Process Integration Methods Process Energy and System Expert

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

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

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

A small Linear Programming (LP) Problem Process, Energy and System Solve the Objective Function and Constraints (a) and (b) as Equations with respect to variable x 2 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

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

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

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

Problems with Mathematical Programming Process, Energy and System Non-Linear Part Binary Part y 1 y 2 y 3 Local Optima 1 1 0 0 1 0 Combinatorial Explosion Optimization Methods T. Gundersen MP 08

Process, Energy and System Stream Ts °C Tt °C H 1 H 2 C 1 C 2 180 130 30 60 80 40 120 100 ST CW 280 15 280 20 m. Cp ΔH k. W/°C k. W 1. 0 2. 0 1. 8 4. 0 100 180 162 160 (var) WS-4 Forbidden Matches Specification: ΔTmin = 10°C Q: What is the effect if H 2 and C 1 are not allowed to exchange heat? Find QH, min , QC, min and the Heat Exchanger Network with and without this forbidden match. Discuss the Degrees of Freedom. Optimization Methods T. Gundersen MP 09

MER Design without Constraints Process, Energy and System Pinch 70° 180° H 1 2 130° H 2 m. Cp (k. W/°C) 80° 1 1. 0 70° 3 43° Cb 40° 2. 0 6 k. W 120° Ha 8 k. W 100° Hb 40 k. W 115. 6° 60° 2 3 1. 8 54 k. W 100 k. W 90° 30° C 1 1 60° C 2 120 k. W U=6 4. 0 60° Optimization Methods T. Gundersen MP 10

“Extended” Heat Cascade ST Process, Energy and System QH 180°C QH 1, 1=50 H 1 170°C 1 130°C 120°C RST, 1 RH 1, 1 QH 1, 2=50 C 2 QC 2, 2=160 2 H 2 QH 2, 2=120 70°C RH 1, 2 RH 2, 2 60°C QC 1, 2=108 RST, 2 QH 2, 3=60 3 QC 1, 3=54 QC 40°C C 1 30°C CW Optimization Methods T. Gundersen MP 11

“Extended” Heat Cascade QP = QPH = 54 k. W ST Process, Energy and System 102 180°C 170°C 50 H 1 50 C 2 130°C 50 102 120°C 54 60°C 40 120 70°C H 2 40°C 60 QC 60 48 54 60 120 C 1 30°C CW Optimization Methods T. Gundersen MP 12

Design with Constraints Process, Energy and System Pinch 70° 180° H 1 140° 3 2 130° H 2 80° QP = QPH = 54 k. W (k. W/°C) 1. 0 70° 1 m. Cp Cb 40° 2. 0 60 k. W 120° Ha 93. 3° 2 48 k. W 100° 3 40 k. W 90° 60° Hb 1. 8 54 k. W 60° C 2 1 30° C 1 120 k. W U=6 4. 0 60° Optimization Methods T. Gundersen MP 13

“Extended” Heat Cascade QP = QPP = 54 k. W ST Process, Energy and System 102 180°C 170°C 50 H 1 50 C 2 130°C 50 70°C 0+x 102 120°C 54 -x 60°C 40 120 60 -x 120 H 2 48+x 54 60 40°C 60 QC C 1 30°C CW Choice: x = 54 k. W Optimization Methods T. Gundersen MP 14

Design with Constraints Process, Energy and System Pinch 70° 180° H 1 140° 3 130° H 2 2 QP = QPP = 54 k. W 80° (k. W/°C) 1. 0 70° 1 m. Cp Cb 40° 2. 0 60 k. W 120° 63. 3° Ha 2 3 40 k. W 90° 1. 8 6+54 k. W 102 k. W 100° 30° C 1 60° C 2 1 120 k. W U=5 4. 0 60° Optimization Methods T. Gundersen MP 15

“Extended” Heat Cascade QP = QPP + QPH = 40 + 14 k. W ST Process, Energy and System 102 180°C 170°C 50 H 1 50 C 2 130°C 50 70°C 0+y 102 120°C 54 -y 60°C 40 -y 0+y 120 60 120 H 2 48 54 60 40°C 60 QC C 1 30°C CW Choice: y = 40 k. W Optimization Methods T. Gundersen MP 16

Design with Constraints Process, Energy and System Pinch 70° 180° H 1 2 130° H 2 QP = QPH + QPP = 54 k. W 80° 70° 1 Cb 40° m. Cp (k. W/°C) 1. 0 2. 0 60 k. W 120° Ha 93. 3° 37. 8° 2 60+40 k. W 48 k. W 100° Hb 40 k. W 90° 60° C 2 1 120 k. W Hc 30° C 1 1. 8 14 k. W U=6 4. 0 60° Optimization Methods T. Gundersen MP 17

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

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

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

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

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