Flows
This file represents all the flows inside the fuel cell system. It is a component of the fuel cell model.
calculate_flows(t, sv, control_variables, i_fc, operating_inputs, parameters)
This function calculates the flows inside the fuel cell system.
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Source code in model/flows.py
def calculate_flows(t, sv, control_variables, i_fc, operating_inputs, parameters):
"""This function calculates the flows inside the fuel cell system.
Parameters
----------
t : float
Time (s).
sv : dict
Variables calculated by the solver. They correspond to the fuel cell internal states.
sv is a contraction of solver_variables for enhanced readability.
control_variables : dict
Variables controlled by the user.
i_fc : float
Fuel cell current density at time t (A.m-2).
operating_inputs : dict
Operating inputs of the fuel cell.
parameters : dict
Parameters of the fuel cell model.
Returns
-------
dict
Flows inside the fuel cell system.
"""
# ___________________________________________________Preliminaries__________________________________________________
# Extraction of the variables
C_v_agc, C_v_acl, C_v_ccl, C_v_cgc = sv['C_v_agc'], sv['C_v_acl'], sv['C_v_ccl'], sv['C_v_cgc']
s_acl, s_ccl = sv['s_acl'], sv['s_ccl']
lambda_acl, lambda_mem, lambda_ccl = sv['lambda_acl'], sv['lambda_mem'], sv['lambda_ccl']
C_H2_agc, C_H2_acl, C_O2_ccl, C_O2_cgc = sv['C_H2_agc'], sv['C_H2_acl'], sv['C_O2_ccl'], sv['C_O2_cgc']
C_N2 = sv['C_N2']
# Extraction of the operating inputs and parameters
Tfc, Pc_des = operating_inputs['Tfc'], operating_inputs['Pc_des']
Hgdl, Hmem, Hcl = parameters['Hgdl'], parameters['Hmem'], parameters['Hcl']
Hgc, Wgc = parameters['Hgc'], parameters['Wgc']
epsilon_gdl, epsilon_c = parameters['epsilon_gdl'], parameters['epsilon_c']
e, kappa_co, n_gdl = parameters['e'], parameters['kappa_co'], parameters['n_gdl']
a_slim, b_slim = parameters['a_slim'], parameters['b_slim']
# Intermediate values
(Pagc, Pcgc, s_agdl_agdl, s_agdl_acl, s_ccl_cgdl, s_cgdl_cgdl, epsilon_mean, theta_c_mean, lambda_acl_mem,
lambda_mem_ccl, Pagc_agdl, Pagdl_agdl, Pagdl_acl, Pccl_cgdl, Pcgdl_cgdl, Pcgdl_cgc, nu_l) \
= flows_int_values(sv, operating_inputs, parameters)
# Inlet and outlet flows
Jv_a_in, Jv_a_out, Jv_c_in, Jv_c_out, J_H2_in, J_H2_out, J_O2_in, J_O2_out, J_N2_in, J_N2_out, \
Wasm_in, Wasm_out, Waem_in, Waem_out, Wcsm_in, Wcsm_out, Wcem_in, Wcem_out, Ware, \
Wv_asm_in, Wv_aem_out, Wv_csm_in, Wv_cem_out \
= auxiliaries(t, sv, control_variables, i_fc, operating_inputs, parameters)
# ________________________________________Dissolved water flows (mol.m-2.s-1)_______________________________________
# Anode side
J_lambda_mem_acl = 2.5 / 22 * i_fc / F * lambda_acl_mem - 2 * rho_mem / M_eq * \
D(lambda_acl_mem) * (lambda_mem - lambda_acl) / (Hmem + Hcl)
# Cathode side
J_lambda_mem_ccl = 2.5 / 22 * i_fc / F * lambda_mem_ccl - 2 * rho_mem / M_eq * \
D(lambda_mem_ccl) * (lambda_ccl - lambda_mem) / (Hmem + Hcl)
# _________________________________________Liquid water flows (kg.m-2.s-1)__________________________________________
# Anode side
Jl_agdl_agdl = [None] + [0] * (n_gdl - 1)
for i in range(1, n_gdl):
Jl_agdl_agdl[i] = - sigma(Tfc) * K0(epsilon_gdl, epsilon_c, epsilon_gdl) / nu_l * abs(np.cos(theta_c_gdl)) * \
(epsilon_gdl / K0(epsilon_gdl, epsilon_c, epsilon_gdl)) ** 0.5 * \
(s_agdl_agdl[i] ** e + 1e-7) * (1.417 - 4.24 * s_agdl_agdl[i] + 3.789 * s_agdl_agdl[i] ** 2) * \
(sv[f's_agdl_{i + 1}'] - sv[f's_agdl_{i}']) / (Hgdl / n_gdl)
Jl_agdl_acl = - 2 * sigma(Tfc) * K0(epsilon_mean, epsilon_c, epsilon_gdl) / nu_l * abs(np.cos(theta_c_mean)) * \
(epsilon_mean / K0(epsilon_mean, epsilon_c, epsilon_gdl)) ** 0.5 * \
(s_agdl_acl ** e + 1e-7) * (1.417 - 4.24 * s_agdl_acl + 3.789 * s_agdl_acl ** 2) * \
(s_acl - sv[f's_agdl_{n_gdl}']) / (Hgdl / n_gdl + Hcl)
# Cathode side
Jl_cgdl_cgdl = [None] + [0] * (n_gdl - 1)
for i in range(1, n_gdl):
Jl_cgdl_cgdl[i] = - sigma(Tfc) * K0(epsilon_gdl, epsilon_c, epsilon_gdl) / nu_l * abs(np.cos(theta_c_gdl)) * \
(epsilon_gdl / K0(epsilon_gdl, epsilon_c, epsilon_gdl)) ** 0.5 * \
(s_cgdl_cgdl[i] ** e + 1e-7) * (1.417 - 4.24 * s_cgdl_cgdl[i] + 3.789 * s_cgdl_cgdl[i] ** 2) * \
(sv[f's_cgdl_{i + 1}'] - sv[f's_cgdl_{i}']) / (Hgdl / n_gdl)
Jl_ccl_cgdl = - 2 * sigma(Tfc) * K0(epsilon_mean, epsilon_c, epsilon_gdl) / nu_l * abs(np.cos(theta_c_mean)) * \
(epsilon_mean / K0(epsilon_mean, epsilon_c, epsilon_gdl)) ** 0.5 * \
(s_ccl_cgdl ** e + 1e-7) * (1.417 - 4.24 * s_ccl_cgdl + 3.789 * s_ccl_cgdl ** 2) * \
(sv['s_cgdl_1'] - s_ccl) / (Hgdl / n_gdl + Hcl)
# _____________________________________________Vapor flows (mol.m-2.s-1)____________________________________________
# Convective vapor flows
# Anode side
Jv_agc_agdl = h_a(Pagc_agdl, Tfc, Wgc, Hgc) * (C_v_agc - sv['C_v_agdl_1'])
# Cathode side
Jv_cgdl_cgc = h_c(Pcgdl_cgc, Tfc, Wgc, Hgc) * (sv[f'C_v_cgdl_{n_gdl}'] - C_v_cgc)
# Conductive vapor flows
# Anode side
Jv_agdl_agdl = [None] + [0] * (n_gdl - 1)
for i in range(1, n_gdl):
Jv_agdl_agdl[i] = - Da_eff(s_agdl_agdl[i], epsilon_gdl, Pagdl_agdl[i], Tfc, epsilon_c, epsilon_gdl) * \
(sv[f'C_v_agdl_{i + 1}'] - sv[f'C_v_agdl_{i}']) / (Hgdl / n_gdl)
Jv_agdl_acl = - 2 * Da_eff(s_agdl_acl, epsilon_mean, Pagdl_acl, Tfc, epsilon_c, epsilon_gdl) * \
(C_v_acl - sv[f'C_v_agdl_{n_gdl}']) / (Hgdl / n_gdl + Hcl)
# Cathode side
Jv_cgdl_cgdl = [None] + [0] * (n_gdl - 1)
for i in range(1, n_gdl):
Jv_cgdl_cgdl[i] = - Dc_eff(s_cgdl_cgdl[i], epsilon_gdl, Pcgdl_cgdl[i], Tfc, epsilon_c, epsilon_gdl) * \
(sv[f'C_v_cgdl_{i + 1}'] - sv[f'C_v_cgdl_{i}']) / (Hgdl / n_gdl)
Jv_ccl_cgdl = - 2 * Dc_eff(s_ccl_cgdl, epsilon_mean, Pccl_cgdl, Tfc, epsilon_c, epsilon_gdl) * \
(sv['C_v_cgdl_1'] - C_v_ccl) / (Hgdl / n_gdl + Hcl)
# __________________________________________H2 and O2 flows (mol.m-2.s-1)___________________________________________
# Hydrogen and oxygen consumption
# Anode side
S_H2_acl = - i_fc / (2 * F * Hcl) - \
R * Tfc / (Hmem * Hcl) * (k_H2(lambda_mem, Tfc, kappa_co) * C_H2_acl +
2 * k_O2(lambda_mem, Tfc, kappa_co) * C_O2_ccl)
# Cathode side
S_O2_ccl = - i_fc / (4 * F * Hcl) - \
R * Tfc / (Hmem * Hcl) * (k_O2(lambda_mem, Tfc, kappa_co) * C_O2_ccl +
1 / 2 * k_H2(lambda_mem, Tfc, kappa_co) * C_H2_acl)
# Conductive-convective H2 and O2 flows
# Anode side
J_H2_agc_agdl = h_a(Pagc_agdl, Tfc, Wgc, Hgc) * (C_H2_agc - sv['C_H2_agdl_1'])
# Cathode side
J_O2_cgdl_cgc = h_c(Pcgdl_cgc, Tfc, Wgc, Hgc) * (sv[f'C_O2_cgdl_{n_gdl}'] - C_O2_cgc)
# Conductive H2 and O2 flows
# Anode side
J_H2_agdl_agdl = [None] + [0] * (n_gdl - 1)
for i in range(1, n_gdl):
J_H2_agdl_agdl[i] = - Da_eff(s_agdl_agdl[i], epsilon_gdl, Pagdl_agdl[i], Tfc, epsilon_c, epsilon_gdl) * \
(sv[f'C_H2_agdl_{i + 1}'] - sv[f'C_H2_agdl_{i}']) / (Hgdl / n_gdl)
J_H2_agdl_acl = - 2 * Da_eff(s_agdl_acl, epsilon_mean, Pagdl_acl, Tfc, epsilon_c, epsilon_gdl) * \
(C_H2_acl - sv[f'C_H2_agdl_{n_gdl}']) / (Hgdl / n_gdl + Hcl)
# Cathode side
J_O2_cgdl_cgdl = [None] + [0] * (n_gdl - 1)
for i in range(1, n_gdl):
J_O2_cgdl_cgdl[i] = - Dc_eff(s_cgdl_cgdl[i], epsilon_gdl, Pcgdl_cgdl[i], Tfc, epsilon_c, epsilon_gdl) * \
(sv[f'C_O2_cgdl_{i + 1}'] - sv[f'C_O2_cgdl_{i}']) / (Hgdl / n_gdl)
J_O2_ccl_cgdl = - 2 * Dc_eff(s_ccl_cgdl, epsilon_mean, Pccl_cgdl, Tfc, epsilon_c, epsilon_gdl) * \
(sv['C_O2_cgdl_1'] - C_O2_ccl) / (Hgdl / n_gdl + Hcl)
# __________________________________________Water generated (mol.m-3.s-1)___________________________________________
# Water produced in the membrane at the CL through the chemical reaction and crossover
# Anode side
Sp_acl = 2 * k_O2(lambda_mem, Tfc, kappa_co) * R * Tfc / (Hmem * Hcl) * C_O2_ccl
# Cathode side
Sp_ccl = i_fc / (2 * F * Hcl) + k_H2(lambda_mem, Tfc, kappa_co) * R * Tfc / (Hmem * Hcl) * C_H2_acl
# Water sorption in the CL:
# Anode side
S_sorp_acl = gamma_sorp(C_v_acl, s_acl, lambda_acl, Tfc, Hcl, Kshape) * rho_mem / M_eq * \
(lambda_eq(C_v_acl, s_acl, Tfc, Kshape) - lambda_acl)
# Cathode side
S_sorp_ccl = gamma_sorp(C_v_ccl, s_ccl, lambda_ccl, Tfc, Hcl, Kshape) * rho_mem / M_eq * \
(lambda_eq(C_v_ccl, s_ccl, Tfc, Kshape) - lambda_ccl)
# Liquid water generated through vapor condensation or degenerated through evaporation
# Anode side
Sl_agdl = [None] + [Svl(sv[f's_agdl_{i}'], sv[f'C_v_agdl_{i}'], sv[f'C_v_agdl_{i}'] + sv[f'C_H2_agdl_{i}'],
epsilon_gdl, Tfc, gamma_cond, gamma_evap) for i in range(1, n_gdl + 1)]
Sl_acl = Svl(s_acl, C_v_acl, C_v_acl + C_H2_acl, epsilon_cl, Tfc, gamma_cond, gamma_evap)
# Cathode side
Sl_cgdl = [None] + [Svl(sv[f's_cgdl_{i}'], sv[f'C_v_cgdl_{i}'], sv[f'C_v_cgdl_{i}'] + sv[f'C_O2_cgdl_{i}'] + C_N2,
epsilon_gdl, Tfc, gamma_cond, gamma_evap) for i in range(1, n_gdl + 1)]
Sl_ccl = Svl(s_ccl, C_v_ccl, C_v_ccl + C_O2_ccl + C_N2, epsilon_cl, Tfc, gamma_cond, gamma_evap)
# Vapor generated through liquid water evaporation or degenerated through condensation
# Anode side
Sv_agdl = [None] + [-x for x in Sl_agdl[1:]]
Sv_acl = - Sl_acl
# Cathode side
Sv_cgdl = [None] + [-x for x in Sl_cgdl[1:]]
Sv_ccl = - Sl_ccl
return {'Jv_a_in': Jv_a_in, 'Jv_a_out': Jv_a_out, 'Jv_c_in': Jv_c_in, 'Jv_c_out': Jv_c_out, 'J_H2_in': J_H2_in,
'J_H2_out': J_H2_out, 'J_O2_in': J_O2_in, 'J_O2_out': J_O2_out, 'J_N2_in': J_N2_in, 'J_N2_out': J_N2_out,
'Jv_agc_agdl': Jv_agc_agdl, 'Jv_agdl_agdl': Jv_agdl_agdl, 'Jv_agdl_acl': Jv_agdl_acl,
'S_sorp_acl': S_sorp_acl, 'S_sorp_ccl': S_sorp_ccl, 'Jv_ccl_cgdl': Jv_ccl_cgdl,
'Jv_cgdl_cgdl': Jv_cgdl_cgdl, 'Jv_cgdl_cgc': Jv_cgdl_cgc, 'Jl_agdl_agdl': Jl_agdl_agdl,
'Jl_agdl_acl': Jl_agdl_acl, 'J_lambda_mem_acl': J_lambda_mem_acl, 'J_lambda_mem_ccl': J_lambda_mem_ccl,
'Jl_ccl_cgdl': Jl_ccl_cgdl, 'Jl_cgdl_cgdl': Jl_cgdl_cgdl, 'Sp_acl': Sp_acl, 'Sp_ccl': Sp_ccl,
'J_H2_agc_agdl': J_H2_agc_agdl, 'J_H2_agdl_agdl': J_H2_agdl_agdl, 'J_H2_agdl_acl': J_H2_agdl_acl,
'J_O2_ccl_cgdl': J_O2_ccl_cgdl, 'J_O2_cgdl_cgdl': J_O2_cgdl_cgdl, 'J_O2_cgdl_cgc': J_O2_cgdl_cgc,
'S_H2_acl': S_H2_acl, 'S_O2_ccl': S_O2_ccl, 'Sv_agdl': Sv_agdl, 'Sv_acl': Sv_acl, 'Sv_ccl': Sv_ccl,
'Sv_cgdl': Sv_cgdl, 'Sl_agdl': Sl_agdl, 'Sl_acl': Sl_acl, 'Sl_ccl': Sl_ccl, 'Sl_cgdl': Sl_cgdl,
'Pagc': Pagc, 'Pcgc': Pcgc, 'Wasm_in': Wasm_in, 'Wasm_out': Wasm_out, 'Waem_in': Waem_in,
'Waem_out': Waem_out, 'Wcsm_in': Wcsm_in, 'Wcsm_out': Wcsm_out, 'Wcem_in': Wcem_in, 'Wcem_out': Wcem_out,
'Ware': Ware, 'Wv_asm_in': Wv_asm_in, 'Wv_aem_out': Wv_aem_out, 'Wv_csm_in': Wv_csm_in,
'Wv_cem_out': Wv_cem_out}