## 2 Experimental
### Statistical Design of Experiment  
A full experiment was carried out in order to study the relationship between the control variables, significant variables, and the interactions between the test parameters. A 2\({}^{5}\)-factorial design was applied in the present investigation. This required a selection of appropriate responses, factors, and levels. Phenol conversion and hydrogen yield were the responses selected. The factors and the experimental levels for each factor were selected based on literature values, available resources, and results from preliminary experiments. The selection of the factors was based on the operating conditions that had a significant influence on the SRP, according to previous experiments (with different feeds) [12; 13; 14; 16; 17; 18]. The considered factors chosen were temperature, \(A\), Feed flow rate, \(B\), Catalyst amount, \(C\), the catalyst ratio of Ni to Co over ZrO\({}_{2}\), \(D\), and phenol concentration in the feed, E. The factors and their levels were assigned in Table 1. A Minitab statistical software (\({}^{\circledapprox}16.2.4\), Minitab, Inc., Pennsylvania State University, University Park, PA, USA, 2013) was used to generate the design matrix and to conduct the data analysis. Also, the interaction among the factors was determined using the same software. The design matrix of un-coded values for the factors and the response for all the experimental runs are given in Table 2. Five factors were analyzed at two levels, using a full factorial 2\({}^{5}\) resulting in 32 runs, in a random order to minimize the effect of uncontrolled variables on responses (\(Y_{1}\) and \(Y_{2}\)). Since one of the aims of this study was to establish a relationship between the five control variables and the responses of the system (phenol conversion and hydrogen yield) a linear regression model  
Figure 1: Schematic flow diagram of experimental setup. was fitted to predict the responses as a function of independent variables and their interactions. In general, the response for the linear regression model is described in Equation (5):  
\[ \begin{array}{ll}Y_{i}=&\beta_{0}+\beta_{1}A+\beta_{2}B+\beta_{3}C+\beta_{4}D+ \beta_{5}E+\beta_{6}AB+\beta_{7}AC+\beta_{8}AD+\beta_{9}AE+\beta_{10}BC+\\ &\beta_{11}BD+\beta_{12}BE+\beta_{13}CD+\beta_{14}CE+\beta_{15}DE+\beta_{16}ABC +\beta_{17}ABD+\beta_{18}ABE+\\ &\beta_{19}ACD+\beta_{20}ACE+\beta_{21}ADE+\beta_{22}BCD+\beta_{23}BCE+\beta_{24 }BDE+\beta_{25}CDE+\\ &\beta_{26}ABCD+\beta_{26}ABCE+\beta_{28}ABDE+\beta_{29}ACDE+\beta_{30}BCDE+ \beta_{31}ABCDE+\varepsilon,\end{array}  \tag{5}\]  
where \(Y_{i}\) is the predicted response, \(\beta_{0}\) is the offset term, \(\beta_{n}\) is the regression coefficient that corresponds to the main effects and interaction effects, and \(\varepsilon\) is the error term. The level of significance of each coefficient was considered by comparing the smaller \(p\)-value (\(<\)0.05) or greater \(t\)-value. The \(t\)-test was used to determine the significance of the regression coefficients of the parameters. The \(p\)-values were used as tools to check the significance of each variable and the interaction among the variables. As the first step, a significance test was performed on all the possible terms in models for phenol conversion (\(Y_{1}\)) and hydrogen yield (\(Y_{2}\)). The results are illustrated in Pareto charts and Normal plots. Then a reduced model was obtained by removing the insignificant terms from the first model with a new reduced regression equation. The \(R^{2}\) value was used to determine the fitting quality of the observed results to the proposed model. The regression analysis, linear regression plots (observed versus predicted value), and Pareto chart of the dependent variables, main effect and interaction plots, surface and contour plots, and response optimizer plots were plotted.