Dynamic Modeling and Optimization
of Large-Scale Cryogenic Processes
Maria Soledad Diaz
Planta Piloto de Ingeniería Química
Universidad Nacional del Sur -CONICET
Bahía Blanca, ARGENTINA
Outline
 Objective
 Natural Gas Processing Plants
 Methodology
 Mathematical Models
 Discussion of Results
 Conclusions and Current Work
2
Objective
 Dynamic optimization model for natural gas processing plant
units
•
•
•
•
•
•
Dynamic energy and mass balances
Phase equilibrium calculations with cubic equation of state
Hydraulic correlations
Phase change in countercurrent heat exchanger
Carbon dioxide precipitation in column
Boiler dynamic optimization for controllers parameters
 Simultaneous dynamic optimization approach
• Discretization of state and control variables
• Resolution of large-scale Nonlinear Programming problem
 Analysis of main variables temporal and spatial profiles
3
Natural Gas Processing Plants
 Provide ethane as raw material for olefin plants (ethylene)
and petrochemical
 Technology: Turboexpansion
 High pressure
 Cryogenic conditions
4
NATURAL GAS PLANT
D
E
H
Y
D
R
A
T
I
O
N
Ethane To
Ethylene Plant
Ethane
Treatment
CO2
TRAIN A
TRAIN B
C2
C3
C4
Propane
Compression/
Recompression
D
E
H
Y
D
R
A
T
I
O
N
TRAIN C
Butane
Separation
Fractionation
Gasoline
To Recompression
TRAIN A
C3
C4
C3
C4
TRAIN B
Pipelines
Absorption Plant
5
Natural gas composition
Component
Feed A
Feed B
Nitrogen
1.44
1.37
Carbon dioxide
0.65
1.30
Methane
90.43
89.40
Ethane
4.61
4.43
Propane
1.76
2.04
Butanes
0.77
0.96
Pentanes+
0.34
0.50
6
Cryogenic Sector
Turboexpander
Demethanizer
Cryogenic Heat
Exchangers
High
Pressure
Separator
to deethanizer
7 column
Previous Work
 Operating conditions optimization in natural gas
processing plants (NLP) (Diaz, Serrani, Be Deistegui, Brignole, 1995)
 Automatic design and debottlenecking of ethane extraction
plants (MINLP) (Diaz, Serrani, Bandoni Brignole, 1997)
 Thermodynamic model effect on the design and
optimization of natural gas plants (NLP) (Diaz, Zabaloy,
Brignole, 1999)
 Flexibility of natural gas processing plants - Dual mode
operation (Bilevel NLP) (Diaz, Bandoni, Brignole, 2002)
8
Dynamic Optimization Problem
General Formulation
min (z(tf),y(tf), u(tf), tf , p)
z(t),y(t), u(t), tf , p
st
F(dz(t)/dt ,(z(t), y(t), u(t), p, t)=0
G(z(t) , y(t), u(t), p, t) = 0
zO = z(0)
z(t)  [zl, zu]
u(t)  [ul, uu]
t, time
z, differential variables
y, algebraic variables
, y(t)  [xl, xu]
, p  [pl, pu]
tf, final time
u, control variables
p, time independent parameters
9
Dynamic Optimization Strategy
Simultaneous Approach
Cervantes, Biegler (1998)
Nonlinear DAE optimization problem
Discretization of Control and State variables
Collocation on finite elements
Large-Scale Nonlinear Programming Problem (NLP)
Interior Point Algorithm
Biegler, Cervantes,
Waechter (2002) 10
Simultaneous Dynamic Optimization
 Discretization of differential equations (DAEs) and state





and control variables
Large-scale nonlinear problem
Profile constraints handled directly
Direct incorporation of equipment design variables
DAE model solved only once
Converges for unstable systems
11
Nonlinear Programming Problem
Application of Barrier method
n
min f ( x)    ln s j
j 1
s.t c( x)  0
s.t c( x)  0
x0
sx0
As   0,
x*()  x*
12
Cryogenic Heat Exchangers and HP Separator
To
Recompression
Rodriguez, Bandoni, Diaz (2004)
Residual Gas
V
To
Turboexpander
ht
Phase
Change
To Demethanizing
Column
Horizontal High
Pressure Separator
Ts2_i = 322 K
Natural Gas
13
HE Model (no Phase Change)
Energy Balances: Partial Differential Equations System
Tube side
ht * Asup
Tt( z ,t )
Tt( z ,t )
Ts( z ,t )  Tt( z ,t )
 vt

t
z
t * Cpt * At * L
Shell side
hs * Asup
Ts( z ,t )
Ts( z ,t )
Tt( z ,t )  Ts( z ,t )
 vs

t
z
s * Cps * As * L
Spatial Discretization: Finite Differences  ODE System
14
HE Model (no Phase Change)
Energy Balance at cell i
ht * Asup
dTti vti
Tsi  Tti 
 Tti1  Tti  
dt
z
ti * Cpt * At * L
Tube side
hs * Asup
dTsi vsi
Tti  Tsi 
 Tsi1  Tsi  
dt
z
si * Cps * As * L
Shell side
Hot stream
at Ts0
Ts1
Ts2
Ts3
Ts4
Ts5
Ts6
Tt1
Tt2
Tt3
Tt4
Tt5
Tt6
Ts1
Ts2
Ts3
Ts4
Ts5
Ts6
Multicell model
E-1
15
Cold stream
at Tt0
Algebraic Equations (no Phase Change)
z j ,i  A j  B j * T j ,i
 j ,i
To
Recompression
Residual Gas
M*P

z j ,i * R * T j ,i
v j ,i 
Fj
 j ,i * A j
G
i = 1, …, N (cells)
j = tube or shell side
HE model
DAE System
Natural Gas
16
HE Model (Partial Phase Change on Shell Side)
Rodriguez, Bandoni, Diaz (2005)
Energy Balance at cell i
dEi
 Lpi*hpi + Vpi*Hpi - Li*hi - Vi*Hi + Qit
dt
Algebraic Equations at cell i
Ki , j
iL, j
 V
i , j
hi  h
ideal
i
Hi  H
yi , j  K i , j xi , j
 hi
ideal
i
 H i
Qt ,i  U 0 * A* Tml ,i
hiideal
H iideal
 yi , j   xi , j  0
j
j
nc
  hiideal
( Ti )xi , j
,j
j 1
nc
  H iideal
( Ti )yi , j
,j
j 1
17
HE Model (Partial Phase Change on Shell Side)
Compressibility factor
ziV  zV ( Tsi , Psi , yi )
ziL  z L ( Tsi , Psi , xi )
Residual enthalpies
H i  H ( Tsi , Psi , yi )
Soave-Redlich-Kwong EoS
hi  h( Tsi , Psi , xi )
Fugacity coefficients
iV, j   V ( Tsi , Psi , yi )
iL, j   L ( Tsi , Psi , xi )
Psi 
siL * ziL * R * Tsi
PmolsiL
Psi 
siV * ziV * R * Tsi
PmolsiV
18
HE Model (Partial Phase Change on Shell Side)
Bell-Delaware method
Pressure drop Pi
2
 V ( i )* Vmax
(i )

Pc ( i )  Ka  N c * Kf ( i )* 
2


Ps (i)   A * Pc (i)  B * Pw (i) * Rl  C * Pc (i)
Pw (i) 
2  0.6 * N cw  * m T2 (i)
Kf (i)  A 
V (i)
B
C
D
E



Re(i) Re(i) 2 Re(i)3 Re(i) 4
V (i) * Vmax (i) * D0
Re(i) 

19
High Pressure Separator
dM
 Lp+ Vp - L - V
dt
Horizontal tank
2
M
D 
VolV    in  Long  L
L
 2 
L  Cv
Ps  Pt    ht
r
M V  V VolV
ML 
M  MV  M L
 L Long 
2 
 ht 
2
2
2

r

r
arccos

h
r

h

  t
t 
PmolL 
r

20
Turboexpander
 Feed: vapor from High Pressure Separator
 Polytropic expansion (η = 1.26, natural gas)
 1

Ts  Ps
  
Te  Pe 
 Assumption: Static mass and energy balances
 Thermodynamic predictions: Soave-Redlich-Kwong
 Algebraic equations: Same as Flash
 Outlet partially condensed stream: to Demethanizing column
21
Demethanizing Column
Diaz, Tonelli, Bandoni, Biegler (2003)
Differential Equations at stage i (1 ≤ i ≤ N)
dM i
 Fi  Vi1  Li1  Vi  Li
dt
dmij
dt
 Fi zi , j  Vi1 yi1, j  Li1 xi1, j  Vi yi , j  Li xi , j
j=1,…,ncomp
dEi
 Vi1 H i1  Li1hi1  Vi H i  Li hi  Fi i H Fi  1  i hFi   Qsri
dt
N = 8; ncomp = 10
22
Demethanizing Column
Algebraic Equations
Compressibility factor ( z iV , z iL)
Residual enthalpies (Hi, hi)
Fugacity coefficients (i , j , i , j)
V
Ki , j
iL, j
 V
i , j
SRK
L
yi , j  K i , j xi , j
 yi , j   xi , j  0
j
j
H i  H iideal  H i
hi  hiideal  hi
23
Demethanizing Column
Algebraic Equations
2
Vapor volume
M iL
 Di 
V
Voli     H plate ,i  L
i
 2
Vapor and liquid holdup
M iV  iV VoliV
M i  M iV  M iL
Component holdup
mi , j  M iV yi , j  M iL xi , j
Internal energy holdup
Ei 
M iV (
Hi 
pi

V
i
)
M iL ( hi

pi

L
i
)
24
Demethanizing Column
Liquid holdup and hydraulic correlations
2
D 
M iL  0.896  i  Hwi  Hwoi iL
2

Vi
4 

Hoi  5.56.10  V
 i AholeiCoi 
2
 iV PmoliV 
 L

L 
 i Pmoli 


Li

L
 i Weirli 
2/ 3
Hwoi  300.01495 Fwi 
2/ 3
Hldropi   Hwi  Hwoi 
Pressure drop
P1  PTOP  KK1V V12
Pi  Pi 1  0.24917 .10 5 Hoi  Hldrop i iL PmoliL
25
Demethanizing Column
Carbon dioxide solubility constraints
Phase Equilibrium
iV,CO 2  iL,CO 2  iS,CO 2
fiV,CO 2  fi ,LCO 2  fi S,CO 2
Assumption
No hydrocarbon in solid phase
To avoid CO2 precipitation
fi S,CO 2  fi S,CO 2
f iV,CO 2  f i ,SCO 2
fi ,LCO 2  fi S,CO 2
But
fi ,LCO 2  fiV,CO 2
from VLE calculations at each stage
26
Demethanizing Column
Carbon dioxide solubility constraints
f iV,CO 2  f i ,SCO 2
Solid phase
f i ,SCO 2  Pi ,SCO 2iV,CO 2
t
 Pi S,CO 2 
 TCO
 T 
 T

2
  14 .480 ln t i   65 .356  t i  1
ln t   14 .568 1 
Ti 
 TCO 2 
 TCO 2 

 PCO 2 
 T  2 
 T 3 
 47 .146  t i   1  14 .540  t i   1
 TCO 2 

 TCO 2 

Vapor phase
f iV,CO 2  yi ,CO 2 Pi iV,CO 2
27
Demethanizing Column: Phase Existence
Raghunathan, Diaz, Biegler (2004)
Gibbs Free Energy minimization at stage i
Karush Kuhn Tucker conditions
Raghunathan, Biegler (2003)
MPECs as additional constraints
IPOPT-C under AMPL
28
Demethanizing Column: Phase Existence
Relaxed Equilibrium Conditions
yi , j   i K i , j ( Ti , Pi , xi , j , yi , j )xi , j
 i 1  siV  siL
siL M iL
0
siV M iV  0
  1  M iL  0 , M iV  0
  1  M iL  0 , M iV  0
  1  M iL  0 , M iV  0 ,
siV  0 , siL  0
29
Demethanizing Column: Phase Existence
Liquid Holdup and Hydraulic correlations
2
 Di 
L
M i  0.896   Hwi iL  M iL  M iL
 2
2
D 
M iL  0.896  i  Hwoi iL
 2
Hwoi  300.01495
2/ 3


L

Fwi  L i
 i Weirli 
2/ 3
3


L

2
 Li  kd ( M i ) 




M iL M iL  0
M iL  0, M iL  0
30
Optimization Problem: HEs + HPS
min 0tf T  TSP  dt
2
T
st .
h
t
L
G
DAE System
60  L  80( kmol / min)
5  G  80( kmol / min)
Tout
303  Tout  306( K )
Alg. Eqns = 403
Differ. Eqns = 37
(10+8 cells in HEs)
31
Numerical Results: HEs + HPS
Optimization variables: G, L
20 elements
65.330
1.00
2 collocation points
65.325
Lt
0.95
h
0.90
0.85
18274 disc. variables
0.80
0.75
65.315
0.70
0.65
65.310
0.60
0.55
65.305
0.50
0.45
65.300
0
5
10
15
20
25
0.40
30
time (min)
32
h (m)
18 iter. (3 barrier problems)
Lt (Kmol/min)
65.320
Numerical Results: HEs + HPS
Optimization variables: G, L
83
307
82
G0
81
Tout
306
80
305
78
304
77
76
303
Tt (K)
Temperature vs time
212.0
75
74
302
211.9
73
72
301
211.8
71
70
0
5
10
15
time (min)
20
25
300
30
Ts (K)
G0 (Kmol/min)
79
211.7
211.6
211.5
211.4
0
5
10
15
20
25
time (min)
33
30
Numerical Results: HEs + HPS
Temp. profile, tubes side
Temp. profile, shell side
(no phase change)
34
Numerical Results: HEs + HPS
58.5
35.0
58.2
30.0
57.9
57.6
25.0
57.3
Ps (bar)
20.0
57.0
L
(Kmol/min)
56.7
15.0
56.4
10.0
56.1
55.8
5.0
55.5
0
2
time (min)
11.5
21
30.5
40 1
2
3
4
5
6
cells
Pressure profile, shell side
6
cells
0.0
5
4
3
2
1 0
2
11.5
21
30.5
40
time (min)
Condensed fraction, shell side
35
Optimization Problem: Demethanizing Column
min
tf
0
ethane   SP  dt
2
st .
DAE System
15  PTOP  22( bar )
99  QREB  200( kJ / min)
0  xB ,CH 4  0.008
f iV,CO 2  0.90 f i ,SCO 2
Alg. Eqns = 385
Differ. Eqns = 97
36
Numerical Results: Demethanizer
Feed A: Low CO2
content (0.65%)
20
82
Optimization variable: PTOP
80
2 collocation points
P
Ethane recovery
18
Ptop (bar)
20 finite elements
76
74
21357 discretized variables
40 iter. (3 barrier problems)
16
72
ss = 80.9 %
70
0
10
20
30
40
50
Time (min)
37
Ethane recovery (%)
78
Numerical Results: Demethanizer
Feed B: High CO2 content
(2%)
Optimization variable: PTOP
Ptop (bar)
2 collocation points
84
80
P
Ethane Recovery
18
43 iter. (3 barrier problems)
76
ss = 74.5 %
16
72
0
10
20
30
40
Time (min)
38
Ethane recovery (%)
20 finite elements
20
Numerical Results: Demethanizer
Feed B (high CO2 content)
84
No solubility constraints
P
Ethane recovery
20
20 finite elements
Ptop (bar)
80
78
18
76
2 collocation points
59 iter. (4 barrier problems)
ss = 76.6 %
74
16
0
10
20
30
40
50
Time (min)
39
Ethane recovery (%)
Optimization variable: PTOP
82
Numerical Results: Demethanizer
Feed A
Optimization variable:
178
Reboiler Heat Duty (QR)
57 iter. (5 barrier problems)
ss = 77.8 % (PTOP=19bar)
Reboiler Heat Duty (MJ/min)
2 collocation points
176
Active constraint: Methane mole
fraction in bottoms
0.007
174
QREBOILER
XCH4
172
0.006
170
0.005
168
166
0.004
164
162
0.003
0
5
10
15
20
25
30
35
40
Time (min)
40
Bottom methane mole fraction
20 finite elements
0.008
Numerical Results: Demethanizer Start Up
MPECs
Optimization variable:
Three time periods
Ideal Gas, Ideal solution
Ethane Recovery
PTOP, QR
Solved in AMPL with
IPOPT-C (Raghunathan,
Biegler, 2003)
Time (min)
41
Numerical Results: Demethanizer Start Up
MPECs
PTOP, QR
3rd time period
17944 discretized
variables
576 complementarity
constraints
Reboiler Heat Duty (MJ/min)
Optimization variable:
97 iterations
Time (min)
42
Boiler Dynamic Optimization
Rodriguez, Bandoni, Diaz (2005b)
Dynamic optimization model to
determine controller parameters
Boiler
Flue Gas
High Pressure Steam
Combustion chamber
Feed Water
Risers
Superheater
E-1
Air
Steam drum
Combustion
Chamber
Superheater
RISER 1
RISER 2
Gas path
Natural Gas
Furnace Gas
PI Controller
(Liquid level in drum)
Manipulated: Feed water flowrate
43
Boiler Dynamic Optimization
DAE Optimization model
 Differential equations
• Momentum and energy balances in risers and superheater
• Mass and energy balances in drum
• Integral part of controller and valve equation
 Algebraic equations
• Friction loss
• Mixture properties
• Vapor holdup in drum
• Gas – temperature distribution equations in furnace, risers ans
superheaters (6 eqns.)
• Air / fuel ratio
• Drum geometric relations (5 eqns.)
• Steam tables correlations for saturated and high pressure steam
density, pressure and enthalpy
44
Boiler Dynamic Optimization
DAE Optimization model
 Optimization variables
• PI controller parameters
 Step change in high pressure steam demand
6
32
4
Liquid Level Drum (ft)
Feed Water Flowrate (lb/s)
30
28
26
24
2
0
-2
22
-4
0
200
400
600
800
1000
0
200
400
Time (s)
600
800
1000
Time (s)
 Additional PI controllers (current work)
• Outlet steam pressure (fuel gas flowrate)
• Superheated steam temperature (air excess)
45
Conclusions
 Rigorous dynamic optimization models for cryogenic train in
Natural Gas Processing plant within a simultaneous approach
 Thermodynamic models with cubic equation of state
 Carbon dioxide solubility addressed in both liquid and vapor phases
as path constraints, at each column stage
 Phase detection through the addition of MPECs for Gibbs free
energy minimization
 Boiler dynamic model for controller parameters
 Simultaneous approach provides efficient framework for the
formulation and resolution of DAE optimization problems for
large-scale real plants
46
References
 Biegler, L. T., A. Cervantes, A. Waechter, “Advances in Simultaneous





Strategies for Dynamic Process Optimization,” Chem. Eng. Sci., 57, 575-593,
2002
Cervantes, A., L. T. Biegler, “Large-Scale DAE Optimization using
Simultaneous Nonlinear Programming Formulations”, AIChE Journal, 44,
1038, 1998
Diaz, S., A. Serrani, R. de Beistegui, E. Brignole, "A MINLP Strategy for the
Debottlenecking problem in an Ethane Extraction Plant"; Computers and
Chemical Engineering, 19s, 175-178, 1995
Diaz, S., A.Serrani, A. Bandoni, E. A. Brignole, “Automatic Design and
Optimization of Natural Gas Plants", Industrial and Engineering Chemistry
Research, 36, 2715-2724, 1997
S. Diaz, M. Zabaloy, E. A. Brignole, “Thermodynamic Model Effect on the
Design And Optimization of Natural Gas Plants”, Proceedings 78th Gas
Processors Association Annual Convention, 38-45, March 1999, Nashville,
Tennessee, USA
Diaz, S., A. Bandoni, E.A. Brignole “Flexibility study on a dual mode natural
gas plant in operation”, Chemical Engineering Communications, 189, 5, 623641, 2002
47
References
 Diaz, S., S. Tonelli, A. Bandoni, L.T. Biegler, “Dynamic optimization for





switching between steady states in cryogenic plants”, Foundations of
Computer Aided Process Operations, 4, 601-604, 2003
Diaz, S., A. Raghunathan , L. Biegler, “Dynamic Optimization of a Cryogenic
Distillation Column Using Complementarity Constraints”, 449d, AIChE
Annual Meeting, San Francisco, Nov. 16-19, 2003. Advances in Optimization
Raghunathan, A., M.S. Diaz, L.T. Biegler, “An MPEC Formulation for
Dynamic Optimization of Distillation Operations”, Computers and Chemical
Engineering, 28, 2037-2052, 2004
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heat exchanger/separator tank system with phase change”, submitted to Energy
Conversion and Management, 2004
Rodríguez, M., J. A. Bandoni, M. S. Diaz, “Dynamic Modelling and
Optimisation of Large-Scale Cryogenic Separation Processes”, IChEAP-7, The
Seventh Italian Conference on Chemical and Process Engineering, 15 - 18
May 2005a Giardini Naxos, Italy
Rodríguez, M., J. A. Bandoni, M. S. Diaz, “Boiler Controller Design Using
Dynamic Optimisation”, PRES05, 8th Conference on Process Integration,
Modelling and Optimisation for Energy Saving and Pollution Reduction, 15 48
18 May 2005b Giardini Naxos, Italy
Appendix: Soave Redlich Kwong Equation of State
z
Compressibility factor
L V
for a mixture ( zi , zi )
V
a

V  b RgT ( V  b )
a   x*j z*j ; x*j x j a j j ; z j   x*k ( 1  k j ,k )
k
b   b j ; b j  0.08664 RgTcj / Pcj
j
 0.42747
aj  

Pcj

Rg2Tcj2




0.5
  T 0.5 
; j  1  m j 1    
  Tcj  
Soave modification
(Temperature
dependence)
m j  0.48  1.574 j  0.176 2j
49
Appendix: Soave Redlich Kwong Equation of State
Fugacity Coeff. for
component j in
mixture ( iV, j ,iL, j )
*

2
a

z
bj   B 
A
j j j
ln  j  ( z  1 )  ln( z  B )  
  ln1  
b
B  a
b 
z
bj
H
1
Residual enthalpy
 B  * *  1  m j 

 1 
ln1   x j z j 

in mixture ( H i , hi ) RgT
bRgT 
zj

j


A
aP
bP
;
B

Rg2T 2
RgT
50