Mesoscale model kinetic equation

The mathematical modeling of polymerization reactions can be classified into three levels microscale, mesoscale, and macroscale. In microscale modeling, polymerization kinetics and mechanisms are modeled on a molecular scale. The microscale model is represented by component population balances or rate equations and molecular weight moment equations. In mesoscale modeling, interfacial mass and heat transfer... 

Mesoscale Model Kinetic equation Euler-Lagrange models... 

In most cases, closure of the terms in the kinetic equation will require prior knowledge of how the mesoscale variables are influenced by the underlying physics. Taking the fluid-drag term as an example, the simplest model has the form... 

The process of formulating mesoscale models from the microscale equations is widely used in transport phenomena (Ferziger Kaper, 1972). For example, heat transfer between the disperse phase and the fluid depends on the Nusselt number, and mass transfer depends on the Sherwood number. Correlations for how the Nusselt and Sherwood numbers depend on the mesoscale variables and the moments of the NDF (e.g. mean particle temperature and mean particle concentration) are available in the literature. As microscale simulations become more and more sophisticated, modified correlations that are based on the microscale results will become more and more common (Beetstra et al, 2007 Holloway et al, 2010 Tenneti et al, 2010). Note that, because the kinetic equation requires mesoscale models that are valid locally in phase space (i.e. for a particular set of mesoscale variables) as opposed to averaged correlations found from macroscale variables, direct numerical simulation of the microscale model is perhaps the only way to obtain the data necessary in order for such models to be thoroughly validated. For example, a macroscale model will depend on the average drag, which is denoted by... 

The transport equations appearing in macroscale models can be derived from the kinetic equation using the definition of the moment of interest. For example, if the moment of interest is the disperse-phase volume fraction, then it suffices to integrate over the mesoscale variables. (See Section 4.3 for a detailed discussion of this process.) Using the velocity-distribution function from Section 1.2.2 as an example, this process yields... 

This example illustrates the classical problem faced when working with macroscale models (Struchtrup, 2005). No matter how the transport equations for the moments are derived, they will always contain unclosed terms that depend on higher-order moments (e.g. Up depends on 0p, etc.). In comparison, the solution to the kinetic equation for the NDF contains information about all possible moments. In other words, if we could compute n t, x, v) directly, it would not be necessary to work with the macroscale model equations. The obvious question then arises Why don t we simply solve the kinetic equation for the mesoscale model instead of working with the macroscale model ... 

Now the whole set of SFM hydrodynamic equations with EMMS mesoscale modeling is estabhshed, where the drag and stress closures are based on the bimodal distribution. However, the mass exchange (or phase change) between the dilute and dense phases is still a big challenge to the kinetic theory derivation. More elaborate efforts are needed on this topic. 

There are two approaches to modeling the SOFC electrochemistry at the mesoscale an elementary kinetics-based model and a modified Butler-Volmer model. In the elementary kinetics-based model, the electrochemical reactions of the SOFC are modeled exactly, whereas in the modified Butler-Volmer model, the phenomenological Butler-Volmer equation is solved based on the local Faradaic current density. 

Here, the stabdity condition redects the compromise of dominant mechanisms at mesoscales and can be used to describe the direction of structure evolution of the system. By contrast, the transport equations for mass and momentum, i.e., Eqs. (18) and (19), describe the dynamics of structure evolution. The relationship of these two approaches is more or less hke the thermodynamics and chemical kinetics. Compared to the traditional CFD models, this integrated approach can be termed the stabdity-constrained multifluid (SCMF) CFD model. 

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