Topic D — A Compressed Felt Electrode in a Redox Flow Battery

large-strain compression of a porous electrode → porosity and permeability → Darcy flow · tier: advanced

Motivation

In a redox flow battery the carbon-felt electrode is squeezed into its frame to lower the contact resistance. Compression also lowers the porosity and the permeability, so the pump has to work harder, and it does so unevenly: the felt is compressed more under the ribs of a flow field than in the channels (intrusion). The trade-off — contact vs pressure drop — is a coupled solid–fluid problem: a large-strain porous solid determines the permeability field of the flow through it.

Related fields: redox flow batteries (electrodes, separators, stacks), water electrolysis, fuel-cell and electrolyser membranes. Course weeks: 1 (hyperelasticity, load steps), 2 (plane strain), 13 (Darcy/Brinkman flow in porous media).

Model

2D cross-section of one rib–channel period of an interdigitated or flow-through cell (plane strain), felt of uncompressed thickness t_0.

Solid: compressible neo-Hookean (or a Blatz–Ko-type foam law) for the felt skeleton; the rib (rigid) pushes the felt down by \delta; frictionless or bonded contact replaced by prescribed displacement under the rib and traction-free under the channel (first model).

Porosity and permeability follow the deformation (solid phase incompressible): \varepsilon=1-\frac{1-\varepsilon_0}{J},\qquad \kappa(\varepsilon)=\kappa_0\Big(\frac{\varepsilon}{\varepsilon_0}\Big)^3\Big(\frac{1-\varepsilon_0}{1-\varepsilon}\Big)^2\ \ \text{(Kozeny–Carman)}, in the deformed configuration (or pulled back to the reference one — say which and transform consistently).

Flow: Darcy in the felt, \mathbf u=-\frac{\kappa}{\mu}\nabla p, \nabla\cdot\mathbf u=0; one-way coupled (the flow does not deform the felt) in the required part, two-way (drag on the skeleton) as a stretch goal.

Design outputs: the pressure drop and the flow distribution (rib vs channel) against the compression ratio \delta/t_0; the mean porosity.

Suggested parameters

t_0=4 mm, \varepsilon_0=0.93, compression 10–50 %, \kappa_0=5\times10^{-11} m², shear modulus of the felt 0.1–1 MPa, \nu = 0.1–0.2 (highly compressible), electrolyte viscosity 4 mPa s, superficial velocity 1–10 mm/s.

Required tasks

  1. Compression of the felt under a flat plate (homogeneous) and under a rib (inhomogeneous): load–displacement curve, J field.
  2. Porosity and permeability fields; check \varepsilon\in(0,1) everywhere.
  3. Darcy flow on the deformed (or pulled-back) domain; pressure drop and flow split between rib and channel.
  4. Study in \delta/t_0; mesh independence of the pressure drop.

Verification specific to this topic

  • Homogeneous confined compression: with lateral confinement and a flat plate, J=1-\delta/t_0 exactly and the stress follows from the law in closed form — the FE solution must be homogeneous and match.
  • Uniform Darcy: in a uniformly compressed felt the pressure drop is \Delta p=\mu uL/\kappa(\varepsilon) (1D, exact).
  • Mass conservation: inflow = outflow through the felt to discretisation error (compute both with consistent fluxes, or use a mixed RT×DG Darcy formulation where it is exact cell by cell).
  • Pull-back check: solving Darcy on the deformed mesh and the pulled-back problem on the reference mesh must give the same pressure drop.
  • MMS for the hyperelastic solid and for Darcy with variable \kappa.

Stretch goals

  1. Contact with friction-free rib edges (penalty or Nitsche) instead of prescribed displacement.
  2. Two-way coupling: the drag -\frac{\mu}{\kappa}\mathbf u acts on the skeleton (Biot/Terzaghi), fixed-point iteration.
  3. Electrochemistry: a first-order electrode reaction in the felt with a mass-transfer coefficient depending on the local velocity.

Milestones

M1 homogeneous compression check · M2 rib compression · M3 permeability field · M4 Darcy with the 1D check and conservation · M5 compression study.

Pitfalls

  • Compression beyond the point where J\to1-\varepsilon_0 makes \varepsilon\to0: the law must stiffen before that, otherwise elements invert. Load steps and a check on \min J (Week 1: reason −4 from \ln J of an inverted element).
  • Mixing reference and deformed configurations in Darcy (the Piola transformation of the flux: \mathbf U=J\mathbf F^{-1}\mathbf u, \mathbb K=J\mathbf F^{-1}\kappa\mathbf F^{-T}).
  • Corner singularity at the rib edge: refine there and report the pressure drop, not the local stress.

References

  • D. S. Aaron et al., “Dramatic performance gains in vanadium redox flow batteries through modified cell architecture”, J. Power Sources 206 (2012) 450–453 — the zero-gap cell with compressed felts.
  • O. Coussy, Poromechanics, Wiley 2004 — Lagrangian porosity and permeability.
  • G. A. Holzapfel, Nonlinear Solid Mechanics, Wiley 2000.