Abstract-- A two-parameter model was applied to describe the activity coefficient for a culture medium, and, compared with Henry’s law for water, Henry's law fitted to experimental data, the extended UNIQUAC model, and experimental data obtained from an experimental setup system, consisting of a liquid culture media Thiobacillus (ATCC 290) with hydrogen sulfide (H2S), dimethyl sulfide (DMS), methyl mercaptan (MM), and dimethyl disulfide (DMDS), separately.
The R2 and R2adj coefficients obtained from the analysis show that Henry's Law for water gives lower or null determinations than Henry’s Law adjusted. Adjusted Henry’s Law shows lower determination coefficients than the two-parameter model (with differences in R2adj between 0.69% and 3.64%), except for DMS (R2adj = 95.88% for adjusted Henry’s Law and R2adj = 94.00% for two-parameter model). The extended UNIQUAC model presents lower determinations than the other models for all compounds except H2S. On the other hand, this species has the worst comparative fits in all the models reviewed, even when Henry's law was adjusted, which may be associated with its effective solubility in the complex culture medium where the study was carried out.
The ANOVA test for model discrimination shows that, for H2S, MM, and DMDS, the two-parameter model significantly represents better the liquid-vapor distribution for such components, precisely when the gas concentration was upper 300 [ppm].
In complex culture media very different from pure water, adjusted models to describe the liquid-vapor equilibrium are necessary due to the high deviations compared to pure water. In the case of the culture medium, using Henry's Law is not sufficient to describe the equilibrium for H2S, MM, and DMDS.
Keywords-- activity coefficient; culture medium; partition coefficient; liquid-vapor balance; liquid-vapor equilibria
I. INTRODUCTION
The mass transfer from the gas to the liquid phase is involved in many bioprocess operations such as fermenters or biofilters. Designing, operating, or scaling up such types of equipment is essential to predict the composition in the liquid phase to determine parameters such as yields or mass transfer coefficients (Garcia-Ochoa and Gomez, 2009).
The distribution of the compounds in the liquid and gas phase is usually modeled by Henry's Law, assuming that the liquid media is pure water; however, in processes using microorganisms; the liquid medium contains salts and organic compounds (polar and non-polar) that allow cellular activity, which means that in this type of process it is no operated under conditions similar to pure water or in infinite dilution (Robles et al., 2018; Solon et al., 2019; Vergara-Fernandez et al., 2019). For this reason, the ionic strength in the solution could cause model deviations (Balomenos et al., 2006). Some mixed models that describe electrolytes and non-electrolytes solutions can be used for a better approximation, i.e., determination of the CO2 and O2 solubility in culture media using the extended UNIQUAC model (Gros et al., 1999; Martis et al., 2013); however, such types of models require a significant number of parameters depending on the interactions considered (Domańska, 2019).
Due to this problem, simplified models have been extended to facilitate calculation (May and Rowland, 2019). For example, if density and dielectric constant are set to the values for pure water, it does not significantly alter the deviation of the results (Faramarzi et al., 2009). Another approach proposed a semi-empirical one-parameter model based on the Debye-Hückel equation, which has determined activity coefficients for solutions over a wide range of ionic strength via experimental data fitting (Lee and Han, 2013).
Based on such work, a two-parameter model that describes the vapor-liquid equilibria in a liquid culture medium is applied for the determination to the vapor-liquid equilibria in a culture media using a new strategy for the adjust to the parameters, taking as reference the partition coefficient for different total reduce compounds (TRS): hydrogen sulphide (H2S), dimethyl sulphide (DMS), methyl mercaptan (MM), and dimethyl disulphide (DMDS) in a Thiobacillus liquid culture medium (ATCC 290).
II. MATERIALS AND METHODS
The distribution of species in equilibrium was determined by using a batch gas-liquid system. Known quantities of gaseous compounds were introduced, and after equilibrium, through a mass balance, the concentrations in the liquid phase were determined using Eq. (1).
(1)
where 
is the
concentration in the liquid phase (PPM), 
is the system
pressure (atm), 
is the ideal gas constant (0.082 atm L mol-1 K-1),
is system
temperature (K), 
is the volume
of liquid phase (L), 
is the
density of the liquid phase (g L-1), 
is the
concentration of the inlet compound (PPM), 
is the
concentration of the compound in the gas phase at equilibrium (PPM), and 
is the volume
of the gas phase of the system (L).
For our purposes, 50 ± 0.1 cm3 of Thiobacilli liquid medium (ATCC 290,) were put in a flask of 130 ± 0.1 cm3. The composition of the liquid medium was, in g L-1: Na2HPO4∙7H2O 2.27; KH2PO4 1.8; MgCl2∙7H2O 0.1; (NH4)2SO4 1.98; MnCl2∙H2O 0.023; CaCl2 0.03; FeCl3∙6H2O 0.033; Na2CO3 1; Na2S2O3∙5H2O 15.69, and pH adjusted to 1.5. A manometer (Winters DPG213) was connected into the flask cap to measure the inside pressure. The flask was hermetically sealed, and known amounts of gaseous H2S, DMS, MM, DMDS were added. The flask was immersed in a thermostatic bath (Tecnal TE-184) at 50 ± 0.1 °C. This temperature was selected as a reference due to many biological process systems use this culture medium for mesophilic and thermophilic microorganisms. The equilibrium was verified when pressure variations of ± 0.02 atmg, according to the instrument precision, on were reached (0.12 atm). 1 ± 0.1 cm3 of gaseous samples were taken to verify that equilibrium. Gaseous concentrations were determined by using a Clarus® 500 Gas Chromatograph (PerkinElmer®) equipped with a flame photometric detector. A Supelpak-S column was used to quantify H2S, DMS, MM, and DMDS. pH was determined by a Hanna Instruments meter (model A20518). The same experiment was done with pure water to verify the reliability of the experimental system, comparing the solubility value obtained with those reported in literature (Iliuta and Larachi, 2007) obtaining differences of 1.5, 3.4, 2.8, and 4.1% for H2S, DMS, MM, and DMDS, respectively.
III. MATHEMATICAL MODEL
The activity coefficient for compound i, considering ideal gas, at equilibrium, is given by Eq. (2):
(2)
where 
is the
fraction of the compound in the gas phase, is the system
pressure, 
is the molar
concentration of the compound i in the liquid phase, and 
is the
Henry´s constant.
A. Extended UNIQUAC Model
The UNIQUAC equation is useful for non-electrolytic multicomponent solutions at different pressures and temperatures. For mixed electrolytic and non-electrolytic solutions, the extended UNIQUAC model (Eq. 3) is a combination of the UNIQUAC and the Debye-Hückel models (Hashemi et al., 2017).
(3)
where 
indicates the
contribution of the UNIQUAC model to the activity coefficient of the compound 
and 
indicates the
contribution of the Debye - Hückel model in the mixture of the same compound. For
mixed electrolytic and non-electrolytic solutions, the extended UNIQUAC model
(Eq. 3) combines the UNIQUAC and the Debye-Hückel models (Hashemi et al.,
2017). The combination of a symmetrical and an asymmetrical model results in an
asymmetrical model. Different reviews have demonstrated its effectivity (Sander
et al., 1986, Loehe and Donahue, 1997; Pinho and Macedo, 1996; Kumar,
1993).
The UNIQUAC contribution is given by (Eq. 4-8).
(4)
(5)
(6)
(7)
(8)
where 
is the
relative surface area; 
, the relative
molecular volume; and 
, an
adjustable interaction parameter.
The Debye - Hückel contribution (Eq. 9-12) is represented by simplifying the higher-order terms of the complete expression, according to the following assumptions:
· Complete electrolyte dissociation
· Each ion is surrounded by oppositely charged ions
· Negligible electrolyte concentrations
(9)
(10)
(11)
(12)
where 
is the solvent permittivity; 
is the
elementary charge number, 
is the molar
concentration; and 
is the ionic
radius, 
is the number
of compounds in the liquid mixture.
Though the Debye - Hückel model depends on several parameters,
some of them could be fixed. In this work, a truncated Debye- Hückel model
(Khan et al., 2016) was used, where 
was set to
1.5 (kg•mol-1)0.5,
and the value of 
(kg•mol-1)0.5 is set as a function of temperatures:
(13)
B. Single parameter model
Lee and Han (2013) developed a model that depends on a proportional
factor (
) and the
effective radius of an ionic sphere (
).
(14)
where 
the activity coefficient, 
is the ionic
strength (Eq.10) and is 
the Debye
screening length (3.0434 Å·I−0.5 m).
The value of is related to
depending on
the type of electrolyte. For example, in a 1:1 electrolyte ratio, relates to as:
(15)
Table1. Henry’s constants, H (M atm-1)
Therefore, this model corresponds to a single-parameter model: the ion radius.
Due to liquid culture mediums are not a 1:1
electrolyte solution, in this work, and are
considered as separate parameters, so their values were fitted using
experimental data.
C. Calculation Method
Initially, the equilibrium was determined using Henry’s law for pure water, taking the values of Henry’s constant from literature (Dobryakov and Vitenberg, 2003; Przyjazny et al., 1983) and fitting the experimental data (Table 1).
In a culture medium, the liquid solution contains several and different dissolved and ionized substances. In our case, we use a Thiobacillus culture medium (ATCC 290) where we consider the following equilibrium reactions:
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
For these equilibria, in the UNIQUAC model, the parameters of interaction were estimated (Pahlevanzadeh and Mohseni-Ahooei, 2005) while the geometric and molecular parameters were taken from the literature (Arrad et al., 2015; Arrad et al., 2016; Arrad et al., 2017; Boulkroune et al., 2013; Raatikainen and Laaksonen, 2005; Thomsen et al., 1996).
IV. RESULTS
The values of and obtained by
the two-parameter model for each compound studied are shown in Table 2. These
values are similar in magnitude to those reported by Lee and Han (2013), who
obtained 1.375
10−10
and 1.88410−10
for and on average,
respectively.
Figures 1 to 4 show the liquid-phase concentrations of the H2S, DMS, MM, and DMDS systems at equilibr-
Table 2. Parameters 
and 
for each studied compound.
Fig. 1: Liquid phase concentration for H2S at different inlet concentrations (● Experimental, ____ Henry water, _ _ _ Henry adjusted, ---- UNIQUAC Ext, …. Two-parameter model).
Fig. 2: Liquid phase concentration for DMS at different inlet concentrations (● Experimental, ____ Henry water, _ _ _ Henry adjusted, ---- UNIQUAC Ext, …. Two-parameter model).
ium, respectively, determined through Henry's law, Henry’s law adjusted from experimental data, the extended UNIQUAC model, and two-parameter model, compared with experimental data which include their respective error bars. In the range of concentrations tested, Henry’s law adjusted, the two-parameters model and the extended UNIQUAC had a good fit with experimental data in contrast with Henry’s law for water that shows a high deviation.
Table 3 shows the R2
and R2adj coefficients for each study case. In general,
Henry's Law for water gives lower or null determinations than Henry's Law
adjusted. The adjusted Henry's Law shows lower determination coefficients than
the two-parameter model, except for DMS. On the other hand, the extended
UNIQUAC model shows lower determinations than these other models for all
compounds except H2S, which could be explained due to the
overparameterization of the model, which worsens the
