Solution of boundary value problems for batteries: Operator-theoretic methods

Batteries with porous electrodes of negligible ionic and electronic conduction resistance are modeled with reaction-diffusion equations in multilayered media. The classical separation of variables becomes inapplicable to battery problems because of nonlinearities in reaction rates and constraints of imposed current. A linear operator-theoretic approach to the diffusive part converts the battery equations into an integral equation and can be efficiently solved by successive approximations. The current density condition is transformed into a restriction and applied to a battery with two porous electrodes and separator. The use of the standard inner product for solution assuming diffusion to be slow in only one electrode introduces nonselfadjointness which is cured by a modification [1]. Example of the lithium battery demonstrates the power of the method to incorporate nonlinear kinetics. This approach is a generic methodology that, combined with computation, will solve a complex variety of problems in battery dynamics in diffusion-reaction controlled regimes.