# Optimizing the Sodium Hydroxide Conversion Using Regression Analysis in CSTR
## 3 Results and Discussion
### Experimental Findings
\[X_{A}=\beta_{o}+\beta_{1}X_{1}+\beta_{4}X_{4}+e \tag{9}\]  
A moderate positive correlation is observed between dependent and independent variable as shown in Table 11. R square demonstrates that the goodness of fit is moderate while adjusted R\({}^{2}\) explains that extraneous variable has moderate influence on response variable. F value, as illustrated in ANNOVA Table 12, explains that the impact of independent variables varies considerably at a significance level of 0.05. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Minimum** & **Maximum** & **Mean** & **Std. Deviation** & **N** \\ \hline Predicted value & 0.349652 & 0.601985 & 0.493438 & 0.0576003 & 16 \\ \hline Residual & \(-\)0.1351061 & 0.0550606 & 0.0000001 & 0.0525465 & 16 \\ \hline Std. predicted value & \(-\)2.496 & 1.884 & 0.000 & 1.000 & 16 \\ \hline Std. residual & \(-\)2.300 & 0.937 & 0.000 & 0.894 & 16 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Residuals statistics with three predictors. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{3}{*}{**Model**} & \multirow{3}{*}{**R**} & \multirow{3}{*}{**Square**} & \multirow{3}{*}{**Adjusted R**} & \multirow{3}{*}{**Std. Error of**} & \multicolumn{5}{c}{**Change Statistics**} & \multirow{3}{*}{**Purbin-Watson**} \\ \cline{3-3} \cline{5-10}  & & & & **R Square** & & **Range** & **df1** & **df2** & **Sig. F** & **Change** \\ \hline
1 & 0.732 & 0.536 & 0.465 & 0.0570525 & 0.536 & 7.507 & 2 & 13 & 0.007 & 0.296 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Model summary with two predictors. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{3}{*}{**Model**} & \multirow{3}{*}{**Model**} & \multicolumn{3}{c}{**Unstandardized**} & \multicolumn{3}{c}{**Standardized**} & \multirow{3}{*}{**Stg.**} & \multirow{3}{*}{**95.0\% Confidence Interval for B**} \\  & & & **Coefficients** & & & & **Lower Bound** & **Upper Bound** \\ \hline \multirow{4}{*}{\(1\)} & \multirow{2}{*}{(Constant)} & \(-\)0.305 & 0.221 & & \(-\)1.377 & 0.194 & \(-\)0.787 & 0.177 \\ \cline{2-9}  & & X\({}_{1}\) & 0.011 & 0.005 & 0.421 & 2.162 & 0.042 & 0.000 & 0.021 \\ \cline{1-1} \cline{2-9}  & X\({}_{4}\) & 0.293 & 0.098 & 0.581 & 2.983 & 0.011 & 0.079 & 0.506 \\ \cline{1-1} \cline{2-9}  & X\({}_{2}\) & 0.001 & 0.001 & 0.099 & 0.510 & 0.619 & \(-\)0.001 & 0.002 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Coefficients of predictors with three predictors. The impact of all independent variables on steady-state NaOH conversion is shown in Table 13. The value of \(p\) is statistically significant for \(X_{1}\) and \(X_{4}\). Equation (10) is formulated applying coefficients shown in Table 13. As observed in Equation (10), the effect of volume and temperature is positive and statistically significant in forecasting the NaOH conversion. \[X_{A}=-0.261+0.011X_{1}+0.293X_{4}+e \tag{10}\]  
Table 14 characterizes the maximal and minimal NaOH conversion as predicted by model Equation (10) applying residual analysis. The highest steady-state NaOH conversion is 0.6019. Whereas, the lowest conversion with respect to NaOH is equal to 0.3496. The findings of response variables with three predictors and with two predictors are almost the same. The two predictors have a positive impact and are statistically significant, hence an attempt was made to check the model based on a 2 nd order polynomial to optimize the reaction conversion. To optimize the product formation, a 2 nd order regression model among important (significant) variables was developed to examine the influence of significant factors (temperature, \(X_{1}\) and volume, \(X_{4}\)) on predicted response (conversion, \(X_{A}\)) as follows (Equation (11)):  
\[X_{A}=\beta_{o}+\beta_{1}X_{1}+\beta_{4}X_{4}+\beta_{3}X_{1}X_{4}+\beta_{16}X_ {1}^{2}+\beta_{19}X_{4}^{2}+e \tag{11}\]  
where, \(\beta_{16}\) and \(\beta_{19}\) are the coefficient of predictors, while \(X_{1}^{2}\) and \(X_{4}^{2}\) are squares of their corresponding predictors. \(X_{1}X_{4}\) are interactions between two significant independent variables.