Ž .Fluid Phase Equilibria 145 1998 53–68

Prediction and correlation of triglyceride–solvent solid–liquidequilibria with activity coefficient models

R.L. Smith Jr. ), G.M. Acosta, K. Arai

Department of Chemical Engineering, Tohoku UniÕersity, Aoba Aza Aramaki, Sendai 980-77, Japan

Accepted 10 October 1997

Abstract

Ž . Ž .Solid–liquid equilibria SLE for solvent–triacylglyceride TAG systems are studied with activity coeffi-cient models. Systems consisted of 29 data sets of C10:0 to C18:0 saturated triglycerides in nine solvents,benzene, acetone, carbon disulfide, carbon tetrachloride, chloroform, diethyl ether, ethanol, ethyl acetate, andn-hexane making up 562 data points. Eighteen activity coefficient models containing 0–2 fitting parameters

Ž . Ž .were evaluated including a number of Flory Free Volume FFV and Entropic Free Volume FEFV modelswevaluated by Coutinho et al. J.A.P. Coutinho, S.I. Andersen, E.H. Stenby, Evaluation of activity coefficient

Ž . xmodels in prediction of alkane solid–liquid equilibria, Fluid Phase Equilibria 103 1995 23–39 , the UNIFACmodel and its variations, the Wilson, UNIQUAC, and van Laar equations and some proposed modifications of

Ž .the FEFV and UNIFAC models. The modifications were as follows: i a Flory type x parameter was added toŽ . Ž .the FEFV model FEFV-X and ii the exponent in the UNIFAC combinatorial term was treated as a fitting

Ž . Ž .parameter UNIFAC-C . Models were compared in terms of absolute average percent deviation AAPD intemperature. The best predictive model for TAG–solvent systems was found to be the Dortmund version ofUNIFAC. Addition of a fitting parameter to the UNIFAC models provided significant improvement. For the

Ž . Ž .UNIFAC-1977, UNIFAC Lyngby and UNIFAC Dortmund , these exponents were found to be 0.897, 0.858,and 0.749, respectively. The best single parameter correlative model was found to be the FEFV-X model, whichgave an AAPD of 0.305%. The Wilson equation was the best two parameter correlation equation and providedan AAPD of 0.154%. q 1998 Elsevier Science B.V.

Keywords: Theory; Excess functions; Methods of calculation

1. Introduction

Ž .The solubility of triacylglycerides TAGs in organic solvents is important for processing materialsw xsuch as collagen for production of sausage casing 1 . Recently, we have reported on the extraction of

) Corresponding author.

0378-3812r98r19.00 q 1998 Elsevier Science B.V. All rights reserved.Ž .PII S0378-3812 97 00303-8

( )R.L. Smith Jr. et al.rFluid Phase Equilibria 145 1998 53–6854

fat from collagen–fat–water mixtures with supercritical carbon dioxide and propane mixtures andw xwith liquid propane 2 . As a result of that study, it became important to compare models for

predicting and correlating TAG solubility in liquid solvents. In this work, we compare a number ofliquid models and show the importance of the entropic and enthalpic contributions.

Several works have evaluated models for predicting and correlating data related to this study.w x Ž .Gmehling et al. 3 applied the UNIFAC model to the prediction of solid–liquid equilibria SLE and

w xeutectic systems with good results. Coutinho et al. 4 evaluated a number of predictive models for theSLE of alkane–alkane mixtures. They concluded that a Flory–Huggins, Regular Solution, modifiedUNIFAC, and original entropic free-volume models could not describe the liquid phase of alkanesystems accurately and that the Flory free volume and modified entropic free-volume models were the

w xsimplest and most reliable. Fornari et al. 5 examined a number of UNIFAC models for the liquidphase activity of vegetable oil–hexane mixtures for vapor–liquid equilibria calculations and con-cluded that the UNIFAC-3r4 model gave a better description of the infinite dilution hexane solventactivity coefficients. Other researchers have evaluated activity coefficient models for specific experi-mental data sets that are related to this work from the point of view of model performance or system

w x Ž .similarity. Domanska 6 reported on the solubility of long-chain C –C ethylene glycol mo-17 21

noesters correlated with the Wilson model and found that the model provided partly satisfactoryw xresults. Roberts et al. 7 reported on the solubility of long chain alkanes in heptane correlated with

regular solution, Flory–Huggins, and Haulait–Pirson models and concluded that the Flory–Hugginsmodel tended to overestimate the solubility while the Haulait–Pirson model gave the best agreement.However, no studies have been performed on evaluation of models for TAG–solvent SLE. Theobjective of this work is to correlate and predict TAG–solvent SLE with a number of liquid activitycoefficient models and to provide a set of recommendations.

2. Thermodynamics

w xThe thermodynamics of SLE is derived in standard texts such as Ref. 8 . For the generalequilibrium between i components at constant T and P:

x sg s f os sx lg l f ol 1Ž .i i i i i i

Ž .For a pure solid phase, the left-hand side of Eq. 1 becomes the fugacity of the pure solid. Theright-hand side contains the activity coefficient referenced to the standard state fugacity, f ol, that is1

the hypothetical state of the solid component in the liquid state at the system temperature and underits own saturation pressure. The solid fugacity and the standard state fugacity in the liquid arenormally written as a ratio. For a binary system with solid component ‘1’:

f osrf ol sx l g l 2Ž .1 1 1 1

Consideration of a thermodynamic path to bring component ‘1’ from the solid state to the hypotheticalliquid state in solution with the solvent at system temperature and assuming that the DC is constantp

and that the triple point can be approximated by the melting point, leads to evaluation of this ratio as:

Dhfus T DC T DC Tt p t p tln x s 1y y 1y y ln y lng 3Ž .1 1ž / ž /RT T R T R T

( )R.L. Smith Jr. et al.rFluid Phase Equilibria 145 1998 53–68 55

Ž .Eq. 3 can be used to determine the solubility of a solid in a liquid. The DC terms are normallypw xsmall and were set equal to zero in this work. Mishra and Yalkowsky 9 showed that this was a better

Ž .approximation than setting DC equal to DS . Eq. 3 contains the assumption that only a single solidp t

phase is present. When the solid goes through a solid–solid transition, additional terms must bew xincluded as discussed in Ref. 10 .

TAGs are known to exist in three main polymorphic states, denoted as the a-modification, theX w xb -modification and the b-modification 11 . The a-modification is unstable and converts on the

order of seconds to the bX-modification. The b

X-modification is the stable modification for odd-chainTAGs and some even number TAGs. The b-modification is the most stable conformation. All threemodifications can be obtained by varying the degree of supercooling from the liquid phase. Forprediction and correlation in this work, only presence of the most stable b-modification was assumed,which is appropriate since the mixtures in this work are even-monoacid TAG solvent systems.Equilibrium between mixed TAG systems in the absence of solvents can be obtained from the work of

w xWesdorp 12 .

3. Activity coefficient models

The activity coefficient can be written in terms of a combinatorial term, g comb, a free volume term,ifv res w xg , and a residual term, g , as shown in Ref. 13 :i i

lng s lng comb q lng fv q lng res 4Ž .i i i i

The combinatorial term corresponds to the entropic contribution to the activity according to size andshape and the free volume term corresponds to the entropic contribution according to size differences.The residual term accounts for energetic interactions that are associated with the enthalpic contribu-tion to the activity coefficient.

Ž .For this work, we evaluated a number of models that can be discussed in terms of Eq. 4 as shownŽ . w xin Table 1. The Flory–Huggins Flory model 14,15 only contains a combinatorial term. The Flory

Ž . w x Ž . w xFree Volume FFV model 16 and the Flory Entropic Free Volume FEFV model 13 containcombinatorial and free volume terms. The Flory–Huggins model relates solution activity to volumefraction whereas the models for the FFV and FEFV include a correction for the free volume by

w xsubtracting off the Bondi 0 K close packed volume 17 from the molar volume of the liquid at agiven temperature. For these models, the solution is assumed to be a thermal and the residual term ofthe activity that corresponds to the enthalpic contribution is set equal to zero. The Flory-x modelcontains a residual term that has a single adjustable parameter. In this work, we also examine adding aFlory-x residual term to the FEFV model and this is denoted FEFV-X.

w xUNIFAC 18,3 are predictive models that contain both combinatorial and residual terms but now xcorrections for free volume. The Lyngby group 19 developed an improved UNIFAC that provided

superior heats of mixing and performance for larger molecules by eliminating the Staverman–Gug-genheim term in the combinatorial portion and introducing temperature dependent interaction

w xparameters. The Dortmund group 20 has developed the latest and most comprehensive versions ofUNIFAC which are continually are being updated with the Dortmund databank. The UNIQUAC-2r3

( )R.L. Smith Jr. et al.rFluid Phase Equilibria 145 1998 53–6856

Table 1Models evaluated

Model Combinatorial and free volume terms Residual term

f fi icomb fv reslng qlng s ln q1y lngi i ix xi i

Ž .Flory–Huggins Flory f s x V rÝ x V yi i i j j2Flory-x f s x V rÝ x V s xfi i i j j j

1r3 1r3 3c 1r3 1r3 3cŽ . Ž . Ž .Flory-Free Volume FFV f s x V yV rÝ x V yV , cs1.1 yi i i w, i j j w, jL LŽ . Ž . Ž .Flory-Entropic Free Volume FEFV f s x V yV rÝ x V yVi i i w, i j j w, j

Ls1yV r Vw,small w,largecomb Ž i. Ž i.Ž . w xUNIFAC-77 lng s ln f r x q1yf r x sn ln G yln Gi i i i i k k k

Ž . Ž Ž .y zr2 q ln u rfi i i.q1yf rui i

f s x r rÝ x ri i j jp p Ž i. Ž i.Ž . w xUNIFAC-Lyngby 1987 f s x r rÝ x r , ps2r3 sn ln G yln Gi i i j j k k k

comb Ž i. Ž i.Ž . Ž . w xUNIFAC-Dortmund 1993 lng s1yf qln f sn ln G yln Gi i i k k kŽ Ž ..y5q 1yV rF qln V rFi i i i i

3r4 3r4f s x r rÝ x ri i j jc c Ž i. Ž i.w xUNIFAC-C f s x r rÝ x r sn ln G yln Gi i i j j k k k

combŽ . Ž . Ž Ž .UNIQUAC r 2r3 lng s ln f r x y q 1yln Ý u ti i i i j j j, iŽ . Ž . .q zr2 q ln u rf Ý u t rÝ u ti i i j j i, j k k k , j

Ž .q l y f r x Ý x li i i j j2r3 2r3f s x r rÝ x ri i i j j

Flory, FFV, FEFV, UNIFAC are predictive models.Flory-x , FEFV-X, UNIFAC-C each contain a single fitting parameter.UNIQUAC-2r3, Wilson, van Laar each contain two fitting parameters.

model contains both combinatorial and residual terms with two fitting parameters in the residual termw xfor each binary pair. Wilson and van Laar models 8 contain two fitting parameters for each binary

pair and are known to provide accurate description of many solution activities. The Wilson modelessentially uses volume fractions in its formulation with a temperature dependence that is similar tothe UNIFAC and UNIQUAC residual terms.

4. Modification of UNIFAC configurational term

Various modifications to the combinatorial portions of the UNIFAC model have been made tow xaccount for large size molecules such as polymers. Oishi and Prausnitz 21 developed the UNIFAC

method for polymer systems by introducing a free volume term based on Flory’s equation into theactivity coefficient framework in addition to the UNIFAC combinatorial term. In their theory, aparameter ‘c1’ was used to modify the number of external degrees of freedom of the solvent alongwith other functions that were based on a methylene subgroup reference. For their systems, ‘c1’ was

w xset equal to 1.1 although they noted that other values of ‘c1’ might be required. Kikic et al. 22developed the combinatorial 2r3 term that was used in the Lyngby version of UNIFAC. Fornari et al.

( )R.L. Smith Jr. et al.rFluid Phase Equilibria 145 1998 53–68 57

w x5 found that for vegetable oil–solvent mixtures, an exponent of 3r4 in the UNIFAC combinatorialterm provided a better description of the boiling temperatures and solvent infinite dilution activitycoefficients. In this work, we have evaluated all versions of UNIFAC and in addition, we have treatedthe exponent in the combinatorial terms as a fitting parameter. This is denoted as UNIFAC-C as

w xshown in the table. The functional form was first suggested by Donohue and Prausnitz 23 in theirinterpolation method between the combinatorial entropy as given by an ideal solution and that givenby Flory–Huggins.

5. TAG–solvent SLE data

w xPractically all TAG–SLE data are dated to the 1950’s and before. Data in Ref. 24 were abstractedw x w xfrom Ref. 25 . The traceable data reference was found to be Ref. 26 . The original reference

w x w xcontained a number of additional data over those abstracted in Ref. 24 or Ref. 25 . The data werew xsupplemented with the measurements in Refs. 27,28 to produce an initial data set in excess of 600

w xx–T points. A small amount of TAG–solvent data for mixed TAGs 29 were not used because oflack of melting point and heat of fusion data. After rejecting data points too close to the melting pointand data which showed extreme scatter in a log x vs. 1rT plot, the final data set consisted of 562

w xdata points for five monoacid TAGs and in nine different solvents. Data in Ref. 27 were found todiffer greatly from the trends of group contribution equations such as UNIFAC and other data.Although these data were included into data analyses, they are listed in the tables as separate datasets.

6. TAG–solvent physical properties

Solvents and TAG properties extracted from the literature are shown in Tables 2 and 3. The stableb form properties were used for the TAGs. Models that are based on volume fraction rather thanmole fraction require a molar volume at a given temperature. The temperature used is frequently takento be either the system temperature or a convenient reference such as 298.15 K. For these cases, the

Table 2Solute physical properties and their literature sources

liq ref ref fusSubstance MW T Acentric V V @T T T D Hc w m3 3w x w x w x w x w x w xK factor v cm rmol cm rmol K K kcalrmol

Ž .Tristearin SSS 891.49 914.3 1.708 604.89 1034.23 353.2 345.7 47.0Ž .Tripalmitin PPP 807.33 903.1 1.630 543.51 932.27 353.2 338.7 41.0Ž .Trimyristin MMM 723.17 892.1 1.542 482.13 817.15 333.2 329.8 35.4

Ž .Trilaurin LLL 639.01 859.7 1.441 420.75 714.79 333.2 319.7 29.5Ž .Tricaprin CCC 554.49 825.9 1.392 359.37 602.45 313.2 304.7 22.3

w x w x liq w xSources: T , v: Ref. 30 ; Bondi volume, V : Ref. 17 ; Molar volume, V : Ref. 31 ; Melting point temperature and heat ofc wfus w xfusion, T , D H : Ref. 32 .m

( )R.L. Smith Jr. et al.rFluid Phase Equilibria 145 1998 53–6858

Table 3Solvent physical properties and their literature sources

liq ref refSubstance MW T Acentric V V @T T Dipole Dielectricc w3 3w x w x w x w x w xK factor v cm rmol cm rmol K moment m D constant ´

Benzene 78.11 562.2 0.212 48.36 88.26 289 0 2.275Acetone 58.08 508.1 0.304 39.04 73.52 293 2.88 20.7Carbon disulfide 76.13 552.0 0.109 31.20 58.88 273 0.06 2.641Carbon tetrachloride 153.82 556.4 0.193 51.40 97.11 298 0 2.238Chloroform 119.38 536.4 0.218 41.64 80.17 293 1.01 4.806Diethylether 74.12 466.7 0.281 51.50 103.96 293 1.15 4.335Ethanol 46.07 513.9 0.644 29.84 58.39 293 1.69 24.55Ethyl acetate 88.11 523.2 0.362 52.77 97.79 293 1.81 6.02n-Hexane 86.18 507.5 0.299 68.26 130.77 293 0.08 1.89

liq w x w x w xSources: T , v, V : Ref. 31 ; Bondi volume, V : Ref. 17 ; m, ´ : Ref. 33 .c w

w xRackett equation 31 was used to extrapolate a reference density to obtain the hypothetical liquidmolar volume.

7. Solution method and objective function

Ž . w xEq. 3 was solved with DZERO routine 34 that uses a combination of bracketing and stepsearches depending on the value and sign of the function. Data were fit with an implementation of

w xNL2SOL 35,34 , which is a gradient search algorithm that takes Newton or Gauss–Newton–MarquardtŽ .steps. The objective function O.F. minimized was:

no . data points1 T yTexp calcErrors 5Ž .Ý

n Texpis1

Statistically, this form has the advantage over other O.F. such as those that compare mole fractions, inthat the calculated and experimental temperatures have the same order of magnitude. This means thatfitted values will not have a bias at higher and lower mole fractions.

8. TAG–solvent combinatorial and residual terms with UNIFAC

Fig. 1 shows TAG–benzene solvent combinatorial and residual terms plotted versus volumeŽ .fraction that were predicted from the UNIFAC 1977 model and calculated from the UNIFAC-C

model after fitting the experimental data. Only CCC and SSS are shown since other TAGs fall inbetween these extremes. For this solvent, the combinatorial term is large compared with the residualterm and therefore entropic effects are much more important than enthalpic effects. At low volumefractions, residual effects become important. As the dielectric constant or dipole moment of the

( )R.L. Smith Jr. et al.rFluid Phase Equilibria 145 1998 53–68 59

Ž . Ž . Ž .Fig. 1. Combinatorial and residual terms dashed lines for tristearin SSS and tricaprin CCC in benzene as predicted byŽ . Ž .the UNIFAC model dash–dot line and correlated by the UNIFAC-C model solid line plotted against volume fraction of

the TAG. Volumes were calculated at the system temperature as described in the text.

solvent increases, as for acetone and ethanol–TAG systems, the magnitude of the combinatorial andresidual terms become comparable. This behavior is important in understanding the performance ofactivity models given in Table 1 and shown in detail below.

9. TAG–solvent SLE with UNIFAC and UNIFAC-C

The section discusses solubility calculations for a representative TAG–nonpolar solvent system anda representative TAG–polar solvent system. Fig. 2a shows solubility calculations for UNIFAC and

Ž .UNIFAC-C models, where the ‘C’ was determined according to Eq. 5 for each binary system. ForCCC-benzene, the difference between the two models was small. However, as the size of the TAGincreased, entropic contributions increased and UNIFAC errors increased. The fitting parameter, ‘C’,in the UNIFAC-C model greatly improved the accuracy. For polar solvents, the residual andcombinatorial terms become comparable as noted previously. Nevertheless, the ‘C’ parameter alsoimproved the accuracy of UNIFAC for TAG–acetone systems as shown in Fig. 2b. For the systems

Ž .studied, the system averaged parameter ‘C’ was found to be close to 0.9 using the UNIFAC 1977w xparameters. For the Lyngby parameters 19 , the system averaged parameter ‘C’ was found to be closew xto 0.858 and for the Dortmund version 20 the system averaged parameter ‘C’ was found to be close

to 0.749.

( )R.L. Smith Jr. et al.rFluid Phase Equilibria 145 1998 53–6860

Ž . Ž .Fig. 2. Predicted and correlated solid–liquid equilibria of TAG–solvent binary mixtures: a benzene solvent; b acetoneŽ . Ž . Ž . Ž .solvent. TAGs: tristearin SSS , triangles; tripalmitin PPP , filled circles; trimyristin MMM , squares; trilaurin LLL , filled

Ž . Ž . Ž . w xsquares; tricaprin CCC circles. Models: UNIFAC dash–dot line ; UNIFAC-C solid line . Data are those of Ref. 26 .

( )R.L. Smith Jr. et al.rFluid Phase Equilibria 145 1998 53–68 61

10. Evaluation of models

We evaluated all of the models in Table 1 and these results are summarized for all solvents inTable 4. The systems were divided into nonpolar, slightly polar and polar groups. The Flory modelgave a qualitative description for most of the systems and greatly overpredicted the temperature asshown by the mainly large positive bias values. The Flory model gave surprisingly good predictionsfor the polar solvent systems. Adding a single parameter, x , to the Flory model that was determinedfor each binary system greatly improved correlation as shown in the tables. There was only a smallimprovement in the calculations when volume fractions were calculated at the system temperature asopposed to 298 K as shown. The FFV and FEFV models gave good predictions for nonpolar systemssuch as benzene but performed poorly as the solvent system became more polar. Addition of a Florytype x parameter to the FFEV model to account for enthalpic effects, provided remarkablecorrelation of the data as shown by the errors in the slightly polar and polar system tables. The

Ž .UNIFAC models provided fair ;3% error predictions for most systems. Adding the fittingparameter ‘C’ greatly improved the performance as shown in the table. By averaging all system dataas discussed below, a universal value of Cs0.897 was determined and this is shown as UNIFAC-0.897 in the tables. The UNIFAC-0.897 model provided good prediction of the SLE data for mostsystems. For the Lyngby and Dortmund versions of UNIFAC, the system averaged values of the

Table 4Overall summary of temperature deviations for all systems

Model All systems Nonpolar Slightly polar Polar

AAPD BIAS AAPD BIAS AAPD BIAS AAPD BIAS

Ž .Flory, Õ1 T 7.828 22.612 8.078 23.771 7.416 20.993 8.198 23.679Ž .Flory-X, Õ1 298K 2.148 3.617 2.096 3.725 1.927 3.133 2.754 4.455Ž .Flory-X, Õ1 T 2.123 3.567 2.070 3.668 1.898 3.078 2.742 4.432

Ž .Flory-Free Volume FFV 2.234 0.790 2.123 1.840 2.485 2.444 1.918 y5.176Ž .Flory-Entropic FEFV 2.185 y2.010 1.854 0.218 2.075 y1.120 3.154 y8.850

Ž .Flory-Entropic-X FEFV-X 0.305 0.135 0.350 0.051 0.275 0.145 0.271 0.297UNIFAC 2.719 y7.655 1.598 y4.474 4.234 y11.947 1.802 y5.062UNIFAC-2r3 3.040 5.203 2.572 6.850 2.667 y0.353 4.887 13.950UNIFAC-3r4 2.606 2.633 1.949 4.561 2.826 y2.631 3.550 10.119UNIFAC-0.897 1.976 y2.949 0.999 y0.364 3.340 y7.646 1.082 1.843UNIFAC-C 0.553 y0.034 0.349 0.117 0.927 y0.191 0.166 y0.017

Ž .UNIFAC L 1.769 3.910 2.243 5.417 1.884 3.507 0.477 1.513Ž .UNIFAC L -C 0.279 0.047 0.331 y0.179 0.333 0.291 0.043 y0.000Ž .UNIFAC D 1.066 y0.197 0.781 y0.520 1.433 0.800 0.870 y1.710Ž .UNIFAC D -C 0.385 y0.026 0.325 y0.120 0.446 0.173 0.380 y0.259

UNIQUAC 0.274 0.351 0.374 0.515 0.229 0.248 0.156 0.223van Laar 0.179 0.063 0.207 0.111 0.144 0.057 0.194 y0.030Wilson 0.154 0.000 0.207 y0.089 0.123 0.049 0.106 0.089Number of data points 562 227 31 104

Nonpolar solvents: benzene, carbon disulfide, carbon tetrachloride, n-hexane.Slightly polar solvents: chloroform, diethyl ether, ethyl acetate.Polar solvents: acetone, ethanol.

( )R.L. Smith Jr. et al.rFluid Phase Equilibria 145 1998 53–6862

Ž .Fig. 3. Variation of UNIFAC exponent ‘C’ in the combinatorial term versus absolute average percent deviation AAPD intemperature for all systems for original UNIFAC, UNIFAC-Lyngby, and UNIFAC-Dortmund.

parameter ‘C’ were found to be 0.858 and 0.749, respectively. The UNIQUAC, Wilson, and van Laarmodels, which contain two adjustable fitting parameters for each binary system, provided excellentcorrelation of all data and all systems.

11. Overall evaluation

Using the UNIFAC-C model, we stepped ‘C’ in 0.005 or so increments and recalculated all SLEdata and the resulting AAPD. Results are shown in Fig. 3 for each version of UNIFAC. The bestpredictive model was found to be Dortmund version of UNIFAC. The Lyngby version of UNIFACand the FEFV, FFV models also provided good performance. The best single parameter model wasfound to be FEFV-X followed by UNIFAC-C. The Wilson equation provided the best two parameterfit of the data.

12. Model parameters

Parameters determined for one and two parameter models are shown in Tables 5 and 6. Within agiven solvent group, the parameters of all models showed some regularity as the TAG increased in

Ž .size from tricaprin to tristearin. Notable differences were observed in the data sets SSSqBZ 2 andŽ .SSSqCF 2 . Only UNIFAC-C values seemed to be have less variation among solvent groups. Some

UNIFAC-C parameter values differed significantly from the proposed 0.897 value and so we explored

()

R.L

.Smith

Jr.etal.r

Fluid

Phase

Equilibria

1451998

53–

6863

Table 5Fitted constants for one-parameter models

System Source Flory-X Õ1 Flory-X Õ1 Flory Entropic-X UNIFAC-77 ‘C’ UNIFAC UNIFAC-Ž .298 K -Lyngby ‘C’ Dortmund ‘C’

SSSqBZ 1 y9.4439Eq00 y9.4344Eq00 2.4593Ey01 8.8365Ey01 9.0975Ey01 8.0374Ey01SSSqBZ 2 y5.1166Eq00 y5.0969Eq00 4.5768Eq00 8.9149Eq02 5.6888Ey01 4.5603Ey01PPPqBZ 1 y1.0532Eq01 y1.0527Eq01 y5.5123Ey01 0.0000Eq00 9.1225Ey01 7.6333Ey01MMMqBZ 1 y9.5303Eq00 y9.5377Eq00 y3.4416Ey01 9.0550Ey01 9.3497Ey01 8.1991Ey01LLLqBZ 1 y1.0322Eq01 y1.0355Eq01 y1.5648Ey01 7.2317Eq02 9.8145Ey01 7.8285Ey01CCCqBZ 1 y1.2691Eq01 y1.2773Eq01 y1.2493Eq00 0.0000Eq00 9.1679Ey01 7.9000Ey01SSSqCS 1 y1.0558Eq01 y1.0544Eq01 1.0974Eq01 8.6328Ey01 n.a. 5.4711Ey01MMMqCS 1 y8.9327Eq00 y8.9415Eq00 7.2222Eq00 5.5850Eq02 n.a. 7.4220Ey01LLLqCS 1 y7.2738Eq00 y7.2889Eq00 5.7892Eq00 0.0000Eq00 n.a. 7.3470Ey01CCCqCS 1 y8.5216Eq00 y8.5616Eq00 4.0607Eq00 9.1271Ey01 n.a. 7.8758Ey01SSSqCT 2 y7.4906Eq00 y7.4715Eq00 4.0758Eq00 7.2317Eq02 6.5225Ey01 6.7787Ey01SSSqHX 2 y3.7629Eq00 y3.7226Eq00 2.7398Eq00 0.0000Eq00 8.3046Ey01 7.1131Ey01SSSqCF 1 y1.6066Eq01 y1.6062Eq01 2.6725Eq00 9.4873Ey01 8.3001Ey01 7.0168Ey01SSSqCF 2 y6.4001Eq00 y6.3716Eq00 2.0275Eq00 5.5850Eq02 8.8099Ey01 7.2048Ey01PPPqCF 1 y1.6451Eq01 y1.6472Eq01 1.6485Eq00 0.0000Eq00 5.9736Ey01 6.4622Ey01MMMqCF 1 y1.4832Eq01 y1.4881Eq01 y4.1652Eq00 6.2854Ey01 9.2747Ey01 8.8211Ey01LLLqCF 2 y1.6200Eq01 y1.6320Eq01 4.4479Eq00 8.9149Eq02 2.5824Ey01 4.5179Ey07SSSqDE 1 y6.8952Eq00 y6.8703Eq00 y4.1584Eq00 0.0000Eq00 9.1522Ey01 8.6797Ey01PPPqDE 1 y5.9607Eq00 y5.9465Eq00 y3.8829Eq00 9.8648Ey01 8.7115Ey01 8.1316Ey01MMMqDE 1 y5.5181Eq00 y5.5183Eq00 y4.6403Eq00 8.9149Eq02 9.0295Ey01 8.4004Ey01LLLqDE 1 y5.0148Eq00 y5.0360Eq00 3.5423Ey01 0.0000Eq00 7.3267Ey01 8.0892Ey01CCCqDE 1 y5.2020Eq00 y5.2521Eq00 7.1976Ey01 9.8310Ey01 6.7029Ey01 7.4585Ey01SSSqEA 2 y2.5403Eq00 y2.5156Eq00 7.3522Ey01 7.2317Eq02 6.3232Ey01 6.9023Ey01SSSqAC 3 3.2605Ey01 3.4705Ey01 4.9535Ey01 0.0000Eq00 6.1915Ey01 6.5807Ey01MMMqAC 3 y5.4936Ey01 y5.5254Ey01 y7.4000Ey04 1.0248Eq00 6.6180Ey01 6.5279Ey01LLLqAC 3 y1.0590Eq00 y1.0666Eq00 1.4875Eq01 8.9149Eq02 8.5402Ey01 7.2491Ey01CCCqAC 3 y1.8469Eq00 y1.8671Eq00 8.7655Eq00 0.0000Eq00 8.3182Ey01 6.4956Ey01SSSqET 1 3.0580Eq00 3.0887Eq00 5.7145Eq00 2.7815Ey01 6.9472Ey01 7.8945Ey01CCCqET 1 1.3023Eq00 1.2984Eq00 2.4693Eq00 7.2317Eq02 3.9692Ey01 6.8418Ey01

Key: BZ, benzene; CS, carbon disulfide; CT, carbon tetrachloride; HX, n-hexane; CF, chloroform; DE, diethyl ether; EA, ethyl acetate; AC, acetone; ET,ethanol. n.a.: parameters not available.

w x w x w xSources: 1, Ref. 26 ; 2, Ref. 27 ; 3, Ref. 28 .

()

R.L

.Smith

Jr.etal.r

Fluid

Phase

Equilibria

1451998

53–

6864

Table 6Fitted constants for two-parameter models

System Source UNIQUAC-2r3 Wilson van Laar

Ž . Ž . Ž . Ž .u Jrmol u Jrmol l Jrmol l Jrmol A B12 21 12 21

SSSqBZ 1 y1.4450Eq03 1.7978Eq03 y1.0997Eq03 9.8454Eq02 y4.6443Eq00 y8.5196Ey01SSSqBZ 2 3.6037Eq03 y1.8444Eq03 y5.6493Eq03 5.1643Eq03 4.0000Ey05 y2.0000Ey05PPPqBZ 1 8.1188Eq02 y8.9941Eq02 y5.9100Eq03 1.6663Eq03 y4.1545Eq00 y1.4606Eq00MMMqBZ 1 7.3295Eq02 y8.0835Eq02 y4.8020Eq03 1.5761Eq03 y3.4298Eq00 y1.2254Eq00LLLqBZ 1 7.7497Eq02 y8.4928Eq02 y3.6906Eq03 1.3275Eq03 y2.9315Eq00 y1.0451Eq00CCCqBZ 1 9.5312Eq02 y1.0504Eq03 y3.7053Eq03 1.1767Eq03 y2.6914Eq00 y1.0789Eq00SSSqCS 1 y1.2526Eq03 1.6768Eq03 1.2145Eq03 1.7571Eq03 y3.9363Eq00 y1.2461Eq00MMMqCS 1 y1.4180Eq03 2.0180Eq03 2.3191Eq03 1.3118Eq03 y3.1467Eq00 y1.2405Eq00LLLqCS 1 y1.0008Eq03 1.3669Eq03 1.8666Eq03 1.3784Eq03 y2.1987Eq00 y1.1138Eq00CCCqCS 1 y1.1388Eq03 1.7116Eq03 3.7781Eq03 1.1560Eq03 y1.6759Eq00 y1.1860Eq00SSSqCT 2 2.6670Eq03 y1.7032Eq03 y2.0407Eq03 1.9024Eq03 y3.0446Eq00 y3.3741Ey01SSSqHX 2 3.0574Eq03 y1.7134Eq03 y7.8938Eq03 5.8695Eq04 1.6555Ey01 6.6158Eq03SSSqCF 1 1.7406Eq03 y1.7314Eq03 2.9675Eq02 y2.1758Eq02 y9.8103Eq00 y1.3399Eq00SSSqCF 2 2.9250Eq03 y1.6526Eq03 y3.9327Eq03 3.7077Eq03 y1.4728Eq00 y1.6004Ey01PPPqCF 1 1.3080Eq03 y1.5140Eq03 y5.3536Eq03 3.8909Eq02 y8.1221Eq00 y1.7330Eq00MMMqCF 1 1.2409Eq03 y1.4486Eq03 y4.8914Eq03 3.4690Eq02 y7.2820Eq00 y1.6459Eq00LLLqCF 1 1.3586Eq03 y1.5881Eq03 y3.3710Eq03 y2.3326Eq02 y7.7688Eq00 y1.5787Eq00SSSqDE 1 1.5962Eq02 y1.8242Eq02 y1.3946Eq03 1.3400Eq03 y2.8596Eq00 y7.6969Ey01PPPqDE 1 y9.4607Eq02 1.1821Eq03 1.0105Eq03 1.1001Eq03 y1.9351Eq00 y8.9839Ey01MMMqDE 1 y1.0284Eq03 1.3446Eq03 1.8665Eq03 9.4203Eq02 y1.4342Eq00 y1.0182Eq00LLLqDE 1 y8.2179Eq02 1.0365Eq03 7.2334Eq02 9.5405Eq02 y1.2657Eq00 y6.8671Ey01CCCqDE 1 y1.2230Eq03 1.6832Eq03 y5.8225Eq03 3.3979Eq03 y1.1316Eq00 y1.1209Eq00SSSqEA 2 2.7076Eq03 y1.4017Eq03 y3.2591Eq03 6.4466Eq03 1.1067Eq00 7.9054Ey01SSSqAC 3 3.8340Eq03 y1.4530Eq03 4.1993Eq03 8.6443Eq03 3.9842Eq00 8.3473Ey01MMMqAC 3 y1.1581Eq02 6.2773Eq02 8.2180Eq03 2.4569Eq03 2.9262Eq00 1.3390Ey02LLLqAC 3 y4.1997Eq02 9.3329Eq02 4.7990Eq03 2.7136Eq03 2.0681Eq00 y5.1734Eq03CCCqAC 3 1.7651Eq03 y8.6467Eq02 y3.2266Eq03 6.2127Eq03 1.1410Eq00 y1.1976Eq05SSSqET 1 y2.2953Eq03 4.5526Eq03 1.2886Eq04 7.8290Eq03 7.3688Eq00 5.0810Ey02CCCqET 1 1.0528Eq03 3.6146Eq01 4.9936Eq03 7.0438Eq03 4.8512Eq00 3.2852Ey01

Key: BZ, benzene; CS, carbon disulfide; CT, carbon tetrachloride; HX, n –hexane; CF, chloroform; DE, diethyl ether; EA, ethyl acetate; AC, acetone; ET,ethanol.

w x w x w xSources: 1, Ref. 26 ; 2, Ref. 27 ; 3, Ref. 28 .

( )R.L. Smith Jr. et al.rFluid Phase Equilibria 145 1998 53–68 65

this further. For each TAG–solvent, we plotted the value of ‘C’ from approximately 0 to 1.2 andcalculated the AAPD. Many of the AAPD curves versus ‘C’ had a distinct minimum like that shownin Fig. 3. For diethyl ether systems, AAPD versus ‘C’ curves appeared differently and monotonicallyincreased as the value of ‘C’ was increased from 0.1 to 1.2. For most TAGqDE systems, the AAPDremained almost constant in for ‘C’ values between 0.1 and 0.5. The SSSqDE system showed aslight minimum at 0.485. Because of this, a more appropriate value of ‘C’ for the diethyl ethersolvent group is probably 0.485. These analyses can be used to develop a set of recommendations forcalculating SLE of TAG–solvent mixtures.

13. Model recommendations

The recommended model depends greatly on the end use and the available data. For predictions ofTAG–solvent SLE, the Dortmund version of UNIFAC can provide SLE temperatures with averagedeviations of around 1.07% or so. If ‘C’ values are used, the error drops to 0.385%. If other UNIFACversions are being used, then the tabulated ‘C’ values can greatly improve the prediction accuracy.Further, if the original UNIFAC is used, then an average value of Cs0.897 works well. For theLyngby and Dortmund versions, average values of ‘C’ that work well are 0.858 and 0.749,respectively. The FEFV and FFV models also have merit due to their moderate accuracy andsimplicity. If only limited data are available, the FEFV-X model is suitable since a single adjustableparameter provides greatly improved accuracy. The FEFV-X model is remarkably simple. For caseswhere data are sufficient and where only correlation is required, the Wilson equation has been shownto provide the most accurate correlations of TAG–solvent systems.

14. Conclusions

The Dortmund version of UNIFAC provides the best predictions of TAG–solvent systems. VariousUNIFAC models can be slightly modified to greatly improve performance by using an exponent in the

Ž .combinatorial term as a fitting parameter. For the UNIFAC-1977, UNIFAC Lyngby and UNIFACŽ .Dortmund , these exponents were found to be 0.897, 0.858, and 0.749, respectively. The FFV andFEFV also gave satisfactory predictions for TAG–nonpolar and slightly polar solvent systems andthese models are considerably simpler than the UNIFAC models. Addition of a Flory-x type residual

Ž .term to a FEFV term gave the best overall one parameter correlation model FEFV-X . The FEFV-Xmodel was also one of the simplest models. The Wilson equation was the best two parametercorrelation model.

15. Nomenclature

DC Heat capacityp

D H fus Enthalpy of fusionc Degrees of freedomC UNIFAC exponent in the combinatorial term

( )R.L. Smith Jr. et al.rFluid Phase Equilibria 145 1998 53–6866

Ž .Ž . Ž .l UNIQUAC parameter defined as zr2 r yq y r -1i i i

L Wilson equation parametern Number of data pointsr UNIFAC segment volumeq UNIFAC segment areaT Triple point temperaturet

V Molar volume, usually at the given system temperatureV Bondi 0 K molar volumew

x Mole fractionz Coordination number

Ž .AAPD Absolute average percent deviation, Eq. 5Ž .BIAS Bias of calculated values determined by: BIASs1rnÝ T yTcalc exp i

CCC TricaprinLLL TrilaurinMMM TrimyristinPPP TripalmitinSSS TristearinTAG Triacylglyceride

Subscriptscalc Calculatedexp Experimentali, j Indexes

Superscriptscomb Combinatorialfv Free volumel Liquido Standard stateres Residuals Solid

Greekb Stable triacylglyceride formg Activity coefficientx Flory parameter, residual activity coefficient fitting parameter´ Dielectric constantG Residual activity coefficient of functional group k in the mixturek

G Ži. Residual activity coefficient of functional group k in the mixture containing onlyk

molecules of type im Dipole momentu Area fraction of group ii

t Energy parameter between molecular segment i and ji, j

n Ži. Number of functional groups of type k in molecule ik

v Acentric factorF Volume fraction

( )R.L. Smith Jr. et al.rFluid Phase Equilibria 145 1998 53–68 67

f Volume, free volume, or entropic fractionL Free volume ration as defined in Table 1

Acknowledgements

ŽWe would like to acknowledge the Research Center of Supercritical Fluid Technology Tohoku.Univ. for support of this work.

References

w x1 K.A. Boni, Process for recovering collagen, US Patent 5,229,497, 1993.w x2 G.M. Acosta, R.L. Smith Jr., J.E. Walsh, K.A. Boni, Beef shank fat solubility in supercritical carbon dioxide–propane

Ž . Ž .mixtures and in liquid propane, J. Food Sci. 60 5 1995 983–987.w x3 J. Gmehling, G. Anderson, T.F. Prausnitz, J.M. Prausnitz, Solid–liquid equilibria using UNIFAC, Ind. Eng. Chem.

Ž . Ž .Fundam. 17 4 1978 269–273.w x4 J.A.P. Coutinho, S.I. Andersen, E.H. Stenby, Evaluation of activity coefficient models in prediction of alkane

Ž .solid–liquid equilibria, Fluid Phase Equilibria 103 1995 23–39.w x Ž . Ž .5 T. Fornari, S. Bottini, E.A. Brignole, Application of UNIFAC to vegetable oil–alkane mixtures, JAOCS 71 4 1994

391–395.w x6 U. Domanska, Solubility of long-chain fatty acid esters in selected organic one- and two-component solvents, J.´

Ž . Ž .Solution Chem. 18 2 1989 173–188.w x7 K.L. Roberts, R.W. Rousseau, A.S. Teja, Solubility of long-chain n-alkanes in heptane between 280 and 350 K, J.

Ž .Chem. Eng. Data 39 1994 793–795.w x8 J.M. Prausnitz, R.N. Lichtenthaler, E. Gomes de Azevedo, Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd

edn., Prentice-Hall, NJ, 1986.w x9 D.S. Mishra, S.H. Yalkowsky, Solubility of organic compounds in nonaqueous systems: polycyclic aromatic hydrocar-

Ž .bons in benzene, Ind. Eng. Chem. Res. 29 1990 2278–2283.w x Ž . Ž .10 P.B. Choi, E. McLaughlin, Effect of a phase transition on the solubility of a solid, AIChE J. 29 1 1983 150–153.w x Ž .11 K. Larsson, in: F.B. Padley Ed. , The Lipids Handbook, Chapmann & Hall, London, 1986.w x12 L.H. Wesdorp, Liquid-multiple solid phase equilibria in fats, Ph.D. Thesis, Delft University, Netherlands, 1990.w x13 H.S. Elbro, Aa. Fredenslund, P. Rasmussen, A new simple equation for the prediction of solvent activities in polymer

Ž .solutions, Macromolecules 23 1990 4707–4714.w x Ž .14 P.J. Flory, J. Chem. Phys. 9 1941 660.w x Ž .15 M.L. Huggins, J. Chem. Phys. 9 1941 440.w x Ž .16 P.J. Flory, Disc. Faraday Soc. 49 1970 7.w x17 A. Bondi, Physical Properties of Molecular Crystals, Liquids and Glasses, Wiley, New York, 1968.w x18 Aa. Fredenslund, J. Gmehling, P. Rasmussen, Vapor–Liquid Equilibria Using UNIFAC, Elsevier, Amsterdam, 1977.w x19 B.L. Larsen, P. Rasmussen, A. Fredenslund, A modified UNIFAC group-contribution model for prediction of phase

Ž .equilibria and heats of mixing, IEC Res. 26 1987 2274–2286.w x20 J. Gmehling, J. Li, M. Schiller, A modified UNIFAC model: 2. Present parameter matrix and results for different

Ž .thermodynamic models, Ind. Eng. Chem. Res. 32 1993 178–193.w x21 T. Oishi, J.M. Prausnitz, Estimation of solvent activities in polymer solutions using a group-contribution method, Ind.

Ž . Ž .Eng. Chem. Proc. Des. Dev. 17 3 1978 333–339.w x22 I. Kikic, P. Alessi, P. Rasmussen, Aa. Fredenslund, On the combinatorial part of the UNIFAC and UNIQUAC models,

Ž .Can. J. Chem. Eng. 58 1980 253–258.w x23 M.D. Donohue, J.M. Prausnitz, Combinatorial entropy of mixing molecules that differ in size and shape. A simple

Ž .approximation for binary and multicomponent mixtures, Can. J. Chem. 53 1975 1586–1592.

( )R.L. Smith Jr. et al.rFluid Phase Equilibria 145 1998 53–6868

w x Ž .24 G.D. Fasman Ed. , Practical Handbook of Biochemistry and Molecular Biology, CRC Press, Boca Raton, FL, 1989.w x25 K.S. Markley, Fatty Acids, 2nd edn., Part 1, Interscience Publishers, New York, 1960.w x Ž . Ž .26 K.Z. Loskit, Zur kenntnis der triglyceride, Phys. Chem. Leipzig 134 1928 135–155.w x Ž .27 C.W. Hoerr, H.J. Harwood, The solubility of tristearin in organic solvents, J. Phys. Chem. 60 1956 1265–1269.w x28 O.S. Privett, E. Breault, J.B. Covell, L.N. Norcia, W.O. Lundberg, Solubilities of fatty acids and derivatives in acetone,

Ž .JAOCS 35 1958 366–370.w x Ž .29 C. Chen, B.F. Daubert, Synthetic triacid triglycerides of saturated fatty acids, J. Am. Chem. Soc. 67 1945 1256–1258.w x30 Z.-R. Yu, S.S.H. Rizvi, J.A. Zollweg, Phase equilibria of oleic acid, methyl oleate, and anhydrous milk fat in

Ž .supercritical carbon dioxide, J. Supercrit. Fluids 5 1992 114–122.w x31 R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids, 4th edn., McGraw-Hill, New York, 1987.w x32 M. Ollivon, R. Perron, Measurements of enthalpies and entropies of unstable crystalline forms of saturated even

Ž .monoacid triglycerides, Themochim. Acta 53 1982 183–194.w x Ž .33 J.A. Dean Ed. , Lange’s Handbook of Chemistry, 13th edn., McGraw-Hill, New York, 1985.w x34 Math 77, Language Systems, Sterling, VA, Release 4.1, April, 1994.w x35 J.E. Dennis Jr., D.M. Gay, R.E. Welsch, An adaptive nonlinear least-squares algorithm, ACM Trans. Math. Software 7

Ž . Ž .3 1981 348–383.