On the well-posedness of a multiscale mathematical model for Lithium-ion batteries

Abstract

We consider the mathematical treatment of a system of nonlinear partial differential equations based on a model, proposed in 1972 by J. Newman, in which the coupling between the Lithium concentration, the phase potentials and temperature in the electrodes and the electrolyte of a Lithium battery cell is considered. After introducing some functional spaces well-adapted to our framework we obtain some rigorous results showing the well-posedness of the system, first for some short time and then, by considering some hypothesis on the nonlinearities, globally in time. As far as we know, this is the first result in the literature proving existence in time of the full Newman model, which follows previous results by the third author in 2016 regarding a simplified case. Keywords: Lithium-ion battery cell, multiscale mathematical model, Green operators, fixed point theory, Browder-Minty existence results, super and sub solutions 2010 MSC: 35M10, 35Q60, 35C15, 35B50, 35B60• The electric potential φ s = φ s (x, t) in the electrodes.• The electric potential measured by a reference Lithium electrode in the electrolyte, ϕ e = ϕ e (x, t).• The temperature T (t) in the cell.The system of equations is given by (5)-(9) below.

Numerical analysis of a finite element formulation of the P2D model for Lithium-ion cells

“…Let D denote a generic open bounded domain in R; hereafter, the closure of a domain D is denoted D. The functional spaces that we use in this paper are the following. The Sobolev spaces H m (D), m being a nonnegative integer, when m = 0, H 0 (D) := L 2 (D); the Lebesgue spaces L p (D), 1 ≤ p ≤ ∞; the spaces of measurable radial functions [4]…”

“…Remark 1 Following the arguments of [4], where its non-isothermal P2D model includes an additional time dependent non linear ordinary differential equation for the bulk temperature T (t), one can formulate an alternative definition of the weak solution to ( 5)-( 8) based on its Definition 2.7 and prove, under the assumptions A1-A3 and for a partition…”

“…being the initial condition in such an interval. Also, Kröner [10] proves a local existence and uniqueness theorem for the weak solution of the isothermal P2D model under less general assumptions than in [4].…”

“…Based on Remark 1 and since H 1 (D i ) → C(D i ), we introduce the following spaces which are used in the error analysis and in the application of the fixed point theorems in Sect. 4.…”

“…By longitudinal and latitudinal symmetry considerations, the diffusion in the active particles is only in the radial direction. (4) The electrochemical reaction of lithium insertion and extraction processes follows the Buttler-Volmer law. (5) The effective transport coefficients are calculated by the Bruggeman relation, i.e., μ e f f = με p (p=1.5), where μ is a generic transport coefficient and ε is the component volume fraction of the material in the composite electrodes and separator.To formulate the equations of the model we distinguish the following domains.…”