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Thermodynamics, Transport, and Fluid Mechanics

Liquid phase density, sound speed, and vapor pressure of linear alkanes using the Mattedi-Tavares-Castier equation of state Davi Hoffmann, Marcelo Castier, Marcio Luis Lyra Paredes, and Silvana Mattedi Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b06346 • Publication Date (Web): 08 Apr 2019 Downloaded from http://pubs.acs.org on April 9, 2019

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Industrial & Engineering Chemistry Research

Liquid phase density, sound speed, and vapor pressure of linear alkanes using the Mattedi-Tavares-Castier equation of state

Davi Homann,† Marcelo Castier,∗ ‡ Márcio L.L. Paredes,¶ and Silvana Mattedi† ,

†Department

of Chemical Engineering, Federal University of Bahia, Salvador, Brazil ‡Chemical Engineering Program, Texas A&M University at Qatar, Doha, Qatar ¶Institute of Chemistry, State University of Rio de Janeiro, Rio de Janeiro, Brazil E-mail: [email protected]

Abstract This work assesses the ability of the Mattedi-Tavares-Castier (MTC) equation of state (EOS) in representing the liquid phase density, sound speed, and vapor pressure of linear alkanes (C6-C20). It is found that its deviations from experimental data for liquid phase density, liquid phase sound speed, and vapor pressure are similar, yet slightly higher than those of the PC-SAFT EOS for the same compounds. Using only experimental data points at reduced pressures smaller than 20 for parameter tting, a condition observed in most commercial chemical processes, leads to smaller deviations from the experimental values. Adopting for the lattice cell volume the value of 15 cm3 /mol, which is the molar volume of the methylene group in the lattice-based UNIQUAC model, all the adjustable MTC EOS parameters follow regular trends with respect to the number of carbon atoms in the linear alkane chains. These parameters could be tted to empirical expressions. When used, they produce deviations in purecomponent liquid phase density, liquid phase sound speed, and vapor pressure equal to 1

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0.6%, 2.8%, and 9.8%, respectively. The results for binary mixtures are generally in good agreement with the experimental data.

Introduction The gradual depletion of traditional sources of oil and gas has triggered the search for deeper and more complex wells, such as oshore and shale deposits, and the use of sophisticated recovery techniques in mature wells. Natural gas has also increased its share of the energy market, either used directly or after its conversion to liquid fuels via Fisher-Tropsch synthesis. To develop such applications and others in oil and gas processing, it is necessary to describe the physical properties of the pure compounds and mixtures involved. Equations of state (EOSs) are convenient for modeling the thermodynamic behavior of these uids because they are, in general, applicable to predict volumetric and calorimetric properties over wide ranges of pressure, temperature, and composition. In the oil and gas industry, cubic EOSs such as the Soave-Redlich-Kwong 1 and PengRobinson 2 models remain the most commonly used despite considerable advances in thermodynamic modeling. An example is the development of the statistical associating uid theory 3 - SAFT - which is a landmark in many ways. Compared to cubic models, the SAFT EOS has a more rened representation of repulsive and attractive interactions and, importantly, includes a term to account for specic interactions via the association contribution. In addition to these terms, specialized versions now exist to account for the presence of electrolytes 46 and uid connement. 7,8 Several variants of the original SAFT model are now available for even more accurate correlations and predictions of physical properties. Examples include the PC-SAFT, 9 soft-SAFT, 10 and SAFT-VR Mie 11 variants, whose underlying assumptions and mathematical details are available in their original references. PC-SAFT currently seems to be the most studied of these variants. SAFT-family EOSs are more accurate than cubic EOS in many calculations, but they

2


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also have more complicated mathematical expressions and impose a heavier computational load. The cubic-plus-association model (CPA) 1214 and the simplied CPA model 15 are compromise solutions that combine the relative simplicity of the physical contribution of cubic models with the association models, thereby decreasing the computational load while retaining the ability to model polar mixtures. Notwithstanding the evidence of their success, the industrial adoption of SAFT-family and CPA-family models is not widespread yet. A possible reason is the need to apply specic association schemes for each polar compound and mixture. In addition, the resolution of the association equations becomes part of the EOS root-nding procedure, increasing its computational load. This work focuses on the Mattedi-Tavares-Castier 16 (MTC) EOS, which is a model developed within the framework of the generalized van der Waals theory. 17 It is based on a lattice theory with void spaces as a way to account for the uid's compressibility. An analogous theory, without void spaces, is the basis of the well-known, well-established, and widely used UNIQUAC model for the excess Gibbs energy, 18 and of the UNIFAC group-contribution model. 19 Likewise, the concept of group contribution can be introduced in the early stages of the MTC EOS derivation. In this way, its framework can be used as a molecular model, as a group-contribution model, or as a region-contribution model (in which polar molecules are assumed to possess electron acceptor and electron donor regions whose intermolecular interaction is interpreted as a hydrogen bond). Very importantly, the MTC EOS can model polar compounds and mixtures without the need for solving association schemes iteratively. Even for pure non-polar compounds, for which there is no need to solve association schemes in SAFT-based models, our tests indicate that the time needed to compute the pressure and the fugacity coecient simultaneously using the MTC EOS is about 55% of the time required by the PC-SAFT EOS. Much of the work on EOSs targets phase equilibrium calculations and, as a consequence, many publications concentrate on vapor pressures of pure compounds, and bubble and dew points of mixtures. Nonetheless, many other properties are necessary for specic applications. 3



An example is the liquid phase density, which is useful for sizing vessels and pipelines, but cubic EOSs notoriously fail in its prediction, unless volume-translation is used. Another example is the thermodynamic sound speed, 2022 which is essential for the design of venting systems from pressure vessels, and yet is a property dicult to predict accurately with cubic EOSs. Besides, sound speed has been recently used in our group 23,24 as input data in the PC-SAFT parameterization of associative compounds, and has led to a large enhancement of the predictive capacity of phase equilibria between ionic liquids and CO2 , 23 and in the parameterization of acetic acid. 24 It was observed that the PC-SAFT model was unable to correlate the isobaric heat capacity accurately together with the other properties, and authors recommended the use of the parameter set obtained by tting vapor pressure, density and speed of sound. So, among the properties cited hitherto, only calorimetric properties are not used, mainly because of our previous results. 24 So, vapor pressure, density and speed of sound are the focus of this work, which assesses the performance of the MTC EOS 16,2527 in their calculation for all linear alkanes in the range C2-C20, after a pre-screening of parameter tting strategies based on n-hexane. These results are compared to those from the literature for the PC-SAFT EOS, 28 which has been recently claimed by the authors to be a good model for the same properties and compounds considered in this paper. In addition, the correlated parameters have been used to predict the liquid phase density and sound speed of binary mixtures of n-hexane with other linear alkanes in the range C7-C20. The next section is a summary of the MTC EOS and the subsequent section discusses parameter tting strategies. The results section shows that the MTC EOS parameters for linear alkanes follow well dened trends. This allows the development of simple correlations for the parameter values as function of the number of carbon atoms in the alkane chain.

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Mattedi-Tavares-Castier equation of state The MTC EOS 16 was derived using the generalized van der Waals theory. 17 It is a latticebased model in which molecular segments can only be at lattice sites, that is, they are not free to occupy any position in space. This introduces an unnatural restriction but simplies the model's derivation and functional form. Unlike several other lattice uid models, the MTC EOS considers the existence of void lattice sites; changes to their number account for system density and pressure variations. Therefore, the model is theoretically capable of describing the behavior of thermodynamic properties and phase equilibrium under high pressure conditions, which is necessary to dierent applications. Details of the EOS derivation can be found elsewhere. 16 A modied version of this EOS was developed based on a holelattice theory augmented with surface charge-densities and applied to pure components 29 and mixtures. 30 In addition, an extended version of the MTC EOS, applicable to electrolyte solutions, exists. 31,32 Originally derived for groups, the equation can be readily written in molecule-based form if each compound is dened as a group, as used by Iglesias et al. 33 Another alternative is to use a region-based approach, in which polar molecules are assumed to have three regions: an electron-donor region, an electron-acceptor region, and a non-polar region. 27,34 A specic interaction occurs when an electron-donor region of a molecule is in contact with an electronacceptor region of another molecule. Here, n-alkanes are assumed to have a single, non-polar region. Therefore, the molecular-based and the region-based approaches are equivalent. Equations 1 and 2 are the expressions for the compressibility factor (Z ) and fugacity coecient of component i in a mixture (φî ) according to the MTC EOS: ng nc X z v˜ − 1 + (q/r) v˜Ψ (q/r) X (Γa − 1) v˜ + v˜rln +l− xi νia Qa Z = v˜rln (1) v˜ − 1 2 v˜ v˜ − 1 + (q/r) i=1 a=1 v˜ − 1 + (q/r) Γa

v ˜ − 1 v ˜ Ψ (q/r) (qi − ri ) lnφî = −ri ln + (1 − li ) ln + v˜ − 1 + (q/r) v˜ − 1 + (q/r) v˜ − 1 + (q/r) (2) Png e e ea ng n g n c X X − ri ) v˜ − 1 + (q/r) ΨX a a( a a e=1 νi Q γ x k νk Q +Ψ νi Q × ln − a v ˜ − 1 + (q/r) Γ r v˜ − 1 + (q/r) Γa a=1 k=1 a=1 5



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In these equations, the lattice coordination number (z) used is equal to 10, 16 which is the value adopted in widely used lattice-based models such as UNIQUAC and UNIFAC. The symbol nc represents the number of components; ng is the number of regions, which is equal to 1 because there is only one region in each component present; νia is the number of regions of type a in component i, which is also equal to 1 in the applications of this paper; and xi is the mole fraction of component i. The symbol Ψ denotes an empirical universal constant of the lattice structure. The number of external contacts (zqi ), the number of segments (ri ), and the bulkiness factor (li ) of a molecule of type i, which takes into account the non-linearity in the molecules, and the reduced volume (˜ v ) are dened as: ng X

νia zQa

(3)

ng 1 X a a ν v ri = ∗ v a=1 i

(4)

z (ri − qi ) − (ri − 1) 2

(5)

v rv ∗

(6)

zqi =

a=1

li =

v˜ =

where v is the molar volume and v ∗ is the volume occupied by one mole of lattice cells. The number of segments, the number of contacts, and the bulkiness factor in a mixture are calculated as:

r=

nc X

xi ri

(7)

xi zqi

(8)

xi li

(9)

i=1

zq =

nc X i=1

l=

nc X i=1

In addition, the following relations are dened: 6


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a

Γ =

ng X

S m γ ma

(10)

νim xi Qm q

(11)

m=1

S

m

Pnc =

i=1

γ ma = exp [−uma /(RT )] where uma , the interaction energy between regions m and a, is given by: uma uma B ma 0 = 1+ R R T

(12)

(13)

In Eq. 13, uma 0 /R is the temperature-independent characteristic energy of interaction between regions a and m and B ma is the temperature-dependence parameter. ma The adjustable parameters of the EOS are Qa , v ∗ , v a , uma . For pure non0 /R, and B

polar compounds that contain a single dispersion region, the cross parameters uma 0 /R and aa B ma reduce to uaa 0 /R and B , respectively.

Thermodynamic sound speed

The thermodynamic sound speed (a) is given by: r 1 a= ks ρM

(14)

where ρ is the molar density, M is the molar mass, and ks is the isentropic compressibility, which is given by:

ks = kT −

α2 T ρcP

(15)

In Eq. 15, cP is the molar heat capacity at constant pressure, kT is the isothermal compressibility, and α is the volume expansivity:

1 ∂v kT = − v ∂P T,x

(16)

1 ∂v α= v ∂T P,x

(17)

7



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res cP = cig P + cP

(18)

In Eq. 18, cres P is the residual molar heat capacity at constant pressure, which is the following derivative of the residual molar enthalpy:

cres P

=

∂hres ∂T

(19) P,x

The residual molar enthalpy is obtained from the MTC EOS as follows: ! ares ∂ RT hres = −RT 2 − RT + P v ∂T

(20)

v,x

where ares denotes the residual molar Helmholtz energy.

Parameter tting strategy aa for the pure components are tted by minimizing the Parameters Qa , v ∗ , v a , uaa 0 /R and B

objective function: N exp sat

Fob =

P X

k=1

Pksat,exp − Pksat,calc Pksat,exp

Nρexp m

!2 +

X k=1

calc ρexp m,k − ρm,k ρexp m,k

!2 +

exp N a X

k=1

calc aexp k − ak aexp k

2

(21)

sat where NPexp ) data points available. The symbols Nρexp sat is the number of vapor pressure (P m

and Naexp have similar interpretations for liquid phase mass density (ρm ) and sound speed (a), respectively. The superscripts exp and calc denote experimental and calculated values, respectively. To assess the performance of the MTC EOS, the average absolute relative deviation (AARD) is used as follows:

N exp 1 X Ωcalc k × 100% %AARD (Ω) = exp − 1 exp Ω N k k=1

(22)

where Ω denotes a physical property among those considered in this paper (liquid phase density, sound speed, or vapor pressure) and N exp is the number of data points available for that physical property. 8


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Sources of experimental data

The references to the publications with the experimental data used in this work are available as a supplementary material. The parameters are tted from liquid phase density, sound speed, and vapor pressure data of linear alkanes. Two data sets are used. One of them comprises data at reduced temperatures between 0.45 and 0.9 for linear alkanes between ethane (C2) and n-decane (C10), obtained from the NIST database and the DIPPR correlations. As this data set was the same used by Liang et al. 28 to assess the performance of the PC-SAFT EOS, it allows for a direct comparison between the results of the MTC EOS and those from the literature 28 for PC-SAFT. The second data set is larger and comprises vapor pressure data at reduced temperatures between 0.45 and 0.9 for linear alkanes between n-hexane (C6) and n-eicosane (C20) obtained using the DIPPR correlations and experimental sound speed and density data obtained from the literature, as shown in Table 1. Table 1: Complete experimental data set Alkanes

C6 - C20

Temperature range (K) Pressure range (MPa) Number of density data points Number of sound speed data points

243.16 - 670.00 0.1 - 700.0 2267 1211

Results and discussion Figure 1 serves as a guide to the analysis of the results because it summarizes the strategies adopted for tting and comparing parameter sets in this work. The parameter values tted using each strategy are available as supplementary material.

9



Initial screening

The parameter tting strategy plays an important role on the overall performance of thermodynamic models. Twenty-four strategies for tting the MTC EOS's parameters were compared on the basis of their eect on the objective function, Eq. 21, in calculations with n-hexane. The three strategies (identied here as P1 , P2 , and P3 ) that stood out as the best alternatives among them have the following common aspects: (a) the empirical universal constant of the lattice structure, Ψ, is set equal to 18; (b) the parameters Qa , v a , and uaa 0 /R are treated as adjustable. In strategy P1 , the value of B aa is tted and the value of v ∗ is set to 10 cm3 /mol. In strategy P2 , B aa is set to zero and the value of v ∗ is tted. In strategy

P3 , B aa and v ∗ are both tted. Parameter sets P1 and P2 have four adjustable parameters; parameter set P3 has ve adjustable parameters.

Comparison with the PC-SAFT EOS

The PC-SAFT EOS represents the sound speed of linear alkanes better than the SRK (Soave Redlich Kwong) and CPA (plus cubic association) EOSs. 28 Thus, the results of the MTC EOS are compared to those of PC-SAFT, using the parameters sets P1 , P2 , and P3 . Tables 2 and 3 show the average absolute relative deviation (% AARD, Eq. 22) in liquid phase density, sound speed, and vapor pressure, obtained from the MTC EOS applying parameter sets P1 ,

P2 , and P3 to data set used by Liang et al. 28 and to the complete data set, respectively. These results are compared with the results of the PC-SAFT EOS. 28 The two tables show that % AARDs with the MTC and PC-SAFT 28 EOS are generally of the same order of magnitude, with the MTC EOS outperforming the PC-SAFT EOS for density predictions, while the opposite happens with vapor pressure predictions. The results with parameter sets

P1 , P2 , and P3 indicate that estimating v ∗ is more eective than estimating B aa . Estimating both lead to a reduction of % AARD in the three properties of 0.88% from P1 to P3 and of 0.30% from P2 to P3 . The typical average uncertainties for the three properties is 0.5% for vapor pressure, 0.1% for density (from DIPPR correlations) and for sound speed is 0.01% 10


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at low pressures and 0.1% at higher pressures, hence at higher pressures the % AARD for the three properties due to uncertainties is of 0.23%. Using this number as a reference, the results with parameter sets P2 and P3 are similar, and even more similar are the results with MTC parameter set P3 and PC-SAFT, which dier 0.24% in % AARD. It was observed that the lattice volume parameter, v ∗ , of dierent hydrocarbons ranged from 6.51 cm3 /mol to 18.65 cm3 /mol for parameter set P2 , with 15 cm3 /mol taken as a representative value. For parameter set P3 , the packing volume parameter ranged from 8.58 cm3 /mol to 27.12 cm3 /mol, with 17 cm3 /mol being a representative value.

a

ρm

Table 2: % AARD for data set used by Liang et al. 28 (data at reduced temperatures between 0.45 and 0.9 for linear alkanes between ethane and n-decane, from the NIST database and the DIPPR correlations) (PC-SAFT EOS results are from Liang et al. 28 )

P sat

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


C2

C3

C4

C5

C6

C7

C8

C9

C10

MTC-P1 MTC-P2 MTC-P3 PC-SAFT

0.40 2.49 0.90 1.75

0.95 1.58 0.74 1.90

1.20 0.82 0.53 1.80

1.48 0.79 0.49 1.59

1.83 0.77 0.44 1.49

1.65 0.61 0.45 1.63

2.02 0.82 0.64 1.56

1.35 0.50 0.49 1.82

1.40 0.51 0.49 1.59


2.43 1.72 1.95 0.90

2.52 2.45 2.60 1.54

2.78 2.80 3.00 1.34

2.85 2.94 3.36 1.68

2.63 2.86 3.48 0.58

2.62 3.12 3.71 1.52

2.37 3.07 3.30 0.41

2.10 3.00 3.21 2.30

1.78 2.73 2.81 1.54


0.38 0.76 0.16 0.68

0.73 1.23 0.55 0.65

2.02 1.64 1.04 0.72

3.68 2.73 1.65 0.58

5.21 3.35 1.91 1.74

5.67 3.25 1.80 1.13

4.67 1.64 0.99 1.16

5.30 1.86 1.28 1.40

4.76 1.07 0.82 1.28

Mean 1.36 0.98 0.58 1.68 2.45 2.74 3.05 1.31 3.60 1.95 1.13 1.04

Comparisons using a reduced data set

Figures 2 and 3 show the eect of pressure on the sound speed and density of n-hexane using the model parameters obtained for the complete data set. The deviations between experimental and calculated results become larger as the pressure increases. Similar patterns appear for components in the range from C7 to C20. Because most chemical processes do not operate at pressure levels as high as those of Figs. 2 and 3, a reduced data set was dened 11



a

ρm

P sat

a

ρm

Table 3: % AARD for the complete data set (as dened in Table 1)

P sat

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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C6

C7

C8

C9

C10

C11

C12

C13


2.38 2.96 2.92 1.49

0.49 1.25 1.20 1.63

0.69 0.68 0.86 1.56

2.5 3.25 3.37 1.82

0.98 1.24 1.15 1.59

1.39 1.93 2.24 1.41

1.09 2.40 1.43 1.41

1.55 0.62 0.66 1.37


3.41 3.94 3.9 2.97

5.47 6.70 6.33 3.47

0.84 1.08 1.32 2.69

1.19 1.79 1.91 2.91

2.96 3.98 3.77 3.93

1.29 2.17 3.25 2.03

1.32 5.66 2.92 2.07

8.76 2.14 2.64 2.24


5.29 3.62 3.70 1.74

5.32 3.32 3.57 1.13

4.79 1.70 0.44 1.16

5.41 1.93 1.49 1.40

5.05 1.33 1.48 1.28

8.68 4.36 1.94 1.52

9.08 4.44 1.52 1.37

6.54 2.24 0.97 1.43

C14

C15

C16

C17

C18

C19

C20

Mean


0.38 2.30 0.71 1.37

1.19 0.91 1.00 1.37

0.69 0.39 0.60 1.33

0.36 1.81 0.67 1.29

2.24 1.97 1.95 1.28

1.29 2.32 2.40 1.26

0.32 1.11 1.21 1.24


2.26 5.41 4.00 2.52

3.76 4.59 4.11 2.94

3.36 4.94 5.62 2.79

11.11 11.41 11.33 3.26

7.29 7.66 7.94 3.47

7.12 8.09 8.58 3.66

5.34 7.66 7.70 3.46


10.96 5.83 1.94 1.11

12.37 5.97 5.50 1.77

10.93 4.32 2.71 1.54

9.24 6.62 3.01 1.49

14.2 5.82 5.20 2.21

11.44 5.85 4.96 2.71

10.15 3.24 3.05 2.57

12


1.17 1.68 1.49 1.43 4.37 5.15 5.02 2.96 8.63 4.04 2.77 1.63

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with only data points below the reduced pressure of 20 for each component. In addition, this reduced data set does not contain n-tetradecane (C14) because only points up to a reduced pressure of 0.06 were available. Use of the reduced data set for parameter tting reduces the % AARD, as summarized in Table 4. The corresponding parameters of sets P1 , P2 , and P3 are reported in Table 5. As it was observed for the complete data set, the packing volume parameter, v ∗ , of dierent hydrocarbons also tends to values substantially higher than 10 cm3 /mol, which is the value adopted for v ∗ in parameter set P1 . It should be noted that, for calculations with mixtures, it is possible to use component-specic v ∗ -values 29,30 but it is more convenient to have a component-independent v ∗ -value, which parameter set P1 has, but P2 and P3 have not. As the ultimate goal of this work was the calculation of mixture properties, a new parameter 0

set P1 was dened with the lattice volume parameter, v ∗ , set equal to 15 cm3 /mol, which is the value for the methylene molecular segment adopted in the UNIQUAC excess Gibbs energy model. Table 4: Component-averaged (C6-C20, except C14) % AARD in physical properties: parameters tted using the reduced data set applied to calculations with the complete and reduced data sets Parameter set P1 Data set Density Sound speed Vapor pressure

Complete 1.23 4.52 8.46

Reduced 0.56 2.72 8.27

Parameter set P2 Complete 1.63 4.92 3.91

Reduced 0.48 2.85 2.21

Parameter set P3 Complete 1.55 5.09 2.79

Reduced 0.51 2.84 1.28

Overall, the results for liquid phase density present the smallest deviation from the experimental data. This may be explained, at least in part, by the fact the experimental data sets used for parameter tting had more density data points than vapor pressure or sound speed data points. However, it should also be noted that accurate sound speed predictions with equations of state are usually more dicult to achieve. Thus, it is possible that both factors, number of data points and property type, have contributed to the outcome. The dierences

13



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in average % AARD between parameter sets P2 and P3 are 0.34% for the complete set and 0.30% for the reduced set, which is about the 0.23% typical uncertainty, but still higher. For parameter sets P1 and P3 these dierences are 1.59% and 2.31%, respectively, and again the estimation of v ∗ is more advantageous than the estimation of B aa , although it is still possible to point out the benets of estimating both. 0

Table 5 also shows the parameters of set P1 , which were obtained using the reduced data set. From the values, it is clear that each parameter follows a well-dened pattern with the respect to the number of carbon atoms in the linear alkane chain. The next subsection presents correlations for these parameters as functions of a single characterizing parameter, which is the number of carbon atoms in the n-alkane chain.

Correlation of the parameters of set

0

P1

aa a a The square symbols in Fig. 4 represent the tted values of parameters uaa 0 , B , Q , and v

as function of the number of carbon atoms in the linear alkane chain. The lines in the same gures represent the results of the following empirical expressions for these parameters as function of the number of carbon atoms (C) in the linear alkane chain, which have coecients of determination ranging from 0.976 to 0.999.

uaa 0 (K) = −8.352 ln C − 13.20 R

(23)

B aa (K) = −37.18 ln C + 84.71

(24)

Qa = 0.8076C + 2.3376

(25)

v a cm3 mol = 15.812C + 19.844

(26)

Table 6 shows that the deviations obtained using the original or correlated parameter 0

values (columns P1c in Table 6) are similar for density and sound speed, while there is a degradation of the vapor pressure results.

14


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0

cm3 mol

Qa

v

a

cm3 mol

B

aa

(K)

uaa 0 R

(K)

v

∗

cm3 mol

Qa

v

a

cm3 mol

B aa (K)

uaa 0 R

(K)

Table 5: Parameters of sets P1 , P2 , P3 , and P1 obtained using the reduced data set.

v

∗

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


C6

C7

C8

C9

C10

C11

C12

P1 P2 P3 0 P1

-28.36 -28.75 -28.46 -28.46

-30.03 -29.85 -29.21 -29.06

-31.18 -31.26 -30.97 -31.17

-32.45 -32.66 -32.5 -32.63

-32.9 -32.4 -32.2 -32.41

-33.96 -34.01 -33.19 -33.99

-34.71 -34.38 -31.95 -34.47

P1 P2 P3 0 P1

-21.89 0 51.67 16.55

-29.32 0 35.81 14.74

-32.86 0 21.69 6.74

-37.67 0 18.07 2

-42.14 0 8.32 -0.13

-42.74 0 41.97 -2.46

-47.33 0 65.9 -6.91

P1 P2 P3 0 P1

110.17 114.08 118.62 115.07

125.53 129.58 133.44 129.89

140.28 146.27 148.64 146.91

156.61 162.53 164.18 162.62

170.97 177.29 178.52 177.36

187.38 194.24 198.16 193.78

202.09 210.11 215.43 209.02

P1 P2 P3 0 P1

9.71 7.88 5.64 7.07

10.83 8.64 6.98 8.06

11.9 9.11 7.99 8.73

12.9 9.54 8.55 9.43

14.37 10.63 10.16 10.63

15.04 10.92 8.87 11.13

16.19 11.6 8.99 12.11

P1 P2 P3 0 P1

10 12.98 19.95 15

10 13.43 18 15

10 14.2 16.9 15

10 14.79 17.04 15

10 15 15.96 15

10 15.39 20.35 15

10 15.91 23.05 15

C13

C15

C16

C17

C18

C19

C20

P1 P2 P3 0 P1

-35.56 -35.01 -34.7 -35.73

-35.8 -35.57 -33.64 -35.84

-36.19 -36.4 -33.86 -36.71

-36.97 -35.69 -33.93 -37.49

-37.26 -36.05 -34.11 -37.72

-35.51 -37.54 -35.62 -38.43

-36.4 -37.78 -36.41 -38.52

P1 P2 P3 0 P1

-53.92 0 25.33 -15.56

-50.08 0 56.86 -11.49

-52.9 0 54.19 -18.51

-61.16 0 61.62 -23.38

-61.7 0 68.32 -23.25

-47.39 0 66.95 -24.72

-52.86 0 42.79 -25.63

P1 P2 P3 0 P1

218.4 227.16 230.58 225.76

251.54 259.74 264.54 256.7

268.49 276.58 280.06 271.89

281.99 288.49 297.69 290.75

297.34 305.05 314.07 303.61

319.63 323.29 331.75 320.41

334.55 340.94 345.86 336.02

P1 P2 P3 0 P1

17.23 11.92 10.42 12.82

19.23 13.76 11.04 14.64

20.03 13.92 11.57 15.43

21.39 15.15 11.95 16.12

22.34 15.64 12.07 16.94

22.42 15.44 12.02 17.34

23.89 16.29 13.84 18.45

P1 P2 P3 0 P1

10 16.96 20.15 15

10 10 10 10 16.48 17.35 17.15 17.67 15 22.47 23.06 23.65 24.93 ACS Paragon Plus Environment 15 15 15 15

10 18.28 25.75 15

10 18.2 22.84 15


Page 16 of 36

Table 6: % AARD for the reduced data set using parameters correlated with the number of 0 0 carbon atoms in the alkane chain (P1c ) and comparison with parameter sets P3 and P1 Density 0

Sound speed 0

0

Vapor pressure 0

0

0

C6 C7 C8 C9 C10 C11 C12 C13 C15 C16 C17 C18 C19 C20

P3 0.25 0.45 0.55 0.23 0.31 1.02 0.60 0.60 0.54 0.48 0.51 0.50 0.60 0.53

P1 0.32 0.57 0.52 0.21 0.31 0.76 0.61 0.51 0.57 0.57 0.40 0.70 0.90 0.47

P1c 0.89 0.62 0.61 0.29 0.38 0.70 0.67 0.50 0.44 0.57 0.51 0.80 0.60 0.81

P3 1.48 2.39 1.35 2.15 3.28 3.37 3.42 2.72 2.74 2.77 3.14 4.33 4.22 2.45

P1 2.30 3.99 1.17 1.84 3.42 2.29 1.92 1.79 2.86 2.75 2.51 2.47 1.89 1.49

P1c 0.66 1.38 1.37 3.48 3.44 3.67 2.08 2.96 2.36 2.91 2.19 5.08 5.23 3.29

P3 1.01 1.06 0.35 0.77 0.27 1.79 0.23 0.76 2.66 1.16 3.70 1.64 1.94 0.60

P1 2.55 2.24 1.24 1.76 0.89 4.57 4.63 3.78 7.38 6.36 6.95 8.51 8.79 6.27

P1c 19.88 9.01 2.14 4.13 7.69 12.68 13.60 7.41 12.80 10.60 6.96 7.44 9.45 13.60

Mean

0.51

0.53

0.59

2.84

2.34

2.81

1.28

4.71

9.82

Predictions for linear alkanes

Figure 5 shows results of the MTC EOS for pure n-tetradecane (C14), n-heneicosane (C21), and n-tetracosane (C24) using parameter values correlated as functions of the number of carbon atoms in the n-alkane chain (Eqs. 23-26). These are predictions for n-tetradecane because its experimental data were not used for parameter tting in the reduced data set. The eect of temperature on the liquid phase density, sound speed, and vapor pressure is satisfactorily predicted. The results for n-heneicosane (C21) represent a slight extrapolation of the range used for parameter tting because this compound has one carbon atom more than the largest n-alkane used for parameter tting (n-eicosane, C20). The results for ntetracosane (C24) are also predictions, but further away from the maximum carbon number used for the MTC EOS parameter correlation. Systematic deviations between the calculated and experimental values are observed for the liquid phase density (Figure 5a), sound speed (Figure 5b), and vapor pressure (Figure 5c) but the overall eect of temperature on these

16


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properties is correctly predicted.

Binary mixtures of linear alkanes

This section presents MTC EOS predictions for the density and sound speed of binary mixtures of n-alkanes using the correlations between parameter values and the number of carbon atoms, Eqs. 23-26. The value of 15 cm3 /mol was adopted for the lattice cell volume 0

in all cases and the parameter set P1 shown in Table 5 was used for the calculations. For the binary mixtures of n-hexane with n-alkanes in the range C7-C12 at 298.15 K, 1 atm, and various concentrations measured by Touriño et al., 35 the % AARD for sound speed and density are smaller than 1.38% and 0.55%, respectively. Figures 6-7 show these results. It is noteworthy that the MTC EOS captures the compositional dependence and that the eect of the carbon chain size dierence on the accuracy of the predictions is minor. Figure 7 shows that the MTC EOS captures the compositional eect also on sound speeds but the deviations between the experimental and predicted sound speeds of the pure components impact the model's ability to predict mixture sound speeds.

Conclusions This study evaluated the ability of the MTC EOS in representing the liquid phase density, sound speed, and vapor pressure of linear alkanes (C6-C20) considering various parameter tting strategies. Among the tested options, it was more advantageous to estimate the characteristic volume (parameter set P2 ) than the temperature-dependence parameter for the characteristic energy of interaction (parameter set P1 ), and estimating both (parameter set

P3 ) was more advantageous but the decrease in deviations was close to the typical experimental uncertainties. The best set of parameters produces deviations in liquid phase density, liquid phase sound speed, and vapor pressure equivalent to the corresponding deviations for the PC-SAFT EOS within typical experimental uncertainties. It was observed that, for 17



the MTC EOS, the deviations between calculated and experimental property values tend to increase as the pressure increases when pressures up to 700 MPa are taken into account. Retting the MTC EOS parameters only with data points whose reduced pressure is smaller than 20, which is a condition observed in most commercial chemical processes, the best set of parameters produced deviations equal to 0.5% in liquid phase density, 2.8% in liquid phase sound speed, and 1.3% in vapor pressure, and this is the parameter set recommended for MTC EOS calculations using ve parameters per compound. Given that the tted lattice cell volumes were within a relatively narrow value range, the xed value of 15 cm3 /mol was adopted, which is the molar volume of the methylene group in the lattice-based UNIQUAC model. Parameters tted using this assumption follow regular trends with respect to the number of carbon atoms in the linear alkane chain and their values were tted to empirical expressions as function of the number of carbon atoms in the linear alkane chain. The purely predictive parameters calculated using these empirical expressions produce deviations in pure-component liquid phase density, liquid phase sound speed, and vapor pressure equal to 0.6%, 2.8%, and 9.8%, respectively. Predictive results for binary mixtures are generally in good agreement with the experimental data. In purely predictive calculations for n-alkanes with the MTC EOS, these empirical expressions provide the recommended parameter set.

Acknowledgement S.M. thanks the hospitality of Texas A&M University at Qatar during her stay in Doha to work on this article. Likewise, M.C. thanks the hospitality of the Federal University of Bahia during his stay in Salvador to work on this article. M.L.L.P. and S.M. acknowledge research productivity fellowships of the Brazilian National Council for Scientic and Technological Development (CNPq grants 306142/2017-1 and 306640/2016-3).

18


Page 18 of 36

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List of symbols Roman Letters

a: thermodynamic sound speed (m/s) ares : molar residual Helmholtz energy (J/mol) AARD: average absolute relative deviation B ma : temperature dependence parameter for the interaction between regions m and a (K) cP : molar specic heat at constant pressure (J/(mol.K)) C : number of carbon atoms in the linear alkane chain h: molar enthalpy (J/mol) ks : isentropic compressibility (1/Pa) kT : isothermal compressibility (1/Pa) l: bulkiness factor M : molar mass (kg/mol) nc : number of components ng : number of regions N : number of points P : absolute pressure (Pa) P sat : vapor pressure (Pa) Q: surface area parameter r: number of segments R: universal gas constant (J/(mol.K)) S m : auxiliary function dened by Eq. 11 T : absolute temperature (K) uma : interaction energy between regions m and a (J/mol) uma 0 : temperature-independent parameter for the interaction between regions m and a (J/mol) 19



v ∗ : volume of one mole of lattice cells (m3 /mol) v˜: reduced volume x: mole fraction x: mole fraction vector z : lattice coordination number zq : number of external contacts Z : compressibility factor

Subscripts

i, k : chemical component r: reduced property

Superscripts

a, m: region type res: residual ig : ideal gas calc: calculated exp: experimental

Greek letters

α: volume expansivity (1/K) γ ea : exponential term of interaction energy between region a and region e dened by Eq. 12 Γma : auxiliary symbol dened by Eq. 13

20


Page 20 of 36

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νia : number of regions a in a molecule type i ρ: molar density (mol/m3 ) ρm : mass density (kg/m3 ) φî : fugacity coecient of a component i in the mixture Ψ: empirical universal constant of the lattice structure Ω: studied property

21



Page 22 of 36

Figure captions Figure 1. Adopted strategies for tting and comparing parameter sets. Figure 2. Experimental sound speed 36 of liquid n-hexane at 298.3 K and calculated values with parameters tted using the complete data set. The curves for parameter sets P2 and

P3 overlap. Figure 3. Experimental density 37 of liquid n-hexane at 298.15 K and calculated values with parameters tted using the complete data set. The curves for parameter sets P2 and P3 overlap. 0

0

Figure 4. MTC EOS parameters: tted (set P1 ) and correlated (set P1c , Eqs. 23-26). Figure 5. Eect of temperature on the properties of n-tetradecane, n-heneicosane, and ntetracosane: experimental data and MTC EOS predictions using parameters correlated with 0

the number of carbon atoms (parameter set P1c ). Figure 6. Density of binary of mixtures of n-hexane + (n-heptane or n-octane or n-nonane or n-decane or n-undecane or n-dodecane) at 298.15 K, 1 atm: experimental data from Touriño et al. 35 and MTC EOS predictions using parameters correlated with the number of carbon 0

atoms (parameter set P1c ). Figure 7. Sound speed of binary of mixtures of n-hexane + (n-heptane or n-octane or nnonane or n-decane or n-undecane or n-dodecane) at 298.15 K, 1 atm: experimental data from Touriño et al. 35 and MTC EOS predictions using parameters correlated with the number 0

of carbon atoms (parameter set P1c ).

22


Page 23 of 36 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 1

23



Figure 2 1900 Experimental Parameter set P1 Parameter set P2 Parameter set P3

1800 1700 Sound speed (m/s)

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 24 of 36

1600 1500 1400 1300 1200 1100 1000 0

20

40

60 80 Pressure (MPa)

24


100

120

Page 25 of 36

Figure 3 900 Experimental Parameter set P1 Parameter set P2 Parameter set P3

850 Density (kg/m3)

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


800

750

700

650 0

100

200

300 400 Pressure (MPa)

25


500

600

700


-28

20 Fitted parameter Correlated parameter

-30

10 5 Baa (K)

-32 u0aa/R (K)

Fitted parameter Correlated parameter

15

-34

0 -5 -10

-36

-15 -20

-38

-25 -40

-30 6

8

10

12 14 Carbon number

16

18

20

6

(a) Temperature-independent interaction

8

10

12 14 Carbon number

16

18

20

(b) Temperature-dependence contribution

20

350 Fitted parameter Correlated parameter

18

Fitted parameter Correlated parameter 300

va (cm3/mol)

16 14 Qa

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 26 of 36

12

250

200

10 150 8 6

100 6

8

10

12 14 Carbon number

16

18

20

6

(c) Molecular surface area parameter

8

10

12 14 Carbon number

16

(d) Molar packing volume

Figure 4

26


18

20

Page 27 of 36

780

Experimental C14 Predicted C14 Experimental C21 Predicted C21 Experimental C24 Predicted C24

Density (kg/m3)

760

740

720

700

680

660 280

300

320

340

360 380 400 Temperature (K)

420

440

460

480

(a) Density at 1 atm 1400


1300

Sound speed (m/s)

1200 1100 1000 900 800 700 600 250

300

350

400 450 Temperature (K)

500

550

(b) Sound speed at 1 atm 100


10 Vapor pressure (kPa)

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


1

0.1

150 125 100 75 50

0.01

25 1.56*10−3

0.001 1.5*10−3

1.60*10−3 2.0*10−3

1.64*10−3 2.5*10−3 1/T (1/K)

3.0*10−3

(c) Vapor pressure Figure 5

27


3.5*10−3


Figure 6 750

C6+C7 C6+C8 C6+C9 C6+C10 C6+C11 C6+C12

740 730 720 3

Density (kg/m )

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 28 of 36

710 700 690 680 670 660 650 0

0.2

0.4 0.6 Mole fraction of n−hexane

28


0.8

1

Page 29 of 36

Figure 7 1350

C6+C7 C6+C8 C6+C9 C6+C10 C6+C11 C6+C12

1300 Sound speed (m/s)

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


1250

1200

1150

1100

1050 0

0.2


29


0.8

1


Supporting information The supporting information le contains references to the experimental data used in this research. It also contains parameters for the Mattedi-Tavares-Castier equation of state obtained from the dierent tting strategies adopted in this work.

30


Page 30 of 36

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TOC Graphic This graphic is for the Table of Contents only.

Figure 8

760

C6+C7 C6+C8 C6+C9 C6+C10 C6+C11 C6+C12

740 3

Density (kg/m )

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 36 of 36

720 700 680 660 640 0

0.2


36


0.8

1

Liquid phase density, sound speed, and vapor ... - ACS Publications

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