6998
Ind. Eng. Chem. Res. 2007, 46, 6998-7007
GENERAL RESEARCH Accurate Global Thermophysical Characterization of Hydrofluoroethers through a Statistical Associating Fluid Theory Variable Range Approach, Based on New Experimental High-Pressure Volumetric and Acoustic Data Thomas Lafitte,*,† Frederic Plantier,† Manuel M. Pin˜ eiro,‡ Jean-Luc Daridon,† and David Bessie` res*,† Laboratoire des Fluides Complexes, UniVersite´ de Pau et des Pays de l’Adour, B.P. 1155, 64013 Pau Cedex, France, and Departamento de Fı´sica Aplicada, Facultade de Ciencias, UniVersidade de Vigo, E-36310, Vigo, Spain
In this work, the ability of a recently proposed statistical associating fluid theory of variable range (SAFT VR) version to estimate thermophysical properties of fluorinated compounds is demonstrated, focusing specially on second-order derivative properties. These properties are fundamental in the simulation and design of the industrial processes where these families of compounds find application as working fluids, but only very recently has their estimation been accomplished with the desired degree of accuracy, as they have been traditionally considered as a severely stringent test to any thermodynamic model. With this aim, a complete thermophysical characterization of two hydrofluoroethers, included among the so-called third generation of chlorofluorocarbon alternatives, with low global warming potential and zero ozone depletion potential, are presented. To achieve this goal, measurements of compressed-liquid densities and the speed of sound were performed in high-pressure conditions and set as the basis for the determination of molecular parameters of a version of the statistical associating fluid theory for chain molecules with attractive potentials of variable range (SAFT VR) equation of state. This coupling between accurate experimental determination of high pressures, density, and the speed of sound and the calculations of characteristic molecular parameters in the framework of a physically sound molecular model allows a complete description of the thermophysical behavior of the pure fluids studied, providing a precise simultaneous estimation of phase equilibria and other first- and second-order derivative properties, ensuring the reliability of the proposed characterization procedure. The relevance of the accurate experimental data is to be emphasized, as without the right experimental input, the applied model, accurate as it eventually is shown to be, would never achieve its maximum performance, as it will be discussed in the following. 1. Introduction The widespread use of chlorofluorocarbons (CFCs) in many industrial applications and processes since the 1930s was shown to carry undesired and harmful side effects in 1974, when Molina and Rowland1 demonstrated that the liberated inorganic chlorine molecules that reached the stratosphere promoted as catalysts the reaction of ozone molecules with the incident ultraviolet radiation. As a result, a remarkable seasonal thinning of the ozone layer could be monitored, mainly in the immediacy of the South Pole. As this effect would carry serious long-term consequences on the atmosphere’s equilibrium, modifying dramatically environmental conditions on earth, an intensive research work started at that moment with the aim to reduce and eventually phase out the use of these substances. In addition to this ozone depleting potential (ODP), CFCs were identified as contributors to the so-called greenhouse effect due to their high global warming potential (GWP), thus favoring the atmosphere heating, a process that has been demonstrated to * To whom correspondence should be addressed. E-mail: t.lafitte@ etud.univ-pau.fr (T.L.);
[email protected] (D.B.). † Universite ´ de Pau et des Pays de l’Adour. ‡ Universidade de Vigo.
be leading to a global climate change which is bound to carry future fearsome consequences. The solution to this problem should entail the identification of working substances suitable to replace CFCs in the applications where they were being applied so far, as for instance in refrigeration, foaming, dry cleaning solvents, aerosol propellants, or fire extinguishers. The Montreal Protocol, first issued in 1987 and amended later in London in 1990 and Copenhagen in 1992,2 established the trends to be followed, as for instance the setting of 1996 as the deadline for the production of CFCs. A first step toward the replacement of CFCs was the adoption of hydrochlorofluorocarbons (HCFCs) as working fluids, but their chlorine content, although less harmful than that of CFCs, forced their international ban, which will be enforced starting in 2020. Other families of halogenated hydrocarbons that have found successful applications are hydrofluorocarbons (HFCs) and perfluorocarbons (PFCs), but their long atmospheric life and still high GWPs prevented their adoption as definitive solutions, reinforcing the need for further research efforts. The so-called third generation of CFC alternatives has been recently proposed, consisting of fluorinated ethers, and they are referred to usually as hydrofluoroethers (HFEs). These substances have been identified as environmentally friendly sub-
10.1021/ie0700462 CCC: $37.00 © 2007 American Chemical Society Published on Web 09/14/2007
Ind. Eng. Chem. Res., Vol. 46, No. 21, 2007 6999 Table 1. Experimental Speed of Sound c (m‚s-1) for HFE 7000 and HFE 7500 T (K) P (MPa)
283.15
293.15
303.15
313.15
323.15
333.15
343.15
353.15
363.15
373.15
539.9 610.2 667.9 718.2 763.1 804.1 841.8 876.9 909.8 940.6
512.2 586.0 646.1 697.8 743.6 785.2 823.6 859.3 892.6 924.1
485.8 563.0 625.0 678.1 724.8 767.3 806.2 842.4 876.3 908.0
460.5 541.0 605.0 659.1 706.9 750.2 789.3 826.0 860.2 892.3
436.7 520.7 586.4 641.7 690.1 733.9 774.0 811.0 845.5 877.7
562.2 631.2 688.6 738.7 783.3 824.2 861.6 896.7 928.9 959.9 989.0
535.3 608.0 667.4 719.0 764.9 806.3 844.4 879.7 913.0 943.9 972.5
509.5 585.9 647.6 700.3 746.9 789.0 828.1 864.0 897.5 929.4 959.3
483.7 564.4 628.0 682.1 729.9 773.1 812.4 848.9 882.9 914.9 945.3
458.1 542.9 608.7 664.6 713.3 757.1 797.0 834.0 868.5 900.8 931.4
HFE 7000 0.1 10 20 30 40 50 60 70 80 90 100
621.8 687.0 742.1 790.8 834.0 873.9 910.7 944.7 976.9 1007.7 1036.0
656.1 713.8 764.7 809.3 850.2 887.7 922.6 955.5 986.6 1015.8
626.1 686.8 739.0 785.3 827.3 865.8 901.6 935.0 966.6 995.9
596.2 659.9 714.1 761.8 805.0 844.3 880.8 915.0 947.0 977.1
0.1 10 20 30 40 50 60 70 80 90 100
703.0 756.8 805.0 848.3 887.2 923.1 956.8 988.4 1017.6 1045.6 1072.6
673.6 730.4 780.3 824.7 865.2 902.3 937.1 969.5 1000.1 1028.9 1055.2
643.8 704.1 755.6 801.6 843.4 881.6 917.1 950.3 981.3 1010.7 1038.4
616.7 679.0 732.5 780.1 822.7 861.6 897.5 931.9 964.0 993.3 1021.3
567.5 634.3 690.3 739.5 783.4 823.4 860.7 895.5 927.9 958.6 HFE 7500 589.1 654.7 710.4 759.1 802.8 842.8 879.4 914.0 946.2 976.3 1003.9
stitutes to other halogenated compounds. Over the past few years, many papers addressing the suitability of HFEs for practical purposes have been published.3-5 Among the applications where this type of products are already being used, the cleaning of electronic and magnetic devices, foaming, dry etching, or low temperature heat exchange refrigerating can be outlined. The main reasons for supporting their use are their zero ODP, low GWP, short atmospheric lifetimes, and very low toxicity. Tsai6 has discussed thoroughly this last fact, including the properties mentioned above together with their low flammability and reduced contribution to photochemical smog. Newsted et al.7 studied the toxicity and ecological risk associated to the HFE 7500, one of the fluids considered in the present study, concluding that this compound poses no risk to either aquatic or terrestrial wildlife and it does not represent a threat to human health. Among this family of compounds, those known as segregated HFEs (where an ether functional group links an aliphatic and a perfluorinated chain) have been pointed out for their useful profile, and some of them are already in use. Nevertheless, accurate experimental thermophysical data are still scarce in the literature, which prevents an adequate description of their complete thermodynamic behavior, an essential aim in order to explore their further practical applications through the usual process simulation techniques. In previous works, our group has presented experimental high-pressure densities8 and speeds of sound9 for methyl and ethyl nonafluorobutyl ether (commercially produced under the denominations of HFE 7100 and HFE 7200). The mixing volumetric behavior of these compounds has been studied as well as for the mixtures methyl nonafluobutyl ether + n-hexane10 and ethyl nonafluobutyl ether + n-hexane.11 From a theoretical perspective, an article describing the saturated and compressed liquid volumetric behavior of pure segregated HFEs using the perturbed chain statistical associating fluid theory (PC-SAFT) equation of state (EOS) has been also published.12 From an experimental point of view, this work is focused on the thermophysical characterization of methyl perfluorobutyl ether (CAS no. 375-03-1) and 2-trifluoromethyl 3-ethoxy dodecofluorohexane (CAS no. 297730-93-9). These chemicals
are produced by 3M under the commercial denominations of Novec engineering fluids HFE 7000 and HFE 7500, respectively, and will be referred to hereafter with these names for the sake of brevity. With the objective of obtaining a complete thermophysical characterization of these fluids, we have considered the combination of the experimental determination of two magnitudes (density and speed of sound), that can be determined with a high degree of accuracy, and a highly accurate molecular based equation of state (EoS) as a theoretical model. The use of this EoS will allow obtaining characteristic molecular parameters from the set of measured experimental data and so estimate accurately other thermophysical properties and phase equilibria of the fluids studied, showing the reliability of this combined experimental and theoretical characterization. The interest of the theoretical approach selected in this case arises from the fact that once their parameters are finely tuned to accurate experimental data, they can be used with confidence to predict the behavior of real fluids over wide temperature and pressure ranges keeping a remarkable accuracy. Among the different approaches developed over the past few years, suitable for characterization of such complex fluids, the statistical associating fluid theory (SAFT)13 stands undoubtedly as one of the most promising tools. A reason to support the interest of this approach in this particular application is that the theory is able to account for the specific effects of both the molecular shape and anisotropies due to directional forces; two features present in segregated HFEs which prevent most classical engineering EOSs from yielding good estimations for these complex substances and their mixtures. Moreover, it should be mentioned that several modifications and improvements of the original SAFT theory have been proposed so far, although all these versions have been developed from the same underlying concept. Among the most recent SAFT versions, the perturbed chain (PC-SAFT),14 soft-SAFT,15 and the variable range (SAFT VR)16 variants may be cited. The reader can find a comprehensive description of these equations in the reviews of Economou17 and Mu¨ller and Gubbins.18 To our knowledge, the only related publications of SAFTVR dealing with HFEs is attributed to Swaminathan et al.19 who
7000
Ind. Eng. Chem. Res., Vol. 46, No. 21, 2007
used the SAFT-VR approach to model the vapor pressure and saturated liquid densities over a wide variety of refrigerant following the same methodology as proposed by Galindo et al.20 for the case of hydrofluorocarbons (HFCs). In both studies, the molecular parameters were fitted to available experimental vapor pressure and saturated liquid densities providing in each cas accurate estimations of phase equilibria of the pure fluids and their mixtures. However, the possibility to extend the modeling to other aspects of the thermodynamic behavior of HFEs, such as derivative properties (isobaric heat capacity, speed of sound, etc.), has not been studied so far. This is then one of the main goals of this work: the analysis of the predictive capability of the SAFT theory to derive a complete thermodynamic description of these complex fluids (phase equilibria and secondderivative properties). Let us remind at this point that HFEs show particular features as very high density and low speed of sound values, which make most classical EoSs unsuitable for their modeling. Related to this it has been recently shown 21 22 that if an accurate theoretical description of the fluid is to be calculated from any theoretical method, the inclusion of acoustic experimental data in the fitting process allows characteristic parameters that yield a remarkably better estimation of secondderivative properties to be obtained, especially those involving pressure derivatives of the Gibbs potential. Following these conclusions, we will show that an experimental determination of high-pressure volumetric and acoustic properties enables a complete thermophysical characterization of HFEs and postulate the feasibility of extension to other fluorinated compounds, opening a wide range of applicability. Concerning the novel estimation of second-derivative properties with a SAFT approach, the works of Llovell et al. 23,24 must be cited as they achieve the same objective through an alternative method, using the well-known soft-SAFT15 EoS. Both approaches, although using different theoretical foundations, stress the critical influence of the potential considered to model the interaction among the chain segments in the molecule, and following both paths, accurate results for the secondderivative properties of both pure compounds and mixtures are obtained. So far, the application of these models to estimation of second-derivative properties of fluorinated compounds had not been envisaged, and in this paper, the ability to capture their peculiar behavior is underlined. 2. Experimental Section Heptafluoropropyl methyl ether and 2-trifluoromethyl-3ethoxydodecafluorohexane (commonly named HFE 7000 and HFE 7500) were supplied by 3M with a purity higher than 99.5% and 99% respectively. Both chemicals were degassed using an ultrasound technique, dried over molecular sieves, and then used without further purification. The speed of sound was measured up to 100 MPa using a fixed path pulse echo technique operating at 3 MHz. The apparatus, which has been described previously in detail,25 is essentially made up of an autoclave cell closed at both ends by two identical piezoelectric transducers. The experimental technique which rests on the measurement by direct chronometry of the travelling time of the wave through the sample and which used calibration with water to determine the length of the sample path has an overall accuracy better of 0.2%. Measurements were carried out in the temperature range 283-373 K from atmospheric pressure up to 100 MPa using 10 K and 10 MPa steps. As the normal boiling temperature of HFE 7000 has been reported to be 307.38 K,26 density values above this temperature
Table 2. Experimental Density G (g‚cm-3) for for HFE 7000 and HFE 7500 T (K) P (MPa)
283.15
293.15
303.15
313.15
323.15
333.15
343.15
1.3434 1.3752 1.4109 1.4401 1.4648 1.486 1.505 1.522 1.538 1.553 1.566
1.3145 1.3496 1.3884 1.4197 1.4460 1.468 1.488 1.506 1.522 1.538 1.552
1.2849 1.3240 1.3664 1.3998 1.4279 1.451 1.472 1.490 1.507 1.523 1.537
1.5666 1.5971 1.6223 1.6442 1.6635 1.681 1.696 1.711 1.724 1.737 1.748 1.5666
1.5450 1.5777 1.6049 1.6277 1.6479 1.666 1.682 1.697 1.711 1.724 1.736 1.5450
1.5229 1.5587 1.5874 1.6118 1.6329 1.652 1.669 1.684 1.698 1.712 1.724 1.5229
0.1 2.5 10 20 30 40 50 60 70 80 90 100
1.4443 1.4519 1.4737 1.4992 1.5213 1.5406 1.558 1.574 1.589 1.602 1.615 1.627
1.4173 1.4254 1.4492 1.4770 1.5005 1.5211 1.539 1.556 1.571 1.586 1.599 1.611
HFE 7000 1.3895 1.3983 1.3709 1.4246 1.4000 1.4548 1.4327 1.4799 1.4598 1.5019 1.4831 1.521 1.503 1.539 1.522 1.555 1.538 1.569 1.553 1.583 1.568 1.596 1.581
0.1 2.5 10 20 30 40 50 60 70 80 90 100
1.6511 1.6731 1.6930 1.7103 1.7262 1.741 1.754 1.766 1.778 1.789 1.799 1.6511
1.6302 1.6542 1.6752 1.6937 1.7103 1.725 1.739 1.752 1.764 1.775 1.786 1.6302
HFE 7500 1.6091 1.5881 1.6350 1.6161 1.6573 1.6398 1.6768 1.6605 1.6943 1.6788 1.710 1.695 1.725 1.710 1.738 1.724 1.751 1.737 1.762 1.749 1.773 1.761 1.6091 1.5881
Table 3. Speed of Sound c (m‚s-1) of HFE 7500 at Atmospheric Pressure as a Function of Temperature T (K) and Resonating Frequency υ (MHz) υ (MHz) T (K)
5
2
0.5
283.15 293.15 303.15 313.15 323.15 333.15 343.15
702.7 673.2 644.1 616.7 589.1 562.5 534.9
701.7 672.7 643.8 615.7 588.0 561.3 535.4
702.3 673.8 644.2 616.8 590.4 561.8 534.5
started from 10 MPa. The results for both HFE 7000 and HFE 7500 are given in Table 1. The complementary density measurements were carried out up to 40 Mpa by means of a U-shaped tube along several isotherms ranging from 283.15 to 353.15 K. The measurements were performed from atmospheric pressure up to 40 MPa in 2.5 MPa steps and were then extended up to 100 MPa from integration of the speed of sound measurement using the Davis and Gordon method27 and an inverse technique11 to initiate the procedure. The resulting values are given every 10 MPa in Table 2. The accuracy of density data evaluated by such a method has been estimated to 0.1% on the basis of comparisons with literature data for n-hexane.28 Ohta et al.29 determined saturated and compressed liquid densities for heptafluoropropyl methyl ether, at temperatures up to 369.983 K and pressures up to 3 MPa, with a magnetic densimeter. Their compressed liquid data were compared with our HFE 7000 experimental data, taking into account only the literature data included in the temperature range of this work, in order to avoid extrapolation. In this comparison, a departure of 0.11 in average percent deviation (APD), corresponding to an absolute average deviation (AAD) value of 1.4 × 10-3 g‚cm-3, was found. APD and AAD were calculated according to the expressions:
Ind. Eng. Chem. Res., Vol. 46, No. 21, 2007 7001
AAD )
APD )
N
1
∑|Fi-exp - Fi-cal| N i)1 ∑| N i)1
100
N
(1)
Fi-exp - Fi-cal Fi-exp
|
(2)
These values might seem high, but if the Bias is calculated according to the expression
Bias )
1
N
∑(Fi-bib - Fi-exp)
(3)
N i)1
where Fi-bib and Fi-exp stand for the literature data and this work, respectively, the result obtained is 2.6 × 10-4 g‚cm-3. This value shows that there is not a systematic discrepancy between both sets of data, and the difference may be attributed only to the different uncertainties of both experimental devices. Before the experimental determination of the speed of sound c was carried out, a test to check the absence of dispersion phenomena in the working frequency range for these compounds was performed for HFE 7500 at atmospheric pressure, as stated before. The c values obtained are shown in Table 3 in order to check the absence of this undesired dispersion phenomena. 3. Modeling Using the SAFT VR Mie EOS EOS,21
In the framework of the so-called SAFT-VR Mie the HFEs molecules are modeled as chains of identical spheres (same diameter σ and dispersive energy ) tangentially bonded together to form chains. The interactions between the spheres are modeled via the generalized Mie potential (λ2, λ1).
uM ) C
[(σr ) - (σr ) ] λ2
λ1
(4)
()
λ 2 λ2 λ2 - λ1 λ1
ARES AMONO ACHAIN AASSOC ) + + NkT NkT NkT NkT
λ1/(λ2-λ1)
(5)
The main feature of this EOS is that λ2 and λ1 are incorporated as two extra pure component parameters establishing the possibility to vary the range of both the repulsive and the attractive terms of the intermolecular potential. It is important to mention that a variable λ2 parameter is used since a rigorous theoretical justification for exponent 12 in the LennardJones potential is not available and initially this value was chosen exclusively on a practical basis with the aim to reduce calculation times. Next, the 1/rλ1 attraction accounts for the van der Waals forces including dispersion as well as permanent and induced polar forces. It is worth noting that the functional form 1/r6 arises from the exact quantum mechanical solution for the nonpolar, neutral atoms with the spherically symmetric electron shell, as for instance the noble gas atoms. Nevertheless, it is always used as an approximation for other atoms (higher order terms arise from the interaction through higher instantaneous moments such as dipole-quadrupole, quadrupole-quadrupole, etc...), and later, we will show the validity of this assumption. Eventually, this λ1 ) 6 value will be adopted in the definitive calculations. Hence in the case of complex molecules like HFEs, λ2 is used in an effective way to model the global long-range interaction. Next, in the framework of the work of Galindo et al.,20 two embedded short-range attractive
(6)
where N is the number of molecules, T is the temperature, and k is the Boltzmann constant. In this equation AMONO, ACHAIN, and AASSOC are the contributions from the monomer segments (M), the formation of chains, and the existence of intermolecular association, respectively. 3.1. Monomer Contribution. For pure components formed by ms tangentially bonded spheres, the monomer Helmholtz free energy can be expressed as
AM AMONO 2 M ) ms ) ms(aHS + βaM 1 + β a2 ) NkT NskT
where
C)
sites (one site of type a and one site of type b with only a-b bonding allowed) representing molecular association are considered to model, in a geometric sense, the short-ranged polar anisotropic effects. It should be noted that bonding is allowed only between the black and white sites corresponding to the 2B associating scheme defined by Huang and Radosz.30 This two-sites model describes the fact that only chains of associated aggregates are allowed but other geometric configurations as for instance treelike structures are forbidden. It is important to emphasize that ring like structures are not permitted at the level of approximation of SAFT. Thus, the SAFT-VR Mie EOS, using a intermolecular potential of variable range, together with short-range attractive sites is able to model the van der Waals long-range forces (in an effective way) as well as the short-range directional effects due to the dipolar nature of the molecule. Note that it is highly probable that the variable repulsive part might also provide a more accurate description of the intermolecular forces involved in these complex substances. Now, we recall here the key expressions for the SAFT-VR Mie EOS.21 The reader can find full details in ref 13. The residual Helmholtz free energy can be written in terms of separate contributions as follows:
(7)
where β ) 1/kT, aHS is the residual free energy of a system of M hard spheres of diameter σBH, and aM 1 and a2 are the first two 16,21 perturbation terms associated with the attractive term of the potential. The value of the hard-core temperature-dependent diameter σBH is derived using the Barker and Henderson perturbation theory for soft-core segments.31
σBH(T) )
(
( ))
∫0σ 1 - exp -u kT
M
dr
(8)
This expression is calculated using Simpson’s method. In addition, the expression of Carnahan and Starling32 is used for the hard-sphere residual free energy,
aHS )
4ηBH - 3ηBH2 (1 - ηBH)2
(9)
where ηBH ) πσBH3Fs/6 is the packing fraction of the system, and Fs ) Ns/V is the monomer density. 16,33 The first perturbation term aM 1 is given by S S aM 1 ) C[-a1 (ηBH, λ2) + a1 (ηBH, λ1)]
(10)
where aS1 stands for the mean attractive energy for a Sutherland-λ system, evaluated with the effective packing fraction ηBH.
7002
Ind. Eng. Chem. Res., Vol. 46, No. 21, 2007
Table 4. SAFT-VR Mie Molecular Parameters for HFE 7000, 7100, 7200, and 7500 APD (%) M
(g.mol-1)
ms
σ (Å)
/k (K)
λ2
λ1
κa,b
Ψa,b/k (K)
Tb
Lv
F (T, P)
c (T, P)
1487.8
0.12
0.15
0.1
2.22
200.05
2.6081
4.2409
205.58
11.548
HFE 7000 6 0.10591
250.06
2.8499
4.3597
217.90
12.123
HFE 7100 6 0.07596
1856.26
0.05
0.26
0.15
2.87
11.85
HFE 7200 6 0.07258
1882.7
0.08
0.30
0.13
2.83
11.909
HFE 7500 6 0.07534
1705.3
0.20
0.14
0.22
3.66
264.09 414.11
3.0783 3.6886
4.4353
218.30
4.7222
250.2
Using the mean-value theorem, Gil-Villegas et al.16 obtained an analytical expression for exponents in the range 3 < λ e 12:
aS1 (ηBH, λ) ) aVDW (ηBH, λ)gHS(1;ηeff(ηBH)) 1
(11)
is the van der Waals mean-field term given by where aVDW 1
(λ -3 3)
(ηBH;λ) ) -4ηBH aVDW 1
(12)
and gHS(1;ηeff) denotes the contact value of the radial distribution function of the hard-sphere reference system evaluated with an effective packing fraction ηeff. This function is calculated as follows:
gHS(1;ηeff) )
1 - ηeff/2 (1 - ηeff)
(13)
3
KHS )
(1 - ηBH)4
(17)
1 + 4ηBH + 4ηBH2
3.2. Chain Contribution. The expression for the contribution to the formation of a chain of ms monomers is expressed as a function of the contact value of the monomer background correlation function:
ACHAIN ) -(ms - 1) ln yM b NkT
(18)
The monomer background correlation function of the Mie potential can be written as follows:21
[
M Cλ1 S 1 ∂a1 M HS ) y (σ ) ) g (σ ) + a (λ ) + β yM b BH BH 4 ∂ηBH 4ηBH 1 1
]
Cλ2 S a (λ ) (19) 4ηBH 1 2
The effective packing fraction ηeff is defined by
ηeff(η;λ) ) c1ηBH + c2ηBH
2
(14)
with
() ( c1
3.3. Association Contribution. The Helmholtz free energy for the association contribution is calculated using the perturbation theory of Werheim for associating fluids. In the case of a pure associating substance, it is given by13
AASSOC
)
NkT
)
c2
-0.943973 0.422543 -0.0371763 0.00116901 0.370942 -0.173333 0.0175599 -0.000572729
() 1 λ × 2 λ λ3
)
(15)
(
S
(16)
where KHS is the Percus Yevick16 hard-sphere isothermal compressibility,
)
Xa 2
+
M 2
(20)
where M denotes the number of associating sites on the molecule and Xa is the mole fraction of molecules not bonded at site a. Note that the sum is over all associating sites on the molecule. The values of Xa are obtained from the solution of the mass balances:
1
Xa ) 1+
As for the second-order perturbation term, aM 2 , it has been shown16,33 that it can be obtained in terms only of the attractive contribution as
∂a1 (ηBH, 2λ1) C HS aM 2 ) K ηBH 2 ∂ηBH
∑A
ln Xa -
(21)
∑b
FXb∆a,b
In this expression, F is the total number density and the quantity ∆a,b is related to the strength of the a-b association bond. The latter value can be approximated by
( (
∆a,b ) σκa,b exp -
) )
ψa,b - 1 gM(σBH) kT
(22)
where κa,b is the effective association volume and ψa,b is the association energy. A detailed discussion of the application of the SAFT-VR Mie EoS to associating substances has been recently published.22
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Figure 1. Representation of the intermolecular (λ2, λ1) Mie potentials obtained for HFE 7000, fitting both parameters (dash-dot lines) or keeping λ1 ) 6 (dashed lines).
3.4. Adjusting the SAFT-VR Mie Parameters. As mentioned before, according to the SAFT-VR Mie model, a complete thermodynamic description of HFEs requires the determination of six molecular parameters: the number of segments ms, the diameter of the segments σ, the depth of the attractive term of the potential /k, the shape of the potential characterized by λ2 (repulsive part), the association energy ψa,b, and the effective association volumeκa,b The classical method to optimize the molecular parameters in SAFT-like EOSs consists of fitting experimental pressure and saturated liquid densities. However, it has been shown21 that, in the case of the SAFT-VR Mie, the inclusion of liquid densities and the speed of sound in the compressed liquid region leads to a better balance of the different estimated properties (volumetric and derivative properties). However, a difficulty arises when studying HFE substances due to the lack of experimental data. Actually, HFE 7000 is the only molecule for which all the above-mentioned properties are known in a wide P-T range. For the other members of the series (HFE 7100, 7200, and 7500), saturated liquid densities are not available and the accuracy of the vapor pressure is generally ensured only at atmospheric pressure. Therefore, since our goal is to describe simultaneously (with a single set of molecular parameters) phase equilibrium and first- and second-derivative properties of HFEs with good accuracy, we first had to find a systematic fitting procedure (i.e., with the same combination of properties for every substance), that could provide reliable characteristic parameters. Taking this into account, a preliminary study has been carried out on HFE 7000 with the aim to test different combinations of properties. Note that the saturated liquid density is a variable not included in the determination of characteristic parameters. Moreover, the normal boiling point is the only common available experimental cloud point for all the members of the HFE series. Hence, the vapor pressure and the saturated liquid density were used as a test for the predictive capability of the different sets of molecular parameters obtained. The fitting procedure was then based on the minimization of an objective function, OF, which can be written as a weighted sum of three departure functions related to individual data groups (VLE stands for vapor liquid equilibrium curve, and L, for liquid state):
Figure 2. Comparison of experimental density at the following temperatures: 283.15 (circles), 303.15 (squares), 323.15 (triangles), and 343.15 K (diamonds); estimated (solid lines) values with SAFT-VR Mie for HFE 7000 (a) and HFE 7500 (b).
OF ) w1,VLEf1,VLE + w1,Lf1,L + w2,Lf2,L where
f1,VLE(ms, σ, , λ2, λ1, ψa,b, κa,b) )
[
N1,L
f1,L(ms, σ, , λ2, λ1, ψa,b, κa,b) )
∑ i)1
[
∑ i)1
Texp b
2
]
(24)
exp calc Fi,liq - Fi,liq
N2,VLE
f2,L(ms, σ, , λ2, λ1, ψa,b, κa,b) )
]
Tbexp - Tcalc b
[
exp Fi,liq
2
(25)
]
exp calc ci,liq - ci,liq exp ci,liq
(23)
2
(26)
Tb, Fliq and cliq being the experimental normal boiling point, density, and speed of sound data, respectively. Several values for the different weight factors w1,VLE, w1,L, and w2,L were tested with values ranging from 0 to 1. Unfortunately, the regression of HFE 7000 parameters through this procedure (using just one experimental cloud point) resulted in poor predictions of the measured vapor pressure and saturated liquid densities. Indeed, we found several parameter sets able to fit the data (liquid densities, speed of sound, and the normal boiling point), but they exhibited little predictive
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Figure 3. Comparison of experimental speed of sound at the following temperatures: 283.15 (circles), 303.15 (squares), 323.15 (triangles), and 343.15 K (diamonds); estimated (solid lines) values with SAFT-VR Mie for HFE 7000 (a) and HFE 7500 (b).
Figure 4. Vapor pressure (a) and saturated liquid density (b) of HFE 7000, 7100, 7200, and 7500; comparison of available experimental data and estimated values with SAFT-VR Mie.
ability. This shortcoming of the first calculation procedure tested addresses the question of the multiplicity of local minima for the objective function for different parameter sets, a common feature to many theoretical approaches. This question must always be handled carefully in order to extract all the possible physical meaning of the final proposed parameter values. However, with the aim to keep a systematic fitting procedure for all the members of the HFE series, an alternative approach was found, consisting of including the heat of vaporization (Lv) in the regression. In this case, a new term is added to the already existing objective function, and this new term is defined as
f2,VLE(ms, σ, , λ2, λ1, ψa,b, κa,b) )
[
]
calc Lexp v - Lv
Lvexp
2
(27)
So, the new objective function OF to minimize becomes now the following:
OF ) w1,VLE f1,VLE + w2,VLE f2,VLE + w1,L f1,L + w2,L f2,L (28) We observed that this method provides excellent agreement with experimental data used in the fitting as well as with the vapor pressure and saturated liquid densities. As a consequence, this approach was the solution adopted to derive the molecular parameters for the HFE series.
Figure 5. Comparison of the experimental (symbols) and estimated (solid lines) atmospheric pressure isobaric heat capacities for HFE 7000 and HFE 7500. In every case, the dashed line shows the ideal gas contribution to the magnitude.
Moreover, it has been noticed that it is possible to obtain a similar accuracy in the representation of experimental data using the Lennard-Jones λ2-6 potentials (i.e. with the fixed value λ1 ) 6 and by adjusting only the repulsive index λ2), if compared to the generalized Mie λ2-λ1 potential. This is explained by the fact that when using the λ2-6 potentials, not only the repulsive part but also the attractive part shape is changed for various λ2 values. The smaller the λ2 value, the wider the attractive part and the weaker the repulsive force. We illustrate this in Figure
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Figure 6. Residual heat capacity (Crp), isobaric thermal expansivity (Rp), and the speed of sound (c) of (a) HFE 7000 and (b) HFE 7500 for different isobars (PR ) 0.1; 0.3; 0.6; 1; 2; 3; 4). PR ) P/PC and TR ) T/TC are the reduced and pressure and temperature. The coordinates of the critical points as calculated with SAFT-VR Mie are (TC ) 455.81 K; PC ) 3.62 MPa) for HFE 7000 and (TC ) 560.07 K; PC ) 2.20 MPa) for HFE 7500.
1 where a plot of the intermolecular potential u(r) obtained for HFE 7000 using both approaches is presented. With the first approach (adjusting λ2 and λ1), u(r) turns out to be the sum of a Sutherland λ ) -10.893 and a Sutherland λ ) -6.3746 potential, while a value of λ2 ) 11.548 is found for the repulsive exponent when using the λ2-6 potential. In both cases the resulting intermolecular potential shape is almost identical, and hence, this explains the same degree of accuracy obtained in the representation of HFE 7000 properties. Therefore, it appears that the repulsion interactions and the van der Waals long range forces involved in HFE molecule can be described together by adjusting only the repulsive exponent.
Thus, all characteristic molecular parameters for HFEs were determined using the λ2-6 potential approach. 3.5. Results and Discussion. Table 4 presents the SAFTVR Mie pure component parameters of HFE 7000 and HFE 7500 together with the APD (%) of the fitted properties (compressed liquid density, speed of sound, heat of vaporization, and normal boiling point). In order to complete this study, we also report the parameters of HFE 7100 and HFE 7200 for which the compressed liquid densities and speed of sound that have been calculated in previous works.8,9 Figures 2 and 3 plot experimental densities and the speed of sound measured in this work and SAFT-VR Mie calculations
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results with the parameters reported in Table 4. Note that the model is able to describe, at the same time and with an excellent accuracy, the high densities of HFEs’ molecules as well as the low values of the speed of sound in a wide range of pressure conditions. However, it should be noted that the equation is less accurate in the case of HFE 7500 where the speed of sound is overpredicted at low pressure and underpredicted at high pressure. This problem might come from the fact that, for such complex and asymmetric molecules, an heteronuclear chain should be envisaged in order to take into account explicitly the different functional groups. Saturation pressure and saturated liquid densities were also calculated using these parameters and compared with experimental data available in the literature. Figure 4 shows the SAFTVR Mie estimations for HFE 7000, 7100, 7200, and 7500. The experimental HFE 7000 vapor pressure and saturated liquid densities reported in Figure 4 were taken from ref 29 and correspond to the reduced temperature range 0.57 < Tr < 0.85. However, for the other HFEs, experimental information is scarce. Actually, the only information available is restricted to simple correlations from ref 34 (vendor specifications), for which the reliability is not ensured. Note that the critical points for these compounds have not been determined so far, so correlations in Figure 5 were plotted for arbitrary temperature intervals. Despite this, the overall agreement can be considered excellent. It is important to recall that the normal boiling point and the heat of vaporization at atmospheric pressure are the only information on vapor-liquid equilibrium that were included in the fitting procedure. This calculation emphasizes the predictive capability of the SAFT-VR Mie model and the consistency of the fitting procedure proposed. The predictive capability can be taken a step further by the calculation of the isobaric heat capacity (Cp), a crucial test for the equation of state since it involves second-order derivatives of the Helmholtz free energy. Figure 5 shows an example for HFE 7000 and 7500 for which correlations at atmospheric pressure are available.34 The prediction agrees well with the correlations, except at very low-temperature conditions. This is actually not surprising since this is the region where the perturbation theory approximation is considered to be less precise. Finally, Figure 6 presents the SAFT-VR Mie predictions for different second-derivative properties (isobaric heat capacity, thermal expansivity, and speed of sound) of HFE 7000 and 7500 in temperature and pressure conditions far away from those used in the fitting procedure. Note that the equation is supposed to generate reliable values due to its inbuilt molecular formulation for the molecular shape and directional forces present in these complex fluids. Related to this, it has been noticed that the present model is able to estimate the heat capacity with the same order of accuracy than the specific EOS for HFE 7000 with 17 coefficients proposed by Widiatmo et al.35 However, it should be mentioned that the correct description of the critical region, as for any analytical equation of state, would require a crossover treatment.23,36 4. Conclusions The results presented in this paper show the high value of experimental high-pressure volumetric and speed of sound data when a complete thermophysical characterization of a fluid is requred. If combined with a reliable EoS, as it is in the case of SAFT-VR Mie, the use of these two properties in the fitting of characteristic molecular parameters has been shown to provide
better global results, allowing not only an accurate VLE phaseequilibrium estimation but also a good representation of other second-derivative properties. The usual parameter fitting method takes only into account experimental saturation pressures and densities, but the parameters obtained this way fail to yield good estimations of other thermophysical properties, especially those involving pressure derivatives of density. Then, the right choice of properties to be used in the fitting, and their relative weights in the objective function minimized, will determine to a great extent the parameters’ range of applicability. This fact is even more relevant when the fluids object of study may be useful in applications such as refrigeration (as it is the case of HFEs, for instance), where secondderivative properties play a key role in the determination of their potential applicability. But, it must be underlined as well that the eventual success of the characterization is not only due to the right choice of experimental information to be included in the fitting but also to the use of a theoretical model that is flexible enough to capture the behavior of all the magnitudes involved. This turns out to be especially difficult when both temperature and pressure derivatives are considered for highly nonideal molecules, as it has been shown here for fluorinated compounds as HFEs, where many theoretical methods fail to yield accurate estimations. This way, the SAFT-VR Mie EoS has proved once more to be physically sound and reliable for the objectives envisaged. Literature Cited (1) Molina, M. J.; Rowland, F. S. Stratospheric sink for chlorofluoromethanes: Chlorine atom-catalysed destruction ozone. Nature 1974, 249, 810. (2) The Montreal Protocol on Substances that Deplete the Ozone Layer; Ozone Secretariat, United Nation Environmental Programme, 2000. (3) Domanski, P. A. Refrigerants for the 21st Century /ASHRAE/NIST Refrigerants Conference. J. Res. Natl. Inst. Stand. Technol. 1998, 103, 529. (4) Bivens, D. B.; Minor, B. H. Fluoroethers and other next generation fluids. Int. J. Refrig. 1998, 7, 567. (5) Sekiya, A.; Misaki, S. J. The potential of hydrofluoroethers to replace CFCs, HCFCs and PFCs. Fluorine Chem. 2000, 101, 215. (6) Tsai, W.-T. Environmental risk assessment of hydrofluoroethers (HFEs). J. Hazard. Mat. 2005, 119 (1-3), 69. (7) Newsted, J. L.; Nakanishi, J.; Cousins, I.; Werner, K.; Giesy, J. P. Predicted Distribution and Ecological Risk Assesment of a “Segregated” Hydrofluoroether in the Japanese Environment. EnViron. Sci. Technol. 2002, 36, 4761. (8) Pin˜eiro, M. M.; Bessie`res, D.; Legido, J. L.; Saint-Guirons, H. PrT Measurements of nonafluorobutyl methyl ether and nonafluorobutyl ethyl ether between 283.15 and 323.15 K at pressures up to 40 MPa. Int. J. Thermophys. 2003, 24, 1265. (9) Pin˜eiro, M. M.; Plantier, F.; Bessie`res, D.; Legido, J. L.; Daridon, J. L. High-pressure speed of sound measurements in methyl nonafluorobutyl ether and ethyl nonafluorobutyl ether. Fluid Phase Equilib. 2004, 222, 297. (10) Cendo´n, J.; Pin˜eiro, M. M.; Bessie`res, D.; Vijande, J.; Legido, J. L. High pressure densities of the binary mixture methyl nonafluorobutyl ether + hexane. J. Chem. Eng. Data 2004, 49, 1368. (11) Cendo´n, J.; Pin˜eiro, M. M.; Vijande, J.; Legido, J. L. Experimental (P,V,T,x) data for the mixture ethyl nonafluorobutyl ether + n-hexane. J. Chem. Eng. Data 2006, 51 (2), 577. (12) Vijande, J.; Pin˜eiro, M. M.; Bessie`res, D.; Saint-Guirons, H.; Legido, J. L. Description of PVT behaviour of hydrofluoroethers using the PC-SAFT EOS. Phys Chem. Chem. Phys. 2004, 6 (4), 766. (13) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. New reference equation of state for associating liquids. Ind Eng. Chem. Res. 1990, 29, 1709. (14) Gross, J.; Sadowski, G. Pertturbed-chain SAFT: An equation of state based on a perturbation theory for chain molecules. Ind. Eng. Chem. Res. 2001, 40, 1244. (15) Blas, F. J.; Vega, L. F. Thermodynamic behaviour of homonuclear and heteronuclear Lennard-Jones chains with association sites from simulation and theory. Mol. Phys. 1997, 92, 135.
Ind. Eng. Chem. Res., Vol. 46, No. 21, 2007 7007 (16) Gil-Villegas, A.; Galindo, A.; Whitehead, P. J.; Mills, S. J.; Jackson, G.; Burgess, A. N. Statistical Associating Fluid Theory for Chain Molecules with Attractive Potential of Variable Range. J. Chem. Phys. 1997, 106, 4168. (17) Economou, I. G. Statistical Associating Fluid Theory: A successful Model for the Calculation of Thermodynamic and Phase Equilibrium Properties of Complex Fluid Mixtures. Ind. Eng. Chem. Res. 2002, 41, 953. (18) Mu¨ller, E. A.; Gubbins, K. E. In Equations of State for Fluids and Fluid Mixtures, Sengers, J. V., Kayser, R. F., Peters, C. J., White, H. J., Jr., Eds.; Elsevier: New York, 2000; pp 435-477. (19) Swaminathan, S.; Visco, D. P., Jr. Thermodynamic Modeling of Refrigerants Using the Statistical Associating Fluid Theory with Variable Range. 1. Pure Components. Ind. Eng. Chem. Res 2005, 44, 4798. (20) Galindo, A.; Gil-Villegas, A.; Whitehead, P. J.; Jackson, G.; Burgess, A. N. Prediction of Phase Equilibria for Refrigerant Mixtures of Difluoromethane (HFE-32), 1,1,1,2-Tetrafluoroethane (HFC-134a), and Pentafluoroethane (HFC-125a) Using SAFT-VR. J. Phys. Chem. B 1998, 102, 7632. (21) Lafitte, T.; Bessie`res, D.; Pin˜eiro, M. M.; Daridon, J.-L. Simultaneous estimation of phase behaviour and second derivative properties using SAFT-VR approach. J. Chem. Phys. 2006, 124, 024509. (22) Lafitte, Th.; Pin˜eiro, M. M.; Daridon, J.-L.; Bessie`res, D. A comprehensive description of chemical association effects on second derivative propertis of alcohols through a SAFT-VR approach, J. Phys. Chem. B 2007, 111 (13), 3447. (23) Llovell, F.; Vega, L. F. Prediction of Thermodynamic Derivative Properties of Pure Fluids through the Soft-SAFT Equation of State. J. Phys. Chem. B 2006, 110 (23), 11427-37. (24) Llovell, F.; Peters, C. J.; Vega, L. F. Second-order thermodynamic derivative properties of selected mixtures by the soft-SAFT equation of state. Fluid Phase Equilib. 2006, 248 (2), 115-122. (25) Plantier, F.; Daridon, J.-L.; Lagourette, B. J. Measurement of the B/A nonlinearity parameter under high pressure: application to water. Acoust. Soc. Am. 2002, 111, 707. (26) Sako, T.; Sato, M.; Nakazawa, N.; Oowa, M.; Yasumoto, M.; Ito, H.; Yamashita, S. Critical Properties of Fluorinated Ethers. J. Chem. Eng. Data 1996, 41 (4), 802.
(27) Davis, L. A.; Gordon, R. B. Compression of mercuryat high pressure. J. Chem. Phys. 1967, 46 (7), 2650. (28) Daridon, J. L.; Lagourette, B.; Grolier, J. P. E. Experimental measurements of the speed of sound in n-hexane from 293 to 373 K and up to 150 Mpa. Int. J. Thermophys. 1998, 19 (1), 145. (29) Ohta, H.; Morimoto, Y.; Widiatmo, J. V.; Watanabe, K. J. LiquidPhase Thermodynamic Properties of New Refrigerants: Pentafluoroethyl Methyl Ether and Heptafluoropropyl Methyl Ether. Chem. Eng. Data 2001, 46 (5), 1020. (30) Huang, S. H.; Radosz, M. Equation of State for small, large, polydisperse, and asscociating molecules. Ind. Eng. Chem. Res. 1990, 29, 2284. (31) Barker, J. A.; Henderson, D. ReV. Mod. Phys.1975, 48, 587. (32) Carnahan, N. F.; Starling, K. E. Equation of State for Nonattracting Rigid Spheres J. Chem. Phys. 1969, 51, 635. (33) Davies, L. A.; Gil-Villegas, A.; Jackson, G. Describing the Properties of Chains of Segments Interacting Via Soft-Core Potentials of Variable Range with the SAFT-VR Approach. Int. J. Thermophys. 1998, 19, 675. (34) 3M Novec Engineered Fluids. http://www.3M.com. (35) Widiatmo, J. V.; Watanabe, K. Equations of state for fluorinated ether refrigerants, pentafluoroethyl methyl ether and heptafluoropropyl methyl ether. Fluid Phase Equilib. 2001, 183, 31. (36) Kiselev, S. B.; Ely, J. F. Crossover SAFT Equation of State. Application for Normal Alkanes. Ind. Eng. Chem. Res. 1999, 38, 4993.
ReceiVed for reView January 9, 2007 ReVised manuscript receiVed May 16, 2007 Accepted May 30, 2007 IE0700462
