Microscale model

The M2UE (Micro-scale Model for Urban Environment Nuterman 2008) is Computational Fluid Dynamics (CFD) microscale model for analysis of atmospheric processes and pollution prediction in the urban environment, which takes into account a complex character of aerodynamics in non-uniform urban relief with penetrable (vegetation) and impenetrable (buildings) obstacles and traffic induced... 

The air quality directive of the European commission demands maps on concentrations and exceedances in different detail. For this purpose numerical models can be used. Some model systems are already adjusted to deliver the corresponding maps. For instance, the model system M-SYS consists of three mesoscale and one microscale model areas and applies one-way-nesting for meteorology and chemistry (Trukenmiiller et al. 2004). 

In microscale models the explicit chain nature has generally been integrated out completely. Polymers are often described by variants of models, which were primarily developed for small molecular weight materials. Examples include the Avrami model of crystallization,- and the director model for liquid crystal polymer texture. Polymeric characteristics appear via the values of certain constants, i.e. different Frank elastic constant for liquid crystal polymers rather than via explicit chain simulations. While models such as the liquid crystal director model are based on continuum theory, they typically capture spatiotemporal interactions, which demand modelling on a very fine scale to capture the essential effects. It is not always clearly defined over which range of scales this approach can be applied. 

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 validation of mesoscale models can be carried out using the numerical solutions to microscale models (Tenneti et al, 2010), in much the same way as that in which DNS is used for model validation in turbulent single-phase flows. A typical mesoscale modeling strategy consists of four steps. 

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. 

As discussed in Chapter 2, the one-particle NDF does not usually provide a complete description of the microscale system. For example, a microscale system containing N particles would be completely described by an A-particle NDF. This is because the mesoscale variable in any one particle can, in principle, be influenced by the mesoscale variables in all N particles. Or, in other words, the N sets of mesoscale variables can be correlated with each other. For example, a system of particles interacting through binary collisions exhibits correlations between the velocities of the two particles before and after a collision. Thus, the time evolution of the one-particle NDF for velocity will involve the two-particle NDF due to the collisions. In the mesoscale modeling approach, the primary physical modeling step involves the approximation of the A-particle NDF (i.e. the exact microscale model) by a functional of the one-particle NDF. A typical example is the closure of the colli-sionterm (see Chapter 6) by approximating the two-particle NDF by the product of two one-particle NDFs. 

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... 

As derived from the microscale model in Chapter 4, the GPBE for a polydisperse multiphase flow has the following form ... 

The diffusion matrices Bpy, and Bfy, are symmetric and, most importantly, conservation of momentum at the microscale will require that they be dependent and, hence, at most only six diffusion coefficients need be determined from the microscale model (see the discussion leading to Eq. (5.17)). The simplest case occurs when the diffusion matrices are isotropic ... 

Note that the dependence of the mesoscale model on the moments of the NDF is used to introduce multi-particle effects through the mean-field variables such as the disperse-phase volume fraction. For this reason, the right-hand sides of Eq. (5.9) are different from the exact microscale models that were introduced in Chapter 4 (i.e. Eqs. (4. l)-(4.3) on page 103). Formally, given these dependences, the mesoscale models on the right-hand sides of Eq. (5.9) can be expressed explicitly (for example) as A p(t, X, ), where... 

The reader should note that the microscale model is used to determine the nonzero terms in B, and thus for the following discussion B can be assumed to be known. Using matrix notation and the properties of the Wiener process (Gardiner, 2004), a symmetric N x N diffusion matrix D can be defined by... 

The microscale model Figure 6 is pseudo onedimensional. Because of random distribution of stacks we assume that the macroscale effective diffusion coefficient is given as D =D, /3 where > 11 is the value directly calculated by (23) using the model of Figure 6. 

The macroscale and microscale models are sown in Figure 8 and 9 with its finite element meshes. The diffusion coefficient in the external pore water is 2.0x10 m /s, and one in the interlayer space 2.62 xl0 ° mVs, which is calculated by MD. The upper stream boundary concentration given at AB in Figure 8 is given as l.OxlO" mol/1. 

---