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T. Gundersen

Hierarchical Analysis. Process Integration Methods. Expert Systems. Rules of Thumb. qualitative. Knowledge Based Systems. Heuristic Methods. Process, Energy and System. automatic. interactive. Optimization Methods. Thermodynamic Methods. quantitative. Stochastic Methods

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T. Gundersen

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  1. Hierarchical Analysis Process Integration Methods Expert Systems Rules of Thumb qualitative Knowledge Based Systems Heuristic Methods Process, Energy and System automatic interactive Optimization Methods Thermodynamic Methods quantitative Stochastic Methods Mathematical Programming Pinch Analysis Exergy Analysis Forward Optimization Methods T. Gundersen MP 01

  2. Limitations in Pinch Analysis & PDM • A lot of “heuristics”, not very rigorous • (N – 1) rule for minimum number of units • Bath formula for minimum total area • Composite Curves cannot handle • Forbidden matches between streams • Limitations in for example distillation • Pinch Design Method is Sequential • Targeting before Design before Optimization • One match at a time, one loop at a time, etc. • Time consuming but gives “good” designs Process, Energy and System Optimization Methods T. Gundersen MP 02

  3. What is Mathematical Programming? • Numerical Optimization Techniques • Can handle various Design Problems • Discrete Decisions related to Equipment • Continuous Decisions related to Operation • Process Constraints can easily be included • Material and Energy Balances, Specifications • Equality and Inequality Constraints • Can handle multivariable Trade-offs • Framework for Automatic Design • “wouldn’t it be nice to have?” Process, Energy and System Optimization Methods T. Gundersen MP 03

  4. A small Linear Programming (LP) Problem Solve the Objective Function and Constraints (a) and (b) as Equations with respect to variable x2 Process, Energy and System The LP Problem can be solved by the well-known and heavily applied Simplex Method, but it can also be solved graphically Optimization Methods T. Gundersen MP 04

  5. Graphical Solution for small LP Problem x2 Optimum: at Vertex Algorithm: Simplex Solution: x1=2 , x2=4 Objective: f = 0 8 7 f=4 f=12 f=8 f=0 6 5 Process, Energy and System 4 3 2 1 0 x1 0 1 2 3 4 5 6 7 8 Optimization Methods T. Gundersen MP 05

  6. Ref.: Papoulias & Grossmann Comput. Chem. Engng, 1983 Mathematical Programming & Superstructure Process, Energy and System Optimization Methods T. Gundersen MP 06

  7. Start General MINLP: Branch & Bound MILP master min f(x,y) s.t. g(x,y) ≤ 0 h(x) = 0 xεRnyε <0,1>m Reduced Gradient NLP sub-problem LB > UB f, g, h linear => MILP (or LP) dim(y) = 0 => NLP (or LP) End Mathematical Programming Process, Energy and System Optimization Methods T. Gundersen MP 07

  8. Non-Linear Part Binary Part y1 1 0 y2 1 0 1 0 y3 Combinatorial Explosion Local Optima Problems with Mathematical Programming Process, Energy and System Optimization Methods T. Gundersen MP 08

  9. Stream Ts Tt mCp ΔH °C °C kW/°C kW H1 180 80 1.0 100 H2 130 40 2.0 180 C1 30 120 1.8 162 C2 60 100 4.0 160 ST 280 280 (var) CW 15 20 (var) WS-4 Forbidden Matches Specification: ΔTmin= 10°C Process, Energy and System Q: What is the effect if H2 and C1 are not allowed to exchange heat? Find QH,min, QC,minand the Heat Exchanger Network with and without this forbidden match. Discuss the Degrees of Freedom. Optimization Methods T. Gundersen MP 09

  10. 2 1 3 3 2 1 U = 6 MER Design without Constraints Pinch 70° mCp (kW/°C) 1.0 2.0 1.8 4.0 180° 80° H1 130° 40° Process, Energy and System 70° 43° H2 Cb 6 kW 30° 120° 115.6° 60° Ha C1 54 kW 8 kW 100 kW 60° 100° 90° Hb C2 40 kW 120 kW 60° Optimization Methods T. Gundersen MP 10

  11. ST H1 C1 H2 C2 CW “Extended” Heat Cascade QH 180°C 170°C QH1,1=50 1 130°C 120°C RST,1 RH1,1 QC2,2=160 QH1,2=50 Process, Energy and System 2 QC1,2=108 QH2,2=120 RH1,2 RH2,2 70°C 60°C RST,2 QH2,3=60 3 QC1,3=54 QC 40°C 30°C Optimization Methods T. Gundersen MP 11

  12. ST H1 QP = QPH = 54 kW C1 H2 C2 CW “Extended” Heat Cascade 102 180°C 170°C 50 130°C 120°C 102 50 120 40 50 60 Process, Energy and System 120 48 70°C 60°C 54 54 60 QC 40°C 30°C 60 Optimization Methods T. Gundersen MP 12

  13. QP = QPH = 54 kW 2 1 3 3 2 1 U = 6 Design with Constraints Pinch 70° mCp (kW/°C) 1.0 2.0 1.8 4.0 180° 80° 140° H1 130° 40° Process, Energy and System 70° H2 Cb 60 kW 30° 120° 60° 93.3° Ha Hb C1 48 kW 54 kW 60 kW 60° 100° 90° C2 40 kW 120 kW 60° Optimization Methods T. Gundersen MP 13

  14. ST H1 QP = QPP = 54 kW C1 H2 C2 CW “Extended” Heat Cascade 102 180°C 170°C 50 130°C 120°C 102 50 120 40 60-x 50 Process, Energy and System 120 48+x 70°C 60°C 0+x 54-x 54 60 QC 40°C 30°C 60 Choice: x = 54 kW Optimization Methods T. Gundersen MP 14

  15. QP = QPP = 54 kW 1 3 3 1 U = 5 Design with Constraints Pinch 70° mCp (kW/°C) 1.0 2.0 1.8 4.0 180° 80° 140° H1 2 130° 40° Process, Energy and System 70° H2 Cb 60 kW 30° 120° 63.3° Ha 2 C1 6+54 kW 102 kW 60° 100° 90° C2 40 kW 120 kW 60° Optimization Methods T. Gundersen MP 15

  16. ST H1 C1 H2 C2 CW “Extended” Heat Cascade QP = QPP + QPH = 40 + 14 kW 102 180°C 170°C 50 0+y 130°C 120°C 102 50 120 40-y 60 50 Process, Energy and System 120 48 70°C 60°C 0+y 54-y 54 60 QC 40°C 30°C 60 Choice: y = 40 kW Optimization Methods T. Gundersen MP 16

  17. 1 1 U = 6 Design with Constraints Pinch 70° QP = QPH + QPP = 54 kW mCp (kW/°C) 1.0 2.0 1.8 4.0 180° 80° H1 2 130° 40° Process, Energy and System 70° H2 Cb 60 kW 30° 120° 37.8° 93.3° Ha 2 Hc C1 60+40 kW 48 kW 14 kW 60° 100° 90° Hb C2 40 kW 120 kW 60° Optimization Methods T. Gundersen MP 17

  18. LP Model for Forbidden Matches Process, Energy and System Easily solved by the Simplex Algorithm Optimization Methods T. Gundersen MP 18

  19. MILP Model for fewest Number of Units Process, Energy and System Logical Constraints relating Discrete & Continuous Variables Optimization Methods T. Gundersen MP 19

  20. Status for Mathematical Programming? • Considerable Research in the 1980’s/90’s • CMU, Princeton, Caltech, Imperial College • One “Road” towards Automatic Design • Math Programming provides the Framework • Has the Potential to identify Superior Solutions • Obstacles against Industrial Use • Lack of Knowledge about the Methods • Lack of user friendly Software • Applications require Expertise • Considerable Numerical Problems • The Advantages are many • Can handle Multiple Trade-offs, Discrete Decisions and Constraints in the Design Process, Energy and System Optimization Methods T. Gundersen MP 20

  21. The Sequential Framework −SeqHENS Process, Energy and System Surprisingly few Iterations are needed to identify the Global Optimum Reason: SeqHENS is strongly based on Insight from PA Optimization Methods T. Gundersen MP 21

  22. UMIST Comments after Sabbatical Process, Energy and System Promoting Mathematical Programming was quite challenging in those Days ! Optimization Methods T. Gundersen MP 22

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