Intermolecular potential model parameters for cyclic ethers and chloroalkanes in the SAFT-VR approach

Fluid Phase Equilibria 255 (2007) 200–206

Intermolecular potential model parameters for cyclic ethers and chloroalkanes in the SAFT-VR approach B. Giner a , F.M. Royo a , C. Lafuente a,∗ , A. Galindo b a

b

Departamento de Qu´ımica Org´anica y Qu´ımica F´ısica, Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain Department of Chemical Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom Received 20 February 2007; received in revised form 3 April 2007; accepted 5 April 2007 Available online 8 April 2007

Abstract The SAFT approach for attractive potentials of variable range (SAFT-VR) has been used to model four cyclic ethers: 1,3-dioxolane, 1,4dioxane, tetrahydrofuran and tetrahydropyran, and six chloroalkanes: 1-chloropropane, 2-chloropropane, 1-chlorobutane, 2-chlorobutane, 1-chloro2-methylpropane and 2-chloro-2-methylpropane. The molecules are represented as chains of m tangentially bonded spherical segments interacting via square-well potentials of variable attractive range. The square-well segments are characterised by a hard-core diameter σ, a well depth ε and a range λ. For each compound, the optimised values of the intermolecular model parameters m, σ, ε and λ are obtained by comparison to experimental vapour pressure and saturated liquid density data. Although all the compounds examined are highly polar, we find that the SAFT-VR approach is capable of providing a good description of the phase behaviour without the need to take into account polar interactions explicitly. The largest errors found are of the order of 2%, both for the pressure and for the density. In future work the models presented here will be used to treat mixture phase behaviour. © 2007 Elsevier B.V. All rights reserved. Keywords: Modelling; SAFT-VR; Cyclic ethers; Chloroalkanes

1. Introduction The design and development of processes such as surfactancy, supercritical fluid extraction and separation procedures in the chemical process industries requires accurate knowledge of mixture properties [1,2]. Dohrn and Pfohl [3] give an excellent discussion of the importance of choosing the right experimental data in model determination and of the impact that errors in the calculation of thermodynamic properties can have in process design decisions. In general, experimental data are more reliable, but are also more expensive, and are not always available. Correlation and prediction methods are then the choice. Correlation methods can be very accurate, but are limited to conditions in which experimental data are available. On the other hand, the quote in which Kammerlingh–Onnes credits van der Waals for his help in achieving the liquefaction of hydrogen: “In what I describe to you, your theory has been my guide. The calculations

∗

Corresponding author. Tel.: +34 976 762295; fax: +34 976 761202. E-mail address: [email protected] (C. Lafuente).

0378-3812/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2007.04.003

were performed entirely on the basis of the law of corresponding states. Guided by the law, I estimated to need 20 litres [of hydrogen]. Had I estimated a few litres fewer, the experiment would not have succeeded” [4], provides one of the first examples of the strength of predictive molecular approaches in studies of fluid phase behaviour. An important advance in the field of molecular equations of state has been the development of the statistical associating fluid theory (SAFT). It was developed by Chapman et al. [5,6] in the late 1980s to model associating chain fluids, and has become the state of the art tool to model and predict the fluid phase behaviour of complex fluids and mixtures. The approach is based on the thermodynamic perturbation theory of Wertheim [7–10] whereby the thermodynamic properties of the chain fluid are obtained from knowledge of the thermodynamic properties of a monomer (un-associated or un-bonded) reference fluid. In the first versions a hard-sphere fluid was used as reference [5,6,11,12], this was later extended to consider a Lennard-Jones fluid as reference [13], as well as fluids of attractive segments of variable range (SAFT-VR) [14,15]. There are now numerous versions and modifications, and the approach, and in its many

B. Giner et al. / Fluid Phase Equilibria 255 (2007) 200–206

forms it has been used to treat a wide variety of fluids and mixtures. An excellent review of most SAFT-related advances and systems considered up to 2001 can be found in [16]. This field has become extremely active and as such it is almost impossible to cite here all the works that have been presented since 2001. Of special interest to the reader may be the development of the a version based on a reference chain fluid [17], a recent work in which the importance of the repulsive shape of the potential highlights the versatility of approaches that allow for variable potential ranges [18], and the extensions to treat critical states accurately [19–21]. In this, the first of a series of papers, we have used the SAFT-VR equation [14,15] to calculate the phase equilibrium of four cyclic ethers: 1,3-dioxolane, 1,4-dioxane, tetrahydrofuran and tetrahydropyran, and six chloroalkanes: 1-chloropropane, 2-chloropropane, 1-chlorobutane, 2-chlorobutane, 1-chloro-2methylpropane and 2-chloro-2-methylpropane. The study of these compounds is of interest due to their differences in structure, shape and size and to the fact that they form new donor– acceptor type interactions in mixtures, which give rise to the appearance of interesting molecular effects. The strength of these new interactions depends on the donor or acceptor ability of each of the chemicals. The acceptor ability of the chloroalkanes studied here is quite similar for all of them. Several studies have however highlighted differences in the donor ability of the cyclic ethers. It has been shown that tetrahydrofuran is the strongest donor, and based on calorimetric and spectroscopic data a sequence tetrahydrofuran > tetrahydropyran > 1,3-dioxolane > 1,4-dioxane has been presented in terms of their electron donor ability [22–24]. Mixtures of these chemicals have been previously studied by one of our groups [25–30] and very interesting molecular phenomena related to the strength of unlike specific interactions that can be established between the compounds of the mixture and to changes in the molecular structure of the pure compounds have been observed. To our knowledge these chemicals have not been modelled previously with the SAFT approach, and the main aim of this work is to assess the reliability of such approach for this type of fluids. In following works, we will present the application of SAFT-VR to the mixtures formed with the compounds studied here and will discuss how the specific molecular interactions affect the results obtained with the approach.

201

2. Molecular models and theory Although all of the molecules considered here are polar (the chloroalkanes more so than the cyclic ethers as can be expected given the symmetry of the latter), we model them using the standard SAFT-VR approach; i.e. we do not explicitly take into account dipolar interactions. Long-range electrostatic interactions can be treated explicitly using, for example, the u expansion for the non-axial multipole interaction as shown by Walsh et al. [31], who combined it with a Wertheim formalism to treat also hydrogen bonding compounds. This approach has been followed by numerous authors, who using different levels of approximation are considering the treatment of polar interactions explicitly within SAFT-like equations, e.g. [32–37]. These approaches can be very successful in modelling real polar and associating fluids, but they require the addition of polar parameters, such as the molecular dipole moments. These are usually taken from experimental data, although care should be taken to note that the value of the dipole moment is state-dependent. A Boltzmann averaging of the dipole–dipole interaction energy over all orientations leads to an angle-averaged (i.e. angle-independent) interaction varying as the sixth inverse power of intermolecular distance, called the Keesom potential [38], which can be treated as contributing to the overall van der Waals intermolecular interaction. We take this view, and treat orientation-independent polar interactions effectively as dispersion forces using square-well potentials of variable range. As we will show below the model is perfectly adequate to describe the phase behaviour of the compounds of interest, and we reduce the need to add polar parameters. In addition, despite the fact that polar interactions between the cyclic monoethers (tetrahydrofuran and tetrahydropyran) are stronger than those in the diethers (1,3-dioxolane and 1,4-dioxane) [39–41], none of these interactions are strong enough to create molecular complexes. Therefore, in terms of the SAFT-VR equation of state, we have considered the studied compounds as non-associating fluids, neglecting the association term of the SAFT equation of state. The molecules are modelled as chains of m tangentially bonded square-well segments of hard-core diameter σ, and characterised by a well depth parameter ε and a range λ. The united atom approach is assumed, so that one segment does not correspond to one atom in the molecule. Instead, in each case

Table 1 Optimised SAFT-VR parameters, m, ε/k, σ and λ and average absolute relative deviation for the pressure and density, P, V Compound

1,3-Dioxolane 1,4-Dioxane Tetrahydrofuran Tetrahydropyran 1-Chloropropane 2-Chloropropane 1-Chlorobutane 2-Chlorobutane 1-Chloro-2-methylpropane 2-Chloro-2-methylpropane

m

1.943 3.222 2.824 3.167 2.593 2.460 2.530 2.112 3.426 1.925

ε/k (K)

397.257 192.540 184.900 170.545 209.516 172.291 226.740 310.063 148.910 282.530

˚ σ (A)

3.804 3.152 3.206 3.289 3.408 3.459 3.722 4.003 3.240 4.126

λ

1.422 1.732 1.738 1.781 1.645 1.762 1.559 1.507 1.799 1.556

AAD% P

ρ

1.72 1.01 1.40 0.44 0.82 2.31 1.05 0.30 2.37 0.78

0.0589 0.2416 1.0249 0.4452 0.2127 0.8093 0.4421 0.0665 1.4421 0.0159

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an optimal value m for the number of segments representing a molecule is obtained by comparison with experimental data as described later. This parameter m treats the non-spherical nature of the molecules, and as it is optimised by comparison with experiment can be applied equally to linear and cyclic molecules. The general form of the SAFT Helmholtz energy A for a fluid of non-associating chain molecules can be written as A Aideal Amono. Achain = + + , NkT NkT NkT NkT

(1)

where N is the total number of molecules, T the temperature and k is the Boltzmann constant. Aideal is the ideal Helmholtz energy, Amono. the residual contribution due to the monomer–monomer interactions, and Achain correspond to the change in Helmholtz energy due to the formation of chains of monomers. The Helmholtz energy of an ideal gas is given by [42]: Aideal = ln(ρΛ3 ) − 1, NkT

(2)

Fig. 1. Vapour pressure (a) and liquid density (b) curves for 1,3-dioxolane. The circles represent the experimental data. The continuous curves represent the SAFT-VR results.

where ρ = N/V is the number density of chain molecules and V is the volume. In the SAFT-VT approach the contribution to the Helmholtz energy due to monomer–monomer interactions is obtained from a Barker and Henderson high-temperature second-order perturbation expansion [14], so that Amono. AHS A1 A2 = + + , NkT NkT NkT NkT

(3)

where the residual Helmholtz energy of the reference hardsphere fluid AHS /NkT is calculated using the Carnahan–Starling equation [43], A1 /NkT corresponds to the mean attractive energy, and the second-order fluctuation term A2 /NkT is calculated using the local compressiblity approximation. Details of each of these terms can be found in the original work [14]. The contribution to the Helmholtz energy due to the formation of a chain of m monomers is obtained from the thermodynamic perturbation theory of Wertheim in terms of the number of contacts in the chain (m − 1) and the cavity distribution function of the

Fig. 2. Vapour pressure (a) and liquid density (b) curves for tetrahydrofuran. The circles represent the experimental data. The continuous curves represent the SAFT-VR results.

B. Giner et al. / Fluid Phase Equilibria 255 (2007) 200–206

reference monomer fluid at contact ymono. (σ). Thus Achain = −(m − 1) ln ymono. (σ). NkT

(4) ymono. (σ) = gmono. (σ),

Noting that for a square-well fluid and the contact value of the monomer–monomer radial distribution function gmono. (σ) is obtained from a self-consistent solution using the Clausius virial theorem. Sear and Jackson [44,45] proposed a modification to account for the extra contact in cyclic molecules, so that in the expression above the factor preceding the radial distribution function is m instead of m − 1. Filipe et al. applied successfully this approach and good results were obtained [46,47]. In our current systems, the obtained results are good enough, and there is no need to apply the modification for the cyclic compounds as m is optimised directly to experimental data. Using standard relationships other thermodynamic properties can be obtained from the Helmholtz energy. The chemical

Fig. 3. Vapour pressure (a) and liquid density (b) curves for 1-chloropropane. The circles represent the experimental data. The continuous curves represent the SAFT-VR results.

potential is given by   ∂A , μ= ∂N T,V and the pressure by   ∂A . P =− ∂V T,N

203

(5)

(6)

The phase equilibrium conditions for the coexisting liquid and vapour phase require the equality of temperature, pressure and chemical potential in each of the two phases to be equal; these are solved using a numerical steepest-descent method [48]. 3. Parameter determination and results In order to obtain SAFT-VR intermolecular potential model parameters, the number of segments in the model chain m, the segments hard-core diameter σ and the depth ε and range λ of the square-well interaction need to be determined for each of the molecules of interest. Here we consider

Fig. 4. Vapour pressure (a) and liquid density (b) curves for 1-chlorobutane. The circles represent the experimental data. The continuous curves represent the SAFT-VR results.

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four cyclic ethers: 1,3-dioxolane, 1,4-dioxane, tetrahydrofuran, tetrahydropyran and six chloroalkanes: 1-chloropropane, 2-chloropropane, 1-chlorobutane, 2-chlorobutane, 1-chloro-2methylpropane, 2-chloro-2-methylpropane, and determine the model parameters by comparison to experimental vapour pressure Pv and saturated liquid density ρl data [49–82]. A least squares objective function of the appropriate residuals: min FObj =

(m,σ,ε,λ)

 Exp.  Calc. (m, σ, ε, λ) 2  Pv,i − Pv,i i

+

Exp.

Pv,i 

Exp.

ρl,i

Calc. (m, σ, ε, λ) − ρl,i Exp.

ρl,i

2 (7)

is constructed. For all the compounds studied the experimental data used in the fitting procedure are restricted up to 90% of the critical point. This cut-off is chosen as the SAFT-VR approach, as any other analytical equation of state, cannot provide an accurate description of the near-critical and critical behaviour. The parameters obtained for the compounds studied here are pre-

sented in Table 1 together with the average absolute relative deviation:   n Exp. 1   Xi − XiCalc.  AAD% = (8)   × 100 Exp.   n Xi i=1 obtained for the pressure and density for each compound. In Figs. 1–5 the vapour pressure (a) and liquid density (b) curves obtained for the 1,3-dioxolane, tetrahydrofuran, 1chloropropane, 1-chlorobutane and 2-chloro-2-methylpropane with the SAFT-VR approach are compared to experimental data. As can be seen, it is possible to obtain a good description of the phase behaviour of the compounds with the SAFT-VR approach, even though the polar nature of the molecules has not been treated explicitly. The AAD% found are of less than 2% in pressure and less than 1% in density, which is of the same order as reported for other compounds with SAFT-like approaches [83]. In a following work these models will be used to study the phase behaviour of binary mixtures composed of one of the cyclic ethers and one of the chloroalkanes. We will show that the approach is predictive for the treatment of these complex mixtures, and that one unlike transferable parameter can be used for the entire family of mixtures. In this sense the model proposed here for the pure compounds is further validated. List of symbols A Helmholtz energy (J molecule−l ) A1 mean attractive Helmholtz energy (J molecule−1 ) second-order fluctuation Helmholtz energy A2 (J molecule−1 ) chain A chain formation Helmholtz energy (J molecule−1 ) HS hard-sphere Helmholtz energy (J molecule−1 ) A ideal A ideal Helmholtz energy (J molecule−1 ) mono. (σ) monomer–monomer interaction Helmholtz energy A (J molecule−1 ) mono. g (σ) radial distribution function of the monomer at contact distance σ k Boltzmann’s constant (k ≈ 1.381 × 10−23 J K−1 ) m number of monomers of a molecule n number of experimental points N number of total molecules P pressure (Pa) Calc. (m, σ, ε, λ) calculated pressure of vapour phase (Pa) Pv,i Exp.

Pv,i T XiCalc.

experimental pressure of vapour phase (Pa) temperature (K) calculated pressure or density in Eq. (8)

Exp.

experimental pressure or density in Eq. (8) Xi ymono. (σ) cavity distribution function of the monomer at contact distance σ

Fig. 5. Vapour pressure (a) and liquid density (b) curves for 2-chloro-2methylpropane. The circles represent the experimental data. The continuous curves represent the SAFT-VR results.

Greek letters ε well depth (J) λ range of the well potential ˚ σ monomer diameter (A) ρ number density (m−3 )

B. Giner et al. / Fluid Phase Equilibria 255 (2007) 200–206 Exp.

ρl,i experimental molar density of liquid phase (mol m3 ) Calc. ρl,i (m, σ, ε, λ) calculated molar density of liquid phase (mol m3 ) Acknowledgements Authors thank the financial assistance support from Ministerio de Educacion y Ciencia and Fondos FEDER (BQU 2003-01765). We are also indebted to DGA and Universidad de Zaragoza for financial support. References [1] [2] [3] [4]

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