Elsevier

Chemical Engineering Science

Volume 201, 29 June 2019, Pages 237-246
Chemical Engineering Science

Poly-dispersed modeling of bubbly flow using the log-normal size distribution

https://doi.org/10.1016/j.ces.2019.02.013Get rights and content

Highlights

  • A pragmatic and efficient poly-disperse method for bubbly flow simulation is presented.

  • Closure is achieved using the log-normal presumed number density function.

  • Good agreement is found with reference data for bubbly flow in the mono-dispersed limit.

  • Poly-dispersity and poly-celerity are shown to play an important role in bubbly flow.

  • Good agreement is found between our moment method and an accurate sectional approach.

Abstract

The bubble size distribution plays an important role in interfacial mass, momentum and energy transfer between bubbles and their carrier liquid in bubbly flow. Accurate modeling of the size distribution is therefore key. A Log-Normal presumed Number Density Function (LNpNDF) approach is proposed, which is embedded into the two-fluid model. Two additional moment transport equations are formulated which are shown to be consistent with the two-fluid model. From the moments, the size distribution can be fully reconstructed using the assumption that its underlying shape is log-normal. This methodology offers closure for the modeling of processes such as bubble coalescence, break-up and bubble poly-celerity. Special attention is paid to the concept of poly-celerity, which is shown to play an important role in the evolution of finite-width size distributions. A new average diameter, which is based on the fifth and third moment of the size distribution, is proposed, and it is shown that this diameter is a more suitable quadrature node for the modeling of bubble–liquid Stokes-like drag. The paper lays the mathematical foundation for a pragmatic, computationally efficient and effective poly-dipsersed method for the modeling of dispersed two-phase flow.

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.

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