Measurement and modeling of the phase behavior of supercritical carbon dioxide + polydisperse non-ionic surfactant systems

Fluid Phase Equilibria 287 (2009) 7–14

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Measurement and modeling of the phase behavior of supercritical carbon dioxide + polydisperse non-ionic surfactant systems Masashi Haruki ∗ , Yuichi Kaida, Kazuhiko Matsuura, Shin-ichi Kihara, Shigeki Takishima Department of Chemical Engineering, Graduate School of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima 739-8527, Japan

a r t i c l e

i n f o

Article history: Received 27 May 2009 Received in revised form 20 August 2009 Accepted 21 August 2009 Available online 28 August 2009 Keywords: Supercritical carbon dioxide Surfactant Phase transition Sanchez–Lacombe Group contribution method

a b s t r a c t Phase transition pressures of the supercritical carbon dioxide (scCO2 ) + polyethylene-2,6,8-trimethyl-4nonyl ether (TMN) systems were measured, using a synthetic method, at temperatures from 313 to 343 K. Three polydisperse TMN samples with different average molecular weights, due to the varying numbers of ethylene oxide groups in the molecules, were used in the experiment: TMN-10, TMN-6 and TMN-3. The molecular weight distribution of each TMN was determined using a mass spectrometer with a time ¯ n, M ¯ w and M ¯ z values [g/mol] were 718.8, 746.9 and 775.7 for TMN-10, 624.2, 646.8 of flight detector. The M and 668.1 for TMN-6, and 397.5, 409.4 and 421.2 for TMN-3, respectively. The phase transition pressures increased with increasing temperature for all systems, and also increased as the average molecular weight increased at a constant temperature and TMN weight fraction. A new prediction model based on the Sanchez–Lacombe equation of state was developed to predict the phase boundaries by introducing the group contribution method. The group parameters, which are the segment interaction parameter, the number of segments per TMN molecule, and the volume occupied by one segment, were determined by correlating the experimental phase transition pressures for the scCO2 + TMN-10 and the scCO2 + TMN-6 systems. Although some deviations were observed between the correlated and experimental results of the scCO2 + TMN-10 system due to the inaccuracy of the correlation of the temperature dependency of the phase transition pressures, the correlated results for both systems reproduced the experimental results moderately well. The predicted results were also in good agreement with the experimental results for the phase transition pressures of the scCO2 + TMN-3 system. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Supercritical carbon dioxide (scCO2 ) is a typical green solvent, and many processes have been developed for the extraction of flavors and pharmaceutical compounds from natural products [1]. However, the application of scCO2 has been limited to processes dealing with weakly polar and low-molecular-weight substances, because highly polar and high-molecular-weight substances are less soluble in scCO2 . Studies of water in CO2 (W/CO2 ) emulsions have been carried out to expand the range of applications of scCO2 . In the semiconductor manufacturing, W/CO2 emulsions distributed in scCO2 have shown distinctive advantages compared with pure scCO2 to remove post-etch residues and drying of the surfaces of wafers [2–5]. Moreover, investigations on the preparation of inorganic nanoparticles using W/CO2 microemulsion have obtained particles applicable to optical use, as well as the catalytic agents [6–8]. Promising engineering applications have been

∗ Corresponding author. Tel.: +81 82 424 7713; fax: +81 82 424 7713. E-mail address: [email protected] (M. Haruki). 0378-3812/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2009.08.019

reviewed as well as basic studies [9,10]. Some experimental investigations have reported on the formation and the size distribution of W/CO2 emulsions by visualizing the phase behavior using either a dynamic light scattering technique [11] or Fourier transform infrared (FT-IR) spectroscopy [12]. Sagisaka et al. [13–16] systematically investigated the effect of the structure of surfactants on the phase behavior of the scCO2 + surfactant + water systems. Sodium bis(1H,1H,2H,2H-heptadecafluorodecyl)-2-sulfosuccinate (8FS(EO)2), an Aerosol-OT (AOT) analogue fluorinated twin-tailstype surfactant, was found to be an excellent surfactant for the formation of W/CO2 microemulsions. The effect of mixed surfactants and the tail length of the surfactant on the ability to form W/CO2 microemulsions was also investigated [17]. Takebayashi et al. [12] investigated the FT-IR spectrum of W/CO2 microemulsions. The results indicate that water molecules are introduced preferentially into CO2 and the interfacial area when the water content is low, and the water molecules are then loaded into the micelle core after the saturation of CO2 with full hydration of the surfactant headgroups. In addition to the phase behavior of scCO2 + surfactant + water ternary systems, investigations on the scCO2 + surfactant systems are also important for the development of engineering

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M. Haruki et al. / Fluid Phase Equilibria 287 (2009) 7–14

processes utilizing W/CO2 microemulsions dispersed in scCO2 . Liu et al. [18] measured the solubility of four types of AOT analogue surfactants (sodium dibutyl sulfosuccinate, sodium dipentyl sulfosuccinate, sodium dihexyl sulfosuccinate, and sodium dioctyl sulfosuccinate) in scCO2 and supercritical 1,1,1,2-tetrafluoroethane (HFC-134a), using a static method coupled with a gravimetric method. The solubilities of these surfactants in HFC-134a were much higher than in scCO2 . Moreover, the solubility of the surfactants in scCO2 increased as the number of carbon atoms in CO2 -philic tails of the surfactant increased, whereas in HFC134a, the solubility decreased as the number of carbon atoms in the surfactant increased. Park et al. [19] also investigated the phase behavior of the scCO2 + fluorinated analogue of AOT, sodium salt of bis(2,2,3,3,4,4,5,5-octafluoro-1-pentanol) sulfosuccinate (di-HCF4). The phase transition pressures increased as the weight fraction of di-HCF4 increased at a constant temperature, and with increasing temperature at a constant di-HCF4 weight fraction. Moreover, they investigated the phase behaviors for the scCO2 + di-HCF4 + water system to clear the effect of the amount of di-HCF4, water and CO2 on phase transition and obtained noteworthy results. Two cloud points were observed: upper and lower cloud points which are defined as the higher pressure and lower pressure boundaries of the possible region for formation of stable W/CO2 emulsions. The pressures of the upper cloud point decreased as the amount of di-HCF4 increased as water content remained constant. On the other hand, the pressure of the lower cloud point was not dependant on the amount of diHCF4. Non-ionic surfactants with CO2 -philic tails formed by hydrocarbon groups have many advantages compared with those formed by fluorocarbon groups, from the viewpoint of economic efficiency, environmental load and the effect on human health. Although some kinds of ionic surfactants and non-ionic surfactants with CO2 -philic tails composed of fluorocarbons can disperse W/CO2 emulsions in scCO2 , few surfactants with hydrocarbon CO2 -philic tails can produce W/CO2 emulsions. Among them, polyethylene2,6,8-trimethyl-4-nonyl ether (TMN), shown in Fig. 1, has received much attention. The balance of the hydrophilic and the CO2 -philic properties of TMN can be controlled by changing the number of ethylene oxide (EO) groups. Ryoo et al. [20] researched the phase behavior of the scCO2 + TMN + water and the scCO2 + surfactant with a linear hydrocarbon in CO2 -philic tails + water systems. They reported that the surfactant with branched hydrocarbon tails such as TMN is better than that with linear hydrocarbon tails for the formation of W/CO2 emulsions. In our previous work, the microscopic phase behavior of the scCO2 + TMN + water system was measured using a synthetic method to develop an understanding of the phase behavior of W/CO2 emulsions formed by non-ionic hydrocarbon surfactants. The TMN used, which was manufactured as TMN-3, had a molec-

Fig. 1. Chemical structure of TMN. HC, EO and OH indicate hydrocarbon, ethylene oxide and hydroxyl, respectively.

ular weight distribution caused by differences in the number of EO groups, and the number- and weight-average molecular weights were 397.5 and 409.4 g/mol, respectively. The measurements were carried out at a constant TMN-3 concentration of 1 wt% and at water concentrations from 0.28 to 1.56 wt%. By gradually decreasing the pressure from about 50 MPa, two different types of phase transitions were observed using a microscope at about 700fold magnification. In the measurement, a transparent phase was observed at high pressures where TMN-3 and water were soluble or formed a W/CO2 microemulsion. The pressure was decreased at constant temperature to the upper phase transition pressure where a light turbidity was observed indicating the formation of an macroemulsion phase. When the pressure was further decreased at constant temperature, the lower phase transition pressure was reached and the emulsion phase separated into two distinct phases [21]. The macroscopic phase transition is also necessary to investigate the phase behavior of solutions containing W/CO2 emulsions as well as the microscopic behavior. Therefore, in this work, the phase behavior of the scCO2 + TMN systems was systematically measured, using a synthetic method, to clarify the effect of the structure of TMN on the phase behavior. The phase behavior of three different TMN samples in scCO2 was investigated. The TMN samples differed in their average molecular weights due to the number of EO units in the TMN molecules as well as the molecular weight distributions. Therefore, each scCO2 + TMN system included multiple kinds of TMNs with different molecular weights. A new prediction model based on the Sanchez–Lacombe equation of state was also developed to predict the phase boundaries of the scCO2 + TMN systems accurately and effectively by introducing the group contribution method.

2. Experiment 2.1. Materials Carbon dioxide was purchased from Iwatani Industrial Gases and had a purity greater than 99.95 mol%. Tergitol® TMN-10, TMN6 and TMN-3 were purchased from FLUKA and their purities were better than 90, 90 and 99 wt%, respectively. Molecular sieves were used to further purify the TMN-6 and TMN-10. The water content of TMN-10, TMN-6 and TMN-3, measured with Karl-Fischer titration (DIA Instruments, KF-21 type), was 0.17, 0.22 and 1.08 wt%, respectively, after purification. The chemical structure of TMN (polyethylene-2,6,8-trimethyl-4-nonyl ether) is shown in Fig. 1. TMNs consist of highly branched CO2 -philic hydrocarbon (HC) groups, hydrophilic ethylene oxide (EO) groups, and a hydroxyl (OH) group. The TMN-10, TMN-6 and TMN-3 used in this study had the molecular weight distributions. TMN-3 used in this work had the same lot number as TMN-3 used in a previous study, and, therefore, the molecular weight distribution and the average molecular weight were the same as previously obtained by mass spectrometric analysis using MALDI-TOF/TOF/MS/MS spectroscopy (BRUKER DALTONICS, Ultra flex I) [21]. On the other hand, the molecular weight distributions of TMN-10 and TMN-6 were estimated in this study from the analytical results of MALDI-TOF/MS spectroscopy (Shimadzu Co., AXIMA-CFR plus). The molecular weight distributions of TMN used in this work are presented in Table 1 and Fig. 2. The molecular weights of TMN-10 and TMN-6 were distributed at 44 g/mol intervals, as was that of TMN-3, as reported in our previous study shown in Fig. 2. The interval of 44 g/mol corresponds to the molecular weight of one EO group. The average number of EO groups for TMN-10, TMN-6 and TMN-3 was 12.1, 9.96 and 4.81, ¯ n, M ¯ w and M ¯ z values of each TMN are listed respectively, and the M in Table 2.

M. Haruki et al. / Fluid Phase Equilibria 287 (2009) 7–14 Table 1 Molecular weight distribution of TMNsa . TMN-10

TMN-6

TMN-3

M

n

x

M

n

x

M

n

x

494 538 582 626 670 714 758 802 846 890 934 978 1022 1066 1110

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

0.054 0.081 0.099 0.119 0.113 0.099 0.095 0.078 0.089 0.053 0.036 0.023 0.018 0.014 0.009

406 450 494 538 582 626 670 714 758 802 846

5 6 7 8 9 10 11 12 13 14 15

0.043 0.072 0.079 0.140 0.122 0.128 0.107 0.097 0.084 0.087 0.041

230 274 318 362 406 450 494 538 582

1 2 3 4 5 6 7 8 9

0.012 0.046 0.143 0.224 0.268 0.191 0.066 0.027 0.025

a

Fig. 2. Molecular weight distributions of TMNs used. n indicates the number of EO group.

2.2. Apparatus A synthetic method was used to measure the phase separation pressures for the scCO2 + TMN systems. Although the composition of a newly appearing phase cannot be determined in multi-component systems, this method can eliminate the error in composition that often occurs in the sampling of components in the cell at high pressure. The schematic diagram of the apparatus is illustrated in Fig. 3. The apparatus consists mainly of a variable-volume optical cell with two sapphire windows (Hikari Koatsu); a CO2 injection system with a small sample cylinder; a hand pump (Hikari Koatsu); a measuring device for the position of a free piston. The internal volume of the cell can be varied from 8 to 15 cm3 by moving the free piston. A Teflon-coated magnetic stirring bar was used to promote mixing to avoid supersaturation of the sample in the cell. The location of the free piston was obtained using both a linear-variable differential transformer (LVDT, Shinko Electric, model 6020) and a digital linear scale (Mitsutoyo, model SD-10B). The cell was heated using four 200 W cartridge heaters arranged in the cell body with a PID controller (Shimaden, SR94). The temperature of the cell was measured with a platinum resistance temperature detector (Okazaki, Table 2 Average molecular weights of TMNs.

a

na

¯ n [g/mol] M

¯ w [g/mol] M

¯ z [g/mol] M

12.1 9.96 4.81

718.8 624.2 397.5

746.9 646.8 409.4

775.7 668.1 421.2

Number of EO groups.

RESIOPAK R35) connected to a thermometer (Yokogawa, Digital thermometer 7563) with an uncertainty of 0.05 K. As the cell has a large heat capacity and was thermally insulated using glass wool, the uncertainty of the temperature of the sample in the cell was estimated to be within ±0.1 K throughout the experiment. The hand pump supplied pressure, measured with a precision Heise gauge (Dresser Industries, model 901A, F.S. 100 MPa), which had been calibrated against a dead weight tester (Pressurements, model M2800, range 0.01–100 MPa, uncertainty 0.025%). The pressure difference between both sides of the free piston was predetermined and did not exceed 0.1 MPa, and the uncertainty of the pressure gauge was 0.015 MPa. Therefore, the uncertainty of the pressure due to the instrument was estimated to be within ±0.1 MPa throughout the experiment. 2.3. Method

M: molecular weight [g/mol], n: number of EO groups, x: molar fraction.

TMN-10 TMN-6 TMN-3

9

TMN was introduced into the high-pressure optical cell using micro-syringes. The amount of TMN introduced was measured with an electric balance. CO2 was first introduced into a small sample cylinder. After replacing the air in the cell with CO2 at atmospheric pressure, CO2 in the sample cylinder was introduced into the cell. The CO2 remaining in the stainless steel tube that was used to connect the sample cylinder and the cell was recovered in the sample cylinder by condensation using liquid nitrogen. The amount of CO2 introduced was determined by measuring the difference in the mass of the sample cylinder before and after the introduction. The amount of atmospheric CO2 initially introduced was estimated with the ideal gas equation of state, and was added to obtain the total amount of CO2 introduced. The total uncertainty in the sample composition was conservatively estimated to be ±0.5 wt% including the uncertainty due to dehydrating the TMN samples. After the cell reached the desired temperature, the pressure in the cell was raised to the point where a transparent homogeneous phase was expected to appear. After the phase in the cell became stable and transparent, the pressure in the cell was gradually decreased and the phase behavior was observed through the sapphire view windows while recording the values of the temperature, T, and pressure, P. In this study, the phase transition pressures were determined by visual observation of the mixture in the cell. As one example of the phase behaviors observed in the measurement, the images of the phase behavior for the scCO2 + TMN-10 system at 323 K and 28.6 wt% of TMN-10 are shown in Fig. 4. The clear transparent phase was observed at higher pressures than the phase transition pressure (images (a) and (b)). At the phase transition pressure, the transparent phase dramatically changed into a cloudy phase (image (c)). As the pressure proceeded to decrease, the new phase was observed clearly (image (d) and (e)). In our measurement, one phase transition was observed multiple times by alternating between pressurizing and depressurizing. The uncertainty of phase transition pressure due to the uncertainty of identification of the phase transition was slightly dependent on the composition of the mixture, which indicates that it is more difficult to identify the phase transition at a significantly low concentration of TMN. Therefore, the measurements were carried out at the composition where the phase transition could be clearly observed. Observational uncertainty in the phase transition pressure measurement was within ±0.1 MPa throughout the experiment. Measurements were carried out at pressures from 10 to 50 MPa and at temperatures from 313 to 343 K. 2.4. Results and discussion The experimental results of the phase transition pressure of the scCO2 + TMN-10 are listed in Table 3 and illustrated in Fig. 5. The phase transition pressure increased as temperature increased at

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M. Haruki et al. / Fluid Phase Equilibria 287 (2009) 7–14

Fig. 3. Schematic diagram of experimental apparatus. (1) Oil reservoir; (2) pump; (3) pressure gauge; (4) pressure damper; (5) water bath; (6) linear scale; (7) linear-variable differential transformer (LVDT); (8) displacement meter; (9) temperature controller; (10) temperature indicator; (11) hotplate stirrer; (12) heat insulator; (13) free piston; (14) cartridge heaters; (15) magnetic stirring bar; (16) sapphire windows; (17) sample cylinder; (18) vacuum pump.

all investigated concentrations of TMN-10. At 19.6 and 21.1 wt% of TMN-10, the color of the transparent phase turned to deep orange in the vicinity of the phase transition pressure, and then the separated phases appeared. On the other hand, the new phase formed a bottom phase below 17.0 wt% of TMN-10 and an upper phase above 28.6 wt% of TMN-10, without the phenomena observed in the vicinity of the phase transition pressures at 19.6 and 21.1 wt% of TMN-10. Therefore, the critical points of the scCO2 + TMN-10 system were approximately 20 wt% of TMN-10 for all measurement temperatures. The critical point of the scCO2 + polydisperse component system is generally not located at the maximum pressure of the cloud point curve, unlike the scCO2 + monodisperse component system which has one binodal curve at a constant temperature. In the scCO2 + polydisperse system, there is a cloud point curve, and a shadow curve, which shows the composition of the new phase at the phase transition pressure, and an infinite number of coexistence curves. The critical point is represented as the intersection of the cloud point and shadow curves. The experimental results for the scCO2 + TMN-6 and the scCO2 + TMN-3 systems are listed in Tables 4 and 5 and described in Figs. 6 and 7, respectively. The phase transition pressures at constant composition of the scCO2 + TMN-6

Table 3 Phase transition pressures for the scCO2 + TMN-10 system. wTMN−10 [wt%]

T [K]

P [MPa]

wTMN-10 [wt%]

T [K]

P [MPa]

9.7

313 323 333 343

38.5 40.9 42.3 44.6

12.1

313 323 333 343

38.3 41.2 43.3 45.2

17.0

313 323 333 343

39.5 42.1 43.9 46.1

19.6

313 323 333 343

40.1 42.5 44.8 47.1

21.1

313 323 333 343

39.4 41.7 44.1 46.0

28.6

313 323 333 343

38.1 40.7 43.2 45.2

42.1

313 323 333 343

24.5 29.9 33.4 36.8

Table 4 Phase transition pressures for the scCO2 + TMN-6 system. wTMN-6 [wt%]

T [K]

P [MPa]

wTMN-6 [wt%]

T [K]

P [MPa]

9.6

313 323 333 343

27.8 31.0 34.1 36.0

17.1

313 323 333 343

30.4 31.8 34.5 36.5

24.2

313 323 333 343

30.4 32.9 35.5 37.8

28.4

313 323 333 343

33.1 35.2 37.2 38.4

45.9

313 323 333 343

24.9 28.1 30.6 32.4

and the scCO2 + TMN-3 systems also increased with increasing temperature. The critical phenomena of both systems were observed at around 30 wt% for all measurement temperatures. The effect of the molecular weight on the phase boundaries at about 17 wt% of TMN is shown in Fig. 8. The phase transition pressures increased linearly with increasing temperature. The experimental results also showed that the phase transition pressure of the scCO2 + TMN system increased as the molecular weight of TMN increased, which was caused by a decreasing CO2 -philic effect due to the inclemency of the EO groups.

Table 5 Phase transition pressures for the scCO2 + TMN-3 system. wTMN-3 [wt%]

T [K]

P [MPa]

wTMN-3 [wt%]

T [K]

P [MPa]

17.9

313 323 333 343

11.8 14.9 17.7 20.2

29.3

313 323 333 343

11.0 13.8 16.3 18.4

36.6

313 323 333 343

11.0 13.8 16.4 18.6

M. Haruki et al. / Fluid Phase Equilibria 287 (2009) 7–14

11

Fig. 4. Images of the phase transition for the scCO2 + TMN-10 system at 323 K and 28.6 wt% of TMN-10.

3. Correlation and prediction

are estimated by the following equations:

3.1. Model

P∗ =

The S-L EOS [22,23] was used to correlate and predict the experimental results of the phase transition pressures for the scCO2 + TMN systems. The EOS is expressed as

where M and R are the molecular weight and the universal gas constant, respectively. ε* and v∗ indicate the interaction energy between segments and the segment volume, respectively. For m-component mixtures, mixing rules of the characteristic parameters are given by the following equations:





 ˜ 2 + P˜ + T˜ ln(1 − ) ˜ + 1− P P˜ = ∗ , P

  ˜ = ∗, 

T T˜ = ∗ T

1 r

 

 ˜ =0 (1)

where P*, * and T* are the S-L characteristic parameters, and r is the number of segments that represents the number of lattice sites occupied by a molecule. The characteristic parameters P*, * and T*

P∗ =

ε∗

v∗

,

∗ =

m m  

M , r v∗

T∗ =

i j (Pi∗ Pj∗ )0.5

ε∗ R

(2)

(3)

i=1 j=1

∗

T =P

∗

m  0 ∗   T i

i=1

Pi∗

i

(4)

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M. Haruki et al. / Fluid Phase Equilibria 287 (2009) 7–14

Fig. 5. Phase transition pressures for the scCO2 + TMN-10 system. Symbol (×) represents the calculated critical point by the S-L EOS.

1  = r m



i0



ri0

i=1

i Pi∗ /Ti∗

i0 =

m

i =

m

j=1

(j Pj∗ /Tj∗ )

wi /i∗

j=1

(wj /j∗ )

Fig. 7. Phase transition pressures for the scCO2 + TMN-3 system. Symbol (×) represents the calculated critical point by the S-L EOS.

3.2. Parameter determination (5)

(6)

(7)

where wi represents the weight fraction of the ith component. In the calculations, the molecular weight distributions of each TMN shown in Table 1 and Fig. 2 were considered. Briefly, TMN10, TMN-6 and TMN-3 were treated as mixtures that have different numbers of EO groups.

Fig. 6. Phase transition pressures for the scCO2 + TMN-6 system. Symbol (×) represents the calculated critical point by the S-L EOS.

The characteristic parameters for CO2 reported by Wang et al. [24] were used: P* = 720.3 MPa and * = 1580 kg/m3 , respectively. T* is temperature dependent, as given by the following equation: T ∗ = 208.9 + 0.459T − 7.56 × 10−4 T 2

(8)

This parameter set was determined by correlating the vapor pressures of CO2 , and the deviation of the vapor pressures at temperatures from 275 to 300 K was 3.7%. Moreover, quite good correlated results of the vapor–liquid equilibria were obtained for the CO2 + benzene and CO2 + toluene systems up to 353 K [24]. In this study, the concept of a group contribution method was used to determine the characteristic parameters of TMN. Specifically, the TMN molecule was divided into three parts: HC, EO and OH groups as shown in Fig. 1. ε∗TMN , v∗TMN and rTMN were estimated from group parameters. ε∗TMN was obtained from the group param-

Fig. 8. P–T diagram for the scCO2 + TMN systems at about 17 wt% of TMN concentration.

M. Haruki et al. / Fluid Phase Equilibria 287 (2009) 7–14

13

Table 6 Characteristic group parameters for TMN. Groupa

ε* [J/mol]

v∗ × 106 [m3 /mol]

r

HC EO OH

1810 3701 3900

4.188 4.188 4.188

11.01 5.334 3.735

a

HC: hydrocarbon; EO: ethylene oxide; OH: hydroxyl.

eters as follows: ε∗TMN =

ε∗HC nHC rHC + ε∗EO nEO rEO + ε∗OH nOH rOH

(9)

nHC rHC + nEO rEO + nOH rOH

where n indicates the number of each group in a TMN molecule. The subscripts TMN, HC, EO and OH indicate TMN molecules, hydrocarbon, ethylene oxide and hydroxyl groups, respectively. The numbers of hydrocarbon and hydroxyl groups in a TMN molecule were assumed to be unity. On the other hand, the number of EO groups was determined based on the results shown in Table 1 and Fig. 2. Moreover, rTMN was estimated using the following equation: rTMN = nHC rHC + nEO rEO + nOH rOH

(10)

In this work, the same value was used for v∗

of the TMN molecule and for all groups, as shown in Eq. (11), to reduce parameters determined by the experimental results because the utilizable vapor pressures and molar volumes of TMNs for determination of group parameters have never been reported

v∗TMN = v∗HC = v∗EO = v∗OH

(11)

All the group parameters for TMN were optimized simultaneously by correlating all the experimental phase transition pressures of both the scCO2 + TMN-10 and scCO2 + TMN-6 systems. The Levenberg-Marquardt method, which is one of the most effective nonlinear least-squares approaches, was used and the initial values of the group parameters were sufficiently changed. The objective function used in the correlation is as follows:





1  Pcalc − Pexp

× 100 [%] O.F. = N Pexp N

(12)

i=1

where N indicates the total number of the experimental data. The subscripts calc and exp show the calculated and experimental values, respectively. 3.3. Results and discussion The group parameters of TMN determined by correlation of the experimental results are listed in Table 6. Moreover, the correlation results for the scCO2 + TMN-10 and the scCO2 + TMN-6 systems are listed in Table 7, and illustrated in Figs. 5 and 6, respectively. There were some deviations in the phase transition pressure due to the difference of temperature dependency between the model correlations and the experimental data, as is evident in the scCO2 + TMN-10 system. The correlated phase transition pressure Table 7 Correlated and predicted results of the phase transition pressures for the scCO2 + TMN systems. T [K]

Dev.[%]a a b c

TMN-10b TMN-6b TMN-3c

Dev.[%] = N1 Correlation. Prediction.

|P calc −P exp | P exp

× 100.

313

323

333

343

11.0 10.9 7.9

7.1 6.3 2.9

10.3 4.1 2.6

2.3 2.1 2.5

Fig. 9. Calculated results of the phase transition pressures for the scCO2 + monodisperse TMN and the scCO2 + polydisperse TMN systems at 343 K.

decreased more gradually with increasing TMN-10 concentration, as compared with the experimental results of the TMN-10 concentrations beyond the critical composition where the correlation errors were larger than those at other concentrations, as shown in Fig. 5. The correlated results reproduced the experimental results moderately well for the phase transition pressures of both the scCO2 + TMN-10 and the scCO2 + TMN-6 systems. The critical compositions of both systems were well correlated, as shown in the figures. The phase transition pressures for the scCO2 + TMN-3 system were then predicted by the model. The model predictions for the phase transition pressures are listed in Table 7, with comparisons to the experimental data in Fig. 7. The predicted results reproduced the experimental results quite well at all temperatures and weight fractions of TMN-3 measured. The effect of the polydispersity of the TMN on the phase behavior of the scCO2 + TMN system was investigated using this prediction model. The calculated phase boundaries at 343 K are shown in Fig. 9. In the calculations for the scCO2 + monodisperse TMN systems, the group parameters listed in Table 6 were used, and the numbers of EO groups were 12.1 for TMN-10, 9.96 for TMN-6, and 4.81 for TMN-3. The calculated phase transition pressures for the scCO2 + polydisperse TMN systems were higher than those of the scCO2 + monodisperse TMN systems at TMN concentrations of less than approximately 50 wt%. The calculated critical concentrations of TMNs for the scCO2 + polydisperse TMN systems were higher than those of the scCO2 + monodisperse TMN systems. 4. Conclusions In this work, phase transition pressures of the scCO2 + TMN-10, the scCO2 + TMN-6 and the scCO2 + TMN-3 systems were measured at temperatures from 313 to 343 K using a synthetic method. The phase transition pressures increased with increasing temperature and the average molecular weight of TMN. In addition, a new prediction model based on the Sanchez–Lacombe equation of state was developed to predict the phase boundaries by introducing the group contribution method. TMN was assumed to consist of three types of groups: hydrocarbon, ethylene oxide and hydroxyl groups. The group parameters were determined by correlating the experimental phase boundaries of the scCO2 + TMN-10 and the

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