Materials and methods Materials Tetracycline hydrochloride [C22H25N2O8Cl] (AR, 99 %), was provided from Sigma–Aldrich. Chemical properties of tetracycline hydrochloride are shown in Table 1 [2]. Sodium persulfate (Na2S2O8, 98 %) was provided from Sigma–Aldrich. All other chemicals were of analytical grade and were used without further purification. The water used in all experiments was purified by a Milli- Q system. Table 1 Chemical properties of tetracycline hydrochloride Molecule Formula Molecular weight (g/mol) Solubility (mol/L) pKa1 pKa2 pKa3 TC C22H24O8N2.HCl 480.9 0.041 3.2 ± 0.3 7.78 ± 0.05 9.6 ± 0.3 Procedure Schematics of the experimental setup applied in this study is demonstrated in Fig. 1. A stock solution of tetracycline was daily prepared with distilled deionized water and diluted as required initial concentration. Sonochemical treatment was carried out with a fixed volume of 100 mL of TC solution in a glass vessel of 200 mL. The vessel was wrapped with tinfoil in order to avoid any photochemical effects. The pH adjustments were conducted with 1 m NaOH or 1 m HCL (Merck Co.) using a pH meter (E520, Metrohm, Tehran, Iran). Sonochemical treatment was performed with an ultrasonic generator at a frequency of 35 kHz and power of 500 W (Elma, Singen, Germany). The reactor was immersed into the ultrasonic bath and its location was always kept similarly. All experiments were conducted at constant temperature using cooling water and temperature controller. At pre-specified time intervals, 2 mL sample was withdrawn, filtered through 0.22 μm syringe filter and mixed with the same volume of methanol to quench the reaction before analysis [3]. Fig 1 Schematic of the experimental used in this study; (1) temperature controller, (2) water-circulating (3) TC solution reactor (4) cooling water inlet, (5) cooling water outlet (6) sampling port Analytical methodology The pH was determined at room temperature using an S-20 pH meter, which was calibrated with pH 4.0 and 7 reference buffer solutions. The concentration of TC in aqueous solution was analyzed by HPLC, with a LC-20 AB pump, Shimadzu, Kyoto, Japan) with a reversed-phase column (VP-ODS-C18 4.6 mm × 250 mm, 5 μm, Shim-Pack, Kyoto, Japan), and UV detector (Shimadzu UV-1600 spectrophotometer). The injection volume was 20 μL; the mobile phase was acetonitrile 0.01 M, oxalic acid solution (31:69, v/v) with a flow rate of 1.0 mL min−1. The detection wavelength and retention time of tetracycline were 360 nm and 2.38 min, respectively. In this study, limit of detection (LOD) were found to be 0.02-0.03 mg/L based on linear regression method. Experimental design A central composite statistical experiment design was used to evaluate the effects of four independent variables (initial solution pH (A), initial TC concentration (B), initial S2O8−2 concentration (C) and reaction time (D)) on the TC degradation. The application of RSM provides a mathematical relationship between variables and experimental data can be fitted to an empirical second-order polynomial model as the following Eq. (8). [55–57].8 Y=β0+β1A+β2B+β3C+β4D+β12AB+β13AC+β14AD+β23BC+β24BD+β34CD+β11A2+β22B2+β33C2+β44D2 Where, y (%) is the predicted response (TC degradation rate), β0 is interception coefficient, β1, β2, β3 and β4 are the linear coefficients, β12, β13, β14, β23, β24 and β34 are interaction coefficients, β11, β22, β33 and β44 are the quadratic coefficients and A, B, C and D are the independent variables. The natural and coded levels of independent variables based on the central composite design are shown in Table 2. The experimental values for each independent variables were chosen according to the results obtained from preliminary analysis. Table 3 indicates the four-factor, five-level CCD and the obtained and predicted values for the TC degradation rate (%) using the developed quadratic model. In RSM analysis, the approximation of y was proposed using the fitted second-order polynomial regression model which is called the quadratic model. A quadratic regression is the process of finding the equation of the parabola that fits best for a set of data [58]. Table 2 Natural and coded levels of independent variables based on the central composite design Independent variable Symbol Coded levels −2 −1 0 +1 +2 Natural level pH A 2.5 5 7.5 10 12.5 Tetracycline (mg/L) B 10 30 50 70 90 Persulfate (Mm) C 1 2 3 4 5 Reaction time (min) D 30 60 90 120 150 Table 3 Four-factor five-level central composite design for RSM Run Experimental conditions TC degradation rate (%) pH (A) Tetracycline (mg/L) (B) Persulfate (mM) (C) Time (min) (D) Observed (%) Predicted (%) 1 7.5 (0) 50 (0) 3 (0) 90 (0) 51.06 49.82 2 2.5 (−2) 50 (0) 3 (0) 90 (0) 55.64 55.74 3 7.5 (0) 50 (0) 3 (0) 90 (0) 48.45 49.82 4 5 (−1) 70 (+1) 4 (+1) 60 (−1) 45.16 44.64 5 5 (−1) 30 (−1) 4 (+1) 120 (+1) 86.62 86.33 6 10 (+1) 70 (+1) 2 (−1) 120 (+1) 61.02 61.32 7 7.5 (0) 50 (0) 3 (0) 150 (+2) 81.85 81.17 8 5 (−1) 30 (−1) 2 (−1) 60 (−1) 34.55 35.25 9 10 (+1) 30 (−1) 2 (−1) 60 (−1) 47.25 46.91 10 10 (+1) 30 (−1) 4 (+1) 60 (−1) 70.44 69.95 11 5 (−1) 30 (−1) 2 (−1) 120 (+1) 61.85 61.79 12 7.5 (0) 50 (0) 1 (−2) 90 (0) 28.72 28.18 13 10 (+1) 70 (+1) 4 (+1) 120 (+1) 85.05 84.34 14 7.5 (0) 50 (0) 3 (0) 90 (0) 49.75 49.82 15 10 (+1) 30 (−1) 2 (−1) 120 (+1) 75.56 76.07 16 10 (+1) 30 (−1) 4 (+1) 120 (+1) 94.25 95.04 17 5 (−1) 70 (+1) 2 (−1) 60 (−1) 12.65 11.98 18 5 (−1) 30 (−1) 4 (+1) 60 (−1) 64.04 63.86 19 7.5 (0) 90 (+2) 3 (0) 90 (0) 41.15 41.31 20 10 (+1) 70 (+1) 4 (+1) 60 (−1) 54.88 55.05 21 7.5 (0) 50 (0) 3 (0) 90 (0) 50.55 49.82 22 7.5 (0) 50 (0) 3 (0) 90 (0) 49.75 49.82 23 7.5 (0) 50 (0) 5 (+2) 90 (0) 79.38 79.82 24 12.5 (+2) 50 (0) 3 (0) 90 (0) 80.62 80.42 25 7.5 (0) 10 (−2) 3 (0) 90 (0) 75.55 75.28 26 7.5 (0) 50 (0) 3 (0) 90 (0) 49.35 49.82 27 7.5 (0) 50 (0) 3 (0) 30 (−2) 24.75 25.33 28 5 (−1) 70 (+1) 2 (−1) 120 (+1) 42.25 42.72 29 10 (+1) 70 (+1) 2 (−1) 60 (−1) 27.68 27.96 30 5 (−1) 70 (+1) 4 (+1) 120 (+1) 70.85 71.31