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On the wellposedness for a fuel cell problem

Luisa Consiglieri

TL;DR

This work establishes the existence of weak solutions for a multiregion PEM fuel cell model that couples a Beavers–Joseph–Saffman fluid–porous interface with a thermoelectrochemical system, including Joule heating and cross-effects. The authors decouple the problem via two auxiliary problems (auxiliary velocity–pressure and auxiliary partial density–temperature–potential triplet) and assemble a fixed-point framework to obtain a solution under small-data conditions, with rigorous a priori estimates guiding the analysis. The main contribution is a quantitative wellposedness result for a physically rich Stokes–Darcy–TEC model, providing a mathematically rigorous foundation for analyzing PEM fuel cell behavior under coupled transport, electrochemical reactions, and thermal effects. This framework supports robust analysis and potential numerical schemes by clarifying dependencies on data and material parameters, and highlights how Joule heating and cross-couplings influence solution control in a realistic, multi-domain setting.

Abstract

This paper investigates the existence of weak solutions to two problems set of elliptic equations in adjoining domains, with Beavers--Joseph--Saffman and regularized Butler--Volmer boundary conditions being prescribed on the common interfaces, porous-fluid and membrane, respectively. Mathematically, the modeling tool is the coupled Stokes/Darcy problem, which consists of the Stokes equation on one part of the domain coupled to the Darcy equation, where the flow velocities are small and mainly driven by the pressure gradient in porous medium, completed by the thermoelectrochemical (TEC) system, which consists of the energy equation and the mass transport associated with electrochemical reactions, where the fluxes are given by generalized Fourier, Fick and Ohm laws, by including the Dufour--Soret and Peltier--Seebeck cross effects, in the multidimensional domain. The present model includes macrohomogeneous models for both hydrogen and methanol crossover. The novelty in the presented model lies in the presence of the Joule effect into the Stokes/Darcy-TEC system altogether to the quasilinear character given by temperature dependence of the physical parameters such as the viscosities and the diffusion coefficients, by the concentration-temperature dependence of cross-effects coefficients, and by the pressure dependence of the permeability. The purpose of the present work is to derive quantitative estimates for solutions to explicit smallness conditions on the data. We use fixed point and compactness arguments based on the quantitative estimates of approximated solutions.

On the wellposedness for a fuel cell problem

TL;DR

This work establishes the existence of weak solutions for a multiregion PEM fuel cell model that couples a Beavers–Joseph–Saffman fluid–porous interface with a thermoelectrochemical system, including Joule heating and cross-effects. The authors decouple the problem via two auxiliary problems (auxiliary velocity–pressure and auxiliary partial density–temperature–potential triplet) and assemble a fixed-point framework to obtain a solution under small-data conditions, with rigorous a priori estimates guiding the analysis. The main contribution is a quantitative wellposedness result for a physically rich Stokes–Darcy–TEC model, providing a mathematically rigorous foundation for analyzing PEM fuel cell behavior under coupled transport, electrochemical reactions, and thermal effects. This framework supports robust analysis and potential numerical schemes by clarifying dependencies on data and material parameters, and highlights how Joule heating and cross-couplings influence solution control in a realistic, multi-domain setting.

Abstract

This paper investigates the existence of weak solutions to two problems set of elliptic equations in adjoining domains, with Beavers--Joseph--Saffman and regularized Butler--Volmer boundary conditions being prescribed on the common interfaces, porous-fluid and membrane, respectively. Mathematically, the modeling tool is the coupled Stokes/Darcy problem, which consists of the Stokes equation on one part of the domain coupled to the Darcy equation, where the flow velocities are small and mainly driven by the pressure gradient in porous medium, completed by the thermoelectrochemical (TEC) system, which consists of the energy equation and the mass transport associated with electrochemical reactions, where the fluxes are given by generalized Fourier, Fick and Ohm laws, by including the Dufour--Soret and Peltier--Seebeck cross effects, in the multidimensional domain. The present model includes macrohomogeneous models for both hydrogen and methanol crossover. The novelty in the presented model lies in the presence of the Joule effect into the Stokes/Darcy-TEC system altogether to the quasilinear character given by temperature dependence of the physical parameters such as the viscosities and the diffusion coefficients, by the concentration-temperature dependence of cross-effects coefficients, and by the pressure dependence of the permeability. The purpose of the present work is to derive quantitative estimates for solutions to explicit smallness conditions on the data. We use fixed point and compactness arguments based on the quantitative estimates of approximated solutions.
Paper Structure (15 sections, 7 theorems, 79 equations, 1 figure, 1 table)

This paper contains 15 sections, 7 theorems, 79 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Statement of the fuel cell problem
  3. In the fluid bidomain ${\Omega}_\mathrm{f}= \Omega_\mathrm{fuel}\cup \Omega_\mathrm{air}$
  4. Number of species $\mathrm{I}$
  5. The anodic fuel compartment $\Omega_\mathrm{fuel}\cup\Omega_\mathrm{a}$
  6. The cathodic air compartment $\Omega_\mathrm{air}\cup\Omega_\mathrm{c}$
  7. In the porous domain $\Omega_\mathrm{p}=\Omega_\mathrm{a}\cup\Omega_\mathrm{m}\cup\Omega_\mathrm{c}$
  8. On the outer boundary $\partial\Omega$
  9. On the fluid-porous interface $\Gamma$
  10. On the membrane interface $\Gamma_\mathrm{CL}=\Gamma_\mathrm{a}\cup\Gamma_\mathrm{c}$
  11. Variational formulation and main result
  12. Auxiliary results
  13. Auxiliary velocity-pressure pair
  14. Auxiliary partial density-temperature-potential triplet solution
  15. Fixed point argument(Proof of Theorem \ref{['tmain']})

Key Result

Theorem 3.1

Let $\Omega$ be a bounded multiregion domain of $\mathbb{R}^n$, $n=2,3$. Under the assumptions (H1)-(H6), the fuel cell problem admits, at least, one solution according to Definition dwt such that if provided by one of the smallness conditions small1 or small2.

Figures (1)

  • Figure 1: The flow region $\Omega_\mathrm{f}= \Omega_\mathrm{fuel}\cup \Omega_\mathrm{air}$ and the porous region $\Omega_\mathrm{p} = \Omega_\mathrm{a}\cup\overline{\Omega}_\mathrm{m}\cup\Omega_\mathrm{c}$ (not in scale), with length $l_\mathrm{a}+l_\mathrm{m}+l_\mathrm{c}<< L$ where $L=1-10\,cm$ denotes each channel length. Left: $xy$ cross-section. Right: $xz$ cross-section.

Theorems & Definitions (14)

  • Theorem 3.1
  • proof
  • Proposition 4.1: Auxiliary velocity-pressure pair
  • proof
  • Proposition 4.2: Continuous dependence
  • proof
  • Proposition 4.3: Auxiliary partial density-temperature-potential triplet
  • proof
  • Proposition 4.4: Continuous dependence
  • proof
  • ...and 4 more