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Synthesis and Optimization of Separation Sequences

Synthesis and Optimization of Separation Sequences. Libin Zhang and Andreas A. Linninger Laboratory for Product and Process Design , Department of Chemical Engineering, University of Illinois, Chicago, IL 60607, U.S.A. A. AB. ABC. B. BC. ABCD. ABCDE. BCD. C. BCDE. CD. CDE. D.

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Synthesis and Optimization of Separation Sequences

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  1. Synthesis and Optimization of Separation Sequences Libin Zhang and Andreas A. Linninger Laboratory for Product and Process Design, Department of Chemical Engineering, University of Illinois, Chicago, IL 60607, U.S.A.

  2. A AB ABC B BC ABCD ABCDE BCD C BCDE CD CDE D DE E Motivation P1 Which SEPARATION configuration is optimal???? PURITY SPECIFICATIONS XA = 0.99, Xb = 0.001,…. X2 P2 FEED Components: A,B, C,D,E Known Compositions X1 P3 Maximum profitand/or Minimum environmental impact X3 P4 X4 structure and operating conditions P5

  3. Outline • Long-term objectives: “Computational synthesis of separation networks” • Algorithmic Approach to determine feasibility of separation specification • Methodology - Minimum Bubble Point Distance Algorithm (MIDI) • Assessing Feasibility of given Separation Tasks • Feasible Range of Column Operations: Minimum & Maximum Refluxes • Toward Computed Aided Distillation Synthesis • Feasibility Test +Genetic algorithm (Stochastic search) • Temperature collocation on finite element • Reduced search space, but rigorous models (non-ideal separations) • Genetic algorithm and feasibility test • Infeasible path algorithm

  4. Quick and Reliable Feasibility Test PRODUCT A PURITY SPECIFICATIONS XA = 0.99, Xb = 0.001,…. SIMPLE DISTILLATION COLUMN ? FEED Components: A,B, C Known Compositions PRODUCT B SEPARATION TASK FEASIBLE????

  5. 1.6 F 50 1.2 Solution Approaches 1. SIMULATION APPROACH 2. DESIGN APPROACH • Performance Calculation • Flowsheet Simulator • Aspen Plus, HYSYS, Pro/II • Design Calculation • Given Feed & 4 D.O.F • SPECIFICATION FEASIBLE?? ??? Given XA,D XB,D DISTILLATE F Fixed Trays ?? Reflux Given Feed Given Feed Given XA,B ??? BOTTOMS • Trial and Error Approach • Does not Assess Feasibility • Direct Feasibility assessment • Evaluate Column Profiles

  6. x2 Feed x1 Design Approach - Underwood’s Equations NUMERICAL PROBLEM ?? OR INFEASIBLE SPECIFICATION?? • Underwood’s Equations • Highly Non-Linear • Difficult to Converge Rectifying Equations (1) Stripping Equations (2) Profile Intersection: (3)

  7. A Robust Column Design Algorithm • Main ideas • 1. Model Column Profile: • Temperature (Not Trays) • Continuous Profile Equations (Doherty, 1985) • 2. Feasibility Check: • Minimum Bubble Point Distance between Rectifying & Stripping Profiles • 3. Solution Strategy • Finite Element Collocation on Orthogonal Polynomials • Supporting Concepts • Pinch Point • Attainable Temperature Window • Bubble Point Distance

  8. x2 d   x1 Reachable Temperatures - Pinch Points • Fixed points in the composition profiles • Pinch, saddle and unstable points • Newton-Horner (Deflation), Continuation Method, Bounded Newton-Raphson algorithm d: distillation : unstable point : saddle point : pinch point (OL) (EQ) At the pinch point: Let in equilibrium withxi,p Pinch equation:    Residual Error d (PL) Temperature, ºF

  9. B P2 Temperature x2 P1 P1 TBottoms Infeasible column Feed B D B Temperature p2 NO OVERLAP x1 Attainable Temperature Window P2 D p1 P2 D P1 TDistillate If , column is infeasible Attainable Temperature Window • Difference between boiling points and the stable points of rectifying and stripping section

  10. TB TA x2 dB dA x1 (3) (1) (2) Acetaldehyde Methanol Water Bubble Point Distance • Bubble Point Distance: • The Euclidean difference of two points on the rectifying and stripping profile whose BP is equal to T; • Feasibility test: p1, p2; p1rectifying ; p2 stripping; (BPD(p1, p2)) Feasible Min d = 0 Infeasible d > 0 T d

  11. d Infeasible Case Feasible Case d  Log(x3) Point of feasible specification Xd3=0.0005, =0.36  2. Minimum Bubble Point Distance(MIDI) Dimensionless Temperature: s.t.

  12. n+1 n Modeling Column Profile:Continuous Differential Equation (Doherty, 1985) • Continuous differential equation for evolution of column profile (Doherty,1985) • Taylor expansion truncated after first term Continuous column profile with independent variable h Stripping Section: Eq. (1) Rectifying Section: Eq. (2)

  13. Temperature Collocation of Column Profiles • Temperature is monotonically increasing in distillation column (normally) • Temperature is bounded : from Distillate and Bottom temperature to the pinch points Implicit Differentiation Eq. (3) Stripping profile: Eq. (4) Rectifying profile: Eq. (5)

  14. x Polynomials · x3[i]= x0[I+1] · · 0th Node Lagrangian polynomials x2[i] · · · x1[i] · x3[I-1]=x0[i] · ´ ´ ´ ´ ´ ´ ´ ´  1st Node Lagrangian polynomials D P1 Collocation nodes Element 1 Element NS Element i Element i+1 2nd Node Lagrangian polynomials 3rd Node Lagrangian polynomials 3. Temperature Finite Element Collocation • Globally collocate entire profile between two temperatures • Using 3-5 finite elements and 2-3 nodes

  15. Branch of saddle pinches c b Branch of maximum nodes of composition profile c’ a b’ x,Composition a’ Temperature, ºF Element Placement • Saddle temperature place an element boundary at the composition profile • The saddle coincide with maximum curvature of the intermediate species

  16. Column Profile Maps Mark Peters, University of the Witwatersrand, South Africa and Lei Huan

  17. Derivation of CPMs Column Section Infinite reflux case (RD ∞): Difference Point Equation (DPE): reflux ratio difference point net molar flow Entire RCM

  18. Derivation of CPMs Finite reflux case: XD = [0.9, 0.05, 0.05] RD = 9 CPM is a simple transform of RCM XDis in the MBT (Region 1)

  19. What happens when XDis in the other Regions?

  20. Pinch Point Loci How do the nodes (pinch points) move as RD changes for a set XD ?

  21. Moving Triangles… A novel concept and design tool…

  22. Personalize Your Column...

  23. Global Terrain Method Global Terrain Method (example) • Equations • Feasible region • Starting point • (1.1, 2.0) 3D space of case 1

  24. Global Terrain Method Global Terrain Method • Basic concept of Global Terrain Method (Lucia and Feng, 2002) • A method to find all physically meaningful solutions and singular points for a given (non) linear system of equations (F=0) • Based on intelligent movement along the valleys and ridges of the least-squares function of the system (FTF) • The task : tracing out lines that ‘connect’ the stationary points of FTF. • Mathematical background • Valleys and ridges in the terrain of FTF could be represented as the solutions (V) to: V = opt gTg such that FTF = L, for all L єL F: a vector function, g = 2JTF, J: Jacobian matrix, L: the level-set of all contours

  25. Global Terrain Method Global Terrain Method • Applying KKT conditions to this optimization problem we get the following Eigen value problem Hi : The Hessian for the i th function • Thus solutions or stationary points are obtained as solutions to an eigen-value problem where the Eigen values are identical to the KKT multipliers • Initial movement • It can be calculated from M or H using Lanzcos or some other eigenvalue-eigenvector technique (Sridhar and Lucia, 2001) • Direction • Downhill: Eigendirection of negative Eigenvalue • Uphill: Eigendirection of positive Eigenvalue

  26. Application 1: Constant Alpha Mixtures • Compute rectifying profile and stripping profile with pseudo-temperature respectively. • Pseudo Temperature: • Then search the minimum distance between profiles. ATW is not empty But Specification is infeasible MIDI = 0.295 B Feasible MIDI 0 P1 Temperature P2 D Xd1= 0.95;xd2=0.049,xb1=0.05;R= 1.0 Xd1= 0.95;xd2=0.049,xb1=0.05;R=2.5

  27. 1.0 0.8 Saddle point Saddle point 0.6 x2 r = 5 x2 r=2.5 0.4 0.2 Feed Distillation Bottoms 1 Methanol 2 Ethanol 3 n-propanol Feed Distillation Bottoms 1 Pentane 2 Hexane 3 Heptane 0.3000 0.9800 5.0•10-4 0.0 0.2500 0.3000 0.3000 0.9500 0.0490 0.0200 0.3513 0.0500 0.3965 0.0 0.2 0.4 0.6 0.8 1.0 Feed x1 x1 0.4500 5.0•10-11 0.6482 0.4000 0.0010 0.5535 Feed (2) (2) (3) (3) (1) (1) Hexane Methanol n-propanol Heptane Pentane Ethanol Application 2: Ideal Mixtures Feasibility test works for both sloppy and sharp split in ideal mixture

  28. (2) (2) (2) B B B (1) (1) (1) A A A (2) (2) (2) (4) (4) (4) Ethanol Ethanol Ethanol D D D (2) (2) (2) Hexane Hexane Hexane (1) (1) (1) Methanol Methanol Methanol (1) (1) (1) Pentane Pentane Pentane (4) (4) (4) Acetic acid Acetic acid Acetic acid (4) (4) (4) Octane Octane Octane Application 3: Quaternary Mixtures Constant alpha mixture Ideal mixture Non-ideal mixture

  29. Random Experiments Random Experiments Random Experiments No. of design specifications No. of design specifications No. of design specifications 10000 10000 10000 No. of feasible designs No. of feasible designs No. of feasible designs 468 533 223 No. of infeasible designs No. of infeasible designs No. of infeasible designs 9532 9777 9463 Convergence failures Convergence failures Convergence failures 0 0 0 Execution time (s) Execution time (s) Execution time (s) 459.8 1017.5 39.4 1.0 (2) (2) (2) Methanol Methanol Methanol 1.0 1.0 0.8 0.8 0.8 0.6 Feed (2) (1) (3) (3) (2) (1) 0.6 0.6 Hexane Pentane Heptane Heptane Hexane Pentane x 0.4 2 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.0 0.0 0.0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1.0 1.0 x (3) (3) (3) (1) (1) (1) 1 Water Water Water Acetaldehyde Acetaldehyde Acetaldehyde Application 4: Feasible Regions Column Specification Search Space (10,000 possibilities) Feasible region Constant alpha 39.4s Feasible region Ideal Mixture Feasible region Non-Ideal Mixture x2 468.8s 1017.5s x1

  30. Synthesis of Separation Sequence

  31. 0 D 1 m F n 1 0 B B Azeotrope F D Challenges • Problem Size of Column Sequences • Large number of state variables (compositions, temperatures,…) • Highly non-linear relationships; • Vapor-liquid equilibrium model • Local convergence • Search for Structural and Parametric Design Variables • Generate structural alternatives • Find optimal parameters without getting trapped in local minima • Converge to global solution in reasonable time; • Solution Approach: • Feasibility test and genetic algorithm • Infeasible path algorithm • Reduced search space and minimum design variable set • Rigorous models

  32. Temperature Collocation Algorithm(Zhang and Linninger, IECR 2004) Coordinate Transformation of Column Profiles into DAE Orthogonal Collocation on Finite Element Bubble points = independent variable BPD (T): Rigorous Feasibility Criterion= : min BPD ~ 0; Infeasible Min bpd > 0 Feasible Min bpd = 0 TB T TA x2 dB dA bpd x1

  33. 1.0 1.0 1.0 0.8 0.8 0.6 0.6 0.8 0.4 0.4 0.2 0.2 0.6 0.0 0.0 0.2 0.4 0.6 1.0 0.8 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 (1) (3) (1) (3) (2) (2) Benzene Benzene Acetone Chloroform Chloroform Acetone Temperature Collocation Algorithm-Results(Zhang and Linninger, 2004) • Column Profiles = DAE • 10,000 specifications in 39.4 CPUs • Robust Convergence for feasible and infeasible specification 10,000 random specifications 39.4 s for finding all feasible specs Azeotropes/ non-ideal mix. s.t.

  34. 1.0 1.0 OCFE result 0.8 0.8 Hysys result OCFE result 0.6 0.6 Hysys result 0.4 x2 x2 0.4 0.2 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 (3) (2) (1) (1) (2) (3) x1 x1 Hexane Diethyl Ether Benzene Octane Toluene Hexane Temperature Collocation-High Accuracy • Size reduction: ~1 Order of Magnitude • Hysys~600 equation TC-OCFE~50 equations • High Fidelity Results: • Small differences attributable continuous vs, tray-by-tray (1st order appr.) • V-L Equilibrium implementation

  35. F3 F1 B2 B3 B1 F1 Reduced Search Space Minimum set of design Variables Each individual different designs (sequence + operating conditions) MASTER GA Genetic Algorithm K individuals [D] [A] [B][C] Feasibility Test Population State variables dramatically reduced by temperature collocation Feasibility test ([A][B])([C][D]) Feasibility test High performance Inverse problem

  36. P1 P2 F1 A(BC) F2 AB(C) B1 (A)B(C) ABC P3 (A)BC (AB)C Problem Representation - Column Sequencing Chromosome: • Only products ordered by relative volatility • Mass flowrate in product of each specie • Structure: Integer string ProductI Product II Product III Reflux Structure Encoding Operating Conditions Product Specs

  37. P1 P2 F1 F2 F2 P3 F3 B2 B1 F1 F3 B2 B3 B3 B1 P4 P2 F2 OFFSPRING F1 B2 P4 B1 F3 P3 P1 B3 Crossover Example 2 Patents -> Offspring with parameter and structure variation Mathematical Formulation Operational Parameters’ Crossover Structural Crossover (2) (4) F1 After Crossover, mass balance is still valid (3) (1)

  38. F2 F3 B2 F1 B3 B1 P2 P4 P3 P1 Structural and Parametrical Mutation (2) Operational parameters (4) F1 (3) (1) Integer parameters P1 P2 F1 F2 P3 B1 F3 B2 B3 P4

  39. 12 10 8 Cost Function 6 12 4 10 2 0 5 10 15 20 25 30 Generation 8 6 Does not stabilize Does not Convergence 4 2 0 5 10 15 20 25 30 Fitness and Convergence 12 10 Mutation & crossover 8 Cost function Op. Cost + Cap. Cost+ Penalty 6 4 2 0 5 10 15 20 25 30 Generation Crossover without mutation Mutation without crossover Cost function Generation

  40. 1.0 1.0 1.0 0.8 0.8 0.8 0.6 0.6 0.6 x2 x2 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 x1 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x1 Initial Population of Ternary Mixture • Different initial population method • Given the composition of all streams • All candidate structure • Only given the composition of products Patterned initial guesses All infeasible designs Random initial guesses

  41. Infeasible Path Approach • Feasible Individual increase at beginning evolution • Trend to stabilize at the end • Optimal Sequences even from all infeasible initial population No of Feasible sequences No of Feasible sequences Generation Generation Only infeasible designs initially

  42. P1 P2 F1 P1 B1 1 Pentane 2 Hexane 3 Heptane F1 D1 P3 1 Pentane 2 Hexane 3 Heptane F1 0.3000 0.3000 0.5221 0.9900 0.0122 0.0063 F2 P1 0.4000 0.4000 0.0027 0.0018 0.9834 0.6375 B1 F2 P3 D1 1.0 1.0 P2 P2 0.3000 0.3000 0.4762 0.0082 0.3503 0.0103 F1 P2 0.8 0.8 P3 0.6 0.6 D1=F2 B1=F2 0.4 0.4 F1 F1 0.2 0.2 P3 P3 P1 P1 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Case study I: Ideal Ternary Mixture Generation Evolution Min. Cost = 2.56 X103 Min. Cost = 3.16 X103

  43. Feed Distillation Bottoms 1 Acetone 2 Chloroform 3 Benzene 0.2000 0.9900 0.0122 0.7000 0.0018 0.6375 0.1000 0.0082 0.3503 (1) (3) (2) Chloroform Benzene Acetone Case study II: Azeotropic Mixture D1(Acetone) 1.0 D2(Chloroform) D2 F1 0.8 F2 II Azeotrope 0.6 B1 I 0.4 B2(Benzene) 0.2 B1 Generation evolution D1 F1 B2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Penetrate the curved boundary

  44. 1.0 0.8 0.6 (2) (3) (1) 0.4 Ethanol Ethylene glycol Water 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Case study III: Entrainer Azeotropic Separation D2 D1(Ethanol) FUpper D2(Water) F FLower F2 F B2 FLower B3(EG) B1 FUpper D1 Flow sheet for separation of water and ethanol with entrainer Break azeotropic with entrainer

  45. 1.0 0.8 R=5.0 0.6 Fr=3.5 Stripping Section 1.0 0.4 D1(Ethanol) 0.8 Middle Section Lower Feed 0.2 FUpper D2(Water) Upper Feed Average Feed M R=4.3 M’ 0.6 Fr=2.14 FLower 0.0 PM 0.0 0.2 0.4 0.6 0.8 1.0 Rectifying Section (2) (1) (3) (2) (3) (1) Stripping Section F2 Water Ethylene glycol Ethanol Water Ethylene glycol Ethanol 0.4 B1 Middle Section Lower Feed 0.2 Upper Feed Average Feed B2(EG) M 0.0 PM 0.0 0.2 0.4 0.6 0.8 1.0 Rectifying Section Case study III: Entrainer Azeotropic Separation Suboptimal solution Best solution

  46. P1 P2 F1 F2 P3 (3) (1) B1 F3 Methanol n-propanol B2 (4) B3 P4 Acetic Acide F3 F1 B2 B3 B1 Case study IV: Quaternary Mixture (2) Ethanol P2 Optimal Cost: 5.98X103 P4 F1 P1 P3

  47. 1.0 P1 Column C1 0.8 0.6 F1 B1 0.4 0.2 P1 1.0 Column C2 B2 0.8 D2 0.6 F2 0.4 B2 0.2 D2 F3 1.0 D2 P2 P3 F1 C3 Column C3 C1 F2 0.8 C1 A(BCDE) C2 F3 (4) (5) (4) (3) (2) (1) (5) (2) (4) (3) (2) (3) (5) 0.6 B1 F4 1-hexanol 1-hexanol 1-octanol 1-heptanol Isobutanol 1-pentanol 1-pentanol 1-heptanol 1-heptanol 1-octanol 1-octanol 1-hexanol 1-pentanol B2 AB(CDE) C4 0.4 F3 (A)B(DE) ABC(DE) 0.2 P2 P3 C3 (A)BC(DE) ABCD(E) 1.0 C3 P4 P5 Column C4 ABCDE C2 (A)BCD(E) (AB)C(DE) F4 0.8 (A)BCDE C1 (AB)CD 0.6 (AB)CDE (ABC)D(E) 0.4 (ABC)DE C4 F4 C2 0.2 C4 P4 P5 (ABCD)E 0.0 480 380 400 420 440 460 Temperature, K Case study V: Five Component Mixture

  48. Define cost function, cost, variables Select GA parameters Generate initial population Find cost for each chromosome Convergence Check No Evaluate sharing factor for each chromosome Yes Calculate shared fitness Select mates Mating Mutation Solution “Niche” Evolutionary methods • Niche methods work on the principles of genetics and natural selection. • They work on a population of possible solutions, while other gradient based methods use a single solution in their iterations. • They are probabilistic (stochastic), not deterministic. • They are capable of detecting multiple optima Flow chart of a continuous GA

  49. Niche Evolutionary Methods • Genetic algorithms for problem with multiple extrema (multi-modal) • Use the concept of fitness sharing • Restricts the number of individuals within a given niche by “sharing” their fitness, so as to allocate individuals to niches in proportion to the niche fitness niche radius No fitness sharing Fitness sharing

  50. Niche Evolutionary Methods (example) • Spots 4 maxima and 1 saddle point

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