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8/19/2019 VLE Methanol- water
1/10
Activity Coefficient Relations in Miscible
and Partially Miscible Multicomponent Systems
Solke Bruin ’
Laboratory for Physical Technology, Eindhoven Uniuersity
of
Technology, Eindhoven, Netherlands
Based on a quasilattice model of a multicomponent solution, three equations are der ived relating component
activity coefficients to the composition of a liquid mixture. The equations also ap ply to systems showing
limited miscibility. Temperature dependence of activity coefficients i s built in. The equations were tested fo r
16 binary systems,
5
of which show limited miscibility, and for 4 ernary systems. The enthalpic Wilson
equation, one of the three, gives the best prediction of binary vapor-liquid equilibrium. Both the enthalpic
Wilson equation and the Orye equations give good results of comparable accuracy for the ternary systems.
PREDICTIONf thermodynamic properties of mixtures from
pure component properties is the main goal of thermo-
dyn amic s of liquid mix tures. Progress in this field has ma de
it possible t o calculate th e properties
of a
mixture
of n
compo-
nents from th e pure component properties supplemented with
experimental vapor-liquid equilibrium (V LE) d ata for all the
binary systems that can be constructed out of the
n
compo-
nents : a tota l of
l / 2n n -
1) (I’rausnitz et al. , 1967). Th e
startin g points of these compu tational methods are thermo-
dynamic models of multicomponent mixtures, which provide
calculational procedures to predict the vapor-liquid equilib-
rium of the mixture from data on the binary sys tems. From
such models relations between the activity coefficients, y
an d th e composition of t he liquid are derived. I n such rela-
tions, parameters characteristic of interactions between
pairs of molecules
of
different kind appear. Formally, such
activ ity coefficient relations have the form
Values for the interaction parameters,
Pi,
a n d
P 3 ( ,
can be
calculated from VL E da ta on th e binary system of compo-
nent s i a n d j .
The best known activity coefficient equations are, un-
doubtedly, the van Laar equations. These equat ions have
proved to be very useful for binary systems b ut less satis-
factory for m ulticomponent systems (Neretnieks, 1968; Orye,
1965; Orye and Prausniti , 1965; Prausnitz
et
al., 1967). Th e
two-parameter Wilson equation gives better results than the
van Laar equat ions for many binary and mult icomponent
mixtures (Nagel and Sinn, 1966, 1967; Prausnitz
et al.,
1967;
Wilson, 1964a, b). No terna ry or higher param eters are needed
in the generalization to multicomponent systems. Tempera-
ture dependence is to some extent bui l t into the Wilson
equations, making them suited to, for example, the isobaric
vapor-liquid equilibrium calculations common ly encountered
in distillation. An undesirab le featu re of th e Wilson equatio ns
is their inapplicability to partially miscible systems. The
modification
of
the equations that Wilson proposed to over-
come this difficulty means introdu ction of a n extra parameter.
I n the present paper three ac tivity coefficient equations
are discussed. These equations follow from the quasilattice
Present address, Western Utilization Research and Develop-
ment Division, Agricultural Research Service, .4lbany, Calif.
94710
model of multicomponent liquid mixtures developed by
Gugg enheim (1935, 1944a, b, 1 952). Th is model yields a ra the r
general relation for the excess Gibbs f~inct ion, n which the
enthalpic and entropic contributions appear as separate
terms. Introduct>ion of van L aar or Wilson parameters i nto
this relation leads to a num ber of possible activity coefficient
equat ions including the van Laar and Wilson
equations
themselves. In Table
I
all poasible combinations of Wilson
paramet,ers (Ai j ) , van Laar parameters
(Aij),
and Gibbs
function are given. The Orye equations (Orye, 1965) result
on introdu ction of Wilson param eters in the general relation
for the excess Gibbs function. Retaining only the enthalpy
part assuming the excess entropy to be ncgligible (regulm
solutions), yields “enthalpic Kilson” equations. The “ex-
tended van Laar” equations (EVL equations) result from
introduction
of
van L aar parameters into the general Gibbs
function,
as
indicated in Table
I.
Introduction
of
van Laar
parameters int’o the e ntropy p art, assuming th e ent,halpy to
be negligible (athermal solutions), gives equations unable to
predict p artial miscibility (Bruin, 1969), and therefore is not
discussed.
The Orye, enthalpic Wilson, and EVL equat ions are
derived and discussed. Th e derivation of th e Orye equations
here differs from th at given originally b y Orye. Subsequently,
the equations are generalized to multicomponent systems.
Th e equations were tested by application to 16 binary an d four
ternary systems, prediction
of
parameters from infinite
dilution activity coefficients, and prediction of temperature
dependence
of
act ivi ty coefficients.
Binary Systems
The activity coefficient,
yl,
in a binary liquid mixture
of
components 1 a n d 2 is given by the exact thermodynamic
relationship
In y1 = [
n l-
N gE/ RT)]
, P , n
1)
where N i s the tota l number
of
moles:
N
= nl +
nz .
T h e
mola r excess Gibbs free energy of m ixing,
g E ,
in Equat io n
1
s
rela ted to the excess molar entrop y and enthalpy according to
=
hE - TsE
= U M +
PvM - TsE
2)
Adopting th e quasilattice model for l iquid mixtures
of
mole-
cules
of
different size, a s developed b y Gu ggenh eim (1935,
Ind. Eng.
Chem.
Fundam.,
Vol.
9 No. 3 , 1970
305
8/19/2019 VLE Methanol- water
2/10
Table
I.
Summary of Relations
fo r g E
Resulting from Different Combinations
o f
Quasilattice
Equations an d Parameters
Type of Parameter gE = hE
-
T ~ E
A i i A i j hE-term
rE-term
(Wilson)
(van
Laar)
included
included Name o f Resulting Equation
for
y i
+ Wilson
- van Laar
+
Orye
- Enthalpic Wilson
+
t
+
+ +
4-
+
+
+
Extended Van Laar (EVL)
-
1
+
2
3 +
4
+
5
6
-
-
-
-
+
-
1944a, b, 1 952), we may w rite the following equat ion for the
energy of mixing, uM, an d the excess entropy:
sE= -R(zl In {rl / (r lz l
+ r 2 4 )
+
2 2
ln{rz/(rlzl+ r s 2 )
1)
(3)
UM =
x12u12
(4)
In Equa t ion 3 r l is the nu mb er of sites in the qua silattice
structure occupied by a molecule of type 1.
u12
represents an
interaction energy:
5)
N is the Avogadro number, w12 denotes the contribution to
the potential energy by
a
pai r of site s, one of wh ich is occu pied
by an element of
a
molecule of type
1,
the other by an ele-
me nt of a different molecule of ty pe 2; an d
z
s the n um ber of
nearest neighbors to a site, the “coordination number.”
In Equa t ion 4
XI2
is defined as the n um ber of pairs of neigh-
boring s ites occupied by different mo lecules, one of which is
of type
1
an d the other of ty pe 2, divided by the to tal number
of
nearest neighbor sites. To calculate X~Z ,uggenheim
introduced the qu asichemical approxim ation
UIZ= ‘ /2~N(2~12 1 1
W Z Z )
which is the m ass-action law for a “reac tion” where
a
molecule
1 s brought from a pure liquid 1 o a (1-2) liquid mixtu re and
simultaneously a molecule 2 is brought from pure 2-liquid t o
the (1-2) liquid mixtu re. I n th is reaction (zX11) an d (zX,)
bonds are broken up, while
2 (2X12 )
bonds are formed. The
nu mb er of 1-1 pairs and 2-2 pairs can be expressed as
22x11 = z rlz1
-
XIS) ( 6 4
22x22 = z ( r s2 -
X12 )
(6b)
I n th e par ticula r case where all energies of mixing are zerc-
al l u12 -c 0 (all molecules distrib uted a t random)-combina-
tion of
5a
an d 6 gives
(7)
8)
When Equat ions 3 and 8 a re subs t i tu t ed in 2 and U is
assumed t o be zero, one obtains for
g E
X 1 2
=
( r 1 4 ( r z a ) / ( r 1 x 1 +
TZXZ )
UM
=
( r l z J ( r s 2 ) u d ( r l z 1 +
rg2
For UM
one obtains
Renon and Prausnitz (1968) recently developed a relation
for g E in which the condition u12 0 on Equatio n 5a was re-
laxed. Moreover, distinction was made between the transfer
of a molecule
1
into the (1-2) liquid mixture [giving
(zX12’)
bond s] and th e transfe r of a molecule 2 into the (1-2) liquid
mixture [giving
(zX2,’)
bonds]. T he resulting relations for th e
acti vity coefficients’ (N R TL equations) were shown to give
excellent representation of a wide variety of binary and
ternar y liquid mixtures.
In the present paper Equ at ion 9 is used as
a
starting point
to derive activity coefficient equations. Guggenheim (1944b)
showed tha t the inaccu racy of Eq uatio n 9 is theoretically of
the order of ( u ~ ~ / R T ) ~ .
If one assumes in Equat ion 2 th at
h E
= 0 (athermal solu-
tions ; Flor y, 1942; Huggins, 1942) one obtains for
g”
g E / R T = In ( r1
)
+ x z In
(
“ ) (10)
rlxl + rsr2 ~1x1
+
ra2
On the other hand, if s E = 0 (regular solutions, van La ar;
Black, 1959; Black and Derr, 1963a, b; Wohl, 1946)
QE
is
given by
Equat ions 9 t o 11give three relations for
QE
a s a fun ctio n of
liquid composition in terms of th e quasilattice param eters,
rl a n d r2, and the interaction energy,
u12.
To obtain two-
parameter equations for y1 a n d y2 from Equat ions 9 to 11,
rl ,
rz,
a n d
u12
have to be combined to two parameters.
Different suggestions have been made. The most obvious
assumes the ratio ( r l / r2 ) o be equal in
a
first approximation
to th e ratio of m olar volumes (ul/u2) :
r l / r J
- VI/VZ) (12)
Another assumption is due to Carlson an d Colburn, in their
modification of the van Laar parameters:
rZu12/RT m/RT
=
12
rluzl/RT = m/RT =
21
(13)
while
u12
=
u21.
Th e implication is th at
r l / r2 )
equals (A21/A12)
:
( r l / rd +
A d A d (14)
Parameters A12 a n d
A21
are sometimes called “effective
molar volumes” (Hildebrand and S cott, 1950).
Wilson suggested another expression for (n /n ) , aking
into accou nt nonrandomness effects by weighting (uI/u2) with
a Boltzm ann factor containing interaction energies
X
:
r 1 / r 2 ) + (u1/v2) exp
{ -
A l 2
-
hm)/RT)
=
Apl
(15a)
306
Ind. Eng.
Chern.
Fundarn.,
Vol.
9,
No. 3,
1970
8/19/2019 VLE Methanol- water
3/10
an d similarly for ( r 2 / T 1 )
:
(r2/r1)- v2/u1) exp - A l z - All)/RT] A12
(15b)
One can expect A12,
A l l ,
a n d A22 to be proportional to zw12,
zwl1,
a n d
zws2
in Equat ion
5 .
Therefore
u12/RT
+ 2A12
- All - An) / RT
= -In ( A 1 2 h )
(16)
Equat ions 14 and
15
provide two reasonable choices (van
Laar and Wilson parameters, respectively) to introduce
parameters into Equat ions 9 to
11,
giving six combinations.
In Table
I
the combinations are indicated. Combining Equa-
tion 10 with Wilson parameters gives the Wilson equation.
Introducing v an Laar param eters into Equation 11 gives the
van Laar equations. When van Laar parameters are intro-
duced in Equatio n 10, a relation obtained is unable
t o
predict
phase separation, in much the same way as the Wilson equa-
tion.
Exten ded van Laar (EVL) Equa t ions . Subs t i tu t ing
A12parameters in to Equatio n 9 an d differentiating the result-
ing equation for g according to Equat ion 1 gives for the ac-
tivi ty coefficient,
y l
In y1 = 1
-
n a+ A1222/A21) -
2
+
A122dA21
(17)
1 A z 1 ~ 2 ~
22 + A ~ ~ Z I / A I Z ) ~
Th e relation for In y~ results when indices are rotate d in the
sequence 1
- -
. Equ ation 13 roughly predicts tempera-
ture dependence of A12 a n d A21,
Entha lp ic Wi l son Equa tions . Subs t i tu t ing Ki l son pa ram -
e te rs in E qua t io n 11 g ives
(18)
Postulating proportionality between the number of sites
occupied by a molecule of ty pe 1 and the ratio between its
molar volume a nd a m ean molar volume (weighted by inter-
action energies), one can write
VI
exp ( -Ad RT )
~1 exp ( - A d R T ) + xzvz exp (-A12lRT)
Comparison
of
Equa t ions 19 and 15 i l lust ra tes tha t Equat ion
19 is exact only in the limit
xl
+ 1. Substitution of Equ ation
19 in Equ ation 18 an d differentiating gives
In y1 = -
qE /RT
=
-r121zz
In (A1&)/(A21n
+
22
(19)
1
-
X
2
In (11121121)
21 + A1252)(22
f A z i ~ i )
For
In
y z
the analogous expression results by rotation of
indices (1
+
+
1).
Orye Equa t ions . Subs t i tu t ion of K i l son pa ra mete r s in
Equa tion 9 gives equations for y derived by O rye (1965), who
used a different approach.
In
y1
= 1 - n (zl +
~ ~ 2 x 2
zz{
~ 1 2 / z l
+ A ~ ~ z ~
Again, In y2 follows from rotation
1
2
+ .
The EVL equations (17), enthalpic Wilson equations (20),
an d Orye equations (21) were tested by
Representation of 16 experimental binary vapor-liquid
equi l ibr ium dat a from l i terature
T-
da ta a t cons tan t
P ) .
Prediction of parameter values for binary systems from ac-
tivity coefficients a t infinite dilution.
Prediction of temperature dependence of activity coeffi-
cients a t infinite dilution (an extremely severe test).
Comparison
o f
Equations
by
Fitting Parameters
to Experimental Data on Binary Systems
The EVL equat ions (17), enthalpic Wilson equations (20),
and Orye equations (21) were tested for 16 binary systems
listed in Table
11.
All equilibria were isobaric (1 at m ), in
order to test the built-in temperature dependence of the
equations; five systems show partial miscibility.
The fit t ing procedure proposed by Prausnitz et
al.
(1967)
was used. In this procedure a correction for vapor-phase
nonideality is incorporated. The objective function,
f, t o
be
minimized was defined as
with
m
the num ber of d ata points. A nonlinear multiple
regression subrou tine which adjus ts the param eters of a
function being fit ted to experimental data in such a manner
as to yield a least squares
fit
was written, following the
method of hfa rqu ard t (1959). Details have been given (Bruin,
1969).
The results are summarized in Tables
I11
to VII. Values
Table
I I .
literature Data for Systems Selected to Test
Acti vity Coefficient-Composition Equations
(Isobaric at
1
Atm)
System
1. hIethano1-water
2. Ethanol-water
3. 1-Butanol-water
4. 2-Butanol-water
5. Acetone-water
6. Butanone-mater
7. Methyl acetate-
8.
Furfural-water
water
9. 2-Propanol-water
10. Icetone-methanol
11. Acetone-ethanol
12. Ethanol-2-propanol
13. Ethanol-benzene
( P
= 750 mm
14. Ethanol-methyl-
15. Methylcyclopen-
16. hlethanol-ethanol
Hg)
cyclopentane
tane benzene
Reference
Uchida, 1934
Carey an d Lewis, 1932
Stockhardt and H ul l ,
1931
Altsybeeva and
Belousov, 1964
Othmer
et
al., 1952
Othmer et al., 1952
hlarshall , 1906
Temp.
Range,
K
338-69
351-68
372-84
364-66
330-61
330-61
329-60
Inte rna tion al Critical 371-432
Wilson an d Simon s, 1952 354-68
Hellwig an d Van Winkle, 330-48
Ballard an d Van Winkle, 351-55
Tables, 1928
Uchida et al., 1950 329-37
1953
1952
Tyr er, 1912 342-47
Sinor an d Weber, 1960 339-49
Griswold an d Ludwig 344-52
1943
h m e r
et al.,
1956 338-49
Ind. Eng. Chem. Fundam., Vol.
9, No
3,1970
307
8/19/2019 VLE Methanol- water
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Table
Ill.
Fitting Results for Wilson Equations to Binary Systems
Parameters, co l /mo le Mo l ar Vo lumes, Cm3/Mole
System A12
-
11 A12 -
A22 v1
V2
F
O K
Methanol-water
198 .113
472.368 42 .11
1 8 . 7 9 3 6 9 . 5 5
Ethanol-water
380.315 910.615 6 3 . 9 8 1 8 . 7 8
368.65
2-Propanol-water 543.845
1 3 5 8 . 3 5 8 2 . 4 8
1 8 , 7 7 3 6 8 . 3 2
Acetone-methanol
-54 .175 472 .74
7 6 . 8 3 4 1 . 2 4
323.15
Acetone-methanol .- 3.454
489.719
7 8 . 4 7 4 2 , 0 7
337.15
Acetone-ethanol
-
4 , 2 3 6 440.50 7 9 . 8 2 6 2 . 2 7 3 4 8 , 2 5
Acetone-water
469 .02 1489.07 81 .42
1 8 . 6 7 3 6 1 . 0
llet 'hanol-ethanol
198 .85 -264 .02 42 .9 3
63 .39 349 .75
Ethanol-2-propanol
215.889
-241.331
62.825 83 .11 355 .05
Ethanol-benzene ( P
=
1540.28
119.80
6 1 . 8 2 9 4 . 6 1
3 4 2 . 6
(isothermal)
750 mm Hg)
Ethanol-methylcyclopentane 2230.23 236.08 61 .58 118.82 339.45
Methylcyclopentane-benzene 13 .68 253.694 120 .66 95.8 9 352.79
Error
in
P
Total,
%
0.3612
1 . 1 5 7 8
2 . 1 5 5
0.3442
0.6474
0 . 8 0 4
2 . 7 0 0
1 . 2 7 9
1 . 2 4 6
0 . 6 2 7
1 . 8 8 4
0.5890
Table
IV.
Fitting Results for van Laar Equations to
Binary Systems
Parameters Error in P
Azi Total, %
ystem
Methanol-water
Ethanol-a a ter
2-Propanol-water
1-Butanol-water
2-Butanol-water
Acetone-methanol
(isothermal)
Acetone-methanol
Acetone-ethanol
Acetone-water
Butanone-water
Methyl acetate-water
Furfural-water
Methanol-ethanol
Ethanol-2-propanol
Ethanol-benzene
( P = 750 mm Hg)
E thanol-methylcylo-
pentane
hfethylcyclopentane-
benzene
Ai2
0 . 8 6 4 9
1 , 6 6 0 1
2.4260
3 . 8 7 7 2
3.5447
0.6166
0.5713
0 , 4 5 4 9
2.3880
3 , 5 0 4 5
3.0565
4 , 6 8 9 5
-0 .5629
-0.01381
1 . 8 3 4 1
2.7404
0.3259
0.5210
0 . 8 6 4 3
1 . 1 6 2 5
1 , 1 7 5 9
1.1387
0.5442
0.5170
0 . 5 8 7 1
1 ,4555
1 ,6525
1
8050
1 . 1 0 0 8
-0 .0580
-
.006945
1.2814
1 . 6 8 8 9
0 . 3 5 0 5
0 . 3 5 2 1
0.7246
1 2026
3 . 3 4 9 1
1 . 4 7 9 5
0 , 3 0 8 3
0 , 6 4 1 0
0 , 7 3 4 0
4 . 4 1 5 1
5 . 1 9 9 6
5 , 5 2 1 2
27.5862
0 . 5 6 9 1
1 . 4 6 2
0 , 3 5 6 5
2 . 7 7 1
0.5796
for
X I Z -
AH),
A ~ z
- XZZ), € 1 2 , and
€21
a re computed a t the
temperatu res indicated in the tables. As a reference, th e Wil-
son an d van Laar equations were also checked. General trends
are that EVL equations (17) give poor results when com-
pared to the van Laar equations. A rem arkable result is tha t
the enthalpic Wilson equation (20) gives very good represen-
tation for most of the systems, in many cases bett,er tha n the
van Laa r or Orye equations (21).
Prediction of Parameter Values from
Infinite Dilution Coefficients
For In yl a t infinite dilution
of
component 1 in a binary
mixture one can derive
F o r
EVL Equations
In ylm
=
1
-
n (AIz/AzI)
- A Z ~ A I Z
AZI
(23)
For Enthalpic Wilson Equations
In
ylm=
-In
( A 1 2 h ) / A n
(24)
F o r Orye
Equations
In ylm= 1 - n 111 2 - AZI - n
hlzA21)/Az1
(25)
Similar relations follow for In
yz
by letting 1 2 1.
If In ylmand In
yZm
re known-e.g., from Pierotti-D eal-Derr
correlations (Pierotti e t al. , 1959) or Helpinstill and Van
Winkle correlations
(1968)-parameter values can be cal-
culated.
Solving Equation 25 and the similar relation for yZm or
A12 a n d A21 is tedious, requiring a double iteration scheme.
Therefore the Orye equ ations were not tested.
For
the EVL and enthalpic Wilson equations convenient
computational methods have been discussed (Bruin, 1969).
Some results for the EVL equation are summarized in Table
VII I . In F igures 1 t o 3 y s and activities, a, are plotted for
acetone and water, 1-butanol and water, and 1-octanol and
System
Methanol-water
Ethanol-water
1-Butanol-water
2-Butanol-water
Acetone-water
Butanone-water
Methyl acetate-water
Furfural-water
Table V.
Ai2
0.6021
0 . 9 8 7 5
1.5696
1 . 5 0 9 3
1.4392
1 . 8 0 4 1
1 . 6 9 7 1
1 ,7366
Fitting Results for EVL Equations to Binary Systems
Parameters
Ail
€21,
cal /mole ell, col /mole r
OK
1 ,0429
442.194
765,917 369 .55
1 . 9 2 2 5
723.459
1408,405 368 .65
4 .6182
1162.336
3419.983 372 .65
4 .2917
1093.13
3 1 0 8 , 2 1
.
3 6 4 . 4 5
2 .5208
1032.47
1808.370 361.00
3 .7793
1282.593
2686.815 361.00
3 .1399
1214,279
2246.585 360.05
5 .4891
1282.762
4054.630 371.71
Error in
P
Total,
%
0.3616
0 , 6 5 0 3
4.0834
1 . 7 8 1 5
5.2401
5 . 5 8 5 8
6.3476
24.1892
308 Ind.
Eng. Chem. Fundom., Vol.
9,
No. 3,
1970
8/19/2019 VLE Methanol- water
5/10
System
Methanol-water
Ethanol-water
2-Propanol-water
1-Butanol-water
2-Butanol-water
Acetone-methanol
Acetone-ethanol
Acetone-water
Butanone-water
Methyl acetate-water
Furfural-n-ater
Methanol-ethanol
E
hanol-2-propanol
Ethanol-benzene ( P
=
Ethanol-methylcyclopentane
Methylcyclopentane-benzene
750 mm Hg)
Table VI.
Fitting Results for Enthalpic Wilson Equations
Parameters , CaI /Mol e Mo lar Volumes, Crn3/Mole
XlZ - i1 XlZ - 22
v1
V2
-208.7681 571.3078
44 ,548 18 .792
- 25,7964
897.1059
63 .977 18 ,779
- 23.065 1150.147
82 .48 18 .77
- 57.8748 1229.9084
98.366 18.837
- 79.327 1098,612
85 .668 18 .721
-216.966 506.093
78 .47 42 .07
-116.696 399.150
79 .82 62 .27
-
36.468 1185.436 81 ,420
18 .675
-396.248 1307.900
8 9 . 5 3 3 1 8 , 6 3 3
-419.1093 1290.6923
87 .373 18 .663
-202.367 1158.676 82 .899
18 .823
188.7053
- 290.622
4 2 . 9 4 6 2 . 3 9
-1329.7304
1267.75 62.82
8 1 . 4 3
745.400
- 6.970 61.82
9 4 . 6 1
1006.585 - 82 .718
61 .58 118 .82
- 5.740 238.011
120 .66
95 .89
1, O K
369 .55
368 .65
368.32
372 .65
364 .45
337.15
3 4 8 , 2 5
361.00
361.00
329 .63
371 .71
349.75
3 5 5 , 0 5
342 .6
339 .45
352 .79
Error in
P
Total, %
0.3473
0 .6506
1 .1245
4.1245
1 .6216
0 . 6 5 4
0 .8046
3 , 2 6 9 5
4.1933
3.4580
23.0276
0 ,4790
0 .3871
0 .3938
2.3874
0 .5921
Table VII.
Fitting Results for Ory e Equations to Binary Systems
Parameters , CaI /Mole Mo lar Volumes, Cm3/Mole
Methanol-Tyater
-211.440
450 .860 44 ,548
18 .791
Ethanol-water
-
48.018 747 ,2929 63 ,977 18 .779
2-Propanol-water -435.954
962 .733 82 ,47
1 8 . 7 7
1-Butanol-water -369.779
1043.374 98.366
18 .837
2-Butanol-water - 79,327 921.84
85 .688 18 .721
Acetone-methanol - 35.9306 408.528 78.47
4 2 , 0 7
Acetone-ethanol
243.84
- 4 ,774 79 .82
62 .27
Acetone-water -479.035
1023 .049 81 ,420
18 .675
Butanone-water -435.140 1122.348 89.533
18 .633
Meth yl acetate-water -419.10 93
1290.692 87.373
1 8 , 6 6 3
Furfural-water -231.3797 970 .618
82 .899 18 .823
Methanol-ethanol
- . 4 2 0 5 1 , 2 1 3 3 4 2 , 9 3 5
62 .392
Ethanol-2-propanol
- 79.812
213 ,857 62 .82
8 1 . 4 3
Ethanol-benzene
654.954 - 97 .128 61 .23
9 4 , 6 1
Ethanol-methylcyclopentane 931 ,4907 - 30.439
6 1 , 5 8 1 1 8 . 8 2
Methylcyclopentane-benzene
-
3 ,107 167.237 120 .66 95 .89
System A12 - XI1 A12 - 22 v1 v2
P = 750 mm Hg)
r
OK
369 .55
368 .65
368.32
372.65
364 .45
337 .15
348 .25
361.00
357 .75
329 .63
3 7 1 , 7 1
349.75
355 .05
342.60
339 .45
352 .79
Error in
Total, %
0 .3479
0 .7323
1 .1650
4 , 4 1 6 1
1 , 7 3 3 1
0 .6566
3 . 3 8 0
4.0037
4 .9331
4.4630
23,2258
1 .3748
1 . 3 7 2
0 .4385
2 .7651
0 .6113
~~ ~~
Table VIII. A-Parameters Estimated from Infinite
Dilution Values for Acti vity Coefficients
No.
of
I t era-
System 1,
O K
Ai2 A n t ions
Acetone-\\ atera
373 15 1 5745
2 5228 8
2-Heptanone-waterb 298 15
2 1590 9 8669 7
1-Butanol-water*
298 15 1 6545
4 6425 7
1-Octanol-wa terb 298 15
2 2763 11 6137 7
Acetone-1-octanolb
373 15 3 6702
0
8224 6
l-Butanol-1- 373 15
3 5548 1 6639 6
2-Heptanon e-l- 373 15 1 8025
0
8354 6
octanolc
octanolb
water. Results for acetone and water are compared with the
da t a of Othmer (1952). Phase separation is predicted for 1-
butanol and w ater and 1-octanol and water a t approximately
the right l iquid composit ions, as can be seen from the activity -
composition plots.
The enthalpic Wilson equation was tested using methanol
and water , ethanol and water , and 1-butanol and water .
Table IX gives values for
A l z
-
l l )
a n d ( A 1 2 - 2 2 ) computed
from
ylm
n d
y 2 .
Reasonable agreem ent exists between values
computed from Equation 24 and those given in Table VI.
In Figure 4 a graph is given for binary systems with posit ive
deviations from ideali ty, from which A 1 2 a n d Azl follow
directly when
y lm
a n d
7 2
are known. The procedure is
i l lustrated for methanol
1)
+ water (2): In y lm
=
0.521,
change of indices gives coordinates of point 2:
A21 =
1-00.
Othmer , 1952*
Pierrotti,
and
Derr
In
yz =
0.865. FromFigure 4: A12 = 0.60 (point 1). Inter-
1959.
c
Wilson an d Deal, 1962.
Table IX.
Estimated A Parameters for Enthalpic Wilson Equations (20) from A ct ivi ty Coefficients at Infinite Dilution
(ENTLAM program)
( X i 2
- 221 Temp., O K
System Aiz Azi (A12 -
zz)
369 .55. Methanol-water. (338-72°K) 0.59 90 0.9 94 4 - 57.553
2. Ethanol-w ater. (351-72'K) 0.483 2 0.928 0
-
64.712 952.309 368.65
3. 1-Butanol-waterb (372-80°K) 0.3 04 9 1.0 054 - 44.979 1220.01 372.65
638.017
a
Stock hardt and Hull , 1931. Pierrott i , Deal, an d Derr correlat ions, 1959.
~~
Ind.
Eng. Chem. Fundam., Vol.
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No.
3,
1970
309
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I-Butanol (1)-Water
(2)
A
1 2
1.6545 T
=
298
O K
X,=aons i , =0 .540 xI+
Activity and activity coefficients for the 1-butanol (1)-water
(2
XI
-
Figure
2.
system
Prediction
o f
Temperature Dependence
of ylm
nd
yzm
Th e temperature dependence of activ ity coefficients is given
by the exact thermodynamic relation
where H i o i s the enthalpy
of
component
i
in the s tandard
s t a t e and Ri s the part ia l molar enthalpy of mixing. At
infinite dilution
of
component i
For the EVL equations one obtains by differentiation
of
Equat ion
17
and taking the l imit z l+0,
assuming e12a n d
to be independent
of
temperature.
Th e enthalpic Wilson equation gives
310
Ind. Eng.
Chem.
Fundam.,
Vol. 9 , No. 3 , 1970
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2.:
x
C
-
I*.
1.5
I c
0.5
0.0
12Or-
System
AIZ=
2.2763
A21
I
I6137
\
~ T =
298 K
loo--
I \ I System
I
2 k
-t-
I-Octanol(l)+Water(2)
I /
X I
- XI
Figure
3.
system
Activity and activity coefficients for
1
-octanol (1)-water
(2 )
In Yla,
_c
Figure 4.
Nomogram giving
.211
and
for
enthalpic Wilson equation from In
ylm
and In
y2m
Ind. Eng. Chem. Fundom.,
Vol.
9,
No.
3,
1970
3
1
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The Orye equation yields
l
@ -
o,8-
Rotat ion
of
indices in the sequence
1
- r
2
-
gives similar
relations for the tem peratu re dependence
of
In 7 2 . Figure 5
VAN LA& EQJATIOK
FOR FOUR
TERNARY SYSTEMS
I : + ccetone;
x
methanol;
e
w t e r
I I : acetone ;
li
methanol: ethanol
a:oethanol ; is0 ptopanol;* water
E ethanol ; * methylcyclopentone
; o
benzene
gives the predicted temperature dependence of In ylm a n d
In yz in the system m ethanol (1) + water 2) for t he EVL,
enthalpic Wilson, and Orye equations. Experimental data
from Per ry (1963) are also given.
It
is evident th at the predicted temperature dependence is
poor for all equations. T he enthalpic Wilson equation ap pears
to give results closer to th e experimentally observed values for
the temperature dependence of In ylm than the Orye equation.
It must be admit ted that the tes t
of
predicting temperature
dependence
of
acti vity coefficients is extremely severe. Th e
conclusion is that
~ I Z A l l
and
(A12
-
A l l )
must have some
dependence on temperature.
WILSON EWATIONS FOR FOUR TERNARY SYSTEMS
I
+
acetone
;
x methanol; wter
I I : acetone
;*
methand;. ethonol
m ethanol ; 1%-popanol: water
E:
uethanol
;
ethylcyclopentane: o benzene
I B
0.6
1
Yexp.
0.5
-
0 4
0.3
2.5
3.0
3.5
4 0
Figure
5
Predicted temperature dependence of In yim
for system methanol (1)-water (2)
- ' / T ) X I O ~
( O K - ' )
--
nthalpic
Wilson
equat ion
- *
-
r y e e q u a t io n
Exper imenta l da ta
_ - _ -
VL
equations
0.6
Yexp.
/
0 0 6 .
//
f
*
0
0 2 3 0.4
0.5
0 6 0 7 0 8 0 9 I0
YCOlC
igure 6.
Fit of van Laar equations to ternary VLE da ta
+
/
Ycolc-
Figure 7. Fit of Wilson equation to ternary VLE data
ENTHALRC WILSON EOUATONS
FOR FOUE
TE3hARY
SYTER'S
1: + acetone: x methanol;
e
water
I I : acetone
: r
methanol; ethanol
E: ethanol
;
iso-propanol;
*
water
E
thonol
; *
methylcyclopentane
:
o benzene
0.67:
Yexp.
0.5
-
0.4
-
0 3
0.2
-
6 I I I I I
0 0.1
0.2
0.3 0.4
0.5
0 6
0.7
0.8
0.9
I
kale
igure 8.
data
Fit of enthalpic Wilson equations to ternary VLE
312
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I
I
.9
CRYE
EQUA; OP.IS FOR FOJR TERN ARY
SYSTEMS
I +
acetzne;
x
re tnono ;
e
wo:er
II ccetoie ;
tr
rne inmol ;
ethanol
m
etccnc
: +
is0
D ~ O ~ G C O ~ : +afer
-
O,s-E .ehcnol
; *
rnethylcyclopentane;
o
benzene
0 7t
G C.
G2
0.3
0 4 0 5 06
G 7
0.8
0.9
k a i c
Figure
9.
Fit of Orye equations to ternary VLE data
Multicomponent Systems
Th e EV L equations (17), enthalpic Wilson equations (20),
and Orye equations (21) can be readily extended to multi-
component systems. In fac t, the quasilattice model of Guggen-
heim (1944a, b, 1952) was developed for multicomponent
systems. In the zeroth approxim ation only two-body inter-
act ions are taken into account and therefore th e Gibbs free
energy is obtained by a summ ation procedure.
Th e EV L equation is not discussed here, as it is clear from
Table
V
that this equation gives rather poor results. The
enthalpic Wilson equation for a multicomponent mixture
becomes
25 111
A1 5A j i )
f f 4 j -
x j xm 1n
AjrnArnj)
a i i m
(31)
a j m
Th e Orye equat ions for a mult icomponent m ixture are
In
Y i
= - n
( ,
A i p x p - Z ( q -
Equat>ions31 and 32 were tested on four ternary systems
listed in Tabl e
IX.
Binary VLE da ta were reduced to p aram-
Table
X.
Ternary Systems Used to Test Enthalpic
Wilson and Orye Equations (Isobaric at 1 Atm)
System Reference
1. Acetone-methanol-water
2. hcetone-methanol-
3.
Ethanol-2-propanol-
4. Ethanol-methylcyclo-
Griswold arid Buford, 1 949
hme r , Paxton, and Van
Kojima, Ochi , and Sa kaw aza,
Sinor and W eber, 1960
ethanol Winkle, 1956
water 1969
pentane-benzene
eters A i j a n d A j i (Tables I11 to VI I) , a nd the values obtai r ied
were used in bubble temperature calculatioiis using the
programs described by Prausnitz et
al.
(1967). Pure compo-
nent properties were t .aken from the li terature. Critical
pressures and temperatures were taken from the li terature or
in some cases calculated with Lyderseri correlations (Reid
and Sherwood, 1958). Acentric factors, dipoles, and vapor
pressure data were taken froni Prausnitz
e t nl.
(1967) and
O'Connell and Prausn itz (1964).
In Figures 6 to 9 the computed vapor compositions for a
num ber of liquid compositions for each syst.ern ar e compared
wit,h t,he experim ental compositions
as
reported in the refer-
ences given in Table IX. As a reference, the results for the
van Laar and Wilson equations for the same four ternaries
are also given.
Conclusions are that the enthalpic Wilsori and
t h e
Orye
equations give
a
considerably bette r
fit
of vapor-liquid equilib-
r ium in ternary sys tems than the va n Laa r equat' ions . Th e
van Laar equation gives a good
fit
for t 'he acetone-methanol-
water system, but poor results for the ethanol-methylcyclo-
pentane -benzen e an d ethanol-2-propanol-water syst.ems.
Th e Wilson equation gives a very good fit
of
VLE d a ta for the
ethanolmethylcyclopentane-benzene system. The results for
the ethanol-2-propanol-water syst.ern, however, are appro xi-
mately
of
the same accuracy as t 'he enthalpic Wilson and
Orye equations. Fo r syst 'ems near t o phase separation o r for
systems showing phase separation, the enthalpic or Orye
equations are recommended. For other systems, the Wilson
equat ions are a t t ract ive t o use .
Acknowledgment
Th e author is grateful to the C omputer Center of the
Eind hov en Univ ersity of T echno logy for th e use of its facili-
ties. Especially th e assistance of M arijke t er Mor sche is grate-
fully acknowledged.
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M.,
IND. NG. HEM.FUNDAM.
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O’g R. V., Ph.D. dissertation, University
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F., Znd. Eng. Chem. 44, 1872 1952).
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A.,
Orye, R. V., O’Connell, J. P.,
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RECEIVEDor review December
23, 1968
ACCEPTED
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1970
Work supported by the
L. E. B.
Foundation, W ageningen.
Thermal Reaction of Propylene
Kinetics
Taiseki Kunugi, Tomoya Sakai, Kazuhiko Soma, and Yaichi Sasaki
Department of Synthetic Chemistry, Faculty of Engineering, University of Tokyo, Hongo, Tokyo, Japan
Kinetics of the thermal reaction of propylene was studied at temperatures ranging from 703’ to 854’C.,
atmospheric pressure, and residence times from 0.078 to
3.3
seconds with and without ni trogen dilut ion.
Main primary products were ethylene, methane, hydrogen, butenes, and butadi ene in the approximate
ratio of
5 :
3 : :
1
: 1 at init ial stages of the reaction. Other primary products were methylcyclopqntene,
hexadienes, acetylene, and ethane. Secondary products were cyclopentadiene, benzene, polycyclic
aromatics, cyclopentene, and toluene. Selectivities of formation of these products, except acetylene and
ethane, showed little dependence on temperature. At higher part ial pressure of propylene, the selec-
tivities of ethylene and methane formation decreased to some extent. The effect of parti al pressure
of propylene
on
the r ate o f propylene disappearance leads to a three-halves-order equation. The rate
constant
i s
given as
k =
10’5*06 cc>/z/(mote’/z sec.).
T H E R M A L REACTION of prop ylene has been studie d exten-
sively (Amano and Uchiyama, 1963; Kallend et aZ., 1967;
Laidler and Wojciechowski, 1960; Sakakibara, 1964; Wheeler
and Wood,
1930).
A few experiments
at
t emperatures f rom
700’
t o
850’
and atmospheric pressure cover the condit ions
of th e industrial m anu factu re of olefins an d aromatics by
cracking hydrocarbon feedstocks. Analyses of th e products
have been l imited to l ighter hydrocarbons below
CCor
t he
products formed in narrow ranges of temperatures and con-
versions. Lack of clear discrimination between the primary
and secondary products is due to these l imited analyses of
the products.
Reaction products were analyzed in d etail to differentiate
the prim ary from the secondary products. The kinetics of th e
therm al reaction of prop ylene was discussed in comparison
with tha t of ethylene. In
a
following paper mechanisms
of
t he
reaction and of formation of higher hydrocarbons above
CS
are to be discussed.
Experimental
Feed propylene was 99.35 mole % pure by gas chromato-
graphic analysis, used without furth er purification. T he im-
purity was propane,
0.65
mole yo.Oxygen content was less
t h a n 1 p.p.m. by weight. Commercially available nitrogen
3
4
Ind. Eng. Chem. Fundam.,
Vol. 9
No.
3, 1970