Analytical solution for electrolyte concentration distribution in lithium-ion batteries

Abstract

In this article, the method of separation of variables (SOV), as illustrated by Subramanian and White (J Power Sources 96:385, 2001), is applied to determine the concentration variations at any point within a three region simplified lithium-ion cell sandwich, undergoing constant current discharge. The primary objective is to obtain an analytical solution that accounts for transient diffusion inside the cell sandwich. The present work involves the application of the SOV method to each region (cathode, separator, and anode) of the lithium-ion cell. This approach can be used as the basis for developing analytical solutions for battery models of greater complexity. This is illustrated here for a case in which nonlinear diffusion is considered, but will be extended to full-order nonlinear pseudo-2D models in later work. The analytical expressions are derived in terms of the relevant system parameters. The system considered for this study has LiCoO 2 -LiC 6 battery chemistry. List of symbolsa Specific interfacial area (m 2 /m 3 ) B, Brugg Bruggeman coefficient c 0 Concentration at initial time t = 0 c i (x, t) Concentration in region i (mol/m 3 ) C i (X, s) Dimensionless concentration in region i D Diffusion coefficient of lithium ions in the electrolyte (cm 2 /s) D eff,i Effective diffusion coefficient of the Li-ion in region i (cm 2 /s) F Faraday's constant (C/mol) i app Applied current density (A/m 2 ) j i Flux density of the Li-ions into the electrode in region i (mol/m 2 s) J i Dimensionless flux density in region i l i Thickness of region i (m) K Ratio of dimensionless flux densities in the electrodes L Total thickness of cell (m) p Dimensionless position of positive electrode/ separator interface q Dimensionless position of separator/negative electrode interface t Time (s) t þ Transference number x Position (m) X Dimensionless position

Mechanical stresses at the cathode–electrolyte interface in lithium-ion batteries

“…Since a half-cell system is simulated and only electrolyte and cathode are included, any increases in lithium concentration in electrolyte and electrode are due to the decrease in lithium content in anode. In addition, our computational simulations have provided a visualized lithium-ion concentration distribution in the half-cell system, and these results are comparable to an analytical study by Guduru et al 48 In the current work, our study limitation lies on the assumption of electrons can reach all the positions where Li 1 ion intercalation takes place, and therefore no polarization of the electrode occurs. 49,50 Thus, we have provided an ideal model that assuming no other resistance occurs (e.g., state of lithiation, current, rate of reaction etc.)…”

Impact of separator's solid‐phase ion conductivity parameter on convection battery performance and modeling

“…Table 4 summarizes dimensionless values of the model parameters derived in a manner similar to that used by Guduru. 16 Few of the parameters have values that are constant throughout the cell. Each parameter that includes a K eff or a D eff value varies from anode to separator to cathode.…”

“…Additionally, this analysis gives way to the ratio of convective to diffusive charge transfer, a dimensionless parameter labeled E i . Table summarizes dimensionless values of the model parameters derived in a manner similar to that used by Guduru …”

A New Method to Solve the Electrolyte Diffusion Equations of Single Particle Model for Lithium-ion Batteries

“…模中 [10,11] 。其中，由于计算量较小，单粒子模型(Single Particle Model, SPM)被 广泛应用 [12,13] 。然而，SPM 由于忽略液相扩散过程而导致其在高放电倍率下误差 较大。为此，许多改进单粒子模型(Enhanced Single Particle Model, ESPM)考虑 了液相扩散扩散过程以提高模型精度 [14,15] 。 ESPM 实时应用的难点在于固相扩散方程和液相物质传输方程的快速而高 精度的计算。 其中， 基于球形粒子的电极固相扩散方程的实时求解问题已经解决， 至少可采用多项式近似、 Padé近似和基于解析解的离散卷积等三种方法高效而可 靠求解 [16] 。 针对 ESPM 的液相扩散方程的求解问题， 主要的求解方法有解析法 [17] 、 微元法、传递函数近似法 [18,19] 和多项式近似法 [14,15,20] 。尽管解析法能够得到准确 的结果，但是该方法的动态工况表达式十分复杂 [17] 。微元法主要有有限元方法、 有限体积法和有限差分法，并已被广泛应用于商业软件中，然而它们的参数众多 且计算量大。 为了降低计算负担， 整数阶传递函数近似和多项式近似被广泛使用， 然而它们在一般工况下的精度较低，并且传递函数和多项式形式的确定方法缺少 一般性 [21]…”