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194 | 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']
T_acl, T_mem, T_ccl = sv['T_acl'], sv['T_mem'], sv['T_ccl']
# Extraction of the operating inputs and parameters
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']
# Intermediate values
(H_gdl_node, Pagc, Pcgc, lambda_acl_mem, lambda_mem_ccl, D_acl_mem, D_mem_ccl, D_cap_agdl_agdl, D_cap_agdl_acl,
D_cap_cgdl_cgdl, D_cap_ccl_cgdl, ha_Da_eff_agc_agdl, hc_Dc_eff_cgdl_cgc, Da_eff_agdl_agdl, Da_eff_agdl_acl,
Dc_eff_cgdl_cgdl, Dc_eff_ccl_cgdl, T_acl_mem_ccl) \
= 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_acl_mem = 2.5 / 22 * i_fc / F * lambda_acl_mem - \
2 * rho_mem / M_eq * D_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_mem_ccl * (lambda_ccl - lambda_mem) / (Hmem + Hcl)
# _________________________________________Liquid water flows (kg.m-2.s-1)__________________________________________
# Anode side
s_agc = 0 # Dirichlet boundary condition (taken at the agc/agdl border).
Jl_agc_agdl = - 2 * Dcap('gdl', sv['s_agdl_1'], sv['T_agdl_1'], epsilon_gdl, e, epsilon_c=epsilon_c) * \
(sv['s_agdl_1'] - s_agc) / H_gdl_node
Jl_agdl_agdl = [None] + [- D_cap_agdl_agdl[i] * (sv[f's_agdl_{i + 1}'] - sv[f's_agdl_{i}']) / H_gdl_node
for i in range(1, n_gdl)]
Jl_agdl_acl = - 2 * D_cap_agdl_acl * (s_acl - sv[f's_agdl_{n_gdl}']) / (H_gdl_node + Hcl)
# Cathode side
s_cgc = 0 # Dirichlet boundary condition (taken at the cgc/cgdl border).
Jl_cgdl_cgdl = [None] + [- D_cap_cgdl_cgdl[i] * (sv[f's_cgdl_{i + 1}'] - sv[f's_cgdl_{i}']) / H_gdl_node
for i in range(1, n_gdl)]
Jl_ccl_cgdl = - 2 * D_cap_ccl_cgdl * (sv['s_cgdl_1'] - s_ccl) / (H_gdl_node + Hcl)
Jl_cgdl_cgc = - 2 * Dcap('gdl', sv[f's_cgdl_{n_gdl}'], sv[f'T_cgdl_{n_gdl}'], epsilon_gdl, e,
epsilon_c=epsilon_c) * \
(s_cgc - sv[f's_cgdl_{n_gdl}']) / H_gdl_node
# _____________________________________________Vapor flows (mol.m-2.s-1)____________________________________________
# Convective vapor flows
# Anode side
Jv_agc_agdl = - 2 * ha_Da_eff_agc_agdl * (sv['C_v_agdl_1'] - C_v_agc) / (H_gdl_node + Hcl)
# Cathode side
Jv_cgdl_cgc = - 2 * hc_Dc_eff_cgdl_cgc * (C_v_cgc - sv[f'C_v_cgdl_{n_gdl}']) / (H_gdl_node + Hcl)
# Conductive vapor flows
# Anode side
Jv_agdl_agdl = [None] + [- Da_eff_agdl_agdl[i] * (sv[f'C_v_agdl_{i + 1}'] - sv[f'C_v_agdl_{i}']) / H_gdl_node
for i in range(1, n_gdl)]
Jv_agdl_acl = - 2 * Da_eff_agdl_acl * (C_v_acl - sv[f'C_v_agdl_{n_gdl}']) / (H_gdl_node + Hcl)
# Cathode side
Jv_cgdl_cgdl = [None] + [- Dc_eff_cgdl_cgdl[i] * (sv[f'C_v_cgdl_{i + 1}'] - sv[f'C_v_cgdl_{i}']) / H_gdl_node
for i in range(1, n_gdl)]
Jv_ccl_cgdl = - 2 * Dc_eff_ccl_cgdl * (sv['C_v_cgdl_1'] - C_v_ccl) / (H_gdl_node + 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 * T_acl_mem_ccl / (Hmem * Hcl) * (k_H2(lambda_mem, T_mem, kappa_co) * C_H2_acl +
2 * k_O2(lambda_mem, T_mem, kappa_co) * C_O2_ccl)
# Cathode side
S_O2_ccl = - i_fc / (4 * F * Hcl) - \
R * T_acl_mem_ccl / (Hmem * Hcl) * (k_O2(lambda_mem, T_mem, kappa_co) * C_O2_ccl +
1 / 2 * k_H2(lambda_mem, T_mem, kappa_co) * C_H2_acl)
# Conductive-convective H2 and O2 flows
# Anode side
J_H2_agc_agdl = - 2 * ha_Da_eff_agc_agdl * (sv['C_H2_agdl_1'] - C_H2_agc) / (H_gdl_node + Hcl)
# Cathode side
J_O2_cgdl_cgc = - 2 * hc_Dc_eff_cgdl_cgc * (C_O2_cgc - sv[f'C_O2_cgdl_{n_gdl}']) / (H_gdl_node + Hcl)
# Conductive H2 and O2 flows
# Anode side
J_H2_agdl_agdl = [None] + [- Da_eff_agdl_agdl[i] * (sv[f'C_H2_agdl_{i+1}'] - sv[f'C_H2_agdl_{i}']) / H_gdl_node
for i in range(1, n_gdl)]
J_H2_agdl_acl = - 2 * Da_eff_agdl_acl * (C_H2_acl - sv[f'C_H2_agdl_{n_gdl}']) / (H_gdl_node + Hcl)
# Cathode side
J_O2_cgdl_cgdl = [None] + [- Dc_eff_cgdl_cgdl[i] * (sv[f'C_O2_cgdl_{i+1}'] - sv[f'C_O2_cgdl_{i}']) / H_gdl_node
for i in range(1, n_gdl)]
J_O2_ccl_cgdl = - 2 * Dc_eff_ccl_cgdl * (sv['C_O2_cgdl_1'] - C_O2_ccl) / (H_gdl_node + 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, T_mem, kappa_co) * R * T_acl_mem_ccl / (Hmem * Hcl) * C_O2_ccl
# Cathode side
Sp_ccl = i_fc / (2 * F * Hcl) + k_H2(lambda_mem, T_mem, kappa_co) * R * T_acl_mem_ccl / (Hmem * Hcl) * C_H2_acl
# Water absorption in the CL:
# Anode side
S_abs_acl = gamma_sorp(C_v_acl, s_acl, lambda_acl, T_acl, Hcl) * rho_mem / M_eq * \
(lambda_eq(C_v_acl, s_acl, T_acl) - lambda_acl)
# Cathode side
S_abs_ccl = gamma_sorp(C_v_ccl, s_ccl, lambda_ccl, T_ccl, Hcl) * rho_mem / M_eq * \
(lambda_eq(C_v_ccl, s_ccl, T_ccl) - 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}'],
sv[f'T_agdl_{i}'], epsilon_gdl, 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, T_acl, epsilon_cl, 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,
sv[f'T_cgdl_{i}'], epsilon_gdl, 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, T_ccl, epsilon_cl, 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_abs_acl': S_abs_acl, 'S_abs_ccl': S_abs_ccl, 'Jv_ccl_cgdl': Jv_ccl_cgdl,
'Jv_cgdl_cgdl': Jv_cgdl_cgdl, 'Jv_cgdl_cgc': Jv_cgdl_cgc, 'Jl_agc_agdl': Jl_agc_agdl,
'Jl_agdl_agdl': Jl_agdl_agdl, 'Jl_agdl_acl': Jl_agdl_acl, 'J_lambda_acl_mem': J_lambda_acl_mem,
'J_lambda_mem_ccl': J_lambda_mem_ccl, 'Jl_ccl_cgdl': Jl_ccl_cgdl, 'Jl_cgdl_cgdl': Jl_cgdl_cgdl,
'Jl_cgdl_cgc': Jl_cgdl_cgc, '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}
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