Figure 3. AAE% values of three variables (T, y AC , and y Et ) obtained from the representation of the vapor-liquid equilibria of ternary systems volatile aroma compound-ethanol-water at 101.3 kPa with different thermodynamic models. (gray bars) Wilson, (blue bars) NRTL, (cyan bars) UNIQUAC, (pink bars) UNIFAC, and (red bars) PSRK. The horizontal dashed lines represented the average AAE% value for each model. Comparison results from Fauńdez et al. 18 and Fauńdez and Valderrama. 20

COMPILATION AND THERMODYNAMIC

 (10 compounds, including one acetal, five alcohols, and sour esters), and Deterre et al., 32 who studied five bitter orange aroma compounds, including two monoterpene hydrocarbons and three oxygenated terpenes.

 This latter study was carried out by the same authors of this paper.

In general, the data are of very variable nature, but in all cases (ternary and multicomponent systems), they are related, directly or indirectly, to the absolute and relative volatilities at 101.3 kPa. No studies presenting experimental data of activity coefficients at infinite dilution and equilibrium conditions were found in the open literature.

One important point in common between the different studies is that the measurements were performed via a dynamic method with recirculating stills. In this method, known for providing rapid and accurate vapor-liquid equilibria data, the

information is fundamental to understand the behavior of aroma compounds during distillation, and therefore to master their concentration to the desired levels, through thermody namic modeling and simulation.

The equilibrium behavior of an aroma compound (AC) in hydro alcoholic medium (solvent), which depends on the physical conditions (T, P) and on the solvent composition, can be characterized by means of three parameters:

• The partition coefficient, equilibrium constant, or absolute volatility (K AC ), which quantifies the distribu tion between the vapor (y AC ) and liquid phases (x AC ). • The relative volatilities with respect to ethanol (Et) (α AC/Et ) and water (W) (α AC/W ), indicator of the repartition of the aroma compounds between the top and bottoms product in distillation. 15 • The activity coefficient at infinite dilution (γ AC ∞ ), a thermodynamic parameter that characterizes the aroma compound (solute)-mixed solvent interactions in the absence of solute-solute interactions, providing accurate information about the deviation from ideality. 16,17 Regarding the thermodynamic modeling, a classical hetero geneous approach can be applied to the systems aroma compounds-ethanol-water because all the chemical species are polar and can develop complex intermolecular interactions (such as van der Waals interactions, hydrogen bonds, and chemical association), and second because the distillation units operate at atmospheric pressure. In this case, the vapor phase is often represented as an ideal gas, whereas the main deviations from ideal behavior are associated with the liquid phase and are described with an excess Gibbs free energy (G E ) model. [START_REF] Fauńdez | Predictive models to describe VLE in ternary mixtures water + ethanol + congener for wine distillation[END_REF] The choice of a suitable thermodynamic model is based on the research work published by Fauńdez and Valderrama concerning the thermodynamic modeling of mixtures found in wine distillation. 7,[START_REF] Fauńdez | Predictive models to describe VLE in ternary mixtures water + ethanol + congener for wine distillation[END_REF][START_REF] Fauńdez | Phase equilibrium modeling in binary mixtures found in wine and must distillation[END_REF][START_REF] Fauńdez | Activity coefficient models to describe vapor liquid equilibrium in ternary hydro alcoholic solutions[END_REF][START_REF] Valderrama | Modeling of vapor-liquid equilibrium in binary and ternary mixtures of interest in alcoholic distillation[END_REF] Their research was focused on three types of mixtures at atmospheric pressure: binary systems volatile aroma compound-ethanol and volatile aroma com pound-water, as well as ternary systems volatile aroma [START_REF] Fauńdez | Phase equilibrium modeling in binary mixtures found in wine and must distillation[END_REF] compound-ethanol-water. Equilibrium data for these systems are relatively scarce in the literature. The authors used data from specialized monographs and databases, including nine aroma compounds from five chemical families: alcohols (methanol, propan 1 ol, 2 methylpropan 1 ol, 3 methylbutan 1 ol, and pentan 1 ol), carbonyl compounds (ethanal), carboxylic acids (ethanoic acid), esters (ethyl ethanoate), and furans (furan 2 carbaldhyde).

The vapor-liquid equilibrium data were represented with two types of models: (i) semiempirical models, in which the experimental data are correlated by adjusting binary interaction parameters, and (ii) predictive models, which include universal parameters (nonspecific to a particular mixture) to represent phase equilibria. The semiempirical models compared in their work belong to the so called local composition approach for the calculation of activity coefficients and include Wilson, [START_REF] Wilson | Vapor liquid equilibrium. XI. A new expression for the excess free energy of mixing[END_REF] NRTL, [START_REF] Renon | Local compositions in thermodynamic excess functions for liquid mixtures[END_REF] and UNIQUAC. [START_REF] Abrams | Statistical thermodynamics of liquid mixtures: A new expression for the excess Gibbs energy of partly or completely miscible systems[END_REF] Regarding the predictive approach, two models were compared: UNIFAC [START_REF] Fredenslund | Group contribution estimation of activity coefficients in nonideal liquid mixtures[END_REF] and PSRK. [START_REF] Holderbaum | A group contribution equation of state based on UNIFAC[END_REF] UNIFAC is a local composition model based on the group contribution concept, while PSRK is an adaptation of the Soave-Redlich-Kwong equation of state [START_REF] Soave | Equilibrium constants from a modified Redlich Kwong equation of state[END_REF] that introduces a mixing rule based on an activity coefficient model (UNIFAC in this case) to compute the attractive parameter.

The results of their comparison are depicted in Figures 1 and2 for binary systems and in Figure 3 for ternary systems. In these figures, the average absolute deviation (AAE%) of the calculated values with respect to the experimental data are presented for two equilibrium variables: temperature (T) and mole fraction in the vapor phase (only for aroma compound (y AC ) in the case of binary systems, and for both aroma compound (y AC ) and ethanol (y Et ) in the case of ternary systems). According to this criteria, the NRTL model provides the best representation of the experimental data, as the overall AAE% values for T, y AC , and y Et are the lowest among the compared models. In the case of the binary systems aroma compound-ethanol (Figure 1), the overall AAE% for T is 1.2% (in comparison to 1.6% with UNIFAC and 1.8% with PSRK) and for y AC is 6.7% (in comparison to 8.8% with UNIFAC and PSRK). Concerning the binary systems aroma compoundwater (Figure 2), the overall AAE% values are slightly lower, 0.9% for T (in comparison to 1.6% with UNIFAC and 1.7% with PSRK), and 5.1% for y AC (in comparison to 10.9% with UNIFAC and 13.1% with PSRK).

Finally, in the case of ternary systems, the global balance for the 8 mixtures is also more favorable with the NRTL model (Figure 3). The overall AAE% value obtained with this model is 0.5% for T (in comparison to 1.0% with Wilson, 0.6% with UNIQUAC, 1.0% with UNIFAC, and 1.2% with PSRK), 18.2% for y AC (in comparison to 20.5% with Wilson, 19.9% with UNIQUAC, 21.8% with UNIFAC, and 25.2% with PSRK), and 3.5% for y Et , equivalent to the value obtained with the Wilson model and lower in comparison to the other models (3.6% with UNIQUAC, 4.8% with UNIFAC, and 4.8% with PSRK).

In the light of these results, the NRTL model is used in this work for modeling purposes. From a theoretical point of view, this choice is consistent regarding two aspects: (i) the nature of the investigated systems and (ii) the operating conditions in alcoholic beverages distillation. In the first case, because this model was developed to represent the phase behavior of moderately and strongly nonideal liquid mixtures by taking into account the effects of both differing molecular size and intermolecular forces. [START_REF] Renon | Local compositions in thermodynamic excess functions for liquid mixtures[END_REF] Regarding the second aspect, NRTL is well adapted to systems at pressures lower than 1000 kPa and temperatures below 150 °C, conditions in which the non ideality can be supposed to be located in the liquid phase. Furthermore, NRTL is appreciated as a performing model in process simulation, not only for its coherence with respect to the Gibbs-Duhem equation but also for a good computing efficiency due to a relatively small set of adjustable parameters. This model has been used to correlate both vapor-liquid and liquid-liquid equilibria with satisfactory results (including ethanol-water [START_REF] Voutsas | Measurements and thermodynamic modeling of the ethanol-water system with emphasis to the azeotropic region[END_REF][START_REF] Lai | Isobaric (vapor + liquid) equilibria for the ternary system of (ethanol + water + 1,3 propanediol) and three constituent binary systems at P = 101.3 kPa[END_REF] as well as some binary [START_REF] Penã Tejedor | Vaporliquid equilibria and excess volumes of the binary systems ethanol + ethyl lactate, isopropanol + isopropyl lactate and n butanol + n butyl lactate at 101.325 kPa[END_REF][START_REF] Vu | Vapor-liquid equilibria in the systems ethyl lactate + ethanol and ethyl lactate + water[END_REF] and multi component aroma systems at high dilution [13][14][15][16][17][START_REF] Fauńdez | Predictive models to describe VLE in ternary mixtures water + ethanol + congener for wine distillation[END_REF][START_REF] Fauńdez | Phase equilibrium modeling in binary mixtures found in wine and must distillation[END_REF][START_REF] Fauńdez | Activity coefficient models to describe vapor liquid equilibrium in ternary hydro alcoholic solutions[END_REF][START_REF] Valderrama | Modeling of vapor-liquid equilibrium in binary and ternary mixtures of interest in alcoholic distillation[END_REF][START_REF] Wilson | Vapor liquid equilibrium. XI. A new expression for the excess free energy of mixing[END_REF][START_REF] Renon | Local compositions in thermodynamic excess functions for liquid mixtures[END_REF][START_REF] Abrams | Statistical thermodynamics of liquid mixtures: A new expression for the excess Gibbs energy of partly or completely miscible systems[END_REF][START_REF] Fredenslund | Group contribution estimation of activity coefficients in nonideal liquid mixtures[END_REF][START_REF] Holderbaum | A group contribution equation of state based on UNIFAC[END_REF][START_REF] Soave | Equilibrium constants from a modified Redlich Kwong equation of state[END_REF][START_REF] Voutsas | Measurements and thermodynamic modeling of the ethanol-water system with emphasis to the azeotropic region[END_REF][START_REF] Lai | Isobaric (vapor + liquid) equilibria for the ternary system of (ethanol + water + 1,3 propanediol) and three constituent binary systems at P = 101.3 kPa[END_REF][START_REF] Penã Tejedor | Vaporliquid equilibria and excess volumes of the binary systems ethanol + ethyl lactate, isopropanol + isopropyl lactate and n butanol + n butyl lactate at 101.325 kPa[END_REF][START_REF] Vu | Vapor-liquid equilibria in the systems ethyl lactate + ethanol and ethyl lactate + water[END_REF][START_REF] Deterre | Vapor-liquid equilibria measurements of bitter orange aroma compounds highly diluted in boiling hydro alcoholic solutions at 101.3 kPa[END_REF] ), as well as mixing heat data and infinite dilution activity coefficients for a great number of polar and nonpolar binary and multicompound systems.

Although the results of research from Fauńdez and Valderrama's research group are useful for general under standing and thermodynamic modeling, they suffer from some drawbacks for simulation purposes: (i) they deal with a limited number of aroma compounds (between 8 and 12) and (ii) the experimental data used for modeling correspond to binary or ternary mixtures in which the ranges of concentration of the aroma compounds are generally much higher than those actually found in alcoholic beverages distillation. For ternary mixtures aroma compound-ethanol-water, the molar fractions in the liquid phase vary between 8 × 10 -4 and 8 × 10 -1 , while for binary mixtures aroma compound-ethanol and aroma compound-water the whole concentration interval in the liquid phase is included (0 < x AC < 1). In both cases, the concentration interval is very different from the case of alcoholic beverages in which the aroma compounds are present at high or infinite dilution.

In this context, the objectives of the current study are to generate a database containing all the information available in the open literature on the vapor-liquid equilibria of aroma compounds highly diluted in ethanol-water mixtures at atmospheric pressure and to generate new binary interaction parameters of the NRTL model for simulation purposes. 44 representative aroma compounds present in distilled beverages such as Armagnac, Calvados, and Cognac are considered. With the purpose of evaluating the extrapolation capability of NRTL model, the study is concluded by a comparison of the equilibrium representation obtained when using the new set of parameters and the one derived from parameters estimated from binary or ternary mixture data at high concentrations.

The paper is organized as follows: a general description of the equilibrium information available in the literature is presented, followed by some elements on the thermodynamic modeling approach. Using the literature data, a set of binary interaction parameters is estimated, followed by a classification of the aroma compounds, and finally by a comparison of the representation obtained with different sets of interaction parameters. vapor, evolved from a liquid phase, is continuously separated under steady state conditions and directed to the condenser, configured to prevent, or at least minimize, any risk of reflux. 8,[START_REF] Soni | Vapour-liquid equilibria and infinite dilution activity coefficient measurements of systems involving diketones[END_REF] The recirculation is maintained until the composition variation of the vapor or liquid phases is no longer appreciable. [START_REF] Othmer | Composition of vapors from boiling binary solutions[END_REF] The recirculating devices can be classified in two categories:

• In the first one, only the vapor phase circulates within the apparatus, while the liquid remains in the boiling chamber. For measurements involving aroma com pounds, all the devices are based on the design of Altsheler and Othmer stills. [START_REF] Othmer | Composition of vapors from boiling binary solutions[END_REF][START_REF] Othmer | Composition of vapors from boiling binary solutions. Improved Equilibrium still[END_REF] • In the second category, the vapor and liquid phased are recirculated and maintained in intimate contact before they are disengaged. This case covers the Gillespie type stills [START_REF] Gillespie | Vapor liquid equilibrium still for miscible liquids[END_REF] and more particularly the Labodest still, developed by i Fischer Engineering GmbH. A detailed description of this latter has been already presented in several experimental works. 13,[START_REF] Puentes | Vapor liquid equilibrium (VLE) of ethyl lactate highly diluted in ethanol water solutions at 101.3 kPa: experimental measurements and thermodynamic modeling with semi empirical models[END_REF][START_REF] Nala | Vapour-liquid equilibrium (VLE) for the systems furan + n hexane and furan + toluene. Measurements, data treatment and modeling using molecular models[END_REF][START_REF] Dias | Vapor-liquid equilibrium of mixtures containing the following higher alcohols: 2 propanol, 2 methyl 1 propanol, and 3 methyl 1 butanol[END_REF] The second category of recirculating stills has been recommended for equilibrium measurements of diluted mixtures at temperatures higher than 298.15 K when coupled to an accurate analysis technique of the liquid and condensed vapor composition. [START_REF] Christensen | Measurement of dilute mixture vapor-liquid equilibrium data for aqueous solutions of methanol and ethanol with a recirculating still[END_REF][START_REF] Rall | Phase Equilibria: Measurement and Computation[END_REF] However, given the highlighted scarcity of vapor-liquid data of aroma compounds-ethanol-water systems, all the information found in the open literature, concerning both the categories of recirculating devices, will be retained in the current work.

In Table 1, the different studies are summarized and classified according to the recirculating method. This synthesis includes information about the experimental measurements (composition analysis, type of equilibrium data, and aroma compounds studied) as well as main features of the thermodynamic modeling when performed.

As a result of this compilation, vapor-liquid equilibrium data at 101.3 kPa for 44 aroma compounds were extracted. The references and some specifications for each chemical species are presented in Table 2, including the number of data points and, if available, the composition ranges of ethanol and of aroma compounds in the vapor and liquid phases.

For most of the chemical species studied in this work, including ethanol and water, the physical properties were available in the Simulis Thermodynamics database, software developed by ProSim and used for the equilibrium calculations presented in the current work. Regarding five missing species, the properties were extracted from literature and subsequently added to the database. These compounds are (Z) hex 3 en 1 ol, ethyl hexanoate, 2 phenylethyl ethanoate, ethyl octanoate, and ethyl decanoate. For equilibrium calculations, the whole set of information provided to characterize each chemical species was: CAS no., molecular formula, molecular mass (MM), boiling temperature at 101.3 kPa (T b ), and vapor pressure (P O ). Most of these values were taken from NIST Chemistry Webbook. 49 Further information about the calculation of vapor pressures for these compounds are presented in section 2.2.

2.2. Thermodynamic Modeling. The principle for vaporliquid equilibrium modeling is the equality of chemical potentials of every species in both phases in conditions of thermal and mechanical equilibrium. For engineering applica tions, the equilibrium condition is written in terms of fugacity, a variable that corresponds to a generalized partial pressure and that depends on the temperature (T), pressure (P), and composition. [START_REF] Prausnitz | Molecular Thermodynamics of Fluid Phase Equilibria[END_REF][START_REF] Rios | Lecture Notes on Chemical Thermodynamics[END_REF] By following a classical heterogeneous approach, also known as the gamma-phi (γ-ϕ) method, the vapor phase fugacity of the component i, f i V (T,P,y), can be expressed as a function of the fugacity coefficient, ϕ i V (T,P,y), and the liquid phase fugacity, f i L (T,P,x), as a function of the activity coefficient, γ i (T,x), and the standard state liquid fugacity, f i OL (T,P). The resulting expressions are

= y x f T P f T P ( , , ) ( , , ) i V i L (1) ϕ γ = y x T P yP T xf T P ( , , ) ( , ) ( , ) i V i i i i OL (2)
At atmospheric pressure, the standard state liquid fugacity can be approximated to the vapor pressure of the pure component at the temperature of the system, P i O (T), and the vapor phase can be considered as an ideal gas mixture, which means that fugacity coefficient is equal to 1. In this way, eq 2 becomes

γ = x yP T xP T ( , ) ( ) i i i i O (3)
which can in turn be rewritten to obtain an expression for the absolute volatility, as follows:

γ = = x x K T P y x T P T P ( , , ) ( , ) ( )
i i i i i O (4)
The relative volatility with respect to ethanol and water are defined as the ratio of absolute volatilities, that is

α γ γ = = = x x x T K K y x y x T P T T P T ( , ) / / ( , ) ( ) ( , ) ( ) i i i i i i O O /Et Et Et Et Et Et
(5)

α γ γ = = = x x x T K K y x y x T P T T P T ( , ) / / ( , ) ( ) ( , ) ( ) i i i i i i O O /W W W W W W (6)
In the case of carboxylic acids, it is necessary to use a supplementary term that account for the chemical equili brium. [START_REF] Detcheberry | Thermodynamic modeling of the condensable fraction of a gaseous effluent from lignocellulosic biomass torrefaction[END_REF] Indeed, theoretical and experimental studies [START_REF] Allen | The association of carboxylic acids[END_REF][START_REF] Vawdrey | Vapor phase association of n aliphatic carboxylic acids[END_REF] have demonstrated that this kind of compounds can exist either as monomer (single molecules) or dimers due to strong hydrogen bonds. This phenomenon is referred to as chemical association or dimerization and essentially takes place in the vapor phase. Higher polymerization and heterodimerization (formation of dimers from monomers of different chemical species) have also been discussed in the literature, [START_REF] Miyamoto | Measurement of isothermal vapor liquid equilibria for binary and ternary systems containing monocarboxylic acid[END_REF][START_REF] Zhu | Geometric structures of associating component optimized toward correlation and prediction of isobaric vapor-liquid equilibria for binary and ternary mixtures of ethanal, ethanol, and ethanoic acid[END_REF] but they will not be considered in this work for the sake of simplicity. For a carboxylic acid (CA x ), the reaction and corresponding chemical equilibrium constant (KD CAx ) can be expressed as follows:

υ ⇌ = + x x T 2CA CA KD ( ) (10 ) x L M T 2 C A / x x CA CA (7)
T is given in K and the parameter υ is a correction factor to express KD CAx in kPa -1 , equal to 7.5. L CAx and M CAx are empirical coefficients to quantify the dependency of the dimerization constant with temperature. On the basis of this representation, Detcheberry et al. [START_REF] Detcheberry | Thermodynamic modeling of the condensable fraction of a gaseous effluent from lignocellulosic biomass torrefaction[END_REF] have proposed the following correction for eq 3:

ϕ γ = = x yP T P T T xP T i x ( , ( )) ( , ) ( ) For CA i i OV i O i i i O (8)
where the term ϕ i OV (T,P i O (T)) can be calculated with the following relation:

ϕ = -+ = T P T T P T T P T i x ( , ( )) 1 [4KD( ) ( )] 2KD ( ) ( ) For CA i OV i O i i O i i O 1/2 (9)
Therefore, the alternative equations for calculating K i , α i/Et , and

α i/W are γ ϕ = = = x x K T P y x T P T T P T P i x ( , , ) ( , ) ( ) ( , ( ))
For CA

i i i i i O i OV i O (10) α γ ϕ γ = = = = x x T x K K y x y x T P T T P T T P T i x ( , ) / / ( , ) ( ) ( , ( )) ( , ) ( ) For CA i i i i i i O i OV i O O /Et Et Et Et Et Et (11) α γ ϕ γ = = = = x x T x K K y x y x T P T T P T T P T i x ( , ) / / ( , ) ( ) ( , ( )) ( , ) ( ) For CA i i i i i i O i OV i O O /W W W W W W (12)
For the first five carboxylic acids studied in this work (from methanoic to 2 methylpropanoic acid), the values of coefficients L CAx and M CAx were taken from the DECHEMA database, available through the Simulis Thermodynamics software. [START_REF] Gmehling | Vapor-liquid Equilibrium Data Collection. Part 5: Carboxylic Acids, Anhydrides, Esters[END_REF] The missing coefficients were estimated by linear interpolation or extrapolation, by using the values available for heptanoic acid (not studied in this work) as supplementary data. The approximation criterion chosen was the number of carbon atoms in the molecule (n C ), for two reasons: (i) the group of compounds corresponds to the homologous series of carboxylic acids from methanoic to octanoic acid, and (ii) the values of L CAx and M CAx for butanoic acid and 2 methylpropanoic acid, both with four carbon atoms, are the same. The estimation equations were established by taking as unique reference points the values of L CAx and M CAx for butanoic acid and heptanoic acid, obtaining:

= - ≥ L n n 0.100 10.501 4 x CA C C (13) = - + ≥ M n n 47.341 3229.2 4 x CA C C (14)
The final set of parameters is summarized in Table 3. No bias of interpolation or extrapolation is associated with this calculation as only two reference points were considered. The values of L CAx and M CAx for the first three carboxylic acids (from methanoic to propanoic acid) were not included because they do not follow a linear trend. This can be explained by the fact that they are small molecules for which chemical association is stronger, hence the dimerized fraction in the vapor phase is more important than for acids with higher molar masses. As a result, one could expect that in such case there is no a simple, direct relation between the dimerization parameters and the number of carbon atoms. Nonetheless, a linear approximation in the interval 4 ≤ n C ≤ 8 remains valid because the known values of L CAx and M CAx are relatively close to each other. Moreover, this estimation is actually not critical as the vaporliquid equilibrium data are represented with a semiempirical model, a case in which the interaction parameters associated with the liquid phase are adjusted by minimizing the deviation between the experimental and calculated values. In the case of carboxylic acids, this fitting procedure implies that the activity coefficients condenses information not only about the energetic interactions in the liquid phase, but also, indirectly, about the association phenomena in the vapor phase.

Returning to phase equilibria modeling, in the region of very low concentration, it is customary to define the activity of the component i with respect to its fugacity at infinite dilution at the temperature and pressure of the mixture. The liquid fugacity is then expressed in an alternative way, the so called Henry law: 58

= x f T P x T P x ( , , ) ( , , ) i L i i s (15)
i , or Henry constant, corresponds to the reference liquid fugacity and is defined by the following relation:

= → x T P x f T P x ( , , ) lim ( , , ) i x i L i s 0 i ( 16
)
i depends not only on temperature and pressure but also on the solvent nature (here ethanol-water) and its composition, x s . By comparison with eq 3, i can also be expressed as the product P i O (T) and γ i (T,x) at infinite dilution or γ i ∞ (T,x s ):

γ = ∞ H T x T x P T ( , ) ( , ) ( ) i i i O s s (17)
Or, by comparison with eq 4, as a function of the absolute volatility and total pressure:

= H T x K P ( , ) i i s (18)
In the current study, the vapor pressure of ethanol, water, and the aroma compounds have been calculated with the Riedel equation, [START_REF] Vetere | The Riedel equation[END_REF] an extended version of the Antoine equation:

= + + + ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ P T A B T C T DT ( ) 1 1000 exp ln( ) i O i i i E i ( 19
)
With P i O (T) given in kPa and T in K. A i , B i , C i , D i , and E i are coefficients specific for each chemical species. Two sources were used to obtain them:

• For the compounds already included in the Simulis Thermodynamics database, the coefficients were taken from the DIPPR database. [START_REF] Rowley | DIPPR Data Compilation of Pure Chemical Properties; Design Institute for Physical Properties[END_REF] • For the five compounds added to the Simulis Thermodynamics database, the coefficients were esti mated by regression of experimental data available in the literature. 49,[START_REF] Dreisbach | Vapor pressure temperature data on some organic compounds[END_REF][START_REF] Verevkin | Determination of vaporization enthalpies of the branched esters from correlation gas chromatography and transpiration methods[END_REF][START_REF] Cervantes | Saturated vapour pressure of aroma compounds at various temperatures[END_REF][START_REF] Zaitsau | Thermodynamics of ethyl decanoate[END_REF][START_REF] Benziane | Experimental vapor pressures of five saturated fatty acid ethyl ester (FAEE) components of Biodiesel[END_REF][START_REF] Kozlovskiy | Vapor pressures and vaporization enthalpies of a series of esters used in flavors by correlation gas chromatography[END_REF][START_REF] Stejfa | Vapor pressures and thermophysical properties of selected hexenols and recommended vapor pressure for hexan 1 ol[END_REF][START_REF]German Social Accident Insurance: Sankt Augusin[END_REF] The minimized objective function (OF) is as follows:

∑ = - = - - - ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ P P P P OF( ) O k N k O k O k 1 exp calc exp 0 2 (20)
and the absolute average relative error (AAE%) between experimental and calculated pressure was obtained with eq 21:

∑ = - = - - - P P P P AAE%( ) 100% O k N k O k O k O 1 exp calc exp (21)
Because of the relatively limited number of experimental values, the coefficients D i and E i were fixed to 0. The values of AAE% vary between 1% for 2 phenylethyl ethanoate and 6% for ethyl decanoate.

For the set of 44 aroma compounds, the coefficients are valid in the temperature range of vapor-liquid equilibrium for hydro alcoholic mixtures at 101.3 kPa, from 351.4 to 373.15 K.

With regard to the activity coefficient, the non random two liquid (NRTL) model was used in this work. [START_REF] Wilson | Vapor liquid equilibrium. XI. A new expression for the excess free energy of mixing[END_REF] It is a pressure independent model of liquid solution based on the concept of local composition introduced by Wilson, [START_REF] Renon | Local compositions in thermodynamic excess functions for liquid mixtures[END_REF] valid at low pressures (less than 1000 kPa) and widely recommended for the description of hydro alcoholic solutions. 7 According to the hypothesis introduced by Wilson, the local concentration around a molecule can be different from the bulk concentration. This phenomenon is due to a difference between the interaction energy of the central molecule with molecules of its own nature and the interaction energy with molecules of other nature. The energy difference also introduces a nonrandomness effect at the local molecular level.

In the NRTL model, the activity coefficient of a component i in a mixture of n components as a function of composition and temperature is given by

∑ γ τ τ τ = ∑ ∑ + ∑ - ∑ ∑ = = = = = = ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ x T G x G x G x G x G x G x ln ( , ) i j n ji ji j k n ki k j n ij j k n kj k ij k n kj kj k k n kj k 1 1 1 1 1 1 (22) with τ = - G c exp( ) ij ij ij (23)
Here, G ij , c ij , and τ ij are binary interaction parameters. Their temperature dependence is evaluated according to the following formalism, included in the Simulis Thermodynamics package: ) account for the non-randomness of the mixture. These variables are specific to each pair of chemical species and must be identified from experimental data of vapor-liquid equilibria and/or other thermophysical properties of the liquid phase. The estimation of these parameters for the different aroma compounds, the object of this paper, is presented in the next section.

= + - c c c T ( 273.15) ij ij ij T 0 (24)

RESULTS AND DISCUSSION

A database including all the experimental vapor-liquid equilibria data previously presented was created and used for the generation of a set of binary interaction parameters of the NRTL model. This set, named NRTL 0, will be used to simulate distillation units involved in the production of alcoholic beverages. In this section, the estimation methodology and the parameters obtained are presented (section 3.1.), followed by a classification of the aroma compounds according to their relative volatilities (section 3.2.) using the model representation over the whole ethanol concentration range in the liquid phase (0 < x Et < 1). The discussion is concluded with a comparison of the equilibrium representation for some aroma compounds using several sets of parameters (section 3.3.): (i) the main set (NRTL 0), (ii) a set estimated from binary data for mixtures aroma compound-ethanol and aroma com pound-water (named NRTL B), and (iii) a set from ternary data for mixtures aroma compound at finite concentrationethanol-water (named NRTL T). Given that data at high dilution are scarce and more difficult to measure, the objective of this comparison is to evaluate if the representation from equilibrium data at finite concentration would be accurate enough for simulation purposes of other aroma compounds, similar in nature to those studied in this work, in the alcoholic beverages field.

3.1. Generation of a Set of Binary Interaction Parameters. The objective of modeling with the NRTL model is to determine a set of interaction parameters for the group of 44 aroma compounds in ethanol-water mixtures, using directly the source of experimental information found in the literature. A critical point before this calculation was the selection of coherent data. Further details on the methodology applied for this purpose are presented as Supporting Information.

In consideration of the big number of chemical species, the following assumptions are considered to simplify the model identification problem:

• The main one is that the interactions between aroma compounds are neglected, as their molar fractions in the liquid phase are equal or lower than x AC < 10 -4 , the limit of infinite dilution defined by Alessi et al. [START_REF] Alessi | Signficance of dilute regions[END_REF] This means that τ ij = 0, when i and j are both aroma compounds. In this way, the only interaction parameters considered are those associated with the solvent binary ethanol (2)water (3) and to the pairs aroma compound (1)-ethanol (2) and aroma compound (1)-water (3). • The nonrandomness parameters, c ij 0 and c ij T , are respectively set to 0.3 and 0 for all binaries. This . Experimental data from ( ◇ ), [START_REF] Lai | Isobaric (vapor + liquid) equilibria for the ternary system of (ethanol + water + 1,3 propanediol) and three constituent binary systems at P = 101.3 kPa[END_REF] ( □ ), [START_REF] Arce | VLE for water+ethanol +1 octanol mixtures. Experimental measurements and correlations[END_REF] (△), [START_REF] Yang | Vapor liquid equilibrium for mixtures of water, alcohols, and ethers[END_REF] (○). [START_REF] Kamihama | Isobaric Vapor-liquid equilibria for ethanol + water + ethylene glycol and its constituent three binary systems[END_REF] () NRTL model using the interaction parameters calculated by Kadir. [START_REF] Kadir | Optimisation du procedéde production d'Alcool surfin[END_REF] Table 5. continued assumption is suitable for polar systems in vapor-liquid equilibrium. [START_REF] Renon | Local compositions in thermodynamic excess functions for liquid mixtures[END_REF] • For the binaries aroma compound (1)-ethanol (2) and aroma compound (1)-water (3), A ij T , the temperature dependent parameter of τ ij is neglected. Two factors justify this approximation: (1) the number of exper imental data is limited and (2) the equilibrium temperature interval is reduced (from 351.44 to 373.15 K).

• The interaction parameters of the binary ethanol (2)water (3) are obtained from the literature. [START_REF] Kadir | Optimisation du procedéde production d'Alcool surfin[END_REF] The values are presented in Table 4. The reliability of these parameters was verified by fitting the experimental data (79 independent points) measured by different au thors. [START_REF] Lai | Isobaric (vapor + liquid) equilibria for the ternary system of (ethanol + water + 1,3 propanediol) and three constituent binary systems at P = 101.3 kPa[END_REF][START_REF] Arce | VLE for water+ethanol +1 octanol mixtures. Experimental measurements and correlations[END_REF][START_REF] Yang | Vapor liquid equilibrium for mixtures of water, alcohols, and ethers[END_REF][START_REF] Kamihama | Isobaric Vapor-liquid equilibria for ethanol + water + ethylene glycol and its constituent three binary systems[END_REF] The average relative deviation between the experimental and the calculated temperatures was 0.2% and that of the absolute volatilities of ethanol and water, 3.1% and 2.2%, respectively. The equilibrium diagram including the experimental data and the NRTL representation is presented in Figure 4. In this way, the problem was reduced to the estimation of a set of four parameters for each aroma compound, that is, 176 parameters:

• 88 associated with each binary aroma compound (1)ethanol (2): A 12 0 , A 21 0 . • 88 associated with each binary aroma compound (1)water (3): A 13 0 , A 31 0 .

The parameters were estimated by minimizing an objective function through the Excel Solver for nonlinear problems. [START_REF]Integrated Analytic Solver[END_REF] The equilibrium property considered was the relative volatility of the aroma compound with respect to ethanol, α AC/Et . This selection is due to the fact that the interaction parameters will be used for the simulation of distillation units, whose separation performance is directly based upon the difference of volatilities between the chemical species. The relative volatility condenses all the information about the equilibrium distribution of an aroma compound and its behavior with respect to ethanol, main component of the distillate, the product of interest.

The objective function is written in terms of an absolute deviation, formulation that gives more weight to high values of α AC/Et :

∑ α α α = - = OF( ) ( ) k N 1 AC/Etexp AC/EtCalc 2 k k ( 26
)
where N is the number of independent data, α AC/Et Expk is the experimental value of α AC/Et and α AC/Et Calck the valued calculated with the NRTL model. For the majority of aroma compounds, the relative volatility is higher when the ethanol concentration in the liquid phase is low, and this corresponds to the region in which distillation of alcoholic beverages takes place.

The calculation of α AC/Et was carried out using the Bubble Temperature algorithm of the Simulis Thermodynamics package. The values of two variables were fixed to the experimental data, namely, pressure (P) and composition in the liquid phase (x). The algorithms allows computation of the temperature (T) and the compositions of the vapor phase (y) in equilibrium. α AC/Et is obtained with eq 5 or eq 11, for carboxylic acids.

The fitting quality is evaluated with respect to three variables (U): K AC , α AC/Et , and α AC/W . Two statistics were calculated: Table 5. continued • Absolute average relative error (AAE%):

∑ = - = N U U U AAE% 1 100% k N k k k 1 exp Calc exp (27)
• Root mean square error (RMSE):

∑ = - = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ N U U RMSE 1 ( ) k N i i 1 exp Calc 2 1/2 (28)
With reference to the interaction parameters already published in the literature, only those proposed for ethyl lactate by the authors [START_REF] Puentes | Vapor liquid equilibrium (VLE) of ethyl lactate highly diluted in ethanol water solutions at 101.3 kPa: experimental measurements and thermodynamic modeling with semi empirical models[END_REF] are directly used in this work. For uniformity reasons, those presented by Athes et al. 13 and Deterre et al. [START_REF] Deterre | Vapor-liquid equilibria measurements of bitter orange aroma compounds highly diluted in boiling hydro alcoholic solutions at 101.3 kPa[END_REF] were recalculated from the corresponding equilibrium data, as in both cases A ij T ≠ 0. In relation to the parameters from Athes et al., 13 it is important to indicate that an alternative formalism was used by the authors to evaluate the temperature dependence of the interaction parameter τ ij . The expression is

τ = + a b T ij ij ij ( 29
)
By comparison with eq 25, the following equivalence with the formalism of the current work is deduced:

+ - = + A RT A T RT a b T ( 273.15) ij ij T ij ij 0 (30)
+ - = + A R A A RT a b T 273.15 ij T ij ij T ij ij 0 (31)
where Category III, heavy compounds: (e) ethanoic acid, (f) methanoic acid. Experimental data at high dilution from (×), 8 ( ◇ ), 9 (△), 13 (○), [START_REF] Ikari | Behavior of a minute amount of furfural in distillation of aqueous ethanol solution under reduced pressure[END_REF] (◐), [START_REF] Ikari | Vapor liquid equilibria of a minute amount of n propyl, isobutyl, and isoamyl alcohols in aqueous ethanol solution under reduced pressure[END_REF] (•), [START_REF] Ikari | Vapor liquid equilibria of trace isobutyraldehyde, ethyl acetate and isoamyl acetate in aqueous ethanol solution under reduced pressure[END_REF] (+). [START_REF] Ikari | Vapor liquid equilibria of a minute amount of methanol, isovaleraldehyde and diacetyl in aqueous ethanol solution under reduced pressure[END_REF] Calculation with NRTL model using () NRTL 0 parameters set (estimated from data at high dilution, Table 5). (blue dotted lines) NRTL B parameters set (estimated from binary data, Table 9). NRTL T parameters set (violet dashed lines) (estimated from ternary data, Table 10).

= A a R ij T ij (32) = + = + A b R A R b a 273.15 ( 273.15 ) ij ij ij T ij ij 0 ( 33
)
and consequently, because a ij ≠ 0, necessarily A ij T ≠ 0, as stated above.

The new set of interaction parameters for the 44 aroma compounds as well as the statistics of fitting are summarized in Table 5. According to these information, the RMSE of K AC varies between 0.0 and 6.9 with an overall average of 1.3, error that remains small regarding the order of magnitude of this property, between 10 0 and 10 2 . For α AC/Et (order of magnitude between 10 0 and 10 1 ), the RMSE is in the range from 0.0 to 1.1 with an overall average of 0.2, and for α AC/Et (order of magnitude between 10 0 and 10 2 ) the overall average is 2.1. Regarding AAE%, the overall average is about 12% for the three equilibrium variables with a variation range between 1% and 33%. The analysis by chemical family indicates that the lowest deviations are associated with acetals, overall AAE% around 7%, while the highest deviation concern the terpenes, with an overall AAE% of the order of 25%.

The evolution of absolute and relative volatilities with the ethanol composition in the liquid phase is presented in Figures 5,6, and 7 for six representative aroma compounds: ethyl ethanoate, ethanal, propan 1 ol, furan 2 carbaldehyde, ethanoic acid, and methanoic acid. These compounds were classified according to the criterion presented in section 3.2: ethyl ethanoate and ethanal as light compounds, propan 1 ol and furan 2 carbaldehyde as intermediary compounds, and ethanoic and methanoic acid as heavy compounds. In some of these figures, the representation with different sets of interaction parameters is also depicted. A comparative interpretation of theses curves is developed later, in section 3.3. Experimental data at high dilution from (×), 8 ( ◇ ), 9 (△), 13 (○), [START_REF] Ikari | Behavior of a minute amount of furfural in distillation of aqueous ethanol solution under reduced pressure[END_REF] (◐), [START_REF] Ikari | Vapor liquid equilibria of a minute amount of n propyl, isobutyl, and isoamyl alcohols in aqueous ethanol solution under reduced pressure[END_REF] (•), [START_REF] Ikari | Vapor liquid equilibria of trace isobutyraldehyde, ethyl acetate and isoamyl acetate in aqueous ethanol solution under reduced pressure[END_REF] (+). [START_REF] Ikari | Vapor liquid equilibria of a minute amount of methanol, isovaleraldehyde and diacetyl in aqueous ethanol solution under reduced pressure[END_REF] Calculation with NRTL model using () NRTL 0 parameters set (estimated from data at high dilution, Table 5). (blue dotted lines) NRTL B parameters set (estimated from binary data, Table 9). NRTL T parameters set (violet dashed lines) (estimated from ternary data, Table 10).

Regarding the experimental data, one can observe that they are globally well represented by the NRTL model, using the interaction parameters calculated in this work (NRTL 0 set). For all compounds, both K AC and α AC/W decrease when the liquid phase is enriched in ethanol, behavior that has already been identified for other aroma compounds. 13,[START_REF] Puentes | Vapor liquid equilibrium (VLE) of ethyl lactate highly diluted in ethanol water solutions at 101.3 kPa: experimental measurements and thermodynamic modeling with semi empirical models[END_REF] Concerning α AC/Et , the evolution with composition are more variable, independently of its order of magnitude. Three trends can be identified: (i) decreasing (for ethyl ethanoate, propan 1 ol, furfural, and most of the aroma compounds studied in this work), (ii) slightly linear increasing (for ethanoic acid), and (iii) nearly constant, in the case of ethanal, after a slight increase of α AC/Et in the region of low ethanol concentration (x Et < 1).

As for the Henry constant of aroma compounds, this parameter can be calculated at boiling conditions by using directly the absolute volatility data, according to eq 18. The values of ln AC as a function of T -1 are depicted in Figure 8 for the same aroma compounds. As in the previous figures, model curves obtained with different parameters sets are plotted (see section 3.3).

In general terms, the activity model represents correctly the experimental data. Regarding ethanal (Figure 8b), even if an important dispersion of the experimental data is observed, the model follows the global trend in most of the temperature interval. In all cases, the continuous representation with the NRTL model shows that the evolution of ln AC is decreasing and concave for all compounds, and not linear, as observed in a pure solvent. This is due to the fact that the temperature and the composition of the liquid phase are not independent at boiling conditions: each bubble temperature T corresponds to a different mole fraction of ethanol.

In this way, considering the great diversity of chemical species and data resources, the whole of results are acceptable Experimental data at high dilution from (×), 8 ( ◇ ), 9 (△), 13 (○), [START_REF] Ikari | Behavior of a minute amount of furfural in distillation of aqueous ethanol solution under reduced pressure[END_REF] (◐), [START_REF] Ikari | Vapor liquid equilibria of a minute amount of n propyl, isobutyl, and isoamyl alcohols in aqueous ethanol solution under reduced pressure[END_REF] (•), [START_REF] Ikari | Vapor liquid equilibria of trace isobutyraldehyde, ethyl acetate and isoamyl acetate in aqueous ethanol solution under reduced pressure[END_REF] (+). [START_REF] Ikari | Vapor liquid equilibria of a minute amount of methanol, isovaleraldehyde and diacetyl in aqueous ethanol solution under reduced pressure[END_REF] Calculation with NRTL model using () NRTL 0 parameters set (estimated from data at high dilution, Table 5). (blue dotted lines) NRTL B parameters set (estimated from binary data, Table 9). NRTL T parameters set (violet dashed lines) (estimated from ternary data, Table 10). and confirm that the NRTL model can correctly represent the vapor-liquid equilibria of aroma compounds in hydroalcoholic mixtures, a conclusion already established in other synthesis works.

3.2. Classification of Aroma Compounds. The set of interaction parameters estimated will be used for simulation purposes at a later stage, with the aim of understanding the behavior of aroma compounds in alcoholic beverages distillation. Given the considerable number of studied species, a systematic classification turns out be useful for identifying general trends.

By following a classic approach of multicomponent distillation, two key components governing the separation are first selected: ethanol as light key and water as heavy key. The first will be recovered to a significant extent in the top product, whereas the second, less volatile, will mainly be recovered in the bottom product. Then, by taking as criterion the relative volatility with respect to both key components over the entire ethanol composition range, three categories of aroma compounds can be defined:

• Category I, light compounds: The minimal value of α AC/Et is higher than the unity (α AC/Et-MIN > 1). They are lighter than light key and will be therefore mainly present in the top product or distillate. • Category II, intermediary or distributed compounds: The minimal value of α AC/Et is lower than the unity (α AC/Et-MIN < 1), and the maximal value of α AC/W is higher than the unity (α AC/W-MAX >1). Their volatility are intermediate between that of light and heavy key. They will be distributed in top and bottom products. • Category III, heavy compounds: the maximal value of α AC/W is lower than the unity (α AC/W-MAX < 1). They are heavier than heavy key and will be therefore mainly recovered in the bottom product. For simulation purposes, an accurate representation of phase equilibrium for intermediary compounds is fundamental to Category III, heavy compounds: (e) ethanoic acid, (f) methanoic acid. Experimental data at high dilution from (×), 8 ( ◇ ), 9 (△), 13 (○), [START_REF] Ikari | Behavior of a minute amount of furfural in distillation of aqueous ethanol solution under reduced pressure[END_REF] (◐), [START_REF] Ikari | Vapor liquid equilibria of a minute amount of n propyl, isobutyl, and isoamyl alcohols in aqueous ethanol solution under reduced pressure[END_REF] (•), [START_REF] Ikari | Vapor liquid equilibria of trace isobutyraldehyde, ethyl acetate and isoamyl acetate in aqueous ethanol solution under reduced pressure[END_REF] (+). [START_REF] Ikari | Vapor liquid equilibria of a minute amount of methanol, isovaleraldehyde and diacetyl in aqueous ethanol solution under reduced pressure[END_REF] Calculation with NRTL model using () NRTL 0 parameters set (estimated from data at high dilution, Table 5). (blue dotted line) NRTL B parameters set (estimated from binary data, Table 9). NRTL T parameters set (violet dashed line) (estimated from ternary data, Table 10).

correctly estimate the ratio of distribution between the top and bottom product. The classification of the aroma compounds studied in this work is presented in Table 6. According to this approach, most of the compounds, 32, are intermediary compounds. Ten compounds, mainly carbonyl compounds, are light compounds, and only two carboxylic acids, methanoic and ethanoic, are heavier than water. The aroma compounds presented in Figures 567are classified on the basis of this criterion, from the more (ethyl ethanoate) to the less volatile (methanoic acid).

It should be noted that the classification is proposed over the whole ethanol composition range in the liquid phase (0 < x Et < 1). Given that the relative volatilities vary with composition, the classification could be different when considering a more restricted composition interval.

3.3. Comparison of the Representation Obtained from Data at High Dilution and That Obtained from Binary and Ternary Data at Higher Concentrations of Aroma Compounds. The discussion concludes with a comparison between: (1) the representation of the equilibrium data using the set of parameters calculated in this work, from data at high dilution regarding the aroma compounds (Table 2 and Table 5), and (2) the one obtained with parameters fitted to data for binary and ternary systems, in which the aroma compounds are present at high concentrations. These parameters are available in the literature for some aroma compounds. 7, [START_REF] Fauńdez | Predictive models to describe VLE in ternary mixtures water + ethanol + congener for wine distillation[END_REF][START_REF] Fauńdez | Phase equilibrium modeling in binary mixtures found in wine and must distillation[END_REF][START_REF] Fauńdez | Activity coefficient models to describe vapor liquid equilibrium in ternary hydro alcoholic solutions[END_REF][START_REF] Valderrama | Modeling of vapor-liquid equilibrium in binary and ternary mixtures of interest in alcoholic distillation[END_REF][START_REF] Tan | Prediction and experimental verification of the salt effect on the vapour-liquid equilibrium of water-ethanol-2 propanol mixture[END_REF][START_REF] Tan | Prediction and experimental verification of the effect of salt on the vapour-liquid equilibrium of ethanol/1 propanol/water mixture[END_REF] However, according to the writing of eqs 22 and 23, they cannot be used directly for two reasons: (i) the formalisms to evaluate the temperature dependence of the interaction parameters c ij and τ ij are not the same of this work, and (ii) the nonrandomness parameters c ij 0 is different from the value set in this work (0.3), which does not make possible the conversion of the parameters G ij and τ ij in the different formalisms.

Consequently, new interaction parameters are estimated from experimental data by following an analogous procedure to that described in section 3.1 Information on the aroma compounds considered, the data references 8,75-107 as well as the temperature and composition intervals are summarized in Table 7 for binary systems (aroma compound-ethanol and aroma compound-water) and in Table 8 for ternary systems [START_REF] Hughes | Binary and Ternary Equilibrium Data[END_REF] Delzenne, 1958 [START_REF] Delzenne | Vapor Liquid equilibrium data for ternary system methanol ethanol water[END_REF] 80 340.0 360.7 0.0 0.9 0.0 0.8 0.0 0.9 0.1 0.9 0.0 0.8 0.0 0.5

Propan-1-ol Ochi and Kojima, 1969; [START_REF] Ochi | Vapor-Liquid Equilibrium Data Collection. Organic Hydroxy compounds: Alcohols[END_REF] 351.2 361.9 0.0 0.9 0.0 0.9 0.0 0.9 0.0 0.9 0.0 0.9 0.0 0.7 2-Methylpropan-1-ol (aroma compound-ethanol-water). Nine aroma compounds from five chemical families are considered. The objective function is expressed in terms of the relative deviation of temperature and absolute volatility data:

∑ ∑ ∑ = - + - = = = ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ T K T T T K K K OF( , ) i k N k N i n i i i 1 Exp calc Exp 2 1 1 Exp Calc Exp 2 k k k k k k ( 34
)
T Calc k and K i Calc k are computed with the Bubble Temperature algorithm of the Simulis Thermodynamics package by fixing the pressure (P) and composition in the liquid phase (x) to the experimental data. In all cases, the interaction parameters of the binary ethanol-water used for the equilibrium calculations were those used previously in this work (Table 4). [START_REF] Kozlovskiy | Vapor pressures and vaporization enthalpies of a series of esters used in flavors by correlation gas chromatography[END_REF] The fitting quality is evaluated with respect to T and K i , by considering the parameters AAE% (eq 27) and RMSE (eq 28).

In comparison to eq 26, used for the parameter estimation from data at high dilution (NRTL 0 set), the formulation of eq 34 was chosen for three reasons: (i) relative deviations were preferred to absolute ones because the function includes two variables with different magnitude orders; (ii) temperature was added because in binary and ternary systems the corresponding intervals are variable, whereas that of high dilution systems is always the same at 101.3 kPa (from 351.4 to 373.2 K), as it is fixed by the ethanol-water binary; (iii) absolute volatilities were used because they do not give more weight to a component with respect to another, which is more appropriate for this estimation because all the species concerned are present at wide concentration intervals (from 0 to 1). The formulation of eq 26 is more adapted to systems with aroma compounds highly diluted, as it privileges this species in relation to ethanol and water.

Thereby, the set of interaction parameters derived from binary data (identified as NRTL B) is given in Table 9 and that from ternary data (identified as NRTL T) in Table 10. In both tables, the fitting statistics, with the temperature and absolute volatilities ranges, are presented. According to these values, all the concerned systems are globally well represented with the NRTL model, with an overall relative error of 1% for temperature (variation between 0% and 3%) and of 7% for absolute volatilities (variation between 1% and 24%).

The comparison with data at high dilution is performed in relation to the fitting quality of the equilibrium data. The values of RMSE and AAE% for the nine aroma compounds are presented in Table 11, using the three sets of interaction parameters: NRTL 0 (from ternary or multicomponent data at low concentration of the aroma compound), NRTL B, and NRTL T.

The evolution of K AC , α AC/Et and α AC/W is also represented with the different sets of interaction parameters in Figures 567. The number of curves for each compound varies according to the binary and ternary data available, reported in Tables 7 and8 (three curves for ethanal, propan 1 ol, and furan 2 carbalde hyde; 2 curves for ethyl ethanoate and ethanoic acid; 1 curve for methanoic acid, for which no binary or ternary data were found in the open literature). These figures show that the experimental data of K AC and α AC/W at high dilution can be represented with reasonable precision using any of the sets. The decreasing tendency obtained is correct. However, as for α AC/Et , Table 9 the representation is only relatively good for ethyl acetate and propanol 1 ol. For ethanoic acid, the curve is systematically shifted from the experimental data, while for ethanal and furan 2 carbaldehyde, the representation with the NRTL B and NRTL T sets is not coherent.

The statistics indicate that the fitting quality is clearly better when using the main parameters estimated in this work (NRTL 0). Only for two aroma compounds, methanol and 1 propanol, the deviations with respect to experimental data from the NRTL B set are acceptable and comparable. When comparing the NRTL B and NRTL T sets, the deviations associated with the first one are always lower (AAE% between 4% and 56% for NRTL B, against AAE% between 22% and 72% for NRTL T).

Two hypothesis could be proposed to explain the reason why the extrapolation of equilibrium data at high concentration to the high dilution region is more accurate when using binary data (NRTL B) than ternary data (NRTL T). First of all, because in the latter case the composition intervals of the solvent components (ethanol and water) in the liquid phase are not strictly the same as those of the high dilution data. An analysis of the distribution of the experimental data over the whole interval of solvent-liquid composition shows that in the case of methanol, 3 methylbutan 1 ol, and furan 2 carbalde hyde, the ternary mixture data do not include experimental points at very low ethanol mole fractions (x Et < 0.1) or very high water mole fractions (x W > 0.9), region where the absolute and relative volatilities at high dilution exhibit the most pronounced variation. As a result, the extrapolation of volatility data at high concentration could be not accurate enough in this interval, in which at least one experimental point from the binary data set and at high dilution are available.

Second, the great difference among the AAE% values could be associated with the quality of experimental data. This aspect cannot be judged precisely by lack of information, but according to Table 10, one could think that the uncertainty of the selected ternary data is more important with respect to the binary data, which could be mainly due to a greater complexity in the quantitative analysis of the coexisting phases. The uncertainty of mole compositions is amplified with the calculation of the absolute and relative volatilities (variables selected to evaluate the quality of the NRTL extrapolation) because they are both defined as composition ratios. This propagation is even more important in the case of α AC/Et and α AC/W , whose definition contains four composition values.

In general terms, these results suggest that an accurate representation of the vapor-liquid behavior of aroma compounds in alcoholic distillation requires data at low concentration. The NRTL model could be used to extrapolate the equilibrium data at high concentration to the high dilution region but only with rough precision. If no data at high dilution are available, binary data in which the aroma compounds are present over the entire composition range could be used for the estimation of interaction parameters used for engineering purposes in the alcoholic beverages field. The reliability of this estimation must be verified with respect to experimental distillation data, by comparing the composition of the output streams (distillate and bottoms) or another equivalent variable (for instance the mass recovery from feed to distillate) predicted by simulation with the data collected during an experimental campaign. These latter data are mandatory to validate the performance of a simulation module. If the deviations are too high, the equilibrium data at high dilution will be strictly required and they should be determined either experimentally or by way of a theoretical estimation with predictive models such as UNIFAC or the quantum based COSMO models. Experimental data at high dilution from: ( ). 9 ( ). 36 ( ). 13 ( ). [START_REF] Martin | A quick method for obtaining partition factor of congeners in spirits[END_REF] The data rejected from the observation of the overall trend are pointed out in , and those rejected by the point-to-point comparison are highlighted in .

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K AC 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Experimental data at high dilution from: ( ). 8 ( ). 9 ( ). 13 ( ). [START_REF] Ikari | Vapor liquid equilibria of trace isobutyraldehyde, ethyl acetate and isoamyl acetate in aqueous ethanol solution under reduced pressure[END_REF] The data rejected from the observation of the overall trend are pointed out in . Ethyl hexanoate. Experimental data at high dilution from: ( ). 13 ( ). [START_REF] Ikari | Vapor liquid equilibria of trace isobutyraldehyde, ethyl acetate and isoamyl acetate in aqueous ethanol solution under reduced pressure[END_REF] ( ). [START_REF] Martin | A quick method for obtaining partition factor of congeners in spirits[END_REF] The data rejected from the observation of the overall trend are pointed out in . Experimental data at high dilution from: ( ). 9 ( ). 13 ( ). [START_REF] Ikari | Behavior of a minute amount of furfural in distillation of aqueous ethanol solution under reduced pressure[END_REF] All data were accepted for model identification.

Figure 1 .

 1 Figure 1. AAE% values of two variables (T and y AC ) obtained from the representation of the vapor-liquid equilibria of binary systems volatile aroma compound-ethanol at 101.3 kPa with different thermodynamic models. (blue bars) NRTL, (pink bars) UNIFAC, and (red bars) PSRK. The horizontal dashed lines represent the average AAE% value for each model. Comparison results from Fauńdez and Valderrama. 19

Figure 2 .

 2 Figure 2. AAE% values of two variables (T and y AC ) obtained from the representation of the vapor-liquid equilibria of binary systems volatile aroma compound-water at 101.3 kPa with different thermodynamic models. (blue bars) NRTL, (pink bars) UNIFAC, and (red bars) PSRK. The horizontal dashed lines represent the average AAE% value for each model. Comparison results from Fauńdez and Valderrama. 19

a

  NP: unpublished data in the original paper but provided by the authors for the current work.

  eq 25, R is the ideal gas constant. The variables A ij 0 , A ij T (nonsymmetric, A ij 0 ≠ A ji 0 and A ij T ≠ A ji T ) are energy parameters describing the i-j interaction, whereas c ij 0 and c ij T (symmetric, c ij 0 = c ji 0 and c ij T = c ji T

Figure 4 .

 4 Figure 4. (a) Vapor-liquid equilibrium diagram of the binary system ethanol-water at 101.3 kPa (T is the temperature, y Et the ethanol mole fraction in the vapor phase, and x Et the ethanol mole fraction in the liquid phase). (b) Evolution of the absolute volatility of ethanol (K Et ) with liquid composition (x Et ). (c) Evolution of the absolute volatility of water (K W ) with liquid composition (x Et ). Experimental data from ( ◇ ),[START_REF] Lai | Isobaric (vapor + liquid) equilibria for the ternary system of (ethanol + water + 1,3 propanediol) and three constituent binary systems at P = 101.3 kPa[END_REF] ( □ ),[START_REF] Arce | VLE for water+ethanol +1 octanol mixtures. Experimental measurements and correlations[END_REF] (△),[START_REF] Yang | Vapor liquid equilibrium for mixtures of water, alcohols, and ethers[END_REF] (○).[START_REF] Kamihama | Isobaric Vapor-liquid equilibria for ethanol + water + ethylene glycol and its constituent three binary systems[END_REF] () NRTL model using the interaction parameters calculated by Kadir.[START_REF] Kadir | Optimisation du procedéde production d'Alcool surfin[END_REF]

a

  Intervals and statistics of fitting quality for absolute and relative volatilities.

Figure 5 .

 5 Figure 5. Evolution of the absolute volatility of aroma compounds (K AC ) with the ethanol composition in the liquid phase (x Et ) at 101.3 kPa for: Category I, light compounds: (a) ethyl ethanoate, (b) ethanal. Category II, intermediary compounds: (c) propan 1 ol, (d) furan 2 carbaldehyde.Category III, heavy compounds: (e) ethanoic acid, (f) methanoic acid. Experimental data at high dilution from (×), 8 ( ◇ ), 9 (△), 13 (○),[START_REF] Ikari | Behavior of a minute amount of furfural in distillation of aqueous ethanol solution under reduced pressure[END_REF] (◐),[START_REF] Ikari | Vapor liquid equilibria of a minute amount of n propyl, isobutyl, and isoamyl alcohols in aqueous ethanol solution under reduced pressure[END_REF] (•),[START_REF] Ikari | Vapor liquid equilibria of trace isobutyraldehyde, ethyl acetate and isoamyl acetate in aqueous ethanol solution under reduced pressure[END_REF] (+).[START_REF] Ikari | Vapor liquid equilibria of a minute amount of methanol, isovaleraldehyde and diacetyl in aqueous ethanol solution under reduced pressure[END_REF] Calculation with NRTL model using () NRTL 0 parameters set (estimated from data at high dilution, Table5). (blue dotted lines) NRTL B parameters set (estimated from binary data, Table9). NRTL T parameters set (violet dashed lines) (estimated from ternary data, Table10).

Figure 6 .

 6 Figure 6. Evolution of the relative volatility of aroma compounds with respect to ethanol (α AC/Et ) with the ethanol composition in the liquid phase (x Et ) at 101.3 kPa. Category I, light compounds: (a) ethyl ethanoate, (b) ethanal. Category II, intermediary compounds: (c) propan 1 ol, (d) furan 2 carbaldehyde. Category III, heavy compounds: (e) ethanoic acid, (f) methanoic acid.Experimental data at high dilution from (×), 8 ( ◇ ), 9 (△), 13 (○),[START_REF] Ikari | Behavior of a minute amount of furfural in distillation of aqueous ethanol solution under reduced pressure[END_REF] (◐),[START_REF] Ikari | Vapor liquid equilibria of a minute amount of n propyl, isobutyl, and isoamyl alcohols in aqueous ethanol solution under reduced pressure[END_REF] (•),[START_REF] Ikari | Vapor liquid equilibria of trace isobutyraldehyde, ethyl acetate and isoamyl acetate in aqueous ethanol solution under reduced pressure[END_REF] (+).[START_REF] Ikari | Vapor liquid equilibria of a minute amount of methanol, isovaleraldehyde and diacetyl in aqueous ethanol solution under reduced pressure[END_REF] Calculation with NRTL model using () NRTL 0 parameters set (estimated from data at high dilution, Table5). (blue dotted lines) NRTL B parameters set (estimated from binary data, Table9). NRTL T parameters set (violet dashed lines) (estimated from ternary data, Table10).

Figure 7 .

 7 Figure 7. Evolution of the relative volatility of aroma compounds with respect to water (α AC/W ) with the ethanol composition in the liquid phase (x Et ) at 101.3 kPa. Category I. light compounds: (a) ethyl ethanoate, (b) ethanal. Category II, intermediary compounds: (c) propan 1 ol, (d) furan 2 carbaldehyde. Category III, heavy compounds: (e) ethanoic acid, (f) methanoic acid.Experimental data at high dilution from (×), 8 ( ◇ ), 9 (△), 13 (○),[START_REF] Ikari | Behavior of a minute amount of furfural in distillation of aqueous ethanol solution under reduced pressure[END_REF] (◐),[START_REF] Ikari | Vapor liquid equilibria of a minute amount of n propyl, isobutyl, and isoamyl alcohols in aqueous ethanol solution under reduced pressure[END_REF] (•),[START_REF] Ikari | Vapor liquid equilibria of trace isobutyraldehyde, ethyl acetate and isoamyl acetate in aqueous ethanol solution under reduced pressure[END_REF] (+).[START_REF] Ikari | Vapor liquid equilibria of a minute amount of methanol, isovaleraldehyde and diacetyl in aqueous ethanol solution under reduced pressure[END_REF] Calculation with NRTL model using () NRTL 0 parameters set (estimated from data at high dilution, Table5). (blue dotted lines) NRTL B parameters set (estimated from binary data, Table9). NRTL T parameters set (violet dashed lines) (estimated from ternary data, Table10).

Figure 8 .

 8 Figure 8. Evolution of ln H with T -1 at 101.3 kPa. Category I, light compounds: (a) ethyl ethanoate, (b) ethanal. Category II, intermediary compounds: (c) propan 1 ol, (d) furan 2 carbaldehyde.Category III, heavy compounds: (e) ethanoic acid, (f) methanoic acid. Experimental data at high dilution from (×), 8 ( ◇ ), 9 (△), 13 (○),[START_REF] Ikari | Behavior of a minute amount of furfural in distillation of aqueous ethanol solution under reduced pressure[END_REF] (◐),[START_REF] Ikari | Vapor liquid equilibria of a minute amount of n propyl, isobutyl, and isoamyl alcohols in aqueous ethanol solution under reduced pressure[END_REF] (•),[START_REF] Ikari | Vapor liquid equilibria of trace isobutyraldehyde, ethyl acetate and isoamyl acetate in aqueous ethanol solution under reduced pressure[END_REF] (+).[START_REF] Ikari | Vapor liquid equilibria of a minute amount of methanol, isovaleraldehyde and diacetyl in aqueous ethanol solution under reduced pressure[END_REF] Calculation with NRTL model using () NRTL 0 parameters set (estimated from data at high dilution, Table5). (blue dotted line) NRTL B parameters set (estimated from binary data, Table9). NRTL T parameters set (violet dashed line) (estimated from ternary data, Table10).

aa

  . NRTL-B Set: Interaction Parameters Calculated from Binary Mixture Data of Aroma Compounds (1) At High Concentrations in Ethanol (2) and Water (3) a Fitting statistics for temperature and absolute volatilities.Table 10. NRTL-T Set: Interaction Parameters Calculated from Ternary Mixture Data of Aroma Compounds at High Concentrations (1) In Ethanol (2)-Water (3) Mixtures a Fitting statistics for temperature and absolute volatilities.

Figure S1 .

 S1 Figure S1. Evolution of the absolute (KAC) and relative (αAC/Et) volatilities with ethanol composition in the liquid phase (xEt) at 101.3 kPa for alcohols: a. Propan-1-ol, b. 2-Methylpropan-1-ol, c. 3-Methylbutan-1-ol.

Figure S2 .

 S2 Figure S2. Evolution of the absolute (KAC) and relative (αAC/Et) volatilities with ethanol composition in the liquid phase (xEt) at 101.3 kPa for carbonyl compounds: a. Ethanal, b. 2-Methylpropanal.

Figure S3 .

 S3 Figure S3. Evolution of the absolute (KAC) and relative (αAC/Et) volatilities with ethanol composition in the liquid phase (xEt) at 101.3 kPa for esters: a. Ethyl ethanoate, b.3-methylbutyl ethanoate, c. Ethyl hexanoate. Experimental data at high dilution from: ( ). 13 ( ).[START_REF] Ikari | Vapor liquid equilibria of trace isobutyraldehyde, ethyl acetate and isoamyl acetate in aqueous ethanol solution under reduced pressure[END_REF] ( ).[START_REF] Martin | A quick method for obtaining partition factor of congeners in spirits[END_REF] The data rejected from the observation of the overall trend are pointed out in .

Figure S4 .

 S4 Figure S4. Evolution of the absolute (KAC) and relative (αAC/Et) volatilities with ethanol composition in the liquid phase (xEt) at 101.3 kPa for furan-2-carbaldehyde.

Figure S5 .

 S5 Figure S5. Evolution of the absolute (KAC) and relative (αAC/Et) volatilities with ethanol composition in the liquid phase (xEt) at 101.3 kPa for terpene 3,7-dimethylocta-1,6-dien-3-ol. Experimental data at high dilution from: ( ). 13 ( ).[START_REF] Deterre | Vapor-liquid equilibria measurements of bitter orange aroma compounds highly diluted in boiling hydro alcoholic solutions at 101.3 kPa[END_REF] The data rejected from the observation of the overall trend are pointed out in , and those rejected by the point-to-point comparison are highlighted in .

Table 1 .

 1 Research Works Published in the Open Literature on Vapor-Liquid Equilibrium Data of Aroma Compounds Highly Diluted in Ethanol Mixtures at 101.3 kPa

	experimental measurements

Aroma compound: Addition reaction of sodium bisulphite in aqueous solution.

Table 1 . continued

 1

	experimental measurements

Table 2 .

 2 Synthesis of the Vapor-Liquid Equilibrium Data Available in the Open Literature for Aroma Compounds Highly Diluted in Ethanol-Mixtures at 101.3 kPa a

	y Et range		min max		0.00 0.99									0.28 0.71						0.28 0.71		0.28 0.71		0.28 0.71	0.18 0.98
	y AC range		min max		1.03 × 10 -3 6.35 × 10 -3									2.80 × 10 -5 4.58 × 10 -4						1.88 × 10 -5 7.16 × 10 -4		1.09 × -8 1.18 × -6		1.39 × 10 -7 6.40 × 10 -6	1.06 × 10 -2 4.82 × 10 -1
	x Et range		min max	0.01 0.85	0.00 0.99	0.00 0.85	0.00 0.86	0.00 1.00	0.00 1.00	0.00 0.86	0.00 0.86	0.00 0.87	0.00 1.00	0.02 0.63	0.00 1.00	0.00 0.88	0.00 0.86	0.00 0.87	0.00 1.00	0.02 0.63	0.00 1.00	0.02 0.63	0.03 0.28	0.02 0.63	0.02 1.00
	x AC range		min max		1.64 × 10 -4 4.19 × 10 -3									1.76 × 10 -5 7.73 × 10 -5						2.68 × 10 -5 1.23 × 10 -4		8.30 × -8 2.85 × -7		6.15 × 10 -6 1.03 × 10 -4	1.70 × 10 -4 8.52 × 10 -2
	T range (K)		min max											351.9 366.4						351.9 366.4		351.9 366.4		351.9 366.4	336.7 367.2
	no. of	independent	data	9	18	12	11	11		11	11	12	11	9		11	12	15	11	9			8	9	28
			ref	Williams, 1962 9	Ikari et al. 1998b 38	Williams, 1962 9	Williams, 1962 9	Ikari et al. 1990 36	Martin et al. 2009 39	Williams, 1962 9	Williams, 1962 9	Williams, 1962 9	Ikari et al. 1990 36	Athes et al. 2008 13	Martin et al. 2009 39	Williams, 1962 9	Williams, 1962 9	Williams, 1962 9	Ikari et al. 1990 36	Athes et al. 2008 13	Martin et al. 2009 39	Athes et al. 13	Williams, 1962 9	Athes et al. 2008 13	Heitz, 1960 8
			CAS no.	105-57-7	67-56-1	107-18-6	71-23-8			67-63-0	71-36-3	78-83-1				75-65-0	137-32-6	123-51-3				928-96-1	111-27-3	60-12-8	75-07-0
	aroma compd		IUPAC name	1,1-Diethoxyethane	Methanol	Prop-2-en-1-ol	Propan-1-ol	Propan-1-ol	Propan-1-ol	Propan-2-ol	Butan-1-ol	2-Methylpropan-1-ol	2-Methylpropan-1-ol	2-Methylpropan-1-ol	2-Methylpropan-1-ol	2-Methylpropan-2-ol	2-Methylbutan-1-ol	3-Methylbutan-1-ol	3-Methylbutan-1-ol	3-Methylbutan-1-ol	3-Methylbutan-1-ol	(Z)-Hex-3-en-1-ol	Hexan-1-ol	2-Phenylethan-1-ol	Ethanal
			common name	Acetal	Methanol	2-Propenol	1-Propanol	1-Propanol	1-Propanol	Isopropanol	1-Butanol	Isobutanol	Isobutanol	Isobutanol	Isobutanol	Tert-butanol	2-Methylbutanol	Isopentanol	Isopentanol	Isopentanol	Isopentanol	Cis-3-hexenol	1-Hexanol	2-Phenylethanol	Acetaldehyde
		chemical	family	Acetals	Alcohols																				Carbonyl	compounds

Table 3 .

 3 Values of the Parameters L CAx and M CAx for the Group of Carboxylic Acids Studied in This Work

			C
	source	aroma compound	atoms	L CAx	M CAx
	DECHEMA database 57	Methanoic acid	1	10.743	3083.0
		Ethanoic acid	2	10.421	3166.0
		Propanoic acid	3	10.843	3316.0
		Butanoic acid	4	10.100	3040.0
		2-Methylpropanoic	4	10.100	3040.0
		acid
	Estimation by	2-Methylbutanoic	5	10.005	2993.0
	interpolation	acid
		3-Methylbutanoic	5	10.005	2993.0
		acid
		Hexanoic acid	6	9.891	2943.0
	DECHEMA database 57 (included	Heptanoic acid	7	9.807	2900.0
	for estimation
	purposes)
	Estimation by	Octanoic acid	8	9.703	2851.3
	extrapolation

Table 4 .

 4 Interaction Parameters of the of the NRTL Model for the Binary Ethanol (2)-Water (3) and Fitting Quality Statistics with Respect to Four Experimental Data Sets Obtained from Literature (cal mol -1 ) A 32 0 ( cal mol -1 ) A 23 T (cal mol -1 K -1 ) A 32 T ( cal mol -1 K -1 ) min max RMSE AAE% min max RMSE AAE% min max RMSE AAE%

	K Et	K W	T
	range	range	range

Table 6 .

 6 Classification of Aroma Compounds According to Their Relative Volatilities with Respect to Ethanol and Water, Over the Whole Ethanol Composition Range in the Liquid Phase

	α AC/Et	α AC/W

Table 7 .

 7 Synthesis of Vapor-Liquid Data for Binary Systems Aroma Compound (1)-Ethanol (2) and Aroma Compound (1)-Water (3) at 101.3 kPa

	T range (K)	x AC range	y AC range

Table 8 .

 8 Synthesis of Vapor-Liquid Data for Ternary Systems Aroma Compound (1)-Ethanol (2 -Water (3) at 101.3 kPa

	T range (K)	x AC range	x Et range	x W range	y AC range	y Et range	y W range

Table 11 .

 11 Statistics of Fitting Quality for Absolute and Relative Volatilities of Aroma Compounds at 101.3 kPa Using Different Three NRTL Parameters Sets: NRTL 0 (Table 5), NRTL B (Table 9), and NRTL T (Table 10) Kojima, K.; Ochi, K.; Nakazawa, Y. Relationship between liquid activity coefficient and composition for ternary systems. Int. Chem. Eng. 1969, 9, 342. Cited in Gmehling, J.; Onken, U. Vapor-Liquid Equilibrium Data Collection. Aqueous-Organic Systems; Dechema: Van Zandijcke, F.; Verhoeye, L. J. Appl. Chem. Biotechnol. 1974, 24, 709. Cited in Gmehling, J.; Onken, U. Vapor-Liquid Equilibrium Data Collection. Aqueous Organic Systems; Dechema: Frankfurt, 1977. Gidroliz. Lesokhim. Prom. 1971, 24, 10. Cited in Gmehling, J.; Onken, U. Vapor-Liquid Equilibrium Data Collection. Aqueous-Organic Systems; Dechema: Frankfurt, 1977.

	K AC	α AC/Et	α AC/W

(107) Kharin, S. E.; Perelygin, V. M.; Polyansky, K. K.

Table S1 . Point-to-point comparison of experimental volatility data from different sources for aroma compounds highly diluted in ethanol-water mixtures. Chemical family

 S1

	Common name	Aroma compound IUPAC name	No. CAS	xEt	Average	KAC	%AE	αAC/Et Average	%AE	Validation Rejection
				0.000±0.000 29.73	57.34%	2.45	57.34%	×
				0.100±0.000	4.38	18.51%	0.99	18.51%	×
				0.200±0.000	3.62		5.65%	0.88	5.65%	×
				0.300±0.000	1.47		1.20%	0.76	1.20%	×
				0.400±0.000	1.08		0.39%	0.70	0.39%	×
	1-Propanol	Propan-1-ol	71-23-8	0.500±0.000 0.600±0.000	0.86 0.71		1.97% 3.37%	0.65 0.61	1.97% 3.37%	× ×
				0.700±0.000	0.63		3.42%	0.58	3.42%	×
				0.800±0.000	0.57		1.64%	0.56	1.64%	×
				0.875±0.001	0.55		1.64%	0.55	2.05%	×
				0.900±0.000	0.54		0.77%	0.54	0.77%	×
	Alcohols			1.000±0.000 0.000±0.000 54.28 0.52	0.58% 60.14%	0.52 4.48	0.58% 60.14%	×	×
				0.100±0.000	6.22	28.81%	1.41	28.81%	×
				0.200±0.000	2.65		8.89%	0.99	8.89%	×
				0.300±0.000	1.50		2.33%	0.78	2.33%	×
				0.400±0.000	1.00		5.98%	0.65	5.98%	×
	Isobutanol	2-Methylpropan-1-ol	78-83-1	0.474±0.001 0.500±0.000	0.63 0.74	47.40% 7.71%	0.47 0.57	44.64% 7.71%	×	×
				0.600±0.000	0.59	10.21%	0.50	10.21%	×
				0.700±0.000	0.49	13.12%	0.46	13.12%	×
				0.800±0.000	0.44	14.73%	0.43	14.73%	×
				0.900±0.000	0.41	14.73%	0.41	14.73%	×
				1.000±0.000	0.39	17.67%	0.39	17.67%	×
											S-4

Table S1 . Continuation. Point-to-point comparison of experimental volatility data from different sources for aroma compounds highly diluted in ethanol-water mixtures.

 S1

	Chemical family	Aroma compound Common name IUPAC name	No. CAS	xEt	Average	KAC	%AE	αAC/Et Average	%AE	Validation Rejection
					0.000±0.000 48.704	43.99%	4.016	43.99%	×
					0.021±0.000 24.325	17.63%	2.241	19.99%	×
					0.075±0.000 7.066	38.90%	1.345	38.91%	×
					0.100±0.000 5.056	34.78%	1.148	34.78%	×
					0.195±0.001 1.707		0.88%	0.615	1.47%	×
					0.200±0.000 1.731		9.41%	0.648	9.41%	×
	Alcohols	Isopentanol	3-Methylbutan-1-ol	123-51-3	0.300±0.000 0.845 0.400±0.000 0.518		4.97% 8.22%	0.437 0.335	4.97% 8.22%	× ×
					0.500±0.000 0.366		9.04%	0.279	9.04%	×
					0.600±0.000 0.284	12.16%	0.243	12.16%	×
					0.700±0.000 0.236	17.60%	0.218	17.60%	×
					0.800±0.000 0.208	17.64%	0.202	17.64%	×
					0.900±0.000 0.194	19.46%	0.193	19.46%	×
					1.000±0.000 0.186	20.04%	0.186	20.04%	×
	Furans	Furfural	Furan-2-carbaldehyde	98-01-1 0.401±0.001	0.32	13.29%	0.45	0.22%	×
	Terpenes Linalool	3,7-dimethylocta-1,6-dien-3-ol	78-70-6	0.062±0.000 18.12	30.82%	2.82	21.17%	×
											S-5

Table S2 . Synthesis of rejected experimental points, before model identification, for aroma compounds highly diluted in ethanol-water mixtures with several data sources available.

 S2

	Chemical family	Common name	Aroma compound IUPAC name	No. CAS	Reference	Number of accepted data	Number of rejected data	xEt	Rejected data KAC αAC/Et Overall trend Criterion Point-to-point comparison
		1-Propanol	Propan-1-ol	71-23-8	Williams, 1962 9 Ikari et al., 1990 36	9 11	2 0	0.000 10.03 0.000 10.04 --	1.48 1.48 -	× ×
					Martin et al., 2009 39	-	-		xEt<0.03	×	×
					Williams, 1962 9	10	2	0.000 17.11 0.000 16.65	2.53 2.47	× ×
		Isobutanol	2-Methylpropan-1-ol	78-83-1	Ikari et al., 1990 36	10	1	0.100 7.26	1.65	×
	Alcohols				Athès et al., 2008 13 Martin et al., 2009 39	8 -	1 -	0.474 0.43 xEt<0.03	0.34	×	× ×
								0.001 18.50	2.85	×
					Williams, 1962 9	11	4	0.001 16.37 0.001 18.93	2.42 2.44	× ×
		Isopentanol	3-Methylbutan-1-ol	123-51-3				0.075 8.77	1.67	×
					Ikari et al., 1990 36	10	1	0.100 6.12	1.39	×
					Athès et al., 2008 13	8	1	0.062 15.11	2.23	×
					Martin et al., 2009 39	-	-		xEt<0.03	×	×
								0.020 52.67	5.88	×
								0.079 43.77	8.75	×
	Carbonyl compounds	Acetaldehyde	Ethanal	75-07-0	Heitz, 1960 8	22	6	0.080 52.35 10.81 0.159 62.33 55.08 0.162 120.67 109.14	× × ×
								0.259 6.26	5.87	×
					Williams, 1962 9	14	0	-	-	-
										S-6

Table S2 . Continuation. Synthesis of rejected experimental points, before model identification, for aroma compounds highly diluted in ethanol-water mixtures with several data sources available.

 S2

	Chemical family	Common name	Aroma compound IUPAC name	No. CAS	Reference	Number of accepted data	Number of rejected data	Rejected data xEt KAC αAC/Et Overall trend Criterion Point-to-point comparison
								0.006 45.44 4.78	×
								0.006 45.15 5.45	×
								0.016 83.93 6.74	×
					Williams, 1962 9	11	7	0.017 60.23 6.29	×
	Carbonyl compounds	Isobutyraldehyde	2-Methylpropanal	78-84-2				0.021 64.02 6.33 0.035 63.45 6.56 0.043 47.72 6.57	× × ×
					Ikari et al., 1998a 37	14	0	-	-	-
								0.021 76.94 5.75	×
					Athès et al., 2008 13	6	3	0.144 19.41 5.21	×
								0.474 3.20	2.48	×
		Ethyl acetate	Ethyl ethanoate	141-78-6	Ikari et al., 1998a 37 Athès et al., 2008 13	14 8	0 1	-0.062 83.90 12.37 --	×
	Esters	Isopentyl acetate	3-Methylbutyl ethanoate	123-92-2	Ikari et al., 1998a 37 Athès et al., 2008 13	14 8	0 1	-0.062 130.45 19.23 --0.021 379.05 28.31	× ×
		Ethyl caproate	Ethyl hexanoate	123-66-0	Athès et al., 2008 13	6	3	0.062 14.59 2.15 0.474 37.90 29.34	× ×
					Martin et al., 2009 39	-	-	-	-	-
					Williams, 1962 9	9	0	-	-	-
	Furans	Furfural	Furan-2-carbaldehyde	98-01-1	Ikari et al., 1984 35	13	0	-	-	-
					Athès et al., 2008 13	9	0	-	-	-
					Athès et al., 2008 13	8	1	0.062 21.42 3.16	×
								0.012 111.47 15.01	×
	Terpenes	Linalool	3,7-dimethylocta-1,6-dien-3-ol	78-70-6	Deterre et al., 2012 32	19	5	0.096 2.40 0.207 0.23 0.208 0.03	0.47 0.09 0.01	× × ×
								0.294 0.67	0.36	×
							S-7

S-2

Selection of experimental coherent data for model identification

One of the critical points for identifying the NRTL interaction parameters was the selection of coherent experimental data. Two criteria were successively applied to accept or reject an experimental value:

-Observation of the overall trend: the evolution of absolute (KAC) and relative volatility (αAC/Et) is expected to be monotonous with respect to ethanol mole fraction in the liquid phase (xEt). Experimental points that do not follow this trend were rejected. -Point-to-point comparison: this criterion was applied when two experimental points from different sources were available at a given xEt (with a maximal difference of ±0.001). The pair of data pass this test if the relative deviation (AE%) between the volatility values (KAC and αAC/Et) is lower than a tolerance, here defined as 20%. Otherwise, one of the points is rejected by considering again the general trend with respect to xEt.

The relative deviation (AE%) between two experimental points Ui and U j is defined as follows:

The tolerance value corresponds to the order of magnitude of the overall relative deviation (AAE%) from NRTL correlation, reported for multicompound systems aroma compoundethanol-water by Athès et al., 13 Faúndez et al., [START_REF] Fauńdez | Predictive models to describe VLE in ternary mixtures water + ethanol + congener for wine distillation[END_REF] and Faúndez and Valderrama. [START_REF] Fauńdez | Activity coefficient models to describe vapor liquid equilibrium in ternary hydro alcoholic solutions[END_REF] The results of the point-to-point comparison are summarized in Table S1 for the 10 aroma compounds with several data sources. A synthesis of the experimental points rejected for this group of aroma compounds is presented in Given the relatively limited number of sources, this methodology was not intended to validate or reject a whole data set, but to choose all the coherent experimental points according to the criteria proposed. In this context, correlation results based on the selected data can be considered as reliable for engineering calculations. Nevertheless, it is important to point out that the uncertainty of equilibrium data used in this work is not negligible, due to the complexity and limited accuracy on the quantification of volatile aroma compounds at high dilution. For simulation purposes in spirits distillation, the interaction parameters here identified must be validated by comparing the simulation results against experimental data, ideally from different distillation units. The aim is not to reproduce perfectly the experimental values but to represent the separation patterns, tendencies and sensitivities, in order to elucidate the influence of design and operating parameters on distillate composition and then on the quality of products