Download Regular Solutions - Mixing Rules for Excess Free Energy Models and more Lecture notes Chemistry in PDF only on Docsity!
J.Serb.Chem.Soc. 66 (4)213–236(2001) UDC 536.77:621. JSCS–2849 Review
R E V I E W
Mixing rules for excess free energy models
BOJAN D. DJORDJEVI] #^ , MIRJANA LJ. KIJEV^ANIN #^ , JADRANKA P. ORLOVI] 1 and SLOBODAN P. [ERBANOVI]#
Faculty of Technology and Metallurgy, University of Belgrade, YU-11120, Karnegijeva 4 and (^1) ”Duga” - Paints and Varnishes Industry, Viline vode 6, YU-11000 Belgrade, Yugoslavia
(Received 30 December 2000)
- Introduction
- Huron-Vidal mixing rule (HV)
- MHV1, MHV2 mixing rules
- Wong-Sandler mixing rule
- LCVM mixing rule (WS)
- Huron-Vidal-Orbey-Sandler mixing rule (HVOS)
- Predictive CEOS/ G E^ models. PSRK model
- Twu et al. mixing rules
- VLE calculation in polymer solutions using EOS/ G E^ models
- Summary Keywords : cubic equation of state, mixing rule, vapour-liquid equilibria, excess molar properties, correlation, prediction.
- INTRODUCTION Cubic equations of state (CEOS) have been the frequent subject of active re- search since van der Waals^1 presented his equation in 1873. Numerous CEOS have been proposed for the phase behavior and thermodynamic properties of pure compo- nents and their mixtures. Due to their simplicity and accuracy, CEOS are the most fre- quently used equations of state for computer-aided design of chemical processes. The accuracy of VLE calculations depends on the accuracy of the vapor pressure calcula- tions of pure compounds and the functional form of the mixing rules. In the past the CEOS approach has been applied successfully to systems containing only non-polar and slightly polar components for all conditions of practical interest. In cases of rela- tively simple mixtures ( i.e., hydrocarbons, hydrocarbons with inorganic gases) the so-called van der Waals one fluid mixing and combining rules perform adequately at the low-density limit as the ideal gas law and at the high-density state (“dense fluid”).
The general form of a two-parameter cubic equations of state (CEOS) used in this review is as follows
213
Serbian Chemical Society active member.
p
RT
v b
a v ub v wb
The constants u and w are EOS dependent. CEOS can describe pure components reasonably well. Remarkable success in the development of a generalized temperature and acentric factor dependent function of the attractive term of CEOS energetic parameters a has been achieved by many authors for example2–10. Alternatively, Xu and Sandler^11 gave polynomial expressions for both the energy and covolumen parameters of the PR EOS which are specific for each fluid. How- ever, for asymmetric non-ideal mixtures where the molecules are dissimilar in size or chemical nature, a number of alternative mixing rules must be applied, first of all compo- sition-dependent and density-dependent mixing rules. The first of them are inconsistent at the low-density limit with the statistical mechanical result that the second virial coeffi- cient must be a quadratic function of composition. To correct this problem attempts have been made to develop density dependent mixing rules. Both rules improve the representa- tion of phase behavior in very complex non-ideal mixtures. Concise reviews of the devel- opment of these two types of rules have been given by a few authors.12–
Very recently some mixing rules combining free energy models ( G E^ or A E) and equations of state (EOS) have been successfully applied to very complex systems of di- versified nature covering wide ranges of temperature and pressure. Among of these models the so-called EOS/ G E^ or EOS/ A E^ have been used for the correlation and predic- tion of vapor-liquid (VLE), liquid-liquid (LLE) equilibria and other thermodynamic properties. These models have been widely studied and an extensive analysis of their applicability has been reviewed in several excellent articles and monographs.12–
In this review, some recent advances in describing phase equilibria and excess properties using CEOS/ G E^ models will be briefly considered.
The starting point for equating excess free energy from activity coefficient mod- els and from equation of state models is the relationship
G CEOS RT [ x i ] i
E (^) = lnj - j ln
å^ (2)
where j and j i are the fugacity coefficients of the mixture and of the pure compo- nent i , both determined from the CEOS at the pressure and temperature of the sys- tem. Thus, one has
[ ] [ ]
G
RT
CEOS z x zi i z x (^) i zi
E
= - å – ln- å ln -
é -
ë
ê ê
ù
û
ú = = ú
ò å ò
z z
v x
z z
v v
v i
i
v i
1 v^ i d
d μ μ
or equivalently, for the excess Helmholtz free energy
A RT
x
z z
z z
v x
z z
i v i (^) v
v i v
v i i
i CEOS
E ln
d
= - d
é -
ë
å ê ò å ò
ê = μ =μ
ù
û
(^) ú ú
214 DJORDJEVI] et al.
where C = C * = C (^) i. The mixing rule (11) does not satisfy the low density boundary condition
B T x x x B x x b
a
RT
b
a R
i i j
i j ij i j
i j ij
ij ( , ) = = -
æ
è
çç
ö
ø
(^) å å å å ÷÷ = - T
æ è
ç
ö ø
÷
A number of authors16–40^ demonstrated the validity of this mixing rule, Eqs. (8) and (9) coupled with various CEOS (RK, SRK, VdW, PRSV, voume-shifted PR, PT) and activ- ity coefficient models (van Laar, Redlich Kister, NRTL, UNIQUAC, UNIFAC, ASOG) to correlate and predict VLE and other thermodynamic properties of complex chemical sys- tems. Among them, Tochigi et al.^27 and Soave et al.^35 investigated a SRK group contribu- tion method to predict high pressure VLE and the infinite pressure activity coefficient, re- spectively.
The poor predictive performance of this model is analyzed in detail and explained by Orbey and Sandler.^16 They concluded that G E^ model parameters obtained by the g – j method at low pressure (for example DECHEMA Chemistry Data Series) could not be used with EOS/ G E^ models. Namely, a main shortcoming of the HV model is the use of the pressure dependent G E^ in the EOS rather than A E^ which is practically pressure independent.
Some modifications of the HV mixing rule were developed and applied in several works.21,27,
- MHV1, MHV2 MIXING RULES More recent mixing rules are based on less restrictive assumptions than those used for the HV mixing rule. Mollerup^41 used the condition that V E^ = 0, but based his approach on the zero pressure limit in a manner similar to the HV model at infinite pressure. Gupte et al.^42 used Eqs. (5) and (7) at the experimental pressure to establish the UNIWAALS model. Some restrictions of this model were eliminated by Gani et al.^43 by reformulating the computational aspects of the resultant model in a more consistent way.
In the VLE calculations of Novenario et al.,^44 the liquid volume was set to be a constant factor K (= v / b ) multiplied by the excluded volume b at the standard state where G g E^ = G EOSE. Using this method, the calculations of a zero pressure liquid volume, as part of any EOS calculations, is not needed. The authors showed that K = 1.15 is suitable for the PR EOS but for each EOS K must be separately determined.
Michelsen^45 proposed the Huron-Vidal approach of matching G E^ using a refer- ence pressure of zero.
Applying the condition for a reference pressure p = 0 to Eq. (5), one obtains
G RT
x
v b
v b i
i i
0,CEOS
E =- æ ln -^0 1 ln^0 è
ç
ö ø
÷ -
æ
è
çç^
ö
ø
÷÷
é
ë
ê^
ù å^ – 1 û
(^) ú - å x + b b
a bRT j C i
where subscript 0 indicates the reference pressure of zero. Eq. (13) can be given in the Michelsen form when rewritten as
216 DJORDJEVI] et al.
q x q
G
RT
x b b i i
,
E i i
( a =) a (+ ) +^0 ln æ è
ç ç
ö ø
÷
å ÷ å
g (14)
where a = a / bRT is the function
q
b b
( ) ln C 0 a =- 0 æ - è
ç
ö ø
÷ +a
n (15)
The zero pressure liquid volume v 0 is determined by solving the CEOS as part of a VLE calculation. However, a problem can arise at temperatures at which there is no liquid root of the EOS. For this reason, Michelsen arbitratily chose a cut-off value of a for which a liquid root exists.
In the first case, with smaller values of a , a linear extrapolation was used q (a ) = q 0 + q 1 a (16)
then Eq. (14) becomes
a= = +
æ
è
çç
ö
ø
÷÷ + a g å å
a RT
G
C RT C
x
b b i x i
i i
0,
E
0 0
ln (^) (17)
or
a= a + +
æ
è
çç
ö
ø
÷÷
é
ë
ê ê
ù
û
ú ú
å å x g q
G
RT
x b b i i i i
ln 1
0,
E (18)
where q 1 is a numerical constant dependent of the EOS. Eqs. (8) and (18) are known as the Modified Huron-Vidal First Order mixing rule (MHV1). Michelsen sets q 1 = – 0.593.
In the second approximation, Dahl and Michelsen^46 used the second-order poly- nomial
q (a ) = q 0 + q 1 a + q 2 a 2 (19)
where the parameters must be chosen to give continuity of the auxiliary function q (a ) and its derivatives. In this way Eq. (14) becomes
q x q x
G
RT
x b b i i i i i i
1 2
2 2 0,
E (a - a + )a - ( a = + ) ln æ è
çç^
ö ø
å å åg^ ÷÷
Eq. (20) is known as the Modified Huron-Vidal Second Order mixing rule (MHV2). For the interval 10 < a < 13, Dahl and Michelsen^46 suggest values of q 1 = – 0.478 and q 2 = – 0.0047 when the RK EOS was used. Huang and Sandler^47 proposed values of q 1 = – 0.4347 and q 2 = – 0.003654 for the PR EOS. When q 2 is set as zero, MHV2 (Eq. (20)) reduces to MHV1 (Eq. (18)). Soave48a^ gave a more accurate expres- sion for a comparable to those from Eq. (19) for the range a = 8 – 18. The results ob- tained in this way remain accurate up to a = 21.
The MHV2 mixing rule is in fact more complex than the HV approach, but G E models obtained by fitting low-pressure data may be used directly by means of the pa- rameters reported in the DECHEMA Chemistry Data Series.
MIXING RULES 217
The parameter f corrects for inadequacies of both the UNIFAC and MHV1 for highly asymmetric systems (CO 2 , CH 4 and C 2 H 6 with alkane systems). In later works59,62^ this method was extended to other gases (C 2 H 4 , CO and H 2 ).
Zhong and Masuoka^65 modified the MHV1 mixing rule in order that it gives al- most identical H E^ predictions to those obtained from the incorporated modified UNIFAC model and which are much better than those obtained from the MHV1 mixing rule.
This model is expressed as follows
a bRT
x
a b RT q
G
RT
x
b b i
i i
i i
= + +h
æ
è
çç
ö
ø
÷÷
é
ë
ê ê
ù
û
(^) ú ú
å^ å
(^1) g ln 1
E (26)
Parameter h is calculated for binary system by solving the equation H SRKE^ = H modEUNIFAC (at p = 0.1013 MPa, x 1 = x 2 = 0.5 and the system temperature T ). The modified MHV1 mixing rule shows significantly improved predictions over the MHV1 mixing rule.
Some comparisons of the cp E^ correlations of the acetone(1)+dodecane(2) system at 288 K by means of the approximate MHV1 and MHV2 models coupled with the PRSV EOS follow. To make these comparisons, the same activity coefficient model (NRTL) was used in all cases. Also, for each of the EOS/ G E^ models, the cp E^ data were fitted with three different NRTL equations: ( i ) two temperature independent parameters t 12 and t 21 and a = 0.3 (MHV1-NRTL2 and MHV2-NRTL2 model), ( ii ) two linear temperature dependent pa- rameters t 12 and t 21 and a = 0.3 (MHV1-NRTL4 and MHV2-NRTL4 model), ( iii ) three
linear temperature dependent parameters t 12 , t 21 and a. (MHV1 – NRTL6 and MHV2 – NRTL6 model). From Fig. 1 it can be seen that the performence of these six models is quite different. The results indicate that only the models with three temperature dependent inter- action parameters (the six optimized coefficients generated from cp E^ data) are very good for correlation. The best results were obtained with the MHV1-NRTL6 model.
- WONG-SANDLER MIXING RULE (WS) Wong and Sandler proposed^69 a new mixing rule which is based on the desired EOS behavior at both low and high densities without being density dependent and of equating free energies at infinite pressure. Namely, Wong and Sandler equate the Helmholtz free energy at infinite pressure from the CEOS with that acquired from any activity coefficient model. In this way the theoretically correct WS mixing rule satisfies the quadratic composition dependence of the second virial coefficient model at high density. On the contrary, the HV, MHV1, MHV2 mixing rules do not satisfy this bound- ary condition. By means of the WS mixing rule, the a and b mixing parameters are ob- tained from the relations
a RT
QD
D
b =
Q
1 - D
MIXING RULES 219
where the quantities Q and D are given by
Q x x b a i j RT
i j ij
= æ - è
ç ö ø
å å ÷
D x
a b RT
G
CRT
i
i i
= å + g
E (30)
where C is a constant that depends on the CEOS (for example, for PR EOS, C = (1+2 1/2^ )/2 1/2^ ).
220 DJORDJEVI] et al.
Fig. 1. c (^) p E^ Correlation of the system acetone (1)+dodecane (2) at 288 K, with the MHV1-NRTL and the MHV2-NRTL models combined with the PRSV EOS. The solid lines represent results cal- culated with the following MHV1-NRTL parameters: NRTL2: t 12 = (0.108549´ 104 )/ RT , t 21 = (0.878439´ 104 )/ RT , a 12 = 0.3; NRTL4: t 12 = (0.245528´ 104 –3.40724 T )/ RT , t 21 = (–0.543588´ 10 3 –6.96601 T )/ RT , a 12 = 0.3; NRTL6: t 12 = (0.17327´ 103 +0.114689 T )/ RT , t 21 = (0.58515´ 10 4 –8.81454 T )/ RT , a 12 = 0.0405539–0.566460´ 10 -3 T. The dashed lines represent results calculated by the MHV2-NRTL parameters: NRTL2: t 12 = (0.553106´ 103 )/ RT , t 21 = (0.8233416´ 10 4 )/ RT , a 12 = 0.3; NRTL4: t 12 = (0.197987´ 104 –7.29184 T )/ RT , t 21 = (0.364533´ 10 3 +2.09463 T )/ RT , a 12 = 0.3; NRTL6: t 12 = (–0.423086´ 104 +3.70220 T )/ RT , t 21 = (0.288382´ 104 +3.38331 T )/ RT , a 12 = –0.0439628–0.509917´ 10 -4 T. The points are expreimental data of Saint-Victor and Patterson.^111
when the size of the molecules is significantly different. The quality of the results of the LCVM are similar to those obtained by the modified MHV1.58,59^ In the work of Orbey and Sandler,^39 a comparison is made of the PSRK, MHV1, MHV2, HVOS and LCVM mixing rules for mixtures of molecules differing largely in size. The results obtained show that the MHV2 model was the least accurate. Further applications of this model are considered in several articles of Tzouvaras^86 and Tassios and coworkers.87–
- HURON-VIDAL-ORBEY-SANDLER MIXING RULE (HVOS) Orbey and Sandler^52 developed an approximate but rigorous model by assuming that for all fluids there is a universal linear algebraic function u that relates the liquid molar volumes and their close packed hard-core volumes such as u = v / b. At infinite pressure, both for the pure liquid and for the mixtures, u has a unique value
A x b b
a bRT
C ub x a b RT
i C ub i
i
i i
EOS i
E (^) =- ln æ ( ) ( ) è
çç
ö ø
å ÷÷ + -å^ (35)
At all conditions one can take u = 1, [ C ( ub ) = C ( ubi ) ], thus the resulting equation is
A RT
G
RT
x b b
C
a bRT
x
a b i i
i
i i
EOS
E E = =- ln
æ
è
çç^
ö
ø
÷÷ + -
æ
è
(^) çç g ö å^ å ø
÷÷ (36)
222 DJORDJEVI] et al.
Fig. 2. H EPrediction of the system acetonitrile(1)+ethanol(2)+benzene(3) at 298 K, with the WS-NRTL-PRSV model. The surface is obtained by the following binary parameters: t 12 = (0.633398´ 104 –0.614310 T )/ RT , t 21 = (0.321367´ 104 –7.25368 T )/ RT , t 13 = (0.321355´ 104 –2.62969 T )/ RT , t 31 = (–0.252335´ 103 –3.46543 T )/ RT , t 23 = (0.331820´ 104 –0.197586´ 102 T )/ RT , t 32 = (0.715713´ 104 +8.79139 T )/ RT ; a 12 = 0.3; a 13 = 0.3, a 23 = 0.47; k 12 = 0.163, k 13 = 0.119, k 23 = 0.390. The points are experimental data of Nagata and Tamura. 112
In the HVOS model, Eqs. (8) and (36) are used to obtain the CEOS parameters a and b. Orbey and Sandler^16 investigated the performance of the HVOS model to corre-
MIXING RULES 223
Fig. 3. H E^ Correlation of the system benzene(1)+methanol(2) at 293, 303 and 308 K, with the HVOS-NRTL models and the PRSV EOS. The dotted lines represent the results of the HVOS-NRTL2 model at T = 293 K: t 12 = (0.562918´ 104 )/ RT , t 21 = (0.100044´ 104 )/ RT ; T = 303 K: t 12 = (0.593072´ 104 )/ RT , t 21 = (0.139647´ 104 )/ RT; T = 308 K: t 12 = (0.604896´ 104 )/ RT , t 21 = (0.157280´ 104 )/ RT. The solid lines denote the results of the HVOS-NRTL4 model at: T = 293 K: t 12 = (0.526860´ 104 +0.904239 T )/ RT , t 21 = (0.109694´ 104 –5.86245 T )/ RT ; T = 303 K: t 12 = (0.657128´ 10 4 –3.69231 T )/ RT , t 21 = (0.108614´ 104 –3.66729 T )/ RT; T = 308 K: t 12 = (0.731127´ 104 –6.14529 T )/ RT , t 21 = (0.103547´ 104 –2.14626 T )/ RT ; In cases of the HVOS-NRTL and the HVOS-NRTL4 models a 12 = 0.47. The dashed lines are from the HVOS-NRTL6 model at T = 293 K: t 12 = (–1.01464´ 104 +0.181202´ 103 T )/ RT , t 21 = (0.125727´ 104 –0.190720´ 102 T )/ RT , a 12 = 0.1838758–0.422773´ 10 -4 T ; T = 303 K: t 12 = (–0.849092´ 104 +0.120972´ 103 T )/ RT , t 21 = (0.468376´ 103 –5.09688 T )/ RT , a 12 = 0.25760–0.330466´ 10 -4 T ; T = 308 K: t 12 = (0.607207´ 104 –0.401049´ 102 T )/ RT , t 21 = (0.172028´ 103 –0.918426´ 102 T )/ RT , a 12 = 0.0914807+0.759089´ 10 –4 T. The points are experimental data of Battler and Rowley^113 : n – 293 K; l – 303 K; s – 308 K.
number of components at the normal boiling point. The PSRK mixing rule can also be used to introduce other G 0 E^ into the SRK EOS. This mixing rule requires only pure component data and the parameters of the chosen G E^ model. The PSRK mixing rule is identical to the MHV1 mixing rule except for the choice of value for q 1 = 0.63.
A comparison with other group contribution EOS, such as MHV2-UNIFAC,^14 UNIWAALS,42,43^ GCEOS^97 and models mentioned above, shows some very impor- tant advantages.96,98^ Also, the PSRK model provides reliable results for VLE and gas solubility of a large number of symmetric and highly asymmetric systems over large pressure and temperature ranges.98–102^ Finally, the parameter matrix for the PSRK model is much larger than that for all other group contribution EOS using a larger range of applicability for the PSRK model compared to the other ones.
- TWU et al. MIXING RULES Twu and coworkers introduced a new class of mixing rules with van der Waals one-fluid mixing rule and second virial coefficient constraint.103–107^ These mixing rules were applied to VLE of various asymmetric, highly non-ideal complex systems.
Twu and Coon^103 treated the ideal solution reference used by Wong and Sandler^69 as only one of many choices that can be used for the reference. They chose a vdW fluid as the reference. In this way, the excess Helmholtz free energy AnR E^ repre- sents the non-random portion of the A E^ given with respect to a vdW fluid instead of an ideal solution.
For the CEOS a and b parameters of mixture, Twu and Coon (TC) developed the following mixing rules at infinite pressure
b
b a
a b C
A
RT
nR
vdW
vdW vdW 1
E 1
æ
è
ç ç
ö
ø
÷
÷
μ
a = b a b C
A
RT
* * (^) vdW nR
vdW 1
* 1 E
æ
è
ç ç
ö
ø
÷
÷
μ (39)
The C 1 constant is characteristic to the EOS used. The a vdW and b vdW parameters are determined from the vdW mixing rules, while a * and b * are defined as
a * = pa / R^2 T^2 b * = pb / RT (40) When AnR E^ is zero, the TC mixing rule reduces to the vdW one-fluid mixing rule. Twu, Coon and Bluck^104 (TCB) extended the TC mixing rule from infinite pres- sure to zero pressure in order to incorporate the UNIFAC group contribution method into the CEOS for high pressure VLE predictions.
The TCB mixing rule in terms of A 0 E^ at zero pressure is given as
b
b a
a b C
A
RT
A
RT
b
v
vdW vdW
vdW vdW 0
0
E (^) 0 vdWE 1 1 ln vdW b
æ è
ç
ö (^) ø
÷
æ
è
ç ç
ö
ø
÷
÷
é
ë
ê ê
ù
û
ú (^) ú
MIXING RULES 225
a = b
a b C
A
RT
A
RT
b b
* v
vdW vdW 0
0
E (^) 0 vdWE +^1 - - lnæ vdW è
ç^ ö ø
÷
æ
è
ç ç
ö
ø
÷
÷
é
ë
ê ê
ù
û
ú ú
where C v0 is a zero pressure function defined as
C w u
v w v u
v 0
0
0
vdW
=- ln
æ
è
ç ç
ö
ø
÷
÷
where w and u are CEOS dependent constants (Eq. (1)).
The reduced liquid volume at zero pressure v * 0 (= v 0 / b ) can be calculated for both the mixture and the pure components from the CEOS using the vdW mixing rule for its a and b parameters. Bearing in mind that A E 0 is at zero pressure, G E^ models such as the NRTL or the UNIFAC can be directly incorporated into A E 0. The same authors^105 ap- plied the TCB mixing rule to predict high pressure VLE using infinite dilution activity coefficients at low temperature. They compared the TCB mixing rule incorporating the Wilson acitvity model with other models such as MHV1 and WS. The TCB model gave
226 DJORDJEVI] et al.
Fig. 4. Simultaneous correlation of H E+ c (^) p^ Edata of the system N , N -dimethylformamide(1)+tetrahydrofuran(2) at 298 K. The lines are correlations obtained by the unique set of parameters (a) H Efrom H E+ c (^) p^ E; (b) cp^ Efrom H E+ c (^) p^ E. The solid lines represent the results calculated with the HVOS-NRTL-PRSV model parameters t 12 = (–0.685303´ 103 +3.91997 T )/ RT , t 21 = (0.214124´ 104 –1.62296 T )/ RT , a 12 = 0.281004–0.299352´ 10 -3 T. The dashed line denote the results us- ing the TC-NRTL-PRSV model parameters t 12 = (–0.188181´ 104 –0.319688 T )/ RT , t 21 = (0.774147´ 104 –0.206451´ 102 T )/ RT , a 12 = –1.55120+0.576814´ 10 -2 T. The points are experimental data of Conti et al..^114
A
RT
A
Rt
v
v
b b
E vdW
E vdW
ln 1 vdW 1
æ
è
ç ç
ö
(^) ø
÷
÷
æ è
ç
ö ø
÷
é
ë
ê ê
ù
(^) û
ú (^) ú
æ
(^) è
ç ç
ö
(^) ø
÷
÷
é
(^) ë
ê (^) ê
ln
w u *
a b
v
v u
w
a b
v w
v
vdW
vdW
vdW
vdW
ln
æ +
è
ç ç
ö
ø
÷
÷
By assuming v * to be the same as v *vdW, they obtained a simplified form where a is based on no reference pressure
a b
a b C
A
RT
A
RT
b
v b
vdW
0
E vdW
E = +^1 - - lnæ vdW è
ç^ ö ø
÷
æ
è
ç ç
ö
ø
÷
÷
é
ë
ê ê
ù
û
(^) ú ú
228 DJORDJEVI] et al.
Fig. 6. VLE Correlation and prediction of the system acetone(1)+water(2) at 523 K, with the TCB-NRTL model and the PRSV EOS. The solid line denotes the results using the TCB-NRTL model parameters: t 12 = 0.686435´10 2 / RT , t 21 = 0.546765´ 104 / RT , a 12 = 0.262017. The dashed line denotes the prediction using the parameters from 298 K given in the legend of Fig. 5. The dot- ted line represents the prediction using the Gmehling et al.^115 NRTL parameters from 298 K. The points are experimental data.^115
where C v0 is given by Eq. (43) with v 0 * = v *. Using the connection between A E^ and G E
A RT
A
RT
G
RT
G
RT
E vdW
E E vdW
E
Eq. (45) becomes
a b
a b C
G
RT
G
RT
b
v b
vdW
E vdW
E (^1) ln vdW
0
= + - - æ è
ç
ö
ø
÷
æ
è
ç ç
ö
ø
÷
÷
é
ë
ê ê
ù
û
ú ú
The b parameter was used both with and without the second virial coefficient constraint.
MIXING RULES 229
Fig. 7. H E^ Correlation of the system 2-butanone(1)+benzene(2) at 298 K, with the TC-NRTL mod- els and the PRSV EOS. The dashed line represents the results of the TC-NRTL2 model with the pa- rameters: t 12 = (–0.137026´ 104 )/ RT , t 21 = (0.225378´ 104 )/ RT , a 12 = 0.3. The solid line reflects the results of the TC-NRTL4 model with the parameters: t 12 = (0.546444´ 104 –0.328042´ 102 T )/ RT , t 21 = (–0.113204 ´ 105 +0.49719´ 102 T )/ RT , a 12 = 0.3. The points are experimental data of Brown and Smith.^116
rule is suitable and comparable with others such as WS-FH and KHFT for the VLE cor- relation of these solutions. The same authors^57 predicted Henry’s constant of liquids and gases in polymer systems using the SRK-MHV1 model coupled with a new modi- fied UNIFAC equation. For most polymers, this model gave good predictions, which were better than both the original UNIFAC and the UNIFAC-FV models.
Tochigi et al.^110 extended the applicability of the PR ASOG-FV group contribu- tion method to predicting the solvent activities in polymer solutions. The accuracy of the PR ASOG-FV model compared with the ASOG-FV and UNIFAC-FV models is very satisfactory.
Louli and Tassios^66 applied the PREOS to the modeling of VLE of poly- mer-solvent systems. Correlation of VLE data is performed by using these mixing rules including the ZM and MHV1-FH ones. Very satisfactory results are obtained with the ZM mixing rule, especially since no phase split is detected with it. Extrapolation with respect to temperature and polymer molecular weight is very good, especially when the ZM mixing rule is employed.
Finally, we would like to emphasize that only at infinite pressure is the CEOS/ G E approach algebraically rigorous and well defined at all temperatures, and that all the zero pressure mixing rules require ad hoc approximations at same conditions as indi- cated by Orbey and Sandler.^52
- SUMMARY The thermodynamic representation of highly non-ideal mixtures containing non-polar and polar compounds that are similar or very dissimilar in size and shape, over large ranges of temperature and pressure, has traditionally been a difficult task.
The CEOS/ G E^ or CEOS/ A E^ models coupled with various EOS and G E^ activity models enable reasonably good correlations and predictions of VLE for these types of systems.
Usually the parameters of these models are slightly temperature dependent, but satisfactory predictions can be obtained when they are assumed to be temperature inde- pendent. In these cases, the already published G E^ model parameters can be taken, for example, from DECHEMA Data Series. If VLE data are available for a very broad temperature interval, fitting the data at all temperatures should provide a single set of parameters for use over the entire temperature range. But, if no experimental data are available, the CEOS/ G E^ models are still capable of providing high quality predictions based on group contribution methods (for example the PSRK). In addition, it has been shown by many authors that VLE can be adequately described with a limited number of interaction parameters.
CEOS/ G E^ models with temperature independent parameters are useful for corre- lations and predictions of LLE of non-associating and self-associating mixtures. But, for cross-associating mixtures that exhibit a closed solubility loop, temperature-de- pendent parameters are needed to reproduce accurately the complex LLE behaviour of such systems.
MIXING RULES 231
Excess properties of liquid mixtures, such as excess enthalpy and excess heat ca- pacity, can be correlated very successfully using the temperature dependent CEOS/ G E models. The functional form of the temperature dependence of the parameters and a number of adjustable coefficients in the multi-parameter CEOS/ G E^ models are very important for the simultaneous fitting of two or more thermodynamic properties (VLE+ H E, VLE+ cp E, H E^ + cp E, VLE+ H E+ cp E, etc .).
The successful use of the CEOS/ G E^ models presented above for a number of complex systems highly recommends them for further development and application. Orbey and Sandler^16 suggest the following systematic investigation of the CEOS/ G E models: ( i ) thermodynamic modeling of mixture behavior at high dilution, ( ii ) simulta- neous correlation and prediction of VLE and other mixture properties such as enthalpy, entropy, heat capacity, etc , ( iii ) polymer-solvent and polymer-supercritical fluid VLE and LLE, ( iv ) simultaneous representation of chemical reaction and phase equilibrium and the evaluation of phase envelopes of reactive mixtures, ( v ) correlation of phase equilibrium for mixtures that form microstructures micellar solutions, ( vi ) LLE and VLLE for non-electrolyte mixtures.
LIST OF SYMBOLS
A E^ – molar excess Helmholtz free energy a , b – equation of state parameters B – second virial coefficient c (^) p E^ – excess heat capacity G E^ – molar excess Gibbs energy H E^ – molar excess enthalpy p – pressure R – gas constant T – absolute temperature v – molar volume V E^ – molar excess volume x – molar fraction z – compressibility factor
Greek letters
j – fugacity coefficient
Subscripts j , i , ij – components 0 – condition for the reference pressure p = 0
¥ – condition for the reference pressure p = ¥ g – activity coefficient model vdW – van der Waals fluid
232 DJORDJEVI] et al.
