# Scale-up and Economic Assessment of Biofunctional Particles Synthesis for Bilirubin Removal
## 3 Results and Discussion  
### Pilot Plant Scale-up  
The first step of the scale-up process is the determination of the shape factors whose value should be constant to assure the geometrical similarity [17]. In the case of CSTRs, the shape factors are S\({}_{1}\) Equation (1), S\({}_{2}\) Equation (2), and S\({}_{3}\) Equation (3) [16]:  
\[\mathrm{S}_{1}=\frac{\mathrm{D}}{\mathrm{d}_{1}} \tag{1}\]  
\[\mathrm{S}_{2}=\frac{\mathrm{h}}{\mathrm{d}_{1}} \tag{2}\]  
\[\mathrm{S}_{3}=\frac{\mathrm{H}}{\mathrm{d}_{1}} \tag{3}\]  
where d\({}_{1}\) refers to the stirrer diameter; D, the reactor diameter; h, the space between the bottom of the tank and the stirrer, and H, the filling reactor level. The reactor and stirrer diameters are shown in Table 2. The distance between the bottom and the stirrer was 44.3 and 70.0 mm for 2 and 10 L, respectively. The filling reactor level was determined considering Equations (4) and (5), giving a value of 111 and 224 mm, for 2 and 10 L, respectively. \[\mathrm{V}=\frac{\mathrm{\pi}}{4}\mathrm{\cdot}\mathrm{D}^{2}\mathrm{\cdot} \mathrm{L} \tag{4}\]  
\[\mathrm{Vreaction}=\frac{\mathrm{mass\ reaction}}{\mathrm{reaction\ density}} \tag{5}\]  
where D represents the tank diameter and L the liquid height, and the reaction density being the average density of aqueous and organic phase (0.98 g cm\({}^{-3}\)). Table 3 summarizes the shape factors of laboratory and pilot plan scale. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Shape Factor** & **2 L Reactor** & **10 L Reactor** \\ \hline S\({}_{1}\) & 3 & 3 \\ S\({}_{2}\) & 1 & 1 \\ S\({}_{3}\) & 2.5 & 3.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Shape factors of laboratory and pilot plant scales. Figure 2: Stirrer design parameters. D: reactor diameter, d\({}_{1}\): impeller diameter, d\({}_{2}\): central disk diameter, W: blade with, L: blade length, H: filling reactor level, h: distance between the tank bottom and the impeller. As can be seen in Table 3, the shape factors are in good agreement with the typical values for turbulent flow regimes for both reactors. These typical values are 3 for S\({}_{1}\); between 0.75 and 1.3 for S\({}_{2}\), and between 2.7 and 3.9 for S\({}_{3}\). Just S\({}_{3}\) of lab scale is slightly smaller than the typical values which were corrected for the pilot plant reactor, whose S\({}_{3}\) is in the typical range. S\({}_{1}\) and S\({}_{2}\) were kept at exactly the same value for the pilot plant reactor. Thus, an acceptable geometrical similarity was reached for the laboratory and pilot scales reactors. Next, the constant Re number was used as partial criterion for the scale up. This criterion is expressed by Equation (6), which allowed to determine the stirring rate in the pilot plant reactor from that applied in the laboratory one and the stirrer diameters [18]. \[\mathrm{N}_{1}\cdot\mathrm{d}_{\mathrm{a,1}}^{2}=\mathrm{N}_{2}\cdot\mathrm{d }_{\mathrm{a,2}}^{2} \tag{6}\]  
where N\({}_{1}\) is the stirring speed of the 2 L reactor; d\({}_{\mathrm{a,1}}\), the stirrer diameter used in the 2 L reactor; N\({}_{2}\), the stirring speed of the 10 L reactor, and d\({}_{\mathrm{a,2}}\), the stirrer diameter of the 10 L reactor. The optimized stirring rate for the St-MMA-GMA-PEGMA particles at laboratory scale (2 L reactor) was 525 rpms since this allowed to obtain the most suitable particle size (304 \(\upmu\)m). Considering this value and Equation (6), the stirring rate for the 10 L reactor was fixed at 210 rpm. To check the suitability of the criterion of constant Re for the scale-up, a reaction in the 10 L reactor with a stirring speed of 210 rpm was carried out and the results were compared to those from the particles obtained in the 2 L reactor. The particle size is a key factor in the application of the developed particles since it is necessary to reach a compromise between the improvement of albumin immobilization, which usually increases as the particles decrease, and the increase in the pressure drop when small particle size is used.