## 1 Introduction
However, these methods require an unconventional experimental design and more complex calculations. In addition, FOS/TAC titration can also be complemented by accurately determining one of the parameters involved (VFA or bicarbonate). [PERSON] and [PERSON] (1961) [2] first developed the \"back titration\" method for the determination of VFA and TAC in an anaerobic digester. The solution is acidified to a low pH and then boiled to remove the CO\({}_{2}\) gas produced by the carbonic acid. The solution is then titrated for the second time with a strong base (i.e., back titration) to obtain the VFA concentration. Different variant methods have been developed, such as the studies of [PERSON] and [PERSON] (1977) [20] and [PERSON] et al. (1990) [21]. TAC is often calibrated by taking into account the concentration of VFAs. An accurate VFA determination could be achieved by physical separation technologies such as steam distillation, direct distillation, chromatography [22] and the total organic carbon (TOC) analysis [23]. Approaches to simulate titration phenomena have also been investigated. [PERSON] et al. (2018) and [PERSON] and [PERSON] (1998) [24,25] have developed nonlinear models based on the acid-base equilibrium theory. Coupling the pH measurement with electrical conductivity [26], the dry matter content [27] and spectrometric property [28] of the titrated solution can give a better accuracy of VFA determination. The buffer intensity integral of different compounds allowed for a better determination of VFA content [26,29]. [PERSON] et al. (2011) [4] gave a multiple linear model by considering the initial pH and the equivalent volumes of two pH points. This method showed a good efficiency of VFA content quantification with a very wide validation range between 38 and 30,056 mg HAc:L\({}^{-1}\). TAC determination, however, has not been discussed. The last 60 years of work have revealed the fact that: (1) The reciprocal impact of TAC and FOS is often omitted, especially for the \"FOS/TAC\" method proposed by [PERSON]; (2) the different techniques developed are only effective for a limited range of FOS and TAC. The determination in reactors treating special substrates that are richer in bicarbonate and VFAs remains to be studied; (3) many studies focus on VFA determination, not on both VFAs and TAC and (4) direct titration techniques are generally easier to perform, but lack accuracy, while techniques that separately determine VFAs and TAC are more accurate, but more sophisticated and require additional equipment or complex experimental protocols. The present study is based on the most common \"FOS/TAC\" method of [PERSON]. It aims to develop a model for the determination of bicarbonate and VFAs that is both accurate and simple with a very wide validation range. The objectives of the present study are (1) the experimental demonstration of the impossibility to independently estimate FOS and TAC in high-loading samples by [PERSON]'s FOS/TAC method; (2) the simulation of the FOS/TAC titration curve and the experimental validation of these simulations in different artificial mixtures of pure sodium bicarbonate and sodium acetate; (3) with the help of the simulation, the establishment of alternative formulas to the Nordmann formulas, allowing a more accurate estimation of bicarbonate and VFA concentration in wider ranges of FOS and TAC; (4) the experimental validation of the proposed formulas using real digestion samples.