## 1 Introduction
In summary, during the TAC titration, VFAs are strongly involved, whereas the effect of TAC on VFA determination exists but can be less significant. [PERSON] et al. (2011) [3] showed that the ratio of VFA concentration to the total alkalinity can be used to assess the adequacy of the organic load in anaerobic digesters. It should be noted that for a successful anaerobic digestion and process stability, the ratio should generally be less than 0.4. Consequently, a wrong result of VFA and TAC contents could lead to reactors being operated incorrectly. In the particular case of two-stage anaerobic digestion, the digestate at the outlet of hydrolysis digesters (55 \({}^{\circ}\)C) could have an extremely high VFA concentration and a disproportionately low alkalinity [4]. Conversely, the digestate at the outlet of the anaerobic digester could have a relatively low VFA concentration compared to the alkalinity, which would be very high [5]. This would likely distort the titration result. Figure 1: Acid–base equilibria of different compounds as a function of pH at 25 \({}^{\circ}\)C.  
The accurate determination of VFAs and TAC by minimizing the interference between them becomes a crucial factor to ensure the right operation of anaerobic digesters. The implementation of VFAs and alkalinity titration has been extensively studied over the past 60 years. Many studies focus on the direct titration of samples by looking for different pH turning points to minimize the effect of mutual interference. The direct titration with two pH equivalent points was first developed by [PERSON] (1977) [6]. It is called the \"FOS/TAC\" method, and is most widely used today. In this method, the VFA concentration is called the FOS for _\"Flichtige Organische Suuren\"_ in German. During FOS/TAC titration, the consumption of a strong acid up to pH = 5 (denoted A, in mL) reflects the buffering capacity of the system carried by bicarbonate, which is also called the TAC (see Figure 1). Between pH = 5 and pH = 4.4, protons are assumed to react exclusively with organic acids [7]. The volume of titrant consumed between these two pH values (denoted B, in mL) is used to calculate the FOS [8]. This method is only valid for a low FOS concentration because the latter could interfere with the TAC determination as mentioned above. The alkalinity determination at pH = 5 is inevitably biased in the case where VFA concentration is much higher. This simple, but partially inaccurate, approach is largely used, because it is easy to implement and allows for the evaluation of the adequacy of the organic loading in an anaerobic digester, especially if it is used to follow the relative evolution of one specific reactor. The [PERSON] method can be implemented in Hach or METTLER TOLEDO titration devices. It is this [PERSON] FOS/TAC method (1977) that the present paper proposes to optimize but keeping the simplicity of the experimental realization by correcting the calculation formulas, considering the reciprocal interferences between the FOS and TAC. Besides [PERSON] (1977), [PERSON] et al. (1983) [9] proposed an alternative method of titrating the solution by a strong acid with a single stop at pH = 5.75. The method remains approximate and is only valid for a restricted range (TAC \(<\) 2000 mg CaCO\({}_{3}\cdot\)L\({}^{-1}\) and VFA \(<\) 500 mg HAC\(\cdot\)L\({}^{-1}\)). The two-pH titration method has been modified or improved since the 1980s. [PERSON] et al. (1986) [10] proposed the distinction between partial alkalinity and intermediate alkalinity. Similarly, additional pH equivalent points have been added for a better evaluation of FOS/TAC such as the four-point [11], five-point [12] and eight-point [13,14] titration. The advantages and disadvantages of these different methods have been described by many authors [3,15,16,17,18,19]. In general, multi-point titration has a satisfactory accuracy for a VFA concentration below 5 g HAC\(\cdot\)L\({}^{-1}\), which covers the FOS content in most traditional anaerobic digesters.