**3. Numerical Model**

The numerical model was developed using the MSMD (multi-scale multi-dimensional) model of Ansys Fluent [25]. The LIR2450 coin cell was modeled with radius of 24.5 mm and height of 5 mm. The mesh was generated with 107,478 nodes and 99,552 elements. The geometry and meshing images are provided in Figure 2. The current density, *jEch*, was calculated from Equation (4), proposed by Newman, Tiedemann, Gu, and Kim (NTGK) [26] as follows:

$$j\_{Ech} = \frac{Q\_{nominal}}{Q\_{ref}Vol} Y(\mathcal{U} - V) \tag{4}$$

where *V* is battery cell voltage; *Qnominal* is the battery capacity in Ampere hours; *Qref* is the battery capacity used in the experiments to obtain *Y* and *U*. *Y* and *U* are LIB depth of discharge functions. In the present study, discharge tests were conducted at different constant currents to obtain *Y* and *U* functions. The obtained *Y* and *U* are fitting parameters with functions of depth of discharge in

Equations (5) and (6), as initially suggested by Gu et al. [27]. The values obtained by curve fitting for the 120 mAh LIR2450 coin cell battery are presented in Table 3. Equations (5) and (6), as initially suggested by Gu et al. [27]. The values obtained by curve fitting for the 120 mAh LIR2450 coin cell battery are presented in Table 3. 1 2 3 4 5 0 1 2 3 4 5 *U a a DOD a DOD a DOD a DOD a DOD* = + + + + + ( ) ( ) ( ) ( ) ( )

$$\mathcal{U} = a\_0 + a\_1 (DOD)^1 + a\_2 (DOD)^2 + a\_3 (DOD)^3 + a\_4 (DOD)^4 + a\_5 (DOD)^5 \tag{5}$$

$$\mathcal{Y} = b\_0 + b\_1 (DOD)^1 + b\_2 (DOD)^2 + b\_3 (DOD)^3 + b\_4 (DOD)^4 + b\_5 (DOD)^5 \tag{6}$$

$$Y = b\_0 + b\_1(DOD)^1 + b\_2(DOD)^2 + b\_3(DOD)^3 + b\_4(DOD)^4 + b\_5(DOD)^5 \tag{6}$$

**Figure 2.** Numerical model geometry and meshing. **Figure 2.** Numerical model geometry and meshing.


**Table 3.** Experimental values for co-efficient of *U* and *Y* functions. **Table 3.** Experimental values for co-efficient of *U* and *Y* functions.

The thermal- and electrical-coupled field equation for battery operation were solved using Equations (7)–(9) [25], where *σ<sup>+</sup>* and *σ<sup>−</sup>* are electrical conductivities of the positive and negative electrode, respectively; *ϕ<sup>+</sup>* and *ϕ<sup>−</sup>* phase potentials for the positive and negative electrodes, respectively; *ECh j* and . *short q* represent volumetric current transfer rate and heat due to *short j* . The thermal- and electrical-coupled field equation for battery operation were solved using Equations (7)–(9) [25], where σ<sup>+</sup> and σ<sup>−</sup> are electrical conductivities of the positive and negative electrode, respectively; φ<sup>+</sup> and φ<sup>−</sup> phase potentials for the positive and negative electrodes, respectively; *<sup>j</sup>ECh* and . *qshort* represent volumetric current transfer rate and heat due to electrochemical reaction, respectively; *<sup>j</sup>short* and . *qshort* represent current transfer rate and heat generation during internal short circuit, respectively. ρ, *k*, and *T* represent density, thermal conductivity, and temperature, respectively.

and

electrochemical reaction, respectively;

*t*

a<sup>5</sup> −3.55203 b<sup>5</sup> −34.5455

*short q*

$$\frac{\partial \rho \mathbf{\hat{C}}\_p \mathbf{T}}{\partial t} = -\mathbf{V} \cdot (\mathbf{k} \nabla T) = \sigma\_+ \left| \nabla \phi\_+ \right|^2 + \sigma\_- \left| \nabla \phi\_- \right|^2 + \dot{q}\_{\text{ECh}} + \dot{q}\_{\text{short}} \tag{7}$$

$$\nabla \cdot (\sigma\_+ \nabla \phi\_+) = -(\dot{j}\_{\text{ECh}} - j\_{\text{short}}) \tag{8}$$

$$
\nabla \cdot (\sigma\_+ \nabla \phi\_+) = -(j\_{\text{ECh}} - j\_{\text{short}}) \tag{8}
$$

$$
\nabla \cdot (\sigma\_+ \nabla \phi\_+) = -(j\_{\text{ECh}} - j\_{\text{short}}) \tag{9}
$$

$$
\nabla \cdot (\sigma\_- \nabla \phi\_-) = j\_{\text{ECh}} - j\_{\text{short}} \tag{9}
$$

represent current transfer rate and heat

(9)

$$
\nabla \cdot (\sigma\_+ \nabla \phi\_+) = -(f\_{\text{ECh}} - f\_{\text{short}}) \tag{8}
$$

$$
\nabla \cdot (\sigma\_- \nabla \phi\_-) = j\_{\text{ECh}} - j\_{\text{short}} \tag{9}
$$

( ) ( ) *ECh short j j* = − − + + (8) In Equation (10), *Qnominal* and *Qref* are the same as the reference battery parameters and are estimated through experimentation. The electrochemical reaction heat was calculated as given in

> *j j* = − − −

( ) *ECh short*

Equation (11). The first term (*U* − *V*) in Equation (11) [25] represents heat generated due to overpotential, and the second term is related to heat generated due to entropic heating.

$$j\_{\rm{ECh}} = \frac{Q\_{\rm{nominal}}}{Q\_{\rm{ref}} Vol} Y[\mathcal{U} - V] \tag{10}$$

$$
\dot{q}\_{\rm ECh} = \dot{j}\_{\rm Ech} \left[ \mathcal{U} - V - T \frac{d\mathcal{U}I}{dT} \right] \tag{11}
$$

During the normal operation of LIB battery, the cathode and anode are separated by separators, which is generally thin polymer material. The separator prevents the direct contact of positive and negative electrodes. In the event of short circuiting, as a result of penetration or crash, the separator gets damaged, which results in a secondary current along with regular current flowing through tabs. The transfer short current density is computed from Equation (12). The heat generated by internal short circuit is computed by volumetric contact resistance (*rc*/*a*), where *r<sup>c</sup>* is contact resistance and *a* is the specific area of the electrode, as shown in Equation (13) [23].

$$j\_{\rm short} = a(\phi\_+ - \phi\_-)/r\_\circ \tag{12}$$

$$\dot{q}\_{\text{short}} = a(\phi\_+ - \phi\_-)^2 / r\_c \tag{13}$$
