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FIuid Phase Equilibria, 65 (1991) 111-133 Elsevier Science Publishers B.V., Amsterdam
111
Simultaneous representation of excess enthalpy and vapor-liquid equilibrium data by the NRTL and UNIQUAC models
Ya$ar Demirel and Hatice Gecegiirmez
Faculty of Art and Sciences, University of Cukurova 01330 Adana (Turkey)
(Received May 29, 1990; accepted in final form February 6, 1991)
ABSTRACT
Demirel, Y. and Gecegijrmez, H., 1991. Simultaneous representation of excess enthalpy and vapor-liquid equilibrium data by the NRTL and UNIQUAC models. Fluid Phase Equilibria, 65: 111-133.
Using data for excess Gibbs energy, g, and enthalpy of mixing, hE, temperature-depen- dent parameters of the NRTL and UNIQUAC models have been estimated for 44 systems of binary mixtures. Thirty-three of them include data for gE and hE at more than one different isotherm. The estimated parameters were later tested by predicting the total pressure, vapor-phase compositions and gE and hE data simultaneously and representing the effect of temperature on such data. System-dependency of the models, non-uniqueness in the parame- ters and the possibility of predicting partial miscibility for completely miscible non-ideal mixtures have been investigated.
INTRODUCTION
The NRTL (Renon and Prausnitz, 1968) and UNIQUAC (Abrams and Prausnitz, 1975) models have found extensive use in the correlation and prediction of fluid phase equilibria and chemical process calculations, such as distillation. They can predict the partial miscibility and are easily ex- tendable to multicomponent mixtures using the parameters obtained from binary data alone. In a recent study, Cairns and Furzer (1988) pointed out that the limitations of these models must be considered in the design and retrofit of distillation columns, although there would appear to be a major difficulty in selecting the best model, since the choice may be system-depen- dent. This aspect needs to be investigated further in order to gain a critical and lasting evaluation on the behavior of the models.
037%3812/91/$03.50 Q 1991 Elsevier Science Publishers B.V.
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112
Renon and Prausnitz (1969) provided charts for estimating the binary adjustable energy parameters of the NRTL model from limiting activity coefficient data and from mutual solubilities. They suggested that, depend- ing on the chemical nature of a mixture, a value of aij must be chosen. But as Tassios (1976) points out, the selection of proper values of clljj is ambiguous and difficult to apply, and the NRTL model performs best when the value of Y;~ is obtained by regression of the available experimental data. Flemr (1976) and McDermott and Ashton (1977) pointed out the theoretical inconsistency of the NRTL and UNIQUAC models and concluded that they should be treated only as empirical. Later Panayiotou and Vera (1981) and Iwai and Arai (1982) have discussed the statistical thermodynamic deriva- tion of the UNIQUAC model.
Tassios (1979) showed that multiple roots are possible for positive devia- tions from Raoults law for the NRTL model. However, he concluded that when (Y~, is allowed to vary, the multiplicity does not seem to represent a problem.
Hanks et al. (1978) have performed a parametric analysis of the NRTL model to determine appropriate limits to its ability to represent gE and hE data simultaneously. This analysis showed that the NRTL model is not capable of correlating both gE and hE data for any system in which the value of hE exceeds a certain limit, A similar study performed by Demirel and Gecegiirmez (1989) indicates that the UNIQUAC model has a similar property. This limitation can be overcome by treating the parameters of the models as functions of temperature and using experimental heats of mixing along with vapor-liquid equilibrium (VLE) data (Murthy and Zudkevitch, 1979; Demirel and Gecegbrmez, 1989).
The NRTL and UNIQUAC models have a moderate built-in temperature dependence. However, Murthy and Zudkevitch (1979) suggest that relying on the NRTL model for representing the effect of temperature on the activity coefficients can be unreliable. Both of the models perform better when the parameters vary with temperature as suggested in the literature (Anderson and Prausnitz, 1978; Demirel and McDermott, 1984). Pablo and Prausnitz (1988) suggest that for the NRTL model a strong temperature dependence is required to describe the coexistence curve in the critical region.
In this study, temperature-dependent parameters for the NRTL and UNIQUAC models were estimated using gE and hE data simultaneously, and the accuracy in predicting gE, VLE and hE data by the equations has been investigated for various types of mixture including alcohols, carboxylic acids, water and esters.
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113
HEAT OF MIXING
The rate of change of excess Gibbs energy, and hence the activity coefficients, y,, with respect to temperature is proportional to the excess enthalpy and is given by the Gibbs-Helmholtz equation
hE ahEm -=- T2 [ 1 aT P,x 0)
Nagata and Yamada (1973) and Nagata et al. (1973) have shown that the NRTL model whose parameters are assumed to be expressed by a linear function of temperature is capable of representing both VLE and hE data with a single set of parameters. Renon and Prausnitz (1968) also assume that the NRTL parameters change with temperature in a linear way.
g,, - g,, = ci + c2( T - 273.15) (2)
g,, - g,, = c3 + c,( T - 273.15) (3) ai2 = c5 + cg( T - 273.15) (4) With the NRTL parameters given in eqns. (2)-(4) the enthalpy of mixing becomes
hE = (x,x2Gd (Xl + X2G2, >
1 _ (hxl) (XI + x2G21)
(c1 _ 273 15~ ) + r,:X,C,RT2 . 2
Xl + XzG21 1 +(
(XlX2GlZ I
x2 + w,2)
where
G2, = exp( - a12721)
Gl2 = exP( - y12712)
721 = (g21 - &J/RT
l- (12712X2 >
(x2 + +I,) ( cg - 273.15~~) + $2y;,,2
1 12 1 (5)
712 = (g,, - g22VRT
Abrams and Prausnitz (1975) state that when both VLE and liquid-liquid equilibrium (LLE) data are used to obtain the UNIQUAC parameters, they appear to be a smooth function of temperature. However, this is not the case for mixtures containing hydrogen bonding, such as water and alcohols (Anderson and Prausnitz, 1978; Murthy and Zudkevitch, 1979). In the present study, the effect of temperature on the characteristic energies is expressed as
azl = d, + d/T (6) al2 = d, + d,/T (7)
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114
These are the expressions also used by Anderson and the UNIQUAC parameters given in eqns. (6) and (7), becomes
Prausnitz (1978). With the enthalpy of mixing
where
72; = exp( -%1/T)
G = exp( - +/T)
8j = Xjqi/Cxiqi
The UNIQUAC model contains pure-component structural parameters r and q. Anderson and Prausnitz (1978) modified the UNIQUAC equation slightly and introduced new values of surface parameters, q, for alcohols and water to be used in the residual part of the equation.
ESTIMATION OF PARAMETERS
In estimating the temperature-dependent parameters, data for gE and hE were used simultaneously. The following objective function, which was also used by Nagata and Yamada (1973), was minimized:
where n and m are, respectively, the number of experimental gE and hE data points at a specified isothermal temperature. N is the number of isothermal system temperatures for the gE data and M is that for the hE data. For minimizing the function F, a package program called MINUIT (James, 1978) was used. The MINUIT program performs minimization and analysis of the shape of a multiparameter function, and incorporates the Fletcher and Simplex techniques. Nagata and Yamada (1973) suggest that the Simplex method is one of the most effective parameter seeking methods. Each technique may be used alone or in combination with the others, according to the behaviour of the function and the requirements of the user. Some global logic is built into the program so that, if one of the techniques fails another technique is automatically called to make another attempt.
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115
An analytical method due to Tassios (1976) is used to determine the ability of the estimated parameters to predict the phase splitting. This consists of the following steps:
(1)
(2)
(3)
(4)
Regress the gE and hE data to obtain the temperature-dependent parameters of the models. Introduce these values into the following expression for G:
T.P
Determine the value of xi = x0 for which G assumes its minimum value, by solving the following equation:
/
G 1,, = 0
T,P (11)
Insert the value of x1 into eqn. (10) and determine the value of G. If G > 0, no partial miscibility will be predicted for the estimated values of the parameters. If G < 0, partial miscibility will be predicted.
RESULTS AND DISCUSSION
Using data for gE and hE, temperature-dependent parameters of the NRTL and UNIQUAC models were estimated for 44 binary systems. These mixtures include hydrocarbon, ester, alcohol, ketone, carboxylic acid and aqueous components in various combinations. Only the values of the param- eters c5 and cs, in the NRTL model, are forced to change between 0.1 and 0.7, and - 0.1 and 0.2, respectively. The estimated parameters for the NRTL and UNIQUAC models and variances of the fit, u, are given in Tables 1 and 2, respectively. The variance of the fit is obtained from
u=
(ffni ) NP 1
+ Ii4
(WNP) 02)
Here CNni and Cmi are the total number of data points for gE and hE, respectively, while NP is the number of parameters. The value of (I provides a measure of how well gE and hE data are represented simultaneously by the NRTL and UNIQUAC models. Temperature-dependent UNIQUAC parameters for the first 24 systems were given elsewhere (Demirel and Gecegormez, 1989).
-
TA
BL
E
1
Tem
per
atu
re-d
epen
den
t p
aram
eter
s of
th
e N
RT
L
mod
el a
nd
th
e va
riat
ion
of
th
e fi
t
Sys
tem
T
win
- T
&w
. (
C)
CI
(cal
mol
-)
::a1
mol
- K
-)
C3
(cal
mol
-)
(cal
mol
- K
-)
C5
g-1,
0V2
(K-
)
Ref
eren
ces
1. M
eth
yl a
ceta
te(l
)-
ben
zen
e(2)
2. M
eth
yl a
ceta
te(l
)-
cycl
ohex
ane(
2)
3. M
eth
anol
(l)-
et
hyl
ace
tate
(2)
4. E
than
ol(l
)-
eth
yl a
ceta
te(2
) 5.
2-P
rop
anol
( l)
- et
hyl
ace
tate
(2)
6. l
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pan
ol(l
)-
eth
yl a
ceta
te(2
) 7.
Eth
yl f
orm
ate(
m
eth
anol
(2)
8. E
thyl
for
mat
e(l)
- et
han
ol(2
) 9.
Eth
yl f
orm
ate(
l)-
1-p
rop
anol
(2)
10.
Eth
yl f
orm
ate(
l)-
2-p
rop
anol
(2)
25-5
0
25-4
5
25-5
5
25-5
5
25-5
5
25-5
5
25-4
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25-4
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4.44
38
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00
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Nag
ata
and
Yam
ada
(197
3)
Nag
ata
and
Yam
ada
(197
3)
Nag
ata
et a
l. (1
975)
Nag
ata
et a
l. (1
975)
Nag
ata
et a
l. (1
975)
Nag
ata
et a
l. (1
975)
Nag
ata
et a
l. (1
976)
Nag
ata
et a
l. (1
976)
Nag
ata
et a
l. (1
976)
Nag
ata
et a
l. (1
976)
-
11.
Met
hyl
ace
tate
(l)-
m
eth
anol
(2)
12.
Met
hyl
ace
tate
(l)-
et
han
ol(2
)
13.
Eth
anol
(l)-
to
luen
e(2)
14
.2-P
ropa
nol
(l)-
n
-hep
tan
e(2)
15
. n
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tan
ol(l
)-
n-h
exan
e(2)
16
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tan
ol(l
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2,3-
dim
eth
yIbu
tan
e(2)
17
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tan
ol(l
)-
2_m
eth
ylpe
nta
ne(
2)
18.
Isop
enta
noI
(l)-
n
-hex
ane(
2)
19.
n-P
enta
nol
(l)-
3-
met
hyl
pen
tan
e(2)
20.
n-P
enta
nol
(l)-
2,
2_di
met
hyl
buta
ne(
2)
21.
Ace
ton
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le(l
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ben
zen
e(2)
22
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enze
ne(
l)-
n-h
epta
ne(
2)
23.
Ace
ton
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le(l
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n-h
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25-4
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Nag
ata
et a
l. (1
972)
Nag
ata
et a
l. (1
972)
Van
Nes
s et
al.
(196
7)
Van
Nes
s et
al.
(196
7)
Ngu
yen
an
d R
atcl
iff
(197
5a)
Say
egh
an
d R
atcl
iff
(197
6)
Ngu
yen
an
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(197
5a)
Say
egh
an
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atcl
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(197
6)
Ngu
yen
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(197
5a)
Say
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6)
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(197
5a)
Say
egh
an
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atcl
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6)
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yen
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d R
atcl
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(197
5a)
Say
egh
an
d R
atcl
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(197
6)
Ngu
yen
an
d R
atcl
iff
(197
5a)
Say
egh
an
d R
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6)
Pal
mer
an
d S
mit
h (
1972
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onfo
rt
(198
3)
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mer
an
d S
mit
h (
1972
) L
un
dber
g (1
964)
P
alm
er a
nd
Sm
ith
(19
72)
-
TA
BL
E
1 (c
onti
nu
ed)
Sys
tem
=
ti,
- =
k,
cl
c3
c5
er/r
R
efer
ence
s
(C
) (C
al m
ol-)
(c
al m
ol-
)
c2
c4
(cal
mol
- K
-l)
( ca
mol
- K
-l)
g-l)
1
(K-l
)
24.
Eth
anol
(l)-
cy
cloh
exan
e(2)
25
. W
ater
(l)-
bu
tyl
glyc
ol(2
) 26
. n
-Bu
tan
ol(l
)-
n-h
epta
ne(
2)
27.
n-B
uta
nol
(l)-
n
-hex
ane(
2)
28.1
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ioxa
ne(
l)-
acet
onit
rile
(2)
29.
Car
bon
te
trac
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ride
(l)-
di
eth
yl s
ulf
ide(
2)
30.
Ch
loro
form
(l)-
di
eth
yl s
ulf
ide(
2)
31.
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uen
e(l)
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chlo
roh
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32
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loro
hex
ane(
l)-
eth
ylbe
nze
ne(
2)
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catc
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0.
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2100
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03
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36
10 -
Sat
kie
wit
cz (
1964
) S
catc
har
d an
d W
ilso
n
(196
4)
Sav
ini
et a
l. (1
965)
; B
erro
an
d P
enel
oux
(198
4)
Ngu
yen
an
d R
atcl
iff
(197
5b)
Ngu
yen
an
d R
atcl
iff
(197
5b)
Ber
ro e
t al
. (1
982)
Fra
nce
scon
i an
d C
omm
elh
(1
988)
G
ray
et a
l. (1
988a
) G
ray
et a
l. (1
988b
) G
ray
et a
l. (1
988a
) G
ray
et a
l. (1
988b
) P
aul
et a
l. (1
988)
Pau
l et
al.
(198
8)
-
33.
l-C
hlo
roh
exan
e(l)
- n
-pro
pylb
enze
ne(
2)
34.
1,3-
Dio
xola
ne(
l)-
met
hyl
cycl
ohex
ane
35.1
-Ch
loro
pen
tan
e(l)
- di
-n-b
uty
l et
her
(2)
36.
1,2-
Dic
hlo
roet
han
e(l)
- di
-n-b
uty
l et
her
(2)
37.
l,l,l-
Tri
chlo
reth
ane(
l)-
di-n
-bu
tyl
eth
er(2
) 38
. E
than
ol(l
)-
acet
one(
2)
39.
Ace
ton
e(l)
- w
ater
(2)
40.
Cyc
loh
exan
e(l)
- m
eth
yl m
eth
acty
late
(2)
41.
Isob
uty
ric
acid
(l)-
cy
cloh
exan
e(2)
42
. T
rim
eth
ylac
etic
aci
d(l)
- cy
cloh
exan
e(2)
43
. Is
obu
tyri
c ac
id(l
)-
n-h
epta
ne(
2)
44.
Tri
met
hyl
acet
ic a
cid(
l)-
n-h
epta
ne(
2)
25-9
0
40 5-50
5-77
10-7
0
25-1
50
50-2
00
25-7
5
25-4
5
25-4
5
25-4
5
25-4
5
- 87
.55
40.7
7 0.
95
- 0.
56
878.
00
1034
.30
- 0.
54
- 0.
84
- 22
.39
69.1
8 -
1.34
1.
52
- 11
8.84
57
6.79
-
1.11
0.
22
464.
84
- 49
6.65
2.
31
- 1.
59
260.
36
398.
08
- 4.
07
1.61
65
3.56
65
7.75
9.
27
- 1.
71
227.
63
600.
09
- 1.
83
- 1.
48
914.
38
417.
11
2.76
0.
51
566.
26
255.
71
1.31
1.
11
1109
.80
508.
12
3.34
0.
83
558.
93
277.
79
1.19
1.
24
0.10
00
0.43
25
- 0.
9886
10
-z
0.59
12
0.09
15
0.54
90
10-z
0.
1001
0.
2266
0.
3295
10
-2
0.64
96
0.18
93
0.60
90
10-s
0.
1072
0.
2507
0.
9571
10
-s
0.41
99
0.04
54
- 0.
2058
10
-2
0.67
25
0.14
95
- 0.
8256
10
-s
0.44
54
0.03
58
- 0.
8637
10
-s
0.55
90
0.06
61
0.10
42
10-a
0.
4235
0.
0868
0.
1868
10
-2
0.67
58
0.02
52
0.93
79
10-s
0.
3111
0.
0370
0.
2009
10
-2
Pau
l et
al.
(198
8)
Cas
tell
ari
et a
l. (1
988)
Pau
l et
al.
(198
6)
Pau
l et
al.
(198
6)
Pau
l et
al.
(198
8)
Nic
olai
des
and
Eck
ert
(197
8)
Cam
pbel
l et
al.
(198
7)
Vil
lam
anan
an
d V
an N
ess
(198
4)
Gri
swol
d an
d W
ong
(194
9)
Lu
o et
al.
(198
7)
Hu
ll a
nd
Lu
(19
84)
Lar
k e
t al
. (1
987)
Lar
k e
t al
. (1
987)
Lar
k e
t al
. (1
987)
Lar
k e
t al
. (1
987)
a P
aram
eter
s ob
tain
ed
by N
agat
a an
d Y
amad
a (1
973)
by
usi
ng
hE
dat
a an
d is
oth
erm
al V
LE
da
ta.
b P
aram
eter
s ob
tain
ed
by N
agat
a et
al.
(197
2)
by u
sin
g h
E d
ata
and
isot
her
mal
V
LE
da
ta.
P
aram
eter
s ob
tain
ed
by B
erro
et
al.
(198
2) b
y u
sin
g h
E d
ata
and
isot
her
mal
VL
E
data
. d
Par
amet
ers
wh
ich
sh
ow p
arti
al m
isci
bili
ty.
i P
aram
eter
s w
hic
h s
how
com
plet
e m
isci
bili
ty.
Par
amet
ers
obta
ined
by
Ber
ro a
nd
Pen
elou
x (1
984)
by
usi
ng
hE
dat
a an
d is
oth
erm
al V
LE
dat
a.
-
120
TABLE 2
Temperature-dependent parameters of the UNIQUAC model and the variation of the fit
System Twin- Max do d, et/2 (C) (K) (K)
d, d, (K2) (K)
25. Water(l)- butyl gfycol(2)
26. n-Butanol(l)- n-heptane(2)
27. n-Butanol(l)- n-hexane(2)
28. 1,4-Dioxane(l)- acetonitrile(2)
29. Carbon tetrachloride(l)- diethyl sulfide(2)
30. Chloroform(l)- diethyl sulfide(2)
31. Toluene(l)- 1 -chlorohexane( 2)
32. l-Chlorohexane(l)- ethylbenzene( 2)
33. l-Chlorohexane(l)- n-propylbenzene(2)
34. 1,3-Dioxalane(l)- methylcyclohexane(2)
35. l-Chloropentane(l)- di-n-butyl ether(2)
36. 1,2-Dichloroethane(l)- di-n-butyl ether(2)
37. l,l,l-Trichloroethane(l)- di-n-butyl ether(2)
38. Ethanol(l)- acetone(2)
39. Acetone(l)- water(2)
40. Cyclohexane(l)- methyl methacrylate(2)
41. Isobutyric acid(l)- cyclohexane(2)
42. Trimethylacetic acid(l)- cyclohexane(2)
43. Isobutyric acid(l)- n-heptane(2)
44. Trimethylacetic acid(l)- n-heptane(2)
5-85
15-90
15-59
40
25
25
15-70
15-70
25-90
40
5-50
5-77
10-70
25-150
50-200
25-75
25-45
25-45
25-45
25-45
377.68 426.31 - 49434.0 - 89183.0
- 940.63 - 389.75 767180.0 62225.0
1196.8 - 347.83 2048.2 52997.0
- 358.85 781.02 50510.0 - 106120.0 -614.33 770.32
167990.0 - 219510.0 737.21 - 487.55
- 204310.0 109680.0 38.61 - 58.47
19241.0 - 12424.0 - 155.44 173.15 21241.0 - 26767.0
- 32.27 29.03 - 2524.3 995.45
240.83 - 175.56 771.0 53095.0 56.77 -41.30
652.8 - 1115.6 33.91 - 19.95
- 7744.9 4126.3 - 102.18 107.72
- 2312.1 8799.0 156.34 - 277.82
69392.0 52280.0 712.49 - 461.03
- 228200.0 252910.0 - 64.94 6.36
15894.0 24213.0 429.29 - 161.10
- 75521.0 42030.0 316.51 - 126.18
- 54611.0 29692.0 682.23 - 284.38
- 105450.0 50801.0 184.01 - 18.96
- 24171.0 5405.33
0.2017
0.2651
0.0943
0.2748
0.2668
0.1098
0.1592
0.2105
0.3107
0.1485
0.3055
0.3046
0.2368
0.1053
0.1638
0.0439
0.0620
0.0817
0.0666
0.0572
-
121
The NRTL model gives multiple roots for the systems 1, 2, 11, 12 and 19, as may be seen from Table 1. The NRTL parameters for the systems n-pentanol-2-methyl pentane and ethanol-cyclohexane predict partial mis- cibility, which is not, however, predicted by the UNIQUAC model. Also, although the system water-butyl glycol shows phase splitting, the NRTL parameters predict complete miscibility, whereas the UNIQUAC model predicts partial miscibility. Multiple roots also result when the UNIQUAC model is used. The parameters that yield the best variation of the fit have been tabulated. For the system acetonitrile-n-heptane, which shows phase splitting, the parameters of both equations predict partial miscibility. Tas- sios (1976) claims that by changing the initial values of parameters it may be possible to obtain the correct values for some systems, as is seen for the system n-pentanol-3-methylpentane, which is a highly non-ideal mixture.
Simultaneous predictions of vapor-phase composition, P, gE and hE, by the models are shown in Table 3. The volume and surface area parameters of the UNIQUAC model are given in Table 4. The average absolute error, S, for gE, P and hE were calculated using
The root mean square deviation (RMSD) in predicting the vapor phase compositions was calculated from the following equation
The values of S( gE), S(P) and RMSD were calculated at each isotherm for gE. Pure-component vapor pressures were obtained from the Antoine equa- tion and the necessary parameters were taken from Reid et al. (1977). The value of S( hE) was calculated at each isotherm for the hE data. The results indicate the sensitivity of the NRTL and UNIQUAC equations in repre- senting gE, VLE and hE data simultaneously at various isotherms, and also show that the models are system-dependent. For a total of 947 data points for gE, the average absolute errors for gE and p calculations from the NRTL model are 7.4% and 3.3%, respectively, while the corresponding values from the UNIQUAC model are 7.2% and 2.9%. The values of RMSD for the NRTL and UNIQUAC models are 1.9% and 1.6%, respectively. For
-
TA
BL
E
3
Sim
ult
aneo
us
rep
rese
nta
tion
of
h
E
and
is
oth
erm
al
VL
E
dat
a by
th
e N
RT
L
and
U
NIQ
UA
C
mod
els
usi
ng
the
tem
per
atu
re-d
epen
den
t p
aram
eter
s
Sys
tem
A
vera
ge a
bso
lute
err
or,
S x
100
an
d R
MS
D x
100
NR
TL
U
NIQ
UA
C
T(g
E>
n
T(h
E)
((3
(
Cl
Met
hyl
ace
tate
(l)-
b
enze
ne(
2)
Met
hyl
ace
tate
(l)-
cy
cloh
exan
e(2)
Met
han
ol(l
)-
eth
yl a
ceta
te(2
) E
than
ol(l
)-
eth
yl a
ceta
te(2
) 2-
Pro
pan
ol(l
)-
eth
yl a
ceta
te(2
) l-
Pro
pan
ol(l
)-
eth
yl a
ceta
te(2
)
Eth
yl f
orm
ate(
l)-
met
han
ol(2
)
Eth
yl f
orm
ate(
l)-
eth
anol
(2)
Eth
yl f
orm
ate(
l)-
1-p
rop
anol
(2)
Eth
yl f
orm
ate(
l)-
2-p
rop
anol
(2)
30
40
50
35
40
55
55
55
55
45
45
50
45
12
25
9 35
11
8 25
9
35
45
11
25
35
10
25
35
10
25
35
11
25
35
45
12
25
35
45
12
25
35
45
8 25
35
45
11
25
35
45
M
S(g
E)
S(h
E)
S(P
) R
MS
D
S(g
E)
S(h
E)
S(P
) R
MS
D
13
12
2.96
8.
73
5.22
6.
58
11
11 9 12
12
12
16
13
18
11
20 9 8 11 9 6 13 9 8 13 9 13
12
11
19.8
8 13
.01
11.3
6 0.
70
3.28
20.0
5 11
.66
14.5
0 1.
98
3.21
0.37
1.
13
0.48
2.
01
1.88
1.22
1.
23
1.52
3.07
2.
61
1.32
22.7
5
1.48
0.59
0.
46
1.16
4.
07
4.10
1.
75
3.10
3.
79
4.70
3.
15
3.42
1.
86
1.85
3.
14
1.46
2.
93
3.54
1.
68
0.84
5 2.
36
2.66
1.
04
1.29
1.
75
0.34
0.
76
1.06
0.
38
0.41
0.
58
2.50
2.
10
2.25
2.
71
1.56
1.
28
1.34
3.
96
6.21
5.
14
1.33
0.
62
1.08
0.
70
4.64
6.
06
22.7
6
6.62
6.
63
6.23
1.35
0.71
0.
40
0.51
2.
31
2.86
0.48
0.90
23.4
3
0.98
1.68
0.
48
2.17
2.
58
5.16
1.
59
2.41
5.
26
4.71
1.
83
2.67
4.
48
4.75
3.
82
4.36
4.
53
5.22
4.
06
4.04
3.
25
4.59
3.
36
2.13
2.
42
10.4
2 4.
71
7.82
10
.54
4.59
7.
85
10.0
4 3.
10
8.35
8.
09
1.43
8.
46
1.15
4.79
1.39
15.8
3
0.31
1.00
8.43
- -
_.
_.
- ._
-
..-
-
-
Met
hyl
ace
tate
(l)-
35
m
eth
anol
(2)
45
13
25
14
4.89
13
35
16
4.
15
45
10
10
25
11
8.78
13
35
12
12
.88
45
7 18
25
26
1.
48
18
45
23
1.18
17
60
26
1.
01
17
30
24
2.57
17
45
22
2.
25
16
60
21
1.87
9
25
9 2.
26
1.54
0.
93
1.92
3.
80
3.71
3.
41
9.06
5.
91
5.05
14
.67
11.6
5 10
.63
15.3
0 15
.59
15.6
6 2.
76
0.75
0.
38
4.94
1.
17
0.68
4.
07
3.58
1.
25
4.02
6.
54
2.55
6.
26
10.4
3 6.
20
8.20
11
.95
9.14
13
.86
3.30
0.77
0.
41
1.14
0.
71
2.63
1.
17
9.13
3.
04
1.44
12
.90
2.79
1.
17
3.02
1.
37
1.32
0.
91
1.27
0.
78
1.33
0.
80
3.89
2.
30
2.98
2.
11
2.22
2.
08
5.52
0.
20
Met
hyl
ace
tate
(l)-
45
et
han
ol(2
) 55
1.67
1.
34
1.97
1.
08
0.87
2.
41
0.70
0.
69
2.78
2.
29
2.18
3.
23
1.30
1.
46
3.16
0.
76
1.05
2.
95
2.06
0.
03
2.14
Eth
anol
(l)-
to
luen
e(2)
30
45
60
30
45
60
25
30
45
25
2-P
ropa
nol
(l)-
n
-hep
tan
e(2)
n-P
enta
nol
(l)-
n
-hex
ane(
2)
8 25
9
2.75
2.
52
0.09
2.
69
4.21
2.
31
0.08
n
-Pen
tan
ol(l
)-
2,3-
dim
eth
yl-
buta
ne(
2)
n-P
enta
nol
(l)-
2_
met
hyl
pen
tan
e(2)
Is
open
tan
ol(l
)-
n-h
exan
e(2)
n
-Pen
tan
ol(l
)-
3_m
eth
ylpe
nta
ne(
2)
n-P
enta
nol
(l)-
2,
2dim
eth
ylbu
tan
e(2)
A
ceto
nit
rile
(l)-
be
nze
ne(
2)
Ben
zen
e(l)
- n
-hep
tan
e(2)
25
9 25
9
3.16
3.
58
2.75
0.
03
2.51
4.
67
2.72
0.
11
25
9 25
9
3.42
7.
21
3.24
0.
04
3.24
5.
18
3.08
0.
19
25
9 25
9
2.63
7.
12
2.54
0.
13
2.58
4.
89
2.55
0.
11
25
9 25
9
2.81
7.
72
2.85
0.
07
2.55
4.
02
2.64
0.
05
45
70
45
9 45
13
7.
60
19
3.79
11
45
14
9.
19
25
8 50
4
0.36
0.
83
0.35
7.
65
1.03
0.
22
4.26
0.
72
2.38
8.
70
2.11
0.
71
1.31
1.
04
0.78
0.
61
0.49
1.
05
1.97
1.
22
1.4
0
1.60
1.
30
-
TA
BL
E
3 (c
onti
nu
ed)
Sys
tem
G
Ave
rage
abs
olu
te e
rror
, S
x 1
00 a
nd
RM
SD
x
100
NR
TL
U
NIQ
UA
C
VgE
) n
T
(h
E)
m
S(g
E)
S(h
E)
S(P
) R
MS
D
S(g
E)
S(h
E)
S(P
) R
MS
D
(C
) (
C)
Ace
ton
itri
le(l
)-
45
8 45
13
7.
09
2.68
11
.65
8.73
7.
31
2128
n
-hep
tan
e(2)
E
than
ol(l
)-
cycI
ohex
ane(
2)
Wat
er(l
)-
buty
l gI
ycol
(2)
n-B
uta
nol
(l)-
n
-hep
tan
e(2)
n-B
uta
nol
(l)-
n
-hex
ane(
2)
l+D
ioxa
ne(
l)-
acet
onit
rile
(2)
Car
bon
tet
rach
Iori
de(l
)-
diet
hyl
su
lfid
e C
hIo
rofo
rm(l
)-
diet
hyl
su
lfid
e(2)
T
olu
ene(
l)-
1-ch
Ioro
hex
ane(
2)
5
20
35
50
65 5
25
45
65
85
60
90
59
40
12
40
10
9.60
25
10
25
11
15.6
8
25
9 25
10
5.
92
50
12
15
19
8.00
70
12
25
17
5.
84
9 5
10
9 20
10
9
35
10
9 50
10
9
65
10
8 5
8 8
25
8 8
45
8 8
65
8 8
85
8 19
30
17
22
45
17
55
10
55
10
23
15
10
1.71
1.
54
2.02
1.
66
1.81
26
.46
26.4
6 25
.98
22.4
2 22
.38
6.47
3.
59
1.97
9.53
3.
90
3.82
4.
60
4.94
31
.42
17.1
1 21
.78
37.8
3 30
.52
14.8
9 14
.48
22.5
8 8.
58
4.13
6.90
3.56
2.86
4.31
7.
43
1.60
0.
98
5.35
2.
56
1.36
4.
35
2.45
1.
60
3.84
2.
24
1.81
4.
03
2.70
1.
83
4.74
15
.44
0.94
9.
50
14.4
8 1.
88
8.90
14
.95
2.08
8.
88
15.3
4 2.
61
8.54
13
.73
2.00
8.
65
2.05
0.
82
4.47
1.
44
0.92
2.
14
2.90
0.
77
1.32
1.
28
0.50
0.
76
1.26
0.
75
0.23
0.
22
0.35
0.
26
3.40
12.6
8
15.4
6
5.93
9.75
12
.31
12.7
3 7.
65
7.32
7.
77
2.11
11
.95
13.0
0 13
.16
10.6
2 12
.87
18.6
0 22
.00
18.0
0 23
.00
5.10
12.3
0
5.60
2.90
1.85
2.
26
12.3
4 9.
22
2.50
0.
98
1.73
1.
03
1.52
0.
92
1.67
1.
04
1.87
1.
30
7.53
0.
55
6.29
0.
39
6.75
0.
49
6.31
0.
70
6.88
1.
04
0.51
0.
83
0.92
0.
83
2.76
0.
56
1.74
1.
64
0.49
0.
55
1.27
0.
75
0.22
0.
19
0.22
0.
09
-
l-C
hlo
roh
exan
e(l)
- et
hyl
ben
zen
e(2)
l-
Ch
loro
hex
ane(
l)-
n-p
ropy
lben
zen
e(2)
1,
3-D
ioxa
lan
e(l)
- m
eth
ylcy
cloh
exan
e l-
Ch
loro
pen
tan
e(l)
- di
-n-b
uty
l et
her
( 2)
1,2-
Dic
hlo
roet
han
e(l)
- di
-n-b
uty
l et
her
(2)
l,l,
l-T
rich
loro
eth
ane(
l)-
di-n
-bu
tyl
eth
er(2
)
Eth
anol
(l)-
ac
eton
e(2)
Ace
ton
e(l)
- w
ater
(2)
Cyc
loh
exan
e(l)
- m
eth
yl m
eth
acry
late
(2)
Isob
uty
ric
acid
(l)-
cy
cloh
exan
e(2)
T
rim
eth
ylac
etic
ac
id(l
)-
cycl
ohex
ane(
2)
Isob
uty
ric
acid
(l)-
n
-hep
tan
e(2)
T
rim
eth
ylac
etic
ac
id(l
)-
n-h
epta
ne(
2)
50
70
90
40
40
50
57
77
97
50
70
100
125
150
100
150
200
45
60
75
25
45
25
45
25
45
25
45
10
15
17
16.5
3 7.
67
14
25
19
9.30
10
.14
8 25
19
17
.31
3.11
15
19
4.
13
26
40
16
6.39
2.
00
12
5 20
18
.09
12
25
19
12.8
4 40
19
13
5
19
14.8
4 12
25
19
10
.11
13
14.0
5 12
25
11
24
.05
13
10
12
12.1
5 35
11
9
25
12
3.00
11
50
12
3.
06
10
1.51
14
50
19
11
.27
14
11.0
2 14
5.
95
17
25
11
2.32
17
2.
55
16
2.55
12
35
9
1.74
11
1.
50
11
35
9 5.
95
12
5.22
12
35
9
0.83
12
0.
80
10
35
8 2.
36
10
2.75
9.49
3.
81
4.01
4.
30
2.11
2.95
3.
00
1.13
1.
97
1.76
1.77
0.38
3.32
0.76
1.16
0.29
0.25
0.
20
22.1
2 0.
25
0.18
12
.04
0.41
0.
50
18.1
0
3.99
4.
79
10.9
3
19.5
9 3.
66
17.3
9 18
.69
2.82
15
.02
0.75
0.
46
16.3
9 0.
36
0.19
9.
21
0.31
0.
25
15.0
5 0.
43
0.08
20
.44
1.08
0.
32
19.9
3
1.01
1.
77
2.41
1.
42
1.17
4.
72
2.98
0.
83
5.14
3.
62
2.50
8.
00
3.27
3.
43
8.80
4.
14
3.69
5.
79
0.26
0.
51
3.39
0.
24
0.50
2.
55
0.46
0.
45
2.75
2.
88
3.96
2.
93
2.34
4.
66
3.25
4.
38
0.98
5.
64
3.60
2.
51
4.45
1.
47
4.90
5.
00
1.44
4.
85
4.36
3.
86
1.63
3.
28
3.72
2.
28
3.89
3.77
5.
18
3.75
4.
93
2.20
17.8
0 10
.46
30.3
3 5.
71
4.36
12.8
9 8.
31
12.7
1 7.
80
7.80
8.90
1.70
2.58
2.42
1.70
1.97
0.33
0.
17
0.09
0.
12
0.38
0.
48
2.38
3.
34
19.6
5 3.
67
18.7
2 2.
82
0.80
0.
43
0.26
0.
17
0.47
0.
25
0.74
0.
14
1.26
0.
34
0.60
1.
49
1.32
1.
08
3.29
0.
95
5.03
3.
32
3.22
3.
05
3.22
2.
99
0.34
0.
44
0.26
0.
48
0.24
0.
49
3.53
3.
98
3.01
4.
71
4.35
0.
61
2.96
2.
16
4.45
4.
98
3.37
4.
99
4.36
1.
40
4.54
2.
35
W
-
126
TABLE 4
Volume and surface area parameters for the UNIQUAC model
r 4 4 Methyl acetate 2.800 2.580 2.580 Benzene 3.190 2.400 2.400 Cyclohexane 3.970 3.010 3.010 Methanol 1.430 1.430 0.960 Ethyl acetate 3.480 3.120 3.120 Ethanol 2.110 1.970 0.920 2-Propanol 3.249 3.124 0.890 1-Propanol 3.249 3.128 0.890 Ethyl formate 2.817 2.576 2.576 Toluene 3.920 2.970 2.970 n-Heptane 5.170 4.400 4.400 n-Pentanol 4.597 4.208 1.150 n-Hexane 4.500 3.860 3.860 2,3-Dimethylbutane 4.500 3.860 3.860 2-Methylpentane 4.499 3.396 3.396 Isopentanol 5.923 4.516 1.150 3-Methylpentane 4.499 3.396 3.396 2,ZMethylpentane 4.498 3.932 3.932 Acetonitrile 1.870 1.724 1.724 Water 0.920 1.400 1.000 Butyl glycol 4.697 4.556 4.556 n-Butanol 3.450 3.050 0.880 l+Dioxane 3.185 2.917 2.917 Carbon tetrachloride 3.330 2.620 2.620 Diethyl sulfide 3.902 3.296 3.296 Chloroform 2.700 2.340 2.340 I-Chlorohexane 5.064 4.272 4.272 Ethylbenzene 4.600 3.510 3.510 n-Propylbenzene 5.272 4.048 4.048 1,3-Dioxolane 2.511 2.100 2.100 Methylcyclohexane 4.640 3.550 3.550 I-Chloropentane 4.390 3.732 3.132 Di-n-butyl ether 6.093 5.176 5.176 1,2-Dichloroethane 2.880 2.520 2.520 l,l,l-Trichloroethane 3.541 3.032 3.032 Acetone 2.570 2.340 2.340 Methyl methacrylate 3.922 3.564 3.564 Isobutyric acid 3.550 3.148 3.148 Trimethylacetic acid 4.224 3.768 3.768
a total of 1101 data points for hE, the average absolute errors for hE calculations from the NRTL and UNIQUAC models are 5.8% and 6.7%, respectively. The predictions of the UNIQUAC model compare well with
-
127
those of the NRTL model, although the latter has six estimated parameters, and the effect of temperature on g E, VLE and hE data is well represented by both models. This shows that eqns. (2)-(4), (6) and (7) are capable of expressing the temperature dependency of the models for various types of mixture, including ones with hydrogen bonding.
Using the error matrix, the off-diagonal elements of the correlation coefficient matrix, which is explained elsewhere in detail (Demirel and Gecegormez, 1989), is calculated as
pjj = Uij/( uiiujj)12 I p;; = 1 Pij = Pji J
) (17)
TABLE 5
The elements of correlation coefficient matrix, v: p,, = 1; pi, = p,,
P IJ System a
2 3 27 22 38 43 25
NRTL
CL21 0.461 P31 0.354 p32 0.645 P41 0.707 IL42 0.087 P43 0.482 P51 0.777 p52 0.671 t453 0.794 CL54 0.723 P61 0.596 CL62 0.330 IL63 0.456 p64 0.476 p65 0.722
UNIQUAC
0.271 0.304 0.744 0.348 0.942 0.778 0.203 0.262 0.946 0.335 0.579 0.589 0.789 0.236 0.535 0.016 0.611 0.951 0.230 0.274 0.758 0.568 0.414 0.904 0.566 0.041 0.970 0.937 0.546 0.598 0.326 0.938 0.531 0.100 0.862 0.449 0.247 0.168 0.336 0.573 0.580 0.001 0.624 0.303 0.419 0.167 0.616 0.014 0.838 0.894 0.153 0.499 0.938 0.002 0.119 0.842 0.530 0.464 0.796 0.027 0.348 0.530 0.693 0.568 0.232 0.459 0.293 0.042 0.922 0.029 0.034 0.893 0.328 0.800 0.530 0.306 0.420 0.987 0.736 0.859 0.820 0.306 0.717 0.312 0.428 0.864 0.357 0.909 0.457 0.007
PZI 0.273 0.923 0.892 0.976 0.960 0.999 0.982 P31 0.356 0.899 0.137 0.993 0.969 0.985 0.990 p32 0.935 0.718 0.424 0.944 0.873 0.989 0.988 1141 0.829 0.935 0.096 0.997 0.923 0.974 0.991 p42 0.355 0.811 0.427 0.987 0.937 0.983 0.989 IL43 0.315 0.979 0.985 0.984 0.986 0.995 0.983
a 2, methyl acetate-cyclohexane; 3, methanol-ethyl acetate; 27, n-butanol-n-heptane; 22, benzene-n-heptane; 38, ethanol-toluene; 43, isobutyric acid-n-heptane; 25, water-butyl gIyco1.
-
TA
BL
E
6
Th
e gl
obal
cor
rela
tion
co
effi
cien
ts
Sys
tem
P
:
NR
TL
Cl
C2
c3
c4
CS
UN
IQU
AC
d,
d,
d3
d4
2 0.
842
0.88
7 0.
833
0.90
2 0.
964
0.72
9 0.
793
0.91
6 0.
916
0.78
6 3
0.83
8 0.
956
0.90
0 0.
981
0.96
1 0.
977
0.96
9 0.
947
0.98
3 0.
983
27
0.91
3 0.
612
0.98
5 0.
984
0.97
7 0.
983
0.94
7 0.
954
0.98
5 0.
988
22
0.99
1 0.
998
0.98
3 0.
997
0.84
1 0.
988
0.99
9 0.
999
0.99
9 0.
999
38
0.85
5 0.
846
0.81
9 0.
953
0.95
8 0.
951
0.96
7 0.
966
0.96
3 0.
974
43
0.95
7 0.
931
0.97
2 0.
971
0.95
5 0.
950
0.99
9 0.
999
0.99
9 0.
999
25
0.99
7 0.
999
0.99
9 0.
997
0.05
5 0.
999
0.94
7 0.
954
0.98
5 0.
988
a 2,
met
hyl
ace
tate
-cyc
loh
exan
e;
3, m
eth
anol
-eth
yl
acet
ate;
27
, n
-bu
tan
ol-n
-hep
tan
e;
22,
ben
zen
e-n
-hep
tan
e;
38,
eth
anol
-tol
uen
e;
43,
isob
uty
ric
acid
-n-h
epta
ne;
25
, w
ater
-bu
tyl
glyc
ol.
-
129
where uii represents the elements of the error matrix. If v is positive definite, p < 1 for all elements. If p = 0, then the parameters are uncorrelated and if p = 1, the par ameters are completely correlated. For a given parameter, the global correlation coefficient is given by
/.A; = 1 - [ UkkVLk] -l (18) and is the correlation between it and that linear combination of the other parameters most highly correlated with it. All such coefficients should be between zero and one for a positive definite error matrix. The values of error matrix correlations and the global correlation coefficients for some of the systems are given in Tables 5 and 6, respectively. The parameters for some of the systems are highly correlated, but for such parameters it is not possible to specify unique values from a given set of data. have arisen when the same free energy function is used phases, as in LLE, at points where the mole fractions in identical.
Such problems for two of the two phases are
CONCLUSIONS
Although the simultaneous fit of Gibbs energy and enthalpy of mixing data for 44 binary systems is satisfactory with the NRTL and UNIQUAC models, choice of the best model is mainly system-dependent. The use of temperature-dependent parameters improves the performance of the models, but care should be exercised in the selection of the best model for a given system, and in the estimation of the parameters for especially non-ideal mixtures.
ACKNOWLEDGMENT
The authors thank the Computer Centre of Cukurova University for the computation facilities provided.
LIST OF SYMBOLS
aij
Cl, c3
c29 c4
UNIQUAC binary interaction parameter related to riT values of (g2, - gir) and (g,, - g,,) at 0C (cal mol-) coefficients of temperature change of (g,, - g,,) and (g,, - g,,) (cal mol- K-)
-
130
c5 value of (Y,~ at 0 C
6 coefficient of temperature change of ai2 (K-l)
d,, d, UNIQUAC parameters related to ajj (K)
6 d, UNIQUAC parameters related to ajj (K) F objective function as defined by eqn. (9)
gE excess molar Gibbs energy (cal mol-) Gi Gh
constant in eqn. (5) Gibbs energy of mixing (cal mol-)
G, G second and third derivative of Gibbs energy of mixing with respect
hE
m, n
M, iv
NP
4; 4,! P
r, R RMSD T s
ij xi Yi
excess enthalpy of mixing (cal mol-) number of experimental hE and gE data points, respectively, at a specified isothermal temperature number of isothermal system temperatures for hE and gE data, respectively number of parameters molecular geometric area parameter for component i molecular interaction area parameter for component i total pressure (Pa) molecular volume parameter for pure component i gas constant (cal mall K-) root mean square deviation absolute temperature (K) average absolute error (eqns. (13), (14), (15)) elements of error matrix liquid-phase mole fraction of component i vapor-phase mole fraction of component i
Greek letters
a12 non-randomness constant for binary l-2 interaction
; activity coefficient of component i area fraction of component i in residual contribution to the activity
coefficient u variance of the fit 7ij NRTL binary parameter 7,; UNIQUAC binary parameter pij elements of correlation coefficient matrix (eqn. (17)) p,, elements of global correlation vector (eqn. (18))
Subscripts
exptl experimental calcd calculated
-
131
4 .i component max maximum min minimum
Superscript
E excess
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