Fuel Cell Cathode with Liquid Water

In low-temperature fuel cells, the produced water can condensate as liquid water if the partial pressure of vapor exceeds the (equilibrium) vapor pressure. The produced liquid water can flood the gas diffusion electrodes, gas diffusion layers, and/or the flow channels and manifolds, resulting in decreased fuel cell performance.

This tutorial expands the model defined in the Mass Transport and Electrochemical Reaction in a Fuel Cell Cathode tutorial to include also the effects liquid water formation in the porous gas diffusion electrode.

Model Definition

Liquid water is produced in the extended model using a user-defined expression for vapor condensation, depending on the relative humidity level in the gas phase:

(1)

where k is a rate constant, pH2O is the partial pressure of water vapor, and pvap the vapor pressure.

The porous gas diffusion electrode is assumed to be hydrophobic. The capillary pressure, pc, is defined as

(2)

where pl and pg are the phase pressures, with the subscripts l and g referring to the liquid and gas phases, respectively.

The capillary pressure depends on the liquid saturation sl in the porous media as depicted in Figure 1. (Note: The capillary pressure curve originally stems from Ref. 1 for measurements on a gas diffusion layer, that is, not a gas diffusion electrode, and is used here solely for tutorial purposes.)

Figure 1: Capillary pressure versus liquid saturation.

The momentum transfer in the liquid phase is modeled using Darcy’s law according to

(3)

with the volume averaged velocity ui in the liquid fluid phase of index l defined as

(4)

where μl is the dynamic viscosity of the fluid and κ the absolute permeability of the porous media.

κr,l in the above equation is the relative permeability, which is defined to depend on the fluid saturation level according to

(5)

(6)

The Phase Transport in Porous Media interface is used to set up the flow formulation of the liquid water, based on Darcy’s law, solving for sl, to the model.

As boundary conditions for the liquid phase transport model, a liquid water saturation according to a capillary pressure of 0 Pa is set at the inlet.

The model is solved in a stationary study consisting of four study steps. In order to facilitate convergence, the potential variables, the pressure and gas phase species, and the liquid saturation are solved for in individual study steps first for a cell voltage of 1 V. The fully coupled problem is then solved in the last study step, using an auxiliary sweep to ramp the voltage from 1 to 0.5 V.

Results and Discussion

Figure 2 compares the relative humidity level in the cell at 0.5 V when and when not including water condensation and liquid water transport in the model. When not including water condensation, the air mixture gets significantly oversaturated to about 160% relative humidity. When including condensation, the relative humidity stays close to 100% throughout the cathode.

Figure 2: Relative humidity assuming no water condensation (left) and when including condensation and water transport (right).

Figure 3 shows the liquid saturation level and the corresponding capillary pressure level when including liquid water transport. Liquid water now forms in the electrode and is transported out toward the inlet hole mainly by the capillary pressure gradient. The water saturation level in the pores does not exceed 10%.

Figure 3: Liquid water saturation level and liquid water flux (left) and corresponding capillary pressure distribution (right).

Finally, the polarization curves for the two cases are compared in Figure 4. Introducing liquid water transport in the model slightly increases the current levels for a given voltage especially for lower voltages. This may seem counterintuitive, but the reason for this is that the condensation of water vapor results in an increased partial pressure of oxygen, which has a positive effect on the cathode potential. The small volume (<10%) of liquid water has only a minor detrimental effect on the overall gas transport rate.