A Preconditioning Mass Matrix to Avoid the Ill-Posed Two-Fluid Model

Abstract

Two-fluid models are central to the simulation of transport processes in two-phase homogenized systems. Even though this physical model has been widely accepted, an inherently nonhyperbolic and nonconservative ill-posed problem arises from the mathematical point of view. It has been demonstrated that this drawback occurs even for a very simplified model, i.e., an inviscid model with no interfacial terms. Much effort has been made to remedy this anomaly and in the literature two different types of approaches can be found. On one hand, extra terms with physical origin are added to model the interphase interaction, but even though this methodology seems to be realistic, several extra parameters arise from each added term with the associated difficulty in their estimation. On the other hand, mathematical based-work has been done to find a way to remove the complex eigenvalues obtained with two-fluid model equations. Preconditioned systems, characterized as a projection of the complex eigenvalues over the real axis, may be one of the choices. The aim of this paper is to introduce a simple and novel mathematical strategy based on the application of a preconditioning mass matrix that circumvents the drawback caused by the nonhyperbolic behavior of the original model. Although the mass and momentum conservation equations are modified, the target of this methodology is to present another way to reach a steady-state solution (using a time marching scheme), greatly valued by researchers in industrial process design. Attaining this goal is possible because only the temporal term is affected by the preconditioner. The obtained matrix has two parameters that correct the nonhyperbolic behavior of the model: the first one modifies the eigenvalues removing their imaginary part and the second one recovers the real part of the original eigenvalues. Besides the theoretical development of the preconditioning matrix, several numerical results are presented to show the validity of the method.

An extended mixture model for the simultaneous treatment of small‐scale and large‐scale interfaces

“…Then, using  as a flag to activate or deactivate certain terms according to the interface scale, which is being resolved, VOF (Weller-VOF) and ASMM can be coupled as in Eqn. (15) 8 < :…”

“…The implementation of the solver for the extended mixture model given by Eqn. (15) is based on the FVM. The starting point is the solver for the Weller-VOF method given in the OpenFOAM r suite [47], which is called interFoam.…”

“…Thus, the relationship between the velocity of center-of-mass, E v m , and the velocity of center-ofvolume, E u, needed in the system given in Eqn. (15) is expressed in terms of face fluxes. From these steps, it is clear that the correct calculation of the flux and its treatment along the solving algorithm plays a central role in the successful implementation of the solver.…”

“…It is worthy to note that, even though the mixture model represents a simplification respect to the multifluid model, the applicability of each model and the results' quality strongly relies on the nature of the problem, giving similar results in many cases [8,15].This scenario leaves an open discussion respect to the development of new models capable to manage several interface scales and/or transitions between them. Examples of such flows are the gas-liquid transitional flows in the nuclear and chemical industries [7] or the interaction of bubble plume with a free surface as is frequent in oceanography [16,17] or siderurgy [15,18].The state of the art in this topic shows that the study of the so called coupled models for the treatment of small and large-scale interfaces (LSI) has been in discussion in the last years. The first work in the topic seems to be that was presented byČerne et al [19], where the authors introduced a coupled method between VOF and multifluid (two-fluid) model.…”

Flow Study and Wetting Efficiency of a Perforated-Plate Tray Distributor in a Trickle Bed Reactor

“…That may be related to the fact that the liquid mass flow rate was more than 1500 times higher than the gas one. As it is well known the multiphase two-fluid model is not mathematically well posed for problems involving large differences in density between phases or where a phase has a volume fraction close to zero (typically in the interface between phases) or high slip velocities between phases (Zanotti et al, 2007). So, for the system analyzed here (the mass flow rate of one fluid is quite smaller than the others), a double precision solver and large CPU time were necessary to get acceptable errors in the global mass balance.…”