# Modelling Methods and Validation Techniques for CFD Simulations of PEM Fuel Cells

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Electrochemistry

## 3. Modelling Methods for CFD Simulation of PEM Fuel Cells

#### 3.1. Governing Equations

- Laminar: A common assumption in the numerical modelling of PEMFC is the consideration of laminar flows. This hypothesis is well justified by the typical velocity ranges in the gas channels (GC), with Reynolds number $Re\sim 1\times {10}^{2}$, further reduced in porous materials due to flow resistance.
- Steady State: The typical time-scales for PEMFC testing are in the order of minutes, thus justifying the assumption. For this reason, the governing equations presented in this paper are in the steady-state form. However, PEMFC testing cycles [9] and related transient simulations [10] are seeing a growing interest, as well as physical phenomena developing on relatively long time-scales ${\delta}_{t}$ (e.g., finite sorption rates with ${\delta}_{t}\sim $ 100–1000 s [11,12]).
- Multi-physics: All the models aiming at simulating processes in a PEMFC (or in a portion of it) include more than one component. Therefore, solid parts with different physics and macroscopic properties (e.g., GDL, CL and/or BPP) will be present alongside fluid continua, requiring dedicated modifications (source terms, material properties) to each equation.
- Multi-phase: Despite the fact that, in the following part of this section, a so-called single-phase approach will be presented, this is to be intended for fluid modelling in GC, GDL and CL. The dissolved water in the membrane (i.e., its presence in isolated clusters of molecules) is always accounted for, therefore introducing more than a single water phase in the simulation.
- Macro-homogeneous: All cell-scale models express the morphological/structural characteristics of solid parts via averaged (or effective) quantities, e.g., porosity, tortuosity, thermal/electrical conductivity, etc. With particular regard to porous parts (GDL and CL), the real fibrous structure is not directly modelled, and its effect is replaced by the use of calibrated integral properties. This allows great computational efficiency when simulating cell/stack domains.

#### 3.1.1. Mixture Multi-Phase (MMP)

#### 3.1.2. Eulerian Multi-Phase (EMP)

#### 3.1.3. Volume of Fluid (VOF)

#### 3.2. Boundary Conditions

#### 3.3. Gas Diffusion Layers: Key Modelling Aspects

#### 3.3.1. Porosity

#### 3.3.2. Tortuosity

#### 3.3.3. Permeability

#### 3.3.4. Thermal Conductivity and Thermal Contact Resistance

#### 3.3.5. Electrical Conductivity and Electrical Contact Resistance

^{−}as the fluid is an insulator. Therefore, it is logical that a relationship between the solid material conductivity (${\kappa}^{eff}$) and the medium porosity ($\u03f5$) must hold. However, electrons travel much easier along the carbon fibre than through the pressed contacts, ultimately resulting in higher ${k}^{eff}$ values for the in-plane direction than for the through-plane one. Whereas the former is associated to the lateral current transport towards the current collectors (e.g., BPP ribs), the latter is related to electrons travel from/to the CL active sites. Several correlations for ${\kappa}^{eff}$ of carbon paper GDL are available in literature, obtained from experimental data fitting and useful for 3D-CFD models as a function of the GDL $\u03f5$. However, many of them do not distinguish between through-plane and in-plane electrical conductivity: the consequence is that an apparent inconsistency originates, hindering the development of physically solid numerical models. Examples are the correlations from Das et al. [63], Looyenga [64] and the common Bruggeman approximation [31]. A significant exception is the correlation from Zamel et al. [65], which was derived from a 3D-reconstructed anisotropic porous material and provided different coefficients for the through-/in-plane direction. All the mentioned correlations are reported in Table 7 and included in Figure 3.

#### 3.3.6. Compression Effect

#### 3.4. Polymeric Membrane: Key Modelling Aspects

^{+}charges from anode to cathode and of separating the two gas streams and half-reactions at CL. Note that the membrane hosts the transport of water and charged species, although their creation/destruction is pertinent to CL component, and for this reason their source term discussion is moved to the next section. From a modelling point of view, the assumption of fluid impermeability reduces to $\overrightarrow{u}=0$ in the continuity and momentum equations in Table 2, Table 3 and Table 4. In the following, the diffusive model and membrane hydration state will be introduced, based on which models for the proton and water transport will be presented.

#### 3.4.1. Diffusive Model

^{+}) and Fickian (for H

_{2}O) diffusive mechanisms.

^{+}transport are closely coupled, and no accurate prediction of one is possible without a careful modeling of the other. The physical concepts that CFD simulations have to capture are as follows:

- H
^{+}transport influences H_{2}O transport via a mechanism called electro-osmotic drag. The entity of it depends on the membrane hydration itself. It is included in simulations via a dedicated coefficient, although a large uncertainty on it is a potential source of inaccuracy. - H${}_{2}$O influences H
^{+}transport via increasing the protonic conductivity ${\sigma}^{eff}$ with its presence, thus facilitating the H^{+}migration through the polymer chain charged sites (SO_{3}^{−}) via multiple mechanisms (direct/vehicular/hopping mechanisms).

#### 3.4.2. Membrane Water Content

_{2}O molecules per SO

_{3}

^{−}group, named water content $\lambda $, and equal to approximately 20 for fully-hydrated Nafion membranes. This is related to the water concentration in the membrane ionomer (${c}_{{H}_{2}O}$, [$\mathrm{kmol}/{\mathrm{m}}^{3}$]), the membrane density (${\rho}_{m,dry}$, [$\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}$]) and the equivalent weight ($EW$, [$\mathrm{k}\mathrm{g}/\mathrm{kmol}$]) as in Equation (17). The membrane water content at equilibrium is usually related to the local water activity via algebraic expression, for which examples are reported in Table 9 and illustrated in Figure 4, showing that a good agreement is found for under-saturated states ($a<1$), whereas different values are reported for over-saturated conditions ($a>1$). The membrane water content is experimentally determined as a function of the local relative humidity and of water activity a as in Equation (18), where ${p}_{{H}_{2}O}$ represents the partial pressure of the water vapour. The relationship between the water concentration and water content $\lambda $ is given by Equation (17), with ${\rho}_{m,dry}$ the dry membrane density and $E{W}_{m}$ the membrane equivalent weight (represented by the dry mass of the membrane (ionomer) over the number of moles of SO

_{3}

^{−}) which takes a value of 1100 $\mathrm{k}\mathrm{g}$/$\mathrm{k}$$\mathrm{mol}$ (for Nafion 112, 115 and 117) or 2100 $\mathrm{k}\mathrm{g}$/$\mathrm{k}$$\mathrm{mol}$ (for Nafion 211 and 212) [5].

#### 3.4.3. Proton Transport in Membrane

^{+}concentration in the membrane is assumed, thus also the second term in Equation (15) is null. Therefore, the H

^{+}flux (i.e., the volumetric current density, ${j}_{e}$) reduces to a potential-gradient Ohm’s law (Equation (19)). Note that (Equation (19)) only describes the H

^{+}flux in the membrane, while the source term for the charged species are included in Table 2.

^{+}migration in the membrane for increasing water content [5], moving from (I) direct-only H

^{+}transfer between charged sites (for $\lambda <2$) to (II) water induced transport of H

^{+}via H

_{3}O

^{+}ions (vehicular diffusion, similar to the the H

^{+}-induced electro-osmotic drag and relevant for $2<\lambda <13$) and (III) H

^{+}migration between adjacent water molecules, as the polymer side chains are occupied by water molecules (hopping mechanism, dominating for $\lambda >13$)

#### 3.4.4. Water Transport in Membrane

^{+}migration to the cathode drags a number of H

_{2}O molecules, opposing to the gradient-oriented transport. This effect is named electro-osmotic drag, and it is quantified by the electro-osmotic drag coefficient ${n}_{d}$ representing the number of transported moles of H

_{2}O for each mole of H

^{+}. Several formulations from literature for ${n}_{d}$ are reported in Table 11 and illustrated in Figure 5. Note that all correlations generally predict an increase in ${n}_{d}$ for increased water content $\lambda $ (correlated with the water concentration), although their disagreement frames a still relevant uncertainty on this effect.

#### 3.5. Catalyst Layers: Key Modelling Aspects

_{2}, e

^{−}and H

^{+}ions combine to produce H

_{2}O.

#### 3.5.1. Modelling Approaches for CL

- Ultra-thin layer model: It is the simplest approach, consisting of neglecting the CL thickness and representing it with an infinitely thin surface where reactions take place as interfacial processes at the contact between membrane and GDL. Clearly, this approach is computationally very efficient and consequently attractive for the simulation of an entire cell. On the other hand, it prevents the analysis of the phenomena occurring within the catalyst layer and the role of the microstructure on cell performance. Berning and Djilali [86] used this model to analyse the effects of different parameters (e.g., GDL porosity and thickness) on the cell performance. However, note that this way to model the CL lead to an overestimation of the current density.
- Macro-homogeneous model (also called pseudo-homogeneous model): Similar to the GDL macro-homogeneous approach previously presented, it is a more accurate approach for CL modelling as it considers the finite CL thickness, with averaged transport coefficients describing the effect of variations in compositional parameters describing platinum catalyst, carbon support, solid GDL matrix and electrolyte materials [87]. However, it cannot assess the complex multi-material structure of the CL.
- Agglomerate model: It is the most complex and sophisticated approach, including both the composition and the structural distribution of CL materials. Generally, in this type of approach the CL is composed by agglomerates, each of them presenting ionomer and Pt/C particles. Agglomerate models results agree with experimental observations showing that a CL is composed by the agglomeration of catalyst particles (Pt and C) and an ionomer [88]. There are two types of pores, primary and secondary: the former are the internal pores within the agglomerates, while the latter are between the different agglomerates. The primary pores inside the agglomerate could be filled by an ionomer phase, as in the model presented in [88]. In this case, within these pores the diffusion of the reactants is permitted only in the dissolved phase, whereas the secondary pores may be partially or fully filled with liquid water. Each agglomerate has a radius and may be covered with an ionomer film of uniform thickness. Another agglomerate model is the one presented by Xing [89], where the generated liquid water occupies the void spaces of both the primary and secondary pores, reducing the void space. The generated liquid water is initially formed inside the primary pores, partially occupying the space until it fills them completely. When the primary pores are fully filled, the liquid water fills the secondary pores as a thin film surrounding the carbon agglomerate. This means that the presence of liquid water in the secondary pores depends on the filling of the primary pores. The effective species diffusivity in the primary pores will therefore be composed of the diffusivity through (I) the ionomer phase, (II) the liquid phase and (III) the void space, whereas the effective species diffusivity in the secondary pores will depend on the volume fraction of liquid.

#### 3.5.2. Electrochemistry Modelling in CL

_{2}/air PEMFC as in Equation (22), where the terms ${S}_{{H}_{2},a}$, ${S}_{{O}_{2},c}$ and ${S}_{{H}_{2}O,c}$, consider the consumption/production of H

_{2}, H

_{2}and H

_{2}O, respectively.

#### 3.5.3. Dissolved Water Treatment in CL

#### 3.5.4. Heat Generation in CL

## 4. Validation Techniques for CFD Simulation of PEM Fuel Cells

#### 4.1. Measurement Techniques for Fuel Cell Stacks

#### 4.1.1. Gas Composition Anode/Cathode

#### Infrared Spectroscopy

#### Mass Spectrometer

#### Gas Chromatography

#### Colorimetric Tubes

#### 4.1.2. Liquid Water Content

#### Transparent Cell

#### Neutron Imaging

#### X-ray Imaging

#### Magnetic Resonance Imaging

#### Raman Spectroscopy

#### 4.1.3. Current Density Distribution

#### Printed Circuit Board (PCB)

#### Segmented Cells

#### Magnetic Resonance

#### Magnetotomography

#### 4.1.4. Temperature Distribution

#### Infrared Thermography

#### Micro-Thermocouples

#### In-Fibre Bragg Grating (FBG) Sensors

#### 4.2. Measurement Techniques for Material Parameters

- high impact: current density deviates by more than 10% (absolute) or
- medium impact: current density deviates between 5% to 10% (absolute)

- (A)
- Membrane thickness
- (B)
- Membrane ionic conductivity
- (C)
- Cathode CL thickness
- (D)
- Cathode CL exchange current density
- (E)
- GDL thickness
- (F)
- GDL porosity
- (G)
- GDL electrical conductivity
- (H)
- Membrane sulfonic acid group concentration
- (I)
- Membrane water diffusion coefficient
- (J)
- Cathode CL transfer coefficient
- (K)
- GDL contact angle

#### 4.2.1. Thickness Measurement

#### 4.2.2. Ionic Conductivity

#### 4.2.3. Water Diffusion Coefficient/Electro-Osmotic Drag Coefficient

#### 4.2.4. Ion-Exchange Capacity

^{+}-ions with Na

^{+}[201,207]. To achieve the ion exchange, the samples are soaked in NaCl for several hours. Afterward, a defined amount of the solution was titrated with NaOH. The IEC can be determined with the following formula [201]:

#### 4.2.5. Porous Media Characterisation

#### Permeability/Porosity

- (1)
- Darcy’s Law [235,236,237]. The pressure drop of flow with low Reynolds number through a porous medium is determined as follows (3D case):$$\nabla P=-\frac{\mu}{K}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}v$$
- (2)
- (3)

#### Electrical Conductivity

#### 4.2.6. (Reference) Exchange Current Density

## 5. Conclusions

- In the first block, the most common modelling approaches for the simulation of fluid/heat/charge transport in PEMFC are presented, namely, the Mixture Multi-Phase (MMP), the Eulerian Multi-Phase (EMP) and the Volume of Fluid (VOF). The multi-part and multi-physics feature of PEMFC and the ubiquitous effect of water transport and phase-change on all the concurring phenomena are the pillars of all the mentioned methods, and their inclusion in model equations is elucidated. Then, a component-based analysis of key properties for multidimensional modelling of GDL, membrane and CL are provided, and the most common correlations from the literature are presented and discussed.
- In the second part, the most advanced measuring methods to measure fluid properties (e.g., gas composition and liquid water content), electrical variables (e.g., current distribution) and material characterisation (e.g., permeability and conductivity) are presented. The variety of the surveyed techniques renders the complexity of the problem and testifies how insufficient is PEMFC testing only based on cell polarisation curves.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ACL | Anode Catalyst Layer |

BPP | Bipolar Plate |

CL | Catalyst Layer |

CCL | Cathode Catalyst Layer |

CCM | Catalyst-Coated Membrane |

CFD | Computational Fluid Dynamics |

CCM | Catalyst-Coated Membrane |

ECR | Electrical Contact Resistance |

EIS | Electrochemical Impedance Spectroscopy |

EMP | Eulerian Multi-Phase |

EW | Equivalent Weight |

FC | Fuel Cell |

FRA | Frequency Response Analyzer |

FTIR | Fourier Transform Infrared Spectroscopy |

GC | Gas Channels |

GDL | Gas Diffusion Layer |

HOR | Hydrogen Oxidation Reaction |

HT-PEMFC | High-Temperature PEMFC |

IEC | Ion Exchange Capacity |

IP | In-Plane |

LSC | Long Side Chain |

MEA | Membrane Electrode Assembly |

MMP | Mixture Multi-Phase |

MPL | Micro-Porous Layer |

NMR | Nuclear Magnetic Resonance |

OCV | Open Circuit Voltage |

ORR | Oxygen Reduction Reaction |

PCB | Printed Circuit Board |

PEM | Polymer Electrolyte Membrane |

PFSA | Perfluoronated Sulfonic Acid |

RH | Relative Humidity |

SEM | Scanning Electron Microscopy |

SSC | Short Side Chain |

STM | Scanning Transmission X-ray Microscopy |

TCR | Thermal Contact Resistance |

TP | Through-Plane |

TPB | Triple-Phase Boundary |

VOF | Volume Of Fluid |

XCT | X-ray Computed Tomography |

a | Activity [-] |

$\alpha $ | Volume Fraction [-] |

${\alpha}_{a/c}$ | Anodic/Cathodic Transfer Coefficient [-] |

c | Species Concentration [kmol/m${}^{3}$] |

D | Diffusion Coefficient [m${}^{2}$/s] |

d | Fiber Diameter [m] |

$\u03f5$ | Porosity [-] |

F | Faraday Constant |

h | Latent Heat of Evaporation [J/kg] |

j | Volumetric Current Density [A/m${}^{3}$] |

k | Thermal Conductivity [W/m/K] |

$\kappa $ | Electric Conductivity [S/m] |

K | Absolute Permeability [m${}^{2}$] |

$\lambda $ | Water Content [-] |

M | Molecular Weight [kg/mol] |

$\mu $ | Molecular Viscosity [Pa s] |

${n}_{d}$ | Electro-Osmotic Drag Coefficient [-] |

$\Phi $ | Electrical Potential [V] |

$\varphi $ | Solid Volume Fraction [-] |

$\rho $ | Density [kg/m${}^{3}$] |

S | Source Term |

s | Liquid Water Volume Fraction [-] |

$\sigma $ | Ionic Conductivity [S/m] |

$\tau $ | Tortuosity [-] |

$\theta $ | Mobility [m${}^{2}$ mol/s/J] |

$\overrightarrow{u}$ | Velocity Vector [m/s] |

x | Mole Fraction [-] |

Y | Mass Fraction [-] |

$\zeta $ | Specific Active Surface Area [1/m] |

a | Anode |

c | Cathode |

e | Electrolyte |

$eff$ | Effective |

g | Gas |

l | Liquid |

m | Membrane |

$mix$ | Mixture |

$rev$ | Reversible |

s | Solid |

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**Figure 4.**Left (

**a**): Correlations for the membrane water content $\lambda $ as a function of the water activity a. Right (

**b**): Membrane protonic conductivity ${\sigma}^{eff}$ from Table 11.

**Figure 6.**Influence of material parameter on current density. Raw values are from the work in [140].

**Figure 7.**XCT-based cross-sectional views at different degradation stages. The thickness of CLs and membrane can be determined. Reprinted from the work in [158].

**Figure 8.**(

**a**) Principle of the four-probe method and complex equivalent circuit. WE: Working electrode; CE: Counterelectrode; RE: Reference electrode. The electrodes are placed on a Teflon bed. For the two-probe method, two RE coincide with WE and RE (based on the work in [162]). (

**b**) Open-ended coaxial probe with equivalent circuit (based on the work in [172]).

**Figure 9.**Typical complex impedance response of a Nafion membrane ($\vartheta =23{}^{\circ}\mathrm{C},\phantom{\rule{4pt}{0ex}}RH=65\%$) with frequency range of 50–1 $\times $. Dashed line: data from measurement; solid line: fitting of all measured points; dotted line: fitting of the first half-circle. The resulting ionic conductivity of the sample is $0.023$ $\mathrm{S}$/$\mathrm{c}$$\mathrm{m}$. (based on the work in [178]).

**Figure 10.**Principle of hydrogen pump cell (based on the work in [196]).

**Figure 11.**Two different GDL fabrication types: (

**a**) carbon paper (Toray 090) and (

**b**) woven carbon cloth (E-Tek Cloth “A”) (reprinted from the work in [50]). (

**c**) Reconstructed 3D view of GDL structure of uncompressed carbon paper using phase contrast tomographic microscopy. Reprinted from the work in [217].

**Figure 12.**Measurement device for permeability measurements on GDLs. Reprinted from the work in [50].

**Figure 13.**Four-point-probe electrode arrangements for TP (

**a**) and IP in square (

**b**) and linear (

**c**) configurations. Reprinted from the work in [247].

Equation | |
---|---|

Mass | $\nabla \left(\right)open="("\; close=")">{\rho}_{mix}{\overrightarrow{u}}_{mix}$ |

Momentum | $\nabla \left(\right)open="("\; close=")">\frac{{\rho}_{mix}{\overrightarrow{u}}_{mix}{\overrightarrow{u}}_{mix}}{{\u03f5}^{2}}+\nabla \left(\right)open="("\; close=")">\frac{{\overrightarrow{u}}_{mix}^{T}}{\u03f5}$ |

Species | $\nabla \left({\rho}_{mix}{Y}_{i}{\overrightarrow{u}}_{mix}\right)=\nabla (\rho {D}_{i}^{eff}\nabla {Y}_{i})+{S}_{i}$ |

Energy | $\nabla \left(\right)open="["\; close="]">{\left({\rho}_{mix}{c}_{p}\right)}^{eff}{\overrightarrow{u}}_{mix}T+{S}_{T}$ |

Charge | $\nabla \left(\right)open="("\; close=")">{\kappa}^{eff}\nabla {\Phi}_{s}$ |

$\nabla \left(\right)open="("\; close=")">{\sigma}^{eff}\nabla {\Phi}_{e}$ |

GC | GDL | CL | Solid Parts | |
---|---|---|---|---|

Mass | ${S}_{m}=0$ | ${S}_{m}=0$ | ${S}_{m}={\sum}_{i}{S}_{i}$ | ${\overrightarrow{u}}_{mix}=0$ |

Momentum | $\u03f5=0$ | ${S}_{u}=\left(\right)open="("\; close=")">-\frac{\mu}{{K}_{GDL}}$ | ${S}_{u}=\left(\right)open="("\; close=")">-\frac{\mu}{{K}_{CL}}$ | ${\overrightarrow{u}}_{mix}=0$ |

${S}_{u}=0$ | ||||

Species | ${S}_{i}=0$ | ${S}_{i}=0$ | Anode H${}_{2}$: ${S}_{{H}_{2},a}=-\frac{{j}_{a}}{2F}{M}_{{H}_{2}}$ | ${Y}_{i}=0$ |

Anode H${}_{2}$O: | ||||

${S}_{{H}_{2}O,a}=-\nabla \left(\right)open="["\; close="]">\frac{{n}_{d}}{F}{\sigma}_{eff}\nabla {\Phi}_{e}$ | ||||

Cathode O${}_{2}$: ${S}_{{O}_{2},c}=-\frac{{j}_{c}}{4F}{M}_{{O}_{2}}$ | ||||

Cathode H${}_{2}$O: | ||||

${S}_{{H}_{2}O,c}=\frac{{j}_{c}}{2F}{M}_{{H}_{2}O}+\nabla \left(\right)open="["\; close="]">\frac{{n}_{d}}{F}{\sigma}_{eff}\nabla {\Phi}_{e}$ | ||||

Energy | ${S}_{T}=0$ | ${S}_{T}=\frac{{i}_{s}^{2}}{{\kappa}^{eff}}$ | Anode CL: | BPP: |

${S}_{T}={j}_{a}\left|{\eta}_{act}\right|+\frac{{i}_{s}^{2}}{{\kappa}^{eff}}+\frac{{i}_{e}^{2}}{{\sigma}^{eff})}$ | ${S}_{T}=\frac{{i}_{s}^{2}}{{\kappa}^{eff}}$ | |||

Cathode CL: | Membrane: | |||

${S}_{T}={j}_{c}\left|{\eta}_{act}\right|+\frac{{j}_{c}T\Delta S}{2F}+\frac{{i}_{s}^{2}}{{\kappa}^{eff}}+\frac{{i}_{e}^{2}}{{\sigma}^{eff}}$ | ${S}_{T}=\frac{{i}_{e}^{2}}{{\sigma}^{eff}}$ | |||

Charge | ${\Phi}_{s}=0$ | ${S}_{{\Phi}_{s}}=0$ | Anode CL: ${S}_{{\Phi}_{s}}=-{j}_{a},\phantom{\rule{1.em}{0ex}}{S}_{{\Phi}_{e}}={j}_{a}$ | ${S}_{{\Phi}_{s}}=0$ |

${\Phi}_{e}=0$ | ${S}_{{\Phi}_{e}}=0$ | Cathode CL: ${S}_{{\Phi}_{s}}={j}_{c},\phantom{\rule{1.em}{0ex}}{S}_{{\Phi}_{e}}=-{j}_{c}$ | ${S}_{{\Phi}_{e}}=0$ |

Equation | |
---|---|

Mass (gas) | $\nabla \left(\right)open="("\; close=")">{\rho}_{g}{\overrightarrow{u}}_{g}$ |

Momentum (gas) | $\nabla \left(\right)open="("\; close=")">\frac{{\rho}_{g}{\overrightarrow{u}}_{g}{\overrightarrow{u}}_{g}}{{\u03f5}^{2}{(1-{\alpha}_{l})}^{2}}+\nabla \left(\right)open="("\; close=")">\frac{{\overrightarrow{u}}_{g}^{T}}{\u03f5(1-{\alpha}_{l})}$ |

Species (gas) | $\nabla \left({\rho}_{g}{Y}_{i}{\overrightarrow{u}}_{g}\right)=\nabla ({\rho}_{g}{D}_{i}^{eff}\nabla {Y}_{i})+{S}_{i}$ |

Water (liquid) | $\nabla \left({\rho}_{l}{\overrightarrow{u}}_{l}\right)=\nabla ({\rho}_{l}{D}_{l}\nabla {\alpha}_{l})+{S}_{l}$ |

Energy (gas) | $\nabla \left(\right)open="["\; close="]">{\left({\rho}_{g}{c}_{p,g}\right)}^{eff}{\overrightarrow{u}}_{g}{T}_{g}+{S}_{T,g}$ |

Energy (liquid) | $\nabla \left(\right)open="["\; close="]">{\left({\rho}_{l}{c}_{p,l}\right)}^{eff}{\overrightarrow{u}}_{l}{T}_{l}+{S}_{T,l}$) |

GC | GDL | CL | |
---|---|---|---|

Mass (gas) | ${S}_{m}=-{S}_{gl}$ | ${S}_{m}=-{S}_{gl}$ | ${S}_{m}=-{S}_{gl}+{\sum}_{i}{S}_{i}$ |

Momentum (gas) | $\u03f5=0$ | ${S}_{u}=\left(\right)open="("\; close=")">-\frac{{\mu}_{g}}{{K}_{GDL}}$ | ${S}_{u}=\left(\right)open="("\; close=")">-\frac{{\mu}_{g}}{{K}_{CL}}$ |

${S}_{u}=0$ | |||

Species (gas) | ${S}_{i}=0$ | ${S}_{i}=0$ | Anode H${}_{2}$: ${S}_{{H}_{2},a}=-\frac{{j}_{a}}{2F}{M}_{{H}_{2}}$ |

Anode H${}_{2}$O: ${S}_{{H}_{2}O,a}=-{S}_{gl}-{S}_{gd}$ | |||

Cathode O${}_{2}$: ${S}_{{O}_{2},c}=-\frac{{j}_{c}}{4F}{M}_{{O}_{2}}$ | |||

Cathode H${}_{2}$O: ${S}_{{H}_{2}O,c}=\frac{{j}_{c}}{2F}{M}_{{H}_{2}O}-{S}_{gl}-{S}_{gd}$ | |||

Water (liquid) | ${S}_{l}={S}_{gl}$ | ${S}_{l}={S}_{gl}$ | Anode H${}_{2}$O: ${S}_{{H}_{2}O,a}={S}_{gl}-{S}_{ld}$ |

Cathode H${}_{2}$O: ${S}_{{H}_{2}O,c}={S}_{gl}-{S}_{ld}$ | |||

Energy (gas) | ${S}_{T}=0$ | ${S}_{T}=\frac{{i}_{s}^{2}}{{\kappa}^{eff}}+{S}_{gl}{h}_{w}$ | ${j}_{a/c}\left(\right)open="["\; close="]">{\eta}_{a/c}+T\frac{\Delta {S}_{a/c}}{nF}$ |

Energy (liquid) | ${S}_{T}=0$ | ${S}_{T}=\frac{{i}_{s}^{2}}{{\kappa}^{eff}}+{S}_{gl}{h}_{w}$ | ${j}_{a/c}\left(\right)open="["\; close="]">{\eta}_{a/c}+T\frac{\Delta {S}_{a/c}}{nF}$ |

$\frac{{i}_{e}^{2}}{{k}^{eff}}+{S}_{gl}{h}_{w}+{S}_{ld}({h}_{w,m}-{h}_{w})$ |

Reference | Correlation | Notes |
---|---|---|

[31] | $\tau ={\left(\right)}^{\frac{1}{\u03f5}}0.5$ | |

[45] | $\tau =1+0.72\frac{1-\u03f5}{{(\u03f5-0.11)}^{0.54}}$ | |

[46] | $\tau ={\left(\right)}^{\frac{1-{\u03f5}_{p}}{\u03f5-{\u03f5}_{p}}}\alpha $ | ${\u03f5}_{p}=0.11$ |

$\alpha =0.521$ (2D parallel flow) | ||

$\alpha =0.758$ (2D normal flow) |

Reference | Correlation | Notes |
---|---|---|

[41] | $K=exp\left(\right)open="["\; close="]">\frac{-12.95+13.9\u03f5}{1+1.57\u03f5-2.22{\u03f5}^{2}}$ | |

[42] | $K=0.012\left(\right)open="("\; close=")">1-\varphi -2\frac{\pi}{4\varphi}+1$ | |

[46] | $K=\frac{\u03f5}{8\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\left(ln\u03f5\right)}^{2}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\left(\right)open="["\; close="]">\frac{{(\u03f5-{\u03f5}_{p})}^{\alpha +2}}{{(1-\u03f5)}^{\alpha}{\left(\right)}^{(\alpha +1)}2}{d}^{2}$ | ${\u03f5}_{p}=0.11$ |

$\alpha =0.521$ (2D parallel flow) | ||

$\alpha =0.758$ (2D normal flow) | ||

[49] | $K=0.0065\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{{\u03f5}^{3.6}}{1-\u03f5}{d}^{2}$ | |

[50] | $K=\frac{{\u03f5}^{3}}{67.2{(1-\u03f5)}^{2}}{d}^{2}$ |

Ref. | Correlation | Notes |
---|---|---|

[63] | ${\kappa}^{eff}={\kappa}_{s}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{2-2\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\u03f5}{2+\u03f5}$ | Bulk conductivity: though-/in-plane directions not distinguished. |

[64] | ${\kappa}^{eff}={\kappa}_{s}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{(1-\u03f5)}^{3}$ | Bulk conductivity: though-/in-plane directions not distinguished. |

[31] | ${\kappa}^{eff}={\kappa}_{s}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{(1-\u03f5)}^{1.5}$ | Bulk conductivity: though-/in-plane directions not distinguished. |

[65] | $\begin{array}{c}{\kappa}^{eff}={\kappa}_{s}\left(\right)open="\{"\; close>1-\left(\right)open="("\; close=")">\frac{3\u03f5}{2+\u03f5}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}A\end{array}\left(\right)open\; close="\}">\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{exp}\left(\right)open="["\; close="]">B\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}(1-\u03f5)& \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{(1-\u03f5)}^{C}$ | Through-plane: |

$A=0.962\pm 0.01$, $B=0.889\pm 0.015$, $C=-0.00715\pm 0.005$ | ||

In-plane: | ||

$A=0.962\pm 0.004$, $B=0.367\pm 0.005$, $C=-0.016\pm 0.002$ |

**Table 8.**Correlations for electrical contact resistance $ECR$ in [$\mathrm{m}\Omega \xb7\mathrm{c}{\mathrm{m}}^{2}$] between GDL and BPP.

Reference | Correlation | Notes |
---|---|---|

[69] | $ECR=2.2163+\frac{3.5306}{{p}_{seal}}$ | Overestimated ECR in the ${p}_{seal}\le $ 2 $\mathrm{M}$$\mathrm{Pa}$ range. |

Ref. | Correlation | Notes |
---|---|---|

[6] | $\lambda =\left(\right)open="\{"\; close>\begin{array}{cc}0.043+17.81a-39.85{a}^{2}+36.0{a}^{3},\hfill & \mathrm{for}0a1\hfill \\ 14+1.4(a-1),\hfill & \mathrm{for}1\le a\le 3\hfill \end{array}$ | |

[78] | $\lambda =\left(\right)open="\{"\; close>\begin{array}{c}0.3+6a\left(\right)open="["\; close="]">1-\mathrm{tanh}(a-0.5)+\hfill \end{array}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}+3.9\sqrt{a}\left(\right)open="["\; close="]">1+\mathrm{tanh}\left(\right)open="("\; close=")">\frac{a-0.89}{0.23}\hfill & +s\left(\right)open="("\; close=")">{\lambda}_{s=1}-{\lambda}_{a=1}\\ ,\mathrm{for}s\le 0\hfill $ | ${\lambda}_{s=1}=16.8$ is the water content at saturation and ${\lambda}_{a=1}$ is the value obtained when $s=1$ and $a=1$. |

[36] | $\lambda =\left(\right)open="\{"\; close>\begin{array}{cc}0.03+18.43a-46.67{a}^{2}+44.36{a}^{3},\hfill & \mathrm{for}0a1\hfill \\ 16.15+5.85(a-1),\hfill & \mathrm{for}1\le a\le 3\hfill \end{array}$ |

**Table 10.**Correlations for membrane protonic conductivity ${\sigma}^{eff}$ in [$\mathrm{S}/\mathrm{m}$].

Ref. | Correlation | Notes |
---|---|---|

[6] | ${\sigma}^{eff}=(0.5139\lambda -0.326){e}^{1268\left(\right)open="("\; close=")">\frac{1}{303}-\frac{1}{T}}$ | |

[36] | ${\sigma}^{eff}=(1.72\lambda -2.26){e}^{2000\left(\right)open="("\; close=")">\frac{1}{303.15}-\frac{1}{T}}$ | |

[79] | ${\sigma}^{eff}=0.5738\lambda -0.7192$ |

Reference | Correlation |
---|---|

[6] | ${n}_{d}=\frac{2.5\lambda}{22}$ |

[80] | ${n}_{d}=\left(\right)open="["\; close="]">\frac{1}{{\left(\right)}^{0.35}}$ |

[81] | ${n}_{d}=0.0028\lambda +0.05\lambda -3.5\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{10}^{-19}$ |

[82] | ${n}_{d}=\left(\right)open="\{"\; close>\begin{array}{cc}1.0,\hfill & \mathrm{if}\lambda \le 14\hfill \\ \frac{1.5}{8}\left(\right)open="("\; close=")">\lambda -14+1.0,\hfill \\ \mathrm{if}\lambda 14\hfill \end{array}$ |

**Table 12.**Correlations for dissolved water diffusion coefficient ${D}_{w}$ in [${\mathrm{m}}^{2}/\mathrm{s}$].

Ref. | Correlation | Notes |
---|---|---|

[6] | ${D}_{w,m}={10}^{-10}{\mathrm{e}}^{2416\left(\right)open="("\; close=")">\frac{1}{303}-\frac{1}{T}}$ | |

[83] | ${D}_{w,m}=\left(\right)open="\{"\; close>\begin{array}{c}3.1\times {10}^{-3}\lambda \left(\right)open="("\; close=")">{\mathrm{e}}^{0.28\lambda}-1\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{e}}^{\left(\right)}\hfill & ,\end{array}\mathrm{for}0\lambda 3\hfill \mathrm{for}3\le \lambda 17\hfill $ | |

[84] | ${D}_{w,m}=\left(\right)open="\{"\; close>\begin{array}{c}\left(\right)open="("\; close=")">0.0049+2.02a-4.53{a}^{2}+4.09{a}^{3}\hfill \end{array}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}{D}_{0}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{e}}^{2416\left(\right)open="("\; close=")">\frac{1}{303}-\frac{1}{T}},\hfill \\ \mathrm{for}a\le 1\hfill \mathrm{for}a1\hfill $ | with ${D}_{0}=5.5\mathrm{e}-7\mathrm{c}\mathrm{m}/\mathrm{s}$ |

[81] | ${D}_{w,m}=\left(\right)open="\{"\; close>\begin{array}{c}{10}^{-10}{\mathrm{e}}^{2416\left(\right)open="("\; close=")">\frac{1}{303}-\frac{1}{T}},\hfill \\ \mathrm{for}\lambda 2\hfill \end{array}{10}^{-10}\left(\right)open="["\; close="]">1+2(\lambda -2){\mathrm{e}}^{2416\left(\right)open="("\; close=")">\frac{1}{303}-\frac{1}{T}}\hfill & ,{10}^{-10}\left(\right)open="["\; close="]">3-1.67(\lambda -3){\mathrm{e}}^{2416\left(\right)open="("\; close=")">\frac{1}{303}-\frac{1}{T}}\hfill & ,$ | |

[26,85] | ${D}_{w,m}=4.1\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{10}^{-10}{\left(\right)}^{\frac{\lambda}{25}}0.15$ |

**Table 13.**Exchange current density and transfer coefficients from literature. The table continues in the bottom part for additional references and variables.

Ref. | ${j}_{0,a}^{ref}[\mathbf{A}/{\mathrm{m}}^{2}]$ | ${j}_{0,c}^{ref}[\mathbf{A}/{\mathrm{m}}^{2}]$ | ${\alpha}_{a}$ | ${\alpha}_{c}$ |

[77] | $1.0\times {10}^{4}$ | 1 | $0.5$ | $0.8$ |

[90] | $3.0\times {10}^{3}$ | $0.3$ | $1.0$ | $0.8$ |

[91] | $3.5$ | $3.5\times {10}^{-4}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{exp}\left(\right)open="["\; close="]">\frac{-7900\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}(353.15-T)}{353.15\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}T}$ | $0.5$ | $0.5$ |

[87] | − | ${10}^{0.037\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}T-16.96}$ | $0.5$ | $1.0$ |

[92] | $3.5\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{exp}\left(\right)open="["\; close="]">\frac{-1400(353.15-T)}{353.15\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}T}$ | $3.5\times {10}^{-4}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{exp}\left(\right)open="["\; close="]">\frac{-1400(353.15-T)}{353.15\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}T}$ | $0.5$ | $0.5$ |

[93] | $1.0\times {10}^{4}$ | $\left(\right)open="["\; close="]">{10}^{\left(\right)}\times {10}^{4}$ | 1 | $0.495+2.3\times {10}^{-3}(T-300)$ |

Ref. | ${j}_{0,a}^{ref}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\zeta}_{a}$ [$\mathrm{A}/{\mathrm{m}}^{3}$] | ${j}_{0,c}^{ref}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\zeta}_{c}$ [$\mathrm{A}/{\mathrm{m}}^{3}$] | ${\alpha}_{a}$ | ${\alpha}_{c}$ |

[28] | $1.0\times {10}^{-9}$ | $3.0\times {10}^{5}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{exp}\left[0.014189(T-353)\right]$ | $1.0$ | $1.0$ |

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**MDPI and ACS Style**

d’Adamo, A.; Haslinger, M.; Corda, G.; Höflinger, J.; Fontanesi, S.; Lauer, T.
Modelling Methods and Validation Techniques for CFD Simulations of PEM Fuel Cells. *Processes* **2021**, *9*, 688.
https://doi.org/10.3390/pr9040688

**AMA Style**

d’Adamo A, Haslinger M, Corda G, Höflinger J, Fontanesi S, Lauer T.
Modelling Methods and Validation Techniques for CFD Simulations of PEM Fuel Cells. *Processes*. 2021; 9(4):688.
https://doi.org/10.3390/pr9040688

**Chicago/Turabian Style**

d’Adamo, Alessandro, Maximilian Haslinger, Giuseppe Corda, Johannes Höflinger, Stefano Fontanesi, and Thomas Lauer.
2021. "Modelling Methods and Validation Techniques for CFD Simulations of PEM Fuel Cells" *Processes* 9, no. 4: 688.
https://doi.org/10.3390/pr9040688