Energy conservation microscale

Hybrid mixture theory is a hybridization of classical volume averaging of field equations (conservation of mass, momenta, energy) and classical theory of mixtures [8] whose theory of constitution results from the exploitation of the entropy inequality in the sense of Coleman and Noll [9], In [4] the microscale field equations for each species of each phase, modified appropriately to include charges, polarization, and an electric field, are averaged to the macroscale, defined to be the scale where the phases are indistinguishable. Thus at the macroscale the porous media is viewed a mixture, with each thermodynamic property for each constituent of each phase defined at each point in space. 

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 above, the stabdity condition is expected to reach extremum in sufficiendy large space (e.g., cross section of a fluidized bed) instead of local cell. The energy to sustain mesoscale structures in a fluidized bed comes largely from the mean relative motion between gas and particles on the macroscale. Furthermore, the dynamic evolution of mesoscale structure and its energy transfer is subject to both macroscale operating conditions and the conservation laws in microscale computational cells. As a result, a two-step scheme was proposed to fulfill the coupHng between EMMS and hydrodynamic conservation equations, called EMMS/matrix (Lu et al, 2009 Wang and Li, 2007). At the macroscale (reactor), the bi-objective optimization method in terms of min was first used to resolve the mesoscale parameters, say, dc and gc. These mesoscale parameters were then incorporated... 

---