*2.4. Ohmic Overvoltage*

Regarding ohmic resistances inside the cell, only the membrane resistance as the most influential factor is considered. Therefore, the overpotential can be calculated with the current density *i*, membrane conductivity σmem and (wet) thickness δmem. Since membrane thickness—at typical PEMFC operating conditions—is only marginally affected by swelling, the thickness is assumed to be constant [17,18]:

$$E\_{\text{ohm}} = \frac{\delta\_{\text{mem}} i}{\sigma\_{\text{mem}}}.\tag{11}$$

To estimate the membrane conductivity σmem, the following correlation is used [9,12,19]:

$$
\sigma\_{\text{mem}} = \; 1.16 \,\text{max} \{ 0, f - 0.06 \}^{1.5} \exp \left[ \frac{15000}{\overline{R}} \left( \frac{1}{T\_{\text{ref}}} - \frac{1}{T} \right) \right] \,\tag{12}
$$

$$
\begin{split}
T\_{\text{ref}} &= 353.15 \,\text{K}; \quad f = \frac{\lambda V\_W}{\lambda V\_W + V\_{\text{ref}}} \\
V\_m &= \frac{EW}{\rho\_{\text{mem}}}; \quad V\_W = \frac{18.01528}{\rho\_W(T)}.
\end{split}
$$

It is mostly dependent on membrane water content λ and temperature *T*mem, whilst also being affected by the material properties of equivalent weight *EW* and membrane density ρmem as well as the density of water ρ*W*(*T*). A simple arithmetic average of anode and cathode site values is used for further calculations.

The density of water ρ*w*(*T*) at 1[bar] as a function of temperature *T* can be approximated by [20]:

$$
\rho\_w(T) = \ 999.972 - 7 \cdot 10^{-3} \ (T - 4)^2 \cdot 10^{-3} \,\mathrm{s} \tag{13}
$$

$$
T \,\,\mathrm{in}\,^\circ \mathrm{C}.
$$

## *2.5. Concentration Overvoltage*

Using the Nernst-equation, concentration overvoltage *E*coni is described as a function of temperature *T*, current density *i* and limiting current density *iL* [11]

$$E\_{\rm con\_i} = \frac{RT\_i}{nF} \ln \left(\frac{\dot{\imath}\_{L\_i}}{\dot{\imath}\_{L\_i} - \dot{\imath}}\right) \tag{14}$$

where *R* is the universal gas constant, *n* the number of electrons involved and *F* the Faraday constant. Since this ideal equation often underestimates the real overvoltage, e.g., because of uneven gas concentration, a correction factor (see Supplementary Materials) is used to adjust the results [11,21].

For the limiting current density *iL*, a simplified expression including the Faraday *F* constant, number of electrons *n*, diffusion coefficient *Di*, gas concentration *Ci* and diffusion distance (here: electrode thickness) δ*<sup>e</sup>* is used [11]:

$$i\_{L\_i} = \frac{nFD\_iC\_i}{\delta\_{\varepsilon}}.\tag{15}$$

To calculate the diffusion coefficient *Di*, the model expects an external reference value *Di*,ref for the gas mixture. It therefore relies on the external calculation of gas concentration and diffusivity. The reference value will then be adjusted for electrode porosity and tortuosity τ as well as temperature *T*, pressure *p* and liquid water volume fraction *s* [9,22,23]:

$$D\_i = \frac{\epsilon}{\tau^2} (1 - s)^3 D\_{i, \text{ref}} \left( \frac{T}{T\_{\text{ref}}} \right)^{1.5} \frac{p\_{\text{ref}}}{p} \,\tag{16}$$

$$T\_{\text{ref}} = 353.15 \text{ K}; \ p\_{\text{ref}} = 1.01325 \text{ bar}.$$

Porosity and tortuosity τ are used to approximately describe the geometry of the electrodes, while a value of 1 for the liquid water volume fraction *s* represents a fully flooded channel. As a simplification, it is assumed that the gas diffusivity in liquid water is 0. The approach of relying on *Di*,ref as a model input for the calculation of *Di* ensures compatibility with arbitrary gas mixtures and composition of external models for fluid mechanics—for instance, when used in a vehicle model and being connected to components for the media supply.

A simple logarithmic average is used to account for nonhomogeneous distribution of the gas concentration *C*lm. If desired, this can be disabled by supplying the same value for input and output [5]:

$$C\_{\rm lm} = \frac{C\_{\rm in} - C\_{\rm out}}{\ln \frac{C\_{\rm in}}{C\_{\rm out}}}.\tag{17}$$
