Process Integration Methods

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

2 Limitations in Pinch Analysis & PDM A lot of heuristics, not very rigorous u (N 1) rule for minimum number of units u Bath formula for minimum total area Composite Curves cannot handle u Forbidden matches between streams u Limitations in for example distillation Pinch Design Method is Sequential u Targeting before Design before Optimization u One match at a time, one loop at a time, etc. Time consuming but gives good designs T. Gundersen MP 0

3 What is Mathematical Programming? Numerical Optimization Techniques Can handle various Design Problems u Discrete Decisions related to Equipment u Continuous Decisions related to Operation Process Constraints can easily be included u Material and Energy Balances, Specifications u Equality and Inequality Constraints Can handle multivariable Trade-offs Framewor for Automatic Design u wouldn t it be nice to have? T. Gundersen MP 03

4 A small Linear Programming (LP) Problem min f( x) = x x subject to: x x x (a) + x 8 (b) x (c) x 1 (d) Solve the Objective Function and Constraints (a) and (b) as Equations with respect to variable x Objective Function: x = x f 1 Constraint (a): x = x + 1 Constraint (b): x = x The LP Problem can be solved by the well-nown and heavily applied Simplex Method, but it can also be solved graphically T. Gundersen MP 04

5 Graphical Solution for small LP Problem x x x x 1+ x 8 f=0 x1 f=4 f= x = x f 1 f=1 x 1 x 1 Optimum: at Vertex Algorithm: Simplex Solution: x 1 =, x =4 Objective: f = 0 T. Gundersen MP 05

6 Mathematical Programming & Superstructure Ref.: Papoulias & Grossmann Comput. Chem. Engng, 1983 T. Gundersen MP 06

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

8 Problems with Mathematical Programming Non-Linear Part y 3 Binary Part y 1 y Local Optima Combinatorial Explosion T. Gundersen MP 08

9 Stream T s T t mcp ΔH C C W/ C W H H C C ST (var) CW 15 0 (var) WS-4 Forbidden Matches Specification: ΔT min = 10 C Q: What is the effect if H and C1 are not allowed to exchange heat? Find Q H,min, Q C,min and the Heat Exchanger Networ with and without this forbidden match. Discuss the Degrees of Freedom. T. Gundersen MP 09

10 MER Design without Constraints 180 H Pinch H 1 3 Cb H a 8 W H b W W 10 W C 3 54 W 6 W U = 6 30 C1 mcp (W/ C) T. Gundersen MP 10

11 Extended Heat Cascade ST H1 H Q H1,1 =50 Q H1, = C 170 C 130 C 10 C R H1,1 Q H R ST,1 Q H, =10 70 C R H1, R H, 60 C 1 Q H,3 =60 3 R ST, C Q C1,3 =54 Q C, =160 Q C1, =108 C1 Q C 40 C 30 C CW T. Gundersen MP 11

12 Extended Heat Cascade H1 H ST 180 C C 130 C C 70 C C 40 C C CW Q C QP = QP H = 54 W C C1 T. Gundersen MP 1

13 Design with Constraints 180 H Pinch H 1 Cb 3 40 W H a 48 W W W 54 W 60 C QP = QP H = 54 W H b 60 W U = 6 30 C1 mcp (W/ C) T. Gundersen MP 13

14 Extended Heat Cascade H1 H ST 180 C C 130 C C 70 C 0+x 54-x 60 C Q C 40 C C QP = QP P = 54 W C x 60-x C1 CW Choice: x = 54 W T. Gundersen MP 14

15 Design with Constraints 180 H Pinch H 1 Cb 3 40 W H a 10 W W C 6+54 W QP = QP P = 54 W 60 W U = 5 30 C1 mcp (W/ C) T. Gundersen MP 15

16 Extended Heat Cascade H1 H ST 180 C C 130 C C 70 C 0+y 54-y 60 C Q C 40 C C QP = QP P + QP H = W 40-y 54 C 10 0+y C1 CW Choice: y = 40 W T. Gundersen MP 16

17 Design with Constraints 180 H Pinch H 1 Cb H a 48 W H b 40 W W W 60 C QP = QP H + QP P = 54 W W H c 14 W U = 6 30 C1 mcp (W/ C) T. Gundersen MP 17

18 min i i j j TI i HU j CU ' i, i, 1 ij i j C, CU R R + Q Q = 0 i HU ' i, i, 1 ij i j C i H i H ' ', HU i,0 i, K ' c Q + c Q + subject to: R R + Q = Q i H R Q = Q j C ij j Q Q = 0 j CU ij j = R = 0 R 0 Q 0 i, Q = 0 ( i, j) P ij ij LP Model for Forbidden Matches Easily solved by the Simplex Algorithm T. Gundersen MP 18

19 min subject to: ' ' ' i H, HU j C, CU R R + Q = Q i H ' i, i, 1 ij i j C, CU ' i, i, 1 ij i j C i H i H, HU R R + Q = Q i HU R = R = 0 R 0 Q 0 i,0 i, K i, ij ij j ij j ij y Q = Q j C Q = Q j CU TI Q U y ij ij ij 0 MILP Model for fewest Number of Units Logical Constraints relating Discrete & Continuous Variables T. Gundersen MP 19

20 Status for Mathematical Programming? Considerable Research in the 1980 s/90 s u CMU, Princeton, Caltech, Imperial College One Road towards Automatic Design u u Math Programming provides the Framewor Has the Potential to identify Superior Solutions Obstacles against Industrial Use u u u u Lac of Knowledge about the Methods Lac 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 T. Gundersen MP 0

21 The Sequential Framewor SeqHENS Surprisingly few Iterations are needed to identify the Global Optimum Reason: SeqHENS is strongly based on Insight from PA T. Gundersen MP 1

22 UMIST Comments after Sabbatical Promoting Mathematical Programming was quite challenging in those Days! T. Gundersen MP