## 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/volarization 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 2\({}^{3}\) 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.