## 3 Results and Discussion
### Experimental Validation of the Proposed Formulas  
After acidification and degasification, the empirical TAC and FOS of the digestate were reduced from 11,009 mg CaCO\({}_{3}\cdot\mathrm{L}^{-1}\) to 2802 \(\pm\) 181 mg CaCO\({}_{3}\cdot\mathrm{L}^{-1}\) in terms of TAC and from 2452 mg HAc\(\cdot\mathrm{L}^{-1}\) to 1152 mg HAc\(\cdot\mathrm{L}^{-1}\) in terms of FOS. These values were both put into new formulas (Equations (11) and (12)) to obtain the corresponding corrected TAC (2598 mg CaCO\({}_{3}\cdot\mathrm{L}^{-1}\)) and FOS (1016 mg HAc\(\cdot\mathrm{L}^{-1}\)). The latter was considered as the X and Y values that were explained in the Materials and Methods section, and taken into account in calculating different true TAC and FOS combinations shown in Table 1.  
Ten conditions with different TAC and FOS combinations were tested using the real anaerobic digestate. Both the empirical FOS and TAC calculated with [PERSON]'s method and the new values corrected with new formulas were recorded and put against their corresponding real concentrations in the digestate (see Figure 5). Figure 5: (**A**) [PERSON]’s FOS and TAC versus real FOS and TAC values in the tested anaerobic digestate; (**B**) FOS and TAC calculated from new formulas versus real FOS and TAC values in the tested anaerobic digestate. Results show that by [PERSON]'s method, the TAC was systematically overestimated as compared to the FOS, of which [PERSON]'s equations showed a relevantly acceptable estimation (Figure 5A). As show in Table 3, the absolute relative errors of TAC and FOS with [PERSON]'s formula varied from 1% to 302% and from 1% to 215% with MAPE of 74% and 46%, respectively. When it came to the TAC and FOS values recalculated with the newly proposed formulas (see Figure 5B), the absolute relative error ranged from 1% to 61% and 1% to 13% for the estimation of TAC and FOS contents (Table 3). The global MAPE for TAC and FOS was, respectively, found at 15.6% and 6%. The major prediction errors for TAC came from the estimation of TAC in very extreme conditions (X/Y + 20,000 and X/Y + 30,000), where highly charged FOS interfered greatly with the determination of a very low TAC. Similarly, the major errors for FOS determination came from the extreme cases where TAC was very high but FOS was very low (X + 11,916/Y and X + 17,874/Y). When these extreme conditions were not considered (X/Y + 20,000, X/Y + 30,000, X + 11,916/Y and X + 17,874/Y), the MAPE covering moderate conditions (still including digestate with a very high loading of both TAC and FOS) dropped to 5.6% for TAC and 3.7% for FOS. The major difference between the modeled and true values may be attributed to the presence of other impacting factors in the digestate, which could interfere with the titration process of bicarbonate and acetic acid (such as particulates and proteins, etc.). The presence of various types of fatty acids (short-chain and other long-chain acids) in complex media could also contribute to the performance of titration. The experimental confrontation of both [PERSON]'s formulas [6,7] and the newly proposed formulas in the present study confirmed the fact that the former would only be efficient and accurate in a very limited range, while the new formulas could give satisfactory estimation of both TAC and FOS up to, respectively, 20,000 mg CaCO\({}_{3}\cdot\)L\({}^{-1}\) and 31,000 HAc\(\cdot\)L\({}^{-1}\), except for certain very extreme conditions. The experiments and the simulation were performed only with acetate. The inclusion of other VFA (propionic, butyric and valeric acids) in the simulation is possible. The original Scilab code is given in Supplementary Materials. One could vary the concentrations of these VFAs to see their respective impact on the final results. The pKa values of propionic, butyric and valeric acids were so close (around 4.7-4.8, cf Table S1) that the consideration of these species in the titration process had a limited