def get_coupled_variables(self, variables):
param = self.param
delta_phi = variables[
f"{self.domain} electrode surface potential difference"]
c_e_n = variables[f"{self.domain} electrolyte concentration"]
T = variables[f"{self.domain} electrode temperature"]
eta_sei = variables[f"{self.domain} electrode SEI film overpotential"]
c_plated_Li = variables[
f"{self.domain} electrode lithium plating concentration"]
j0_stripping = param.j0_stripping(c_e_n, c_plated_Li, T)
j0_plating = param.j0_plating(c_e_n, c_plated_Li, T)
phi_ref = param.U_n_ref / param.potential_scale
eta_stripping = delta_phi + phi_ref + eta_sei
eta_plating = -eta_stripping
prefactor = 1 / (2 * (1 + self.param.Theta * T))
j_stripping = j0_stripping * pybamm.exp(
prefactor * eta_stripping) - j0_plating * pybamm.exp(
prefactor * eta_plating)
variables.update(
self._get_standard_overpotential_variables(eta_stripping))
variables.update(self._get_standard_reaction_variables(j_stripping))
# Update whole cell variables, which also updates the "sum of" variables
if ("Negative electrode lithium plating interfacial current density"
in variables and
"Positive electrode lithium plating interfacial current density"
in variables and "Lithium plating interfacial current density"
not in variables):
variables.update(
self._get_standard_whole_cell_interfacial_current_variables(
variables))
return variables
def electrolyte_diffusivity_Capiglia1999(c_e, T):
"""
Diffusivity of LiPF6 in EC:DMC as a function of ion concentration. The original data
is from [1]. The fit from Dualfoil [2].
References
----------
.. [1] C Capiglia et al. 7Li and 19F diffusion coefficients and thermal
properties of non-aqueous electrolyte solutions for rechargeable lithium batteries.
Journal of power sources 81 (1999): 859-862.
.. [2] http://www.cchem.berkeley.edu/jsngrp/fortran.html
Parameters
----------
c_e: :class:`pybamm.Symbol`
Dimensional electrolyte concentration
T: :class:`pybamm.Symbol`
Dimensional temperature
Returns
-------
:class:`pybamm.Symbol`
Solid diffusivity
"""
D_c_e = 5.34e-10 * exp(-0.65 * c_e / 1000)
E_D_e = 37040
arrhenius = exp(E_D_e / constants.R * (1 / 298.15 - 1 / T))
return D_c_e * arrhenius
def electrolyte_conductivity_Landesfeind2019_base(c_e, T, coeffs):
"""
Conductivity of LiPF6 in solvent_X as a function of ion concentration and
Temperature. The data comes from [1].
References
----------
.. [1] Landesfeind, J. and Gasteiger, H.A., 2019. Temperature and Concentration
Dependence of the Ionic Transport Properties of Lithium-Ion Battery Electrolytes.
Journal of The Electrochemical Society, 166(14), pp.A3079-A3097.
----------
c_e: :class: `numpy.Array`
Dimensional electrolyte concentration
T: :class: `numpy.Array`
Dimensional temperature
coeffs: :class: `numpy.Array`
Fitting parameter coefficients
Returns
-------
:`numpy.Array`
Electrolyte diffusivity
"""
c = c_e / 1000 # mol.m-3 -> mol.l
p1, p2, p3, p4, p5, p6 = coeffs
A = p1 * (1 + (T - p2))
B = 1 + p3 * sqrt(c) + p4 * (1 + p5 * exp(1000 / T)) * c
C = 1 + c**4 * (p6 * exp(1000 / T))
sigma_e = A * c * B / C # mS.cm-1
return sigma_e / 10
def electrolyte_diffusivity_Landesfeind2019_base(c_e, T, coeffs):
"""
Conductivity of LiPF6 in solvent_X as a function of ion concentration and
Temperature. The data comes from [1].
References
----------
.. [1] Landesfeind, J. and Gasteiger, H.A., 2019. Temperature and Concentration
Dependence of the Ionic Transport Properties of Lithium-Ion Battery Electrolytes.
Journal of The Electrochemical Society, 166(14), pp.A3079-A3097.
----------
c_e: :class:`pybamm.Symbol`
Dimensional electrolyte concentration
T: :class:`pybamm.Symbol`
Dimensional temperature
coeffs: :class:`pybamm.Symbol`
Fitting parameter coefficients
Returns
-------
:class:`pybamm.Symbol`
Electrolyte diffusivity
"""
c = c_e / 1000 # mol.m-3 -> mol.l
p1, p2, p3, p4 = coeffs
A = p1 * exp(p2 * c)
B = exp(p3 / T)
C = exp(p4 * c / T)
D_e = A * B * C * 1e-10 # m2/s
return D_e
def electrolyte_diffusivity_Kim2011(c_e, T, T_inf, E_D_e, R_g):
"""
Diffusivity of LiPF6 in EC as a function of ion concentration from [1].
References
----------
.. [1] Kim, G. H., Smith, K., Lee, K. J., Santhanagopalan, S., & Pesaran, A.
(2011). Multi-domain modeling of lithium-ion batteries encompassing
multi-physics in varied length scales. Journal of The Electrochemical
Society, 158(8), A955-A969.
Parameters
----------
c_e: :class: `numpy.Array`
Dimensional electrolyte concentration
T: :class: `numpy.Array`
Dimensional temperature
T_inf: double
Reference temperature
E_D_e: double
Electrolyte diffusion activation energy
R_g: double
The ideal gas constant
Returns
-------
:`numpy.Array`
Solid diffusivity
"""
D_c_e = (5.84 * 10**(-7) * exp(-2870 / T) * (c_e / 1000)**2 -
33.9 * 10**(-7) * exp(-2920 / T) * (c_e / 1000) +
129 * 10**(-7) * exp(-3200 / T))
return D_c_e
def nco_diffusivity_Ecker2015(sto, T):
"""
NCO diffusivity as a function of stochiometry [1, 2, 3].
References
----------
.. [1] Ecker, Madeleine, et al. "Parameterization of a physico-chemical model of
a lithium-ion battery i. determination of parameters." Journal of the
Electrochemical Society 162.9 (2015): A1836-A1848.
.. [2] Ecker, Madeleine, et al. "Parameterization of a physico-chemical model of
a lithium-ion battery ii. model validation." Journal of The Electrochemical
Society 162.9 (2015): A1849-A1857.
.. [3] Richardson, Giles, et. al. "Generalised single particle models for
high-rate operation of graded lithium-ion electrodes: Systematic derivation
and validation." Electrochemica Acta 339 (2020): 135862
Parameters
----------
sto: :class:`pybamm.Symbol`
Electrode stochiometry
T: :class:`pybamm.Symbol`
Dimensional temperature
Returns
-------
:class:`pybamm.Symbol`
Solid diffusivity
"""
D_ref = 3.7e-13 - 3.4e-13 * exp(-12 * (sto - 0.62) * (sto - 0.62))
E_D_s = 8.06e4
arrhenius = exp(-E_D_s / (constants.R * T)) * exp(E_D_s / (constants.R * 296.15))
return D_ref * arrhenius
def electrolyte_conductivity_Kim2011(c_e, T):
"""
Conductivity of LiPF6 in EC as a function of ion concentration from [1].
References
----------
.. [1] Kim, G. H., Smith, K., Lee, K. J., Santhanagopalan, S., & Pesaran, A.
(2011). Multi-domain modeling of lithium-ion batteries encompassing
multi-physics in varied length scales. Journal of The Electrochemical
Society, 158(8), A955-A969.
Parameters
----------
c_e: :class:`pybamm.Symbol`
Dimensional electrolyte concentration
T: :class:`pybamm.Symbol`
Dimensional temperature
Returns
-------
:class:`pybamm.Symbol`
Solid diffusivity
"""
sigma_e = (
3.45 * exp(-798 / T) * (c_e / 1000) ** 3
- 48.5 * exp(-1080 / T) * (c_e / 1000) ** 2
+ 244 * exp(-1440 / T) * (c_e / 1000)
)
return sigma_e
def electrolyte_diffusivity_Capiglia1999(c_e, T, T_inf, E_D_e, R_g):
"""
Diffusivity of LiPF6 in EC:DMC as a function of ion concentration. The original data
is from [1]. The fit from Dualfoil [2].
References
----------
.. [1] C Capiglia et al. 7Li and 19F diffusion coefficients and thermal
properties of non-aqueous electrolyte solutions for rechargeable lithium batteries.
Journal of power sources 81 (1999): 859-862.
.. [2] http://www.cchem.berkeley.edu/jsngrp/fortran.html
Parameters
----------
c_e: :class: `numpy.Array`
Dimensional electrolyte concentration
T: :class: `numpy.Array`
Dimensional temperature
T_inf: double
Reference temperature
E_D_e: double
Electrolyte diffusion activation energy
R_g: double
The ideal gas constant
Returns
-------
:`numpy.Array`
Solid diffusivity
"""
D_c_e = 5.34e-10 * exp(-0.65 * c_e / 1000)
arrhenius = exp(E_D_e / R_g * (1 / T_inf - 1 / T))
return D_c_e * arrhenius
def test_advanced_functions(self):
a = pybamm.StateVector(slice(0, 1))
b = pybamm.StateVector(slice(1, 2))
y = np.array([5, 3])
#
func = a * pybamm.exp(b)
self.assertAlmostEqual(func.diff(a).evaluate(y=y)[0], np.exp(3))
func = pybamm.exp(a + 2 * b + a * b) + a * pybamm.exp(b)
self.assertEqual(
func.diff(a).evaluate(y=y), (4 * np.exp(3 * 5 + 5 + 2 * 3) + np.exp(3))
)
self.assertEqual(
func.diff(b).evaluate(y=y), np.exp(3) * (7 * np.exp(3 * 5 + 5 + 3) + 5)
)
#
func = pybamm.sin(pybamm.cos(a * 4) / 2) * pybamm.cos(4 * pybamm.exp(b / 3))
self.assertEqual(
func.diff(a).evaluate(y=y),
-2 * np.sin(20) * np.cos(np.cos(20) / 2) * np.cos(4 * np.exp(1)),
)
self.assertEqual(
func.diff(b).evaluate(y=y),
-4 / 3 * np.exp(1) * np.sin(4 * np.exp(1)) * np.sin(np.cos(20) / 2),
)
#
func = pybamm.sin(a * b)
self.assertEqual(func.diff(a).evaluate(y=y), 3 * np.cos(15))
def graphite_ocp_Ecker2015_function(sto):
"""
Graphite OCP as a function of stochiometry [1, 2, 3].
References
----------
.. [1] Ecker, Madeleine, et al. "Parameterization of a physico-chemical model of
a lithium-ion battery i. determination of parameters." Journal of the
Electrochemical Society 162.9 (2015): A1836-A1848.
.. [2] Ecker, Madeleine, et al. "Parameterization of a physico-chemical model of
a lithium-ion battery ii. model validation." Journal of The Electrochemical
Society 162.9 (2015): A1849-A1857.
.. [3] Richardson, Giles, et. al. "Generalised single particle models for
high-rate operation of graded lithium-ion electrodes: Systematic derivation
and validation." Electrochemica Acta 339 (2020): 135862
Parameters
----------
sto: :class:`pybamm.Symbol`
Electrode stochiometry
Returns
-------
:class:`pybamm.Symbol`
Open circuit potential
"""
# Graphite negative electrode from Ecker, Kabitz, Laresgoiti et al.
# Analytical fit (WebPlotDigitizer + gnuplot)
a = 0.716502
b = 369.028
c = 0.12193
d = 35.6478
e = 0.0530947
g = 0.0169644
h = 27.1365
i = 0.312832
j = 0.0199313
k = 28.5697
m = 0.614221
n = 0.931153
o = 36.328
p = 1.10743
q = 0.140031
r = 0.0189193
s = 21.1967
t = 0.196176
u_eq = (
a * exp(-b * sto)
+ c * exp(-d * (sto - e))
- r * tanh(s * (sto - t))
- g * tanh(h * (sto - i))
- j * tanh(k * (sto - m))
- n * exp(o * (sto - p))
+ q
)
return u_eq
def test_exp(self):
a = pybamm.InputParameter("a")
fun = pybamm.exp(a)
self.assertIsInstance(fun, pybamm.Exponential)
self.assertEqual(fun.children[0].id, a.id)
self.assertEqual(fun.evaluate(inputs={"a": 3}), np.exp(3))
h = 0.0000001
self.assertAlmostEqual(
fun.diff(a).evaluate(inputs={"a": 3}),
(pybamm.exp(pybamm.Scalar(3 + h)).evaluate() -
fun.evaluate(inputs={"a": 3})) / h,
places=5,
)
def electrolyte_diffusivity_Nyman2008(c_e, T, T_inf, E_D_e, R_g):
"""
Diffusivity of LiPF6 in EC:EMC (3:7) as a function of ion concentration. The data
comes from [1]
References
----------
.. [1] A. Nyman, M. Behm, and G. Lindbergh, "Electrochemical characterisation and
modelling of the mass transport phenomena in LiPF6-EC-EMC electrolyte,"
Electrochim. Acta, vol. 53, no. 22, pp. 6356–6365, 2008.
Parameters
----------
c_e: :class: `numpy.Array`
Dimensional electrolyte concentration
T: :class: `numpy.Array`
Dimensional temperature
T_inf: double
Reference temperature
E_D_e: double
Electrolyte diffusion activation energy
R_g: double
The ideal gas constant
Returns
-------
:`numpy.Array`
Solid diffusivity
"""
D_c_e = 8.794e-11 * (c_e / 1000) ** 2 - 3.972e-10 * (c_e / 1000) + 4.862e-10
arrhenius = exp(E_D_e / R_g * (1 / T_inf - 1 / T))
return D_c_e * arrhenius
def test_diff(self):
a = pybamm.StateVector(slice(0, 1))
b = pybamm.StateVector(slice(1, 2))
y = np.array([5])
func = pybamm.Function(test_function, a)
self.assertEqual(func.diff(a).evaluate(y=y), 2)
self.assertEqual(func.diff(func).evaluate(), 1)
func = pybamm.sin(a)
self.assertEqual(func.evaluate(y=y), np.sin(a.evaluate(y=y)))
self.assertEqual(func.diff(a).evaluate(y=y), np.cos(a.evaluate(y=y)))
func = pybamm.exp(a)
self.assertEqual(func.evaluate(y=y), np.exp(a.evaluate(y=y)))
self.assertEqual(func.diff(a).evaluate(y=y), np.exp(a.evaluate(y=y)))
# multiple variables
func = pybamm.Function(test_multi_var_function, 4 * a, 3 * a)
self.assertEqual(func.diff(a).evaluate(y=y), 7)
func = pybamm.Function(test_multi_var_function, 4 * a, 3 * b)
self.assertEqual(func.diff(a).evaluate(y=np.array([5, 6])), 4)
self.assertEqual(func.diff(b).evaluate(y=np.array([5, 6])), 3)
func = pybamm.Function(test_multi_var_function_cube, 4 * a, 3 * b)
self.assertEqual(func.diff(a).evaluate(y=np.array([5, 6])), 4)
self.assertEqual(
func.diff(b).evaluate(y=np.array([5, 6])), 3 * 3 * (3 * 6) ** 2
)
# exceptions
func = pybamm.Function(
test_multi_var_function_cube, 4 * a, 3 * b, derivative="derivative"
)
with self.assertRaises(ValueError):
func.diff(a)
def graphite_entropic_change_Moura2016(sto, c_n_max):
"""
Graphite entropic change in open circuit potential (OCP) at a temperature of
298.15K as a function of the stochiometry taken from Scott Moura's FastDFN code
[1].
References
----------
.. [1] https://github.com/scott-moura/fastDFN
Parameters
----------
sto: double
Stochiometry of material (li-fraction)
"""
du_dT = (
-1.5 * (120.0 / c_n_max) * exp(-120 * sto)
+ (0.0351 / (0.083 * c_n_max)) * ((cosh((sto - 0.286) / 0.083)) ** (-2))
- (0.0045 / (0.119 * c_n_max)) * ((cosh((sto - 0.849) / 0.119)) ** (-2))
- (0.035 / (0.05 * c_n_max)) * ((cosh((sto - 0.9233) / 0.05)) ** (-2))
- (0.0147 / (0.034 * c_n_max)) * ((cosh((sto - 0.5) / 0.034)) ** (-2))
- (0.102 / (0.142 * c_n_max)) * ((cosh((sto - 0.194) / 0.142)) ** (-2))
- (0.022 / (0.0164 * c_n_max)) * ((cosh((sto - 0.9) / 0.0164)) ** (-2))
- (0.011 / (0.0226 * c_n_max)) * ((cosh((sto - 0.124) / 0.0226)) ** (-2))
+ (0.0155 / (0.029 * c_n_max)) * ((cosh((sto - 0.105) / 0.029)) ** (-2))
)
return du_dT
def graphite_electrolyte_exchange_current_density_Ramadass2004(
c_e, c_s_surf, T):
"""
Exchange-current density for Butler-Volmer reactions between graphite and LiPF6 in
EC:DMC.
References
----------
.. [1] P. Ramadass, Bala Haran, Parthasarathy M. Gomadam, Ralph White, and Branko
N. Popov. "Development of First Principles Capacity Fade Model for Li-Ion Cells."
(2004)
Parameters
----------
c_e : :class:`pybamm.Symbol`
Electrolyte concentration [mol.m-3]
c_s_surf : :class:`pybamm.Symbol`
Particle concentration [mol.m-3]
T : :class:`pybamm.Symbol`
Temperature [K]
Returns
-------
:class:`pybamm.Symbol`
Exchange-current density [A.m-2]
"""
m_ref = 4.854 * 10**(-6) # (A/m2)(mol/m3)**1.5
E_r = 37480
arrhenius = exp(E_r / constants.R * (1 / 298.15 - 1 / T))
c_n_max = standard_parameters_lithium_ion.c_n_max
return (m_ref * arrhenius * c_e**0.5 * c_s_surf**0.5 *
(c_n_max - c_s_surf)**0.5)
def test_exp(self):
a = pybamm.Scalar(3)
fun = pybamm.exp(a)
self.assertIsInstance(fun, pybamm.Exponential)
self.assertEqual(fun.children[0].id, a.id)
self.assertEqual(fun.evaluate(), np.exp(3))
self.assertEqual(fun.diff(a).evaluate(), np.exp(3))
def NMC_entropic_change_PeymanMPM(sto):
"""
Nickel Manganese Cobalt (NMC) entropic change in open circuit potential (OCP) at
a temperature of 298.15K as a function of the OCP. The fit is taken from [1].
References
----------
.. [1] W. Le, I. Belharouak, D. Vissers, K. Amine, "In situ thermal study of
li1+ x [ni1/ 3co1/ 3mn1/ 3] 1- x o2 using isothermal micro-clorimetric
techniques",
J. of the Electrochemical Society 153 (11) (2006) A2147–A2151.
Parameters
----------
sto : :class:`pybamm.Symbol`
Stochiometry of material (li-fraction)
"""
# Since the equation uses the OCP at each stoichiometry as input,
# we need OCP function here
u_eq = (4.3452 - 1.6518 * sto + 1.6225 * sto**2 - 2.0843 * sto**3 +
3.5146 * sto**4 -
0.5623 * 10**(-4) * pybamm.exp(109.451 * sto - 100.006))
du_dT = (-800 + 779 * u_eq - 284 * u_eq**2 + 46 * u_eq**3 -
2.8 * u_eq**4) * 10**(-3)
return du_dT
def graphite_mcmb2528_ocp_Dualfoil1998(sto):
"""
Graphite MCMB 2528 Open Circuit Potential (OCP) as a function of the
stochiometry. The fit is taken from Dualfoil [1]. Dualfoil states that the data
was measured by Chris Bogatu at Telcordia and PolyStor materials, 2000. However,
we could not find any other records of this measurment.
References
----------
.. [1] http://www.cchem.berkeley.edu/jsngrp/fortran.html
"""
u_eq = (
0.194
+ 1.5 * exp(-120.0 * sto)
+ 0.0351 * tanh((sto - 0.286) / 0.083)
- 0.0045 * tanh((sto - 0.849) / 0.119)
- 0.035 * tanh((sto - 0.9233) / 0.05)
- 0.0147 * tanh((sto - 0.5) / 0.034)
- 0.102 * tanh((sto - 0.194) / 0.142)
- 0.022 * tanh((sto - 0.9) / 0.0164)
- 0.011 * tanh((sto - 0.124) / 0.0226)
+ 0.0155 * tanh((sto - 0.105) / 0.029)
)
return u_eq
def get_coupled_variables(self, variables):
L_sei_inner = variables["Inner " + self.domain.lower() +
" electrode sei thickness"]
phi_s_n = variables[self.domain + " electrode potential"]
if self.domain == "Negative":
C_sei = pybamm.sei_parameters.C_sei_inter_n
j_sei = -pybamm.exp(-phi_s_n) / (C_sei * L_sei_inner)
alpha = 0.5
j_inner = alpha * j_sei
j_outer = (1 - alpha) * j_sei
variables.update(
self._get_standard_reaction_variables(j_inner, j_outer))
# Update whole cell variables, which also updates the "sum of" variables
if ("Negative electrode sei interfacial current density" in variables
and "Positive electrode sei interfacial current density"
in variables
and "Sei interfacial current density" not in variables):
variables.update(
self._get_standard_whole_cell_interfacial_current_variables(
variables))
return variables
def set_algebraic(self, variables):
phi_s_n = variables[self.domain + " electrode potential"]
phi_e_n = variables[self.domain + " electrolyte potential"]
j_sei = variables["Outer " + self.domain.lower() +
" electrode sei interfacial current density"]
L_sei = variables["Outer " + self.domain.lower() +
" electrode sei thickness"]
c_ec = variables[self.domain + " electrode EC surface concentration"]
# Look for current that contributes to the -IR drop
# If we can't find the interfacial current density from the main reaction, j,
# it's ok to fall back on the total interfacial current density, j_tot
# This should only happen when the interface submodel is "InverseButlerVolmer"
# in which case j = j_tot (uniform) anyway
try:
j = variables["Total " + self.domain.lower() +
" electrode interfacial current density"]
except KeyError:
j = variables["X-averaged " + self.domain.lower() +
" electrode total interfacial current density"]
if self.domain == "Negative":
C_sei_ec = self.param.C_sei_ec_n
R_sei = self.param.R_sei_n
# need to revise for thermal case
self.algebraic = {
j_sei:
j_sei + C_sei_ec * c_ec *
pybamm.exp(-0.5 * (phi_s_n - phi_e_n - j * L_sei * R_sei))
}
def graphite_LGM50_diffusivity_Chen2020(sto, T):
"""
LG M50 Graphite diffusivity as a function of stochiometry, in this case the
diffusivity is taken to be a constant. The value is taken from [1].
References
----------
.. [1] Chang-Hui Chen, Ferran Brosa Planella, Kieran O’Regan, Dominika Gastol, W.
Dhammika Widanage, and Emma Kendrick. "Development of Experimental Techniques for
Parameterization of Multi-scale Lithium-ion Battery Models." Journal of the
Electrochemical Society 167 (2020): 080534.
Parameters
----------
sto: :class:`pybamm.Symbol`
Electrode stochiometry
T: :class:`pybamm.Symbol`
Dimensional temperature
Returns
-------
:class:`pybamm.Symbol`
Solid diffusivity
"""
D_ref = 3.3e-14
E_D_s = 0 # to be implemented
arrhenius = exp(E_D_s / constants.R * (1 / 298.15 - 1 / T))
return D_ref * arrhenius
def graphite_LGM50_electrolyte_reaction_rate_Chen2020(T, T_inf, E_r, R_g):
"""
Reaction rate for Butler-Volmer reactions between graphite and LiPF6 in EC:DMC.
References
----------
.. [1] Chang-Hui Chen, Ferran Brosa Planella, Kieran O’Regan, Dominika Gastol, W.
Dhammika Widanage, and Emma Kendrick. "Development of Experimental Techniques for
Parameterization of Multi-scale Lithium-ion Battery Models." Submitted for
publication (2020).
Parameters
----------
T: :class: `numpy.Array`
Dimensional temperature
T_inf: double
Reference temperature
E_r: double
Reaction activation energy
R_g: double
The ideal gas constant
Returns
-------
:`numpy.Array`
Reaction rate
"""
m_ref = 6.48e-7
arrhenius = exp(E_r / R_g * (1 / T_inf - 1 / T))
return m_ref * arrhenius
def graphite_electrolyte_exchange_current_density_Dualfoil1998(
c_e, c_s_surf, T):
"""
Exchange-current density for Butler-Volmer reactions between graphite and LiPF6 in
EC:DMC.
References
----------
.. [2] http://www.cchem.berkeley.edu/jsngrp/fortran.html
Parameters
----------
c_e : :class:`pybamm.Symbol`
Electrolyte concentration [mol.m-3]
c_s_surf : :class:`pybamm.Symbol`
Particle concentration [mol.m-3]
T : :class:`pybamm.Symbol`
Temperature [K]
Returns
-------
:class:`pybamm.Symbol`
Exchange-current density [A.m-2]
"""
m_ref = (1 * 10**(-11) * constants.F
) # (A/m2)(mol/m3)**1.5 - includes ref concentrations
E_r = 5000 # activation energy for Temperature Dependent Reaction Constant [J/mol]
arrhenius = exp(E_r / constants.R * (1 / 298.15 - 1 / T))
c_n_max = Parameter(
"Maximum concentration in negative electrode [mol.m-3]")
return (m_ref * arrhenius * c_e**0.5 * c_s_surf**0.5 *
(c_n_max - c_s_surf)**0.5)
def beta(T):
T_inf = pybamm.FunctionParameter("T_inf", {"x": self.x})
h = pybamm.Parameter("h")
eps0 = pybamm.Parameter("eps0")
return (1e-4 * (1.0 + 5.0 * pybamm.sin(3 * np.pi * T / 200.0) +
pybamm.exp(0.02 * T)) / eps0 + h * (T_inf - T) /
(T_inf**4 - T**4) / eps0)
def nmc_LGM50_electrolyte_exchange_current_density_Chen2020(c_e, c_s_surf, T):
"""
Exchange-current density for Butler-Volmer reactions between NMC and LiPF6 in
EC:DMC.
References
----------
.. [1] Chang-Hui Chen, Ferran Brosa Planella, Kieran O’Regan, Dominika Gastol, W.
Dhammika Widanage, and Emma Kendrick. "Development of Experimental Techniques for
Parameterization of Multi-scale Lithium-ion Battery Models." Journal of the
Electrochemical Society 167 (2020): 080534.
Parameters
----------
c_e : :class:`pybamm.Symbol`
Electrolyte concentration [mol.m-3]
c_s_surf : :class:`pybamm.Symbol`
Particle concentration [mol.m-3]
T : :class:`pybamm.Symbol`
Temperature [K]
Returns
-------
:class:`pybamm.Symbol`
Exchange-current density [A.m-2]
"""
m_ref = 3.42e-6 # (A/m2)(mol/m3)**1.5 - includes ref concentrations
E_r = 17800
arrhenius = exp(E_r / constants.R * (1 / 298.15 - 1 / T))
c_p_max = standard_parameters_lithium_ion.c_p_max
return (
m_ref * arrhenius * c_e ** 0.5 * c_s_surf ** 0.5 * (c_p_max - c_s_surf) ** 0.5
)
def lico2_diffusivity_Dualfoil1998(sto, T):
"""
LiCo2 diffusivity as a function of stochiometry, in this case the
diffusivity is taken to be a constant. The value is taken from Dualfoil [1].
References
----------
.. [1] http://www.cchem.berkeley.edu/jsngrp/fortran.html
Parameters
----------
sto: :class:`pybamm.Symbol`
Electrode stochiometry
T: :class:`pybamm.Symbol`
Dimensional temperature, [K]
Returns
-------
:class:`pybamm.Symbol`
Solid diffusivity [m2.s-1]
"""
D_ref = 5.387 * 10**(-15)
E_D_s = 5000
T_ref = Parameter("Reference temperature [K]")
arrhenius = exp(E_D_s / constants.R * (1 / T_ref - 1 / T))
return D_ref * arrhenius
def graphite_diffusivity_PeymanMPM(sto, T):
"""
Graphite diffusivity as a function of stochiometry, in this case the
diffusivity is taken to be a constant. The value is taken from Peyman MPM.
References
----------
.. [1] http://www.cchem.berkeley.edu/jsngrp/fortran.html
Parameters
----------
sto: :class:`pybamm.Symbol`
Electrode stochiometry
T: :class:`pybamm.Symbol`
Dimensional temperature
Returns
-------
:class:`pybamm.Symbol`
Solid diffusivity
"""
D_ref = 5.0 * 10**(-15)
E_D_s = 42770
arrhenius = exp(E_D_s / constants.R * (1 / 298.15 - 1 / T))
return D_ref * arrhenius
def _get_dj_ddeltaphi(self, variables):
"See :meth:`pybamm.interface.kinetics.BaseKinetics._get_dj_ddeltaphi`"
_, delta_phi, j0, ne, ocp, T = self._get_interface_variables_for_first_order(
variables)
eta_r = delta_phi - ocp
return (2 * j0 * (ne / (2 * (1 + self.param.Theta * T))) * pybamm.exp(
(ne / 2) * eta_r))
def NMC_electrolyte_exchange_current_density_PeymanMPM(c_e, c_s_surf, T):
"""
Exchange-current density for Butler-Volmer reactions between NMC and LiPF6 in
EC:DMC.
References
----------
.. Peyman MPM manuscript (to be submitted)
Parameters
----------
c_e : :class:`pybamm.Symbol`
Electrolyte concentration [mol.m-3]
c_s_surf : :class:`pybamm.Symbol`
Particle concentration [mol.m-3]
T : :class:`pybamm.Symbol`
Temperature [K]
Returns
-------
:class:`pybamm.Symbol`
Exchange-current density [A.m-2]
"""
m_ref = 4.824 * 10 ** (-6) # (A/m2)(mol/m3)**1.5 - includes ref concentrations
E_r = 39570
arrhenius = exp(E_r / constants.R * (1 / 298.15 - 1 / T))
c_p_max = Parameter("Maximum concentration in positive electrode [mol.m-3]")
return (
m_ref * arrhenius * c_e ** 0.5 * c_s_surf ** 0.5 * (c_p_max - c_s_surf) ** 0.5
)
def graphite_LGM50_diffusivity_Chen2020(sto, T, T_inf, E_D_s, R_g):
"""
LG M50 Graphite diffusivity as a function of stochiometry, in this case the
diffusivity is taken to be a constant. The value is taken from [1].
References
----------
.. [1] Chang-Hui Chen, Ferran Brosa Planella, Kieran O’Regan, Dominika Gastol, W.
Dhammika Widanage, and Emma Kendrick. "Development of Experimental Techniques for
Parameterization of Multi-scale Lithium-ion Battery Models." Submitted for
publication (2020).
Parameters
----------
sto: :class: `numpy.Array`
Electrode stochiometry
T: :class: `numpy.Array`
Dimensional temperature
T_inf: double
Reference temperature
E_D_s: double
Solid diffusion activation energy
R_g: double
The ideal gas constant
Returns
-------
: double
Solid diffusivity
"""
D_ref = 3.3e-14
arrhenius = exp(E_D_s / R_g * (1 / T_inf - 1 / T))
return D_ref * arrhenius