The inhomogeneous MUSIG model for the simulation of polydispersed flows

Abstract

A generalized inhomogeneous multiple size group (MUSIG) model based on the Eulerian modeling framework was developed in close cooperation of ANSYS-CFX and Forschungszentrum Dresden-Rossendorf and implemented into the CFD code CFX. The model enables the subdivision of the dispersed phase into a number of size groups regarding the mass balance as well as regarding the momentum balance.

In this work, the special case of polydispersed bubbly flow is considered. By simulating such flows, the mass exchanged between bubble size classes by bubble coalescence and bubble fragmentation, as well as the momentum transfer between the bubbles and the surrounding liquid due to bubble size dependent interfacial forces have to be considered. Particularly the lift force has been proven to play an important role in establishing a certain bubble size distribution dependent flow regime.

In a previous study [Krepper, E., Lucas, D., Prasser, H.-M., 2005. On the modeling of bubbly flow in vertical pipes. Nucl. Eng. Des. 235, 597–611] the application of such effects were considered and justified and a general outline of such a model concept was given. In this paper the model and its validation for several vertical pipe flow situations is presented. The experimental data were obtained from the TOPFLOW test facility at the Forschungszentrum Dresden-Rossendorf (FZD). The wire-mesh technology measuring local gas volume fractions, bubble size distributions and velocities of gas and liquid phases were employed.

The inhomogeneous MUSIG model approach was shown as capable of describing bubbly flows with higher gas content. Particularly the separation phenomenon of small and large bubbles is well described. This separation has been proven as a key phenomenon in the establishment of the corresponding flow regime. Weaknesses in this approach can be attributed to the characterization of bubble coalescence and bubble fragmentation, which must be further investigated.

Introduction

Reliable simulations of two-phase flow situations are a key topic of thermohydraulics related to nuclear reactor safety research. In general, one can distinguish between separated and dispersed two-phase situations, which of course may also occur in parallel within a single flow domain. This work considers dispersed flows, for which the gas (non-condensables or vapour of the liquid phase) is dispersed in the continuous liquid phase. Flow regimes found in vertical pipes are dependent on the void fraction of the gaseous phase, which vary from bubbly flows at low fractions to higher void fraction regimes of slug flow, churn turbulent flow, annular flow and finally to droplet flow.

In the regime of bubbly and slug flow the multiphase flow in general shows a spectrum of different bubble sizes. While disperse bubbly flows with low gas volume fractions are mostly mono-disperse, an increase of the gas volume fraction leads to a broader bubble size distribution due to breakup and coalescence of bubbles. The bubbles are also subject to lateral migration due to forces acting in lateral direction, which is different from the direction of main drag force. These lateral bubble forces as well as the drag and the virtual mass force depend on the bubble size (Lucas et al., 2007). Even the bubble lift force was found to change its sign as the bubble size varies (Tomiyama, 1998, Bothe et al., 2006). Consequently, this lateral migration leads to a radial de-mixing of small and large bubbles and to a further coalescence of large bubbles that migrate towards the pipe center into even larger Taylor bubbles or slugs.

An adequate modeling approach has to account for all these phenomena. The simulation of bubble coalescence and breakup generally requires the solution of a population balance equation. Different model concepts based on the two-fluid model approach can be found in the literature. One of the most popular is the discretised population balance method, which however leads to a large additional numerical effort. Other methods try to reduce this effort. Kocamustafaogullari and Ishii (1995) proposed the solution of an additional transport equation for the interfacial area concentration. The moments model first proposed by Hulburt and Katz (1964) is based on the solution of the population balance equation through the moments of the particle size distribution. The tracking of only a few lower order moments of this function with a very limited additional numerical effort could be shown to be adequate for several applications. To reduce the limitations of applicability of this method the quadrature method of moments was proposed, e.g. by McGraw (1997) in which all integrals involving the number density function are solved through an ad hoc quadrature approximation. All these approaches are able to simulate the described phenomena provided they are able to model the radial separation of bubbles of different size.

The necessity to consider different velocity fields for bubbles of different size was investigated by a multi bubble size class test solver, first introduced by Lucas et al. (2001). In the result of these investigations and simulations were incorporated into the CFD code CFX via the concept of the inhomogeneous multiple size group (MUSIG) model based on the Eulerian modeling framework was proposed by Krepper et al. (2005). This concept is an enhancement of the homogeneous MUSIG model previously introduced by Lo (1996). It has been and implemented into CFX-5.7 in close cooperation of ANSYS-CFX and Forschungszentrum Dresden-Rossendorf. It is now available in all the upgraded CFX releases. Within the inhomogeneous MUSIG model the dispersed gaseous phase is divided into N inhomogeneous velocity groups (phases) and each of these groups is subdivided into M_j bubble size classes. Bubble breakup and coalescence processes between all bubble size classes M_j are taken into account by appropriate models. Details of the models and its validation for vertical pipe flows are presented in this paper.