Coupled Thermal-Electrochemical Analysis

The intercalation process at the solid-liquid interface is governed by electrochemical kinetics and is defined by the Butler-Volmer equation. The Butler-Volmer equation describes the relationship between the electric current density through the solid electrode-electrolyte interface, $I_{\mathrm{BV}}$ , and the voltage difference between the electrode and electrolyte according to:

where $I_0$ and $\eta$ are the exchange current density and overpotential, respectively; ${\sigma }_m$ is the mean stress or hydrostatic pressure; and $\mathrm{\Omega }$ represents the partial molar volume of lithium in the electrode material. The term ${\sigma }_m\mathrm{\Omega }$ captures the effects of mechanical stress on the electrical response of the electrode material. The overpotential, $\eta$ , is defined as:

The exchange current density, $I_0$ , can be written as:

In the above equations,

$F$

is Faraday's constant;

$R$

is the gas constant;

$z$

is the charge number of the lithium ion battery;

$\theta$

is temperature;

${\theta }^z$

is the absolute zero temperature,

$k_c,k_a$

are the cathodic and anodic rate constants, respectively;

${\alpha }_c,{\alpha }_a$

are the cathodic and anodic transfer coefficients, respectively;

$C_s^{\max }$

is the theoretical maximum lithium capacity;

${\hat{C}}_{s,P}^{\mathrm{surf}}$

is the normalized surface lithium concentration in the microscale particle;

$C_e^{\mathrm{ref}}$

is the reference value of lithium ion concentration in the electrolyte; and

$E_{\mathrm{ref}}$

is the open circuit potential (OCP) as a function of ${\hat{C}}_{s,P}^{\mathrm{surf}}$ and $\theta$ .

The normalized concentration, ${\hat{C}}_{s,P}$ , is defined as ${\hat{C}}_{s,P}=\frac{C_{s,P}}{C_{s,P}^{\max }}$ , where $C_{s,P}$ is the lithium concentration in the particle. The normalized surface concentration, ${\hat{C}}_{s,P}^{\mathrm{surf}}$ , is defined as ${\hat{C}}_{s,P}^{\mathrm{surf}}=\frac{C_{s,P}^{\mathrm{surf}}}{C_{s,P}^{\max }}$ , where $C_{s,P}^{\mathrm{surf}}$ is the lithium concentration on the surface of the particle.

The equilibrium equation for the current density, ${\mathbf{I}}_s$ , in the solid phase is given by:

where

${\mathbf{K}}_s^{\mathrm{eff}}$ is the effective electrical conductivity of the solid phase computed using the Bruggeman relationship for tortuosity ( ${\tau }_s={\epsilon }_s^{\mathrm{-\alpha }}$ ). Tortuosity represents an intrinsic property of a porous material characterizing the ratio of the actual flow path length to the straight distance between the ends.

For isotropic tortuosity:

For anisotropic tortuosity, the above expression can be written as:

where

$K_s$

is the electric conductivity of the bulk solid material (which can be a function of normalized concentration);

${\tau }_s$

is the tortuosity of the solid porous electrode;

${\epsilon }_s$

is the volume fraction of the solid phase;

$\mathbf{I}$

is the unit tensor;

$\alpha$

is the Bruggeman exponent for isotropic tortuosity;

${\alpha }_x,{\alpha }_y,{\alpha }_z$

are the Bruggeman exponents for anisotropic tortuosity; and

$a_s$

is the wetted particle surface area per unit volume, computed by default as $a_s=\frac{3{\epsilon }_s}{R_p}$ , where $R_p$ is the outer radius of the spherical particle associated with the microscale model.

The current density in the electrolyte, ${\mathbf{I}}_e$ , and the lithium ion concentration, $C_e$ , in the liquid phase are solved for using the following governing equations:

where

In the above equations,

${\mathbf{K}}_e^{\mathrm{eff}}$

is the effective electrical conductivity in the electrolyte phase,

${\mathbf{D}}_e^{\mathrm{eff}}$

is the effective ion diffusivity in the electrolyte phase,

$t_{+}$

is the transference number that defines the fraction of the total electrical current that is carried by lithium ions in the electrolyte, and

$\left(\frac{{\mathrm{d\; ln\; f}}_{\pm }}{{\mathrm{d\; ln\; C}}_e}\right)$

is the molar activity coefficient that accounts for deviations from the ideal behavior in a mixture of chemical substances.

${\mathbf{K}}_e^{\mathrm{eff}}$ and ${\mathbf{D}}_e^{\mathrm{eff}}$ are computed using the Bruggeman relationship for tortuosity.

For isotropic tortuosity:

For anisotropic tortuosity, the above expression can be written as:

where

$K_e,D_e$

are the electric conductivity and ion diffusivity of the electrolyte, respectively;

$\mathbf{I}$

is the unit tensor;

${\tau }_e$

is the tortuosity of the liquid (electrolyte) phase;

${\epsilon }_e$

is the volume fraction of the liquid phase;

$\alpha$

is the Bruggeman exponent for isotropic tortuosity; and

${\alpha }_x,{\alpha }_y,{\alpha }_z$

are the Bruggeman exponents for anisotropic tortuosity.

The active material at the electrodes is assumed to consist of spherical microparticles. The particles are assumed to have electrical contact between themselves but no interparticle diffusion. It is assumed that the microparticle material is a good electronic conductor (transport number=1) and has spherical symmetry. The conservation of lithium inside the particle is governed by Fick’s second law of diffusion, written in spherical coordinates:

where

$r$

is the radial coordinate of the spherical particle,

$C_{s,P}$

is the lithium concentration in the particle, and

$D_s\left(C_{s,P}\right)$

is the concentration-dependent diffusion constant of the solid material.

The time evolution of lithium concentration in the particle is determined by diffusion and the intercalation current density, $I_{\mathrm{BV}}$ , from the electrochemical model in the continuum scale. The microscale particle is modeled as a sphere over which a one-dimensional finite element mesh in spherical coordinates is created internally by Abaqus, and the diffusion problem is solved. You can specify the discretization for the internal mesh and the type of meshing over the microscale particle (see Particle Layer Mesh).

The coupling of the microscale with the continuum scale happens through the Butler-Volmer current density, $I_{\mathrm{BV}}$ , from the electrochemical model:

The total heat generation in a battery is attributed to the contribution of five different sources: flow of current at the solid-liquid interface, flow of current in the solid phase, flow of current in the liquid phase, flow of ions, and entropy generation. The amount of energy converted into heat from each of the source terms can be scaled using the conversion factors, ${\beta }_{\alpha }$ ( $\alpha$ =1–5):

where

$Q_1=a_s{I}_{\mathrm{BV}}\eta$

is the ohmic loss at the solid-liquid interface;

$Q_2=a_s{I}_{\mathrm{BV}}(\theta -{\theta }^z)\frac{\mathrm{\partial U}}{\partial \theta }$

is the entropy generation, where $\frac{\mathrm{\partial U}}{\partial \theta }$ is the derivative of the open circuit potential with respect to temperature that you can specify in tabular form;

$Q_3={\mathbf{K}}_s^{\mathrm{eff}}\nabla {\Phi }_s\cdot \nabla {\Phi }_s$

is the ohmic loss in the electrode;

$Q_4={\mathbf{K}}_e^{\mathrm{eff}}\nabla {\Phi }_e\cdot \nabla {\Phi }_e$

is the ohmic loss in the electrolyte; and

$Q_5=\frac{2R(\theta -{\theta }^z)}{F}{\mathbf{K}}_e^{\mathrm{eff}}(1+\frac{\mathrm{d\; ln}{f}_{\pm }}{\mathrm{d\; ln}{C}_e})(1-t_{+})\nabla \ln C_e\mathrm{\cdot \nabla }{\Phi }_e$

is the ohmic loss due to ion diffusion.