# Thermo-Electric Battery Solution

- 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.

- 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

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

Where ${\text{U}}_{\text{T}}$ is the terminal voltage, soc is the state of charge, T is the temperature, ${\text{V}}_{\text{ocv}}$ is the open circuit voltage, ${\text{R}}_{0}$ is the ohmic resistance, ${\text{R}}_{1}$ and ${\text{R}}_{2}$ are the polarization resistances of the first and second RC pairs respectively, ${\text{C}}_{1}$ and ${\text{C}}_{2}$ are the polarization capacitance of the first and second RC pairs respectively, $i\left(\text{t}\right)$ is the input current, and t is the time.

The output terminal voltage is therefore dependent on the input current and the ECM parameters (${\text{R}}_{0},\text{\hspace{0.17em}}{\text{R}}_{1},\text{\hspace{0.17em}}{\text{R}}_{2},{\text{C}}_{1}$ and ${\text{C}}_{2}$ ) 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:

where $I$ is the current flux. State of charge (soc) represents the amount of charge in a battery relative to its nominal capacity (${\text{Q}}_{\text{Ah}}$ ).

## 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
1^{st} order ECM model this would be:

where
${I}_{\varphi}$
is the current flux on the negative terminal of the
battery (from the charge conservation PDE solution),
${\text{V}}_{\text{ocv}}$
is the open circuit voltage,
${\text{R}}_{0}$
is the ohmic resistance,
${R}_{1}$
is the polarization resistance of the 1^{st}
RC pair,
${C}_{1}$
is the polarization capacitance of the
1^{st} RC pair,
${I}_{{R}_{1}}^{k-1}$
is diffusion resistor current at time step k-1, and
$\text{\Delta}t$
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:

where ${q}_{B}$ is the heat generation (see later for a description of this heat source) from the battery and ${q}_{J}$ is the joule heat generated in electrically conducting components, $\text{\sigma}$ is the conductivity and $\varphi $ 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 $j$ and electric field ( $E=-\nabla \varphi $ ):

## 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.

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:

Equivalently at the positive current collector the governing equation is:

where $j$ is the transfer current density and is modeled through an electrochemical sub-model, for example, the ECM model. Additionally, $j$ is subject to the constraint:

where ${I}_{\text{Cell}}$ 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:

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 ${\varphi}_{1}$ and ${\varphi}_{2}$ electrical field. Depending on battery connectivity, ${\varphi}_{1}$ and ${\varphi}_{2}$ 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 ${\varphi}_{1}$ and ${\varphi}_{2}$ on the left indicates which field was solved in the tabs/busbars in the line. The governing equations for the cell scale become:

For ${\varphi}_{1}$ in an odd stage:

For ${\varphi}_{1}$ in an even stage:

For ${\varphi}_{2}$ in an odd stage:

## 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

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.