Reliability-Based Design Optimization of the PEMFC Flow Field with Consideration of Statistical Uncertainty of Design Variables
Abstract
:1. Introduction
2. Multi-Scale, Multi-Phase 3D PEMFC Model
2.1. Model Assumptions
- (1)
- The present study was guided by a set of specific assumptions outlined below.
- (2)
- Ideal gas mixtures in the gas phase were considered due to the low operating pressures involved.
- (3)
- Flow velocity was assumed to be low, maintaining a laminar condition.
- (4)
- Negligible effects of gravity were taken into account.
- (5)
- The influence of immobile liquid saturation in porous regions was considered to be negligible.
2.2. Governing Equations and Source Terms
2.3. Microscale CL Model
2.4. Boundary Conditions and Numerical Implementation
3. Reliability Based Design Optimization
3.1. General Formulation of RBDO
3.2. Sampling-Based RBDO
3.3. Metamodeling (Surrogate)
4. Results and Discussion
4.1. Experimental Validation of a 3D PEMFC Model and Construction of the MLP Models
4.2. Comparative Analysis of Single-Objective and Multi-Objective Optimization Methods
4.3. RBDO Accounting for Output Constraints and Production Tolerances of Design Variables
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Nomenclature
a | Ratio of active surface area per unit electrode volume, m2/m3 or water activity | |
A | Area, m2 | |
C | Molar concentration of species, mol/m3 or capability | |
COP | Constrained optimal point | |
D | Species diffusivity, m2/s | |
DOP | Deterministic optimal point | |
d | Diameter or vector of design variables or number of input variables or depth | |
E | Activation energy, kJ/mol or expectation operator | |
EW | Equivalent weight of a dry membrane, kg/mol | |
F | Faraday’s constant, 96,487 C/mol | |
f | Objective function | |
G | Global best or deterministic constraint function | |
H | Joint cumulative density function | |
h | Joint probability density function or output of the | |
hidden layer upon activation | ||
i0 | Exchange current density, A/cm2 | |
I | Operating current density, A/cm2 or indicator function | |
j | Transfer current density, A/cm3 | |
k | Thermal conductivity, W/m·K or relative permeability | |
K | Hydraulic permeability, m2 | |
M | Number of Monte Carlo samples | |
MW | Molecular weight, kg/mol | |
N | Number of training points | |
n | Number of electrons transferred in the electrode reaction | |
nc | Number of probabilistic constraints | |
nd | Number of design variables | |
net | Product of weights and the training data | |
nr | Number of random variables | |
OP | Optimal point | |
P | Pressure, Pa | |
P | Waste product or probability function or power density, kW/L | |
p | Number of weight coefficients in each second-order equation or number of | |
weights in the hidden layer | ||
q | Interpolation coefficient | |
R | Real space | |
s | Liquid saturation or the first-order score function | |
S | Source term in the transport equation | |
T | Temperature, K | |
u | Concentration of chemical species U | |
Fluid velocity and superficial velocity in a porous medium, m/s | ||
V | Voltage | |
v | Concentration of chemical species V | |
W | Matrix of weights | |
w | Width or weight | |
wt | Weight ratio | |
X | Vector of random variables or matrix of design variables | |
X | Random variable | |
x | Input variable | |
z | Transport resistance coefficient | |
Greek symbols | ||
α | Transfer coefficient | |
γ | Reaction order or local density | |
δ | Thickness, m | |
ε | Volume fraction | |
η | Surface overpotential, V | |
Contact angle of the gas diffusion layer | ||
κ | Proton conductivity, S/m | |
Water content | ||
Mean | ||
ξ | Stoichiometry flow ratio | |
ρ | Density, kg/m | |
σ | Electronic conductivity, S/m or standard deviation | |
τ | Viscous shear stress, N/m2 | |
Standard normal cumulative density function or phase potential | ||
Ω | Oxygen transport resistance | |
Ω | Failure set | |
Superscripts | ||
c | Catalyst coverage coefficient | |
eff | Effective | |
g | Gas | |
l | Liquid | |
mem | Membrane | |
ref | Reference value | |
Tar | Target | |
Subscripts | ||
a | Anode | |
C | Carbon | |
CL | Catalyst layer | |
c | Cathode | |
cell | Cell | |
ch | Gas channel | |
e | Electrolyte | |
F | Failure | |
gdl | Gas diffusion layer | |
I | Current density | |
i | Species or ith random variables | |
in | Channel inlet | |
int | Interface | |
j | jth constraint | |
k | kth objective function | |
L | Lower | |
land | Land | |
mem | Membrane | |
multi | Multi objective optimization | |
p | Process | |
s | Solid or Surface | |
sgl | Single objective optimization | |
stack | Stack | |
T | Temperature | |
U | Upper | |
u | Momentum equation | |
w | Water | |
0 | Initial conditions or standard conditions, i.e., 298.15 K and 101.3 kPa (1 atm) |
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Governing Equations | ||
---|---|---|
Mass | (1) | |
Momentum | (2) | |
Species | Flow channels and porous media: | (3) |
Water transport in membrane: | (4) | |
Charge | Proton transport: | (5) |
Electron transport: | (6) | |
Energy | (7) |
Description | Expression | ||
---|---|---|---|
Momentum | Porous media | ||
Species | H2 in anode CL | ||
O2 in cathode CL | |||
Water in anode CL | |||
Water in cathode CL | |||
Energy | In anode CL | ||
In cathode CL | |||
In membrane | |||
Charge | In CLs: | ||
Electrochemical reactions HOR on the anode side: ORR on the cathode side: | |||
Transfer current density, | HOR in anode CL: | (8) | |
ORR in cathode CL: | (9) | ||
Overpotential | where | (10) |
Description | Value/Expression | Ref. |
---|---|---|
Activation energy of anode, | 10.0 kJ/mol | [24] |
Activation energy of cathode, | 70.0 kJ/mol | [24] |
Transfer coefficient of HOR, | 1 | [28] |
Transfer coefficient of ORR, | 1 | [28] |
Reference H2/O2 molar concentration, | 40.88 mol/m3 | [28] |
Permeability of GDL/CL, | 1.0 × 10−12/1.0 × 10−13 m2 | [24] |
Equivalent weight of electrolyte in the membrane, | 1.1 kg/mol | [24] |
Faraday’s constant, | 96,485 C/mol | [28] |
Universal gas constant, | 8.314 | |
H2 diffusivity in the anode gas channel, | 1.1028 × 10−4 m2/s | [28] |
H2O diffusivity in the anode gas channel, | 1.1028 × 10−4 m2/s | [28] |
O2 diffusivity in the cathode gas channel, | 3.2348 × 10−4 m2/s | [28] |
H2O diffusivity in the cathode gas channel, | 7.35 × 10−5 m2/s | [28] |
Binary gas diffusivity ( | For nonporous regions | (11) |
Effective diffusivity ( | For porous regions | (12) |
Description | Expression | |
---|---|---|
Mixture density ( | (13) | |
Gas mixture density | (14) | |
Mixture velocity ( | (15) | |
Mixture mass fraction | (16) | |
Relative permeability | (17) | |
(18) | ||
Kinematic viscosity of the two-phase mixture | (19) | |
Kinematic viscosity of the gas mixture | (20) | |
where and | (21) | |
, T in kelvin | (22) | |
Relative mobility | (23) | |
(24) | ||
Diffusive mass flux | (25) | |
Capillary pressure Pc | (26) | |
Leverett function J(s) | (27) |
Description | Expression | |
---|---|---|
Membrane water content (λ) | Water | (28) |
Water activity, | (29) | |
Electro-osmotic drag (EOD) coefficient of water | (30) | |
Proton conductivity () | (31) | |
Water diffusion coefficient () | (32) | |
Interfacial resistance of the water film | (33) |
Description | Expression | |
---|---|---|
Carbon volume fraction () | (34) | |
Pt particle volume fraction () | (35) | |
Ionomer volume fraction () | (36) | |
CL thickness () | (37) | |
Thickness of the ionomer film () | (38) | |
Thickness of water film () | (39) | |
Oxygen balance in a single Pt/C particle domain | (40) | |
Volumetric surface area of ionomer ( | (41) | |
Volumetric surface area of Pt ( | (42) | |
(43) | ||
Total oxygen transport resistance ( | (44) | |
Total transport resistance | (45a) | |
Interfacial resistance in ionomer | (45b) | |
Interfacial resistance in water film | (45c) | |
Interfacial resistance in platinum particle | (45d) | |
Effective ionomer thickness | (46) | |
Effective water thickness | (47) | |
Number of Pt particles on a single carbon particle | (48) | |
Oxygen concentration on the Pt particle surfaces | (49) |
Description | Expression | |
---|---|---|
Anode inlet velocity | (50) | |
Cathode inlet velocity | (51) | |
Constant temperature on side walls | (52) | |
Heat flux on the top and bottom surfaces | (53) | |
Electric potential on anode outer BP surface | (54) | |
Electric potential on cathode outer BP surface | (55) | |
Pressure(anode/cathode), | 2/2 bar | |
Operating temperature, | 333.15 K | |
Stoichiometry(anode/cathode), | 1.5/2.0 |
Design Cases | Design Variables | [kW/L] | [Pa/cm] | Reliability | |||
---|---|---|---|---|---|---|---|
[mm] | [mm] | [mm] | PG1 [%] | PG2 [%] | |||
Ref. point | 1.00 | 1.00 | 1.00 | 1.667 | 2.832 | 100 | 0.00 |
Deterministic Optimum point (DOP) | 0.30 | 1.48 | 0.30 | 3.252 | 30.209 | 0.00 | 100 |
Constraint Optimum point (COP) | 0.30 | 1.51 | 0.50 | 3.008 | 8.000 | 49.79 | 100 |
RBDO | 0.30 | 1.46 | 0.53 | 2.974 | 6.918 | 80.04 | 99.99 |
RBDO | 0.30 | 1.51 | 0.54 | 2.958 | 6.453 | 90.21 | 99.99 |
RBDO | 0.30 | 1.55 | 0.58 | 2.918 | 5.519 | 99.13 | 99.99 |
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Heo, S.; Choi, J.; Park, Y.; Vaz, N.; Ju, H. Reliability-Based Design Optimization of the PEMFC Flow Field with Consideration of Statistical Uncertainty of Design Variables. Energies 2024, 17, 1882. https://doi.org/10.3390/en17081882
Heo S, Choi J, Park Y, Vaz N, Ju H. Reliability-Based Design Optimization of the PEMFC Flow Field with Consideration of Statistical Uncertainty of Design Variables. Energies. 2024; 17(8):1882. https://doi.org/10.3390/en17081882
Chicago/Turabian StyleHeo, Seongku, Jaeyoo Choi, Yooseong Park, Neil Vaz, and Hyunchul Ju. 2024. "Reliability-Based Design Optimization of the PEMFC Flow Field with Consideration of Statistical Uncertainty of Design Variables" Energies 17, no. 8: 1882. https://doi.org/10.3390/en17081882