Mass transfer mesoscale model

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... 

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 remaining chapters in this book are organized as follows. Chapter 2 provides a brief introduction to the mesoscale description of polydisperse systems. There, the mathematical definition of a number-density function (NDF) formulated in terms of different choices for the internal coordinates is described, followed by an introduction to population-balance equations (PBE) in their various forms. Chapter 2 concludes with a short discussion on the differences between the moment-transport equations associated with the PBE and those arising due to ensemble averaging in turbulence theory. This difference is very important, and the reader should keep in mind that at the mesoscale level the microscale turbulence appears in the form of correlations for fluid drag, mass transfer, etc., and thus the mesoscale models can have non-turbulent solutions even when the microscale flow is turbulent (i.e. turbulent wakes behind individual particles). Thus, when dealing with turbulence models for mesoscale flows, a separate ensemble-averaging procedure must be applied to the moment-transport equations of the PBE (or to the PBE itself). In this book, we are primarily... 

In Section 5.3 we will discuss mesoscale models for the change in momentum due to mass transfer, where we shall see that Afp contains a contribution proportional to Gp. 

Given T, the expression for is closed, thereby fixing the mass-transfer rate. The discussion above is applicable to single-component droplets. In many applications, the liquid/gas phase will contain multiple chemical species, for which additional internal coordinates will be necessary in order to describe the physics of evaporation (Sazhin, 2006). In the context of a single-particle model for a multicomponent droplet, the simplest mesoscale model must include the particle mass Mp, the component mass fractions Yp and Yf, and the temperatures Tp and Tf. 

As can be seen from Eq. (5.100), the virtual-mass force reduces the drag and lift forces by a factor of 1 /y. The buoyancy force is not modified because we have chosen to define it in terms of the effective volume Vpy. We remind the reader that the mesoscale acceleration model for the fluid seen by the particle A j must be consistent with the mesoscale model for the particle phase A p in order to ensure that the overall system conserves momentum at the mesoscale. (See Section 4.3.8 for more details.) As discussed near Eq. (5.14) on page 144, this is accomplished in the single-particle model by constraining the model for Apf given the model for Afp (which is derived from the force terms introduced in this section). Thus, as in Eqs. (5.98) and (5.99), it is not necessary to derive separate models for the momentum-transfer terms appearing in Apf. 

Gwo, J.P., R, O Brien, and P.M. Jardine. 1998. Mass transfer in stmctured porous media Embedding mesoscale structure and microscale hydrodynamics in a two-region model. J. Hydrol. 208 204-222. 

In a gas—sohd CFB with heterogeneous reactions and mass transfer, in Hne with the structural characteristics of the SFM model (Hong et al, 2012), as shown in Fig. 12, the mass transfer and reaction in any local space can be divided into components of the dense cluster (denoted by subscript c), the dilute broth (denoted by subscript f), and in-between (denoted by subscript i), respectively. And these terms can be represented by Ri (1 = gc, gf, gi, sc, sf, si). Both the dense and dilute phases are assumed homogenous and continuous inside, and the dense phase is fiarther assumed suspended uniformly in the dilute phase in forms of clusters of particles. Then the mass transfer terms can be described with Ranz-Marshall-hke relations for uniform suspension of particles (Haider and Basu, 1988). In particular, the mesoscale interaction over the cluster will be treated as is for a big particle with hydrodynamic equivalent diameter of d. Due to dynamic nature of clusters, there are mass exchanges between the dilute and dense phases with rate ofTk (k = g, s), pointing outward from the dilute to the dense phase. 

In this chapter, the multiscale analysis of FTS process is illustrated, and the effects of every scale are discussed on the product distribution of FTS reaction. Flowever, constraint will be exploded in mesoscales viewpoint because there are many combined factors to influence FTS reaction, such as complex reaction mechanism, mass transfer of reactants and products, and flow type of reactor. Therefore, the ASF distribution is always improper in the FTS system. On the other hand, the flow model of the reactor has been investigated, but how the reactor influence the reaction performance of catalyst is still not clear. Therefore, if we want to know the product distribution of different reactors, a lot of experiment should be taken place. Based on these points, the mesoscale phenomena and effects for products of FTS reaction should be investigated in detail. 

Microscale fluid turbulence is, by deflnition, present only when the continuous fluid phase is present. The coefficients Bpv describe the interaction of the particle phase with the continuous phase. In contrast, Bpvf models rapid fluctuations in the fluid velocity seen by the particle that are not included in the mesoscale drag term Ap. In the mesoscale particle momentum balance, the term that generates Bpv will depend on the fluid-phase mass density and, hence, will be null when the fluid material density (pf) is null. In any case, Bpv models momentum transfer to/from the particle phase in fluid-particle systems for which the total momentum is conserved (see discussion leading to Eq. (5.17)). 

In order to complete our discussion on momentum transfer, we must consider the final forms of the mesoscale acceleration models in the presence of all the fluid-particle forces. When the virtual-mass force is included, the mesoscale acceleration models must be derived starting from the force balance on a single particle ... 

This polymerization process can be separated into three different levels as proposed by Ray [22]. First this is the microscale level, modeling all processes at the surface and inside the growing polymer particle. The next level is the mesoscale level, describing all mass and heat transfer processes inside the three-phase slurry containing gas bubbles, hydrocarbon diluent with the dissolved aluminumalkyl compound, and the solid growing polymer particles loaded with the active sites. Finally, there is the macroscale level comprising the polymerization vessel as a whole, with sensors to control this slurry polymerization process. These three levels are shown in Fig. 4. 

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