Resulta and Disscusion Analysis of variance (ANOVA) The results of the analysis of variance test is summarized in Table 4. The probability > F for the model is less than 0.05 which implies that the model is significant and the terms in the model have significant effects on the response. In this case A, B, C, D, AB, AC, AD, BC, BD, CD, A2, B2, C2, D2 are significant model terms at the 95 % confidence level (α =5 %). The model F-value of 1387.59 and P-value of < 0.0001 implies that the model is highly significant. Based on the ANOVA results, the values of R2, Adjusted R2 and Predicted R2 were 0.9992, 0.9985 and 0.9971, respectively. This result suggests that the regression model is well interpreted the relationship between the independent variables and the response. Furthermore, the adequate precision ratio of 149.08 in the study shows that this model could be applied to navigate the design space defined by the CCD. Table 4 ANOVA results for Response Surface Quadratic Model Source Sum of squares df Mean square F -Value P- value model 12060.48 1 861.46 1387.59 <0.0001 A-pH 914.15 1 914.15 1472.45 <0.0001 B-TCcon. 1730.94 1 1730.94 2788.08 <0.0001 C-PScon. 3999 1 3999 6441.32 <0.0001 D-Time 4676.04 1 4676.04 7531.85 <0.0001 AB 18.66 1 18.66 30.06 <0.0001 AC 30.97 1 30.97 49.88 <0.0001 AD 6.84 1 6.84 11.01 0.0047 BC 16.4 1 16.4 26.42 0.0001 BD 17.64 1 17.64 28.41 <0.0001 CD 16.61 1 16.61 26.75 0.0001 A2 571.64 1 571.64 920.76 <0.0001 B2 123.3 1 123.3 198.6 <0.0001 C2 29.96 1 29.96 48.26 <0.0001 D2 20.18 1 20.18 32.5 <0.0001 Residual 9.31 15 0.62 Lack of Fit 5.13 10 0.51 0.61 0.7607 Pure Error 4.18 5 0.84 Cor Total 12069.8 29 R2 = 0.9992; Adjusted R2 = 0.9985 and Predicted R2 = 0.9971 Evaluation of model adequacy There are many statistical techniques for the evaluation of model adequacy, but graphical residual analysis is the primary statistical method for assessment of model adequacy [59]. The normal probability plot indicates that the points on this plot are formed a nearly linear pattern (Fig. 2 (a)). Therefore, the normal distribution is a good model for this data set. Random scattering of the points of internally studentized residual (the residual divided by the estimated standard deviation of that residual) versus predicted values between −3 and +3 emphasizes highly accurate prediction of the experimental data through the derived quadratic model (Fig. 2 (b)). Fig 2 The plot of: (a) Normal Plot of Residuals; (b) residuals vs Predicted Response; (c) predicted vs Actual values; (d) Residuals vs Run Order The plot of predicted vs Actual values (Fig. 2 (c)) indicate a higher correlation and low differences between actual and predicted values. Hence, the predictions of the experimental data by developed quadratic models for the TC degradation is perfectly acceptable and this model fits the data better. Also, the random spread of the residuals across the range of the data between −3 and +3 implies that there are no evident drift in this process and the model was a goodness fit (Fig. 2 (d)). The Box-cox plot is used for determine the suitability of a power low transformation for the selected data (Fig. 3 (a)). In this study, the best lambda values of 0.92 was obtained with low and high confidence interval 0.73 and 1.11, respectively. Therefore, recommend the standard transformation by the software is 'None'. The plot of points Leverage vs Run order is shown in Fig. 3 (b). The factorial and axial points have the most influence with a leverage of approximately 0.59, while the center points have the least effect with a leverage of 0.16. Fig 3 The plot of: (a) Box-Cox Plot for Power Transforms; (b) The points Leverage vs Run order for the CCD Design Design matrix evaluation for response surface quadratic model Design matrix evaluation implies that there are no aliases for the quadratic model. In general, a minimum of degrees of freedom 3 and 4 has been recommended for lack-of-fit and pure error, respectively. Therefore, degrees of freedom obtained in this study ensured a valid lack of fit test (Table 5). Table 5 Degrees of freedom for evaluation Model 14 Residuals 15 Lack of Fit 10 Pure Error 5 The standard error (SE) used to measure the precision of the estimate of the coefficient. The smaller standard error implies the more accurate the estimate. The variables of A, B, C and D have a standard errors = 0.16. The interceptions of AB, AC, AD, BC, BD and CD have slightly high standard errors = 0.2, while A2, B2, C2 and D2 have standard errors = 0.15. An approximate 95 % confidence interval for the coefficient is given by the estimate plus and minus 2 times the standard error. For example, with 95 % confidence can be said that the value of the regression coefficient A is between 6.49 and 5.85 (6.17 ± 2 × 0.16). The quadratic model coefficients for the CCD are shown in Table 6. This results suggested that the variables coefficients and their interactions are estimated adequately without multicollinearity. The low Ri-squared for independent variables and their interactions imply that the model is a good fit. In general, power should be approximately 80 % for detecting an effect [60]. In this study, there are more than 99 % chance of detecting a main effect while it is twice the background sigma. Table 6 The Quadratic model coefficients for the CCD Term StdErr** VIF Ri-Squared Power at 5 % Power at 5 % Power at 5 % SN = 0.5 SN = 1 SN = 2 A 0.16 1 0 20.90 % 63.00 % 99.50 % B 0.16 1 0 20.90 % 63.00 % 99.50 % C 0.16 1 0 20.90 % 63.00 % 99.50 % D 0.16 1 0 20.90 % 63.00 % 99.50 % AB 0.2 1 0 15.50 % 46.50 % 96.20 % AC 0.2 1 0 15.50 % 46.50 % 96.20 % AD 0.2 1 0 15.50 % 46.50 % 96.20 % BC 0.2 1 0 15.50 % 46.50 % 96.20 % BD 0.2 1 0 15.50 % 46.50 % 96.20 % CD 0.2 1 0 15.50 % 46.50 % 96.20 % A2 0.15 1.05 0.0476 68.70 % 99.80 % 99.90 % B2 0.15 1.05 0.0476 68.70 % 99.980 % 99.90 % C2 0.15 1.05 0.0476 68.70 % 99.80 % 99.90 % D2 0.15 1.05 0.0476 68.70 % 99.80 % 99.90 % **Basis Std. Dev. = 1.0 Final equation and model graphs The values of regression coefficients were determined and the experimental results of CCD were fitted with second order polynomial equation. The quadratic model for TC degradation rate in terms of coded were determined using as following Eq. (9): Final equation in terms of coded factors 9 Y = + 49.82 + 6.17 * A − 8.49 * B + 12.91 * C + 13.96 * D + 1.08 * A * B − 1.39 * A * C + 0.65 * A * D + 1.01 * B * C + 1.05 * B * D − 1.02 * C * D + 4.57 * A 2 + 2.12 * B 2 + 1 .05 * C 2 + 0.86 * D 2 The factors in the quadratic equation were coded to produce the response surface with limiting the responses into a range of −1 to +1. The ramp function graph for the maximum TC degradation rate is shown in Fig. 4. The optimization of experimental conditions was conducted for maximize the TC degradation at defined criteria of the variable. The developed quadratic model for the TC degradation (Eq. (8)) was applied as an objective function to the optimization of operating conditions. Consequently, the optimum parameters were achieved using the numerical technology based on the predicted model and the variable in their critical range. The maximum degradation of 95.01 % was achieved at pH = 9.9, TC concentration = 30.19 mg/L, PS concentration = 3.97 mM and reaction time = 119.98 min. in order to evaluation of the model validity, the experiments were carried out under the optimal operating conditions. 93.45 % TC degradation was obtained under the optimum operating conditions, which supported the results of the developed model. Fig 4 Ramp function graph for the numerical optimization of TC degradation The perturbation Plot of independent variables implies that reaction time (D) has the most significant effect (steepest slope) on the TC degradation rate, followed by S2O8−2 concentration (C) and TC concentration (B), whereas pH (A) has the lowest effect on the TC degradation. (Fig. 5). Fig 5 The perturbation Plot of independent variables Interactive effect of independent variables on the TC degradation Three-dimensional surfaces and contour plots are graphical representation of regression equation for the optimization of reaction Status. The results of the interactions between four independent variables and dependent variable are indicated in Figs. 6 and 7. Fig 6 Contour and 3-D plots showing Interactive effect of: (a) TC concentration (mg/L) and PS concentration (mM); (b) TC concentration (mg/L) and sonication time (min) Figure 6(a) indicates the interaction effect of TC concentration and PS concentration on the TC degradation rate with reaction time of 120 min. with the increasing PS concentration, the TC degradation rate significantly enhanced. With increasing PS concentration from 2 to 4 mM, the TC degradation rate increased from 75.56 % to 94.25 % at TC concentration of 30 mg/L. These results suggest that with increasing PS concentration, more sulfate radicals are produced which leads to more quickly TC degradation [32]. Figure 6(b) indicates the interaction effect of initial TC concentration and reaction time on the TC degradation rate. The TC degradation rate strongly increased with increase of sonication time from 60 to 120 min. with increasing reaction time from 60 to 120 min, TC concentration of 30 and 70 mg/L, the TC degradation rate increased from 70.44 % to 94.25 % at TC concentration of 30 mg/L. With increasing the TC concentration from 30 to 70 mg/L, the TC degradation rate decreased from 94.25 % to 85.05 %. In the constant conditions, with the increasing TC concentration, possibility of reaction between TC molecules and reactive species were declined. Moreover, the higher concentration of TC may lead to the creation of resistant byproducts and consequently decreases the degradation rate of TC [14, 61]. However, the total amount of degraded TC increased with the increasing initial TC concentration. This results are in agreement with the results obtained by other researchers [50]. Figure 7 indicates the interaction influence of pH value and initial TC concentration on the TC degradation rate. With increasing pH from acidic (5) to natural (7.5), the degradation rate slightly decreased, whereas with increasing pH from neutral (7.5) to alkaline (10), the degradation rate significantly enhanced. The TC degradation rate increased from 86.62 % to 94.25 % with increasing pH from 5 to 10, at TC concentrations of 30 mg/L. Under alkaline conditions (pH ≥10), alkaline-activated persulfate is the primary responsible for the production of SO4-•, O2-• and HO• radicals as following equations: [62, 63].10 S2O82−+2H2O→OH−HO2−+2SO42−+3H+11 HO2−+S2O82−→SO4−•+SO42−+H++O2−•12 SO4−•+OH−→SO42−+HO• Also, at alkaline pH, sulfate radicals can react with hydroxyl anions to generate hydroxyl radicals (HO•) according to Eq. (3). In addition, a theory was introduced by other researchers that with increasing pH, the PS degradation into HO• and SO4-• increased [64]. The SO4-• is the predominant radical responsible for TC degradation at acidic pH, whereas both SO4-•and OH• are contributing in TC degradation at natural pH. Thus, three reactions compete with each other in natural pH: the reaction between SO4-• and HO•, the reaction between SO4-• and TC, and the reaction between HO• and TC, the simultaneous occurrence of these reactions may reduce the TC degradation rate [37, 65]. Kinetics of tetracycline degradation The sonochemical degradation process typically follows pseudo first-order kinetics as shown in the following Eqs. (13) and (14). Many studies have suggested that oxidation of organic pollutants by ultrasound follows pseudo first-order kinetics [42, 47, 52].13 −dTC/dt=kTC Eq. (13) can be rewritten as:14 lnCi/Ct=kt Where Ci is the initial TC concentration, Ct is the TC concentration at time t, k is the pseudo first order reaction rate t is constant (min−1) and the reaction time (min). To study the TC degradation by US/S2O82− process, the data obtained was investigated using the pseudo first order kinetics. The effect of different parameters such as initial TC concentration, initial PS concentration, pH and temperature on the kinetic of TC degradation was evaluated. In all the experiments, TC degradation well-fitted to the using the pseudo first order kinetics with higher correlation coefficients (R2). The values of kinetic rate constants (k) related to the different parameters, with their regression coefficients R2 are shown in Table 7. Table 7 Effect of operation parameters on the kinetics degradation of TC parameter Value k 0 (min−1) × 10−2 R2 t 1/2 (min) TC concentration (mg/L) 25 2.29 0.9973 30.2 50 1.75 0.9952 39.6 75 1.23 0.9956 56.3 PS concentration (mM) 2 1.15 0.9816 60.6 3 1.52 0.9946 45.9 4 2.29 0.9973 30.2 pH 5 1.62 0.9937 42.7 7.5 1.12 0.9942 62.8 10 2.29 0.9973 30.2 Temperature (°C) 25 2.29 0.9973 30.2 45 5.70 0.9127 12.1 55 7.87 0.921 8.8 65 10.42 0.9824 6.6 The effect of temperature on the degradation of tetracycline To investigate the effect of temperature on the TC degradation rate, experiments were done with various temperature varying from 25 to 65 °C. With increasing temperature from 25 to 65 °C, the degradation rate constant increased from 0.0229 to 0.1042 min −1. Complete TC degradation occurs after 40, 60 and 75 min of reaction at 65, 55 and 45 °C respectively. The activation of S2O82− can be done under heat to form SO4-• radical as following Eq. (15). Therefore, complete removal of TC by high temperature could be as a result of thermally activated S2O82− oxidation. Moreover, the increase of temperature significantly enhanced the cavitation activity and chemical effects, resulting in greater degradation rate of TC by US/S2O82− process [22, 60].15 S2O8−2+→Termal−activation2SO4−•30°C