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Optimal Synthesis of Complex Distillation Columns Using Rigorous

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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)
                Rigorous Models


        Computational Challenges
 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


Goal: Review previous work
     Propose decomposition strategy for complex columns
               MINLP Algorithms

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)
          Methods Generalized Disjunctive Programming

                                       GDP



            Logic based methods                        Reformulation MINLP
                                                         Outer-Approximation
                                                         Generalized Benders
                                                        Extended Cutting Plane


Branch and bound            Decomposition
(Lee & Grossmann, 2000)     Outer-Approximation        Convex-hull Big-M
                            Generalized Benders         Direct
                          (Turkay & Grossmann, 1997)
                                                        Cutting plane
                                                         (Lee & Grossmann, 2000)
                    Remarks on methods

MINLP and GDP can be applied to optimize discrete
and continuous decision in distillation design
        Discrete: 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
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
fi

                          Min cost
                       st MESH eqtns
                          ‡”f    i   = F
                           ¸
                         i LOC



                      NLP VMP: Variable-Metric Projection
                                Optimal Feedtray Location (Cont)

                                              Viswanathan & Grossmann (1990)
                                       D
                      1    1                    MINLP Formulation
                 L1             V2
                           2
                                                    Min cost
                 L2             V3

    fi           L3
                           3
                                                st MESH eqtns
                                V4
    zi                     .                        ‡”z        i   =1
F                                                    ¸
                                                   i LOC
                           .
                                                   ‡”f     i       =F
                                                    ¸
                                                  i LOC
             LN-                VN-2
             3            N-2
                                                   f i - F zi ¡Ü 0 i ¸LOC
         LN-                    VN-1               zi = 0,1 i ¸LOC
                          N-1
         2
             LN-1               VN         MINLP DICOPT: AP-Outer Approximation-ER
                           N



                                       B   Remark: MINLP solves as relaxed NLP!

                                              Feed tray composition tends to
                                              match composition of feed
                     Optimization of Number of Trays
Viswanathan & Grossmann (1993)
                                                                                    Non-existing tray



         zri = 0,1                                                                        Vapor Flow
                                                      zrm = 1   No liquid on tray
                             Number                                                           Existing trays
         MINLP =>            trays
                                                    zbn = 1                              Liquid Flow

                                                                                      Non-existing tray
         zbi = 0,1
                                                                No vapor on tray



  Discrete variables: Number of trays, feed tray location.
  Continuous variables: reflux ratio, heat loads, exchanger areas, column diameter.

        Zero flows- Discontinuities appear, convergence difficulties.
            Redundant equations are solved- Increases CPU time.
                      Optimal Design Columns with Multiple Feeds

                               Air Products & Chemicals                      Viswanthan & Grossmann (1993)
                       Separation Methanol - Water with 3 Feeds

                                                                               MINLP model
                                                                  60             Virial/UNIQUAC
                                           59                                  115 0-1 binary variables

                                                                               1683 continuous variables
            3
     F
                (0.85, 0.15)
                                                                               1919 constraints

            2                                       ri
     F                                                                     Solved with DICOPT on a HP 9000/730
                (0.5, 0.5)
                                                                                  (5 major iterations, 45 min)

    F
        1                                                                        Optimal solution
                (0.15, 0.85)

                                                                                 Number of trays = 53

                                                                                 Feed location:
                                            2                                    Feed 1      Tray 4
                                                                                 Feed 2      Tray 6
700,000 alternatives!                                                            Feed 3      Tray 12
                                                                       1
     Continuous Optimization Approach
                                                         Lang & Biegler (2001)
Basic idea: continuous approximation of 0-1 variables

            Differentiable Distribution Function




                        σ     parameter
                        Nc     variable

            If σ   ¨0   Nc = i         =>      ¨
                                            di 1
    di used to multiply flows into tray fidi
  Highly nonconvex: requires good initial guess
                            See Neves, Silva, Oliveira
     Disjunctive Programming Model
Yeomans & Grossmann (2000)

Permanent trays:                            Permanent
                                            trays
Feed, reboiler, condenser                                        Conditional
                                                                 trays
Conditional trays:
Intermediate trays might
be selected or not.
               Trays not allowed to “disappear” from
               column:
                    VLE mass transfer if selected.
                Disjunction   VLE        -OR-
                   No VLE, trivial mass/energy balanceNOTnotVLE
                                                        if bypass)
                                                      (tray
                   selected
                Single Column GDP Model
Light
Product
                          Condenser Tray
                          (permanent)               • Permanent and
                                                      conditional trays:
                 -OR-


                 -OR-
                        }     Rectification Trays
                              (conditional)
                                                       – MESH equations
                                                         for condenser,
                                                         reboiler and feed
 Feed
                         Feed Tray
                         (permanent)
                                                         trays
                                                       – Mass & energy
                 -OR-


                 -OR-
                        }    Stripping Trays
                             (conditional)



                        Reboiler Tray
                                                         balances for
                                                         rectification and
                                                         stripping trays.
Heavy Product           (permanent)                      Equilibrium Stage      Vapor Flow


                                                    • Conditional trays
                                                        Non-equilibrium Stage   Liquid Flow



                                                      only:
                     Which model is better?

                           Barttfeld, Aguirre & Grossmann (2003)
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

        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.
                              D               D              D
                                       F              F
                         F
  Variable feed and
                                                             B
  reboiler location
                              B               B



                              D               D


  Variable feed and      F
                                                             D

  condenser location                  F               F


                              B               B              B



                              D               D
                                                             D
Variable condenser and   F            F               F
   reboiler location
                                                             B
                              B               B
                   GDP Representation Alternatives
Permanent Trays (top and bottom stages) are fixed stages in the structure. Existence
   of each Intermediate tray modeled with a disjunction
                                        D               D
                                                                        D

         Fixed feed location   F                F               F

        (Yeomans, Grossmann,
                                                                        B
               2000)
                                        B               B

                                        D                   D



                                                                            D
                               F
          Variable feed
                                                    F               F
          location (bot)

                                        B                   B               B



                                        D                   D
                                                                            D
          Variable feed                             F
                                                                    F
          location (top)       F

                                                                            B


                                        B                   B
                   Solution approaches

              MINLP                                               GDP


                General                                         General
             Preprocessing
                               Aggregate NLP                 Preprocessing
                               NLP fixed max number trays        Phase
                 Phase



              RMINLP                                         NLP1 solution:
             Preliminary                                    All trays existing
              Solution


                                                            NLP2 solution:
Heuristic     Reduction of                                                       Optional
                                                             Subset trays
            Candidates Trays




            Reduced MINLP                                        GDP
               Solution                                        Algorithm
                   Logic-based OA Algorithm

                                                   Turkay, Grossmann (1996)


OA Algorithm    Selected Equations

                                                 Subproblem
                                                    (NLP)                    Solution

Linearization
of Nonlinear    Selection of Continuous                                             YES
 Equations      Disjunctions variables for                          NO
                                       Initialization
                                                                            Converge?
                  Discrete variables                                                    Initialization
 Big-M form
                                              Master Problem                               Initial
  of linear
                MILP form of                      (MILP)                                Subproblems
disjunctions
                Disjunctive                                                                (NLP)
                equations
Pre-
Processing


                             Data flow                         Algorithm cycle

                                 Implemented in GAMS CPLEX/CONOPT
                 General trends of results

Best MINLP Model: Variable feed/reboiler
Best GDP Model: Fixed feed Yeomans & Grossmann (2000)
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
    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                                              D (98% Benzene)
     – Composition: 0.33/0.33/0.34                               1        242.65 kW
     – Feed: 10 mol/sec
     – Upper number trays: 35                                    9
                                                             F
     – Recovery, purity distillate: 98%
                      Preprocessing (NLP)
       Continuous Variables                   3273                        258.95 kW
                                                                 20
       Constraints                            2674
                                                                          B
       Time [CPU s]                            0.68
                                                       Relaxed Solution RMINLP - 79,223 $/yr
                     Rigorous Model (MINLP)
                                                                           D (98% Benzene)
       Continuous Variables                   1507
                                                                 1        241.7 kW
       Binary Variables                         33
       Constranits                            1830
                                                                 9
       Iterations                               17           F

       Time RMINLP [CPU s]                     0.52
       Time MINLP [CPU min]                    10.81             18       258 kW
       Total Cost [$/año]                   79,962                         B
                                                       Integer Soluiton MINLP – 79,962 $/yr
                                    Example GDP

                                          Methanol/ethanol/water - GDP: fixed tray location
          GDP Formulation                             Preprocessing Phase: NLP tray-by-tray Models
                                          Continuous Variables                              1597
Mixture: Methanol/Ethanol/water           Constraints                                       1544
Feed Flow= 10 mol/sec                     Total CPU time (s)                                1.12

Feed composition= 0.2/0.2/0.6                                          Model Description
                                          Continuous Variables                              2933
P = 1.01 bar                              Binary Variables                                   60
                                          Constraints                                       2862
Product Specification:                    Nonlinear nonzero elements                        5656
products composition reversible model     Number of iterations                               10
                                          NLP CPU time (s)                                  9.14
Upper bound No. Trays: 60                 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

  Extension Single Column GDP Model                Jackson & Grossmann (2001)

• Conditional Trays:

     Active Trays
         Separation with reaction may take place
           – Positive liquid holdup
         Separation only make take place
           – Liquid holdup equals zero
                                                      Active Trays
    OR

     Inactive Trays
         Input-Output operation with no mass
         transfer and no reaction


                                                     Inactive Trays
               Example: Metathesis of Pentene
Conversion of 2-cis-pentene into
2-cis-butene and 3-cis-hexene:   2C5 H 10 ⇔ C 4 H 8 + C 6 H 12

     GDP Model: 25 discrete variables
              731 continuous variables
              730 constraints
•    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
            Synthesis of complex distillation systems

                                              Mariana Barttfeld, Pio Aguirre/INGAR

Superstructure Representation
–      Suitable for zeotropic and azeotropic mixtures
–      General and automatically generated
–      Includes thermodynamic information
–      Embeds many possible alternative designs

Superstructure Formulation
              GDP formulation

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 for Synthesizing Configurations

             Sargent and Gaminibandara (1976)
Generated with the State-Task-Network (STN) (Sargent, 1998)
               STN Representation                    Sargent-Gaminibandara Superstructure
            (4 Component Zeotropic Mixture)                  (4 Component Zeotropic Mixture)
                                                                                       A
                                      A                                 AB

                          AB
                                                                ABC
                ABC                   B
                                                                                       B

     ABCD                 BC
                                                                         BC
                                                      ABCD
                BCD                   C


                          CD
                                                                                      C

                                     D
                                                                BCD

                         States                                         CD

                         Tasks                                                         D




                                  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
                                                                          A
    RDSM-based STN Representation
                                                                     AB
    (4 Component Zeotropic Mixture)
                                            A

                                                                          B
                                  AB
                                                               ABC
                       ABC
                                            B
                                                                     BC
                                  BC
                                                                          C
           ABCD                                         ABCD
                                                                          B

                                  BC                                 BC
                                            C
                       BCD

                                                                          C
                                  CD                           BCD

                  States
                                            D
                  Tasks                                              CD

                           Avoid mixing intermediates                     D
             Modification for Azeotropic Mixtures
                                        A




Product
                                                              • RDSM-based STN cannot be defined a
                                                                priori
                                        ABC
Azeotrope
Mass Balance                        F                         • Composition diagram needed
Distillation Boundary
                                                              • Azeotrope recycled
                              BC
                  C                                       B
                                              BC-Azeo


                                                                                                       A
      RDSM-based STN Representation
          (4 Component Azeotropic Mixture)                    A                             AB

                                                   AB
                                                               B                 ABC                   B
                                        ABC

                                                              Azeo
                                                   BC
                                                                                            BC
                        ABC                                               ABC
                                                               B                                       B

                                                   Azeo

                                        BC
                                                                                       BC            Azeotrope
                                                    C
                               States
                                                                                                 C
                               Tasks
                    Superstructures

                           A


               AB
                                                            A



                                                 AB
                           B

         ABC
                                      ABC                   B

               BC
                           C
ABCD
                           B                     BC
                               ABC
               BC                                           B




                           C
         BCD                                BC            Azeotrope


               CD
                                                      C
                           D




       Zeotropic Mixture              Azeotropic Mixture
                     Mapping to Specific Designs
                                      A
                                                                               A
                     AB   4
                                                            ABC
                                                                                        B
                                                                  2

                 2                    B
           ABC                                                                 5
                          5
                                                                      BC
                     BC                                                                 C
ABCD                                  C   ABCD          1
       1                                                                                B
                                      B
                     BC                                               BC
                          6                                                    6


                 3                    C
           BCD                                                    3                     C
                                                            BCD
                          7
                     CD
                                                                               D
                                      D

                              A                                            A

           ABC
                 2                B                                                         B

                     BC
                                                                               BC
                              5                                                     5

ABCD   1
                     BC                          ABCD                          BC
                              6                                                     6




                 3
                                  C                                                         C
           BCD

                              D                                            D
     Discrete Decisions

Two hierarchical levels
1. Selection sections
2. Selection Trays




section s
              Selection of sections    Configuration
                 If section selected Ys = True
                 If section not selected  Ys = False
section s+1
              Configuration Model Formulation


                                           min        z = TAC                           Objective Function
                                           s.t. g( x ) ≤ 0
                                                                                         Overall Process
                                                   h( x ) = 0                             Constraints

                                                              ¬ Ys
                                                        f nL,i = 0
                                                                                             Section Boolean
                                                             V
                                                        f   n ,i   =0                           Variables
                                                       TnV = TnV+1
DISJUNCTION              Ys
                                                       TnL = TnL 1
                  ntrays =              stg n     ∨            −         ∀ s ∈ S,i ∈C
                             n ∈ secs                  Vn = 0
                                                       Ln = 0
                                                       xn ,i = xn −1,i
                                                       yn ,i = yn +1,i
                                                       ntrays = 0

                                                Ω( Y ) = True                       Logic Relationships

                                         x ∈ X ,Ys ∈ {True, False}
               Selection of Trays


                   • Permanent Trays

Permanent             – Fixed stages: condenser, reboiler and
   Tray                 feed trays

Intermediate          – Interconnect columns
    Tray

                      – Heat exchange takes place

                   • Intermediate Trays
                      – Use DISJUNCTIONS for modeling
                         Wn = True         Wn = False
                         apply VLE   OR apply by − pass
                         equations        equations

                      – If section selected (Ys = True)
                 Configuration Model Formulation

                                                       min     z = TAC                                  Objective Function
                                                       s.t. g( x ) ≤ 0
                                                                                                        Overall Process
                                                              h( x ) = 0                                 Constraints
  Tray Boolean
   Variables                                      Ys                                                              Section Boolean
                                                             ¬ Wn                   ¬ Ys                             Variables
                               Wn                         f nL = 0
                                                             ,i                  f nL = 0
                                                                                    ,i

                   f nL = f ( Tn ,Pn ,xn ,i )
                      ,i                                  f nV,i = 0             f nV,i = 0
                   f nV,i = f ( Tn ,Pn , yn ,i )         TnV = TnV+1             TnV = TnV+1
DISJUNCTION        f nL = f nV,i
                      ,i                                 TnL = TnL 1             TnL = TnL 1
                                                   ∨             −           ∨           −         ∀ s ∈ S,i ∈C
                   TnV = TnL                             Vn = Vn +1              Vn = 0
                   LIQn ,i = Ln xn ,i                    Ln = Ln +1              Ln = 0
                   VAPn ,i = Vn yn ,i                    xn ,i = xn −1,i         xn ,i = xn −1,i
                   stg n = 1                             yn ,i = yn +1,i         yn ,i = yn +1,i
                                                         stg n = 0               ntrays = 0
                  ntrays =                stg n          ∀ n ∈ secs
                               n ∈ secs

                                                             Ω( Y ) = True                          Logic Relationships
                                                             Ω(W ) = True
                                                       x ∈ X ,Ys ,Wn ∈ {True, False}
           Detailed Cost Functions

   Annual Cost                              Cinv
                     TAC = Cop +
                                            Tdep
                                   Qc                Qh
Operating Cost       Cop =                  Cagua +        Cvapor
                               Cpagua ∆Tcon         ∆H vap
Investment Cost      Cinv = Ccol + Ctray + Creb + Ccond
    Column Cost      Ccol = kcol nt Dcol1.066 htray 0.802

       Tray costs    Ctray = ktray nt Dcol1.55 htray

    Reboiler cost    Creb = kreb Areb 0.65
  Condenser Cost     Ccond = kcond Acond 0.65
                       Dcol ≥ Dtrayn
                                            0.5              0.5
                                                  Tvapor R
          Dtrayn = kd Vn       PM i yn ,i
                           i                         p
                Solution Strategy


Preprocessing    -Initialization Phase-
                                          Aggregate NLP
                           NLP            NLP fixed max number trays
   Phase
                       Problems



GDP Section
                -Selection of Sections-
                                               Fixed Max
                         MILP                  Number Trays
  Problem              Problem


                 -Selection of Trays-
                        MILP                    Fixed Number
 GDP Tray                                       Sections
                      Problem
  Problem


                     Reduced NLP
                       Problem
                                                    Algorithm Cycle
                               Zeotropic Example (1)

                                                                                                            PP1
               Problem specs              Superstructure
Mixture: N-pentane/ N-hexane/ N-heptane
Feed composition: 0.33/ 0.33/ 0.34
Feed: 10 moles/s
                                                                              F
Pressure: 1 atm                                                                                             PP2
Max no trays: 15 (each section)
Min purity: 98%
Ideal thermodynamics
                                            Initialization
                                                                                                            PP3
                                                1.0

 GDP Model                                      0.9

                                                0.8

                                                0.7
 Discrete Variables              96
                                                0.6

 Continuous Variables           3301            0.5

 Constraints                    3230            0.4

                                                0.3

                                                0.2

                                                0.1

                                                0.0
                                                      0.0   0.1   0.2   0.3   0.4   0.5   0.6   0.7   0.8    0.9   1.0
                                                                              Zeotropic Example (2)

                                                                                                                     PP1                                                   PP1
Optimal Configuration                                                                    1
                                                                                                                     98% n-pentane                                         98% n-pentane
                                                                                                                                                  1
     $140,880 /yr                                                                                                                                                Qc = 271.3 kW
                                                                                                       Dcrect2 = 0.6 m
                                                                                        14
                                                                                                                                                                 Qc = 52.4 kW
                                                                                                                                               14
                                                                  1                                    Dcstrip2 = 0.45 m
                                                                                                                                                       1    12
                                                                                                  23                                                                  1
                                                        F       12                                                                         F   26
                                                                                                                                                                                9
                                                                                                                                                                                              PP2
                                                 Dc1 = 0.45 m                                1                       PP2                                                                      98% n-hexane
                                                                                                                     98% n-hexane              36
                                                                                                                                                                 48.8 kW
                                                                22
                                                                                                       Dcrect3 = 0.45 m
                                                                                                                                                                     19
                                                                                        10             Dcstrip3 = 0.63 m
                                                                                                                                                                     32             QH = 298.8 kW
                                               All sections selected                    23
                                                                                                                                                                                             PP3
                             1.0                                                                                   PP3                                                                       98% n-heptane
                                                                                                                   98% n-heptane
                             0.9
   Mole Fraction n-pentane




                             0.8
                                                                                  Feed                                                                     Optimal Design
                             0.7
                                                                                  Col 1 (tray 1 to 14)
                             0.6                                                  Col 1 (tray 15 to 34)                              Annual cost ($/year)                                           140,880
                             0.5
                                                                                  Col 2 (tray 1 al 9)
                                                                                  Col 2 (tray 10 al 32)                              Preprocessing(min)                                               2.20
                             0.4

                             0.3                                                                                                     Subproblems NLP (min)                                            6.97
                             0.2
                                                                                                                                     Subproblems MILP (min)                                           2.29
                             0.1

                             0.0                                                                                                     Iterations                                                       5
                                   0.0   0.1    0.2   0.3   0.4       0.5   0.6   0.7    0.8     0.9     1.0
                                                                                                                                     Total solution time (min)                                       11.46
                                               Mole Fraction n-hexane
                                                                                                                                                      667MHz. Pentium III PC
                                 Zeotropic Example (3)

                                                              PP1                                                   PP1
Optimal Configuration             1
                                                              98% n-pentane                                         98% n-pentane
                                                                                           1
     $140,880 /yr                                                                                         Qc = 271.3 kW
                                                Dcrect2 = 0.6 m
                                 14
                                                                                                          Qc = 52.4 kW
                                                                                        14
                            1                   Dcstrip2 = 0.45 m
                                                                                                1    12
                                           23                                                                  1
                   F        12                                                      F   26
                                                                                                                         9
                                                                                                                                       PP2
             Dc1 = 0.45 m             1                       PP2                                                                      98% n-hexane
                                                              98% n-hexane              36
                                                                                                          48.8 kW
                            22
                                                Dcrect3 = 0.45 m
                                                                                                              19
                                 10             Dcstrip3 = 0.63 m
                                                                                                              32             QH = 298.8 kW
                                 23
                                                                                                                                      PP3
                                                            PP3                                                                       98% n-heptane
                                                            98% n-heptane


                                                                                                    Optimal Design
                                                                              Annual cost ($/year)                                           140,880

Configuration Side-Rectifier              $143,440 /yr                        Preprocessing(min)                                               2.20
                                                                              Subproblems NLP (min)                                            6.97
Direct Sequence                           $145,040 /yr                        Subproblems MILP (min)                                           2.29
                                                                              Iterations                                                       5
                                                                              Total solution time (min)                                       11.46
                                                                                               667MHz. Pentium III PC
                        Azeotropic Example (1)

              Problem Specs                                 methanol
                                     Superstructure
Mixture: Methanol/ Ethanol/ Water                             ehtanol
Feed composition: 0.5/ 0.3/ 0.2
Feed: 10 moles/s
Pressure: 1 atm                                               ethanol
                                                        F
Max no. trays: 20 (per section)
 Min purity: 95%
Ideal/Wilson models
                                                               Azeotrope

                                       Initialization
                                                            Water
GDP Model


Discrete Variables             210
Continuous Variables          9025
Constraints                   8996
                  Azeotropic Example (2)

                                                                       PP1 = 5.158 mole/sec
 Product Specifications 95%                                            95% Methanol


      Optimal Configuration
          $318,400 /yr                             38      260 kW
                                                                       PP4 = 0.836 mole/sec
                                 F                                     95% Ethanol


                                                  622 kW
                                        39                  PP5 = 2.376 mole/sec
                                                            Azeotrope

Profiles Optimal Configuration                      35
                                                            200 kW

                                                                     PP6 = 1.292 mole/sec
                                                                     95% Water

                                     4 out of 10 sections deleted

                                                     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
                                  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 optimization
       See Floudas

				
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