Decoupled finite element methods for general steady two-dimensional Boussinesq equations

This work presents two kinds of decoupled finite element methods for the steady natural convection problem in two dimensions. Firstly, the standard Galerkin finite element method is derived in detail stating algorithms needed for the realization in MATLAB. A numerical example verifies the error convergence. Secondly, using iteration, the Boussinesq equations are decoupled into the Navier-Stokes equations and a parabolic problem. The resulting problems are solved either in parallel or sequentially. Finally, the same numerical example as before is used to confirm the convergence and analyze the methods in terms of iteration performance. In addition to a higher flexibility and the convenience of exploiting existing solvers, the new decoupled finite element methods can be realized with less iteration steps, and thus more efficiently, if the focus is only on some of the unknowns or more information is provided --Abstract, page iii

Simulation of Li-Plating in Si/Graphite Composite Electrodes

It is a common perception that the demand for high-performance batteries is constantly increasing. While the primary characteristics depend on the application, decisive criteria are energy density, safety, cost, and sustainability of the batteries. Here, Li-ion batteries play a key role especially for electric vehicles and portable electronics. However, additional improvements are needed, for example, to achieve sufficient driving ranges. The performance characteristics of the batteries such as high energy or power density can be tuned by the electrode microstructure, composition, or choice of materials. One such promising material for the negative electrode is Silicon: Si exhibits a high theoretical capacity and is very abundant. On the other hand, Si shows a large volume expansion and low Li mobility. Thus, to take advantage of the high theoretical capacity and to limit the deformation during cycling, Si is mixed with Graphite to produce more practical Si/Graphite composite electrodes. In order to increase the cycle life of Si containing electrodes, it is critical to trace the degradation processes responsible for their performance loss. One major aging mechanism causing fast degradation and fundamental safety risks is lithium plating. The deposition of a metallic lithium phase on the surface of Si/Graphite anodes is barely studied in the literature yet crucial to improve the performance and safety of state-of-the-art Li-ion batteries. In our contribution we present simulation results of Si/Graphite composite electrodes including models for plating on Graphite [1] and Si particles in 3D microstructure-resolved simulations. Moreover, we developed a homogenized model of composite electrodes which also includes the volume changes during lithiation and delithiation of representative Si particles and the effect on transport processes [2]. The models serve as a starting point to understand Li plating in Si containing electrodes and, eventually, the design of better Si/Graphite composite electrodes. [1] Hein et.al., ACS Appl. Energy Mater. 2020, 3, 8519−8531, DOI: 10.1021/acsaem.0c01155 [2] Chandrasekaran et. al., J Electrochem Society 2010, 157, A1139-A1151, DOI: 10.1149/1.347422

Simulation of Li-Plating in Si/Graphite Composite-Electrodes

It is a common perception that the demand for high-performance batteries is constantly increasing. While the primary characteristics depend on the application, decisive criteria are energy density, safety, cost, and sustainability of the batteries. Here, Li-ion batteries play a key role especially for electric vehicles and portable electronics. How-ever, additional improvements are needed, for example, to achieve sufficient driving ranges. The per-formance characteristics of the batteries such as high energy or power density can be tuned by the electrode microstructure, composition, or choice of materials. At the negative electrode, Si compounds are mixed with graphite to increase the practical specific energies. On the one hand, Si exhibits a high theoretical capacity and is very abundant. On the other hand, Si shows a large volume expansion and low Li mobility. In order to increase the cycle life of Si containing electrodes, it is critical to trace the degradation processes responsible for their performance loss. One major aging mechanism causing fast degradation and fundamental safety risks is lithium plating. The deposition of a metallic lithium phase on the surface of Si/graphite anodes is barely studied in the literature yet crucial to improve the performance and safety of state-of-the-art Li-Ion batteries. In our contribution we present results of a homogenized model of a Si/Graphite composite electrode including models for plating on representative graphite [1] and Si particles. Moreover, we include the volume changes during lithiation and delithiation of representative Si particles and the effect on transport processes [2]. The model serves as a starting point to understand Li plating in Si containing electrodes and, eventually, the design of better Si/graphite composite-electrodes. [1] Hein et.al., ACS Appl. Energy Mater. 2020, 3, 8519−8531, DOI: 10.1021/acsaem.0c01155 [2] Chandrasekaran et. al., J Electrochem Society 2010, 157, A1139-A1151, DOI: 10.1149/1.347422

A Decoupled, Parallel, Iterative Finite Element Method for Solving the Steady Boussinesq Equations

In this work, a decoupled, parallel, iterative finite element method for solving the steady Boussinesq equations is proposed and analyzed. Starting from an initial guess, an iterative algorithm is designed to decouple the Naiver-Stokes equations and the heat equation based on certain explicit treatment with the solution from the previous iteration step. At each step of the iteration, the two equations can be solved in parallel by using finite element discretization. The existence and uniqueness of the solution to each step of the algorithm is proved. The stability analysis and error estimation are also carried out. Numerical tests are presented to verify the analysis results and illustrate the applicability of the proposed method

Simulation of Li Plating in Si/Graphite Composite Electrodes

It is a common perception that the demand for high-performance batteries is constantly increasing. While the primary characteristics depend on the application, decisive criteria are energy density, safety, cost, and sustainability of the batteries. Here, Li-ion batteries play a key role especially for electric vehicles and portable electronics. How-ever, additional improvements are needed, for example, to achieve the fast charging criteria given by the automotive industry. The performance characteristics of the batteries such as high energy or power density can be tuned by the electrode microstructure, composition, or choice of materials. One such promising material for the negative electrode is Silicon: Si exhibits a high theoretical capacity and is very abundant. On the other hand, Si shows a large volume expansion and low Li mobility. Thus, to take advantage of the high theoretical capacity and to limit the deformation during cycling, Si is mixed with Graphite to produce more practical Si/Graphite composite electrodes. In order to increase the cycle life of Si containing electrodes, it is critical to trace the degradation processes responsible for their performance loss. One major aging mechanism causing fast degradation and fundamental safety risks is Li plating. This deposition of a metallic lithium phase on the surface of Si/Graphite anodes is barely studied in the literature yet crucial to improve the performance and safety of state-of-the-art Li-ion batteries. In our contribution we present simulation results of Si/Graphite composite electrodes including mod-els for Li plating on Graphite and Si particles in 3D microstructure-resolved simulations. While focus-sing on the differing lithiation behaviours of Graphite and Si, the findings are validated with experimental results from our project partners. More specifically, we compare our simulations to multiple complemen-tary techniques such as neutron depth profiling (NDP), post-mortem glow discharge optical emission spectroscopy (GD-OES) depth profiling, and X-ray diffraction analysis (XRD). Under examination are commercial and self-manufactured full and half cells containing varying amounts of Si or SiC blend. Furthermore, we developed a homogenized p2D model of composite electrodes which also includes the volume changes during lithiation and delithiation of representative Si particles and the effect on transport processes. The studies serve as a starting point to understand Li plating in Si containing electrodes and, eventually, the design of better Si/Graphite composite electrodes

Charge Relaxation within Silicon/Graphite Anodes – A Multi-Method Study

As silicon/graphite (SiG) composites are more commonly used as the anode active material in commercial Li-ion batteries, investigation of the (de-)lithiation behavior of the blended anodes becomes increasingly important. In this study, the charge redistribution between graphite and silicon was investigated in graphite-NMC 622 and SiG (23 wt.-% Si) – NMC 622 bilayer pouch cells using in situ and operando X-ray diffraction (XRD). In addition to XRD, ex situ and in situ optical microscopy (IOM), as well as microstructural resolved simulations using digital twins of the cells, were used. Different SOC values (0%, 25%, 50%, 75%, and 100%) and two different C-rates (0.1C and 0.5C) were compared in cells during operation and in the relaxed state. Insights into the relaxation process at 75% SOC were gained by tracking of the charge redistribution in IOM cells. Ex situ optical microscopy measurements reinforced the findings of the IOM measurements. Both XRD and optical microscopy showed the disappearance of charge in the graphite component of the SiG anode during the relaxation period (≥24h) at SOC ≤75%, indicating a redistribution of Li from graphite into Si in the anode. The simulations allowed tracking of the concentration of Li in both active material components, verifying the observations on the charge relaxation processes observed in the XRD and microscopy experiments. The gained insights can support a better understanding of aging of blended SiG anodes during operation

Simulation of Li Plating in Si/Graphite Composite Electrodes

Li-ion batteries play a key role especially for electric vehicles and portable electronics. However, additional improvements are needed, for example, to achieve the fast charging criteria given by the automotive industry. The performance characteristics of the batteries such as high energy or power density can be tuned by the electrode microstructure, composition, or choice of materials. One such promising material for the negative electrode is Silicon: Si exhibits a high theoretical capacity and is very abundant. On the other hand, Si shows a large volume expansion and low Li mobility. Thus, to take advantage of the high theoretical capacity and to limit the deformation during cycling, Si is mixed with Graphite to produce more practical Si/Graphite composite electrodes. In order to increase the cycle life of Si containing electrodes, it is critical to trace the degradation processes responsible for their performance loss. One major aging mechanism causing fast degradation and fundamental safety risks is Li plating. This deposition of a metallic Li phase on the surface of Si/Graphite anodes is barely studied in the literature yet crucial to improve the performance and safety of state-of-the-art Li-ion batteries. In our contribution we present simulation results of Si/Graphite composite electrodes including models for Li plating on Graphite and Si particles in 3D microstructure-resolved simulations. While focusing on the differing lithiation behaviors of Graphite and Si, the findings are validated with experimental results from our project partners. More specifically, we compare our simulations to multiple complementary techniques such as neutron depth profiling (NDP), post-mortem glow discharge optical emission spectroscopy (GD-OES) depth profiling, and X-ray diffraction analysis (XRD). We examine commercial and self-manufactured full- and half-cells containing varying amounts of Si or SiC blend. Since inhomogeneities in the amount of Li plating were observed, studies on simplified half-cells are conducted to clarify the impact of relevant material parameters. Furthermore, we developed a homogenized p2D model of composite electrodes which also includes the volume changes during lithiation and delithiation of representative Si particles and the effect on transport processes. Not only provides the complementarity of sophisticated experimental and simulative studies a better understanding of how Li plating takes place in Si containing electrodes, but also enables an improved possibility to optimize the design of Si/Graphite composite electrodes