def open_circuit(c_n_surf, c_p_surf, T0, T1, param):
"""
Computes the open circuit voltage.
Parameters
----------
c_n_surf: array_like
The value of the concetration at the surface of the negative electrode
particle.
c_n_surf: array_like
The value of the concetration at the surface of the negative electrode
particle.
c_e: array_like
Array of the electrolyte concetration.
T0: array_like
Array of the leading-order temperature.
T1: array_like
Array of the first-order temperature.
param: object
Object containing model parameters.
Returns
----------
float
The open circuit voltage.
"""
U_eq = (
ocp.U_p(c_p_surf, T0, param) - ocp.U_n(c_n_surf, T0, param) +
param.delta *
(ocp.dUdT_p(c_p_surf, param) * T1 - ocp.dUdT_n(c_n_surf, param) * T1))
return U_eq
def SPM_fun(I, c_n, c_p, param):
g_n = param.m_n * param.C_hat_n * np.sqrt(c_n) * np.sqrt(1 - c_n)
g_p = param.m_p * param.C_hat_p * np.sqrt(c_p) * np.sqrt(1 - c_p)
result = (ocp.U_p(c_p, param.T_0, param) -
ocp.U_n(param.c_n, param.T_0, param) -
(2 / param.Lambda) * np.arcsinh(I / (g_p * param.L_p)) -
(2 / param.Lambda) * np.arcsinh(I / (g_n * param.L_n)))
return result
def plot_OCP(soln, mesh, param):
# Get variables
c_n, c_p, c_e_n, c_e_s, c_e_p, T0, T1 = get_vars_time(soln.y, mesh)
t = soln.t
# Surface concentration for BV
c_n_surf = c_n[-1, :] + (c_n[-1, :] - c_n[-2, :]) / 2
c_p_surf = c_p[-1, :] + (c_p[-1, :] - c_p[-2, :]) / 2
# LIONSIMBA results
# t column 0, OCP 14 - 17
t_LION, U_n, U_p, dUdT_n, dUdT_p = np.loadtxt('LIONSIMBA001_t.txt',
usecols=(0, 14, 15, 16, 17),
unpack=True)
# Font stuff
plt.rc('text', usetex=True)
plt.rc('font', family='serif')
# Plot OCP and entropic coefficient at a fixed T
fig = plt.figure()
plt.subplot(2, 2, 1)
plt.plot(t * param.tau_d_star,
ocp.U_n(c_n_surf, T0, param) * param.Phi_star)
plt.plot(t_LION, U_n, 'o')
plt.xlabel(r'$t$ [s]', fontsize=11)
plt.ylabel(r'$U_{{\mathrm{{n}}}}$', fontsize=11)
plt.subplot(2, 2, 3)
plt.plot(t * param.tau_d_star,
ocp.dUdT_n(c_n_surf, param) * param.Phi_star / param.Delta_T_star)
plt.plot(t_LION, dUdT_n, 'o')
plt.xlabel(r'$t$ [s]', fontsize=11)
plt.ylabel(r'$\mathrm{{d}}U_{{\mathrm{{n}}}}$'
r'$ / \mathrm{{d}}T$', fontsize=11)
plt.subplot(2, 2, 2)
plt.plot(t * param.tau_d_star,
ocp.U_p(c_p_surf, T0, param) * param.Phi_star)
plt.plot(t_LION, U_p, 'o')
plt.xlabel(r'$t$ [s]', fontsize=11)
plt.ylabel(r'$U_{{\mathrm{{p}}}}$', fontsize=11)
plt.subplot(2, 2, 4)
plt.plot(t * param.tau_d_star,
ocp.dUdT_p(c_p_surf, param) * param.Phi_star / param.Delta_T_star)
plt.plot(t_LION, dUdT_p, 'o')
plt.xlabel(r'$t$ [s]', fontsize=11)
plt.ylabel(r'$\mathrm{{d}}U_{{\mathrm{{p}}}}$'
r'$ / \mathrm{{d}}T$', fontsize=11)
fig.tight_layout()
def voltage_cutoff(t, y, mesh, param, I_app):
# Get variables
c_n, c_p, c_e_n, c_e_s, c_e_p, T0, T1 = get_vars(y, mesh)
# Surface concentration for BV
c_n_surf = c_n[-1] + (c_n[-1] - c_n[-2]) / 2
c_p_surf = c_p[-1] + (c_p[-1] - c_p[-2]) / 2
# OCV
OCV = ocp.U_p(c_p_surf, T0, param) - ocp.U_n(c_n_surf, T0, param)
# Overpotential
g_n = (param.m_n * param.C_hat_n * c_n_surf**(1 / 2) *
(1 - c_n_surf)**(1 / 2))
g_p = (param.m_p * param.C_hat_p * c_p_surf**(1 / 2) *
(1 - c_p_surf)**(1 / 2))
eta_r = (-2 * (1 + param.Theta * T0) / param.Lambda *
np.arcsinh(I_app / g_p / param.L_p / param.Ly) - 2 *
(1 + param.Theta * T0) / param.Lambda *
np.arcsinh(I_app / g_n / param.L_n / param.Ly))
voltage = OCV + eta_r
TOL = 1e-3
return voltage - param.V_min - TOL
def plot_OCP(c, T, param):
# Font stuff
plt.rc("text", usetex=True)
plt.rc("font", family="serif")
# Plot OCP and entropic coefficient at a fixed T
fig = plt.figure()
plt.subplot(2, 2, 1)
plt.plot(c, ocp.U_n(c, T, param) * param.Phi_star)
plt.xlabel(r"$c_{{\mathrm{{n}}}}$")
plt.ylabel(r"$U_{{\mathrm{{n}}}}$")
plt.subplot(2, 2, 3)
plt.plot(c, ocp.dUdT_n(c, param) * param.Phi_star / param.Delta_T_star)
plt.xlabel(r"$c_{{\mathrm{{n}}}}$")
plt.ylabel(r"$\frac{{\mathrm{{d}} U_{{\mathrm{{n}}}}}}{{\mathrm{{d}}T}}$")
plt.subplot(2, 2, 2)
plt.plot(c, ocp.U_p(c, T, param) * param.Phi_star)
plt.xlabel(r"$c_{{\mathrm{{p}}}}$")
plt.ylabel(r"$U_{{\mathrm{{p}}}}$")
plt.subplot(2, 2, 4)
plt.plot(c, ocp.dUdT_p(c, param) * param.Phi_star / param.Delta_T_star)
plt.xlabel(r"$c_{{\mathrm{{p}}}}$")
plt.ylabel(r"$\frac{{\mathrm{{d}} U_{{\mathrm{{p}}}}}}{{\mathrm{{d}}T}}$")
fig.tight_layout()