Fluid Phase Equilibria Modification of NRTL-NRF

Fluid Phase Equilibria 278 (2009) 20–26

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Modification of NRTL-NRF model for computation of liquid–liquid equilibria in aqueous two-phase polymer–salt systems Ali Haghtalab ∗ , Marzieh Joda Department of Chemical Engineering, Tarbiat Modares University (TMU), P.O. Box 14115-111, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 20 September 2008 Received in revised form 5 December 2008 Accepted 18 December 2008 Available online 27 December 2008 Keywords: Activity coefficient Excess Gibbs energy Electrolyte solutions Aqueous two-phase system NRTL-NRF Polyethylene glycol Sodium sulfate Potassium sulfate Ammonium sulfate

a b s t r a c t The aqueous two-phase system (ATPS), a polymer and a salt, is widely used as an extraction process for separation of biomolecules in down stream processing. At this work, using local area fraction, the local composition non-random factor (NRF) model of Haghtalab and Vera is extended for aqueous binaries of electrolyte, polymer and ternaries of aqueous polymer–salt solutions. Using area fraction of species, a new function of excess Gibbs energy has been developed for multicomponent aqueous polymer–electrolyte systems. The modified NRTL-NRF model is applied to correlate the phase behavior of aqueous two-phase polymer–salt systems of PEG + Na2 SO4 + H2 O, PEG + K2 HPO4 + H2 O and PEG + (NH4 )2 SO4 + H2 O at 25 ◦ C. The effect of pH solution and molecular weight of polyethylene glycol (PEG) is investigated and the tie lines with binodal curves for the fifteen systems are shown. The comparison of the results of the present model with the experiment shows that the modified NRTL-NRF model is more accurate. © 2008 Elsevier B.V. All rights reserved.

1. Introduction Presently, industrial demands and economic downstream processes for extraction and purification of biomolecules with high yield purity of the product are growing fast [1,2]. The aqueous twophase system (ATPS) provides a powerful method to separating mixtures of biomolecules by extraction in down stream processing. Using ATPS as a practical process allows one to integrate clarification, concentrating, and partial purification of biomolecules in one step. The ATPS consists of two immiscible aqueous solutions that it contains two different polymers such as polyethylene glycol (PEG) and Dextran or one polymer and one salt such as PEG and ammonium sulfate. Also polymer–salt aqueous two-phase systems have several advantages such as low price, low viscosity, and short time for phase separation. The basis of partitioning depends upon surface properties of the particles and molecules that include size, charge, and hydrophobicity. Moreover, the most characteristic feature of a two-phase system is that the water content in such system is as high as 85–99%, when is complemented with suitable buffers and salts allows one to provide a suitable milieu for bio-

∗ Corresponding author at: Department of Chemical Engineering, Tarbiat Modares University (TMU), P.O. Box 14115-143, Tehran, Iran. Tel.: +98 21 82883313; fax: +98 21 82883381. E-mail address: [email protected] (A. Haghtalab). 0378-3812/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2008.12.006

logical materials [1]. A wide range of biomolecules (protein, lipids, nucleic acids, viruses and whole cells) have been separated using this technique [2,3], specially for proteins, since it takes less time consuming and has the potential to give high yield and high purity, involving low investment, less energy, and lower labor costs. On the other hand, a liquid–liquid extraction process requires knowledge of the phase behavior of the system for engineering design and process optimization. To design and operate an aqueous two-phase extraction process, availability of the predictive models to construct the phase diagrams is of great importance [4,5]. As a consequence, some successful models in this specialized area have been developed. The first group of the models, based on osmotic virial expansion, was derived from knowledge of the osmotic pressure of a solvent in a solution. King et al. [6], Sassi et al. [7], Wu et al. [8] and KabiriBadr and Cabezas [9] developed a virial model which was based on Hill theory. Haynes et al. [10,11] developed a McMillan-Mayer type model that was derived in the semi-grand ensemble at constant temperature and pressure. The second group of the models, based on the extension of the local composition concept, includes the NRTL, UNIQUAC, UNIFAC [12–16], UNIQUAC-NRF and UNIFAC-NRF models [17,18] that have been used for correlation and prediction liquid–liquid equilibria and phase diagrams of polymer–polymer and polymer–salt ATPS. Haghtalab and Vera [19] proposed the NRTL-NRF model for mean activity coefficient of the binary aqueous electrolyte sys-

A. Haghtalab, M. Joda / Fluid Phase Equilibria 278 (2009) 20–26

tems. At this model, the random state of species is considered as a reference state and the nonrandom factor, , is used to show the deviation of local composition from bulk composition. Using the assumptions of NRTL-NRF model, Haghtalab and Asadollahi [20] developed the UNIQUAC-NRF model which was used for polymer–polymer ATPS. Haghtalab and Mokhtarani [17] applied the UNIQUAC-NRF model for polymer–salt ATPS. Based on the group contribution method, Haghtalab and Mokhtarani developed the UNIFAC-NRF model [21,22] for ATPS of polymer–salt with and without proteins. At this work on extension of the NRTL-NRF model, the new version of local area fraction model is developed for aqueous polymer–salt solutions.

i and j. The Gibbs energy of the same cell in the reference state, random state, is expressed as

gE RT

=

gE RT

+ DH

gE RT

+ C

gE RT

(1) R

where the first term accounts for the contribution of long-range electrostatic interactions; the second and third terms denote the contribution of combinatorial and residual parts, respectively. So by proper differentiation of the above equation, the activity coefficient of species i can be expressed as ln i =

ln iDH

+ ln iC

+ ln iR

3

ln iC =

ln

i=1

i

xi

+

z q ln 2 i

where z li = (ri − qi ) − (ri − 1) 2

i i

+ li −

3 i

xi

xj lj

(3)

j=1

ionic species and ji = ji for molecular species, where z is the charge number of an ion. As one can see, we assume the random state as a reference state. So using the two-fluid theory, the molar residual excess Gibbs energy of a solution is written as

and i =

ri xi

3

gE RT

=

(5)

rx i=1 i i

gE i xi

R

(9)

RT

i

At this study, the local area fraction is represented by means of nonrandom factors (NRF). Thus, the effective local area fraction of i–j interactions can be written as ji = j ji

(10)

Similarly, ii = i ii

(11)

where “ ” is called the nonrandom factor. So by dividing the above relations, ji ii

=

j

ji

i

(12)

where ji

ji =

(13)

ii

On the other hand, using Boltzmann factor as was proposed by Wilson [23],

a ji

ji = exp −

(14)

Z

where gji

aji = (4)

(8)

where j is the effective area fraction of species j and ji = zj ji for

(2)

The long-range activity coefficient, i.e., Debye-Hückel equation, is given in Appendix A. Using Guggenheim equation, the combinatorial activity coefficient can be written as

j gji

j=1 j= / i for i = 1, 2

n

gi0 =

2. The modified NRTL-NRF model with area fraction At this work, we follow Haghtalab and Vera’s model, NRTL-NRF [19], with assumption of the existence of four types of cells for polymer–salt solutions that it depends on number of species in a solution. The excess Gibbs energy and activity coefficient functions are derived on the basis of the assumptions of the NRTL-NRF model and local area fraction of the species. Using the general approach of the local composition models, the excess Gibbs energy is written as a sum of three contributions,

21

(15)

RT

where aij is the binary interaction adjustable parameter of i–j pair and Z is the coordination number. By combining Eqs. (6–8) and using Eqs. (10–13), the residual molar excess Gibbs energy, Eq. (9), for a multicomponent system is obtained as

gE RT

n

=− R

Zxi

n j=1

j (ji

− 1)ln

i=1

ji

(16)

ii

The short-range contribution of activity coefficient can be developed as following. For a multicomponent system the configurational excess Gibbs energy for each central cell can be written as

Consequently, by proper differentiation of Eq (16), the equation of activity coefficient for a multicomponent system can be obtained:

giE = gi − gi0

ln i = Z(1 − i )

(i = a, c, w, p)

(6)

where gi and gi0 are Gibbs energy of the cell “i” in the nonrandom and random states, respectively. Also the configurational Gibbs energy of the each cell in the nonrandom state is expressed as

n

gi =

ij gji

j

is effective local area fraction of species j surrounding where central species i, and gji is interaction energy between two species

ln

ji ii

+

ij qi ln qj jj

j=1 j= / i

j k

ji ln

ji ii

+

ij qi ij kj ln qj jj

+Z

−Z

n

n

j=1 j= / i

k=1 k= / i for k = j, k = ion

n n

kj qi ln (ij kj − 1) qj j k jj

(17)

j=1 k=1 j= / i k= / i, j

(7)

j=1 j= / i for j = 1, 2

ji

n

where i =

x qi

n i

xq j=1 j j

(18)

22


ji

ji =

;

n

1

ii =

n

j .ji

j=1 i= / j for ion

(19) j .ji

j=1 i= / j for ion

For a binary electrolyte system with three species, an anion (1), a cation (2) and a molecular solvent (3), using Eq. (16) the molar excess Gibbs energy can be written as [24]:

gE RT

= qE xE E (E − 1)aE − q3 x3 3 (3 − 1)aE3

(20)

R

where the area parameter of an electrolyte pair-ion is expressed as qE = 1 q1 + 2 q2

(21)

The unsymmetrical mean activity coefficient for a binary electrolyte system is derived as [24]: (ln ±∗ )R =

qE aE E +

(E − 1)(1 + 3 ) +E2 E 3

qE −1 E ZE E qE

qE 2 qE 2ZE E qE 3 3 (3 − 1) − aE3 1 − qE

(32 3 − 1) 3

(22) where aE and aE3 are the binary adjustable parameters for a binary electrolyte system. The subscripts “E” and “3” denote electrolyte and solvent, respectively. 3. Aqueous two-phase system At this work, the extended NRTL-NRF model is applied for computation of liquid–liquid equilibria of the aqueous two-phase systems. The system consists of four species as anion (1), cation (2), polyethylene glycol (PEG) (3) and water (4). The standard state is chosen in a conventional way, i.e., the pure liquid is taken for water and infinite dilution for ions and PEG. So the activity coefficient of the ionic species and polymer compound is easily normalized to those based on the infinite dilution standard state. Thus, the activity coefficient of the anion and cation at infinite dilution is obtained as q1 ln 1∞ = Z (1 − 14 )ln 14 (23) q4 ln 2∞ = Z

q2 (1 − 24 )ln 24 q4

(24)

Similarly the activity coefficient of the polymer at infinite dilution is expressed as ln 3∞ = −Z

43 E3

2

−1

ln 43 − Z

q3 (34 − 1)ln 34 q4

(25)

Finally, the unsymmetrical residual molar excess Gibbs energy can be expressed as

g E,∗ RT

= R

gE RT

− x1 ln 1∞ − x2 ln 2∞ − x3 ln 3∞

(26)

R

4. Results and discussion The structural parameters, i.e., the surface and volume of ions, polymer, and water, are needed in the MNRTL-NRF equation. So dividing the molecular form of PEG into two groups: CH2 OH and CH2 OCH2 , the surface and volume parameters of PEG and water molecules are obtained using the group division that was used by Gao et al. [15,16]. The volume and surface area parameters of the groups were denoted by “R” and “Q”, respectively, which are shown at Table 1. The radii of ions are given by Marcus [27] and the “R”

Table 1 The group parameters for segments of PEG and water [15]. Group

H2 O

CH2 OH

CH2 OCH2

R Q

0.92 1.400

1.674 1.740

1.593 1.320

and “Q” parameters for the ions were calculated by Bondi’s equation [26]. The structural parameters for all components are given at Table 2. The mean activity coefficient data for binary aqueous systems of K2 HPO4 , (NH4 )2 SO4 and Na2 SO4 are given by Stokes and Robison [25]. The computation of liquid–liquid equilibria for the ternary systems PEG–Na2 SO4 –H2 O, PEG–K2 HPO4 –H2 O and PEG–(NH4 )2 SO4 –H2 O is investigated for various molecular weights of PEG and different pH at 25 ◦ C. Gao et al. [15,16], Peng et al. [28] and Haghtalab and Mokhtarani [17] reported the experimental data for those systems. Using computation of the LLE with the equilibrium experimental data, the interaction parameters of the species are obtained by optimization of the following objective function [15,16]: 1 =

n 4

(xi i∗ )Ij − (xi i∗ )˘ j

2 (27)

(xi i∗ )IIj

i=1 j=1

where I and II represent upper and lower phases, respectively. The subscripts i and j denote the species and tie lines, respectively. The interaction parameters are used for calculation of the composition of species in the two phases. However, to obtain the accurate values of the parameters, those again should be optimized by using the following objective function: 2 =

M

Min

2 3

k

i

exp

cal xijk − xijk

2 (28)

j

where xexp and xcal are the experimental and calculated mol fractions, respectively. The subscripts j, i and k denote the numbers of phases, species and tie lines, respectively. The calculated values of the compositions in the two equilibrium phases are obtained by flash calculation through Newton-Rafson algorithm. The present model has six adjustable parameters aE, aE3, aE4, a3E, a4E and a34 as follows: a12 = a21 = aE

(29)

with using assumption: 13 = 23 , one can write a13 = a23 + aE3 + aE3

and

a31 = a32 = a3E

(30)

a41 = a42 = a4E

(31)

Similarly, one can conclude, a14 = a24 + aE4 + aE3

and

Table 2 The volume and surface parameters of the different compounds. Component

r (pm)

q (cm2 /mole)

Reference

H2 O (NH4 )2 SO4 Na2 SO4 K2 HPO4 PEG(1000) PEG(1500) PEG(1450) PEG(1540) PEG(2000) PEG(3350) PEG(4000) PEG(6000)

0.92 2.70 1.68 3.144 37.26 55.34 53.53 56.79 73.42 122.4 145.7 218.7

1.40 2.226 1.914 2.362 31.59 46.58 44.71 47.77 61.55 102.2 121.5 181.4

[15] [27] [27] [27] [15] [15] [15] [15] [15] [15] [15] [15]


23

Fig. 1. The activity of water in the PEG–H2 O system versus weight fraction of the polymer at different molecular weight of polymer, experimental data [29].

It should be noted that the parameters a34 and a43 are not equivalent; however, one can obtain a43 in terms of the other parameters as a43 = a34 + a4E + aE3 − a3E − aE4

(32)

In correlation of ATPS, the binary interaction parameters for the salt–polymer–water systems are obtained. Before applying the MNRTL-NRF model for calculation of LLE in ATPS, the activity of water was correlated for the binary PEG–H2 O systems at different molecular weight of PEG [29]. Table 3 shows the binary interaction parameters for aqueous PEG solution at different molar masses. Also Fig. 1 shows the change of activity of water respect to the mole fraction of PEG at various molar masses. As one may consider the results of correlation of the water activity is in very good agreement with experiment. Using Eq. (22), the mean activity coefficient of the binary electrolyte systems of (NH4 )2 SO4 –H2 O, K2 HPO4 –H2 O and Na2 SO4 –H2 O were correlated and the two binary interaction parameters aE and aE4 were adjusted as shown in Table 4. By fixing four parameters and adjusting only two parameters, attempts to correlate the ternary systems in ATPS were not successful because the data for the binaries were from different source. So by fixing the two electrolyte parameters, only four parameters were optimized for ternary systems at each molecular weight of PEG and for given pH. Using Simplex method of the Nelder-Mead Table 3 The interaction parameters for binary PEG(3)–H2 O(4) systems at 25 ◦ C using MNRTLNRF, experimental data [29]. MW of PEG

a34

a43

RMSDa

1000 1450 3350 6000

25.01 26.404 26.634 26.104

−0.998 −1.098 −1.248 −1.648

0.218 0.095 0.067 0.079

a

RSMD = 1/N

i

1/2

algorithm, the values of interaction parameters were optimized with regression of the experimental data. These parameters are shown in Table 5, respectively, for the systems PEG–Na2 SO4 –H2 O, PEG–K2 HPO4 –H2 O and PEG–(NH4 )2 SO4 –H2 O. The average relative deviation is defined as cal 1 (xi − xi ) × 100% exp n xi n

ε=

exp

where n is the number of experimental points; the subscripts ‘exp’ and ‘cal’ denote the experimental and calculated values, respectively. The overall average deviation is defined as follows: OARD =

1 εj 3

−

acal ) i

× 100; ai = activity of water.

(34)

j

where εj presents the relative deviation for each component. It should be noted that the missing pHs at Table 5 were not reported in the original papers [28,15]. The results of four systems at different molecular weight and pH are presented at Table 6. As one can see the results are in very good agreement with experiment. Also one may observe that the overall average relative deviation of the calculated mole fraction from the experiment is very low for the systems PEG–Na2 SO4 –H2 O, PEG–K2 HPO4 –H2 O and PEG–(NH4 )2 SO4 –H2 O at different molecular weight and pH. The results of the present model are compared with those obtained using the UNIFAC, UNIQUAC and UNIFAC-NRF Table 4 The interaction parameters for binary electrolyte systems at 25 ◦ C using MNRTL-NRF, experimental data [25]. Binary electrolyte solution

aE

aE4

RMSDa

(NH4 )2 SO4 –H2 O K2 HPO4 –H2 O Na2 SO4 –H2 O

−2.000 18.058 74.000

4.000 63.906 59.800

0.359 0.327 0.199

2 exp (ai

(33)

i=1

a

RSMD = 1/N

electrolyte.

exp

(ln i

− ln ical )

2

1/2

× 100; i = mean activity coefficient of

24


Table 5 The interaction parameters for the ternary PEG(3)–Salt(E)–H2 O(4) systems at different molecular weight and pH, at 25 ◦ C. MW of PEG

pH

aE3

a3E

a4E

a34

PEG–Na2 SO4 –H2 O [17] 1500 4000 4000

4.7 9.2 5.3

−69.975 −530.983 −560.983

−59.975 −106.983 −126.983

−113.975 −140.983 −178.983

45.017 66.709 80.709

PEG–K2 HPO4 –H2 O [17] 1500 4000 4000 4000

9.1 7.2 9.1 10.8

−85.983 −519.983 −500.983 −460.983

−96.983 −89.983 −94.983 −101.083

−127.983 −129.983 −130.115 −125.083

54.709 57.009 56.709 62.709

– – –

−887.993 −839.975 −880.030 −896.405

−130.000 −48.950 −44.120 −96.647

−1.001 −4.277 −1.002 −1.265

139.730 11.017 0.999 0.998

– – – –

−394.975 −11.938 −14.975 −24.975

−260.977 20.061 16.038 76.024

146.024 146.061 108.024 105.024

203.017 101.054 136.017 72.017

PEG–K2 HPO4 –H2 O [28] 1000 2000 4000 6000 PEG–(NH4 )2 SO4 –H2 O [15] 1000 1540 2000 4000

Table 6 The percent of average relative deviation (ARD%) of the calculated mole fraction from the experiment using MNRT-NRF model for PEG, salt and water.

Table 8 The overall average relative deviation for the K2 HPO4 –PEG–H2 O Na2 SO4 –PEG–H2 O systems using the UNIFAC-NRF and MNRTL-NRF models.

MW of PEG

PEG

Salt

Water

MW of PEG

PEG–Na2 SO4 –H2 O [17] 1500 4000 4000

0.010 0.153 0.100

2.732 0.970 0.171

3.262 2.826 2.200

PEG–Na2 SO4 –H2 O 1500 4000 4000

PEG–K2 HPO4 –H2 O [17] 1500 4000 4000 4000

0.269 0.574 0.458 0.219

0.177 1.833 0.794 1.470

1.849 0.869 1.399 1.579

PEG–K2 HPO4 –H2 O 1500 4000 4000 4000

PEG–K2 HPO4 –H2 O [28] 1000 2000 4000 6000

5.815 0.053 0.012 9.909

2.160 1.651 1.597 0.298

0.172 1.210 3.180 1.280

PEG–(NH4 )2 SO4 –H2 O [15] 1000 1540 2000 4000

0.085 0.104 0.118 0.140

0.940 1.691 3.242 1.465

3.572 2.162 2.866 2.645

pH

and

UNIFAC-NRF [21]

MNRTL-NRF

4.7 9.2 5.3

4.300 4.240 5.800

1.316 2.002 4.123

9.1 7.2 9.1 10.8

2.920 2.840 5.100 2.370

0.764 1.090 1.552 1.089

be noted that experimental data of LLE from different sources might induce more deviation. This may result from the difference in the characteristics of the polymer, such as polydispersity and distribution of molecular weight (Table 8). To present the accuracy of the model, a comparison of the tie lines for the systems were performed. Figs. 2–4 show the binodal curves and the tie lines for the systems PEG2000–(NH4 )2 SO4 –H2 O, PEG4000–Na2 SO4 –H2 O and PEG4000–K2 HPO4 –H2 O, respectively.

models. As shown at Table 7, the results of the MNRTL-NRF model show a better accuracy, particularly at higher molecular weight of PEG, in correlation of the experimental data of liquid–liquid equilibria. The missing deviations at Table 7 have not been reported for the system PEG–K2 HPO4 –H2 O using the UNIQUAC model. It should Table 7 The overall average relative deviation for the PEG–(NH4 )2 SO4 –H2 O and PEG–K2 HPO4 –H2 O systems using the UNIQUAC, UNIFAC and MNRTL-NRF models. MW of PEG

UNIQUAC [15]

UNIFAC [16,28]

MNRTL-NRF

PEG–(NH4 )2 SO4 –H2 O 1000 3.686 1540 2.793 2000 3.656 4000 2.906

2.07 1.85 1.73 –

1.513 1.319 2.043 1.417

PEG–K2 HPO4 –H2 O 1000 2000 4000 6000

2.160 2.890 2.913 4.486

2.716 0.971 1.597 3.829

– – – –

Fig. 2. The binodal curve and tie lines for the PEG2000–(NH4 )2 SO4 –H2 O system at 25 ◦ C using MNRTL-NRF model, the symbols are experimental data [15].


Fig. 3. The binodal curve and tie lines for the PEG4000–Na2 SO4 –H2 O system at 25 ◦ C and pH: 9.2 using NRTL-NRF model, the symbols are experimental data [17].

25

Fig. 6. The equilibrium phase diagram of the PEG–Na2 SO4 –H2 O system at different pH, the solid lines and symbols show the calculated and experiment, respectively [17].

One can see the calculated tie lines are in excellent agreement with the experimental data. The advantage of the present model demonstrates a very good agreement between the calculated and experiment and coincidence of feed composition at respected point on the tie line between the top and bottom compositions. Fig. 5 shows the phase diagram of the PEG–(NH4 )2 SO4 –H2 O system and the effect of molecular weight of PEG on the ATPS. As one can observe, at higher molecular weight of the PEG, the lower concentration of the polymer is required for phase separation. Fig. 6 shows the binodal curve of the PEG–Na2 SO4 –H2 O system and the effect of pH on the phase diagram, so with increasing of pH, i.e., at higher pH more concentration of the polymer is required for phase separation. The points are the experiment and the curves represent the calculated values using the MNRTL-NRF model. As a result, using the binary interaction parameters from binary aqueous electrolyte systems, only four adjustable parameters are needed to correlate the phase equilibria in the ATPS of ternary aqueous-polymer–salt systems. Fig. 4. The binodal curve and tie lines for the PEG1500–K2 HPO4 –H2 O system at 25 ◦ C and pH 9.1 using NRTL-NRF model, the symbols are experimental data [17].

Fig. 5. The equilibrium phase diagram of the PEG–(NH4 )2 SO4 –H2 O system at different molecular weight of PEG, the solid lines and symbols show the calculated and experiment, respectively [15].

5. Conclusion At this study, a new version of local composition model, MNRTLNRF, based on area fraction has been developed for binaries of electrolyte–solvent, polymer–solvent and ternary aqueous twophase polymer–salt systems. The extended model for binaries of aqueous electrolyte and polymer systems has only two adjustable parameters and four adjustable parameters were used to correlation of aqueous ternary polymer–salt systems. The local area fraction, expressed in terms of nonrandom factor, is used to represent the effect of short-range interactions in aqueous two-phase polymer–salt systems. The Debye-Hückel theory based on FowlerGuggenheim was applied for long-range interactions of ionic species. Calculation of the LLE for the systems PEG–Na2 SO4 –H2 O, PEG–K2 HPO4 –H2 O and PEG–(NH4 )2 SO4 –H2 O was performed and the effects of pH and molecular weight of polyethylene glycol (PEG) were investigated. Also the tie lines with binodal curves for those systems were constructed. In addition, the binodal curves and tie lines at different molecular weight and pH for those systems were obtained in which the estimated results are in very good agreement with the experiment. As a result, the MNRTL-NRF model could be considered as a correlation model that can be successfully applied for calculation of phase behavior of ATPS with very good accuracy. Finally, the results showed that the present model is more accurate than the UNIQUAC and UNIFAC-NRF models.

26


List of symbols aij interaction parameter between molecule i and j A Debye-Hückel constant B distance of closest approach parameter D dielectric constant of water di density of substance i d density of a mixed solution molar excess Gibbs energy gE I ionic strength m molality MW molecular weight n number of experimental points qi surface parameter of molecule i volume parameter of molecule i ri R gas constant RMSD root mean square deviation T absolute temperature V total volume of the solution v¯ j partial molar volume of component j x mole fraction Z coordination number ZE absolute value of ionic charge Greek letters nonrandom factor i activity coefficient for component i ± mean ionic activity coefficient for electrolyte ε error percent i surface area fraction i effective surface area fraction ij local area fraction effective local area fraction ij 1, 2 objective function Boltzmann factor Superscripts C combinatorial cal calculation exp experimental LR Long-range R Residual I, II two aqueous phase in equilibria * unsymmetrical convention ∞ infinite dilution Subscripts i, j, k component index m neutral species s salt ± mean ionic 1, 2, 3, 4 anion, cation, polyethylene glycol and water Appendix A Using Fowler and Guggenheim [30], the contribution of longrange electrostatic interactions for ionic species is written as

ln iLR =

√ −zi2 A I √ 1+b I

(A.1)

where the subscript of i stands for the ion species, zi the absolute charge number of ionic species i and “I”, the ionic strength of the mixture, is defined as I=

n

mi zi2

(A.2)

i=1

where mi is the molality of species i. For neutral species, the contribution of the long-range forces to the activity coefficient takes the form [13]: LR ln m =

√ √ 2AVm d 1 1+b I− √ − 2 ln(1 + b I) 3 b 1+b I

(A.3)

where m stands for the neutral species, vm is the molar volume of the species m, d is the density of the water. The constants of A and b are the Debye-Hückel parameters that were given as [13]: A = 1 · 327757 × 105 × d0.5 /(D · T )1.5 b = 6 · 359696 × d0.5 × d0.5 /(D · T )0.5

(A.4)

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