Empirical Modeling and Optimization by Active Central Composite Rotatable Design: Brilliant Red HE-3B Dye Biosorption onto Residual Yeast Biomass-Based Biosorbents

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An empirical model using an active central composite rotatable design of 2³ order was studied, and the optimum values of all variables involved in the mathematical model were found by classical optimization method for obtaining the maximum removal of reactive Brilliant Red HE-3B dye onto residual Saccharomyces pastorianus yeast biomass encapsulated and/or immobilized in sodium alginate.

Abstract

(1) Introduction: Natural polymers can be successfully used as a matrix to immobilize residual yeast-based biomass in a form that is easy to handle and can be used as biosorbent capable of removing persistent polluting species from different aqueous systems such as reactive azo dyes. (2) Experimental: Two types of new biosorbents were prepared based on residual Saccharomyces pastorianus yeast biomass immobilized in sodium alginate (using two different practice techniques) and studied in the biosorption process of reactive Brilliant Red HE-3B dye using certain experimental planning matrices according to the active central composite rotatable design of 2³ order. The experimental data obtained under certain selected working conditions were processed considering the influence of three independent variables (biosorbent concentration—X₁, initial dye concentration—X₂ and biosorption time—X₃) onto the dependent variable (Y = f(X₁,X₂,X₃)) expressing the performance of reactive dye biosorption onto the new prepared biosorbents (i.e., dye removal degree, %). (3) Results: Two mathematical models were proposed for each prepared biosorbent. The maximum dye removal was 52.878% (Y₁) when 18 g/L biosorbent 1 (micro-encapsulated form) was applied in 70 mg/L dye-containing solution for at least 8 h, and 75.338% (Y₂) for 22.109 g/L biosorbent 2 (immobilized form) in 48.49 mg/L dye-containing solution for at least 8.799 h. (4) Discussion: The optimal values achieved for the two tested biosorbents were compared, and we investigated the possibility of using this residual biomass as a biosorbent for the reactive dye removal, supported by the experimental results with the recommended variation domains of each influencing variable. The results are sufficient to permit performing dye removal higher than 50% (biosorbent 1) or 70% (biosorbent 2), working with more than 18–22 g/L biosorbent after at least 8 h (as an exchange at work). (5) Conclusions: The proposed models are in good agreement with the experimental data and permit the prediction of dye biosorption behavior onto the experimental variation domain of each independent variable.

1. Introduction

Among the new biotechnologies for environmental protection, biosorption remains an open subject for researchers and specialists in the field. Biosorption is based on a process of retaining persistent chemical pollutants on biological or natural materials as biosorbents. It is a method achieved through extracellular and intracellular bonding, interactions that depend on the nature of organic compound, on the structure of adsorptive materials, on microbial metabolism and on transport process. Similar to the biodegradation of organic compounds, biosorption sometimes implies the breakage and formation of new chemical bonds, which degrades the origin molecular structure of the pollutants [1,2,3,4,5].

Experimental research shows the expansion of a new category of biosorbents with a fairly high efficiency in the process of retaining/removing persistent chemical pollutants. This relates to capitalizing/valorization of residual biomass, including microbial biomass, industrial and agricultural biomass (wood waste, fruit, plants or shells, which contain cellulose, hemicellulose, pectin and lignin), in free form [5,6,7]. In order to improve the ease of handling of residual microbial biomass, it has been immobilized by various techniques (especially two practice techniques, i.e., encapsulation and immobilization) in polysaccharide matrices [7,8,9].

Transposing the process from the laboratory scale to the real world, industrial systems require studies of the sorption equilibrium, thermodynamic and process kinetics, and the application of optimization models and algorithms, in order to obtain adequate mathematical models for further technological and mathematical optimization [8].

Brilliant Red HE-3B is a reactive dye and its biosorption from wastewater resulting from the process of dyeing textiles, by using a residual Saccharomyces pastorianus yeast (available at large scale, easy and safe to use) immobilized in sodium alginate (using two different practice techniques) as biosorbent has been analyzed and reported, as corresponding to a maximum biosorption capacity of around 224.47–555.55 mg/g [10,11]. The process is influenced by some operating factors: temperature, pH, S/L phases contact time (biosorption time period), biosorbent and adsorbate concentration and other particular characteristics such as the diameters of the biosorbent granules depending on the immobilization techniques used. In order to select the best operating process conditions, a modeling and optimization procedure is beneficial, as it could provide the scientific basis for biosorption scaling-up.

The validated biosorption process of the studied reactive dye onto immobilized residual biomass is mathematically modelled considering some specific variation domains of three important independent variables with significant influence on the biosorption efficiency, i.e., biosorbent concentration, dye concentration and biosorption contact time. In this sense, different experimental modeling design methods can be applied for empirical modeling, but we selected an active central composite rotatable design of 2³ order in association with the maximum finding by the classical optimization methodology. The optimum values performed for two new prepared biosorbents were compared, and the possibility was investigated of using the residual encapsulated and/or immobilized biomass as biosorbent for the studied reference reactive dye sustained with the performed experimental results.

The main goal of this paper is to propose pertinent mathematical empirical models for the biosorption process of reactive Brilliant Red HE-3B dye onto the two new prepared biosorbents based on residual biomass of Saccharomyces pastorianus immobilized/encapsulated in sodium alginate. To this purpose, the biosorption will be studied according to the specific experiments’ planning design, by analyzing the experimental data related to three important influencing variables and dye biosorption performance, and also considering the reported analysis of specific adsorption isotherms, and thermodynamic and kinetic models in association with the predicted biosorption mechanism and its controlling rate [9,10]. Moreover, the prepared biosorbent material was commonly physico-chemically characterized before and after the biosorption process of the reactive Brilliant Red HE-3B dye to demonstrate that dye biosorption takes place and to underline the biosorption progress and efficiency. These experimental data were already reported in previous authors’ articles [10,11].

For the modeling of experimental data a well-known design was used, based on an active empirical design applied in numerous scientific reports [12,13,14,15,16,17,18,19,20,21]. This mathematical model is obviously required for each technological process to estimate the final result, if some variations of operating parameters take place, and to anticipate the best value when multiple operating variables are considered, or to remediate some technical operating problems in order to maintain acceptable process results.

2. Materials and Methods

2.1. Materials

Biomass. In the brewing processes, a few by-products are formed, which are available in large amounts, e.g., residual Saccharomyces pastorianus yeasts (Saccharomycetaceae family) [11]. We tested the residual Saccharomyces pastorianus biomass at the finishing of brewing process organized at an industrial company (Albrau, Onești, Romania). The residual yeast-based biomass was separated by centrifugation (8000 rpm), dried at 80 °C and then immobilized in sodium alginate (using two different practice techniques) [10,11].

Biosorbent. The two biosorbent types used in the experimental biosorption studies are based on the immobilization of residual biomass (Saccharomyces pastorianus) in sodium alginate, using two techniques: (1) a Buchi microencapsulator technique application (encapsulation for preparation of biosorbent 1) and (2) a simple dropping technique application (immobilization for preparation of biosorbent 2) following the methodologies presented in our previous paper [10,11].

Adsorbate. A reactive dye, Brilliant Red HE-3B (BRed) (MW = 1430 g/mol, λ_max = 530 nm, from Bezema) with chemical structure showed in Figure 1, is selected as chemical polluting species (reference model of reactive dye) of aqueous system for this study.

It was used the commercial form of reactive dye (powder) for preparation of a stock dye solution (500 mg dye/L) and further of the working solutions by adequate dilution of stock solution with distilled water.

2.2. Experimental Methods

2.2.1. Batch Biosorption Method

The biosorption experiments were developed after the contact of an established amounts of new prepared biosorbent (biosorbent 1—by micro-encapsulation, and biosorbent 2—by a simple dropping technique, in sodium alginate of the residual biomass of Saccharomyces pastorianus (5% dry matter (d.w.), after the traces of calcium chloride were removed by washing with distilled water) with 25 mL of dye solution with different initial concentrations (in the range 10.88–174.08 mg/L for biosorbent 1, and 10.64–170.24 mg/L for biosorbent 2). The pH values were adjusted with 0.1N HCl solution at the value of pH = 3 (measured with waterproof Combo pH/EC/TDS Testers, Hanna Instruments Inc.) for the BRed dye biosorption (highest adsorption capacities were preliminarily obtained at pH 3) [10,11]. The working temperature (kept constant) was that of the room environment, around 25 °C, and the contact time of the phases was varied by no more than 10 h. The remaining dye concentration in the aqueous solution was determined using the spectrophotometer-based method at the maximum dye wavelength of 530 nm, using the MeterTech SP-830 Plus spectrophotometer (MeterTech Inc., version 1.06). The values of dye removal (R, or Y, (%)), or biosorption efficiency (q, (mg/g)) were calculated with the Equations (1a) and (1b):

$$R \left(or Y\right)=\frac{C0-Ct}{C0}\cdot 100$$

(1a)

$$q=\frac{C0-C}{G}\cdot V$$

(1b)

where C₀ and C_t—the initial and remaining dye (at t biosorption time) concentration in the aqueous solution (mg/L), G—the biosorbent amount (g) and V—the aqueous solution volume (L).

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 (Z₁, (g/L)), BRed dye concentration (Z₂, (mg/L)) and biosorption contact time (Z₃, (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 2³ 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 = b₀ + ∑b_ix_i + ∑b_iix_i² + ∑b_ijx_ix_j

(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 b₀, 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 (Zi₀) and imposed variation step (ΔZ_i0) corresponding to the coded variable values (X_i = 0, ±1, or α = ±1.682). The Fisher constant (F), multiple correlation coefficient (R_Yx1x2x3), 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_Yx1x2x3 to establish the correlation between the dependent variable (Y) and the three independent variables (Xi) 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 (ν₁ = n − 1 = 19 and ν₂= 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* = f(X₁*,X₂*,X₃*). 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 X₁* − z₁*, X₂* − z₂*, X₃* − z₃*. Graphical representations (3D surface and 2D contour) of response function (Y) variation vs. two or one independent variables (X_i) (all other variables considered at their basic values) were illustrated using WinSurf and Excel programs.

3. Results

3.1. Biosorption Process Performance

The principal influencing factors of reactive Brilliant Red-HEB 3B dye biosorption onto residual biomass of Saccharomyces pastorianus immobilized in sodium alginate by using a Buchi micro-encapsulator (with biosorbent 1) and/or by a simple dropping technique (with biosorbent 2) were studied and reported related to the significant dye removal efficiencies and highest dye biosorption capacities (i.e., q > 80–220 mg of dye/g biosorbent in the case of simple dropping technique, or 312.5–2500 mg dye/g biosorbent in the case of encapsulation with Buchi micro-encapsulator) [10,11]. For the best experimental biosorption results, the operating conditions that must be considered are as follows: pH 3, temperature of 25–30 °C with a biosorbent concentration of minimum 2.60 g/L or 3.2 g/L (with 5% d.w.) according to the diameter of the biosorbent granules (in this case, 4 mm for biosorbent granules obtained by a simple dropping technique, and 1500 μm for biosorbent granules obtained by using a Buchi micro-encapsulator), and an S/L phases contact time for biosorption of at least 7.4 h or 12 h at a dye concentration in the aqueous solution of 10.64–170.24 mg/L and 16.88–174.08 mg/L, respectively [10,11]. All biosorption experiments were performed according to the experimental planning matrix (of the active central compositional rotatable design of 2³ order).

In our previous papers, it was concluded that the mechanism of biosorption process, referring the biosorption energy value (E) (8.28–11.23 KJ/mol) (biosorbent 1) [11] and (8.45–13.61 KJ/mol) (biosorbent 2) [10], is based on physical bonding established between the positively charged surface of the biosorbent and the functional groups of reactive Red Brilliant HE-3B dye.

A few new experimental data of studied dye biosorption in a static working regime were selected in Figure 2 after the S/L phases contact period of 24 h (biosorption equilibrium attained) but the reference experimental data were reported in our previous works, in which it was demonstrated by advanced analytical analysis (SEM, FTIR-EDX) that the dye was retained in the molecular structure of both biosorbents [10,11]. In the present modeling and optimization study, the dye biosorption was evaluated at pH 3 for a period of no more than 10 h, referring to the biosorption capacity and efficiency in retaining of reactive BRed dye onto residual immobilized biomass.

High BRed dye biosorption capacities were performed for dye concentrations more than 75 mg/L, i.e., >130.56 mg/L for biosorbent 1 (3.2 g/L of biosorbent) and >127.68 mg/L for biosorbent 2 (2.60 g/L of biosorbent). The highest dye biosorption capacity was performed at the highest dye concentration of around 130 mg/L of BRed dye (Figure 2a), i.e., a dye biosorption capacity of around 83.275 mg/g for biosorbent 1 and around 213.870 mg/g for biosorbent 2.

As shown in Figure 2b, for the initial dye concentration of 31.95 mg/L, the dye biosorption capacity varied in the range of around 29.23–33.06 mg/g working with biosorbent 1 and 19.505–35.313 mg/g working with biosorbent 2, considering the biosorbent amount of 0.00285–0.00915 g d.w. (d.w.—dry mass of biosorbent per 25 mL of dye solution), corresponding to a biosorbent concentration of 2.4–3.2 g/L.

Better results were performed for higher reactive BRed dye concentration in the aqueous system (>45–50 mg/L) working with the biosorbent concentration of more than 12 g/L in the same operating conditions of pH 3 and T = 20–30 °C (causing the residual biomass degradation to be easily controllable).

3.2. 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. Codification of independent variables in the active central compositional rotatable design of 2³ order for reactive BRed dye biosorption onto residual biomass (encapsulated/immobilized type).

Variable/Value	Real Variable (Zi)	Coded Variable (Xi)	Real Basic Variable (Zi0)	Variation Step (∆Zi0)
Biosorbent 1: Residual yeast biomass prepared by using a Buchi microencapsulator
Dye concentration, (mg/L)	Z1	X1	50	20
Residual encapsulated biomass, (g/L)	Z2	X2	12	6
Biosorption (S/L) contact time, (h)	Z3	X3	6	2
Biosorbent 2: Residual biomass immobilized in sodium alginate by simple dropping technique
Residual immobilized biomass, (g/ L)	Z1	X1	12	6
Dye concentration, (mg/L)	Z2	X2	50	20
Biosorption (S/L) contact time, (h)	Z3	X3	6	2

, in association with their variation steps (∆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 °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) (Y₁) and residual immobilized yeast biomass (biosorbent 2) (Y₂) are as follows:

Y₁ = 27.3415 + 7.008X₁ + 8.5806X₃ − 1.4085X₁² + 1.0799X₂² + 0.6971X₃² − 0.7284X₁X₂ + 2.7969X₁X₃ +
1.8257X₂X₃

(3)

Y₂ = 60.6002 + 9.3796X₁ + 5.6881X₂ + 10.1079X₃ − 3.3381X₁² − 6.461X₂² − 4.5396X₃² + 1.3353X₁X₃ − 4.6628X₂X₃

(4)

without X₂ term (specifically, the term of ‘+0.0772X₂’) in relation (3) and without X₁X₂ 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. Experimental planning matrix in the active central compositional rotatable design of 2³ order for static BRed dye biosorption onto biosorbent 1 (residual yeast biomass encapsulated in sodium alginate).

Exp. No.	Z1, (mg/L)	Z2, (g/L)	Z3, (h)	X1	X2	X3	Yei (%)	Yci (%)	Deviation (A)
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

for BRed dye biosorption onto biosorbent 1 and

Table 3. Experimental planning matrix in the active central compositional rotatable design of 2³ order for static BRed dye biosorption onto biosorbent 2 (residual yeast biomass immobilized in sodium alginate).

Exp. No.	Z1, (g/L)	Z2, (mg/L)	Z3, (h)	X1	X2	X3	Yei (%)	Yci (%)	Deviation (A)
1	6	30	4	−1	−1	−1	13.6243	17.9134	−0.3148
2	18	30	4	1	−1	−1	34.7884	33.6922	0.0315
3	6	70	4	−1	1	−1	33.3531	38.3054	−0.1485
4	18	70	4	1	1	−1	48.4867	54.7038	−0.1282
5	6	30	8	−1	−1	1	52.6316	44.7842	0.1491
6	18	30	8	1	−1	1	72.4868	65.9042	0.0908
7	6	70	8	−1	1	1	47.0588	46.5250	0.0113
8	18	70	8	1	1	1	74.1839	68.2646	0.0798
9	1.908	50	6	−1.682	0	0	36.6774	35.3798	0.0354
10	22.092	50	6	1.682	0	0	63.3226	66.9328	−0.0570
11	12	16.36	6	0	−1.682	0	34.6505	41.5375	−0.1988
12	12	83.64	6	0	1.682	0	51.9091	49.6046	0.0444
13	12	50	2.64	0	0	−1.682	44.9438	30.7559	0.3157
14	12	50	9.36	0	0	1.682	69.6629	65.8538	0.0547
15	12	50	6	0	0	0	56.7416	60.6002	−0.0680
16	12	50	6	0	0	0	55.3772	60.6002	−0.0943
17	0.60	50	6	0	0	0	56.8218	60.6002	−0.0665
18	0.60	50	6	0	0	0	58.1059	60.6002	−0.0429
19	0.60	50	6	0	0	0	54.8957	60.6002	−0.1039
20	0.60	50	6	0	0	0	56.5008	60.6002	−0.0726

for Bred dye biosorption onto biosorbent 2.

Mathematical model analysis. The value of Fisher constant is found to be F = 9269 for Y₁ (BRed dye removal onto biosorbent 1) and, respectively, F = 2438 for Y₂ (Bred dye removal onto biosorbent 2), and the statistic value (from table) is F_tab = 4.60 (for α = 99, ν₁ = n − 1 = 19, ν₂ = b − 1 = 2, where n is the number of experiments (n = 20), and b is the number of independent variables (b = 3)). This is because both values of F > F_tab, the deviation of experimental data (Y_ei) referring to the average experimental value ($\overline{Y_{ei}}$), 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 Y₁ (BRed dye removal onto biosorbent 1) and R_YX1X2X3 = 0.9232 for Y₂ (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_C = 31.92545 for Y₁ (BRed dye removal onto biosorbent 1) and F_C = 30.7766 for Y₂ (BRed dye removal onto biosorbent 2) which was much higher than the statistical value (from table) of F_C,tab = 6.59 for the freedom degree of ν = n-k − 1 = 16 and ν = k=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 Y₁ and 25.5418 for Y₂, and the coefficients dispersion was 0.2027 for Y₁ and 0.0652 for Y₂.

The processed data indicate an acceptable accordance between the experimental and model-calculated data, the mean/average deviation being of −2.081% for Y₁ and −2.414% for Y₂, both in the admissible limit range of −10% ÷ +10%. From the experimental planning matrices (Table 2 and Table 3), it seems that for the selected experimental variation field of all independent variables exists a local maximum value (Y* = 52.878%) for Y₁ and a local maximum (Y* = 74.184%) for Y₂, corresponding to X₁* = +1.0, X₂* = +1.0 and X₃* = +1.0 for both functions (Y₁/Y₂), and to a residual encapsulated yeast biomass concentration of 18 g/L (0.45 g per 25 mL), and, respectively, a residual immobilized yeast-based biomass concentration of 18 g/L (0.45 g per 25 mL), a BRed dye concentration of 70 mg/L, and a S/W contact time for biosorption of 8 h.

The analysis of the response functions (Y₁ and Y₂) indicate that all X_i independent variables (residual encapsulated or immobilized yeast biomass concentration, dye concentration and biosorption contact time of S/L phases) have an important influence on the reactive Bred dye removal efficiency, a fact which is demonstrated by the values of X₁, X₂ and X₃ coefficients much higher than unity, except in the case of biosorbent 1 (residual yeasts encapsulated biomass) for which the BRed dye concentration (X₂) had an insignificant effect in the experimental X₂ variation field (19–170 mgL) toward dye biosorption efficiency.

The influence of the X₃ variable (S/L phases contact time for biosorption) in Y₁ response function is quite similar to that of X₁ (biosorbent concentration) (i.e., X₃ influence is 1.2244 times higher than that of X₁) for the BRed dye removal (Y₁ decision function), and its effects are similar (e.g., the biosorbent concentration (X₁) increasing and the S/L contact time (X₃) increasing in the biosorption process increases the dye removal efficiency, Y₁).

The influence of X₃ variable (S/L phases contact time for biosorption) in Y₂ response function is almost two times (exactly ≅ 1.78) higher than that of X₂ (BRed reactive dye concentration), and quite similar to that of X₁ (biosorbent concentration) (i.e., X₃ influence is of 1.078 times higher than of X₁) for the BRed dye removal (Y₂ optimization criterion, or response/decision function), and its effects are similar (e.g., the biosorbent concentration (X₁) increasing and the S/L contact time (X₃) increasing in the biosorption process increases the dye removal efficiency, Y₂), except for X₂ and X₃ which had the opposite effects (negative sign), meaning that the biosorption time increasing (X₃) and decrease in the dye concentration (X₂) increases the dye removal degree (Y₂). Thus, it was recommended to work in the biosorption with a dye concentration in range of 45–50 mg/L.

3.3. Optimization of the Proposed Mathematical Models

The application of classical optimization method for finding the maximum of the response function/proposed optimization criterion, i.e., Y* = f(X₁*,X₂*,X₃*), consists of the solving of specific equations’ systems with first- and second-order derivates equalized with zero.

The result for Y₁ response/decision function optimization leads us to the conclusion that the Y₁* response function (maximum dye removal by biosorption onto biosorbent 1) has no distinct maximum and only a local maximum experimentally found as corresponding to X₁* = +1.00, X₂* = +1.00 and X₃* = +1.00, meaning a dye removal of 52.878%, working at least 8 h with 70 mg/L BRed dye and 18 g/L of biosorbent 1.

In the case of Y₂ function optimization, the result leads to the conclusion that the Y₂* function (maximum dye removal by biosorption onto biosorbent 2) has a distinct maximum (Y = 75.3375%), corresponding to the following values of three independent values: X₁* = +1.6849, X₂* = −0.0755 and X₃* = +1.3999. Transposed to real values, these independent variables for maximum Y₂* function correspond to a biosorbent 2 concentration of 22.1094 g/L (0.553 g per 25 mL), a BRed dye concentration of 48.49 mg/L, and a S/L contact time for biosorption of at least 8.80 h.

4. Discussions

Figure 3a–f present the influence of two independent variables (one independent variable kept at the basic value) vs. reactive Bred dye removal (Y₁) (i.e., Y = Y₁(X₁,X₂,0), Y = Y₁(X₁,0,X₃) and Y = Y₁(0,X₂,X₃)).

Figure 4a–f illustrate the dependence of reactive dye removal (Y₂) vs. two independent variables (one independent variable kept at the basic value) (i.e., Y = Y₂(X₁,X₂,0), Y = Y₂(X₁,0,X₃) and Y = Y₂(0,X₂,X₃)).

The graphical representations of the BRed dye removal (Y_i, %) variation in the experimental field of two independent variables (on isolines) in Figure 3 for biosorbent 1 and Figure 4 for biosorbent 2 indicate the following aspects:

(1)

The biosorbent concentration (X₁) had one of the highest levels of significance in obtaining a high dye removal degree, with higher values recommended than the basic (X₁ > 0) where the local maximum dye removal was no more than 23% for Y₁ (Figure 3c,d) and 65% for Y₂ (Figure 4a,b);

(2)

High local dye removal (Y) close to the value of 52% for Y₁ function and 80% for Y₂ function can be performed for an initial dye concentration close to 50 mg/L in the aqueous system studied (Figure 3a,b and Figure 4c,d);

(3)

A biosorption time higher than 8 h (as a possible exchange period at work) is beneficial but at least 6 h (basic value of X₃) are obligatory for dye removals of at least 35% for Y₁ (or > 52.878% experimentally performed, Figure 3e,f) and 50% for Y_2, or much more than the obtained experimental value of 72.4868% (Figure 4e,f);

(4)

If the biosorption operating regime is discontinuous, higher values of the dye removal degree will be performed after a S/L contact time greater than 20 h (as found in a previous authors’ report after 24 h) [10,11].

Other aspects can be underlined when two independent variables are constant in the biosorption process, especially at their basic values (X_i,j = 0). In this context, the increase in BRed reactive dye removal percentage (dye biosorption efficiency) can vary with the increase in each X_i independent value, as follows:

-

for Y₁ function (BRed dye biosorption efficiency onto biosorbent 1): the influence of initial dye concentration (X₁) vs. Y₁ is illustrated in Figure 5a. It seems that it is no distinct maximum but only an extreme local maximum for BRed dye removal (Y₁* = 35.724%) at X₁*= +2.00 corresponding to a BRed dye concentration of 100.00 mg/L, for a biosorbent 1 concentration of 12 g/L and t_biosorption of 6 h (X₂ and X₃ kept constant at their basic values). The dependence of BRed dye removal (Y₁) vs. X₂ (biosorbent 1 concentration) (X₁ and X₃ coded values are 0) indicates no distinct minimum of dye removal degree; thus, an extreme local maximum can be considered (Y = 31.661%) at X₂* = +2.00, corresponding to a biosorbent concentration of 24 g/L (0.60 g per 25 mL), a dye concentration of 50.00 mg/L and t_sorption of 6 h (Figure 5b). The dependence of reactive BRed dye removal (Y₁) vs. X₃ (biosorption time) (X₁ and X₂ coded values are 0) indicates no distinct maximum dye removal degree, thus only an extreme local maximum can be considered (Y = 47.291%) for X₃*= +2.00 (i.e., a local maximum point after a S/W contact time in biosorption step of 12 h, a dye concentration of 50 mg/L and a biosorbent concentration of 12 g/L (i.e., around 0.3 g per 25 mL) (Figure 5c). The X₃ variable (biosorption time) has the most significant influence on the maximum value Y₁.

-

for Y₂ function (BRed dye biosorption onto biosorbent 2): the influence of biosorbent 2 concentration (X₁) vs. Y₂ is evidenced in Figure 6a. It seems that there exists a distinct maximum of the BRed dye removal (Y₂* = 67.189%) at X₁* = +1.4048 corresponding to a biosorbent 2 concentration of 20.4288 g/L (0.5107 g per 25 mL), for a BRed dye concentration of 50 mg/L and t_biosorption of 6 h (X₂ and X₃ kept constant at their basic values). The dependence of BRed dye removal (Y₂) vs. X₂ (dye concentration) (X₁ and X₃ coded values are 0) indicates a distinct maximum dye removal degree (Y = 61.852%) at X₂* = +0.4402, corresponding to a biosorbent concentration of 14.6412 g/L (0.366 g per 25 mL), a dye concentration of 50.00 mg/L and t_sorption of 6 h (Figure 6b). The dependence of reactive BRed dye removal (Y₂) vs. X₃ (biosorption time) (X₁ and X₂ coded values are 0) indicates a distinct maximum dye removal degree (Y = 66.207%) for X₃*= +1.1133 (i.e., a maximum point after a S/W contact time in biosorption step of 8.226 h, a dye concentration of 50 mg/L and a biosorbent concentration of 12 g/L (i.e., around 0.30 g per 25 mL) (Figure 6c).

Considering two independent variables at their basic values, we can conclude that the optimum values for the dye removal performance (Y*) by biosorption onto residual micro-encapsulated (biosorbent 1) or immobilized yeasts (biosorbent 2) biomass corresponded to a relatively high quantity of biosorbent (more than 14.64–24.00 (g/L)) and thus high dye removal efficiency can be performed working with a relatively high biosorbent concentration (i.e., >20 g/L, or >0.50 g per 25 mL of sample), a dye concentration around of 45–50.00 mg/L or more, and a biosorption contact time of at least 8 h (or ca. 12 h).

The application of other direct empirical optimization methodologies (i.e., univariant search or gradient methods) [13,14,15,16,17,18,19,20] for the maximum findings in the case of the proposed mathematical model leads to optimum values close to that found by the application of the classical optimization method for the proposed mathematical model, i.e., Y₁* = 52.878% or Y₂* = 75.338%.

The graphical representations (Figure 3, Figure 4, Figure 5 and Figure 6) allowed us to visualize the 3D and 2D variation of the selected response/decision function and its maximal points for the selected experimental variation field on isolines (2D), and also when some of the independent variables were constantly kept at their basic values (3D and/or 2D). The WinSurf and Excel data processing programs were used.

5. Conclusions

The biosorption onto two new prepared biosorbents based on residual biomass of Saccharomyces pastorianus micro-encapsulated (biosorbent 1) or immobilized (biosorbent 2) in sodium alginate can be applied with good efficiencies in the reactive BRed dye removal from aqueous systems.

Two empirical models were proposed by an active central compositional rotatable design of 2³ order, considering the biosorbent concentration (X₁), dye concentration (X₂) and biosorption time (X₃) as independent variables and the Bred dye removal as decision/response function, or optimization criterion (Y₁ and Y₂, %). The maximum values of response/decision function (Y₁ and Y₂) considered as optimization criterion were determined by using the classical optimization methodology and appreciated, in association with the significance and importance of each variable (i.e., Fisher constant (F), test (F_c), multiple correlation coefficient (R_Yx1x2x3), Student t-test, experimental and coefficients’ dispersion, among others).

The Y₁ decision function (for biosorbent 1) was found to have only a local maximum of 52.878%, working with 70 mg/L BRed dye, 18 g/L of residual yeasts (Saccharomyces pasturianus) biomass micro-encapsulated in sodium alginate and after at least 8 h. A distinct maximum for Y₂ function (with biosorbent 2) was found to correspond to an immobilized biosorbent concentration of 22.109 g/L (0.553 g per 25 mL), a dye concentration of 48.49 mg/L and biosorption time of 8.799 h for a maximum Bred dye removal of 75.338%.

The graphical representation of the dye removal variation vs. one or two selected independent variables permits the localization of the optimal variation field of each studied variable. The maximum solutions are encouraging (BRed reactive dye removal > 52% onto biosorbent 1 and >75% onto biosorbent 2), and thus the continuity of this modeling and optimization study is recommended when the utilization of other improved, empirical and programmed (controlled) modeling and optimization methods is possible.