## 2 Materials and Methods
### Mathematical Model
Furthermore, it is a challenge to study the interactions of more than five variables, and large variations in the factors can give misleading results, possibly due to errors, biases, or lack of reproducibility [2,20]. Therefore, similarity models are required, which can be obtained with mathematical analysis tools, such as [PERSON]'s Pi theorem, to scale-up. This theorem is applied when there is little knowledge of the system's behavior, but the variables that are related to it are known. The grouping in dimensionless packages simplifies the experimental runs. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Factor**} & \multicolumn{2}{c}{**Factor Bioleaching of Gold Using**} & \multicolumn{2}{c}{**Factor Bioleaching of Gold Using**} \\  & **the Infrared Spectrum at 1060 cm\({}^{-1}\)** & **the Infrared Spectrum at 1456 cm\({}^{-1}\)** \\  & **(R\({}^{2}\) = 0.9895)** & **(R\({}^{2}\) = 0.8931)** \\ \hline  & **Coded** & **Level** & **Coded** & **Level** \\ \hline FN (g/L) & 0.96 & Urea & 1.0 & Urea \\ FC (g/L) & 0.97 & 120 & 0.97 & 120 \\ FK (g/L) & \(-\)1.0 & K\({}_{2}\)HPO\({}_{4}\) & \(-\)1.0 & K\({}_{2}\)HPO\({}_{4}\) \\ Traces & 1.0 & Yes & \(-\)0.99 & No \\ Agitation & 0.92 & Mechanical & 0.98 & Mechanical \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Optimal values of the studied factors, according to the \(2^{5-2}\) fractional factorial design of experiments. ### Dimensional Analysis Using the Buckingham Pi Theorem

In this work, [PERSON]'s Pi theorem was applied because it allows defining correlations between variables of a specific physical problem through its dimensional analysis. In this way, a physical law can be reformulated that interconnects some physical quantities as a function of dimensionless numbers. The objective of the Pi theorem is to define significant non-dimensional parameters, which can be representative of the problem considered and can be used to scale the variables involved on specific physical assumptions [30,31]. [PERSON] et al. [16] applied [PERSON]'s Pi theorem in a biological system concerning a microbial fuel cell (MFC) because they are complex and involve numerous variables. They also require countless data collection and calculations. Repeating the experiment is expensive, time-consuming, and labor intensive, with misleading results. In this study, the composition of the fermented broth was correlated in terms of carbon source (\(C\)), nitrogen (\(N\)), potassium (\(K\)), traces (\(T\)), the stirring speed, and the airflow rate. The oxygen was included in the correlation according to the findings of the design of experiments. The models based on the K\({}_{\text{L}}\)a coefficient show that the airflow and agitation speed have a statistically representative influence on biotechnological processes, for example, to produce fermented broths to be used in bioleaching stages [32,33]. From the point of view of dimensional analysis, this engineering problem refers to a system of three-dimensional parameters: [\(M\), \(L\), \(T\)], as presented in Table 5.

The seven parameters (n), shown in Table 5, were represented with three dimensions (m) to obtain n \(-\) m = 7 \(-\) 3 = 4 mutually independent dimensionless parameters. For the formulation of these parameters, the following variables were selected as repeated variables: \(C\), \(Q\), and \(A\), since these represent together all the fundamental dimensions present in the study. The obtaining of each dimensionless parameter is shown in Equations (2)-(11)

\[\Pi_{1}=\mathbb{C}^{t_{1}}*Q^{b_{1}}*A^{c_{1}}*\lambda_{1} \tag{2}\]

\[M^{0}*L^{0}*T^{0}=\left(\frac{M}{L^{3}}\right)^{4}*\left(\frac{L^{3}}{T} \right)^{b_{1}}*\left(\frac{1}{T}\right)^{c_{1}}*L^{1} \tag{3}\]

\[0=a_{1}\qquad a_{1}=0 \tag{4}\]

\[0=-3a_{1}+3b_{1}+1\qquad\quad b_{1}=-\frac{1}{3} \tag{5}\]

\[0=-b_{1}-c_{1}\qquad\quad c_{1}=\frac{1}{3} \tag{6}\]

\[\Pi_{1}=\frac{A^{\
icefrac{{1}}{{3}}}*\lambda_{1}}{Q^{\
icefrac{{1}}{{3}}}}= \sqrt[3]{\frac{A}{Q}}*\lambda_{1} \tag{7}\]

\[\Pi_{2}=\mathbb{C}^{2}*Q^{b_{2}}*A^{c_{2}}*N \tag{8}\]