RESUMO Diferentes tipos de modelos matemáticos têm sido usados para prever o comportamento dinâmico do novo coronavírus (COVID-19). Muitos deles envolvem a formulação e solução de problemas inversos. Esse tipo de problema geralmente considera que o modelo, o vetor de variáveis de projeto e os parâmetros do sistema são todos valores determinísticos. Nesta contribuição, uma metodologia baseada em um processo de iteração de loop duplo é considerada para avaliar a influência de incertezas durante a resolução do problema inverso, proposto para estimar os parâmetros do modelo. No loop intermo, a solução associada ao maior valor de probabilidade associada as variáveis que apresentam incerteza é determinada. Já no loop externo, os valores das variáveis determinísticas é obtida. Para essa finalidade, a Análise de Confiabilidade Inversa é considerada como ferramenta de confiabilidade e o algoritmo de Evolução Diferencial é usado como ferramenta de otimização. Para fins de ilustração, a metodologia proposta é aplicada para estimar os parâmetros do modelo SIRD (SusceptívelInfeccioso-Recuperado-Morto) associado ao comportamento dinâmico da pandemia de COVID-19, considerando dados reais da epidemia da China e incertezas no número básico de reprodução (ℛ0). Os resultados obtidos demonstram que o aumento da confiabilidade implica no aumento do valor da função objetivo.

ABSTRACT Different types of mathematical models have been used to predict the dynamic behavior of the novel coronavirus (COVID-19). Many of them involve the formulation and solution of inverse problems. This kind of problem is generally carried out by considering the model, the vector of design variables, and system parameters as deterministic values. In this contribution, a methodology based on a double loop iteration process and devoted to evaluate the influence of uncertainties on inverse problem is evaluated. The inner optimization loop is used to find the solution associated with the highest probability value, and the outer loop is the regular optimization loop used to determine the vector of design variables. For this task, we use an inverse reliability approach and Differential Evolution algorithm. For illustration purposes, the proposed methodology is applied to estimate the parameters of SIRD (Susceptible-Infectious-RecoveryDead) model associated with dynamic behavior of COVID-19 pandemic considering real data from China’s epidemic and uncertainties in the basic reproduction number (ℛ0). The obtained results demonstrate, as expected, that the increase of reliability implies the increase of the objective function value.

RESUMEN Los problemas de optimización multimodal se encuentran comúnmente en problemas de ingeniería y su solución puede ser muy desafiante para los enfoques metaheurísticos. En este trabajo se analizó el uso de un método metaheurístico multimodal recientemente propuesto - el Algoritmo Multimodal de Polinización de la Flor - en problemas de equilibrio de fase fluida en dos etapas: (i) el cálculo de azeótropos dobles y (ii) la estimación de parámetros en un modelo termodinámico. También se consideran dos formulaciones diferentes en el problema de doble azeotropía. En el cálculo de azeótropo, se realizó un análisis estadístico para verificar si el desempeño del algoritmo se ve afectado por la formulación del problema. Los resultados computacionales indican que la metodología proporciona resultados robustos y que la función objetivo empleada afecta el rendimiento computacional.

ABSTRACT Multimodal optimization problems are commonly found in engineering problems, and their solution can be very challenging for metaheuristic approaches. In this work, the use of a recently proposed multimodal metaheuristic method was analyzed - the Multimodal Flower Pollination Algorithm - in two fluid phase equilibrium problems: (i) the calculation of double azeotropes and (ii) parameter estimation in a thermodynamic model. Two different formulations were also considered in the double azeotropy problem. In the azeotrope calculation, a statistical analysis was conducted in order to verify if the algorithm performance is affected by the the problem formulation. The computational results indicate that the methodology provides robust results and that the objective function employed affects the computational performance.

RESUMO Neste trabalho investigou-se a eficiência computacional na solução do problema de equilíbrio termodinâmico sólido-líquido (ESL) em óleos parafínicos. Foram avaliados diversos modelos termodinâmicos para a fase líquida: a equação de estado de Peng-Robinson (PR-EOS) e os modelos de coeficiente de atividade Wilson preditivo, UNIversal QUAsiChemical (UNIQUAC) preditivo e UNIQUAC Functionalgroup Activity Coefficient (UNIFAC), além do modelo de solução ideal. A fase sólida é caracterizada pelo modelo multisólido e a previsão de formação de fase sólida é inicialmente avaliada por um teste de estabilidade termodinâmica. Posteriormente, o sistema de equações não-lineares (SENL) formado pelas equações de equilíbrio termodinâmico e de balanço material é resolvido pelo método de Newton multivariável. Até onde pudemos verificar, não há estudos sistemáticos sobre o efeito do tipo de modelo (e sua complexidade) no tempo de computação necessário para a solução do problema. Por outro lado, tendo em conta o elevado número de componentes nas frações de petróleo estudadas, aspectos como a robustez (frente a diferentes estimativas iniciais) e tempo de computação são aspectos importantes neste contexto e que foram analisados neste trabalho.

ABSTRACT In this paper, the computational efficiency in the solid-liquid equilibrium (SLE) problem for paraffinic oils was examined. Several different thermodynamic models were evaluated for liquid phase: the Peng-Robinson equation of state (PR-EOS), and the activity coefficient models for ideal solution, the predictive Wilson model, the predictive Universal Quasi-chemical (UNIQUAC) model, and UNIQUAC Functional-group Activity Coefficient (UNIFAC) model. The solid phase is characterized by the multisolid model and the prediction of solid phase formation is initially estimated by a thermodynamic stability test. Afterwards, the systems of non-linear equations (SNLE) formed by the thermodynamic equilibrium and the material balance equations was solved using Newton-Raphson method (multivariate). As far as we can see, there are no systematic studies addressing the effect of the model (and its complexity) on the computational effort of the algorithm. On the other hand, considering the high number of different components in the oil fractions studied, the robustness (due to different initial estimates), and the computational time are important aspects and were analyzed in this work.

In this work, we present the calculation of critical coordinates for the esterification of acetic acid with ethanol in compressed carbon dioxide. Determination of the critical pressure for this system is useful, since the conversion of this reaction increases with pressure in the two-phase region, reaching a maximum at the critical point. We used a calculation framework based on a coordinate transformation for molar fractions, producing a new compositional domain. For a system with five components (acetic acid + ethanol + ethyl acetate + water + carbon dioxide) and one equilibrium reaction, the compositional domain is entirely described by three independent transformed coordinates. The results obtained were compared with experimental observations presented in the literature. The results illustrate the capability of the framework used to determine critical coordinates for reactive systems, and thus its usefulness as a tool for pressure tuning for this esterification reaction in compressed carbon dioxide.

Continuous Mixtures are mixtures of so many components that is no longer useful to distinguish among chemical species. Instead, an index is chosen to characterize each molecule, such as carbon atoms in each substructure, chromatographic retention time or normal boiling point and the continuity of this index is admitted. The present work aims the mathematical formulation and the simulation of a distillation column in which a continuous mixture is fed. This is performed with the assumption of a mixture of linear paraffins, in which ideal mixture hypothesis in both liquid and gas phases are reasonable. A quadrature method is employed to calculate the integrals which turns the functional-algebraic problem into a numerical one.

In this paper, we examine the resolution of some phase rule problems within the context of multiple chemical equilibrium reactions, using cubic equations of state and an activity coefficient model. Bubble and dew reactive surfaces, reactive azeotropic loci and reactive critical loci are generated and presented in graphical form. Also isobaric bubble and dew reactive enthalpy loci, which may be useful in the modeling of reactive distillation operations, are depicted. All the formalism here employed is developed within the coordinate transformation of Ung and Doherty, which is appropriate for equilibrium reactive or multireactive systems. The major contribution of this work is the determination of critical loci for reactive or multireactive equilibrium systems. Since it is known that for some class of chemical reactions the kinetics and product distribution exhibit high sensitivity to pressure near criticality, the present study may be useful as a predicting tool in these cases if the chemical equilibrium condition is not too far from the real phenomenon.