#
# Class for one- and two-dimensional potential pair current collector models
#
import pybamm
from .base_current_collector import BaseModel
[docs]
class BasePotentialPair(BaseModel):
"""A submodel for Ohm's law plus conservation of current in the current collectors.
For details on the potential pair formulation see :footcite:t:`Timms2021` and
:footcite:t:`Marquis2020`.
Parameters
----------
param : parameter class
The parameters to use for this submodel
"""
def __init__(self, param):
super().__init__(param)
pybamm.citations.register("Timms2021")
[docs]
def get_fundamental_variables(self):
phi_s_cn = pybamm.Variable(
"Negative current collector potential [V]", domain="current collector"
)
variables = self._get_standard_negative_potential_variables(phi_s_cn)
# TODO: grad not implemented for 2D yet
i_cc = pybamm.Scalar(0)
i_boundary_cc = pybamm.Variable(
"Current collector current density [A.m-2]",
domain="current collector",
scale=self.param.Q / self.param.A_cc,
)
variables.update(self._get_standard_current_variables(i_cc, i_boundary_cc))
return variables
[docs]
def set_algebraic(self, variables):
phi_s_cn = variables["Negative current collector potential [V]"]
phi_s_cp = variables["Positive current collector potential [V]"]
i_boundary_cc = variables["Current collector current density [A.m-2]"]
self.algebraic = {
phi_s_cn: (self.param.n.sigma_cc * self.param.n.L_cc)
* pybamm.laplacian(phi_s_cn)
- pybamm.source(i_boundary_cc, phi_s_cn),
i_boundary_cc: (self.param.p.sigma_cc * self.param.p.L_cc)
* pybamm.laplacian(phi_s_cp)
+ pybamm.source(i_boundary_cc, phi_s_cp),
}
[docs]
def set_initial_conditions(self, variables):
phi_s_cn = variables["Negative current collector potential [V]"]
i_boundary_cc = variables["Current collector current density [A.m-2]"]
self.initial_conditions = {
phi_s_cn: pybamm.Scalar(0),
i_boundary_cc: pybamm.Scalar(0),
}
[docs]
class PotentialPair1plus1D(BasePotentialPair):
"""Base class for a 1+1D potential pair model."""
def __init__(self, param):
super().__init__(param)
[docs]
def set_boundary_conditions(self, variables):
phi_s_cn = variables["Negative current collector potential [V]"]
phi_s_cp = variables["Positive current collector potential [V]"]
applied_current_density = variables["Total current density [A.m-2]"]
total_current = applied_current_density * self.param.A_cc
# In the 1+1D model, the behaviour is averaged over the y-direction, so the
# effective tab area is the cell width multiplied by the current collector
# thickness
positive_tab_area = self.param.L_y * self.param.p.L_cc
pos_tab_bc = -total_current / (self.param.p.sigma_cc * positive_tab_area)
# Boundary condition needs to be on the variables that go into the Laplacian,
# even though phi_s_cp isn't a pybamm.Variable object
self.boundary_conditions = {
phi_s_cn: {
"negative tab": (pybamm.Scalar(0), "Dirichlet"),
"no tab": (pybamm.Scalar(0), "Neumann"),
},
phi_s_cp: {
"no tab": (pybamm.Scalar(0), "Neumann"),
"positive tab": (pos_tab_bc, "Neumann"),
},
}
[docs]
class PotentialPair2plus1D(BasePotentialPair):
"""Base class for a 2+1D potential pair model"""
def __init__(self, param):
super().__init__(param)
[docs]
def set_boundary_conditions(self, variables):
phi_s_cn = variables["Negative current collector potential [V]"]
phi_s_cp = variables["Positive current collector potential [V]"]
applied_current_density = variables["Total current density [A.m-2]"]
total_current = applied_current_density * self.param.A_cc
# Note: we divide by the *numerical* tab area so that the correct total
# current is applied. That is, numerically integrating the current density
# around the boundary gives the applied current exactly.
positive_tab_area = pybamm.BoundaryIntegral(
pybamm.PrimaryBroadcast(self.param.p.L_cc, "current collector"),
region="positive tab",
)
# cc_area appears here due to choice of non-dimensionalisation
pos_tab_bc = -total_current / (self.param.p.sigma_cc * positive_tab_area)
# Boundary condition needs to be on the variables that go into the Laplacian,
# even though phi_s_cp isn't a pybamm.Variable object
# In the 2+1D model, the equations for the current collector potentials
# are solved on a 2D domain and the regions "negative tab" and "positive tab"
# are the projections of the tabs onto this 2D domain.
# In the 2D formulation it is assumed that no flux boundary conditions
# are applied everywhere apart from the tabs.
# The reference potential is taken to be zero on the negative tab,
# giving the zero Dirichlet condition on phi_s_cn. Elsewhere, the boundary
# is insulated, giving no flux conditions on phi_s_cn. This is automatically
# applied everywhere, apart from the region corresponding to the projection
# of the positive tab, so we need to explitly apply a zero-flux boundary
# condition on the region "positive tab" for phi_s_cn.
# A current is drawn from the positive tab, giving the non-zero Neumann
# boundary condition on phi_s_cp at "positive tab". Elsewhere, the boundary is
# insulated, so, as with phi_s_cn, we need to explicitly give the zero-flux
# condition on the region "negative tab" for phi_s_cp.
self.boundary_conditions = {
phi_s_cn: {
"negative tab": (pybamm.Scalar(0), "Dirichlet"),
"positive tab": (pybamm.Scalar(0), "Neumann"),
},
phi_s_cp: {
"negative tab": (pybamm.Scalar(0), "Neumann"),
"positive tab": (pos_tab_bc, "Neumann"),
},
}
