## Void Fraction Correlations for Vertical Upward Two-Phase Flow in Pipes

Afshin J. Ghajar and Clement C. Tang July 2, 2010

Description of Case

Vertical upward two-phase flow in pipes is found commonly in industries involving oil and gas production, water treatment, nuclear reactors, and geothermal systems. Whether the two-phase flow exists in the form of different components (e.g., air and water) or occurs as a result of phase change caused by evaporation or condensation of a single fluid, the void fraction is an important parameter in the analysis of pressure drop, heat transfer, and mass transfer. For example, void fraction is significant in estimating the reactivity of a boiling water reactor (BWR), in which light water is used as neutron moderator and coolant. To predict the void fraction in vertical upward two-phase flow with reliable accuracy, methods to estimate void fraction correctly and accurately are essential.

Statement of the Problem
Researchers have made considerable efforts to develop void fraction correlations for vertical upward two-phase flows. Currently, dozens of void fraction correlations for different flow patterns are available in the literature. Although Woldesemayat and Ghajar¹ have proposed a general void fraction correlation that is robust and suitable for various flow patterns, gas-liquid combinations, and pipe inclination angles, that correlation has a minor drawback. From the point of view of a general void fraction correlation, their correlation has been verified to be reliably accurate when compared with experimental data obtained from various sources with different experimental facilities. However, when engineers are dealing specifically with vertical upward two-phase flow, other correlations are available to provide predictions with an improvement in accuracy. Since vertical upward two-phase flows are relatively common in industry, void fraction correlations that can provide the greatest degree of accuracy need to be sorted out.

Description of Solution
A comprehensive comparison of more than 50 void fraction correlations with a total of 1,208 experimental data points compiled from 10 independent sources for gas-liquid combinations and pipe diameters has yielded two correlations that deliver excellent results.² The first correlation was proposed by Nicklin and associates,³ and the other by Rouhani and Axelsson.4 Both correlations were based on the drift flux model and have the following expression:

The two-phase distribution coefficient (C0) and gas drift velocity (ugm) are given as follows:

Nicklin and associates³:

Rouhani and Axelsson4:

where the subscripts l and g refer to the liquid phase and the gas phase and:
C0 = two-phase distribution coefficient
D = tube diameter, m
g = gravitational acceleration, m/s²
ugm = gas drift velocity, m/s
Vsg = superficial gas velocity, m/s
Vsl = superficial liquid velocity, m/s
x = flow quality
α = void fraction
ρ = density, kg/m³
σ = surface tension, N/m

Description of Results
The correlations by Nicklin and associates³ and Rouhani and Axelsson4 were validated and compared with a total of 1,208 experimental data points. The experimental data were compiled from various sources with different experimental facilities, which included various gas-liquid combinations and pipe diameters (ranging from 12.7 to 76.0 mm) for vertical upward flow. The results of the comparison of void fraction correlations with experimental data are summarized in Table 1. Although that table indicates that the correlation by Nicklin and associates³ has predicted more data points within the error bands of ±15% and ±20% than has the correlation by Rouhani and Axelsson,4 further scrutiny has indicated that the Nicklin and associates correlation did not perform as well in the range of 0.8 to 1.0 void fraction. The Rouhani and Axelsson correlation, in contrast, was found to perform satisfactorily for the entire range of void fraction (0 < α < 1).

Figures 1 and 2 show the comparison of the Rouhani and Axelsson and Nicklin and associates correlations with the entire experimental database of 1,208 data points. Figure 1 shows that the Rouhani and Axelsson correlation performed satisfactorily for the entire range of void fraction, with a slight tendency toward underprediction in the range of 0.8 to 1.0. By contrast, Figure 2 shows that the Nicklin and associates correlation conspicuously underpredicted the experimental data in the range of 0.8 to 1.0. Hence, the Nicklin and associates correlation is expected to work satisfactorily, possibly with better accuracy than the Rouhani and Axelsson correlation, in the 0.0 to 0.8 range. The Rouhani and Axelsson correlation, however, is expected to perform satisfactorily for the entire void fraction range (0 < α < 1).

Figure 1: Comparison of Void Fraction Correlation by
Rouhani and Axelsson4 with 1,208 Experimental Data Points

Figure 2: Comparison of Void Fraction Correlation by
Nicklin and associates³ with 1,208 Experimental Data Points

Table 1: Results of the Correlations When
Compared with All 1,208 Experimental Data Points

Wider Applicability of Results
The void fraction correlations shown in Equations (1) to (3) are applicable for estimating void fraction of vertical upward two-phase flow in pipes, which may include boiling and nonboiling flows. Vertical upward two-phase flows are used in offshore exploration for oil and gas. Offshore oil fields are being developed in deeper water, where vertical pipelines are used for transporting oil and gas from the seabed to the floating production vessel. In such cases, accurate void fraction results are important in the estimation of pressure drop of the two-phase flow in the pipelines. Another possible application of the void fraction correlations is in the membrane bioreactors (MBRs) used in the wastewater treatment process. A technique to reduce fouling on the membrane surface involves gas sparging, in which gas is injected to generate a gas-liquid two-phase cross-flow (typically slug flow).5 Slug flow induces an increase in surface shear stress, thus removing the fouling layers. In ozonation systems, void fraction is an important parameter in analyzing the influence of the ozone mass transfer on water treatment. Other wider applications in vertical upward two-phase flows can be found in nuclear reactor technology and pumping systems for vertical geothermal wells.

References
1. Woldesemayat, M. A., and Ghajar, A. J. Comparison of Void Fraction Correlations for Different Flow Patterns in Horizontal and Upward Inclined Pipes. International Journal of Multiphase Flow, vol. 33, no. 4, pp. 347–370, 2007.
2. Godbole, P. V. Study of Flow Patterns and Void Fraction in Vertical Upward Two-Phase Flow. M.S. thesis, Oklahoma State University, Stillwater, 2009.
3. Nicklin, D. J., Wilkes, J. O., and Davidson, J. F. Two-Phase Flow in Vertical Tubes. Chemical Engineering Research and Design, vol. 40, pp. 61–68, 1962.
4. Rouhani, S. Z., and Axelsson, E., Calculation of Void Volume Fraction in the Subcooled and Quality Boiling Regions. International Journal of Heat and Mass Transfer, vol. 13, no. 2, pp. 383–393, 1970.
5. Psoch, C., and Schiewer, S. Long-Term Study of an Intermittent Air Sparged MBR for Synthetic Wastewater Treatment. Journal of Membrane Science, vol. 260, no. 1–2, pp. 56–65, 2005.