Using Volume Shifts To Improve the Description of Speed of Sound

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Thermodynamics, Transport, and Fluid Mechanics

Using volume shifts to improve the description of speed of sound and other derivative properties with cubic Equations of State. André M. Palma, Antonio J. Queimada, and Joao A.P. Coutinho Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.9b00817 • Publication Date (Web): 02 May 2019 Downloaded from http://pubs.acs.org on May 3, 2019

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Industrial & Engineering Chemistry Research

Using volume shifts to improve the description of speed of sound and other derivative properties with cubic Equations of State. André M. Palma1, António J. Queimada2,* and João A. P. Coutinho1 1

CICECO, Chemistry Department, University of Aveiro, Campus de Santiago, 3810-193 Aveiro, Portugal. 2

KBC Advanced Technologies Limited (A Yokogawa Company), 42-50 Hersham Road, Walton-onThames, Surrey, United Kingdom, KT12 1RZ. *Corresponding author. E-mail address: [email protected] Abstract The simultaneous description of phase equilibria, volumetric properties and derivative properties is of great relevance to the industry. However, most thermodynamic models are unable to describe all these properties with the same set of parameters. Equations of state are no exception, especially cubic equations of state where this behavior is even more relevant. When considering the classic fitting of these equations to critical data, a volume shift is often required for an accurate description of density, but this approach fails to provide a satisfactory description of derivative properties. In this work, we analyze the influence of a volume shift within the modified CPA, PR and SRK equations of state, when fitted to speed of sound data instead of density data. An analysis is conducted on the effect for other derivative properties such as isothermal compressibility and isobaric expansivity. The analysis is based on water, alkanols/diols and a group of hydrocarbons. A critical assessment is conducted on how this approach affects other properties. Keywords: CPA, Volume Shift, Equation of State, Speed of Sound, Derivative Properties.

1. Introduction It is well known that, for many properties, cubic equations of state (EoS) good performance is based on cancellation of errors. However, there are some properties where these errors do not compensate and are instead amplified. One such property is the speed of sound. Most equations of state need to fit this property directly in order to obtain a correct description and are known to be unable to provide a correct, simultaneous, description of the speed of sound, liquid density and saturation pressure with the same set of parameters. 1 SAFT variants tend to present better estimates of speed of sound even when this property is not fitted to the experimental data. Studies on the capacity to describe this property for different classes of compounds have been reported using different SAFT variants, including the original SAFT, SAFT-BACK, PC-SAFT and SAFT-VR-Mie. 2–9 1 ACS Paragon Plus Environment

Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Despite the better performance of SAFT type equations, cubic equations of state are still the most applied in the industry, due to their simplicity and lower calculation times. Another advantage of the cubic equations, using their classic parameterization, is the correct description of critical temperatures and pressures. Some SAFT variants, as is the case of CP-PC-SAFT, 10 are capable of a correct description of these properties, but tend to incorrectly predict saturation pressures at conditions far from the critical point. 2 In previous works 11–13, we analyzed the description of pure component properties of associative compounds with a modified version of the CPA equation of state. Among other differences described elsewhere, 12,13 this version uses a constant Péneloux type volume shift

14

to fit the

liquid density instead of regressing this property in the main parameterization routine. In these studies, most properties analyzed were independent of the volume shift, except for density. In a recent study, Jaubert et al.

15

presented the properties modified by the use of a constant

volume shift in a cubic EoS. The same can be done for other EoS such as CPA 16 or PC-SAFT. 17 In the specific case of water, a large group of properties presents an anomalous behavior, even at atmospheric conditions. These include, for example, maxima with temperature, in the liquid speed of sound, density and isochoric heat capacity and a minimum with temperature, in the isobaric heat capacity. This type of behavior is not exclusive to water, and can be observed in simpler molecules, as is the cases of alkanes, when under higher pressures. parameters for water with the modified CPA

13

18,19

When the

were presented, the study concerned mostly

binary systems and the description of both isobaric and isochoric heat capacity, with the former respecting the above mentioned minima. A maxima in respect to the speed of sound is also obtained at temperatures close to the experimental. However, for this to be quantitatively correct the description of both density and the Joule-Thomson coefficient are greatly deteriorated. For this compound an analysis on the effect of the volume shift on gas properties was also carried. In this work, the differences between using volume/density or the speed of sound to optimize the volume shift are analyzed with the modified CPA, SRK, PR 20 and modified versions of these two cubic EoSs, which use the same Mathias-Copeman alpha function as the modified CPA. Some comparisons are also conducted with the original CPA and two PC-SAFT variants. A critical assessment on how this approach affects other properties is reported. 2.1 Models In a generalized form, Peng-Robinson (PR) and Soave-Redlich-Kwong (SRK) equations of state 20 can be written for a pure component, in terms of pressure, as: 2 ACS Paragon Plus Environment

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𝑅𝑇

𝑃 = (𝑉−𝐵) − (𝑉+𝛿

𝐴(𝑇) 1 𝐵)(𝑉+𝛿2 𝐵)

(1)

Where T is temperature, P is pressure, V is volume and R is the gas constant. The compound specific parameters/functions of these equations are the co-volume (𝐵 = 𝑛𝑏), the value of the energy parameter (𝐴(𝑇) = 𝑛2 𝑎𝑐 𝛼(𝑇))), as well as the alpha function (𝛼(𝑇)). 𝛿1 and 𝛿2 are specific of each of these equations (𝛿1 = 1 ; 𝛿2 = 0 for SRK and 𝛿1 = 1 + √2 ; 𝛿2 = 1 − √2 for PR). In this work, only the SRK based version of the modified CPA 12 is analyzed. Thus, to the cubic term of SRK we need to add the following association term: 1

𝑃𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑖𝑜𝑛 = 2𝑉 (𝑉

𝜕𝑙𝑛𝑔 − 𝜕𝑉

1) ∑𝑖 𝑚𝑖 (1 − 𝑋𝑖 )

(2)

Here mi is the mole number of sites of type i and Xi is the non-bonded fraction of sites of type i. g is a simplified hard-sphere radial distribution function: 1

𝑔(𝜌) = 1−0.475𝑏⁄𝑉

(3)

The non-bonded site fractions are obtained from:

𝑋𝑖 =

1 𝑔 1+ ∑𝑗 𝑚𝑗 𝑋𝑗 Δ𝑖𝑗 𝑉

(4)

The association strength (Δ𝑖𝑗 ) is given by: 𝜀𝑖𝑗

Δ𝑖𝑗 = 𝑏 𝑖𝑗 𝛽 𝑖𝑗 (𝑒 𝑅𝑇 − 1)

(5)

With, 𝜀 𝑖𝑗 and 𝛽 𝑖𝑗 being the energy and volume of association, respectively for interactions between sites i and j. Two association schemes 21 are applied in this work: 2B for the hydroxyl group and 4C for water, for diols two 2B schemes, with the same parameters, are considered, which is technically the same approach as considering the 4C scheme, as the parameters for both hydroxyl groups are considered as being the same. Figure 1 presents these association schemes.

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Figure 1 – Association schemes applied in this work When using CPA to describe these compounds, these are the most commonly used association schemes, as discussed by Kontogeorgis and co-workers. 16 Two alpha functions are applied in this work, the Soave alpha function 22 (which has only one parameter) and a modified Mathias-Copeman

23

alpha function, which can have up to five

parameters. These functions can be represented from the general equation bellow:

𝛼(𝑇) =

𝑛 (1 + ∑1 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠 𝑐𝑛

(1 −

𝑛 2 𝑇 √𝑇 ) ) 𝑐

(6)

Where, cn are fitting parameters and Tc is the critical temperature. The Soave alpha function is used when considering the classical SRK and PR. For the modified version of these equations, as well as for the version of CPA in study, the modified MathiasCopeman alpha function is used instead. It is important to note that the present alpha function is based on a Mathias-Copeman function and thus does not cope with all of the consistency requirements proposed by le Guennec et al. 24

. This may lead to some pitfalls beside those presented by Segura et al.

25

for Soave type

equations of state. Nevertheless, for the range of temperatures analyzed in this work the alpha functions for the non-associative compounds are consistent with the requirements of le Guennec et al. 24. For the associative compounds there are some more complex behaviors (case of water, see previous article

13

). However, in the range of temperatures of the study, the

presence of the associative term compensates for most of the inconsistencies presented by le Guennec et al. 24. For example in the presence of an inflection in the alpha function for these compounds cvres will not be 0 due to the presence of the association term. For all these equations, the co-volume and energy parameter of the cubic term are fitted directly to the critical pressures and temperatures, so that: 𝑒𝑥𝑝

𝑃𝑐𝑎𝑙𝑐 (𝑇𝑐

𝑒𝑥𝑝

, 𝑉𝑐𝑐𝑎𝑙𝑐 ) = 𝑃𝑐

(7) 4

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𝜕𝑃𝐶𝑎𝑙𝑐(𝑇𝑐𝑒𝑥𝑝 ) 𝜕𝑉

=0

𝜕2 𝑃𝐶𝑎𝑙𝑐(𝑇𝑐𝑒𝑥𝑝 )

=0

𝜕𝑉 2

(8)

(9)

Thus, these models reproduce the experimental critical temperatures and critical pressures, for pure compounds. For the modified versions, the alpha function parameters are then fitted to saturation pressure data, while for the classic SRK and PR these are obtained from the classic equations for the alpha function parameters. 26 A constant Péneloux type volume shift 14 affects both volume and co-volume so that: 𝑉 = 𝑉0 − 𝑐𝑣𝑠 { 𝑡 𝑏𝑡 = 𝑏0 − 𝑐𝑣𝑠

(10)

Vt and bt are the translated volume and co-volume, whereas V0 and b0 are the original volume and co-volume obtained by the equation. The parameter c is the volume shift. 2.2 Properties in study As discussed by Jaubert and co-workers, 15 diverse properties remain unaffected by the use of a constant volume shift. These include, saturation pressures, the enthalpy of vaporization (both the vapor and liquid enthalpies change, but the difference remains unchanged) Cp, and Cv. It is also important to note, that, despite the chemical potential of a given phase changing with the introduction of a volume shift, this change will be equal for all phases, if the adequate mixing rule is applied (𝑐𝑚𝑖𝑥𝑡𝑢𝑟𝑒 = ∑ 𝑥𝑖 𝑐𝑖 ). Thus, phase equilibria will remain unchanged. Alternatively to volume/density, the volume shift in this work was also fitted to the liquid speed of sound. This property can be obtained from other thermodynamic properties using: 𝑢 = 𝑉√𝐶

𝐶𝑝

𝑣 𝑀𝑤 𝑘𝑇 𝑉

(11)

Here, Cp, Cv and the molar weight (Mw) are all independent of the volume shift. The result of volume multiplied by the isothermal compressibility (kt) is also independent of the volume shift: 𝑑𝑉

𝑘𝑡 𝑉 = − (𝑑𝑃)

𝑇

(12)

Which, as the volume shift does not vary with pressure, is a constant. Thus, all terms inside the square root term above are independent of a constant volume shift.



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Other properties in analysis on this work are the isobaric expansivity coefficient (αp), the JouleThomson Coefficient (µJT) and the isentropic compressibility (ks), which are given by the following equations: 1 𝑑𝑉 𝑉 𝑑𝑇 𝑃

𝛼𝑝 = ( ) 1

(13)

𝑑𝑃

𝑑𝑃

𝜇𝐽𝑇 = − 𝐶 (𝑉 + 𝑇 (𝑑𝑇 ) ⁄(𝑑𝑉) ) 𝑉

𝑝

𝑇

𝑉

𝑘𝑠 = 𝑢 2 𝑀

(14) (15)

𝑤

3 Results The use of constant, Péneloux type, volume shifts is well-established in the literature as a method to improve the description of volumes, without affecting phase equilibria. The use of such methods enables improvements on the saturation pressures, as well as, some derivative properties, by focusing the parameterization on the saturation pressures and correcting the volume afterward. However, cubic equations of state, are known to fail not only in the description of volume, but also on volume derivative properties. Nevertheless, if the process of interest does not require a correct description of density (or does requires it, but obtains that description from another model), it should be possible to compensate partially the errors on these derivatives, to obtain a reasonable description of other properties, as is the case of the speed of sound. 𝑑𝑃

Speed of sound depends on Cp, Cv, volume and (𝑑𝑉) , as can be seen in equation 11. Thus, by 𝑇

having a good representation of both Cp and Cv the contributors to deviations in the speed of 𝑑𝑃

sound can only be volume and its pressure derivative. But (𝑑𝑉) , is not affected by the volume 𝑇

shift. Thus, by changing the volume shift (if the volume shift is a constant), the only term affected in the speed of sound expression is the volume. This study will start from simpler molecules (hydrocarbons) to more complex molecules with association (alcohols and water). One of the topics will be looking at the description of the speed of sound for water in the liquid phase, as this is a very specific case, where its speed of sound presents a maximum with respect to temperature, even at atmospheric conditions, which has proved very difficult to represent by different models. 27 The pure compound parameters, except for volume shift are presented on Table A1 of the supporting information. When the volume shift is fitted to volume, this is always conducted at 0.7 Tr, while the fitting temperatures to the speed of sound are presented on Table A2 of the supporting information. 6 ACS Paragon Plus Environment

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3.1 Hydrocarbons For hydrocarbons, the use of the modified versions of SRK and PR available from Multiflash, 28 improves the saturation pressure and some derivative properties, including Cp. It is important to note that, when considering a volume shift for both versions, the volumetric results for the classical cubic EoS and their modified versions are very similar, as can be seen in Figure 2 for hexadecane. This remains true for the remaining compounds in this study, as well as for PR. Thus, we will focus on the results from the modified versions.

Figure 2 – Liquid density (left) and speed of sound (right) of n-hexadecane with SRK. Full lines – Classical SRK; Dashed lines – Modified SRK. Blue lines – VS fitted to speed of sound; Green lines – VS fitted to density. Data values are from Bolotnikov et al. 29 and Multiflash. 30 In the case of n-hexane, the results for the liquid density, speed of sound, kT and αp are presented in Figure 3.



8000

1600 SRK vshift ssound SRK vshift density PR vshift ssound PR vshift density

1300

6000

u/m.s-1

ρ/mol.m-3

7000

5000

3000

270

320

370

T/K

420

1000

700

SRK vshift ssound SRK vshift density PR vshift ssound PR vshift density

4000

400 270

470

0.005

320

370

T/K

420

15 SRK vshift ssound

SRK vshift ssound

SRK vshift density

SRK vshift density

0.004

PR vshift ssound

PR vshift ssound

10

PR vshift density

0.003

PR vshift density

kT/GPa-1

αP/K-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.002

5 0.001

0

0 270

320

T/K

370

270

420

320

370

T/K

420

Figure 3 – Influence of volume shifts on the calculated properties of hexane with SRK and PR. Data from Cerdeiriña et al., 31 Garbajosa et al., 32 Bolotnikov et al. 29 and Multiflash. 30 In this case, Peng-Robinson presents better liquid densities, while also presenting smaller 𝑑𝑃

differences in the two volume shifts (due to a more accurate description of (𝑑𝑉) by this EoS). 𝑇

However, for both models, the isobaric expansivity coefficient is less accurate if we consider a 𝑑𝑃 𝑑𝑉 𝑇

𝑑𝑉 𝑑𝑇 𝑃

volume shift fitted to the speed of sound. The ( ) and ( ) description of both models is analyzed in Figure 4. Besides these equations, the results from the parameters from Gross and Sadowski 33 with PC-SAFT and the ones from the VS-PC-SAFT are presented. 17


0.0E+00

4.00E-07

-1.0E+00

3.50E-07

-2.0E+00

(dV/dT)P/m3.K-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(dP/dV)T/TPa.m-3

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-3.0E+00 -4.0E+00 SRK PR VS-PC-SAFT PC-SAFT original CPA

-5.0E+00 -6.0E+00

299

324

349

3.00E-07 2.50E-07 2.00E-07 1.50E-07 1.00E-07

-7.0E+00 274

SRK PR VS-PC-SAFT PC-SAFT original CPA

274

374

374

T/K

T/K 𝑑𝑃

324

𝑑𝑉

Figure 4 – (𝑑𝑉) and (𝑑𝑇 ) of hexane, for atmospheric conditions, using different equations of 𝑇

𝑃

state. Data calculated from Cerdeiriña et al., 31 Garbajosa et al. 32 and Multiflash. 30 𝑑𝑃 𝑑𝑉 𝑇

As mentioned above, there is a significant improvement in the description of ( ) from the 𝑑𝑉 𝑑𝑇 𝑃

cubic term of SRK to the one from PR, while for ( ) the results are similar. When considering the two sets from PC-SAFT, the results are improved, when comparing to the cubic EoSs. Nevertheless, it is not possible to describe both derivatives correctly with a single set of parameters. Another important aspect is that when not applying a fit to the critical temperature and pressure, and fitting pressure and liquid density simultaneously, these derivatives tend to improve, as observed with the original CPA parameters from Oliveira et al. 34 Figure 5 presents results for the speed of sound and liquid density of n-decane, toluene and benzene. For all of these compounds Peng-Robinson presents a more accurate description of 𝑑𝑃

(𝑑𝑉) , while the description of Cp and Cv by both EoS are similar. This leads to a better 𝑇

description of the speed of sound, when considering the volume shift fitted to density with this EoS. Thus, the differences between the volume shift needed to describe density and the one used to describe the speed of sound are smaller with PR than with SRK. Nevertheless, care is needed when looking at very small compounds. For these molecules SRK tends to present a better description of volume than PR. However, for lighter alkanes, when a volume shift is applied to both equations, PR still presents a better description of volume and a smaller difference between the volume shift for density and for the speed of sound.




1400


5500

ρ/mol.m-3

u/m.s-1

1200

1000

4500

3500 800

a1)

b1)

2500

600 290

310

330

T/K

350

270

370

370


1400

470

T/K

13000

1600


11000

ρ/mol.m-3

u/m.s-1

1200 1000

800

9000

7000 600

a2)

b2)

400 270

320

370

420

5000

470

270

T/K

370

470

T/K

11000 SRK vshift ssound SRK vshift density PR vshift ssound PR vshift density


1500

9000

ρ/mol.m-3

1300

u/m.s-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1100

7000 900

a3)

b3)

700

5000 270

290

310

T/K

330

350

370

270

370

T/K

470

Figure 5 – Speed of sound (a) and liquid density (b) for n-decane (1), benzene (2) and toluene (3). Data from Paredes et al., 35 Fortin et al., 36 the TRC database 37 and Multiflash. 30


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The cubic EoSs are known to fail in the description of volumetric properties and derivative properties along an isotherm. For these, the change on volume shift fitting does not bring significant advantages. The qualitative description of speed of sound along the isothermals is far from accurate even when a correct value is applied at saturation. While PR tends to be more accurate both models fail in the description of density and speed of sound for high pressures. This is in large extent due to the cubic term and particularly the repulsive term employed in these equations, which is far simpler, than those of the SAFT variants. Examples for the liquid density and speed of sound in the cases of n-hexane and benzene are presented in Figure 6. 8500 313 K 323 K 333 K 373 K

1700

8300 8100

1500

7700

u/m.s-1

ρ/mol.m-3

7900

7500 7300

1300 1100

7100 6900

a1)

6700

313 K

323 K

348 K

373 K

900

b1) 700

6500 0

10

20

30

40

0

50

10

20

P/MPa 1800

298 K

12.4

318 K

12.2

30

40

50

P/MPa

12.6

283 K 313 K 353 K

308 K 1700

328 K

1600

12

298 K 333 K

1500

11.8

u/m.s-1

ρ/kmol.m-3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


11.6 11.4 11.2

1400 1300

1200

11 10.8

1100

a2)

10.6 0

10

b2)

1000 20

P/MPa

30

40

0

10

20

30

P/MPa

Figure 6 – High pressure liquid density (a) and speed of sound (b) for hexane (1) and benzene (2). Full lines – PR; Dashed lines – SRK. The volume shifts were used according to the property in study. Data are from Kiran and Sen, 38 Daridon et al., 39 Colin et al. 40 and Wegge et al. 41 It is important to note, when analyzing Figure 6, that the volume shifts for density and speed of sound, were not fitted at the same temperatures, being the density fitted at Tr = 0.7. Speed of



Page 12 of 35

sound presents a better description at saturation, as the volume shift for this property was fitted near the temperature of the experimental data (see Table A2 of the supporting information). 3.2 Associating compounds After analyzing the results for hydrocarbons, it is now important to study what happens for associating compounds. In the first place, it is important to note that the use of the association term in CPA tends to compensate partially for the errors on the volumetric properties, and the densities are usually more accurate than for non-associating compounds. In this work these compounds are studied with the modified version of CPA. Beside density, it is expected that speed of sound will be easier to describe for some alkanols, than they are for hydrocarbons. The modelling of both density and speed of sound for 1-butanol and for 1-octanol are presented in Figure 7.


Page 13 of 35

12000

1500 vshift density vshift ssound

11000

vshift density 1300

9000

u/m.s-1

ρ/mol.m-3

10000

8000 7000 6000

vshift ssound

1100

900

700

5000

a1)

b1)

4000

500

270

370

T/K

470

570

270

7000

320

370

420

T/K

470

1600 vshift density vshift ssound

6500

vshift density 1400

vshift ssound

6000 5500

u/m.s-1

ρ/mol.m-3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


5000

1200

1000

4500 800 4000

a2)

3500

600 270

370

T/K

470

570

b2) 270

320

370

420

T/K

470

Figure 7 – Description of density (a) and speed of sound (b) for 1-butanol (1) and 1-octanol (2), using different volume shifts. Data from Safarov et al. 42, Troncoso et al., 43 Multiflash 30 and Nain. 44 𝑑𝑃

The introduction of the association term improves the description of (𝑑𝑉) and thus the 𝑇

difference between the two volume shifts is reduced, when compared to hydrocarbons. It also appears that for alkanols, the higher the association relevance, the better the description of speed of sound. This is visible from the temperature trends for 1-butanol, where the association is more prevalent and so the temperature dependence is better described than for 1-octanol. The association volume and energy are the same for 1-butanol and 1-octanol, as used in a previous work, 11 however, the weight of the contribution of the association term is smaller for 1-octanol. Some other properties of 1-butanol are analyzed in Figure 8.



8

5

7

4.5

vshift density

vshift density

4

6

vshift ssound

3.5

αP/kK-1

5

kT/Pa-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 35

4

3

2.5

3

2

1.5

2

1

1

0.5

0

0

270

320

370

T/K

420

470

270

320

370

T/K

420

470

Figure 8 – Description of the isothermal compressibility (left) and isobaric expansivity coefficient (right) for 1-butanol using two volume shifts. Data from Safarov et al. 42 Using a volume shift fitted to the speed of sound improves the quantitative description of kT. For αP the description at lower temperatures is better when considering the volume shift fitted to liquid density. However as both approaches fail for higher temperatures, there is a large range of temperatures, where this property is better estimated with the volume shift fitted to the speed of sound. To complete the analysis of the primary alkanols, Figure 9 presents results for the speed of sound, isentropic compressibility and density of methanol and ethanol.


Page 15 of 35

1400

vshift ssound

vshift density vshift ssound original CPA

1200

original CPA

1200

1100

1100

u/m.s-1

u/m.s-1

1300

vshift density

1300

1000

1000 900

900 800

800

a1)

a2)

700

700

270

290

310

330

T/K

4.0

350

270

290

310

T/K

3.0


3.5

330

350


2.5 2.0

2.5

ks/GPa-1

ks/GPa-1

3.0

2.0 1.5

1.5

1.0

1.0 0.5

0.5

b1)

0.0

0.0 270

290

310

330

T/K

26000

350

b2) 270

290

310

T/K

330

350

18000


vshift density vshift ssound

16000

22000

original CPA

ρ/mol.m-3

ρ/mol.m-3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


18000

14000

14000

12000

10000

c2)

c1)

10000 270

8000 370

T/K

470

270

370

T/K

470

570

Figure 9 – Results for speed of sound (a), isentropic compressibility (b) and liquid density (c) for methanol (1) and ethanol (2). Original CPA parameters from Oliveira et al. 34 Data from Pereira et al. 45 and Salinas et al. 46



Page 16 of 35

In the smaller alkanols while the trend with temperature is improved for the speed of sound, the difference in volume shift is similar to the one observed in 1-butanol (see Table A2). The results using the original CPA for these compounds are accurate. As discussed on Figure 4, by not fitting the critical temperatures and pressures, while fitting density and pressure simultaneously, the original CPA provides a quantitatively better description of the derivatives in analysis. This is here verified when considering the fit of volume shift to density, where the results for the three properties are not as accurate as when applying the original CPA, without volume shift. Nevertheless qualitatively, when considering the volume shift fitted to speed of sound, it seems that the model is capable of a better description of the speed of sound. The isentropic compressibility, as presented in equation 15 can be easily calculated from the speed of sound and volume. As this property depends on the volume divided by the squared speed of sound, the results are improved when the volume shift is fitted to the speed of sound. It is also relevant to investigate how this approach works for diols, in this case ethylene glycol (1,2-ethanediol) and 1,3-propanediol. Figure 10 presents liquid density, speed of sound, kT and 𝑑𝑃

(𝑑𝑉) for ethylene glycol. 𝑇


Page 17 of 35

19000

1800

1700

vshift ssound

17000

vshift ssound

1600

u/m.s-1

16000

ρ/mol.m-3

vshift density

vshift density

18000

15000

14000 13000

1500 1400 1300

12000 11000

1200

a)

10000

b)

1100

270

370

T/K

470

570

270

0.8

290

310

T/K

330

350

-20

vshift density 0.7

-25

(dP/dV)T/TPa.m-3

vshift ssound

0.6

kT/MPa-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.5 0.4 0.3

-30 -35 -40 -45 -50 -55

c) 0.2

d)

-60 290

295

300

305

310

315

290

T/K

300

310

T/K 𝑑𝑃 𝑑𝑉 𝑇

Figure 10 – Liquid density (a), speed of sound (b), isothermal compressibility (c) and ( ) (d, independent of the volume shift) for liquid ethylene glycol. Data/correlations are from the TRC 37 database and Multiflash. 30 The difference in the results using the two volume shifts is higher than those for the alkanols. However, the description of the speed of sound is accurate, according to the available 𝑑𝑃

experimental data. As for the other properties, (𝑑𝑉) is highly overestimated, leading to an 𝑇

overestimation of kT, even when the volume shift is fitted to the speed of sound. The results for 1,3-propanediol are presented in Figure 11.



14500

1750 vshift density

14000

vshift density

1700

vshift ssound

13500

vshift ssound

1650 1600

u/m.s-1

13000

ρ/mol.m-3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 35

12500

12000

1550

1500

11500

1450

11000

1400

10500

1350

10000

1300

290

340

390

T/K

440

490

540

290

310

T/K

330

350

Figure 11 – Liquid density (left) and speed of sound (right) for 1,3-propanediol. Experimental data are from George and Sastry 47 and the DIPPR database. 48 The difference between the volume shifts in this case is smaller than for ethylene glycol. Nevertheless, there are higher deviations for both density and the speed of sound. This should 𝑑𝑃

come mainly from the description of (𝑑𝑉) , which is quantitatively closer to the experimental 𝑇

data than that of ethylene glycol, however is not qualitatively correct. 3.3 Water. In a previous study of water, 13 we observed that, between 273.15 K and 303.15K, the difference between Cp and Cv, obtained with the modified CPA, was very close to the experimental value. In this range of temperatures, the values for Cp and Cv in water are very close to each other and most equations of state are unable to correctly describe both properties simultaneously. However, there is another interesting result when considering temperatures below 373.15 K. The results obtained with the modified CPA for the speed of sound, although far from being quantitatively correct, could represent the experimentally observed maximum. This is shown in Figure 12, where, in addition to the previously obtained volume shift from fitting liquid density at Tr = 0.7, we also plot the results using a second volume shift, fitted to the liquid speed of sound around Tr = 0.45.


Data Gupta et al. Data NIST vshift Ssound vshift density original CPA

1900 1700

u/m.s-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1500 1300

62 57 52 47

ρ/kmol.m-3

Page 19 of 35

42 37 32 data critical density vshift density vshift Ssound original CPA

27

1100

22 900

17

700

12 280

380

480

580

260

360

460

560

660

T/K

T/K

Figure 12 – Results for liquid water speed of sound (left) and liquid density (right), using two different volume shifts. Data are from Gupta et al. 49, Multiflash 30 and REFPROP . 50 As discussed above, the Cp and Cv of water are well described with this version of CPA, thus the 𝑑𝑃

volume and (𝑑𝑉) have the main impact on the description of the speed of sound. As in the case 𝑇

𝑑𝑃

of ethylene glycol, it seems that the description of (𝑑𝑉) is qualitatively correct, this will be 𝑇

analyzed further below. However, as the accuracy of the descriptions of Cp and, in particular, Cv are not correct for a large group of compounds, 1 the quality of the description of the speed of sound can be inferior to that presented on Figure 12. This was previously shown for diverse compounds. As discussed for ethanol, the quantitative description of the speed of sound with the original CPA is superior to that of the present model when considering the fit of the volume shift to density. Nevertheless in this case the qualitative description is better, presenting a maxima in the speed of sound, where none is observed in the other approach. It is important to verify how this change on the fitting of the volume shift affects other water derivative properties. Figure 13 presents the results for both the isobaric expansivity coefficient and the isothermal compressibility. In the case of the modified CPA, the use of a volume shift, is one of the factors that improved derivative properties (along with the changes in alpha function and of values for the association parameters), due to a focus on the accurate description of properties independent of the volume shift. This enabled, specifically in the case of water, a 𝑑𝑃

𝑑𝑃

better description of (𝑑𝑇 ) , and affected the description of (𝑑𝑉) . For this second derivative, 𝑉

𝑇

the obtained trend with temperature is qualitatively closer to the trend observed experimentally, when compared to that obtained with the original CPA. However, the overall 𝑑𝑃

value of (𝑑𝑉) is incorrect. It is due to these improvements that a maxima is observed in the 𝑇



description of the speed of sound in water. Thus, by fitting the volume shift to the speed of sound, instead of density, it is possible to describe that property for a large range of temperatures. 1.2

1.1 vshift density

1

vshift density

1

vshift ssound

vshift Ssound

0.9

kT/GPa-1

0.8

αp /kK-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 35

0.6 0.4

0.8 0.7 0.6 0.5

0.2

0.4

0

0.3 280

295

310

325

T/K

340

355

370

280

295

310

325

T/K

340

355

370

Figure 13 – Results for the isobaric expansivity coefficient (left) and the isothermal compressibility (right) of water, both at atmospheric pressure, using two different volume shifts. Data from Kell. 51 Both αp and kT are calculated from the volume and its derivatives which do not vary with a constant volume shift. Therefore, as in the case of the speed of sound, the differences in the two sets of results are only due to the different value for the volume term present in equations 12 and 13. Figure 14 presents results for the Joule-Thomson coefficient. Here the effect is opposite to that of αp (at low temperatures) and kT. As this property is influenced by both the derivative of pressure in relation to volume and in relation to temperature, the influence of the quantitatively 𝑑𝑃

incorrect (𝑑𝑉) is partially compensated, thus by fitting the volume shift to the speed of sound, 𝑇

we are increasing the deviation on this property.


Page 21 of 35

-1E-07 vshift density

-1.5E-07

μJT/K.Pa-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


vshift Ssound

-2E-07 -2.5E-07 -3E-07 -3.5E-07 280

305

330

355

380

T/K

Figure 14 – Results for the Joule-Thomson coefficient of water, using two different volume shifts calculated from the data of Kell 51 and Multiflash. 30 𝑑𝑃

𝑑𝑉

𝑑𝑃

In Figure 15, the accuracy of the derivatives (𝑑𝑇 ) , (𝑑𝑇 ) and (𝑑𝑉) for water, is analyzed at 𝑉

𝑃

𝑇

atmospheric conditions.



190

a)

-50

b)

170

150

-70

(dP/dT)V/bar.K-1

(dP/dV)T/TPa.m-3

-90 -110 -130

-150

130 110 90 70 50

This work

30

original CPA

10

-170

This work original CPA

-10 274

299

324

349

374

274

299

324

T/K

349

374

T/K

17

c)

15

(dV/dT)P/mm3.K-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 35

13 11 9 7 5

This work

3

original CPA

1 -1 274

294

314

334

354

374

T/K 𝑑𝑃

Figure 15 – Description of (𝑑𝑇 )

𝑉,𝑛

𝑑𝑃

(a), (𝑑𝑉)

𝑑𝑉

𝑇,𝑛

(b) and (𝑑𝑇 )

𝑃,𝑛

(c) for water with both the

modified CPA (this work) and original CPA 52. Derivative data calculated from the data of Kell 51 and Multiflash. 30 The behavior of the Mathias-Copeman alpha function, as well as, the higher value for the 𝑑𝑃

association energy (see our previous study 13 and Figure 16) enables (𝑑𝑇 ) to be increasing with 𝑉

temperature ant to have values closer to the experimental data near 300 K. However, the behavior of the curve is still not qualitatively correct. Nevertheless, due to a compensation of 𝑑𝑃

errors between these deviations and those of (𝑑𝑉) the representation of both Cp and Cv for 𝑇

water are good. To complete this analysis for liquid water, it is interesting to look at the monomer fractions with both the present version of CPA and the most commonly used original CPA. 52 These results are presented in Figure 16.


Page 23 of 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 16 – Description of free site fraction (left) and % monomer fraction (right) of pure water, using both the modified CPA and original CPA. Data from Luck. 53 At higher temperatures, the free site fraction is too high with the modified CPA, leading to an underestimation of the association contribution at these temperatures. However, at lower temperatures (up to 400 K) a good representation is obtained. Table 1 presents the values of β, b and ε for the two CPA versions. The set used for water with the original CPA, as many sets with some SAFT variants,

27

tends to overestimate association in the whole temperature range.

Nevertheless, these inaccurate behaviors (both overestimation and underestimation) tend to compensate deviations in the cubic/physical terms. It is important to note that Liang et al.

54

recently analyzed these data values and suggest that care should be taken when considering them, at least until these measurements are repeated. Table 1 – Parameters with influence on the associative term for both CPA variants. b.105 (m3.mol-1)

ε (J.mol-1)

β.102

Modified CPA

2.388

22013

0.48

Original CPA

1.452

16655

6.92

The description of volume dependencies when using cubic EoSs is far from perfect. Thus, it is not possible to describe simultaneously volume and speed of sound, while keeping an accurate representation of the saturation pressures and correct Tc and Pc. This is a problem shared with modern EoSs, as even the more robust versions of SAFT struggle to describe simultaneously all these properties, for many compounds. Nevertheless, if the coefficient Cp/Cv is well described by the EoS and the saturation pressure results are accurate, the user should be able to adjust a volume shift to improve at least one of the derivative properties. 17 23 ACS Paragon Plus Environment


Page 24 of 35

It is also important to note that the volume shift also affects the properties of the vapor phase. Nevertheless, the differences between the results with the different volume shifts are not relevant, except at high temperatures. Some examples of this behavior are presented in Figure 17. The results of this work are compared to the sets of Gross and Sadowski 55 with PC-SAFT and the set of the VS-PC-SAFT. 17


Page 25 of 35

0.0050

1.0E-06 vshift liq.dens

vshift liq.dens

vshift liq.ssound

8.0E-07

vshift liq.ssound

0.0045

VS-PC-SAFT

VS-PC-SAFT PC-SAFT

0.0040

6.0E-07

αP/K-1

kT/Pa-1

PC-SAFT

4.0E-07

0.0035

2.0E-07

0.0030

a) 450

500

550

T/K

600

280

650

200

380

480

580

T/K

80

180 160

this work

140

VS-PC-SAFT

70

CV/J.mol-1.K-1

CP/J.mol-1.K-1

b)

0.0025

0.0E+00

PC-SAFT

120

100 80

this work

VS-PC-SAFT

60

PC-SAFT 50 40

d)

c)

60

30

40 20

20 250

350

450

550

650

250

350

450

T/K

550

650

T/K

540

5.0E-04 vshift liq.dens vshift liq.ssound VS-PC-SAFT PC-SAFT

520 4.0E-04

μJT/K.Pa-1

500

u/m.s-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


480 460 vshift liq.dens vshift liq.ssound VS-PC-SAFT PC-SAFT

440 420

e)

3.0E-04

2.0E-04

1.0E-04

f)

400

0.0E+00 280

380

480

T/K

580

300

400

T/K

500

600

Figure 17 – Description of kT (a), αP (b), CP (c), Cv (d), speed of sound (e) and the Joule-Thomson coefficient (f) for gaseous water. Data are results using the IAPWS95 model 56 as present on Multiflash. 28 25 ACS Paragon Plus Environment


Page 26 of 35

The results for kT in the vapor phase seem to benefit from the correct description of the critical temperatures and pressures. When compared to the PC-SAFT variants, the results are very similar at low temperatures, while close to the critical point the modified CPA performs better, due to the description of the critical point. As for αP the other models present a better description at low temperatures. Nevertheless, these highly underestimate the results for high temperatures, where CPA performs a better description for this property. For the Cp and Cv of the gas phase all models in analysis describe incorrectly the trend with temperature. As can be seen in the Figure the VS-PC-SAFT performs well for the low temperatures on these properties, but fails the description above 400 K, while CPA and the parameter set from Gross and Sadowski 55 fail for the whole range of Cv. The modified CPA tends to underestimate the speed of sound at lower temperatures and closer to the critical temperature to find a minimum before the critical point, which is not observed in the IAPWS95 results. The PC-SAFT and VS-PC-SAFT results do not present these minima, however, they still overestimate the speed of sound at higher temperatures. For the JouleThomson coefficient, the different methods tend to present an accurate description above 500 K, below this temperature most models overestimate this property, while the VS-PC-SAFT underestimates it. Results for the second virial coefficient of some of the compounds analyzed in this study are presented in the supporting information.

Conclusions In this work we investigated how second derivative properties can be improved, when calculated from SRK, PR and a modified version of CPA, using a constant volume shift fitted to the liquid speed of sound instead of fitted from liquid density data. The major advantage of this approach is that phase equilibria calculations are not affected, while we now have the option to improve properties such as the speed of sound, which is usually not well represented by cubic equations of state. Our analysis shows that there is no overall improvement for properties such as liquid density and speed of sound as function of pressure, but there is a clear improvement for the data at near atmospheric pressure or pressures close to the saturation pressure. Most water derivative properties were improved when using a modified version of CPA: isothermal compressibility, isobaric expansivity and notably the speed of sound, for which our approach can represent the maximum observed experimentally, although the model still fails to 26 ACS Paragon Plus Environment

Page 27 of 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


represent the magnitude of the decrease with decreasing temperature below the maximum. Some alkanols and diols were also investigated with similar conclusions. Similarly, for non-associating components there is a clear improvement for the speed of sound. For isothermal compressibility, the behavior for hexane is similar to that observed for 1-butanol and is improved with this volume shift. Nevertheless, due to the higher amplitude between the two volume shifts in hexane than with 1-butanol, the isothermal expansivity presents higher deviations when considering this second volume shift. All these improvements are achieved while keeping the phase equilibria intact and solely at the expense of a decrease in the performance of modelling liquid densities.

Supporting information Tables with pure component parameters and information on the fitting of volume shifts and results for the description of second virial coefficients.

Funding and acknowledgments This work was funded by KBC Advanced Technologies Limited (A Yokogawa Company) under project "Extension of the CPA model for Polyfunctional Associating Mixtures”. André M. Palma Acknowledges KBC for his Post-Doctoral grant. This work was developed within the scope of the project CICECO-Aveiro Institute of Materials, FCT Ref. UID/CTM/50011/2019, financed by national funds through the FCT/MCTES.

Nomenclature 𝐴, 𝑎 = energy parameter of CPA. 𝑎 (Pa·m6·mol-2), 𝐴 = 𝑛2 𝑎. 𝑎𝑐 = value of the energy parameter at the critical point. (Pa·m6·mol-2). B, b = co-volume. b (m3.mol-1), 𝐵 = 𝑛𝑏. (cn)= alpha function parameters Cp = Isobaric heat capacity (J.mol-1.K-1) Cv = Isochoric heat capacity (J.mol-1.K-1) cvs = volume shift (m3.mol-1) kij = binary interaction parameter for the cubic energy term ks = Isentropic compressibility (Pa-1) kT = Isothermal compressibility (Pa-1) P = vapor pressure (Pa) T = temperature (K) 27 ACS Paragon Plus Environment


Page 28 of 35

v0, vt = volume before and after translation, respectively (m3.mol-1) b0, bt = co-volume before and after translation, respectively (m3.mol-1) x, y = liquid/vapor mole fraction

Greek symbols α(T) = alpha function αP = Isobaric expansivity coefficient (K-1) β = association volume βij = cross-association volume ε = association energy (J.mol-1) µJT = Joule-Thomson Coefficient (K.Pa-1) ρ = molar density (mol.m-3)

Abbreviations CPA = Cubic Plus Association EoS = equation of state PC-SAFT = Perturbed Chain Statistical Fluid Association Theory VS-PC-SAFT = Volume Shifted PC-SAFT PR = Peng-Robinson SRK = Soave-Redlich-Kwong VS = Volume shift References (1)

de Villiers, A. J.; Schwarz, C. E.; Burger, A. J.; Kontogeorgis, G. M. Evaluation of the PCSAFT, SAFT and CPA Equations of State in Predicting Derivative Properties of Selected Non-Polar and Hydrogen-Bonding Compounds. Fluid Phase Equilib. 2013, 338, 1–15. https://doi.org/10.1016/j.fluid.2012.09.035.

(2)

Lubarsky, H.; Polishuk, I.; Nguyenhuynh, D. The Group Contribution Method (GC) versus the Critical Point-Based Approach (CP): Predicting Thermodynamic Properties of Weakly- and Non-Associated Oxygenated Compounds by GC-PPC-SAFT and CP-PC-SAFT. J. Supercrit. Fluids 2016, 110, 11–21. https://doi.org/10.1016/j.supflu.2015.12.007.


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(3)

Papaioannou, V.; Lafitte, T.; Avendaño, C.; Adjiman, C. S.; Jackson, G.; Müller, E. a.; Galindo, A. Group Contribution Methodology Based on the Statistical Associating Fluid Theory for Heteronuclear Molecules Formed from Mie Segments. J. Chem. Phys. 2014, 140 (5), 1–30. https://doi.org/10.1063/1.4851455.

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Dufal, S.; Papaioannou, V.; Sadeqzadeh, M.; Pogiatzis, T.; Chremos, A.; Adjiman, C. S.; Jackson, G.; Galindo, A. Prediction of Thermodynamic Properties and Phase Behavior of Fluids and Mixtures with the SAFT-γ Mie Group-Contribution Equation of State. J. Chem. Eng. Data 2014, 59 (10), 3272–3288. https://doi.org/10.1021/je500248h.

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Sadeqzadeh, M.; Papaioannou, V.; Dufal, S.; Adjiman, C. S.; Jackson, G.; Galindo, A. The Development of Unlike Induced Association-Site Models to Study the Phase Behaviour of Aqueous Mixtures Comprising Acetone, Alkanes and Alkyl Carboxylic Acids with the SAFT-γ Mie Group Contribution Methodology. Fluid Phase Equilib. 2016, 407, 39–57. https://doi.org/10.1016/j.fluid.2015.07.047.

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Salimi, M.; Bahramian, A. The Prediction of the Speed of Sound in Hydrocarbon Liquids and Gases: The Peng-Robinson Equation of State versus SAFT-BACK. Pet. Sci. Technol. 2014, 32 (4), 409–417. https://doi.org/10.1080/10916466.2011.580301.

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Diamantonis, N. I.; Economou, I. G. Evaluation of Statistical Associating Fluid Theory (SAFT) and Perturbed Chain-SAFT Equations of State for the Calculation of Thermodynamic Derivative Properties of Fluids Related to Carbon Capture and Sequestration. Energy and Fuels 2011, 25 (7), 3334–3343. https://doi.org/10.1021/ef200387p.

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