Source code for pybamm.models.submodels.electrode.ohm.full_ohm

#
# Full model of electrode employing Ohm's law
#
import pybamm

from .base_ohm import BaseModel




class Full(BaseModel):
    """This submodel implements Ohm's law for electrodes, accounting for full resistance effects.

    The model solves for the electrode potential distribution using the effective conductivity,
    which is the product of the material conductivity and the transport efficiency.

    .. math::
        \\nabla \\cdot (\\sigma_{\text{eff}} \\nabla \\phi_s) = -a_j

    where:
        - :math:`\\sigma_{\text{eff}} = \\sigma \\cdot \\tau_{\text{or}}` is the effective conductivity
        - :math:`\\phi_s` is the electrode potential
        - :math:`a_j` is the volumetric interfacial current density

    """

    def __init__(self, param, domain, options=None):
        super().__init__(param, domain, options=options)



    def get_fundamental_variables(self):
        domain, Domain = self.domain_Domain
        # potential is scaled with reference potential
        if domain == "negative":
            reference = 0
        else:
            reference = self.param.ocv_init

        phi_s = pybamm.Variable(
            f"{Domain} electrode potential [V]",
            domain=f"{domain} electrode",
            auxiliary_domains={"secondary": "current collector"},
            reference=reference,
        )
        phi_s.print_name = f"phi_s_{domain[0]}"
        variables = self._get_standard_potential_variables(phi_s)

        return variables




    def get_coupled_variables(self, variables):
        Domain = self.domain.capitalize()

        phi_s = variables[f"{Domain} electrode potential [V]"]
        tor = variables[f"{Domain} electrode transport efficiency"]
        T = variables[f"{Domain} electrode temperature [K]"]

        sigma = self.domain_param.sigma(T)

        sigma_eff = sigma * tor
        i_s = -sigma_eff * pybamm.grad(phi_s)

        variables.update({f"{Domain} electrode effective conductivity": sigma_eff})

        variables.update(self._get_standard_current_variables(i_s))

        if self.domain == "positive":
            variables.update(self._get_standard_whole_cell_variables(variables))

        return variables




    def set_algebraic(self, variables):
        domain, Domain = self.domain_Domain

        phi_s = variables[f"{Domain} electrode potential [V]"]
        i_s = variables[f"{Domain} electrode current density [A.m-2]"]

        # Variable summing all of the interfacial current densities
        sum_a_j = variables[
            f"Sum of {domain} electrode volumetric "
            "interfacial current densities [A.m-3]"
        ]

        # multiply by Lx**2 to improve conditioning
        self.algebraic[phi_s] = self.param.L_x**2 * (pybamm.div(i_s) + sum_a_j)




    def set_boundary_conditions(self, variables):
        Domain = self.domain.capitalize()

        phi_s = variables[f"{Domain} electrode potential [V]"]
        phi_s_cn = variables["Negative current collector potential [V]"]
        tor = variables[f"{Domain} electrode transport efficiency"]
        T = variables[f"{Domain} electrode temperature [K]"]

        if self.domain == "negative":
            lbc = (phi_s_cn, "Dirichlet")
            rbc = (pybamm.Scalar(0), "Neumann")

        elif self.domain == "positive":
            lbc = (pybamm.Scalar(0), "Neumann")
            sigma_eff = self.param.p.sigma(T) * tor
            i_boundary_cc = variables["Current collector current density [A.m-2]"]
            rbc = (
                i_boundary_cc / pybamm.boundary_value(-sigma_eff, "right"),
                "Neumann",
            )

        self.boundary_conditions[phi_s] = {"left": lbc, "right": rbc}




    def set_initial_conditions(self, variables):
        Domain = self.domain.capitalize()

        phi_s = variables[f"{Domain} electrode potential [V]"]

        if self.domain == "negative":
            phi_s_init = pybamm.Scalar(0)
        else:
            phi_s_init = self.param.ocv_init

        self.initial_conditions[phi_s] = phi_s_init