Two Phase Flows
6. Void Fraction Correlations
In order to calculate the change of static pressure along a two-phase flow system, it is essential to obtain the void fraction of the mixture at every point in the flow.
The number of void fraction correlations that have been developed is extensive.
At present, there are over thirty different methods of correlating void fraction with other flow parameters. Most of these relationships are based on one and two component adiabatic flow data. Systematic comparisons have been made between accepted correlations and data banks containing void fraction or density measurements. Ten parameters may be considered to affect the void fraction of the mixture for two-phase flow in a round, straight pipe under adiabatic correlations:
a) the liquid and gas (or vapor) mass flow rates (Wf,Wg), b) the liquid and gas (or vapor) densities (ρf, ρg),
c) the liquid and gas (or vapor) viscosities (µf, µg), d) the surface tension (σ),
e) the pipe inside diameter (D), f) the pipe roughness (ε),
g) the orientation of the pipe (θ).
The effects of heat flux and possible mass transfer between the phases
(evaporation or condensation), together with the effects of complex geometries, bends and sudden flow changes should be considered for more general
conditions.
po– pi= w2
Ai2 1 – AiAo
2 1 – x2 ρf 1 – α + xρgα2
6.1 Homogeneous model The void fraction is given by:
The homogeneous void fraction correlation applies best for flows where the two phases are almost similar in density, single component flows near critical
pressure. This correlation predicts, on average, higher values of void than are found experimentally. As expected, the correlation holds near the conditions of 100% void for all flows.
The prediction of mean density using the homogeneous theory always gives an underestimated value. This error increases with increase in the ratio of the density of the phases. Under all conditions the error on predicted mean density decreases with increase in mass velocity.
6.2 Von Glahn correlation The Von Glahn correlation has the form:
The exponents were derived empirically to obtain the best fit to available steam-water data for flow in unheated pipes.
The predicting ability of the Von Glahn correlation gets worse as the phase density ratio decreases and with increased velocity.
6.3 Rooney correlation The Rooney correlation has the form
α = 1
1 + 1 – x x y0.67
y0.1
where y =ρg
ρf
α = x νg 1 – x νf+ x νg
The Rooney correlation does not necessarily fulfil the end condition that α → 1 as β → 1. Unless very low mass velocity flows are considered, the correlation will give correct values at 100% void. Mass velocity has little effect on the prediction of this correlation except for large phase density ratios when an increase in mass velocity improves the predicted value.
6.4 Armand and Massena correlation
This modified homogeneous void fraction correlation has the following form:
The Armand and Massena correlation is a modification of Armand’s correlation to allow void fraction α → 1 as x → 1, based on steam-water flow data. The Armand and Massena correlation only predicts correctly for high voids for high pressure steam flows.
6.5 Smith Correlation
The Smith correlation is based on an annular flow model with liquid and mist phases, each having the same momentum flux. The entrainment of the liquid into the gas phase was found empirically. The entrainment coefficient is defined as:
The slip ratio takes the form e = liquid entrained
The Smith correlation predictions are more accurate in horizontal flow than in vertical flow. The predicted void is generally underestimated for high pressure steam flows. In all conditions the predictions improve at high mass velocity.
6.6 Chisholm correlation
The Chisholm correlation has the following form for the slip ratio:
Figures 16 shows the calculated voids versus quality for a series of void correlations. It can be noted that the homogeneous model predicts the highest void for any given quality.
S = 1 + x νg
νf – 1
1 / 2
for β < 0.9 S = e + 1 – e
νg
νf + e 1 – xx 1 + e 1 – xx
1 / 2
Figure 16
Void versus quality for different correlations
Two phase flow is an area of continuous research and development. The
introduction of new calculation tools does not prevent the necessary knowledge of experimental results on which the numerous correlations are based on. The careful selection of the solution scheme along with the use of appropriate correlations is of utmost importance when solving two phase phenomena.
1.0
0.8
0.6
0.4
0.2
0.0
0.0 10.0 20.0 30.0 40.0 50.0
Quality (%) VHOMO
VROON
VARMAN
VGLAHN
Void Fraction
60.0 70.0 80.0 90.0 100.0
7. Notation
Symbol Description SI Units
A Flow area m2
Ag Flow area occupied by gaseous phase m2
Af Flow area occupied by liquid phase m2
D Pipe diameter m
Dh Hydraulic diameter m
e Entrainment coefficient f Single-phase friction factor ftp Two-phase friction factor
g Acceleration due to gravity m/s2
G Mass flux kg/s.m2
hfg Latent heat of vaporization J/kg
j Volumetric flux (superficial velocity) m/s
jf Superficial velocity of liquid phase m/s
jg Superficial velocity of vapor phase m/s
k Coefficient in modified homogeneous model (α = kβ)
P Fluid pressure Pa
q Heat flux W/m2
Q Volumetric rate of flow m3/s
Qf Volumetric rate of flow of liquid phase m3/s Qg Volumetric rate of flow of vapor phase m3/s
Rc Radius of curvature of bend m
S Slip ratio
T Temperature °C
TF, TL Bulk liquid temperature °C
Tg Vapor temperature °C
Tsat Saturation temperature °C
u Velocity m/s
Uf Actual velocity of liquid phase m/s
Ug Actual velocity of vapor phase m/s
V Velocity of fluid m/s
νf Specific volume of liquid m3/kg
νg Specific volume of vapor m3/kg
w Angular velocity kg/s
W Mass rate of flow kg/s
Wf Mass rate of flow of liquid phase kg/s
Wg Mass rate of flow of vapor phase kg/s
x, xIN Thermodynamic quality XLM Lockhart-Martinelli parameter
z Length in direction of flow m
Greek
αf Homogeneous liquid volume fraction
α Void fraction
β Volumetric quality
ε Pipe roughness m
δ1 Nominal thickness of viscous sublayer m
ρ Density kg/m3
ρf Liquid density kg/m3
ρg Vapor density kg/m3
ρH Mean density of homogeneous fluid kg/m3
ρm Mean density of two-phase fluid kg/m3
σtp Two-phase density kg/m3
φ heat flux W/m2
φ2 Two-phase friction multiplier
φ2LO Two-phase frictional multiplier based on pressure gradient for total flow assumed liquid φ2MN Martinelli-Nelson multiplier
µ Absolute viscosity N.s/m2
µf Liquid viscosity N.s/m2
µg Vapor viscosity N.s/m2
µTP Two-phase viscosity N.s/m2
σ Surface tension for planar interface N/m
θ Orientation of pipe: angle to horizontal deg.
Dimensionless Numbers
Fr Froude Number
Re Reynolds Number
Subscripts
f liquid phase
g vapor phase
lo liquid only
sc subcooled
tp two-phase
8. References
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Carver, M.B., Carlucci, L.N. and Inch, W.W.R., “Thermal-Hydraulics in Recirculating Steam Generators - THIRST Code User’s Manual”, AECL-7254, April 1981.
Banerjee, S. Hetsroni, G., Hewitt, G.F. and Yadigaroglw, G., “Multiphase Flow and Heat Transfer: Bases and Applications”, Short Course held at the University of California (Santa Barbara), January 9-13, 1989.
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Taitel, Y. and Dukler, A.E., “A Model For Predicting Flow Regime Transitions in Horizontal and Non Horizontal Gas-Liquid Flow”, ASME 75-WA/HT-29, 1975.
Carlucci, L.N., “Review of Fluid Flow Correlations For Steam Generator Thermal-Hydraulic Analysis”, CRNL-1999, 1980 January.
Freidel, L., “Mean Void Fraction and Friction Pressure Drop: A Comparison of Some Correlations With Experimental Data”, Paper A7, European Two Phase Flow Group Meeting, Grenoble, 1977.
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Taitel, Y., and Duckler, A.E., “Flow Pattern Transitions in Gas-Liquid Systems:
Measurement and Modelling”, Multi-phase Science and Technology, Vol. II, pp. 1-94, 1986.
Jones, A.B., and Dight, D.G., “Hydrodynamic Stability of a Boiling Channel”, Part 2, KAPL-2208, 1962.
Zuber, N., and Findlay, J., “Average Volumetric Concentration in Two-Phase Flow Systems”, Trans. ASME J. Heat Transfer, 87, 453 (1965).
Wallis, G.B., “One Dimensional Two-Phase Flow”, McGraw-Hill Book Company, 1969.