optimal synthesis of complex distillation columns using ...optimal synthesis of complex distillation...
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Optimal Synthesis of Complex Distillation Columns
Using Rigorous Models
Ignacio E. Grossmann
Department of Chemical Engineering
Carnegie Mellon University
Pittsburgh, PA 15213
Pío A. Aguirre and Mariana Barttfeld
INGAR
3000 Santa Fe, Argentina
Motivation
1. Synthesis of complex distillation systems non-trivial task
2. Complex physical phenomena requires rigorous models
3. Potential for finding innovative and improved designs
Research area pioneered by Roger Sargent !
Mathematical Programming Approaches
Linear Models (MILP/MINLP)
Andrecovich & Westerberg (1985), Paules & Floudas (1988), Aggarwal &
Floudas (1990), Raman & Grossmann (1994), Kakhu & Flower (1998),
Shah &Kokossis (2002)
Short-cut / Aggregated Models (MINLP)
Bagajewicz & Manousiouthakis (1992), Novak, Kravanja & Grossmann (1996),
Papalexandri & Pistikopoulos (1996), Caballero & Grossmann (1999, 2003),
Proios & Pistikopoulos (2004)
Rigorous Models (MINLP/GDP)
Sargent & Gaminibandara (1976), Viswanathan & Grossmann (1993),
Smith & Pantelides (1995), Bauer & Stichlmair (1998), Dunnebier &
Pantelides (1999), Yeomans & Grossmann (2000), Lang & Biegler (2002),
Barttfeld, Aguirre & Grossmann (2004)
Highly nonlinear and nonconvex
Large-scale problem
Singularities introduced by internal flows when sections (or whole)
columns disappear
Convergence difficult to achieve
Rigorous Complex Columns Models Difficult to Optimize
Computational Challenges
Goal: Review previous workPropose decomposition strategy for complex columns
Rigorous Models
Branch and Bound method (BB)Ravindran and Gupta (1985) Leyffer and Fletcher (2001)
Branch and cut: Stubbs and Mehrotra (1999)
Generalized Benders Decomposition (GBD)Geoffrion (1972)
Outer-Approximation (OA)Duran & Grossmann (1986), Yuan et al. (1988), Fletcher & Leyffer (1994)
Extended Cutting Plane (ECP)Westerlund and Pettersson (1995)
MINLP Algorithms
GDP
Logic based methods
Branch and bound(Lee & Grossmann, 2000)
Decomposition
Outer-Approximation
Generalized Benders
(Turkay & Grossmann, 1997)
Reformulation MINLPOuter-Approximation
Generalized Benders
Extended Cutting Plane
Methods Generalized Disjunctive Programming
Convex-hull Big-MDirect
Cutting plane
(Lee & Grossmann, 2000)
MINLP and GDP can be applied to optimize discrete
and continuous decision in distillation designDiscrete: Configuration, number of trays
Continuous: Reflux ratio, heat loads, flows, compositions
MINLP:
Greater availability of software (DICOPT, MINOPT, BARON, SBB, -ECP)
Difficulty of requiring full space solutions
Remarks on methods
GDP:
LOGMIP only software; special tailored solutions needed
Decomposition does not require full space solutions
NLP only:
Variety of codes available (CONOPT, SNOPT, IPOPT, LOQO, etc.)
Requires continuous approximations
Full space solutions
In all cases nonconvexity is major issue !
Optimal Feedtray Location
Sargent & Gaminibandara (1976)
NLP Formulation
Min cost
st MESH eqtns
Ff
LOCi
i =‡”¸
NLP VMP: Variable-Metric Projection
if
1
.
.
L1 V2
L2 V3
L3 V4
LN-1
1
2
3
N
N-1
N-2
VN
VN-1LN-
2
VN-2LN-
3
B
D
1
.
.
L1 V2
L2 V3
L3 V4
LN-1
1
2
3
N
N-1
N-2
VN
VN-1LN-
2
VN-2LN-
3
B
D
LOCiz
LOCizFf
Ff
z
i
ii
LOCi
i
LOCi
i
¸1,0=
¸0¡Ü-
=
1=
‡”
‡”
¸
¸
if
Optimal Feedtray Location (Cont)
MINLP Formulation
Min cost
st MESH eqtns
Viswanathan & Grossmann (1990)
MINLP DICOPT: AP-Outer Approximation-ER
F
Remark: MINLP solves as relaxed NLP!
Feed tray composition tends to
match composition of feed
iz
Optimization of Number of Trays
Discrete variables: Number of trays, feed tray location.
Continuous variables: reflux ratio, heat loads, exchanger areas, column diameter.
No liquid on tray
No vapor on tray
Existing trays
Vapor Flow
Liquid Flow
Viswanathan & Grossmann (1993)
Zero flows- Discontinuities appear, convergence difficulties.
Redundant equations are solved- Increases CPU time.
Non-existing tray
Non-existing tray
1=mzr
1=nzb
1,0=izb
MINLP => Number
trays
1,0=izr
Optimal Design Columns with Multiple Feeds
60
59
F2
F1
ri
1
(0.15, 0.85)
F3
(0.5, 0.5)
(0.85, 0.15)
2
Separation Methanol - Water with 3 Feeds
MINLP model
Virial/UNIQUAC
115 0-1 binary variables
1683 continuous variables
1919 constraints
700,000 alternatives!
Air Products & Chemicals
Solved with DICOPT on a HP 9000/730
(5 major iterations, 45 min)
Optimal solution
Number of trays = 53
Feed location: Feed 1 Tray 4
Feed 2 Tray 6
Feed 3 Tray 12
Viswanthan & Grossmann (1993)
Differentiable Distribution Function
σ
0¨σ
Continuous Optimization Approach
parameter
cN variable
If iNc = => 1¨id
Basic idea: continuous approximation of 0-1 variables
Lang & Biegler (2001)
id used to multiply flows into tray
Highly nonconvex: requires good initial guess
iidf
See Neves, Silva, Oliveira
Disjunctive Programming Model
Permanent trays:
Feed, reboiler, condenser
Conditional trays:
Intermediate trays might
be selected or not.
Conditional trays
Permanent trays
Trays not allowed to “disappear” from
column:
VLE mass transfer if selected.
No VLE, trivial mass/energy balance if not selected
-OR-VLE NOT VLE
(tray bypass)
Disjunction
Yeomans & Grossmann (2000)
• Permanent and conditional trays:– MESH equations
for condenser, reboiler and feed trays
– Mass & energy balances for rectification and stripping trays.
• Conditional trays only:
Condenser Tray
(permanent)
Rectification Trays
(conditional)
Feed Tray
(permanent)
Stripping Trays
(conditional)
Reboiler Tray
(permanent)Heavy Product
Feed
Light
Product
-OR-
-OR-
-OR-
-OR-
Vapor Flow
Liquid Flow
Equilibrium Stage
Non-equilibrium Stage
Single Column GDP Model
Which model is better?
Objectives:
-Comprehensive comparison MINLP and GDP models
-Increase robustness optimization
Initialization (Aguirre, Barttfeld, 2001)
Two step optimization procedure:
1. Adiabatic approximation of reversible column (NLP)
minimize energy
2. Fixed maximum number trays (NLP)
minimize deviations adiabatic compositions
Barttfeld, Aguirre & Grossmann (2003)
Provides good initial guess for rigorous model
Other MINLP Representations
The number of trays is selected by optimizing the
condenser, reboiler and/or feed stream locations.
F
D
B
F
B
D
F
B
D
Variable feed and
reboiler location
F
D
B
F
B
DF
B
D
Variable feed and
condenser location
F
B
D
F
D
B
F
B
DVariable condenser and
reboiler location
Permanent Trays (top and bottom stages) are fixed stages in the structure. Existenceof each Intermediate tray modeled with a disjunction
GDP Representation Alternatives
F
B
D
B
D
F F
B
D
Fixed feed location
(Yeomans, Grossmann,
2000)
F
B
D
B
D
F F
D
B
Variable feed
location (bot)
F
B
D
B
D
FF
D
B
Variable feed
location (top)
General
PreprocessingPhase
RMINLP
Preliminary Solution
Reduction of
Candidates Trays
Reduced MINLP
Solution
MINLP
General
PreprocessingPhase
NLP1 solution:All trays existing
NLP2 solution:
Subset trays
GDP
Algorithm
GDP
Solution approaches
Heuristic Optional
Aggregate NLP
NLP fixed max number trays
Logic-based OA Algorithm
Master Problem(MILP)
Subproblem(NLP)
Selection ofDisjunctions
Converge?
Solution
InitialSubproblems
(NLP)
Linearizationof NonlinearEquations
Big-M formof linear
disjunctions
Pre-
Processing
MILP form ofDisjunctiveequations
Continuousvariables forInitialization
Discrete variables
Selected Equations
YESNO
Initialization
OA Algorithm
Data flow Algorithm cycle
Implemented in GAMS CPLEX/CONOPT
Turkay, Grossmann (1996)
General trends of results
Trade-offs MINLP vs. GDP
MINLP tended to find somewhat lower cost solutions due to the
reduction of candidate trays from MINLP relaxation
More sensitive to initialization (thermo model essential)
Easier implement: DICOPT
Best MINLP Model: Variable feed/reboiler
Best GDP Model: Fixed feed Yeomans & Grossmann (2000)
GDP was typically one order of magnitude faster and more robust
Less sensitive to initialization
Algorithm implemented within GAMS
–Future LOGMIP should help
Example MINLP
• Benzene, Toluene, Oxylene
– Composition: 0.33/0.33/0.34
– Feed: 10 mol/sec
– Upper number trays: 35
– Recovery, purity distillate: 98%
Preprocessing (NLP)
Continuous Variables 3273
Constraints 2674
Time [CPU s] 0.68
Rigorous Model (MINLP)
Continuous Variables 1507
Binary Variables 33
Constranits 1830
Iterations 17
Time RMINLP [CPU s] 0.52
Time MINLP [CPU min] 10.81
Total Cost [$/año] 79,962
D (98% Benzene)
F
B
1
9
20
242.65 kW
258.95 kW
Relaxed Solution RMINLP - 79,223 $/yr
D (98% Benzene)
F
241.7 kW
258 kW
B
1
9
18
Integer Soluiton MINLP – 79,962 $/yr
Example GDP
GDP Formulation
Mixture: Methanol/Ethanol/water
Feed Flow= 10 mol/sec
Feed composition= 0.2/0.2/0.6
P = 1.01 bar
Product Specification:
products composition reversible model
Upper bound No. Trays: 60
Methanol/ethanol/water - GDP: fixed tray location Preprocessing Phase: NLP tray-by-tray Models
Continuous Variables 1597 Constraints 1544 Total CPU time (s) 1.12
Model Description
Continuous Variables 2933 Binary Variables 60 Constraints 2862 Nonlinear nonzero elements 5656 Number of iterations 10 NLP CPU time (s) 9.14 MILP CPU time (s) 16.97 Total CPU time (s) 401
Optimal Solution Total number of trays 41 Feed tray 20 Column diameter (m) 0.51 Condenser duty (KJ/s) 387.4 Reboiler duty (KJ/s) 386.5 Objective value ($/yr) 117,600
GAMS PIII, 667 MHz. with 256 MB of RAM.
CONOPT2 NLP subproblems/ CPLEX MILP subproblems.
Reactive Distillation
• Conditional Trays:
Active Trays
Separation with reaction may take place
– Positive liquid holdup
Separation only make take place
– Liquid holdup equals zeroActive Trays
Inactive Trays
Inactive Trays
Input-Output operation with no mass transfer and no reaction
Extension Single Column GDP Model Jackson & Grossmann (2001)
OR
Example: Metathesis of Pentene
• Annualized Cost: $1.167x106 per year
• Design/Operating Parameters:
21 Trays; 5 Feeds
Column Diameter = 3.8ft
Column Height = 107ft
Boilup = 0.374
Reflux = 0.811
Reboiler Duty = 153 kW
Condenser Duty = 984 kW
• Reaction Zone:
Trays 1 – 18
Total Liquid Holdup = 752 ft3
90% Conversion of Pentene
126841052 HCHCHC +⇔Conversion of 2-cis-pentene into
2-cis-butene and 3-cis-hexene:
GDP Model: 25 discrete variables
731 continuous variables
730 constraints
Superstructure Representation – Suitable for zeotropic and azeotropic mixtures
– General and automatically generated
– Includes thermodynamic information
– Embeds many possible alternative designs
Synthesis of complex distillation systems
Mariana Barttfeld, Pio Aguirre/INGAR
Solution Procedure–Decomposition algorithm (decision levels)
•First level: selection of sections
•Second level: selection of trays in existent sections
–Initialization phase: reversible sequence approximation
–Robust and effective solutions
Superstructure Formulation GDP formulation
Superstructure for Synthesizing Configurations
Sargent and Gaminibandara (1976)
Generated with the State-Task-Network (STN) (Sargent, 1998)
ABCD
ABC
BCD
AB
BC
CD
A
B
C
D
States
Tasks
STN Representation
(4 Component Zeotropic Mixture)
B C
C D
A B
A B C
B C D
A
A B C D
B
C
D
Sargent-Gaminibandara Superstructure
(4 Component Zeotropic Mixture)
GDP Model: Yeomans & Grossmann (2000)
Simultaneous selection sections & trays
Superstructure Zeotropic Mixtures
• Based on the Reversible Distillation Sequence Model (RDSM) (Fonyo, 1974)
Motivated by thermodynamic initialization scheme
• Automatically generated with the State-Task-Representation (STN)
• Contains 2NC-1-1 columns and NC-1 level
B C
C D
A B
A B C
B C D
A
A B C D
B C
B
C
B
C
D
States
Tasks
ABCD
BCD
ABC
AB
BC
CD
A
B
C
D
BC
RDSM-based STN Representation
(4 Component Zeotropic Mixture)
Avoid mixing intermediates
Modification for Azeotropic Mixtures
• RDSM-based STN cannot be defined a priori
• Composition diagram needed
• Azeotrope recycled
ABC
ABC
BC
AB
BC
A
B
B
Azeotrope
C
A
BC
F
ABC
BC
BC-Azeo
Product
Azeotrope
Mass Balance
Distillation Boundary
ABC
BC
ABC
AB
BC
C
A
B
Azeo
B
Azeo
States
Tasks
RDSM-based STN Representation
(4 Component Azeotropic Mixture)
ABC
ABC
BC
AB
BC
A
B
B
Azeotrope
C
B C
C D
A B
A B C
B C D
A
A B C D
B C
B
C
B
C
D
Zeotropic Mixture Azeotropic Mixture
Superstructures
B C
A B C
B C D3
6
1
5
2
A
A B C D
B C
B
C
B
C
D
6
5
B C
B C
C
A
A B C D
D
B
B C D3
6
5
B C
A B C
A
A B C DB C
C
D
B2
1
B C
C D
A B
A B C
B C D
A
A B C D
B C
B
C
B
C
D
1
2
3
4
5
6
7
Mapping to Specific Designs
section s+1
section sSelection of sections Configuration
If section selected Ys = True
If section not selected Ys = False
Discrete Decisions
Two hierarchical levels1. Selection sections2. Selection Trays
Configuration Model Formulation
1
1
1
1
0
0
0
0
0
s
s
L
n,i
V
n,i
V V
n ns
L L
n ns n
n secn
n
n,i n ,i
n ,i n ,i
s
Y
f
f
T TY
T Tntray stg s S , i C
V
L
x x
y y
ntray
+
−
∈
−
+
¬
= = =
= = ∨ ∀ ∈ ∈ = =
= = =
min z TAC=
0s.t. g( x ) ≤
0h( x ) =
(Y ) TrueΩ =
sx X ,Y True, False∈ ∈
Objective Function
Overall Process
Constraints
DISJUNCTION
Logic Relationships
Section Boolean
Variables
Selection of Trays
• Permanent Trays
– Fixed stages: condenser, reboiler and feed trays
– Interconnect columns
– Heat exchange takes place
• Intermediate Trays
– Use DISJUNCTIONS for modeling
– If section selected (Ys = True)
Intermediate Tray
Permanent Tray
n nW True W False
apply VLE OR apply by pass
equations equations
= =
−
Configuration Model Formulation
1
1
1
1
1
1
0
0
1
0
s
n
Ln n,i
L Vn,i n n n ,i n ,i
V V Vn,i n n n,i n n
L V L Ln,i n,i n n
V L
n nn n
n nn,i n n,i
n ,i n ,in ,i n n ,i
n ,i n ,in
n
Y
W
W f
f f (T ,P ,x ) f
f f (T ,P , y ) T T
f f T T
V VT T
L LLIQ L x
x xVAP V y
y ystg
stg
+
−
+
+
−
+
¬
=
= = = = = =
∨ = =
==
==
== =
1
1
1
1
0
0
0
0
0
s
s
L
n,i
V
n,i
V V
n n
L L
n n
n
n
n,i n ,i
n ,i n ,i
s
s n s
n sec
Y
f
f
T T
T T
V
L
x x
y y
ntray
ntray stg n sec
+
−
−
+
∈
¬ = = =
= ∨ = = = = =
= ∀ ∈
s S , i C
∀ ∈ ∈
min z TAC=
0s.t. g( x ) ≤
0h( x ) =
(Y ) TrueΩ =
(W ) TrueΩ =
s nx X ,Y ,W True, False∈ ∈
Objective Function
Overall Process Constraints
Logic Relationships
Section Boolean
Variables
Tray Boolean
Variables
DISJUNCTION
Detailed Cost Functions
dep
CinvTAC Cop
T= +Annual Cost
agua vapor
agua con vap
Qc QhCop C C
Cp T H= +
∆ ∆Operating Cost
Cinv Ccol Ctray Creb Ccond= + + +
1 066 0 802. .
colCcol k nt Dcol htray=
1 55.
trayCtray k nt Dcol htray=
0 65.
rebCreb k Areb=
0 65.
condCcond k Acond=
Investment Cost
Column Cost
Tray costs
Condenser Cost
Reboiler cost
nDcol Dtray≥
0 50 5 ..
vapor
n d n i n,i
i
T RDtray k V PM y
p
=
Solution Strategy
GDP Section
Problem
-Selection of Sections-MILP
Problem
-Selection of Trays-MILP
Problem
Reduced NLPProblem
-Initialization Phase-NLP
Problems
GDP Tray
Problem
Preprocessing
Phase
Algorithm Cycle
Fixed Max
Number Trays
Fixed Number
Sections
Aggregate NLP
NLP fixed max number trays
Problem specs
Mixture: N-pentane/ N-hexane/ N-heptaneFeed composition: 0.33/ 0.33/ 0.34Feed: 10 moles/sPressure: 1 atmMax no trays: 15 (each section)Min purity: 98%Ideal thermodynamics
SuperstructurePP1
PP2
F
PP3
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Initialization
Zeotropic Example (1)
GDP Model
Discrete Variables 96
Continuous Variables 3301
Constraints 3230
Optimal Configuration$140,880 /yr
Optimal Design
Annual cost ($/year) 140,880
Preprocessing(min) 2.20
Subproblems NLP (min) 6.97
Subproblems MILP (min) 2.29
Iterations 5
Total solution time (min) 11.46
667MHz. Pentium III PC
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Mol
e F
ract
ion
n-pe
ntan
e
Mole Fraction n-hexane
FeedCol 1 (tray 1 to 14)Col 1 (tray 15 to 34)Col 2 (tray 1 al 9)Col 2 (tray 10 al 32)
PP3
98% n-heptane
36
9
32
PP2
98% n-hexane
26
19
PP1
98% n-pentane
F
Qc = 52.4 kW
QH = 298.8 kW
48.8 kW
1
11 12
14
Qc = 271.3 kW
PP3
98% n-heptane
12
23
PP2
98% n-hexane
10
PP1
98% n-pentane
F
1
22
Dc1 = 0.45 m
Dcrect2 = 0.6 m
Dcstrip2 = 0.45 m
1
14
23
1
Dcstrip3 = 0.63 m
Dcrect3 = 0.45 m
Zeotropic Example (2)
All sections selected
Optimal Configuration$140,880 /yr
Optimal Design
Annual cost ($/year) 140,880
Preprocessing(min) 2.20
Subproblems NLP (min) 6.97
Subproblems MILP (min) 2.29
Iterations 5
Total solution time (min) 11.46
667MHz. Pentium III PC
PP3
98% n-heptane
36
9
32
PP2
98% n-hexane
26
19
PP1
98% n-pentane
F
Qc = 52.4 kW
QH = 298.8 kW
48.8 kW
1
11 12
14
Qc = 271.3 kW
PP3
98% n-heptane
12
23
PP2
98% n-hexane
10
PP1
98% n-pentane
F
1
22
Dc1 = 0.45 m
Dcrect2 = 0.6 m
Dcstrip2 = 0.45 m
1
14
23
1
Dcstrip3 = 0.63 m
Dcrect3 = 0.45 m
Configuration Side-Rectifier $143,440 /yr
Direct Sequence $145,040 /yr
Zeotropic Example (3)
Azeotropic Example (1)
Problem Specs
Mixture: Methanol/ Ethanol/ WaterFeed composition: 0.5/ 0.3/ 0.2Feed: 10 moles/sPressure: 1 atmMax no. trays: 20 (per section)Min purity: 95%Ideal/Wilson models
F
methanol
ehtanol
Azeotrope
Water
ethanol
Superstructure
Initialization
GDP Model
Discrete Variables 210
Continuous Variables 9025
Constraints 8996
Azeotropic Example (2)
Product Specifications 95%
Optimal Configuration$318,400 /yr
Optimal Solution
Annual Cost ($/year) 318,400
Preprocessing (min) 6.05
Subproblems NLP (min) 36.3
Subproblems MILP (min) 3.70
Iterations 3
Total Solution Time (min) 46.01
667MHz. Pentium III PC
Profiles Optimal Configuration
F
PP6 = 1.292 mole/sec95% Water
PP1 = 5.158 mole/sec95% Methanol
PP4 = 0.836 mole/sec95% Ethanol
39
38
35
PP5 = 2.376 mole/secAzeotrope
622 kW
260 kW
200 kW
4 out of 10 sections deleted
Conclusions
1. Distillation optimization with rigorous models remains major
computational challenge
2. Optimal feed tray and number of trays problems are solvable
Keys: Initialization, MINLP/GDP models
3. Synthesis of complex columns produces novel designs (non-trivial)
Progress with initialization, GDP, decomposition
Future challenges: General azeotropic problem See Bruggemann, Marquardt; Wasylkiewicz; Vasconcelos, Maciel
Simultaneous design and heat integration See Caballero et al; Gani, Jorgensen; Alstad et al., Rong et al.
Reactive Distillation See Sand et al; Thery et al., ; Alstad et al., Rong et al.; Dragomir, Jobson; Al-Araf; Urdaneta et al.; Bonet
Global optimizationSee Floudas