A modified mixing model for vapor-liquid equilibrium calculations

ELSEVIER

~uid PhasvEquilibria | 15(1996) 95- 112

A modified mixing model for vapor-liquid equilibrium calculations Y o - L i C h o u , Y a h - P i n g Chert * Deparmwm of Chemical Engineering, National Taiwan University, Taipei. Taiwan

Received 20 September 1994, accepted 13 July 1995

Abstract Vapor-liquid equilibria (VIE) on binary and ternary mixtures are calculated using the Peng-Robi~son equation of stale (EOS). The mixture parameters of the EOS are evaluated in a similar approach to the Wong-Sandler model, Besides the Helmhollz free energy and the second virial coefficient equations in the Wong-Sandler model, the EOS mixture parameters are determined in this study by also using a coordination number model. The EOS parameters satisfy the theoretical requirements at low and high density limits. At an intermediate density condition, the coordination number expressed by the EOS is equal to that determined from a theoretically-based model. The group contribution UNIFAC activity coefficient model and a modified coordination number equation are employed in this work, Without any empirically adjusted parameter, we obtain comparably good VLE calculation results, as those from other existing EOS + G E models, from low to high pressures. With this modified mixing model, the Henry's law constants are predicted with significantly better results than those from other approaches. Keywords: Theory; Vapor-liquid equilibria; Mixture: Nonpolar, Polar, Coordination number model; Mixing roles

1. Introduction Vapor-liquid equilibrium (VLE) calculations are essential to the design of chemical processes. Simple cubic type equations of state (EOS) are usually used in VLE computations, and the key point of the EOS method on V L E calculations is the choice of appropriate mixing rules. The conventional van der Waals one-fluid mixing model is widely used owing to its simplicity. Single or nmiliple unlike pair interaction parameters are usually needed to yield accurate phase equilibrimn calculation results. It is difficult to correlate these binary parameters by a generalized equation. An alternative mixing model by equating the excess Gibbs free energy calculated from an EOS at an infinite pressure that o f an activity coefficient model was presented by Huron and Vidal (1979). Various * Corresponding author. 037g-3g12/gfi/$[5.0~ ~ 1996ElsevierScienceB.V. All rights reserved $5DI 0378-3812(95)02825-0

Y.-L. Chou, Y.-P. (:hen / Fluid Phase Equilibria 115 (1996) 95-112

96

modifications of the Huron-Vidal algorithm have been suggested in literature by changing the infinite pressure reference state (e.g. Michelsen, 1990a, Michelsen, 1990b; Boukouvalas et al., 1994), or by allowing the existence of an excess volume (Sheng et al., 1992). Wong and Sandler (1992), Wong et al. (1992), and Orbey et al. (1993) further proposed a new mixing model which gives consistent agreement to statistical mechanics where the second virial coefficient should follow the quadratic mixing rule. A binary parameter was introduced in their model. They determined the binary parameter by equating the excess Helmholtz free energy calculated from an EOS to that from an activity coefficient model at a fixed composition for each isotherm. Satisfactory VLE calculation results were obtained by using the Wong-Sandter mixing model where the binary parameters were taken as temperature-independent. A modified mixing model similar to that of Wong and Sandier is developed in this study, We retain the Helmholtz free energy and the second virial coefficient equations of the Wong-Sandler model. The EOS parameter is determined in an alternative way by additionally applying the coordination number model. The EOS parameters are evaluated in this work by solving the simultaneous equations where no empirically adjusted factor is needed. The modified mixing model presented in this work gives comparably good VLE calculation results as those from other existing methods from low to high pressures. It also yields better results at extreme conditions such as the infinite dilution states.

2. Theory The Peng-Robinson EOS (Peng and Robinson, 1976) is used in this work for VLE calculations: e =

-

RT ,~-b -

-

a(T) ,,(,,+ b) + b(,,- b)

(1)

where the pure component parameters are calculated by: a t = 0.45724

R2r~ Pci

~(T) = [1 + K ( l -

o~(T)

(T/Tc)'/~)] 2

K = 0.37464 + 1.54226¢o - 0.26992 w:

(2) (3) (4)

Rrci b~ = 0.07780

(5)

For mixture calculations, mixing rules are used to evaluate the mixture parameters. As mentioned earlier, a mixing model suggested by Huron and VidaI (1979) can be used in phase equilibrium calculations as a predictive method without adjustable parameters. The Huron-Vidal model gives good VLE calculation results, however, the theoretical consistency of mixture parameters at low densities is not satisfied. Wong and Sandier (1992) proposed an improved mixing model where the second virial coefficient calculated from an EOS follows the quadratic mixing rule from statistical mechanics. Wong and Sandier and their co-workers showed satisfactory VLE calculation results but a binary parameter is still needed in their model. According to Sandier and co-workers, the binary

E-L. Chou, I'.-P. Chen / ~'luid Phase Equilibria !15 11996~95-112

97

parameter is determined using an excess Gibbs free energy model at a fixed composition, say x 1 equals to 0.5 (Orbey et al., 1993). For a fluid mixture, the EOS parameters determined by the Huron-Vidal or the Wong-Sandler mixing rules are temperature-dependent but are independent of density of the mixture. In this study, we propose an alternative method, also without any empirically fitted binary factor, to calculate the EOS mixture parameters. In our method, we retain the advantage of the Wong-Sat, dler model, but additionally allow the EOS parameters to be both temperature and density dependent. At a given temperature, we claim that the EOS parameters a m and bm should satisfy the following requirements at the limiting pressure conditions: (1) at an infinitely high pressure, the excess Helmholtz free energy calculated from the EOS is equal to that from a group contribution activity coefficient model, like UN1FAC, and (2) for the second virial coefficient, B, of a mixture, which is a thermodynamic property determined from the PVT data at a low density limit, the statistical mechanics requirement on the quadratic mixing rule must be obeyed, as described by Wong and Sandier (1992). At these two limiting conditions, the following equations can be written for the Peng-Robinson EOS: ( am

ai)

- -~m--~"xl-~ii

l

'2+vr~'

"~ln[2-----~J

=RTExilnT~a'/~¢i

(6)

and

Bm= ~ExtxjB~

(7)

ij

where a

B ffi b -

--

(s)

RT

and the cross term of the second viriat coefficient is expressed as:

1

Bij- ~[(b i -

ai/RT ) +

(bj-

aj/RT)]

(9)

F.xlS. (6) and (7) describe the two requirements that the appropriate EOS parameters must satisfy at the two limiting conditions at any isotherm+ The EOS parameters solving from only these two equations are not dependent on density. Theoretically, the EOS parameters determined in this manner at a fixed temperature are correct at limiting pressure conditions, and they can be used on mixture calculations at the system pressure which is in between the two limiting states. A binary parameter may be included in the expression of the cross term of the second virial coefficient (Wong and Sandier, 1992):

! B|j == "~ [(b i -

ai//RT ) ..t-(bj - aj/RT)](I -

kgi)

(10)

The binary parameter is temperature-independent and should be determined not in an empirical way, in order to keep the mixing model predictive in phase equilibrium calculations. According to Orbey et

98

Y.-L. Chou, Y.-P. Chen / Fluid Phase Equilibriu 115 (1996) 95-112

al. (1993), the binary parameter can be determined at each temperature with a fixed composition of x t =- 0.5 for VLE calculations of a binary mixture. This method of evaluating the binary parameter works well for some systems but may become complex, for example, on the computations of multicomponent mixtures, where several binary interaction factors have to be determined before calculating the EOS mixture parameters. It will not be suitable to evaluate the binary parameter from a composition at x~ = 0.5 on the calculation of the Henry's law constant where the infinite dilution state is involved. In this work, we propose an alternative method to determine the EOS mixture parameters. We add one more constraint to the Wong-Sandter modal that the EOS parameters, not only satisfying the low and high limiting pressure requirements at each isotherm, should also meet a theoretical equation at the pressure of the system. With this additional equation, we can solve for the EOS mixture parameters at any temperature and pressure. The EOS parameters evaluated in this way not only satisfy the theoretical equations at both limiting pressures, but also would be correct at a density that is actually employed in our phase equilibrium calculations. This extra requirement that we apply in this study is the expression of the coordination number, N¢. The coordination number is the number of molecules interacting with a central molecule. From the discussion of the generalized van der Waals partition function (Sandier et al., 1986), it is illustrated that the coordination number plays an important role in the expressions of an EOS, a local composition and an activity coefficient model. The coordination number is dependent on density, temperature and the interaction forces between molecules. Molecular simulation results on the coordination number have been presented for square-well fluids from low to high densities (Lee and Chao, 1987; Lee et al., 1985; Guo et al., 1990) and those models can be used in our present evaluation for the EOS parameters. In this work, taking the mixture as a pseudo-component with proper EOS parameters, we set the coordination number determined from a theoretically-based model equal to that from an EOS: 4~

3

e

ara(l+K/

1 I

[Vm + (! + ~ - ) b .

The right hand side of Eq. (1 !) is obtained according to the generalized van der Waals theory applying to a square-well fluid (Sandier et al., 1986) where the coordination number expression built into the Peng-Robinson t~.OS is evaluated. It includes the EOS mixture parameters at a given temperature and a molar volume. The left hand side of Eq. (11) is a modified expression of the coordination number from its value at a low density limit. Usually, the coordination number at any given density is written as a product of a low density limit term and a correction term which is function of reduced density (e.g. Lee and Chao, 1988; Guo et al., 1990). We employ a parameter c in the coordination number expression. This correction parameter is allowed to be temperature and density dependent and is determined in a predictive way in this study. The EOS mixture parameters a m and bin, and the correction parameter c are solved from the simultaneous Eqs. (6), (7) and (11). No empirically adjusted factor is needed and these parameters are calculated in a totally predictive manner. In our solutions, the molar volume (v m in Eq. (i 1)) of a fluid at any given state is solved from the Peng-Robinson EOS itself. In this work, we determine the c parameter in Eq. (11) as a temperature and pressure dependent factor for a specific fluid mixture. We have examined the numerical values of the c parameters and hence our calculated coordination numbers in the V I E calculations, and this ~. ;il be discussed in the next section of this work. The potential parameters used

Y..L. Chou, Y..P. Chen / Fluid Pha,~e Equilibria I I~ (1996) 95 - 112

99

in Eq. (11) are taken as functions of real properties of pure fluids through the simple mixing rules, as were usually employed in previous researches: 0-3 = ~xi0-i 3

(12)

3 ,~, = ~ b ,

(13)

,19= ~-'~ Xi X j( ~'i &'j) 0'5

(14")

~'ffi 1.3

(15)

and

We apply numerical method to solve simultaneously the four non-linear equations. Eqs. (1), (6), (7) and (11), and obtain the EOS parameters at any given temperature and pressure. The EOS parameters determined from this method are thus dependent on temperature and pressure. These EOS parameters are then used in VLB calculations where the fugacities of component i in the equilibrium phases are equal:

~"=f?

(16)

The fugacity of component i in a mixture is calculated using the PR EOS:

In

Z

i la bo

ld _e o i (z-')- t ' [ "m ;[I (l 1 - , ItO.b. l ---I/in!

--

-

-.-.-~

-

(17)

The EOS parameters of a mixture are determined from our new mixing model, and numerical method is used to evaluate the derivatives of these parameters in calculating the fugaeity coefficient. We calculate the bubble point pressure of binary systems from low to high pressure conditions and compare our calculated results with literature data, We also compare our results with those from two other mixing models of Huron-Vidal and Wong-Sandler. Besides the VLE calculation results, we calculate the Henry's law constants for solutes (component 2) in heavy solvents (component 1) as well: U2. , -- e ~ " =

(l 8)

In E.q. (18), the fugacity coefficient of the solute at infinite dilution is calculated using the Peng-Robinson EOS with our proposed mixing model. The limiting value of the fugacity coefficient is calculated by numerical method. The comparison between the experimental data and the calculated Henry's law constant would indicate the applicability of the mixing model a~ the infinite dilution limiting conditions. We apply this as a severe test of our proposed model and also compare our calculated results with those computed from other existing models.

IOO

Y.-L. ehou, Y.-P. Chert~Fluid Phase Equilibria 115 fl996) 95-112

3. Results and discussion VLE calculations are made in this work using our proposed mixing model. An example of the calculated Pxy curves is shown in Fig. 1 for the binary mixture of methanol and benzene at various temperatures. These results show that our mixing model are satisfactory over a wide range of temperature and pressure. Bubble point pressure calculations on binary and ternary systems are carried out in this study and the results from our mixing model and those from the Huron-Vidal and Wong-Sandler models are compared in this work. The EOS mixture parameters are evaluated by using the UNIFAC group contribution activity coefficient model (Larsen et al., 1987) in the thrce mixing models. In the Wens-Sandier mixing model, the binary interaction parameter can be determined following the method suggested by Orbey et al. (1993) where the known group interaction parameters of the UNIFAC model are employed. We have also tried to neglect the binary parameter in the Wens-Sandier model and carried out the similar VLE calculations in this study. All the calculation results of bubble point pressures and compositions on binary mixtures from low to high pressures are shown in Table 1. It is observed that three mixing models give comparable calculation results, however, our proposed model has the minimum absolute average errors in both bubble point pressures and compositions. Our mixing model al,~,a yields comparable results to those from other mixing models on the asymmetric systems such as ~ater + propanol, water + 2-pentanol and benzene + l-butanol. It is noticed that even without the binary parameter in the Wong-Sandler method, the calculation results are still satisfactory. This suggests an alternative way to employ the Wong-Sandler mixing rules in multicomponent calculations and to avoid the complexity of using the binary parameters. Table 2 gives the values of the c parameters and Nc determined in our VLE calculations using our proposed mixing model. It is shown that these c values arc in reasonable range and are close to unity for both the vapor and liquid phase computations. The calculated Nc values from our 10' [[" " ' ' ' o P-X expl. data (Butcher and Nedanl. 1968) I- . s s . ~ p - y expl, data this work

k

i0 m

0,00

0.20

0.40 0.60 Xl o r Yl

0.80

1.00

Fig. I. P r y diagram o f the binary mixture of methanol(1) and benzene(2) at various temperatures calculated from the mixing

model proposedin this work.

F.-L. Chou, Y.-P. Chert/Fluid Phose Eqnilihriu I IS (1996} 9.,<-112

I01

proposed mixing model are also reasonable because a mean value of 10 is usually assumed for the coordination number in activity coefficient models. The VLE calculation results are also demonstrated ~aphicatly in this work. In the graphical comparisons, the binary parameters in the Wong-Sandler model are determined by the method presented by Orboy et al. (1993). Fig. 2 shows the calculated Pxy curves of the binary system of the polar components of dimethyl ether and methanol at 373.15 K from the three mixing models where our calculation results show good agreement with experimental data. Fig. 3 shows the calculation results on ethanol and nitromethane binary mixtures at 348.15 K. This mixture has a highly nonideal phase behavior where an azeotrope exists. Our mixing model again yields good agreement with the experimental data. We extend our model to ternary system calculations and the results are presented in Table 3. In the ternary calculations, we neglect the binary parameters in the Wong-Sandler model. It is indicated that all models, including the Wong-Sandler model without binary parameters, can calculate the bubble point pressure well. It is again demonstrated that our model gives smaller absolute average error on ternary calculations. We have computed the saturated specific volumes of the equilibrium phases and compared our calculated results with the experimental data. The results are shown in Table 4. On calculating the satur:~ted specific volumes with the Wong-Sandler mixing model, we observe that the use of the binary parameters according to the method of Orbey et al. (1993) yields less satisfactory results on the calculated volumetric properties. It is found preferable to neglect the binary parameters in these calculations on applying the Wong-Sandler mixing model and reasonable agreement with the experimental data is obtained, as shown in Table 4. It is demonstrated in Table 4 that all models give satisfactory calculated saturated volumes where our proposed mixing model yields slightly better r~sults. The accuracy in predicting the volumetric properties is useful to the improvement on phase equilibrium calculations. Graphical comparisons of the specific volume calculations are shown in Fig. 3,5 "''°"

P - x expl. data (Chand et at., 1982)

• ' ' Q " P~y eXpl. data _ - - Ht2ron-~idal model .,~,~ . . . . . . Wo~-Sand|e.r mode.] {kij=0.160) . ~ .~/ thig work . . .-" J yr /

3.0

~

z /-

,* 1.5

,/

.'"

/

/ ' /

/"

/"

,7

0.5 r 0.0

II ........

0.00

I,,,1

0.20

.....

Iz,,,i

....

h .....

0.40 0,60 Xt o r Yl

h,,l

LILt

0.80

......

1,00

Fig. 2. pxy diagram of the binary mixture of dimethyl e~e~ i) and methanol(2) at 373.15 K calcu]a~dfrom various mixing models.

Y.-L. Chou, Y.-P. Chen / Pluid Phase Equilibria 115 (1996)95-112

102

~

~

~

~

~ ~

I

b

N

~

I

-

-

~

I

~

~

o

I

-

~

I

-

~

~

~

~

o

I

I

-

~

i

I

I

i

I

~

o

~ ~

I

I

I

~

~

~

I

~

I

r

"

I

I

I

~°~ ~

I

~

o

i

I

~

I

_

~

I

I

I

~

r~

I

~

i

G.

Y..L Chou, Y.-P. Chert / Fluid Phase Equilibria 115 f 1996) 95- 112

t~

.--,

0

•*

-

~

v

~

v

I

I

I

1

|

~

~

;~., -..,.,

I [~-

_'.2,,',.2.'

-

-

~ i

~x

~

. ".---"

.v

~

'~

~,,~

103

104

Y..L. Choa, Y.-P. Own ~Fluid Phase Equilibria t 15 (1996195-112

Table 2 The calculated values of c and N¢ from our mixing model of selected binary systems Component I Component2 Temp. Pressure No. of c c Nc range range (kPa) da~.a (liquid phase) (vaporphase) (liquid phase) (K) points acetone acetone cyclobexane dichloromethane dichloromethane n-pentane benzene ethanol ethyl acetate

chloroform 298-328 cyclohexane 298-325 ten-butyl alcohol 328-343 acetone 298-398 dichloromethane 298-398 ehlorobenzene 348-398 acetonitrile 343-393 aeetonitrile 313-393

benzene

I-ehlorobutane

toluene

t-chlorobutane

benzene

n-pentane

1-chlorobutaue

298-348

348-398 348-398 348-397

24-39 15-107 35-95 15-295 31-999 63-1187 23-315 80+449 24-360 39-334 87- 353 127-929

36 60 31 38 57 69 I~ 18 27 18 18 18

0.886~0,971 0.867-0.972 0,919-1,067 0,861-0.979 0.903-1.085 0.903-1.118 0.891 - 1.002 0,978-1.138 0,925- 1,072 0.924-1.022 0.935 - 1.023 0.962- I.i 12

0.886-0.974 0.879-0,972 0,936-1,059 0.869-0,980 0.903-1.084 0.905- I. 116 0.907- 1.004 0.987-1.136 0.926-1.072 0.929-1.024 0.935- 1.023 0.982-1. 117

12.35-13.93 I 1.94-13.8 t 11.58-13.46 I 1,02-13.99 9.28-13.43 8.24- t2.72 10.68-13.72 10.61-13.10 10,36- I3.70 10.23-12.96 I0.13 - 12.13 8.42- t 159

4. Only limited experimental data of the saturated specific volumes are available in literature. It is found that the satisfactory results from each mixing model shown in Table 4 do not imply that they can prediet the excess volume of fluid mixtures with good accuracy. Table 5 presents the calculated results of the Henry's law constants of various solute molecules in

heavy solvents. In this calculation, we need to evaluate the fugacity coefficient of the solute at an infinite dilution condition. We examine the validity of the composition dependence of the three mixing models by these calculations. The absolute average deviations from the experimental data show quite different results from the three models. The Huron-Vidal and Wong-Sandler model give larger deviations, and our results are obviously superior to those from the other models. Our model

yield the minimum absolute average error, and most of our calculated results are within the range of experimental accuracy, In the Henry's law constant calculations, the binary parameter of the Wong-Sandler model is either evaluated by the method presented by Orbey et al. (1993) or is assigned to be zero, depending on which way gives the smaller error. Graphical comparisons of the

calculated Henry's law constants are presented in Figs. 5 and 6, respectively. The good agreement of the results from our model with the experimental data again testifies the applicability of our mixing model.

Notes to Table 3: I00 ' AAD

I pc,,i _ p~,p I N - No. o data poi° , I

in the order of volume/part/page.

I

series, numberscorrespond to the pages

¥..L. ClJoa. F.-P. Chen / Fluid Pha.~eEquilibria 115 f1996195-112

1o5

Table 3 VLE calculation results for ternary systems using the Peng-Robinson EO$ and different mixing models Ternary system

methanol + tetrachloromethane + benzene methanol+ benzene+ eyclohexane tetrnchlorornethane + ethanol + benzene cyclohexane + benzene+ aniline n-hexane + methylcyelopentamc+ benzene l-heptene + n-heptane + n-octane methylcyclohexane+ toluene+ aniline benzene+ eyclohexaae + ethylenediamine cyclohexane + n-heptane + toluene I-hexene + n-hexane + n-octane ethanol+water+ 2-propanol ethyl acetate+ ethanol + water

benzene+ 2-propanol + water

Grand average

Temp.

Pressure

No. of

AAD P(%) '~

range (K)

range (kPa)

data points

Huron-Vidal

This work

Data soal'ee$ b

308 328

38.8--4 I. 1 88.6--95.6

6 8

3.55 3.30

4.87 6.63

4.4 I 4,37

1/2a/607 I/2a/608

312 328

44.5--52.5 85.0--100.4

10 9

6.12 5.99

7.40 8.52

5.19 3.87

I/2a/639 I/2a/640

323

47.3--56.8

30

2.48

3.88

3. I 0

i/2a/6~

343

35.3--70.8

10

I 1.40

14.39

8.20

I/6a/639

333

60.0--71.8

37

3.51

3.54

3.13

I/6a/654

328

16.0--25.5

12

6.38

6.70

6.09

1/6b/463

353 363 373 313 333

32.8--50.1 52.7--68,6 60.0--92.1 27.5--28.4 57.6--57.7

7 9 9 9 5

4.61 7.52 7.59 15.25 17.78

5.31 8.38 8.41 6.44 7.69

2.85 5.13 5.26 6.66 7.44

1/6b/464 /6b/465 1/6b1466 I/6c/587 I/6c/588

298

5.6-- 11.7

16

1.22

1.74

1.36

1/ 6c / 591

328

20.7--66.9

12

0.56

2.07

1.34

/6cl597

403

399.2--578.3

27

8.77

7.18

3.27

I/1/609

313 328 343 343 353 303 318 333 303 318 333

19.0--29.3 35.7~-55.5 71.8--99.0 52.6--96.8 g0.1-- 137.4 19.1--22.2 40.8--43.8 74.6--80.6 9,4--20.1 23, f--39.8 45.3--71.0

8 10 9 27 27 14 II 19 21 23 22 407

13.88 10.70 t 1.61 6.49 6.74 18.83 13.72 22.31 10.10 6.46 5.43 7.90

13.53 10.08 t 1.53 5.58 5.58 10.36 8.88 I i .58 4.91 2.20 2.87 6.22

9.20 9.06 8.33 5.26 4.77 7.89 9.56 8.28 3.70 3.16 2.7 I 4.77

t/1/621 1/ i / 6 2 2 i/1t623 111a/561 t/!a/562 I/la/6~9 l/la/621 1/ l a i 6 2 3 1/ 1a/620 1/ i a / 6 2 2 I / 1a/624

Wong-Sandler (ki] ~ O)

t06

Y.-L. Cttou, Y--P. C h e n . / F l u i d Phtt.ve Equilibri¢~ 115 4t 9 9 6 ) 9 5 - 1 1 2 -.-"

I~

~d °

~

~

l

i

l

l

l

l

t

l T..

• ~

. =

.

~

l

- l

l

l

l

l

l

E t-,-

~

~

'

~

I

I

I

I

F

I

I

I

-,.

0

~= "2 "-"

_

~-~

e--: --: ~-.-i,.~--~,-~

~-~

"

"

",-,4 ~i

o.~

..

N

~'~

~

~ E

~a

--

Z

n

~

,~N

Y.-L. Chou. Y.-P, Chert/ Fluid Phase Equilibria I/5 (1996) 95-112

107

tO0.O

ao.o

P.,

40.0~

20.0

..... P - X expl. d a ~ ¿Khurma e t al., 1 9 8 ~ o • . . . . P - y expl. data _ _ Kur~n-Vtda}. model ..... W0nl[-Sandler mode| (kij=O,1lO) - - _ t h i s work ........ ' ..... ~'~'" ........ ' ......... ' .........

0.00

0.2.0

0.40 Xl

0.60

or

O.BO

I.O0

Yi

Fig. 3. Pxy diagram of the binary mixture of ethanol(I) and nitromethane(2) at 348.15 K calcula|ed from various mixing

models.

We have examined the VLE calculations using the three mixing models and with different EOS, like Soave-RK EOS, or other activity coefficient model like NRTL. It is indicated that our proposed mixing model yields smaller average deviations on the VLE calculations in those cases, No significant

B,O *'---expl. d a t a (SaJze a n d Lncey, 1 9 4 0 ) _ _ Hurc,n-Vida| model . . . . . . W o a g - S s n d l e r m o d e ] {ki~=O.0) t h i s work .."

0

8e.o .*°/ ,// .'f ,/J

/// .-f

o

~2.0

0.00

/

.

iz

0.20

0.40

0.60

O.00

Yl

Fig. 4. Specific volumes of the saturated vapor phase of the binary mixture n-pentane(i) and propane(2) at 344.25 K calculated from various mixing models.

108

¥.-L. Cbou. Y.-P. Chen / Fluid Phase Equilibria l l S ~1996) 9 5 - 1 1 2

Table 5 Results of the Henry's law constant calculations using different mixing models Solute

Solvent

T (K)

AAD (%) ~ Huron-Vidal

Wang-Sandier

This work

propane

n-hexadecane

propane

n-octadecane

propane

n-eicosane

propane

diphenylmethane

propylene

n-octadeeane

propylene

n-eicosane

n-butane

n-hexadecane

n-butane

n-eicosane

isobutane

n-eicosane

n-pentane

n-hexadecane

n-pentane

n-eicosane

300 325 350 303 323 343 308 323 343 325 350 375 323 343 373 300 325 350 308 323 343 323 343 373 300 325 350 375 303 323 343 325 350 375 325 350 375 303 323 343 310 315 320 313 318 323

55.6 51.6 48.2 52.8 50.3 49.6 53,6 52.7 52.3 62.2 59.4 58.5 56.0 55.4 54,8 47.5 41.6 36.7 56.0 55.8 55.6 59.4 59.2 58.6 53.2 43.0 41.5 39.1 44.2 4 I. I 41.7 52.4 48,7 48. I 54.7 51,9 49.4 35. I 34,3 30,7 33.7 34.5 35.O 40.2 40.9 40.9

62.7 50.8 35.6 59.5 50.1 40.7 75.1 74,9 61,6 80.3 77.7 66.4 77.0 79,8 63.4 68.6 65. I 62.3 76.7 74,8 62.2 79.0 79.9 66.8 64.2 46.9 33.0 14.2 56.3 46.O 36.8 73.4 71.3 61.8 75,1 73.5 63,0 53.3 44.1 29.7 49.5 47.9 46.3 64-.~ 64.8 64.8

2.7 13.0 29.1 4.3 15.6 23.7 10.7 17.4 24.4 2.3 [6.9 26.5 18.4 26.2 37.3 2.6 16.7 33.9 8.3 13,0 19.1 12.6 19.0 29.5 23.0 0.9 11.3 23.7 7.3 3.8 9.0 3.6 11.7 20.7 0.2 13.3 27.0 12.5 5.3 6.5 8.4 7.9 6.6 6.2 5.1 3.6

Data sources

109

F..L. Chou. Y..P. Cht.n/ Fluid Phu:arEquilibria tl5 (1996)95-112

Table 5 (continued) Solute

Solvent

n-hexane

n-hexadecane

n-hexane

n-eieosane

n-heptane

n-hexadecane

n-heptane

n-eicosane

n-octane

n-hexadeeane

n-octane

n-eicosane

Grand average (67 data points)

T (K) 310 315 320 313 318 323 328 310 315 320 313 3!8 323 328 310 315 320 313 318 323 328 41,3

Data

AAD (%) ~ Huron-Vidal

Wong-Sandler

This work

22,8 25,2 25,2 37.2 34.9 36.2 35.9 19.6 18.3 19.3 33.5 30.3 30.4 32. I 11.2 I4.8 15.2 26.0 27.0 27.2 26,8 56.2

45,7 44.9 42.2 60.3 58.8 59.6 59.4 40.2 39.2 42.0 55.4 53,3 53.3 54.4 29.9 32.8 33.0 47.6 48.3 48.5 48. I 14.4

9. I 10.3 8.3 16.7 t 1.6 I 1,9 9.1 16.5 13.5 12.5 22.8 17.1 15.8 15.7 16.4 18.1 16.6 22,8 22,0 20,7 18.0

SOUl-tee S

4

4

4

4

4

4

° A A I ~ % ) = - N - ~ ' | ' ~ - H"~'-ff~ I; H - Henry's law constant; N - No. of dola points. Data sources: (I) Chappelow am~ ! Prausnitz (1974); (2) Xuliani et al. (1993); (3) Ng et al. (1969); (4) Donohue et al. (1985)

difference in computer time is observed by using different mixing models, EOS or activity coefficient models.

4. Conclusion A new mixing model is suggested in this work that calculates the vapor-liquid equilibrium results satisfactorily on binary and ternary mixtures. Our proposed mixing model, like other predictive methods, evaluates the EOS parameters without any empirically adjusted constant, and yields comparably good VLE and volumetric results to the Huron-Vidal and Wong-Sandler mixing models. Our model shows much better results on the calculation of the Henry's law constants than those from the other two mixing models. It is indicated that our work yields reasonably good EOS mixture parameters in a predictive manner from low to high densities and from infinite dilution to concentrated compositions.

II0

Y.-L. Chou, Y.-P. Chen / Fluid Phase Equilibria 115 (1996) 95-112

3.00 • - . . - ExpL data Huron-Vidal model ...... Wong-San~er model

this work

-~-3,60 .,¢1

"~ 3,00 o

1.50

~L~I.O0

0,50

0.00 300.0

PP'lelPlllll'll~4(lllhlhnm,I

310.0

.....

I I ~ l l l l ,

320.0 330.0 340.0 T e m p e r a t u r e (K)

350.0

Fig. 5. Henry's law constants of the binary mixture of n-pcntane and n-hexadecane at six temperatures calculated using various mixing models (expl. data: Donohue et al., 1985 and Zuliani et al., 1993).

1,60 • , * - * Expl. data H~ron-V~da[ model ...... Wong-Sandler mQdel this work ~.~1.30

o

"_/-1

/

I

J

O.gO

~ 0.40

0.00

31B+0

n l i i l l i l l i i t n e t i l l ~ ! L p [ l l L l l l i n n n n

315,0

318.0 Temperature

321.0 (K)

....

3~4,0

Fig. 6. Henry's law constants of the binary mixture of n-pcntane and n-eicosane at thrce temperatures calculated using various mixing models (expl. data: Donohue et al., 1985).

F.-L. Chou. Y.-P. Chen / Fluid Phase Equilibria 115 ql996~ 95-112

Ill

5. List of symbols a,b c f G H k k~j n Nc P R T v x,y Z

EOS parameter correction factor fugacity Gibbs free energy Henry's law constant Boltzmann constant binary interaction parameter number of moles coordination number pressure gas constant temperature molar volume mole fraction compressibility factor

5.1.1. Greek letters a 3' K A p o~b ra

parameter defined in Eq. (3) activity coefficient square-well depth parameter defined in Eq. (4) a constant of 1.5 used in Eq. ( I 1) density molecular diameter fugacity coefficient acentric factor

5.1.2. Superscripts I v ao ^

liquid state vapor state infinite dilution property of a component in a mixture

5.1.3. Subscripts 1,2 c i,j m

component l(so|ven0 or 2(solute) in a mixture critical values component i or j in a mixture mixture

Acknowledgements The authors are grateful to the National Science Council, Taiwan, Republic of China for supporting this research.

112

Y.-L. Clrou. Y.-P. Chen / Irhad Phase Equilibria 115 ~1996195-112

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