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On the smallness conditions for a PEMFC single cell problem

Luisa Consiglieri

TL;DR

This work develops a rigorous existence theory for a steady, multi-region PEMFC model that couples a Stokes–Darcy–Fourier framework with thermoelectrochemical effects in a two-dimensional setting. The authors split the nonlinear system into two auxiliary problems and apply a fixed-point argument to obtain a weak solution under explicit smallness conditions on the data, together with quantitative Poincaré–Sobolev estimates. The main contribution lies in providing explicit, computable smallness thresholds (via a root of a quadratic) that ensure the fixed-point mapping is well-defined and closed, making the result physically interpretable. The findings advance the mathematical understanding of PEMFC models with strong cross-couplings and discontinuous coefficients, and offer a pathway toward guaranteed existence under realistic operating parameters.

Abstract

The aim of the present paper is to prove whose smallness conditions being necessary in order to get the final result of existence of a solution. In the first part, we present the model for a proton exchange membrane fuel cell (PEMFC) single cell and we clarify the interactions of the different components namely, velocity, pressure, density, temperature and potential. The final mathematical model is a quasilinear elliptic system where the cross effects have a strong interlink. It consists of the Stokes--Darcy system altogether with thermoelectrochemical system under some non-standard interface and boundary conditions. The proof of existence of weak solutions relies on the Tychonof fixed point theorem, by providing some regularity and some smallness conditions. The actual system is divided into two systems of equations and they are separately studied. The novelty of the present work is to establish quantitative estimates for improving the technical hypotheses and, in particular, the smallness conditions in the two-dimensional case. Indeed, the smallness conditions only can be explicit if quantitative estimates are established. To this aim, we establish quantitative estimates for the Poincaré and Sobolev inequalities and for some trilinear terms.

On the smallness conditions for a PEMFC single cell problem

TL;DR

This work develops a rigorous existence theory for a steady, multi-region PEMFC model that couples a Stokes–Darcy–Fourier framework with thermoelectrochemical effects in a two-dimensional setting. The authors split the nonlinear system into two auxiliary problems and apply a fixed-point argument to obtain a weak solution under explicit smallness conditions on the data, together with quantitative Poincaré–Sobolev estimates. The main contribution lies in providing explicit, computable smallness thresholds (via a root of a quadratic) that ensure the fixed-point mapping is well-defined and closed, making the result physically interpretable. The findings advance the mathematical understanding of PEMFC models with strong cross-couplings and discontinuous coefficients, and offer a pathway toward guaranteed existence under realistic operating parameters.

Abstract

The aim of the present paper is to prove whose smallness conditions being necessary in order to get the final result of existence of a solution. In the first part, we present the model for a proton exchange membrane fuel cell (PEMFC) single cell and we clarify the interactions of the different components namely, velocity, pressure, density, temperature and potential. The final mathematical model is a quasilinear elliptic system where the cross effects have a strong interlink. It consists of the Stokes--Darcy system altogether with thermoelectrochemical system under some non-standard interface and boundary conditions. The proof of existence of weak solutions relies on the Tychonof fixed point theorem, by providing some regularity and some smallness conditions. The actual system is divided into two systems of equations and they are separately studied. The novelty of the present work is to establish quantitative estimates for improving the technical hypotheses and, in particular, the smallness conditions in the two-dimensional case. Indeed, the smallness conditions only can be explicit if quantitative estimates are established. To this aim, we establish quantitative estimates for the Poincaré and Sobolev inequalities and for some trilinear terms.
Paper Structure (12 sections, 8 theorems, 91 equations, 1 figure, 1 table)

This paper contains 12 sections, 8 theorems, 91 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. In the porous domain $\Omega_\mathrm{p}=\Omega_\mathrm{a}\cup\overline{\Omega}_\mathrm{m}\cup\Omega_\mathrm{c}$
  5. On the outer boundary $\partial\Omega$
  6. On the fluid-porous interface $\Gamma$
  7. On the membrane interface $\Gamma_\mathrm{CL}=\Gamma_\mathrm{a}\cup\Gamma_\mathrm{c}$
  8. Variational formulation and main result
  9. Strategy
  10. Auxiliary results
  11. Existence of auxiliary solutions
  12. Fixed point argument(Proof of Theorem \ref{['tmain']})

Key Result

Theorem 3.1

Let $\Omega$ be a bounded multiregion domain of $\mathbb{R}^n$, $n=2$. Under the assumptions (H1)-(H9), the fuel cell problem admits, at least, one solution according to Definition dwt such that if provided by the smallness condition where $root_2$ is the positive root of the quadratic polynomial 2polynomial, $C_K >1$ is the Korn constant in korn and $C_0, \mathcal{B}_0, a_\#>0$ are constants de

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 width $l_\mathrm{a}+l_\mathrm{m}+l_\mathrm{c}<< L$ where $L=1-10\,cm$ denotes each channel length.

Theorems & Definitions (16)

  • Theorem 3.1
  • Proposition 5.1
  • proof
  • Proposition 5.2: n=2
  • proof
  • proof
  • proof
  • Proposition 6.1: Auxiliary velocity-pressure pair
  • proof
  • Proposition 6.2: Continuous dependence
  • ...and 6 more