A model for droplet entrainment in churn flow
Introduction
Churn flow is a highly turbulent mixed flow of gas and liquid and generally characterized by the presence of a very thick and unstable liquid film, with the liquid frequently oscillating up and down. As one of the least understood of gas–liquid flow regimes, churn flow has been enduring efforts to be defined (Zuber and Findlay, 1965, Hewitt and Hall-Taylor, 1970, Taitel et al., 1980, Mao and Dukler, 1993) and generally be considered as an intermediate flow regime between the slug flow and annular flow. This occurs after break down of slug flow as velocity increases (Hewitt and Hall-Taylor, 1970, Jayanti and Hewitt, 1992) and transits to annular flow associated with the flow reversal point in counter-current flow (Wallis, 1969). The criterion for the churn–annular transition is expressed as (Wallis, 1969, Hewitt et al., 1985)where , , dT, , and are the dimensionless gas velocity, superficial gas velocity, pipe diameter, gravitational acceleration, liquid density and gas density, respectively.
The process of the droplet entrainment is very complex and the entrainment fraction E is defined to characterize the distribution of liquid phase flow between the liquid film and the entrained droplets. Although studies have shown that the entrained fraction is high in churn flow and reaches the minimum around the churn–annular transition (Wallis, 1962, Barbosa et al., 2002), the underlying mechanisms of droplet entrainment in churn flow is still not well explored and few studies have been reported to investigate the entrainment mechanism in churn flow. Barbosa et al. (2002) suggested that two droplet entrainment mechanisms (bag breakup and ligament breakup) were the reasons for the entrained droplets in churn flow based on the atomization theory proposed by Azzopardi (1983). Subsequently, Wang (2012) verified that bag breakup and ligament breakup were coexistent in churn flow based on his observation: bag breakup (under-cut) played a dominant role at low gas superficial velocity, but the ligament breakup (shear-off) came to gain greater importance with the increase of gas flowrate. However, to the best knowledge, there is no mathematical model that has been developed to understand the entrainment mechanism in churn flow. At present, theoretical studies of entrainment in annular flow are mostly based on Kelvin–Helmholtz instability and force balance on interfacial waves, including Holowach et al. (2002) and Ryua and Park (2011), and serve as good references in present study.
In this paper, the author established an analytical model to characterize the process of the droplet entrainment in churn flow based on the theory of Kelvin–Helmholtz instability. Also, analyzed in detail, the forces acting on the wave crest in order to comprehensively investigate the impact of forces including gravity, surface tension force and drag force. The present model was compared and verified carefully with the existing experimental data. On the basis of present mathematical model, the effects of parameters such as pipe diameter, gas and liquid flowrate and pressure on drop entrainment were investigated. With the aid of Ahmad et al. (2010)'s suggestion, a more accurate and reasonable formula for the entrained rate in churn flow was devised.
Section snippets
Control volume
The wave distribution in the axial and circumferential directions and the control volume for the present model are shown in Fig. 1. The interfacial wave initially forms along the axisymmetric direction and its further growth leads to secondary instabilities which give rise to variations around the periphery. Therefore, the interfacial wave is three-dimensional. In addition, the droplet entrained rate me is defined, as Holowach et al. (2002) suggested, as a function of the number of waves in the
Model verification and modification
Fig. 6 shows the comparison of the calculation results with the experimental data from Ahmad et al. (2010). Although the predicted entrained rate seems to match well with the experimental data in lower gas velocity, they reveal quite different trends as the gas velocity increases: the experimental entrained rate decreases with the dimensionless gas velocity increasing, however, the calculation shows the contrary results. The reason for these opposite trends is considered as the unreasonable
Conclusions
In summary, we established a mechanical model for droplet entrainment in churn flow based on the theory of Kelvin–Helmholtz instability and a fundamental force analysis over the control volume in this paper. By comparing with existing experimental data, the proposed model was verified carefully and showed an accurate prediction of entrained rate under churn flow condition. On the basis of the model, the author investigated in detail the influence of pipe diameter, gas and liquid flowrate and
Nomenclature
Projected area of a droplet (m2)
Wave amplitude (m)
Volume of a droplet (m3)
Bond number
Droplet drag coefficient
Interfacial drag coefficient (kg/m4)
Droplet concentration
Velocity profile coefficient in the liquid slug
Capillary number
Drag coefficient
Small bubble diameter (m)
Drop diameter (m)
Large bubble diameter (m)
Pipe diameter (m)
Entrained fraction
Etövös number
Friction factor
Drag force (N)
Gravity (N)
Surface tension force (N)
Gravitational acceleration (m/s2)
Acknowledgments
This work was financially supported by the National Nature Science Foundation of China under the Contract no. 51276140.
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