Dynamic Modeling and Optimization of Large-Scale Cryogenic...
Transcript of Dynamic Modeling and Optimization of Large-Scale Cryogenic...
Dynamic Modeling and Optimizationof Large-Scale Cryogenic Processes
Maria Soledad Diaz
Planta Piloto de Ingeniería QuímicaUniversidad Nacional del Sur -CONICET
Bahía Blanca, ARGENTINA
2
Outline
Objective
Natural Gas Processing Plants
Methodology
Mathematical Models
Discussion of Results
Conclusions and Current Work
3
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• Phase existence in column
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
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Natural Gas in Argentina
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
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Natural Gas Processing Plants
Provide ethane as raw material for olefin plants (ethylene) and petrochemicalHigh ethylene yieldMinimum by-products
EP II EP I PEA D
PEBDL PEBD
PEBDL
Pta. Fracc. de LGN
Pta. NH3 y Pta. UreaAlmacen.
de GN I
VCM PVC
Pta.CloroSoda
Posta de Inflamables
a Bahía Blanca a Bahía Blanca
Ingeniero White
Puerto Galván
EP II EP I PEA D
PEBDL PEBD
PEBDL
Pta. Fracc. de LGN
Pta. NH3 y Pta. UreaAlmacen.
de GN I
VCM PVC
Pta.CloroSoda
Posta de Inflamables
a Bahía Blanca a Bahía Blanca
Ingeniero White
Puerto Galván
EP II EP I PEA D
PEBDL PEBD
PEBDL
Pta. Fracc. de LGN
Pta. NH3 y Pta. UreaAlmacen.
de GN I
VCM PVC
Pta.CloroSoda
Posta de Inflamables
a Bahía Blanca a Bahía Blanca
Ingeniero White
Puerto Galván
GN I
a Bahía Blanca
acce
so a
Gen
eral
Cer
ri
General D. Cerri
Ruta Nº3 Sur
ferrocarril
ferro
carr
il
a Médanos
Esta
ción
D. C
err i
Esta
ción
Agu
ara
Polo Petr
oquímico B
ahía Blan
ca
GN I
a Bahía Blanca
acce
so a
Gen
eral
Cer
ri
General D. Cerri
Ruta Nº3 Sur
ferrocarril
ferro
carr
il
a Médanos
Esta
ción
D. C
err i
Esta
ción
Agu
ara
Polo Petr
oquímico B
ahía Blan
ca
NGL Plant
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Natural Gas Plants at Bahia Blanca
LPG Gasoline
Residual GasResidual Gas
C3 + C3=
C4
C5+
EthaneEthane
EthyleneEthylene
Natural GasNatural Gas NGLNGL
LPG Gasoline533533355355
700 700 MtMt/y/y
24 24 MMsmMMsm33/d/d 5 5 MMsmMMsm33/d/d
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Natural Gas Processing Plants
Cryogenic sector (demethanizing), Conventional fractionation (deethanizing, depropanizing, debutanizing, gasoline stabilization), CO2 removalCurrent technology: Turboexpansion (Old technology: Absorption)High pressure Cryogenic conditions
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NATURAL GAS PLANT
Pipelines
DEHYDRATION
TRAIN B
TRAIN A
TRAIN C
EthaneTreatment CO2
Propane
Butane
GasolineSeparation Fractionation
Ethane To Ethylene Plant
Absorption Plant
TRAIN A
TRAIN B
Compression/Recompression
DEHYDRATION
C2 C3 C4
C3
C3
C4
C4
To Recompression
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Cryogenic Sector
Turboexpander
HighPressureSeparator
to deethanizer column
Cryogenic HeatExchangers
Demethanizer
10
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)
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Dynamic Optimization Problem
min Φ(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)=0G(z(t) , y(t), u(t), p, t) = 0
zO = z(0)
z(t) ∈ [zl, zu] , y(t) ∈ [xl, xu]u(t) ∈ [ul, uu] , p ∈ [pl, pu]
t, time tf, final timez, differential variables u, control variablesy, algebraic variables p, time independent parameters
General Formulation
12
Dynamic Optimization Strategy
Nonlinear DAE optimization problem
Discretization of Control and State variables
Collocation on finite elements
Large-Scale Nonlinear Programming Problem (NLP)
Interior Point Algorithm
Simultaneous Approach Cervantes, Biegler (1998)
Biegler, Cervantes, Waechter (2002)
13
rSQP AlgorithmInitialization
LinearizationObjective function, constraints
and gradients at xk
Step for dependent variables (dY)(Linear system)
Step for independent variables (dZ)QP in null space (Inequality constraints)
Line search or Trust region
Check ConvergenceOptimal Solution
dx = YdY + ZdZ
Basis selection
QP
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Nonlinear Programming Problem
Application of Barrier method
)(xfmin
0)(s.t =xc
0 ≥x
∑−=
n
jjxlnxfmin
1)( µ
0)(s.t =xc
As µ 0, x*(µ) x*
Sequence of barrier problems for decreasing µ values
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Barrier Method Algorithm: Primal-Dual ApproachInitialization
Step for dependent variables (dY)(Linear system)
Step for independent variables (dZ)Unconstrained QP in null space
Line search or Trust region
Check Barrier Problem Convergence
Check NLP Convergence
dx = YdY + ZdZ
Step for dual variables dv
Update µ, ε
Update B
B, µ,ε
No
Update B
OptimalSolution
QP
BP
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TurboexpanderArea 203
Area 204 Area 501
High pressureseparator
HeatExchangers
Columm
Cryogenic Sector
17
Ts2_i = 322 K
ht
V
ToRecompression
ToTurboexpander
Natural Gas
Residual Gas
To DemethanizingColumn
Cryogenic Heat Exchangers and HP Separator
Phase Change Horizontal High
Pressure Separator
Rodriguez, Bandoni, Diaz (2004)
Baffled shell-and-tube single-pass countercurrent heat exchangers
Tubes: residual gas
Shell: natural gas feed
Partial condensation: Second heat exchanger
18
Energy Balances: Partial Differential Equations System
( ))t,z(Tt)t,z(TsL*A*Cp*t
A*hz
)t,z(Ttvtt
)t,z(Tttt
supt −=∂
∂+
∂∂
ρ
HE Model (no Phase Change)
( ))t,z(Ts)t,z(TtL*A*Cp*s
A*hz
)t,z(Tsvst
)t,z(Tsss
sups −=∂
∂−
∂∂
ρ
Tube side
Shell side
First-order hyperbolic PDE
Method of Lines
Spatial Discretization: Backward Finite Differences
for Convection Term ODE System
z z+∆z
Flowing Fluid
)z(*Ts),z(Ts)z(*Tt),z(Tt)t(Tso)t,L(Ts
)t(Tto)t,(Tt
====
00
0
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Energy Balance at cell i
( ) ( )iissi
supsii
ii TsTtL*A*Cp*s
A*hTsTs
zvs
dtdTs
−+−∆
= − ρ1
HE Model (no Phase Change)
( ) ( )iitti
suptii
ii TtTsL*A*Cp*t
A*hTtTt
zvt
dtdTt
−+−∆
−= − ρ1Tube side
Shell side
Multicell model
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ji,j
ji,j A*
Fv
ρ=
i = 1, …, N (cells)j = tube or shell side
Algebraic Equations (no Phase Change)
G
ToRecompression
Natural Gas
Residual Gas
HE model DAE System
i,ji,ji,j T*R*z
P*M=ρ
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HE Model (Partial Phase Change on Shell Side)
Mass Balance at cell i
i,VLiii,V - m - VV
dtdM
1−=Vapor Phase
i,VLiii,L m - LL
dtdM
+= −1
j,iij,iij,iij,iiij xLyVxLyV
dtdm
−−+= −−−− 1111
Liquid Phase
Component
Energy Balance at cell i
tiiiiiiiii
i + Q*H - V*h - L*H + V*hLdt
dE1111 −−−−=
mVL,i: interfacial mass-transport rate from vapor to liquid phase
Rodriguez, Bandoni, Diaz (2005)
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HE Model (Partial Phase Change on Shell Side)
Momentum Balance at cell i
i,Li,VLi,erinti,Pi,Vii,Vii,Vi,V vmF F v - VvV
dt)vM(d
−−+= −− 11Vapor Phase
i,wi,Li,VLi,erinti,Pi,Lii,Lii,Li,L FvmF F v - LvL
dt)vM(d
−+++= −− 11Liquid Phase
Algebraic Equations at cell i
Void fraction based on cross-section area
i,ViiTi,V LeAM ρθ=
T
i,Vi A
A=θ
i,LiiTi,L )(LeAM ρθ−= 1
i,Vi,V
ii,V A
Vvρ
=
i,Li,V
ii,L )A(
Lvρ−
=1
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HE Model (Partial Phase Change on Shell Side)
Driving Force ( ) Tiiiii,P APsPsF θθ −= −− 11
Interfacial shear stress
( )22
4i,Li,V
i,Li,f
ii,erint vvf
DF −=
ρθ
Interfacial friction factor
+=
Dff
/
i,V
i,Lvi,f
δρρ
31
241
>
≤≤=
−
−
300000460
3000040000790200
250
Re,Re.
Re,Re.f
.
.
v
Friction resistance on wall surface
wi,w DF τ4
=
24
HE Model (Partial Phase Change on Shell Side)
j,ij,ij,i xKy =
,HHH iidealii ∆−=
,hhh iidealii ∆−=
Vj,i
Lj,i
j,iKφφ
=
j,inc
ji
idealj,i
ideali x)T(hh ∑
==
1
j,inc
ji
idealj,i
ideali y)T(HH ∑
==
1
∑∑ =−j
j,ij
j,i xy 0
Equilibrium ratio
Summation Eqns
Internal Energy ii,Lii,Vi hMHME +=
25
Compressibility factor
Residual enthalpies
Fugacity coefficients
HE Model (Partial Phase Change on Shell Side)
)x,Ps,Ts(zz
)y,Ps,Ts(zz
j,iiiLL
i
j,iiiVV
i
=
=
)x,Ps,Ts(
)y,Ps,Ts(
j,iiiLL
j,i
j,iiiVV
j,i
φφ
φφ
=
=
Soave-Redlich-Kwong EoS
Li
iLii,L
i PmolsTs*R*z*
Psρ
= Vi
iVii,V
i PmolsTs*R*z*
Psρ
=
)x,Ps,Ts(hh
)y,Ps,Ts(HH
j,iiii
j,iiii
∆=∆
∆=∆
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- VLp+ Vp - Ldt
dM=
L
LV
MLongrVolρ
π −= 2
−+
−= 2222
ttt
L
LL hrh
rharccosrr
PmolLongM πρ
VVV VolM ρ= LV MMM +=
( )wL
tLstv /
)rh(gPPxCLρρρ ++−
=
High Pressure Separator
Horizontal tank
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Turboexpander
Feed: vapor from High Pressure Separator
Polytropic expansion (η = 1.26, natural gas)
Assumption: Static mass and energy balances
Thermodynamic predictions: Soave-Redlich-Kwong
Algebraic equations: Same as Flash
Outlet partially condensed stream: to Demethanizing column
ηη 1−
=
e
s
e
s
PP
TT
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iiiiii LVLVF
dtdM
−−++= −+ 11
j,iij,iij,iij,iij,iiij xLyVxLyVzF
dtdm
−−++= −−++ 1111 j=1,…,ncomp
( )[ ] iFiiFiiiiiiiiiiii QsrhHFhLHVhLHV
dtdE
+−+−−−+= −−++ ϕϕ 11111
Differential Equations at stage i (1 ≤ i ≤ N)
Demethanizing Column
Diaz, Tonelli, Bandoni, Biegler (2003)
N = 8; ncomp = 10
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Algebraic Equations
Vj,i
Lj,i
j,iKφφ
= j,ij,ij,i xKy =
iidealii HHH ∆−=
iidealii hhh ∆−=
Demethanizing Column
Compressibility factor ( )
Residual enthalpies (∆Hi, ∆hi)
Fugacity coefficients ( )
Li
Vi zz ,
Lji
Vji ,, , φφ
SRK
∑∑ =−j
j,ij
j,i xy 0
30
Vapor volume
Demethanizing Column
Li
Li
i,plateiV
iMHDVolρ
π −
=
2
2
Vi
Vi
Vi VolM ρ=
Li
Vii MMM +=
j,iLij,i
Vij,i xMyMm +=
Vapor and liquid holdup
Component holdup
)ph(M)pH(ME Li
ii
LiV
i
ii
Vii ρρ
−+−=Internal energy holdup
Algebraic Equations
31
Liquid holdup and hydraulic correlations
Demethanizing Column
( ) Liii
iLi HwoHwD.M ρπ +
=
2
28960
= −
Li
Li
Vi
Vi
iiVi
ii Pmol
PmolCoAhole
V..Hoρρ
ρ
2410565
( )32
3201495030/
iLi
ii
/i Weirl
LFw.Hwo
=
ρ
( )iii HwoHwHldrop += β
2111 VKKPP V
TOP ρ−=Pressure drop
( ) Li
Liiiii PmolHldropHo..PP ρ++= −
−5
1 10249170
Equivalent height of vapor free liquid
Pressure drop through aerated liquid
Dry hole pressure drop
i
i+1
HwHwo
32
Phase Equilibrium
Carbon dioxide solubility constraints
SCO,i
LCO,i
VCO,i fff 222 ==
Demethanizing Column
SCO,i
LCO,i
VCO,i 222 µµµ ==
Assumption
No hydrocarbon in solid phase
To avoid CO2 precipitation
But from VLE calculations at each stage
SCO,i
SCO,i ff 22 =
SCO,i
VCO,i ff 22 ≤
SCO,i
LCO,i ff 22 ≤
VCO,i
LCO,i ff 22 =
33
Solid phase
VCO,i
SCO,i
SCO,i Pf 222 φ=
−
+
−
−
−+
−
−=
154014114647
13566548014156814
3
2
2
2
22
2
2
2
tCO
it
CO
i
tCO
it
CO
i
i
tCO
tCO
SCO,i
TT.
TT.
TT.
TTln.
TT.
PP
ln
Vapor phase
VCO,iiCO,i
VCO,i Pyf 222 φ=
Carbon dioxide solubility constraints SCO,i
VCO,i ff 22 ≤
Demethanizing Column
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Gibbs Free Energy minimization at stage i
Demethanizing Column: Phase Existence
Raghunathan, Diaz, Biegler (2004)
Karush Kuhn Tucker conditions
Raghunathan, Biegler (2003)
MPECs (Mathematical Program with Equilibrium Constraints)
as additional constraints ( x.y = 0; x,y≥0)
IPOPT-C under AMPL (x.y ≤ δµ)
35
Demethanizing Column: Phase Existence
Relaxed Equilibrium Conditions
j,ij,ij,iiij,iij,i x)y,x,P,T(Ky γ=
Li
Vii ss −=−1γ
00 ≥⊥≤ Li
Li sM
,M,M
M,M
M,M
Vi
Li
Vi
Li
Vi
Li
001
001
001
=>⇒>
>=⇒<
>>⇒=
γ
γ
γ
00 ≥⊥≤ Vi
Vi sM
ComplementarityConstraints
∑∑ =−j
j,ij
j,i xy 0
36
Demethanizing Column: Phase Existence
Liquid Holdup and Hydraulic correlations
00 ≥⊥≤ −+ Li
Li MM
−+ −+
= L
iLi
Lii
iLi MMHwD.M ρπ
2
28960
Lii
iLi HwoD.M ρπ
2
28960
=+
( )32
3201495030/
iLi
ii
/i Weirl
LFw.Hwo
=
ρ
= + 2
3
)M(kL Lidi
HwHwo
i
i+1
Numerical Results
38
( )
{ }
)K(Tmin)/kmol(Gmin)/kmol(L
System DAE
.stdtTTmin
out
tfSP
3063038058060
20
≤≤≤≤≤≤
−∫
Optimization Problem: HEs + HPS
G
ht
L
T
Tout
(8+6 cells in HEs)
39
Optimization variables: G, L
20 elements
2 collocation points
18274 disc. variables
18 iter. (3 barrier problems)
Numerical Results: HEs + HPS
0 5 10 15 20 25 3065.300
65.305
65.310
65.315
65.320
65.325
65.330
h
Lt
Lt (K
mol
/min
)
time (min)
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
h (m
)
40
Numerical Results: HEs + HPS
Optimization variables: G, L
0 5 10 15 20 25 307071727374757677787980818283
Tout
G0
G0 (
Kmol
/min
)
time (min)
300
301
302
303
304
305
306
307
Tt (K
)
0 5 10 15 20 25 30
211.4
211.5
211.6
211.7
211.8
211.9
212.0
Temperature vs time
Ts (K
)
time (min)
41
Temp. profile, tubes side Temp. profile, shell side(no phase change)
Numerical Results: HEs + HPS
42
Pressure profile, shell side Liquid flowrate, shell side
Numerical Results: HEs + HPS
0 211.5 21
30.540 1
23
45
6
55.5
55.8
56.1
56.4
56.7
57.0
57.3
57.6
57.9
58.2
58.5
Ps (bar)
time (min) cells 0 2 11.5 21 30.5 40
12
34
56 0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
L (Kmol/min)
time (min)cells
43
( )
{ }
SCO,i
VCO,i
CH,B
REB
TOP
tfSPethane
f.f
.xmin)/kJ(Q
)bar(P
System DAE
.stdtmin
22
4
20
900
0080020099
2215
≤
≤≤≤≤≤≤
−∫ ηη
Alg. Eqns = 385Differ. Eqns = 97
Optimization Problem: Demethanizing Column
44
Numerical Results: Demethanizer
0 10 20 30 40 50
16
18
20
P Ethane recovery
Time (min)
Pto
p (b
ar)
70
72
74
76
78
80
82
Ethane recovery (%)
Optimization variable: PTOP
20 finite elements2 collocation points21357 discretized variables40 iter. (3 barrier problems)ηss = 80.9 %
Feed A: Low CO2content (0.65%)
45
Numerical Results: Demethanizer
0 10 20 30 4016
18
20
P Ethane Recovery
Time (min)
Pto
p (b
ar)
72
76
80
84
Ethane recovery (%
)
Feed B: High CO2 content (2%)
Optimization variable: PTOP
20 finite elements2 collocation points43 iter. (3 barrier problems)
ηss = 74.5 %
46
Numerical Results: Demethanizer
0 10 20 30 40 5016
18
20 P Ethane recovery
Time (min)
Pto
p (b
ar)
74
76
78
80
82
84
Ethane recovery (%
)
Feed B (high CO2 content)
No solubility constraints
Optimization variable: PTOP
20 finite elements2 collocation points59 iter. (4 barrier problems)
ηss = 76.6 %
47
Numerical Results: Demethanizer
0 5 10 15 20 25 30 35 40162
164
166
168
170
172
174
176
178
QREBOILER XCH4
Time (min)
Reb
oile
r Hea
t Dut
y (M
J/m
in)
0.003
0.004
0.005
0.006
0.007
0.008
Bottom m
ethane mole fraction
Feed A
Optimization variable:
Reboiler Heat Duty (QR)
20 finite elements2 collocation points57 iter. (5 barrier problems)
ηss = 77.8 % (PTOP=19bar)
Active constraint: Methane mole fraction in bottoms
48
Numerical Results: Demethanizer Start Up
MPECs
Optimization variable:
PTOP, QR
Three time periods
Ideal Gas, Ideal solution
Solved in AMPL with IPOPT-C (Raghunathan, Biegler, 2003)
Time (min)
Etha
ne R
ecov
ery
49
Numerical Results: Demethanizer Start Up
MPECs
Optimization variable:
PTOP, QR
3rd time period17944 discretizedvariables576 complementarityconstraints97 iterations
Time (min)
Reb
oile
rHea
t Dut
y (M
J/m
in)
50
Boiler Dynamic Optimization
Rodriguez, Bandoni, Diaz (2005b)
E-1
CombustionChamber
Natural Gas Furnace Gas
Air
Superheater RISER 1 RISER 2
High Pressure Steam
Flue Gas
Feed Water
Dynamic optimization model to determine controller parameters
Boiler
Combustion chamber
Risers
Superheater
Steam drum
Gas path
PI Controller
(Liquid level in drum)
Manipulated: Feed water flowrate
51
Boiler Dynamic Optimization
DAE Optimization modelDifferential 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
52
Boiler Dynamic Optimization
DAE Optimization modelOptimization variables• PI controller parameters
Step change in high pressure steam demand
Additional PI controllers (current work)• Outlet steam pressure (fuel gas flowrate)• Superheated steam temperature (air excess)
0 200 400 600 800 1000
22
24
26
28
30
32
Feed
Wat
er F
low
rate
(lb/
s)
Time (s)0 200 400 600 800 1000
-4
-2
0
2
4
6
Liqu
id L
evel
Dru
m (f
t)
Time (s)
53
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 first order optimality conditions for Gibbs free energy minimization (MPECs)
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
54
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, 1997S. 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, 623-641, 2002
55
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, AIChEAnnual Meeting, San Francisco, Nov. 16-19, 2003. Advances in OptimizationRaghunathan, A., M.S. Diaz, L.T. Biegler, “An MPEC Formulation for Dynamic Optimization of Distillation Operations”, Computers and Chemical Engineering, 28, 2037-2052, 2004 M. Rodríguez, J. A. Bandoni, M. S. Diaz, “Optimal dynamic responses of a heat exchanger/separator tank system with phase change”, submitted to Energy Conversion and Management, 2004Rodríguez, 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 -18 May 2005b Giardini Naxos, Italy
56
Appendix: Soave Redlich Kwong Equation of State
)bV(TRa
bVVz
g +−
−=
α
∑∑ −===k
k,j*kjjjj
*j
*j
*j )k(xz;axx;zxa 1αα
∑ ==j
cjcjgjj P/TR.b;bb 086640
−+=
=
505022
11427470
.
cjjj
.
cj
cjgj T
Tm;P
TR.a α
217605741480 jjj ...m ωω −+=
Compressibility factor for a mixture
Soave modification (Temperature dependence)
)z,z( Vi
Li
57
Appendix: Soave Redlich Kwong Equation of State
+
−−−−−=
zBln
bb
aza
BA)Bzln()z(
bb
ln j*jjjj
j 12
1αα
φ
∑
+
+−−=
∆−
j j
j*j
*j
gg
mzx
zBln
TbRTRH
α1
111
TRbPB;
TRPaA
gg== 22
α
Fugacity Coeff. for component j in mixture ),( L
j,iV
j,i φφ
Residual enthalpy in mixture )h,H( ii ∆∆