Thesis for degree: Licentiate of Engineering

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intrinsic rate constant of each event. These methods are most suited for computational calculations when it is unfeasible to compute an exact result with a deterministic algorithm [19]. The MC methods is especially useful for simulating systems with many coupled parameters, such as fluids and disordered materials, but unfortunately also tend to have a high computational cost. MC methods vary, but tend to follow a particular pattern: Define a domain of possible inputs, generate inputs randomly from a probability distribution over the domain, perform a deterministic computation on the inputs and aggregate the results. MC is a modeling method for the dynamic behavior of molecules by comparing the rates of individual steps with random numbers. The method is used to investigate non-equilibrium systems or the time evolution of some specific processes occurring in nature, typically processes that occur with a known rate. The probability needs to be defined prior to the simulation [19]. MC was proven functional by a study by Lau et al. [19], conducted on a cathode of an SOFC for the oxygen reduction reaction. Lau et al. [19] found that the temperature had a great effect throughout the simulation which in this case was on the ionic current density. One difference between Monte Carlo and Molecular dynamics (MD) modeling is the different time scales. MC can handle longer time scales, typically seconds, whereas MD handles time scales around microseconds or even smaller [19]. To make statistically valid conclusions from the simulations, the time span simulated should match the kinetics of the natural process. The modeling should allow for different reaction pathways [17]. The relevant parameters are possible to be captured by both MC and MD. However, one would prefer to use the models for different processes depending on the needed elapsed time. MD simulates physical movements of atoms and molecules by computer modeling based on statistical mechanics. MD allows insight into molecular motion on an atomic scale and detailed time and space resolution into representative behavior for carefully selected systems [20]. MD can be used to model local structures at elevated temperatures where it is challenging to perform experiments with conclusive outcomes. MD has been used to perform simulations on diffusion and ion transport with successful outcome [20]. 2.2.4 Modeling integration issues Physical problems can often be described with a set of partial differential equations. The coupled partial differential equations can be solved simultaneously in physical domains for corresponding physical phenomena. The integration issues occur because the physical and chemical processes are linked to each other and so also the equations in the model [11]. The mass, heat, momentum transport as well as ionic/electronic transport and the reaction rate are dependent on each other. The fluid properties and the momentum transport (flow field) depend on the temperature and concentrations and so does the chemical reactions. The chemical reactions generate and consume heat, i.e., the temperature distribution depends on the chemical reaction rate, as well as on the solid and the gas properties, for example the specific heat and the thermal conductivity [1, 11]. Until now, multicomponent gas diffusion has been modeled in the continuum flow regime for the porous electrodes using the Stefan-Maxwell equations by several research groups. Some of the equations used in our CFD-model, e.g., the Stefan-Maxwell model (SMM), are described based on a few empirical parameters, which are difficult to measure [14]. To enhance the knowledge of the impact of the transport process on the performance within the electrodes, the microstructure needs to be modeled in detail. Recent advances have made it possible to evaluate the microstructure using LBM without any modification of empirical parameters. Models have been developed for all the three spatial dimensions, but so far the main focus has been on particular parts of the fuel cell, e.g., anode, and also simple geometries [14-16]. 8
2.3 Lattice Boltzmann method LBM has shown promising simulation results of fluid flows and mass diffusion through complex geometries and this is an attractive characteristic for fuel cell modeling. Conventional CFD methods use fluid density, velocity and pressure as their primary variables, while the LBM uses a more fundamental approach with a so-called particle velocity distribution function (PDF) [16]. The PDF is here denoted f a and originates from the basic Boltzmann gas concepts where a derived simplified form of the Boltzmann equation is described by classical mechanics with statistical treatment based on the high number of particles. The distribution is described by the coordinates of the position and momentum vectors, and the time step [3]. The LBM framework is built on lattice points, which are given locations placed all over the solution domain. The lattice unit (lu) is a fundamental measure of length and the time step (ts) is the measure of the time unit. The neighboring particles are connected to the main focused particle at the time with the velocity magnitude e a , schematically described in Figure 2.3 from a 2D point of view. This type of structural framework is called D2Q9 and stands for two dimensions with nine velocities marked as e a , where a represents the direction (= 0, 1, 2... 8) [3, 16]. For the 2D case, there exist two appropriate choices of system structure, namely D2Q5 and D2Q9 with 5 and 9 directional velocities, respectively. The D2Q9 framework has the possibility to capture more information but will be computationally heavier than the smaller one D2Q5. Figure 2.3: Schematic structural framework for the D2Q9 lattice and velocities. The PDF is defined as the number of particles of the same species travelling along a particular direction with a particular velocity. The single-particle distribution function f a can essentially be seen as a histogram representing a frequency of occurrence. The frequencies can be considered to be direction specific fluid densities. The LBM is described by two different actions taking part at each lattice point (site); namely streaming and collision. Streaming describes the movement of the particles of each species and the collision describes the interactions between the particles of the same or different species. Furthermore, these actions are combined in the LB equation called the distribution function [3, 14-16]. 9
Page 1 and 2: SOFC modeling from micro to macrosc
Page 3 and 4: Abstract The purpose of this work i
Page 5 and 6: katalytisk omvandling av naturgas o
Page 7 and 8: List of publications Journal public
Page 9 and 10: 3.2 Governing equations for the mac
Page 11 and 12: Q source term (heat), W/m3 q heat f
Page 13 and 14: 1 Introduction After some time of d
Page 15 and 16: 2 SOFC modeling at smaller scales T
Page 17 and 18: etween 0.3 and 1.5 mm. In this conf
Page 19: Volume Method (FVM) which adopt the
Page 23 and 24: streaming [3]. The concentration ρ
Page 25 and 26: ease the correspondence between the
Page 27 and 28: applicable for a mixture of two spe
Page 29 and 30: Figure 2.6: Effective boundary arra
Page 31 and 32: Table 2.1: Comparison of previous w
Page 33 and 34: Figure 2.7: Schematic illustration
Page 35 and 36: model is presented after the data c
Page 37 and 38: 3.2.1 Mass transport In the electro
Page 39 and 40: where h v is the volume heat transf
Page 41 and 42: Achenbach [43], it was found that t
Page 43 and 44: cathode are assumed to be similar t
Page 45 and 46: (3.48) where, besides the parameter
Page 47 and 48: for an isothermal spherical particl
Page 49 and 50: Velocity carried out to check wheth
Page 51 and 52: Figure 4.4: Velocity field [lu/ts]
Page 53 and 54: Table 4.2: Parameters in the SOFC a
Page 55 and 56: eactions are safely fulfilled. This
Page 57 and 58: Figure 4.8: Mole fraction distribut
Page 59 and 60: Table 4.5: Inlet mole fractions for
Page 61 and 62: Figure 4.12: Mole fraction distribu
Page 63 and 64: 5 Conclusions The physics and the t
Page 65 and 66: 7 References [1] Andersson M., Yuan
Page 67 and 68: [25] Latt J., Hydrodynamic Limit of

Thesis for degree: Licentiate of Engineering

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