Poly-dispersed modeling of bubbly flow using the log-normal size distribution
Introduction
In bubbly flow, the size of a bubble plays an important role on the heat, mass and momentum transfer from bubble to carrier fluid and vice versa. Usually, bubble size is not a uniform property; bubbles are often formed through a chaotic dynamic of break-up of slugs and simultaneous coalescence of smaller structures. Rather than a mono-dispersed size distribution, which can be conveniently modeled as a Dirac delta function with one bubble size, real bubbly flows always contain a poly-dispersed collection of bubbles with a corresponding finite bandwidth of the size distribution in the size space. As the bubbles move through space, this size distribution undergoes changes through mechanisms like bubble compressibility, evaporation or boiling, bubble break-up or bubble-bubble coalescence. These mechanisms should be modeled accurately, in order to predict the evolution of the bubble size distribution. In turn, the heat, mass and momentum transfer between bubbles and carrier fluid can be better modeled once the bubble size distribution is known.
To model the evolution of a bubble size distribution with finite width, two main families of methods have been developed: sectional methods (Gelbard and Seinfeld, 1980) and moment methods (Friedlander, 1983). The first family of methods aims to capture the size distribution by means of a discretization of the bubble size distribution in size space. This often leads to a set of bubble number concentration fields each representing the frequency with which a certain bubble size or size range is present as a function of time and position. The evolution of the size distribution is modeled by solving transport equations for each concentration field. In spatially inhomogeneous settings a fixed pivot technique (Kumar and Ramkrishna, 1996a) in which representative sizes remain fixed in time is convenient, but also moving pivot techniques (Kumar and Ramkrishna, 1996b) exist. Applications of sectional methods to bubbly flows can be found in (Krepper et al., 2005, Krepper et al., 2008, Hänsch et al., 2012, Liao et al., 2015, Liao et al., 2018). A major benefit of sectional methods is that no a priori assumptions are imposed on the shape of the bubble size distribution. Moreover, if each size concentration field is allowed to adhere to its own momentum equation, poly-celerity, i.e., the observation that bubble velocity is size-dependent, is automatically accounted for. In particular in this situation, but also in the situation of a single momentum equation for all bubble sizes, sectional modeling can be computationally expensive. The accuracy of sectional methods is directly proportional to the number of sections, which, in turn, carries a proportionality to the computational effort.
Alternative to this, the family of moment methods aims to reduce the complexity of the description of the size distribution by considering only integral moments of the distribution in size space. Transport equations are developed for these moments, accounting for the physical mechanisms which influence the distribution and its corresponding moments. The size distribution can then be reconstructed either directly using a presumed shape of the size distribution (Lee et al., 1984) or indirectly using quadrature-based methods (McGraw, 1997, Marchisio and Fox, 2005, Marchisio and Fox, 2013) in order to establish mathematical closure of terms in the moment transport equations which depend on unknown moments. Moment methods are attractive due to their more modest computational requirements, but may fail to capture distinct features of size distributions, such as multi-modal shapes, due to the integral approach. In many practical cases, however, the processes of interest are mostly dependent on just the integral moments, like the total interfacial area (Ishii et al., 2005), and not so much on the exact distribution of the underlying properties in size space. This invites not only from a computational, but also from a physical point of view, to adopt and investigate moment methods in the modeling of bubbly flow.
The most straight-forward way of achieving the necessary closure for condensation, coalescence, break-up and poly-celerity processes, is to use a presumed shape of the size distribution, i.e., a presumed Number Density Function (pNDF). Examples are the Rosin-Rammler, Nukiyama-Tanasawa, power law, exponential, Khrgian-Mazin or gamma distribution (Hinds, 2012). Mainly in aerosol science, another very common pNDF is the log-normal distribution. This distribution is defined only in positive space, which is a convenient mathematical property as bubbles have positive size. Moreover, the distribution fits observed size distributions reasonably well, and has a mathematically convenient form for dealing with moments of the distribution (Hinds, 2012).
In this work, we explore the Log-Normal pNDF (LNpNDF) approach developed mainly for aerosol droplet distributions, in order to achieve mathematical closure for the processes playing a key role in bubbly flow instead. We investigate primarily mathematically the integration of the log-normal distribution with the two-fluid model, while leaving investigation of the physical relevance of the log-normal assumption in bubbly flow for further research. The feasibility and effectiveness of the LNpNDF approach applied to bubbly flow is shown, and forms the main contribution of this paper. Special attention is paid to the development of a log-normal mathematical framework upon which further bubbly flow modeling, such as coalescence and break-up, can be built. We focus on the consistent derivation of three moment transport equations, which are embedded inside the two-fluid model (Ishii and Hibiki, 2010), in order to mathematically close the LNpNDF approach. The topic of poly-celerity is addressed by developing algebraic relations for the unique transport velocity of each moment, alongside the volume-average bubble velocity which follows from the solution of the two-fluid model. It is shown that for bubble distributions with a finite bandwidth, poly-celerity plays an important role and can be captured accurately inside the LNpNDF approach. We also focus on the derivation of the appropriate mean bubble diameter which is used to compute the effective momentum transfer between the bubble distribution and the carrier fluid, inside the two-fluid model. Rather than the often-used Sauter mean diameter, an alternative diameter is proposed, which is based on the fifth and third moment of the bubble size distribution. The work will be tested in the context of the bubbly pipe flow experiments of Liu and Bankoff, 1993a, Liu and Bankoff, 1993b, using the Bubbly And Moderate void Fraction (BAMF) model of Sugrue et al. (2017). Validation of our implementation of the model in OpenFOAM is achieved by cross-code comparison using the results of Sugrue et al. (2017), showing good agreement. Although more recent experimental results are available and several important effects are not included in the model, the goals of the paper are to explore the consistent integration of the log-normal size distribution with the two-fluid model, and to study the role of the modeling of a poly-dispersed size distribution in bubbly flow. The current paper lays the mathematical foundation for a pragmatic, computationally efficient and effective poly-dispersed method for the modeling of dispersed flow, with bubbly flow in particular.
The layout of this paper is as follows. In Section 2 we discuss the two-fluid model and the embedded LNpNDF approach. In Section 3 the LNpNDF approach is applied to the modeling of upward bubbly pipe flow, in order to validate the method and to study the behavior of the method in scenarios with non-zero width of the bubble size distribution. Finally, in Section 4 we present our conclusions.
Section snippets
The two-fluid model and the method of moments
In this section, the mathematical framework of the method of moments for the description of the bubble size distribution is presented. Special attention is paid to the derivation of additional moment transport equations which are consistently related to the standard two-fluid model through a bubble size distribution. Closure is achieved using the log-normal size distribution function.
Simulation of bubbly pipe flow
In order to test the new poly-dispersed methodology, we apply it to the simulation of upward bubbly pipe flow. The experimental setting of Liu and Bankoff, 1993a, Liu and Bankoff, 1993b is selected, and in particular, the closely related numerical datasets of Sugrue et al. (2017) are used as a point of reference. The upward bubbly pipe flow setting offers a non-trivial test platform as it is essentially two-dimensional while not being too computationally expensive, allowing for reasonably quick
Conclusions
In this paper, a Log-Normal presumed Number Density Function (LNpNDF) approach was proposed, which was embedded inside the two-fluid model, for the modeling of two-phase bubbly flow. Two additional moment transport equations were proposed. The LNpNDF offers a mathematically convenient way of achieving closure in the modeling of processes such as bubble coalescence, break-up and poly-celerity. Special attention was paid to poly-celerity, by deriving algebraic expressions for the moment transport
Conflict of interest
The authors declared that there is no conflict of interest.
Acknowledgments
The authors wish to thank Dr. B. Magolan and Prof. E. Baglietto, both of the Nuclear Science and Engineering faculty at MIT, for their support and assistance.
References (25)
- et al.
Simulation of multicomponent aerosol dynamics
J. Colloid Interface Sci.
(1980) - et al.
A multi-field two-fluid concept for transitions between different scales of interfacial structures
Int. J. Multiph. Flow
(2012) - et al.
On the modelling of bubbly flow in vertical pipes
Nucl. Eng. Des.
(2005) - et al.
The inhomogeneous musig model for the simulation of polydispersed flows
Nucl. Eng. Des.
(2008) - et al.
On the solution of population balance equations by discretization–I. A fixed pivot technique
Chem. Eng. Sci.
(1996) - et al.
On the solution of population balance equations by discretization–II. A moving pivot technique
Chem. Eng. Sci.
(1996) The simulation of multidimensional multiphase flows
Nucl. Eng. Des.
(2005)- et al.
Baseline closure model for dispersed bubbly flow: bubble coalescence and breakup
Chem. Eng. Sci.
(2015) - et al.
Structure of air-water bubbly flow in a vertical pipe–I. Liquid mean velocity and turbulence measurements
Int. J. Heat Mass Transf.
(1993) - et al.
Structure of air-water bubbly flow in a vertical pipe–II. Void fraction, bubble velocity and bubble size distribution
Int. J. Heat Mass Transf.
(1993)
Solution of population balance equations using the direct quadrature method of moments
J. Aerosol Sci.
CFD modeling of bubble-induced turbulence
Int. J. Multiph. Flow
Cited by (11)
Simulation of noble metal particle growth and removal in the molten salt fast reactor
2023, Nuclear Engineering and DesignComparison of population balance models for polydisperse bubbly flow
2023, Chemical Engineering ScienceWeakly nonlinear focused ultrasound in viscoelastic media containing multiple bubbles
2023, Ultrasonics SonochemistryWeakly nonlinear propagation of focused ultrasound in bubbly liquids with a thermal effect: Derivation of two cases of Khokolov–Zabolotskaya–Kuznetsoz equations
2022, Ultrasonics SonochemistryCitation Excerpt :Initially, the bubbly liquid is at rest and spatially uniform, except for the bubble distribution. The initial polydispersity of the bubbles [73–76] is not considered. Only the bubble distribution is spatially nonuniform in the initial state [53–56] because medical applications, such as HIFU treatment, often only use the bubbles at the focus.
Modeling of bubble coalescence and break-up using the Log-normal Method of Moments
2022, Chemical Engineering ScienceCitation Excerpt :Simulating bubbly flows using a two-fluid formulation for mass, momentum and energy transport is a pragmatic and efficient way to predict flow characteristics which are relevant to the safe design of nuclear reactors. In this work, LogMoM, introduced by Frederix et al. (2019) as a simple way to track bubble populations with only three moment transport equations, was extended to account for bubble coalescence and break-up. New source terms were added to the moment transport equations for coalescence and binary break-up, and Gauss-Hermite and Gauss–Legendre quadrature were used to approximate the integrals of these source terms in a computationally efficient manner.
All-regime two-phase flow modeling using a novel four-field large interface simulation approach
2021, International Journal of Multiphase FlowCitation Excerpt :Recently, efforts in the direction of addressing this challenge have been undertaken by the development of a ‘baseline model’ (Lucas et al., 2016). In the current paper, a simplified set of momentum closure models is selected (Sugrue et al., 2017; Mathur et al., 2019; Frederix et al., 2019). These momentum closures were shown to capture distinct two-phase flow features relatively well.