The optimum conditions for maximising the percent TOC removal were determined by means of a four-factor three-level Box–Behnken experimental design combined with the response surface methodology to correlate experimentally obtained criteria and experimental conditions given by Box–Behnken experimental design. The Box–Behnken design, a modified central composite design, is known as an independent and rotatable quadratic design having no fractional factorial points. In this type of design, the variable combinations are at the centre and the midpoints of the edges of the variable space. Also, compared to other types of experimental designs such as full factorial design, the Box–Behnken experimental design needs fewer experimental trials. Therefore, the effects of four independent variables on the response functions were investigated. The independent variables were initial concentrations of PAA (X1) and H2O2:Fe3+(X2), pH (X3), and recirculation rate (X4) that were coded as −1, 0 and +1 as shown in Table 1. The total number of experimental trials was 27 based on three levels and a four factor experimental design, with three replicates at the centre of the design, to estimate a pure error sum of squares. The percent TOC removal and the ‘pseudo’-second order rate constant were considered as the dependent factors (process responses). The independent variables and their critical experimental levels as shown in Table 1 were chosen based on the preliminary experimental results and the values reported in the open literature.[34-36] In the RSM, as the initial step, an appropriate approximation which is generally a first-order model is applied to find the true functional relationship between the response function and the set of factors. However, in the first-order model, there is a lack of fit due to the existence of the surface curvature. Therefore, the first-order model was upgraded by adding higher order terms.[37, 38] Consequently, in the next step, the experimental data from the Box–Behnken experimental design was fitted to the following quadratic equation:
where Y, βo, βi, βii and βij are the predicted response, the constant coefficient (intercept term), the linear coefficients, the quadratic coefficients and the interaction coefficients, respectively. The parameters Xi and Xj are independent variables, where k (in this case, k = 4) and e are the number of factors and the residual term allowing uncertainties between observed and predicted values, respectively. The statistical softwares, STATISTICA (trial version 10.0) and Design-Expert (trial version 8.0) were used for the regression analysis and the parameter estimation of the response functions, respectively. The statistical significance of the model equations was completely analysed using analysis of variance (ANOVA) at 95% confidence intervals. Three-dimensional response surface plots and two-dimensional contour plots were developed while holding a variable constant in the quadratic model. The experimental and predicted values were compared to validate the developed models. The optimal operating conditions to maximise the percent TOC removal were also determined using a numerical technique built in the software Design-Expert 8.0. The ‘pseudo’-second order rate constant was calculated at the determined optimal conditions. Also, another experimental trial was carried out to verify the obtained optimal conditions by the developed models for both response functions.