Analytical Considerations for Hydraulic Resistance Models of Electrochemical Cells
Authors: Beale Steven B. , Zhang Shidong , Kulikovsky Andrei 
Published: December 2024

The distributed resistance analogy (DRA) was originally developed to build three-dimensional (3-D) computational fluid dynamics (CFD) models of heat exchangers [1], and subsequentially substantially modified/extended for the physicochemical hydrodynamics occurring in electrochemical devices such as fuel cell and electrolyzer stacks [2]. DRA based macrohomogeneous models eliminate the requirement to construct very large computational meshes by replacing viscous/diffusion terms in the governing Navier-Stokes equations with simple rate terms for pressure drop, heat, and mass transfer. This results in the required computational effort being reduced by two (2) orders of magnitude. Typically, macrohomogeneous hydraulic resistance models are validated by comparison with experimental results, and/or with those obtained with detailed numerical simulations (DNS) for any given design.
This paper examines the theoretical foundation of the methodology by simultaneously considering both macroscopic and microscopic force and mass balances within an idealized polymer electrolyte membrane fuel/electrolyzer cell in the form of a plane duct with an electric current being applied along one or more boundaries. Both constant and variable current density are considered.
For constant current density, the analytical solutions of Berman [3] and Jorne [4], for plane ducts, are derived in integral form from the well-known differential forms, and then compared to CFD results based both on the DRA and DNS formulations, which the authors employ within the OpenFOAM code suite.
For variable current density, it is presumed that the oxygen/current density, j(x), and associated wall velocity, follow a power law distribution as a function of λ [5, 6], where λ is the (oxygen or water) stoichiometric number. Using Berman’s approach, Kulikovsky [7] solved the problem under conditions of variable current density. Comparisons are made for the present numerical calculations based on both DRA and DNS CFD methods with those of Kulikovsky [7]. Both fuel cell and electrolyzer modes of operation are considered.
The results of the microscopic and macroscopic balances are presented and discussed in detail, in terms of pressure drop and shear stresses. The presence of mass injection/suction leads to both boundary (shear) and inertial (added mass) effects in the momentum/pressure equation, which have not be considered before. However, it is shown that these effects are second-order, and can often be neglected locally, at least for the current densities encountered today in polymer electrolyte membrane fuel and electrolysis cells. It is thus shown, by comparison to analytical and semi-analytical results, that the DRA approximation is based on sound reasoning, and leads to reliable solutions for flow-field simulations in the channels of electrochemical cells and stacks.
References

Patankar, S.V. and D.B. Spalding, A Calculation Procedure for the Transient and Steady-state Behavior of Shell-and-tube Heat Exchangers, in Heat Exchangers: Design and Theory Sourcebook, N. Afgan and E.U. Schlünder, Editors. 1974, Scripta Book Company: Washington. D.C. p. 155-176.

Beale, S.B. and S.V. Zhubrin, A distributed resistance analogy for solid oxide fuel cells. Numerical Heat Transfer, Part B, 2005. 47(6): p. 573-591.

Berman, A.S., Laminar Flow in Channels with Porous Walls. Journal of Applied Physics, 1953. 24(9): p. 1232-1235.

Jorne, J., Mass-Transfer in Laminar-Flow Channel with Porous Wall. Journal of the Electrochemical Society, 1982. 129(8): p. 1727-1733.

Kulikovsky, A.A., The effect of stoichiometric ratio λ on the performance of a polymer electrolyte fuel cell. Electrochimica Acta, 2004. 49(4): p. 617-625.

Kulikovsky, A.A., A. Kucernak, and A.A. Kornyshev, Feeding PEM fuel cells. Electrochimica Acta, 2005. 50(6): p. 1323-1333.

Kulikovsky, A., Laminar Flow in a PEM Fuel Cell Cathode Channel. Journal of The Electrochemical Society, 2023. 170(2): p. 024510.