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Ecker2015_params.py
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from pybamm import exp, tanh, constants, Parameter, FunctionParameter, ParameterValues
def electrolyte_conductivity_Ecker2015(c_e, T):
"""
Conductivity of LiPF6 in EC:DMC as a function of ion concentration [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
----------
c_e: :class:`pybamm.Symbol`
Dimensional electrolyte concentration
T: :class:`pybamm.Symbol`
Dimensional temperature
Returns
-------
:class:`pybamm.Symbol`
Solid diffusivity
"""
# mol/m^3 to mol/l
cm = 1e-3 * c_e
# value at T = 296K
sigma_e_296 = 0.2667 * cm**3 - 1.2983 * cm**2 + 1.7919 * cm + 0.1726
# add temperature dependence
E_k_e = 1.71e4
C = 296 * exp(E_k_e / (constants.R * 296))
sigma_e = C * sigma_e_296 * exp(-E_k_e / (constants.R * T)) / T
return sigma_e
def electrolyte_diffusivity_Ecker2015(c_e, T):
"""
Diffusivity of LiPF6 in EC:DMC as a function of ion concentration [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
----------
c_e: :class:`pybamm.Symbol`
Dimensional electrolyte concentration
T: :class:`pybamm.Symbol`
Dimensional temperature
Returns
-------
:class:`pybamm.Symbol`
Solid diffusivity
"""
# The diffusivity epends on the electrolyte conductivity
inputs = {"Electrolyte concentration [mol.m-3]": c_e, "Temperature [K]": T}
sigma_e = FunctionParameter("Electrolyte conductivity [S.m-1]", inputs)
D_c_e = (constants.k_b / (constants.F * constants.q_e)) * sigma_e * T / c_e
return D_c_e
def graphite_diffusivity_Ecker2015(sto, T):
"""
Graphite 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 = 8.4e-13 * exp(-11.3 * sto) + 8.2e-15
E_D_s = 3.03e4
arrhenius = exp(-E_D_s / (constants.R * T)) * exp(E_D_s / (constants.R * 296))
return D_ref * arrhenius
def graphite_electrolyte_exchange_current_density_Ecker2015(c_e, c_s_surf, T):
"""
Exchange-current density for Butler-Volmer reactions between graphite and LiPF6 in
EC:DMC.
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
----------
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]
"""
k_ref = 1.995 * 1e-10
# multiply by Faraday's constant to get correct units
m_ref = constants.F * k_ref # (A/m2)(mol/m3)**1.5 - includes ref concentrations
E_r = 53400
arrhenius = exp(-E_r / (constants.R * T)) * exp(E_r / (constants.R * 296.15))
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 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 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 nco_electrolyte_exchange_current_density_Ecker2015(c_e, c_s_surf, T):
"""
Exchange-current density for Butler-Volmer reactions between NCO and LiPF6 in
EC:DMC [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
----------
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]
"""
k_ref = 5.196e-11
# multiply by Faraday's constant to get correct units
m_ref = constants.F * k_ref # (A/m2)(mol/m3)**1.5 - includes ref concentrations
E_r = 4.36e4
arrhenius = exp(-E_r / (constants.R * T)) * exp(E_r / (constants.R * 296.15))
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 nco_ocp_Ecker2015_function(sto):
"""
NCO 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`
Stochiometry of material (li-fraction)
"""
# LiNiCo from Ecker, Kabitz, Laresgoiti et al.
# Analytical fit (WebPlotDigitizer + gnuplot)
a = -2.35211
c = 0.0747061
d = 31.886
e = 0.0219921
g = 0.640243
h = 5.48623
i = 0.439245
j = 3.82383
k = 4.12167
m = 0.176187
n = 0.0542123
o = 18.2919
p = 0.762272
q = 4.23285
r = -6.34984
s = 2.66395
t = 0.174352
u_eq = (
a * sto
- c * tanh(d * (sto - e))
- r * tanh(s * (sto - t))
- g * tanh(h * (sto - i))
- j * tanh(k * (sto - m))
- n * tanh(o * (sto - p))
+ q
)
return u_eq
parameter_values = ParameterValues(
{
"1 + dlnf/dlnc": 1.0,
"Ambient temperature [K]": 298.15,
"Cation transference number": 0.26,
"Cell cooling surface area [m2]": 0.0172,
"Cell volume [m3]": 1.52e-06,
"Current function [A]": 0.15652,
"Electrode height [m]": 0.101,
"Electrode width [m]": 0.085,
"Electrolyte conductivity [S.m-1]": electrolyte_conductivity_Ecker2015,
"Electrolyte diffusivity [m2.s-1]": electrolyte_diffusivity_Ecker2015,
"Initial concentration in electrolyte [mol.m-3]": 1000.0,
"Initial concentration in negative electrode [mol.m-3]": 26120.05,
"Initial concentration in positive electrode [mol.m-3]": 12630.8,
"Initial temperature [K]": 298.15,
"Lower voltage cut-off [V]": 2.5,
"Maximum concentration in negative electrode [mol.m-3]": 31920.0,
"Maximum concentration in positive electrode [mol.m-3]": 48580.0,
"Negative current collector conductivity [S.m-1]": 58411000.0,
"Negative current collector density [kg.m-3]": 8933.0,
"Negative current collector specific heat capacity [J.kg-1.K-1]": 385.0,
"Negative current collector thermal conductivity [W.m-1.K-1]": 398.0,
"Negative current collector thickness [m]": 1.4e-05,
"Negative electrode Bruggeman coefficient (electrode)": 0.0,
"Negative electrode Bruggeman coefficient (electrolyte)": 1.6372789338386007,
"Negative electrode OCP [V]": graphite_ocp_Ecker2015_function,
"Negative electrode OCP entropic change [V.K-1]": 0.0,
"Negative electrode active material volume fraction": 0.372403,
"Negative electrode conductivity [S.m-1]": 14.0,
"Negative electrode density [kg.m-3]": 1555.0,
"Negative electrode diffusivity [m2.s-1]": graphite_diffusivity_Ecker2015,
"Negative electrode electrons in reaction": 1.0,
"Negative electrode exchange-current density [A.m-2]": graphite_electrolyte_exchange_current_density_Ecker2015,
"Negative electrode porosity": 0.329,
"Negative electrode specific heat capacity [J.kg-1.K-1]": 1437.0,
"Negative electrode thermal conductivity [W.m-1.K-1]": 1.58,
"Negative electrode thickness [m]": 7.4e-05,
"Negative particle radius [m]": 1.37e-05,
"Nominal cell capacity [A.h]": 0.15625,
"Number of cells connected in series to make a battery": 1.0,
"Number of electrodes connected in parallel to make a cell": 1.0,
"Positive current collector conductivity [S.m-1]": 36914000.0,
"Positive current collector density [kg.m-3]": 2702.0,
"Positive current collector specific heat capacity [J.kg-1.K-1]": 903.0,
"Positive current collector thermal conductivity [W.m-1.K-1]": 238.0,
"Positive current collector thickness [m]": 1.5e-05,
"Positive electrode Bruggeman coefficient (electrode)": 0.0,
"Positive electrode Bruggeman coefficient (electrolyte)": 1.5442267190786427,
"Positive electrode OCP [V]": nco_ocp_Ecker2015_function,
"Positive electrode OCP entropic change [V.K-1]": 0.0,
"Positive electrode active material volume fraction": 0.40832,
"Positive electrode conductivity [S.m-1]": 68.1,
"Positive electrode density [kg.m-3]": 2895.0,
"Positive electrode diffusivity [m2.s-1]": nco_diffusivity_Ecker2015,
"Positive electrode electrons in reaction": 1.0,
"Positive electrode exchange-current density [A.m-2]": nco_electrolyte_exchange_current_density_Ecker2015,
"Positive electrode porosity": 0.296,
"Positive electrode specific heat capacity [J.kg-1.K-1]": 1270.0,
"Positive electrode thermal conductivity [W.m-1.K-1]": 1.04,
"Positive electrode thickness [m]": 5.4e-05,
"Positive particle radius [m]": 6.5e-06,
"Reference temperature [K]": 296.15,
"Separator Bruggeman coefficient (electrolyte)": 1.9804586773134945,
"Separator density [kg.m-3]": 1017.0,
"Separator porosity": 0.508,
"Separator specific heat capacity [J.kg-1.K-1]": 1978.0,
"Separator thermal conductivity [W.m-1.K-1]": 0.34,
"Separator thickness [m]": 2e-05,
"Total heat transfer coefficient [W.m-2.K-1]": 10.0,
"Typical current [A]": 0.15652,
"Typical electrolyte concentration [mol.m-3]": 1000.0,
"Upper voltage cut-off [V]": 4.2,
}
)