## 3 Results  
### Raceway bioreactor Scaling-Up  
The results of the dimensional analysis showed that the following magnitudes are involved: flow velocity, characteristic diameter, density, viscosity, gravitational acceleration, and surface tension. The following dimensionless numbers predominated: Reynolds (\(Re\)), Froude (\(Fr\)), and Euler (\(Eu\)). Since the model-based scaling method used involves the effects of the bioreactor geometry on the hydrodynamic behavior [23; 24], it is of utmost importance to maintain relations of gravitational, inertial, and viscous force relationships. However, it is often impossible to keep all dimensionless groups constant during scaling. Because of this, dimensional analysis helps to determine the most important groups [25]. With the application of the geometric similarity model, the dimensions of the pilot-scale bioreactor were obtained: 1.62 m long, 0.15 m deep, and 0.462 m wide (Figure 1). The Raceway is constructed of fiberglass, is provided with a stainless steel six curved blade turbine stirrer, and has a total capacity of 100 L.  
Table 1 shows the experimental data of average velocities (\(V\)) obtained in the Raceway bioreactor. Due to the nature of the system, the speed variations in the different sections are very pronounced. Therefore, the standard deviations were not considered.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\omega\) (\(\mathbf{rpm}\)) & \(V\) (\(\mathbf{m/s}\)) & \(Re\) & \(Fr\) & \(Re^{\prime}\) & \(\tau\) (\(\mathbf{Pa}\)) \\ \hline
20 & 0.2452 & 99.008 & 0.2858 & 205.251 & 0.1344 \\
25 & 0.2870 & 115.887 & 0.3345 & 256.564 & 0.1856 \\
30 & 0.3515 & 141.952 & 0.4098 & 307.877 & 0.2430 \\
35 & 0.4058 & 163.881 & 0.4731 & 359.190 & 0.3047 \\
40 & 0.4563 & 184.275 & 0.532 & 410.502 & 0.3707 \\
45 & 0.5068 & 201.669 & 0.5908 & 461.815 & 0.4408 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Hydrodynamic parameters in the laboratory Raceway bioreactor.  
Figure 1: Pilot raceway bioreactor with agitation system. (**a**) Top view layout with raceway dimensions (**b**) 100 L raceway pilot unit.  
Depending on the speed and the geometric characteristics, hydrodynamic parameters were determined, such as: \(Re\) and \(Fr\), Reynolds in the stirrer (\(Re^{\prime}\)), as well as the hydrodynamic stress value (\(\tau\)) that will influence the microalgae. Table 1 shows these values, where there are hydrodynamic stress values from 0.1344 to 0.4408 Pa that correspond to the values determined in the 10 L bioreactor.  
When scaling involving dynamic and kinematic characteristics is performed, it is recommended to determine velocity correspondences in both systems [23]. Therefore, the speed correspondence was determined in both systems to verify that the scaling-up was carried out correctly. This correspondence (\(V^{\prime}\), \(V^{\prime\prime}\)) was carried out from 30 to 45 rpm, where it was found that, in all cases, the correspondence was close to unity (Table 2).  
The velocity and correspondence values of the systems ensured that the behavior of the fluid in the pilot Raceway will be the same as the laboratory system. There are studies on medium and large-scale Raceway bioreactors, such as [PERSON] et al. [26], which addressed a 20 m\({}^{2}\) Raceway bioreactor, which maintains a varying work volume between 3000 and 4000 L and flow velocities between 0.15 and 0.25 m/s, in addition to a 500 m\({}^{2}\) industrial Raceway bioreactor that maintains a flow velocity of 0.20 m/s. The previous speeds, contrasted with Table 1, correspond to a stirring of 20 rpm. This shows that the pilot bioreactor scaled-up in this project is comparable with larger bioreactors, even industrial ones.