As can be seen Table 1, there was a considerable variation in the degradation efficiency at different values of the chosen variables. The analysis of variance (ANOVA) was performed to specify the significance of the statistical method, as given in Table 4. F-value of model should be greater than the tabulated value of the F-distribution for a certain number of degrees of freedom at a level of significance, ?=5%. F-values of the degradation of Eosin B dye were reported as 42 which are significant. P-values of model for Eosin B degradation were significant.

The insignificant lack of fit, P-value of 0.3504 (more than 0.05) for the degradation percentage of Eosin B, showed that the quadratic model was valid for the present study. The minimum value of standard error design (0.479) around the centroid indicates that the model can be used to behavior the design space (Fig. 2).The significance of each coefficient for the degradation percentage of Eosin was determined by F-values and P-values as listed in Table 5.

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Based on these results, a relationship between the degradation percentage of Eosin B and selected variables was explained by the second-order polynomial equation. The regression equation obtained after the ANOVA showed that the coefficient of determination (R2) was 0.9855 for Eosin B dye degradation, meaning that more than 99.91 % of the data deviation can be expressed by the model. The r2 value for the sample size and the number of terms in the model is modified by using the degrees of freedom on its calculation. If there are many terms in a model and not a very large sample size, adjusted R2 may be smaller than R2 30. Therefore the high value of adjusted R2 (0.

9982) indicates a high degree of correlation between the experimental and predicted values and consequently a good predictability of the model. It was observed that the predicted R2 was 0.9324. Thus, predicted R2 is in agreement with the adjusted R2. Hence, the quadratic model can be used to navigate the design space. The low values of coefficient of variation CV (1.13%) and standard deviation SD (1.01) showed high reliability.

In this case, low CV and SD values indicate the capability with which the experiment was conducted. The low predicted residual sum of squares (PRESS) (70.87) is a measure of how well the model fits each point in the model 32.

The smaller the PRESS statistic, the better the model fits the data points. Therefore, the high R2 value, significant F-value, insignificant lack-of-fit P-value and low PRESS indicate high suitability and reliability of models in predicting the Eosin B dye degradation. Therefore, these models were used for further analysis.The 3D surface plots expose the type of interaction between the variables and allow to obtain the optimum conditions 32.

These plots of the second order polynomial equation, with two variables constant and the two variables within the experimental data, are given in Fig. 3. The maximum predicted value is shown via the surface hump in the surface plots 33. The interactive effect of the ultrasound irradiation power, ranging from 50 to 250 watt, and the ultrasound irradiation time, ranging from 10 to 90 min, is shown in Fig. 5(a), where the dye concentration and the Catalyst dosage were 15 mg/L and 1.

0 g/l, respectively. These two factor causes the generation of more OH radicals because of the increased of the adsorption of heat and energy produced from the cavitation phenomenon. Also, the maximum dye removal efficiency was obtained with an increasing irradiation power of dye as shown in Fig.

3(b and c). The ultrasound irradiation provides an important driving force to overcome all mass transfer resistances of the dye between the aqueous and solid phases. Hence a higher irradiation power will enhance the degradation process.As can be seen in Fig.

3(a, d and e), the degradation efficiency of Eosin B dye rapidly increases with an increase in ultrasound irradiation time from 10 to 30 min. In the present study, the irradiation time of solution plays a very important role on the dye degradation efficiency, which may be attributed to the production of more oxidative •OH radicals owing to the higher ultrasonic time in the experimental conditions 22,23. Also, the ultrasonic time enhancement is helpful to produce turbulence in the solution and to enhance the degree of cavitation, which favors the degradation of dye 24.The surface plots in Fig. 3(b, d and f) for catalyst dosage (ZnO nanoparticles) indicated that the removal efficiency of Eosin B dye increased with an increasing Catalyst dosage. The degradation efficiency (%) of Eosin B rose with increasing the catalyst dosage from 0.1 to 0.3 and then it began to decrease.

So, the determination of the optimum catalyst dosage for enhancing sonocatalytic degradation efficiency and avoiding excess addition of the nanocatalyst is important. As shown in Fig. 5(b), the degradation efficiency (%) increased with increasing the dosage of ZnO up to a specified value. Increasing the catalyst dosage causes more growth and collapse of bubbles for the cavitation phenomenon to enhance the generation of free radicals. The aggregation of nanoparticles in the solution at excessive concentrations, led to in the loss of active surficial sites and generating OH radicals in the solution.

Fig. 9(c, e and f) exhibited the effect of dye concentration on Eosin B dye degradation efficiency in dye solution. The removal efficiency was decreased with an increase in dye concentration. The extreme adsorption of dye molecules on the surface of particles due of the increased dye concentration causes a significant drop in the sonocatalysis.