Membrane Crossover Theory
In a membrane-based fuel cell or electrolyzer, the reacting gases may dissolve into the membrane and diffuse through the membrane to the other side of the cell. This cross-membrane transport of species is termed crossover.
For hydrogen and oxygen, these species will be quickly oxidized/reduced when reaching the other side of the cell. These reactions give rise to small parasitic current densities which will have an effect on, for instance, the open circuit voltage of the cell. For nitrogen, typically diffusing from the oxygen to the hydrogen side in air-operated fuel cells, the crossover may result in a dilution effect of the hydrogen fuel.
Hydrogen Crossover Reactions
On the hydrogen side of the membrane, hydrogen dissolves into the membrane according to
The dissolved hydrogen then diffuses through the membrane, and in the gas diffusion electrode on oxygen side, the hydrogen gets oxidized according to (depending on charge-carrying ion):
Proton exchange:
Hydroxide exchange:
Due to the highly anodic overpotential potential for the hydrogen oxidation reaction on the oxygen side, the reaction operates at a limiting current density, completely governed by the molar flux of hydrogen though the membrane.
Oxygen Crossover Reactions
Similarly as for hydrogen, on the oxygen side of the membrane, oxygen dissolves into the membrane according to
The dissolved oxygen then diffuses through the membrane and in the gas diffusion electrode on hydrogen side, oxygen gets reduced according to (depending on the charge-carrying ion):
Proton exchange:
Hydroxide exchange:
Due to the highly cathodic overpotential potential for the oxygen reduction reaction on the hydrogen side, the reaction operates at a limiting current density, completely governed by the molar flux of oxygen though the membrane.
Nitrogen Crossover Reactions
On either side of the membrane, nitrogen may dissolve into and out of the membrane according to
Membrane Crossover Transport Model
The membrane crossover model defines either one or several thermodynamical activities of hydrogen and oxygen in the membrane, aH2, aO2, and aN2, respectively, as dependent variables. The governing equation for the membrane species is based on Fick’s law of diffusion in combination with Henry’s law
where Ji,mem (mol/(m2·s)) is the molar flux of the species and Ψi (mol/(m·s Pa)) is the corresponding permeation coefficient. pref is a reference pressure, arbitrarily set to 1 atm.
The species concentration in the membrane ci,mem (mol/m3) used in the time derivative term is calculated based on Henry’s law according to
where Hi (1) is the Henry’s law coefficient of the species.
Note that, for isothermal conditions, the transport equation is equivalent to Fick’s law of diffusion according to
with the diffusion coefficient Di (m2/s) defined as
Hydrogen boundary conditions:
On Membrane boundaries adjacent to hydrogen Gas Phase domains, the hydrogen activity aH2 is set as follows
where pH2(g) is the hydrogen partial pressure in the adjacent gas phase. Hydrogen is assumed to be instantly oxidized when it reaches an electrode on the oxygen gas side. On Membrane boundaries adjacent to oxygen Gas Diffusion Electrode domains (or oxygen Thin Gas Diffusion Electrode boundaries), the hydrogen activity is set to
Oxygen boundary conditions:
Similarly, on Membrane boundaries adjacent to oxygen Gas Phase domains, the oxygen activity aO2 is set as follows
where pO2(g) is the oxygen partial pressure in the adjacent gas phase. Oxygen is assumed to be instantly reduced when it reaches an electrode on the hydrogen gas side. On Membrane boundaries adjacent to hydrogen Gas Diffusion Electrode domains (or hydrogen Thin Gas Diffusion Electrode boundaries), the oxygen activity is set to
Nitrogen boundary conditions:
On Membrane boundaries adjacent to any Gas Phase domain, the nitrogen activity aN2 is set to
Contributions to Current Distribution Model
For the adjacent oxygen gas diffusion electrode boundaries where aH2,mem is set to 0, the corresponding oxidation charge transfer reaction will give rise to a local crossover current density iloc according to
Similarly, the local crossover current density iloc on adjacent hydrogen gas diffusion electrode boundaries becomes
The local crossover current density is added as normal current density contributions for the electronic and electrolyte potentials, respectively:
Parasitic reactions not involving the H2/O2 species are not added by default, but may be added as ordinary electrode reactions in gas diffusion electrodes.
Heat Of Reactions
The parasitic reactions of the H2/O2 species also give rise to boundary heat sources. The thermoneutral potential Etherm is defined as
where n is the number of electrons included in the electrode reaction, and the corresponding heat source Qb is defined as
Contributions to Gas Diffusion and Momentum Transfer Model
The species crossover flux will have an impact on the gas phase flux. For hydrogen, on the Membrane boundary adjacent to the hydrogen Gas Phase domain, the following flux contribution is added. Similarly, for oxygen, on the Membrane boundary adjacent to the oxygen Gas Phase domain, the following flux contribution is added For nitrogen, on Membrane boundaries adjacent to any Gas Phase domain, the following flux contribution is added
The above contributions are also added to the momentum transfer equation and Stefan velocity as
For any gas species participating in the parasitic hydrogen oxidation or oxygen reduction reaction, the corresponding mass flux contribution, on Membrane boundaries adjacent to H2/O2 Gas Diffusion Electrode domains or H2/O2 Thin Gas Diffusion Electrode boundaries, is
The above contribution is also added to the momentum transfer equation and Stefan velocity as