

Mass flow rate of both phases, [kg/s]

Dfloat

Diameter of the channel, [m]

gfloat, optional

Acceleration due to gravity, [m/s^2]

Returns

alphafloat

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.

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.

Examples

>>> Xu_Fang_voidage(0.4, 800., 2.5, m=1, D=0.3)
0.9414660089942093

fluids.two_phase_voidage.Woldesemayat_Ghajar(x, rhol, rhog, sigma, m, D, P, angle=0, g=9.80665)

Calculates void fraction in two-phase flow according to the model of [1].

$$\alpha = \frac{v_{gs}}{v_{gs}\left(1 + \left(\frac{v_{ls}}{v_{gs}} \right)^{\left(\frac{\rho_g}{\rho_l}\right)^{0.1}}\right) + 2.9\left[\frac{gD\sigma(1+\cos\theta)(\rho_l-\rho_g)} {\rho_l^2}\right]^{0.25}(1.22 + 1.22\sin\theta)^{\frac{P}{P_{atm}}}}$$

$$v_{gs} = \frac{mx}{\rho_g \frac{\pi}{4}D^2}$$

$$v_{ls} = \frac{m(1-x)}{\rho_l \frac{\pi}{4}D^2}$$

Parameters

xfloat

Quality at the specific tube interval []

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

sigmafloat

Surface tension of liquid [N/m]

mfloat

Mass flow rate of both phases, [kg/s]

Dfloat

Diameter of the channel, [m]

Pfloat

Pressure of the fluid, [Pa]

anglefloat

Angle of the channel with respect to the horizontal (vertical = 90), [degrees]

gfloat, optional

Acceleration due to gravity, [m/s^2]

Returns

alphafloat

Void fraction (area of gas / total area of channel), [-]

Notes

Strongly recommended.

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.

Examples

>>> Woldesemayat_Ghajar(0.4, 800., 2.5, sigma=0.2, m=1, D=0.3, P=1E6, angle=45)
0.7640815513429202

Utilities

fluids.two_phase_voidage.Lockhart_Martinelli_Xtt(x, rhol, rhog, mul, mug, pow_x=0.9, pow_rho=0.5, pow_mu=0.1, n=None)

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.

$$X_{tt} = \left(\frac{1-x}{x}\right)^{0.9} \left(\frac{\rho_g}{\rho_l} \right)^{0.5}\left(\frac{\mu_l}{\mu_g}\right)^{0.1}$$

$$X_{tt} = \left(\frac{1-x}{x}\right)^{(2-n)/2} \left(\frac{\rho_g} {\rho_l}\right)^{0.5}\left(\frac{\mu_l}{\mu_g}\right)^{n/2}$$

Parameters

xfloat

Quality at the specific tube interval [-]

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

mulfloat

Viscosity of liquid [Pa*s]

mugfloat

Viscosity of gas [Pa*s]

pow_xfloat, optional

Power for the phase ratio (1-x)/x, [-]

pow_rhofloat, optional

Power for the density ratio rhog/rhol, [-]

pow_mufloat, optional

Power for the viscosity ratio mul/mug, [-]

nfloat, optional

Number to be used for calculating pow_x and pow_mu if provided, [-]

Returns

Xttfloat

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.

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.

Examples

>>> Lockhart_Martinelli_Xtt(0.4, 800, 2.5, 1E-3, 1E-5)
0.12761659240532292

fluids.two_phase_voidage.two_phase_voidage_experimental(rho_lg, rhol, rhog)

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.

$$\alpha = \frac{\rho_m - \rho_l}{\rho_g - \rho_l}$$

Parameters

rho_lgfloat

Two-phase effective density [kg/m^3]

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

Returns

alphafloat

Void fraction (area of gas / total area of channel), [-]

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.

Examples

>>> two_phase_voidage_experimental(481.0, 800, 2.5)
0.4

Gas/Liquid Viscosity

fluids.two_phase_voidage.Beattie_Whalley(x, mul, mug, rhol, rhog)

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].

$$\mu_m = \mu_l(1-\alpha_m)(1 + 2.5\alpha_m) + \mu_g\alpha_m$$

$$\alpha_m = \frac{1}{1 + \left(\frac{1-x}{x}\right)\frac{\rho_g}{\rho_l}} \text{(homogeneous model)}$$

Parameters

xfloat

Quality of the gas-liquid flow, [-]

mulfloat

Viscosity of liquid, [Pa*s]

mugfloat

Viscosity of gas, [Pa*s]

rholfloat

Density of the liquid [kg/m^3]

rhogfloat

Density of the gas [kg/m^3]

Returns

mu_lgfloat

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.

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.

Examples

>>> Beattie_Whalley(x=0.4, mul=1E-3, mug=1E-5, rhol=850, rhog=1.2)
1.7363806909512365e-05

fluids.two_phase_voidage.McAdams(x, mul, mug)

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].

$$\mu_m = \left(\frac{x}{\mu_g} + \frac{1-x}{\mu_l}\right)^{-1}$$

Parameters

xfloat

Quality of the gas-liquid flow, [-]

mulfloat

Viscosity of liquid, [Pa*s]

mugfloat

Viscosity of gas, [Pa*s]

Returns

mu_lgfloat

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.

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(1,2)

Kim, Sung-Min, and Issam Mudawar. “Review of Databases and Predictive Methods for Pressure Drop in Adiabatic, Condensing and Boiling Mini/Micro-Channel Flows.” International Journal of Heat and Mass Transfer 77 (October 2014): 74-97. doi:10.1016/j.ijheatmasstransfer.2014.04.035.

Examples

>>> McAdams(x=0.4, mul=1E-3, mug=1E-5)
2.4630541871921184e-05

fluids.two_phase_voidage.Cicchitti(x, mul, mug)

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].

$$\mu_m = x\mu_g + (1-x)\mu_l$$

Parameters

xfloat

Quality of the gas-liquid flow, [-]

mulfloat

Viscosity of liquid, [Pa*s]

mugfloat

Viscosity of gas, [Pa*s]

Returns

mu_lgfloat

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.

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.

Examples

>>> Cicchitti(x=0.4, mul=1E-3, mug=1E-5)
0.0006039999999999999

fluids.two_phase_voidage.Lin_Kwok(x, mul, mug)

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].

$$\mu_m = \frac{\mu_l \mu_g}{\mu_g + x^{1.4}(\mu_l - \mu_g)}$$

Parameters

xfloat

Quality of the gas-liquid flow, [-]

mulfloat

Viscosity of liquid, [Pa*s]

mugfloat

Viscosity of gas, [Pa*s]

Returns

mu_lgfloat

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.

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.

Examples

>>> Lin_Kwok(x=0.4, mul=1E-3, mug=1E-5)
3.515119398126066e-05

fluids.two_phase_voidage.Fourar_Bories(x, mul, mug, rhol, rhog)

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].

$$\mu_m = \rho_m\left(\sqrt{x\nu_g} + \sqrt{(1-x)\nu_l}\right)^2$$

Parameters

xfloat

Quality of the gas-liquid flow, [-]

mulfloat

Viscosity of liquid, [Pa*s]

mugfloat

Viscosity of gas, [Pa*s]

rholfloat

Density of the liquid, [kg/m^3]

rhogfloat

Density of the gas, [kg/m^3]

Returns

mu_lgfloat

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:

$$\mu_m = \rho_m\left(x\nu_g + (1-x)\nu_l + 2\sqrt{x(1-x)\nu_g\nu_l} \right)$$

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).

Examples

>>> Fourar_Bories(x=0.4, mul=1E-3, mug=1E-5, rhol=850, rhog=1.2)
2.127617150298565e-05

fluids.two_phase_voidage.Duckler(x, mul, mug, rhol, rhog)

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].

$$\mu_m = \frac{\frac{x\mu_g}{\rho_g} + \frac{(1-x)\mu_l}{\rho_l} } {\frac{x}{\rho_g} + \frac{(1-x)}{\rho_l} }$$

Parameters

xfloat

Quality of the gas-liquid flow, [-]

mulfloat

Viscosity of liquid, [Pa*s]

mugfloat

Viscosity of gas, [Pa*s]

rholfloat

Density of the liquid, [kg/m^3]

rhogfloat

Density of the gas, [kg/m^3]

Returns

mu_lgfloat

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:

$$\mu_m = \rho_m \left[x\left(\frac{\mu_g}{\rho_g}\right) + (1 - x)\left(\frac{\mu_l}{\rho_l}\right)\right]$$

According to the homogeneous definition of two-phase density.

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).

Examples

>>> Duckler(x=0.4, mul=1E-3, mug=1E-5, rhol=850, rhog=1.2)
1.2092040385066917e-05