Verification of methods for calculating the gas volume fraction in the vertical descending flow of two-phase gas-liquid mixtures

: The paper presents the results of research on the vertical falling flow and their analysis. Methods for calculating the gas volume fraction, which are characterized by high accuracy, and are often proposed in the literature. Their accuracy was presented, as well as the methods with the highest computational usefulness when designing devices in which two-phase gas - liquid flow is used.


INTRODUCTION
Two-phase flow occurs in many branches of industry. It is also used in many technological processes such as sedimentation, fluidization, filtration, etc. For this reason, two-phase flows are the goal of many tests and analyzes. In the two-phase flow there is simultaneous flow of the continuous phase, which is a gas or liquid and a dispersed phase, where it is a solid, liquid or gas.
A characteristic feature of two-phase flow is the phase separation boundary, which forms and changes during movement. The individual phases of the twophase mixture should be able to be separated mechanically by, e.g. centrifugation, filtration or percolation.
The basic parameters that characterize two-phase flow are: − two-phase flow resistance, − two-phase flow structure, − volume fraction of one of the phases.
Determining the volume fraction of the gas is very complicated due to the density of the flowing mixture, which is related to the value of the gas volume fraction R G or the value of the volume fraction of the liquid 1-R G . The value of the volume fraction is influenced by, inter alia, occurrence of phase slip phenomenon, which should be taken into mind that both phases flow at different velocity.
If no phase slip is taken into account, the volume fraction takes the form (1): .
However, the volume share, taking into account the phase slip, generally takes the form (2): , (2) where: ρ g -gas density, kg/m 3 , ρ c -liquid density, kg/m 3 , w g -gas velocity, m/s, w c -liquid velocity, m/s. In the literature various calculation models are presented: non-slip models, where value w g /w c = 1 and slip models, in which w g /w c > 1. The calculation models of the average value of gas volume fraction presented by the authors often differ not only in the form itself, but also in the ranges of use as well as the types of two-phase mixtures.
The experimental data used for the analysis of calculation methods was taken from the dataset of Department Process Engineering of University of Technology in Opole [17]. The tests were carried out in vertical cannels with a diameter of 20, 24, 25, 32, 44 and 50 mm, with co-current descending two-phase gas and liquid flow. The range of apparent gas velocities (w g ) was 0.01-75 m/s and the liquid (w c ) 0.01-2 m/s. The preliminary assessment allowed for the selection of 12 calculation methods (Tab. 1), determining the gas volume fraction characterizing the interfacial slip (e.g. the Lockhart-Martinnelli model) and the different validity of using only in narrow ranges or for selected two-phase mixtures, e.g. the Zuber-Findlay method.
The assessment of the accuracy of individual methods and their usefulness in determining the gas volume participation consisted in determining the characteristic statistical parameters including the determination of the mean value of the relative error δR g as well as the mean value of the absolute error │δR g │(Tab. 2).  Figure 1 shows a graphic interpretation of comparisons of computational methods that were characterized by high accuracy of calculations. The best accuracy with experimental data is characterized by the Armand and Chen method (Figs. 2-3) for which approximately 70% of the points are within ± 30% of the absolute error. Both methods are characterized by high accuracy, because they are based on a slip model that best reflects the effect of viscosity changes and liquid density on the value of gas volume fraction. In addition, the Armand and Chen methods include interfacial slip, which contains the ratio of the actual velocities of the individual phases. The impact of selected parameters on the calculation results (distribution of points) in the R g,obl. -R g,zm system was evaluated. Graphical interpretations of selected calculation methods for gas volume fraction for various mixtures are shown in the Figures 2-13.

CONCLUSIONS
Determining the volume fraction of gas in twophase flow as one of the three most important parameters is immensely important, so can be found so many calculation methods that have been proposed by different researchers. The authors of individual methods make their accuracy dependent on hydrodynamic parameters and ranges of applicability. Determining the volume fraction of gas is necessary to determine other parameters, including densities of a two-phase mixture that guarantee the correct design of equipment and apparatus, where two-phase flows are used.
After analyzing the results of the volume fraction of gas using the methods proposed by the authors, it should be noted that in different ranges the volume fraction of gas does not coincide with the obtained experimental results. In a wide range of changes in flow parameters as well as physicochemical properties of two-phase mixture components, the highest accuracy of results is obtained using the methods of Armand and Chen, and therefore they can be recommended to calculate the gas volume fraction.