construction of global phase equilibrium diagrams martín cismondi universidad nacional de córdoba...
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Construction of Global Phase Equilibrium Diagrams
Martín CismondiUniversidad 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)
0
20
40
60
80
100
120
220 270 320 370 420
Temperature [K]
Pre
ssu
re [
Bar
]
Pv2
A
Pv1
Type II: Also a LL critical line and LLV
0
50
100
150
200
250
200 250 300 350 400 450 500 550
Temperature [K]
Pre
ssu
re [
Bar
]
BA
LLV Ps2
Ps1
Type IV: Discontinuity in the LV critical line and second LLV region
0
50
100
150
200
250
300
350
400
200 250 300 350 400 450 500 550
Temperature [K]
Pre
ssu
re [
Bar
]
81
83
85
87
89
91
93
95
316 318 320 322 324 326 328Temperature [K]
B
D
E
D
E
UCEP
LCEP
LLV
LLV
LLV
Type IV: T-x projection
220
270
320
370
420
470
520
0.4 0.5 0.6 0.7 0.8 0.9 1
Composition
Tem
per
atu
re [
K]
B D
E
UCEP
LCEP
UCEP
Type III: “rearrangement” of critical lines
0
50
100
150
200
250
300
350
400
200 250 300 350 400 450 500 550
Temperature [K]
Pre
ssu
re [
Bar
]
C
DUCEP
Type III: T-x projection
220
270
320
370
420
470
520
0.4 0.5 0.6 0.7 0.8 0.9 1
Composition
Tem
per
atu
re [
K]
C
D
UCEP
Type V: Just like type IV but without LL immiscibility at low T
0
50
100
150
200
250
300
120 170 220 270 320 370 420 470
Temperature [K]
Pre
ssu
re [
Bar
]
30
35
40
45
50
55
60
65
70
180 182 184 186 188 190 192 194 196 198
Temperature [K]
Pre
ssu
re [
Bar
]
D
E D
E
UCEP
LCEPLLV
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)
0
20
40
60
80
100
200 220 240 260 280 300 320 340 360 380
Temperature [K]
Pre
ssu
re [
Bar
]
CAEP
HAEP
The same example in T-x
150
200
250
300
350
400
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Molar Fraction of CO2 DNN
Tem
per
atu
re [
K]
CAEP
HAEP
System: Ethanol - n-Hexane
120
170
220
270
320
370
420
470
520
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Molar Fraction of Ethanol
Tem
per
atu
re [
K]
Cases with two azeotropic lines!
PH
C
H
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.
Algorithm:
Basic Structure
Critical line from C2 to…
Critical line Dfrom CP1 to UCEP
Critical Line Buntil UCEP
type I or II type III
LCEP
type I or V
not found
High PressureC1
E CA
Search for a high pressure critical point
found
type II or IV
type IV or V
Critical line from C2 to…
Critical line Dfrom CP1 to UCEP
Critical Line Buntil UCEP
type I or II type III
LCEP
type I or V
not found
High PressureC1
E CA
Search for a high pressure critical point
found
type II or IV
type IV or V
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
tpd2=0 b = smallest eigenvalue λ1=0
tpd3=0 c = = 0
211
0
1
ss
s s
VTj
ijiij n
fzzB
,
ˆln
n1 = z1 + s u1; n2 = z2 + s u2 122
21 uu1z 2z
How to locate an isolated LL critical line?
Tc
Tem
pera
ture
Xc Composition
P = 2000 bar
P = 2000 bar
0
200 K T
300 K
Composition Xc
PTn
,2
1ˆln1
Must be 0 and min at (T, P)
LLV equilibrium and CEP’s
Use of stability analysis in the search for a Critical End Point (CEP)
0.0 0.2 0.4 0.6 0.8 1.0-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
0.10
tpd
Reduced tangent plane distance (tpd) curvesat four consecutive critical pointsat conditions close to an UCEP
Molar fraction of component 1
Calculation of a Critical End Point
Calculation of LLV lines
Examples: type II
100
120
140
160
180
200
220
240
260
280
300
320
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
A
B
Critical lines LLVE lines
CH4 Molar Fraction
Tem
pera
ture
(K)
100 120 140 160 180 200 220 240 260 280 300 3200
20
40
60
80
100
120
140
160
180
200
A
B
Critical lines LLVE lines Vapour pressure
CH4 + CO
2
SRK EOS
kij = 0.120
Pre
ssur
e (b
ar)
Temperature (K)
100
120
140
160
180
200
220
240
260
280
300
320
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
A
B
Critical lines LLVE lines
CH4 Molar Fraction
Tem
pera
ture
(K)
100 120 140 160 180 200 220 240 260 280 300 3200
20
40
60
80
100
120
140
160
180
200
A
B
Critical lines LLVE lines Vapour pressure
CH4 + CO
2
SRK EOS
kij = 0.120
Pre
ssur
e (b
ar)
Temperature (K)
minimum composition
Split of LV critical line in type IV or V
140 150 160 170 180 190 200 210 220-80
-60
-40
-20
0
20
40
60
80
100
LCEPUCEP
unstablecritical line
E
D
Critical lines LLVE lines Vapour pressure
CH4 + C
6H
14
SRK EOS
kij = 0.00
Pre
ssur
e (b
ar)
Temperature (K)
140
160
180
200
220
240
0.90 0.92 0.94 0.96 0.98 1.00
unstablecriticalline
UCEP
LCEP
E
Critical lines LLVE lines
CO2 Molar Fraction
Tem
pera
ture
(K)
140 150 160 170 180 190 200 210 220-80
-60
-40
-20
0
20
40
60
80
100
LCEPUCEP
unstablecritical line
E
D
Critical lines LLVE lines Vapour pressure
CH4 + C
6H
14
SRK EOS
kij = 0.00
Pre
ssur
e (b
ar)
Temperature (K)
140
160
180
200
220
240
0.90 0.92 0.94 0.96 0.98 1.00
unstablecriticalline
UCEP
LCEP
E
Critical lines LLVE lines
CO2 Molar Fraction
Tem
pera
ture
(K)
Transition
III
IV
II
200 300 400 500 600 7000
100
200
300
400
500
600
A
B
Critical lines LLVE lines Vapour pressure
kij = 0.078
Pre
ssur
e (b
ar)
Temperature (K)
240
260
280
300
320
340
360
380
400
420
440
0.85 0.90 0.95 1.00
A
B
kij = 0.078
Critical lines LLVE lines
CO2 Molar Fraction
Te
mp
era
ture
(K
)
200 300 400 500 600 7000
100
200
300
400
500
600
300 305 310 315 320
70
75
80
85
90
95
100
E
B
kij = 0.084
Pre
ssur
e (b
ar)
Temperature (K)
D
E
240
260
280
300
320
340
360
380
400
420
440
0.85 0.90 0.95 1.00
B
E
D
E
B
kij = 0.084
CO2 Molar Fraction
Te
mp
era
ture
(K
)
200 300 400 500 600 7000
100
200
300
400
500
600
300 305 310 315 320
70
75
80
85
90
95
100
C
C
kij = 0.090
Pre
ssur
e (b
ar)
Temperature (K)
D
C
240
260
280
300
320
340
360
380
400
420
440
0.85 0.90 0.95 1.00
C
D
C
kij = 0.090
CO2 Molar Fraction
Te
mp
era
ture
(K
)
200 300 400 500 600 7000
100
200
300
400
500
600
A
B
Critical lines LLVE lines Vapour pressure
kij = 0.078
Pre
ssur
e (b
ar)
Temperature (K)
240
260
280
300
320
340
360
380
400
420
440
0.85 0.90 0.95 1.00
A
B
kij = 0.078
Critical lines LLVE lines
CO2 Molar Fraction
Te
mp
era
ture
(K
)
200 300 400 500 600 7000
100
200
300
400
500
600
A
B
Critical lines LLVE lines Vapour pressure
kij = 0.078
Pre
ssur
e (b
ar)
Temperature (K)
240
260
280
300
320
340
360
380
400
420
440
0.85 0.90 0.95 1.00
A
B
kij = 0.078
Critical lines LLVE lines
CO2 Molar Fraction
Te
mp
era
ture
(K
)
200 300 400 500 600 7000
100
200
300
400
500
600
300 305 310 315 320
70
75
80
85
90
95
100
E
B
kij = 0.084
Pre
ssur
e (b
ar)
Temperature (K)
D
E
240
260
280
300
320
340
360
380
400
420
440
0.85 0.90 0.95 1.00
B
E
D
E
B
kij = 0.084
CO2 Molar Fraction
Te
mp
era
ture
(K
)
200 300 400 500 600 7000
100
200
300
400
500
600
300 305 310 315 320
70
75
80
85
90
95
100
E
B
kij = 0.084
Pre
ssur
e (b
ar)
Temperature (K)
D
E
240
260
280
300
320
340
360
380
400
420
440
0.85 0.90 0.95 1.00
B
E
D
E
B
kij = 0.084
CO2 Molar Fraction
Te
mp
era
ture
(K
)
200 300 400 500 600 7000
100
200
300
400
500
600
300 305 310 315 320
70
75
80
85
90
95
100
C
C
kij = 0.090
Pre
ssur
e (b
ar)
Temperature (K)
D
C
240
260
280
300
320
340
360
380
400
420
440
0.85 0.90 0.95 1.00
C
D
C
kij = 0.090
CO2 Molar Fraction
Te
mp
era
ture
(K
)
200 300 400 500 600 7000
100
200
300
400
500
600
300 305 310 315 320
70
75
80
85
90
95
100
C
C
kij = 0.090
Pre
ssur
e (b
ar)
Temperature (K)
D
C
240
260
280
300
320
340
360
380
400
420
440
0.85 0.90 0.95 1.00
C
D
C
kij = 0.090
CO2 Molar Fraction
Te
mp
era
ture
(K
)
Our Classification for Adding Azeotropy
Line (a) Azeotropy (b) Usual Types 0 to P P, N or D I, II, V 0 to C N V P to P P, N or D I, II, V P to C P, N or D I H to P P II, IV H to C P II, IV C to C P or D I, II P to H H to P
P II
P to H H to C
P II
Detection of AEP’s
• PAEP: compute
along each vapour pressure line
• CAEP: Pseudocritical point compute 1st
derivative along the LV critical line
• HAEP: crossing between L and V composition
compute y1 – x1 along LLV line
)0(ˆln)0(ˆln iVii
Li zz
0,,
2
2
TzTzv
P
v
P
Calculation of azeotropic lines: variables and equations
Illustration: Negative Azeotropy
0
10
20
30
40
50
60
70
80
140 160 180 200 220 240 260 280 300 320
Temperature [K]
Pre
ssu
re [
Bar
]
Double Azeotropy: Minimum T in the azeotropic line
70
120
170
220
270
320
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Molar Fraction of CO2
Tem
per
atu
re [
K]
0
5
10
15
20
25
30
190 200 210 220 230 240 250 260 270
Temperature [K]
Pre
ssur
e [B
ar]
PAEP
PAEP
Tmin
Bancroft point
(Pv1 = Pv2)
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.
T specified (NVP=2, NC=2, NLLV=1)
Composition
Pre
ssur
e
Translating from limiting points to diagrams
variables and equations…
0
20
40
60
80
100
120
140
160
0.0 0.2 0.4 0.6 0.8 1.0
300 K
Ethane molar fraction
Pre
ssur
e (b
ar)
150
200
250
300
350
400
450
500
0.0 0.2 0.4 0.6 0.8 1.0
50 bar
Ethane molar fraction
Tem
pera
ture
(K
)
0
20
40
60
80
100
120
140
160
0.0 0.2 0.4 0.6 0.8 1.0
330 K
Ethane molar fraction
Pre
ssu
re (
bar)
150
200
250
300
350
400
450
500
0.0 0.2 0.4 0.6 0.8 1.0
120 bar
Ethane molar fraction
Tem
pera
ture
(K
)
0
20
40
60
80
100
120
140
160
0.0 0.2 0.4 0.6 0.8 1.0
300 K
Ethane molar fraction
Pre
ssur
e (b
ar)
150
200
250
300
350
400
450
500
0.0 0.2 0.4 0.6 0.8 1.0
50 bar
Ethane molar fraction
Tem
pera
ture
(K
)
0
20
40
60
80
100
120
140
160
0.0 0.2 0.4 0.6 0.8 1.0
330 K
Ethane molar fraction
Pre
ssu
re (
bar)
150
200
250
300
350
400
450
500
0.0 0.2 0.4 0.6 0.8 1.0
120 bar
Ethane molar fraction
Tem
pera
ture
(K
)
Examples: Closed loops in Pxy diagrams
200 300 400 500 600 700 8000
100
200
300
400
500
600
700
800
900
770 775 780 785 7900
50
100
150
CO2 + n-Docosane
RK-PR EOS (kij=0.10)
Pre
ssu
re (
ba
r)
Temperature (K)
0
20
40
60
80
100
120
140
160
0.0 0.2 0.4 0.6 0.8 1.0
10
20
30
0.0 0.1 0.2 0.3
CO2 + n-Docosane
RK-PR EOS (kij=0.10)
CO2 Molar Fraction
Pre
ssur
e (b
ar)
774 K 776 K 778 K 780 K
200 300 400 500 600 700 8000
100
200
300
400
500
600
700
800
900
770 775 780 785 7900
50
100
150
CO2 + n-Docosane
RK-PR EOS (kij=0.10)
Pre
ssu
re (
ba
r)
Temperature (K)
0
20
40
60
80
100
120
140
160
0.0 0.2 0.4 0.6 0.8 1.0
10
20
30
0.0 0.1 0.2 0.3
CO2 + n-Docosane
RK-PR EOS (kij=0.10)
CO2 Molar Fraction
Pre
ssur
e (b
ar)
774 K 776 K 778 K 780 K
• 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.
Generation of Complete Isopleths
Location of intersection points
100
200
300
400
500
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Ethane + Methanol
RK-PR EOS kij = 0.02
lij = 0.20
Critical lines LLVE lines
Ethane Molar Fraction
Te
mp
era
ture
(K
)
z=0.
45
z=0.
71
z=0.
94
z=0.
97C
L
C
L2
L1
L
V
100
200
300
400
500
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Ethane + Methanol
RK-PR EOS kij = 0.02
lij = 0.20
Critical lines LLVE lines
Ethane Molar Fraction
Te
mp
era
ture
(K
)
z=0.
45
z=0.
71
z=0.
94
z=0.
97C
L
C
L2
L1
L
V
Number and nature of segments
NV
NL
NC
RI Phase
Behav. Type
Segments of the isopleth homogeneity boundary
Portions of LLV line to print
Isopleth Case
0 0 1 III ( C | LTDP)y ( C | HPLP)x All 3 0 1 1 II/III/IV ( C | LTDP)y ( L | C)x ( L | HPLP)x/y Tmin to L 7
0 2 0 III/IV ( L1 | LTDP)y ( L2 | L1)x ( L2 | HPLP)y
Tmin to L2 L1 to K
11
1 1 1 II/III (V | LTDP)y (C | V)y (L | C)x (L | HPLP)y Tmin to V 17
Calculation of each segment
Numerical continuation method• Sensitivities are used to
– Choose which variable to specify for next point– Estimate values for all variables
P
T
vv
y
x
Xy
x
ln
ln
lnln
ln
ln
2
1
0
)(
ln)(),,(ˆln),,(ˆln
),,(ˆln),,(ˆln
ln),,(ln
ln),,(ln
22
11
SXg
zXgvTyfvTxf
vTyfvTxf
PvTyP
PvTxP
F
spec
iphase
yy
xx
yy
xx
yy
xx
Global Diagram: P-T projection
200 250 300 350 400 450 5000
20
40
60
80
100
120
140
160
180
200
Ethane + Methanol
D
C
Critical lines LLVE lines Vapour pressure
RK-PR EOS kij = 0.02
lij = 0.20
Pre
ssur
e (
bar
)
Temperature (K)
100
200
300
400
500
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Ethane + Methanol
RK-PR EOS kij = 0.02
lij = 0.20
Critical lines LLVE lines
Ethane Molar Fraction
Te
mp
era
ture
(K
)
z=0.
45
z=0.
71
z=0.
94
z=0.
97C
L
C
L2
L1
L
V
100
200
300
400
500
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Ethane + Methanol
RK-PR EOS kij = 0.02
lij = 0.20
Critical lines LLVE lines
Ethane Molar Fraction
Te
mp
era
ture
(K
)
z=0.
45
z=0.
71
z=0.
94
z=0.
97C
L
C
L2
L1
L
V
Global Diagram: T-x projection
200 250 300 350 400 4500
50
100
150
200
250
Critical
point
Isopleth for z = 0.45(case 7 in Table 1)
LLVE
LLE LVE
Vapour
phase
Liquid phase
P
ress
ure
(bar)
Temperature (K)
200 250 300 350 400 4500
50
100
150
200
250
Critical
point
Isopleth for z = 0.45(case 7 in Table 1)
LLVE
LLE LVE
Vapour
phase
Liquid phase
P
ress
ure
(bar)
Temperature (K)
Ethane-Methanol. RK-PR EOS.
200 250 300 350 400 4500
50
100
150
200
250
300
Critical point
LLVE
LLE
LVEVapour
phase
Dense phase
Pre
ssure
(bar)
Temperature (K)
Isopleth for z = 0.71(case 3 in Table 1)
( C | LTDP)y
( C | HPLP)x
200 250 300 350 400 4500
50
100
150
200
250
300
Critical point
LLVE
LLE
LVEVapour
phase
Dense phase
Pre
ssure
(bar)
Temperature (K)
Isopleth for z = 0.71(case 3 in Table 1)
( C | LTDP)y
( C | HPLP)x
Ethane-Methanol. RK-PR EOS.
200 250 300 3500
50
100
150
200
320 321 322 323 324 325 32656
57
58
59
60
61
62
LLE
LVE Vapour
phase
Liquid phase
P
ress
ure
(bar)
Temperature (K)
Isopleth for z = 0.94(case 11 in Table 1)
LVE
LLVE
LLVE
LLE
LVE Vapour
phase
Liquid phase
Ethane-Methanol. RK-PR EOS.
200 250 3000
50
100
150
314 315 316 317 318 319 320 321 322 323
52
53
54
55
56
57
58
LLE
P
ress
ure
(bar)
Temperature (K)
Isopleth for z = 0.97(case 17 in Table 1) Critical point
LIIVE LIVE
LLVE
LLE
LVE Vapour
phase
Liquid phase
Ethane-Methanol. RK-PR EOS.
Modular Approach
• One general subroutine for calculation of P, 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
if̂ln
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.arwww.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.