Multi-fluid Modeling Framework

The average multi-fluid model equations are outlined in the following together with the conventional interfacial closures that are frequently adopted in gas-liquid bubbly flow analyzes. The average multi-fluid continuity equation for phase k reads  

In a consistent manner the momentum balance for phase k yields  

The limiting steps in the model development are the formulation of closure relations or closure laws determining the interfacial transfer fluxes, the bubble coalescence and breakage processes, and turbulence effects. 

When sufficiently dilute dispersions are considered, only particle-fluid interactions are significant and the two-fluid closures can be employed. In these particular cases, only the interaction between each of the dispersed gas phases d) and the continuous liquid phase (c) is considered parameterizing the last term on the RHS of (8.12)  

The steady drag-, added mass-, lift-, turbulent diffusion- and wall forces, respectively, are presented in Sect. 5.1. Moreover, the force terms for dilute dispersions 

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To gain insight on the capability of the present models to capture physical responses to changes in the bubble size distributions, a few preliminary analyzes have been performed adopting the multi-fluid modeling framework. 

The multi-fluid model framework is required to simulate chemical processes containing dispersed phases of multiple sizes. Two different designs of the multi-fluid model have emerged over the years representing very different levels of complexity. For dilute flows the dispersed phases are assumed not to interact, so no population balance model is needed. For denser flows a population balance equation is included to describe the effects of the dispersed phases interaction processes. Further details on the multi-fluid model formulations are given in chap 8 and chap 9. 

Chapter 3 contains a survey of a large number of books and journal papers dealing with the basic theory of multi-fluid flow modeling. Emphasis is placed on applying the multi-fluid model framework to describe reactive flows. In the more advanced textbooks the basic multi-component multiphase theory is introduced in a rather mathematical context, thus there is a need for a less demanding presentation easily accessible for chemical reaction engineering students. 

Unfortunately, the present models are still on a level aiming at reasonable solutions with several model parameters tuned to known flow fields. For predictive purposes, these models are hardly able to predict unknown flow fields with reasonable degree of accuracy. It appears that the CFD evaluations of bubble columns by use of multi-dimensional multi-fluid models still have very limited inherent capabilities to fully replace the empirical based analysis (i.e., in the framework of axial dispersion models) in use today [63]. After two decades performing fluid dynamic modeling of bubble columns, it has been realized that there is a limit for how accurate one will be able to formulate closure laws adopting the Eulerian framework. In the subsequent sections a survay of the present status on bubble column modeling is given. 

The term macroscale will be used to denote multiphase models that employ a hydro-dynamic description of the disperse phase. Such models are also called multi-fluid models (because the disperse phase is treated as an effective fluid), or Euler-Euler models. The name of the latter comes from the numerical treatment of the disperse phase (i.e. discretization on a fixed grid), as opposed to Euler-Lagrange models wherein the disperse phase is tracked in a Lagrangian framework as discrete entities. We should note that, in the... 

The EMMS model was proposed for the time-mean behavior of fluidized beds on the reactor scale. A more extensive application of the EMMS model to gas-solid flow is through its coupling with the two-fluid CFD approaches, which brings about an EMMS-based multi-scale CFD framework for gas—solid flow. For this purpose, Yang et al. (2003) introduced an acceleration, a, into the EMMS model to account for the... 

The overall reactor model comprises, as the heart of it, the single catalyst pellet model which is formulated in an overall framework that includes the changes in the bulk fluid phase. The equations for the catalyst pellet coupled with the equations for the bulk fluid phase represent what we may call in certain cases, the overall reactor model or in a more restricted sense, the catalyst bed module. This catalyst bed module may represent the overall reactor model in certain cases such as the single adiabatic catalytic packed bed reactor. In other cases, this module may represent only the essential part of the overall reactor model such as in non-adiabatic and multi-bed reactors. 

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