## 2 Experimental Section
### Modeling of Experimental Bisosportion Data in Dynamic Regime  
In order to evaluate some characteristic biosorption parameters in a dynamic biosorption system, the experimental data were processed using two of the most well-known models from the scientific literature, i.e., the [PERSON] and Yoon-Nelson kinetic models [27; 32; 33; 34]. These selected models were applied for the investigated biosorption system; the removal BRed dye using a fixed/packed bed of immobilized residual biomass operating in a continuous dynamic regime.  
One of the most used sorption models in packed-bed column studies is considered to be the [PERSON] model, which assumes the Langmuir kinetics of biosorption-desorption without axial dispersion, and the hypothesis of the rate driving force corresponding to second-order reversible reaction kinetics [28]. The [PERSON] model does not rely on resistance to internal and external mass transfer, and does not consider the axial dispersion phenomenon [35], and is characterized as in Equation (4):  
\[ln(\frac{C_{0}}{C_{t}}-1)=\frac{k_{T}\cdot q_{0(T)}\cdot m}{F_{V}}-\frac{K_{T} \cdot C_{0}}{F_{V}}\cdot V_{t} \tag{4}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & **Internal** & & **Height of** \\
**Biosorbent** & **Biosorbent** & **Diameter of** & **Height of** & **Biosorption** \\
**Mass, [kg]** & **Volume, [m\({}^{3}\)]** & **Biosorption** & **Packed Bed, [m]** & **(Column)** \\  & & **Reactor, [m]** & & **[m]** \\ \hline \(M_{ads}=\frac{F_{vi}\cdot E_{vi}}{V_{opt}}\) & \(V_{ads}=\frac{M_{ads}}{\rho_{ads}}\) & \(D_{R}=\sqrt[3]{\frac{4\cdot M_{ads}}{\rho_{R}+\rho_{ads}}}\) & \(H_{ads}=\beta\cdot D_{R}\) & \(H_{R}=\lambda\cdot H_{ads}\) \\  & & & & \(\beta\) = 1–3 & (\(\lambda\) = 1–3) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Relations used in the experimental design of the biosorption (column) reactor.  
where \(C_{0}\) and \(C_{t}\) are the dye concentration at the initial moment (\(t=0\)) and time \(t\) (mg/L); \(k_{T}\) is the Thomas constant (L/min:mg); \(F_{v}\) is the effluent flow rate (L/min); \(q_{0(T)}\) is the maximum adsorptive capacity (mg/g); \(m\) is the weighted adsorbent mass (g) and \(V_{t}\) is the treated effluent volume at time \(t\) (L). The linear model corresponds to the graphical representation of \(ln(C_{0}/C_{t}-1)\) versus \(V_{t}\).  
A simpler kinetic model was proposed by [PERSON] [32]. This model assumed that the dye amount adsorbed in a packed bed is half of the total initial dye amount passing through the packed bed a within \(t_{1/2}\) period [36], and the decrease in the adsorption rate for each adsorbed species is proportional to the probability of adsorbate breakthrough onto the adsorbent bed [37], according to Equation (5):  
\[ln(\frac{C_{t}}{C_{0}-C_{t}})=k_{YN}\cdot t-t_{1/2}\cdot k_{YN} \tag{5}\]  
where \(k_{YN}\) is the Yoon-Nelson rate constant (min\({}^{-1}\)); \(t_{1/2}\) is the time required for 50% dye breakthrough; and \(t\) is the biosorption time (min). The linear model corresponds to the graphical representation of \(ln[C_{t}/(C_{0}-C_{t})]\) versus \(t\).  
The specific biosorption parameters of these two models are determined from the slope and intercepts of the characteristic graphical representations of their linearized forms, considering the specific theoretical assumptions for each one. For comparison of the experimental and calculated data with the proposed dynamic models, linear regression coefficients (\(R^{2}\)) were used and their values must be close to unity for the best data accordance/agreement.