Gas–liquid flows in medium and large vertical pipes
Introduction
Two-phase gas–liquid flows are featured in numerous industrial applications such as in areas of chemical, civil, nuclear, mineral, energy, food, pharmaceutical and metallurgy. Because of the inherently complex two-phase flow structures that are within these technological systems as well as they are dynamically evolving with respect to different flow regimes, the phenomenological understanding of bubble size and its dispersion behaviors is of paramount importance. In the bubbly flow regime, while gas volume fraction and bubble number density are considerably low, the spectrum of bubble size usually shows a single-peaked distribution due to moderate bubble coalescence and break-up. With increasing gas volume fraction and bubble number density, the bubble size distribution gradually becomes bimodal or double-peaked, which is caused by rigorous bubble interactions. Nevertheless, such dynamical changes of the bubble size spectrum are further complicated by the presence of lateral forces acting on the bubbles. Recent findings (Tomiyama, 1998, Bothe et al., 2006) have led to conclude that the lateral lift force, which acts in the perpendicular direction to the main flow in a vertical pipe, significantly affects the bubble size distribution. Small bubbles driven by positive lift forces are separated from those opposite directed large bubbles which migrate towards the pipe centre turning into cap or even larger Taylor bubbles via additional coalescence.
Recently, population balance modelling has become a rather useful approach in synthesizing the behaviors of bubble interactions in gas–liquid flows. Based on the framework of two-fluid modelling approach, different numerical algorithms have been proposed to obtain practical solutions to the population balance equation (PBE). The class method (CM), due to its rather straightforward implementation, is receiving greater attention and increasingly being adopted as the preferred method for population balance modelling. Research studies based on the MUSIG model typified the application of CM in gas–liquid flow simulations (Pochorecki et al., 2001, Yeoh and Tu, 2004, Wang et al., 2006). By describing the spectrum of bubble size through a series of discrete bubble size classes, the dynamical changes of size distribution can thus be tracked. The method then becomes more viable with appropriate coalescence and break-up mechanisms to dictate the inter-group mass transfer. Lately, the inhomogeneous MUSIG model (Krepper et al., 2005) has been proposed which further extended the model capability to consider different velocity fields for different bubble classes, specifically to address the co-existence of small and large bubbles travelling within the gas–liquid flows. However, a main disadvantage of CM is the requirement of substantial number of transport equations to be solved for different bubble classes especially for the treatment of gas–liquids flows where the range of bubble sizes may be significantly wide.
In Cheung et al., 2007a, Cheung et al., 2007b, a simple population balance approach – the ABND model – has been proposed in handling isothermal gas–liquid flows. Model predictions were compared and validated against predictions from MUSIG model as well as experimental data by Liu and Bankoff (1993) and Hibiki et al. (2001). Encouraging results demonstrated the performance of ABND model in measuring up with the more sophisticated MUSIG model. Nonetheless, limited by the experiment setup, flow conditions deployed for validation exhibited a rather narrow bubble size range due to the considerably moderate bubble interaction rates occurring within the two-phase flow system. Theoretically speaking, the muted bubble interaction behavior that had been experienced in the two experiments reduced the non-linearity of the problem thereby producing a less challenging prospect in fully assessing the model's capability. The exercise allows the model to be validated to some degree but may conceal the possible potential of predicting complex gas–liquid flows where rigorous bubble interactions would be significant.
The objectives of this paper are twofold: (i) to present a comprehensive model validation study to assess the ABND model in simulating flow conditions with wider range of bubble sizes and more rigorous bubble interactions and (ii) to determine the relative merits and capabilities of applying the ABND model in comparison to the inhomogeneous MUSIG model under the same gas–liquid flow situation within the two-fluid modelling framework. Particular attention is directed towards evaluating the performance of the two models in capturing the transition from “wall peak” to “core peak” radial void fraction distribution, corresponding to the prevalence of lift forces acting on the small and large bubbles. Predictions by the ABND and inhomogeneous MUSIG models are validated against gas–liquid flow experiments in vertical pipes of medium size by Lucas et al. (2005) and large size of Prasser et al. (2007) measured in the Forschungszentrum Dresden-Rossendorf FZD facility.
Section snippets
Experiments
Two individual set of experiments – MTLOOP (Lucas et al., 2005) and TOPFLOW (Prasser et al., 2007) – that have been performed in the Forschungszentrum Dresden-Rossendorf FZD facility are briefly described below. This section exemplifies the main configuration details and the effect on the overall bubble coalescence and break-up processes occurring within both experiments.
Mathematical models
The three-dimensional two-fluid model solves the ensemble-averaged of mass and momentum transport equations governing each phase (Drew and Lahey, 1979). Denoting the liquid as the continuum phase (αl) and the gas (i.e. bubbles) as disperse phase (αg), these equations can be written as:
Continuity equation of liquid phase
Continuity equation of gas phase
Momentum equation of liquid phase
Momentum
Numerical details and results
Numerical calculations were achieved through the use of the generic computational fluid dynamics code ANSYS-CFX11 (CFX-11, 2007). The average bubble number density transport equation with appropriate source and sink terms describing the coalescence and break-up rate of bubble was implemented through the CFX Command Language (CCL). Radial symmetry was assumed in both experimental conditions thereby allowing the computational geometry to be simplified through consideration of a 60° radial sector
Conclusions
The assessment of two population balance models – Inhomogeneous MUSIG and ABND models – has been performed against the experimental data of Lucas et al. (2005) and Prasser et al. (2007) measured in Forschungszentrum Dresden-Rossendorf FZD facility. With two different gas injection methods, bubble coalescence was found to be dominant feature in the MTLOOP experiment (Lucas et al., 2005) while bubble break-up prevailed as the main characteristic in the TOPFLOW experiment. Predictions from both
Nomenclature
- CL
Lift coefficient
- CRC
Random collision coefficient
- CTI
Turbulent impact coefficient
- dH
Maximum bubble horizontal dimension
- Ds
Bubble Sauter mean diameter
- DB
Mass loss rate due to breakage
- DC
Mass loss rate due to coalescence
- Eo
Eötvos number
- Edg
Modified Eötvos number
- f
Bubble size distribution
- F
Total interfacial force
Drag force
Lift force
Wall lubrication force
Turbulent dispersion force
- g
Gravitational acceleration
- n
Average number density of gas phase
- PB
Mass production rate
Acknowledgement
The financial support provided by the Australian Research Council (ARC project ID DP0877734) is gratefully acknowledged.
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