2.7.2. Diffusion

In terms of diffusion *j*diff, the abovementioned functions to differentiate between the liquid and gaseous phase are utilized. The basis for these calculations is given by [17]

$$j\_{\rm diff} = \frac{AD\_{\lambda} \nabla \mathbb{C}\_{w}}{\delta\_{\rm mem}} \,, \tag{24}$$

which considers the cell area *A*, (average) diffusive coefficient of water *D*λ, water concentration gradient ∇*Cw*—here: difference between cathode and anode—and membrane thickness δmem. The latter is treated as a constant (=201 μm) in the above equation and variations are instead considered by the adjustment-functions (Equations (26)–(32)).

To estimate the diffusion coefficient for water through the membrane *D*λ, the following expression dependent on membrane temperature *T*mem and (average) water content λ is applied [9,26]:

$$D\_{\lambda} = \frac{3.842 \ \lambda^3 - 32.03 \ \lambda^2 - 67.74 \ \lambda}{\lambda^3 - 2.115 \ \lambda^2 - 33.013 \ \lambda + 103.37} \cdot 10^{-6} \exp\left[20 \cdot \frac{20000}{R} \left(\frac{1}{T\_{\text{ref}}} - \frac{1}{T\_{\text{mean}}}\right)\right] \tag{25}$$
 
$$T\_{\text{ref}} = 353.15 \text{K}. \tag{25}$$

Subsequently, the diffusive flow *j*diff is adjusted for the thickness of the membrane. VVP-correction is applied in the complete absence of liquid water, whilst LVP-correction is used if at least one side is flooded with liquid water. For mixtures of gaseous and liquid phases, a linear scaling is applied:

$$j\_{\rm diff\_{VVP}} = 0.9178 \cdot j\_{\rm diff}(\delta\_{\rm mem}|\_{0.0201}) \Big( -947.5 \,\delta\_{\rm mem}^2 - 6.198 \,\delta\_{\rm mem} + 1.508 \,\), \tag{26}$$

$$j\_{\rm diff\_{NP}} = 3.592 \cdot j\_{\rm diff} (\delta\_{\rm mem} |\_{0.0201}) \left( -687 \,\delta\_{\rm mem}^2 - 21.73 \,\delta\_{\rm mem} + 1.714 \right). \tag{27}$$
