## 3 Results and Discussion
### Proposing New Formulas for FOS and TAC Estimation  
Since the determination of both [NaHCO\({}_{3}\)] and [HAc] depends on each other, it was found that the formulas according to the FOS/TAC method had a lower accuracy when the bicarbonate and/or VFAs concentration was in the average conditions (either low or high). Two new formulas (Equations (6) and (7)) were proposed to recalculate FOS and TAC in a more accurate way. Using the equivalent volumes at pH = 5.0 and 4.4 obtained by the simulation at different conditions (12 \(\times\) 12 = 144 datasets of [NaHCO\({}_{3}\)] and [HAc] ranging from 0 to 40 g:L\({}^{-1}\)), a multiple linear regression with Scilab was performed in search of their relationships with the true values of [NaHCO\({}_{3}\)] and [HAc]. The two new formulas with the identified model parameters are shown in Equations (8) and (9). \[\left[\text{NaHCO}_{3}\right]_{\text{new}} =1822\times\text{A}-2010\times\text{B}+8.166 \tag{8}\] \[\text{FOS}_{\text{new}} =\left[\text{HAc}\right]_{\text{new}} =-125.8\times\text{A}+3763\times\text{B}-22.05 \tag{9}\]  
Figure 4A,B shows the recalculated values, denoted [NaHCO3]\({}_{\text{new}}\) and [HAc]\({}_{\text{new}}\), using the new formulas compared to the theoretical values. In these figures, all of the points at the same bicarbonate or acetic acid level were mixed up, superimposed and, thus, not visible. It appeared that the two proposed formulas made possible the accurate determination of bicarbonate and acetic acid concentrations by equivalent volumes at pH = 5.0 and 4.4. The mean absolute percentage error (MAPE) was reduced from 1346% to 0.33% for the bicarbonate content and from 159% to 0.37% for the acetic acid content in the simulation ranges concerned (see Table 2). This implied that without modifying the experimental protocol of the FOS/TAC method, it would be possible to more precisely estimate the concentration of bicarbonate and acetic acid and, therefore, FOS and TAC, for wider ranges (from 0 to 40 g:L\({}^{-1}\)) thanks to these two formulas. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**MAPE** & **[NaHCO\({}_{3}\)]\({}_{\text{Norm}}\)** & **[NaHCO\({}_{3}\)]\({}_{\text{new}}\)** & **[NaHCO\({}_{3}\)]\({}_{\text{corr}}\)** \\ \hline \(\left[\text{NaHCO}_{3}\right]_{\text{real}}\) & 1346\% & 0.33\% & 0.33\% \\ \hline MAPE & [HAc]\({}_{\text{Norm}}\) & [HAc]\({}_{\text{new}}\) & [HAc]\({}_{\text{corr}}\) \\ \hline \(\left[\text{HAc}\right]_{\text{real}}\) & 159\% & 0.37\% & 0.41\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: MAPE (mean absolute percentage error) of the simulated concentrations with [PERSON]’s formulas versus respective real values and MAPE of the simulated concentrations with new formulas versus respective real values ([NaHCO\({}_{3}\)] and [HAc] ranging from 0 to 40,000 mg:L\({}^{-1}\) in water solutions). Figure 4: (**A**) Recalculated sodium bicarbonate concentrations (points) using Equation (8) versus true sodium bicarbonate concentrations at different acetic acid levels; (**B**) recalculated acetic acid concentrations (points) using Equation (9) versus true acetic acid concentrations at different sodium bicarbonate levels. We could also, from the empirical FOS and TAC calculated with the Nordmann equations, recalculate the equivalent volumes A and B from previous experiments and then reinject these values into Equations (8) and (9) so as to obtain the corresponding new FOS and TAC, as shown in Equations (10) and (11). This allowed us to correct previous values without repeating the experiments. The dilution level can change the acid-base equilibrium of acetate and bicarbonate. Therefore, it has to be noted that one should apply the same dilution protocol as we did (5 mL of sample + 15 mL of water) when using the new formulas.