Transport Properties of Mixtures by the Soft-SAFT + Free-Volume

Article pubs.acs.org/JPCB

Transport Properties of Mixtures by the Soft-SAFT + Free-Volume Theory: Application to Mixtures of n‑Alkanes and Hydrofluorocarbons F. Llovell,*,† R. M. Marcos,‡ and L. F. Vega†,§ †

MATGAS Research Center (Carburos Metálicos/Air Products, CSIC, UAB), Campus de la UAB, 08193 Bellaterra, Barcelona, Spain Departament d’Enginyeria Mecànica, ETSE, Universitat Rovira i Virgili, 43007 Tarragona, Spain § Carburos Metálicos/Air Products Group, C/Aragón, 300, 08069 Barcelona, Spain ‡

ABSTRACT: In a previous paper (Llovell et al. J. Phys. Chem. B, submitted for publication), the free-volume theory (FVT) was coupled with the soft-SAFT equation of state for the first time to extend the capabilities of the equation to the calculation of transport properties. The equation was tested with molecular simulations and applied to the family of nalkanes. The capability of the soft-SAFT + FVT treatment is extended here to other chemical families and mixtures. The compositional rules of Wilke (Wilke, C. R. J. Chem. Phys. 1950, 18, 517−519) are used for the diluted term of the viscosity, while the dense term is evaluated using very simple mixing rules to calculate the viscosity parameters. The theory is then used to predict the vapor−liquid equilibrium and the viscosity of mixtures of nonassociating and associating compounds. The approach is applied to determine the viscosity of a selected group of hydrofluorocarbons, in a similar manner as previously done for n-alkanes. The soft-SAFT molecular parameters are taken from a previous work, fitted to vapor−liquid equilibria experimental data. The application of FVT requires three additional parameters related to the viscosity of the pure fluid. Using a transferable approach, the α parameter is taken from the equivalent n-alkane, while the remaining two parameters B and Lv are fitted to viscosity data of the pure fluid at several isobars. The effect of these parameters is then investigated and compared to those obtained for n-alkanes, in order to better understand their effect on the calculations. Once the pure fluids are well characterized, the vapor−liquid equilibrium and the viscosity of nonassociating and associating mixtures, including n-alkane + n-alkane, hydrofluorocarbon + hydrofluorocarbon, and n-alkane + hydrofluorocarbon mixtures, are calculated. One or two binary parameters are used to account for deviations in the vapor−liquid equilibrium diagram for nonideal mixtures; these parameters are used in a transferable manner to predict the viscosity of the mixtures. Very good agreement with available experimental data is found in all cases, with an average absolute deviation ranging between 1.0% and 5.5%, even when the system presents azeotropy, reinforcing the robustness of the approach. Due to industrial requirements, the viscosity of many fluids has been determined experimentally in different manners. However, a systematic modeling approach is needed for process design. Molecular analysis can help in the determination of specific viscosity measurements, helping to discriminate which substances will be of better use in a given application. In a previous work,1 we referred to some reviews on a variety of empirical, semiempirical, and theoretical methods to estimate the viscosity.2−4 Depending on the nature of the approach, they can be used in a wider or narrower range of conditions. However, predictive approaches or methodologies with extrapolation capacity are very scarce. Moreover, the calculation of the viscosity of binary and multicomponent mixtures adds

1. INTRODUCTION Viscosity is probably the most important transport property in process design. It describes the internal friction of a fluid or gas when pressure is applied, measuring how the substance will respond to an external force. In other words, it describes the resistance of a liquid to penetration. Hence, it is of particular interest in many chemical engineering processes. For example, in oil piping, it is necessary to address the issue of the force needed by continuously measuring the viscosity and determining if greater or lesser pressure must be added to keep the flow of oil constant and steady. Motor oil is also subject to changing viscosity when heated by an engine. Oil that becomes too thin from the engine’s heat will not work properly in the car engine. In a different and equally important area, the viscosity of refrigerants is a basic property in order to design a refrigeration process. © XXXX American Chemical Society

Received: February 19, 2013 Revised: April 5, 2013

A

dx.doi.org/10.1021/jp401754r | J. Phys. Chem. B XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry B

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reported important deviations for the mixture (AAD of 27.51%). Zéberg-Mikkelsen et al.37 measured and modeled the viscosity the ethanol + n-heptane and ethanol + toluene mixtures. They also compared different methods, with FVT providing very good agreement for both mixtures (AAD of 5.16% and 7.75%, respectively), with the density obtained from experimental data. Finally, Cain et al.23 studied the thermodynamic and transport properties of the decahydronaphthalene + propane mixture. They used the Sanchez−Lacombe EoS for the phase behavior and obtained the density from this equation. In that contribution, an adjustable binary parameter was used for the α mixture parameter value of the FVT, making the extension to mixtures not entirely predictive. The authors obtained good agreement when the adjustable parameter was used (AAD of 5.7%) and qualitative agreement without the binary parameter (AAD of 14.5%). The goal of the present work is 2-fold: on one side, we intend to apply the free-volume theory coupled with the softSAFT to describe the viscosity of the hydrofluorocarbons (HFCs) family, validating the approach for associating fluids. On the other side, the soft-SAFT + FVT treatment is extended to fluid mixtures for the prediction of the viscosity of binary mixtures, including n-alkanes and hydrofluorocarbons. The crossover soft-SAFT version,38−40 including a specific treatment to account for the fluctuations in the critical region, is used for the thermodynamic calculations. The crossover softSAFT equation has already been applied with success to the thermodynamic characterization of hydrofluorocarbons,41 and mixtures of n-alkanes among them,40 and with other compounds.42−44 The rest of the paper is organized as follows: the soft-SAFT + FVT approach is described in the next section, highlighting only the main features with some additional details for the extension to mixtures. Then, the approach is used to describe the viscosity of a list of common hydrofluorocarbons used as refrigerants and a series of symmetric and asymmetric mixtures. The influence of an accurate description of the vapor−liquid equilibrium in nonideal azeotropic mixtures between n-alkanes and hydrofluorocarbons is discussed in detail. The last section is devoted to summarizing the major findings.

complexity to the problem, while industrial needs require modeling tools able to accurately reproduce the thermodynamic and transport properties of fluid mixtures. Considering that viscosity and density are properties somehow related (according to several theoretical approaches), it would be of interest to find or develop equations of state (EoS) able to simultaneously describe phase and transport properties, with the same degree of accuracy, and using the same underlying model. Pointing in this direction, molecular-based EoSs are a well-known class of equations that have proven to be very accurate in the representation of thermodynamic properties. The idea of using an accurate EoS in conjunction with a viscosity approach has already been explored in the literature for some viscosity models, such as the hard-sphere scheme,5,6 the free-volume theory (FVT),7,8 the friction theory (f-theory) and further developments,9−11 or the kinetic theory for chains, 12,13 just to mention some popular theoretical approaches. Other interesting methodologies are based on scaling relations with the residual (or excess) entropy,14−16 and on the development of an accurate equation from molecular dynamics simulations.17−20 For the particular case of combining a statistical mechanics based equation with a viscosity approach for mixtures, the number of contributions decreases drastically, although some challenging contributions can be found in the open literature. Tan et al. used the f-theory to estimate the viscosity of mixtures of n-alkanes with the SAFT1 and PCSAFT EoSs,21 while Quiñones-Cisneros and collaborators developed a generalized friction theory model for the PCSAFT EoS and applied it to n-alkanes and to mixtures between n-alkanes and with carbon dioxide.22 The free-volume theory was also used with the Sanchez−Lacombe equation for the decahydronaphthalene + propane mixture.23 In a very recent paper,1 we have summarized a list of contributions where the free-volume theory had been used to calculate the viscosity of pure compounds.7,8,24−31 As far as mixtures are concerned, the published work is not abundant and a short summary is provided here. Canet et al.32 measured and correlated the viscosity of 1-methylnaphthalene + 2,2,4,4,6,8,8-heptamethylnonane using several methods, including the FVT. The density was obtained from the Tait equation. With FVT, they obtained very good agreement for the viscosity of the pure fluids (average absolute deviation (AAD) of 2.43% and 1.84%, respectively) and reasonable agreement for the mixture (AAD of 10.6%). In another work from the same research group,33 the dynamic viscosity of two hydrocarbon mixtures representative of some heavy petroleum distillation cuts was measured and modeled with several treatments, including FVT. The density was measured and also correlated with the Tait equation. The agreement reached with the experiments was very good (less than 2% for a ternary mixture and less than 3.5% for a quinary mixture). Baylaucq and coworkers did a similar comparative study for the viscosity of methane + toluene,34 and methane + n-decane mixtures,34,35 reaching good agreement with the experimental data when the FVT was used (AAD of 4.95% and 5.25%, respectively). In this case, the density was obtained using the Lee-Kesler EoS. Monsalvo et al.36 also performed a similar analysis of different methods for the 1,1,1,2-tetrafluoroethane (R134a) + tetraethylene glycol dimethyl ether (TEGDME) mixture. In their work, the density values were taken from interpolated experimental results obtained in their laboratory. Again, the application of FVT provided very good agreement for the viscosity of the pure fluids (0.63% and 1.46%, respectively) but

2. THEORY In our previous contribution,1 the soft-SAFT EoS was briefly summarized and the free-volume theory was described in detail. In order to avoid repetition, only the main equations and the details concerning the extension to mixtures are explained here. 2.1. Soft-SAFT Equation of State. The soft-SAFT equation is an equation based on Wertheim’s first-order thermodynamic perturbation theory (TPT1),45−48 from which the total Helmholtz energy of the system is obtained as a sum of microscopic contributions a res = a − a id = a ref + achain + aassoc

(1)

where ares is the residual Helmholtz energy density, a and aid are the total and ideal Helmholtz energy density of the fluid of interest, and the superscripts ref, chain, and assoc represent, respectively, the reference, the chain, and the association contribution to the Helmholtz energy. The reference term in the soft-SAFT EoS accounts for the interactions among the monomers using a Lennard−Jones (LJ) intermolecular potential as the reference fluid, with two parameters characterizing the system: the segment (monomer) diameter σii and the dispersive energy between segments εii. B



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This term is described following the EoS of Johnson et al.49 For the case of mixtures, the same equation is used by applying the van der Waals one-fluid theory with generalized Lorentz− Berthelot mixing rules for the size and energy parameter of the monomers forming the chains:

⎛ σii + σjj ⎞ σij = ηij′⎜ ⎟ ⎝ 2 ⎠

(2)

εij = ξij(εiiεjj)1/2

(3)

mixtures is the total density of the system. The crossover treatment includes two additional parameters, the cutoff length, L, and the average gradient of the wavelet function, ϕ, who are extended to mixtures following compositional mixing rules: n

Lmixt =

i

⎛ Xα⎞ M aassoc = ρkBT ∑ xi ∑ ⎜ln Xiα − i ⎟ + i ⎝ 2 ⎠ 2 i α

φmixt =

1 1 + Navogρ ∑j xj ∑β Xjβ Δαiβj

(4)

(5)

η = η0 + Δη

εabHB, ij

=

(εabHB, iiεabHB, jj)1/2

(11)

where η0 is the viscosity of a dilute gas, where the intermolecular effects are neglected, and Δη is the dense-state correction term. The correction term vanishes when the fluid system approaches the dilute gas limit. In this work, viscosity is given in mPa·s. The dilute gas term can be calculated by means of Chung’s expression,58 a modification of the Chapman−Enskog kinetic theory59

(6)

The term Δ is related to the strength of the association bond between site α in molecule i and site β in molecule j, and contains the association energy parameter, εαiβj, and the association bonding volume parameter Kαiβj. For binary mixtures, cross-associating interactions are possible. The cross-interaction values for the volume and energy of association are described by the classical Lorentz−Berthelot combining rules using the mean arithmetic and geometric average of the pure component values, respectively: 3 HB ⎞ Kab , jj ⎟ ⎟ ⎠

(10)

The reader is referred to the original crossover soft-SAFT papers for more details about the theory and its implementation.39,40 2.2. Free-Volume Theory. The free-volume theory (FVT) used in this work was first formulated by Allal and co-workers in two publications that appeared in 2001.7,8 In the first one,7 the approach was only conceived to reproduce the viscosity of liquids, while the second contribution added a specific term for the diluted-gas phase.8 The principle of FVT was based on the connection between viscosity and the concept of empty space among the molecules and the diffusion models.56,57 This theory has been extensively described in our previous paper,1 and the reader is referred to it and to the original contributions of Allal et al. for more details.1,7,8 Only the main equations and details about the extension to mixtures are shown here. This approach expresses the viscosity as a sum of two contributions

αiβj

3

∑ φixi i=1

where ρ is the molecular density of the fluid, T is the temperature, kB is the Boltzmann constant, mi is the number of segments in component i, xi is the corresponding mole fraction, gLJ is the radial distribution function of the reference fluid at contact,50 Mi is the number of associating sites of component i, and Xαi the mole fraction of component i not bonded at site α, which accounts for the contributions of all associating sites in each species:

HB Κab , ij

(9)

n

achain = ρkBT ∑ xi(1 − mi) ln gLJ

⎛ 3 HB ⎜ Kab , ii + =⎜ 2 ⎝

∑ Li 3xi i=1

where ηij′ (do not confuse with the viscosity symbol) and ξij are the size and energy binary adjustable parameters, respectively. In the case of ideal or quasi-ideal mixtures, both values are set to unity. The chain term, achain, and the association term, aassoc, come from Wertheim’s TPT1 theory and they account for the energy of formation of chains from units of the reference fluid and for hydrogen bonding and other short-range, attractive interactions, respectively. Their general expressions are

Xiα =

3

η0 = 40.785 × 10−2

vc

M wT Fc Ω*(T *)

2/3

(12)

where Mw is the molecular weight (g/mol), vc is the critical volume (L/mol), and Ω* is the reduced collision integral. This integral can be obtained from the expression of Neufeld et al.60 for the LJ case evaluated at a dimensionless temperature T* = 1.2593Tr, with Tr being the reduced temperature respect to the critical temperature (Tc) of the compound. Finally, Fc is the empirical factor introduced by Chung et al.58

(7)

Fc = 1 − 0.2756ω − 0.059035μr 4 − κ

(8)

(13)

The correction includes three different factors that account for nonsphericity (ω), polarity (μr), and hydrogen bonding (κ), respectively. ω is the acentric factor of the compound, and κ is an empirical parameter to account for hydrogen-bonding formation. μr is a dimensionless dipole moment of the molecule, which is reduced using the critical volume (νc) and temperature (Tc) of the fluid with the following expression:

No binary adjustable parameters are used for the association contribution. The association term, summarized in eqs 5−8, is solved using the numerical procedure proposed by Tan et al.51 Finally, an additional contribution that considers the longrange fluctuations in the vicinity of the critical point is added by means of the renormalization group treatment based on White’s work.52−54 The extension to mixtures is done under the hypothesis of the isomorphism assumption,55 considering that the order parameter to describe vapor−liquid equilibria in

μr = C

13.13μ 0.1vcTc

(14) dx.doi.org/10.1021/jp401754r | J. Phys. Chem. B XXXX, XXX, XXX−XXX


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where μ is the dipole moment of the compound (D). All the molecules of this work are considered with κ = 0, either because they are nonassociating (n-alkanes), or because the polarity is dominant respect to hydrogen bonding (hydrofluorocarbons) and the parameter can be neglected. The dilute gas term is extended to mixtures using the expression proposed by Wilke,61 which only depends on the molecular weight and the composition of each compound. For the particular case of a binary mixture, the expression reads η0,1 η0mixt = (x 2 / x1)[1 + (η0,1 / η0,2)1/2 (M w2 / M w1)1/4 ]2 1+ 1/2

10−3M w 3RT 3/2 ⎤ ⎡ ⎛ 3 10 P + αρ2 M w ⎞ ⎥ ⎟ exp⎢B⎜ ⎢ ⎝ ρRT ⎠ ⎥⎦ ⎣

Δη = Lv (0.1P + 10−4αρ2 M w )

or, in an equivalent condensed free-volume-dependent expression ⎡ B⎤ 10−3RTM w −2/3 fv exp⎢ ⎥ ⎢⎣ fv ⎥⎦ 3

Δη = 10−4ρLv

(4 2 )[1 + (M w1 / M w2)] 1/2

1+

(x1 / x 2)[1 + (η0,2 / η0,1)

where Lv (Å) is L /bf. The thermodynamic variables, such as the pressure, the temperature, and the density, are obtained from soft-SAFT. The final equation includes three adjustable parameters: α describes the proportionality between the energy barrier and the density, B corresponds to the free-volume overlap, and Lv is a length parameter related to the structure of the molecules and the characteristic relaxation time. These parameters are fitted to available experimental viscosity data and related to the molecular weight of members of the same chemical family. The extension to mixtures depends on the evaluation of those three parameters (or the free-volume fraction f v) for mixtures. In the literature, there is not agreement about the best mixing rules to apply.23,32−37 As a consequence, we have decided to work with the simplest one, a linear compositional mixing rule of the Lorentz type for the three viscosity parameters of the dense fluid term:

1/4 2

(M w1 / M w2)

(4 2 )[1 + (M w2 / M w1)]1/2

]

(15)

with η0,1 and η0,2 being the viscosity of compounds 1 and 2, respectively, calculated from eq 12. The dense-state term is obtained through an expression that relates the viscosity to the microstructure of the fluid using a generalized dumbbell model Δη = 10−14ρNaζL2

(16)

and to the free space among the molecules, defined as a freevolume fraction through an exponential relation originally proposed by Doolittle:56 ⎛ B⎞ Δη = A exp⎜⎜ ⎟⎟ ⎝ fv ⎠

(17)

n

The combination of eqs 16 and 17 provides the following expression ⎛ B⎞ Δη = 10−14ρNaL2ζ0 exp⎜⎜ ⎟⎟ ⎝ fv ⎠

αmixt =

103P + αρM w ρ

(23)

n

Bmixt = (18)

∑ Bi xi i=1

(24)

n

Lv ,mixt =

∑ Lv,ixi i=1

(25)

The appropriateness of these simple rules will be assessed when comparing with experimental data. Hence, eqs 21 and 22 are readily applicable to mixtures using the mixture viscosity parameters from eqs 23−25. Here, it is important to notice that no binary adjustable parameters are used for the viscosity treatment of mixtures.

3. RESULTS AND DISCUSSION This section is divided in two parts. In the first subsection, the viscosity of a group of 12 hydrofluorocarbons is described in detail with the new approach. Subsection 3.2 is devoted to the study of three different types of binary mixtures. 3.1. Viscosity of Selected Hydrofluorocarbon Refrigerants. As done in our previous paper for the n-alkanes,1 the free-volume theory is applied here to reproduce the viscosity properties of the hydrofluorocarbon refrigerants (HFCs) family. We have chosen a group of 12 HFCs, including fluoromethanes, fluoroethanes and fluoropropanes, because of their interest in refrigeration processes and the considerable amount of experimental data available to validate the theory. In the original work of Allal and co-workers,7 the FVT approach was already used to successfully model the viscosity of one hydrofluorocarbon (R152) and two hydrochlorofluorocar-

(19)

with P being the pressure of the system (MPa) and α being the energy barrier that the molecule has to cross to diffuse (J·m3/ mol·kg). The ζ0 friction coefficient is related to the energy of dissipation E for a given length of dissipation bf (Å): 1/2 −3 E ⎛ 10 M w ⎞ ⎟ ⎜ ζ0 = 10 Nabf ⎝ 3RT ⎠

∑ αixi i=1

where ρ is the density (mol/L), Na is Avogadro’s number (mol−1), ζ0 is a friction coefficient related to the diffusion process and the mobility of the molecule (kg/s), L2 is an average quadratic length related to the size of the molecule (Å2), and B is a parameter related to the free-volume overlap among the molecules. As it can be observed, the friction coefficient is expressed in the exponential form and depends on the free-volume fraction f v. The free-volume fraction is defined as the ratio between the free molecular volume available vf and the total molecular volume v. This ratio is related to the potential energy of interaction E (J), considered as a sum of two terms, an ideal gas term and a second term directly related to the density

E=

(22)

2

η0,2

+

(21)

10

(20)

Combination of eqs 18−20 provides a final expression for the dense fluid term, in mPa·s D



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Table 1. Optimized Characteristic Parameters of the Free-Volume Theory for Some Selected Hydrofluorocarbonsa

a

HFC

formula

T range (K)

α (J·m3/mol·kg)

B

Lv (Å)

AAD (%)

R41 R32 R23 R152a R143a R134a R125 R245fa R245ca R236fa R236ea R227ea

CH3F CH2F2 CHF3 CHF2−CH3 CF3−CH3 CF3−CH2F CF3−CHF2 CF3−CH2−CHF2 CHF2−CF2−CH2F CF3−CH2−CF3 CF3−CHF−CHF2 CF3−CHF−CF3

180−300 200−340 180−295 200−300 200−300 200−300 200−300 250−420 250−440 220−370 250−400 200−300

34.60 34.60 34.60 41.43 41.43 41.43 41.43 53.90 53.90 53.90 53.90 53.90

0.01155 0.007194 0.003855 0.008938 0.006642 0.007173 0.004938 0.006187 0.007420 0.004578 0.005576 0.003516

0.6660 0.4078 0.3887 0.6107 0.4427 0.4185 0.4136 0.4876 0.4876 0.4454 0.4454 0.4288

0.987 0.839 0.964 0.430 1.920 0.879 0.280 1.030 1.930 2.170 1.920 1.370

The temperature range of optimization and the average absolute deviation (AAD) are also included.

(AAD) for each compound are indicated in Table 1. The results obtained deviate less than 2.20% from experimental data64 in all cases, with an overall AAD of 1.23%. As an example, the performance of soft-SAFT with FVT is depicted for some compounds and conditions in Figure 1. The viscosity at 1 and 5 MPa for R23, R32, and R41, three fluoromethanes, as a function of temperature, is shown in Figure 1a, while the viscosity of the fluoroethanes R125, R134a, and R152 at the same conditions is presented in Figure 1b. The description of the viscosity with the theory is in excellent

bons (R11 and R12), among other compounds. In addition, the very popular R134a has been studied with FVT,36 with the ftheory,11,36,62 and with the modified Yarranton-Satyro correlation.30 The f-theory was also used to describe the viscosity of R125, R152a, R32, and R143a.62 Here, we intend to broaden the modeling with FVT to a more complete list of common HFCs (see Table 1). First, a molecular model within the soft-SAFT framework has to be proposed for the HFCs. Following previous work,41 these molecules are described as homonuclear chainlike molecules, modeled as mi LJ segments of equal diameter σii, and the same dispersive energy εii, bonded to form the chain. Due to the electronegativity of the fluorine atom, these molecules show a high dipole moment. Since the associating interactions are due to the permanent dipoles, the directional forces between opposite partial charges can be mimicked considering two sites that interact with each other with a certain energy, εHB ij , and volume, KHB ij , of association. All the soft-SAFT parameters are taken from that contribution, as it was shown they are able to accurately reproduce the vapor−liquid phase diagram of those systems.41 The reader is referred to that work to see the description of the vapor−liquid equilibrium of those compounds with crossover soft-SAFT. Once the information required for an accurate representation of the thermodynamic properties is known, we proceed to optimize the viscosity parameters of the FVT using single-phase viscosity data. The information necessary to estimate the dilute term includes the critical constants, the dipole moment, and the acentric factor. This information has been taken from the NIST Reference Fluid Thermodynamic and Transport Properties Database, REFPROP.63 Concerning the dense term, and in order to search for transferability and decrease the multivariability of solutions, we decided to transfer the α parameter from that of the equivalent n-alkane with the same carbon number. The same hypothesis was also done for the segment number soft-SAFT parameter mi,41 without prejudicing the final results. Hence, only two parameters, B and Lv, have been fitted to viscosity data. Additionally, the correlation length parameter Lv has been set constant for the compounds with the same molecular weight (i.e., isomers R236ea-R236fa and R245caR245fa). In all cases, the range of temperature and pressure of the data used for the fitting has been chosen according to the conditions of applicability for refrigeration processes. In this case, two isobars at 1 and 5 MPa have been used for their optimization. The final set of viscosity parameters, the range of temperature conditions and the average absolute deviation

Figure 1. Viscosity as a function of temperature at a constant pressure of 1 MPa (lower lines) and 5 MPa (upper lines) of (a) R32 (○), R41 (△), and R23 (◊), and (b) R134a (○), R125a (△), and R143a (◊). Symbols represent the correlated experimental data,64 while the curves correspond to the soft-SAFT + FVT modeling. E



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effect of the fluorine atom is also decreasing the Lv parameter, although the values tend to reach a plateau. In all cases, the Lv values are significantly lower than the values obtained for the nalkanes family. 3.2. Viscosity of Binary Mixtures. We next present results concerning the ability of the soft-SAFT + FVT approach to predict the viscosity of binary mixtures of fluids. Although the free-volume theory has been applied for the calculation of the viscosity of mixtures,23,32−37 the amount of contributions is still limited and further assessments are needed. Moreover, as mentioned in the Theory section, there is not agreement about the appropriate mixing rules for the extension of the theory. Here, we propose very simple compositional rules for the three involved parameters (see eqs 23−25). 3.2.1. n-Alkane Mixtures. In this section, we have evaluated several mixtures composed of two n-alkanes of different chain lengths. In all cases, the soft-SAFT + FVT parameters that characterize these compounds are taken from previous work.1,39 First, we have calculated the viscosity of binary mixtures of nalkanes of similar size. In this case, the vapor−liquid equilibrium can be described from pure component parameters, without the use of any binary adjustable parameter (η′ and ξ in eqs 2 and 3 are equal to unity). We have verified that the density (not shown here) is accurately represented without further fitting. In Figure 3a, the viscosity of a n-pentane + noctane mixture as a function of temperature and a constant pressure of 10 MPa is depicted. The lines represent the viscosity calculated using the free-volume theory. The top and bottom lines correspond to the pure n-octane and n-pentane compounds, respectively, while the intermediate lines are, from top to bottom, calculations with a 50.7% and 81.0% of npentane, respectively. Good agreement between the predictions and the experimental data65 is achieved in the whole range of temperatures, with an AAD of 5.26%. The soft-SAFT + FVT values are slightly lower than the experiments, but still they represent a very good approximation, more taking into account that this is a full prediction from the pure component parameters. It is interesting to note that the methodology is able to capture the behavior of the viscosity reproducing the same shape experimentally observed. A similar conclusion can be obtained from the results shown in Figure 3b, representing the viscosity of a n-octane + n-decane mixture as a function of temperature and a constant pressure of 10 MPa. Once again, the intermediate calculated compositions at 45.2% and 82.1% of n-octane are in very good agreement with the experimental data (AAD of 3.55%).66 We have also performed other calculations at lower (5 MPa) and higher (15 MPa) pressures for these two mixtures (not shown here), and the results are of the same nature as those described in Figure 3. The soft-SAFT + FVT approach is now used to calculate the viscosity of some additional n-alkane mixtures where the difference in chain length between the two compounds is more significant. Previous studies have shown that usually one or two binary parameters are necessary for an accurate description of the vapor−liquid equilibrium of asymmetric mixtures and, consequently, for the density. First of all, we have modeled the n-pentane + n-decane system, for which viscosity data were available.67 The study of the vapor−liquid equilibria has revealed that the application of the Lorentz−Berthelot combining rules still provides good agreement with the experimental data,68 (i.e., no fitting to mixture is needed) as it can be concluded from observation of Figure 4a, where the vapor−liquid equilibrium of this mixture at two different

agreement with the experimental data at both isobars and in all the range of temperatures. An analysis of the trends of the viscosity parameters has been done for all the hydrofluorocarbons. Here, it has not been possible to establish correlations with the molecular weight for each parameter, as it was done for the n-alkanes in previous work,1 as the compounds involved differ in the number of fluorine atoms; but some trends can be identified (see the values in Table 1). In Figure 2, the parameters B and Lv are

Figure 2. Trends of the viscosity parameters (a) B and (b) Lv with the molecular weight for the list of hydrofluorocarbons studied in this work. Blue circles correspond to fluoromethanes, red diamonds to fluoroethanes, and green squares are fluoropropanes. Dashed curves are included to guide reader’s eye.

shown as a function of the molecular weight. The parameter α is not included as it has been transferred from the n-alkanes. In both graphics (Figure 2, a and b), the blue circles represent the fluoromethanes, the red diamonds are the fluoroethanes, and the green squares are the fluoropropanes. In general, the freevolume overlap parameter B decreases as the molecular weight of the compound increases (as observed for the n-alkanes). It is worth noting that the decrease of the B parameter is more pronounced because of the inclusion of the fluorine atoms in the molecule than due to the increase of the number of carbon atoms. This observation is in agreement with the concept of the free-volume overlap, which should decrease drastically if a hydrogen atom is substituted by a fluorine atom in the molecule. Similar conclusions arise from the study of Figure 2b, which shows the correlation length Lv parameter. As for the nalkanes, this value decreases with the molecular weight. The F



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Figure 3. Viscosity as a function of temperature at a constant pressure of 10 MPa of (a) n-pentane + n-octane at a composition, from top to bottom, of 0 (○), 0.5066 (◊), 0.8097 (□), and 1 (∗) of n-pentane; and (b) n-octane + n-decane binary mixture, at a composition, from top to bottom, of 0 (○), 0.4520 (◊), 0.8208 (□), and 1 (∗) of noctane. Symbols represent the experimental data,65,66 while the curves correspond to the soft-SAFT + FVT modeling.

Figure 4. Study of the n-pentane + n-decane mixture. (a) Vapor− liquid equilibrium at 317.7 K (□) and 333.7 K (○), and (b) singlephase liquid viscosity at 10 MPa and a composition, from top to bottom, of 0 (○), 0.5061 (◊), 0.8040 (□), and 1 (∗) of n-pentane. Symbols represent the experimental data,67,68 while the curves correspond to the soft-SAFT + FVT modeling.

temperatures is plotted. Hence, even if the degree of asymmetry is higher than in the previous cases, it is not necessary to use any binary parameter. The viscosity of the n-pentane + ndecane mixture at 10 MPa at two intermediate compositions of 50.6% and 80.4% of n-pentane is shown in Figure 4b. Interestingly, the predicted viscosity is again in good agreement (AAD of 5.32%) with the experimental data,67 and the methodology does not apparently suffer from a loss of accuracy. In Figure 5, the viscosity of the asymmetric n-decane + neicosane mixture at atmospheric pressure is represented. In this case, the source of viscosity experimental data69 also contains the density value of the mixture at the same conditions. Hence, we have used this information to fit the binary parameters parameters η′ and ξ to the density of three isopleths containing 20%, 50%, and 80% of n-decane. Only one binary parameter, η′, was needed to obtain a good description of the density (η′ = 1.018). In Figure 5a, the temperature−density diagram at these compositions is shown. Once the density has been adjusted, the viscosity is predicted for the same three isopleths with the FVT and compared to the experimental data (see Figure 5b). Even if the degree of asymmetry is noticeable, the predictions obtained still provide good results (AAD of 5.16%). In fact, it is important to remind that we had already been able to identify a trend of the viscosity parameters with the molecular weight for the calculation of heavier compounds.1 The current results

confirm the validity of the approach to estimate the viscosity of binary mixtures of n-alkanes of different lengths. 3.2.2. Hydrofluorocarbon Mixtures. The ability of the theory to predict the viscosity of binary mixtures of hydrofluorocarbons is presented next. The study of these mixtures, independently of their industrial interest, will also allow addressing the validity of the assumptions that have been made previously (value of α transferred from that of the equivalent n-alkane with the same number of carbons, and independent of the number of fluorine atoms). In Figure 6, we show some results for two mixtures of refrigerants. Figure 6a depicts the saturated liquid viscosity for a mixture of two fluoroethanes, R125 + R134a. The figure includes the viscosity at equilibrium of both pure fluids (top and bottom lines) and at two intermediate compositions of 30% and 70% of R125. In a previous work,41 it was shown that the vapor−liquid equilibrium of this mixture was well predicted with the current soft-SAFT models, and no binary parameters were necessary to account for asymmetry, differences in size and energy between both molecules. In terms of viscosity, there is also an excellent agreement with the experimental data70 and the predictions from the model, with an AAD of only 1.76%, even if no adjustable mixture parameters were used. G



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Figure 5. Study of the n-decane + n-eicosane mixture. (a) Density− temperature and (b) viscosity−temperature diagrams at atmospheric pressure, at a composition of 0 (○), 0.20 (◊), 0.50 (□), 0.80 (△), and 1 (∗) of n-decane. Symbols represent the experimental data,69 while the curves correspond to the soft-SAFT + FVT modeling.

Figure 6. Saturated liquid viscosity as a function of temperature at different isopleths for binary mixtures: (a) R125 (1) + R134a (2), and (b) R32 (1) + R134a (2). Using compound 1 as a reference, isopleths are plotted, from top to bottom, at a composition 0 (○), 0.30 (◊), 0.70 (□), and 1 (∗). Symbols represent the experimental data,70 while the curves correspond to the soft-SAFT + FVT modeling.

Additionally, we have also studied the viscosity of two refrigerants with different number of carbons. Figure 6b is devoted to the saturated liquid viscosity of a mixture of R32 + R134a, at the same conditions of the previous figure. The mixture is calculated without any binary parameter correction as the vapor−liquid equilibrium (not shown here) is accurately reproduced from the pure molecular soft-SAFT parameters. The reader is referred to our work on refrigerants,41 where the VLE of a very similar mixture (R23 + R134a) was reproduced in excellent agreement with the experimental data. Similar conclusions to Figure 6a can be highlighted: the viscosity is accurately predicted at both compositions (AAD of 1.54%) in the whole range of temperatures studied. 3.2.3. n-Alkane + Hydrofluorocarbon Mixtures. Finally, in this section we show the capability of the method to reproduce the viscosity of binary mixtures of two compounds of different chemical families. For that purpose, we have studied two mixtures of propane with R32 and R134a, respectively. Results concerning the mixture of propane and R32 are provided in Figure 7. As this is a mixture we had not studied previously with soft-SAFT, we have first calculated the vapor− liquid equilibria before proceeding to the calculation of the viscosity. Those mixtures are nonideal, and they require the help of binary adjustable parameters to quantitatively describe the VLE behavior.41

In Figure 7a, the VLE of this mixture at two equilibrium isotherms (298 and 313 K) is depicted, while in Figure 7b predictions for the liquid saturated viscosity of the same mixture at two isopleths of 35% and 65% of R32 are shown. Given the highly nonideal behavior of the mixture, the viscosity of the mixture falls below the viscosity of both pure fluids in a wide range of compositions. This feature is accurately reproduced with soft-SAFT + FVT using two binary parameters (η′ and ξ) obtained from adjusting to phase equilibrium data. In a previous work,41 it was observed that mixtures of hydrofluorocarbons with n-alkanes were well described by fitting only the energy binary parameter ξ. In this case, it was also possible to obtain excellent agreement with the experimental data71 for the VLE with a value of ξ = 1.050, the same one at both temperatures. However, the predicted viscosities (not shown here) were not describing the correct experimental behavior. In a second attempt, the size binary adjustable parameter η′ was also used and an accurate description of the VLE was achieved using η′ = 1.040 (with ξ = 1.035) as shown in Figure 7a. In this second case, the predicted viscosities were in very good agreement with the experimental information (Figure 7b), with an overall AAD of 1.93%.70 This highlights the direct relation between the density and the viscosity. The η′ parameter accounts for corrections on H



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Figure 7. R32 (1) + propane (2) mixture: (a) vapor−liquid equilibrium at 254.15 K (□) and 273.15 K (○) and (b) saturated liquid viscosity as a function of temperature at different isopleths at a composition of 0 (○), 0.30 (◊), 0.70 (□), and 1 (∗) of propane. Symbols represent the experimental data for the VLE71 and the viscosity,70 while the curves correspond to the soft-SAFT + FVT modeling.

Figure 8. R134 (1) + propane (2) mixture: (a) vapor−liquid equilibrium at 263.15 K (□) and 273.15 K (○), and (b) saturated liquid viscosity as a function of temperature at different isopleths at a composition of 0 (○), 0.35 (◊), 0.65 (□), and 1 (∗) of propane. Symbols represent the experimental data for the VLE72 and the viscosity,70 while the curves correspond to the soft-SAFT + FVT modeling.

the average segment diameter of the mixture and it affects the density of the system in a direct manner. As expected, a more accurate representation of the density of the mixture allows a better representation of the viscosity. Finally, Figure 8 shows the phase equilibrium and the viscosity of a mixture of R134 + propane. The same procedure described for R32 + propane was used here, and similar results were obtained. Two binary parameters ξ and η′ were used, prioritizing the use of η′ to fit the VLE of this mixture, described in Figure 8a. Good agreement with the experimental values72 is reached using a value of η′ = 1.075 (with ξ = 1.015) for the VLE. Then, the saturated liquid viscosities at two isopleths of 35% and 65% of R134 are predicted (see Figure 8b), finding again good agreement with the experimental data,70 although the deviation is slightly higher (AAD = 5.48%) than in the previous mixture.

The FVT has been first used to model the viscosity of a group of 12 hydrofluorocarbons, commonly used in refrigeration processes. The soft-SAFT molecular parameters describing the vapor−liquid equilibrium were taken from previous work and they were used to calculate the density and pressure of the system, as an input for the viscosity evaluation. In order to exploit the transferability of the approach, only two of the three FVT viscosity parameters were fitted to viscosity data, as the barrier energy α parameter could be transferred from that of the equivalent n-alkane (the one with the same carbon number). Very good agreement with the experimental data has been found for all cases in a wide range of temperature and pressure, with an overall AAD of 1.23%. Then, the theory has been applied to the prediction of the viscosity of n-alkanes and hydrofluorocarbons mixtures. Very good agreement with the experimental data has been found for nearly symmetric (n-pentane + n-octane and n-octane + ndecane) and asymmetric (n-pentane + n-decane and n-decane + n-eicosane) mixtures of n-alkanes, with AADs ranging between 3.55% and 5.32%. In a similar manner, excellent agreement with experimental data (AADs below 2%) has been obtained for mixtures between two hydrofluorocarbons, without using any information from the mixture. Finally, nonideal n-alkane + nhydrofluorocarbon mixtures have also been described. In this case, it was necessary to adjust the binary size and energy

4. CONCLUSIONS The soft-SAFT + free-volume-theory approach has been extended to the calculation of mixtures. The methodology of Allal and co-workers has been implemented and extended using a compositional mixing rule for the diluted term and simple mixing rules for the viscosity parameters of the dense term. No fitting binary parameters have been added in the FVT, making the theory predictive for transport properties. I



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(14) Goel, T.; Patra, C. N.; Mukherjee, T.; Chakravarty, C. Excess Entropy Scaling of Transport Properties of Lennard-Jones Chains. J. Chem. Phys. 2008, 129, 164904. (15) Galliero, G.; Boned, C.; Fernández, J. Scaling of the Viscosity of the Lennard-Jones Chain Fluid Model, Argon and Some Normal Alkanes. J. Chem. Phys. 2011, 134, 064505. (16) Novak, L. T. Fluid Viscosity-Residual Entropy Correlation. Int. J. Chem. React. Eng. 2011, 9, A107. (17) Galliero, G.; Boned, C.; Baylaucq, A. Molecular Dynamics Study of the Lennard−Jones Fluid Viscosity: Application to Real Fluids. Ind. Eng. Chem. Res. 2005, 44, 6963−6972. (18) Galliero, G.; Boned, C. Shear Viscosity of the Lennard-Jones Chain Fluid in its Gaseous, Supercritical, and Liquid States. Phys. Rev. E 2009, 79, 021201. (19) Zabaloy, M. S.; Machado, J. M. V.; Macedo, E. A. A Study of Lennard-Jones Equivalent Analytical Relationships for Modeling Viscosities. Int. J. Thermophys. 2001, 22, 829−858. (20) Zabaloy, M. S.; Vasquez, R. V.; Macedo, E. A. Viscosity of Pure Supercritical Fluids. J. Sup. Fluids. 2005, 36, 106−117. (21) Tan, S. P.; Adidharma, H.; Towler, B. F.; Radosz, M. Friction Theory Coupled with Statistical Associating Fluid Theory for Estimating the Viscosity of n-Alkane Mixtures. Ind. Eng. Chem. Res. 2006, 45, 2116−2122. (22) Quiñones-Cisneros, S. E.; Zéberg-Mikkelsen, C. K.; Fernández, J.; García, J. General Friction Theory Viscosity Model for the PCSAFT Equation of State. AIChE J. 2006, 52, 1600−1610. (23) Cain, N.; Roberts, G.; Kiserov, D.; Carbonell, R. Modeling the Thermodynamic and Transport Properties of Decahydronaphthalene/ Propane Mixtures: Phase Equilibria, Density, and Viscosity. Fluid Phase Equilib. 2011, 305, 25−33. (24) Tan, S. P.; Adidharma, H.; Fowler, B. F.; Radosz, M. Friction Theory and Free-Volume Theory Coupled with Statistical Associating Fluid Theory for Estimating the Viscosity of Pure n-Alkanes. Ind. Eng. Chem. Res. 2005, 44, 8409−8418. (25) Boned, C.; Allal, A.; Baylaucq, A.; Zéberg-Mikkelsen, C. K.; Bessières, D.; Quiñones-Cisneros, S. E. Simultaneous Free-Volume Modeling of the Self-Diffusion Coefficient and Dynamic Viscosity at High Pressure. Phys. Rev. E 2004, 69, 031203. (26) Baylaucq, A.; Comuñas, M. J. P.; Boned, C.; Allal, A.; Fernández, J. High Pressure Viscosity and Density Modeling of Two Polyethers and Two Dialkyl Carbonates. Fluid Phase Equilib. 2002, 199, 249−263. (27) Zéberg-Mikkelsen, C. K.; Baylaucq, A.; Barrouhou, M.; Boned, C. A Study of cis-Decalin and trans-Decalin Versus Pressure and Temperature. Phys. Chem. Chem. Phys. 2003, 5, 1547−1551. (28) Comuñas, M. J. P.; Baylaucq, A.; Plantier, F.; Boned, C.; Fernández, J. Influence of the Number of CH2−CH2−O Groups on the Viscosity of Polyethylene Glycol Dimethyl Ethers at High Pressure. Fluid Phase Equilib. 2004, 222−223, 331−338. (29) Reghem, P.; Baylaucq, A.; Comuñas, M. J. P.; Fernández, J.; Boned, C. Influence of the Molecular Structure on the Viscosity of Some Alkoxyethanols. Fluid Phase Equilib. 2005, 236, 229−236. (30) Polishuk, I. Modeling of Viscosities in Extended Pressure Range Using SAFT + Cubic EoS and Modified Yarranton-Satyro Correlation. Ind. Eng. Chem. Res. 2012, 51, 13527−13537. (31) Burgess, W. A.; Tapriyal, D.; Gamwo, I. K.; Morreale, B. D.; McHugh, M. A.; Enick, R. M. Viscosity Models Based on the Free Volume and Frictional Theories for Systems at Pressures to 276 MPa and Temperatures to 533 K. Ind. Eng. Chem. Res. 2012, 51, 16721− 16733. (32) Canet, X.; Daugé, P.; Baylaucq, A.; Boned, C.; ZébergMikkelsen, C. K.; Quiñones-Cisneros, S. E.; Stenby, E. H. Density and Viscosity of the 1-Methylnaphthalene + 2,2,4,4,6,8,8-Heptamethylnonane System from 293.15 to 353.15 K at Pressures up to 100 MPa. Int. J. Thermophys. 2001, 22, 1669−1689. (33) Boned, C.; Zéberg-Mikkelsen, C. K.; Baylaucq, A.; Daugé, P. High-Pressure Dynamic Viscosity and Density of Two Synthetic Hydrocarbon Mixtures Representative of Some Heavy Petroleum Distillation Cuts. Fluid Phase Equilib. 2003, 212, 143−164.

parameters to VLE equilibria mixture data to later proceed to the evaluation of the viscosity. It has been noted that, if the size binary parameter η′ is used to fit the VLE, a better description of the nonideal viscosity behavior is achieved. In summary, this work shows that the combination of a sound molecular-based equation of state such as soft-SAFT, together with FVT, allows an accurate prediction of both thermodynamic and transport properties of industrially relevant mixtures using relatively simple molecular models.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: fl[email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS F.L. acknowledges a TALENT fellowship from the Generalitat de Catalunya. This work was partially financed by the Spanish Government under projects CTQ2008-05370/PPQ and CENIT SOST-CO2, CEN-2008-1027. Additional support from Carburos Metálicos, Air Products Group, and the Catalan Government was also provided (2009SGR-666).

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REFERENCES

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Transport Properties of Mixtures by the Soft-SAFT + Free-Volume

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