## 2 Materials and Methods
### Mathematical Model
A pH of 2.2 was optimal for fungus growth, as well as citric acid production, while a higher pH, around 5.4 to 6.5, is optimal for the production of citric acid in molasses medium [27]. In a study of citric acid production from beet molasses in continuous culture, [PERSON] and [PERSON] found that, at a pH value of 2.5, gluconic acid is the main product and citric acid is the main product product at low pH [28]. #### 3.1.1 Estimated Effects

All statistical calculations were generated using the STATGRAPHICS Centurion XVIII software to analyze the estimated effects, the \(t\)-test (Pareto chart), the regression coefficient, and the optimal values of the factors. To analyze data in the STATGRAPHICS software it is necessary to provide quantitative factors (parameters or variables). When there is at least one qualitative factor, it is required to be encoded by a number: \(+1.0\) or \(-1.0\). These values correspond to the levels, that is, the range in which the factor is evaluated (upper and lower); thus \(+1.0\): upper level and \(-1.0\): lower level. In the case of this study, the qualitative variables were N source, K source, traces, and agitation. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**Experim-ment**}} & \multicolumn{6}{c}{**Factors**} & \multicolumn{6}{c}{**Response Variables**} \\ \cline{2-13}  & **N Source** & **C Source** & **K Source** & **Trace** & **Ele-** & **Aigation** & **pH** & **A1** & **A2** & **A3** & **A4** & **\(\lambda_{1}\)** & \(\lambda_{2}\)** \\  & **(FN)** & **(FC)** & **(FK)** & **(FFK)** & **melents** & **Origin** & **(g/L)** & **(g/L)** & **(g/L)** & **(g/L)** & **(g/L)** & **(g/L)** & **(100)** & **1456** \\ \hline \(1\) & Urea (0.75 g/L) & Sucrose (120 g/L) & K\({}_{2}\)HPO\({}_{4}\) (0.128 g/L) & No & Mechanical & 3.192 & 0.222 & 0.252 & 6.503 & 0.034 & 0.095 & 0.0720 \\ \(2\) & (NH\({}_{4}\))\({}_{2}\)\({}_{3}\)\({}_{5}\)\({}_{6}\)\({}_{1}\) (0.428 g/L) & Sucrose (40 g/L) & K\({}_{2}\)HPO\({}_{4}\) (0.8 g/L) & No & Mechanical & 2.820 & 0.009 & 0.291 & 0.964 & 0.011 & 0.066 & 0.0630 \\ \(3\) & (NH\({}_{4}\))\({}_{2}\)\({}_{5}\)\({}_{6}\)\({}_{1}\) (0.428 g/L) & Sucrose (120 g/L) & K\({}_{2}\)HPO\({}_{4}\) (0.8 g/L) & Yes & Mechanical & 2.942 & 0.002 & 0.101 & 5.627 & 0.024 & 0.098 & 0.0686 \\ \(4\) & Urea (0.75 g/L) & Sucrose (40 g/L) & K\({}_{2}\)HPO\({}_{4}\) (0.8 g/L) & Yes & \begin{tabular}{c} mixing \\ \end{tabular} & 3.119 & 0.002 & 0.019 & 3.470 & 0.102 & 0.053 & 0.0599 \\ \(5\) & (NH\({}_{4}\))\({}_{2}\)\({}_{5}\)\({}_{6}\)\({}_{1}\) (0.428 g/L) & Sucrose (120 g/L) & K\({}_{2}\)HPO\({}_{4}\) (0.128 g/L) & Yes & \begin{tabular}{c} mixing \\ \end{tabular} & 3.252 & 0.006 & 0.107 & 6.091 & 0.131 & 0.096 & 0.0683 \\ \(6\) & Urea (0.75 g/L) & Sucrose (40 g/L) & K\({}_{2}\)HPO\({}_{4}\) (0.128 g/L) & Yes & Mechanical & 3.152 & 0.007 & 0.784 & 0.972 & 0.030 & 0.063 & 0.0614 \\ \(7\) & Urea (0.75 g/L) & Sucrose (120 g/L) & K\({}_{2}\)HPO\({}_{4}\) (0.8 g/L) & No & \
\begin{tabular}{c} mixing \\ \end{tabular} & 2.771 & 0.015 & 0.428 & 3.446 & 0.036 & 0.066 & 0.0672 \\ \(8\) & (NH\({}_{4}\))\({}_{2}\)\({}_{5}\)\({}_{6}\)\({}_{1}\) (0.428 g/L) & Sucrose (40 g/L) & K\({}_{2}\)HPO\({}_{4}\) (0.128 g/L) & No &
\begin{tabular}{c} mixing \\ \end{tabular} & 2.472 & 0.007 & 0.140 & 1.177 & 0.117 & 0.068 & 0.0638 \\ \hline \hline \end{tabular} Note: in all the tests, the presence of proteins was evidenced. FN: N Source; FC: C Source; FK: K Source.

\end{table} 
Table 2: Laboratory scale \(2^{5-2}\) fractional factorial design matrix with results. A1 (Oxalic acid), A2 (Citric acid), A3 (Malic acid), A4 (Acetic acid). The effect of a factor is defined as the change in response produced by a change in its level. This also refers to the main effect because it refers to the primary factors of interest in the experiment.