Mesoscale model energy conservation

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. 

Mesoscale modeling of SOFCs focuses on modeling the transport and reactions of gas species in the porous microstructures of the electrodes [3, 34, 56-59]. In these models, the porous microstructure is explicitly resolved, which negates the need for the effective parameters of macroscale models. The transport and reactions of species in mesoscale models are described by the species [Eq. (26.1)], momentum [Eq. (26.5)], and energy [Eq. (26.7)] conservation equations, which are solved at the pore scale. At the pore scale, the conservation equations are solved in two separate domains the solid domain of the tri-layer and the gas domain of the pore space within the tri-layer. Mesoscale models aim to understand the effects of microstructure and local conditions near the electrode-electrolyte interface on the SOEC physics and performance. These models have been used to investigate a number of design and degradation issues in the electrodes such as the effects of microstructure on the transport of species in the anode [19, 56] and the reactions of chromium contaminants in the cathode [34]. 

Cluster/mesoscale To develop a general theory to link the discrete and continuum approaches, so that particle scale heat transfer information, generated from DEM-based simulation, can be quantified in terms of (macroscopic) energy conservation equations, constitutive relations, and boundary conditions that can be implemented in continuum-based process modeling of thermochemical behaviors. 

Cell-level models solve the species [Eq. (26.1)], momentum [Eq. (26.5)], and energy [Eq. (26.7)] conservation equations using the effective properties of the electrodes and can include the electrochemistry using a continuum-scale (Section 26.2.4.1) or a mesoscale (Section 26.2.4.2) approach. Traditionally, cell-level models use a continuum-scale electrochemistry approach, which includes the electrochemistry as a boundary condition at the electrode-electrolyte interface [17, 51, 54] or over a specified reaction zone near the interface. The electrochemistry is modeled via the Nernst equation [Eq. (26.12)] using a prescribed current density and assumptions for the polarizations in the cell. The continuum-scale electrochemistry is then coupled to the species conservation equation [Eq. (26.1)] using Faraday s law ... 

Figure 4 also compares the single-bubble-size (SBS) model and DBS model. In the former case, only one bubble class is introduced in the resolution of structure and energy dissipation, and hence there are only three structure parameters and two conservation equations. The three energy dissipation terms also only hold for one bubble class. The SBS model calculation shows only a monotonous increase of gas holdup and there is no jump change. This can be understood since the mesoscale mechanism, i.e., the compromise between the two dominant mechanisms pertinent to the TBCs, cannot be reflected in the SBS model, and therefore, the capabUity of reflecting the structure heterogeneity and evolution at macroscale is lost. 

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