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.

SimLab battery solutions for thermal runaway provide a virtual experimental platform to test pack designs for thermal safety, including the assessment of propagation characteristics and heat shield effectiveness. Three models are available to simulate thermal runaway:
  • 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 ( S TR MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4ua8aadaWgaaWcbaWdbiaabsfacaqGsbaapaqabaaaaa@38E9@ ) to the energy equation.

ρ c p dT dt = κT + S TR MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyWdiaadogapaWaaSbaaSqaa8qacaWGWbaapaqabaGcpeWaaSaa a8aabaWdbiaadsgacaWGubaapaqaa8qacaWGKbGaamiDaaaacqGH9a qpcqGHhis0cqGHflY1daqadaWdaeaapeGaaeOUdiabgEGirlaadsfa aiaawIcacaGLPaaacqGHRaWkcaWGtbWdamaaBaaaleaapeGaamivai aadkfaa8aabeaaaaa@4B78@

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 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaiOFaiaaiIdacaaIWaGaeyOeI0IaaGymaiaaicdacaaIWaGaaiiO a8aadaahaaWcbeqaa8qacaWGVbaaaOGaam4qaaaa@3EDD@ ); 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 T R = S S E I + S a + S c + S e l e c MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ua8aadaWgaaWcbaWdbiaadsfacaWGsbaapaqabaGcpeGaeyyp a0Jaam4ua8aadaWgaaWcbaWdbiaadofacaWGfbGaamysaaWdaeqaaO WdbiabgUcaRiaadofapaWaaSbaaSqaa8qacaWGHbaapaqabaGcpeGa ey4kaSIaam4ua8aadaWgaaWcbaWdbiaadogaa8aabeaak8qacqGHRa WkcaWGtbWdamaaBaaaleaapeGaamyzaiaadYgacaWGLbGaam4yaaWd aeqaaaaa@49B6@ , 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:

d x s dt = x s · A s ·exp E s k b T MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiaadsgacaWG4bWdamaaBaaaleaapeGaam4CaaWd aeqaaaGcbaWdbiaadsgacaWG0baaaiabg2da9iabgkHiTiaadIhapa WaaSbaaSqaa8qacaWGZbaapaqabaGcpeGaai4TaiaadgeapaWaaSba aSqaa8qacaWGZbaapaqabaGcpeGaai4TaiaadwgacaWG4bGaaeiCam aabmaapaqaa8qacqGHsisldaWcaaWdaeaapeGaamyra8aadaWgaaWc baWdbiaadohaa8aabeaaaOqaa8qacaWGRbWdamaaBaaaleaapeGaam OyaaWdaeqaaOWdbiaadsfaaaaacaGLOaGaayzkaaaaaa@4F6C@

Where x s MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiEa8aadaWgaaWcbaWdbiaadohaa8aabeaaaaa@385C@ is the fraction of lithium in the anode, A s MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyqa8aadaWgaaWcbaWdbiaadohaa8aabeaaaaa@3825@ the frequency factor for anode decomposition, E s MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyra8aadaWgaaWcbaWdbiaadohaa8aabeaaaaa@3829@ the activation energy for anode decomposition, T MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamivaaaa@36E6@ the temperature, and k b MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Aa8aadaWgaaWcbaWdbiaadkgaa8aabeaaaaa@383E@ the Boltzmann constant.

SEI decomposition exothermic heat:

S SEI = H SEI W a d x s dt MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ua8aadaWgaaWcbaWdbiaadofacaWGfbGaamysaaWdaeqaaOWd biabg2da9iaadIeapaWaaSbaaSqaa8qacaWGtbGaamyraiaadMeaa8 aabeaak8qacaWGxbWdamaaBaaaleaapeGaamyyaaWdaeqaaOWdbmaa laaapaqaa8qacaWGKbGaamiEa8aadaWgaaWcbaWdbiaadohaa8aabe aaaOqaa8qacaWGKbGaamiDaaaaaaa@4619@

Anode decomposition reaction rate:

d x a dt = x a · A a ·exp E a k b T ·exp z z 0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiaadsgacaWG4bWdamaaBaaaleaapeGaamyyaaWd aeqaaaGcbaWdbiaadsgacaWG0baaaiabg2da9iabgkHiTiaadIhapa WaaSbaaSqaa8qacaWGHbaapaqabaGcpeGaai4TaiaadgeapaWaaSba aSqaa8qacaWGHbaapaqabaGcpeGaai4TaiaadwgacaWG4bGaaeiCam aabmaapaqaa8qacqGHsisldaWcaaWdaeaapeGaamyra8aadaWgaaWc baWdbiaadggaa8aabeaaaOqaa8qacaWGRbWdamaaBaaaleaapeGaam OyaaWdaeqaaOWdbiaadsfaaaaacaGLOaGaayzkaaGaai4Taiaadwga caWG4bGaaeiCamaabmaapaqaa8qacqGHsisldaWcaaWdaeaapeGaam OEaaWdaeaapeGaamOEa8aadaWgaaWcbaWdbiaaicdaa8aabeaaaaaa k8qacaGLOaGaayzkaaaaaa@5948@

Where x a MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiEa8aadaWgaaWcbaWdbiaadggaa8aabeaaaaa@384A@ is the fraction of lithium in the anode, A a MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyqa8aadaWgaaWcbaWdbiaadggaa8aabeaaaaa@3813@ the frequency factor for anode decomposition, E a MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyra8aadaWgaaWcbaWdbiaadggaa8aabeaaaaa@3817@ 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 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOEaaaa@370C@ represents the relative SEI thickness:

dz dt = x a · A a ·exp E a k b T ·exp z z 0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiaadsgacaWG6baapaqaa8qacaWGKbGaamiDaaaa cqGH9aqpcaWG4bWdamaaBaaaleaapeGaamyyaaWdaeqaaOWdbiaacE lacaWGbbWdamaaBaaaleaapeGaamyyaaWdaeqaaOWdbiaacElacaWG LbGaamiEaiaabchadaqadaWdaeaapeGaeyOeI0YaaSaaa8aabaWdbi aadweapaWaaSbaaSqaa8qacaWGHbaapaqabaaakeaapeGaam4Aa8aa daWgaaWcbaWdbiaadkgaa8aabeaak8qacaWGubaaaaGaayjkaiaawM caaiaacElacaWGLbGaamiEaiaabchadaqadaWdaeaapeGaeyOeI0Ya aSaaa8aabaWdbiaadQhaa8aabaWdbiaadQhapaWaaSbaaSqaa8qaca aIWaaapaqabaaaaaGcpeGaayjkaiaawMcaaaaa@5722@

Anode decomposition exothermic heat:

S a = H a W a d x a dt MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ua8aadaWgaaWcbaWdbiaadggaa8aabeaak8qacqGH9aqpcaWG ibWdamaaBaaaleaapeGaamyyaaWdaeqaaOWdbiaadEfapaWaaSbaaS qaa8qacaWGHbaapaqabaGcpeWaaSaaa8aabaWdbiaadsgacaWG4bWd amaaBaaaleaapeGaamyyaaWdaeqaaaGcbaWdbiaadsgacaWG0baaaa aa@42F3@

Where H a MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamisa8aadaWgaaWcbaWdbiaadggaa8aabeaaaaa@381A@ is the negative solvent enthalpy, W a MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4va8aadaWgaaWcbaWdbiaadggaa8aabeaaaaa@3829@ is the specific carbon content in the jelly roll.

Cathode conversion reaction rate:

dα dt =α 1α · A c ·exp E c k b T MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiaadsgacaqGXoaapaqaa8qacaWGKbGaamiDaaaa cqGH9aqpcqGHsislcaqGXoWaaeWaa8aabaWdbiaaigdacqGHsislca qGXoaacaGLOaGaayzkaaGaai4TaiaadgeapaWaaSbaaSqaa8qacaWG JbaapaqabaGcpeGaai4TaiaadwgacaWG4bGaaeiCamaabmaapaqaa8 qacqGHsisldaWcaaWdaeaapeGaamyra8aadaWgaaWcbaWdbiaadoga a8aabeaaaOqaa8qacaWGRbWdamaaBaaaleaapeGaamOyaaWdaeqaaO WdbiaadsfaaaaacaGLOaGaayzkaaaaaa@518E@

Cathode conversion exothermic heat:

S c = H c W c dα dt MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ua8aadaWgaaWcbaWdbiaadogaa8aabeaak8qacqGH9aqpcaWG ibWdamaaBaaaleaapeGaam4yaaWdaeqaaOWdbiaadEfapaWaaSbaaS qaa8qacaWGJbaapaqabaGcpeWaaSaaa8aabaWdbiaadsgacaqGXoaa paqaa8qacaWGKbGaamiDaaaaaaa@41F8@

Where H c MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamisa8aadaWgaaWcbaWdbiaadogaa8aabeaaaaa@381C@ is the positive solvent enthalpy, W c MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4va8aadaWgaaWcbaWdbiaadogaa8aabeaaaaa@382B@ is the specific cathode active material content in the jelly roll.

Electrolyte decomposition exothermic heat:

d c elec dt = c elec · A elec ·exp E elec k b T MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiaadsgacaWGJbWdamaaBaaaleaapeGaamyzaiaa dYgacaWGLbGaam4yaaWdaeqaaaGcbaWdbiaadsgacaWG0baaaiabg2 da9iabgkHiTiaadogapaWaaSbaaSqaa8qacaWGLbGaamiBaiaadwga caWGJbaapaqabaGcpeGaai4TaiaadgeapaWaaSbaaSqaa8qacaWGLb GaamiBaiaadwgacaWGJbaapaqabaGcpeGaai4TaiaadwgacaWG4bGa aeiCamaabmaapaqaa8qacqGHsisldaWcaaWdaeaapeGaamyra8aada WgaaWcbaWdbiaadwgacaWGSbGaamyzaiaadogaa8aabeaaaOqaa8qa caWGRbWdamaaBaaaleaapeGaamOyaaWdaeqaaOWdbiaadsfaaaaaca GLOaGaayzkaaaaaa@5A16@

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):

dSoC dt = T isc ·SoC· A ec ·exp E ec k b T MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiaadsgacaWGtbGaam4Baiaadoeaa8aabaWdbiaa dsgacaWG0baaaiabg2da9iabgkHiTiaabsfapaWaaSbaaSqaa8qaca qGPbGaae4Caiaabogaa8aabeaak8qacaGG3cGaam4uaiaad+gacaWG dbGaai4TaiaadgeapaWaaSbaaSqaa8qacaWGLbGaam4yaaWdaeqaaO WdbiaacElacaWGLbGaamiEaiaabchadaqadaWdaeaapeGaeyOeI0Ya aSaaa8aabaWdbiaadweapaWaaSbaaSqaa8qacaWGLbGaam4yaaWdae qaaaGcbaWdbiaadUgapaWaaSbaaSqaa8qacaWGIbaapaqabaGcpeGa amivaaaaaiaawIcacaGLPaaaaaa@56E3@

In which T ISC MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiva8aadaWgaaWcbaWdbiaadMeacaWGtbGaam4qaaWdaeqaaaaa @39AE@ 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:

i dt = α i n 1 α i m · A a,i ·exp E a,i / k b T MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiaabsgacaqGXoWdamaaBaaaleaapeGaaeyAaaWd aeqaaaGcbaWdbiaabsgacaqG0baaaiabg2da9iaabg7apaWaa0baaS qaa8qacaqGPbaapaqaa8qacaqGUbaaaOWaaeWaa8aabaWdbiaaigda cqGHsislcaqGXoWdamaaBaaaleaapeGaaeyAaaWdaeqaaaGcpeGaay jkaiaawMcaa8aadaahaaWcbeqaa8qacaqGTbaaaOGaai4Taiaadgea paWaaSbaaSqaa8qadaqadaWdaeaapeGaamyyaiaacYcacaWGPbaaca GLOaGaayzkaaaapaqabaGcpeGaai4TaiaadwgacaWG4bGaamiCamaa bmaapaqaa8qacqGHsislcaWGfbWdamaaBaaaleaapeWaaeWaa8aaba WdbiaadggacaGGSaGaamyAaaGaayjkaiaawMcaaaWdaeqaaOWdbiaa c+cadaqadaWdaeaapeGaam4Aa8aadaWgaaWcbaWdbiaadkgaa8aabe aak8qacaWGubaacaGLOaGaayzkaaaacaGLOaGaayzkaaaaaa@5F8A@

Where A a , i MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyqa8aadaWgaaWcbaWdbmaabmaapaqaa8qacaWGHbGaaiilaiaa dMgaaiaawIcacaGLPaaaa8aabeaaaaa@3B59@ is the frequency factor for a specific stage (1/s), E a , i MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyra8aadaWgaaWcbaWdbmaabmaapaqaa8qacaWGHbGaaiilaiaa dMgaaiaawIcacaGLPaaaa8aabeaaaaa@3B5D@ is the activation energy (J), k b MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Aa8aadaWgaaWcbaWdbiaadkgaa8aabeaaaaa@383E@ is the Boltzmann constant, T MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamivaaaa@36E6@ 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,

m c p dT dt = h i · i dt MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyBaiaadogapaWaaSbaaSqaa8qacaWGWbaapaqabaGcpeWaaSaa a8aabaWdbiaadsgacaWGubaapaqaa8qacaWGKbGaamiDaaaacqGH9a qpcqGHris5caWGObWdamaaBaaaleaapeGaamyAaaWdaeqaaOWdbiaa cEladaWcaaWdaeaapeGaaeizaiaabg7apaWaaSbaaSqaa8qacaqGPb aapaqabaaakeaapeGaaeizaiaabshaaaaaaa@48FF@

Where h i MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiAa8aadaWgaaWcbaWdbiaadMgaa8aabeaaaaa@3842@ is the enthalpy of the component defined by,

h i = m i c p,i ΔT MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiAa8aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacqGH9aqpcaWG TbWdamaaBaaaleaapeGaamyAaaWdaeqaaOWdbiaadogapaWaaSbaaS qaa8qacaWGWbGaaiilaiaadMgaa8aabeaak8qacaqGuoGaamivaaaa @4198@

The heat contribution for each stage combines to form the heat source, S T R MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ua8aadaWgaaWcbaWdbiaadsfacaWGsbaapaqabaaaaa@38EF@ , 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:

S TR = ρ cell c p,cell dT dt MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ua8aadaWgaaWcbaWdbiaadsfacaWGsbaapaqabaGcpeGaeyyp a0JaeqyWdi3damaaBaaaleaapeGaam4yaiaadwgacaWGSbGaamiBaa WdaeqaaOWdbiaadogapaWaaSbaaSqaa8qacaWGWbGaaiilaiaadoga caWGLbGaamiBaiaadYgaa8aabeaak8qadaWcaaWdaeaapeGaaeizai aadsfaa8aabaWdbiaabsgacaqG0baaaaaa@4A98@

Where ρ cell MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyWdi3damaaBaaaleaapeGaam4yaiaadwgacaWGSbGaamiBaaWd aeqaaaaa@3BDB@ is the effective density of the cell, c p,cell MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ya8aadaWgaaWcbaWdbiaadchacaGGSaGaam4yaiaadwgacaWG SbGaamiBaaWdaeqaaaaa@3CA8@ is the effective specific heat of the cell and dT/dt MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamizaiaadsfacaGGVaGaamizaiaadshaaaa@3A64@ is the heat rate data directly read from the ARC test.