Mesoscale validation

Because mesoscale methods are so new, it is very important to validate the results as much as possible. One of the best forms of validation is to compare the computational results to experimental results. Often, experimental results are not available for the system of interest, so an initial validation calculation is done for a similar system for which experimental results are available. Results may also be compared to any other applicable theoretical results. The researcher can verify that a sulficiently long simulation was run by seeing that the same end results are obtained after starting from several different initial configurations. 

The applicability of mesoscale techniques to systems difficult to describe in any other manner makes it likely that these simulations will continue to be used. At the present time, there is very little performance data available for these simulations. Researchers are advised to carefully consider the fundamental assumptions of these techniques and validate the results as much as possible. 

In studies of reactions in nanomaterials, biochemical reactions within the cell, and other systems with small length scales, it is necessary to deal with reactive dynamics on a mesoscale level that incorporates the effects of molecular fluctuations. In such systems mean field kinetic approaches may lose their validity. In this section we show how hybrid MPC-MD schemes can be generalized to treat chemical reactions. 

Sewell and co workers [145-148] have performed molecular dynamics simulations using the HMX model developed by Smith and Bharadwaj [142] to predict thermophysical and mechanical properties of HMX for use in mesoscale simulations of HMX-containing plastic-bonded explosives. Since much of the information needed for the mesoscale models cannot readily be obtained through experimental measurement, Menikoff and Sewell [145] demonstrate how information on HMX generated through molecular dynamics simulation supplement the available experimental information to provide the necessary data for the mesoscale models. The information generated from molecular dynamics simulations of HMX using the Smith and Bharadwaj model [142] includes shear viscosity, self-diffusion [146] and thermal conductivity [147] of liquid HMX. Sewell et al. have also assessed the validity of the HMX flexible model proposed by Smith and Bharadwaj in molecular dynamics studies of HMX crystalline polymorphs. 

We think that judicious application of molecular simulation tools for the calculation of thermophysical and mechanical properties is a viable strategy for obtaining some of the information required as input to mesoscale equations of state. Given a validated potential-energy surface, simulations can serve as a complement to experimental data by extending intervals in pressure and temperature for which information is available. Furthermore, in many cases, simulations provide the only realistic means to obtain key properties e.g., for explosives that decompose upon melting, measurement of liquid-state properties is extremely difficult, if not impossible, due to extremely fast reaction rates, which nevertheless correspond to time scales that must be resolved in mesoscale simulations of explosive shock initiation. By contrast, molecular dynamics simulations can provide converged values for those properties on time scales below the chemical reaction induction times. Finally,... 

Roulet, Y.-A., Martilli, A., Rotach, M W and Clappier, A.(2005) Validation of an Urban Surface Exchange Parameterization for Mesoscale Models - ID Case in a Street Canyon, Journal of Applied Meteorology Vol. 44, No. 9,1484-1498. 

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. 

Use the microscale simulation results to validate the predictions of the mesoscale model. 

Return to step 1 to improve the mesoscale model by applying new insights gained from discrepancies seen during validation. 

Note that the order of these steps is important. Indeed, it is very rare for a successful mesoscale model to result from simply studying the results from a microscale simulation that is not specifically designed to test a particular assumption or hypothesis. In general, a microscale simulation will produce an enormous quantity of data if all flow variables are stored at every time step, and it is not realistic to expect that such data sets will reveal the correct form of the mesoscale models without some a priori understanding of the physics. On the other hand, microscale simulations offer an invaluable and often unique tool for sorting out the validity of a proposed mesoscale model because they contain all of the microscale variables, many of which are impossible to measure in laboratory experiments. 

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... 

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... 

Alternatively, if one argues that all particles are statistically identical, then Ajp )i and should not depend on n. Although this assumption is not necessary for deriving Eq. (4.39), it is usually invoked to simplify the physical modeling. In other words, if all particles are statistically identical, the mesoscale models can be validated by considering single-particle statistics from DNS for randomly selected particles. 

It is important to remind the reader that U is the velocity of the fluid phase seen by the particle, U - U is the slip velocity, dp is the particle diameter, and Vf is the kinematic viscosity of the fluid phase. Note that Eq. (5.33) depends on the particle velocity U and is valid in the zero-Stokes-number limit where U = U so that particles follow the fluid. The correlation in Eq. (5.31) is valid only for RCp < 1 and Sc > 200. For larger particle Reynolds numbers the following correlations can be used Sh = 2 -i- 0.724Rep Sc, which is valid for 100 < RCp < 2000, and Sh = 2 -i- 0.425RCp Sc, which is valid for 2000 < RCp <10. Among the other correlations available, it is important to cite the one proposed by Ranz Marshall (1952) for macroparticles Sh = 2.0 -i- O.bReJ Sc. These expressions assume that the fluid velocity U is known. For micron-sized (or smaller) particles moving in turbulent fluids for which only the ensemble-mean fluid velocity (Uf) is known, it is instead better to employ the mesoscale model derived by Armenante Kirwan (1989) Sh = 2.0 -i- 0.52(Re ) Sc, where Re = is the modi-... 

The mesoscale models for momentum transfer between phases differ quite substantially depending on the multiphase system under investigation, and different semi-empirical relationships have been developed for different systems. Since the nature of the disperse phase is particularly important, the available mesoscale models are generally divided into those valid for fluid-fluid and those valid for fluid-solid systems. The main difference is that in fluid-fluid systems the elements of the disperse phase are deformable particles (i.e. bubbles or droplets), whereas in fluid-solid systems the disperse phase is constituted by particles of constant shape. Typical fluid-fluid systems for which the mesoscale models reported below apply are gas-liquid, liquid-liquid, and liquid-gas systems. The mesoscale models reported for fluid-solid systems are valid both for gas-solid and for liquid-solid systems. As a general rule, the mesoscale model for Afp should be derived starting from a single-particle momentum balance ... 

These relationships are valid for isolated bubbles moving under laminar flow conditions. In the case of turbulent flow, the effect of turbulent eddies impinging on the bubble surface is to increase the drag forces. This is typically accounted for by introducing an effective fluid viscosity (rather than the molecular viscosity of the continuous phase, yUf) defined as pi.eff = Pi + C pts, where ef is the turbulence-dissipation rate in the fluid phase and Cl is a constant that is usually taken equal to 0.02. This effective viscosity, which is used for the calculation of the bubble/particle Reynolds number (Bakker van den Akker, 1994), accounts for the turbulent reduction of slip due to the increased momentum transport around the bubble, which is in turn related to the ratio of bubble size and turbulence length scale. However, the reader is reminded that the mesoscale model does not include macroscale turbulence and, hence, using an effective viscosity that is based on the macroscale turbulence is not appropriate. 

The method was validated by hydraulic calculations carried out in a pleistocene aged watershed covering 220 km (Stoebber river) and applied for the calculation of groundwater residence times in the mesoscale watershed of the Ucker river (covering 2400 km ) and planned for modelling substance balance in the Oderbruch region (800 km2). 

It will be assumed that the polymerization conditions at the active site are known in this present section we will worry about them in a later section. In Section 2.4, mesoscale models will be developed that will tell us how the concentration of reagents, as well as temperature, varies as a function of radial position in the polymer particle. If radial gradients in the polymer particles are significant, the models developed in this section are still valid for the mesoscale, but only locally, at a given radial position in the particle. Finally, at the end of Section 2.5, a simple way to connect micro-, meso- and macroscale in one unified approach for modeling olefin polymerization reactors will be proposed. 

Dudhia, J. (1993). A nonhydrostatic version of the Penn State/NCAR mesoscale model Validation tests and simulations of an Atlantic cyclone and cold frorA Monthly Weather Rev. 121, 1493-1513. 

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