# Battery Thermal Runaway Model

Thermal runaway in battery packs is a major safety concern with potentially catastrophic outcomes, such as battery pack fires. This phenomenon occurs in batteries due to exothermic degradation reactions when a battery is subjected to abuse conditions, such as physical damage, internal short circuits, overcharging, or overheating. These conditions, along with the associated decomposition of battery components, for example, anode, cathode, and separator, lead to significant heat release and an uncontrollable rise in temperature. When a single cell enters these thermally unstable conditions, the heat release can cause adjacent cells to heat up and enter thermal runaway, eventually propagating through and consuming the entire battery pack. Additionally, the decomposition reactions generate flammable gases that can ignite, causing fires to spread through the battery pack.

- NREL model
- Arc reaction heat model
- Heat rate model

The first two models are chemical kinetic models based on the Arrhenius equation. The NREL model takes a mechanistic approach to model the kinetics and heat release of individual cell components, while the arc reaction heat model adopts a more phenomenological approach, deriving the kinetics directly from ARC data. The heat rate model also uses ARC data but directly reads this data to determine the volumetric heat source.

All of the mathematical models for thermal runaway add an additional heat source ( ${\text{S}}_{\text{TR}}$ ) to the energy equation.

The form of the heat source (aggregated or direct calculation) depends on the model. A brief description of these models and the heat source term is given in the following sections.

## NREL Thermal Runaway Model Formulation

The mathematical model for thermal abuse involves heat generation resulting from series of exothermic decomposition reactions of various battery components. The heat generation from these individual reactions are aggregated and incorporated into the energy equation as a combined source term.

The decomposition reactions within the battery are characterized using Arrhenius equations, accounting for: 1) Solid electrolyte interface (SEI) decomposition at the anode-electrolyte interface ( $~80-100{}^{o}C$ ); 2) Anode decomposition (intercalated lithium reacting with electrolyte, facilitated by SEI decomposition); 3) Cathode conversion (active material decomposition releasing oxygen), which is highly exothermic; and 4) Electrolyte decomposition at very high temperatures. The final source term from the above is written: ${S}_{TR}={S}_{SEI}+{S}_{a}+{S}_{c}+{S}_{elec}$ , representing the heat release from the exothermic decomposition reactions of the SEI, anode, cathode, and electrolyte, respectively.

The governing equations for these reactions and each exothermic heat source contributing to the energy equation are summarized below.

SEI decomposition reaction rate:

Where ${x}_{s}$ is the fraction of lithium in the anode, ${A}_{s}$ the frequency factor for anode decomposition, ${E}_{s}$ the activation energy for anode decomposition, $T$ the temperature, and ${k}_{b}$ the Boltzmann constant.

SEI decomposition exothermic heat:

Anode decomposition reaction rate:

Where ${x}_{a}$ is the fraction of lithium in the anode, ${A}_{a}$ the frequency factor for anode decomposition, ${E}_{a}$ the activation energy for anode decomposition. As part of anode decomposition, the tunneling effect is considered due to the reduction in thickness of the SEI, where $z$ represents the relative SEI thickness:

Anode decomposition exothermic heat:

Where ${H}_{a}$ is the negative solvent enthalpy, ${W}_{a}$ is the specific carbon content in the jelly roll.

Cathode conversion reaction rate:

Cathode conversion exothermic heat:

Where ${H}_{c}$ is the positive solvent enthalpy, ${W}_{c}$ is the specific cathode active material content in the jelly roll.

Electrolyte decomposition exothermic heat:

Finally, short circuit events can also be included, for example, separator melting leading to direct connection between the anode and cathode. The short circuit leads to charge depletion and is given as a function of state of charge (SOC):

In which ${T}_{ISC}$ becomes active when temperature is above a pre-defined internal short circuit temperature.

## ARC Reaction Model Formulation

The ARC reaction model is a staged Arrhenius-based kinetic model that can be directly fitted to Accelerating Rate Calorimetry (ARC) data. This enables fitting N number of stages to the data based on the desired level of accuracy. Up to five ODEs (i = 1-5) are employed to represent the data, each with distinct parameters for activation and decay, characterizing a specific stage. The summation of these heat sources represents the ARC heat rate data. The general form of the equation is as follows:

Where ${A}_{\left(a,i\right)}$ is the frequency factor for a specific stage (1/s), ${E}_{\left(a,i\right)}$ is the activation energy (J), ${k}_{b}$ is the Boltzmann constant, $T$ the cell temperature (K), n and m are related to the form of the solid reaction model. Depending on the value of m this reaction can be nth order (when m=0) or an auto-catalytic type, that is, the reaction increases as the product is generated (m>0).

The above equation is a more phenomenological modeling approach compared to the NREL model; however, it can still be motivated from a physical perspective providing the stages are divided into bands that represent an approximate physical reaction process occurring, for example, anode decomposition.

The total exothermic heat generation for this model is given by,

Where ${h}_{i}$ is the enthalpy of the component defined by,

The heat contribution for each stage combines to form the heat source, ${S}_{TR}$ , in the energy equation.

## Direct Reading of ARC Data Formulation

A direct reading of thermal runaway ARC data provides a source term directly in the energy equation derived from the heat rate-temperature data. To calculate this heat source using ARC data the following equation is utilized:

Where ${\rho}_{cell}$ is the effective density of the cell, ${c}_{p,cell}$ is the effective specific heat of the cell and $dT/dt$ is the heat rate data directly read from the ARC test.