Fully Coupled Thermal-Electrochemical-Structural–Pore Pressure Analysis

The primary example of a fully coupled thermal-electrochemical-structural–pore pressure analysis in 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, which in turn leads to an increase in porosity as well as flow of electrolyte into the cathode. 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, which in turn leads to a decrease in porosity as well as flow of electrolyte out of the anode. 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 more information, 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, electrolyte fluid pressures, and electrolyte movement can be important to understand their 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, thermal strains, and ion depletion. Large deformations can cause failure of the separator, resulting in thermal runaways in batteries. In such applications, the coupling between temperature and displacement fields might 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.

The effects of fluid flow and fluid pressure are important to determine localized regions of low electrolyte concentrations within the battery that can degrade battery performance. It is possible to model the different regions in the battery as either fully saturated or partially saturated to get a better understanding of electrolyte flow effects during a charge-discharge cycle.

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 at the microscale, caused by particle swelling, results in macroscopic eigenstrains (also referred to as inherent strain, assumed strain, or "stress-free" strain) that affects the deformation of the porous electrodes, solid volume fraction, ${\epsilon }_s$ , pore fluid pressure in the electrolyte, and electrolyte flow. Abaqus treats the continuum (porous electrodes containing electrolyte) as a multiphase material and adopts an effective stress principle to describe its behavior (see Effective stress principle for porous media). In this approach, the total stress at a material point is assumed to be made up of an effective stress, $\overline{\sigma }$ , which represents the stress in the skeleton of the porous electrode, and an average pressure stress, $u_w$ , in the electrolyte, such that:

The above expression assumes partially saturated electrolyte flow, with saturation level, $s$ , where $s=1.0$ for fully saturated flow and $s<1.0$ represents partially saturated flow. The response of such a porous medium is based on satisfying:

• mechanical equilibrium based on all internal and external forces in the system, as well as
• fluid continuity for the electrolyte phase.

The porous medium is modeled with a Lagrangian finite element mesh that keeps track of the evolving volume fractions of the solid and electrolyte phases at each material point, as well as the flow of the electrolyte phase. The electrolyte is assumed to be relatively incompressible. The effective response of the porous medium can be described in terms of any mechanical constitutive model in Abaqus. For the special case of a linear elastic effective response, the effective stress induced by an eigenstrain can be represented as:

$\overline{\sigma }$

is the effective Cauchy stress;

$D^{el}$

is the elasticity matrix;

$\epsilon$

is the total strain;

${\epsilon }^{*}$

is the eigenstrain; and

${\epsilon }^{el}$

is the elastic strain.

The continuity equation for the electrolyte flow is described in Continuity statement for the wetting liquid phase in a porous medium. The formulation of the continuity equation is in terms of pore pressure as the basic nodal variable (degree of freedom 8 at each node). The conjugate flux is the nodal volumetric flow rate.

The ion flux equation (see Diffusion and Migration of Lithium Ions in the Liquid Phase) can optionally include advection contributions resulting from the velocity field of the electrolyte. If advection effects are included in the model, any electrolyte flux at the external boundaries is associated with a corresponding ion flux (that is, the electrolyte leaves or enters the domain with ions). If advection effects are not included in the model, only "clean electrolyte" (that is, electrolyte without any ions) can leave or enter through the external boundaries.

A fully coupled solution scheme is required when the stress analysis is strongly dependent on the other fields involved in an electrochemical analysis, such as temperature, pore pressure, 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–pore pressure analysis in Abaqus uses an exact implementation of Newton’s method, leading to an unsymmetric Jacobian matrix in the form:

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.

A steady-state analysis neglects the transient terms in all the macroscopic field equations. These include:

• internal energy term in heat transfer;
• transient term in the diffusion equation for the lithium ion concentration in the electrolyte;
• electrical transients in the conduction equations for both the solid and electrolyte phases (characteristic time scales associated with the electrical transient effects are much smaller compared to the characteristic times of thermal and ion-diffusion); and
• electrolyte flow continuity equation. When the modeling of electrolyte flow includes gravity effects, a steady-state step ensures the loads and initial stresses equilibrate exactly.

A steady-state analysis has no effect on the microscale solution; the transient terms are always included in the solution of the lithium concentration in the solid particle.

*COUPLED TEMPERATURE-DISPLACEMENT, ELECTROCHEMICAL, PORE PRESSURE, STEADY STATE, ADVECTION=YES 

*COUPLED TEMPERATURE-DISPLACEMENT, ELECTROCHEMICAL, PORE PRESSURE, STEADY STATE, ADVECTION=NO 

In a transient analysis, the transient effects in the heat transfer, pore fluid continuity, and ion-diffusion equations are included in the solution. Electrical transient effects are always omitted because they are rapid compared to the characteristic times of thermal and ion-diffusion effects.

*COUPLED TEMPERATURE-DISPLACEMENT, ELECTROCHEMICAL, PORE PRESSURE, ADVECTION=YES 

*COUPLED TEMPERATURE-DISPLACEMENT, ELECTROCHEMICAL, PORE PRESSURE, ADVECTION=NO 

In a fully coupled thermal-electrochemical-structural–pore pressure analysis, you can model the presence of an external reservoir of fluid electrolyte with a prescribed ion concentration value of $C_e^{amb}$ .

When you model fluid velocity effects in conjunction with the presence of an external reservoir:

• If the pore pressure inside the domain is higher than the prescribed pore pressure at a boundary node, fluid electrolyte leaves the active domain through the node. The exiting fluid carries an ionic concentration of value equal to that of the ion concentration degree of freedom at that boundary node at that instance in time.
• If the pore pressure inside the domain is lower than the prescribed pore pressure at the boundary node, fluid electrolyte enters the active domain through the node. The entering fluid carries an ionic concentration equal to the prescribed ambient ion concentration value of $C_e^{amb}$ .

*COUPLED TEMPERATURE-DISPLACEMENT, ELECTROCHEMICAL, PORE PRESSURE, ADVECTION=YES, AMBIENT ION CONCENTRATION=$C_e^{amb}$

The integration procedure Abaqus/Standard uses for fluid flow analysis with the pore pressure degree of freedom introduces a relationship between the minimum usable time increment and the element size. If time increments smaller than these values are used, spurious oscillations can appear in the solution.

The integration procedure Abaqus/Standard uses for the transient heat transfer equation also results in similar limitations on the minimum usable time increment. 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 more information about time integration involving the pore pressure and temperature degrees of freedom, see Spurious Oscillations due to Small Time Increments in Uncoupled Heat Transfer Analysis.