## 2 Materials and Methods
### Experimental Methods
#### 2.2.2 Empirical Modeling and Optimization of Biosorption Process  
In all biosorption experiments of BRed reactive dye onto residual biomass (immobilized in sodium alginate), the independent variables were considered to be the following: the biosorbent concentration (\(Z_{1}\), (g/L)), BRed dye concentration (\(Z_{2}\), (mg/L)) and biosorption contact time (\(Z_{3}\), (h)). As the decision/response function (dependent variable), also considered as an optimization criterion, the dye removal was chosen (\(Y\), or \(R\), (%)). The mathematical biosorption model was proposed by application of the empirical modeling using an active central composite rotatable design of \(2^{3}\) order which is based on specific experimental planning matrices with imposed values for all coded variables (\(X_{i}\)). The proposed mathematical model was of \"b = 3\" independent variables, which was found after the experimental biosorption testing according to the specific experimental planning matrix, and is expressed by the Equation (2) [12; 13; 14; 15; 16; 17; 18; 19; 20; 21]. \[Y=b_{0}+\sum b_{i}x_{i}+\sum b_{i}x_{i}x^{2}+\sum b_{i}x_{i}x_{j} \tag{2}\]  
where \(Y\) represents the decision/response function, or optimization criterion; \(x_{i}\), \(x_{j}\), \(x_{ii}\), \(x_{ij}\)--the coded independent variables of the biosorption-based treatment system, and \(b_{0}\), \(b_{i}\), \(b_{j}\), \(b_{ij}\)--the model coefficients (\(i\), \(j\) = 1, 2, 3). The model coefficients were calculated using statistical analysis based on the least square fitting model obtained in the experimental design points [12; 13; 14; 21]. In all biosorption experiments values (levels) were attributed to each independent variable \(Z_{i}\), in accordance with its basic value (\(Z_{i0}\)) and imposed variation step (\(\Delta Z_{i0}\)) corresponding to the coded variable values (\(X_{i}\) = 0, \(\pm\)1, or \(\alpha\) = \(\pm\)1.682). The Fisher constant (\(F\)), multiple correlation coefficient (\(R_{Y\times 1\times 2\times 3}\)), or Fisher test (\(F_{C}\)), defined by well-known statistic relations, were calculated for establishment of the correlation between the decision/response function (\(Y\)) and the three coded independent variables (\(X_{i}\)) as in the relation 2 [12; 13; 14]. The dispersion of all experiments and coefficients were also statistically evaluated. The mean deviation between the calculated and experimental data must be in the range of \(-\)10% and \(+\)10% for a very good accordance. The model validation was carried out by an appropriate analysis of variance, consisting of: (1) the calculation of the \(F\) constant and its comparison with the statistical value to underline the significance of all independent variable vs. experimental errors; (2) the calculation of the \(R_{Y\times 1\times 2\times 3}\) to establish the correlation between the dependent variable (\(Y\)) and the three independent variables (\(X_{i}\)) as a whole; (3) the determination of the coefficient's significance using the Student \(t\)-test, for a certain significance level (\(p\) = 0.05) and degrees of freedom (v\({}_{1}\) = n \(-\) 1 = 19 and v\({}_{2}\)= k \(-\) 1) [12; 13; 14; 21]; (4) the calculation of \(Fc\) value to compare with its statistical value; and (5) the calculation of mean/ average deviation (A) between the model-calculated data and the experimental ones (for a very good data agreement, \(A\) must be between \(+\)10% and \(-\)10%). The mathematical model optimization was permitted by classical method application for finding the maximum of response function/optimization criterion, \(Y^{\star}\) = f(\(X_{1}\)*\(X_{2}\)*\(X_{3}\)*). In this sense, the optimization method consists of the finding of the optimum values of all independent variables related to maximum value of the dependent response function/optimization criterion (Y*, %), i.e., the maximum dye removal by biosorption onto residual biomass, meaning the corresponding values of \(X_{1}\)* \(-\)\(z_{1}\)*, \(X_{2}\)* \(-\)\(z_{2}\)*, \(X_{3}\)* \(-\)\(z_{3}\)*.