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ESCAPE-16 & PSE 2006
Garmisch-Partenkirchen, Germany
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
Anantharaman & Gundersen, PSE/ESCAPE ’06
1
Overview
1.
Introducing the Sequential Framework
1.
2.
3.
Motivation
Our Goal
Our Engine
1.
2.
2.
Challenges
1.
Combinatorial Explosion – MILP
1.
2.
2.
Automated starting values
Modal trimming method
Examples
1.
2.
4.
Temperature Intervals
EMAT as an area variable
Non-convexities - NLP
1.
2.
3.
Subproblems
Loops
7 stream problem
15 stream problem
Concluding remarks
Anantharaman & Gundersen, PSE/ESCAPE ’06
2
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
Anantharaman & Gundersen, PSE/ESCAPE ’06
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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
Anantharaman & Gundersen, PSE/ESCAPE ’06
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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
Anantharaman & Gundersen, PSE/ESCAPE ’06
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Our Engine –
A Sequential Framework
Adjust EMAT
New HLD
Preoptim.
HRAT
LP
QH
MILP
U
QC
(EMAT=0)
EMAT
Vertical
MILP
HLD
1
Final
NLP
Adjust Units
Adjust HRAT
2
Network
3
4
Compromise between Pinch Design and MINLP Methods
Anantharaman & Gundersen, PSE/ESCAPE ’06
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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
Anantharaman & Gundersen, PSE/ESCAPE ’06
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Anantharaman & Gundersen, PSE/ESCAPE ’06
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Temperature Intervals (TIs)
in the VertMILP model
Objective function is minimizing pseudo area
Qim, jn
min
i
j
m
n
U ij TLM ,mn
H1
m-1
n-1
C1
VertMILP model works best when the pseudo
area accurately reflects the actual HX area
i
m
n
j
This happens when the number of TIs approaches infinity
HH
m+1
n+1
CC
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
Anantharaman & Gundersen, PSE/ESCAPE ’06
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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
EMAT
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)
Anantharaman & Gundersen, PSE/ESCAPE ’06
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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
Anantharaman & Gundersen, PSE/ESCAPE ’06
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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
Anantharaman & Gundersen, PSE/ESCAPE ’06
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Serial/Parallel mCp Generator
Simple & flexible method
Little physical insight needed
Parallel arrangement gives
feasible solution to most
problems (90%)
Anantharaman & Gundersen, PSE/ESCAPE ’06
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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
Anantharaman & Gundersen, PSE/ESCAPE ’06
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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
Anantharaman & Gundersen, PSE/ESCAPE ’06
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Modal Trimming Method for Global
Optimization of NLP subproblem
Search for local optimal solution
Objective function f(x)
Search for feasible solutions
f*0, x*0
f*1, x*1
f*2, x*2
x0
x2
x1
Variables x
Anantharaman & Gundersen, PSE/ESCAPE ’06
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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
Anantharaman & Gundersen, PSE/ESCAPE ’06
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Illustrating Example 1
Stream
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 ($) = 8,600 + 670A0.83 (A is in m2)
References:
Example 3 in Colberg, R. D. and Morari M., Area and Capital Cost Targets for Heat Exchanger Network Synthesis with Constrained
Matches and Unequal Heat Transfer Coefficients, Computers chem. Engng. Vol. 14, No. 1, 1990
Example 4 in Yee, T. F. and Grossmann I. E., Simulataneous Optimization Models for Heat Integration - II. Heat Exchanger Network
Synthesis, Computers chem. Engng. Vol. 14, No. 10, 1990
Anantharaman & Gundersen, PSE/ESCAPE ’06
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Example 1 – Initial Values
Adjust EMAT
2
New HLD
HRAT
Q
MILP
LP H (EMAT=0)
QC
U
EMAT
Vertical
MILP
HLD
1
Final
NLP
Network
3
Adjust Units
HRAT fixed at 20K
QH = 244.1 kW
QC = 172.6 kW
Absolute Minimum
Number of Units = 8
Anantharaman & Gundersen, PSE/ESCAPE ’06
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Example 1 – Looping to Solution
Adjust EMAT
2
New HLD
HRAT
LP
QH MILP
QC (EMAT=0)
U
EMAT
Vertical
MILP
1
Final
HLD NLP
Network
3
Adjust Units
Soln. No
U
EMAT (K)
HLD#
INVESTMENT COST ($)
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
Anantharaman & Gundersen, PSE/ESCAPE ’06
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Example 1 – `Best´ Solution
HRAT = 20, EMAT = 2.5, ΔTsmall= 3
Anantharaman & Gundersen, PSE/ESCAPE ’06
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Example 1 – Solution Comparisons
No of Units
Area (m2)
Cost
Remarks
Colberg & Morari
(1990)
22
173.6
-
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
$147,861
MILP optimized w.r.t ”area”
NLP optimized w.r.t cost
Our work
9
189.7
Optimized w.r.t area
Spaghetti design
Anantharaman & Gundersen, PSE/ESCAPE ’06
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Illustrating Example 2
Stream
Tin
(°C)
Tout
(°C)
mCp
(kW/°C)
ΔH
(kW)
h
(kW/m2 °C)
H1
H2
H3
H4
H5
H6
H7
H8
C1
C2
C3
C4
C5
C6
C7
ST
CW
180
280
180
140
220
180
200
120
40
100
40
50
50
90
160
325
25
75
120
75
40
120
55
60
40
230
220
290
290
250
190
250
325
40
30
60
30
30
50
35
30
100
20
60
35
30
60
50
60
3150
9600
3150
3000
5000
4375
4200
8000
3800
7200
8750
7200
12000
5000
5400
2
1
2
1
1
2
0.4
0.5
1
1
2
2
2
1
3
1
2
Exchanger cost ($) = 8,000 + 500A0.75 (A is in m2)
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
Anantharaman & Gundersen, PSE/ESCAPE ’06
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Example 2 – Initial Values
Adjust EMAT
2
New HLD
HRAT
LP
QH MILP
QC (EMAT=0)
U
EMAT
Vertical
MILP
HLD
1
Final
NLP
Network
3
Adjust Units
HRAT fixed at 20.35 C
QH = 11539.25 kW
QC = 9164.25 kW
Absolute Minimum
Number of Units = 14
Anantharaman & Gundersen, PSE/ESCAPE ’06
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Example 2 – Looping to Solution
Adjust EMAT
2
New HLD
HRAT
LP
QH MILP
QC (EMAT=0)
U
EMAT
Vertical
MILP
1
Final
HLD NLP
Network
3
Adjust Units
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
Anantharaman & Gundersen, PSE/ESCAPE ’06
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Example 2 – `Best´ Solution
HRAT = 20.35
EMAT = 5
ΔTsmall= 4.9
Anantharaman & Gundersen, PSE/ESCAPE ’06
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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
Anantharaman & Gundersen, PSE/ESCAPE ’06
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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
Anantharaman & Gundersen, PSE/ESCAPE ’06
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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
Anantharaman & Gundersen, PSE/ESCAPE ’06
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