New Thermodynamic Approach for Nonspherical Molecules Based on

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Thermodynamics, Transport, and Fluid Mechanics

A new thermodynamic approach for nonspherical molecules based on a perturbation theory for ellipsoids Joyce Tavares Lopes, and Luis Fernando Mercier Franco Ind. Eng. Chem. Res., Just Accepted Manuscript • Publication Date (Web): 25 Mar 2019 Downloaded from http://pubs.acs.org on March 29, 2019

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Industrial & Engineering Chemistry Research

A new thermodynamic approach for nonspherical molecules based on a perturbation theory for ellipsoids Joyce T. Lopes and Lu´ıs F. M. Franco∗ School of Chemical Engineering, University of Campinas, Av. Albert Einstein 500, CEP:13083-852, Campinas, Brazil E-mail: [email protected]

Abstract We propose a new thermodynamic approach for nonspherical molecules by applying a perturbation theory in which an anisotropic intermolecular potential, the Hard Gaussian Overlap (HGO), is the reference system. The new Equation of State (EoS) modifies the usual SAFT (Statistical Associating Fluid Theory) approach by combining both segment and chain contributions as a single anisotropic term. Fluid particles are represented as ellipsoids rather than a set of few tangential spherical segments. The perturbed potential is taken as a square well, following the original formulation of SAFT with attractive potential of variable range (SAFT-VR SW). The parameters of the proposed model were optimized to fit vapor pressures and saturated liquid densities for ethane and carbon dioxide. Derivative properties, such as isobaric and isochoric heat capacities, speed of sound, Joule-Thomson coefficient, thermal expansion coefficient, and isothermal compressibility, were evaluated at supercritical conditions up to 70 MPa for ethane and 200 MPa for carbon dioxide. The proposed EoS outperforms the original SAFT-VR SW EoS for many of these properties. This implies that

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an ellipsoidal geometry is an adequate representation of such nonspherical molecules, avoiding the approximations usually applied in Wertheim’s first-order thermodynamic perturbation theory (TPT1) for the calculation of the Helmholtz free energy of chain formation.

Introduction Volumetric equations of state (EoS) are fundamental to process design. For the calculation of thermodynamic and phase equilibrium properties, most of the models routinely used in engineering are cubic equations of state, 1–3 which are empirical extensions of van der Waals equation of state. Although they are simple and reliable when applied to conditions they were designed to describe, they are sensitive to the experimental data used to fit their parameters. Hence, they might be inadequate to extrapolated conditions, 4 e.g. carbon dioxide in oil and gas reservoirs, which requires handling fluids up to 100 MPa. 5 Given the limitations of those empirical equations, theoretical and semi-empirical models based on statistical mechanics have been gaining ground lately, as an attempt to improve thermodynamic fluid properties calculations. From a molecular point of view, all the thermophysical properties of fluids are intrinsically connected to the potential energy function governing the fluid particles interactions. An exact solution to this problem by means of the classical statistical mechanics is impossible to be obtained even for a hard-sphere fluid, for this problem constitutes a many-body problem. With the exception of a numerical solution, e.g. molecular simulations, approximations with reasonable physical arguments must be made. Perturbation theory has been used, therefore, as an alternative route to find an approximate solution. Longuet-Higgins 6 proposed an ablation of the total potential energy function into two contributions: a reference, and a perturbed potential. 7,8 Since the liquid structure is mainly dominated by repulsive interactions, 9 the idea of a repulsive reference potential has been the common basis for the thermodynamic perturbation theory. Zwanzig 10 used the 2


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hard-sphere fluid as the reference, or unperturbed, potential and Lennard-Jones potential as the perturbed one. His perturbation theory emerges as an expansion of the inverse of the temperature. Nonetheless, except for the first order, the other terms are quite difficult to evaluate since third and higher orders density functions are introduced. Barker and Henderson 11 proposed a way to calculate the second-order term, introducing the local compressibility approximation, and the macroscopic compressibility approximation. Zhang 12 has improved the macroscopic compressibility approximation by introducing a correlation coefficient in the formulation, instead of assuming that the number of molecules in neighbouring shells would be uncorrelated as proposed by Barker and Henderson. For a square-well potential, the inclusion of higher-order terms in the expansion using molecular simulations significantly improves the predictions of the critical points and heat capacities. 13–15 van Westen and Gross 16 also showed that higher-order perturbation theory improves the description of thermodynamic properties of a Lennard-Jones fluid. Although Zwanzig and Barker-Henderson perturbation theories might be adequate for isotropic potential functions, association and bond formation are a result of highly directional attractive forces. Wertheim 17–22 developed a first-order perturbation theory that takes into account such directional forces in the perturbed potential. Wertheim’s theory provided the basis for the Statistical Associating Fluid Theory Equation of State (SAFT EoS). 23,24 Within SAFT framework, the residual Helmholtz free energy, AR , is expressed as a sum of different contributions: Aseg (segment-segment interactions), Achain (chain formation by covalent bonds between two segments), and Aassoc (hydrogen-bonding interactions). Modern variations of SAFT EoS, e.g. PC-SAFT - where PC stands for perturbed chain, 25 and SAFT-VR Mie - where VR stands for variable range, 26 have been widely used to describe complex fluids due to their accuracy and versatility. In the early SAFT EoS works, 27 hydrocarbons used to be modeled as chains of tangential spherical segments considering the equivalence between the number of segments per chain and the number of carbons per molecule. Nevertheless, for the sake of a better correlation

3



with the experimental data, in the more recent applications of SAFT EoS, the number of segments for nonspherical molecules has usually been fitted as a non-integer number. 25,26 Some molecules, however, when considered to be spherical, are modeled as a single segment, e.g. methane, 25,26,28 water, 29 and hydrogen. 30 A non-integer number of segments fitted for nonspherical molecules weakens its physical meaning and precludes a more predictive use of it. Gil-Villegas et al. 28 have shown that, for chains of square-well monomers, as the number of segments increases, the overprediction of the vapor-liquid coexistence curve obtained with SAFT-VR SW when compared to results obtained with Monte Carlo simulations is magnified. These observations might lead to the conclusion that something is rather missing in the theory, or that some approximations used in the chain formation contribution are inadequate. For nonspherical molecules, one might suppose that, instead of having spherical segments in a chain, such molecules could be represented as single ellipsoids. Thus, in this contribution, we propose a model based on a perturbation theory in which the reference potential is given by the Hard Gaussian Overlap (HGO) potential, and the perturbed potential is a squarewell potential. Similar strategies but representing molecules as spherocylinders have been attempted. Williamson and del R´ıo 31 have presented two different models to describe the isotropic and nematic phases in a fluid of spherocylinders. The resulting equation of state was applied, in a subsequent work, to calculate liquid-vapor coexistence curves for n-alkanes. 32 A second-order term was later developed by Garc´ıa-S´anchez et al. 33 Wu et al. 34 have recently developed a theoretical equation of state for hard spherocylinders with an anisotropic squarewell potential.

Theoretical Framework The literature devoted to the treatment of anisotropic potentials within statistical mechanics presents some different approaches, frequently dedicated to liquid crystals. 35,36 One step before developing an adequate statistical mechanical formulation for a specific model is the

4


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proposition of the potential energy function itself. The most popular isotropic intermolecular potentials are approximate effective potentials that take into account many physical aspects implicitly. The main advantage of an effective potential energy function is its mathematical tractability. A balance between a deep physical description of the system and a simple mathematical form is essential in developing new models. To reduce the computational demands in simulating nonspherical and polyatomic molecules, Berne and Pechukas 37 developed an orientation-dependent intermolecular potential: the Gaussian Model Potential. In this model, molecules are represented as a rigid union of a set of ellipsoids, and the potential between ellipsoids are associated to the mathematical overlap of two Gaussian distributions. The mathematical structure of the resulting model is quite similar to the Lennard-Jones potential, but the main difference is that the contact distance depends on the particles orientations. Gay and Berne 38 pointed out that, comparing a multisite approach to the Gaussian Overlap Potential in a four-site molecule, the model developed by Berne and Pechukas had some unrealistic features. The potential was then modified in such a way that the well depth of a side-by-side configuration between two ellipsoids differs from the end-by-end one. Later Velasco et al. 39 derived a thermodynamic perturbation theory for the Gay-Berne intermolecular potential in which the perturbed contribution is the Gay-Berne potential, and the reference term is calculated using the Hard Gaussian Overlap model (HGO).

Hard Gaussian Overlap For intermolecular potentials based on the Gaussian overlap model 37 , particles are taken as ellipsoids characterized by the parameters σs and σe , which are defined as the distances at which the potentials between two side-by-side and end-by-end ellipsoids become zero, respectively. For HGO, the parameters are the distances at which both ellipsoids touch each other, as illustrated in Figure 1. If the position coordinates and the orientation of two ellipsoids coincide, σs and σe become the symmetry axes. 5



(a) Side-by-side

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(b) End-by-end

Figure 1: Ellipsoids configurations. From σs and σe , the elongation, κ, and the shape anisotropy parameter, χ, can be defined as: κ=

σe σs

and χ =

κ2 − 1 κ2 + 1

(1)

For spherical particles, κ → 1 and χ → 0, for long rods κ → ∞ and χ → 1, and for very thin disks κ → 0 and χ → −1. The contact distance, σ, between two ellipsoids will be constrained between the values of σs and σe depending on the orientation of the particles, as shown in Equation 2: σHGO = σs

χ 1− 2

(Ω1 · r + Ω2 · r)2 (Ω1 · r − Ω2 · r)2 + 1 + χ(Ω1 · Ω2 ) 1 − χ(Ω1 · Ω2 )

−1/2 (2)

where σHGO is the contact distance calculated by the HGO model, r is the unit vector along the vector connecting two ellipsoids centers of mass, and Ω1 and Ω2 are the unit vectors along the axis of the ellipsoids, which represent the molecules orientations, as illustrated in Figure 2. HGO model simplifies the calculation of the contact distance between two ellipsoids to a great extent, when compared to the Hard Ellipsoid of Revolution (HER) model, as calculated by Vieillard-Baron. 40 For a configuration in which two ellipsoids are perpendicular to each other, however, the contact distance between these two ellipsoids is slightly overestimated by HGO. 41 Despite its intrinsic limitations, HGO model provides a good balance between

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Figure 2: Two ellipsoids with different orientations at a certain distance r. accuracy and simplicity. The HGO potential has a similar structure to the hard-sphere potential, i.e., the potential is purely repulsive, as described in Equation 3:

uHGO =

   +∞,

if

  0,

if

r ≤ σHGO

(3)

r > σHGO

Once the potential is formally defined, the calculation of the Helmholtz free energy can be pursued. For a homogeneous fluid with orientational degrees of freedom, considering the virial expansion introduced by Onsager, 42 the Helmholtz free energy for cylindrically symmetric particles can be calculated as: 43

βa = ln

ρν 4π

Z −1+

" Z # +∞ n+1 X Y ρn f (Ω) ln [4πf (Ω)] dΩ+ f (Ωj )Bn+1 (Ω1 , · · · , Ωj ) dΩj (4) n! n=1 j=1

where β = 1/(kB T ), kB is Boltzmann constant, T is the absolute temperature, a is the molar Helmholtz free energy, ρ is the number density, ν is the de Broglie volume incorporating rotational and translational degrees of freedom, f (Ω) is the orientational distribution function, and Bn is the n-th orientation-dependent virial coefficient.

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For an isotropic distribution, f (Ω) is constant:

f (Ω) =

1 4π

(5)

Therefore, Equation (4) can be further simplified to: " Z # +∞ n+1 n X Y 1 ρ βa = ln −1+ Bn+1 (Ω1 , · · · , Ωj ) dΩj 4π n! 4π n=1 j=1 ρν

(6)

Velasco et al. 39 applied the “decoupling approximation” (DA) within Onsager’s theory to formulate an expression for the Helmholtz free energy of the HGO fluid. The DA, which was proposed by Parsons, 44 consists of mapping the system of ellipsoids into a system of hard-spheres interacting with an orientation dependent potential. Parsons approximates ˆ 1, Ω ˆ 2 ) to g(r/σHGO ), decoupling the orientational and the position degrees of freedom, g(~r, Ω hence DA is only exact at low densities. The HGO Helmholtz free energy for an isotropic distribution can then be written as:

βaHGO

(4 − 3η)η −1+ = ln 4π 2(1 − η)2 ρν

arcsin χ 1+ p χ 1 − χ2

! (7)

where η = ρπσs3 κ/6 is the packing fraction. If χ = 0, i.e. spherical particles, aHGO becomes equal to the Carnahan-Starling 45 expression for the molar Helmholtz free energy of a hardsphere fluid.

Equation of State Formulation Taking Longuet-Higgins perturbation theory, 6 one may split the Helmholtz free energy in two contributions: the reference Helmholtz free energy, and the perturbed Helmholtz free energy: βa = βa(0) + βa(1)

8


(8)

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where a(0) is the molar Helmholtz free energy of the reference fluid, and a(1) is the molar Helmholtz free energy of the perturbed potential. Assuming that the reference potential is the HGO potential, Equation (8) becomes:

βa = βaHGO + βa(1)

(9)

where βaHGO can be calculated by Equation (7). For the perturbed potential, we have chosen the spherical square-well potential:

uSW =

   +∞,     −ε,       0,

if

r≤σ

if

σ < r ≤ λσ

if

r > λσ

where ε is the well depth, λ is the attractive range, and σ =

√ 3

(10)

σs 2 σe , which is calculated by

equating the hard core volumes of a sphere and an ellipsoid of revolution: πσs2 σe πσ 3 = 6 6

(11)

Taking Barker-Henderson second-order perturbation theory, the perturbed contribution to the Helmholtz free energy averaged over a hard-sphere fluid, following Gil-Villegas et al. approach, 28 becomes:

βa

(1)

= −4βε λ − 1 η g HS + 3

βε(1 − η)4 2 (1 + 4η + 4η 2 )

g

HS

∂g HS +η ∂η

(12)

where g HS =

1 − ηeff /2 (1 − ηeff )3

9


(13)


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where ηeff is the effective packing fraction calculated as: ηeff = c1 η + c2 η 2 + c3 η 3

(14)

where the coefficients c1 , c2 , and c3 are given by:       c1   2.25855 −1.50349 0.249434   1        c  = −0.669270 1.40049 −0.827739 ×  λ   2           c3 10.1576 −15.0427 5.30827 λ2

(15)

Applying Gil-Villegas et al. 28 formulation for the perturbed potential contribution to the Helmholtz free energy, using HGO as the reference fluid, means that the perturbed potential is independent of the orientation. This is a strong assumption, since even the contact distance depends on the relative orientation between two molecules. Moreover, the perturbed potential is averaged over the hard-sphere fluid instead of the HGO fluid. Such an approximation may have low impact at low densities, but it becomes quite relevant for compressed fluids.

Results and Discussion The proposed equation of state was applied to ethane and carbon dioxide, both small nonspherical molecules. For spherical molecules like methane, the proposed EoS and SAFT-VR SW are completely equivalent. The model parameters were optimized to fit vapor pressure and saturated liquid density data obtained in NIST. 46 Although taking into account supercritical derivative properties such as heat capacity and speed of sound improves the applicability of the fitted parameters, 26 we have maintained the same properties used in the original optimization of SAFT-VR SW EoS 28 for the sake of a fair comparison. The optimized parameters are shown in Table 1.

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Table 1: Optimized parameters for ethane and carbon dioxide. Compound ethane

EoS This work SAFT-VR SW 28 carbon dioxide This work SAFT-VR SW 47 a b

λ 1.597 1.448 1.626 1.516

ε/kB / K 243.5 241.8 275.0 179.3

σs a / ˚ A 2.968 3.788 2.144 2.786

σe / ˚ A 6.864 7.494 -

mb 1.3 2.0

for SAFT-VR SW, σs = σ. m stands for the number of spherical segments in a chain.

Figures 3 and 4 present the vapor-liquid equilibrium of ethane and carbon dioxide. The proposed equation of state correlates the coexistence curves of both ethane and carbon dioxide better than SAFT-VR SW when compared to NIST data. Both the proposed equation of state and SAFT-VR SW 28 overpredict the critical point of ethane and carbon dioxide. Nevertheless, our proposed EoS predicts more accurately the values of critical temperature and critical pressure when compared to NIST data, 46 as shown in Table 2. 330

5.5

320

5

310

4.5

300

4

pvap / MPa

T/K

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


290 280

3.5 3 2.5

270

2

260

1.5 250 0

75

150

225

300

375

450

1 250

ρ / kg · m-3

260

270

280

290

300

T/K

(a)

(b)

Figure 3: Vapor-liquid equilibrium for pure ethane: (a) coexistence curve, (b) vapor pressure as a function of temperature. Open symbols, NIST data. 46 Continuous lines, our proposed equation of state. Dotted lines, SAFT-VR SW. 28 The reason behind the overprediction of the critical properties (temperature and pressure) for both models is twofold: the choice of a discrete perturbed potential as the square-well potential, and the truncation of the high temperature series expansion on the second term. Lafitte et al. 26 showed that, with the inclusion of the third term in the expansion with a 11



330

8

320

7

310 6

300

pvap / MPa

290

T/K

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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280 270 260

5 4 3 2

250 240

1

230 0

200

400

600

800 -3

1000 1200

0 230 240 250 260 270 280 290 300

ρ / kg · m

T/K

(a)

(b)

Figure 4: Vapor-liquid equilibrium for pure carbon dioxide: (a) coexistence curve, (b) vapor pressure as a function of temperature. Open symbols, NIST data. 46 Continuous lines, our proposed equation of state. Dotted lines, SAFT-VR SW 28 with optimized parameters from Galindo and Blas. 47 Table 2: Critical properties of ethane and carbon dioxide. Compound ethane

NIST 46 This work SAFT-VR SW carbon dioxide NIST 46 This work SAFT-VR SW

Tc / K 305.33 320.57 325.33 304.12 315.49 322.97

pc / MPa 4.87 6.10 7.43 7.38 8.52 11.18

Mie potential, the prediction of the critical point is much more accurate. Likewise, using molecular simulations, the same improvement in the prediction of the critical point appyling higher-order terms in the free energy expansion was observed for square-well 15 and LennardJones 16 fluids. Another strategy would be to take into account the critical point in the fitting procedure, but the correlation of the saturated liquid density would certainly be deteriorated. For the sake of a quantitative comparison, the Average Absolute Relative Deviation (AARD) was calculated: Np 1 X ϕNIST − ϕEoS i i × 100% AARD(%) = NIST Np i=1 ϕi 12


(16)

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where Np is the number of calculated points, ϕNIST is the NIST 46 value for a certain property ϕ, and ϕEoS is the property value calculated by the Equation of State. Table 3 presents the AARD values for vapor pressure, saturated liquid density, and saturated vapor density of both ethane and carbon dioxide calculated with the proposed EoS and with SAFT-VR SW. The vapor pressure of ethane is better correlated with the proposed equation of state; whereas the opposite is observed to carbon dioxide. Nevertheless, the saturated liquid density for both fluids is better correlated with the proposed EoS. Table 3: Average Absolute Relative Deviation (%) for vapor pressure, saturated liquid density, and saturated vapor density. pvap This work 1.96 SAFT-VR SW 5.07 carbon dioxide This work 3.53 SAFT-VR SW 0.76

Compound ethane

ρl 1.41 6.08 0.78 2.26

ρv 8.78 7.96 6.41 7.95

The parameters of the proposed equation of state were adjusted solely to correlate saturated properties. The supercritical properties of ethane and carbon dioxide, however, can be used to assess the predictive power of the proposed model. Figure 5 presents the results for supercritical density of ethane and carbon dioxide. Overall the proposed EoS predictions are more accurate than those obtained with SAFT-VR SW, with the exception of ethane at a low temperature (350 K). SAFT-VR SW generally overpredicts pressure at a given density and temperature for both ethane and carbon dioxide. Supercritical derivative properties of ethane and carbon dioxide, such as isochoric and isobaric heat capacities, speed of sound, Joule-Thomson coefficient, isothermal compressibility, and thermal expansion coefficient, were also investigated. Figures 6 and 7 show the results for the proposed EoS and SAFT-VR SW, 28 compared to NIST data. 46 The ideal gas isobaric heat capacity was calculated according to the empirical expression proposed by Passut and Danner. 48 The proposed EoS captures the trends observed for all thermodynamic derivative properties, with the exception of the isochoric heat capacity. The original SAFTVR SW, however, describes qualitatively well only the speed of sound and the isothermal 13



70

200

625 K 500 K 400 K 350 K

60

700 K 600 K 500 K 360 K

150

p / MPa

50

p / MPa

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

40 30 20

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100

50

10 0

0 0

50 100 150 200 250 300 350 400 450 -3

0

200

ρ / kg · m

400

600 -3

800

1000

ρ / kg · m

(a) Ethane

(b) Carbon Dioxide

Figure 5: Pressure versus density at constant temperature for: (a) ethane, and (b) carbon dioxide. Open symbols, NIST data. 46 Continuous lines, our proposed equation of state. Dotted lines, SAFT-VR SW. 28 compressibility. Even keeping a square-well potential as the perturbed potential, the sole change in the repulsive contribution from a spherical to an ellipsoidal representation resulted in a significant improvement in the prediction of derivative properties. This is somehow a surprising result, since it has been implied that the failure in describing these properties with SAFT-VR SW could be the choice of a square-well potential. 49 Larger deviations are observed for the proposed EoS at high pressures and low temperatures. At these thermodynamic conditions, a high dense fluid is found, and the approximation made in the formulation of the equation of state in which the reference and the perturbed potential are treated with different molecular geometries is challenged. The replacement of the square-well potential by an anisotropic intermolecular potential in the attractive part of the perturbation theory might improve the prediction of derivative properties at these specific conditions. The decoupling approximation applied to formulate the HGO Helmholtz free energy also has a relevant impact on the prediction of properties at higher densities, since DA is only exact at low densities. The AARD values for the calculation of supercritical density and derivative properties for ethane and carbon dioxide are shown in Table 4. With the exception of the carbon dioxide

14


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180 625 K 500 K 400 K 350 K

cp / J · mol-1 · K-1

90

625 K 500 K 400 K 350 K

160

-1

cv / J · mol · K

-1

100

80

70

60

140 120 100 80 60

50

40 0

10

20

30

40

50

60

70

0

10

20

30

p / MPa

40

50

60

70

p / MPa

(a)

(b)

1100 625 K 500 K 400 K 350 K

1000

625 K 500 K 400 K 350 K

8 7 6

µJT / K · MPa-1

cs / m · s-1

900 800 700 600 500 400

5 4 3 2 1

300

0

200

-1 0

10

20

30

40

50

60

70

0

10

20

30

p / MPa

40

50

60

70

p / MPa

(c)

(d)

1

0.022 625 K 500 K 400 K 350 K

625 K 500 K 400 K 350 K

0.02 0.018 0.016

0.1

0.014

α / K-1

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


0.012 0.01 0.008

0.01

0.006 0.004 0.002 0.001

0 10

20

30

40

50

60

70

0

p / MPa

10

20

30

40

50

60

70

p / MPa

(e)

(f)

Figure 6: Derivative properties of supercritical ethane: (a) isochoric heat capacity, (b) isobaric heat capacity, (c) speed of sound, (d) Joule-Thomson coefficient, (e) isothermal compressibility, and (f) thermal expansion coefficient. Open symbols, NIST data. 46 Continuous lines, our proposed equation of state. Dotted lines, SAFT-VR SW. 28

15



55

120 700 K 600 K 500 K 360 K

700 K 600 K 500 K 360 K

110

cp / J · mol-1 · K-1

cv / J · mol-1 · K-1

50

45

40

35

100 90 80 70 60 50 40

30

30 0

50

100

150

200

0

50

p / MPa

150

200

(b)

1300

7 700 K 600 K 500 K 360 K

1200 1100

700 K 600 K 500 K 360 K

6 5

µJT / K · MPa-1

1000

cs / m · s-1

100

p / MPa

(a)

900 800 700 600 500

4 3 2 1

400 0

300 200

-1 0

50

100

150

200

0

50

p / MPa

100

150

200

p / MPa

(c)

(d) 0.014 700 K 600 K 500 K 360 K

0.1

700 K 600 K 500 K 360 K

0.012

α / K-1

0.01

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

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0.01

0.008 0.006 0.004 0.002

0.001

0 20 40 60 80 100 120 140 160 180 200

0

p / MPa

20 40 60 80 100 120 140 160 180 200

p / MPa

(e)

(f)

Figure 7: Derivative properties of supercritical carbon dioxide: (a) isochoric heat capacity, (b) isobaric heat capacity, (c) speed of sound, (d) Joule-Thomson coefficient, (e) isothermal compressibility, and (f) thermal expansion coefficient. Open symbols, NIST data. 46 Continuous lines, our proposed equation of state. Dotted lines, SAFT-VR SW. 28

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isothermal compressibility, the proposed EoS gives better predictions than SAFT-VR SW for all supercritical properties. The highest deviations are observed for the Joule-Thomson coefficient predictions. A thorough analysis shows that such high deviations are exclusively related to the deviation in the prediction of the inversion point. Table 4: Average Absolute Relative Deviation (%) for supercritical density, isochoric heat capacity, isobaric heat capacity, Joule-Thomson coefficient, speed of sound, thermal expansion coefficient, and isothermal compressibility. Compound ethane

ρ This work 1.69 SAFT-VR SW 3.86 carbon dioxide This work 2.43 SAFT-VR SW 6.88

cv cp µJT cs α 3.39 1.90 114.13 2.00 5.55 3.63 7.10 157.34 2.34 10.25 7.75 2.79 131.45 2.44 11.05 9.29 24.91 724.19 6.43 36.32

kT 4.74 5.28 8.26 3.64

The results obtained here by the proposed equation of state shows that somehow the assumption of an ellipsoidal geometry seems to be more adequate to represent these molecules than the original SAFT-VR SW 28 approach for which the Helmholtz free energy of spherical segments forming a chain is calculated with Wertheim’s first-order thermodynamic perturbation theory. The proposed formulation has also the benefit of eliminating the apparent physical issue of a non-integer number of segments. Nevertheless, the fitted parameters for the proposed EoS assuming an ellipsoidal geometry must be physically sound. A simple way to check this is to compare the shape and the volume of the fitted ellipsoid with the molecular models for ethane and carbon dioxide. The Transferable Potential for Phase Equilibria (TraPPE) 50 is a united atom molecular model using Lennard-Jones potential frequently applied in molecular simulations to calculate phase equilibrium and thermodynamic properties. 5 For TraPPE, 50 ethane is represented as two spherical particles. The distance between these two particles is kept fix as 1.54 ˚ A. Assuming the diameter of such spherical particles as the distance at which the intermolecular potential is zero, then the volume of a single ethane molecule is 43.7 ˚ A3 . The volume of the ellipsoid, calculated with the fitted parameters shown in Table 1, is 31.7 ˚ A3 . Therefore, 17



the ratio between the volume of the fitted ellipsoid and the volume obtained with TraPPE ˚3 , molecular model is 0.73. For carbon dioxide, TraPPE model 51 gives a volume of 22.9 A ˚3 , giving a ratio between the volume of the while the fitted ellipsoid has a volume of 18.0 A fitted ellipsoid and the volume obtained with TraPPE molecular model of 0.79. Being the calculated volume ratios for ethane and carbon dioxide close to 1, one might conclude that the fitted parameters provide a physically meaningul geometry for both ethane and carbon dioxide, as also illustrated in Figure 8.

(a) Ethane

(b) Carbon Dioxide

Figure 8: Illustrative comparison between TraPPE molecular model for ethane 50 and carbon dioxide 51 and the ellipsoidal geometry obtained with the fitted parameters for the proposed EoS. . An explanation for why the ellipsoids volumes are lower than the ones calculated with TraPPE model might reside in a compensation for the use of the square-well potential as the perturbed term. Taking the potential well-depth proposed by Berne and Pechukas 37 and analyzing the four site molecule example in Gay and Berne, 38 one may see that the welldepth of nonspherical particles tend to be larger than that for spherical particles. Thus, the attractive term on perturbation theory might be larger if one applies an anisotropic potential. Therefore, a smaller volume in the proposed EoS reduces the repulsive contribution to be compatible to the attractive contribution given by the square-well perturbed potential.

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Conclusion In this proof-of-concept work, we have formulated an alternative equation of state for nonspherical molecules based on a perturbation theory, in which the Hard Gaussian Overlap model is used as the reference potential and the perturbed contribution is given by a squarewell potential. The vapor-liquid equilibrium for ethane and carbon dioxide was successfully correlated with the proposed EoS. Moreover, the proposed EoS predicts more accurate critical properties, when compared to SAFT-VR SW. For suprecritical thermodynamic derivative properties, the proposed EoS generally provides better estimates than the original SAFT-VR SW for both ethane and carbon dioxide. The results obtained with the proposed EoS might imply that the choice of a single ellipsoid to represent such small molecules is an adequate alternative as to fitting a non-integer number of segments, as commonly done in SAFT framework using Wertheim’s first-order thermodynamic perturbation theory. The new proposed approach also significantly improves the prediction of derivative properties while keeping the square-well potential as the attractive contribution. The comparison between the ellipsoid volume resulting from the fitted parameters with molecular models such as TraPPE shows that the proposed EoS parameters are physically meaningful. Finally, the extension of the proposed model for larger and associating molecules remains to be addressed.

Acknowledgement The authors gratefully acknowledge the financial support from the Brazilian Coordination for the Improvement of Higher Education Personnel (CAPES).

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New Thermodynamic Approach for Nonspherical Molecules Based on

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