Phase Equilibria in Ternary Systems Carbon ... - ACS Publications

Article Cite This: J. Chem. Eng. Data XXXX, XXX, XXX−XXX

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Phase Equilibria in Ternary Systems Carbon Dioxide + 1‑Hexanol + n‑Pentadecane and Carbon Dioxide + 1‑Heptanol + n‑Pentadecane: Modeling of Holes in Critical Surface and Miscibility Windows Dan Geană* Department of Inorganic Chemistry, Physical Chemistry, and Electrochemistry, Faculty of Applied Chemistry and Materials Science, University Politehnica of Bucharest, 1-7 Gh. Polizu Street, 011061 Bucharest, Romania ABSTRACT: In this paper, the modeling of holes in the critical surface and miscibility windows, in the ternary carbon dioxide + 1-hexanol + npentadecane and carbon dioxide + 1-heptanol + n-pentadecane systems, was made. A software for phase equilibrium (PHEQ) data management and applications was used for calculations with the cubic general equation of state (GEOS), developed by the author. The software PHEQ was extended for calculations of critical lines (module GEOS CRITHK), using the Heidemann and Khalil method, and liquid−liquid−vapor (llg) lines (module GEOS LLVE) in binary and ternary mixtures. These modules were used for the calculation of holes in critical surfaces, and of miscibility windows. Binary and quasi-binary measured phase equilibrium data by Scheidgen [Ph.D. Thesis, Ruhr-Universität Bochum, 1997] were necessary for understanding and modeling of ternary systems, regarding holes in critical surface and miscibility windows. For modeling, the equation of state Soave−Redlich−Kwong, integrated in the cubic GEOS form, coupled with the two-parameters conventional mixing rule was used. A simple semipredictive approach was adopted for the estimation of the two binary interaction parameters by a trial and error procedure for reproducing the temperature and pressure values of the two experimental double critical end points (DCEPs). A unique set of parameters obtained by this procedure was used to predict the critical and three phases llg lines for all quasi-binary systems at reduced mass fractions of n-pentadecane in the range between the two experimental DCEPs. The calculated holes in the critical surface and miscibility windows are in satisfactory agreement with the available experimental data.

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Patton et al. (1993) have reported a special fluid phase behavior in the ternary mixture carbon dioxide + 1-decanol + ntetradecane.5 They observed, in the three-phase surface liquid− liquid−vapor (l1l2g), a two-phase region liquid−vapor (lg) which is completely bounded by a closed loop critical end point locus of the type liquid = liquid + vapor (l1 = l2 + g). Very important contributions have been added by two groups of research, almost simultaneously, one of Cor Peters and co-workers at Delft University, and the second of Gerhard Schneider and co-workers at University Bochum. Peters et al.6 (1995) confirmed the original finding of Patton et al. (1993), and have presented new experimental results of some ternary mixtures showing quite similar fluid phase behavior, two-phase liquid−gas (lg) holes in the three-phase liquid−liquid−gas (llg) surface, and/or one-phase miscibility windows surrounded by heterogeneous states of equilibria. Two doctorial theses7,8 have added valuable contributions with experimental data and modeling of cosolvency effect,

INTRODUCTION

Systematic experimental and theoretical studies of multiphase equilibria behavior in fluid binary and multicomponent mixtures are of considerable interest for both science and application.1−4 The first step in the knowledge of phase equilibria in multicomponent mixtures is the study of ternary systems. Mixtures of three components can be of interest as model systems for many processes in chemical engineering, as for example in supercritical fluid extraction (SFE). An important type of ternary model system for the study of phase behavior is a mixture of one supercritical solvent and two low-volatile components, for example, carbon dioxide and two low-volatile organic compounds, since carbon dioxide is the most used solvent in SFE and supercritical fluid chromatography (SFC). Ternary mixtures consisting of carbon dioxide + 1-alkanol + n-alkane have received special attention for both theory and applications. Such fluid mixtures show a large pallet of types of phase behavior and thermodynamic properties. Among various aspects of phase behavior, cosolvency, and related effects have been studied. Selected carbon dioxide binaries, exhibiting gas−gas like fluid phase behavior, have been combined to ternary systems, showing large cosolvency effects, due to improved mutual miscibility. © XXXX American Chemical Society

Special Issue: In Honor of Cor Peters Received: August 31, 2017 Accepted: November 30, 2017

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DOI: 10.1021/acs.jced.7b00781 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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miscibility windows, and holes in the critical surface. Limited parts of both of these were published in journals, with experimental data presented only in graphical form.9−11 Our research group has investigated the phase equilibria of the binary systems carbon dioxide + 1-heptanol, carbon dioxide + 1-hexanol, and carbon dioxide + n-pentadecane.12−14 The cosolvency effect was studied in ternary carbon dioxide + 1heptanol + n-pentadecane mixture.15 The modeling of holes in critical surfaces was reported in the literature by Peters et al.8,9 The algorithm of Heidemann and Khalil16 for the critical points calculation was modified for direct calculation of critical end points (CEPs), based on Gibbs tangent plane criterion in the formulation of Michelsen.2 Calculations with the Peng−Robinson (PR) equation of state for the ternary systems carbon dioxide + 1-hexanol + ntridecane and carbon dioxide + 1-pentanol + n-tridecane were compared to experimental critical end point data.8,9 The authors have discussed some difficulties related to the binary interaction parameters (BIPs) used in modeling, and concluded that only “qualitative agreement between the calculated and the experimental data” was obtained.9 In this paper, the modeling of holes in the critical surface and miscibility windows was done with computer programs developed by the author and co-workers. The program for phase equilibrium (PHEQ) database management and application, was used for calculations with the cubic general equation of state (GEOS), developed by the author.17−20 The software PHEQ21 was extended with module GEOS CRITHK for calculation of critical lines, using the Heidemann and Khalil method16 and llg lines (module GEOS LLVE) in binary and ternary mixtures. These modules were used for the calculation of holes in critical surfaces and of miscibility windows. Experimental binary and quasi-binary phase equilibrium data measured by Scheidgen7 were necessary for understanding and modeling of the holes in the critical surface and miscibility windows in the ternary systems carbon dioxide + 1-hexanol + npentadecane and carbon dioxide + 1-heptanol + n-pentadecane. The equation of state Soave−Redlich−Kwong (SRK), integrated in the cubic GEOS form, was used for modeling, coupled with the two-parameters conventional mixing rule (2PCMR). The calculations of the critical lines and the threephase lines were done at constant reduced mass fractions of npentadecane, using a quasi-binary representation of the ternary systems. The results of the model calculation of holes in the critical surface and miscibility windows are in satisfactory agreement with the available experimental data. The following sections are dedicated to graphical presentations of experimental data from the literature7 used in modeling of ternary systems, to modeling equations and methodology, to results and discussion and conclusions.

Figure 1. Experimental pressure−temperature−reduced mass fraction diagram of the CO2 + 1-heptanol + n-pentadecane system.7 Reduced mass fractions are 0.0, 0.33, 0.62, 0.72, and 1.0 in n-pentadecane.

In the ternary system A + B + C, the solubility of a certain solute B (1-alkanol), in a supercritical solvent A (CO2), may strongly be affected by the addition of a second low-volatile component C (n-pentadecane), by the so-called cosolvency effect. This means that a mixture of two components B and C is more soluble in a supercritical solvent A, than each of the pure components B or C separately. For the quasi-binary representation of a ternary system, one can define the solvent-free, reduced mole fraction or reduced mass fraction red of component C, xred C or WC , respectively: xC WC xCred = ; WCred = x B + xC WB + WC (1) Experimentally or calculating from a model, one can trace critical curves for constant reduced mole or mass fractions of component C, which is n-pentadecane. Many such critical curves, traced for reduced mass fractions between 0 and 1, enable the visualization of the critical surface of the ternary system. Several critical curves measured experimentally are presented in Figure 1, the critical surface has the shape of a seat sagging at medium reduced mass fractions. The data for pressures between 40 and 1000 bar were taken from the literature.7 As can be seen in Figure 1, the critical curves for ternary systems show two types of phase behavior: type III at reduced mass fraction 0.33, and type IV with a discontinuity at reduced mass fractions 0.62 and 0.72. Figure 2 shows the range of reduced mass fractions, where the critical curves have a type IV phase behavior (discontinuities). In the critical surface, these discontinuities determine the occurrence of a hole, which is a two-phase liquid−gas (lg) range in a three-phase liquid− liquid−gas (llg) surface of the ternary system. In the Figure 3 the hole in the critical surface is shown in a temperature-reduced mass fraction projection of critical end points. Critical end points LCEP (lower critical end point) and UCEP (upper critical end point), and the DCEP (double critical end point) data were taken from the literature.7 Between the two DCEP points, the ternary system shows a type IV phase behavior, as pointed out above. Alternatively, the hole in the critical surface is shown in a pressure-reduced mass fraction projection of critical end points. Such a projection will be presented in the section on results of modeling of the paper.

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PHASE EQUILIBRIUM DATA IN TERNARY SYSTEMS CO2 + 1-HEPTANOL + N-PENTADECANE AND CO2 + 1-HEXANOL + N-PENTADECANE: HOLES IN CRITICAL SURFACE AND MISCIBILITY WINDOWS The binary systems carbon dioxide + 1-heptanol and carbon dioxide + n-pentadecane show a type III phase behavior in the classification of Scott and van Konynenburg.4,22−25 The critical curves of binary and quasi-binary mixtures are plotted as critical pressures versus the corresponding temperatures, as shown in Figure 1. B



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Figure 2. Experimental pressure−temperature−reduced mass fraction diagram of the CO2 + 1-heptanol + n-pentadecane.7 Reduced mass fractions are 0.59, 0.62, 0.67, and 0.72 in n-pentadecane. Figure 4. P,T projection of the binary critical curves for CO2− heptanol, CO2−pentadecane, and of the minimum range of the critical curve of CO2 + heptanol + pentadecane at Wred C = 0.67. Experimental data.7 Isobaric miscibility windows, closed, solid line; open to CO2 + 1-heptanol system, dash line; open to both binaries with CO2, dash dot line.

miscibility windows will be presented in the section on results of modeling of the paper. A similar phase equilibrium behavior was experimentally measured in the ternary system CO2 + 1-hexanol + npentadecane.7 The critical curves for ternary systems show two types of phase behavior: type III at reduced mass fraction lower than 0.09 and higher than 0.82, and type IV with a discontinuity at reduced mass fraction in the range between 0.33 and 0.79. Figure 5 shows the range of reduced mass fractions, where the critical curves have a type IV phase behavior (discontinuities), and the critical curves of the pure components, 1-hexanol and n-pentadecane.7 In the critical surface, the discontinuities of type IV critical curves determine the occurrence of a hole,

Figure 3. Hole in the critical surface: temperature-reduced mass fraction projection of critical end points for the system CO2 + 1heptanol + n-pentadecane. Experimental LCEP, UCEP, and DCEP data of the literature.7

In addition to a noncritical lg hole in the critical surface of the ternary system, one observes the phenomenon called a miscibility window. Sections at constant pressures through the critical surface can lead to isobaric critical lines, separating a homogeneous one-phase region from a surrounding two-phase region. Figure 4 shows the range at which isobaric miscibility windows can be observed for the CO2 + 1-heptanol + npentadecane system. In this ternary system, one observes both isobaric miscibility windows and a noncritical lg hole in the critical surface of a ternary system. Two kinds of miscibility windows have been observed: closed, section at solid line pressure of Figure 4, and open to one of binary mixtures (CO2 + 1-heptanol, section at dash line pressure) or to both binaries with CO2 (section at dash dot line pressure). Closed isobaric miscibility windows can be found at pressures between the highest LCEP of the quasi-binary mixtures and the lower pressure minimum of the two binary critical lines. Miscibility windows open to one of the binary mixtures can be found at pressures between the minima of the two binary critical lines. Miscibility windows open to both binaries appear at pressures higher than minima of the two binary critical lines. Examples of

Figure 5. Pressure−temperature−reduced mass fraction diagram of the CO2 + 1-hexanol + n-pentadecane. Reduced mass fractions are 0.0, 0.33, 0.50, 0.67, 0.79, and 1.0 in n-pentadecane.7 C



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which is a two-phase liquid−gas range in three-phase liquid− liquid−gas surface of the ternary system. With decreasing carbon number of the solutes, the critical surface shifts toward lower temperatures and pressures, as miscibility increases, leading to a larger hole in the critical surface of the system with 1-hexanol comparative to that containing 1-heptanol. The hole in the critical surface can be shown in a temperature-reduced mass fraction projection of critical end points, and in a pressure-reduced mass fraction projection of critical end points. Such projections will be presented in the section on modeling results of the paper for the system CO2 + 1-hexanol + n-pentadecane.

The cubic GEOS is a general form for all the cubic EoSs with two, three, and four parameters.17−19 Thus, to obtain the SRK EoS from eq 2, the following restrictions were set: c = −(b/2)2 ; Ωc = −(Ω b/2)2 ;

P=

c = Ωc

ac = Ωa

R2Tc2

d = Ωd

Pc2

Pc

b = Ωb

RTc Pc

(4)

Tr

B(SRK) = (6)

3

Ωb =

2 1 ⎛⎜ 1 − 3B ⎞⎟ ; 36 ⎝ 1 − B ⎠

Zc(SRK) = 1/3

⎛ ∂P ⎞ αc = ⎜ r ⎟ ⎝ ∂Tr ⎠V

r

Ω b = Zc − B

a=

(17)

Ω b = 0.08664

2− 3 2 3

Ωc = − 0.0018766 (18)

Ωa =

1 9( 2 − 1) 3

2 −1 3

∑ ∑ xixjaij ; i

(8)

j

aij = (aiaj)1/2 (1 − kij);

Ωc ≡ C = (1 − B)2 (B − 0.25); D = (1 − B)/2

(16)

(19)

The Ωa, Ωb parameters for SRK EoS are universal. The cubic SRK EoS applied to mixtures was coupled with two parameters conventional mixing rules (2PCMR), derived from the van der Waals one fluid approximation, introducing two binary interaction parameters (BIPs) for a and b:

at Tr = 1 and Vr = 1, leading to eqs 8 and 9:

Ωd = Zc − (1 − B)/2;

(15)

Alternatively, the first eq 16 can be rearranged to a cubic in B and solved giving:

(5)

(7)

Ωa ≡ A = (1 − B)3 ;

mSRK = 0.480 + 1.574ω − 0.176ω 2

Ωd = −0.04332

with the notation U = Zc(Vr − 1) . The expressions of the parameters Ωa, Ωb, Ωc, and Ωd and A, B, C, and D are obtained1,17 by setting four critical conditions in eq 5, with αc the Riedel’s criterion

r

(14)

Ωa = 0.42748

Tr Aβ(Tr) Pr = − U+B (U + D)2 + C

⎛ ∂ 2P ⎞ ⎜ 2r ⎟ = 0 ⎝ ∂Vr ⎠

β(Tr) = [1 + mSRK (1 − Tr0.5)]2

The equation for B can be iteratively solved, starting with an initial approximation of B in the right-hand term. The corresponding values for Ωa, Ωb, Ωc, Ωd are given by eqs 8 and 9:

or

⎛ ∂Pr ⎞ ⎟ =0 ⎜ ⎝ ∂Vr ⎠T

(13)

giving

(3)

Ωaβ(Tr) Tr − ZcVr − Ω b (ZcVr − Ωd)2 + Ωc

Pr = 1

R2Tc2 β(Tr) Pc

B = 0.25 −

The GEOS form in reduced variables is Pr =

(12)

B(SRK) = 0.2467

RTc Pc

26

The methodology to obtain the parameters of the Soave− Redlich−Kwong (SRK) equation of state from eqs 8 and 9 were previously presented, based on restrictions from relations (10, 11).20,27 This results in

where the four parameters a, b, c, and d are given by following relations a(T ) = acβ(Tr)

(11)

a(T ) RT − V−b V (V + b)

a(Tr) = Ωa

(2)

R2Tc2

Ωd = −(Ω b/2)

with the temperature function:

MODELING The cubic general equation of state (GEOS) reduced to Soave−Redlich−Kwong (SRK) EoS, coupled with conventional (van der Waals) mixing rules, was used to model the hole in the critical surface and miscibility windows in CO2 + 1-heptanol + n-pentadecane and CO2 + 1-hexanol + n-pentadecane ternary systems. The cubic GEOS form1,17−20 is a(T ) RT − V−b (V − d)2 + c

(10)

These restrictions lead to the well-known SRK equation:

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P=

d = −b/2

b=

∑ ∑ xixjbij i

(20)

j

bij =

bi + bj 2

(1 − lij)

(21)

and c, d given by eq 10. The values of critical data (Tc, Pc) and acentric factor (ω) of the pure components28 are given in Table 1. The software PHEQ21 (Phase Equilibria Database and Applications), developed by Geană and Rus, and the GPEC29,30 (Global Phase Equilibrium Calculations) package

(9)

A generalized diagram in coordinates c/b − d/b having constant B and Zc as parameters, based on cubic GEOS, was presented in Figure 3.1, p. 27 in literature.1 2

D



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Table 1. Critical Data (Tc, Pc) and acentric factor (ω) of the pure components28 compounds

Tc/K

Pc/bar

ω

carbon dioxide 1-hexanol 1-heptanol n-pentadecane

304.12 611.4 631.9 708

73.74 35.1 31.5 14.8

0.225 0.573 0.588 0.6863

Cismondi et al.37 proposed a simple procedure to obtain optimized rescaled parameters of PC-SAFT EoS from only Tc, Pc, and the acentric factor, and implemented the model in GPEQ software.30 The approach of Polishuk37 requires for the numerical solution of the PC-SAFT parameters the critical constants and the triple-point liquid density. The implementation, named CP- PC-SAFT, was made by re-evaluating part of the PC-SAFT universal parameters matrix and some additional revisions. Cubic EOSs remained, in spite of these advances, important tools in modeling of phase equilibria in complex mixtures. Moreover, cubic EOSs have been extended to the semiempirical hybrid model, as perturbed hard sphere PHSGEOS36,39,40 and SAFT + cubic EOS.41

were used for calculation of critical curves and three phases llg lines. PHEQ package was extended with module GEOS CRITHK for the calculation of critical lines, using the Heidemann and Khalil method16 with the numerical derivatives given by Stockfleth and Dohrn.31 Three phase llg lines were calculated with module GEOS LLVE. A modification of the algorithm of Heidemann and Khalil enables the calculation of all critical points in the P−T stability limit equation, at constant mole fraction. The computer programs, written in FORTRAN, were used in numerous papers, for example, refs 27 and 32−36, and for various mixtures of components. Comparative calculations with the SRK EoS were made with GPEC software, for confirming the results obtained with GEOS modules. The modules GEOS CRITHK and GEOS CRITHK were used in this paper for the calculation of holes in critical surfaces, and of the miscibility windows. Since the early 1990s, advanced models in the form of the statistical association fluid theory (SAFT) were applied in equations of state.3 These equations of state have the advantage of a relationship between molecular and macroscopic levels of pure substances and mixtures. But these SAFT-type models do not reproduce well the critical behavior of pure components and mixtures.33 To remove this weakness, two papers of Cismondi et al.37 and Polishuk38 proposed approaches that require the critical constants of compounds for the numerical solution of the PC-SAFT parameters, following the procedure used for a long time in cubic EoSs.

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RESULTS AND DISCUSSION The modeling of holes in the critical surface and miscibility windows was made with the cubic SRK EoS, integrated in the GEOS form. Two-parameters conventional mixing rule (2PCMR) was used for mixtures. The calculation of the critical lines and the three-phases lines at constant reduced mass fractions was adopted for a quasi-binary representation of the ternary systems. As far as we know, this approach was not used previously in the literature, to model the hole in the critical surface of such systems. The quasi-binary systems were mixtures of CO2 and a pseudocomponent consisting of 1alcohol + n-pentadecane of constant reduced mass fraction (Wred C ). The reduced mass fractions were those used in the experimental measurements. 7 The critical data of the pseudocomponent, necessary in the calculations of critical curves and llg lines of the quasi-binary systems, were estimated as linear combinations of pure component critical data and acentric factors through reduced mole fractions: Ycpc = (1 − xCred)YcB + xcred Y cC

(22)

with Y = T, P, ω for critical temperature, critical pressure, and acentric factor.

Figure 6. Calculated critical lines and three-phases line at reduced mass fraction 0.573 in n-pentadecane, for the system CO2 + 1-heptanol + npentadecane (a). Enlarged DCEP range (b). Calculated DCEP is compared with measured data of the literature.7 Estimated parameter set k12 = 0.078 and l12 = −0.065. E



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curves at DCEPs reduced mass fraction were not available. Reproducibility of the type of phase diagram at the DCEP, a calculated critical curve tangent at the three phases line, was also implicitly imposed. The set of parameters obtained by this procedure was used to predict the critical and three phases llg lines for all quasi-binary systems at reduced mass fractions in the range between the two experimental DCEPs. The experimental DCEPs of the system CO2 + heptanol + pentadecane were taken from the literature.7 Figures 6 and 7 show the critical lines calculated for quasibinary systems at the reduced mass-fraction of 0.573 and 0.721 in n-pentadecane. As can be seen the calculated critical curve is tangent at the three phases line, in the double critical end point (DCEP, filled circle). Also given in Figures 6 and 7 is the threephase line, as well as the experimental measured DCEP (quadrat). The unique parameter set estimated by the trial and error procedure is k12 = 0.078 and l12 = −0.065. Using the set of parameters k12 = 0.078 and l12 = −0.065, predictive calculations were made at experimental reduced mass-fractions in the range between the two DCEPs. The diagrams in Figure 8 display the type IV behavior of the calculated critical line and three-phases line at reduced mass fraction 0.67 in n-pentadecane, for the system CO2 + 1heptanol + n-pentadecane. Calculated LCEP and UCEP are marked on the diagram. The SRK EoS calculations with parameter set k12 = 0.078 and l12= −0.065 are compared with the measured data in the literature.7 As can be seen the calculated critical line shows a discontinuity between the two critical end points (UCEP and LCEP). Similar calculations were made at all experimental reduced mass-fractions of n-pentadecane (notation C), which lead to the hole in the critical surface obtained from the SRK equation of state. This is also a two-phases liquid−gas (lg) range in three-phases liquid−liquid−gas (llg) surface of the ternary system, as shown in Figure 8 (discontinuity both in the critical and three-phase line). Figures 9 and 10 depict the

Figure 7. Calculated critical line and three phases line at reduced mass fraction 0.721 in n-pentadecane, for the system CO2 + 1-heptanol + npentadecane. Calculated DCEP is compared with measured data of the literature.7 Estimated parameter set k12 = 0.078 and l12 = −0.065.

For calculating the critical lines and the three-phase lines in quasi-binary systems, the values of the two-parameters conventional mixing rule (2PCMR) are necessary. After several options tested, a simple semipredictive approach (SPA) was adopted for the estimation of the two interaction parameters by a trial and error procedure. Only the deviations in calculated temperature and pressure of the two double critical end points (DCEPs) versus experimental values were used in the optimization criteria. Other experimental points of the critical

Figure 8. Calculated critical line and three phases line at reduced mass fraction 0.67 in n-pentadecane, for the system CO2 + 1-heptanol + npentadecane. Calculated LCEP and UCEP are marked on the diagram (a). Enlarged LCEP and UCEP range (b). The SRK EoS calculations with parameter set k12 = 0.078 and l12= −0.065 are compared with measured data in the literature.7 F



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Figure 9. Calculated critical line and three phases line at reduced mass fraction 0.592 in n-pentadecane, for the system CO2 + 1-heptanol + npentadecane (a). Enlarged LCEP and UCEP range (b). The SRK EoS calculations with parameter set k12 = 0.078 and l12= −0.065 are compared with measured data of the literature.7

Figure 10. Calculated critical line and three phases line at reduced mass fraction 0.625 in n-pentadecane, for the system CO2 + 1-heptanol + npentadecane (a). Enlarged LCEP and UCEP range (b). The SRK EoS calculations with parameter set k12 = 0.078 and l12 = −0.065 are compared with measured data of the literature.7

taken from the literature.7 Between the two DCEP points, the ternary system shows a type IV phase behavior, as pointed out above. In Figure 12 the hole in the critical surface, calculated with SRK EoS, is shown in a pressure-reduced mass fraction projection of critical end points. Experimental critical end points LCEP (lower critical end point), UCEP (upper critical end point), and the DCEP (double critical end point) data are taken from the literature.7

representative results for reduced mass-fractions of npentadecane of 0.592, and 0.625, respectively. The agreement with experimental points is satisfactory considering the predictive calculations. In Figure 11 the hole in critical surface, calculated with SRK EoS, is shown in a temperature-reduced mass fraction projection of critical end points. Experimental critical end points LCEP (lower critical end point), UCEP (upper critical end point), and the DCEP (double critical end point) data are G



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Figure 13. Closed miscibility window in the system CO2 + 1-heptanol + n-pentadecane: temperature-reduced mass fraction projection of critical curves. The SRK EoS calculations with parameter set k12 = 0.078 and l12 = −0.065 are compared with interpolation at 80 bar between experimental points measured in the literature.7

Figure 11. Hole in the critical surface: temperature-reduced mass fraction projection of critical end points for the system CO2 + 1heptanol + n-pentadecane. The SRK EoS calculations with parameter set k12 = 0.078 and l12 = −0.065 are compared with experimental LCEP, UCEP, and DCEP data.7

semipredictive approach for obtaining the BIPs. Table 2 gives the deviations of SRK EoS calculated values of CEPs versus experimental data. The average absolute deviations (AAD) are 1.69 K and 2.86 bar. Ternary systems can exhibit both hole in critical surface and miscibility windows; it is the case of the system CO2 + 1heptanol + n-pentadecane. A miscibility window is a homogeneous one-phase region, which is separated from the surrounding heterogeneous states by a closed loop of critical states. In Figures 8−10, at pressures above the LCEP, a section gives two points of intersection with the critical lines, for the quasi-binary system at fixed reduced mass fraction. The locus of such points for a lot of critical quasi-binary curves, at different reduced mass fractions, is an isobaric miscibility window, for the ternary system. Figure 13 shows a closed isobaric miscibility window calculated with SRK EoS (filled quadrat and drawn line) at 80 bar (section at a pressure of the solid line in Figure 4), and the parameter set k12 = 0.078 and l12 = −0.065. The experimental data measured in the literature7 for the system CO2 + 1-heptanol + n-pentadecane (filled circles and drawn curve), are shown in the Figure 13 for comparison. As can be seen the calculated miscibility window is in satisfactory agreement with the measured data, considering the semipredictive approach for obtaining the BIPs. Similar calculations were made for the system CO2 + 1hexanol + n-pentadecane. For estimating the unique set of BIPs, the two DCEPs measured in the literature7 at reduced mass fractions of n-pentadecane (notation C) of 0.092 and 0.815 were used. As example, Figure 14 shows the critical lines calculated for the quasi-binary system at the reduced massfraction of 0.815 in n-pentadecane. As can be seen the calculated critical line is tangent at the three phases line, in the double critical end point (DCEP, filled circle). Also given in the Figure 14 is the three phases line, as well as the experimental measured DCEP (quadrat). The parameter set estimated by the trial and error procedure is k12 = 0.076 and l12 = −0.05. Using the unique set of parameters k12 = 0.076 and l12 = −0.05, predictive calculations were made at experimental reduced mass-fractions of n-pentadecane (notation C), in the range between the two DCEPs. To exemplify, Figures 15 display the type IV behavior of the calculated critical line and three-phases line at reduced mass fraction 0.67 in n-

Figure 12. Hole in the critical surface: pressure-reduced mass fraction projection of critical end points for the system CO2 + 1-heptanol + npentadecane. The SRK EoS calculations with parameter set k12 = 0.078 and l12 = −0.065 are compared with experimental LCEP, UCEP, and DCEP data.7

Table 2. T and P Deviationsa of Calculated CEPs in Ternary CO2 + 1-Heptanol + n-Pentadecane System

a

CEP

Wcred

ε(T)/K

ε(P)/bar

DCEP LCEP LCEP LCEP LCEP DCEP UCEP UCEP UCEP UCEP

0.573 0.592 0.625 0.675 0.715 0.721 0.715 0.675 0.625 0.592 AAD

1.31 1.99 −0.81 −2.05 3.25 −1.85 −1.70 0.94 1.20 −1.84 1.69

−1.52 0.14 −2.64 −5.32 1.21 −5.45 −4.23 −1.49 −0.74 −5.84 2.86

Difference between calculated and experimental values.

As can be seen the calculated projections of the hole are in good agreement with the measured data, considering the H



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Figure 14. Calculated critical lines and three-phases line at reduced mass fraction 0.815 in n-pentadecane, for the system CO2 + 1-hexanol + npentadecane (a). Enlarged DCEP range (b). Calculated DCEP is compared with measured data of the literature.7 Estimated parameter set k12 = 0.076 and l12 = −0.05.

Figure 15. Calculated critical line and three-phases line at reduced mass fraction 0.67 in n-pentadecane, for the system CO2 + 1-hexanol + npentadecane (a). Enlarged LCEP and UCEP range (b). The SRK EoS calculations with parameter set k12 = 0.076 and l12 = −0.05 are compared with measured data of the literature.7

pentadecane, for the system CO2 + 1-hexanol + n-pentadecane. The SRK EoS calculations with parameter set k12 = 0.076 and l12 = −0.05 are compared with measured data of the literature.7 As can be seen the calculated critical line shows a discontinuity, between the two critical end points (UCEP and LCEP). The agreement with experimental points is satisfactory considering the predictive calculations. The calculations made at all experimental reduced massfractions lead to the hole in the critical surface obtained from the SRK equation of state. This is also a two-phases liquid−gas (lg) range in three-phases liquid−liquid−gas (llg) surface of the ternary system.

In Figure 16 the hole in critical surface, calculated with SRK EoS, is shown in a temperature-reduced mass fraction projection of critical end points. Experimental critical end points LCEP (lower critical end point), UCEP (upper critical end point), and the DCEP (double critical end point) data are taken from the literature.7 Between the two DCEP points, the ternary system shows a type IV phase behavior, as pointed out above. In Figure 17 the hole in the critical surface, calculated by SRK EoS with the unique set of BIPs, is shown in a pressurereduced mass fraction projection of critical end points. Experimental critical end points LCEP (lower critical end I



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Table 3. T and P Deviationsa of Calculated CEPs in Ternary CO2 + 1-Hexanol + n-Pentadecane System

Figure 16. Hole in the critical surface: temperature-reduced mass fraction projection of critical end points for the system CO2 + 1hexanol + n-pentadecane. The SRK EoS calculations with parameter set k12 = 0.076 and l12 = −0.05 are compared with experimental LCEP, UCEP, and DCEP data.7

a

CEP

Wcred

ε(T)/K

ε(P)/bar

DCEP LCEP LCEP LCEP LCEP LCEP LCEP DCEP UCEP UCEP UCEP UCEP UCEP UCEP

0.092 0.133 0.167 0.333 0.5 0.667 0.79 0.815 0.79 0.667 0.5 0.333 0.167 0.133 AAD

−5.84 −0.37 −2.95 −10.1 −12.7 −9.65 −0.67 −3.18 −7.35 −0.18 4.78 2.04 −4.45 −9.24 5.26

−11.1 −4.55 −7.92 −18.9 −21.4 −16.2 −2.83 −6.46 −10.1 −4.73 1.35 −1.25 −9.31 −13.8 9.29

Difference between calculated and experimental values.

Figure 18. Miscibility window in the system CO2 + 1-hexanol + npentadecane: temperature-reduced mass fraction projection of critical curves. The SRK EoS calculations with parameter set k12 = 0.076 and l12 = −0.05 are compared with interpolation at 90 bar between experimental points measured in the literature.7

Figure 17. Hole in the critical surface: pressure-reduced mass fraction projection of critical end points for the system CO2 + 1-hexanol + npentadecane. The SRK EoS calculations with parameter set k12 = 0.076 and l12= −0.05 are compared with experimental LCEP, UCEP, and DCEP data.7

calculated projection curves, as compared to that of experimental data. This fact might be determined by the linear combination rule used for calculating the critical values and the acentric factor of the pseudocomponents in eq 22, and/or unique set of the binary interaction parameters in all range of reduced mass fraction (Wred C ). The system CO2 + 1-hexanol + n-pentadecane can exhibit both hole in critical surface and miscibility windows. In Figure 15, at pressures above the LCEP, a section gives two points of intersection with the critical lines, for the quasi-binary system at fixed reduced mass fraction. The locus of such points for many critical quasi-binary curves, at different reduced mass fractions, gives an isobaric miscibility window for the ternary system. Figure 18 shows an isobaric miscibility window open to the binary subsystem CO2 + 1-hexanol, calculated with the SRK EoS (filled quadrat and drawn line) at 90 bar (section at a pressure between the two critical curves of the binary subsystems with CO2 in Figure 4), and the parameter set k12 = 0.076 and l12 = −0.05. The experimental data measured in the literature,7 for the system CO2 + 1-hexanol + n-pentadecane

point), UCEP (upper critical end point), and the DCEP (double critical end point) data are taken from the literature.7 Table 3 gives the deviations of SRK EoS calculated values of CEPs versus experimental data. The average absolute deviations (AAD) are 5.26 K and 9.29 bar. As compared with the AAD values from Table 2 for the ternary CO2 + 1-heptanol + npentadecane system, the deviations of calculated CEPs are higher. A possible explanation of the higher deviations in this system might be that the parameters in mixing rules are dependent on the reduced mass fraction, and a constant set of parameter values is an approximation. This approximation leads to higher deviations in the wide range of the reduced mass fraction of the hole in the critical surface of the system containing 1-hexanol, in comparison with the system containing 1-heptanol. The calculated projections of the hole are in satisfactory agreement with the measured data, considering the semipredictive approach for obtaining the BIPs. At the same time, both ternary systems present a change of the convexity in the J



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(filled circles and drawn curve), are shown in the Figure 18 for comparison. As can be seen the calculated miscibility window is in satisfactory agreement with the measured data, considering the semipredictive approach for obtaining the BIPs.

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CONCLUSIONS The modeling of holes in the critical surface and miscibility windows was done with computer programs developed by the author and co-workers. An intelligent database and calculation software, for phase equilibrium (PHEQ) data management and application, was used for calculations with the cubic general equation of state (GEOS), developed by the author.17−20 The software package PHEQ was extended for calculations of critical lines (module GEOS CRITHK), using the Heidemann and Khalil method, and llg lines (module GEOS LLVE) in binary and ternary mixtures. These extensions of the software enable the calculation of holes in critical surfaces, as well as of miscibility windows. Binary and quasi-binary measured phase equilibrium data by Scheidgen7 were necessary for understanding and modeling of ternary systems carbon dioxide + 1-hexanol + pentadecane and carbon dioxide + 1-heptanol + pentadecane, regarding holes in critical surface and miscibility windows. For modeling, the equation of state Soave−Redlich−Kwong (SRK), integrated in the cubic GEOS form, coupled with two parameters conventional mixing rule (2PCMR) was used. A quasi-binary representation of the ternary systems was adopted. A simple semipredictive approach (SPA) was used for the estimation of the two binary interaction parameters (BIPs) by a trial and error procedure for reproducing the temperature and pressure values of the two experimental DCEPs. A unique set of parameters obtained by this procedure was used to predict the critical and three phases llg lines for all quasi-binary systems at reduced mass fractions of pentadecane in the range between the two experimental DCEPs. The calculated holes in the critical surface and miscibility windows are in satisfactory agreement with the available experimental data.

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R = universal gas constant SPA = semipredictive approach SRK = Soave−Redlich−Kwong equation of state T = temperature U = notation in GEOS UCEP = upper critical end point, terminating a three-phase locus toward higher temperatures and pressures V = volume in EoS vdW = van der Waals VLE = vapor−liquid equilibria W = mass fraction Y = notation for critical data of pseudocomponent a, b, c, d = GEOS parameters k, l = binary interaction parameters lg = liquid−gas (vapor) phases llg = liquid−liquid-gas (vapor) phases m = parameter in the equation of state x = mole fraction

GREEK LETTERS αc = Riedel’s criterion β(Tr) = temperature function in equation of state Ωa, Ωb, Ωc, Ωd = parameters in GEOS ω = acentric factor ε(T), ε(P) = deviations in T or P, differences between calculated and experimental values SUBSCRIPTS A, B, C = component of ternary mixture c = critic r = reduced SUPERSCRIPTS red = reduced REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Dan Geană: 0000-0003-2532-1575 Notes

The author declares no competing financial interest.

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LIST OF SYMBOLS A, B, C, D = GEOS parameters A, B, C = components of ternary system AAD = average absolute deviation BIPs = binary interaction parameters CEP = critical end point DCEP = double critical end point EoS = equation of state GEOS = cubic General Equation of State LCEP = lower critical end point, terminating a three-phase locus toward lower temperatures and pressures LLE = liquid−liquid equilibria LLVE = liquid−liquid−vapor equilibria P = pressure 2PCMR = two parameters conventional (van der Waals) mixing rules K



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