*2.3. Membrane Water Content*

For estimating the water content λ inside of a Nafion-membrane, a function of water activity *a* is used [15]:

$$
\lambda = \begin{cases}
0.043 + 17.81 \, a - 39.85 \, a^2 + 36 \, a^3 \\
\text{for } a \le 1 \\
14 + 1.4(a - 1) \\
\text{for } 1 < a \le 3
\end{cases} \tag{7}
$$

As a simplification, ab- and desorption of water were neglected. Furthermore, the distribution of water along the membrane geometry was assumed to be uniform.

The water activity *a* can be expressed as [16]

$$a = RH + \text{2 }s,\tag{8}$$

where *RH* denotes the relative humidity (for ideal gas properties) and *s* the liquid water volume fraction. For this model, it is assumed that liquid water is only present in the catalyst layer. For *RH* and *s*, a logarithmic average is used to account for nonhomogeneous distribution inside the channels (see Equation (17)). This can be disabled by suppling the model with equal values for input and output.

In case of a cold start at subzero temperatures, the behavior of frozen water inside of a Nafion-membrane is approximated using the following terms [16]:

$$\begin{aligned} \lambda\_{\text{stat}} & \begin{cases} = 4.837 \\ \text{for } T < 223.15 \text{ K} \\ = [-1.304 + 0.01479 \text{ T } - 3.594 \cdot 10^{-5} \text{ T}^2]^{-1} \\ \text{for } 223.15 \text{ K} \le T < T\_{\text{first}} \\ > \lambda \\ \text{for } T \ge T\_{\text{first}} \end{cases} \end{aligned} \tag{9}$$

To estimate the water concentration *Cw*, a simple proportional correlation between the membrane density ρmem, equivalent weight and water content λ is used [11]:

$$\mathcal{C}\_{\mathcal{W}} = \frac{\rho\_{\text{mcm}}}{EW} \lambda. \tag{10}$$
