## 2 Materials and Methods
### Mathematical Model
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model Coefficient** & **Mean Value** & **Min Value** & **Max Value** \\ \hline \(a\) & \(6.307\times 10^{-1}\) & \(5.32997\times 10^{-1}\) & \(7.28467\times 10^{-1}\) \\ \(b\) & \(-8.941\times 10^{-4}\) & \(-1.12021\times 10^{-3}\) & \(-6.68058\times 10^{-4}\) \\ \(c\) & \(-5.181\) & \(-6.06245\) & \(-4.29899\) \\ \(d\) & \(4.233\times 10^{-7}\) & \(1.64967\times 10^{-7}\) & \(6.81627\times 10^{-7}\) \\ \(f\) & \(11.07\) & \(9.14895\) & \(13.0008\) \\ \(g\) & \(4.950\times 10^{-3}\) & \(3.98539\times 10^{-3}\) & \(5.91491\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Model coefficients obtained for the mathematical model representing the functional relationship between the absorbance at 1060 cm\({}^{-1}\) with the dimensionless parameter C/N and C/K. Figure 5: Response surface of the mathematical model of absorbance as a function of the C/N and C/K ratio, frontal view. ### Bioprocess Scale-Up

In scaling bioprocesses, dimensionless parameters serve as the basis for formulating scale factors. Similarity models can be developed based on equality of the \(\pi\) parameters found or combined with other parameters used in the hydrodynamic study of systems such as the Reynolds number, Froude number, and Weber number, among others [16]. Given that, in this biotechnological process, it was decided to operate the bioreactor at a low Reynolds number to avoid significant damage to the fungal biomass and considering

Figure 6: Response surface of the mathematical model of absorbance as a function of the C/N and C/K ratio, back view. Figure 7: Contour lines of the response surface of the mathematical model of absorbance as a function of the C/N and C/K ratios. the solubility of oxygen in the culture medium used in the experimental designs, the mass transfer of oxygen into the fluid was used as the controlling phenomenon for scaling, as shown in Equations (16)-(18) [37]. \[K_{L}a_{Experimental}=K_{L}a_{Pilot} \tag{16}\]

\[K_{L}a=0.99\left(\frac{Q^{2.56}}{D^{5}}\right)^{0.778} \tag{17}\]

\[Q_{2}=Q_{1}\left(\frac{D_{2}}{D_{1}}\right)^{2.56} \tag{18}\]

where:

\(D\): Bioreator diameter (m);

\(Q\): Bioreactor Airflow (m\({}^{3}\)/s). One refers to the experimental bioreactor, and two refers to the pilot bioreactor

Equation (16) shows the criteria for bioreactor scale-up. The equality of the mass transfer of oxygen. Equation (17) represents the common correlation for mass transfer of oxygen in cylindrical bioreactors, which derives in Equation (18), as a guide to estimate the bioreactor airflow during the process scale-up. A similar conclusion was developed by [30] in the scaling of photobioreactors for the cultivation of microalgae, in which, using [PERSON]'s \(\pi\) theorem, the authors managed to develop a mathematical model of similarity between the laboratory and pilot scales. In this study, the dimensionless parameters were related directly to the agitation speed and the aeration flow through Equation (5), previously presented. ## 4 Conclusions

In this study, a methodology was proposed that combines statistical analysis by the design of experiments with [PERSON]'s Pi theorem to formulate a predictive model for bioprocess scale-up. This bioprocess, as a system, was found to be complex due to its multivariate nature, and, through a \(2^{5-2}\) fractional factorial design, the ANOVA analysis of variance showed that the interactions between the factors are highly relevant as opposed to the individual factors. Using [PERSON]'s Pi theorem, it was found that the _C/N_, _C/K_, and _T/C_ ratios are dimensionless factors that explain the potential of the fermented broth for the bioleaching process and allow an analytical basis for scaling-up. Using the Pearson Product-Moment bivariate technique, the dimensionless factors _C/N_ and _C/K_ correlate with the leaching potential of the fermented broth at 1060 cm\({}^{-1}\). A non-linear logarithmic model was formulated based on dimensionless parameters explaining the behavior of the system.