Effective temperature bridges length-scales by facilitating lumped models of heterogeneous battery cells

The kinetics of a battery cell vary with temperature, which causes changes to electrical properties such as power capability and accessible capacity. Knowledge of cell temperature is therefore essential for battery design and operation. However, complications arise since electrochemical systems generate heat, causing thermal gradients to emerge. No single measure of this temperature distribution exists, which hinders battery design and management, and means pack, cell, and coin cell data cannot be directlycompared. Here we address this issue, by proposing a temperature measurement of a heterogeneous system that accounts for the effects of the heterogeneity on overall system kinetics. Previous authors have noted that a cell with a temperature gradient shows the same impedance as an identical lumped cell, at some uniform temperature. We refer to this as the effective temperature of a heterogeneous cell. We derive an explicit distributed cell model, consisting of parallel connected Thevenin circuits, from which a formula for effective temperature is defined. This results in a single temperature measurement for a heterogeneous battery system, which gives the temperature value that best represents the overall electrical dynamics, and correctly accounts for uneven temperature distributions. We show how the parameters of a thermal model can be extracted from electrical data, and note that the calibrated model recovers the effective temperature of the cell, despite having not been fitted to temperature data. This shows that our proposed temperature metric is an intrinsic and fundamentally meaningful measurement. Our results enable accurate lumped models of heterogeneous cells and packs, by capturing heterogeneity with effective temperature. In turn, this acts as a bridge to enable direct comparisons of electrochemical systems, regardless of length-scales and heterogeneity.