Fully coupled thermal-electrochemical-structural analysis is intended for the
analysis of battery electrochemistry applications that require solving simultaneously
for displacements, temperature, electric potentials in the solid electrodes, electric
potential in the electrolyte, concentration of ions in the electrolyte, and
concentration in the solid particles used in the electrodes.
The primary example of a battery electrochemistry application is the charging and
discharging of lithium-ion battery cells. During the charging cycle, the lithium
ions are extracted (deintercalated) from the active particles of the positive
electrode (cathode). This process results in a reduction of the volume of the active
particles. The ions move through the electrolyte by migration and diffusion from the
positive electrode to the negative electrode (anode). At the anode, the ions
intercalate into the active particles. This process results in an increase of the
volume of the active particles and induces significant variation in tortuosities on
both electrodes, thus strongly influencing the overall electrochemical behavior.
Heat is generated during the flow of current in the solid and liquid phases, flow of
current in the solid-liquid interface, and flow of ions in the electrolyte. During
discharging, the cycle is reversed.
Rechargeable lithium-ion batteries are widely used in a variety of applications,
including portable electronic devices and electric vehicles. The performance of a
battery highly depends on the effects of repeated charging and discharging cycles,
which can cause the degradation of the battery capacity over time. The porous
electrode theory (Newman et al., 2004) is commonly accepted as the
leading method for modeling the charge-discharge behavior of lithium-ion cells. The
method is based on a homogenized Newman-type approach that does not consider the
details of the pore geometry. The porous electrode theory is based on a concurrent
solution of a highly coupled multiphysics-multiscale formulation. For further
details on the thermal-electrochemical analysis, see Coupled Thermal-Electrochemical Analysis.
In some applications, a detailed understanding of the effects of the
thermal-electrochemical fields on the mechanical state (deformations) of the
lithium-ion cell is important and can have a strong influence on the overall
performance of the cell. Battery performance characteristics (such as energy storage
capacity and discharge voltages) can degrade with deformations caused by particle
swelling and thermal strains. Large deformations can cause failure of the separator,
resulting in thermal runaways in batteries. In such applications, the coupling
between temperature and displacement fields may be due to temperature-dependent
material properties, internal heat generation, or thermal expansion. The volume
changes in the active electrode particles resulting from the
intercalation/deintercalation of lithium ions are modeled as particle
concentration–dependent eigenstrains at the macroscale level. These volume changes
affect the displacement field in the cell and porosity and tortuosity evolutions. In
addition, they generate a convective transport term in the electrolyte that
influences the movement of lithium ions through the electrolyte.
Governing Equations
The governing equations for the thermal-electrochemical process are based on the
porous electrode theory and are described in detail in Governing Equations. The governing equations for particle swelling are also described in Particle Swelling. The volumetric strain that is computed based on particle swelling affects the
macroscale quantities, such as the solid volume fraction, ; porosities; tortuosities; and eigenstrain.
Eigenstrain (also referred to as inherent strain, assumed strain, or "stress-free"
strain) is an engineering concept used to account for all sources of inelastic
deformation that lead to residual stresses and distortions in manufactured
components. For example, thermal strains are an example of eigenstrains. Abaqus solves for mechanical equilibrium of all internal and external forces in the
system.
In linear elastic deformation, the stress induced by an eigenstrain can be
represented as
where
is the Cauchy stress;
is the elasticity matrix;
is the total strain;
is the eigenstrain; and
is the elastic strain.
Using constitutive equations (such as that shown above), eigenstrains can be used to
compute the stresses coming from mechanical, thermal, and microstructural sources.
It is possible to include the effects of particle swelling in both the coupled
thermal-electrochemical and the coupled thermal-electrochemical-structural analyses.
In a purely coupled thermal-electrochemical analysis, particle swelling results in
convection of the electrolyte, which leads to changes in the porosity and the
tortuosity of the electrodes. In a coupled thermal-electrochemical-structural
analysis, particle swelling also results in mechanical deformations that are modeled
as eigenstrains and results in stresses in the medium. Such mechanical deformations
can also have an impact on the performance of a battery cell.
Fully Coupled Solution Scheme
A fully coupled solution scheme is needed when the stress analysis is dependent on
the other fields involved in an electrochemical analysis, such as temperature,
electric potentials in the solid and electrolyte, and ion concentration. In Abaqus/Standard, the temperature is integrated in time using a backward-difference scheme. The
nonlinear coupled system is solved using Newton's method. The coupled
thermal-electrochemical-structural analysis in Abaqus uses an exact implementation of Newton’s method, leading to an unsymmetric
Jacobian matrix in the form:
Steady-State Analysis
Steady-state analysis provides the steady-state solution by neglecting the transient
terms in the continuum scale equations. It can be used to achieve a balanced initial
state or to assess conditions in the cell after a long storage period.
In the thermal equation, the internal energy term in the governing heat transfer
equation is omitted. Similarly, the transient term is omitted in the diffusion
equations for the lithium ion concentration in the electrolyte. Electrical transient
effects are not included in the equations because they are very rapid compared to
the characteristic times of thermal and diffusion effects. A steady-state analysis
has no effect on the microscale solution; the transient terms are always considered
in the solution of the lithium concentration in the solid particle.
In a transient analysis, the transient effects in the heat transfer and diffusion
equations are included in the solution. Electrical transient effects are always
omitted because they are very rapid compared to the characteristic times of thermal
and mass diffusion effects.
Spurious Oscillations due to Small Time Increments
The integration procedure Abaqus/Standard uses for the transient heat transfer equation introduces a relationship between
the minimum usable time increment and the element size. Time increments below this
minimum can lead to spurious oscillations in the temperature field. By default, Abaqus/Standard uses nodal integration for the heat capacity term for first-order elements in a
transient analysis, resulting in a lumped treatment for this term. This treatment
eliminates nonphysical oscillations; however, it can lead to locally inaccurate
solutions, especially in terms of the heat flux for small time increments. If
smaller time increments are required, you should use a finer mesh in regions where
the temperature changes occur. For additional details for time integration involving
temperature degree of freedom, see Spurious Oscillations due to Small Time Increments in Uncoupled Heat Transfer Analysis.
Controlling the Analysis Based on Solution Variables
A battery analysis typically consists of four (or more) steps, in the following order:
A charging step, in which the battery is subjected to a constant charging
current. The voltage is monitored,and the step is terminated when the voltage
increases to a specified cut-off value.
A constant voltage step, in which the battery is subjected to a constant
prescribed voltage. The reaction current is monitored, and the step is
terminated when the reaction current drops below a specified threshold
value.
A resting step, in which the battery is not subjected to any prescribed current
or voltage.
A discharge step, in which a constant current is drawn from the battery. The
voltage is monitored, and the step is terminated when the voltage drops to a
specified cut-off value.
You can use the step control mechanism (see Controlling the Analysis Based on the Simulation State in Abaqus/Standard) together with sensors (see Defining Sensors) to model
the workflow described above. In particular, you can associate the history output of
the voltage or the current at specific locations in the model with sensor
definitions, and terminate a step when the sensor reaches a predefined cut-off value
or threshold.
Initial Conditions
By default, the initial values of electric potential in the solid, temperature,
electric potential in the electrolyte, and ion concentration of all nodes are set to
zero. You can specify nonzero initial values for the primary solution variables (see
Initial Conditions).
Boundary Conditions
You can prescribe the following boundary conditions:
Displacement degrees of freedom (degrees of freedom 1, 2, and 3).
Electric potential in the solid, (degree of freedom 9).
Electric potential in the electrolyte, (degree of freedom 32).
Temperature, (degree of freedom 11).
Ion concentration in the electrolyte, (degree of freedom 33) at the nodes.
You can specify boundary conditions as functions of time by referring to amplitude
curves.
A boundary without any prescribed boundary conditions corresponds to an insulated
(zero flux) surface.
The typical boundary condition consists of only grounding (setting to zero) the solid
electric potential at the anode. Thermal boundary conditions vary.
Loads
You can apply mechanical, thermal, electrical, and electrochemical loads in a coupled
thermal-electrochemical-structural analysis.
Concentrated nodal forces on displacement degrees of freedom.
Distributed forces.
You can prescribe the following types of thermal loads (as described in Thermal Loads):
Concentrated heat flux.
Body flux and distributed surface flux.
Convective film and radiation conditions.
You can prescribe the following types of electrical loads on the solid (as described
in Electromagnetic Loads):
Concentrated current.
Distributed surface current densities and body current densities.
You can prescribe the following types of electrical loads on the electrolyte (as
described in Electromagnetic Loads):
Concentrated current.
Distributed surface current densities and body current densities.
You can prescribe the following types of ion concentration loads (as described in
Thermal Loads):
Concentrated flux.
Distributed body flux.
The typical loads include specification of a solid electric flux (current) at the
cathode. Thermal boundary conditions vary but typically include convective film on
the exterior surfaces. Customarily, no loads are applied on the concentrations and
electrolyte potential.
Predefined Fields
Predefined temperature fields are not allowed in coupled
thermal-electrochemical-structural analyses. Instead, you can use boundary
conditions to prescribe degree of freedom 11. You can specify other predefined field
variables in a fully coupled thermal-electrochemical-structural analysis. These
values affect only field variable–dependent material properties, if any.
Material Options
The material definition in a fully coupled thermal-electrochemical-structural
analysis must include thermal, electrical, electrochemical, and mechanical
properties.
The electrochemistry framework requires that the material definition contain the
complete specification of properties required for the porous electrode theory, as
described in Material Options. In
addition, the material name must begin with
"ABQ_EChemPET_" to enable the coupled micro-macro
solution at the different electrodes. Special-purpose parameter and property tables
of type names starting with “ABQ_EChemPET_” are
required in these material definitions (see Parameter Table Type Reference and Property Table Type Reference). For more details about the material definitions for the
thermal-electrochemical behavior, see Material Options.
You can define anisotropic swelling of the particle to be used in eigenstrain
computations.
Input File Usage
Use the following options to specify properties for the porous electrode
theory:
A fully coupled thermal-electrochemical-structural analysis requires the use of
elements that have displacement (degrees of freedom 1, 2, 3), electric potential in
the solid (degree of freedom 9), temperature (degree of freedom 11), electric
potential in the electrolyte (degree of freedom 32), and ion concentration in the
electrolyte (degree of freedom 33) as nodal variables. The coupled
thermal-electrochemical-structural elements are available in Abaqus/Standard only in three dimensions (see Coupled Thermal-Electrochemical-Structural Elements).
A fully coupled thermal-electrochemical-structural analysis can also include regions
modeled using the following categories of nonelectrochemistry elements:
three-dimensional stress/displacement solid elements, three-dimensional
stress/displacement shell elements, three-dimensional coupled
temperature-displacement solid elements, three-dimensional coupled
temperature-displacement shell elements, as well as rigid elements
R3D3 and
R3D4.
Output
In addition to the output quantities available for the coupled thermal-electric and
the coupled thermal-electrochemical procedures, you can request the Abaqus/Standard output variables for mechanical degrees of freedom.