# Evolutionary Design Optimization of an Alkaline Water Electrolysis Cell for Hydrogen Production
## 2 Physics and Economy of the Alkaline Water Electrolysis Cell  
### Theory  
An alkaline water electrolysis cell produces hydrogen and oxygen using electric energy via an electrochemical process. The Figure 1 presents a classical water electrolysis cell. It consists of a positive pole called the anode (where the oxidation of hydroxyl ions occurs Equation (2)), a negative pole called the cathode (where the reduction of water occurs Equation (3)) and a membrane (brown part) that prevents the mixing of hydrogen and oxygen. \[2 OH^{-}(_{aq})\to H_{2}O_{(liq)}\ +\ \frac{1}{2}\ O_{2(g)}\ +\ 2e^{-} \tag{2}\] \[2H_{2}O_{(liq)}\ +\ 2e^{-}\to 2 OH^{-}\ _{(aq)}\ +\ H_{2(g)} \tag{3}\]  
The reactions of Equations (2) and (3) occur at alkaline pH (\(>\)7). Thus, KOH or NaOH salts must be added. The classical operating conditions of alkaline electrolyzer are: Temperature around 80 \({}^{\circ}\)C, atmospheric pressure and 0.30 KOH. In the next paragraph, we will explain why those choices have been made. As explained in the introduction section, the minimum voltage that must be applied to a water electrolysis cell is called reversible voltage. [PERSON] et al. [10] developed a model (Appendix A) that simulates the temperature, pressure, and mass fraction sensitivity of the reversible voltage. The reversible voltage decreases with temperature while mass fraction and the pressure increase it. The ohmic electrolyte resistance \(R\) is depending on the electrolyte conductivity \(\sigma\), the diphasic boundary thickness \(\delta\) and the void fraction \(\varepsilon\), see Equation (4) [11]. \[R(\varepsilon,\delta,T,Y)=\frac{\delta}{\sigma(\varepsilon,T,Y)}\ +\ \frac{h-\delta}{ \sigma(T,Y)} \tag{4}\]  
Figure 1: Design of the electrochemical cell. With \(h\) the thickness of the electrolyte (catholyte or anolyte) in m, \(\sigma\) the electrical conductivity in S m\({}^{-1}\). [PERSON] et al. [12] showed the electrical conductivity dependency on void fraction can be simulated using the [PERSON]'s correlation Equation (5). \[\frac{\sigma(\varepsilon,T,Y)}{\sigma(T,Y)}=(1\ -\ \varepsilon)^{\frac{3}{2}} \tag{5}\]  
The Equation (6) can be simplified by assuming that the gas is dispersed throughout the electrolyte thickness. \[R(\varepsilon,T,Y)=\frac{h}{\sigma(\varepsilon,T,Y)} \tag{6}\]  
Thus, ohmic resistance increases with void fraction. The quantity of gas injected in the electrolyte is calculated via [PERSON]'s law Equation (7). According to the Equation (8), the pressure decreases the volumetric hydrogen flow and thus the void fraction. Moreover, [PERSON] et al. [13] and [PERSON] et al. [14] showed that the pressure decreases the detachment bubble radius, thus decreasing the surface coverage. [PERSON] et al. [15] and [PERSON] et al. [16] measured the electrical conductivity depending on the KOH mass fraction and temperature. They showed that there exists an optimum KOH mass fraction between 0.2 and 0.4 that depends on the temperature. The different correlations for electrical conductivity, viscosity, density of KOH and NaOH have been summarized by [PERSON] et al. [17]. The temperature increases the efficiency of the process but a high temperature and an extremely alkaline (pH \(>\) 14) condition led to material deterioration and electrolyte boiling. Thus, the temperature should not exceed 100 \({}^{\circ}\)C.  
\[Q_{vH_{2}}=\frac{j\ S\ v_{mol}}{2\ F} \tag{7}\]  
\[v_{mol}=\frac{R_{G}\ T}{p} \tag{8}\]  
\(Q_{vH_{2}}\) the volumetric hydrogen flow m\({}^{3}\) s\({}^{-1}\), \(S\) the electrode surface in m\({}^{2}\), \(v_{mol}\) the molar volume m\({}^{3}\) mol\({}^{-1}\), \(F\) = 96,500 C mol\({}^{-1}\) the Faraday's constant in C mol\({}^{-1}\), \(R_{G}\) = 8.314 J kg K\({}^{-1}\) the ideal gas constant in J mol\({}^{-1}\) K\({}^{-1}\), \(T\) the temperature in K, \(p\) the pressure in Pa. Therefore, this analysis explained the choice of mass fraction and temperature but not pressure choice. Actually, high pressure increases the manufacturing cost. However, according to [PERSON] et al.