Mesoscale modeling reaction

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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... 

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]. 

Mesoscale models provide valuable insight into the operation of SOFCs and how the micrometer-scale phenomena translate into the macroscale behavior of the SOFC. By discretely modeling the gas phase and solid phase of the SOFC electrodes, they can investigate the surface reactions and transport in SOFCs, which could lead to advances in the design of the electrodes to improve the electrochemical performance of the SOFC. They are also able to provide macroscale models with effective properties for the transport and reaction parameters based on the local microstracture and physics of the SOFC. 

Mesoscale Modeling for Reaction-Diffusion in Catalyst Pellet 296... 

Apparendy, there lacks an expHcit hnk between the microscale and macroscale models discussed above. In this section, a mesoscale model is introduced to describe the reaction—diffusion in a single catalyst pellet. The significance of this model can be embodied at least in two aspects a necessary link between the microscale model and macroscale model and the theoretical basis for MTO catalyst design optimization. 

It is meaningful to examine the relation between microscale model, mesoscale model, and micromodel. For reaction kinetics, microscale and mesoscale models adopt the same kinetics that based on element reaction system. For diffusion, mesoscale model embodies two diffusion mechanisms (one for micropores and another for mesopores and macropores), and microscale model considers one diffusion mechanism since it only has micropores. No diffusion was considered within the macropores. It is obvious that the mesoscale model possesses the same theoretical foundation as the microscale model, but its application scope has been enlarged compared to the microscale model. Therefore, it could be reliably used as a tool to derive some parameters, such as effective chemical kinetics and effective diffusion parameters, for macroscale model. In the section following, we discuss the method on how to link the microscale kinetics to the lumped macroscale kinetics via the mesoscale modeling approach. 

In all of the above cases, a strong non-linear coupling exists between reaction and transport at micro- and mesoscales, and the reactor performance at the macroscale. As a result, the physics at small scales influences the reactor and hence the process performance significantly. As stated in the introduction, such small-scale effects could be quantified by numerically solving the full CDR equation from the macro down to the microscale. However, the solution of the CDR equation from the reactor (macro) scale down to the local diffusional (micro) scale using CFD is prohibitive in terms of numerical effort, and impractical for the purpose of reactor control and optimization. Our focus here is how to obtain accurate low-dimensional models of these multi-scale systems in terms of average (and measurable) variables. 

In the last section, we discussed the use of QC calculations to elucidate reaction mechanisms. First-principle atomistic calculations offer valuable information on how reactions happen by providing detailed PES for various reaction pathways. Potential energy surfaces can also be obtained as a function of electrode potential (for example see Refs. [16, 18, 33, 38]). However, these calculations do not provide information on the complex reaction kinetics that occur on timescales and lengthscales of electrochemical experiments. Mesoscale lattice models can be used to address this issue. For example, in Refs. [25, 51, 52] kinetic Monte Carlo (KMC) simulations were used to simulate voltammetry transients in the timescale of seconds to model Pt(l 11) and Pt(lOO) surfaces containing up to 256x256 atoms. These models can be developed based on insights obtained from first-principle QC calculations and experiments. Theory and/or experiments can be used to parameterize these models. For example, rate theories [22, 24, 53, 54] can be applied on detailed potential energy surfaces from accurate QC calculations to calculate electrochemical rate constants. On the other hand, approximate rate constants for some reactions can be obtained from experiments (for example see Refs. [25, 26]). This chapter describes the later approach. 

As shown in Figure 26.1, the wide gap opens up between the particle and continuum paradigms. This gap cannot be spanned using statistical mechanical methods only. The existing theoretical models to be applied in the mesoscale are based on heuristics obtained via downscaling of macroscopic models and upscaling particle approach. Simphfied theoretical models of complex fluid flows, e.g., flows in porous media, non-Newtonian fluid dynamics, thin film behavior, flows in presence of chemical reactions, and hydrodynamic instabilities formation, involve not only vah-dation but should be supported by more accurate computational models as well. However, until now, there has not been any precisely defined computational model, which operates in the mesoscale, in the range from 10 A to tens of microns. 

In recent years, new discrete-particle methods have been developed for modeling physical and chemical phenomena occurring in the mesoscale. The most popular are grid-type techniques such as cellular automata (CA), LG, LBG, diffusion- and reaction-limited aggregation [37] and stochastic gridless methods, e.g., DSMC used for modeling systems characterized by a large Knudsen number [41], and SRD [39]. Unlike DSMC, in SRD collisions are modeled by simultaneous stochastic rotation of the relative velocities of every particle in each cell. 

This is followed in Section 26.4 by a discussion of mesoscale modehng of the SOFC electrodes in which the SOFC electrodes are explicitly resolved and the detailed reactive transport and electrochemistry is modeled. Section 26.5 briefly describes nanoscale approaches for modeling the transport and reactions of species in the SOFCs, which are suitable for elucidating kinetic and mechanistic issues relevant to SOFC performance. 

In SOFC modeling, the electrochemistry of the fuel cell can be included in the model at various levels of detail. In a continuum-scale approach, empirical current-potential (f-V) relations are typically used to model the electrochemistry of the SOFC. In a mesoscale approach, the electrochemical reactions and the transport of electrons and ions in the SOFC can be modeled explicitly. The continuum-scale approach allows for a quick evaluation of the I-V performance of the SOFC by assuming that the electrochemistry occurs only at the interface of the electrode and electrolyte. In the mesoscale approach, the electrochemistry of the SOFC is resolved through the thickness of the electrodes based on the local conditions in the cell. In this section, we discuss the details of both approaches. 

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