## 3 Results
### Empirical Experimental Modeling  
The basic values, real (\(Z_{i0}\)) and coded (\(X_{i0}\)) values, for the three tested independent variables of the reactive dye biosorption process onto the two prepared biosorbents (encapsulated and/ or immobilized yeast biomass in sodium alginate) are presented in Table 1, in association with their variation steps (\(\Delta Z_{i0}\)). The reactive dye solution samples of 25 mL were contacted with different amounts of residual immobilized biomass at adequate pH (pH = 3), room temperature (25 \({}^{\circ}\)C), under initial continuous stirring (30 rpm) of 1-3 min, and analyzed after a specific biosorption time (t, (h)) of static biosorption for finding the dye removal efficiency/performance, based on the experimental planning design presented in Table 1.  
The proposed mathematical models for reactive BRed dye biosorption onto residual encapsulated yeast biomass biosorbent 1) (\(Y_{1}\)) and residual immobilized yeast biomass (biosorbent 2) (\(Y_{2}\)) are as follows:  
\[ \begin{array}{l}Y_{1}=27.3415+7.008X_{1}+8.5806X_{3}-1.4085X_{1}^{2}+1.0799X_ {2}{}^{2}+0.6971X_{3}{}^{2}-0.7284X_{1}X_{2}+2.7969X_{1}X_{3}+\\ 1.8257X_{2}X_{3}\end{array}  \tag{3}\]  
Figure 2: The influence of the initial concentration of dye (**a**) and the amount of dry biomass per 25 mL aqueous dye solution (**b**) on the biosorption of Brilliant Red HE-3B dye onto the residual biomass of _Saccharomyces pastorisiumus_ immobilized in sodium alginate. Working conditions: pH =3, contact time = 24 h, and (**a**) 3.2 g/L of biosorbent 1 and 2.4 g/L biosorbent 2, T = 30 \({}^{\circ}\)C; (**b**) C\({}_{0}\) = 31.95 mg/L of dye, T = 20 \({}^{\circ}\)C for both two biosorbents; biosorbent 1 (f 1 = 1.5 mm, prepared by using a Buchi micro-encapsulator) and biosorbent 2 (f 2 = 4 mm, prepared by a simple dropping technique). \[Y_{2}=60.6002+9.3796X_{1}+5.6881X_{2}+10.1079X_{3}-3.3381{X_{1}}^{2}-6.461{X_{2}}^{ 2}-4.5396{X_{3}}^{2}+1.3353X_{1}X_{3}-4.6628X_{2}X_{3} \tag{4}\]  
without \(X_{2}\) term (specifically, the term of '+0.0772X2') in relation (3) and without \(X_{1}X_{2}\) term (term of \"+0.1548X1X2\") in relation (4) found as insignificant after the application and results analysis of the Student's \(t\)-test. The model coefficients were calculated with the well known statistical formula reported in other authors' reports [11; 12; 13]. The correlation between the experimental and modelled data is presented in Table 2 for BRed dye biosorption onto biosorbent 1 and Table 3 for Bred dye biosorption onto biosorbent 2. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Variable/Value** & **Real Variable** & **Coded Variable** & **Real Basic Variable** & **Variation Step** \\  & **(Z\({}_{i}\))** & **(X\({}_{i}\))** & **(Z\({}_{i0}\))** & **(\(\Delta Z_{i0}\))** \\ \hline \multicolumn{5}{c}{biosorbent 1: Residual yeast biomass prepared by using a Buchi microencapsulator} \\ \hline Dye concentration, (mg/L) & \(Z_{1}\) & \(X_{1}\) & 50 & 20 \\ \hline Residual encapsulated biomass, (g/L) & \(Z_{2}\) & \(X_{2}\) & 12 & 6 \\ \hline Biosorption (S/L) contact time, (h) & \(Z_{3}\) & \(X_{3}\) & 6 & 2 \\ \hline \multicolumn{5}{c}{biosorbent 2: Residual biomass immobilized in sodium alginate by simple dropping technique} \\ \hline Residual immobilized biomass, (g/ L) & \(Z_{1}\) & \(X_{1}\) & 12 & 6 \\ Dye concentration, (mg/L) & \(Z_{2}\) & \(X_{2}\) & 50 & 20 \\ Biosorption (S/L) contact time, (h) & \(Z_{3}\) & \(X_{3}\) & 6 & 2 \\ \hline \end{tabular}
\end{table} 
Table 1: Codification of independent variables in the active central compositional rotatable design of \(2^{3}\) order for reactive BRed dye biosorption onto residual biomass (encapsulated/immobilized type).