BINARY PHASE DIAGRAMS
FROM A CUBIC EQUATION OF STATE
by
GLENN THOMAS HONG
B.S., State University of New York at Albany
(1976)
SUBMITTED TO THE DEPARTMENT OF
CHEMICAL ENGINEERING IN PARTIAL
FULFILLMENT
OF THE
REQUIREMENTS FOR THE
DEGREE OF
DOCTOR OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
August, 1981
Massachusetts Institqte of Technology 1981
Signature of Author .
,
Department of Ch ical Engineering
August 27, 1981
Certified by
Michael Modell
Thesis Supervisor
Accepted by
Glenn C. Williams
Archives Chairman,
MASSACHUSETTS
INSTITUTE
OF TECHNOLOGY
OCT 2
1981
LIBRARIES
Department Committee
2
BINARY PHASE DIAGRAMS
FROM A CUBIC EQUATION OF STATE
by
GLENN THOMAS HONG
Submitted to the Department of Chemical Engineering
on August 27, 1981 in partial fulfillment of the
requirements for the Degree of Doctor of Science in
Chemical Engineering
ABSTRACT
A methodology is developed for generating binary P-T-x
diagrams over extensive ranges of temperature and pressure.
The Peng-Robinson equation of state is used with a constant
interaction parameter to generate semiquantitative P-T-x
diagrams over the entire fluid range for four binary systems.
The systems considered are n-hexadecane - carbon dioxide,
naphthalene - carbon dioxide, naphthalene - ethylene, and
benzene - water. These systems exhibit various types of
liquid-liquid immiscibilities and critical phenomena.
In
addition, the naphthalene systems involve formation of a
solid phase.
Predictions include two-phase and three-phase
equilibrium, azeotropic points, critical lines, critical end
points, and spinodal curves. Fugacity-composition plots are
shown to be a' valuable
tool
in
phase
equilibrium
calculations,
and for defining stable, metastable, and
unstable equilibria.
Failure to consider metastable and
unstable equilibria can result in erroneous predictions. The
method is useful in the evaluation and design of phase
equilibrium
processes,
especially
those
involving
supercritical fluids.
It is also of value in experimental
studies, since it points out regions of the P-T-x space which
merit detailed attention.
Thesis Supervisor:
Title:
Dr; Michael Modell
Associate Professor of
Chemical Engineering
3
Acknowledgements
Professor Michael Modell is gratefully acknowledged
his
superb
for
guidance throughout the course of this work.
It
has indeed been a pleasure knowing him as a teacher, advisor,
and friend.
Helpful suggestions along the way were
provided
by
Fred Putnam, R.C.Reid, and Costas Vayenas, the members of
my
thesis
committee.
appreciated.
Their
assistance
is
very
much
Jeff Tester provided some much needed editorial
comment down the home stretch, and deserves a special note of
thanks.
My
stay
acquaintance
at
with
MIT
has
many
been
faculty
made
and
enjoyable
fellow grad students,
especially those involved with the Chem E
intramural
I will miss them when my athletic card expires.
from
SUNY
Albany,
through
teams.
My colleague
Frank Bates, has always been the best of
friends, and lent some measure of excitement to an
otherwise
drab existence.
Finally, I would like
encouragement
of
dedicated to them.
my
to
academic
thank
my
family
endeavors.
My deepest love and
for
This
thanks
are
their
thesis is
for
my
wife Karen, without whom this thesis would be handwritten and
minus about 300 diagrams.
Patience is one of her virtues.
4
List of
Tables
6
List of Figures
7
Chapter
1 - Summary
1.1 - Introduction
18
1.2 - Solubility in Supercritical Fluids
18
1.3 - Phase Diagrams
22
1.4 - The Equation of State
26
1.5 - Theoretical Aspects
27
1.6 - The n-Hexadecane - Carbon
Dioxide
34
Dioxide
44
- Ethylene System
52
System
1.7 - The Naphthalene - Carbon
System
1.8 - The Naphthalene
1.9 - The Benzene - Water System
58
1.10 - Conclusions
65
Chapter 2 - Introduction
71
2.1 - Supercritical Fluids
72
2.2 - Phase Diagrams
83
2.3 - Equations of State
137
2.4 - Scope and Context of Present Work
153
Chapter 3
- Theoretical Considerations
157
Chapter 4 - Pure Component Phase Diagrams
Chapter 5 -
The
n-Hexadecane
-
Carbon
212
Dioxide
234
5
System
Chapter 6 -
The
Naphthalene
-
Carbon
Dioxide
286
System
6.1
Chapter 7
- Qualitative Predictions
286
6.2 - Quantitative Predictions
321
- The Naphthalene - Ethylene
System
345
Chapter 8 - The Benzene - Water System
383
Chapter 9 - Conclusions and Recommendations
427
Appendix A
-
Equation
of
State
and
Derived
433
Expressions
Appendix B - Fugacity - Composition Plots
440
Appendix
469
C - Computer Programs
and Documentation
Appendix D - Notation
521
References
523
Biographical Note
532
6
List of Tables
Description
1.1
Physical Constants of Pure Components
19
2.1
Proposed
Supercritical
73
2.2
Physical
Constants
Fluid Proces ses
of
Supercr itical
75
Solvents
2.3
Physical
and
2.4
Properties
Supercritical
Literature
of
Gases,
quids,
Lib
76
Supercr itical
82
Fluids
Sources
for
Water
2.5
Equation of
3.1
Criteria
4.1
Pure
5.1
Physical
State
Comparison
for Phase Diagram
141
Features
211
Component Constants
Constants
213
of
n-Hexadecane
and
235
of
Naphthalene
and
287
Carbon Dioxide
6.1
Physical
Carbon
Constants
Dioxide
6.2
Naphthalene
7.1
Physical
- CO 2
Constants
Critical
of
End
Points
Naphthalene
and
334
346
Ethylene
7.2
Naphthalene
8.1
Physical Constants
- C2H 4
Critical
End
Points
of Benzene and Water
356
384
7
Page
Description
1.1
Experimental solubility map, naphthalene
-
21
miscible
24
carbon dioxide.
1.2
P-T projection for a system
with
liquid phases and immiscible solid phases.
1.3
P-T projection for a system
liquid
phases
and
a
with
miscible
25
supercritical fluid
region.
1.4
fH-xH plot at 300 K and 0.1 atm.
28
1.5
fH-xH plot showing closed loop.
31
1.6
Upper
critical
line
for
n-hexadecane
-
35
line
for
n-hexadecane
-
37
Qualitative T-x section for n-hexadecane
-
38
-
40
carbon
41
carbon dioxide.
1.7
Lower
critical
carbon dioxide.
1.8
carbon dioxide at 70 atm.
1.9
Predicted T-x section
for
n-hexadecane
carbon dioxide at 70 atm.
1.10
T-x
section
dioxide
at
for
70
n-hexadecane
-
atm showing metastable and
unstable binodals.
1.11
f -x plot showing the
H H
monotonic segments.
1.12
P-x
section
for
dioxide at 463.1 K.
numbering
n-hexadecane
-
of
the
42
carbon
43
8
1.13
in
supercritical
46
Solubility of naphthalene in
supercritical
46
supercritical
47
Solubility
of naphthalene
carbon dioxide at 45 C.
1.14
carbon dioxide at 55 C.
1.15
Solubility of naphthalene in
0
carbon
dioxide at 55 C, P-R metastable and
unstable solutions included.
1.16
Solubility of naphthalene in carbon dioxide
at
64.90 C,
and
metastable
P-R
48
unstable
solutions included.
1.17
Predicted solubility map for naphthalene -
50
carbon dioxide.
1.18
P-T projection
for
naphthalene
-
carbon
51
dioxide.
1.19
P-T projection for naphthalene - ethylene.
53
1.20
P-x projection for naphthalene - ethylene.
54
1.21
P-T projection of the lower
line
56
ethylene
at
57
ethylene
at
57
P-T projection of the benzene - water upper
59
critical
for naphthalene - ethylene.
1.22
P-x section for naphthalene 318.85 K.
1.23
P-x section for naphthalene 313.85 K.
1.24
critical line.
1.25
Experimental
lower
critical
line
for
60
-
62
benzene - water.
1.26
Experimental
P-x
section
for
benzene
9
water at 553.15 K.
1.27
Predicted lower critical line for benzene -
63
water.
1.28
Predicted P-x section for benzene
-
water
64
-
78
a
86-87
at 554 K.
2.1
Experimental solubility map, naphthalene
carbon dioxide.
2.2
P-T-x
diagram
and
P-T
for
sections
completely miscible system.
2.3
P-T projections for six types
of
critical
diagram
for
90
curves.
2.4
P-T projection
and
P-T-x
a
92-93
system with a liquid immiscibility dome.
2.5
P-T projection
and
P-T-x
diagram
for
a
95-96
diagram
for
a
97-98
system with a UCST line.
2.6
P-T projection
and
P-T-x
system with immiscibility near the critical
point of the light component.
2.7
P-T projection
and
P-T-x
diagram
for
a
100-101
system with immiscibility near the critical
point of the light component.
2.8
P-T projection
system
whose
and
upper
P-T-x
diagram
critical
line
for
a
102-103
has a
pressure maximum and a pressure minimum.
2.9
P-T projection
system
whose
and
upper
P-T-x
diagram
critical
line
for
a
has a
pressure maximum, a pressure minimum, and a
104-105
10
temperature minimum.
2.10
and
P-T projection
system
whose
P-T-x
upper
diagram
for
line
critical
a
106-107
has a
.pressure minimum and a temperature minimum.
2.11
P-T projection
P-T-x
and
diagram
for
a
108-209
system whose upper critical line has only a
temperature minimum.
2.12
P-T projection
system
and
whose
P-T-x
upper
diagram
critical
for
a
110-11
line has no
maxima or minima.
2.13a
P-x sections for azeotropic behavior in the
113
critical region.
2.13b
Azeotropic P-x and T-x sections.
114
2.14
Different types of azeotropic behavior.
116
2.15
P-T projection
system
with
and
P-T-x
miscible
diagram
liquid
for
phases
a
118-119
and
immiscible solid phases.
121
2.16
P-x sections for 2.15.
2.17
P-T projection for a system
liquid
phases
and
a
with
miscible
123
supercritical fluid
region.
2.18
P-T
P-T-x
projection,
section
for
immiscibility
a
diagram,
system
dome
with
interrupted
and
a
by
P-x
125-127
liquid
solid
formation.
2.19
P-T projection
of
immiscibility
near
a
system
the
with
light
liquid
component
129
11
critical
point
formation.
2.20
interrupted
by
solid
No SCF region.
P-T projection and P-T-x diagram similar to
130-131
2.19, but with an SCF region.
2.21
P-T projection for
critical
a
system
whose
upper
132
line has a pressure maximum and a
pressure minimum,
and
is
interrupted
by
solid formation.
2.22
P-T projection
system
whose
maxima or
and
upper
minima
solid formation.
2.23
P-T-x
and
diagram
critical
is
for
a
133-134
line has no
interrupted
by
An SCF region exists.
P-T projection for a system whose SLF curve
135
has a temperature minimum.
3.1
Stable and unstable critical points.
167
3.2
P-v isotherm for a pure component.
170
3.3
fH-xH plot at 300 K and 0.1 atm.
172
3.4
fH-H
175-178
3.5
fH-xH plot showing metastable states.
181
3.6
fH-XH plot showing closed loop.
184
3.7
fH-XH plots.
187-194
3.8
fH-H
197
3.9
Ll-xH plots.
199-206
4.1
P-v diagrams for pure components.
215-220
4.2
P-T diagrams for pure components.
224-229
4.3
Naphthalene triple point.
232
5.1
Upper critical line for n-hexadecane - CO 2 .
236
plots.
plot showing open loop.
I
12
5.2
Temperature
dependence
interaction
of
237
parameter.
5.3
T-x sections for n-hexadecane - CO 2 showing
239-245
L1=0 and M1=0 curves.
5.4
fH-XH plots
showing
stable
and
unstable
249-251
regions.
2.
253
210
255
at
70
257
at
70
258
5.5
Lower critical line for n-hexadecane - C
5.6
T-x section for n-hexadecane - CO
2
at
atm.
5.7
T-x sections for n-hexadecane - CO2
atm.
5.8a
T-x section for n-hexadecane showing
atm
metastable
CO 2
and
unstable
binodals.
5.8b,c fH-xH plots showing the
numbering
of
the
259-260
monotonic segments.
plot showing the three-phase tieline.
5.8d
fH-x
5.8e
T-x section for n-hexadecane -
H
H
CO 2
261
at
70
262
atm near the CO 2 axis.
5.9
P-x sections
for
n-hexadecane
-
CO 2
at
266-267
for
n-hexadecane
-
CO 2
at
268-269
for
n-hexadecane
-
CO 2
at
271-272
for
n-hexadecane
-
CO 2
at
274
300 K.
5.10
P-x sections
305 K.
5.11
P-x sections
311 K.
5.12
P-x
section
313.5 K.
13
5.13
P-x
section
for
n-hexadecane
-
CO
for
n-hexadecane
-
CO
2
at
275
at
276
at
277
at
278
at
280-281
400 K.
5.14
P-x
section
2
463.1 K.
for
n-hexadecane
-
CO
for
n-hexadecane
-
CO
5.17a,b P-x sections for
n-hexadecane
-
CO
5.15
P-x
section
542.9
5.16
2
K.
P-x
section
2
623.6 K.
650 K.
5.17c,d f -x
H
5.18
P-x
282-283
plots at 650 K.
H
section
for
-
n-hexadecane
at
285
at 75 atm
289
CO
2
663.8
6.1
K.
T-x section for naphthalene - CO
2
showing L1=0 and M=0
curves.
6.2
P-T projections for naphthalene - CO
291-292
6.3
Qualitative P-x sections for naphthalene
294
CO 2
6.4a ,b P-x sections for naphthalene -CO
at 300 K.
299-300
2
6.4c
f -x plot showing metastable states.
N N
6.5a
P-x section
for naphthalene
-
301
at 290 K.
305
CO
at 304 K.
306
CO
2
6.5b
P-x section
for naphthalene 2
6.6
P-x
sections
for
naphthalene
-
CO
at
307-308
2
305.3 K.
6.7
P-x section for naphthalene - CO
at 311 K.
310
2
6.8
P-x
section
for
naphthalene
-
CO
at
2
325.3 K.
312
14
6.9
P-x
at
313-314
at
316
at 360 K.
317
-
naphthalene
for
sections
CO
2
340 K.
6.10
P-x
section
-
naphthalene
for
CO
2
354.5 K.
6.11
P-x section for naphthalene - CO
2
6.12
region
319-324
supercritical
322-324
P-x and T-x projections of the
SLF
for naphthalene - CO .
6.13
Solubility of naphthalene in
CO.
2
6.14
projection
of
the
SLG
region
for
326
Experimental solubility map for naphthalene
327
T-x
naphthalene - CO 2.
6.15
- CO2 .
6.16
Predicted solubility map for naphthalene
-
328
-
330
CO 2'
6.17
T-x section at 65
atm
for
naphthalene
co21
CO 2
6.18
Solubility of naphthalene in CO
near
the
331-332
near
the
336-338
2
UCEP.
6.19
Solubility of naphthalene in CO2
2
UCEP, P-R metastable and unstable solutions
included.
6.20
Solubility of naphthalene in compressed CO 2
341-343
gas.
7.1
supercritical
347-351
P-T projections for naphthalene - ethylene.
353-354
Solubility of naphthalene in
ethylene.
7.2
15
7.3
P-x and T-x projections for
358-362
naphthalene
ethylene.
7.4
P-x sections for naphthalene - ethylene
at
364-366
255 K.
7.5
N-xN plots at 255 K and low pressures.
7.6
P-x sections for naphthalene - ethylene
368-370
at
372-373
ethylene
at
374
ethylene
at
375
ethylene
at
377
ethylene
at
378
ethylene
at
379
ethylene
at
381
ethylene
at
382
line
386
lower
388
lower
390
278.5 K.
7.7
P-x section for naphthalene 283.2 K.
7.8
P-x section for naphthalene 288.7 K.
7.9
P-x section for naphthalene 313.85 K.
7.10
P-x section for naphthalene 318.85 K.
7.11
P-x section for naphthalene 340 K.
7.12
P-x section for naphthalene 354.5 K.
7.13
P-x section for naphthalene 360 K.
8.1
P-T projection of the upper
critical
for benzene - water.
8.2
Experimental P-T projection
of
the
critical line for benzene - water.
8.3
Predicted
P-T
projection
of
the
critical line for benzene - water.
16
8.4
for
sections
Experimental P-x
-
benzene
392-397
water.
8.5
T-x
for
projection
a
heteroazeotropic
401
system.
8.6
P-x
528.15
8.7
P-x
for
sections
benzene
-
water
at
402-403
-
water
at
404
-
water
at
406-407
K.
section
for
benzene
541.45 K.
8.8
P-x
sections
553.15
for
benzene
K.
8.9
P-x section for benzene - water at 554 K.
8.10
P-x
sections
408
for
benzene
-
water
at
409-411
for
benzene
-
water
at
413-414
568.15 K.
8.11
P-x
sections
573.15 K.
8.12
P-x section for benzene - water at 580 K.
8.13
P-x
sections
603.15
for
benzene
-
water
415
at
416-417
K.
8.14
fB-xB plots
B.1
Matrix of fH-xH plots; 7
419-424
at 568.15 K.
temperatures
and
441-468
f-x
470
16 pressures.
C.1
Numbering
of
monotonic
segments
in
plots.
C.3
Subroutine zer.
474
C.4
Main program pure.
477
C.5
Function zrp.
481
C.6
Subroutine Zvalp.
483
17
C.7
Function fp.
486
C.8
Main program pr.
488
C.9
Function zr.
496
C.10
Subroutine Zv.
498
C.11
Function LM.
501
C.12
Subroutine sol.
505
C.13a
Subroutine peq.
508
C.13bc
Phase equilibrium search.
509
C.14
Function f.
517
C.15
Typical printouts.
519-520
18
Chapter 1
1.1.
Summary
ntroduction
of
This thesis is concerned with the generation
phase
binary
diagrams from a cubic equation of state over extensive
Research on
ranges of temperature and pressure.
topic
this
was motivated by the unique solvent properties exhibited by a
substance
pressure, known
fluid
above
somewhat
as
a
technology
its
critical
supercritical
a
is
fluid.
relatively
and
temperature
Supercritical
recent
Effective evaluation and use of the technology
development.
is
aided
by
knowledge of a system's overall phase behavior, and it is the
object
of
acquired
this
for
illustrate
work
to
binary
the
need
show
how such a knowledge may be
systems.
The
for
careful
mathematical criteria which define
systems
covered
are
techniques
application
reference
in
by
a
the
Solubility
the
The
n-hexadecane - C0 2, naphthalene - C0 2,
set
complete
following
of
Each
discussion,
n upercritical
Fluift
has
P-T-x diagrams.
Table
physical constants of the compounds considered.
1.2.
of
equilibrium.
phase
naphthalene - ethylene, and benzene - water.
characterized
developed
1.1
been
For
lists
19
4
3
I
o
O
c4
CS
c
a
or-0
C.
-
u
o
-T
N
_:-r
-T-
0
0
o(N
CN
ll~r'
o
0
'CN
sN
N
o_
o-
o
o
r-
q0
N
N
11 a
Q
t05
0
o
\o
S
cr
N
U
L.
-_E
,a rg=
-.
T
4.
,O
Q)
CL
0
E
43
0
0.
-O
o
rl
a
UVI
aD
.6
U
a
cl
.~*
_
0
s
E
02E l
4)
L.a.
co
-
N
r0\
V0
O
m
cN
m
I
0
m
UN
C,%
r-
N
0
o
0
C."_
-
c-~
N
O
O
>o
0
02
I-
I-
H
U
0
C
-2 0
C
'a
U
0
u-
.
4.4
U-
0f
Eu
N-
E
(O
1*
_
0
'a >
0
0
O
O
0Vl
O
0
o
O
U
4.1
IE
u
U
r,
cc
.-
a.
co
r-
-
_:
o
0
C-
-wT
C
0.
\
'0
U-
_-_
o
O
C9
0
VI-w
cc
N
co
CO
.
I_.
r_-
0
C-f
v
z0
-r
'D
N
0
CI
Q
O0
Q)
)
C
a
C
0
U
>.
-o
C
oJ
N
r
*u
(a
Q0
I
4
-
L-
XC-
-
4)
Sv
r
m
:
I
C
zz
4
-
20
Solubility effects in a
illustrated
by example.
supercritical
fluid
are
best
Figure 1.1 gives a solubility "map"
for the system naphthalene - carbon dioxide near the critical
point of carbon dioxide.
Tracing a particular isobar in this
diagram shows how the concentration
carbon
dioxide
fluid
phase
varies
lines in the diagram
represent
there
pure
is
always
a
of
naphthalene
in
the
with temperature.
The
saturated
solid
solutions,
naphthalene
(naphthalene's normal melting point is 80°C).
i.e.,
phase present
Above about 75
atm an isobar gives the composition of a saturated CO 2
fluid
in
lower
equilibrium
pressure,
with
there
is
solid
a
naphthalene.
temperature
at
For
which
any
three phases
coexist - pure solid, saturated liquid, and saturated
The
composition
vapor.
of the saturated liquid and vapor at 65 atm
are shown in the figure by a tieline; the locus of all
tieline
compositions
dashed line.
for
these
various pressures is given by the
If 'the solid-liquid-vapor (SLV or SLG) locus is
traced to higher temperatures (and pressures),
it
is
found
that the liquid-vapor tielines converge to the critical point
labelled
in
the
liquid
and
diagram.
At this point the near-critical
near-critical
indistinguishable,
or,
in
vapor
other
words,
phases
the
critical phenomenon is observed in the presence
become
vapor-liquid
of
a
solid
phase.
The occurrence of the critical phenomenon between two
phases
in
the
presence
of a third characterizes a special
class of critical points known as critical end
points.
The
critical point in Figure 1.1 is known as a lower critical end
21
T(°C)
- 1
0lo
0
'10
!
I
20
30
40
50
I
I
I
!
)
Liquid
I
10
Critical Point(LEP·
90
10- 3
80
L
70 atm.
/
tie line
/
/
/
J
Figure 1.1.
/
The experimental solubility map for the
naphthalene (N) - carbon dioxide system.
Isobars indicate
the mole fraction of naphthalene in solution.
The dashed
line corresponds to a three-phase solid-liquid-vapor
equilibrium.
This equilibrium terminates at LCEP, the
lower critical end point.
22
point
(LCEP),
a
designation which will become clear in the
following sections.
The effects of temperature and
may
be
observed
in
Figure 1.1.
pressure
on
solubility
Along the 300 atm isobar,
solubility increases continuously with temperature due to the
increasing
activity
of
the
solid
phase.
At
55 C,
the
concentration of naphthalene in the supercritical fluid phase
is
about
5
mole percent, or 15 weight percent.
compares unfavorably with traditional nonpolar
While this
solvents,
it
is nevertheless a significant quantity of naphthalene.
At 80 atm, solubility decreases sharply with temperature
at about 35 C.
slight
increase
the density
of
concentration
power.
This is the near
Moving
temperatures,
critical
region,
where
a
in temperature leads to a large decrease in
the
drops
beyond
fluid.
The
fluid
as
CO 2
loses much of its solvent
the
the
critical
phase
region
naphthalene
toward
higher
the density changes again become small and the
isobar reverts to a positive slope.
Phase Diagram
1.3.
A
full
supercritical
understanding
fluid
of
phase
behavior
in
the
region requires an understanding of its
23
These diagrams may
context, the binary P-T-x space.
a
exhibit
multitude of different shapes, some of them quite complex.
A P-T projection
for
simplest
the
behavior is shown in Figure 1.2.
not
give
type
binary
phase
A diagram of this type does
information
compositional
of
on
the
The
mixture.
figure contains the P-T diagrams of the two pure components A
and B which comprise the system, and these are of
usual
The sublimation, fusion, and vaporization curves for
shape.
each component meet at the triple
curves
the
terminate
at
pure
point.
component
other lines in the diagrams represent
mixtures of A and B.
represents
the
and S LG.
vaporization
critical points.
univariant
mixture
critical
A
to
lines shown, SASG
line.
pure
The
states
The dashed line connecting CPA
along this line from pure
three-phase
The
of
and CPB
Composition varies
B.
There
are
four
S SBL (eutectic line), SALG,
These lines all intersect at the
quadruple
point
B
Q, where the four phases S , G, L, and S
B
A
same temperature and pressure.
For systems where the
liquid
phase
is
very
solubility
low,
all coexist at the
of
solid
B
in
and the temperature of TPB
higher than that of CP , the S LG
A
B
quadruple point is almost pure A.
liquid
phase
near
the
is
the
When this low solubility
persists to temperatures above the critical point of
A,
the
SLG
line cuts the LG critical locus to give a lower critical
end
point (LCEP) as shown in Figure 1.3.
SBLG
line
starting
from
TPB
similarly
The segment of the
intersects
the
24
P
T
Figure
1.2.
The P-T projection for a system in which the liquid
phases are completely miscible, and the solid phases are completely
immiscible.
Triple points are denoted by TP and the quadruple
point by Q.
SA and SB are the pure solids.
25
P
Figure
1.3. The P-T projection of a system in which the gas-liquid
critical line is cut by the SBLG line.
The length of the lower
critical line has been greatly exaggerated.
within a few
C of CPA .
The LCEP is typically
26
critical
line
at
an
upper
critical end point (UCEP).
At
temperatures between the critical end points, the fluid phase
phenomenon
critical
is
not
observed
at
any
pressure.
Consequently,
this region is designated as the supercritical
fluid region.
The critical
a
noted,
lower
critical
point
in
end point.
1.3
Figure
was,
as
An upper critical end
point did not appear in that diagram.
1.4.
f State
The Euation
Generation of phase diagrams of the type
previous
shown
in
the
section requires fugacity expressions for the solid
phase and for a component in a fluid
phase.
The
first
of
these is easily obtained from an equation for the sublimation
pressure
of the 'solid. To represent the gaseous, fluid, and
liquid phases present, the Peng-Robinson
state was used.
compromise
(P-R)
equation
of
This equation was felt to represent the best
between
simplicity and accuracy in attaining the
desired goal.
The P-R equation is
RT
v-b
v(v+b)
a
+ b(v+b
(1.1)
27
It is a modified van der Waals type equation.
b
is
associated
forces
parameter
between molecules.
a = a(T)
and
temperature
are
specified.
may
the
determined
Using
(1.1), an expresssion for the fugacity of a
mixture
with
The equation is cubic
in volume, so that volume may be analytically
pressure
parameter
with the volume occupied by the molecules,
and the temperature dependent
attractive
The
if
equation
component
in
a
be derived and used in the calculation of phase
equilibria.
For a binary system,
parameter, 612
an
empirical
interaction
must often be employed to obtain reasonable
agreement with experimental data.
1.5.Theoretical Aects
Equilibrium of phases requires that
equal
temperature
and
pressure,
the
phases
be
of
and that the fugacity (or
chemical potential) of a given component be the same in every
phase.
It has been
found
that
plots
of
fugacity
versus
composition, at constant temperature and pressure, provide an
excellent
means
of
visualizing
Figure
the
1.4,
criterion
which
is
equal
fugacities.
Consider
n-hexadecane
(subscript H) fugacity versus n-hexadecane mole
fraction for the n-hexadecane -carbon dioxide
temperature
a
of
system,
of 300 K and a pressure of 0.1 atm.
represented by this diagram are a
result
of
plot
of
at
a
The systems
mixing
liquid
28
log fH
V1
X
V2
X
X
log
Figure 1.4. An f-x
xH
plot for n-hexadecane - carbon dioxide.
Segment G is the gaseous root, segment U is the mechanically
unstable root, and segment L is the liquid root. Segments U
and L start at the slope infinity and continue to the pure
hexadecane axis.
29
n-C 1 6
with gaseous C
are
components
2,
present
and VLE is to be expected if the two
in
The most striking feature of
actually
separate
two
a certain range of proportions.
the
plot
is
Segment
curves.
there
that
are
is essentially
G
linear over the entire range of composition, with a slope
and
unity
an
intercept
the pure C16
at
of
axis of 0.1 atm.
This is the equivalent of ideal gas behavior, since
fH
where PH
(1.2)
HP =PH
is the partial pressure of C16.
when the
appears
cubic equation
The
second
gives
(1.4)
curve
three
real
df
In the composition range from the infinity in
roots.
xH= 1, segment G corresponds to
the
volume
high
dxH
to
(gaseous)
segment L to the low volume (liquid) root, and segment
root,
U to the middle volume root.
Between the slope infinity and
df
x
man
,
-
dxH
for
liquid
phase is
range.
For xH
segment L is
materially
in
unstable
greater than xmin, dx H
represents a stable or metastable liquid
appears
to
be
indicating that the
negative,
this
and
phase.
materially stable in this plot.
middle volume root for pure component
systems,
composition
segment
L
Segment
U
As with the
however,
violates the criterion of mechanical stability (i.e.,
it
30
(p)
not
does
thus
and
> 0)
stable
represent
As indicated by the crossing of segments G and U
conditions.
in the graph, the middle volume root does not always give the
middle fugacity root.
of
Suppose now that suitable quantities
C
and
are
compositions x
that xL
that
the
in
present
system
so
is
fH
H
that two phases exist with
and x , and fugacity fH
between
The f-x plot shows
furthermore,
and xH = 1, and
must lie between xmin
f H (xmin)
min
CO
2
16
and
fH(XH
H H = 1).
For
straight
coexisting vapor phase, which must lie on the
the
line
segment, equality of fugacities requires that
V
<
(1.3)
<
This limiting procedure illustrates the usefulness of the f-x
plots in determining two phase equilibrium, and is one of the
steps in the phase equilibrium algorithm
To
thesis.
course
this
the actual compositions for VLE, it is of
find
necessary
in
developed
to
match
partial
fugacities
of
carbon
dioxide as well.
Fugacity-composition plots may have
than
that
given
in
many
shapes
other
Figure 1.5 depicts an f-x
Figure 1.4.
curve with a closed loop, chosen to illustrate a special case
of
phase
speaking,
stability
if
one
discovered
starts
with
in
this
a
stable
work.
Generally
system and moves
31
f
Un
2
0
x
0
.w~
l.0010
.0015
.0020
XH
Figure
1.5.
A Cartesian plot illustrating the
closed loop
behavior of the fH- xH curve at conditions
near CO2's vapor
pressure.
.0025
32
continuously toward the limit of stability, the highest order
stability criterion will be the first (or among the first) to
be violated.
For binary systems, this means
system
violate
will
the
criterion
that
a
stable
material stability
of
before the criterion of mechanical stability, so that
only
the
former
is
ascertain phase
to
sufficient
to
stability.
This is shown in Figure 1.5 by tracing the stable
gaseous
root
stability
check
it
increasing x.
with
criterion is
At fmax'
dfH
as dx
dxH
violated
the
becomes
material
negative.
The mechanical stability criterion is not violated until
A special case
composition of the slope infinity is reached.
arises,
however,
the stable gaseous or liquid root is
when
Since all three roots join at the
traced with decreasing xH .
origin, it is possible to follow segment G or
and
then
proceed
with
increasing xH
L
to
criterion
a
material
without
violation
of
xH = 0,
along segment U.
this instance, the mechanical stability
violated
the
the
has
In
been
stability
criterion.
In light of this possibility, a discontinuity in
df
dH
dxH
be
must
criterion.
treated
as
Upon tracing
discontinuity,
it
is
a
a
violation
stable
necessary
criteria, not just the one of
of
system
to
highest
the
stability
through
such
a
evaluate all stability
order,
to
determine
phase stability.
Fugacity-composition
plots
may
be
used
in
the
I
33
determination
of critical points, since an f-x plot exhibits
a
inflection
horizontal
convenient,
however,
this purpose.
The
at
such
a
point.
It
is
more
to evaluate two alternate criteria for
first
criterion
of
criticality,
which
corresponds to the material stability criterion, is given by:
Axx
xv
2
L1 = A
A x
Ax
XV AVV
A
A
the
f-x
equivalent to the
plot.
dfH
H
dXH
reason,
it
is
criterion is not strictly
criterion since
best
i.e.,
Spinodal points correspond to
The L1
differ in sign in a region of
this
(1.4)
At the limit of stability,
spinodal curve, Ll=O.
extrema in an
> 0
XV
For a stable system, L>O.
on
- A
these two
mechanical
to
treat
quantities
instability.
For
material stability as
undefined in a region of mechanical instability.
The
second
criterion of criticality is
AVV AXV
Ml=a1 Ml
Ll aL
av
2
L=A
AXXXAVV
A VVVXXXV
- AvvvAxxAxv
3A VVXVVXX
Avv
xvAvxx
ax
2
+ 2Ax + AwAvvxAxx-
Points of M=0
plot.
correspond to
inflection
At a critical point, both L1 and M
(1.5)
points
in
an
equal zero.
f-x
This
34
criterion may be replaced by the
work has shown that the M=0
in critical point
= 0
aL1
criterion
determinations.
T,P
This must be so because a critical point is a terminus
a
As the critical temperature and pressure
surface.
spinodal
of
are approached, two points of L1=0 may be brought arbitrarily
point,
At the critical
close to one another in composition.
the two L1 zeroes coincide (see Figure 1.10).
1.6. The n-Hxadecane - Carbon Dioxide System
The considerations of the
overall
determine
to
possible
sections
preceding
behavior
phase
systems, provided sufficient data is available to
choice
of
C 1 6 - CO 2
an
appropriate
line
and
(using the L1=0
effect
of
generated
M1=0
experimental upper critical line
point
of
rises
C 1 6,
pressure minimum,
pressures
612
and
and
to
a
temperatures.
critical
values of 612
the
on the predictions.
The
and
starts
pressure
thereafter
For the
illustrates
criteria),
parameter
this
the
Figure 1.6 shows the
various
for
it
binary
allow
system, P-T measurements along the upper
critical
marked
in
parameter.
interaction
line provide the requisite information.
upper
make
runs
from
critical
the
maximum,
steeply
falls to a
to
higher
On the basis of this curve, a
of 0.081 was selected as giving the
best
overall
fit.
35
C1 6 H 3 4 - CO2
40
30
E
O-
20
10
300
400
500
600
700
T, K
Figure
1.6. The P-T projection for the C1 6 -C0 2 system showing
the effect of 612 on the predicted upper critical line.
pure C
2
The
critical point has also been included, although it is
not a part of this line.
36
No solids form in this pressure and temperature regime.
Figure 1.6 includes the critical
serves
as
a
starting
point
for
point
The
C02 ,
which
the lower critical line.
Figure 1.7 is an enlargement of the P-T
region.
of
projection
in
this
lower critical line intersects the three-phase
L1 L2 G line at an LCEP analogous to that shown in Figures
and
1.3.
1.1
Now, however, the liquid-vapor critical phenomenon
occurs in
critical
the
presence
line
of
continues
a
second
past
liquid
phase.
the
LCEP
in
a
fashion, until being terminated at a
cusp
by
the
critical line.
The
metastable
unstable
The unstable critical line is the terminus of
an unstable binodal surface.
The three-phase line
noting
the
intersection
temperature
or
qualitatively
basis of the
regions
in
of
the
shape
measured
intersect
G-L 2
critical
terminates
region
line.
at
the
a
boiling point of C2
off
G-L 1
CO2
predicted
at
1.8
a
by
given
illustrates
of the T-x section expected on the
loci.
common
at
a
region,
vapor
at 70 atm).
The
three
temperature
At the
terminates
The
is
curves
Figure
critical
at
1.7
binodal
pressure.
three-phase L 1 -L 2 -G tieline.
the
Figure
high
give
temperature
point
on
to
binodal
The L1-L 2
end,
along the upper
the
pressure
a
other
point
hand,
(i.e., the
region
extends
the plot to lower temperatures, where it will eventually
terminate due to solid formation.
37
85
80
P(atm)
75
70
65
300
305
310
315
T(K)
Figure
C 16 -C02 .
1.7. The P-T projection of the lower critical line for
The stable critical line is metastable between LCEP
and the cusp.
38
P
XH
Figure
1.8. A qualitative T-x section for C16-C02 at 70 atm.
39
Figure 1.9 gives the P-R prediction for the T-x
section
The concentration
at temperatures near the three-phase line.
axis
is given on a logarithmic scale to allow the very small
G-L 1
region
at
predicted
be
to
Although
304 K.
qualitatively correct, it
equation
Three-phase
visible.
is
equilibrium
predicted
the
incomplete,
diagram
the
because
is
is
P-R
generates metastable and unstable phase equilibrium
solutions in addition
to
the
stable
solutions
which
are
shown.
Figure
equilibrium
1.10
gives
the
solutions
the
from
section
T-x
phase
P-R equation are drawn in.
which
The binodal curves are labelled by number pairs
to
all
when
refer
particular monotonic segments of the f-x curve from which
Figure 1.11 gives a typical f-x plot
the equilibrium arises.
for this
the
region' showing
of
Comparison
segments.
the
Figure
region
G-L1
1 2
portion of the
bounded
by
stable
(1,5)
binodals
two spinodal curves.
roots
to
the
1.9
Figure
the
(1,7)
Note that a significant
falls
cubic.
in
a
region
Stable solutions appear in
this nominally unstable region because of
multiple
with
monotonic
by the (1,5) binodals, and the
region by the (5,7) binodals.
L -L
1.10
the
region is represented by
indicates that the G-L2
binodals,
of
numbering
the
existence
of
Each of the stable binodals
(1,5), (1,7), and (5,7) continue in a metastable fashion past
the
three-phase
line
to
reach
a
maximum
or
minimum
40
306
T,K
304
302
.00001
.0001
.01
.001
.1
1
XH
Figure 1.9.
The predicted T-x section for C16-C02 at 70 atm, in the
vicinity of the three-phase line.
41
P
70 atm
307
306
305
304
303
302
301
. OD0I
.0001
.001
.01
.1
x
H
Figure
1.10.The T-x section at 70 atm showing the continuation of
the predicted binodal curves.
Each binodal is labelled with a
number pair to indicate which segments of the f-x curve the
equilibrium is derived from.
To determine the compositions of
coexisting phases, a tieline is drawn at constant temperature
joining binodals labelled with the same number pair.
Note that the
three-phase tieline (dashed) joins the two curves labelled
the two curves labelled (1,5), and the two curves
The
labelled (5,7).
spinodal curve (Ll=0) is given by the heavy lines. At
/ a(L1=0)
unstable critical point, both
ax
)
and
T,P
(1,7),
-
ax TP
the
are zero.
42
-3
-4
-5
-6
-7
-8
-9
0.00001
0.0001
0.001
0.01
0.1
1
XH
Figure 1.11.
An f-x plot in the three-phase region, showing the
numbering of the seven monotonic segments.
0
43
E
4-
QO
v
0.00001
0.0001
Figure 1.12.
0.001
0.01
0.1
The P-x section at 463.1 K.
I
44
temperature.
At
a
locus.
binodal joins an unstable binodal
separate
curves
continuous
in
are
There
three
The first two
1.9.
Figure
metastable
the
extremum,
temperature
follow the course (1,7)-(3,7)-(4,7)-(5,7), and the third
course (1,5)-(1,6)-(3,6)-(4,6)-(4,6)-(3,6)-(1,6)-(1,5).
last
the
This
curve exhibits an unstable critical point where the two
(4,6) branches join.
Reference to
Figure
shows
1.11
that
both segments 4 and 6 are materially unstable.
A small amount of low pressure P-T-x data
for
the
C 1 6 - C0 2
comparison
612 = 0.081.
system.
between
On
this
Figure
data
1.12
and
is
available
gives a typical
predictions
with
the basis of the lean component, errors in
the C02-rich phase are less than fifteen percent, while those
On
-rich phase are less than twenty-five percent.
16
basis of the rich component, the errors are less than 1
in the C
the
and 5 percent, respectively.
considering
that
no
is
This agreement
composition
data
was
quite
used
good
in
the
adds
the
determination of the interaction parameter.
1.7.
The Naphthalene - Carbon Dioxide System
The naphthalene (N) - carbon
complication
of
dioxide
system
pure solid formation to the phase diagrams.
45
Predictions were carried out using an
determined
data in the supercritical fluid
solubility
from
parameter
interaction
region, the same data used in the construction of Figure 1.1.
data,
Figure 1.13 gives some of the supercritical solubility
shows
and
parameter.
four
for
predictions
values of the interaction
The predicted curves all have the correct general
shape and for the most part are within a factor of two of the
It is evident that a pressure dependent
experimental values.
interaction parameter is needed for
Figure
1.14
experiment
at
agreement.
quantitative
shows
the
comparison
550C.
The
three
of
lowest
predictions
the
of
values
and
interaction parameter predict a sudden jump in the solubility
of
naphthalene
measurements.
equation
show a large discrepancy with the
thus
and
The solubility jump occurs
predicts
a
phase transition.
of 612 - 0.11.
The
solubility
correspond to the S-L-F tieline, as
shifts
the
from solid-fluid to solid-liquid.
P-R
Figure 1.15 gives a
more complete picture of the P-R predictions at
value
the
because
jump
55 C
is
for
seen
equilibrium
a
to
system
Note that the data
are well correlated by the metastable S-F line
to
pressures
over 300 atm.
Figure
The solubility jump of
1.14
is
experimentally
above 550 C.
Figure 1.16 compares
predictions and measurements at 64.90 C.
Data taken after the
observed
at
temperatures
solubility jump exhibited a great deal of scatter due to
presence
of
more
than
one
phase in the CO 2
the
stream being
46
log x N
10n
250
seo
]o
P (atm)
Figure
1.13.
The solubility of naphthalene in supercritical
0
CO2 at 45 c.
log
xH
P (atm)
Figure 1.14.
CO2 at 550 C.
The solubility of naphthalene in supercritical
For
12'
0.09,
0.10,
is predicted by the P-R equation.
and 0.11, a solubility jump
47
a)
4
U-'
U'
1I.
O
A
U)
0C
.
D
C
--
o/
rlI
m
Cb
CY
054
.4
OUN
N
0
O
rn
Q-U)
II
)
54
-
L
I
I
I
0
o
Hl
I
_
N
48
0
0cq
CD
c
-o
c
El
e4-
0
-o
-0
el
7a)
L0
r
I
Nr
r
I
t~3
0
Hl
I
!
I
49
measured; the three points shown represent a maximum
observed
solubilities.
Nevertheless,
they
of
the
probably still
represent a mixture of stable saturated liquid and metastable
unsaturated liquid.
The true liquid phase solubility
should
thus be higher, and quite close to the predicted value.
With the same interaction parameter, the
generates
equation
the solubility map shown in Figure 1.17 This is to
be compared with Figure
phase
P-R
behavior
is excellent.
1.1.
The
qualitative
picture
in the S-L-G and supercritical fluid regions
The P-R prediction
of
an
S-L-F
three-phase
region terminated by an upper critical end point (cf.
1.3)
of
has been included.
Figure
The UCEP pressure and mole fraction
of naphthalene are both much higher than for the LCEP.
Figure 1.18 shows the naphthalene - CO 2
which
is
to
be
compared
with
P-T projection,
Figure
1.3.
For
naphthalene - CO2, the critical line is tending toward higher
pressures
at
the
UCEP.
This indicates that, even had the
solid phase not precipitated out, liquid-liquid immiscibility
would preclude the occurrence of a continuous
between
the
two
pure component critical points.
shows the course of the lower critical
terminates
Figures
available
at
1.1
the
and
over
LCEP.
line
of
1.18.
line
The inset
(dashed)
which
This is the LCEP which appears in
1.17. Unfortunately,
most
covered by Figure
critical
there
is
no
data
the pressure and temperature range
To
allow
some
judgment
of
its
50
T (K)
280
290
300
310
320
330
340
350
360
0
-1
-2
log
XN
-3
-4
-5
Figure 1.17.
The predicted solubility map for naphthalene - CO2 .
51
800
700
600
500
400
300
200
100
n
300
400
500
600
700
T(°K)
Figure 1.18. The P-T projection for the naphthalene - carbon
dioxide system.
The range of temperatures between the LCEP
and the UCEP is the supercritical fluid region.
52
accuracy, the naphthalene-ethylene system was studied.
- Ethvlene System
The Nahthalene
1.8.
is
data
Considerably more experimental
for
available
naphthalene - ethylene than for naphthalene - CO2 , allowing a
of the correctness of the predicted P-T-x
better
evaluation
space.
In order to draw an analogy between
for
naphthalene - ethylene
the
predictions
and those for naphthalene - CO2 ,
the interaction parameter for this system was selected on the
612
of
A value
basis of data in the supercritical fluid region.
= 0 is found to give predictions and agreement similar to
range
entire
of
naphthalene - CO2
line
is
tending
and
temperatures
system, at the
toward
quite
is
agreement
shows that qualitative
liquid-liquid
be
The qualitative
present.
indication
of
the
Figure 1.20, the
shows
the
the
UCEP
= 0,
over
the
Unlike the
upper
critical
As shown by the
pressures.
lower
again
there
however,
immiscibility if the solid were not
of
accuracy
projection
semiquantitative
experimental phase compositions
line and the L-F critical line.
for
of
is
Figure
an
1.18.
naphthalene-ethylene,
agreement
for
1.19
Figure
accuracy
qualitative
P-x
good
pressures.
metastable continuation of this line,
would
612
Figure 1.19, generated with
1.13 and 1.14.
Figures
the
of
predicted
S-L-F
and
three-phase
53
,
JI
200
-.
100
300
400
500
600
700
T,K
Figure 1.19. The P-T projection of the naphthalene - ethylene
system.
The inset shows the region near ethylene's critical
point (see also Figure 1.21).
Welie and Diepen (1961).
Experimental curves are from van
54
300
200
P (atm)
100
CPN
TPN
0
.1
.2
.3
.4
.6
.5
.7
.8
.9
1.0
XN
Figure 1.20.
The P-x projection of the upper critical line and the
SLF region for naphthalene - ethylene.
55
Figure
1.21
an
is
P-T
naphthalene-ethylene
projection
in
the
vicinity
terminated
the unstable critical line at a cusp.
meets
also depicts the
metastable
critical
The
line.
of
C02 , the lower critical line
As with C6-
starts at the solvent critical point and is
it
the
of
Agreement with experimental S-L-G
ethylene's critical point.
data is excellent.
enlargement
LLG
metastable
line
and
LLG
line
when
The diagram
metastable
L-F
terminates at a
metastable CEP at either end.
Figure 1.22 illustrates a P-x section for this system at
a temperature
point.
The
the
between
S-L-F
solid-liquid
and
and
line
in
Figure
1.19.
the
S-F
At
The L-F binodal
but
is
except
at
inflection.
This is shown in Figure
in
binodal
the
the
point
S-F
for
intersects
the
critical
a
This
1.23.
is
true
of
horizontal
The
horizontal
stable binodal is a necessary and
of
a
critical
end
not only for SLF or SLG critical end
points, but also for LLG critical end points in the
C
16
system.
the
which
point,
point
sufficient criterion for the occurrence
point.
below
metastable
the UCEP the L-F binodal falls entirely within
binodalt
inflection
the
binodals are part of a single
solid-fluid
has a stable critical point,
that
Note
curve with metastable and unstable portions.
tieline.
triple
naphthalene's
tieline corresponds to a point along the
S-L-F
three-phase
UCEP
- CO
2
56
Mu
50
40
Pi
30
20
250
260
270
280
T(K)
Figure 1.21.
The P-T projection of the naphthalene - ethylene
system in the vicinity of ethylene's critical point.
290
57
P(atm)
Figure 1.22. A P-x section between the UCEP and naphthalene's
triple point.
P(atm)
XN
Figure 1.23. The P-x section at the upper critical end point
temperature (approximately). The S+F binodal exhibits a point
of horizontal inflection.
58
The presence of the LLG line
in
Figure
1.21
and
binodals in Figure 1.22 emphasize the fact that
liquid-fluid
the P-R equation predicts a complete fluid phase diagram
the
the
system,
does
and
for
not indicate the formation of solid.
Solid phase equilibrium lines only result when the expression
and
for solid fugacity is introduced,
these
lines
are
in
essence superimposed on the fluid phase diagram.
1.9.
The Benzene
ater System
-
An interaction parameter for the benzene - water
was
determined
matching the P-T coordinates of a single
by
upper
point along the experimental
1.24
shows
12;
values of
the
critical
the
line
critical
line.
Figure
predicted for two different
line
the value chosen
critical
upper
system
was
apparently
Although
= 0.052.
612
exhibits
behavior, some qualitative discrepancies are
the correct
encountered
in
the region of the lower critical line.
Figure 1.25 shows the
the
vicinity of benzene's
experimental
P-T
critical point.
benzene
The
critical
in
projection
lower
point,
critical
line
begins
at
reaches a minimum temperature, and
shortly thereafter intersects the LLG line at a critical
point.
The
continuation
dashed
of
the
the
line
critical
represents
line.
This
the
end
metastable
"hypothetical"
59
-
800" -
700.
b
C
H6 -
H2 0
600,
500
P(atm)
400.
300.
H2 0
200.
2-.052
100- I
_
0critical
n
.
.
_
O Alwani
C6 H6
.
point
_
VZ
_
560
570
580
590
600
610
620
630
640
T(K)
Figure 1.24. The P-T projection of the upper critical locus for
benzene - water.
The predicted curves are actually two separate
segments (see Figures 125
and 1,27),
60
a)
0
o
ra f
Um
0
0
*H
-4
-
4
cJ
r0
0 O
)
0
U
H
I
)
P4
N
H
a)
o
,
*,
,-*H
a o
*r
4 4
O~
a)
a)
a
0
CN
04)
*H =
II
I
P
*
H
61
line
critical
has
not
actually
to
represent
system,
this
the
of
intersection
line
this
1.26,
Figure
in
phases
three
such,
is
binodal
two
1.8.
regions, rather than three binodal regions as in Figure
As
a
of the LLG line past the LCEP has
As shown
been determined.
For
measured.
been
continuation
"metastable"
reported
been suggested by experimental work, but
has
not coexist along this line.
would
However, the line is continuous with the stable LLG line, and
the
represents
course
that
line
line
LLG
metastable
point,
simultaneously
the L-L and L-G critical lines.
The transition
terminates at a critical solution end
intersecting
critical
the
had
The
intervened.
not
phenomenon
of
between the L-L and L-G critical
lines
has
been
drawn
as
smooth, although experimental measurements indicate that this
is
not
actually
true.
No measurements of the L-L critical
line are available in this region
to
indicate
its
correct
shape, however.
Figure 1.27 gives the calculated P-T projection
vicinity
of
benzene's
starting from the pure
critical
water
point.
critical
in
the
The critical curve
point
(cf.
Figure
1.24) does not intersect the liquid-liquid critical line, but
rather
reaches
a minimum in temperature and then returns to
the benzene critical point.
on
the
other
hand
reaches
The liquid-liquid critical
a
minimum in pressure shortly
before being terminated by an unstable
figure
also
line
critical
line.
The
shows the three-phase LLG line which terminates
62
P (atm
0
10
30
20
40
50
60
70
80
90
100
wt % H20
Figure 1.26.
A P-x section between the LCEP and benzene's
critical point.
63
%4
0
o
.)
al
-,4
G;
o
.)
Ln
N
ia
ID
ur
4Z
O
u
o
U
4
0
Pi
o
&
0
0*
n UE
o
LA
D
0
ea
4.
o
-4
'O
0
S.)
04
X
0 v-
.
4.)
2.
64
130
120
110
100
90
P(atm)
80
70
60
50
40
0
10
20
30
40
50
60
70
80
90
wt ; H20
2
Figure 1.28. A P-x section slightly above the temperature
minimum in the G-L critical line of Figure 1.27.
100
65
at a critical end point on the LL critical line.
short
The
segment of the LL critical line between the CEP and the point
of
the
the unstable critical line is joined is
where
cusp
metastable.
The experimental and predicted P-T projections differ in
that the former depicts a metastable portion of the LLG line,
of
corresponding to the intersection
binodal
two
regions.
However, data were taken on a fairly coarse grid, and a third
undetected.
very
have
gone
The equation of state predictions indicate
that
of
region
binodal
limited
are
measurements
careful
extent
well
may
to determine the
necessary
stability or metastability of the LLG line up to the critical
solution end point.
At present it is
not
possible
to
say
which of the two P-T projections is correct.
Figure 1.28 'gives a
system,
at
a
predicted
temperature
slightly
P-x
above
minimum in the G-L critical line of Figure
prediction
of
a stable azeotrope.
1.10.
Conclusions
for
this
the temperature
1.27.
Note
the
Although the experiments
were not detailed enough to verify this, a
necessarily exists in this system.
section
stable
azeotrope
66
Knowledge of the overall phase behavior of systems is of
importance in process design, since it indicates
conditions
a
out.
carried
equilibrium
phase
For
example,
under
what
is most favorably
process
extraction
supercritical
of
naphthalene with ethylene should probably be carried out near
the
UCEP
rather
the
than
LCEP.
Although
the pressures
required are 3.5 times as high (174 atm versus 51
of
solubility
naphthalene
0.002 mole fraction).
the
is 85 times as high (0.17 versus
it
As a further example, suppose
was
The shape of the upper
desired to dissolve benzene in water.
line for this system shows that conditions of about
critical
300 C
atm),
and
170
atm
are
to
sufficient
yield
complete
miscibility of these two substances.
In developing a methodology
diagrams,
a
number
of
to
predict
significant
overall
P-T-x
results were obtained.
These are as follows:
-Fugacity-composition plots provide an excellent means of
representing
metastable
the
phase
possible
types
of
stable
and
equilibrium at a given temperature
and pressure.
point
-If the material stability criterion reaches a
discontinuity
that
point
stability.
in
may
This
the
tracking
represent
is
true
a
of
of a stable system,
limit
even
if
of
mechanical
the
material
67
either
stability criterion is satisfied on
of
side
the discontinuity.
in
-Material stability should be considered undefined
of
region
since alternate
instability,
mechanical
criterion
stability
statements of the material
a
may
give conflicting information in such a region.
-When the cubic equation has multiple roots, P-x and
sections
by
bounded
region
stable states within a
existence of
the
indicate
correctly
may
T-x
spinodal
curves.
-In critical point determinations, the criterion Ml=O may
be replaced by the alternate criterion
aLl\
T,P
0.
-An unstable critical point
terminus
of
an
may
be
unstable
identified
binodal
TP
Alternatively, ( x )is
as
the
locus.
greater than zero for
a stable critical point and less
than
zero
for
an
unstable critical point.
-A necessary and sufficient criterion for
existence
the
point
is
the
occurrence
horizontal inflection point
in
the
stable
of
a
critical
end
of a
binodal
locus in a P-x or T-x section.
-Solution
of
sufficient
the
two-phase
for
equilibrium, since
the
problem
equilibrium
determination
three-phase
of
is
three-phase
equilibrium
results
68
from the intersection of binodal loci.
From the systems investigated, the following conclusions
have been drawn:
-The Peng-Robinson
with
equation,
constant
single,
a
parameter, gives a qualitatively correct
interaction
representation of
the
over
behavior
phase
entire
The system
fluid region for nonpolar binary systems.
may be nonideal with respect to molecular size of the
components.
-The Peng-Robinson
representation of
fluid
region
component.
for
one
with
systems
binary
entire
the
over
behavior
phase
constant
single,
reasonably correct
a
gives
parameter,
interaction
a
with
equation,
polar
incorrect
The qualitative picture may be
in certain minor details.
by
interest
of
region
supercritical
fluid
an
choosing
is
region
any
appropriate
particular,
In
parameter.
interaction
in
obtained
be
may
-Semiquantitative predictions
the
to
such
allows
the
amenable
treatment.
-Analysis
of
fugacity-composition
development
plots
of phase equilibrium algorithms suitable
for any region of the phase diagram.
-Metastable
and
equilibrium
unstable
equations
solutions
are
always
to
the
associated
phase
with
69
three-phase equilibrium
solutions
in
a
system.
binary
The
may exist over a substantial region of the
P-T-x space, and if not accounted
for
can
lead
to
conclusions
for
computer
must
be
strongly
erroneous predictions.
The significance of the last
of
calculations
emphasized.
behavior
phase
the
equilibrium
which
Algorithms
of
two
fugacity
fail
to
for
account
functionality
are
the
liable
to
experience convergence problems when this functionality takes
on a complex shape.
regions,
as
well
Complex shapes are found in
as
near
vapor pressure curves.
three-phase
azeotropic coordinates and pure
Similarly, algorithms which
fail
to
account for all phase equilibrium solutions to an equation of
state
may
converge
to
an
prediction of both binodal
occur.
Careful
curves
development
algorithms
equilibrium
incorrect
is
and
and
result.
critical
application
necessary
to
Erroneous
lines
of
may
phase
avoid
these
difficulties.
Taken as a whole, the above conclusions indicate that
simple,
cubic
equation
of
state
is
well
suited
to
orienting or preliminary study of binary phase behavior.
a
an
The
minimal computing time required by these equations makes them
an excellent choice for such an application.
The preliminary
study maps out regions where more detailed investigation,
both
the
theoretical
in
and experimental sense, is warranted.
70
From the theoretical standpoint, once these regions have been
defined, it may in some cases be desirable
accurate
more
of state.
equation
to
switch
the
interaction
parameter.
a
In other cases the cubic
equation will provide sufficient accuracy with only a
in
to
From
the
change
experimental
standpoint, the preliminary study indicates the regions where
further data should be taken, and how carefully
taken.
It
also
serves
experimental results.
as
a
it
need
be
check on the consistency of
71
Chapter 2
Introdction
This thesis is concerned with the generation
phase
of
binary
diagrams from a cubic equation of state over extensive
ranges of temperature and pressure.
Research on
this
topic
was motivated by the unique solvent properties exhibited by a
substance
somewhat
pressure, known
fluid
as
technology
above
a
its
critical
supercritical
is
a
temperature
fluid.
relatively
Supercritical
recent
development.
Effective evaluation and use of the technology
knowledge
of
a
system's
overall
the
diagrams,
is
aided
phase behavior.
generation of phase diagrams requires firstly
with
and
a
by
The
familiarity
and secondly, a mathematical model, the
equation of state, as a means of
prediction.
This
chapter
includes a survey of these topics.
Chapter 2 considers
equilibrium
predictions,
applied to pure component
present
the
the
theoretical
basis
for
phase
and Chapter 3 illustrates these as
systems.
Chapters
4
through
7
results of this work, calculated phase diagrams
covering the complete range of fluid phase behavior for
binary systems.
four
Chapter 8 sums up the conclusions drawn from
these results, and gives recommendations for future work.
72
2.1
Supercritical Fluids
The first published observation
solids
in
supercritical
Hogarth in 1879.
and
they
to
Geochemists,
was
the
given
by
noted
changes
the
in
concerned
high
temperature
and
to
realize
phase behavior.
1912,
and
solutes
of
the
presssure.
with the transport of minerals below
probably
the
the practical importance of supercritical
Niggli discussed
by
of
Hannay and
sensitivity
the earth's surface by pressurized steam, were
first
solubility
Working with a number of different
solvents,
solubility
fluids
of
this
topic
as
early
as
the 1930s, considerable experimental work had
been undertaken.
Table 2.1 lists
fluids
which
industries.
are
some
of
proposed
interest
Messmore (1943)
was
to
uses
of
supercritical
the chemical processing
apparently
the
first
to
suggest an engineering application of supercritical solvents,
a process for the removal of heavy components from crude oil.
Since
that
time
a
number
of
other
supercritical
processes have been proposed, but only within
decades
have
the
first
past
two
these proposals been economically practicable.
A milestone was reached in June of 1978, with
of
the
fluid
symposium
on
fluids (Wilke, 1978).
That
commercialization
a
of
extraction
year
also
the
convening
with supercritical
saw
the
industrial
coffee decaffeination process using
supercritical carbon dioxide, a process discovered
by
Zosel
73
Table 2.1
Proposed Supercritical Fluid Processes
Process
Supercritical Solvent
Removal of asphaltenes
Hydrocarbons
Reference
Messmore (1943)
from oil
Removal of asphaltenes
Propane
from oil
Extraction of lanolin
from wool grease
Zhuse & Kapelyushnikov
(1955)
Hydroca rbons
Zhuse, Yushkevic, &
Gekker (1958)
Supercritical fluid
chromatography
Carbon Dioxide
Sie, et al. (1966)
Decaffeination of
coffee
Carbon Dioxide
Zosel (1970)
Extraction of coal
Toluene
Wise (1970)
Carbon Dioxide
Enhanced oil recovery
Huang & Tracht (1974)
Supercritical fluid
chromatography
Various solvents, e.g.
Pentane, Isopropanol
Removal of nicotine
from tobacco
Carbon Dioxide
Hubert & Vitzthum (1978)
Extraction of 'flavor
components from hops,
spices & cocoa
Carbon Dioxide
Hubert & Vitzthum (1978)
Extraction of oil seeds
Carbon Dioxide
Hubert & Vitzthum (1978)
Regeneration of
Carbon Dioxide
Modell (1978)
Purification of dredge
Water
Modell (1979)
Reformingof forest produc:ts
Water
Modell (1980)
activated
Klesper (1978)
carbon
spoils
74
in 1970.
In theory, any solvent with a chemically stable critical
point may be used as
includes
the
hydrocarbons.
supercritical
examples
substances
Table
solvent.
water,
of
However, the class of
includes
also
a
toluene,
supercritical
which
gaseous
are
conditions, such as carbon dioxide and ethylene.
(Klesper,
lists
1978)
and
critical
data
for
2.1
other
solvents
at
normal
Table
2.2
of
the
some
substances which have been suggested for use as supercritical
solvents.
The state of each substance at ambient
conditions
is indicated by its boiling point.
Supercritical
ranging
between
solvents
those
of
possess
physical
gases and liquids, and it is for
this reason that the term fluid is used.
Jentoft, 1972) lists
gases,
liquids,
some
and
typical
supercritical
Table 2.3 (Gouw and
physical
fluids.
table only gives ranges for the properties
fluids;
properties
of
properties
of
Note that the
supercritical
an important characteristic of a supercritical fluid
is that its properties are highly variable for small
changes
in temperature and pressure.
Table 2.3 indicates that supercritical solvents
substantially
higher
Higher rates of
supercritical
mass
solvent,
self-diffusivity
transfer
and
are
such
to
have
a
than normal liquids.
be
expected
in
a
an effect has indeed been
75
Table 2.2
Physical Constants of Supercritical Solventsa
Boiling Point
-
(at 1 atm)
(°c)
Critical Point Data
T
c
(°C)
P
c
(atm)
P
Pc
(kg/m3 )
CO2
-78.5
31.3
72.9
448
NH3
H2 0
-33.4
100.0
132.3
111.3
240
374.4
226.8
344
Methanol
64.7
240.5
78.9
272
Ethanol
78.4
243.4
63.0
276
Isopropanol
82.5
253.3
47.0
273
Ethane
-88.0
32.4
48.3
203
n-Propane
-44.5
96.8
42.0
220
n-Butane
- 0.5
152.0
37.5
228
n-Pentane
36.3
196.6
33.3
232
n-Hexane
69.0
234.2
29.6
234
2,3 Dimethylbutane
58.0
226.8
31.0
241
Benzene
80.1
288.9
48.3
302
-29.8
111.7
39.4
558
Dich 1orofl uoromethane
8.9
178.5
51.0
522
Trichlorofluoromethane
23.7
196.6
41.7
554
1,2-Dichlorotetrafluoroethane 3.5
146.1
35.5
582
Dichlorodifluoromethane
Chlorotrifluoromethane
-81.4
28.8
39.0
580
Nitrous oxide
-89.0
36.5
71.4
457
Diethyl ether
34.6
193.6
36.3
267
7.6
164.7
43.4
272
Ethyl methyl
ether
a - Klesper (1978)
76
(jl
Q)
a-r
L
cO
U
0
L
I
CU
Ci
r0'
I ·
O
O
O
I
I
O
-
I
O
-I
O
L
LL
U
LI
U
cn
Q
0
O
I.-
U
0.
:3
O
._
-3
00o
0,_
C
X
o
Ln
l
,
.I Q)
L
r.
)CU
u,
-
U,
U,
E
rC
4-
;L
E
E
0
4t,
0y
0
I-
E
~ ~
0o
0.
I-
i
4-
-
O
a
Q
C
e mo
.,
t4_
a,
U
.-
,
.>
0 0
C
9
77
measured (Hubert
and
Vitzthum,
1978).
In
an
process, this leads to a reduced contact time.
extraction
Likewise, the
viscosity of supercritical solvents results in a lower
lower
pressure
drop
important
for
flow
continuous
of
characteristic
a
systems.
solvent,
The
though,
affinity for a given solute, which determines the
of
the
solute.
its
is
solubility
affinity, a result of intermolecular
This
attractive forces, is
most
highly
dependent
upon
distance
the
Near the critical point, small changes in
between molecules.
temperature or pressure lead to sizeable changes in the fluid
with
density,
concomitant
in
changes
solubility
by
This phenomenon is best illustrated
characteristics.
an
example.
Figure 2.1 (Modell, 1979) is a solubility "map" for
system
naphthalene
diagram
shows
how
Tracing a particular isobar
in
there is
the
phase
a
varies
with
temperature.
pure
solid
naphthalene
(naphthalene's normal melting point is 800 C).
phase
equilibrium
pressure, there
coexist
with
is
a
The
present
Above about 75
atm an isobar gives the composition of a saturated CO 2
in
this
represent saturated solutions, i.e.,
diagram
always
in
concentration of naphthalene in the
the
carbon dioxide fluid
lines
- carbon dioxide (CO2 ) near the
(C0Hs)
critical point of CO 2.
the
solid
naphthalene.
temperature
at
which
For
any
three
fluid
lower
phases
- pure solid, saturated liquid, and saturated vapor.
The composition of the liquid and vapor at 65 atm
are
shown
78
T(°C)
0
10- 1
_
I
10
20
30
I
I
I
40
50
I
I
)
10-2
XN
]o
10
-3
-
tm.
/
tie line
I
Figure 2.1.
The solubility map for the naphthalene (N) -
carbon dioxide system.
naphthalene in solution.
Isobars indicate the mole fraction of
The dashed line corresponds to a three-
phase solid-liquid-vapor equilibrium.
terminates at LCP,
This equilibrium
the lower critical end point.
79
in
by a tieline; the locus of all these tieline
figure
the
compositions for various pressures is
by
dashed
the
the solid-liquid-vapor (S+L+V) locus is traced to
If
line.
given
higher temperatures (and pressures), it
is
the
that
found
liquid-vapor tielines converge to the critical point labelled
in
the
diagram.
At this point the near-critical liquid and
in
or,
near-critical vapor phases become indistinguishable,
other words, the vapor-liquid critical phenomenon is observed
in
a
of
presence
the
solid phase.
The occurrence of the
of
critical phenomenon between two phases in the presence
third
characterizes a special class of critical points known
The critical point in Figure 2.1
as critical end points.
known
a
a
as
lower
is
critical end point (LCEP), a designation
which will become clear in the discussion of
diagrams
phase
(Section 2.2).
The effects of temperature and pressure on solubility may
be observed
in
2.1.
Figure
Along
the
300
atm
isobar,
solubility increases continuously with temperature due to the
increasing
activity
of
the
solid
At
phase.
55
C, the
concentration of naphthalene in the supercritical fluid phase
is about 5 mole percent, or 15 weight
compares
unfavorably
percent.
While
this
with traditional nonpolar solvents, it
is nevertheless a significant quantity of naphthalene.
At 80 atm, solubility decreases sharply with temperature
at about 35°C.
This is the near
critical
region,
where
a
80
of
the density
the
drops
concentration
power.
in temperature leads to a large decrease in
increase
slight
The
fluid
as
C02
loses much of its solvent
the
higher
toward
region
critical
the
beyond
Moving
naphthalene
phase
fluid.
the density changes again become small and the
temperatures,
isobar reverts to a positive slope.
region
the
provide
in
phenomena
The solubility
With the aid of Figure 2.1, it
fluid
supercritical
to
compressed
is
of
450 C,
At a
is
70
atm
or
at
SCF
carries
which
The
less.
the
point the
naphthalene
is lowered by a factor of 50, and it precipitates
solubility
out as a pure solid.
is
CO2
example,
The
vessel,
collection
a
pressure is reduced to
recycled.
work.
a
atm where it dissolves 2 mole percent of
150
to
how
imagine
might
for
naphthalene from an inert substrate.
naphthalene
to
easy
process
extraction
supercritical temperature
fluid
the processes in Table 2.1.
for
basis
supercritical
the
The CO2
may
now
be
pressurized
and
The fact that no distillation or purification step
required
for
solute recovery often results in an energy
savings over conventional extraction operations.
It
is
important
to
note
the
modest
temperatures
at which a supercritical CO2
carried out.
In the example of Figure 2.1,
300
atm
pressures
and
extraction can be
below
pressures
and temperatures in the range of 300 C to 600 C would
be suitable.
The low temperature of this process is
a
main
81
is one of the most promising of supercritical
reason why C2
fluid
are
solvents.
unstable
nontoxicity
It can be used to extract substances which
at
high
Combined
temperatures.
with
and ready availability, supercritical CO2
is an
products.
ideal extracting agent for food and pharmaceutical
This fact is evidenced in Table 2.1.
its
is
Availability of CO
2
also
a
consideration
prime
process of
Table
researching
is
the
American
application
natural underground
intend
reservoirs
enhanced oil recovery
petroleum
to
companies
utilize CO2
(Stalkup,
from
The
1978).
C02
pumped from the reservoir to an oil field, where it
be
will
The
2.1.
this
in
injected
into
oil
the
bearing
substrate.
it
There
solubilizes and displaces the oil to bring it to the surface.
Water
is
another
role
supercritical
in
geochemical
with
solvent
great
has
phenomena
been
promise.
Its
mentioned.
Two applications, which appear in Table 2.1, have
Table 2.4
recently been suggested by Modell.
number
of
literature
properties of
sources
supercritical
pertaining
water.
also
to
gives
solvent
the
solubility
The
a
of
a
hydrocarbon in water will be investigated in Chapter 7, which
deals with the benzene-water system.
The preceding paragraphs have
and
potential
of
indicated
supercritical
engineering applications.
Evaluation
the
importance
fluids
for
chemical
of
and
existing
new
82
Table 2.4
Literature Sources for Supercritical Water
Reference
Application
Solubility of wood
Woerner (1976)
Solubility of glass (silica)
Martynova
Solubility of hydrocarbons
Connolly (1966)
Solubility of argon
Lentz & Franck (1969)
Solubility of nitrogen
Tsiklis & Maslennikova
Solubility of salts
Niggli
Geochemical aspects
Morey (1957)
(1976)
(1912)
(1965)
83
processes
would
be
greatly
if
facilitated
a
goal
Before
predictive
considering
necessary
gain
to
in
behavior
the
a
the
methods,
however,
it
is
complete understanding of phase
more
region.
critical
through
graphically
It
thesis to investigate one such method.
this
of
means were
behavior.
available to predict supercritical solubility
is
a
use
of
is
This
best
done
phase diagrams, the topic
treated in the next section.
2.2
PhaseDiagrams
The term "phase diagram" is
in
generic
being
nature,
applicable to a graphic depiction of any of the properties of
one
or
more
phases
in
systems, P-T, P-V, and P-P
binary
systems,
diagrams is the
one
P-T-x
For single component
equilibrium.
diagrams are
most
For
common.
of the most informative types of phase
space
diagram,
a
three-dimensional
representation of the relation between pressure, temperature,
and
composition.
While
such
a diagram gives an excellent
overview of the system's phase equilibrium
difficult
to
extract
actual
reason,
two-dimensional
three-dimensional space diagram
These
would
be
T-x
is
P-T-x coordinates from such a
diagram, and it is relatively time-consuming
this
it
behavior,
slices
are
or
most
to
For
draw.
sections
frequently
of
the
used.
(constant P), P-x (constant T), or P-T
t
84
(constant
x)
diagrams.
Another
useful
two-dimensional
diagram results when lines in the space diagram are projected
onto
a
plane.
This
results
in either a P-T, P-x, or T-x
projection, all of which have a particular utility.
"P-T-x diagram" is
projections
and
space diagram.
used
to
sections,
refer
to
all
The term
two-dimensional
as well as the three-dimensional
For binary systems, V-T-x diagrams
are
also
quite common, but will not be considered in this work.
Phase diagrams may be drawn for various combinations
solid,
liquid,
probably
or
received
importance
in
vapor phases.
the
the
most
field
Solid phase diagrams have
attention,
of
being
materials
of
science.
by Reisman (1970) and Gordon (1968).
journal Calphad is a
publication
devoted
primary
A
introduction to these types of diagrams may be found
books
of
good
in
the
In addition, the
entirely
to
the
computer calculation of solid phase diagrams.
In the chemical processing
involving
"fluid"
fluids
includes
supercritical
phase
are
diagrams
Rowlinson
more
both
fluids.
for
(1969).
industries,
often
gases
and
of
phase
interest.
liquids,
as
diagrams
The term
well
A relatively complete survey of fluid
binary
mixtures
Schneider
has
(1970)
been
given
interest
to
this
study.
presented by Rowlinson and Schneider
by
has also reviewed the
literature with emphasis on the critical region, which is
particular
as
The
are
phase
derived
of
diagrams
from
the
85
experimental
work
diagrams
these
of
is
numerous
investigators.
necessary
to
obtain
A study of
a
complete
understanding of the phase behavior of binary systems, and is
indispensible
in
the
interpretation
have
theoretical phase
Most of the phase diagrams in
equilibrium predictions.
section
of
been abstracted from the above references.
all diagrams, species B is
the
component
with
this
In
the
higher
a
binary
critical temperature.
The simplest type of fluid phase behavior for
system is depicted in the P-T-x space diagram of Figure 2.2a.
The
lines
dashed
indicate the course of the pure component
vapor pressure curves.
At any given temperature, component A
is more volatile (has a higher vapor pressure) than component
B.
The vapor pressure curves terminate at the pure component
critical points CP
and CP . As illustrated, component B
A
B
has the higher critical temperature, but component A has the
higher critical pressure.
Systems in which the more volatile
component has the lower critical pressure
The
two
pure
component
critical
points
are
also
common.
joined by a
are
continuous curve known as the critical line, which gives
locus
of
critical points of mixtures of the two components.
Since for a given
above
the
those
at
temperature
which
liquid
exists
at
pressures
a vapor exists, it is clear that the
region above the P-T-x surface of Figure 2.2a must correspond
to a single phase liquid.
Likewise,
the
region
P-T-x surface represents a single gaseous phase.
below
the
86
G-Lcritical line
CPA-
Ps
-k
P
is*
Figure 2 .2a.
The P-T-x space diagram for a system with completely miscible
fluid phases. vapor pressure curves, sown as dashed lines, terminate at the
pure componentcritical points. These points are joined by the gas-liquid
critical line for the mixture.
T-x sections are shown.
Three unshaded P-x sections and two shaded
87
p
T
Figure 2.2b.
P-T sections for a completely miscible
system at compositions xl and x2.
Bubble point curves
represent conditions at which the liquid first begins
to vaporize.
condensation.
Dew point curves represent the onset of
88
Figure
(unshaded)
2.2a
includes
and
two
three
isobaric
isothermal
T-x
sections
illustrate vapor-liquid equilibrium (VLE).
at
T1
lies
below
the
P-x
critical
vapor
pressure
(shaded)
The
P-x
temperature
curve.
To
to
section
of
both
a
pure
determine
the
components, and thus terminates at either end
component
sections
along
compositions of the coexisting liquid and vapor, a tieline is
drawn at constant pressure connecting the two branches of the
VLE envelope.
The points of intersection
compositions.
Again,
since
liquid
give
must
the
lie
desired
at
higher
pressures and gas at lower pressures, it is evident from
figure
that
the
VLE
liquid
phase
contains
a
proportion of component B than the VLE vapor phase.
the
greater
This
is
envelope
no
consistent with the lower volatility of component B.
At T , a temperature above
3
longer
CP ,
extends to the pure A axis.
tielines
maximum
the
VLE
A
are
drawn
pressure
at
If at such a temperature
successively
higher
pressures,
will be reached where the liquid and vapor
legs join and the tieline converges to a point.
point
on
the
critical
This is
envelope
requires
extreme.
At temperatures above the critical points
longer exist.
only
that
one
the
line for the specified temperature.
Since it is at a maximum pressure, the smoothness of the
components,
a
VLE
it does not fall at a compositional
of
both
phase is present and VLE envelopes no
For the T-x sections, VLE tielines
are
drawn
89
at
constant
T.
It follows that for a T-x section above the
critical pressure of component B
the critical point falls at
a maximum temperature and is not a compositional extreme.
A third possibility is to take
space
a
P-T
diagram at constant composition.
section
be
Tielines
may
drawn in a P-T section and, in general, the critical
point is not located at either the maximum
maximum
the
Figure 2.2b presents
two such sections, at compositions xl and x2.
not
of
temperature of the VLE loop.
locus
of
the mixture.
or
the
As indicated in Figure
2.2b, however, the envelope of these loops
critical
pressure
does
define
the
This fact is often used in
the experimental determination of critical lines.
The critical curve in Figure 2.2a
maximum.
While
universal.
corresponds
which
the
illustrates
have
pure
been
component
found
for
critical
of
critical
points.
two
P-T
lines
Type
to the critical behavior of Figure 2.2a.
1
Type 2
components
fairly similar characteristics such as size, shape, and
polarity.
If the components are very
line
be
may
nearly
5
possessing
similar
the
critical
straight as with type 3 in the figure.
Types 4 and 5 are critical
type
pressure
types
six
is a monotonic critical line, found when the
have
a
behavior is common, it is by no means
Figu're 2.3
projections
connecting
such
exhibits
as
lines
well
a
with
temperature
pressure
maximum.
exhibits maxima in both temperature and pressure.
minima,
Type 6
Analogous
90
(Schneider,
5
1970)
I
(.P
-'A
P
.10
sr-Ba
--.
-
-
1-
-
T
Figure 2.3.
P-T projections of six types of critical curves
which may connect the two pure component critical points.
91
critical
curves
are
when
found
component
A
has a lower
critical pressure than component B.
The systems considered thus far have been simple in that
the liquid phases
of
two
components
were
completely
Liquid phase immiscibilities may be manifested in
miscible.
Figures 2.4 - 2.12 illustrate
a variety of ways.
of
the
behavior
for
cases
such
P-T-x space diagrams.
the
range
with both P-T projections and
The figures
are
roughly
ordered
by
degree of miscibility.
Figure 2.4 shows
immiscibility,
where
entirely bounded by
critical
line.
the
simplest
the
region
the
VLE
type
of
of coexistence is a dome,
surface
and
a
lower
critical
liquid-liquid
critical line represents the locus of
This
upper critical solution pressures (at constant
and
liquid-liquid
solution
T)
or
upper
temperatures (at constant P).
The immiscibility dome meets the VLE surface along a line
in
the P-T projection; this is the three-phase L-L-G line, which
in
turn intersects the L-L critical line at two critical end
At a critical end point (CEP), the three-phase
points.
two
of
the
phases
it represents become
terminates
since
identical.
The critical locus also terminates in
that
it
Chapter
becomes
metastable,
a
line
the
sense
concept to be discussed in
2.
The liquid-liquid critical line in Figure 2.4
does
not
92
gas-liquid critical line
liquid-liquid critical
ine
_
^
N1,LPB
f1D0
P
_
T
Figure 2.4a.
The P-T projection for a liquid-liquid
immiscibility dome.
The liquid-liquid (L-L) critical line
defines the top of the dome.
It intersects the bubble point
surface at two critical end points (CEPs).
a terminus
line.
of both
the critical
Each CEP represents
line and the three-phase
LLG
93
P
Figure 2.4b.
The P-T-x space diagram for Figure 2 .4a.
94
the VLE surface, but may continue to
to
return
necessarily
The line may be
higher pressures as indicated in Figure 2.5.
temperatures
considered to represent upper critical solution
(UCST's),
and
decrease as pressure increases.
increase or
either
may
temperatures
these
The figure shows the former
dP
case, i.e.,
dT
is positive in the P-T projection, so that
the liquid phase immiscibility extends to very high, possibly
dP
negative, the UCST curve will
For dT
infinite, pressures.
eventually be terminated by the formation of a solid phase.
just
Substances less miscible than those
exhibit
immiscibility
volatile component.
in
region of the more
critical
Figure 2.6 illustrates such a situation.
This is the first instance
connecting
the
will
treated
of
the
L-G
critical
the two pure component critical points.
line
not
Instead,
the lower L-G critical line beginning from CP , terminates at
the lower critical end point or LCEP.
critical
line
starting
at
critical end point or UCEP.
what
is
traditionally
CPB
Similarly,
terminates
are
Since the CEP's
labelled
at
the
upper
the upper
joined
by
an L-L-G three phase line,
this upper critical line must be designated as liquid-liquid.
Near CPB , however, the upper critical line
the
L-G
behaves
much
critical line in a completely miscible system.
this reason, itiis
most
accurate
to
refer
to
the
critical line as a liquid-fluid or L-F critical locus.
as
For
upper
95
P
T
Figure 2.5a.
The P-T projection for a system in
which the liquid-liquid immiscibility extends to
very high pressures.
96
P
T
Figure 2.5;b.
The space diagram for Figure 2.5a.
97
P
Figure
2 6
. a.
The P-T projection for a system exhibiting immiscibility
in the region of CPA, the critical point of the more volatile component.
The upper critical line terminates at the upper critical end point
(UCEP), while the lower critical line terminates at the lower critical
end point (LCEP).
The L-L-G line also terminates at the CEP's.
The diagram arbitrarily shows component B with a higher critical
pressure than component A.
98
(Rowlinson,
1969)
Upper critical line
.1-11
-Z-;%\
N.A
LCEP
P
Lower critical
T
X
Figure 2.6b.
The space diagram for Figure 2.6a.
99
Figure 2.7 is quite similar to 2.6, but
with
a
second
region of liquid-liquid immiscibility of the type illustrated
in
Figure
curves and
2.5.
This
three
critical
still
increases
temperatures,
continuation
finally
two separate L-L-G
exhibits
end
points.
the
further,
UCST
intersecting
As
line
and
immiscibility
moves
to higher
forming
to the upper critical locus.
depicted in Figure 2.8.
curve,
system
a
This situation is
The system again has a single
but now only one CEP in the fluid region.
shows the behavior when the
upper
smooth
high
pressure
critical curve has a positive slope.
L-L-G
Figure 2.9
portion
of
the
The n-hexadecane
- carbon dioxide system exhibits features of both Figures 2.8
and 2.9 (see Chapter 5).
A further variation of this general
type is found when the upper critical line
exhibits
minimum
a
in
minimum.
pressure,
than
both
maximum
This case is shown in Figure 2.10, again
three-phase
pressures.
rather
only
a
and a
with
the
line at pressures above the pure component vapor
The benzene-water system (see Chapter 8)
belongs
to this category.
For
highly
immiscibility
immiscible
persists
up
point of the less volatile
line
may
exhibit
to
systems,
the
region
of
the vicinity of the critical
component.
The
upper
critical
a temperature minimum, as shown in Figure
2.11, or proceed directly to higher temperatures as in Figure
2.12.
Both types of upper critical lines have been
referred
100
P
T
Figure
2
.7a.
The P-T projection
for a system of the type shown
in Figure 2.6, but with a separate region of liquid-liquid
immiscibil i ty.
101
(Schneider, 1970)
P
T
Figure 2.7b..
The space diagram for Figure 2.7a.
102
P
T
Figure 2.8a.
The P-T projection for a system in which the L-L
critical line has become part of the upper critical line.
The
L-L critical
is
decreased.
line moves
to higher
pressures
as temperature
103
P
T
Figure 2.8b.
The space diagram for Figure 2.8a.
104
P
T
Figure 2.9a.
critical
The P-T projection
for a system in which the L-L
line has becomepart of the upper critical
line.
The L-L
critical
ine moves to higher pressures as temperature is increased.
Unlike previous diagrams, the LLG line now falls at pressures
above the vapor pressure curve of A.
105
(Schneider, 1970)
P
T
Figure 2.9b.
The space diagram for Figure 2.9a.
106
P
T
Figure 2.10a.
The P-T projection for a system of the type shown
in Figure 2.9, but with no pressure maximum in the upper critical
line.
Once again the LLG equilibrium is found at pressures higher
than the vapor pressure of either component.
A proposed metastable
critical line has been drawn in, showing the course of the lower
critical line had liquid-liquid immiscibility not interfered.
107
(Schneider, 1970)
P
T
Figure 2.10b.
The space diagram
for Figure 2.10a.
108
P
T
Figure 2.11a.
The P-T projection for a system exhibiting gas-gas
immiscibility of the first type.
shown
in this diagram.
Only the upper critical line is
109
/
I
I
I
P
Figure 2.11b.
\
Upper critical
"Uppercritical
The space diagram for Figure 2.11a.
110
(Schneider, 1976)
r critical
P
line
1 ine
.- II
.-O'
T
Figure 2.12a.
The P-T projection for a system exhibiting gas-
gas immiscibility of the second type.
111
(Schneider,
1976)
P
T
Figure 2.12b.
The space diagram for Figure 2.12a.
112
to
defining gas-gas equilibria (Schneider, 1970), though
as
again the term fluid-fluid is to be preferred.
Another measure of complexity is added to the
preceding
At
classification scheme if azeotropic lines are considered.
an
azeotropic point, distinguishable liquid and vapor phases
coexist,
have
but
coordinates
azeotropic
of
locus
one degree of freedom, and thus
has
forms a line in a P-T-x space diagram.
sections
A
composition.
identical
A
of
sequence
P-x
an azeotropic system is shown in Figure 2.13a.
for
In this example, component B has the higher critical pressure
of the two components.
component
At temperature
envelope
below
the
pure
critical temperatures, the VLE envelope terminates
at either extreme at a pure component
azeotrope
T1 ,
The
pressure.
vapor
which occurs at a pressure maximum divides the VLE
two
into
branches.
There
exists
a
of
range
pressures for which the gaseous phase may be richer or leaner
in
component
B
than the coexisting liquid phase, depending
upon the overall composition of
relative
compositions
on
a
system.
the
Whatever
of vapor and liquid, the liquid phase
is the denser of the two.
made
the
molar basis.
(The density distinction
must
be
In the hydrogen-helium system, for
example, the gaseous phase is gravimetrically more dense than
the liquid phase.)
At temperature T2 , above the critical point of component
A, the A-rich branch of the
VLE
envelope
terminates
at
a
113
CPB
P
CPA
1
0
XB
Figure 2.13a.
P-x sections for azeotropic behavior
in the critical region.
This is a maximum pressure
or minimum temperature (boiling) azeotrope.
A cusp
is produced in the VLE envelope when the azeotrope
line intersects
the critical
line.
114
(Rowlinson,
1969)
/
-Azeotrope
1 ine
P
Figure 2.13b.
This diagram illustrates
the relation between
azeotropic P-x and T-x sections.
The T-x section is shown
shaded.
The equivalence
of maximum pressure
boiling azeotropes is evident.
and minimum
115
point
along
the critical line.
but
occurs at a minimum pressure,
extreme.
This mixture critical point
not
at
compositional
As temperature increases further, the critical line
the azeotrope line and the A-rich branch shrinks.
approaches
At T3 , the lines meet and the VLE envelope
cusp.
a
in
terminates
a
The tip of this cusp is the azeotropic critical point,
occurs at extremes in both pressure and composition.
and
it
For
temperatures
T3 ,
above
the
cusp
to
reverts
normal
behavior, as illustrated by the section at T.
Figure 2.13b shows the relationship between P-x and
It is seen that a pressure
sections in an azeotropic system.
maximum
in
the
minimum in the
encountered
behavior
P-x
T-x
in
of
section.
constant
a
system
classification.
corresponds
section
As
Since
pressure
serves
T-x
as
to a temperature
azeotropes
are
distillations,
the
often
the
basis
for
T-x
its
the azeotrope in the T-x section occurs
at a minimum temperature, the present system is designated
a
minimum boiling azeotrope.
Other types of
Figure
2.14.
azeotropic
Figure
over
region
of
composition,
a
portion
P-x
section for a
liquid-liquid
immiscibility
the
of the compositional range.
immiscibility
the
presented
shows
2.14a
minimum boiling azeotrope where
exists
are
behavior
behavior
extends
of
beyond
Figure
Figures 2.9 and 2.10 (benzene-water)
the
2.14b
exhibit
in
If the
azeotropic
will
this
result.
type
of
116 -
Figure 2.14a.
A system with both a minimum
boiling azeotrope and liquid-liquid
p
immiscibility at the same temperature.
Xs
Figure 2.14b.
A heteroazeotropic system.
The azeotropic composition, which is the
composition of the gas phase along the
three-phase line, now lies between the
compositions of the immiscible liquids.
p
Heteroazeotropes revert to the behavior
shown in Figure 2.14a as the UCST is
approached and the region of liquid-liquid
immiscibility shrinks.
Xs
Figure
2
.14 c.
A maximum boiling azeotrope
which lies outside the region of liquidP
liquid immiscibility.
Maximum boiling
azeotropes may exist when no immiscibility
is present,
"
as well
as at the same
as a minimum boiling azeotrope.
(Rowlinson,
1969)
temperature
117
known
behavior,
as
Figure 2.14c shows a
heteroazeotropy.
maximum boiling azeotrope outside the range of immiscibility.
often
Supercritical fluid extraction processes
the
of
dissolution
involve
P-T-x
a solid, as Table 2.1 indicates.
diagrams for systems in which pure solid phases occur are now
attention
with
considered,
the
of
solidification
less
directed
mainly
toward
volatile
component.
These
diagrams may be thought of as fluid phase diagrams
with
the
precipitation of a solid phase superimposed.
The simplest case again occurs
temperature scale to include the
of
diagrams
result.
2.15
Figure
triple
point
In
S BLG.
and
A
B
the
Extending the
region,
the
the
P-T
SASBG,
SASBL,
These lines all intersect at the quadruple
point Q, where the four phases SA, G, L and S B
In
phase
critical
2.15a,
projection, there are four three-phase lines:
SALG,
under
system
liquid
any
been given in Figure 2.2.
have
region
exhibit
the
The P-T and P-T-x diagrams in the
immiscibilities.
point
not
does
consideration
when
all
coexist.
P-T-x diagram, these points are labelled QA' QG' %'
and QB' respectively, and they all possess the same
and temperature coordinates.
S S L eutectic
A B
Q
pressure
is also the terminus of the
line.
The SBLG line in the P-T diagram is a projection of
lines
TP -Q
B
G
and
TP -Q
B
L
in
the
P-T-x diagram.
the
TP -Q
B
G
118
(Rowlinson,
1969)
P
T
Figure 2.15a.
The P-T projection for a system in which the liquid
phases are completely miscible, and the solid phases are completely
immiscible.
by Q.
Triple points are denoted by TP and the quadruple point
SA and SB are the pure solids.
119
.
P
T
Figure 2.15b.
The P-T-x space diagram for Figure
2
.15a.
The four phases at the quadruple point, QA' QG' QL' and
QB are now evident. The top face of the diagram is a
typical eutectic diagram.
120
gives
the
vapor
composition
TPA-QG
all
for
composition
all
and TPA-%
S LG
SBLG
and
three-phase
systems.
three-phase
in
other
the
composition
the
systems.
give the corresponding
diagram indicates that the TPA-%
each
TPB-Q L
liquid
The lines
compositions
for
An examination of the P-T-x
and
lines
TPB-QG
cross
This fact is
coordinate.
illustrated more clearly by the P-x sections given in
Figure
Starting at the quadruple point temperature in 2.16a,
2.16.
there exist two regions of gas-solid equilibrium at pressures
pressure
to
coexist
as
condensation occurs, and the gas-solid equilibrium shifts
to
below the quadruple point tieline.
that
the
of
quadruple
one of liquid-solid.
point,
Increasing
four
phases
Above the quadruple point pressure, for
a eutectic line with positive slope, only two pure solids are
in equilibrium.
Increasing
point,
the
temperature
section
P-x
shape illustrated
in
tieline
been
has
now
B
SBLG
2.16b.
The
quadruple
point
replaced by the three-phase tielines
As depicted in
Figure
gas phase.
2.15a,
the
or
in
than
For pressures between the SALG tieline
and the eutectic pressure P , liquid may exist
phase,
quadruple
The S LG liquid phase is richer in component B
second.
the
the
these tielines falls at a higher pressure than the
of
first
AB
above
takes on the much more complicated
Figure
S LG and S LG and S S L.
A
slightly
equilibrium
overall system composition.
with S
A
Above the
or S
B
as
a
single
according to the
three-phase
line
at
121
A
C
B
PE
A SB
B
SA+
P
P
?G
!
pQ
QB
G
0
1
0
1
XB
0
1
xB
Figure 2.16.
P-x sections for Figure 2.15b. TA is the quadruple point
temperature.
At TB, slightly above TA, the liquid phase in
equilibrium with SA is richer in component B than the gas phase in
equilibrium with SB.
has inverted.
At TC, further above TA,
this relationship
122
P , only two pure solid phases coexist.
E
Increasing the temperature further, but still below TP ,
the
P-x
section
of
results.
than the SBLG gas phase.
a
liquid
single
this
Compared to Figure 2.16b, then, the
has
relative composition of these two phases
TC ,
At
SALG liquid phase is leaner in component B
the
temperature,
2.16c
Figure
phase
may
switched.
to
persist
much
At
higher
pressures than at TB.
For systems where the
liquid
phase
is
very
low,
higher than that of CPA ,
quadruple
point
solubility
of
B
solid
in
and the temperature of TPB
the
SBLG
is almost pure A.
liquid
phase
near
the
is
the
When this low solubility
persists to temperatures above the critical point of
A,
the
SBLG line cuts the LG critical locus to give a lower critical
end point.
This is shown in Figure 2.17.
SBLG
starting
line
critical
locus
at
from
the
TP
B
upper
The segment of the
similarly intersects the LF
critical
end
point.
At
temperatures between the critical end points, the fluid phase
critical
phenomenon
Consequently,
this
is
region
not
is
observed
often
at
any
designated
pressure.
as
the
supercritical fluid region.
Consider next the case of a liquid
as
shown in Figure 2.4.
immiscibility
dome,
Precipitation of a solid phase at a
temperature in the mid-region of the dome leads to the
phase
123
(Rowlinson, 1969)
P
T
Figure 2.17.
The P-T projection of a system in which the gas-
liquid critical line is cut by the SBLG line.
The length of the
lower critical line has been greatly exaggerated.
typically within a few °C of CPA.
The LCEP is
124
diagrams
of
Figure 2.18.
For temperatures below the SBL 1 L2
line, the behavior is essentially the same as in Figure 2.15,
with a eutectic quadruple point.
For temperatures above this
line, however, phase diagrams of the
2.18c
are encountered.
type
shown
also
evident
Figure
This P-x section has two three-phase
tielines, L1 L 2 G and SBL2G, as required by Figure
is
in
2.18a.
It
from Figure 2.18a that the L1 L2 G and SBL 2 G
lines intersect at a second quadruple point.
in equilibrium with two liquid
phases
and
Here solid B is
a
vapor
phase.
Figure 2.18a has retained the higher temperature critical end
point from Figure 2.4 (CEP1 ) , and generated a second critical
end
point
where
the
SBL L
line meets the liquid-liquid
B1 2
critical locus (CEP ).
3
The UCST line of Figure 2.5 will
same
phase
diagrams
a
the
the
P-T
projection.
positive slope of this critical line, a CEP along the
S L1L
line still occurs.
phases
only
B 12
essentially
as in Figure 2.18 if the liquid-liquid
critical locus has a negative slope in
For
give
to Figure
Now,
however,
the
two
liquid
coexist at pressures above the CEP, in contrast
2.18.
Below
the
pressure
of
the
CEP,
phase
diagrams similar to those shown in Figure 2.15 are found.
Figures
liquid-liquid
2.6
and
2.7
illustrated
critical
case
of
immiscibility causing a relatively minor break
in the gas-liquid critical line.
the
the
region,
these
When solid precipitates
in
system types will usually give
125
(Rowlinson,
1969)
P
T
Figure
2 .18a.
immiscibility
The P-T projection
dome interrupted
for a system with
by formation of a solid
diagram is to be compared with Figure 2.4a.
figure.
The second L 1 L 2 G critical
metastable,
and no longer appears.
end point,
diagram.
line.
phase. This
CEP
1 appeared in that
CEP2,
A new critical
is now the terminus of the L-L critical
Q1 and Q2 ' appear in this
a liquid
has become
end point,
CEP3 ,
Two quadruple points,
126
(Rowlinson,
1969)
P
T
Figure 2.18b.
The space diagram of Figure 2.18a.
The points
QA' QB' QL' and QG all coincide with Q1 in that diagram.
line joining the points labelled Q2 gives the compositions
the second quadruple point.
The
at
127
p
Figure 2.18 c.
A P-x section of Figure 2.18b, at a
temperature above the SBL1L2 surface but below TPB.
128
In some instances,
similar phase diagrams.
formation
solid
occurs below the critical temperature of the light component,
in
this temperature.
above
others
given by Figures 2.19
and
2.20,
These possibilities are
In
respectively.
Figure
as in Figure 2.17, there exists a supercritical fluid
2.20a,
observed.
region within which the critical phenomenon is not
2.20b gives a P-T-x space diagram for a system of the
Figure
based
is
The space diagram
type shown in Figure 2.20a.
on
the naphthalene-ethylene system (see Chapter 7), with certain
features exaggerated for clarity.
formation
Figure 2.20 would also be valid for solid
the
in
phase diagrams of Figures 2.8 through 2.10, if the upper
critical line terminates
the
however,
Frequently,
liquid-liquid-like
Examples
before
at
the
upper
the
at
UCEP
on
possibility
Figure 2.22a gives the
higher
temperatures,
the
on
fall
line.
critical
The
the
that
UCEP
P-T
with
projection
for
concomitant
the
This diagram
is
theoretical predictions for the naphthalene-carbon
dioxide system (see Chapter 6).
interesting
situation
Although
the
of
Figure 2.22b gives the P-T-x
Finally Figure 2.23
space diagram for this case.
line.
slope.
2.21, 2.22, and 2.23.
occurrence of a supercritical fluid region.
based
steep
a
a lower temperature than the LCEP, as was the case
in Figure 2.19.
the
does
Figures
in
first of these illustrates
falls
UCEP
of
portion
given
are
attaining
shows
the
a temperature minimum in the SBLF
critical
phenomenon
is
not
observed
129
(Rowlinson. 1969)
P
T
Figure 2.19.
The P-T projection for a system with liquid immiscibility
in the vicinity of CPA
.
The upper critical line is terminated by the
formation of a solid phase at the UCEP.
In this example, the UCEP is
at a lower temperature than the LCEP and no supercritical fluid
region exists.
Two quadruple points are found in the diagram.
130
(Rowlinson,
1969)
P
T
to that depicted in Figure 2.19,
but with the UCEPat a higher temperature than the LCEP. A
fluid region now exists between the LCEPand the
supercritical
UCEP. Only one quadruple point is found in this diagram.
Figure 2.20a.
A system similar
131
P
T
The space diagram for Figure 2.20a. This diagram
on the naphthalene (B) - ethylene (A)
is based qualitatively
Figure 2.20b.
system.
132
(BUchner,
1906b)
P
T
Figure 2.21.
A system of the type shown in Figure 2.8, with
precipi
tation
of solid at temperatures below the LCEP.
133
0
rTr
v
(U
44)
._C0O
co
Q
a)
0
._
LL
3
C
Q 0L
ur
0aa
r
4I
<
r~~~a
< a)
La)
To
4Lhr.
, ._
VN
* )
X N
4(U
N
uo
-a)- -o
C Lz~U
* OO
a
i
Us)
~~LL
134
(BUichner, 1906b)
P
T
Figure
2.22b.
The P-T-x space diagram of Figure
Eight P-x sections are shown.
2.22a.
135
(BUchner,
1906b)
~ ~~~~~~~ ~ ~ ~ ~
I~~~~~
ii
Uppercritical line
P
Lower critical
line
CPA
T
Figure 2.23.
A system similar to that of Figure 2.22, but
with a temperature minimum in the SBL2 F curve.
I
136
between
LCEP
UCEP, it is now possible to encounter two
and
liquids coexisting in this temperature range.
In Figures 2.11 and 2.12, the entire upper critical line
is
found
at
temperature
near
temperatures
of
B.
A
UCEP
the
critical
to solid formation is not
due
usually found in these systems.
above
or
Behavior near
the
critical
point of A is normal, with an LCEP of the type illustrated in
previous diagrams.
preceding
The
classification
of
binary
systems
was
intended not only to provide an understanding of the types of
behavior which exist, but also to emphasize the need for such
an
The complexity of the diagrams can easily
understanding.
lead to a misinterpretation of experimental
theoretical
measurements
Furthermore,
predictions.
knowledge
system's overall P-T-x behavior is useful when
phase
dealing
or
of
a
with
For example, the P-T-x behavior
equilibrium problems.
of the naphthalene - carbon dioxide system indicates that the
extraction
effectively
illustrated
carried
pressure regime.
thesis
to
in
Figure
in
out
a
2.1
might
investigate
prediction
behavior of binary systems.
temperature
different
For these reasons, it is
The means
of
a
goal
of
more
and
this
overall
phase
obtaining
these
the
for
be
predictions are discussed in the following section.
137
2.
Eaguations f State
A formula relating the temperature, pressure, volume (or
equation
density) and composition of a system is known as an
of
even
The accurate correlation of these properties for
state.
a
pure
component
often
requires
state for water which has been developed by
This
(1978).
equation,
used
to
Keenan,
et
al.
generate steam table PVT
data, contains 50 empirical constants.
It is
applicable
to
fluid phases of water, but contains no information about
Furthermore, the equation is solely for use
the solid phase.
with
complex
An extreme example is provided by the equation of
equation.
all
fairly
a
pure water;
aqueous
it is of no use in determining
mixtures.
requires
equations
properties
of
diagrams
phase
Prediction
of
binary
applicable
to
both pure components and
mixtures.
Development of suitable equations of state
for
gaseous
and liquid phases is a problem which has intrigued scientists
for several hundred years.
The oldest and most well-known of
these is the ideal gas law,
PV = NRT
(2.1)
which resulted from the work of Boyle in
Gay-Lussac
in
the
late
eighteenth
and
1662,
early
Charles
and
nineteenth
138
in
centuries, and finally Avogadro
equation
of
1812.
Thus
the
first
took 150 years to formulate, and was not
state
fully accepted until about 1860.
It is
applicable
only
to
low density gases.
In his doctoral dissertation
Waals
of
1873,
J.D.
van
proposed a semiempirical modification of the ideal gas
equation which attempted to account for nonidealities of
pure
der
gaseous
state.
nonidealities (see
While
Brush,
not
the first model for these
it
1961),
equation to see widespread usage.
the
was
the
first
such
This equation is stated in
intensive form as
(P +2)
(2.2)
(v-b) =RT
or
RT
P = -
a
2
Here a and b are constants, with the term
(2.3)
- 2
v
accounting for
the attractive forces between the molecules and b the
volume
occupied by these molecules.
finite
Equation (2.3) provides
a good representation of the behavior of gases up to moderate
densities, but at
quite
large.
higher
More
densities
of
the
deviations
become
importantly, though, the nonideal terms
introduced by van der Waals cause
condensation
the
the
equation
gas at higher densities.
to
predict
As a result,
the van der Waals equation may be used to model both gas
and
139
liquid
In
phases.
application
equation to permit
This
paved
step
van
1890,
to
way
the
der
for
generalized his
Waals
systems.
multicomponent
great
understanding of fluid phase behavior of
advances
mixtures.
in
the
In
the
decade of this century, the Dutch schools at Amsterdam
first
mapped
experimentally
and Leiden
presented
diagrams
in
the
out
previous
many
of
the
section.
phase
time
The
required for manual phase equilibrium calculations prohibited
the
actual
analysis
prediction
the
of
of
phase
Waals equation, however, allowed
der
van
Mathematical
diagrams.
these workers to check the consistency of their
and
results
explain unexpected findings (see e.g., Smits, 1903 and 1905).
Van
der
himself
Waals
was
an
active participant in this
Katz and Rzasa (1946) have prepared a useful
interpretation.
bibliography giving references to the work in this period.
Another significant result of the
formulation
the
of
Onnes in 1902.
Dutch
work
This equation is expressed as
V
+
(
(2.4)
V
V
Originally proposed on empirical grounds, it has
from
statistical
Spurling, 1969).
theoretical
the
virial equation of state by Kammerlingh
RTC
P = RT
RT + RTB
RTB + 3
derived
was
mechanical
This fact allows
expressions
for
the
for
principles
the
constants
virial equation is the only equation of state
been
since
(Mason and
development
B, C, etc.
with
a
I
of
The
sound
140
theoretical
basis
for
both
pure
components and mixtures.
to
Unfortunately, it is applicable only
gases
of
low
and
moderate density.
Since the time of van
der
Waals
and
his
colleagues,
numerous authors have proposed gas-liquid equations of state.
As
of
1944,
literature
100 such equations could be found in the
over
(Dodge,
1944).
total
The
undoubtedly several times that number.
at
present
is
The large majority of
these equations have essentially been modifications of either
(2.3) or (2.4).
equation
Recently,
Oellrich,
et al.
(1977)
have surveyed the equations of state most commonly
used
VLE
require a
calculations.
These
equations,
which
all
for
single empirical interaction parameter when used for a binary
mixture, are given in Table 2.5.
The authors
used
each
of
these equations to model a variety of binary systems in order
to
ascertain
pressure
and
temperatures
the
gas
and
Lee-Kesler
Lee-Kesler
is
which
minimal
substances.
gave
the
composition
pressures.
equation
equation
since
gave
most accurate predictions of
over
a
wide
the
of
In general, a modified form of
the
best
results.
The
has 24 constants, but this complication
the
constants
are
the
same
for
all
Unfortunately, as with many Benedict-Webb-Rubin
type equations, this equation is prone to anomalous
in
range
critical
Paulaitis, 1981).
behavior
region (Pl6cker, et al., 1978; Francis and
141
Table 2.5
Equation of State Comparison by Oellrich, et al. (1977)
Parent Equations
Equation
Soave (1972)
van der Waals (1873); Redlich-Kwong
Starling (1973)
Vir ial (1901);
Lee-Kesler (1975),
Benedict-Webb-Rubin
Virial; Benedict-Webb-Rubin
modification of Plicker,
Knapp, & Prausnitz (1977)
Peng-Robinson (1976)
Lu, et al.
(1977)
van der Waals; Redli ch-Kwong
'van der Waals; Redl ich-Kwong
(1949)
(1940)
142
For the modified van der Waals type equations, the Soave
and Lu equations generally provided results nearly as good as
the Plocker-Lee-Kesler
notably
CO 2
correlation.
the
binaries,
In
results
some
were
cases,
most
superior.
The
not
Peng-Robinson equation (Peng and Robinson, 1976) was
as
satisfactory as the Soave and Lu equations.
Frequently, van der Waals type equations are
for
used
more
The two constants
their simplicity than their accuracy.
in the equations are always chosen so that the pure component
Often
critical point criteria (see Chapter 3) are satisfied.
virial
this procedure is not followed with the multiconstant
type
equations,
region.
equations
It
to
is
leading
also
to
difficulties
possible
with
in
van
the critical
der
Waals
type
specify temperature and pressure and solve for
volume analytically, in contrast to the virial type equations
for which a trial
mixtures,
and
extension
error
of
procedure
the
two-constant
semiempirical and straightforward.
specifying
"mixing
rules"
The
which
the mixture.
more
to
For
equations
how
obtain
is
involves
the
pure
constants
Since the constants do have an approximate
physical meaning, these rules
For
required.
extension
dictate
component constants are to be combined
for
is
complicated
not
entirely
arbitrary.
equations, however, constants with no
physical significance
are
suitable
of
combination
are
often
these
parameters is a difficult task.
employed.
Determining
a
constants to obtain mixture
143
A weakness of both the van der
is
equations
Waals
al.
(1976)
have
shown
that
Levelt
real
nonanalytic at the critical point, a fact
the
to
scaling equations
equations
are,
which
Sengers,
substances
are
reflected
not
by
These authors recommend the use of
equations.
previous
type
they cannot accurately model behavior in
that
the immediate vicinity of a critical point.
et
virial
and
accurately
model
unfortunately,
critical
not
behavior,
very useful over
wider temperature and pressure ranges.
The gas-liquid equations of state
(with
the
exception
of
the
viewed the liquid state as
state.
At
equation.
common
low
a
equation
thus
far
for water) have all
perturbation
they
densities
considered
simplify
of
the
gaseous
to the ideal gas
At high densities, on the other hand, there is
form and 'errors are more significant.
no
An alternative
approach is to use an equation developed specifically for the
The higher degree of molecular interaction
liquid phase.
the
condensed
state
as
in
compared to the gaseous state does
make a theoretical description of liquids substantially
more
liquids
have
Equations
difficult.
of
state
for
nevertheless been proposed, most notably
Woods
(1966)
and
Chueh
and
pure
those
of
Yen
Prausnitz (1967, 1969).
result from correlations of experimental data rather
theoretical
model
and
Both
than
a
of the liquid state, a fact which becomes
particularly evident in extending the equations to
mixtures.
144
The
Yen and Woods correlation is not suitable for use in the
critical region, while the Chueh and Prausnitz correlation is
applicable only if the actual mixture critical temperature is
already known.
is
reader
The
to
referred
Reid,
et
al.
(1977) for a review of these formulations.
removed
For conditions
the
from
gas-liquid
critical
locus, volume and density are of secondary importance for the
state.
liquid
Phase equilibrium calculations may be carried
out without regard to pressure
volume,
solution"
a
given
used
effects
temperature
review
of
Scatchard
"theories
of
These include the regular
for this purpose.
solution theory of
various
the
on
Prausnitz
an equation of state unnecessary.
making
(1969) has
and
Hildebrand,
and
Guggenheim's
lattice theory, and the athermal solution theory of Flory and
Huggins,
among others.
results for the
gas-liquid
Solution theories give more accurate
'liquid state
of
equations
those
than
They
state.
obtainable
from
are also useful for
liquid-liquid critical phenomena occurring in regions removed
from the gas-liquid critical locus.
The
example,
lattice
is
a
particularly
in
equations
contrast
discussed
to
the
(1970) and Schouten, et al.
approach
previously.
Kleintjens and Koningsveld (1980; see
al.
interesting
as it represents the liquid state as a perturbation
of the solid state,
gas-liquid
theory
also
of
the
Recently,
Trappeniers,
et
(1974)) have extended the
145
lattice theory to the gaseous
of
existence
treatment, a pure
mixture
or
holes
vacancies
substance
by
region
as
that
The
have
authors
"lattice-gas model" correctly predicts the
this
occurrence of gas-gas equilibrium (see Section
methane-water system.
in
2.2)
the
Beris (1981) has also demonstrated the
of
correctness
qualitative
quasi-binary
a
Similarly, a binary mixture
of holes and molecules.
is considered as a quasi-ternary mixture.
shown
In this
in the lattice.
viewed
is
the
for
allowing
benzene-water
diagrams
phase
generated with this model.
The preceding discussion indicates that the choice of an
equation of state for a particular fluid phase application is
In this work, prediction of
not always an easy task.
P-T-x
diagrams over wide ranges of temperature and pressure,
with emphasis on the critical region, was the
The
binary
(P-R)
Peng-Robinson
equation
was
goal.
desired
this
for
selected
purpose for the following reasons:
a) In
an
exploratory
appropriate
equation
is
to
a
use
study
of
this
the simplest approach.
cubic,
two-constant
it
nature,
is
The P-R
the
equation,
simplest type that can provide realistic results.
b) The critical region is the most difficult
the
P-T-x
space to model.
the P-R.
of
In this region, only the
scaling equations provide results superior
of
region
to
those
As noted, the scaling equations are not
146
applicable for conditions removed from
the
critical
locus.
These
reasons
unambiguous choice.
recently
proposed
do
not
make
the
P-R
the
P-R
an
In fact, a three constant cubic equation
by
Martin
(1979)
holds some promise of
being at least as good as the P-R equation in
However,
equation
all
respects.
equation is among the simplest, and among
the most accurate, of the equations
which
can
be
used
to
attain the stated goal.
The Peng-Robinson equation is expressed as:
RT
Pv-b
a
v(v+b) + b(v-b)
Note the similarity between this equation
van
der
Waals
equation (2.3).
(2.5)
and
the
The parameter b retains the
meaning of the van der Waals covolume, while the
dependent
original
temperature
parameter a = a(T) still represents the attractive
forces between molecules.
The
equation
may
be
rearranged
into cubic form to give
z3
-
3) = 0
(1-B)Z2 + (A-3B2-2B)Z - (AB-B2-B
(2.6)
where
Z = Pv
RT
(2.7)
147
aP
A
B =
For
bP
(2.9)
RT
of
definition
further
(2.8)
a2
RT
terms,
Appendix
A
should
be
The pressure explicit form, equation (2.5) is to
consulted.
be preferred when the quantities T and V are specified, while
the cubic form is used to analytically determine V when P and
T are specified.
have
can
equation
form
The cubic
also
indicates
one or three real roots.
that
the
Equation (2.5)
predicts a critical compressibility factor of
Pv
Z
c
-C
RT
= 0.307
(2.10)
Note that v is determined from equation (2.5), and
in
general
substance.
equal
v,
the
actual
critical
does
not
volume of the
For a real substance,
Pv
c
with Z
RT
RT
(2.11)
c
being less than 0.307 for most materials.
der Waals type equations predict Z
which
is
the
's
greater
Other van
than
0.307,
major reason for their inferiority to the P-R
equation in the critical region.
The more realistic Z
leads to a better prediction of liquid densities by
the
also
P-R
148
equation (Peng and Robinson, 1976a; Martin, 1979).
equilibrium
prime importance.
fugacity
the
calculations,
For phase
is of
of a material
As shown in Appendix A, the
P-R
equation
to the following expression for the fugacity of a pure
leads
compound:
f = Pexp
(Z-)
- ln(Z-B)
A
Aln
Z + (/+l)B
2SB \z - (-l)S J
(2.12)
In contrast to the equation of state for water which was
previously cited, the P-R equation may be applied to any pure
substance provided the
according
constants
a
b
and
are
determined
to that substance's critical temperature, critical
pressure, and Pitzer acentric factor
w
(see
A).
Appendix
The hypothesis that such a procedure is valid is known as the
theory
It is a reasonably accurate
of corresponding states.
assumption for many substances, as Chapter 4 will show.
The
states
corresponding
may
theory
be
applied
to
mixtures as well as pure components, if the appropriate a and
b
parameters
for
the
can
mixture
be
determined.
It is
generally presumed that these parameters can be expressed
a
suitable
combination
comprising the mixture.
by
of the parameters of the components
This
combination
is
carried
out
according to "mixing rules" which have been specified for the
system.
The
mixing
rules
are
necessarily semiempirical,
since a and b are themselves semiempirical.
The mixing rules
149
used with the P-R equation are:
a =
ij
x.x.a..ij
1
ix.b.
b =
where aii=
(2.14)
ai , the single subscript
component i.
Equations
(2.13)
1J
The cross terms aij
(2.13)
referring
to
the
pure
will be discussed shortly.
and (2.14) are the mixing rules originally
suggested by van der Waals in
1890.
Considering
a
binary
system, and keeping in mind the meaning of the parameters, it
is
easy
to
see
the logic behind these rules.
system of components 1 and 2, a must
represent
attractive force between the molecules.
to
the
attractive
parameter
In a binary
the
average
This should be equal
between
any
weighted by the number of times that such a
two
molecules
pairing
occurs.
Thus, for random mixing,
2
2
(2.15)
a = x 1 a + 2Xlx2 a12 + X2 a2
which
is seen to
Similarly,
be
a
special
case
of
equation
(2.13).
b must represent the average volume occupied by a
molecule in the mixture, so that
(2.16)
b = Xlb 1 + x 2 b 2
in accordance with equation (2.14).
been
specified,
it
Once mixing
rules
have
is possible to derive an expression for
the fugacity of a component in a mixture (see Appendix A):
150
f
= xiPexp
b (Z-1)
-
2 r2B
2
n(Z-B)
a
b
n Z + (v'+l)]
Z- (--1)
(2.17)
B
The occurrence of the cross term aij
has been noted.
This term is defined as
1 /iT
a.ij= (1-6.)
where
6ij
is
parameter.
in equation (2.13)
an
empirically
Equation
(2.18)
(2.18)
determined
is
a
interaction
modification
to
the
geometric mean rule
a.. =
(2.19)
a.a.
which was originally
used
by
theoretical
Some
justification for this expression
basis.
van
der
Waals
was provided by the work of London in 1937.
at
large
distances,
the
with
He showed
potential energy r
spherically symmetric molecules
having
little
the
that,
between two
same
size
and
ionization potential is given approximately by
r
= vr-
.2
(2.20)
Although equation (2.20) is an approximate derivation for the
simplest case, equation
complicated
systems.
(2.19)
has
been
applied
an
interaction
many
Good results are not always obtained,
and it is now common practice to modifiy the
with
to
parameter
geometric
mean
when necessary, as shown in
151
equation (2.18).
and
Eckert
would
suggested
first
was
by
in 1965 for a regular solution model.
Prausnitz
It was based on the
molecules
(2.18)
equation
The form of
mixtures
that
premise
complicated
of
deviations from the geometric mean
exhibit
rule and that the geometric mean would actually represent the
the
of
interaction
energy
maximum
value
earlier
(1959) suggested the form aij=
Chueh and Prausnitz adopted the
parameter
interaction
significance,
means of
is
and
determining
geometric
6ij
always
how
mean should be.
j /aiaj ).
formula
While
equation.
Redlich-Kwong
(Prausnitz
on
for
have
does
to
at
a
constant,
composition, and density.
This
physical
from
the
6ij are always chosen
Values for
least one data point.
as
the
deviations
the
large
equation
of
Use of a number of data
the
sensitive to the value of the parameter.
6i j
grounds
between 0 and 1, there are no
points is often recommended, since
take
the
some
empirically, by determining the best fit of the
state
In 1967,
with
use
these
had
equations
are
very
It is also usual to
independent
of
temperature,
convention
is
followed
in
this thesis.
Since the interaction parameter
from
represents
a
deviation
a mixture of molecules possessing spherically symmetric
interactions, one would
expect
systems approaching this ideal.
small
values
of
6ij
for
Thus, for the ternary system
152
Peng
methane-ethane-propane,
For systems which are asymmetric, polar, or otherwise
6ij=0.
of 6ij
value
the
nonideal,
may be large or small, or even
of
negative, depending on the interplay
nonidealities.
the
6ij is not
is usually observed that the optimum value of
constant in
and
Zudkevitch
systems.
these
function of temperature and/or pressure for
a
over
vary
conditions
sizeable
Robinson (1980) used a temperature dependent
in
cases
Peng and
range.
6
j
which
with
their
the calculation of aqueous-organic equilibria.
for
equation
(1970)
Joffe
interaction parameter be established as a
the
that
suggest
these
all
i.e.,
rule,
mixing
suitable to use the geometric mean
It
(1976) found it
Robinson
and
The interaction parameter, expressed in the form of
equation
(2.18), was allowed to vary with temperature to give the best
fit
the
to
indicating
Its
data.
the
that
value
increased with temperature,
energy
attraction
between
molecules decreased with increasing temperature.
did
not
parameter.
a
derive
an
unlike
The authors
explicit temperature dependence for the
(1980) have
Similarly, Lazalde et al.
suggested
temperature dependent correction to the geometric mean for
systems of polar molecules.
Eij =(l
where
eij
accounts
is
a
This expression has the form
+
v
characteristic
(1.21)
.
energy.
The
constant
c
for orientational effects, and may be identified as
an interaction parameter.
Lazalde,
et
al.
have
proposed
153
this
since
form
important at low
temperatures
should
effects
orientational
than
high
at
be
more
temperatures.
Energetically stronger interactions would be favored at lower
ij should rise as temperature falls.
temperatures, so that
In
contrast
to
the
many
the solid phase is quite standard.
often
is
found
that
considered
in
little of the more volatile
very
this
systems,
For solid-fluid
component dissolves in the solid phase.
systems
for
fluid phase behavior, thermodynamic analysis of
representing
it
available
choices
This is true of
the
to
the
thesis,
and
leads
assumption that the solid is present in a pure form.
4, which deals
with
pure
component
phase
Chapter
diagrams,
will
discuss the treatment of the solid phase.
2.A
Scope and'Context Qf Present Work
The
objectives
of
this
thesis
are,
in
order
of
importance:
a) To illustrate a method for
phase behavior of binary systems.
important
overall
This capability is
in process design feasibility studies.
experimental phase equilibrium
of
the
determining
results
studies,
In
consistency
may be checked, and regions of the P-T-x
space requiring special attention are indicated.
b) To illustrate the application
of
an
algorithm
for
154
phase equilibrium calculations and the interpretation
of
results.
its
Computational algorithms sometimes
experience convergence problems in the
and
critical
azeotropic regions, and can also give erroneous phase
predictions.
equilibrium
these problems should
careful examination of
A
indicate
how
they
might
be
avoided.
c) To investigate the applicability of the Peng-Robinson
equation of state to the supercritical fluid
region.
If sufficiently accurate results can be obtained, the
will
equation
be
useful
in
process
design
and
correlation of experimental data.
Certain aspects of this study have been treated by other
Scott
see
The work of van Konynenburg (1968;
investigators.
also
and van Konynenburg, 1970) is most closely akin to the
first objective.
of
Using the van der Waals equation
state
(2.3), van Konynenburg showed that by varying the constants a
and
b,
most
types
known
of binary fluid-fluid equilibria
could be qualitatively described.
according
to
the
presence
Systems
were
classified
and location of critical lines,
three-phase lines, and azeotropic lines.
Van Konynenburg did
not undertake predictions for particular
binaries,
and
did
not consider the formation of solid phases.
Predictions of fluid phase equilibria have been
carried
out since the time of van der Waals by numerous workers.
For
155
of
treatment
systems,
binary
critical region is also
the
In contrast, prediction of the solubility
fairly common.
of
solids in supercritical fluids has received limited attention
The earliest work on this topic, which
until quite recently.
(1959).
Rowlinson and Richardson
In 1965, Prausnitz reviewed
treating the
equation,
Redlich-Kwong
modified
a
fluid as a dense gas, while the second treated the
an
these
of
The first
two standard approaches to the problem.
utilized
been covered by
has
state,
of
equation
virial
the
used
fluid
as
expanded liquid and was based on regular solution theory.
Neither of these methods employed
predict
Peng and Robinson (1978) used their equation to
up
locus
S-L-G
to
been
determined
at
Excellent results
parameter
interaction
were obtained using a constant binary
had
the
region of methane in the
critical
the
cryogenic methane - carbon dioxide system.
which
parameter
interaction
results were only semiquantitative.
surprisingly,
not
and,
an
vapor-liquid
conditions removed from the three-phase line.
equilibrium
authors
These
have also given references to several similar studies.
Paulaitis
In 1979, Mackay and
predictions
solubility
accuracy.
although
in
the
fluid
The
an
which
is
proposed
gives
treated
a
method
for
reasonable quantitaive
as
an
expanded
liquid,
equation of state (Redlich-Kwong) is still used
calculations.
correlation,
a
coefficient
at
Two
parameters
are
used
in
the
binary interaction parameter and an activity
infinite
dilution,
the
latter
being
156
temperature
dependent.
Modellr
et
al.
(1979) presented
semiquantitative solubility predictions using the
approach
with
the
Peng-Robinson
equation
constant binary interaction parameter.
diagrams
well
as
as
of
gas
state and a
That paper summarizes
some of the early work on this thesis, and
P-T-x
dense
includes
calculations
several
critical end
of
points.
Within
the
supercritical
et
al.
year,
a
number
have
shown
that
reasonably
on
Kurnik,
quantitative
result with the dense gas approach if the binary
interaction parameter is allowed to
Kurnik
studies
of
fluid solubility have been published.
(1981)
predictions
past
vary
with
temperature.
(1981) has also extended these predictions to systems
of two solids in equilibrium with a supercritical fluid.
The
P-R equation correctly predicts an increase in the solubility
of both components over what
Francis
with
and
a
would
be
found
individually.
Paulaitis (1981) have used the dense gas model,
modified
semiquantitative
Lee-Kessler
results
equation,
with
and
constant
a
obtained
interaction
These authors have also predicted upper
parameter.
critical
of success.
Johnston and
Eckert (1981) have used a perturbed hard sphere
modification
end
of
points
the
with
van
der
a
fair
Waals
degree
equation to calculate solubilities.
Reasonable results were obtained with
dependent
interaction
parameter.
however, to fairly low solubilities.
a
The
single
temperature
method was limited,
157
Chater 3
Theoretical Considerations
concisely
Phase equilibrium in chemical systems is most
by
defined
The first of these
four mathematical relations.
is the criterion of equilibrium, which establishes whether or
not phase equilibrium is theoretically possible.
second
The
of these determines if the theoretically possible equilibrium
is
stable or unstable, and is thus known as the criterion of
stability.
Stable equilibrium can be realized
while unstable equilibrium cannot.
criterion
of
criticality.
This
in
practice,
The third relation is the
criterion determines if a
type
theoretically possible equilibrium is of a limiting
which
two
coexisting
phases have become indistinguishable.
Such an equilibrium is said to represent
The
final
in
a
critical
point.
relation, that of critical stability, defines the
stability or instability of a critical point.
The
phase
consideration
equilibrium
of
equilibrium state
relations
the
variations
might
undergo.
a
system
The
it
will
suffice
workers and show how these
component systems.
in
a
a
stable
derivations
become
fairly involved, and are best treated elsewhere.
purposes,
from
result
For present
to cite the results of previous
results
apply
to
one
and
two
158
A typical derivation of the criterion of equilibrium
given
by Modell and Reid (1974).
Taylor series expansion of a
energy
is
These authors consider the
perturbation
about an equilibrium state.
in
the
internal
The internal energy is
given by the Fundamental Equation
U = U(S, YV, N 1 ,...N
(3.1)
n)
The expansion is written as
AU =u
+ ~.
+ 1
2u +
(3.2)
m
where
n+2/aU
6z(3.3)
sU =
2
n+2
n+2/
a2U
6 i-l
Ez E
=l
--
j
zi
1
J
zi 6Zz
(3.4)
Z
etc.
The symbol 6 represents a
quantity,
while
zi
set s, V, N1 , ..., Nn
small
but
finite
change
in
a
represents a particular variable of the
For a system at
equilibrium,
it
is
found that
6U = 0
Equilibrium states are
energy functionality.
t
therefore
(3.5)
extrema
in
the
internal
159
For a one-component system, equation (3.5) becomes
6V +
S
+
S
6U =
N
6N
S,V
-N
_,N-
6N
= T6S - P6V +
(3.6)
criterion
If only one phase is present, the
does
not
lead
to
any
constraints
of
equilibrium
If two
on the system.
phases are present, however, it is necessary that
6V = 6v
I
+
II
II
= 0
(3.7)
= 0
(3.8)
and
6N =
N I + 6N
The superscripts I and II denote the two
coexisting
phases.
By equation (3.6), assuming T # 0,
aS = 6S
as well.
TI S
Since 6UI
=
+ 11II
_ pI VI +
6N
-PV
+ S
I
(3.9)
=
S-UII, equation
_(TIIII
= -(T
s
(3.6) gives
-P II 6VII +
II SN
NII)
(3.10)
Rearranging and using equations (3.7) to (3.9), it
is
found
that
CTI -T.TII )S
I
-(P
6vI + (vi-i
pI -PpII )6V
+ (I
II)6N I = 0
)SN =0
(3.11)
160
Since S I, V I, and N I
follows
are independently variable, it
that the constraints for two-phase equilibrium are
TI = T
p
I
= P
JI =
(3.12)
II
(3.13)
(3.14)
II
Were a third phase present, it too would have equal T, P, and
(For a pure
chemical potential.
occur
at
the
easily extended to a
can
only
temperature
and
of all phases present are equal, and any particular
pressure
same
component must have the
phase.
The
system.
binary
this
These well-known results are
point.)
triple
component,
chemical
potential
every
in
An equivalent expression for the equality of chemical
potentials
is
the
equality
of
fugacities.
From
the
definition of fugacity, it follows that
o
ii-
V
= RTln
f
(3.15)
f
f-
where the superscript 0 denotes a reference state at the same
temperature as the system.
Applying equation
(3.14)
for
a
component i in phases I and II, one may write
1fhs1I I=
uI ,
1
fII
(3.16)
fII
1
(3.17)
161
of
States which satisfy the criterion
said
to
locus may
be
equilibrium
state,
the
or
internal
(3.5), then,
to equation
addition
stable
either
For
unstable.
a
lowest
mu is the
(3.2)
equation
stability.
(m>l).
it is required
that
(3.18)
order
nonvanishing
form of equation
> 0
in
(3.19)
point,
the
proper
(3.18) is
62u
> 0
(3.20)
The full expression for
(3.4).
Beegle,
the
et al.
42 U
term
squares.
was
given
in
(1974a,b) have shown that
the right hand side of that equation may be
of
variation
Equation (3.18) is the criterion of
Excepting the special case of a critical
sum
In
It is equivalent to the statement:
AU
equation
stable
is a minimum.
energy
6mu >
where
are
A region of the binodal
on the binodal locus.
lie
equilibrium
expressed
as
a
The condition that the coefficient of each
squared term must be positive (to be certain of stability for
all possible variations in the zi ) leads to an expression
the
criterion
of
stability
of
in terms of Legendre transform
theory:
(k-l)
> 0,
k
=
.....
,
n+l
(3.21)
162
Here y (k-1)
see
is the
Beegle,
second
et
order
(k - 1) Legendre
al.,
partial
transform
of
1974b) and the subscript kk denotes a
derivative
with
respect
to
independent varible k in the Fundamental Equation.
of
stability
kk
as
the
the
The limit
is reached when any one of the y(k-l)
equal to zero.
(y = U;
y
becomes
The locus of the limit of stability is
spinodal (or spinoidal) curve.
known
If one starts with a
stable system, the criteria for stability may
be
simplified
to
Y (n_1) (n-l)
n
(n-)(n)
since the transform
become zero.
if
the
necessary
(3.'22)
-
is always among the first to
To determine a priori if a system is stable, or
term
y(n
to
(n-
evaluate
has
all
become
of
the
undefined,
it
is
partial derivatives in
equation (3.21).
For
a
pure
component,
the
stability
criteria
from
equation (3.21) are:
)
/23T
V
Y11
·
al
2
,N
as
_=
,N
>
323)
(3.23)
i
1ap\>
NkT
(3.24)
I
N
NN
and
/2
1
Y22I= --Avv
faA
(ap\
2N T,N
-
_
163
where A is the total Helmholtz free
energy.
For
a
binary
system, a third condition must be satisfied:
2=G
=
Y33
G
>0
--
where G is the total Gibbs free energy.
as
known
of
Equation
(3.23)
state,
Equation (3.24) is
need not be considered further here.
the criterion of mechanical stability and equation (3.25)
the criterion of material stability.
from
an
to
Equilibrium predictions
Otherwise, they are unstable.
Two other forms of the
relevant
material
the present work.
stability
criterion
For a pressure explicit
equation of state, it is more convenient to work with A
E.
Beegle,
is
of state which satisfy these inequalities
equation
are deemed stable.
are
is
The heat
stability.
thermal
cannot be determined from an equation of
capacity C
and
criterion
the
(3.25)
=
et al.
than
(1974b) have shown that the appropriate
stability criterion in this case is
All AlV
L1 =
-
AllAv
-AV
>
(3.26)
A-v V
In this notation, L
terms
of
the
symbolizes
Legendre
a
transform
stability
criterion
in
of the first independent
variable in the Fundamental Equation (3.1) (
in this case).
164
(1974a) have also shown that by use of a
Beegle, et al.
step
down
directly
energy.
operator,
into
(3.25)
equation
may
be
transformed
expression in terms of the Helmholtz free
an
This expression is
A2
Either
(3.26) or
equation
(3.27) is
material stability, provided A W
AW
is
negative,
one
of
(3.27)
>
Gll = All
a
for
is a positive quantity.
the
equations
stability and the other instability.
Thus,
mechanical
A W<0)
instability
criterion
valid
(i.e., when
will
in
indicate
regions
it
If
of
is best to
consider material stability as being undefined.
Reid and Beegle (1977) have shown that the mole
in
equations
fractions.
(3.25)
to
(3.27)
may
be
numbers
replaced
by mole
Another equivalent criterion for equation
(3.25)
therefore is
---xf =RT
) >0
ax
,P
\
aXl
T,P
At the limit of stability, these
zero.
partial
Thus,
at
fugacity
constant
versus
T
mole
(3.28)
derivatives
and
P,
are
equal
to
extrema in a plot of
fraction
correspond to points on the spinodal curve.
of
a
component
165
point
As with the spinodal criteria, critical
may
be
for
Modell (1977) and
the
mathematical
at which a stable system reaches
point
the
that
shown
have
(1977)
Reid and Beegle
conditions
a variety of ways.
in
expressed
criteria
criticality are given by:
=
)(n+l)
Y(
n
(3.29)
(3.29)
Y(n+l)(n+l)(n+l)
Yn+l)
(also
in this context),
criterion
is
(3.29)
Equation
criterion
(n+l)
0
(3.30)
(330)
(3.31)
(n+l) > 0
the
again
known as the first criterion of criticality
while
equation
(3.30)
is
result
variations in
(3.31),
the
critical
second criterion of criticality) and equation
(or
(3.31) is the critical stability criterion.
criteria
stability
material
last
two
from considering the third and fourth order
equation
should
These
the
(3.2),
respectively.
In
equation
equal sign apply, the greater than sign
must hold for the next nonvanishing derivative.
For a pure component, it has been
(3.29) becomes
P) -0
IQV T N
shown
that
equation
(3.32)
166
Equations (3.30) and (3.31) then become:
(a2)o
= 0
(3.33)
>0
(3.34)
3
_
-
- T,N
For a stable binary system, the criteria.become
X(T
1 )
(3.35)
-0
N1 TP,N 2 \N1 T,P,N 2
=
-=
1 T,P,N 2
1 ,PN2
--T =P ,N2
As
by
implied
criterion,
the
unstable
>
,;,iŽ
T,P,N 2
TPN
(3.37)
critical
stability
points may exist.
Figure 3.1
of
necessity
critical
and
a
critical
unstable
(see also discussion of Figure 5.8a).
a
0
\3N 1 T,P,N2
gives a comparison of stable
derivative
(3.36)
points
Figure 3.la shows the
evaluated along the
coexistence curve
in the region around a stable critical point.
The derivative
critical
point where it
is always
positive,
except at the
167
0
3N
N
/T,P,N2
0
N1
N
1
a.
b.
Stable critical point
Figure 3.1.
Unstable critical point
The two types of critical points.
evaluated along the binodal locus.
The ordinate is
The stable critical point is a
terminus of a stable binodal locus, so that
-
is always > 0.
aN1
2
a lap\
aIl
At the critical point,
-
= 0,
-
=0, and
-
aNI
3NI
aN1
aN
The unstable critical point is a terminus of an unstable binodal
all
locus, so that
-
is always < 0.
At the critical point,
aN 1
a
a= 0,
3N1
=
= 0, and
=0,
NaN
2
G
.
aNI
< 0.
> 0.
168
is zero as
At the critical point, (2_
equals zero.
?,N2
stab ility criterion,
, the critical
well, while
--
is positive.
Figure 3.lb shows the
spinodal
the region around an unstable critical point.
along
the
coexistence
point,
critical
is
curve
negative
At the
where it is zero.
T,P,N
in
The derivative
except
at
the
critical point,
is ne,gative.
is zero and
1
derivative
2
2
may be replaced by f
In equations (3.35) to (3.37),1
(or RTlnf1 ), and N1
by
xl,
without
loss
of
generality.
Thus, equation (3.36) corresponds to an inflection point in a
plot
of f
versus xl.
(1977) equation
(3.36)
Helmholtz
energy,
free
Using the results of Reid and Beegle
can
be
for
written
use
in
terms
of
the
a pressure explicit
with
equation of state, as follows:
AVV A1V
Ml =
=Ll L1
V
=A
-ll
2 - AVVVA
Vl
-
3
VVAlA-
V
N1
+ 2A
-IV2 +
A
VV-1
_11lV
l = 00
M1 symbolizes a critical point
Legendre
v
transform
criterion
(3.38)
in
terms
of
the
of the first independent variable in the
Fundamental Equation (3.1).
Again,
this expression is also valid.
the
intensive
form
of
169
above
There are many ways to use the criteria developed
to
desired goal.
a
achieve
criteria need be employed.
In most cases only some of the
For example, to generate
binodal
only the criterion of equilibrium need be applied.
loci,
only
critical lines are to be calculated, on the other hand,
the
criteria of stability and criticality are necessary.
the generation of binary phase diagrams, it
the
that
shape
of
the
determine regions of
unstable
well
out
as
stable
and
The specific application of the
points.
critical
as
instability,
turns
In
sufficient to
is
curves
binodal
even
If
criteria in the present work will now be described.
Pure
phase
component,
equilibrium
tielines
are
conventionally drawn for isotherms in a P-v diagram, as shown
in
Figure
3.2.
The
curve
has two extrema which define a
region of three possible volumes for a given
lowest
the
of
vapor
unstable
The
phase.
solution
since
TV= 0, so these must be
extrema<,
The tieline connects the volumes
liquid
and vapor phases.
a
volume is a
T > 0.
of
the
At
the
spinodal
coexisting
It is distinguished from the other
constant pressure (horizontal) lines
the
middle
points on the
curve.
between
The
these volumes corresponds to the liquid phase and
highest to the
mechanically
pressure.
which
might
be
drawn
liquid and vapor segments in that the phases it
170
P
Oft
V
Figure 3.2.
A typical
The tieline,
which gives the volumes of the coexisting
and vapor phases,
P-v isotherm for a pure component.
is shown dashed.
liquid
171
.connects have equal fugacity.
For a binary system, an
chemical
or
potential
This might be anticipated by comparing
analog of Figure 3.3.
(3.32)
the pure component critical point criteria (equations
to
with those for a binary system (equations (3.35)
(3.34))
a
contain
curves
potential-composition
information about the phase behavior of a
the
thesis,
a
curves were used to develop
point,
of
examples
it
is
H)
subscript
the
(n-C
-
carbon
for
in detail in
discussed
with gaseous C
components
to
(f-x)
H 34,
are
2
Figure 3.3
plots.
as
abbreviated
C16
system,
dioxide
present
separate
of
result
at
a
The systems
mixing
liquid
, and VLE is to be expected if the two
in
The most striking feature of
two
some
consider
of 300 K and a pressure of 0.1 atm.
represented by this diagram are a
actually
algorithm
fugacity versus n-hexadecane mole fraction
n-hexadecane
temperature
n-C16
this
of fugacity-composition
instructive
fugacity-composition
gives a plot of n-hexadecane
for
In
system.
of
C.
At this
or
as
deal
great
computational
equilibrium,
phase
determining
Appendix
shapes
characteristic
chemical
and
Fugacity-composition
(3.37)).
to
of
composition is the
versus
fugacity
plot
isobaric
isothermal,
a certain range of proportions.
the
curves.
plot
that
is
Segment
G
there
are
is essentially
linear over the entire range of composition, with a slope
of
172
log fH
V1
X
V2
X
X .
log xH
Figure 3.3.
mn
An fH- xH plot for n-hexadecane - carbon dioxide.
Segment G is the gaseous root, segment U is the mechanically unstable
root, and segment L is the liquid root.
Segments U andL
start at
the slope- infirity and-.cont-inueto the pure hexadecane axis.
173
unity
and
an
intercept
at
the pure C
axis of 0.1 atm.
16
This is the equivalent of ideal gas behavior, since
fH
where PH
which
=
XHP = PH
(3.39)
is the partial pressure of C16.
The second
curve,
is found for compositions greater than the composition
of the
df
infinity
in
H
,
gives
the
mechanically
unstable
dx
H
(segment U) and liquid (segment L) fugacities.
of this curve corresponds to
roots
to
the
cubic
the
equation
existence
(2.6).
The existence
of
three
real
The partial fugacity
expression, equation (2.17), is evaluated at a given xH
each
for
of the three volumes resulting from the cubic equation.
As usual,
the
highest
fugacity,
the
middle
fugacity,
and
the
fugacity.
As
volume
root
gives
the
gas
phase
volume root the mechanically unstable
lowest
volume
root
the
liquid
phase
indicated by the crossing of segments G and U
in the graph, the middle volume root does not always give the
middle fugacity root.
The mechanically unstable root may be recognized in
plots
f-x
since it is associated with the joining of the gas and
liquid roots.
This
will
become
pressures (as in Figure 3.4).
The
more
evident
actual
at
higher
violation of the
174
mechanical
stability criterion, i.e.,
aV T > 0, may
that
not be determined from the plot.
On the other hand, material
stability
in
is
readily
apparent
dfH
dx > 0, the system is
segment
L
between
materially unstable.
f-x
materially stable,
the system is materially unstable.
of
the
the
Note
and when
At 0.1 atm,
infinity
that
plots.
and
the
the
the
When
dfH <
( 0,
portion
minimum
mechanically
is
unstable
segment U appears as materially stable in this plot.
Suppose now that suitable quantities
are
present
in
the
system
so
and xL, and fugacity fE
that
lie
must
furthermore,
For the
that
coexisting
between
f
H
is
vapor
straight line segment,
x
between
phase,
equality of
C16
and
CO2
that two phases exist with
compositions xv
xL
of
The f-x plot shows
= 0.15
f (x
H
and
min
which
xH = 1,
) and f (x
H
must
lie
and
= 1).
H
on
the
fugacities requires that
xV l < xV <V2
(3.40)
This limiting procedure illustrates the usefulness of the f-x
plots in determining two phase equlibrium, and is one of
the
steps in the phase equilibrium algorithm detailed in Appendix
C.
To find the actual compositions for VLE, it is of course
necessary to match partial fugacities of
well.
carbon
dioxide
as
175
Consider now Figure 3.4, which
0.5
atm,
the
fugacity
(mechanically
unstable
circumstances
is
that
phase
C
axis.
and
U
L
solutions,
effect
One
slightly.
of
unstable and vapor phase
the
on
fugacity
solutions now terminate at nearly the same
pure
to
increased
segments
of
liquid
and
respectively) has also increased
these
a
the vapor phase (segment G) has
of
The fugacity
increased accordingly.
is
If pressure
atm, has just been discussed.
in
plots
The first of these, at 0.1
of 16 pressures at 300 K.
series
f-x
gives
the
A second effect is that the composition of
16
the vapor phase is at a lower x
than it
was
at
0.1
atm.
This must be so because the liquid phase fugacity can only be
matched
if
the
vapor
phase composition is less than x
=
10 -5
At 1 atm, the vapor and unstable solutions
at
a
have
joined
peak, so that the region of three roots for a given xH
no longer extends all the way to
xH
1.
second infinity and
continuous, and a
The
curve
is
second zero appear in
dfH
.
The two zeroes, i.e., the maximum and the minimum, give
H
two points on the spinodal curve at this T and P.
phase
solution
is
no
longer
a
The
vapor
straight line, indicating
deviations from ideal gas behavior.
At 5 atm, the peak has rounded
trend
toward lower C16
substantially,
and
the
mole fraction in the VLE vapor phase
176
2
0
-2
log
fH
-4
-6
-8
.
2
0
-2
log fH
-4
-6
-8
-5
-4
-3
-2
-1
log xH
0
-5
-4
-3
equation with 612=0.081.
-1
0
log xH
Figure 3.4. (First of four pages) fH-xH plots
carbon dioxide system.
-2
for
the
n-hexadecane-
The curves are calculated from the P-R
Segments labelled G represent gas fugacities.
Segments labelled U are mechanically unstable.
are liquid phase fugacities.
Segments labelled L
177
2
0
-2
log fH
-4
-6
-8
r
300 K
60 att
0
-2
log
-4
-6
-8
-5
-4
-2
-3
log
Figure 3.4.
-1
0
-5
xH
-4
-3
-2
-1
log xH
(Second of four pages) f-x plots for the n-
hexadecane - carbon dioxide system.
0
fH
178
2
300 K
65 atm
300 K
64.3 atm
0
-2
log XH
-4
-6
-8
300 K
70 atm
300 K
67.2 atm
2
-2
log fH
-4
•'~~~~~~
-6
-8
-5
-4
-3
log
-2
-1
0
-5
-4
xH
-3
-2
log xH
Figure 3.4. (Third of four pages) f-x plots for the
n-hexadecane - carbon dioxide system.
-1
0
179
2
300 K
75 atm
300 K
73 atm
0
-2
log fH
-4
-6
-8
2
300 K
85 atm
300 K
0
300 atm
-2
log
-4
-6
-8
.
.
I
-5
-4
-3
-2
-1
0
-5
-4
log xH
Figure 3.4.
-3
-2
-1
log xH
(Fourth of four pages) f-x plots for the
n-hexadecane - carbon dioxide system.
0
fH
180
By the time 20
continues.
has
trend
begun
to
is
atm
reached,
reverse itself.
This is an indication
that, for a P - x phase diagram at 300 K, there is a
with respect to xH
this
however,
minimum
At
in the vapor branch of the VLE curve.
20 atm, it might also be noticed that the materially unstable
This is more
segment is beginning to show a plateau.
liquid
At these pressures the
evident at 40 and 50 atm.
phase
is
xH
coming
back
into
the
vapor
VLE
range covered by the
concentration axis.
It
is
worthwhile
interpretation
the
this
point
redrawn
side,
present
atm
50
has
in Figure 3.5 with a hypothetical VLE tieline.
but
this
discussion).
compositions
the
f-x plot over the entire composition
(The actual VLE tieline would extend off
vapor
consider
to
For illustrative purposes, the plot for
range.
been
of
at
are
graph
on
the
not affect the validity of the
does
two - phase
The
xV
by
denoted
the
and
equilibrium
xL.
VLE
For an overall
system composition of xH < xv, only one phase is present vapor,
with a fugacity as given on the f-x plot (extended to
lower xH
between
as need be).
For
an
overall
composition
system
x V and xL, the stable system is two phases of these
compositions.
affects
a
the
Varying composition between xv
proportion
of
two phases.
the
beyond xL, there would again be
a
single
liquid with the fugacity shown on the plot.
and
xL
only
Increasing xH
phase
system,
a
181
log fH
x
V
x
max
x .
min
X
log xH
Figure 3 .5.
phases.
An f-x plot illustrating the possibility of metastable
and xmax,
Between x
Between x
L
max
a metastable vapor phase may exist.
n and x , a metastable liquid
mln
phase may exist.
182
The f - x plot, however, indicates all possibilities for
For
example,
increasing
xH
beyond x
considered.
just
phase behavior, not only the stable states
will not invariably
that
It is possible
lead to the formation of two phases.
a
single, metastable vapor phase will instead persist, with the
A metastable vapor phase is
fugacity given by the f-x curve.
for
possible
becomes
vapor
two-phase
system.
x
At
is
to x
This
zero.
a
mechanical
the
an example of the stability
is
et
al.
(1974b)
and moves toward a region of instability, the
criterion
to
reverts
-
if
one
a stable system (metastable vapor in this case)
with
starts
and
than
rather
criterion as stated by Beegle,
. At this point, the
is the material stability
it
max
violated,
criterion.
max
unstable
intrinsically
criterion which
stability
up
compositions
highest
order
is always the first, or among the first, to become
As has been seen, the region of mechanical instability
does not begin until the infinity is reached.
An argument similar to that above may be applied to
compositions
phase
liquid
in
Figure
3.5.
with only a single liquid phase, it is possible
xH
to
phase.
,
xmin
all
Starting at xL
to
decrease
the while maintaining a single liquid
the system
At x
the
intrinsically
becomes
unstable
min
and must revert to two-phase behavior.
At 60 atm, two interesting developments have taken place
(see
Figure
3.4).
The
plateau
after
the
low
fugacity
183
rise to a minimum and maximum, making a
given
has
infinity
There is
total of four spinodal points at this T and P.
one
of
material instability, and three regions
latter
The
stability.
phases
three
having
of
possibility
the
indicate
of
regions
three
instability,
mechanical
of
region
now
simultaneously in equilibrium, as the range of fugacities for
required,
It
degree.
each segment overlap to some
is
no
by
means
though, that three phases coexist at this T and P.
For this binary, there is only one
at
pressure
K
300
for
liquid-liquid-vapor equilibrium can occur (see Figures
which
Other possibilities at
2.8 and 2.9).
pressure
this
a
are
single phase, two-phase equilibrium, or two sets of two-phase
Which of
equilibrium (i.e., liquid-liquid and liquid-vapor).
in
these
rising
is
composition
phase
composition
the
on
depends
of about
liquid
phase
vapor phase fugacity.
now
constrained
= 2 x 10- 5
x
fugacity,
A system
,
system
overall
It is also seen at this pressure that
composition.
vapor
occurs
fact
due
VLE
the
to a minimum
partly
to
a
but mostly to the depressed
of
lower
xH
would
be
a
single vapor phase, with a fugacity as given by the f-x plot.
Considering
the series of f-x plots from 20 to 60 atm, it is
evident that the rising pressure
solubility
of C
has
led
to
an
in CO .
16
2
Between 64.3 and 65 atm, a closed loop develops
f-x
plot.
increased
The
closed
coordinates, so the 65 atm
loop
does
section
not
has
in
the
show up in log-log
been
replotted
in
184
Cartesian
All three roots of the
coordinates in Figure 3.6.
fugacity equation converge at the origin, as required by
form
xH
of
0.
(2.17) as x H
equation
of fCO
(A plot
would have three different real roots at x
the
versus
= 0, as
was
= 1). With the collapse onto
true for f
at 0.1 atm and x
H
H
the origin, the curve loses an infinity and a minimum, and is
no
statement of
(1974b),
as
the
stability
given
applied carefully.
phase
vapor
in
criterion
this case the
et
Beegle,
by
al.
conjunction with Figure 3.5, must be
Conceivably, one
metastable
or
In
differentiable.
continuously
longer
track
could
stable
a
liquid phase back to xH
then continue, with increasing x H , along the middle
0 and
=
fugacity
solution without violating a criterion of material stability.
Beegle's
Using
definition,
it
might
mechanical stability criterion could
in
when
fact it has been.
non-differentiable
procedure
be
a
yet
be
violated,
It is therefore important that a
encountered
poin
treated
not
thought that the
be
violation
in
Df
the
racking
the
stability
criterion.
Returning to the sequence of f-x plots in Figure 3.4, it
is seen that the loop shrinks considerably in moving from
to
67.2 atm.
Upon reaching 70 atm, the loop has disappeared
entirely, and there are no longer
three
mole
solutions to the cubic form exist.
fractions
are
at
which
In general, a loop
is found if and only if the P and T coordinates
plot
65
of
the
f-x
near the vapor pressure curve of a pure component.
185
3
o
0
.uuu3
.0010
.0015
.0020
.0025
XH
Figure 3.6.
A Cartesian plot
illustrating
the closed
loop
behavior of the fH- XHcurve at conditions
near C02's vapor
pressure.
186
At 300 K, the vapor
while
atm,
0.1
at
found
axis
closed loop terminating at the pure C2
a
After
was found between 64.3 and 70 atm.
a
pressure
vapor
pure component critical point, loops
a
at
terminates
line
Thus, an open
was
loop bounded by the pure hexadecane axis
10- 6
x
4
is
n-hexadecane
that of carbon dioxide is 66.3 atm.
and
atm,
of
pressure
terminating at that axis will not occur.
From 70 atm on up, the f-x curves are all
possessing
same,
the
those
with
curves
Simple fugacity-composition
one minimum.
and
maximum
one
only
qualitatively
two
and
extrema,
which are monotonic in composition, comprise the large
majority of curves at the various temperatures and pressures.
evident
entirely
not
is
This
and
temperature
from
were
pressures
chosen
carefully
illustrate the various shapes that are encountered.
though
realistic,
somewhat
still
biased,
a
for
matrix
seven
of
temperatures
pressures are given there, the large majority of
at
most
two
extrema.
As
might
may be
The
and
to
more
A
picture
obtained by studying the f-x curves in Appendix B.
plots
where
3.4,
Figure
f-x
sixteen
which
have
be expected, the simpler
curves are the easiest to analyze and solve for VLE.
Until now only the shapes of hexadecane
have
been
considered.
fugacity
Figure 3.7 shows how carbon dioxide
fugacity compares with hexadecane fugacity for
series at 300 K.
curves
the
pressure
It is not possible to show all details with
187
300 K
2
L
L
300 K
0.1 atm
0.5 atm
G
log
fco2
log
C16
0
G
-1.0
-2
0
-2
-4
-6
-8
-5
-4
-3
-2
-1
0
-5
-4
-3
log x
-1
log x
C 16
16
Figure 3.7.
-2
(First of eight pages) fo
n-hexadecane - carbon dioxide system.
-xH
plots for the
Note the one-to-one
correspondence between slope infinities and extrema.
Segments are numbered as in Figure 3.4.
0
188
2
1
log fco
Co 2
0
-1
2
0
-2
log fC
16
-4
-6
-8
-5
-4
-3
-2
-1
0
-5
log Xc
16
Figure 3.7. (Secondof eight pages)
-4
-3
-2
log x C
16
-1
0
189
I
I
log fco
20
-2
-2
log fC
C16
-4
-6
-8
-5
-4
.3
-2
log Xc
-1
0
-5
-4
-3
log x
16
C16
F igure
-2
3. 7
(Third of eight pages).
-1
0
190
I.
b3
.64
1.61
.63
300 K
50 atm
log fo
20
r
1.59
.62
1.57
.61
I
300 K
0
60 atm
*-2
log f
-4
-6
-8
L
-5
-4
-3
-2
-1
0
-5
-4
-3
·
1
-2
-1
log x
log x
16
Figure 3.7. (Fourth of eight pages)
C 16
0
C1 6
191
1.639
1.637
log fco
1.635
2
1.635
1.633
2
300 K
65 atm
300 K
64.3 atm
0
-2
log fc
C1 6
-4
-6
-8
-5
-4
-3
-2
-1
0
-5
-4
log Xc
C 16
Figure 3.7. (Fifth of eight pages)
-3
-2
log X
16
-1
0
192
1.643
1.65
1.641
1.63
log fco
1.639
1.61
1.637
1.59
2
300 K
70 atm
300 K
67.2 atm
0
-2
log
f16
16
-4
-6
-8
-5
-4
-3
log
-2
-1
0
-5
-4
Xc
C1 6
Figure 3.7.
(Sixth of eight pages)
-3
-2
log Xc
-1
0
193
65
63
log fco
20
61
59
2
300 K
75 atm
300 K
73 atm
0
-2
log fc
16
-4
-6
/I~
-8
-5
-4
-3
-2
-1
0
-5
-4
log x
-3
log
C1 6
Figure 3.7. (Seventh of eight pages).
-2
Xc
16
-1
0
194
1.84
1
67
65
1.84
log fco
63
1.82
61
1.80
I
300 K
85 atm
0
300 K
300 atm
-2
log fc
16
-6
,
-8
-5
-4
-3
-2
-1
0
-5
log XC16
16
Figure 3.7. (Eighth of eight pages).
-4
-2
-3
log x
C 16
-1
0
195
a
uniform fugacity axis for the CO
In a region
there must be noted.
lower
C
plots, so scale changes
of
three
solutions,
the
root is associated with the higher CO
fugacity
2
16
and
vice
root,
unstable
root connects the two.
and the f
curves
have
while
versa,
fugacity
a
the
mechanically
As expected, the f
one-to-one
H
curves
correspondence
in
CO2
That this must be so is evident upon
extrema and infinities.
writing
Gibbs-Duhem equation for the partial fugacities
the
in a binary
system
T and P:
at constant
x1 dlnf1 + x 2 dlnf 2 =
(3.41)
Rearranging and dividing by dx2 , one has
dlnfl
x
1
dx
Clearly,
dlnf 2
=
2
dx 2
-x
2
(3.42)
if
dlnf2
dlnfl
CX then
-
dx 2
-a
dx 2
and if
dlnf
dlnf1
2
0O, then
dx 2
0
dx 2
as well, unless x - O.
If x -0
2
2
,
then either
dlnf 2
dlnf 1
0
dx 2
-
CO
or
dx 2
196
The latter possibility is exemplified by the
plots.
f
versus
H
x
H
For the plot at 1 atm
dlnf H
1
as
- xH
0.
dlnxH
Since
dlnfH
dlnfH
XHdXH
dlnxH
it is required
that
dlnfH
as
X
-
'
-O0
xH
dxH
to obtain the finite limit of 1.
The
curves
one-to-one
implies
that
correspondence
they
may
be
between
f
H
analyzed
in
and
C02
virtually
identical fashion, so that only a few comments are in
At
f
order.
the lower pressures, the presence of the ideal gas regime
where fi=
xP
is again noted.
Since xc
2
is near 1 in
the
left hand portion of the plots, this reduces approximately to
= P,
f
or
a
line at the system
horizontal
pressure.
It
CO 2
was previously mentioned
values
would
that
be found for f
0D 2
at
65
atm,
three
at the pure CO2
distinct
axis.
The
2
logarithmic plot of Figure 3.7 has been redrawn in
Cartesian
coordinates in Figure 3.8 to illustrate this behavior.
197
43.5
43.4
fco
20
43.3
.0005
.0010
.0015
.0020
.0025
XH
Figure 3.8.
A Cartesian plot illustrating the loop in
at conditions near C02's vapor pressure.
f
02
2~
198
Attention is now turned to the shape of the L
versus x
curves at constant T and P, L1 being the determinant
by
equation
This
(3.26).
stability
spinodal
curves
and
critical
Figure 3.9 shows L1 versus xH
at 300 K.
points
in Chapters 5-8.
curves for the pressure series
As L1 is
spinodal surface has been located.
H, the
its
determination
crosses the zero axis, a point on
Whenever L
derivative
in
criterion,
intensive (mole fraction) form, is used in the
of
defined
the
related to the
included in
f-x plots have also been
dohxH
the figure.
For the L
curves, the
vertical
changes
scale
between 64.3 and 65 atm.
For the determination of the spinodal curve,
and f-x curves are equivalent.
The
Ll-x
In mathematical terms,
dInfH
= 0
when
L
= 0.
dx H
As long as
~/T
< 0,
these two quantities agree in sign
seen,
for
example,
at
P
= 0.1 atm.
ideal gas solution (segment G) is
line
across
the
as
well.
may
be
The L1 curve for the
essentially
entire range of composition.
slightly above the zero axis, i.e.,
This
positive,
a
horizontal
It is always
but
at
this
199
10
6
106
0
0
L1
-106
2
-2
log fC
-4
-6
-8
-5
-4
-3
log
-2
-1
0
C16
C1 6
Figure 3.9. (First of eight pages)
-5
-4
-3
-1
0
log XC
C16
Plots of the stability criterion
matrix, L1, versus mole fraction of n-hexadecane.
of L,
-2
In the calculation
compositional derivatives of the Helmholtz free energy are
taken with respect to n-hexadecane.
Figure 3.4.
Segments are labelled as in
16
200
6
10
L
io
G
0
0
300 K
5 atm
-
L1
U
io6
-10
6
2
0
-2
log f
16
-6
-8
-5
-4
-3
-2
log XC16
C16
-1
0
-5
-4
-3
-2
log xc
C16
Figure 3.9. (Second of eight pages).
-1
0
201
106
L1
_106
2
log f
C16
-5
-4
-3
-2
-1
0
-5
-4
-3
-2
log xc
log xc
16
16
Figure 3.9. (Third of eight pages).
-1
0
202
6
10
300 K
50 atm
106
300 K
60 atm
G
G
L
L
0
L1
-106
-_o6
U1
I
300 K
60 atm
2
-2
log fc
16
-4
G
-6
L
-8
l
5
-4
-3
log xc
-2
-1
0
-5
-4
l
-3
log x c
16
l
-2
-1
16
Figure 3.9. (Fourth of eight pages).
0
203
300 K
65 atm
G
-
~
\I
_-
0
f
}.
11
L1
-107
!-
-2
300 K
65 atm
300 K
64.3 atm
.-2
log fC
C16
-4
.-6
L
·-8
·
.....
-5
-4
-3
-2
-1
0
-5
-4
-3
-2
log XC
log XC
16
Figure 3.9. (Fifth of eight pages).
Note
change of scale in the L-x plots between
64.3
and 65 atm.
-1
0
204
7
7
10
10
300 K
70 atm
L
L1
-
-107
300 K
70 atm
300 K
67.2 atm
-2
log fc
C16
-4
-6
G
-8
.5
-4
-3
-2
log XC
C16
-1
0
-5
-4
-3
-2
-
0
-3
-2
-1
0
log Xc
16
3.9. (Sixth of eight pages).
Figure
205
300 K
75 atm
300 K
73 atm
l
j
I
7
107
0
|
L1
-10
300 K
75 atm
300 K
73 atm
0
-2
log fc
16
-4
-6
-8
.5
-4
-3
-2
log Xc
16
-1
0
-5
-4
-3
.
.
.
-2
log Xc
16
Figure 3.9. (Seventh of eight pages).
-1
0
206
300 K
85 atm
10
300 K
101
300 atm
I
0
0
-107
-107
I
0
300 K
300 atm
-2
log fC
-4
-6
-8
-5
-4
-3
log
-2
Xc
c 16
-1
0
-5
-4
-3
-2
log XC
c 16
Figure 3 .9. (Eighth of eight pages).
-1
0
L1
207
scale
The liquid root
from the axis.
indistinguishable
is
(segment L) starts out at a negative value, crosses the
zero
values.
The
and
axis,
to
on
continues
positive
more
mechanically unstable root (segment U) meanwhile
at
a
value
negative
and
crosses, the zero axis.
dlnf
approaches,
closely
out
starts
but
never
In this case,
H
<
0
while
Ll
> 0.
dlnxH
The mechanically unstable
smoothly
at
a
minimum
root
value
and
of
the
liquid
root
join
xH , although this is not
readily visible at the scale of this plot.
The
minimum
in
x H which corresponds with the infinity on the f-x plot, does
not
coincide
with
the minimum in Ll, which appears to be a
cusp (not continuously differentiable).
In fact, the minimum
is smooth, and formed entirely by the liquid root.
in L
At 1 atm, a second apparent cusp is formed as the region
of three solutions moves away from the pure C16
root
vapor
now
crosses
the
zero
axis.
The
axis to give a spinodal
point, and smoothly joins the mechanically unstable root at a
maximum value of x
H
A comparison of the Ll-x curves at 5 and 20
atm
points
out an occurrence that is not particularly evident in the f-x
curves.
xH
While at 5 atm the high fH
infinity fell at higher
than the liquid root minimum, at 20 atm it has moved to a
208
lower x
has
Also at 20 atm, the inflection in the liquid
become
quite
noticeable.
When a pressure of 40 atm is
reached, the inflection gives rise to a maximum and
At
root
minimum.
60 atm, the maximum has crossed the zero axis to give two
more points on the spinodal surface, for
Although
not
shown
on
the
graphs,
a
the
total
liquid
mechanically unstable root continue to join at a
x , while the associated minimum in L
of
four.
root and
minimum
in
remains smooth.
H
At 64.3 atm, the L
maximum in the liquid root has moved
off the graph, and at 65 atm it is entirely gone.
of three roots now extends to the pure CO2
three
spinodal
points
are
found.
axis,
and
only
Finally, at 70 atm, the
vapor and metastable roots have disappeared
there are only two points where L=0.
The region
and
once
again
This behavior persists
to very high pressures.
It was stated earlier that the quantity L
was
related
to the derivative
dlnfH
dlnxH
It is helpful to define this relationship more precisely.
It
may be shown that
(-1-
=
x 1 /T,P
L1
A
(3.43)
209
where A
between
and L1 are in intensive
form.
The
relationship
slope in the logarithmic f-x plots and L
can now be
stated:
dlnfH
xH RTdlnfH
xH d
=
dlnxH
...-
RT
dx H
xH
H
RT dx H
_ L1
(3.44)
RTAvv
XH
Thus, if the dimensionless quantity
RTA
were plotted
versus
log x H
Li
vv
for a set of conditions
in Figure
3.4, a diagram showing the derivative of the f-x curve
would
result.
Ml, the determinant defined in equation (3.38), was used
for the second criterion of criticality in this work.
not
over
convenient
many
It
is
to plot this quantity since its value ranges
orders
of
magnitude.
No
encountered in its evaluation, however.
difficulties
are
It may be shown, for
M1 expressed in intensive form, that
=
Together
critical
equations
point
(3.43)
(i.e., when
2M1
and
(3.45)
(3.45)
imply
that,
at
a
L1 = M1 = 0),
(
tL1)
aX1/T ,P
-°
(3.46)
210
Table 3.1 summarizes the features of the phase
generated
in
this
thesis,
identify each feature.
criterion
was
stability
of
associated
not
a
Note
used
critical
binodal
and
locus
the
that
in
an
point
(see
criterion
the
stability
form,
apparent
Figure
applied to
critical
explicit
was
diagrams
3.1).
since the
from
its
As mentioned
earlier, the shape of the binodal locus alone can be used
determine
critical points.
the shape of
interested
the
binodal
This is a reasonable approach if
locus
is
desired.
criteria.
lists three features of binary phase
one
is
from
the
criteria
introduced
in
Chapter
discussed
2,
and
Table 3.1 also
diagrams
Azeotropy, three phase lines, and critical
been
If
only in locating a critical point, however, it is
more efficient to use the L1 and M
indirectly
to
which
in
this chapter.
end
will
thoroughly in Chapters 5-8 as the need arises.
result
points
have
be treated more
211
-
4
-
01
. O * .l
_
V*-u
n
·-
.
7
4-J
0
'
'3
E
C
O
I-
a).
4
-.
0
"
O
II
E
0
0 o
0
a)
~0
CU
2
..
c
4Q
*-
C4
CU
II
0
04-,
C
C
0
)
,~
L)
a)
4-J
C
0)
*uC
n
)V
-
Q)0)
Io - L C~
O~·
_L.C0 4-J 4-J
4J
aC
CC-
0-C
a) 4- - E
C
4-J
u
-Ia)I
c
4-
E
aj
3
c
L
. _,
-
7
u
U
L
L
4-
CU
O)
LL
O
Q
~ ~
-~
U
..
0
-0
0
U)- 0 -
C-
C
O
0
Q.
C-U
U
0
C
*-
mz
(n
0.
mCU
0
U
O
0
C
4-
.t
u
4-
_
L
u
(
a
c0
,
M Uf
0
N
c:
a
C4-'
4JL
.
4-'
0
C
U,
oa)
4J
.-o- *-4. -u
.*C
4 U
.- ).-
e-4-,
e
. CU
=3
1
L....-
a CU a ) .- 4S
oL>- - (A.
. 4( cU
CU
-
()-*
4. J
C
4-.0-
*-t
(D
4)
0
-
; O0O
4J 0 4-J
c C C.-
': E.
i
.0
U)
a-)
*-E
~3--c, J si:
W
0
0
.-
0'
0 C
C r
.--
I-C4U
a0
0 C
)4
>C
C
C.-- -.-
4J
4J
-;
.c0
tS
'
DCU4
·-C
04-J
4
0
>,
e-
CU
U N UC
0 r'OC
.O
" )
00
Q
C O
.-
--
-
-0
C.--
Ca)
0) C
O
CL
cU
U 4-
0)
>
0
C
cU
0
-
4-'
L.
xL
IL
'._
4-
r1~
Q 0cI-
L
N
a)
0
U')m,
0
S.
cO4-:
4-O C..-
*
0
*-..
-
0
C
3,-*-_
W
-
4-J
0
a)
_.
I--
U
) CL
4-J
l
*
L_
C
1
u
212
Pure Component Phase Diagrams
A
Chapt
the
In
generation
diagrams
component
binary
of
become
of
in
interest
pure
diagrams,
phase
respects.
two
of
Firstly, the pure component diagrams are an integral part
binary
the
These
compositional extremes.
regions
of
phase
giving
space,
P-T-x
sections
at
termini
of
define
sections
For example, VLE envelopes
equilibrium.
terminate at points along a
P-T
the
vapor
component
pure
pressure
of
pure
component predictions is that equations of state are in
most
cases
second
The
curve.
derived
on
for
reason
the
of
basis
importance
the
component behavior.
pure
of
Extension to mixtures is effected through the application
mixing
which describe the system in terms of the pure
rules
components
which
multicomponent
the
comprise
systems
the
form
proper
mixture
properties
is
therefore
accurate than the prediction of
It
pure
most
of these rules is
unknown, and simple approximations are used.
of
For
system.
prediction
The
to
unlikely
be more
properties.
component
thus becomes important to know how well a chosen equation
of state mimics the pure
Fortunately,
component
thermodynamic
data
properties
for
pure
of
concern.
components
is
usually available.
Table 4.1 lists the critical temperatures and
pressures
of the six components dealt with in this thesis, and compares
213
Table 4.1
Constants
of Pure Components
ac tua I
3
predicted
actual
predicted
actual
Substance
T (K)
Pc(atm)
v(
c m /kgmol)
Ethylene (C2H4 )
282.4
49.7
0.129
0.143
217
196
0.276
0.085
Carbondioxide
304.2
72.8
0.094
0.105
468
419
0.274
0.225
Benzene (C6 H 6 )
562.1
48.3
0.259
0.293
301
266
0.271
0.212
Water (H20)
647.3
217.6
0.056
0.075
321
240
0.229
0.344
n-Hexadecane
717.0
14.0
748.4
40.0
v (m3/kgmol)
3
pc(kg/m )
3
Pc(kg/m)
, Zcc
w
(Co2 )
0.742
175
1.290
(C16 H 34 )
Naphthalene
)
(CloH8
0.410
0.471
Note - The P-R equation predicts a universal Zc of 0.307.
312
272
0.267
0.302
214
the
calculated
and
measured
critical
volumes
and
compressibility factors.
Pure component P-v-T data is easily
and concisely
by
isotherms
plotting
in
a
P-v
These diagrams have been drawn up in Figure 4.1 for
diagram.
the
expressed
Figure 4.1a gives the P-v diagram
species in Table 4.1.
for carbon
dioxide
equation.
The
as
are
a
phase solution,
the
of
range
predicted.
from
the
Peng-Robinson
at 200, 250, and 298 K illustrate
isotherms
the fact that for
volumes
determined
pressures,
low volume root is the liquid
The
volume
high
different
three
root
is
the
gas
phase
solution, and the middle root is mechanically unstable.
At each temperature, a horizontal tieline may
the
between
liquid
gas
and
phase
volumes of the coexisting phases.
determining
solutions
the
yield
at
isotherm
pressure
equal
298
K,
at
be
giving the
solutions,
The tielines are found
which
fugacities.
liquid
As
minimum
stability
and
limit
evidenced
in
spinodal
to vaporize (boil) when the
this
curve.
curve.
curve are points on the
Tracing
the
occurs
at
binodal
about 63 atm.
curve
is
298
K
intersected.
It is possible under certain
conditions, however, to reduce pressure below 63 atm
a transition to the vapor phase.
liquid
the
from high pressures, the liquid will normally begin
isotherm
This
by
the liquid and gas phase solutions are
maximum
or
by
gas phase
and
actually joined by the unstable root to form a smooth
The
drawn
is
I
formed.
without
In this case, a superheated
This metastable phase may persist until
215
rI
1
,.,
I, ,,
SWUv
_
-
SOLID
100
+
I
LIQUID\
P(atm)
10'
1.01
.1
1
10
P-v diagram for carbon dioxide.
Between the critical
v (liters/mole)
Figure 4.la.
point and the SLV (triple point) tieline, liquid and vapor coexist.
Below the SLV tieline (light shading) solid and vapor coexist,
although the P-R equation continues to predict (metastable) vaporliquid equilibrium.
The darker shaded region gives approximate
boundaries for solid-liquid equilibrium.
216
P(at
10
1
0.'
V (liters/mole)
Figure4
lIb.
P-v diagram for n-hexadecane.
100
217
l-nnn
£UUU
100
P (atm)
10
1
0.1
100
10
1
V (liters/mDle)
Figure 4.lc.
P-v diagram for naphthalene.
Note the anomalous
behavior of the experimental data for the saturated vapor.
218
nnn
IUU
1
P(at
0.01
1
0.1
v (liters/mole)
Figure 4.1d. P-v diagram for ethylene.
LU
219
1nnn
100
P (atn)
10
1
.01
1
.1
V (liters/mole)
Fi~gure 4 le.
P-v diagram for water.
10
220
rF(n
1-vvv
100
P (atm
10
1
.1
V (liters/-ole)
Figure 4.1f.
P-v diagram for benzene.
10
221
the isotherm intersects the spinodal curve at about
at
59
atm,
point there must occur a spontaneous transition to
which
the gaseous state.
At 200 and 250 K, the three roots still
curve
as they did at 298 K.
so
is
not shown on the plot.
the isotherms converge at
represents
terminus
the
a
single
At these temperatures, however,
the stability limit for the liquid is located
and
form
the
of
below
atm,
1
The maxima and minima in
critical
which
point,
thus
Above the
the spinodal locus.
critical temperature, the isotherms no longer exhibit extrema
(see, however, Gomez-Nieto, 1979).
point
represents
Similarly
critical
the
the terminus of the binodal locus, and the
liquid-vapor tielines converge to zero length.
Figure 4.1a also includes
comparison.
The
some
data
experimental
binodal VLE curve is represented very well
by the equation of states except for liquid volumes
reduced
pressure
for
of
about
0.5.
The
above
prediction
critical volume is in error by about 10 percent.
a
for the
Similarly,
the 298 K isotherm agrees with experimental gas phase volumes
quite
closely.
In
the liquid region, though, the slope of
the isotherm is not steep enough and
significant
deviations
are found.
At low temperatures,
point
tieline
and
pressures
below
the
triple
(about 5 atm), experimental data indicate the
222
formation of a solid phase in equilibrium with a vapor phase.
Points on the solid-vapor binodal
An
figure.
has
equation
Peng-Robinson
given
been
in
the
representation of the solid-liquid
approximate
region
coexistence
have
been
no
has
as
included
The
well.
for predicting solid
means
phase boundaries, so the predicted VLE dome has been extended
Metastable VLE below the triple
into the solid-vapor region.
however,
point is actually possible,
of
labelling
200
the
K
tieline.
indicated
as
by
the
The limit to which VLE
metastability may persist is unknown.
Figure 4.lb shows
P-v
the
diagram
for
n-hexadecane.
There is no data available for comparison in the region shown
and, in fact, even the critical volume has not been measured.
The
naphthalene
diagram
in
Figure
4.1c shows the limited
The data does
amount of data available for this system.
seem
phase
not
particularly reliable in view of the inconsistent vapor
measurements.
predictions
for
The
quantitative
accuracy
of
the
and n-hexadecane is uncertain,
naphthalene
but it is probably comparable to that of
Figure
4.la.
For
ethylene, water, and benzene, Figures 4.1d, e, and f, data is
available
for
Except for the absence of solid
comparison.
formation, the diagrams are quite similar to Figure 4.1a.
For the actual construction of
pure
component
diagrams
just
P-T
diagrams
considered.
As
are
binary
P-T-x
diagrams,
more useful than the P-v
mentioned
previously,
P-T
223
diagrams are actually part of the binary P-T-x space diagram,
falling
at
along
work,
measurements.
in
pressure
vapor
experimental
with
interest
of
compounds
gives these P-T diagrams for the
this
Figure 4.2
end of the composition axis.
either
accuracy
Except for naphthalene, quantitative
As was true with Figure 4.1c, the naphthalene
is quite good.
data are of questionable validity.
The P-T-x diagrams which are covered in Chapters 6 and 7
include a region of
formation
of
solid
The
naphthalene.
is to an excellent approximation a pure phase, and its
solid
fugacity is thus simply expressed.
phase
fugacity
in
turn
Knowledge
of
the
solid
calculation of the desired
allows
phase equilibria.
To derive an expression for the pure solid fugacity, one
starts
with
the
Gibbs-Duhem
equation
written
the
for
coexisting solid and vapor phases:
du c = -SCdT + vCdP
(4.1)
du g = -SgdT + vgdP
(4.2)
Here the supercript c denotes the
phase
and
pure
solid
(crystalline)
the superscript g the pure vapor (gaseous) phase.
At equilibrium, d
g
= d ,
,
so
equated and rearranged to yield
the
two
equations
may
be
224
80
70
60
50
285
290
295
300
T (K)
Figure 4.2a.
P-T diagram for carbon dioxide.
305
225
300
400
500
600
T(K)
Figure 4.2b.
P-T diagram for n-hexadecane.
700
800
226
40
30
B
i
.I-
20
10
300
400
500
600
T,K
Figure 4.2c.
P-T diagram for naphthalene.
700
800
227
en
45
P (atn)
40
35
265
270
275
T,K
Figure 4.2d. P-T diagram for ethylene.
280
285
228
7cn
ZDU
200
150
100
5(
2(
500
600
550
T (K)
Figure 4.2e.
P-T diagram for water.
650
229
50
40
B
(d
04
30
20
470
480
490
500
510
520
530
T (K)
Figure 4.2f.
P-T diagram for benzene.
540
550
560
570
230
- SC
dP
S
dT
vg - VC
(4.3)
Applying the identity G=H-TS to both phases
allows
equation
(4.3) to be rewritten as
Hg
dP
dT
the
H
AHsub
T(v g - v )
superscript
sublimation
gas equation
sub
curve.
is
(4.4)
TAvsub
indicating
conditions
along
the
Under sublimation conditions, the ideal
usually
applicable
to
the
vapor
phase.
Furthermore, the crystalline volume is negligible compared to
that
of
the
gas.
Incorporating these two assumptions, one
finds
The quantity AHsub
temperature
dlnP
AHsub
d(l/T)
R
is relatively constant
range,
so
that
equation
(4.5)
over
(4.5)
a
is
moderate
readily
integrated to give
sub
AH sub(4.6)
RT
This equation expresses the relationship between temperature
sub
and pressure along the sublimation curve.
The values of AH
R
231
and C are usually evaluated
Since
some
(4.6),
by
fitting
experimental
data.
approximations are involved in deriving equation
Hsub
will be slightly different than the
true
heat
of sublimation.
Prausnitz (1969) has shown that the fugacity of
a
pure
solid at a given temperature and pressure may be stated as
fC
sub
vCdp
dP
sat expv
fexp
RT
Jpsub
Here
(4.7)
(4.7)
J
sat is the fugacity coefficient of the saturated vapor
phase.
For the low
equilibrium,
molar
volume
wide
ranges
pressures
it is almost
associated
always
with
very close
to 1.
solid-vapor
v
is the
of the solid, and is essentially constant over
of
pressure.
Incorporating
these
two
simplifications, equation (4.7) becomes
fC
psub exp [v (P -
=
P
b)
(4.8)
RT
Equation (4.8) is used
in
the
calculation
of
solid-fluid
equilibria.
Application of equation (4.6) in
P-R
equation
for
with
the
a pure substance allows the prediction of
naphthalene's triple point.
picture of the C01
conjunction
8
Figure
4.3
gives
a
detailed
triple point region, in comparison with
232
i
Cd
350
355
360
T (K)
Figure 4.3.
A comparison of experimental and predicted
triple points for naphthalene.
The vapor pressure curve
is calculated using the P-R equation, the sublimation
curve is calculated using equation
(4.9).
233
The calculated vapor pressure curve is in
experimental data.
error
about
by
for
curve
pressure
when
the
calculated
intersects the calculated sublimation
curve
pressure
vapor
predicted
The triple point is
shown.
over the range of temperatures
atm
0.0006
The
solid.
the
expression
the
for
sublimation curve is given by
psub
=
0.0001 exp [-8719/T + 29.3]
which stems from a reduction of data taken at
(Ambrose,
et
experimental
curve
covered
temperatures
expressions
Fowler,
et
by
as
al.
as
much
Figure
in
(1968))
(4.9)
gives
a
better
the
atm
at
4.4.
More
accurate
al.
et
(1975)
or
but they do not yield significant
differences in the binary phase diagrams.
P-R vapor pressure curve
333 K
0.0004
(see Ambrose,
are available
to
Equation (4.9) deviates from the
1975).
al.,
273
(4.9)
is
somewhat
prediction
In fact, since the
equation
inaccurate,
of the triple point than
these other expressions.
To summarize, a
phase
consideration
of
the
pure
component
diagrams gives every indication that the Peng-Robinson
equation should be suitable
P-T-x diagrams.
for
the
generation
of
binary
234
Chaster
The n-lHexaecane
- Carbon Dioxide System
The n-hexadecane (C1 6 ) - carbon dioxide (C02) system
similar to the types shown in Figures 2.8 and 2.9.
is
The upper
critical line starts at the critical point of C 1 6, rises to a
pressure maximum, falls to a pressure minimum, and thereafter
runs
to
very high pressures with a slightly positive dP/dT.
The lower critical line starts at
with
the
CO2
constants for C 1 6
For ease
and CO 2
of
reference,
the
determination
of
P-T-x
diagrams
is
a value for the interaction parameter
For this system, the
612.
physical
are given in Table 5.1.
The first step in the prediction of
pressure-temperature
coordinates
the upper critical line have been measured (Schneider, et
al., 1967)
Figure
5.1
and
provide
shows
various values of
a
the
612,
basis
upper
data
for
this
critical
determination.
lines generated for
and illustrates the marked
this parameter on the predictions.
effect
of
It is possible to fit the
precisely by allowing the interaction parameter to vary
with temperature (and possibly pressure).
the
point
a positive dP/dT, and after a short distance intersects
the three-phase LLG line.
of
critical
values of 612
systems.
similar
5.2
shows
needed to match the data in Figure 5.1 as
a function of temperature.
found
Figure
results
Although 612
Peng
in
and
working
Robinson
with
(1980)
have
aqueous-organic
increases monotonically with
T,
the
235
CN
I
L
.13
u
x
r,
r-4
N
CN
$C
C)
CL
'C
ro
ed
r.
r
CZe
LO1
%O
N,-4
'IO
r-,
r.
r-I
N-
,-
C
I-i
(N
I-
(a
0
c
O
4
m
.,4-4
-H
v
,H
P4.
--
0
c
u
_U
PL
E-
iU
Pi
)
3
a)
236
400
300
E
-0
200
100
n
300
400
500
600
700
T, K
Figure
5.1.
P-T projection
for
the C16 - C02 system showing
the effect of 612 on the predicted upper critical line.
The pure CO2 critical point has also been included,
although
it is not a part of
this
line.
237
.20o
.16
.12
612
.08
.04
350
550
450
650
750
T(K)
Figure 5.2.
The temperature dependenceof the interaction
parameter
Allowing 612 to vary with temperature along the curve shown,
an accurate matching of the data in Figure 5.1 is obtained.
612.
238
functionality
is
not
simple
in
shape.
More
than
one
numerical constant would be required to express this function
analytically.
parameter
The pressure functionality of the
is
more
complicated
than
interaction
the
temperature
functionality.
It was
not
considered
desirable
in
this
thesis
to
introduce more than one empirical parameter into the equation
of
state.
For
this
reason,
the conventional approach of
using a constant interaction parameter was followed.
A value
of 612=0.081 was selected, and is seen from Figure 5.1 to
a
reasonably
composition
optimal
data
choice.
along
the
It
should
upper
be
critical
noted
line
be
that
is
not
available, and thus is not a factor in this determination.
Having selected an appropriate interaction parameter, it
is helpful to backtrack a bit and see how the critical points
in Figure 5.1 are predicted.
of
T-x
sections
Figure 5.3 presents a
sequence
which illustrate the shape of the spinodal
(Ll=0) and second critical criterion (Mi=0) curves.
Critical
points are located at the intersections of these two curves.
Starting at
300
atm,
relatively simple shapes.
and
is
actually
critical point.
intrinsic
Figure
5.3a,
the
curves
have
The spinodal curve is dome shaped,
composed of two branches which meet at the
The right hand branch
stability
for
liquid
gives
phases,
the
limit
of
and the left hand
239
P = 300 atm
.W
1
6
6
5
-
MI
O
4
LI-O
3(
0
.1
.2
.3
.4
.5
.6
.7
X
H
Figure 5.3a.
the L=0
The T-x section at 300 atm showing
and M1=0 curves.
point at this pressure.
There is one critical
240
P
MI
250
oatm
0
P
I-
LI
0
P
LI 0
.1
.2
.3
.4
X
Figure 5.3b.
.5
.6
H
The T-x section at 250 atm showing
the L1=O and M1=0 curves.
points at this pressure.
There are three critical
241
P = 210 atm
700
MI =0
600
500
LI 0
I-
LI =O
7
XH
5
Figure
the L=O
.3c.
The T-x section at 210 atm showing
and MI=0 curves.
242
P = 100 atm
7r/n
MI
=-
cP.
I-
LI =O
0
.I
.2
.3
.4
.5
.6
.7
XH
Figure 5.3d.
The T-x section at 100 atm showing
the L1=O and M=O curves.
The two separate L=O
curves present at higher pressures have merged.
243
P 75 atm
tical
line)
h-
ne)
X
H
Figure 5.3e,
The T-x section at 75 atm showing the
L1=0 and M1=0 curves.
Once again three critical points
are present.
The two low temperature critical points
in this figure, however, are not related to the two
low temperature critical points present at 210 and
250 atm.
244
P = 70 atm
· A
(
6
=0
5
4
3
7
X
H
Figure 5.3f.
the L=O
point
The T-x section at 70 atm showing
and M=O
is visible
curves.
Only one critical
at this scale.
245
312
310
308
306
304
302
300
0.00001
0.0001
0.001
0.01
1.0
0.1
Figure
3g
ATsection
at
70
atm
showing
the
Figure 5.3g.
A T-x section
at 70 atm showing the L=0 and M=0 curves.
The
four separate branches of the spinodal (Ll=O) curve have been labelled with
subscripts.
The shaded area indicates the region of three roots.
Of the
three intersections of the L1 and Ml curves, only one, the unstable CP,
actually satisfies the criteria of criticality.
246
branch the limit of intrinsic stability for
A
system
homogeneous
spinodal
zero.
dome.
At the
as
Also,
exist
cannot
within the
points
Chapter
in
phases.
point, both L1 and M
critical
discussed
at
gaseous
=
3,
are
The
.
dx
to
critical point in this diagram corresponds
the
critical
point at 300 atm in Figure 5.1.
At 250 atm, Figure 5.3b, a
appeared
in
form
the
of
a
spinodal is crossed by the M
second
spinodal
The closed loop
closed loop.
= 0
curve
twice,
to
a constant pressure line
position
of
these
line is evident.
spinodal
has
at
250
atm
in
give
a
By tracing
of three critical points at this pressure.
total
has
curve
5.1,
Figure
the
critical points along the upper critical
At 210 atm (Figure 5.3c), the
increased
in
size,
and
closed
loop
finally, at 100 atm
(Figure 5.3d), the two spinodals have merged.
There is again
as
confirmed
by
= 0 and Ml = 0 curves
at 75 atm
is
but one critical point at
this
pressure,
Figure 5.1.
The course
of the L
shown in Figure 5.3e.
upper
critical
pressures.
At
differentiable
Behavior for the critical point on the
similar
locus
is
lower
temperatures,
point
has
developed
to
that
at
though,
in
the higher
sharp
but
curve.
The
a
the L
region of this point has been enlarged in the inset
its
two intersections with the Ml = 0 curve.
to
show
As labelled in
247
the figure, one of these critical
points
is
the
on
lower
locus, which starts from CO 2 's critical point.
critical
second of these
is
surface,
it is an unstable critical point.
i.e.,
the
terminus
of
an
unstable
binodal
The shape
presented
of the unstable binodal surface will be
The
later
in
this chapter.
Figure 5.3f illustrates the L1=0 and M1=0 curves
This
atm.
pressure
at
70
is below the critical pressure of CO 2.
Figure 5.3g gives the enlargement for the region near the CO 2
axis, with
a
logarithmic
concentration
scale.
The
four
segments of the spinodal curve have been labelled separately.
Once
also two more points of
graph
as
false
L-Ml
critical
intersection,
points.
noted
the
the
cubic
of state, as indicated by the shaded portion of the
figure.
They are not true critical points because
criterion
is satisfied for one root while the M=0
is satisfied for another root.
(e.g.,
on
These false intersection
points lie within a region of three real roots to
equation
There are
there is an unstable critical point.
again,
the
unstable
CP
in
For
the
a
true
the
L=0
criterion
critical
point
figure) the L1=0 and M1=0
criteria must be simultaneously satisfied for the
same
root
of the cubic.
The occurrence of 3 roots leads also to
of
two branches of the L1=0 curve.
to define the intrinsically unstable
an
intersection
It is no longer accurate
region
as
the
region
248
two branches of the spinodal curve.
the
between
The stable
and unstable regions in Figure 5.3g are not readily apparent,
and it is best to return to the fugacity-composition plots to
Figure 5.4 shows a
gain a proper understanding.
of f-x plots at 70 atm.
sequence
In light of the discussion
in Chapter 3, the general shape of
these
should
curves
is
above
temperature at which the L
the
cross in Figure 5.3g.
Although there is a small
= 0 curves
composition
range of three roots, all mole fractions between Lll
give
L1 2
unstable
Once they
two branches in Figure 5.3g cross.
stable solution may be found for any
and
and
L1 2,
formed
Figure
provided
mole
a
3.4).
previously
closed
In
a
implied,
region
of
greater
unstable.
phase
than
a
section
T-x
has
moved
must
roots
be
With respect to Figure
L1 1 =
Likewiser for xH
solution
fraction
multiple
vapor phase solution is stable for xH
xH
crossed,
between
to
xH=O
loop, as at 302.6 K (Figure 5.4c; cf.
separately for each root.
for
have
the proper leg of the f-x curve is
This also holds true after L1 2
chosen.
i.e., until the
shown at 302.8 K (Figure 5.4b) is valid, and a
behavior
L1
1
and L1 2
This will continue to be so until
solutions.
moves to a lower mole fraction than L1 1,
the
be
Consider first Figure 5.4a, drawn for T = 303.6 K,
familiar.
which
temperature
then,
as
interpreted
5.3g,
the
less than Lll=0, while
0, the vapor phase solution is
less
than
is unstable, and for xH
the liquid phase solution is stable.
L12 =0,
the
liquid
greater than L12 =0,
249
-2
-3
-5
0
-6
.7
9
0.0001
0.00001
0.001
0.01
0.1
1
XH
Figure
5
.4a.
where L1 = 0.
f-x plot.
The labelled extrema indicate points
A single phase system with a composition between
L11 and L12 would be either materially or mechanically unstable.
250
-2
-3
-4
-5
-6
7
8
9
0.00001
0.0001
0.001
0.01
X
Figure 5.4b.
f-x plot.
0.1
H
A single phase vapor is stable or
metastable at compositions up to L
1.
A single phase liquid is
stable or metastable between L12 and L3.
The only composition
range in which a single phase system cannot exist is from L13
to L14.
1
0r
Q
251
2o
3
4
H
0
5
0.00001
0.0001
0.001
0.01
0.1
XH
H
Figure 5.4c.
f-x plot.
L12 has moved to xH=0. Behavior is
otherwise the same as in Figure 5 .4b.
1
4
252
prediction
clarified,
now
Here
5.5.
Figure
this
The P-T projection for
possible.
the
lower critical locus is
the
of
dashed
criteria
point
With the interpretation of the critical
line
shown
is
locus
the
indicates
vapor
critical
pressure curve of CO2 , which terminates at the CO 2
From this point, the lower critical locus begins, and
point.
in
in
conjunction with the unstable critical locus forms a cusp
terminating at about 86 atm and 315 K.
T-x
below
locus extends to pressures and temperatures
CO 2
phase
critical
point.
L1 -L2 -G
line,
shortly.
considered
phase
Beyond
the
metastable
phases.
been
and
the
calculation
which
of
will
be
It intersects the lower critical locus
since
there
the
one of the liquid phases become identical.
intersection
with
pure
the
Also shown in Figure 5.5 is the three
at the lower critical end point, and stops
vapor
the
70 atm (Figure 5.3g), the unstable critical
at
section
by
As indicated
respect
point,
to
the
the
experimentally
determined,
this
is
two liquid
of
formation
The lower critical locus for
locus
critical
system
has
not
but its behavior is of the
type expected from the discussion of Chapter
2
(see
Figure
2.8).
The correct shape of the critical lines
predicted
with
61z-.0.081 indicates that a qualitatively accurate picture of
the P-T-x space for this system may be obtained
use
of the P-R equation.
through
the
A complete definition of the P-T-x
space requires the calculation of the binodal surfaces.
The
253
85
80
P(atm)
75
70
65
300
305
310
315
T(K)
Figure 5.5.
The P-T projection of the lower critical
locus for C16- CO2
The stable critical line is metastable
between LCEP and the cusp.
254
used
algorithm
for
purpose is explained in detail in
this
Appendix C.
Briefly, for a
composition
is
given
up
and
P
search
a
The
curve
f-x
may
satisfying
mentioned
in
the
curves
3,
Chapter
of
criterion
large
which
of
majority
P-T
no
slope
are
found
have
coordinates
in the space
Figure 5.6, the T-x section at 210 atm, exemplifies
diagram.
this fact.
The spinodal curves in this diagram have
already
Since a point of Ll=0 corresponds
seen in Figure 5.3c.
been
be
As
equilibrium.
infinities, and either no extrema or two extrema,
the
then
into monotonic sections which are easily searched
for points
for
in
out to locate all extrema and slope
carried
infinities in the f-x curve.
broken
T
to a maximum or a minimum in an f-x curve, it is clear
that,
there are either two extrema or none.
It is
for any given T
clear
as
well
that there cannot be phase equilibrium for a
since
monotonic f-x function,
different
Thus
fugaclty.
every
mole
fraction
has
a
is required that the binodal
it
curve intersect the spinodal curve tangentially at a critical
Figure 5.6 also indicates that either the spinodal or
point.
binodal
curve
point,
since
alone
T
= 0
Alternatively, for the binodal
when
used to
could be
for
both
curve,
predict a
at
one
such
could
critical
a
point.
determine
the compositions of the coexisting phases became equal,
i.e., when the tielines became of zero length, to locate
critical point.
the
criterion
the
These two procedures emphasize the fact that
M1
=
0
is
not
so much a prerequisite
for
255
16-C02
600
500
400
300
0
.1
.2
Figure 5 .6.
.3
metastable.
.5
.6
.7
A T-x section at 210 atm showing the binodal
and spinodal curves.
unstable.
.4
Areas enclosed by the spinodals are
Areas between the spinodals and binodals are
256
criticality as a consequence of criticality.
complicated.
more
shape of the
measured
at
a
T-x
tieline.
the
high
the
on
expected
section
basis
of
give
to
a
temperature
three-phase
L1 -L 2 -G
the G+L 2
end,
upper
terminates at a point along the
critical
pressure
The L1 +L 2
(i.e., the boiling point of C0 2
point
region
The
line.
region, on the other hand, terminates at the C0 2
G+L 1
the
The three binodal regions intersect
temperature
At
much
Figure 5.7a illustrates qualitatively the
critical loci.
common
is
curves
At 70 atm, the behavior of the binodal
vapor
at 70 atm).
region extends off the plot to lower temperatures,
where it will eventually terminate due to solid formation.
Figure 5.7b gives the P-R prediction for the T-x section
The concentration
at temperatures near the three-phase line.
axis is given on a logarithmic scale to allow the very
G+L1
region
predicted at
to
be
304
K.
correct,
qualitatively
Three-phase
visible.
Although
it
is
the
equilibrium is
predicted
incomplete,
in
addition
to
the
stable
diagram
is
because the P-R
equation generates metastable and unstable phase
solutions
small
equilibrium
solutions which are
shown.
Figure
equilibrium
5.8a
shows
solutions
the
from
T-x
the
section
when
all
phase
P-R equation are drawn in.
The spinodal curves, which were already given in Figure 5.3g,
257
4)
0
4._
co
0
-C
x
0u
4)
I
-I
-
C
o
__,
o
U
L.
0.
o
o0~
o
0cr"
0c
%O
_:
L
4
E.
LA
.
00m
I-
a
uo
a'
.-44
co
E
e
.,-
r0
=
r
.
to
C,)
.
I
cn
o·s SS
0LY
.
258
C16-C02
P = 70 atm
307
306
305
304
303
302
301
.00001
.0001
.001
.01
x
Figure 5.8a.
.1
H
A T-x section at 70 atm showing the continuation of
the predicted binodal curves.
Each binodal is labelled with a number
pair to indicate which segments of the f-x curve the equilibrium
is derived from.
To determine the compositions of coexisting phases,
a tieline is drawn at constant temperature joining binodals labelled
with the same number pair.
Note that the three phase tieline, given
by the dashed line, joins the two curves labelled (1,7), the two
curves labelled (1,5), and the two curves labelled (5,7).
spinodal curve (Ll=0) is given by the heavy lines.
The
259
-3
-4
-5
-6
-7
-8
.9
0.00001
0.0001
0.001
0.01
1
0.1
XH
Figure 5.8b.
An f-x plot in the three-phase region, showing the
numbering of the seven monotonic segments.
0
260
n
-1
-2
-3
-4
-5
-6
-7
0.00001
0.0001
0.001
0.01
0.1
1
XH
Figure 5.8c.
f-x plot.
segments are present.
At this temperature, only five monotonic
Comparison with Figure 5.8b shows that
segment 2 (or, alternatively, segment 4) and segment 3 no longer
exist.
261
.
n
I
T = 304 K
O
34
:M
0.0001
0.00001
V
0.01
0.001
X
L
x
0.1
1
xLC
x
H
Figure 5.8d.
f-x plot at the three-phase temperature.
The three-
phase tieline is shown connecting the compositions of the coexisting
phases in segments 1, 5, and 7.
L2
x
, respectively.
L1
, and
Although the tieline crosses segments 3 and 6,
these compositions are not in equilibrium.
the different partial C
plot.
v
These are denoted as x , x
2
This would be shown by
fugacities of these points in an fCo - x
2
262
C 1 6-CO 2
302.6
302.5
302.4
T,K
302.3
302.2
302.1
0
.0001
.0002
.0003
XH
Figure 5.8e.
coordinates.
A T-x section at 70 atm in Cartesian
The VLE region terminates at the C02
vapor pressure. The zR=Oline is the boundary
between the regions of one and three roots.
263
are also displayed in the figure.
labelled
by
monotonic
number
pairs,
segments
segments.
Not all of
are
region
the
f-x
segment.
Figure
by
5.8c.
three-phase
from
curves
in
the
which
the
is
temperature
the
following
This
to the particular
three-phase
varied.
It
evolution
of
a
particular
illustrated by the f-x curve at 306 K in
is
Figure
5.8d
temperature,
304
shows
the
f-x
plot
at
the
K, with the tieline drawn in.
The three-phase equlibrium is seen to be between segments
5,
and 7.
1,
This is also evident from the intersection of the
binodal curves (1,7), (1,5), and (5,7) in Figure 5.8a.
These
binodal curves do not terminate at the three-phase line,
continue
is
to assign each binodal curve a consistent
though,
number pair
curve
are
of seven segments since some segments
comprised
may appear or disappear as
possible,
refer
curves
showing the numbering of the seven
region,
three-phase
f-x
binodal
Figure 5.8b gives a typical f-x plot for
equilibrium arises.
the
which
the
of
The
Upon
crossing
the three-phase line,
however, a Gibbs free energy
analysis
will
binodal
beyond
represents
1981).
al.,
it.
but
Thus,
show
that
metastable equilibrium (see Baker, et
a
at
temperatures
below
304
K
(the
three-phase temperature) the (1,7) binodal is metastable.
its
minimum
the
temperature, the metastable
At
(1,7) binodal joins
the unstable (3,7) binodal.
There are three separate
5.8a.
The
first
two
continuous
curves
in
Figure
follow the course (1,7)-(3,7)-(4,7)-
264
(5,7).
Of these two, the one at low x H
crosses the spinodal
between (1,7) and (2,7) while the one at
cusp at the spinodal and turns back.
the
course
XH,
(1,5)-(1,6)-(3,6)-(4,6)-(4,6)-(3,6)-(1,6)-(1,5).
but
not at high X H.
5.8c
point.
Reference
Figure
at
to
5.8a
also
that, in tracing the three curves, the occurrence of a
cusp is accompanied by the occurrence of a
For
(3,6)
shows that both segments 4 and 6 are materially
unstable, in accordance with Figure 3.1.
shows
and
The two branches of the (4,6)
binodal meet at the unstable critical
Figure
a
The third curve follows
This curve crosses the spinodal between (1,6)
low
x H reaches
high
example,
following
smooth
extremum.
the two branches of the VLE binodal
(1,5), it is seen that the vapor leg ends in a cusp while the
liquid leg ends in a smooth maximum.
A significant portion of the G+L 1
5.7b)
defined
by
the
(1,5)
region
binodal
falls
(see
Figure
within
the
"apparently unstable" region between two spinodal curves.
previously discussed, the (1,5) binodal
region.
axis
vapor
is
stable
in
As
this
The termination of the (1,5) binodal at the pure CO2
is
shown in Figure 5.8e in Cartesian coordinates.
and
liquid
temperature
of
CO2
roots
converge
at
the
The
vaporization
at 70 atm, and remain within the region
of three roots.
As evidenced by the phase diagrams in Chapter 2,
more
common
it
is
to define the binodal surfaces in a P-T-x space
265
diagram by P-x sections, rather than by T-x or P-T
A
Figure
P-x diagrams will now be considered.
of
sequence
sections.
the
5.9a shows the P-x section at 300 K, a temperature below
points
critical
Thus, the very narrow
both components.
of
region terminates at the vapor pressure of
G+L 1
axis.
C 1 6 's
pressure
vapor
region terminates at
Likewise, the G+L 2
the pure CO 2
the
along
along
CO 2
pure
pressures, the gaseous branch of the G+L 2
At low
axis.
C16
binodal has a very
The
plot.
region
L1 +L 2
to
extends
very high pressures
The three binodal regions all intersect simultaneously
5.1.
to give a three-phase L 1 -L 2 -G line
at
about
seen to be on the L 1 -L 2-G line in Figure 5.5.
section
64
The
atm.
and temperature of this three-phase equilibrium are
pressure
at
the enlargement of Figure 5.9b.
in
binodals are both part of the same
L 1 +L 2
P-x
As in the
70 atm, metastable and unstable binodals connect
the stable binodals shown in Figure 5.9a, and
the
Figure
with
without reaching a critical point, in agreement
seen
the
on
low concentration of hexadecane, and does not appear
may
these
The G+L 1
and G+L 2
while
curve,
continuous
be
binodal is separate, terminating at low pressure
at an unstable critical point.
Figure
5.10a
gives
temperature
above
the
the
P-x
critical
section
point
region has now detached from the pure CO 2
is more
inclusion
evident
of
the
in
the
enlargement,
of
305
at
CO2 .
K,
a
The G+L 1
axis, a fact which
Figure
5.10b.
The
metastable and unstable phase equilibrium
266
C16-CO
C16
2
300
200
rt
1-1
100
0
.00001
.0001
Figure 5.9a.
.001
.01
.1
The P-x section below C2 's critical point.
1
267
C16-C02
7A
74
70
66 -
62
58
54
50
0.00001
0.0001
0.001
0.01
0.1
1
XH
The P-x section at 300 K in the region of the three-
Figure 5.9b.
phase line (dashed) showing all metastable and unstable binodals.
The stable regions in this graph are essentially what would result
if the T-x section at 70 atm (Figure 5.8a) were flipped over.
the present
rather
case,
however,
the L 1 + L2 binodal
than the L1+ G binodal.
is a separate
In
curve,
-
268
C1 6-C02
300
200
100
0
0.00001
0.0001
0.01
0.001
0.1
XH
Figure 5.10a.
the LCEP.
critical point and
The P-x section between CO2's
2
1
269
C1 6-C02
76
74
72
70 riD
68 -
66
64
62
0.00001
0.0001
0.001
0.01
0.1
1
XH
Figure 5.10b.
phase line.
The P-x section at 305 K in the region of the three-
The G + L
binodl
is now a separate curve, exhibiting
both a stable and an unstable critical point.
270
binodal
solutions shows that the G+L 1
curve.
critical point, in accordance with Figure
to
reference
5.5,
Figure
5.
The meaning of this metastability is
noted.
P-x
a
generating
a
upon
clarified
a temperature in this range.
at
section
of
presence
the
the LCEP and the cusp is
between
line
critical
metastable
separate
a
binodal exhibits both a stable and an unstable
This
With
now
is
This is done in Figure 5.11, at 311 K.
shows
5.11a
Figure
the large scale P-x section, including stable binodal curves,
atm,
At about 80
curves, and the region of three roots.
spinodal
the
binodal
curve
low
displays
xH
a
nearly
This might be regarded as the influence
segment.
horizontal
at
of the lower critical end point, which occurs a few tenths of
a degree below
a
exhibits
311
At
K.
diagram
three-phase
line
of
no
LCEP,
5.11b
Figure
present,
longer
regions are not clearly distinguishable.
thus
collectively
305 K, the G+L 1
been
the F+L 2
labelled
the
the
incipient
binodal
at
(1,3)
region.
critical
metastability,
its
the
results.
With
the
and L+L
G+L 2
have
These regions
region.
a
stable
2
As at
and
an
Now, however, the entire curve lies
Under these conditions, the stable
(1,3) segments represent a metastable
LCEP,
curve
Enlarging this
F+L 2
both
exhibits
binodal
unstable critical point.
within
binodal
the
point of inflection.
horizontal
region, the
the
point
and
equilibrium.
would
would
just
be
in
touch
horizontal inflection point.
At
the
a state of
the
F+L 2
At 311 K, the
271
280
260
240
220
200
180
160
140
P (atm)
120
100
80
60
40
20
0
0.00001
0.0001
0.01
0.001
0.1
XH
Figure 5.11a.
The P-x section at 311 K.
The spinodal
curve does not actually intersect the binodal curve at
80 atm, although it appears so at the scale of this drawing.
272
81.6
81.4
81.2
81.0
E4 80.8
80.6
80.4
80.2
80.0
79.8
0.00001
0.0001
0.001
0.01
0.1
1
XH
Figure 5.11b.
The P-x section at 311 K in the vicinity of 80 atm.
The metastable G+L 1 region lies entirely within the F+L 2 region.
This diagram shows that the spinodal curve does not intersect the
F+L 2 binodal curve.
273
G+L 1
critical points lie much closer together than they
at 305 K.
It
is
G+L 1
did
This circumstance is also indicated in Figure 5.5.
now evident that the cusp in that figure forms as the
region shrinks to a
point
where
the
metastable
and
unstable critical points coincide.
Figure 5.12 shows the P-x section predicted at
313.5 K.
The necked shape of Figures 5.9 through 5.11 has given way to
a
splitting
of
the
F+L 2
region.
The
high
coexistence region may be unambiguously labelled
L+L
equilibrium.
2
The
occurrence
of
pressure
as
the
one
of
split
is
necessitated by the temperature minimum in the upper critical
line, as shown in Figure 5.1.
The existence of a temperature
minimum has been experimentally confirmed.
Increasing temperature to 400
Figure
5.13
is
found.
K,
the
P-x
section
The VLE region no longer extends to
very low hexadecane concentrations, and is beginning to
more
like
system.
at
a
normal
VLE
of 463.1 K, 542.9 K, and 623.1 K.
temperatures, some low pressure
available
and
are
phase
composition
is included in the diagrams.
the concentration of the lean
predictions
within
component,
This
to
5.16,
At these
data
is
On the basis of
the
liquid
phase.
fifteen percent and the vapor phase
predictions are within twenty-five percent
values.
look
envelope in a completely miscible
This trend continues through Figures 5.14
temperatures
of
of
the
measured
agreement is considered quite good in view of
274
C1 6-C02
300
200
-S
VI_
100
0
0.00001
0.0001
0.001
0.01
0.1
1
XH
Figure 5.12.
The P-x section at 313.5 K. Two separate regions of
fluid-fluid equilibrium exist.
275
C16-C02
300
200
a)
a,
rt
:q
100
0
0.00001
0.0001
0.001
0.01
XH
Figure 5.13.
The P-x section at 400 K.
0.1
1
276
C16-C02
30
20
E
4-,
0
L
CL
10
0.00001
0.0001
0.001
0.01
XH
Figure 5.14.
The P-x section at 463.1 K.
0.1
277
16-C 2
E
Co
4~J
vL
0.00001
0.0001
0.001
0.01
XH
Figure 5.15.
The P-x section at 542.9 K.
0.1
278
C6-C 2
300
200
E
v 4J
_(O
0L
100
0
0.00001
0.0001
0.001
0.01
XH
Figure 5.16.
The P-x section at 623.6 K.
0.1
279
the
no
that
fact
compositional
was
data
in
used
the
determination of the interaction parameter.
includes
the
and
L1=0
P-x
the
gives
Figure 5.17a
at
section
K,
650
and
The behavior of these
M1=0 curves.
below
curves is quite simple, except at pressures
about
15
Figure 5.17b gives an enlargement of this low pressure
atm.
three
of
region.
The shaded area corresponds to the region
roots.
Within this region, two false critical points of the
A
type encountered in Figure 5.3g are found.
of
the
curves helps clarify the meaning of the various
f-x
at
9
the
at
entered
zR=0
line.
composition of the second infinity
Shortly
an
thereafter,
corresponds to
fmin
(fmax
encountered
in
concentrations
the
L1=0
Figure
0.95,
about
of
fraction
Figure 5.17b, if a line of
is followed at 9 atm, a region of three
increasing xH
is
to
Referring
atm.
f-x
Figure 5.17c shows the Cartesian
lines in Figure 5.17b.
plot
consideration
in
a
5.17c),
This
is the
Figure
5.17c.
crossed,
which
in
is
5.17c.
and
At
a
point
slightly
at
to
one
mole
C1 6
spinodal
second
revert
equations
composition
point
line
roots
root
is
higher
(first
infinity point in 5.17c).
Following a constant pressure line at 7 atm,
root
region
is
entered
infinity point in Figure
(fmin
in
a
point
5.17d.
Only
at
5.17d) is encountered as xH
the
three
corresponding to the
one
is
spinodal
point
increased at this
280
C 1 6-co
2
120
100
80
P (a tm)
60
40
20
0o
.I
.2
.3
.4
.6
.5
.7
.8
XH
Figure 5.17a.
The P-x section at 650 K.
.9
281
C16-Co
C16
2o
20
15
P(atm)
10
5
.85
.90
.95
XH
Figure 5.17b. The low pressure P-x section at 650 K.
The region of three roots, shown shaded, is bounded
by the two zR=O curves.
Within this region, two
false critical points are found.
1.00
282
5.5
5.0
4.5
4.0
0.75
0.8
0.85
0.9
0.95
XH
Figure
5
.17c.
The f-x plot for Figure 5.17b at 9 atm.
The
region of three roots is bounded at either end by a slope
infinity.
Two spinodal points, fmin and fmax' are present.
283
5.0
fH
4.5
4.0
0
0.75
0.8
0.85
0.90
0.95
1.0
xl
Figure 5.17d.
The f-x plot for Figure 5.17b at 7 atm.
The
region of three roots extends to the pure hexadecane axis.
Only one spinodal point is present.
284
pressure, however, since the region of three roots extends to
the pure C 1 6
axis.
A final P-x section, at 663.8
5.18.
the
predictions
is
given
in
Figure
data is available, and the accuracy of
pressure
Low
is
K,
to
comparable
that
at
the
lower
temperatures.
This chapter has shown that
constant
interaction
parameter
qualitative picture of the
n-hexadecane
-
carbon
semiquantitative accuracy
binary
has
the
overall
dioxide
may
P-R
equation
with
yields
an
excellent
phase
behavior
system.
be
In
obtained.
of
a
the
some regions,
Of
the
four
systems treated in this thesis, the C 1 6 - CO2 - system
the
simplest
illustrates
many
phase
behavior.
Nevertheless,
it
of the characteristics of the more complex
diagrams which will be studied.
One of these characteristics
is the critical end point phenomenon.
Analysis
of
the
P-x
diagrams has shown that such a point may be located using the
criterion of a horizontal inflection point.
indicated
that
a
solution
problem is sufficient for the
equilibrium,
since
curves.
the
two-phase
determination
of
equilibrium
three-phase
the binodal curves mutually intersect at
the three-phase conditions.
curves
of
It has also been
Furthermore, the stable
binodal
are interconnected by metastable and unstable binodal
285
C
16
-CO
2
300
200
-0
v
100
0
0.00001
0.0001
0.001
0.01
XH
Figure 5.18.
The P-x section at 663.8 K.
0.1
1
286
Chapter 6
6.1
The Naphthalene-Carbn
Oualitative
predictions
The naphthalene (N) - carbon
complication
diagram.
given
of
Digzide System
pure
solid
dioxide
formation
system
to
adds
the
the P-T-x space
The physical constants of these two substances
in
Table
6.1.
The
melting point of naphthalene is
nearly 50 K higher than the critical temperature
circumstance
are
of
C
2,
a
which, as pointed out in Chapter 2, often leads
to the interruption of
the
critical
lines
by
three-phase
S-L-G lines.
The predictions in this section are carried out using an
interaction parameter of 612=0.11.
on
the
region,
basis
as
quantitative
of
will
This value
P-T-x
be
discussed
predictions
space,
n-hexadecane and
selected
data available in the supercritical fluid
are
more
as
carbon
was
done
dioxide.
thoroughly
considered.
interaction parameter, it is possible
phase
was
to
when
With
generate
this
a
fluid
in the last chapter with
More
specifically,
the
Peng-Robinson equation represents only the gaseous and liquid
phases,
and
formation.
P-T-x
space
does
not
indicate
Introduction of
requires
the
solid-fluid
possibility
equilibria
of solid
to
the
the use of an equation describing the
287
-Table 6 .1
Physical Constants of Naphthalene and Carbon Dioxide
CloH8
T (K)
c
P (atm)
w(acentric factor)
Melting Point (K)
CO 2
748.4
304.2
40.0
72.8
0.302
353.3
0.225
288
the
of
fugacity
for
developed
naphthalene
pure
the
matching
by
fugacity of
partial
the
with
fugacity
naphthalene
The solid-fluid
4.
Chapter
determined
equilibria may be simply
solid
in
was
relation
This
phase.
solid
pure
naphthalene in the fluid phase.
As with C 1 6 - C0 2 , the first features of interest in the
C 1 0H 8 - CO 2
involve
P-T-x diagram
the
critical
loci.
These
phases, and are thus generated from the
fluid
only
are
To reiterate, the equation of state
equation of state alone.
has no means of defining under what conditions a solid
will
form, and so predicts fluid phase P-T-x relations as it
any
would for
parameters
all-fluid
other
for
C10H8
diagrams
is
for C 01 H 8 - C0 2
not
is given
in
compared
with Figure 5.3e.
unstable
and
temperature
two
reduced
pressure
the
phase
which
is
to
be
Again note the prediction of one
point
The
points.
critical
for the C 10
system occurs at
8
of
high
the
corresponding
This is mainly due to the difference in
the two systems - P r ( = P/PC) for C 1 6
for
is about 5, while Pr
that
6.1,
almost twice the solute concentration
critical point.
critical
the
A T-x section at 75 atm
Figure
stable
critical
surprising
attributes.
similar
have
As
system.
are not too far removed from those of
n-hexadecane (C16 ), it
C16
phase
for the C01 H8
is
less
than
2.
For
naphthalene - C0 2, the critical line is still fairly close to
its
origin
at
the
pure naphthalene critical point.
also seen in the figure
that
the
M1
=
0
curve
has
It is
two
289
P = 75 atm
CP--
700
600
L1 = 0
·
500
400
314
312
310
308
306
304
300
0
0.1
0.2
0.3
0.4
0
.01
.02
.03
0.5
0.6
0.7
0.8
xI
Figure 6.1.
The T-x section at 75 atm showing the L1=0 and M1=0 curves.
Three critical points are predicted.
290
separate
one of these being the closed loop shown
branches,
This
in the inset.
behavior
was
not
in
found
the
T-x
sections for C16 - CO 2.
Upon predicting critical points for a number of different
temperatures and pressures, the P-T projection of the
be
may
hexadecane - CO2
each
lines,
in
beginning
at
6.2a.
Figure
system, there
The hexadecane - CO 2
end
as
constructed
two
are
points
with
As
the
critical
separate
a pure component critical point.
system, however, had only one
critical
point, where liquid and vapor phases became identical in
the presence of a third, liquid phase.
end
critical
are
There
two
now
points, where the liquid-vapor or liquid-fluid
critical phenomenon occurs in the presence of a solid
phase.
Another way of stating this is that, while the critical lines
in
general
indicate the critical phenomenon for unsaturated
solutions, the critical end
points
the
represent
critical
phenomenon for saturated solutions.
Figure
6.2a
also
indicates
that
the
prediction
critical lines does not stop at the critical end points.
upper
critical
locus,
critical
of
The
locus continues indefinitely while the lower
shown
in
more
detail
in
Figure
6.2b,
continues until it meets with an unstable critical locus in a
cusp.
A
continuation
of
the
three-phase lines after the
CEP's is not possible, on the other hand,
critical
since
beyond
the
line the equation of state predicts only one fluid.
291
800
700
600
500
400
300
200
100
0
300
400
500
600
700
T(OK)
Figure
6
.2a. The P-T projection for the naphthalene - carbon
dioxide system.
A portion of this diagram appeared in Chapter
2 as Figure 2.22a.
The range of temperatures between the lower
critical end point (LCEP) and the upper critical end point(UCEP)
is the supercritical fluid region.
292
85
80
P(atm)
75
70
65
300
305
310
315
T(K)
Figure 6.2b.
An enlargement of the naphthalene - carbon
dioxide P-T projection in the vicinity of the lower critical
end point.
The stable/metastable
unstable critical line at a cusp.
critical line meets the
293
In addition to the critical and three-phase
projection
in
Figure
6.2a
depicts
the
vaporization curves and the naphthalene
vaporization
Chapter 4
curves
are
at
experimentally
predicted
this
by
scale
measured
lines,
pure
fusion
the
P-R
P-T
component
curve.
The
equation as in
indistinguishable
curves.
the
from
the
The predicted naphthalene
fusion curve, on the contrary, shows a significant deviation.
This is a result
densities
by
of
the
poor
modelling
of
liquid
the P-R equation (see Chapter 4).
phase
Figure 6.2b
illustrates several other features of the C 1 0 H 8 - CO2
system
which will be discussed in the context of the P-x sections to
be considered next.
Before calculating the P-x sections for the
system,
be
it is helpful
shown
to see what
experimental
(1906b)
is to be expected.
As will
in Chapter 7, predictions of phase behavior in the
naphthalene-ethylene
lines in
C1 0 H8 - CO2
data
Figure
system
is
6.2
(for
available)
are
which
considerable
suggest that the critical
qualitatively
correct.
Buchner
has given a P-T-x space diagram for this case, which
was presented in Figure 2.22b.
On the basis of this diagram,
it is possible to draw up the P-x sections which are expected
to be found in the
presents
a
series
naphthalene - CO 2
of
eight
system.
Figure
6.3
schematized P-x sections, for
temperatures ranging from below CO2's critical
the near critical region of naphthalene.
point
up
to
Certain features of
these sections have been exaggerated for clarity.
294
B.
A.
C.
II+S
or
Pb
P?:
P
LCEP
Pa
PVP
V.8
or O.S
_ ~PS
P3
I; sub
Psub
I
0 XI X2
0
1
0
1
0
1
F.
S.
D.
Psub
P
0
1
1
0
H.
G.
Pfus
P
F+L 2 __
PV
0
Figure 6.3.
CO2'
1
Pp
0
A series of qualitative P-xN diagrams for naphthalene -
Pvp signifies a pure component vapor pressure, P3 a three-phase
pressure, Psub
a naphthalene sublimation pressure, Pfus a naphthalene
fusion pressure, and PTP the naphthalene triple point pressure.
295
Figure 6.3a is drawn at a temperature below the critical
point of the C0 2
solvent.
There
is
a
small
region
of
vapor-liquid
equilibrium starting at Pvp' the vapor pressure
of pure CO 2.
Adding more CH
vapor
to the system depresses
8
the
pressure of the unsaturated solution until finally, at
P3' the solid no longer dissolves but begins
This
is
the
three-phase
to
accumulate.
S-L-V (or S-L-G) pressure at this
temperature.
Suppose
that
instead
of
varying
independently, an overall mole fraction of x
the
system
pressure
varied.
Below
composition
is chosen, and
P sub' the sublimation
pressure of naphthalene, there is only a single vapor
phase.
Upon increasing the pressure, solid begins to precipitate out
and
a
two-phase
reached.
liquid
system
forms,
persisting
until
P3
is
of
solid
to
At this pressure, the transformation
occurs.
liquid and
increase
When
vapor
in
remain
equilibrium.
liquid
leg
of
the
VLE
At higher pressures, only the
remains.
If a composition of x 2
x1
was chosen, the three-phase pressure P 3
in
the
same
For
a
further
section
is
At this point, the vapor condenses to a liquid
of composition x
composition x 1
in
transformation is complete only
pressure, the liquid and vapor phases remain in
equilibrium until the
intersected.
this
manner.
liquid
of
rather than
would be reached
Now, however, the vapor condenses to
liquid rather than the solid melting to
liquid.
Proceeding
296
to
higher pressures, the solid is in equilibrium with liquid
at first, but dissolves entirely at Pa.
The
phase
point
persists
precipitates
until
out.
Pb,
Thus,
at
the
which
system
single
at
liquid
solid
x2
again
exhibits
retrograde solidification.
Figure 6.3b is drawn at the
critical
end
point.
Here
detached from the pure CO 2
CO 2
critical
process
a
point,
temperature
the
VLE
axis at
of
loop,
the
which
temperatures
lower
becomes
above
the
has just disappeared, yielding in the
horizontal
inflection
point.
The
distinction
between liquid and vapor is no longer clear in the absence of
the
three-phase
referred
to
temperature
line,
as
in
so
the two-phase equilibrium is now
solid-fluid.
the
The
LCEP
is
the
supercritical fluid regime, where it is
possible to increase pressure indefinitely without
liquid
phase
immiscibility.
of a single
fluid
system.
P-x
A
phase,
section
observing
The system is comprised either
or
is
a
two-phase
solid-fluid
in the middle of the supercritical
fluid region is given in Figure 6.3c.
where
lowest
There are two
regions
a large increase in solubility is observed for a small
increase in pressure, a fact which is used
supercritical
fluid
extraction
to
processes.
advantage
in
The two regions
are associated with the critical end points,
as
upon
In Figure 6.3d,
comparison
of Figures 6.3b, c, and d.
the high pressure region of solubility increase
into
a
horizontal
inflection
point
which
is
evident
has
is
the
evolved
upper
297
critical end point.
Figure 6.3e illustrates the situation after leaving
supercritical
fluid
the
regime on the high temperature side.
three-phase line, the S-L-F
line
present.
concentration range for which an
There
exists
a
increase in pressure leads to
immiscibility.
It
is
in
Figure
6.2a,
A
liquid-liquid
clear,
however,
is
now
or
fluid-fluid
that
sufficient
pressure increase will lead to either complete miscibility of
the two
components,
or
a
two-phase
system
equilibrium with a concentrated solution.
can
attain
significantly
possible in the SCF regime.
dashed
line
connecting
equilibrium.
and
the
equilibrium,
equilibrium
lines
continuation
of
is
the
in
(1906b).
Figure
The
analogous
reached.
to
the
unstable
is
a
maximum
solid-fluid
portions
connection
fluid-fluid
6.3e
are
represent
of the solid-fluid
the
connection
equilibrium
The shape
or
lines in the
of
the
dashed
was suggested by the work of Bchner
It is not an experimentally measured line.
Increasing
Figure
an
between
remaining
hexadecane-carbon dioxide system.
curve
than
the two curves defining solid-fluid
the
conditions.
in
In either case one
Also shown in Figure 6.3e
represents
while
solid
solubilities
The portion of this line
minimum
metastable
higher
of
6.3f,
temperature
the
This is a
to
the
C 01 H 8
triple
point,
terminus of the S-L-F three-phase line is
result
of
the
melting
of
the
solid
298
naphthalene.
Above
the
triple
phase is encountered only at
shown
in
Figure
6.3g.
point temperature, a solid
elevated
The
L 2 +S
pressures.
This
region moves quickly to
higher pressures as temperature is raised, due to
slope of the naphthalene fusion curve.
Figure
6.3h,
diagram.
the
L 2 +S
region
no
is
the
steep
At the temperature of
longer
appears
in the
The phase diagram has reverted to the familiar
and
more simple form found for vapor-liquid systems.
A sequence of
equation
will
P-x
now
sections
be
presented,
temperatures below the C0 2
sections
critical
from
again
point.
the
starting
The
6.4a
of
predicted
gas-liquid
equilibrium
to
be
visible.
Except for a small region at about
70
the S+F binodal represents the stable equilibrium state.
fluid-fluid
binodal
falls
largely
within
tieline
has
been
easily
The
L1 L2 G
metastable
This tieline
line
in
Figure
The SLG tieline, also indicated in Figure 6.2b, is not
drawn
until
the
70 atm region is enlarged.
6.4b gives
this
enlargement,
tielines.
This
diagram
roots,
The
shown in the figure.
represents a point on the metastable
atm,
the S+F region,
where it represents metastable equilibrium.
6.2b.
at
shows the P-x section at 300 K, and corresponds
to Figure 6.3a.
L1L2G
P-R
must be plotted on a logarithmic concentration axis
for the region
Figure
calculated
the
fluid-fluid
and
also
spinodal
curves,
binodals.
The
shows
both
Figure
three-phase
includes the region of three
and
L1 +G
the
labelling
binodals,
of
the
which were not
299
C10H8-C0
2
11
0.00001
0.01
0.001
0.0001
0. 1
XN
Figure
6
.4 a.
A P-x section below C02's critical point.
corresponds to Figure
6
.3a.
occur until around 1000 atm.
This figure
Retrograde solidification does not
T
in this and all following figures
is given with respect to C02. The L 1+G region does not appear at
this scale.
300
T =
T
r
300
K
= 0.986
0
Figure
6.4b.
The dot-dash
The P-x section
at 300 K in the vicinity
line is the SLG tieline,
metastable LLG tieline.
while
the dotted
of 70 atm.
line
is the
Limit points along the solid-fluid
binodal have been circled.
The region of three roots is shaded.
301
log f N
0.00001
X
(see text).
AN
G+L2 tieline.
is a metastable
1
N
An f-x plot illustrating
Figure 6.4c.
0.1
0.01
0.001
0.0001
various metastable states
S+G tieline
and BL is a metastable
GN is the stable S+L tieline.
302
visible in Figure 6.4a, form a closed loop which follows
path
(1,5)-(1,6)-(3,6)-(3,6)-(1,6)-(1,5).
below CO2 s critical temperature,
the
(1,5) would extend to the pure CO2
as
in
Figure
6.3a.
The
metastable
L+L
stable
300
L1 +G
K is
binodal
axis on a Cartesian plot,
metastable
represented by the (1,7) segments,
Since
the
and
L 2 +G
are
binodals
joined
are
to
the
binodals, (5,7), by the unstable (3,7) and
2
(4,7) segments.
Solid-fluid equilibria are represented in Figure 6.4b by
a single curve.
rises
to
a
The S+G binodal starts at low pressures
smooth
maximum
in
pressure.
designated as a limit point, and is circled
The
curve
then
the
S+L
in
point is
the
figure.
enters an unstable region, reaches a smooth
pressure minimum, and finally proceeds
along
This
and
binodal.
The
to
pressure
higher
pressures
minimum
is
also
noted
upon
designated as a limit point.
An interesting feature of this
following
the
S+G
diagram
binodal more carefully.
is
Starting at low
pressure, the three-phase S-L-G pressure is reached at
66.6
atm.
In continuing to higher pressures, the metastable
S+G curve passes entirely through
the
G+L 1
region
becoming intrinsically unstable at the limit point.
have
previously
true
before
Examples
been given where a single phase could exist
in a metastable state, with a two-phase
the
about
stable equilibrium.
system
representing
The present case demonstrates
303
may
phases
that the opposite can also occur, i.e., that two
It
exist in a metastable state in a stable one-phase region.
is
instructive
consider how the S+G metastable state is
to
defined in an f-x plot.
Figure 6.4c gives
an
f-x
at
plot
66.8 atm, which is slightly above CO2's vapor pressure (i.e.,
region), and below the maximum pressure limit
above the G+L 1
(see
point
6.4b).
Figure
This pressure is also below the
pressure maximum in the gaseous branch of the (1,7)
so
L1+G
region.
low xN
the
fugacity
the
that
binodal,
plot must also indicate a metastable
from
Referring now to Figure 6.4c, the segment
to A represents a metastable vapor phase system since
is
pressure
naphthalene
fugacity of
region.
above the G+L 1
in
vapor
the
At A, the partial
phase
the
reaches
fugacity of the pure solid, and the system may split into two
S+G, as indicated by tieline AN.
phases,
The points F and G
are not in equilibrium with A since they do not have matching
CO 2
Alternatively, the vapor may persist
fugacities.
single
phase
to
point B.
Again at B, the system may split
into two phases, this time G+L 2
points
and L.
B
a
as
with compositions
given
by
If the system at B does not split into two
phases, it may continue as a single phase until point C, when
it must revert to the metastable G+L2
S+G
state,
G
It has already
systems
could
metastable
the stable, single liquid phase system found
or
along segment OG.
composition
state, the
The stable system
for
xN
greater
than
is given by the tieline GN, representing S+L.
been
shown
that
metastable
S+G
and
G+L
exist for this range of compositions as well.
304
These are not the only possibilities, however.
H, metastable L1
beyond
H,
L1
might persist.
is
in
At a
drawn
B-L.
If L 1
composition
slightly
metastable equilibrium with L 2
composition slightly greater than L.
been
Between G and
This
tieline
of a
has
not
since it would be indistinguishable from tieline
does
not
split
to
an
L 1 +L 2
system
at
H
(approximately), it may exist until composition I is reached,
at
which
point
it reverts to a two-phase system.
for compositions greater than K, a single phase
is possible.
Finally,
etastable L2
Analysis of the f-x plot thus demonstrates
complexity
of
the
phase
equilibrium
metastable states are accounted for.
H and I, either the stable system L+S
predictions
the
when
Between compositions of
or
one
of
the
four
metastable systems L1 , L1 +L 2, L 2 +Gr and S+G could exist.
f-x
plot
also
makes
clear
how
each
of
the
The
different
metastable states arises.
The metastable S+G and L 2 +G
increase
in
size
as
regions
temperature
illustrated in Figure 6.5a, at 290 K.
is
present
at
decreased.
300
K
This is
On the other hand,
at
304 K, these regions no longer protrude into the single phase
liquid region.
still
below
This is shown in Figure 6.5b.
CO2's
critical
extends to the pure CO2
temperature,
Since 304 K is
the
L+G
region
axis.
Figure 6.6a depicts the overall P-x section at 306.3
This
K.
is very nearly the LCEP temperature, so the diagram may
305
T =
290
K
T = 0.953
r
bu
50
40
30
0.00001
Fiaure 6.5a.
0.0001
0.001
0.01
0.1
1
The P-x section at 290 K in the vicinity of the three-
phase lines. The metastable S+G and L2 +G binodals extend about 5 atm
into the single phase liquid region.
306
2
10HC8
T - 304 K
Tr
=
0.999
p.0
0.00001
0.0001
0.001
0.01
0.1
1.0
XN
Figure 6.5b.
phase lines
The P-x section at 304 K in the vicinity
(not shown).
The metastable
not extend into the single phase liquid
of the three-
S+G and L2+ G binodals
region.
do
307
C1 H8-Co
2
r
pi
0.00001
0.0001
Figure
6 6
LCEP.
The fluid-fluid
. a.
0.001
0.01
0. 1
A P-x section between CO2's critical point and the
binodal
(corresponding
to L 1 + L 2 at high
pressures and G+L2at low pressures) no longer crosses the solidfluid binodal.
308
10H8C
T
2
305.3 K
T = 1.004
r
m
XN
Figure 6.6b.
The P-x section at 305.3 K in the vicinity of the
three-phase region.
The L 1 + G binodal crosses the solid-fluid
binodal to yield a three-phase SLG tieline.
exhibit two critical points.
The L+
G binodals
This temperature is slightly
below the lower critical end point temperature.
309
be compared with Figure 6.3b.
The section is similar to that
at 300 K (Figure 6.4a), except that the
no
fluid-fluid
longer crosses the solid-fluid binodal.
The L 1 +G binodal
still intersects the solid-fluid binodal, thuiigh, a
the enlargement of Figure 6.6b.
pure
CO 2
axis,
critical point.
Having
binodal
detached
shown in
from
the
the L1 +G region now terminates in a stable
At a temperature slightly above 306.3 K, the
stable critical point
will
remain
as
the
last
point
of
contact between the L,+G binodal and the solid-fluid binodal.
This
is
the
solid-fluid
lower
critical end point.
binodal
inflection.
It
is
exhibits
also
a
predicted
as
the
a
metastable
metastable L1 L2 G line.
horizontal
evident
temperature is raised further,
At this point, the
point
of
from the figure that, as
metastable
critical
LCEP
point
The meaning of the two
will
be
meets
the
critical
end
points in Figure 6.2b should now be clear.
Figure 6.7 gives a
P-x
section
fluid
region,
6.3c.
The solid-fluid binodal is now
system
at 311 K.
in
the
supercritical
This diagram corresponds to Figure
the
stable
throughout the entire pressure range.
increase between about 40 and
80
atm
is
two-phase
The solubility
actually
not
as
dramatic as it was at lower temperatures (see Figure 6.4a and
6.6a),
although
at
higher
pressures
somewhat greater than previously.
associated
with
The
the
solubility
solubility
is
increase
the upper critical end point is not visible
in this diagram since it occurs above 700 atm.
310
C10H8
-C
2
DI
0.00001
0.0001
0.001
0.01
0.1
XN
Figure 6.7. A P-x section in the supercritical
fluid region.
The S+F binodal represents the stable two-phase equilibrium
-at allpressures.
-
-
....-
-
311
Increasing temperature to 326.3 K, the
Figure
6.8
is
found.
critical end point
similar
to
intersects
now
a
slightly
and
6.3d.
exists
A
between
the
of
above the upper
diagram
region
650
section
of
and
is
thus
liquid-liquid
680
atm.
It
the solid-fluid binodal in two places to form the
three-phase SLF line.
again
is
temperature,
Figure
equilibrium
This
P-x
single
Note that the solid-fluid
curve
suggests, at the UCEP
with
two
extrema.
temperature
the
binodal
is
As the diagram
L-L
critical
point
intersects the solid-fluid binodal at a horizontal inflection
point.
Figure
6.8
also
illustrates
the
course
of
the
spinodal curve and the region of three roots.
The P-x section at 340 K is given in Figure 6.9a.
diagram
S-L-F
is
the
line
pressure,
is
stable
predicted
depending
liquid-liquid
systems.
equivalent
or
of
at
upon
Figure 6.3e.
about
65
overall
solid-liquid
are
A three-phase
atm.
system
the
This
stable
Above
this
composition,
two - phase
Below this pressure, the S+F binodal represents the
two-phase
system.
Consequently,
the
low pressure
portion of the L+F envelope is metastable.
The connection of the S+F and S+L
binodals, shown as a
hypothetical curve in Figure 6.3e, is not at all apparent
6.9a.
in
In this case, the continuation is discernable only at
very low pressures, as depicted in Figure 6.9b.
As is
seen,
312
10H8
-CO2
0.00001
0.0001
0.001
0.01
0.1
xN
Figure 6.8.
A P-x section at a temperature slightly above the
upper critical end point.
region of three roots
The SLF tieline is shown dashed.
is shaded.
The
313
IUU
600
500
400
00
200
100
0
0.00001
Figure
0.0001
6
.9a.
triple point.
0.001
0.01
0. 1
A P-x section between the UCEP and naphthalene's
The SLF tieline is shown dashed.
1
314
C10H8C2
T
340 K
(atm)
0
.1
.2
.3
.4
.5
.6
.7
.8
XN
Figure 6.9b.
The low pressure
P-x section
at 340 K.
.9
1
315
metastable S+L line connects to the unstable S+F line at
the
about 3 x 10
the
-6
lower
The unstable S+F line then proceeds
to
point where it joins the metastable S+F line (see
limit
turn
The metastable S+F segment in
Figure 6.9a).
to
atm.
pressures,
fluid branch of the L+F
the
crossing
continues
and
curve at the three-phase pressure, thus becoming stable,
attaining
a
minimum
Thereafter,
atm.
increases
concentration
naphthalene
in
fluid
the
the
both
crossing
pressure,
decreasing
with
concentration at about 30
naphthalene
unstable S+F and metastable S+L lines before intersecting the
pure naphthalene axis at the
pressure.
It
is
that the metastable S+L line extends to pressures below
seen
the C
has
sublimation
8
not
sublimation pressure.
been
experimentally
Such
a
observed.
and
binodal is also shown in the diagram,
metastable
region
The metastable L+F
terminates
at
a
metastable naphthalene vapor pressure.
Figure 6.10 gives the calculated P-x section at 354.5 K,
This
the predicted triple point temperature of naphthalene.
diagram corresponds to Figure 6.3f.
binodal
S+L 2
The L-F envelope and the
meet at the triple point.
situation at 340 K, the S+F binodal
In contrast to the
represents
now
only
a
metastable equilibrium.
Figure 6.11 gives the predicted P-x section
at
somewhat above naphthalene's triple point temperature.
to
be
compared
with
Figure
6.3g.
At
360
K,
It is
sufficiently high
316
C10H8
0.00001
0.0001
2-CO
0.01
0.001
0.1
XN
Figure 6.10.
The P-x section at the triple point temperature.
Each of the binodal curves terminates at the triple point.
317
C10Hs-C 2
---
700
600
500
400
00
OO
200
100
0
0.00001
0.0001
0.001
0.01
0. 1
XN
Figure 6.11.
temperature.
A P-x section above naphthalene's triple point
-
318
pressures, it is still possible to have solid present in
system,
in
equilibrium
with
the
a very concentrated solution.
This two-phase region begins at a point along the naphthalene
fusion curve, labelled as the C 1 0H 8
Such a point
in the figure.
at
pressure
which
fN = fN
is
crystallization pressure
found
determining
by
x N =1.
when
Knowledge
crystallization pressure at several temperatures
to
plot
the
Figure 6.2a.
the
of the
allows
one
predicted naphthalene fusion curve depicted in
As at 354.5 K, the S+F binodal
is
metastable.
A metastable S+F tieline has been included in this diagram to
emphasize
that
the
limit
point
is
not a critical point.
Although two S+F binodals meet at the limit
phases
in
equilibrium
(solid
and
point,
fluid)
loci
do
the
not
cross,
two
become
identical.
At low pressure, the S+F
with
the
metastable
segment intersecting the pure naphthalene axis at
a metastable sublimation pressure.
The
important
system.
preceding
features
It
is
sequence
of
the
convenient
of
P-x
sections
naphthalene
to
summarize
-
covers
carbon
some
the
dioxide
of
the
compositional information contained in these sections through
P-x
and
T-x
projections.
These
projections are given in
Figures 6.12a and b for the upper critical line and
the
SLF
three-phase region.
The P-x and T-x projections of the lower
critical
the SLG region are very similar to those
line
and
for the naphthalene - ethylene system, which will be shown in
Chapter 7 (Figure 7.3).
319
P4
CPN
x
Figure
6
.12a.
N
The P-x projection of the upper critical line
and the SLF region.
A sample SLF tieline is shown dashed.
320
357
353
349
345
3
E- 341
337
333
329
325
0
.1
.2
.3
.4
.6
.5
.7
.8
.9
1.0Io
XN
Figure 6.12b.
The T-x projection of the upper critical line
and the SLF region.
A sample SLF tieline is shown dashed.
321
6.2
Ouantitative Predictions
A limited amount of actual
available
for
the
phase
equilibrium
naphthalene - C0 2
system.
data
The
is
most
frequently cited measurements are those of Russian origin, by
Tsekhanskaya,
recently
et
been
al.
(1964),
within
the
(1981).
supercritical
plotted as shown in Figure 6.13.
the
effect
observations
of
the
The Russian
data,
general
solubility
to
which
fluid region, are typically
Also illustrated
there
is
value of the interaction parameter.
At
35 C and 45 0 C, Figures 6.13a and b, the curves all
correct
have
confirmed by the work of McHugh and Paulaitis
(1980) and Kurnik, et al.
fall
whose
shape,
within
and
a
for
factor
the
of
most
two.
have
part
The
the
give the
predicted
solubility increase with pressure due to the proximity of the
lower
critical end point is not quite rapid enough, which is
particularly evident at 35 C
and
about
80
atm
calculated values are off by a factor of 4 or 6.
where
the
The largest
discrepancy, however, arises at 55 C (Figure 6.13c), when the
Peng-Robinson
equation
predicts
S+L+F region for the three lowest
entry into the three-phase
12
values.
The
vertical
lines in Figure 6.13c represent SLF tielines of the type seen
in Figures 6.3e and 6.9a.
due
to
the
After predicting a solubility jump
phase change of fluid to liquid, the calculated
solubility for 612 = 0.11 is eight times too high.
According
322
Z
x
r-I
150
100
200
250
300
P (atm)
The solubility of naphthalene in supercritical
Figure 6.13a.
CO at35 0 C.
2
Predicted solubilities for four different values of
the interaction
parameter are shown.
323
0
-1
xZ
0
I-f
-2
-7
100
150
200
250
300
P (atm)'
Figure 6.13b.
CO2 at 45 0C.
The solubility of naphthalenein supercritical
324
C 10Hs-CO
2
xt
0
Hl
100
150
200
250
300
P (atm)
Figure
6
.13c.
CO2 at 550 C.
The solubility of
For
1 2=
0.09, 0.10,
is predicted by the P-R equation.
naphthalene in supercritical
and 0.11, a solubility jump
325
to experimental observations, a three-phase pressure will not
be encountered until somewhat higher temperatures.
predicting
the
three-phase
temperature, a
line
Although
and the UCEP at too low a
of 0.11 gives better agreement over
612
of the SCF region than a 612
most
of 0.12, and was therefore used
for all further calculations.
Saturated liquid solubilities in
been measured by Quinn (1928).
(1954)
is
in
close
the
SLG
end
(1964).
within
point
agreement
as
with
determined
The Peng-Robinson
a
factor
of
2,
Quinn's observations.
appropriate
with
are
again
except near the LCEP.
sole
the
lower
by Tsekhanskaya, et al.
solubilities
point at 250 C is used as the
have
A measurement made by Francis
These data are plotted in Figure 6.14 along
critical
region
basis
for
largely
If the data
selecting
an
612 = 0.09 is
interaction parameter, a value of
found.
Using the data of Quinn and Tsekhanskaya, Modell, et al.
(1979) have developed the "solubility
which
was presented in Chapter 2.
in Figure 6.15, is essentially
a
map"
for
C 10 H 8 - CO 2
This diagram, shown again
projection
of
solubility
isobars on the T-x plane, and is a particularly concise means
for
summarizing the data in the SCF region.
region represented by the dashed
saturated
Figure
vapor
6.2.
+
Figure
solid)
6.16
line
corresponds
gives
the
The three-phase
(saturated
to
liquid
+
the SLG line in
solubility
map
as
326
35
30
25
20
15
10
T(
C)
5
0
-5
-10
-15
-20
-25
0
.001
.002
.003
.004
.005
.006
.007
.008
X
N
Figure 6.14.
The T-x projection of the SLG region.
327
T(°C)
10- 1
0
10
20
30
40
50
i
I
J
I
I
I
10- 2
)
XN
)
(
90
10-3
80
70 atm.
tie line
/
/ /
/
/
I
Figure 6.15. The solubility map for the naphthalene
carbon dioxide system.
naphthalene in solution.
(N) -
Isobars indicate the mole fraction of
The dashed line corresponds to a three-
phase solid-liquid-vapor equilibrium.
This equilibrium
terminates at LCEP, the lower critical end point.
328
T (K)
280
290
300
310
320
330
340
350
360
0
-1
-2
log xN
-3
-4
-5
Figure 6.16.
The predicted solubility map for
- C2.
naphthalene
329
predicted
the P-R equation, with the SLF region and UCEP
by
now appearing.
It has already been noted that the UCEP seems
that
in
solubilities
the
region may be off by a
SCF
the
equation
It is nevertheless clear that the
factor of 2.
and
temperatures,
higher
somewhat
to occur in practice at
of
state gives an excellent qualitative description of the phase
As
region.
this
in
behavior
a
to
aid
further
the
interpretation of the solubility map, a T-x section at 65 atm
heavy line.
atm
appear in Figure 6.11 have been denoted by the
which
isobar
65
the
of
Portions
has been included in Figure 6.17.
corresponds
Figure 6.17 also
to
horizontal
a
plane at 65 atm in Figure 6.2a.
naphthalene
determined
experimentally
(1980)
have
solubilities
in (or
Paulaitis
and
McHugh
Recently,
near) the SCF region at 60.4 0 C and 64.9°C and
coordinates of the UCEP.
the
P-R
Their data, and its comparison with
for 612=0.11, is shown in Figure 6.18.
predictions
In both cases the predicted solubilities are too
large
change.
equation
temperature
the
pressures
230
with
high,
of state predicts a phase
interesting,
The data at 64.9°C is particularly
the
between
the
when
errors
the
estimated
as
is high enough to intersect the SLF line at
The
studied.
solubility
jump,
occurring
and 236 atm, still falls significantly short of
the predicted solubility level.
be
discrepancy
will
data allows
estimation
A possible reason
considered shortly.
of
the
for
this
In any event, the
coordinates
of
the
upper
330
T,K
280
290
300
310
320
330
340
350
360
1
10-1
10-2
XN
-
10
Figure 6.17. A T-x section at 65 atm. Phase
equilibrium lines which appear as the isobar in
Figure 6.16 have been accentuated.
331
2
C 10H8-C
A
-1.
00
0
0
0
0
o
z
x
0
0
o
.H
T= 60.4C
0
O
McHugh and Paulaitis (1980)
I
3. I-
50
200
150
100
250
P (atm)
Figure
6 .1 8 a.
For the predicted
naphthalene in CO2 at 60.40 C.
The solubility
of
curve,
12 = 0.11.
300
332
C10H8-CO2
__
0
0
o
-1.
o0
0
z
0
0
0
0
T
O
=
0
64.9 C
-2 -
O McHugh and Paulaitis (1980)
-3-
I
I
!
.
!
200
250
.
j
50
100
150
P (atm)
Figure 6.18b.
The solubility of naphthalene in CO 2 at 64.90 C.
For the predicted curve, 6 1 2 = 0.11.
300
333
critical end point.
Since a solubility jump was not observed
at 60.4 0C, but was at 64.9 0C, the UCEP must lie between these
two temperatures (assuming that the UCEP falls below 290 atm,
the
highest
pressure
investigated).
Since Bchner
(1906a)
has observed the formation of a liquid phase in the
presence
of solid and vapor at 64 0 C, the estimate should actually fall
below
this
temperature.
McHugh and Paulaitis' estimate of
the UCEP temperature is shown in Table
calculated
and
experimental
6.2,
critical
which
end
compares
points.
These
authors have also estimated the UCEP pressure on the basis of
the shape of the solubility isotherms.
provide
the
a very rough estimate.
estimated
calculated
UCEP
only
It is seen in Table 6.2 that
composition
agrees
well
with
the
value, in marked contrast to the temperature, and
especially the pressure, coordinates.
LCEP
This method can
The situation
at
the
is just the opposite, with reasonable agreement between
pressure and temperature
and
a
predicted
composition
2.5
times too small.
Up
to
this
point,
it
has
been
presumed
that
the
experimental data available represent true stable equilibrium
conditions.
There
is
little
reason
to
question
this
assumption at the lower temperatures, where agreement between
prediction and experiment
55 0C
is
relatively
At
and above, however, when the equation of state predicts
a phase change, there are some grounds
the
satisfactory.
calculated
values
may
be
for
contending
that
closer to the truth than at
334
o
ro
Bd
0
(N
m
%O
',CT
co
*4
U)
r{
.,o
_
0
O-4
'O
H
.,
rN
rI
0
0
(t
E0
0
U
N
U
H
Ln
0
_ 4dC
0
U
H
0co
n 4)
44
0
O
a,
rl
LA
dX
N
U
C)
U)
,
Xd
(1
r-A
Q
r.
-4
_
U
0
Hz
r
X H
U)
- n~
LA
.,,
_H
U)
C
U
O
4
-4
.1-t
R Od
r-- 0z a
Cv
11
C,
4-'
,l
0
U)
x
Cd
cd
0
0
0
4
>4
ra
4.4
0
O
4
Eq
)4
U
M
4.fdr
C'
II
04
a)
Un
a4
x
N,
4i
E-rU)
.,~
I
a~
I
I
I
C)
ro
335
first appears.
and
(66.80C,
A comparison of Figure 6.9a
(550 C, 328 K), which give qualitatively identical
6.13c
P-x sections, shows that the solubility jump
the
three-phase
complete picture.
experimental
gives a
are
corresponds
Figure 6.19 shows again the predicted
for
solubilities
as
included,
and
cases where the P-R equation
solid-fluid
all
however,
Now,
jump.
to
that Figure 6.13c is not a
and
line,
SLF
solubility
binodals
340 K)
as the stable fluid-fluid
well
binodals.
moment
Consider Figure 6.19a, at 55°C, ignoring for the
the
L+F
It
equilibrium.
is
seen that the metastable S+F
solution matches the data quite well,
pressure
of
320
atm.
not
highest
the
At 60.40 C (Figure 6.19b) and 64.9°C
(Figure 6.19c), on the other hand, the
does
at
even
metastable
S+F
line
extend to high enough pressures to match the data.
It is nonetheless plausible, given the
limited
accuracy
of
the P-R equation of state, that the experimental points do in
actuality represent metastable, subsaturated S+F equilibrium.
The
likelihood of the existence of such metastable states is
uncertain, but it should be
pointed
out
that
examples
of
metastable supersaturated solutions are quite common.
Some
evidence
equilibrium
at
64.9°C.
may
for
the
existence
of
metastable
be supplied by the solubility jump observed
McHugh
and
Paulaitis
(1980)
indicate
that
considerable uncertainty was involved in the determination of
336
4-,
u
, UL'
LL
,
c
Cn
0m
o-0
C -0
r
"-
CO
i
u
a 4
-
wII
rP4
$4
o
C
0O
4J
n
C
l
0
0O
U)
V C
337
CJ
O
o
4J
o
oC4
Cm
Co
c.i
-c
o
-P
P4
u
0.
C.
ret
49
0
-o
0
-
L._
0
I
cm
I
N
I
0
CV
o
K
I
338
0
Cd
D
C
._
uC
I
QS
o
0
-,
.c
-
U
r
a)
0
X0
bo
0
r--
U.
t
339
the solubility points at pressures above the solubility jump.
One possible explanation of this scattering is that more than
one
stream being measured.
phase was present in the CO 2
between
occurs
pressure
three-phase
is easiest to imagine that the
230 and 236 atm, and that the scattered measurements
represent the combined solubility of naphthalene in a
liquid and a metastable fluid phase.
phases
in
stream
CO2
the
would
representative of
true
the
For this
variable.
be
of
the
pressure
as
solubility.
If
particular
liquid
phase
was still present in these cases, however,
fluid
metastable
a
at
obtained
solubilities
stable
The proportion of these
reason, McHugh and Paulaitis have chosen the maximum
the true
It
actually
is
solubility
liquid
the
than
higher
reported values, and closer to the predicted values.
could
The existence of metastable phases in this system
be
readily
ascertained
by
approaching
the
SLF line from
pressures above the limit point (i.e., 400 atm or more).
The
carried
out
6.19
experiments of Figure
entirely
below
experimentally
actual UCEP
unresolved
most
likely
limit point, as 320 atm was the highest
the
pressure attained.
were
In
light
of
the
uncertainty
in
the
measured solubility jump, the question of the
temperature
and
pressure
must
until more data becomes available.
be
considered
Most probably
they lie somewhere between the experimental estimates and the
predicted coordinates.
340
- CO 2
One other region of the C1 0 H 8
been
experimentally
investigated.
phase diagram
Najour
has
and King (1966)
have taken measurements at 9 temperatures from 24 C to 73
with
pressures
ranging
between 2 and 60 atm, yielding what
might be called compressed gas
data
solubilities
appeared in Figure 6.19a).
(some
of
this
A sampling of their results
in comparison with predictions is given in Figures 6.20a,
and c.
for
C,
b,
Although the interaction parameter of 0.11 was chosen
compatibility
with
the
SCF
actually better in the compressed
region data, agreement is
gas
region.
Predictions
are well within a factor of two in all cases (note the use of
a
linear concentration axis).
Also shown in the figures are
the solubilities which would be found in the ideal case, when
Dalton's law would apply, i.e.,
sub
XNPN =
(5.1)
PN
Deviations from the ideal case are significant even below
10
atm.
This
chapter
has
indicated
that
the
Peng-Robinson
equation, with a constant binary interaction parameter, gives
at
best
fluid
semiquantitative
region
parameter
is
of
solubilities in the supercritical
naphthalene - CO 2 .
allowed
If
the
interaction
to vary with temperature, a review of
Figure 6.13 shows that predicted
solubilities
within approximately a factor of 2.
may
be
kept
A pressure functionality
341
C10H8
-C
2
3
,.
:
z
X
2
0
0
10
20
30
40
50
60
70
P (atm)
Figure 6.20a.
The solubility of naphthalene in CO2 at 260 C.
342
C10H8-C 2
19
18
17
16
15
LU
C
z
14
13
X
12
11
10
0
P (atm)
Figure 6.20b.
The solubility of naphthalene in CO2 at 550 C.
2
343
C H8-CO
2
34
32
30
LO
-it
X
28
z
26
24
0
10
20
30
40
50
60
70
P (atm)
Figure 6.20c.
The solubility of naphthalene in CO2
at 64°C.
344
of
would
612
predictions.
be
Kurnik
necessary
(1981)
to
has
obtain
reported
quantitative
similar
results
using the P-R equation.
More importantly, though,
allowed
the
construction
qualitatively correct
system.
Knowledge
of
P-T-x
of
the
the
equation
what
space
is
for
overall
of
state
believed
the
P-T-x
be
-
8
behavior
C02
is of
process
design.
show,
that
supercritical fluid extraction
example,
a
P-T-x
a
substantial value in
for
The
CH
to
has
process run near the UCEP rather than near the LCEP
advantage
of
much higher solute solubilities.
also suggest that a two-phase
desirable
(see
Figure
6.16).
extraction
has
might
interpreted
may be necessary.
with
be
the predicted
phase diagrams indicate regions where experimental data
be
the
The diagrams
process
Furthermore,
diagrams
must
care, and in what regions further data
345
Chapter 7
The
Naphthalene - Ethylene System
A drawback to the modelling
dioxide
system
of
the
is that data are available over only a small
range of pressures and temperatures.
line,
and
distance
the
entire
from
the
determination
naphthalene-carbon
of
L-F
region
A portion of the
critical
which
locus,
can
be
fall
used
the interaction parameter.
at some
in
the
The only basis
for judging the accuracy or qualitative correctness
extrapolated
S-L-F
of
such
predictions is by analogy with similar systems.
One such system is that of naphthalene-ethylene, for which
fairly
complete
reported.
These
set
of
include
experimental
accurate
results
measurements
a
have
been
of
the
critical end points, as well as data along the S-L-F line and
L-F
critical
line.
For
these
reasons
naphthalene-ethylene system was chosen for
constants
for
naphthalene
and
study.
the
Physical
ethylene are given in Table
7.1.
In order to gauge the reliability of the naphthalene-CO
predictions
by
the
reliability
of
the
2
naphthalene-C2 H4
predictions, only data in the supercritical fluid regime were
used
in
the
determination of the interaction parameter
was done with C 1P 8-CO
2 ).
A value of 612=0 was found to give
reasonably good fit, as shown in Figure 7.1.
The behavior is
essentially the same as in Figures 6.13 and 6.19
the
analogous
plots
for
(as
C 1C8 - CO2 .
In
which
gave
that system, a
346
Table 7.1
Physical Constants of Naphthalene and Ethylene
C10H 8
T (K)
c
P (atm)
c
w(acentric factor)
Melting Point (K)
2
748.4
282.4
40.0
49.7
0.302
353.3
0.085
347
C10H8-C2H4
-1
-2
z
x
O
-4
50
100
150
200
250
P (atm)
Figure 7.1a.
The solubility of naphthalene in ethylene at
285.2 K (120 C).
For the predicted curve, 612= O.
The reduced
temperature in this and all following figures is given with
respect to ethylene.
348
Clo H8-C2H4
-1
-2
x
0o
rH4
-3
-4
-5
0
50
100
150
200
250
P (atm)
Figure 7.lb.
298.2
The solubility of naphthalene in ethylene at
K (250°C).
300
349
C
H8-C2H
4
-1
-2
Z
r-i
-3
-4
-5
0
50
100
150
200
250
P(atm)
Figure 7.1c.
308.2
The solubility of naphthalene in ethylene at
K (350 C).
300
350
C 10H8-C2H
4
0
-1
log xN
-2
-3
-4
50
100
200
150
250
300
P (atm)
Figure 7.1d.
A solubility
The solubility of naphthalene
0
in ethylene at 318.2 K (45 C).
jump is predicted, so the complete
stable L+F binodals have been included.
solid-fluid
binodal and the
351
C1 0 H 8 -C 2 H4
0
-1
log xN
-2
-3
-4
50
100
200
150
250
P (atm)
Figure 7.1e.
The solubility of naphthalene
A solubility jump is again predicted
0
in ethylene at 323.2 K (50 C).
at this temperature.
352
a solubility jump is predicted at 450 C,
naphthalene-ethylene
around Pr= 6 , as compared to a Pr
to
range,
pressure
6.8.
in Figure
respect to C0 2
no
data
With respect to ethylene, the
as shown in Figure 7.ld.
extend
For
55 C.
at
predicted
was
jump
solublity
solubility
Even
with
jump
is
of about 4 with
extensive
this
by the
evidenced
experimental points, an indication that 450 C is
below
truly
the UCEP temperature.
With
are
7.2
Figure
of
and
temperatures
projections
P-T
good
quite
the
over
from
indistinguishable
the
entire
The calculated pure
pressures.
naphthalene
component vapor pressure curves for ethylene and
are
experimental
curves.
calculated fusion line, however, has a steeper slope than
Figure
should.
naphthalene-C0 2
predicted
P-T
from
differs
7.2a
means
universal.
line speaks in favor of the qualitative
naphthalene-C02
critical
line.
The
once again shown continuing past its
with
line at the UCEP.
the
the
maximum
is
Rather,
the
correctness of the naphthalene-ethylene critical
qualitative
S-L-F
it
As indicated in Chapter
in the L-F critical line.
no
a
The
6.2a,
Figure
that
in
projection,
2, such a maximum is by
of
Qualitative agreement with the
generated.
experimental phase diagram is
range
the
zero,
set equal to
612
metastable
enlargement,
Figure
correctness
of
the
L-F critical line is
intersection
with
the
This metastable portion forms a CEP
L-L-G
7.2b.
locus,
as
depicted
in
the
There is a second CEP formed by
353
C
H8-C2 H4
300
200
-i
_11
100
0
300
400
500
600
700
T,K
Figure 7.2a.
system.
point.
The
The P-T projection of the naphthalene--ethylene
inset shows the region near ethylene's critical
(See also Figure 7.2b.)
354
C
H8-C2H
4
60
50
40
30
20
250
260
270
280
T(K)
Figure 7.2b.
The P-T projection of the naphthalene - ethylene
system in the vicinity of ethylene's critical point.
290
355
the intersection
of
L-G
metastable
metastable
the
locus
the
and
figure also shows the
The
line.
critical
L-L-G
generally excellent agreement between calculated and measured
Note that the S-L-G line is
S-L-G loci.
terminated
not
by
the metastable L-F critical line.
this
A comparison of the stable critical end points for
is
system
in
given
As with naphthalene-C0 2,
7.2.
Table
(Table 6.2), the temperature and pressure coordinates for the
LCEP agree quite closely, while the predicted
is
fraction
mole
At the
low, in this instance by a factor of 6 or 7.
too
too
UCEP the predicted temperature is about 20 percent
low,
and
the predicted naphthalene mole fraction about 35 percent
too
low.
This
is
The
naphthalene-CO 2.
to
similar
what
UCEP
predicted
UCEP
pressure
in
Table
found
for
is
only
pressure
6.2.
In
that table,
however, the calculated UCEP pressure was almost three
the
slopes
of the L-F critical line and the S-L-F line in
P-T
naphthalene-CO 2
temperatures near the UCEP.
an
inaccuracy
change
the
of
From
(Figure
projection
only
by
at
From a calculational standpoint,
pressure
an
by
almost
one
hundred
experimental point of view, assuming
that a region of steep slopes does indeed exist, it
detected
6.2a)
a few degrees in the predicted UCEP could
predicted
atmospheres.
times
This is perhaps due to the
the estimated experimental value.
steep
the
of
reliability
slightly low, which might speak for the
calculated
was
measurements
at
carefully
will
be
selected
356
0o
0
rZ
i-
o
P4
w
0
.-W
U
H
E-
L-
0
N
.H
0
r.
ro
H
.H
u
0o0o
0
-W
r-
f-I
u
a
w,
a)
I
xZI
0
_
CN
O
0
m
0)
r-4
9-
'.0
(c
Z
U
_
>
0
o
co
L
H)
_
a)
r04
r.
*
.3-4
(a
zft
z
=rd
,
a
co
a,
a
o
(D
II
I
I
x,
R
3
357
temperatures and pressures.
system, van Welie and Diepen
For the naphthalene - C 2 H4
have also measured composition coordinates of the S-L-F three
the
T-x and P-x projections in addition to the
of
plotting
allows
This
phase line and part of the L-F critical locus.
The T-x
data compare with predictions in the P-x projection.
the
of
projection
a portion of the L-F
and
region
S-L-F
Figure 7.3c.
described
as
crtical
L-F
stable
entire
the
other
line is illustrated in
Figures 7.3d and e show the
semiquantitative.
than the LCEP is available.
and
for
projections.
In
two
the
contrast
to
bend
pressures.
and
thereafter
The
S-L-F
runs
locus,
as
fairly high pressures before it can
line at the UCEP.
and
P-x
T-x
The only
the
P-x
naphthalene - C2 H4
L-F
figures
the
critical line, the naphthalene - CO 2
sharp
Figure 7.3
system.
naphthalene - CO 2
major difference between
highest
the
pressure along the S-L-G locus.
the
no
As evident from the
is to be compared with Figure 6.7, which gives
projections
which
for
region,
P-T projection of Figure 7.2b, the LCEP lies at
temperature
best
are
results
For all three diagrams, the
P-x and T-x projections of the S-L-G
data
projection
T-x
The
critical line are given in Figure 7.3b.
of
these
Figure 7.3a shows how
P-T projection just considered.
is
in
L-F critical line has a
very
steeply
to
higher
a result, must extend to
intersect
the
critical
358
C 10H8-C
H4
300
r
200
100
0
0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
XN
Figure 7.3a.
The P-x projection of the upper critical line and the
SLF region for naphthalene - ethylene.
compared with Figure
6
.12a.
This diagram is to be
359
C
H8-C 2H 4
80
70
T( C)
60
50
40
.2
Figure 7.3b.
.4
.6
1.0
.8
The T-x projection of the SLF region and a portion
of the upper critical line for naphthalene - ethylene.
diagram is to be compared with Figure 6.12b.
This
360
0.1
0.2
0.3
0.4(
0.5(
xN
XN
0.6(
0.7(
0. 8
0.9
1.0C
100
200
300
400
T(K)
Figure 7.3c.
The T-x projection of the upper critical line
for naphthalene - ethylene.
500
361
C 10H8-C2H 4
0
.001
.002
.003
XN
Figure 7.3d.
The P-x projection of the SLG region and a portion
of the lower critical line for naphthalene - ethylene.
A sample
SLG tieline, which continues off the graph to the pure
naphthalene axis, has been included.
The corresponding diagram
for naphthalene - CO2 is qualitatively the same.
362
C 10H8-C2 H4
0
.001
.002
X
Figure 7.3e.
.003
N
The T-x projection of the SLG region and a portion
of the lower critical
line for naphthalene
- ethylene.
A sample
SLG tieline, which continues off the graph to the pure
naphthalene axis, has been included.
The corresponding diagram
for naphthalene - CO2 is qualitatively the same.
is to be compared with Figure 6.14.
This diagram
363
In order
to
provide
naphthalene - ethylene
further
and
comparison
between
naphthalene - CO 2
systems, P-x
sections at nine different temperatures have been
Some
of
the
drawn
up.
sections have been chosen at the same absolute
temperature, and some at the same reduced
respect
the
to
the
counterparts.
solvent,
as
temperature,
their
with
naphthalene - C02
Experimental data is not available
for
these
diagrams.
Figure
temperature
7.4a
shows
below
the
metastable LLG
between
the
scale.
7.4a.
line
various
the
low
(see
P-x
section
temperature
Figure
at
The
a
connections
binodal curves are not evident at this
Figure 7.4b expands the low pressure range of
Figure
The presence of the SL1 G tieline is now evident, along
behavior
regions.
The metastable
is in most respects similar to that of Figure 6.5a.
The metastable L1+L 2
critical point is
a
point
metastable L-F critical line in Figure 7.2b.
and
7.4b,
previously
it
appears
given
sublimation
in
the
In both Figures
unstable
the
S+F
usual
and
fashion,
7.4c,
met
with
An
computational
which were not encountered in the generation of
6.9b.
metastable
the
along
for very low pressures in Figure 6.9b.
attempt to show this, Figure
difficulties
that
segments will join
metastable S+L 1
Figure
K,
termination of the
7.2b).
with the stable S+G, L1 +G, and S+L 1
7.4a
255
L1 +G
Although
region
pressure
and
the
stable
terminate
the
S+G
segment
normally
metastable
at
and
the
C 1 0H 8
the
C1 0 H8
vapor
364
20
30
P (atm)
00
C
XN
Figure 7. 4 a.
Figure 7.2b.
A P-x section below the lower metastable CEP in
365
C
H8-C2H
4
30
20
P (atm)
10
10-7
10-6
10-
10-5
3
10-2
10-1
XN
Figure 7.4b.
The P-x section
G+L2 region is present.
at 255 K and low pressures.
No
366
C 10H8-C2
H4
108 214~~~~~~~~~~~~~~~~~~~~
-10i
10-2
10-3
P (atm)
10- 4
10-5
10
1-6
-7
-6
-5
-41
-3
-2
-1
0
log xN
Figure 7.4c.
The P-x section
at 255 K and very low pressures.
367
pressure,
respectively,
legs
diverging
the S+F and S+L 1 legs still do not
-6
join for pressures as low as 10
atm.
In fact, the two
are
at
this
point.
binodals are traced down to 10
convergence or divergence.
in
the
defining
atm without any significant
By 10
equations
The metastable L1+L 2
had
atm, certain
become
quantities
extremely
small,
resulting in unreliable calculations even in double precision
mode.
Since this low pressure region
is
not
of
practical
significance, the problem was not considered further.
In the course of tracing the low pressure
Figure
7.4c,
investigated.
have
not
the
form
of
the
Figure 7.5 illustrates
been previously presented.
f-x
ideal
gas
root
is
system
pressure.
shapes
range.
essentially
a
which
are
also
which
The
high
straight line,
approximately
the
The low volume, liquid root has two
spinodal points and is quite similar to the curves
root
was
in
All of the plots yield
intersecting the pure naphthalene axis at
total
curves
several
three roots across the entire composition
volume,
solutions
for
presented in Chapter 3 and Appendix B.
this
The
liquid solution is affected very little by the small absolute
pressure difference
between
the
three
figures.
The
f-x
curves shown previously at 0.1 and 1 atm for n-hexadecane-C0 2
showed
the
unstable
root
continuous with the liquid root.
This behavior will also be found in the
system
naphthalene-ethylene
at temperatures higher than 255 K.
Figure 7.5 that when the unstable root
does
It can be seen in
join
with
the
368
C
H8-C
H4
1
0
1
-2
-3
-4
-5
0
z-h
-6
-7
-8
-9
0.00001
0.0001
Figure 7.5a.
range.
0.001
0.01
0.1
1
An f-x plot with three roots over the entire composition
The three roots would converge at xN= 0 in a Cartesian plot.
369
C
H8-C2H
4
2
1
0
-1
-2
-3
-4
-h
-5
-6
-7
-8
-9
0.00001
0.0001
Figure 7.5b.
0.001
0.01
0.1
An f-x plot with three roots over the entire
composition range.
1
370
C10H8-C2H
4
log fN
0.00001
0.0001
0.001
0.01
X
Figure 7.5c.
range.
An fx
0.1
N
plot with three roots over the entire composition
The mechanically unstable root has the highest fugacity over
much of this range.
371
liquid
root,
will
it
a shape similar to the C 16 -C0 2
give
plots at these pressures.
Figure
7.6a
and L+L
shows
only
the
b
S+F
region.
which
solubility
in
have
been
at
drawn
K,
a
The metastable
The
L +G
stable
the
same
reduced
In Figure 6.4a, the
the S+G region is considerably higher than in
This is due to the lower
Figure 7.6a.
Figure
278.5
on an expanded scale, as depicted in
up
temperature with respect to the solvent.
in
at
Figures 7.6a and b may be compared with Figures
Figure 7.6b.
6.4a and
section
P-x
regions are now of a familiar shape, and are
2
completely enclosed by
region
the
below ethylene's critical point.
temperature
G+L 2
gives
7.6a,
leads
which
absolute
temperature
to a lower naphthalene vapor
pressure, and hence a lower solubility.
At 283.2 K, Figure 7.7, the LCEP
reached.
The
S+F
equilibrium
horizontal inflection.
has
temperature
curve
Coincident with
exhibits
this
been
a point of
point
is
the
stable critical point of the metastable L1 -G region which is,
too
however,
small
to
appear in the figure.
The shape of
this region has previously been given in Figure 6.6b.
Increasing temperature further, the supercritical
region
is entered.
Figure 7.8 is drawn at 288.7 K, the same
reduced temperature with respect to
6.7.
The
stable
fluid
two-phase
the
solvent
as
Figure
system for the entire pressure
372
P (atm)
xXN
N
Figure 7.6a.
A P-x section below ethylene's
critical
point.
This diagram is at the same reduced temperature, with respect
to the solvent, as Figure 6.4.
373
P (atm)
0.00001
0.0001
0.001
0.01
0.1
1
XN
Figure 7.6b.
The P-x section at 278.5 K in the vicinity of the
three-phase region.
This diagram is to be compared with Figure
6
.4 .
374
P4
rJ
CU
0
0
1.. cr
-C
o
O0C0
Cr
a
0
CU
N
4U-J
U.W
0
a
0-J
CU
C
Q)
0o
4-a
0.
-e
4-
a
U.
0
0
0
IC
.
r-.
C
0
0
0
Q)
0
OUU.
LaI
0
0
0
9
0
375
P (atm)
0
x
Figure 7.8.
N
A P-x section in the supercritical fluid region
of naphthalene - ethylene.
The metastable G+L 2 and L1+L 2
regions have become a single, metastable L+F region.
376
This temperature is above the upper metastable
range is S+F.
CEP in Figure 7.2b, so that separate L2 +G and L 1 +L 2
no
The
exist.
longer
L1+G
binodals, though too small to
the
L+F
region,
and
entirely
They fall
appear in the figure, are still present.
within
regions
the usual stable and
exhibit
These
unstable critical points (see Figure 6.11b).
critical
points persist up to the cusp in Figure 7.2b.
which
Figure 7.9 is drawn at 313.85 K,
the
upper
critical
is
essentially
The solid-fluid
end point temperature.
binodal now exhibits a horizontal inflection point, which
coincident
with
L+F region.
the stable critical point of the metastable
This diagram is to be compared with Figure 6.8.
Figure 7.10 gives the P-x section at 318.85 K,
the
above
UCEP
temperature.
The
points,
joined.
where
Below
solid-fluid
the
associated
SLF
now possesses two limit
binodal
metastable
somewhat
in Figure 6.9a, a stable
As
region of L+F is now present, along with the
tieline.
is
and
unstable
segments
are
the SLF tieline, S+F is the stable two-phase
tieline,
on
depending
the
system.
At pressures above the
overall
system composition, the stable two phase system will
be L+F or S+L.
Figure 7.11 gives the P-x section
temperature
as
Figure
6.9a.
temperature lies between the UCEP
As
at
with
and
340 K,
Figure
the
same
7.10, this
naphthalene's
triple
377
P (atm)
XN
Figure 7 .9.
The P-x section at the upper critical end point
temperature (approximately).
of horizontal inflection.
The S+F binodal exhibits a point
378
P (atm)
0o.
XN
Figure 7.10.
triple point.
A P-x section between the UCEP and naphthalene's
379
P (atm)
x
N
Figure 7.11.
triple point.
A P-x section between the UCEP and naphthalene's
380
and
point,
the
of
features
phase
the
point
limit
joined at very low pressures, so a second
does
The L+F critical point occurs at a
in the plot.
appear
are
The S+L and S+F binodal branches are
qualitatively the same.
not
diagram
with
agreement
higher pressure than at 318.15 K, in
Figure
7.2a.
At naphthalene's triple point, the P-x section of Figure
6.10.
The
pressure.
7.13,
compared
This diagram is to be
7.12 is found.
S+L
and
L+F
S+L
region
Figure
binodals meet at the triple point
Upon increasing the temperature to
the
with
360 K,
Figure
As seen in
has moved off the graph.
Figure 6.11, naphthalene will not precipitate out until about
390 atm.
The conclusions to
naphthalene - ethylene
chapter.
be
drawn
system
from
the
study
measurements
have
proven
validity of the predicted P-T-x space.
that the P-T-x predictions for
are qualitatively correct.
the
include those of the previous
The most significant result, however, is
experimental
of
the
that
the
qualitative
This is an indication
naphthalene - carbon
dioxide
381
i
04
o
1
o
0~~~~~~~o
o
4
0
o,-4
I
c-
,,
U)
0.
4-
C
Q)
x
X
-C
4-;
-
C
X
C
.
0
X
0I
o
*g
0o
0
-~O
382
.. 4K
P (atm)
0.00001
0.0001
0.001
0.01
0.1
XN
Figure 7.13.
A P-x section above naphthalene's triple point.
383
The Benzene - Water System
Chapter 8
The preceding chapters have demonstrated the utility
the
liquid-liquid
nonidealities
self-affinity,
polar
this
in
polar
intermolecular
manifested
as
of
-
forces
state
description
reasonably accurate qualitative
of
constant
50
seen,
be
will
still
such
model
accurately
(1978) - but, as
of
bonding of
hydrogen
the
strong
a
have
witness
polarity
and
size
will
the
case,
present
the
both
to
task
equation
cubic
Peng
nonidealities
compound
equation of Keenan, et al.
system.
in
instance
It is a formidable
simple
different
In
result
a
are
water.
the
considerably
immiscibilities.
The
differences.
to
leading
size,
molecular
were
components
the
systems
In these
modelling nonpolar systems.
in
equation
P-R
of
provides
the
a
binary
Robinson (1976b) have in fact used their
and
equation to predict three-phase equilibrium in multicomponent
hydrocarbon - water systems with some degree of success.
was
indicated
in
It
Chapter 4, as well, that the P-R equation
gives a reasonable picture of
the
phase
behavior
pure
of
water.
The physical constants of water and benzene are given in
Table 8.1.
At 455.2 K, the vapor pressures
water are equal.
volatile
benzene
and
Below this temperature, benzene is the more
component,
more volatile.
of
while
at
The crossing of
higher temperatures water is
the
vapor
pressure
curves
384
Table 8.1
Physical Constants of Benzene and Water
C6H 6
T (K)
c
P (atm)
C
( acentric factor)
H20.
562.1
647.3
48.3
217.6
0.212
0.344
385
requires
the
occurrence
of
an
azeotrope
although not necessarily a stable one.
considering
in
the system,
This is evident
a binary VLE P-x section, which is terminated at
compositional extremes by the pure component vapor
(see e.g., Figure 2.2).
the
upon
pressures
Since the pressures at either end of
envelope are the same, or nearly the same, any deviation
from ideality must lead to an
envelope
with
a
maximum
or
minimum pressure, i.e., an azeotrope.
A fairly extensive amount of experimental data has
gathered
for
the
investigators.
depicted
in
benzene-water
The
system
Figure
2.10.
system,
belongs
to
by
the
a
number
general
low
to
of
class
No single study has covered the
entire fluid region, but between the work of Rebert
(1959) at
been
and
Kay
moderate pressures, the work of Connolly
(1966) at moderate to high pressures, and the mapping of
upper
critical
locus
the
by Alwani and Schneider (1967) a good
overall picture of the phase behavior is obtained.
A value of
612
for this system was selected by matching
a single point along the P-T projection of the upper critical
locus.
Since the critical locus is univariant, choice
temperature
fixes
one
composition along the locus.
or
more
sets
of
of
pressure
the experimental locus fixes the composition and 612.
shows
the
and
In using the equation of state,
choosing the pressure and temperature of a critical point
8.1
a
on
Figure
experimental upper critical locus along with
386
P(atn
560
570
580
590
600
610
620
630
640
T(K)
Figure 8.1.
The P-T projection of the upper critical locus
for benzene - water.
The curves are actually composed of
two separate segments (see Figures 8.2 and 8.3).
387
two predicted loci.
optimal
in
The value
of 61.2=0.052
is
reasonably
reproducing the course of the critical line, and
was used for all further
predictions
in
the
benzene-water
system.
The critical curves in Figure 8.1 have been depicted
smooth and continuous.
near
the
this
benzene
It is necessary to examine the region
critical
behavior.
as
Figure
point more closely to elucidate
8.2
illustrates
the
phase
relationships as deduced from experimental measurements.
The
lower critical locus begins at the benzene critical point and
tends
toward
reaches
a
lower
temperatures
temperature
temperatures,
and
minimum,
shortly
lower critical end point.
suggested
a
and higher pressures.
turns
thereafter
toward
It
higher
is terminated at the
Alwani and Schneider
(1967)
have
hypothetical continuation of the lower critical
line to indicate its course had a third phase not intervened.
This line might be thought of as a metastable critical
It
line.
is presumed to intersect the upper critical line near its
pressure minimum.
Rebert
and
Kay's
three-phase
LLG
line
measurements
also
has
a
indicate
locus
will
be
discussed
sections for the benzene - H20
terminates
at
the
metastable continuation
beyond the lower critical end point.
metastable
that
system
The
meaning
of
this
shortly, when the P-x
are
considered.
It
a critical solution end point, simultaneously
388
1;
1
1
1
P (at
530
54G
550
570
560
580
590
600
T,K
The experimental P-T projection of the benzene- water system in the vicinity
Figure 8.2.
point.
pure benzenecritical
of the
389
intersecting the L-L and L-G critical lines.
between
the
L-L
and
L-G
The
transition
critical lines has been drawn as
smooth, although experimental measurements indicate that this
No measurements of
is not actually true.
line
available
are
the
critical
L-L
this region to indicate its correct
in
shape, however.
Figure 8.3 gives the calculated P-T
vicinity
of
starting
from
transform
critical
benzene's
pure
the
the
into
water
point.
critical
to
the
the
The critical curve
point
not
does
liquid-liquid critical line (see Figure
8.1), but rather reaches a minimum in
returns
in
projection
benzene
and
temperature
critical point.
then
The liquid-liquid
critical line on the other hand reaches a minimum in pressure
shortly before being terminated by an unstable critical line.
Figure
8.3
terminates
also
at
gives
the
three-phase
line
LLG
which
a critical end point on the LL critical line.
The short segment of the LL critical line between the CEP and
the point of the cusp where the
unstable
critical
line
is
joined is metastable.
The experimental and predicted P-T projections differ in
that the former depicts a metastable portion of the LLG line.
Rebert and Kay assert that this metastable
to
the
intersection
were taken on a fairly
region
of
line
of two binodal regions.
coarse
grid,
and
a
corresponds
However, data
third
binodal
limited extent may well have gone undetected.
As
390
p
520
530
540
550
560
570
580
590
T,K
Figure 8.3. The predicted P-T projection of the benzene- water system in the vicinity
pure benzenecritical point.
of the
391
consideration of the P-x sections will show, the equation
state
predictions
indicate that a third binodal region does
exist, and that very careful measurements will
to
determine
of
be
necessary
the stability or metastability of the LLG line
up to-the critical solution end point.
At present it is
not
possible to say which of the two P-T projections is correct.
The P-x sections for the benzene - water system will now
be considered.
been
A fairly complete set of these
published
by
Rebert
reproduced in Figure
8.4.
diagrams
and Kay (1959).
Some
high
has
These have been
pressure
data
from
Connolly (1966) has been included in Figures 8.4d and e.
Figure
temperature
8.4a
shows
the
P-x
section
lies
either
528.15 K,
below the critical end point in Figure 8.2.
diagram is typical for systems in which the
line
at
at
a
The
LLG
higher pressure than the vapor pressure of
component. - Systems
heteroazeotropes
three-phase
a
(see
of
Figure
this
type
2.14).
are
known
as
In this instance the
azeotropic composition at 528.15 K is given by x.
When
two
coexisting liquid phases of overall composition x are brought
to
the boiling point, the vapor phase which forms also has a
composition
x.
separation
may
Thus,
be
as
with
effected
Heteroazeotropes also exhibit the
a
by
normal
azeotrope,
simple
property
of
no
distillation.
vapor
phase
compositions intermediate to two liquid phase compositions at
a
given
pressure and temperature.
Despite this composition
392
JOU
90
80
70
60
P (atm) 50
40
30
20
10
0
10
20
30
40
50
60
70
80
90
wt %H20
Figure 8. 4 a.
An experimental P-x section below the
temperature minimum in the lower critical line (see
Figure 8 .2).
100
393
120
110
100
90
80
P (atm)
70
60
50
40
30
0
10
20
30
40
50
60
70
80
90
wt % H20
Figure 8.4b.
Figure 8.2.
A P-x section slightly above the LCEP in
100
394
P (atr
0
10
30
20
40
50
60
70
80
90
100
wt % H20
Figure 8.4 c.
A P-x section between the LCEP and benzene's
critical point.
395
600
500
400
300
P (atm)
200
100
0
0 '
20
40
60
80
100
wt % H20
Figure 8.4d.
A P-x section between benzene's critical
point
and the temperature minimum in the upper critical locus.
396
600
T
573.15K
500
0
Connolly (1966)
Other data from
Rebert and Kay (195,
400
C6H6 -
H20
6.
300
P (atm)
200 -
CP
I
L1
L2
G +
100 -
0
E
0
_ I
20
I.
'
40
60
2
80
100
wt % H2 0
Figure 8.4e.
A P-x section between the temperature minimum
in the upper critical locus and the critical solution end point.
397
600
500
400
300
P (atm)
200
100
0
0
20
40
60
80
100
wt %H20
Figure 8.4f.
end point.
A P-x section above the critical
solution
398
inversion, the liquid is more dense than the coexisting vapor
along both branches of the VLE envelope.
Figure 8.4b is drawn at 541.45 K, a temperature slightly
from the water-rich G+L 2
separated
has
This separation occurs at
the
the
between
to
attached
the
the
three-phase
be
must
This
so,
a
benzene-rich
of
in addition to the
line
pressure
of
pure
since at any temperature just
above the minimum in the critical locus
critical points.
of
minimum and the
locus
critical
lower
separate "island" stemming from the vapor
benzene.
in
In the few tenths
critical end point, there is a small region
VLE
regions.
minimum
temperature
end point as might at first appear.
degree
and L1 +L 2
locus in Figure 8.2, and not at the critical
critical
lower
region
The benzene-rich G+L 1
above the critical end point.
must
there
be
two
A diagram of this type will be presented in
Figure 8.9.
The L+L2
intersect
and G+L 2 regions in Figure 8.4b are
along
a
horizontal
on
corresponds to a point
Figure
8.2.
Thus,
this
the
tieline
metastable
three-phase
line
shown
(dashed)
LLG
to
which
in
tieline
represents the
intersection of two, rather than three binodal regions, as is
normally the case.
At the metastable three-phase pressure, a
discontinuity in the slope of
indicated.
Rebert
and
Kay
the
binodal
locus
has
been
have drawn the section in this
manner, although their data are insufficient to prove such
a
399
discontinuity.
state
does
As seen in previous chapters, the equation of
not
yield
slope
discontinuities when only two
binodal regions merge.
Increasing
Figure
8.4c
temperature
is found.
to
553.15 K,
above benzene's critical
no
longer
point,
exists.
leg
so
The
higher pressure data along the
water-rich
section
This temperature is
that
figure
L 1 +L 2
the
benzene-rich
also includes some
envelope.
Only
of the curve has been measured.
The
maximum in water
benzene
leg
would
composition.
At
the
As expected
from Figure 2.10, it exhibits a pronounced minimum
composition.
of
The benzene-rich island is now quite
small. Figure 8.4d is drawn at 568.15 K.
island
the
in
water
have a corresponding
573.15 K,
as
shown
in
Figure 8.4e, the pinching effect created by these extrema has
been
completed,
yielding
two
separated by a miscibility gap.
increased,
the
disappears
at
temperatures
lower
the
above
L 1 +L 2
critical
distinct
As
L +L 2
temperature
region
shrinks
solution
end
regions
is
further
and
finally
point.
At
this end point, the VLE envelopes are of
the usual shape, as illustrated in Figure 8.4f.
The P-x sections generated from the
now
be considered.
temperatures
Reference
to
of
equation
will
This series of diagrams includes all the
Figure
Figure
P-R
8.3
8.4
as
should
well
as
be
made
relationship between the various sections.
several
to
others.
clarify the
To reiterate, the
400
major qualitative difference between the predicted Figure 8.3
and the experimental Figure 8.2 is that
the
former
has
no
critical end point along the critical locus starting from the
pure
benzene
critical
point.
This
leads
fact
to
prediction of normal azeotropes because, as Rowlinson
has
normal azeotropes as
approached.
the
critical
8.6a
528.15 K.
Three
As
in
gives
the
stable
previous
calculated
binodal
P-x
section
regions
chapters,
and G+L 2
with one another, while the L+G
fact
the
are
stable
binodal
8.7,
binodals are continuous
curve.
A metastable normal azeotropic
This
point
At 541.45 K, as shown in
the predicted behavior is essentially the same.
A benzene-rich island has not broken off as it did in
Figure
will not occur until the temperature minimum in
the vapor-liquid critical locus has been
curves
at
correctly
locus forms a closed
is visible in this diagram as well.
8.4b.
is
is more evident in Figure 8.6b, an enlargement of
the azeotropic region.
Figure
point
are interconnected by metastable and unstable curves.
In this instance the L+L2
This
end
solution
Figure 8.5 illustrates this point.
Figure
curves
(1969)
out, heteroazeotropes must in general revert to
pointed
predicted.
the
reached.
Spinodal
have also been given in this figure, and indicate the
location of an unstable critical
point.
along the unstable locus in Figure 8.3.
This
is
a
point
401
T
X
Figure 8.5.
The T-x projection for a heteroazeotrope system.
Only the compositions of the three coexisting phases at each
temperature have been plotted.
The azeotropic composition must
fall outside the composition range of the coexisting liquids as
the critical solution temperature, CP,
is
approached.
402
P (atm)
wt % H20
Figure
8
.6 a.
A predicted P-x section below the temperature
minimum in the G-L critical line.
.
(See Figure 8.3.)
403
72
71
70
69
-
68
B
n3
P4 67
66
65
64
63
20
30
40
50
70
60
wt % H2 0
Figure 8.6b. The P-x section at 528.15 K in the vicinity
the azeotrope.
of
404
,9n
1iU
110
100
90
80
70
60
50
40
30
0
10
20
30
40
50
60
70
80
90
100
wt % H2O
Figure 8.7.
A P-x section below the temperature minimum in the
G-L critical line.
405
Figure 8.8a shows the P-x
section
for
553.15 K.
benzene-rich
VLE
envelope has continued to narrow.
temperature,
the
previously
point
moved
has
into
the
enlargement of Figure 8.8b.
metastable
stable
normal
The
At this
azeotropic
as seen in the
region,
Thus, a normal, minimum
boiling
azeotrope is now predicted rather than a heteroazeotrope.
Increasing
temperature
temperature
minimum
in
just been exceeded.
slightly,
to
554 K,
the vapor-liquid critical locus has
From Figure
8.3,
it
is
evident
there must now be a separate benzene-rich region.
illustrates
these
features.
just
above
the
temperature
the
region.
chance
presence
of
a
normal
lower
critical
heteroazeotrope-normal
situation
is
locus
azeotrope
of
two
detailed
is
transition
important
system's
was
which
unless
coincident
of state has thus brought out an
behavior
enough
however,
to
not
by
with the
point.
analogous to that of Figure 8.5.
phase
critical
azeotrope in this small
A normal azeotrope must exist,
the
Figure 8.9
minimum in Figure 8.2.
Experimental measurements have not been
verify
that
As mentioned previously, there
must be a similar break with the formation
points
the
This
The equation
feature
evident
of
the
from the
experimental results.
Figure 8.10a
azeotropic
region
is
has
the
P-x
section
at
568.15 K.
diminished in size considerably.
interesting change has also taken place in
the
The
An
continuation
406
.nZ
JU
120
110
100
"
.-
90
90
80
70
60
50
40
wt % H20
Figure 8.8a.
A P-x section below the temperature minimum in the
G-L critical line.
407
90
89.9
89.8
89.7
89.6
35
40
45
50
55
60
wt % H20
Figure 8.8b.
the azeotrope.
The P-x section at 553.15 K in the vicinity of
The azeotrope is now stable, outside the region
of liquid-liquid immiscibility.
408
130
120
110
100
E
90
v
_
80
70
60
50
40
0
10
20
30
40
50
60
70
80
90
wt % H20
Figure 8.9.
A P-x section slightly above the temperature
minimum in the G-L critical line.
100
409
E
a.
a-
0
10
20
30
40
50
60
70
80
90
100
wt % H 20
Figure 8.10a. A P-x section between benzene's critical point
the temperature minimum in the L-L critical line.
and
4i0
C6H6
H2 0
109
108
fr
.4
107
106
40
50
60
70
80
90
wt % H2 0
Figure 8.10b.
The P-x section at 568.15 K in the vicinity of
the three-phase line.
with one another.
The G+L 1 and G+L2 binodals are continuous
411
600
500
400
300
200
100
0
10
20
30
40
50
60
70
80
90
100
wt % H 20
Figure 8.10c.
p ress u res.
The P-x section at 568.15 K extended to high
412
of
stable
the
binodal
It is now the L+L
curves.
and G+L 2
which of itself forms a closed loop, while the G+L 1
curves
these
shows
8.10b
Figure
interconnected.
are
curve
2
interconnections in an enlargement of the three-phase region.
For
comparison
with
the high pressure data in Figure 8.4d,
the
in
effect
The necking
Figure 8.10c has been included.
region, due to the proximity of the critical solution
L1+L 2
temperature, is evident.
Eventually, the
branch of the azeotrope continues to shrink.
critical
azeotrope
point
azeotrope point in a cusp.
above
slightly
the
near
very
appears
in
This behavior was illustrated
cusp
the
temperature.
tip of the G+L 1
A
comparison
with
Figure
8.4e.
point
critical
In Figure
region.
8.11b, the section is redrawn to include high
for
meet the
will
Figure 8.11a, at 573.15 K, is at a temperature
2.13.
Figure
8.10b
Figure
of
benzene-rich
the
increased,
As temperature is further
data
pressure
Although the predicted
compositions are not accurate, the miscibility gap
been
has
correctly predicted.
Figure 8.12 is a P-x section at a temperature just above
in
that of the critical end point
critical
at
8.2.
The
L+L
2
point has now become metastable, and will disappear
as the L1 +L2
drawn
Figure
region converges to
603.15 K,
is
critical line termination.
at
a
a
point.
temperature
Figure
8.13a,
above the L1+L 2
The VLE envelope is
now
of
the
413
E
4-;
(U
2o
CL
0
10
20
30
40
50
60
70
80
90
100
wt % H20
2
Figure
8
.11a.
the L-L critical
A P-x section between the temperature minimum in
line and the critical end point.
414
Bit
10
20
30
40
50
60
70
80
90
100
wt % H20
Figure 8.11b.
The P-x section at 573.15 K extended
to high pressures.
415
I
I
1
I
I
0
10
20
30
40
50
60
70
80
90
100
wt % H20
2
Fi gure 8 .12.
A P-x section slightly
above the critical
end point.
416
170
160
150
140
130
120
110
100
0
10
20
30
40
50
wt
Fiqure
8
.13a.
60
70
80
90
100
H20
A P-x section above the L-L critical line cusp.
417
600.
500.
C6H 6 - H20
T = 603.15
K
40(
B
P4
3010.
D.
20C
CP
100n
10
I
i.
mI.
20
30
40
50
I
I
60
70
I
..
80
90
100
wt % H2 0
Figure 8.13b.
The P-x section at 603.15 K extended
to high pressures.
418
usual
type.
8.13b
Figure
is
the
same section on a high
pressure axis, for comparison with Figure 8.4f.
In generating the preceding P-x
types
of
curves
fugacity
of Cartesian
sequence
Figure
f-x
Figure
8.14.
8.14a
different in several respects
in Chapter 3.
discussed
several
A
encountered.
were
at
plots
sections,
568.15
is
K
new
pressure
given
in
is a plot at 100 atm, which is
from
the
closed
plots
loop
The mechanically unstable or middle
volume root no longer gives the middle value of fugacity.
As
shown in the figure, it yields the minimum fugacity over
the
entire
root range.
multiple
Also, the two stable roots are
no longer conveniently identified as liquid and gaseous.
For
most of the multiple root range, the monotonic solution gives
the intermediate fugacity value, but it becomes
the
maximum
point is approached.
The
second stable solution, which forms the top of the loop,
has
fugacity
two
value
maxima
points.
and
as
the
one
infinity
minimum
for a total of three spinodal
Continuation of the plot to the
pure
benzene
axis
does not yield any further spinodal points.
The f-x plots at this temperature exhibit an interesting
transition in the region near
8.14b
focuses
on
the
infinity
point.
Figure
this region at a pressure of 108.158 atm.
The first infinity point is equivalent to the infinity
in Figure 8.14a, i.e., it is at the tip of the loop.
point
The tip
is now quite pointed, and the spinodal point is very close to
419
I
Iw
~1
tU
0
0.05
0.10
0.15
0.20
HH
C6H6
x
Figure
8
.14 a.
An f-x plot.
The mechanically unstable root
always gives the minimum fugacity.
0.25
420
11.8
11.7
11.6
11.5
11.4
U
44
11.3
11.2
11.1
11.0
10.9
.24
.245
.25
.255
.26
.265
C6H6
Figure .14b.
An f-x plot
in the region of
the infinity
points.
421
,,
.
%.0
U
tw-
.258
.259
.260
.261
X
Figure 8.14c.
.262
.263
H
6 6
An f-x plot in the region of the infinity points.
422
to
U
'44
.2585
.2595
.2590
.2600
.2605
X
C6H 6
Figure 8.14d.
An f-x plot.
in Figure 8.14c. have merged.
The first and third infinity points
423
12
10
8
-I
6
4
2
A)
0
.05
.10
.15
.20
.25
xc H
6H6
Figure 8.14e.
water axis.
The f-x plot of Figure 8.14d extended to the pure
424
12
10
8
U
6
2
0
.05
.10
.15
xc
Figure 8.14f.
An f-x plot.
.20
H
6H6
The region of three roots is
shrinking toward the origin with increasing pressure.
.25
425
the
monotonic solution now
formerly
The
point.
infinity
spinodals
exhibits a kink, which adds two infinities and two
for
of three infinities and five spinodals at this
total
a
The kink is associated with
pressure and temperature.
the
along
solutions
equilibrium
phase
azeotropic
benzene-rich
vapor
(At pressures and temperatures near benzene's
branch.
over to the pure benzene axis to
moves
it
curve,
pressure
is
pressure
the
atm,
108.175
to
increased
3.3).
Figure
give a discontinuity of the type shown in
As
kink becomes
sharper, and the first and third infinity points approach one
This is
another.
Figure
in
shown
Upon
8.14c.
reaching
108.22 atm, the two infinities have merged, as illustrated by
Figure
8.14e shows the course of the entire
Figure
8.14d.
The curve is quite
loop at this pressure.
to
similar
that
in Figure 8.14a, with a total of three spinodal points
given
Now, however, the top half of the loop has
and one infinity.
This is more easily seen in
only one spinodal point.
8.14f,
Figure
the loop shrinks toward the pure water axis.
as
The
loop will disappear as pressure is further increased.
a
To summarize,
P-x
predicted
provided by
correct.
the
equation
the
experimental
and
that the qualitative picture
shows
sections
the
of
comparison
state
of
in
is
measure
large
The calculated P-x sections clarify the behavior of
normal
experimental
three-phase
azeotrope
The
work.
line
which
and
the
was
not
correctness
G-L
investigated
of
the
in the
predicted
critical line is uncertain,
426
pointing out the-need for more careful experimental
this region.
work
in
427
Chapter
Conclusions and Recommendations
Knowledge of the overall phase behvaior of systems is of
importance in process design, since it indicates
conditions
carried
a
phase
out.
equilibrium
For
example,
process
under
what
is most favorably
supercritical
extraction
of
naphthalene with ethylene should probably be carried out near
the
UCEP
rather
than
the
LCEP.
Although
the pressures
required are 3.5 times as high (174 atm versus 51
solubility
of
naphthalene
0.002 mole fraction ).
300 C
the
is 85 times as high (0.17 versus
As a further example, suppose it
desired to dissolve benzene in water.
critical
atm),
was
The shape of the upper
line for this system shows that conditions of about
and
170
atm
are
sufficient
to
yield
complete
miscibility of these two substances.
In developing a methodology
diagrams,
a
number
of
to
predict
significant
overall
P-T-x
results were obtained.
These are as follows:
-Fugacity-composition plots provide an excellent means of
representing
metastable
the
phase
possible
types
of
stable
and
equilibrium at a given temperature
and pressure.
-If the material stability criterion reaches a
discontinuity
that
point
in
may
the
tracking
represent
a
point
of
of a stable system,
limit
of
mechanical
428
is
This
stability.
true
if
even
the
either
stability criterion is satisfied on
material
side
of
the discontinuity.
-Material stability should be considered undefined
of
region
statements of the material
a
since alternate
instability,
mechanical
in
criterion
stability
may
give conflicting information in such a region.
-When the cubic equation has multiple roots, P-x and
sections
may
correctly
stable states within a
the
indicate
region
T-x
existence of
by
bounded
spinodal
curves.
-In critical point determinations, the criterion Ml=O may
be replaced by the alternate criterion
0.
-An unstable critical point
may
be
identified
as
the,
terminus of an unstable binodal locus.
(2a 2
Alternatively,
ax
is
greater
than zero for
T,P
a stable critical point and less
zero
than
for
an
unstable critical point.
-A necessary and sufficient criterion for
the
existence
point
is
the
occurrence
horizontal inflection point
in
the
stable
of
a
critical
end
of a
binodal
locus in a P-x or T-x section.
-Solution
of
sufficient
the
for
two-phase
the
eqilibrium
determination
problem
of
is
three-phase
429
equilibrium, since
three-phase
equilibrium
results
from the intersection of binodal loci.
From the four systems investigated, the following conclusions
have been drawn:
-The Peng-Robinson
interaction
equation,
with
a
single,
constant
parameter, gives a qualitatively correct
representation of
phase
behavior
over
the
fluid region for nonpolar binary systems.
entire
The system
may be nonideal with respect to molecular size of the
components.
-The Peng-Robinson
interaction
equation,
parameter,
representation of
fluid
region
component.
for
with
gives
phase
a
a
single,
reasonably correct
behavior
binary
constant
systems
over
the
with
entire
one
The qualitative picture may be
polar
incorrect
in certain minor details.
-Semiquantitative predictions
region
of
interest
temperature
dependent
particular,
the
may
by
be
obtained
choosing
interaction
supercritical
an
in
any
appropriate
parameter.
fluid
In
region
is
amenable to such treatment.
-Analysis
of
fugacity-composition
development
plots
allows
the
of phase equilibrium algorithms suitable
for any region of the phase diagram.
-Metastable
and
equilibrium
unstable
equations
solutions
are
always
to
the
associated
phase
with
430
in
three-phase equilibrium
solutions
a
system.
binary
The
may exist over a substantial region of the
for
can
lead
conclusions
for
computer
must
be
strongly
P-T-x space, and if not accounted
to
erroneous predictions.
The significance of the last
of
calculations
Algorithms
emphasized.
of
behavior
phase
the
two
equilibrium
which
fugacity
fail
to
for
account
are
functionality
the
liable
to
experience convergence problems when this functionality takes
on a complex shape.
Complex shapes are found in
three-phase
regions as well as near azeotropic coordinates and pure vapor
Similarly, algorithms which fail to account
pressure curves.
for
all
phase equilibrium solutions to an equation of state
Erroneous prediction of
may converge to an incorrect result.
both binodal curves and critical lines
development
occur.
may
Careful
application of phase equilibrium algorithms
and
is necessary to avoid these difficulties.
Taken as a whole,
cubic
simple,
these
equation
of
conclusions
state
is
indicate
well
suited
that
to
orienting or preliminary study of binary phase behavior.
a
an
The
minimal computing time required by these equations makes them
an obvious choice for such an application.
study
The
preliminary
maps out regions where more detailed investigation, in
both the theoretical and experimental
sense,
is
warranted.
From the theoretical standpoint, once these regions have been
defined,
it
may
in
some cases be desirable to switch to a
431
more accurate equation of state.
equation
in
In other
cases
the
cubic
will provide sufficient accuracy with only a change
the
interaction
parameter.
From
the
experimental
standpoint, the preliminary study indicates the regions where
further
data
should
be taken, and serves as a check on the
consistency of results.
Continued work on the topic of this thesis would be most
worthwhile.
would
Generation of P-T-x diagrams for
provide
conclusions.
polar
verification
be
of
interest.
be
A
categorization
very
useful.
the
The
value
of
above
of
systems.
the
of the
employed,
612
categorization
possible to rapidly estimate the overall
many
the
P-T projections according to T , P , and w
species involved, as well as
would
of
systems
Treatment of a system where both components are
should
predicted
additional
other
would make it
phase
behavior
of
This is along the lines of van Konynenburg's
(1968) work with the van der Waals equation.
Avenues for further development are several.
lines
suggested
by
numerous
temperature and pressure
workers,
functionality
Along
determination
of
the
the
of a
interaction
parameter would lead to a more quantitative prediction of the
P-T-x diagrams.
P-T-x
space
It would also be instructive to generate the
with
a
accurate equation to
computing
time.
more
allow
complicated, and presumably more
a
comparison
of
the
required
For a non-cubic, pressure explicit equation
432
of state, use of T and V as the independent variables may
be
an efficient approach.
Automation of the methodology, including
provision
for
computer graphics, would increase its utility and convenience
greatly.
techniques
systems.
Finally,
developed
an
attempt should be made to extend the
here
to
ternary
and
multicomponent
433
Appendix A
The Equation of State and Derived Expressions
The Peng-Robinson equation of state is
P=
(A.1)
a(T)
RT
v - b
v(v+b) + b(v-b)
where a(T) is a temperature dependent attraction energy parameter.
This equation may be rewritten in cubic form as
Z3 - (1-B)Z 2 + (A-3B2-2B)Z - (AB-B-B
)
=
(A.2)
where
A
B =
aP
(A.3)
2
2 2T
RT
bP
(A.4)
RT
Z = P
RT
(A.5)
By requiring equation (A.1) to satisfy the pure component
critical point criteria, equations (2.32) and (2.33), it is
found that
R2T2
a(T ) = 0.45724
(A.6)
c
pc
434
RT
b = 0.07780
(A.7)
P
Z
= 0.307
C
(A.8)
For temperatures other than T c , let
a(T) = a(T c)
a (Tr ,
)
(A.9)
with
a
= 1 +
(A.10)
(1-Tr)
r
K = 0.37464 + 1.54226
- 0.26992w2
(A. 1)
The fugacity of a pure component is derived from the
relation
p
=v
RT
P
1
dP
(A.12)
P
Using equation (A.1), the integration yields
f = Pexp
Z - 1 - ln(Z-B)
A
-
2I8B
in (Z+2.414By\ (A.13)
\z-0.414B/J
For a mixture,
a =
Z xx.a.
ij
(A.14)
435
b =
x.b.i
ij ( ij
(A.15)
i
aij = (1-6i.)(aiaj)
2
(A.16)
is an empirically determined interaction parameter.
where 6..ij
The form of equations (A.14) and (A.15) for binary systems was
The fugacity of a
given in equations (2.15) and (2.16).
component in a mixture is found from the relation
f
RTln
V
RT
=
1
=
P
RT i
V400
(T
V
d
(aN-)T,V,{Ni
- RTlnZ
(A.17)
-
Substituting equation (A.1), the result for component 1 in a
binary system is
f
x Pexp
1
1
[b
(Z-l)
-
ln(Z-B)
B
2
*Z +
(V-
2(
1 1
+ a12 2 )
a
b
bJ
)B
- (-1)l
(A.18)
Use of the above fugacity expressions allows the calculation
of phase equilibrium.
The mechanical stability of a phase is found from equation
(A.1).
For a stable system,
=-
x
T
RT
(v-b)
2a(v+b)2
-
v
+2bv-b 2
<
(A.9
436
Determination of material stability spinodal curves was
carried out using the L1 determinant of Beegle, et al. (1974b).
This is given in intensive form for a binary system as
Aw
Avx
~=|
Ll
2
(A.20)
AvvAxx - Avx
A
A
vx Xx
Compositional derivatives (subscript x) are taken with respect
to component 2.
This quantity is zero at a point on the stability limit.
The M
determinant of Reid and Beegle (1977) was used as the
second criterion of criticality.
This is expressed in intensive
form for a binary system as
Avv
M1 =
Avx
= A A
1 = |(aL\
v
T,x
A
VV XX
x vvX
x
+A
vv
2A
T,v
vv vx vxx
xx vxAvv
vx
At a critical point, this quantity is zero.
vvx
(A21)
Evaluation of L1
and M1 requires an expression for the Helmholtz free energy
A = A(T, V, x).
Using their equation, Peng and Robinson (1977)
have formulated this expression as
437
XiRT
A = RT
x il n
J
a
v-b
2
T
iv+(/Z+1)B)
/ln
i
b
+
v-(/2-1)3B
T
xiCpidT- T
To
E x.
Pi dT - RT
dT - RT
(A.22)
T0
where the asterisk denotes an ideal gas state.
Using equation (A.22), the derivatives in equations (A.20)
and (A.21) are as follows:
ax =
axx =
2 (-ax
1
2
(a
'+a12(1-2x2
-
2 a1
2
b x = b2 - b 1
r = a/b
axb - abx
r
=
b2
axx - 2rxx
b
r
b
xx
-3 rxxbx
r
b
W = v2 + 2bv - b 2
Wx = 2b (v-b)
W
xx
= -2b 2
x
+ a2 )
+a
2 2)
438
X = v++ (1+)
b
X = (1+v/)bX
Y = v + (1-2)b
Y
= (-V)bx
2a (v+b)
RT
AW
w2
(v-b) 2
-RTbx
ax
aW x
(v-b)2
W
W2
X
A
+
=xxRT
xx
x
X
(v-b)
2
r xxy
n- + 2rx \Xx
A
2
2
r
2a
-2RT
3
vvv
X
x
- 1
2
W
(v-b)
W
2RTb
2 (ax
X (v+b) + abxX )
AVVX
4aW (v+b)
+
x
W3
(v-b)3
-2RTb
2
(v-b)3
(v-b)
(y
a
W-
R+xxx
2axWx
xx
W2
aW
xx
2aW
x
2
2
yx
)
439
xxx
v-b
xlx2
!
rxxx In
(
X
y
3
3
x
+ 2r
-·
·· · · · ix
x
x
X
y
·
2
2
440
Appendix B
This appendix presents plots of n-hexadecane fugacity versus
n-hexadecane mole fraction for a matrix of seven temperatures
and sixteen pressures.
Plots of naphthalene fugacity versus
naphthalene mole fraction in the naphthalene - carbon
dioxide and naphthalene - ethylene systems are very similar
to those shown here.
44A
2
0
-2
log fH
-4
-6
-8
2
0
-2
log fH
-4
-6
-8
-5
-4
-3
-2
log
log xH
0
-1XH
-5
-4
-3log
-2
XH~
log xH
-1
~~~~-1
0
,44
7
-
/
-
290 K
40 atm
0
-2
log
fH
log
fH
-4
-6
-8
2_
290 K
60 atm
I0
-2
-4
-6
-8
I
-5
-4
-3
-2
-1
l o g~
log XH
0
-5
~ ~~~~~
.
-4
,
-3 o~~~~-2
0 f
log x H
i,
-1
0
443
290 K
290 K
65 atm
64.3 atm
-2
log fH
-4
-6
-8
l
l
I
:
2
290 K
70 atm
290 K
67.2 atm
0
-2
log fH
-4
-6
/~~
-8
-5
-4
'-3
log xH
-2
-1
0
-5
-4
-3
log xH
-2
-1
0
444
2
290 K
75 atm
290 K
73 atm
0
-2
log fH
-4
-6
-8
.
,
i
J,
,
2
290 K
300 atm
290 K
85 atm
0
-2
log
-6
I
-8
-5
-4
-3
log xH
-2
-1
0
-5
I
,
-4
-3
log x H
,j
-2
-1
0
fH
A
,3
2
0
-2
log fH
-4
-6
-8
2
0
-2
log
fH
-4
l0
g~~~
lo
-6
-8
-5
-4
-3
log xH
-2
-1
0
-5
-4
-3
-2
~~~~~~ ~X~~
log x H
-1
0
446
2
295 K
20 atm
295 K
40 atm
o
-2
log
fH
-4
,t~~~~~~~
-8
.
.
w
2
295 K
0
60 atm
-2
log fH
-4
-6
-8
-5
-4
-3
log xH
-2
-1
0
-5
-4
-3
log xH
-2
-1
0
447
2
295 K
295 K
65 atm
64.3 atm
-2
log
fH
-4
-6
,~·/
:--8
2
295 K
70 atm
295 K
67.2 atm
0
-2
log fH
-4
A/
-6
-8
,,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
.
-5
-4
-3
-2
log x H
-1
0
-5
-4
-3
-2
log x H
-1
0
448
2
295 K
73 atm
295 K
75 atm
0
-2
log fH
_A
-6
-8
l
1.
2
295 K
300 atm
295 K
85 atm
-2
log fH
-4
-6
-8
.
-5
-4
.
.
.
-3
-2
-1
log x H
c
0
c
-5
-4
-3
-2
log x H
-1
0
449
-2
log fH
-4
-6
-8
2
0
-2
log
-4
-6
. :1*
-8
-5
-4
-3
-2
log x H
-1
0
-5
log~~~~~~~~~~-
-4
-3log
X0
-2
log x H
-1
0
fH
450
2
-2
log
fH
log
fH
-4
-6
-8
2
300 K
0
60 atm
-2
-4
-6
-8
I
-5
-4
-3
log xH
-2
-1
0
-5
-4
' III
-3
log xH
-2
-1
0
H
451
2
300 K
0
64.3 atm
-2
log xH
-4
-6
-8
2
300 K
300 K
70 atm
67.2 atm
0
-2
log
-4
-6
-8
-5
-4
-3
log xH
-2
-1
-5
-4
-3
log xH
-2
-1
0
fH
452
2
300 K
75 atm
300 K
73 atm
0
-2
log
fH
log
fH
-4
-6
/~~
-8
2
300 K
85 atm
0
300 K
300 atm
-2
-4
-6
-8
/
.
-5
-4
-3
-2
log x H
-1
log
0
XH
-5
-4
-1
-2
-3log x~~~~~~1
log xH
0
53
2
305 K
0
305 K
0.1 atm
0. 5 atm
-2
log fH
-4
-6
-8
,
l
2
-2
log fH
-4
-6
-8
-5
-4
-3
-2
log xH
-1
0
-5
-4
-3
-2
log XH
-1
0
454
2
305 K
20 atm
305 K
40 atm
-2
log
fH
-4
-6
-8
l
@
l
l
2
305 K
0
60 atm
-2
log fH
-4
-6
-8
-5
-4
-3
log xH
-2
-1
0
-5
-4
-3
-2
log xH
-1
0
455
2
305
K
305 K
65 atm
64.3 atm
-2
log
fH
log
fH
-4
-6
-8
l
v
g-
305 K
70 atm
305 K
67.2 atm
0
-2
-4
-6
-8
l
-5
-4
-3
log x H
-2
-1
Q
-5
-4
-3
log xH
-2
-1
0
456
2
305 K
75 atm
305 K
73 atm
-2
log fH
.- 4
-6
-8
|
l
l
l
C
'2
305 K
300 atm
305 K
85 atm
-2
log
-4
-6
-8
-4
I
l
-
-5
-3
log xH
-2
-1
(I
[
.....
-5
-4
-3
log xH
I
l
-2
-1
0
fH
457
2
log
fH
log
fH
-4
-6
-8
2
0
-2
-4
-6
-8
-5
-4
-3
log x H
-2
-1
0
-5
-4
-3
-2
log x
H
-1
0
45
2
311 K
20 atm
311 K
40 atm
-2
log
fH
log
fH
-4
-6
-8
l
2
311 K
60 atm
311 K
50 atm
-2
-4
-6
-8
l
-5
-4
-3
l
l
-2
-1
0
...
-2
log x H
-1
0
-5
-4
-3
log xH
u-
0
459
2
311 K
311 K
65 atm
64.3 atm
0
-2
log
fH
-4
-6
-8
l
l
l
l
-2
311 K
70 atm
311 K
67.2 atm
-2
log fH
-4
-6
-8
F-
-5
,l
- -
l
-4
,
l
l
-2
-1
I
-3
log xH
l,
t
I
0
-5
-4
l"
lI
-3
-2
log xH
-1
0
4,50
2
311 K
75 atm
311 K
73 atm
0
-2
log
fH
log
fH
-4
-6
-8
2
311 K
300 atm
311 K
85 atm
0
-2
-4
-6
-8
.~~~~~~~~~~~~~~~~~~~~~~
l
-5
-4
-3
log xH
-2
-1
0
-5
-4
-3
log xH
-2
-1
0
461
2
0
-2
log fH
-4
-6
-8
2
0
-2
log fH
-4
-6
-8
-5
-4
-3
log xH
-2
-1
0
-5
-4
-3
log xH
-2
-1
0
462
2
400 K
40 atm
iL
400 K
20 atm
0
-2
log
fH
-4
-6
-8
l
2
0
-2
log fH
-4
-6
-8
.5
-4
-3
log xH
-2
-1
-5
-4
-3
-2
log xH
-1
0
463
2
400 K
65 atm
400 K
64.3 atm
-2
log
fH
log
fH
-4
-6
-8
·
l
__
I
Rh
400 K
70 atm
400 K
67.2 atm
0
-2
-4
-6
-8
,.
-5
-4
I
i
-2
-3
log
I
v
X
H
-1
0
]
-5
-4
·
i
I
I
-3
-2
log XH
I'
-1
'
0
2
400 K
73 atm
400 K
75 atm
0
-2
log fH
-4
-6
-8
-
-
l
.
2
400 K
85 atm
400 K
300 atm
-2
log
-4
-6
-8
tI
-5
-4
I
-3
log x H
i
I~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-2
-1
0
-5
i
-4
-3
l
-2
log x H
l
-1
0
fH
465
2
log
fH
log
fH
-4
-6
-8
2
-2
-4
-6
-8
-5
-4
-3
log XH
-2
-1
0
-5
-4
-3
log XH
-2
-1
0
466
650 K
20 atm
650 K
40 atm
2
-2
log fH
-4
-6
-8
r
l
l
|
2
650 K
50 atm
650 K
60 atm
0
-2
log fH
-4
-6
-8
I
lI
-5
-4
-3
log xH
-2
-1L
0
-5
-4
-3
-2
log x H
~
I
-1
0
467
2
650 K
650 K
64.3 atm
65 atm
0
-2
log
fH
-4
-6
-8
l
2
650 K
650 K
70 atm
67.2 atm
0
-2
log fH
-4
-6
-8
I
I
-5
-4
-3
log xH
l
-2
l
-1
0
-5
-4
B
l
I
-3
-2
log XH
-1
0
';
2
650 K
650 K
75 atm
73 atm
-2
log
fH
-4
-6
-8
C
l
,
X
I
2
650 K
300 atm
650 K
85 atm
0
·2
log fH
-4
-6
XH~~~
log
x.~~~~
log
-8
L
~
-5
-4
-3
log xH
-2
I-
:
-1
l
L
0
-5
l
-4
l
l
-3
-2
log xH
l
-1
0
469
Algorithm and Documentation of Programs
Appendix C
C.1
Phase equilibrium algorithm
P-T-x
The basic algorithm used in the generation of the
diagrams
is
in
quite
principle
and involves the
simple,
prediction of the equilibrium phase compositions at
temperature
and
The
pressure.
f-x
segments
monotonic
routine.
slope
into
divided
first
for convergence of the search
allow
to
given
for component 2
plot
is
(which may denote either substance)
a
and
This division consists of locating the extrema
infinities
in
fugacity functionality across the
the
Figure C.1
entire composition range.
example
an
gives
of
this procedure.
solubility
If prediction of a solid
is
being
carried
out, each segment is checked for a composition at which
f
These
points
fluid-fluid
with
another
correspond
F
C
fC
f2
=
2
to
equilibrium
is
segment,
and,
(C.1)
solid-fluid
desired,
if
equilibrium.
If
a segment is compared
possible,
a
pair
of
compositions obtained for which
f I2
f II2
(C2)
470
Figure C.la.
x 2 is incremented monotonically,
starting from near 0, to locate the two
extrema.
The curve is then divided into the
three segments shown.
r'~~~~~~~~~~~~~
Figure C.lb.
x 2 is increased along segments
1 and 2 until the first infinity is reached.
(The second infinity is actually encountered
first, but is skipped over).
i
rst
In.tni
y
decreased along segment 3 until the second
infinity is reached.
/ Icnd
x 2 is then
The search now continues
in a positive direction until x2= 1 is reached.
infinity
1
g
Altogether
seven segments are determined.
Figure C.lc.
A closed loop f-x curve.
The
analysis of this curve is similar to that
in Figure C.3b, with seven segments determined.
,- ,,
st
, " f~ , ,t ,
2~~~~
In this case, however, segment 4 is of null
length.
471
Here I and II refer to the two segments being
these
same compositions,
a check
I
fI
1
If
equation
this
is
not
compared.
is made to see if
II
f1 1
=
At
(C.3)
1
another
satisfied,
of
set
compositions in these segments for which equation (C.2) holds
is
If equation
examined.
(C.3) is satisfied,
fluid-fluid equilibrium have been determined.
the circumstances (see documentation
same
two
segments
of
conditions
Depending upon
program
All
segments.
the
peq),
will be searched for an additional phase
equilibrium point, or the program moves on to the
of
of
next
pair
combinations of segments are compared in
it
this manner, except for adjacent segments, for which
has
been found that no equilibrium solutions can exist.
Once all segments have been compared, the
complete
the
for
given
temperature
and
procedure
pressure.
construct a P-x section, the procedure is repeated for
Binodal
pressures.
curves
are
appropriate equilibrium points.
present
at
the
then
other
If a three-phase tieline
is
temperature chosen, it will be indicated by
at
a
common
Critical points are located by noting when binodal
tielines become of zero length.
sections
To
drawn connecting the
the intersection of two or more binodal curves
pressure.
is
in
this
fashion,
By analyzing a number of P-x
a P-T projection or P-T-x space
diagram for the system may be constructed.
472
C.2
General comments on programs
The twelve Fortran programs used for
calculations
are
phase
documented in this appendix.
equilibrium
The first of
these, zer, is a trial and error search
subroutine,
used
component and binary
in
calculations
system diagrams.
and
for
both
pure
The next four programs, pure,
and
zrp,
fp pertain only to pure component computations.
Zvalp,
Program
pure is the main program, Zvalp is a subroutine, and zrp
fp
are
functions.
sol,
and
functions.
this
was
peq
Program pr is the main
program,
are subroutines, while zr, LM, and f are
All programs are in double precision mode,
found
Computations
time-sharing
were
to
be
since
necessary in a number of instances.
carried
system
Processing Center.
and
The last seven programs pertain only to
binary system computations.
Zv,
is
out
operated
on
by
the
the
Honeywell
MIT
Multics
Information
473
C.3
Subroutine zer
This subroutine searches a given composition interval to
determine where some function of composition attains a
value.
function
This
be the fugacity, L1, M,
may
cubic determinant zr, and must be
interval.
The
and
trial
monotonic
error
search
for
is
given
or the
the
given
by
performed
bisection (interval halving).
Variable list:
fx - Function under consideration.
Possibilities were listed
above.
i - Function definition.
For
A dummy variable for
function
function LM, it specifies selection of L
For fugacity,
it specifies
component
zr.
or M1.
1 or 2.
nrt - Root specifier when multiple real roots
to
the
cubic
exist.
xlo - First endpoint of the composition range to be searched.
xhi - Second
endpoint
of
the
composition
range
be
to
searched.
fxm - The value which the function under consideration is
to
attain.
xapp - The composition at which the function has attained the
desired value.
This is the result which is
returned
to the calling program.
lox - Interval endpoint.
Varies as the search
the composition range.
narrows
down
474
COMPILATIOON
LISTIN(G UF zer (>user_di
r_ci r>hO>Hong>zer.fortran)
Compilea y: .AuLtics ew Fortran Compiler,
Combiteo on: 37/22/dJl 2L49. eGt wea
Options:
ansi 66 ap
Retease
a
Subroutine
xapp,*)
1
subroutine zer(fx,i,nrtsxLoxhi,fx,
2
3
impLicit aouoLe precision(a-ho-zA-Z)
ox
aouoLe precision
4
5
o
7
LoxsxLo
hi xxi
asigmtx(lox/2*hix/2,inrt)
co 4U j1,60
9
x tox/2+*hix/2
if ((t x(loxi,nrt)-fxm)/
12
13 30
14 4U
15
l1 5U
go to 40
hixx
continue
return 1
xapp x
11 20
Lox-x
17
return 2
lo
end
Figure
/10
if (abs(f x(xinrt)-fxam)-ats(asig))
1U10
d 3.
zer
50,5,G10
( fx(x ,i,nrt)-fxm))3U,2t,2U
Subroutine zer.
475
hix - Interval endpoint.
Analogous to lox.
asig - Convergence criterion variable.
j - Do loop iteration variable.
x - Trial composition, midway between the
endpoints
of
the
interval being searched.
Flow scheme:
Line number
-
Operation
4,5 - Transfer input variables to working variables.
6 - Establish appropriate convergence accuracy.
7 - Enter or continue search loop.
If 60 iterations
already
performed, go to line 15.
8 - Guess
x.
9 - Test for convergence.
line 16.
If convergence sufficient,
go
to
If not, go to line 10.
10 - If value of function is too high at x, go
to
line
13.
If too low, go to line 11.
11 - Set new endpoint for interval to be searched.
12 - Go to line 14.
13 - Set new endpoint for interval to be searched.
14 - Proceed to line 7 to continue search.
15 - Contingency return to calling program.
The
convergence
criterion has not been satisfied after 60 iterations.
16 - Set value to be returned to calling program.
17 - Return to calling program.
18 - End program.
476
C.4
Main Program pure
This is the main program
component diagrams.
P-V
plot,
but
for
the
generation
of
pure
It is set up to calculate isotherms in a
also determines a phase equilibrium pressure
when appropriate.
Thus
the
program
is
also
useful
for
determining tielines and P-T diagrams.
Variable list:
a - Peng-Robinson (P-R) attraction parameter.
b - P-R covolume.
T - Temperature.
v - Molar volume.
R - Gas constant.
Z - Compressibility factor.
Tc - Critical temperature.
Pc - Critical pressure.
w - Acentric factor.
ac - Value of attraction parameter at critical point.
Tlow - Lowest temperature isotherm to be calculated.
Thigh - Highest temperature isotherm to be calculated.
Tstep - Temperature increment between isotherms.
Plow - Lowest pressure for which volume is to be calculated.
Phigh - Highest
pressure
for
which
volume
is
to
be
calculated.
P - Pressure.
nrp - Root specifier when multiple real roots
exist.
to
the
cubic
477
C OMPILATION L!S T ING OF pure
(>lse
ri
r_ai r>H)>Hrcng>ure.fortran)
Compilea by:
uLtics New Fortran Compiler,
Comupite on: 07/22/o1
2U49.7 eat Weu
Options: ansio6 map
Release
a
4adinProgram
implicit douole precision(a-ho-z,A-Z)
common/pure/a,b,T , vR Z
aata TcPc,/304..,Z.,.225/
1
2
3
4
RH.i082057
S
ac.45724R' Z*Tc d/Pc
6
ca.0778?R*Tc/Pc
print "Input TowiThi 9h,Tstep,PLowP nignFstep"
7
read,T ow,Thighs step,PLowPhigh.Pstep
oo 100 TaTLouwThigh,Tstep
8
9
10
a-ac*( 1+(.37464+1.5426*w-.Z6992*w'2)*(1-(T/Tc)
12
print,"Tm".T,"
13
14
ao 100 nroxl,2-sign(1,zrp(P))
caLL Zvatp(Pnrp)
11
oo 100 PPlow,PhighPstep
P=',*P
15
print, v" ,v
16
if(nrp-2)100,100,10
if(fp(Pll,)/fp(P-.99Pstepi,1))2J,100,10U
1? 10
ld 2U
19
20 1UO
21
22
call zer(fp,l1l,P-.99-Pstep,P,G,PvLe.,lbO)
print,'vLeat P,PvLe
continue
end
Figure C.4. Main program pure.
'.5))2
478
See Section C.5.
zrp - Determinant function.
See Section C.6.
Zvalp - Compressibility factor subroutine.
fp - Fugacity function.
See Section C.7
zer - Bisection subroutine.
of
Pvle - Pressure
See Section C.3
vapor-liquid
a
at
equilibrium
given
temperature.
Flow scheme:
Line number
-
Operation
5-7 - Assign values to constants.
8 - Request input of desired temperature and pressure grid.
9 - Read in temperature and pressure grid.
10 - Pick a new temperature.
If greater than
Thigh,
go
to
line 22.
11 - Assign value to temperature dependent constant.
12 - Pick a new pressure.
If greater than Phigh, go to
line
10.
13 -Print
out temperature and pressure.
14 - Enter or continue loop to calculate volume.
the
second
iteration
If
iteration
three
there
is
and there is one real root to
the cubic, go to line 12.
and
If this
are
this
is
real
the
fourth
roots to the
cubic, go to line 12.
15 - Call subroutine to calculate volume.
16 - Print out volume.
17 - If this
is
the
third
iteration,
Otherwise, go to line 21.
go
to
line
18.
479
18 - Evaluate function fp at
(approximately)
quotient of these
the
two
in
the
current
previous
values
this
pressure
pressure.
is
and
at
If
the
a
VLE
negative,
pressure interval.
pressure
exists
line 19.
If the quotient is nonnegative, go to
Go to
line
21.
19 - Call subroutine zer to determine more accurately the VLE
pressure.
If VLE pressure cannot be
to line 21.
20 - Print the VLE pressure.
t
21 -
Go to line 14.
22 -
End program.
determined,
go
480
C.5
Function zrp
This function calculates zrp,
the
determinant
of
cubic
equation.
root.
If zrp is negative, the cubic has three real roots.
the
If zrp is positive, the cubic has one real
Variable list:
a, b, T, v, R
zb, zc, zd,
Z
zp,
P - Defined
zq
calculation
-
in main
program
Intermediate
pure.
variables
of zrp.
A - Dimensionless attraction parameter.
B - Dimensionless covolume.
Flow scheme:
Line number
-
Operation
6-7 - Assign values to constants.
8-12 - Assign values to intermediate variables.
13 - Calculate zrp.
14 - End program and return to calling program.
in
the
481
COrMPILA.
TIOJ LISTING OF zrp ( >user_ i r_uir>HD.>hon>zrp.fortran)
Compilec by: uLtics
.Jew Fortrdn Compiler,
Compileu on: J7/Z2,ol
LJ4).7 eut eu
uptions:
ansi6Q mao
elease td
Function
1
couole recision function
3
c ommon/ u re/ ar, T,v,R,Z
5
ccmmon/fp/A,8
A=ia*P/(R*T) '2
Eb o*P/RIT
7
implicit touble precision(a-n,o-z,A-Z)
common/ZvaLp/zp,zq,zO
bb=t-1
9
1:
11
1Z
z ^c-A3*e' 2-2*d
zrd8' 2+e' 3-A *u
zpZC-ab'2/3
zqzzd-zb*zc/3+2zo'3/27
1
zrpa(zp/3)
14
ru(P)
end
3+( zq/2)
Figure C.5.
2
Function zrp.
zrp
482
C.6
Subroutine Zvalp
This
subroutine
calculates
the
value
of
the
compressibility factor, Z, and the molar volume v.
Variable list:
Z, P, nrp - Defined in main program pure.
a, b, T, v, R
zp, zq, zb - Defined
in function
zrp.
z - Array which stores the three values of Z when
the
cubic
has three real roots.
zrp - Determinant function.
See Section C.5.
zC, zD, zy - Intermediate variables in the calculation
of
Z
when cubic has one real root.
n - Do loop iteration variable.
phi - Angle used in calculation of Z
when
cubic
has
three
real roots.
Flow scheme:
Line number
-
Operation
6 - If three real roots to cubic, go to line 12.
go to line 7.
7-9 - Assign values to intermediate variable.
10 - Calculate
Z.
11 - Go to line 21.
Otherwise,
483
COMPILATIJ.NLISTING UF val( (user_.ir.ir>HOt>hon;)la
Cowmiled oy: 4ultics
New Fortran Comailers
Compltea on: J7/22/31
2u49.6 eat Wea
Options: ansio6 mao
Lp. fortran)
Release oa
Subroutine
I
Z
3
4
5
suDroutine Zvatp(P,nro)
implicit douole recision(a-n,o-z,A-Z)
commonpureiabT,v,?,Z
common/ZvaLp/z; ,qz
dimension z(3)
o
7 13
if(zro(P))2O,10,1j
zC-abs(-zqi2+zrp(P)*.5)'(4./3)/(-zq/2+zrp(P)'.5)
0
zD- dos(zq/2+zrp(P)'
9
1
zy-zC+zD
2=zzy-zb/3
11
12 2U
go to 7U
co 3U n1,3
.5)(4./3)/(z;/2+zr;(P)
13
phi-acos((-27*zq'2/4/zp'3).5)
14 3C
z(n)-(zqlabs(zq))*2*(-zp/3)'.5*cos(phil3+2.0943951*(n-1))-zb/3
15
16 4G
go to (40,o,50)
Z-ma(z(1),zU))
17
Go to
1 50
19
Zumin(z(1),z(2))
to to 70
2U oU
Z z( 3)
21 73
22
Zvalp
.5)
nrp
70
vZ*R*T/P
end
Figure
C. 6.
Subroutine Zvalp.
484
12 - Enter or continue do loop to calculate three
Z.
values
of
If this is the fourth iteration, go to line 15.
13 - Calculate angle phi.
14 - Calculate a value for array z.
15 - Select the desired value of Z.
(large Z) go to line 16.
Z)
go
to
line
18.
For
the
gaseous
value
For the liquid value
(small
For
the
unstable
(intermediate Z) go to line 20.
16 - Assign Z the maximum value in the array z.
17 - Go to line 21.
18 - Assign Z the minimum value in the array z.
19 - Go to line 21.
20 - Assign Z the intermediate value in the array z.
21 - Calculate molar volume v.
22 - End program and return to calling program.
value
485
C.7
Function fp
This function
fugacity
at a
calculates
the
difference
between
the
of a pure vapor and the fugacity of the pure liquid
given
important
temperature
since,
if
and
equal
pressure.
to
This
quantity
is
zero, the criterion of two-
phase equilibrium is satisfied.
Variable list:
a, b, T
v, R
j,k - Dummy
Z
P, nrp - Defined in main program pure.
variables,
included
for
compatibility
with
subroutine zer.
A, B
-
Defined in function zrp.
f - Array which stores the values of
fugacities.
The
value
the
vapor
and
liquid
of the unstable fugacity is
not stored.
Zvalp - Compressibility factor subroutine.
See Section C.6.
Flow scheme:
Line number
-
Operation
6 - Enter or continue do loop to calculate vapor
fugacities.
If
this
and
liquid
is the third iteration, go to
line 9.
7 - Call subroutine
Zvalp
to
assign
values
to
necessary
variables.
8 - Calculate fugacity.
Go to line 6.
9 - Calculate fp, the fugacity difference between
liquid.
10 - End program and return to calling program.
I
vapor
and
486
CO PILATIOlJ LISTING OF fp (>user_ dir_
ir>HD>Hong>fp.fortran)
CompiLea y: .uLtics
ew Fortran Compiler,
2J49.s et wed
Compilea on: 7/22/51
Options: dnsi66 map
ReLease a
Function
1
2
aouo e precision function fp(P,j k)
impLicitdouble precision(a-h,o-z,A-Z)
3
c ommon/pure/aoTiv,RZ
4
5
o
7
common/fp/AB
aimension f(3)
ao 10 nrpal,3,2
caLL ZvaLo(Pnrp)
a 1G
9
10
f(nrp)-P*exp(Z-l-aLo(Z-a)-A/'.5/*aLoq((Z+(2'.5+1)*d)/(Z-(2'.5-1)*)))
fpf (1)-f (3)
ena
Figure
C. 7.
Function fp.
fp
487
C.8
Main Program pr
This is the main program for the calculation
of
phase equilibrium, both fluid-fluid and solid-fluid.
coding
the
extrema
Most of
with analysis of the fugacity-composition
deals
Infinity
functionality at a given temperature and pressure.
points,
binary
or
spinodal
(Ll=O) and inflection
points
The fugacity curve is
points (Ml=O) are calculated.
divided
into monotonic segments, as described in Figures C.la, b, and
c,
and
these
the
for
subroutines
are
segments
compositional
determining equilibrium along
called.
derivatives
refer
In
following,
all
differentiation
with
the
to
respect to mole fraction of component 2.
Variable list:
d - Array which stores
segments.
the
end
points
of
the
monontonic
These occur at mole fractions of 0 and 1,
as well as at slope infinities and extrema.
nr - Array which stores a root specifier for
An
element
in
each
interval.
this array is meaningful only if the
cubic has three real roots in the given interval.
the cubic has one real root, the value
of
the
specifier is immaterial.
bl, b2 - Covolumes for components 1 and 2.
bx - Compositional derivative of mixture covolume.
T - Temperature.
al, a2 - Attraction parameters for components 1 and 2.
If
root
488
COIPILATIO:4
LISTING OF pr
>uJ>HDN)>Horn
;pr.tcrtrdn)
Compiled oy: .ulticsNew Fortran Comoiler, Reledse ba
Compile on: 7/22/11 212.3 et wed
3t ions: ansio dap
.ain Pro,ram
I
2
3
4
implicit double precision(a-h,o-zA-Z)
dimension d(15),nr(14)
common/pr/Ol~o2bsoxTa,aaalaa2,P,Rvc
aata Tc1,Pcl,wl,Tc2,Pc2,w2,d1
2/3U4.2,72.6,.25,74b.4,40,..02,.11
/
aata cl1c2d' 1), nr(1) / .uOJ 1,.99999V,1d-6,1/
R'.082J57
vc.1 248
5
6
7
alc=.45724*R'2*Tc1' 2/Pcl
/
&
9
10
a2c".45724*R 2*Tc2 2 Pc2
bl .0778R*Tc1/Pc 1
11
o2.0
12
13
14
bxub2-bl
print,"Input Tlow,Thigh,Tstep,Plow,Phigh,Pstepocoarse
readTlowThighTstepPlowPhighPstep,c,
fti
15
oo 200 TTowTh
1o
17
1S
19
alaalc*(1 (.37464+1.5422t,
w1-.26992*w1'2)*
(1-(T/Tc1)
'.5))'2
aaaZc(1+(.37464+ 1.5422o*w2-.2699d*w2
2) (1-(T/Tc) .5)) 2
a12"(1-d12)*(al*a2)
.5
ao 230 P=PlowPhigh,Pstep
778*R*Tc2P/P c 2
i h,ste
20
print,"T#,*T,"
22
23 40
if(zr(0o0,11).Lt.0ad)kf1
yux
21
24
23
2e
27
26 50
29
oO
3gfg; l1;inca1:k=o;cf
if(x.gt..
p.',
0; xld-6
15)gcg
if(L.,je.3)g=cg
xix+inc*g
if(x)65,65,50
i f (x-1) 70.0,60
if(zr(dlaOl1).gt.U.or.kf.gt.o)go
to ld80
30 65
31
32
ksk+l
kf'kf+l
d(k+ l)aidint
34
35
36
37 70
3b a0
39
nr (k+l):k+1 -2*(k/3+k /4)
d (k+l)-idint
(x)
go to 100
d0,80,11J
if(zr(y,1,1)/(zr(x,11)))
kf-kf+l
go to (11090O,9U,11e11J,90,9U,1¶10) kf
33
4
41
90
U
42
43 100
44
45
l=t+1
(x)
kk+l
call zer(r, 1,1, x,yJ,d(L+k),$11U)
nr(k+l)ak+1-2*(k/J+k/4)
inc--inc
if(mod(k,2) ..
1) xc2*d (k +t )
if(rnod(k,2).eq.0) xmax(c-1,c*(k+L)
4o
printp"zRzero at x2",d(k+l)
47
4o 11U
49 140
go to 40
if(L.I(y,2,nr(k+L))/
l=L+l
5u
51
LL,(x,Z,nr(k+L)))
)
14l0,15u,15U
nr(k+l)s+l1-2* (k/3+/4 )
call zer(LM42,nr(k+L),xyUd(k+l),S150)
52
53 150
54 1O
55
50
print,"L1 zero at x2=",a(k+L),"
nr="rr (k+l)
lc0L,4L,4U
if(L1(y,3,nr(k+L))/Ll.(x,3,nr(k+l)))
caLL zer(LM,3nr(+l),xy,U,..lu,$4O)
print,"F-1
zero at x2='"L'"
nr=",nr(k+l)
go to 4L
57 1B3
d(k+t+l)=1
56
59 190
60
61
ao 190 il,k+l
call
ol(max(d(i),d(l+1)),min(a(i),d(i+l)),nr(i))
do 200 ilk+L-2
do 200 ji+2,k+t
62
6i 200
64
ridfine
p
call pea(min(d(i),d(i+1)),max(a(i),d(i+1)),min(a(j),a(j+1)),max(o(j),d(j+1)),nr(i),nr(j))
continue
end
Figure C.8.
Main program pr.
rid"
489
a12 - Parameter for attraction between components 1 and 2.
P - Pressure.
R - Gas constant.
vc - Molar volume of solid phase.
Tcl, Tc2 - Critical temperatures.
Pcl, Pc2 - Critical pressures.
wl, w2 - Acentric factors.
d12 - Interaction parameter.
cl - Constant slightly greater than 1.
c2 - Constant slightly less than 1.
alc, a2c - Attraction parameters at critical points.
Tlow - Lowest temperature for which phase equilibrium
is
to
be calculated.
Thigh - Highest temperature for which phase equilibrium is to
be calculated.
Tstep-
Increment
between
temperatures
for
which
phase
equilibrium is to be calculated.
Plow - Lowest pressure for which phase equilibrium is
to
be
calculated.
Phigh - Highest pressure for which phase equilibrium is to be
calculated.
Pstep - Increment
between
pressures
for
which
equilibrium is to be calculated.
cg - Large composition increment (coarse grid).
fg - Small composition increment (fine grid).
g - Current compositional increment.
1 - Associated with number of spinodal points located.
t
phase
490
inc - Sets
direction
increasing
of
mole
compositional
fraction
of
search.
component
+1
for
2;
-1 for
of
slope
decreasing mole fraction of component 2.
k - Associated with number of infinities located.
kf - Infinity
flag.
infinities
Associated
with
encountered.
A
number
given
infinity
is
sometimes encountered twice.
x - Current mole fraction of component 2.
zr - Determinant function.
See Section C.9.
y - Previous mole fraction of component 2.
zer - Bisection subroutine.
See Section C.3.
LM - Helmholtz derivative function.
See Section C.11.
MO - Mole fraction of an inflection point.
i - Do loop iteration variable.
sol - Solid solubility subroutine.
See Section C.12.
j - Do loop iteration variable.
peq - Fluid-fluid equilibrium subroutine.
See Section C.13.
Flow Scheme:
Line number
-
Operation
6-12 - Assign values to constants.
13 - Request input of temperature, pressure, and
composition
grid.
14 - Read in temperature, pressure, and composition grids.
15 - Enter or continue do loop to calculate phase
at a new temperature.
to line 65.
equilibria
If T is greater than Thigh, go
491
16-18 - Assign values to temperature dependent constants.
19 - Enter or continue do loop to calculate phase
than Phigh,
If P is greater
at a new pressure.
equilibria
go to
line 15.
20 - Print out T and P.
21 - Initialize variables.
22 - Determine if three roots are present at x=0.
If so, set
the infinity flag to 1.
23 - Assign current value of x to previous value of x.
24-26 - Set
composition
current
on
depending
step
necessary to search a
much
closely
more
size
to
particular
than
fine,
It is sometimes
of x.
value
or
coarse
composition
range
the rest of the range.
It
becomes evident in the generation of a P-x diagram if
and where such a fine search is necessary.
The range
given here for the fine search is x less than 0.15 or
until the second extremum is
different
for
The
found.
range
is
different temperatures and pressures,
and is often in essence deleted by inputting a
value
of fg equal to cg at line 14.
27 - Assign new current value of x.
the
28 - If x is nonpositive (this occurs during
segment
3 in Figure C.lc) go to line 31.
search
of
Otherwise,
go to line 29.
29 - If x is greater
than or
equal
to
1
go
to
line
30.
Otherwise, go to line 38.
30 -
If there is only one root at x=l, or if the three
roots
492
at x=l have already been accounted for, the search at
this T and P is complete.
Go to line 58.
31 - Increment number of infinities located.
32 - Increment infinity flag.
33 - Assign an interval endpoint the value of 0
if
x<O,
or
the value of 1 is x>l.
34 - Increment number of extrema located.
35 - Assign proper value to root specifier.
36 - Same as line 33.
effect
of
Together, these
assigning
a
statements
have
the
null length to segment 4 in
Figure C.lc, or to an analogous segment
occuring
at
x=1.
37 - Go to line 44.
38 - Test for a
zero
corresponds
in
the
cubic
determinant.
A
zero
to an infinity in the fugacity plot.
If
a zero is present in the interval between x and y, go
to line 39.
Otherwise, go to line 49.
39 - Increment infinity flag.
40 - The value of the
infinity
flag
determines
infinity
is
to be located and stored.
line 41.
If not, go to line 49.
if
a
new
If so, go to
41 - Increment number of infinities located.
42 - Call subroutine to accurately
location
in array d.
locate
infinity.
Store
If infinity cannot be located,
go to line 49.
43 - Assign proper value to root specifier.
44 - Change direction of composition search.
493
45, 46 -
Bring x back into the range of 0 to 1.
47 - Print out location of infinity.
48 - Go to line 23.
49 - Test for an extremum (spinodal) in the x-y interval.
If
Otherwise, go
to
an extremum exists, go to line 50.
line 54.
50 - Increment number of extrema located.
51 - Assign proper value to root specifier.
52 - Call subroutine to accurately
location
in array d.
locate
extremum.
Store
If extremum cannot be located,
go to line 54.
53 - Print out location of extremum and root specified.
54 - Test for an inflection point in the x-y interval.
inflection exists, go to line 55.
If an
Otherwise,
go
to
inflection
point.
line 23.
55 - Call subroutine to accurately locate
If inflection point cannot be located, go to line 23.
56 - Print out location of inflection point.
57 - Go to line 23.
58 - Set final segment end point at x=l.
59 - Enter
or
continue
loop
to
calculate
solid
If all segments have been checked, go
solubilities.
to line 61.
do
This statement is deleted for systems in
which no solid phase is present.
60 - Call subroutine to
given segment.
calculate
solid
solubility
for
a
This statement is deleted for systems
in which no solid phase is present.
494
61 - Enter or continue subroutine
calculate
to
fluid-fluid
equilibria.
Specify the first segment of the two to
be compared.
If all
necessary
segments
have
been
checked, go to line 65.
62 - Enter or continue subroutine
equilibria.
be
compared.
to
calculate
fluid-fluid
Specify the second segment of the two to
If
all
necessary segments have been
specified, go to line 61.
63 - Call
subroutine
to
calculate
between two segments.
64 - Go to line 62.
65 - End program.
fluid-fluid
equilibria
495
C.9
Function zr
This function calculates
If
zr,
the
determinant
of
cubic
equation.
root.
If zr is negative, the cubic has three real roots.
the
zr is positive, the cubic has one real
Variable lists:
x2 - Mole fraction of component 2.
dl,d2 - Dummy arguments
for
compatibility
with
subroutine
zer.
bl, b2, bx, T, al, - Defined in main program pr.
a2, a12, P, R
vc
a - Mixture attraction parameter.
b - Mixture covolume.
v - Molar volume of mixture.
Z - Compressibility factor of mixture.
A - Dimensionless mixture attraction parameter.
B - Dimensionless mixture covolume.
zb, zc, zd,
zp,
zq
-
Intermediate
variables
calculation of zr.
Flow scheme:
Line number
-
Operation
6-9 - Assign values to constants.
10-14 - Assign values to intermediate variables.
15 - Calculate zr.
16 - End program and return to calling program.
in
the
496
frtran)
LISTI:4uOF zr (>user_ir cir>HD.J>hong>zr.
COMCPILATION
Compilea
y: Multics New Fortran Commpiler, Release
Compilea o: J7/22/l61 25u.o
ptions: ansio6 mao
et
ed
Function zr
1
douole orecision function zr(x2,al,J2)
implicit dcounle precision(a-h,o-z,A-Z)
3
common/pr/bl,:2,bx,Tala2sa2,P,R,
4
5
common/Zv/a, v,Z ,A,u
common/zr/zpzqz
O
7
d
dd1*(1-x2)'2+2*a12*(
'b1-(1-x2)+b2*(x2)
vc
1-x2)*(x2)+a2*(x 2)'2
Aa*P/(R*T)'2
9
10I
Bb*P/R/T
zb=8-1
11
12
13
14
15
zcA-3*8'2-2*1
ZcY'2+t'3-A*u
zp'zc-zo'2/3
zqza-zo*zc/3+2*zOD3/e?
zr=(zp/3)'3+(zq/2)'2
16
eno
Figure C.9.
Function zr.
497
C.10
Subroutine Zv
This subroutine calculates
factor
compressibility
Z,
and
the
value
the
molar
of
the
volume
mixture
the
of
mixture, v.
Variable list:
x2 - Mole fraction of component Z.
nrt - Root specifier.
bl, b2, bx, T al, - Defined in main program pr.
a2, a12, P R, vc
a, b, v, Z
- Defined in function zr.
A
B, zp, zq, zb
z - Array which stores the three vaues of Z
when
the
cubic
has three real roots.
zR - Value of the determinant zr at a given x2.
zr - Determinant function.
zC, zD, zy - Intermediate
See Section C.9.
variables
in the calculation
of
Z
when the cubic has one real root.
n - Do loop iteration variable.
phi - Angle used in calculation of Z
when
cubic
has
three
real roots.
zl, z2, z3 - Intermediate variables in the
the
middle
(mechanically
factor.
Flow scheme:
Line number
-
Operation
determination
of
unstable) compressibility
498
COMPILATIO;4 LISTING
OF Zv (>user_ir_ir>r1HD>ong>Zv.ftrtran)
ComJiea oy: 4uLtics
New Fortran ComiLer,
Compilea on: )7/2 2 /o1 2;5u.3 ect eu
options:
ansi6o map
elease
Sucroutine
1
2
suoroutine Zv(x2,nrt,*)
implicit
douoLe precision(a-h,o-z.A-Z)
4
5
common/Zva ,b ,v,Z,A ,d
common/zr /zp,z q,zo
7
zkZzr(x2,1,1)
3
6
6
a
Zv
common/pr/bl,b2,ox,T,al,a2l 12,P,R,vc
aimension z(3)
if(zR) 20,10,10
9 10
1u
11
12
zCabs(-zq/2+zR'.S)'(4./3)
(-zq/2+zR'.5)
zD-abs(zq/2+zR'.5)'(4./3)/(zq/2+zR'.5)
zyzC +zD
Zzy-zb/3
13
Vo to
70
14'20
ao 30 nl,3
15
16 33
17
pnidacos((-27*zq'4/4/zp'3)'.5)
z(n)u-(zq/abs(zq))*2*(-zp/3)'.5*cos(phij3+2.U943951*(n-1))-zb/3
go to (60,50,40) nrt
1s 40
19
20 50
21
22
23
24
.25 60
Z-min(z(1),z(2),z(3))
go to 70
zl=max(z(1).z(2))
z2mmax(z(1),z(3))
c3zmax(z(2),z(3))
Zinin(zlz2,zS)
go to 70
Z-max(z(1),z(2),z(3))
Zo 70
it((L-8)
27
26
vZ*
end
.Lt.
0) return
1
*T/P
Figure C.10.
Subroutine
Zv.
499
7 - Evaluate function zr at the input composition and
assign
this value to zR.
8 - If three real roots to cubic, go to line 14.
Otherwise,
go to line 9.
9-11 - Assign values to intermediate variables.
12 - Calculate Z.
13 - Go to line 26.
14 - Enter or continue do loop to calculate three
Z.
values
of
If this is the fourth iteration, go to line 17.
15 - Calculate angle phi.
16 - Calculate a value for array z.
17 - Select the desired value of Z.
to line 18.
to
line
For the lowest value, go
For the mechanically unstable value,
20.
go
For the highest value, go to line 25.
Z's must be chosen so that the monotonic segments are
ordered as shown in Figure C.l.
18 - Assign Z the minimum value in the array z.
19 - Go to line 26.
20-22 - Assign values to intermediate variables.
23 - Assign Z the middle value in the array z.
24 - Go to line 26.
25 - Assign Z the maximum value in the array z.
26- Contingency return.
The calculated value
physical meaning.
27 - Calculate molar volume v.
28 - End program and return to calling program.
of
Z
has
no
500
C.ll
Function LM
This function calculates values of the mechanical
(3)(=Pv)
stability term
, the material stability term
The mechanical stability
L1, and the critical point term Ml.
term is no longer used in any of the other programs since
was
found
to
change
sign
infinities in the f-x
at
only
it
functionality.
Variable list:
x2 - Mole fraction of component 2.
lorm - Function definition.
value
whether
Specifies
of the function
the
return
is Pv, Ll, or Ml.
nrt - Root specifier.
bl, b2, bx, T, al, - Defined in main program pr.
a2, a12, P
P, vc
a, b, v, Z
A, B - Defined
Zv - Compressibility factor function.
ax - Compositional
zr.
in function
derivative
See Section C.10.
of
mixture
attraction
parameter.
axx - Compositional derivative of ax.
W, X, Y - Intermediate variables in the
calculation
of
Pv,
L1, and M1.
Wx, Xx, Yx -
Compositional
derivatives
of
respectively.
r - Ratio of attraction parameter to covolume.
rx - Compositional derivative of r.
W,
X,
and
Y,
501
It
II
II
II-
II
X
_
43
4
I
.
4
"IL
34'
,
_
0
1N>
-
rrj
-34
4
44
Ng
.2.
*I 3.
I
0x
43+%-
r
0
N
4
.4
r-i
4r
C
03
I
N
0
34443
3-N-x
4
x
C
.2
I,40.2
4
C-.
4-4%
4'
.
E
-
I>
oI
00
0
-I
.2.-34
o4
,
004
X
U4.0
4.44
4.N
4
I
I
C
03 C
.. N
_
N
^
+45
-H+3-.
-
D^
- ^_
C
vc
00
ou.)
0
f
1
N
I
D _
"
4
_4I
DxO
-OO
4
r
I
Q
-
I
X
I
, N
N4_ 4 c . E
D N
*
.
Dr 0
;
00
C
>
X
_
aN__
._
>W c
eu
o
.
_,
_
_ *X
.s *
40
O
cv
O
s
O
>
*
I
.·;~
4-414
s
D
v 0
_
I
r
F~ _U I
)
>
_
N.,E
x
N_
"
Y 1
x ;1;X
N
aE
O E
0
9
._ ,,
0
"X
(XU XU 11
-r
N
4
4
-
Io
3
1,
1
v 11
X
3U X X
>11X
11
X
+
V
x
1
w
_te
>
vX
v
1
N44'-v
Yx
_O
II
N
43
unt
U
I
N43.NN
flN.2N
iirr II
N>NNN
N4343N
'...4
0
I
'O
v sx+
cr
4
4
.. f434IA
-0444XU~4N
.
o
.e
4
I..-
4*a 4('
4444
4
Nfl%
X N(%hI-4J
_3- ."l 14
_u004
.43xN
%N443
.
4
4
rQn
*_^a
4
'
:Cv_IN
C N 0 0,^,
I 44
0 I.0%
3
_4.4-
*0.
_,
0O.
~-,
'*.
O0~.
0
C
_ N
C
4
43N(4
4.
t
-N
- a
F4
I
*
N
0
43~0
30
N
~
"
'
-
N
C
4
X_ N- 0
N
4.
X
N
3
_
N
1
II
X
_
<:
11
3D
I
r
~
N43343
4- 4
4 .
42
11-4
4
r4~
I
.-.
j4
X)Nr
I
_ rv
_ '_7 N_· N _nLn _O
0 I_ 2_ r_ _3 c ^
I
N
z11 OX
21
X
2
4
EI
1
w
Z
.4
- nI -· re-
K
I
C
Cz
-
<
I
UG
I -N
^sX
-as
sr
_N
>
x.o
°>N
4>
502
rxx - Compositional derivative of rx.
Avv, Avx, Axxr Avvv, Avvx, Axxx - Partial derivatives of
Helmholtz free energy.
the
See Appendix A.
Wxx - Compositional derivative of Wx.
rxxx - Compositional derivative of rxx.
Flow scheme:
Line number
-
Operation
5 - Call subroutine to assign proper values to constants
calculate
v.
and
If a value of v cannot be calculated,
go to line 34.
6-16 - Assign values to constants and intermediate variables.
17 - Calculate Avv.
18 - If Pv is the desired return value of the function, go to
line 19.
If not, go to line 21.
Branch to
line
34
is not used.
19 - Calculate -Pv (=Avv).
20 - Return to calling program.
21, 22 - Calculate derivatives of Helmholtz free energy.
23 - If L1 is the desired return value of the function, go to
line 24.
If not, go to line 26.
Branch to
line
is not used.
24 - Calculate
L1.
25 - Return to calling program.
26, 27 - Assign values to intermediate variables.
28-31 - Calculate derivatives of Helmholtz free energy.
32 - Calculate Ml.
34
503
33 - Return to calling program.
34 - Set LM equal to an error code value of -1.
35 - End program and return to calling program.
504
C.12
Subroutine sol
This subroutine searches a given monotonic segment of an
f-x curve for points
These
points
of
equality
correspond
to
with
a
compositions
solid
of
fugacity.
solid-fluid
equilibrium.
Variable list:
yl - High composition end point of segment to be searched.
y3 - Low composition end point of segment to be searched.
nj - Root specifier.
y - Array to store modified segment end points.
bl, b2, bx, T, al, - Defined in main program pr.
a2, al2, P R vc
js - Slope parameter.
mole fraction.
+1 if f2
increases with increasing x2
-1 if f2
decreases
with
increasing
x2 .
f - Fugacity function.
See Section C.13.
xloj, xhij - Modified segment end points.
P2sat - Vapor pressure of pure solid phase.
fc - Fugacity of pure solid phase.
asig - Convergence criterion variable.
zer - Bisection subroutine.
See Section C.3.
xj - Composition of fluid phase
phase.
Refers to x2 .
in
equilibrium
with
solid
505
COPILATION LIST ING OF sol (>user_Uir_ir>H3r.>Hor
ComniLeoRby: :'ultics
New Fortran Co.piler
Comr;iLea on: 07/22/al
2'J.7
eat wea
Opt ions: ansi66 map
g>soL.fortran)
Release
a
Sucrout ine sc
1
3
4
5
7
9
suoroutine soL(ylyJ,nj)
aouole jrecision(a-n,o-z,A-Z)
oimension y(3)
common/pr/ol,b2,ox,T,a1 ,d2,a12,P,svc
implitit
if(yl.eq.y3)joto 140
y (1)(1-1 d-9)*yl;y()
s i n(1, f (y(1)
xoj"y(2+ j)
( 1+1a-)y
,nj )-f(y(3),n
3
j) )
xhijuy(2-js)
10
11
12
13
P2sats0.0 01 *exp(-?1 9/T+29.)
R/T)
fcsP2sat*exp(vc*(P-P2sat)/
if(max(f(xLoj,2,nj),f(xhij,2,nj)).Le.fc)co
to 140
if(min(f(xLoj,2snj),f(xhij,2,nj)).3e.fc)jo
to 140
15 90
Print,"sfeq
17
end
14
16 140
call zer( f,2,nj,x Loj,x hi j,fc,xj,S140,9)
return
at x2'",xj,nj
Figure C. 12.
Subroutine sol.
506
Flow scheme:
Line number
-
Operation
5 - If segment is a null segment, go to line 17.
6 - Modify segment
end
to
points
insure
monotonicity
of
segment.
7 - Determine slope of segment in
8 - Set xloj as the
f2
vs
segment
modified
x2
end
coordinates.
point
of
lower
fugacity.
9 - Set xhij as the modified
segment
end
point
of
higher
fugacity
range
fugacity.
10 - Calculate solid vapor pressure.
11 - Calculate solid fugacity.
12, 13 - Determine
if
segment
spans
If not, go to line 17.
including the solid fugacity.
14 - Call
subroutine
equilibrium
to
a
accurately
composition.
If
determine
phase
no convergence, go to
line 17.
15 - Print
out
phase
equilibrium
specifier.
16 - Return to calling program.
17 - End program.
composition
and
root
507
C.13
Subroutine peq
This subroutine searches a pair of monotonic segments of
an
f2 - X2
curve,
compositions
The
first
considering
which
step
nominally
satisfy
is
only
to
the
limit
the
overlapping
of these new segments, the
j
and
k,
for
the phase equilibrium criteria.
illustrated in Figure C.13b.
equal.
designated
original
range
segments
by
of fugacities, as
At the corresponding end points
fugacities
of
component
2
are
At the end points of lower fugacity, the difference
=
dflo
fl(xloj) - fl(xlok)
is calculated, and at the end points of higher fugacity, dfhi
is likewise calculated.
If dflo and
dfhi
are
of
opposite
sign, then a set of compositions must exist for which
f
=
2
k
f2
2
and
f
1
=
k
fk
1
This set of compositions represents fluid-fluid
To
determine
these
compositions
equilibrium.
accurately,
a
trial
composition xj in the middle of segment j is chosen, and
composition
the
xk in segment k which matches the fugacity of xj
is determined.
This step is
illustrated
Now the difference
df
=
fl(xj)
-
fl(xk)
in
Figure
C.13c.
508
CO0iPILATIONLISTING F p eq (>uJo>HDN>hcn>eq.tortran)
Compiled oy: lultics
ei Fortran Compiler,
elease
CompiLea on: J7/22/61
2125.9 eat eu
ansi66 map
Options:
d
Suoroutine eq
1
subroutinepeq(y1,y3yy2,y4
.jik)
3
dimension
4
5
0
if((yl.eq.y3).or.(y2.e.y4))go
to 200
y(l1)(+ld-6)*yl;y(2)s(C1+d-6)*y2;y(3)"(1-1d-6)*y3;y(4)1-1a-6)*y4
j smsign(lt(y(1l),2,j)-f(y(3),Z,j))
6
x j y( 2+js)
xhjay(2-js)
xlky (3+ks)
xhk"y(3-ks)
2
implicit
7
douote
)recision(a-hn,o-z,A-Z)
4
y( )
kssiwn(lof(y(2),2,k)-f(y (4),,k))
9
10
11
12
asi f(txLk/2+xhkl21,k)/1U'8
13
do 10 n1,2
14
if(max(f(xLj,n.j),f(xhj.n,j)).le.min(t(xLk,n,k),ftxhk,n,k)))go
lo
if(ft(xLj,2,j)-t(xk,2Zk))
15 13
17
1l
19
2U
21
20
30
40
U50
oO
22 70
23
24
25 80
20
27
26 90
29
if(max(fCxLk,n,k),f(xhkonk)).Le.min(ftxLjnj),f(xhjnj)))go
caLL zer(f,2,j,xijixhj,tf(xLk,2,k),xLj,S2uO,$40)
caLL zer(f,2,k,xLkxnkf(xLj.2,j).xlkS2UO)
if(f(xhj,2,j)-f(xnkrZx))
50,200,60
call zer(f2,kxtlksxnkf(xhj*2,j),xhkS2CGUS?0)
jSS2u)
call zer(f.2,jxtjoxlhj.(xxhk,2,k),xh
f Lof (xljtj)-f
(xLklk)
athif (xhj,l,j)-f(xhk,l,k)
if(dfLo/dfhi)130,200,dO
0o 100 n"1,19
xjaxlj+n*(xhj-xlj)/2U
call zer (fr2,kxLkxhksf(xj2,j),xx$200,$90)
df t(x j,1, j)-f(Xkl1,)
ift(df/df tlo)110,100100
.SJ 100
continue
31
go to 230
32 110
33
34
35
30
xjm"xj
xkmaxk
xjhaxhj
xkhuxhx
xhjmxj
37
kft2
36
go to 140
39 120
xLjaxja
4U
xhjaxjh
41
x Lkx km
42
43
xhkxkh
dflot(Cxjml1,j)-f(xxmlk)
44 133
45 140
kftl
x j'xhj/2+xlj/2
46
47 150
4o
49 10o
50 17U
51
52
53 i1c
54
55
50 190
57
5b
59 200
60
2,4G,30
caLL zer(f,2,kxLk,xnk,f
ufsf(xj,l,j)-f(xk,1,k)
if(aosCdf)-asig)190,190,1o0
if Cf/df Lo) 17C,7u,1 U
xhj xj
xhkmxk
go to
xtjxij
xj,2,j)xk,S20,S1
50)
14C
xlkxk
go to 140
x " ,xj," y =",xk
print,
to 12U
it(kf.eq.2)go
return
"1 phase"
print,
end
Figure C.13a.
Subroutine peq.
to 200
to 200
509
xloj
0
xhij
xlok
xhik
1
x2
Figure C.13b.
j and k.
fugacity.
The search for fluid phase equilibrium between segments
Equilibrium can only be found in regions of overlapping
Thus, the equilibrium composition in segment j must lie
between xloj and xhij, and the equilibrium composition in segment k
must lie between xlok and xhik.
f2
xloj
0
xhij
xlok
xhik
1
X2
Figure C.13c.
To determine the equilibrium compositions in segments
j and k, a trial value of xj = (xloj + xhij)/2 is chosen.
is chosen
so that f2(xk)
= f 2 (xj).
Then xk
510
If
is calculated.
xj
compositions
has
df
xk
and
sign
same
the
the
become
new
as
this
fashion
df
until
meets the convergence
When this occurs, xj and xk are
criterion.
and
The program
xk become the new upper limits for the segments.
in
the
lower limits for
If df and dfhi have the same sign, xj
segments j and k.
proceeds
dflo,
printed
out
as
the phase equilibrium compositions.
When dflo and dfhi have the same sign, there is
not
between the chosen pair of segments.
equilibrium
phase
usually
However, in the vicinity of an azeotrope, the function df may
have two zeroes so that dflo and dfhi have the same sign
two
sets
segments.
in
of
phase
equilibrium
exist
between
When this situation is encountered,
question
a
the
pair of
segments
are split into two parts, so that two new pairs
of segments with dflo and dfhi of opposite sign result.
preceding
method
search
Furthermore,
more
The
for phase equilibrium compositions
No instance has been found of two segments
may then be used.
exhibiting
and
than
two
equilibrium
sets
never
of
phase
equilibrium.
exists between two adjacent
segments.
Variable list:
yl - Low composition end point of first segment (segment j).
y3 - High composition end point of segment j.
y2 - Low composition end point of second segment (segment k).
y4 - High composition end point of segment k.
511
j - Root specifier for segment j.
k - Root specifier for segment k.
y - Array to store modified segment end points.
js - Slope parameter for segment j.
increasing x2.
+1 if f2
decreases with
-1 if f2
decreases
with
increasing
x2 .
ks - Slope parameter for segment k.
Same sign convention
as
js.
f - Fugacity function.
See Section C.14.
xlj, xhj - Modified end points for segment j.
xlk, xhk - Modified end points for segment k.
asig - Convergence criterion variable.
n - Do loop iteration variable.
zer - Bisection subroutine.
dflo - Fugacity
See Section C.3.
difference
for
component
1
for
the
1
for
the
composition pair xlj and xlk.
dfhi - Fugacity
difference
for
component
composition pair xhj and xhk.
xj - Trial composition within segment j.
xk - Composition in segment k which has the same fugacity
as
xj.
df - Fugacity difference for component 1 for the
composition
pair xj and xk.
xjm - New
segment
end
point
within
segment
j.
In
an
azeotropic region, segment j must be divided into two
segments.
xkm - Variable which saves a value of xhj.
512
xkh - Variable which saves a value of xhk.
kf - Condition
indicator.
segments
If
equal
to
2,
the
pair
of
is to be searched for a second set of phase
equilibrium compositions.
If equal to 1,
both
sets
of compositions have already been determined, or only
one such set exists.
Flow scheme:
Line number
-
Operation
4 - If either segment is a null segment, go to line 59.
5 - Modify segment
end
points
to
insure
monotonicity
of
segments.
6 - Determine slope of segment j in
f 2 vs x2
coordinates.
7 - Determine slope of segment k in
f, vs x
coordinates.
8 - Set xlj as the modified segment
j
end
point
of
low er
fugacity.
9 - Set xhj as the modified segment j
end
point
of
higher
fugacity.
10 - Set xlk as the modified segment k
end
point
of
lower
fugacity.
11 - Set xhk as the modified segment k end
point
of
higher
fugacity.
12 - Establish appropriate convergence accuracy.
13 - Enter or continue
do
loop
to
check
fugacity ranges between the segments.
for
overlapping
If ranges have
been checked for both components, go to line 16.
14, 15 - Check
for
overlapping
fugacity
ranges
between
513
segments.
If no overlap, go to line 59.
ranges.
16-21 - Limit segments to overlapping fugacity
See
Figure C.13b.
22 - Calculate dflo,
the
fugacity
at
difference
the
low
fugacity end points for component 1.
23 - Calculate dfhi, the
fugacity
difference
at
the
to
line
high
fugacity end points for component 1.
24 - If dflo and dfhi are of opposite sign, go
If
of
the
same
dflo
and
dfhi
If not in an
sign, go to line 25.
azeotropic region, this statement
of
is
44.
altered.
For
the same sign, the program would
proceed to line 59 rather than line 25.
In this case
lines 25 through 43 are skipped.
25 - Enter or continue do loop to find new segment end point.
If end point has not been found after 20
iterations,
go to line 31.
26 - Pick a trial composition xj in segment j.
27 - Call subroutine to determine composition xk in segment k
which matches
f2
at xj.
If xk cannot be
found,
go
to line 59.
28 - Calculate df, the fugacity difference for component 1 at
xj and xk.
29 - If df and dflo are of opposite sign, go to line 32.
of the same sign, go to line 30.
30 - Go to line 25.
31 - Go to line 59.
32 - Set new end point xjm at xj.
If
514
33 - Set new end point xkm at xk.
34 - Assign high fugacity end point of segment j to xjh.
35 - Assign high fugacity end point of segment k to xkh.
j
segment
at
j
segment
at
40 - Set high fugacity end point of second new j
segment
at
k
segment
at
42 - Set high fugacity end point of second new k
segment
at
36 - Set high fugacity end point in first new
xjm.
37 - Set condition indicator to 2.
38 - Go to line 45.
39 - Set low fugacity end point of second new
xjm.
the original high fugacity end point xjh.
41 - Set low fugacity end point of second new
xkm.
the original high fugacity end point xkh.
43 - Calculate dflo for second new set of segments.
44 - Set condition indicator to 1.
45 - Pick a trial composition xj in current segment j.
46 - Call subroutine to determine composition xk in segment k
which matches f 2
at xj.
If xk cannot be
found,
go
to line 59.
47 - Calculate df, the fugacity difference for component 1 at
xj and xk.
48 - Check convergence
sufficiently
criterion.
close
If
df
is
to zero, go to line 56.
If not,
go to line 49.
49 - If df and dflo are of opposite sign, go to line 50.
If
515
not, go to line 53.
50 - Set high fugacity end point of segment j to xj.
51 - Set high fugacity end point of segment k to xk.
52 - Go to line 45.
53 - Set low fugacity end point of segment j to xj.
54 - Set low fugacity end point of segment k to xk.
55 - Go to line 45.
56 - Print out phase equilibrium compositions.
57 - If condition indicator equals 2, go to line 39.
58 - Return to calling program.
59 - Print out message of no phase equilibrium.
60 - End program and return to calling program.
516
C.14
Function f
This function calculates the fugacities of components
1
and 2.
Variable list:
x2 - Mole fraction of component 2.
icomponent - Component specifier.
nrt - Root specifier.
bl, b2, bx, T, al, - Defined in main program pr.
a2, a12, P, .R, vc
a, b, v, Z
A, B - Defined in function zr.
Zv - Compressibility factor function.
See Section C.10.
Flow scheme:
Line number
-
Operation
5 - Call subroutine to assign proper values to constants
calculate
Z.
and
If Z cannot be calculated, go to line
11.
6 - If fugacity
of component 1 desired,
go
to
line
fugacity of component 2 desired, go to line 7.
7 - Calculate fugacity of component 2.
8 - Return to calling program.
9 - Calculate fugacity of component 1.
10 - Return to-calling program.
11 - Assign error code value of 0 to f.
12 - End program and return to calling program.
9.
If
517
_0 · -
_
41
4'J
-I
In
u
N
0
a*
.
_,
ru
-0
-_
_*
ro_
N_
a. C.
_Da,
-
O_
_
I
0r
I
_
00
_
M-
.
0
O
O
0
0
_
0 4I
_
_I
I
I
N
4
*_
*_
.
o
O
O
0
0
O0
I
0
I
a_
I
_
ra 0_D
u~~~~
-
-r'
a4._*_
a.
*
0 -
C
X
_0'
-
I a.
_
a
X
0 3
*
_
I
o0
E
2 -
-
E-
NJ
_
Ou
N
>4sX
a
*
518
C.15
Typical Printouts
Figure C.15a gives a
carbon
dioxide.
printout
seconds
of
CPU
time.
program
pr
for
the
infinities,
reported.
printed
two-phase
naphthalene - carbon
L1
zeroes,
of
and
equilibrium
Ml
solid-fluid
if appropriate.
segments being compared.
of CPU time.
for
was
1.4
C.15b gives a printout from
dioxide
zeroes
system.
are
first
equilibrium
are
Finally, the compositions of
coexisting fluid phases are given.
no
pure
This run required about
Figure
Compositions
next,
program
Molar volumes and the vapor pressure at the
given temperature are reported.
Slope
from
"1 phase" signifies
found
between
that
the
two f-x
This run required about 43
seconds
519
fi hr ;ter
Ir-PJ
t Tow Ti
' h, lowr
Tste1F
250,250,l,0,50,10
T=
.25000000000000C00t'l03
v-
.104739654974425603r+01
v=
v=
.136009322667388695D+00
.41342984945296837D-01
T=
*2500000000000000000.
v=
.800238944381563493D+00
v=
.157720416125397523rf+00
v=
.410769959094028311D-01
3
P=
.205900244119081288D1+0)O
v=
.40827237519666378411-01
T=
.25000000000
v=
.405918843345159761D 01
v=
0;oooo
0 (;ooo0(ooooo
rl0-O"
.JOOOOOOOOOOOOOOOOO-i-0
.410404707797054515D+00
v=
T-=
F:
21D+02
.17422979660117
vle at P=
T=
*25000000000000000D03
v=
.100000000000000000D02
0
250000000000000000D+03
.40369413?533852214D
--
T 19:44
1.417
.300000000000000000D;02
0000000000000D+3
F
=
.5400000000000000000D+02
*50O00000000000000ori +02'
1
1
Figure C.15a. Printout from main program pure.
run is for CO 2 at 250 K and 10-50 atm.
This
520
r
Inut Tlow Thigh, Tste PPlow F';liihl
F'teP.coarse
300,l1,65,
.
rid f ire
rid
01,r .0001
T=
.300000000000000000D+03
F';:
. 650000000000000000D+02
L1 zero at x2=
.36546687332151.8302D-02
r' r=
zR zero at x.2=
.415134530320599676D-02
zR zero
at x2=
OOOOOOOOOOOOOOOOOOOD+00
L1 zero at x2=
Mh zero
L1 zero
sfeo at
sfea at
sfee at
1
.409125412086248397D-o01
at x2=
.143327527737081051D+0OO
at x2=
.300956465137958527DO00
x2=
* 143700206057755996D 03
x<2=
.832927592691383789rD-03
X2=
.296782347844118646D-02
3
3
35
r=
lir =
irl i':=
1
3
1 phase
1 phase
x =
=
.256201805869170037D--03
*412069181002753919D-03
x =
1
1
1
1
1
*313266558200543859D-03
=
=
.701273947117438119D-02
.846595315247294190D-o01
-
.47129856894355734ID+00
Phase
Phase
Phase
phase
phase
.181178579638753206D-02
>x =
V=
.152479788766619985D-02
.'4313747018272560Di- 0
·
..466622118852607962D
00O
.*4733340230171
8D*,To00o
1 Phase
1 phase
=
.1012747846462524D-01
r 20:36 42.905 24
=
Figure C.15b. Printout from main program pr.
run is for C1 0H8 - CO2 at 300 K and 65 atm.
This
521
ADendix
Notation
A - Helmholtz free energy, parameter in cubic equation of
state.
B - Parameter in cubic equation of state.
CEP - Critical end point.
CP - Critical point.
F - Fluid.
G - Gas (or vapor); Gibbs free energy.
H - Enthalpy; Hexadecane.
L - Liquid.
L1 - Spinodal (first criticality criterion) determinant.
LCEP - Lower critical end point.
LP - Limit point.
M1 - Critical (second criticality criterion) determinant.
N - Mole number; Naphthalene.
P - Pressure.
Q - Quadruple point.
R - Gas constant.
S - Solid; entropy.
T - Temperature.
TP - Triple point.
U - Internal Energy.
UCEP - Upper critical end point.
V - Vapor (or gas).
Z - Compressibility factor.
522
a,b - Parameters in cubic equation of state.
f - Fugacity.
v - Molar volume.
x - Mole fraction.
Subscrits
c - Critical property.
i,
j, 1,
2,
- Component
indices.
r - Reduced property.
Superscripts
c - Crystalline.
g - Gas.
- Reference state.
I, II - Phase or segment indices.
Greek Letters
6 - interaction parameter
w - acentric factor
p - chemical potential
An underbar denotes a total property.
523
References
Alwani, Z.
and
G.M. Schneider.
"Phasengleichgewichte,
kritische Erscheinungen und PVT - Daten in binaren
Mischungen von Wasser mit aromtischen Kohlenwasserstoffen
bis 420UC und 2200 bar." Be.
Bunsenges., 73, 294
(1969).
Ambrose, D., I.J. Lawrenson and C.H.S. Sprake.
"The Vapour
Pressure of Naphthalene." I.
Chem. Thermo., 1, 1173
(1975).
Baker, L.E., A.C. Pierce and K.D. Luks.
"Gibbs
Energy
Analysis of Phase Equilibrium." Paper presented at the
SPE/DOE Second Joint Symposium on Enhanced Oil Recovery
of the Society of Petroleum Engineers, Tulsa, Oklahoma,
April 5-8, 1981.
Beegle, B.L., M. Modell and R.C. Reid. "Legendre Transforms
and Their Application in Thermodynamics." AIChE J., 20,
1194 (1974a).
"Thermodynamic
Stability
Criterion
for
Pure
Substances and Mixtures." AIChE
., 20, 1200 (1974b).
Benedict, M., G.B. Webb and L.C. Rubin.
"An
Empirical
Equation
for
Thermodynamic
Properties
of
Light
Hydrocarbons and Their Mixtures." J.
Chem.
PRhv,
8,
334 (1940);
.
Che.
Phs., l, 747 (1942); . Chem
Phs.., 47, 419 (1951).
Beris, A.
personal communication (1981).
Brush, S.G. "Development of the Kinetic Theory of Gases. V.
The Equation of State." Amer. J. Phys., 29, 593 (1961).
Bichner, E.H. "Flussige Kohlensaure als
Phy.
Chem., 54, 665 (1906a).
Losungsmittel."
Z.
"Die beschrankte Mischbarkeit von Flussigkeiten das
System Diphenylamine und Kohlensaure."
. Phys.
Che.,
56, 257
Calphad.
(1906b).
Pergamon Press, New York, 1981.
Campbell. A.N. and R.M. Chatterjee.
"Orthobaric Data of
Certain Pure Liquids in the Neighborhood of the Critical
Point." Can. J.
Chem., A46,575 (1967).
Chueh, P.L. and J.M. Prausnitz. "Vapor-Liquid Equilibria at
High Pressures.
Vapor-Phase Fugacity Coefficients
in
Nonpolar and Quantum-Gas Mixtures." Ind. Eng. Chem.
524
Fund., 6, 492 (1967).
Chueh, P.L. and J.M. Prausnitz. "Vapor-Liquid Equilibria at
High Pressures: Calculation of Partial Molar Volumes in
Nonpolar Liquid Mixtures." AIChE J., 13, 1099 (1967).
Chueh, P.L. and J.M. Prausnitz. "A Generalized Correlation
for the Compressibilities of Normal Liquids." AIChE J.,
15,
471
(1969).
Connolly, J.F. "Solubility of Hydrocarbons in Water Near the
Data,
Ens.
Chem.
Critical Solution Temperature." J.
11, 13 (1966).
Diepen, G.A.M. and F.E.C. Scheffer. "On Critical Phenomena
of Saturated Solutions in Binary Systems." . Am. Chem.
Soc., 70, 4081 (1948a).
"The Solubility of
Ethylene." i. Am. Chem.
Naphthalene in Supercritical
Sc., 70, 4085 (1948b).
"The Solubility of
Ethylene - II." J. Phys.
Naphthalene in Supercritical
Chem., 57, 575 (1953).
Engineerin g
Chemical
B.F.
Dodge,
McGraw-Hill, New York, 1944.
Thermodynamics.
"Phase Equilibria for
and J.M. Praunitz.
Eckert, C.A.
Strongly Nonideal Liquid Mixtures at Low Temperatures."
AIChE J., 2i, 462 (1980).
Egloff, G., Physical Constants
Reihold, New York, 1939.
Qf
Hydrocarbons
Vol
I.
Francis, A.W. "Ternary Systems of Liquid Carbon Dioxide." J.
Phys. Chem., 58, 1099 (1954).
"Solid Solubilities in
Francis, D.C. and M.E. Paulaitis.
Supercritical Fluids - High Pressure Phase Equilibrium
Calculations Near the Upper Critical End Point Using the
Lee-Kesler Equation of State." submitted for publication
(1981).
"Vapor Pressure
13, 209 (1968).
Fowler, L., W.N. Trump and C.E. Vogler,
Data,
Eng.
Naphthalene." . Chem.
Gibbs,
J.W.
"On
Substances." Trans.
the
Equilibrium
Conn
ad., III,
of
of
Heterogeneous
108 (1876).
Goldman, K. "Thermodynamic Properties of Ethylene Gas." in
and its Industrial Derivatives. S.A. Miller,
Ethylene
ed.,r Ernest Benn, London, 1969.
Gomez-Nieto
M.
"Comments on: 'A New
Two-Constant
Equation
525
Eng.
of State'." Ind.
Gordon,,
P.
Sy.tems.
Gouw,
Fund,
Chem.
rinciles Qf Phase
18, 197 (1979).
Materials
Diagramsin
McGraw-Hill, New York, 1968.
T.H.
and
R.E. Jentoft.
Chromotography."J.
"Supercritical
Fluid
Chromoator., 68, 303 (1972).
"The
Displacement
of
Huang, E.T.S.
and J.H. Tracht.
Residual Oil by Carbon Dioxide." SPE Paper 4735, Improved
Oil Recovery Symposium, Tulsa, Oklahoma, April 22-24,
1974.
Hubert, P. and O.G. Vitzthum. "Fluid Extraction of Hops,
Spices, and Tobacco with Supercritical Gases." Angew.
Chem. Internat. Ed., 17, 710 (1978).
"An
Analytical
K.P.
and
C.A. Eckert.
Johnston,
Carnahan-Starling-van der Waals Model for Solubility of
submitted
Hydrocarbon Solids in Supercritical Fluids."
to AIChE J. (1981).
Kammerlingh Onnes, H.
Comm.
Phys.
Lab.
Leiden.,
Eq.71
(1901).
Katz,
D.L.
and
Behavior
Qf
Phenomena.
M.L. Rzasa.
Bibliography
fLOr
Hydrocarbons Under Pressure and
Physical
Related
Edwards, Ann Arbor (1946).
Keenan, J.H., F.G. Keyes, P.G. Hill and J.G. Moore.
Tables. John Wiley and Sons, New York, 1978.
Steam
Kleintjens, L.A. and R. Koningsveld. "Lattice Gas Treatment
of Supercritical Phase Behavior in Fluid Mixtures." Paper
presented at the 157th Meeting, The Electrochemical
Society, St. Louis, May, 1980.
"Chromatography with Supercritical
Klesper, E.
Angew. Chem. Iternat.
Ed., 12, 738 (1978).
Fluids."
Kurnik, R.T. "Supercritical Fluid Extraction: A Study of
Binary and Multicomponent Solid-Fluid Equilibria." Sc.D.
Dissertation, M.I.T. (1981).
"Solubility of
Kurnik, R.T., S.J. Holla and R.C. Reid.
Solids in Supercritical Carbon Dioxide and Ethylene." i.
Chem. Eng. Data., 26, 47 (1981).
Lazalde-Crabtree, H., G.J.F. Breedveld and J.M. Prausnitz.
Solubility of
Losses
in Gas Absorption.
"Solvent
Methanol in Compressed Natural and Synthetic Gases."
26, 462 (1980).
AIChE i.,
Lee
B.
and M.
Kessler.
"A Generalized Thermodynamic
526
Correlation
Based
on
Three-Parameter
States." AChE J, 26, 462 (1980).
Lentz, H.
and E.U. Franck, Ber.Bunsenges.
Corresponding
.
Chem., 73,
28 (1969).
Levelt Sengers, J.M.H., W.L. Greer and J.V. Sengers. "Scaled
Equation of State Parameters for Gases in the Critical
Region." J.
Phys. Chem. Ref. Data., .5, 1 (1976).
Lu, B.C.
and S.S. Woods.
"A Generalized Equation
Computer Calculation of Liquid Densities." AIChE J.,
for
12,
95 (1966).
Mackay, M.E. and M.E. Paulaitis. "A Method to Calculate the
Solubility of a Heavy Component in a Supercritical
Solvent." Ind. Eng. Chem. Fund., 18, 149 (1979).
Martin, J.J.
Chem.
"Cubic Equations of State - Which?" Ind.
Fund., 1,
Eng.
81 (1979).
Martynova, O.I.
"Solubility of Inorganic Compounds
in
Subcritical
and
Supercritical
Water."
in
High
Temperature, High Pressur
Electrochemistry in Agueus
Solutions.
(NACE-4), (1976).
Mason, E.A.
and T.H. Spurling.
The Virial
State. Pergamon Press, New York, 1969.
Equation
of
McHugh, M.
and M.E. Paulaitis.
"Solid Solubilities of
Naphthalene
and
Biphenyl
in
Supercritical Carbon
Dioxide." . Chem. Enig. Data., 2, 326 (1980).
Messmore, H.E.
U.S.
- Pat.
2420185, Phillips Petroleum Co.
(1943).
Modell, M. "Criteria of Criticality." A
SmD.
Si.
e,
60, Estimation and Correlation of Phase Eguilibria and
Fluid Properties in the Chemical Industry. S.I. Sandler
and T. Storvick, ed. (1977).
"Decontamination of Dredge Spoils by ·Extraction with
Water Under Near Critical Conditions." Sea Grant Proposal
R/T-1, M.I.T. (1979).
"Wastewater Treatment with Desorbing of an Adsorbate
from an Adsorbent with a Solvent in the Near Critical
State." U.S.
Patent 4147624, Arthur D. Little, Inc.
(1979).
"Gasification and Liquefaction of Forest Products in
Supercritical Water." paper presented at
the
89th
National Meeting, AIChE, Portland, Oregon, August 20,
1980.
527
the
"The Use of
Modell, M., G.T. Hong and A. Heiba.
in Determining Supercritical
Equation
Peng-Robinson
Behavior." paper presented at the 72nd Annual Meeting,
AIChE, San Francisco, California, November 26, 1979.
and
Thermodynamics
R.C. Reid.
and
Modell, M.
Englewood Cliffs,
Prentice-Hall,
Applications.
(1974).
Jersey.
Ita
New
Modell, M., R.J. Robey, V. Krukonis, R.P. de Filippi and
Fluid Regeneration of
"Supercritical
D. Oestreich.
Activated Carbon." paper presented at the 87th National
Meeting, AIChE, Boston (1979).
Morey, G. "The Solubility of Solids in Gases." Econ.
52, 227 (1957).
GP.1.,
Najour, G.C. and A.O. King, Jr. "Solubility of Naphthalene
in Compressed Methane, Ethylene, and Carbon Dioxide.
Evidence for a Gas-Phase Complex Between Naphthalene and
Carbon Dioxide." J. gChem. Phy., 45, 1915 (1966).
"Die Gase im Magma." Centr.
Niggli, P.
Min.,
321 (1912).
H. Knapp.
and
Oellrich, L., U. Pl6cker, J.M. Prausnitz
"Methoden zur Berechnung von Phasengleichgewichten und
Enthalpien mit Hilfe von Zustandsgleichungen." Chem.-Iag.
Tech.,
49, 955
(1977).
Peng, D.Y. and D.B. Robinson. "A New Two-Constant Equation
of State." Ind. Eng. Chem. Lundam., 1L, 59 (1976a).
"Two and Three Phase Equilibrium Calculations for
Systems Containing Water." Can. J. Chem. Eng., 54, 595
(1976b).
"A Rigorous Method for Predicting the Critical
Properties of Multicomponent Systems from an Equation of
State." AIChE a., 23, 137 (1977).
Solid-Liquid-Vapor
Three-Phase
"Calculation of
Equilibrium Using an Equation of State." paper presented
Beach,
Florida,
at the 176th Meeting, ACS, Miami
September 11-14, 1978; in Advances in Chemistry Series
182., American Chemical Society, Washington, D.C., 1979.
"Two- and Three-Phase Equilibrium Calculations for
Coal Gasification and Related Processes." paper presented
at AIChE-NBS-NSF symposium, Warrenton, Virginia, October
22-25, 1979; see Thermodynamics of Agueous systems with
Industrial
Applications. ACS Sympsium Serie., i32, 393
(1980).
528
Plocker, U.J., H. Knapp and J.M. Prausnitz. "Calculation of
High-Pressure
Vapor-Liquid
Equilibria
from
a
Corresponding-States
Correlation
with
Emphasis
on
Asymmetric Mixtures. " Ind.
Eng.
Chem.
Proc. Des.
Dev., 17, 324 (1978).
Prausnitz, J.M. "Fugacities in High-Pressure Equilibria
in Rate Processes." AIChE J., 5., 3 (1959).
and
"Solubility of Solids in Dense Gases." NBS Technical
Note 316, July, 1965.
Prausnitz, J.M.,
R. Hsieh and
T.F. Anderson, E.A. Grens,
C.A. Eckert,
J.P. O'Connell. Computer Calculations for
Multicomponent
Vaor-Liauid
and Liuid-Lui
Eguilibria.
Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1980.
Quinn, E.L. "The Internal Pressure of Liquid Carbon Dioxide
from Solubility Measurements." I. Am. Chem. S.,
50,
672
(1928).
Rebert, C.J.
Solubility
J.,
and W.B. Kay.
"The Phase
Behavior
and
Relations of the Benzene-Water System." AIChE
5i,285 (1959).
Redlich, 0. and J.N.S. Kwong.
"On the Thermodynamics of
Solutions.
V.
An Equation of State. Fugacities of
Gaseous Solutions." Chem. Rev., 44, 233 (1949).
Reid, R.C. and B.L. Beegle.
"Critical Point Criteria in
Legendre Transform Notation." AIChE J., 2a, 726 (1977).
Reid, R.C., J.M. Prausnitz and T.K. Sherwood. The Properties
of Gases and Liguids. McGraw-Hill, New York, New York
(1977).
Reisman, A.
1970.
Phase Equilibria.
Academic
Press,
New
York,
Ricci, J.E. The Phase Rule and Heterogeneous
Van Nostrand, New York, 1951.
Eguilibrium.
Rowlinson, J.S. Ligidas
London, 1969.
Butterworth,
nd Liguid
Rowlinson, J.S. and M.J. Richardson.
in Compressed Gases." Adv. Chem.
Mixtures.
"Solubility of Solids
Phys.,
2, 85 (1959).
Schneider, G.M. "Phase Equilibria in Fluid Mixtures at
Pressures." Adv. Chem. Phys., 17, 1 (1970).
Schneider,
G.M.,
Z. Alwani,
W. Heim,
E.U. Franck.
'Phasengleichgewichte
Erscheinungen in binaren Michsystemen
High
E. Horvath
and
und
kritische
bis 1500 bar."
529
Chem.
Ing.
Tech.,
39, 649
(1967).
Schouten, J.A., C.A. ten Seldam and N.J. Trappeniers.
"The
Two-Component Lattice Ga
Model." Phsica.,
13, 556
(1974).
Scott, R.L.
Related
SAc.,
and P.H. van Konynenburg. "Van der Waals and
Models for Hydrocarbon Mixtures." Disc. Faraday
49, 87 (1970).
Sebastian, H.M., J.J. Simnock, H. Lin and K. Chao.
"VaporLiquid Equilibrium in Binary Mixtures of Carbon Dioxide
and n-Decane and Carbon Dioxide and n-Hexdecane." J.
Chem. Eng. Data., 25, 138 (1980).
Sie, S.T.,
W. Van
Beersum
and
G.W.A. Rijinders.
"High-Pressure Gas Chromatography and Chromatography with
Supercritical Fluids.
I.
The Effect of Pressure on
Partition Coefficients in Gas-Liquid Chromatography with
Carbon Dioxide as a Carrier Gas." Separation Sci., 1, 459
(1966).
Smits. A. "The Course of the Solubility Curve in the Region
of Critical Temperatures of Binary Mixtures I." Proc.
]at. Acad. Sci.. Amsterdam,
, 171 (1903a).
"The Course of the Solubility Curve in the Region of
Critical Temperatures of Binary Mixtures.
II." Proc.
BQ.
Acad. Sci. Amterdam,
6, 484 (1903b).
"On the Hidden Equilibria in the P-x Diagram of a
Binary System in Consequence of the Appearance of Solid
Substances." Proc. BRQ. Acad. Sci.
Amsterdam, 8,
196
(1905a).
"On the Hidden Equilibria in the P-x Sections Below
Acad. Sci. Amsaterdam,
the Eutectic Point." Proc. Roy
j, 568,
(1905b).
Constants
from
a
Soave,
G.
"Equilibrium
Redlich-Kwong Equation of State." Chem. Eng.
1197 (1972).
Modified
Sci., 27,
Stalkup, F.I.
"Carbon Dioxide Miscible Flooding: Past,
et.
Tech.,
Present, and Outlook for the Future." J.
Aug. 1978, 1102.
Starling, K.E.
Fluid Thermodynamic Properties
of Light
Petroleum Systems. Gulf Publishing Co., Houston, 1973.
Timmermans, J.
Compounds.
Trappeniers,
Physico-hemical Constants af Pure Organic
Elsevier, Vol. I, 1950; Vol. II, 1965.
N.J.,
J.A. Schouten
and
C.A. ten
Sel dam.
530
"Gas-Gas
and
Equilibrium
the Two-Component Lattice-Gas
.
Letters.,
Model." Chem. Phv.
5,
541 (1970).
E.V. Mushkina.
and
M.B. Iomtev
Y.V.,
Tsekhanskaya,
"Solubility of Diphenylamine and Naphthalene in Carbon
Chem., 36, 117
Phys.
Dioxide Under Pressure." R ss.
(1962).
"Solubility of Naphthalene in
Dioxide Under Pressure." Russ.
1173 (1964).
Ethylene and Carbon
J. Phys. Chem., 38,
and
E.V. Mushkina.
N.G. Roginskaya
Tsekhanskaya, Y.V.,
"Volume Changes in Naphthalene Solutions in Compressed
.
Chem., 40., 1152
Phy.
Carbon Dioxide." Russ.
(1966).
Mutual
"Limited
and V.Y. Maslennikova.
Tsiklis, D.A.
Solubility of the Gases in the Water-Nitrogen System."
Akad. Nauk. SSSR Proc., Phys. Chem.
Sec., 1£1, 262
(1965).
A.W. Sigthoff,
van der Waals, J.D. Thesis.
ed.); Barth, Leipzig (German ed.), 1873.
"Molekulartheorie
eines
K6rpers,
verschiedenen Stoffen besteht." z.
PhYv.
Leiden
der
(Dutch
zwei
aus
Chem.,
5r
133
(1890).
"On
Van Gunst, C.A., F.E.C. Scheffer and G.A.M. Diepen.
Critical Phenomena of Saturated Solutions in Binary
Chem., 57, 578 (1953).
Systems. II." Ji. Phs.
Van Konynenburg, P.H. "Critical Lines and Phase Equilibria
Dissertation, U.C.L.A.,
Mixtures." PhD.
Binary
in
(1968).
Van Welie, G.S.A. and G.A.M. Diepen. "The P-T-x Space Model
Trav.
(III)." Rec.
of the System Ethylene-Naphthalene
Chim., 80, 673 (1961).
Vargaftik, N.B. TableA
Liguids and Gases.
f
n the Thermophysical Properties
John Wiley and Sons, New York, 1975.
Thermohvysical
and
V.V. Altunin.
M.P.
Vukalovich,
Collets, London, 1968.
Properties of Carbon DiZxid.
A
Gases
"Extraction with Supercritical
Wilker G.
Chem Internat. Ed., 17, 701 (1978).
Foreword." Angew.
"Solvent Extraction
Wise, W.S.
(London) , 950 (1970).
Woerner,
G.A.
"Thermal
of
Decomposition
Coal."
and
Chem.
Reforming
Ind.
of
531
Glucose and Wood at the Critical Conditions of Water."
M.S. thesis, M.I.T., August, 1976.
and S.S. Woods, "A Generalized Equation
Yen, L.C.
Computer Calculation of Liquid Densities." AIChE .
for
12,
95 (1966).
Zhuse, T.P.
and A.A. Kapelyushnikov,
USSR
-
Pat.
113325
(1955).
Zhuse, T.P., G.N. Yushkevic and
Prom.,
Zosel, K.
J.E. Gekker.
Mas b .-Zhi .
24, 34 (1958).
DBP 2005293, Studiengesellschaft Kohle (1970).
Zudkevitch, D. and J. Joffe. "Correlation and Prediction of
Vapor-Liquid Equilibria with the Redlich-Kwong Equation
of State." AIChE J., i, 112 (1970).
532
Bioarahical Sketch
Glenn Thomas Hong was born on March
Meadow,
thereafter
He
a
Bachelor
of
Science
East
York,
in
June
of
the State University of New
attended
York at Albany, as a National Merit
with
in
He graduated as salutatorian
Long Island, New York.
from Freeport High School, Freeport, New
1972.
1954
28,
Scholar,
degree
and
graduated
in biology, summa cum
laude, in May of 1976.
Mr.
Hong entered the chemical engineering department at
MIT in September of 1976.
Sullivan
on
July
He was married to
31, 1977.
Institute of Chemical
Karen
He is a member of the American
Engineers,
Sigma
Xi,
and
Kappa, and is the coauthor of one technical paper.
graduation, Mr.
Lorraine
Phi
Beta
Following
Hong will continue his work on supercritical
water at Modar, Inc. in Natick, Massachusetts.