Lithium-ion batteries are an important technology to facilitate efficient energy storage and enable a shift from petroleum based energy to more environmentally benign sources. Such systems can be utilized most efficiently if good understanding of performance can be achieved for a range of operating conditions. Mathematical models can be useful to predict battery behavior to allow for optimization of design and control. An analytical solution is ideally preferred to solve the equations of a mathematical model, as it eliminates the error that arises when using numerical techniques and is usually computationally cheap. An analytical solution provides insight into the behavior of the system and also explicitly shows the effects of different parameters on the behavior. However, most engineering models, including the majority of battery models, cannot be solved analytically due to non-linearities in the equations and state dependent transport and kinetic parameters. The numerical method used to solve the system of equations describing a battery operation can have a significant impact on the computational cost of the simulation. In this paper, a model reformulation of the porous electrode pseudo three dimensional (P3D) which significantly reduces the computational cost of lithium ion battery simulation, while maintaining high accuracy, is discussed. This reformulation enables the use of the P3D model into applications that would otherwise be too computationally expensive to justify its use, such as online control, optimization, and parameter estimation. Furthermore, the P3D model has proven to be robust enough to allow for the inclusion of additional physical phenomena as understanding improves. In this paper, the reformulated model is used to allow for more complicated physical phenomena to be considered for study, including thermal effects. There is an increasing societal pressure to utilize alternative energy sources to supplant the high use of fossil fuels. As energy and power demand is continually increasing, both in terms of grid usage and for transportation, there has been more interest in developing renewable energy sources. One problem with renewable energy sources is the intermittent nature and short-term unpredictability of supply of sources such as wind and solar. Thus, in order to match supply and demand, some form of energy storage is required, and lithium-ion battery technologies are one possible solution. Furthermore, electric vehicles are increasing in popularity as the price of liquid fuels generally increase. Lithium-ion batteries are a popular choice for electric vehicles because of their high energy and power density compared to other battery chemistries.The performance of lithium-ion batteries is highly dependent on the conditions at which they are exposed as well as the state of the internal variables. This has led to the development of several mathematical models to simulate battery behavior, ranging from simple empirical-based models or circuit based models 1,2 to computationally expensive mole...

The graphite anode in lithium-ion batteries is vulnerable to capacity fade due to several mechanisms. Advancement in understanding of such capacity fade mechanisms has paved the way for selecting design parameters that consider these effects. This paper shows the effect of porosity, thickness, and tortuosity of the anode on capacity fade mechanisms. Three main capacity fade mechanisms are analyzed in this paper: (1) solid electrolyte interface (SEI) side reaction, (2) lithium plating side reactions and (3) mechanical degradation due to intercalation induced stresses. Moreover, for a given thickness and porosity of anode, the effect of porosity variation on capacity fade mechanisms is also presented. Lithium-ion chemistries are attractive for many applications due to high cell voltage, high volumetric and gravimetric energy density (100 Wh/kg), high power density (300 W/kg), good temperature range, low memory effect, and relatively long battery life.1-3 Capacity fade, underutilization, and thermal runaway are the main issues that need to be addressed in order to use a lithium-ion battery efficiently and safely over a long life.Research on various fronts is underway to address the issues mentioned above. While finding better materials and improving their properties is one approach, the use of system level strategies to reach better efficiency in existing and emerging systems is another approach. The true potential of battery materials cannot be realized due to system level efficiencies, especially where transport effects become dominant (e.g. higher rates of charging/discharging at normal temperature or low temperatures operations).One of the many problems that can be addressed by continuum level modeling approaches is finding the optimum thicknesses and porosities of anode and cathode materials while keeping various processes and objectives in mind. These objectives may be discharge capacities at higher rates, charging time, mechanical degradation due to intercalation induced stresses, loss of active lithium due to parasitic side reaction (SEI layer and lithium plating), safety etc. While one would like to maximize energy density by packing the solid phase material compactly with larger thickness; rate capacity, safety and capacity fade may cause such an approach to be impractical.How should one choose the porosity and length of anode and cathode is an interesting research problem. Design optimization (porosity and thickness) for lithium-ion battery can be traced back to the work done by Prof. Newman using the reaction zone model 4 and with the pseudo two dimensional model. 5 Work on determining the optimal porosity distribution by considering the ohmic drop has been done by Ramadesigan et al. 6 Effect of low temperature and porosity on the performance of lithium-ion batteries is also studied by Ji et al. 7 While these works are based on maximizing the energy/power density of lithium-ion batteries by choosing optimal design parameters, no work has been done in quantifying the effect of design parameters on capaci...

Lithium-ion batteries are typically modeled using porous electrode theory coupled with various transport and reaction mechanisms with an appropriate discretization or approximation for the solid phase diffusion within the electrode particle. One of the major difficulties in simulating Li-ion battery models is the need for simulating solid-phase diffusion in the second radial dimension r within the particle. It increases the complexity of the model as well as the computation time/cost to a great extent. This is particularly true for the inclusion of pressure induced diffusion inside particles experiencing volume change. A computationally efficient representation for solid-phase diffusion is discussed in this paper. The operating condition has a significant effect on the validity, accuracy, and efficiency of various approximations for the solid-phase transport governed by pressure induced diffusion. This paper introduces efficient methods for solid phase reformulation-(1) parabolic profile approach and (2) a mixed order finite difference method for approximating/representing solid-phase concentration variations within the active materials of porous electrodes for macroscopic models for lithium-ion batteries.

Oriented one-dimensional nanostructures have been of substantial interest as electrodes for lithium-ion batteries due to the better performance both in terms of initial capacity and lower capacity fade compared to powder pressed electrodes. This paper focuses on a model driven approach to understanding the relationship between the morphology of these oriented nanostructures to the performance of the battery. The Newman-type P2D modeling technique is applied to a porous electrode made up with solid continuous cylinders that extends from the current collectors to separator. TiO 2 columnar nanostructures of varying heights were synthesized using the aerosol chemical vapor deposition (ACVD) and their performance as electrodes in a lithium-ion battery was measured. This electrochemical transport model was validated with the experimental data. This model was used to understand the role of transport parameters, including the diffusivity of lithium in the TiO 2 and the electronic conductivity of the TiO 2 columns, and structural parameters, including the height of the columns and the porosity of the electrode, on the areal capacity of a lithium ion battery at different rates of discharge. The model enables for the prediction of optimized structural parameters of one-dimensional electrodes tailored to the desired application of lithium and sodium-ion batteries. Lithium-ion batteries (LiBs) have emerged as the dominant power source for most electronic applications today, as well as the most suitable candidates for electric vehicles and hybrid electric vehicles. The diverse range of applications for which LIBs are used demand both high energy densities and high power densities, although they are inversely related.1 Several research approaches have been adopted for increasing both the energy density and the power density of lithium ion batteries, and controlling the nanostructure of the electrode material has been one such widely adopted approach. One-dimensional (1D) nanostructures in particular have received considerable attention for both cathode and anode materials 3-5 due to the several advantages provided by the 1D nanostructures, which can enhance both the energy and the power density of the battery. These advantages include (1) the efficient electron transport pathway provided by the nanostructure, 6 (2) shorter ion diffusion path owing to the less tortuous path and the larger surface to volume ratio, 7 and (3) better strain relaxation due to the accommodation space in between the nanostructures.8 Recent research has focused on the direct growth of the 1D nanostructures on the current collector to obtain oriented nanostructures which further provide improved performance due to the direct attachment of each 1D nanostructure to the current collector ensuring their participation in the electrochemical reaction and obviating the need for any binding agent. Different applications of LiBs demand the optimization of their energy density and power density. While nanostructuring aims to maximize both the densities, further ...

Porous electrode theory coupled with transport and reaction mechanisms is a widely used technique to model Li-ion batteries employing an appropriate discretization or approximation for solid phase diffusion with electrode particles. One of the major difficulties in simulating Li-ion battery models is the need to account for solid phase diffusion in a second-radial-dimension r, which increases the computation time/cost to a great extent. Various methods that reduce the computational cost have been introduced to treat this phenomenon, but most of them do not guarantee mass conservation. The aim of this paper is to introduce an inherently mass conserving yet computationally efficient method for solid phase diffusion based on Lobatto III A quadrature. This paper also presents coupling of the new solid phase reformulation scheme with a macro-homogeneous porous electrode theory based pseudo 2D model for Li-ion battery. Lithium-ion chemistry has been identified as a good candidate for high-power/high-energy secondary batteries which are expected to play a vital role in the future of automobile, power storage, military, mobile, and space applications. Significant efforts have been made and reported in literature regarding the modeling and understanding of Lithium-ion batteries using physics based first-principles models. The most widely used first principles model for the lithium-ion battery is the porous electrode pseudo two dimensional (P2D) model, 1 which is based on the fundamentals of electrochemistry and transport phenomena. These models are represented by coupled nonlinear PDEs in 1-2 dimensions, are typically solved numerically and require few minutes to hours to simulate.For the P2D model, the diffusion of Lithium ion into the solid electrode particles is solved in a pseudo dimension r, which is coupled to the macro-homogenous model at the surface of the intercalation particles. This pseudo 2 dimensional approximation avoids the need for a solution of a full 2 dimensional model and hence the name. Accurate predictions of the concentration at the surface of the particle are therefore important, as it contributes to the exchange current density for the reaction at the particle-electrolyte interface. Typically, solid phase diffusion in the micro-scale is modeled using Fick's law of diffusion. More detailed schemes involving pressure induced diffusion along with solid phase diffusion have also been reported in literature. 2These models are important especially for high capacity materials where the stress developed affects the concentration profile inside the intercalation particle. For phase changing materials, the shrinking core model 3 has been used and approximate solutions have been proposed. 4 Cahn-Hilliard models 5 have also been employed to track the phase boundary within active material particles during Lithium intercalation.6,7 One of the major difficulties in the electrochemical engineering models is that even the inclusion of a simple Fickian model for solid phase diffusion in a second dimension r increa...