1 Anantharaman & Gundersen, PSE/ESCAPE ’06 Developments in the Sequential Framework for Heat...

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Anantharaman & Gundersen, PSE/ESCAPE ’06

Developments in the Developments in the Sequential Framework Sequential Framework

for Heat Exchanger Network for Heat Exchanger Network Synthesis of industrial Synthesis of industrial

size problemssize problems

Rahul Anantharaman and Truls Gundersen

Dept of Energy and Process EngineeringNorwegian University of Science and Technology

Trondheim, Norway

ESCAPE-16 & PSE 2006Garmisch-Partenkirchen, Germany

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Overview1. Introducing the Sequential Framework

1. Motivation2. Our Goal3. Our Engine

1. Subproblems2. Loops

2. Challenges1. Combinatorial Explosion – MILP

1. Temperature Intervals2. EMAT as an area variable

2. Non-convexities - NLP1. Automated starting values2. Modal trimming method

3. Examples1. 7 stream problem2. 15 stream problem

4. Concluding remarks

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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

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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

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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 ΔTmin

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

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Our Engine – A Sequential Framework

Vertical MILP

LP NLP

Adjust Units

Adjust HRAT

MILP U HLD Final

Network

QH

QC (EMAT=0)

New HLD1

4

3

EMAT

Adjust EMAT2

Pre-optim.

HRAT

Compromise between Pinch Design and MINLP Methods

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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

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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

,

,

min im jn

i j m n ij LM mn

Q

U T

H1 m-1

m

m+1

n-1

n

n+1

i

HH

C1

CC

j

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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

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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 ΔTLM,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

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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

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Serial/Parallel mCp Generator

Simple & flexible method

Little physical insight needed

Parallel arrangement gives feasible solution to most problems (90%)

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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

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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

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Modal Trimming Method for Global Optimization of NLP subproblem

Obj

ecti

ve f

unct

ion

f(x)

Variables x

x0 x1x2

f*0, x*0

f*2, x*2

f*1, x*1

Search for local optimal solution

Search for feasible solutions

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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

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Illustrating Example 1Stream

Tin

(K)

Tout

(K)

mCp

(kW/K)ΔH

(kW)h

(kW/m2 K)

H1 626 586 9.802 392.08 1.25

H2 620 519 2.931 296.03 0.05

H3 528 353 6.161 1078.18 3.20

C1 497 613 7.179 832.76 0.65

C2 389 576 0.641 119.87 0.25

C3 326 386 7.627 457.62 0.33

C4 313 566 1.690 427.57 3.20

ST 650 650 - - 3.50

CW 293 308 - - 3.50

Exchanger cost ($) = 8,600 + 670A0.83 (A is in m2)

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

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Example 1 – Initial Values

Vertical MILP

LP NLP

Adjust Units

MILP U HLD Final

Network

QH

QC(EMAT=0)

New HLD1

EMAT

Adjust EMAT

HRAT

2

3

HRAT fixed at 20KQH = 244.1 kWQC = 172.6 kW

Absolute MinimumNumber of Units = 8

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Example 1 – Looping to Solution

Vertical MILP

LP NLP

Adjust Units

MILP U HLD Final

Network

QH

QC(EMAT=0)

New HLD1

EMAT

Adjust EMAT

HRAT

2

3

Soln. No U EMAT (K) HLD# INVESTMENT COST ($)

1 8 2.5 A 199,914

2 8 2.5 B Not feasible

3 9 2.5 A 147,861

4 9 2.5 B 151,477

5 9 5.0 A 147,867

6 9 5.0 B 151,508

7 10 2.5 A 164,381

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Example 1 – `Best´ Solution

HRAT = 20, EMAT = 2.5, ΔTsmall= 3

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Example 1 – Solution Comparisons

No of Units Area (m2) Cost Remarks

Colberg & Morari (1990) 22 173.6 -

Optimized w.r.t areaSpaghetti design

Colberg & Morari (1990) 12 188.9 $177,385 Synthesized network by evolution

Yee and Grossmann (1990) 9 217.8 $150,998 Optimized w.r.t. cost

Our work 9 189.7 $147,861

MILP optimized w.r.t ”area”

NLP optimized w.r.t cost

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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

StreamTin Tout mCp ΔH h (°C) (°C) (kW/°C) (kW) (kW/m2 °C)

H1 180 75 30 3150 2H2 280 120 60 9600 1H3 180 75 30 3150 2H4 140 40 30 3000 1H5 220 120 50 5000 1H6 180 55 35 4375 2H7 200 60 30 4200 0.4H8 120 40 100 8000 0.5C1 40 230 20 3800 1C2 100 220 60 7200 1C3 40 290 35 8750 2C4 50 290 30 7200 2C5 50 250 60 12000 2C6 90 190 50 5000 1C7 160 250 60 5400 3ST 325 325 1CW 25 40 2

Exchanger cost ($) = 8,000 + 500A0.75 (A is in m2)

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Example 2 – Initial Values

Vertical MILP

LP NLP

Adjust Units

MILP U HLD Final

Network

QH

QC(EMAT=0)

New HLD1

EMAT

Adjust EMAT

HRAT

2

3

HRAT fixed at 20.35 CQH = 11539.25 kWQC = 9164.25 kW

Absolute MinimumNumber of Units = 14

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Example 2 – Looping to Solution

Vertical MILP

LP NLP

Adjust Units

MILP U HLD Final

Network

QH

QC(EMAT=0)

New HLD1

EMAT

Adjust EMAT

HRAT

2

3

Soln. No U EMAT (K) HLD# TAC ($)

1 14 2.5 A 1,545,375

2 15 2.5 A 1,532,148

3 15 2.5 B 1,536,900

4 15 5.0 A 1,529,968

5 15 5.0 B 1,533,261

6 16 2.5 A 1,547,353

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Example 2 – `Best´ Solution

HRAT = 20.35EMAT = 5ΔTsmall= 4.9

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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

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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

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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

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1 Anantharaman & Gundersen, PSE/ESCAPE ’06 Developments in the Sequential Framework for Heat...

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