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The Impact of Reaction on the Effective Properties of Multiscale Catalytic Porous Media: A Case of Polymer Electrolyte Fuel Cells

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Abstract

Reaction–diffusion processes in multiscale catalytic porous media are found in a wide range of scientific areas as, for example, electrochemical energy conversion and storage devices, geological systems and bioengineering. The dependency of effective transport properties on reaction rate has been long debated in the literature, and traditionally ignored in emerging fields, such as polymer electrolyte fuel cells (PEFCs). In this work, a 1D upscaling method is presented to evaluate the effective properties (effective diffusivity and catalyst utilization) of PEFC catalyst layers featuring first-order kinetics. Unlike Whitaker’s closure method, the present algorithm is easy to implement and well suited for porous media with arbitrarily complex 3D geometries. The numerical results show that the normalized effective diffusivity and catalyst utilization are not passive geometrical properties but are influenced by the reaction–diffusion coupling when the Thiele modulus is higher than 1. This effect can be important at high current densities in the cathode catalyst layer of state-of-the-art PEFCs.

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Abbreviations

\( A_{\gamma \kappa } \) :

Solid–fluid interface

\( {\mathbf{b}} \) :

Closure variable vector solves Eq. (8) (m)

\( {\mathbf{b}}_{rx} \) :

Closure variable vector solves Eq. (9) (m)

\( {\mathbf{b}}_{\gamma } \) :

Closure variable vector solves Eq. (10) (m)

\( c \) :

Mole concentration (mol/m3)

\( C \) :

Macroscopic superficial concentration (mol/m3)

\( c^{*} \) :

Dimensionless concentration

\( \left\langle c \right\rangle^{\gamma } \) :

Volume-averaged concentration in \( \gamma \) phase (mol/m3)

\( c_{{{\text{O}}_{2} }}^{\text{cl}} \) :

Oxygen concentration in catalyst layer (mol/m3)

\( c_{{{\text{O}}_{2} }}^{\text{in}} \) :

Oxygen concentration in the inlet of the channel (mol/m3)

\( c_{\text{g}} \) :

Overall gas concentration in the inlet of the channel (mol/m3)

\( c_{{{\text{H}}_{2} {\text{O}}}} \) :

Vapor concentration in the inlet of the channel (mol/m3)

\( D \) :

Bulk diffusivity (m2/s)

\( D_{\text{solid}} \) :

Diffusivity in solid phase (m2/s)

\( D_{\text{eff}} \) :

Effective diffusivity (m2/s)

\( {\mathbf{D}}_{\text{diff}} \) :

Passive effective diffusion coefficient which is independent of reaction (m2/s)

\( {\mathbf{D}}_{\text{eff}} \) :

Effective diffusivity tensor (m2/s)

\( {\mathbf{D}}_{rx} \) :

Diffusion-like coefficient which depends on reaction rate (m2/s)

\( F \) :

Faraday’s constant (C/mol)

\( {\mathbf{I}} \) :

Identity tensor

\( j \) :

Current density (A/m2)

\( j_{\text{critical}} \) :

Critical current density (A/m2)

\( k \) :

Reaction rate coefficient (s−1)

\( k_{\text{eff}} \) :

Effective reaction rate coefficient (s−1)

\( {\mathbf{l}}_{i} \) :

Unit cell lattice vectors, \( i \) = 1, 2, 3 (m)

\( L \) :

Thickness between inlet and outlet boundaries (m)

\( L_{\text{eff}} \) :

Length of the tortuous pathway (m)

\( {\mathbf{n}}_{\gamma \kappa } \) :

Unit normal vector directed from the fluid to the solid phase

\( M \) :

Molecular weight (kg/mol)

\( N \) :

Superficial diffusive flux (mol/m2s)

\( N_{{{\text{O}}_{2} }} \) :

Diffusive oxygen flux (mol/m2s)

\( p_{\text{g}} \) :

Gas pressure (Pa)

\( p_{{{\text{H}}_{2} {\text{O}}}}^{\text{sat}} \) :

Vapor pressure of water (Pa)

\( {\mathbf{r}} \) :

Position vector directed from the origin to a point in the averaging domain (m)

\( R \) :

Reaction rate (mol/m3s)

\( R_{{{\text{O}}_{2} }} \) :

Overall oxygen mass transport resistance between inlet of the channel and the catalyst layer (s/m)

\( R_{u} \) :

Gas constant (J/mol K)

\( T \) :

Temperature (K)

\( U_{\text{eff}} \) :

Utilization factor

\( V_{\text{total}} \) :

Total volume of the porous material (m3)

\( V_{\text{fluid}} \) :

Volume of fluid space (m3)

\( V_{\gamma } \) :

Volume of \( \gamma \) phase (m3)

\( {\mathbf{x}} \) :

Position vector directed from the origin to the averaging domain center (m)

\( z \) :

Coordinate in z-direction (m)

\( z^{*} \) :

Dimensionless z-coordinate

\( \rho \) :

Gas mixture density (kg/m3)

\( \omega \) :

Mass fraction

ε :

Porosity

\( \tau \) :

Tortuosity factor

\( \phi \) :

Thiele modulus

\( \phi_{\text{eff}} \) :

Effective Thiele modulus

\( \varOmega_{\gamma } \) :

Domain occupied by \( \gamma \) phase

\( \partial \varOmega_{\gamma \kappa } \) :

Domain occupied by solid–fluid interface

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Acknowledgements

The work was supported by NSF CAREER award #1652445. This research used resources of the Advanced Photon Source, a US Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC02-06CH11357. Pablo A. García-Salaberri thanks support from Project ENE2015-68703-C2-1-R (MINECO/FEDER, UE), the research grant ‘Ayudas a la Investigación en Energía y Medio Ambiente’ of the Spanish Iberdrola Foundation and the US-Spain Fulbright Commission.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Tufts University, Medford, MA, 02155, USA

    Jiangjin Liu & Iryna V. Zenyuk

  2. Departamento de Ingeniería Térmica y de Fluidos, Universidad Carlos III de Madrid, 28911, Leganés, Spain

    Pablo A. García-Salaberri

  3. Department of Chemical and Biomolecular Engineering, National Fuel Cell Research Center, University of California, Irvine, CA, 92697, USA

    Iryna V. Zenyuk

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  1. Jiangjin Liu
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Correspondence to Pablo A. García-Salaberri or Iryna V. Zenyuk.

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Liu, J., García-Salaberri, P.A. & Zenyuk, I.V. The Impact of Reaction on the Effective Properties of Multiscale Catalytic Porous Media: A Case of Polymer Electrolyte Fuel Cells. Transp Porous Med 128, 363–384 (2019). https://doi.org/10.1007/s11242-019-01252-8

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