In Situ Bio-Methanation Modelling of a Randomly Packed Gas Stirred Tank Reactor (GSTR)

: In situ Bio-Methanation (BM) is a recently developed biogas upgrading technique which ﬁnds application also in the Power to Gas (P2G) ﬁeld. In this study a novel conﬁguration of BM digester, the randomly packed Gas Stirred Tank Reactor (GSTR), was modelled. A 49 L reactor, in thermophilic conditions (55 ◦ C) and at atmospheric pressure, was ﬁlled up with random packing on which the microbial populations could adhere. The feedstock used was Second Cheese Whey (SCW), liquid waste of cheese factories, rich in lactose (38 g/L), and its ﬂowrate was chosen to obtain a Hydraulic Retention Time (HRT) of 30 days. The process was analyzed for different hydrogen inlet ﬂowrates of 10 mL/min and 50 mL/min. The produced biogas was also recirculated in the reactor in order to transfer, into the liquid phase, as much hydrogen as possible. The model parameters were estimated by means of stationary state information of the reactor working without hydrogen injection, while a dynamical ﬁtting was necessary to evaluate the value of the hydrogen mass transfer coefﬁcient during BM. The model well described the reactor behavior and, by means of a dimensionless analysis in which the numbers of Stanton (St) and β were deﬁned, it was found out that the mass transfer coefﬁcient is the limiting step of the process. already adopted by Bernard et al. [19]. In situ BM was reproduced including the biomass of Methanobacterium thermoautotrophicum ( X 3 ) as responsible for H 2 and CO 2 consumption and CH 4 generation. Such microbial species, belonging to the Methanobacteriaceae family, was found to be, in previous analysis made on the same reactor [17], the most abundant among all the hydrogenotrophic community. A block diagram of the whole biological model is provided in Figure 2.


Introduction
Anaerobic digestion (AD) is a biological process used to convert organic waste material into biogas, a gaseous mixture mainly composed of CH 4 (~50-60%) and CO 2 (~50-40%). Due to its abundance in methane, biogas is suitable for energy and heat generation purposes [1,2]. Moreover, the removal of CO 2 from biogas by means of upgrading processes makes AD a sustainable source of CH 4 . In such cases, the methane produced is directly injected in the already existent natural gas grids [3]. Currently, the most used biogas upgrading techniques such as membrane separation and adsorption are performed in specific units downwards the anaerobic digester. To avoid the costs related to the presence of additional units, an attractive solution which exploits the ability of anaerobic microorganisms to convert hydrogen and carbon dioxide in methane was proposed: the in situ bio methanation. The in situ bio-methanation consists of injecting hydrogen into an anaerobic digester in order to reduce and convert CO 2 , produced during AD, to CH 4 . This process finds a possible application also in the Power to Gas (P2G) field for the chemical storage of electrical energy produced by renewable sources [4,5]. In many studies, it was observed that the limiting step of the process is the hydrogen mass transfer to the liquid phase [6,7], but to achieve a complete comprehension of the system behavior, the development of a mathematical model is crucial. The AD is composed of a series of different syntrophic biological pathways. Several AD models were developed with a different degree of complexity. Andrews et al. [8] developed a model considering only two distinct types of microbial populations which could predict the reactor failure for the decrease in pH due to Volatile Fatty Acids (VFAs) accumulation. Angelidaki et al. [9] realized a model which included the presence of three different types of bacteria and ammonia inhibition on the acetoclastic step. Zaher et al. [10] considered in the simulation of AD the hydrogenotrophic biomass. The most complete and complex models for the high numbers of parameters and for considering almost all the phenomena that take place in the AD are the "BioModel" developed by Angelidaki et al. [11] and the "ADM1" proposed by Batstone et al. [12]. In literature only a few models are available for the description of in situ biological methanation (BM). In particular, Lovato et al. [13] integrated the "BioModel" with the presence of the hydrogenotrophic pathway proposed by Hill et al. [14] for the description of a conventional CSTR with a working volume <2 L. Bensmann et al. [15] used the ADM1 model together with the characteristic growth kinetics of the Methanobacterium thermoautotrophicum obtained from the work of Shill et al. [16] to detect the limit working conditions (related to biological and mass transfer phenomena) of an anaerobic digester performing the biological methanation.
In order to reduce the computational complexity due to the high number of parameters that usually occur in biological models, the aim of this work is to propose a simple biological methanation model for the description of a randomly packed Gas Stirred Tank Reactor (GSTR) including in the model the presence of suspended attached growth media. Once estimated all the parameters of the model, a dimensionless analysis is performed to obtain information about the limiting steps of the process.

The Influent
The experiments were performed with a Second Cheese Whey (SCW) gotten from Caseificio Santa Maria s.r.l., a small dairy factory located in Rome, Italy. SCW is a lactose rich wastewater stream obtained from the production process of ricotta (a typical soft cheese of the Mediterranean region). According to this, the collected SCW had some fluctuations in its composition, but for modelling purposes, the average concentration of lactose was used as input of the system, and it was set to 38 g/L. Further details on the SCW used are reported in Lembo et al. [17].

The Process
The pilot plant is made of a 61 L randomly packed Gas Stirred Tank Reactor (GSTR) working in thermophilic (55 ± 1 • C) and atmospheric pressure conditions [17]. The liquid feed of SCW is introduced in the lower part of the reactor, while the digestate is collected in the upper part of the reactor in order to maintain a constant the liquid working volume of 49 L. The liquid flow rate was set to achieve a Hydraulic Retention Time (HRT) of 30 days. Part of the biogas stream is recirculated into the reactor to guarantee a homogeneous mixing of the nutrients into the liquid phase and to enhance the mass transfer between the liquid and the gaseous phases. Random packing material (SAGM 600, HDPE), with a specific surface area of 800 m 2 /m 3 , was placed in the central part of the reactor. The immobilized area occupied a volume of about 30 L.
From the bottom of the reactor, pure H 2 (≥99%) obtained from an electrolyzer (DBS, model PGH2 100, Linde Gases Division, Pullach, Germany) was injected. Depending on whether H 2 was flowed into the reactor or not, in situ Bio-Methanation (BM) mode or Anaerobic Digestion (AD) mode were run respectively in separate steps. The AD was conducted until stationary conditions were reached (working time > 3 HRT (day)) setting the recirculation biogas flow to 4 L/min. For the BM a period of 43 days was investigated: 10 mL/min of H 2 was injected and the recirculation biogas flow was fixed to 4 L/min for the first 7 days while 50 mL/min of H 2 was introduced, and the recirculation biogas flow was fixed to 6 L/min for other 34 days. A complete sketch of the process is given in Figure 1.
Anaerobic Digestion (AD) mode were run respectively in separate steps. Th ducted until stationary conditions were reached (working time > 3 HRT (da recirculation biogas flow to 4 L/min. For the BM a period of 43 days was in mL/min of H2 was injected and the recirculation biogas flow was fixed to 4 first 7 days while 50 mL/min of H2 was introduced, and the recirculation bi fixed to 6 L/min for other 34 days. A complete sketch of the process is give

Analytical Methods
Biogas composition was measured by an online Micro Gas Chromato (GC4900). Biogas and H2 flow rate were monitored by a digital online flo Flow select series, Bronkhorst High-Tech B.V, Ruurlo, The Netherlands). T SCW, of the reactor effluent and of the immobilized part were analyzed fo ards methods [18].

Bio-Methanation Modeling
The complexity of the model is directly related to the number of bacter taken into account [19]. In a conventional anaerobic digester, it was observed contribution to methane generation is given by acetoclastic methanogens hydrogenotrophic methanogens played a minor role. The scenario change gen is directly injected into the digester as happens with in situ bio-meth hydrogenotrophic methanogens are responsible of the conversion of CO2 an In order to well represent the two working operation modes of the reactor (in situ BM and AD), it was decided to develop a suitable model for (AD) validity toward in situ BM by introducing a little modification. According eration, the model developed takes into account two main groups of mic tions for the description of AD and considers an additional biomass to deal w consumption during in situ BM. More in detail, AD is completely repro phases: acidogenesis is the first phase of the process in which acidogeni consume the organic substrate available (S1, in this case lactose) and pro VFAs (S2); methanogenesis is the last phase where acetoclastic methanoge consume the volatile fatty acids, which are assumed to behave like pure

Analytical Methods
Biogas composition was measured by an online Micro Gas Chromatograph Varian (GC4900). Biogas and H 2 flow rate were monitored by a digital online flow meters (EL-Flow select series, Bronkhorst High-Tech B.V, Ruurlo, The Netherlands). The VSS of the SCW, of the reactor effluent and of the immobilized part were analyzed following standards methods [18].

Bio-Methanation Modeling
The complexity of the model is directly related to the number of bacteria populations taken into account [19]. In a conventional anaerobic digester, it was observed that the main contribution to methane generation is given by acetoclastic methanogens archaea while hydrogenotrophic methanogens played a minor role. The scenario changes when hydrogen is directly injected into the digester as happens with in situ bio-methanation where hydrogenotrophic methanogens are responsible of the conversion of CO 2 and H 2 into CH 4 . In order to well represent the two working operation modes of the reactor under analysis (in situ BM and AD), it was decided to develop a suitable model for (AD) and extend its validity toward in situ BM by introducing a little modification. According to this consideration, the model developed takes into account two main groups of microbial populations for the description of AD and considers an additional biomass to deal with hydrogen consumption during in situ BM. More in detail, AD is completely reproduced in two phases: acidogenesis is the first phase of the process in which acidogenic bacteria (X 1 ) consume the organic substrate available (S 1 , in this case lactose) and produce CO 2 and VFAs (S 2 ); methanogenesis is the last phase where acetoclastic methanogen archaea (X 2 ) consume the volatile fatty acids, which are assumed to behave like pure acetic acid, to generate CO 2 and CH 4 . A crucial aspect of AD is the substrate inhibition of acetoclastic methanogen archaea which may cause a strong pH decrease and eventually may lead to process failure [8]. It is important to underline that no kind of ion inhibition was assumed in this model, thus the ion balances were not considered. Since lactose is a dimer, the hydrolytic step, carried out by extracellular enzymes [20], was assumed to be faster than all the others and therefore neglected. The same assumptions for AD modeling were already adopted by Bernard et al. [19]. In situ BM was reproduced including the biomass of Methanobacterium thermoautotrophicum (X 3 ) as responsible for H 2 and CO 2 consumption and CH 4 generation. Such microbial species, belonging to the Methanobacteriaceae family, was found to be, in previous analysis made on the same reactor [17], the most abundant among all the hydrogenotrophic community. A block diagram of the whole biological model is provided in Figure 2. rocesses 2021, 9, x FOR PEER REVIEW the others and therefore neglected. The same assumptions for AD modeling adopted by Bernard et al. [19]. In situ BM was reproduced including the bio anobacterium thermoautotrophicum (X3) as responsible for H2 and CO2 consum generation. Such microbial species, belonging to the Methanobacteriaceae fam to be, in previous analysis made on the same reactor [17], the most abund the hydrogenotrophic community. A block diagram of the whole biologica vided in Figure 2.

Biological Reactions: Stoichiometry and Kinetics
The biological reactions that occur in the system are related to the met different types of microorganisms considered.
Acidogenesis stoichiometry is described in Equation (1) as follows: where S1 is the organic substrate consumed by the biomass X1, S2 are the vola which were assumed to behave like pure acetic acid, and ki are the reaction to the biomass growth. The reaction rate of Equation (1) is (r1), and it w follow the Monod kinetics depending on the availability of substrate S1 in t thus (Equation (2)) where , is the maximum specific growth rate, is the half satura and is the acidogenic biomass concentration. Acetoclastic methanogenesis stoichiometry is reported in Equation (3) as The reaction rate of Equation (3) is (r2), and it was modelled by the H sion which is commonly utilized for substrate inhibited kinetics: Outline of the biogas model.

Biological Reactions: Stoichiometry and Kinetics
The biological reactions that occur in the system are related to the metabolism of the different types of microorganisms considered.
Acidogenesis stoichiometry is described in Equation (1) as follows: where S 1 is the organic substrate consumed by the biomass X 1 , S 2 are the volatile fatty acids which were assumed to behave like pure acetic acid, and k i are the reaction yields referred to the biomass growth. The reaction rate of Equation (1) is (r 1 ), and it was assumed to follow the Monod kinetics depending on the availability of substrate S1 in the liquid bulk, thus (Equation (2)) where µ 1,max is the maximum specific growth rate, K S 1 is the half saturation constant, and X 1 is the acidogenic biomass concentration. Acetoclastic methanogenesis stoichiometry is reported in Equation (3) as follows: The reaction rate of Equation (3) is (r 2 ), and it was modelled by the Haldane expression which is commonly utilized for substrate inhibited kinetics: where µ 2,max is the maximum specific growth rate, K S2 is the half saturation constant, K I is the inhibition constant, and X 2 is the acetocastic methanogen concentration.
Hydrogenotrophic methanogenesis results from the metabolism of Methanobacterium thermoautotrophicum which is composed of a reaction for cellular growth (ν G ) and a reaction for cellular maintenance [21] (ν M ) that are reported in Equation (5) and Equation (6), respectively.
H 2 + k 9 CO 2 → k 8 X 3 + k 10 CH 4 (5) The overall hydrogen rate of consumption was assumed to have a saturation kinetics as adopted by Schill et al. [16] where q max is the maximum hydrogen specific growth rate, K H 2 is the half saturation constant, C L H 2 is the hydrogen concentration dissolved in the liquid phase, and X 3 is the hydrogenotrophic methanogen bacteria concentration. The rate of maintenance is expressed as follows: where m is the maintenance coefficient. According to Equation (7) and Equation (8) the rate of CH 4 production, CO 2 consumption and biomass growth are: The kinetic parameters used in this work referred to all the biological reactions are listed in Table 1.

Mass Balances
In this section are provided all the mass balances of the system which are crucial to evaluate the performances of the process. Both the liquid and the gaseous phases are assumed to be perfectly mixed as happens in a Continuous Stirred Tank Reactor (CSTR). The macroscopic mass balances were written including in the control volume the gas recirculation stream which influences only the gas-liquid mass transfer coefficient.
Liquid phase. All the biochemical reactions take place in the liquid phase. In order to consider the presence of random packing, it was assumed that only a fraction of the total biomass is homogeneously dispersed in the liquid phase while the other part is immobilized on the packing rings. Such fraction of "free" biomass was identified introducing the parameter α as done in the work of Bernard et al. [19]. According to such consider-Processes 2021, 9, 846 6 of 13 ation, the mass balances for biomass growth, one for each type of microbial population (X 1 , X 2 , X 3 ), are: where D represents the dilution factor which is the reciprocal of the HRT. The material balances for the substrates and the dissolved gaseous components are expressed in Equation (13): where k i,j is the yield coefficient of the component i referred to the reaction j, N is the total number of biological reactions and N L−G i is the volumetric transfer rate for the component i from the liquid to the gaseous phase.
Gaseous phase. The mass balances are written in Equation (14), one for each component present in the gaseous phase (CO 2 , CH 4 , H 2 ): where, n i are the moles of component i in the gaseous phase, V L is the liquid volume. The inlet and outlet molar flow rates are F in i and F out i , respectively. According to the control volume considered, only for hydrogen F in i = 0 when it is injected in the system during BM. The volumetric transfer rate N L−G i is defined as follows where k L,i a is the overall volumetric mass transfer coefficient referred to the component i in the liquid phase, C L i is the concentration of the gaseous component i dissolved in the liquid, and C L * i represents the hypothetic liquid concentration of component i in equilibrium with the gas bulk. The concentration C L * i is linked to the partial pressure of the related component P i by means of the Henry's law where H i is the Henry's constant referred to the component i. Assuming valid the penetration theory for the gas-liquid mass transfer, for each component the mass transfer coefficient is proportional to the square root of its diffusivity. The latter consideration permits to link all the mass transfer coefficients as described in Equation (17): where D i is the diffusivity of the component i. The values of the diffusivities and of the Henry's constants for every component are reported in Table 2.

Parameter Estimation
The model was developed in order to be suitable to both Anaerobic digestion and Bio-Methanation operation mode. For this reason, it was possible to split the parameter estimation in two steps:

1.
Parameter Estimation related to the Anaerobic Digestion by evaluating the stationary state of the system. As will be explained in the result section, the unknown parameters to determine are: all the yield parameters related to the reactions r 1 and r 2 , the mass transfer coefficients k L,i a and the parameter α.

2.
Evaluation of the remaining parameters by a dynamic simulation fitting during Biomethanation. It will be later explained how the only parameter to fit will be the hydrogen mass transfer coefficient k L,H 2 a.

Determination of the AD Stationary State
The equations that describe the stationary state of the reactor are crucial to build the algebraic system necessary for the parameter evaluation. The same strategy was applied by Bernard et al. [19].
Liquid phase. Starting from setting to zero the time derivative of Equation (16), it is possible to derive the values of the concentration of lactose (S 1 ) and VFA (S 2 ) at stationary state.
Equation (19) shows that there are two possible values of S * 2 . The difference between them depends on whether the system is working or not in a VFAs inhibited mode. Since in the system under analysis this situation did not occur, the minimum value of S 2 solving Equation (19) was taken in consideration as S * 2 . Once obtained S * 1 and S * 2 , from Equation (17), it is possible to obtain the expression for the biomass X 1 and X 1 concentration at the stationary state: In order to determine k 1 , the knowledge of α and one between X * 1 or S * 1 is essential. For this reason, it was assumed (as suggested in the work of Bernard et al.) as a (rough) estimation of the total biomass concentration the total amount of Volatile Suspended Solids (VSS) in the digestate at stationary conditions. Moreover, the ratio between the acidogenic microbial population respect to the total one (γ) was taken from literature [24] and fixed to γ = 0.2. According to the above-mentioned considerations and to Equations (20) and (21) it follows: At this point, only the value of α must be evaluated to calculate k 1 . For this purpose, it was exploited the definition of α previously given in Section 2.4.3. Considering the liquid volume filled with random packing (V 1~3 0 L) and the volume of the reactor occupied only by liquid (V 2~2 0 L), α is the solution of the system reported below: where X im and X L are the concentration of biomass in the immobilized and liquid part, respectively. Once calculated k 1 , from Equation (21), k 3 was obtained by fixing k 2 from literature. In fact, from the sensitivity analysis developed in the work of Bernard et al. [19], k 2 was find out to have a minor influence on the system. Gaseous phase. The information on the reactor behavior that is available by means of the experimental operations is biogas flowrate and composition. The utilization of such information in the stationary state Equations of the system is crucial for the evaluation of the parameters k 4 , k 5 , k 7 . Those parameters do not depend on the values of k L,CO 2 a and k L,CH 4 a for two different reasons that will be verified in the result section:

•
The concentration of CO 2 in the liquid phase was supposed to be at equilibrium conditions since it is a highly soluble gas [23]. Combining Equations (13) and (14) and setting to zero the time derivative terms, the following equation was obtained were k 4 was fixed from literature since it is low impact on the system sensitivity again considering the work of Bernard et al. [19].
The concentration of CH 4 in the liquid solution happens to be so low that it was neglected the term that it takes it into account for the evaluation of the parameter k 6 . The latter was calculated by combining Equations (13) and (14) and setting to zero the time derivative terms which led to the expression reported below.

Dynamic Fitting
All the yield parameters related to Methanobacterium thermoautotrophicum bacteria metabolism (reported in Table 3) were taken from the work of Schill et al. [16]. Since all the other yield parameters were calculated using the stationary state of the Anaerobic Digestion, the only parameter to evaluate remained k L,H2 a in the case of 4 L/min and 6 L/min biogas flow recirculation. A non-linear regression, based on the minimization of the Mean Square Error, was performed with the aid of the process simulator gProms (Process System Enterprises, London, UK) to fit the experimental data.

Anaerobic Digestion Mode
During the Anaerobic Digestion Mode, the system reached stable conditions already after 10 days. The amount of biogas produced was of 30.8 ± 2.1 mL/min with a volumetric composition of 51.21% ± 1.04 and 49.28% ± 2.36 for CH 4 and CO 2 , respectively. The value of VSS calculated from liquid samples of the digestate was of 6.3 ± 0.1 (g/L), and it corresponds to the value used for X L . The concentration of biomass in the immobilized part of the reactor (X im ) was calculated to be 7.51 ± 0.3 (g/L). As explained in the previous section, those experimental data (reported in Table 4) were fundamental for the estimation of the yield parameters related to the acidogenesis and acetoclastic methanogenesis reactions. By means of the Equation (23), it was calculated α = 0.36. Subsequently, from Equations (18) and (19) S * 1 and S * 2 were found to be 71 mg/L and 0.15 mmol/L respectively. According to the Equations (20)- (22) were evaluated the yield parameters k 1 and k 3 , while from Equations (24) and (25), the yield parameters k 5 and k 6 were determined. The values of all the yield parameters related to the Anaerobic Digestion Mode are reported in Table 5. Yield for CO 2 formation from acidogenesis mmol/g 50.6 [19] k 5 Yield for CO 2 formation from acetoclastic methanogenesis mmol/g 208 This work k 6 Yield for CH 4 formation from acetoclastic methanogenesis mmol/g 275 This work

Bio-Methanation Mode
Before running the simulation, the initial conditions of the system should be provided. The gaseous phase initial conditions were fixed according to the biogas composition at the beginning of the experiment. Different considerations were necessary to select the appropriate initial conditions for the liquid phase. The initial substrate concentrations were set as S * 1 and S * 2 of the Anaerobic Digestion mode. The microbial population analysis, carried out by Lembo et al. [17] on the same reactor in previous experiments, showed that the 0.1% of the total biomass during AD mode was made of methanogens mainly belonging to the methanobacteriaceae family and the latter was mainly composed by hydrogenotrophic Methanothermobacter thermoautrophicus. For this reason, X 3 was estimated as 0.1% of the total VSS of the reactor liquid effluent during Anaerobic Digestion. Since the ratio between the acidogenic microbial population respect to the total one (γ) was kept to 0.2, the following initial conditions for the biomass were adopted: The results of model fitting are reported in Figure 3, and the values of the hydrogen mass transfer coefficient, derived from fitting, are reported in Table 6. As expected, enhancing the recirculation flowrate, the mass transfer coefficient increases. In particular, modifying the recirculation rate from 4 to 6 L/min, the K L,H 2 a changes from 100 to 220 d −1 . This phenomenon is explained by considering the reduction of the gas-liquid film resistance due to the induction of higher turbulence conditions. The results of model fitting are reported in Figure 3, and the values of the hydrogen mass transfer coefficient, derived from fitting, are reported in Table 6. As expected, enhancing the recirculation flowrate, the mass transfer coefficient increases. In particular, modifying the recirculation rate from 4 to 6 L/min, the , changes from 100 to 220 d −1 . This phenomenon is explained by considering the reduction of the gas-liquid film resistance due to the induction of higher turbulence conditions. Figure 3. Experimental data fitting related to the composition of (a) hydrogen, (b) methane and (c) carbon dioxide and to (d) the outlet biogas flowrate. At day 7 the inlet flowrate of hydrogen in the system was switched from 10 mL/min to 50 mL/min, and the recirculation biogas flowrate was switched from 4 L/min to 6 L/min. As it is possible to notice in Figure 3, the model well represents the dynamic of the system even when a step change in the inlet hydrogen stream is applied. The experimental data fluctuation observed in Figure 3 related to the biogas composition may be caused by the variation of lactose content in the Second Chees Whey (SCW), which was not considered in the model. In order to obtain further information about the system behavior, the dimensionless number of Stanton and β were calculated as follows: where is volumetric recirculation flowrate, is the characteristic time of reaction, is the characteristic residence time of the gas into the reactor, and is the characteristic time of material transfer from the gaseous to the liquid phase [15]. Both St and depend  As it is possible to notice in Figure 3, the model well represents the dynamic of the system even when a step change in the inlet hydrogen stream is applied. The experimental data fluctuation observed in Figure 3 related to the biogas composition may be caused by the variation of lactose content in the Second Chees Whey (SCW), which was not considered in the model. In order to obtain further information about the system behavior, the dimensionless number of Stanton and β were calculated as follows: where Q R is volumetric recirculation flowrate, τ r is the characteristic time of reaction, τ p is the characteristic residence time of the gas into the reactor, and τ t is the characteristic time of material transfer from the gaseous to the liquid phase [15]. Both St and β depend on time. According to this, the model was used to calculate St and β during time as reported in Figure 4.
zation of a gas sparger which may reduce the dimensions of the flowing bubbles. Another important factor is the influence of biomass immobilization on the system performances.
In this work such aspect was taken into account by the introduction of the parameter α.
The yield parameters of the Anaerobic Digestion are highly sensible to the value attributed to α. Bearing this in mind, it is possible to affirm that the model it is able to reproduce the behavior of the system, but further experiments should be carried out to really understand the role of bacteria immobilization, and its effect on the yield parameters of the model.

Conclusions
In this work, the model proposed was able to reproduce the experimental data of a 49 L randomly packed Gas Stirred Tank Reactor (GSTR). The parameter estimation was performed in two steps applied in the following order: (1) calculation of the yield parameters for the acidogenic and acetoclastic microorganisms using the stationary state of the reactor during anaerobic digestion and (2) evaluation of the hydrogen mass transfer coefficient by means of a dynamical fitting. It was found out that the hydrogen mass transfer coefficient strictly depends upon the recirculation flowrate. Moreover, by defining the dimensionless numbers of St and β, it was understood that the system could be capable of treating a higher amount of hydrogen as long as the hydrogen mass transfer coefficient is increased sufficiently. Further investigations are necessary to better understand how the As shown in Figure 4, β >> 1 for the working period. This condition indicates that the consumption of hydrogen in the liquid phase is much higher than its supply. On the other hand, St ≈ 1, which means that the residence time of hydrogen in the reactor is comparable with the characteristic time for its transfer to the liquid phase. More precisely, the value of St changes from 0.92 to 1.35 with the recirculation rate of 4 L/min and 6 L/min, respectively. According to the consideration made on β, a possible strategy to increase the reactor performances is to enhance the liquid to gas mass transfer coefficient in order to exploit the capacity of biomass in converting H 2 and CO 2 into CH 4 . It was observed that the recirculation flowrate plays a fundamental role for the regulation of K L,H 2 a, but the values of St are still too low to obtain a high conversion of the hydrogen injected in the reactor. Since St grows once increased the recirculation flowrate, it is possible to conclude that the benefits obtained by enhancing the recirculation flowrate (increase of K L,H 2 a) are higher than the drawbacks (decrease of the residence time of the gas in the reactor). However, a significant increase of St could be obtained only if K L,H 2 a is adjusted by modifying other design parameters rather than the recirculation rate. In order to clarify this aspect, further studies are necessary to investigate deeper the dependence of K L,H 2 a with the biogas recirculation flow. In addition, the mass transfer coefficient can be enhanced with the utilization of a gas sparger which may reduce the dimensions of the flowing bubbles. Another important factor is the influence of biomass immobilization on the system performances. In this work such aspect was taken into account by the introduction of the parameter α. The yield parameters of the Anaerobic Digestion are highly sensible to the value attributed to α. Bearing this in mind, it is possible to affirm that the model it is able to reproduce the behavior of the system, but further experiments should be carried out to really understand the role of bacteria immobilization, and its effect on the yield parameters of the model.

Conclusions
In this work, the model proposed was able to reproduce the experimental data of a 49 L randomly packed Gas Stirred Tank Reactor (GSTR). The parameter estimation was performed in two steps applied in the following order: (1) calculation of the yield parameters for the acidogenic and acetoclastic microorganisms using the stationary state of the reactor during anaerobic digestion and (2) evaluation of the hydrogen mass transfer coefficient by means of a dynamical fitting. It was found out that the hydrogen mass transfer coefficient strictly depends upon the recirculation flowrate. Moreover, by defining the dimensionless numbers of St and β, it was understood that the system could be capable of treating a higher amount of hydrogen as long as the hydrogen mass transfer coefficient is increased sufficiently. Further investigations are necessary to better understand how the recirculation flowrate affects the hydrodynamics of the reactor and to quantify the influence that the introduction of a gas sparger may induce to the hydrogen mass transfer. In addition, also the benefits given by the presence of random packing should be deeply evaluated.  Half saturation constant associated with H 2 (µmol/L) K L,H2 Hydrogen mass transfer coefficient (1/d) K S1 Half saturation constant associated with S 1 (g/L) K S2 Half saturation constant associated with S 2 (mmol/L) K I Inhibition constant associated with S 2 (mmol/L) k 1 Yield for lactose consumption k 2 Yield for VFA formation (mmol/g) k 3 Yield for VFA consumption (mmol/g) k 4 Yield for CO 2 formation from acidogenesis (mmol/g) k 5 Yield for CO 2 formation from acetoclastic methanogenesis (mmol/g) k 6 Yield for CH 4 formation from acetoclastic methanogenesis (mmol/g) k 8 Yield for Methanobacterium thermoautotrophicum growth (g VSS /mol) k 9 Yield for CO 2 consumption from idrogenotrophic methanogenesis k 10 Yield for CH 4 production idrogenotrophic methanogenesis m Maintenance coefficient (mol/g d) q max Maximum hydrogen specific production rate (mol/g d) St Stanton dimensionless number S 1 Organic substrate consumed by the biomass X 1 (g/L) S 2 Volatile Fatty Acids (mmol/L) V L Reactor liquid volume (L) X 1 Acidogenic bacteria (g/L) X 2 Acetoclastic methanogen archaea (g/L) X 3 Methanobacterium thermoautotrophicum biomass (g/L) Greek symbols α Fraction of the total non-immobilized biomass β Dimensionless number γ Biomass fration of the acidogenic microbial population µ 1,max Maximum acidogenic biomass growth rate (1/d) µ 2,max Maximum acetoclastic methanogenic biomass growth rate (1/d)