Construction of Global
Phase Equilibrium Diagrams
Martín Cismondi
Universidad Nacional de Córdoba - CONICET
Introduction
• A real binary system show one of 5 (or 6)
different types of phase behaviour.
• EOS modelling leads to the same possible
types (+ other) .
• Correspondence between real and predicted
type depends on the model and parameters.
Type I: Unique Critical Line (LV)
120
100
A
Pressure [Bar]
80
60
40
Pv1
Pv2
20
0
220
270
320
Temperature [K]
370
420
Type II: Also a LL critical line and LLV
250
Pressure [Bar]
200
150
A
100
B
Ps1
50
Ps2
LLV
0
200
250
300
350
400
Temperature [K]
450
500
550
Type IV: Discontinuity in the LV
critical line and second LLV region
400
350
E
300
UCEP
Pressure [Bar]
250
LLV
D
200
LCEP
150
E
B
100
D
LLV
50
LLV
0
200
250
300
350
400
Temperature [K]
450
500
550
Type IV: T-x projection
520
470
Temperature [K]
420
E
370
UCEP
320
LCEP B
270
D
UCEP
220
0.4
0.5
0.6
0.7
Composition
0.8
0.9
1
Type III: “rearrangement” of critical lines
400
350
300
Pressure [Bar]
250
200
C
150
100
D
UCEP
50
0
200
250
300
350
400
Temperature [K]
450
500
550
Type III: T-x projection
520
470
Temperature [K]
420
C
370
UCEP
320
D
270
220
0.4
0.5
0.6
0.7
Composition
0.8
0.9
1
Type V: Just like type IV but
without LL immiscibility at low T
300
250
E
150
70
65
60
100
55
Pressure [Bar]
Pressure [Bar]
200
E
50
D
UCEP
45
D
50
LLV
40
35
30
180
LCEP
182
184
186
188
190
192
194
196
198
Temperature [K]
0
120
170
220
270
320
Temperature [K]
370
420
470
Azeotropic lines and…
Azeotropic End Points (AEP)
• PAEP (Pure, meeting a vapour pressure line)
• CAEP (Critical, meeting a critical line)
• HAEP (Heterogeneous, meeting a LLV line)
One example of azeotropic line (P-T)
100
80
Pressure [Bar]
CAEP
60
40
HAEP
20
0
200
220
240
260
280
300
Temperature [K]
320
340
360
380
The same example in T-x
400
350
Temperature [K]
CAEP
300
250
HAEP
200
150
0
0.1
0.2
0.3
0.4
0.5
0.6
Molar Fraction of CO2 DNN
0.7
0.8
0.9
1
Cases with two azeotropic lines!
System: Ethanol - n-Hexane
520
C
470
Temperature [K]
420
370
320
H
270
H
220
P
170
120
0
0.1
0.2
0.3
0.4
0.5
0.6
Molar Fraction of Ethanol
0.7
0.8
0.9
1
Objectives
• Identification of predicted type
• Automated calculation of global phase
equilibrium diagrams
• Automated calculation of Pxy, Txy and
isoplethic diagrams from limiting points
What do we need?
• Strategy for construction of a GPED
without knowing the type in advance.
• General method for CRIT lines calculation.
• Location of isolated LL critical lines.
• General methods for LLV and AZE lines.
• Detection of CEP’s and AEP’s (critical and
azeotropic end points).
• Classifications of Pxy, Txy and isoplethic
diagrams in terms of limiting points.
• Methods for calculation of Pxy, Txy and
isoplethic segments.
Critical line from C2 to…
Algorithm:
Basic
Structure
A
E
C
C1
LCEP
High Pressure
type I or II
type IV or V
type III
Search for a high
pressure critical point
found
not
found
Critical Line B
until UCEP
type II or IV
type I or V
Critical line D
from CP1 to UCEP
Some remarks about the methods…
• Formulation in T, v and x, y, w…
• Solve using Newton
J ΔX = -F ; Fn= XS - S
• Michelsen’s procedure for tracing lines
J (dX/dS) = (dF/dS) → Xnew= Xold + (dX/dS) ΔS
• ΔSnew = min (4 ΔSold / Niter , ΔSmax)
• The variable to be specified depends on dX/dS
Calculation of critical points:
Criticality conditions
  ln fˆi 

Bij  zi z j 
 n 
j T ,V

tpd2=0
tpd3=0
b = smallest eigenvalue λ1=0
c=
1 s     1 s   
 1 

 

s
2

 s 0
n1 = z1 + s z1 u1; n2 = z2 + s z 2 u2
=0
u12  u 22  1
How to locate an
isolated LL critical line?
  ln ˆ1 

  1  
 n2  T , P
P = 2000 bar
P = 2000 bar
300 K

Tc
Temperature
Must be 0 and min at (T, P)
T
200 K
0
Xc
Composition
Composition
Xc
LLV equilibrium and CEP’s
Use of stability analysis in the search
for a Critical End Point (CEP)
0.10
Reduced tangent plane distance (tpd) curves
at four consecutive critical points
at conditions close to an UCEP
0.08
0.06
0.04
tpd
0.02
0.00
-0.02
-0.04
0.0
0.2
0.4
0.6
Molar fraction of component 1
0.8
1.0
Calculation of a Critical End Point
Calculation of LLV lines
Examples: type II
320
200
300
180
CH4 + CO2
280
160
140
B
SRK EOS
260
kij = 0.120
240
Temperature (K)
Pressure (bar)
120
100
Critical lines
LLVE lines
Vapour pressure
80
60
A
A
220
200
B
180
160
40
140
20
0
100
Critical lines
LLVE lines
120
100
120
140
160
180
200
220
240
Temperature (K)
260
280
300
320
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
CH4 Molar Fraction
minimum composition
0.8
0.9
1.0
Split of LV critical line in type IV or V
100
E
80
Critical lines
LLVE lines
Vapour pressure
D
LCEP
UCEP
40
Pressure (bar)
Critical lines
LLVE lines
220
unstable
critical line
20
Temperature (K)
60
240
0
CH4 + C6H14
-20
E
UCEP
200
180
LCEP
unstable
critical
line
SRK EOS
-40
160
kij = 0.00
-60
-80
140
150
160
170
180
190
Temperature (K)
200
210
220
140
0.90
0.92
0.94
0.96
CO2 Molar Fraction
0.98
1.00
440
600
Transition
100
500
kij = 0.090
420
C
95
C
90
400
85
80
C
75
300
70
300
200
305
310
315
320
C
Temperature (K)
Pressure (bar)
III
380
D
400
360
340
C
320
D
300
280
kij = 0.090
100
260
0
240
200
300
400
500
600
0.85
700
0.90
0.95
1.00
CO 2 Molar Fraction
Temperature (K)
440
600
100
E
95
500
420
90
kij = 0.084
E
400
85
380
B
B
D
80
75
300
70
300
200
305
310
315
320
E
Temperature (K)
IV
Pressure (bar)
400
360
340
E
320
D
300
B
280
kij = 0.084
100
260
0
240
200
300
400
500
600
0.85
700
0.90
0.95
1.00
CO 2 Molar Fraction
Temperature (K)
440
600
kij = 0.078
420
Critical lines
LLVE lines
Vapour pressure
500
Critical lines
LLVE lines
400
380
B
Temperature (K)
II
Pressure (bar)
400
300
A
200
360
340
A
320
300
B
280
kij = 0.078
100
260
0
240
200
300
400
500
Temperature (K)
600
700
0.85
0.90
0.95
CO 2 Molar Fraction
1.00
Our Classification for Adding Azeotropy
Line (a)
0 to P
0 to C
P to P
P to C
H to P
H to C
C to C
P to H
H to P
P to H
H to C
Azeotropy (b)
P, N or D
N
P, N or D
P, N or D
P
P
P or D
P
Usual Types
I, II, V
V
I, II, V
I
II, IV
II, IV
I, II
II
P
II
Detection of AEP’s
• PAEP: compute ln ˆiL ( zi  0)  ln ˆiV ( zi  0)
along each vapour pressure line
• CAEP: Pseudocritical point
 2P 
 P 
 2      0
 v  z ,T  v  z ,T
compute 1st derivative along the LV critical line
• HAEP: crossing between L and V composition
compute y1 – x1 along LLV line
Calculation of azeotropic lines:
variables and equations
Illustration: Negative Azeotropy
80
70
60
Pressure [Bar]
50
40
30
20
10
0
140
160
180
200
220
240
Temperature [K]
260
280
300
320
Double Azeotropy:
Minimum T in the azeotropic line
320
Temperature [K]
270
220
170
120
70
0
0.1
0.2
0.3
0.4
0.5
Molar Fraction of CO2
0.6
0.7
0.8
0.9
1
30
PAEP
25
Pressure [Bar]
20
15
Bancroft point
(Pv1 = Pv2)
10
PAEP
5
0
190
Tmin
200
210
220
230
Temperature [K]
240
250
260
270
Automated construction of complete
Pxy and Txy diagrams
• Reading and storing the lines and points of the Global
Phase Equilibrium Diagram. Identification of type.
• Detection of local temperature and pressure minima
or maxima in critical lines.
• Determination of the pressures (or temperatures) at
which the different lines intersect at the specified
temperature (or pressure).
• Deduction, from the points obtained, of how many
and which zones there will be.
• Calculation of each zone or two-phase region.
(NVP=2, NC=2, NLLV=1)
Pressure
T specified
Composition
Translating from limiting points to diagrams
variables and equations…
160
500
140
450
400
100
Temperature (K)
Pressure (bar)
120
50 bar
300 K
80
60
350
300
250
40
200
20
0
150
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.6
0.8
1.0
0.8
1.0
Ethane molar fraction
Ethane molar fraction
160
500
140
450
120
400
100
Temperature (K)
Pressure (bar)
0.4
330 K
80
60
350
300
120 bar
250
40
200
20
0
150
0.0
0.2
0.4
0.6
Ethane molar fraction
0.8
1.0
0.0
0.2
0.4
0.6
Ethane molar fraction
Examples: Closed loops in Pxy diagrams
160
900
800
CO2 + n-Docosane
700
RK-PR EOS (kij=0.10)
140
120
20
600
Pressure (bar)
Pressure (bar)
30
500
400
150
300
100
80
10
0.0
0.1
0.2
0.3
774 K
776 K
778 K
780 K
60
100
40
200
50
CO2 + n-Docosane
20
100
RK-PR EOS (kij=0.10)
0
770
0
200
775
780
785
790
0
300
400
500
600
Temperature (K)
700
800
0.0
0.2
0.4
0.6
CO2 Molar Fraction
0.8
1.0
Generation of Complete Isopleths
•
•
•
•
Detection of composition local minima or
maxima in critical lines, as well as in vapour
or liquid branches of LLV lines.
Location of intersection points at specified
composition.
Deduction of the number and nature of the
segments the isopleth will be constituted of.
Calculation of each segment of the isopleth.
Location of intersection points
500
Ethane + Methanol
C
z=0.45
L1
V
300
L
C
z=0.71
200
L2
L
Critical lines
LLVE lines
z=0.94
Temperature (K)
400
z=0.97
RK-PR EOS kij = 0.02
lij = 0.20
100
0.3
0.4
0.5
0.6
0.7
Ethane Molar Fraction
0.8
0.9
1.0
NV
NL
NCRI
Number and nature of segments
Phase
Behav.
Type
0
0
0
1
1
1
III
( C | LTDP)y ( C | HPLP)x
II/III/IV ( C | LTDP)y ( L | C)x ( L | HPLP)x/y
0
2
0
III/IV
( L1 | LTDP)y ( L2 | L1)x ( L2 | HPLP)y
1
1
1
II/III
(V | LTDP)y (C | V)y (L | C)x (L | HPLP)y
Segments of the isopleth homogeneity
boundary
Portions of
LLV line
to print
All
Tmin to L
Tmin to L2
L1 to K
Tmin to V
Isopleth
Case
3
7
11
17
Calculation of each segment
 ln x1 
ln y 
2

 ln v x 
X 
ln v y 


 ln T 


ln
P





ln
F
ln



ln Px ( x, T , v x )  ln P


ln Py ( y, T , v y )  ln P

fˆ1x ( x, T , v x )  ln fˆ1 y ( y, T , v y )
ˆf x ( x, T , v )  ln fˆ y ( y, T , v )  0
2
x
2
y 

g phase ( X )  ln z i

g spec ( X )  S

Numerical continuation method
• Sensitivities are used to
– Choose which variable to specify for next point
– Estimate values for all variables
Global Diagram: P-T projection
200
Ethane + Methanol
180
Critical lines
LLVE lines
Vapour pressure
160
RK-PR EOS kij = 0.02
lij = 0.20
Pressure (bar)
140
C
120
100
80
60
D
40
20
0
200
250
300
350
400
Temperature (K)
450
500
Global Diagram: T-x projection
500
Ethane + Methanol
C
z=0.45
L1
V
300
L
C
z=0.71
200
L2
L
Critical lines
LLVE lines
z=0.94
Temperature (K)
400
z=0.97
RK-PR EOS kij = 0.02
lij = 0.20
100
0.3
0.4
0.5
0.6
0.7
Ethane Molar Fraction
0.8
0.9
1.0
250
Pressure (bar)
Isopleth for z = 0.45
(case 7 in Table 1)
x/y
PLP)
(L | H
200
Ethane-Methanol. RK-PR EOS.
150
Critical
point
100
Liquid phase
50
(L
x
)
|C
LLE
LVE
LLVE
(C
y
)
DP Vapour
T
|L
phase
0
200
250
300
350
Temperature (K)
400
450
Ethane-Methanol. RK-PR EOS.
300
Pressure (bar)
250
200
( C | HPLP)x
Isopleth for z = 0.71
LLE
(case 3 in Table 1)
Critical point
150
Dense phase
100
( C | LTDP)y
LLVE
50
Vapour
phase
LVE
0
200
250
300
350
Temperature (K)
400
450
200
Ethane-Methanol. RK-PR EOS.
62
61
Isopleth for z = 0.94
150
60
(case 11 in Table 1)
Pressure (bar)
59
LLVE
58
57
100
LVE
56
320
LLE
321
322
323
324
325
326
50
Liquid phase
LLVE
0
200
250
LVE
Vapour
phase
300
350
Temperature (K)
Ethane-Methanol. RK-PR EOS.
150
58
Isopleth for z = 0.97
(case 17 in Table 1)
57
Critical point
56
Pressure (bar)
55
LIVE
LIIVE
54
100
53
LLVE
52
314
315
316
317
318
319
320
321
322
323
LLE
50
Liquid phase
Vapour
LVE
phase
0
200
250
Temperature (K)
300
Modular Approach
• One general subroutine for calculation of
P, ln fˆi and derivatives wrt T, V and n
(given T, V and n)
• Model specific subroutines for calculation
of Ar and derivatives wrt T, V and n
Conclusions
• We have provided strategies for constructing
GPED’s from scratch.
• Types I to V, with or without azeotropy.
• Pxy, Txy and Isopleths can be derived.
• Strength: based on the GPED
• Weakness: everything is based on the GPED
www.gpec.plapiqui.edu.ar
www.gpec.efn.uncor.edu
References
•
Global phase equilibrium calculations
–
–
–
•
Cismondi, M., Michelsen, M. “Global Phase Equilibrium Calculations: Critical Lines,
Critical End Points and Liquid-Liquid-Vapour Equilibrium in Binary Mixtures”. The
Journal of Supercritical Fluids, Vol. 39, 287-295. 2007.
Cismondi, M., Michelsen, M. “Automated Calculation of Complete Pxy and Txy
Diagrams for Binary Systems”. Fluid Phase Equilibria, Vol. 259, 228-234. 2007.
Cismondi, M., Michelsen, M. L., Zabaloy, M.S. “Automated generation of phase
diagrams for binary systems with azeotropic behavior”. Industrial and Engineering
Chemistry Research, Vol. 47 Issue 23, 9728–9743. 2008.
GPEC (the program)
–
•
Cismondi, M., Nuñez, D. N., Zabaloy, M. S., Brignole, E. A., Michelsen, M. L.,
Mollerup, J. M. “GPEC: A Program for Global Phase Equilibrium Calculations in Binary
Systems” (Oral Presentation). EQUIFASE 2006. Morelia, Michoacán, México. October
21-25, 2006.
Models and their pure compound parameters
–
–
Cismondi, M., Mollerup, J. “Development and Application of a Three-Parameter RK-PR
Equation of State”. Fluid Phase Equilibria, Vol. 232, 74-89. 2005.
Cismondi, M., Brignole, E. A., Mollerup, J. “Rescaling of Three-Parameter Equations
of State: PC-SAFT and SPHCT”. Fluid Phase Equilibria, Vol. 234, 108-121. 2005.