Macroscale model

Ivers-Tiffee E., Weber A., Schmid K., Krebs V. (2004) Macroscale modeling of cathode formation in SOFC. Solid State Ionics 174, 223-232. 

M.F. Horstemeyer et al Using a micromechanical finite element parametric study to motivate a phenomenological macroscale model for void/crack nucleation in aluminum with a hard second phase. Mech. Matls. 35, 675-687 (2003)... 

During the last decade considerable attention has been put on the macroscale modeling of bubble breakage in gas-liquid dispersions (e.g., [92, 43, 99,... 

Macroscale Model Hydrodynamic description Euler-Euler models... 

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 mentioned above, macroscale models are written in terms of transport equations for the lower-order moments of the NDF. The different types of moments will be discussed in Chapters 2 and 4. However, the lower-order moments that usually appear in macroscale models for monodisperse particles are the disperse-phase volume fraction, the disperse-phase mean velocity, and the disperse-phase granular temperature. When the particles are polydisperse, a description of the PSD requires (at a minimum) the mean and standard deviation of the particle size, or in other words the first three moments of the PSD. However, a more complete description of the PSD will require a larger set of particle-size moments. 

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 ... 

Figure 1.5. From the kinetic equation to macroscale models using moment methods that are based on reconstruction of the NDF.

Chapter 1 introduces key concepts, such as flow regimes and relevant dimensionless numbers, by using two examples the PBE for fine particles and the KE for gas-particle flow. Subsequently the mesoscale modeling approach used throughout the book is explained in detail, with particular focus on the relation to microscale and macroscale models and the resulting closure problems. 

The importance of developing appropriate quantitative descriptions of microstructure that not only can be obtained by well-established stereological measurements but also can be expressed in a manner compatible with continuum descriptions of matter is fundamental to connecting the micro- and mesoscopic to the macroscopic descriptions of materials. Furthermore, such descriptions must be incorporated into macroscale models of material behavior in a manner that permits the calculation of macroscopic engineering properties in terms of the mesoscopic attributes of materials. 

Figure 8. The macroscale model of through-diffusion experiment.

Several macroscale models with varying levels of complexity and covering macroscale phenomena such as the reactor residence time effects and micro- and macromixing behavior have been developed for olefin polymerization but is not discussed here in detail for the sake of brevity [149-157],... 

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