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Rahul - thesis - final version - 30 May 2011

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Energy Efficiency in Process Plants with
emphasis on Heat Exchanger Networks
Optimization, Thermodynamics and Insight
Rahul Anantharaman
Department of Energy and Process Engineering
Norwegian University of Science and Technology
A thesis submitted for the degree of
Philosophæ Doctor (PhD)
2011 May, Trondheim
Preface
The thesis is submitted in partial fulfillment of the requirements for the
degree of philosophiæ doctor (PhD) at the Norwegian University of Science
and Technology (NTNU). The work was carried out at the Department of
Energy and Process Engineering at the Faculty of Engineering Science and
Technology, with Prof. Truls Gundersen as supervisor.
The research was funded by the Norwegian Research Council.
Abstract
This thesis focuses on energy recovery system design and energy integration
to improve the energy efficiency of process plants. The objectives of this work
are to (a) develop a systematic methodology based on thermodynamic principles to integrate energy intensive processes and (b) develop a mathematical
programming based approach using thermodynamics and insight for solving
industrial sized HENS problems.
A novel energy integration methodology, Energy Level Composite Curves
(ELCC), has been developed that is a synergy of Exergy Analysis and Composite Curves. ELCC is a graphical tool which provides the engineer with
insights on energy integration and this work represents the first methodological attempt to represent thermal, mechanical and chemical energy in
a graphical form similar to composite curves for the thermal integration of
energy intensive processes. This method provides physical insight to integrate energy sources with sinks. The methodology is useful as a screening
tool, functioning as an idea generator prior to the heat and power integration step. A simple energy targeting algorithm is developed to obtain utility
targets. The ELCC was applied to a methanol plant to show the efficacy of
the methodology.
The Sequential Framework, an iterative and sequential methodology for
Heat Exchanger Network Synthesis (HENS), is presented in this thesis. The
main objective of the Sequential Framework is to solve industrial size problems. The subtasks of the design process are solved sequentially using Mathematical Programming. There are two main advantages of the methodology.
First, the design procedure is, to a large extent, automated while keeping
significant user interaction. Second, the subtasks of the framework (MILP
and NLP problems) are much easier to solve numerically than the MINLP
models that have been suggested for HENS. Application of the Sequential
Framework to literature examples showed that the methodology generated
solutions with total annualized costs lower than those presented in the literature. The examples showed the efficiency of the Sequential Framework in
that even though there a four nested loops in the framework, the “best” solution is reached within a few iterations. This is primarily due to the capability
of the stream match generator to identify superior Heat Load Distributions
(HLDs) leading to low total heat transfer area and low Total Annualized
Cost.
The three sub-problems in the Sequential Framework, minimum number of
units (MILP model), stream match generator (“vertical” MILP model) and
network generation and optimization (NLP model), are described with details on their formulation. In the minimum number of units sub-problem, it
is shown that stream supply temperature are sufficient to define temperature intervals. The importance and role of Exchanger Minimum Approach
Temperature (EMAT) in the stream match generator model is shown and
motivated the addition of an EMAT loop in the Sequential Framework.
One of the limiting factors in the methodology is related to the computational
complexity of the two MILP sub-problems where significant improvements
are required to prevent combinatorial explosion. To ease this problem for the
minimum number of units MILP sub-problem, it is modified to reduce the
gap using physical insights and heuristics. Another novel approach tested
was to reformulate some parts of the model by use of some ideas from set
partitioning problems. Results show that even though both methods succeed in tightening the LP relaxation, the model solution times remain too
long to overcome the size in the Sequential Framework. A problem difficulty indicator is explored to identify computationally expensive problems
prior to solution. For the stream match generator MILP sub-problem, the
model is modified to reduce the gap using physical insights. The objective
is changed to include binary variables and priorities were set for these variables. Though these modifications showed improvement in solution time,
orders of magnitude improvement are required to solve large models. Another limiting factor in the methodology is that the network generation and
optimization sub-problem is formulated as a non-convex NLP leading to local optima. Clever starting value generators based on physical insight were
developed to mitigate this issue.
vi
To my teachers.
Agyana timir-andhasya Gyananjana Shalakaya.
Chakshur-oonmeelitam yena tasmai Shri Gurave Namaha.
Acknowledgements
I would like to thank my supervisor Prof. Truls Gundersen at the Department of Energy and Process Engineering for his patient, friendly and skilled
guidance over the course of this work and who always had time for me despite having many places to run to (literally!). A special thanks for making
me and my family feel welcome and at home in Trondheim.
I would also like to thank Prof. Bjørn Nygreen from the Department of Industrial Economics and Technology Management for collaborating with us
and providing insights to tackle the tough optimization problems. Contributions by Ola Sørås, Stein Erik Hilmersen, Atle Stokke and Ivar Nastad
through master projects and theses are greatfully acknowledged.
Special thanks to all my friends during the course of this long and winded
journey, especially Michaël Becidan and Morten Seljeskog for the wonderful
lunch and “stripa” breaks, Lars Nord for, as he put it, discussions, bike rides
and once-in-a-blue-moon chess game, Ravikiran Kota for the stimulating and
heated discussions and Sarin Kumar and Ajit Bopardikar for the wonderful
hours we spent making music, Rehan Naqvi for helping us get settled and
everyone associated with the Three Lions Cricket Club.
I express my sincerest gratitude to SINTEF Energy and my colleagues at
the Energy Process department for being supportive while I was working
towards finishing this thesis. Tusen takk.
Thank are due to the administrative staff for their assistance on all practical
matters.
I express my gratitude to the Norwegian research Council for funding my
PhD project.
My family has always been behind me, supporting me in all my endeavors.
No words can describe my gratitude for them.
I want to thank my wife Arthi for always being there for me. This thesis
owes a lot to her patience, support, drive and energy. Finally I would like to
thank my daughter Ananya for teaching me the importance of living this life
to its fullest!
I would be remiss in not expressing my gratitude to the United States Immigration Services department for my move to NTNU and Trondheim.
iv
Contents
List of Figures
xi
List of Tables
xiii
Nomenclature
xv
1 Introduction
1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.4.1 Exergy based method for energy integration . . . . . . . . . . . . . .
6
1.4.2 Heat exchanger network synthesis review . . . . . . . . . . . . . . .
6
1.4.3 Sequential Framework for heat exchanger network synthesis . . . .
7
1.5 Thesis Structure and Guidelines . . . . . . . . . . . . . . . . . . . . . . . . .
8
2 Energy Issues in Process Synthesis
11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.2 Process Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.2.1 Decomposition and hierarchy-based approach to process synthesis .
13
2.2.2 Methodologies for process synthesis . . . . . . . . . . . . . . . . . . .
14
2.3 Process Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.4 Energy Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.4.1 Energy integration methods . . . . . . . . . . . . . . . . . . . . . . . .
19
2.4.2 A note on Energy and Exergy . . . . . . . . . . . . . . . . . . . . . . .
20
v
CONTENTS
3 Energy Level Composite Curves
25
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.2 Energy Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
3.2.1 Evaluating Energy Level . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.3 Construction and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.3.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.3.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.4 Minimum Energy Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.4.1 Pressure - Temperature relationship . . . . . . . . . . . . . . . . . . .
33
3.4.2 First Law and Target for available energy . . . . . . . . . . . . . . .
33
3.4.3 Optimal Path Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.4.4 Algorithm for Energy Targeting . . . . . . . . . . . . . . . . . . . . . .
36
3.5 Methanol Plant Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.5.1 Energy Integration study using ELCC . . . . . . . . . . . . . . . . . .
38
3.5.2 Energy targeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.6 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.7 Conclusions and further work . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
4 The Heat Exchanger Network Synthesis Problem - Review of the stateof-the-art in the new millenium
45
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
4.2 The history of HENS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
4.2.1 Overview of the general timeline . . . . . . . . . . . . . . . . . . . . .
46
4.2.2 Developmental milestones . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.3 HENS literature in the new millenium . . . . . . . . . . . . . . . . . . . . . .
49
4.3.1 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
4.3.2 Topics in HENS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
4.3.3 Heat Integration Topics . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
4.3.4 HENS solution methods . . . . . . . . . . . . . . . . . . . . . . . . . .
54
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
HENS Bibliography 2000-2008
59
vi
CONTENTS
5 The Sequential Framework for Heat Exchanger Network Synthesis
75
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
5.2 Ultimate goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
5.3 The Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
5.3.1 Sequential synthesis of HENS using Mathematical Programming .
77
5.3.2 The Sequential Framework for HENS . . . . . . . . . . . . . . . . . .
78
5.3.3 Minimum Utilities Targeting . . . . . . . . . . . . . . . . . . . . . . .
79
5.3.4 Calculating the absolute Minimum Number of Units . . . . . . . . .
80
5.3.5 Stream Match Generator . . . . . . . . . . . . . . . . . . . . . . . . . .
80
5.3.6 Network Generation and Optimization . . . . . . . . . . . . . . . . .
80
5.3.7 Rationale for loops in the framework . . . . . . . . . . . . . . . . . . .
80
5.3.8 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
5.3.9 Loop sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
5.4 Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
5.5 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
5.6 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
5.7 A semi-automatic design tool - SeqHENS . . . . . . . . . . . . . . . . . . . .
84
5.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
5.8.1 Example 1 (7TP1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
5.8.2 Example 2 (15TP1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
5.9 Conclusions and further work . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
6 Minimum Number of Units Sub-problem
93
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
6.2 The minimum number of units sub-problem in the Sequential Framework
94
6.2.1 Temperature Intervals in the transshipment model . . . . . . . . . .
97
6.3 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.4 Model modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.4.1 Sharpening the LP relaxation by decreasing the big M . . . . . . . . 108
6.4.2 Integer cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.4.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.5 Model reformulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
vii
CONTENTS
6.5.1 New formulation with integer variables representing hot stream
matches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.5.2 New reformulation with integer variables representing cold stream
matches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.5.3 New formulation with integer variables representing both hot and
cold stream matches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.5.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.6 A problem difficulty index? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.7 Conclusions and further work . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7 Stream Match Generator Sub-problem
127
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.2 Stream Match Generator model formulation . . . . . . . . . . . . . . . . . . 128
7.2.1 Development history . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.2.2 MILP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.2.3 Temperature intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.2.4 EMAT as an optimizing variable . . . . . . . . . . . . . . . . . . . . . 136
7.3 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.3.1 Pre-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.3.2 Model modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.3.3 Improving efficiency of the B&B method . . . . . . . . . . . . . . . . 145
7.4 Conclusions and further work . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8 Network Generation and Optimization
149
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.2 Network generation and optimization model formulation . . . . . . . . . . . 150
8.2.1 Superstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.2.2 NLP formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
8.3 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8.3.1 Causes of Local Optima . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8.4 Starting Value Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.4.1 Basic Serial/Parallel heuristic . . . . . . . . . . . . . . . . . . . . . . . 161
8.4.2 Serial H/H Heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.4.3 Stream Match Generator based heuristic . . . . . . . . . . . . . . . . 162
viii
CONTENTS
8.4.4 Combinatorial heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
8.5 NLP solvers in the Sequential Framework . . . . . . . . . . . . . . . . . . . 165
8.6 Conclusion and further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
9 Conclusions and further work
167
9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
9.2 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
References
173
A Test Problems
183
A.1 7TP1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
A.2 15TP1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
A.3 21TP1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
A.4 21TP2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
A.5 22TP1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
ix
CONTENTS
x
List of Figures
1.1 Global anthropogenic green house gas emissions in 2004 [2] . . . . . . . . .
2
1.2 World primary energy consumption 1984-2009 [1] . . . . . . . . . . . . . . .
3
1.3 Reductions in CO2 emissions in the ACT Map scenario by technology area
[3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.4 CO2 emissions and energy intensities from 1970 - 2004 [2] . . . . . . . . . .
5
2.1 Onion model of process design . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.2 Classification of process synthesis . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.3 Classification of process integration methods . . . . . . . . . . . . . . . . . .
17
2.4 Modified onion model of process design to include Energy Recovery Systems 19
2.5 Illustration of different exergy components . . . . . . . . . . . . . . . . . . .
21
3.1 Composite Curves of Pinch Analysis . . . . . . . . . . . . . . . . . . . . . . .
26
3.2 Relative roles of heat load and temperature in evaluating exergy . . . . . .
28
3.3 Algorithm for energy targeting using optimal path heuristics . . . . . . . .
36
3.4 Methanol plant case study process flow diagram with stream numbers . .
38
3.5 Energy Level Composite Curves for the Methanol plant case study . . . . .
39
4.1 Number of HENS journal papers published annually . . . . . . . . . . . . .
50
4.2 HENS journal papers published divided among journals . . . . . . . . . . .
51
4.3 Number of HENS journal papers published by country . . . . . . . . . . . .
51
4.4 Word cloud of the journal paper titles . . . . . . . . . . . . . . . . . . . . . . .
52
5.1 Three way trade-off in HENS problems . . . . . . . . . . . . . . . . . . . . .
76
5.2 The Sequential Framework for heat exchanger network synthesis . . . . .
79
5.3 SeqHENS interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
xi
LIST OF FIGURES
5.4 The best heat exchanger network for Example 1 (7TP1) - Solution no. 4 . .
87
5.5 The best obtained heat exchanger network for Example 2 (15TP1) - Solution no. 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
6.1 Transshipment formulation for minimum number of units sub-problem . .
96
6.2 Temperature intervals for Example . . . . . . . . . . . . . . . . . . . . . . . .
97
6.3 Combinatorial explosion in a binary search tree as a function of the total
number of process streams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.4 Complexity classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.5 Maximum heat transfer between a hot stream i and a cold stream j . . . . 109
7.1 Vertical heat transfer between composite curves . . . . . . . . . . . . . . . . 129
7.2 Transportation formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.3 Polynomial increase in the number of Q im jn variables with the number
of temperature intervals. The number of hot temperature intervals is assumed equal to the number of cold temperature intervals in this figure. . . 133
7.4 Primary temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.5 Primary and Secondary temperatures . . . . . . . . . . . . . . . . . . . . . . 135
7.6 Primary and tertiary temperatures . . . . . . . . . . . . . . . . . . . . . . . . 135
7.7 Solution times as a function of number of heat exchanger units in the
stream match generator model . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.1 Stream superstructure for a stream with 3 matches . . . . . . . . . . . . . . 152
8.2 Starting value generator implemented as part of SeqHENS . . . . . . . . . 160
8.3 Serial/Parallel starting value generator . . . . . . . . . . . . . . . . . . . . . 162
8.4 VertMILP based generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
8.5 VertMILP based generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
xii
List of Tables
2.1 Energy versus Exergy [142] . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.1 Stream data for the methanol plant case study . . . . . . . . . . . . . . . . .
40
3.2 Stream data for streams 13, 14 and 16 split into heat and work streams .
41
3.3 Base case actual and theoretical energy targets . . . . . . . . . . . . . . . .
41
3.4 Energy targets for base case and after process modification . . . . . . . . .
42
4.1 HENS Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
4.2 Topics in HENS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
4.3 Heat Integration Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.4 HENS methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
4.5 Mathematical Programming formulations . . . . . . . . . . . . . . . . . . . .
57
4.6 Deterministic optimization models . . . . . . . . . . . . . . . . . . . . . . . .
57
4.7 Metaheuristic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
5.1 Stream data and heat exchanger cost data for Example 1 . . . . . . . . . .
83
5.2 TAC at each step of the Sequential Framework for Example 1(7TP1) . . .
85
5.3 Comparison of the results of Example 1 (7TP1) . . . . . . . . . . . . . . . . .
86
5.4 Match details of best heat exchanger network Example 1 (7TP1) . . . . . .
86
5.5 Stream data and heat exchanger cost data for Example 2 (15TP1) . . . . .
88
5.6 TAC at each step of the Sequential Framework for Example 2 (15TP1) . .
89
5.7 Match details of best heat exchanger network Example 2 (15TP1) . . . . .
90
6.1 Temperature intervals for Example Implementations 1 and 2 for EMAT = 0 98
6.2 Root node LP relaxation value with different measures for 22TP1 with IP
solution 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
xiii
LIST OF TABLES
6.3 Root node LP relaxation value with different measures for 21TP1 with IP
solution 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.4 Root node LP relaxation value and total solution time with different measures for 21TP2 with IP solution 22 . . . . . . . . . . . . . . . . . . . . . . . . 112
6.5 Root node LP relaxation value with different model reformulations for
22TP1 with IP solution 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.6 Root node LP relaxation value with different model reformulations for
21TP1 with IP solution 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.7 Feasibility matrix for 21TP1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.8 Feasibility matrix for 21TP2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.9 Feasibility matrix for 22TP1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.10 Feasibility matrix for 15TP1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.11 Feasibility matrix for 7TP1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.12 Problem difficulty metrics for the test cases . . . . . . . . . . . . . . . . . . . 124
7.1 Temperature Intervals for Example 7TP1 with EMAT = 2.5 K. . . . . . . . 137
7.2 Number of temperature intervals, model solution time and heat exchanger
network cost for 15TP1 problem with EMAT = 2.5 using the three TI generation methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.3 HLD for 15TP1 problem with EMAT = 2.5 using the TI generation method
presented in this work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.4 HLD for 15TP1 problem with EMAT = 2.5 using the TI generation method
presented in Linnhoff and Flower [89] and Jez̆owski et al. [77]. . . . . . . . 139
7.5 Heat Load Distributions calculated for Example 7TP1 with EMAT = 1K
and EMAT = 2.5K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.6 Percentage increase in ∆T LM ,mn values with EMAT = 2.5K compared to
EMAT = 1K for Example 7TP1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.7 Effect of various improvement measures for model solution time - Example
15TP1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.1 Temperature bounds for 7TP1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
xiv
η
efficiency
Superscripts
Nomenclature
ch
chemical
clt
cumulative
k
kinetic
p
potential
ph
physical
Subscripts
Roman Symbols
c
Carnot
eq
equilibrium conditions
min
minimum
total not of process and utility streams
s
supply conditions
U
number of heat exchanger units
t
target conditions
D
direction factor
0
standard/reference conditions
E
exergy, kJ
Acronyms
I
information
ACT
Accelerated Technology
k
Boltzmann constant, 1.38e−26 kJ/K
ASME
American Society of Mechanical Engi-
m
mass, kg
P
pressure, bar
Q
heat, kJ
S
A
availability factor
c
speed of light, 299,792,458 m/s
N
neers
CC
Composite Curves
CCS
CO2 Capture and Storage
EA
Exergy Analysis
entropy, kJ/K
EGCC
Exergy Grand Composite Curve
T
temperature, K
EI
Enthalpy Interval
U
internal energy, kJ
ELCC
Energy Level Composite Curves
v
velocity, m/s
EM AT
Exchanger Minimum Approach Tem-
W
work, kJ
x
composition vector
z
height, m
perature
Greek Symbols
EU D
Energy Utilization Diagram
GDP
Gross Domestic Product
GHG
GreenHouse Gas
HENS
Heat Exchanger Network Synthesis
∆
change in property value
HLD
Heat Load Distribution
Ω
energy level
HR AT
Heat Recovery Approach Temperature
δ
infinitesimal change in property value
IE A
International Energy Agency
xv
NOMENCLATURE
IPCC
Intergovernmental Panel on Climate
PI
Process Integration
ppp
purchasing power parity
PSE
Process Systems Engineering
Change
LP
Linear Program
M ILP
Mixed Integer Linear Program
M I NLP
Mixed Integer Non-Linear Program
T AC
Total Annualized Cost
NLP
Non-Linear Program
TI
Temperature Interval
PA
Pinch Analysis
TPES
Total Primary Energy Supply
PDM
Pinch Design Method
xvi
1
Introduction
“Man’s long adventure with knowledge has been a climb up the heat ladder. . . The creature that crept furred through the blue glacial nights now
lives surrounded by the hiss of steam, the roar of engines, and the bubbling
of vats. And he is himself a great flame, a great roaring wasteful furnace,
devouring irreplaceable substances of the earth.”
Loren Eiseley
1.1
Background
Climate change is a serious problem facing our generation. The IPCC fourth assessment
report [2] concludes, with a very high confidence level, that this is caused by increasing emissions of Greenhouse Gases (GHG) into the earth’s atmosphere due to human
activities. Carbon dioxide is the most significant anthropogenic GHG that accumulates
in the earth’s atmosphere and enhances the greenhouse effect (Figure 1.1). Over 50%
of the greenhouse gas emissions are attributable to CO2 emissions from fossil fuel use.
Fig 1.2 shows the increase in world primary energy1 consumption from 1981-2006. The
world’s primary energy needs are projected to grow by 55% between 2005 and 2030 with
fossil fuels remaining the dominant source of primary energy accounting for 84% of the
increase [4]. This increase is mainly due to the expected economic growth in China and
India.
1 Primary energy is energy that has not been subjected to any transformation process
1
1. INTRODUCTION
Figure 1.1: Global anthropogenic green house gas emissions in 2004 [2]
Climate change mitigation methods for sustainable development can be broadly classified into the following three options:
Option 1: Decrease consumer energy demand
Option 2: Increase energy efficiency
Option 3: Decarbonization of primary energy (including CCS)
More information on the different mitigating options can be found in [2].
Figure 1.3 [3] presents the relative emission reductions attributed to the different mitigating technology options from 2003-2050 in an specific scenario (ACT Map) with optimistic projections in all technology areas and a 2% p.a. increase in end use energy
efficiency.
Improved energy efficiency is seen as the top priority. This, in the above scenario, halves
the expected growth of electricity demand1 and reduces the need for new generation
capacity by a third. CO2 Capture and Storage (CCS) is also expected to play an important
role in mitigating climate change.
1 Note that the 2% efficiency improvement is on the total end-use energy while the 50% reduction is only
on the incremental increase in energy demand
2
1.1 Background
Figure 1.2: World primary energy consumption 1984-2009 [1]
Three Es in the global warming discussion
The interplays between the Economy, Energy and the Environment are at the crux of
the climate change debate.
• Economic growth causes an increase in energy consumption.
• Market forces could drive the price of primary energy upwards if there is a supply
side constraint caused by the increase in fossil fuel consumption. This in turn
could cause an economic slowdown.
• It is common knowledge that an increase in energy consumption adversely affects
the environment.
• Economic growth affects the environment due to other factors such as growth of
industries, increased emissions from transport, deforestation, etc.
GDP/capita1 and population were the main drivers for the increase in CO2 emissions in
the last three decades of the 20 th century (Figure 1.4).
1 GDP is an indicator of the size of an economy while GDP/capita is an indicator of the standard of living
in an economy
3
1. INTRODUCTION
Figure 1.3: Reductions in CO2 emissions in the ACT Map scenario by technology area [3]
. . . and the fourth E
Economic growth is of course desirable and is in fact the driving force that keeps the
gears of the world turning. Efficiency, however, is a key and the only factor that can
break (or loosen) the relationship between energy and economy.
Figure 1.3 shows that improved energy efficiency is expected to be the greatest contributor to reduced CO2 emissions. It is often more cost-effective to invest in energy efficiency
improvement than in increasing energy supply to satisfy demand for energy services.
Efficiency improvement has a positive effect on energy security, local and regional air
pollution abatement, and employment [2].
1.2
Motivation
Energy supply companies and the process industry (including oil refineries) were responsible for approximately 26% and 20% respectively of the total GHG emissions in 2004 [2].
Committment to corporate social responsibility has resulted in moving the process industry towards a more consious way of performing business and requires reporting on the
triple bottom line (people, planet and profit). Improving the efficiency in these sectors
will have a high impact in GHG mitigation while also allowing for profitable businesses.
4
1.2 Motivation
Figure 1.4: CO2 emissions and energy intensities from 1970 - 2004 [2]
Energy integration of a process involves minimizing the consumption of external utilities1 thus leading to an increased system efficiency. Heat integration is a branch of
energy integration where only heat effects (temperature considerations) are taken into
account.
Energy integration, beyond heat integration, of energy intensive processes are mainly
carried out using experience-based heuristics or trial and error methods. Existing systematic methodologies based on thermodynamics, such as Exergy Analysis, only identify
the causes of thermodynamic imperfections in thermal and chemical processes. They do
not provide energy targets or guidelines on how to integrate the process that provide the
basis for an efficient design.
Heat integration or Heat Exchanger Network Synthesis (HENS) has been a subject of
extensive research over the past four decades. One of the key motivations for the use
of a Mathematical Programming based approach is that multiple and complex economic
trade-offs are involved in HENS that simply cannot be properly addressed and solved in
a manual way. Simultaneous Mixed Integer Non-Linear Programming (MINLP) models
can, in theory, address and solve the trade-offs in the HENS problem. These models
have demonstrated severe numerical problems related to the non-linear (nonconvex) and
discrete (combinatorial) nature of the HENS problem. Even with the rapid advancement
1 Electricity (or mechanical work), heating medium, steam, cooling water and refrigerants are commonly
used utilities
5
1. INTRODUCTION
in computing power and optimization technology, the size of the problems solved using
these models does not meet industrial needs.
Stochastic optimization techniques have also been used to solve the HENS problem.
These methods are, however, non-rigorous and the quality of the solution depends on the
time spent on the search.
1.3
Objectives
The primary objectives of this work have been to:
• Develop a systematic methodology based on thermodynamic principles to integrate
energy intensive processes while serving as a screening tool for subsequent heat
integration.
• Develop a mathematical programming based approach using thermodynamics and
insight for solving industrial sized HENS problems while including industrial realism and avoiding heuristics and simplifications.
• Develop a semi-automatic design tool that allows the significant user interaction
to identify near-optimal and practical networks.
1.4
Contributions
The main contributions of this thesis can be divided into three parts:
1.4.1
Exergy based method for energy integration
A novel methodology, “Energy Level Composite Curves”, for energy integration of energy intensive processes based on a graphical tool providing thermodynamic insight was
developed.
1.4.2
Heat exchanger network synthesis review
A brief review of important developments in Heat Exchanger Network Synthesis is presented along with a bibliography of published literature in the period 2000-2008.
6
1.4 Contributions
1.4.3
Sequential Framework for heat exchanger network synthesis
The Sequential Framework for heat exchanger network synthesis has been in development by Prof. Truls Gundersen and his group for a few years. The contributions related
to this work have thus been to take the methodology to a “new level” while adressing the
challenges in its sub-problems. The specific contributions related to this are:
1. Identified and rationalized the loops in the Sequential Framework for heat exchanger network synthesis in terms of the three-way energy, heat transfer area
and the number of heat exchanger units trade-off.
2. It is well known that stream supply temperatures are sufficient to define temperature intervals in the transshipment formulation for minimum energy consumption
and corresponding heat recovery pinch. This work showed that stream supply temperatures are also sufficienct for the corresponding formulation for the minimum
number of units.
3. Novel formulation of the minimum number of units sub-problem was developed.
4. Developed a problem difficulty index for the minimum number of units sub-problem
to identify problems that will be computationally expensive.
5. The importance of Exchanger Minimum Approach Temperature (EMAT) in the
stream match generator sub-problem and its role in obtaining a ranked sequence
of Heat Load Distributions (HLDs) identified. A new EMAT loop added to the
Sequential Framework as part of this work.
6. Automated starting value generators based on physical insight were developed for
the network generation and optimization sub-problem to ensure a “good” solution
for the non-convex Non Linear Programming (NLP) problem.
7. An Excel add-in “SeqHENS” was developed as a front-end tool for user inputs in
generating near-optimal and practical heat exchanger networks using the Sequential Framework.
7
1. INTRODUCTION
1.5
Thesis Structure and Guidelines
The content of this thesis is organized in 9 chapters and 1 appendix.
Chapter 2 presents a brief description of energy issues in process synthesis. Process
Synthesis and Process Integration are introduced and the concept of Energy Integration is defined. The relative merits of exergy and energy are explored in the
context of energy integration.
Chapter 3 presents a novel methodology for energy integration of energy intensive processes based on a graphical tool providing thermodynamic insight. The possibility
of integrating the process with pressure exchangers in addition to heat exchangers is explained. Heuristics are presented for energy integration to obtain energy
targets.
Chapter 4 introduces the HENS problem and presents a brief review of solution methods along with historic milestones. An annotated bibliography of papers related to
HENS published after 2000 is listed.
Chapter 5 is the chapter that introduces and explains the Sequential Framework for
HENS. The motivation for the framework is presented along with the rationale
for the loops in the framework. Advantages and limitations of the framework are
explained. Two literature examples are solved and results presented to evaluate
and compare the framework with other methods.
Chapter 6 presents the minimum number of units subproblem in the framework. The
numerical problems associated with the discrete variables in the model are explained. Strategies to mitigate this issue are dealt with in detail with examples
from literature.
Chapter 7 presents the stream match generator in the framework. The importance
and history of this model is detailed. The numerical problems associated with the
discrete variables in the model are explained. Strategies to mitigate this issue are
dealt with in detail with examples from literature.
Chapter 8 presents the network generation and optimization subproblem in the framework. Numerical problems associated with the non-linear nature of this model are
presented. Strategies to overcome this issue are presented in detail.
8
1.5 Thesis Structure and Guidelines
Chapter 9 summarizes the conclusions from the previous chapters, highlights the contributions and presents suggestions for further work.
9
1. INTRODUCTION
10
2
Energy Issues in Process
Synthesis
“Energy the Eternal Destiny”
Willam Blake
“Nothing in life is certain except death, taxes and the second law of thermodynamics. All three are processes in which useful or accessible forms of some
quantity, such as energy or money, are transformed into useless, inaccessible
forms of the same quantity. That is not to say that these three processes
don’t have fringe benefits: taxes pay for roads and schools; the second law of
thermodynamics drives cars, computers and metabolism; and death, at the
very least, opens up tenured faculty positions. ”
Seth Lloyd
2.1
Introduction
Efficiency of a process is important in ensuring competitive advantage over other viable
processes achieving the same goal. Process Systems Engineering (PSE) involves the
understanding and development of systematic procedures for the design and operation
of efficient process systems, ranging from microsystems to industrial-scale continuous
and batch processes. A broader definition of PSE [53] is:
11
2. ENERGY ISSUES IN PROCESS SYNTHESIS
Process Systems Engineering is concerned with the improvement of decisionmaking processes for the creation and operation of the chemical supply chain.
It deals with the discovery, design, manufacture, and distribution of chemical
products in the context of many conflicting goals.
A key feature of PSE is the discovery of concepts and models for the prediction of performance and for decision-making in an engineered system. The process involves creating
representations and models to generate reasonable alternatives to achieve a goal, and
then select from among them a solution that meets constraints and ideally optimizes an
objective.
The developments in PSE have been closely related to the developments in computing
starting in the 1960s. Computing efficiency is an important intellectual challenge in the
PSE area [53]. Thus an alternative term covering much of the same field is Computer
Aided Process Design and Operation. In fact, the first international conference on PSE
was arranged much later, in 1982, in Kyoto, Japan.
There are two parts to the requirement of computational efficiency. Firstly, engineering
problems of industrial interest in the PSE area are quite often N P -hard. This implies
that in a worst case scenario, computational requirements will increase exponentially
with problem size. Secondly, if the models are being used in a real-time environment,
computations must be completed in a short time frame.
Developing novel representations and models that capture the non-trivial features, as
well as developing computationally efficient solution methods and software tools that
provide new capabilities, are all considered to be original contributions in PSE [53].
The four main fields of PSE are (1) Process synthesis/design, (2) Process control, (3)
Process operations and (4) Supporting tools. This work focuses on a particular process
synthesis problem, discussed in Section 2.4, and its supporting tools.
2.2
Process Synthesis
A process, for the purposes of this thesis, is defined as the transformation of raw materials into products. The synthesis of a process involves two main activities [124]. First,
individual transformation steps are selected. Second, these individual transformations
are interconnected to form a complete process that achieves the required overall transformation.
12
2.2 Process Synthesis
Process synthesis is the systematic generation of alternative process flowsheets1 and selection of a design whose configuration and parameters optimize a given objective function.
Rudd and co-workers introduced the term synthesis, in the design context, in 1969 [95].
The first review paper on process synthesis was published in 1973 [64] and there have
been many subsequent review papers [65, 100, 138, 146]. The most recent ‘review’ papers are describing the trends in conceptual process synthesis [82], a retrospective [147]
and prospective [20] of process synthesis.
2.2.1
Decomposition and hierarchy-based approach to process synthesis
The design of complete processes is a difficult task complicated by the large number
of possible unit operations and their inter-connections. These complex processes, for
which design procedures did not exist, were proposed to be broken down into manageable
subsystems with established design procedures [115]. Rudd and co-workers ordered the
design decisions in a hierarchy that influenced all subsequent process design methods
[123].
Linnhoff and co-workers illustrated the decomposition of the design process by the onion
diagram [92]. The design process starts with selecting the reactor system, followed by
the separation system, followed by the compressors and expanders and finally the heat
recovery system. A more common version of the onion diagram [124] shown in Figure
2.1 ignores the compression and expansion layer while including 2 new outer layers in
the heating and cooling utilities and finally the waste and effluent treatment. An onion
model for process design incorporating the compressors and expanders layer for subambient processes has been proposed [15]. This onion model will be discussed in more
detail in Section 2.4.
Douglas proposed a hierarchical decomposition approach in his design text [32] where
decomposition is ordered, similar to earlier work. The order proposed was: (1) choosing
between continuous and batch, (2) selecting input-output structure of the flowsheet or
selecting the raw materials and products, (3) choosing the recycle structure of the flowsheet, (4) designing the general structure of the vapour and liquid separation systems,
and (5) designing the heat recovery system.
1 Flowsheet is a diagrammatic representation of the process steps with their interconnections
13
2. ENERGY ISSUES IN PROCESS SYNTHESIS
Figure 2.1: Onion model of process design
The formulation, coordination, integration, and overall control of subproblem solutions
have a major impact on the performance of the current generation of process synthesis
methods. Based on the above discussion, it can be seen that process synthesis can be
further classified into flowsheet synthesis and subsystem synthesis as shown in Figure
2.2. The subsystems shown in Figure 2.2 are representative, but not all subsystems are
shown.
2.2.2
Methodologies for process synthesis
Process synthesis methods and tools have been evolving in response to challenges faced
by the chemical process industry. The three broad classes of process synthesis methods
are briefly described below.
1. Heuristics: Heuristic rules and assumptions based on experience and engineering
judgment have been used in all aspects of engineering, and process synthesis is no
exception. Heuristics are required to solve many of the problems industry poses
as they can be used to reduce the solution space. As opposed to the other methods
of process design, heuristics will have to be constantly updated. It is very likely
that the relative importance of many design factors will be radically altered in the
future. The source and cost of energy, the avoidance of climatic impact, materials
14
2.2 Process Synthesis
Figure 2.2: Classification of process synthesis
of construction, and the portfolio of available unit operations may be very different
from today. As a result, many of the common design heuristics based on tradeoffs among factors in today’s context may not be applicable in the future. It will
be necessary to continually reevaluate heuristics and other design assumptions in
light of changing practices, constraints, and economics before they are used.
2. Thermodynamics: Thermodynamic methods have been used to identify design
targets before the design process. Insight obtained while developing concepts and
procedures for targeting is used in the design phase by providing guidelines for the
design. In addition, knowledge about target values can be used to check the quality of the design. An example of such a thermodynamic method is Pinch Analysis
[92] used for targeting the heating and cooling requirements in a process plant.
This methodology has also been extended to mass pinch, thus moving it away from
the realm of thermodynamics. Nevertheless, it is commonly accepted that the extensions of heat pinch are also classified under thermodynamic methods. Thermodynamic methods, such as Exergy Analysis, can also be used to identify causes of
imperfections in the design [81]. Exergy Analysis gives information on the flow of
useful energy through the various steps in a process and has been developed as a
process synthesis methodology [126].
15
2. ENERGY ISSUES IN PROCESS SYNTHESIS
3. Optimization: The first step in this method is to create a superstructure that
embeds all feasible process options and interconnections that are possible optimal
candidates. The design problem is formulated as a mathematical model with an
objective function and a set of constraints. There are two major problems with this
methodology [20] : (1) generating a superstructure that contains the optimal solution and (2) how to solve the large optimization problem inherent in all practical
process synthesis problems. Optimization in process synthesis normally involves
extremely difficult mathematical programming problems, formulated as Mixed Integer Non-Linear Programming (MINLP) models, even for simple design cases. For
all classes of mathematical programming models, there are two major challenges
when applied to process design [54] (a) Non-convex models that lead to local optima and (b) Combinatorial explosion that makes it difficult to solve industrial
problems. There has been considerable development in the use of optimization
methods in process synthesis including stochastic optimization techniques.
Heuristics, thermodynamic methods and optimization are rarely used as stand-alone
methodologies in an industrial design process. It is common to define the synthesis
problem as an optimization model and use thermodynamics and heuristics to reduce the
solution space and hence numerical complexity.
2.3
Process Integration
Process Integration (PI) originated from the heat recovery pinch concept in the 1970s
[66, 89, 90, 94, 139, 140] and the term emerged in the 1980s. It is a dynamic field used
to describe certain system related activities pertaining to process design. Though the
definition of process integration varies, the general definition given by the International
Energy Agency (IEA) [55] is
Systematic and general methods for designing integrated production systems, ranging from individual processes to total sites, with special emphasis
on the efficient use of energy and reducing environmental effects.
Process integration is similar to process synthesis with an emphasis on energy efficiency
and sustainability. The scope of PI has expanded (from its pinch concept base) to total
16
2.3 Process Integration
Figure 2.3: Classification of process integration methods
process design. A key aspect of PI is establishing performance targets before the design
phase. The main features of these targets are [55]:
1. Any design can be objectively compared with the best possible
2. The way some targets are calculated also provides guidelines for the design
The heat recovery pinch concept at the core of PI, has been expanded to other areas by
using various analogies. This made it possible to move from heat transfer systems to, for
example, mass transfer systems [35, 36], waste water and effluent treatment systems
[143, 144] and hydrogen management in oil refineries [7, 135].
Process integration methods can be classified into the same categories as the synthesis
methodologies detailed earlier. One possible classification of Process Integration methods is to use the two-dimensional (automatic vs. interactive and quantitative vs. qualitative) representation in Figure 2.3 [55]. Hierarchical Analysis is placed in the middle
of the figure to indicate that all sensible design methods are based on this idea in order
to make the complete design problem tractable by systematic methods.
The subject area of this thesis is process integration and most of the work presented here
is influenced by the pinch concept.
17
2. ENERGY ISSUES IN PROCESS SYNTHESIS
2.4
Energy Integration
Energy specifications arising from the heat or temperature effects in the first two steps
of the design hierarchy (Figure 2.1) are:
Reactor system design: Heat effects arising from the exothermic/endothermic nature
of the reactions involved, temperature requirements for suitable equilibrium condition or reaction kinetics, etc.
Separation system design: Reboiler and condenser duties for mass transfer columns,
temperature effects in membrane processes, etc.
The heat recovery system is designed based on these inputs. Although temperature and
heat effects are the most conspicuous energy issues in reactor and separation system
design, pressure and shaft-work also play an important role.
In separation systems such as distillation columns, the pressure of the system impacts
heat integration. By changing the pressure levels in such systems, the temperature levels of large heat sources or sinks will change and this may have an impact on direct heat
integration or heat pumping. Reactor system pressure affects the reaction equilibrium
and thus heat effects due to the reaction. Apart from the interaction between the subsystems, reactor or separation system pressure specifications will also determine the extent
of shaft-work requirement in a process.
The onion diagram has been suggested to be modified [15] for sub-ambient processes to
include compression and expansion process design as a third step prior to heat recovery
system design. In this work, we propose yet another version of the onion diagram to
replace the heat recovery design step in Figure 2.1 with an Energy Recovery System
design phase as shown in Figure 2.4.
The Energy Recovery System combines the compression and expansion process and heat
recovery system design phases. This is essential to deal with the interplay between
pressure and temperature in energy issues related to process design.
Energy integration can be defined as systematic methods for generating integrated energy recovery systems.
Energy integration of a process involves minimizing the consumption of external utilities
thus leading to an increased system efficiency. Heat integration is a branch of energy
integration where only heat effects (temperature considerations) are taken in account.
18
2.4 Energy Integration
Figure 2.4: Modified onion model of process design to include Energy Recovery Systems
2.4.1
Energy integration methods
This thesis focuses on Energy Recovery System design and Energy Integration. The
two energy recovery system design methods developed as part of this work are briefly
described below.
Energy Level Composite Curves
Energy Level Composite Curves (ELCC) - a synergy of Exergy Analysis and the Composite Curves of Pinch Analysis - is a novel method for energy integration that incorporates
pressure and composition changes in the process in addition to temperature. This is the
first methodological attempt to represent thermal, mechanical and chemical energy in a
graphical form similar to Composite Curves. This method provides physical insight on
how to integrate energy sources with sinks. The methodology is useful as a screening
tool by functioning as an idea generator prior to the heat integration step.
This methodology is discussed in detail in Chapter 3.
Sequential Framework for Heat Exchanger Network Synthesis
The Sequential Framework is a compromise between Pinch Analysis and simultaneous MINLP models for Heat Exchanger Network Synthesis (HENS). It is an iterative
19
2. ENERGY ISSUES IN PROCESS SYNTHESIS
framework with the main objective of finding near optimal heat exchanger networks for
industrial size problems. The method is based on the recognition that the selection of
matches between hot and cold streams (as well as their duties), referred to as heat load
distributions, impacts both the quantitative (network cost) and the qualitative aspects
such as network complexity, operability and controllability. The Vertical MILP model for
selection of matches and the subsequent NLP model for generating and optimizing the
network form the core engine of the framework. There are two main advantages of the
proposed methodology. First, the design procedure is, to a large extent, automated while
keeping significant user interaction. Second, the subtasks of the framework (MILP and
NLP problems) are much easier to solve numerically than the MINLP models that have
been suggested for HENS.
The various aspects of this methodology are discussed in Chapters 5 - 8.
2.4.2
A note on Energy and Exergy
Energy has been at the forefront of our collective consciousness in this generation. The
energy crisis that has been described may actually be referring to another crisis - an
exergy crisis [114].
Thermodynamics provides the concepts of temperature, pressure, heat, work, energy,
entropy and four laws of thermodynamics.
The first law is a conservation law for energy stating that the energy of a system and
its surroundings, considered together, is constant. This law can also be thought of as an
energy analysis and generally fails to identify losses of work and potential improvements
or the effective use of resources. It treats work and heat interactions as equivalent forms
of energy in transit and offers no indication about the possibility of a spontaneous process
proceeding in a certain direction.
The second law states that the entropy of an isolated system can never decrease. This
can be used to predict what processes can and cannot occur and is the basis for thermodynamic equilibrium calculations. The second law of thermodynamics shows that, for
some energy forms, only a part of the energy is convertible to work, i.e. the exergy1 or
energy quality.
1 Exergy, an International Journal, was established in 2001 but was eventually merged with Energy, an
International Journal. Life imitating nature in that exergy is dispersive while energy is conserved?
20
2.4 Energy Integration
Figure 2.5: Illustration of different exergy components
Exergy and Exergy Analysis
Exergy is the standard for energy quality and can be viewed as the capacity to cause
change. A formal definition is [130]:
Exergy is the amount of work obtainable when some matter is brought to
a state of thermodynamic equilibrium with the common components of its
surrounding nature by means of reversible processes, involving interaction
only with the above mentioned components of nature.
In the absence of nuclear, magnetic, electrical and interfacial effects, the total exergy
consists of [137]:
Kinetic exergy E k : due to the system velocity measured relative to the environment
Potential exergy E p : due to system height measured relative to the environment
Physical exergy E ph : due to the deviation of the temperature and pressure of the system from those of the environment. This is also referred to as thermo-mechanical
exergy.
Chemical exergy E ch : due to the deviation of the chemical composition of the system
from that of the environment.
21
2. ENERGY ISSUES IN PROCESS SYNTHESIS
Energy
Exergy
The first law of thermodynamics
The second law of thermodynamics
Nothing disappears
Everything disperses
Energy is ability to do work
∆U = Q − W
Exergy is the ability to produce useful work
´
³
tot
− S tot
E = T0 S eq
Ener g y = mc2
E = k ln 2T0 I
Energy and matter m is the ‘same thing’
Exergy and information I is the ‘same thing’
Energy is always conserved
Exergy is never conserved in a real process
Energy is a measure of quantity
Exergy is a measure of quality and quantity
Table 2.1: Energy versus Exergy [142]
Physical exergy consists of a temperature component associated with the system temperature and a pressure component associated with the system pressure [81, 137]. Chemical exergy can also be split into a reactive component, associated in its calculation with
chemical reactions, and an non-reactive component, associated in its calculation with
non-reactive processes such as expansion, compression, mixing and separation [81, 137].
The splitting of thermo-mechanical and chemical exergy is mainly useful for defining
better exergetic efficiencies.
An illustration of the different exergy components is shown in Figure 2.5 where the
kinetic and potential components of exergy are grouped together as mechanical exergy
and physical and chemical components of exergy are grouped together as thermal exergy
[131]. It must be noted that thermal and mechanical exergies as defined in [131] can be
misleading as the temperature component of physical exergy is called thermal exergy
and the pressure component is called mechanical exergy in [137].
Exergy balances are written similar to energy balances except that while the energy
balance is a law of conservation of energy, the exergy balance can be taken to be a law
of degradation of energy. The exergy balance for a system thus includes terms for irreversibilities or exergy losses in the system. This is the basis of Exergy Analysis. A
comparison between exergy and energy is provided in Table 2.1 [49].
In the context of process synthesis, the use of exergy is gaining significance. Sama
presents thirteen guidelines for process synthesis based on the second law [118]. One of
the reasons for the use of exergy in process integration is exemplified in the last entry
22
2.4 Energy Integration
of Table 2.1. Exergy can be used as a quantity replacing energy, e.g. minimizing exergy
losses rather than energy consumption of a process, or as a quality parameter. The
ELCC methodology presented in Chapter 3 utilizes exergy as an instrument for energy
integration.
23
2. ENERGY ISSUES IN PROCESS SYNTHESIS
24
3
Energy Level Composite Curves
This chapter presents a novel methodology for energy integration of energy intensive
processes based on a graphical tool providing thermodynamic insight [8, 13].
3.1
Introduction
The efficient use of energy in the process industry is one of the keys to pollution prevention and sustainability. The methodologies for process synthesis described in Section
2.2.2 can be used to improve the energy efficiency of a process in a systematic way. The
earliest approach was to explore all possible plant configurations, leading to the development of a) heuristics to minimize the set of possible configurations to manageable levels
and b) mathematical optimization or search techniques that explore the various configurations efficiently using computers. Thermodynamic methods in process synthesis were
a later development that resulted from a need to better understand the process and lead
to the design of efficient processes.
Pinch Analysis (PA) [92] is a thermodynamic approach to energy integration that is
based on simple, yet powerful, graphical representations. The Composite Curves (CC) of
Pinch Analysis are temperature versus enthalpy curves (one for hot streams that require
cooling and the other for cold streams that require heating) that are used to identify targets for heat exchange. Figure 3.1 shows an example of Composite Curves.
Though Pinch Analysis laid the foundation for energy integration, the fact that it is
based on the first law of thermodynamics and only utilizes temperature as a quality
25
3. ENERGY LEVEL COMPOSITE CURVES
Figure 3.1: Composite Curves of Pinch Analysis
parameter, restricted its application to heat integration. The second law of thermodynamics must be utilized to include work in energy integration studies.
Exergy Analysis (EA) has been used to identify causes of thermodynamic imperfection
in thermal and chemical processes [81, 131]. As exergy takes into account the quality
of energy as well as the quantity, opportunities for efficiency improvement can be explored. Sorin and co-workers proposed an exergy based approach to process synthesis
[126] where exergy load distribution is used to improve process efficiency and sustainability.
In response to a challenge problem issued by Professor B. Linnhoff at the Advanced Energy Systems Division Symposium of the ASME Winter Annual Meeting, Boston, Massachusetts, December 13-18, 1987, the relative merits of Pinch Analysis and Exergy
Analysis methodologies to improve energy utilization of a nitric acid process plant were
explored [47, 87]. These show that while Exergy Analysis is heavily dependent on judgment and past experience to realize improvements in efficiency [47], efficiency improvement using Pinch Analysis is enhanced by visualization of the problem using Composite
and Grand Composite Curves and targets established at the start of the design process.
There have been many approaches to include the concept of exergy in Pinch Analysis
for heat and power integration and to account for exergy losses arising from pressure
and composition gradients in addition to temperature gradients. Umeda and co-workers
[139, 140] used (1 − T0 /T) instead of T in the heat-temperature diagram to make the area
represent exergy annihilations for heat exchanger operations. The Energy Utilization
26
3.1 Introduction
Diagrams (EUD) developed by Ishida and co-workers [73, 74] are used to show exergy
annihilations in all processes, including, but not limited to, heat exchange. Linnhoff and
Dhole developed the Exergy Grand Composite Curve (EGCC) to obtain shaftwork targets
in sub-ambient processes [88]. Staine and Favrat [128] proposed an extension of Pinch
Analysis that takes into account the complete heat transfer exergy losses, the pressure
drop exergy losses and the exergy associated with the fabrication of the heat exchangers.
Feng and Zhu [37] developed a combined pinch and exergy analysis similar to the EUD.
Sorin and Paris [127] incorporate Pinch Analysis in Exergy Analysis by identifying the
heat exchanger network as one unit operation in the exergy load distribution diagram.
Homšak and Glavič [68] proposed the temperature vs. power availability diagram in
addition to normal Pinch Analysis as extended composite curves to incorporate pressure
effects in Composite Curves.
The different approaches listed above present methods to improve efficiency of the process by suggesting process plant modifications based on graphical approaches. Holiastos
and Manousiouthakis [67] present a math programming approach to target for minimum hot, cold and electric utility that can be visualized in temperature vs. enthalpy
and temperature vs. entropy diagrams. Patel and co-workers [106] present a simple
graphical method to set mass, heat and work targets for a process from a mass, energy
and entropy perspective. These two methods only provide targets for the overall process
but do not provide insight or suggestions on process design. Aspelund and co-workers
[15] developed the Extended Pinch Analysis and Design (ExPAnD) procedure where possibilities for converting pressure exergy in process streams to temperature exergy for
heat integration in sub-ambient processes are explored. This methodology utilizes exergy analysis for targeting purposes and provides heuristics for process design.
Most of the methodologies presented in the earlier paragraphs, combining Pinch Analysis and Exergy Analysis lean towards Exergy Analysis in that while sources of imperfection are identified for improvement, integration schemes are not presented. The
exceptions [68, 88] are limited in application.
A novel method for energy integration, Energy Level Composite Curves (ELCC), developed to provide a physical insight to integrate energy sources with sinks is detailed in
this chapter. It is a synergy of Exergy Analysis and Composite Curves of Pinch Analysis
and incorporates pressure and composition changes in the process in addition to temperature. This is the first methodological attempt to represent thermal, mechanical and
27
3. ENERGY LEVEL COMPOSITE CURVES
Figure 3.2: Relative roles of heat load and temperature in evaluating exergy
chemical energy in a graphical form similar to Composite Curves. This method is a useful screening tool functioning as an idea generator prior to the detailed heat integration
step.
3.2
Energy Level
The first step in developing a graphical methodology to represent temperature, pressure and composition similar to the CC of Pinch Analysis involves identifying a quality
parameter. Exergy is unique since it represents both the quality of energy as well as
its quantity. Figure 3.2 [84] shows the role of heat load and temperature in evaluating
exergy where a large heat load with a low temperature can have the same exergy as a
small heat load with a high temperature. This shows that exergy alone cannot be used
as the quality parameter.
The direction factor for a process, D, defined as [73]:
D=
T0 ∆S
∆H
(3.1)
is a quality that can be used in a graphical methodology [73]. As D can take negative
values, the methodology was modified to use the availability factor, A, defined as [74]:
A=
∆E
T0 ∆S
= 1−
∆H
∆H
28
(3.2)
3.2 Energy Level
The availability factor versus enthalpy diagram is called the Energy Utilization Diagram
[74]. As mentioned in Section 3.1, the EUD leans towards exergy analysis in that the
pairing of energy donating and energy accepting streams is fixed and the EUD helps in
reducing the exergy losses by reducing the area between the energy donating and energy
accepting curves.
The availability factor has been used as a quality parameter related to the pricing of
energy carriers [50]. Feng and Zhu [37] called the availability factor energy level, Ω , and
defined it as:
Ω=
exer g y
ener g y
(3.3)
Thus for work:
Ω=1
(3.4)
and for heat
Ω = ηc = 1 −
T0
T
(3.5)
and for steady flow systems
Ω=
∆E
∆H
(3.6)
The energy level fits the requirements of the quality parameter to represent temperature,
pressure and composition. The energy level concept was introduced [37] to visualize
energy quality loss in a process unit and screen for potential process modifications. This,
similar to the EUD, assumes matches between energy donors and acceptors.
Thus, rather than evaluating the energy level of a process unit, it was observed that
evaluating energy level at the supply and target conditions of a process stream gives a
better understanding of the energy requirements and behavior of each stream, and thus
by extension the entire process. This is similar to Pinch Analysis where stream supply
and target temperatures are considered to evaluate the heat content of the stream.
A stream with increasing energy level is an energy sink and a stream with decreasing
energy level is an energy source. Further extending the analogy to PA, an energy source
at higher energy level can be integrated with an energy sink at lower energy level. Thus,
energy level in ELCC is equivalent to temperature in Composite Curves of Pinch Analysis.
29
3. ENERGY LEVEL COMPOSITE CURVES
3.2.1
Evaluating Energy Level
Energy level at stream supply and target conditions is evaluated as:
Ω=
(H − H0 ) − T0 (S − S 0 )
H − H0
(3.7)
Exergy can be broken down in many components as shown in Figure 2.5. In this paper,
only pressure and temperature contributions to exergy, the thermo-mechanical exergy,
are taken into account for evaluating energy level as the focus is mainly on identifying
opportunities for pressure exchange (in the form of shaftwork) in addition to heat exchange. This simplification is a necessary first step to develop a systematic method for
temperature and pressure exchange before considering composition contributions to exergy. The implication is that currently streams undergoing composition change cannot
be properly analyzed by this method.
This simplification would be unacceptable in the "traditional" Exergy Analysis of a chemical plant where exergy and energy accounting is carried out. Evaluating energy level
incorporating chemical exergy, involves developing a new and common reference state
calculation procedure for enthalpy and exergy that ensures positive values for Ω. This
is detailed later in this chapter.
3.3
Construction and Analysis
The new energy level versus enthalpy diagram is constructed by plotting energy level
intervals of process units against cumulative values of enthalpy differences - very similar
to the construction of CC in Pinch Analysis. A stepwise procedure for the construction of
the curves follows.
3.3.1
Construction
Step 1 Evaluate the energy levels (Ω) at the supply and target conditions of each process stream - Ωs and Ω t .
Step 2 To evaluate enthalpy, H, at any given energy level between Ωs and Ω t , calculate
the slope and intercept for each process stream:
Slope =
30
Ω t − Ωs
Ht − Hs
(3.8)
3.3 Construction and Analysis
I nterce pt = Ωs − Slope · H s
(3.9)
Thus
Ω − I nterce pt
(3.10)
Slope
This is a simplified form of the relationship between these two entities - analogous
H(Ω) =
to using constant heat capacity to calculate enthalpy.
Step 3 Divide the streams into two categories - energy acceptors (streams with Ω increasing between supply and target state) and energy donors (streams with Ω decreasing between supply and target state).
Step 4 For energy donors, the energy levels are sorted in ascending order and recurring
Ω values are dropped to give a set of n unique energy levels. This defines the Ω
intervals.
Step 5 If a process stream operates (is present) in the Ω interval [Ω i , Ω i+1 ], its enthalpy
H is calculated at Ω i and Ω i+1 .
Step 6 For this Ω interval, calculate the enthalpy difference using the following equation.
∆H i = [
X
HΩ i −
all streams
X
HΩi−1 ]
∀ i ∈ 2... n
(3.11)
all streams
The total number of intervals is n − 1.
Step 7 Steps 5 and 6 are repeated for all Ω intervals.
Step 8 Calculate cumulative enthalpy difference ∆ H clt , as
∆ H iclt = ∆ H iclt
−1 + ∆ H i −1
∀ i ∈ 2... n
(3.12)
Note that ∆ H1clt = 0
Step 9 At the end of this step there will be n values of Ω and ∆ H clt . These are plotted
to give the energy source curve.
Step 10 Steps 4-9 are repeated for energy acceptors. This gives the energy sink curve.
Step 11 The two curves are moved horizontally relative to one another such that the
energy source curve is above the energy sink curve. This is done for clarity only
and to mimic the Composite Curves of Pinch Analysis.
31
3. ENERGY LEVEL COMPOSITE CURVES
3.3.2
Analysis
The analysis of an ELCC is similar to Composite Curves in Pinch Analysis. The ELCC
provides physical insight to the engineer regarding opportunities for energy integration between the streams on the energy source curve and the energy sink curve. When
considering heat transfer between streams, Equation (3.5) shows that the principle of
transferring energy from a higher energy level value to a lower value is valid as higher
temperature streams will have higher energy level.
Similar to the idea of vertical heat transfer between Composite Curves to minimize heat
transfer area, one would expect that streams on the energy source curve should be integrated with streams placed adjacently below on the energy sink curve to maximize
total energy integration. Further, one would also expect that integration between energy sources and sinks should start where the vertical distance is the least.
An important aspect of the ELCC is that it functions as an idea generator rather than
a design generator. This methodology cannot give any explicit recommendation for integration between process units. A high energy level of a stream can be caused by high
pressure or high temperature or a combination of these. Obviously, one has to distinguish between pressure exchange and heat exchange.
To alleviate this drawback, and to identify the scope for integration, a preliminary targeting methodology has been developed that can be used together with the ELCC.
3.4
Minimum Energy Targets
The minimum energy targets consists of four components - hot utility, cold utility, shaftwork consumed and shaftwork obtained. Linnhoff and Dhole [88] extended Pinch Analysis for the design of low-temperature processes to yield shaftwork targets directly from
basic process data. The EGCC is used to generate shaftwork targets for retrofit schemes
in refrigeration systems. Sorin and Hammache [125] present a modified Site Utility
Grand Composite Curve for shaftwork targeting on total sites. This targeting is based
on thermodynamic insight on co-generation and the Rankine cycle. These procedures
cannot be extended to chemical process systems such as the one studied in Section 3.5.
The primary interest in chemical process systems is how changes in stream pressure
translates into shaftwork.
32
3.4 Minimum Energy Targets
The following is a preliminary utility target procedure based on heuristics and Pinch
Analysis.
3.4.1
Pressure - Temperature relationship
Before developing an algorithm to obtain minimum energy requirement, it would be
worthwhile to refresh the pressure-temperature relationship in process plants.
1. Temperature change from supply to target temperature is possible with very small
pressure drop - one dimensional in temperature. ⇒ Heat transfer
2. For incompressible fluids (liquids), pressure change does not cause any significant
change in temperature - one dimensional in pressure. ⇒ Pumping liquids
3. For compressible fluids (gases), pressure change results in a considerable change
in temperature - two dimensional (temperature and pressure). ⇒ Adiabatic expansion and compression of gases
4. Simultaneous pressure and temperature change is possible without any change in
enthalpy ⇒ Isenthalpic expansion of fluids through a valve
Further, vapour streams undergoing a pressure change almost always (except in the
isenthalpic expansion case) exchange energy in the form of shaftwork in addition to heat.
Streams undergoing temperature change only exchange energy in the form of heat.
3.4.2
First Law and Target for available energy
The enthalpy change (∆ H) of a process stream, neglecting changes in kinetic and potential energy, is given by the first law of thermodynamics for steady state flow processes
as
∆ Ḣ = Q̇ − Ẇ
(3.13)
Q̇ is defined to be positive when heat is added to the system and Ẇ is defined to be positive when work is done by the system. The enthalpy change for a stream undergoing
a process between two thermodynamic states is fixed and independent of process path.
Earlier work [141] approaches the criteria of the optimal process path from an exergy
perspective. Work Ẇ is pure exergy while the exergy of heat Q̇ is η c · Q̇, where η c is
33
3. ENERGY LEVEL COMPOSITE CURVES
the Carnot efficiency. For a process with ∆ Ḣ > 0, e.g. a compression process, the thermodynamic criterion for optimal path between supply and target conditions would be to
minimize the following function:
Z
tar get
su p pl y
(η c · δQ̇ − δẆ)
(3.14)
As a simple illustration, consider a compression process with two process paths. In
process path 1, the stream is pre-cooled and then compressed to the target pressure and
temperature. We have for this process path:
∆ Ḣ = −Q˙1c + Ẇ1
(3.15)
To reduce work, the stream can be cooled to a lower temperature than in process path 1.
Adding a heating duty term, Q˙h , to ensure constant ∆ Ḣ, we have for process path 2:
2
∆ Ḣ = −Q˙2c + Q˙2h + Ẇ2
(3.16)
Note that all terms in Equations 3.15 and 3.16 are positive and do not follow the traditional sign convention in thermodynamics. Subtracting Equation 3.15 from Equation
3.16 we have
¢ ¡
¢
¡
Q˙2h = Q˙2c − Q˙1c + Ẇ1 − Ẇ2
(3.17)
¯ ¯ ¯ ¯
As ¯Q˙2c ¯ > ¯Q˙1c ¯ and Ẇ1 > Ẇ2 , one can conclude that Q˙2h > 0.
As work is done on the system in a compression process and Carnot efficiency is less
than 1, paths involving more heat and less work are optimal. The opposite is true for
processes where ∆ Ḣ < 0.
This is valid from a simple economic perspective too. Electricity is usually the most
expensive utility and hence its consumption must be minimized and, conversely, its production must be maximized.
If work is done by the system for a process with ∆ Ḣ > 0, minimization of (3.14) is unconstrained and would require additional process constraints for solution. Similarly to the
above discussion, the opposite is true for processes with ∆ Ḣ < 0.
This work presents a completely new stream data situation, compared to Pinch Analysis,
as it deals with varying pressures and temperatures. There are many possible paths
from supply to target conditions. This is elaborated in the next section.
34
3.4 Minimum Energy Targets
3.4.3
Optimal Path Heuristics
As seen from the previous section, optimal path heuristics imply minimizing or maximizing work for a given process stream from its supply to target conditions. For a stream at
supply conditions (T s , P s ) and target conditions (T t , P t ) there are four possible situations
above atmospheric pressure [141]:
1. T s < T t , P s < P t (heating + compression)
2. T s > T t , P s > P t (cooling + expansion)
3. T s > T t , P s < P t (cooling + compression)
4. T s < T t , P s > P t (heating + expansion)
Outlined below are heuristics for each of the above cases.
1. To increase pressure from P s to P t , shaftwork is always necessary - either by
pumps or compressors. For vapour streams, compression at lower temperatures
is optimal since this requires lesser work. Further, temperature is also increased
in the process. To obtain minimum work, interstage cooling can be considered, particularly in cases where temperature at the outlet of a single stage compression is
greater than T t .
2. A supply pressure higher than the target pressure offers a potential to gain shaftwork. In the case of vapour streams, expansion must be performed at the highest
possible temperature to extract maximum work. Liquid streams can be expanded
isenthalpically in a valve. Alternatively, a liquid stream can be vapourised and
superheated, passed through an expander and cooled to the target temperature
(depending on heating available at suitable levels).
3. In case of vapour streams cool to temperature below T t and then compress for
minimum shaftwork. For liquid streams, the sequence of pumping and cooling is
not important.
4. A supply pressure higher than the target pressure offers a potential to gain shaftwork. To get maximum work for a vapour stream, heat the stream to a temperature greater than T t and then expand the stream. A liquid stream can be heated
35
3. ENERGY LEVEL COMPOSITE CURVES
Figure 3.3: Algorithm for energy targeting using optimal path heuristics
and then expanded isenthalpically or vapourised and superheated, expanded and
then cooled. The choice of path depends on heating and cooling availability in the
system under consideration.
In all the above instances, it is clear that by assuming optimal path heuristics based on
shaftwork, opportunities for process heat integration are ignored. For a rigorous calculation, optimization methods will have to be employed.
There are many possible paths for going from the supply state to the target state. Only a
few paths, relevant to the case study in Section 3.5, are dealt with here. As the methodology is applied to other case studies, the path options can be expanded.
3.4.4
Algorithm for Energy Targeting
The algorithm for energy targeting based on the optimal path heuristics developed in
the previous section is shown in Figure 3.3 and detailed below.
1. For each stream
36
3.5 Methanol Plant Case Study
(a) If P s < P t , use the suitable optimal path heuristic 1 or 3 to evaluate the minimum shaftwork. Based on information from the optimal path heuristics, split
the stream into one or more work streams and one or more heat streams.
(b) If P s > P t and (P s − P t ) > 5 bar1 , use the suitable optimal path heuristic 2 or 4
to evaluate the maximum shaftwork. Based on information from the optimal
path heuristics, split the stream into one or more work streams and one or
more heat streams.
(c) If P s > P t and (P s − P t ) < 5 bar , treat the entire stream as a heat stream.
2. At the end of Step 1, the minimum work consumed and maximum work obtained
can be evaluated.
3. Use the Heat Cascade method of Pinch Analysis for all the heat streams to evaluate the minimum hot and cold utilities.
3.5
Methanol Plant Case Study
The methanol process is ideally suited for energy integration studies using this methodology as it is an energy intensive process with an enormous interplay of thermal, mechanical and chemical energy in the plant. A methanol plant in Norway [102] is used
to test the efficacy of the methodology for industrial cases. The utility system was not
included in these energy integration studies to simplify the case under consideration.
Figure 3.4 shows the process flow diagram for the methanol plant under consideration.
The 900,000 tonnes/year plant has a reforming section consisting of combined reforming
with a pre-reformer and a Lurgi low pressure methanol synthesis section.
A HYSYS simulation model of the plant was used to provide process stream data for
the energy integration studies. As the simulation model did not have detailed distillation column models for the purification section, the energy integration studies did not
consider that section.
37
3. ENERGY LEVEL COMPOSITE CURVES
Figure 3.4: Methanol plant case study process flow diagram with stream numbers
3.5.1
Energy Integration study using ELCC
An Excel Add-In, HYSYSLink, developed during the course of this work is used to construct the ELCC for the plant as shown in Figure 3.5. The stream data are given in
Table 3.1.
From the curves and stream data, it can be deduced that Streams 14 (exhaust from
burner), 10 (secondary reformer product) and 11 (vapour from Separator 1) are large
energy sources and it appears to be possible to transfer energy to the energy sinks. As
discussed earlier, starting where the vertical distance between the curves is minimum
(Ω pinch), Stream 14 (exhaust from burner) can potentially be integrated with Streams
3 (Primary reformer feed), 4 (Pre-reformer feed), 7 (Natural gas feed) etc in that order.
A study of all the potential matches suggested by Figure 3.5 suggests that the plant
is possibly well integrated with minimum scope for further integration. A targeting
method is required in such cases to verify if the plant is sufficiently integrated and if
further scope for integration exists.
1 The maximum pressure drop through a set of heat exchangers for a stream is taken to be 5 bar for
the case study in Section 3.5. This will change depending on the case on hand and can be evaluated as a
function of (T t − T s ) and P t /P s . This function has not been developed in this work.
38
3.5 Methanol Plant Case Study
Figure 3.5: Energy Level Composite Curves for the Methanol plant case study
Streams 2 (MeOH synthesis reactor recycle), 12 (Syngas from reforming section to MeOH
reactor), 13 (combustion air) and 19 (fuel to burner) involve an increase in pressure.
Electricity is imported from the utility section to supply their needs. Many streams
undergo a small change in pressure that can be attributed to pressure drops in heat
exchangers.
Stream 16 (feed to MeOH purification section) represents an interesting possibility. It
is the only stream in the process undergoing a substantial pressure drop (77.5 bar to 11
bar). The stream is a liquid and hence cannot be expanded to generate shaftwork that
could have been used to satisfy pressure increasing streams. This is indicated by the low
energy level of Stream 16.
It could, however, be possible to vapourise the liquid stream and superheat it (increase
its energy level), expand the stream to generate electricity and then further cool it to the
target temperature.
The effectiveness of this scheme depends on whether the excess energy required for
vapourising and cooling the stream is available in the process at the required level -
39
3. ENERGY LEVEL COMPOSITE CURVES
Supply
Target
Temp
Pressure
Enthalpy
Omega
Temp
Pressure
Enthalpy
Omega
(°C)
(bar)
(MW)
(°C)
(bar)
(MW)
Stream 1
247.8
79.0
173.2
0.27
45.0
77.5
19.6
0.08
Stream 2
45.0
77.5
17.2
0.08
54.5
83.5
21.2
0.09
Stream 3
460.5
40.0
122.2
0.40
587.0
39.0
139.1
0.43
Stream 4
329.0
41.0
111.6
0.38
487.0
40.0
131.8
0.41
Stream 5
117.8
45.0
20.1
0.18
328.0
41.0
107.7
0.38
Stream 6
422.9
46.0
18.8
0.41
238.0
45.0
9.1
0.29
Stream 7
48.0
50.0
0.7
0.08
435.0
46.0
18.5
0.42
Stream 8
45.0
77.5
0.1
0.08
136.0
77.4
0.3
0.18
Stream 9
25.0
40.0
0.1
0.04
220.0
39.4
2.3
0.27
Stream 10
972.9
35.5
222.0
0.51
158.6
32.6
62.9
0.26
Stream 11
158.6
32.6
55.3
0.27
22.6
32.0
2.6
0.04
Stream 12
22.6
32.0
1.9
0.04
148.0
80.5
14.3
0.20
Stream 13
0.0
1.0
0.0
0.00
125.0
1.0
8.2
0.18
Stream 14
1029.6
1.0
106.8
0.51
237.6
1.0
38.0
0.20
Stream 15
72.2
80.5
35.5
0.11
212.5
80.0
111.1
0.26
Stream 16
45.0
77.5
8.4
0.08
46.7
11.0
8.4
0.05
Stream 17
242.6
35.0
1.7
0.41
241.8
34.5
0.6
0.29
Stream 18
245.9
35.0
105.8
0.41
194.7
34.5
30.1
0.25
Stream 19
40.0
50.1
0.1
0.06
73.8
50.0
0.3
0.11
Table 3.1: Stream data for the methanol plant case study
without any additional heating or cooling utilities. This can also be verified with a targeting method.
3.5.2
Energy targeting
The HYSYSLink Excel Add-in is modified to evaluate the energy targets for a HYSYS
simulation. The pertinent streams separated into heat and work streams are given in
Table 3.2.
Applying the optimal path heuristics detailed in Section 3.4 to Streams 12 (Syngas from
reforming section to MeOH reactor), 13 (combustion air) and 14 (exhaust from burner),
it is seen that
• Stream 12 (Syngas from reforming section to MeOH reactor) uses heuristic 2 and
requires no subsequent heating. It is entirely a work stream.
40
3.5 Methanol Plant Case Study
Supply
Target
Temp
Pressure
Enthalpy
Omega
Temp
Pressure
Enthalpy
Omega
(°C)
(bar)
(MW)
(°C)
(bar)
(MW)
Stream 13 A
0.0
1.0
0.0
0.00
9.2
1.1
0.6
0.02
Stream 13 B
9.2
1.1
0.6
0.02
125.0
1.0
8.2
0.18
Stream 14 A
1029.6
1.0
106.8
0.51
233.4
1.0
37.7
0.19
Stream 14 B
233.4
1.0
37.7
0.19
237.6
1.0
38.0
0.20
Stream 16 A
45.0
77.5
8.4
0.08
300.0
77.5
70.6
0.21
Stream 16 B
300.0
77.5
70.6
0.21
154.2
11.0
65.1
0.17
Stream 16 C
154.2
11.0
65.1
0.17
46.7
11.0
8.4
0.05
Table 3.2: Stream data for streams 13, 14 and 16 split into heat and work streams
• Stream 13 (combustion air) follows heuristic 2 and requires subsequent heating.
• Stream 14 (exhaust from burner) follows heuristic 3.
Base case - existing plant
Table 3.3 shows the energy targets obtained for the plant (as in Figure 3.4) using the
new algorithm (with a ∆T min of 40°C for heat cascade calculations) and the actual consumption from the simulation model. Detailed examination of the process flowsheet and
Target
Actual
Hot Utility (MW)
0
1.84
Cold Utility (MW)
292.12
293.98
Work Consumed (MW)
17.34
17.37
Work Obtained (MW)
-
-
Table 3.3: Base case actual and theoretical energy targets
the ELCC shows that the target is not achieved since energy source, Stream 10 (secondary reformer product), is not integrated with any energy sink. This stream is instead
used to generate HP steam and supplies energy to reboilers in the methanol purification
section. In addition, a few other streams are not integrated because of locational and
process constraints.
The results show that the energy targeting model can be applied to generate good estimates for utility consumption that can be realized.
41
3. ENERGY LEVEL COMPOSITE CURVES
Proposed modification from the ELCC
The energy source Stream 16 (feed to MeOH purification section) is considered for integration as indicated in the case study section. The optimal path heuristic for Stream 16
is 2. The stream is divided into three parts - two heat streams and one work stream.
Table 3.4 shows the energy targets obtained after process modification with a ∆T min of
40°C for heat cascade calculations.
Target after modifications
Base Case Target
Hot Utility (MW)
0
0
Cold Utility (MW)
286.58
292.12
Work Consumed (MW)
17.34
17.34
Work Obtained (MW)
5.5
-
Table 3.4: Energy targets for base case and after process modification
The results show that additional shaft power can be generated while no additional hot
or cold utility is consumed as compared to the base case. The proposed modification is
theoretically possible. The plant is tightly integrated with the utility system and unless
a detailed heat exchanger network synthesis study is performed, it is not possible to
evaluate the actual effect on the overall plant energy consumption.
3.6
Limitations
A major limitation of this methodology is that it cannot give any explicit recommendation
for integration between process units and streams; rather it gives a ``feel, insight and
understanding´´of the energy levels in the various process units and guides the engineer
towards potential benefits. This methodology functions as an idea generator rather than
a design generator.
The methodology presented here tries to represent the various facets of energy in a plant
using the two dimensions of the ELCCs. Energy level is a function of the temperature,
pressure and composition of the streams. A high energy level of a stream can be caused
by high pressure or high temperature or a combination of these. The ELCCs thus cannot
distinguish between opportunities for heat exchange or pressure exchange.
42
3.7 Conclusions and further work
To calculate the enthalpy at a particular Ω for a process stream, a linear function was
used while Ω actually is a non-linear function of enthalpy. This simplification could
result in misleading suggestions about integration opportunities.
3.7
Conclusions and further work
A new energy integration methodology has been developed that is a synergy of Exergy
Analysis and Composite Curves. The ELCC is a graphical tool which provides the engineer with insights about energy integration. As pressure, temperature and composition
changes are taken into account when developing the theory for this method, it can be applied to a wide range of processes and in particular to energy intensive chemical plants.
Despite the limitations mentioned earlier, this work represents the first methodological attempt to represent thermal, mechanical and chemical energy in a graphical form
similar to composite curves for the integration of energy intensive processes.
The targeting methodology must be modified to take heat integration into consideration
while developing the work targets. An optimization scheme would be best suited for this.
Finally, the methodology should be expanded to include composition changes in addition
to pressure and temperature changes to ensure that the entire chemical plant can be
analyzed for energy integration.
This energy integration strategy, still in its early phase of development, has shown considerable promise when applied to an industrial case study. Substantial work is required
to develop a complete systematic framework that incorporates thermal, mechanical and
chemical energies.
43
3. ENERGY LEVEL COMPOSITE CURVES
44
4
The Heat Exchanger Network
Synthesis Problem - Review of the
state-of-the-art in the new
millenium
This chapter presents a brief introduction to the Heat Exchanger Network Synthesis
problem and provides a categorized review of the literature published in this area from
2000-2008. This can be considered to be an extension of the review published by Furman
and Sahinidis [46].
4.1
Introduction
Heat Exchanger Network Synthesis (HENS) involves designing the heat recovery system
of a process to improve energy efficiency. It is a component of the Energy Recovery
System layer in the onion model of process design shown in Figure 2.4.
HENS is a challenging task and the most commonly studied problem in process synthesis. The first HENS related work was presented by Ten Broeck [133] in 1944 while
the first attempts to systematically solve the HENS problem were by Westbrook [145]
in 1961 and Hwa [70] in 1965. Rudd and coworkers did pioneering work in this area,
among them the first rigorous definition of the HENS problem [95].
45
4. THE HEAT EXCHANGER NETWORK SYNTHESIS PROBLEM - REVIEW OF
THE STATE-OF-THE-ART IN THE NEW MILLENIUM
The HENS problem can be defined as Given:
• a set H of hot process streams to be cooled,
• a set C of cold process streams to be heated,
• supply and target temperatures, heat capacities, flow rates and heat transfer coefficients of the hot and cold process streams,
• a set of hot and cold utilities available,
• temperatures or temperature ranges, costs and film heat transfer coefficients of
the utilities, and
• heat exchanger cost data,
Develop:
• a network of heat exchangers with minimum Total Annualized Cost (TAC), where
TAC is the sum of the annualized investment and operating costs.
Reducing energy consumption in the early stages of methodology development, has given
way to developing cost-efficient heat exchanger networks as being the main objective
in HENS. This involves a three way trade-off between energy consumption (E), heat
transfer area (A), and how this total area is distributed among the actual number of
heat exchangers (U).
More information on the fundamentals of HENS can be found in texts such as [21, 119,
124]. Exhaustive and thorough reviews on HENS were presented by Gundersen and
Naess [59] in 1988, Ježowski [75, 76] in 1994 and Furman and Sahinidis [46] in 2002. In
fact, this chapter can be considered an extension of the latter incorporating papers from
2000-2008.
4.2
4.2.1
The history of HENS
Overview of the general timeline
Early efforts in HENS, similar to other areas of Process Synthesis, started with the understanding of there being multiple designs for a HENS problem. This led to the generation and evaluation of a large number of these networks on a computer. Research in the
46
4.2 The history of HENS
1960s was focused on generating computer algorithms and heuristics or rules-of-thumb
to efficiently identify the best network(s). Limitations in these techniques prompted the
use of thermodynamic methods in the 1970s, where the methodology led to a better physical understanding of the process. Targeting ahead of design was a terminology born out
of thermodynamic methods.
Developments in Process Synthesis have been closely related to developments in computation. With new developments in computer hardware and solution algorithms (pioneered by researchers in the field of synthesis), there was a renewed interest in mathematical programming based approaches to HENS in the late 1980s and early 1990s.
Sequential methods gave way to simultaneous synthesis methods and the development
of global optimization techniques. Metaheuristic based HENS methods were used in the
1990s with the development of stochastic search algorithms and the failed promise of
the MINLP methods to solve industrial size problems.
The HENS problem was proved to be N P -hard in the strong sense by Furman and
Sahinidis [43] in 2001. Since then there has been a renewed interest in metaheuristics,
hybrid optimization techniques coupling metaheuristics with mathematical programming approaches and evolutionary synthesis.
4.2.2
Developmental milestones
The list of chronological milestones in HENS advances is a modification and extension
of a timeline presented in Furman and Sahinidis [46]. Here the timeline is extended to
2008 with a few modifications before 2000.
• 1944: Ten Broeck [133]. First HENS related paper.
• 1961: Westbrook [145]. First use of dynamic programming for HENS.
• 1965: Hwa [70]. First grassroots HENS. First use of separable programming. First
use of superstructure.
• 1968: Rudd [115]. Decomposition of the process synthesis problem into sub-problems
with HENS being one of them.
• 1969: Masso and Rudd [95]. First formal definition of the HENS problem.
• 1969: Kesler and Parker [80]. The first assignment method for HENS.
47
4. THE HEAT EXCHANGER NETWORK SYNTHESIS PROBLEM - REVIEW OF
THE STATE-OF-THE-ART IN THE NEW MILLENIUM
• 1971: Hohmann [66]. Hohmann-Lockhart composite curves allow calculation of
minimum utilities requirement. The (N −1) estimate for minimum number of units
is first proposed. Groundwork for the Pinch Design Method (PDM) laid in this
thesis. First annotated bibliography for HENS.
• 1972: McGalliard and Westerberg [96]. First paper incorporating sensitivity issues
into HENS.
• 1976: Huang and Elshout [69]. Discovery of point where maximum heat can be
transferred.
• 1978: Umeda et al. [140] and Linnhoff and Flower [89, 90]. Formalization of the
heat recovery pinch point.
• 1978: Westerberg and Shah [148]. First deterministic method for global optimization of HENS.
• 1982: Colbert [29]. First Dual Temperature Approach Method (DTAM) is presented.
• 1983: Linnhoff and Hindmarsh [91]. The PDM is proposed.
• 1983: Cerda et al. [24] and Cerda and Westerberg [25]. The minimum utilities
and minimum number of matches problems are mathematically formulated using
a transportation model.
• 1983: Papoulias and Grossmann [105]. The minimum utilities and minimum number of matches problems are mathematically formulated using a transshipment
model.
• 1984: Linnhoff and Vredeveld [93]. The first HENS paper involving a retrofit is
presented.
• 1986: Tjoe and Linnhoff [134]. A HENS retrofit method based on pinch design is
presented.
• 1986: Li and Motard [83]. A supertargeting method is first developed.
48
4.3 HENS literature in the new millenium
• 1986: Duran and Grossmann [33]. First mathematical formulation to allow one to
embed the minimum cost heat exchange problem within a superstructure model
for entire processes.
• 1986: Floudas et al. [40]. The first fully automated HENS design using an exhaustive superstructure is proposed and implemented in MAGNETS.
• 1986: Rév and Fonyó [111, 112]. Pseudo-pinch point identified.
• 1987: Jones [79]. First use of vertical heat transfer model in HENS.
• 1989: Dolan et al. [31]. Simulated annealing is first used in HENS. This was the
first use of a meta-heuristic method for HENS.
• 1989: Floudas and Ciric [38, 39]. The simultaneous match-network HENS formulation is presented.
• 1989-91: Yuan et al. [152], Yee and Grossmann [150] and Ciric and Floudas [27].
Fully simultaneous HENS formulations are proposed.
• 1997: Athier et al. [16]. First hybrid optimization method for HENS.
• 2002: Furman and Sahinidis [43]. HENS problem was proven to be N P -hard in
the strong sense.
• 2005: Pettersson [108]. First deterministic optimization approach shown to solve
problem with over 30 streams.
4.3
HENS literature in the new millenium
The main purpose of this chapter is to compile and categorize HENS papers from 2000 to
2008. The papers covered here are limited to journal papers in English, published Ph.D.
theses, some conference papers and text books.
225 references have been published in the period 2000-2008 where 216 are journal papers, 4 are papers from conferences, 10 are Ph.D. theses and 4 are text books. Figure 4.1
shows the number of journal papers published by year and indicates that after a period
of decreasing activity from 2000 to 2006 the trend seems to be an increase in research
activity after 2006. The papers have been published in 48 different journals and Figure
49
4. THE HEAT EXCHANGER NETWORK SYNTHESIS PROBLEM - REVIEW OF
THE STATE-OF-THE-ART IN THE NEW MILLENIUM
45
40
35
30
25
20
15
10
5
0
2000
2001
2002
2003
2004
2005
2006
2007
2008
Figure 4.1: Number of HENS journal papers published annually
4.2 shows some of the journals that have contributed to this body of literature. Demographically, researchers and research groups from 43 countries have contributed to the
literature. Figure 4.3 shows countries that have been involved in five or more papers.
A word cloud based on the the titles of the journal articles is shown in Figure 4.4. The
words that appear more frequently in the titles and abstracts will be more prominent in
the respective word clouds. This provides a good overview of the main focus of the papers
by highlighting the “buzz-words”.
The journal title word cloud indicates the prevalence of synthesis or methodology based
papers with a focus on optimization. Pinch also figures prominently. Retrofit methods
and water networks appear to be common themes in the papers. Further, MINLP and
genetic algorithm based methods seem to be equally common areas of focus based on the
word cloud.
While the word cloud provides an overview, the next sections sort the literature based
on different classification schemes. The classification schemes are based on Furman and
Sahinidis [46] but have been somewhat modified to represent the reviewed literature appropriately. Note that the references cited in this section refer to the HENS Bibliography
2000-2008 at the end of this chapter and numbered from R1 onwards.
50
4.3 HENS literature in the new millenium
Figure 4.2: HENS journal papers published divided among journals
50
45
40
35
30
25
20
15
10
5
0
Figure 4.3: Number of HENS journal papers published by country
51
4. THE HEAT EXCHANGER NETWORK SYNTHESIS PROBLEM - REVIEW OF
THE STATE-OF-THE-ART IN THE NEW MILLENIUM
52
Figure 4.4: Word cloud of the journal paper titles
4.3 HENS literature in the new millenium
4.3.1
Software
Table 4.1 lists articles that are partly or completely devoted to describing HENS software
package(s).
AtHENS
[109]
BatcHEN
[28]
DarboTEK
[130]
EXSYS
[81]
HES
[111]
Hint
[117]
SeqHENS
[9]
Table 4.1: HENS Software
4.3.2
Topics in HENS
Table 4.2 lists citations by the various topics of interest in this field. General analysis
includes citations for papers dealing with the basic analysis of the HENS problem or a
particular detail of HENS. Exergy analysis uses thermodynamic second law analysis in
HENS. Flexibility/controllability includes citations that consider controllability aspects
of HENS and the degree of flexibility of HENS in case of parameter variations. Multipass
heat exchangers allow streams to exchange heat in multiple passes. Multistream heat
exchangers include citations that involve heat exchangers with more than two streams.
Pressure drop is work that incorporate pressure drop effects in HENS and are important
for networks with large distances. Multi-period is citations that involve multi-period
HENS. Detailed HX design refer to papers that include detail heat exchanger design
along with network synthesis. Retrofit design includes citations that involve modification
of existing networks.
4.3.3
Heat Integration Topics
Citations related to heat integration topics in process synthesis are listed in the Table
4.3. Heat integration involves synthesizing the HEN and the process as a whole. Specific
topics of process and heat integration covered are batch, distillation, reactors, heat engines/pumps and refrigeration systems, mass exchanger networks and separations, water
53
4. THE HEAT EXCHANGER NETWORK SYNTHESIS PROBLEM - REVIEW OF
THE STATE-OF-THE-ART IN THE NEW MILLENIUM
General analysis
Exergy analysis
Controllability/flexibility
[79], [208]
[78], [79], [89], [98], [105], [114], [123], [173]
[1], [6], [12], [17], [32], [70], [88], [113]
[129], [142], [192], [193], [194], [217]
Multipass heat exchangers
Multistream heat exchangers
Pressure drop
[103], [148]
[42], [150], [207], [212], [213]
[5], [29], [50], [78], [135], [152], [177]
[178], [204], [214], [223]
Multi-period
Detailed HX design
Retrofit design
[1], [77], [113], [221]
[29], [78], [108], [122], [146], [152], [153], [184]
[7], [8], [20], [30], [29], [56], [90], [91]
[93], [110], [112], [127], [128], [132], [134]
[139], [143], [145], [151], [200], [198]
[202], [211], [214], [225]
Other
[52], [80], [51]
Table 4.2: Topics in HENS
networks and utility systems. There is an overlap in the categories of mass exchanger
and water networks, but given the prevalence of references to water networks (see Figure 4.4), water networks are included as a separate classification. Process integration
refers to general heat integration in process synthesis. The plant/site integration includes citations for heat integration across multiple processes in a plant or site.
4.3.4
HENS solution methods
The solution methods for HENS problems can be broadly classified into Pinch Analysis and other thermodynamics based evolutionary methods, deterministic optimization
methods, metaheuristic methods and hybrid solution techniques. The citations related to
these methods are given in Table 4.4. The citations for Pinch Analysis based methods
are are categorized into methodology papers, where new methods based on Pinch Analysis are developed, and application or case studies where Pinch Analysis based methods
have been used to solve industrial problems.
Deterministic optimizations methods are those that solve to an optimum without probabilistic or stochastic methods. Mathematical Programming is deterministic. Table
tab:HENSdet gives citations for the four major classes of Mathematical Programming
54
4.4 Conclusions
Batch processes
Reactors
Distillation Processes
[3], [28], [34], [47], [92], [197]
[95], [165]
[12], [11], [17], [49], [55], [77], [115], [116]
[151], [162]
Heat Engines/pumps and refrigeration systems
MENS and separations
[20], [82], [123], [211]
[27], [26], [37], [38], [60], [64], [67], [68]
[118], [154], [209]
Water networks
[16], [27], [26], [72], [101], [106], [154]
[169], [170], [171], [189], [222]
Utility system
Process integration
[39], [87], [133], [147], [201]
[20], [59], [60], [62], [81], [102]
[210], [224]
Plant/site integration
[15], [13], [160], [161]
Table 4.3: Heat Integration Topics
problems - Linear Program LP, Mixed Integer Linear program MILP, Non-linear program NLP and Mixed Integer Non-Linear program MINLP. Citations for Mathematical
Programming based formulations for HENS is given in Table 4.5. Simultaneous HENS
citations are split into citations that are based on the stage-wise superstructure of Yee
and Grossmann [150] and other superstructures. Similarly, the retrofit HENS is also
split into those that are based on the stage-wise superstructure or not.
Metaheuristic methods are further categorized into the different solution techniques.
Hybrid methods are those that use both deterministic and metaheuristic based methods
to solve the HENS problem.
Citations for reviews and books are also given in Table ??.
4.4
Conclusions
HENS as a research field has continued to be an active area of research in the new
millennium. The number of journal papers (Figure 4.1) published in the period 20002008 is a testimony to this. It has also attracted researches from many countries (Figure
4.3).
55
4. THE HEAT EXCHANGER NETWORK SYNTHESIS PROBLEM - REVIEW OF
THE STATE-OF-THE-ART IN THE NEW MILLENIUM
Pinch analysis and other
evolutionary methods
(Methodology)
[130], [5], [11], [31], [31], [39], [47], [55]
[58], [78], [79], [87], [94], [108], [120]
[128], [140], [164], [167], [166], [168], [172], [175], [176]
[179], [186], [141], [203], [208], [209], [210]
Pinch analysis and other
evolutionary methods
Applications/Case studies
[126], [7], [8], [10], [20], [41], [60]
[65], [82], [85], [86], [89], [93], [96]
[99], [98], [97], [115], [119], [125]
[134], [139], [162], [163], [182], [195], [198]
[205], [216], [219], [131]
Deterministic methods
[1], [2], [3], [6], [15], [13]
[14], [16], [17], [19], [19], [25], [27]
[26], [32], [36], [38], [45], [50], [51]
[56], [64], [66], [67], [69], [84], [100], [118], [121]
[156], [157], [158],
Metaheuristics
[4], [33], [34], [40], [42], [72], [92], [102]
[104], [107], [113], [136], [146], [155], [184], [206], [212]
[213], [215], [218]
Hybrid
Review/Book
[24], [35], [42], [49], [195]
[43], [44], [46], [53], [59], [61], [62], [83], [63]
[174], [187], [188], [224]
Table 4.4: HENS methodologies
Publications in this field are varied in topic as can be observed from Table 4.2. Further,
the varied nature of citations necessitated an increase in the number of classifications
since Furman and Sahinidis [46] published their review.
There has been sustained interest in simultaneous synthesis using mathematical programming, albeit for smaller test problems. Most of the simultaneous synthesis references are based on the superstructure of Yee and Grossmann [150] or a variant thereof.
While most of the papers published were methodology oriented papers, over 25% of the
papers were devoted to case studies. Most of the case studies applied Pinch Analysis
based evolutionary methods.
Though there has been significant developments in HENS using mathematical programming methods, synthesis of large scale HENS problems without simplifications and
heuristics have been lacking. This is an area that requires more research before mathe-
56
4.4 Conclusions
Utilities cost
[12] , [13], [14], [39], [66], [70], [73]
[74], [75], [181]
Number of matches/units/shells
[12], [54], [57], [71], [180]
Area/Network topology
[9], [45], [76], [124], [181]
Simultaneous HENS (Yee and Grossmann)
[1], [2], [22], [25], [32], [36], [50], [69]
[84], [88], [100], [122], [138], [144], [152]
[190]
Simultaneous HENS (other)
Simultaneous Process and HENS Synthesis
Retrofit HENS (Yee and Grossmann)
Retrofit HENS (other)
[6], [18], [19], [28], [51], [137]
[37], [38], [125]
[145], [191]
[56], [112], [185], [220]
Table 4.5: Mathematical Programming formulations
LP
MILP
[13], [66], [73], [76], [75], [132], [161]
[9], [13], [14], [16], [18], [19], [71], [28]
[37], [56], [57], [64], [70], [74], [138], [137]
[180], [181]
NLP
MINLP
[9], [90], [91], [124], [183], [185]
[1], [2], [21], [22], [23], [25]
[26], [28], [34], [36], [50], [51], [67], [69]
[84], [106], [100], [112], [122], [149], [138], [141]
[142], [144], [145], [147], [152], [159]
[177], [190], [196], [221]
Table 4.6: Deterministic optimization models
matical programming based approaches can be used in the industry.
??table.caption.5186
table.caption.5186
table.caption.5186 table.caption.51, 87table.caption.5187 ta-
ble.caption.51, 87table.caption.5187 table.caption.51, 87table.caption.5187
ta-
ble.caption.441, 139table.caption.441139table.caption.441139 table.caption.441, 139table.caption.441139table.caption.441139
57
4. THE HEAT EXCHANGER NETWORK SYNTHESIS PROBLEM - REVIEW OF
THE STATE-OF-THE-ART IN THE NEW MILLENIUM
Genertic algorithms
Simulated annealing
[4], [33], [40], [72], [92], [102], [104], [146], [155], [206], [207]
[42], [113], [206]
Tabu search
[107]
Random search
[136]
Particle swarm
[184]
Differential evoution
[215]
Table 4.7: Metaheuristic methods
58
HENS Bibliography 2000-2008
[R1] J. Aaltola. Simultaneous synthesis of flexible heat exchanger network. Applied Thermal Engineering,
22(8):907–918, June 2002.
[R2] C. S. Adjiman, I. P. Androulakis, and C. A. Floudas. Global optimization of mixed-integer nonlinear
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5
The Sequential Framework for
Heat Exchanger Network
Synthesis
This chapter presents the Sequential Framework, a sequential and iterative framework
for the near-optimal synthesis of heat exchanger networks [9, 10, 11, 57, 58, 61, 63].
5.1
Introduction
The Heat Exchanger Network Synthesis (HENS) problem involves solving a three way
trade-off between energy (E), heat transfer area (A), and how this total area is distributed into a number of heat exchangers (U). Figure 5.1 pictorially represents the
trade-offs in HENS problems. Chapter 4 provides a brief introduction to the HENS
problem and recent trends in solution methods.
While heuristic approaches based on experience have governed the process industries
since its inception, they have, increasingly, given way to systematic design methods. This
is particularly true in the case of HENS. The discovery of the heat recovery pinch, driven
by thermodynamic analysis, provided the basis for advancement of synthesis techniques
for HENS. Pinch Analysis (PA) [85, 91, 94] is a sequential solution strategy based on
the thermodynamic approach, where the HENS problem is solved in stages. Though
thermodynamic based methods offer physical insight using graphical diagrams and user
interaction, the complex and multiple trade-offs involved in the HENS problem simply
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5. THE SEQUENTIAL FRAMEWORK FOR HEAT EXCHANGER NETWORK
SYNTHESIS
Figure 5.1: Three way trade-off in HENS problems
cannot be addressed and solved in a manual way. Further, all the methods mentioned
above are evolutionary and one of the drawbacks of such methods is that they may end
up in topology traps [60].
Optimization methods have been routinely applied in an effort to solve complex and multiple trade-offs that are inherent to the HENS problem. Simultaneous MINLP methods
(for example [27] and [150]) can, in theory, address and solve the trade-offs in the HENS
problem. These models have demonstrated severe numerical problems related to the
non-linear (non-convex) and discrete (combinatorial) nature of the HENS problem. Even
with rapid advancements in computing power and optimization technology, the size of
the problems solved with these methods does not meet industrial needs. In addition,
these models are restricted by too many simplifying assumptions.
While deterministic approaches have experienced numerical problems, as discussed above,
stochastic search algorithms or metaheuristic methods (such as Simulated Annealing
[31] and Genetic Algorithms [14]) have been applied to try to overcome some of these
problems. A limitation with such methods, however, is that user interaction is minimal
(there has been some work towards rectifying this recently [42]) and search towards the
(possible) optimum moves (as the name indicates) in stochastic ways. There is a trade-off
between the quality of the solutions and the time spent in the search - the search may
take hours and even days. Further, there is no way to identify how far away from (or
close to) the global optimum the obtained solution is.
The HENS problem has been proven to be N P -hard in the strong sense [43] and has
prompted renewed interest in synthesis methods for HENS that utilize the strategy of
dividing the HENS problem into a series of sub-problems to reduce the computational
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5.2 Ultimate goal
complexity of obtaining a network design. One such methodology is the Sequential
Framework presented in this chapter.
5.2
Ultimate goal
The goal of the Sequential Framework is to develop a methodology for HENS that
• solves industrial size problems: Industrial size problems, in this work, are defined
to be those with 30 or more process streams. Most deterministic methodologies
deal with small 5-10 stream examples and cannot be applied to large problems due
to numerical issues.
• includes industrial realism: In addition to solving large problems, the methodology
should be able to incorporate multiple utilities, constraints in heat utilization, include heat exchanger models beyond pure countercurrent, and allow multiple cost
laws.
• avoids heuristics and simplifications: The methodology must have no global or
fixed ∆T min , pinch decomposition or simplified cost laws.
• incorporates a semi-automatic design tool: A user friendly software that allows significant user interaction and control while identifying near-optimal and practical
networks.
5.3
5.3.1
The Methodology
Sequential synthesis of HENS using Mathematical Programming
Sequential synthesis methods divide the HENS problem into a series of subproblems
that are solved sequentially in order to reduce the computational complexity of the problem. Typically, sequential synthesis via mathematical programming involves solving
three subproblems:
1. The first subproblem is the minimum utilities cost problem (or utilities targeting) where the operating cost of the network (sum of hot and cold utility costs) is found.
This is an LP problem that can be formulated as a transshipment (such as [105]) or
transportation model (such as [24]) to include the possibility of forbidden matches.
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5. THE SEQUENTIAL FRAMEWORK FOR HEAT EXCHANGER NETWORK
SYNTHESIS
2. With the utility targets obtained from the previous problem, a MILP formulation,
again either as a transshipment (such as [105]) or transportation (such as [25])
model is solved for minimum number of matches or heat exchanger units. The
main purpose of this model is to determine Heat Load Distributions (HLDs). HLDs
can be thought of as a matrix, with each column representing a cold process stream
or utility and each row representing a hot process stream or utility. A zero entry
in the matrix indicates that there is no heat exchange between the corresponding
streams, while a non-zero entry indicates a heat exchange between the streams
with duty equal to the entry value. The number of heat exchanger units is the
total number of non-zero elements in the matrix.
3. An NLP model (such as [40]) is applied for the development of a HEN based on
a network superstructure solving for the minimum capital cost with respect to
exchanger area, given the HLD from the previous subproblem.
Even though the methodology detailed above uses mathematical programming, it follows
the thermodynamic approach in requiring the partitioning of temperature ranges into
temperature intervals. This is important to ensure that heat exchange follows the laws
of thermodynamics.
MAGNETS [40] was the first software for the sequential synthesis of HENS using the
subproblems listed above. The Sequential Framework described in this work is an extension of such a procedure that is a combination of thermodynamic methods and mathematical programming. Other sequential synthesis methods that are based on a combination of thermodynamic methods and mathematical programming are the block decomposition method [154] and the hypertargeting methodology [23].
5.3.2
The Sequential Framework for HENS
As a compromise between Pinch Analysis and simultaneous MINLP methods, a sequential and iterative framework has been under development [9, 10, 11, 57, 58, 61, 63] with
the main objective of finding near optimal heat exchanger networks for industrial size
problems. The Sequential Framework is based on the recognition [58] that the selection
of HLDs impacts both the quantitative aspects (cost) and qualitative aspects (complexity,
operability and controllability) of networks.
78
5.3 The Methodology
Figure 5.2: The Sequential Framework for heat exchanger network synthesis
The subtasks of the Sequential Framework involve: establishing the minimum energy
consumption (LP), determining the minimum number of units (MILP), finding sets of
matches and corresponding Heat Load Distributions (HLDs) for the minimum or a given
number of units (MILP), and network generation and optimization (NLP) as shown in
Figure 5.3.
The Vertical MILP model for the selection of matches and the subsequent NLP model for
generating and optimizing the network constitute the core engines of the framework.
It is important to note that all HENS solution methods break down the problem. While
the Pinch Design Method is sequential and evolutionary, simultaneous MINLP methods
allow mathematical considerations to decompose the problem. In the Sequential Framework, the problems are decomposed based on knowledge about the HENS problem. The
methodology allows significant user interaction with the engineer acting as a top level
optimizer making judgments based on quantitative and qualitative considerations.
A brief description of the four subtasks, rationale for the loops in the framework and
initialization and loop sequence are given in the subsequent sub-sections.
5.3.3
Minimum Utilities Targeting
Utilities targeting is done using a transshipment model based on the model presented
by Papoulias and Grossmann [105] and extended to include multiple utilities. Given a
value of the Heat Recovery Approach Temperature (HRAT), the minimum hot and cold
utility requirement are evaluated.
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5. THE SEQUENTIAL FRAMEWORK FOR HEAT EXCHANGER NETWORK
SYNTHESIS
5.3.4
Calculating the absolute Minimum Number of Units
The minimum number of units problem is formulated as a MILP transshipment problem
based on the model presented by Papoulias and Grossmann [105]. This units target is
calculated for a given energy target using the Exchanger Minimum Approach Temperature (EMAT) of zero. The model has been modified subsequently as part of this work to
improve solution time and deal with the combinatorial explosion issue. This model and
its role in the Sequential Framework is looked at in depth in Chapter 6.
5.3.5
Stream Match Generator
The Stream Match Generator subproblem generates HLDs for a given energy target and
number of units. This subproblem is formulated as an MILP transportation model based
on the model published by Cerda and Westerberg [25]. The objective function of the
model in [25] is changed to minimize the “pseudo-area” of the heat exchanger network.
This model is the key feature of the Sequential Framework presented in this work and
is based on vertical heat exchange considerations. The “Vertical” MILP model in the
framework has been in constant development and is presented in detail in Chapter 7.
5.3.6
Network Generation and Optimization
The network generation and optimization subproblem of the framework generates a cost
optimum heat exchanger network given a set of heat load distributions. This is formulated as an NLP problem [40] and does not include any specification for EMAT or HRAT.
This non-convex model as well as efforts to generate a global solution are presented in
detail in Chapter 8.
5.3.7
Rationale for loops in the framework
The loops in the framework simulate the three way trade-off indicated in the introduction. Loops 1 and 2 can be thought of as area loops, loop 3 as the unit loop and finally
loop 4 as the energy loop.
5.3.8
Initialization
The level of heat recovery (represented by HRAT, the Heat Recovery Approach Temperature) is initialized by a pre-optimization procedure such as SuperTargeting (ST) pre-
80
5.3 The Methodology
sented by [86]. The number of units (U) is initialized for the corresponding HRAT to
be the absolute minimum number of units (Umin ) using an MILP Transshipment model
allowing the Exchanger Minimum Approach Temperature (EMAT) to be zero. Using an
EMAT in addition to HRAT, where EMAT ≤ HRAT, allows heat exchange across pinch
points both ways and hence a larger number of feasible solutions. The EMAT for the
Stream Match Generator in the core of the framework is initialized to a small value (ex.
HRAT/8), a more thorough discussion is provided in Section 5.3.9 and Chapter 7.
5.3.9
Loop sequence
The logical sequence of actions is indicated in Figure 5.3 as the following nested loops:
1. Derive networks for the second or the third best HLDs, keeping U, EMAT and
HRAT unchanged: Experience has shown that the Vertical MILP Transportation
model identifies an almost perfectly ranked sequence of HLDs that leads to networks with increasing cost. The HLD loop is mainly relevant for the qualitative
aspects of the network as described earlier.
2. Adjust the value of EMAT slightly above the earlier value: Choosing EMAT is
not straightforward in the Vertical MILP Transportation model as the value of
EMAT is used to create additional enthalpy intervals and has to be balanced. If it
is set too low, the HLDs with non-vertical heat transfer will face large penalties.
On the other hand, if the EMAT is set too high, potentially good HLDs may be
excluded from the feasible set of solutions. EMAT can be varied in two or three
steps between HRAT/8 and HRAT/2. It is worth noting that EMAT performs a
similar function as the +X/-X rule when optimizing networks in the Pinch Design
Method using heat load loops and paths.
3. Increase the number of units by one: For a given value of HRAT, the best solution
is one where U is close to the corresponding Umin . Hence starting at Umin ensures
that the number of loops the designer has to explore to obtain the best solution is
minimal. Also, in the first run of the framework, with U = Umin , experience shows
that EMAT does not affect the HLDs. This could be due to the absence of any
degree of freedom in the model.Thus loop 2 can be ignored in the vast majority of
cases. The units loop is terminated when increasing the value of U does not lead
to a decrease in the TAC after fully exploring the two inner loops.
81
5. THE SEQUENTIAL FRAMEWORK FOR HEAT EXCHANGER NETWORK
SYNTHESIS
4. Adjust the value of HRAT: With a good pre-optimization procedure, only minor
adjustments are expected here.
From the above discussion it is evident that, though there are a number of loops in the
framework, the best solution is arrived at early in the synthesis process.
5.4
Advantages
There are two main advantages of the Sequential Framework. Firstly, the subtasks of
the framework (MILP and NLP problems) are much easier to solve numerically than the
MINLP models suggested for HENS. This is a key feature of all sequential methods, and
is discussed in detail in the chapters for the individual subproblems - Chapters 6, 7 and 8.
The second advantage is that the design procedure is, to a large extent, automated while
keeping significant user interaction. The design engineer acts as a top level optimizer
making judgments based on quantitative and qualitative considerations.
The loops in the framework enable the methodology to explore a large part of the solution
space with respect to the trade-off between E, U and A. This enables the generation of
multiple designs that can be evaluated, as mentioned earlier, on a quantitative and/or
qualitative basis. This is a feature that is missing in other sequential methods.
The Sequential Framework does not contain any model simplification, assumptions or
heuristics. This is advantageous in generating networks with industrial realism. Another advantage is the fact that the search for good designs by exploring the loops of the
framework will focus on the most promising part of the feasible solution space. This is
the result from using domain knowledge in setting up the loop structure and initializing
parameters.
5.5
Challenges
As the number of streams is increased while using the sequential framework, the first
bottleneck occurs in the minimum number of units sub-problem, where the MILP formulation is unable to handle large problems due to “combinatorial explosion”. This
is experienced in the stream match generator sub-problem as well. Chapters 6 and 7
present work done to mitigate this problem. Significant improvements are required to
solve industrial size problems.
82
5.6 Limitations
Ts
Tt
mC p
∆H
h
K
K
kW/K
kW
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
1.69
427.57
3.20
ST
650
650
-
-
3.50
CW
293
308
-
-
3.50
Stream
0.83
Exchanger cost ($) = 8,600 + 670A
(A is in m2 )
Table 5.1: Stream data and heat exchanger cost data for Example 1
As mentioned earlier, the NLP model is non-convex and global optimization methods
have to be employed to this sub-problem. This is detailed in Chapter 8. It must be noted
that this challenge is only weakly dependent on the size of the HENS problem and it
does not represent a bottleneck with respect to time.
5.6
Limitations
The Sequential Framework does not generate networks that include cyclic matches where
a pair of streams are matched against each other more than once. The limitation arises
in the stream match generator sub-problem, where HLDs are generated for a given number of units and only one match is allowed between a pair of streams.
By virtue of being a sequential method for HENS, the methodology can not guarantee
global optimum for the overall HENS problem. The Sequential Framework detailed in
this work generates near-optimal networks. Even though the loops allow exploring a
large region of the solution space, it may not always be practically feasible to generate a
whole range of networks to identify the “best” solution.
83
5. THE SEQUENTIAL FRAMEWORK FOR HEAT EXCHANGER NETWORK
SYNTHESIS
Figure 5.3: SeqHENS interface
5.7
A semi-automatic design tool - SeqHENS
The Sequential Framework methodology, as described in the earlier sections, requires
information transfer from one sub-problem to the subsequent sub-problem. Each of the
sub-problems are modeled separately in GAMS1 . In addition, user inputs are required
to solve each sub-problem. To ease the transfer of data between the sub-problems and
interface with the user, an Excel add-in SeqHENS was developed as part of this work.
The user inputs data, such as stream data, number of units etc., in Excel and this data
is passed to GAMS and the solution from GAMS is sent back to Excel. GAMS runs in
the background in this set-up.
SeqHENS has been developed as a semi-automatic design tool for synthesis of heat exchanger network synthesis using the Sequential Framework. This allows for significant
user interaction in the synthesis process.
1 General Algebraic Modeling System (http://www.gams.com) - a high-level modeling system for mathe-
matical programming and optimization
84
5.8 Examples
Soln. No
U
EMAT (K)
HLD
TAC ($)
1
8
2.5
A
199,914
2
8
5
A
199,914
3
8
7.5
-
No Soln
4
9
2.5
A
147,861
5
9
2.5
B
151,477
6
9
5
A
147,867
7
9
5
B
151,508
8
9
7.5
A
149,025
9
9
7.5
B
149,224
10
10
2.5
A
164,381
11
10
5
A
167,111
12
10
7.5
A
164,764
Table 5.2: TAC at each step of the Sequential Framework for Example 1(7TP1)
5.8
5.8.1
Examples
Example 1 (7TP1)
The Sequential Framework is used to design a heat exchanger network for Example 3 in
Colberg and Morari [28] and Example 4 in Yee and Grossmann [150] with the problem
data given in Table 5.1. This is an interesting problem as the streams involved have a
large difference in heat transfer coefficient. Further, the two earlier papers where this
example is presented approach the problem from very different perspectives - minimizing the total annualized cost as compared to minimizing area. This is discussed in detail
subsequently.
HRAT is fixed to be 20 K [150]. The Umin for this level of heat recovery is 8. U is adjusted
between 8 and 10 in loop 3, and EMAT is adjusted to be 2.5, 5 and 7.5 in loop 2. Table 5.2
presents loop parameters and Total Annualized Cost (TAC) at each step of the network
generation process using the Sequential Framework.
For the case where U = 8, both EMAT = 2.5 and EMAT = 5 give the same (and only)
solution with a TAC of $199,914 with no split streams. This is in agreement with the
arguments made in Section 5.3.9. The cost of this solution is 32.4% above the solution
from Yee and Grossmann [150] where an optimized network of 9 units with a cost of
$150,998 is presented. For EMAT = 7.5 there is no feasible solution to the Vertical MILP
85
5. THE SEQUENTIAL FRAMEWORK FOR HEAT EXCHANGER NETWORK
SYNTHESIS
No. of units
Area (m2 )
Colberg and Morari [28]
22
173.6
Colberg and Morari [28]
12
188.9
177,385
Yee and Grossmann [150]
9
217.8
150,998
Isiafade and Fraser [72]
10
251.5
168,700
Sequential Framework
9
189.7
147, 861
Cost ($)
Table 5.3: Comparison of the results of Example 1 (7TP1)
Match
Duty
Area
kW
m2
H1-C1
323.6
14.33
H1-C4
68.4
1.83
H2-C1
176.2
74.71
H2-C2
119.9
45.68
H3-C1
88.8
13.11
H3-C3
457.6
17.72
H3-C4
359.1
12.19
ST-C1
244.1
8.54
H3-CW
172.6
1.56
Table 5.4: Match details of best heat exchanger network Example 1 (7TP1)
Transportation model. For the case where U = 9 and EMAT = 2.5, the “best” solution
from SeqHENS is obtained and presented in Figure 5.4. Match details for this “best” solution is given in Table 5.4. The TAC of $147,861 compares favourably with the solution
presented in Yee and Grossmann [150]. For the case where EMAT = 5, the matches are
the same as that when EMAT = 2.5, only the heat loads are slightly shifted giving a total
cost of $147,867. For the case where EMAT = 7.5, the matches are completely different
and the resultant network resembles the solution of Yee and Grossmann [150] with a
TAC $149,025.
Table 5.3 shows a comparison of the results from Colberg and Morari [28], Yee and
Grossmann [150], Isafiade and Fraser [72] and the Sequential Framework. Colberg and
86
5.8 Examples
Figure 5.4: The best heat exchanger network for Example 1 (7TP1) - Solution no. 4
Morari [28] optimize the area using a spaghetti network1 . They have a solution with
22 units and an area of 173.6 m2 . From this solution, they synthesize the network by
evolution to present a network with 12 units, area of 188.9 m2 and TAC of $177,385. The
solution presented by Yee and Grossmann [150] has a total area of 217.8 m2 , which is
higher than that of Colberg and Morari [28], but has a much lower investment cost. The
total investment cost of the network generated by the Sequential Framework compares
favourably in cost with the solution presented in Yee and Grossmann [150] and the area
of 189.7 m2 compares favourably to that of the result from Colberg and Morari [28]. An
explanation for this is that in the Sequential Framework, the stream match generator
optimizes with respect to “pseudo-area”and the network generation and optimization
phase is optimized with respect to cost.
A more recent approach to synthesizing heat exchanger networks is the Interval Based
MINLP Superstructure (IBMS) of Isafiade and Fraser [72]. The IBMS method, for the
same problem, gave an optimum solution with 10 units and TAC of $168,700.
It is worth noticing that the “best” solution from SeqHENS is obtained after only 4 iterations.
1 A spaghetti network is one with a very large number of exchangers with each exchanger mimicking the
temperuature profile of it respective enthalpy interval, defined by kinks in the composite curve, to minimize
the area of network.
87
5. THE SEQUENTIAL FRAMEWORK FOR HEAT EXCHANGER NETWORK
SYNTHESIS
Ts
Tt
mC p
∆H
h
°C
°C
kW/°C
kW
kW/m2 °C
H1
180
75
30
3150
2
H2
280
120
60
9600
1
H3
180
75
30
3150
2
H4
140
40
30
3000
1
H5
220
120
50
5000
1
H6
180
55
35
4375
2
H7
200
60
30
4200
0.4
H8
120
40
100
8000
0.5
C1
40
230
20
3800
1
C2
100
220
60
7200
1
C3
40
290
35
8750
2
C4
50
290
30
7200
2
C5
50
250
60
12000
2
C6
90
190
50
5000
1
C7
160
250
60
5400
3
ST
325
325
ă
ă
1
CW
25
40
ă
ă
2
Stream
0.75
Exchanger cost ($) = 8,000 + 500A
(A is in m2 )
Table 5.5: Stream data and heat exchanger cost data for Example 2 (15TP1)
5.8.2
Example 2 (15TP1)
The Sequential Framework is also used to design a heat exchanger network for the Example from Björk and Nordman [22] with the problem data given in Table 5.5. This is
a medium sized example with 15 process streams. This problem has only been solved
using stochastic optimization or hybrid optimization methods [22]. For comparison purposes, the operating cost of the solution, 1,014,323 $/yr , presented in [22] corresponds
to an HRAT of 20.35°C and this value was kept unchanged.
Table 5.6 presents different loops and total cost at each step of the network generation
process using the Sequential Framework. The second step of the framework generates
the best solution with a TAC of $1,511,047 and 15 units. This compares well with the
solution for base case given in [22] with a TAC of $1,530,063. The network generated is
shown in Figure 5.5 and match details are given in Table 5.7. A simpler network with
88
5.9 Conclusions and further work
Soln. No
U
EMAT (C)
HLD
TAC ($)
1
14
2.5
A
1,565,375
2
15
2.5
A
1,511,047
3
15
2.5
B
1,522,000
4
15
5
A
1,529,968
5
15
5
B
1,532,148
6
16
2.5
A
1,547,353
Table 5.6: TAC at each step of the Sequential Framework for Example 2 (15TP1)
a total cost similar to the one reported by Björk and Nordaman [22] is also generated in
the third step of the framework. It is also worth noting that the Vertical MILP transportation model for selecting HLDs in the Sequential Framework allows only one match
between streams. When the same “simplification” constraint is applied in [22], the resulting network has a TAC of $1,568,745, thus 2.5% above the TAC from the Sequential
Framework.
5.9
Conclusions and further work
The Sequential Framework for heat exchanger network synthesis, presented in this
chapter, is a sequential and iterative framework with the main objective of finding near
optimal heat exchanger networks for industrial size problems. The Sequential Framework is a compromise between Pinch Analysis and simultaneous MINLP methods. There
are two main advantages of the Sequential Framework:
1. The subtasks of the framework (MILP and NLP problems) are much easier to solve
numerically than the simultaneous MINLP models suggested for HENS.
2. The design procedure is, to a large extent, automated while keeping significant
user interaction. The design engineer acts as a top level optimizer making judgments based on quantitative as well as qualitative considerations.
Two test problems are solved using the Sequential Framework showing the ability to
generate networks with lower Total Annualized Costs compared to other solutions in
the literature. The Sequential Framework arrives at the best solution efficiently in a
small number of iterations, despite the four loops in the framework. This is due to the
89
5. THE SEQUENTIAL FRAMEWORK FOR HEAT EXCHANGER NETWORK
SYNTHESIS
Match
Duty
Area
kW
m2
H1-C4
3150
200.01
H2-C1
1350
29.39
H2-C2
7200
556.98
H2-C4
1050
58.28
H3-C5
3150
251.35
H4-C5
1835.75
141.29
H5-C6
5000
333.33
H6-C3
4375
437.50
H7-C1
2450
202.36
H7-C5
1750
290.23
ST-C3
875
8.92
ST-C5
5264.25
69.72
ST-C7
5400
63.08
H4-CW
1164.25
42.85
H8-CW
8000
354.67
Table 5.7: Match details of best heat exchanger network Example 2 (15TP1)
fact that the search for good designs by exploring the loops of the framework will focus
on the most promising part of the feasible solution space; a result from using domain
knowledge in setting up the loop structure and initializing parameters.
90
5.9 Conclusions and further work
Figure 5.5: The best obtained heat exchanger network for Example 2 (15TP1) - Solution no.
2
91
5. THE SEQUENTIAL FRAMEWORK FOR HEAT EXCHANGER NETWORK
SYNTHESIS
92
6
Minimum Number of Units
Sub-problem
This chapter presents the minimum number of units sub-problem in the Sequential
Framework, its formulation, challenges and approaches to ease computational issues
in this subproblem [12, 98].
6.1
Introduction
A consequence of recognizing the three way trade-off in HENS has been to explore network designs at the limits, i.e. designs with minimum energy consumption, designs
with minimum heat transfer area and designs with minimum number of heat exchanger
units. Hohmann [66] identified that among the contributions to the network’s capital
cost, the number of exchanger units is more important as most network designs have
similar total area. The minimum number of heat exchanger units in a network has thus
become an important step in most evolutionary and sequential methods for HENS. This
section gives a brief overview on the search for quantifying the minimum number of
units in a network as an introduction to the minimum number of units sub-problem of
the Sequential Framework.
Without the use of multi-stream heat exchangers, the absolute minimum number of
units is the maximum of the sum of all hot process streams and utilities, and the sum
of all cold process streams and utilities. The absolute minimum number of units does
not hold much practical significance in the design of networks. Hohmann [66] defined a
93
6. MINIMUM NUMBER OF UNITS SUB-PROBLEM
quasi-minimum number of units as:
U quasi−min = N − 1
(6.1)
where N is the total number of streams (sum of hot and cold process streams and utilities). The “quasi” qualifier demonstrates that Hohmann realized that this was not true
for all cases. Linnhoff et al. [94] showed that this was a special case of Euler’s general
network theorem [52]:
U = N +L−S
(6.2)
where U is the number of heat exchanger units, N is the total number of streams (similar
to Equation 6.1), L is the number of independent loops in the network and S the number
of sub-networks (separate components in a network). Equation 6.2 is used to evaluate
the actual number of units in a network rather than the minimum number. Grimes
et al. [51] showed that Equation 6.1 can be used to calculate the minimum number
of units above and below the pinch and was incorporated in the Pinch Design Method
(PDM) by Linnhoff and Hindmarsh [91]. Equation 6.2 indicates that for a given set
of streams, decreasing the number of loops in the network or increasing the number
of sub-networks minimize the number of units. Evolutionary methods in HENS were
subsequently developed taking this into account for breaking loops (e.g. Zhu et al. [155],
Han et al. [62], Jez̆owski et al. [78]) and identifying sub-networks (e.g. Mocsny and
Govind [97], Shethna and Jez̆owski [122]) to minimize the number of units.
The Mathematical Programming based sequential methods minimize the number of
matches (or heat exchanger units), in one of the subtasks, to get a Heat Load Distribution (HLD) for the network. These are typically formulated as network flow problems.
Two such formulations in the HENS literature are the transshipment model of Papoulias
and Grossmann [105] and the transportation model of Cerdá and Westerberg [25].
6.2
The minimum number of units sub-problem in the Sequential Framework
The minimum number of matches problem in the Sequential Framework can be defined
as follows
94
6.2 The minimum number of units sub-problem in the Sequential Framework
- Given:
• a set H of hot process streams to be cooled and hot utilities,
• a set C of cold process streams to be heated and cold utilities,
• supply and target temperatures, heat capacities and flow rates of the hot and cold
process streams,
• temperatures or temperature ranges and fixed heat loads of the utilities, and
• Exchanger Minimum Approach Temperature (EMAT) set to zero.
- Calculate the minimum number of matches between hot process streams and utilities
and cold process streams and utilities such that the heating and cooling requirements
for each stream are met.
EMAT is set to zero in the Sequential Framework to determine the absolute minimum
number of units for a given level of heat recovery. The Sequential Framework is an iterative procedure and includes a loop for number of units (Loop 3) as detailed in Chapter
5. To explore the number of units solution space systematically would require starting
at one of the bounds. Based on discussions in Section 6.1, the lower bound of absolute
minimum number of units is selected. This initialization for the units loop results in the
least number of iterations as almost the entire body of HENS literature suggests that
minimum capital cost for a network occurs when the number of heat exchanger units is
close to the minimum.
The minimum number of units sub-problem is formulated as an MILP Transshipment
model from Operations Research (see Figure 6.1). The basic transshipment model for
minimum number of units used in the Sequential Framework is the model presented
by Papoulias and Grossmann [105]. Our model (P1) shown below differs from the model
presented in [105] since no sub-networks are considered in (P1) (no pinch decomposition).
Let H be the set of all hot process streams and utilities, while C be the set of all cold
process streams and utilities. HP and CP are the sets of hot and cold process streams
respectively, and HU and CU are the sets of hot and cold utilities respectively. The entire
temperature range of all streams are partitioned into K temperature intervals based on
the inlet temperatures, with the temperature intervals in set TI being labeled from the
highest temperature level (k = 1) to the lowest (k = K). The heat load of hot process
95
6. MINIMUM NUMBER OF UNITS SUB-PROBLEM
Figure 6.1: Transshipment formulation for minimum number of units sub-problem
stream or utility i entering temperature interval k is represented by Q H
, while Q Cjk is
ik
the heat load flowing to cold process stream or utility j from temperature interval k.
These heat loads can be evaluated from stream data and utility targeting.
H k = { i | i ∈ H, hot process stream or utility i is present in interval k̄ ≤ k; k̄, k ∈ T I }
C k = { j | j ∈ C, cold process stream or utility j is present in interval k; k ∈ T I }
The residual of hot process stream or utility i from interval k is represented as R ik and
Q i jk represents heat exchanged between hot process stream or utility i and cold process
stream or utility j in interval k. The binary variable yi j denotes the existence of a match
between hot process stream or utility i and cold process stream or utility j. U i j is a large
number (upper bound) sometimes referred to as the big M, linking the binary variables
yi j to the continuous variables Q i jk and is discussed in detail in Section 6.4.1
min z =
X X
yi j
i ∈ H j ∈C
96
(P1)
6.2 The minimum number of units sub-problem in the Sequential Framework
Figure 6.2: Temperature intervals for Example
s.t.
R i,k − R i,k−1 +
X
Q i jk = Q H
ik
∀ i ∈ Hk , k ∈ T I
(P1.1)
Q i jk = Q Cjk
∀ j ∈ Ck, k ∈ T I
(P1.2)
∀ i ∈ H, j ∈ C
(P1.3)
∀ i ∈ Hk , k ∈ T I
(P1.4)
∀i∈H
(P1.5)
∀ i ∈ Hk , j ∈ Ck , k ∈ T I
(P1.6)
∀ i ∈ H, j ∈ C
(P1.7)
j ∈C k
X
i∈H k
X
Q i jk − U i j yi j ≤ 0
k∈T I
R ik ≥ 0
R i0 = R iK = 0
Q i jk ≥ 0
yi j = {0, 1}
6.2.1
Temperature Intervals in the transshipment model
It is obvious that only supply temperatures are needed to establish pinch and minimum
utility consumption, since only supply temperatures are pinch candidates. For the minimum number of units case, it is not evident.
The temperature intervals in the transshipment model P1 are developed based on the
97
6. MINIMUM NUMBER OF UNITS SUB-PROBLEM
supply temperatures only. The target temperatures are not required which is shown
using a conceptual example below.
Example
Let us consider a simple example with 1 hot stream, 2 cold streams and 1 hot and 1
cold utility. Both utilities are point utilities (i.e. constant temperature). Temperature
ranges of the streams are shown in Figure 6.2. Let us now consider two different implementations of Model P1, EX-I1 where the temperature intervals are partitioned based on
stream supply temperatures, and EX-I2 where the temperature intervals are partitioned
based on stream supply and target temperatures. As seen in Table 6.1, the temperature
intervals in EX-I1 are numbered from 1 through 4 while those in EX-I2 are numbered 1’
through 7’.
EX - I1
EX - I2
T sHU
1
T sHU
1’
T tC 1
T sH 1
2
2’
T sH 1
T sC 1
3
3’
T tC 2
T sC 2
4
4’
T sCU
T sC 1
5’
T tH 1
6’
T sC 2
7’
T sCU
Table 6.1: Temperature intervals for Example Implementations 1 and 2 for EMAT = 0
The objective function will be the same for both implementations. The constraints for
implementation 1 are presented below. Only heat balance constraints are included and
the following constraints are not included:
98
6.2 The minimum number of units sub-problem in the Sequential Framework
• Constraint P1.3: This constraint is used to set the binary variable yi j to 1 when
there is any heat exchanged between hot process stream or utility i and cold process stream or utility j
• Constraint P1.4
• Constraint P1.6
• Constraint P1.7
R ( HU , 1) − R ( HU , 0) + Q ( HU , C 1, 1) + Q ( HU , C 2, 1) − Q H ( HU , 1) = 0
H
(I1.1)
R ( HU , 2) − R ( HU , 1) + Q ( HU , C 1, 2) + Q ( HU , C 2, 2) − Q ( HU , 2) = 0
(I1.2)
R ( HU , 3) − R ( HU , 2) + Q ( HU , C 1, 3) + Q ( HU , C 2, 3) − Q H ( HU , 3) = 0
(I1.3)
H
R ( HU , 4) − R ( HU , 3) + Q ( HU , C 1, 4) + Q ( HU , C 2, 4) − Q ( HU , 4) = 0
H
(I1.4)
R ( H 1, 1) − R ( H 1, 0) + Q ( H 1, C 1, 1) + Q ( H 1, C 2, 1) + Q ( H 1, CU , 1) − Q ( H 1, 1) = 0
(I1.5)
R ( H 1, 2) − R ( H 1, 1) + Q ( H 1, C 1, 2) + Q ( H 1, C 2, 2) + Q ( H 1, CU , 2) − Q H ( H 1, 2) = 0
(I1.6)
H
(I1.7)
H
R ( H 1, 4) − R ( H 1, 3) + Q ( H 1, C 1, 4) + Q ( H 1, C 2, 4) + Q ( H 1, CU , 4) − Q ( H 1, 4) = 0
(I1.8)
Q ( HU , C 1, 1) + Q ( H 1, C 1, 1) − Q C (C 1, 1) = 0
(I1.9)
R ( H 1, 3) − R ( H 1, 2) + Q ( H 1, C 1, 3) + Q ( H 1, C 2, 3) + Q ( H 1, CU , 3) − Q ( H 1, 3) = 0
C
(I1.10)
C
Q ( HU , C 1, 3) + Q ( H 1, C 1, 3) − Q (C 1, 3) = 0
(I1.11)
Q ( HU , C 1, 4) + Q ( H 1, C 1, 4) − Q C (C 1, 4) = 0
(I1.12)
Q ( HU , C 1, 2) + Q ( H 1, C 1, 2) − Q (C 1, 2) = 0
C
(I1.13)
C
Q ( HU , C 2, 2) + Q ( H 1, C 2, 2) − Q (C 2, 2) = 0
(I1.14)
Q ( HU , C 2, 3) + Q ( H 1, C 2, 3) − Q C (C 2, 3) = 0
(I1.15)
Q ( HU , C 2, 1) + Q ( H 1, C 2, 1) − Q (C 2, 1) = 0
C
Q ( HU , C 2, 4) + Q ( H 1, C 2, 4) − Q (C 2, 4) = 0
C
Q ( H 1, CU , 1) − Q (CU , 1) = 0
(I1.17)
Q ( H 1, CU , 2) − Q C (CU , 2) = 0
(I1.18)
C
(I1.19)
C
Q ( H 1, CU , 4) − Q (CU , 4) = 0
(I1.20)
R ( HU , 0) = R ( HU , 4) = 0
(I1.21)
R ( H 1, 0) = R ( H 1, 4) = 0
(I1.22)
Q ( H 1, CU , 3) − Q (CU , 3) = 0
The following simplifications can be made to the constraint equations.
99
(I1.16)
6. MINIMUM NUMBER OF UNITS SUB-PROBLEM
• Appropriate residuals can be set to zero in constraints I1.1 - I1.8 based on constraints I1.21 and I1.22.
• Point utilities imply that Q H (HU, 2) = Q H (HU, 3) = Q H (HU, 4) = 0 and Q C (CU, 1) =
Q C (CU, 2) = Q C (CU, 3) = 0.
• Using stream temperature ranges we have Q H (H1, 1) = Q H (H1, 4) = 0, Q C (C1, 3) =
Q C (C1, 4) = 0 and Q C (C2, 1) = Q C (C2, 4) = 0.
From the above simplifications it follows that Q(HU, C1, 3) = Q(HU, C1, 4) = 0, Q(HU, C2, 1) =
Q(HU, C2, 4) = 0, Q(H1, C1, 1) = Q(H1, C1, 3) = Q(H1, C1, 4) = 0, Q(H1, C2, 1) = Q(H1, C2, 4) =
0 and Q(H1, CU, 1) = Q(H1, CU, 2) = Q(H1, CU, 3) = 0. The implementation reduces to
R ( HU , 1) + Q ( HU , C 1, 1) − Q H ( HU , 1) = 0
(I1.23)
R ( HU , 2) − R ( HU , 1) + Q ( HU , C 1, 2) + Q ( HU , C 2, 2) = 0
(I1.24)
R ( HU , 3) − R ( HU , 2) + Q ( HU , C 2, 3) = 0
(I1.25)
H
(I1.26)
H
R ( H 1, 3) − R ( H 1, 2) + Q ( H 1, C 2, 3) − Q ( H 1, 3) = 0
(I1.27)
−R ( H 1, 3) + Q ( H 1, CU , 4) = 0
(I1.28)
R ( H 1, 2) + Q ( H 1, C 1, 2) + Q ( H 1, C 2, 2) − Q ( H 1, 2) = 0
C
(I1.29)
C
(I1.30)
C
(I1.31)
C
(I1.32)
Q ( HU , C 1, 1) − Q (C 1, 1) = 0
Q ( HU , C 1, 2) + Q ( H 1, C 1, 2) − Q (C 1, 2) = 0
Q ( HU , C 2, 2) + Q ( H 1, C 2, 2) − Q (C 2, 2) = 0
Q ( HU , C 2, 3) + Q ( H 1, C 2, 3) − Q (C 2, 3) = 0
C
Q ( H 1, CU , 4) − Q (CU , 4) = 0
(I1.33)
Q H (ST, 1) is the minimum hot utility requirement, Q H ,min and Q C (CU, 4) is the minimum cold utility requirement, Q C ,min . Heat available from hot streams and heat required by the cold streams in each interval (Q H
and Q Cjk ) are known quantities and can
ik
be replaced by the product of their mass flow, heat capacity and temperature difference.
100
6.2 The minimum number of units sub-problem in the Sequential Framework
Q H ,min − R ( HU , 1) = Q ( HU , C 1, 1)
(I1.34)
R ( HU , 1) − R ( HU , 2) = Q ( HU , C 1, 2) + Q ( HU , C 2, 2)
(I1.35)
R ( HU , 2) − R ( HU , 3) = Q ( HU , C 2, 3)
(I1.36)
1
H1
C1
mC H
p (T s − T s ) − R ( H 1, 2) = Q ( H 1, C 1, 2) + Q ( H 1, C 2, 2)
1
C1
H1
mC H
p (T s − T t ) + R ( H 1, 2) − R ( H 1, 3) = Q ( H 1, C 2, 3)
R ( H 1, 3) = Q ( H 1, CU , 4)
(I1.37)
(I1.38)
(I1.39)
H1
1
C1
mC C
p (T t − T s ) = Q ( HU , C 1, 1)
(I1.40)
1
H1
C1
mC C
p (T s − T s ) = Q ( HU , C 1, 2) + Q ( H 1, C 1, 2)
(I1.41)
C1
2
C2
mC C
p (T t − T s ) = Q ( HU , C 2, 2) + Q ( H 1, C 2, 2)
(I1.42)
2
C1
C2
mC C
p (T s − T s ) = Q ( HU , C 2, 3) + Q ( H 1, C 2, 3)
(I1.43)
Q C ,min = Q ( H 1, CU , 4)
(I1.44)
Constraints I1.34 to I1.44 represent the constraint set under consideration of Implementation 1 of the example. Similarly, the heat balance constraints in Implementation 2 of
the example is shown below:
101
6. MINIMUM NUMBER OF UNITS SUB-PROBLEM
R ( HU , 1 0 ) − R ( HU , 0 0 ) + Q ( HU , C 1, 1 0 ) + Q ( HU , C 2, 1 0 ) − Q H ( HU , 1 0 ) = 0
0
0
0
H
0
0
(I2.1)
R ( HU , 2 ) − R ( HU , 1 ) + Q ( HU , C 1, 2 ) + Q ( HU , C 2, 2 ) − Q ( HU , 2 ) = 0
(I2.2)
R ( HU , 3 0 ) − R ( HU , 2 0 ) + Q ( HU , C 1, 3 0 ) + Q ( HU , C 2, 3 0 ) − Q H ( HU , 3 0 ) = 0
(I2.3)
0
0
0
0
H
0
(I2.4)
0
0
0
0
H
0
R ( HU , 5 ) − R ( HU , 4 ) + Q ( HU , C 1, 5 ) + Q ( HU , C 2, 5 ) − Q ( HU , 5 ) = 0
(I2.5)
R ( HU , 6 0 ) − R ( HU , 5 0 ) + Q ( HU , C 1, 6 0 ) + Q ( HU , C 2, 6 0 ) − Q H ( HU , 6 0 ) = 0
(I2.6)
R ( HU , 4 ) − R ( HU , 3 ) + Q ( HU , C 1, 4 ) + Q ( HU , C 2, 4 ) − Q ( HU , 4 ) = 0
0
0
0
H
0
0
(I2.7)
0
R ( H 1, 1 ) − R ( H 1, 0 ) + Q ( H 1, C 1, 1 ) + Q ( H 1, C 2, 1 ) + Q ( H 1, CU , 1 ) − Q ( H 1, 1 ) = 0
(I2.8)
R ( H 1, 2 0 ) − R ( H 1, 1 0 ) + Q ( H 1, C 1, 2 0 ) + Q ( H 1, C 2, 2 0 ) + Q ( H 1, CU , 2 0 ) − Q H ( H 1, 2 0 ) = 0
(I2.9)
R ( HU , 7 ) − R ( HU , 6 ) + Q ( HU , C 1, 7 ) + Q ( HU , C 2, 7 ) − Q ( HU , 7 ) = 0
0
0
0
0
H
0
0
0
0
0
0
H
0
(I2.10)
0
0
0
0
0
H
0
R ( H 1, 4 ) − R ( H 1, 3 ) + Q ( H 1, C 1, 4 ) + Q ( H 1, C 2, 4 ) + Q ( H 1, CU , 4 ) − Q ( H 1, 4 ) = 0
(I2.11)
R ( H 1, 5 0 ) − R ( H 1, 4 0 ) + Q ( H 1, C 1, 5 0 ) + Q ( H 1, C 2, 5 0 ) + Q ( H 1, CU , 5 0 ) − Q H ( H 1, 5 0 ) = 0
(I2.12)
R ( H 1, 3 ) − R ( H 1, 2 ) + Q ( H 1, C 1, 3 ) + Q ( H 1, C 2, 3 ) + Q ( H 1, CU , 3 ) − Q ( H 1, 3 ) = 0
0
0
0
0
0
H
0
(I2.13)
0
0
0
0
0
H
0
R ( H 1, 7 ) − R ( H 1, 6 ) + Q ( H 1, C 1, 7 ) + Q ( H 1, C 2, 7 ) + Q ( H 1, CU , 7 ) − Q ( H 1, 7 ) = 0
(I2.14)
Q ( HU , C 1, 1 0 ) + Q ( H 1, C 1, 1 0 ) − Q C (C 1, 1 0 ) = 0
(I2.15)
R ( H 1, 6 ) − R ( H 1, 5 ) + Q ( H 1, C 1, 6 ) + Q ( H 1, C 2, 6 ) + Q ( H 1, CU , 6 ) − Q ( H 1, 6 ) = 0
0
0
C
0
(I2.16)
0
0
C
0
Q ( HU , C 1, 3 ) + Q ( H 1, C 1, 3 ) − Q (C 1, 3 ) = 0
(I2.17)
Q ( HU , C 1, 4 0 ) + Q ( H 1, C 1, 4 0 ) − Q C (C 1, 4 0 ) = 0
(I2.18)
Q ( HU , C 1, 2 ) + Q ( H 1, C 1, 2 ) − Q (C 1, 2 ) = 0
0
0
C
0
(I2.19)
0
0
C
0
Q ( HU , C 1, 6 ) + Q ( H 1, C 1, 6 ) − Q (C 1, 6 ) = 0
(I2.20)
Q ( HU , C 1, 7 0 ) + Q ( H 1, C 1, 7 0 ) − Q C (C 1, 7 0 ) = 0
(I2.21)
Q ( HU , C 1, 5 ) + Q ( H 1, C 1, 5 ) − Q (C 1, 5 ) = 0
102
6.2 The minimum number of units sub-problem in the Sequential Framework
Q ( HU , C 2, 1 0 ) + Q ( H 1, C 2, 1 0 ) − Q C (C 2, 1 0 ) = 0
(I2.22)
Q ( HU , C 2, 2 0 ) + Q ( H 1, C 2, 2 0 ) − Q C (C 2, 2 0 ) = 0
(I2.23)
0
0
C
0
(I2.24)
0
0
C
0
Q ( HU , C 2, 4 ) + Q ( H 1, C 2, 4 ) − Q (C 2, 4 ) = 0
(I2.25)
Q ( HU , C 2, 5 0 ) + Q ( H 1, C 2, 5 0 ) − Q C (C 2, 5 0 ) = 0
(I2.26)
Q ( HU , C 2, 3 ) + Q ( H 1, C 2, 3 ) − Q (C 2, 3 ) = 0
0
0
C
0
(I2.27)
0
0
C
0
Q ( HU , C 2, 7 ) + Q ( H 1, C 2, 7 ) − Q (C 2, 7 ) = 0
(I2.28)
Q ( H 1, CU , 1 0 ) − Q C (CU , 1 0 ) = 0
(I2.29)
Q ( HU , C 2, 6 ) + Q ( H 1, C 2, 6 ) − Q (C 2, 6 ) = 0
0
C
0
(I2.30)
0
C
0
Q ( H 1, CU , 3 ) − Q (CU , 3 ) = 0
(I2.31)
Q ( H 1, CU , 4 0 ) − Q C (CU , 4 0 ) = 0
(I2.32)
Q ( H 1, CU , 2 ) − Q (CU , 2 ) = 0
0
C
0
(I2.33)
0
C
0
Q ( H 1, CU , 6 ) − Q (CU , 6 ) = 0
(I2.34)
Q ( H 1, CU , 7 0 ) − Q C (CU , 7 0 ) = 0
(I2.35)
Q ( H 1, CU , 5 ) − Q (CU , 5 ) = 0
0
0
(I2.36)
0
(I2.37)
R ( HU , 0 ) = R ( HU , 7 ) = 0
0
R ( H 1, 0 ) = R ( H 1, 7 ) = 0
This can be simplified, similarly to Implementation 1 as follows.
Q H ,min − R ( HU , 1 0 ) = 0
0
(I2.38)
0
0
R ( HU , 1 ) − R ( HU , 2 ) = Q ( HU , C 1, 2 )
(I2.39)
R ( HU , 2 0 ) − R ( HU , 3 0 ) = Q ( HU , C 1, 3 0 )
0
0
0
0
0
0
(I2.40)
0
R ( HU , 3 ) − R ( HU , 4 ) = Q ( HU , C 1, 4 ) + Q (ST , C 2, 4 )
(I2.41)
R ( HU , 4 ) − R ( HU , 5 ) = Q ( HU , C 2, 5 )
(I2.42)
R ( HU , 5 0 ) − R ( HU , 6 0 ) = Q ( HU , C 2, 6 0 )
(I2.43)
C2
0
0
1
H1
mC H
p (T s − T t ) − R ( H 1, 3 ) = Q ( H 1, C 1, 3 )
(I2.44)
0
0
0
0
1
C2
C1
mC H
p (T t − T s ) + R ( H 1, 3 ) − R ( H 1, 4 ) = Q ( H 1, C 1, 4 ) + Q ( H 1, C 2, 4 )
(I2.45)
1
C1
H1
0
0
0
mC H
p (T s − T t ) + R ( H 1, 4 ) − R ( H 1, 5 ) = Q ( H 1, C 2, 5 )
(I2.46)
0
(I2.47)
0
0
R ( H 1, 5 ) − R ( H 1, 6 ) = Q ( H 1, C 2, 6 )
0
0
R ( H 1, 6 ) = Q ( H 1, CU , 7 )
103
(I2.48)
6. MINIMUM NUMBER OF UNITS SUB-PROBLEM
1
C1
H1
0
mC C
p (T t − T s ) = Q ( HU , C 1, 2 )
(I2.49)
1
H1
C2
0
0
mC C
p (T s − T t ) = Q ( HU , C 1, 3 ) + Q ( H 1, C 1, 3 )
(I2.50)
1
C2
C1
0
0
mC C
p (T t − T s ) = Q ( HU , C 1, 4 ) + Q ( H 1, C 1, 4 )
(I2.51)
2
C2
C1
0
0
mC C
p (T t − T s ) = Q ( HU , C 2, 4 ) + Q ( H 1, C 2, 4 )
(I2.52)
2
C1
H1
0
0
mC C
p (T s − T t ) = Q ( HU , C 2, 5 ) + Q ( H 1, C 2, 5 )
(I2.53)
2
H1
C2
0
0
mC C
p (T t − T s ) = Q ( HU , C 2, 6 ) + Q ( H 1, C 2, 6 )
(I2.54)
0
Q C ,min = Q ( H 1, CU , 7 )
(I2.55)
Adding constraints1 I2.38 and I2.39, I2.40 and I2.41, I2.42 and I2.43, I2.44 and I2.45,
I2.46 and I2.47, I2.50 and I2.51 and, I2.53 and I2.54, we have
Q H ,min − R ( HU , 2 0 ) =Q ( HU , C 1, 2 0 )
0
0
(I2.56)
0
0
R ( HU , 2 ) − R ( HU , 4 ) =Q ( HU , C 1, 3 ) + Q ( HU , C 1, 4 )+
Q ( HU , C 2, 4 0 )
0
0
(I2.57)
0
0
R ( HU , 4 ) − R ( HU , 6 ) =Q ( HU , C 2, 5 ) + Q ( HU , C 2, 6 )
(I2.58)
1
H1
C1
0
0
0
mC H
p (T s − T s ) − R ( H 1, 4 ) =Q ( H 1, C 1, 3 ) + Q ( H 1, C 1, 4 )+
Q ( H 1, C 2, 4 0 )
(I2.59)
1
C1
H1
0
0
0
0
mC H
p (T s − T t ) + R ( H 1, 4 ) − R ( H 1, 6 ) =Q ( H 1, C 2, 5 ) + Q ( H 1, C 2, 6 )
0
0
R ( H 1, 6 ) = Q ( H 1, CU , 7 )
1
C1
H1
0
mC C
p (T t − T s ) =Q ( HU , C 1, 2 )
(I2.60)
(I2.61)
(I2.62)
1
H1
C1
0
0
mC C
p (T s − T s ) =Q ( HU , C 1, 3 ) + Q ( HU , C 1, 4 )+
Q ( H 1, C 1, 3 0 ) + Q ( H 1, C 1, 4 0 )
2
C2
C1
0
0
mC C
p (T t − T s ) =Q ( HU , C 2, 4 ) + Q ( H 1, C 2, 4 )
(I2.63)
(I2.64)
C2
0
0
2
C1
mC C
p (T s − T s ) =Q ( HU , C 2, 5 ) + Q ( HU , C 2, 6 )+
Q ( H 1, C 2, 5 0 ) + Q ( H 1, C 2, 6 0 )
Q C ,min =Q ( H 1, CU , 7 0 )
(I2.65)
(I2.66)
Constraints I2.56 to I2.66 represent the constraint set under consideration in Implementation 2 of the example and are equivalent to constraints I1.34 to ?? in Implementation
1. Comparing the two constraints of Implementation 1 and Implementation 2 we have
1 Summing two constraints in a model does not lead to a reduction in solution space
104
6.3 Challenges
R ( HU , 1) = R ( HU , 2 0 )
Q ( HU , C 1, 1) = Q ( HU , C 1, 2 0 )
R ( HU , 2) = R ( HU , 4 0 )
Q ( HU , C 1, 2) = Q ( HU , C 1, 3 0 ) + Q ( HU , C 1, 4 0 )
Q ( HU , C 2, 2) = Q ( HU , C 2, 4 0 )
R ( HU , 3) = R ( HU , 6 0 )
Q ( HU , C 2, 3) = Q ( HU , C 2, 5 0 ) + Q ( HU , C 2, 6 0 )
R ( H 1, 2) = R ( H 1, 4 0 )
Q ( H 1, C 1, 2) = Q ( H 1, C 1, 3 0 ) + Q ( H 1, C 1, 4 0 )
Q ( H 1, C 2, 2) = Q ( H 1, C 2, 4 0 )
R ( H 1, 3) = R ( H 1, 6 0 )
Q ( H 1, C 2, 3) = Q ( H 1, C 2, 5 0 ) + Q ( H 1, C 2, 6 0 )
Q ( H 1, CU , 4) = Q ( H 1, CU , 7 0 )
The optimum solution of Implementation 2 is also the optimum solution of Implementation 1. Thus the supply temperatures are the only ones required to set up temperature
intervals in the minimum units model formulation.
6.3
Challenges
As the number of streams increases, the MILP formulation for the minimum number
of units sub-problem becomes hard and eventually impossible to solve due to “combinatorial explosion”. With an increase in the number of streams, the binary search tree
increases exponentially (see Figure 6.3). Intuitively it is expected that increasing the
number of process streams (and binary variables) would lead to an exponential increase
in solution time. Experience indicates that this is typically seen for problems with more
than 20 streams even though Egeberg [34] solves problems with over 20 streams.
Williams [149] states that the number of binary variables in a model is a poor indicator
of its difficulty. Due to developments in Branch & Bound or Branch & Cut methods,
an exponential increase in the number of binary variables does not necessarily result in
prohibitive solution times. Thus the number of streams alone is not a good indicator of
problem difficulty. This is discussed in detail in Section 6.6.
105
6. MINIMUM NUMBER OF UNITS SUB-PROBLEM
Figure 6.3: Combinatorial explosion in a binary search tree as a function of the total number of process streams
The amount of time and number of calculation steps are important characteristics of an
optimization problem. Even though there exist algorithms to solve any instance of the
problem exactly, excessive time requirements can render the algorithm useless. The computational complexity theorem provides a rigorous method to classify problems based on
their susceptabilty to efficient algorithmic solution. An efficient algorithm is defined
in Papadimitriou and Steiglitz [103] as an algorithm requiring a number of steps that
grows as a polynomial in the size of the input: O(n c ), where c is a constant and n is the
size of the input. A brief overview of the different complexity classes follows.
A problem is said to be in complexity class P if there exists a deterministic algorithm
that solves it in polynomial time. The complexity class N P consists of problems that
can be solved with a non-deterministic polynomial time algorithm. N P -complete algorithms are those problems within the set N P for which no polynomial time algorithms
exist, assuming P 6= N P . A problem to which an N P -complete problem can be reduced
to in polynomial time is termed N P -hard. N P -hard problems are defined to be at least
as hard as the hardest problems in N P and may or may not be in set N P (see Figure
6.4. A problem is said to be N P -hard or N P -complete in the strong sense if it remains
so even when all of its numerical parameters are bounded by a polynomial in the length
106
6.3 Challenges
Figure 6.4: Complexity classes
of the input.
Furman and Sahinidis [43] proved that the minimum number of units problem is N P hard in the strong sense. This implies that there exists no polynomial time algorithm
for solving the minimum units problem for all inputs. As a consequence of this result,
Furman and Sahinidis [44] develop approximation algorithms for the minimum units
problem. However, in the Sequential Framework, the number of units loop must start at
the absolute minimum number of units to ensure that potentially good solutions are not
ignored. The remainder of this chapter is devoted to approaches to alleviating the issues
in the minimum number of units problem.
The three major ways to improve the model solution time (by alleviating the combinatorial explosion problem) are:
1. Pre-processing to reduce model size using insight and heuristics
2. Model modification/reformulation
3. Improving efficiency of the B&B method
The different options are tested on a set of 3 problems with more than 20 streams 21TP1, 21TP2 and 22TP1. The test problem stream data are given in Appendix A.
Pre-processing to reduce model size could involve fixing binary variables based on knowledge about the problem. The B&B method may also be improved by specifying priorities
to the variables, modifying parameters in the solvers, etc. Focus in this work, however,
has been on model modification and reformulation.
107
6. MINIMUM NUMBER OF UNITS SUB-PROBLEM
6.4
6.4.1
Model modification
Sharpening the LP relaxation by decreasing the big M
Constraint P1.3 in model P1 is a logic relation between the continuous variable Q i jk
and the binary variable yi j , connected using the so-called big M formulation (here U i j ).
The gap, i.e. the difference between the LP relaxation and the actual binary solution, is
dependent on the value of U i j , the upper limit on the amount of heat transfer between
streams i and j. A smaller value of U i j corresponds to a smaller gap and thus reduced
computing times.
The value of U i j in [105] was set to be the upper bound on the heat that can be exchanged
between a hot process stream or utility and a cold process stream or utility, only using
information about heat delivered from hot process stream or utility i and heat needed
by cold process stream or utility j:
(
U i j = min
)
X
QH
ik ,
k∈T I
X
Q Cjk
(6.3)
k∈T I
A drawback with Equation (6.3) is that the temperatures of the streams are not taken
into account when setting the U i j . A more recent relation based on thermodynamic
information (temperatures and heat capacity flow rates) is given by [61]:
(
U i j = min
X
k∈T I
where
mC p H
i
QH
ik ,
X
Q Cjk , max
h
³
´ ³
´ i
C
H
C
min mC p H
i , mC p j · T s i − T s j − EM AT , 0
k∈T I
)
(6.4)
is the heat capacity flow rate of hot process stream i,
mC p Cj
the heat
capacity flow rate of cold process stream j, T s H
the supply temperature of hot process
i
stream i and T sCj the supply temperature of cold process stream j.
Equation (6.4) gives the maximum amount of heat that is possible to transfer between a
hot process stream i and a cold process stream j given a value of EMAT (taken to be 0 in
this work, thus the thermodynamic limit). An illustration of the importance of temperatures and heat capacity flow rates is given in Figure 6.5. If the hot supply temperature
is less than the cold supply temperature plus EMAT, then the last term in Equation (6.4)
becomes negative and thus set to 0 by the max operator since no heat exchange is possible between these two streams. The value of U i j obtained from Equation (6.4) is always
108
6.4 Model modification
Figure 6.5: Maximum heat transfer between a hot stream i and a cold stream j
less than or equal to the value of U i j when Equation (6.3) is used. This gives a tighter
bound for the LP relaxation.
Another way of specifying the maximum amount of heat transfer between two streams
is on a temperature interval basis. The maximum amount of heat that can be exchanged
between streams i and j in interval k is given by:
(Ã
!
)
X H
C
U i jk = min
Q i k̄ ,Q jk
(6.5)
k̄≤ k
Equation (6.5) defines a ‘local U’ referred to as U i jk and the logical constraint utilizing
this, similar to P1.3, is given by Equation (6.6). Operations Research theory points to the
fact that the local logical constraints from Equation (6.6) will reduce the feasible region
more than Constraint P1.3 and hence a tighter formulation is achieved by reducing the
gap.
Q i jk − U i jk yi j ≤ 0
6.4.2
∀ i ∈ Hk , j ∈ Ck , k ∈ T I
(6.6)
Integer cuts
The difficulty of solving the MILP can be traced by the gap between the LP relaxation
based lower bound and the optimal integer solution. This gap can be reduced by employing integer cuts to the model. A potential drawback of adding such cuts is the increase in
model size and hence computation time. Thus there is a trade-off between reducing the
gap and increasing the model size. This section details two kinds of integer cuts added
to the model.
109
6. MINIMUM NUMBER OF UNITS SUB-PROBLEM
Compulsory matches
This constraint specifies that at least one hot process stream or utility must heat each
cold process stream to its target temperature and vice versa for the hot process streams.
Defining sets M H
to be the set of hot process streams or utilities i that can heat a cold
j
process stream j to its target temperature and M iC the set of cold process streams or
utilities j that can cool a hot process stream i to its target temperature, we can define
the integer cuts as:
X
yi j ≥ 1
∀ i ∈ HP
(6.7a)
yi j ≥ 1
∀ j ∈ CP
(6.7b)
j ∈ M iC
X
i∈ M H
j
Minimum matches per stream
The value of U i j gives the maximum heat that can be exchanged between process streams
i and j. If the total heat available in hot process stream i, Q H
, is greater than U i j for
i
each of the cold process streams j, it implies that stream i must exchange heat with at
least two cold streams. This does not hold true if Q H
is less than the minimum cold
i
utility consumption, Q C ,min , since the cooling of stream i can be satisfied by cold utility.
Equivalent conditions hold true for the cold process streams. The following sets can be
defined based on the preceding discussion:
H
M I N H X H = { i | i ∈ HP if Q H
i > U i j ∀ j ∈ CP and Q i ≤ Q C , min }
H
M I N2H X H = { i | i ∈ HP if Q H
i > U i j ∀ j ∈ CP and Q i > Q C , min }
M I N H X C = { j | j ∈ CP if Q Cj > U i j ∀ i ∈ HP and Q Cj ≤ Q H ,min }
M I N2H X C = { j | j ∈ CP if Q Cj > U i j ∀ i ∈ HP and Q Cj > Q H ,min }
The integer cuts associated with these sets are:
2 · yi,CU +
X
yi j ≥ 2
∀ i ∈ MINH XH
(6.8a)
yi j ≥ 2
∀ i ∈ M I N2H X H
(6.8b)
yi j ≥ 2
∀ j ∈ M I N H XC
(6.8c)
yi j ≥ 2
∀ j ∈ M I N2H X C
(6.8d)
j ∈CP
X
j ∈C
2 · yHU , j +
X
i ∈ HP
X
i∈H
110
6.4 Model modification
6.4.3
Results and discussion
The 21TP1 and 22TP1 problems shown in Appendix A cannot be solved in less than 12
hours with the basic model P1 or with the modifications presented in this paper. The
LP relaxation value is taken as an indication of gap size and problem solution time. The
larger the LP relaxation value, the smaller the gap and it is expected that the model
solution time will be less. However, as mentioned earlier, the model solution time depends on the gap as well as the size of the model which is increased by adding the extra
constraints. The 21TP2 problem can be solved to optimality and results for the LP relaxation and solution times are given. Results for the three test problems, modeled in
GAMS with CPLEX (version 10) as the MILP solver, using the modifications presented
in this section are given below in Tables 6.2, 6.3 and 6.4.
U definition
No integer
Compulsory
Minimum matches
Both cuts
cuts
matches
per stream
Eq 6.3
12.21
-
-
-
Global Eq 6.4
15.17
16.27
16.82
18.62
Local Eqs 6.5, 6.6
15.78
16.56
17.63
18.62
Table 6.2: Root node LP relaxation value with different measures for 22TP1 with IP solution
23
U definition
No integer
Compulsory
Minimum matches
Both cuts
cuts
matches
per stream
Eq 6.3
11.93
-
-
-
Global Eq 6.4
14.30
15.14
14.49
15.21
Local Eqs 6.5, 6.6
14.87
15.39
14.95
15.40
Table 6.3: Root node LP relaxation value with different measures for 21TP1 with IP solution
22
The results show that the global and local U definition presented in this paper give
tighter lower bounds than the original U definition given by [105]. From Table 6.4 it can
be seen that the model with the global U definition solves slightly faster than the original
111
6. MINIMUM NUMBER OF UNITS SUB-PROBLEM
U definition
Eq 6.3
Global Eq 6.4
Local Eqs 6.5, 6.6
No integer
Compulsory
Minimum matches
Both cuts
cuts
matches
per stream
16.00
-
-
-
20 s
-
-
-
17.67
18.80
18.13
18.83
19 s
27 s
23 s
21 s
18.80
18.80
18.81
18.83
36 s
46 s
45 s
42 s
Table 6.4: Root node LP relaxation value and total solution time with different measures
for 21TP2 with IP solution 22
U definition, whereas the same is not true for the model with the local U definition. It is
expected that this is because the model size increases.
The compulsory matches integer cuts always improve the lower bound whereas models
with only the minimum matches integer cuts have varying results. Models with both
these integer cuts implemented always have a better lower bound. When using both
integer cuts for problems 22TP1 and 21TP2, the local U definition does not give a tighter
bound than the global U definition. However, the lower bound provided with the local U
is as good as the model with the global U and hence does not contradict our prediction.
As mentioned earlier, even with these modifications to the model, the 22TP1 and 21TP1
problems do not solve within 12 hours.
These modifications may have improved the model, but does not have an impact of reducing the solution times significantly. It is worth noting from these results that the gap
is not the only measure of the complexity of an integer problem. Two of the main issues
in this particular integer problem are the number of feasible solutions and the number
of multiple optima.
6.5
Model reformulation
Set covering and set partitioning models have been used for a long time to reformulate whole or some parts of routing problems, see [30], and location problems, see [19].
Some constraints that might be difficult to formulate efficiently before reformulation,
112
6.5 Model reformulation
can easily be taken care of during the creation of the sets used in set covering and set
partitioning formulations.
The minimum number of units problem can also be partly reformulated by use of elements from set partitioning formulations by defining, for each stream, a new binary
variable for each set of possible matches with streams of the other type - out of which
only one set of matches can be chosen. The new binary variables can be defined for hot
stream matches, cold stream matches or both. This section presents the reformulated
models.
6.5.1
New formulation with integer variables representing hot stream
matches
Let λ is be binary variables that are defined to represent all feasible sets of matches,
s ∈ S i , between a hot process stream i and all cold process streams and utilities. The
individual set of matches for a hot stream i, SS is , is defined as:
SS is = { j | j ∈ C if stream j is in set of matches s for stream i }
For a problem with hot process stream H1, two cold process streams C1 and C2, and one
cold utility CW, the incidence matrix for the total sets of matches involving hot stream
H1 can be visualized as:
s
1
2
3
4
5
6
7
C1
1
0
0
1
1
0
1
C2
0
1
0
1
0
1
1
CW
0
0
1
0
1
1
1
The maximum number of sets of matches for each hot process stream is 2nc − 1, where
nc is the number of cold process streams and utilities in the problem. The case where
no cold process streams or utilities match with the hot process stream is excluded by
subtracting 1 from the total number of sets of matches 2nc . If by using insight into the
physical problem, the model user may declare that some of the 2nc − 1 matches certainly
are non-optimal, then the number of sets of matches in the model will be reduced and
the feasible region of the LP relaxation will also be reduced. By definition,
λ is =
(
1,
0,
if set of matches s is chosen for hot process stream i
otherwise
113
6. MINIMUM NUMBER OF UNITS SUB-PROBLEM
The set partitioning constraint will then be
X
λ is = 1
∀i∈H
(6.9)
s∈ S i
Potentially a large number of binary variables will be introduced into the model, where
one of the main challenges is dealing with combinatorial explosion. It is important to
notice that it was expected that the reformulated model would ensure better branching
characteristics that can overcome the negative effects of the increase in the number
of binary variables. As indicated earlier, thermodynamics and insight can be used to
reduce the set of matches from the maximum set of feasible matches. The constraints
for defining the set of feasible matches are:
1. At least one cold process stream or utility in the set of matches must have a supply
temperature lower than or equal to the hot process stream’s target temperature
and satisfy the hot process stream’s duty in this temperature range.
2. The total heat demand for the set of cold process streams or utilities below the
hot process stream’s supply temperature must be greater than or equal to the hot
process stream’s total heat duty.
3. The total number of cold process streams and utilities in a set of matches should
not exceed a user-specified maximum value.
4. For cases with streams having large duties, the number of streams in a set of
matches for a utility must be larger than one.
It should emphasized that constraints 1 and 2 are thermodynamically based while constraints 3 and 4 are heuristics based on insights gained from testing various problems.
Constraint 3 is user specified to allow the user to impart their knowledge about the problem on hand to reduce the problem size. This constraint has an impact on the solution
time. For the purposes of this work, a general guideline was not developed and is left to
engineering judgment.
The binary variable λ is specifies possible matches for the hot side only. To develop the
model it must be ensured that all cold process streams are also matched with a hot process stream or utility and that the model must include the possibility for a cold process
114
6.5 Model reformulation
stream to match with multiple hot process streams or utilities. This is done by a set covering constraint that sums across sets of matches of all hot process streams or utilities
such that each cold process stream is matched at least once.
X X
λ is ≥ 1
∀ j∈C
(6.10)
i ∈ H s∈ P i j
where P i j represents feasible sets of matches where hot stream i and cold stream j are
involved.
The reformulated model for the minimum number of units sub-problem with λ is as the
integer variables is given below as P2.
min z =
X X
c is λ is
(P2)
i ∈ H s∈ S i
s.t.
R i,k − R i,k−1 +
Q i jk = Q H
ik
∀ i ∈ Hk , k ∈ T I
(P2.1)
Q i jk = Q Cjk
∀ j ∈ Ck, k ∈ T I
(P2.2)
U i j λ is ≤ 0
∀ i ∈ H, j ∈ C
(P2.3)
X
λ is = 1
∀i∈H
(P2.4)
λ is ≥ 1
∀ j∈C
(P2.5)
λ is ≤ max value
∀ j∈C
(P2.6)
c is = card (SS is )
∀ i ∈ H, s ∈ S i
(P2.7)
∀ i ∈ Hk , k ∈ T I
(P2.8)
∀i∈H
(P2.9)
X
j ∈C k
X
i∈H k
X
k∈T I
Q i jk −
X
s∈ P i j
s∈ S i
X X
i ∈ H s∈ P i j
X X
i ∈ H s∈ P i j
R ik ≥ 0
R i0 = R iK = 0
Q i jk ≥ 0
λ is = {0, 1}
∀ i ∈ Hk , j ∈ Ck , k ∈ T I
(P2.10)
∀ i ∈ H, s ∈ S i
(P2.11)
Constraint P2.6 specifies the maximum number of heat exchangers in a set of matches,
while constraint P2.7 defines the value of c is - the coefficient that has a value equal to
the number of heat exchangers in a set of matches.
115
6. MINIMUM NUMBER OF UNITS SUB-PROBLEM
Surprisingly, preliminary testing of this formulation showed a weaker LP relaxation
with slower solution times for solvable problems, and significantly worse bounding for
large problems as compared to model P1 with added constraints given by Equations
(6.6)-(6.8). Thus, as a next attempt, the new binary variables λ is and the associated
constraints are incorporated into model P1 to give model P3 as shown below.
min z =
XX
i
yi j
(P3)
j
s.t.
R i,k − R i,k−1 +
X
Q i jk = Q H
ik
∀ i ∈ Hk , k ∈ T I
(P3.1)
Q i jk = Q Cjk
∀ j ∈ Ck, k ∈ T I
(P3.2)
∀ i ∈ H, j ∈ C
(P3.3)
λ is = 1
∀i∈H
(P3.4)
λ is ≥ 1
∀ j∈C
(P3.5)
λ is ≤ max value
∀ j∈C
(P3.6)
λ is = yij
∀ i ∈ H, j ∈ C
(P3.7)
R ik ≥ 0
∀ i ∈ Hk , k ∈ T I
(P3.8)
∀i∈H
(P3.9)
j ∈C k
X
i∈H k
X
Q i jk − U i j yi j ≤ 0
k∈T I
X
s∈ S i
X X
i ∈ H s∈ P i j
X X
i ∈ H s∈ P i j
X
s∈ P i j
R i0 = R iK = 0
Q i jk ≥ 0
∀ i ∈ Hk , j ∈ Ck , k ∈ T I
(P3.10)
yi j = {0, 1}
∀ i ∈ H, j ∈ C
(P3.11)
λ is = {0, 1}
∀ i ∈ H, s ∈ S i
(P3.12)
In addition to these constraints, Equations (6.6)-(6.8) can also be used as additional
constraints.
6.5.2
New reformulation with integer variables representing cold stream
matches
Similar to the λ is binary variables, new variables µ jt are defined to represent all feasible
sets of matches, t ∈ T j , between a cold process stream j and all hot process streams and
116
6.5 Model reformulation
utilities.
(
µ jt =
1,
if set of matches t is chosen for cold process stream j
0,
otherwise
The constraints defining the feasible sets of matches for µ jt are similar to the four constraints for defining λ is in Section 6.5.1. Model P4 is the minimum number of units
problem reformulated to include cold stream sets of matches as binary variables µ jt .
This model is similar to P3. The set V ji represents feasible sets of matches of cold process stream j with hot process stream or utility i. Equations (6.6)-(6.8) can be added as
optional constraints to the model P4 shown below.
min z =
XX
i
yi j
(P4)
j
s.t.
R i,k − R i,k−1 +
X
Q i jk = Q H
ik
∀ i ∈ Hk , k ∈ T I
(P4.1)
Q i jk = Q Cjk
∀ j ∈ Ck, k ∈ T I
(P4.2)
∀ i ∈ H, j ∈ C
(P4.3)
µ jt = 1
∀ j∈C
(P4.4)
µ jt ≥ 1
∀i∈H
(P4.5)
µ jt ≤ max value
∀i∈H
(P4.6)
µ jt = yi j
∀ i ∈ H, j ∈ C
(P4.7)
R ik ≥ 0
∀ i ∈ Hk , k ∈ T I
(P4.8)
∀i∈H
(P4.9)
j ∈C k
X
i∈H k
X
Q i jk − U i j yi j ≤ 0
k∈T I
X
t∈ T j
X X
j ∈C t∈V ji
X X
j ∈C t∈V ji
X
t∈V ji
R i0 = R iK = 0
Q i jk ≥ 0
∀ i ∈ Hk , j ∈ Ck , k ∈ T I
(P4.10)
yi j = {0, 1}
∀ i ∈ H, j ∈ C
(P4.11)
µ jt = {0, 1}
∀ j ∈ C, t ∈ T j
(P4.12)
117
6. MINIMUM NUMBER OF UNITS SUB-PROBLEM
6.5.3
New formulation with integer variables representing both hot
and cold stream matches
A model formulation for the minimum number of units sub-problem with both binary
variables λ is and µ jt representing the hot and cold process stream sets of matches is
shown below as model P5.
min z =
XX
i
yi j
(P5)
j
s.t.
R i,k − R i,k−1 +
X
Q i jk = Q H
ik
∀ i ∈ Hk , k ∈ T I
(P5.1)
Q i jk = Q Cjk
∀ j ∈ Ck, k ∈ T I
(P5.2)
∀ i ∈ H, j ∈ C
(P5.3)
λ is = 1
∀i∈H
(P5.4)
µ jt = 1
∀ j∈C
(P5.5)
λ is = yi j
∀ i ∈ H, j ∈ C
(P5.6)
µ jt = yi j
∀ i ∈ H, j ∈ C
(P5.7)
R ik ≥ 0
∀ i ∈ Hk , k ∈ T I
(P5.8)
∀i∈H
(P5.9)
j ∈C k
X
i∈H k
X
Q i jk − U i j yi j ≤ 0
k∈T I
X
s∈ S i
X
t∈ T j
X
s∈ P i j
X
t∈V ji
R i0 = R iK = 0
Q i jk ≥ 0
∀ i ∈ Hk , j ∈ Ck , k ∈ T I
(P5.10)
yi j = {0, 1}
∀ i ∈ H, j ∈ C
(P5.11)
λ is = {0, 1}
∀ i ∈ H, s ∈ S i
(P5.12)
µ jt = {0, 1}
∀ j ∈ C, t ∈ T j
(P5.13)
As both the hot and cold stream matches are represented by binary variable in Model P5,
the set covering constraints P3.5 and P4.5 are superfluous in this model. The constraints
P3.6 and P4.6 are also superfluous. Equation (6.6) can be used as an optional constraint
in this model. Note that Equations (7.5) and (6.8) are not used here as the sets of matches
for both hot and cold side are already defined using λ is and µ jt .
118
6.5 Model reformulation
Model
Binary
U model
Additional
LP relaxation
constraints
value
Global Eq-2
Eqs 5,6
18.62
Local Eqs 3,4
Eqs 5,6
18.62
Global Eq-2
Eqs 5,6
18.62
Local Eqs 3,4
Eqs 5,6
18.62
Global Eq-2
Eqs 5,6
18.67
Local Eqs 3,4
Eqs 5,6
18.67
Global Eq-2
None
18.67
variables
P1
P3
143
2500
with λ is
P4
2625
with µ jt
P5
4999
with λ is & µ jt
Local Eqs 3,4
18.67
Table 6.5: Root node LP relaxation value with different model reformulations for 22TP1
with IP solution 23
6.5.4
Results and discussion
Prioritizing the branching variables (yi j , λ is and µ jt ) and using SOS1 sets for these
variables are discussed in detail by Nastad [98]. The results presented here are for
models with variables suitably prioritized.
Inclusion of the new λ is and/or µ jt variables increases the number of binary variables
as discussed earlier and might therefore increase the number of nodes in the B&B tree.
Since all the sets of matches are unique, it follows from Equations (P5.4)-(P5.7) that if
all yi j variables have binary values, the same is true for λ is and µ jt . This means that
by using priorities to force the optimizer to first branch on the yi j variables one would
expect the number of branches to be unchanged. If not all 2nc − 1 matches are defined,
then that might change the LP relaxations and the branching sequence as well as the
number of branches.
With Equations (P5.4) and (P5.5) in the model, we are able to branch on special ordered
sets of type 1, SOS1, as discussed by [98]. If it had been easy to order the sets of matches
linearly, one would expect SOS1 branching to be effective, but since [98] did not find it
easy to order sets of matches linearly the results are not as good as initially hoped for.
Tables 6.5 and 6.6 present results for the LP relaxation value obtained by the three
reformulated models P3, P4 and P5 for the test problems 22TP1 and 21TP1.
The results show that the reformulated models may result in strengthening the LP re-
119
6. MINIMUM NUMBER OF UNITS SUB-PROBLEM
Model
Binary
U model
Additional
LP relaxation
constraints
value
Global Eq 6.4
Eqs 6.7,6.8
15.21
Local Eqs 6.5,6.6
Eqs 6.7,6.8
15.40
Global Eq 6.4
Eqs 6.7,6.8
15.74
Local Eqs 6.5,6.6
Eqs 6.7,6.8
15.82
Global Eq 6.4
Eqs 6.7,6.8
16.33
Local Eqs 6.5,6.6
Eqs 6.7,6.8
16.45
Global Eq 6.4
None
16.86
variables
P1
P3
131
4645
with λ is
P4
5435
with µ jt
P5
with λ is & µ jt
9879
Local Eqs 6.5,6.6
16.93
Table 6.6: Root node LP relaxation value with different model reformulations for 21TP1
with IP solution 22
laxation. While the improvement is marginal for 22TP1 it is larger for 21TP1. The
reason for these tighter relaxations is that the reformulated models contain more information regarding the thermodynamics of the matches than the basic model P1 as
stream matches are explicitly incorporated only for feasible cases. Even though the reformulated models lead to a strengthened LP relaxation, the test problems could not be
solved to optimality within 12 hours. One possible explanation is that the reformulated
models are larger with a lot more binary variables, thus counteracting the potential savings in solution time from the reduced gaps. However, it can also be argued that the
reductions in gap are too small to significantly reduce solution times.
6.6
A problem difficulty index?
The results from earlier sections show that the degree of difficulty in solving a problem
does not depend only on the number of streams. As an illustration, 21TP1 does not solve
within 12 hours while 21TP2 solves in 20 seconds. Is it possible to develop a problem
difficulty index, that permits identifying difficult problems before solving them?
A Feasibility Matrix (FM) can be contructed for each problem. The value of cell F M(i, j) =
0 implies that a match between hot process stream i and cold process stream j is not
feasible (based on stream temperature ranges), while a value of 1 indicates a feasible
match. The matrices for test problems 21TP1, 21TP2 and 22TP1 are given in Tables 6.7,
120
6.6 A problem difficulty index?
6.8 and 6.9. For comparison purposes, the matrices for test problems 7TP1 and 15TP1
used in Chapter 5 are given in Tables 6.10 and 6.11.
C01
C02
C03
C04
C05
C06
C07
C08
C09
C10
H01
1
1
1
1
1
1
1
1
1
1
H02
1
1
1
1
0
0
0
0
1
1
H03
1
1
1
1
1
1
1
1
1
1
H04
1
1
1
1
0
1
1
1
1
1
H05
1
1
1
1
1
1
1
1
1
1
H06
1
1
1
1
1
1
1
1
1
1
H07
1
1
1
1
0
1
1
1
1
1
H08
1
1
1
1
1
1
1
1
1
1
H09
1
1
1
1
1
1
1
1
1
1
H10
1
1
1
1
0
0
0
1
1
1
H11
1
1
1
1
1
1
1
1
1
1
Table 6.7: Feasibility matrix for 21TP1
Comparing Tables 6.7, 6.8 and 6.9 shows that the feasibility matrix for 21TP2 is much
sparser than those for 21TP1 and 22TP1. A Sparsity index (Sparsity f ) for the feasibility
matrix can be evaluated for each problem as:
Sparsity f =
Total number of feasible matches
n H · nC
(6.11)
Note that n H · n C is the total number of possible matches.
Sparsity f can be viewed as a problem level metric. Another problem level metric is
the total number of feasible matches (or the number of binary variables in the model).
Algorithm level metrics such as number of nodes etc. could, potentially, be important in
identifying difficult problems. The following algorithm level metrics are considered (see
Figure 6.1):
Nodes = n H + n C + nodestrans
Shape =
max(nodestrans , arcscold )
min(nodestrans , arcscold )
121
(6.12)
(6.13)
6. MINIMUM NUMBER OF UNITS SUB-PROBLEM
C01
C02
C03
C04
C05
C06
C07
C08
C09
C10
H01
1
1
1
1
1
1
1
1
1
1
H02
1
1
1
1
1
1
1
1
1
1
H03
1
0
1
0
0
0
0
1
0
1
H04
1
0
1
0
0
0
0
1
0
1
H05
1
0
1
0
0
0
0
1
0
0
H06
0
0
0
0
0
0
0
1
0
0
H07
1
0
1
0
0
0
0
1
0
0
H08
1
0
1
0
0
0
0
1
0
0
H09
1
0
1
0
0
0
0
1
0
0
H10
1
0
1
0
0
0
0
1
0
0
H11
1
0
1
0
0
0
0
1
0
1
Table 6.8: Feasibility matrix for 21TP2
Sparsityarcs =
Total number of feasible arcs
Total number of possible arcs
(6.14)
Table 6.12 lists the metrics for the different test problems. The Sparsity f of 21TP2 is
significantly lower than that of 21TP1 and 22TP1. Hence, 21TP2 can be thought of as
being a simpler problem to solve. 15TP1 and 7TP1 have very high values for Sparsity f ,
but solve faster because they are smaller problems. The number of feasible matches is
possibly the most important factor as 21TP2 and 15TP1 have similar values and also
have similar solution times. The number of feasible matches for 21TP1 and 22TP1 are
twice that of 21TP2.
Two of the algorithm level metrics, Nodes and Shape, do not give much insight. Sparsityarcs
for 21TP2 has a lower value compared to 21TP1 and 22TP1 and the Sparsityarcs for
15TP1 and 7TP1 are greater than those for the 21TP1, 21TP2 and 22TP1. This is similar to that discussed for Sparsity f .
From the results of the various indices, it appears that the number of binary variables,
defined as the number of feasible matches, is the defining criteria for problem difficulty.
122
6.7 Conclusions and further work
C01
C02
C03
C04
C05
C06
C07
C08
C09
C10
C11
H01
1
1
1
1
0
1
0
1
1
1
0
H02
1
1
1
1
0
1
0
1
1
1
0
H03
1
1
1
1
1
1
1
1
1
1
1
H04
1
1
1
1
1
1
1
1
1
1
1
H05
1
1
1
1
1
1
1
1
1
1
1
H06
1
1
1
1
0
1
0
1
1
1
0
H07
1
1
1
1
1
1
1
1
1
1
1
H08
1
1
1
1
0
1
0
1
1
1
0
H09
1
1
1
1
0
1
0
1
1
1
0
H10
1
1
1
1
1
1
1
1
1
1
1
H11
1
1
1
1
1
1
1
1
1
1
1
Table 6.9: Feasibility matrix for 22TP1
C1
C2
C3
C4
C5
C6
C7
H1
1
1
1
1
1
1
1
H2
1
1
1
1
1
1
1
H3
1
1
1
1
1
1
1
H4
1
1
1
1
1
1
0
H5
1
1
1
1
1
1
1
H6
1
1
1
1
1
1
1
H7
1
1
1
1
1
1
1
H8
1
1
1
1
1
1
0
Table 6.10: Feasibility matrix for 15TP1
6.7
Conclusions and further work
The minimum number of units sub-problem was introduced and the MILP transshipment formulation of the sub-problem was discussed. It was shown that only stream
supply temperatures are required to set up temperature intervals in the formulation of
minimum number of units sub-problem. The chapter presented improvements to the existing minimum number of units sub-problem and a novel method for reformulating the
model to mitigate the combinatorial explosion associated with the MILP model. Results
for large problems show that the proposed modifications strengthen the LP relaxation,
however, the model solution times remain too long to be of interest in the Sequential
123
6. MINIMUM NUMBER OF UNITS SUB-PROBLEM
C1
C2
C3
C4
H1
1
1
1
1
H2
1
1
1
1
H3
1
1
1
1
Table 6.11: Feasibility matrix for 7TP1
21TP1
21TP2
22TP1
15TP1
7TP1
Sparsity f
0.92
0.44
0.88
0.96
1.00
No feasible matches
101
48
106
54
12
Nodes
177
106
144
72
22
Shape
2.20
2.30
1.94
1.06
1.36
Sparsityarcs
0.49
0.36
0.47
0.67
0.53
Table 6.12: Problem difficulty metrics for the test cases
Framework. Orders of magnitude improvements are required to solve large problems.
Model solution times depend on the the number of binary variables in the model and on
specific model characteristics. In this paper, the models with solution times less than 50
seconds have fewer than 150 binary variables while models with more than 5000 binary
variables do not solve within 12 hours. The computing times reported are for for the
minimum number of units sub-problem only, and are not sensitive to initial estimates.
To help identify problems that are computationally expensive, the concept of a problem
difficulty index was explored. From the results of the various indices, it appears that the
number of binary variables is the defining criteria for problem difficulty.
Further work
Experience from running the models indicate that the optimum value is reached early in
the solution process and most of the effort is expended in proving optimality. This is the
main reason for the focus on strengthening the LP relaxation. Future work can involve
identifying heuristics to stop the search after an appropriate solution time.
124
6.7 Conclusions and further work
Identifying subnetworks by relaxing stream temperatures and flow rates using a methodology similar to Shethna and Jez̆owski [122] it may be possible to get a good initial bound
on the minimum number of units. This value could be used by CPLEX as the initial lower
bound thus tightening the gap.
Since the 1990s, work has been done (e.g. Taylor et al. [132]) to show that all NPcomplete problems have a phase separation i.e. both hard and easy regions, with a
sharp boundary between them. On crossing that frontier, the problem undergoes a phase
transition, analogous to the boiling or freezing of water. Identifying the phase transition
of the minimum units problem would enable the user to decide on using deterministic
methods for solving it or other non-deterministic methods depending on which phase the
problem happens to be in.
125
6. MINIMUM NUMBER OF UNITS SUB-PROBLEM
126
7
Stream Match Generator
Sub-problem
This chapter presents the Stream Match Generator model in the Sequential Framework
where “optimal” heat load distributions (HLDs) for a specified level of heat recovery and
number of units are determined using an area targeting model [10].
7.1
Introduction
The Total Annualized Cost (TAC) of a heat exchanger network, given the Heat Recovery
Approach Temperature (HRAT) and number of heat exchanger units, depends on the
heat transfer area. Area considerations in heat exchanger network synthesis were first
elaborated by Hohmann [66] and Nishida et al. [99]. The Bath formula proposed by
Townsend and Linnhoff [136] represents the first targeting method for area of heat exchanger networks. This widely used formula, later extended in Ahmad et al. [5], is based
on the concept of vertical heat transfer between (see Figure 7.1) composite curves and
the resultant “spaghetti” design of the network. Ahmad et al. [5] modified the original
Bath formula [136] by incorporating the variation in heat transfer coefficients by using
stream contributions to ∆T min developed by Nishimura [101]. Rév and Fonyó [113] proposed the diverse pinch concept which incorporated a generalized version of individual
stream contributions to ∆T min and calculated area targets based on modified composite
curves. This was further updated by Serna and Jiménez [120] taking into account shifts
in duties between matches in the spaghetti design due to shift in temperatures.
127
7. STREAM MATCH GENERATOR SUB-PROBLEM
Zhu et al. [156] and Briones and Kokossis [23] utilize the diverse pinch concept to develop area targets. These form the basis for the block decomposition method [154] and
the hypertargeting [23] method for heat exchanger network design.
Colberg and Morari [28] and Yee et al. [151] developed NLP models for area targeting.
While the NLP model of Colberg and Morari [28] is based on a transshipment model for
heat exchange with spaghetti structure incorporating temperature and enthalpy intervals, the model of Yee et al. [151] is based on another approach to the spaghetti structure
using a stage-wise model with the limitation of isothermal mixing. Shethna et al. [121]
formulate an MILP transportation problem that simultaneously optimizes for heat exchanger units, area and loads on utilities. Jez̆owski et al. [77] present an LP model for
area targeting based on the transportation formulation.
This chapter presents the Stream Match Generator model in the Sequential Framework
where “optimal” heat load distributions (HLDs) for a specified level of heat recovery
and number of units are determined using an area targeting model. The stream match
generator model used in this work is based on an MILP transportation formulation.
7.2
Stream Match Generator model formulation
The stream match generator sub-problem in the Sequential Framework can be defined
as follows
- Given:
• a set H of hot process streams to be cooled and hot utilities,
• a set C of cold process streams to be heated and cold utilities,
• supply and target temperatures, heat capacities and flow rates of the hot and cold
process streams,
• temperatures or temperature ranges and fixed heat loads of the utilities, and
• a specified number of heat exchanger units.
- Determine a ranked sequence of HLDs between hot process streams and utilities and
cold process streams and utilities that leads to networks with increasing cost and such
that the heating and cooling requirements for each stream are met.
128
7.2 Stream Match Generator model formulation
Figure 7.1: Vertical heat transfer between composite curves
7.2.1
Development history
The stream match generator model, in development since 1990 as the Vertical MILP
model, has been motivated by the insight to include area considerations in the selection
of HLDs [58].
Gundersen and Grossmann [58] developed the first Vertical MILP model based on the
idea that vertical heat transfer between the Composite Curves will improve the use of
available driving forces and thus reduce total area. The Composite Curves are divided
into Enthalpy Intervals (EIs) at every kink of the curves in addition to the Temperature
Intervals (TIs) required for the MILP transshipment model for the fewest number of
units as shown in Figure 7.1. The EIs based on the HRAT (kinks in the composite
curves) determine the quality of potential matches within thermodynamically feasible
TIs based on EMAT. A penalty term (S i j ) is added to the minimum number of heat
exchangers problem objective to indicate the deviation in actual heat transfer to the
theoretical maximum that could be transferred vertically between the pair of streams.
The penalty term gives only a lower bound on non-vertical heat transfer but has been
shown to provide a good indication of the ability of the network to transfer heat vertically
and reduce total network area.
129
7. STREAM MATCH GENERATOR SUB-PROBLEM
When the stream film heat transfer coefficients are significantly different, strict vertical
heat transfer no longer guarantees minimum total heat transfer area. Gundersen and
Grossman [58] explain two effects that are important in such cases. The first effect,
shifting, occurs, for example, when a hot stream with high film heat transfer coefficient
should be used to heat up a cold stream at higher temperatures than strict vertical heat
transfer indicates. The use of individual stream contributions to ∆T min and the use of
modified temperatures can account for shifting. This is incorporated in the model of
Gundersen and Grossmann [58].
The second effect, pairing, is related to the fact that it is often beneficial to isolate
streams with poor film heat transfer coefficients in separate exchangers and allow larger
driving forces for these units. This was incorporated in the model presented by Gundersen et al. [56]. A new penalty term was added to the objective function where designs
involving matches between streams with similar heat transfer conditions have a lower
penalty term value. The penalty related to the matches is based on the principle was put
forward by Umeda et al. [140]. The objective function proposed by Gundersen et al. [56]
is:
XX
XX
© XX
£
¡
¢¤ª
min α ·
yi j + β ·
Si j + γ ·
yi j 1 − min h i /h j , h j /h i
(7.1)
where h i and h j are the hot and cold stream film heat transfer coefficients respectively.
α is set to a small value, while β and γ are set to values that make the second and third
terms in the objective function normalized (0, 1). Even though the Vertical MILP model
was run with a fixed number of units, the first term in the objective is kept to improve
the search.
The drawback of the extended model based on the transshipment formulation presented
by Gundersen et al. [56] is that due to the nature of the transshipment model it is unable
to account for where non-vertical heat transfer occurs. To do this it is required to keep
track of where heat is introduced and removed from the cascade. This is however possible in a transportation model. Further, another disadvantage of the extended vertical
transshipment model was the need to assign heuristic values to the weights β and γ.
Taking this into consideration, Gundersen et al. [61] presented a Vertical MILP model
formulated as a transportation model. Here the objective function is:
130
7.2 Stream Match Generator model formulation
Figure 7.2: Transportation formulation
min
XXXX
i
j m n
Q im, jn
µ
¶
U i j · ∆T LM ,mn
(7.2)
This transportation formulation of the Vertical MILP model is the basis for the Stream
Match Generator Model and is detailed in the next sections.
7.2.2
MILP model
The Stream Match Generator sub-problem is formulated as an MILP Transportation
model from Operations Research (see Figure 7.2). The basic transportation model for
minimum number of units presented by Cerdá and Westerberg [24] is modified by changing the objection function to Equation 7.2. Further, the model (R1) shown below differs
from the model presented in [24] since no sub-networks are considered in (R1) (no pinch
decomposition).
The heat supplied by process stream or utility i in interval m is represented as Q im and
Q jn represents the heat demand of cold process stream or utility j in temperature interval n. Q im, jn is the heat exchanged between hot process stream or utility i in interval
m and cold process stream or utility j in interval n. The binary variable yi j denotes the
131
7. STREAM MATCH GENERATOR SUB-PROBLEM
existence of a match between hot process stream or utility i and cold process stream or
utility j. U i j is a large number (upper bound) sometimes referred to as the big M, linking
the binary variables yi j to the continuous variables Q im, jn and is discussed in detail in
Section ??.
min z =
Q im, jn
XXXX
j m n
i
(R1)
U i j · ∆T LM ,mn
s.t.
N
X X
Q im, jn = Q im
∀ i ∈ H m , m ∈ T I H, m, n ∈ F mn
(R1.1)
Q im, jn = Q jn
∀ j ∈ C n , n ∈ T IC, m, n ∈ F mn
(R1.2)
∀ i ∈ H, j ∈ C, m, n ∈ F mn
(R1.3)
j ∈C n n=1
M
X X
i ∈ H m m=1
X
X
Q im, jn − U i j yi j ≤ 0
m∈T I H n∈T IC
X X
yi j = H XU
(R1.4)
i ∈ H j ∈C
Q im, jn ≥ 0
yi j = {0, 1}
∀ i ∈ H m , j ∈ C n , m ∈ T I H, n ∈ T I H (R1.5)
∀ i ∈ H, j ∈ C
(R1.6)
U i j is the overall heat transfer coefficient for the match between hot process stream or
utility i and cold process stream or utility j. ∆T LM ,mn is the log mean temperature
difference for heat transfer between intervals m and n. U i j and ∆T LM ,mn are constants
and can be calculated ahead of time. As regards constraint R1.5, the equality hold when
m, n ∉ F mn . The constraint R1.4 sets the total number of heat exchanger units to a user
specified value H XU.
The objective function of Model R1 can be thought of as “pseudo-area” and the model
gives a ranked sequence of increasing network area when the “pseudo-area” replicates
the actual area of heat exchangers in the network. This is possible when the sizes of the
TIs are small (or large number of TIs). This can be visualized as: creating more intervals
allows matching corresponding to the spaghetti structure - and thus minimum area.
Note that the formulation of the stream match generator does not allow for cyclic matches
where a pair of streams are matched against each other more than once. This is due to
the fact that the HLDs are generated for a given number of units and only one match is
allowed between a pair of streams.
132
7.2 Stream Match Generator model formulation
16000
14000
Number of Qimjn variables
12000
10000
8000
6000
4000
2000
0
0
5
10
15
20
25
30
Number of hot temperature intervals
Figure 7.3: Polynomial increase in the number of Q im jn variables with the number of temperature intervals. The number of hot temperature intervals is assumed equal to the number of cold temperature intervals in this figure.
7.2.3
Temperature intervals
The number of Q im jn variables in Model R1 increases polynomially with the number of
hot and cold temperature intervals (Figure 7.3). As mentioned earlier, the smaller the
size of the TI, the better “pseudo-area” represents actual area. However, the transportation model is a polynomial time algorithm [43] and the size of the problem affects the
solution time. The number of TIs must thus be limited to reduce computational time
while ensuring that the model predicts the accurate ranked sequence. The procedure
below describes the procedure to generate Temperature Intervals for the stream match
generator model based on EIs of the balanced composite curves - a Vertical model.
Step 1. Establish the balanced composite curves, using HRAT, stream and utility data.
Step 2. Supply and target temperatures of all streams, including utility streams, are set
to be the Primary Hot/Cold Temperatures.
Step 3. For all cold supply and target temperatures, find adjacent hot temperatures placed
vertically above the kinks of the cold composite curve. These are the Secondary Hot
Temperatures. Similarly, for all hot supply and target temperatures, find adjacent
133
7. STREAM MATCH GENERATOR SUB-PROBLEM
Figure 7.4: Primary temperatures
cold temperatures placed vertically below the kinks of the hot composite curve.
These are the Secondary Cold Temperatures.
Step 4. For all cold supply temperatures, find the corresponding Tertiary Hot Temperatures by adding EMAT. Disregard any hot temperature that is colder than the
coldest hot target temperature. Similarly, for all hot supply temperatures, find the
corresponding Tertiary Cold Temperatures by subtracting EMAT. Disregard any
cold temperature that is hotter than the hottest cold target temperature.
Step 5. Quaternary Hot/Cold Temperatures are calculated by adding/subtracting EMAT
to/from the Secondary Cold/Hot Temperatures.
Step 6. The hot/cold temperatures from Steps 2 to 5 are merged. They are then sorted and
duplicate temperatures removed to give the corresponding hot and cold TIs.
It is important to note that the number of hot TIs need not equal the number of cold
TIs. Thus each pair of hot and cold temperature intervals have to be checked for thermodynamic feasibility (see set Fmn in model R1). Figures 7.4 to 7.6 show the steps in
generating the temperature intervals.
134
7.2 Stream Match Generator model formulation
Figure 7.5: Primary and Secondary temperatures
Figure 7.6: Primary and tertiary temperatures
135
7. STREAM MATCH GENERATOR SUB-PROBLEM
The temperature intervals for Example 7TP1 (for stream data, see Appendix A.1) with
EMAT = 2.5 K is shown in Table 7.1. The letters in parenthesis next to the temperatures indicate the type of temperature: P - primary, S - secondary, T - tertiary and Q quaternary. As can be seen in Table 7.1, the secondary and quarternary temperatures
constitute the bulk of the additional temperatures in the TIs.
Extensive testing of the alternative combinations of temperatures shows that these set
of TIs in the stream match generator model accurately predicts the ranked sequence
of HLDs with increasing total heat transfer area. Two other means of generating TIs
are compared to the method described above. One of them is a simple method to generate temperature intervals based on stream supply and target temperatures presented
by Linnhoff and Flower [89] while the second is a heuristic approach for creating TIs
described by Jez̆owski et al. [77] for an area targeting model.
A comparison of the number of temperature intervals, the eventual HEN capital cost
with the model solution times for the test problem 15TP1 (see Appendix A.2) with EMAT
= 2.5 °C are presented in Table 7.2. The HLDs obtained for the three TIs generation
methods are given in Tables 7.3 and 7.4. The results show that the model gives the
lowest cost network when the TI generation method described in this work are used.
However, the simplest method of using just the supply and target temperatures for the
TIs solves in a quarter of the time used for the elaborate TIs presented in this work
while having a network with 2% more capital cost. Utilizing the TI generation method
presented by Jez̆owski et al. [77] does not provide much advantage. These results are
representative of the different examples tested as part of the work.
Utilizing the TI generation method presented in this work is used in the Sequential
Framework to ensure that the best HLD with respect to cost is obtained. However, the
simple supply and target temperature method can be used if computational resource is
of importance.
7.2.4
EMAT as an optimizing variable
In the original Vertical MILP transshipment model described by Gundersen and Grossmann [58], EM AT ≤ HR AT was used to develop enthalpy intervals and temperature
intervals, and it was demonstrated that EMAT is not an optimizing variable.
EMAT is used to develop the TIs for the stream match generator model as discussed in
Section 7.2.3. A large value of EMAT could lead to potential HLDs being excluded from
136
7.2 Stream Match Generator model formulation
Hot TI
Hot Temp (K)
Cold Temp (K)
650.0 (P)
613.0 (P)
626.0 (P)
579.0 (S)
1
Cold TI
1
2
2
576.0 (P)
623.8 (S)
3
3
571.2 (S)
620.0 (P)
4
4
616.8 (S)
566.0 (P)
615.5 (Q)
525.5 (T)
586.0 (P)
524.8 (S)
581.5 (Q)
514.5 (Q)
5
5
6
6
7
7
8
8
573.7 (Q)
506.9 (S)
9
9
498.3 (S)
528.0 (P)
10
10
527.3 (Q)
497.0 (P)
519.0 (P)
472.8 (Q)
517.0 (S)
389.0 (P)
11
11
12
12
13
13
509.4 (Q)
386.0 (P)
14
14
500.8 (Q)
382.1 (Q)
15
15
378.5 (Q)
499.5 (T)
16
16
475.3 (S)
350.5 (Q)
391.5 (T)
326.0 (P)
384.6 (S)
313.0 (P)
17
17
18
18
19
19
381 (S)
308.0 (P)
20
20
353 (P)
293.0 (P)
Table 7.1: Temperature Intervals for Example 7TP1 with EMAT = 2.5 K.
137
7. STREAM MATCH GENERATOR SUB-PROBLEM
TI generation method
TIs (hot/cold)
Solution Time (mins)
Capital Cost ( $)
This work
30/30
83
496,724
Linnhoff and Flower [89]
22/22
21
507,679
Jez̆owski et al. [77]
27/27
58
507,679
Table 7.2: Number of temperature intervals, model solution time and heat exchanger network cost for 15TP1 problem with EMAT = 2.5 using the three TI generation methods.
C1
C2
ST
C3
875
H1
H2
C4
C5
5264.25
C7
CW
5400
3150
1350
7200
1050
H3
3150
H4
1835.75
1164.25
5000
H5
H6
H7
C6
4375
2450
1750
H8
8000
Table 7.3: HLD for 15TP1 problem with EMAT = 2.5 using the TI generation method presented in this work.
the feasible set of solutions. However, if EMAT is set too low, the non-vertical values of
∆T LM ,mn become very small and such non-vertical heat transfer will face large penalties.
Experience indicates that defining an EMAT that balances the inclusion of promising
solutions and the exclusion of poor candidates (HLDs) is far from straightforward.
Let us consider Example 7TP1 discussed in Section 5.8.1. The best solution was found
to be an HLD evaluated with EMAT = 2.5 K. The HLD for this case is given in Table
7.5. Setting EMAT = 1.0 K to increase the number of feasible solutions in the stream
generator model, the HLD shown in Table 7.5 is evaluated as the “optimum”. Notice
that while using EMAT=1K rather than EMAT=2.5K, the HLD for 1K results in a more
expensive network (TAC = $ 151,559) compared to the 2.5K (TAC= $ 147,861). The HLD
for 1K has a different set if matches as compared to the HLD generated with EMAT =
2.5K. The reason for the preferring the H2C1 match in the HLD when EMAT = 2.5K is
that the ∆T LM ,mn for the feasible H2C1 temperature intervals in the case with EMAT =
2.5K is greater than that when EMAT = 1K. This match is penalized when evaluating
138
7.2 Stream Match Generator model formulation
C1
ST
C2
C3
C4
1489.25
C5
C6
4650
H1
CW
5400
3150
7200
H2
1350
H3
1050
3150
1835.75
H4
1164.25
H5
H6
C7
5000
2310.75
2064.25
4200
H7
H8
8000
Table 7.4: HLD for 15TP1 problem with EMAT = 2.5 using the TI generation method presented in Linnhoff and Flower [89] and Jez̆owski et al. [77].
EMAT = 1
C1
EMAT = 2.5
H1
H2
H3
392.08
106.495
90.058
C2
119.867
C3
C4
69.669
C1
H1
H2
H3
323.635
176.164
88.834
C2
457.62
C3
357.901
C4
119.867
457.62
68.445
359.125
Table 7.5: Heat Load Distributions calculated for Example 7TP1 with EMAT = 1K and
EMAT = 2.5K
the HLDs with EMAT = 1K. Table 7.6 gives the percentage increase in ∆T LM ,mn using
EMAT=2.5K as compared to EMAT =1K in the feasible intervals for the H2C1 match.
While the increase appears marginal, it is significant enough to cause a shift in the
structure - the H2C4 match for EMAT=2.5K is replaced by H1C4 match for EMAT=1K.
As ∆T LM ,mn is a term included in the objective function (see R1) and depends directly on
EMAT, it follows that EMAT is an optimizing variable in this formulation. As seen from
the example evaluated earlier, the value of EMAT chosen affects the HLD generated
and thus the TAC of the network. A loop for EMAT is thus included in the Sequential
Framework as a contribution of this work to explore the solution space with respect to
EMAT. As mentioned earlier, the value of EMAT selected affects the objective function
and is thus considered as part of the “area” loop in the framework.
EMAT has an another interesting characteristic related to the HLDs generated in the
139
7. STREAM MATCH GENERATOR SUB-PROBLEM
n/m
4
5
6
7
8
9
10
11
1
0.0
2.9
41.7
2
0.0
2.0
2.6
16.0
3
0.0
1.8
2.3
8.1
75.9
4
0.0
1.6
2.0
5.2
20.4
5
0.9
1.9
2.8
3.3
8.3
62.1
6
0.8
1.7
1.8
2.6
4.4
4.6
150.0
7
0.7
1.5
1.9
2.3
4.0
6.7
17.5
86.9
8
0.7
1.4
1.6
2.1
3.5
3.1
10.0
14.1
9
0.0
0.7
0.7
0.9
2.0
1.3
2.8
3.9
10
0.0
0.6
0.7
0.9
1.9
1.2
2.6
2.8
Table 7.6: Percentage increase in ∆T LM ,mn values with EMAT = 2.5K compared to EMAT
= 1K for Example 7TP1
Stream Match Generator model. When the number of heat exchangers is greater than
the absolute minimum, EMAT adjusts the HLDs similar to the +X/-X rule when optimizing networks in the Pinch Design Method using heat load loops and paths.
7.3
Challenges
Similar to the minimum number of units sub-problem, as the number of streams increases, the MILP formulation for the stream match generator sub-problem becomes
hard and eventually impossible to solve due to “combinatorial explosion”. With an increase in the number of streams, the binary search tree increases exponentially (see
Figure 6.3). Intuitively it is expected that increasing the number of process streams
(and binary variables) would lead to an exponential increase in solution time.
Furman and Sahinidis [43] proved the minimum number of units sub-problem and the
simultaneous stage-wise synthesis of Yee and Grossmann [150] to be N P -hard in the
strong sense. While the complexity class of the stream match generator model with the
modified objective function was not evaluated, Furman and Sahinidis [43] showed that
even simple special cases of HENS were shown to be N P -hard in the strong sense.
This implies that a computationally efficient exact solution algorithm is highly unlikely
to exist for this problem.
140
7.3 Challenges
The stream match generator sub-problem takes much longer to solve than for the corresponding minimum number of units sub-problem. One of the reasons is the larger model
size due to the increased number of variables arising out of the larger number of temperature intervals. As discussed earlier, these temperature intervals are necessary to
ensure that a ranked sequence of HLDs are obtained from the stream match generator
sub-problem. Another possible reason is that the branching is not effective in the stream
match generator sub-problem because the binary variables are not part of the objective
function.
The solution time of the model for a given problem, with fixed utility targets and predefined temperature intervals, depends on the number of units H XU. It is observed that
model solution time is high for the absolute minimum number of units (H XU = Umin ).
Solution time for the model with H XU = Umin + 1 drops appreciably and then proceeds
to increase monotonically with the number of units. The increase in model solution time
with the number of units can be explained by the fact that the degree of freedom in the
model is increased thus opening up more match options. As far as the increased solution
time for the case when the number of units is set to the absolute minimum, this could be
due to the model being tightly constrained. Model solution times as a function of number
of heat exchanger units is shown for test problem 15TP1 in Figure 7.7
The stream match generator sub-problem and the minimum utilities sub-problem share
a common limitation of combinatorial explosion and the steps to alleviate this issue will
also be the similar. As discussed in Chapter 6, the three major ways to improve the
model solution time (by alleviating the combinatorial explosion problem) are:
1. Pre-processing to reduce model size using insight and heuristics
2. Model modification/reformulation
3. Improving efficiency of the Branch and Bound (B&B) method
7.3.1
Pre-processing
Fixing binary variables
The “combinatorial explosion” in the stream match generator sub-problem is caused by
the increasing number of binary variables in the model. Each binary variable, yi j in
141
7. STREAM MATCH GENERATOR SUB-PROBLEM
Figure 7.7: Solution times as a function of number of heat exchanger units in the stream
match generator model
Model R1 represents a potential match between a hot stream of utility i and a cold
stream or utility j. The binary variables can be fixed to either 0 or 1, indicating the
absence or presence of a match, in the pre-processing stage based on knowledge of the
problem. This effectively reduces the number of binary variables in the model.
All binary variables representing matches between hot process utilities and cold process
utilities are, by definition, set to zero.
yi j = 0 ∀ i ∈ HU, j ∈ CU
(7.3)
A match between a hot process stream or utility i and a cold process stream or utility j
is not thermodynamically feasible when T is − EM AT ≤ T sj + EM AT and yi j can be fixed
to zero. In the test problem 15TP1, the matches H4C7 and H8C7 are not thermodynamically feasible. The binary variables associated with these two matches can be set
to zero. The solution time is improved by 2% while the number of iterations is reduced
by 10% (see Table 7.7 - Fix binary variables (1)). Additionally, other binary variables
can be fixed to zero using heuristics. For example in test problem 15TP1, the matches
H1C7 and H3C7 can have a maximum duty of 525 kW while the H6C7 match can have a
142
7.3 Challenges
maximum duty of 612.5 kW for EMAT = 2.5, corresponding to 17%, 17% and 14% of the
total heat available in hot streams H1, H3 and H6. This corresponds to 10% and 11% of
the total heat required for stream C7. When the number of units is close to the absolute
minumum number of units, exchangers with small duties are not expected. Thus the binary variables representing the matches H1C7, H3C7 and H6C7 can be set to zero. The
improvements are similar in magnitude and are shown in Table 7.7 (Fix binary variables
(2)).
When a hot process stream can be cooled down to its target temperature by only one cold
process stream or utility, the match between this hot process stream and the cold process
stream or utility is a “required match” and the binary variable related to this match is
set to 1. Similarly, when a cold process stream can be heated up to its target temperature
by only one hot process stream or utility, the match between this cold process stream and
the hot process stream or utility is a “required match” and the binary variable related to
this match is set to 1.
Lower bound on objective value
The B&B algorithm tries to reduce the gap between lower bound evaluated by the algorithm and the best integer solution found thus far. The lower bounds and the best
integer solution are updated progressively. Setting a good lower bound based on physical understanding of the problem can help reduce this gap. The Bath formula [136] or
any of the other area targeting methods mentioned in the Introduction section can be
used to get a lower bound on the area. These methods provide a lower bound to area as
the area is calculated using HRAT rather than EMAT. As the more advanced methods
provided lower area targets than the simple Bath formula, the latter is used to set the
lower bound to the objective value of the stream match generator sub-problem.
The results with test problems show a marked increase in both solution times and iteration count. The lower bound of the objective function for 15TP1 test problem was
calculated to be 2680.65 m2 based on the Bath formula, while the actual objective calculated is 3354.48 m2 . The lower bound set corresponds to a gap of approximately 25%.
Values for the test problem 15TP1 is given in Table 7.7. The results were unexpected and
an understanding of the reason behind this should be explored as part of future work.
143
7. STREAM MATCH GENERATOR SUB-PROBLEM
7.3.2
Model modification
Model modification is an important option to improve model solution time. In this work,
sharpening the LP relaxation by decreasing the big M similar to what was done in Chapter 6 is considered in addition to adding integer cuts.
Based on results from the minimum number of units sub-problem, the value of U i j in
Constraint R1.3 is set based on thermodynamic information (temperatures and heat
capacity flow rates) and is given by:
(
U i j = min
X
QH
ik ,
k∈T I
X
Q Cjk , max
h
³
´ ³
´ i
C
H
C
min mC p H
,
mC
p
·
T
s
−
T
s
−
EM
AT
,0
i
i
j
j
k∈T I
)
(7.4)
Integer cuts are expected to be more important in the stream match generator subproblem as compared to the minimum units sub-problem as the cuts will add constraints
on the binary variables enabling better branching. Of the integers cuts discussed in
Chapter 6, compulsory matches and minimum matches per stream, only the compulsory
matches cuts are considered for the stream match generator sub-problem. For the sake
of clarity, the of compulsory matches integer cuts are reproduced below.
Compusory matches
This constraint specifies that at least one hot process stream or utility must heat each
cold process stream to its target temperature and vice versa for the hot process streams.
Defining sets M H
to be the set of hot process streams or utilities i that can heat a cold
j
process stream j to its target temperature and M iC the set of cold process streams or
utilities j that can cool a hot process stream i to its target temperature, we can define
the integer cuts as:
X
yi j ≥ 1
∀ i ∈ HP
(7.5a)
yi j ≥ 1
∀ j ∈ CP
(7.5b)
j ∈ M iC
X
i∈ M H
j
The results for modifications discussed above are given in Table 7.7. The updated U i j
results in a 30% improvement in solution time while the integer cuts did not show a
noticable improvement.
144
7.3 Challenges
Improvement (%)
Solution time (mins)
Iterations
Solution time
Iterations
120
16173213
Fix binary variables (1)
118
14521804
2
10
Fix binary variables (2)
116
11714409
3
28
Set objective LB
128
28315582
-7
-75
Updated U i j
83
15189095
31
6
Compulsory matches
119
15176509
1
6
Modified objective
115
15003594
4
7
Setting priorities to yi j
101
10362633
16
36
Table 7.7: Effect of various improvement measures for model solution time - Example
15TP1
Modifying the objective function
Improving branching characteristics of the model is important in ensuring better solution times. The objective function of the stream match generator sub-problem does not
include any binary variables. It is expected that modifying the objective to include the
binary variables will improve solution time by promoting better branching in the B&B
process. The objective function is modified to:
min z =
XXXX
i
j m n
Q im, jn
U i j · ∆T LM ,mn
+
XX
i
yi j
(7.6)
j
As the total number of heat exchanger units is fixed the term added to the objective
P P
function, i j yi j , is a constant. The HLDs generated by the stream match generator
sub-problem are not expected to be affected. Modifying the objective function improves
the solution time improves by 4%.
Based on the results from applying the three model modifications, the stream match
generator model is updated to include the modified objective and updated U i j values.
7.3.3
Improving efficiency of the B&B method
Setting priorities to binary variables defines how high up in the search tree these will be
used as branching variables. Thus it is possible, using priorities, to ensure that important variables are branched early in the process. This could potentially lead to searching
only the most promising part of the binary tree.
145
7. STREAM MATCH GENERATOR SUB-PROBLEM
Strategy for setting the priorities of variables used in this work is based on how much
heat can be exchanged between a pair of matches and the driving force between them.
The maximum heat that can be exchanged between a pair of streams is the modified
thermodynamic U i j given by Equation 7.4 and the driving force is ∆T LM . The idea
behind setting these priorities is based on the Pinch Design Method. In the Pinch Design Method network design is started at pinch where the driving force is minimum.
Similarly, in the stream match generator sub-problem, it is expected that matches with
smaller driving forces should be higher up as branching variables than matches of similar size but larger driving force. The ∆T LM for a potential match can be calculated based
on the temperature intervals defined for the model.
The matches are divided into five groups, based on the magnitude of U i j - a larger value
of U i j indicating a potentially larger match between the two streams. Bigger potential
matches are prioritized over smaller ones. Thus the five groups of U i j are ordered based
on decreasing value. Within each group of U i j , lower ∆T LM is prioritized over larger
values. Thus, within each group of U i j , the matches are prioritized based on increasing
value of ∆T LM . Adding priorities to binary variables reduces the solution time by 16%
for test problem 15TP1 as shown in Table 7.7.
Setting cutoff value to the objective function
CPLEX allows the user to set a cutoff value for the objective function such that parts
of the tree with an objective worse that the specified cutoff are deleted. Unlike the
minimum number of units sub-problem, where the optimum solution is reached very
early in the search, the optimum value in the stream match generator model is reached
after a significant time. This cutoff could potentially speed up the initial phase of the
algorithm weeding out unfruitful parts of the search tree.
The value of this cutoff can be an expected upper bound on the objective. The solution
from the minimum number of units problems gives HLDs for the absolute minimum
number of units. The area associated with such a network will be greater than networks
with a larger number of units. Using the HLD from the minimum number of units
sub-problem for the 15TP1 test problem, this upper bound for area was calculated to be
9218.83 m2 . This value is not of much use from a cutoff perspective as it is 300% greater
than the optimum objective value.
146
7.4 Conclusions and further work
Testing the effect of setting a cutoff value on solution time shows, as expected, model
solution times are greatly reduced when the cutoff value is close to the eventual objective
value. While different options for setting the cutoff were explored, no systematic method
gave a reasonable cutoff value. This is envisaged to be as part of future work.
7.4
Conclusions and further work
The stream match generator sub-problem is presented and a procedure for determining the temperature intervals for the MILP transportation model is established. The
importance and role of Exchanger Minimum Approach Temperature (EMAT) in achieving Heat Load Distributions (HLDs) with minimum area was identified and explained
along with the rationale for adding a new EMAT loop in the Sequential Framework.
The stream-match generator is computationally more expensive than the corresponding
minimum number of units sub-problem. Different strategies to reduce model solution
time were explored. The modifications reduce solution times by approximately 30% for
test problem 15TP1. Similar to the minimum number of units sub-problem, orders of
magnitude improvements are required to solve larger problems.
Further work
Unlike the minimum number of units sub-problem, where the optimum value is reached
early in the solution process, the optimum value in the stream match generator subproblem is not reached early in the solution process. Cutting off bad solutions by using
the Cutoff method of CPLEX showed to be beneficial. Developing good cut off values for
the objective using physical understanding of the model could potentially help reduce
solution times. Setting lower bounds on the objective function, contrary to expectation,
showed an increase in solution time. Further work could involve understanding and
setting appropriate bounds on the objective.
147
7. STREAM MATCH GENERATOR SUB-PROBLEM
148
8
Network Generation and
Optimization
This chapter presents the network generation and optimization sub-problem of the Sequential Framework [10, 11, 129].
8.1
Introduction
The final step in the Sequential Framework involves generating a heat exchanger network with minimum investment cost for a given set of Heat Load Distributions (HLDs)
obtained from the Stream Match Generator sub-problem discussed in Chapter 7. A superstructure of all feasible alternatives is required in synthesis problems using mathematical programming. Hwa [70] presented the first use of a superstructure for the heat
exchanger network synthesis problem by proposing that all the alternatives configurations can be considered systematically by including them in a processing scheme that
contains a finite number of units with all their possible interconnections.
Floudas et al. [40] proposed a superstructure that has embedded heat exchanger network configurations that satisfy the criterion of minimum utility cost and contains as
units the minimum number of matches predicted by the MILP transshipment model
proposed by Papoulias and Grossmann [105]. The synthesis strategy in the Sequential
Framework is similar to that proposed by Floudas et al. [40] with HLDs fixed prior to
the network generation. The difference between the strategies however is that the HLDs
generated in the Sequential Framework are for a given number of units rather than the
149
8. NETWORK GENERATION AND OPTIMIZATION
minimum number of units as in Floudas et al. [40]. A nonlinear programming (NLP)
formulation of the all-inclusive superstructure proposed by Floudas et al. is used in the
Sequential Framework for network generation and optimization.
While Floudas and Ciric [27, 38, 39] used the superstructure of Floudas et al. [40] for simultaneous synthesis, other superstructures have also been proposed for simultaneous
synthesis of HENS. Yuan et al. [152] proposed one of the first superstructures for simultaneous synthesis. Yee and Grossmann [150] proposed a stage-wise superstructure for
simultaneous synthesis that has been the basis for a lot of published literature related to
simultaneous HENS. Papalexandri and Pistikopoulos [104] extended the superstructure
of Ciric and Floudas [27] to allow for multiple matches between a pair of streams. Furman [45] proposed a formulation to encompass all possible network configurations while
Isafiade and Fraser [71] present an interval based superstructure. These superstructures developed for simultaneous synthesis are not suitable for use with a sequential
synthesis strategy.
8.2
Network generation and optimization model formulation
The network generation and optimization sub-problem in the Sequential Framework can
be defined as follows
- Given:
• a set H of hot process streams to be cooled and hot utilities HU ∈ H,
• a set C of cold process streams to be heated and cold utilities CU ∈ C,
• supply and target temperatures, heat capacities and flow rates of the hot and cold
process streams,
• temperatures or temperature ranges and fixed heat loads of the utilities, and
• a specified number of heat exchanger units and the heat load distribution.
- Obtain a heat exchanger network configuration that minimizes the investment cost.
150
8.2 Network generation and optimization model formulation
8.2.1
Superstructure
The network topology is extracted from the stream superstructure proposed by Floudas
et al. [40] where all possible network structures are included, given a specified number
of exchanger units and the respective stream matches. Floudas et al. [40] present the
method for derivation of the superstructure in detail with illustrative examples. However, for the sake of clarity, the salient aspects of the superstructure are repeated in this
section.
Each of the matches in the HLD from the stream match generator sub-problem represents a heat exchanger unit in the proposed superstructure. An independent superstructure is developed for each stream where all possible configurations for matches
in the stream are included. The interconnections, represented by flow rates and temperatures, between the matches are unknowns. The individual stream superstructures
are then combined into an overall exhaustive superstructure where matches between
streams provide the link between the individual stream superstructures. Each stream
superstructure consists of:
• An initial splitting point for the inlet stream where the process stream is split into
a number of branches that correspond to the number of matches for the actual
stream. Each split branch contains an exchanger related to the given match.
• Splitters at the outlet of each exchanger to enable recycle streams to the other
exchangers that the actual stream is involved in. One of these split streams is fed
to the final mixer for the outlet stream.
• Mixers at the inlet of each exchanger.
• A final mixing point for the outlet stream.
An example of a stream superstructure for a stream that has three matches is shown in
Figure 8.1.
To derive the superstructure of a hot or cold point utility, the utility stream is segregated
into a number of sub-streams equal to the number of matches for the utility. Each substream is then assigned a separate “simple” superstructure with one heat exchanger
associated with a match for the utility. The flow rate and temperatures of the utility
151
8. NETWORK GENERATION AND OPTIMIZATION
Figure 8.1: Stream superstructure for a stream with 3 matches
sub-streams are the same as that of the original utility. For non-point utilities, the
superstructure is derived similar to a process stream as detailed above.
The procedure for the derivation of the total superstructure for the given set of matches
in a particular subnetwork is:
1. Derive a process stream superstructure for each process stream.
2. Derive a utility stream superstructure for each utility. For point utilities it consists
of one match where the inlet and outlet temperatures of the utility stream are
those provided.
3. Define the total superstructure as the aggregate of all process stream and utility
stream superstructures. The heat loads in the exchangers of this network are given
by the heat exchange predicted by the stream match generator model.
As mentioned earlier, this superstructure allows all possible network structures to be included, given a specified number of exchanger units and the respective stream matches.
However, the drawback of this superstructure is that each stream is limited to exchanging heat with another stream only once. In other words cyclic matches are not possible.
Papalexandri and Pistikopoulos [104] extended this superstructure to allow for cyclic
matching of streams using “sub-networks”. In the Sequential Framework the number of
matches are fixed with only one match allowable between a pair of streams. This is a limitation in the framework and stems from the stream match generator sub-problem. Thus
152
8.2 Network generation and optimization model formulation
the drawback in the superstructure of not allowing multiple matches does not affect the
network generation and optimization sub-problem.
8.2.2
NLP formulation
The network generation and optimization sub-problem is formulated as a nonlinear programming (NLP) problem. The heat exchangers mixers and splitters are nodes in the
superstructure as shown in Figure 8.1 with the connections between these nodes (split
stream flows) representing arcs. The total number of nodes, including a start node and
an end node, in a stream superstructure for streams with more than one match is given
as:
Total nodes = 3 ∗ (No. of matches) + 4
For streams with more than one match the nodes in the stream superstructure are identified as follows:
• The start node and end node are nodes 1 and |Total Nodes|.
• The common splitter and common mixer nodes are 2 and |Total nodes -1|.
• The heat exchanger nodes are 4 + 3 ∗ (Match No.-1)
• Mixer nodes prior to heat exchangers are numbered as 3 + 3 ∗ (Match No.-1)
• Splitter nodes after heat exchangers are 5 + 3 ∗ (Match No.-1)
Let H be the set of all hot process streams and utilities, while C be the set of all cold
process streams and utilities. Let ST be the set of all streams w such that ST = H ∪ C.
s
t
The flow rates, supply and target temperatures for the streams are FC p w , T w
and T w
.
The set of matches provided from the stream match generator model are:
M A = {(i, j) |hot stream or utility i exchanges heat with cold stream or utility j, i ∈ H, j ∈ C }
A more generic set of matches are defined as:
M A = {(w, v) |process stream or utility w exchanges heat with process stream or utilityv, w, v ∈ ST }
It is clear from the definition of matches in set M A that w 6= v. The heat exchanged for
each match is Q wv . Let Nw be the set of nodes 1 to L w for the superstructure of stream
153
8. NETWORK GENERATION AND OPTIMIZATION
w. S w is the set of all splitter nodes in the superstructure for stream w, X w is the set
of all heat exchanger nodes in the superstructure for stream w and M w is the set of all
mixer nodes in the superstructure for stream w. S w , M w , X w ⊂ Nw . The set of arcs in the
superstructure for stream w is defined to be A w . Let Q wk be the heat exchanged in node
k of the superstructure for stream w and Uwv be the overall heat transfer coefficient for
heat exchange between streams w and v. A parameter, U TL w , is defined to identify if a
stream w is a utility stream or not. Thus
½
U TL w =
1
0
if w is a utility stream
if w is a process stream
Similarly, SI Nw is a parameter defined to identify if stream w has a only one match with
other streams, i.e. it has a one heat exchanger superstructure.
½
SI Nw =
1
0
if w matches with only one other stream
if w matches with more than one stream
The variables f wkl represent the heat capacity flow rate from node k to node l in the
superstructure for stream w. The variables t wkl represent the temperature in the arc
from node k to node l in the superstructure for stream w. Similarly the variables t vno
represent the temperature in the arc from node n to node o in the superstructure for
stream v. ar wv are variables representing the area required for heat exchange between
stream w and stream v, while ∆T LM ,wv is the variable for log mean temperature difference in the heat exchanger unit for the match between stream w and stream v.
The objective function for minimizing the investment cost is given by:
min
C
X
B wv · ar wvwv
(8.1)
( v , w )∈ M A
where B wv and C wv are cost coefficients and C wv < 1. The cost of heat exchanger unit
includes a fixed charge term (for example refer A.1 and A.2). This is neglected in the
objective function as the number of heat exchanger units are fixed when solving for the
minimum cost network in this sub-problem. The network generation and optimization
154
8.2 Network generation and optimization model formulation
model based on the superstructure and set defined can now be written as
Q wv
min
B wv ·
Uwv · ∆T LM ,wv
( v , w )∈ M A
µ
X
¶C wv
(N1)
s.t.
X
f wkl −
( k , l )∈ A w
X
X
f wlm = 0 ∀w, l ∈ Nw /{1, L w },U TL w = 0, SI Nw = 0
(N1.1)
( l , m )∈ A w
t wkl − t wlm = 0 ∀w, l ∈ S w , (k, l) ∈ A w , (l, m) ∈ A w , SI Nw = 0
(N1.2)
( f wkl · t wkl ) − f wlm · t wlm = 0 ∀w, l ∈ M w , (l, m) ∈ A w , SI Nw = 0
(N1.3)
( k , l )∈ A w
Q wk + f wkl · t wkl − f wlm · t wlm = 0
∀w, l ∈ H w , (k, l) ∈ A w , (l, m) ∈ A w , w ∈ X ,U TL w = 0, SI Nw = 0
(N1.4)
q¡
¢
¤
¢
1 £¡
2
t vop − t wkl (t vno − t wlm ) − · t vop − t wkl + (t vno − t wlm ) = 0
∆T LM ,wv − ·
3
6
∀(w, v) ∈ M A, l ∈ X w , o ∈ X v , (k, l) ∈ A w , (l, m) ∈ A w , (n, o) ∈ A v , (o, p) ∈ A v (N1.5)
t vop − t wkl ≥ EM AT (w, v) ∈ M A, l ∈ X w , o ∈ X v , (k, l) ∈ A w , (o, p) ∈ A v
(N1.6)
t vno − t wlm ≥ EM AT (w, v) ∈ M A, l ∈ X w , o ∈ X v , (l, m) ∈ A w , (n, o) ∈ A v
(N1.7)
Constraint N1.1 represents the mass balance for all nodes in the superstructure. Constraints N1.3 and N1.4 are the heat balances for mixer and heat exchanger nodes respectively while Constraint N1.2 sets the temperatures in the arcs associated with each splitter node. Constraints N1.6 and N1.7 ensure thermodynamic feasibility, where EMAT is
set to be 1.
The logarithmic mean temperature different for a heat exchanger in the superstructure,
∆T LM ,wv is given by:
¡
∆T LM ,wv =
¢
t vop − t wkl − (t vno − t wlm )
(8.2)
t vop − t wkl
ln t vno
− t wlm
³
´
Q
The area in Equation 8.1 is replaced by the Uwv ·∆TwvLM ,wv . Equation 8.2 is not suitable
for optimization applications due to numerical issues related to division by zero when
¡
¢
t vop − t wkl = (t vno − t wlm ). Paterson [107] and Chen [26] have developed approximations for the log mean temperature difference to overcome this numerical problem. Paterson’s approximation, based on the realization that the logarithmic mean is bounded
by the arithmetic and the geometric means, is used in the NLP model. It is modeled
as Constraint N1.5 in Problem N1. Paterson’s approximation tends to under-estimate
155
8. NETWORK GENERATION AND OPTIMIZATION
area while Chen’s approximation tends to over-estimate area [41]. Arithmetic mean and
geometric mean have also been used in HENS literature [109, 153] to replace the logarithmic mean temperature difference.
Floudas et al. [40] have proven the existence of a one-to-one correspondence between
the matches predicted by the stream match generator sub-problem and the units of a
feasible network embedded in the proposed superstructure.
Variable bounds
Lower and upper bounds for all the variables in the model, f wkl , t wkl , ∆T LM ,wv and
ar wv , are added as constraints in addition to the constraints N1.1 to N1.7. Good bounds
on variables are often a premise in order to obtain feasible solutions for nonlinear problems. Reasonable bounds may contribute to the exclusion of solution space that are
physically impossible, and will in addition make the search more efficient. However, in
some cases, tight variable bounds can make a problem harder to solve as the model becomes too constrained. Different routines implemented to put bounds on the variables
based on an understanding of the nature of the HENS problem are described below.
Temperature bounds
Initial Bounds
Updated Bounds
Stream
Stream Data
Ts
Tt
Lower
Upper
Lower
Upper
Lower
Final Bounds
Upper
H1
626
586
293
650
294
626
314
626
H2
620
519
293
650
294
620
390
620
H3
528
353
293
650
294
528
294
528
C1
497
613
293
650
497
649
497
649
C2
389
576
293
650
389
576
389
576
C3
326
386
293
650
326
386
326
386
C4
313
566
293
650
313
649
313
625
Table 8.1: Temperature bounds for 7TP1
The upper bounds on hot streams and lower bounds on cold streams are set to their respective supply temperature. In addition, information regarding heat exchange is used
to set the lower bound on hot temperature and upper bound on cold temperatures. The
maximum temperature of a cold stream is set to be the maximum temperature of any of
156
8.2 Network generation and optimization model formulation
the hot streams that exchanges heat with the cold stream −9/10 · EM AT. Similarly, the
lower bound on a hot stream is equal to the lowest temperature of any of the cold streams
that it exchanges heat with +/ − 9/10 · EM AT. Setting the bound to be +/ − EM AT rather
than +/ − 9/10 · EM AT leads to numerical difficulties in some test cases. It is expected
that this is due to constraining the problem by setting some of the values in Constraints
N1.6 and N1.7 to equalities. For streams with a single match, the target temperatures
are used to set these bounds. The temperature bounds on a stream originating from
the initial splitter node of any superstructure is set to the supply temperature of the
respective stream as it does not exchange heat at this stage. The column “Final Bounds”
in Table 8.1 presents the temperature bounds set as part of this work for test problem
7TP1. The column “Initial Bounds” shows crude bounds on the temperatures set by the
minimum and maximum temperatures possible in the system while the column “Updated Bounds” shows updated bounds by setting the upper and lower bounds of hot and
cold streams respectively to their supply temperatures. The table shows the process in
tightening temperature bounds with using progressively more insight from the problem.
Heat capacity flow rate bounds
Setting bounds for the heat capacity flow rate is more straight forward than for temperatures as there is not enough information available to set bounds based on physical
knowledge. The upper bound is set to the value from the stream data table and the lower
bound is set to zero.
Log mean temperature difference (∆T LM )
The lower bound on possible temperature difference at each end of the exchanger (from
Constraints N1.6 and N1.7) is EMAT. Using these values in Constraint N1.5, the lower
bound on log mean temperature difference is set to EMAT. No upper bound on the log
mean temperature difference is set.
Area
Only lower bounds on the area variables are required as the optimizer tries to minimize
the area. “Pseudo-area” for each match calculated in the stream match generator subproblem is used as a lower bound for area in the model P1.
157
8. NETWORK GENERATION AND OPTIMIZATION
8.3
Challenges
The network generation and optimization sub-problem formulated as an NLP model N1
and detailed in 8.2.2 is non-convex. This means that the problem may have several local
optima and unless the non-convexities are handled or a global solver is utilized, the final
solution depends on the starting point. The challenge here thus is that the quality of the
solution depends on the starting point. Further, NLP solvers fail to provide solutions
without a good starting point.
The model involves the following sources of non-convexities that may result in local optima:
1. Products of variable flow rates and temperatures in the heat balances for heat
exchanger and mixer nodes (Constraints N1.3 and N1.4).
2. Equations that define the log mean temperature differences used to calculate heat
transfer area(Constraint N1.5).
3. The economy of scale type cost equation that relates investment cost to the heat
transfer area (Objective model N1).
8.3.1
Causes of Local Optima
Non-convexities due to economy of scale type cost equation
Generally, power laws with an exponent lesser than one introduce non-convexities. In
the objective of model P1, the heat exchanger or match duties, Q w v, are fixed from the
stream match generator sub-problem and ∆T LM ,wv are variables in the model. The
∆T LM ,wv terms occur in the denominator and hence do not introduce non-convexities
in the model. Convexities of these terms can be shown by two times differentiation. This
is an advantage of the Sequential Framework compared to simultaneous HENS models
where the heat exchanger duties are unknown.
Non-convexities due to log mean temperature difference calculations
The Paterson equation for evaluating ∆T LM is:
µ
¶
2
1 θ1 + θ2
0.5
∆T LM = (θ1 · θ2 ) +
3
3
2
158
(8.3)
8.3 Challenges
where θ1 and θ2 are temperature differences in the hot and cold end of the heat exchanger respectively.
A multivariable function is convex if the eigenvalues of its Hessian matrix are convex
implying that the Hessian matrix is positive definite. For Equation 8.3, there are two
variables θ1 and θ2 . Floudas and Ciric [38] show that to prove both eigenvalues are
positive, it is sufficient to prove that the quantities C 1 and C 2 are positive, where:
C 1 = λ1 + λ2 = a 11 + a 12
(8.4)
C 2 = λ1 λ2 = a 11 a 22 − a212
(8.5)
a 11 , a 22 and a 12 are defined to be second order derivatives of ∆T LM with respect to θ1
and θ2 . Calculating these values we have:
à !0.5
1 θ2
a 11 = −
6 θ13
à !0.5
1 θ1
a 22 = −
6 θ23
µ
¶
1 0.5
1
a 12 =
6 θ1 θ2
(8.6)
(8.7)
(8.8)
From the above values it is clear that C 1 is always negative. This means that Equation
8.3 gives rise to non-convexities in the model.
Floudas and Ciric [38] prove that incorporating Equation 8.2 into the objective in the
model N1 leads to a convex objective function. Their work also mentions that this hold
true for the Paterson approximation as well. This non-convexity can thus be handled by
removing the variable ∆T LM ,wv from the model and incorporating Constraint N1.5 for
∆T LM ,wv directly in the objective.
Non-convexities due to heat balance constraints in the heat exchanger and
mixer nodes
The heat balance equations, Constraints N1.3 and N1.4, form constraints that are bilinear with different signs for flow rates and temperatures. This gives rise to nonconvexities in the constraints and thus the model N1.
159
8. NETWORK GENERATION AND OPTIMIZATION
Figure 8.2: Starting value generator implemented as part of SeqHENS
Floudas and Ciric [38] present an approach to global optimization of non-convex NLP
problems based on Generalized Benders Decomposition [48]. The variables set is decomposed into two sets - complicating and non-complicating variables - resulting in the
decomposition of the constraint set leading to two convex subproblems. A series of these
subproblems are solved to determine the global optimum. The heat capacity flow rates
are chosen as the complicating variables for the NLP formulation model N1.
Sahinidis and Grossmann [116] and Bagajewicz and Manousiouthakis [17] present some
the the problems associated with using the Generalized Benders Decomposition for NLPs
where it is shown that in certain cases the convergence is very slow [116] and, more importantly, the algorithm converges to the local optimum [17].
Another approach is the use of convex relaxations for the bilinear terms that cause the
non-convexities. Hashemi-Ahmady et al. [63] describe such relaxations that can be
used in conjunction with the Sequential Framework using the method of Al-Khayyal
and Falk [6]. Solution of the relaxed models provides a lower bound for the value of the
total investment cost, while evaluating the objective for known feasible network designs
(local solutions of the NLP model N1) provide the upper bound.
The former approach is the more desirable method as it mainly involves solving a set
of linear problems. It is less computationally expensive than the latter method, but is
certainly more expensive than the basic NLP formulation.
160
8.4 Starting Value Generators
8.4
Starting Value Generators
For the numerical solution of the NLP formulation, it is important to start with a “good”
initial guess for deriving the network configuration. This is particularly true for large
industrial sized problems where a good initial guess is a prerequisite for getting a solution, not to mention a globally optimum one. Multiple starting points allow the user to
explore the solution space, and in the case of a difficult problem, ensure a feasible solution. This section details five automated starting value generators developed for this
NLP formulation in an Excel/GAMS environment for Sequential Framework called SeqHENS (as discussed in Chapter 5). The guiding light has been to use physical insight to
ensure “good” local optima. The starting value generators are described and then summarized as regards their efficacy in solving a set of 10 heat load distributions from the
test problems.
Heat capacity flow rates and temperatures of the streams in the superstructure are the
optimizing variables, with heat capacity flow rates being identified as the decision variables that are in turn used to calculate the temperatures. Thus, the starting value
generators mainly involve setting the heat capacity flow rates.
8.4.1
Basic Serial/Parallel heuristic
This is a simple and very flexible method of setting the starting values. For each stream,
the user decides if the stream configuration should be pure serial or parallel. In case of
serial configuration, the user has a further choice on wether the sequence of matches are
in terms of increasing or decreasing heat exchanger loads as shown in Figure 8.3. This
method is not based on physical insight but provides the user with flexibility and hence
a large number of starting values.
8.4.2
Serial H/H Heuristic
As the name suggests, this starting value generator is based on the hottest/highest
heuristic proposed by Ponton and Donaldson [110]. This is based on the intuition and
engineering practice that hottest final cold stream temperature results from exchange
with hottest stream available. For a given set of matches (i, j) ∈ M A for a hot stream i,
the hot supply end of the stream is matched with a ranked set of cold stream matches
161
8. NETWORK GENERATION AND OPTIMIZATION
Figure 8.3: Serial/Parallel starting value generator
³ ´
such that cold stream j with max T tj is matched with hot stream i at the hot supply
end. This generator includes physical insight in the use of temperature driving forces
but does not consider the match duty and stream heat capacity flow rates.
8.4.3
Stream Match Generator based heuristic
This starting value generator uses results from the VertMILP model that generates
the HLDs. The temperature range (∆T) for the hot and cold streams of each match
(i, j) ∈ M A is available from the VertMILP model and the starting value generator
tries to replicate this temperature range in the stream superstructure (Figure 8.4. This
method is based on the initialization procedure described in Floudas et al. [40], and has
been modified based on the fact that there is no pinch decomposition in the Sequential
Framework.
8.4.4
Combinatorial heuristic
The combinatorial generator utilizes all stream and match information to generate a
feasible network as the starting point. The first step in this method is to allocate the
162
8.4 Starting Value Generators
Figure 8.4: VertMILP based generator
utilities - they are set to match with process streams at their target end and a new
modified target temperature is calculated. The next step is to check streams with one
match for feasibility with all other streams - exchange at the modified target end of the
process streams is only permitted. This ensures that the possibility of feasible exchanges
increases progressively. The next step is to check for feasible exchanges for hot streams
with multiple matches. This is also done similar to the earlier cases where a match is
allowed only at the target end of the cold stream. Once all hot streams are done, the
procedure involves looping through these steps (checking feasibility of all single match
streams and hot multiple match streams), as opportunities may have opened up for
fixing exchangers, until there are no more exchangers that can be fixed (Figure 8.5). The
remaining matches are set up, using split streams, as parallel exchangers where the
splits are calculated based on the temperature range and match duty. This procedure is
resource intensive, but ensures a feasible starting point.
8.4.5
Results
The starting value heuristics were tested on 10 heat load distributions from test problems 7TP1 (see A.1) and 15TP1 (see A.2). The combinatorial heuristic always ensured
that an optimum was found and this optimum was the lowest compared to runs with
163
8. NETWORK GENERATION AND OPTIMIZATION
Figure 8.5: VertMILP based generator
other starting values. The Stream Match Generator based heuristic and the parallel
heuristic performed second best and ensured that a feasible solution was found in 90%
of the cases, with the parallel configuration performing better for larger problems. The
Stream Match Generator based heuristic does not assign parallel heat exchangers and
for larger problems this is important short comming. The serial H/H heuristic produced
a feasible solution in 50% of the cases, while the pure serial configuration gave a feasible solution in only 10% of the cases. The results show a considerable difference in the
performance of the various starting value generators.
The combinatorial starting value heuristic was set to be the standard method for gener-
164
8.5 NLP solvers in the Sequential Framework
ating starting values in the SeqHENS. This can, however, be over-ridden by the user to
use the starting value generator of their choice.
8.5
NLP solvers in the Sequential Framework
The networks are generated using the combinatorial heuristic for starting value generators is solved in GAMS version 23.5 with CONOPT3 from ARKI Consulting and Development A/S as the NLP solver. For the problems tested CONOPT gave feasible solutions
in all cases while MINOS gave feasible solution in only 40% of the cases.
BARON [117] solver can be used to achieve global optimum solution. BARON uses MINOS as NLP solver and CPLEX as LP solver. BARON, because of its branch and bound
solution algorithm, is computationally intensive. Experience with BARON for the NLP
formulation of the network generation and optimization sub-problem however indicates
that it does not provide better solutions than those obtained using the starting value
generators detailed in this work.
8.6
Conclusion and further work
The network generation and optimization phase of the Sequential Framework is one of
the core sub-problems. The NLP formulation of the network generation and optimization sub-problem is presented with details as to the source of the non-convexities in
the model. It is seen from the discussion that the NLP formulation in the Sequential
Framework is much easier to solve than the MINLP formulations for HENS since the
non-convexities involved are substantially reduced. Non-convexity is still an issue for
this NLP formulation. Automated starting value generators based on physical insight
are developed to ensure that the base NLP formulation solves to a “good” local optimum.
Two methods of dealing with the non-convexities, Generalized Benders Decomposition
and convex relaxations are briefly presented. Both these methods require a good starting point, which is provided by the optimum found using the starting value generators.
Experience running the model for various examples show that the combinatorial starting
value generator, based on physical insight, gives very good initial starting values that
165
8. NETWORK GENERATION AND OPTIMIZATION
provides the global optimum in each of the cases that were tested. However, it does not
guarantee global optimum.
166
9
Conclusions and further work
9.1
Conclusions
A new energy integration methodology has been developed that is a synergy of Exergy
Analysis and Composite Curves. The Energy Level Composite Curves (ELCC), detailed
in Chapter 3, is a graphical tool which provides the engineer with insights on energy
integration and this work represents the first methodological attempt to represent thermal, mechanical and chemical energy in a graphical form similar to composite curves
for the integration of energy intensive processes. As pressure, temperature and composition changes are taken into account when developing the theory for this method, it can
be applied to a wide range of processes and in particular to energy intensive chemical
plants. A simple energy targeting algorithm is developed to obtain work, heating and
cooling targets. The ELCC was applied to a methanol plant to show the efficacy of the
methodology.
A review of published literature in Heat Exchanger Network Synthesis (HENS) between
2000 and 2008, presented in Chapter 4, shows that HENS as a research field has continued to be an active area of research in the new millennium. The number of journal
papers published in the period 200-2008 is a testimony to this. It has also attracted
researchers from many countries. There has been sustained interest in simultaneous
synthesis using mathematical programming, albeit for smaller test problems. Most of
the simultaneous synthesis references are based on the superstructure of Yee and Grossmann or a variant thereof. While most of the papers published were methodology oriented papers, over 25% of the papers were devoted to case studies. Most of the case
167
9. CONCLUSIONS AND FURTHER WORK
studies applied Pinch Analysis based evolutionary methods. Though there has been significant developments in HENS using mathematical programming methods, synthesis
of large scale HENS problems without simplifications and heuristics have been lacking.
This is an area that requires more research before mathematical programming based
approaches can be used in the industry.
The Sequential Framework for heat exchanger network synthesis, presented in Chapter
5, is a sequential and iterative framework with the main objective of finding near optimal
heat exchanger networks for industrial size problems. The Sequential Framework is a
compromise between Pinch Analysis and simultaneous MINLP methods. There are two
main advantages of the Sequential Framework:
1. The subtasks of the framework (MILP and NLP problems) are much easier to solve
numerically than the simultaneous MINLP models suggested for HENS.
2. The design procedure is, to a large extent, automated while keeping significant
user interaction. The design engineer acts as a top level optimizer making judgments based on quantitative as well as qualitative considerations.
Two test problems are solved using the Sequential Framework showing the ability to
generate networks with lower Total Annualized Costs compared to other solutions in
the literature. The Sequential Framework arrives at the best solution efficiently in a
small number of iterations, despite the four loops in the framework. This is due to the
fact that the search for good designs by exploring the loops of the framework will focus
on the most promising part of the feasible solution space; a result from using domain
knowledge in setting up the loop structure and initializing parameters.
The challenges in the Sequential Framework are
1. As the number of streams is increased in the Sequential Framework, the first
bottleneck occurs in the minimum number of units sub-problem, where the MILP
formulation is unable to handle large problems due to “combinatorial explosion”.
This is experienced in the stream match generator sub-problem as well.
2. The network generation and optimization NLP model is non-convex and global optimization methods have to be employed to this sub-problem. This, however, is not
dependent on the size of the HENS problem and is not a bottleneck with respect to
168
9.1 Conclusions
time.
The minimum number of units sub-problem was presented in Chapter 6 and the MILP
transshipment formulation of the sub-problem was discussed. It was shown that only
stream supply temperatures are required to set up temperature intervals in the formulation the of minimum number of units sub-problem. Improvements to the existing
minimum number of units sub-problem and a novel method for reformulating the model
to mitigate the combinatorial explosion associated with the MILP model were also presented. Results for large problems show that the proposed modifications strengthen the
LP relaxation, however, the model solution times remain too long to be of interest in
the Sequential Framework. At present, this methodology is able to solve problems with
around 20 streams, however, the computational resource required varies considerably
among problems of equal size. Orders of magnitude improvements are required to solve
larger problems. Model solution times depend on the the number of binary variables in
the model and on specific problem characteristics. To help identify problems that are
computationally expensive, the concept of a problem difficulty index was explored. From
the results of the various indices explored, it appears that the number of binary variables
is the defining criteria for problem difficulty.
Chapter 7 presented the stream match generator sub-problem and established a procedure for determining the temperature intervals for the MILP transportation model. The
importance and role of Exchanger Minimum Approach Temperature (EMAT) in achieving Heat Load Distributions (HLDs) with minimum area was identified and explained
along with the rationale for adding a new EMAT loop in the Sequential Framework.
The stream-match generator is computationally more expensive than the corresponding
minimum number of units sub-problem. Different strategies to reduce model solution
time were explored. The modifications reduce solution times by approximately 30% for
test problem 15TP1. Similar to the minimum number of units sub-problem, orders of
magnitude improvements are required to solve larger problems.
The NLP formulation of the network generation and optimization sub-problem is presented in Chapter 8 with details as to the sources of the non-convexities in the model.
The NLP formulation in the Sequential Framework is much easier to solve than the
MINLP formulations for HENS since the non-convexities involved are substantially reduced. Non-convexity is still an issue for this NLP formulation. Automated starting
169
9. CONCLUSIONS AND FURTHER WORK
value generators based on physical insights are developed to ensure that the base NLP
formulation solves to a “good” local optimum. Two methods of dealing with the nonconvexities, Generalized Benders Decomposition and convex relaxations are briefly presented. Both methods require a good starting point, which is provided by the optimum
found using the starting value generators. Experience running the model for various examples show that the combinatorial starting value generator, based on physical insight,
gives very good initial starting values that provides the global optimum in each of the
cases that were tested. However, it does not guarantee global optimum.
9.2
Further work
The Energy Level Composite Curves, detailed in Chapter 3, is in its early phase of development. Significant improvement is required to develop a complete systematic framework that incorporates thermal, mechanical and chemical energies. The methodology
developed thus far focuses only on the thermal and mechanical aspects - temperature
and pressure. Incorporating compositional changes in the form of chemical exergy is required to ensure that the entire chemical plant can be analyzed for energy integration.
The targeting methodology must be modified to take heat integration into consideration
while developing the work targets. An optimization scheme would be best suited for this.
The Sequential Framework that has been in development for quite a few years was taken
to the “next level” as part of this work by showing its relevance and competitiveness to
other heat exchanger network synthesis methods. However, the framework is limited by
combinatorial explosion issues due to binary variables in its two MILP sub-problems and
local optima caused by the non-convex NLP sub-problem. Further work in the Sequential
Framework should focus on mitigating these issues in the sub-problems.
Experience from running the minimum number of units models indicates that the optimum value is reached early in the solution process and most of the effort is expended
in proving optimality. This is the main reason for the focus on strengthening the LP relaxation as part of this work. Future work can involve identifying heuristics to stop the
search after an appropriate solution time. Identifying subnetworks by relaxing stream
temperatures and flow rates using a methodology similar to Shethna and Jez̆owski [122]
it may be possible to get a good initial bound on the minimum number of units. This
value could be used by CPLEX as the initial lower bound thus tightening the gap.
170
9.2 Further work
Since the 1990s, work has been done (e.g. Taylor et al. [132]) to show that all N P complete problems have a phase separation i.e. both hard and easy regions, with a
sharp boundary between them. On crossing that frontier, the problem undergoes a phase
transition, analogous to the boiling or freezing of water. Identifying the phase transition
of the minimum units problem would enable the user to decide on using deterministic
methods for solving it or other non-deterministic methods depending on which phase
the problem happens to be in. The location of the problem in the phase diagram would
indicate the its problem difficulty index.
Unlike the minimum number of units sub-problem, where the optimum value is reached
early in the solution process, the optimum value in the stream match generator subproblem is not reached early in the solution process. Cutting off bad solutions by using
the Cutoff method of CPLEX showed to be beneficial. Developing good cut off values for
the objective using physical understanding of the model could potentially help reduce
solution times. Setting lower bounds on the objective function, contrary to expectation,
showed an increase in solution time. Further work could involve understanding and
setting appropriate bounds on the objective.
Non-convex NLPs require good starting values to find even a local solution. The starting
value generators, in addition to providing starting values for ensuring a solution, generate very “good” solutions that are close to, if not, globally optimal. BARON can be used
to generate globally optimum solutions for the NLP model. No further work is envisaged
in the NLP sub-problem within the Sequential Framework.
171
9. CONCLUSIONS AND FURTHER WORK
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global MINLP optimization algorithm for the
each stage for best operation. Hydrocarbon
synthesis of heat exchanger networks with
Processing & Petroleum Refinery, 40(9):201–
no stream splits.
206, 1961. 45, 47
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182
Appendix A
Test Problems
This appendix presents stream data for the test problems used in the work.
A.1
7TP1
This test problem was first presented in Colberg and Morari [28]. The stream data has
been modified for consistency in units.
Ts
Tt
mC p
∆H
h
K
K
kW/K
kW
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
1.69
427.57
3.20
ST
650
650
-
-
3.50
CW
293
308
-
-
3.50
Stream
Exchanger cost ($) = 8,600 + 670A0.83 (A is in m2 )
HRAT = 20°C, QH ,min = 244.13 kW, QC ,min = 172.60 kW
183
A. TEST PROBLEMS
A.2
15TP1
This test problem was first presented in Björk and Nordman [22].
T in
T out
mC p
∆H
h
°C
°C
kW/°C
kW
kW/m2 °C
H1
180
75
30
3150
2
H2
280
120
60
9600
1
H3
180
75
30
3150
2
H4
140
40
30
3000
1
H5
220
120
50
5000
1
H6
180
55
35
4375
2
H7
200
60
30
4200
0.4
H8
120
40
100
8000
0.5
C1
40
230
20
3800
1
C2
100
220
60
7200
1
C3
40
290
35
8750
2
C4
50
290
30
7200
2
C5
50
250
60
12000
2
C6
90
190
50
5000
1
C7
160
250
60
5400
3
ST
325
325
CW
25
40
Stream
1
2
0.75
Exchanger cost ($) = 8,000 + 500A
(A is in m2 )
HRAT = 20.35°C, QH ,min = 11539.25 kW, QC ,min = 9164.25 kW
184
A.3 21TP1
A.3
21TP1
This test problem was first presented in Egeberg [34]
Stream
Ts
Tt
o
o
C
mCp
Q
C
kW/ o C
MW
H01
240.5
131.8
25.8
2804
H02
136.0
24.0
213.7
23934
H03
310.0
207.0
30.2
3111
H04
201.0
165.0
239.0
8604
H05
233.0
17.0
42.4
9158
H06
281.0
34.9
128.9
31722
H07
181.0
160.0
91.2
1915
H08
287.8
245.0
185.3
7931
H09
340.0
192.0
19.3
2856
H10
151.0
35.0
15.2
1763
H11
465.0
29.0
100.1
43644
C01
15.0
335.0
171.0
54720
C02
26.9
282.2
124.3
31734
C03
106.5
486.0
98.0
37191
C04
29.0
172.0
41.8
5977
C05
228.0
246.0
185.7
3343
C06
170.0
184.0
485.9
6803
C07
174.0
178.0
1983.8
7935
C08
150.0
153.0
3004.0
9012
C09
92.0
120.0
172.8
4838
C10
104.0
105.0
601.8
602
ST
496.0
496.0
CW
7.0
10.0
HRAT = 10°C, QH ,min = 38066.56 kW, QC ,min = 133355.16 kW
The hot utility, ST, has a very high temperature for a point utility like steam. This is
done to simplify the problem.
185
A. TEST PROBLEMS
A.4
21TP2
This test problem was first presented in Egeberg [34]
Stream
Ts
Tt
o
o
C
mCp
Q
C
kW/ o C
MW
H01
207.9
30.0
177.6
31.60
H02
207.9
30.0
177.6
31.60
H03
62.0
22.5
652.3
25.77
H04
62.0
40.6
850.8
18.21
H05
40.6
22.5
416.7
7.54
H06
-0.2
-10.5
1073.0
11.05
H07
35.0
30.1
6165.3
30.21
H08
30.4
-45.0
81.5
6.15
H09
38.3
21.6
1812.8
30.27
H10
21.9
8.0
20.9
0.29
H11
51.4
8.0
36.2
1.57
C01
2.0
75.2
241.1
17.65
C02
75.2
120.4
416.9
18.84
C03
2.0
75.2
241.1
17.65
C04
75.2
120.4
416.9
18.84
C05
206.1
226.0
1435.7
28.57
C06
206.1
226.2
1421.4
28.57
C07
87.0
88.7
13216.5
22.47
C08
-10.7
50.0
7.1
0.43
C09
82.9
82.9
579275.0
23.17
C10
51.0
51.2
135995.0
27.20
ST
236.2
236.2
R1
-55.0
-50.0
HRAT = 10°C, QH ,min = 125667.7 kW, QC ,min = 116524.3 kW
186
A.5 22TP1
A.5
22TP1
This test problem was first presented in Bagajewicz et. al. [18]
Stream
Ts
Tt
o
o
C
mCp
Q
C
kW/ o C
MW
H01
137.8
51.7
5.3
0.45
H02
160.0
51.7
10.1
1.10
H03
232.2
160.0
12.9
0.93
H04
187.8
87.8
9.6
0.96
H05
232.2
148.9
11.0
0.91
H06
160.0
93.3
21.1
1.41
H07
248.9
187.8
7.9
0.48
H08
154.4
93.3
10.5
0.64
H09
160.0
65.6
15.0
1.42
H10
232.2
115.6
13.2
1.54
H11
243.3
160.0
4.4
0.37
C01
37.8
115.6
6.9
0.54
C02
54.4
137.8
8.2
0.69
C03
65.6
148.9
14.9
1.24
C04
43.3
173.9
9.2
1.21
C05
173.9
215.6
18.5
0.77
C06
82.2
204.4
6.9
0.84
C07
173.9
260.0
22.6
1.94
C08
60.0
148.9
9.1
0.81
C09
85.0
173.9
6.3
0.56
C10
65.6
204.4
12.2
1.70
C11
176.7
260.0
19.8
1.65
ST
270
270
CW
30
60
HRAT = 5.6°C, QH ,min = 2144.96 kW, QC ,min = 422.91 kW
187
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