Multiphase Flow Dynamics: A Perspective from the Brazilian Academy and Industry (Lecture Notes in Mechanical Engineering) 3030934551, 9783030934552

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ABCM Series on Mechanical Sciences and Engineering

Marcio Ferreira Martins Rogério Ramos Humberto Belich   Editors

Multiphase Flow Dynamics A Perspective from the Brazilian Academy and Industry

Lecture Notes in Mechanical Engineering

ABCM Series on Mechanical Sciences and Engineering Series Editors Ricardo Diego Torres, Rio de Janeiro, Rio de Janeiro, Brazil Marcello Augusto Faraco de Medeiros, Departamento de Engenharia de Materiais, USP, Escola de Engenharia de Sao Ca, Sao Carlos, Brazil Marco Bittencourt, Faculdade de Engenharia Mecancia, Universidade de Campinas, Campinas, Brazil

More information about this subseries at https://link.springer.com/bookseries/14172

Marcio Ferreira Martins Rogério Ramos Humberto Belich •

•

Editors

Multiphase Flow Dynamics A Perspective from the Brazilian Academy and Industry

123

Editors Marcio Ferreira Martins Universidade Federal do Espírito Santo Vitoria-ES, Espírito Santo, Brazil

Rogério Ramos Universidade Federal do Espírito Santo Vitoria-ES, Espírito Santo, Brazil

Humberto Belich Universidade Federal do Espírito Santo Vitoria-ES, Espírito Santo, Brazil

ISSN 2195-4356 ISSN 2195-4364 (electronic) Lecture Notes in Mechanical Engineering ISSN 2662-3021 ISSN 2662-303X (electronic) ABCM Series on Mechanical Sciences and Engineering ISBN 978-3-030-93455-2 ISBN 978-3-030-93456-9 (eBook) https://doi.org/10.1007/978-3-030-93456-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Isothermal and non-isothermal multiphase flows, with and without phase change or chemical reactions, are of interest to researchers and engineers working in mining, oil-and-gas, power, nuclear, chemical-process, space, food, bio-medical, micro- and nanotechnology, and other industries. Over the years, the “Jornada de Escoamentos Multifásicos” (JEM), a journey promoted by the Brazilian Society of Mechanical Sciences and Engineering (ABCM), has continuously evolved through meetings and courses, reflecting on the academy and Industry’s ongoing progress and developments. In the last 10 years, JEM had been successfully organized by the leader Brazilian univiersities UFSC (Universidade Federal de Santa Catarina), as EBECEM, EESC-USP (Universidade de São Paulo), UTFPR (Universidade Tecnológica Federal do Paraná), FEM/Unicamp (Universidade Estadual de Campinas). Most of the famous experts and scholars from around the world have participated in our Journey. In the previous journeys, more than 100 speakers and hundreds of attendees had presented their up-to-date research. They made a fulfilling communication on timely issues in the field of multiphase flow. Meanwhile, it has promoted the communication of colleagues and collaboration of partners in this field. With generous praise and suggestions from our participants, we from the Universidade Federal do Espirito Santo organized the 6th Multiphase Flow Journey, JEM 2021, which was successful! The first time JEM was held virtually from 17 to 20 May 2021, the format follows the trend of the major global events due to the COVID-19 pandemic situation of 2020/2021. The journey aims were to promote 2 days of short courses and 2 days of keynotes, session talks, and poster presentations on an online interactive exchange of scientific knowledge and experience under specific themes related to multiphase flows. The book chapters were grouped into themes (parts) related to multiphase flows mainly practiced in the Brazilian Academy and Industry. Each part here was represented by we calling fellows of the Brazilian Multiphase Flow community that, in a friendly spirit of contribution, accepted our invitation to participate and collaborate in JEM 2021 pre-, peri-, and post-activities, this book. v

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Preface

Each JEM edition is organized around main courses delivered by renowned Brazilian and international scientists. On this last occasion, there were four 8-h courses, for which the lecturers were Professores Branimir Cvetkovic, André Mesquita, Rigoberto Morales, and Humberto Belich. Together with four 1-h talks by Dr. Anselmo Pereira, Prof. Daniel Cruz, Dr. Fatemeh Salehi, and Mr. Luiz Octavio. In addition, we recognize the JEM award 2021 for Ms. Stella Cavalli among the young investigators’ talks. Finally, some of our former students gave us precious help to the organization, or direct scientific disciples, who acted as chairman of the session or brief lectures in the courses, Bruno de Moura, Leandro Amorim, Marco Zanoni, and Ramon Martins. The editors warmly thank Profs. Oscar Rodrigues and Renato Siqueira for their contributions to the success of the sections. We extend our thanks and gratitude to all sponsors and supporting institutions for their valuable contributions: 2Solve, PPGEM/UFES, NEMOG, LFTC. Finally, the editors acknowledge ABCM Springer Series and their editors-in-chief, Ricardo Diego Torres, Marcello Augusto Faraco de Medeiros, Marco Bittencourt, interest in publishing the most representative scientific and industrial material presented in the 6th Journey of Multiphase Flow. When we learned that we were responsible for organizing JEM 2021, we suggested to the Springer ABCM series editor Marcello Faraco that it would be interesting to publish a volume with hand-selected original contributions crossing a wide range of topics. For more accessible admittance, we decided on a rough categorization of the contributions: Multiphase Flow in Industry, Multiphase Flow Measurement & Instrumentation, Multiphase Flow With Phase Change & Chemical Reactions, Multiphase Flow Modeling, Experimental Multiphase Flow, and Wet and Dry Particulate Systems. Although we were well aware that no categorization would ever be absolute, we state the following few words about each part.

Part I: Introduction It seems that the relationship between Academy and Industry has gained more and more attention from the perspective of Higher Education scientists. At that time, we invited Dr. Erick Quintella from Oil & Gas Industry to reflect under his in-exercise perspective on this sensitive subject.

Part II: Multiphase Flow in Industry This part is opening by the chapter of Prof. Branimir Cvetkovic on Multifractured Horizontal Well Production and Drainage. A multidisciplinary field that the word multiphase gains real significance, where gas, liquid, and solid matter at

Preface

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temperature and pressure variations and under different flow domains have been challenging generations of engineers and scientists. The part was complemented by chapters written by young researchers working in offshore oil systems and industrial devices such as valves and multiphase meters.

Part III: Multiphase Flow Measurement and Instrumentation In the opening chapter of this part, Prof. Oscar Rodriguez told us about his substantial contribution over the last 17 years to the Brazilian multiphase flow field, from academic to industrial developments. Flow Measurement and Instrumentation area is undoubtedly one of the primary sources of Brazilian scientific and industrial production, from gamma-ray, wire-mesh, and electrical impedance tomographs development, passing by differential pressure meters methods, to machine learning applied to multiphase flow.

Part IV: Multiphase Flow with Phase Change and Chemical Reactions The multidisciplinary aspects of multiphase phenomena again come out through Dr. Fatemeh Salehi’s brilliant contribution to multiphase flows with heat and mass transfer. The readers will find applied Eulerian–Eulerian and Lagrangian–Eulerian methods for Computational Fluid Dynamics and the finesse of the experimental techniques to investigate the flow boiling, measure liquid-film thickness, or cryopreserve cells by phase change thermal aspects.

Part V: Multiphase Flow Modeling In this part, our invited Dr. Anselmo Pereira transcribed his outstanding keynote talk to a text where readers interested in non-Newtonian fluids, specifically, the viscoplastic materials in multiphase flow, can be instructed on fundamental modeling aspects. The other chapters, in this part, introduce new trends in multiphase flow simulations and revise the traditional CFD modeling of Newtonian fluids to investigate flow patterns.

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Part VI: Experimental Multiphase Flow All chapters in this part show the main robust multiphase flow laboratory structures existent in Brazil. Collaboration of the research groups of Profs. Oscar Rodriguez (Industrial Multiphase Flow Laboratory (LEMI), University of São Paulo), Rigoberto Morales (Multiphase Flow Research Center (NUEM), Federal University of Technology—Paraná), and Rogério Ramos (Research Group for Oil and Gas Flow and Measurement (NEMOG), Federal University of Espirito Santo) showed the Brazilian groups’ commitment to the multiphase flow field.

Part VII: Wet and Dry Particulate Systems Adding one more layer of the diversity of how complex the multiphase flow can be dealt, but bring lots of insights, crossing modeling, and experiments for Newtonian and non-Newtonian fluids in multiphase flows, Prof. André Mesquita summarized in his chapter the Discrete Element Method nuances for granular flows, from the fundamental to the large industrial cases. Espirito Santo, Brazil May 2021

Marcio Ferreira Martins Rogério Ramos Humberto Belich

Contents

Part I 1

What Is the Importance of the Brazilian Academy for the Brazilian Industry? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Erick Quintella

Part II 2

3

Introduction

Multiphase Flow in Industry

Screening a Multi-fractured Horizontal Well Production and Drainage in an Unconventional Reservoir . . . . . . . . . . . . . . . . Branimir Cvetkovic Thermal Effects in Multiphase Flow Analysis for Offshore Oil Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thierry Caique Lima Magalhães, Gabriel Romualdo de Azevedo, Ivanilto Andreolli, and Jorge Luis Baliño

4

Flow Simulation in Choke Valves for Offshore Oil Industry . . . . . . Enzo D. Giustina, Nikolas Lukin, and Raquel J. Lobosco

5

Automated Quick Closing Valve System for Three-Phase Flow Holdup Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Niederauer Mastelari, Eugênio S. Rosa, Ricardo A. Mazza, and Jordan V. Leite

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3

Determination of the Discharge Coefficient of Multiphase Meters Through Computational Simulation . . . . . . . . . . . . . . . . . . Matheus Constança de Oliveira and Eugenio F. Simões

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25

37

43

53

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Contents

Part III 7

8

9

Multiphase Flow Measurement and Instrumentation

State-of-the-Art Instrumentation and Experimental Methods Developed at the Industrial Multiphase Flow Laboratory over the Last 17 Years: From Gamma-Ray and Wire-Mesh Tomography to Physics-Informed Machine Learning . . . . . . . . . . . Oscar M. H. Rodriguez, André M. Quintino, Marlon M. Hernández-Cely, and Francisco J. Nascimento Wet Gas Metering by Differential Pressure Meters: A Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Renan Fávaro Caliman and Rogério Ramos

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Implementation of a Phase-Sensitive Detector with CORDIC Algorithm in Microcontrollers for Low-Cost EIT Demodulation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Bruno F. de Moura, Adriana M. M. da Mata, Marcio F. Martins, Francisco H. S. Palma, and Rogério Ramos

10 EIT Performance Criteria According to Variations in Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Adriana M. M. da Mata, Bruno F. de Moura, Marcio F. Martins, Francisco H. S. Palma, and Rogério Ramos 11 Data-Driven Machine Learning Applied to Liquid-Liquid Flow Pattern Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Lívia O. Zampereti, André M. Quintino, and Oscar M. H. Rodriguez Part IV

Multiphase Flow with Phase Change and Chemical Reactions

12 Multiphase Flows with Heat and Mass Transfer . . . . . . . . . . . . . . . 133 Fatemeh Salehi 13 Investigation of Vorticity Fields During Two-Phase Flow in Flow Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Jeferson D. de Oliveira, Jacqueline B. Copetti, and Elaine M. Cardoso 14 Liquid-Film Thickness Measurements During Convective Condensation of R290 in a Horizontal Channel . . . . . . . . . . . . . . . 155 Tiago A. Moreira and Gherhardt Ribatski 15 Thermal Resistance Model of a Polymeric Pulsating Heat Pipe . . . 165 Sabrina dos S. Ferreira, Fernando N. Q. dos Santos, and Cristiano B. Tibiriçá 16 A Critical Review on Cryopreservation of Cells Technics: Thermal Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Guilherme D. Steffenon, Jacqueline B. Copetti, Jeferson D. de Oliveira, Mario H. Macagnan, and Elaine M. Cardoso

Contents

Part V

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Multiphase Flow Modeling

17 Multiphase Flows with Viscoplastic Materials . . . . . . . . . . . . . . . . . 187 Anselmo Pereira, K. Isukwem, J. Gatin, E. Hachem, and R. Valette 18 New Trends on Multiphase Flow Simulations Using Slug Capturing Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Farhad Nikfarjam, Hamidreza Anbarlooei, and Daniel O. A. Cruz 19 Transient CFD Simulation of Vortex Formation in a Stirred Multiphase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 André Lourenço Nogueira 20 Numerical Simulation of a Downward Two-Phase Vertical Flow: A Preliminary Analysis of Convection Intensity and Gas Flow Rate Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Lucca D. V. Melo, Vitor P. Pinheiro, Ramon S. Martins, and Vinicius P. Franco 21 ANN and CFD-DPM Modeling of Alumina-Water Nanofluid Heat Transfer in a Double Synthetic Jet Microchannel . . . . . . . . . 231 Javad Mohammadpour, Zhaleh Ghouchani, Fatemeh Salehi, and Ann Lee Part VI

Experimental Multiphase Flow

22 Evaluation of Multiphase Flow Pattern and Friction Loss Prediction Models Applied to NEMOG’s Multiphase Flow Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Tiago G. S. Lima, Francisco J. do Nascimento, Oscar M. H. Rodriguez, and Rogério Ramos 23 Engineering Aspects on Flow Similarity for Design Water-in-oil Emulsion Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Edimilson Kempin Jr., Juliana T. A. Roberti, Ligia G. Franco, and Rogério Ramos 24 Experimental Analysis of Three-Phase Solid-Liquid-Gas Slug Flow with Hydrate-Like Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Stella Cavalli, Rafael F. Alves, Carlos L. Bassani, Eduardo Nunes dos Santos, Marco da Silva, Moises A. Marcelino Neto, Amadeu K. Sum, and Rigoberto E. M. Morales 25 Influence of Liquid Viscosity on Horizontal Two-Phase Slug Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Bruna P. Naidek, Marco G. Conte, Cristiane Cozin, Marco J. da Silva, and Rigoberto E. M. Morales

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Contents

26 Experimental and Numerical Two-Phase Slug Flow Evolution Analysis with a Slightly Downward Direction Change . . . . . . . . . . 283 Hedilberto A. A. Barros, Fernando Czelusniak, Cristiane Cozin, Eduardo N. dos Santos, Marco J. da Silva, Moisés A. M Neto, and Rigoberto E. M. Morales Part VII

Wet and Dry Particulate Systems

27 DEM Simulation: From Granular Crystal Modeling to Large Industrial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 André L. A. Mesquita, Luís Paulo S. Machado, and Alexandre L. A. Mesquita 28 Computational Model for Path Mapping of Spherical Particles in a Continuous Medium Applied to Iron Ore Transfer Chute . . . . 319 Rodrigo X. A. Leão, Enrico Sarcinelli, Leandro Amorim, Humberto Belich, and Marcio F. Martins 29 A CFD Model for Free Falling Bulk Materials Analysis . . . . . . . . . 335 Leandro F. B. Lima, Maciel C. Furtado, and André L. A. Mesquita 30 A Theoretical Framework for a Toroidal Vortex as a Dust Scattering Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 Humberto Belich, Marcio F. Martins, Leandro Amorim, Enrico Sarcinelli, and Rodrigo X. A. Leão

Part I

Introduction

Chapter 1

What Is the Importance of the Brazilian Academy for the Brazilian Industry? Erick Quintella

It’s a very broad question! I will try to answer it from the particular case of the industry that I know a little more about, the oil industry, and rely on the generosity of readers to extract together, from this window of analysis, the elements that can be generalized to the entire landscape of Brazilian industry. Join me on the “technical visit” pictured below: Imagine that a student participating in the 6th Multiphase Flow Journey climbs an oil platform and finds a large pressure vessel that receives at its entrance, always in a mixed form, gas, oil, and water, usually carrying suspended solids. In addiction, has internally elements distributed in a precise geometric configuration to efficiently guarantee the separation of the four phases at the output of the equipment. Impressed by the size of the equipment and the complexity of the multiphase flow, as well as the efficiency of the equipment in separating the phases, the student would ask the technician accompanying him on this visit: “Who supplied you with this very complex equipment? And how did you know upon delivery that this equipment would efficiently perform its function?”

The technician would respond that an industry, that is, a facility that has a large manufacturing capacity, is the one who supplied the equipment. It would explain that its efficiency was first demonstrated on a semi-commercial scale. In addition, by E. Quintella (B) CENPES/PDIDP/ESUP/TPMF - Tecnologia de Processamento e Medição de Fluidos, Rio de Janeiro, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_1

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E. Quintella

realizing that there would be demand in the market, the industry was economically stimulated to persist in the development of a commercial prototype. With the partnership of the platform technicians, the industry subjected this prototype to long-term tests in order to prove its robustness against the platform’s weathering and guaranteeing safety for the operation, considering, finally, the technology proved to be acquired by the oil industry as a definitive equipment of the phase separation process. The student would continue: “But at the advanced semi-commercial scale, this industry already knew, as “by magic”, what the best geometric distribution of the internal elements was?”

If it did not know, did it stop supplying the equipment and invested in mobilizing a team to carry out more studies in order to improve its product, causing its profit, in the short term, to be postponed to some point in the medium term? The technician would say that very few industries work with margins to invest in more in-depth studies at this advanced level of technological and commercial readiness, when it is only expected to test the robustness of the equipment as defined in the project. However, within the technological development plan, the improvement of this development would be up to another partner, namely a small or medium-sized company that is a mix of academia and industry. In other words, a company that has a small manufacturing capacity and can handle the aggressiveness of the real fluids of the platforms, and that employs masters and doctors for more complex investigations in its production processes. The technician would add that because these prototypes and pilots, developed and tested in this intermediate scale of technological and commercial readiness, do not yet constitute a perfectly “saleable” product, the market has only a marginal interest in financing this stage of the cycle. Leaving the State to complete the investment in this mix of academia and industry to continue the development of the national technology. The technician would add that there is a shortage of this type of partner in Brazil, and would regret the gap caused by this true “death valley of technologies” right in the middle of the scientific-technological development cycle, opining that it contributes a lot to stagnation of the national oil industry. The student would ask: “Do you mean, then, that this ‘mix of academia and industry’ is the one who carries out analytical and experimental studies aiming to prove the concept that will continue the development of technology?”

The technician would correct the student by stating that PROOFING THE CONCEPT TO THE INDUSTRY IS THE ROLE OF THE ACADEMY! Contracted (for mixed phase separation studies or for any other study) directly by a Petroleum Industry R&D Center, or financed by the State by research development agencies, uni-

1 What Is the Importance of the Brazilian Academy for the Brazilian Industry?

5

versities would fulfill their role as natural institutions to carry out initial studies, establish the technological concept, and, finally, to prove the concept object of its contract on a laboratory scale. Giving security to the subsequent partner to continue the technological development cycle of that industry.

Assuming that the student would be reasonably satisfied with the answers, the technician would conclude that these three actors (scaled, now, in the correct order) preferably carry out R&D in the oil industry: 1. Academy: responsible for the technological proof of concept. 2. “Mix of academia and industry”: responsible for the demonstration and testing of technology at scales with an intermediate vision between academia and industry. 3. Industry: responsible for demonstration and testing on a commercial scale.

Which allows me, as proposed at the beginning, to rely on the generosity of readers to save the particularities of each industry and generalize, in common understanding, that THE BRAZILIAN ACADEMY IS THE FIRST PARTNER OF THE BRAZILIAN INDUSTRY in the research and development of all national technology. Moreover, events like this 2021 Multiphase Flow Journey only reinforce the importance of the partnership between Academy and Industry! May they be perennial!

Part II

Multiphase Flow in Industry

Chapter 2

Screening a Multi-fractured Horizontal Well Production and Drainage in an Unconventional Reservoir Branimir Cvetkovic

Abstract Nowadays, industry firmly establishes the use of fractured horizontals wells for exploiting oil and gas reservoirs. In 2016, hydraulically fractured horizontal wells accounted for 69% of all oil and natural gas wells drilled in the United States. This work investigates the transient well responses of a multi-fractured horizontal well producing an Unconventional Reservoir. The created individual fracture responses originate from an infinitive reservoir and are considered diffusion equation full-time rate responses. The analytical screening process is helpful for prognosis, diagnosis, and improved modeling of hydrocarbon production and drainage. Screening analyses can generate valuable information for fracture diagnosis in addition to a well and fracture production prognosis, mainly when limited input data are available. Multi-fractured horizontal well rate time and pressure time responses represent the solutions to a diffusion equation with varying boundary conditions and fracture options (i.e., various fracture orientations, various fracture lengths, etc.). The well response solutions are analytical and the model screens a horizontal well with multi-fractured productions. The transient model calculates individual fracture rates, productivity indexes, and equivalent wellbore radius. Individual fractures are acid or proppant. Each fracture to a well inflow includes choking pressure effects. This presentation studies the transient decline, emphasizing a horizontal well with fracture wellbore responses positioned in Unconventional Reservoirs. We want to find out the most optimal number of fracture stages and support management in quickly making fast decisions, mainly when most G&G and reservoir data are limited or unavailable for a complete reservoir flow simulation study. To optimize a horizontal well production rate, we investigate the most significant effect of a reservoir, fracture, and well on such wells’ productivity. Further, we use an analytical model that quickly assesses the impact of various well or reservoir properties on well performance. The model is three-dimensional, heterogeneous, and considers full-time rate–pressure time solutions. It assumes single-phase flow and fractures positioned on a horizontal well are transverse or longitudinal of varying types (uniform flux, infinite conB. Cvetkovic (B) Head of the Reservoir Engineering and Management Department, National University Yuri Kondratyuk Poltava Polytechnic, NUPP Poltava, Ukraine e-mail: [email protected] Reservoir Management and Engineering Lead, Petro Brelle Energy Norway, Lillestrøm, Norway © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_2

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B. Cvetkovic

ductivity, finite conductivity). We use various constraints and multi-run sensitivity for the prognosis of multi-fractured horizontal well productivity and its diagnosis. For screening and optimization of a multi-fractured horizontal well, we use a fast and robust numerical algorithm. This paper investigates a horizontal well-fractured design and proposes the most optimal number of created transversal fractures economically valuable. A semi-analytical screening efficiently generates rate–pressure time monophase transient wellbore responses. Such wellbore responses complement finite differences multi-phase model solutions that handle heterogeneous and complex geometry but include numerical dispersion when solving partial derivatives with the algebraic equation. The study estimates hydrocarbon production profiles for various fractures, including the most optimal fracture number (fracture height, fracture half-length, fracture conductivity). Further, it provides a workflow for optimizing a fractured horizontal well production and drainage with the risk assessments in an Unconventional North Sea Reservoir. Keywords Multilateral fracture · Horizontal well · Semi-analytical · Flow simulations · Proppant fracture

2.1 Introduction The growing demand for gas in the North Sea hydrocarbon market and the availability of multilateral fractured horizontal well MFHOW technology spurs tight gas reservoirs production throughout the North Sea. Draining the reservoir efficiently using MFHOW depends on fracture spacing and the optimal number of fractures, especially for low-permeability reservoirs. Presuming that we know how many fractures on a horizontal well exist, we justify that number with the screening semi-analytical flow simulations study. The MFHOW model is heterogeneous and monophase. It has both production and injection liquid solutions for a horizontal well with fractures. For gas, we use inner boundary conditions of constant pressure. This paper gives a procedure for calculating the optimal number of finite conductive (proppant fracture) of a horizontal well in the North Sea gas reservoir environments. In this Greenfield reservoir screening study, we do not use any history matches available features. Further, we use only full-time analytical solutions for well pressure and rates calculations (not any early or late time solutions).

2 Screening a Multi-fractured Horizontal Well Production and Drainage …

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2.2 Mathematical Model 2.2.1 A Horizontal Well with Transversal Fractures (Transient Solutions) The general assumptions and limitations for a model of a multi-fractured horizontal well include an infinite (or geometry SLAB) reservoir that is isotropic or anisotropic, with no-flow boundaries above and below. The well passes through the center of transversal, rectangular, and fully penetrating fractures. Flow to a well occurs only through fractures that are either uniform flux or finite conductivity type (proppant fractures). The finite conductivity uses the “equivalent pressure point” method from Refs. [1–4]. For the general case, fractures are uniformly spaced and sized. However, it is possible to use a limited number of variations of sizing and spacing additionally. Inner boundary conditions include a constant rate or constant pressure. Nevertheless, variable rate inner boundary conditions may be employed. Figure 2.1 presents the top view of a model. The basic pressure (diffusion) equation can be expressed:  ∇

 ∂p k ∇ p − φc =q μ ∂t

(2.1)

We use the common simplifying assumptions: one-phase Darcy flow, Newtonian fluids, isothermal conditions, negligible gravity, small pressure gradients, constant compressibility, viscosity, porosity, and permeability.

Fig. 2.1 A fractured horizontal well of length, L, with three transversal fractures of half-lengths, Lf. The reservoir is non-bounded or infinite in the x and y directions (Top view)

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B. Cvetkovic

Here, µ is fluid viscosity; k, permeability tensor; , the porosity of the medium, constant; q, source volumetric production rate per unit volume. For the principal axes of permeability (anisotropy) coinciding with the coordinate the axes, we may write 1 ∂p μ − q=0 (2.2) ∇2 p η ∂t k k where η = φcμ is the diffusivity coefficient of the porous medium. We have written  3 k x k y k, to obtain invariance of volume elements and the fluxes, while the new k=  coordinate variables are x  = kkx x, etc., reducing to isotropy. By introducing the 



usual dimensionless variables x D = xl ,y D = yl , z D = obtain the expression in the dimensionless form: ∇D 2 p −

z , l

tD =

η t, l2

qD =

∂ pD qD − =0 ∂t D φc

l2 q, η

we

(2.3)

The application of Laplace transform, defined by 

∞

f (s) = L { f (t D )} =

e−st D f (t D ) dt D

(2.4)

0

Given the simpler equations: ∇D 2 p − s p =

qD − pi φc

(2.5)

Further, we avoid the derivation of the solutions and present solution p = Bn (s) q

(2.6)

We have to solve expression for B for a various number of fractures defined by n. Both p and q are in Laplace space. Thus, for the defined inner boundary condition of pressure, p, we can calculate rates q for a well with n fractures in Laplace space. With the Stehfest algorithm, we covert the Laplace to real space solutions. More details on solutions presented by Eq. 2.6 can be found in SPE papers by Cvetkovic et al. [5–7].

2.2.2 Fracture–Well Limited Communication (Choking Effect) Following the work of [8], which discussed the influence of a limited communication interval on the transient pressure behavior and the long-term productivity of a frac-

2 Screening a Multi-fractured Horizontal Well Production and Drainage …

13

tured well, Eq. 2.8 presents the dimensionless pressure (drop) or the “skin factor,” Slc. This option is available for the IBC of constant rate. Derivation details were published by Cvetkovic et al. [9]. From the formula for the pressure (drop) p D = p

πk 2π kh = z N qμ Nk f b f

(2.7)

and after averaging the total inflow over N fractures, the expression for the skin factor, Slc, or the dimensionless pressure (drop) becomes: Slc = p D =

hk π erw log sin Nk f b f h

(2.8)

2.3 Case Study We chose to use several options from the fractured horizontal well model to demonstrate screening analysis capabilities with data from a North Sea well. In cases where well data were lacking, properties that we believed to be appropriate for the sake of this review we assumed. The model is a fast mini-simulator, modeling one-phase (slightly compressible) liquid flow into a multi-fractured horizontal well in an open (slab) or closed (box) reservoir. The fractures are rectangular and vertical and either transversal or longitudinally relative to the well direction. They are also alternatively of finite conductivity, infinite conductivity, or uniform flux type. Further, the fractures are fully or partially penetrating, of equal or unequal length and spacing. There is no limitation to the number of fractures. The well is either open or perforated only at the fractures; however, one option is a partially perforated well with no fractures. The well produces a single-layered reservoir. The simulation setup is made easy for the user through a screen input interface design. Special features include the multi-run option, linear or logarithmic scales on-axis, and a choice of units (field, metric, or dimensionless).

2.3.1 Input Data As an input, we use various relevant geometric and physical parameters for the reservoir–well–fracture system, including the alternative of well rate or pressure, each with the option of being constant or variable or varying in time in a feely stepwise manner. The output given is correspondingly the well pressure or rate as a function of time, in addition to individual fracture rates. Further output parameters comprise pressure derivatives and all cumulative rates. Also, productivity indices and effective wellbore radii are calculated. Output (and input) data are available, both in tabular form and through a thorough display of curves by a specially developed

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B. Cvetkovic

Table 2.1 Reservoir, well, and fracture data Reservoir, well, and fracture data Model IBC Slab/box Constant pressure Reservoir

Well

Fracture

Fracture character

Name Length in X direction (m) Length in Y direction (m) Formation thickness (m) Porosity Permeability (mD) System compressibility (bar−1 ) Initial reservoir pressure (bar) Gas viscosity (cp) Gas formation volume factor (Rm3 /Sm3 ) Well length (m) Wellbore diameter (m) Bottomhole flowing pressure (bar) N. of fractures Fracturing spacing Fracture height (m) Fracture half-length (m) Uniform flux Infinite conductivity Finite conductivity fracture width (m) Finite conductivity fracture permeability (mD)

OBC No flow Data Infinity Infinity 460 6 0.05 6.25 × 10−5 900 0.022 0.004 1981.2 0.091 800 Varying Varying 57.91 45.72 – – 0.00508 100000

application of the Excel software. Well data required for analysis using the fracturedhorizontal well model are presented in Table 2.1. Model Input data are: 1. Wellbore, Well length; Direct flow to a wellbore; and a Fracture to well friction (choking effect). 2. Fracture, Conductivity of Unform flux, Infinite conductivity and Finite conductivity; Fracture Half-length (equal, unequal), and Fracture penetration (fully penetrated, partially penetrated).

2 Screening a Multi-fractured Horizontal Well Production and Drainage …

15

3. Reservoir, Initial pressure, Pi ; Reservoir height, h; Viscosity, µ; Compressibility and FVF, Bl (l = oil or gas).

2.3.2 Simulation Results Simulation study objectives are to estimate an optimal number of fractures, target production of 1 mil Sm3 /D of a horizontal well with fracture, assess the productivity of created fractures, and predict fracture-type various output (for proppant and acid fractures). The case study presented in Fig. 2.2 shows eight fractures that the operator wanted to create. The well was undulating, and the transversal fractures were positioned along a trajectory path. We first increased the number of transversal fractures from eight to ten during flow simulation to investigate the optimal number of fractures, as shown in Fig. 2.3. Many transverse fractures have different forms around a horizontal well after being fractured multi-stage insight gas reservoirs. This fracturing greatly increases the contact area between the gas well and the formation. We simplify fracture shape and presume equal fracture spacing as in Figs. 2.4 and 2.5. Additionally, we simulate ten simulations of a horizontal well with fractures. Well simulation with transversal fractures results are given in time: for Cumulative pro-

Fig. 2.2 A multifratured horizontal well positioned in a tight gas reservoir with eight transversal proppant fractures

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B. Cvetkovic

Fig. 2.3 How many proppants transversal fractures are optimal?

Fig. 2.4 Model assumption horizontal well penetrating eight transversal fractures each having equal height and equal half-length Fig. 2.5 A transversal fracture height and half-length with the fracture width, omega, and the fracture half-lengths, Lf of fracture permeability, Kf

2 Screening a Multi-fractured Horizontal Well Production and Drainage …

17

Fig. 2.6 Cumulative gas versus time for a horizontal well with ten proppant transversal fractures

Fig. 2.7 Gas rate versus time for a horizontal well with ten proppant transversal fractures

duction, Qg , in Fig. 2.6, Gas rate, qg , in Fig. 2.7, and Productivity index, PI, in Fig. 2.8. Well with fractures Cumulative gas production, Qg , Gas Rate, qg , and Productivity Index, P.I., at 1560 days presents Figs. 2.9, 2.10, and 2.11. Gas production percentage for various fractures gives Fig. 2.12. The same figure marks the gas production for the optimal number of a horizontal well with fractures. Figure 2.13 presents the cumulative gas production (Sm3) for each of the eight fractures positioned along a horizontal well after 1560 days. Due to the fracture interference, it is evident that fractures 8 and 1, 7 and 2, 6 and 3, and 4 and 5 have the same production. Thus, we will ideally have only four cumulative production curves as in Fig. 2.14. Income

18

B. Cvetkovic

Fig. 2.8 Productivity index versus time for a horizontal well with ten proppant transversal fractures

Fig. 2.9 Cumulative gas (Sm3) versus fracture number at 1560 days

versus the number of horizontal well fractures (for various fracture costs) in USD and NOK presents Fig. 2.15. The figure also investigates the cumulative production difference. We cut off the additional fracture once the production difference value is minor or equal to the fracturing costs. The additional options of MFHOW model allow simulating a horizontal well with fractures with various inner boundary conditions of constant rate (line in green). Figure 2.16 presents the influence of fracture well-limited communication, to well with fracture pressure, and the well with fracture productivity index (with synthetic data).

2 Screening a Multi-fractured Horizontal Well Production and Drainage …

19

Fig. 2.10 Gas rate (Sm3/D) versus fracture number at 1560 days

Fig. 2.11 Productivity index (m3/Bar*D) versus fracture number at 1560 days

Fig. 2.12 The Gas production percentage for a various number of proppant-transversal fractures

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B. Cvetkovic

Fig. 2.13 The Cumulative gas production, Qg (Sm3), versus fracture number (at the end of simulation time)

Fig. 2.14 Cumulative Gas Production verses Time (eight proppant finite-conductive fractures)

2.4 Conclusion and Future Perspective The analytical and numeric simulation in evaluating the well productivity and reserve recovery of hydraulic stimulated wells are common today. Although both approaches are still widely used analytical techniques are considered more commonly used to perform initial stimulation option screening exercises or well performance quick look common sense checks rather than comprehensive, detailed engineering studies. This study considers the transient rate decline and brings solutions and interpretation techniques that can help analyze a fractured-horizontal well. A flow simulation model comprises a series of solutions corresponding to various wellbores with fractures

2 Screening a Multi-fractured Horizontal Well Production and Drainage …

21

Fig. 2.15 Income versus a number of horizontal well fractures (for various fracture costs in USD and NOK)

Fig. 2.16 Fracture-Well Limited Communication (Choking Effect) (for variable rate IBC) on pressure difference and well with fractures Productivity Index, PI

conditions. Once all input data are available, it is possible to match the well data with the options implemented in a module. The individual fracture rate, the cumulative production, and the productivity index responses represented additional valuable available information. A fine-tuning of the variable rate and variable pressure would have improved the fitting of the well data. Additionally, the method was shown to be ideally suited for solving multiple fractured horizontal wells in an oil or gas reservoir when available data was limited,

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B. Cvetkovic

and analysis could not be carried out with conventional reservoir simulation tools so fast. This study illustrates the use of a screening tool for improved modeling of gas production from horizontal or near-horizontal wells with induced fractures. MFHOW models have the potential of being further developed. For a horizontal well with fractures, the model developments recommendations are: 1. To be extended to handle any fracture orientation, determined by the stress orientation of the reservoir, a fracture wall damage or fracture skin and wellbore skin. 2. To include heterogeneity as layering and naturally fractured reservoir complexity. 3. To be able to handle gas flow with the non-Darcy flow. 4. To couple screening features of a fractured horizontal well model to state-oftechnology risking software (commercial software). 5. To improve fracture diagnosis by coupling microseismic modeling devices to the fractured horizontal well screening features. 6. To include geomechanical modeling in flow simulation. 7. To verify the numerical model solutions with the semi-analytical solutions (commercial software). A horizontal well with fractures has become the industry standard for unconventional and tight formation gas reservoirs. The horizontal well programs require integrating petrophysical, reservoir, completion, and fracture stimulation disciplines for success in tight formation and unconventional gas reservoirs. Today several joint industry projects, JIP, are related to unconventional well pressure–rate time performance analysis, both numerical and analytical. This study uses slightly modified input due to the confidentiality of data. For future flow simulation studies, we find the available data repository of horizontal wells with fractures. The SPE Bleeding Edge of RTA Group (BERG) [10] posts are dedicated to promoting the value of Rate Transient Analysis and petroleum engineering. Acknowledgements I owe a special thanks to Dr. Gotskalk Halvorsen for his mathematical support and all our valuable discussions. Thanks to Jan Sagen for his contribution in programming the model options.

References 1. A.C. Gringarten, H.J. Ramey Jr et al. (1972) 2. C.C. Chen, E. Ozkan, R. Raghavan, in SPE Annual Technical Conference and Exhibition (OnePetro, 1991) 3. R. Raghavan, S.D. Joshi, SPE Form. Eval. 8(01), 11 (1993) 4. T. Blasingame, B. Poe, in SPE Annual Technical Conference and Exhibition (OnePetro, 1993) 5. B. Cvetkovic, G. Halvorsen, J. Sagen, in SPE/CIM International Conference on Horizontal Well Technology (OnePetro, 2000) 6. B. Cvetkovic, G. Halvorsen, J. Sagen, E. Rigatos, in SPE Rocky Mountain Petroleum Technology Conference (OnePetro, 2001)

2 Screening a Multi-fractured Horizontal Well Production and Drainage …

23

7. B. Cvetkovic, in SPE Production and Operations Symposium (OnePetro, 2009) 8. W. Schulte, SPE Product. Eng. 1(05), 333 (1986) 9. B. Cvetkovic, G. Halvorsen, J. Sagen. Howif-oil box model. the final and the half-year report for b.p., conoco, phillips (1999) 10. Bleeding edge of rta group (berg) (spe.org). https://www.spe.org/en/industry/berg/. Accessed: 2021-08-23

Chapter 3

Thermal Effects in Multiphase Flow Analysis for Offshore Oil Systems Thierry Caique Lima Magalhães, Gabriel Romualdo de Azevedo, Ivanilto Andreolli, and Jorge Luis Baliño

Abstract In this paper, a numerical study on the stationary state for the multiphase flow of oil in a pipeline-riser system is presented. The developed model considers that the flow is one-dimensional and three-phase throughout the system. The liquid and gas phases are considered compressible. Furthermore, it is assumed that oil and water have the same speed and are homogenized. The flow pattern in the pipeline and riser is determined based on the local state variables and inclination angle. The characterization of fluids is done according to the black-oil model. The conservation equations were solved numerically using MATLAB software [1]. For a constant separator pressure as boundary condition, the pressures calculated by the model with the thermal effects were higher than those obtained by the isothermal model. The average difference between them was 2.3%. The deviations occurred due to the effect of temperature on the mass transfer, compressibility, expansion, and the thermophysical properties of the phases. The stationary analysis shows that the temperature has a secondary effect on the flow in offshore systems. However, deviations about the isothermal model must be taken into account in order to improve the reliability of numerical results and better production planning. Keywords Multiphase flow · Thermal effects · Stationary state · Pipeline-riser systems · Petroleum production technology

T. C. L. Magalhães (B) · G. R. de Azevedo · J. L. Baliño Departamento de Engenharia Mecânica, Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil e-mail: [email protected] J. L. Baliño e-mail: [email protected] I. Andreolli Engenharia de Petróleo, Universidade Santa Cecília, UNISANTA, Santos-SP, Brazil © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_3

25

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T. C. L. Magalhães et al.

3.1 Introduction The relevance of oil to society is vast and essential. In this context, studies about the flow of oil, especially in the production and exploration systems of the wells, are vital. To perform the multiphase flow analysis, it is necessary to develop a model that describes its physical behavior, consisting of the equations of mass balance, linear momentum, and energy, in addition to models for the set of fluid properties. Multiphase flows involve several complexes and coupled phenomena, such as slip between phases, heat transfer, thermodynamic equilibrium processes, variation in fluid properties, and flow. Therefore, simplifications must be adopted [2]. Then, empirical, analytical correlations based on experimental results and field data are used, such as closing laws in the modeling of multiphase flows, to determine the properties of fluids. Petroleum is a complex substance, which can be present in the three phases of matter (solid, liquid, and gas), forming emulsions, hydrates, paraffin, and inorganic incrustations, making it extremely difficult to characterize the flow. Thermal analysis is important in assuring minimum temperatures in order to prevent hydrates, and paraffin formations in the pipelines [3]. Offshore oil production systems are designed to operate in steady-state regime. However, it is possible that this condition does not exist [4]. Incorporating the energy equation and, consequently, the thermal effects search make the model analyze the flows even more realistic. This study proposal makes it possible to obtain fast and concise information about the flow in offshore systems through computer simulations.

3.2 Model The developed model considers that the flow is one-dimensional and three-phase throughout the system. The liquid and gas phases are considered compressible. Furthermore, it is assumed that oil and water have the same speed and are homogenized. The temperature drop in the biphasic flow (gas-liquid) is being evaluated. The flow pattern in the pipeline and riser is determined based on the local state variables and inclination angle. The characterization of fluids and the transfer of mass between the phases is done according to the black-oil model. At the separator, constant pressure is assumed, while at Wet Christmas Tree (WCT), a constant temperature is considered. Mass flow of the different phases is constant at the inlet, where the gas and oil distribution depends on the equilibrium condition for the local pressure and temperature.

3 Thermal Effects in Multiphase Flow Analysis for Offshore Oil Systems

27

Fig. 3.1 General geometry of the production system

3.2.1 System Geometry Figure 3.1 presents the general geometry of the production system considered, to formulate the balance equations. This geometry consists of a flow line connecting the Wet Christmas Tree (WCT), located at the seabed, through a variable path s with the platform. The local inclination angle θ depends on the local position. The origin of the coordinate system (x, z) is set at the WCT.

3.2.2 Conservation Equations The multiphase flow model adopted is based on the mixture model and considers four conservation equations: mass balance for the gas and liquid phases and simplified equations for linear momentum and energy balance for the gas-liquid mixture. The mass balance equations for the gas and liquid phases have been adapted [5]: ∂ ∂ (ρg α) + (ρg jg ) =  ∂t ∂s

(3.1)

∂ ∂ [ρl (1 − α)] + (ρl jl ) = − ∂t ∂s

(3.2)

In Eqs. 3.1 and 3.2, ρg and jg are the density and superficial velocity for gas, while ρl and jl are the density and superficial velocity for liquid,  the mass transfer term, α the void fraction, and t the time. For most transients that happen in the segment of oil and gas, the system’s response proves to be sufficiently slow. Proves to be sufficiently slow. Furthermore, pressure waves have no dominant effect compared to continuity waves. The pressure drop is

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T. C. L. Magalhães et al.

modeled, according to the NPW (No Pressure Wave) model, in which the terms of inertial are neglected [6]: ∂P = ∂s



∂P ∂s

 − pgsinθ

(3.3)

p = ρg α + ρo αo + ρw αw

(3.4)

α + αo + αw = 1

(3.5)

F

where ( ∂∂sP ) F is the frictional pressure drop; P is the pressure; ρ, ρo , and ρw are, respectively, the density of the mixture, oil, and water; g represents the acceleration of local gravity; while αo and αw are, respectively, volumetric fractions of oil and water. The energy conservation equation that is implemented was based on [7] and neglects axial heat flux, dissipative sources, and normal stress powers. To be consistent with the linear momentum conservation equation, the kinetic energy terms are also neglected: −

∂ ∂ q0 Pc − Ggsinθ = [αρg uˆ g + (1 − α)ρl uˆ l ] + [ρg jg hˆ g + ρl jl hˆ l ] A ∂t ∂s

(3.6)

G = ρg jg + ρl jl

(3.7)

where q0 is the heat flux on inner walls of the duct; Pc the heated perimeter; G is total mass flux, given by Eq. 3.7; uˆ g and hˆ g are, respectively, the internal energy and enthalpy of gas; while uˆ l and hˆ l the internal energy and enthalpy of the liquid phase.

3.2.3 Closure Laws 3.2.3.1

Frictional Pressure Drop

The friction term in the linear momentum equation Eq. 3.3 was modeled using the two-phase multiplier ϕ 2f0 , as presented in [7], as being the ratio between the frictional pressure drop and the pressure drop, assuming that the total mass flux is composed only of liquid:   − ∂∂sP F 2 (3.8) ϕ f0 =  ∂ P  − ∂s l

3 Thermal Effects in Multiphase Flow Analysis for Offshore Oil Systems

  ∂P 1 G2 − = fl ∂s l 2 ρl D

29

(3.9)

where fl is the Darcy friction factor for the liquid phase and D is the pipe inner diameter. Different correlations may be used in this approach. Here, the correlation for the homogeneous two-phase multiplier ϕ 2f0 is adopted according to [8]. Therefore, Eq. 3.3 becomes 

3.2.3.2

∂P ∂s

 + ϕ 2f0

1 G2 fl + ρgsinθ = 0 2 ρl D

(3.10)

Void Fraction

To determine the void fraction, it is assumed that there is an algebraic relation between the void fraction and the local flow conditions: α( jg , jl , P, T, θ )

(3.11)

where T represents the local temperature of gas-liquid mixture. Equation 3.11 allows the application of many empirical correlations for the void fraction, especially the correlations based on the drift flux model [9], being used locally. The stratified and intermittent flow patterns for different inclination angles are common in offshore systems in the stationary condition, and then specific correlations should be used. When the flow pattern is stratified, an approach based on local equilibrium condition is used [10], while when the flow pattern is intermittent, the void fraction correlation of [11] is considered. The transition between the stratified and intermittent flow patterns was determined using the approach [2].

3.2.3.3

Heat Flux

The term related to inner heat flux in Eq. 3.6 was modeled as follows: q0 =

q 2πrint

(3.12)

q  = U A(T − TW )

UA =

(3.13)

2π 1 rint h int

+

ln( rrext ) int

ke f f

(3.14) +

1 rext h ext

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T. C. L. Magalhães et al.

where Tsw represents the temperature of the seawater on the outer side; while rint and rext are, respectively, the inner and outer radii; UA the thermal conductance; and q  the thermal power per unit length. ke f f =

ln( rrext ) int

 N −1

ln(

1

r1+1 ri

)

(3.15)

Ki

Radiation heat transfer was neglected in the determination of the global heat transfer coefficient U since this process is relevant at high temperatures. The coefficient for forced convection on the inner side hint depends on the flow pattern. Given these considerations, the correlation of [12] was adopted. The convective coefficient for the external flow hext can be obtained through the Nusselt number correlation presented in [13] for single-phase free convection in long and inclined cylinders.

3.2.3.4

Fluid Characterization

According to [14], oil can be characterized by a compositional model, which considers different components, or through the black-oil model, which defines oil as being composed of liquid and gas phases, with the same composition. In this study, the black-oil model was adopted for the characterization of fluids. The black-oil approach considers the existence of two components in the mixture, liquid and gas. For a specific condition P, T , the black-oil model admits that the liquid phase contains dissolved gas, and when this phase is stabilized in the standard condition SC, a liquid phase is separated into gas and liquid. The set of input variables that characterize the fluids are as follows: the standard densities of oil, gas, and water, represented, respectively, by ρo0 , ρg0 , and ρw0 ; gas-oil ratio GOR; and water-oil ratio WOR. The densities of oil, gas, water, and liquid under local pressure and temperature conditions are calculated by ρo0 + ρg0 R So Bo

(3.16)

ρg =

ρg0 Bg

(3.17)

ρw =

ρw0 Bw

(3.18)

ρl0 + ρg0 R Sl Bl

(3.19)

ρo =

ρl =

3 Thermal Effects in Multiphase Flow Analysis for Offshore Oil Systems

31

where Bo (P, T ), Bg (P, T ), Bw (P, T ), and Bl (P, T ) are the volume factors of formation of oil, gas, water, and liquid, respectively, while R So (P, T ) and R Sl (P, T ) are, respectively, the solubility ratios: gas-oil and gas-liquid. Several black-oil correlations can be used to calculate these variables as a function of pressure, temperature, and oil API grade.

3.2.3.5

Mass Transfer Term

The mass transfer model selected was the one developed in [15], in which the blackoil approach was considered. Assuming that mass transfer occurs, due to changes in gas solubility as the oil flows with speed: u l = αjoo , the term of mass transfer between the phases can be expressed by   = −ρg0 jl0

1 − α ∂ Rsl ∂ Rsl + jl ∂t ∂S

 (3.20)

where jl0 is the liquid superficial velocity at standard condition. > 0 (there must be oil in vaporization), It is important to note that for  > 0 → ∂α ∂S while for  < 0 → ∂α < 0 (there must be condensing gas). ∂S

3.2.4 Stationary State The stationary state can be obtained by setting to zero the time derivatives in the dynamic equations. Variables at stationary state are denoted with superscript ∼. Therefore, the following equations must be solved simultaneously: G O R − R˜so ˜ j˜g = Bg jl0 1 + WOR B˜o jl0 1 + WOR

(3.22)

WOR ˜ Bw jl0 1 + WOR

(3.23)

j˜o = j˜w =

(3.21)

The superficial velocity of the liquid can be written as B˜o + W O R ˜ j˜l = j˜o + j˜w = Bw jl0 1 + WOR

(3.24)

Integrating with Eq. 3.10 local pressure is obtained throughout the system, then:

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P˜ = PS −



S



0

∂P ∂s

˜ ds

(3.25)

where PS represents the constant pressure at the separator. The local temperature along with the system is obtained through the integration of Eq. 3.6 resulting in   S ∂T ˜ ds (3.26) T˜ = TW C T + ∂s 0 where TW C T is the constant temperature at the WCT. Void fraction, liquid and gas density, and any other dependent variable are calculated from the corresponding relations evaluated with the stationary values.

3.2.5 Discretization The length of the pipe is discretized in N nodes and the equations of balance in the stationary condition integrated into the interval si ≤ s ≤ si+1 . Representative values for any φ function or any variable within the integration range are calculated using the average value between adjacent nodes: φi+1/2 ≈

1 [(v˜i ) + φ(vi+1 ˜ )] 2

vi+1 ˜ ≈

(3.27)

1 [(v˜i + vi+1 ˜ )] 2

(3.28)

The spatial derivatives were approximated by the finite difference method: 

∂ v˜i ∂s

 ≈ i+1/2

1 2



vi+1 ˜ + v˜i

si

 (3.29)

3.3 Results and Discussion Figures 3.2 and 3.3 present the pressure and temperature profiles along the flow obtained through computer simulations, for analysis of an offshore pipeline-catenary riser, with input parameters shown in Table 3.1. In Fig. 3.2, the blue curve represents the results for the model with thermal effects, while the orange line represents the constant temperature in the WCT used in the isothermal model. Inlet temperature was fifty-one degree Celsius, and the temperature drop in the system was about twenty-eight degree Celsius.

3 Thermal Effects in Multiphase Flow Analysis for Offshore Oil Systems

33

Fig. 3.2 Temperature distribution obtained through the parameters from Table 3.1

Fig. 3.3 Pressure distribution obtained through the parameters from Table 3.1

The results achieved in Fig. 3.3 were compared to those found using the same approach for an isothermal analysis, similar to those carried [16]. The pressures at the WCT calculated by the model that includes the thermal effects were higher than those obtained by the isothermal model. The average difference between them was 2.3%. The deviations occurred due to the effect of temperature on the mass transfer (vaporization), compressibility, expansion, and the thermophysical properties of the phases.

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T. C. L. Magalhães et al.

Table 3.1 General input parameters for the simulations Symbol Variable Value γ API γg0 ρw TW C T ε Ps ρsw Tsw GOR WOR D Dext ke f f N L β Xr Zr

◦ API

Gas specific gravity Water density Temperature at the WCT Roughness Pressure at the separator Seawater density Temperature of the seawater Gas-oil ratio Water-oil ratio Inner diameter Outside diameter Effective thermal conductivity Number of nodes Pipeline length Pipeline inclination angle Horizontal length of the top of the riser Height of the top of the riser

Unit

19 0.6602 999.014 323.15

– – kg/m3 K

4.5 x 10−5 25

m bar

1024 277.15

kg/m3 K

145 30 0.1524 0.3693 2.7808

sm3 /sm3 % m m W/(m K)

101 1000 2

– m

845

m

1300

m

◦

3.4 Conclusion and Future Perspective The stationary analysis, through the numerical simulation of the multiphase flow of an oil system, shows that the temperature has a secondary effect on the flow in offshore systems. However, deviations from the isothermal model must be taken into account to improve the reliability of numerical results and better production planning. Acknowledgements This work was granted by Petróleo Brasileiro S.A. (Petrobras). The authors wish to thank Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Brazil), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES, Brazil), and Agência Nacional de Petróleo (ANP, Brazil).

3 Thermal Effects in Multiphase Flow Analysis for Offshore Oil Systems

35

References 1. E. Magrab, An Engineer’s Guide to MATLAB: With Applications from Mechanical, Aerospace, Electrical, Civil, and Biological Systems Engineering. Engineer’s Guide To (Prentice Hall, 2011). https://books.google.com.br/books?id=mOftRwAACAAJ 2. O. Shoham, (2006) 3. I. Andreolli, Editora Interciência 1, 594 (2016) 4. I. Andreolli, Estabilidade linear aplicada ao escoamento multifásico de petróleo. Ph.D. thesis, Universidade de São Paulo 5. J.G. Collier, J.R. Thome, Convective Boiling and Condensation (Clarendon Press, 1994) 6. J. Masella, Q. Tran, D. Ferre, C. Pauchon, Int. J. Multiph. Flow 24(5), 739 (1998) 7. G. Wallis, One-Dimensional Two-Phase Flow (McGraw-Hill, 1969). https://books.google. com.br/books?id=xvFQAAAAMAAJ 8. M. Ishii, T. Hibiki, Thermo-Fluid Dynamics of Two-Phase Flow (Springer Science & Business Media, 2010) 9. N. Zuber, J.A. Findlay, (1965) 10. Y. Taitel, A.E. Dukler, AIChE J. 22(1), 47 (1976) 11. K.H. Bendiksen, Int. J. Multiph. Flow 10(4), 467 (1984) 12. C.C. Tang, A.J. Ghajar, in Heat Transfer Summer Conference, vol. 42754 (2007), pp. 205–218 13. S.W. Churchill, H.H. Chu, Int. J. Heat and Mass Transf. 18(9), 1049 (1975) 14. W.D. McCain Jr, (1973) 15. R.H. Nemoto, J.L. Baliño, Int. J. Multiph. Flow 40, 144 (2012) 16. I. Andreolli, M. Zortea, J.L. Baliño, J. Pet. Sci. Eng. 157, 14 (2017)

Chapter 4

Flow Simulation in Choke Valves for Offshore Oil Industry Enzo D. Giustina, Nikolas Lukin, and Raquel J. Lobosco

Abstract Flow control valves are essential to the oil industry through exploration and production phases. These valves must be capable of operating in offshore wells under an extremely high-pressure regime. In addition to contact with abrasive fluids, the flow occurs at high speed, which leads the equipment to suffer significant abrasive wear due to the high flow rates and pressure drops. This work aims to evaluate the flow through a drilling choke, using Computational Fluid Dynamics (CFD) open-source software, OpenFOAM. The flow was evaluated for five stages of the valve’s opening and flow conditions (25.0, 26.0, 28.0, 30.5, and 35.0 percent open). The results were compared with the experimental data obtained from a water test proceeded in the oil industry. It was found that good adherence between numerical and experimental results for pressure drop and the simulations provided an extensive understanding of pressure and velocity fields, allowing to predict the areas of cavitation on choke valves. Keywords Computational fluid dynamics · Cavitation · Choke valve · Oil and gas industry

4.1 Introduction Initially, pressure drop on choke valves was modeled analytically using empirical correlations based on data adjustments obtained experimentally. The numerical and analytical models have evolved to predict the effect of multiphase flow (gas and liquid mixture) through the valves. In order to account for multiphase flow effects, a recursive term is defined in an expression for the flow rate value, which allows predicting a relationship between the critical and the sub-critical effects. Homogeneous E. D. Giustina (B) · R. J. Lobosco Federal University of Rio de Janeiro, Macaé, Brazil e-mail: [email protected] N. Lukin Petrobras University, Rio de Janeiro, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_4

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models were subsequently proposed, such as, for example, by [1] and evolved into a non-homogeneous fluid slip treatment. The use of computational fluid dynamics can contribute to the operational purpose of controlling the well and for measuring the equipment integrity [2]. Numerous damage mechanisms, such as corrosion, flashing, erosion, and cavitation, also contribute to the decrease of the expected life service [3]. However, fatigue caused by cyclic loads from the flow induces vibrations, amplifying the loss of equipment integrity due to the high working pressures of Choke valves. The presence of flow strangulation and great pressure drops can reach vapor pressure causing steam bubble nucleation, and thus the cavitation phenomena [4, 5]. Predicting the regions of cavitation occurrence aids for better equipment project and design, [6]. Novel models for multiphase flow behavior of choke valve predictions, especially in critical conditions, have been recently developed using advanced and new technologies, such as Artificial Neural Network for flow rate optimization [7]. The analysis allows predicting the flow rate performance for optimizing the operational control of the valve.

4.2 Methodology Flow analysis in the critical strangulation region enables optimization of the valve geometry and automation of operational procedures aimed for reducing the equipment wear [8]. There are different operational analytical models for the multiphase flow in choke valves on the open literature, [9, 10]; however, detailing the complex distribution of pressure along the valve is still on state of the art. The valve mesh is illustrated in Fig. 4.1 where the fluid is injected in the upper left conduit. The equipment was tested with water as in chokes calibration tests performed in offshore drilling rigs. The geometry was initially created in CAD, software SolidWorks, and imported to OpenFOAM version 7. Two main parts essentially constitute the choke valve: a fixed body and a mobile valve that can adjust its linear position by an actuator used to regulate the restriction to flow. This allows the choke to control the pressures on the well, avoiding well collapse and the influx of undesirable fluid from the well. In this work, five opening levels of the choke valve were conducted in the numeric simulation, each representing different operational stages of the valve: 25.0, 26.0, 28.0, 30.5, and 35.0% opened. That consequently varies the flow conditions. The water flows at a fixed rate of 200 gallons per minute defines the inlet boundary condition, accounting for the reduced circulation flow rate for good control. The valve initially set to the completely open setup is gradually closed, and the opening percentage is recorded when the upstream pressure sensor indicates preset pressures of 1000, 1500, 2000, 2500, and 3000 psi as per Petrobras standard. This test allows checking the choke performance in a good control situation experimentally.

4 Flow Simulation in Choke Valves for Offshore Oil Industry

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Fig. 4.1 Lateral and frontal illustration of the choke valve numerical mesh

4.3 Theoretical Foundations The mathematical model used for the OpenFOAM is a derivation of the numerical approach of the comprehensibility for the two-phase flow in which the subscript l denotes liquid and the subscript v is used for the vapor phase, as described by [11]. The mass conservation for the liquid phase is given by Eq. 4.1 while the vapor one is given by Eq. 4.1. ∂ρl αl + ∇ · (ρl αl Ul ) = m˙ (4.1) ∂t ∂ρv αv + ∇ · (ρv αv Uv ) = −m˙ (4.2) ∂t αl + αv = 1

(4.3)

The sum of the volumetric fraction must be equal to one as shown by Eq. 4.3. The multiphase transport equation, as described in [11], is given by Eq. 4.4.      ∂αl ρ˙l 1 1 ρ˙v 1 + ∇ · (αl U ) + ∇ · [αl αv Ur ] = αl αv − − αl − + αl [∇ · U ] + m˙ ∂t ρv ρl ρl ρl ρv

(4.4)

For an incompressible flow, the Momentum Equation originates from the Newton Second Law and can be written as Eq. 4.5.

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 m˙

1 1 − ρl ρv



 +∇

H (U ) ap



 −∇ ·

1 ∇p ap

 (4.5)

SimpleFoam solver with the no-slip hypothesis at the wall’s boundaries condition and the fixed flow for each geometry representing their different openness states was used in each simulation. The numerical results for pressure drop along the choke valve were compared with experimental data for model validation. The incompressible and monophase flow hypotheses were used [12]. The pressure was taken when the flow achieved a permanent regime which was verified through numeric residuals analysis, according to the computational fluid dynamic techniques described by [2]. Note that the current data uses the incompressible hypothesis. Thus, the pressure output from the OpenFOAM solver is divided by the water density.

4.4 Results and Data Analysis The choke valve flows’ numerical results present good adherence with the experimental data. The results of each opening stage are shown in Table 4.1, indicating a good agreement with the experimental results. The 25% opening percentage, which

Table 4.1 Comparison of the differences in computational results and experimental values of pressure Openess (%) Experimental results (psi) Numerical results (psi) Deviation (%) 35.0 30.5 28.0 26.0 25.0

1000 1500 2000 2500 3000

1064 1598 2018 2409 2756

6.4 6.5 0.9 3.7 8.1

Fig. 4.2 Velocity profile of the shaded blue section (x = 200 mm) in the strangulation region for the operational stages of the valve: 25.0, 26.0, 28.0, 30.5, and 35.0% opened

4 Flow Simulation in Choke Valves for Offshore Oil Industry

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Fig. 4.3 Detail of pressure distribution in choke valve in the strangulation region for the operational stages of the valve: 25.0, 26.0, 28.0, 30.5, and 35.0% opened

is the most critical stage due to the flow high pressure and speed, resulted in 2756 psi, an error of approximately 8.1%. Such high deviation from experimental results is attributed to compressible effects on the flow. The velocity profile of the constriction area can be seen in Fig. 4.2 for the same flow rate, varying the openness percentage of the valve. Numerical results also predict the flow velocity field and the occurrence of cavitation within low-pressure regions in the strangulation area, as shown in Fig. 4.3.

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4.5 Conclusion and Future Perspective The present study concluded that there is a good agreement between numerical and experimental results for pressure drop in choke valve up to 25.0% open condition, indicating that adopted boundary conditions are adherent to flow modeling. At such a restriction, the maximum flow velocity was calculated to be 12 m/s. For further restriction in the valve, compressible effects are no longer negligible and must be considered for simulation accuracy. Simulations also predicted high-pressure gradient fields, which causes cavitations from 35.0% open condition. A potential addition to the results would be implementing compressible solvers for the simulations to verify its influence in higher deviations in the pressure gradient as noted in the 25% of openness.

References 1. W. Gilbert, in Drilling and Production Practice (OnePetro, 1954) 2. H.K. Versteeg, W. Malalasekera, An Introduction to Computational Fluid Dynamics: the Finite Volume Method (Pearson education, 2007) 3. M. Ejaz, E.R. Mahmoud, S.Z. Khan, Eng. Fail. Anal. 110, 104434 (2020) 4. R.T. Knapp, J.W. Daily, F.G. Hammitt, Cavitation (McGraw-Hill Education, 1970) 5. P.L. Skousen, Valve Handbook (McGraw-Hill Education, 2011) 6. S. Tattersall, Meas. Control 49(3), 104 (2016) 7. S. Rashid, A. Ghamartale, J. Abbasi, H. Darvish, A. Tatar, Flow Meas. Instrum. 69, 101579 (2019) 8. G. Hauke, R. Moreau, An Introduction to Fluid Mechanics and Transport Phenomena, vol. 86 (Springer, 2008) 9. F.K. Buffa, J.L. Baliño, in IV Journeys in Multiphase Flows (2017) 10. E.M. Alsafran, M.G. Kelkar, SPE Prod. Oper. 24(02), 249 (2009) 11. C. Yuan, J. Song, M. Liu, Eng. Appl. Comput. Fluid Mech. 13(1), 67 (2019) 12. A. Prosperetti, G. Tryggvason, Computational Methods for Multiphase Flow (Cambridge University Press, 2009)

Chapter 5

Automated Quick Closing Valve System for Three-Phase Flow Holdup Measurement Niederauer Mastelari, Eugênio S. Rosa, Ricardo A. Mazza, and Jordan V. Leite

Abstract Quick closing valve (QCV) is a widely applied technique in laboratory multiphase flow studies. It can be used as a reliable calibration method to pick up the volumetric fraction of the phases on the vertical column when the steady state of multiphase vertical streams is established. In laboratory Three-Phase Flow studies, thousands of trials are performed. Hence, it is suitable to automate QCV systems for holdup measurement. The automation improves costs, standardizes, and minimizes spending time with tiring repetitive measurements. This work presents the methodology and design of an automated QCV for Holdup measurement system for three-phase flow: air, kerosene, and water. For this, direct column heights measurements are obtained by an optical sensor precisely moved up/down. Furthermore, two pressure sensors are used to measure the weight of the three-phase column. Merging the data with the Maximum Likelihood Estimation algorithm, the methodology and its uncertainties are presented. Data Fusion reduces uncertainties, making it possible to check inconsistencies and detect failures. Automated validation tests show that it is possible to accurately measure small columns of less than 1% of the total column. Keywords Quick closing valves · Multiphase flow systems · Holdup measurement · Maximum likelihood estimation algorithm

N. Mastelari (B) · E. S. Rosa · R. A. Mazza · J. V. Leite Faculty of Mechanical Engineering, University of Campinas, Campinas, Brazil e-mail: [email protected] E. S. Rosa e-mail: [email protected] R. A. Mazza e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_5

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5.1 Introduction Quick Closing Valves (QCV) is one of the most used and reliable ways to measure steady-state holdup in multiphase flow systems techniques. QCV picks up the mean of the phases that represents the steady-state measurement of non-intermittent streams associated. Furthermore, it provides a reliable online calibration of in situ phase fraction [1, 2]. The QCV system is formed by a transparent branch delimited by fastacting valves, creating an alternate clear section for measurement and visualization on the experimental loop. It is based on pneumatic ball valves that are usually open, which enable the trapping of the fluid flowing in the pipe. When the valves are triggered, the flow is trapped, which in turn allows for the accurate measurement of the absolute volumetric fraction (holdup) of each phase. A third normally closed valve bypasses the flux to an alternative circuit to avoid damages to pipes bombing and other loop devices [3]. The QCV has been used over time by many researchers for vertical upward and downward two-phase and three-phase flow studies [1, 4, 5]. Flow studies use thousands of QCV points gained in operating hours of the experimental apparatus [6, 7]. Although it is a very accurate method, the direct measurement of liquid heights or weights takes much time working, it is subject to noise, so it is repetitive and costly. With the availability of motion control technology of precision axis by microcontrollers, sensors, and supervisory systems, it is possible to automate experimental setup with reliability, especially Three-Phase Flow QCV systems measurements. This work presents the design of an automated QCV for three-phase flow holdup measurement. This experimental apparatus operates with kerosene, tap water, and air. For this, an optical assembly is precisely moved up/down around the glass pipe, measuring the fluids column heights directly, based on the Refraction Index, IR, of the fluid inside the tube. However, water-Kerosene interface identification is subject to uncertainties due to mixtures and emulsions at the interface and unforeseen wallattached bubbles. To overcome these effects, redundant measurements are performed to obtain the water height from column pressure indirectly. These direct and indirect data are fused using the Maximum Likelihood Estimation (MLE) algorithm to reduce this uncertainty. The QCV System controlled by a supervisory system shows that it is possible to measure precisely small columns of less than 1% of the total column.

5.2 Maximum Likelihood Estimation Algorithm The Maximum Likelihood Estimation Algorithm, MLE, [8, 9] consists of maximizing the probability function of a redundant sample set of redundant measures x1 , x2 , ... , xn , independent and equally distributed, with normal distribution and knowing their uncertainties u1 , u2 , ... , un . By MLE, it is possible to combine this information

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45

to obtain a more probable value and uncertainty. This algorithm is based on the analysis of the likelihood function that is the product of the independent distributions of their average, and it can be synthesized by XF =  =

uXF =

n  1 xi u i2 i=1

(5.1)

1 1 1 + 2 + ... + 2 u12 u2 un

 n   ∂ XF i=1

∂ xi

2 0.5 ui



 =

(5.2)

1 

(5.3)

5.3 QCV Apparatus The QCV system was designed to operate on a mainframe with a 30 m long test three-phase line, vertical z-axis, for upward flow. The system is totally automated and controlled by a supervisory system. It has a section of a 2000 mm long transparent glass pipe with an inner diameter of 26 mm representing the global phenomena involved in the three-phase flow. The glass tube has a pressure tap at the bottom end with 3 mm of inner diameter. Two pressure transmitters are plugged onto the tap and measure the manometric column pressure concurrently. They are set to a scale of 0–2000 mm with 0.5 % of full-scale error. There is a movable structure with a degree of freedom in the z-axis around the glass tube. This mobile platform allows measuring liquid column heights. It slides precisely guided by the precision ball screw and two steel rectified shafts. This recirculating ball screw has a length of 2000 mm, 16 mm in diameter, and 10 mm pitch around. This laminated Ball Screw is specified on Precision Class C07 of Japanese Industrial Standards—JIS with a travel distance error of 0.05 mm/315 mm. It has an inherent negligible backlash. The optical device formed by diode laser and phototransistors, FTs, are located on this mobile platform. It is moved precisely up/down on a vertical axis driven by a step motor with static torque of 6 N.m and coils configured in series. It performs one revolution every 1600 pulses. Two end-stop switches, upper and lower, determine the range of displacement. The distance between them is 1844 mm. For each one test, the mobile platform goes to a lower position zeroing the measurement.

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Fig. 5.1 Optical assembly plant view

5.3.1 Optical Sensor The optical sensor, Fig. 5.1, is composed of a red laser diode, 650 nm, and a set of six phototransistors type BPW85c. FT6 is the most clockwise and positioned at 21◦ , angular reference for the optical sensor. The diode laser is 28 mm from the center of the glass pipe. The centerline of the diode laser and the centerline of the phototransistors belong to the same XY plane. A red laser beam crosses the glass wall of the pipe and fluids, being detected on the other side of the tube by FTs. The index of refraction, IR, of the fluids was used to identify each phase. It is estimated by data fusion of all FTs signals using the MLE algorithm. Each one of FTs is associated to a specific IR: FT1 = 0.990; FT2 = 1.052; FT3 = 1.127; FT4 = 1.217; FT5 = 1.333; FT6 = 1.472. The noise observed in the measurement of light on the phototransistors has a normal distribution, and its standard deviation is inversely proportional to the light intensity received for each photodiode concerning the whole set: Ik 1 = c. u F Tk I

(5.4)

where c is a constant of proportionality canceled in MLE calculus. The parameter Ik is the intensity of light measured by the FTk that is acquired with a 1 kHz sample rate by a commercial microcontroller Freescale Kl25z. The value I is the total intensity of light measured. Therefore, we can estimate IR: IR =

6  k=1

F Tk .u F Tk 2

(5.5)

Figure 5.2 shows the scanning of the three-phase column with FT’s voltage on the left and the data fused on the right. It is possible to identify the water and kerosene column heights, although there are liquid bubbles adhered to the pipe, both in the

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47

Fig. 5.2 Data Fusion of Photo Transistors measurements of the liquid column with 800 mm, 536 mm of water and 264 mm of Kerosene

part that is determined with water and the one determined with air. This causes interference in the measurement. The height of the liquid column, L L , is easier to obtain and less prone to errors. It is determined when there is a rapid transition of the refractive index, from IR>1.45 to IR 1.45. The experimental uncertainty of the water column is determined by the last detection of water and the first detection of kerosene.

5.4 Methodology of Three-Phase Holdup Measurement To obtain Three-Phase Holdups and their uncertainties, the QCV apparatus makes the measurements: L L , Lwso ,and u L wso . Simultaneously, two equal industrial pressure transmitters measure the manometric pressure, L p1 and L p2 , redundantly sharing the same pressure tap on at the bottom end of the glass tube. MLE merges these results to obtain the height of the water column with less uncertainty.

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5.4.1 Holdup of Water Fusing Indirect and Direct Measurements Two indirect measurements of the water column height, L W p1,2 , are obtained by L L , L p1 , and L p2 . Here, we are considering that the pressure taps are located at the ends of the piping, and the effect of the air column on gauge pressure is negligible. The manometric pressure, P, due to the liquid column is calculated by Eq. 5.6. Indices 1 and 2 refer to the pressure sensors. P1,2 = L k .g.ρk + L w .g.ρw

(5.6)

L L = Lk + Lw

(5.7)

where ρk and ρw are the kerosene and water density, respectively; g is the gravitational acceleration; and Lk is the height of kerosene column. LT is the total length of the column, its value, and uncertainty, u L T , is a well-known fixed value. The pressure transmitter is configured to express the pressure variation in mmH2 O, so the equivalent water column, measured by the pressure transmitters, is defined by Lp1 , Lp2 . P1,2 L p1,2 = (5.8) g.ρw We can calculate two independent water column heights and their uncertainties: L W P 1,2 =  u P1,2 2 =

ρw ρw − ρk

2

 u L p1,2 2 +

L P 1,2 − 1−

ρw ρw − ρk

ρk ρw ρk ρw

LL

(5.9)

2

 uLL 2 +

(L P1,2 − L L ).ρw (ρw − ρk )2

2 u ρk 2

(5.10) The Lwso and its uncertainty, u Lwso , are directly measured by optical sensor. Applying the MLE algorithm,  =

1 u Lwso 2

+

1 u P1 2

+

1



u P2 2

(5.11)

We can obtain the fused value of water column height and it respective uncertainty: Lw =

  1 1 1 1 . L + L + L W W P1 W P2  u S O 2 S O u P1 2 u P2 2  u Lw =

1 

(5.12)

(5.13)

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49

The water holdup and its uncertainty Hw =  u HW =

1 LT

Lw LT

2

(5.14) 

.u L W + 2

LW LT 2

2 .u L T 2

(5.15)

As for the total column height values, the liquid height is known, with its respective uncertainties, from the value of the water column height. The other values such as air holdup, Ha , kerosene holdup, Hk , and their uncertainties are consequently obtained.

5.5 Validation Tests and Results To validate the operation of the QCV system, tests were performed controlled by the supervisory system. Table 5.1 shows three different arrangements of volumetric fractions measured by the Automated QCV system. It can be seen that the system measures low-height columns precisely. In the last test, the holdup of kerosene was less than 1% of the total column. A set of measures was performed after ten minutes of air injection, flow rate of Q = 15 l/min, in the column initially well-defined with L L = 337 mm; Lw = 302 mm; Lk = 35 mm. The system was programmed to obtain measurements every two and a half minutes until fifteen minutes after quick closing valves. This is presented in Table 5.2. Due to the intense agitation, it is possible to observe that kerosene particles mix with the water and vice versa. In addition, liquid particles adhere to the surface of the glass column, so there is a decrease in the liquid columns. The values of the heights of water and kerosene vary with time, which is expected with the coalescence of the drops. There is an agreement between the measures.

Table 5.1 Three-phase holdup data obtained from supervisory system with separate fluids, two sets from each trial Measured data Results Lp1 Lp2 LL Lwso u Lwso Ha Hw uH w Hk uH k (mmH2 O) (mm) (%) 507 506 310 311 132 132

517 517 299 301 142 142

572,7 572,9 307,1 308,6 140,5 140,5

301,4 300,1 262,6 262,6 124,0 124,6

2,30 2,30 2,30 2,30 2,00 2,70

0,7093 0,7092 0,8441 0,8434 0,9287 0,9287

15,30 15,23 13,34 13,34 6,30 6,33

0,12 0,12 0,12 0,12 0,10 0,14

13,77 13,85 2,25 2,33 0,84 0,81

0,13 0,13 0,13 0,13 0,11 0,15

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Table 5.2 Three-phase Holdup data after air injection data obtained from supervisory system Measured data Results Time Lp1 Lp2 LL Lwso u Lwso Ha Hw uH w Hk uH k (min) (mmH2 O) (mm) (%) 2,5 5,0 7,5 10,0 12,5 15,0

319 319 320 320 320 321

329 330 330 330 330 330

329,1 329,6 329,9 330,1 330,4 330,4

306,4 305,1 305,1 303,9 303,9 303,9

3,30 3,30 3,30 3,30 3,30 3,30

83,29 83,27 83,25 83,24 83,23 83,23

15,55 15,49 15,49 15,43 15,43 15,43

0,17 0,17 0,17 0,17 0,17 0,17

1,15 1,24 1,26 1,33 1,34 1,34

0,17 0,17 0,17 0,17 0,17 0,17

5.6 Conclusion and Future Perspective The objective of this work was to present a QCV Automated system for Holdup measurement on a three-phase vertical experimental setup. An optical device was developed to scan measuring the heights by the IR of each phase distributed internally to the glass tube. However, these direct measurements are affected by the foam and droplets coalescing at the interface or rushing to the tube wall. Thereby, two more redundant pressure measurements of the liquid column were made. The MLE algorithm was used to merge these data to calculate holdups with an appropriate compromise between reliability and accuracy. Validation tests controlled by a supervisory system show that it is possible to use this method to automatically measure small columns of less than 1% of the total column accurately. Since the three-phase optic sensor was developed for a program on three-phase research, the optic sensor will test in an experimental apparatus 30 m high for upward vertical water-kerosene-air. The experimental setup uses stainless steel tube with a 24.5 mm ID. The QCVs have valves apart of 2000 mm in a pipe of glass. The inner glass diameter matches the stainless steel pipes. The bottom position of the low QCV is at 5000 mm from discharge to the atmosphere. The tests of the optic sensor for three phases are going to be: (a) how many cycles of the QCV are necessary to have a stable average within an uncertainty; (b) each action of the QCV, the main flow is diverted to an auxiliary line with the same diameter. After the measurement of the holdups, the mainline is aligned again. We have to test how much time is on the test section to return to the steady state. (c) lastly, the uncertainty of the holdup of each phase. The concluding phase of tests is over; the laboratory has a steady state for three phases with uncertainties, with some cycles and the time out to re-establish the steady flow state.

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References 1. H. Wu, C. Tan, X. Dong, F. Dong, Design of a conductance and capacitance combination sensor for water holdup measurement in oil–water two-phase flow. Flow Meas. Instrum. 46, 218 (2015) 2. M. Rocha, J. Simões-Moreira, Void fraction measurement and signal analysis from multipleelectrode impedance sensors. Heat Transf. Eng. 29(11), 924–935 (2008) 3. W.L. Cook, An experimental apparatus for measurement of pressure drop, void fraction, and nonboiling two-phase heat transfer and flow visualization in pipes for all inclinations (Oklahoma State University, 2008) 4. Y. Xue, H. Li, C. Hao, C. Yao, Investigation on the void fraction of gas–liquid two-phase flows in vertically-downward pipes. Int. Commun. Heat Mass Transf. 77, 1 (2016) 5. H. Qian, P. Hrnjak, Void fraction measurement and flow regimes visualization of R134a in horizontal and vertical ID 7 mm circular tubes. Int. J. Refrig. 103, 191 (2019) 6. G. Oddie, H. Shi, L. Durlofsky, K. Aziz, B. Pfeffer, J. Holmes, Experimental study of two and three phase flows in large diameter inclined pipes. Int. J. Multiph. Flow 29(4), 527 (2003) 7. S.M. Bhagwat, A.J. Ghajar, Similarities and differences in the flow patterns and void fraction in vertical upward and downward two phase flow. Exp. Therm. Fluid Sci. 39, 213 (2012) 8. J.M.M. Villanueva, S.Y.C. Catunda, R. Tanscheit, Maximum-likelihood data fusion of phasedifference and threshold-detection techniques for wind-speed measurement. IEEE Trans. Instrum. Meas. 58(7), 2189 (2009) 9. G.C. Jena, in Handbook of Research on Advanced Wireless Sensor Network Applications, Protocols, and Architectures (IGI Global, 2017), pp. 29–61

Chapter 6

Determination of the Discharge Coefficient of Multiphase Meters Through Computational Simulation Matheus Constança de Oliveira and Eugenio F. Simões

Abstract In oil production, multiphase flow, composed of oil, gas, and water, is often found in the production system. The conventional method to measure the flow rates of multiphase flow is by measuring the flow of the individual phases after separation, which is a method commonly used in host platforms. However, there are several cases in which this approach is not feasible or practical, for example, in subsea systems, for the measurement of individual wells in a commingled system. In this context, multiphase flowmeters are usually employed. These meters can measure multiphase flow without phase separation and typically have a Venturi as the primary element. Although Venturi tubes are standardized by International Organization for Standardization (ISO) 5167-4:2003, the standard does not cover the Discharge Coefficient (Cd) for Multiphase Flow. Moreover, there is little academic work devoted to investigating the Discharge Coefficient of Venturis with Multiphase flow. In this work, numerical simulations were performed in Computational Fluid Dynamics based on the finite volume technique in three dimensions, using a non-homogeneous model (i.e., gas-liquid slip ratio =1), through the ANSYS Fluent software, for a Venturi meter in vertical and horizontal positions. Finally, the results are discussed and compared to other academic works. Keywords Discharge coefficient · Multiphase meters through · Computational fluid dynamics (cfd)

6.1 Introduction Multiphase flowmeters are used in the oil and gas industry for well flow measurement. In Brazil, allocation measurement with multiphase meters is ruled by ANP (National Brazilian Petroleum Agency) in resolution n◦ 44 of 10/15/2015 [1]. These meters usually have Venturi tubes as the primary element. However, the ISO standard for Venturis (ISO-5167-4:2003, [2]) does not cover the Discharge Coefficient (Cd) for M. C. de Oliveira (B) · E. F. Simões Universidade Veiga de Almeida, Rio de Janeiro, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_6

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Multiphase Flow. Moreover, there is little academic work devoted to investigating the Cd of Venturis with Multiphase flow. In this work, numerical simulations were performed in CFD (Computational Fluid Dynamics) for a Venturi meter in vertical and horizontal positions, using data from tests performed in the laboratory. Results are discussed and compared to other academic works. The mass flow through a Venturi tube is defined in ISO-5167-4:2003 [2]:   π β 2 D2  2ρ P m = Cd  4 1 − β4 β=

d D

(6.1)

(6.2)

where Cd is the Discharge Coefficient, d is the internal diameter at the throat of the Venturi, D is the internal diameter in the entrance of the Venturi, ρ is the density, and  P is the differential pressure between upstream and the throat of the Venturi. Figure 6.1 shows the Venturi shape and dimensions. The Cd can be interpreted as a correction for the energy loss due to viscous flow and is typically obtained in flow loop tests, using Eq. 6.3. Cd =

π 4

m∗ (β 2 D2 ) √2ρ P √

(6.3)

1−β 4

where m∗ is the measured mass flow [kg/s] (in the flow loop tests). In this work, a gas-liquid bi-phase model was used, and Eq. 6.3 for obtaining the Cd is slightly modified:

Fig. 6.1 SpaceClaim geometry—transversal section

6 Determination of the Discharge Coefficient of Multiphase Meters …

Cd =

π 4

(β √

2

D

2

mm ∗ ) 2ρ  P m 4

55

(6.4)

1−β

1 1−x x = + ρm ρl ρg x=

mg ∗ mm ∗

m m ∗ = m g ∗ + ml ∗

(6.5)

(6.6) (6.7)

where m m is the mixture mass flow [kg/s], m g is the gas phase mass flow [kg/s], m l is the liquid phase mass flow [kg/s], ρ m is the mixture density [kg/m3 ], ρ g is the gas density [kg/m3 ], and ρ l is the liquid phase density [kg/m3 ]. Superscript * denotes that the measured mass flows (inputs for obtaining the Cd). It is worth mentioning the assumptions of the above model: • No mass transfer between phases; • Isothermal flow across the Venturi (no heat transfer between phases and to the external environment); • Both phases have the same temperature; • Radial pressure gradient is negligible. Considering the above arguments, this work aims to determine the discharge coefficient of multiphase flowmeters, destined to monitoring/measurement of the production flow of oil wells composed mostly of oil, gas, and water. The specific objective of this work is to attain the discharge coefficient through computational numerical simulation, using the finite volume method for discussion and comparison to other works.

6.2 Numerical Methodology CFD numerical simulations were performed using the ANSYS Fluent software. The simulation was based on the finite volume technique in three dimensions, using a non-homogeneous model (i.e., gas-liquid slip ratio can be different from 1). Figures 6.1 and 6.2 illustrate the dimensions of the created geometry. In this work, it was decided to use a structured mesh with tetrahedral elements. The unstructured mesh (commonly tetrahedral) allows adaptation to irregular geometries and automatic generation in most commercial CFD software [3, 4]. It discretizes the computational domain in second-order tetrahedral geometries (elements), elements of high quality. In the modeling of the fluid-structure interface, the FLUID142 element was used. The element FLUID142 is defined with the coordinate system composed of eight nodes, widely used to model fluid systems in the permanent or transient

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Fig. 6.2 SpaceClaim Geometry Table 6.1 Characteristics of the mesh in the domain Parameter Value Total number of nodes Total number of elements

81252 447670

regimes and fluid-solid interaction analysis problems through the parameters: velocity, pressure, and temperature. When using the FLUID142 element, velocities are obtained from the moment conservation principle, and pressure is obtained from the mass conservation principle (ANSYS Fluent-Solver Theory Guide, [5]). The different geometric configurations of the FLUID142 element are shown in Fig. 6.3, and Table 6.1 presents the characteristics of the mesh in the domain. The viscosity model chosen was the K-ε [6]. This model is composed of two equations and is based on the concept of turbulent viscosity. The boundary conditions and inputs adopted were according to Table 6.2. This corresponds to test cases #10, #12, and #14 of the experimental work of Pereira [7], where a multiphase meter was

Fig. 6.3 FLUID142 element geometry, Ref. [5] ANSYS fluent-solver theory guide

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Table 6.2 Boundary conditions and inputs for numerical simulations Parameter Value U sl at Venturi inlet (#10) U sl at Venturi inlet (#12) U sl at Venturi inlet (#14) U sg at Venturi inlet (#10) U sg at Venturi inlet (#12) U sg at Venturi inlet (#14) P out ρl ρg D d

0.498 m/s 0.627 m/s 0.539 m/s 0.202 m/s 0.042 m/s 0.326 m/s 10 bar 985.85 kg/m3 8.56 kg/m3 0.137 m 0.07 m

tested in a flow loop with oil, water, and gas. It was considered a homogeneous flow between the liquid phases (oil and water). Usl is the liquid superficial velocity, Usg is the gas’s superficial velocity, and Pout is the pressure at the Venturi outlet. Superficial Velocities are defined according to Eq. 6.8, where in this equation, A is the cross-section area at the Venturi inlet. ⎧ ⎨ Usl =

ml ρl A

⎩ Usg =

mg ρg A

(6.8)

In the simulation, the SIMPLE algorithm was used due to its excellent performance in various problems. The integration method Second-Order Upwind was used to evaluate the convective flow on the faces of the control volume. The Courant number used was equal to 1, intending to accelerate numerical convergence.

6.3 Results and Discussion The results of the discharge coefficients are presented in Table 6.3 along with Fig. 6.4 from [8] to guide the interpretation. For comparison, ISO 5167-4:2013 defines the Cd for a Venturi with machined convergent section, with single-phase flow, as 0,995 (constant) over the Reynolds range (2 × 105 to 1 × 106 ). Reynolds for the mixture flow can be calculated as below: 4m m (6.9) Rem = μm π D

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μm =

μl m l + μg m g mm

(6.10)

where μ is the dynamic viscosity (Pa.s) and Rem is the Reynolds number for the bi-phase mixture flow. Rem for the current case falls below the range of the standard mentioned above, with a value of 6.67E4. Hollingshead et al. [8] found that the Cd value decreases for low Reynolds number; see Fig. 6.4. However, the present values for bi-phase flow are much lower than those presented in Fig. 6.4—suggesting that the recommended Cd from ISO 5167-4:2013 may not be suitable for multiphase flow. Additionally, Cd values on Table 6.3 for vertical and horizontal Venturi were different, with the horizontal being lower than the vertical. This suggests that, when vertical, the gas-liquid interface geometry would lead to less energy loss. However, more simulations with other Venturi sizes and different flow conditions need to be performed to understand the phenomena better. The results from Table 6.3 can also be compared to the results in [9], which also performed CFD simulations to obtain the Cd. However, the work mentioned above used a homogeneous model (i.e., gas and liquid flows with the same velocity), leading to a discharge coefficient above 1. Since the Cd is a correction for the energy loss due to viscous flow, its value shall be below 1. Another work that evaluated the Cd is the experimental work of [10], which also presents Cd values higher than 1 for some cases. However, uncertainties in the measurement system could be the cause of these results.

Table 6.3 Results of discharge coefficients Test point Horizontal 10 12 14

0.546 0.598 0.642

Fig. 6.4 Cd behavior with Re from [8]

Vertical 0.610 0.587 0.642

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Fig. 6.5 Typical examples of pressure contour with streamlines

Typical examples of pressure contour with streamlines are illustrated from Fig. 6.5a–f for Vertical and Horizontal flows.

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6.4 Conclusion and Future Perspective A numerical simulation was performed to investigate the Cd of a Venturi meter with the multiphase flow. The results obtained show a lower value than the one recommended by ISO 5167-4:2003 for single-phase flow. This is because Reynolds number is lower than the value recommended by ISO (2 × 105 to 1 × 106 ). Also, Cd values for vertical and horizontal Venturi were different, with the horizontal being lower than the vertical. This suggests that, when vertical, the gas-liquid interface geometry would lead to less energy loss. However, more simulations with other Venturi sizes and different flow conditions need to be performed to understand the phenomena better. Future work recommends running more cases for vertical, horizontal, and inclined Venturis with several Reynolds in the range recommended by ISO 5167-4:2003, in different flow regimes (e.g., stratified, slug).

References 1. Agência Nacional do Petróleo, Gás Natural e Biocombustíveis - ANP. Resolução anp no 44 de 15/10/2015 (2015). https://www.in.gov.br/materia/-/asset_publisher/ Kujrw0TZC2Mb/content/id/33274114/do1-2015-10-16-resolucao-n-44-de-15-de-outubrode-2015-33274110 2. ISO 5167-4:2003, Measurement of fluid flow by means of pressure differential devices inserted in circular cross-section conduits running full - part 4: Venturi tubes (2003). https://www.iso. org/standard/30192.html 3. M. Vakili, M.N. Esfahany, Chem. Eng. Sci. 64(2), 351 (2009) 4. A.S. Ito, W. Martignoni, 5. U. Manual, Theory Guide (2009) 6. B.E. Launder, D.B. Spalding, in Numerical Prediction of Flow, Heat Transfer, Turbulence and Combustion (Elsevier, 1983), pp. 96–116 7. L.O. Pereira, Metodologia de verificação de desempenho de medidores de vazão de fluido multifásico na medição para apropriação na indústria de óleo e gás natural. Ph.D. thesis, PUCRio (2018) 8. C.L. Hollingshead, M.C. Johnson, S.L. Barfuss, R.E. Spall, J. Pet. Sci. Eng. 78(3–4), 559 (2011) 9. N. Barton, A. Parry, Meas. Control 46(2), 50 (2013) 10. A. Hall, M. Reader-Harris, in Proceedings of the North Sea Flow Measurement Workshop (1999)

Part III

Multiphase Flow Measurement and Instrumentation

Chapter 7

State-of-the-Art Instrumentation and Experimental Methods Developed at the Industrial Multiphase Flow Laboratory over the Last 17 Years: From Gamma-Ray and Wire-Mesh Tomography to Physics-Informed Machine Learning Oscar M. H. Rodriguez, André M. Quintino, Marlon M. Hernández-Cely, and Francisco J. Nascimento

Abstract Research devoted to understanding, modeling, and engineering multiphase flows have been the aims of the Industrial Multiphase Flow Laboratory (LEMI), located in the countryside of the state of São Paulo for nearly two decades now. The accurate determination of the spatial distribution of the phases in multicomponent and turbulent flows in big pipes using a homemade wire-mesh tomographic system, through which the phase fraction and the flow pattern involved can be obtained, the synchronized use of state-of-the-art equipment (PIV, wire-mesh sensor, and highspeed video camera) for the detailed study of turbulence in annular ducts, and the application of collimated gamma-ray densitometry for measurement of interfacial geometrical properties of stratified flows are among the achievements of the LEMI’s team that are described in the paper. In addition, full-scale experiments and recent developments in which data-driven and physics-informed machine learning techniques were used to predict two-phase flow parameters are discussed. The use of artificial intelligence is a new trend in our field of investigation and the results, although promising, indicate that physics is still necessary as big data are not available in many important engineering applications.

O. M. H. Rodriguez (B) · A. M. Quintino · M. M. Hernández-Cely · F. J. Nascimento Industrial Multiphase Flow Laboratory (LEMI), Department of Mechanical Engineering, São Carlos School of Engineering (EESC), University of São Paulo (USP), av. Trabalhador São-carlense 400, 13566-970 São Carlos, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_7

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7.1 Introduction One of the key parameters for multiphase flow predictions is the spatial distribution of the phases, through which the phase fraction and the flow pattern involved can be determined. In order to obtain the spatial distribution of liquid and gas phases, the Industrial Multiphase Flow Laboratory (LEMI) at the campus of São Carlos from the University of São Paulo, Brazil, has implemented multiple state-of-theart instrumentation over the last 17 years. They can be listed as: new development of the Wire-Mesh Tomography [1–4] associated with Particle Image Velocimetry [5–7], spatial resolution enhancement for Gamma-Ray Densitometry [8–10], and the Machine Learning Data Analysis for better prediction and enhancement of the measured parameters [11, 12]. Velasco-Peña and Rodriguez [2] and Tompkins et al. [13] performed extensive literature reviews on the Wire-Mesh Tomography with several applications of this technique in more than 20 years since its introduction by Prasser et al. [14]. The WMT is presented, analyzed, and contrasted with other measurement principles, and recommendations for use are established. Some of the conclusions of these studies are that: • the TWM presents high adaptability to different types of flow; • it has good agreement with other methods for measuring void fraction (on average 11% error) and bubble velocity; • it presents an advantageous cost-benefit relationship; • the measurement is direct, without the need for reconstruction by inverse methods; • it is mostly found in the literature on liquid-gas two-phase flow applications, with little information on liquid-liquid-gas three-phase flows; • it presents two ways of measuring that are by electrical conductivity and by electrical permittivity, being recommended the application of each one depending on the properties of the phases involved; • caution should be taken regarding the intrusiveness in the flow, when one of the phases has high electrical conductivity, when the flow has high turbulence, and when there is reduced phase speed (resulting in a drop or bubble trapped in an intersection of wires); • and piston/agitating, stratified, and annular flow patterns show better measurement agreement. For these last two, it is necessary that the thickness of the layer close to the tube wall is greater than the spacing between the measuring cells.

Measurement techniques applied to multiphase flow are commonly based on the attenuation of radiation [15] and it should not be different with Gamma-Ray attenuation, with the advantage that this radiation has high penetration capacity, low fluctuation intensity, and suffers no refraction when there is a material interface [16],

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not being necessary spatial reconstruction algorithms. Gamma radiation is emitted spherically by a source, and therefore, even with the use of a collimator (element that absorbs radiation in unwanted directions) in the source itself, the beam that leaves the gamma-ray emitter is conical. Therefore, for improvements in spatial resolution, it is recommended to use a collimator also close to the gamma-ray detector [17–19]. GRD is a non-intrusive method, with high spatial resolution and precision, that can be applied to the determination of the phase distribution and also can be attributed to the calibration of other methodologies for the same purpose, but with less restriction on the use of ionizing radiation sources, as is the case of wire-mesh sensors. As two tomographic techniques were listed above, the consequence of their use is obtaining the special distribution of the phases, from which the flow patterns can be extracted [20]. With those information, new techniques can be developed to predict the transitions between flow patterns, specially considering conditions with a low amount of data, as is the case of the liquid/dense-gas flow. For that reason, researchers at LEMI have been developing Hybrid Physics and Machine Learning studies, presenting superior results when considering only Physics models or only Machine Learning-based predictions [12]. This chapter is then dedicated to the description of the most recent conclusions and experimental work developed at LEMI with the concern of flow pattern (instrumentation and prediction).

7.2 Homemade Wire-Mesh Sensor Applied to Two- and Three-Phase Flows Wire-Mesh Tomography (WMT) consists of inserting a mesh of transmitter and receiver wires (spaced apart and arranged perpendicularly) in a cross section of the flow, between which (at each wire intersection, called measurement cell) is performed the electrical impedance measurement, Z MC , as shown in Fig. 7.1. This measurement returns a matrix of electrical impedance values that is translated as a percentage of each of the phases in each measurement cell. The transmitter wires are individually enabled by an electronic selector, which is an electronic circuit composed of logic gates that allow the passage of the excitation signal depending on the bit sequence received from the acquisition system. When a transmitter wire is enabled, the input signal excites it and the receiver wires receive an electrical signal relative to the electrical impedance, Z MC , of the medium present in each measurement cell. The other transmitter wires are grounded by the electronic selector so that there is no leakage of electrical current to other measurement cells [14]. In this way, after the signal of each receiver wire is amplified and acquired by an acquisition system, it is guaranteed to obtain the signal related to the electrical impedance of the medium that composes each measurement cell. By switching the input signal enabled from the first to the last of the transmitter wires and storing the reading of the signals from the receiving wires for each alternation, the matrix

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Fig. 7.1 Schematic arrangement of a Wire-Mesh Tomograph a on a cross section of the flow and b together with electronic elements for measurement

Fig. 7.2 Representation of the electronic circuit for signal amplification from a measurement cell, adapted from [21]

of electrical signals reading of the measuring cells of the WMT of the section is obtained. The signal amplifier located between the receiving wire and an analog input channel of the acquisition system is composed of a self-balanced bridge [21] which is shown in Fig. 7.2. It is composed of an operational amplifier with feedback (composed of a resistor, R S B B , and a capacitor, C S B B , connected in parallel). As the measuring cell has electrical characteristics of a parallel connection between a resistor, R MC , and a capacitor, C MC , the self-balanced bridge provides the gain (ratio between the output signal, V out [t], and the signal of input, V in [t], proportional to the ratio between the feedback impedances, Z S B B , and the measuring cell, Z MC .

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Calibration Procedure The WMT calibration procedure consists of flooding the measurement section with only one of the phases that composes the flow. By alternating the working fluids and obtaining the electrical impedance for each measuring cell, the minimum and maximum values are established, whose intermediate impedance value will be associated with the fraction of the phases that compose the medium of a given measuring cell. For two-phase flow, only one of the properties listed above is necessary to distinguish the fraction of each of the phases in each measurement cell, being associated with the ratio of the sum of the fractions of the two phases to be equal to 1. For three-phase flow, one variable (added phase fraction value) is added in relation to the two-phase flow, being solved by determining both properties (electrical resistivity and electrical permittivity). The distinction between the properties of electrical resistivity (determined by measuring R MC ) and electrical permittivity (determined by measuring C MC ) for different phases of the flow requires that they present sufficient contrast for an adequate reading of the electrical signal by the acquisition system. Signal Analysis Prasser et al. [14] used a square wave excitation signal, V in (t), as illustrated in Fig. 7.3a to distinguish water from air. Note in this figure that the output signal, V out (t), has an accommodation time, after which the measurement is performed. As V out (t) is obtained by the acquisition system only at the point of interest (acquisition period), there is no need for its detailed characterization, being able to work with an acquisition frequency close to the excitation frequency. Da Silva [21] added a logarithmic detector (electronic circuit that provides the magnitude of the oscillating signal in the form of voltage) to the V out (t) output, as shown in Fig. 7.3b, resulting in a voltage signal reading V log (t) and therefore without the need for the acquisition rate to be linked to the excitation frequency of the transmitter wires. Using this solution, the author performed measurements of phase fractions of the three-phase flow water, oil, and air, in which he excited the transmitting wires alternately with a low frequency (75 kHz) to distinguish water from other fluids and a high frequency (1.85 MHz) to distinguish oil from the air. The disadvantage of this type of approach is the need to accommodate the signal after enabling a transmitter wire to occur for each frequency, increasing the time needed to acquire the tomography. Velasco-Peña [3] performed measurements of the phase distribution of the water, oil, and air three-phase flow from the distinction between the real part and the complex part of the output signal of the self-balanced bridge. This distinction can be obtained from V out (t) with the insertion of an IQ demodulator, an electronic circuit that provides two voltage output signals: V Q (t) proportional to the real part of V out (t) and V I (t) proportional to the imaginary part of V out (t), as shown in Fig. 7.3c. The real part of V out (t) is determined by the electrical resistance of the phases present in the measuring cell, R MC , while the complex part of V out (t) is determined by the capacitance of the measuring cell, C MC . The effect of the capacitance of the measuring cell

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Fig. 7.3 Representation of voltage input and output signals for a reading in a TWM measurement cell. a Prasser et al. [14], b da Silva [21], and c Velasco-Peña [3]

is verified in the phase shift of the phase angle, ∅, of V out (t) with respect to V in (t), and the oscillation amplitude of V out (t) (which is the magnitude of this signal) represents the sum of the resistive and capacitive portions of the measuring cell. By subtracting the modulus of the complex part of the signal magnitude for a given measurement cell, the real part of V out (t) is obtained. The proportionalities with the readings taken are adjusted in the calibration procedure. It is important to say that this approach needs twice the acquisition channels in relation to the number of receiver wires or the use of alternating acquisition between V Q (t) and V I (t), performed by an electronic switch. As an alternative for increasing the number of acquisition channels and maintaining a high acquisition rate, Velasco-Peña [3] implemented the IQ demodulator circuit in an acquisition module within the system used. However, it should be pointed out that this solution requires an acquisition rate significantly higher than the oscillation frequency of the V out (t) signal, which was 1 MHz. The determination of the electrical properties of the medium flowing in a given measurement cell must be submitted to electrical permittivity models so that they can be associated with the correct phase fraction. This is due to the fact that for the same fraction value of a given phase in the measuring cell, its arrangement can take different forms that affect the flow of electrons between the transmitter wire and the receiver wire involved. The electrical permittivity models are summarized by [2] in a literature review on TWM applications and, according to these authors, for each flow condition, there is a more adequate model to correlate the electrical permittivity of the measuring cell to the fractions of the phases involved. Evidently, the flows patterns with the presence of dispersions confer greater errors in the determination of the phase fractions due to the fact that the size of a bubble or droplet of the dispersed phase can be in the same order of magnitude or inferior to the gap between the wires of a measurement cell. Experimental Results for the WMT at LEMI The development of the Velasco-Peña [3] study was performed at LEMI and resulted recently in the wire-mesh tomography of three-phase flow and two-phase flow in annular duct associated with PIV (particle image velocimetry) measurements. Those results are as follows.

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7.2.1 Three-Phase Flow WMT Measurement Velasco-Peña [3] performed experiments for a 2 inches internal diameter vertical three-phase flow of water, oil, and air (gas). The superficial velocity varied from 0 to 1.3 m/s for the water (J W ), 0.04–0.6 m/s for the oil (J O ), and 0.01–1.85 m/s for the air (J G ). Essentially, the flow pattern obtained was the Intermittent gas-Bubbly oil according to Bannwart et al. [22]’s classification, consisting of air Taylor bubbles followed by pistons of oil dispersed into small bubbles in the water. A qualitative result of the three-phase vertical flow WMT for a 16 × 16 wire-mesh sensor (WMS) is shown in Fig. 7.4. The tomographic images were taken at the rate of 1000 frames/s. It can be seen from Fig. 7.4a the longitudinal phase distribution over time, obtained from the measurement cells at a central chord (diameter) of the cross section where the WMS is located. Figure 7.4b shows the phase distribution over four cross sections obtained from the WMT relative to the images obtained from a high-speed camera as shown in Fig. 7.4c.

Fig. 7.4 WMT during the passage of a Taylor bubble for vertical flow of water (blue), oil (pink) and air (green). (a) Longitudinal WMT measurement, (b) cross section WMT measurement, (c) lateral high-speed imaging and location of the wire-mesh sensor (WMS) in one of the images. Velasco-Peña [3]

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The obtained fractions of each phase, determined by its time average at the cross section where the WMS was placed in the flow, were compared against the phase fractions measured by the quick-closing-valve method. This method consists of closing both ends of a length section where the flow occurs by pneumatic actuated valves at the same time while a third valve (opened) allows the flux to continue in a by-pass tube. At a given time after the flow interruption, due to the gravity, the three phases are separated and the length that each phase occupies relative to the total length of this section translates their phase fraction. These quantitative results (not presented here for being under peer-review) are extremely dependent on which mixture permittivity model is used. In general, for all the mixture permittivity models used, the values of water and oil fractions are overpredicted by the WMS while the void fraction is underpredicted. It is also concluded from those results that the mixture permittivity models are more adequate depending on the spatial arrangement of the phases at the WMS for a given instant, achieving more precise measurements of each phase fraction.

7.2.2 Two-Phase Flow WMT and PIV in Annular Duct Analysis of velocity profiles, turbulence characteristics, wall law, and Reynolds stresses for annular geometry ducts are found in [5, 7, 23–28]. Using the PIV technique, analyzed turbulence characteristics in single-phase water flow and water-air two-phase flow [6, 7, 28]. Velocity profiles obtained through the PIV system were synchronized with the tomography obtained by a sensor based on the permittivity and electrical conductivity of the fluid [6]. Analysis of the different turbulence characteristics for two concentric geometries of annular ducts was carried out in [7]. The following topics present the analyses performed at LEMI by [5–7], for two concentric annular ducts with radius ratio (K = 0.80 and K = 0.39—internal annular radius over the external annular radius), where the velocity profiles and Reynolds stress are obtained and an analysis of the wall law is made in the bottom side of the duct, for both laminar and turbulent flow in single-phase flow. After that there is the analysis carried out by [6] for water-air two-phase flow in slug pattern, where the synchronism results for the annular wire-mesh sensor (AWMS), PIV, and high-speed camera are shown, highlighting the counter flow, presented in the bottom side of the duct, at the passage of the bubble in the symmetric geometry of radius ratio K = 0.39.

7.2.2.1

PIV Analysis for Laminar and Turbulent Flow in Annular Duct

In laminar flow it was performed a comparison between two configurations of annular (concentric) ducts, where it shows that for the lower radius ratio (K = 0.39), the radial location of the maximum velocity is closer to the inner pipe 0.58 [.] compared to

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geometry radius ratio (K = 0.80) located in 0.52 [-]. For a detailed description of duct laminar flows, please refer to [7]. On the other hand, for turbulent flow, the maximum velocity in the two geometries of radius ratio of K = 0.39 and K = 0.80 is located at a distance of 27.5 mm and 7.8 mm, respectively, in relation to the external wall of the pipe (please refer to the [7] p. 5). Compared to the laminar profiles, the difference in its shape is observed, where the laminar profile is more pointed. While they have similarities with profiles observed in pipes [7].

7.2.2.2

Reynolds Stress Profiles

The effect of the geometry in the Reynolds stress profiles, they are equivalent to those found in pipes, but the radial location of zero Reynolds stress is slightly dislocated toward the inner pipe, Fig. 7.5 [7].

7.2.2.3

Law of the Wall

Considering the analysis of the law of the wall, analyzed in [7], for both radius ratio (K = 0.39 and K = 0.80) and near the inner pipe wall (from Y+ = 40 to Y+ = 300 in

Fig. 7.5 Comparison of Reynolds stress between the two geometries, K = 0.39 (circles) and K = 0.80 (crosses), for several Reynolds numbers (Rebuilt figure using data from [7])

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Fig. 7.6 PIV and log-law velocity profiles near the inner (a) and outer (b) pipe walls of the annular duct for radius ratio α = 0.39 (circle) and α = 0.80 (asterisk) (Rebuilt figures using data from [7])

Fig. 7.6), the dimensionless PIV velocity profile is rather similar to that predicted by the log-law. With Y+ being the wall coordinate: distance of the flow from the wall, made dimensionless with the friction velocity (square root of the shear stress over the density of the fluid) and kinematic viscosity; and U+ being the dimensionless velocity: velocity of the flow in the mainstream direction for a given distance from the wall over the friction velocity.

7.2.2.4

AWMS and PIV Synchronization

Hernández Cely [6] presented the different velocity profiles in the bottom of the annular duct when the bubble passed in different positions and observed the counter flow that occurs and the great impact of the bubble that it generates on the flow, Figs. 7.7 and 7.8.

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Fig. 7.7 Bubble and liquid slug crossing the visualization section at the top of the annular duct and mean axial velocity profiles at the bottom of the annular duct at different positions with respect to the bubble nose for JW = 0.154 m/s and JG = 0.044 m/s [6]

Fig. 7.8 Bubble crossing the visualization section at the top and mean axial velocity profiles over the diametrical vertical plane at the bottom side of the annular duct at different positions with respect to the bubble nose for JW = 0.154 m/s and JG = 0.044 m/s [6]

Synchronized tomographic images obtained by an annular wire-mesh sensor (AWMS) with the velocity profiles obtained using the PIV technique, in different positions to the passage of the bubble and the slug, are shown in Fig. 7.7. In this figure, the influence of the passage of a bubble in the velocity of the lower part of the annular duct and presenting even back flow can be observed.

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7.3 Collimated Gamma-Ray Densitometry for Measurement of Detailed Interfacial Geometrical Properties in Stratified Flow As described above, phase fraction is an extremely important parameter for multiphase flow studies, and measuring it with precision is still a challenge. Therefore, the Gamma-Ray Densitometry (GRD) is a technique that can be applied to multiphase flow to identify the mean density of the chord (when the distance covered by the gamma-ray beam in the flow is greater than its section) or in a given volume (when the distance traveled by the gamma-ray beam in the flow is of the same order as its section). This mean density measurement can be correlated with the phase fraction in the flow section. It uses the principle of attenuation of radiation emitted by a source when crossing a medium, with this attenuation being directly related to its density [15]. GRD has high applicability in situations where there is no access to the flow, as well as in the absence of a transparent section, common situations in flows with high conditions of temperature and pressure, and the presence of chemical agents. Adding to the advantages of this technique is that gamma radiation does not diffract when changing its medium, so the measurement position is precisely known. When the average density of the two-phase flow is determined and considering that the values of the densities of the phases are known, the proportion of each of them is estimated. These values can also be expressed in terms of the gamma-ray intensity, I, that reaches the meter. According to [29], the determination of the void fraction, α, in a chord perpendicular to the flow (x direction) and at a given position on the l (y)) , where I l y axis (axes illustrated in Fig. 7.9), is expressed by α(y) = lnln(II (y)−I ( g (y)−Il (y)) and I g are, respectively, the gamma-ray intensities previously detected by the meter when in the given position y there is only liquid or gas. From this relationship, it can be seen that the fraction of the gas phase at a given position in y is independent of the absorption of gamma rays by other means such as pipe walls or collimators, as

Fig. 7.9 Schematic illustration of the arrangement of elements for measuring the void fraction by GRD in two-phase liquid-gas flows. Flow direction occurs in z

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these are present in the determination of the gamma-ray intensity at that position for liquid-phase and gas-only conditions. Analogously, it is possible to determine the fraction of liquid-liquid phases. Figure 7.9 schematically illustrates the usual arrangement of the gamma-ray emitter, the pipe in which the two-phase flow occurs, a collimator used for gamma-ray beam chord selection for measurement, the gamma-ray detector, and the coordinate axes. The direction of flow occurs on the z-axis.

7.3.1 Static Interface Experiment A static interface experimental apparatus was used to improve the spatial resolution and obtain the know-how of this technique applied to horizontal flow (schematically presented in Fig. 7.9). The set emitter (cesium 137), detector (sodium iodide scintillator with photomultipliers), and signal amplifier are from Berthold, model LB 444. As a first configuration of this apparatus, it did not have a collimator close to the gamma radiation detector. The tube used is steel, and the positioner for micrometric displacement was manufactured in extruded aluminum profiles and its movement through the rotation of threaded rods. The entire system is fixed by a structure made of extruded aluminum profiles. Measurements along the vertical were obtained for the case where there is no collimator close to the detector. Figure 7.10 shows the data obtained for half the tube filled with water and half filled with air. In it, the data are listed in terms of the water fraction, α W , and the normalized height of the tube y/r (where y is the vertical position relative to the tube’s center and r is its radius). It can be seen from this figure that the water/air interface at the center of the tube is presented in terms of water fraction as a transition region rather than abrupt detection. Therefore, there is a need to use a collimator to block radiation that crosses unwanted regions for detection. A first lead collimator was then fabricated (cast and machined) to increase the resolution of the phase fraction measurement. The slit through which the gamma rays would pass was a rectangular section. 1 mm high and 52 mm wide (wide in relation to height to compensate for the loss of energy in the collimator), the lead piece being 25 mm thick. By repeating the tests mentioned above, an increase in the determination of the water/air interface was obtained, as can be seen in Fig. 7.10. It is verified that the interface detection for the case of Fig. 7.10 for the first collimator is still unsatisfactory, presenting a smooth transition region of the water fraction values between 0 and 1 in the interface region. It is estimated that the lead thickness of the collimator could be insufficient to block all the radiation coming from the gamma-ray beam emitted by the source and that the height of the collimator slit should be reduced. Therefore, a second collimator was manufactured, with a thickness of 67 mm (the same value as the thickness of the lead layer present in the gamma-ray source) and with a slit height of 0.5 mm. Figure 7.10 shows the water fraction for different heights

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Fig. 7.10 Water Fraction verses Relative Height (y/r) for different collimators [9]

of the tube, and an abrupt transition between phases in the central region of the tube is verified, where the water/air interface is located. With this second configuration of the collimator, it was possible to satisfactorily determine the phases in a tube with the presence of water and air with a static interface. It was also intended to evaluate the detection of the water/oil interface from the condition of 50% of the height of the tube filled with water and 50% with oil. Figure 7.10 shows the fraction of water in relation to oil. From it, it is concluded that the detection of the water/oil interface also took place satisfactorily based on the new geometry of the collimator.

7.3.2 Dynamic Interface Experiment In addition to the static determination of the phase arrangement in a horizontal tube (water/air and water/oil), it is desired to determine the characteristic of the signal obtained by the gamma-ray detector (statistical arrangement of the phases) when the same tube horizontal is filled with water/air and water/oil and subjected to a wave generator. This solution was established to contemplate dynamic tests prior to application in an experimental facility designed for the study of liquid/dense-gas flows for the detection of phase distribution.

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Fig. 7.11 Time averaged water fraction distribution for relative height (y/r) obtained from the Gamma-Ray Densitometry and an image from the generated wave (water and air) [10]

This solution for dynamic tests proved to be viable from the considerations of (i) maintenance of the average level equivalent to the level in the static condition, so the average position of the interface is invariable for the same water/air and water/oil ratios and (ii) absence of wave reflection at the pipe ends due to the presence of wave absorbers. Although the average interface could be determined through static measurement, it was decided to use transparent tubing so that the waveform, as well as its amplitude and frequency, were determined from a high-speed camera. For this purpose, a mirror was inserted at 45◦ in relation to the horizontal (with gamma radiation passing through it), and a high-speed camera was positioned vertically (so that it obtains images of the same flow segment and is not subjected to gamma radiation). Dye is added to the water to increase contrast during filming. A millimetric pattern is placed inside the tube for the camera calibration procedure so the pixels from the peak and valley of the generated wave can determine its amplitude. Chagas [9] and Chagas et al. [10] present results for three different wave frequencies and three different heights of the interface using water/air and water/oil in a 34 mm diameter horizontal tube. Comparisons between the wave amplitudes from the GRD technique and from the analysis of high-speed camera images obtained an average error of 8% with a maximum value of 12%. Figure 7.11 shows the result of the time average water fraction over the relative height from the GRD technique in comparison with the image obtained from the high-speed camera, from which is extracted the wave amplitude, A.

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Fig. 7.12 Inclinable structure for liquid/dense-gas experiments at LEMI and the project for the Gamma-Ray Densitometry phase distribution measurement

With those dynamic results, the next step of this study is to implement the GRD in the horizontal, inclined, and vertical liquid/dense-gas experiment for phase fraction distribution measurement, as shown in Fig. 7.12.

7.4 Data-Driven and Hybrid Machine Learning Techniques for Prediction of Two-Phase Flow Properties Machine learning is ever-present in everyday life, whereas it is in text categorization [30], handwriting recognition [31], speech recognition, and face detection on social networks. In industrial applications, the use of machine learning occurs alongside the digital transformation of processes. With the wider use of sensors and digitalization of the systems, the available database is improved in terms of volume, variety, and veracity. Those three dimensions are essential for the prediction improvement of data-driven machine learning [32]. In multiphase flow applications, machine learning is being used for flow pattern identification in experiments [33–35], flow pattern transition boundaries in maps [36], and more efforts with multiphase computational fluid dynamics [37]. More

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recently, there are efforts for using data-driven machine learning in liquid-liquid and gas-liquid flow pattern predictions [11, 12, 38] and for hold-up and pressure drop predictions [38]. Those results show that data-driven approaches had good accuracy for the range in which experimental data were available. However, it also reinforces two challenges when developing a data-driven model: the difficulty in generalization and the high demand for experimental data [39]. The integration of physics-based models and machine learning in the so-called physics-informed machine learning is being proposed as an alternative to reduce the dependence on the availability of big data [40]. This approach shows promising results in several research fields: thermodynamics, climate science, aerospace, and turbulence [40–43]. In two-phase flow, Quintino et al. [12] evaluated the benefits of using hybrid-physics-data machine learning in gas-liquid flow pattern prediction, comparing it with experimental data, phenomenological model, and data-driven machine learning.

7.4.1 Data-Driven Technique Applied to Pilot-Scale Experiments The liquid-liquid annular flow pattern, also known as core-annular flow, is observed when the conditions are such that both fluids form continuous and parallel phases. One phase forms a thin film adjacent to the pipe wall, and the second phase forms the flow core. An interesting feature of this flow configuration is that the frictional pressure drop is comparable to that of single-phase flow of the thinner fluid under equivalent flow conditions [44]. The existence of a thin film adjacent to the pipe wall and surrounding a viscous phase has been the explanation for low frictional pressure gradients observed in liquid-liquid pipe flow, and it is attractive for the artificial lift in the oil and gas industry due to its reduced frictional pressure drop. Bannwart et al. [35] investigated the use of artificial neural networks as an alternative to monitoring the core-annular flow pattern using high-frequency pressuresignature with the Gabor transform. A pilot-scale upward-vertical oil-water flow experiment was conducted in a 2 7/8-inch 300 m deep well. The experimental results were used to train and test the artificial neural network, and the model shows good accuracy for the core-annular flow pattern detection.

7.4.1.1

Methodology

The tests were performed in the multiphase flow test facilities of the Research Center (CENPES) of Petrobras, Brazil (for a detailed description of facilities, please refer to [35]). Water and oil were pumped from individual tanks to the test well. A set of Coriolis flow meters delivered the individual water and oil flow rates. After the

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test well, the mixture flows to a 50 m3 separator tank, from which water and oil are pumped to their respective tanks. A control system keeps constant the water level of the separator tank. The fluids used in the tests were dead crude oil of about 2000 mPa.s of viscosity and 950 kg/m3 of density at test temperature (35 ◦ C) and water. The ranges of oil and water superficial velocities were chosen after analysis of a flow map specially drawn for the specific condition of the test [22, 45]. Three measurements of pressure were taken: bottom-hole, wellhead, and return line. The bottom-hole temperature was also measured. A high-frequency pressure transducer (ValidyneTM ) was installed at the wellhead with pressure taps connected 1 m apart from each other. The pressure-signature based only on its Fourier transform may not be sufficient to reveal every characteristic of a flow pattern. The Gabor transform delivers a 2D time-frequency spectral analysis from a raw 1-D signal in time, showing power density spectra related to each time instant. Therefore, the Gabor transform’s result was chosen to train machine learning. A feed-forward back-propagation neural network (FFBP) was implemented. The database contains 819 vectors of 320 elements, and more than 30.000 periods were used for training a FFBP with the following architecture: [20x16] – [10x8] – [1x1]. The chosen activation function was the tangent sigmoid, and the neural network initial weights were randomly chosen. The output label was the detection of the core-annular flow pattern.

7.4.1.2

Results

Figure 7.13 presents pressure-signature signals in time and time-frequency domains for core-annular flow and oil-in-water dispersion. The time-domain signal can be seen at the topmost side, and the other two pictures describe the top view and topological view of the time-frequency signal. Despite some subtle differences, the time-domain signals related to core-annular flow and oil-in-water dispersed flow are similar. On the other hand, the timefrequency maps obtained via the Gabor transform present significant differences. They provided confidence that the machine learning could identify patterns in the data and detect the annular flow pattern. This was confirmed by the medium square error (MSE) convergence of the neural network training, which converged and tended to the value of 0.098. The core-annular flow pattern recognition rate was of the order of 90%.

7.4.1.3

Summaring

A pilot-scale experimental campaign was carried out in a 2 7/8-inch 300 m deep well to assess the practicality of using high-frequency pressure-signature to identify the core-annular flow pattern. However, no significant difference between the annular

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Fig. 7.13 Pressure-signature signals in time and time-frequency domains for core-annular flow (a) and disperse flow (b). Please refer to [35] for further pressure-signature signals

and disperse flow was observed pattern in the time-domain pressure signal. With the Gabor transform, the data pattern was more evident, and using this information for the machine learning training, a core-annular flow pattern recognition rate of 90% was obtained. This technique is suggested as a promising solution for monitoring and control of the water-assisted artificial lift. This study shows the benefits of machine learning in two-phase flow monitoring. Another use for machine learning is for flow parameter prediction. The following work indicates the promising field of using both physical models with the machine learning tools in the gas-liquid flow pattern prediction.

7.4.2 Physics-Informed Technique Applied to Gas-Liquid and Liquid-Liquid Two-Phase Flows Quintino et al. [12] compared the performance of three different modeling approaches for gas-liquid flow pattern transition: phenomenological, data-driven machine learning, and hybrid-physics-data machine learning. The methods were trained with experimental data collected from the literature and evaluated through graphical analysis by direct comparison with experimental data. Lastly, the effects of reducing available data for training on each algorithm’s performance were also assessed.

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Methodology

The machine learning supervised training was performed using experimental twophase flow results from the literature. In total, 8788 experimental data points were collected, each containing the features: duct internal diameter (d), pipe inclination (θ ), viscosities (μ L and μG ), densities (ρ L and ρG ), interfacial tension (σ ), and superficial velocities of each of the phases (JL and JG ). The output label was the observed flow pattern. It is important to note that the flow pattern nomenclature is a subjective choice of the authors. Hence, a unified flow pattern classification was adopted: stratified, intermittent, annular, and dispersed. Random forest was chosen to evaluate whether a data-driven model can predict the flow pattern transition. The architecture of the random forest was composed of 50 decision trees, and each decision tree was trained with a subset of the database. During the training process, the number of decision nodes and terminal leaves of each tree is the result of the impurity minimization on each branch, where the impurity is measured through the Gini index, and it is a measure of the fraction of the data that corresponds to a single flow pattern. The phenomenological model was based on the classical models developed by Taitel and Dukler [46] and Taitel et al. [47] and the unified model for all pipe inclinations of Barnea [48]. For each flow pattern transition boundary, the transitions’ physical phenomena were modeled, and analytical transition criteria were calculated. The hybrid-physics-data machine learning was built using as features both the experimental database and the phenomenological model prediction. Random forest was also chosen as an algorithm with the same architecture as previously discussed. The decision to use the same architectures in both data-driven and hybrid-physicsdata approaches was taken to assure a fair comparison between approaches and evaluate the benefits of the inclusion of physics in the algorithms’ predictions. The chosen metric for evaluating the model’s performance was graphical. A flow pattern map is selected from the database as a reference for comparison purposes; the remaining data is used to train the data-driven and hybrid machine learning. The trained models are then used to predict the transition lines in operational conditions similar to that of the reference flow pattern map. The transition lines predicted by the phenomenological model are also plotted on the map. The graphical process is suggested as a performance appropriate to typical two-phase flow applications in industry, i.e., where the prediction of flow patterns is necessary, but only a few experimental data are available. Figure 7.14 shows a schematic for generating the transition lines for the phenomenological, data-driven, and hybrid-physics-data models in flow pattern maps.

7.4.2.2

Results

Figure 7.15 shows the transition boundaries comparison between the data-driven, phenomenological, and hybrid models for horizontal air-water flow, 25.4 mm pipe at 101325 Pa and 15 ◦ C. One can observe that the three models are capable of predict-

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Fig. 7.14 Flow pattern map comparison by extracting a reference flow pattern map from the database, the remaining data is used to train the algorithms, and transition boundaries are compared [12]

Fig. 7.15 Flow pattern map for horizontal air-water flow, 25.4 mm pipe at 101325 Pa and 15 ◦ C. The lines are the model’s transition boundaries (to see the experimental data points, please refer to [12])

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ing the transition boundaries, with the hybrid model with slightly higher accuracy. It should be noticed that the hybrid model’s behavior tends to follow both the phenomenological and data-driven transition lines. In addition, an analysis was carried out to evaluate the prediction performance of the models tested in this work as a function of the amount of experimental data used to train the machine learning algorithms. The same set of data was used for the tests, representing a percentage of 25% of the database. However, for the training, it started with 75% of the data, but the percentage was gradually decreased to a minimum value of about 2% of the database. This process was performed five times with different randomly selected training and test data. One of the expected benefits of the inclusion of physics-based models in machine learning is to reduce the dependency of the prediction method on a large amount of data. According to [12], when a large amount of data is available, both machine learning approaches, data-driven and hybrid, present similar results and are superior in comparison with the phenomenological model. As the amount of data used for training decreases, the accuracy of the hybrid model becomes progressively better in comparison with the data-driven model. It is worth noticing that when the percentage of the database used for training is less than 5%, only the hybrid model presents an accuracy similar to that offered by the phenomenological model. In contrast, the accuracy of the data-driven model drops significantly. On one hand, it is clear that the accuracy of the data-driven model is more sensitive to the reduction of the amount of data. On the other hand, the hybrid-physics-data model is more robust and keeps the accuracy from dropping to very unsatisfactory levels of accuracy.

7.5 Conclusion and Future Perspective This paper’s goal is to present the main achievements of the Industrial Multiphase Flow Laboratory (LEMI) of the University of São Paulo over the last 17 years regarding Measurement and Instrumentation applied to Multiphase Flow. Our independent developments of the wire-mesh tomography are described. The detailed flow information presented and that has been extracted with this technology is of great scientific value, especially for gas-oil-water three-phase pipe flow due to the evident literature’s lack of data. A fine example of result discussed in this paper that can be got from wiremesh technology was the velocity profiles measured via PIV at various positions with respect to a Slug flow’s bubble passing through an annular duct. The synchronization of the wire-mesh sensor with the PIV system allowed a complete 3-D description of the flow in time and space. A gamma-ray collimator was specially designed to reach micrometric spatial sensitivity never seen before in the area for the study of stratified flow hydrodynamic instability. It was possible not only to plot phase fraction distributions with great accuracy, but also interfacial wave geometrical properties were acquired, as the wave amplitude. It was the first time that gamma-ray densitometry was used for the measurement of dynamic parameters of wavy structures in stratified flow. The data collected by the techniques described above have been

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used by the LEMI’s team for the development of deterministic phenomenological models to predict in-situ volumetric fraction and frictional pressure gradients aiming to design pipelines and equipment, and many papers have been published about it. Nevertheless, the data set has been also used in the last years in machine learning studies. Data-driven algorithms represented as neural networks, random forest, and others have been used to predict flow patterns using laboratory and field data. More recently, hybrid-physics-data or physics-informed machine learning was applied to predict gas-liquid and liquid-liquid flow patterns, volumetric fraction, and pressure drop in pipe flow. One of the most interesting findings is that phenomenological models add robustness to the AI tool. They keep the AI’s prediction quality from getting poorer with the diminishing of the data set’s size when it has the help of physics to guide it. Acknowledgements The authors would like to thank Petróleo Brasileiro S.A. and Equinor Brasil, for financing several parts of the experimental work, and FAPESP, CAPES, and CNPq (311057/2020-9) for the research grants.

References 1. M. Da Silva, E. Dos Santos, U. Hampel, I. Rodriguez, O. Rodriguez, Meas. Sci. Technol. 22, 104020 2. H. Velasco-Pena, O. Rodriguez, Applications of wire-mesh sensors in multiphase flows. Flow measurement and instrumentation, v 3. H. Velasco-Peña, Estudo Topológico de Escoamento Trifásico Óleo-Água-Ar através de Sensor de Impedância de Resposta Rápida do Tipo Wire-Mesh (Tese (Doutorado), Universidade de São Paulo, São Carlos, Brasil) 4. I. Rodriguez, H. Velasco Peña, A. Bonilla Riaño, R. Henkes, O. Rodriguez, Int. J. Multiphase Flow v., 70–113 5. M. Hernández Cely, A. Oliveira, V. Baptistella, O. Rodríguez, in Instrumentation and Measurement Technology Conference (I2MTC), 2017 IEEE International. Date Added to IEEE Xplore, ed. by I.E.E.E. (Italy), pp. 1–6 6. M. Hernández Cely, V. Baptistella, O. Rodriguez, Exp. Therm. Fluid Sci. 98, 563–575 7. M. Hernández-Cely, V. Baptistella, O. Rodriguez, J. Fluids Eng. 141(6), 061102 8. D. Chagas, T. Bicudo, F. do Nascimento, O. Rodriguez, in 10th International Conference on Multiphase Flow, ICMF 2019 (Rio de Janeiro, Brazil) 9. D. Chagas, "estudo experimental do escoamento estratificado gás-denso/líquido em tubulação usando densitometria por raios gama". dissertação de mestrado. escola de engenharia de são carlos, universidade de são paulo 10. Collimated gamma-ray densitometry application for interfacial wave amplitude study in stratified interfaces, Jornada de Escoamentos Multifásicos, JEM, ES (Brazil, Vitória), 2021) 11. A. Quintino, D. da Rocha, O. Rodriguez, in 10th International Conference on Multiphase Flow (Rio de Janeiro) 12. A.M. Quintino, D.L.L.N. da Rocha, R. Fonseca Jr., O.M.H. Rodriguez, J. Fluids Eng. 143, 1–11 13. C. Tompkins, H.M. Prasser, M. Corradini, Nucl. Eng. Des. 337, 205–220 14. H.M. Prasser, A. Böttger, J. Zschau, A new electrode-mesh tomograph for gas-liquid flows. Flow measurement and instrumentation 15. C. Tibiriçá, F. Do Nascimento, G. Ribatski, Exp. Therm. Fluid Sci. 34, 463–473

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16. S. Jayanti, G. Hewitt, Review of literature on dispersed two-phase flow with a view to CFD modelling, AEA-APS-0099 (AEA Petroleum Services, Harwell Laboratory, Oxon, UK) 17. B.G. B., A.M. E, Nucl. Instrum. Methods 158, 443–451 18. Braz. J. Phys. 35(3) 19. R. Hanus, M. Zych, M. Kusy, M. Jaszczur, L. Petryka, Flow Meas. Instrum 60, 17–23 20. G. Elseth, An Experimental Study of Oil-Water Flow in Horizontal Pipes (Norway Telemark University College) 21. M. Da Silva, Impedance sensors for fast multiphase flow measurement and imaging 22. A. Bannwart, O. Rodriguez, C. de Carvalho, I. Wang, R. Vara, J. Energy Res. Technol 126(3), 184 23. Ind. Eng. Chem. 42, 2511–2520 24. Trans. ASME J. Basic Eng. 86, 835–844 25. A. Quarmby, Int. J. Mech. Sci. 9, 205–221 26. A. Quarmby, Appl. Sci. Res. 19, 1–24 27. K. Rehme, J. Fluid Mech. 64, 263 28. F. Rodriguez-Corredor, M. Bizhani, M. Ashrafuzzaman, E. Kuru, J. Fluids Eng. 136, 051203 29. P. Stahl, P. Von Rohr, Exp. Therm. Fluid Sci. 28, 533–544 30. T. Joachims, Text categorization with support vector machines: Learning with many relevant features, 31. C. Cortes, V. Vapnik, Mach. Learn. 297, 273–297 32. A. Gandomi, M. Haider, Int. J. Inf. Manag. 35(2), 137–144 33. H. Wang, L. Zhang, Meas. Sci. Technol. 20(11), 114007 34. M. Al-Naser, M. Elshafei, A. Al-Sarkhi, J. Pet. Sci. Eng. 145, 548–564 35. A. Bannwart, O. Rodriguez, J. Biazussi, F. Martins, M. Selli, C. De Carvalho, Water-assisted flow of heavy oil in a vertical pipe: Pilot-scale experiments 36. T. Trafalis, O. Oladunni, D. Papavassiliou, Ind. Eng. Chem. Res. 44(12), 4414–4426 37. Int. J. Multiph. Flow 131, 103378 38. E. Kanin, A. Osiptsov, A. Vainshtein, E. Burnaev, J. Pet. Sci. Eng. 180, 727–746 39. D. Lazer, R. Kennedy, G. King, A. Vespignani, Science 343(6176), 1203–1205 40. A. Karpatne, G. Atluri, J. Faghmous, M. Steinbach, A. Banerjee, A. Ganguly, S. Shekhar, N. Samatova, V. Kumar, IEEE Trans. Knowl. Data Eng. 29(10), 2318–2331 41. J. Faghmous, V. Kumar, Big Data 2(3), 155–163 42. A.D. Dourado, F. Viana, in AIAA Scitech Forum, American Institute of Aeronautics and Astronautics (Reston, Virginia, 2020) 43. J. Wang, J. Wu, H. Xiao, Phys. Rev. Fluids 2(3), 1–36 44. R.V.A. Oliemans, G. Ooms, Core-Annular Flow of Oil and Water through a Pipeline (Springer, Berlin, 1986), pp. 427–476. https://doi.org/10.1007/978-3-662-01657-2_6 45. O. Rodriguez. Forma da interface e gradiente de pressão no padrão líquido-líquido anular vertical ascendente 46. Y. Taitel, A. Dukler, AIChE J. 22(1), 47–55 47. Y. Taitel, D. Barnea, A. Dukler, AIChE J. 26(3), 345–354 48. D. Barnea, Int. J. Multiph. Flow 13, 1–12

Chapter 8

Wet Gas Metering by Differential Pressure Meters: A Literature Review Renan Fávaro Caliman and Rogério Ramos

Abstract Flow measurement is an essential task to production management, well monitoring, custody transfer, and legislation matters for natural gas production plants. Differential pressure (DP) meters are an economically attractive and reliable alternative for gas flow measurements. On the other hand, wet gas flows occur due to the nature of processes, especially at upstream gas production. Although the performance of DP meters in single-phase flows measurement is well known and consolidated in the literature, for wet gas applications, the wetness produces a positive bias in the differential pressure, called over-reading, inducing an erroneous gas flow rate reading, if not corrected. Motivated to know and manage such over-reading, several authors have investigated this phenomenon, proposing empirical correlations to correct flow readings. In this work, the most relevant correlations are investigated, comparing their performance and limits of each one. Keywords Wet gas · Orifice plate · Venturi · v-cone · Over-reading · Differential pressure meters

8.1 Introduction Wet gas flows represent a particular case of multiphase flows, commonly present in many productive sectors, such as power generation, food, chemical, and mainly in oil and gas industries. In oil and gas production, wet gas is a common matter that engineering must deal with, especially in the case of oil wells operating on the later stage of their production lives, a stage where the water content of the multiphase flow increases and the condensation of heavier hydrocarbon components due to the pressure drop in production lines rises. Additionally, depending on its efficiency,

R. F. Caliman (B) · R. Ramos Research Group for Oil and Gas Flow and Measurement (NEMOG), Department of Postgraduate Studies in Mechanical Engineering, Federal University of Espírito Santo, 514 Fernando Ferrari Av, Vitoria, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_8

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separator vessels leave remains of liquid on the gas outlet. Therefore, to measure the flow rates, expensive two-phase meters are required in those situations. Measurement of gas flow rate is essential to industry, making possible appropriate production management, well monitoring, property transfer, and legislation matters. However, in most gas production fields, the use of complex multiphase meters is economically unviable. Given this scenario, the response of differential pressure (DP) meters to wet gas flows becomes an important research topic, mainly because in most of the productions sights, a single-phase DP meter is already installed to measure dry gas flows. Furthermore, DP meters demand low installation and operation costs, are based on simple principles, are reliable, have a repeatable response, and have many years of operation history on single-phase measurement. Even though the performance of DP meters in single-phase flows measurement is well known and consolidated in the literature, in the case of two-phase applications, the liquid loading causes a positive bias on the pressure differential readings due to phase interactions, called over-reading, and resulting in an erroneous gas flow rate prediction. Aiming to correct this shift, since 1949, with Lockhart and Martinelli [1], authors are researching and developing empirical correlations based on experimental data. This paper reviews the most relevant contributions available in the literature about orifice, Venturi, and inverted cone meters operating with wet gas, comparing each correlation by performance and parameters range limits, aiming the improvement of these correlations in the future with new experimental data for air-water flows and a potential new correlation for wet gas over-reading correction.

8.2 Wet Gas Flow Parameters The most critical parameter in wet gas flow measurement by DP meters is the overreading (OR), representing the positive bias caused by liquid mixed in the flow. It is defined as a false prediction total gas mass flowrate (m f p ) to real dry gas mass flowrate (mg ) ratio, as shown in Eq. 8.1. An approximated practical way to calculate the OR is by the square root of the two-phase differential pressure (PT P ) to gas differential pressure (Pg ) ratio.  m f p ∼ PT P OR = (8.1) = mg Pg In the literature, there are many definitions of “wet gas” based on parameters that represent the amount of liquid on the mixture, as the gas volume fraction (GVF), as Eq. 8.2, where Qg and QT P are the gas and the two-phase volume flowrates, respectively. Based on the GVF, the ISO TR 11583 [2] defines wet gas flow as a two-phase mixture with a minimal GVF of 95%. Furthermore, another parameter largely used to delimit the wet gas flow is the Lockhart-Martinelli (X L M ). Steven et

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al. [3] state that a wet gas flow is any combination of gas and liquid with X L M less or equal to 0.3, i.e., X L M ≤ 0.3. However, the limit of X L M ≤ 0.3 is not consensual in some regulatory texts, like API [4] and NSFOGM [5]. GV F =

Qg QT P

(8.2)

Hall et al. [6] detail the history of X L M parameter, starting with the first definition proposed by [1]. They state that Lockhart and Martinelli [1], studying the pressure losses in two-phase flow, proposed a parameter denoted as X, defined as Eq. 8.3 for low single-phase Reynolds number, where Pl and Pg are the pressure drop of the liquid and gas phases, respectively, if they were flowing alone. For Reynolds number above 2000, the authors proposed Eq. 8.4, where m, ρ, and μ express the mass flow rate, density, and viscosity, respectively, and subscripts l and g mean the liquid and gas phases. It is clear that the first definitions of X L M parameter were not a liquid loading parameter. X2 = Pl = X = Pg 2



Pl Pg

ml mg

1.8 

(8.3) ρg ρl



μl μg

(8.4)

In sequence Murdock [7] proposes a parameter expressed by Eq. 8.5, where Cd,l and Cd,g are the liquid and gas discharge coefficient. This equation was erroneously denoted by X and confused by the Lockhart-Martinelli parameter in some derivative works, even though Murdock has never used it in this fashion.     ρg Cd,g ml Pl = (8.5) X Mur dock = Pg Cd,l mg ρl Lately, Chisholm [8] derived a new parameter as the square root of the ratio of the gas and liquid phase flows inertia showed in Eq. 8.6 and called erroneously by him the “Lockhart-Martinelli correlating group” although it is entirely different. This parameter has no geometrical dependence like the original Lockhart-Martinelli parameter and the Murdock parameter, being a useful non-dimensional tool to compare the liquid loading on different flows. Steven [9], in an inspired text, deduced this parameter represented by Eq. 8.6 as one of the dimensionless groups generated by Buckingham-Pi theorem applied to two-phase flow metering by Venturi. By the force of the everyday use in literature, in the present paper, the Chisholm parameter (Eq. 8.6) will be called Lockhart-Martinelli parameter.     ρg ml (8.6) XLM = mg ρl

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Fig. 8.1 Lockhart-Martinelli definition comparison

The use of these different definitions leads to different values of over-reading, resulting in significant differences in the gas flow correction and the measurement processes. To illustrate these shifts, consider an air-water flow varying only the airflow rate. The airflow rate increase results in a liquid load decrease; in other words, the Lockhart-Martinelli parameter decreases. Considering the Steven et al. [10] correlation and Chisholm’s X L M as base values, the use of different definitions results in 1.2–41% of relative shift on the over-reading estimation, as shown in Fig. 8.1. Kinney and Steven [11] state that industry does not have an universal definition for “wet gas” flows, but a precise definition seems to be unnecessary, whereas the wet gas meter only needs to be capable of estimating the liquid loading and consequently the dry gas flow rate in the project wetness range. Steven et al. [3] explain that to express the line pressure in a two-phase flow, a dimensionless parameter called density ratio (DR) emerges, as defined in Eq. 8.7. It is a simple ratio between the gas density (ρ g ) and the liquid density (ρ l ). This correlation is possible because, unlike gases that carry the pressure effect in the form of density variation, density is not affected by the pressure in most liquid flows. DR =

ρg ρl

(8.7)

The gas densiometric Froude number (Frg ), as shown in Eq. 8.8, is a nondimensional parameter of the gas flowrate, with considerable importance to evaluate the flow pattern in two-phase flows. The term A represents the pipe’s cross-sectional area, D is the inlet diameter, g is the gravitational acceleration, and the other parameters are the same as defined previously.

8 Wet Gas Metering by Differential Pressure Meters: A Literature Review

Fr g =

mg √ ρg A g D



ρg ρl − ρg

91

(8.8)

Until now, every definition was based only on one liquid and one gas. However, there are some situations where the liquid content is a mix of two liquids, such as water and light hydrocarbon (LHC). Consequently, in these situations, the liquid density is a mixture density. Considering that the liquid compounds may be considered as homogeneously mixed, Steven et al. [3] says that the liquid mix density (ρ l,mi x ) is defined as Eq. 8.10, where WLR is the water-to-liquid ratio, as defined in Eq. 8.9 (mw and m L H C are the water and light hydrocarbon mass flow rates respectively), ρ L H C and ρ w are the light hydrocarbon and water density, respectively. mw (m w + m L H C )

(8.9)

ρw ρ L H C ρ L H C × W L R + ρw (1 − W L R)

(8.10)

WLR = WLR =

8.3 Wet Gas Correlations History Throughout history, many researchers studied the liquid loading consequences in gas flows. One of the pioneers in this field was Lockhart and Martinelli [1], evaluating the straight pipe pressure drop for air-liquid flow. On the other hand, Chisholm [12] investigated the orifice plates’ response to steam-water flow, but not from the flow measurement perspective. The first important contribution to wet gas flow measurement field was driven by Murdock [7], the Associated Technical Director for Applied Physics at the Naval Boiler and Turbine Laboratory, Philadelphia. Murdock published a wet gas meter correlation for orifice plates based on experimental data of air-water, steam-water, natural gas-water, and natural gas-distillate flows. The correlation proposed presents a linear behavior of liquid loading, as shown in Eq. 8.11, resulting in a reported uncertainty of ±1.5%. Besides that, Murdock recognized the significance of the term (Pl /Pg )1/2 to describe the relative amount of liquid in the gas flow, which later was called Lockhart-Martinelli parameter, by other authors.   PT P Pl = O R Mur dock = 1.26 +1 (8.11) Pg Pg The enormous relevance on Murdock’s work is based on the publication of 90 experimental data points with a 2.5–4 inches pipe diameter range, a 0.26–0.5 orifice plate beta range, and a liquid loading range, expressed by X L M , from 0.041 to 0.25, based on Eq. 8.6 definition. Despite the significance of his work in wet gas knowledge, Steven et al. [10] state that Murdock assumed a separated flow, although some of

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the data set had other flow patterns. Additionally, he did not consider important parameters for a two-phase flow, such as line pressure and slip ratio, resulting in a limited over-reading correlation, depending only on gas wetness. Chisholm [8] continuing 1958s two-phase flow investigation and, motivated by the limitations of Murdock’s work, studied the pressure line influence and the slip ratio on the orifice plate over-reading in wet gas flows. According to [13], Chisholm was a National Engineering Laboratory (NEL) member, where he developed his experiments using water-vapor combinations at 10, 30, 50, and 70 bar of pressure in 21, 32, and 44 mm pipe diameter. His paper introduced a new liquid loading parameter for orifice plates, called by him as Lockhart-Martinelli parameter, defined by Eq. 8.6. Such definition was applied to develop a new over-reading correction correlation, represented by Eqs. 8.12 and 8.13, where the pressure influence on the over-reading is implicit on the gas-to-liquid density ratio (DR), characterized by the CCh term. He claims a ±2% uncertainty performance comparing to experimental data.  (8.12) O RChisholm = 1 + CCh × X L M + X 2L M  CCh =

ρl ρg

1/4

 +

ρg ρl

1/4

 =

1 DR

1/4 + (D R)1/4

(8.13)

Steven et al. [10] state that Chisholm’s considerations to develop the correlation were an incompressible and stratified flow, with a constant phase velocity ratio (or slip ratio), only dependent on the gas-to-liquid density ratio (DR). These assumptions limited the correlation in a low densiometric Froude number, where the flow pattern is predominantly stratified, i.e., limited in a low gas flow rate. Chisholm did not notice that his assumption of dependence on over-reading with slip ratio was the influence of flow pattern on the over-reading. After Chisholm’s publication, [8], a small amount of research was done on the field of wet gas metering with differential pressure meters. However, with the rising interest in natural gas flows by the industry, [14], a Shell International Exploration and Production employee released research on wet gas metering with a 4 in, 0.4 beta ratio Venturi, showing that the flow pattern governed Venturi’s over-reading in addition with Lockhart-Martinelli parameter and DR relation. Accordingly, with de Leeuw, the flow pattern was a gas densiometric Froude number function and hence the over-reading, with a directly proportional relation, i.e., as gas densiometric, Froude number rises, the over-reading rises, for all other parameters kept unchanged. Another observation was that Venturi’s over-reading was higher than orifice meter over-reading, needing a higher correction factor. A new data set was acquired from the SINTEF Multiphase Flow laboratory by a 4 in, 0.4 beta ratio Venturi in a NitrogenDiesel oil flow, covering a 15 to 90 bar pressure range, gas velocities up to 17 m/s, 1.5 ≤ Frg ≤ 4.8, and 0 ≤ X L M ≤0.3. With these combinations, the minimal gas density tested was 17 kg/m3 , becoming a limitation of the algorithm. The Venturi meter correlation proposed is shown in Eqs. 8.14–8.16, with a stated uncertainty of ±2%.

8 Wet Gas Metering by Differential Pressure Meters: A Literature Review

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Fig. 8.2 Sketch of traditional pressure (Pt), permanent pressure loss (P P P L ), and recovered pressure (Pr) taps, for wet gas estimation

O RChisholm =  CCh = 

ρl ρg

n



 +

1 + C × X L M + X 2L M

ρg ρl

n

 =

n dl = 0.41 n dl = 0.606 × [1 − ex p (−0.746×Frg ) ]

1 DR

(8.14)

n + (D R)n

for 0.5  Fr g < 1.5 for Fr g  1.5

(8.15)

(8.16)

Another important de Leeuw [14]’s contribution was the PLR x X L M relation. The pressure loss ratio (PLR) is the ratio between the permanent pressure loss (P P P L ) and the traditional pressure differential (Pt ), exemplified in Fig. 8.2. According to de Leeuw, the PLR could be used as a Lockhart-Martinelli parameter estimator without external methods. However, the sensitivity of PLR with the amount of liquid was not constant, being quality (x) and pressure dependent. Its use would be suitable for Lockhart-Martinelli values below 0.15, where the PLR x X L M sensitivity is noticeable. Ting [15] studying the orifice plate measurement performance in wet gas flows with low wetness (typically X L M < 0.01) noted an under-reading phenomenon, diverging from the expected behavior of wet gas flows. It has been postulated that the pipe wetted internal surface reduces the wall friction, consequently reducing the pressure drop. Chisholm’s correlation did not take it into account because the under-reading is less than −2%, staying within the correlation’s range of uncertainty. Following up the development of differential pressure technology, [16], members of the NEL 1999-2002 Flow Programme investigated the V-cone meter’s performance on wet gas flows. Two V-cone meters, with 0.55 and 0.75 beta ratios, were used to collect new experimental data in three pressure levels 15, 30, and 60 bar, at a range of Nitrogen and Kerosene flowrates, resulting in a 0.4 to 4.0 gas densiometric Froude number range. The results indicated a solid over-reading dependence on the LockhartMartinelli parameter, a pressure, and a gas densiometric Froude number effect similar to that in Venturi meters. To develop a new correlation applied to V-cone meters, the authors first tested the available data with existing Venturi correction correlations,

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noting that Venturi meters’ over-reading was higher than in V-cone. Hence, the gas flow rate error was overcorrected. Based on these results, new correlations were proposed, one for each beta ratio. Steven et al. [17] unified all the V-cone data available at that time and slightly improved the Stewart et al. [16] correlation as shown in Eqs. 8.17–8.19, resulting in a gas flow rate prediction to ±2% uncertainty. O Rstw = For a DR < 0.027, then

1 + A ∗ X L M + B ∗ Fr g 1 + C ∗ X L M + B ∗ Fr g A = 2.431 B = −0.151 C =1

(8.17)

(8.18)

For DR ≥ 0.027, then: 0.3997 A = −0.013 + √ DR 0.0317 B = 0.0420 − √ DR 0.2819 C = −0.7157 + √ DR

(8.19)

Steven [18] proceeded a theoretical derivation of Chisholm’s model for a homogeneous flow getting a correction correlation, which equation set is the same of Chisholm’s publication (Eqs. (8.12) and (8.13)) changing only the exponent from 1/4 to 1/2. Such a homogeneous model works for all differential pressure meters, dependent only on the Lockhart-Martinelli parameter and the gas-to-liquid density ratio (DR). Steven et al. [19] motivated by Stewart’s [16] observation about beta ratio influence on Venturi’s over-reading studied this behavior in orifice plates, constating that as the beta ratio rises, the over-reading decreases. This effect is far less sensitive in orifice plates than in Venturi, smaller enough to be negligible. Chisholm did not report this beta effect in his publication, and de Leeuw [14] said it was irrelevant in Venturis. Reader-Harris and Graham [20], continuing the de Leeuw [14]’s studies, propose a new correction correlation for Venturis taking into account the Froude and beta effect. They collected new wet gas data from National Engineering Laboratory 4 in a loop using Nitrogen-Exxsol 80, Argon-Exxsol 80, and Nitrogen-water as twophase fluid with 0.4 to 0.75 beta range, 15–60/pressure range, and 0 ≤ X L M ≤0.3. The resulting correlation is shown in Eqs. (8.14), (8.15), and (8.20), where β is the diameter ratio (beta ratio) and H is a function of the surface tension, i.e., a fluid function being 1 for hydrocarbon fluids, 1.35 for ambient temperature water, and 0.79 for hot water (in a wet-steam flow).

8 Wet Gas Metering by Differential Pressure Meters: A Literature Review

 0.8Fr g n R H = max 0.583 − 0.18β 2 − 0.578e H , 0.392 − 0.18β 2

95

(8.20)

Another important observation made by [20] was the fact that the discharge coefficient (Cd ) for wet gas flows differs from dry gas flows. The wetness resulted in a Cd decrease. So, they proposed an appropriated way to estimate the Cd in wet gas flows, given by Eqs. (8.21) and (8.22). 



X L M Cd = 1 − 0.0463e−0.05Frg,th × min 1, (8.21) 0.016 Fr g,th =

Fg β 2.5

(8.22)

Within the data set range, [20] stated a ±3% uncertainty for X L M ≤ 0.15 and ±2.5% uncertainty for 0.15 1.5 ⎩ n S H = √2 Fr g

Testing this correlation with a 0–100% WLR data, which was not used to develop the correlation, [21] found a slight shift on the correction up to −3%, an overcorrection result. Steven et al. [10] with more two-compound liquid loading (water + hydrocarbon) data observed that the water content on the liquid mixture reduced the OR in an almost linear manner. It was a result of transition gas densiometric Froude number increase from stratified to annular mist flow pattern. As the WLR increases, i.e., the water content on the liquid loading and the mixture surface tension also increase, making the flow pattern tend to separated flow. After this finding, Steven et al. [10] developed a new correlation, including the WLR effect as shown in Eqs. (8.14), (8.15), (8.24)– (8.26), where Frg,strat is the transitional gas densiometric Froude number between stratified and annular mist flow. Fr g,strat = 1.5 + (0.2 × W R L)

(8.24)

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  W = 0.4 + (−0.1) × [ex p −W L R ] ⎧ 2   ⎪ 1 W ⎪ √ √ n = ⎨ SH 2 Fr g,star t 2   ⎪ ⎪ 1 W ⎩n S H = √ √ 2 Fr g

(8.25)

for Fr g  Fr g,star t (8.26) for Fr g > Fr g,star t

Accordingly with [10] this algorithm corrected the data within a ±2% uncertainty at a 95% confidence level. Additionally to the new correlation, they brought up again the PLR ×X L M relation proposed by [14] but this time for orifice plates, stating that PLR was sensitive to X L M only in orifice plates with beta larger than 0.5 and this relation was extremely DR dependent. As the DR increases the wet pressure loss ratio (PLRwet ), obtained in wet gas flows, approaches the dry pressure loss ratio (PLRdr y ), obtained in dry gas flows, reducing the PLR to X L M sensitivity. In face of such finding and with the Steven and Hall [21] and Reader-Harris and Graham [20]’s correction correlations for orifice plates and Venturis, respectively, the International Organization for Standardization (ISO) released the ISO TR 11583 [2], recommending a wet gas measurement methodology using the differential pressure meters, including equations to estimate the Lockhart-Martinelli parameter (X L M ) by means of PLR as proposed by [10], as shown in Eqs. (8.27)–(8.29), limited by X L M < (0.45 × D R 0.46 ) and D R  (0.21β − 0.09).   

1 − β 4 1 − Cd2 − Cd β 2 P L Rdr y =  (8.27)  

1 − β 4 1 − Cd2 + Cd β 2 Y = P L Rwet − P L Rdr y

(8.28)

6, 41Y (D R)0.92 β 4.9

(8.29)

XLM =

However, Steven et al. [3] state that the PLRdr y Eq. 8.27, proposed by ISO 5167, had some shifts from the experimental data available and the X L M Eq. 8.29 did not behave well for β>0.55 and was developed only for hydrocarbon liquid loading not for water content. To correct these limitations, Steven et al. [3] proposed a new equation set to estimate the X L M employing PLR. Unfortunately, for confidentiality matters, they did not expose their algorithm but stated less than ± 2% uncertainty for a WLR = 1 and all data set tested, a global ± 4% uncertainty at a 95% confidence level.

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8.4 Results In order to compare the most relevant and available contributions on wet gas correlations published in the literature, each correlation was plotted in a OR verses X L M graphic for an air-water flow with fixed gas flow, pressure, geometry, and DR. The results are shown in Fig. 8.3. All the curves have their own range of uncertainty on the OR correction, as reported by the respective authors. Murdock [7] reported an uncertainty of ±1.5%, Chisholm [8] a range of ± 2%, de Leeuw [14] ± 2% to ±4% depending on the liquid content, Steven et al. [17] a ± 2%, Reader-Harris [20] ±2.5% to ±3%, Steven and Hall [21], Steven et al. [10], and ISO TR 11583 [2] all reported a minimal uncertainty of ± 2%. Analyzing Fig. 8.3, it is clear that the Venturi meters are more sensitive to the liquid content than inverted cone and Orifice meters, presenting a higher over-reading, as reported by de Leeuw [14], up to 35% of absolute difference. Another important constatation is that different installation data results for different correlations even if the fluids and geometries are similar, emphasizing the restricted application range of those correlations. To compare the orifice plate correlation performance, experimental data points were extracted from Murdock [7]’s work, and due to lack of data published on literature, an image data point extraction was done on Steven et al. [10] work using specific software, resulting in an associated uncertainty on those numbers. Figure 8.4 shows all the experimental data extracted and the orifice plate OR correlations. Despite being one of the first contributions on wet gas correction and all limitations of his correlation, the Murdock [7] curve predicts the OR for orifice plates with the lowest fundamental shift, varying from a minimum of 2,3% relative to Precision Flow 8”, 0.72β data to 4,8% relative to Murdock’s data, while the newest Steven et al. [10]

Fig. 8.3 Correlation’s performance for differential pressure technologies operating in wet gas

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Fig. 8.4 Comparison of orifice plate correlations and experimental data for wet gas flow

correlation had a minimum shift of 4,8% relative to Murdock’s data to a maximum of 19% relative to Emerson 4” 0.65β data. Those results show the importance and relevance of Murdock’s work. Furthermore, it can be noticed that Chisholm [8], de Leeuw [14], Steven and Hall [21], and ISO TR 11583 [2] correlations predict the same OR for entire range of Lockhart-Martinelli, over the full range of wet gas, considering the uncertainty of ± 3%. Therefore, for an air-water flow point of view, those correlations have the same response, since the difference between them is based on different parameter ranges like fluids, pipe diameter, diameter ratio, gas densiometric Froude number, and consequently flow pattern as exposed in Sect. 8.3.

8.5 Conclusions and Future Perspectives Despite the vast effort to develop over-reading correction correlations for wet gas flows, the performance of these correlations, at their respective best intervals, is a maximum of ± 2% of error for a 95% confidence level. Such error is out of the exigencies of Brazil’s National Petroleum Agency [22] for natural gas flows, which is ± 1,5%. This points to the need for more research in this field to develop know-how and improve the accuracy of the wet gas flow meters. The international community recognizes the field of studies for solving multiphase flows as a technological challenge that is still current and hard to solve, considering the variables and all phenomena involved. This knowledge becomes even more specific when measuring the flow of these flows, which are restricted by legal requirements. In this sense, the Research Group for Oil and Gas Flow and Measurement (NEMOG, in Portuguese), associated with the Federal University of Espirito Santo, Brazil, joins this effort to develop

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more reliable and more accurate correlations from both experimental and by numerical simulation. Studies related to the independence of installation configurations and flux compositions must be carried out in the future.

References 1. L.R. W, M.R. C, Chem. Eng. Prog. 45, 39–48 2. I.S. Organisation, Measurement of Wet Gas Flow by Means of Pressure Differential Devices Inserted in Circular Cross-Section Conduits (ISO TR 11583, In) 3. R. Steven, S. C, R. Kutty, Orifice meter multiphase wet gas flow performance – the pressure loss ratio solution to the ‘ill-posed’ problem, in North Sea Flow Measurement Workshop 4. State of the Art Multiphase Flow Metering Report (API Publication 2566, Measurement coordination, USA) 5. S. Corneliussen, et al., Handbook of multiphase flow metering (Norwegian Society for Oil and Gas Measurement, Oslo, 2005), vol. 55 6. A. Hall, G. D, R. Steven, A discussion on wet gas flow parameter definition, in North Sea Flow Measurement Workshop 7. J. Basic Eng. 84, 4–419 8. D. Chisholm, J. Mech. Eng. Sci. 19, 128–130 9. R. Steven, Flow Meas. Instrum. 19, 342–349. https://doi.org/10.1016/j.flowmeasinst.2008.05. 004 10. R. Steven, G. Stobie, H. A, B. Priddy, Horizontally installed orifice plate meter response to wet gas flows, in North Sea Flow Measurement Workshop 11. K. J, S. R, 12. D. Chisholm, The flow of steam/water mixtures through sharp-edged orifices. Eng. Boile House Rev 13. C. A, C. S., Evolution of wet gas venturi metering and wet gas correlation algorithims. measurement + control 46 14. de Leeuw R, Liquid correction of venturi meter readings in wet gas flow, in North Sea Flow Measurement Workshop (coordinated by nel) 15. V. Ting. Effects of nonstandard operating conditions on the accuracy of orifice meters. society of petroleum engineers production and facilities 16. D. Stewart, D. Hodges, S. R, P.R. W, 17. R. Steven, K. T, C. Britton. An update on v-cone meter wet gas flow metering research 18. R. Steven, (Andrews, Scotland, UK) 19. R. Steven, T.V. C, G. Stobie, 20. R.H.M. J, G. E, 21. S. R, H. A, Flow Meas. Instrum. J. 20, 141–151 22. Agência Nacional do Petróleo, Gás Natural e Biocombustíveis - ANP. Resolução anp n◦ 44 de 15/10/2015 (2015). https://www.in.gov.br/materia/-/asset_publisher/Kujrw0TZC2Mb/content/id/33274114/do12015-10-16-resolucao-n-44-de-15-de-outubro-de-2015-33274110

Chapter 9

Implementation of a Phase-Sensitive Detector with CORDIC Algorithm in Microcontrollers for Low-Cost EIT Demodulation Procedure Bruno F. de Moura, Adriana M. M. da Mata, Marcio F. Martins, Francisco H. S. Palma, and Rogério Ramos Abstract Electrical impedance tomography is being widely applied to multiphase flow investigation in techniques such as void fraction and dispersed phase flow velocity measurements. The technique is desirable since it presents no ionizing radiation, has a high time resolution, and is relatively low cost. The system works by injecting bidirectional electrical currents into the domain. It is necessary to measure effectively the outcome of the injection, which is commonly a sinusoidal excitation. It is crucial to demodulate this voltage effectively to allow a good accuracy and speed of the system. A way to do this demodulation is by applying a digital phase-sensitive detector, which can divide the signal into amplitude and phase information. It is necessary to multiply a sine and cosine reference to retrieve the signal information, using the Coordinate Rotation Digital Computer (CORDIC) algorithm in this work, to the digitized signal. For that matter, the objective is to implement a digital demodulation procedure utilizing a phase-sensitive detector, generating the reference signals by the CORDIC algorithm using an open-source microcontroller prototyping board, such as the Arduino Due. The accuracy and the noise are investigated further by measuring a reference signal.

B. F. de Moura (B) Faculty of Engineering, Universidade Federal de Catalão, Goiás-BR, Catalão, Brazil e-mail: [email protected] A. M. M. da Mata · M. F. Martins Laboratory for Computational Transport Phenomena (LFTC), The Federal University of Espirito Santo, Vitoria, Brazil e-mail: [email protected] F. H. S. Palma Laboratorio de Metrología Térmica, Universidad de Santiago de Chile, Santiago, Chile e-mail: [email protected] R. Ramos Research Group for Oil and Gas Flow and Measurement (NEMOG), The Federal University of Espirito Santo, Vitoria, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_9

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9.1 Introduction The EIT technique is an imaging procedure that aims to estimate the electrical conductivity of the medium where it is connected to the pipe. Recently, it is being considered for multiphase flow applications, such as the void fraction and flow pattern identification [1, 2], and dispersed phase flow measurement [2, 3]. The method combines the use of a forward problem, modeling the domain effectively to be applied the signal, and an inverse problem. Thus, the conductivity is given by minimizing the difference between the modeled signal and experimental signal [4]. The data acquisition that provides the necessary set of voltages from the current injection must be as much as possible absent of noise and possess a good accuracy [5, 6] since the inverse problem is highly ill-posed [4]. There are several architectures of the EIT data acquisition system in the literature over the years [7–9]. Recently, low-cost designs with low-complexity implementation are being developed to allow easy access to the technology for the researchers. The demodulation procedure is critical in EIT systems since the time resolution can be related to this step. Some low-cost systems encountered in the literature utilize external integrated circuit (IC) [10], sample-and-hold technique [11, 12]. De Moura [13–15] presented a low-cost structure that utilizes an analog rectifying demodulator. Although the system presented satisfactory results, the system’s speed was heavily harmed by the demodulator’s low-pass filter settling time, delivering a speed of 6.4 images/s when reconstructing with 16 electrodes. It is desirable to implement digital demodulation to allow a higher speed for the data acquisition since it does not rely on the settling time of the low-pass filter [16]. Several authors presented this demodulation over time for electrical impedance tomography [16, 17]. Although it is widely applied to EIT system design, it is not implemented in low-cost systems, as mentioned before. To implement microcontrollers, one must have special care when computing the amplitude and phase of the signal since it may have an unwanted burden and may use a good portion of the memory to store the reference signals. For that matter, the CORDIC algorithm, proposed by Volder [18], can output the sin, cos, and atan2 results only by adders/subtractors/shifters and a small angle look-up table (LUT) [19]. Given the reasons above, the objective of this work is to implement a phasesensitive detector combining the CORDIC algorithm for the reference signal to demodulate the sine wave from EIT measurement, giving the amplitude and phase information.

9.2 Digital Phase-Sensitive Demodulator The methodology is summarized in Fig. 9.1. The experimental procedure consists of a sine wave analog input signal Vin (t) of frequency f , amplitude A, phase reference φ, and time t defined by Eq. 9.1 [20].

9 Implementation of a Phase-Sensitive Detector with CORDIC Algorithm …

103

Fig. 9.1 Phase-sensitive detector with CORDIC references

Vin (t) = Asin (2π f t + φ)

(9.1)

In the digital-to-analog (DAC), it is digitized the signal Vd obtained from Vin by sampling at a frequency f s , where f s is multiple of the frequency. The digitized signal is given by Eq. 9.2.   f (9.2) Vin (n) = Asin 2π n + φ , n = 0, 1, . . . , N − 1 fs where N is the number of samples, Nfs is the measurement time, which must be multiple of 1f , being the number of cycles an integer. The sine reference (in-phase, Vref,sin ) must possess the same frequency as the signal and unit amplitude/zero phase. The same trends are necessary for the cosine reference (quadrature, Vref,cos ), and it is defined as Eqs. 9.3 and 9.4.   f (9.3) Vref,sin (n) = sin 2π n , n = 0, 1, . . . , N − 1 fs   f Vref,cos (n) = cos 2π n , n = 0, 1, . . . , N − 1 fs

(9.4)

The output in-phase (IP) and quadrature (QD) are given after multiplying the signal by the reference, according to Eqs. 9.5 and 9.6. IP=

N 1  Vin (n)Vref,cos (n) N n=0

(9.5)

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QD =

N 1  Vin (n)Vref,sin (n) N n=0

(9.6)

The outcome of the multiplication is a direct current (DC) signal given by Eqs. 9.7 and 9.8. IP=

1 Acosφ 2

(9.7)

QD =

1 Asinφ 2

(9.8)

From Eqs. 9.7 and 9.8, the amplitude and phase can be calculated according to Eqs. 9.9 and 9.10.  A = 2 I P 2 + Q D2 (9.9)  φ = atan

QD IP

 (9.10)

9.3 CORDIC Algorithm The CORDIC algorithm is derived from the basic rotation transform equation, given in Eqs. 9.11 and 9.12 [21]. In the algorithm, the rotation angle tanθ is restricted to the integers multiple of 2, so it can be implemented solely on shifting operations. The angle θ = atan(2−1 ) is given by an LUT according to Table 9.1. Combining the sum and differences of the angles in the LUT makes it possible to obtain any angle. xnew = xold cosθ − yold sinθ

(9.11)

ynew = yold cosθ + xold sinθ

(9.12)

The rotation’s equation can be rewritten based on the tan, according to Eqs. 9.13 and 9.14 xnew = cosθ [xold − yold tanθ ]

(9.13)

ynew = cosθ [yold + xold tanθ ]

(9.14)

9 Implementation of a Phase-Sensitive Detector with CORDIC Algorithm … Table 9.1 LUT from the CORDIC algorithm Iteration (i) 2−1 0 1 2 3 4 5 6 7 8 9 10 11 12

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Angle(◦ )

1 0.5 0.25 0.125 0.0625 0.03125 0.015625 0.007812 0.003906 0.001953 0.000976 0.000488 0.000244

45 26.57 14.04 7.125 3.5763 1.78991 0.895173 0.447614 0.223810 0.111905 0.055952 0.027976 0.013988

Table 9.1 LUT from the CORDIC algorithm. The iterative rotation, expressed by additions and shifting operations, is given in Eqs. 9.15–9.17.   xi+1 = K [xi − yi · di · 2−i ]

(9.15)

  yi+1 = K [yi + xi · di · 2−i ]

(9.16)

  z i+1 = z i − [di · atan 2−i ]

(9.17)

where i is the iteration, di is the direction (−1 when z i < 0 and +1 otherwise), and K is the CORDIC gain, given by Eq. 9.18. K =

n 

cos(atan(2−i ))

(9.18)

i =1

The CORDIC gain is K ≈ 0.6073. The sine and cosine outputs are obtained when the x- and y-values are initialized with 1 and 0, respectively (Kumar, 2019). However, the output is scaled with the CORDIC gain. The value of x should be initialized with 1/K to avoid scaling.

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9.4 Results and Discussion The experimental procedure consists of the Arduino Due microcontroller prototyping board based on the microcontroller Atmel SAM3X8E. It has produced a sine wave with 1.5 kHz from the digital-to-analog (DAC) of 1 MHz sampling frequency. The output sine wave is measured by the ADC and the oscilloscope DS203, as shown in Fig. 9.2. As it is shown, the amplitude of the wave is close to 325 mV. The oscilloscope measure is a value between 320 and 296 mV from the output of the DAC. There is some difference between the oscilloscope and the ADC, which may be related to the DS203 low accuracy. From this point, the digitized signal is multiplied by the reference sin/cos signal from the CORDIC algorithm and divided by the number of samples. The output of the algorithm is shown in Fig. 9.3. It is demonstrated that the algorithm outputs a good reference signal. The in-phase (IP)/quadrature (QD) signal is obtained from the reference and the measurement. This value makes it possible to calculate the wave’s amplitude and the phase using the CORDIC algorithm. The calculated amplitude is 335 mV, observing that the method ignores the offset of the signal. The noise produced by this procedure from 100 subsequent measurements is investigated further. The signal-to-noise ratio can be calculated by Eq. 9.19. μ

(9.19) S N R = 20 · log σ Fig. 9.2 DAC output measured by the ADC and an oscilloscope

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Fig. 9.3 CORDIC algorithm output

where μ is the mean and σ is the standard deviation of the signal. It is observed that results for eight samples presented a higher amount of noise, although, for one sine wave cycle, it could be tolerated. The number of samples is directly related to the sampling frequency and, since the microcontroller’s ADC is limited, one must take special care if sampling sine waves at higher frequencies are

Fig. 9.4 SNR of the signal digitized and demodulated

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necessary. One technique possible is to aliasing sample [22]. Therefore, it is tested more cycles of the sine wave. From Fig. 9.4, it is observed that it is possible to sample in 1 sine wave cycle with 16 samples and get satisfactory results. For two sine wave cycles, it required 32 samples. On the following number of cycles, it is necessary for a higher number of samples. It was measured the time of the demodulation process, more specifically 104 measurements (the same as independent adjacent pattern measurement number), taking approximately 9015 µs and a system assembled based on this demodulation may allow over 100 images/s.

9.5 Conclusion and Future Perspective A phase-sensitive detector was implemented with the CORDIC algorithm to demodulate the sine wave amplitude and phase information. Results have shown that the DAC and ADC sampling of the signal is satisfactory. The CORDIC algorithm outputs good results. The demodulation results are according to an amplitude value of 335 mV, while the measurement is between 296 and 320 mV. The SNR presented results about 56 dB using 16 samples and 1 sine wave cycle. Sampling with 8 samples only presented higher noise. It is possible to sample in 2 sine wave cycles to allow aliasing sampling. Thus, the demodulation procedure should significantly improve low-cost process tomography temporal resolution compared to other applications and should be applied in future systems projects. Regarding the measurement of the phase information, it is yet to determine a way to synchronize the signal to the measurement since it does not recover this information appropriately in the current state. Acknowledgements This work was supported by Petrobras and the Brazilian agency Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES). Additionally, I would like to thank the Arduino community for their outstanding support.

References 1. Z. Meng, Z. Huang, B. Wang, H. Ji, H. Li, Y. Yan, Flow Meas. Instrum. 21(3), 268–276. https:// doi.org/10.1016/j.flowmeasinst.2010.02.006 2. M. Wang, J. Jia, Y. Faraj, Q. Wang, C. gang Xie, G. Oddie, C. Qiu, Flow Meas. Instrum. 46, 204–212. https://doi.org/10.1016/j.flowmeasinst.2015.06.022 3. X. Deng, F. Dong, L. Xu, X. Liu, L. Xu, Meas. Sci. Technol. 12(8), 1024–1031. https://doi. org/10.1088/0957-0233/12/8/306 4. M. Vauhkonen, D. Vadasz, P. Karjalainen, E. Somersalo, J. Kaipio, IEEE Trans. Med. Imaging 17(2), 285–293. https://doi.org/10.1109/42.700740 5. A. da Mata, B. de Moura, M. Martins, F. Palma, R. Ramos, Measurement 178, 109401 6. A. da Mata, B. de Moura, M. Martins, F. Palma, R. Ramos, Measurement 174, 108992 7. M. Wang, Y. Ma, N. Holliday, Y. Dai, R. Williams, G. Lucas, IEEE Sens. J. 5(2), 289–299. https://doi.org/10.1109/JSEN.2005.843904

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8. C. Tan, S. Liu, J. Jia, F. Dong, IEEE Trans. Instrum. Meas. 69(1), 144–154 9. T. Oh, H. Wi, D. Kim, P. Yoo, E. Woo, Physiol. Meas. 32(7), 835–849. https://doi.org/10.1088/ 0967-3334/32/7/s08 10. S. Wang, Y. Liu, K. Andrikopoulos, W. Yin, in 2016 IEEE International Conference on Imaging Systems and Techniques (IST) (IEEE, 2016), pp. 273–277 11. M. Soleimani, Electrical impedance tomography system: an open access circuit design (Biomedical engineering online, 5). https://doi.org/10.1186/1475-925X-5-28 12. M. Khalighi, B. Vosoughi Vahdat, M. Mortazavi, W. Hy, M. Soleimani, pp. 1259–1263. https:// doi.org/10.1109/I2MTC.2012.6229173 13. B. de Moura, M. Martins, F. Palma, W. Da Silva, J. Cabello, R. Ramos, Design of a low-cost acquisition system to reconstruct images through electrical resistance tomography (IEEE Latin America Transactions,18(09)) 14. B.F. de Moura, M.F. Martins, F.H.S. Palma, W.B. da Silva, J.A. Cabello, R. Ramos, Measurement 110216 (2021) 15. B.F. de Moura, A.M.M. da Mata, M.F. Martins, F.H.S. Palma, R. Ramos, Meas. Sci. Technol. (2021) 16. R. Smith, I. Freeston, B. Brown, A. Sinton, Meas. Sci. Technol. 11(1054–1062) 17. S. Sun, L. Xu, Z. Cao, J. Sun, W. Yang, IEEE Sens. J. 1(1), 10–1109 18. J. Volder, The cordic computing technique (Association for Computing Machinery, New York, NY, USA). https://doi.org/10.1145/1457838.1457886. In Papers presented at the the march 3–5 19. T. Vladimirova, H. Tiggeler, in Military and Aerospace Applications of Programmable Devices and Technologies Conference 20. C. He, L. Zhang, B. Liu, Z. Xu, Z. Zhang, in 2008 World Automation Congress, pp. 1–4 21. P. Kumar, in 2019 5th International Conference on Advanced Computing Communication Systems (ICACCS), pp. 894–900. https://doi.org/10.1109/ICACCS.2019.8728315 22. D. Holder, Electrical Impedance Tomography: Methods, History and Applications (CRC Press)

Chapter 10

EIT Performance Criteria According to Variations in Conductivity Adriana M. M. da Mata, Bruno F. de Moura, Marcio F. Martins, Francisco H. S. Palma, and Rogério Ramos

Abstract Multiphase flow measurements allow monitoring and control of flows in real time. The extraction of oil in new fields has resulted in the demand for technologies capable of acting in extreme conditions, such as the high conductivity of saline water. Conductivity variation can impact EIT image reconstruction, and an alternative to ensure image reconstruction is to make online calibration viable. To support online calibration solutions, the study of the impact of conductivity on performance criteria can contribute to safety and control. Therefore, this article aims to assess the behavior of EIT performance criteria according to variations in conductivity, through the analysis of the conductivity effects on measurement errors, through the most common indexes for the analysis of measurement errors, for example, signalto-noise ratio (SNR), and reciprocity error (RE). Through this study, it is possible to observe that in low concentrations, it is necessary to increase the signal conditioning gain according to the behavior of the solution conductivity. After a specific concentration, the conductivity stabilizes, and there is no need to adjust the gain. Keywords EIT · Conductivity · Measurement · Error

A. M. M. da Mata (B) · B. F. de Moura Laboratory for Computational Transport Phenomena (LFTC), The Federal University of Espirito Santo, Vitoria, Brazil e-mail: [email protected] M. F. Martins Universidade Federal de Catalão, Goiás-BR, Catalão, Brazil e-mail: [email protected] F. H. S. Palma Laboratorio de Metrología Térmica, Universidad de Santiago de Chile, Santiago, Chile e-mail: [email protected] R. Ramos Research Group for Oil and Gas Flow and Measurement (NEMOG), The Federal University of Espirito Santo, Vitoria, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_10

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10.1 Introduction In the oil industry, the need to explore new oil fields has brought challenges with the increase in gas and water fractions [1]. It was then that multiphase flow measurements gained prominence in the search for solutions for the demand for technologies capable of acting in more unstable flow conditions [1, 2]. In a multiphase flow, flow measurement techniques focus on estimating essential parameters such as individual flow rates and phase fractions [3], such as the volumetric percentage of gas [4, 5]. Multiphase flow measurement (MFM) allows monitoring and control of the flow in real time [1]. Electrical Impedance Tomography (EIT) is a non-invasive technique that allows high temporal resolution and high frame rates [6]. Wang et al. [7] proposed the visualization and measurement technology for multiphase continuous water flows. In a water domain, the EIT technique makes it possible to reconstruct images of the non-conducting phases (oil and gas), but differentiating oil from gas is still a challenge [8]. To measure three-phase flow, dual modality methods are an alternative: EIT-UTT [8], EIT-UT [9], and EIT-URT [10]. In the majority of the tests and applications, the conductivity range utilized is from the tap water, usually between 0.01 and 0.1 S m−1 [11]. In the search for new oil fields, the presence of saline water reaches high conductivities that can vary between 3.0 and 5.3 S m−1 [11]. EIT consists of image reconstruction based on estimating flow’s conductivity through voltage measurements on surrounding electrodes. Tests carried out with two different concentrations of the saline solution indicated that the increase in concentration resulted in low amplitudes in the measurement signals between electrodes [12]. In general, an EIT system may be divided into voltage measurements from an acquisition system and mathematical problem (Forward Problem (FP) e Inverse Problem (IP)). Measurement errors are related to data acquisition, while modeling errors are associated with the mathematical problem. According to Dimas, Uzunoglu, and Sotiriadis [12], increased concentration leads to an increase in measurement errors, which may lead to an increase in modeling errors, but not necessarily. Also, at high concentrations, the reciprocity error (measurement error index) increases above the acceptable level (4–5%) for EIT [13]. Therefore, the study of the conductivity influence becomes relevant. The high temporal resolution of the technique (EIT) enables the study of the dynamic behavior of the system [14]. In the oil industry, real-time imaging becomes essential for the safety control of systems and flow measurement without phase separation. The online calibration [7] guarantees the performance of the EIT system, even with the conductivity variation of the multiphase flow. To support online calibration solutions, the study of the impact of conductivity on performance criteria can contribute to safety and control. Therefore, this article aims to assess the behavior of EIT performance criteria according to variations in conductivity, through the most common indexes for the analysis of measurement errors, for example, signal-to-noise ratio (SNR) and reciprocity error (RE).

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10.2 Methodology The study of the behavior of the performance criteria was based on an EIT measurement system, including body interface (i.e., cables and electrodes), electronic stimulation and measurement hardware, and image reconstruction software [15, 16]. First, we provide an overview of the procedure and describe the objectives of the assessment, Fig. 10.1. The EIT system will be evaluated with the phantom in the central position (with its appropriate markings designed to assess the system noise and accuracy) for all conductivity tested. The measurements will be carried out with the homogeneous tank and variation of the NaCl concentration up to high conductivities. The tests are designed to measure Signal-to-Noise Ratio (SNR) and Reciprocity Error (RE), based on the average measurements for each channel.

10.2.1 EIT Measurement Protocol The following measurement protocol requires an EIT system with electrode connections around a cylindrical tank filled with saline water. In position, Z =0, the radius of the tank is r0 and corresponds to the central region about the positioned electrodes. With the number of electrodes (Ne ) equal to 16, a sequence of EIT data is acquired by repeated measurements, where each frame contains a set of stimulation and measurement. Each data acquisition provides Ne × (Ne − 3) measurements for the electrode pairs, according to the adjacent current injection protocol. We used a tank with saline solution 144.00 mm in diameter and 152.4 mm high, with different conductivity of a saline solution, filled at the height of 92.8 mm with a single electrode, plane placed so that the depth of the saline solution above and below the plane electrode is the same. The target users will be described later. To ensure accurate measurements, the test environment has been controlled to assure that the temperature of the saline solution is kept constant and that the liquid surface is not moving. Two measurement protocols were used.

Fig. 10.1 Methodology of the study of the behavior of EIT performance criteria

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Homogeneous Tank Measurements

In the homogeneous tank filled with saline, an acquisition sequence of at least 300 frames was performed, based on the protocol of Yasin et al. [17]. This acquisition was sequentially repeated at 5 min intervals for 2 h.

10.2.1.2

Non-conductive Target

A single conductivity contrast target of volume, VT , was placed in the tank in the central position. First, a tank measurement sequence is performed as a reference for differential images (homogeneous tank), with at least 100 frames, based on the protocol of Yasin et al. [17]. Then, the target is tested for each solution, and the interval of 1 min is taken to make the measurements. The starting position was in the center of the cylinder. The diameter of the phantom is 21.25 mm, and its insulating material (conductivity ≈ 0).

10.2.2 Quantitative Indexes The performance criteria were calculated from indexes of measurement errors commonly used in EIT systems analysis. 1. SNR. The SNR estimates the degree to which the repeated measurements under unaltered conditions show the same results. The SNR is calculated for each measurement channel as Yasin et al. [17]: S N Ri =

Vi S D(Vi )

(10.1)

2. where S D(Vi ) is the noise amplitude and Vi is the mean voltage measurement for each channel. 3. RE. The RE is related to the difference between the measurement of the channel and its reciprocal measurement (pair of electrodes corresponding to inverting the current direction) according to [17]    V −V  r,i   i (10.2) R Ei =   × 100%  Vi  4. where Vi is the mean voltage measurement for each channel and Vr,i is the mean reciprocal measurement for each channel.

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10.3 Results At first, the measurements were collected in a homogeneous tank according to the first data collection protocol (300 frames, for 2 h). Subsequently, the results of different salt concentrations were measured with a target object in two different measurements, according to the non-conductive target protocol (Sect. 10.2.1.2) using only one representative measure from each solution. Overall, it takes approximately 6 h to perform all measurements after setting up the system. Figure 10.2 shows measurement results for the evaluation of a single-system performance. Seventeen sets of results are obtained in terms of SNR and RE calculated from the measured (homogeneous tank) data. Results in Fig. 10.2 show the performance of the EIT system homogeneous based on the SNR with the behavior of specific channels with varying noise levels. When calculating the average of all channels, the set of measurements that presented the highest SNR average (45.70 db) and median value equal to 43.48 db was evaluated according to their descriptive statistics, as shown in Table 10.1.

Fig. 10.2 Homogeneous tank—300 frames with repeated measurements, removing the measure that oscillated with outliers

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Table 10.1 Descriptive statistics Parameters SNR (dB) Maximum value Average Median value Smallest value Most frequent value Standard deviation Variance

RE

60.46 45.70 43.48 37.01 37.01 6.58 43.27

28.50 1.80 1.74 –29.21 –29.21 10.84 117.40

10.3.1 Need for Gain Adjustment The gain must be adjusted to take full advantage of the ADC’s dynamic range. This result is multiplied by the signal measured at the EIT to increase the signal. As the Arduino has a 3.3 V limit, the signal multiplied by the gain cannot go beyond this. For security, the signal is usually kept close to 1.0 V because the reconstruction may include signal peaks [6]. The measurements shown in Fig. 10.2 (homogeneous tank with tap water) were taken with the 12.2 k gain resistor in signal conditioning, resulting in a gain equivalent to 5.10. For distilled water, the gain of 5.10 has already reached the safety limit in the voltage measurement. Therefore, the 12.2 k resistor was maintained. Afterward, the prepared solutions were described in Table 10.2. According to their respective concentrations, saline solutions were prepared in this study, which increased up to the value of 35g NaCl/L. The INA128P gain equation used as Inamp, according to datasheet: Gain = 1 +

50k R

(10.3)

where R represents the gain resistor. For solution 1 (0.005 mol L−1 ), a 1.68 k resistor was used, which corresponds to a gain of 30.76, as seen in Fig. 10.3. It was necessary to adjust the resistor until the gain of 120.05. From the solution with a molar concentration of 0.020 mol L−1 , the gain stabilized the measurements with measurements close to the safety peak of the Arduino. It was possible to observe the behavior of the solutions from strong electrolytes in low concentration, according to Kohlrausch’s law [18]. We see as a whole that as the salt concentration increases, the conductivity of the solution increases, generating smaller measurement signals. Therefore, when increasing the saline concentration, we need to increase the gain. However, at low concentrations, molar conductivity has a maximum limit according to the electrolyte under analysis, as it can be interpreted through Kohlrausch’s law equation [18]. As the molar concentration increases, the

10 EIT Performance Criteria According to Variations in Conductivity Table 10.2 Saline solutions prepared in this study Molar concentration (mol/L) Mass of NaCl (g/L) 0.005 0.010 0.051 0.100 0.176 0.343 0.599

0.304 0.584 2.980 5.862 10.305 20.068 35.009

117

Mass of NaCl (g/1500 mL) 0.456 0.876 4.469 8.793 15.457 30.102 52.514

Fig. 10.3 Gain adjustment for solutions of NaCl

molar conductivity decreases. That is, the total conductivity of the solution increases less and less until it is not necessary to increase the gain. σm =σm◦ −κ×c1/2

(10.4)

where σm◦ is the limit molar conductivity (for c = 0) and κ is a constant mainly dependent on the stoichiometry of the electrolyte.

10.3.2 Low Conductivities Versus High Conductivities It was observed through Fig. 10.4 that the distilled water (lower conductivity) presented larger measurement errors (lower SNR and higher RE). When increasing the conductivity for tap water (Fig. 10.5), the measurement errors present the most representative data (with the highest SNR). However, the reciprocity error for tap water was larger than 2% (median value = 2.24%). According to EIT hardware standards literature RE < 2% for typical performance, RE < 1% for high performance [12].

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Fig. 10.4 Plot of measurement errors for each measurement channel. a 208 measurements, b 104 pairs of reciprocal measurements

Fig. 10.5 Conductivity of the analyzed solutions

Finally, in the solutions with saline concentration, it was observed that although the SNR has decreased, the reciprocity error has reduced to values that indicate high performance. The behavior of measurement errors can be characterized as the salt concentration increases, according to Table 10.3 (SNR) and Table 10.4 (RE). Measurement errors tend to stabilize from a saline concentration, as noted in the table data, through the median value.

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Table 10.3 Descriptive statistics of SNR Saline solutions Maximum value Average Median value Smallest value Most frequent Standard Variance

0.005 41.21

0.010 48.89

0.051 41.27

0.100 39.75

0.176 37.85

0.343 39.73

0.599 39.06

27.73 25.18

30.37 27.71

27.86 27.11

27.87 27.16

27.71 27.20

27.73 27.09

27.62 26.95

23.12

26.17

19.91

24.42

23.17

23.87

21.73

23.12

26.17

19.91

24.42

23.17

23.87

21.73

5.21 27.12

5.21 27.18

3.00 8.99

2.89 8.35

2.13 4.53

2.67 7.13

2.59 6.70

Table 10.4 Descriptive statistics of RE for saline solutions Saline solutions 0.005 Maximum 7.14 value Average 1.04 Median 1.35 value Smallest –6.14 value Most –6.14 frequent Standard 2.59 Variance 6.71

0.010 6.36

0.051 14.41

0.100 19.68

0.176 17.23

0.343 15.74

0.599 16.23

0.89 1.05

2.19 0.92

0.81 0.52

0.87 0.49

0.79 0.40

0.98 0.58

–4.14

–9.72

–5.69

–4.95

–4.71

–3.25

–4.14

–9.72

–5.69

–4.95

–4.71

–3.25

2.17 4.72

4.59 21.07

3.11 9.66

2.58 6.64

2.38 5.67

2.46 6.03

Through Fig. 10.5 and Table 10.5, it is possible to observe that although the conductivity of the solutions has increased significantly, the molar conductivity tends to stabilize. Thus, the molar conductivity may have a more significant impact on the analysis in EIT.

10.4 Conclusion and Future Perspective It was possible to observe the behavior of measurement errors with the increase in the concentration of NaCl solution up to concentrations close to seawater. Although the total conductivity of the solution increases, the molar conductivity decreases due to

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Table 10.5 Stabilization of the molar conductivity of saline solutions Molar concentration Molar conductivity (mol L−1 ) (S m2 mol−1 ) 0.005 0.010 0.020 0.051 0.100 0.176 0.343 0.599

0.012 0.012 0.012 0.011 0.011 0.010 0.008 0.008

interionic attractions that delay ion movement. The increase in saline concentration stabilized the SNR and reciprocity errors, which may be related to the stabilization of molar conductivity (Table 10.5). The molar conductivity tends to stabilize. Thus, the molar conductivity may have a more significant impact on the analysis in EIT. However, although we observe the stabilization of measurement errors, it is impossible to guarantee that there will be the same behavior in the quality analyses of image reconstruction. Therefore, further investigation is necessary to assess the effects of increased concentration on image quality and to ensure the study of the EIT equipment’s operating range and its performance in saline waters. Understanding the behavior of saline solutions to EIT reconstruction can aid the development of new techniques mainly because the multiphase flow with saline water is a reality that demands practical answers. For future work, identifying the parameters that most interfere in these measurements is fundamental for the application of computational intelligence and improvements in the modeling of the problem. The calculated gain resistors can explore the trend of the gain concerning the increase in electrical conductivity.

References 1. G. Falcone, G. Hewitt, C. Alimonti, B. Harrison, JPT, J. Pet. Technol. 54(4), 77–84. https:// doi.org/10.2118/74689-JPT 2. R. Thorn, G. Johansen, B. Hjertaker, Meas. Sci. Technol. 24(1). https://doi.org/10.1088/09570233/24/1/012003 3. Y. Yan, L. Wang, T. Wang, X. Wang, Y. Hu, Q. Duan, Flow Meas. Instrum. 60, 30–43. https:// doi.org/10.1016/j.flowmeasinst.2018.02.017 4. Y. Zhao, Q. Bi, R. Hu, Appl. Therm. Eng. 60(1–2), 398–410. https://doi.org/10.1016/j. applthermaleng.2013.07.006 5. M. Roshani, Flow Meas. Instrum. 75 (2020). https://doi.org/10.1016/j.flowmeasinst.2020. 101804

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6. B. de Moura, A process of technology appropriation: electrical resistance tomography for multiphase pattern measurements 7. M. Wang, J. Jia, Y. Faraj, Q. Wang, C. gang Xie, G. Oddie, C. Qiu, Flow Meas. Instrum. 46, 204–212. https://doi.org/10.1016/j.flowmeasinst.2015.06.022 8. X. Duan, P. Koulountzios, M. Soleimani, IEEE Access 8, 14523–14530. https://doi.org/10. 1109/ACCESS.2020.2966668 9. G. Liang, S. Ren, S. Zhao, F. Dong, Sensors (Switzerland) 19(9) (2019). https://doi.org/10. 3390/s19091966 10. G. Liang, S. Ren, F. Dong, IEEE Trans. Image Process 29, 4099–4113. https://doi.org/10.1109/ TIP.2020.2969077 11. J. Jia, M. Wang, H. Schlaberg, H. Li, Flow Meas. Instrum. 21(3), 184–190. https://doi.org/10. 1016/j.flowmeasinst.2009.12.002 12. C. Dimas, N. Uzunoglu, P. Sotiriadis, Technologies 8(1), 13. https://doi.org/10.3390/ technologies8010013 13. J. Kourunen, T. Savolainen, A. Lehikoinen, M. Vauhkonen, L. Heikkinen, Meas. Sci. Technol. 20(1). https://doi.org/10.1088/0957-0233/20/1/015503 14. V. Torsani, Estudo da influencia do esforço e da posição corporal no esvaziamento pulmonar regional em individuos saudaveis por meio da tomografia de impedância eletrica (Universidade de São Paulo) 15. B. de Moura, M. Martins, F. Palma, W. Da Silva, J. Cabello, R. Ramos, Design of a low-cost acquisition system to reconstruct images through electrical resistance tomography. IEEE Lat. Am. Trans. 18(09) 16. A. da Mata, B. de Moura, M. Martins, F. Palma, R. Ramos, Measurement 174, 108992 17. M. Yasin, S. Böhm, P. Gaggero, A. Adler, Physiol. Meas. 32(7), 851–865. https://doi.org/10. 1088/0967-3334/32/7/S09 18. P. Atkins, J. Paula. ‘fisicoquímica’. Technical report, fondo educativo interamericano

Chapter 11

Data-Driven Machine Learning Applied to Liquid-Liquid Flow Pattern Prediction Lívia O. Zampereti, André M. Quintino, and Oscar M. H. Rodriguez

Abstract The two-phase flow pattern prediction in pipes is a crucial design factor for the energy industry, given its influence on the system’s pressure drop and hold-up. A common approach is to use phenomenological models to predict those parameters as a function of the two-phase flow pattern. As more experimental data became available in the literature, the machine learning methodologies also became an option. In this work, we evaluate the use of data-driven machine learning to predict the liquid-liquid flow pattern transition. The database comprises data from the open literature. Although there is not the same amount of liquid-liquid flow data available, unlike for gas-liquid flow, this study shows that it is possible to predict the liquidliquid flow patterns using the data-driven approach regardless of the viscosity ratio. Dimensionless parameters derived from the two-phase flow’s governing equations are used to train XGBoost. Four main groups of flow patterns were used in this work: stratified, intermittent, annular, and dispersed. The algorithm’s hyperparameters are tuned using cross-validation and accuracy as the target metric. In addition, a perflow-pattern-map accuracy is also shown. Keywords Machine learning · Liquid-liquid · Flow pattern

11.1 Introduction Liquid-liquid flow in pipes is typical in many industrial processes, especially in the oil-and-gas industry, where the flow of oil and water can be observed in the pipelines. The two-phase flow pattern directly affects many vital parameters for the pipeline L. O. Zampereti (B) · A. M. Quintino · O. M. H. Rodriguez Industrial Multiphase Flow Laboratory (LEMI), Department of Mechanical Engineering, São Carlos School of Engineering (EESC), University of São Paulo (USP), av. Trabalhador São-carlense, 400, 13566-970 São Carlos, SP, Brazil e-mail: [email protected] A. M. Quintino e-mail: [email protected] O. M. H. Rodriguez e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_11

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systems’ design, such as pressure drop, hold-up, corrosion, and structural vibration [1], and a common approach to predict the flow pattern is using phenomenological models supported by experimental data through closure relations [2]. In horizontal flow, the basic flow patterns include stratified, dispersed, and annular, and for vertical upward flow, the flow patterns include dispersed, intermittent, and annular [3]. Bannwart et al. [3], and Brauner and Maron [4] also proposed relations of the Eotvös number with the occurrence of the core annular flow for those conditions. As more experimental data became available in the literature, the same data could train a machine learning algorithm. Machine learning has been applied for gasliquid flow pattern identification and drawing of boundary lines for flow pattern map prediction [5–7], and for liquid-liquid flow pattern map prediction [8]. This work aims to predict the two-phase flow pattern transition boundaries using data-driven machine learning for liquid-liquid flow in pipes, and special attention will be given to identifying the occurrence of core annular flow.

11.2 Methodology A database was compiled with 8411 experimental points available in the literature containing information about the fluids (density, viscosity, superficial velocities, and interfacial tension), the experimental apparatus (tube diameter and inclination), and the flow patterns defined by the paper’s authors. It is important to note that the flow pattern nomenclature is not standardized. For this work, the flow patterns were categorized into three major groups: (i) stratified, (ii) annular, and (iii) dispersed. It is important to note that some authors also observed the intermittent flow pattern [3]; however, this flow pattern was a rare occurrence in the database, occurring 3.46%, and could also be classified as a dispersed flow pattern [9]. This imbalance and subjective classification could lead to poor accuracy of the intermittent flow pattern [10]. Therefore, for this work, we also considered the intermittent flow pattern in the dispersed category. The data were standardized by subtracting the mean of the training set and dividing the result by the standard deviation. For the input features used for the machine learning training, dimensionless parameters derived through the two-phase flow governing equations were used. Seven parameters were selected, containing information on the dynamic of the flow, fluidsproperties ratio, and geometry [11]. ρo μo σρw D Uw + Uo Uw gρw2 D 3 ; ; ; ; ; ; β √ 2 Uo μw ρw μw μ2w gD

(11.1)

U w and U o are the superficial velocities of the water and oil, respectively, g the gravity, D the pipe internal diameters, p and μ the fluids’ density and viscosity, σ the interfacial tension, and β the pipe inclination.

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The chosen machine learning algorithm was gradient boosting, more specifically the extreme gradient boosting (XGBoost) proposed by [12]. It is a gradient tree boosting regularized, which means that the model is based on regression trees ensemble model. The regression tree is a decision table, where each leaf or each prediction is given by a series of if-else decision conditions with a continuous score, so the XGBoost model trains different tree models in sequence, where each tree corrects the error of the last tree until no other possible improvements can be made. A regularized function and more techniques are applied to prevent overfitting and improve the performance compared to gradient tree boosting. A train/test ratio of 75 and 25% were used and randomly chosen. The train set was used to tune the model through the GridSearch approach, which means the model was trained several times with different combinations of hyperparameters and returns the combination that better fits the database. In other words, the combination of hyperparameters that produces better results to the database: subsample = 0.9, n_estimators = 150, min_child_weight = 10, max_depth = 45, learning_rate = 0.4, gamma = 1.5, and colsample_bytree = 0.9. The algorithms were implemented with Python’s library Scikit-learn [13]. As Quintino [7] proposed, the model should be evaluated graphically to observe the predicted boundary transition in each flow map in comparison with the experimental data. Therefore, the algorithms were evaluated through a per-flow-pattern map, which means that the whole database, except the tested map, was used to train the algorithm. Its prediction is then compared with the tested map experimental data. Another way to evaluate the model is by calculating each flow map’s accuracy and then calculating the mean accuracy. In terms of post-processing, as gradient boosting is an algorithm based on the decision tree, the predicted boundary lines have sharp edges; overfitted support vector machine (SVM) was used to smooth the transition lines. The stratified flow pattern predictions for high inclination cases were also eliminated, as they are seldom observed in practice.

11.3 Results Table 11.1 shows the confusion matrices for the test data, 25% of the database randomly chosen. The diagonal numbers are the correct predictions, and each row represents the experimental reference data, and the column is the predicted flow pattern by the machine learning model. The overall accuracy was 90.1%, which is a high accuracy compared with phenomenological models; nevertheless, more importantly, it is to evaluate the results’ physical coherence, which can be done through inspection of flow pattern maps. Five diverse cases in terms of fluids properties, inclination, and diameters were chosen to assess the coherence of the machine learning algorithm (Table 11.2). It is important to reinforce that none of the test data was used during the training process. The flow pattern maps are presented in Figs. 11.1, 11.2, 11.3, 11.4, and 11.5.

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Table 11.1 Confusion matrix for the test set Annular Annular Dispersed Stratified

408 14 36

Table 11.2 Confusion matrix for the test set Figure μwater μoil (mPa.s) (mPa.s) 1 2 3 4 5

0.970 1.000 1.300 1.000 1.000

28.712 488 799 280 3800

Dispersed

Stratified

21 516 17

12 10 180

Duct inclination (◦ )

Source

1 90 10 0 0

[9] [3] [14] [1] [15]

Fig. 11.1 Prediction of the boundary line compared to experimental data [9]. The solid line shows the data-driven model prediction boundary, the colored dots the experimental data from the literature, and the letters the predicted flow pattern region stratified (S) and dispersed (D)

Figures 11.1 and 11.2 show two cases of flow pattern maps with only two patterns, both showing good accuracy: 99.41 and 97.77%. Figure 11.1 is an example of the interpolation capability of data-driven machine learning, given that the database used in this work has a high concentration of data of low viscosity oil in low pipe inclinations. Figures 11.3 and 11.4 represent cases where the core annular flow occurs, and the dispersed and stratified flow patterns could also be observed. Both have shown high accuracy: 91.49% and 96.08%, respectively. Both figures highlight that the model can predict the stratified, dispersed, and annular transition boundaries. However, in

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Fig. 11.2 2 Prediction of the boundary line compared to experimental data [3]. The solid line shows the data-driven model prediction boundary, the colored dots the experimental data from the literature, and the letters the predicted flow pattern region dispersed (D) and annular (A)

Fig. 11.3 Prediction of the boundary line compared to experimental data [14]. The solid line shows the data-driven model prediction boundary, the colored dots the experimental data from the literature, and the letters the predicted flow pattern region stratified (S), dispersed (D), and annular (A)

Fig. 11.4, one can also observe some inconsistency at the bottom right region of the graph with the experimental data available. The transition boundaries are not consistent with what is commonly predicted by traditional phenomenological models regarding stratified flow. In Fig. 11.5, on the other hand, one can see a misclassification of the stratified region, resulting in low accuracy in this map: 51.56%. This can be explained by the high viscosity of the oil used in experiment 3.8 Pa.s, which is a condition where the database lacks experimental data. It highlights the poor capabilities of the data-driven model in terms of extrapolation beyond the available experimental data.

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Fig. 11.4 Prediction of the boundary line compared to experimental data [1]. The solid line shows the data-driven model prediction boundary, the colored dots the experimental data from the literature, and the letters the predicted flow pattern region stratified (S), dispersed (D), and annular (A)

Fig. 11.5 Prediction of the boundary line compared to experimental data [15]. The solid line shows the data-driven model prediction boundary, the colored dots the experimental data from the literature, and the letters the predicted flow pattern region stratified (S), dispersed (D), and annular (A)

11.4 Conclusion and Future Perspective In this work, the use of data-driven machine learning for flow pattern transition in liquid-liquid flow was evaluated. The model’s training used a database from open literature, and the predictions were evaluated using accuracy, confusion matrix, and graphical representation. It was shown that the liquid-liquid flow pattern could be predicted using datadriven machine learning, even for those conditions where there is an occurrence of (i) only stratified and dispersed; (ii) stratified, dispersed, and annular; and (iii) dispersed and annular. Nevertheless, this work also highlights the data-driven machine

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learning strength and weakness, the former being its capability of high accuracy for interpolation and the latter the low accuracy for extrapolations. For future work, in addition to the flow pattern, the data-driven machine learning algorithm will also be used to predict the liquid-liquid phase inversion; it can provide a complete view of the liquid-liquid flow transitions. Acknowledgements The authors are grateful to USP (the University of São Paulo), CNPq (311057/2020-9), PETROBRAS, and ANP for the funding and research grants.

References 1. O.M. Rodriguez, M.S. Castro, Int. J. Multiph. Flow 58, 114–126 2. D. Barnea, Int. J. Multiph. Flow 13, 1–12 3. A. Bannwart, O. Rodriguez, C. de Carvalho, I. Wang, R. Vara, J. Energy Resour. Technol. 126(3), 184 4. N. Brauner, D. Moalem Maron, Int. J. Multiph. Flow 18(1), 123–140 5. M. Al-Naser, M. Elshafei, A. Al-Sarkhi, J. Pet. Sci. Eng. 145, 548–564 6. T. Trafalis, O. Oladunni, D. Papavassiliou, Ind. Eng. Chem. Res. 44(12), 4414–4426 7. A.M. Quintino, D.L.L.N. da Rocha, R. Fonseca Jr., O.M.H. Rodriguez, J. Fluids Eng. 143, 1–11 8. A. Quintino, D. da Rocha, O. Rodriguez, in 10th International Conference on Multiphase Flow (Rio de Janeiro) 9. J. Trallero, Oil-water flow patterns in horizontal pipes 10. S. Yanmin, A. Wong, M. Kamal, Int. J. Pattern Recognit. Artif. Intell. 23(4), 687–719 11. J.R. Thome, Encyclopedia of Two-Phase Heat Transfer and Flow I: Fundamentals and Methods (A 4-Volume Set) (World Scientific, 2015) 12. T. Chen, C. Guestrin, Proc. ACM SIGKDD Int. Conf. Knowl. Discov. Data Min. 13(17), 785– 794. https://doi.org/10.1145/2939672.2939785 13. F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, . Duchesnay, J. Mach. Learn. Res. 12, 2825–2830. https://doi.org/10.1007/s13398-014-01737.2 14. Grassi, Experimental validation of theoretical models in two-phase high-viscosity ratio liquid– liquid flows in horizontal and slightly inclined pipes 15. Shi, An experimental investigation of high-viscosity oil-water flow in a horizontal pipe

Part IV

Multiphase Flow with Phase Change and Chemical Reactions

Chapter 12

Multiphase Flows with Heat and Mass Transfer Fatemeh Salehi

Abstract Developing new engineering technologies relies on the understanding of complex multiphase flows with heat and mass transfer. Such flows involve two- and four-way interactions between particles/droplets and the continuous career phase in both laminar and turbulent spectrums. This paper provides an overview of these flows in a number of engineering applications including (i) spray flows in diesel and gas turbines, (ii) biomass pyrolysis, and (iii) nanofluid microchannels. The methods addressing such multiphase flows are classified into two main groups including Eulerian–Eulerian and Lagrangian–Eulerian methods. Eulerian–Eulerian methods treat the particle/droplet phase as a continuous phase along with the carrier phase. Lagrangian–Eulerian techniques track individual particles and droplets while treating the fluid phase surrounding the particles/droplet as a continuum. Eulerian–Eulerian models are more common when the particle/droplet mass loading is high such as biomass pyrolysis that can be modelled in the concept of the Granular flows while Eulerian–Lagrangian methods are more suitable for the secondary breakup and dilute spray regions. For nanofluid applications with low mass loading (4%. It can be seen that the local temperature distribution becomes more uniform for the in-phase configuration at a higher particle concentration.

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Fig. 12.7 Average temperature for different particle concentrations for MCHS with multiple synthetic jets [41]

12.5 Conclusion and Future Perspective The multiphase flows are complex multi-scale, multi-physics flows with heat and mass transfers which are relevant to a wide range of applications such as (i) reacting spray flows, (ii) biomass pyrolysis, and (iii) nanofluid microchannels. While the first application is highly turbulent, the second and third applications are dominantly laminar. Computational fluid dynamic (CFD) simulations of such flows have provided a better understanding of two- and four couplings between the particle/droplet and the carrier phase. The use of both Eulerian–Eulerian and Lagrangian–Eulerian techniques were discussed for such flows. Brief discussions were provided on the main sub-models required to address the physical inter-related phenomena such as dispersion, chemical reaction, thermal decomposition, and phase change. It was also discussed that for nanofluid heat transfer, the mixture model can be effectively applied where it considers one set of governing equations for the mixture. This is a costeffective approach compared to the discrete phase model that tracks particles in Lagrangian frameworks where both rotational and translational movements are considered for the particles. Despite previous efforts in modelling multiphase flows, there are still requirements to develop accurate and cost-effective models for such flows that effectively predict the complex multi-scale multi-physics inter-related phenomena. In particleladen and droplet-laden flows, there is a lack of accurate models for heat and mass transfers between dispersed and continuous phases, particularly when non-spherical droplets/particles and ligaments are presented. Models based on a population balance equation (PBE) and probability density function (PDF) have shown to be effective for modelling poly-size and poly-shape droplets. However, the development of submodels in the concept of turbulent flows is still challenging. In modelling nanofluids, the effects of inertia are small and hence it is common to assume that particles follow the carrier fluid or even adopt mixture models that solve one set of governing equations while computing the properties such as viscosity for the mixture. Nevertheless, there are still challenges in modelling phase change and heat transfer coefficients

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in such flows. The use of machine learning algorithms has created a new pathway to address the challenges in computational fluid dynamic modelling. Access to detailed measurement and direct numerical simulation datasets has fuelled interest in machine learning to extract information from these data, allowing to understand, predict, optimise, and control flows. Data-driven physical models can be developed in both concepts of a posteriori analysis and a priori analysis. They not only provide more accurate physical models but also resolve the case-to-case adjustment issue which is required in existing models. There have been significant efforts in advancing data-driven models for turbulent single-phase reacting and non-reacting flows although there are limited studies for multiphase flows due to the complexity of such flows. The key barrier in advancing data-driven models is the lack of detailed reliable datasets which are necessary for model training. It has been also found that there are no guarantees for performance, robustness, or convergence of ML algorithms. For future efforts, better interactions between fluids researchers with the ML community will help to achieve the interpretability and generalizability of the results.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

J. Lasheras, E. Hopfinger, Annu. Rev. Fluid Mech. 32(1), 275–308 M. Linne, M. Paciaroni, T. Hall, T. Parker, Exp. Fluids 40(6), 836–846 S. Subramaniam, Prog. Energy Combust. Sci. 39(2–3), 215–245 M. Razak, F. Salehi, M. Chishty, A, Flow, Turbul. Combust. 103(3), 605–624 N. Peters, in Progress in Energy and Combustion Science, vol. 10, pp. 319–339 A. Klimenko, R.W. Bilger, Prog. Energy Combust. Sci. 25(6), 595–687 F. Salehi, M. Talei, E. Hawkes, C. Yoo, T. Lucchini, G. D’Errico, S. Kook, Flow, Turbul. Combust. 95(1), 1–28 R. Grout, W. Bushe, C. Blair, Combust. Theory Model. 11(6), 1009–1028 S. Pope, in Progress in Energy and Combustion Science, vol. 11, pp. 119–192 M. Cleary, A. Klimenko, Phys. Fluids 23(11), 115102 F. Salehi, M. Cleary, R.A. Masri, SAE International F. Salehi, M. Cleary, A. Masri, J. Fluid Mech. 831, 719 F. Salehi, M. Cleary, A. Masri, Y. Ge, A. Klimenko, in Proceedings of the Combustion Institute, vol. 36, pp. 3577–3585 Z. Huo, F. Salehi, S. Galindo-Lopez, M. Cleary, A. Masri, in Proceedings of the Combustion Institute, vol. 37, issue 2, pp. 2191–2198 J. Dukowicz, J. Comput. Phys. 35(2), 229–253 S. Rigopoulos, Prog. Energy Combust. Sci. 36(4), 412–443 R. Reitz, Atomisation Spray Technol. 3(4), 309–337 J. Lasheras, C. Eastwood, C. Martinez-Bazan, J. Montanes, Int. J. Multiph. Flow 28(2), 247 (2002). https://doi.org/10.1016/S0301-9322(01)00046-5 W. Sirignano, Fluid Dynamics and Transport of Droplets and Sprays (Cambridge university press) W. Ranz, R.W. Marshall, Chem. Eng. Prog. 48(3), 141–146 L. Pickett, C. Genzale, G. Bruneaux, L.M. Malbec, L. Hermant, C. Christiansen, J. Schramm, SAE Int. J. Engines 3(2), 156–181 Y. Ong, F. Salehi, M. Ghiji, V. Garaniya, Combust. Theory Model. 25(2), 208–234 F. Salehi, M. Ghiji, L. Chen, 82, 108551 C. Gong, M. Jangi, X.S. Bai, Appl. Energy 136, 373–381

146

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25. S. Skeen, J. Manin, L. Pickett, E. Cenker, G. Bruneaux, K. Kondo, T. Aizawa, F. Westlye, K. Dalen, A. Ivarsson, SAE Int. J. Engines 9(2), 883–898 26. F. Payri, J. García-Oliver, R. Novella, J.E. Pérez-Sánchez, Combust. Flame 208, 198–218 27. T. Kan, V. Strezov, J.T. Evans, Renew. Sustain. Energy Rev. 57, 1126–1140 28. R. Miller, J. Bellan, Combust. Sci. Technol. 126(1–6), 97–137 29. X. Ku, T. Li, T. Løvås, Chem. Eng. Sci. 122, 270–283 30. A. Sharma, S. Wang, V. Pareek, D.H. Yang, Chem. Eng. Sci. 123, 311–321 31. H. Zhong, Q. Xiong, L. Yin, J. Zhang, Y. Zhu, S. Liang, B. Niu, X. Zhang, Renew. Energy 152, 613–626 32. J. Clissold, S. Jalalifar, F. Salehi, R. Abbassi, M. Ghodrat, Fuel 273, 117791 33. S. Jalalifar, M. Masoudi, R. Abbassi, V. Garaniya, M. Ghiji, F. Salehi, Energy 191, 116414 34. G. Liang, I. Mudawar, Int. J. Heat Mass Transf. 136, 324–354 35. J. Mohammadpour, A. Lee, M. Mozafari, M. Zargarabadi, S.A. Mujumdar, Int. J. Therm. Sci. 161, 106705 36. O. Mahian, L. Kolsi, M. Amani, P. Estellé, G. Ahmadi, C. Kleinstreuer, J. Marshall, M. Siavashi, R. Taylor, H. Niazmand, S. Wongwises, T. Hayat, A. Kolanjiyil, A. Kasaeian, I. Pop, Phys. Rep. 790, 1–48 37. O. Mahian, L. Kolsi, M. Amani, P. Estellé, G. Ahmadi, C. Kleinstreuer, J. Marshall, R. Taylor, E. Abu-Nada, S. Rashidi, H. Niazmand, S. Wongwises, T. Hayat, A. Kasaeian, I. Pop, 791, 1–59 38. R. Mashayekhi, E. Khodabandeh, O. Akbari, D. Toghraie, M. Bahiraei, M. Gholami, J. Therm. Anal. Calorim. 134(3), 2305–2315 39. M. Sadiq Al-Baghdadi, Z. Noor, A. Zeiny, A. Burns, D. Wen, Therm. Sci. Eng. Prog. 20, 100685 40. J. Mohammadpour, A. Lee, M. Mozafari, M. Zargarabadi, S.A. Mujumdar, Int. J. Therm. Sci. 106705 41. J. Mohammadpour, F. Salehi, M. Sheikholeslami, M. Masoudi, A. Lee, Int. J. Therm. Sci. 167, 107008 42. A. Lee, G. Yeoh, V. Timchenko, A.J. Reizes, Appl. Therm. Eng. 48, 275–288

Chapter 13

Investigation of Vorticity Fields During Two-Phase Flow in Flow Boiling Jeferson D. de Oliveira, Jacqueline B. Copetti, and Elaine M. Cardoso

Abstract Flow patterns are found in many processes involving multiphase flows in the industry. This experimental investigation focuses on the behavior of the vorticity fields found in the liquid-vapor interface during flow boiling of R-600a with heat flux varying from 5 to 20 kW/m2 and fixed mass flux of 890 kg/(m2 s) at a saturation temperature of 17 ◦ C. Tests were performed in a horizontal tube with 1.0 mm ID. Four different flow patterns were identified based on images, and the vorticity fields were investigated using an optical flow method. Keywords Flow patterns · Optical flow · Flow boiling · Vorticity

13.1 Introduction Several experimental studies have been performed to investigate characteristics of flow patterns due to their important influence on the performance of steam generators, evaporators, and condensers. Thus, the study of velocity and vorticity fields contributes to understanding the flow patterns and instabilities observed in two-phase flow systems. In particular, many authors have focused their studies on the flow patterns’ dynamic, including superficial velocities and behavior of the liquid-vapor interface. For example, studies using PIV/PTV have been performed to analyze twoJ. D. de Oliveira (B) FSG-University Center, Center of Innovation and Technology, Department of Mechanical Engineering, Os Dezoito do Forte, 2366, Caxias do Sul 95020-472, RS, Brazil e-mail: [email protected] J. B. Copetti The University of Vale do Rio dos Sinos, Mechanical Engineering Graduate Program, LETEF, Laboratory of Thermal and Fluid Dynamic Studies, Unisinos Avenue, 950, São Leopoldo 93022-750, RS, Brazil e-mail: [email protected] E. M. Cardoso UNESP—São Paulo State University, Post-Graduation Program in Mechanical Engineering, Av. Brasil, 56, 15385-000 Ilha Solteira, SP, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_13

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phase solid-liquid flow [1], cavitation phenomena [2], and others. Nevertheless, such methods are limited to dispersed phases (small bubbles or particles, for example). On the other hand, many other studies have been reported on literature focusing on the behavior of flow patterns [3–5], without considering the characteristics of interface velocity. For this reason, the main contribution of the present investigation is to analyze vorticity fields from liquid-vapor interface velocity using an optical flow method. The flow patterns were identified using a high-speed camera. In turn, characteristic vorticity fields and enstrophy were investigated according to the influence of parameters such as heat flux, slip ratio, vapor quality, and void fraction.

13.2 Experimental Facility and Methodology An experimental facility was developed to investigate the flow boiling, pressure drop, and flow patterns in a small horizontal channel, Fig. 13.1. The experimental system consists of a loop that provides controlled mass velocity, and it was designed to test different fluids under a wide range of flow conditions. The main part of the loop has a pre-heater PH, a test section TS, and a visualization section VS. The secondary part consists of a condenser, a refrigerant reservoir, a liquid refrigerant vessel, a mass flow meter, a magnetic gear pump, and a sub-cooler.

Fig. 13.1 Schematic of the experimental apparatus

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13.2.1 Data Reduction The heat flux on the test section (Eq. 13.1) is based on the Joule effect, considering the electric power qT S imposed on its superficial area Aheated : qT S =

qT S U×I ≡ Aheated π Di L heated

(13.1)

where U and I correspond to the voltage and electric current applied by the power supply, Di is the internal diameter, and L heated is the heated length. The vapor quality in the test section inlet is calculated from the energy balance in the pre-heater section, according to Eq. 13.2:   qT S + i (i−P H ) − il m˙ x(i−T S) = (13.2) ilv where il and ilv correspond to the liquid enthalpy and latent heat of vaporization as a function of saturation pressure. The local enthalpy is considered as a linear function of the location z in the test section, according to Eq. 13.3:   qT S (z) + i (i−T S) (13.3) i (z) = m˙ Finally, along the test section, the local vapor quality is calculated according to Eq. 13.4: x (z) =

i (z) − il ilv

(13.4)

The void fraction was evaluated with the relation developed by Rouhani and Axelsson [6]. Thermodynamics and transport properties used in data reduction and pressure drop correlations were obtained from REFPROP v. 9.1 [7].

13.2.2 Optical Flow Method The optical flow is based on the method developed by Liu and Shen [8] considering optical flow equation for different flow visualizations, which is given in terms of image coordinates: ∂g + ∇ · (gu) = f (x, y, z) ∂t

(13.5)

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where g represents the normalized image intensity that is proportional to the radiance received by the camera, u = u(ux , u y ) the velocity in the image plane referred to as the optical flow, and f (x,y,g) corresponds to a boundary and diffusion term. The optical flow u is proportional to the light-path-averaged velocity weighted with the field quantity ψ related to a visualization medium. A variational formulation with a smoothness constraint is used to determine the optical flow, according to a functional given by Eq. 13.6,   j (u) =



∂g + ∇ · (gu) ∂t

2



  2 + λ |∇u x |2 + ∇u y  d xd y

(13.6)

where λ corresponds to the Lagrange multiplier and  is an image domain. By minimizing Eq. 13.6, the Euler-Lagrange equation is obtained:  ∂g (13.7) + ∇ · (gu) − f + λ∇ 2 u = 0 g∇ ∂t The solution of Eq. 13.7 is found using the standard difference method with Neumann condition ∂u/∂n = 0 on the image domain.

13.3 Results To identify the flow patterns during the tests, a high-speed camera was used to capture the images. In this case, 200 images were captured to ensure the consistency of each pattern analysis with a time interval between the two images of 1.111 × 10−4 s (9,000 Hz). Four different flow patterns were identified: bubby, plug, slug, and churn flow. The sequence of two images of bubbly flow is exemplified in Fig. 13.2a– b and used to generate both the global interface velocity field and, consequently, the normalized vorticity field; see Fig. 13.2c. Variations of vorticity can occur in the upper and lower regions of the bubbles caused by the liquid phase’s different velocities along the tube’s radial direction. Oscillations are also observed at the liquid-vapor interface generated by the different local vorticity intensities. As heat flux increases, vapor quality and void fraction also increase, leading to the plug flow pattern. However, the vorticity field has the same characteristics observed in the bubbly pattern. An example of a slug flow pattern with the vorticity field obtained from the optical flow method is presented in Fig. 13.3. Figure 13.3c shows the vorticity field obtained from the analysis of both images. The magnitude of normalized vorticity of the interface is observed along with the flow. The increase of the slip ratio (from 1.21 to 1.34) creates instabilities in the liquid-vapor interface, thus generating vorticity zones at the rear of the slug. As noted, the instability in such areas of vorticity can lead to the vapor phase detaching.

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Fig. 13.2 a–b Sequence of images of bubbly flow pattern for a heat flux of 5 kW/m2 and vapor quality of 0.02; c vorticity field from optical flow

Fig. 13.3 a–b Sequence of images of slug flow pattern for a heat flux of 10 kW/m2 and vapor quality of 0.11; c vorticity field from optical flow

On the other hand, the vorticity at the front of the slug is caused by velocity fluctuations near the interface; such fluctuations can be due to pressure fluctuations generated by the previous slug. Dispersed bubbles are also observed, indicating that the vapor phase does not present a constant velocity. The analysis of the velocity field also indicates vorticity zones between slugs and among dispersed bubbles (Fig. 13.3). Churn flow can be considered the transition between the slug and annular patterns, and it was observed in all conditions with a slip ratio ranging from 1.32 to 1.61 with a void fraction from 0.43 and vapor quality ranging from 0.17 to 0.35. Due to the more significant Kelvin-Helmholtz instabilities and high-velocity vapor core, this pattern presents more pronounced vorticity fields, as exemplified in Fig. 13.4. The radial component of the velocity of the liquid-vapor interface is mainly responsible for the greater intensity of the vorticity, as can be seen by the red arrows. In order to estimate the dissipation of turbulent kinetic energy in the radial direc 2 tion, the enstrophy ε =  |∇ × u| d was evaluated for each example of the flow regime. According to the results, bubbly flow (Fig. 13.5a) presents a greater dispersion of kinetic energy in intermittent regions to the bubbles, as expected. The highest intensities of enstrophy are found at the backside of the bubbles and may indicate the emergence of the vortex stretching in the radial direction. The distribution of enstrophy along the flow for the plug regime is shown in Fig. 13.5b. The comparative analysis between these results and the images indicates that the greatest dispersion

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Fig. 13.4 a–b Sequence of images of churn flow for a heat flux of 20 kW/m2 and vapor quality of 0.25; c vorticity field resulted from optical flow

Fig. 13.5 Enstrophy of a bubbly flow with 20 kW/m2 ; b plug flow with 20 kW/m2 ; c slug flow with 10 kW/m2 , and d churn flow with 10 kW/m2

of turbulent kinetic energy is found between the tail of the pistons and the bubbles released. The velocity fluctuations in the tail of the plugs, caused by the dispersion of the turbulence, generate radial and axial oscillations in the movement of the bubbles over time, as observed by the sequence of images. The slug flow (Fig. 13.5c), on the other hand, presents the highest intensity of the enstrophy both in the intermittent regions between the phases and at the liquid-vapor interface along the axial direction. This characteristic is related to the increase in the slip ratio and the non-slip condition on the wall. Finally, a smaller dispersion of turbulent kinetic energy in the radial direction is observed in the churn flow (Fig. 13.5d) due to the effect of the gushes in the axial direction.

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13.4 Conclusion and Future Perspective The present investigation analyzed the liquid-vapor interface vorticity of flow patterns from images captured by a high-speed camera during flow boiling of R-600a for a saturation temperature of 17 ◦ C performed in a horizontal tube with an internal diameter of 1.0 mm. A method based on optical flow was used to determine the vorticity fields. Besides the vorticity components, the enstrophy was analyzed along the flows caused by different velocities on the interface, which determines the dynamics of flow patterns.

References 1. 2. 3. 4. 5.

J. Kolaas, D. Drazen, A. Jensen, J. Disp. Sci. Tech. 36, 1473–1482 K. Harada, M. Murakami, T. Ishii, II and He I. Cryogenics 46, 648–657 F. Klein, P. Junior, E. Hervieu, J. Brazilian Soc. Mech. Sci. Eng. 28, 174–179 R. Revellin, V. Dupont, T. Ursenbacher, J.R. Thome, I. Zun, Int. J. Multiph. Flow 32, 755–774 L. O’Neill, I. Mudawar, M. Hasan, H. Nahra, R. Balasubramaniam, J. Mackey, Int. J. Heat Mass. Trans. 125, 1240–1263 6. S. Rouhani, E. Axelsson, Int. J. Heat Mass. Trans. 13, 383–393 7. E. Lemmon, M. Hube, M. McLinden, Physics and Chemical Properties Division (REFPROP 9.1, NIST Standard Reference Database 23, Version 9.1) 8. T. Liu, L. Shen, Int. J. Fluid. Mech. 614, 253–291

Chapter 14

Liquid-Film Thickness Measurements During Convective Condensation of R290 in a Horizontal Channel Tiago A. Moreira and Gherhardt Ribatski

Abstract Heat transfer and pressure drop during convective boiling and condensation are intimately related to the behavior of the liquid-film thickness along the tube perimeter. In this context, the present paper provides liquid-film thickness measurements at the top of the conduit during in-tube convective condensation inside a horizontal channel of 9.43 mm internal diameter. Experiments are performed for R290 at mass velocities ranging from 50 to 250 kg/m2 s, vapor qualities from 0 to unity, and heat fluxes from 5 to 60 kW/m2 . In general, a negligible effect of heat flux was noticed. Also, the behavior of the liquid-film thickness with increasing quality varies according to the mass velocity due to flow pattern changes. Keywords Liquid-film thickness · R290 · Horizontal channel

14.1 Introduction The liquid-film thickness, LFT, and liquid/vapor interface features directly affect the heat transfer and pressure drop behaviors in annular flows for both convective boiling and condensation [1–3]. Moreover, in stratified and wavy flow patterns during convective condensation inside horizontal channels, the heat transfer occurs mainly by condensation of a descending liquid film from the upper region of the conduit [4]. Therefore, the characterization of the liquid film is of fundamental relevance in developing accurate phenomenological prediction methods for the heat transfer coefficient and pressure drop. T. A. Moreira (B) Thermal Hydraulics Laboratory, Department of Mechanical Engineering, University of Wisconsin-Madison, 1500 Engineering Drive, Madison, WI 53706, USA e-mail: [email protected] G. Ribatski Heat Transfer Research Group—HTRG, São Carlos School of Engineering (EESC), University of Sao Paulo (USP), Av. Trabalhador São-Carlense, 400, Parque Arnold Schimidt, CEP: 13566-590 São Carlos, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_14

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In this context, the present study concerns an experimental evaluation of the liquidfilm thickness and its dynamic behavior during convective condensation inside a horizontal conventional-sized channel of 9.43 mm internal diameter of R290, a null Ozone Depletion Potential (ODP), and low Global Warming Potential (GWP) fluid. Experiments are performed at a saturation temperature of 35 ◦ C, mass velocities, G, from 50 to 250 kg/m2 s, vapor qualities, x, from 0 to unity, and heat fluxes from 5 to 60 kW/m2 s.

14.2 Experimental Apparatus Figure 14.1 illustrates a schematic of the experimental apparatus used in the present study. It consists of a closed loop, at which the fluid is driven by a gear oil-free micropump. The refrigerant is boiled at the pre-heater and partially condensed at the test section. The pre-heater and test sections comprise tube-in-tube heat exchangers using water as secondary fluid in the annular region while the refrigerant flows inside the inner tube. Downstream the test section is affixed the visualization section, see Fig. 14.1, composed of a borosilicate glass tube with an inner diameter equal to the one of the test section (9.43 mm ID), at which LFT measurements (see details in Sect. 14.2.1) are performed and flow pattern images obtained through a high-speed video camera (Phantom v2012, 1000000 images/s with resolution of 128 × 16 pixels and 22500 images/s with resolution of 1280 × 800 pixels). The following reference Moreira and Ribatski [5] is recommended for more details about the experimental apparatus, test section, and data regression procedures.

Fig. 14.1 Schematic of the experimental apparatus

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14.2.1 Liquid-Film Thickness Measurement Principle and Setup In the present study, the LFT is measured at the top of the visualization section through the non-intrusive optical measurement technique developed by Shedd and Newell [6], and modified by Moreira et al. [3]. The liquid-film thickness is obtained from the light pattern created by reflections of an incident diffuse light source at the liquid/vapor interface of the liquid film. Figure 14.2a shows the optical path taken by light beams originating from the diffuse light source at various angles. At low angles of incidence (concerning the normal to the liquid/vapor interface), most light is transmitted through the liquid/vapor interface, and only a fraction of it is reflected. However, beyond a critical value of incidence angle, all the light hitting the interface is reflected, creating a pattern at the external surface of the transparent wall. Then, the liquid-film thickness can be estimated by solving the geometrical problem shown in Fig. 14.2b, as follows: LFT =

L dr y L − 2tanθcl 2tanθwl

(14.1)

where L = L dr y + L l and L dr y is given by L dr y = 2tw tanθwl

(14.2)

Fig. 14.2 a Optical paths of light beams originating from a diffuse source of light at the bottom of a transparent wall and reflected at the liquid/vapor interface of a liquid film at the top surface of the wall [6] and b geometrical relations used to calculate the LFT [3]

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Fig. 14.3 Rendering of the LFT measurement setup

θcl is the critical angle at the liquid/vapor interface; θwl is the refraction angle at the wall-liquid interface; tw is the thickness of the wall; Ll is the distance that the light travels while inside the liquid; and Ldr y is the distance the light travels while inside the wall. The distance L is measured from images captured by the high-speed video camera at a frequency 15000 Hz and a resolution of 680 × 480 pixels of the light pattern through an image processing procedure developed in MATLAB [7]. Details about the image processing procedure are found in [3]. A rendering of the liquid-film thickness setup assembled to the visualization section is illustrated in Fig. 14.3. This figure notices that two lasers (Z-LASER Inc., ZM18, 532 nm, 40 mW) are used, one at the top (for measuring the liquid-film thickness) and another at the bottom of the test tube. As previously mentioned, only the data obtained at the top of the test section are presented in this study. For the measurements in both regions (top and bottom), a mirror with a hole in the center is positioned inclined at 45◦ relative to the horizontal plane and with the reflective side facing the glass tube. The centerline of the hole is aligned to be perpendicular to the centerline of the test section. The laser is positioned directly centered with the hole of the mirror for the measurements. A slice of translucent tape is affixed in both the top and bottom sides of the glass tube. This tape acts to diffuse the laser beams and serve as a screen for the image generated by reflection on the liquid-vapor interface used to estimate the LFT. This image is then reflected in the mirrors and captured by the high-speed camera.

14 Liquid-Film Thickness Measurements During Convective … Table 14.1 Experimental uncertainties Parameter Uncertainty Temperature Tube diameter x* LFT* p p Mass velocity

0.05 0.1 0.06 15.6 0.231 0.075 0.1

159

Unit ◦C

mm – µm kPa %FS %

*Mean value; FS: Full-scale

14.2.2 Uncertainties The method of sequential perturbation by Taylor and Kuyatt [8] is used to estimate the uncertainty related to calculated parameters, such as x, G, and LFT. Directly measured variables, e.g., temperature and pressure, have their uncertainties estimated by calibration procedures as suggested by Abernethy and Thompson [9]. Table 14.1 presents a summary of the experimental uncertainties related to the measurements in the present study.

14.3 Results Liquid-film thickness measurements are performed at the top of the test tube for propane (R290) at mass velocities ranging from 50 to 250 kg/m2 s, vapor qualities from 0 to unity, heat fluxes from 5 to 60 kW/m2 , and for a saturation temperature of 35 ◦ C. As a general result, a negligible effect of varying heat flux on the LFT is noticed. Figure 14.4 illustrates time-traces of the LFT for G = 100 kg/m2 s and the behavior of the base liquid film with increasing vapor quality. In the time-traces, the red dotline represents the base liquid film, characterized in the present study as the median of the liquid-film thickness data for each experimental condition discarding values higher than 700 µm. The LFT measurements are threshold-limited by the technique to values below the one just mentioned. The sensor does not capture Thicker liquid layers, which returns a random value between 700 and 800 µm. Also, flow pattern images obtained under similar experimental conditions are presented in the time-trace figures. Figure 14.4a shows that the base LFT varies only marginally for 0 7 (magenta area), the filament deformation/breakup is given by a capillaryinertial balance. The flow cases illustrated here indicate that the viscoplastic dripping is a complex problem related to a diversity of stretching/breakup regimes involving gravitational, capillary, inertial and viscoplastic effects. Hence, would be interesting to consider

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in future works deeper analyses aiming to highlight not only the variety of different regimes that can emerge, but also the critical dimensionless number values related to each one of them.

17.3.2 Bubble Rising in a Viscoplastic Medium In this Subsection, we analyse the bubble physical mechanisms related to the stoppage of a gas bubble of density ρair , viscosity ηair , and initial diameter d0 immersed in a viscosplastic medium of density ρ, consistency k, and yield-stress τ0 . This problem is illustrated in Fig. 17.4, in which the vertical displacement of a 5 mm diameter air bubble in water (Fig. 17.4a; τ0 = 0 Pa) is compared to those of a similar air bubble in two viscoplastic materials: τ0 = 2 Pa (Fig. 17.4b); and τ0 = 7 Pa (Fig. 17.4c). The contours in the x-y plane indicate the magnitude of the horizontal component of the velocity vector u x , while the pink dash-dotted line denotes the initial position of the centre of the air bubbles (sky blue part). For all three cases shown in this Subsection, ρair = 1 kg/m3 , ηair = 10−5 Pa s, ρ = 1000 kg/m3 , k = 0.001 Pa.s, and σ = 0.072 N/m. As expected, the buoyancy force [∝ (ρ − ρair ) gd0 ] drives the upwards movement of the air bubble in water (Fig. 17.4a), which, in turn, perturbates the liquid creating a number a rotational regions, as indicated by the x-velocity contours. Such a rotational movement is considerably attenuated by the increase of τ0 from 0 to 2 Pa (Fig. 17.4b) and eventually vanishes when the complex material stops the air bubble at τ0 = 7 Pa (Fig. 17.4c). Based on a simple force balance analysis, one could say that the air bubble stops when the yield-stress overpasses buoyancy, e.g. τ0  (ρ − ρair ) gd0 . Hence, the bubble displacement would cease when Pl ≥ k1 ,

(17.13)

where Pl = (ρ−ρτair0 )gd0 denotes the Plastic number associated to the problem, and k1 indicates a critical Pl value from which the bubble stops. In other words, the bubble rising stoppage cloud be predicted just by considering the Plastic number. The force balance arguments presented above are confirmed by the results displayed in Fig. 17.5a-d, where both yielded (flowing; ||τ || > τ0 ; in grey) and unyielded (non-flowing; ||τ || ≤ τ0 ; in black) regions are shown for flow cases at four different Plastic numbers: Pl = 0 (Newtonian); Pl = 0.04; Pl = 0.08 and Pl = 0.1. The magenta dash-dotted lines indicate the initial bubble centre positions. Cleary, the unyielded region is an increasing function of Pl. At Pl = 0.1, the latter appears entirely dominated by plasticity and, as a results, the bubble does not move. Thus, there is a critical Plastic number value k1 between 0.08 and 0.1 from which the viscoplastic material behaves like a solid. The critical Pl is shown in Fig. 17.5e, in which the horizontal position of the centre of the bubble ycentr e , made dimensionless by its

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Fig. 17.4 The vertical displacement of a 5 mm diameter air bubble in water (a; τ0 = 0 Pa) is compared to those of a similar air bubble in two viscoplastic materials: τ0 = 2 Pa (b); and τ0 = 7 Pa (c). The contours in the x-y plane indicate the magnitude of the horizontal component of the velocity vector u x , while the pink dash-dotted line denotes the initial position of the centre of the air bubbles (sky blue part). For all shown cases ρair = 1 kg/m3 , ηair = 10−5 Pa s, ρ = 1000 kg/m3 , k = 0.001 Pa.s, and σ = 0.072 N/m. A movie concerning these flow cases is available on https:// anselmopereira.net/videos/

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Fig. 17.5 a–d Both yielded (flowing; τ > τ0 ; in grey) and unyielded (non-flowing; ||τ || ≤ τ0 ; in black) regions are shown for flow cases at four different Plastic numbers: (a) Pl = 0 (Newtonian); (b) Pl = 0.04; (c) Pl = 0.08; and (d) Pl = 0.1. The magenta dash-dotted lines indicate the initial bubble centre positions. (e) Horizontal position of the centre of the bubble ycentr e , made dimensionless by its initial location yinitial , plotted as a function of time t. For all the considered cases, d0 = 5 mm, ρair = 1 kg/m3 , ηair = 10−5 Pa s, ρ = 1000 kg/m3 , k = 0.001 Pa.s, and σ = 0.072 N/m

initial location yinitial , is plotted as a function of time t. According to these curves, k1 ≈ 0.099, which is close to experimental/numerical results previously reported in literature [24–26]. For future works on this topic, it would be interesting to investigate different techniques through which one could trigger the bubble rising movement at highly plastic mediums (Pl >> 0.1), such as the vibration-induced bubble rising method [26].

17.4 Concluding Remarks In the present work, we analysed two major multiphase flows with viscoplastic materials: the viscoplastic dripping and the bubble rising in a viscoplastic medium. These problems were explored by performing two-dimensional numerical simulations based on an adaptive variational multi-scale method for two materials (viscoplastic medium/air) whose main characteristics were also presented. The ability of the presented numerical framework in capturing the physical mechanism driving complex multiphase systems was stressed through energy budget analyses and scaling laws. As shown, our findings are rather in line with experimental/numerical/theoretical results reported in literature [5, 23–26], which encourage us to move towards three-dimensional industrial-like scenarios involving more complex configurations and supplemental non-Newtonian components, such as elasticity, solid particle interactions (granular materials and dense suspensions) and thixotropy.

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

N. Balmforth, I. Frigaard, G. Ovarlez, Ann. Rev. Fluid Mech. 46, 121 (2014) R. Thompson, L. Sica, P. de Souza Mendes, J. Non-Newt. Fluid Mech. 261, 211 (2018) X. Huang, M.H. Garcia, J. Fluid Mech. 374, 305 (1998) C. Ancey, J. Non-Newt. Fluid Mech. 142, 4 (2007) P. Coussot, F. Gaulard, Phys. Rev. E 72(031409), 1 (2005) F. Rasschaert, E. Talansier, D. Blésès, A. Magnin, M. Lambert, Transport phenomena and fluid mechanics. AIChE J. 64, 1117 (2018) J. Venkatesan, D.S. Sankar, K. Hemalatha, Y. Yazariah, J. Appl. Math. 2013, 583809 (2013) M. Cremonesi, L. Ferrara, A. Frangi, U. Perego, J. Non-Newt. Fluid Mech. 165, 1555 (2010) R. Valette, A. Pereira, S. Riber, L. Sardo, A. Larcher, E. Hachem, J. Non-Newt. Fluid Mech. 287, 104447 (2021) T. Coupez, E. Hachem, Comput. Methods Appl. Mech. Eng. 267, 65 (2013) R. Valette, E. Hachem, M. Khalloufi, A. Pereira, M. Mackley, S. Butler, J. Non-Newt. Fluid Mech. 263, 130 (2019) A. Pereira, A. Larcher, E. Hachem, R. Valette, Comput. Fluids 190, 514 (2019) A. Pereira, E. Hachem, R. Valette, J. Non-Newt. Fluid Mech. 282 (2020) A. Pereira, N. Valade, E. Hachem, R. Valette, Under Rev. Phys. Rev. E (2021) E. Bingham, Bull. Bureau Stand. 13, 309 (1916) E. Bingham, (1922) T. Papanastasiou, J. Rheol. 31, 385 (1987) S. Riber, R. Valette, Y. Mesri, E. Hachem, Comput. Fluids 138, 51 (2016) E. Hachem, M. Khalloufi, J. Bruchon, R. Valette, Y. Mesri, Comput. Methods Appl. Mech. Eng. 308, 238 (2016) D.H. Peregrine, G. Shoker, A. Symon, J. Fluid Mech. 212, 25 (1990) J. Eggers, Phys. Rev. Lett. 71, 3458 (1993) A. Deblais, M.A. Herrada, I. Hauner, K.P. Velikov, T. van Roon, H. Kellay, J. Eggers, D. Bonn, Phys. Rev. Lett. 121, 254501 (2018) N.J. Balmforth, N. Dubash, A.C. Slim, J. Non-Newt. Fluid Mech. 165, 1147 (2010) N. Dubash, I.A. Frigaard, J. Non-Newt. Fluid Mech. 142, 123 (2007) J. Tsamopoulos, Y. Dimakopoulos, N. Chatzidai, G. Karapetsas, M. Pavlidis, J. Fluid Mech. 601, 123 (2008) G. Karapetsas, D. Photeinos, Y. Dimakopoulos, J. Tsamopoulos, J. Fluid Mech. 865, 381 (2019)

Chapter 18

New Trends on Multiphase Flow Simulations Using Slug Capturing Approach Farhad Nikfarjam, Hamidreza Anbarlooei, and Daniel O. A. Cruz

Abstract It is well known that the 4-equation formulation of the two-fluid model is ill-posed. As a result, it is impossible to differentiate between the errors originating from uncertainty in the empirical closure models and the nonphysical oscillations due to ill-posedness of the equations. The present work uses a 5-equation formulation, by adding a volume fraction evolution equation, which is unconditionally hyperbolic. To solve the resulting system of equations, a high-resolution Roe’s scheme is developed, and to keep the solution time practical, an adaptive mesh refinement algorithm is implemented. To obtain the wall shear stresses, a novel set of analytically developed friction equations is used for the non-Newtonian liquid phase. Finally, the capability of the developed code has been examined for the terrain slugging cases. Keywords Slug capturing approach · Two-fluid model · Ill-posed

18.1 Introduction Slug capturing methods, using a minimal set of empirical closure equations, can predict stratified, slug and transitional regimes. They can be used to simulate transient phenomena such as start-up or shutdown processes and are able to handle terrain and severe slugging naturally. It is in contrast to the “unit cell” or “slug tracking” methods, which generally are not usable in these cases and have several limitations.

F. Nikfarjam · H. Anbarlooei Department of Applied Mathematics, Institute of Mathematics, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil e-mail: [email protected] H. Anbarlooei e-mail: [email protected] D. O. A. Cruz (B) Mechanical Engineering Program, Coppe, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_18

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It is common to simulate the intermittent two-phase flow using two-fluid model, where the conservation of mass and momentum will be solved for each phase:

∂(αk ρk u k ) ∂t

+

∂(αk ρk ) + ∂(αk∂ρxk u k ) = 0, ∂t ∂(αk ρk u 2k ) + αk ∂∂pxk = −αk ρk g sin θ ∂x

(18.1) − Fkw − k Fi .

(18.2)

Here, subscript k (= g, l) indicates the phase (gas or liquid) and αk , ρk , u k , pk , Fkw and Fi represent volume fraction, density, velocity, pressure, external frictional forces and interfacial friction force, respectively. Also, θ is the inclination angle and g is the acceleration due to gravity. However, this model becomes ill-posed (non-hyperbolic) for many important practical conditions, and could result in nonphysical results [1, 2]. Techniques such as adding a regularizing term (interfacial pressure) to the momentum equation have been explored to tackle this problem. Assuming difference between phase pressure ( pk ) and phase interface pressure ( pki ), one could obtain the following momentum balance for the two-fluid model: ∂ pi ∂(αk ρk u k ) ∂(αk ρk u 2k + αk ( pk − pki )) + + αk k = Sk . ∂t ∂x ∂x

(18.3)

Here, Sk is the source term (gravitational and friction) on the right-hand side of the Eq. (18.2). Several models (corrections) have been introduced for the interfacial pressure such as pk = pk − pki = δ

αg αl ρg ρl (u g − u l )2 αg ρl + αl ρg

(18.4)

which has been used in CATHARE code [3] or the following which is introduced by Moalem [4]: ∂h ∂ (αk pk ) = αk ρk g cos θ ∂x ∂x

(18.5)

Here, h represents the liquid height. However, this technique is not always successful and still nonphysical waves could form during simulation [2, 5]. In such a case, it is almost impossible to differentiate between the errors due to the uncertainty in the incorporated closure models and nonphysical oscillations generated due to the ill-posedness. The mathematical regularization technique introduced by Baer and Nunziato [6] can completely remove this issue, by adding an extra (gas) volume fraction evolution equation, ∂αg ∂αg + ui = r p ( pg − pl ), (18.6) ∂t ∂x

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where u i and r p stand for the interfacial velocity and pressure relaxation coefficient which model the inter-phase interactions. Equations (18.1), (18.2) and (18.6) form our new governing equations. It is possible to recast these equations in to the following form: ∂q ∂q + A(q) = s(q), ∂t ∂x

(18.7)

 T where q = αg , αg ρg , αg ρg u g , αl ρl , αl ρl u l is the vector of conservative variables and s(q) = [r p ( pg − pl ), 0, Sg , 0, Sl ]T is the source terms. Without loss of generality, if one considers the equation of state for each phase as pk = ck2 (ρk − ρk0 ) − pk0 ,

(18.8)

where ck is the speed of sound, and ρk0 is the reference density at reference pressure pk0 , then matrix A becomes ⎛ ⎞ ui 0 0 0 0 ⎜ 0 0 1 0 0 ⎟ ⎜ ⎟ 2 2 2 ⎜ 0 0 ⎟ (18.9) A = ⎜−ρg cg cg − u g 2u g ⎟. ⎝ 0 0 0 0 1 ⎠ 0 0 cl2 − u l2 2u l ρl cl2 It is easy to show that the eigenvalues of A are λ = [u i , u g − cg , u g + cg , u l − cl , u l + cl ].

(18.10)

This means that the 5-equation system considered above is always well-posed (hyperbolic). To compare, 4-equation system of (18.1) and (18.3) with interfacial pressure (18.4) has the following approximate eigenvalues [5] ˆ u v ± γˆ ], λ = [u p ± c,

(18.11)

where u p and u v stand for speed of pressure and material waves and cˆ is the average Wood’s sound speed [7]. Also, γˆ is defined as

p(αg ρl + αl ρg ) − αg ρg αl ρl (u g − u l )2 . (18.12) γˆ = (αg ρl + αl ρg )2 It is evident that γˆ and as a result eigenvalues of the 4-equation system can be complex, and therefore, the 4-equation system in not unconditionally hyperbolic. The present research is based on the 5-equation formulation of the two-fluid model to simulate the intermittent flows. As discussed above, this formulation is always hyperbolic (well-posed), even in the vertical configuration, and therefore, has an advantage over the traditional 4-equation formulations. In the following sections,

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we first review the numerical method used to solve the resulting system, and then the closure equations necessary to complete our system are discussed in Sect. 18.3. In Sect. 18.4, some numerical results are presented. Finally, Sect. 18.5 contains the conclusion of present work and discusses future extensions.

18.2 Numerical Method To solve Eq. (18.7), the finite volume method in conjunction with the fractional stepping method of Strang [8] (see also Saurel and Abgrall [9]) has been used. In this context, the updated solution can be obtained as Qin+1 = L S L R Lh Qin ,

(18.13)

where Qin and Qin+1 are solutions at current and next time steps at cell i. The operator Lh is related to the solution of the hydrodynamic part of Eq. (18.7), ∂q ∂q + A(q) = 0, ∂t ∂x

(18.14)

while operator L R represent the solution of dαg = r p ( pg − pl ), dt

(18.15)

which will be called the relaxation step in what follows. Finally, the operator L S applies the source term effects on momentum Eqs. (18.2), as ∂q = s(q). ∂t

(18.16)

Equation (18.14) has been discretized using Godunov’s method in conjunction with the Roe linearization to define the approximate Riemann solutions [10]. The semi-discrete form of this equation will be ∂Qi 1 ˆ− m ˆ + 1 Qm 1 ). = Lh (Qim ) = ( A 1 Qi+ 1 + A i− 2 i− 2 2 ∂t x i+ 2

(18.17)

Here, superscript m indicates time (step), and AQ represents the effect of the moving wave into or out of cell i, which can be calculated as 1 ˆ ± ∗ ∗ ˆ i+ 1 )(Qi+1 ( A 1 ± | A| − Qi∗ ). Aˆ i+ 1 Qi+ 1 = 2 2 2 2 i+ 2

(18.18)

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Here, Aˆ i+ 21 indicates the Roe matrix which approximates the flux at the interface between Qi and Qi+1 . For more details, see Ferrari et al. [11]. In the relaxation step, operator L R , one solves Eq. (18.15). In the present work, we assume r p ≈ ∞, which means instantaneous pressure equilibrium between phases. If the result of this step shown with superscript †, then pg† = pl† . It must be noticed that during this step, (αk ρk )† = (αk ρk )∗ and (αk ρk u k )† = (αk ρk u k )∗ . Considering αg + αl = 1 and the equation of state in (18.8), one can obtain an equation for the pressure at the end of the relaxation step as αg + αl =

(αg ρg )∗ ρg† ( p)

+

(αl ρl )∗ ρl† ( p)

= 1.

(18.19)

After obtaining p † from the above equation, one could calculate the density and volume fraction of each phase too. Finally, the L S operator could be simply formulated as Qn+1 = Q† + s(Qm )tm

(18.20)

For the temporal discretization, second order BDF2 method is used as  4 1 2 L S L R Lh (Qin+1 ) . Qin+1 − Qin + Qin−1 = 3 3 3

(18.21)

To solve (18.21), Jacobian free Newton–Krylov (JFNK) method is used. If one rearranges Eq. (18.21) as a system of residual equations, R(Q∗ ) = 0, then the Newton method can be formulated as Jk δQk = −R(Qk ),  ∂R  where Jacobian matrix is Jk = ∂Q  and the k + 1-th iteration will be

(18.22)

k

Q

k+1

= Qk + δQk

(18.23)

In JNFK method, Eq. (18.22) will be solved using the Krylov method. This method does not require to calculate the Jacobian matrix explicitly and instead approximates the product of the Jacobian matrix and a vector, for example, Jk δQk in (18.22), as Jv ≈

R(Qk + v) − R(Qk ) , 

(18.24)

where v is the Krylov vector and  is a small number. In order to keep the execution time practical (especially for industrial applications), the adaptive mesh refinement technique is implemented based on the method proposed by Khokhlov [12]. In this method, the refined cells are organized in a fully threaded tree. Also, an adaptive time stepping algorithm is used for different grid

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levels present in the domain. Both gradient- based and Kelvin–Helmholtz stability [13] refinement indicators have been used in the simulations. The described method above is first-order accurate in space and second-order accurate in time. Higher order spatial extensions have also been explored using the WENO method. Also, it is well known [1] that upwind finite volume methods, as the one described in the present work, suffer from loss of accuracy and efficiency at low Mach numbers. Therefore, to complement the developed numerical method, a low Mach number preconditioning techniques have also been developed. The latter discussed are out of the scope of the present paper and will not discussed in detail here.

18.3 Closure Models To close our system of governing equations (Eqs. 18.1, 18.2 and 18.6), the closure models same as [14] have been used. The wall frictional forces for each phase and the interfacial force between gas and the liquid are expressed as Fkw =

τkw Sk , A

Fi =

τi Si A

(18.25)

where A, Sk and Si are the area of the pipe, the wetted perimeter by each phase and the interfacial surface between phases, respectively. The shear stresses are given by τkw =

1 f k ρk u 2k , 2

τi =

1 f i ρg (u g − u l )2 2

(18.26)

In the present work, the closure models have been extended to handle the purely viscous non-Newtonian liquid phase based on the friction factor equations developed in [15]. For the simplest case of the power-law type fluid, the laminar friction factor is f = 16/Re M R , where the Metzner–Reed Reynolds number is defined as Re M R =

ρU 2−n D n . K 8n−1 ( 1+3n )n 4n

(18.27)

Based on the theory developed in [15], the turbulent friction factor is given by −1

f = C f ReG2(n+1) ,

(18.28)

where C f is proportionality constant which assumed to be equal to the Blasius constant (0.079). The generalized Reynolds number in this equation is defined as ReG =

ρ U 2−n D n . K

(18.29)

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Anbarlooei et al. [15] indicated that the Eq. (18.28) predicts the experimental single-phase data very accurately. The transition Reynolds number (from laminar flow to turbulent flow) is modelled using the correlation suggested by Guillot [16], as Retrans M R = 3250 − 1150 n,

(18.30)

where n is the flow exponent of the power-law model.

18.4 Numerical Test Cases The developed code has been evaluated by simulating different flows reported in literature [11, 13, 14, 17]. Some of them have been reported in this section. The selected examples are just to demonstrate the capabilities of the developed numerical method and precise comparison with experimental data has been skipped.

18.4.1 Water Faucet Problem Figure 18.1 shows the results of the water faucet problem. In this problem, water enters from the top into a vertical pipe of length 12m with fixed velocity (u l = 10 m/s) and volume fraction (αl = 0.8). Air can not enter from the top inlet, and the bottom outlet condition is open to the ambient pressure p = 105 Pa. At the beginning, the gas is at rest and liquid occupies the pipe with the same condition as the inlet. Figure 18.1 show the time evolution of the results, compared to the analytical solution [18]. The results are obtained on the grid with 1000 cells and for C F L = 0.85. In Fig. 18.2, the results of the developed method are compared with and without the low Mach number preconditioning (discussed in Sect. 18.2). As evident, without preconditioning the numerical dissipation totally smears the discontinuities. Limiting numerical dissipation is crucial in long duration integration, such as slug flow simulation, in which high amount of artificial dissipation could flatten the slug flow into stratified. This matter will be discussed in more detail in future work.

18.4.2 Slug Flow in Horizontal Pipe The numerical method developed in the present work has been examined here in the case of slug flow. The two-phase (air-water) flow in a horizontal, 36 m long pipe is considered. The diameter of the pipe is 0.078 m and viscosities of air and water are μa = 1.79 × 10−5 Pa and μw = 1.14 × 10−3 Pa. The gas and liquid velocities and volumes fractions are fixed at the inlet (αg = 0.5, u g = 4.0, u l = 3.0), while the

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0.65 Analytical Sol. t=0.2 s t=0.4 s t=0.6 s t=0.8 s t=1.0 s

0.6

18

Liquid velocity (m/s)

Gas volume fraction

0.55

Analytical Sol. t=0.2 s t=0.4 s t=0.6 s t=0.8 s t=1.0 s

0.5 0.45 0.4 0.35 0.3

16

14

12

0.25 10

0.2 4

2

0

8

6

12

10

0

2

6

4

10

8

12

x (m)

x (m)

Fig. 18.1 Results of the Water faucet problem. Time evolution of the gas volume fraction (left) and of the liquid phase velocity (right) obtain on a uniform grid with 1000 cells at CFL = 0.85 0.55 Analytical Sol. t=0.2 s (with Preconditioning) t=0.6 s (with Preconditioning) t=0.2 s (without Preconditioning) t=0.6 s (without Preconditioning)

Gas volume fraction

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0

2

4

6

8

10

12

x (m)

Fig. 18.2 Comparison of the results with and without low Mach number preconditioning in water faucet problem

outlet opens into the ambient condition ( p = 1 atm). A computational grid consists of 800 cells and implicit time integration with CFL = 1.5 has been used for the simulations. Figure 18.3 shows the time trace of the liquid volume fraction for this problem. As evident, the current method successfully predicts the initiation and propagation of the slugs in the domain. Figure 18.4 shows the time trace of liquid holdup at 30 m from the beginning of the pipe. Each peak in this figure represents a slug passing this point (probe).

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Liquid volume fraction

10 8 6 4 2 0

0

5

10

15

20

25

30

35

x (m)

Fig. 18.3 Time trace of liquid volume fraction profiles for the slug flow problem in a horizontal pipe

Liquid volume fraction

1 0.9 0.8 0.7 0.6 0.5 0.4 0

20

40

60

80

100

120

140

Time (s)

Fig. 18.4 Time trace of liquid holdup at 30 m form the beginning of the pipe, in horizontal pipe slug flow problem

It is very interesting to note that after 100 s, some form of statistically developed behaviour (frequency) appears in the results. While the same simulation using 4equation system (ill-posed models) usually does not show such behaviour and several randomly distributed waves will be visible in the results, e.g. [19, 20]. Considering the fact that there is no source of randomness in the governing equations, the validity of such results is questionable.

18.4.3 Vertical Pipe The present governing equations are well-posed even in the vertical flow case. In this section, vertical flow of water and air in a 10 m long pipe, with an inner diameter of

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Fig. 18.5 Liquid holdup in vertical pipe problem. Blue represents the liquid and white is the gas

0.05 m is examined. The volume fraction and superficial velocities are fixed at the inlet (αg = 0.5, u sg = 0.5, u sl = 0.01), while the outlet is the ambient pressure. The result of this problem is shown in Fig. 18.5. These results are in agreement with the flow map of e Taitel et al. [21] and also the simulations of Ferrari et al. [11].

18.4.4 Terrain Slugging Fig. 18.6 shows a terrain slugging test case. It is an air-high viscous oil flow pipe with an inner diameter of D = 0.0508 m which composed of three different consecutive zones: the first zone is a L 1 = 20 m long upward inclination pipe with θ = 2◦ , the second zone is a horizontal pipe with a length of L 2 = 1 m, and the last zone is a L 3 = 20 m long downward inclined pipe with θ = −2◦ . Inlet liquid and gas superficial velocities are set to 0.08 m/s and 0.6 m/s, respectively. According to Ferrari et al. [14], the experimental results show the formation of slug flows in both upward and downward inclined branches of pipe in contrast to the theoretical model proposed by Brauner and Moalem Maron [22] which predicts the

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Fig. 18.6 Air-high viscous oil slug flow evolution

slug flow only in the upward inclined pipe. In our simulation, the slug flow forms also in both upward and downward inclined pipes which is consistent with [14] (see Fig. 18.6).

18.5 Conclusion and Future Perspective A well-posed (always hyperbolic) 5-equation formulation of the two-phase flow is studied in this work. The governing equations are the mass and momentum conservation equations for each phase. To ensure the hyperbolicity, an extra volume fraction evolution equation is also added to the system. The resulting system is solved using the Godunov type finite volume method, where the Riemann problem is approximated using Roe’s method. Low Mach preconditioning is also introduced to tackle the related issues. Several test cases are solved with the developed method, and it is observed that the method can precisely capture the desired behaviours. As mentioned, the low Mach preconditioning needs more investigation and is the topic of our future work. Another issue which must be addressed is the extension to higher order spatial discretization. Although this has been done at this stage, the developed methods need more precise evaluation. Also, its integration with low Mach preconditioner must be examined thoroughly.

References 1. S.T. Munkejord, Comput. Fluids 36(6), 1061 (2007). https://doi.org/10.1016/j.compfluid.2007. 01.001 2. M. Bonizzi, Transient one-dimensional modelling of multiphase slug flows. Ph.D. thesis, Mechanical Engineering Deptartment, Imperial College London (2003) 3. D. Bestion, Nucl. Eng. Des. 124(3), 229 (1990) 4. N. Brauner, D.M. Maron, Int. J. Mult. Flow 18(1), 103 (1992) 5. S. Evje, T. Flåtten, J. Comput. Phys. 192(1), 175 (2003) 6. M.R. Baer, J.W. Nunziato, Int. J. Mult. flow 12(6), 861 (1986) 7. R. Saurel, F. Petitpas, R.A. Berry, J. Comput. Phys. 228(5), 1678 (2009). https://doi.org/10. 1016/j.jcp.2008.11.002

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8. G. Strang, SIAM J. Numer. Anal. 5(3), 506 (1968) 9. R. Saurel, R. Abgrall, J. Comput. Phys. 150(2), 425 (1999). https://doi.org/10.1006/jcph.1999. 6187 10. R.J. LeVeque, et al., Finite Volume Methods for Hyperbolic Problems, vol. 31 (Cambridge University Press, 2002) 11. M. Ferrari, A. Bonzanini, P. Poesio, Int. J. Numer. Methods Fluids 85(6), 327 (2017) 12. A.M. Khokhlov, J. Comput. Phys. 143(2), 519 (1998) 13. M. Gourma, N. Jia, C. Thompson, Int. J. Multi. flow 49, 83 (2013) 14. M. Ferrari, A. Bonzanini, P. Poesio, Petroleum 5(2), 171 (2019) 15. H. Anbarlooei, D.O.D.A. Cruz, F. Ramos, C.M.M. Santos, A.S. Freire, Physica D: Nonlinear Phenomena 376, 69 (2018) 16. D. Guillot, Schlumberger Educational Services (Houston, 1990) 17. J. Carneiro, R. Fonseca Jr., A. Ortega, R. Chucuya, A. Nieckele, L. Azevedo, J. Brazilian Soc. Mech. Sci. Eng. 33(SPE1), 251 (2011) 18. J.A. Trapp, R.A. Riemke, J. Comput. Phys. 66(1), 62 (1986) 19. M. Gourma, N. Jia, J. Comput. Multi. Flows 7(2), 57 (2015) 20. A. Nieckele, J. Carneiro, R. Chucuya, J. Azevedo, J. Fluids Eng. 135(12) (2013) 21. Y. Taitel, D. Bornea, A.E. Dukler, AIChE J. 26(3), 345 (1980). https://doi.org/10.1002/aic. 690260304 22. N. Brauner, D.M. Maron, Int. J. Multi. Flow 18(4), 541 (1992)

Chapter 19

Transient CFD Simulation of Vortex Formation in a Stirred Multiphase Flow André Lourenço Nogueira

Abstract The present study simulated the dynamic behavior of the water–air interface in a cylindrical tank stirred mechanically. CFD techniques were applied using the software ANSYS-Fluent. The Volume of Fluid (VOF) model with explicit formulation coupled to the κ-ω SST turbulence model was employed to numerically represent the transient displacement of the water–air interface starting from a situation of stagnated flow. An isothermal operation was considered for the case studied. A simple experimental apparatus composed of a glass vessel filled with dyed water and a mechanical stirrer with a pitched-bladed impeller was used to generate the experimental data of the interface position along the time. The model accuracy was evaluated at stirring rates of 300 and 400 RPM. According to the results, the geometric shape profile of the vortex at the pseudo-stationary state could be reasonably well reproduced by the model. However, the model could not generate accurate results regarding the transient behavior of the vortex formation. Additional strategies should be tested to enhance the model capability to reproduce the transient displacement of the water–air interface due to the rotational flow. Keywords Computational fluid dynamics · Two-phase interface · Stirred tank

19.1 Introduction Industries of different areas such as chemical, food, pharmaceutical, cosmetic, oil, mining, wastewater treatment use stirring systems as an essential unit operation in their processes. According to Junior et al. [1], the stirring operation can be considered as the heart of many sorts of processes. For instance, to guarantee the quality of the products, polymerization [2], crystallization [3], solid–liquid reactions [4], and flocculation [5] processes are strongly affected by the efficiency of the stirring system. Among distinct types of stirring systems, the most common are cylindrical tanks containing a rotating device with one or more impellers installed along A. L. Nogueira (B) Department of Chemical Engineering and Post-Graduation Program in Process Engineering, University of the Region of Joinville, UNIVILLE, Joinville 89219-710, Brazil © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_19

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a shaft connected to an electric engine. The most common impellers found in the market include the marine propeller, Rushton turbine (90◦ blades), pitched blade (45◦ blades), and anchor. Each of these impellers promotes a preferential flow inside the tank. The type of impeller or stirring system can be selected according to the viscoelastic characteristics of the fluid to be mixed [1]. The development of customized stirring systems for specific applications may require prototype fabrication, which delays the project execution and increases its costs. In this way, using simulation tools based on computational fluid dynamics (CFD) appears as an attractive strategy to develop more efficient conceptual projects of specific stirring systems. The virtual project of such systems allows engineers and designers to evaluate quickly and efficiently the performance of different concepts of stirrers and optimize them to reach the desired results. Only after achieving the optimized virtual project, the prototype may be constructed to validate the simulated performance. However, since numerical validation of stirred fluid flow fields requires expensive techniques, such as Particle Image Velocimetry—PIV [6] or Laser Doppler Anemometry—LDA [7], alternative and cheaper approaches might be welcome even if they only provide qualitative information of the system. Although not so accurate, monitoring the vortex generated in the central part of stirring tanks can be considered an attractive alternative way to qualitatively evaluate the rotational fluid flow patterns in stirred tanks. Based on this idea, CFD simulations of stirring systems designs can be performed. The generated data could be qualitatively validated by visualizing the formation and geometrical shape of the vortex experimentally. Many CFD studies about stirred tanks assume that the tank is filled only with liquid and, consequently, the influence of the central vortex in the fluid flow pattern is neglected. This assumption may lead to errors when dealing with real laboratory, and industrial stirred tanks [8]. In this scenario, the present study performed CFD simulations of a simple stirred tank using a multiphase turbulent model at ANSYS-Fluent to evaluate the model’s capability to reproduce the dynamics of the vortex formation and its geometrical shape at the pseudo-steady state.

19.2 Experimental Techniques and Simulation Details The experimental apparatus was composed of a 600 mL flat-bottom glass vessel with no chicanes (beaker, 10 cm height, 8.5 cm diameter) and a mechanical stirrer with a pitched-blade impeller (four blades inclined in 45o ). The images in Fig. 19.1 show a schematic representation of the simulation domain and an image of the experimental set before the stirrer was turned on (t = 0). Water was used in the experimental runs which were filmed to capture the deformation of the water–air interface since the condition of stagnated flow. A water volume of 400 mL was used in each experimental run, representing a water level of 7.0 cm height. Potassium permanganate (KMnO4 , 2.0 mL) was used to color the water and enhance the contrast to facilitate the interface visualization. As seen in Fig. 19.1 (right image), a frame made with graph paper was

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Fig. 19.1 Schematic representation of the simulation domain and image of the experimental set

positioned around the beaker to locate the fluid level in a cartesian coordinate system. Each mark in this frame represents 0.5 cm. Two different stirring frequencies were used in the tests. Images were taken from the footage, and the position of the interface along the time was transformed in a set of x,y points using the open-source software WebPlot Digitizer. Experimental data of the interface position were compared with the simulated ones. The left image of Fig. 19.1 shows the simulation domain created in the CAD Space Claim, which is composed of two regions: a stationary and rotating domain. The stationary domain is represented by the tank walls, while the rotating one consists of the region that encloses the impeller body. The software ANSYS-Fluent Academic Student 19.2 was used in the simulations. Three levels of refinement were used to generate the numerical mesh and evaluate the mesh independence of the results. Hybrid meshes composed of tetrahedral and prismatic elements were used regardless of the mesh refinement. Tetrahedral elements were used in the bulk of the fluid and surface of the stirring system, while three layers of prismatic elements with a growth ratio of 1.1 were used in the tank wall. The first layer thickness varied according to the mesh refinement used. A more refined mesh was generated in the impeller region due to the higher velocity gradients expected for this region. Elements sizes of 1.0 mm were used in the surface of the shaft and blades of the stirring system, regardless of the mesh refinement. In the physical situation of the present study, the vessel was partially filled with water, and the remaining space was filled with air. The liquid phase (water) flows in contact with a gaseous one (air) during the process operation. The multiphase Volume of Fluid (VOF) model with two Eulerian phases was used in the computational simulations to reproduce the dynamic displacement of the water–air interface due to

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the stirred rotating flow. The explicit formulation for the VOF model was selected to track the volume fraction of each fluid throughout the domain. Although this mathematical approach is time-dependent, the volume fraction at the current time step is directly calculated based on known quantities at the previous time step. Therefore, the explicit formulation does not require an iterative solution of the transport equation at each time step [9]. Furthermore, the explicit formulation exhibits better numerical accuracy to reproduce the interface between fluids instead of the implicit formulation. However, the time step size is limited by a Courant-based stability criterion [10]. The equations shown below represent the complete formulation of the volume fraction transport differential equation (19.1) and its explicit algebraic formulation Eq. 19.2: ⎤ ⎡ n        1 ⎣∂ αq ρq + ∇ · αq ρq vq = Sαq + m˙ pq − m˙ q p ⎦ (19.1) · ρq ∂t p=1

αqn+1 ρqn+1 − αqn ρqn t

⎡ ⎤ n      n n ⎣ Sαq + V+ αq, m˙ pq − m˙ q p ⎦ V f ρq U f = f

(19.2)

p=1

The VOF setup also considered a sharp interface modeling with the interfacial antidiffusion to suppress possible numerical diffusions at the interface. Additionally, the Geo-Reconstruct volume fraction advection scheme was selected. The Continuum Surface Model was also enabled to include the effects of surface tension at the water– air interface. A surface tension coefficient of 0.072 N/m was used in the simulations. The κ-ω SST (Shear Stress Transport) model was used in the present study to estimate the turbulent properties of the fluid flow. This turbulence model was selected because it can accurately predict the transport and dissipation of the specific turbulent kinetic energy close to wall regions. The turbulent kinetic energy (κ) and the specific dissipation rate (ω) were calculated by Eqs. 19.3 and 19.4, respectively. The mathematical model still considered the mass and momentum conservation through the Continuity and the Reynolds-Averaged Navier–Stokes equations (Eqs. 19.5 and 19.6, respectively). The moving reference frame model was adopted in the rotating domain to consider the rotational flow around the Z-axis. Finally, the PISO algorithm (pressure-velocity coupling scheme) was used to perform the transient calculations.

 τij ∂U i μ ∂κ μt ∂κ ∂ ∂κ ∗ (19.3) + uj + = − β κω + ∂t ∂x j ρ ∂x j ∂x j ρ ρσκ ∂ x j

19 Transient CFD Simulation of Vortex Formation …



  μ ∂ω μt ∂ω ∂ω ω τi j ∂ U¯ i ∂ 2 ¯ + Uj + =α − βω + ∂t ∂x j κ ρ ∂x j ∂x j ρ ρσω ∂ x j 2 ∂κ ∂ω + (1 − F1 ) σω,2 ω ∂ x j ∂ x j ∂ρ ∂ U¯ i =0 +ρ ∂t ∂ xi

¯ 

¯  ∂ U¯ j U¯ i ∂ U¯ j ∂ Ui ∂ Ui ∂ P¯ ∂ =ρ + ρgi − + + (μ + μt ) ∂ xi ∂x j ∂x j ∂ xi ∂t ∂x j

215

(19.4)

(19.5)

(19.6)

19.3 Results and Discussion As mentioned in Sect. 19.2, three meshes were generated to evaluate their effect on the quality of the results. These data were assessed based on the geometrical shape and position of the water–air interface. A stirring rate of 300 RPM was considered in this first investigation. Figure 19.2 displays the simulated air volume fraction distribution obtained after 8.0 s of stirring. This time was long enough for the system to achieve the pseudo-stationary operation condition. According to these results, the geometrical shapes of the water–air interfaces were similar regardless of the mesh refinement. However, the lower the mesh refinement, the higher the numerical diffusion at the interface due to the larger mesh elements used to calculate the fluid flow in this region. The dynamic behavior of the vortex formation due to the turbulent rotational flow was evaluated at 300 and 400 RPM. Although the refined mesh requires a high computational effort, the experimental data were compared with the simulation results generated with the best mesh to minimize numerical diffusion at the interface, as seen in Fig. 19.2. Figures 19.3a and b compare the experimental and simulated position of the water–air interface after 2, 3, 4, and 5 s of stirring at 300 RPM, and after 1, 2, and 3 s of stirring at 400 RPM, respectively. Both conditions (experimental and simulation at 300 and 400 RPM) considered that the fluid was stagnated at zero time. As seen for both stirring rates, the simulated results are advanced in time compared to the experimental observations, which means that the simulated position of the water–air interface changed faster than the experimental interface during the system’s transient behavior. These findings suggest that the simulated dynamics of the vortex formation occurred with a velocities field more intense than the experimental. The results shown in Fig. 19.4 demonstrate that the simulations could reasonably well reproduce the position of the interface. The error in predicting the interface position increased slightly by raising the stirring rate, but it can still be considered acceptable, considering the simplicity of the procedure to acquire the experimental data. The estimated error of the interface position at the peripheral region of the

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Fig. 19.2 Influence of mesh refinement on the geometrical shape of the water–air interface after 8.2 s of stirring at 300 RPM: a air volume fraction field; b XY plot of the interface position

vortex was about 7.0 and 8.5% for the situations considering 300 and 400 RPM, respectively. One possible reason for the deviation between simulated and experimental results might be attributed to the simplifications made in the geometrical model of the impeller set. Because of the lack of the twisted part of the blades that connect them with the shaft in the CAD model, as seen in Fig. 19.1, the surface area that pushes the fluid is a little larger than the real one. Consequently, the simulated velocities generated by the impeller’s rotational movement were probably higher than the experimental one, which would explain the flow advanced in time observed in the simulations. Additionally, due to the size of the elements at the impeller surface (approximately 1.0 mm), the maximum y+ values were 19.5 and 24.7 at 300 and 400 RPM, respectively (Fig. 19.5). These results indicate that the boundary layer

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Fig. 19.3 Dynamic behavior of the experimental and simulated water–air interface shape at a 300 RPM; b 400 RPM

Fig. 19.4 Comparison of the simulated and experimental position of the water–air interface at the pseudo-steady state: a 300 RPM and 5.0 s; b 400 RPM and 3.0 s

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Fig. 19.5 Distribution of y+ values over the impeller surfaces: a 300 RPM; b 400 RPM

at the impeller surface is modeled instead of resolved, possibly contributing to the deviation of the simulated results from the real ones.

19.4 Conclusion and Future Perspective This study showed that monitoring the geometrical shape of the central vortex developed in stirring tanks can be used as a cheap and straightforward strategy to qualitatively validate the rotational fluid flow patterns of virtual conceptual projects of impeller systems. The model provided good information regarding the position of the water–air interface at the pseudo-steady state when using stirring frequencies of 300 and 400 RPM. However, it could not reproduce suitably the transient behavior of the vortex formation. In this case, the numerical results showed a faster deformation of the interface than the real one. More simulations still need to be performed considering a more precise geometrical model of the impeller and a numerical mesh capable of resolving the fluid flow inside the boundary layer over the impeller surface. Additionally, the use of a refinement-adaptive mesh at the interface based on the phases volume fraction could also contribute to a better prediction of the water–air interface position along the time.

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References 1. C. Joaquim Jr, E. Cekinski, J. Nunhez, L. Urenha, LTC Editora, Rio de Janeiro (2007) 2. A.L. Nogueira, M.B. Quadri, R.A. Machado, C.A. Claumann, Macromol. React. Eng. 11(3), 1600040 (2017) 3. B.D. Crittenden, M. Yang, L. Dong, R. Hanson, J. Jones, K. Kundu, J. Harris, O. Klochok, O. Arsenyeva, P. Kapustenko, Heat Trans. Eng. 36(7–8), 741 (2015) 4. G. Kasat, A. Khopkar, V. Ranade, A. Pandit, Chem. Eng. Sci. 63(15), 3877 (2008) 5. V. Vajihinejad, J.B. Soares, Chem. Eng. J. 346, 447 (2018) 6. V. Ranade, M. Perrard, N. Le Sauze, C. Xuereb, J. Bertrand, Chem. Eng. Res. Des. 79(1), 3 (2001) 7. J.B. Joshi, N.K. Nere, C.V. Rane, B. Murthy, C.S. Mathpati, A.W. Patwardhan, V.V. Ranade, Canad. J. Chem. Eng. 89(1), 23 (2011) 8. E. Nogueira, J. Pinto, A. Vianna Jr., Canad. J. Chem. Eng. 90(4), 983 (2012) 9. A. Fluent, et al., ANSYS Inc., USA, vol. 15317, p. 724 (2011) 10. Ansys fluent user’s guide (2021). https://ansyshelp.ansys.com. Accessed 17 May 2021

Chapter 20

Numerical Simulation of a Downward Two-Phase Vertical Flow: A Preliminary Analysis of Convection Intensity and Gas Flow Rate Effects Lucca D. V. Melo, Vitor P. Pinheiro, Ramon S. Martins, and Vinicius P. Franco Abstract In the context of research on turbulent multiphase flows, the experimental and numerical reports available in the literature on downward bubble columns are still relatively scarce. Substantial advances in multiphase models, turbulence models, and new constitutive proposals for interfacial forces prove that the line of research is active and requires more scientific investigation. In this line, the current work aims to carry out a preliminary numerical analysis of a downward bubble column in a vertical tube with the ANSYS Fluent commercial package. A vertical pipe comprises the geometry used in the numerical experiment with a 20mm inner diameter. The multiphase flow is characterized by two average bubble diameters, 1.75 and 2.2 mm, and a gas volume fraction ranging from 2 to 10% at the inlet. Regarding the analysis, initially, the convective intensity was varied to establish its influence on the axial velocity profile of the continuous phase and the void fraction distribution in the radial direction. At last, the gas flow rate was gradually increased to observe its effects on the radial void distribution. The results achieved were validated by comparison with well-established experimental data and presented satisfactory agreement in general. Keywords Downward vertical flow · Two-fluid multiphase model · Convection intensity · Gas flow rate L. D. V. Melo (B) · R. S. Martins Laboratory for Computational Transport Phenomena (LFTC), Universidade Federal do Espìrito Santo, Vitoria, Brazil e-mail: [email protected] R. S. Martins e-mail: [email protected] V. P. Pinheiro Postgraduate Program in Mechanical Engineering, Universidade Federal do Espìrito Santo, Vitoria, Brazil e-mail: [email protected] V. P. Franco Blast Furnace II and III - Maintenance Management, ArcelorMittal Tubarão, Bairro Polo Industrial, Vitoria, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_20

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20.1 Introduction The study of multiphase flow dynamics has become increasingly relevant in various industrial sectors such as oil and gas, steel, food, paper, and pulp, among others [1]. The main challenge resulting from the theme lies in the difficulty to properly understand the physical phenomenology, as well as developing mathematical modeling suitable for each condition [2]. Experimental and numerical reports available concerning the downward bubble column are still relatively scarce compared to the scientific material of upward flows. The state of the art of downward two-phase flow has a significant contribution to the Russian engineering school [3, 4]. Among the several critical physical behaviors in this type of flow stands out the intensification of wall shear stresses in comparison to single-phase configuration that generates damping effects of the turbulent pulsations [5]. It is also essential to highlight the presence of a local maximum in the axial liquid velocity and void fraction profiles, clearly reported in the works of Kashinsky et al. in turbulent regime [6–8] and also in laminar flows [9–11]. All of these effects end up altering the mechanical balance between the forces of inertia, buoyancy, lift, drag, and other forces of a turbulent nature, which act at the interface between the dispersed and continuous phase [12–14]. The current article proposes a preliminary study of a downward air-water bubble column flow in a 20 mm-diameter geometry. The main guideline is to perform a parametric analysis of the axial velocity and void fraction distribution concerning the velocity and gas flow rate that characterizes the flow inlet boundary condition. The numerical simulations are performed with the assistance of the ANSYS Fluent computational tool and validated with experimental data of [6, 7] aiming to achieve both qualitative and quantitative adherence.

20.2 Mathematical Modeling The mathematical description of the hydrodynamic behavior of multiphase flow is given, as in the single-phase case, by the set of physical balances of mass conservation and linear momentum conservation [15]. Mass conservation, also frequently known as the continuity equation of fluid mechanics, is given below for a generic q phase. n       ∂  αq ρq + ∇ · αq ρq vq = m˙ pq − m˙ q p + Sq ∂t p=1

(20.1)

Note that, in parallel to the single-phase continuity equation, which is identically null, the Eq. (20.1) is equivalent to a net balance of mass flow between the phases, added of a source term [1]. Regarding momentum conservation, the two-fluid model uses robust mathematics, whereas it solves a momentum equation for each phase present in the flow. This

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formulation brings a structure already known by the fundamental fluid mechanics until the term is inherent to the divergent of the viscous tensor τq . In addition to these, two new terms are appropriate for describing the influence of interfacial effects that should be modeled accordingly [1].    ∂  → → → → αq ρq − u q + ∇ · αq ρq − uq − u q = −αq ∇ p + αq ρq − g + ∇ · τ¯q + ∂t n   −  −→ → − → − → − → − →  R pq + m˙ pq − u→ u→ ˙ qp− pq − m q p + F q + F l,q + F wl,q + F vm,q + F td,q p=1

(20.2) The first term inside the summation counts the interaction force between phases R pq and the net balance of linear momentum exchanged between phases due to the relative velocity between them [1, 16]. The second term refers to some of the main interfacial forces existing in multiphase flows, namely, lift Fl,q , wall lubrication Fwl,q , virtual mass Fvm,q , and turbulent dispersion Ftd,q . In order to take the turbulent phenomenon into account in a flow, a specific mathematical treatment is required, which should be chosen following the desired accuracy in the numerical simulation and the computational resource available. Therefore, a simplified approach to turbulence is achieved through the RANS models. The RANS models are originated from the average processes on Navier-Stokes equations. Among the available RANS models, the Shear Stress Transport (SST) k-ω [17] stands out, given its broad applicability in engineering issues, which is mainly justified by its remarkable ability to encompass the good performances of the k-ω model in the near-wall region and the k-ε model in the far-from-the-wall region.

20.3 Methodology The geometry is characterized by a tube of diameter D = 20 mm and a length of 100D, adequately represented in Fig. 20.1. All the results analyzed in sequence are correlated to the axial position y/D = 100. The strategy used in the current paper took advantage of the geometry axis-symmetric to increase the computational power, according to Fig. 20.1. An important step, which precedes the generation of results, is the mesh test with progressive refinement to check whether the mesh refinement no longer interferes with the results. After a few tests, the structured mesh was selected for the simulations, with volumes in the radial direction (50) and axial direction (2000), with a clustering closer to the duct wall. The value of the dimensionless wall distance, y + , for the first element was 1.3. It is worth noticing that, in order to achieve quantitative adherence in the sequence, this

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Fig. 20.1 Geometry sketch and boundary conditions

kind of refinement level will be needed, such as Kashinsky et al. [7] in their work, which uses unitary values of y + . The validation procedure of the mesh used was successfully performed using the single-phase log law. The main parameters of the mesh test were a bubble diameter of 1.75 mm and inlet velocity equal to 0.5 m/s. In order to calibrate the wall shear stress levels, the present work uses Armand’s correlation [18], which shows efficiency in adjusting the velocity profiles, Fig. 20.2, accordingly to the expected shapes [7]. This correlation was selected mainly due to its simplicity, despite underestimating the multiphase wall shear stresses in some situations, as reported in [5] and [19]. According to Fig. 20.1, at the domain inlet, a uniform velocity profile is imposed on both phases, with a gas volume flow rate of 2, 5, and 10% and mean bubble diameter of 1.75 and 2.2 mm, in alignment with Kashinsky’s work [6, 7]. The domain outlet is physically described by atmospheric pressure conditions and a reverse flow model that uses the neighboring cells’ parameters. There is a demand for selecting constitutive models to describe each interfacial force of the multiphase flow, where the main ones are: drag, lift, turbulent dispersion, wall lubrication, surface tension, and the virtual mass. The choice of suitable force models depends on the flow parameters in each analysis. The virtual mass force is kept constant with a coefficient of 0.5. The drag force is accounted for in Tomiyama’s model [20] and the lift force in another proposal by Tomiyama. Frank’s formulation [21] was used to model the wall lubrication force using the Hosokawa coefficient.

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Fig. 20.2 Log-Law Adherence

Regarding the forces related to turbulence, the Lopez de Bertodano model [22] was used to treat the turbulent dispersion and the Sato model to represent the turbulent interaction.

20.4 Results and Discussions This section is divided into two stages of analysis, namely, the first consists of measuring the impact of the flow convection intensity on the axial velocity profile of the continuous phase and void fraction distribution. Finally, the gas volume flow rate at the inlet is increased, and the effects of this parameter on the radial distribution of the dispersed phase are observed. All performed numerical simulations are compared with experimental data well established in the literature [6, 7]. The mechanical balance of forces acting on the dispersed phase in a downward flow is illustrated in Fig. 20.3. It is possible to realize in the axial direction an opposition between the inertia and buoyancy forces. The result of this dispute defines whether the bubbles follow the flow or take an upward direction. In addition, forces linked to the lateral migration of the bubbles, such as the lift force, tend to push the bubbles away from the wall, increasing the concentration of the dispersed phase in a core region, which consequently causes the establishment of a bubble-free region close to the wall. In order to compare the intensity of the inertia forces concerning buoyancy forces that act on the bubble due to fluid flow, the magnitude of the velocity of the continuous phase at the domain inlet was varied, as can be seen in Fig. 20.4, for three hydrodynamic levels, 0.3, 0.5, 1.0 m/s. The numerical simulations presented in Figs. 20.4

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Fig. 20.3 Mechanical balance of forces acting on a bubble in downward two-phase flow Fig. 20.4 Influence of the Convection Flow on the Radial Distribution of Liquid Axial Velocity

and 20.5 use a gas volume flow rate of β = 3% and a mean bubble diameter equal to db = 2.2 mm. A careful observation of Fig. 20.4 reveals that the numerical simulation related to the minor level of convection intensity, Vliq = 0.3m/s, correctly captures the maximum velocity peaks in the near-wall region. Increasing the liquid inlet velocity, the hydrodynamic radial distribution becomes flattened, similar to the single-phase configuration. For all velocity levels, it can be seen that the results adhere to the trend of the experimental data.

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Fig. 20.5 Influence of the Convection Flow on the Radial Distribution of the Void Fraction

The expected physical trend, in general, of the gradual increase in the liquid velocity injection is to lead to a thinner bubble-free zone, which tends to slow down the flow in the near-wall region, which in turn mitigates the presence of the velocity peaks [12]. This effect can be well explained by the balance between inertia and buoyancy forces [7]. In the sequence, an analysis of the influence of the convection flow on the void fraction distribution is evaluated, as can be seen in Fig. 20.5. All flow parameters of the previous results are preserved. For the lowest velocity level tested, the agreement with experimental data is clear, and it also can be noted that a bell shape characterizes the void fraction distribution, widely provided in the literature [6]. In Fig. 20.5, it is also particularly worth noticing the region near the wall, which presents zero volume fraction that conceptualizes and confirms the presence of the bubble-free zone in downward flows [5]. The intermediate level of velocity tested presents a significant difference in comparison to the experimental data of Kashinsky et al. [6]. Despite that, a careful analysis of other contributions of the same authors, see [7], shows that for a fixed mean bubble diameter and gas volume flow rate, an increase of liquid velocity toward a void distribution with higher mean value [5]. This last argument, in turn, is aligned with the numerical results presented. On the other hand, for the highest velocity level, the increase of the inertial effects on the flow generates higher levels of shear stresses on the interfacial surfaces. Consequently, a strong tendency to bubble breaking [12], which causes a monotonic distribution of the dispersed phase along the radial direction. In this case, it is noticed a satisfactory adherence of the numerical results with the experimental prediction again, mainly in the near-wall zone, and a minor mis-

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Fig. 20.6 Influence of the Gas Flow Rate on the Radial Distribution of the Void Fraction

match in the core region, probably due to the breaking mechanism, duly captured by the experimental results, but not foreseen in the configurations of the numerical simulations. The second analysis was carried out to determine the influence of the gas injection flow rate on the distribution of the void fraction in the radial direction of the tube, as can be seen in Fig. 20.6. In this case, the parameters used are Vliq = 5m/s and db = 1.75 mm. It is possible to note that the numerical results showed a good adherence with the experimental data of Kashinsky et al. [7] for all fraction levels tested, based on an average α, see [5]. It is also noticed that the curve with the highest level of β did not reproduce the local peak of the void distribution as well the numerical simulations performed by the same authors of the experimental references [7].

20.5 Conclusions and Future Perspective It is essential to highlight that the mesh validation stage reveals the fundamental importance of adequate wall refinement in the general numerical treatment of turbulent flows. Such importance is amplified in the approach of multiphase flows, given the interaction of interfacial forces with the wall. In this way, the level of refinement in this region is a critical aspect and must be observed. The simulations presented a satisfactory agreement compared to experimental data for low to moderate gas volume fractions and all levels of inlet velocity tested. It is essential to highlight that the comparison of numerical simulations with experimental

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data is not always balanced since the empirical data can capture some details of the physical phenomenon, which CFD codes may not account for according to the selected configurations. The preliminary results presented in this work suggest a more robust mesh study should be conducted using other wall stresses correlations. There is a demand for testing other interfacial force models and the impact of using breaking and coalescence models to improve the physical representation of the dispersed phase, which can probably improve the relative adherence with the experimental data available. To carry out future simulations, it is necessary to parameterize its setup to achieve a quantitative adherence with experimental data, and consequently allowing to extrapolate the analyzes toward a downward bubble column in a vertical tube to higher gas volume fraction scenarios, different average bubble diameter situations, heat transfer coupling, aiming to describe some industrial phenomena of interest in the oil and gas and steel industry.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

E.S.E.M.I. Rosa, (2009) N.I.M.F.D. Kolev, (2005) R.S. Gorelik, O.N. Kashinskii, V.E. Nakoryakov, J. Appl. Mech. Tech. 28, 1 (1987) B.G. Ganchev, V.G. PeresadKo, J. Eng. Phys. 49(2), 879 (1986) O.N. Kashinsky, V.V. Randin, Int. J. Multi. Flow 25(1), 109 (1999) O.N. Kashinsky, P.D. Lobanov, V.V. Randin, J. Eng. Thermophys. 17, 120 (2008) O.N. Kashinsky et al., Int. J. Heat Mass Trans. 49, 3717 (2006) T. Hibiki, H. Goda, S. Kim, M. Ishii, J. Uhle, Int. J. Heat Mass Trans. 47(8–9), 1847 (2004) O.N. Kashinsky, V.V. Randin, M.A. Vorobyev, J. Phys.: Conf. Ser. 1105 (2018) N. Riviere, A. Cartellier, Eur. J. Mech.-B/Fluids 18(5), 823 (1999) N. Riviere, A. Cartellier, L. Timkin, O. Kashinsky, Eur. J. Mech.-B/Fluids 18(5), 847 (1999) S.M. Bhagwat, A.J. Ghajar, Exp. Thermal Fluid Sci. 39, 213 (2012) T. Hibiki, H. Goda, S. Kim, M. Ishii, J. Uhle, Exp. Fluids 35(1), 100 (2003) M. Ishii, S.S. Paranjape, S. Kim, S., Interfacial structures and interfacial area transport in downward two-phase bubbly flow 30(7–8), 779 H. Schlichting, G.K.B.L., Theory 2016 A.A.F.T.G.V. Fluent, C. Ansys Inc. (2013) F.R. Menter, AIAA J. 30(8), 2066 (1992) A.A. Armand, Izvestia Vsesoyuznogo Teplotekh. Inst 1, 16 (1946) P.D. Lobanov, J. Eng. Thermophys. 27(2), 232 (2018) T. Takamasa, A. Tomiyama, in Ninth International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-9) (San Francisco, CA, 1999) T. Frank, J. Shi, A.D. Burns, in Proceeding of the Third International Symposium on Two-Phase Modelling and Experimentation (Citeseer, Pisa, Italy, 2004), pp. 22–25 M.L. de Bertodano, R. Lahey Jr., O. Jones, Nucl. Eng. Des. 146(1–3), 43 (1994)

Chapter 21

ANN and CFD-DPM Modeling of Alumina-Water Nanofluid Heat Transfer in a Double Synthetic Jet Microchannel Javad Mohammadpour, Zhaleh Ghouchani, Fatemeh Salehi, and Ann Lee

Abstract In this paper, integrating different cooling techniques into a microchannel heat sink (MCHS) is evaluated to enhance the heat transfer rate. A parametric study is conducted using the computational fluid dynamics simulation consisting of the discrete phase model (CFD-DPM) to understand the effect of operational and geometrical parameters on cooling performance. The CFD-DPM simulations for varying different influential parameters are conducted to obtain the local temperature over the silicon wafer wall, creating a large set of samples. The artificial neural network (ANN) method is then employed to discover an accurate model to predict the local temperature. The ANN model includes two hidden layers and 24 neurons in each layer, showing a precise estimation of temperature with an overall mean square error (MSE) value and correlation coefficient (R) of 6.974810−7 and 0.9952, respectively. The temperature distributions along the silicon wall predicted by both CFD-DPM and ANN models verify that smaller sizes of alumina nanoparticles improve heat transfer remarkably compared to larger particles. Keywords Computational fluid dynamics (CFD) · Artificial neural network (ANN) · Discrete phase model (DPM) · Nanofluid heat transfer · Synthetic jet (SJ)

J. Mohammadpour (B) · F. Salehi · A. Lee School of Engineering, Macquarie University, 2109 Sydney, NSW, Australia e-mail: [email protected] F. Salehi e-mail: [email protected] A. Lee e-mail: [email protected] Z. Ghouchani Department of Industrial Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_21

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21.1 Introduction Recent breakthroughs in electronic cooling systems have drawn significant attention to producing compact microprocessors with higher efficiency as transferring the heat fluxes over 1 MW/m2 is a serious challenge in designing new microdevices [1]. Maintaining the maximum operating temperature below 85◦ C is a design requirement, however, it causes compromising in the cooling performance. To overcome this challenge, various strategies were suggested. Using a synthetic jet (SJ) was proposed [2] to enhance heat transfer in MCHSs in laminar flows. A pair of synthetic jets was used at the bottom of an MCHS resulted in more uniform and higher heat transfer [3]. The orifice distance and phase actuation of double SJs showed significant impacts on the heat transfer of an MCHS, Mohammadpour et al. [4]. The addition of nanoparticles into base fluids was also suggested [5]. The effect of different nanoparticles dispersed into the water was investigated in an MCHS equipped with an SJ [6]. It was reported that Al2 O3 -water nanofluids at the volume fraction of 5% improved the cooling performance by 38%. The effect of alumina nanoparticle sizes and concentrations on heat transfer was studied in an MCHS equipped with an SJ [7]. It was revealed that smaller particles and higher particle concentrations resulted in higher heat transfer enhancement. It was further shown that increasing the frequency and amplitude of the membrane enhanced the heat transfer rate. Various numerical methods such as single-phase models (SPMs) [6] and Eulerian– Lagrangian approaches [7] were applied to study the nanofluid heat transfer. However, they are computationally expensive. Alternatively, machine learning algorithms can be used in the study of such flows. The thermal conductivity and dynamic viscosity of nanofluids were predicted by an ANN model and two sets of correlations [8]. The ANN was more accurate than empirical correlations in predicting thermophysical properties [8, 9]. In the present paper, a novel study using a discrete phase model (DPM) is first conducted to understand the effect of influential parameters on heat transfer in an MCHS equipped with multiple SJs. Different neural network (NN) structures are then analyzed to propose the most accurate ANN model for the prediction of temperature distribution along the silicon wall that will significantly reduce the computational cost associated with the DPM simulations.

21.2 Problem Definition A schematic configuration of a 2D MCHS with a pair of SJs in a crossflow is shown in Fig. 21.1. The crossflow enters the microchannel from the inlet and leaves that from the outlet. Table 21.1 gives a summary of the MCHS dimensions. The expulsion or suction of flow occurs through jet orifices to/from the crossflow.

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Fig. 21.1 Schematic diagram of the MCHS with multiple synthetic jets Table 21.1 Dimensions of the MCHS Parts Symbol Channel Length Orifice Diameter Channel Height Top Silicon Height Bottom Silicon Height Boundaries spacing

Dimension (mm)

L Do hc hs H ts

4 0.05 0.2 0.5 0.6 0.025

Thermophysical properties of water and alumina particles are listed in Table 21.2. Table 21.2 Thermophysical properties [7] Materials ρ Water Al2 O3

997.1 3970

k

Cp

0.613 40

4179 765

The motion of the left and right membranes is modeled through the implementation of user-defined functions (UDFs), including temporal parabolas varying in a sinusoidal fashion written by Eqs. 21.1 and 21.2, respectively.     x +b 2 Ym,le f t = A 1 − sin(2π f t) (21.1) D/2

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Table 21.3 Variation range of parameters Parameters Symbol Frequency of left membrane Frequency of right membrane Amplitude of left membrane Amplitude of right membrane Orifice distance Particle concentration Particle diameter Diaphragm length Actuation phase

fL fR AL AR S ϕ dp D Phase

 Ym,right = A 1 −



x −b D/2

Unit

Range of variation

Hz Hz μm μm mm % nm mm -

280∼560 280∼560 20∼40 20∼40 1∼3 0∼5 50∼100 0.95∼1.96 In-phase & 180◦ Out-of-phase

2  sin(2π f t)

(21.2)

where A and f represent the amplitude and frequency of oscillation, respectively. “b” is the distance between the middle of the left/right orifice and the microchannel. Ten parameters are identified as influential for heat transfer enhancement. They are orifice distance, particle concentration, particle diameter, the length of diaphragms, the arrangements of both jets (being in-phase (I) or 180◦ out-of-phase (O)), and the amplitude and frequency of oscillation. The variation range of these parameters is given in Table 21.3.

21.3 Methodology A total number of 130 simulations are carried out by the DPM. The simulations are conducted using Ansys/Fluent. A summary of governing equations for DPM is presented in Sect. 21.4, and more details can be found in our previous study [4]. After 20 cycles of membrane oscillation, more than 32,570 samples are exported from the CFD-DPM solver, creating the dataset for the ANN analysis. ANN is a new approach providing advancements in computer sciences. The ANN approach is inspired by the brain learning mechanism. Figure 21.2 shows a feed-forward multilayer perceptron (MLP) network as one of the most practical ANN architectures [10], which is employed in this study. As can be seen, there are two hidden layers in the ANN structure. Tansig and purelin are two transfer functions used for the hidden and output layers, respectively. In this study, through random sampling, 70% of the total dataset is considered for training, and the remaining data are equally divided into validation and test sets. The purpose of using the training dataset is to find the weight

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Fig. 21.2 ANN structure

and bias values. The validation dataset is used to avoid overfitting and adjust loss functions or the learning rate. The final evaluation of the trained model is performed by the test dataset. All variables are normalized because they are on different scales. A “structured trial and error” technique is employed to find a suitable number and arrangement of neurons in hidden layers. The Levenberg–Marquardt (LM) learning algorithm is applied for the network’s training process [10]. The performance of the network is then evaluated by the comparison of predicted ANN values with numerical results.

21.4 Governing Equations Governing equations of the Eulerian-Lagrangian approach are briefly presented in this section.

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21.4.1 Eulerian Phase Equations Conservation equations of mass, momentum and energy are expressed as Mirzaei et al. [11]: ∇ · (ρ(1 − ϕ)u) = 0

(21.3)

Du + u · ∇(ρ(1 − ϕ)u) Dt = ρ(1 − ϕ)g − (1 − ϕ)∇ P + ∇ · (μ(1 − ϕ)∇u) + S

(21.4)

  DT + u.∇ ρc p (1 − ϕ)T = ∇ · (k(1 − ϕ)∇T ) + St Dt

(21.5)

The two source terms on the right-hand side of Eqs. 21.4 and 21.5 are momentum and energy exchange between water and nanoparticles, respectively, which are defined in the next section.

21.4.2 Lagrangian Phase Equations Translational and rotational motions of particles are govern using Eqs. 21.6 and 21.7.   →   g ρp − ρ d− up − → = m p FD u − u p + m p + Fother (21.6) mp dt ρp   ρ f dp 5 dω p Ip = Cω || ·  = T (21.7) dt 2 2 The first and second terms on the right-hand side of Eq. 21.7 represent drag and gravity forces, respectively. Based on previous studies [7, 12], all other significant forces applied on nanoparticles are Saffman and Magnus lift forces, pressure gradient force, virtual mass force, Brownian force, and thermophoretic gradient force, which are included in the third term (Fother ) [4].

21.4.3 Energy Equation of Nanoparticles The first law of thermodynamics is employed for nanoparticles as m pcp

  dT p = π h D 2p T f − T p dt

(21.8)

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where the heat transfer coefficient (h) is calculated by the Ranz and Marshall correlation [13] Nup =

hd p 1/3 = 2.0 + 0.6Re1/2 p Pr k∞

(21.9)

21.5 Numerical Set-Up and Validation The governing equations are discretized using the finite volume method (FVM). A second-order central scheme is employed for the pressure gradient and diffusion terms. The velocity and pressure fields are coupled with the SIMPLE pressure correction algorithm [14]. The convergence criterion is met when residuals are less than 10−6 . The pressure difference between both the ends of the microchannel is set so that the average velocity remains at 0.45 m/s when both membranes are inactive. Based on a grid independence study, the structured mesh with 160,000 cells is found accurate for the simulations. The inlet temperature is 298 K, and the flow is assumed to be laminar. The DPM model used in this study is validated with the experimental study [15]. The maximum deviation between the numerical and experimental data is almost 2% for 1% alumina nanofluids [4].

21.6 Results and Discussion Various structures of NNs are designed and trained using the CFD-DPM dataset. For the evaluation of different ANN structures, a statistical analysis is conducted through the mean square error (MSE), correlation coefficient (R), and standard deviation, which are defined, respectively, as follows: MSE =

n

n 2 1   C F D−D P M xi − xiAN N n i=1

(21.10)



  xiC F D−D P M − x C F D−D P M xiAN N − x AN N  R =   2 2

n C F D−D P M n AN N C F D−D P M AN N −x −x i=1 x i i=1 x i i=1

σ =

n i=1

(21.11) (ei − μ) n

(21.12)

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Fig. 21.3 Validation performance

where n represents the number of samples, which is approximately 32,570 in this study. x C F D−D P M and x AN N are the temperature values obtained by the CFD-DPM or predicted by the ANN, respectively. Also, x C F D−D P M and x AN N are the average values of x C F D−D P M and x AN N , e is the deviation value between actual and estimated values, and μ is the mean of errors. The NN structure with two hidden layers and 24 neurons in each layer shows the most accurate prediction. Figure 21.3 presents the variation of the MSE against iteration numbers for training, validation, and test datasets of the best NN structure. As can be seen, the best validation performance with the minimum MSE of 6.472410−7 is obtained after 487 iterations. The network shows precise behavior without overfitting due to similar trends for all three datasets. Indeed, overfitting occurs when the test curve increases remarkably while the validation curve is not that much [10]. Due to the importance of both test and validation sets, 30% of the total dataset is allocated for this purpose. The accuracy of ANN structures can also be assessed through the correlation coefficients between the predicted and actual values. The correlation coefficients of both validation and test sets are 0.99466 and 0.99518, respectively. As shown in Fig. 21.4 for the test dataset, there is good agreement between the predicted NN data and CFD-DPM results for the normalized local temperature

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Fig. 21.4 Comparison between CFD-DPM and ANN results in the prediction of normalized local temperatures for the test set Table 21.4 General performance of the NN R MSE Validation set Test set

0.99466 0.99518

7.631910−07 6.974810−07

μ

σ

3.285810−06 5.661910−06

0.00087369 0.00083522

on the MCHS upper wall. Table 21.4 shows a summary of the statistical analysis parameters for the test and validation sets, confirming the excellent performance of the selected NN structure. Temperature distributions over the silicon wafer wall at the end of the 20th cycle of in-phase actuation are shown in Fig. 21.5. There is excellent agreement between the ANN and the CFD-DPM data. ANN results show that decreasing the size of particles results in higher heat transfer enhancement, which is consistent with the observation in the CFD-DPM simulations. This is due to the higher rate of momentum and energy exchange between particles and water since the surface per volume ratio is higher [7].

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Fig. 21.5 Comparison between temperature distribution of different nanoparticle sizes predicted ANN and CFD-DPM models at f = 280 Hz, A = 20 μm, S = 3 mm, D = 1.96 mm, in-phase actuation and τ ∗ = 1

21.7 Conclusion and Future Perspective A numerical analysis is conducted to combine different cooling techniques to enhance heat transfer in an MCHS. In this regard, a parametric study is carried out to understand the effect of different parameters on heat transfer. Ten parameters identified as influential are orifice spacing (S), particle volume fraction (ϕ), the diameter of nanoparticles (d p ), the length of membranes (D), phase actuation (being in-phase (I) or 180◦ out-of-phase (O)), and the amplitude and frequency of oscillation (A and f ). Despite the high accuracy of CFD-DPM results, this model is very costly and time-consuming. A feed-forward multi-layer perceptron (MLP) network is adopted to save computational time and cost in predicting local temperatures. More than 32,570 samples exported from numerical simulations are considered as the dataset for the ANN analysis. The training process of the network is performed through the LM learning algorithm. Various NN structures are analyzed, and then a two-hiddenlayer NN structure with 24 neurons in each layer is found as the best model for predicting local temperature over the silicon wafer wall. The MSE and correlation coefficient (R) of this model for the test set are 6.974810−7 and 0.9952, respectively.

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Finally, the effect of nanoparticle sizes on heat transfer of the MCHS is investigated by the CFD-DPM and the proposed NN model. The estimated values show the acceptable level of agreement with the numerical findings. The obtained results from both models verify that smaller nanoparticles have a better cooling performance compared to larger ones. This study can be extended into a 3D configuration with more variables. Given that the 3D simulation of nanofluid heat transfer is time-consuming, future works should use different machine learning techniques in numerical studies to save computational time and cost.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

I. Mudawar, IEEE Trans. Compon. Pack. Technol. 24(2), 122 (2001) V. Timchenko, J. Reizes, E. Leonardi, Int. J. Numer. Methods Heat Fluid Flow (2007) A. Lee, G. Yeoh, V. Timchenko, J. Reizes, Appl. Thermal Eng. 48, 275 (2012) J. Mohammadpour, F. Salehi, M. Sheikholeslami, M. Masoudi, A. Lee, Int. J. Thermal Sci. 167, 107008 (2021) S.U. Choi, J.A. Eastman, Enhancing thermal conductivity of fluids with nanoparticles. Technical report, Argonne National Lab., IL (United States) (1995) G. Lau, J. Mohammadpour, A. Lee, Appl. Thermal Eng. 188, 116624 (2021) J. Mohammadpour, A. Lee, M. Mozafari, M.R. Zargarabadi, A.S. Mujumdar, Int. J. Thermal Sci. 161, 106705 (2021) M.H. Esfe, S. Saedodin, N. Sina, M. Afrand, S. Rostami, Int. Commu. Heat Mass Trans. 68, 50 (2015) M.H. Esfe, H. Rostamian, S. Esfandeh, M. Afrand, Physica A: Stat. Mech. Appl. 510, 625 (2018) H. Ansari, M. Zarei, S. Sabbaghi, P. Keshavarz, Int. Commun. Heat Mass Trans. 91, 158 (2018) M. Mirzaei, M. Saffar-Avval, H. Naderan, Canad. J. Chem. Eng. 92(6), 1139 (2014) O. Mahian, L. Kolsi, M. Amani, P. Estellé, G. Ahmadi, C. Kleinstreuer, J.S. Marshall, M. Siavashi, R.A. Taylor, H. Niazmand et al., Phys. Rep. 790, 1 (2019) W.E. Ranz, Chem. Eng. Prog. 48, 141 (1952) S. Patankar, Google Scholar J. Lee, I. Mudawar, Int. J. Heat Mass Tras. 50(3–4), 452 (2007)

Part VI

Experimental Multiphase Flow

Chapter 22

Evaluation of Multiphase Flow Pattern and Friction Loss Prediction Models Applied to NEMOG’s Multiphase Flow Circuit Tiago G. S. Lima, Francisco J. do Nascimento, Oscar M. H. Rodriguez, and Rogério Ramos

Abstract Over the years, the multiphase flow has been playing an essential role in many industry sectors. However, prediction of flow pattern and pressure drop with accuracy is still challenging. For this reason, this paper presents a bibliographic review of mechanistic models applied to multiphase flow. Then, flow pattern and friction loss models were selected to be validated and applied to an experimental circuit in the Research Group for Oil and Gas Flow and Measurement (abbreviated as NEMOG, in Portuguese), considering its operational capacity. Furthermore, operational limits could be estimated for multiphase flow. Keywords Multiphase flow · Multiphase closed-loop · Flow pattern · Friction loss · NEMOG

22.1 Introduction The industry has intensively demanded the prediction of flow patterns based on operational conditions. Taitel and Barnea [1] brought some examples in need for multiphase modeling: heat exchangers, oil and gas transport, geothermal and solar power plants, microelectronic components, nuclear power plants, and even climate control systems in space vehicles.

T. G. S. Lima (B) · F. J. do Nascimento · R. Ramos Research Group for Oil and Gas Flow and Measurement (NEMOG), The Federal University of Espirito Santo, Vitoria, Brazil e-mail: [email protected] F. J. do Nascimento · O. M. H. Rodriguez Department of Mechanical Engineering, School of Engineering (EESC), Industrial Multiphase Flow Laboratory (LEMI), University of São Paulo (USP), São Carlos, SP, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_22

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Jerez-Carrizales et al. [2] presented a literature review of models applied to predict multiphase flow pattern and friction losses. In that review study, it was highlighted that the complexity associated with mathematical models does not always imply accuracy, and sometimes the simple and mechanistic ones return satisfactory results. In a complementary way, empiric models may be restricted to operational conditions while mechanistic models are based on the fundamentals of physics laws. As a consequence, more reliable data can be obtained [3]. The first author to present a complete mechanistic model to predict two-phase flow patterns was Taitel and Dukler [4] for horizontal flow, classifying patterns into a dispersed bubble (DB), intermittent (I), stratified smooth (SS), stratified wavy (SW), and annular-dispersed liquid (AD). In sequence, Taitel et al. [5] developed a model also for flow pattern prediction but in vertical flows. Since the transition mechanisms for horizontal and vertical flow are different, Barnea et al. [6] investigated these transitions in inclined pipes from 0◦ to 90◦ . In a later study based on the presented models, Barnea [7] developed a unified model adaptable to any slope. Along with the flow pattern prediction models, Xiao et al. (1990) incorporated friction losses prediction to a slightly modified version of Taitel and Dukler’s [4] model, which is capable of estimating the pressure drop of each flow pattern individually for horizontal pipes and with small slopes (±15◦ ). Furthermore, Jerez-Carrizales et al. [2] highlighted the consistent performance of a model developed by Ansari et al. [8] for flow pattern and pressure drop predictions; however, this model is restricted for upward vertical flows. The latest studies, as the one accomplished by Petalas and Aziz [3], count on more complex transition criteria for each situation and is capable of predicting flow pattern and friction loss to any slope. This model compiles techniques used by other authors [4, 6, 7, 9] and includes new classifications to flow patterns such as dispersed bubble (DB), stratified smooth (SS), stratified wavy (SW), annular-mist (AM), elongated bubble (EB), slug (S), and froth (F). Besides that, it was pointed out by the author that this model performed better when compared to other models presented in the literatures [9–12]. This comparison was made by applying all models to the same conditions as 5951 experimental data points from the Stanford Multiphase Flow Database [13]. Besides that, Shuard et al. [14] compared predictions of flow pattern and friction losses made by this model with 60 CFD simulations reporting a satisfactory agreement between model and simulation for flow pattern and friction losses. At last, Krumrick et al. [15] also compared the model and 17 CFD simulations obtaining considerably consistent results. The objective of the present work is to implement mechanistic flow pattern and friction loss models for gas-liquid and then predict the operational range of a multiphase circuit of the Research Group for Oil and Gas Flow and Measurement (NEMOG). Based on the bibliographic review of gas-liquid flow models mentioned above, the Taitel and Dukler [4] flow pattern prediction model was chosen for being validated with experimental data provided by Mandhane et al. [16] and being the one that other authors based their work to develop new models. Also, the model developed by Xiao et al. [9] was chosen because it is based on the work of Taitel and Dukler [4] and presents a tool to calculate friction losses. In order to compare

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results obtained by different models, the latest developed model presented by Petalas and Aziz [3] was also implemented, and both flow pattern and friction losses were estimated. This model was selected due to the author’s reported performance and its capabilities to predict flow patterns and friction losses at any slope. Therefore, soon and using data generated by the NEMOG’s test facility, it will be possible to experimentally validate the predictions made by currently selected models and evaluate its performance in a wide flow range.

22.2 Methodology In general, mechanistic models are widely applicable, which means these models can be used for many fluids with a broad range of flow rates and flow patterns. For this implementation, the models developed by Taitel and Dukler [4] for flow pattern prediction, Xiao et al. [9] for frictional pressure drop, and Petalas and Aziz [3] for both flow pattern, and friction losses prediction were chosen. The first model was validated against the authors’ experimental results, while the second one was compared with a data bank provided by Payne et al. [17]. For the third model validation, the flow pattern prediction mechanism was also confronted with the author’s results, and friction losses calculations were compared with the same data set. Such data set was selected due to its geometrical and operational similarity compared to NEMOG’s multiphase circuit. Both multiphase circuits operate with the air-water flow in 52.48 mm (2" Schedule 40) diameter pipes. After validating the selected models, they were implemented to predict NEMOG’s circuit operational range for air-water flow, considering at first the flow rate domain given by the water pump and the air compressor capacities. A new domain is established from the obtained frictional pressure drop for the flow rates, and the expected flow patterns are presented.

22.3 Preliminary Results 22.3.1 Implementation and Validation of Models The validation of selected models was made by a Visual Basic for Applications (VBA) code implemented on Microsoft Excel. The ones proposed by Taitel and Dukler [4] and Petalas and Aziz [3] for flow pattern prediction were validated by applying the same superficial velocities, Usl and Usg , used by the authors. Figure 22.1 presents the comparison between the flow pattern transition lines presented by the authors and the implementation of the present work (dots). It is also important to highlight that the units of superficial velocity were selected according to the results presented by each author.

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Fig. 22.1 Pattern flow map Taitel and Dukler (1976). Usg = 0.1 − 500 m/s, Usl = 0.01 − 50 m/s at 25 ◦ C, 1 atm e Dint = 25 mm (left)—Petalas; Aziz (2000). Usg = 0.01 − 200 ft/s, Usl = 0.01 − 100 m/s at 25 ◦ C, μg = 0.01 cP, μl = 1.0 cP, ρ g = 1.28 kg/m3 , ρl = 999.55 kg/m3 , e Dint = 52 mm (right)

It can be observed that the regions representing each flow pattern, as predicted by current computational implementation, agree with delimited regions highlighted by each respective author. In sequence, the validation of Xiao et al. [9] and Petalas and Aziz [3] models for friction losses was made by plotting the calculated pressure drop versus the experimental data presented in the databank of Payne et al. [17], which consists of 70 experimental results with different pressures, temperatures, and superficial velocities. The comparison can be seen in Figs. 22.2 and 22.3. In general, it can be concluded that the prediction made with both models followed the 45◦ line considering the entire data bank. From the results obtained by Xiao et al.’s [9] model, the average absolute relative error of the whole flow range is E AT = 52.90%, while the average absolute relative errors for each flow patterns are E AI = 33.34% for intermittent, E ASW = 61.24% for stratified wavy, and E A AD = 66.67% for annular-dispersed liquid. The author also reported that the frictional pressure drop for the intermittent flow presented inferior error than the other flow patterns, which is also noted in the present study. Besides that, it was noticed that the model also performed better in water flow ranges higher than 3 m3 /h with an average absolute relative error limited to this range of E A3 = 22.32%. On the other hand, the results obtained from the model developed by Petalas and Aziz [3] showed more precise results on entire Payne et al.’s [17] data bank, presenting an average absolute relative error of E AT = 36.80%. Besides that, the average absolute relative errors for each flow patterns are E AI = 39.40% for intermittent, E ASW = 79.80% for stratified wavy, and E A AD = 20.7% for annular mist. As calculated, the Petalas’ model performance is notably better considering the average absolute relative error for all predictions. It can also be observed that the calculated standard deviation for each model is σ Xiao = 46.38% and σ Petalas = 25.36%,

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Fig. 22.2 Pressure drop prediction by Xiao et al.’s [9] model versus Payne et al.’s [17] database. D = 52.48 mm, L = 139.2 m and α = 0.14 ◦

Fig. 22.3 Pressure drop prediction by Petalas and Aziz’s [3] model versus Payne et al.’s [17] database. D = 52.48 mm, L = 139.2 m and α = 0.14 ◦

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which indicate that Petalas’ model may result in better quality predictions for applied conditions. However, Xiao’s model’s performance is distinguishable compared to the second one for stratified wavy and intermittent regimes, considering only the average error. Even with these results, it is possible to verify that the dispersion of the first model is greater when confronted with the second model, and the maximum absolute relative error reaches 228.74% on the annular dispersed regime, which indicates that for a single measurement, the error can be significant. On the other hand, the second model presented a maximum absolute relative error of 93.43% on the stratified wavy regime, which supports this model’s lower error standard deviation. At last, linear trend lines were plotted, and both models’ courses are following the 45 ◦ line with the entire data bank. According to the results, Xiao et al.’s [9] model slightly deviates from the 45 ◦ line in higher friction losses situations. As a result, an overprediction can be observed on these points. For Petalas and Aziz’s [3] model, an almost parallel trend line is shown; however, underestimating friction losses results along the 45 ◦ line. The overall evaluation supporting the better results of Petalas and Aziz’s [3] model may be explained by the high concentration of experimental points in the annular regime, where this model clearly showed more precise predictions. Additionally, the relatively small error standard deviation of this model may lead to trend lines that follow the 45 ◦ line on annular mist and intermittent regimes, while results of stratified wavy trend lines are inconclusive. For these reasons, it can be concluded that a more extensive data set is required to precisely evaluate prediction in specific flow regimes since the present study explored only three flow patterns on a total of 70 points. From observation of Figs. 22.1 to 22.3, it can be concluded that the implementation of the referred predictive methods are considered validated.

22.3.2 NEMOG’s Multiphase Circuit Application The multiphase closed-loop is dedicated to evaluating the performance of multiphase flow meters by comparing the single-phase flow metering of each component: mineral oil, water, and compressed air. Each section of the closed-loop has its respective functionality, and details can be observed in Fig. 22.4. Single-phase fluids are stored in tanks for tap water and mineral oil (Sector I). In addition, after individual measurement and control in Section III, the multiphase flow is generated by mixing single-phase components (Sector IV). The reference technologies for single-phase flow metering are independent of each other. Besides flow rate, fluid densities, BS&W, pressure, and temperature are monitored as well. The tubing of the test branch is made of steel, #150 psi class (2” × 12 m each straight branch). Additionally, translucid tube spools allow the visualization of multiphase flow patterns (Sector V). After passing by the test branch, the components are separated by a sizeable gravimetric vessel (Sector II).

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Fig. 22.4 Overview of NEMOG’s multiphase flow loop Table 22.1 Operational conditions based on NEMOG’s facility Fluid Operating conditions Minimum flow rate Water Air Water Air

P = 101.3 kPa, T 25 ◦ C P = 101.3 kPa, T 25 ◦ C P = 335.5 kPa, T 25 ◦ C P = 335.5 kPa, T 25 ◦ C

Maximum flow rate

=

Q min =

=

Q min = 7.2 m3 /h

Q max = 120 m3 /h

=

Q min = 2 m3 /h

Q max = 80 m3 /h

=

Q min = 7.2 m3 /h

Q max = 120 m3 /h

2 m3 /h

Q max = 80 m3 /h

Considering the flow loop showed in Fig. 22.4, the capacity of the circuit was evaluated in terms of operational conditions of water pump, air compressor, and single-phase flow meters. It is important to bring out that the water pump delimited the maximum operational pressure increment, which reaches  Pmax = 234.2 kPa. This capacity is shown in Table 22.1. In sequence, Xiao et al. [9] and Petalas and Aziz [3] models were implemented in order to calculate the maximum possible flow rate in which the pressure drop would not overcome the value of Pmax . It was found that water flow rates equal to 42.5 m3 /h and airflow rates equal to 70 m3 /h are the limiting operational conditions related to the first model. On the other hand, the second model resulted in a stricter water flow rate, limited to 35 m3 /h and a broader air flow rate, limited to 120 m3 /h. Figure 22.5 shows all operational conditions calculated and the respective pressure drop, considering only combinations included in this limit. Alternatively, patterns that will be present in future experiments can be estimated by restricting flow ranges in flow pattern maps generated using Taitel and Dukler [4], and Petalas and Aziz [3] models. Figure 22.6 demonstrates ranges delimited by rectangles for operational conditions and the calculated pressure drop. According to the results, the presented pressure variation implies a displacement of flow regimes for both models. However, neither of them indicates that a new

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Fig. 22.5 Pressure drop prediction of NEMOG’s circuit. D = 52.48 mm, L = 25.07 m and α = 0 ◦

flow pattern will be observed or suppressed. On the other hand, each model shows that different flow patterns will be observed in the operational calculated range. For the Taitel and Dukler’s [4] model, it is verified that annular dispersed liquid, intermittent, and possibly dispersed bubble regimes may be present on NEMOG’s facility. When compared to Petalas and Aziz’s [3] model results, it can be seen that more flow patterns are described, including dispersed bubble, froth, annular mist, stratified wavy, and slug.

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Fig. 22.6 Pattern flow prediction of NEMOG’s circuit. D = 52.48 mm and α = 0 ◦

22.4 Conclusion and Future Perspective It was possible to validate the implementation of Taitel and Dukler [4], Xiao et al. [9], and Petalas and Aziz [3] models. Also, two flow pattern maps for each pressure value were elaborated considering the operational conditions of NEMOG’s multiphase circuit, and possible limitations of pressure drops are exposed accordingly to the model applied. Both models predict similar results. However, Petalas and Aziz’s [3] model demonstrated some more flow pattern configurations, reduced standard deviation, and lowest average absolute relative error for pressure drop prediction. It is possible to assimilate those results because most mechanistic models require empirical correlations to describe some flow patterns completely. Since the latest model is based on previously developed and tested models then preliminary published in 1996, being continually refined and tested against large data banks until four years later, it is expected that its results may show the best agreement with observations and measurements compared to other models. Considering the limited availability of experimental data, only 70 readings found in the literature were used to compare both models’ performance to predict fric-

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tion losses. This experimental lack of data does not lead to a precise analysis of pressure drop on each flow pattern and a comparison of each model’s performance. Furthermore, it is expected to compare the results obtained by each model with the experimental data generated on the NEMOG’s flow circuit. In this way, the performance of each model can be determined based on a large sampling of experimental readings in terms of flow pattern prediction and pressure drop, as well. Consequently, it may be possible to identify accuracy, limitations, discontinuities, or even conditions for each model. Finally, it is expected to contribute and improve mechanisms adopted by those models on pattern flow transition and friction losses calculations. Acknowledgements The authors gratefully acknowledge the financial support under Contract number 2018/00194-3 given by Petrobras and the research scholarship number 88887.496555/202000 given by CAPES (Coordination for the Improvement of Higher-Level Personnel), which has been made possible the work described in this paper.

References 1. Y. TAITEL, D. BARNEA, Encyclopedia of Two-Phase Heat Transfer and Flow I (World Scientific Publishing Co. Pte. Ltd, Lausanne) (2015) 2. M. Jerez-Carrizales, J. Jaramillo, D. Fuentes, Review. Ingeniería y Ciencia v., 11, 213–233 (2015) 3. N. Petalas, K. Aziz, J. Canad. Petrol. Technol. 39, 43–55 (2000) 4. Y. Taitel, A. Dukler, AIChE J. 22(1), 47–55 (1976) 5. Y. Taitel, D. Barnea, A. Dukler, AIChE J. 26(3), 345–354 (1980) 6. D. Barnea, O. Shoham, Y. Taitel, A. Dukler, Chem. Eng. Sci. 40(1), 131 (1985) 7. D. Barnea, Int. J. Multi. Flow v., 13, 1–12 (1987) 8. A. Ansari, SPE Prod. Facil. v., 9, 143–152 (1994) 9. J. Xiao, O. Shoham, J. Brill, in Proceedings—SPE Annual Technical Conference and Exhibition, vol. v. Pi, pp. 167–180 (1990) 10. A. Dukler, M. Wicks, R. Cleveland, Frictional pressure drop in two-phase flow: a. A comparison of existing correlations for pressure loss and holdup (1964) 11. H. BEGGS, J. Brill, JPT, J. Petrol. Technol. (1973) 12. H. Mukherjee, J. Brill, J. Energy Res. Technol. Trans. ASME v., 107, 549–554 (1985) 13. N. Petalas, K. Aziz, Stanford Multiphase Flow Database—Users Manual, Version 0.2 Petroleum Engineering (Dept., Standford University) 14. A. Shuard, H. Mahmud, A. King, IOP Conf. Ser.: Mater. Sci. Eng. 121 (2016) 15. E. Krumrick, E. Lopez, A. Camacho, Mecánica Computacional 34, 2101–2114 (2016) 16. J. Mandhane, G. Gregory, K. Aziz, A flow patten map for gas-liquid flow in horizontal pipes (1974) 17. G. Payne, JPT, J. Petrol. Technol. v., 9–1198 (1979)

Chapter 23

Engineering Aspects on Flow Similarity for Design Water-in-oil Emulsion Circuit Edimilson Kempin Jr., Juliana T. A. Roberti, Ligia G. Franco, and Rogério Ramos

Abstract This study aims to show some aspects of flow similarity among an actual extraction plant of oil production and the design of a lab flow circuit. Emulsions are formed when oil and water are pumped from a reservoir to a separator vessel in oil production. In Brazil, oil production is mainly offshore and presents kilometric dimensions. In order to design a laboratory-scale flow circuit while maintaining some dynamic resemblance to the production lines, technical challenges must be faced, such as the balance among the increase in pressure drop as a consequence of the decrease in tube diameter due to the geometric similarity. Even so, the Weber number fits flow similarity requirements. Keywords Emulsion · Droplet break up · Multiphase flow · Flow similarity

23.1 Introduction Hydrocarbons such as oil, natural gas, and coal are important sources of energy over renewable energy, and nuclear energy [1]. Thus, it is desired to improve the efficiency of the overall process of oil and gas extraction. That way, it reduces the chemical consumed in the phase separation process. In oil production, emulsions are formed when oil and water are pumped from an oil reservoir to an upside separator vessel [2]. Emulsions are mixtures of two or more immiscible liquids in which one liquid is a continuous phase, and the others are droplets distribuition. The dynamics of emulsion flow through curves, tubes, pumps, valves break such droplets into tiny sizes of very stable emulsion [3]. The physical mechanisms that break the droplets over the several pipe fittings from the reservoir through the separator are not completely known, such as choke valve, submersible pump, and Wet Christmas Tree (WCT). Crude oil emulsions are stabilized by the presence of the natural surfactants in petroleum, such as asphaltenes E. Kempin Jr. · J. T. A. Roberti (B) · L. G. Franco · R. Ramos Research Group for Oil and Gas Flow and Measurement (NEMOG), Federal University of Espirito Santo, Vitoria, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_23

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and resins [4]. Demulsifiers are employed in the process of emulsion breaking into separate phases, i.e., oil and water. It is important to avoid such emulsions before transportation through pipelines and before refining [2] or to obtain emulsions with larger droplet sizes. Those emulsions may cause several operational problems from production to refinings, such as in the demulsification of crude oil and brine and also the oxidation of the facilities [5]. In addition, separation of emulsions requires expensive chemical products as well. The knowledge of the two-phase flow similarity of oil and water under a set of conditions, such as thermodynamic state and daily amount of oil production, constitutes a relevant industrial challenge for the petroleum industry. Initial studies in oil-water flow used the experimental approach investigating vertical [6] and horizontal flow [7]. The works from Govier et al. [6] and Trallero et al. [7] are used as a reference in further studies for vertical, horizontal, and inclined pipes [8–11]. In those researches that investigated liquid-liquid flow, oil and water were pumped to the test loop as separate phases, which does not guarantee the formation of emulsions but only a two-phase flow. In the other way, it is found in literature articles about experiments with stabilized oil-water emulsions. Wang et al. [12] studied the flow of crude oil-water and mineral oil-water emulsions, where both oils present similar properties. Hence, the objective of this paper is to design a laboratory flow circuit representing in scale, an actual offshore installation, from downhole to choke valve. This test bench aims to evaluate the distribution of droplets’ size of water-in-oil emulsions due to the presence of pumps, pipe fittings, and valves.

23.2 Methodology For the aims of this work, a circuit, on a reduced scale, must be designed in a way that represents offshore oil production and, at the same time, be installed in a laboratory environment. Then, the dimensionless parameters and pressure drop are estimated and compared to numerical data from numerical simulation of an actual offshore hydrocarbon extraction plant.

23.2.1 Scale The actual production line is sketched in Fig. 23.1 showing its shape and primary devices, such as Permanent Downhole Gauge (PDG) and WCT. In Fig. 23.1 the devices are PDG, that is, a pressure and temperature gauge; gas lift valve (GLV) that injects a high-pressure gas continuously down the annulus; WCT, which is an assembly of valves to regulate the flow; Flowline and Riser are flexible pipelines that cover the seabed and rise the fluid to the platform, respectively.

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-500

VERTICAL COORDINATE (m)

-1000

-1500 TPT WCT

FLOWLINE + RISER WELL

-2000

WCT PLATFORM

-2500

PDG GLV

GLV -3000

TPT

PDG -3500

-7000

-6000

-5000

-4000

-3000

-2000

-1000

0

-4000

HORIZONTAL COORDINATE (m)

Fig. 23.1 Sketch of an actual oil exploitation tubing line Table 23.1 Maximum dimensions allowed in the laboratory Dimension Value [m] Height Length

7 10

The laboratory scaled flow circuit should satisfy the space available at the Flow Laboratory of Research Group for Oil and Gas Flow and Measurement (NEMOG, in Portuguese), located at the Federal University of Espírito Santo (UFES). Table 23.1 shows the maximum available space inside the lab. One can notice that the flow circuit must have its geometrical scale reduced and keep its flow similarity. Table 23.2 shows different dimensions of the actual extraction plant and proposed scales for the flow circuit. After analyzing Tables 23.1 and 23.2, the best scale can be selected.

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Table 23.2 Proposed scales for the flow circuit Section Dimension Scale 1:1 1:8 PGD - WTC Flowline Riser a

La

[m] Db [m] L [m] D [m] L [m] D [m]

1,593.0 1.2×10−1 5,830.0 1.8×10−1 1,862.0 1.8×10−1

199.1 1.6×10−2 728.8 2.2×10−2 232.8 2.2×10−2

1:10

1:100

1:600

159.3 1.2×10−2 583.0 1.8×10−2 186.2 1.8×10−2

15.9 1.2×10−3 58.3 1.8×10−3 18.6 1.8×10−3

2.7 2.0×10−4 9.7 3.0×10−4 3.1 3.0×10−4

Length, b Diameter

Table 23.3 Oil and brine properties Data Oil [kg/m3 ]

Density Viscosity [Pa.s]

Brine

877.9 1.4×10−1

1,022.1 9.0×10−4

23.2.2 Emulsion In order to reach the objectives of these studies, emulsions composed of mineral oil (Mobil AW-68) as the continuous phase, brine (35g/L NaCl), and surfactants are prepared at the Laboratory of Research and Development of Methodologies for the Analysis of Oils (LabPetro/UFES). The properties of oil and brine are shown in Table 23.3. The properties of emulsions, such as density ρe , viscosity μe , interfacial tension σ , and the water drop diameter distribution ddr are obtained experimentally using analyzers and standard test method detailed at Table 23.4. After several tests, stabilized emulsions are composed of 10% of brine and 0.1% of surfactants.

Table 23.4 Equipments employed to obtain emulsions properties Properties Devices Density (ρe ) Viscosity (μe ) Interfacial tension (σ ) Water droplet size distribution (dd )

ASTM D5002 ASTM D4402 Tensiometer SEO (model Phoenix MT) Betterziser ST

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23.2.3 Dimensionless Parameters Dimensionless parameters can yield insight into the underlying physical phenomena and thus indicate the dominant forces for specific flow conditions [13]. Those that are important for this study are analyzed by the aspects of flow similarity. To assure the flow similarity among the lab emulsion flow and hydrocarbon oil emulsions flow, dimensionless parameters must present similar values or, at least, be reported at a reasonable order of magnitude. Weber number stands for the ratio among the inertial force and interfacial tension amongst phases. This is important for droplet breakup because inertial forces deform droplets, and when it exceeds Weber’s critical value, the droplet breaks [14]. Assuming a homogeneous no-slip model [15], Reynolds and Weber numbers for emulsion flow are computed by definitions:

Ree =

We =

ρe Ve D μe

(23.1)

ρe Ve2 ddr σ

(23.2)

where Ve is the emulsion velocity and D is the tube diameter.

23.2.4 Pressure Drop The pressure drop can be estimated after defining the length scale, tube diameter, emulsions, and dimensionless parameters. It is essential to know how much pressure is lost by the friction resistance on the flowing fluid, and it is computed by the classical definition of pressure drop [13]. Pressure drop is estimated for different fluid compositions (brine, oil, and emulsions) to compare among them. This analysis is essential once the pressure drop cannot exceed the lab pressure class of 150 psi. Moreover, pressure drops are required for the correct sizing pumps, valves, and tubes.

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23.3 Results and Discussion 23.3.1 Scale Analyzing the lengths, the scale 1:600, as shown in Table 23.2, is considered as the best fit when compared to the available space. However, the diameter at this scale is a capillary tube, which does not represent the physical phenomenon of emulsion flow in current research. With that in mind, the diameter must be on a larger scale, considering commercially available tubes. The scale that provides a reasonable emulsion flow similarity is 1:8. Even though the sections at PDG, WCT, flowline, and riser present different diameters, those can be neglected. By standardizing the diameter of the tube, the one commercially available near these dimensions is a 3/4 inch, stainless steel made, schedule 40. Figure 23.2 shows the sketch of the laboratory flow circuit. As can be seen in Fig. 23.2, the primary devices have been inserted into the circuit designed in a simplified way. In addition, its layout guarantees changes in the future, such as a device to represent the GLV. 5

4

1)PDG 2)WCT 3)Flowline

4)Riser 5)Choke Valve 6)GLV*

3 2

1

6

Fig. 23.2 Sketch of laboratory flow circuit project

23.3.2 Dimensionless Parameters For the analysis of dimensionless parameters, simulated data at PDG position are used. Such data are shown in Table 23.5.

23 Engineering Aspects on Flow Similarity for Design … Table 23.5 Simulated offshore data Data

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PDG

Pressure [bar] Temperature [◦ C] Density (ρ) [kg/m3 ] Viscosity (μ) [Pa.s] Interfacial Tension (σ ) [N/m] Droplet Diameter (ddr ) [m] Reynolds Number Weber Number

221.6 65.8 922.20 6.20×10−4 2.04×10−2 1.00×10−4 3×105 6.04

As introduced before, the Weber number is an important parameter to define the droplet breakup phenomenon. Therefore, the Weber number is the reference for this study, and, consequently, the emulsion properties must be adapted to yield the Weber number as close as possible to its value on the offshore extraction process. Even though the properties can be changed, Weber’s number also depends on the emulsion superficial’s velocity, as shown in Equation (23.2). It is considered that the maximum workable superficial velocity is 3.9 m/s in liquid flow loops [16]. Since Weber’s number must be kept, the Reynolds number is computed with the same flow velocity and fluid properties manipulated to yield the most suitable Weber number.

23.3.2.1

Offshore x Laboratory Circuit

Table 23.6 shows properties of emulsions A, B, C, and D, and Table 23.7 compares their performance in circuit.

Table 23.6 Emulsions characterization Properties Emulsions A B Density (ρ) [kg/m3 ] Viscosity (25 ◦ C) (µ) [Pa · s] Droplet diameter (ddr ) [µm] Interfacial tension (σ ) [mN/m]

C

D

875.3

879.0

882.3

883.2

0.1581

0.1721

0.1722

0.1631

16.20

9.40

8.86

17.62

10.22

2.50

2.50

9.48

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Table 23.7 Operational parameters for a settled Weber number (We=6.04) Operational Emulsions parameters A B C Reynolds number Superficial velocity [m/s] Flow rate [m3 / h] Experimental time [min]

D

220 2.09

132 1.35

136 1.39

198 1.92

2.14 10.97

1.39 16.93

1.43 16.47

1.97 11.93

From Table 23.7, one can notice that all emulsions are suitable once the superficial velocity does not exceed the maximum velocity (Vmax ) for safety operation, which for the current project is 3 m/s. Emulsions A and D determine the highest Reynolds number, but their experimental batch time is lower than those of emulsions B and C. This parameter is important to the operation once the flow circuit will mostly operate in an open-loop. Thereupon, emulsions B and C are the most suitable. Although this may be true, analyzing the Reynolds number, the value obtained for emulsion B is three orders of magnitude smaller than the actual Reynolds number at PDG. There are some explanations for that, with emphasis on the high viscosity of the mineral oil selected for emulsions (0.1315 Pa.s) over the viscosity data at PDG, as shown in Table 23.5. Additionally, PDG data present high pressure and high temperature (221 bar and 65 ◦ C), impacting density and viscosity. High pressure dissolves gases in oil which substantially reduces the oil viscosity. Furthermore, high temperatures reduce the oil viscosity and also might decline emulsion stabilization. To reduce the gap in Reynolds number, one can increase the diameter of the pipe, the velocity, and the ratio among density and viscosity. The first option cannot be done due to the geometric similarity. The velocity can be modified, but this one is critical since it affects Weber’s number and pressure drop, explained starting now.

23.3.3 Pressure Drop Pressure drop is estimated, as a function of Reynolds number, for brine, mineral oil, emulsions A, B, C, and D, as shown in Fig. 23.3. In Fig. 23.3, one can notice that in the case of neat brine flowing in the circuit, there is no restriction for Reynolds number once the pressure drop is lower than 1 bar. To better understand its behavior, Table 23.8 shows the computed data. From Table 23.8, it is noted that the required superficial velocity to reach the borderline Reynolds number for laminar flow Re = 2,300 is around 0.1 m/s, while the Reynolds number for Vmax is 65,611. For both cases, the estimated pressure drop is low due to the low viscosity of brine.

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80 A B C D Brine AW68

Pressure Drop [bar]

60

40

20

0 0

500

1000

1500

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Reynolds Number - Ree

Fig. 23.3 Pressure drop estimations for brine, oil, and emulsions A, B, C, and D Table 23.8 Behavior of brine flow Reynolds Velocity [m/s] 2,300 65,611

0.105 Vmax

Pressure drop [bar] 0.603 2.124

In contrast, the pressure drop predicted for AW-68 samples (neat or emulsified) increases significantly, restricting the acceptable flow velocity and, consequently, restricting Reynolds number as well. Emulsions B and C present a similar pressure drop curve and the highest pressure drop due to their higher viscosities than emulsions A and D. Although this drawback, these emulsions have low interfacial tension, which fits Weber’s number at low superficial velocity. Furthermore, in order to chose tube diameter, Fig. 23.4 shows the pressure drop of emulsion B, which is the one with the best operational characteristics, for distinct diameters and the same Reynolds number range. Other than that, Table 23.9 highlights some relevance points from Fig. 23.4. As can be seen in Fig. 23.4 and Table 23.9, tube diameters under 3/4 inch exceed the maximum pressure drop of 10.3 bar with low Reynolds number. In contrast, greater diameters present lower pressure drops but divert even more from the geometric similarity. Even though it seems to increase the Reynolds number from 300 to 500 at 10 bar, the superficial velocity is higher than Vmax . This analysis validates the chosen diameter of 3/4 inch.

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Pressure drop [bar]

100

10

1 0

400

800

1200

1600

2000

2400

Reynolds Number - Ree

Fig. 23.4 Pressure drop for distinct pipe diameters Table 23.9 Detailed points of pressure drop of emulsion B Reynolds number Diameter 1" 3/4" Superficial Pressure drop Superficial Velocity [m/s] [bar] Velocity [m/s] 300 500

2.31 3.85

6.05 9.73

3.08 5.14

Pressure drop [bar] 10.34 16.89

23.4 Conclusion and Future Perspective Considering all discussions through this paper, a test bench is designed with flow similarity with an actual extraction plant guaranteed by the Weber number. Complete flow similarity is not reached for several reasons, such as space limitations and a laboratory pressure class of 150 psi. A complete flow similarity between the actual oil extraction plant and test circuit is of enormous complexity. The laboratory pressure class cannot be exceeded for safety issues even though an experimental flow circuit was designed to study the droplet distribution in the water-in-oil emulsions along with the production plant. The Reynolds number by simulation is 300,000, while the designed flow circuit allows Reynolds number less than 300. This considerable difference occurs because the pressure of hydrocarbons right after a petroleum reservoir is higher than 220 bar. In this pressure, fluid is in the supercritical state; thus, both density and viscosity of

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supercritical liquid are low, such as in the order of magnitude of gases. Furthermore, at high pressures, the lighter components of petroleum are dissolved in liquid phases. One way to increase the Reynolds number of lab flow circuits is the addition of GLV. As shown in Fig. 23.2, the circuit presents a space to inject gas into the flow. Another important conclusion is about the pressure drop. Emulsions B and C presented a restricted operation range. It is difficult to increase this range once there are pressure drop limitations due to the lab pressure class. To increase the similarity among actual production lines and laboratory circuits, the lab pressure class must be at least four times higher, i.e., 600 psi. All aspects considered, the designed flow circuit fit the Weber number that describes the droplet breakup phenomenon. Emulsions B and C can be taken as a parameter to develop others with better performance. Acknowledgements The authors would like to express their gratitude to Petróleo Brasileiro Company (Petrobras) and Agência Nacional do Petróleo, Gós Natural e Bio Combustíveis (ANP) by financial support. Thanks to the Research Group for Oil and Gas Flow and Measurement (NEMOG/UFES) and the Laboratory of Research and Development of Methodologies for the Analysis of Oils (LabPetro/UFES).

References 1. T. Paulauskiene, Petroleum Extraction Engineering (Recent Insights in Petroleum Science and Engineering, 2018). https://doi.org/10.5772/intechopen.70360 2. A.M. Alsabagh, Egypt. J. Petrol. 25(4), 585 (2016) 3. M.J. Van der Zande, W.M.G.T.V. den Broek, in Proceedings of ASME Energy Sources Technology Conference and Exhibition (Houston, 1998) 4. C.G. Quintero, Rheologica Acta 47(4), 417 (2008) 5. M. Stewart, K.E. Arnold, Surface Production Operations, Volume 1: Design of Oil Handling Systems and Facilities, vol. 1 (Elsevier, 2011) 6. G.W. Govier, G.A. Sullivan, R.K. Wood., Canad. J. Chem. Eng. 39.2, 67 (1961) 7. J.L. Trallero, C. Sarica, J.P. Brill, SPE Product. Facilit. 12(03), 165 (1997) 8. J.G. Flores, Ph.D. thesis, vol. 950 (1997) 9. J.G. Flores, SPE Product. Facil. 14(02), 94 (1999) 10. O.M.H. Rodriguez, R.V.A. Oliemans., Int. J. Multi. Flow 32.3, 323 (2006) 11. T. Al-Wahaibi, J. Petrol. Sci. Eng. 122, 266 (2014) 12. W. Wang, J. Gong, P. Angeli., Int. J. Multi. Flow 37.9, 1156 (2011) 13. R.W. Fox, A.T. McDonald, J.W. Mitchell, Fox and McDonald’s Introduction to Fluid Mechanics (10th edn., Wiley, 2020), p. 205 14. R.Q. Duan, S. Koshizuka, Y. Oka., J. Nucl. Sci. Technol. 40.7, 501 (2003) 15. G.B. Wallis, One-Dimensional Two-Phase Flow (Dover Publications, 2020) 16. W.L. Loh, V.K. Premanadhan., J. Petrol. Sci. Eng. 147, 87 (2016)

Chapter 24

Experimental Analysis of Three-Phase Solid-Liquid-Gas Slug Flow with Hydrate-Like Particles Stella Cavalli, Rafael F. Alves, Carlos L. Bassani, Eduardo Nunes dos Santos, Marco da Silva, Moises A. Marcelino Neto, Amadeu K. Sum, and Rigoberto E. M. Morales Abstract Gas hydrates formation and agglomeration represent a significant challenge in flow assurance during oil and gas production operations. An alternative to managing this problem is the use of anti-agglomerants (AAs), allowing hydrates to flow as solid particles dispersed in the liquid phase. Although the particles are henceforward not susceptible to agglomeration, it is still essential to understand how those particles affect and are affected by the gas-liquid multiphase flow structures. This study evaluates the influence of the introduction of particles, with properties similar to gas hydrates, into the multiphase flow, focusing on the slug flow pattern. The experiments were conducted in a 9-m length, 26-mm ID flow loop, with air and water at ambient conditions as working fluids. The particles were made of polyethylene with a density of 937 kg/m3 , similar to gas hydrates. The particle sizes tested were 100 µm, 200 µm, 300 µm, and 400 µm, with volumetric concentrations of 1%, 2.5%, and 5%. This study discusses how the particles influence the velocities and lengths of the structures and how the multiphase flow affects the transportability of the particles. Keywords Slug flow · Hydrates · Flow with particles

24.1 Introduction Gas hydrates formation and agglomeration represent one of the most significant challenges in flow assurance in oil and gas production operations [1]. An alternative to managing this problem is the use of anti-agglomerants (AAs), allowing hydrates to S. Cavalli (B) · R. F. Alves · C. L. Bassani · E. N. dos Santos · M. da Silva · M. A. M. Neto · R. E. M. Morales Multiphase Flow Research Center (NUEM), Federal University of Technology - Paraná (UTFPR), Curitiba, Brazil e-mail: [email protected] A. K. Sum Phases to Flow Laboratory, Chemical and Biological Engineering Department, Colorado School of Mines, Golden, CO, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_24

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flow as inert solid particles dispersed in the liquid phase. Nevertheless, should the particles be inert agglomeration-wise, one still needs to understand their transportability and how those particles affect and are affected by the gas-liquid multiphase flow structures. This comes from the microfluid dynamics occurring around the particles, which modulate the turbulent field of the liquid phase [2] and then interact with the larger structures of the flow [3]. Most studies that assess slurry flows analyze sand particles and focus on transportability only. Unlike the others, Rosas et al. [3] evaluated the influence of the particles on the gas-liquid slug flow structures. The authors analyzed inert polyethylene particles of 500 µm, with hydrate-similar properties, in volumetric concentrations under 1%. They reported an increase in the turbulent intensity, which explained the increase in the slug aeration and the reduction of the length of the elongated bubble. The present study extends the grid test of Rosas et al. [3] by considering the influence of different particle sizes and higher concentrations.

24.2 Experimental Techniques The experiments were conducted in a 9-m length, 26-mm ID flow loop made of a transparent acrylic pipeline to allow flow visualization. The fluids used were air and water at ambient temperature and pressure. The particles were made of polyethylene with a density of 937 kg/m3 , mimicking the one present in gas hydrates. The particle sizes tested were 100 µm, 200 µm, 300 µm, and 400 µm, with volumetric concentrations of 1%, 2.5%, and 5%. A total of twelve (12) combinations (5.1.2) of liquid and gas flow rates were performed, with superficial velocities ranging within 0.25 < JG < 1.5 m/s and 0.5 < J L < 1.5 m/s. All combinations correspond to the slug flow pattern, as shown in Fig. 24.1, where the flow rate combinations are plotted on the Taitel and Dukler [4] flow map. The measurements were also carried out for gas-liquid flows (without particles) in the same conditions, for the sake of comparisons. The particles were thoroughly mixed inside the water tank, assuring their homogeneity. The slurry and the gas were then injected at the inlet of the test section as a stratified flow, which naturally transitioned to a slug flow pattern a few meters downstream. A High-Speed Camera was used to visualize the slug flow morphology and the slurry homogeneity along the slug flow’s unit cells. The camera was positioned at 6 m from the inlet of the test section. Resistivity sensors, specially developed in a non-intrusive split-ring configuration, were used to monitor the slug flow parameters. Split-ring sensors monitored the resistivity of the medium in order to estimate the in-situ phase fraction. The sensors were used in pairs, allowing the estimation of the elongated bubble translational velocity through the transit time of the signals between two sensors, as well as the elongated bubble and slug lengths.

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Fig. 24.1 Pairs of superficial velocities inside the Taitel and Dukler [4] flow map

24.3 Results and Discussions The insertion of solid particles causes several effects in the slug flow, whereas the slug flow hydrodynamics influences the distribution of the particles along the unit cell. For cases with high superficial liquid velocities, the slurry remains homogeneous. In the case of low superficial liquid velocities, the particles tend to float close to the gas-liquid interface in the film region, thus causing a heterogeneous slurry along the unit cell, as presented in Fig. 24.2. With the increase of the concentration of particles in the flow, there is the formation of a film around the elongated bubble, which becomes increasingly thicker for increasing particle concentrations. This film influences the displacement of the elongated bubble and the coalescence between elongated bubbles and dispersed bubbles along the flow. The presence of the particles causes the number of dispersed bubbles to decrease, in both the film and the slug regions, occurring mainly for the larger particles, as shown in Fig. 24.3. This shows that the presence of the solid particles disturbs the detachment of dispersed bubbles from the elongated bubble tail. In turn, the presence of the solid particles does not significantly affect the elongated bubble profile. Rosas et al. [3] observed an average increasing trend of the elongated bubble velocity with particle concentration, explained by the turbulence promoted when particles are introduced. The present study, covering a wider range of particle sizes and concentrations, shows that deceleration is actually most prone to occur. Figure 24.4 maps the unit cell’s translational velocity behavior when the particles are introduced in the gas-liquid slug flow. The red dots indicate that UT increases because of the presence of the particles in the flow; the blue ones that UT decreases and the green ones that it is not substantially affected by the particles.

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Fig. 24.2 Wake region of the elongated bubble for 1% particle concentration and for different particle diameters a JL = 1.5 m/s and JG = 0.5 m/s and b JL = 0.5 m/s and JG = 1.5 m/s

Fig. 24.3 Elongated bubble for different particle diameters and for the case without particles. The photos are for 1% particle concentration, JL = 1.5 m/s, and JG = 0.5 m/s

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Fig. 24.4 Map showing the variation of the elongated bubble velocity when a case of gas-liquid-solid flow is compared to a case of gas-liquid flow. Red: increase, blue: decrease, and green: remains the same. Cutting factor: 3%

Notably, the elongated bubble acceleration is related to liquid loadings below 60% and mainly for small particle sizes and concentrations. This behavior can be explained by the process of turbulence modulation, caused by the particles [2]. Larger particles and higher concentrations promote turbulence, so the velocity profile presents a flatter centerline. As the liquid velocity ahead the elongated bubble dictates the unit cell’s translational velocity [5], a flatter liquid velocity profile causes the bubble to decelerate. The particles influence on the elongated bubble and slug lengths can be divided into three groups: the particles can reduce the elongated bubbles’ and slugs’ lengths, increase them, or induce a slight rearrangement of the unit cell geometry. The first group is explained by the turbulence modulation caused by the presence of the particles in the flow. For larger particles and higher concentrations, the turbulence promotion produces a flatter liquid velocity profile. Thus, the wake region becomes less agitated. Consequently, coalescence tends to decrease, thus decreasing the elongated bubbles and slugs lengths along the flow. The second group, in which the particles increase the lengths, is related to slug flow formation. In these particular liquid and gas flow rate combinations, the presence of the particles dissipates the Kelvin-Helmholtz instabilities that form the slug and bubble regions. The slug flow, in this case, evolves from larger bubble and slug lengths since the inlet of the pipe. The unit cells with larger elongated bubbles and slugs tend to coalesce at a higher rate along the flow, and therefore, the second group also shows a trend to increase the coalescence rate. Finally, in the third group, as the elongated bubble velocity tends to decrease, the unit cell structure experiences a rearrangement to balance the liquid mass distribution in both the film and the slug regions. The lengths of the structures are, however, not affected as strongly as in the other groups.

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Fig. 24.5 Map showing the variation of the coalescence rate when a case of gas-liquid-solid flow is compared to a case of gas-liquid flow. Red: increase, blue: decrease, and green: remains the same. Cutting factor: 30%

Most of the scenarios analyzed in the present work correspond to the third case: the elongated bubble and slug lengths remain approximately the same as the ones of the gas-liquid slug flow, presenting only slight variations with the introduction of particles. The green dots represent these scenarios in the map of Fig. 24.5. Both first and second groups concentrate in the lower mixture velocities (JG + J L ), where the liquid phase inertia is not sufficient to overcome the influence of the particles in the liquid velocity profile. The parameter that separates the trends is the intermittency factor (the ratio between the elongated bubble length and the whole unit cell length, L B /LU ). For L B /LU < 0.6, the coalescence increases, as Fig. 24.5 shows, by the red dots. For L B /LU > 0.6, the coalescence rate decreases, as indicated by the blue dots. As the frequency represents the ratio between the elongated bubble translational velocity and the unit cell length, their results depend on the trends presented earlier. In most cases, the elongated bubble and slug lengths remain nearly the same, as demonstrated previously. So, most of the results for the slug flow frequency depend on the elongated bubble velocity variation: for high liquid loadings (above 60%), the velocity tends to decrease, and so does the frequency.

24.4 Conclusion and Future Perspective This study presents experimental data of gas-liquid-solid flow in the slug flow pattern, covering wider ranges of particle sizes and higher concentrations than are the ones currently available in the literature. As a new observation, it was found out that the turbulence promotion caused by the particles acts toward decreasing the elongated bubble velocity due to the flatter liquid velocity profile. This effect is observed mainly in cases with liquid loadings above 60%.

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The elongated bubble and slug lengths remain approximately the same compared to those of the gas-liquid slug flow. For some particular scenarios, mainly for lower mixture velocities, the particles can increase or decrease these lengths along the flow. The decreasing trend comes from the turbulence damping in the wake zone, thus affecting the coalescence between elongated bubbles. The increasing trend occurs when the particles influence the slug formation, generating larger lengths from the inlet of the test section. Finally, variations in the slug flow frequency are a consequence of the variations observed in the elongated bubble velocity and the structures’ lengths. With the advance of oil exploration through long subsea tiebacks (in deeper wells farther and farther away from the coast), an increased number of production lines will be exposed to favorable hydrate formation and agglomeration conditions, especially during shutdowns, scheduled or otherwise. With this insight, the study of solidliquid-gas three-phase multiphase flows shall become increasingly important so that the influence of these solid particles on multiphase flow hydrodynamics and the influence of the multiphase flows on the transportation of these particles can be better understood. The present work introduces an alternative insight into this field of study: using model particles to simulate the hydrates’ hydrodynamics in the flow. New approaches aimed at tackling this problem may quite well arise in the future. Acknowledgements The authors acknowledge the financial support of the Coordination for the Improvement of Higher Education Personnel, Brazil (CAPES).

References 1. C.A. Cardoso, M.A. Goncalves, R.M. Camargo, J. Chem. Eng. Data 60(2), 330 (2015) 2. R. Gore, C.T. Crowe, Int. J. Multiph. Flow 15(2), 279 (1989) 3. L.M. Rosas, C.L. Bassani, R.F. Alves, F.A. Schneider, M.A. Marcelino Neto, R.E. Morales, A.K. Sum, AIChE J. 64(7), 2864 (2018) 4. Y. Taitel, A.E. Dukler, AIChE J. 22(1), 47 (1976) 5. K.H. Bendiksen, Int. J. Multiph. Flow 10(4), 467 (1984)

Chapter 25

Influence of Liquid Viscosity on Horizontal Two-Phase Slug Flow Bruna P. Naidek, Marco G. Conte, Cristiane Cozin, Marco J. da Silva, and Rigoberto E. M. Morales

Abstract Slug flow is present in many industrial processes, including the ones related to the petroleum industry. Such flow pattern is characterized by the intermittent repetition of liquid slugs that may or may not be aerated and elongated bubbles that flow atop a liquid film. Most of the existing models for slug flow have been developed for two-phase water-air flows, but in oil and gas production, the liquid phase can be substantially more viscous than water. This article aims to evaluate the effect of liquid viscosity increase on slug flow parameters, such as bubble velocity and frequency. An experimental study on liquid-gas flows in a 26 mm ID and the 8.65 m long horizontal pipe was developed to achieve this goal. Water and mixtures of water and glycerin with a viscosity of 5.46, 10.27, 15.39, 20.33, and 30.37 cP comprise the working liquids. The slug flow parameters were measured by a resistivity sensor located at one measuring station. Results show that the increase in the viscosity of the liquid results in an increase in the velocity of the elongated bubble. The effects on the slug frequency depend on the superficial velocities of the fluids. Keywords Two-phase slug flow · Viscous slug flow · Influence of the liquid viscosity

25.1 Introduction Multiphase flows are characterized by the simultaneous flow of two or more phases, be those different gases, liquids, or solids. This kind of flow is commonly found in the nuclear and oil industries. During the production of reservoir fluids, the simultaneous flow of gases, liquids, and solid particles is commonly found. However, this flow is usually modeled as a liquid-gas two-phase flow to facilitate analysis and modeling. In a pipe, the interfaces of a two-phase liquid-gas mixture may assume several spatial distributions, depending on the physical properties of the fluids and the pipe B. P. Naidek · M. G. Conte · C. Cozin · M. J. da Silva · R. E. M. Morales (B) Multiphase Flow Research Center (NUEM), Federal University of Technology - Paraná (UTFPR), Curitiba, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_25

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specs, such as its diameter and inclination. Those distributions are called flow patterns or flow regimes. In the oil industry, slug flow is the most commonly reported regime [1]. Slug flows are characterized by the intermittent passage of a liquid slug—which may or may not be aerated—followed by an elongated gas bubble filling most of the cross-sectional area of the pipeline. In a flow in a horizontal pipeline, the elongated bubble concentrates at the top of the pipe due to buoyancy and slides on the top of a liquid film. On evaluating the influence of fluid viscosity on the transitions between flow patterns, the literature has shown that the increase in the liquid viscosity yields slug flows with smaller liquid superficial velocities when compared to the air-water transitions, thus enlarging the envelope of occurrence of this pattern [2, 3]. It is possible to associate this escalated occurrence of the slug flow with fluids whose viscosity is higher than the water, with reports from oil companies accounting that this pattern is the most frequent one. The increase in viscosity also affects intrinsic slug flow parameters such as the elongated bubble and the liquid slug lengths, the elongated bubble velocity, and the slug frequency. Upon evaluating the effect of the liquid viscosity, some authors showed that an increase in the liquid viscosity would reduce the length of the liquid slug region [4, 5]. This reduction in the slug length is associated with the fact that there is less recirculation at the bubble tail, and thus the slug needs a shorter length to develop [6]. The slug frequency increases with the increase in viscosity of the liquid phase since there is a reduction in the lengths of the liquid slug and of the elongated bubble [7, 8]. In the case of the elongated bubble velocity, the studies are split between the effect of viscosity on the drift and the translational velocities of the bubble. As for the drift velocity, it was observed that the increased viscosity of the liquid phase causes its reduction [9, 10]. Nevertheless, the translational velocity of the bubble increases with an increasing liquid phase viscosity [8, 11, 12]. Some authors added a viscosity number to the distribution parameter to include the effect of the liquid viscosity on the bubble velocity [13, 14], thus taking the influence of the liquid viscosity in the elongated bubble nose into account. This study proposes an experimental study on the evolution of two-phase gasliquid slug flows along with a 26 mm ID, 8.65 m long horizontal pipe. The authors will analyze the effects of different viscosities of the liquid on the slug flow parameters. Resistivity sensors were used to measure the elongated bubble velocity and slug frequency. With the variation of those parameters, the phenomena occurring under different viscosities will be discussed.

25.2 Materials and Methods The experimental apparatus used to evaluate the influence of the viscosity of the liquid phase on the characteristic parameters of the slug flow is presented in Fig. 25.1.

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Fig. 25.1 Experimental flow loop

The two-phase circuit has approximately 8.65 m in length and 26 mm in internal diameter. The measuring station is located at 5.57 m (214 D) downstream of the mixer. The station has a pair of resistive sensors developed at the multiphase flow research center—NUEM to extract the intrinsic slug flow parameters [15] and a high-speed camera that allows flow visualization. The flow loop was built in a controlled temperature room. The experimental circuit had two temperature sensors to monitor the liquid viscosity: the first was part of the Coriolis liquid flowmeter, whereas the second was a PT-100 sensor installed at the end of the two-phase circuit. Mixtures of water and glycerin made the viscous liquids used in the experiments. Data for pure water to gauge the instruments and to make the comparisons possible were also gathered. Those mixtures were characterized in a Brookfield viscometer. In the experiments, six different liquid phase viscosities were used. The first fluid was pure water, whereas the other five fluids consisted of mixtures of water and glycerin with viscosities of 5.46, 10.27, 15.39, 20.33, and 30.37 cP. The flow starts in a mixer where the two phases enter parallel, with the liquid at the bottom and the gas at the top, initially producing a stratified flow, which develops into slug flow soon after.

25.3 Results and Discussion Twenty pairs of liquid and gas superficial velocities, varying from 0.25 to 2.0 m/s, were investigated. The experimental cases were defined in order to ensure the slug flow pattern along the entire pipe. The superficial velocities of the liquid and gas directly impact the flow characteristics, such as the detachment of the elongated bubble nose with the increasing superficial velocity of the mixture. This change in the position of the bubble nose reflects the importance of the effects of inertial and gravitational forces. With the

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Fig. 25.2 Bubble nose position as a function of the liquid viscosity and the mixture Froude number

increase in the mixture Froude number, the inertia effects start to dominate the flow, and the bubble nose moves toward the center of the pipe. Figure 25.2 presents the bubble nose images as a function of the liquid viscosity (µ) and the mixture Froude number (Fr J ). The bubble nose begins to take on a narrower appearance with the increase in the mixture Froude number. For the same experimental point, that is, the same superficial velocities of the phases and approximately the same mixture Froude numbers (not quite the same because of the effect of the gas expansion and the increase in gas’s superficial velocity), a narrowing trend of the bubble nose is observed, and the interface becomes smoother. These effects can be associated with the Reynolds numbers that decrease with the increase in liquid viscosity. For mixture Froude numbers associated with elongated bubbles flowing at the upper portion of the pipe and air-water flows, it was observed that the elongated bubble initiates the detachment process at lower mixture Froude numbers for fluids whose liquid viscosity is greater than that of water. This change in the position of the bubble nose with the increase in the mixture Froude number is one of the reasons why, with higher superficial velocities of the mixture, the drift velocity of the bubble can thenceforth be considered null [16]. In addition, it is also one of the reasons for the increase in the elongated bubble velocity since the elongated bubble velocity is a function of the position of the bubble nose and the superficial velocities of the phases. Figure 25.3 shows the mean values obtained for the elongated bubble velocity (VB) for all the 20 experimental points. The main phenomena involved during the slug flow development are the gas expansion and the coalescence between neighboring elongated bubbles. The gas expansion occurs because of the pressure drop along the pipeline. Coalescence owes to the interaction between two consecutive unit cells, where elongated bubbles with higher velocity relative to preceding ones tend to coalesce into a single structure.

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Fig. 25.3 Mean values of the bubble velocity as function of mixture Froude number (Fr J )

There was an increase in the elongated bubble velocity for all analyzed viscosities with an increasing liquid phase viscosity. The greater the mixture Froude number (Fr J ), the greater the increase in the elongated bubble velocity with the increase in the liquid viscosity. This phenomenon can be associated with the increase in the pressure drop, intensifying the expansion of the gas phase. This expansion induces an increase in the volumetric flow rate of the gas phase, thus increasing the elongated bubble velocity. This increase in the bubble velocity with the increase in the liquid viscosity is also associated with the changes observed in the bubble nose, as shown in Fig. 25.2. The comparison between the experimental elongated bubble velocity and the one calculated according to Bendiksen [16] for all experimental points in the six analyzed liquid viscosities is shown in Fig. 25.4. For the elongated bubble velocity in air-water flows, errors of less than 10% were found. However, with the increase in the liquid viscosity, the errors increase, and the higher the liquid viscosities, the greater the errors. Although the correlation of [16] presents a good approximation for the air-water data, experiments with higher liquid viscosities require a correlation that considers the liquid phase viscosity. The Strouhal number as a function of the ratio between the gas and the liquid superficial velocities, shown in Fig. 25.5, reveals no clear trend for the influence of the liquid viscosity on slug frequency. Most of the experiments have shown that an increase in the liquid viscosity increases the slug frequency. However, this trend reverts for the experimental points with mixture Froude numbers greater than 3 and the ratio between the superficial velocities of the liquid and the mixture above is 0.5, with a reduction in slug frequency and increase in the liquid viscosity. This lack of a unique trend in the case of the influence of liquid viscosity on the slug frequency is associated with the effects that occur during the formation of the slugs when the onset of slug flow springs from the stratified flow. The height of the liquid (related to the liquid holdup, R L ) before slug formation in stratified flow

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Fig. 25.4 Comparison between experimental elongated bubble velocity and calculated using correlation of Bendiksen [16]

Fig. 25.5 Strouhal number (St ) in function of the ratio between the superficial velocities of the gas (JG ) and the liquid (JL ) phase

and the difference between the velocities of the gas (UG ) and liquid (U L ) phases are parameters that facilitate the formation of the liquid slugs. In all superficial velocities of the experimental points, the height of liquid in the stratified flow increases with an increasing liquid viscosity. This increase in liquid height can be associated with the increase in viscous drag on the tube walls, which reduces the liquid velocity and requires a larger area (bigger liquid holdup) for the liquid to flow to satisfy the continuity equation. This difference between the velocities of the phases is associated with the Bernoulli effect. The bigger this difference, the easier for a wave to grow in amplitude. There is an increase in this difference with an increase in liquid viscosity for all experimental points. This implies that the gas needs less energy to raise the liquid interface to the top of the pipeline, forming the liquid slug.

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However, along with the increase in the liquid viscosity in the experiments, a phase density increase was observed. Such an increase attenuates the waves caused by the disturbance on the interface because the density difference between the phases increases. Another effect that influences the dissipation of waves is the fluid viscosity, since the higher the liquid viscosity, the greater the effect of dissipative forces of the waves due to viscosity. Because of these concurrent effects, there is no single trend for the slug frequency.

25.4 Conclusion and Future Perspective This work presented an experimental study on the influence of the liquid viscosity on the two-phase gas-liquid slug flow in a horizontal pipe. The experiments were performed in a 26 mm ID pipe with 8.65 m of length. The experiments were performed with air and water and with mixtures of air and water-glycerin solutions with viscosities of 5.46, 10.27, 15.39, 20.33, and 30.37 cP. A metering station was positioned in the pipe, with a pair of resistivity sensors installed in it. Those sensors were used to extract some characteristic slug flow parameters. The influence of the liquid viscosity on the slug flow was evaluated based on the elongated bubble velocities and slug frequencies. In the elongated bubble velocity, it was noticed that the increase in the liquid viscosity increases the elongated bubble velocity for all the experimental points. This latter increase is associated with a greater pressure drop and a consequent increase in the mixture’s superficial velocity and greater gas expansion. As for the slug frequency, no single trend was found. The liquid viscosity effect on the slug frequency is a competition of forces, in which one has the easiness of formation of the slugs with the increase in the liquid height and increase in the difference between the gas and liquid velocities. At the same time, there is an increased difficulty of forming slugs with the increased liquid phase density and dissipative forces due to the liquid viscosity. The majority of the experiments showed an increase in the slug frequency, except for experimental points with a mixture Froude number greater than 3 and superficial liquid velocities much larger than the superficial gas ones, where a trend to reduce the slug frequency with the increase in the liquid viscosity was found. To improve the analysis performed, it is recommended to increase the test grid and liquid viscosity. This increase in the test grid will allow us to investigate with more detail the influence of the fluid viscosity in the slug flow parameters. Perform experiments on pipes with different inclinations will allow analyzing the effect of the liquid viscosity in the flow parameters. It is also recommended to perform experiments with a pipe of different diameters to understand the slug flow dynamics better. Acknowledgements The authors acknowledge the financial support from PETROBRAS/ CENPES/TE.

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

16.

O. Shoham, (2006) Y. Taitel, A. Dukler, Int. J. Multiph. Flow 13(4), 577 (1987) J. Weisman, D. Duncan, J. Gibson, T. Crawford, Int. J. Multiph. Flow 5(6), 437 (1979) J. Colmenares, P. Ortega, J. Padrino, J. Trallero, in SPE International Thermal Operations and Heavy Oil Symposium (OnePetro, 2001) E.S. Rosa, J. Fagundes Netto, in 5th International Conference on Multiphase Flow, ICMF, vol. 4 (2004) E. Al-Safran, B. Gokcal, C. Sarica, in 15th International Conference on Multiphase Production Technology (OnePetro, 2011) Y. Zhao, H. Yeung, L. Lao, in 16th International Conference on Multiphase Production Technology (OnePetro, 2013) G. Losi, D. Arnone, S. Correra, P. Poesio, Chem. Eng. Sci. 148, 190 (2016) R. Ben-Mansour, A. Al-Sarkhi, A. Sharma, B. Jeyachandra, C. Sarica, B. Gokcal, in 7th North American Conference on Multiphase Technology (OnePetro, 2010) B.C. Jeyachandra, C. Sarica, H.Q. Zhang, E. Pereyra, in SPE Annual Technical Conference and Exhibition (OnePetro, 2012) R. Brito, E. Pereyra, C. Sarica, in 9th North American Conference on Multiphase Technology (OnePetro, 2014) T. Kim, T. Aydin, E. Pereyra, C. Sarica, Int. J. Multiph. Flow 106, 75 (2018) E.M. Al-Safran, B. Gokcal, C. Sarica, in SPE Heavy Oil Conference and Exhibition (OnePetro, 2011) Y.D. Baba, A. Archibong-Eso, A.M. Aliyu, O.T. Fajemidupe, J.X. Ribeiro, L. Lao, H. Yeung, Fluids 4(3), 170 (2019) D. Machado, F. Hildebrando, L. Lipinski, Multichannel resistive sensors system for measurement of two-phase flows. Master’s thesis, Graduation in Electrical Engineering - Federal Technology University - Paraná (2013). 64 p K.H. Bendiksen, Int. J. Multiph. Flow 10(4), 467 (1984)

Chapter 26

Experimental and Numerical Two-Phase Slug Flow Evolution Analysis with a Slightly Downward Direction Change Hedilberto A. A. Barros, Fernando Czelusniak, Cristiane Cozin, Eduardo N. dos Santos, Marco J. da Silva, Moisés A. M Neto, and Rigoberto E. M. Morales Abstract The intermittent transit of gas and liquid pockets in a pipe is the most distinctive characteristic of the gas–liquid two-phase slug flow pattern. This flow regime is found mainly in oil and gas production and transportation lines. In deepwater oil operations, the pipes must conform to the seafloor, thus causing direction changes in the flow. In this context, the present work presents an experimental and numerical study on gas–liquid two-phase slug flows in ducts with slight direction changes. The pipeline consisted of a horizontal stretch followed by a downward inclined one with −3◦ and −5◦ concerning the horizontal in a 26-mm ID, 35.6-m long pipe. Along the experimental circuit, the phases are detected by employing resistive sensors installed in four measuring stations that monitor the characteristic of slug flow parameters, namely the elongated bubble velocity and the unit cell frequency. In the numerical part of this study, a transient methodology based on a simplified two-fluid Lagrangian model was used. In addition, this methodology simulates the gas–liquid regime transitions in pipes with a direction change from horizontal to downward inclined flow. Both the complete and the partial slug dissipations were observed in the downward section. The mean values of the slug flow parameters and the stratification process found both experimentally and numerically presented a good agreement. Keywords Two-phase flow · Slug flow · Direction change · Slug dissipation

26.1 Introduction Two-phase gas–liquid flows are often observed in natural and industrial applications alike. In pipelines, the spatial distribution of the phases, commonly known as flow patterns, are governed by flow and geometry conditions. The slug flow is reputed H. A. A. Barros · F. Czelusniak · C. Cozin · E. N. dos Santos · M. J. da Silva · M. A. M. Neto · R. E. M. Morales (B) Multiphase Flow Research Center (NUEM), Federal University of Technology - Paraná (UTFPR), Curitiba, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_26

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as the most common flow pattern in the petroleum industry [1]. This flow pattern is characterized by the alternate passage of two structures: a liquid slug and an elongated bubble. United, those two structures form that what is known as a unit cell [2]. Some characteristic parameters explain the slug flow behavior: the elongated bubble velocity, the unit cell frequency, the slug, elongated bubble lengths, and the phase fractions. In deep-water production, the pipelines must conform to the seafloor, which may cause direction changes for long distances with a slight inclination. In this case, the slug flow tends to modify itself by changing its parameters or even transitioning to stratified flow [3]. Predictions of both slug flow parameters and their behavior, in this scenario, are important for the correct equipment design and safety of oil and gas transportation. In this line of work, recent studies [4–7] focused on experimental and numerical approaches. The slug flow mathematical modeling goes from simple steady-state models to complex models for transient regimes. In the two-fluid approach, the conservation laws are applied to each phase separately. This scheme needs solving four partial differential equations, considering the continuity and momentum equations only, or six if the energy equation is included. This approach can predict the slug initiation, which is why it is known as a slug capturing approach. Renault [8] presented a simplified two-fluid Lagrangian model capable of simulating the transition from the stratified-to-slug flow. The equations for the gas in the elongated bubble and liquid in the liquid slug are solved through a finite-difference scheme. An analytical solution for the Riemann problem is used to solve the equation for the liquid in the film, similar to what is done by solving the shallow water equations. Based on this model, Conte et al. [9] developed a computational routine to simulate slug initiation. The results were compared to experimental data, and a good agreement was found. In this context, the present article introduces an experimental and numerical study on gas–liquid two-phase slug flows in ducts with slight direction changes. The experimental layout consists of a horizontal pipe, followed by an inclined section. Three different configurations were used: two downward ones with −3◦ and −5◦ , and a fully horizontal one. A transient methodology based on Conte et al. [9] was used to numerically simulate the gas–liquid slug-to-stratified regimes in pipes with a direction change from horizontal to downward inclination based on [9].

26.2 Experimental Methodology The experimental tests were made in an experimental rig in the Multiphase Flow Research Center (NUEM) at the Federal Technological University of Paraná (UTFPR). Figure 26.1 illustrates the experimental facility. Air and water at ambient conditions were used as the gaseous and liquid phases, respectively. The liquid phase was stored in an open atmospheric tank and pumped into the system by a centrifugal pump, controlled by a frequency inverter on the electric motor.

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Fig. 26.1 Schematic representation of the test facility

A Coriolis-type flowmeter was used to measure the liquid flow rate, density and, temperature, with a relative standard uncertainty of 0.1%. For the gaseous phase, the air was compressed and stored in pressure vessels. The gas flow rate was measured by Coriolis-type flowmeters with a relative standard uncertainty of 0.1%, in addition to a manual valve placed between the flow meter and the mixer. Besides, each phase flow rate was controlled in conformity with the liquid (J L ) and gas (J G ) superficial velocities. The J L and J G relative standard deviations (estimated using the uncertainties propagation) were ±0.21% and ±0.23%, respectively. The test section used had 26 mm ID transparent acrylic pipe for three different scenarios. The first scenario had 35.6m (∼1369D) long complete horizontal pipe. Furthermore, the two other scenarios counted on a 15-m (∼577D) long horizontal section followed by 20.6-m (∼792D) long downward inclined section, with inclinations (θ) of −3◦ and −5◦ to the horizontal, as shown in Fig. 26.1. The two-phase flow was formed inside the parallel plate mixer and developed along the pipe. The mixture flowed through the inclined section until the test section outlet was reached, where the gravity-driven phase separation occurred. The water returned to the tank, thus closing the circuit, while the air was discarded into the atmosphere. Four metering stations were placed at 14.15 m (∼544D), 16.15 m (∼620D), 23 m (∼888D) and 30 m (∼1163D) from the mixer, as shown in Fig. 26.1. A pair of resistive sensors and a pressure gauge composed each metering station. The resistive sensor was used for the extraction of the slug flow main characteristic parameters. Meanwhile, the pressure gauge monitored the pressure along the pipe and corrected the gas’s superficial velocity. The resistive sensor acquired voltage values and correlated them to the liquid and gas fractions along the pipe. The sensors were developed by Dos Santos et al. [10],

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Fig. 26.2 Computational cell used to solve the simplified two-fluid model [9]

and their signals were processed according to Vicencio et al. [11] algorithm. This processing provided slug flow characteristic parameters, such as elongated bubble velocity (V B ) and unit cell frequency (f ), elongated bubble (L B ), and liquid slug lengths (L S ).

26.3 Mathematical and Numerical Model A simplified two-fluid model presented by Conte et al. (2014) uses the continuity and momentum equations for the gas and the liquid. The goal is to model the transition from stratified to slug flow. Those are the model’s assumptions: the liquid is incompressible, the gas is ideal, and the temperature is constant. The Lagrangian solution uses a grid divided into cells, see Fig. 26.2, and each cell can be either a stratified section or a slug. An elongated bubble may be divided into several sections. The boundaries are free to move, but each section is restricted to a maximum length defined by the mesh refinement. The complete solution methodology and validation are described in Conte et al. [9]. The solution of this model determines the slug flow characteristic parameters along the pipe. This study will compare the numerical results for the elongated bubble velocity (VB ) and the unit cell frequency ( f ) against the experimental data.

26.4 Results and Discussion In this section, experimental and numerical results for the slug flow parameters will be presented and discussed. The experimental points were defined as pairs of superficial velocities, varying from 0.5 to 1.0 m/s for the gas and 0.5 to 1.3 m/s for the liquid, being four (4) points in total as Table 26.1 shows. This was done to ensure the slug flow pattern occurred along the horizontal test section. The slug flow pattern, in a downward inclined section, may transition to stratified flow. Figure 26.3 shows the test grid on a Taitel and Dukler [12] flow map for 0◦ , −3◦ , and − 5◦ . The stratified

26 Experimental and Numerical Two-Phase Slug Flow … Table 26.1 Test grid Pair P01 P02 P03 P04

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JG (m/s)

JL (m/s)

0.75 0.5 1.0 0.7

0.75 1.0 0.5 1.3

Fig. 26.3 Test grid on a Taitel and Dukler [12] flow map for 0◦ , −3◦ and, −5◦

region grows with a downward inclination because the inclined downward pipeline presence causes the liquid acceleration and, with that, a higher liquid flow rate is necessary to cause the interface instabilities that generate the slug flow [13]. Analyzing the experimental points in Fig. 26.3, we can observe that the points P01, P02 and, P03, which are within the slug flow region in the horizontal case, pass to the stratified region in the inclined downward case. The point P01 (JL = JG = 0.75 m/s) should transition from slug to stratified flow for the −3◦ and −5◦ inclinations. Experimentally, we observe that a transition does not happen for −3◦ , but only for −5◦ . According to Alves et al. [7], this divergence occurs because the Taitel and Dukler [12]’s model considers a stratified flow at the inlet, and the Kelvin-Helmholtz instabilities generate the slug flow. In this work, the flow pattern is slug flow before the downward inclined region, contrary to the model. The slug flow parameters tend to change in the downward region before the slug dissipation. The evolution of the P01 parameters for the 0◦ , −3◦ , and −5◦ inclinations can be observed in Fig. 26.4. The slug flow characteristic parameters in the four metering stations can be observed along the pipe. The elbow is located between the first and second metering stations and is represented by the dashed lines. The

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Fig. 26.4 Point P01 (JL = JG = 0.75 m/s) a VB , b L B /D, c L S /D and, d f evolutions for 0◦ , −3◦ and, −5◦

complete slug dissipation occurs only for the −5◦ inclination, after the third station. Because of that, values for the last station are not available for this inclination. In Fig. 26.4a, VB tends to increase along the pipe in the horizontal case (rounded black points). This occurs because of the gas expansion due to the pressure drop. For the downward cases, we can observe that VB had a slight decrease after the elbow. The buoyancy starts to act contrary to the flow direction in the downward section, decreasing VB . The gravity accelerates the liquid along the pipe, which in turn accelerates the gas. That is why VB increases again in the inclined section but always lags in speed regarding the horizontal case. Figure 26.4b shows the evolution of the dimensionless bubble length (L B /D) for the three scenarios. In the −5◦ case, where one of the flow patterns changes from slug-to-stratified flow, it can be observed that L B /D tends to infinity in the third

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station. This characterizes the stratified trend because, at this point, there is almost one infinite bubble in the pipe. The evolution of the dimensionless slug length (L S /D), in Fig. 26.4c, tends to increase along the pipe in the horizontal case because of the bubble coalescence. When two bubbles coalesce, the liquid in the slug between them tends to migrate to the near slugs, thus increasing the size of those latter. In the inclined downward cases, the L S /D tends to decrease after the elbow. This occurs because of the slug dissipation phenomenon caused by the liquid film acceleration. The faster-moving liquid film removes liquid from the slug, decreasing the slug length. In the −3◦ inclination, the slug dissipation is partial; therefore, the slug flow remains with a high coalescence rate, increasing the L S /D along the pipe. On the other hand, in the −5◦ case, a complete slug dissipation was observed, and, with that, the L S /D decreases until the complete stratification after the third station occurs. The slug frequency tends to decrease along the pipe because of bubble coalescences and the gas expansion, as Fig. 26.4d shows for the horizontal case. In the inclined cases, the slug dissipation causes a high coalescence rate, as discussed before, decreasing the slug frequency even more when compared to the horizontal case. In the −5◦ inclination, it can be observed that the slug frequency tends to zero in the third station before stratification occurs. In other words, there are virtually no slugs in the pipe at that station. Figure 26.5 shows the comparisons between the numerical simulation of the entire pipe with photos in three different positions (horizontal section, elbow, inclined section) for the point P02 (JL = 1.0 m/s; JG = 0.5 m/s). Following Taitel and Dukler [12], the point P02 should not stratify for the −3◦ inclination; however, it should stratify for the −5◦ case. In Fig. 26.5a, in experimental terms, the slug flow pattern remains the same before and after the −3◦ elbow according to the Taitel and Dukler [12] map. The numerical results agree with the experiments and the model prediction, keeping the slug flow pattern along the pipeline. It can be observed that the bubble length tends to increase when the slug flow passes from the horizontal section to the inclined one. This happens because of the higher coalescence rate caused by the partial slug dissipation in the −3◦ inclined section.

Fig. 26.5 Comparison between the simulation and the experimental results for the point P02 (JL = 1.0 m/s; JG = 0.5 m/s) in a −3◦ and b −5◦ inclinations

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Figure 26.5b shows the same comparison for the −5◦ inclination. In this scenario, a transition from the slug-to-stratified flow was expected. Experimentally, the transition has occurred. The numerical simulation is again in agreement with the experiments, and the complete slug dissipation occurs when the slug flow passes through the elbow, causing the flow pattern to transition from slug to stratified. A quantitative comparison between the experiments and the numerical simulations can be observed in Fig. 26.6a. In this case, for the point P02 (JL = 0.5 m/s; JG = 1.0 m/s), the experimental slug frequency is compared with the numerical simulation for the −5◦ inclination. This point stratifies right after the second station. The numerical slug frequency tends to decrease drastically in the first 200D segment because of the high coalescence rate due to the entrance effect. Along the horizontal section, the slug frequency tends to stabilize as the coalescence rate decreases. After the elbow, the coalescence rate increases quickly because of the slug dissipation and decreasing the slug frequency to zero, and the flow pattern changes from the slug to stratified flow. The pipe representation can be observed in Fig. 26.6a, where the stratification process can be observed right after the elbow. It is possible to see the waves in the stratified flow caused by the Kelvin-Helmotz interface instabilities. The gravitational effects caused by the inclination angle are significant for low flow rates. For higher flow rates, the inertial effects are dominant when compared to the gravitational effects. Figure 26.6b shows a higher flow rate example. The point P04 (JL = 1.3 m/s; JG = 0.7 m/s) is practically independent concerning the angle. The evolution of VB shows that the experimental trend remains the same before and after the elbow. In other words, VB tends to increase along the pipe independently

Fig. 26.6 a Slug frequency evolution comparison between experiment and numerical simulation for the point P03 (JL = 0.5 m/s; JG = 1.0 m/s) for the −5◦ inclination; b Bubble velocity evolution comparison between experiment and numerical simulation for the point P4 (JL = 1.3 m/s; JG = 0.7 m/s) for the −5◦ inclination

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of the elbow and the inclined section’s presence. Besides, the numerical simulation agrees with this trend for all the analyzed scenarios.

26.5 Conclusion and Future Perspective In this work, experimental and numerical studies on a two-phase gas–liquid slug flow with a slight change of direction were presented. The two downward pipe segments with angles −3◦ and −5◦ and a fully horizontal one were used. The mean slug flow parameters and the stratification process found experimentally were compared with the slug capturing simulation, and a good agreement between them was found. The slug flow parameters show a changing trend after the elbow and before the stratification process due to the buoyancy acting opposite the flow’s direction. The elongated bubble velocity tends to decrease because of that. The slug frequency tends to decrease abruptly because of the coalescence process in the downward section, which leads to stratification. The stratification process between the experimental photos and a simulation was compared. In both present cases, the simulation fits in, capturing both the partial and the total coalescence, where the slug-to-stratified transition occurs. The future perspective of this work resumes in two aspects. The first one is the presence of multiple elbows in the experimental circuit. That is important because some effects happen in a single elbow that is not propagated along the pipe. However, the presence of several valleys and peaks must change the parameters in slug flow. Therefore, it is necessary to adapt the simulation model to consider the presence of multiple elbows as in the experimental circuit. Furthermore, a relevant improvement for the slug-capturing model is considering the aeration in the liquid slug region. Acknowledgements The authors wish to express their gratitude for all the technical and financial support of PETROBRAS/CENPES.

References 1. L. Shemer, Int. J. Heat Fluid Flow 24(3), 334 (2003) 2. G. Wallis, One-Dimensional Two-phase Flow (McGraw-Hill, 1969). https://books.google.com. br/books?id=xvFQAAAAMAAJ 3. H. BEGGS, J. BRILL, JPT, J. Pet. Technol. 4. V. De Henau, G. Raithby, Int. J. Multiph. Flow 21(3), 365 (1995) 5. Y. Taitel, D. Barnea, SPE J. 5(01), 102 (2000) 6. T. Mandal, M. Bhuyan, G. Das, P. Das, Chem. Eng. Res. Des. 86(3), 269 (2008) 7. R.F. Alves, F.A. Schneider, F.A. Barbuto, P.H. Santos, R.E. Morales, Int. J. Multiph. Flow 119, 155 (2019) 8. F. Renault, (2007) 9. M.G. Conte, C. Cozin, F.A. Barbuto, R.E. Morales, in Fluids Engineering Division Summer Meeting, vol. 46261 (American Society of Mechanical Engineers, 2014), p. V002T20A003

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10. E.N. Dos Santos, A.D.N. Wrasse, T.P. Vendruscolo, N.S. Reginaldo, G. Torelli, R.F. Alves, B.P. Naidek, R.E. Morales, M.J. Da Silva, IEEE Access 7, 5374 (2018) 11. F.E. Vicencio, F.A. Schneider, C. Cozin, F.A. Barbuto, M.J. Da Silva, R.E. Morales, in ASME International Mechanical Engineering Congress and Exposition, vol. 57472 (American Society of Mechanical Engineers, 2015), p. V07BT09A020 12. Y. Taitel, A.E. Dukler, AIChE J. 22(1), 47 (1976) 13. D. Barnea, O. Shoham, Y. Taitel, A. Dukler, Int. J. Multiph. Flow 6(3), 217 (1980)

Part VII

Wet and Dry Particulate Systems

Chapter 27

DEM Simulation: From Granular Crystal Modeling to Large Industrial André L. A. Mesquita, Luís Paulo S. Machado, and Alexandre L. A. Mesquita

Abstract Granular materials are solid particles whose collective dynamics produce unique phenomena. They can be uncounted for in industrial applications as dry or wet bulk materials. The most straightforward systems, in dry configuration, can be modeled as granular crystals formed by orderly arrangements of particles without significant relative movement. Possible applications of granular crystals are devices to control wave propagation. Proposals to attenuate impacts and filter mechanical waves are discussed. For particles with more energy and degrees of freedom, with cohesive forces, as van der Waals forces and capillary forces, in the case of wet granular material, a more general 3D modeling is needed. Granular flow is critical in the mining industry, where industrial steps process can be optimized. The different forms of numerical simulation of granular flows are presented in this chapter. A review of some investigations from the authors is briefly discussed. In the general DEM method, examples are presented for dry particles: granular crystal modeling and wet granular flow. For industrial mining cases, numerical and experimental results are provided for transfer chutes and hopper feedings cases studies. Modeling discussion, calibration steps, practical applications, and perspectives are presented. Keywords Granular crystal modeling · Large industrial · DEM-CFD · Cohesive forces · Dry particles

A. L. A. Mesquita (B) · L. P. S. Machado · A. L. A. Mesquita Laboratory of Fluid Dynamics and Particulate (FluidPar), Amazon Development Center in Engineering (NDAE), Federal University of Pará, Campus of Tucuruí, Tucuruí, Brazil e-mail: [email protected] L. P. S. Machado e-mail: [email protected] A. L. A. Mesquita e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_27

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27.1 Introduction Granular materials are present in many human activities. Examples are food grains, mineral ores, biomass fuel, pharmaceutical, makeup powder, soils, etc. It is estimated that • the processing of granular materials consumes about 10% of all electricity produced on the planet [1]; • the world annual production of grains is close to 10 billion tons [1]; • 80% of all chemical products are particulate matters [2]; and • grains are the second-most-manipulated matter in the industry [3]. Thus, granular materials are commercially important, are directly related to our quality of life, and are observed in scenarios ranging from industrial sectors to environmental issues. The principal characteristics of granular systems consist of a large number of macroscopic solid particles, with well-defined borders, that act collectively [4]. Their interactions occur mainly via inelastic collisions, where negligible thermal variations and gravity are important in dynamics. The typical grain size varies from one micron (powder) to kilometers (asteroids and icebergs). Below this lower size limit, the Brownian motion can be noted. The variety of grain sizes explains the different contexts where they are observed [5]. What makes a handful of grains so different from ordinary solids? Why do they deserve special treatment? For example, granular materials can behave similarly to regular solids, liquids, and gases depending on the average energy per grain and boundary conditions. We can even see phase transitions [6]. Some industrial steps explore these behaviors. Granular fluids are best suited to transport granular material from one point to another within factories, where we can use conveyor belts, chutes, pipes, and funnels. Particularly, according to the world ranking [7], Brazil is the (i) fourth-largest producer of grains and (ii) second-largest grain exporter. In 2020, ∼US$ 20 billion in ore was exported from Pará (a Brazilian State) alone [8]. Also, typical solid behaviors are useful for granular storage, wave propagation, impact absorption, and avoiding clogging of pipes. This paper presents a brief review into two areas of research in granular materials: • (i) wave propagation in granular crystals (solids) • and (ii) granular flows applied to the mining industry. Granular crystals are the simplest granular systems. Here, they are reported to present some basic properties and to implement some foundational techniques. These systems can be used for learning in the granular area. On the other hand, studying granular flow applied to the mining industry requires more sophisticated algorithms. As the mining industry in Brazil continues to grow and maximize its productivity is

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a challenge, there is a need for research and investments in the area. We have been studying how to increase efficiency in some industrial stages, such as chute clogging, belt misalignment, inclined screen, and silo drainage. The remainder of this paper is organized as follows. Section 27.2 details the equations of motion, initial conditions, and typical discrete particle simulation models. Section 27.3 discusses the granular crystals and applications concerning wave propagation and shows simple scenarios where the modeling can be implemented. Section 27.4 presents a series of outcomes (numerical, analytical, and experimental) for granular flows of great interest in the mining industry. Finally, Sect. 27.5 concludes with potential applications and perspectives.

27.2 DEM Modeling 27.2.1 Granular Crystal Modeling Modeling of granular materials is based on Molecular Dynamics, i.e., we solve the classical many-body problem. The first step is to consider grain as a massive point particle, regardless of its shape and size. In a simple scenario, dry grains interact through collisions (Fig. 27.1). When particles are in contact, the force is purely repulsive. There is no interaction when the particles are separated. A general force is given as follows [9]: Ftotal =

⎧ ⎨

FN ,el = k N (α N )n , 0, for α N ≤ 0 where FN ,d = −k d (α N)β vrel , ⎩ FN ,el + FN ,d + FT ,for α N > 0 FT = −μ  FN ,el  tanh (k T |α˙ T |) tˆ.

(27.1)

FN ,el is the normal elastic force where the grains interact via nonlinear elastic springs. α N is the normal compression between neighboring grains. The elastic coefficient k N depends on Young’s modulus, Poisson’s ratio, and geometric parameters. For n = 3/2, the interaction follows the Hertzian contact law, and the grains are spherical. FN ,d is a typical dissipative normal contact force, where the grains are considered to be immersed in a viscous liquid. kd is a viscous damping coefficient that depends on the material and shape of the grains, vr el is the relative velocity between two grains, and β = 1/2 for spherical grains [10]. FT is a tangential frictional force according to a continuous smooth Coulombtanh model [11]. Friction is important when rotation is considered. μ and k T are control parameters related to the smoothness of the frictional law, and α  T represents the tangential relative velocity between grains at the contact point, given by

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Fig. 27.1 Schematic of contact model between two dry grains. Forces are purely repulsive. Springs represent elastic deformation. Dashpots and slider depict friction

δT =

    − → − →



→ − →



i j − − · tˆ N ji r j + Rj θ j ×

r i + Ri θ i × N

(27.2)

− → → being − r the location vector of the center of grain  i, Rthe radius, θ the angular  − → − → − → − → i j = r j − ri / r j − ri , and tˆdenotes the unit vector in displacement vector, N the tangential direction of the contact interface between grains. Finally, we solve Newton’s equations of motion for a system of several particles. Standard methods for the numerical solution of ordinary differential equations are the Runge-Kutta fourth-order method, Verlet algorithm, and Gear Predictor-corrector. The basic idea is as follows. Given the dynamic information, the equations are solved on a step-by-step basis. This is the basic idea behind the Discrete Element Method. In a general scenario, there may be attractive forces between the grains. Wet grains and electrostatic forces are common in the mining industry. More details next.

27.2.2 General DEM Modeling The Discrete Element Method (DEM), developed by Cundall and Strack [12], is a reliable and powerful tool to study and predict the flow of particles. The DEM captures the contacts between individual particles in an explicit manner. In contrast to continuum methods that smear out the individual particles into a smooth plenum, the

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Fig. 27.2 Illustration of the forces acting on a particle

discrete element method captures each particle’s individual geometry and dynamics, including dissipative effects of frictions. The DEM method takes into account the translational and rotational motion equations of the particles. The mathematical model (Eqs. 27.3 and 27.4) includes the interaction between particles and between particle and wall, as well as the van der Waals forces, electrostatic forces, liquid bridges, particle-fluids interaction force, etc. (Fig. 27.2).   dv i f g nc = Fi,c j + Fi,k + Fi + Fi , (27.3) mi dt j j  dωi = Ti, j , dt j=1 k

Ii

(27.4)

where m i and Ii are, respectively, the mass and moment of inertia of particle i; vi and ωi are the translational and rotational velocities of particle i, respectively; Fi,c j and Ti, j are the contact force and the torque acting on particle i by particle j or wall; f nc Fi,k is the non-contact force acting on particle i by particle k or other sources; Fi is g the particle-fluid interaction force on particle i; and Fi is the gravitational force. Various models have been proposed to calculate these forces and torques [13]. Once forces and torques are known, Eqs. 27.1 and 27.2 can be readily solved numer-

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Fig. 27.3 Workflow in EDEM and its main modules

ically. Thus, the trajectories, velocities, and the transient forces of all particles in a system considered can be determined. There are several commercial (e.g., EDEM, Bulk Flow Analyst, and Rocky DEM) and non-commercial (open source) (e.g., LAMMPS and LIGGGHTS) DEM software packages available. The software EDEM has three main components or modules: Creator (preprocessing), Simulator (solver), and Analyst (post-processing). These modules are represented in Fig. 27.3.

27.3 Granular Crystal Application Metamaterials are artificial structures produced in the laboratory, with properties not observed in materials taken directly from nature. Granular crystals are metamaterials that are easy to build and maintain, inexpensive, scalable in size, and have many possible applications. The general configuration consists of ordered arrays of grains that only interact with the nearest neighbors. The grains move around their equilibrium positions. Each grain has a low degree of freedom. Thus, granular crystals are the most straightforward systems to solve numerically and can also be used for learning in the granular area. Extensive studies have reported granular crystals as candidates for components of nonlinear acoustic devices [14–16]. The idea is simple, as the material medium can control the properties of mechanical waves, and granular crystals can be used to

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tune nonlinear wave propagation. We will discuss some results we have gotten over the years for both one-dimensional and two-dimensional systems.

27.3.1 Granular Chains: 1D Systems Pioneering works on 1D systems were done by Nesterenko [17]. The most fundamental systems are alignments of identical spheres, so-called monodispersed granular chains (see Fig. 27.4a). We can perturb a chain by colliding an extra grain against the end of the chain. Then, we observe the propagation of a spatially localized, coherent, and shape-preserving, strongly nonlinear solitary pulse. This phenomenon opened a new field: wave propagation in granular crystals. Variations in the monodispersed chains have been proposed. Tapered granular chains are alignments of spherical granules whose radii vary down the chain (see Fig. 27.4b). We have studied several tapering configurations, such as forward linear, forward exponential, backward linear, and backward exponential tapering [18]. We applied an analytic binary collision approximation to predict the propagation of momentum and energy along the chain. The approximation explores the fact that most of the pulse energy is concentrated in two grains. The forward exponential tapering is interesting for shock absorption because the momentum decrease is exponential along the chain (see Fig. 27.5).

Fig. 27.4 Examples of granular chains. a Monodispersed. b Tapered

Fig. 27.5 Comparison between the binary collision approximation momentum (plus symbols) and those obtained from the numerical integration of the equations of motion (circles) in forward exponential tapered chains. From top to bottom, the tapering parameter increases

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Fig. 27.6 The left panels present a quasi-unidimensional chain formed by a central tapered chain decorated with side grains. a Three-dimensional view. b Sideways view. The right panel shows the decay of maximum momentum along the central chain

Later, we have proposed a quasi-unidimensional granular chain that was dramatically mitigating impacts using very short chains [19]. The system consists of central tapered granular chains, with the radii decreasing in geometrical progression, decorated with four extra grains laterally positioned on the central chain (see the left panels in Fig. 27.6). The findings reveal that 90% of the momentum amplitude of solitary pulses are attenuated for chains made of ten grains (see the right panel in Fig. 27.6).

27.3.2 Two-Dimensional Granular Crystals Arrays in 2D and 3D present potentially richer possibilities in making novel nonlinear acoustic systems. Shukla et al. [20] first reported such investigations in 2D arrangements. As a continuation of our previous work, we have presented a way to control the shape of the wavefront in a square granular crystal (bidimensional system) [21]. We have used two species of grains (see the left panel in Fig. 27.7): • (i) the main grains form a square array, and they are responsible for the perturbation propagation; • (ii) the interstitial grains work as localized non-uniformities in the medium. Their role is to trap part of the perturbation as vibration and scatter the perturbation into other degrees of freedom. Material combinations affect the decay of the wave amplitude. We had reported that when we included tapered main grains ∼ 90% impulse attenuation was observed using a system thickness of five granular layers (see the right panel in Fig. 27.7). It should be emphasized that the majority of these works have focused on the propagation of solitary waves. A new frontier concerns the propagation of continuous waves. In a recent collaboration, we have studied granular crystals as a filter for low-

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Fig. 27.7 The left panel shows a square two-dimensional granular crystal decorated with interstitial grains. The right panel presents the decay of momentum along the impact direction

Fig. 27.8 The left panels show monodispersed chains with fixed side decoration. The right panel presents the cutoff frequencies as a function of the driving amplitude for chains with several tapering parameters (q) and restitutional losses (w)

frequency ultrasonic signals [22]. We showed that using 2D decorated crystals, it is possible to enhance the low-frequency filtration properties. Filtration properties are relevant to control structural vibrations and reduce noise that can harm the human body. Finally, we have proposed a fixed granular decoration, attached to a simple monodispersed chain (see the left panels in Fig. 27.8), to act as an impact attenuator and a low-pass filter [23]. We have shown that regular chains are low-pass filters depending on the driven amplitude, restitutional losses, and tapering (see the right panel in Fig. 27.8). However, the granular chain with fixed decoration filters all the input signals.

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We believe that these studies could be used to build impact absorber panels to protect buildings against seismic shocks, vehicle surfaces, bulletproof vests, safer helmets, areas to control explosions, and thin impact dispersion systems in general. The same systems could filter harmful signals or noise found in road traffic, industries, war, telecommunication, medicine, and domestic applications. Next, we discuss the results of granular flows applied to the mining industry, where richer phenomena are observed.

27.4 DEM Simulation for Large Industrial Applications 27.4.1 DEM Calibration 27.4.1.1

The Problem

In the application of the DEM model to solve bulk solid flow problems, the calibration step is identified as a key component for the success of the modeling results [24]. It is a subject covered by the international literature, where different devices are employed to achieve the DEM model calibration. The experiment which is very used for this objective is the flow characterization through a small flow box, where the repose angle is the parameter observed as a calibration target. Zhou et al. [25] and Li et al. [26] used glass bed to investigate the influence of the friction sliding and rolling friction coefficients on the calibration process. Some researchers use shear tester as experiment for DEM calibration [27, 28]. However, this experiment is adequate for silo flow or similar slow flow. For bulk solid flow problems with more high dynamics, as in transfer chutes, this is not recommended [29]. Other devices are employed for DEM calibration purpose, such as rotating drum [30], swing-arm slump tester [31], silo discharge, and others [32]. The particle shape has a strong effect on the bulk solid flow, and several authors have performed studies using overlapping spheres model to represent the real particles [33, 34] or techniques more advanced as digital image segmentation to obtain particle shape parameters for the generation of irregular shaped DEM particles [35]. In mineral processing, the particles are not homogenous, and, therefore, this approach is not practical. In this view, the use of the rolling friction coefficient as a control parameter to provide the macro flow behavior of irregular particles using spherical particles is applied with success [36–38]. A detailed review on the several aspects of the DEM calibration was performed by Coetzee [39], such as particle shape and size modeling, experiments to the calibration process, and some attempts to reduce computation time, which is very important in order for the DEM simulation to be feasible for engineering design purpose. Despite the large number of publications covering the devices and techniques employed for DEM model calibration, there is much less information on the scaling laws in the calibration process for DEM simulations. The solid flow mass in the

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calibration experiments is, in general, much smaller than in industrial applications, in particular for the industrial mining handling process. In this way, the DEM particle diameter in the calibration process is also smaller than the particle diameter employed in the industrial simulations. If the DEM particle size is kept constant in both calibration laboratory and industrial cases, the computational time could be too large, making the simulations for engineering analysis impractical.

27.4.1.2

Similarity Laws for Cohesive Granular Flows

For wet granular flow, Nase et al. [40] define two dimensionless numbers, which are used for the characterization of the granular flow. They are the Granular Bond Number, Bog , which represents the ratio of maximum capillary force and the weight of the particle, and the Collision Number, Co, which relates the capillary force to the collisional force, as defined by Bagnold [41]. Thus, Bog and Co are expressed as [40] 2R 3 , (27.5) = Bog = 4 2 3 2R ρs g π R ρs g 3 Co =

2π Rγ γ δ2  2 = 2ρ, S λ2 R 3 u2 πρs λ2 R 4 du dy

(27.6)

where R the is the radius of the particle, γ is the surface tension of the fluid, ρs is the density of the solid, g the gravitational acceleration, u is the velocity of the granular flow, and the shear rate, within the shearing region of a granular flow with average velocity , can be approximated as [40] 2 u du ≈ . dy δ

(27.7)

δ is the width of the flowing region perpendicular to , and λ is the linear concentration, which is a function of the granular material porosity and its maximum possible concentration, as defined by Bagnold [41]. Performing the same analysis for the flow of granular material in transfer chutes from rest and using the similarity of the collision number, one obtains 

δp δm

2 

Lp Lm



 =

dp dm

3 

λp λm

2 .

(27.8)

Since we have the same dimensional ratio in the width as in the length of the chute, we obtain the following relation: 

Lp Lm

3

 =

dp dm

3 

λp λm

2 .

(27.9)

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It turns out that the particle diameter follows the same dimensional scale only if the following condition is observed: λ p = λm .

(27.10)

The linear concentration is a function of the porosity of the medium and its maximum porosity. Thus, assuming that this relationship is satisfied when the diameter ratio is not too large, the particle diameter can then be scaled for cohesive flow. This is the basis for the calibration methodology.

27.4.1.3

Methodology for Large Industrial Applications

The methodology consists of using a large compartmentalized rectangular box (henceforth referred to as flow box) to simulate particles of diameters compatible with industrial applications. Thus, the same calibration diameter can be maintained in the application with high mass flow rates, ensuring the method’s reliability. A rule of thumb is always to have the virtual DEM particle with a maximum radius of 1/20 of the tiniest opening for the particulate flow. The flow box used consists of a 3,000 mm high structure with a 500 mm × 1,000 mm cross-section, containing a 1,000 × 496 mm material storage box, the bottom of which has a 300× 496 mm gate. The methodology consists of adjusting the parameters of rolling friction, cohesion/adhesion, and interparticle friction until the simulation results provide the exact angles of repose as the tests. In needing even larger particles, the methodology allows scaling the virtual geometry for the calibration process, allowing the particle to be scaled up with the same experiment. As previously discussed, a rule of not exceeding four times this scaling was established, aiming to maintain the similarity conditions. Tests were performed with two flow boxes to evaluate the methodology, as shown in Fig. 27.9. One, called prototype, is used in the tests, and the other, called model, is four times smaller. Tests were performed to verify the methodology, and the results were satisfactory, as shown in Fig. 27.10.

27.4.2 Mining Applications 27.4.2.1

Transfer Chutes

In applying DEM modeling to solve ore flow problems in transfer chutes, two case studies are presented. The first case is a solution to the misalignment problem of iron ore sinter feed discharge from a chute onto a belt conveyor. Iron ore sinter feeds the main raw material used in steel production. Iron ore is traded under three primary forms: fines, lump ore, and pellets. The fines comprise the sinter feed and the pellet

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Fig. 27.9 Flow boxes used in the calibration procedure: prototype flow box (left) and model flow box (right)

Fig. 27.10 Results from tests on the prototype flow box and from calibrated DEM simulation

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Fig. 27.11 a CAD drawing of the transfer chute that generates a misalignment problem on the conveyor belt. b DEM Simulation of iron ore sinter feed flow

feeds, where sinter feed are particles ranging from 0.15 mm to 6.35 mm in diameter, and for usage, in a blast furnace, they must be sintered (agglomerated). The misalignment problem causes premature wear on one side of the belt. The current chute was drawn in CAD and is shown in Fig. 27.11a, and the characteristics of the material are shown in Table 27.1. The mass flow rate is 16,000 t/h. The Chute CAD model was exported to EDEM to perform the flow simulation. After particles model calibration, the simulation result is presented in Fig. 27.11b. In DEM software, it is possible to count the number of particles that fall on each side of the belt conveyor. Thus, we determine the number of particles on the right side (NPRS) and the number of particles on the left side (NPLS), as shown in Fig. 27.12a. According to NPRS, NPLS, and a total number of articles, the percentage of particles in each side of the belt conveyor can be determined. Figure 27.12b shows a high difference (21%) between those percentages, resulting in premature wear on the right side of the belt. The proposed solution using the methodologies of Roberts [42] and Benjamim [43] is presented in the CAD drawing in Fig. 27.13 (left). A hood and spoon chute was proposed to replace the chute using the rock boxes. This hood and spoon chute

Table 27.1 Material data Material Bulk density (kg/m3 ) Solid density (kg/m3 ) Moisture (%)

Iron ores sinter feed 2.450 4670 10

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Fig. 27.12 a Receiving belt conveyor presenting NPRS and NPLS. b Percentages of particles falling on each side of belt conveyor from current chute and their difference

Fig. 27.13 Proposed chute design (left) and the flow DEM simulation result (right)

drawing was exported to EDEM, and the result of the simulation is presented in Fig. 27.13 (right). According to the results of flow simulations in the proposed chute, the difference between the numbers of particles falling on each side of the belt conveyor decreased to 0.8% (Fig. 27.14). Another case study was the resizing of a central transfer chute of a reclaimer, with a loading rate of 8,000 t/h, shown in Fig. 27.15. In the iron ore transfer operation through the chute, a high misalignment was observed on the receiving belt. Figure 27.16 shows the belt misalignment and the DEM numerical simulation of this misalignment. The simulations of flow through the chute and misalignment quantification are shown in Fig. 27.17 when the reclaimer boom is positioned 135◦ horizontally and −13◦ vertically.

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Fig. 27.14 Percentages of particles falling on each side of belt conveyor from the proposed chute and their difference

Fig. 27.15 Iron ore reclaimer illustration

Fig. 27.16 Image of the misalignment (left) and DEM reproduction

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Fig. 27.17 a Simulation of the iron ore flow in the reclaimer’s central chute. Boom rotation 135◦ and vertical position −13◦ . b Fig. 27.16. Misalignment result

The solution was found by introducing two rock boxes and a lower deflector plate in the bulk solid channel flow. The solution is presented in Fig. 27.18. This change has solved the misaligned flow on the conveyor belt (Fig. 27.19). In addition, the ore now falls onto the belt from a lower height, reducing impact and increasing belt life (Figs. 27.20 and 27.21). The second case study concerns the design of a hopper for discharging coal ore into a truck. A 10 m3 capacity grab fed the hopper, so three grab operations were required to fill the truck (Fig. 27.22), which had to move during the operations. The

Fig. 27.18 Illustrations of the rock boxes and deflector plate inserted inside the transfer chute

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Fig. 27.19 Ore transportation on the conveyor belt before and after the solution

Fig. 27.20 Discharge of manganese ore to a hopper. b Original grid formed by a set of I-beams

grab did not maintain a fixed position during the three operations because a steel cable suspended it. The project aimed to resize the hopper so that the truck would not need to move during ore loading. Figure 27.23 shows the DEM simulation with the resized hopper during the three grab loading operations, totaling 30 m3 . In this case, no truck movement is required during the operations. Finally, case 3 shows the discrete element method assisting the resizing of three bauxite discharge hoppers. The main problem of the hoppers was that they were in configurations that generate funnel-type flows [44], which favor blockage. Therefore, the project aimed at sizing a new hopper to replace the three original ones, which would provide a mass flow (uniform flow), thus not having a blockage tendency and allowing a faster discharge.

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Fig. 27.21 a DEM simulation for initial hopper grid geometry. b DEM simulation with modified grid

Fig. 27.22 Three configurations of grab positioning during ore discharge operation

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Fig. 27.23 DEM simulation of the three loading operations with the new hopper design

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Fig. 27.24 Comparison of the flow between the hoppers

Fig. 27.25 Comparison of discharge flow between hoppers

Figure 27.24 presents a comparison of the flows between the DEM models of the hoppers for the time interval of 6 s after the start of the discharge (see Fig. 27.25). This figure shows the better performance of the new hopper (hopper design 4).

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27.5 Final Remarks and Perspective The DEM methodology is a powerful tool for granular materials flow simulation from simple dry particle collision problems to the complex wet bulk material flow in mining handling operations. The numerical approach can be accomplished from granular crystal modeling, in the case of 1D/2D dry particle contact, or the general form of 3D wet particle contact with the interaction of capillary forces and other cohesive forces. In the most general problem, the key of the DEM simulation is in the model calibration step, which is necessary to take into account the effect of particle sphericity, size distribution, and complex cohesive forces. Additionally, and fundamental, calibration must be mandatorily considered since the DEM virtual particle is, in general, more significant than the actual particle due to computational time considerations. Even with supercomputers or big computer clusters architecture, it is impractical to simulate, for example, the fine ore handling with the true mean particle size. For a larger flow rate in the mining industry, attaining up to 16,000 t/h, the experimental devices for calibration in the laboratory must be large sufficient to be able to calibrate the DEM model with the same virtual particle size, which will be employed in the industrial simulation. When this is impossible, it is necessary to use a scaling model calibration methodology, as presented in this work. The challenge in DEM simulation for mineral operation is the change in material characteristics as mining advances, requiring new calibration parameters. Its mean inline or in-situ experiments for calibration purposes. In this view, a new methodology or new experimental devices are needed to be developed. Other situations, such as ore liquefaction, due to critical moisture contents, or other operations, such as mineral particle coating, conduct the simulation for CFDDEM modeling, where fluid and solid particle flows are considered together. In this case, calibration is also mandatory. We extend the DEM approach to mining operations to other industries such as food, pharmaceuticals, power generation, and grains. DEM or, more general, CFDDEM is a powerful engineering tool for solving industrial problems, always with a necessary and adequate calibration methodology.

References 1. J. Duran, powders and Grains: An Introduction to the Physics of Granular Materials (Springer, Ed) 2. H. Merkus, G. Meesters, Production, Handling and Characterization of Particulate (Springer International Publishing, Switzerland) 3. G.P. G, Rev. Mod. Phys. 71(2), 374–382 4. O. Mouraille, Sound propagation in dry granular materials: discrete element simulations, theory, and experiments 5. S. Franklin, M. Shattuck, Handbook of Granular Materials (CRC Press) 6. D. Vescovi, I. Prisco, C., Int. J. Solids Struct. 202, 495–510

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7. A. Aragão, E. Contini, Emprapa 8. A.B.d.M. ABM. Indústria de mineração do pará registra 15% de crescimento no volume de exportações em (2020). https://www.abmbrasil.com.br/por/noticias 9. E. Santos, L. Carvalho, A. Mesquita, L. Gomes, K. Pinheiro, A. Mesquita, REM - Int. Eng. J. 73(3), 361–369 10. N. Brilliantov, F. Spahn, J.M. Hertsch, T. Pöschel, Phys. Revi. 53(5), 5382–5392 11. A.S., A. Söderberg, S. Björklund, Tribol. Int. 40, 580–587 12. P. Cundall, O. Strack, Geotechnique 29, 47–65 13. H. Zhu, Z. Zhou, R. Yang, A. Yu, Chem. Eng. Sci. 62, 3378–3396 14. S. Sen, J. Hong, J. Bang, E. Avalos, R. Doney, Phys. Rep. 462(2), 21–66 15. M. Tiwari, T. Mohan, S. Sen, Int. J. Mod. Phys. B 31(10), 1742012 16. A. Rosas, K. Lindenberg, Int. J. Mod. Phys. B 31(10), 1742016 17. V. Nesterenko, Dynamics of Heterogeneous Materials (Springer, New York) 18. L. Machado, A. Rosas, K. Lindenberg, Granul. Matter 15, 735–746 19. L. Machado, A. Rosas, K. Lindenberg, Eur. Phys. J. 37(119) 20. A. Shukla, C. Damania, Exp. Mech. 27, 268–281 21. L. Machado, S. Sen, Phys. Rev. 98(32907) 22. L. Machado, S. Sen, Granul. Matter 22(1), 1–6 23. L. Machado, S. Sen, Phys. Rev. 103(42904) 24. A. Levy, J. Y, Granul. Matter 13, 107 25. Y. Zhou, B. Xu, A. Yu, P. Zulli, Powder Technol. 125, 45–54 26. Y. Li, Y. Xu, C. Thornton, Powder Technol. 260, 219–228 27. I. Keppler, F. Safranyik, I. Oldal, Eng. Comput. 33, 742–758 28. T. Simons, R. Weiler, S. Strege, S. Bensmann, M. Schilling, A. Kwade, Procedia Eng. 102, 741–748 29. T. Gröger, A. Katterfeld, in 16th European Symposium on Computer Aided Process Engineering, pp. 533–538 30. M. Marigo, E. Stitt, Kona Powder Part. J. 32, 236–252 31. A. Grima, P. Wypych, Granul. Matter 13, 127–132 32. C. Coetzee, D. Els, Comput. Electron. Agric. 65, 198–212 33. C. Li, W. Xua, Q. Meng, Powder Technol. 286, 478–487 34. C. Coetzee, Powder Technol. 297, 50–70 35. K. Williams, W. Chen, S. Weeger, T. Donohue, Particuology 12, 80–89 36. N. Estrada, E. Azéma, F. Radjai, A. Taboada, Phys. Rev. 84(11306) 37. C. Wensrich, A. Katterfeld, Powder Technol. 217, 409–417 38. J. Irazábal, F. Salazar, E. Oñate, Comput. Geotech. 85, 220–229 39. C. Coetzee, Powder Technol. 310, 104–142 40. S. Nase, W. Vargas, A. Abatan, J. McCarthy, Powder Technol. 116, 214–223 41. R. Bagnold, Proceedings of the Royal Society of London. Ser. A, Math. Phys. Sci. 225, 49–63 42. A. Roberts, Chem. Eng. Technol. 26(2), 163–170 43. C. Benjamim, The transfer chute design manual for conveyor belt systems. Conveyor Transfer Design (Pty Ltd) 44. D. Schulze, Powders and Bulk Solids - Behavior, Characterization, Storage and Flow, 1st edn. (Springer)

Chapter 28

Computational Model for Path Mapping of Spherical Particles in a Continuous Medium Applied to Iron Ore Transfer Chute Rodrigo X. A. Leão, Enrico Sarcinelli, Leandro Amorim, Humberto Belich, and Marcio F. Martins Abstract During the fall of bulk materials, the air around the material mainstream is dragged by the non-slip condition modifying the near pressure and velocity fields generating air currents around the bulk material fall. A particle that comes off the mainstream will be subject to the drag force due to the moving air and can be transported away, characterizing the formation and dispersion of dust. This work applies a computational model in an iron ore conveyor belt transfer chute to evaluate the forces acting on microparticles that can be carried away from the control volume. The computational model is easy to implement, has low computational effort, and allows to map the particle’s path making it possible to estimate the particle diameters with a greater propensity for dust formation than observing the preferred dust escape paths. The capacity of an extractor fan to absorb suspended particles was also evaluated. Analytical solutions for the particle velocity in continuous medium considering the Stokes flow are presented and used in model validation. Physical parameters relevant to flows involving particle displacement (specifically iron ore particles) are presented and discussed: terminal velocity, terminal time, relative Reynolds number, drag coefficient, and suspension time. Keywords Bulk materials · Belt transfer chute · Stokes’ flow · Particle displacement

R. X. A. Leão (B) · E. Sarcinelli · L. Amorim · H. Belich · M. F. Martins Laboratory for Computational Transport Phenomena (LFTC), The Federal University of Espirito Santo, Vitoria, Brazil e-mail: [email protected] H. Belich e-mail: [email protected] M. F. Martins e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_28

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28.1 Introduction Transportation of bulk material will eventually involve launching it from a certain height (e.g., from one conveyor belt to another); in these cases, the falling material drags the air around it due to the noon slip condition. This changes the pressure field around the material mainstream (core), inducing vortex formation near the impact zone and upward air currents close to the falling material. These air currents can be sufficient to transport micro and nanoparticles characterizing the formation and dispersion of dust, Leão et al. [1]. An excellent computational view of the velocity field and the dust boundary layer next to a falling bulk material can be found in Leão et al. [2]. According to the International Standardization Organization (ISO 2006, [3]) “Dust: small solid particles, conventionally taken as those particles below 75 μm in diameter, which settle out under their weight, but which may remain suspended for some time”. Microparticles of bulk material can be generated from mechanical shocks between larger particles or any process through which the material goes. Particles around 10 μm in diameter, considered as primary sources of dust formation [4], are easily transported by air and, once absorbed by the airways, can reach the lungs and cause severe damage to health depending on its chemical composition, [5]. On industrial scales, uncontrolled dust can become a social problem going beyond the physical boundaries of the industry, causing damage to the entire community [6]. In this work, a computational model to predict the displacement of sphere particles within a continuous medium velocity field is proposed. The model consists of evaluating the forces acting on the particle (weight, drag, and buoyancy) at discretized time intervals; it is applied in an iron ore conveyor belt chute situation and allows to estimate the particle diameters prone to form dust; the ability of an extractor fan to remove suspended particles is also evaluated. This work does not have as the main objective to reproduce an actual situation faithfully, but to present a computational model of easy implementation and low computational effort to predict the displacement of spherical particles inside a continuous medium, as well as show its usefulness in making decisions related to the reduction of dust formation. In addition, critical physical parameters related to particle displacement are presented and discussed: terminal velocity, terminal time, relative Reynolds number, drag coefficient, and suspension time.

28.1.1 Drag Force The first analytical solution for the flow of a fluid around a sphere for low Reynolds numbers (less than the unit) in a steady and uniform flow regime was developed by Stokes in 1851 [7]; after Stokes, Basset [8], Boussinesq [9], and Oseen [10] studied the behavior of a falling sphere inside a quiescent fluid, and traditionally the equation

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of motion for a spherical particle in a fluid is called the Basset-Boussinesq-Oseen (BBO) equation in honor of the authors above. A derivation of the particle motion equation for non-uniform and non-permanent flows was proposed by Maxey and Riley [11] and presented in Eq. 28.1 [12], where the subscript “c” refers to the continuous phase (e.g., air) and the subscript “d” refers to the disperse phase (e.g., solid particle); the variables of interest are as follows: gravity (g) particle mass (m), particle diameter (D), particle velocity (u), continuous medium velocity (v), continuous medium dynamic viscosity (μc ), and continuous medium specific mass (ρc ). The undisturbed flow term contains the forces acting on the particle due to pressure gradient (buoyancy) and shear stress; this term can be neglected for gas-solid flows where the specific mass ratio (ρc / ρd ) is too small but should be considered for flows where the density ratio is the order of unity (e.g., liquid-solid flows). The virtual mass term and the Basset term should be considered only in unsteady flows [12].

undisturbed flow

steady state drag

        ∂p du i ∂T i j D2 2 + 3π μc D (vi − u i ) + = mgi + Vd − ∇ vi m + dt ∂ xi ∂x j 24 virtual or apparent mass term

    1 d D2 2 ρc Vd ∇ vi (vi − u i ) + 2 dt 24

(28.1)

Basset or history term

 

 

t 2 2 d/dτ v − u + D /24 × ∇ v 3 i i i + π μc D 2 dτ 2 π vc (t − τ )1/2 0 Doing the above consideration for a small heavy particle (e.g., iron ore powder) displacing in a steady flow, only the steady-state drag remains and Eq. 28.1 becomes Fi = 3π μc D (vi − u i ) + π μc

D3 2 ∇ vi 8

(28.2)

The Laplacian term is the Faxen force which is a correction to the Stokes drag for the curvature of the velocity field and can be neglected for flows which effects of Reynolds number are more significant as well as for straight fields. Besides that, the Faxen force is proportional to D3 , and for nano and microparticles, it is expected to be of little significance. In this way, the force acting on a heavy small particle displacing in a steady flow can be expressed in term of the drag factor (f ) in Eq. 28.3. F = 3π μc D f (v − u)

(28.3)

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Or in term of the drag coefficient (C D ) in Eq. 28.4. F=

1 ρc C D A p |v − u|(v − u) 2

(28.4)

where A p is the projected area of the particle, for a sphere, which is given by Eq. 28.5. Ap =

π 2 D 4

(28.5)

Relationship between the drag factor (f ) and the drag coefficient (C D ) can be obtained by equating Eqs. 28.3 and 28.4: 1 ρc C D A p |v − u|(v − u) = 3π μc D f (v − u) 2 And f =

C D Rer C D ρc |v − u|D = 24 μc 24

(28.6)

(28.7)

where Rer is the relative Reynolds number presented in Eq. 28.8: Rer =

D|v − u| vc

(28.8)

where (νc = μc /ρc ) is the kinematic viscosity of the continuous medium. Correlations between the drag factor and the Reynolds number are presented in [13–15]. Figure 28.1 presents the behavior of C D with Rer for a non-rotating sphere. Graphic in Fig. 28.1 can be divided into three regions: • Stokes’ flow regime (Rer < 1) (Stokes, 1851): the flow is regarded as a creeping flow in which the inertial terms in the Navier-Stokes equations are less important and the viscous forces are imperative. In this region, the drag coefficient varies inversely with Reynolds number, and the classic drag Stokes relation is presented in Eq. 28.9. Applying relation 28.9 in Eq. 28.7, leads to a constant value of (f = 1)

Fig. 28.1 Variation of the drag coefficient (C D ) of a sphere with relative Reynolds number (Rer ), see [12]

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for Stokes regime. Extensions of the Stokes analysis include the HandamardRybczynski drag law (Clift et al., 1978) and (Oseen, 1910). CD =

24 Rer

(28.9)

• Newton’s drag law (50 < Rer < 3 × 105 ): the flow around the particle separates forming vortices and viscous forces are less significant. Drag coefficient can be approximated by a constant value of (C D = 0.445). • Critical Reynolds (Rer > 3 × 105 ): the boundary layer becomes turbulent, and the separation point is moved rearwards reducing the drag coefficient.

28.1.2 Particle Velocity in a Continuous Medium for Stokes’ Flow Particles motion in the Stokes flow can be described by Eq. 28.10. Weight

Buoyancy

Drag

         du m = ρd Vd g − ρc Vd g + 3π μc D f (v − u) dt

(28.10)

where ρd is the specific mass of the particle material (disperse phase) and Vd is the particle volume. Defining: f = 1, b = (ρd – ρc ) Vd , and k = 3π μc D, Eq. 28.10 can be written like du = bg + k(v − u) (28.11) m dt Rearranging du k bg kv + u= + dt m m m

(28.12)

Solution of Eq. 28.11 considering u(0) = u 0 gives u(t) = u 0 e

−(k/m)t

 +



bg + v 1 − e−(k/m)t k

(28.13)

If t → ∞, Eq. 28.13 tends to the final particle velocity u(t) →u∞ : u∞ =

bg +v k

(28.14)

From Eq. 28.14, it is noted that if no body forces are present (g = 0), the disperse phase tends to the continuous phase velocity; on the other hand, if body forces are present, the first term should be considered, depending on the (bg / k) value. Another

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good observation is that when t → ∞, the relative Reynolds number always tends to the same value regardless of continuum velocity; the relative Reynolds number when t → ∞ is determined by Eq. 28.15.    u→ D v − − D|bg| ∞ Rer = = vc vc k

(28.15)

Terminal velocity (Eq. 28.16) is the final velocity reached by a particle that falls through a quiescent fluid suffering the action of the fluid’s resistance and is obtained doing (t → ∞) and (v→ 0) in Eq. 28.13: vter m =

b g k

(28.16)

If (ρd >> ρc ) so that (b ≈ m) and Eq. 16 takes the most usual form in Eq. 28.17, vter m =

m g k

(28.17)

Particles flow in a fluid is a subcategory of multicomponent, multiphase flows in which it is usual to consider air as a mixture whose properties (e.g., viscosity and thermal conductivity) are representative of the mixture [12]. In bulk materials fall, the air close to the material mainstream can be flooded with nano and microparticles that come off the bulk material so that the fluid (air + nanoparticles) close to the material mainstream can be interpreted as a mixture of the continuous and the disperse phase with specific mass (ρm ) determined by Eq. 28.18: ρm = ρd φ + ρc (1 − φ)

(28.18)

Equation 28.18 is a weighted average where φ is the volume fraction of solids.

28.2 The Computational Model Equations applied in the computational model are described below. Displacement equation is obtained from Newton’s second law (Eq. 28.19). m r¨i = mai

(28.19)

For initial conditions: r (0) = r0 and dr/dt(0) = 0, Eq. 28.19 gives r˙i = v0i + ai t ri = r0i + v0i t +

(28.20) ai t 2 2

(28.21)

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Discretized solution in a time interval t is presented in Eqs. 28.22 and 28.23. u i(t+t) = u i(t) + ai(t) t ri(t+t) = ri(t) + u i(t) t +

ai(t) 2 t 2

(28.22) (28.23)

Forces acting on the particle are gravitational and drag presented in Eq. 28.24.  −−−−−→ → gi + 3πρc vc D f V (x, y)i − Ui Fi = m −

(28.24)

where V(x, y) is the velocity of the continuous phase assessed in the current position of the particle and U is the current particle velocity at time t. Particle acceleration (Eq. 28.25) is obtained dividing Eq. 28.23 by the particle mass (m = ρd π D 3 /6). ai(t) =

Fi ρc vc −−−−→ −−→ f − U = gi + 18 V (x, y) i i(t) m ρd D 2

(28.25)

The combination of Eqs. 28.22, 28.23, and 28.25 can be used to map the path of a particle within a steady-state continuous phase flow. Note that the greater the particle specific mass and diameter the smaller the viscous acceleration term (inversely proportional to ρd and D2 ) so that the displacement of larger and heavy particles is preferably due to gravity, and on the other hand, the continuum viscosity strongly influences smaller and lighter particles.

28.2.1 Validation and Application Methodology Four cases were solved analytically and computationally for the model validation; results for particle velocity and relative Reynolds number were compared. The cases consist of the fall from the rest of an iron ore particle (ρd = 5000 kg/m3 ) diameter (D = 50 μm) in the pure air of specific mass (ρc = 1.3 kg/m3 ) and kinematic viscosity (νc = 16 cStoke) in four situations: (a) quiescent air; (b) air has a lower velocity than the particle terminal velocity in the direction of gravity (particle displacement direction); (c) air has a lower velocity than the particle terminal velocity in the opposite direction of gravity; and (d) air has a greater velocity than the particle terminal velocity in the opposite direction of gravity. In the analytical solution, the vertically downward direction is positive. Figure 28.2 illustrates the direction of vector quantities involved in the analytical solution. For the conveyor belt cases, the control volume of the computational model consists of a 1.5 m high and 0.6 m long region. A conveyor belt is positioned at a height of 1 m and has a speed of 1 m/s, so that the bulk material leaves the conveyor belt at position r = [0.1i + 1 j] (m) of the control volume with initial velocity u0 = [1i] (m/s) presented in Figs. 28.3 and 28.4.

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Fig. 28.2 Direction of vector quantities in analytical solution: vertical axis, gravity, u (particle velocity), and v (continuous medium velocity)

Fig. 28.3 Control volume with upward internal flow of velocity V = [0.1i + 0.5 j] (m/s)

Four cases were analyzed: • (a) quiescent air is considered in the entire control volume; • (b) an upward airflow v = [0.1i + 0.5 j] (m/s) appears inside the control volume due to the bulk material fall itself (Fig. 28.3) [1, 2]; • (c) an increase in the air specific mass is considered due to the presence of nano and micro particles forming the dust; considering a concentration of (Ø = 0.0025) [16] dust specific mass (ρm = 13.8 kg/m3 ) is obtained through Eq. 28.18; and • (d) an extractor fan is added over the belt that induces a velocity airflow at the top of the control volume vext = [−0.5i + 0.5 j] (m/s), and the extractor fan ability to absorb suspended particles is evaluated (Fig. 28.4). A computational model was developed in the open-source software Octave. Studied cases use time step Δt = 0.0005(s) in a 0.04(m) 2D square mesh.

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Fig. 28.4 Control volume in presence of extractor fan over the belt vex = [−0.5i + 0.5 j] (m/s). Upward internal flow of velocity V = [0.1i + 0.5 j] (m/s)

28.2.2 Validation and Preliminary Results Figures 28.5 and 28.6 present the agreement between analytical results (continuous line) and the computational model results (hollow points) for the particle velocity and relative Reynolds number, respectively. The model validation case consists in a particle fall from rest in pure air considering four different velocities of the continuous phase: (a) V = [0](m/s) (yellow); (b) V = [0.1 j] (m/s) (green); (c) V = [−0.1 j] (m/s) (red); and (d) V = [−0.4] (m/s) (purple). In Fig. 28.5, for the quiescent air case (yellow), the particle reaches 99% of its terminal velocity uter m = 0.327(m/s) in the time tter m = 0.154(s); see Table 28.1. In the other cases, the particle also tends to its final velocity as predicted by Eq. 28.14. In Fig. 28.6, note that the particle has a characteristic relative Reynolds number that is achieved in all cases; for an iron ore particle of diameter (D = 50 μm) and specific mass (ρd = 5000 kg/m3 ), the value of relative Reynolds number when the particle reaches 99% of its terminal velocity is Rer −ter m = 1.023. In the case where there is an airflow down v = [0.1 j] (m/s) (green), the relative Reynolds number has a “V” shape. This occurs since the particle starts from rest and initially accelerates up to the velocity of the continuous medium, reducing the velocity difference and consequently the relative Reynolds number. When the parity between the velocity of the medium and the particle is reached, the relative Reynolds value is zero; as the velocity of the medium is less than the final velocity of the particle, the particle continues to accelerate until reaching its final velocity, increasing the velocity difference and consequently increasing the relative Reynolds number until its final value. Figures 28.7, 28.8, 28.9, and 28.10 present the results of velocity (u), relative Reynolds number (Rer ), drag factor (C D ), and acceleration obtained analytically for

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Fig. 28.5 Comparison between the computational model (hollow points) and analytical (continuous line). Velocity of an iron ore particle (ρd = 5000 kg/m3 ) and diameter (D = 50 μm) starting from rest in: quiescent pure air (yellow), airflow down (v = +0.1 m/s) (green), airflow up (v = −0.1 m/s) (red), and airflow up (v = −0.4 m/s) (blue)

Fig. 28.6 Comparison between the computational model (hollow points) and analytical (continuous line). Relative Reynolds number of an iron ore particle (ρd = 5000 kg/m3 ) and diameter (D = 50 μm) starting from rest in: quiescent pure air (yellow), airflow down (v = + 0.1 m/s) (green), airflow up (v = −0.1 m/s) (red), and airflow up (v = −0.4 m/s) (blue)

iron ore particles (ρd = 5000 kg/m3 ) of diameters (D = 10, 20, 30, 40, 50 μm) falling from rest in quiescent pure air. In Fig. 28.7, the vertical dotted lines mark the terminal time (t∞ ) when the particle reaches 99% of its terminal velocity, while the horizontal ones mark the terminal velocity itself (u∞ ); for a computational model, it is important to observe the terminal time and keep the model time step well below this value to observe variations in particle acceleration. In this work, the smallest particle has diameter 10 μm with terminal time t∞ = 6 × 10−3 (Table 28.1) and the model time step was set to t = 0.5 × 10−3 being less than 10% of the terminal time. In Fig. 28.8, each particle tends to its characteristic relative Reynolds number, as previously discussed. Figure 28.9 shows the behavior of the drag coefficient (C D ) acting on the particles during the fall; the drag coefficient starts from a high value

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Table 28.1 Values as a function of the diameter for iron ore particles (ρd = 5000 kg/m3 ): mass (m), terminal time (t∞ ) or time to reach 99% of the terminal velocity, particle velocity at terminal time (u∞ ), relative Reynolds number at the terminal time (Rer∞ ), drag coefficient at the terminal time (C D ∞ ), and suspension time (tsusp ) for a drop of 1 m in height D (10−6 m) m (10−12 kg)

t∞ (s)

u∞ (m/s)

Rer∞

C D∞

tsusp (s)

10 20 30 40 50 100 150 200

0.0061501 0.0246 0.055351 0.098401 0.15375 0.61501 1.3838 2.45

0.013098 0.065488 0.22102 0.5239 1.0232 8.186 27.628 64.833

0.008186 0.065488 0.22102 0.5239 1.0232 8.186 27.628 64.833

2931.8 366.48 108.59 45.81 23.455 2.9318 0.86869 0.445

76.351 19.093 8.4952 4.793 3.0871 0.89571 0.63681 0.525

2.618 20.944 70.686 167.55 327.25 2618 8836 20944

Fig. 28.7 Velocity in time for iron ore particles (ρd = 5000 kg/m3 ) of diameters (D = 10, 20, 30, 40, 50 μm) falling in quiescent pure air

and tends to a constant value. This final value will be higher for smaller particles that suffer greater action from the air resistance; from Fig. 28.10, it is observed that the acceleration tends to zero as the gravitational and drag forces are balanced. Table 28.1 presents results obtained analytically and computationally for iron ore particles of different diameters: particle mass (m), terminal time (t∞ ) (time required for the particle to reach 99% of its terminal velocity), velocity (u∞ ), relative Reynolds number (Rer ), and drag coefficient (C D ) obtained in the terminal time. The suspension time (t∞ ) is the time required for a particle that is dropped from a certain height to reach the ground, in Table 28.1 (h = 1 m). Analysis of Table 28.1 and, more specifically, the terminal velocity and the suspension time helps to understand the problem of nano and microparticles suspension. Let us take the case of an iron ore particle of 10 μm in diameter, if it is launched in

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Fig. 28.8 Relative Reynolds number (Rer ) in time for iron ore particles (ρd = 5000 kg/m3 ) of diameters (D = 10, 20, 30, 40, 50 μm) falling in quiescent pure air Fig. 28.9 Drag coefficient (C D ) in time for iron ore particles (ρd = 5000 kg/m3 ) of diameters (D = 10, 20, 30, 40, 50 μm) falling in quiescent pure air

Fig. 28.10 Acceleration in time for iron ore particles (ρd = 5000 kg/m3 ) of diameters (D = 10, 20, 30, 40, 50 μm) falling in quiescent pure air

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the air from a 1 m height, it will remain floating for more than 1 minute, and during this time, any movement of air mass with velocity bigger than the particle terminal velocity of u∞ = 13.3 mm/s can carry it away.

28.3 Conveyor Belt Cases The computational model was applied to predict the displacement of particles in a transfer chute from a conveyor belt. Figure 28.11 presents the displacement of iron ore particles of different diameters from a conveyor belt with a speed of 1 m/s at the height of 1 m from the ground in pure quiescent air. The larger the particle diameter, the more it tends to a Newtonian displacement suffering action mainly by gravity and inertia. Smaller particles, on the other hand, suffer great action of drag force; in the horizontal axis, this is reflected in a smaller displacement, and in the vertical axis, it means a longer suspension time as shown in Table 28.1. Particles of 10 μm in diameter practically do not move when leaving the conveyor belt and will remain in suspension for a long time or until be carried by an air current. Figure 28.12 presents the case considering an upward pure air current of velocity V = [0.1i + 0.5 j] (m/s) (Fig. 28.3). As a reference, the displacements of the particles in quiescent air and vacuum were maintained and represented by the continuous lines, while the displacements of the particles under the action of the airflow are shown in dashed lines. Under the action of an upward flow of pure air, particles with diameters less than 50 μm fluctuate and are transported away from the control volume. Particles of diameters between 50 μm and 150 μm suffer considerable drag force action but are not loaded; particles with a diameter of 200 μm or greater suffer a small action of the drag force and preferably have a Newtonian behavior.

Fig. 28.11 Displacement of iron ore particles (ρd = 5000 kg/m3 ) of diameters (D = 10, 50, 100, 150, 200 μm) falling in quiescent pure air from a conveyor belt at 1 m height and 1 m/s velocity

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Fig. 28.12 Displacement of iron ore particles (ρd = 5000 kg/m3 ) of diameters (D = 10, 50, 100, 150, 200 μm) falling from a conveyor belt at 1 m height and 1 m/s velocity. Continuous lines: falling in quiescent pure air; Dashed lines: in pure airflow upward of velocity v = [0.1i + 0.5 j] (m/s) (Fig. 28.3)

Fig. 28.13 Displacement of iron ore particles (ρd = 5000 kg/m3 ) of diameters (D = 10, 50, 100, 150, 200 μm) falling from a conveyor belt at 1 m height and 1 m/s velocity. Continuous lines: falling in quiescent pure air; Dashed lines: in dust mixture (ρm = 13.8 kg/m3 ) considering upward flow of velocity v = [0.1i + 0.5 j] (m/s) (Fig. 28.3)

Figure 28.13 presents the case in which the continuous medium in the control volume is a mixture of air with nanoparticles of iron ore in concentration (Ø = 0.0025) and specific mass (ρm = 13.8 kg/m3 ). The increase in the specific mass of the continuous medium causes the drag force to be felt over larger particles with greater intensity than in pure air. In this case, the computational model indicates that particles up to 200 μm would be suspended and carried out. Simulations of 10 μm were unstable in this case, but it can be concluded that these particles will preferentially follow the continuous flow.

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Fig. 28.14 Displacement of iron ore particles (ρd = 5000 kg/m3 ) of diameters (D = 10, 50, 100, 150, 200 μm) falling from a conveyor belt at 1 m height and 1 m/s velocity. Continuous lines: falling in quiescent pure air; Dashed lines: in dust mixture (ρm = 13.8 kg/m3 ) considering upward flow of velocity v = [0.1i + 0.5 j] (m/s) in presence of an extractor fan at a height of 1.2 m inducing an airflow of velocity v = [−0.5i + 0.5 j] (m/s) at the top of the control volume

Figure 28.14 presents the effect of inserting an extractor fan at the top of control volume which induces a velocity airflow V = [−0.5i + 0.5 j] (m/s) from the height of 1.2 m (Fig. 28.4). The extractor fan would suck particles up to 150 μm, preventing them from being carried out of the control volume.

28.4 Conclusion and Future Perspective A computational model to predict the displacement of spherical particles in the presence of a continuous moving medium has been proposed and proved to be according to the proposed analytical solutions. The model was applied to an iron ore conveyor belt case to analyze the behavior of microparticles. The model indicated that when dumping the material from a 1 m/s velocity and 1 m high, once considering the increase in specific mass of the continuous medium, particles with a diameter of up to 200 μm can suspend and be taken out of the control volume by the fall dynamics itself. The capacity of an extractor fan installed on the belt at the height of 1.2 m to absorb suspended particles was also evaluated; the extractor fan has been shown to suck particles up to 150 μm. Analytical and computational results referring to physical parameters of iron ore particles in the air were presented, including terminal velocity, terminal time, relative Reynolds number at a terminal time, drag factor at a terminal time, and suspension time. These parameters can be helpful in decision-making regarding the mitigation of micro and nanoparticles suspension with consequent dust formation in bulk material transport. Analyzing Table 28.1, it is considered that the 20 μm particles are prone to form dust as the 10 μm particles are.

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It is worth mentioning that the greatest difficulty in applying the model to real situations consists in defining the velocity field of the continuous medium, which can be challenging since these fields present high gradients of velocity and vorticity. In the present work, the most complex velocity field consists of the simple union of two uniform fields without vorticity and interaction between them. Acknowledgements The authors would like to thank the Foundation of Support and Research Development (FADESP) and mining company VALE.

References 1. R.X. de Almeida Leão, L.S. Amorim, M.F. Martins, H.B. Junior, E. Sarcinelli, A.L.A. Mesquita, Powder Technol. 385, 1 (2021) 2. R.X. de Almeida Leão, L.S. Amorim, M.F. Martins, H.B. Junior, E. Sarcinelli, A.L.A. Mesquita, Powder Technol. 371, 190 (2020) 3. I.O. for Standardization, Air quality-Particle size fraction definitions for health-related sampling (ISO, 2006) 4. P. Cooper, P. Arnold, KONA Powder Part. J. 13, 125 (1995) 5. A.F. Tena, P.C. Clarà, Archivos de Bronconeumología (English Edition) 48(7), 240 (2012) 6. L. PINHEIRO, A construção de um problema social: o caso do “pó preto” e seu debate nas audiências públicas de licenciamento ambiental em vitória-es. 2012. Ph.D. thesis, Dissertação (Mestrado em Ciências Sociais)-Centro de Ciências Humanas e ...(2016) 7. G. Stokes, Trans. Camb. Phil. Soc. 9(8), 106 (1850) 8. A.B. Basset, A Treatise on Hydrodynamics: with Numerous Examples, vol. 2 (Deighton, Bell and Company, 1888) 9. J.B.J. Baron Fourier, Théorie analytique de la chaleur (Chez Firmin Didot, père et fils, 1822) 10. C.W. Oseen, Hydrodynamik (Akademische Verlagsgesellschaft, 1927) 11. M.R. Maxey, J.J. Riley, Phys. Fluids 26(4), 883 (1983) 12. C. Crowe, J. Schwarzkopf, M. Sommerfeld, Y. Tsuji, DOI 10, b11103 (2011) 13. L. Schiller, Z. Vereines Deutscher Inge. 77, 318 (1933) 14. A. Putnam, Integratable form of droplet drag coefficient (1961) 15. R. Clift, Proc. Chemeca’70 1, 14 (1970) 16. R. Ansart, A. De Ryck, J.A. Dodds, Chem. Eng. J. 152(2–3), 415 (2009)

Chapter 29

A CFD Model for Free Falling Bulk Materials Analysis Leandro F. B. Lima, Maciel C. Furtado, and André L. A. Mesquita

Abstract This work aims to present a CFD Multiphase in order to calculate the induced airflow rate produced by a free-falling bulk material. The numerical solution is obtained through the DDPM model (Dense Discrete Phase Model) implemented in the ANSYS Fluent commercial code. This analysis tool will enable an improvement of dedusting projects in industries that work with granular materials. The simulations performed were compared with other computational methods and experiments and semi-empirical models available in the literature, obtaining good results. Keywords CFD multiphase · DDPM model · Free-falling bulk material

29.1 Introduction The handling of bulk solids often involves free-falling particle streams in the air, such as filling particles from a hopper into a silo. The free-falling particles can induce the flow of still air due to particle-fluid interaction. The induced air will usually detour from the mainstream of the solids, and at the same time, fine particles contained in the particle stream will be entrained into the airflow, causing dust emission into the air. Dust emission is a hazard to both environments and the health of the operating personnel if not adequately controlled. Exposure may happen when the dust concentration reaches a certain value [1]. Since the work of Hemeon [2], several studies have been conducted to calculate and determine the capacity of air entrained during the free fall of granular flows [3, 4]. Liu et al. [5] found that the velocity profile of air entrained by the free-falling particles can be modeled as a Gaussian distribution. They also reported that the spread angle L. F. B. Lima · M. C. Furtado (B) · A. L. A. Mesquita Laboratory of Fluid Dynamics and Particulate (FluidPar), Amazon Development Center in Engineering (NDAE), Federal University of Pará, Campus of Tucuruí, Brazil e-mail: [email protected] A. L. A. Mesquita e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_29

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of the particle-driven plumes was much smaller than other plumes, such as miscible plumes arising from heat sources. The air entrainment pattern of a free-falling stream of bulk material can be influenced by bulk material properties such as particle size and density as well as operating conditions such as drop height, particle mass flow rate, and the other environmental factors [6, 7]. Uchiyama [8] used a numerical two-dimensional vortex method to analyze the flow of the particulate stream generated by solid particles falling from an orifice at a predefined control volume. The effects of particle diameter and density on the flow are investigated and compared with an analytical model. As a result, it was verified that the values obtained for the airflow followed the prediction from the massive particle model. A recent study was performed by Esmaili et al. [9]. Experiments were presented to measure the air induced by free-falling particle flow. From these results, a new semi-empirical for predicting the induced air flow rate is developed and presented in analytical equations. Significant effort has been devoted to improving numerical tools, such as Computational Fluid Dynamics (CFD) tool, to predict such complex flows. However, it has been identified that systems containing one or more particulate phases are the most complex and challenging in multiphase flow modeling. To accurately predict the behavior of the solids, it is necessary to choose a numerical method capable of accounting for not only particle-fluid interactions but also particle-wall and particleparticle interactions in three dimensions and across any particle size distribution, [10]. In dealing with modeling of gas-solids flows, the Eulerian-Eulerian and the Eulerian-Lagrangian methods are the frequently used approaches [11]. In thebreak Eulerian-Eulerian approach, all the phases are treated as continuous phases, while in the Eulerian-Lagrangian approach, the fluid phase is treated as a continuous phase, but the solid phase is treated as a discrete phase [12]. According to Chu et al. [13], in recent years, numerical modeling has proven to be a helpful tool to improve the fundamental understanding of fluid flows in the literature. The authors developed a study via CFD-DEM (Discrete Element Method) analyzing the rate of induced air and the drag force both feed linearly according to the height of fall, but the results varied, and the airflow was challenging to be estimated only using the drag force. Schulz et al. [14] also used CFD-DEM coupling modeling to investigate the release of dust from the bulk solid during handling. Another work developed by Leão et al. [15, 16] presents a model for the propagation of dust particles, driven only by free fall bulk material. The work applied CFD using a model similar to a fluid with a mixture of air and dust to propagate material dust in an open field, based on the principle that the movement of dust occurs almost statically. As a result, flow pattern zones were presented where they could potentially serve as a source of dust propagation when subjected to an external force Sun et al. [17] presented a model based on the air jet theory for dropping the flow of unconfined and semi-confined particles. Theoretical formulas for the trapped air flow rate were developed, and a numerical simulation was performed to analyze the particle’s speed and the air entrained under falling processes. As a result, it was

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analyzed that the airflow of the confined particle is less than the one in a flow of non-confined particles. They concluded that as the distance of the particles in the wall increases, the induced airflow also increases. The objective of the work is to propose a computational model of CFD using the DDPM tool (Dense Discrete Phase Model) of the Ansys–Fluent Eulerian multiphase model to predict the amount of air entrained during the free fall of particles taking as model Esmaili et al. [9]. The results obtained in the simulation, using data processing tools (2D plan and graphs), were compared with the experimental procedures, analytical equations, and another numerical simulation model.

29.2 CFD Model The Dense Discrete Phase Model (DDPM) is an extension of the DPM model, available in the program code ANSYS FLUENT, which considers particles’ volumetric fraction in the solution of the continuous phase equations, allowing a better transfer of momentum in the coupling between the phases. Therefore, the DDPM is called a hybrid model, which combines the advantages of the Euler Granular (including particle interactions) and the DPM (easy including of different particle size classes) approach [18]. Equations 29.1 and 29.2 are showing the continuity equation and the momentum equation (standard form) for the continuous phase, respectively. Kαβ in Eq. 29.2 is the momentum exchange coefficient between the phases due to the drag force (Popoff and Braum [19]). In the case that the influence of the dispersed phase on the primary phase cannot be neglected (common for DDPM), an additional source term S D P M can be included on the right side of the equation to account for these effects (two-way coupling) [19–21]:     ∂εβ ρβ m ˙ βa + m + ∇ · εβ ρβ uβ = ˙ aβ ∂t

(29.1)

    ∂εβ ρβ u β + ∇ · εβ ρβ uβ uβ = −εβ ∇ pβ + ∇ · εβ u β ∇ · uβ + εβ ρβ g+ ∂t (29.2)     K aβ ua − uβ + m ˙ βa uβa − m ˙ aβ uaβ + F D P M + S D P M Since  is the volume fraction of the fluid phase, m˙ βα and m˙ αβ represent the mass transfer, respectively, from phase β to phase α and from phase, α to phase β, both per unit volume of the primary phase β (fluid), u and u are the transport speeds between the phases, determined as a function of the direction of the mass transfer rate (e.g., for m > 0 it means that phase α transfers mass to phase β with velocity u = u), K is the coupling coefficient between the phases. F D P M is the coupling term for changing the amount of movement due to the forces of the discrete phase, and the S D P M is the

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source term of the discrete phase due to the displacement of the continuous phase concerning the input of the discrete phase (particle) in a given volume. The coupling between the phases is done through the term F D P M , which incorporates the change (gain or loss) of the amount of movement, due to the passage of the discrete phase through each control volume of the computational mesh of the continuous phase. The coupling term of the amount of movement is expressed by Eq. 29.3 [19]:     ˙ p t K pβ up − uβ + Fpβ m (29.3) FDPM = where Kpβ the coupling coefficient of the drag force (Fd ) and F p the other forces of the discrete phase, incorporated in the DDPM, which have to couple with the fluid. For the current study, F p can be represented by the virtual mass forces Fvm and lift Fls .

29.3 Case Study The model configuration is schematically shown in Fig. 29.1. Initially, it is assumed that the air velocity that moves along the cone equals the velocity of a single free-fall particle with drag. It is also assumed that the pressure is constant within the system and gravity as 9.81 m/s2 . According to work by Esmaili et al. [22], the flow of falling material is modeled, as shown in Eq. 29.4. Q ind =

Vcone = ts

  π h db2 + db ds + d S2    3ρa cd h 4ρ p d p −1 ρ p d p +V0 e 12 3ρa gcd cosh

(29.4)

The main dimensions of the cone are the height of 1200 mm representing the drop height with the 26 mm hopper outlet, 300 mm representing the diameter of the top of the cone 500 mm to the bottom. The materials used in the experimental program are summarized in Table 29.1, the same ones used in the work of Esmaili et al. [9], assuming the air density of 1.185 kg/m3 . The materials and their properties were selected to analyze the particle’s diameter and density on the amount of air entrained. The development of the mesh is one of the main steps to obtain a practical simulation. The quality of the mesh directly reflects the accuracy of the results and directly influencing the convergence of the model under study. Figure 29.2 shows the mesh generated by the Ansys software (v. 19.0). In this work, the discretization domain was conducted in a 3D subdomain of the analyzed control volume (mesh) structured, using hexahedral elements. The mesh data used in the development of this work are presented in Table 29.2, in which four different mesh sizes were used in order to verify the convergence of the results based on the size variation of its elements without undergoing significant changes in

29 A CFD Model for Free Falling Bulk Materials Analysis

Fig. 29.1 Model configuration

Fig. 29.2 Mesh 4 generated for analyses

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Table 29.1 Material settings Material Density (kg/m3) Plastic pellets Glass beads Glass beads

870 2450 2450

Particle diameter (mm)

26 mm hopper outlet (kg/s)

Input speed (m/s)

3.0 3.0 6.0

0.09 0.34 0.26

0.35 0.47 0.37

Table 29.2 Mesh convergence analysis for 3 mm plastic pellets Mesh Element size N◦ of N◦ of nodes Average air Average solid (mm) elements speed at speed at 1200 mm(m/s) 1200 mm(m/s) 1 2 3 4

10 5 4 3

1536473 4034121 7853467 9503131

Table 29.3 CFD modeling configuration Parameters CFD Model Continuous phase Disperse phase Turbulence Input condition Outlet condition

275843 743300 1165982 1707450

1,13 1,31 1.35 1,37

3,57 3,84 3.91 4,05

Model Eulerian - DDPM Air Spherical particles k-ω/SST Inlet speed Outlet pressure

the results, thus acquiring a mesh that can represent the result with the least possible errors. Convergence tests were done using air velocity and solid at the control volume outlet (3 mm plastic spheres). With a mesh quality parameter of maximum Skewness of 0.76 (Good), mesh 4 with a 3 mm element size was selected for the simulations. The software applies the Finite Volume Method to discretize the equations. The model used here was the DDPM (Discrete and Dense Phase Model) to analyze the fluid dynamics of particulate materials in a fluid medium. The stationary approach was used to achieve convergence, assuming that the side walls are open to the environment. Thus, the modeling configurations used to follow in the Table 29.3 respecting the same molds of the simulation generated by Esmaili et al. [9] in their experimental work in Esmaili et al. [22]:

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29.4 Results Figure 29.3 represents the simulated air velocity contour from the influence of drag caused by the fall of the analyzed particles. The plastic pellets with the same diameter as the glass beads showed a lower maximum velocity of the surrounding air. This is due to the fact of the influence of a greater mass of the glass beads. The influence of the particle size for the same material can be observed when comparing the results of the 3 and 6 mm particles for the glass, where it is possible to notice that the smaller glass particle with 3 mm has a higher velocity of the surrounding airflow than a particle larger than 6 mm. Based on the graph in Fig. 29.4, it is possible to observe that the particle speed in the simulation carried out in this work was very close to the experimental work carried out by Esmaili et al. [22]. Furthermore, based on the simulation results shown in Fig. 29.5, it is possible to notice a substantial dispersion of the induced air velocity field (from inside to outside the flow), causing a detachment about the surrounding air as the height drop is increasing. This result corroborates with the data demonstrated in the works of Esmaili et al. [9, 22]. The graph in Fig. 29.6 shows the results for air velocity in the 900 mm drop section. This velocity profile can be related to the inconstant shape of the flow of granular material in free fall, concentrating its greatest speed in the flow center. Figure 29.7 shows air displacement with the material drop height for the 3 mm plastic pellets of the result predicted by Eq. 29.4 and those of the simulation using the Eulerian DDPM model. When analyzing the amount of air entrained by the 3 mm plastic particles, the simulation performed here maintains a good correlation with the simulation by Esmaili et al. [9] and Eq. 29.4, although it slightly overestimates the amount of air when compared to the experimental data performed by Esmaili et al. [22], the same occurs with Fig. 29.8 for the 3 mm glass beads and Fig. 29.9 for the 6 mm glass beads.

29.5 Conclusion and Future Perspective The ANSYS Fluent–DDPM model used to simulate free-falling bulk materials has been demonstrated to be efficient, being validated with results of simulations and experiments available in the literature. Through the simulations presented here, it was possible to verify how the entrained air can vary depending on the diameter and properties of the analyzed material. In addition, the variation in the drop height of the material directly influences the variation in the amount of air induced by the falling material. With these results, it was possible to conclude that in a flow of granulated material with larger particles, the surrounding air will have a lower speed than with the smaller particles (same material). In the same way, increasing the density of the particles to

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Fig. 29.3 Air speed of 3 mm plastic pellets (a); Air speed of 3 mm glass beads (b); Air speed of 6 mm glass beads (c)

the same diameter will increase the speed of the air induced by the particles in free fall. Therefore, the use of CFD techniques is essential to save time and money, but always with the care of doing a practical case study so that the modeling is consistent with generating the most accurate possible results.

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Fig. 29.4 Velocity of 3 mm plastic pellets, 3 mm glass beads and 6 mm glass beads compared to the practical experiment

Fig. 29.5 3D demonstration of the air velocity field for 3 mm glass beads

344 Fig. 29.6 Air velocity distribution in the 900 mm high cross section

Fig. 29.7 Induced air results from simulations for 3 mm plastic pellets

Fig. 29.8 Induced air results from simulations for 3 mm glass beads

Fig. 29.9 Induced air results from simulations for 6 mm glass beads

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References 1. L.L. Waduge, S. Zigan, L. Stone, A. Belaidi, P. García-Triñanes, Process Saf. Environ. Prot. 105, 262 (2017) 2. D.J. Burton, Hemeon’s Plant & Process Ventilation (CRC Press, 2018) 3. R. Ansart, A. De Ryck, J.A. Dodds, Chem. Eng. J. 152(2–3), 415 (2009) 4. R. Ansart, J.J. Letourneau, A. De Ryck, J.A. Dodds, Powder Technol. 212(3), 418 (2011) 5. Z. Liu, P. Cooper, P. Wypych, Part. Sci. Technol. 25(4), 357 (2007) 6. P. Cooper, P. Arnold, KONA Powder Part. J. 13, 125 (1995) 7. P. Wypych, D. Cook, P. Cooper, Chem. Eng. Process. Process Intensif. 44(2), 323 (2005) 8. T. Uchiyama, Powder Technol. 145(2), 123 (2004) 9. A. Esmaili, T. Donohue, C. Wheeler, W. McBride, A. Roberts, Int. J. Miner. Process. 142, 82 (2015) 10. J. Parker, K. LaMarche, W. Chen, K. Williams, H. Stamato, S. Thibault, Powder Technol. 235, 115 (2013) 11. X. Chen, J. Wang, Powder Technol. 254, 94 (2014) 12. W.H. Ariyaratne, E. Manjula, C. Ratnayake, M.C. Melaaen, in Proceedings of The 9th EUROSIM Congress on Modelling and Simulation, EUROSIM 2016, The 57th SIMS Conference on Simulation and Modelling SIMS 2016 (Linköping University Electronic Press, 2018), 142, pp. 680–686 13. K. Chu, Y. Wang, Q. Zheng, A. Yu, R. Pan, Powder Technol. 361, 836 (2020) 14. D. Schulz, N. Schwindt, E. Schmidt, R. Jaseviˇcius, H. Kruggel-Emden, Powder Technol. 355, 37 (2019) 15. R.X. de Almeida Leão, L.S. Amorim, M.F. Martins, H.B. Junior, E. Sarcinelli, A.L.A. Mesquita, Powder Technol. 371, 190 (2020) 16. R.X. de Almeida Leão, L.S. Amorim, M.F. Martins, H.B. Junior, E. Sarcinelli, A.L.A. Mesquita, Powder Technol. 385, 1 (2021) 17. H. Sun, A. Li, J. Wu, J. Zhang, Granul. Matter 22(2), 1 (2020) 18. T.J. Niemi, et al., (2012) 19. B. Popoff, M. Braun, in International Conference on Multiphase Flow, Leipzig, Germany (2007) 20. S. Cloete, S. Johansen, M. Braun, B. Popoff, S. Amini, in 7th International Conference on Multiphase Flow (2010) 21. S. Cloete, S. Amini, in 9th International Conference on Multiphase Flow, Firenze, Italy (2016) 22. A. Esmaili, T. Donohue, C. Wheeler, W. McBride, A. Roberts, Part. Sci. Technol. 31(3), 248 (2013)

Chapter 30

A Theoretical Framework for a Toroidal Vortex as a Dust Scattering Center Humberto Belich, Marcio F. Martins, Leandro Amorim, Enrico Sarcinelli, and Rodrigo X. A. Leão

Abstract The citizens of urban regions constantly suffer from dust particles present in the air that are dispersed mainly from the local industries’ facilities. To understand the dispersion of dust caused by the free fall of granular material, we have observed that the dust dispersion is made by forming a toroidal vortex. We studied a vertical vortex to carry out an analytical study of the velocity field formed by the vortex. Then by this study, it was possible to establish the velocity field of a toroidal vortex by the contribution of two vortices with different orientations. Keywords Free-falling bulk material · Dust spreading mechanism · Toroidal vortex · Analytical velocity field

30.1 Introduction A theoretical framework for a toroidal vortex as a dust scattering center The citizens of Vitoria-ES city, in Brazil, constantly suffer from dust particles present in the air that are dispersed mainly from the local industries’ facilities [1] and from the civil construction industry [2]. The wind, as an external force agent (direction and speed), promotes the particles spreading. In the present work, we are studying the case of free-falling bulk material since this is the mechanism that originates dust spreading in many industries [3, 4], and this is the most reported experimental setup in literature [5–10]. The mechanism of free-falling bulk particles involves the relative movement between the falling particles and the cycling air [7, 11–13]. Also, due to the different H. Belich (B) · M. F. Martins · L. Amorim · E. Sarcinelli · R. X. A. Leão Laboratory for Computational Transport Phenomena (LFTC), The Federal University of Espirito Santo, Vitoria, Brazil e-mail: [email protected] M. F. Martins e-mail: [email protected] E. Sarcinelli e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Ferreira Martins et al. (eds.), Multiphase Flow Dynamics, ABCM Series on Mechanical Sciences and Engineering, https://doi.org/10.1007/978-3-030-93456-9_30

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Fig. 30.1 Velocity streamlines of airborne dust particles spreading caused by free-falling bulk materials, [10]

shapes and sizes, collisions among particles often occur. Some relevant matters of the dragging flow in a body of homogeneous fluid can be found in the field theory of plume flow [14–21] and free air jet theory in [22–24]. The first studies on free-falling material particles modeled the core of free-falling particles as a shower-water geometry. In this case, the entrained air volume flow rate caused by the falling particles is the main variable investigated through various methods leading to its determination [6–9]. For the entrained air zone that wraps the dense particle core, Liu et al. [5] developed an experimental study providing an interesting understanding of the physical process and concluded that the air-entrained velocity could be modeled as a Gaussian or uniform distribution for small or large drop-heights, respectively. In our case, to understand the dynamics of dust dispersion, we assume that the core of the falling particles provides the necessary drag for the lighter particles outside the core to disperse [10], see Fig. 30.1. Therefore, we consider that particles capable of moving along with the wind (a mixed fluid-like [25], thereby behaving as an incompressible fluid) are the ones that escape when the particles fall vertically from a conveyor. Then, only thin-sized particles are considered components of the dustformed fluid. The particles smaller than 5 µm in diameter are the main components [11], which is an appropriated particle size for the treatment of the incompressible fluid assumption [26, 27]. Toward the recent contributions of the continuum modeling of granular flows, by doing computer simulations [10], we have concluded that there is a dispersing center formed by a velocity field that can be approached by toroidal vortex geometry. Flows influenced by a torus presence are not only an intrinsic theoretical extension in the field of classical fluid mechanics. In the present work, it acquires potential applicability to dust propagation. It is of theoretical interest to investigate whether this geometric nature of the torus gives rise to different flow interactions that are

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not yet reported for dust-spreading dynamics. Therefore, the paper’s objective is to obtain the velocity field of the toroidal vortex as a contribution of two vortices with different orientations. For that, the theory of vertical vortices [28] is adapted as a start methodology to describe the flow pattern formed around the core of free-falling particles. The structure of this paper is as follows. In Sect. 30.2, we investigate the necessary velocity field to generate a constant vorticity vector inside a vertical vortex. This allowed us to develop an analytical expression for the type of velocity field inside and outside this vortex; in Sect. 30.3, the vertical vortex extremities are joined by an axial symmetry. That leads to the fact that to find the velocity field generated by the toroid. It would be enough to analyze the field generated by two vortices with opposite orientations; Sect. 30.3 presents our conclusions.

30.2 Navier–Stokes Equation and Vorticity The modeling execution was subjected to the main assumption that the heavier particles participate indirectly in fluid transport. As the core falls, a toroidal vortex appears around it. To arrive at the solution that describes these toroidal vortices that cause more dispersion in free fall, we begin our study by describing the movement of a fluid by the Navier–Stokes equation, Eq. 30.1: d v = −∇ P + νv. dt

(30.1)

Equation 30.1 written with index notation becomes ∂t vα + vβ ∂β vα = −∂α P + f α + ν∂ 2 vα

(30.2)

∂α vα = 0, α = 1, 2, 3.

(30.3)

In Eqs. 30.2 and 30.3, vα is the velocity field, P is the pressure, f α is an external gravitational force and ν is the viscosity. The role of pressure is to ensure the link of incompressibility, given by Eq. 30.3. Applying the divergent operator to the Navier– Stokes equation and using the fluid incompressibility link, we have ∂t vα + αγ vβ ∂β vγ = αγ f γ + ν∂ 2 vα

(30.4)

where αγ = δαγ − ∂α ∂γ /∂ 2 is the projector over transverse modes. The nonlinear term of (Eq. 30.4) is associated with the convention, representing interactions between vortices at different lengths of scale, and the term viscosity is responsible for the dissipation of energy through diffusion. In this line, we can also write the Helmholtz equation Eq. 30.5

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∂t ωα + vβ ∂β ωα − ωβ ∂β vα = v∂ 2 ωα + αβλ ∂β f λ ,

(30.5)

ωα = αβλ ∂β vλ ,

(30.6)

where we define → In recent years, the vorticity − ω has established itself as the major object of study in the analysis of turbulent phenomena in fluids. It is worth remembering the relationship between vorticity and circulation, which, according to Stokes’ theorem can be written as follows: 

− → v · dx=

= c

where



˙ → − → ω · d− s,

(30.7)

s

− → → − → ω = ∇ ×− v.

(30.8)

The so-called vortex lines represent the field lines formed by the vorticity field, − → ω , which is similar to the magnetic field, with zero-divergence and always tending to closed loops, as can be obtained from the definition of Eq. 30.8. However, it is common in the literature to deal with vortex filament models. The study of vortex dynamics has been one of the main targets for understanding turbulence [29]. In this sense, it is essential to mention the works of Chorin, with his kinematic models [30–33], and also contributions from Saffman’s vortex filament models and others [29, 34].

30.2.1 Vortex in the Z-Axis To do a more didactic approach to the problem, we will quickly solve the velocity field of a stationary vortex with the axis of symmetry on the Z -axis [35]. A vortex with this configuration will have a constant ω vorticity vector, located inside a cylinder, which is a core of the vortex. We need to remember the rotational formula in cylindrical coordinates   1 ∂(r vθ ) ∂vr − zˆ , (30.9) ωz = r ∂r ∂θ In our case, we only have the component vθ not null, so   1 ∂(r vθ ) zˆ , ωz = r ∂r

(30.10)

A stationary vortex to have this configuration must have the following functional form at its core: r |ω z | vθ = , (30.11) 2

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In the outer part of the cylinder, using Stokes’ theorem (for this we will establish the vorticity flow as = ω z S = π (r0 )2 |ω z |), we will have the form vθ =

, 2πr

(30.12)

Taking the equation into account (Eq. 30.2) with f α = 0, we can rewrite it as    1− − → → − − → v2 → → v, + ∇ ×− v ×→ v + ∇ P = ν∇ 2 − ∂t v + ∇ 2 ρ

(30.13)

− → applying the differential operator ∇ × in Eq. 30.13, Eq. 30.14 is found.  − →

→ ∂t ω + ∇ × ω × − v = ν∇ 2 ω,

(30.14)

In the particular case developing here, ω = |ω z | zˆ and the velocity field stays in

 − → → the X Y plane, then ω ⊥ v and ω × − v = f (r )ˆr is radial. The expression ∇ × f (r )ˆr = 0 therefore, Eq. 30.14 becomes ∂t ω = ν∇ 2 ω,

(30.15)

A detailed discussion is analyzed in the Ref. [36] and here we present the solution of this equation as:    r2

1 − exp − , (30.16) vθ = 2πr 4νt In this way, we have a correction of (Eq. 30.12), taking into account the dissipation. In Fig. 30.2, it is shown that the vertical vortex, with the vorticity vector ω, is in Z direction. Noting that,    

r2 1 ∂(r vθ ) zˆ = exp − zˆ , (30.17) ω= r ∂r 4π νt 4νt and from Eq. 30.17, we can verify that Eq. 30.15 is the actual solution which we were searching. In the next section, we will arrive at the establishment of the velocity field for a vertical vortex and then build the velocity field of a toroidal vortex.

30.2.2 Toroidal Vortex To find the velocity field v(X, Y, Z ) of the toroidal vortex, we will use the Ref. [30]. Let’s put the axis of symmetry of the toroid on the axis Z , that is, in the plane (X, Y ) perpendicular to the Z -axis.

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wZ

Fig. 30.2 The vortice is shown with the vorticity in the Z direction

vθ

Z X

r0 − → → → Let’s put the vector − ω = ∇ ×− v tangent to the circumference in the Z = 0 plane, with center in (0, 0). As the system has axial symmetry, we can analyze only what happens in theX Z plane and then rotate around the Z axis and describe the velocity field in the entire space. As we already have the shape of a velocity field for a vortex, it is enough to make the vectorial sum of two vortices of the same intensity in the X Z plane with opposite orientations. As it appears in Fig. 30.3, we have a → vectorial sum of the two sources of circulation vector fields − v . To exemplify the velocity field in the external part of the toroid, in the point (0, 0, 0), we have

vθ = − π R0



  R02 1 − exp − , 4νt

(30.18)

In Fig. 30.3, we sketch the curves of the velocity field external to the toroid. So we have two distinct regions in the configuration of field lines: • the region internal to the toroid that traps any dust particles, • and the outer region would make the dispersion of particles in the process of falling particulate material. Therefore, the general expression of the total velocity vector field is given by Eq. 30.19.  vθ = t

1 2π R1

       R2 R2 1 1 − exp − 1 1 − exp − 2 θˆ1 + θˆ2 . 4νt 2π R2 4νt (30.19)

30 A Theoretical Framework for a Toroidal Vortex …

353 Longitudinal plane Z

Z e

lan

tp

Cu

r0 X

v

X

-Z

X

θ2 R2

θ1

v2

φ

v1

R1

v

R0

ω

Y Fig. 30.3 The toroidal vortice is shown and a cut in the XZ plane with an external point showing the vector sum of the contributions of two vortices in the XZ plane

In Eq. 30.19, we called the vortex of side left of 1 and the side right of 2. Then (R1 , θ1 ) and (R2 , θ2 ) are the coordinates in relation of the two vortices, and t = ωθ S = π (r0 )2 ωθ is the flux of vorticity in the inner part of the toroid. To generate the entire vector field, it is enough to remember the cylindrical symmetry and rotating the field in the X Z plan by π around the Z axis.

30.3 Conclusions To understand the dispersion of dust caused by the free-falling of granular material, we have observed that the dust dispersion is made by forming a toroidal vortex. We have then studied a vertical vortex to carry out an analytical study of the velocity field created by this vortex. The field generated by this vortex is found, and then we understand what velocities field is necessary to develop a vertical vortex and then obtain the velocity field of such vortex. The dissipation effect is taken into account by the vorticity equation (Eq. 30.15), and we find out the solution by the velocity field expression (Eq. 30.16). Then by this study, we can stipulate the velocity field of a toroidal vortex by the contribution of two vortices with different orientations. The analytical expression is found out through a cut in the Y Z plane. Then, we could

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find the velocity field of the toroidal vortex through the contribution of two vortices with the opposite rotation speed field.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36.

E.Z. Monte, Ciência e Natura 38(2), 588 (2016) J. Santos, H. Lisboa, A. Leon, T. Albuquerque, N. Reis Jr, V. Reisen, (2018) R. Junemann, R. Holzhauer, Bulk Solids Handl. 12(2), 217 (1992) S. Chakravarty, M. Fischer, P. García-Triñanes, D. Parker, O. Le Bihan, M. Morgeneyer, Powder Technol. 322, 54 (2017) Z. Liu, P. Wypych, P. Cooper, Powder Handl. Process. 11(4), 421 (1999) R. Ansart, A. De Ryck, J.A. Dodds, M. Roudet, D. Fabre, F. Charru, Powder Technol. 190(1–2), 274 (2009) R. Ansart, J.J. Letourneau, A. De Ryck, J.A. Dodds, Powder Technol. 212(3), 418 (2011) H. Sun, A. Li, J. Wu, Part. Sci. Technol. 0(0), 1 (2019). https://doi.org/10.1080/02726351. 2019.1666948 K. Chu, Y. Wang, Q. Zheng, A. Yu, R. Pan, Powder Technol. 361, 836 (2020) R.X. de Almeida Leão, L.S. Amorim, M.F. Martins, H.B. Junior, E. Sarcinelli, A.L.A. Mesquita, Powder Technol. (2020). https://doi.org/10.1016/j.powtec.2020.05.087 P. Cooper, P. Arnold, KONA Powder Part. J. 13, 125 (1995) L. Chen, P. Li, G. Liu, W. Cheng, Z. Liu, J. Loss Prev. Process Ind. 55, 232 (2018) J. Wu, G. Li, Z. Luo, M. Zhu, X. Wu, Appl. Opt. 57(7), B59 (2018) G.K. Batchelor, The Theory of Homogeneous Turbulence (Cambridge university press, 1953) B. Morton, G.I. Taylor, J.S. Turner, Proceedings of the royal society of London. Ser. A, Math. Phys. Sci. 234(1196), 1 (1956) G. Rooney, P. Linden, J. Fluid Mech. 318, 237 (1996) T.J. McDougall, J. Fluid Mech. 85(4), 655 (1978) J. Milgram, J. Fluid Mech. 133, 345 (1983) B. Gebhart, Y. Jaluria, R.L. Mahajan, B. Sammakia, (1988) S. Kakac, W. Aung, R. Viskanta, Washington, DC, Hemisphere Publishing Corp., 1985, 1191 p. No individual items are abstracted in this volume (1985) S. Ghosh, J. Hunt, J. Phillips, P. Miller, Appl. Sci. Res. 51(1–2), 257 (1993) F.P. Ricou, D. Spalding, J. Fluid Mech. 11(1), 21 (1961) H.B. Fischer, J.E. List, C.R. Koh, J. Imberger, N.H. Brooks, Mixing in Inland and Coastal Waters (Elsevier, 2013) E. List, Annu. Rev. Fluid Mech. 14(1), 189 (1982) C.T. Crowe, J.D. Schwarzkopf, M. Sommerfeld, Y. Tsuji, Multiphase Flows with Droplets and Particles (CRC press, 2011) D.G. Schaeffer, J. Differ. Equ. (Print) 66(1), 19 (1987) T. Barker, J. Gray, J. Fluid Mech. 828, 5 (2017) Y.l. Chen, C. Wu, M. Ye, X.m. Ju, J. Hydrodyn. 19(2), 143 (2007) D. Pullin, P. Saffman, Annu. Rev. Fluid Mech. 30(1), 31 (1998) A.J. Chorin, Vorticity and Turbulence, vol. 103 (Springer Science & Business Media, 2013) A.J. Chorin, Commun. Math. Phys. 83(4), 517 (1982) A.J. Chorin, J. Comput. Phys. 91(1), 1 (1990) A.J. Chorin, Phys. Rev. Lett. 60(19), 1947 (1988) D.W. Moore, P.G. Saffman, Proceedings of the royal society of London. A. Math. Phys. Sci. 338(1615), 535 (1974) M. Cheng, J. Lou, T. Lim, Phys. Fluids 22(9), 097101 (2010) R. Prud’homme, Flows of Reactive Fluids, vol. 94 (Springer Science & Business Media, 2010)