Modeling the Impact of Electrolyte Flow on Heat Management in a Li-Ion Convection Cell

To illustrate how the thermal effects impacts convection cell performance, we begin by repeating a base case analysis described in our prior work, ³⁶ where the galvanostatic discharge of a single LIB cell with a stagnant electrolyte (v = 0 μm s⁻¹) was simulated at 5.7C (150 A m⁻² for a 26 Ah m⁻² cell) under isothermal conditions. As shown in Fig. 2, we perform the same simulation while relaxing the isothermal assumption and allowing the cell temperature to vary freely. A small cell convective heat transfer coefficient (h_cell = 0.5 W/(m²K)) is used to represent any background heat exchange between the cell and its surroundings. The discharge curve, Fig. 2a, shows that the accessible capacity without flow is just 54% of theoretical capacity. A large electrolyte concentration gradient develops across the cell during discharge due to the competing effects of diffusion and electromigration (Fig. 2b). However, with the inclusion of thermal processes, complete electrolyte salt depletion in the positive electrode is not realized by the end of the discharge, indicating that electrolyte mass transport limitation is not the sole cause of the abbreviated runtime. Rather, in the absence of electrolyte flow, the cell temperature increases rapidly upon discharge and reaches the cutoff (safety) temperature prior to electrolyte depletion (Fig. 2c). The rate of temperature rise can be slowed by convective flow, and for the single cell considered here, a superficial electrolyte velocity of 0.7 μm s⁻¹ prevents the cell from hitting the cutoff temperature during discharge. For perspective, this flowrate for the single cell corresponds to a residence time of ∼290 s. Greater velocities further suppress cell temperature rise, and gradually reduce the electrolyte concentration gradient as well (Fig. 2b). Note that while the model simulates the cell temperature as a function of axial position, the temperature variation is negligible throughout the cell due to its thinness (Fig. S2). Thus, the temperature shown in Fig. 2c represents both the average cell temperature and the temperature at all positions and in all phases within the cell. Overall, Fig. 2 shows that the introduction of electrolyte flow simultaneously enhances mass and heat transport, both driving electrolyte concentration uniformity and mitigating cell temperature rise. It is noteworthy that, for the single cell analyzed in this study, overcoming the mass and thermal limitations requires electrolyte convection at similar scales, on the order of 1–10 μm s⁻¹, for a near-uniform electrolyte concentration profile (Fig. 2b) and a temperature trajectory close to the initial cell temperature (Fig. 2c). In a battery format of greater thickness, such as multiple layers of the same single cell, the flow rate required to eliminate mass transport limitation would remain in the same order of magnitude as that for the single cell. However, a larger flow rate would be necessary to overcome thermal limitations. Nevertheless, for the scope of the current study, the corresponding pressure drop and resultant pumping losses across a single cell follow the estimates described in our previous study ³⁶ and remain negligible compared to the energy gain on a per cell basis (Supplementary Note 1).

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Having demonstrated the case of benefit to runtime with both heat and mass transport effects, we present a generalized analysis of cell and component dimensions, component properties, and operational parameters where convection may prove advantageous. Compact and meaningful representation of cell performance as a function of individual cell properties and operating conditions can be challenging, as it often requires comprehensive parametric sweeps. Alternatively, combining the relevant physical parameters into dimensionless groups can provide insight on how the relative scales of different physical processes influence performance. For the single convection cell considered in this study, the temperature profile across the cell remains uniform. Hence, we limit our focus to dimensionless groups that describe the average cell temperature rise. To derive the dimensionless groups, we examine the overall cell heat balance equation: ^a

$$\begin{eqnarray}&&\begin{array}{c}\bar{\rho }{\bar{C}}_{p}{A}_{\mathrm{cell}}{L}_{\mathrm{cell}}\frac{d{T}_{\mathrm{cell}}}{dt}={\bar{Q}}_{\mathrm{total}}{A}_{\mathrm{cell}}{L}_{\mathrm{cell}}\\ \,\begin{array}{ccc}\begin{array}{ccc}\begin{array}{ccc}\begin{array}{c}\end{array} & & \end{array} & & \end{array}\begin{array}{ccc} & & \end{array} & & \end{array}-\,{A}_{\mathrm{cell}}{\rho }_{e}{C}_{p,e}v\left({T}_{\mathrm{cell}}-{T}_{\mathrm{tank}}\right)\\ \,\begin{array}{ccc}\begin{array}{ccc}\begin{array}{ccc}\begin{array}{c}\end{array} & & \begin{array}{ccc} & & \end{array}\end{array} & & \end{array} & & \end{array}-\,2{A}_{{\rm{c}}\mathrm{.c}.}{h}_{\mathrm{cell}}\left({T}_{\mathrm{cell}}-{T}_{ambient}\right)\end{array}\end{eqnarray} \tag{ 8 }$$

Next, we identify appropriate scales for the terms in Eq. 8. Applied current density and areal capacity provide a timescale of discharge, ${t}_{\mathrm{dis}}=\tfrac{{Q}_{A}}{{I}_{\mathrm{app}}}.$ Then, by approximating the cell temperature change from its initial temperature, T_init, at t = 0, to a safety cutoff temperature, T_max, at t = t_dis, and the tank temperature with its initial temperature, ${T}_{\mathrm{tank},\mathrm{init}}$, several useful quantities emerge: (1) the magnitude of ambient heat exchange rate of the cell, $2{A}_{{\rm{c}}{\rm{.c}}.}{h}_{{\rm{cell}}}\left({T}_{{\rm{\max }}}-{T}_{ambient}\right)$– in this case, the cell exchanges heat with the surrounding environment through the current collectors at either end of the device only; (2) the magnitude of convective heat exchange rate with the tank, ${A}_{{\rm{cell}}}{\rho }_{e}{C}_{p,e}v({T}_{{\rm{\max }}}-{T}_{{\rm{tank}},{\rm{init}}});$ (3) average heat generation rate, ${\bar{Q}}_{total}{A}_{{\rm{cell}}}{L}_{{\rm{cell}}};$ and (4) Average heat storage rate, $\tfrac{\bar{\rho }{\bar{C}}_{p}{A}_{{\rm{cell}}}{L}_{{\rm{cell}}}\left({T}_{{\rm{\max }}}-{T}_{{\rm{init}}}\right)}{{t}_{{\rm{dis}}}}.$ The average volumetric heat generation rate, ${\bar{Q}}_{total},$ can be estimated using the cell parameters and operating conditions, as detailed in Supplementary Note 4; the multiplicative product of the average density, $\bar{\rho },$ and the average heat capacity, ${\bar{C}}_{p},$ can be calculated by averaging the ρC_p products of each cell component weighted by their respective thicknesses. Comparing these quantities yields the list of dimensionless groups that can be used to evaluate the average cell temperature gain (shown in Table I), which are subsequently denoted with the subscript H. To prevent any temperature rise, the heat removal rate from the cell must balance the heat generation rate within the cell. That is, the ratio of heat generation to heat removal must be small. In the case of an enclosed device (e.g., a conventional LIB cell) where heat removal relies on ambient heat exchange alone, this ratio is captured by the dimensionless parameter, γ_H . A large γ_H value indicates that the heat removal rate is insufficient, which, in turn, leads to temperature increases during charge or discharge. In the case of a convection cell, the electrolyte flow provides an additional mode of heat removal, and ξ_H represents the ratio of the heat generation to the sum of heat removal via ambient cooling and electrolyte convection. In the absence of convection, γ_H = ξ_H . Similar to γ_H , a large ξ_H value indicates insufficient combined heat removal as compared to heat generation. Finally, the inherent buffering ability of the cell against heat generation is captured by β_H , where a large value indicates the cell is more prone to temperature increase for a given amount of heat generation. The dimensionless groups for mass transport, derived in our prior work, ³⁶ are also included in Table I for reference and comparison, and are subsequently denoted with the subscript M. These groups are analogous to those used to describe heat transport. Specifically, γ_M compares the electromigrative and diffusive fluxes, where a large value of this parameter indicates an increased likelihood for electrolyte salt depletion in the positive electrode during discharge due to insufficient diffusive transport. ξ_M compares the electromigrative flux to the sum of diffusive and convective fluxes, where large values of this variable indicates that the combination of diffusive and convective fluxes are slower than the electromigrative flux that removes anions from the cathode. β_M measures the buffering ability of the cell against electrolyte salt depletion, and a large value of this quantity indicates that the initial amount of salt in the electrolyte is insufficient compared to the depletion by the electromigration of the anions.

Table I. Dimensionless groups quantifying the extent of bulk mass and thermal transport limitations.

Dimensionless group	Meaning	Mass transport [M]	Heat transfer [H] (${\bar{T}}_{cell}\,{\rm{vs}}\,t$)
γ	Inherent transport ability	$\tfrac{{I}_{{\rm{app}}}(1-{t}_{+})L}{F{D}_{{\rm{eff}}}{c}_{{\rm{initial}}}}$	$\tfrac{{\bar{Q}}_{total}{L}_{{\rm{cell}}}}{2\tfrac{{A}_{{\rm{c}}{\rm{.c}}.}}{{A}_{{\rm{cell}}}}{h}_{{\rm{cell}}}\left({T}_{{\rm{\max }}}-{T}_{ambient}\right)}$
ξ	Transport ability with convection	$\tfrac{{I}_{{\rm{app}}}(1-{t}_{+})}{\tfrac{F{D}_{{\rm{eff}}}{c}_{{\rm{initial}}}}{L}+Fv{c}_{{\rm{initial}}}}$	$\tfrac{{\bar{Q}}_{total}{L}_{{\rm{cell}}}}{2\tfrac{{A}_{{\rm{c}}{\rm{.c}}.}}{{A}_{{\rm{cell}}}}{h}_{{\rm{cell}}}\left({T}_{{\rm{\max }}}-{T}_{ambient}\right)+{\rho }_{e}{C}_{p,e}v\left({T}_{{\rm{\max }}}-{T}_{{\rm{tank}},{\rm{init}}}\right)}$
β	Buffering ability	$\tfrac{{I}_{{\rm{app}}}{t}_{{\rm{dis}}}(1-{t}_{+})}{F{c}_{{\rm{initial}}}\varepsilon L}$	$\tfrac{{\bar{Q}}_{total}{t}_{{\rm{dis}}}}{\bar{\rho }{\bar{C}}_{p}\left({T}_{{\rm{\max }}}-{T}_{{\rm{init}}}\right)}$

The corresponding dimensionless group values of the Base Case are shown in Table II, column (a). The large values for all the four dimensionless groups indicate that, under these conditions and in the absence of convection, the inherent mass and thermal transport properties are insufficient to support the high discharge rate of 5.7C in a cell of these dimensions. This deficiency explains the large electrolyte concentration gradient and rapid temperature increase shown in Figs. 2b and 2c, respectively.

While the Base Case demonstrates that the introduction of electrolyte convection enables simultaneous enhancement of mass and heat transport, further investigation is needed to decouple and elucidate the impact of convection on each process. This can be achieved by contemplating two extensions of the Base Case: (1) Diffusive Transport Limited Case, where without convection, the cell has adequate heat transfer capabilities but the same poor mass transport properties and conditions used in the Base Case, and (2) Thermal Transport Limited Case, where the mass transport in the cell without convection is sufficiently facile, but the heat transfer rates are restricting. The following sections discuss the findings from these two alternative cases.

In the first scenario, to focus on a diffusion limited case, we consider a cell with a convective heat transfer coefficient (h_cell = 500 W/(m²K)) large enough to facilitate thermal transport to the point that the increase in cell temperature during discharge is negligible (Fig. S4), while all other conditions remain identical to the Base Case. The corresponding dimensionless group values are shown in Table II, column (b). The heat transfer coefficient value chosen here is representative of an indirect liquid cooling system. ⁴⁵ This case corresponds to a cell with external heat removal capability sufficient to maintain near-isothermal operation (e.g., a highly effective thermal regulation system), which approaches the isothermal base case described in our prior work. ³⁶ In the absence of an elevated average cell temperature, which increases diffusion rates, electrolyte mass transport presents significant limitations, but this can be alleviated by increasing electrolyte convection (or decreasing ξ_M value) as shown in Fig. 3a. Enhanced electrolyte mass transport rates also positively impact the thermal regulation of the cell. As shown in Fig. 3b, reduced electrolyte concentration gradients facilitate more uniform reversible heat generation, suggesting improved reaction homogeneity particularly within the Li-ion-consuming electrode (here, the positive electrode). Importantly, this observation implies that electrolyte convection may help mitigate reaction maldistribution that limits power output and capacity utilization of thick electrodes in energy-dense cells. ⁴⁶ Furthermore, electrolyte convection can reduce irreversible reaction heat generation in the positive electrode and overall Joule heating as shown in Figs. 3c and 3d, respectively. As convection enables higher electrolyte concentration in the positive electrode facilitating the charge transfer reaction, the activation overpotential required to sustain a desired current output is reduced thus lowering irreversible reaction heat generation. Moreover, a uniform ∼1 M electrolyte concentration across the cell decreases Joule heating by lowering ohmic losses (high ionic conductivity). Collectively, these results demonstrate that electrolyte flow can enable greater spatial uniformity in heat generation as well as reduce overall heat generation within a cell. In this case, the average volumetric heat generation rate (${\bar{Q}}_{total}$) is halved at 10 μm s⁻¹ as compared to stagnant operation. As previously mentioned, the cell temperature gain is determined by both the rate of heat generation and the rate of heat removal. It is this ability to decrease heat generation via electrolyte flow that differentiates the convection approach from other external and internal thermal regulation methods that are solely designed to enhance heat removal rates. The reduction in heat generation through convection appears unique amongst reported BTMS and relaxes the requirement for heat removal capability in the first place.

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In the second scenario, we consider a cell without mass transfer limitations by invoking a very large electrolyte diffusivity (D = normal value × 10³ ) with all other conditions the same as the Base Case. While, to the best of our knowledge, such diffusive rates are infeasible in practical electrolytes, the simulated conditions are instructive as they enable isolation of the effects of electrolyte convection on heat transfer. The corresponding dimensionless group values are shown in Table II, column (c). The γ_H and β_H values are both slightly lower than Base Case values as the augmented diffusivity leads to more uniform electrolyte concentration across the cell, reducing the activation overpotential and thus diminishing the heat generation rate, ${\bar{Q}}_{total}.$ This result is analogous to that obtained when simulating increased electrolyte flow rate in the previous case. This effect is clearly observable in Fig. 4, where even without electrolyte flow, the cell can discharge for a longer time as compared to its counterpart in Fig. 2c. With the introduction of electrolyte flow, the temperature rise is further suppressed, and when ξ_H $\lesssim \,$1, the cell can be fully discharged, accessing almost all of the available charge storage capacity, without hitting the upper threshold. Thus, Fig. 4 evinces the second effect of electrolyte flow on thermal management, namely that bulk flow can remove heat from the cell to prevent temperature increase (i.e., boosting the heat removal capability of the cell). Additionally, consistent with physical intuition, a transition value of ξ_H $\lesssim$ 1 means that the heat generation rate $\lesssim$ the heat removal rate. As a heuristic, the amount of flow required to prevent the cell from hitting the cutoff temperature can be estimated from ξ_H = 1, and the remaining parameters defining this dimensionless group in Table I. Note that although a single cell (∼200 μm) is demonstrated in this study, ξ_H is also applicable to a larger battery format, which suggests that the flowrate needed to suppress average temperature rise roughly scales with the system dimension in the direction of flow. Also note that while convection can effectively prevent the cell temperature gain, it may be desirable for the cell to operate at a set temperature above or below the ambient conditions. ^47,48 This may be to promote favorable reaction kinetics and/or mass transport characteristics, to heat a device in cold climes, or to cool a device subjected to excessive heat due to local weather conditions or proximity to an external heat source. A non-ambient target temperature can be achieved by regulating the flow rate and/or introducing electrolyte pre-conditioned at a different inlet temperature.

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The analyses above illustrate the unique advantages that a flowing electrolyte offers through simultaneous improvement of mass and heat transport with one of the two limitations significantly relaxed. Now we directly compare the impact of convection (hereafter referred to as "internal cooling") and the typical external thermal management methods (hereafter referred to as "external cooling") on temperature regulation and cell performance, with no upfront relaxation of any transport limitation. We use the same cell configuration and discharge conditions as in the Base Case. For external cooling, the cell exchanges heat with the surrounding environment through the axial ends (i.e., the current collectors), with a total representative heat exchange rate of $2{A}_{{\rm{cell}}}{h}_{{\rm{cell}}}\left({T}_{{\rm{\max }}}-{T}_{ambient}\right).$ For internal cooling, electrolyte flows through the cell in the direction shown in Fig. 1b (from the negative electrode to the positive electrode) and exchanges heat with the tank at a representative heat exchange rate, ${A}_{{\rm{cell}}}{\rho }_{e}{C}_{p,e}v({T}_{{\rm{\max }}}-{T}_{{\rm{tank}}}).$ Equivalent heat removal rates are anticipated for the external and internal modes when the two heat exchange rates are equal, as is shown in Supplementary Note 5. However, as Fig. 5 demonstrates, an "equivalent" set of v and h_cell values yields the same heat removal capability yet different impact on cell temperature regulation and thus performance. With external cooling, the first increments in h_cell value help extend the cell runtime (Fig. 5a) by inhibiting temperature rise (Fig. 5c) and prolonging or avoiding the 325 K cutoff. However, without internal convection, mass transfer relies upon temperature-dependent diffusion alone, and further increases in h_cell values have an adverse effect on mass transfer. A lower average cell temperature results in slower electrolyte mass transport (Fig. 5b) leading to lower cell operating voltage (reduced power density), and even shortens the cell runtime as electrolyte salt depletion increases at higher h_cell values. Hence, for external cooling, increasing heat removal capability (i.e., increasing h_cell) improves thermal transport but compromises mass transport; if the cell has a large electrolyte mass transport resistance (i.e., large γ_M and β_M values), such as the case shown in Fig. 5, increasing h_cell values will ultimately cause the cell to transition from thermal-transport-limited behavior to diffusive-transport-limited behavior. This compromise is in contrast with the internal cooling, where increasing heat removal capability (i.e., increasing v) results in simultaneous heat and mass transport enhancement, as shown in Figs. 5d–5f, ensuring that neither bulk transport mode hinders cell performance at sufficient v.

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The contrasting impact of the two cooling strategies on electrolyte mass transport has further implications on their efficacy. As discussed in the Diffusive Transport Limited Case, improved electrolyte mass transport, such as with internal cooling, leads to diminished heat generation rates due to reduced overpotential losses. Conversely, heat removal with external cooling decreases electrolyte mass transport and therefore increases heat generation as shown in Fig. S6. Consequently, internal cooling suppresses cell temperature rise more effectively than external cooling even with the same heat removal capability, as observed by comparing Figs. 5c with 5f. This implies that compared to external cooling, the heat removal capability required of internal cooling is lowered due to the reduced amount of heat generation. This finding further supports the argument that through enhanced electrolyte mass transport, the internal cooling method offers unique advantages over external cooling methods by both removing heat (heat transfer) and reducing heat generation (mass transfer), resulting in greater effectiveness.

The comparison between internal and external cooling discussed here is not limited to a single cell of ∼200 μm thickness but is also applicable to larger systems, such as multiple cell layers. However, in these cases, the effect of temperature gradient becomes significant and should be considered for the design with both cooling strategies. Regarding the internal cooling strategy, the temperature gradient is mainly present in the direction of flow and is closely related to the system design/configuration and resultant flow path, which are beyond the scope of this initial work. Nevertheless, the temperature gradient can be significantly reduced or become negligible at high flow rates. As such, the findings presented here represent the full potential of electrolyte flow (i.e., the performance of an optimized system) and provide a basis for future work on system development.

The ability to modulate electrolyte flow rate during convection cell operation offers new opportunities to dynamically regulate cell performance. While many different and specific scenarios could be considered, we contemplate two general approaches in Fig. 6. In the first scenario, as shown in Fig. 6a, we illustrate a reactive approach where electrolyte convection is triggered by a cell safety temperature (in this example 325 K), and a sufficient flow rate (that satisfies ξ_H $\lesssim \,$1) suppresses the temperature rise almost instantaneously. In practice, this scenario may correspond to urgent rescue situations, such as thermal runaway protection. Interestingly, the higher flow rates (0.7–10 μm s⁻¹) have convex temperature profiles, where the temperature drops immediately upon the introduction of flow, but given sufficient time, gradually increases again. This is because as the cell temperature reduces, the heat removal rate decreases to the point where it is less than the heat generation rate and the cell temperature begins to increase again. In a second scenario, we show a proactive approach where the flow rate is changed based on operating conditions. As shown in Fig. 6b, electrolyte convection is simultaneously introduced when there is a change in C-rate. This corresponds to a situation where a rapid change in battery power is required during operation, such as, a sudden acceleration in a BEV. The ξ_H values calculated for this scenario are based on 5.7 C and a cutoff temperature of 325 K, so a flow rate that suffices ξ_H $\lesssim \,$1 suggests that the cell will not reach the cutoff temperature of 325 K. If minimal temperature rise is desired, a smaller cutoff temperature (e.g., 300 K) could be used to calculate a corresponding flow rate that yields ξ_H = 1.

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Importantly, variable "on-demand" convection instead of continuous flow allows the system to forgo convection when not needed, such as for low current charge or discharge, for intentional self-heating, or for pumping energy savings. Reactive mode (Fig. 6a) can tune the use of flow to critical situations only, but precise effectiveness of this mode may rely on accurate temperature measurements of the cells within a battery pack. The proactive use of flow coordinated with a significant increase in heat generation, for example, in Fig. 6b, can prevent temperature escalation in advance or when difficult to reliably measure. However, the proactive scenario may call for flow prematurely or when unnecessary, such as, if high heat generation operation is aborted.

$$\begin{eqnarray*}\begin{array}{c}{Q}_{{\rm{ohm}}}=\,\left\{\begin{array}{lll}{\sigma }_{{\rm{eff}},i}{\left(\displaystyle \frac{\partial {{\rm{\Phi }}}_{s}(x,t)}{\partial x}\right)}^{2}+{\kappa }_{{\rm{eff}},i}{\left(\displaystyle \frac{\partial {{\rm{\Phi }}}_{e}(x,t)}{\partial x}\right)}^{2}-\displaystyle \frac{2{\kappa }_{{\rm{eff}},i}RT(x,t)}{F}(1-{t}_{+})\displaystyle \frac{\partial \,\mathrm{ln}\,{c}_{e}(x,t)}{\partial x}\displaystyle \frac{\partial {{\rm{\Phi }}}_{e}(x,t)}{\partial x} & {\rm{if}} & i\in \{p,n\}\\ {\kappa }_{{\rm{eff}},i}{\left(\displaystyle \frac{\partial {{\rm{\Phi }}}_{e}(x,t)}{\partial x}\right)}^{2}-\displaystyle \frac{2{\kappa }_{{\rm{eff}},i}RT(x,t)}{F}(1-{t}_{+})\displaystyle \frac{\partial \,\mathrm{ln}\,{c}_{e}(x,t)}{\partial x}\displaystyle \frac{\partial {{\rm{\Phi }}}_{e}(x,t)}{\partial x} & {\rm{if}} & i\in \{s\}\end{array}\right.\end{array}\end{eqnarray*}$$

where ${\sigma }_{{\rm{eff}},i}={\sigma }_{i}\left(1-{\varepsilon }_{i}-{\varepsilon }_{{\rm{filler}},{\rm{i}}}\right)$

$$\begin{eqnarray*}&&{Q}_{{\rm{rxn}}}=F{a}_{i}j\left(x,t\right){\eta }_{i}\left(x,t\right)\end{eqnarray*}$$

where ${\eta }_{i}(x,t)={{\rm{\Phi }}}_{s}(x,t)-{{\rm{\Phi }}}_{e}(x,t)-{U}_{i},$ $i\in \{p,n\}$

$$\begin{eqnarray*}&&{Q}_{{\rm{rev}}}=F{a}_{i}j\left(x,t\right)T\left(x,t\right){\left.\displaystyle \frac{\partial {U}_{i}}{\partial T}\right|}_{{T}_{ref}},i\in \left\{p,n\right\}\end{eqnarray*}$$

A·2·1. Open circuit potential

$$\begin{eqnarray*}&&{U}_{p}={U}_{p,ref}+\left(T\left(x,t\right)-{T}_{ref}\right){\left.\displaystyle \frac{\partial {U}_{p}}{\partial T}\right|}_{{T}_{ref}}\end{eqnarray*}$$

$$\begin{eqnarray*}&&{U}_{n}={U}_{n,ref}+\left(T\left(x,t\right)-{T}_{ref}\right){\left.\displaystyle \frac{\partial {U}_{n}}{\partial T}\right|}_{{T}_{ref}}\end{eqnarray*}$$

$$\begin{eqnarray*}&&\begin{array}{c}{\left.\displaystyle \frac{\partial {U}_{p}}{\partial T}\right|}_{{T}_{ref}}=-\,0.001\times \,\left(\displaystyle \frac{0.199521039-0.928373822{\theta }_{p}+1.364550689000003{{\theta }_{p}}^{2}-0.6115448939999998{{\theta }_{p}}^{3}}{1-5.661479886999997{\theta }_{p}+11.47636191{{\theta }_{p}}^{2}-9.82431213599998{{\theta }_{p}}^{3}+3.048755063{{\theta }_{p}}^{4}}\right)\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}&&\begin{array}{c}{\left.\displaystyle \frac{\partial {U}_{n}}{\partial T}\right|}_{{T}_{ref}}=\,0.001\times \,\left(\displaystyle \frac{\begin{array}{c}0.005269056+3.299265709{\theta }_{n}-91.79325798{{\theta }_{n}}^{2}+1004.911008{{\theta }_{n}}^{3}-5812.278127{{\theta }_{n}}^{4}\,+\\ 19329.7549{{\theta }_{n}}^{5}-37147.8947{{\theta }_{n}}^{6}+38379.18127{{\theta }_{n}}^{7}-16515.05308{{\theta }_{n}}^{8}\end{array}}{\begin{array}{c}1-48.09287227{\theta }_{n}+1017.234804{{\theta }_{n}}^{2}-10481.80419{{\theta }_{n}}^{3}+59431.3{{\theta }_{n}}^{4}\,-\\ 195881.6488{{\theta }_{n}}^{5}+374577.3152{{\theta }_{n}}^{6}-385821.1607{{\theta }_{n}}^{7}+165705.8597{{\theta }_{n}}^{8}\end{array}}\right)\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}&&\begin{array}{c}{U}_{p,ref}=\,\displaystyle \frac{-4.656+88.669{{\theta }_{p}}^{2}-401.119{{\theta }_{p}}^{4}+342.909{{\theta }_{p}}^{6}-462.471{{\theta }_{p}}^{8}+433.434{{\theta }_{p}}^{10}}{-1+18.933{{\theta }_{p}}^{2}-79.532{{\theta }_{p}}^{4}+37.311{{\theta }_{p}}^{6}-73.083{{\theta }_{p}}^{8}+95.96{{\theta }_{p}}^{10}}\end{array}\end{eqnarray*}$$

Where ${\theta }_{p}=\tfrac{{c}_{s,p}^{* }\left(x,t\right)}{{c}_{s,p}^{{\rm{\max }}}}$

$$\begin{eqnarray*}&&\begin{array}{c}{U}_{n,ref}=0.7222+0.1387{\theta }_{n}+0.029{{\theta }_{n}}^{0.5}-\,\displaystyle \frac{0.0172}{{\theta }_{n}}+\displaystyle \frac{0.0019}{{{\theta }_{n}}^{1.5}}+0.2808{e}^{0.9-15{\theta }_{n}}-\,0.7984{e}^{0.4465{\theta }_{n}-0.4108}\end{array}\end{eqnarray*}$$

Where ${\theta }_{n}=\tfrac{{c}_{s,n}^{* }\left(x,t\right)}{{c}_{s,n}^{{\rm{\max }}}}$

A·2·2. Effective electrolyte diffusion coefficient

$$\begin{eqnarray*}&&{D}_{{\rm{eff}},i}={\varepsilon }_{i}^{brugg,i}\times {10}^{-4}\times {10}^{-4.43-\displaystyle \frac{54}{T(x,t)-229-5\times {10}^{-3}{c}_{e}(x,t)}-0.22\,\times \,{10}^{-3}{c}_{e}\left(x,t\right)}\end{eqnarray*}$$

A·2·3. Effective electrolyte conductivity

$$\begin{eqnarray*}&&\begin{array}{c}{\kappa }_{{\rm{eff}},i}={\varepsilon }_{i}^{brugg,i}\times {10}^{-4}\times {c}_{e}(x,t){\left(\begin{array}{l}\begin{array}{c}-\,10.5+0.668\times {10}^{-3}\cdot {c}_{e}(x,t)\\ +\,0.494\times {10}^{-6}\cdot {{c}_{e}}^{2}\left(x,t\right)\,+\end{array}\\ \begin{array}{c}\left(0.074-1.78\times {10}^{-5}\cdot {c}_{e}(x,t)\right.\\ \left.-\,8.86\times {10}^{-10}\cdot {{c}_{e}}^{2}(x,t)\right)\cdot T(x,t)\,+\end{array}\\ \begin{array}{c}\left(-6.96\times {10}^{-5}+2.8\times {10}^{-8}\cdot {c}_{e}(x,t)\right)\\ \cdot {T}^{2}(x,t)\end{array}\end{array}\right)}^{2}\end{array}\end{eqnarray*}$$

A·2·4. Effective reaction rate

$$\begin{eqnarray*}&&{{\bf{k}}}_{{\text{eff}}{\boldsymbol{,}}i}={k}_{i}{e}^{-\tfrac{{E}_{a}^{{k}_{i}}}{R}\left(\tfrac{1}{T(x,t)}-\tfrac{1}{{T}_{ref}}\right)}\end{eqnarray*}$$

A·2·5. Effective solid-phase diffusion coefficient

$$\begin{eqnarray*}&&{D}_{{\rm{eff}},i}^{s}={D}_{i}^{s}{e}^{-\tfrac{{E}_{a}^{{D}_{i}^{s}}}{R}\left(\tfrac{1}{T(x,t)}-\tfrac{1}{{T}_{ref}}\right)}\end{eqnarray*}$$

Table A·I. Simulation parameters used in this study.

	Units	Positive CC	Positive electrode	Separator	Negative electrode	Negative CC
	—	Al	Liθ CoO2	—	Liθ C6	Cu
a	m2/m3	—	862500	—	851100	—
brugg	—	—	2.5	2.5	2.5	—
Cp	J/kg/K	903	1269	1978	1437	385
Cp,e	2055 J kg−1 K−1
cinitial	mol m−3	—	1000	1000	1000	—
cs max	mol m−3	—	51554	—	30555	—
${D}_{i}^{s}$	m2 s−1	—	1 × 10–14	—	3.9 × 10–14	—
${E}_{a}^{{D}_{i}^{s}}$	J mol−1	—	5000	—	5000	—
${E}_{a}^{{k}_{i}}$	J mol−1	—	5000	—	5000	—
F	96487 C mol−1	—	—	—	—	—
Iapp	150 A m−2
ki	m2.5/(mol0.5s)	—	2.334 × 10–11	—	5.031 × 10–11	—
L	m	1 × 10–5	8 × 10–5	4 × 10–5	8 × 10–5	1 × 10–5
n	—	12	100	50	100	12
R	8.314 J mol−1 K−1	—	—	—	—	—
Rp	m	—	2 × 10–6	—	2 × 10–6	—
t+	0.37
ε	—	—	0.4	0.4	0.4	—
εfiller	—	—	0.025	—	0.0326	—
Θ100%	—	—	0.4955	—	0.8551	—
Θ0%	—	—	0.9917	—	0.0066	—
λ	W/m/K	238	1.58	0.33	1.04	398
ρ	kg m−3	2702	2329	1009	1347	8933
ρe	1130 kg m−3
σ	S m−1	3.55 × 107	100	—	100	5.96 × 107