## 3 Test Data Analysis
### Process Optimization  
In recent years, halloysite nanotubes have been widely researched as a \"green resource\" for their applications as drug carriers, for sustained drug release, and catalysis, to name a few [47; 48]. Their function as an adsorptive material [49] is an application of great interest. To optimize the extent of adsorption in HNT for BG remediation statistically, a quadratic model in Fractional Factorial Experimental Design under RSM was used [50]. The following general equation can explain the model. \[Y=\beta_{0}+\sum\beta_{i}X_{i}+\sum\beta_{ii}X^{2}+\sum\beta_{ij}X_{i}X_{j}\]  
where \(Y\) addresses the reliant reaction variable, \(\beta_{0}\) is a relapse coefficient, \(\beta i\) is the direct impact, \(\beta ii\) is the squared impact, and \(\beta ij\) is the collaboration impact of autonomous variable \(X\). Plan master (7.0.0), measurable programming was utilized for RSM study and graphical portrayal (3D and shape plots) for the impact of the free factors on the reaction [51; 52; 53; 54; 55; 56]. The quadratic regression equation derived from ANOVA shows the possible individual and combined effect of the factors for the BG-HNT system (Table 7) [57; 58]. The _p-esteem_ < 0.05% with 95% certainty span was considered huge. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Initial** & **Temp** & \(\Delta\mathbf{G}^{\circ}\) & \(\Delta\mathbf{S}^{\circ}\) & \(\Delta\mathbf{H}^{\circ}\) & **In  
The regression Equation obtained for the BG-HNT system is shown below:  
\[\text{BG}-\text{HNT System}=4.2+(26.1\times\text{A})+(1.4\times\text{B})+(37.1 \times\text{C})-\] \[\left(40.9\times\text{D})(10.2\times\text{E})+(3.2\times\text{AB} )+(11.4\times\text{AC})-(0.4\times\text{BC})-\right.\] \[\left(11.3\times\text{A}^{2}\right)+\left(3.3\times\text{B}^{2} \right)+\left(8.3\times\text{C}^{2}\right)+\left(46.5\times\text{D}^{2}\right) +\left(16.8\times\text{E}^{2}\right)\]  
In this examination A, C, D, AC, A\({}^{2}\), and D\({}^{2}\) are significant factors, and the other variables are inconsequential. Cross-items AD, AE, BD, BE, BF, CD, CE, CF, DE, DF, and EF are zero and, thus, are avoided to build the previously mentioned relapse condition. The RSM format is exceptionally critical concerning the F-value model of 30.12. The examination diagram for real versus anticipated qualities (Figure 11) demonstrate a solid connection between the experimental and anticipated reactions. The regression coefficient values indicate the effect of the parameter(s) on the adsorption capacity. The graphs showing the contour and surface response illustrate the combined influence caused by two factors in the process of adsorption, and the results were graphically presented in Figures 12 and 13. Based on the selected results obtained by employing the conditions of the study evolved through statistical modeling, a value of \(q_{e}\) = 203 mg/g was obtained for an initial dye concentration of 344 mg/L at pH 12, adsorbent quantity = 0.52 g/L for the duration, 180 min along 165 rpm orbital shaking at 27.7 \({}^{\circ}\)C.  
Figure 11: Actual versus predicted values of BG-HNT. Figure 12: _Cont._  
Figure 12: 3D surface plot and contour plot depicting a capacity variation of the adsorption with (**a**) time versus temperature, (**b**) time versus concentration, (**c**) time versus the adsorbent dosage, (**d**) time versus pH, and (**e**) temperature versus concentration. Figure 13: Cont. Figure 13: 3D surface plot and contour plot depicting the capacity variation of adsorption with (**a**) the temperature versus adsorbent dosage, (**b**) temperature versus pH, (**c**) concentration versus adsorbent dosage, (**d**) concentration versus pH, and (**e**) adsorbent dosage versus pH. The measurable cycle enhancement in a given scope of boundary esteems permits for ascertaining the ideal condition, yet, deciding the impact of the interaction conditions on the adsorption. Figures 12 and 13 presents the collaborated impact of two boundaries of the process.