2.7.3. Hydraulic Permeation

Beyond that, hydraulic permeation is only considered if liquid water is present on both sides of the membrane, since the pressure difference has a minor impact on water vapor transport [27]. The hydraulic flow *j*hyd is estimated by a linear correlation with the pressure gradient ∇*p*—here: difference between cathode and anode—affected by the cell area *A*, dynamic viscosity of water μH2O as well as water concentration inside the membrane *Cw*, its thickness δmem and hydraulic permeability *K*<sup>λ</sup> [16]:

$$j\_{\rm hyd} = \frac{A\mathcal{C}\_w K\_\lambda}{\mu\_{\rm H\_2O} \delta\_{\rm mem}} \nabla p \cdot 10^5. \tag{28}$$

For the hydraulic permeability *K*λ, a direct dependency on the membrane water content λ is assumed [16]:

$$K\_{\lambda} = K\_{w}\lambda.\tag{29}$$

Furthermore, the dynamic viscosity of water is approximated by a function of temperature *T* [28]

$$\mu\_{\rm av} = \mu\_0 \exp\left| a\_{\mu} p + \frac{d\_{\mu} - b\_{\mu} p}{R \left( T - \partial\_{\mu} - c\_{\mu} p \right)} \right|, \tag{30}$$

where μ<sup>0</sup> denotes a reference value while *a*μ, *b*μ, *c*<sup>μ</sup> and *d*<sup>μ</sup> are constants. The pressure *p* has been neglected, since it has a minor impact on μ*<sup>w</sup>* at typical PEMFC operating pressures.

Experimental data suggest a nonlinear relation between the hydraulic flow and membrane thickness [24], hence an adjustment-function is applied for the hydraulic permeability *K*<sup>λ</sup> at a reference pressure difference of 0.025 [bar]:

$$\begin{aligned} \, \_{K\_{\text{LLP}}} \begin{cases} = 0.1158 \, K\_{\text{\lambda}} (5.749 \cdot 10^{-3} \, \delta\_{\text{mem}} \exp[-1.326]) \\ \quad \text{for } \delta\_{\text{mem}} \ge 0.0056 \text{ cm} \\ = 0.1158 \, K\_{\text{\lambda}} (2.518 \cdot 10^{-4} \, \delta\_{\text{mem}} \exp[-1.872]) \\ \quad \text{for } \delta\_{\text{mem}} < 0.0056 \text{ cm} \end{cases} \end{aligned} \tag{31}$$

Subsequently, the hydraulic flow is corrected for the actual pressure difference:

$$j\_{\rm hyd\_{LIP}} = j\_{\rm hyd\_{wt}}(32.41 \,\Delta p + 0.06016). \tag{32}$$
