Multi-fluid model

Laurent F, Massot M, ViUedieu P (2004) Eulerian Multi-fluid modeling for the numerical simulation of coalescence in polydisperse dense liquid sprays. J Comp Phys 194 505-543... 

Modeling Concepts for Multiphase Flow 343 Averaged Eulerian-Eulerian Multi-fluid Models... 

The averaged Eulerian-Eulerian multi-fluid model denotes the averaged mass and momentum conservation equations as formulated in an Eulerian frame of reference for both the dispersed and continuous phases describing the time-dependent motion. For multiphase isothermal systems involving laminar flow, the averaged conservation equations for mass and momentum are given by ... 

In the jump-condition formulation the physical problem is generally decomposed into k bulk phase domains where the continuity and momentum equations for isothermal incompressible flows holds, and at the interface between these domains boundary conditions are specified using the interface jump conditions. That is, across the interface some quantities are required to be continuous, while others are required to have specific jumps. The discontinuous (singular) momentum jump condition can be derived by use of the surface divergence theorem (see e.g., [63] p 51 [26]). A rigorous derivation of the jump balances for the multi-fluid model is given in sect 3.3. 

Basic Principles and Derivation of Multi-Fluid Models... 

The Microscopic Transport Equations for a Finite Number of Dispersed Phases - the Multi-Fluid Model... 

Fig. 3.6. General procedure for formulating a multi-fluid model.

In problems in which the dispersed phase momentum equations can be approximated and reduced to an algebraic relation the mixture model is simpler to solve than the corresponding multi-fluid model, however this model reduction requires several approximate constitutive assumptions so important characteristics of the flow can be lost. Nevertheless the simplicity of this form of the mixture model makes it very useful in many engineering applications. This approximate mixture model formulation is generally expected to provide reasonable predictions for dilute and uniform multiphase flows which are not influenced by any wall effects. In these cases the dispersed phase elements do not significantly affect the momentum and density of the mixture. Such a situation may occur when the dispersed phase elements are very small. There are several concepts available for the purpose of relating the dispersed phase velocity to the mixture velocity, and thereby reducing the dispersed... 

In this section we derive the algebraic-slip mixture model equations for cold flow studies starting out from the multi-fluid model equations derived applying the time- after volume averaging operator without mass-weighting [204, 205]. The momentum equations for the dispersed phases are determined in terms... 

In this section we derive the diffusion mixture model equations starting from the time averaged multi-fluid model expressed in terms of phase- and mass weighted variables [112]. The relative movement of the individual phases is given in terms of diffusion velocities. 

The well-posedness of the two-fluid model has been a source of controversy reflected by the large number of papers on this issue that can be found in the literature. This issue is linked with analysis of the characteristics, stability and wavelength phenomena in multi -phase flow equation systems. The controversy originates primarily from the fact that with the present level of knowledge, there is no general way to determine whether the 3D multi-fluid model is well posed as an initial-boundary value problem. The mathematical theory of well posedness for systems of partial differential equations describing dispersed chemical reacting flows needs to be examined. 

In reaction engineering the 2D and 3D multi-fluid model equations used are obviously solvable although with certain struggle, hence to some extent affected by the mathematical model properties reflected by the simplified model equations examined in the above mentioned work. Besides, in some cases numerical problems occur due to the inadequate mathematical properties reflected by the constitutive equations used in particular in the limit as —> 0... 

One can thus state that the constitutive equations for the interfacial terms are the weakest link in a multi-fluid model formulation because of considerable difEculties in terms of experimentation and modeling on a macroscopic level. The main difEculties in modeling arise from the existence of interfaces between phases and discontinuities associated with them. 

Future work might consider extensions of these interfacial transfer concepts to ameliorate the simulation accuracy by utilizing the local information provided by the multi-fluid models. For multiphase reactive systems these processes can be rate determining, in such cases there are no use for advanced flow calculations unless these fluxes can be determined with appropriate accuracy. 

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 interfacial and turbulence closures suggested in the literature also differ considering the anticipated importance of the bubble size distributions. It thus seemed obvious for many researchers that further progress on the flow pattern description was difficult to obtain without a proper description of the interfacial coupling terms, and especially on the contact area or projected area for the drag forces. The bubble column research thus turned towards the development of a dynamic multi-fluid model that is extended with a population balance module for the bubble size distribution. However, the existing models are still restricted in some way or another due to the large cpu demands required by 3D multi-fluid simulations. 

Multi-Fluid Models and Bubble Size Distributions... 

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. 

Tomiyama A, Shimada N (2001) A Numerical Method for Bubbly Flow Simulation Based on a Multi-Fluid Model. Journal of Pressure Vessel Technology-Trans ASME 123 510-516... 

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