Extending application of Dehaghani Association Equation of State for Modeling Phase Behavior of Water-Alcohol Systems

Accepted Manuscript Extending application of Dehaghani Association Equation of State for modeling phase behavior of water-alcohol systems Amir Hossein Saeedi Dehaghani, Mohammad Hasan Badizad PII:

S0378-3812(18)30486-2

DOI:

https://doi.org/10.1016/j.fluid.2018.11.027

Reference:

FLUID 12014

To appear in:

Fluid Phase Equilibria

Received Date: 7 November 2017 Revised Date:

21 November 2018

Accepted Date: 22 November 2018

Please cite this article as: A.H.S. Dehaghani, M.H. Badizad, Extending application of Dehaghani Association Equation of State for modeling phase behavior of water-alcohol systems, Fluid Phase Equilibria (2018), doi: https://doi.org/10.1016/j.fluid.2018.11.027. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Short Communication

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Extending Application of Dehaghani Association Equation of

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State for Modeling Phase Behavior of Water-Alcohol Systems

Abstract

This paper concerns extending applicability of Dehaghani Association equation of state (in short

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DA-EOS) to model non-ideality in phase behavior caused by hydrogen bonding. Advantageously, this equation only needs two and one parameters to treat self- and cross association, respectively, which were determined for methanol/water and ethanol/water

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mixture in this study. In this respect, DA was coupled with PR and SRK EOS to account for physical interaction between molecules. Notably, DA-PR persistently provides more accurate

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prediction than DA-SRK for both aqueous methanol and ethanol solutions. Results of this work encourage researchers to use DA as a promising EOS for associating fluids.

Keywords: Alcohol; Association; Equation of State; Thermodynamic model; Aqueous solution.

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1. Introduction Among various mixtures dealt with in industry, those forming hydrogen-bond are of

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particular interest [1]. Due to strong polarity, such solutions tend to form molecular clusters which leads to large non-idealities of their thermodynamic behavior [2]. In conventional (usual) mood, the constituent molecules of a simple system interact

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through dispersion forces which is known as physical van der Waals interactions [3]. In this case, the mixture phase behavior could be simply predicted using cubic Equations

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of State (EOS) [4]. However, hydrogen bonding, just by itself, renders a mixture large chemical complexity, for instance, water with four hydrogen bonding sites per molecule, has a high boiling point compared to methane with nearly equal molecular weight [5]. Such special non-ideality could occur for species of same kind, which is

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called self-association, or more generally taking place between different components, known as cross-association [6]. To date, predicting phase behavior of associating solutions has been subject of numerous studies, reflecting importance of those

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substances in chemical and petroleum processes [7]. Regarding underlying philosophy

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for representing non-ideality of hydrogen bonding, most association equations of states (AEOS) fall in three categories: (1) chemical theory, (2) lattice-fluid (quasichemical) theory, and (3) perturbation theory [8]. Historically, chemical theory laid the foundation of most AEOS developed so far [7]. Dolezalek pioneered derivation of mathematical framework to account for non-ideality of associating solutions [9]. In his approach, association was supposed to be 2

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hypothetical chemical reactions to form oligomers through complexion of constituent monomers [9]. In this fashion, association strength is directly proportional to the extent

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of linkage reaction taking place among molecules, which is represented via equilibrium constant(s). Clearly, this approach necessitates solving many chemical and phase equilibrium equations simultaneously, which has proved to be numerically tedious [7]!

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On the other hand, Dolezalek’s approach leads to an iterative numerical procedure rather than a convenient closed-form EOS [10]. This method, while being of

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theoretically valuable, are in contrast to practical engineering calculations which demand analytical expressions for straightforward calculation of phase behavior. To achieve a convenient mathematical model, Heidemann and Prausnitz made attempts to provide analytical expressions for chemical equilibria of an associating system, so

for each one [5].

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that physical and association effects could be treated through different formulations

In a worthy effort, Vafaei-Sefti et al. developed a novel AEOS (called VSA-EOS) by

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presuming infinite extent of associating reactions, corresponding to an unbounded

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spectrum of oligomers [11]. Albeit giving accurate prediction by a simple mathematical expression, however, VSA-EOS is restricted to mixtures solely containing a single associating component encompassed by non-associating (inert) ones [12]. To relax underlying assumption of VSA-EOS, recently, Dehaghani et. al. put forward a new expression by taking finite length for oligomers formed by hydrogen bonding, either in pure fluids or mixtures with any number of associating species [13]. This new model, which was further known as Dehaghani Association Equation of State (in short DA-EOS), 3

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demonstrated excellent capability for handling association in various types of mixtures. For instances, DA-EOS coupled with PR-EOS gives accurate prediction for co-

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precipitation of resin and asphaltene from heavy crude oils once blending with light gases [14, 15]. Additionally, Dehaghani and Badizad recently evaluated performance of DA-EOS for predicting incipient pressure of gas hydrates in presence of thermodynamic

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inhibitors [12]. Recently, they pointed out ability of DA-PR to represent strength of diverse dispersants for retarding onset of asphaltene precipitation [16]. Clearly, DA-EOS

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is in its infancy, needing further investigations to reveal its concealed strength. For this purpose, present communication aims to evaluate capability of DA-EOS for predicting phase behavior of methanol/water and ethanol/water mixtures. In the following, first, we will briefly explain formulation of DA-EOS and then, its prediction performance will

2. Model and Theory

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be thoroughly evaluated.

As explained earlier, to represent phase behavior of an associating mixture, both

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physical (van der Waals) and chemical (hydrogen bonding) interactions should be

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regarded. To incorporate both non-idealities into a mathematical model, Lambert suggested splitting second virial coefficient (B) into two main terms as follows [17]: B = B ph + B ch

(1)

Where superscripts ph and ch stand for physical and chemical contributions, respectively. Based on foregoing expression, compressibility factor of an associating fluid is expressed as [12]: 4

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(2) Z = Z ph + Z ch − 1 Where Z ph stands for physical contribution of compressibility factor acquired readily

using a cubic equation of state. In this respect, Peng-Robbinson (PR) and Soave-

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Redlich–Kwong (SRK) equations of state were chosen to calculated physical term of compressibility factor in this study. For detailed formulation of those cubic EOS refer to [18]. To handle physical interactions between dissimilar components, e.g., i and j,

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Chueh-Prausnitz’s equation was used to calculate binary interaction parameters kij ,

θ

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given by [19]:

 16 16   2υc υc  kij = 1 −  1 i 1i  (3)  υc3 + υc3  j   i Where υc denotes critical molar volume. θ is an adjustable parameter obtained via

regression of equilibrium data.

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To treat chemical compressibility factor Zch appeared in Eq. (2), chemical theory comes in use by showing that [8]: nT n0

(4)

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Z ch =

Here nT and n0 represent true and apparent (superficial) number of moles, respectively,

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in presence and absence of association reactions. Dehaghani et al. proposed following expression for chemical contribution in compressibility factor of a single associating fluid [13]: Z ch =

C K RT C+ 0 P υ

(5)

Where K is association constant and C represents extent of an association reaction. Also, P0 denotes standard pressure. 5

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Further, Dehaghani et al. generalized foregoing formulation for solutions containing several associating species in presence of non-associating compounds [14]:

i =1

nb C i xi + ∑ xk n 1 RT    Ci + 0 ∑ j =1K ij x j  k =1 P υ 

(6)

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na

Z ch = ∑

Here, first and second summations go over total number of associating (na) and nonassociating components (nb), respectively. Advantageously, preceding expression

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(known as DA-EOS) could be employed to calculate chemical compressibility factor of any mixtures containing both associating and non-associating components. Respecting

Z ch =

Ca xa Ca +

1 RT  Ka ,a xa + K a ,w xw  P0 υ 

+

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focus of this study, preceding equation could be written for alcohol-water mixture as: Cw xw

Cw +

1 RT  K w,w xw + K w,a xa  P0 υ 

+ 1 − xa − xw

(7)

Where subscripts a and w denote alcohol and water components, respectively. Kaa and Kww are constants of self-association for alcohol and water, respectively. Kwa represents

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chemical constant of cross-association between water and alcohol which is considered to be symmetric, i.e., Ka,w=Kw,a.

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Once having compressibility factor, fugacity coefficient φ is obtained simply through following exact thermodynamic expression [16]: V 1 RT  ∂P  ( − )dV  RT ∫∞ V  ∂ni T ,V , j ≠i

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ln(φi Z ) =

(8)

For an associating mixture, Eq. (8) could be decomposed as [12]: ln(φi Z ) = ln(φi ph Z ph ) + ln(φich Z ch )

(9)

Using a cubic EOS, physical part of above relation could be simply derived. Also, by introducing DA-EOS into Eq. (8), one will derive chemical part of preceding equation for any associating compound, for example, alcohol in presence of water: 6

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ln(φach Z ch ) = ln

Ca υ + ( K aa xa + K aw xw ) RT K aa RTxa K wa RTxw + + − ln Ca υ Ca υ + ( K aa xa + K aw xw ) RT Cw υ + ( K wa xa + K ww xw ) RT

(10)

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Detailed mathematics of DA-EOS was described in our previous works [12-14, 16]. Given fugacity coefficients detailed above, one could perform flash calculation to obtain P-xy and T-xy diagrams for alcohol-water mixture, which is discussed in the

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following section. 3. Results and Discussion

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DA-EOS and its scheme to formulate effect of association on compressibility factor were introduced in foregoing section. At this stage, we will examine accuracy of DAEOS coupled with PR or SRK for predicting phase behavior of methanol or ethanol/water system. Fortunately, there are enough equilibrium data reported in

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open literature for alcohol-water mixture, which facilitates analysis. On the other hand, since involvement of both types of self- and cross-association, studying alcohol-water

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system is a matter of choice.

As mentioned earlier, few fluids have ever treated by DA-EOS, hence, before

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proceeding to any prediction, one should inevitably obtain unknown parameters, i.e., K, C and θ, for a given mixture. With this in mind, vapor-liquid equilibrium (VLE) data at 14.18 kPa (equivalently 0.14 atm) were employed to correlate model’s parameters. Tables 1 and 2-3 summarize regressed parameters and corresponding average absolute percent deviation (AAD%), respectively. Table 1

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Regressed parameters of DA-PR and DA-SRK obtained by correlating VLE data of methanol-water and ethanol-water mixtures. Equilibrium θ Ca Cw Kww Kaa Kaw -5

2.32×10 -5 1.34×10

-4

2.11×10 -4 1.47×10

-10

1.03×10 -10 5.50×10

-10

1.03×10 -10 5.50×10

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2.50×10 -9 1.37×10

PR -0.0883 -0.0138

SRK -0.0884 -0.0162

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Phases Methanol/Water Ethanol/Water

Table 2

AAD% values for correlation and prediction of VLE methanol-water data by DA-PR and DA-SRK.

18 18 18 18 18 18 50

Temperature, K

0.94 0.96 1.21 1.02 0.88 0.69

1.26 2.62 2.21 1.30 1.89

2.16 2.27 4.03 2.89 4.08 1.93

1.78 1.91 1.80 1.23 0.93 1.58

0.76

1.07

1.35

0.89

2.31

1.48

4.28 2.08 3.51

1.84 2.56 2.09

1.90 1.42 1.73

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Overall 1

3.61 3.80 3.65 2.64 3.40 2.41

DA-SRK AADx% AADy%

P-xy

30 16

312 322

DA-PR 2 3 AADx% AADy%

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14.18 (0.14) 28.37(0.28) 42.56 (0.42) 55.73 (0.55) 66.87 (0.66) 95.24 (0.94) 101.32 (1.00) Overall

T-xy 1

Np

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Pressure, kPa (atm)

Number of data points 2 AAD% for mole fraction in liquid phase. 3 AAD% for mole fraction in vapor phase

Table 3

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AAD% values for correlation and prediction of VLE ethanol-water data by DA-PR and DA-SRK. Pressure, kPa (atm)

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6.66 (0.066) 11.99 (0.125) 25.33 (0.250) 50.66 (0.500) 101.32 (1.000) Overall

1

Np

19 25 20 20 36

DA-PR 2 3 AADx% AADy%

DA-SRK AADx% AADy%

3.78 3.41 3.22 2.34

0.85 0.78 1.05 1.02

1.97 2.11 3.86 2.67

1.59 1.83 1.12 1.24

1.48

0.87

1.83

0.94

2.53

0.65

2.42

1.20

1

Number of data points 2 AAD% for mole fraction in liquid phase. 3 AAD% for mole fraction in vapor phase

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As shown in Figs. 1 (a) and 2 (a), one could notice impact of association on correlating T-xy data, particularly in case of SRK, for aqueous methanol and ethanol mixtures,

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respectively. Also, y-x diagram is of similar trend, as shown in Figs. 1 (b) and 2 (b). In this case, DA-PR correlation is slightly better than DA-SRK. It should be mentioned that values of θ, and accordingly resultant binary interaction parameters, obtained through

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correlation of VLE data by DA-PR and DA-SRK, are identical to those acquired using PR and SRK, respectively, as presented in Table 1. This observation highlights the

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consequence of separating chemical and physical contribution in compressibility factor, which was earlier pointed out by Lambert and described in Eq. (2). Regarding this fact, one might expect different correlation (and prediction) while coupling DA-EOS with different cubic EOS, which is evident in Figs. 1 and 2. As a result, selecting a proper

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compeer cubic EOS is necessary to achieve greatest accuracy of DA-EOS.

Figure 1 (a) T-xy, and (b) y-x diagrams obtained by correlating VLE data of methanol-water mixture at 0.14 atm. All experimental data were adopted from [19].

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Figure 2 (a) T-xy, and (b) y-x diagrams obtained by correlating VLE data of ethanol-water mixture at 1

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atm. All experimental data were adopted from [19].

Given adjustable parameters, we proceed to analyze DA-EOS prediction at higher

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pressures and different temperatures. To this end, Figs. 3 and 4 visually compares performance of different models for predicting T-xy diagram of aqueous methanol and

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ethanol mixtures, repectively, at 14.18 to 101.32 kPa. Additionally, Figs. 5 and 6 shows corresponding results in y-x diagram. Evidently, incorporating DA-EOS to phase behavior calculation improves prediction accuracy. Apart from this observation, one could observe a subtle descending trend in AAD% values presented in Tables 2 and 3, which is consistent with previous studies [14].

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Figure 3 T-xy plot for methanol-water systems at constant pressure of: (a) 0.28 atm; (b) 0.42 atm; (c) 0.55 atm; (d) 0.66 atm; (e) 0.94 atm; and (f) 1 atm. All experimental data were adopted from [20].

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Figure 4 T-xy plot for ethanol-water systems at constant pressure of: (a) 0.066 atm; (b) 0.125 atm; (c) 0.25 atm; and (d) 0.5 atm. All experimental data were adopted from [19].

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Figure 5 y-x plot for methanol-water system in constant pressure of: (a) 0.28 atm; (b) 0.42 atm; (c) 0.55 atm; (d) 0.66 atm; (e) 0.94 atm; and (f) 1 atm. Note Figure 2 shows T-xy diagrams corresponding to above figures. All experimental data were adopted from [20].

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Figure 6 y-x plot for ethanol-water system in constant pressure of: (a) 0.066 atm; (b) 0.125 atm; (c) 0.25

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atm; and (d) 0.5 atm. Note Figure 2 shows T-xy diagrams corresponding to above figures. All

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experimental data were adopted from [19].

Fig. 7 shows P-xy and respective y-x plots at 312 and 322 K. As seen, dew point curve in P-xy plot demonstrates slightly positive deviation from Raoult’s law, because selfassociation of water and methanol molecules is comparable to their cross-hydrogen bonding [21, 22]. Likewise, respective y-x plots exhibit persistent positive deviation from unit slop straight line, implying that mole fraction of methanol in vapor phase is

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much greater than liquid phase which is a consequence of more volatile nature of

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methanol than water [23].

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Figure 7 P-xy (left) and respective y-x (right) plot for methanol-water system at constant temperature of: (a) 312 K; and (b) 322 K. All experimental data were adopted from [20].

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The predictions given by our proposed EOS was compared to results obtained by other well-known models proposed in the literature, including UNIQUAC, NRTL and Wilson [24]. The AAD% values obtained by DA-EOS, as presented in Table 4, are comparable prediction given by other models. Noteworthy, the prediction of DA-EOS is even more accurate than UNIQUAC. Therefore, irrespective to kind of cubic EOS used, DA should

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be regarded an accurate expression for repressing phase behavior of alcohol-water solutions.

DA-PR 1.02 0.87

DA-SRK 1.24 0.94

UNIQUAC 0.72 1.64

NRTL 0.43 0.56

Wilson 0.54 0.80

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P, kPa (atm) 50.66 (0.5) 101.32 (1.0)

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Table 4 absolute average error of vapor phase composition for ethanol-water mixture using different models [24].

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4. Conclusions

Applicability of DA-EOS for predicting phase behavior of methanol/water and ethanol/water mixture was explored in this mixture. For this purpose, DA was coupled with PR and SRK to account for physical part of the compressibility factor, and former gave more accurate predicting regarding both T-xy and P-xy plots. Therefore, it was

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conclude that performance of DA-EOS depends on accompanying cubic EOS.

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another outcome, unknown parameters of DA-EOS were obtained for methanol and

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water to be used in future studies. In general, modeling results were in good consistency with the experimental data, with maximum deviation for liquid and vapor

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mole fraction of (4.03, 1.91) and (3.86, 1.83) for aqueous methanol and ethanol solutions, respectively. References

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