Mesoscale model formulation

Yamartino, R. J., J. S. Scire, G. R. Carmichael, and Y. S. Chang, The CALGRID Mesoscale Photochemical Grid Model. I. Model Formulation, Atmos. Environ., 26A, 1493-1512(1992). 

Of course, nanocomposites are not the only area where mesoscale theories are being used to predict nanostructure and morphology. Other applications include—but are not limited to—block copolymer-based materials, surfactant and lipid liquid crystalline phases, micro-encapsulation of drugs and other actives, and phase behavior of polymer blends and solutions. In all these areas, mesoscale models are utilized to describe—qualitatively and often semi-quantitatively—how the structure of each component and the overall formulation influence the formation of the nanoscale morphology. 

In summary, the microscale description provides two important pieces of information needed for the development of mesoscale models. First, the mathematical formulation of the microscale model, which includes all of the relevant physics needed to completely describe a disperse multiphase flow, provides valuable insights into what mesoscale variables are needed and how these variables interact with each other at the mesoscale. These insights are used to formulate a mesoscale model. Second, the detailed numerical solutions from the microscale model are directly used for validation of a proposed mesoscale model. When significant deviations between the mesoscale model predictions and the microscale simulations are observed, these differences lead to a reformulation of the mesoscale model in order to improve the physical description. Note that it is important to remember that this validation step should be done by comparing exact solutions to the mesoscale model with the microscale results, not approximate solutions that result... 

In the formulation of a mesoscale model, the number-density function (NDF) plays a key role. For this reason, we discuss the properties of the NDF in some detail in Chapter 2. In words, the NDF is the number of particles per unit volume with a given set of values for the mesoscale variables. Since at any time instant a microscale particle will have a unique set of microscale variables, the NDF is also referred to as the one-particle NDF. In general, the one-particle NDF is nonzero only for realizable values of the mesoscale variables. In other words, the realizable mesoscale values are the ones observed in the ensemble of all particles appearing in the microscale simulation. In contrast, sets of mesoscale values that are never observed in the microscale simulations are non-realizable. Realizability constraints may occur for a variety of reasons, e.g. due to conservation of mass, momentum, energy, etc., and are intrinsic properties of the microscale model. It is also important to note that the mesoscale values are usually strongly correlated. By this we mean that the NDF for any two mesoscale variables cannot be reconstructed from knowledge of the separate NDFs for each variable. Thus, by construction, the one-particle NDF contains all of the underlying correlations between the mesoscale variables for only one particle. 

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... 

Different particulate processes will behave very differently, and, in each case, a microscale physical scenario must be hypothesized in order to formulate the mesoscale model. For more details on these and other examples, readers are referred to the work of Lee et al. (1962), Marchisio Barresi (2009), and Zucca et al. (2006). 

This is a very important point because the added-mass term will modify the model for when other forces are included. In fact, for the general formulation, one should start with the single-particle momentum balance in Eq. (5.82) and add the other forces on the right-hand side. The final mesoscale model for Afp will have all of the terms on the right-hand side multiplied by the added-mass factor CvmPf/(pp + C vmPf)- In other words, due to the added mass, Afp cannot be found by simply adding together the models for the individual forces. See Section 5.3.4 for more details. 

Schayes, G., Thunis, R, and Bornstein, R. (1996) Topographic Vorticity-Mode Mesoscale-K (TVM) Model. Parti Formulation, Journal of Applied Meteorology Vol. 35, 1815-1824. 

This formulation is particularly convenient when Euler-Lagrange simulations are used to approximate the disperse multiphase flow in terms of a fimte sample of particles. As discussed in Sections 5.2 and 5.3, although some of the mesoscale variables are intensive (i.e. independent of the particle mass), it is usually best to start with a conserved extensive variable (e.g. particle mass or particle momentum) when deriving the single-particle models. For example, in Chapter 4 we found that must have at least one component, corresponding to the fluid mass seen by a particle, in order to describe cases in which the disperse-phase volume fraction is not constant. 

Laprise, R., Caya, D., Bergeron, G., and Giguere, M. (1997). The formulation of the Andre Robert MC (mesoscale compressible community) model, Atmos. Ocean XXXV, 195-220. 

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