Analysis of the Heat Generation Rate of Lithium-Ion Battery Using an Electrochemical Thermal Model

The heat generation in a battery can be explained with the first law of thermodynamics. Any differential change of internal energy in a closed system is equal to the sum of heat gained or lost and the work done by the system. The first law can be expressed as follows;

$$\begin{eqnarray}&&dU=dQ-dW,\end{eqnarray} \tag{ 2 }$$

where dU is a change of the internal energy, dQ is a change of the heat that can be added to the system (positive) or given off by the system (negative), and dW is the work done by the system. If some of the reactions induce any pressure or volume changes, the amount of heat exchange is no longer equal to the change of the internal energy because some of the heat is converted into work. Hence, enthalpy as a new property is introduced due to the limitation above, which is defined as the sum of the internal energy and the product of pressure and volume of the system.

$$\begin{eqnarray}&&H=U+PV\end{eqnarray} \tag{ 3 }$$

At a constant temperature, pressure, and volume, the change of the internal energy can be replaced by the change of the enthalpy. The change of the enthalpy can be further expressed via several terms that are related to enthalpy of reaction, phase change, heat capacity, and enthalpy of mixing.² By neglecting the phase change and enthalpy of mixing, the change of enthalpy in a single phase is expressed as follows;

$$\begin{eqnarray}&&\displaystyle \frac{dH}{dt}=I{T}^{2}\displaystyle \frac{d}{dT}\left(\displaystyle \frac{{U}_{OC}}{T}\right)+m{c}_{p}\displaystyle \frac{dT}{dt}\end{eqnarray} \tag{ 4 }$$

The first and second terms on the right side of the equation above represent the change of enthalpy caused by reaction and the change of heat energy of the battery by the change of temperature, respectively. Combining Eq. 4 with the first law yields the equation as follows;

$$\begin{eqnarray}&&\displaystyle \frac{dQ}{dt}=\displaystyle \frac{dW}{dt}-I{U}_{OC}+IT\displaystyle \frac{d{U}_{OC}}{dT}+m{c}_{p}\displaystyle \frac{dT}{dt},\end{eqnarray} \tag{ 5 }$$

where dQ/dt is the heat transfer rate with the surroundings and dW/dt is the electric power of the battery that is generally expressed as the product of terminal voltage and current. Under the assumption that the process is isothermal, the HGR given off to the system is the same as that from the battery. Therefore, the equation above becomes the same as Eq. 1, where the sign of the heat transfer rate switches to express the heat released from the battery is positive.

The electric power of the battery in the conventional thermal model is determined only by the product of terminal current and voltage, which does not consider internal chemical reaction processes and detailed prediction of heat distribution in components and the tabs or welds of a cell. Therefore, a detailed formulation is derived.

The internal processes during charging are summarized as a schematic diagram in Fig. 1. In this study, only heat generated inside the cell is considered. During charging and discharging, oxidation and reduction processes take place when electrons and ions are involved. Those charges are transported via migration and diffusion which are driven by gradients in potential and concentration, respectively. During charge transport, some of the power from the charger is dissipated due to internal resistances of the battery and the remaining power is converted to increase or decrease chemical energy of the battery by changing ion concentration in the active materials, which is the actual realized power to the battery cell.

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The HGR from the migration of the electrons in the solid phase can be expressed as³;

$$\begin{eqnarray}&&{\dot{Q}}_{s}=\displaystyle \int -{i}_{s}{\rm{\nabla }}{\phi }_{s}dv,\end{eqnarray} \tag{ 6 }$$

where ${i}_{s}$ is an averaged current density through the solid phase as expressed⁴;

$$\begin{eqnarray}&&{i}_{s}=-{\sigma }^{eff}{\rm{\nabla }}{\phi }_{s},\end{eqnarray} \tag{ 7 }$$

where ${\sigma }^{eff}$ is effective conductivity of active material.

Due to the high mobility of the electrons, the overall magnitude of the HGR is relatively small. However, when the currents flow through interfaces between the current collectors and active material particles in composite electrodes, the current can be concentrated at the area of microcontact spots, where the real contact area is reduced to only a small portion of the nominal contact area. Accordingly, the reduced area increases a contact resistance and the generated heat is expressed as;

$$\begin{eqnarray}&&{\dot{Q}}_{CR}={I}^{2}{R}_{CR},\end{eqnarray} \tag{ 8 }$$

where ${R}_{CR}$ is the contact resistance that is experimentally determined from the Ohmic voltage drop during charging and discharging operations.^12–14

Similarly, the HGR by ion migration in the electrolyte phase is expressed as;

$$\begin{eqnarray}&&{\dot{Q}}_{e}=\displaystyle \int -{i}_{e}{\rm{\nabla }}{\phi }_{e}dv,\end{eqnarray} \tag{ 9 }$$

where ${i}_{e}$ is an averaged current density through the electrolyte phase. Due to the lower mobility of the ions, the HGR by electrolyte is relatively large compared to that of the electrode.

On the other hand, most of the charge transport is determined by migration and as a result, the heat generation from diffusion is negligible. However, the gradient of the ion concentration in the electrolyte induces a concentration overpotential, which decreases the potential across the electrolyte and consequently the current density through the electrolyte phase as follows;

$$\begin{eqnarray}&&{i}_{e}=-{\kappa }^{eff}{\rm{\nabla }}{\phi }_{e}+{\kappa }^{eff}\displaystyle \frac{2RT}{F}\left(1-{t}_{+}^{0}\right){\rm{\nabla }}\,\mathrm{ln}\,{c}_{e},\end{eqnarray} \tag{ 10 }$$

where ${\kappa }^{eff}$ is effective ionic conductivity and ${t}_{+}^{0}$ is transference number. The first and the second terms represent the current density from the potential gradient and that from the concentration overpotential, respectively.

At the surface of the active material, a solid-electrolyte interphase (SEI) layer deposited on the active materials of anode electrode acts as an additional ionic resistance prior to intercalation. The associated HGR is expressed as follows;

$$\begin{eqnarray}&&{\dot{Q}}_{SEI}=\displaystyle \int {j}^{Li}{V}_{SEI}dv,\end{eqnarray} \tag{ 11 }$$

where ${V}_{SEI}$ is a potential drop across the SEI layer and expressed as;

$$\begin{eqnarray}&&{V}_{SEI}=\displaystyle \frac{{R}_{SEI}}{{a}_{s}}{j}^{Li}\end{eqnarray} \tag{ 12 }$$

During the reaction, the activation overpotential generates heat expressed as;

$$\begin{eqnarray}&&{\dot{Q}}_{act}=\displaystyle \int {j}^{Li}\eta dv,\end{eqnarray} \tag{ 13 }$$

where ${j}^{Li}$ is a reaction rate and $\eta$ is the activation overpotential.

Since all reactions take place at the surface of the active materials, the amount of the actual power applied to the battery can be expressed using the change of surface ion concentration of the active materials as follows;

$$\begin{eqnarray}&&{\dot{W}}_{act}=\displaystyle \int {j}^{Li}{U}_{eq}\left({c}_{s,surf}\right)dv\end{eqnarray} \tag{ 14 }$$

The sum of all equations from (6) to (14) results in the electric power of the battery.

The internal energy of the battery can be rewritten considering reaction rate and heat of mixing, as expressed below;

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{dU}{dt}={C}_{p}\displaystyle \frac{dT}{dt}-\displaystyle \int {j}^{Li}T\displaystyle \frac{d{U}_{OC}}{dT}dv\\ \,\,\,\,+\,\displaystyle \int {j}^{Li}{U}_{eq}\left({c}_{s,ave}\right)dv-\displaystyle \int \left(\overline{H}-{\overline{H}}^{ave}\right)\displaystyle \frac{\partial c}{\partial t}dv\end{array}\end{eqnarray} \tag{ 15 }$$

The difference between the actual power and the change of the internal energy based on Eqs. 14 and 15 represents an additional heat generated in the active materials, which yields a new term that expresses the difference of the lithium-ion concentration at the surface of the particle and averaged ion concentration in the particle as follows;

$$\begin{eqnarray}&&{\dot{Q}}_{concentration}=\displaystyle \int {j}^{Li}\left[{U}_{eq}\left({c}_{s,surf}\right)-{U}_{eq}\left({c}_{s,ave}\right)\right]dv\end{eqnarray} \tag{ 16 }$$

Lastly, there is a resistance to mass transport in the battery, which leads to the formation of the concentration gradient as ions are transported. When the current is turned off, the concentration gradients relax, which results in the heat of mixing. Most researchers do not consider the heat of mixing because the magnitude of the heat of mixing is very small.³ However, if the heat of mixing is not considered, it is not possible to model the heat that is generated after the current is turned off. For example, when using Eq. 1, the HGR becomes zero if the current is zero. In order to derive an equation for the heat of mixing, it is assumed that there are two components in the system and the total enthalpy is a sum of enthalpies of each component as follows;¹⁵

$$\begin{eqnarray}&&H=\displaystyle \int \left({c}_{A}{\bar{H}}_{A}+{c}_{B}{\bar{H}}_{B}\right)dv,\end{eqnarray} \tag{ 17 }$$

where c and $\bar{H}$ are ion concentration and partial molar enthalpy of the components A and B. Partial molar enthalpy is a function of temperature, pressure, and concentration. Then the partial molar enthalpy can be expanded from the 2nd order Taylor series.

$$\begin{eqnarray}&&\begin{array}{l}{\bar{H}}_{A}={\bar{H}}_{A,ave}+{\left.\displaystyle \frac{\partial {\bar{H}}_{A}}{\partial {c}_{A}}\right|}_{ave}\left({c}_{A}-{c}_{A,ave}\right)+\displaystyle \frac{1}{2}{\left.\displaystyle \frac{{\partial }^{2}{\bar{H}}_{A}}{\partial {{c}_{A}}^{2}}\right|}_{ave}{\left({c}_{A}-{c}_{A,ave}\right)}^{2}\\ \,\,+\,{\left.\displaystyle \frac{\partial {\bar{H}}_{A}}{\partial T}\right|}_{ave}\left(T-{T}_{ave}\right)+{\left.\displaystyle \frac{\partial {\bar{H}}_{A}}{\partial P}\right|}_{ave}\left(P-{P}_{ave}\right)+E\left({\rm{\Delta }}{c}^{3}\right)\\ {\bar{H}}_{B}={\bar{H}}_{B,ave}+{\left.\displaystyle \frac{\partial {\bar{H}}_{B}}{\partial {c}_{A}}\right|}_{ave}\left({c}_{A}-{c}_{A,ave}\right)+\displaystyle \frac{1}{2}{\left.\displaystyle \frac{{\partial }^{2}{\bar{H}}_{B}}{\partial {{c}_{A}}^{2}}\right|}_{ave}{\left({c}_{A}-{c}_{A,ave}\right)}^{2}\\ \,\,+\,{\left.\displaystyle \frac{\partial {\bar{H}}_{B}}{\partial T}\right|}_{ave}\left(T-{T}_{ave}\right)+{\left.\displaystyle \frac{\partial {\bar{H}}_{B}}{\partial P}\right|}_{ave}\left(P-{P}_{ave}\right)+E\left({\rm{\Delta }}{c}^{3}\right)\end{array}\end{eqnarray} \tag{ 18 }$$

Under the assumptions of constant temperature and pressure, the equations above can be simplified to an equation for the heat of mixing in a single component as follows¹⁶;

$$\begin{eqnarray}&&{\dot{Q}}_{mix}=\displaystyle \frac{\partial }{\partial t}\left[\displaystyle \frac{1}{2}\displaystyle \frac{\partial {\bar{H}}_{s}}{\partial {c}_{s}}\right]\displaystyle \int {\left({c}_{s}-{c}_{s,ave}\right)}^{2}dv\end{eqnarray} \tag{ 19 }$$

Using the derivative of the partial molar enthalpy, the ion concentration can be rewritten from an enthalpy potential that is defined as the voltage corresponding to reaction¹⁷ and is expressed as enthalpy per the unit of charge, $\partial {\bar{H}}_{s}/\partial {c}_{s}=-F\partial {U}_{H}/\partial {c}_{s}.$ The enthalpy voltage is further expressed from the Gibbs–Helmholz equation.¹⁸

$$\begin{eqnarray}&&{U}_{H}=-{\rm{\Delta }}H/nF={U}_{OC}-T\displaystyle \frac{\partial {U}_{OC}}{\partial T}\end{eqnarray} \tag{ 20 }$$

Therefore, the heat of mixing can be calculated from the entropy coefficient and ion concentration as shown;

$$\begin{eqnarray}&&{\dot{Q}}_{mix}=-\displaystyle \frac{\partial }{\partial t}\left[\displaystyle \frac{1}{2}F\displaystyle \frac{\partial \left({U}_{OC}-Td{U}_{OC}/dT\right)}{\partial {c}_{s}}\right]\displaystyle \int {\left({c}_{s}-{c}_{s,ave}\right)}^{2}dv\end{eqnarray} \tag{ 21 }$$

All heat source terms are summarized in the appendix. The theoretical calculation of the HGR of a battery requires information on internal variables and parameters of the battery. The internal variables are calculated from a physics-based electrochemical model and the parameter set for the thermal model is obtained by comparing the electrochemical model prediction to the experimental voltage profiles.⁹

Estimation of reversible heat generation rate based on an empirical method

Measurement of the entropy coefficient

There are two methods to measure the entropy coefficient experimentally depending upon if the battery temperature or SOC is kept constant. The first method involves keeping the temperature constant and measuring voltage by varying SOC at different temperatures. The other method requires keeping SOC constant while varying temperature.^11,15,18,19 Since OCV varies linearly with temperature within the reference temperature of 25 °C^11,20 but nonlinearly with SOC, the second method is more preferred due to its high accuracy.

The procedure for the measurement of the entropy coefficient is summarized in Fig. 2a. Initially, a battery is charged or discharged to a target SOC point, which is repeated for 21 points from 0% to 100 SOC with a 5% interval. Then, the battery is rested for at least 10 hours until an equilibrium state is reached. At the equilibrium state, the temperature of the battery is changed using the calorimeter. Figure 2b shows the change of the battery temperature with ±10 °C over time, at the temperature change rate of 0.5 °C min⁻¹. At the same time, the change of terminal voltage due to the temperature change is measured as shown in Fig. 2c, where the voltage is measured at the 25% SOC point. Then, the entropy coefficient at given SOC can be obtained by linearly fitting the slope of the voltage to the temperature.

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The entropy coefficient obtained from two different methods are compared in Fig. 2d. The blue line is the empirical coefficient and the red dots and line are the measured data and fitted curve. The two results show a good agreement except at 0% and 100% SOC. In the case of empirical calculation, CC charging and discharging are applied, and the cutoff voltage is reached earlier than 0% and 100% SOC, which results in the inaccuracy of the measurement. Therefore, the measured coefficient is used for the thermal model. It can be assumed that one fitted curve can be used for the other temperature ranges because the slope of the voltage is constant as a function of temperature as shown in Fig. 2(c). On the other hand, the coefficient is highly dependent on the SOC and especially at low SOC, it has a high negative value that can cause the reversible HGR to be endothermal and exothermal during charging and discharging respectively.

Battery temperature is changed by the heat generated and the heat transferred to the surrounding according to the energy equation, which is expressed as;

$$\begin{eqnarray}&&m{c}_{p}\displaystyle \frac{dT}{dt}={\dot{Q}}_{gen}-{\dot{Q}}_{transfer}\end{eqnarray} \tag{ 22 }$$

If the battery is operated in an adiabatic condition, the heat transferred to the environment is zero and all heat generated directly leads to an increase of the battery temperature. If the battery is under isothermal operation, the change of the battery temperature is zero and all heat generated is transferred to the environment. Therefore, a calorimeter that maintains the surface temperature of the testing cell as constant is required for the measurement of the HGR. The calorimeter is developed using thermal electric modules (TEM) that work as a heat pump and a thermostat at the same time. A testing cell is placed between two TEMs and its surface temperatures are measured using thermocouples. When the battery generates heat during charging or discharging, the surface temperature of the battery changes. Then, the TEM absorbs or releases heat to control the battery temperature to a preset reference in a feedback loop.^21,22 In this work, it is assumed that the effects of a cross-plane temperature gradient are negligible.

Theoretical calculation of the HGR of a battery requires knowledge of internal variables such as the ion concentration, the potentials in solid and electrolyte phase, or activation overpotential that are immeasurable at the terminal of the battery. Those variables can be calculated from the governing nonlinear partial differential equations (PDE) that describe ion dynamics considering chemical reactions, which is called a full order model (FOM). Even with high accuracy, the FOM is still inappropriate for control purposes in real-time because of computational time. A possible approach is to reduce the order of the FOM by converting PDEs into ordinary differential equations (ODE) and linearize the nonlinear equations, which is called a reduced-order electrochemical model (ROM). There are mainly two types of ROM, which are a single particle model (SPM) and a pseudo-2-dimensional model (P2D). The P2D ROM is used in this study because the SPM cannot calculate the gradients of potential and concentration, which is necessary for the study of the heat generation behavior along the thickness direction. Details of the equations are summarized in the appendix.

The thermal model including ROM is experimentally validated with respect to HGR using pouch type LMO (30%)—NMC (70%) / Graphite cell with a capacity of 26Ah under various C-rates and temperatures. The HGRs from experiments and simulations at 25 °C during charging and discharging with different C-rates are compared in Figs. 3a and 3b, and those during 1C discharging at different temperatures are compared in Figs. 3c and 3d. Generally, the HGRs during charging are smaller than those during discharging at the same C-rate. Besides, the profiles of the HGR from 15 °C to 45 °C are similar but rapidly increase when the temperature is below 0 °C. The detailed analysis of the effect of C-rate and temperature will be discussed in the next section.

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The corresponding root-mean-square (RMS) errors of the simulation data are calculated and summarized in Tables I and II. The overall profiles of the HGR from the simulation are in good agreement with those from experiments. However, the difference between the simulated and measured heat generation rate tends to increase at high C rates or low temperatures, which can be caused by the neglected cross-plane temperature gradient from the measurements and the assumptions of the model. In addition, another reason for the difference at low temperatures is the neglected temperature dependency of the entropy coefficient.

The HGR of the battery is dependent on many factors such as SOC, applied C-rate, temperature, and state of degradation. In this study, the effects of the SOC, C-rate, and temperature on the heat sources are analyzed.

Heat source analysis

The detailed heat sources including the heat from charge transport, overpotentials, and reversible entropic heat are compared during both charging and discharging. The profiles of the HGR from each heat source during 2C constant current (CC) charging and discharging at 25 °C are plotted in Fig. 4 as a function of time and SOC. The most significant heat sources are; (1) change of the entropy, (2) migration from the electrolyte phase, (3) contact resistance, and (4) difference of surface and averaged ion concentration within particles. Each heat source has a different magnitude and tendency as a function of SOC.

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As seen in Fig. 4, the entropic HGR is highly dependent on SOC and mirrored at charging and discharging because of its reversible nature. Additionally, the HGR from the electrolyte phase is much larger than that from the electrode phase because of the different mobility of charges. However, it is reduced by the concentration overpotential, and accordingly, the sign of the HGR resulting from the concentration overpotential is negative. Due to different chemistries composing the anode and cathode active materials, the magnitudes of the HGR by migration during charging and discharging are different at a given C-rate. The HGR due to the contact resistance has the same profile as that of the current, which dominantly contributes to the total HGR. Lastly, the HGR due to the difference of ion concentration within particles is also dependent on SOC and its magnitude is seen to be the largest, especially at low SOC during discharging. In fact, impedances measured by EIS have shown that the charge transfer resistance of the cell is the largest when the voltage is at 0% SOC.²³ In addition, the internal cell resistance is dependent upon charging and discharging and the resistance during discharging is larger than that during charging.²⁴ There are other negligible heat sources such as the activation overpotential, SEI resistance at the Beginning of Life (BoL), and the heat of mixing.

The total irreversible HGRs during 1 C charging and discharging for each component are plotted in Figs. 5a and 5b under the assumption that the contact resistance at the interface of the anode electrode and current collector is same as that of the cathode electrode and current collector. The magnitude of the HGR from the cathode electrode for both charging and discharging has a peak at a low SOC where the difference of ion concentration within particles is larger, which leads to a large potential difference. At the anode electrode, the potential does not significantly change with the ion concentration. At the separator, the SOC dependency of the HGR is similar, and the magnitude is small because the only heat generated is due to ion transport. The ratio of magnitude of each heat source to the total heat at the anode and cathode is compared in Figs. 5c and 5d, where the heat from concentration overpotential (yellow) is negative. At the anode electrode, the heat from the contact resistance is dominant with the addition of the heat from SEI resistance, which is unique to the anode electrode. In addition, the heat from the difference in ion concentration within particles is not significant. However, at the cathode electrode in Fig. 5d, the heat from the difference in ion concentration is significant heat source. The migration heat at the electrolyte phase is also a dominant heat source although some amount of heat is reduced by the concentration overpotential. Other heat sources, migration from the electrode, activation overpotential, heat of mixing, are small enough to neglect.

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Effect of the C-rates on the heat generation rates

In order to analyze the effects of C-rates, the total heat is calculated, which provides a reference for the calculation of efficiency during operations. The total heat during charging and discharging at 35 °C is plotted as a function of C-rates in Fig. 6a. When C-rate increases, Joule heating, which is the main heat source in the battery, increases proportionally to the square of applied current, but the charging or discharging time decreases. As a result, the total heat increases almost linearly as the C-rate increases. In addition, the total heat at charging is always smaller than that at discharging at the given C-rate because of the contribution of the reversible heat. In Fig. 6b, the ratio of the heat from each source to the total heat during charging and discharging is compared as a function of C-rates. The entropic heat is seen to be the largest at a low C-rate but rapidly decreases as C-rate increases. In addition, the profiles at charging and discharging are similar but mirrored because of the reversible process, which is the reason for higher heat during discharging. On the other hand, the irreversible heat sources have a similar trend regardless of charging and discharging. Those heat sources are small at a low C-rate but become dominant as C-rate increases because of the overpotentials.²⁵ Above 1.5 C, the ratio of irreversible heat sources is seen to be relatively constant because the increasing rates of irreversible heat sources are identical.

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Effect of temperature on the heat generation rate

Since the temperature of a battery can vary during operation, the effect of the operating temperature on the HGR is analyzed with the validated model. For the analysis, the results during discharging are compared due to the higher HGR compared to charging. Different heat sources during 1 C CC discharging at different temperatures are plotted in Fig. 7. The results show that the most dominant heat source at high temperatures is the change of entropy because the battery internal resistance is small at high temperatures, which leads to a small irreversible HGR. On the other hand, the contribution of the contact resistance, ion concentration difference within particles, and ion migration to the overall heat generation rate rapidly increases at low temperatures. The area of microcontact spots of the interface between active material particles and current collectors is caused by a deformation of materials, which is directly related to the strength of the material. At low temperatures, the material strength increases, and the deformation is reduced. Accordingly, the contact area is reduced and the contact resistance increases.²⁶ In addition, the gradient of ion concentrations in particles increases the charge transfer resistance at low temperatures.²³ Ion transport at low temperature gets hindered and heat generation rate increases, however some amount of heat is reduced by concentration overpotential.

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The two most dominant irreversible heat sources at low temperatures are plotted in Figs. 8a and 8c. Since the HGR due to the contact resistance follows the same profile as the applied current, only the magnitude of the HGR is plotted as a function of temperature. In addition, the magnitude of the HGR due to the SEI resistance is plotted in Fig. 8b and is seen to increase rapidly at low temperatures even though the HGR is small enough to neglect at the beginning of life. Both resistances exponentially increase at low temperatures.

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The discharging time is also significantly affected by the operating temperature. Therefore, it is more effective to compare results as a function of SOC. The HGR caused by the difference in the ion concentration within the particles is plotted at various temperatures in Fig. 8c. The overall HGR increases as temperature decreases. However, the peak HGR at −30 °C is smaller than that at −15 °C because the same terminal voltage cutoff occurs at a higher SOC at lower temperatures because of a larger ohmic overpotential at low temperatures, which results in the smaller peak.

The entropic HGRs at various temperatures are plotted in Fig. 8d, where a consistent trend in the entropic HGR is seen. In this study, the entropy coefficient is measured at 25 °C as a reference temperature and used for all temperature ranges. As a result, the absolute temperature does not affect the overall HGR significantly at the given C-rate. On the other hand, at low temperatures, the voltage reaches the cutoff voltage earlier due to increased ohmic overpotential. Therefore, the total heat decreases at lower temperatures due to the reduced discharging time.

Temperature distribution through the plane of a single cell

The thermal model previously introduced assumed that a microcell represents a single cell. In fact, a single cell is composed of multiple microcells that are connected in parallel. Therefore, the validated thermal model for a microcell is extended to predict temperature profiles through the plane of a single cell, which schematic diagram is depicted in Fig. 9. In this study, the number of anode and cathode electrodes is 20 and 19, respectively. Besides, it is assumed that the reactions taking place at all microcells are identical and there are thermal contact resistances between separators and electrodes. The value of thermal contact resistance is assumed to 80% of total thermal resistance in this study.^27,28 The dimensions and thermal properties of each component are summarized in Table III.

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Table III.  List of model parameters.

Electrochemical parameter (a: manufacture; b: model validation).
	Negative electrode	Separator	Positive electrode	Unit
Thicknessa, δ	66 × 10–6	31 × 10–6	73 × 10–6	m
Particle radius,29 RS	8 × 10–6	—	9.8 × 10–6	m
Stoichiometry at 0%b SOC	0.28	—	0.84
Stoichiometry at 100%b SOC	0.92	—	0.46
Average electrolyte concentration, ce	1.2 × 103	1.2 × 103	1.2 × 103	mol m−3
Exchange current density coefficient, i0	13.2 × 104	—	6.79 × 104	A m−2
Charge transfer coefficient, αa, αc	0.5, 0.5	—	0.5, 0.5
Solid phase diffusion coefficientb, Ds	2.28 × 10–13	—	1.04 × 10–13	m2 s−1
Solid phase conductivity, σ	5.4	—-	0.2	S m−1
Bruggeman's porosity exponent, p	1.5	1.5	1.5
Electrolyte phase ionic conductivity, κ	$\kappa =15.8{c}_{e}\exp \left(-13472{c}_{e}^{1.4}\right)$			S m−1
Li+ transference number, t0+	0.363	0.363	0.363
Equilibrium potential of anode	$\begin{array}{l}{U}_{eq-}\left(x\right)=8.00229+5.0647x-12.578{x}^{0.5}\\ -\,8.6322\times {10}^{-4}{x}^{-1}+2.1765\times {10}^{-5}{x}^{-1.5}\\ -\,0.46016\exp \left(15\times \left(0.06-x\right)\right)\\ -\,0.55364\exp (-2.4326\times (x-0.92))\end{array}$
Thermal parameter.30
Components	Material	Density [g cm−3]	Heat capacity [J g−1 K−1]	Conductivity [W cm−1 K−1]
Negative CC	Copper	8.96	0.386	3.97
Anode	Graphite	2.1	0.71	0.1511
Separator	Microporous polymer	1.4	1.551	0.0035
Cathode	LMO—NMC	4.89	0.85	0.00687
Positive CC	Aluminum	2.7	0.9	2.3
Pouch	Aluminum/PET/PPE/Nylon	1.7	1.15	0.61

In addition, forced air convection is assumed at both surfaces, where the convection coefficient is set as 30 W m⁻² K⁻¹.³¹ Discharging current and temperature profiles for the single cell level during 2C CC discharging at 25 °C are calculated and plotted in Figs. 10a–10c. The temperature profile is symmetric due to the symmetrical structure and HGR profile. Small ripples in the profile through the plane are caused by thermal contact resistance as well as different HGR profile and thermal properties of the components.

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A thermal model is developed from the physics-based reduced-order electrochemical model to estimate the HGR of the battery and the results are validated against experimental data. Then a detailed analysis of the HGR is conducted using the validated thermal model. The analysis has shown that the dominant heat sources of the battery are caused by the contact resistance, migration of the ions within the electrolyte phase, change of entropy, and the difference in the ion concentration within active material particles. Those sources vary with different operating conditions. Therefore, the effects of the C-rates and temperatures on the overall HGR are investigated. The entropic HGR is proportional to the C-rate and its sign depends upon the charging or discharging process. Additionally, the irreversible HGR has a quadratic dependency with the C-rate. Consequently, the entropic heat is dominant at low C-rates, while the irreversible heat becomes dominant as the C-rate increases. In addition, at the same C-rate, the magnitude of total heat during discharging is always greater than that during charging due to the entropic heat and higher charge transfer resistance. It has also been shown that lowering temperature rapidly increases the irreversible HGR because of increased overpotentials, however, the entropic HGR is not affected significantly by the temperature. This paper demonstrates that a physics-based electrochemical thermal model can be used to analyze the effects of operating conditions of various C-rates and temperatures on the HGR which could be used to optimize thermal management systems with respect to cost and efficiency. Future work includes an extension of the thermal model to a multi-dimensional model considering degradation effects at the middle-of-life and end-of-life.