A Critical Review of Thermal Issues in Lithium-Ion Batteries

Bernardi et al.⁴⁵ derived an expression for battery heat using a thermodynamic energy balance on a complete cell. Discrete phases inside the battery interact with each other by means of electrochemical reaction, phase changes, and mixing. By applying the first law of thermodynamics around the cell control volume (not including current collectors) and making numerous simplifications, they determined the following expression for heat generation inside the battery

Here, the first term is the electrical power produced by the battery. The second term is the sum of producible reversible work and entropic heating from the reaction and is summed over all simultaneously occurring reactions. The third term is heat produced from mixing. Because the reaction rates are not uniform, concentration variations across the battery are developed as the reaction proceeds. When the current is interrupted, the concentration gradients developed inside the battery relax, causing heat to be released or absorbed. Hence, as the concentration profile inside the battery is developed during operation, an apparent relaxation heat will occur and be opposite in sign but equal in magnitude to the heat observed when the current is cut off. This term may be significant if the enthalpy of the mixture as a function of concentration is nonlinear. The final term in the energy balance is heat from material phase changes. The equation proposed by Bernardi et al. is cited frequently in the literature in its simplified form⁴⁶

This form has been reported previously (e.g., Sherfey and Brenner⁴⁷); some key features are as follows. The first term is the overpotential due to ohmic losses in the cell, charge transfer overpotentials at the interface, and mass transfer limitations. The electrode potential is determined at the average composition. The second term is the entropic heat, and the potential derivative with respect to temperature is often referred to as the entropic heat coefficient. Phase change and mixing effects are neglected in this expression.

In a subsequent study, Rao and Newman⁴⁸ noted that the use of average composition for the electrode open circuit potentials can lead to significant errors when predicting heat generation. They presented two different methodologies for estimating heat generation: a thermodynamic energy balance and a local heat generation method. For the energy balance, they neglected phase change and mixing effects but used the average local concentration in each phase to determine the rate of change of the enthalpy of the species and phases inside the battery. After neglecting the concentration dependence of the reference enthalpy and applying Faraday's law (ignoring the electrical double layer), Rao and Newman arrive at the following expression for the battery heat generation

The first term in the above expression is the average enthalpy potential (which can include entropic heat effects) integrated where the reactions are actually taking place across the thickness of the battery. They assert that this equation should be used instead of simply using the average composition to determine the open circuit potential.

The local heat generation method derived by Rao and Newman is the summation of all the relevant local thermal effects occurring inside the battery. These include heat generated at the electrochemical interfaces and in the bulk material in different parts of the cell. For this method, Rao and Newman neglect phase changes, concentration gradients in the electrolyte phase, and thermal effects from lithium diffusion. The temperature of the cell is assumed to be uniform. The necessary integration is carried out only normal to the electrode stack (i.e., this is a one-dimensional analysis). The thermal effects include only the Joule heating in the separator and in the electrolyte and matrix in the porous positive electrode (a lithium metal negative electrode was used), as well as the kinetic surface overpotential in both electrodes, resulting in the following

Rao and Newman show after integrating this expression (over the positive electrode) and applying the appropriate boundary conditions that this expression is equivalent to the energy balance method. (This expression does not include entropic heat effects, but, if inserted, should not change the result due to the appearance of the Peltier coefficient in the interface overpotential, which is equal to the entropic heating term if both electrodes have the same temperature.) They conclude that assuming no concentration gradients in the electrolyte appears to be similar to assuming no mixing effects. However, as discussed below, using the average local concentration for determining the effective open circuit potential is similar to accounting for the apparent mixing in the bulk electrode.⁴⁹ Furthermore, they show that using the open circuit potential at the average composition causes the cell overpotential to be poorly predicted in their specific lithium polymer battery design.

Recently, Thomas and Newman⁴⁹ studied the heat of mixing effect inside a battery containing a porous insertion electrode. They noted that although Rao and Newman stated they neglected mixing effects, they really did account for the heat of mixing inside the bulk electrode through variation in local current density on the effective electrode open circuit potential. However, Thomas and Newman pointed out that this was only one of four possible ways mixing effects occur inside the cell. The remaining modes of mixing heat are from concentration gradients inside the spherical particles, bulk electrolyte, and inside the electrolyte pores of the insertion electrode. Thomas and Newman illustrated the relative magnitudes of these terms using representative calculations on data collected for a battery discharged at the and rates for and , respectively. In both cases, the mixing heat from the cylindrical electrolyte pores was negligible due its small concentration gradients. The mixing heat across the bulk electrodes and electrolyte was more significant, but small compared to the irreversible and reversible heat. This was a different conclusion than that of Rao and Newman because the ionic conductivity used in this study was an order of magnitude higher, which is more representative of commercial organic solvent-based electrolyte batteries.⁴⁹ Although the heat of mixing in the spherical particle at the lower discharge rate was small at , the mixing heat across the spherical particles was significant compared to the sum of irreversible and reversible heats at the higher discharge rate.

To calculate the enthalpy of mixing in each case, Thomas and Newman determined expressions for the difference in enthalpy from the operating state to the relaxed state using a Taylor-series expansion for the molar enthalpy of each species, while neglecting density and temperature changes and concentration dependence on the second derivative of partial molar volume with respect to partial molar enthalpy. For the solid spherical particles, they assumed that the rate of reaction is approximately constant with time (pseudo-steady-state) and the particles had uniform current distribution on the surface. For a constant diffusivity, the enthalpy of mixing for the particle per unit separator area is calculated as follows

The relaxed state is denoted by ∞. This expression determines the energy per unit separator area released/absorbed when the concentration gradients inside the particles are allowed to relax. The constant 1050 arises from integration of species concentration. Thomas and Newman pointed out that their tested battery design will have a higher heat of mixing than for commercially made batteries due to their large particle size . Thus, it is generally safe to neglect this when estimating volumetric heat generation. However, one can use the above expression to determine if the heat of mixing in the solid particle is important for a particular electrode design.

In addition to electrochemical heat generation, Joule heating is produced from bulk electron movement in the current collectors (see Fig. 1). In small cells, this term may be insignificant. However, increased attention is now being paid to larger cells used in HEVs and EVs, where the electrical distribution in the current collectors may have a large impact on the overall heat generation rate. For example, Kim et al.^{50, 51} have investigated the impact of current distribution on several different electrode configurations for a polymer electrolyte battery. The two-dimensional (2D) model (variation in the thickness was neglected) consisted of two electrode current collectors coupled via a temperature-independent parameterized electrochemical model, which uses local overpotential to estimate local current production. The current fields in each collector are determined via Ohm's law, and the local volumetric heat generation rate is determined as follows

The first term is the electrochemical heat generation (per unit volume), which was discussed in detail above. The last two terms are the resistive heating due to current movement in the positive and negative metal current collectors, respectively. If the tab locations are designed appropriately, the distribution of local current generation from the electrochemical reaction may be minimally impacted. However, even though current production and overpotential may be similar, increased current is passing through the battery locally, which increases resistive heating in the current collectors. For example, in Kim et al. 's first study, a wide and tall cell with wide current collector tabs placed on the top end of the current collectors caused less than 0.5% maldistribution in current when discharged at the rate for . However, both the simulation and experiments show a more than temperature variation, with the hottest portions of the battery near the current collection tabs. The recent work at the National Renewable Energy Lab (NREL) that includes solving electrochemical transport locally has also shown similar results.^{52, 53}

In summary, the heat generation estimated by Bernardi et al. (Eq. 3) is the most commonly used equation to estimate battery heat generation. This equation may be readily applied to estimate the amount of electrochemical heat generation in small lithium-ion batteries if there is no heat from mixing or phase change, no spatial variation in temperature or SOC, only one electrochemical reaction occurring at each electrode, and negligible Joule heating in the current collectors. In lithium-ion batteries without side reactions, there is only one reaction occurring at each electrode, and no phase change effects exist. Neglecting the heat of mixing terms is acceptable for low discharge rates, and will be for high discharge rates when the particle size is sufficiently small, which is representative of commercial battery designs. In large battery simulations, the electrochemical reactions may be sufficiently nonuniform for poorly placed current collection tabs, which result in both SOC and temperature nonuniformities. However, resistance heating in the current collectors of large batteries may be significant relative to the electrochemical heat generation rate for even well designed batteries that have uniform current production and working potential across the cell throughout the battery. In these cases, discretization of the battery into smaller cells and accounting for current collector Joule heating is imperative. Thus, it is recommended that Eq. 7 be used to simulate heat generation in batteries.

Table V shows a summary of different techniques used to measure heat generation rate in lithium-ion battery half and full cells during charge and discharge. This table includes the experimental technique along with a short description of how heat is removed and measured from the battery during the test. The primary experimental methods are accelerated-rate calorimetry (ARC) and isothermal heat conduction calorimetry (IHC). The ARC method consists of measuring the heat rejected from the battery surface through air or Styrofoam to a constant temperature sink. In this method, the temperature of the battery is allowed to rise during operation, and the heat generation rate is estimated using an energy balance on the battery as follows⁵⁶

The first term is the amount of heat stored inside the battery, while the latter is the heat transferred from the surface of the battery to the constant temperature well. The calorimeter constant and the effective heat capacity of the cell are determined experimentally.^{55, 56} For IHC, the battery remains at one temperature throughout operation using an isothermal well in close contact with the surface of the battery (e.g., liquid or a metal heat sink). High accuracy thermopiles either embedded inside the heat sink or placed near the surface of the battery are used to measure the heat rate. As in all calorimetric methods, special data processing or experimental procedures are usually required due to the long instrument time constants (i.e., time elapsed from heat generation to measurement). For example, Thomas and Newman⁴⁹ numerically integrated a convolution integral using an approximate Dirac delta function estimated from a previous experiment and also added the time lag to the deconvoluted heat generation rate to facilitate comparison with their model results. In contrast, Kobayashi et al.⁶⁰ simply discharged at slow rates and neglected the time constant. Self-discharge can be important for these long discharge times and complicate the analysis. For example, Lu et al.⁶² corrected their data from the self-discharge observed at high positive electrode potentials.

Table V. Summary of battery heat generation test methods.

Investigators	Heat rejection from battery	Special data processing
ARC — measurement of battery and calorimeter surface temperature
Al Hallaj et al.54	Conduction through Styrofoam	Experimentally determined calorimeter constant
Al Hallaj et al.55	Natural convection and radiation through air	Experimentally determined calorimeter constant
Hong et al.56	Conduction through Styrofoam	Experimentally determined calorimeter constant
IHC — measurement of thermopile temperature difference
Bang et al.57	Conduction through isothermal heat sink (bottom only)	Difference between test cell and dummy cell is the heat rate
Kim et al.58	Conduction through aluminum heat sink	time lag correction
Kobayashi et al.59	Conduction through isothermal heat sink (bottom only)	Neglected instrument time lag
Kobayashi et al.60	Conduction through isothermal heat sink (bottom only)	None given
Lu and Prakash61	Not given (isothermal bath)	Time lag correction (no specific time value given)
Lu et al.62	Not given (isothermal bath)	Self-discharge correction
Lu et al.63	Not given	None given
Lu et al.64	Not given	None given
Onda et al.65	Convection through isothermal water bath	None given
Saito et al.66	Conduction through temperature- controlled heat sink	time lag correction
Saito et al.67	Not given	time lag correction
Song and Evans68	Not given (isothermal bath)	None given
Thomas and Newman49	Conduction through copper shot	Numerical convolution of data, equipment offset correction, time lag correction
Yang and Prakash69	Not given	Subtracted constant value instrument offset
RC — measurement of applied heater power, cell temperature, and lead-wire temperature
Vaidyanathan et al.70	Radiation through vacuum through copper chamber ( by liquid )	External heat applied and wire lead heat loss correction

Radiation calorimetry (RC) is achieved by suspending the battery in an evacuated constant-temperature container. The energy balance on the battery is as follows

The first three terms on the right side are the heat accumulated in the battery, radiation heat lost from the battery to the constant temperature container, and conductive loss through the lead wires, respectively. The last term is the heat applied to maintain a constant battery temperature while at rest.

To facilitate discussion of these methods, Table VI shows a gap analysis that includes the geometry, chemistry, and discharge/charge rates tested. As shown, the majority of previous studies on insertion electrodes for lithium-ion batteries have been conducted on small coin cells for low to moderate discharge rates (i.e., to ). There are few studies conducted for discharge rates . This emphasizes the difficulty in measuring transient high rate heat generation. The fast heating of the cells can rapidly increase the internal temperature of the battery. This makes the heat storage term the most significant value in the energy balance, thus requiring the battery heat capacity to be precisely measured. In addition, for large batteries, significant thermal gradients inside the cell may develop, thus requiring more careful experimental techniques. Moreover, the multiple studies on low rate discharge of coin cells are attempts to measure the entropic heat coefficient. This is achieved relatively easily because the low heating rates minimize temperature rise. In addition, the coin cells are thermally thin, thus substantially reducing internal temperature gradients that can influence the results.

Table VII. Summary of battery tests: battery geometry, material, tested currents, and temperatures.

Investigators	Battery description		Currents	Temperatures
	Geometry	Material
Al Hallaj et al.54	Cylindrical cell: OD, height Coin cell: OD, height	Half coin cell:Full cylindrical cells: (graphite) (A—graphite, B—hard carbon, C—graphitized carbon fiber)	to
Al Hallaj et al.55	Cylindrical cell: OD, height Prismatic cell: (D) (E)	Full cells: (A—graphite, B—hard carbon, C/D/E—graphitized carbon fiber)	to
Hong et al.56	Cylindrical cell: OD, height	Full cell: (hard carbon, Sony)	to
Bang et al.57	Coin cell: OD, height	Half cells: and	to
Kim et al.58	Coin cell: OD, height	Half cell:	to
Kobayashi et al.59	Prismatic cell:Cylindrical cell: OD, height	Full prismatic cell:Full cylindrical cell:	,
Kobayashi et al.60	Cylindrical cell: OD, height Coin cell: OD, height	Half coin cells: and Full cylindrical and coin cells:	, ,	not given
Lu and Prakash61	Coin cell: OD, height	Half cell: (MCMB)	to
Lu et al.62	Coin cell: OD, height	Half cell:
Lu et al.63	Coin cell: OD, height	Half cells: and Full cell: (MCMB)	to
Lu et al.64	Coin cell: OD, height	Half cells:,, and cells
Onda et al.65	Cylindrical cell: OD, height	Full cells: (A—hard carbon, B— graphite)	to
Saito et al.66	Cylindrical cell: OD, height	Full cells: (hard carbon)
Saito et al.67	Cylindrical cell: OD, height	Full cells: (A/B/C—graphite, D—hard carbon)	,
Song and Evans68	unit cell and area	Half cell:	to
Thomas and Newman49	Coin cell: OD, 2.5 height	Half cell:	to
Yang and Prakash69	Coin cell: OD, height	Half cells: and Full cells: (natural graphite Mag-10)	to
Vaidyanathan et al.70	Cylindrical cell: OD, height	Full cells: and (75%) and (25%)	to	,

Table VIII. Summary of battery tests: charging heat generation.

Investigators	Cell type	Heat generation rates (W/L)
Al Hallaj et al.54	Cylindrical	to 0.26
Hong et al.56	Cylindrical			to 1.63
Bang et al.57	Coin 1		to 0.63 to 1.11	0.03 to 2.65	0.11 to 7.51
	Coin 2		to 0.82 0.00 to 1.32	0.00 to 4.41	0.00 to 14.9
Kim et al.58	Coin		to 0.49 to 0.58	0.47 to 0.97	to 3.21
Kobayashi et al.59	Cylindrical		to 0.97
	Prismatic	to 0.15	to 0.76
Kobayashi et al.60	Coin 1	0.00 to 0.02
	Coin 2	to 0.01
	Coin 3	to 0.00
Lu and Prakash61	Coin		to 0.04	0.00 to 0.08	0.00 to 0.35
Lu et al.62	Coin		0.01 to 0.02
Lu et al.63	Coin 1		0.00 to 0.02	0.02 to 0.10	0.24 to 0.73
	Coin 2		to 0.03	to 0.07	0.00 to 0.32
	Coin 3		to 0.02	to 0.07	0.00 to 0.40
Lu et al.64	Coin 1		0.00 to 0.02
	Coin 2		0.00 to 0.01
	Coin 3		0.00 to 0.01
Saito et al.67	Cylindrical 1	to 0
	Cylindrical 2	to 0.45
	Cylindrical 3	to 0.45
	Cylindrical 4	to 0.11
Thomas and Newman49	Coin		to 1.82	to 5.69		9.01 to 18.8
Yang and Prakash69	Coin 1	0.00 to 0.01	0.00 to 0.02	0.00 to 0.04	0.00 to 0.60
	Coin 2	to 0.02	to 0.05	to 0.12	to 1.07
	Coin 3	to 0.00	to 0.02	to 0.06	0.00 to 0.41
Vaidyanathan et al.70	Cylindrical 1		to 1.95
	Cylindrical 2	to 0.00	1.00 (average)

Table IX. Summary of battery tests: discharging heat generation.

Investigators	Cell type	Heat generation rates (W/L)
Al Hallaj et al.54	Cylindrical	to 0.59
Al Hallaj et al.55	Cylindrical 1		0.00 to 1.89	0.00 to 5.29	0.00 to 7.68 0.00 to 15.5
	Cylindrical 2		0.00 to 6.55	0.00 to 10.3	0.00 to 15.1 0.00 to 27.7
	Cylindrical 3		0.00 to 2.27	0.00 to 4.53	0.00 to 6.58 0.00 to 13.2
	Cylindrical 4		0.00 to 2.50	0.00 to 4.38	0.00 to 14.1	0.00 to 30.3
	Cylindrical 5		0.00 to 2.74	0.00 to 9.08	0.00 to 19.6	0.00 to 26.8
Hong et al.56	Cylindrical			0 to 12.1	0 to 16.1 0 to 33.3
Bang et al.57	Coin 1		to 1.15 to 1.97	to 4.75	to 10.2
	Coin 2		to 0.57 0.00 to 0.77	0.00 to 1.81	0.00 to 8.11
Kim et al.58	Coin		to 0.41 to 0.95	to 1.77	0.00 to 5.71
Kobayashi et al.59	Cylindrical
	Prismatic	to 0.24
Kobayashi et al.60	Coin 1	to 0.05
	Coin 2	0.01 to 0.06
	Coin 3	0.00 to 0.05
	Cylindrical	to 0.32
Lu and Prakash61	Coin		to 0.03	to 0.07	0.00 to 0.31
Lu et al.62	Coin		to 0.00
Lu et al.63	Coin 1		0.00 to 0.08	0.91 to 0.20	0.13 to 0.80
	Coin 2		to 0.03	0.00 to 0.08	0.00 to 0.37
	Coin 3		to 0.04	to 0.13	0.00 to 0.59
Lu et al.64	Coin 1		0.00 to 0.02
	Coin 2		0.00 to 0.01
	Coin 3		0.00 to 0.02
Onda et al.65	Cylindrical		to 11.0	to 27.5 4.12 to 44.0	13.3 to 84.5
Saito et al.66	Cylindrical		0.79 to 3.90
Saito et al.67	Cylindrical 1	0.08 to 1.80
	Cylindrical 2	to 2.04
	Cylindrical 3	to2.47
	Cylindrical 4	0.11 to 1.95
Song and Evans68	Coin				1.42 to 9.05 2.38 to 16.4	5.00 to 31.4
Thomas and Newman49	Coin		to 2.38	0.00 to 7.85		13.1 to 16.6
Yang and Prakash69	Coin 1	0.00 to 0.02	0.00 to 0.06	0.00 to 0.07	0.00 to 0.40
	Coin 2	to 0.04	to 0.07	to 0.15	0.00 to 0.61
	Coin 3	to 0.04	to 0.09	to 0.20	0.00 to 0.62
Vaidyanathan et al.70	Cylindrical 1		0.00 to 2.67		0.00 to 11.2 0.00 to 22.0
	Cylindrical 2			0.00 to 10.1	0.00 to 17.0 0.00 to 40.1

Tables VII, VIII and IX show a summary of the battery tests and the measured volumetric heat generation rates for charge and discharge, respectively, as a function of rate for the reviewed studies. The most important term for thermal simulation of batteries is the volumetric heat generation rate. However, because no investigations actually report the heat rate on a volume basis, cross-study comparison requires significant manipulation of the reported data. This includes estimating the heat rate from the total cell volume,^{54–56, 59–68, 70} from the thickness of the unit cell⁴⁹ or the entire battery,⁶⁹ or from standard densities of full cells.^{57, 58} Unfortunately, these assumptions can greatly influence the estimated volumetric heat generation rate. For example, Yang and Prakash⁶⁹ reported the heat generation rate per unit separator area for a 2032 coin cell (i.e., outside diameter (OD) and height). If one uses the total cell thickness , the heat generation rate for the discharge ranges from . In reality, the unit cell may be a factor of 10 (or more) smaller, thus changing the results to . Furthermore, the values reported contain wide variability from study to study, even for the same chemistry. For example, Onda et al.⁶⁵ and Al Hallaj et al.⁵⁵ gathered data on the same commercial cylindrical cell (Sony 18650, ) using two different techniques: IHC and ARC, respectively. Onda and co-workers reported heat generation rates ranging from for a discharge rate, whereas Al Hallaj and co-workers reported for a discharge rate.

Cross-comparison of chemistry, rates, and measurement techniques is a particularly difficult task for the following reasons: large discrepancies appear from study to study (even using the same batteries), and no investigation reports volumetric heat generation rate directly. Nevertheless, a few general statements about the previously reported results can be made. First, the heat rates are a strong function of the discharge rate. For example, in the study of Onda et al.⁶⁵ the heat rate ranges from and from for the and discharge rates, respectively. The negative value reflects the impact of entropic heating, which is discussed below. Second, there is an extremely wide range of reported peak values at the same discharge rate. For example, peak rates ranging from are reported for the discharge. Without knowledge of the geometric details or the uncertainty in the reported values, it cannot be known if these differences are real or artifacts of testing method and/or data reduction choices. Third, the heat rate increases as the battery is discharged further for moderately high discharge rates, but the profile differs between the studies. For example, the studies by Al Hallaj and co-workers^54–56 show that the heat rates increases monotonically as the battery is discharged. However, the studies of Bang et al. ,⁵⁷ Kim et al. ,⁵⁸ and Onda et al.⁶⁵ show that the heat rate follows an S-shaped curve as the battery is discharged. The profiles appear to be dependent on the relative magnitude of the entropic and overpotential heats, which is discussed in more detail below. Finally, the majority of studies has been conducted at one temperature (near nominal ambient: ), and few studies have investigated the impact of temperature on heat generation. For those that have investigated temperature influences, the range of temperatures is typically small. The studies of Saito et al.⁶⁶ and Kobayashi et al.⁵⁹ have the largest temperature ranges ( and , respectively), but they only tested batteries discharged at low current . Thomas and Newman⁴⁹ and Hong et al.⁵⁶ tested higher rates, but with a more narrow temperature spread ( and , respectively). In both of these studies, no appreciable difference in heat generation was observed for the temperatures tested. However, it is expected that as the test temperature range broadens, the heat generation rate will be seen as a strong function of temperature as the discharge rate increases. As discussed below, the cell overpotential generally decreases with cell temperature due to improved mass transport and kinetics, which should reduce its heat generation rate. However, at low rates, the reversible heat can be relatively large, and it is not a strong function of temperature. Thus, an investigation that measures heat generation rate over a wider range of test temperatures at higher rates is warranted.

As stated above, there have been many investigations that measure the entropic heat generation. In general, a variety of conclusions have been drawn about the importance of including entropic heat in the energy balance in thermal models. In this section, the different techniques used to estimate the entropic heat coefficient are reviewed, and the magnitudes for several relevant lithium-ion battery chemistries are analyzed.

Several methods are used to estimate the entropic heat coefficient (i.e., ). The most common method is to measure the OCP variation with temperature at a fixed SOC.^{54–57, 61, 62, 65, 69, 71, 72} This is the most straightforward and easiest method to implement, but it does have a few minor drawbacks that have led other investigators to explore other methods. First, long times are required to ensure that the OCP is steady. For example, Onda and co-workers^{65, 71} waited at each SOC tested (e.g., six test points in Ref. ⁶⁵) at four different test temperatures (10, 20, 30, and ). This results in at least of required testing for one battery. Yang and Prakash,⁶⁹ Lu et al. ,⁶³ and Lu and Prakash⁶¹ considered the voltage stabilized within , but the relaxation time was still approximately .⁶⁹ Furthermore, self-discharge can be important at these long wait times, especially at a high SOC. Lu et al. ,⁶⁴ Hong et al. ,⁵⁶ and Onda et al.⁷¹ all had difficulties maintaining stable voltages for a high SOC. Subsequently, Thomas et al.⁷³ developed another technique to minimize the effect of self-discharge. In their study, they discharged a fully charged cell to a specific SOC, followed by a charge/discharge cycle at a very slow rate with rest periods in between ( after the charge and after the discharge). During the rest period, the temperature of the cell was varied between 21 and at a ramp rate of approximately using a water bath. The slope of the voltage versus temperature curve (and, hence, the entropic heat coefficient) was estimated during the rest period. The process is repeated for multiple SOCs. Even for this short duration test, the problem of self-discharge for this cell (doped and undoped positive electrodes) was evident for OCP's greater than . To compensate for self-discharge, they subtracted a voltage profile for a battery held at constant temperature from the variable temperature data, which yielded a linear relationship between voltage and temperature.

The other methods are calorimeter-based. First, several authors assumed that irreversible heat remained constant upon charge and discharge. Hence, they subtracted the charge calorimeter data from the discharge data, which cancels the overpotential heat, and allows for the entropic heat to be determined as follows

Thomas et al.⁷³ showed that this method provided similar results to their SOC cycling method described above. Onda et al.⁶⁵ also showed that this method produced results similar to those of the direct measurement of OCP versus temperature. However, Hong et al.⁵⁶ observed that this method produced entropic heat coefficients that were a function of rate, which may be attributable to inaccuracies in their measurement technique. Another calorimetric method is to subtract an estimated irreversible heat from the total heat. The overpotential heat is typically (and most accurately) estimated by direct calculation of the overpotential using OCP and operating voltage data.^61–64

Table X summarizes previous investigations that estimate entropic heat for electrodes used in lithium-ion batteries. Each material investigated is also listed, and half cells mean that lithium metal is the counter electrode. The data show that the entropic heat coefficients of each material tested lie within and that the values can switch from negative (exothermic on discharge) to positive (endothermic on discharge) as the electrode is cycled from 0 to 100% SOC. For the positive electrode materials (i.e., the metal oxides), the switch from negative to positive can coincide with crystalline phase changes or different lithium insertion/deinsertion staging. For example, Al Hallaj et al.⁵⁴ used the phase diagram of Reimers and Dahn⁷⁴ for to show that the increase–decrease–increase in a surface temperature trace of a Panasonic 18650 cell was caused by the crystalline phase change from hexagonal to monoclinic to hexagonal. In addition, the order at which the sublattice in is filled with lithium ions is dependent on the SOC, which affects the observed heat: endothermic when nearly empty and exothermic when the nearest sublattice is full for discharging.⁷³ Doping agents can affect the nature of the order/disorder in various materials.^{62, 64, 73} In the carbon negative electrode, the type of carbon can affect the entropic heat coefficient. For example, the full cells with coke-based carbon negative electrodes tested by Hong et al.⁵⁶ have primarily negative coefficients whereas other investigations with the same positive electrode material but different types of negative electrodes have coefficients that can switch from positive to negative. Lu and Prakash⁶¹ and Thomas and Newman⁷² have both suggested that the entropic coefficient for the MCMB's negative electrode was mostly negative but became positive at the end of discharge due to the different lithium staging.

Table X. Summary of entropic heat test methods.

Study	Material	Capacity	Entropic Heat
			Methodology	(mV/K)
Al Hallaj et al.54	Half cell:		Measure directly	to 0.48
Al Hallaj et al.55	Full cell: (graphite)		Measure directly	to 0.23
	Full cell: (graphitized carbon fiber)			to
	Full cell: (graphitized carbon fiber)			to
Hong et al.56	Full cell: (hard carbon)		(a) Measure directly (b) Subtract charge from discharge heat	(a) to (b) to
Bang et al.57	Half cell:	Not given	Measure directly	to 0.18
	Half cell:			to 0.05
Lu and Prakash61	Half cell: (MCMB)	Not given	(a) Measure directly (b) Subtract overpotential heat	(a) to 0.28 (b) to 0.30
Lu et al.62	Half cell:		(a) Measure directly (b) Subtract overpotential heat	(a) to 0.06 (b) to
Lu et al.63	Half cell:	Not given	Subtract overpotential heat	to 0.02
	Half cell: (MCMB)			to 0.28
	Full cell: (MCMB)			to 0.20
Lu et al.64	Half cell:	Not given	Subtract overpotential heat	to 0.11
	Half cell:			to 0.13
	Half cell: cells			to 0.08
Onda et al.65	Full cell: (hard carbon)		(a) Measure directly (b) Substract charge from discharge heat	(a) to (b) to
	Full cell: (graphite)			(a) to 0.21
Onda et al.71	Full cell:		Measure directly	to 0.02
Thomas and Newman49	Half cell:	Not given	(a) Measure directly (b) Substract charge from discharge heat	(a) to 0.10 (b) to 0.10
	Half cell:			(a) to 0.25
Yang and Prakash69	Half cell:		Measure	to 0.02
	Half cell: (natural graphite, MAG-10)			to 0.55
	Full cell: (natural graphite Mag-10)			to 0.18

Table XI presents a comparison between heat generation from overpotential at the rate and the entropic heat at a nominal ambient temperature . This is accomplished by comparing the overpotential directly with the effective potential difference from entropic heat effects (i.e., ). Negative values reported for the entropic heat denote exothermic on discharge and endothermic on charge, but positive values for the resistance indicate exothermic heat generation. Hence, the negative of the entropic heat is presented in Table XI. The results show that most of the heat at the rate is due to overpotential, but the entropic heat is significant, although it is difficult to assess the impact of the entropic heat using only the data in the table. This is because the overpotential increases steadily throughout most of the charge/discharge followed by a dramatic increase at the end, whereas the entropic heat may be a maximum (or minimum) at any SOC. On the other hand, battery surface temperature traces can be used to qualitatively assess the impact of entropic heating, and the data show evidence that the entropic heat is significant, even at high discharge rates. For example, the data presented by Al Hallaj et al.⁵⁴ showed that the Panasonic cell temperature did not increase steadily but showed oscillations during the beginning of discharge even for rates as high as a . This clearly coincided with the cooling effect observed from the crystalline phase change. Similarly, the temperature trace for the battery tested by Onda et al.⁶⁵ showed a distinct oscillatory temperature profile during the beginning of discharge. In addition, the temperature profile presented by Thomas and Newman⁴⁹ appeared to be a strong function of the entropic heat for the rate, and the data on the spinel presented by Kim et al.⁵⁸ also showed a strong influence of the entropic heat for the rate. In contrast, the impact of the entropic heat can also be much less. For instance, the heat rate profiles shown by Lu and Prakash⁶¹ appear to be impacted by the entropic heat for the and rates, but there is almost no effect on the rate data, except that the heat rates for charge and discharge appear to be slightly offset. Similarly, Yang and Prakash⁶⁹ observed that the entropic heat for their full cell contributed to only 5% of the total heat generation.

Table XI. Comparison of overpotential and effective entropic overpotential at .

Study	Comparison at discharge at
	(mV)	(mV)
Al Hallaj et al.54	Not calculated	to 26
Al Hallaj et al.55	381 to 1560	to 225
	311 to 756	48 to 294
	276 to 822
Hong et al.56	(a) 311 to 581 (b) 284 to 513 (c) 284 to 446	(a) 129 to 225 (b) 126 to 252
Bang et al.57	Current not given	to 132
		to 153
Lu and Prakash61	Current not given	(a) to 30 (b) to 48
Lu et al.62	(a) 31.6 to 191 (b) 52.2 to 191	(a) to 39 (b) 24 to 45
Lu et al.63	Current not given	to 24
		to 30
		to 90
Lu et al.64	235 to 469	to 66
	213 to 226	to 38
	126 to 150	to 30
Onda et al.65	(a) 189 to 432 (b) 230 to 378 (c) 203 to 378 (d) 203 to 284	(a) 57 to 156 (b) 24 to 198
	(a) 450 to 1098 (b) 594 to 990 (c) 270 to 810 (d) 216 to 270	(a) to 243
Onda et al.71	(a) Charge: 252 to 504  Discharge: 306 to 576 (b) Charge: 252 to 306  Discharge: 288 to 630 (c) Charge: 270 to 288  Discharge: 270 to 504 (d) Charge: 162 to 252  Discharge: 180 to 252	to 197
Thomas et al.73	Overpotential not given	(a) to 120 (b) to 66
		(a) to 105
Yang and Prakash69	52.8 to 1050	to 24
	94.9 to 338	to 78
	(a) 37.9 to 471 (b) 94.7 to 412 (c) 28.4 to 128 (d) 150 to 353	to 114

This section describes some conclusions and observations drawn from previous investigations on experimental methods used to measure heat generation. First, heat is measured primarily at the outer surface of the battery. For the ARC method, this is complicated by the need to estimate the amount of heat stored. Moreover, substantial instrument time lags exist for all of the reported measurement techniques. Both of these drawbacks may have contributed to the fact that very few studies measure heat generation at high rates (i.e., above ). Furthermore, increased rates will cause concentration gradients to develop, which can change the local heat generation rate at high discharge rates. This can be observed in the heat rate measurements of Al Hallaj et al. ,⁵⁵ Kim et al. ,⁵⁸ and Hong et al. ,⁵⁶ which show significant heat rejection after the current is stopped. Similarly, Thomas and Newman⁴⁹ observed an overprediction by their model of the experimental data for short-duration pulse discharge at , which they attributed to neglecting heat of mixing within the solid particles, but this was attributed to large particle sizes that probably will not be utilized in commercial EV and HEV battery applications. Furthermore, their analysis showed that the heat of mixing is a small fraction at lower discharge rates. Thus, this observed effect in other studies may simply be an artifact of the testing procedure (i.e., cell cooling). Moreover, the IHC test methods may not be capable of maintaining isothermal conditions during testing due to limited surface heat rejection, which was implied by Lu and Prakash.⁶¹ Also, all of the above conclusions are complicated by the fact that only one investigation has reported the experimental uncertainty for the testing methods (Thomas and Newman: with a offset). Finally, all of the reported data in the open literature are for relatively small cells (which may not have substantial Joule heating in the current collectors) that will not be used in HEV and EV applications. Clearly, there is an opportunity to develop an improved heat collection method to discern the relative impact of mixing, entropic, overpotential, and current collector resistance heat and to measure heat rates above the nominal discharge/charge rates, which will be important for designing thermal management systems for lithium-ion battery packs in EVs and HEVs.

From a review of the experimental data, entropic heat should not be neglected in simulation. As shown in Table XI, the product of temperature and the entropic heat coefficient appears to be of the same order of magnitude as the actual overpotential at the moderately high rate. This has also been shown in a recently published review paper to be true across a variety of cell chemistries.⁷⁵ In addition, some studies show that the heat rate profile has an S-shaped curve even at the rate, which can be attributed to entropic heating (e.g., Bang et al.⁵⁷ and Kim et al.⁵⁸).

The measured overall heat generation for these batteries is not large (at most a peak of for the rate at the end of discharge) in comparison to the power delivered ( in this case,⁶⁵ with the assumption of a nominal voltage of ). This emphasizes the fact that heat rejection from lithium-ion batteries is not primarily a heat flux issue: it is a low conductivity, high thermal mass problem. In other words, a significant amount of heat is accumulated inside the battery for small temperature rises, but this heat is not easily transported to the surface due to the low effective radial thermal conductivity of the battery. This is clearly demonstrated by Hong et al. ,⁵⁶ who noted that the measured thermal energy accumulated inside the cell was approximately equal to one half the estimated heat generated when discharged at the rate.

In experimental studies, overpotential and entropic heat coefficients gathered from experiments are used to predict the volumetric heat generation rate. As shown in Table XIII, all reviewed simulation studies use the simplified form to predict the heat rate (Eq. 3), which is inserted into the heat equation on a per unit volume basis as follows

The first term is the heat stored by the battery, followed by the heat conduction and generation terms. In each study, the battery is divided into multiple cells. Using appropriate boundary and initial conditions, the heat equation is applied to these cells and solved to determine the temperature distribution throughout the battery. The only means of coupling the thermal field to the electrochemical heat generation is through the entropic heat, which is the product of temperature and the entropic heat coefficient.

Table XIII. Summary of experimental thermal simulation studies.

Investigation	Heat generation		Cell voltage and OCP	Temperature- dependent parameters
	Equation	Temperature dependency
Al Hallaj et al.28	Equation 3	Entropic only	Experimental data in paper	None
Chen et al.84	Equation 3	Entropic only	Experimental data in paper	Surface convection
Onda et al.71	Equation 3	Entropic heat and current distribution	Experimental data in paper	Heat capacity, surface convection
Chen and Evans85	Equation 3	Entropic only	Hooper and North89	None
Chen and Evans86	Equation 3	Entropic only	: Hooper and North89 : Gauthier et al.90	None
Chen and Evans87	Equation 3	Entropic only	Gauthier et al.90	None
Chen and Evans30	Equation 3	Entropic only	Au and Sulkes91 (discharge) Fuller et al.92 (OCP)	None
Chen et al.88	Equation 3	Entropic only	Experimental data in paper	Surface convection
Kim et al.50, 51	Equation 7	Entropic only	Curve fit to experimental data	None

The thermal simulation studies reviewed here all used direct measurement of the overpotential, and most assume constant and uniform current distribution throughout operation. However, caution must be exercised when collected cell voltage data are used to predict heat generation. For high discharge rates, the battery can heat up substantially, which can allow for a significant cell voltage increase due to the improved electrochemical performance. This can be seen clearly in the experimental data presented by Chen and co-workers^{84, 88} for discharge, which shows significant voltage recovery. In contrast, Onda et al.⁷¹ presented a simplified electrochemical model to predict current distribution along their spirally wound cylindrical cell (, ). They assumed that current flows only perpendicular to the wind direction and that heat is transported in the radial direction only. Their calculated results show that the temperature difference between the surface and center of the cell was only after of discharge at the rate. Their experimental results confirmed that this temperature difference was after 90% discharge. However, no evidence of current (or SOC) maldistribution was presented. In addition, there may also be substantial resistive heating that arises from current collection in a single battery. As discussed above, the work of Kim et al. ,^{50, 51} who used a simplified electrochemical parameterization using experimental data, has shown that this resistive heating may be large, even in well distributed batteries.

Detailed electrochemical models have been used to calculate local heat generation in response to temperature changes and vice versa. The studies reviewed here are summarized in Table XIV. A detailed description of electrochemical modeling is outside the scope of this document. An introduction to and details of electrochemical modeling can be found in the chapter on battery modeling in the textbook by Newman and Thomas-Alyea.⁹⁷ The electrochemical models of Doyle et al. ,⁹³ Fuller et al. ,⁹² Doyle et al. ,⁹⁶ and Smith and Wang⁷⁹ are similar. In each of these models, lithium-ion transport through the electrolyte is modeled using concentrated solution theory in which the driving force for mass transport is the gradient in the electrochemical potential.⁹² Furthermore, the porous insertion electrodes consist of spherical particles with diffusion of lithium ions into the solid. The primary equations in these models are species and charge balances in the electrolyte and solid particles. In addition, Butler–Volmer expressions are used to model the charge transfer kinetics at the solid/electrolyte interface. All of these models are one-dimensional (i.e., current flows only perpendicular to the current collectors), and the primary boundary conditions are as follows:

• No concentration or potential gradients in the electrolyte at the current collector/electrolyte interface
• Lithium concentration symmetry at the center of the spherical particles
• Proportionality between the lithium pore wall flux and the concentration gradient at the solid particle surface through the solid diffusion coefficient
• No potential gradients in the solid matrix at the separator/electrode interface⁷⁹
• Applied constant current discharge

Other expressions are required to close the system of equations: Faraday's law relating electrolyte current to pore wall flux,^{92, 93, 96} conservation of current density in the two phases,^{92, 93, 96} and values for the SEI resistance (included in Butler–Volmer kinetics expression).⁷⁹ These models allow determination of the local potential and reaction current density variation in the stack thickness for a fixed total current. The operation of the cell can be sensitive to temperature variations due to its influence on various transport properties in the battery. As shown in Table XV, the primary temperature-dependent transport properties are electrolyte ionic conductivity and diffusion coefficient and solid electronic conductivity and diffusion coefficient. Other possible temperature-dependent properties include the transference number⁷⁷ and the mean molar activity coefficient (affects the current–potential relationship).⁸²

Table XIV. Summary of thermal simulations studies using electrochemical models to predict heat generation.

Investigation	Electrochemical model	Heat generation		OCP	Operation type
		Equation	Temperature dependency
Pals and Newman76, 83	Doyle et al.93	Equation 3	Overpotential	Not given	Constant
Botte et al.77	Fuller et al.92	Rao and Newman Equation 4 plus negative electrode decomposition equation 17	Decomposition reaction rate	Dahn et al.94Doyle95	Constant
Thomas and Newman49	Doyle et al.93	Given formulation with or without mixing or reaction heat	Overpotential and entropic, but simulations are isothermal	Thomas et al.73	Constant
Srinivasan and Wang78	Doyle et al.96	Gu and Wang46	Overpotential and entropic	Doyle et al.96	Constant
Smith and Wang79	1D model in paper	Given formulation: integrate overpotential, ohmic heat in matrix and electrolyte, contact resistance at collector	Overpotential and ohmic	Equations in paper	Constant
Gomadam et al.81	Doyle et al.93	Gu and Wang46	Overpotential, ohmic, and entropic	Not given	Constant
Kumaresan et al.82	Doyle et al.93	Gu and Wang46	Overpotential, ohmic, and entropic	Measured in paper	Constant
Verbrugge29	Applied voltage use to solve for and fields	Equation 3	Overpotential, ohmic, and entropic	Data given in Paper	Constant
Song and Evans68	Doyle et al.93	Given formula in Paper includes activation/concentration overpotential in the electrolyte and positive electrode, ohmic drop across SEI, electrolyte, and electrode	Overpotential, ohmic, and entropic	Doyle et al.93	Constant

Table XV. Summary of temperature dependency in electrochemical model-based thermal simulation studies.

Investigation	Temperature-dependent parameters	Current/temperature feedback simulated?
Pals and Newman76, 83	Electrolyte ionic conductivity and diffusion coefficient	No—although transport properties are temperature dependent, feedback is on the potential only
Botte et al.77	Ion diffusion coefficients in the solvent, transference number, solid diffusion coefficients, ionic conductivity of electrolyte	No—transport properties are evaluated at only
Thomas and Newman49	None given	No—simulation was isothermal
Srinivasan and Wang78	All diffusion coefficients and conductivities	Yes—galvanostatic, but reaction distribution is temperature dependent
Smith and Wang79	Solid and electrolyte diffusion coefficients, electrolyte conductivity	No—simulations are isothermal
Gomadam et al.81	Same as Pals and Newman76	Yes—galvanostatic, but reaction distribution is temperature dependent
Kumaresan et al.82	Solid-phase diffusion coefficient for negative electrode, mean molar activity, coefficient of the salt, ionic conductivity and salt-diffusion coefficient of the electrolyte	No—temperature dependent properties cause voltage changes, not current distribution
Verbrugge29	Interfacial exchange current density coefficient, ionic conductivity of the electrolyte, thermal conductivity of the electrodes and electrolyte, heat capacity of electrodes	Yes
Song and Evans68	Ionic conductivity and diffusion coefficient of electrolyte	No—voltage profile was modified for fixed current density

In the model by Verbrugge,²⁹ current is generated in each subdivided battery section using a relationship between the applied cell voltage and the composite charge conductivity of the entire cell (see Eq. 1). The composite charge conductivity is calculated as follows

The ionic conductivities of both the electrode and the electrolyte are calculated as follows

The interfacial resistance is modeled using the exchange current density at each electrode, which is assumed to follow an approximate temperature-dependent relationship

The exchange current density multiplier is used as an adjustable parameter in the model. For his simulation, Verbrugge²⁹ solved the coupled electrochemical and thermal fields simultaneously for a fixed SOC and applied voltage for a bipolar battery stack. Portions of the cell that are more insulated from the environment rise in temperature, thus allowing more current to pass. This increases the local heat generation rate and thus further increases the temperature of these internal cells.

A description of the form of the heat generation equation for each of these studies is listed in Table XIV. Pals and Newman^{76, 83} and Verbrugge²⁹ apply the simplest form to calculate heat generation rate (Eq. 3) using the average cell overpotential, whereas the other methods take advantage of the detailed electrochemical information to use variations of the expressions presented by Gu and Wang,⁴⁶ Rao and Newman,⁴⁸ or Thomas and Newman.⁴⁹ Smith and Wang⁷⁹ integrate the local reaction and ohmic overpotentials to estimate the heat rate. They also introduce a contact resistance between the current collectors and the electrodes, which is modeled as follows

This resistance is an empirical parameter for which Smith and Wang⁷⁹ assume a constant value of , which dominates other heat generation mechanisms. Other parameters can also be included in the heat generation equation, including the temperature-dependent expression for heat generated due to side reactions (e.g., decomposition of the negative electrode⁷⁷)

In all cases, the temperature dependency of the electrochemical performance is directly coupled to the thermal model through these expressions for the volumetric heat generation, which represents a significant difference from the majority of experimentally based thermal simulations. However, only a few of these studies (Srinivasan and Wang,⁷⁸ Gomadam et al. ,⁸¹ and Verbrugge²⁹) allow the current distribution inside a battery to vary with temperature (see Table XV).

The thermal properties of batteries exhibit interesting characteristics. The battery is a layered cell containing multiple materials with different thicknesses and thermal properties (see Fig. 2). As a result, the thermal conductivity of the battery is anisotropic. Heat flowing perpendicular to the wound direction must flow through each sheet in series, while heat flowing along the stack flows through each layer in parallel. Because the thickness of each layer is very small, significant computational time is consumed when attempting to resolve heat flowing in each layer. Instead, the battery unit cell is typically modeled as a uniform layer with anisotropic thermal conductivity (and uniform heat generation) as follows

Chen et al.⁸⁴ presented an example double-coated unit cell with double-coated current collectors. Using the reported data in Table XVI, the parallel and perpendicular thermal conductivities are 33.9 and , respectively. The substantially lower perpendicular thermal conductivity is primarily due to the electrolyte-soaked separator and the electrodes, which govern the heat transport from the battery. The heat capacity for the equivalent cell is volume-averaged as follows

Using the same data in Table XVI, the heat capacity for this lithium-ion cell is . This method has been shown to give results similar to those obtained from calculating the contribution from each layer separately.⁸⁵ In general, these bulk properties are assumed temperature independent in the simulation studies.

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Table XVI. Example unit cell dimensions and properties from Chen et al.⁸⁴

Layer		ρ	[J/(kg K)]	[W/(m K)]
Coated carbon negative electrode	55	1347	1437	1.04
Cu current collector	14	8933	385	398
Electrolyte soaked separator	30	1008	1978	0.334
Coated positive e lectrode	55	2328	1269	1.58
Al current collector	20	2702	903	238

One interesting note about the literature-reported values is that their appears to be only one study (Maleki et al.⁹⁸) that reports the technique used to measure the necessary thermal properties in sufficient detail. The remainder of investigations simply report the thermal properties used. In their study, Maleki et al. measured the composite thermal conductivity (in stack parallel and perpendicular directions) and specific heat of the negative and positive electrodes ( with thickness of 0.198 and , respectively) and of a three layer stack consisting of the negative electrode, plastic separator, and positive electrode both with and without an electrolyte [ in EC–DMC (dimethyl carbonate) in ] using a xenon flash technique. The tests were conducted at two thermodynamic open circuit potentials (2.45 and ) to elucidate any effects of SOC on the measured thermal properties. The thermal conductivity in the perpendicular to the stack was estimated to be irrespective of the SOC. However, there appears to be something incorrect with this measurement technique for estimating the thermal conductivity perpendicular to the stack, which is critical for designing thermal management systems. For example, the thermal conductivities at for the positive and negative electrodes are 2.49 and (possibly without an electrolyte), respectively, but the composite thermal conductivities for a negative electrode/separator/positive electrode assembly without and with an electrolyte are 2.36 and , respectively. Neither of these values is mathematically possible: just these two layers by themselves would produce an effective thermal conductivity of using Eq. 19. Furthermore, the thermal conductivity of a polypropylene separator should be between 0.15 and ,⁹⁹ and the electrolyte conductivity should be similar [e.g., the thermal conductivity of DMC is near (Ref. ¹⁰⁰)]. In addition, their stack assembly is incomplete for an actual battery because an additional separator is required for a double-side coated current collector (see Fig. 2). However, because the addition of the electrolyte does appear to have a significant effect on the measured conductivity, there must be significant thermal contact resistances between the separator and the electrodes and possibly between the electrode material and the current collectors. This investigation was done with special fixtures that ensured adequate contact pressure during the tests. In actual cells, the contact pressure may not be the same, and further investigation is warranted. In addition, there have been no reported values for the positive electrode (although recently Guo et al.¹⁰¹ cite , but no explicit reference was given), which is known to be a poor electrical conductor requiring special processing techniques for use in commercial batteries.¹⁰² Thus, there appears to be a significant need to understand the thermal conductivity of assembled cells.

This section summarizes observations and conclusions made from the thermal simulation studies on batteries with lithium-based insertion electrodes presented above. Because these models provide insights into battery performance, the focus here is on implications for measuring battery heat generation and impacts on thermal management strategies.

The investigations that calculate heat generation using experimental polarization data use only one temperature during their experiment. However, as shown in the simulation studies that use electrochemical models, that assumption may not hold while the battery is operating because at higher temperatures, the battery operates at a lower polarization due to increased efficiency. For example, in their single-cell study on a titanium disulfide polymer electrolyte cell, Pals and Newman⁸³ showed that the polarization curves are a strong function of temperature. As the temperature increases, both the ionic conductivity and the diffusion coefficient in the electrolyte increase and improve the lithium-ion mass transport, thus allowing the cell to operate more efficiently, generating less heat. At high discharge rates ( and higher for a initial battery temperature) and adiabatic operation, their cell potential reached a local minima near the beginning of discharge because increased cell temperatures offset cell polarization. For the highest discharge rate studied , the cell potential drops almost to the cutoff potential prior to rebounding. This explains the unusual voltage-SOC data for 3C discharge reported by Chen et al.⁸⁴ For discharge with heat transfer, Pals and Newman showed that the temperature decreased as the rate of convection increased. This causes the cell potential to decrease and, therefore, heat generation rate to increase. Similarly, Srinivasan and Wang⁷⁸ showed that for a system, the rate of heat generation is higher at lower discharge rates for an isothermal cell than for a cell that accumulates heat adiabatically or with heat transfer to the environment. Again, this is because the cell operated adiabatically or with heat rejection undergoes an increase in temperature as it discharges, thus operating more efficiently than for isothermal operation. Because it is a strong function of temperature, more studies measuring the heat generation rate should be conducted. Furthermore, these temperature-dependent measurements should be incorporated into the simplified simulations that use experimental data.

There also appears to be disagreement between simulated and experimental results, especially at higher discharge rates. Al Hallaj et al.²⁸ reported that their experimentally based predictions on a 18650 cell simulate the cell surface temperature for discharge rates below . However, at the rate, the temperature rise is not predicted well. Although essentially the same at the end, they underpredict the cell temperature by about when the cell is about halfway discharged. This is significant because the total temperature rise is . In addition, although they predicted that the cell would rise for a discharge, because they poorly predicted the rate, this estimate should be used with caution. Song and Evans⁶⁸ showed that their detailed electrochemical–thermal model did not predict high rate discharge heat generation for a coin cell: both path and magnitude are off considerably, as much as 19% underpredicted at the end of discharge. In the study summarized by Gomadam et al.⁸¹ on the 18650 cell, the overall trend of temperature for discharge rates ranging from is predicted reasonably well but the shape of the simulated curve is different from the collected experimental data. In the study by Thomas and Newman⁴⁹ on a coin cell, the complex heat profile, which is very dependent on the entropic heat, is captured fairly well for the and discharge and charge rates using their detailed isothermal electrochemical model, even though there is as much as 50% overprediction in some stages. In contrast, the discharge and charge heat generation is consistently overpredicted, by as much as 67%. In the study by Onda et al.⁷¹ on a 18650 cell, there are discrepancies in the surface temperature predictions, especially at the end of charge or discharge. For discharge, the and temperatures at 0.2 SOC are underpredicted by 2.8 and , respectively, while the temperature is overpredicted by at the same discharged state. (The temperature rises for the , , and discharge rates are 23.1, 43.8, and , respectively.) For charge, the rate data are underpredicted by (at a total temperature rise of ). However, the charge data are predicted very well, but this is probably an artifact of the short duration—only 0.1 is possible prior to reaching the cutoff voltage.

In the few studies where comparison is possible, the simulated heat generation rates also appear to be inconsistent with the measured rates. The following studies examined the Sony 18650 cell: Al Hallaj et al. ,²⁸ Chen and Evans,³⁰ Hong et al.⁵⁶ and Al Hallaj et al.⁵⁵ The simulation results for the discharge from Al Hallaj et al.²⁸ show that the volumetric heat generation rate varied from , with most of the data (i.e., from 0.1 to 0.7 DOD) falling in the range from . Similarly, the expected heat generation rates at a discharge rate for a similar cell reported in Chen and Evans³⁰ use only the overpotential data range from , with the majority of rates near . In contrast, the experimental studies of Al Hallaj et al.⁵⁵ and Hong et al.⁵⁶ show that for the same Sony 18650 cell, the heat rate increases linearly from and from , respectively. This difference in magnitude between the experimental and simulation studies can be explained if the volumes used by the reviewers were incorrectly estimated. However, this does not account for the significant difference in observed trends, even for the same battery brand, size, and chemistry. Additional comparisons across various studies could be made if the simulation studies reported volumetric heat generation rate (e.g., Srinivasan and Wang⁷⁸ and Botte et al.⁷⁷) and if a wider range of currents were tested for various experimental studies (e.g., Kobayashi et al.⁵⁹).

In summary, simulation results do not adequately predict the trend of heat generation reported by various experiments, which manifests itself in incorrectly predicting the temperature rise. Some of the discrepancies may be attributable to differences in the battery design and the simulations could be considered adequate, but the experimental methods mostly do not report enough key parameters to enable assessment, often do not adequately control relevant parameters while others are varied, do not address wide enough ranges of discharge rates, times, and other parameters, do not measure internal temperatures in batteries with acceptable spatial and temporal resolution, and do not report measurement uncertainties. Therefore, it is difficult to gauge the adequacy of the models on the one hand and also the accuracy of the measurements on the other hand.

It was stated above that internal portions of large batteries poorly transport heat to the cooling media. For low surface convection, this results in increased overall temperatures for the more thermally insulated internal portions of the battery. Furthermore, increasing the surface convection can mitigate the peak temperature rise, but does so at the expense of producing a substantial thermal gradient. For example, using a 2D simulation, Chen and Evans⁸⁶ showed that for discharge in a long by thick polymer battery, as the surface convection increased from , the peak temperature decreased from , but the thermal gradient increased from . Similar results were shown for their subsequent lithium-ion battery study:³⁰ an increase in temperature difference from for a surface convection increase from . The thermal gradients were greatest in the cell stack direction due to the low conductivity polymer electrolyte [e.g., ] and separator . Al Hallaj et al.²⁸ also obtained similar results on a 18650 lithium-ion cell discharge at the rate. For an applied surface convection of , the peak temperature of the battery rose more than at the end of discharge, but with almost no difference inside the battery. In contrast, the battery peak temperature rose almost for an applied surface convection of , while a thermal gradient was induced.

As mentioned above, the cell operating temperature affects the performance of the cell by improving its electrochemical performance. Parts of batteries in closer thermal proximity to the cooling environment will generate more heat than insulated interior parts of the battery when current is uniformly distributed across the battery. As a result, the colder cells that generate more heat flatten the temperature gradient inside the battery. This is clearly shown by the works of Pals and Newman⁷⁶ and Song and Evans.⁶⁸ Pals and Newman simulated the temperature response of a multicell prismatic lithium–titanium disulfide polymer electrolyte battery using time–temperature lookup tables for heat generation produced by their previous single-cell isothermal model.⁸³ This allowed for each cell to generate heat, depending only on its temperature and discharged state (which was uniformly applied). For the 576 cell stack discharged at the rate, the temperature and heat generation was essentially uniform throughout the first of discharge. At the end of discharge, the heat generation rate at the cooler cells was more than 2 times higher than in the warmer interior cells. This phenomenon tended to flatten out the temperature profile. Song and Evans⁶⁸ compared a coupled electrochemical–thermal model to a decoupled thermal model for a multicell polymer battery. The thermal model showed a roughly parabolic temperature profile at the end of a discharge with a temperature gradient. The coupled model showed a slight decrease in the temperature gradient to , but the interior cell temperature profile was flatter.

Colder cells will generate more heat than warmer cells for the same applied current. However, the cell performance increases at higher temperatures due to improved mass transport. For a larger battery, this allows additional current to pass through hotter sections of the battery, thus counteracting increased polarization for colder cells. This is clearly shown by the work of Verbrugge,²⁹ who presented a 3D coupled electrochemical–thermal mode for a lithium vanadium oxide polymer electrolyte bipolar battery. The battery contained 35 parallel modules, with each module containing 35 cells. Each cell in the module was in cross section, and the total depth of the module was . As mentioned above, the composite conductivity increases with temperature. Hence, the cooler cells allow less current to pass for the same voltage. For example, the locations at the center of the cell can be more than higher than at the cooled edges after of discharge at an applied voltage of for each cell ( was the OCP). This resulted in a doubling of the current passing through the center cells versus the surface cells ( vs ). In another study, Gomadam et al.⁸¹ summarized a previous investigation on lithium-ion 18650 cells and noted that for a cell cooled preferentially at the bottom opposite the cell terminals, more current passes through the top of the cell because the colder cells have become more resistive. However, as pointed out by Kim et al. ,^{50, 51} there appears to be a significant link between the current field and the local heat generation. In their simulation, the simplified electrochemical model had no temperature feedback. Thus, the current field was entirely dependent on the design of the current collectors. In light of the simulations by Verbrugge, Srinivasan and Wang, and Pals and Newman showing the coupled nature of temperature and local current generation, it would be best that their electrochemical model be modified to further elucidate the impact of temperature on the local cell performance. This is apparently what is being currently investigated at the NREL.^{52, 53}

Several investigations show that as the discharge rate increases, the cell temperature at the end of discharge reaches a maximum, followed by a subsequent decrease. With a fixed cutoff potential, the discharge time decreases faster than the heat generation rate as the applied current increases. Botte et al.⁷⁷ investigated the impact of increased discharge rate on the peak cell temperature for a lithium-ion cell using a detailed electrochemical model. For rates higher than , the time elapsed before the cell reached the cutoff potential decreased enough to limit the temperature rise. For example, the elapsed time for a discharge rate of was of that for , thus reducing the temperature rise from at to at . Therefore, as the rate was increased, the peak temperature achieved by the cell decreased despite an increase in the heating rate. Srinivasan and Wang⁷⁸ also showed that as the discharge rate increased, the total heat energy peaked and subsequently decreased at higher rates. The peak occurs because at high rates, cell utilization decreased. This was further substantiated by observing that the reversible heat, which was directly linked to the amount of reaction, decreased for higher discharge rates. For example, if the discharge rate increases from to , the cell utilization drops quickly from 80 to 36%. This caused a reduction in the total amount of energy generated from . However, if a battery is cycled repeatedly, as in an HEV application, without adequately removing the accumulated heat, the cell temperature can continue to rise. For example, Chen and Evans³⁰ simulated ten repeated cycles of constant current discharge and constant current, constant voltage charge (, unknown voltage) for a lithium-ion battery cube. The results showed that the temperature increased to a maximum of and the thermal gradient increased to .

From the simulation studies, it is clear that the thermal management strategies will significantly impact the performance of batteries. The negative feedback produced by colder cells generating more heat should tend to stabilize temperature maldistribution inside the battery. Conversely, the positive feedback produced by hotter cells generating more current (and thus heat) will exacerbate inherent temperature gradients generated from uneven cooling, which will be more pronounced for multiple cells connected in parallel instead of in series, where all current must pass through each cell. The more serious consequences of the current/temperature positive feedback are the increased chances of thermal runaway stemming from overcharge, overdischarge, or dangerously high temperatures (e.g., separator melting temperatures) resulting from repeated, rapid cycling or extreme environments. Thus, it is imperative that the cooling strategies cool the battery uniformly and mitigate thermal accumulation by decreasing the thermal resistance between the hotter cells of the battery and the cooling media.

Another observation is that the entropic heat is applied inconsistently to the thermal models. Table XVII shows the entropic heat coefficients used in the simulation studies reviewed here. In these studies, the entropic heat is neglected,^{76, 77, 79, 83} assumed constant,^{30, 68, 84–88} assumed linear between two SOC measurements,²⁸ a nonlinear function of SOC,^{49, 71, 78, 82} and not reported.^{29, 50, 51, 81} As the previous section on experimental methods for reversible heat measurement shows, the entropic heat can be of the same order of magnitude as the reversible heat, which is consistent with the reported values in the thermal simulation studies. (One exception is the study of Chen et al. ,⁸⁴ which reports an entropic heat coefficient of . This is probably a misprint since they referred to the study of Chen and Evans⁸⁵ for this value.) Smith and Wang⁷⁹ argued that for HEV applications, the battery cycles around a specific SOC and that the reversible nature of the entropic heat should cancel when cycled repeatedly. However, because the maximum entropic heat may not occur when the overpotential heat is at a maximum, the impact of the reversible heat is not straightforward. Thus, because the entropic heat is significant, even at moderately high charge and discharge rates, it is best to include entropic heating as a function of SOC in all thermal simulations.

Table XVII. Summary of entropic heat coefficients used in the reviewed thermal simulation studies.

Investigation	Entropic heat coefficient
Pals and Newman76, 83	Neglected
Botte et al.77	Neglected
Thomas and Newman49	Function of SOC in Thomas et al.73
Srinivasan and Wang78	Function of SOC using Thomas et al.73 and Al Hallaj et al.54
Smith and Wang79	Neglected
Gomadam et al.81	Not mentioned, but included in heat generation formulation
Kumaresan et al.82	Thomas and Newman72
Verbrugge29	Not given, but could be determined from data in Paper
Song and Evans68
Al Hallaj et al.28	Linear fit between 4.044 and:
Chen et al.84
Onda et al.71	Function of SOC:
Chen and Evans85
Chen and Evans86
Chen and Evans87
Chen and Evans30
Chen et al.88
Kim et al.50, 51	Not given