A finite element implementation of the transient nonlinear Nernst-Planck-Poisson (NPP) and Nernst-Planck-Poisson-modified Stern (NPPMS) models is presented. The NPPMS model uses multipoint constraints to account for finite ion size, resulting in realistic ion concentrations even at high surface potential. The Poisson-Boltzmann equation is used to provide a limited check of the transient models for low surface potential and dilute bulk solutions. The effects of the surface potential and bulk molarity on the electric potential and ion concentrations as functions of space and time are studied. The ability of the models to predict realistic energy storage capacity is investigated. The predicted energy is much more sensitive to surface potential than to bulk solution molarity.

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A planar electrode containing cylindrical pores with semi-circular ends is modeled using a finite element implementation of the transient nonlinear 2D NernstPlanck-Poisson-modified Stern (NPPMS) model. The model uses a modified Stern layer to account for finite ion size. The study includes the effects of pore radius and depth on the predicted electric potential, ion concentration, surface charge density, surface energy density, and charging time. The ion concentration and electric potential are found to be sensitive to the change in radii of the pore and insensitive to the pore depth. The surface charge density is slightly higher within the pore than along the vertical flat regions of the electrode. The increase in surface area due to porosity increases the charging time.