optimal synthesis of complex distillation columns using ...optimal synthesis of complex distillation...

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