## 3 Results
### Empirical Experimental Modeling
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Exp. No.** & \(Z_{1}\)**, (mg/L)** & \(Z_{2}\)**, (g/L)** & \(Z_{3}\)**, (h)** & \(X_{1}\) & \(X_{2}\) & \(X_{3}\) & \(Y_{\text{el}}\) (\%) & \(Y_{\text{cl}}\) (\%) & **Deviation (A)** \\ \hline
1 & 30 & 6 & 4 & \(-1\) & \(-1\) & \(-1\) & 13.624 & 16.023 & \(-0.176\) \\
2 & 30 & 18 & 4 & \(1\) & \(-1\) & \(-1\) & 31.878 & 25.887 & 0.188 \\
3 & 70 & 6 & 4 & \(-1\) & 1 & \(-1\) & 14.362 & 13.828 & 0.037 \\
4 & 70 & 18 & 4 & 1 & \(1\) & \(-1\) & 16.795 & 20.779 & \(-0.237\) \\
5 & 30 & 6 & 8 & \(-1\) & \(-1\) & 1 & 31.217 & 23.939 & 0.233 \\
6 & 30 & 18 & 8 & 1 & \(-1\) & 1 & 47.751 & 44.991 & 0.058 \\
7 & 70 & 6 & 8 & \(-1\) & 1 & 1 & 26.350 & 29.047 & \(-0.102\) \\
8 & 70 & 18 & 8 & 1 & 1 & 1 & 52.878 & 47.186 & 0.108 \\
9 & 50 & 1.908 & 6 & \(-1.682\) & 0 & 0 & 11.557 & 11.581 & \(-0.002\) \\
10 & 50 & 22.092 & 6 & 1.682 & 0 & 0 & 30.498 & 35.132 & \(-0.152\) \\
11 & 16.36 & 12 & 6 & 0 & \(-1.682\) & 0 & 31.266 & 29.240 & 0.064 \\
12 & 83.64 & 12 & 6 & 0 & 1.682 & 0 & 31.270 & 30.907 & 0.012 \\
13 & 50 & 12 & 2.64 & 0 & 0 & \(-1.682\) & 19.823 & 14.881 & 0.249 \\
14 & 50 & 12 & 9.36 & 0 & 0 & 1.682 & 49.278 & 40.721 & 0.174 \\
15 & 50 & 12 & 6 & 0 & 0 & 0 & 23.756 & 27.342 & \(-0.151\) \\
16 & 50 & 12 & 6 & 0 & 0 & 0 & 24.318 & 27.342 & \(-0.124\) \\
17 & 50 & 0.60 & 6 & 0 & 0 & 0 & 23.274 & 27.342 & \(-0.175\) \\
18 & 50 & 0.60 & 6 & 0 & 0 & 0 & 24.478 & 27.342 & \(-0.117\) \\
19 & 50 & 0.60 & 6 & 0 & 0 & 0 & 23.596 & 27.342 & \(-0.159\) \\
20 & 50 & 0.60 & 6 & 0 & 0 & 0 & 23.917 & 27.342 & \(-0.143\) \\ \hline \end{tabular}
\end{table} 
Table 2: Experimental planning matrix in the active central compositional rotatable design of \(2^{3}\) order for static BRed dye biosorption onto biosorbent 1 (residual yeast biomass encapsulated in sodium alginate). Mathematical model analysis. The value of Fisher constant is found to be \(F=9269\) for \(Y_{1}\) (BRed dye removal onto biosorbent 1) and, respectively, \(F=2438\) for \(Y_{2}\) (Bred dye removal onto biosorbent 2), and the statistic value (from table) is \(F_{tab}=4.60\) (for \(\alpha=99\), \(\
u_{1}=n-1=19\), \(\
u_{2}=\text{b}-1=2\), where \(n\) is the number of experiments (\(n=20\)), and \(b\) is the number of independent variables (\(\text{b}=3\)). This is because both values of \(F>F_{tab}\), the deviation of experimental data (\(Y_{ci}\)) referring to the average experimental value (\(\overline{Y_{ci}}\)), are due to the influence of independent variables towards the decision/response function, not of the experimental errors. The value of the multiple correlation coefficient is found to be \(R_{YX1X2X3}=0.9257\) for \(Y_{1}\) (BRed dye removal onto biosorbent 1) and \(R_{YX1X2X3}=0.9232\) for \(Y_{2}\) (Bred dye removal onto biosorbent 2) closed to the unity; thus, the high importance of all independent variables (\(X_{i}\)) is demonstrated in the variation of decision/response function (Bred dye removal) for the experimental variation field of each variable \(Z_{i}\) or coded \(X_{i}\). The calculated Fisher test value is \(F_{\text{C}}=31.92545\) for \(Y_{1}\) (BRed dye removal onto biosorbent 1) and \(F_{\text{C}}=30.7766\) for \(Y_{2}\) (Bred dye removal onto biosorbent 2) which was much higher than the statistical value (from table) of \(F_{\text{C},tab}=6.59\) for the freedom degree of \(\
u=n\text{-k}-1=16\) and \(\
u=\text{k}\text{=3}\), demonstrating that the independent variables had a significant influence on the decision/response function (dye biosorption efficiency). Moreover, for these proposed mathematical models, the experimental dispersion was \(27.5943\) for \(Y_{1}\) and \(25.5418\) for \(Y_{2}\), and the coefficients dispersion was \(0.2027\) for \(Y_{1}\) and \(0.0652\) for \(Y_{2}\). The processed data indicate an acceptable accordance between the experimental and model-calculated data, the mean/average deviation being of \(-2.081\%\) for \(Y_{1}\) and \(-2.414\%\) for \(Y_{2}\), both in the admissible limit range of \(-10\%\div+10\%\).