Volatile Fatty Acids from Household Food Waste: Production and Kinetics

Abstract

Household food waste (HFW), which is rich in organic matter, is a good candidate for producing added-value bio-based chemicals, such as volatile fatty acids (VFAs), by acidogenic fermentation processes. However, the lack of design tools, such as appropriate kinetic models, hinders the implementation of this technology because the results of these processes are affected by operational factors. In this work, VFA production by the acidogenic fermentation of HFW under uncontrolled pH levels (4–5) was studied at thermophilic (55 °C) and mesophilic (35 °C) temperature conditions. Batch reactors were used to digest HFW, and VFA production and the individual acid distributions were measured at different fermentation times from 0 to 624 h. The results showed higher individual and total VFA production at 35 °C and 120 h of fermentation time as a consequence of the competition between the VFA production and decomposition reactions. Acetic and valeric acids were VFAs mainly produced as a result of a high content of proteins in the initial substrate, and a small amount of propionic and butyric acids were present. A simplified kinetic model was successfully developed to represent the complex process of VFA formation from the acidogenic fermentation of HFW. A simple mechanism for the production–decomposition of VFAs, corresponding to a zero-order reaction for the first 48 h and a single consecutive reaction from that time on, was proposed. For both mesophilic and thermophilic conditions, the suggested kinetic model was able to predict the individual and total concentrations of VFAs along the fermentation time.

1. Introduction

As a result of the high rate of human population growth (around 8.1 × 10⁹ people currently) [1], huge amounts of waste are generated from the food produced around the world (nearly 59 million tons of food waste in the European Union, EU) [2]. Household food waste (HFW) made up the biggest proportion of food waste (54%), with more than 31 million tons of production per year in the EU [2].

HFW is usually landfilled, composted, or incinerated [3], but it consists mainly of organic matter such as proteins, carbohydrates, and lipids [4], so it is a good source of bio-based chemicals. According to a report on the European market for bio-based chemicals, only 3 percent of Europe’s overall chemical manufacturing comes from bio-based production techniques [5]. Specifically, no more than 0.3% of the platform chemicals, such as volatile fatty acids (VFAs), lactic acid, ethanol, etc., are manufactured via bio-based processes [5].

However, there is a growing daily need for bio-based chemicals: in 2023, the size of the global market for bio-based chemicals was USD 73,160 million, with an expected increase from USD 99,860 million in 2024 to 207,950 million in 2032 with a compound annual growth rate (CAGR) of 9.6% during the forecast period (2024–2032) [6].

Acidogenic fermentation has been proposed as an alternative method for manufacturing bio-chemicals from concentrated organic waste [7]. HFW is a perfect substrate for the acidogenic fermentation process because of its high solid content (>20%) and high level of fermentability [8,9].

The acidogenic fermentation of HFW is an anaerobic bioprocess that results in the production of valuable products such as short-chain VFAs, like acetic acid, propionic acid, butyric acid, and valeric acid [7,10], which have a wide range of applications in different industries such as the pharmaceutical, cosmetic, and tanning industries [11]. They also serve in biotechnological processes as a precursor for different bioproducts and biofuels, such as bioplastics [12], biopolymers [13], and biodiesel [14], or in bioenergy and electricity generation [15].

The production of VFAs from HFW has attracted a great deal of attention due to the high market value of VFAs (

Table 1. Market and compound annual growth rate (CAGR) of VFAs.

Acid	Market in 2023 (USD million)	CAGR 2024–2032 (%)	Reference
Acetic	22,220	4.4	[17]
Propionic	1800	2.7	[18]
Butyric	235	12.3	[19]
Valeric	17,000	5.3	[20,21]

) as well as their easy and safe storage and transportation [16]. Additionally, the conventional manufacturing of VFAs from non-renewable petrochemical sources can be replaced by the production of VFAs from HFW, supporting environmental sustainability and the circular economy [14]. Therefore, a future increase in demand for VFAs is expected (see the CAGR for VFAs in Table 1).

Under these circumstances, acidogenic fermentation is an eco-friendly alternative to treat HFW and valorise it through bioconversion to high-value products [22], reducing the use of non-renewable carbon sources and the operation of industries with traditional petrochemical routes.

However, there is still work to be done to develop sustainable and economical ways to produce VFAs from HFW. The production of VFAs by acidogenic fermentation is a hot topic right now, requiring precise control over the kinetics of acidogenesis to maximize recovery efficiency.

Factors that affect acidogenic fermentation efficiency must be considered when designing a digester. Operating parameters such as the percentage of total solids and volatile solids in the HFW, time, and operational temperature can all be important parameters in the design of digesters [23].

Kinetic-based models are an effective tool to utilize in the design phase to increase process productivity. The use of mathematical modelling can greatly reduce the time and cost required for optimizing the selection of operating conditions [24,25].

These tools have been investigated for total anaerobic digestion to optimize the production of biogas from food waste [26,27,28]. However, kinetic modelling studies that evaluate the acidogenic fermentation of organic waste, especially household food waste, from the point of view of the carboxylic platform are still incipient in the literature.

Thus, the objective of this study is to evaluate the potential of anaerobic VFA production, perform mathematical modelling, and estimate the kinetic parameters that describe the process of the consumption of substrates and VFA production under acidogenic anaerobic conditions using HFW as a substrate. In this work, a batch reactor was chosen because batch studies are useful for supplying information about a substance’s capability for acidogenic fermentation, in addition to the fact that higher VFA concentrations are obtained using batch fermentation [15]. VFA production and the individual acid distributions were measured at different fermentation times and temperature conditions; as said previously, the fermentation time and operating temperature are parameters that directly affect the acidogenic fermentation of organic solid waste in batch reactors [29].

2. Materials and Methods

2.1. Food Waste and Inoculum

Food waste used in this study was collected from local households (Salamanca, Spain). The collected food waste mainly consisted of fruit, vegetable, bread, fish, and meat remains; non-food materials such as paper, glass, plastic, metal, etc., were manually removed from the food waste. This source-selected organic fraction was ground in a Retsch SM 2000 blade mill (Retsch Ltd., Haan, Germany) with a 1 mm particle diameter and then stored in a refrigerator at 4 °C. The characteristics of the food waste are presented in

Table 2. Food waste and inoculum characteristics.

Parameter	Food Waste	Inoculum
Moisture (g/g fresh matter)	0.80 ± 0.014	0.97 ± 0.012
pH	4.94 ± 0.10	7.01 ± 0.10
Density (g/mL)	1.04 ± 0.04	1.00 ± 0.01
Total Solids (g/g fresh matter)	0.20 ± 0.006	0.033 ± 0.0007
Volatile Solids (g/g fresh matter)	0.18 ± 0.006	0.031 ± 0.0006
Total Organic Carbon (g/g dry matter)	0.57 ± 0.017	0.58 ± 0.021
Total Kjeldahl Nitrogen (g/g dry matter)	0.020 ± 0.0005	0.023 ± 0.0006

.

Anaerobic consortium collected from anoxic reactor of Salamanca wastewater treatment plant was used as parent inoculum. Table 2 shows the characteristics of the inoculum.

2.2. Experimental Setup

Glass reactors with a working volume of 250 mL were used and operated in batch mode. About 30 g inoculum and 100 g food waste were put into each reactor as initial substrate following the ratio already used in the literature [30]. The main characteristics of the initial substrate fed to the reactors were analysed and are shown in

Table 3. Initial substrate characteristics.

Parameter	Value
Moisture (g/g fresh matter)	0.84 ± 0.015
pH	4.94 ± 0.10
Density (g/mL)	1.03 ± 0.04
Total Solids (g/g fresh matter)	0.16 ± 0.004
Volatile Solids (g/g fresh matter)	0.14 ± 0.004
Total Organic Carbon (g/g dry matter)	0.57 ± 0.017
Total Kjeldahl Nitrogen (g/g dry matter)	0.021 ± 0.0005
Volatile fatty acids:
Acetic acid (g/L)	2.36 ± 0.16
Propionic acid (g/L)	0.019 ± 0.0012
Butyric acid (g/L)	0.65 ± 0.0042
Valeric acid (g/L)	0.56 ± 0.036
Total (g/L)	5.83 ± 0.41 1

¹ Total volatile fatty acids expressed as g/L of acetic acid.

.

The reactors were provided in a suitable arrangement to prevent the entry of oxygen and to allow for the observation of gas production (Figure 1). After addition of initial substrate (food waste and inoculum), the reactors were placed in an orbital shaker incubator. At the start of the experiment, nitrogen gas was flushed through the reactors to create anaerobic conditions.

The pH level was not controlled in the experiments. The effect of temperature was investigated by operating the reactors either at 35 °C or at 55 °C, and the effect of fermentation time was investigated by operating the reactors from 0 to 624 h. Different reactors were incubated at different temperatures for different times. One set of reactors operated at 35 °C and other one at 55 °C, and they were sacrificed after 48 h, 120 h, 384 h, and 624h. All the experiments were performed in triplicate.

2.3. Analytical Methods

Density was determined using gravimetric displacement [31]: a sample with a known weight was placed in a volumetric flask containing water, the volume of displacement was measured, and density was calculated by dividing the sample weight by the displaced volume. The moisture content was estimated by heating the samples with anhydrous sand until constant weight was obtained at 103 °C [32]. The pH levels, total solids (TS), volatile solids (VS), and total Kjeldahl nitrogen (TKN) were determined according to the standard methods described by American Public Health Association (methods 4500H⁺, 2540B, 2540E, and 4500-N_org C, respectively) [33]. The total organic carbon (TOC) was calculated using Equation (1) [34]:

$$TOC (g TOC/g \mathrm{d}\mathrm{r}\mathrm{y} matter)={\displaystyle \frac{1-ash (g ash/g \mathrm{d}\mathrm{r}\mathrm{y} matter)}{1.8}}$$

(1)

where ash ($g ash/g \mathrm{d}\mathrm{r}\mathrm{y} matter$) = TS ($g TS/g fresh matter$) − VS ($g VS/g fresh matter$) [35].

Volatile fatty acids (VFAs) were analysed by using a Perkin Elmer 8500 gas chromatograph (Perkin Elmer Corp., Norwalk, CT, USA) equipped with a flame ionization detector (FID) and a packed column (Porapak Q with a length of 150 cm and internal diameter of 3 mm) [36]. The temperatures of the column, injector, and detector were 200, 220, and 240 °C, respectively [36]. Nitrogen was used as carrier gas at a flow rate of 40 mL/min [36]. The samples were initially filtered through a 0.45 µm cellulosic filter membrane, acidified with concentrated sulphuric acid, and injected (1 µL) to measure free acids [36]. The VFAs analysed included acetic, propionic, butyric, and valeric acids because acidogenic bacteria promote the formation of these four acids [4].

For calculating the total volatile fatty acid (TVFA) concentration, all volatile fatty acids other than acetic acid present in the samples were reported as their equivalent g/L of acetic acid:

Total volatile fatty acid concentration = Acetic acid concentration + Propionic acid concentration in acetic equivalents + Butyric acid concentration in acetic equivalents + Valeric acid concentration in acetic equivalents

(2)

Chemical oxygen demand (COD) was used as correction factor for TVFA calculations.

Table 4. COD calculations.

Acid	Chemical Reaction	COD Calculation
Acetic	$C_2{H}_4{O}_2+2O_2\to 2{CO}_2+2H_2$	${\displaystyle \frac{O_2}{C_2{H}_4{O}_2}}={\displaystyle \frac{2·32}{60}}=1.066{\displaystyle \frac{gCOD}{g acid}}$
Propionic	$C_3{H}_6{O}_2+{\displaystyle \frac{7}{2}}{O}_2\to 3{CO}_2+3H_2$	${\displaystyle \frac{O_2}{C_3{H}_6{O}_2}}={\displaystyle \frac{3.5·32}{74}}=1.512{\displaystyle \frac{gCOD}{g acid}}$
Butyric	$C_4{H}_8{O}_2+5O_2\to 4{CO}_2+4H_2$	${\displaystyle \frac{O_2}{C_4{H}_8{O}_2}}={\displaystyle \frac{5·32}{88}}=1.816{\displaystyle \frac{gCOD}{g acid}}$
Valeric	$C_5{H}_{10}{O}_2+{\displaystyle \frac{13}{2}}{O}_2\to 5{CO}_2+5H_2$	${\displaystyle \frac{O_2}{C_5{H}_{10}{O}_2}}={\displaystyle \frac{6.5·32}{102}}=2.037{\displaystyle \frac{gCOD}{g acid}}$

shows COD calculations for individual VFAs.

Individual VFA concentrations in acetic acid equivalents were calculated using the following equations:

Propionic acid concentration in acetic equivalents =
Propionic acid concentration · (COD of propionic acid/COD of acetic acid) =
Propionic acid concentration · (1.512/1.066) = Propionic acid concentration · 1.418

(3)

Butyric acid concentration in acetic equivalents =
Butyric acid concentration · (COD of butyric acid/COD of acetic acid) =
Butyric acid concentration · (1.816/1.066) = Butyric acid concentration · 1.704

(4)

Valeric acid concentration in acetic equivalents =
Valeric acid concentration · (COD of valeric acid/COD of acetic acid) =
Valeric acid concentration · (2.037/1.066) = Valeric acid concentration · 1.911

(5)

All determinations were carried out by triplicate. Results are expressed as a mean ± standard deviation.

Kinetic analysis and model building were carried out using nonlinear regression analysis with the Marquardt 342 algorithm in STATGRAPH plus 5.1 software.

3. Results and Discussion

3.1. Volatile Fatty Acid (VFA) Production

The pH level remained uncontrolled during the experiments. A decrease in pH (from 4.94 to 4.07 for mesophilic conditions and from 4.94 to 4.90 for thermophilic conditions) was observed in the first 48 h of the fermentation time, and it was constant from 48 to 624 h (Figure 2). Parawira et al. [37] and Sarkar et al. [38] obtained a pH profile similar to that obtained in our work when they carried out studies on the production of volatile fatty acids from potato waste using mesophilic fermentation and from brewery spent grains using thermophilic fermentation, respectively; they also verified an inhibition of methanogenesis when working at a pH around four to five. The higher decrease in pH for mesophilic conditions seems to be due to the greater production of VFAs [39,40].

The influence of the fermentation time on the production of total fatty acids calculated as the acetic acid equivalent concentration (g acetic acid equivalents/L) was determined for mesophilic and thermophilic conditions. A maximum value for a time of 120 h (5 days) was found for both operating conditions (Figure 3).

The acclimation period of microorganisms to the operating conditions (the lag phase) would justify the low production of acids during the first days of digestion. In the exponential phase, when the acidogenic bacteria experience fast growth, they are in contact with high concentrations of dissolved degradable substrate and generate the production of high levels of fatty acids [41]. The decrease in dissolved substrate concentration hinders the production of new amounts of fatty acids and promotes the breakdown of fatty acids, causing them to decrease.

The curves obtained differ from the usual ones generated in the processes for obtaining fatty acids, in which the production of acids shows an asymptotic increase over time [42,43,44], resembling, however, those produced in methane generation processes [45,46,47]. This seems to indicate the existence of reactions for the decomposition of acids despite the low operating pH. VFAs’ decomposition into methane during methanogenesis is known; however, in this work, methane production was not expected because of the pH of the reaction [37,48]. Then the decomposition observed for the VFAs could only be a consequence of acetogenesis. It is well known that during this stage, propionic, butyric, and valeric acids decompose into acetic acid, and acetic acid could decompose into carbonates and H₂ [49]:

$$Acetat{e}^{-1}+4H_{2}O\to H^{+}+4H_2+2HC{O}_3^{-}$$

(6)

$$Propionat{e}^{-1}+3H_{2}O\to Acetat{e}^{-1}+HC{O}_3^{-}+H^{+}+3H_2$$

(7)

$$Butyrat{e}^{-1}+2H_{2}O\to 2Acetat{e}^{-1}+H^{+}+2H_2$$

(8)

$$Valerat{e}^{-1}+3H_{2}O\to 3Acetat{e}^{-1}+2H^{+}+4H_2$$

(9)

Our current understanding is that methanogenesis is not expected to be significant at this low pH; therefore, acetogenesis would be the only explanation for the obtained results. Jiang et al. [50] and Ding et al. [46] obtained analogous results for VFA production from food waste when a high organic load, similar to that used in this work, is present in the reactor.

Other authors have found similar results using substrates different from food waste. Huang et al. [51] studied the anaerobic fermentation of a secondary sludge at 30 °C and found a maximum VFA production at 120 h of fermentation time. Sukphun et al. [52] reported that, in batch acidogenic fermentation of organic waste, the maximum VFA yield is typically achieved in 4–10 days.

The individual VFA production (Figure 4) showed curves similar to those of the total VFA production with a maximum value at 120 h.

From these curves, it can be seen that acetic and valeric acids are the favored product of the acidogenesis process. Acetic acid’s predominance is showed in the literature [50,53] but the high valeric acid concentrations are not usual in the anaerobic digestion process. This behavior can be explained by the high content of proteins in the substrate (133.85 g TKN/kg TS); the valeric acid is mainly associated with the fermentation of proteins and it can be formed via the reductive deamination of single amino acids or by oxidation–reduction between pairs of amino acids via the Stickland reaction [54,55,56,57]. Propionic acid predominance on butyric acid is usual in anaerobic digestion carried out in batch reactors with ratios inoculum/substrate and pH similar to these used in this study [58].

Temperature is also one of the fundamental factors in the anaerobic fermentation of food waste because it influences both bacterial growth and metabolic activity [59,60]. Since each type of microbial organism has a range of temperatures within which it can replicate, alterations in working temperature can also affect the microbial composition of the consortium involved in acidogenic fermentation [60,61].

Figure 3 and Figure 4 show the influence of temperature on acidogenesis. As seen in the figures, the total VFA concentration at 35 °C was mostly higher than that at 55 °C, and the production of acetic and valeric acids rather than propionic and butyric acids was always favored. These results seem opposite to what one would expect from traditional theories: thermophilic regimes (>50 °C) have been adopted for anaerobic digestion because they provide several advantages, when compared with mesophilic regimes (30–40 °C), such as an increased destruction rate of organic solids [62]. However, the studies carried out by Kim et al. [63], Komemoto et al. [64], Mata-Álvarez et al. [65], He et al. [66], Jiang et al. [50] and Fernández [4] with conditions similar to our operating conditions agree with a higher production of VFAs for mesophilic than for thermophilic temperatures. The total VFA productivity at 35 °C of 0.53 gVFA/gVS was similar to the result of Khosroshahi [67] who obtained 0.53 gVFA/gVS from the anaerobic digestion of food waste at a mesophilic temperature but using a mixture of food waste and sewage sludge as the substrate.

The temperature did not have a remarkable effect on the concentration of the propionic and butyric acids during the experimentation time, and similar curves were obtained for both temperatures (Figure 4). For fermentation times other than 120 h, the acetic and valeric production is also almost not affected by the temperature, but at 120 h, a higher concentration is obtained at 35 °C, which agrees with the result for the total volatile fatty acids (Figure 4). Comparable results were obtained by Fernández [4] in his study of temperature’s influence on VFA production from food waste at MBT plants; he observed a maximum acetic acid percentage at 144 h for both 35 and 55 °C but slight differences in acetic acid percentages for lower and higher fermentation times. No notable effect of temperature on the curves for the propionic acid percentage was observed for the fermentation times [4]. However, the pattern observed by Fernández [4] was different for butyric and valeric acids: butyric acid percentages prevailed over valeric acid percentages at both 35 °C and 55 °C. The difference in compositional characteristics of the food waste used in Fernández’s study [4] and in our study justifies the different percentages of these acids that were obtained [68]. As noted above, the high valeric acid concentrations found for our anaerobic digestion process can be explained by the high content of proteins in the substrate, and acetic acid’s prevalence in both mesophilic and thermophilic conditions is supported by results in the literature. Jankowska et al. [69] and Faucher [70] showed that the optimal fermentation time for VFA production differs depending on the substrate characteristics and complexity but verified that the advantage of using a short retention time to maximize acidification is evident for all substrate types.

3.2. Kinetic Model for Batch Anaerobic Digestion of Food Waste

The analysis of the VFA concentration versus the fermentation time (Figure 3 and Figure 4) shows that the concentration of the VFAs gradually increased with the fermentation time up to 120 h, and above this time, the VFA concentration gradually decreased with the fermentation time. These results indicate that VFA production from the food waste was not stable under the reaction conditions used after 120 h of fermentation time. Therefore, the kinetic model for explaining the food waste degradation in acidic conditions for VFA production must include both the production and the decomposition of VFAs.

The kinetic models typically used for representing the VFA production from food waste anaerobic digestion consider the same kinetic behaviour along the whole reaction time; the consequence of this supposition is a discrepancy between the experimental and calculated data [71,72,73].

The analysis of the VFA concentration versus the fermentation time (Figure 3 and Figure 4) showed a linear relationship up to 48 h that is explained by zero-order kinetics. In the first stage of the reaction, when an excess of the substrate (organic matter expressed as g/L of volatile solids) is present, the reaction rate is independent of the initial concentration of substrate:

$$Organic Matter \stackrel{k_A}{\to } VFA+C{O}_2+H_2$$

(10)

The CO₂ and H₂ produced during the process of fermentation were not quantified but the presence of gas bubbles in the hydraulic seal (used in the experimental device of Figure 1 to prevent air from entering the reactor and maintain an anaerobic environment during the process) led us to presume the existence of these compounds because, as said previously, methane’s presence is not anticipated to be significant at a low operating pH level.

According to this scheme, the reaction rate is given as follows:

$$v_a={\displaystyle \frac{d[VFA]}{dt}}=k_A$$

(11)

The integration of Equation (11) yields the following:

$$\left[VFA\right]-{[VFA]}_0=k_{A}t$$

(12)

where [VFA] is the volatile fatty acid concentration (g/L), [VFA]₀ is the initial volatile fatty acid concentration (g/L), t is the fermentation time (h), and k_A is the kinetic constant (g·L⁻¹·h⁻¹).

The amount of VFAs after the first 48 h increased significantly up to 120 h, the time from which the concentration of fatty acids decreases as a result of their decomposition.

VFA production during acidogenesis is a well-known process and justifies the increase in VFA concentration in the reactor [63]. It is also known that, during the methanogenesis process, VFAs break down into methane, but as said before, due to the reaction’s pH, methane generation was not expected in this study [37,48]. Therefore, as has been previously stated, the VFA decomposition observed could only result from the acetogenesis process. This theory seems to contradict the usual approaches in anaerobic degradation according to which neither acetogenic nor methanogenic activity occurs below a pH of 5.8 in the first case and 6.5 in the second one. However, Paulo et al. [74] noted the degradation of methanol in strongly acidic conditions (a pH of 4). The study lasted for 160 days, in the course of which they obtained a reduction in the amount of methanol, but no methane was detected. They noted the emergence of acetic acid during the first days of the experiment followed by a further decline during the last 5 days of the experiment, indicative of a transformation of the acetate in a compound that is not CH₄. This behavior, similar to that observed in our research, can be explained by the phenomenon known as “acid habituation” or “adaptive acid tolerance response” described by Hall et al. [75]. According to their description, bacterial cells grown at a moderately acidic pH or that are temporarily exposed to a low pH resist being killed at a low pH much better than the cells grown at a pH of 7. For our substrate, we do not know at what extent the pH inside the granules was affected, but probably it was lower than the optimum pH range for the growth of methanogens (6.5–8.0) and homoacetogens (5.8–7.0) [76]. Several mechanisms to maintain the intracellular pH and minimize the stress from a non-optimal extracellular pH are cited in the literature [77].

When the kinetics becomes dependent on the concentration, this behavior can be described by assuming that VFAs decompose according to consecutive reactions as shown below:

$$Organic Matter\stackrel{k_B}{\to }VFA+C{O}_2+H_2\stackrel{k_C}{\to }Decomposition products$$

(13)

Assuming that each stage is an irreversible first-order reaction, the model outlined above can be described both in terms of the degradation rate of the substrate as shown below:

$$v_S=-{\displaystyle \frac{d[S]}{d\tau }}=k_B[S]$$

(14)

and in terms of the production rate of VFAs as shown below:

$$v_a={\displaystyle \frac{d[VFA]}{d\tau }}=k_B[S]-k_C[VFA]$$

(15)

where the substrate concentration of organic matter [S] is expressed as g/L of volatile solids, the VFA concentration is expressed as g/L of volatile fatty acids, and k_B and k_C are kinetics constants (h⁻¹).

The time t in Equation (11), where kinetics is independent of the substrate concentration, is different from the time $\tau$ in Equations (14) and (15), which indicates the time for the consecutive reactions in which the kinetics relies on the substrate concentration.

The integration of Equation (14) shows the dependence of the substrate concentration on the time:

$$[S]=[S{]}_0{e}^{-k_B \tau }$$

(16)

where [S]₀ is the initial substrate concentration at t = 0 because its variation is negligible compared to the variation in the VFA concentration.

A first-order linear differential equation is obtained by inserting Equation (16) into Equation (15).

$${\displaystyle \frac{d[VFA]}{d\tau }}+k_C[VFA]=k_B[S{]}_0{e}^{-k_B\tau }$$

(17)

After integrating Equation (17), the kinetic model is expressed as follows:

$$e^{k_{C }\tau }[VFA]={\displaystyle \frac{k_B[S{]}_0}{k_C-k_B}}{e}^{(k_C-k_B) \tau }+\Theta$$

(18)

where $\Theta$ is the integration constant whose value can be determined by considering that, when $\tau$ is zero, the concentration of VFAs is the one generated after 48 h of digestion:

$$\Theta =[VFA{]}_{48h}-{\displaystyle \frac{k_B[S{]}_0}{k_C-k_B}}$$

(19)

Combining the last two equations, an expression for the evolution of the VFA concentration is obtained as a function of the time and the initial concentration of organic matter:

$$[VFA]=[VFA{]}_{48h}{e}^{-k_C\tau }+{\displaystyle \frac{k_B[S{]}_0}{k_C-k_B}}(e^{-k_{B }\tau }-e^{-k_{C }\tau })$$

(20)

This equation is valid starting from a reaction time of 48 h, so $\tau$ = t − 48. By substituting this value and rearranging Equation (20), it can be rewritten as follows:

$$[VFA]=[VFA{]}_{48h}{e}^{-k_C(t-48)}+{\displaystyle \frac{k_B[S{]}_0}{k_C-k_B}}(e^{-k_{B }(t-48)}-e^{-k_{C }(t-48)})$$

(21)

Therefore, the following two equations should be used for the kinetic model:

$$\left[VFA\right]=k_{A}t+{[VFA]}_0 \mathrm{a}\mathrm{t} 0\le \mathrm{t}\le 48 \mathrm{h}$$

(22)

$$[VFA]=[VFA{]}_{48h} e^{-k_C\left(t-48\right)}+{\displaystyle \frac{k_B}{k_C-k_B }}\left(e^{-k_B\left(t-48\right)}-e^{-k_C\left(t-48\right)}\right)[S{]}_0 \mathrm{a}\mathrm{t} \mathrm{t}\ge 48 \mathrm{h}$$

(23)

Figure 3 and Figure 4 show the fit of the experimental data based on Equations (22) and (23). The kinetics constants obtained from the fitting are listed in

Table 5. Kinetic constants for the volatile fatty acids and coefficients of determination (R²) for equations fitting theoretical and experimental data.

VFA	Temperature
	35 °C				55 °C
	kA (g·L−1·h−1)	kB (h−1)	kC (h−1)	R2	kA (g·L−1·h−1)	kB (h−1)	kC (h−1)	R2
Total VFA	8.05 × 101	1.71 × 10−2	2.15 × 10−2	1.0	2.66 × 102	1.17 × 10−2	3.81 × 10−2	0.99
Acetic acid	5.52 × 101	1.30 × 10−2	3.15 × 10−2	1.0	2.14 × 102	1.22 × 10−2	4.88 × 10−2	0.99
Propionic acid	4.67 × 100	9.00 × 10−3	6.70 × 10−1	0.99	1.72 × 101	1.20 × 10−2	4.10 × 10−1	1.0
Butyric acid	8.15 × 10−1	7.13 × 10−3	1.24 × 100	1.0	8.95 × 100	8.20 × 10−3	1.09 × 100	0.98
Valeric acid	7.63 × 101	1.28 × 10−2	4.08 × 10−2	0.99	5.56 × 101	2.07 × 10−2	1.42 × 10−1	1.0

. A good agreement between the experimental (the dots in Figure 3 and Figure 4) and the calculated values (the solid lines in Figure 3 and Figure 4) was found for the concentration of total and individual VFAs (R² > 0.99 and R² ≥ 0.98, respectively, according to Table 5).

The results display how k_A is dependent on the increase in concentration of VFAs ([VFA] − [VFA]₀), showing higher values of k_A for VFAs with higher increases in concentrations in food waste, such as acetic and valeric acids. As can be seen, the k_A of all acids, except valeric acid, is lower at 35 °C than at 55 °C; this implies a higher production rate at 55 °C for the initial reaction stages and a higher concentration of these acids (and consequently of the total VFA) in the reactor for t ≤ 48 h. Moreover, the higher values of k_B and the lower values of k_C for the acetic and valeric acids compared with those for the propionic and butyric acids support the predominance of the production versus the degradation rate of these acids and consequently the high concentrations of acetic and valeric acids obtained in the experiments.

It could be argued that the proposed model would not be valid for inoculum concentrations different from the one used in this work. However, it has been proven that, in anaerobic digestion, the inoculum concentration does not influence the percentage conversion of the substrate [78,79,80] and does not exert an effect on the extension of the product’s production but on its rate [79,81].

The kinetic model proposed in this paper is supported by the study of Esteban et al. [82] who used a similar kinetic model to represent the transformation of organic slaughterhouse waste made up of complex substrates (such as proteins) into simpler products (peptides and amino acids) that, in turn, evolved into decomposition products.

4. Conclusions

The optimization of VFA production from household food waste in batch anaerobic mode under an uncontrolled pH (4.0–5.0) was investigated in the present research. It was proven that different fermentation times and temperature values influenced the total and individual VFA productivity. The conditions that led to the best VFA yield were a fermentation time of 120 h and a mesophilic temperature (35 °C). Under these conditions, the VFA yield was 0.53 gVFA/gVS. Regarding the VFA composition, acetic acid was the main product followed by valeric acid; propionic and butyric acids were produced in minor amounts. The high levels of valeric acid obtained can be attributed to the high content of proteins in the initial substrate (133.8 g NTK/kg ST).

To explain the complex process of VFA production from household food waste acidification, a kinetic model was proposed. The reactions were described by a simple mechanism of the production–decomposition of VFAs, which showed a strong correlation between experimental and predicted outcomes. The process followed a zero-order reaction for the first 48 h and a single consecutive reaction from that time on. The kinetic model proposed was able to predict the individual and the total concentrations of VFAs for the reaction times for both mesophilic and thermophilic conditions.