cideMOD.models.PXD.electrochemical.equations

Classes

ElectrochemicalModelEquations()

class cideMOD.models.PXD.electrochemical.equations.ElectrochemicalModelEquations

Bases: BaseCellModelEquations

ButlerVolmer_equation(alpha, F, R, T, eta)

Implements the Butler-Volmer equation

Parameters:

• alpha (Constant or similar) – Charge transfer coefficient

• F (Constant or similar) – Farady’s constant

• R (Constant or similar) – Universal Gas constant

• T (Constant or similar) – Absolute temperature

• eta (dolfinx.fem.Function) – Activation overpotential

Returns:

Right hand side

Return type:

ufl.Operator

build_weak_formulation(eq: ProblemEquations, var: ProblemVariables, cell: BatteryCell, mesher: BaseMesher, DT: TimeScheme, W: BlockFunctionSpace, problem) → None

This method builds the weak formulation of the electrochemical model.

Parameters:

• equations (ProblemEquations) – Object that contains the system of equations of the problem.

• var (ProblemVariables) – Object that store the preprocessed problem variables.

• cell (BatteryCell) – Object where cell parameters are preprocessed and stored.

• mesher (BaseMesher) – Object that store the mesh information.

• DT (TimeScheme) – Object that provide the temporal derivatives with the specified scheme.

• W (BlockFunctionSpace) – Object that store the function space of each state variable.

• problem (Problem) – Object that handles the battery cell simulation.

build_weak_formulation_stationary(eq: ProblemEquations, var: ProblemVariables, cell: BatteryCell, mesher: BaseMesher, W: BlockFunctionSpace, problem)

This method builds and adds the weak formulation of the electrochemical model that will be used to solve the stationary problem.

Parameters:

• equations (ProblemEquations) – Object that contains the system of equations of the stationary problem.

• var (ProblemVariables) – Object that store the preprocessed problem variables.

• cell (BatteryCell) – Object where cell parameters are preprocessed and stored.

• mesher (BaseMesher) – Object that store the mesh information.

• W (BlockFunctionSpace) – Object that store the function space of each state variable.

• problem (Problem) – Object that handles the battery cell simulation.

build_weak_formulation_transitory(eq: ProblemEquations, var: ProblemVariables, cell: BatteryCell, mesher: BaseMesher, W: BlockFunctionSpace, problem)

This method builds and adds the weak formulation of the electrochemical model that will be used to solve the transitory problem.

Parameters:

• equations (ProblemEquations) – Object that contains the system of equations of the transitory problem.

• var (ProblemVariables) – Object that store the preprocessed problem variables.

• cell (BatteryCell) – Object where cell parameters are preprocessed and stored.

• mesher (BaseMesher) – Object that store the mesh information.

• W (BlockFunctionSpace) – Object that store the function space of each state variable.

• problem (Problem) – Object that handles the battery cell simulation.

c_e_equation(var, dx, DT, cell, component, scale_factor=1)

Implements variational formulation for electrolyte concentration c_e

Parameters:

• cell (BatteryCell) – Object where cell parameters are preprocessed and stored.

• var (ProblemVariables) –

  Object that store the preprocessed problem variables. Requires:

  • c_e_0 : dolfinx.fem.Function - Electrode Potential last timestep

  • c_e : dolfinx.fem.Function - Electrode Potential

  • test : TestFunction - Electrode Potential Equation Test Function

  • j_Li : dolfinx.fem.Function - Intercalation reaction Flux

• d (Measures) – Measure of the domain over the integral must integrate

• DT (TimeScheme) – Instance of the TimeScheme class

• cell –

  Object where cell parameters are preprocessed and stored. Requires:

  • F : Constant or similar - Faraday Constant

  • electrolyte.t_p : Constant or similar - Transference number for the electrolyte

• component (BaseCellComponent) –

  Object where component parameters are preprocessed and stored. Contains:

  • D_e : Constant or similar - Effective Diffusivity of the electrolyte

  • eps_e : Constant or similar - Volume fraction occupied by the electrolyte in the domain

  • grad : dolfinx.fem.Function - python function that returns the UFL gradient of the argument

  • L : Constant - Thickness used to normalize the domain, by default None

Returns:

Electrolyte Concentration Equation

Return type:

Form

explicit_update(problem) → None

This method updates some stuff after the implicit timestep is performed.

Parameters:

problem (Problem) – Object that handles the battery cell simulation.

get_solvers_info(solvers_info, problem) → None

This method get the solvers information that concerns the electrochemical model.

Parameters:

• solvers_info (dict) – Dictionary containing solvers information.

• problem (Problem) – Object that handles the battery cell simulation.

i_n_equation(k, c_e, c_s, c_s_max, alpha)

Implements the interface for electrode potential phi_s

Parameters:

• lagrange_multiplier (dolfinx.fem.Function) – Lagrange multiplier on the interface between the elctrode and the cc associated to the continuity of the potential in the solid face

• k (Constant or similar) – Kinetic constant of the active material

• c_e (dolfinx.fem.Function) – Electrolyte concentration

• c_s (dolfinx.fem.Function) – Electrode concentration

• c_s_max (TestFunction) – Maximum Electrode concentration

• alpha (Constant or similar) – Charge transfer coefficient

Returns:

Regularisation of i_0

Return type:

ufl.Operator

j_Li_equation(var, dx, cell, material)

Exchange between the electrolyte and the electrode by lithium intercalation

Parameters:

• var (ProblemVariables) – Object that store the preprocessed problem variables. Requires:

• d (Measures) – Measure of the domain over the integral must integrate

• cell (BatteryCell) – Object where cell parameters are preprocessed and stored.

• material (BaseCellComponent) – Object where active material parameters are preprocessed and stored.

Returns:

Butler-Volmer equation of the specified material

Return type:

Form

phi_e_equation(var, dx, component, scale_factor=1)

Implements variational formulation equation for electrolyte potential phi_e

Parameters:

• var (ProblemVariables) –

  Object that store the preprocessed problem variables. Contains:

  • phi_e : dolfinx.fem.Function - Electrolyte Potential

  • test.phi_e : TestFunction - Electrolyte Potential Equation Test Function

  • j_Li : dolfinx.fem.Function - Intercalation reaction Flux (Optional)

  • c_e : dolfinx.fem.Function - Electrolyte Concentration

• dx (Measures) – Measure of the domain over the integral must integrate

• component (BaseCellComponent) –

  Object where component parameters are preprocessed and stored. Contains:

  • kappa : Constant or similar - Effective electric conductivity of the electrolyte

  • kappa_D : Constant or similar - Effective concentration induced conductivity (More or less)

  • grad : dolfinx.fem.Function - python function that returns the UFL gradient of the argument

  • L : Constant - Thickness used to normalize the domain, by default None

Returns:

Electrolyte Potential Equation

Return type:

Form

phi_s_bc(var, cell, ds)

Implements boundary conditions for electrode potential equation phi_s

Parameters:

• var (ProblemVariables) –

  Object that store the preprocessed problem variables. Contains:

  • lm_app : Constant, Function or similar - Current applied to the boundary of the electrode

  • test : TestFunction - Electrode Potential Equation Test Function

• cell (BatteryCell) – Object where cell parameters are preprocessed and stored.

• ds (Measure) – Measure of the boundary domain over the integral must integrate

Returns:

Electrode Potential boundary condition Equation

Return type:

Form

phi_s_continuity(var, dS_el, dS_cc)

Implements the continuity equation for electrode potential phi_s

Parameters:

• phi_s_electrode (dolfinx.fem.Function) – Electrode Potential Equation

• phi_s_cc (dolfinx.fem.Function) – Current Collector Potential Equation

• lm_test (TestFunction) – Test Function

• dS_el (Measure) – Measure of the boundary domain over the integral must integrate in the electrolyte

• dS_cc (Measure) – Measure of the boundary domain over the integral must integrate in the cc

Returns:

Potential Continuity Equation

Return type:

Form

phi_s_equation(var, d, cell, component)

Implements variational formulation for electrode potential phi_s

Parameters:

• var (ProblemVariables) –

  Object that store the preprocessed problem variables. Contains:

  • phi_s : dolfinx.fem.Function - Electrode Potential

  • j_Li : dolfinx.fem.Function - Intercalation reaction Flux

  • test : TestFunction - Electrode Potential Equation Test Function

  • lm_phi_s : dolfinx.fem.Function - Lagrange multiplier (Optional)

• d (Measures) –

  • dx : Measure - Measure of the domain over the integral must integrate

  • dS : Measure - Measure of the boundary domain over the integral must integrate

• cell (BatteryCell) – Object where cell parameters are preprocessed and stored.

• component (BaseCellComponent) –

  Object where component parameters are preprocessed and stored. Contains:

  • sigma : Constant or similar - Effective electric conductivity of the electrode material

  • grad : dolfinx.fem.Function - python function that returns the UFL gradient of the argument

  • L : Constant - Thickness used to normalize the domain, by default None

Returns:

Electrode Potential Equation

Return type:

Form

phi_s_interface(lagrange_multiplier, dS, phi_s_test=None, phi_s_cc_test=None, scale_factor=1)

Implements the interface for electrode potential phi_s

Parameters:

• lagrange_multiplier (dolfinx.fem.Function) – Lagrange multiplier on the interface between the elctrode and the cc associated to the continuity of the potential in the solid face

• dS (Measure) – Measure of the boundary domain over the integral must integrate

• phi_s_test (TestFunction) – Electrode Potential Equation Test Function

• phi_s_cc_test (TestFunction) – Electrode Potential and cc Equation Test Function

Returns:

Electrode Potential interface Equation

Return type:

Form