1. 1 Anantharaman & Gundersen, PSE/ESCAPE ’06 Developments in the Sequential Framework for Heat Exchanger Network Synthesis of industrial size problems Rahul Anantharaman and Truls Gundersen Dept of Energy and Process Engineering Norwegian University of Science and Technology Trondheim, Norway ESCAPE-16 & PSE 2006 Garmisch-Partenkirchen, Germany

2. 2 Anantharaman & Gundersen, PSE/ESCAPE ’06 Overview  Introducing the Sequential Framework 1.Motivation 2.Our Goal 3.Our Engine 1.Subproblems 2.Loops  Challenges 1.Combinatorial Explosion – MILP 1.Temperature Intervals 2.EMAT as an area variable 2.Non-convexities - NLP 1.Automated starting values 2.Modal trimming method  Examples 1.7 stream problem 2.15 stream problem  Concluding remarks

3. 3 Anantharaman & Gundersen, PSE/ESCAPE ’06 Motivation for the Sequential Framework  Pinch Methods for Network Design  Improper trade-off handling  Time consuming  Several topological traps  MINLP Methods for Network Design  Severe numerical problems  Difficult user interaction  Fail to solve large scale problems  Stochastic Optimization Methods for Network Design  Non-rigorous algorithms  Quality of solution depends on time spent on search

4. 4 Anantharaman & Gundersen, PSE/ESCAPE ’06 Motivation for the Sequential Framework  HENS techniques decompose the main problem  Pinch Design Method is sequential and evolutionary  Simultaneous MINLP methods let math considerations define the decomposition  The Sequential Framework decomposes the problem into subproblems based on knowledge of the HENS problem  Engineer acts as optimizer at the top level  Quantitative and qualitative considerations

5. 5 Anantharaman & Gundersen, PSE/ESCAPE ’06 Our Ultimate Goal  Solve Industrial Size Problems  Defined to involve 30 or more streams  Include Industrial Realism  Multiple Utilities  Constraints in Heat Utilization (Forbidden matches)  Heat exchanger models beyond pure countercurrent  Avoid Heuristics and Simplifications  No global or fixed ΔT min  No Pinch Decomposition  Develop Semi-Automatic Design Tool  A tool SeqHENS is under development  EXCEL/VBA (preprocessing and front end)  MATLAB (mathematical processing)  GAMS (core optimization engine)  Allow significant user interaction and control  Identify near optimal and practical networks

6. 6 Anantharaman & Gundersen, PSE/ESCAPE ’06 Our Engine – A Sequential Framework Vertical MILP LP NLP Adjust Units Adjust HRAT MILP U HLD Final Network QHQH QCQC (EMAT=0) New HLD 1 4 3 EMAT Adjust EMAT 2 Pre- optim. HRAT Compromise between Pinch Design and MINLP Methods

7. 7 Anantharaman & Gundersen, PSE/ESCAPE ’06 Challenges  Combinatorial explosion (binary variables)  Problem proved to be NP -complete in the strong sense  Any algorithm may take exponential number of steps to reach optimality  Use physical/engineering insights based on understanding of the problem  Will not remove the problem but help mitigate it  MILP and VMILP are currently the bottlenecks w.r.t. time (and thus size)  Local optima (non-convexities in the NLP model)  Convex estimators developed for MINLP models are computationally intensive  Only very small problems have been solved  Explore other options  Time to solve the NLP is not a problem  Relatively easier to solve than MINLP formulations

8. 8 Anantharaman & Gundersen, PSE/ESCAPE ’06

9. 9 Temperature Intervals (TIs) in the VertMILP model  Objective function is minimizing pseudo area  VertMILP model works best when the pseudo area accurately reflects the actual HX area  This happens when the number of TIs approaches infinity  Size of the VertMILP model increases exponentially with the number of temperature intervals  The transportation model has a polynomial time algorithm → Keep number of TIs to a minimum while ensuring the model achieves its objective H1H1 m-1 m m+1 n-1 n n+1 i H C1C1 C j

10. 10 Anantharaman & Gundersen, PSE/ESCAPE ’06 Temperature Intervals (TIs) in the VertMILP model  Original philosophy of the VertMILP model  Minimum area is achieved by vertical heat transfer  Temperature intervals must facilitate vertical heat transfer  Use Enthalpy Intervals to develop the vertical TIs  The Normal and Enthalpy based (vertical) TIs are the basis for the VertMILP model  Elaborate testing show that the VertMILP model achieves its objective with this set of TIs  Size of the model is reduced, on an average, by 10% (more for larger models) EMAT

11. 11 Anantharaman & Gundersen, PSE/ESCAPE ’06 EMAT as an Area Variable  Choosing EMAT is not straightforward  EMAT set too low (close to zero)  non-vertical heat transfer (m=n) will have very small ΔT LM,mn and very large penalties in the objective function  EMAT set too high (close to HRAT)  Potentially good HLDs will be excluded from the feasible set of solutions  HLDs are affected by the choice of EMAT  EMAT comes into play only when there is an extra degree of freedom in the system : U > Umin

12. 12 Anantharaman & Gundersen, PSE/ESCAPE ’06 Automated Starting Values and Bounds for the NLP subproblem  Multiple starting values for the NLP subproblem  Ensure a feasible solution  Explore different local optima  Use physical insight to ensure `good´ local optima  Heat Capacity Flowrates (mCps) identified to be the decision variables  Lower Bounds for Area were found to be crucial in getting a feasible solution  Information from the VertMILP subproblem is utilized  4 different strategies for starting values were explored  Ref.: Hilmersen S. E. and Stokke A., M.Sc Thesis, NTNU 2006

13. 13 Anantharaman & Gundersen, PSE/ESCAPE ’06 Serial/Parallel mCp Generator  Simple & flexible method  Little physical insight needed  Parallel arrangement gives feasible solution to most problems (90%)

14. 14 Anantharaman & Gundersen, PSE/ESCAPE ’06 Clever Serial mCp Generator  Serial configuration assumed for all streams  Assigns demanding exchangers at the supply end  Only stream temperatures are considered  Heat exchanger duties & stream mCp values are not considered  Assumed sequence of heat exchangers  Hot supply end matched with ranked set of cold targets & vice versa  Similar to the Ponton/Donaldson heuristic synthesis approach  Only serial configuration is limiting in many cases  Feasible solution in 50% of cases tested

15. 15 Anantharaman & Gundersen, PSE/ESCAPE ’06 Combinatorial mCp Generator  Utilizes heat loads, temperatures and overall mCp values to assign stream flows  Uses physical insight to determine flows  Provides a feasible solution to the NLP subproblem in all cases tested

16. 16 Anantharaman & Gundersen, PSE/ESCAPE ’06 Modal Trimming Method for Global Optimization of NLP subproblem

17. 17 Anantharaman & Gundersen, PSE/ESCAPE ’06 Modal Trimming Method for Global Optimization of NLP subproblem  Search for feasible solutions is the most important step Testing showed the Modal Trimming method to be inefficient and computationally expensive for solving the NLP model

18. 18 Anantharaman & Gundersen, PSE/ESCAPE ’06 Illustrating Example 1 Stream T in (K) T out (K) mC p (kW/K) ΔH (kW) h (kW/m 2 K) H16265869.802392.081.25 H26205192.931296.030.05 H35283536.1611078.183.20 C14976137.179832.760.65 C23895760.641119.870.25 C33263867.627457.620.33 C43135661.690427.573.20 ST650 --3.50 CW293308--3.50 Exchanger cost ($) = 8,600 + 670A 0.83 (A is in m 2 ) References: Example 3 in Colberg, R. D. and Morari M., Area and Capital Cost Targets for Heat Exchanger Network Synthesis with Constrained Matches and Unequal Heat Transfer Coefficients, Computers chem. Engng. Vol. 14, No. 1, 1990 Example 4 in Yee, T. F. and Grossmann I. E., Simulataneous Optimization Models for Heat Integration - II. Heat Exchanger Network Synthesis, Computers chem. Engng. Vol. 14, No. 10, 1990

19. 19 Anantharaman & Gundersen, PSE/ESCAPE ’06 Example 1 – Initial Values Vertical MILP LP NLP Adjust Units MILP UHLD Final Network QHQH QCQC (EMAT=0) New HLD 1 EMAT Adjust EMAT HRAT 2 3 HRAT fixed at 20K Q H = 244.1 kW Q C = 172.6 kW Absolute Minimum Number of Units = 8

20. 20 Anantharaman & Gundersen, PSE/ESCAPE ’06 Example 1 – Looping to Solution Vertical MILP LP NLP Adjust Units MILP UHLD Final Network QHQH QCQC (EMAT=0) New HLD 1 EMAT Adjust EMAT HRAT 2 3 Soln. NoUEMAT (K)HLD#INVESTMENT COST ($) 182.5A199,914 282.5BNot feasible 392.5A147,861 492.5B151,477 595.0A147,867 695.0B151,508 7102.5A164,381

21. 21 Anantharaman & Gundersen, PSE/ESCAPE ’06 Example 1 – `Best´ Solution HRAT = 20, EMAT = 2.5, ΔT small = 3

22. 22 Anantharaman & Gundersen, PSE/ESCAPE ’06 Example 1 – Solution Comparisons No of UnitsArea (m 2 )CostRemarks Colberg & Morari (1990)22173.6 - Optimized w.r.t area Spaghetti design Colberg & Morari (1990)12188.9$177,385Synthesized network by evolution Yee and Grossmann (1990)9217.8$150,998Optimized w.r.t. cost Our work9189.7$147,861 MILP optimized w.r.t ”area” NLP optimized w.r.t cost

23. 23 Anantharaman & Gundersen, PSE/ESCAPE ’06 Illustrating Example 2 Reference: Björk K.M and Nordman R., Solving large-scale retrofit heat exchanger network synthesis problems with mathematical optimization methods, Chemical Engineering and Processing. Vol. 44, 2005 Stream TinToutmCpΔHh (°C) (kW/°C)(kW)(kW/m2 °C) H1180753031502 H22801206096001 H3180753031502 H4140403030001 H52201205050001 H6180553543752 H7200603042000.4 H81204010080000.5 C1402302038001 C21002206072001 C3402903587502 C4502903072002 C55025060120002 C6901905050001 C71602506054003 ST325 1 CW2540 2 Exchanger cost ($) = 8,000 + 500A 0.75 (A is in m 2 )

24. 24 Anantharaman & Gundersen, PSE/ESCAPE ’06 Example 2 – Initial Values Vertical MILP LP NLP Adjust Units MILP UHLD Final Network QHQH QCQC (EMAT=0) New HLD 1 EMAT Adjust EMAT HRAT 2 3 HRAT fixed at 20.35 C Q H = 11539.25 kW Q C = 9164.25 kW Absolute Minimum Number of Units = 14

25. 25 Anantharaman & Gundersen, PSE/ESCAPE ’06 Example 2 – Looping to Solution Vertical MILP LP NLP Adjust Units MILP UHLD Final Network QHQH QCQC (EMAT=0) New HLD 1 EMAT Adjust EMAT HRAT 2 3 Soln. NoUEMAT (K)HLD#TAC ($) 1142.5A1,545,375 2152.5A1,532,148 3152.5B1,536,900 4155.0A1,529,968 5155.0B1,533,261 6162.5A1,547,353

26. 26 Anantharaman & Gundersen, PSE/ESCAPE ’06 Example 2 – `Best´ Solution HRAT = 20.35 EMAT = 5 ΔT small = 4.9

27. 27 Anantharaman & Gundersen, PSE/ESCAPE ’06 Example – Solution Comparison  The solution given here with a TAC of $1,529,968, about the same cost as the solution presented in the original paper (TAC $1,530,063)  When only one match was allowed between a pair of streams the TAC is reported as $1,568,745 - Björk & Nordman (2005)  The Sequential Framework allows only 1 match between a pair of streams  Solution at Iteration 2 (TAC $ 1,532,148) provides a slightly more expensive but slightly less compless network  Unable to compare the solutions apart from cost as the paper did not present the networks in their work

28. 28 Anantharaman & Gundersen, PSE/ESCAPE ’06 Global vs Local Optimum  Global optima in each of the subproblems does not, by itself, ensure overall global optimum for the HENS problem  Inherent feature of any problem decomposition  The emphasis has been on utilizing knowledge of the problem and engineering insight to achieve a network close to global optimum

29. 29 Anantharaman & Gundersen, PSE/ESCAPE ’06 Concluding Remarks  Sequential Framework has many advantages  Automates the design process  Allows significant User interaction  Numerically much easier than MINLPs  Progress  EMAT identified as an optimizing `area variable´  Improved HLDs from VertMILP subproblem  Algorithm for generating optimal TIs for the VertMILP  Significantly better and automated starting values for NLP subproblem  Limiting elements  NLP model for Network Generation and Optimization  Enhanced convex estimators are required to ensure global optimum  VertMILP Transportation Model for promising HLDs  Significant improvements required to fight combinatorial explosion  MILP Transhipment model for minimum number of units  Similar combinatorial problems as the Transportation model