## 5 Transport Analysis in Support of the Design
### Gas Phase Operation  
The permeation of hydrogen through a membrane is called gas-driven permeation because the permeation is induced by the pressure gradient of the hydrogen gas between upstream (high-pressure side) and downstream (low-pressure side). This permeation is influenced by at least four factors: diffusivity, solubility, surface adsorption and desorption, and hydrogen-defect interaction. For the purposes of this study, the presence of structural defects or trapping sites is neglected, so it is possible to identify two limiting regimes: the Diffusion-Limited Regime (DLR) and the Surface-Limited Regime (SLR). The term _limited_ refers to the slowest process that dominates the process. The key parameter for identifying the regime condition is the permeation parameter \(W\)[15]. It can be shown that in the case  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Parameter** & **Symbol** & **Value** & **Units** \\ \hline LiPb Sieverts’ constant & \(k_{S,l}\) & \(2.35\cdot 10^{-2}\) & mol m\({}^{-3}\)Pa\({}^{-0.5}\) \\ LiPb diffusivity & \(D_{l}\) & \(1.78\cdot 10^{-7}\) & m\({}^{2}\) s\({}^{-1}\) \\ LiPb mass transfer coefficient & \(k_{m,l}\) & \(5.48\cdot 10^{-6}\) & m s\({}^{-1}\) \\ OIVb recombination constant & \(k_{r,s}\) & \(9.32\cdot 10^{-10}\) & m\({}^{4}\)mol\({}^{-1}\)s\({}^{-1}\) \\ OIVb dissociation constant & \(k_{d,s}\) & \(8.82\cdot 10^{-11}\) & mol m\({}^{-2}\) s\({}^{-1}\) Pa\({}^{-1}\) \\ OIVb permeability & \(\Phi_{s}\) & \(6.74\cdot 10^{-12}\) & mol m\({}^{-1}\) s\({}^{-1}\) Pa\({}^{-0.5}\) \\ Pipe thickness & \(t_{s}\) & 2.8 & mm \\ Pipe length & \(l_{s}\) & 0.9 & m \\ \hline \hline \end{tabular}
\end{table}
Table 2: Input data for hydrogen transport analysis in support to WCLL mock-up.  
of permeation of a single isotope, for example hydrogen, the permeation parameter can be written as follows:  
\[W=\frac{k_{r,s}k_{S,s}}{D_{s}}\cdot t_{s}\cdot\sqrt{p_{H_{2}}} \tag{2}\]  
where \(p_{H_{2}}\) is the hydrogen partial pressure at the high-pressure side of the pipe. The permeation parameter is a property of the material through which permeation occurs. According to [10], the intermediate regime can be adopted for \(1/10<W<10\). In general, a mass balance at both pipe interfaces can be written in terms of the following set of algebraic equations:  
\[v=\sqrt{1-u^{2}} \tag{3}\] \[W^{2}u^{4}+2 Wu^{3}+2u^{2}=1\]  
Here, \(u=c_{h}/k_{S,s}\sqrt{p_{H_{2}}}\) and \(v=c_{l}/k_{S,s}\sqrt{p_{H_{2}}}\) are the dimensionless concentrations, whereas \(c_{h}\) and \(c_{l}\) (mol m\({}^{-3}\)) are the hydrogen concentrations; these terms are evaluated at the high-pressure side and at the low-pressure side of the pipe. It should be observed that analytical solutions for the two limiting cases exist. However, considering a pressure range \(p_{H_{2}}=10\)-\(1000\) Pa, the permeation parameter evaluated with Equation (2) gives \(W=0.11\)-\(1.14\), hence a typical mixed regime characterises the permeation phenomenon. In this case, only numerical solutions for Equation (3) can be obtained. The permeation flux for the gas phase case, \(J_{gp}\) (mol m\({}^{-2}\) s\({}^{-1}\)), can be written as:  
\[J_{gp}=k_{d,s}\cdot p_{H_{2}}\cdot\left(1-v^{2}\right) \tag{4}\]  
The calculations were carried out considering different values of the \(PRF\) in the range 1-10,000. The case \(PRF=1\) represents the case of bare pipe. The results are reported in Figure 4 in terms of the molar flux \(\dot{n}\) (mol s\({}^{-1}\)), i.e., the product of permeation flux and external pipe permeation area.