Heat generation and a conservation law for chemical energy in Li-ion batteries

Abstract

Present theories of irreversible energy losses and heat generation within Li-ion cells are unsatisfactory because they are not compatible with energy conservation and typically give rise to significant errors in the estimation of these quantities. This work aims to provide a consistent theoretical treatment of energy transport and losses in such devices. An energy conservation law is derived from the Doyle–Fuller–Newman (DFN) model of a Li-ion cell using a rigorous mathematical approach. The resulting law allows irreversible chemical energy losses to be located to seven different regions of the cell, namely: (i) the electrolyte, (ii) the anode particles, (iii) the cathode particles, (iv) the solid parts of the anode (ohmic losses), (v) the solid parts of the cathode (ohmic losses), (vi) the surfaces of the anode particles (polarisation losses), and (vii) the surfaces of the cathode particles (polarisation losses). Numerical solutions to the DFN model are used to validate the conservation law in the cases of a drive cycle and constant current discharges, and to compare the energy losses occurring in different locations. It is indicated how cell design can be improved, for a specified set of operating conditions, by comparing the magnitude of energy losses in the different regions of the cell.

Introduction

The drive to eliminate carbon based fuels from transportation systems, and the resulting legislation to phase out the internal combustion engine across large parts of the world before 2040, has led to rapid growth interest in Li-ion battery technology. Currently this technology is used in most portable consumer electronics, and is increasingly being used in home energy storage units, but it is its use in electric vehicles (EVs) that is set to see the biggest growth in its market, being predicted to increase from 45 GWh/year (in 2015) to around 390 GWh/year in 2030 [37]. The prime reason for its dominance of the automotive industry is its unrivalled high power and energy densities. It also has the advantages of discharging slowly when not in use, little or no need for maintenance and the ability to undergo a large number of charge/discharge cycles without significant degradation.

The Doyle Fuller Newman (DFN) model [1], [5], [12], [13], [23], which is also often called the Newman model or the P2D model, has proved itself to be an extremely useful, and versatile, tool for understanding Li-ion battery performance. Recent works have shown that, providing that lithium ion transport within the electrode particles is modelled by an appropriately calibrated nonlinear diffusion equation, the model is capable of accurately predicting battery performance [7], even when subjected to highly non-uniform drive cycles [38]. While these predictions of the DFN model provide an accurate relation between the cell voltage and the current draw they have not, to date, been used to provide a consistent picture of the irreversible energy losses occurring within the cell. While there are many works that use DFN to estimate irreversible energy loss and heating within Li-ion cells none of them uses a theory of energy dissipation that is consistent with the DFN model. In this context we note the following works that are based on DFN simulation [4], [6], [9], [10], [14], [16], [18], [19], [22], [24], [25], [31], [32], [33], [35] and [2], which is based on a single particle model (a simplification of the DFN model). All of these works predict energy losses without accounting for the enthalpy of mixing (or heat of mixing) in the electrode particles and only partially account for the irreversible energy loss in the electrolyte, again neglecting the enthalpy of mixing. This method of estimating the energy dissipation is based on thermodynamic treatments by Rao and Newman [27] and Gu and Wang [15], which are ultimately based on the work of Bernardi [3]. We note also the works Tranter et al. [34] and Farag et al. [11], which are both based on the DFN model, but use alternative methods for estimating heat production, neither of which are consistent with overall energy conservation within the DFN model, although Farag et al. do approximate the heat of mixing within the electrode particles, noting that it is often significant. Finally, we remark that Latz and Zausch [20], [21] have used a thermodynamic method to estimate energy dissipation in a lithium cell and, as we shall show, obtain expressions for irreversible energy losses within the device that are consistent with the energy conservation law that we derive here from the DFN model. However, as far as we are aware, their work has never been applied to the DFN model. The relative lack of attention that their estimate of energy dissipation has received can be attributed to (I) the large number of other works in the literature that use thermodynamic arguments to arrive at incomplete estimates of the irreversible energy losses and (II) the fact that there is no independent theoretical procedure for determining which of these thermodynamic approaches are correct. The current work aims to address the second of these points and thereby fill a significant void in the literature.

Given the widespread use, and overall utility, of the DFN model of Li-ion battery behaviour it would be a major step forward to unequivocally establish the form of the irreversible energy dissipation law that is consistent with this model. This will not only allow accurate modelling of heat generation in composite cells (such as cylindrical and pouch cells), in which inadequate cooling can lead to significant temperature heterogeneities, but could also be used as a design tool in order to identify the components of the cell in which irreversible losses are most significant under the cell’s characteristic operational conditions. To date, a unifying theory of energy transport and dissipation in the DFN model remains to be established and it is precisely this omission that we aim to address here. In order to accomplish this we restrict our attention to a single cell, which we can consider to have spatial uniform temperature $T=T\left(t\right)$, given its small width. Rather than adopt a thermodynamic approach we directly derive an energy conservation law from the DFN model. This has the advantage that it avoids the pitfalls and intricacies of non-equilibrium thermodynamics, which, in this application, have led to a number of incorrect (or at best approximate) results. In fact we rigorously prove that the conservation law is a consequence of the DFN model, and hence demonstrate unequivocally that the irreversible energy loss terms derived here are the only ones consistent with the DFN model. This result is validated against numerical solutions to the DFN model, both for constant current discharge and drive cycles. These solutions are computed using the fast, second-order accurate, DFN solver DandeLiion [17] and are, in turn, used to evaluate each term in the energy conservation equation.

The paper is set out as follows. In Section 2 we recap the Doyle–Fuller–Newman model. In Section 3 we summarise the energy conservation law in a concise form, which can be easily applied to the DFN model, and then in Section 4 we validate this law against full simulations of the model, both for constant current discharge and for a drive cycle. The derivation of the energy conservation law from the DFN model is presented in Section 5 before finally, in Section 6, we draw our conclusions.