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 problems. AIChE Journal, 46(9):1769–1797, 2000. [R3] R. Adonyi, J. Romero, L. Puigjaner, and F. Friedler. Incorporating heat integration in batch process scheduling. Applied Thermal Engineering, 23(14):1743–1762, October 2003. [R4] A. Agarwal and S. K. Gupta. Multiobjective optimal design of heat exchanger networks using new adaptations of the elitist nondominated sorting genetic algorithm, NSGA-II. Industrial & Engineering Chemistry Research, 47(10):3489–3501, 2008. [R5] A. Akbari, M. R. Omidkhah, and M. R. Hojjati. Heat exchanger network area targeting considering stream allocation to shell or tubes. Computers & Chemical Engineering, 32(12):3143–3152, 2008. [R6] U. Akman, K. Uygun, D. Uztürk, and A. E. Konukman. HEN optimizations without using logarithmicmean-temperature difference. AIChE Journal, 48(3):596–606, 2002. [R7] B. A. Al-Riyami, J. Klemes, and S. Perry. Heat integration retrofit analysis of a heat exchanger network of a fluid catalytic cracking plant. Applied Thermal Engineering, 21(13-14):1449–1487, October 2001. [R8] H. Y. Alhammadi. A systematic procedure for optimizing crude oil distillation systems. In 18th European Symposium on Computer Aided Process Engineering, volume 25 of Computer Aided Chemical Engineering, pages 169–174. Elsevier, 2008. [R9] R. Anantharaman and T. Gundersen. Developments in the sequential framework for heat exchanger network synthesis of industrial size problems. In W. Marquardt and C. Pantelides, editors, 16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering, volume Volume 21, Part 1 of Computer Aided Chemical Engineering, pages 725–730. Elsevier, 2006. [R10] A. S. Ashaibani and I. M. Mujtaba. Minimisation of fuel energy wastage by improved heat exchanger network design - an industrial case study. Asia-Pacific Journal of Chemical Engineering, 2(6):575–584, 2007. 59 HENS BIBLIOGRAPHY 2000-2008 [R11] M. J. Bagajewicz and S. Ji. Rigorous procedure for the design of conventional atmospheric crude fractionation units. Part I: Targeting. Industrial & Engineering Chemistry Research, 40(2):617–626, 2001. [R12] M. J. Bagajewicz, S. Ji, and J. Soto. Heat exchanger networks for petroleum fractionation units handling crudes of different density. Latin American Applied Research, 31(4):339–343, 2001. [R13] M. J. Bagajewicz and H. Rodera. Energy savings in the total site heat integration across many plants. Computers & Chemical Engineering, 24(2-7):1237–1242, July 2000. [R14] M. J. Bagajewicz and H. Rodera. Corrigendum to "Energy savings in the total site. heat integration across many plants". Computers & Chemical Engineering, 25(2-3):493, March 2001. [R15] M. J. Bagajewicz and H. Rodera. Multiple plant heat integration in a total site. AIChE Journal, 48(10):2255–2270, 2002. [R16] M. J. Bagajewicz, H. Rodera, and M. Savelski. Energy efficient water utilization systems in process plants. Computers & Chemical Engineering, 26(1):59–79, 2002. [R17] M. J. Bagajewicz and J. Soto. Rigorous procedure for the design of conventional atmospheric crude fractionation units. Part II: heat exchanger network. Industrial & Engineering Chemistry Research, 40(2):627–634, 2001. [R18] A. Barbaro and M. J. Bagajewicz. New rigorous one-step MILP formulation for heat exchanger network synthesis. Computers & Chemical Engineering, 29(9):1945–1976, August 2005. [R19] A. Barbaro and M. J. Bagajewicz. Corrigendum to New rigorous one-step MILP formulation for heat exchanger network synthesis" [Comput. chem. eng. 29 (2005) 1945-1976]. Computers & Chemical Engineering, 30(8):1310–1313, June 2006. [R20] C. Bengtsson, R. Nordman, and T. Berntsson. Utilization of excess heat in the pulp and pa- per industry–a case study of technical and economic opportunities. Applied Thermal Engineering, 22(9):1069–1081, June 2002. [R21] M. L. Bergamini, I. E. Grossmann, N. Scenna, and P. Aguirre. An improved piecewise outer- approximation algorithm for the global optimization of MINLP models involving concave and bilinear terms. Computers & Chemical Engineering, 32(3):477–493, 2008. [R22] M. L. Bergamini, N. J. Scenna, and P. A. Aguirre. Global optimal structures of heat exchanger networks by piecewise relaxation. Industrial & Engineering Chemistry Research, 46(6):1752–1763, 2007. [R23] K. M. Björk, P. O. Lindberg, and T. Westerlund. Some convexifications in global optimization of problems containing signomial terms. Computers & Chemical Engineering, 27(5):669–679, May 2003. [R24] K. M. Björk and R. Nordman. Solving large-scale retrofit heat exchanger network synthesis problems with mathematical optimization methods. Chemical Engineering & Processing, 44(8):869–876, August 2005. 60 HENS BIBLIOGRAPHY 2000-2008 [R25] K. M. Björk and T. Westerlund. Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption. Computers & Chemical Engineering, 26(11):1581–1593, November 2002. [R26] M. Bogataj and M. J. Bagajewicz. Design of non-isothermal process water networks. In 17th European Symposium on Computer Aided Process Engineering, volume 24 of Computer Aided Chemical Engineering, pages 377–382. Elsevier, 2007. [R27] M. Bogataj and M. J. Bagajewicz. Synthesis of non-isothermal heat integrated water networks in chemical processes. Computers & Chemical Engineering, 32(12):3130–3142, 2008. [R28] M. Bozan, F. Borak, and I. Or. A computerized and integrated approach for heat exchanger network design in multipurpose batch plants. Chemical Engineering & Processing, 40(6):511–524, November 2001. [R29] I. Bulatov. Plate fin heat exchanger network retrofit procedure. International Journal of Heat Exchangers, 5(2):315–336, 2004. [R30] I. Bulatov. Retrofit optimization framework for compact heat exchangers. Heat Transfer Engineering, 26(5):4–14, 2005. [R31] M. Castier. Pinch analysis revisited: New rules for utility targeting. Applied Thermal Engineering, 27(8-9):1653–1656, June 2007. [R32] C. L. Chen and P. S. Hung. Multicriteria synthesis of flexible heat-exchanger networks with uncertain source-stream temperatures. Chemical Engineering & Processing, 44(1):89–100, 2005. [R33] D. Chen, S. Yang, X. Luo, Q. Wen, and H. Ma. An explicit solution for thermal calculation and synthesis of superstructure heat exchanger networks. Chinese Journal of Chemical Engineering, 15(2):296– 301, 2007. [R34] G. L. Chen and Y. J. Ciou. Design and optimization of indirect energy storage systems for batch process plants. Industrial & Engineering Chemistry Research, 47(14):4817–4829, 2008. [R35] X. Chen, Z. Li, J. Yang, Z. Shao, and L. Zhu. Nested tabu search (TS) and sequential quadratic programming (SQP) method, combined with adaptive model reformulation for heat exchanger network synthesis (HENS). Industrial & Engineering Chemistry Research, 47(7):2320–2330, 2008. [R36] K. Y. Cheung and C. W. Hui. Heat exchanger network optimization with discontinuous exchanger cost function. Applied Thermal Engineering, 21(13-14):1397–1405, October 2001. [R37] L. A. Cisternas, J. Cueto, and R. E. Swaney. Flowsheet synthesis of fractional crystallization processes with cake washing. Computers & Chemical Engineering, 28(5):613–623, May 2004. [R38] L. A. Cisternas, C. P. Guerrero, and R. E. Swaney. Separation system synthesis of fractional crystallization processes with heat integration. Computers & Chemical Engineering, 25(4-6):595–602, May 2001. [R39] S. W. A. Coetzee and T. Majozi. Steam system network synthesis using process integration. Industrial & Engineering Chemistry Research, 47(13):4405–4413, 2008. 61 HENS BIBLIOGRAPHY 2000-2008 [R40] J. Dipama, A. Teyssedou, and M. Sorin. Synthesis of heat exchanger networks using genetic algorithms. Applied Thermal Engineering, 28(14-15):1763–1773, 2008. [R41] M. Djaeni, P. Bartels, J. Sanders, G. van Straten, and A. J. B. van Boxtel. Process integration for food drying with air dehumidified by zeolites. Drying Technology, 25(1):225–239, 2007. [R42] H. G. Dong, C. Y. Lin, and C. T. Chang. Simultaneous optimization strategy for synthesizing heat exchanger networks with multi-stream mixers. Chemical Engineering Research & Design, 86(3):299– 309, 2008. [R43] R. F. Dunn and M. M. El-Halwagi. Process integration technology review: background and applications in the chemical process industry. Journal of Chemical Technology & Biotechnology, 78(9):1011– 1021, 2003. [R44] M. M. El-Halwagi. Process Integration, Volume 7. Academic Press, 1 edition, February 2006. [R45] M. Errico, S. Maccioni, G. Tola, and P. Zuddas. A deterministic algorithm for the synthesis of maximum energy recovery heat exchanger network. Computers & Chemical Engineering, 31(7):773–781, 2007. [R46] C. A. Floudas, I. G. Akrotirianakis, S. Caratzoulas, C. A. Meyer, and J. Kallrath. Global optimization in the 21st century: Advances and challenges. Computers & Chemical Engineering, 29(6):1185–1202, May 2005. [R47] D. C. Y. Foo, Y. H. Chew, and C. T. Lee. Minimum units targeting and network evolution for batch heat exchanger network. Applied Thermal Engineering, 28(16):2089–2099, 2008. [R48] E. S. Fraga, R. Patel, and G. W. A. Rowe. A visual representation of process heat exchange as a basis for user interaction and stochastic optimization. Chemical Engineering Research & Design, 79(7):765– 776, October 2001. 76 [R49] E. S. Fraga and A. Zilinskas. Evaluation of hybrid optimization methods for the optimal design of heat integrated distillation sequences. Advances in Engineering Software, 34(2):73–86, February 2003. [R50] S. Frausto-Hernández, V. Rico-Ramírez, A. Jiménez-Gutiérrez, and S. Hernández-Castro. MINLP synthesis of heat exchanger networks considering pressure drop effects. Computers & Chemical Engineering, 27(8-9):1143–1152, September 2003. [R51] K. C. Furman. Analytical investigations in Heat Exchanger Network Synthesis. PhD thesis, University of Illinois at Urbana-Champaign, 2002. [R52] K. C. Furman and N. V. Sahinidis. Computational complexity of heat exchanger network synthesis. Computers & Chemical Engineering, 25(9-10):1371–1390, September 2001. [R53] K. C. Furman and N. V. Sahinidis. A critical review and annotated bibliography for heat exchanger network synthesis in the 20th century. Industrial & Engineering Chemistry Research, 41(10):2335– 2370, May 2002. 62 HENS BIBLIOGRAPHY 2000-2008 [R54] K. C. Furman and N. V. Sahinidis. Approximation algorithms for the minimum number of matches problem in heat exchanger network synthesis. Industrial & Engineering Chemistry Research, 43(14):3554–3565, July 2004. [R55] M. Gadalla, Z. Olujic, L. Sun, A. De Rijke, and P. J. Jansens. Pinch Analysis-Based approach to conceptual design of internally Heat-Integrated distillation columns. Chemical Engineering Research & Design, 83(8):987–993, August 2005. [R56] M. R. Galli and J. Cerdá. Retrofit of heat exchanger networks with topology changes under designer control. Latin American Applied Research, 31(4):247–254, 2001. [R57] M. R. Galli and Jaime Cerdá. Synthesis of heat exchanger networks featuring a minimum number of constrained-size shells of 1-2 type. Applied Thermal Engineering, 20(15-16):1443–1467, October 2000. [R58] J. Geldermann, M. Treitz, and O. Rentz. Integrated technique assessment based on the pinch analysis approach for the design of production networks. European Journal of Operational Research, 171(3):1020–1032, June 2006. [R59] P. Glavič. Complex integration of processes. The Canadian Journal of Chemical Engineering, 79(4):643–654, 2001. [R60] M. Grabowski, J. Klemes, K. Urbaniec, G. Vaccari, and X. X. Zhu. Minimum energy consumption in sugar production by cooling crystallisation of concentrated raw juice. Applied Thermal Engineering, 21(13-14):1319–1329, October 2001. [R61] I. E. Grossmann, J. A. Caballero, and H. Yeomans. Advances in mathematical programming for the synthesis of process systems. Latin American Applied Research, 30(4):263–284, 2000. [R62] I. E. Grossmann, J. A. Caballero, and H. Yeomans. Mathematical programming approaches to the synthesis of chemical process systems. Korean Journal of Chemical Engineering, 17(SUPPL.):407– 426, 2000. [R63] N. Hallale. Burning bright - Trends in process integration. Chemical Engineering Progress, 97(7):30– 41, 2001. [R64] I. Heckl, F. Friedler, and L. T. Fan. Integrated synthesis of optimal separation and heat exchanger networks involving separations based on various properties. Heat Transfer Engineering, 26(5):25–41, 2005. [R65] A. Herrera, J. Islas, and A. Arriola. Pinch technology application in a hospital. Applied Thermal Engineering, 23(2):127–139, February 2003. [R66] K. Holiastos and V. Manousiouthakis. Minimum hot/cold/electric utility cost for heat exchange networks. Computers & Chemical Engineering, 26(1):3–16, 2002. [R67] A. J. Isafiade. Interval based MINLP superstructure synthesis of heat and mass exchange networks. PhD thesis, University of Cape Town, 2008. 63 HENS BIBLIOGRAPHY 2000-2008 [R68] A. J. Isafiade and D. Fraser. Optimization of combined heat and mass exchanger networks using pinch technology. Asia-Pacific Journal of Chemical Engineering, 2(6):554–565, 2007. [R69] A. J. Isafiade and D. M. Fraser. Interval-based MINLP superstructure synthesis of heat exchange networks. Chemical Engineering Research & Design, 86(3):245–257, 2008. [R70] J. M. Ježowski, R. Bochenek, and A. Jezowska. Pinch locations at heat capacity flow-rate disturbances of streams for minimum utility cost heat exchanger networks. Applied Thermal Engineering, 20(15-16):1481–1494, October 2000. [R71] J. M. Ježowski, R. Bochenek, and A. Jezowska. Loop breaking in heat exchanger networks by mathematical programming. Applied Thermal Engineering, 21(13-14):1429–1448, October 2001. [R72] J. M. Ježowski, R. Bochenek, and G. Poplewski. On application of stochastic optimization techniques to designing heat exchanger- and water networks. Chemical Engineering & Processing: Process Intensification, 46(11):1160–1174, 2007. [R73] J. M. Ježowski and A. Jezowska. Some remarks on heat exchanger networks targeting. Chemical Papers, 56(6):362–368, 2002. [R74] J. M. Ježowski, H. K. Shethna, R. J. Bochenek, and F. J. L. Castillo. On extensions of approaches for heat recovery calculations in integrated chemical process systems. Computers & Chemistry, 24(5):595–601, July 2000. [R75] J. M. Ježowski, H. K. Shethna, F. J. L. Castillo, and A. Jezowska. Novel optimization approaches for targeting heat exchanger networks. Chemical Papers, 55(6):369–375, 2001. [R76] J. M. Ježowski, Hiren K. Shethna, and Francisco J. L. Castillo. Area target for heat exchanger networks using linear programming. Industrial & Engineering Chemistry Research, 42(8):1723–1730, April 2003. [R77] S. Ji and M. J. Bagajewicz. Design of crude distillation plants with vacuum units. II. Heat exchanger network design. Industrial & Engineering Chemistry Research, 41(24):6100–6106, November 2002. [R78] Z. Jin, Q. Dong, and M. Liu. Heat exchanger network synthesis with detailed heat exchanger design. Chemical Engineering & Technology, 31(7):1046–1050, 2008. [R79] Z. L. Jin, Q. W. Dong, and M. S. Liu. Exergoeconomic analysis of heat exchanger networks for optimum minimum approach temperature. Chemical Engineering & Technology, 31(2):265–269, 2008. [R80] W. R. Johns. Computational complexity of heat exchanger networks (Furman & sahinidis). Computers & Chemical Engineering, 25(9-10):1391–1393, September 2001. [R81] B. Kalitventzeff and F. Maréchal. Optimal insertion of energy saving technologies in industrial processes: a web-based tool helps in developments and co-ordination of a european R&D project. Applied Thermal Engineering, 20(15-16):1347–1364, October 2000. [R82] P. O. Kapustenko, L. M. Ulyev, S. A. Boldyryev, and A. O. Garev. Integration of a heat pump into the heat supply system of a cheese production plant. Energy, 33(6):882–889, 2008. 64 HENS BIBLIOGRAPHY 2000-2008 [R83] I. C. Kemp. Pinch Analysis and Process Integration, Second Edition: A User Guide on Process Integration for the Efficient Use of Energy. Butterworth-Heinemann, 2 edition, April 2007. [R84] P. Kesavan, R. J. Allgor, E. P. Gatzke, and P. I. Barton. Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs. Mathematical Programming, 100(3):517–535, July 2004. [R85] S. Khan and C. Riverol. Performance of a pinch analysis for the process of recovery of ethanol from fermentation. Chemical Engineering & Technology, 30(10):1328–1339, 2007. [R86] M. Kijevčanin, B. Djordjević, O. Ocić, M. Crnomarković, M. Marić, and S. Šerbanović. Energy and economy savings in the process of methanol synthesis using pinch technology. Journal of the Serbian Chemical Society, 69(10):827–837, 2004. [R87] J. K. Kim and R. Smith. Cooling water system design. Chemical Engineering Science, 56(12):3641– 3658, June 2001. [R88] A. E. Konukman, M. Ca̧murdan, and U. Akman. Simultaneous flexibility targeting and synthesis of minimum-utility heat-exchanger networks with superstructure-based MILP formulation. Chemical Engineering & Processing, 41(6):501–518, July 2002. [R89] C. Koroneos and D. Rovas. A solar thermal power system in the city of Thessalonica with the use of the pinch method for entropy minimisation. International Journal of Exergy, 4(2):134–150, 2007. [R90] A. K. Kralj and P. Glavić. Simultaneous retrofit of complex and energy intensive processes-III. Computers & Chemical Engineering, 24(2-7):1229–1235, 2000. [R91] A. K. Kralj, P. Glavić, and Z. Kravanja. Retrofit of complex and energy intensive processes II: Stepwise simultaneous superstructural approach. Computers & Chemical Engineering, 24(1):125–138, April 2000. [R92] P. Krummenacher. Contribution to heat integration of batch processes (with or without heat storage). PhD thesis, Ecole Polytechnique Federale de Lausanne, 2001. [R93] H. M. S. Lababidi, Imad M. Alatiqi, and Lutfi J. Nayfeh. Energy retrofit study of an ammonia plant. Applied Thermal Engineering, 20(15-16):1495–1503, October 2000. [R94] R. Lakshmanan and E. S. Fraga. Pinch location and minimum temperature approach for discontinuous composite curves. Computers & Chemical Engineering, 26(6):779–783, June 2002. [R95] V. Lavric, V. Plesu, and J. De Ruyck. Chemical reactors energy integration through virtual heat exchangers–benefits and drawbacks. Applied Thermal Engineering, 25(7):1033–1044, May 2005. [R96] A. Lazzaretto and D. Daniele. Thermodynamic and pinch analyses for improving efficiency and structure of a CRCC plant with natural gas reforming and CO2 absorption. ASME Conference Proceedings, 2004(41723):199–210, 2004. [R97] A. Lazzaretto and F. Segato. Thermodynamic optimization of the HAT cycle plant Structure—Part i: Optimization of the “Basic plant configuration”. Journal of Engineering for Gas Turbines and Power, 123(1):1–7, 2001. 65 HENS BIBLIOGRAPHY 2000-2008 [R98] A. Lazzaretto and F. Segato. Thermodynamic optimization of the HAT cycle plant Structure—Part II: structure of the heat exchanger network. Journal of Engineering for Gas Turbines and Power, 123(1):8–16, 2001. [R99] A. Lazzaretto and F. Segato. A thermodynamic approach to the definition of the HAT cycle plant structure. Energy Conversion and Management, 43(9-12):1377–1391, 2002. [R100] S. Lee and I. E. Grossmann. A global optimization algorithm for nonconvex generalized disjunctive programming and applications to process systems. Computers & Chemical Engineering, 25(1112):1675–1697, November 2001. [R101] B. Leewongtanawit and J. K. Kim. Synthesis and optimisation of heat-integrated multiple- contaminant water systems. Chemical Engineering & Processing: Process Intensification, 47(4):670– 694, 2008. [R102] G. B. Leyland. Multi-objective optimisation applied to industrial energy problems. PhD thesis, Ec, 2002. [R103] S. Li and P. Yao. Synthesis of heat exchanger network considering multipass exchangers. Chinese Journal of Chemical Engineering, 9(3):242–246, 2001. [R104] Z. Li. Modeling and optimization for heat exchanger networks synthesis based on expert system and genetic algorithm. Chinese Journal of Chemical Engineering, 10(3):290–297, 2002. [R105] Z. H. Li and B. Hua. Modeling and optimizing for heat exchanger networks synthesis based on expert system and exergo–economic objective function. Computers & Chemical Engineering, 24(27):1223–1228, July 2000. [R106] Z. Liao, J. Wu, B. Jiang, J. Wang, and Y. Yang. Design energy efficient water utilization systems allowing operation split. Chinese Journal of Chemical Engineering, 16(1):16–20, 2008. [R107] B. Lin and D. C. Miller. Solving heat exchanger network synthesis problems with tabu search. Computers & Chemical Engineering, 28(8):1451–1464, July 2004. [R108] F. S. Liporace, F. L. P. Pessoa, and E. M. Queiroz. The influence of heat exchanger design on the synthesis of heat exchanger networks. Brazilian Journal of Chemical Engineering, 17(4):735–750, 2000. [R109] F. S. Liporace, F. L. P. Pessoa, and E. M. Queiroz. AtHENS (automatic heat exchanger network synthesis) performance. Latin American Applied Research, 31(5):383–390, 2001. [R110] F. S. Liporace, F. L. P. Pessoa, and E. M. Queiroz. An alternative procedure to retrofit an industrial plant. A case study. Latin American Applied Research, 32(2):161–170, 2002. [R111] L. M. F. Lona, F. A. N. Fernandes, M. C. Roque, and S. Rodrigues. Developing an educational software for heat exchangers and heat exchanger networks projects. Computers & Chemical Engineering, 24(2-7):1247–1251, July 2000. [R112] K. L. Ma, C. W. Hui, and T. F. Yee. Constant approach temperature model for HEN retrofit. Applied Thermal Engineering, 20(15-16):1505–1533, October 2000. 66 HENS BIBLIOGRAPHY 2000-2008 [R113] X. Ma, P. Yao, X. Luo, and W. Roetzel. Synthesis of multi-stream heat exchanger network for multi-period operation with genetic/simulated annealing algorithms. Applied Thermal Engineering, 28(8-9):809–823, 2008. [R114] M. Maiorano and E. Sciubba. Heat exchangers networks synthesis and optimization performed by an exergy-based expert assistant. International Journal of Applied Thermodynamics, 3(1):1–19, 2000. [R115] M. Markowski. Reconstruction of a heat exchanger network under industrial constraints – the case of a crude distillation unit. Applied Thermal Engineering, 20(15-16):1535–1544, October 2000. [R116] M. Markowski, M. Trafczynski, and K. Urbaniec. Energy expenditure in the thermal separation of hydrocarbon mixtures using a sequence of heat-integrated distillation columns. Applied Thermal Engineering, 27(7 SPEC. ISS.):1198–1204, 2007. [R117] A. Martín and F. A. Mato. Hint: An educational software for heat exchanger network design with the pinch method. Education for Chemical Engineers, 3(1), 2008. [R118] L. L. Martin and V. Manousiouthakis. Total annualized cost optimality properties of state space models for mass and heat exchanger networks. Chemical Engineering Science, 56(20):5835–5851, October 2001. [R119] L. Matijaševiæand H. Otmaéiæ. Energy recovery by pinch technology. Applied Thermal Engineering, 22(4):477–484, March 2002. [R120] R. K. C. Mehta, S. K. Devalkar, and S. Narasimhan. An optimization approach for evolutionary synthesis of heat exchanger networks. Chemical Engineering Research & Design, 79(2):143–150, March 2001. [R121] J. Mikkelsen and B. Qvale. A combinatorial method for the automatic generation of multiple, NearOptimal heat exchanger networks. Chemical Engineering Research & Design, 79(6):663–672, September 2001. [R122] F. T. Mizutani, F. L. P. Pessoa, E. M. Queiroz, S. Hauan, and I. E. Grossmann. Mathematical programming model for Heat-Exchanger network synthesis including detailed Heat-Exchanger designs. 2. network synthesis. Industrial & Engineering Chemistry Research, 42(17):4019–4027, 2003. [R123] T. Morosuk, C. Morosuk, and M. Feidt. New proposal in the thermodynamic analysis of complex heat regeneration systems. Energy, 29(12-15):2517–2535, 2004. [R124] W. Morton. Optimization of a heat exchanger network superstructure using nonlinear programming. Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering, 216(2):89–104, 2002. [R125] C. Mou and B. Qvale. The consequences of unpredictable development of economic conditions on heat exchanger network configurations and economic results. Energy Conversion and Management, 43(9-12):1549–1562, 2002. [R126] C .M. Vali no, Í. C. Irízar, A. A. Báscones, A. Heins, and S. Toccaceli. Improving crude unit heat integration. Petroleum Technology Quarterly, 13(2):107–111, 2008. 67 HENS BIBLIOGRAPHY 2000-2008 [R127] R. Nordman. New process integration methods for heat-saving retrofit projects in industrial systems. Ph.D., Chalmers Institute of Technology, 2005. [R128] R. Nordman and T. Berntsson. New pinch technology based hen analysis methodologies for costeffective retrofitting. The Canadian Journal of Chemical Engineering, 79(4):655–662, 2001. [R129] S. G. Oliveira, F.S. Liporace, O. Q. F. Araújo, and E. M. Queiroz. The importance of control considerations for heat exchanger network synthesis: A case study. Brazilian Journal of Chemical Engineering, 18(2):195–210, 2001. [R130] Semra Özkan and Salih Dinçer. Application for pinch design of heat exchanger networks by use of a computer code employing an improved problem algorithm table. Energy Conversion and Management, 42(18):2043–2051, December 2001. [R131] F. Palazzi, N. Autissier, F. M. A. Marechal, and D. Favrat. A methodology for thermo-economic modeling and optimization of solid oxide fuel cell systems. Applied Thermal Engineering, 27(16 SPEC. ISS.):2703–2712, 2007. [R132] M. H. Panjehshahi and M. M. Nouzari. Retrofit of heat exchanger networks considering existing structure: A new targeting procedure. Iranian Journal of Chemistry & Chemical Engineering, 20(1):44–52, 2001. [R133] M. H. Panjeshahi and A. Ataei. Application of an environmentally optimum cooling water system design to water and energy conservation. International Journal of Environmental Science and Technology, 5(2):251–262, 2008. [R134] M. H. Panjeshahi, E. G. Langeroudi, and N. Tahouni. Retrofit of ammonia plant for improving energy efficiency. Energy, 33(1):46–64, 2008. [R135] M. H. Panjeshahi and N. Tahouni. Pressure drop optimisation in debottlenecking of heat exchanger networks. Energy, 33(6):942–951, 2008. [R136] A. Pariyani, A. Gupta, and P. Ghosh. Design of heat exchanger networks using randomized algorithm. Computers & Chemical Engineering, 30(6-7):1046–1053, May 2006. [R137] F. Pettersson. Synthesis of large-scale heat exchanger networks using a sequential match reduction approach. Computers & Chemical Engineering, 29(5):993–1007, April 2005. [R138] F. Pettersson. Heat exchanger network design using geometric mean temperature difference. Computers & Chemical Engineering, 32(8):1726–1734, 2008. [R139] M. Phipps and A. Hoadleyf. Experiences from using heat integration software to determine retrofit opportunities within a refinery process. Korean Journal of Chemical Engineering, 20(4):642–648, July 2003. [R140] Z. N. Pintarič and P. Glavič. Direct enthalpy exchange between process utilities. Acta Chimica Slovenica, 49(2):387–400, 2002. [R141] Z. N. Pintarič and P. Glavič. Integration of flue gas into the process flowsheet by combined Pinch & MINLP approach. Chemical Engineering Research & Design, 80(6):606–614, September 2002. 68 HENS BIBLIOGRAPHY 2000-2008 [R142] Z. N. Pintarič and Z. Kravanja. A strategy for MINLP synthesis of flexible and operable processes. Computers & Chemical Engineering, 28(6-7):1105–1119, June 2004. [R143] G. T. Polley and M. Amidpour. Don’t let the retrofit pinch pinch you. Chemical Engineering Progress, 96(11):43–48, 2000. [R144] J. M. Ponce-Ortega, A. Jiménez-Gutiérrez, and I. E. Grossmann. exchanger networks involving isothermal process streams. Optimal synthesis of heat Computers & Chemical Engineering, 32(8):1918–1942, 2008. [R145] J. M. Ponce-Ortega, A. Jiménez-Gutiérrez, and I. E. Grossmann. Simultaneous retrofit and heat integration of chemical processes. Industrial & Engineering Chemistry Research, 47(15):5512–5528, 2008. [R146] J. M. Ponce-Ortega, M. Serna-González, and A. Jiménez-Gutiérrez. Heat exchanger network synthesis including detailed heat exchanger design using genetic algorithms. Industrial & Engineering Chemistry Research, 46(25):8767–8780, 2007. [R147] J. M. Ponce-Ortega, M. Serna-González, and A. Jiménez-Gutiérrez. MINLP synthesis of optimal cooling networks. Chemical Engineering Science, 62(21):5728–5735, 2007. [R148] J. M. Ponce-Ortega, M. Serna-González, and A. Jiménez-Gutiérrez. Synthesis of multipass heat exchanger networks using genetic algorithms. Computers & Chemical Engineering, 32(10):2320–2332, 2008. [R149] R. Pörn, K. M. Björk, and T. Westerlund. Global solution of optimization problems with signomial parts. Discrete Optimization, 5(1):108–120, February 2008. [R150] L. M. Pua and X. X. Zhu. Integrated heat exchanger network and equipment design using compact heat exchangers. Heat Transfer Engineering, 23(6):18–35, 2002. [R151] A. Querzoli, A. Hoadley, and T. Dyson. Identification of heat integration retrofit opportunities for crude distillation and residue cracking units. Korean Journal of Chemical Engineering, 20(4):635–641, July 2003. [R152] M. A. S. S. Ravagnani and J. A. Caballero. Optimal heat exchanger network synthesis with the detailed heat transfer equipment design. Computers & Chemical Engineering, 31(11):1432–1448, 2007. [R153] M. A. S. S. Ravagnani, A. P. da Silva, and A. L. Andrade. Detailed equipment design in heat exchanger networks synthesis and optimisation. Applied Thermal Engineering, 23(2):141–151, February 2003. [R154] M. A. S. S. Ravagnani, A. R. Righetto, and M. F. Marquini. Improving energetic performance and water usage in an industrial ethanol distillery. Process Safety and Environmental Protection, 85(6 B):526–532, 2007. [R155] M. A. S. S. Ravagnani, A. P. Silva, P. A. Arroyo, and A. A. Constantino. Heat exchanger network synthesis and optimisation using genetic algorithm. Applied Thermal Engineering, 25(7):1003–1017, May 2005. 69 HENS BIBLIOGRAPHY 2000-2008 [R156] Y. Ren. A recursive design method for heat exchanger networks. PhD thesis, Adelaide University, 2000. [R157] Y. Ren, B. K. O’Neill, and J. R. Roach. A recursive synthesis method for heat exchanger networks. i. the algorithm. Industrial & Engineering Chemistry Research, 40(4):1168–1175, February 2001. [R158] Y. Ren, B. K. O’Neill, and J. R. Roach. A recursive synthesis method for heat exchanger networks. II. case studies. Industrial & Engineering Chemistry Research, 40(4):1176–1185, February 2001. [R159] H. M. Reza, O. M. Reza, and P. S. M. Hassan. Cost effective heat exchanger network design with mixed materials of construction. Iranian Journal of Chemistry & Chemical Engineering, 23(2):89–100, 2004. [R160] H. Rodera and M. J. Bagajewicz. Multipurpose Heat-Exchanger networks for heat integration across plants. Industrial & Engineering Chemistry Research, 40(23):5585–5603, November 2001. [R161] H. Rodera and M. J. Bagajewicz. Targeting procedures for energy savings in the total site. Latin American Applied Research, 31(5):477–482, 2001. [R162] H. Rodera, M. J. Savelski, M. J. Bagajewicz, F. Hess, and T. Seidel. Energy retrofit with simultaneous optimization for a crude fractionation unit. Latin American Applied Research, 31(5):483–486, 2001. [R163] G. J. Roelant, A. J. Kemppainen, and D. R. Shonnard. Assessment of the automobile assembly paint process for energy, environmental, and economic improvement. Journal of Industrial Ecology, 8(1-2):173–191, 2004. [R164] M. C. Roque and L. M. F. Lona. The economics of the detailed design of heat exchanger networks using the bell delaware method. Computers & Chemical Engineering, 24(2-7):1349–1353, July 2000. [R165] J. De Ruyck, V. Lavric, D. Baetens, and V. Plesu. Broadening the capabilities of pinch analysis through virtual heat exchanger networks. Energy Conversion and Management, 44(14):2321–2329, August 2003. [R166] A. I. A. Salama. Heat exchanger network synthesis based on minimum rule variations. Applied Thermal Engineering, 28(10):1234–1249, 2008. [R167] A. I.A. Salama. Numerical techniques for determining heat energy targets in pinch analysis. Computers & Chemical Engineering, 29(8):1861–1866, 2005. [R168] L. C. Santos and R. J. Zemp. Energy and capital targets for constrained heat exchanger networks. Brazilian Journal of Chemical Engineering, 17(4):659–669, 2000. [R169] L. Savulescu, J. K. Kim, and R. Smith. Studies on simultaneous energy and water minimisation Part I: Systems with no water re-use. Chemical Engineering Science, 60(12):3279–3290, June 2005. [R170] L. Savulescu, J. K. Kim, and R. Smith. Studies on simultaneous energy and water minimisation - Part II: Systems with maximum re-use of water. Chemical Engineering Science, 60(12):3291–3308, June 2005. 70 HENS BIBLIOGRAPHY 2000-2008 [R171] L. E. Savulescu, M. Sorin, and R. Smith. Direct and indirect heat transfer in water network systems. Applied Thermal Engineering, 22(8):981–988, June 2002. [R172] K. Schack. Straightforward solutions of heat exchanger network (HEN) problems. Chemical Engineering & Technology, 25(11):1107–1114, 2002. [R173] E. Sciubba. Beyond pinch technology: Synthesis of heat exchanger networks performed by an expert design assistant based on second law reasoning. International Journal of Heat & Technology, 22(1):37–49, 2004. [R174] W. D. Seider, J. D. Seader, and D. R. Lewin. Product and Process Design Principles: Synthesis, Analysis, and Evaluation. Wiley, 2 edition, July 2003. [R175] M. Serna-González and A. Jiménez-Gutiérrez. An area targeting algorithm for the synthesis of heat exchanger networks. Chemical Engineering Science, 59(12):2517–2520, June 2004. [R176] M. Serna-González, A. Jiménez-Gutiérrez, and J. M. Ponce-Ortega. Targets for heat exchanger network synthesis with different heat transfer coefficients and non-uniform exchanger specifications. Chemical Engineering Research & Design, 85(10 A):1447–1457, 2007. [R177] M. Serna-González, J. M. Ponce-Ortega, and A. Jiménez-Gutiérrez. Two-Level optimization algorithm for heat exchanger networks including pressure drop considerations. Industrial & Engineering Chemistry Research, 43(21):6766–6773, October 2004. [R178] M. H. P. Shahi and A. Khoshgard. Heat exchanger networks targeting and design with unequal heat transfer coefficient regarding allowable pressure drop of streams. Heat Transfer Engineering, 27(9):36–43, 2006. [R179] P. Shahi, M. Hassan, M. Amidpour, and H. Ahmadi Danesh. Generalization of decomposed integration methods for cost effective heat exchanger networks with multiple cost laws. Iranian Journal of Chemistry & Chemical Engineering, 24(1):7–19, 2005. [R180] H. K. Shethna and J. M. Ježowski. Near independent subnetworks in heat exchanger network design. Industrial & Engineering Chemistry Research, 45(13):4629–4636, June 2006. [R181] H. K. Shethna, J. M. Ježowski, and F. J. L. Castillo. A new methodology for simultaneous optimization of capital and operating cost targets in heat exchanger network design. Applied Thermal Engineering, 20(15-16):1577–1587, October 2000. [R182] H. Shihn and R. K. Shah. Process integration and optimization of molten carbonate fuel cell system. ASME Conference Proceedings, 2003(36681):363–371, 2003. [R183] K. Shivakumar and S. Narasimhan. A robust and efficient NLP formulation using graph theoretic principles for synthesis of heat exchanger networks. Computers & Chemical Engineering, 26(11):1517–1532, November 2002. [R184] A. P. Silva, M. A. S. S. Ravagnani, and E. C. Biscaia Jr. Particle swarm optimisation in heat exchanger network synthesis including detailed equipment design. In 18th European Symposium on Computer Aided Process Eng, volume 25 of Computer Aided Chemical Engineering, pages 713–718. Elsevier, 2008. 71 HENS BIBLIOGRAPHY 2000-2008 [R185] M. L. Silva and R. J. Zemp. Retrofit of pressure drop constrained heat exchanger networks. Applied Thermal Engineering, 20(15-16):1469–1480, October 2000. [R186] H. Singh and F. Castillo. Process life cycle solutions for the case of automated heat exchanger network retrofit. Applied Thermal Engineering, 22(8):949–958, June 2002. [R187] R. Smith. State of the art in process integration. Applied Thermal Engineering, 20(15-16):1337– 1345, October 2000. [R188] R. M. Smith. Chemical Process: Design and Integration. Wiley, 2 sub edition, April 2005. [R189] M. Sorin and L. Savulescu. On minimization of the number of heat exchangers in water networks. Heat Transfer Engineering, 25(5 SPEC. ISS.):30–38, 2004. [R190] A. Sorsak and Z. Kravanja. Simultaneous MINLP synthesis of heat exchanger networks comprising different exchanger types. Computers & Chemical Engineering, 26(4-5):599–615, May 2002. [R191] A. Sorsak and Z. Kravanja. MINLP retrofit of heat exchanger networks comprising different exchanger types. Computers & Chemical Engineering, 28(1-2):235–251, 2004. [R192] L. Tantimuratha. Automated design of flexible and operable Heat Exchanger Networks. PhD thesis, University of Manchester Institute of Science and Technology, 2001. [R193] L. Tantimuratha, G. Asteris, D. K. Antonopoulos, and A. C. Kokossis. A conceptual programming approach for the design of flexible HENs. Computers & Chemical Engineering, 25(4-6):887–892, May 2001. [R194] L. Tantimuratha and A. C. Kokossis. Flexible energy management and heat exchanger network design. Annals of Operations Research, 132(1):277–300, November 2004. [R195] L. Tantimuratha, A. C. Kokossis, and F. U. Müller. The heat exchanger network design as a paradigm of technology integration. Applied Thermal Engineering, 20(15-16):1589–1605, October 2000. [R196] Z. G. Ugray. OQCRG: A Multi-Start Algorithm for Global Solution of Nonlinear and Mxed Integer Programs. PhD thesis, The University of Texas at Austin, 2001. [R197] S. Uhlenbruck, R. Vogel, and K. Lucas. Heat integration of batch processes. Chemical Engineering & Technology, 23(3):226–229, 2000. [R198] K. Urbaniec, P. Zalewski, and X. X. Zhu. Decomposition approach for retrofit design of energy systems in the sugar industry. Applied Thermal Engineering, 20(15):1431–1442, 2000. [R199] V. Václavek, A. Novotná, and J. Dedková. Pressure as a further parameter of composite curves in energy process integration. Applied Thermal Engineering, 23(14):1785–1795, October 2003. 33, 35 [R200] J. van Reisen. A structured approach to heat exchanger network retrofit design. PhD thesis, Technical University of Delft, 2008. [R201] P. Varbanov, S. Perry, Y. Makwana, X. X. Zhu, and R. Smith. Top-level analysis of site utility systems. Chemical Engineering Research & Design, 82(6):784–795, June 2004. 72 HENS BIBLIOGRAPHY 2000-2008 [R202] P. S. Varbanov and J. Klemes. Rules for paths construction for HENs debottlenecking. Applied Thermal Engineering, 20(15-16):1409–1420, October 2000. [R203] J. Varghese and S. Bandyopadhyay. Targeting for energy integration of multiple fired heaters. Industrial & Engineering Chemistry Research, 46(17):5631–5644, 2007. [R204] A. J. M. Vieira, F. L. P. Pessoa, and E. M. Queiroz. Fluid dynamical considerations on heat exchanger networks. Brazilian Journal of Chemical Engineering, 17(1):19–27, 2000. [R205] C. Wallmark and P. Alvfors. Design of stationary PEFC system configurations to meet heat and power demands. Journal of Power Sources, 106(1-2):83–92, April 2002. [R206] G. Wei, P. Yao, X. Luo, and W. Roetzel. A parallel genetic Algorithm/Simulated annealing algorithm for synthesizing multistream heat exchanger networks. Journal of the Chinese Institute of Chemical Engineers, 35(3):285–297, 2004. [R207] G. Wei, P. Yao, X. Luo, and W. Roetzel. Study on multi-stream heat exchanger network synthesis with parallel genetic/simulated annealing algorithm. Chinese Journal of Chemical Engineering, 12(1):66–77, 2004. [R208] Y. Wen and D. R. Shonnard. Environmental and economic assessments of heat exchanger networks for optimum minimum approach temperature. Computers & Chemical Engineering, 27(11):1577– 1590, November 2003. [R209] D. L. Westphalen and M. R. Wolf Maciel. Pinch analysis of evaporation systems. Brazilian Journal of Chemical Engineering, 17(4):525–537, 2000. [R210] U. Wising. Process integration in model kraft pulp mills technical, economic and environmental implications. PhD thesis, Chalmers Institute of Technology, 2003. [R211] G. Wu and X. X. Zhu. Retrofit of integrated refrigeration systems. Chemical Engineering Research & Design, 79(2):163–181, March 2001. [R212] W. Xiao, H. Dong, X. Li, P. Yao, X. Luo, and R. Wilfried. Synthesis of large-scale multistream heat exchanger networks based on stream pseudo temperature. Chinese Journal of Chemical Engineering, 14(5):574–583, 2006. [R213] W. Xiao, P.-J. Yao, X. Luo, and W. Roetzel. A new and efficient NLP formulation for synthesizing large scale multi-stream heat exchanger networks. Journal of the Chinese Institute of Chemical Engineers, 37(4):383–394, 2006. [R214] B. L. Yeap, D. I. Wilson, G. T. Polley, and S. J. Pugh. Retrofitting crude oil refinery heat exchanger networks to minimize fouling while maximizing heat recovery. Heat Transfer Engineering, 26(1):23– 34, 2005. [R215] K. M. Yerramsetty and C. V. S. Murty. Synthesis of cost-optimal heat exchanger networks using differential evolution. Computers & Chemical Engineering, 32(8):1861–1876, 2008. [R216] S. G. Yoon, J. Lee, and S. Park. Heat integration analysis for an industrial ethylbenzene plant using pinch analysis. Applied Thermal Engineering, 27(5-6):886–893, 2007. 73 HENS BIBLIOGRAPHY 2000-2008 [R217] B. R. Young, R. Tellez, and W. Y. Svrcek. Towards integrated process and control system synthesis for heat-integrated plants. Canadian Journal of Chemical Engineering, 84(2):219–229, 2006. [R218] H. Yu, H. Fang, P. Yao, and Y. Yuan. A combined genetic algorithm/simulated annealing algorithm for large scale system energy integration. Computers & Chemical Engineering, 24(8):2023– 2035, September 2000. [R219] J. Yuan, B. Zhu, R. K. Shah, and B. Sunden. Modeling and analysis of a Bio-Fuelled ceramic fuel cell stack. ASME Conference Proceedings, 2004(41650):453–459, 2004. [R220] J. Zhang and X. X. Zhu. Simultaneous optimization approach for heat exchanger network retrofit with process changes. Industrial & Engineering Chemistry Research, 39(12):4963–4973, December 2000. [R221] W. V. N. Zhang. Design of flexible heat exchanger network for multi-period operation. Chemical Engineering Science, 61(23):7754–7765, 2006. [R222] T. K. Zhelev. Water conservation through energy management. Journal of Cleaner Production, 13(15):1395–1404, December 2005. [R223] X. X. Zhu and X. R. Nie. Pressure drop considerations for heat exchanger network grassroots design. Computers & Chemical Engineering, 26(12):1661–1676, December 2002. [R224] X. X. Zhu and L. Vaideeswaran. Recent research development of process integration in analysis and optimisation of energy systems. Applied Thermal Engineering, 20(15-16):1381–1392, October 2000. [R225] X. X. Zhu, M. Zanfir, and J. Klemes. Heat transfer enhancement for heat exchanger network retrofit. Heat Transfer Engineering, 21(2):7–18, 2000. 74 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 75 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 76 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. 77 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. 79 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: Ã !0.5 1 θ2 a 11 = − 6 θ13 Ã !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 < ??table.caption.4568 table.caption.4568 table.caption.45, 69table.caption.4569 table.caption.45, 69table.caption.4569table.caption.4569 table.caption.45, 69table.caption.4569 table.caption.435, 121table.caption.435121table.caption.435121 table.caption.435, 121table.caption.435121table.caption.435121 172 networks. In J. Klemes (Editor), Chemical Engineering Transactions, volume 7 of PRES ’05, pages 67–72. Giardini Naxos, Italy, 2005. 75, 78, 57, 60 [10] Anantharaman, R. and Gundersen, T. Developments in the sequential framework References for heat exchanger network synthesis of industrial sized problems. In W. Marquardt and C. Pantelides (Editors), 16th European Symposium on Computer Aided Process [1] BP statistical review of world energy 2010. xi, Engineering 3 [2] Climate and 9th International Symposium on Process Systems Engineering, Change 2007 - IPCC volume 21A of Computer Aided Chemical fourth Engineering, pages 725–730. 2006. 75, 78, assesment report (AR4). xi, 1, 2, 4, 5 127, 149, 57, 60, 109, 131 [3] IEA Energy Technology Perspectives 2006. xi, [11] Anantharaman, R. and Gundersen, T. The 2, 4 sequential framework for heat exchanger network synthesis - network generation and [4] IEA World Energy Outlook 2007. 1 optimization. In J. Klemes (Editor), Chemical [5] Ahmad, S., Linnhoff, B. and Smith, R. Engineering Transactions, volume 12 of Cost optimum heat exchanger networks - 2. PRES ’07, pages 19–24. Ischia, Italy, 2007. 75, targets and design for detailed capital cost 78, 149, 57, 60, 131 models. Computers & Chemical Engineering, [12] Anantharaman, R., Nastad, I., Nygreen, 14(7):751–767, 1990. 127, 109 B. and Gundersen, T. [6] Al-Khayyal, F.A. and Falk, J.E. Constrained Biconvex Mathematics of Jointly framework Programming. Operations for heat The sequential exchanger network synthesis–the minimum number of units Research, sub-problem. 8(2):273–286, 1983. 160, 142 Engineering, Computers & Chemical 34(11):1822 – 1830, 2010. 93, 75 [7] Alves, J.J. Analysis and design of refinery hydrogen distribution systems. Ph.D. thesis, [13] Anantharaman, R., Sørås, O. and Gundersen, The University of Manchester, Department of T. Energy level composite curves - from theory Process Integration, 1999. 17 to practise. In Proceedings of ECOS 2005. 2005. Trondheim, Norway. 25 [8] Anantharaman, Gundersen, T. R., Abbas, O.S. and Energy level composite [14] Androulakis, I. and Venkatasubramanian, V. curves - a new graphical methodology for A genetic algorithmic framework for process the integration of energy intensive processes. design and optimization. Applied Thermal Engineering, 26(13):1378– Chemical Engineering, 15(4):217–228, 1991. 1384, 2006. 25 76, 58 [9] Anantharaman, R. and Gundersen, T. Computers & [15] Aspelund, A., Berstad, D.O. and Gundersen, Revisiting the sequential framework for T. near-optimal synthesis of heat exchanger procedure utilizing pressure based exergy 173 An extended pinch analysis and design REFERENCES for subambient cooling. Applied Thermal 54:519–539, 1999. 78, 128, 60, 110 [24] Cerdá, J., Westerberg, A.W., Mason, D. and [16] Athier, G., Floquet, P., Pibouleau, L. and Domenech, S. Chemical Engineering Science, complexity. Engineering, 27:2633–2649, 2007. 13, 18, 27 Linnhoff, B. Synthesis of heat-exchanger Minimum utilty usage in network by simulated annealing and NLP heat exchanger network synthesis. Chemical AIChE Journal, 43(11):3007– Engineering Science, 38(3):373–387, 1983. 48, procedures. 77, 131, 59, 113 3020, 1997. 49 [25] Cerdá, J. and Westerberg, A.W. Synthesizing [17] Bagajewicz, M. and Manousiouthakis, V. On the Generalized Computers & Benders Decomposition. heat exchanger networks having restricted Engineering, stream/stream matches using transportation Chemical problem formulations. Chemical Engineering 15(10):691–700, 1991. 160, 142 Science, 38(10):1723–1740, 1983. 48, 78, 80, [18] Bagajewicz, M.J., Pham, Manousiouthakis, V. R.P. and 94, 60, 62, 76 On the state space [26] Chen, J. approach to mass/heat exchanger network Chemical Engineering Science, design. Comments on improvements on a replacement for the logarithmic mean. Chemical Engineering Science, 42(10):2489– 53(14):2595–2621, 1998. 187, 169 2488, 1987. 155, 137 [19] Baldacci, R., Hadjiconstantinou, E., Maniezzo, V. and Mingozzi, A. A new [27] Ciric, A.R. and Floudas, C.A. Heat exchanger method location network synthesis without decomposition. for problems solving based on capacitated a set Computers partitioning & Chemical approach. Computers & Operations Research, 15(6):385–396, 1991. 29:365–386, 2002. 112, 94 132 Engineering, 49, 76, 150, 58, Process [28] Colberg, R.D. and Morari, M. Area and capital synthesis prospective. Computers & Chemical cost targets for heat exchanger network Engineering, 28:441–446, 2004. 13, 16 synthesis with constrained matches and [20] Barnicki, S.D. and Siirola, J.J. unequal heat transfer coefficients. Computers [21] Biegler, L.T., Grossmann, Westerberg, A.W. I.E. and & Chemical Engineering, 14(1):1–22, 1990. Systematic methods of 85, 86??, 87, 128, 183, 67, 68??, 69, 110, 165 chemical process deisgn. Prentice Hall, 1997. [29] Colbert, R.W. 46 networks. [22] Björk, K. and Nordman, R. scale retrofit synthesis heat problems Solving large- exchanger with Industrial heat exchange Chemical Engineering Progress, 78(7):47–54, 1982. 48 network [30] Desrochers, M., Desrosiers, J. and Solomon, mathematical optimization methods. Chemical Engineering M. & Processing, 44(8):869–876, 2005. 88, 89, vehicle routing problem with time windows. 184, 70, 71, 166 Operations Research, 40:342–354, 1992. 112, A new optimization algorithm for the 94 [23] Briones, V. and Kokossis, A.C. Hypertargets: [31] Dolan, W.B., Cummings, P.T. and LeVan, a conceptual programming approach for the optimization of industrial heat exchanger M.D. networks - i. grassroots design and network annealing: 174 Process optimization via simulated Application to network design. REFERENCES AIChE AIChE Journal, 35(5):725–736, 1989. 49, 76, exchanger network configurations. 58 Journal, 32(2):276–290, 1986. 49, 78, 80, 149, 150, 151, 156, 162, 60, 62, 131, 132, 133, 138, [32] Douglas, J.M. Conceptual design of chemical processes. 144 McGraw-Hill, New York, N.Y., [41] Floudas, C.A. Nonlinear and Mixed-Integer 1988. 13 Optimization. Oxford University Press, 1995. [33] Duran, M.A. Simultaneous and Grossmann, optimization I.E. and integration of chemical processes. 156, 138 heat [42] Fraga, E., Patel, R. and Rowe, G. A visual AIChE representation of process heat exchange as Journal, 32(1):123–138, 1986. 49 a basis for user interaction and stochastic [34] Egeberg, L.E. Efficient MILP models optimization. Chemical Engineering Research for design of heat exchanger networks. & Design, 79(7):765–776, 2001. 76, 58 Master’s thesis, Department of Energy and [43] Furman, Process Engineering, Norwegian University of Science and Technology, 1998. K.C. and Sahinidis, N.V. Computational complexity of heat exchanger (In network synthesis. Computers & Chemical Norwegian). 105, 185, 186, 87, 167, 168 Engineering, 25:1371–1390, 2001. 47, 49, 76, [35] El-Halwagi, V. M.M. and Manousiouthakis, 107, 133, 140, 58, 89, 115, 122 Synthesis of mass exchanger networks. [44] Furman, AIChE Journal, 35(8):1233–1244, 1989. 17 K.C. and Sahinidis, N.V. Approximation algorithms for the minimum [36] El-Halwagi, V. M.M. and Manousiouthakis, number Automatic synthesis of mass exchange of matches problem in networks with single component targets. heat Ind. Eng. exchanger network synthesis. Chem. Res., 43:3554–3565, 2004. 107, 89 Chemical Engineering Science, 45(9):2813– [45] Furman, K.C. 2831, 1990. 17 Analytical Investigations in Heat Exchanger Network Synthesis. [37] Feng, X. and Zhu, X.X. Combining pinch Ph.D. thesis, University of Illinois at Urbana- and exergy analysis for process modifications. Champaign, 2002. 150, 132 Applied Thermal Engineering, 17(3):249–261, [46] Furman, K.C. and Sahinidis, N.V. A critical 1997. 27, 29 review and annotated bibliography for heat [38] Floudas, C.A. and Ciric, A.R. Strategies for exchanger network synthesis in the 20th overcoming uncertainties in heat exchanger century. Industrial & Engineering Chemistry network synthesis. Computers & Chemical Research, 41(10):2335–2370, 2002. 45, 46, 47, Engineering, 13(10):1133–1152, 1989. 50, 56, 55 49, 150, 159, 160, 132, 141, 142 [47] Gaggioli, R.A., Sama, D.A., Qian, S. and El[39] Floudas, C.A. and Ciric, A.R. Corrigendum Sayed, Y.M. Integration of a new process - strategies for overcomming uncertainities into an existing site: A case study in the in application of exergy analysis. heat exchanger network synthesis. Journal of Computers & Chemical Engineering, 14(8):I, Engineering for Gas Turbines and Power, 1990. 49, 150, 132 113(2):170–180, 1991. 26 [48] Geoffrion, [40] Floudas, C.A., Ciric, A.R. and Grossmann, I.E. A.M. decomposition. Automatic synthesis of optimum heat 175 Generalized Benders Journal of Optimization REFERENCES Theory and Applications, Computers 10(4):237–260, & Chemical Engineering, 20(Suppl.):97–102, 1996. 75, 78, 57, 60 1972. 160, 142 [49] Gong, M. and Wall, G. On exergetics, [58] Gundersen, T. and Grossmann, I.E. Improved economics and optimization of technical optimization processes heat exchanger network synthesis through to meet environmental In International Conference on conditions. strategies for automated Computers & Chemical physical insights. Thermodynamic Analysis and Improvement Engineering, 14(9):925–944, 1990. of Energy Systems. 1997. 22 129, 130, 136, 57, 60, 111, 112, 118 [50] van Gool, W. The value of energy carriers. [59] Gundersen, T. and Naess, L. The synthesis Energy, 12(6):509–518, 1987. 29 of cost optimal heat exchanger networks an industrial review of the state of the [51] Grimes, L.E., Rychener, M.D. and Westerberg, art. A.W. The synthesis and evolution of networks Chemical Engineering [60] Gundersen, T., Sagli, B. and Kiste, K. Communications, 14:339–360, 1982. 94, 76 Problems in sequential and simultaneous strategies [52] Gross, J.L. and Yellen, J. Graph Theory and Its Applications. Chapman & Hall, 2005. I.E. and Westerberg, heat exchanger Computer-Oriented network Process Engineering, 1991. 76, 58 [61] Gundersen, T., Traedal, P. and Hashemi- A.W. Ahmady, A. Research challenges in process systems engineering. for synthesis. ISBN 158488505X. 94, 76 [53] Grossmann, Computers & Chemical Engineering, 12(6):503–530, 1988. 46 of heat exchange that feature the minimum number of units. 75, 78, for AIChE Journal, 69(9):1700– the Improved sequential strategy synthesis of near-optimal heat exchanger networks. Computers & Chemical 1703, 2000. 11, 12 Engineering, 21:S59–S64, 1997. 75, 78, 108, [54] Gundersen, T. 130, 57, 60, 90, 112 Achievements and future challenges in industrial design applications [62] Han, Z., Zhu, J., Rao, M. and Chuang, K.T. of process systems engineering. In PSE’91. Determination of independent loops in heat Quebec, Canada, 1991. 16 exchanger networks. Chemical Engineering [55] Gundersen, T. International Energy Agency. Implementating agreement on Communications, 164:191–204, 1998. 94, 76 process [63] Hashemi-Ahmady, A., Zamora, J.M. and integration. Annex I - A Process Integration Primer. SINTEF Energy Research, 2002. 16, Gundersen, T. 17 for optimal synthesis of industrial size heat A sequential framework exchanger networks. [56] Gundersen, T., Duvold, S. and Hashemi- In Proceedings from PRES ’99. 1999. 75, 78, 160, 57, 60, 142 Ahmady, A. An extended vertical MILP model for Heat Exchanger Network Synthesis. [64] Henry, J.E., Rudd, D.F. and Seader, J.D. Computers & Chemical Engineering, 20:S97– Synthesis in the design of chemical processes. S102, 1996. 130, 112 AIChE Journal, 19(1):1–15, 1973. 13 [57] Gundersen, T., Duvold, S. and Hashemi- [65] Hlavacek, V. Synthesis in the design of Ahmady, A. An extended vertical milp model chemical processes. Computers & Chemical for Engineering, 2(1):67–75, 1978. 13 heat exchanger network synthesis. 176 REFERENCES Optimum networks for [66] Hohmann, E.C. heat exchange. utilization diagram. Journal of Membrane Science, 24(3):271–283, 1985. 27, 28, 29 Ph.D. thesis, University of Southern California, 1971. 16, 48, 93, 127, [75] Jez̆owski, 75, 109 J. Heat exchanger network grassroot and retrofit design. the review of V. the state-of-the-art: Part i. heat exchanger Minimum hot/cold/electric utility cost for network targeting and insight based methods Computers and of synthesis. Hung. J. Ind. Chem., 22:279– [67] Holiastos, K. and Manousiouthakis, heat exchange networks. Chemical Engineering, 26(1):3–16, 2002. 27 [68] Homšak, M. and Glavič, P. 284, 1994. 46 [76] Jez̆owski, Pressure J. Heat exchanger network exchangers in pinch technology. Computers grassroot and retrofit design. the review of & Chemical Engineering, 20(6-7):711–715, the state-of-the-art: Part ii. heat exchanger 1996. 27 network synthesis by mathematical methods and approaches for retrofit design. Hung. J. [69] Huang, F. and Elshout, R. Optimizing the heat recovery of crude units. Ind. Chem., 22:295–308, 1994. 46 Chemical Engineering Progress, 72:68–74, 1976. 48 [77] Jez̆owski, J.M., Shethna, H.K. and Castillo, F.J.L. [70] Hwa, C.S. Mathematical formulation and optimization of heat exchanger networks using separable programming. Area target for heat exchanger networks using linear programming. Ind. Eng. Chem. Res., 42:1723–1730, 2003. xiv, In AIChE- 128, 136, 138??, 110, 118, 120??, 129 IChemE Symposium Series, volume 4. 1965. [78] Jez̆owski, J., Bochenek, R. and Jez̆owska, A. 45, 47, 149, 131 Loop breaking in heat exchanger networks [71] Isafiade, A.J. and Fraser, D.M. Interval-based exchange networks. Applied by mathematical programming. MINLP superstructure synthesis of heat Thermal Engineering, 21:1429–1448, 2001. Chemical Engineering 94, 76 Research and Design, 86(3):245–257, 2008. [79] Jones, S.A. 150, 132 Methods for the generation and evaluation of alternative heat exchanger [72] Isafiade, A. and Fraser, D. Interval-based networks. Ph.D. thesis, ETH Zürich, Zürich, MINLP superstructure synthesis of heat exchange networks. 1987. 49 Chemical Engineering Research and Design, 86:245–257, 2008. 86??, [80] Kesler, M.G. and Parker, R.O. networks of heat exchange. 68??, 69 Engineering [73] Ishida, M. and Kawamura, K. Energy and Progress Optimal Chemical Symposium Series, 65(92):111–120, 1969. 47 exergy analysis of a chemical process system [81] Kotas, T.J. with distributed parameters based on the The exergy method of thermal enthalpy-direction factor diagram. Industrial plant analysis. Krieger Publishing Company, & Engineering Chemistry Process Design and Malabar, Florida, 1995. 15, 22, 26 Development, 21(4):690–695, 1982. 27, 28 [82] Li, X. and Kraslawski, A. Conceptual process [74] Ishida, M. and Nakagawa, N. Exergy analysis synthesis: past and current trends. Chemical of a pervaporation system and its combination Engineering & Processing, 43(5):589–600, with a distillation column based on an energy 2004. 13 177 REFERENCES Optimal pinch [92] Linnhoff, B., Townsend, D.W., Boland, D., approach temperature in heat-exchanger Hewitt, G.F., Thomas, B.E.A., Guy, A.R. and [83] Li, Y. and Motard, R.L. Industrial and Engineering networks. Chemistry Fundamentals, A user guide on process Marsland, R.H. integration for the efficient use of energy. 25(4):577–581, IChemE, Rugby, UK, 1982. 13, 15, 25 1986. 48 [84] Linnhoff, B. New concepts in thermodynamics [93] Linnhoff, B. and Vredeveld, D.R. Pinch for better chemical process design. Chemical technology has come of age. Engineering Research and Design, 61:207– Engineering Progress, 80(7):33–40, 1984. 48 Chemical 223, 1983. 28 [94] Linnhoff, [85] Linnhoff, B. Pinch analysis - a state-of-the-art B., Mason, D.R. and Wardle, I. Understanding heat exchanger networks. overview. Chem. Eng. Res. Des., 71(A):503– Computers & Chemical Engineering, 3:295– 522, 1993. 75, 57 302, 1979. 16, 75, 94, 57, 76 [86] Linnhoff, B. and Ahmad, S. Cost optimum heat exchanger networks - 1. minimum [95] Masso, A.H. and Rudd, D.F. The synthesis of energy and capital using simple models systems design: Heuristic structuring. AIChE for capital cost. Journal, 15(10), 1969. 13, 45, 47 Computers & Chemical Engineering, 14(7):729–750, 1990. 81, 63 [96] McGalliard, R.L. and Westerberg, A.W. Structural sensitivity analysis in design [87] Linnhoff, B. and Alanis, F.J. Integration of a The Chemical Engineering new process into an existing site: A case study synthesis. in the application of pinch technology. Journal Journal, 4(2):127–138, 1972. 48 of Engineering for Gas Turbines and Power, [97] Mocsny, D. and Govind, R. 113(2):159–168, 1991. 26 Decomposition strategy for the synthesis of minimum-unit [88] Linnhoff, B. and Dhole, V.R. Shaftwork AIChE Journal, heat exchanger networks. targets for low-temperature process design. 30:853–856, 1984. 94, 76 Chemical Engineering Science, 47(8):2081– 2091, 1992. 27, 32 [98] Nastad, I. Design of heat exchanger networks. [89] Linnhoff, B. and Flower, J.R. Master’s thesis, Department of Industrial Synthesis Economics and Technology Management, of heat exchanger networks. I. Systematic generation of energy optimal Norwegian networks. AIChE Journal, 24:633–642, 1978. xiv, 16, 48, 136, 138??, 118, 120?? [90] Linnhoff, B. and Flower, J.R. University of Science and Technology, 2008. 93, 119, 75, 101, 102 [99] Nishida, N., Kobayashi, S. and Ichikawa, A. Synthesis of Optimal synthesis of heat exchange heat exchanger networks. II. Evolutionary systems Necessary conditions for minimum generation of networks with various criteria heat transfer area and their application to of optimality. systems synthesis. AIChE Journal, 24:642–654, Chemical Engineering Science, 26(11):1841–1856, 1971. 127, 109 1978. 16, 48 [100] Nishida, [91] Linnhoff, B. and Hindmarsh, E. The pinch N., Stephanopoulos, design method for heat exchanger networks. Westerberg, A.W. Chemical Engineering Science, 38(5):745– synthesis. 763, 1983. 48, 75, 94, 57, 76 1981. 13 178 G. and A review of process AIChE Journal, 27(3):321–351, REFERENCES [101] Nishimura, H. [109] Pettersson, F. A theory for the optimal Heat exchanger network synthesis of heat-exchanger systems. Journal design using geometric mean temperature of Optimization Theory and Applications, difference. 30(3):423–450, 1980. 127, 109 Engineering, 32(8):1734–1726, 2008. 156, 138 Computers & Chemical [102] Olsvik, O., Vada, S. and Hansen, R. Statoil’s [110] Ponton, J. and Donaldson, R. A fast method methanol plant in Tjelbergodden (Statoils for the synthesis of optimal heat exchanger methanolanlegg på Tjelbergodden). 29(12):2375–2377, 1974. 161, 143 57:42–44, 1997. 37 [103] Papadimitriou, C.H. and Steiglitz, [111] Rév, E. and Fonyó, Z. K. heat exchange networks. Hung. J. Ind. Chem, Dover Publications Inc., 2000. 14:181–201, 1986. 49 ISBN 0486402584. 106, 88 [104] Papalexandri, E.N. K.P. and Additional pinch phenomena providing improved synthesis of Combinatorial Optimization: Algorithms and Complexity. Chemical Engineering Science, networks. Kjemi, [112] Rév, E. and Fonyó, Z. Hidden and pseudo- Pistikopoulos, pinch phenomena and relaxation in the Synthesis and Retrofit Design of synthesis Operable Heat Exchanger Networks. 1. of Computers Flexibility and Structural Controllability heat-exchange & Chemical networks. Engineering, 10(6):601–607, 1986. 49 Aspects. Industrial & Engineering Chemistry Research, 33(7):1718–1737, 1994. 150, 152, [113] Rév, E. and Fonyó, Z. Diverse pinch concept 132, 134 for heat exchange network synthesis: the case of different heat transfer conditions. [105] Papoulias, S.A. and Grossmann, I.E. A Chemical Engineering Science, 46(7):1623– structural optimization approach in process 1634, 1991. 127, 109 synthesis - II. heat recovery networks. Computers & Engineering, [114] Rosen, M.A. Energy or exergy crisis? Exergy, 48, 77, 78, 79, 80, an International Journal, 2:125–127, 2002. 20 Chemical 7(6):707–721, 1983. 94, 95, 108, 111, 149, 59, 60, 61, 62, 76, 90, [115] Rudd, D.F. The synthesis of system designs, 93, 131 I: Elementary decomposition theory. AIChE Journal, 14(2):343–349, 1968. 13, 47 [106] Patel, B., Hildebrandt, D., Glasser, D. and Hausberger, B. Synthesis and integration of [116] Sahinidis, chemical processes from a mass, energy, and entropy perspective. & Engineering Chemistry Convergence Industrial benders Research, logarithmic mean. and Grossmann, properties decomposition. of I.E. generalized Computers & Chemical Engineering, 15(7):481–491, 1991. 46(25):8756–8766, 2007. 27 [107] Paterson, W.R. N.V. 160, 142 A replacement for the [117] Sahinidis, Chemical Engineering N.V. and Tawarmalani, M. Accelerating Branch-and-Bound through a Science, 39(11):1636–1635, 1984. 155, 137 Modeling Language Construct for RelaxationSpecific Constraints. [108] Pettersson, F. Synthesis of large-scale heat Journal of Global Optimization, 32(2):259–280, 2005. 165, 147 exchanger networks using a sequential match reduction approach. Computers & Chemical [118] Sama, D.A. The Use of the Second Law of Thermodynamics in Process Design. Journal Engineering, 29(5):993–1007, 2005. 49 179 REFERENCES of Energy Resources Technology, 117(3):179– [128] Staine, F. and Favrat, D. Energy integration of industrial processes based on the pinch 185, 1995. 22 analysis method extended to include exergy [119] Seider, W.D., Seader, J.D. and Lewin, factors. D.R. Product and Process Design Principles: Synthesis, Analysis, and Evaluation. Applied Thermal Engineering, 16(6):497–507, 1996. 27 John Wiley & Sons, New York, NY, 2nd edition, [129] Stokke, A. and Hilmersen, S.E. Design of 2003. 46 Heat Exchanger Networks - A Mathematical Programming Approach. [120] Serna, M. and Jiménez, A. An area targeting Department algorithm for the synthesis of heat exchanger networks. 149, 131 [121] Shethna, H.K., Jez̆owski, J.M. and Castillo, [130] Szargut, J. International progress in second F.J.L. A new methodology for simultaneous law analysis. Energy, 5(8-9):709–718, 1980. optimization of capital and operating cost 21 targets in heat exchanger network design. Applied Thermal Engineering, 20(15-16):1577 [131] Szargut, J., Morris, D.R. and Steward, – 1587, 2000. 128, 110 F.R. [122] Shethna, H.K. and Jez̆owski, J.M. Near 26 Chemistry Research, 45(13):4629–4636, 2006. [132] Taylor, W.M., Cheeseman, P., Cheeseman, 94, 125, 170, 76, 107, 152 P., Kanefsky, B. and Kanefsky, B. Where In In J. the really hard problems are. [123] Siirola, J.J., Powers, G.J. and Rudd, D.F. Mylopoulos and R. Reiter (Eds.), Proceedings Synthesis of system designs III: Towards a of 12th International Joint Conference on AI process concept generator. AIChE Journal, (IJCAI-91),Volume 1, pages 331–337. Morgan 17(3):677–682, 1971. 13 Kauffman, 1991. 125, 171, 107, 153 Chemical process design and [133] Ten Broeck, H. integration. Wiley, 2005. 12, 13, 46 exchanger sizes. [125] Sorin, M. and Hammache, A. model Hemisphere Publishing Corporation, New York, 1988. 22, Industrial & Engineering targeting on total sites. Exergy analysis of thermal, chemical and metallurgical processes. independent subnetworks in heat exchanger thermodynamic Economics University of Science and Technology, 2006. 59(12):2517–2520, 2004. 127, 109 [124] Smith, R. Master’s thesis, Industrial and Technology Management, Norwegian Chemical Engineering Science, network design. of for Economic selection of Industrial & Engineering Chemistry, 36(1):67–64, 1944. 45, 47 A new shaftwork [134] Tjoe, T.N. and Linnhoff, B. Applied Thermal Using pinch technology for process retrofit. Engineering, 25(7):961–972, 2005. 32 Chemical Engineering (New York), 93(8):47–60, 1986. 48 [126] Sorin, M., Hammache, A. and Diallo, O. Exergy based approach for process synthesis. [135] Towler, G.P., Mann, R., Serriere, A.J. Energy, 25(2):105–129, 2000. 15, 26 and Gabaude, C.M.D. [127] Sorin, M. and Paris, J. management: Combined exergy Refinery hydrogen cost analysis of chemicallyIndustrial & and pinch approach to process analysis. integrated Computers Engineering Chemistry Research, 35(7):2378– & Chemical Engineering, facilities. 2388, 1996. 17 21(Supplement 1):S23–S28, 1997. 27 180 REFERENCES [146] Westerberg, [136] Townsend, D.W. and Linnhoff, B. Surface A.W. Review of process area targets for heat exchanger networks. synthesis. In 11th Annual Research Meeting. IChemE, (Editors), Computer applications to chemical Bath, UK, 1984. 127, 143, 109, 125 engineering, number 124 in ACS Symposium [137] Tsatsaronis, nomenclature G. Definitions in exergy exergoeconomics. analysis Series, pages 53–87. 1980. 13 and and [147] Westerberg, A.W. A retrospective on design Energy, 32(4):249–253, and process synthesis. Computers & Chemical 2007. 21, 22 [138] Umeda, Engineering, 28:447–458, 2004. 13 T. Computer aided process Computers synthesis. & [148] Westerberg, A.W. and Shah, J.V. Assuring Chemical a global optimum by the use of an upper Engineering, 7(4):279–309, 1983. 13 [139] Umeda, K. T., Harada, T. and bound on the lower (dual) bound. Computers & Chemical Engineering, 2(2-3):83–92, 1978. Shiroko, 48 A thermodynamic approach to the synthesis of heat integration systems in [149] Williams, chemical processes. Computers & Chemical T., Itoh, Model H.P. Mathematical Programming. Engineering, 3:273–282, 1979. 16, 26 [140] Umeda, In R. Squires and G. Reklaitis Building & Sons, 4th edition edition, 1999. J. and Shiroko, in John Wiley ISBN 0471997889. 105, 87 K. Heat exchange system synthesis. Chemical [150] Yee, T.F. and Grossmann, I.E. Simultaneous Engineering Progress, 74:70–76, 1978. 16, 26, optimization models for heat integration 48, 130, 112 - II. heat exchanger network synthesis. [141] Vàclavek, V., Novotnà, A. and Dedkovà, J. Computers & Chemical Engineering, Pressure as a further parameter of composite 14(10):1165–1184, 1990. curves in energy process integration. Applied 85, 86??, 87, 140, 150, 58, 67, 68??, 69, 122, Thermal 132 Engineering, 23(14):1785–1795, 49, 55, 56, 76, 2003. 33, 35 [151] Yee, T., Grossmann, I. and Kravanja, Z. [142] Wall, G. and Gong, M. On exergy and Simultaneous optimization models for heat sustainable development -Part 1: Conditions integration - I. Area and energy targeting Exergy, An International and modeling of multi-stream exchangers. and concepts. Journal, 1(3):128–145, 2001. xiii, 22 [143] Wang, Y.P. and Smith, R. Computers Design of distributed effluent treatment systems. Chemical Engineering, [152] Yuan, X., Pibouleau, L. and Domenech, S. Chemical Engineering Science, 49(18):3127– Experiments in process synthesis via mixed- 3145, 1994. 17 integer programming. Chemical Engineering [144] Wang, Y.P. and Smith, R. and Processing, 25(2):99–116, 1989. 49, 150, Wastewater 132 minimization. Chemical Engineering Science, 49(7):981–1006, 1994. 17 [145] Westbrook, G.T. & 14(10):1151–1164, 1990. 128, 110 [153] Zamora, J.M. and Grossmann, I.E. A Use this method to size 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 Engineering, 22(3):384–367, 1998. 156, 138 181 Computers & Chemical REFERENCES Chemical [154] Zhu, X.X. Automated design method for heat Engineering Communications, 126:141–153, 1993. 94, 76 exchanger network using block decomposition and heuristic rules. Computers & Chemical Engineering, 21(10):1095–1104, 1997. [156] Zhu, X., O’neill, B., Roach, J. and Wood, 78, R. 128, 60, 110 Area-targeting methods for the direct synthesis of heat exchanger networks with unequal film coefficients. [155] Zhu, X.X., O’Neill, B.K., Roach, J.R. and Computers & Wood, R.M. Kirchhoff’s law and loop-breaking Chemical Engineering, 19(2):223–239, 1995. for the design of heat exchanger networks. 128, 110 182 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