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integrated_conductivity.py
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#
# Composite electrolyte potential employing integrated Stefan-Maxwell
#
import pybamm
from .base_electrolyte_conductivity import BaseElectrolyteConductivity
class Integrated(BaseElectrolyteConductivity):
"""Base class for conservation of charge in the electrolyte employing the
Stefan-Maxwell constitutive equations.
Parameters
----------
param : parameter class
The parameters to use for this submodel
higher_order_terms : str
What kind of higher-order terms to use ('composite' or 'first-order')
domain : str, optional
The domain in which the model holds
**Extends:** :class:`pybamm.electrolyte_conductivity.BaseElectrolyteConductivity`
"""
def __init__(self, param, domain=None):
super().__init__(param, domain)
def _higher_order_macinnes_function(self, x):
return pybamm.log(x)
def get_coupled_variables(self, variables):
c_e_av = variables["X-averaged electrolyte concentration"]
i_boundary_cc_0 = variables["Leading-order current collector current density"]
c_e_n = variables["Negative electrolyte concentration"]
c_e_s = variables["Separator electrolyte concentration"]
c_e_p = variables["Positive electrolyte concentration"]
c_e_n0 = pybamm.boundary_value(c_e_n, "left")
delta_phi_n_av = variables[
"X-averaged negative electrode surface potential difference"
]
phi_s_n_av = variables["X-averaged negative electrode potential"]
tor_n = variables["Negative electrolyte tortuosity"]
tor_s = variables["Separator tortuosity"]
tor_p = variables["Positive electrolyte tortuosity"]
T_av = variables["X-averaged cell temperature"]
T_av_n = pybamm.PrimaryBroadcast(T_av, "negative electrode")
T_av_s = pybamm.PrimaryBroadcast(T_av, "separator")
T_av_p = pybamm.PrimaryBroadcast(T_av, "positive electrode")
param = self.param
l_n = param.l_n
l_p = param.l_p
x_n = pybamm.standard_spatial_vars.x_n
x_s = pybamm.standard_spatial_vars.x_s
x_p = pybamm.standard_spatial_vars.x_p
x_n_edge = pybamm.standard_spatial_vars.x_n_edge
# x_s_edge = pybamm.standard_spatial_vars.x_s_edge
x_p_edge = pybamm.standard_spatial_vars.x_p_edge
chi_av = param.chi(c_e_av)
chi_av_n = pybamm.PrimaryBroadcast(chi_av, "negative electrode")
chi_av_s = pybamm.PrimaryBroadcast(chi_av, "separator")
chi_av_p = pybamm.PrimaryBroadcast(chi_av, "positive electrode")
# electrolyte current
i_e_n = i_boundary_cc_0 * x_n / l_n
i_e_s = pybamm.PrimaryBroadcast(i_boundary_cc_0, "separator")
i_e_p = i_boundary_cc_0 * (1 - x_p) / l_p
i_e = pybamm.Concatenation(i_e_n, i_e_s, i_e_p)
i_e_n_edge = i_boundary_cc_0 * x_n_edge / l_n
i_e_s_edge = pybamm.PrimaryBroadcastToEdges(i_boundary_cc_0, "separator")
i_e_p_edge = i_boundary_cc_0 * (1 - x_p_edge) / l_p
# i_e_edge = pybamm.Concatenation(i_e_n_edge, i_e_s_edge, i_e_p_edge)
# electrolyte potential
indef_integral_n = (
pybamm.IndefiniteIntegral(
i_e_n_edge / (param.kappa_e(c_e_n, T_av_n) * tor_n), x_n
)
* param.C_e
/ param.gamma_e
)
indef_integral_s = (
pybamm.IndefiniteIntegral(
i_e_s_edge / (param.kappa_e(c_e_s, T_av_s) * tor_s), x_s
)
* param.C_e
/ param.gamma_e
)
indef_integral_p = (
pybamm.IndefiniteIntegral(
i_e_p_edge / (param.kappa_e(c_e_p, T_av_p) * tor_p), x_p
)
* param.C_e
/ param.gamma_e
)
integral_n = indef_integral_n - pybamm.boundary_value(indef_integral_n, "left")
integral_s = (
indef_integral_s
- pybamm.boundary_value(indef_integral_s, "left")
+ pybamm.boundary_value(integral_n, "right")
)
integral_p = (
indef_integral_p
- pybamm.boundary_value(indef_integral_p, "left")
+ pybamm.boundary_value(integral_s, "right")
)
phi_e_const = (
-delta_phi_n_av
+ phi_s_n_av
- (
chi_av
* (1 + param.Theta * T_av)
* pybamm.x_average(self._higher_order_macinnes_function(c_e_n / c_e_n0))
)
+ pybamm.x_average(integral_n)
)
phi_e_n = (
phi_e_const
+ (
chi_av_n
* (1 + param.Theta * T_av_n)
* self._higher_order_macinnes_function(c_e_n / c_e_n0)
)
- integral_n
)
phi_e_s = (
phi_e_const
+ (
chi_av_s
* (1 + param.Theta * T_av_s)
* self._higher_order_macinnes_function(c_e_s / c_e_n0)
)
- integral_s
)
phi_e_p = (
phi_e_const
+ (
chi_av_p
* (1 + param.Theta * T_av_p)
* self._higher_order_macinnes_function(c_e_p / c_e_n0)
)
- integral_p
)
# phi_e_n = 0 * phi_e_n
# phi_e_s = 0 * phi_e_s
# phi_e_p = 0 * phi_e_p
# concentration overpotential
eta_c_av = (
chi_av
* (1 + param.Theta * T_av)
* (
pybamm.x_average(self._higher_order_macinnes_function(c_e_p / c_e_av))
- pybamm.x_average(self._higher_order_macinnes_function(c_e_n / c_e_av))
)
)
# average electrolyte ohmic losses
delta_phi_e_av = -(pybamm.x_average(integral_p) - pybamm.x_average(integral_n))
variables.update(
self._get_standard_potential_variables(phi_e_n, phi_e_s, phi_e_p)
)
variables.update(self._get_standard_current_variables(i_e))
variables.update(self._get_split_overpotential(eta_c_av, delta_phi_e_av))
return variables