1 s2.0 S1359431123017817 main

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Available online 7 October 2023

aHeat Transfer and RAC Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India

bHeat Transfer and Thermal Power Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600036, India

A R T I C L E I N F O

Keywords:

Artificial neural network Bayesian method Li-ion battery MH-MCMC

Orthotropic thermal properties Volumetric heat generation

A B S T R A C T

Accurate estimation of temperature-dependent orthotropic thermal properties and volumetric heat generation of a Li-ion battery is crucial for thermal modeling, thermal safety, and the design of thermal management systems for electric vehicles. Though various studies are available on estimating thermophysical properties and heat generation, simple and easily applicable methods are rare. Moreover, these studies failed to report temperature sensitivity and standard deviation of the estimates. In this study, the temperature-dependent orthotropic thermal conductivities (𝑘𝑟, 𝑘_𝜃, 𝑘_𝑧), specific heat (𝑐𝑝), and volumetric heat generation (𝑞𝑣) of Panasonic NCR18650BD cylindrical battery are estimated using an inverse approach (Metropolis Hastings- Markov Chain Monte Carlo algorithm based Bayesian method) with the help of experimentally obtained surface temperatures measured at convenient locations on the battery. From the estimation, the average values of 𝑘_𝑟, 𝑘_𝜃, 𝑘_𝑧and𝑐_𝑣are observed to be 3.18±0.19, 20.34±1.26, 19.89±1.29 (W/mK), and 3180±202 (J/kgK), respectively. The average heat generation rates from the same battery obtained using the same methodology are 0.1±0.005, 0.34±0.012, and 1.51±0.026 W for 0.5, 1, and 2C discharge rates, respectively. The estimated thermophysical properties and heat generation rates are in good agreement with the results obtained from both in-house experiments and literature. In addition to the estimation of thermophysical properties and heat generation, the proposed methodology opens vistas to predict the strength and location of hotspots in the battery domain, which helps in designing appropriate and effective thermal management systems for battery packs.

1. Introduction

Lithium-Ion Batteries (LIBs) have been widely used as energy storage and conversion systems in a wide range of applications, including Electric Vehicles (EVs), Hybrid Electric Vehicles (HEVs), consumer electronics such as cell phones and laptops, and aerospace systems [1].

LIBs enable high energy storage densities and faster conversion rates in energy storage and conversion applications compared to its competing technologies. In automobile applications, for example, EVs and HEVs, LIBs enhance fuel efficiency, reduce emission of greenhouse gases, and unfasten fossil fuels depletion rate [2–4]. At present, LIBs are the leading power source technologies in EVs and HEVs owing to their high energy density, high power, low rate of self-discharge, high nominal voltage, and long cycle life compared to other secondary battery technologies [5].

However, the advantages of LIBs and their safety are highly sen- sitive to their operating temperatures. Several studies show that the

∗ Corresponding author.

E-mail address: [email protected](S. Kumar).

1 Post-Doctoral Fellow.

optimal temperature range of LIBs is between 20 and 40 ^◦C [6,7].

Despite starting the battery operations (discharging) at room temperature, its operating temperature rises due to internal heat generation throughout the volume caused by ionic resistance and exothermic reactions. The temperatures beyond the mentioned limits adversely affect their performance and stability. For example, operating LIBs at higher temperatures enhance the risk of fire and explosion [8], and at lower temperatures and high discharge rates deteriorates its capacity due to lithium plating [9]. The limits on operating temperatures restrain bulk deployment of LIBs in EVs. In addition to the absolute temperatures, large thermal gradients in a pack affect battery performance. The maximum allowable thermal gradient in any battery pack is 5 ^◦C [10]. With these notes on the thermal limitations of LIBs, it is of great importance to design a Battery Thermal Man- agement System (BTMS) either by using active, or passive, or both cooling methodologies [11–13] to keep the operating temperatures of LIBs within the safety limits during all events of their operation.

https://doi.org/10.1016/j.applthermaleng.2023.121752

Received 12 June 2023; Received in revised form 6 September 2023; Accepted 4 October 2023

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Nomenclature

𝐴 Acceptance ratio

𝐶 Discharge rate

𝑐_𝑝 Specific heat capacity of the active material, J∕KgK

𝑐_𝑝,𝑚 Specific heat capacity of the mandrel, J∕KgK

𝑐_{𝑝,𝑡𝑏} Specific heat capacity of the tabs,J∕KgK 𝑐_{𝑝,𝑜𝑐} Specific heat capacity of the outer can,

J∕KgK

𝑐_{𝑝,𝑠𝑠} Specific heat capacity of the stainless steel, J∕KgK

𝐸 Estimates vector

𝑓 Function

ℎ Heat transfer coefficient,W∕m²K

𝑘_𝑟 Radial thermal conductivity of the active material,W∕mK

𝑘_𝜃 Angular thermal conductivity of the active material,W∕mK

𝑘_𝑧 Axial thermal conductivity of the active material,W∕mK

𝑘_𝑚 Thermal conductivity of the mandrel,

W∕mK

𝑘_𝑡𝑏 Thermal conductivity of the tabs,W∕mK 𝑘_𝑜𝑐 Thermal conductivity of the outer can,

W∕mK

𝑘_𝑠𝑠 Thermal conductivity of the stainless steel, W∕mK

𝑛 Number of locations of temperature measurements

𝑃 Probability

𝑞_𝑉 Volumetric heat generation,W∕m³

𝑅 Correlation coefficient

𝑅^′ Radius of the active material, mm

𝑅₀ Radius of the battery, mm

𝑟 Radial direction

𝑡 Time, s

𝑇 Temperature,^◦C

𝑣 Random number

𝑌 Unsteady measured temperature data,^◦C

𝑧 Axial direction

Greek letters

𝜇_𝑝 Mean of the prior

𝜎_𝑝 Standard deviation of the prior

𝜃 Angular direction

𝜌 Density of the active material,kg∕m³ 𝜌_𝑚 Density of the mandrel,kg∕m³ 𝜌_𝑡𝑏 Density of the tabs,kg∕m³ 𝜌_𝑜𝑐 Density of the outer can,kg∕m³ 𝜌_𝑠𝑠 Density of the stainless steel,kg∕m³ Subscripts

sim Simulated

meas Measured

𝑉 Volume

Abbreviation

ANN Artificial neural network

BTMS Battery thermal management system

EVs Electric vehicles

HGR Heat generation rate

LIB Lithium ion battery

LFP Lithium Ferro Phosphate

LTO Lithium Titanium Oxide

MH Metropolis Hastings

MCMC Markov Chain Monte Carlo

MAP Maximum a posteriori

MSMD Multi-Scale Multi-Domain

NMC Nickel Manganese Cobalt

PPDF Posterior probability density function

SD Standard deviation

TIS Thermal impedance spectroscopy

The traditional approach to design any BTMS is based on the surface temperature distribution of the battery for different discharge rates.

However, this method does not consider the thermal inertia of the battery, which decides the temperature distribution within its volume.

As a result, the BTMS design based on the surface temperature alone may not provide an acceptable quality of thermal protection [14].

Given this, the thermal modeling of LIBs, which elucidates the un- derlying mechanism for heat generation and temperature evolution in the battery domain, is essential for designing an effective BTMS.

Several numerical approaches have been developed in the literature for the thermal modeling of batteries. The electrochemical and lumped thermal models are two important and likely used thermal models [15].

Electrochemical and thermal models can accurately estimate the heat generation and temperature distribution of batteries; however, they are complex and time-consuming. Lumped thermal models are simple and cost-effective in computation time but with relatively low accuracy compared to electrochemical thermal models. The accuracy of the lumped capacity models can be improved by the correct estimation of the thermophysical properties of the battery material and heat generation under different operating conditions [16].

The LIBs are composed of three major components, namely, electrode layers, porous separator, and electrolyte, and they are rolled or folded into the required shape. As batteries are composed of het- erogeneous materials, their overall thermal conductivity and specific heat are often not well known in advance and not provided by LIB manufacturers. The morphology and chemical composition of electrode and separator layers and their packing within the battery shell influence thermal conductivity and specific heat. As a result, these properties change with changes in battery chemistries (e.g., LFP, NMC, LTO) and shapes (cylindrical, pouch, prismatic). In addition, these properties are often strong functions of the electrochemical state of the battery and temperature, and they vary significantly in space within the battery volume [15].

With the necessity of knowing the values of anisotropic thermal conductivity and specific heat, and their criticality in measuring or estimating, researchers worldwide have developed several methods in recent years. Shah et al. [17] provided an overview of techniques for measuring the thermal characteristics of a Li-ion cell. In Maleki et al.

[18], Nieto et al. [19], and Spinner et al. [20], the calorimeter method was used to measure the heat capacity of a Li-ion battery. However, the complex methodology and requirement of an expensive calorimeter make this method less preferable. A method of thermocouple insertion in the battery volume to measure thermal conductivity and specific heat capacity was used in the study by Forgez et al. [21]. Despite getting accurate results with this method, thermocouple insertion in the battery severely affects its functionality. The specific heat capacity and thermal conductivity of a Li-ion battery were suggested to be

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Fig. 1. Schematic view of the complete experimental setup for estimating the temperature-dependent orthotropic thermal properties of the active material of the Li-ion battery used in the present study.

measured using the thermal impedance spectroscopy (TIS) technique in Fleckenstein et al. [22]. This method uses the thermal impedance curves, which depict the cell temperature response to the excitation signal, to estimate the thermophysical properties. The major drawback of the TIS method is that the frequency of battery thermal response is very low (typically in the scale of mHz), and as a result, it takes longer times for estimation [23]. Using experimental temperature measurements, Murashko et al. [14] analytically calculated the through-plane thermal conductivity and average specific heat capacity of 18650 and 26650 cylindrical Li-ion batteries. In addition to the above methods, the anisotropic thermal conductivity and specific heat of the battery were also calculated by Zhang et al. [23] using optimization technique based on least-square regression and by Ruan et al. [16] using the reduced wide-temperature-range electro-thermal coupled model, and by other researchers in their studies [15,24].

In addtion to the studies on identifying thermophysical properties, researchers have also putforth different methodologies to estimate heat generation from Li-ion batteries. The Bernardi technique [25] is the most well-known theoretical approach for calculating the heat generation rate (HGR). The HGR in the Bernardi method is the result of adding reversible and irreversible heat. The irreversible heat caused by ohmic losses and the reversible heat due to entropy changes were directly determined by heat flow measurements using a novel experimental technique in Christen et al. [26]. The studies conducted in [27–30], and [31], an electrochemical model named Multi-Scale Multi-Domain (MSMD) approach is built to calculate the heat generation from battery for various discharge rates. However, these theoretical and electrochemical methods are not appropriate for actual applications due to the intricacies involved in calculating the parameters required to run the model. In other studies, Wang et al. [32] introduced an empirical heat source model, Cao et al. [33] and Sheng et al. [34] used calorimetric approach, Esmaeili and Jannesari [35] implemented artificial neural networks (ANNs), and Parhizi et al. [36] developed an analytical heat transfer model to estimate or calculate heat generation from the battery.

From the above literature, it is seen that many methods have been implemented to measure the specific heat capacity, thermal conductivity, and volumetric heat generation of Li-ion batteries, which are expensive and time-consuming. Moreover, the standard deviation of estimation, sensitivity of thermophysical properties and heat generation

for various operating conditions, and strength and location of hotspots in the battery have not been determined in the previous studies.

To fill the above-mentioned research gaps, the present study im- plements an effective inverse technique called the Bayesian inference- based Metropolis Hastings-Markov Chain Monte Carlo (MH-MCMC) method [37–42]. This inverse methodology combines measurements, modeling, probabilities, and sampling to estimate parameters in heat transfer problems with an additional advantage of reporting uncer- tainties in the parameters. In essence, the innovation in this study lies in developing a methodology that uses experimental temperatures measured at convenient locations to estimate temperature-dependent orthotropic thermal properties, variation in volumetric heat generation, and the occurrence of hot spots within a cylindrical Li-ion battery. An added benefit of this approach is its efficiency in terms of computational time coupled with its independence from the need for detailed electrochemical knowledge to predict heat generation. The identifi- cation of these temperature hot spots holds significant implications by providing valuable insights for the optimal design and effective implementation of thermal management systems within battery packs.

2. Experimental methodology

In the present study, the experimental methodology is divided into two parts: One is to estimate thermophysical properties and the other is to estimate volumetric heat generation of Panasonic NCR18650BD battery.

2.1. Experimental methodology for thermal characterization

Laminar natural conjugate convection experiments are conducted on the active material of the Panasonic NCR18650BD Li-ion battery to estimate the thermophysical properties.Fig. 1shows a schematic view of the complete experimental setup for estimating the temperature- dependent orthotropic thermal properties of the active material of the Li-ion battery. The setup consists of a cylindrical active material of the battery, hot fluid circulator (Julabo), chamber, hot plate, copper plate, stand, fixtures and fasteners, insulating material, T-type thermocouples, data logger, and computer. An interior view of the chamber considered in the experimental setup is explicitly shown inFig. 2. The cylindrical active material of the battery is placed on the top of the copper plate,

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Fig. 2. An image of the interior view of the chamber of the experimental setup used in the present study.

and both are clamped together on top of the hot aluminum plate using fixtures and fasteners. The hot plate is placed inside the chamber and fixed to a stand for support. This hot plate is integrated with cylindrical copper tubes for the supply of hot fluid. Five T-type thermocouples (uncertainty = ±0.25 ^◦C) are attached at different locations on the surface of the active material, and one K-type thermocouple (uncertainty =±0.25^◦C) is used to measure the hot plate temperature (see Fig. 3). The body of the active material is heated from its base by supplying hot fluid in the hot plate. The data logger is connected to thermocouples to read temperatures, and the temperature data is saved in a computer. Low emittance tape is pasted over the active material to eliminate the radiation heat transfer from its outer surface. As a result, only conduction and natural convection are present inside and on the outer surface of the active material of the battery, respectively.

The initial temperature of the system and the ambient temperature are the same and equal to 26± 1^◦C. The temperature of the hot fluid circulator is set at 38^◦C. The unsteady temperatures are measured on the outer curved surface of the active material (Fig. 4(a)) and at the hot plate (i.e., bottom surface of the active material) (Fig. 4(b)). The measured temperatures (𝑇₁, 𝑇₂, 𝑇₃, 𝑇₄, 𝑇₅, 𝑇₆) are used in the inverse model to estimate the temperature-dependent orthotropic thermal properties of the active material. From the in-house experiments, the temperatures measured using thermocouple𝑇₆ (hot plate temperature) are used as a boundary condition in the forward model. Full details of the use of forward model in the estimation of thermophysical properties of the active material are reported in the ensuing sections. The heat transfer coefficient (h), to be also used as a boundary condition on the outer curved surface of the active material in the forward model, is calculated using the average surface temperature measured from the temperature data obtained by thermocouples (𝑇₁–𝑇₅) and a correlation [43] for the Nusselt number. The calculated experimental heat transfer coefficient (h) is presented inFig. 4(c), which is used as a boundary condition to solve the forward model of the active material.

2.2. Experimental methodology for estimating volumetric heat generation A cylindrical Li-ion battery of model, Panasonic NCR18650BD, is considered and discharged at 0.5, 1, and 2C discharge rates using a battery charge–discharge cyclic tester (NEWARE, BTS-4000). Five T- type thermocouples are attached on its outer surface to measure the temperature variation during these discharge rates (see Fig. 5). For more details on the experimental testing of the same Li-ion battery, please refer to Ref. [44]. Heat generation is a byproduct of chemical to electrical energy conversion. Estimation of heat generation in the

Fig. 3.Locations of thermocouples for the temperature measurements.

battery is critical for effective thermal management. In view of this, an inverse heat transfer methodology is employed in the present study to estimate the volumetric heat generation in the Li-ion battery accurately.

For solving the inverse heat transfer problem, the experimental temperatures measured on the surface of the battery for various discharge rates of 0.5, 1, and 2C are used. The thermocouple locations for measuring temperatures on the surface of the battery are shown in Fig. 5(b).

The unsteady temperatures measured at the given five locations are shown inFigs. 6(a), (b), and (c) for discharge rates of 0.5, 1, and 2C, respectively. The ambient temperature of 22^◦Cis maintained during these experiments.

The heat transfer coefficient (ℎ_𝐷) on the surface of the battery is also calculated using the average temperatures obtained from thermo- couples𝑇₇–𝑇₁₁ and the Churchill-Chu correlation [43] of the Nusselt number variation for natural convection. The variation of the calculated heat transfer coefficient with time is shown inFig. 6(d). It is used as a boundary condition in the forward model simulations to obtain the simulated temperatures. It is to be noted that the forward model discussed in the previous section to estimate the thermophysical properties of the active material is different from the present forward model. In the previous forward model, only the base of the active material is heated with constant temperature boundary condition using a hot fluid circulator. Whereas, the forward model discussed in this

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Fig. 4. (a) and (b) variations of the unsteady temperatures measured on the side (𝑇₁, 𝑇₂, 𝑇₃, 𝑇₄, 𝑇₅) and bottom (𝑇₆) surface of active material, respectively (c) variation of the heat transfer coefficient on the surface of the active material.

Fig. 5. (a) An image of the experimental setup and (b) locations of the thermocouples for temperature measurements to estimate the volumetric heat generation in the battery.

section has a full Li-ion battery, which is discharged using a battery charge–discharge cyclic tester. More details about both the forward models are presented in the ensuing sections. The measured unsteady temperatures are used in the inverse model to estimate the volumetric heat generation variation in the battery.

Experiments performed to estimate the thermophysical properties and heat generation are conducted twice to ensure repeatability. From the repeatability test, the highest temperature deviation at any location and at any time instant is less than 0.5^◦C. As mentioned earlier, this study combines experimentation, modeling, and inverse methodology principles to estimate the temperature-dependent orthotropic thermal properties of the active material of the Li-ion battery and the volumetric heat generation for 0.5, 1 and 2C discharge rates. A detailed flow chart of the entire study is shown inFig. 7.

3. Forward model

The forward model includes the governing equations of the problem along with the respective boundary conditions. The solution of the forward model gives the simulated temperatures for various values of the unknown parameters. The simulated data corresponding to the multiple values of the unknown parameters are used in an artificial

neural network to develop the surrogate model, which is implemented in the inverse methodology to estimate the unknown parameters.

3.1. Forward model for thermal characterization

Fig. 8shows the three-dimensional geometry of the active material of the battery for the unsteady-state conduction problem. The governing equation for the above is presented in Eq.(1). The boundary conditions, shown inTable 1, are used to solve the following governing equation in COMSOL [45].

1 𝑟

𝜕

𝜕𝑟 (

𝑘_𝑟(𝑇)𝑟𝜕𝑇

𝜕𝑟 )

+ 1 𝑟²

𝜕

𝜕𝜃 (

𝑘_𝜃(𝑇)𝜕𝑇

𝜕𝜃 )

+ 𝜕

𝜕𝑧 (

𝑘_𝑧(𝑇)𝜕𝑇

𝜕𝑧 )

=𝜌𝑐_𝑝𝜕𝑇

𝜕𝑡 (1) Grid independence study is conducted on the computation domain by comparing the variation of simulated temperature–time histories at location (8.7, 0, 40 (all in mm)) of the forward model for extra coarse, fine, and extra fine mesh types. The number of mesh elements generated for extra coarse, fine, and extra fine mesh types are 2197, 56408, and 256385, respectively. The thermophysical properties 𝑘_𝑟 = 2.5 W/mK,𝑘_𝜃= 16W/mK,𝑘_𝑧= 16W/mK, and𝑐_𝑝= 2600J/kgK are taken for simulations. The temperature–time history for the computational meshes with fine and extra-fine are very close, which can be clearly

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Fig. 6. Variations of unsteady temperatures measured at five locations (𝑇₇, 𝑇₈, 𝑇₉, 𝑇₁₀, 𝑇₁₁) on the side surface of a Li-ion battery for the discharge rates of (a) 0.5C (b) 1C, and (c) 2C and (d) an experimental heat transfer coefficient variation for discharge rates of 0.5, 1, and 2C.

Table 1

Boundary conditions for solving the governing equations of the active material of the battery.

Boundary Condition Description

Outer side surface of the battery (𝑟= 8.7, 0≤𝜃≤360,0≤𝑧≤60)

−𝑘𝑟(𝑇)^𝜕𝑇_𝜕𝑟 =ℎ(𝑇𝑠−𝑇_∞),

−𝑘𝜃(𝑇)^𝜕𝑇

𝑅^′𝜕𝜃=ℎ(𝑇𝑠−𝑇_∞),

−𝑘_𝑧(𝑇)^𝜕𝑇

𝜕𝑧=ℎ(𝑇_𝑠−𝑇_∞)

Natural convection

Bottom surface

(𝑧= 0, 0≤𝜃≤360,0≤𝑟≤8.7)

𝑇(𝑟, 𝜃, 𝑧, 𝑡) =𝑇_𝑏(𝑡) Constant base temperature Top surface

(𝑧= 60, 0≤𝜃≤360,0≤𝑟≤8.7)

𝜕𝑇

𝜕𝑧= 0 Insulated

Inner surface of the mandrel (𝑟= 1, 0≤𝜃≤360,0≤𝑟≤8.7)

𝜕𝑇

𝜕𝑟 = 0,^𝜕𝑇

𝜕𝜃= 0,^𝜕𝑇

seen inFig. 9. Based on the grid independence tests, 56408 fine-size elements are employed for simulations.

Simulations are conducted on the active material of the battery using forward model for the ranges of unknown orthotropic thermal properties (radial thermal conductivity (𝑘_𝑟), angular thermal conductivity (𝑘_𝜃), axial thermal conductivity (𝑘_𝑧), specific heat capacity (𝑐_𝑝)). The ranges of the unknown orthotropic thermal properties are found from literature [30,46–50], and the same are shown in Table 2. For these ranges, the simulated temperatures are obtained at each time interval of 100 s. The total time intervals and samples of the unknown parameters (𝑘_𝑟, 𝑘_𝜃, 𝑘_𝑧, 𝑐_𝑝) are 12 and 201, respectively. The simulated temperatures obtained for the given ranges of the unknown parameters at each time interval are used to develop the surrogate model.

Table 2

Ranges of the unknown parameters (𝑘𝑟,𝑘_𝜃,𝑘_𝑧,𝑐_𝑝) at each time interval to obtain the simulated temperatures.

Time (t), s Ranges of unknown parameters

at each time interval

100≤𝑡≤1 200 Time step (𝛥t) = 100

0.5≤𝑘_𝑟≤5 (W/mK), 2≤𝑘_𝜃≤32 (W/mK), 2≤𝑘_𝑧≤32 (W/mK), 600≤𝑐_𝑝≤5 000 (J/kgK) (samples = 201)

3.2. Forward model for estimating the volumetric heat generation

Fig. 10shows the three-dimensional geometry of the cylindrical Li- ion battery considered in the forward model to estimate the volumetric heat generation (𝑞_𝑉). The governing equations of the active material, mandrel, positive and negative tabs, and outer can of the Li-ion battery are presented in Eqs.(2),(3),(4), and(5), respectively. In the governing equation (shown in Eq.(2)) of the active material, the orthotropic thermal properties (𝑘_𝑟(𝑇), 𝑘_𝜃(𝑇), 𝑘_𝑧(𝑇), 𝑐_𝑝(𝑇)) are temperature dependent.

The values of these properties are estimated first, hence, the volumetric heat generation (𝑞_𝑉) is the only unknown term in Eq.(2).Table 3shows the boundary conditions used to solve the above governing equations in COMSOL [45].

Active material of the Li-ion battery:

1 𝑟

𝜕

𝜕𝑟 (

𝑘_𝑟(𝑇)𝑟𝜕𝑇

𝜕𝑟 )

+1 𝑟²

𝜕

𝜕𝜃 (

𝑘_𝜃(𝑇)𝜕𝑇

𝜕𝜃 )

+ 𝜕

𝜕𝑧 (

𝑘_𝑧(𝑇)𝜕𝑇

𝜕𝑧 )

+𝑞_𝑉 =𝜌𝑐_𝑝𝜕𝑇

𝜕𝑡 (2)

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Fig. 7. A flow chart of the entire study.

Mandrel of the battery:

1 𝑟

𝜕

𝜕𝑟 (

𝑘_𝑚𝑟𝜕𝑇

𝜕𝑟

)=𝜌_𝑚𝑐_𝑝,𝑚𝜕𝑇

𝜕𝑡 (3)

Positive and negative tabs of the battery:

1 𝑟

𝜕

𝜕𝑟 (

𝑘_𝑡𝑏𝑟𝜕𝑇

𝜕𝑟 )

=𝜌_𝑡𝑏𝑐_{𝑝,𝑡𝑏}𝜕𝑇

𝜕𝑡 (4)

Outer can of the battery:

1 𝑟

𝜕

𝜕𝑟 (

𝑘_𝑜𝑐𝑟𝜕𝑇

𝜕𝑟 )

=𝜌_𝑜𝑐𝑐_{𝑝,𝑜𝑐}𝜕𝑇

𝜕𝑡 (5)

Where 𝑘_𝑚 =𝑘_𝑡𝑏 =𝑘_𝑜𝑐 =𝑘_𝑠𝑠 = 14.5 W∕mK[51],𝑐_𝑝,𝑚= 𝑐_{𝑝,𝑡𝑏}=𝑐_{𝑝,𝑜𝑐} = 𝑐_{𝑝,𝑠𝑠}= 560 J∕kgK[52], and𝜌_𝑚=𝜌_𝑡𝑏=𝜌_𝑜𝑐=𝜌_𝑠𝑠= 8000 kg∕m³.

As this geometry is composed of active material, positive and negative tabs, and outer casing, the grid independence study is again conducted to identify the optimum number of mesh elements required for simulation.Fig. 11depicts the variation of the simulated temperature–

time histories of the battery at location (9, 0, 13 (all in mm)) for coarse,

fine, and extra fine mesh types when subject to a volumetric heat of 5000W∕m³. From the figure, it is clear that 259880 number of elements are sufficient for all the simulations conducted on this geometry.

The range of the unknown ‘𝑞_𝑉’ at the given discharge rates of 0.5, 1, and 2C are taken from literature [53] and are shown inTable 4. The simulated temperatures are obtained at each time interval for this range of the unknown parameter𝑞_𝑉 at the given discharge rates. The total time intervals for 0.5, 1, and 2C are 16, 15, and 16, respectively (see Table 4). The total number of samples of ‘𝑞_𝑉’ for each time interval and each discharge rate is equal to 201. The simulated temperatures thus obtained for the given range of the unknown parameter (𝑞_𝑉) at each time interval are used to develop the surrogate model.

4. Surrogate model

Artificial Neural Networks (ANNs) are created to act as a surrogate model to reduce computational time for solving the present inverse

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Table 3

Boundary and domain conditions of the Li-ion battery for solving the governing equations.

Boundary Condition Description

Outer surface of the battery −𝑘𝑟(𝑇)^𝜕𝑇_𝜕𝑟 =ℎ_𝐷(𝑇𝑠−𝑇_∞),

−𝑘𝜃(𝑇) ^𝜕𝑇

𝑅0𝜕𝜃=ℎ_𝐷(𝑇𝑠−𝑇∞),

−𝑘𝑧(𝑇)^𝜕𝑇

𝜕𝑧=ℎ_𝐷(𝑇𝑠−𝑇∞)

Natural convection

Inner surface of the mandrel (𝑟= 1, 0≤𝜃≤360,0≤𝑟≤8.7)

𝜕𝑇

𝜕𝑟 = 0,^𝜕𝑇

𝜕𝜃= 0,^𝜕𝑇

Domain Condition Description

Active material of the battery 𝑞_𝑉=𝑓(𝑡) Volumetric heat generation

Fig. 8. Three-dimensional geometry of the active material of the battery for the unsteady-state conduction problem.

Fig. 9. Simulated temperature–time history at location (8.7, 0, 40 (all in mm)) of the forward model with assumed values parameters (𝑘_𝑟= 2.5W/mK,𝑘_𝜃= 16W/mK, 𝑘_𝑧= 16W/mK, and𝑐_𝑝= 2600J/kgK) for different mesh size elements.

problems. To estimate the temperature-dependent orthotropic thermal properties (𝑘_𝑟(𝑇), 𝑘_𝜃(𝑇), 𝑘_𝑧(𝑇), 𝑐_𝑝(𝑇)), 12 ANNs corresponding to 12 time intervals, are generated (shown inTable 2). At each time instant, each neural network is trained with input data of 201 samples of 𝑘_𝑟(𝑇), 𝑘_𝜃(𝑇), 𝑘_𝑧(𝑇), 𝑐_𝑝(𝑇)and an output data of corresponding simulated temperatures obtained from forward model. The trained neural network

Fig. 10. Three-dimensional geometry of the cylindrical Li-ion battery with volumetric heat generation.

Fig. 11. Variation of simulated temperature–time history at location (9, 0, 13 (all in mm)) of the forward model with assumed value parameter (𝑞_𝑉 = 5000 W∕m³) for different mesh size elements.

connects the simulated temperatures (𝑇_{𝑠𝑖𝑚,1}, 𝑇_{𝑠𝑖𝑚,2}, 𝑇_{𝑠𝑖𝑚,3}, 𝑇_{𝑠𝑖𝑚,4}, 𝑇_{𝑠𝑖𝑚,5}) as a function of𝑘_𝑟, 𝑘_𝜃, 𝑘_𝑧, 𝑐_𝑝.

The surrogate model developed in the case of estimating volumetric heat generation, a total of 16, 15, and 16 ANNs are created corresponding to total time intervals (shown inTable 4) for 0.5, 1, and 2C discharge rates, respectively. Again, each ANN is used to train the simulated temperatures (𝑇_{𝑠𝑖𝑚,7}, 𝑇_{𝑠𝑖𝑚,8}, 𝑇_{𝑠𝑖𝑚,9}, 𝑇_{𝑠𝑖𝑚,10}, 𝑇_{𝑠𝑖𝑚,11}) as

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Fig. 12. ANN architectures for estimating (a) the temperature-dependent orthotropic thermal properties (𝑘𝑟(𝑇), 𝑘𝜃(𝑇), 𝑘𝑧(𝑇), 𝑐𝑝(𝑇)) of the active material and (b) the volumetric heat generation (𝑞𝑉) with time in the battery.

Table 4

Range of the unknown parameter (𝑞𝑉) at each time interval to obtain the simulated temperatures.

Discharge rate Time (t), s Ranges of unknown parameters at each time interval

0.5C 437≤𝑡≤6992

Time step (𝛥t) = 437

4000≤𝑞_𝑉 ≤12000 (W/m³) (samples = 201)

1C 231≤𝑡≤3465

15000≤𝑞_𝑉 ≤60000 (W/m³) (samples = 201)

2C 200≤𝑡≤1700

15000≤𝑞_𝑉 ≤180000 (W/m³) (samples = 201)

a function of𝑞_𝑉.Figs. 12(a) and (b) show the ANN architectures for estimating the temperature-dependent orthotropic thermal properties (𝑘_𝑟(𝑇), 𝑘_𝜃(𝑇), 𝑘_𝑧(𝑇), 𝑐_𝑝(𝑇)) of the active material and the volumetric heat generation (𝑞_𝑉) with time in the battery, respectively. Based on the neuron independence study, a total of 10 neurons in each hidden layer are used to develop ANNs for both estimations. The Levenberg–

Marquardt optimization method is used to minimize the weights of each network. The output layer of the activation function uses the sigmoid logistic function while training the ANNs. For developing each ANN in this study, 70%, 15%, and 15% of samples are used for training, validating, and testing, respectively.

5. Inverse model

The study employs the Bayesian inference method to retrieve the temperature-dependent orthotropic thermal properties (𝑘_𝑟,𝑘_𝜃,𝑘_𝑧, and 𝑐_𝑝) and volumetric heat generation (𝑞_𝑉) separately from unsteady experiments carried out on (i) the active material of the Li-ion battery arrangement connected to a hot fluid circulator and (ii) the Li-ion battery for various discharge rates (0.5, 1, and 2C), respectively. The Bayesian inference method depends on the Bayes theorem, which states that the posterior probability density function of an estimate is directly proportional to its likelihood function.

Given the unsteady measured data (Y), the posterior probability density function (PPDF) of the estimates vector𝐸is defined by 𝑃(𝐸∕𝑌) =𝑃(𝑌∕𝐸) ×𝑃(𝐸)

𝑃(𝑌) = 𝑃(𝑌∕𝐸) ×𝑃(𝐸)

∫ 𝑃(𝑌∕𝐸) ×𝑃(𝐸) d𝐸 (6) where P(𝐸/Y), P(Y/𝐸), and P(𝐸) are the PPDF, likelihood, and prior density functions, respectively. The P(Y/𝐸) acquired by comparing the experimental and simulated temperatures (obtained from the surrogate

model) for the given parameters as 𝑃(𝑌∕𝐸) = 1

(√2𝜋𝜎²)𝑛𝑒⁻⁽

𝜒2

2) (7)

where𝜒² =

∑𝑛

𝑖=1(𝑌_{𝑚𝑒𝑎𝑠,𝑖}−𝑌_{𝑠𝑖𝑚,𝑖})²

𝜎² , 𝜎 is the uncertainty between the measurement and forward model, 𝑛 is a dimension of the temperature measurement (𝑌_{𝑚𝑒𝑎𝑠}), and𝑌_𝑠𝑖𝑚is the simulated temperatures obtained from the surrogate model with given the parameter vector (𝐸). In the study, n (the number of locations where temperatures were measured) is 5. The prior density function (P(𝐸)) is written as follows in the case of a normal prior, where the mean and standard deviation of the estimates are𝜇_𝑝 and𝜎_𝑝, respectively.

𝑃(𝐸) = 1

√ 2𝜋𝜎_𝑝²

𝑒

−^{(𝐸−𝜇𝑝)2} 2𝜎2

𝑝 (8)

Using Eqs.(7)and(8)in Eq.(6), the PPDF was calculated as

𝑃(𝐸∕𝑌) = 𝑒

− [𝜒2

2+^{(𝐸−𝜇𝑝)2} 2𝜎2

𝑝 ]

∫ 𝑒

− [

𝜒2 2+^{(𝐸−𝜇𝑝)2}

2𝜎2 𝑝

]

d𝐸

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Here,𝐸denotes (𝑘_𝑟,𝑘_𝜃,𝑘_𝑧,𝑐_𝑝) or𝑞_𝑉, as the case may be.

The Mean, Maximum a posteriori (MAP), and Standard deviation (SD) employed in the study are described in Kumar et al. [54].

Metropolis Hastings-Markov Chain Monte Carlo (MH-MCMC) algo- rithm

To dynamically produce the samples of the parameters (𝐸), the MH-MCMC algorithm is used in Bayesian inference. In the MH-MCMC algorithm, the new sample is produced using a normal density function N(𝐸) around the mean of the old sample, a standard deviation of 5%

of the mean, and a random number between 0 and 1. More details on the MH-MCMC sampling algorithm have been presented in Kumar and Balaji [52]. In this study, 10% samples are used for burn-in.

For both single-parameter estimation and multi-parameter estimation, the sampling procedure of the MH-MCMC algorithm is as follows:

1. Initialize𝐸¹=(𝐸₁¹,𝐸¹

2, . . . ..,𝐸¹

𝑛^′) 2. For i=1, 2, . . . ., M or j=1, 2, . . . .,𝑛^′

(a) Generate a random number, v∼R(0,1) (b) Evaluate the next sample𝐸^∗_𝑗 ∼𝑁(𝐸_−𝑗^𝑖+1, 𝐸^𝑖_𝑗, 𝜎²

𝐸_𝑗^𝑖)

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Fig. 13. (a) Estimated orthotropic thermal conductivities (𝑘_𝑟,𝑘_𝜃,𝑘_𝑧) and (b) specific heat capacity (𝑐_𝑝) variations with time (t) using ‘‘offline’’ Bayesian approach.

(c) If v<A(𝐸^∗_𝑗,𝐸^𝑖

𝑗), accept𝐸^𝑖+1

𝑗 =𝐸^∗

𝑗

(d) Else go to step 2 with𝐸_𝑗^𝑖+1=𝐸_𝑗^𝑖

In the aforementioned procedure,𝐸stands for the parameter or group of parameters that need to be retrieved, depending on the situation.

M and𝑛^′are the number of samples and parameters, respectively, and 𝐸_−𝑗^𝑖+1 = (𝐸₁^𝑖+1,…, 𝐸_𝑛^𝑖)^𝑇. ‘A’ denotes the acceptance ratio and has been discussed in Kumar and Balaji [52].

6. Results and discussion

6.1. Estimation of the temperature-dependent orthotropic thermal properties (𝑘_𝑟,𝑘_𝜃,𝑘_𝑧,𝑐_𝑝)

As mentioned earlier, accurate estimations of the temperature- dependent orthotropic thermal properties (𝑘_𝑟,𝑘_𝜃,𝑘_𝑧,𝑐_𝑝) of the active material and the volumetric heat generation in the active material of Li- ion battery are important for the thermal management. In this section, the estimation of temperature-dependent orthotropic thermal properties are presented first. Following this, the estimation of volumetric heat generation with time in a Li-ion battery is reported for 0.5, 1, and 2C discharge rates of Panasonic NCR18650BD battery.

The orthotropic thermal properties (𝑘_𝑟, 𝑘_𝜃, 𝑘_𝑧, 𝑐_𝑝) of the active material are estimated in terms of the mean and standard deviation (SD) at each time interval using unsteady surface temperatures obtained from the in-house experiments. These estimations are performed using two inverse methods, namely (i) ‘‘offline’’ and (ii) ‘‘online’’

Bayesian approaches. In the ‘‘offline’’ inverse method, the samples of the estimates (𝑘_𝑟,𝑘_𝜃,𝑘_𝑧,𝑐_𝑝) are generated by dividing the ranges of estimates into a set of equal intervals. The ‘‘online’’ method means samples of the estimates are generated dynamically using the Metropolis Hasting-Markov Chain Monte Carlo (MH-MCMC) sampling algorithm.

The samples of 2000 and 4000 are considered in ‘‘offline’’ and ‘‘online’’

Bayesian approaches, respectively, for the estimations. The variation of the estimated orthotropic thermal conductivities 𝑘_𝑟, 𝑘_𝜃, and 𝑘_𝑧 with time are shown in Fig. 13(a) by employing the inverse method of

‘‘offline’’ Bayesian approach. Fig. 13(b) shows the estimated specific heat capacity (𝑐_𝑝) variation with time of the active material using the same inverse method. The estimated parameters (𝑘_𝑟,𝑘_𝜃,𝑘_𝑧,𝑐_𝑝) using

‘‘offline’’ inverse approach are given in the ‘‘online’’ Bayesian approach as priors knowledge for better estimations. The results of the estimated 𝑘_𝑟, 𝑘_𝜃, and𝑘_𝑧 variation with time using ‘‘online’’ Bayesian approach with priors are presented in Fig. 14(a). From Fig. 14(a), it can be observed that the predicted values for axial thermal conductivity (𝑘_𝑧) and angular thermal conductivity (𝑘_𝜃) are nearly the same.Fig. 14(b) shows the results of the estimated𝑐_𝑝variation with time using the same

Table 5

Validation of the estimated average values of the thermal properties (𝑘_𝑟,𝑘_𝜃,𝑘_𝑧,𝑐_𝑝) with literature.

Average values of thermal properties Present study Literature [30,46,55]

𝑘_𝑟(W/mK) 3.18±0.19 3

𝑘_𝜃(W/mK) 20.34±1.26 28.05

𝑘_𝑧(W/mK) 19.89±1.29 28.05

𝑐_𝑝(J/kgK) 3180±202 2400

‘‘online’’ inverse approach. The estimated variation of the standard deviation with time for the orthotropic thermal conductivities (𝑘_𝑟,𝑘_𝜃, and𝑘_𝑧) and the specific heat capacity (𝑐_𝑝) of the active material using

‘‘offline’’ and ‘‘online’’ Bayesian methods are shown inFigs. 15(a) and (b), respectively. From these figures, it is observed that the standard deviation (SD) of the orthotropic thermal properties (𝑘_𝑟, 𝑘_𝜃, 𝑘_𝑧, 𝑐_𝑝) estimated using ‘‘online’’ Bayesian method with priors is much lower compared to ‘‘offline’’ Bayesian method. A higher reduction in the SD of the estimates (𝑘_𝑟,𝑘_𝜃,𝑘_𝑧,𝑐_𝑝) using priors denotes the superiority of the Bayesian method and the significance of the priors in the estimation process. The average values of the estimated orthotropic thermal properties (𝑘_𝑟,𝑘_𝜃,𝑘_𝑧,𝑐_𝑝) of a cylindrical Li-ion battery obtained from the present study and literature [30,46,55] are shown inTable 5for comparison. In view of the close agreement of the estimated thermophysical properties against literature, the properties estimated in the present study can be deemed to be accurate.

To obtain the temperature-dependent orthotropic thermal properties, the average measured temperature on the surface of the active material is considered corresponding to each time interval. The Gaus- sian curve fits of the estimated𝑘_𝑟, 𝑘_𝜃, 𝑘_𝑧, and𝑐_𝑝 with respect to the average measured temperatures are shown inFigs. 16(a), (b), (c), and (d), respectively. The temperature-dependent functions of the estimated 𝑘_𝑟,𝑘_𝜃,𝑘_𝑧, and𝑐_𝑝are obtained using Gaussian curve fit and are shown inTable 6. The temperature-dependent functions of the estimated𝑘_𝑟, 𝑘_𝜃,𝑘_𝑧, and𝑐_𝑝are then given to the forward model of the active material to obtain the simulated temperatures at locations shown in Fig. 3.

These simulated temperatures are compared against experimental temperatures for validation, and are presented in a parity plot shown in Fig. 17. From this figure, it is observed that the discrepancy between the experimental and the simulated temperatures (obtained by giving the estimated temperature-dependent orthotropic thermal properties (𝑘_𝑟, 𝑘_𝜃, 𝑘_𝑧, 𝑐_𝑝) in the forward model) is observed to be less than

±1.5^◦C. The small deviation (±1.5^◦C) demonstrates that the procedure employed in the present study is quite effective in accurate estimation of the thermal properties of the active material.

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Fig. 14. (a) Estimated orthotropic thermal conductivities (𝑘_𝑟,𝑘_𝜃,𝑘_𝑧) and (b) specific heat capacity (𝑐_𝑝) variations with time (t) using ‘‘online’’ Bayesian approach.

Fig. 15. Estimated standard deviation variation with time of (a) the orthotropic thermal conductivities (𝑘_𝑟,𝑘_𝜃,𝑘_𝑧) (b) specific heat capacity (𝑐_𝑝) using offline and online Bayesian approaches.

Fig. 16. Gaussian fits of the estimated (a) radial thermal conductivity (𝑘𝑟) (b) angular thermal conductivity (𝑘𝜃) (c) axial thermal conductivity (𝑘𝑧) and (d) specific heat capacity (𝑐𝑝).