The main purpose of the Plan Composite Centered (PCC) is to mathematically model the studied responses in the form of a 2nd order polynomial equation and to optimize them. This method is also called the Box-Wilson type design which uses the response surface methodology and is used for continuous variables. The Design-Expert software 18 is used.

In our study, the answer is the flow of hydrogen produced. The number of factors chosen is five: the concentration of the electrolyte in the electrolyser, the distance between the electrodes, the height of the electrode immersed in the solution, the total supply voltage, the temperature of the solution and the type of electrolytes. Therefore, the total number of experiments is Ntotal=50 experiences ((N1=32, N2=10 et N3=8). We repeated this test matrix twice, once with potassium hydroxide (KOH) electrolyte and once with sodium hydroxide (NaOH) electrolyte (50+50=100 experiences).

The variation interval of each factor is deduced on the one hand from the bibliographical study and on the other hand from the experimental constraints. The five factors and their ranges of variation are listed in Table 1:

After we fixed the factors and the necessary experimental plan, we carried out the experimental tests to fill the two matrices (with KOH and with NaOH). Figure 1 gives a schematic representation of the experimental device. The response studied is the rate of hydrogen production. This flow rate is calculated indirectly by calculating the time (t) necessary for the production of a fixed volume of hydrogen for all the tests. Figure 2 and Figure 3 respectively represent the electrodes used and the distances between them. The hydrogen recovery tube is placed above the cathode.

To carry out the necessary experiments, we first prepared a tank with the five distances between the electrodes (- α, -1, 0, +1 et + α) where we placed crocodile clips. Then we prepared the solutions with the requested concentrations. We heated the solution to the required temperatures. We carried out the experiments following an order where there is the minimum of the chemicals used.

The experimental results found of the volume flow of hydrogen produced are grouped in Table 2.

The mathematical model is a polynomial model according to the coded factors and applicable to the defined experimental domain. To choose the suggested model, the R² value must be on the one hand close to 1 and on the other hand it is necessary that R² and R² adjusted do not differ considerably; if not then there is a strong chance that there are insignificant terms in the model. Table 3a and Table 3b respectively give the analysis results for the choice of the suitable model for the tests with KOH and for the tests with NaOH. We note that the quadratic model is the suggested model for both types of electrolytes.

Table 4a and Table 4b present the ANOVA analysis of variance to judge the performance of the model obtained successively for KOH and for NaOH. Based on the ANOVA analysis, we can conclude that:

The two models (for KOH and for NaOH) are validated by the Fischer test.

The significant parameters if the electrolyte is KOH are A, B, C, D, E, 𝐵2 and 𝐷2

The significant parameters if the electrolyte is NaOH are A, B, D, E, DE, 𝐷2and 𝐸2

Effects and Interactions

The effects and interactions on the response are grouped in Table 5a and Table 5b.

The histograms of figure 4a and figure 4b give a comparison between the different effects and interactions respectively with the electrolyte KOH and NaOH.

From the ANOVA analysis, we can note that:

Concentration has a positive and significant effect regardless of the electrolyte used. This shows that if the concentration of the electrolyte increases the flow rate increases. This is explained by the fact that if the concentration increases then the charge transfer increases. These results are in agreement with the results of Fatima ezzahra Chakik et al 19, who studied the effect of electrolyte concentration on hydrogen production. They used a solution of NaOH with different concentrations, and they found that as long as the concentration matters so much the production.

The distance between the electrodes has a negative and significant effect for both types of electrolytes. That is, if the distance between the electrodes increases then the hydrogen production rate decreases. Indeed, increasing the distance between the electrodes increases the path traveled by the charges, so it limits the production speed. N. Nagai et al 20 have shown that if the space between the electrodes increases then the electrical resistance increases and the efficiency of electrolysis decreases.

The supply voltage has a positive effect and is the most important parameter regardless of the electrolyte used (which is noticeable in Figures 4-a and 4-b). Dayana D'arc of Fatima Palhares et al 21 found that if the tension increases the production also increases. The same result is found by Kenji Kikuchi et al 22

The operating temperature has a positive and significant effect whatever the electrolyte used. Yangyang Li et al 23, Boissonneau et al 24 and Damien le Bideau 25 studied the effect of temperature on the efficiency of electrolysis, and they found that if the temperature increases then the production increase. Yangyang Li et al 26experimentally studied the effect of temperature and pressure under different current densities, they found that if the operating temperature increases the voltage required, for the same amount of hydrogen production, decreases. We can explain this by the fact that increasing the operating temperature increases the activity of the catalyst.

The height of the electrodes has a positive and significant effect if the electrolyte is KOH. If the electrolyte is NaOH, the effect of the height of the electrodes is also positive but not significant.

From our results, we can notice very clearly that electrolysis with KOH is more efficient than electrolysis with NaOH. As long as the diameter of the electrolyte is large as long as the transfer is faster. These results are in good agreement with the results of M. Hassen sellami et al 27. Who experimentally studied the effect of the nature of the electrolyte on the volume of hydrogen produced. We can increase the efficiency of electrolysis by using KOH (since it is more efficient) not in pure water but in another solution (for example water from the air conditioning system or wastewater) 28. This shows that the number of valence electrons in the electrolyte has an important effect, and the application of a magnetic field can promote the transfer 29. The concern that: in a stoichiometric approach, it takes 9 kg of water to produce 1 kg of hydrogen; however, after taking into consideration the inefficiencies of the process the water consumption amounts to 18-24 kg per kg of hydrogen ^30-31; will no longer be the case, especially if wastewater is used in the electrolyser 32.

According to the ANOVA analysis, we notice that no interaction of order 2 is significant with the KOH electrolyte. The only significant interaction found is DE (voltage-temperature) with the electrolyte NaOH. Figure 5 gives a representation of the response surfaces of these interactions with the NaOH electrolyte. For five factors we have ten interactions of order 2 (AB, AC, AD, AE, BC, BD, BE, CD, CE, DE).

The quadratic complete model used is given by the following equation:

𝑞 = 𝑥₀ + 𝑥₁ 𝐴 + 𝑥₂ 𝐵 + 𝑥₃ 𝐶 + 𝑥₄ 𝐷 + 𝑥₅ 𝐸 + 𝑥₁₂ 𝐴𝐵 + 𝑥₁₃ 𝐴𝐶 + 𝑥₁₄ 𝐴𝐷 + 𝑥₁₅ 𝐴𝐸

+ 𝑥₂₃ 𝐵𝐶 + 𝑥₂₄ 𝐵𝐷 + 𝑥₂₅ 𝐵𝐸 + 𝑥₃₄ 𝐶𝐷 + 𝑥₃₅ 𝐶𝐸 + 𝑥₄₅ 𝐷𝐸 + 𝑥₁₁ 𝐴²

+ 𝑥₂₂ 𝐵2 + 𝑥₃₃ 𝐶2 + 𝑥₄₄ 𝐷2 + 𝑥⁵⁵ 𝐸²

with:

𝑥₀, 𝑥₁, 𝑥₂, 𝑥₃, 𝑥₄, 𝑥₅ : average effects of variables respectively A, B, C, D, E.

𝑥₁₂, 𝑥₁₃, 𝑥₁₄, 𝑥₁₅, 𝑥₂₃, 𝑥₂₄, 𝑥₂₅, 𝑥₃₄, 𝑥₃₅, 𝑥₄₅ : effect of variable interactions.

𝑥₁₁, 𝑥₂₂, 𝑥₃₃, 𝑥₄₄, 𝑥₅₅ : quadratic effects of variables.

We can improve the complete model by eliminating insignificant factors from the complete model and we obtain a so-called reduced model.

if the electrolyte is KOH the reduced model is:

𝑞 = 0.2249 + 0.0207 𝐴 − 0.0194 𝐵 + 0.0152 𝐶 + 0.0908 𝐷 + 0.0150 𝐸 + 0.0278 𝐵2

+ 0.0143 𝐷²

if the electrolyte is NaOH the reduced model is:

𝑞 = 0.1556 + 0.0238 𝐴 − 0.0162 𝐵 + 0.0757 𝐷 + 0.0279 𝐸 + 0.0094 𝐷𝐸 + 0.0167 𝐷2

+ 0.0107 𝐸²

To study the adequacy of the found model, we draw the curve which represents the values given by the model according to the experimental values. Figure 5-a and Figure 5-b respectively show the adequacy of the model if the electrolyte is KOH and if the electrolyte is NaOH.

Optimization seeks a combination of factor levels that simultaneously meet the established criteria for a best desired response. Table 6 represents the objectives, the importance of each objective and the results.

In this study three optimizations are carried out:

Optimization 1: The objective in this optimization is to maximize hydrogen production independently of other parameters. That is, to maximize the hydrogen production rate and the other parameters are random.

Optimization 2: In this optimization we will keep the same objective as the first optimization, and we add the condition to minimize the supply voltage of the electrolyser.

Optimization 3: n this optimization we will keep the same objective as the first optimization, and we add the condition to minimize the supply voltage of the electrolyzer and the condition to also minimize the concentration of the electrolyte.

Three optimizations of ascending constraints on the operating parameters to have a maximum hydrogen production are carried out. We notice that if we increase the constraints on the optimized parameters the value of the response (Hydrogen flow rate) decreases.