# Practical Methodology for a Three-Dimensional-Printed Hybrid Desalination System
## 2 Materials and Methods
### Fundamental Equations of the RED Process  
The performance of the RED reactor is characterized through theoretical equations that estimate the potential difference across the membranes, the power output of the cell, and the contribution of resistance from various reactor components, among other aspects. [PERSON] and Micale [8] indicate that the membrane potential difference (\(E_{IEM}\)) can be represented by a modified form of the [PERSON] equation, as detailed in Equation (3):  
\[E_{IEM}=\alpha_{IEM}\frac{RT}{zF}ln\left(\frac{\gamma_{c}\times C_{c}}{\gamma_ {d}\times C_{d}}\right)\ (V), \tag{3}\]  
where \(\alpha_{IEM}\) represents the membrane selectivity (%), \(R\) denotes the ideal gas constant (8.314 J/mol\(\cdot\)K), \(T\) is the absolute temperature in Kelvin, \(z\) is the ion valence in solution, \(F\) stands for the Faraday constant, and \(\gamma_{c}\), \(\gamma_{d}\), \(C_{c}\), and \(C_{d}\) refer to the activity coefficients and concentrations of the concentrated and diluted solutions (mol/L), respectively.  
The electrical potential of a RED cell (\(E_{cell}\)) can be defined by the sum of the potentials from each membrane (\(E_{AEM}\) and \(E_{CEM}\)) and the number of repeated unit cells (\(N\)) present in the system, as shown in Equation (4). The resistance of a RED unit cell (\(R_{cell}\)) is calculated based on the resistances of the membranes (\(R_{AEM}\) and \(R_{CEM}\)) along with the resistance generated in the water compartments of the stack (\(R_{c}\) and \(R_{d}\)), as illustrated in Equation (5). The electric current (\(I\)) is determined using Equation (6), where \(R_{ext}\) represents the external resistance applied [8].  
\[E_{cell}=N\times(E_{AEM}+E_{CEM})\ (V), \tag{4}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Specifications** & **AEM** & **CEM** \\ \hline Thickness dry (\(\mu\)m) & 125 & 135 \\ Electrical Resistance 1 & 1.7 & 2.0 \\ Perm selectivity 2 & 95 & 99 \\ IE Capacity 3 & 1.8 & 1.5 \\ Water permeation 4 & 6.5 & 6.5 \\ Burst strength 5 & 2.8 & 2.8 \\ pH stability & pH 1–13 & pH 1–13 \\ Temp stability 6 & 60 & 60 \\ \hline \hline \end{tabular} 1 Electrical resistance measured \(\Omega\) cm\({}^{2}\) at 2 M NaCl; 2 NaCl permselectivity measured at 0.05–0.5 M KCl; 3 ion exchange capacity (meq/g); 4 water permeation (ml/bar-m\({}^{2}\cdot\)h); 5 burst strength (kg/cm\({}^{2}\)); 6 temperature (\({}^{\circ}\)C).  
\end{table}
Table 1: Main features of Fujifilm Type 10 membranes.  
\[R_{cell}=R_{AEM}+R_{CEM}+R_{c}+R_{d}\ \ (\Omega), \tag{5}\]  
\[I=\frac{E_{cell}}{R_{cell}+R_{ext}}\ \ (\mathrm{A}), \tag{6}\]  
The maximum power (\(P_{max}\)) generated by the RED reactor was calculated using Equation (7). To evaluate the power and current relative to the active membrane area (\(A_{mem}\)), the power density (\(P_{d}\)) and current density (\(J\)) were computed using Equations (8) and (9):  
\[P_{max}=\frac{E_{cell}}{4R_{cell}}\ \ (\mathrm{W}), \tag{7}\]  
\[P_{d}=\frac{P}{A_{mem}}\ \ \left(\frac{\mathrm{W}}{\mathrm{m}^{2}}\right), \tag{8}\]  
\[J=\frac{I}{A_{mem}}\ \ \ \left(\frac{\mathrm{A}}{\mathrm{m}^{2}}\right). \tag{9}\]