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heat_generation.py
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import numpy as np
import open_circuit_potentials as ocp
def rxn_n_0(T0, c_n_surf, param, I_app):
"""
Computes leading-order heating due to electrochemical reactions in the
negative electrode.
Parameters
----------
T0: float
The leading-order bulk temperature.
c_n_surf: float
The value of the concetration at the surface of the negative electrode
particle.
param: object
Object containing model parameters.
I_app: float
The applied current.
Returns
----------
float
The leading-order heating due to electrochemical reactions.
"""
# Compute surface flux
g_bar_n_0 = (param.m_n * param.C_hat_n
* c_n_surf ** (1/2) * (1 - c_n_surf) ** (1/2))
# Compute heating
result = ((I_app / param.Ly)
* ((2 * (1 + param.Theta * T0) / param.Lambda)
* (np.arcsinh(I_app / (g_bar_n_0 * param.L_n * param.Ly)))))
return result / param.L
def rxn_p_0(T0, c_p_surf, param, I_app):
"""
Computes leading-order heating due to electrochemical reactions in the
positive electrode.
Parameters
----------
T0: float
The leading-order bulk temperature.
c_p_surf: float
The value of the concetration at the surface of the positive electrode
particle.
param: object
Object containing model parameters.
I_app: float
The applied current.
Returns
----------
float
The leading-order heating due to electrochemical reactions.
"""
# Compute surface flux
g_bar_p_0 = (param.m_p * param.C_hat_p
* c_p_surf ** (1/2) * (1 - c_p_surf) ** (1/2))
# Compute heating
result = ((I_app / param.Ly)
* ((2 * (1 + param.Theta * T0) / param.Lambda)
* np.arcsinh(I_app / (g_bar_p_0 * param.L_p * param.Ly))))
return result / param.L
def rev_n_0(T0, c_n_surf, param, I_app):
"""
Computes leading-order reversible heating in the negative electrode.
Parameters
----------
T0: float
The leading-order bulk temperature.
c_n_surf: float
The value of the concetration at the surface of the negative electrode
particle.
param: object
Object containing model parameters.
I_app: float
The applied current.
Returns
----------
float
The leading-order reversible heating.
"""
result = ((I_app / param.Ly) * (1 / param.Theta + T0)
* (ocp.dUdT_n(c_n_surf, param)))
return result / param.L
def rev_p_0(T0, c_p_surf, param, I_app):
"""
Computes leading-order reversible heating.
Parameters
----------
T0: float
The leading-order bulk temperature.
c_p_surf: float
The value of the concetration at the surface of the positive electrode
particle.
param: object
Object containing model parameters.
I_app: float
The applied current.
Returns
----------
float
The leading-order reversible heating.
"""
result = (-(I_app / param.Ly) * (1 / param.Theta + T0)
* (ocp.dUdT_p(c_p_surf, param)))
return result / param.L
def ohmic_cc_1(R, param, I_app):
"""
Computes first-order Ohmic heating in the current collectors.
Parameters
----------
R: float
The current collector effective resistance.
param: object
Object containing model parameters.
I_app: float
The applied current.
Returns
----------
float
The first-order Ohmic heating in the current collectors.
"""
result = I_app ** 2 * R
return result / param.L
def ohmic_n_1(c_e_n_bar, c_e_neg_sep, param, I_app):
"""
Computes first-order Ohmic heating in the negative electrode.
Parameters
----------
c_e_n_bar: float
The x-averaged electrolyte concetration in the negative electrode.
c_e_neg_sep: float
The value of the electrolyte concetration at the boundary of the
negative electrode and separator.
param: object
Object containing model parameters.
I_app: float
The applied current.
Returns
----------
float
The first-order Ohmic heating in the negative electrode.
"""
solid = (I_app * param.Ly) ** 2 * (param.L_n / 3 / param.sigma_n_prime)
electrolyte = ((I_app * param.Ly) ** 2
* (param.nu / param.Lambda
/ (param.epsilon_n ** param.brug)
/ param.electrolyte_conductivity(1))
* (param.L_n / 3)
- (I_app * param.Ly)
* (2*(1 - param.t_plus) / param.Lambda)
* (c_e_neg_sep - c_e_n_bar))
result = solid + electrolyte
return result / param.L
def ohmic_s_1(c_e_neg_sep, c_e_pos_sep, param, I_app):
"""
Computes first-order Ohmic heating in the separator.
Parameters
----------
c_e_neg_sep: float
The value of the electrolyte concetration at the boundary of the
negative electrode and separator.
c_e_pos_sep: float
The value of the electrolyte concetration at the boundary of the
positive electrode and separator.
param: object
Object containing model parameters.
I_app: float
The applied current.
Returns
----------
float
The first-order Ohmic heating in the separator.
"""
result = ((I_app * param.Ly) ** 2
* (param.nu / param.Lambda
/ (param.epsilon_s ** param.brug)
/ param.electrolyte_conductivity(1))
* param.L_s
- (I_app * param.Ly) * (2*(1 - param.t_plus) / param.Lambda)
* (c_e_pos_sep - c_e_neg_sep))
return result / param.L
def ohmic_p_1(c_e_p_bar, c_e_pos_sep, param, I_app):
"""
Computes first-order Ohmic heating in the positive electrode.
Parameters
----------
c_e_p_bar: float
The x-averaged electrolyte concetration in the positive electrode.
c_e_pos_sep: float
The value of the electrolyte concetration at the boundary of the
positive electrode and separator.
param: object
Object containing model parameters.
I_app: float
The applied current.
Returns
----------
float
The first-order Ohmic heating in the positive electrode.
"""
solid = (I_app * param.Ly) ** 2 * (param.L_p / 3 / param.sigma_p_prime)
electrolyte = ((I_app * param.Ly) ** 2
* (param.nu / param.Lambda
/ (param.epsilon_p ** param.brug)
/ param.electrolyte_conductivity(1))
* (param.L_p / 3)
- (I_app * param.Ly)
* (2*(1 - param.t_plus) / param.Lambda)
* (-c_e_pos_sep + c_e_p_bar))
result = solid + electrolyte
return result / param.L
def rxn_n_1(T0, T1, c_n_surf, c_e_n_bar, param, I_app):
"""
Computes first-order heating in the negative electrode due to
electrochemical reactions.
Parameters
----------
T0: array_like
Array of the leading-order temperature at the current time.
T0: array_like
Array of the first-order temperature at the current time.
c_n_surf: float
The value of the concetration at the surface of the negative electrode
particle.
c_e_n_bar: float
The x-averaged electrolyte concetration in the negative electrode.
param: object
Object containing model parameters.
I_app: float
The applied current.
Returns
----------
float
The first-order heating in the negative electrode due to
electrochemical reactions.
"""
g_bar_n_0 = (param.m_n * param.C_hat_n
* c_n_surf ** (1/2) * (1 - c_n_surf) ** (1/2))
eta_n_1 = ((- c_e_n_bar / param.Lambda / param.L_n)
* np.tanh((np.arcsinh(I_app
/ (g_bar_n_0 * param.L_n * param.Ly))))
+ param.Theta * T1 / (1 + param.Theta * T0))
result = (I_app / param.Ly)*eta_n_1
return result / param.L
def rxn_p_1(T0, T1, c_p_surf, c_e_p_bar, param, I_app):
"""
Computes first-order heating in the positive electrode due to
electrochemical reactions.
Parameters
----------
T0: array_like
Array of the leading-order temperature at the current time.
T0: array_like
Array of the first-order temperature at the current time.
c_p_surf: float
The value of the concetration at the surface of the positive electrode
particle.
c_e_p_bar: float
The x-averaged electrolyte concetration in the positive electrode.
param: object
Object containing model parameters.
I_app: float
The applied current.
Returns
----------
float
The first-order heating in the positive electrode due to
electrochemical reactions.
"""
g_bar_p_0 = (param.m_p * param.C_hat_p
* c_p_surf ** (1/2) * (1 - c_p_surf) ** (1/2))
eta_p_1 = ((- c_e_p_bar / param.Lambda / param.L_p)
* np.tanh(-(np.arcsinh(I_app
/ (g_bar_p_0 * param.L_p * param.Ly))))
+ param.Theta * T1 / (1 + param.Theta * T0))
result = -(I_app / param.Ly)*eta_p_1
return result / param.L
def rev_n_1(T, c_n_surf, param, I_app):
"""
Computes first-order reversible heating in the negative electrode.
Parameters
----------
T: float
First-order temperature at the current time.
c_n_surf: float
The value of the concetration at the surface of the negative electrode
particle.
param: object
Object containing model parameters.
I_app: float
The applied current.
Returns
----------
float
The first-order reversible heating in the negative electrode.
"""
result = ((I_app / param.Ly) * T
* ocp.dUdT_n(c_n_surf, param))
return result / param.L
def rev_p_1(T, c_p_surf, param, I_app):
"""
Computes first-order reversible heating in the positive electrode.
Parameters
----------
T: float
First-order temperature at the current time.
c_p_surf: float
The value of the concetration at the surface of the positive electrode
particle.
param: object
Object containing model parameters.
I_app: float
The applied current.
Returns
----------
float
The first-order reversible heating in the positive electrode.
"""
result = -((I_app / param.Ly) * T
* ocp.dUdT_p(c_p_surf, param))
return result / param.L