A numeric symbolic solution technique is introduced for the simulation of ac impedance response of electrochemical devices. The proposed method is numerical in the spatial coordinates and yields a closed form symbolic solution in the system parameters. The system of algebraic equations obtained by the spatial discretization is written in matrix form and solved symbolically. Although this method is capable of simulating ac impedance data for systems with multiple coupled partial differential equations, the method and its advantages over classical methods are illustrated using diffusion in a planar electrode.

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