UNIVERSIDADE FEDERAL DE MINAS GERAIS MÁRCIO RIBEIRO VIANNA NETO CHEMICAL AND PHASE EQUILIBRIA THROUGH DETERMINISTIC GLOBAL OPTIMIZATION BELO HORIZONTE 2016 MÁRCIO RIBEIRO VIANNA NETO CHEMICAL AND PHASE EQUILIBRIA THROUGH DETERMINISTIC GLOBAL OPTIMIZATION Dissertação apresentada ao Curso de Pós-Graduação em Engenharia Química como parte dos requisitos exigidos para a obtenção do título de MESTRE EM ENGENHARIA QUÍMICA. Orientador: Prof. Dr. Éder Domingos de Oliveira Belo Horizonte 2016 ii To my parents, Marcos Vianna and Rita Vianna, to whom I owe all that is best in me. iii ACKNOWLEDGEMENTS I would like to thank my parents, Marcos and Rita, for all their love, patience and never-ending support. My good friend Bernardo who has always been there and supported all my endeavors. Also, my thanks to Professor Éder Domingos for his continuous support, for all his trust, for all the freedom that he has given me during my studies, and lastly, for his seemingly endless patience. My thanks to Professor Marcelo Cardoso for his friendship, for all the opportunities with which he has provided me and for his invaluable advice. My thanks to all my colleagues at the Industrial Processes Laboratory for being such an engaging, welcoming and supportive team. My dearest thanks to all the people who have indirectly contributed to this work, either by giving me bits of advice or by keeping me sane. My many thanks to Professor Ívina Paula for being my first chemistry teacher, and also Professor Miriam Miranda for encouraging me to do research. Lastly, I would like to thank CNPq for their financial support. iv “Valeu a pena? Tudo vale a pena Se a alma não é pequena Quem quer passar além do Bojador Tem que passar além da dor Deus ao mar o perigo e o abismo deu, Mas nele é que espelhou o céu” Fernando Pessoa v ABSTRACT Chemical and phase equilibrium calculations are commonly performed by solving a constrained optimization problem known as Gibbs energy minimization. This problem is, in general, nonconvex, which implies that it is not a trivial task to solve for its global minimum, as many local minima may exist. The global minimum is the only solution that bears physical significance. Among the various techniques found in the literature that attempt to solve this problem, the 훼퐵퐵 algorithm with interval analysis seems particularly interesting due to its generality and to the fact that it mathematically guarantees global optimality. However, in order to apply it directly to the equilibrium problem, it is necessary to circumvent somehow the fact that in its original formulation, lower bounds for mole numbers that are too close to zero may cause numerical underflow, leading the algorithm to fail. An algorithm based on the original 훼퐵퐵 is presented and is used to evaluate 8 benchmark equilibrium problems extracted form the literature. The algorithm, despite no longer being able to mathematically guarantee global optimality, was capable of solving all problems correctly and with relative efficiency. vi RESUMO Cálculos de equilíbrio químico e de fases são comumente realizados através da resolução de um problema de otimização restrita conhecido como minimização da energia de Gibbs. O problema é, em geral, não-convexo, o que faz com que a busca pelo mínimo global não seja trivial, já que podem existir vários mínimos locais. O mínimo global é a única solução que tem significado físico. Dentre as várias técnicas encontradas na literatura, o algoritmo 훼퐵퐵 com análise de intervalos parece ser particularmente interessante devido à sua generalidade e ao fato de que ele é capaz de garantir matematicamente que o mínimo encontrado será o mínimo global. Apesar disso, para que seja possível aplicá-lo diretamente ao problema de equilíbrio, é necessário contornar de alguma forma o fato de que, em sua formulação original, cotas inferiores muito próximas de zero podem causar underflow numérico, fazendo com que o algoritmo não seja bem-sucedido. Um algoritmo baseado no 훼퐵퐵 original é apresentado e usado para resolver 8 problemas-teste de equilíbrio extraídos da literatura. O algoritmo, apesar de não mais garantir matematicamente que o ótimo global é alcançado, foi capaz de corretamente resolver todos os problemas com relativa eficiência. vii TABLE OF CONTENTS Acknowledgements .......................................................................................................................iv Abstract .........................................................................................................................................vi Resumo ......................................................................................................................................... vii Table of contents ......................................................................................................................... viii List of figures ................................................................................................................................. xi List of tables ................................................................................................................................ xiv List of symbols .............................................................................................................................. xv 1 Introduction ........................................................................................................................ 18 2 Foundations ......................................................................................................................... 20 2.1 Mathematical definitions ............................................................................................ 20 2.1.1 Set theory and topology ...................................................................................... 20 2.1.2 Linear algebra ...................................................................................................... 20 2.1.3 Multivariable calculus ......................................................................................... 21 2.1.4 Convex analysis ................................................................................................... 21 2.2 Chemical and phase equilibria .................................................................................... 22 2.2.1 Problem formulation ........................................................................................... 22 2.2.2 Alternative formulations ..................................................................................... 25 2.2.3 Activity coefficient models .................................................................................. 29 2.3 Mathematical optimization ......................................................................................... 34 2.3.1 Basic definitions .................................................................................................. 34 2.3.2 Duality ................................................................................................................. 42 2.3.3 Conditions for optimality .................................................................................... 43 2.3.4 Interior point methods for constrained optimization ......................................... 44 2.4 Interval analysis ........................................................................................................... 45 3 Bibliographic review ............................................................................................................ 48 3.1 Grid-based methods .................................................................................................... 48 3.2 Linear programming-based methods .......................................................................... 51 3.3 Interior point methods ................................................................................................ 55 3.4 Global optimization methods ...................................................................................... 59 3.4.1 Stochastic approaches ......................................................................................... 59 3.4.2 Deterministic approaches ................................................................................... 60 4 Methodology ....................................................................................................................... 73 viii 4.1 Benchmark equilibrium problems ............................................................................... 73 4.2 Cost function and constraints ..................................................................................... 75 4.3 Algorithms and software architecture ........................................................................ 77 4.3.1 Solver class .......................................................................................................... 78 4.3.2 Interval class ........................................................................................................ 81 4.3.3 Vertex class .......................................................................................................... 82 4.3.4 PriorityQueue class.............................................................................................. 82 4.3.5 The solve method ................................................................................................ 83 4.4 Testing ......................................................................................................................... 85 5 Results and discussion ........................................................................................................