Thermo-Electric Battery Solution

The design and performance of lithium-ion battery cells, modules and packs is important for a wide variety of battery applications. The thermo-electric battery solution enables detailed analysis of thermal and electric design and performance for these applications under a wide array of scenarios. The analysis is enabled through four core modeling components:
Battery cell core modeling
The cell is modeled as a homogenized region, where the battery voltage response and associated heat generation is determined using the solution of an ODE system. The solution utilizes two approaches: single potential and multi-scale multi-dimensional (MSMD).
Charge conservation
In electrically conducting components, for example, connectors, busbars, and terminals, the charge conservation equation is solved and coupled to the energy equation as a source term.
Transient thermal conduction
Transient thermal conduction in all cell/module/pack components is solved using the energy equation.
Cooling
Optionally, the Navier-Stokes equation can be included to design and enhance cooling systems.

Collectively, the components allow comprehensive analysis and optimization of the thermal and electrical performance of battery systems.

Two overarching methods are employed in the thermo-electric battery solution: single potential and multi-scale multi-dimensional (MSMD). The former solution is applicable for all battery types, for example, cylindrical-cells, prismatic and pouch-type cells, while the latter is only applicable for pouch cells. Each approach solves a variant of charge conservation and the energy equation for the batteries:
Single potential
This approach uses equivalent circuit models (ECM) to characterize the voltage-current response. From the ECM, a source term can be derived and applied to the energy equation and boundary conditions based on current and terminal voltage drop for charge conservation in the connected components of the batteries, for example, terminals/tabs, busbars and any other electrically conducting component.
MSMD
This approach employs a multi-scale method, with a cell scale and a sub-domain scale. On the cell scale, two potential fields represent the potentials of the negative and positive current collectors. The sub-domain model, currently only supporting ECM models, represents the voltage-current response of the battery derived from the solution of an ODE system. Inter-domain coupling between the sub-domain and cell scale is achieved via averaging source terms to eliminate any spatial dependence, in the dual electric potential fields and the energy equation. Cell scale to sub-domain coupling is achieved using spatially resolved variables directly in the sub-domain equations.

The ECM models supported are first, second and third order, where the former has the advantage of simplicity, for example, fewer parameters, and the latter two give a more accurate battery cell voltage response.

A more detailed summary of the models is given in the following sections.

Equivalent Circuit Modeling

Figure 1. 2nd Order Equivalent Circuit Battery Model


In an equivalent circuit model (ECM), the electric behavior, for example, the voltage-current response, is modeled using a phenomenological electric circuit approach. For example, in a 2nd order ECM (see Figure 1), that is, two RC pairs, the governing set of ODEs are given by

U T = V ocv soc , T  V 1 V 2 R 0 soc , T i t MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeyva8aadaWgaaWcbaWdbiaabsfaa8aabeaak8qacqGH9aqpcaqG wbWdamaaBaaaleaapeGaae4BaiaabogacaqG2baapaqabaGcpeWaae Waa8aabaWdbiaabohacaqGVbGaae4yaiaacYcacaqGubaacaGLOaGa ayzkaaGaeyOeI0IaaeiOaiaabAfapaWaaSbaaSqaa8qacaaIXaaapa qabaGcpeGaeyOeI0IaaeOva8aadaWgaaWcbaWdbiaaikdaa8aabeaa k8qacqGHsislcaqGsbWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbm aabmaapaqaa8qacaqGZbGaae4BaiaabogacaGGSaGaaeivaaGaayjk aiaawMcaaiaaykW7caaMc8UaamyAamaabmaapaqaa8qacaWG0baaca GLOaGaayzkaaaaaa@59F1@
dV 1 d t = 1 R 1 soc , T C 1 soc , T V 1 1 C 1 soc , T i t MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiaabsgacaqGwbWdamaaBaaaleaapeGaaGymaaWd aeqaaaGcbaWdbiaabsgacaWG0baaaiabg2da9iabgkHiTmaalaaapa qaa8qacaaIXaaapaqaa8qacaqGsbWdamaaBaaaleaapeGaaGymaaWd aeqaaOWdbmaabmaapaqaa8qacaqGZbGaae4BaiaabogacaGGSaGaae ivaaGaayjkaiaawMcaaiaaboeapaWaaSbaaSqaa8qacaaIXaaapaqa baGcpeWaaeWaa8aabaWdbiaabohacaqGVbGaae4yaiaacYcacaqGub aacaGLOaGaayzkaaaaaiaabAfapaWaaSbaaSqaa8qacaaIXaaapaqa baGcpeGaeyOeI0YaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaboeapa WaaSbaaSqaa8qacaaIXaaapaqabaGcpeWaaeWaa8aabaWdbiaaboha caqGVbGaae4yaiaacYcacaqGubaacaGLOaGaayzkaaaaaiaadMgada qadaWdaeaapeGaamiDaaGaayjkaiaawMcaaaaa@5D79@
dV 2 d t = 1 R 2 soc , T C 2 soc , T V 2 1 C 2 soc , T i t MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiaabsgacaqGwbWdamaaBaaaleaapeGaaGOmaaWd aeqaaaGcbaWdbiaabsgacaWG0baaaiabg2da9iabgkHiTmaalaaapa qaa8qacaaIXaaapaqaa8qacaqGsbWdamaaBaaaleaapeGaaGOmaaWd aeqaaOWdbmaabmaapaqaa8qacaqGZbGaae4BaiaabogacaGGSaGaae ivaaGaayjkaiaawMcaaiaaboeapaWaaSbaaSqaa8qacaaIYaaapaqa baGcpeWaaeWaa8aabaWdbiaabohacaqGVbGaae4yaiaacYcacaqGub aacaGLOaGaayzkaaaaaiaabAfapaWaaSbaaSqaa8qacaaIYaaapaqa baGcpeGaeyOeI0YaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaboeapa WaaSbaaSqaa8qacaaIYaaapaqabaGcpeWaaeWaa8aabaWdbiaaboha caqGVbGaae4yaiaacYcacaqGubaacaGLOaGaayzkaaaaaiaadMgada qadaWdaeaapeGaamiDaaGaayjkaiaawMcaaaaa@5D7E@

Where U T MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeyva8aadaWgaaWcbaWdbiaabsfaa8aabeaaaaa@3816@ is the terminal voltage, soc is the state of charge, T is the temperature, V ocv MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOva8aadaWgaaWcbaWdbiaab+gacaqGJbGaaeODaaWdaeqaaaaa @3A11@ is the open circuit voltage, R 0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOua8aadaWgaaWcbaWdbiaaicdaa8aabeaaaaa@37F6@ is the ohmic resistance, R 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOua8aadaWgaaWcbaWdbiaaigdaa8aabeaaaaa@37F7@ and R 2 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOua8aadaWgaaWcbaWdbiaaikdaa8aabeaaaaa@37F8@ are the polarization resistances of the first and second RC pairs respectively, C 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4qa8aadaWgaaWcbaWdbiaaigdaa8aabeaaaaa@37E8@ and C 2 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4qa8aadaWgaaWcbaWdbiaaikdaa8aabeaaaaa@37E9@ are the polarization capacitance of the first and second RC pairs respectively, i t MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAamaabmaapaqaa8qacaqG0baacaGLOaGaayzkaaaaaa@399A@ is the input current, and t is the time.

The output terminal voltage is therefore dependent on the input current and the ECM parameters ( R 0 , R 1 ,  R 2 , C 1 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOua8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaGGSaGaaGPa VlaabkfapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiaacc kacaaMc8UaaeOua8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacaGG SaGaae4qa8aadaWgaaWcbaWdbiaaigdaa8aabeaaaaa@443D@ and C 2 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4qa8aadaWgaaWcbaWdbiaaikdaa8aabeaaaaa@37E9@ ) which can depend on both state of charge and temperature.

In AcuSolve, the sign convention dictates that current is positive during discharge. AcuSolve supports three types of ECM models: 1st, 2nd, and 3rd order. The choice of model order depends on the desired dynamics, such as dynamic current conditions. The 2nd and 3rd order models offer enhanced accuracy for pulse-discharge experiments and dynamic current scenarios. Typically, the 2nd order model is considered the optimal choice, striking a balance between accuracy and ease of parameter determination.

The battery state of charge (soc) is updated using coulomb counting:

dsoc dt = I 3600 Q Ah MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiaabsgacaaMc8Uaae4Caiaab+gacaqGJbaapaqa a8qacaqGKbGaamiDaaaacqGH9aqpdaWcaaWdaeaapeGaamysaaWdae aapeGaaG4maiaaiAdacaaIWaGaaGimaiaaykW7caqGrbWdamaaBaaa leaapeGaaeyqaiaabIgaa8aabeaaaaaaaa@46F6@

where I MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysaaaa@36DB@ is the current flux. State of charge (soc) represents the amount of charge in a battery relative to its nominal capacity ( Q Ah MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeyua8aadaWgaaWcbaWdbiaabgeacaqGObaapaqabaaaaa@38EA@ ).

Single Potential Model

In the single potential model, the equivalent circuit model (ECM) computes the voltage drop and current within the homogenized section of the battery cell, automatically applying associated boundary/interface conditions. Briefly, the ECM model is stepped in time and the terminal voltage at time k is updated. For a 1st order ECM model this would be:

U T k = V O C V I ϕ k R 0 R 1  exp Δ t R 1 C 1 I R 1 k 1 + R 1   1 exp Δ t R 1 C 1 I ϕ k 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyva8aadaqhaaWcbaWdbiaadsfaa8aabaWdbiaadUgaaaGccqGH 9aqpcaWGwbWdamaaBaaaleaapeGaam4taiaadoeacaWGwbaapaqaba GcpeGaeyOeI0Iaamysa8aadaqhaaWcbaWdbiabew9aMbWdaeaapeGa am4AaaaakiaadkfapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaey OeI0IaamOua8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaqGGcGa aeyzaiaabIhacaqGWbWaaeWaa8aabaWdbmaalaaapaqaa8qacqGHsi slcaqGuoGaamiDaaWdaeaapeGaaeOua8aadaWgaaWcbaWdbiaaigda a8aabeaak8qacaqGdbWdamaaBaaaleaapeGaaGymaaWdaeqaaaaaaO WdbiaawIcacaGLPaaacaqGjbWdamaaDaaaleaapeGaaeOua8aadaWg aaadbaWdbiaaigdaa8aabeaaaSqaa8qacaqGRbGaeyOeI0IaaGymaa aakiabgUcaRiaadkfapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGa aeiOamaabmaapaqaa8qacaaIXaGaeyOeI0IaaeyzaiaabIhacaqGWb WaaeWaa8aabaWdbmaalaaapaqaa8qacqGHsislcaqGuoGaamiDaaWd aeaapeGaaeOua8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaqGdb WdamaaBaaaleaapeGaaGymaaWdaeqaaaaaaOWdbiaawIcacaGLPaaa aiaawIcacaGLPaaacaWGjbWdamaaDaaaleaapeGaeqy1dygapaqaa8 qacaWGRbGaeyOeI0IaaGymaaaaaaa@7330@

where I ϕ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysa8aadaWgaaWcbaWdbiabew9aMbWdaeqaaaaa@38FD@ is the current flux on the negative terminal of the battery (from the charge conservation PDE solution), V ocv MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOva8aadaWgaaWcbaWdbiaab+gacaqGJbGaaeODaaWdaeqaaaaa @3A11@ is the open circuit voltage, R 0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOua8aadaWgaaWcbaWdbiaaicdaa8aabeaaaaa@37F6@ is the ohmic resistance, R 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOua8aadaWgaaWcbaWdbiaaigdaa8aabeaaaaa@37F9@ is the polarization resistance of the 1st RC pair, C 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4qa8aadaWgaaWcbaWdbiaaigdaa8aabeaaaaa@37EA@ is the polarization capacitance of the 1st RC pair, I R 1 k 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysa8aadaqhaaWcbaWdbiaadkfapaWaaSbaaWqaa8qacaaIXaaa paqabaaaleaapeGaam4AaiabgkHiTiaaigdaaaaaaa@3BC7@ is diffusion resistor current at time step k-1, and Δ t MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiaadshaaaa@3820@ is the time step size.

The terminal voltage is applied to the positive terminal of the battery for the solution of charge conservation in the electrically conducting components of the battery pack. Proper distribution of current is determined from the pack load and how this distributes to individual cells in the modules that make up the pack. This distribution is based on the properties of a cell, for example, SOC level or temperature. The current distribution is applied as a flux condition at the negative terminals of the battery cells.

The governing equation for the thermal (energy equation) and electrical solution (charge conservation) are given by:

ρ c p d T d t = κ T + q B + q J MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyWdiaadogapaWaaSbaaSqaa8qacaWGWbaapaqabaGcpeWaaSaa a8aabaWdbiaadsgacaWGubaapaqaa8qacaWGKbGaamiDaaaacqGH9a qpcqGHhis0cqGHflY1daqadaWdaeaapeGaaeOUdiabgEGirlaadsfa aiaawIcacaGLPaaacqGHRaWkcaWGXbWdamaaBaaaleaapeGaamOqaa WdaeqaaOWdbiabgUcaRiaadghapaWaaSbaaSqaa8qacaWGkbaapaqa baaaaa@4DC8@
σ ϕ = 0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaey4bIeTaeyyXIC9aaeWaa8aabaWdbiabgkHiTiaabo8acqGHhis0 cqaHvpGzaiaawIcacaGLPaaacqGH9aqpcaaIWaaaaa@42C9@

where q B MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyCa8aadaWgaaWcbaWdbiaadkeaa8aabeaaaaa@3824@ is the heat generation (see later for a description of this heat source) from the battery and q J MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyCa8aadaWgaaWcbaWdbiaadQeaa8aabeaaaaa@382C@ is the joule heat generated in electrically conducting components, σ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4Wdaaa@3756@ is the conductivity and ϕ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dygaaa@37D5@ is the electric potential.

This source term, derived from Joule’s first law, denotes the heat generated per unit volume, calculated as the product of current density j MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa8NAaaaa@3704@ and electric field ( E = ϕ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyraiabg2da9iabgkHiTiabgEGirlabew9aMbaa@3C18@ ):

q J =j·E=σ ϕ 2 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyCa8aadaWgaaWcbaWdbiaadQeaa8aabeaak8qacqGH9aqpieWa caWFQbGaeS4JPFMaamyraiabg2da9iaabo8adaabdaWdaeaapeGaey 4bIeTaeqy1dygacaGLhWUaayjcSdWdamaaCaaaleqabaWdbiaaikda aaaaaa@4763@

MSMD Model

For pouch batteries the thermal-electric response can be modeled using a MSMD approach which bridges the scale between the cell domain and the electrode scale. In this approach electric potential fields are modeled in a cell composite, or more specifically the current collectors, together with the thermal field. This is then coupled using a multi- scale approach down to the sub-scale, where the electrochemical phenomena, that is, the inner workings of the battery, are modeled.

Cell scale model

At the cell scale, the temperature and a pair of electric potential fields are solved. More specifically, the 3D battery geometry is treated as a homogenized medium with orthotropic electrical and thermal conductivities. Positive and negative electric potential is computed in the battery from charge conservation at the current collectors of the cell composite. For the negative current collector, the governing equation is:

σ ϕ j = 0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaey4bIe9aaeWaa8aabaWdbiaabo8apaWaaSbaaSqaa8qacqGHsisl a8aabeaak8qacqGHhis0cqaHvpGzpaWaaSbaaSqaa8qacqGHsisla8 aabeaaaOWdbiaawIcacaGLPaaacqGHsislcaWGQbGaeyypa0JaaGim aaaa@4430@

Equivalently at the positive current collector the governing equation is:

σ + ϕ + + j = 0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaey4bIe9aaeWaa8aabaWdbiaabo8apaWaaSbaaSqaa8qacqGHRaWk a8aabeaak8qacqGHhis0cqaHvpGzpaWaaSbaaSqaa8qacqGHRaWka8 aabeaaaOWdbiaawIcacaGLPaaacqGHRaWkcaWGQbGaeyypa0JaaGim aaaa@440F@

where j MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOAaaaa@36FC@ is the transfer current density and is modeled through an electrochemical sub-model, for example, the ECM model. Additionally, j MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOAaaaa@36FC@ is subject to the constraint:

I Cell = Ω   j d Ω MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysa8aadaWgaaWcbaWdbiaaboeacaqGLbGaaeiBaiaabYgaa8aa beaak8qacqGH9aqpdaqfWaqabSWdaeaapeGaaeyQdaWdaeaapeGaai iOaaqdpaqaa8qacqGHRiI8aaGccaWGQbGaaGPaVlaadsgacaqGPoaa aa@455D@

where I Cell MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysa8aadaWgaaWcbaWdbiaaboeacaqGLbGaaeiBaiaabYgaa8aa beaaaaa@3AC1@ is the current through the cell. The above system of equations comes with associated boundary conditions such as current, for example, constant current discharge, or voltage.

The energy equation in AcuSolve is modified by an additional source term that couples both sources of ohmic (or joule) heating from the current collectors and electrochemical reaction heat. The total heat source term is given by:

q = σ ϕ 2 + σ + ϕ + 2 + q B MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyCaiabg2da9iaabo8apaWaaSbaaSqaa8qacqGHsisla8aabeaa k8qadaabdaWdaeaapeGaey4bIeTaeqy1dy2damaaBaaaleaapeGaey OeI0capaqabaaak8qacaGLhWUaayjcSdWdamaaCaaaleqabaWdbiaa ikdaaaGccqGHRaWkcaqGdpWdamaaBaaaleaapeGaey4kaScapaqaba GcpeWaaqWaa8aabaWdbiabgEGirlabew9aM9aadaWgaaWcbaWdbiab gUcaRaWdaeqaaaGcpeGaay5bSlaawIa7a8aadaahaaWcbeqaa8qaca aIYaaaaOGaey4kaSIaaeyCa8aadaWgaaWcbaWdbiaabkeaa8aabeaa aaa@5320@

where the first two terms are ohmic heating and the last term, q B MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyCa8aadaWgaaWcbaWdbiaabkeaa8aabeaaaaa@3820@ , is the heat generated by the ECM model.
Note: The ECM model can be closer tied to an electrochemical model by fitting parameters to virtual experimental data provided by a DFN (Doyle Fuller Newman) model.
MSMD multi-cell formulation

In a battery module, batteries are connected in a nSmP pattern, for example, 8s6p, where m batteries are connected in parallel and then n of these parallel connected units are connected in serial to form a pack. In the MSMD method, two potential equations are solved for the ϕ 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dy2damaaBaaaleaapeGaaGymaaWdaeqaaaaa@38EA@ and ϕ 2 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dy2damaaBaaaleaapeGaaGOmaaWdaeqaaaaa@38EB@ electrical field. Depending on battery connectivity, ϕ 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dy2damaaBaaaleaapeGaaGymaaWdaeqaaaaa@38EA@ and ϕ 2 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dy2damaaBaaaleaapeGaaGOmaaWdaeqaaaaa@38EB@ could be the potential from the positive current collector or the negative one. Therefore, in a battery module simulation, the two potential equations are solved with the transfer current density changing sign dependent on the location in the module. The cells are alternating as odd and even stages, see Figure 2. In tabs and busbars, you only solve for one of the fields. In Figure 2, the ϕ 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dy2damaaBaaaleaapeGaaGymaaWdaeqaaaaa@38EA@ and ϕ 2 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dy2damaaBaaaleaapeGaaGOmaaWdaeqaaaaa@38EB@ on the left indicates which field was solved in the tabs/busbars in the line. The governing equations for the cell scale become:

σ ϕ = j MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaey4bIe9aaeWaa8aabaWdbiaabo8acqGHhis0cqaHvpGzaiaawIca caGLPaaacqGH9aqpcaqGQbaaaa@3FC5@

For ϕ 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dy2damaaBaaaleaapeGaaGymaaWdaeqaaaaa@38EA@ in an odd stage:

σ = σ +        j =  I / V MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4Wdiabg2da9iaabo8apaWaaSbaaSqaa8qacqGHRaWka8aabeaa k8qacaqGGcGaaeiOaiaabckacaqGGcGaaeiOaiaabckacaqGGcGaae OAaiabg2da9iabgkHiTiaabckacaqGjbGaai4laiaabAfaaaa@494B@

For ϕ 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dy2damaaBaaaleaapeGaaGymaaWdaeqaaaaa@38EA@ in an even stage:

σ = σ j = I / V MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4Wdiabg2da9iaabo8apaWaaSbaaSqaa8qacqGHsisla8aabeaa kiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7peGaaeOAaiabg2 da9iaabMeacaGGVaGaaeOvaaaa@6143@

For ϕ 2 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dy2damaaBaaaleaapeGaaGOmaaWdaeqaaaaa@38EB@ in an odd stage:

σ = σ           j =  I / V MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4Wdiabg2da9iaabo8apaWaaSbaaSqaa8qacqGHsisla8aabeaa k8qacaqGGcGaaeiOaiaabckacaqGGcGaaeiOaiaabckacaqGGcGaae iOaiaabckacaqGGcGaaeOAaiabg2da9iaabckacaqGjbGaai4laiaa bAfaaaa@4BD2@

For ϕ 2 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dy2damaaBaaaleaapeGaaGOmaaWdaeqaaaaa@38EB@ in an even stage:
σ= σ + j=I/V MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4Wdiabg2da9iaabo8apaWaaSbaaSqaa8qacqGHRaWka8aabeaa kiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7peGaaeOAaiabg2da9iabgkHiTiaabMeaca GGVaGaaeOvaaaa@5D84@
Figure 2. Odd and Even Stages for Connected Cells


Bernardi’s Heat Generation Equation

Bernardi’s heat generation equation represents the irreversible and reversible heat generation from the battery and is included in the energy equation as a source term derived from the solution of the ECM model, given by

q B = i t Volume cell V ocv U T T dV oc dT + σ + 2 σ + + σ 2 σ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyCa8aadaWgaaWcbaWdbiaadkeaa8aabeaak8qacqGH9aqpdaWc aaWdaeaapeGaaeyAamaabmaapaqaa8qacaqG0baacaGLOaGaayzkaa aapaqaa8qacaqGwbGaae4BaiaabYgacaqG1bGaaeyBaiaabwgapaWa aSbaaSqaa8qacaWGJbGaamyzaiaadYgacaWGSbaapaqabaaaaOWdbm aabmaapaqaa8qadaqadaWdaeaapeGaaeOva8aadaWgaaWcbaWdbiaa b+gacaqGJbGaaeODaaWdaeqaaOWdbiabgkHiTiaabwfapaWaaSbaaS qaa8qacaqGubaapaqabaaak8qacaGLOaGaayzkaaGaeyOeI0Iaamiv amaalaaapaqaa8qacaqGKbGaaeOva8aadaWgaaWcbaWdbiaab+gaca qGJbaapaqabaaakeaapeGaaeizaiaabsfaaaaacaGLOaGaayzkaaGa ey4kaSIaae4Wd8aadaWgaaWcbaWdbiabgUcaRaWdaeqaaOWdbiabgE Gir=aadaahaaWcbeqaa8qacaaIYaaaaOGaae4Wd8aadaWgaaWcbaWd biabgUcaRaWdaeqaaOWdbiabgUcaRiaabo8apaWaaSbaaSqaa8qacq GHsisla8aabeaak8qacqGHhis0paWaaWbaaSqabeaapeGaaGOmaaaa kiaabo8apaWaaSbaaSqaa8qacqGHsisla8aabeaaaaa@6A78@

The first term of the RHS represents the cell over potential and includes irreversibilities, such as ohmic losses and mass transport limitations, while the second term represents entropic heat generation. The last two terms represent ohmic losses and are enabled only for the MSMD approach. Bernardi’s heat generation equation is automatically included in the energy equation as a source term for all battery models. The entropic heat can be both temperature and state_of_charge dependent.