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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