Liquid Holdup Correlations for Inclined Two-Phase Flow
Authors: Mukherjee Hemanta , Brill James P. 
Published: December 2007

Summary
Liquid holdup behavior in two-phase inclined flow was studied in an inclined pipe-flow simulator. Two sets of empirical equations, one each for uphill and downhill flow, are presented. For downhill stratified flow, a third equation is presented. The liquid holdup equations are functions of dimensionless liquid and gas velocity numbers in addition to liquid viscosity number and angle of inclination. These four parameters uniquely define the flow-pattern transitions in inclined two-phase flow. Consequently, the holdup equations are also implicitly flow-pattern dependent.

Introduction
An accurate prediction of liquid holdup is required to compute the hydrostatic head loss in two-phase inclined flow. In this case, hydrostatic head may be the most important of the pressure gradient components. There are many liquid holdup correlations in the literature, but almost all are for horizontal or vertical uphill flow. Eaton et al. proposed a holdup correlation based on data for natural gas, water, crude oil, and distillate oil mixtures in 2-and 4-in. (5- and 10-cm) diameter horizontal pipes. This correlation is based on five dimensionless groups reflecting various physical properties, flow rates, system pressures, and pipe diameters. A study by Vohra et al. showed that this correlation performed best on a collection of horizontal data taken by Eaton et al. and Beggs and Brill. Cunliffe found that using the Eaton et al correlation to predict the total liquid volume in a wet gas pipeline was quite successful for determining the incremental volume of liquid removed from rate increases. Using dynamic similarity analysis, Dukler et al developed a holdup correlation for horizontal two-phase flow. This holdup correlation is implicit in liquid holdup, requiring an iterative calculation. Experience has shown that most wet gas-transmission applications will result in a no-slip liquid holdup calculation when the Dukler et al. correlation is used. The Beggs and Brill correlation was developed from data obtained in an air/water flow system with 1- and 1 1/2-in. (2.5-and 3.8-cm) diameter pipes. They considered a range of inclination angles from 0 to 1900. Use pipes. They considered a range of inclination angles from 0 to 1900. Use of the correlation requires first determining the holdup for horizontal flow according to predicted horizontal flow patterns. The horizontal holdup is then corrected for angle of inclination. Palmer found that the Beggs and Brill liquid holdup was overpredicted for both uphill and downhill flow and suggested proper correction factors. For uphill flow from 0 to 9, Guzhov et al. proposed a holdup correlation that is independent of inclination angle. Hughmark and Pressburg developed a general holdup correlation for gas/liquid flow covering a wide range of physical properties and diameters. This correlation is based on data taken physical properties and diameters. This correlation is based on data taken in 1-in. (2.54-cm) diameter pipe for vertical uphill flow of air, water, oils of different viscosities, and carefully selected data of other investigators. In addition to these empirical liquid holdup correlations, at least two analytical holdup correlations deserve mention. Bonnecaze et al. developed a slug flow model for inclined pipe based on a mass and force balance around a simplified slug unit. The pressure drop contributions caused by the liquid film and the gas bubble were neglected. Using this holdup correlation, they correlated pressure drop data obtained in 1 1/2-in. (3.8-cm) diameter pipe inclined at various angles around 1100 to find an expression for friction factor. The holdup equation and friction factor correlation were compared with field data taken in a 6-in. (15.24-cm) diameter, 10,000-ft (3048-m) long pipe with a maximum deviation of 5%. With a very similar mechanistic approach, Singh and Griffith proposed a simple model for two-phase slug flow in inclined pipes. Most of their model parameters were experimentally determined using five different diameters of copper pipe at 5, 10, and 150 inclinations with an air/water system. For stratified flow, the authors developed a holdup model based on Chezy's open-channel flow equation.
JPT
p. 1003