Computational Methods in Multiphase Flow V (WIT Transactions on the Engineering Sciences) (Wit Transactions on Engineering Sciences)

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COMPUTATIONAL METHODS IN MULTIPHASE FLOW V

WIT Press publishes leading books in Science and Technology. Visit our website for new and current list of titles. www.witpress.com

WITeLibrary Home of the Transactions of the Wessex Institute. Papers presented at Multiphase Flow V are archived in the WIT eLibrary in volume 56 of WIT Transactions on Engineering Sciences (ISSN 1743-3533). The WIT eLibrary provides the international scientific community with immediate and permanent access to individual papers presented at WIT conferences. http://library.witpress.com

FIFTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL AND EXPERIMENTAL METHODS IN MULTIPHASE AND COMPLEX FLOW

MULTIPHASE FLOW V CONFERENCE CHAIRMEN A.A. Mammoli University of New Mexico, USA C.A. Brebbia Wessex Institute of Technology, UK

INTERNATIONAL SCIENTIFIC ADVISORY COMMITTEE J. Adilson de Castro M. Asuaje A. Doinikov R. Groll R. Klasinc C. Koenig N. Mahinpey J. Mls P. Montgomery A. Rychkov L. Skerget R. van der Sman Y. Yan M. Zadravec

Organised by Wessex Institute of Technology, UK and University of New Mexico, USA Sponsored by WIT Transactions on Engineering Sciences

WIT Transactions Transactions Editor Carlos Brebbia Wessex Institute of Technology Ashurst Lodge, Ashurst Southampton SO40 7AA, UK Email: [email protected]

Editorial Board B Abersek University of Maribor, Slovenia Y N Abousleiman University of Oklahoma, USA P L Aguilar University of Extremadura, Spain K S Al Jabri Sultan Qaboos University, Oman E Alarcon Universidad Politecnica de Madrid, Spain A Aldama IMTA, Mexico C Alessandri Universita di Ferrara, Italy D Almorza Gomar University of Cadiz, Spain B Alzahabi Kettering University, USA J A C Ambrosio IDMEC, Portugal A M Amer Cairo University, Egypt S A Anagnostopoulos University of Patras, Greece M Andretta Montecatini, Italy E Angelino A.R.P.A. Lombardia, Italy H Antes Technische Universitat Braunschweig, Germany M A Atherton South Bank University, UK A G Atkins University of Reading, UK D Aubry Ecole Centrale de Paris, France H Azegami Toyohashi University of Technology, Japan A F M Azevedo University of Porto, Portugal J Baish Bucknell University, USA J M Baldasano Universitat Politecnica de Catalunya, Spain J G Bartzis Institute of Nuclear Technology, Greece A Bejan Duke University, USA

M P Bekakos Democritus University of Thrace, Greece G Belingardi Politecnico di Torino, Italy R Belmans Katholieke Universiteit Leuven, Belgium C D Bertram The University of New South Wales, Australia D E Beskos University of Patras, Greece S K Bhattacharyya Indian Institute of Technology, India E Blums Latvian Academy of Sciences, Latvia J Boarder Cartref Consulting Systems, UK B Bobee Institut National de la Recherche Scientifique, Canada H Boileau ESIGEC, France J J Bommer Imperial College London, UK M Bonnet Ecole Polytechnique, France C A Borrego University of Aveiro, Portugal A R Bretones University of Granada, Spain J A Bryant University of Exeter, UK F-G Buchholz Universitat Gesanthochschule Paderborn, Germany M B Bush The University of Western Australia, Australia F Butera Politecnico di Milano, Italy J Byrne University of Portsmouth, UK W Cantwell Liverpool University, UK D J Cartwright Bucknell University, USA P G Carydis National Technical University of Athens, Greece J J Casares Long Universidad de Santiago de Compostela, Spain, M A Celia Princeton University, USA A Chakrabarti Indian Institute of Science, India

A H-D Cheng University of Mississippi, USA J Chilton University of Lincoln, UK C-L Chiu University of Pittsburgh, USA H Choi Kangnung National University, Korea A Cieslak Technical University of Lodz, Poland S Clement Transport System Centre, Australia M W Collins Brunel University, UK J J Connor Massachusetts Institute of Technology, USA M C Constantinou State University of New York at Buffalo, USA D E Cormack University of Toronto, Canada M Costantino Royal Bank of Scotland, UK D F Cutler Royal Botanic Gardens, UK W Czyczula Krakow University of Technology, Poland M da Conceicao Cunha University of Coimbra, Portugal A Davies University of Hertfordshire, UK M Davis Temple University, USA A B de Almeida Instituto Superior Tecnico, Portugal E R de Arantes e Oliveira Instituto Superior Tecnico, Portugal L De Biase University of Milan, Italy R de Borst Delft University of Technology, Netherlands G De Mey University of Ghent, Belgium A De Montis Universita di Cagliari, Italy A De Naeyer Universiteit Ghent, Belgium W P De Wilde Vrije Universiteit Brussel, Belgium L Debnath University of Texas-Pan American, USA N J Dedios Mimbela Universidad de Cordoba, Spain G Degrande Katholieke Universiteit Leuven, Belgium S del Giudice University of Udine, Italy G Deplano Universita di Cagliari, Italy I Doltsinis University of Stuttgart, Germany M Domaszewski Universite de Technologie de Belfort-Montbeliard, France J Dominguez University of Seville, Spain

K Dorow Pacific Northwest National Laboratory, USA W Dover University College London, UK C Dowlen South Bank University, UK J P du Plessis University of Stellenbosch, South Africa R Duffell University of Hertfordshire, UK A Ebel University of Cologne, Germany E E Edoutos Democritus University of Thrace, Greece G K Egan Monash University, Australia K M Elawadly Alexandria University, Egypt K-H Elmer Universitat Hannover, Germany D Elms University of Canterbury, New Zealand M E M El-Sayed Kettering University, USA D M Elsom Oxford Brookes University, UK A El-Zafrany Cranfield University, UK F Erdogan Lehigh University, USA F P Escrig University of Seville, Spain D J Evans Nottingham Trent University, UK J W Everett Rowan University, USA M Faghri University of Rhode Island, USA R A Falconer Cardiff University, UK M N Fardis University of Patras, Greece P Fedelinski Silesian Technical University, Poland H J S Fernando Arizona State University, USA S Finger Carnegie Mellon University, USA J I Frankel University of Tennessee, USA D M Fraser University of Cape Town, South Africa M J Fritzler University of Calgary, Canada U Gabbert Otto-von-Guericke Universitat Magdeburg, Germany G Gambolati Universita di Padova, Italy C J Gantes National Technical University of Athens, Greece L Gaul Universitat Stuttgart, Germany A Genco University of Palermo, Italy N Georgantzis Universitat Jaume I, Spain P Giudici Universita di Pavia, Italy F Gomez Universidad Politecnica de Valencia, Spain R Gomez Martin University of Granada, Spain D Goulias University of Maryland, USA

K G Goulias Pennsylvania State University, USA F Grandori Politecnico di Milano, Italy W E Grant Texas A & M University, USA S Grilli University of Rhode Island, USA R H J Grimshaw, Loughborough University, UK D Gross Technische Hochschule Darmstadt, Germany R Grundmann Technische Universitat Dresden, Germany A Gualtierotti IDHEAP, Switzerland R C Gupta National University of Singapore, Singapore J M Hale University of Newcastle, UK K Hameyer Katholieke Universiteit Leuven, Belgium C Hanke Danish Technical University, Denmark K Hayami National Institute of Informatics, Japan Y Hayashi Nagoya University, Japan L Haydock Newage International Limited, UK A H Hendrickx Free University of Brussels, Belgium C Herman John Hopkins University, USA S Heslop University of Bristol, UK I Hideaki Nagoya University, Japan D A Hills University of Oxford, UK W F Huebner Southwest Research Institute, USA J A C Humphrey Bucknell University, USA M Y Hussaini Florida State University, USA W Hutchinson Edith Cowan University, Australia T H Hyde University of Nottingham, UK M Iguchi Science University of Tokyo, Japan D B Ingham University of Leeds, UK L Int Panis VITO Expertisecentrum IMS, Belgium N Ishikawa National Defence Academy, Japan J Jaafar UiTm, Malaysia W Jager Technical University of Dresden, Germany Y Jaluria Rutgers University, USA C M Jefferson University of the West of England, UK P R Johnston Griffith University, Australia

D R H Jones University of Cambridge, UK N Jones University of Liverpool, UK D Kaliampakos National Technical University of Athens, Greece N Kamiya Nagoya University, Japan D L Karabalis University of Patras, Greece M Karlsson Linkoping University, Sweden T Katayama Doshisha University, Japan K L Katsifarakis Aristotle University of Thessaloniki, Greece J T Katsikadelis National Technical University of Athens, Greece E Kausel Massachusetts Institute of Technology, USA H Kawashima The University of Tokyo, Japan B A Kazimee Washington State University, USA S Kim University of Wisconsin-Madison, USA D Kirkland Nicholas Grimshaw & Partners Ltd, UK E Kita Nagoya University, Japan A S Kobayashi University of Washington, USA T Kobayashi University of Tokyo, Japan D Koga Saga University, Japan A Konrad University of Toronto, Canada S Kotake University of Tokyo, Japan A N Kounadis National Technical University of Athens, Greece W B Kratzig Ruhr Universitat Bochum, Germany T Krauthammer Penn State University, USA C-H Lai University of Greenwich, UK M Langseth Norwegian University of Science and Technology, Norway B S Larsen Technical University of Denmark, Denmark F Lattarulo, Politecnico di Bari, Italy A Lebedev Moscow State University, Russia L J Leon University of Montreal, Canada D Lewis Mississippi State University, USA S lghobashi University of California Irvine, USA K-C Lin University of New Brunswick, Canada A A Liolios Democritus University of Thrace, Greece

S Lomov Katholieke Universiteit Leuven, Belgium J W S Longhurst University of the West of England, UK G Loo The University of Auckland, New Zealand J Lourenco Universidade do Minho, Portugal J E Luco University of California at San Diego, USA H Lui State Seismological Bureau Harbin, China C J Lumsden University of Toronto, Canada L Lundqvist Division of Transport and Location Analysis, Sweden T Lyons Murdoch University, Australia Y-W Mai University of Sydney, Australia M Majowiecki University of Bologna, Italy D Malerba Università degli Studi di Bari, Italy G Manara University of Pisa, Italy B N Mandal Indian Statistical Institute, India Ü Mander University of Tartu, Estonia H A Mang Technische Universitat Wien, Austria, G D, Manolis, Aristotle University of Thessaloniki, Greece W J Mansur COPPE/UFRJ, Brazil N Marchettini University of Siena, Italy J D M Marsh Griffith University, Australia J F Martin-Duque Universidad Complutense, Spain T Matsui Nagoya University, Japan G Mattrisch DaimlerChrysler AG, Germany F M Mazzolani University of Naples “Federico II”, Italy K McManis University of New Orleans, USA A C Mendes Universidade de Beira Interior, Portugal, R A Meric Research Institute for Basic Sciences, Turkey J Mikielewicz Polish Academy of Sciences, Poland N Milic-Frayling Microsoft Research Ltd, UK R A W Mines University of Liverpool, UK C A Mitchell University of Sydney, Australia

K Miura Kajima Corporation, Japan A Miyamoto Yamaguchi University, Japan T Miyoshi Kobe University, Japan G Molinari University of Genoa, Italy T B Moodie University of Alberta, Canada D B Murray Trinity College Dublin, Ireland G Nakhaeizadeh DaimlerChrysler AG, Germany M B Neace Mercer University, USA D Necsulescu University of Ottawa, Canada F Neumann University of Vienna, Austria S-I Nishida Saga University, Japan H Nisitani Kyushu Sangyo University, Japan B Notaros University of Massachusetts, USA P O’Donoghue University College Dublin, Ireland R O O’Neill Oak Ridge National Laboratory, USA M Ohkusu Kyushu University, Japan G Oliveto Universitá di Catania, Italy R Olsen Camp Dresser & McKee Inc., USA E Oñate Universitat Politecnica de Catalunya, Spain K Onishi Ibaraki University, Japan P H Oosthuizen Queens University, Canada E L Ortiz Imperial College London, UK E Outa Waseda University, Japan A S Papageorgiou Rensselaer Polytechnic Institute, USA J Park Seoul National University, Korea G Passerini Universita delle Marche, Italy B C Patten, University of Georgia, USA G Pelosi University of Florence, Italy G G Penelis, Aristotle University of Thessaloniki, Greece W Perrie Bedford Institute of Oceanography, Canada R Pietrabissa Politecnico di Milano, Italy H Pina Instituto Superior Tecnico, Portugal M F Platzer Naval Postgraduate School, USA D Poljak University of Split, Croatia V Popov Wessex Institute of Technology, UK H Power University of Nottingham, UK D Prandle Proudman Oceanographic Laboratory, UK

M Predeleanu University Paris VI, France M R I Purvis University of Portsmouth, UK I S Putra Institute of Technology Bandung, Indonesia Y A Pykh Russian Academy of Sciences, Russia F Rachidi EMC Group, Switzerland M Rahman Dalhousie University, Canada K R Rajagopal Texas A & M University, USA T Rang Tallinn Technical University, Estonia J Rao Case Western Reserve University, USA A M Reinhorn State University of New York at Buffalo, USA A D Rey McGill University, Canada D N Riahi University of Illinois at UrbanaChampaign, USA B Ribas Spanish National Centre for Environmental Health, Spain K Richter Graz University of Technology, Austria S Rinaldi Politecnico di Milano, Italy F Robuste Universitat Politecnica de Catalunya, Spain J Roddick Flinders University, Australia A C Rodrigues Universidade Nova de Lisboa, Portugal F Rodrigues Poly Institute of Porto, Portugal C W Roeder University of Washington, USA J M Roesset Texas A & M University, USA W Roetzel Universitaet der Bundeswehr Hamburg, Germany V Roje University of Split, Croatia R Rosset Laboratoire d’Aerologie, France J L Rubio Centro de Investigaciones sobre Desertificacion, Spain T J Rudolphi Iowa State University, USA S Russenchuck Magnet Group, Switzerland H Ryssel Fraunhofer Institut Integrierte Schaltungen, Germany S G Saad American University in Cairo, Egypt M Saiidi University of Nevada-Reno, USA R San Jose Technical University of Madrid, Spain F J Sanchez-Sesma Instituto Mexicano del Petroleo, Mexico

B Sarler Nova Gorica Polytechnic, Slovenia S A Savidis Technische Universitat Berlin, Germany A Savini Universita de Pavia, Italy G Schmid Ruhr-Universitat Bochum, Germany R Schmidt RWTH Aachen, Germany B Scholtes Universitaet of Kassel, Germany W Schreiber University of Alabama, USA A P S Selvadurai McGill University, Canada J J Sendra University of Seville, Spain J J Sharp Memorial University of Newfoundland, Canada Q Shen Massachusetts Institute of Technology, USA X Shixiong Fudan University, China G C Sih Lehigh University, USA L C Simoes University of Coimbra, Portugal A C Singhal Arizona State University, USA P Skerget University of Maribor, Slovenia J Sladek Slovak Academy of Sciences, Slovakia V Sladek Slovak Academy of Sciences, Slovakia A C M Sousa University of New Brunswick, Canada H Sozer Illinois Institute of Technology, USA D B Spalding CHAM, UK P D Spanos Rice University, USA T Speck Albert-Ludwigs-Universitaet Freiburg, Germany C C Spyrakos National Technical University of Athens, Greece I V Stangeeva St Petersburg University, Russia J Stasiek Technical University of Gdansk, Poland G E Swaters University of Alberta, Canada S Syngellakis University of Southampton, UK J Szmyd University of Mining and Metallurgy, Poland S T Tadano Hokkaido University, Japan H Takemiya Okayama University, Japan I Takewaki Kyoto University, Japan C-L Tan Carleton University, Canada M Tanaka Shinshu University, Japan E Taniguchi Kyoto University, Japan

S Tanimura Aichi University of Technology, Japan J L Tassoulas University of Texas at Austin, USA M A P Taylor University of South Australia, Australia A Terranova Politecnico di Milano, Italy E Tiezzi University of Siena, Italy A G Tijhuis Technische Universiteit Eindhoven, Netherlands T Tirabassi Institute FISBAT-CNR, Italy S Tkachenko Otto-von-GuerickeUniversity, Germany N Tosaka Nihon University, Japan T Tran-Cong University of Southern Queensland, Australia R Tremblay Ecole Polytechnique, Canada I Tsukrov University of New Hampshire, USA R Turra CINECA Interuniversity Computing Centre, Italy S G Tushinski Moscow State University, Russia J-L Uso Universitat Jaume I, Spain E Van den Bulck Katholieke Universiteit Leuven, Belgium D Van den Poel Ghent University, Belgium R van der Heijden Radboud University, Netherlands R van Duin Delft University of Technology, Netherlands P Vas University of Aberdeen, UK W S Venturini University of Sao Paulo, Brazil

R Verhoeven Ghent University, Belgium A Viguri Universitat Jaume I, Spain Y Villacampa Esteve Universidad de Alicante, Spain F F V Vincent University of Bath, UK S Walker Imperial College, UK G Walters University of Exeter, UK B Weiss University of Vienna, Austria H Westphal University of Magdeburg, Germany J R Whiteman Brunel University, UK Z-Y Yan Peking University, China S Yanniotis Agricultural University of Athens, Greece A Yeh University of Hong Kong, China J Yoon Old Dominion University, USA K Yoshizato Hiroshima University, Japan T X Yu Hong Kong University of Science & Technology, Hong Kong M Zador Technical University of Budapest, Hungary K Zakrzewski Politechnika Lodzka, Poland M Zamir University of Western Ontario, Canada R Zarnic University of Ljubljana, Slovenia G Zharkova Institute of Theoretical and Applied Mechanics, Russia N Zhong Maebashi Institute of Technology, Japan H G Zimmermann Siemens AG, Germany

COMPUTATIONAL METHODS IN MULTIPHASE FLOW V EDITORS A.A. Mammoli University of New Mexico, USA C.A. Brebbia Wessex Institute of Technology, UK

Editors: A.A. Mammoli University of New Mexico, USA C.A. Brebbia Wessex Institute of Technology, UK Published by WIT Press Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK Tel: 44 (0) 238 029 3223; Fax: 44 (0) 238 029 2853 E-Mail: [email protected] http://www.witpress.com For USA, Canada and Mexico Computational Mechanics Inc 25 Bridge Street, Billerica, MA 01821, USA Tel: 978 667 5841; Fax: 978 667 7582 E-Mail: [email protected] http://www.witpress.com British Library Cataloguing-in-Publication Data A Catalogue record for this book is available from the British Library ISBN: 978-1-84564-188-7 ISSN: 1746-4471 (print) ISSN: 1743-3533 (on-line) The texts of the papers in this volume were set individually by the authors or under their supervision. Only minor corrections to the text may have been carried out by the publisher. No responsibility is assumed by the Publisher, the Editors and Authors for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. © WIT Press 2009 Printed in Great Britain by MPG Book Group. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the Publisher.

Preface Professionals in the energy, automotive, manufacturing, construction, food processing, and pharmaceuticals industries, to name a few, are frequently faced with problems associated with multiphase flow. Scientists in environmental sciences, biology, and medicine also encounter multiphase flows regularly. Despite the efforts of several generations of fluid mechanicsists who tackled multiphase flows, including Einstein in the early 20th century, a complete description of the behavior of multiphase materials still eludes us. Constitutive equations have been developed for some cases, and numerical models based on these have been successful at predicting phenomena such as particle migration in dense suspensions. Much of the inspiration for the development of models comes from experimental observation. In some cases, the phenomena of interest are recorded directly, while in other cases, simplified models of the flows are necessary to restrict and control the parameter space. Direct numerical simulation is becoming increasingly useful for providing insight to constitutive modelers, and in some cases even for calibrating models. However, it remains true that very specialized models are developed for individual areas. In the first part of this book, several examples of modeling approaches, each well suited to a particular application, are grouped together, to present a view of the breadth of the field. The second part of the book contains papers describing experimental observation of multiphase flows at scales ranging from rivers and lakes to laboratory experiments. Topics include turbulence, interfaces and fluidized beds. Experimental techniques, such as PIV and other imaging methods, are also represented. We are confident that conference participants will find a fruitful exchange of ideas, and that readers of this book will find many insights. The contents of this book reflect the quality of the submissions and the diligence of the reviewers, whom we wish to thank. The Editors New Forest, 2009

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Contents Section 1: Multiphase flow simulation 3D Eulerian simulation of a gas-solid bubbling fluidized bed: assessment of drag coefficient correlations E. Esmaili & N. Mahinpey ................................................................................... 3 Computational and experimental methods for the on-line measurement of the apparent viscosity of a crystal suspension C. Herman, F. Debaste, V. Halloin, T. Leyssens, A. Line & B. Haut ................ 15 Two-phase flow modelling within expansion and contraction singularities V. G. Kourakos, P. Rambaud, S. Chabane, D. Pierrat & J. M. Buchlin............ 27 Numerical simulation of gas-solid flow in ducts by CFD techniques R. K. Decker, D. Noriler, H. F. Meier & M. Mori ............................................. 45 Modelling of solidification of binary fluids with non-linear viscosity models T. Wacławczyk, D. Sternel & M. Schäfer........................................................... 55 A simulation of the coupled problem of magnetohydrodynamics and a free surface for liquid metals S. Golak & R. Przyłucki ..................................................................................... 67 Ammonia concentration analysis for the steam condenser by combining two phase flow CFD simulation with condensation and process simulation K. Karube, M. Maekawa, S. Lo & K. Mimura ................................................... 77

An approach to the prediction of wax and asphaltene deposition in a pipeline based on Couette device experimental data D. Eskin, J. Ratulowski, K. Akbarzadeh & T. Lindvig ....................................... 85 Large amplitude waves in a slug tracking scheme A. De Leebeeck & O. J. Nydal ........................................................................... 99 Numerical simulation of an intermediate sized bubble rising in a vertical pipe J. Hua, S. Quan & J. Nossen ........................................................................... 111 Modelling of stratified two phase flows using an interfacial area density model T. Höhne & C. Vallée ...................................................................................... 123 Multi-phase mixture modelling of nucleate boiling applied to engine coolant flows V. Pržulj & M. Shala........................................................................................ 135 On the application of Mesoscopic Eulerian Formalism to modulation of turbulence by solid phase Z. Zeren & B. Bédat......................................................................................... 147 VOF-based simulation of conjugate mass transfer from freely moving fluid particles A. Alke, D. Bothe, M. Kroeger, & H.-J.Warnecke ........................................... 157 Computational fluid dynamic as a feature to understand the heat and mass transfer in a vacuum tower K. Ropelato, A. V. Castro, W. O. Geraldelli & M. Mori.................................. 169 Understanding segregation and mixing effects in a riser using the quadrature method of moments A. Dutta, J. Raeckelboom, G. J. Heynderickx & G. B. Marin.......................... 181 Numerical simulation of heavy oil flows in pipes using the core-annular flow technique K. C. O. Crivelaro, Y. T. Damacena, T. H. F. Andrade, A. G. B. Lima & S. R. Farias Neto................................................................... 193 Simulation of flow and modelling the residence time distribution in a continuous two impinging liquid-liquid streams reactor using the Monte Carlo Technique M. Sohrabi & E. Rajaie ................................................................................... 205

Section 2: Interaction of gas, liquids and solids Vortex study on a hydraulic model of Godar-e-Landar Dam and Hydropower Plant R. Roshan, H. Sarkardeh & A. R. Zarrati ........................................................ 217 Minimum fluidization velocity, bubble behaviour and pressure drop in fluidized beds with a range of particle sizes B. M. Halvorsen & B. Arvoh............................................................................ 227 Section 3: Turbulent flow Mathematical modelling on particle diffusion in fluidised beds and dense turbulent two-phase flows R. Groll ............................................................................................................ 241 A numerical study of the scale effects affecting the evolution and sediment entrainment capacity of a gravity current, propagating over a loose bed containing large-scale roughness elements T. Tokyay & G. Constantinescu ....................................................................... 251 Effect of relative motion between bubbles and surrounding liquid on the Reynolds stress as a mechanism controlling the radial gas holdup distribution K. Ueyama ....................................................................................................... 263 Velocity and turbulence measurements of oil-water flow in horizontal and slightly inclined pipes using PIV W. A. S. Kumara, B. M. Halvorsen & M. C. Melaaen ..................................... 277 Section 4: Environmental multiphase flow Meandering of a particle-laden rivulet P. Vorobieff, A. Mammoli, J. Coonrod, V. Putkaradze & K. Mertens ............. 295 Experimental study on the rheological behaviour of debris flow material in the Campania region A. Scotto di Santolo, A. M. Pellegrino & A. Evangelista................................. 305 Experimental and numerical investigation of mixed flow in a gallery S. Erpicum, F. Kerger, P. Archambeau, B. J. Dewals & M. Pirotton.............. 317

Sediment transport via dam-break flows over sloping erodible beds M. Emmett & T. B. Moodie.............................................................................. 329 Section 5: Bubble and drop dynamics Hydrodynamic drag and velocity of micro-bubbles in dilute paper machine suspensions A. Haapala, M. Honkanen, H. Liimatainen, T. Stoor & J. Niinimäki.............. 343 Effects of physical properties on the behaviour of Taylor bubbles V. Hernandez-Perez, L. A. Abdulkareem & B. J. Azzopardi............................ 355 Numerical simulation of one-dimensional mixed flow with air/water interaction F. Kerger, S. Erpicum, P. Archambeau, B. J. Dewals & M. Pirotton.............. 367 Geometry effects on the interaction of two equal-sized drops in simple shear flow at finite Reynolds numbers S. Mortazavi & M. Bayareh ............................................................................. 379 Section 6: Flow in porous media Modelling the tide effects in groundwater J. Mls ............................................................................................................... 391 Modelling of diffusion in porous structures E. du Plessis & S. Woudberg ........................................................................... 399 Measurement and prediction for air flow drag in different packing materials C. Rautenbach, B. M. Halvorsen, E. du Plessis, S. Woudberg & J. P. du Plessis ....................................................................... 409 CFD simulation with multiphase flows in porous media and open mineral storage pile S. Torno, J. Toraño, I. Diego, M. Menéndez, M. Gent & J. Velasco ............... 421 Powered addition applied to the fluidisation of a packed bed P. D. de Wet, B. M. Halvorsen & J. P. du Plessis............................................ 431 Section 7: Heat transfer Tube bundle’s cooling by aqueous foam J. Gylys, S. Sinkunas, T. Zdankus, M. Gylys & R. Maladauskas ..................... 445

Desulfurization of heavy crude oil by microwave irradiation A. Miadonye, S. Snow, D. J. G. Irwin, M. Rashid Khan & A. J. Britten .......... 455 Section 8: Image processing Reconstruction of a three-dimensional bubble surface from high-speed orthogonal imaging of dilute bubbly flow M. Honkanen ................................................................................................... 469 Experimental investigation on air entrainment below impinging jets by means of video observations and image processing D. V. Danciu, M. J. da Silva, M. Schmidtke, D. Lucas & U. Hampel.............. 481 Section 9: Interfacial behaviour LBM simulation of interfacial behaviour of bubbles flow at low Reynolds number in a square microchannel Y. Y. Yan & Y. Q. Zu ........................................................................................ 495 Experimental investigation of a 2d impinging jet on a liquid surface R. Berger, S. Depardon, P. Rambaud & J. M. Buchlin.................................... 507 Author Index .................................................................................................. 521


Section 1 Multiphase flow simulation


Computational Methods in Multiphase Flow V

3

3D Eulerian simulation of a gas-solid bubbling fluidized bed: assessment of drag coefficient correlations E. Esmaili & N. Mahinpey Department of Chemical and Petroleum Engineering, Schulich School of Engineering, University of Calgary, Canada

Abstract Fluidized beds have been widely used in power generation and in the chemical, biochemical, and petroleum industries. The 3D simulation of commercial scale fluidized beds has been computationally impractical due to the required memory and processor speeds. However, in this study, 3D Computational Fluid Dynamics simulation of a gas-solid bubbling fluidized bed is performed to investigate the effect of using different inter-phase drag models. The drag correlations of Syamlal–O’Brien, Gidaspow, and Wen–Yu are reviewed using a multiphase Eulerian–Eulerian model to simulate the momentum transfer between phases. Comparisons are made with both a 2D Cartesian simulation and experimental data. The experiments are performed on a Plexiglas rectangular fluidized bed consisting of spherical glass beads and ambient air as the gas phase. The aim of this work is to present an optimum drag model to simulate the momentum transfer between phases and compare the results using 3D versus 2D simulation of gas-solid bubbling fluidized beds. Comparisons were made based on solid volume fractions, expansion height, and pressure drop inside the fluidized bed at different superficial gas velocities. The results were found to agree well with experimental data. Keywords: multiphase flow, fluidized bed, computational fluid dynamics, interphase drag model.

1

Introduction

Two approaches are typically used for Computational Fluid Dynamics (CFD) modeling of gas–solid fluidized beds. The first one is Lagrangian–Eulerian WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090011

4 Computational Methods in Multiphase Flow V modeling [1,2,7,10–12], which solves the equations of motion for each particle individually and uses a continuous interpenetrating model (Eulerian framework) for modeling the gas phase. As a consequence, the Lagrangian–Eulerian model requires large computational resources for large systems of particles. The second method is Eulerian–Eulerian modeling [4,9,12–16], which assumes both phases can be considered as fluid and also considers the interpenetrating effect of each phase by using drag models. Therefore, applying a proper drag model in Eulerian–Eulerian modeling is of a great importance. Many researchers have applied 2D Cartesian simulations to model pseudo-2D beds [1,4,6,15]. Peirano et al. [5] have investigated the importance of three dimensionality in the Eulerian approach simulations of stationary bubbling fluidized beds. The results of their simulations show that two-dimensional simulations should be used with caution and only for sensitivity analysis, whereas three-dimensional simulations are able to reproduce both the statics (bed height and spatial distribution of particles) and the dynamics (power spectrum of pressure fluctuations) of the bed. In addition, they believe that the issue of accurate prediction of the drag force (the force exerted by the gas on a single particle in a suspension) does not seem to be the most important problem when dealing with bubbling beds where accurate empirical correlations are available. In contrast, in the present work, it is found that using a proper drag model can increase the accuracy of results in 3D simulation of bubbling fluidized beds. Goldschmidt et al. [6] applied a two-dimensional multi-fluid Eulerian CFD model to study the influence of the coefficient of restitution on the hydrodynamics of dense gas-solid fluidized beds. They showed that, in order to obtain realistic bed dynamics from fundamental hydrodynamic models, it is of prime importance to correctly take the effect of energy dissipation due to nonideal particle-particle encounters into account. van Wachem et al. [8] implemented a CFD model for a free bubbling fluidized bed in the commercial code CFX of AEA Technology to verify experimentally Eulerian-Eulerian gassolid model simulations of bubbling fluidized beds with existing correlations for bubble size and bubble velocity. They concluded that smaller bubbles and a lower rise velocity are usually observed with 2D beds. Cammarata et al. [9] compared the bubbling behavior predicted by 2D and 3D simulations of a rectangular fluidized bed using commercial software, CFX. The bed expansion, bubble hold-up, and bubble size calculated from the 2D and 3D simulations were compared with the predictions obtained from the Darton equation [3]. A more realistic physical behavior model of fluidization was obtained using 3D simulations. They also indicate that 2D simulations could be used to conduct sensitivity analyses. Xie et al. [14] compared the results of 2D and 3D simulation of slugging, bubbling, and turbulent gas-solid fluidized beds. They also investigated the effect of using different coordinate systems. Their results show that there is a significant difference between 2D and 3D simulations, and only 3D simulations could predict the correct bed height and pressure spectra. Behjat et al. [15] applied a two dimensional CFD technique in order to investigate hydrodynamic and heat transfer phenomena. They conclude that a Eulerian-Eulerian model is WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


5

suitable for modeling industrial fluidized bed reactors. Their results indicate that considering two solid phases, particles with smaller diameters have lower volume fraction at the bottom of the bed and higher volume fraction at the top of the bed and also that the gas temperature increases as it moves upward in the reactor due to the heat of polymerization reaction leading to the higher temperatures at the top of the bed. Li et al. [16] conducted a three-dimensional numerical simulation of a single horizontal gas jet into a laboratory-scale cylindrical gas–solid fluidized bed. They proposed a scaled drag model and implemented it into the simulation of a fluidized bed of FCC particles. They also obtained the jet penetration lengths of different jet velocities and compared them with published experimental data, as well as with predictions of empirical correlations. No previous works in the literature have investigated the effect of using different drag models in 3D simulation of fluidized beds to present an optimum drag model for simulation of bubbling gas-solid fluidized beds. In this respect, the underlying objective of this study is to present an optimum drag model to simulate the momentum transfer between phases and compare the results using 3D versus 2D simulation of gas-solid bubbling fluidized beds.

Figure 1:

2

Geometry of 3D Plexiglas fluidized bed.

Experimental setup

Experiments were carried out in the Department of Chemical and Biological Engineering at the University of British Columbia. The Column is a psudo-2D Plexiglas of 1.2 m height, 0.28 m width, and 0.025 m thickness. Spherical glass beads of 250–300 μm diameter and density 2500 kg/m3 were fluidized with air at ambient conditions. Pressure drops were measured using three differential pressure transducers located at elevation 0.03, 0.3, and 0.6 m above the gas distributor, respectively. The static bed height of 0.4 m with a solid volume fraction of 0.6 was used in all the experiments. Pressure drop and bed expansion were monitored at different superficial gas velocities ranging from 0 to 0.8 (m/s).

WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

6 Computational Methods in Multiphase Flow V

3

Governing equations

The proposed model’s equations, which are solved numerically, are presented in Table 1.

4

Numerical simulation

Governing equations of mass and momentum conservation are solved using finite volume method employing the Semi Implicit Method for Pressure Linked Equations (PC-SIMPLE) algorithm, which is an extension of the SIMPLE algorithm to multiphase flow. A multi fluid Eulerian-Eulerian model, which considers the conservation of mass and momentum for the gas and solid phases, was applied. The kinetic theory of granular flow, which considers the conservation of solid fluctuation energy, was used for closure of the solids stress terms. The three-dimensional (3D) geometry has been meshed using 336,000 structured rectangular cells. Volume fraction, density, and pressure are stored at the main grid points that are placed in the center of each control volume. A staggered grid arrangement is used, and the velocity components are solved at the control volume surfaces. A pressure correction equation is built based on total volume continuity. Pressure and velocities are then corrected so as to satisfy the continuity constraint. A grid sensitivity analysis is performed using different mesh sizes and 2 mm mesh interval spacing was chosen for all the simulation runs. Second-order upwind discretization schemes were used for discretizing the governing equations. Based on the estimation of the truncation error, an adaptive time-stepping algorithm with 100 iterations per each time step and a minimum value of order 10-5 for the lower domain of time step was used to ensure a stable convergence. The convergence criteria for other residual components associated with the relative error between two successive iterations has been specified in the order of 10-5. Three different drag models are studied in this work to simulate the momentum transfer between phases (Gidaspow, Syamlal–O’Brien, and Wen–Yu.). FLUENT 6.3 on a 20 AMD/Opteron 64bit processor Sun Grid Microsystems workstation W2100Z with 4 GB RAM is employed to solve the governing equations. Computational model parameters are listed in Table.2.

5

Results and discussion

Simulation results were compared with the experimental data in order to validate the model. Figure 2 shows the time average pressure drop inside the bed between two specific elevations (i.e. 0.03 m and 0.3 m as demonstrated in Fig. 1) for different studied cases and experimental results. In order to calculate the pressure at each pressure sensor (i.e. y=0.03 m), two kinds of averaging have been applied. The first one is the spatial averaging, which is the average value of pressure for all nodes in the plane of first pressure sensor (plane y=0.03). The second one is the time averaging of spatial-averaged pressure values in the period of 3-10 sec real time. As indicated in Fig. 2, the pressure drop for all the WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


Table 1:

Governing equations.

Continuity: .

0

.

0

Momentum equations: .

.

.

.

Solid pressure 1

2

1

Radial distribution function 1

,

Granular temperature

2 1 1

√

2

√ 3 3

1

1

3

8

1

1

5√

12 1 √

Stress tensors ,

. . . . . . 4 5

1

5√ 48 1

√

1


1

7


Table 1:

Continued.

4 3 Inter-phase momentum exchange Gidaspow [17]

1

150

.

.

.

24 . 1 0.15 . . 0.44

0.2 .

1000

.

1000

Syamlal and O’Brien [18] 3 4 0.63 0.5

0.06

4.8

0.06

0.12

2

. .

0.8

2.65

0.85 0.85

Wen Yu [19] 3 4

.

24 . 1 0.15 . . 0.44


.

1000

.

1000


Table 2:

Computational model parameters.

Parameter Particle density Gas density Mean particle diameter Initial solid packing Superficial gas velocity Bed dimension Static bed height Grid interval spacing Inlet boundary condition type Outlet boundary condition type Under-relaxation factors

Figure 2:

9

Value ⁄ 2500 ⁄ 1.225 275 0.6 ⁄ 11.7,21,38,46 1.2 0.025 0.4 0.002

0.28

Pressure Momentum Volume fraction Granular temperature

Pressure drop inside the bed

0.6 0.4 0.3 0.2

P  P 1

Z 0.03 m

 PZ 0.3m  .

models showed a declining trend with increase of the superficial gas velocity, which is in good qualitative agreement with the experimental data. It can be easily seen that 3D simulations show their superiority in predicting the pressure drop inside the bed compared with 2D simulation. Also, it can be concluded that the Gidaspow drag model in a 3D simulation will give results closer to the experimental data.


10 Computational Methods in Multiphase Flow V The experimental data of the time-average bed expansion ratio were compared with corresponding values predicted by the model, using Syamlal– O'Brien, Gidaspow, and Wen–Yu drag functions for various velocities as depicted in Figure 3. All drag models demonstrate a consistent increase in bed expansion with gas velocity and predict the bed expansion reasonably well. Figure 3 shows the considerable relative increase in bed expansion as the fluidizing velocity increases; a 5% increase was obtained at 0.11 m/s, a 20% increase at 0.21 m/s, and 42% at 0.38 m/s, and up to a 50% increase in bed height was measured at 0.46 m/s, the highest fluidized velocity investigated. It can be seen that using a 3D simulation, especially for lower superficial gas velocities, will increase the accuracy of the results. The reason can be the effect of participating governing equations in the z direction (depth of the bed) in simulating the fluid flow when the gas velocity increases. It is also seen that using the Gidaspow drag model in 3D simulation of a gas-solid fluidized bed will give better results for predicting the bed expansion ratio than the other two drag models.

Figure 3:

Comparison of simulated bed expansion ratio with experimental data.

Figure 4 shows a contour plot of solid volume fraction for the three drag models studied in this work for a superficial gas velocity of 0.38 m/sec at 10 sec real-time simulations. As can be observed from the plots, the Syamlal model represents the lowest bed expansion and gas void fraction. This fact could have been foreseen from the minimum fluidization velocity prediction of this model, which is almost five times larger than experimental data reported in the literature. The rest of the models showed approximately the same range of bed expansion. Expansion of the bed started with the formation of bubbles for all the



(a) Figure 4:

(b)

11

(c)

Contours of solid volume fraction at t=10s and u=0.38m/s (a) Syamlal (b) Wen Yu (c) Gidaspow.

models and eventually reached a statistically steady-state bed height. After this point, an unsteady chaotic generation of bubbles was observed after almost 3 seconds of real time simulation.

6

Conclusion

Numerical simulation of a bubbling gas-solid fluidized bed have been performed in a three dimensional solution domain using the Eulerian-Eulerian approach to investigate the effect of using three dimensional analysis versus two dimensional simulation of fluidized beds. FLUENT 6.3 was used to perform the calculations. The results show that although three-dimensional simulation takes more time and computing processors than two-dimensional simulation, it gives more accurate results when the models are compared with experimental data. Also, a comparison between three common drag models, Syamlal, Wen Yu, Gidaspow, was performed to develop an optimized drag model for simulation of momentum transfer between phases in a 3D bubbling gas-solid fluidized bed. It is concluded that the Gidaspow drag model achieved better results in predicting the bed expansion ratio and pressure drop inside the bed than Syamlal drag correlations. However, further modeling efforts are required to study the influence of using other drag models, which have not been studied, and optimizing existing drag models based on minimum fluidization velocity in three dimensional simulations will be performed in future work.

Nomenclature Single particle drag function, dimensionless Rate of strain tensor WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

12 Computational Methods in Multiphase Flow V Solid diameter, m Restitution coefficient, dimensionless Acceleration due to gravity, ⁄ Radial distribution coefficient, dimensionless Gas/solid momentum exchange coefficient, dimensionless Pressure, Pa Solid pressure, Pa Velocity, m/s Reynolds number, dimensionless Greek symbols Density, ⁄ ⁄ Granular temperature, Stress tensor, Pa Shear viscosity, ⁄ Bulk viscosity, ⁄ Volume fraction, dimensionless Subscripts Gas Solid

References [1] B.P.B. Hoomans, J.A.M. Kuipers, W.J. Briels, V.W.P.M. Swaaij, Discrete particle simulation of bubble and slug formulation in a two-dimensional gas-fluidized bed: a hard-sphere approach, Chem. Eng. Sci. 51 (1996) 99. [2] B. Xu, A. Yu, Numerical simulation of the gas–solid flow in a fluidized bed by combining discrete particle method with computational fluid dynamics, Chem. Eng. Sci. 52 (1997) 2785. [3] R.C. Darton, R.D. LaNauZe, J.F. Davidson, D. Harrison, Bubble growth due to coalescence in fluidized beds, Trans. Inst. Chem. Eng. 55 (1977) 274. [4] B.G.M. van Wachem, J.C. Schouterf, R. Krishnab, and C.M. van den Bleek, Eulerian Simulations of Bubbling Behavior in Gas-Solid Fluidized Beds, Computers chem. Engng Vol. 22, Suppl., pp. S 299-S306. 1998. [5] E. Peiranoa, V. Delloumea, B. Lecknera, Two- or three-dimensional simulations of turbulent gas–solid flows applied to fluidization, Chemical Engineering Science 56 (2001) 4787–4799. [6] M. J. V. Goldschmidt, J. A. M. Kuipers, W. P. M. van Swaaij, Hydrodynamic modeling of dense gas-fluidized beds using the kinetic theory of granular flow: effect of coefficient of restitution on bed dynamics, Chemical Engineering Science 56 (2001) 571-578. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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[7] K.D. Kafui, C. Thornton, M.J. Adams, Discrete particle-continuum fluid modeling of gas–solid fluidized beds, Chem. Eng. Sci. 57 (2002) 2395. [8] W.B.G.W. Wachem, A.E. Almstedt, Methods for multiphase computational fluid dynamics, Chem. Eng. J. 96 (2003) 81. [9] L. Cammarata, P. Lettieri, G.D.M. Micale, D. Colman, 2d and 3d cfd simulations of bubbling fluidized beds using Eulerian–Eulerian models, Int. J. Chem. Reactor Eng. 1 (2003) (Article A48). [10] M.J.V. Goldschmidt, R. Beetstra, J.A.M. Kuipers, Hydrodynamic modeling of dense gas-fluidized beds: comparison and validation of 3d discrete particle and continuum models, Powder Technol. 142 (2004) 23. [11] J.S. Curtis, V.B. Wachem, Modeling particle-laden flows: a research outlook, AIChE J. 50 (2004) 2638. [12] M. Chiesa, V. Mathiesen, J. A. Melheim, B. Halvorsen, Numerical simulation of particulate flow by the Eulerian–Lagrangian and the Eulerian–Eulerian approach with application to a fluidized bed. Comput. Chem. Eng. 29 (2005) 291. [13] J. Sun, F. Battaglia, Hydrodynamic modeling of particle rotation for segregation in bubbling gas-fluidized beds, Chem. Eng. Sci. 61 (2006) 1470. [14] N. Xie, F. Battaglia, S. Pannala, Effects of using two-versus threedimensional computational modeling of fluidized beds Part I, hydrodynamics, Powder Technology 182 (2008). [15] Y. Behjat, S. Shahhosseini, S. H. Hashemabadi, CFD modeling of hydrodynamic and heat transfer in fluidized bed reactors, International Communications in Heat and Mass Transfer 35 (2008) 357–368. [16] T. Li, K. Pougatch, M. Salcudean, D. Grecov, Numerical simulation of horizontal jet penetration in a three-dimensional fluidized bed, Powder Technology 184 (2008) 89–99. [17] D. Gidaspow, Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions, Academic Press, Boston, 1994. [18] M. Syamlal, T.J. O'Brien, Computer simulation of bubbles in a fluidized bed, AIChE Symposium Series 85 (1989) 22–31. [19] C.Y. Wen, Y.H. Yu, Mechanics of fluidization, Chemical Engineering Progress Symposium Series, 1966, pp. 100–111.




15

Computational and experimental methods for the on-line measurement of the apparent viscosity of a crystal suspension C. Herman1, F. Debaste1, V. Halloin1, T. Leyssens2, A. Line3 & B. Haut1 1

Transferts, Interfaces and Processes (TIPs), Chemical Engineering Unit, Université Libre de Bruxelles, Bruxelles, Belgium 2 Automation and Process Control Laboratory (APC), UCB Pharma, Braine l’Alleud, Belgium 3 Laboratoire de l’Ingénierie des Systèmes Biologiques et des Procédés (LISBP), Institut National des Sciences Appliquées (INSA) de Toulouse, Toulouse, France

Abstract This paper proposes an experimental method, based on the Metzner and Otto concept, for the on-line measurement of the apparent viscosity of a crystal slurry during a crystallization process. The first step of this procedure consists of the determination of the Np–Re–Fr relation for Newtonian liquids, for two impellertank configurations, chosen such that this relation is a bijective one. This is achieved both experimentally and numerically, using Computational Fluid Dynamics. In the second step of the procedure, the same impeller-tank configurations are used for the determination of the evolution of the apparent viscosity of the crystal slurry (non-Newtonian liquid) during a reference pharmaceutical crystallization process (Etiracetam – UCB). The paper concludes on the influence of the Particle Size Distribution of the crystals on the apparent viscosity of the suspension. For a given crystal mass fraction, the bigger the crystals are – and then the less abundant they are – and the smaller the span is, the smaller the apparent viscosity of the suspension is. Keywords: apparent viscosity, non-Newtonian liquid, suspension, on-line measurement, Computational Fluid Dynamics, crystallization process, Metzner and Otto concept, process rheometer. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090021


1

Introduction

The follow up of a crystallization process can be achieved using several probes. These probes often give information about the Particle Size Distribution (PSD) and form, or the solution concentration, but do not allow estimating any physical characteristics of the suspension such as its apparent viscosity. This physicochemical parameter can be used to characterize and control a crystallization process. Very few on-line experimental techniques exist to directly determine the evolution of the apparent viscosity of a crystal slurry during a crystallization process. Moreover, very few theoretical or empirical correlations of the apparent viscosity of a suspension can be found in the literature. Actually, the correlations proposed are based on the crystal volume fraction [1]. More recent studies show the influence of the PSD on the apparent viscosity of a suspension [2, 3]. Therefore, this paper proposes an experimental method to follow the apparent viscosity of a crystal suspension during a crystallization process, based on the Metzner and Otto [4] concept. Using an agitation system as a process rheometer, the method consists of determining the apparent viscosity of a non-Newtonian liquid, as equal to the one of a Newtonian liquid whose agitation, in the same impeller-tank configuration and operational condition, leads to an equal dissipated power. As presented in Fig.1, the link between the dissipated power, P, and the apparent viscosity, µ, consists of a relation between three dimensionless numbers, the Reynolds number, Re, the Froude number, Fr, and the Power number, Np, which are respectively defined as follows:

Re =

ρNd 2 µ

N 2d g P Np = ρd 5 N 3 Fr =

(1) (2) (3)

where ρ is the density of the suspension, d the diameter of the impeller, N the rotational speed of the impeller and g the gravitational acceleration. For a given impeller-tank configuration, the first step of this procedure consists of the determination of the Np – Re – Fr relation for Newtonian liquids. It is achieved using several reference Newtonian liquids of known density, ρ, and viscosity, µ. The determination of the power dissipated in the liquid when it is agitated with an impeller of diameter d, at a fixed rotational speed, N, allows determining the three dimensionless numbers (Fig.1). These determinations are realized experimentally and numerically, using Computational Fluid Dynamics (CFD). The relation between Re, Fr and Np, in the laminar and intermediate Reynolds range, is proposed such as [1, 5]:

N p = c Re x Fr y WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

(4)


17

The unknown parameters, c, x and y, which depend on the impeller-tank configuration, are identified by a least square optimisation. In the turbulent Reynolds range, the Power number is constant (independent on Re and Fr). PT P' N

P = PT − P '

Determination of the Np – Re – Fr relation for Newtonian liquids

Experimentally Numerically

ρ d g

N p = c Re x Fr y

Re =

ρNd 2 µ

Fr =

N 2d g

P = PT − P '

P Np = ρN 3 d 5

P' N

ρ d g

µ

ρNd 2

µ=

P x

Figure 1:

PT

ρd 5 N 3

c N d  g  2

Determination of the apparent viscosity of a non-Newtonian liquid (crystal slurry)

y

Schematical presentation of method to determine the apparent viscosity of a suspension, based on the Metzner and Otto [4] concept.

In the second step of the procedure, the same impeller-tank configuration is used for the determination of the apparent viscosity of a non-Newtonian liquid. For this purpose, the power dissipated in the liquid is measured when it is agitated with the impeller of diameter d, rotating at a fixed rotational speed, N. Knowing the density, ρ, of the suspension, the inversion of the Np – Re – Fr relation allows determining its apparent viscosity. This method can only be used in the Reynolds range in which the Np – Re – Fr relation is a bijective one.

ρNd 2

µ=

P x

ρd 5 N

(5) 3

 Nd 2   c  g   

y

The method is then applied in the second part of the paper for the determination of the evolution of the apparent viscosity of the crystal slurry during the crystallization process of a reference pharmaceutical compound.



2

Materials and methods

2.1 Introduction The experimental device consists of an unbaffled double–jacketed glass tank of 1l (Fig.2). The mixing of the liquid is ensured by a stirrer equipped with two impellers types: an Anchor or a 4-arms Flat Blade Turbine. The characteristics about the two impeller-tank configurations are reported in Tab.1. ammeter

D d

H b

Figure 2:

Schematical presentation of the experimental installation. Table 1:

Impeller-tank configurations.

Impeller

Anchor

4-arms Flat Blade Turbine

D (cm) H (cm) d (cm) b (cm)

10 10 8.9 2.5

10 10 7.9 3.1

As the two impeller-tank configurations used are not standard ones, very few results about the Np – Re – Fr relation of these systems can be found in the literature. Therefore, this relation (Eq.(4)) is here determined, both experimentally and numerically, for each of the two impeller-tank configurations used. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


19

2.2 Experimental determination of the Np – Re – Fr relation for Newtonian liquids In order to determine this relation for the two impeller-tank configurations in an extended range of Reynolds number (Re), different Newtonian liquids are used: 1. In order to investigate the low Re range, a mixture of glycerine and water (97%) at different temperatures (10°C, 15°C, 20°C, 25°C, 30°C, 35°C, 40°C, 50°C and 60°C) is used. Several rotational speeds (100 rpm, 200 rpm, 300 rpm, 400 rpm and 500 rpm) are used. The density and the viscosity of the mixture are determined experimentally with a falling sphere viscosimeter (Tab.2). 2. A mixture of salt and water (300 g of salt per kg of water) at different temperatures (5°C, 10°C, 15°C and 20°C) is used to investigate the intermediate Re range. 300 rpm is used as the rotational speed of the impeller device. The density and the viscosity of the mixture are provided by Solvay S.A. (Tab.2). 3. Information for the high Re range is obtained using the methanol solvent at ambient temperature (20°C). The rotational speeds used extend between 100 rpm and 450 rpm by 50 rpm steps. The density and the viscosity of methanol are obtained from literature (Tab.2). For each experiment, the tank is filled with 10 cm of the reference Newtonian liquid. The power dissipated in the liquid is determined using an on-line ammeter (APPA 350) (Fig.2). The dissipated power is obtained deducing the value measured at no load (P’) from the observed total power (PT) value (Fig.1). 2.3 Numerically determination of the Np – Re – Fr relation for Newtonian liquid in the turbulent range The CFD simulations are realized with the commercial codes Gambit 2.3 and Fluent 6.3. The geometries of the two impeller-tank configurations are created. They are meshed with approximately 500.000 tetrahedral elements: the size of the elements in the rotor zone and the stator zone are 1.5 mm and 3 mm, respectively. The Sliding Meshes (SM) model is preferred to the Moving Reference Frame (MRF) model as it has been shown that it determines the dissipated power with more accuracy. As the tank is unbaffled, in order to take up the effect of the free surface deformation, the Volume Of Fluid (VOF) multiphase approach is selected. The standard k-ε approach is selected to model the turbulence. The tank is filled with 10 cm of methanol. The first order-upwind discretization scheme is selected for the momentum, the turbulent kinetic energy and the turbulent dissipation range. The discretization scheme selected for the pressure and the volume fraction are the PRESTO! and the geo-reconstruct, respectively. The pressure-velocity coupling scheme recommended for this kind of simulation is the PISO. In order to reach the convergence and conserve a courant number inferior to 0.25, the time-step used is 5.10-5 sec. For each simulation, the dissipated power is determined by calculating the total moment forces on the impeller shaft. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

20 Computational Methods in Multiphase Flow V Table 2:

Density and viscosity of the reference Newtonian liquids used.

Substance Glycerine – Water Glycerine – Water Glycerine – Water Glycerine – Water Glycerine – Water Glycerine – Water Glycerine – Water Glycerine – Water Glycerine – Water Salt – Water Salt – Water Salt – Water Salt – Water Methanol

3

Temperature °C 10 15 20 25 30 35 40 50 60 5 10 15 20 20

Density, ρ kg/m3 1230 1230 1230 1220 1220 1220 1220 1210 1210 1239 1237 1234 1231 780

Viscosity, µ mPa.s 1372 1015 751 556 411 304 225 123 67 3.09 2.01 2.32 1.53 0.55


The experimental results of the Power number as a function of the Reynolds number and the Froude number are obtained for the two impeller-tank configurations in the extended Reynolds number range. The results indicate that there is not a significant influence of the Froude number on the Power number. Therefore, in the intermediate range, the relation links the Power number to the Reynolds number only. The least square optimisation in the intermediate range leads to two relations. The first one is valid for Re < 500, and the second one for 500 < Re < 104 – 105. The Power number in the turbulent range (Re > 104 – 105) is constant. The relations Np – Re are reported in Tab.3. Table 3: Impeller Anchor 4-arms FBT

Np –Re – Fr relation for the two impeller-tank configurations. Range Intermediate (Re < 500) Intermediate (500 < Re < 104 – 105) Turbulent ( Re > 104 – 105) Intermediate (Re < 500) Intermediate (500 < Re < 104 – 105) Turbulent ( Re > 104 – 105)

c 0.499 83.811 0.0025 0.465 58.343 0.005

x -0.541 -1.369 0 -0.465 -1.275 0

The experimental results of the Power number as a function of the Reynolds number are presented in Fig.3 for the 4-arms Flat Blade Turbine. The curve adjusting on the experimental results related to the Anchor is also presented. It can be observed that the curves of both impeller-tank configurations in the intermediate range are quite similar. However, as it can also be seen in the WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

21


literature [5], the Power number in the turbulent range is bigger for the 4-arms Flat Blade Turbine than for the Anchor. 1,E+00

Power Number

1,E-01

1,E-02

Anchor 1,E-03 1,E+01

1,E+02

1,E+03

1,E+04

1,E+05

1,E+06

Reynolds Number Glycerine-Water

Figure 3:

Salt-Water

Methanol

Num

Power number as a function of the Reynolds number for the two impeller-tank configurations.

Fig.4 (left) presents the comparison of the Power number as a function of the Reynolds number in the turbulent range obtained both experimentally and numerically by CFD. Fig.4 (right) shows that at least 5 revolutions of the impeller are required to reach the numerical stationary conditions. 1,E-02

0,008

Power number

Power Number

0,006

0,004

0,002

1,E-03 1,E+04 Salt-Water - Turbine Salt-Water - Anchor

Figure 4:

1,E+05 Reynolds Number Methanol - Turbine Methanol - Anchor

1,E+06 Num -Turbine Num - Anchor

0 0

2

4 6 Number of rotational tours

4-arms Flat Blade Turbine - 150 rpm

8

10

Anchor -350 rpm

(Left) Comparison between the experimental and the numerical results of the Np vs. Re in the turbulent range. (Right) Numerical evolution of the Power number with the increase of the number of the rotational tours.

The Power numbers obtained are smaller that those generally found in the literature. Indeed, the important deformation of the free surface leads to a limitation of the dissipated power [5]. Moreover, with these impeller-tank configurations, the turbulent range is reduced. This characteristic is useful for the WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

22 Computational Methods in Multiphase Flow V method proposed. Indeed, Eq.(5) can be used to determine the apparent viscosity of a suspension assuming that the relation between the Power number and the Reynolds number is a bijective one.

4

Application

The method is then applied for the determination of the evolution of the apparent viscosity of the crystal slurry during a crystallization process. The process chosen consists of the re-crystallization of the Etiracetam product (racemic compound (R-S product)): an intermediate product of the Levetiracetam (UCB Keppra). In this crystallization process, the drug can crystallize into two crystallographic forms, called morph I and morph II. The morph I is the stable crystallographic form below the transition temperature (30°C) while the morph II is the stable crystallographic form beyond this temperature. The reference pharmaceutical crude compound is initially dissolved (0.6 gproduct/gsolution) in the methanol solvent at high temperature (~ 60°C) and stirred in a tank. The cooling of this solution (15.5°C/h, started at time t = 0) induces the crystallization of both morphs. Nevertheless, the non-desired crystallographic form (morph II) nucleation kinetics is dominant. The massive primary nucleation of morph II is observed at the so-called induction time t = t1 by the first exothermic peak on the temperature curve (curve 1) in Fig.5. In order to obtain the pharmaceutical desired crystallographic form (morph I), the temperature is further lowered and the system is kept at a constant ripening temperature (Trip). At the so called latency time t = t2, a polymorphic transition from the unstable morph II to the stable morph I is observed. The transition is characterized by a second exothermic peak on the temperature curve (curve 1) in Fig.5. This transition, mediated by the solvent, is controlled by the morph I nucleation. Temperature (°C)

Power (W)

t1 t2 = latency time 10

60

2

Trip

0 phase a 1

2

phase b 3

…

…

phase c …

1

Time (h)

Figure 5: Evolution of the temperature (curve 1) and the power dissipated (curve 2) in the suspension during the polymorphic crystallization process.



23

In order to determine the evolution of the apparent viscosity of the suspension during the crystallization process (curve 2 in Fig.5), it is realized in the impeller-tank configurations presented in Tab.1 (H = 10 cm). Several rotational speeds and ripening temperatures are investigated in order to discuss the evolution of the apparent viscosity of the suspension during the crystallization process as a function of the operational conditions. On the one hand, the influence of the mixing is investigated. For this purpose, two rotational speeds are investigated for both configurations (250 rpm and 350 rpm) fixing the ripening temperature equal to -2°C. On the other hand, the influence of the ripening temperature is investigated. For this purpose, three temperatures are investigated (-10°C, -2°C and 10°C) fixing the agitation (Anchor – 250 rpm). The power dissipated in the suspension during the crystallization process (curve 2 in Fig.5) is recorded for each experimental condition. The evolution of the apparent viscosity of the suspension is determined using the method proposed (Eq.(5)) assuming that the density of the suspension is constant (960 kg/m3). For each experiment, the morph II and morph I PSD, recorded at the end of the phase a and c, respectively, are off-line determined on dry powder with a MasterSizer. The average particle size (D[4,3]), the median (D(v,0.5)) and the span distribution ([D(v,0.9) – D(v,0.1)] / D(v,0.5)) are reported in this paper. 4.1 Influence of the Particle Size Distribution of the crystals of morph I and morph II Fig.6 (left) presents the dissipated power measured during the crystallization process using the Anchor rotating at 350 rpm. The evolution of the apparent viscosity of the suspension, determined by the method, is also presented. The curves present a drastic change between the end of phase a and the end of phase c. In both cases, there are approximately 50% in mass of solid crystals in the suspension. Therefore, this change may be explained by the difference in the PSD and number of morph II and morph I crystals in suspension. At the end of the phase a, the density and the size of the morph II solid particles are large and they quickly settle down after stopping the rotation of the impeller. In this case, the flow is in a intermediate state and the suspension is a Newtonian liquid. At the end of the phase c, the density and the size of the morph I solid particles are smaller than those of the morph II crystals (Fig.6 (right)) and the amount of solid suspended is more abundant, so that the particles do not settle down quickly. In this case, the flow is in a laminar state, and the suspension is a non-Newtonian liquid. The analysis of both Fig.6 seems to indicate that, for a same crystal mass fraction in the suspension, the smaller the median is (and then the smaller and the more abundant the crystals are) and the bigger the span is, the bigger the apparent viscosity of the suspension is.


24 Computational Methods in Multiphase Flow V 1

Latency time Induction time

Dissipated Power (W)

0,8

3,5

350

0,9

4 phase b

phase a

phase c

0,7

3

0,6

2,5

0,5

2

0,4

1,5

0,3

1

0,2

0,5

0,1

300 250

µm

200 150 100 50

9,32

8,60

7,88

7,17

6,45

5,73

5,02

4,30

3,58

2,87

2,15

1,43

0,72

0

0,00

0

Apparent viscosity (Pa.s)

5 4,5

Time (h) Dissipated power

Figure 6:

Apparent viscosity

0

morph II

morph I

D[4,3]

304,23

85,77

D(v,0,5) Span * 10

274,17 20,82

39,60 59,47

(Left) Evolution of the dissipated power and the apparent viscosity of the suspension (Anchor – 350rpm). (Right) Average morph II (end of phase a) and morph I (end of phase c) PSD.

4.2 Influence of the mixing The influence of the mixing on the evolution of the apparent viscosity of the crystal slurry is investigated by 4 experimental conditions. Both impeller-tank configurations (Tab.1) are used with two rotational speeds (250 rpm and 350 rpm). Fig.7 (left) presents the apparent viscosity at the end of the phase a (morph II) and at the end of the phase c (morph I) for the four mixing experimental conditions. For each of them, the total mass of morph II crystals and morph I crystals at the end of the phase a and c, respectively, is the same. Moreover, they are approximately 50% in mass of crystals in the suspension. It can be observed that the apparent viscosity at the end of each phase depends on the impeller-tank configuration but is quite independent on the rotational speed investigated. These results can be discussed with the results of the morph I PSD (Fig.7 (right)) for the four experimental conditions. It can be seen that, for the two impeller-tank configurations, D[4,3] and D(v,0.5) are reduced when the rotational speed is increased while the span does not present a significant change. Fig.7 also indicate that, for a given rotational speed, the morph I PSD related to the Turbine is more extended (bigger span) than those related to the Anchor while they present a same median. This induces a bigger heterogeneity of the particle size of the morph I crystals in the suspension. As previously, this may explain the difference in the apparent viscosity of the suspension obtained when agitating with the Anchor or the Turbine. For a same crystal mass fraction and a same median, the bigger the span is, the bigger the apparent viscosity of the suspension is.


Computational Methods in Multiphase Flow V 100

0,9

90

0,8


25

80

0,7

70

0,6

60

0,5

µm

50 40

0,4

30

0,3

20

0,2

10

0,1

0

0 Anchor 250rpm

Anchor 350rpm

Turbine 250rpm

Turbine 350rpm

Experimental conditions

End phase a - morph II

Figure 7:

End phase c - morph I

250 rpm

350 rpm

D[4,3] Anchor

90,07

77,25

D(v,0,5) Anchor Span * 10 Anchor

43,40 54,65

38,11 52,15

D[4,3] Turbine D(v;0,5) Turbine

99,98 44,71

92,32 37,50

Span * 10 Turbine

63,15

72,11

For the four experimental conditions investigated. (Left) Apparent viscosity at the end of phase a and phase c. (Right) Morph I PSD.

4.3 Influence of the ripening temperature


Three ripening temperatures are investigated in order to study their influence on the time evolution of the apparent viscosity of the crystal slurry. Fig.8 (left) presents the dissipated power in the suspension at the end of the phase a and at the end of the phase c for the three ripening temperatures investigated. Fig.8 (right) present the morph I PSD at the end of the phase c. 1

90

0,9

80

0,8

70

0,7

60

0,6

50 µm

0,5

40

0,4 30

0,3

20

0,2

10

0,1 0

0

-10

-2

10

Ripening Temperature (°C) End phase a - morph II

Figure 8:

End phase c - morph I

D[4,3] D(v,0,5) Span * 10

-10 °C

-2 °C

10 °C

88,35

85,77

80,02

32,56 83,16

39,60 59,47

59,30 29,43

For the three experimental conditions investigated. (Left) Apparent viscosity at the end of phase a and phase c. (Right) Morph I PSD.

As previously, these Fig.8 allow one to conclude that the bigger the crystals are and the smaller the span is, the smaller the apparent viscosity of the suspension is. Moreover, when reducing the ripening temperature from 10°C to -10°C, as the solubility decreases, the mass and number of the crystals in suspension increase leading to an augmentation of the apparent viscosity .



5

Conclusion

This paper presents an experimental method to follow the apparent viscosity of a crystal suspension during a crystallization process, based on the Metzner and Otto concept. The first part of this paper focuses on the experimental and numerical (by CFD) determination of the Np – Re – Fr relation for Newtonian liquid for two impeller-tank configurations (unbaffled tank, Anchor or 4-arms Flat Blade Turbine). These one are chosen in order to obtain a bijective relation between the Power number and the Reynolds number in an extended Reynolds range. The method proposed is then described and applied to on reference pharmaceutical crystallization process. In order to analyse the influence of the solid phase characteristics on the apparent viscosity of the suspension, several operational conditions (mixing and temperature) are investigated. The paper concludes on the influence of the Particle Size Distribution of the crystals on the apparent viscosity of the suspension. For a given crystal mass fraction, the bigger the crystals are - and then the less abundant they are - and/or the smaller the span is, the smaller the apparent viscosity of the suspension is.

Acknowledgements Christelle Herman acknowledges the Fonds National de la Recherche Scientifique (FNRS), UCB Pharma and the Hubert Currien (Tournesol) partnership for their financial support.

References [1] Mullin, J.W., Crystallization, Elsevier, Oxford, UK, 2001 [2] Roscoe, R., The viscosity of suspension of rigid spheres, J. Appl. Phys., 3, pp. 267-269, 1952 [3] Snabre, P. and Mills, P., Rheology of concentrated suspensions of viscoelastic particles, Colloïd and Surfaces A., Physicochemical and engineering aspects, 152(1-2), pp. 79-88, 1999 [4] Metzner, A.B. and Otto, R.E., Agitation of Non-Newtonian Fluids, AIChE Journal, 3(1), pp. 3-10, 1957 [5] Nagata, S., Mixing: Principles and applications, A. Halsted press book, Tokyo, Japan, 1975



27

Two-phase flow modelling within expansion and contraction singularities V. G. Kourakos1, P. Rambaud1 , S. Chabane2 , D. Pierrat2 & J. M. Buchlin1 1 von

Karman Institute for Fluid Dynamics, Belgium Technique des Industries Mecaniques, ´ France

2 Centre

Abstract An experimental study is performed in order to describe the single- and twophase (air-water) horizontal flow in the presence of pipe expansion and contraction. Three types of singularities are investigated; smooth convergence and sudden and progressive enlargement. The opening angles for progressive singularities are 5, 8, 9 and 15 degrees. The surface area ratios tested are σ = 0.43, 0.64, 0.65 and 1.56. Bubbly flow is the dominant flow regime that is investigated for volumetric quality up to 30%. The pressure distribution for both single and two-phase horizontal flow is examined versus axial position. For expansion geometries, it is found that the smaller the enlargement angle, the larger the recovery pressure for the same flow conditions; the pressure drop caused by the singularity is higher in the case of a sharper expansion. The comparison of the experimental results to published models leads to a proposed corrective coefficient for Jannsen’s correlation. Flow visualization is also performed; the flow patterns downstream from the different singularities are identified in each configuration and plotted in Baker’s map for horizontal flow. Keywords: two-phase flow, singularity, sudden expansion, contraction, pressure drop, bubbly flow, flow visualization.

1 Introduction Two-phase flow can be frequently met in nuclear, chemical or mechanical engineering where gas-liquid reactors, boilers, condensers, evaporators and combustion systems are often used. The presence of geometrical singularities in pipes may significantly affect the behaviour of two-phase flow and subsequently the WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090031

28 Computational Methods in Multiphase Flow V resulting pressure drop. Therefore, it is an important subject of investigation in particular when the application concerns industrial safety valves. The studies of two-phase flow in straight pipes existing in the literature are numerous. However, investigations of two-phase flow in divergence, convergence, bends and other types of singularities are rather sparse. The aim of studying these geometries is to find how these geometrical accidents influence the two-phase flow pattern and pressure distribution. In particular, the understanding of the flow in such basic geometries can lead to a better design of safety systems. Some of the authors that have analyzed two-phase flow in expansion geometries are Jannsen and Kervinen [1], McGee [2], Chisholm [3], Chisholm [4] and Lottes [5]. Correlations for estimating the pressure change in two-phase flow in this type of piping geometry are reported by these authors. These correlations can be extracted from the conservation equations applied downstream of the sudden expansion. The equations used take into account different parameters of the geometry and the flow such as surface area ratio σ , mass quality x and mass velocity G. More recently, Aloui and Souhar [6], Aloui et al. [7], Schmidt and Friedel [8], Hwang and Pal [9], et al. Ahmed et al. [10] and Ahmed et al. [11] have evaluated the pressure change in a sudden expansion duct. Moreover, some of them (Aloui and Souhar [6]; Ahmed et al. [10]) have measured the bubble velocities and local void fraction to characterize the flow regime downstream from the singularity. The lack of studies in progressive enlargements in two-phase flow in the literature makes such an investigation more appealing. In this paper, progressive contraction and divergence geometry of different opening angles is considered. The latter is compared to the case of sudden expansion. The two fluids are air and water in isothermal conditions. The volumetric quality of the air varies from 0–30% and bubbly flow is the dominant regime. Four surface area ratios, σ = 0.43, 0.64, 0.65 and 1.56, are tested. The opening angles for the case of progressive singularities are 5, 8, 9 and 15 degrees. The Reynolds number Re of the liquid ranges from 8 · 104 to 23 · 104 . The determination of the recovery pressure for each of the aforementioned geometries is one of the main objectives of this investigation.

A) Flow d

Į

D

Reattachment length-L/d

B) Flow

D

d

Reattachment length-L/d

Figure 1: A) Progressive expansion of different opening angles-reattachment length L/d. B) Sudden expansion-reattachment length L/d. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


29

In Figure 1, the two different types of expansion geometries tested in this paper are presented. Figure 1A shows the divergent pipe with the opening angle α and Figure 1B the sudden expansion. The normalized reattachment length L/d, noticed in Figure 1, denotes the eventual recirculation zone. In the case of convergence geometry, a contraction region can be observed; a vena contracta is formed in the pipe downstream from the singularity.

2 Experimental facility and conditions 2.1 Experimental facility A schematic of the horizontal air-water flow facility used for the present study is shown in Figure 2. A centrifugal pump (1) with a maximum flow rate of 65 m3 /h is sucking water from a reservoir and is controlled with a frequency inverter. During the experiments, an air release valve (11) connected to the tank is kept continuously open to the atmosphere to avoid bubbles entering the circuit. A by pass valve (12) is used to prevent facility from the water hammer phenomenon. A temperature sensor is placed in the reservoir to monitor the temperature for each measurement. Two electronic flow meters are used to measure the water flow rate (2 and 3); their maximum capacity is 12 m3 /h (3) and 32 m3 /h (2), respectively. In the case of the desired maximum flow rate, which is 40 m3 /h, the two flow meters are used in series. A bourdon tube pressure gauge (4) is placed upstream in the pipe to obtain the wall static pressure relative to the atmosphere. This indication helped to prevent excessive pressure that could lead to a breaking of the test section (made in Polymethyl Methacrylate, PMMA). Moreover, the pressure has to be high enough to allow the necessary purging of the pressure transducers. Therefore, the pressure is held constant at around 200 kPa. The setup has an upstream calming section (5) consisting of a stainless steel pipe length of 50 diameters (50d). This ensures a fully developed flow after the bend. Close to the test section, the injection of the air is performed through a gas injector (6) as indicated in Figure 2. A regulation valve (7)

11

Inverter

1 Pump

T

2 Big electronic water flow meter 3 Small electronic water flow meter

Water tank V

M

O

4 Bourdon tube pressure gauge

12

5 Calming length

N

6 Air injector

Water discharge

U

7 Regulation valve 8 Electronic air mass flow meter

P

P

9 Heat exchanger

R

Q

Test section T

S

10 Pressure regulation valve 11 Air release valve 12 By pass valve T Temperature measurement

Compressed air

Figure 2: Schematic of the experimental facility. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

30 Computational Methods in Multiphase Flow V controls the air that is supplied from a compressor. The air flow rate is measured by an electronic mass flow meter (8). The design and the positioning of the air injection devices are such that uniform bubbly flow is produced at the inlet of the test section. It is found that the most suitable distance for the air injection is 20 pipe diameters upstream from the singularity. After the test section, a heat exchanger (9) is placed for maintaining the temperature constant at around 21◦ C during the experiments. A draining valve is also located at the bottom of the reservoir. Finally, a pressure regulation valve (10) controls the pressure of the system. A detailed view of the test section is presented in Figures 3, 4 and 5. The case of a DN 40/65 (σ =0.43) divergent section with an opening angle of 8◦ is exemplified. At each section of measurement, four pressure taps are placed with an angle of 45◦ between them as shown in Figure 3. Thus, any three dimensionality of the flow could be identified from pressure measurement. The four taps are named as A, B, C and D according to Figure 3. Figure 4 depicts an overview of the test section. The setup is built in PMMA to allow optical access. Pressure taps are placed along the tube in several points as is shown in Figure 5. The distance between pressure holes is normally equal to one tube diameter but becomes smaller when approaching the singularity. The pressure taps are also more dense inside and downstream from the singularity. This allows better tracking of the flow behaviour in the singularity.

A

B

4 pressure taps (45° angle between them)

D

C Aluminum table

Figure 3: Pressure taps placed in four Figure 4: Overview of the test section. different points of the tube with 45 ◦ between them.

Figure 5: Detailed view of the test section with the pressure taps and their position. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


31

Table 1: Different test cases studied. Singularity d1 [m] D2 [m] σ [-] Ls [m] Ls /d1 Angle α [◦ ] Smooth contraction 0.04 0.032 1.56 0.025 0.63 9 Divergence 0.032 0.04 0.64 0.025 0.78 9 Divergence 0.041 0.0627 0.43 0.041 1 15 Divergence 0.041 0.0627 0.43 0.07503 1.83 8 Divergence 0.041 0.0627 0.43 0.1238 3 5 Divergence 0.0627 0.078 0.65 0.0529 0.8 8 Sudden expansion 0.041 0.0627 0.43 90 Sudden expansion 0.0627 0.078 0.65 90

Table 2: Upstream conditions for pressure measurements and flow visualization. ¯ [kg/m2 s] ReL1 ·104 d1 [m] Fluid Q¯ [l/s] J [m/s] β [%] G Water 2 2.5 2500 9 Water 4.7 5.8 5850 20 0.032 1-40 Air 0.017 0.02 0.03 0.005 Air 1.8 2.2 2.61 0.46 Water 2.3 1.8 1750 8 Water 7 5.4 5300 23 0.041 5-30 Air 0.4 0.3 0.38 0.09 Air 2.8 2.2 2.73 0.58 Water 6 1.9 1950 13 Water 10.5 3.4 3400 23.5 0.0627 5-25 Air 0.4 0.1 0.15 0.05 Air 3.4 1.1 1.29 0.45

Flow regime Laminar Min Turbulent Max Laminar Min Turbulent Max Turbulent Min Turbulent Max Laminar Min Turbulent Max Turbulent Min Turbulent Max Laminar Min Turbulent Max

Pressure distribution is measured upstream and downstream from the divergence. The test matrix is summarized in Table 1. 2.2 Flow conditions and measurement devices The flow conditions of the experimental campaigns are listed in Table 2. Table 2 presents the test conditions for the pressure measurements and for flow visualization. It should be pointed out that the ReL1 number of the liquid is based on the upstream pipe diameter d. For the comparison between single and two-phase flow, ReL1 is kept constant. This is obtained by adjusting the water flow rate when increasing the air to reach a higher volumetric quality β. Consequently, we can assume that the total mass flux is constant, since the mass of the air compared to that of water is negligible. Differential pressure transducers of the Rosemount type are used. The uncertainty associated with the pressure transducers varies from a minimum of 0.35% to a maximum of 0.75%, depending on the range of the WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

32 Computational Methods in Multiphase Flow V measurement (100-20% of the scale of the range respectively). To obtain the best accuracy possible, four different pressure transducers are selected: 1. Calibrated at 0–1.6 kPa 2. Calibrated at 0–4 kPa 3. Calibrated at 0–8 kPa 4. Calibrated at 0–16 kPa Every transducer is used in a range that gives the best accuracy in all the conditions covered. Prior to the measurements, predictions of regular pressure drop are performed by means of Blasius and Colebrook-White formulas for single-phase and Lockhart and Martinelli [12] for two-phase flow. Thus, this P estimation allows the selection of the appropriate pressure transducers for each test. Additionally, for the prediction of the singular pressure change in singlephase, the coefficients given by Idel’cik [13] are used. The uncertainty related to the flow rate measurements varies from a minimum of 0.5% to a maximum of 1.10%. The temperature variation during the experiments is of the order of ±4◦ C with an average value of 21◦ C. Although a heat exchanger is used for reducing this variation, a small fluctuation of the temperature could not be avoided. A variation of ±5◦ C will change ρ and ν by 0.1% and 11% respectively. Therefore, a correction of the liquid density and viscosity is performed. The sampling frequency of the measurements is fsampling = 2 Hz and the acquisition time for each measurement point is tacq. = 1 minute with the aim of assuring a more accurate average. In some cases (for sudden and progressive enlargement of σ = 0.65), a higher fluctuation of the signal is observed; in this occurrence an acquisition time of 2 minutes is chosen.

3 Results and discussion 3.1 Pressure measurements One of the main objectives of the study is the determination of the pressure distribution through the different singularities. Figure 6 indicates how the measurements are performed and how the singular single and two-phase pressure change is determined (the case of divergence is chosen). As the graph of Figure 6 shows, following a normal decrease upstream from the geometrical accident, the pressure will increase to a maximum value inside the divergent section and will start decreasing after a certain length in a regular way. We can split the whole phenomenon into three regions; the upstream fully developed flow, the transitional region with a recirculation zone and the downstream fully developed flow. The length of the transitional region varies with ReL1 , σ , and the type of the singularity. In all the tests, the measurement of the regular and singular static pressure changes refers to the pressure measured at ≈ 10d upstream from the singularity (Figure 6). The singular pressure change P can be finally determined by extrapolating the regular static pressure drop from the start of the singularity to the reattachment point. Since the points downstream from the singularity are not enough to obtain fully established flow, the regular pressure drop is WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


33

ǻPregular (calculated)

ǻP [mbar]

Pmax ǻPsingular

ǻPSINGULAR-FINAL

(measured)

Preference

ǻPregular (measured)

ǻP=0 Singularity

Axial position [z/d] z/d=0

Flow

Fully developed flow Inlet

Transitional flow Fully developed flow Outlet

Figure 6: Explanation of the way to determine the singular pressure change in expansion geometry.

calculated by means of the Blasius and Colebrook-White formulas for singlephase and the model of Lockhart and Martinelli [12] for two-phase. The final singular pressure change is calculated by a simple summation of these three terms (|Pregular-measured|+|Psingular-measured|+|Pregular-calculated|). The reattachment length is determined as the location of the maximum recovery pressure. 3.1.1 Expansion singularities 3.1.1.1 Sudden expansion In Figure 7, the two-phase pressure change along the pipe and the singularity is plotted for sudden expansion of σ = 0.43 and at ReL1 = 1.82 · 105. The single-phase result is also drawn on the same graph. The pressure is measured at the four peripheral taps on the tube sections close to the singularity (points A, B, C and D) as well as their average (point M). The two-phase experimental data are compared with prediction of the singular pressure change for axisymmetric sudden expansion geometry obtained from the two following models: Jannsen and Kervinen [1]:

Pt ot

G21 ρL 2 =− −1 , (1 − σ ) 1 + x 2ρL ρG

(1)

where G1 is the mass flux upstream from the singularity, ρL is the density of water, σ is the area ratio, x is the mass quality of air and ρG is the density of air. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

34 Computational Methods in Multiphase Flow V 45 40 35

Sudden enargement ı=0.43 Two-phase-20%air-ReL1=1.82E5

30

ǻP [mbar]

25

A

B

D

C

20 15 10

Single-phase-Experimental L-M&Chisholm (1969) L-M&Jannsen(1966) Point M Point A Point B Point C Point D

5 0 -5 -10 -15 -20 -10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

z/d [-]

Figure 7: Two-phase static pressure change versus axial position for sudden enlargement of σ = 0.43 and for ReL1 = 1.82 · 105 -comparison with experimental single-phase and with the models of Jannsen and Kervinen [1] and Chisholm [4].

Chisholm [4]: G21 1 C 2 σ (1 − σ ) (1 − x) 1 + + 2 , Pst = − 2ρL X X where

(2)

1 − x 2 ρG , X x ρL 0.5 0.5 ρL ρG ρL − ρG 0.5 . + C = 1 + 0.5 ρL ρG ρL

2

Both models rely on the assumption of a homogeneous flow. Figure 7 shows that Jannsen’s model [1] fits satisfactorily with the experimental results while Chisholm’s [4] model overestimates the pressure change. This was also reported by Velasco [14]. To better emphasize the effect of two-phase flow we define the dimensionless pressure change L as follows: L =

TP PSingular SP PSingular

,

(3)

TP where PSingular is the singular two-phase pressure change as explained in SP Figure 6 and PSingular the single-phase one. Figure 8 displays the evolution of the experimental L versus volumetric quality at ReL1 = 2.0 · 105 . The data are compared to the model of Jannsen and Kervinen [1] and Chisholm [4], respectively. As it was previously mentioned, Jannsen’s [1] correlation agrees better than Chisholm’s [4] correlation with the experimental results. The comparative graphs given in Figures 9 and 10 indicate that the maximum deviation from the WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


35

1.40

Sudden enargement ı=0.43 ReL1=2.0E5

1.35 1.30

ĭL [-]

1.25 1.20 1.15 1.10 1.05 Experimental

1.00 Chisholm (1969)

0.95

Jannsen (1966)

0.90 0

5

10

15

20

25

30

35

Volumetric quality ȕ [%]

Figure 8: Dimensionless singular pressure change L versus volumetric quality. Comparison with the models of Jannsen and Kervinen [1] and Chisholm [4]. 80

80


5%

70

10% 65 60

Single-phase Air 5% Air 10% Air 15% Air 20% Air 25% Air 30% Air 35%

55 50 45 40 40

45

50

55

60

65

ǻPsingular experimental [mbar]

70

75

ǻPsingular Chisholm [mbar]

ǻPsingular Jannsen [mbar]

75

75


5%

70

10%

65 60


55 50 45 40

80

40

45

50

55

60

65

70

75

80

ǻPsingular experimental [mbar]

Figure 9: Deviation of Jannsen and Figure 10: Deviation of Chisholm [4] Kervinen [1] model from model from experimental experimental results. results.

experimental data for the model of Jannsen and Kervinen [1] is limited to 5% while it reaches 10% for Chisholm [4] model. Measurements with the same flow conditions are repeated for a sudden enlargement of surface area ratio σ = 0.65. A summarizing graph of the static pressure recovery measured in both geometries of σ = 0.43 and 0.65 for different ReL1 and for volumetric quality, β, varying from 0 to 35% is presented in Figure 11. The singular pressure change is increasing for higher β and ReL1 . Furthermore, for the same ReL1 lower σ results in a lower P (up to three times smaller). 3.1.1.2 Progressive and sudden enlargement: comparison Compared to sudden expansion, a progressive enlargement will create for the same flow conditions, less pressure loss and accordingly will exhibit a higher pressure recovery as depicted in Figures 12 and 13. Figure 12 shows a single-phase P diagram along sudden expansion and divergent of angles 5, 8 and 15◦, of surface area ratio σ = 0.43 and at ReL1 = 1.8 · 105 . In Figure 13, the same type of plot is built for β = 20% of air. It can be seen that, for single-phase, the pressure drops 17% passing from divergent section of 5◦ to 15◦ and 29% from 5◦ to sudden WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)



80

ǻPSingular [mbar]

70 60

Sudden enargement ı=0.43

50 40 Sudden enargement ı=0.65

30 20 10 0 80000

100000

120000

140000

160000

180000

200000

220000

240000

ReL1 [-]

ǻP [mbar]

Figure 11: Psingular for several ReL1 from 0–35% of air for sudden enlargement of surface areas σ = 0.43 and σ = 0.65. 50 45 40 35 30 25 20 15 10 5 0 -5 -10 -15 -20

Singularity ı=0.43 Single-phase-ReL1=1.8E5

Sudden enlargement Divergent-angle 5 Divergent-angle 8 Divergent-angle 15

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

16

z/d [-]

Figure 12: Pressure recovery diagram for single-phase flow and the same ReL1 , a singularity of σ = 0.43 and for sudden enlargement and divergence of angles 5, 8 and 15◦ .

expansion. For two-phase flow, the pressure drop is 11% and 21% respectively. Additionally, we can notice that all the curves in Figure 13 are shifted to the right, meaning that the flow becomes fully developed further downstream from the singularity and thus the recirculation zone is longer in two-phase flow. In the case of sudden enlargement, contrary to smooth divergence, the pressure before starting to increase slightly decreases at 1d and starts increasing again at 2d upstream of the singularity. This is due to the presence of a secondary recirculation zone. 3.1.1.3 Proposed correlation for expansion singularities The proposed correlation relies upon Jannsen [1] formulation. By fitting this model to the experimental values, a corrective coefficient is defined. It turns out that this parameter C is a function of the opening angle α and ReL1 as shown by the 3D representation proposed in Figure 14. Although Jannsen’s [1] model is chosen as the most accurate, attempts are made with Chisholm [4] model as well. Hence, the corrective coefficient C for Chisholm’s [4] correlation is represented in a 3D plot WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

ǻP [mbar]

Computational Methods in Multiphase Flow V 50 45 40 35 30 25 20 15 10 5 0 -5 -10 -15 -20

37

Singularity ı=0.43 Two-phase 20 % air-ReL1=1.8E5

Sudden enlargement Divergent-angle 5 Divergent-angle 8 Divergent-angle 15

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

16

z/d [-]

Figure 13: Pressure recovery diagram for two-phase flow (20% of air) and the same ReL1 , a singularity of σ = 0.43 and for sudden enlargement and divergence of angles 5, 8 and 15◦ . Z

Jannsen (1966) model

Z

Chisholm (1969) model

X 0.6 5

5 0.5

1.4 1.38

6 1.3 1.36

1.2 6

1.3 4

1.3

1.3 2

2 1.2

C [-]

1.42

1.4

4

1.2

0.45

0.4

1.1

0.4

8

0.4

1.8x10+05

1.2

0.5

8 1.2

0.6

1.3

0.45

0.7

1.32

0.5

0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3

Y

1.34

0.6

C [-]

X C [-] 1.42 1.4 1.38 1.36 1.34 1.32 1.3 1.28 1.26 1.24 1.22 1.2 1.18

C [-]

Y

0.35

+05

Re [-]

2.2x10

0.3

0.3

2.0x10

+05

14

12

10

[°] Angle

8

6

1.8x10

Re

1.2

+05

2.0x10 [-]

+05

2.2x10+05 14

12

10

8

6

[°] Angle

Figure 14: Coefficient C in function of Figure 15: Coefficient C in function α and ReL1 for Jannsen and of α and ReL1 for the Kervinen [1] model. Chisholm [4] model.

in Figure 15. In Table 3, the coefficients that are calculated for both models and for the different parameters tested in progressive expansion are given. The corrective coefficient C for Jannsen and Kervinen [1] formulation can be modelled as follows: −0.8283 C = 0.061 · α 0.8917 − 10717 · ReL1 + 0.378.

(4)

This coefficient when applied to Jannsen’s [1] model gives a maximum deviation from the model fit of 58% for the case of σ = 0.43, ReL1 = 1.84 · 105 and 5◦ and minimum of 1.4% for ReL1 = 2.36 · 105 and 15◦ . It should be stressed that further experimental data are needed to refine the C modelling and improve the validation. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


Table 3: Coefficient for adaption of the Jannsen and Kervinen [1] and Chisholm [4] models to fit to the experimental results for divergence geometry and for several α, σ and ReL1 . Jannsen [1] ReL1 ·105 α [◦ ] 1.84 5 2.3 5 1.78 8 σ =0.43 2.36 8 1.76 15 2.36 15 1.79 8 σ =0.65 2.26 8

C [-] Chisholm [4] ReL1 ·105 α [◦ ] 0.4 1.84 5 0.26 2.3 5 0.48 1.78 8 σ =0.43 0.38 2.36 8 0.7 1.76 15 0.69 2.36 15 0.3 1.79 8 σ =0.65 0.24 2.26 8

C [-] 1.34 1.445 1.3 1.365 1.155 1.163 1.187 1.365

Finally, the modified Jannsen’s [1] correlation can be written as:

−0.8283 Pt ot = − 0.061 · α0.8917 − 10717 · ReL1 + 0.378 G21 ρL 2 −1 . · (1 − σ ) 1 + x 2ρL ρG

(5)

3.1.2 Contraction singularity 3.1.2.1 Measurements in progressive contraction Convergence geometry of σ = 1.56 and angle 9◦ is studied. The geometry is identical to the test section shown in Figure 5 with a scaling factor of 1/2 (DN40/32). The experimental facility and flow conditions are described in section 2.2. Pressure transducers of type Validyne are used for this experimental campaign with the same acquisition time (tacq. = 1 min) and sampling frequency (fsampling = 2 Hz). The different membranes that cover all the range of the pressure measurements are: 1. Calibrated at 0–2.2 kPa 2. Calibrated at 0–8.6 kPa 3. Calibrated at 0–35 kPa Additionally, numerical simulations are carried out with the commercial CFD code Fluent. The test parameters and conditions are: 2D axisymmetric computation, realizable k − turbulence model with enhanced wall treatment and second order discretization scheme. Convergence criterion is set at 10−7 . In Figure 16, the experimental and numerical static pressure drop is plotted against axial position for several ReL1 in single and two-phase flow. The pressure is decreasing in a regular way before the singularity; the contraction creates a high pressure drop step and then starts decreasing regularly downstream. The flow is observed fully developed close to the singularity (at ≈2d upstream and downstream) contrary to the case of divergence for which the reattachment length is detected at ≈10d. Therefore, the singular pressure change Psingular for convergence geometry is determined by measuring the static pressure at equal WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


39

ǻP [mbar]

distance upstream and downstream from the singularity (2d). A summarizing graph of all experimental and numerical results obtained for single and two-phase flow is shown in Figure 17. The results concerning the case of sudden contraction for several σ and G (Guglielmini et al. [15]) are compared to the experimental data. The experimental results for smooth contraction are plotted in terms of the dimensionless pressure change L , defined by eqn. (3). In Figure 17 Jannsen’s [1] correlation for sudden contraction is adapted with a correction coefficient of C = 0.81 to fit with the results (G = 1990 kg/m2s).

140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 0.00

Single-phase-Exp-Re=136000 Single-phase-Exp-Re=79300 Single-phase-CFD-Re=739000 Two-phase-11% air-Exp-Re=95100 Two-phase-10%air-CFD-Re=66500

Smooth convergence ı=1.56, angle 9°

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

L [m]

Figure 16: Experimental and numerical single and two-phase static pressure change versus axial position for convergence of σ = 1.56 and angle 9◦ for several ReL1 .

1.55 Exp-G=1990 kg/m^2s

1.5

Exp-G=2786 kg/m^2s

CFD-G=1300-1700 kg/m^2s

1.4

Guglielmini et al.-G and ı varying

1.35

ĭL [-]

G=1990 kg/m^2s

Exp-G=1990-3424 kg/m^2s

1.45

Janssen(1966) correlation-C=0.81

1.3 1.25 1.2 1.15 1.1 Smooth convergence ı=1.56, angle 9°

1.05 1 0

5

10

15

20

25

30

35

Volumetric quality ȕ [%]

Figure 17: Experimental and numerical dimensionless singular pressure change L versus volumetric quality. Comparison to literature (Guglielmini et al. [15]) and to the adapted (C = 0.81) Jannsen and Kervinen [1] model. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


Table 4: Coefficient for adaption of Jannsen’s [1] formulation to fit to the experimental and numerical results for progressive contraction for several G. GL1 [kg/m2 s] Correction coefficient C [-] 1592 0.835 1990 0.800 ◦ Convergence-angle 9 , σ =1.56 2786 0.770 4378 0.754 Jannsen [1] correlation

3.1.2.2 Proposed correlation for progressive contraction The correlation for sudden convergence, as described from Jannsen [1], is recalled: 2 G22 1 1 ρL PTP = 1+x . (6) −1 +1− 2 2ρL CC ρG σ where Cc is the contraction coefficient defined as Cc = Ac /A1 where Ac the flow area in the vena contracta. A typical value of this parameter equal to 0.64 is considered for this investigation. This correlation can be modified and then applied for the case of smooth contraction. The parameter varying is the mass flux of water upstream of the singularity GL1 . A fit to the present results is made and the resulting corrective coefficients are listed in Table 4. A correlation to calculate the correction coefficient C is obtained as a function of GL1 . C = 2 · 10−8 G2L1 − 0.0001 · GL1 + 0.9913. (7) The relative discrepancy between experimental-numerical data and model fit, when eqn. (7) is applied, varies from 5.72% to a maximum of 24.25%. The final corrected correlation for the case of smooth convergence of angle 9◦ is:

PTP = 2 · 10−8 G2L1 − 0.0001 · GL1 + 0.9913 2 (8) G22 1 1 ρL 1+x . · −1 +1− 2 2ρL CC ρG σ 3.2 Flow pattern maps and visualization Flow regime maps are often considered in two-phase flow. A common chart is the one proposed by Baker [16]. It has been established for horizontal flows in pipes of constant cross section. In the present study, the flow is visualized both upstream and downstream from the singularity. As it is illustrated in Figure 18, four different flow patterns are identified downstream of the divergence; Bubbly, Plug, Disperse and Annular flow. For sudden and progressive enlargement (angles 5◦ and 8◦ ) with σ = 0.43 and σ = 0.65, a normal video camera is used to determine the condition for transition WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


41

from bubbly flow to other types of flow just after the singularity. The results are plotted on Baker’s map and are reported in Figure 19. However, since the departure from bubbly flow is decided on visual information, the transition criterion remains rather subjective and the results given in Figure 19 are only indicative. The second campaign of visualization is performed, using a high-speed camera, in a fully transparent setup that allows better optical access (without pressure taps). Consequently, distinction between flow regimes is more straightforward. In this facility, a progressive enlargement of σ = 0.64 for an opening angle of α = 9◦ is tested. The flow conditions for which these regimes are visualized are reported in Figure 20. Finally, we should draw attention to the fact that all flow conditions calculated refer to the upstream position. Indeed, for these test cases, the flow regime upstream from the singularity corresponds to bubbly flow (Baker map) while downstream three additional flow patterns occur (plug, disperse and annular).

Bubbly

Plug

Disperse

Annular

Figure 18: Flow patterns identified downstream of the divergence geometry of α = 9◦ and σ = 0.64. p p

100

100

Singularities ı=0.43 and ı=0.65

Wavy

Annular

-2 -1

GG1 / Ȝ [kg.m s ]

-2 -1

GG1 / Ȝ [kg.m s ]

Wavy 10

Stratified

Slug Bubbly

1

Sud.enl.-ı=0.43

Divergence ı=0.64 Angle 9°

Annular

10

Bubbly Slug Stratified

1

Sud.enl.-ı=0.65

Plug

Bubbly

Plug

Div.angle 5 ı=0.43

Disperse

Div.angle 8 ı=0.43 Div. angle 8 ı=0.65

0.1 1

10

100

1000

10000

100000

Plug Annular

0.1 1

10

GL1ȥ [kg.m-2s-1]

100

1000

10000

100000

GL1ȥ [kg.m-2s-1]

Figure 19: Modified Baker [16] map Figure 20: Modified Baker [16] map for progressive expansion of for progressive and sudden σ = 0.64 and α = 9◦ . expansion of σ = 0.43 and 0.65. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


4 Conclusions An investigation of horizontal air-water flow in sudden and progressive enlargements and smooth contraction is performed. The static pressure evolution along these geometrical accidents is measured and flow visualization is performed. The results are expressed in terms of the dimensionless singular pressure change L . Compared to literature, a deviation of 5% is found with Jannsen’s [1] model and 10% with Chisholm’s [4] model for axisymmetric sudden expansion. For progressive enlargement of the same surface area ratio σ , the smallest the opening angle, the highest the pressure recovery. For the same flow conditions, the minimum pressure recovery occurs for sudden enlargement geometry. A modified version of Jannsen’s [1] correlation is suggested for both progressive expansion and contraction. A corrective parameter taking into account the different effects of the divergent angle and the liquid Reynolds number of the divergent section and the upstream mass flux for convergence, is introduced. The proposed correlation gives satisfactory results but needs further validation. In the convergence configuration, the single and two-phase static pressure drop along the pipe is compared with literature and CFD simulations; a satisfactory agreement is found. Finally, flow visualization shows that departure from bubbly flow to plug, disperse or annular flow may occur in the downstream section of a divergent singularity.

Acknowledgements The support of the French company CETIM (Centre Technique des Industries Me´ caniques) is gratefully acknowledged. Mr. E.C. Bacharoudis and Miss R. Delgado-Tardáguila are thanked for their contribution to the numerical and experimental study in this paper.

Nomenclature A surface area [m2 ] C correction coefficient [-] d upstream diameter [m] D downstream diameter [m] f frequency [Hz] G mass velocity [kg/m2 s] J superficial velocity [m/s] L length of the pipe [m] P pressure [Pa] Q¯ volumetric flow rate [l/s] ReReynolds number [-] t time [s] x mass quality [-] z axial position [m]

Greek symbols α opening angle [◦ ] β volumetric quality [-] ν kinematic viscosity [m2 /s] ρ density [kg/m3 ] σ surface area A1 /A2 [-] dimensionless P [-] Sub-Superscripts c contraction G gaseous phase L liquid phase 1 upstream 2 downstream s singularity

Abbreviations SP single-phase st static tot total TP two-phase



43

References [1] Jannsen, E. & Kervinen, J.A., Two-phase pressure drop across contractions and expansions of water-steam mixture at 600 to 1400 psia. Technical Report Geap 4622-US, 1966. [2] McGee, J., Two-phase flow through abrupt expansion and contraction. Ph.D. thesis, North Carolina State University, Raleigh, 1966. [3] Chisholm, D., Prediction of pressure losses at changes of sections, bends and throttling devices. Technical Report NEL rept. 388, 1968. [4] Chisholm, D., Theoretical aspects of pressure changes at changes of section during steam-water flow. Technical Report NEL rept. 418, 1969. [5] Lottes, P., Expansion losses in two-phase flow. Nucl Sci Eng, 9, pp. 26–31, 1960. [6] Aloui, F. & Souhar, M., Experimental study of a two-phase bubbly flow in a flat duct symmetric sudden expansion. Part I: Visualization, pressure and void fraction. Int J Multiphase Flow, 4, pp. 651–665, 1996. [7] Aloui, F., Doubliez, L., Legrand, J. & Souhar, M., Bubbly flow in an axisymmetric sudden expansion: pressure drop, void fraction, wall shear stress, bubble velocities and sizes. Exp Therm Fluid Sci, 18, pp. 118–130, 1999. [8] Schmidt, J. & Friedel, L., Two-phase pressure change across sudden expansions in duct areas. Chem Eng Commun, 141, pp. 175–190, 1996. [9] Hwang, C.Y. & Pal, R., Flow of two-phase oil/water mixtures through sudden expansions and contractions. Chem Eng J, 68, pp. 157–163, 1997. [10] Ahmed, W., Ching, C. & Shoukri, M., Pressure recovery of two-phase flow across sudden expansions. Int J Multiphase Flow, 33, pp. 579–594, 2008. [11] Ahmed, W., Ching, C. & Shoukri, M., Development of two-phase flow downstream of a horizontal sudden expansion. Int J Heat and Fluid Flow, 29, pp. 194–206, 2008. [12] Lockhart, R.W. & Martinelli, R.C., Proposed correlation of data for isothermal two-phase two-component flow in pipes. Chem Eng Prog, 45, pp. 39–48, 1949. [13] Idel’cik, I.E., Memento des pertes de charge. Editions Eyrolles: 61 Bd SaintGermain Paris, 5th edition, 1986. [14] Velasco, I., L’ ecoulement ´ diphasique a` travers un elargissement ´ brusque, 1975. [15] Guglielmini, G., Muzzio, A. & Sotgia, G., The structure of two-phase flow in ducts with sudden contractions and its effects on the pressure drop. Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, 1997. [16] Baker, O. Oil Gas J, 53, p. 185, 1954.




45

Numerical simulation of gas-solid flow in ducts by CFD techniques R. K. Decker1, D. Noriler2, H. F. Meier2 & M. Mori1 1

Chemical Process Department, State University of Campinas, Brazil Chemical Engineering Department, Regional University of Blumenau, Brazil

2

Abstract Numerical models are nowadays used on a large scale for the simulation of two-phase flow processes in the chemical industry, due mainly to their low implementation cost and important results. However, to find the best numerical models in such processes, validation with experimental data is required. In this sense, the main objective of this work is to apply a CFD model under Eulerian-Eulerian framework for the gas-particle flow, with the capability to predict the fluid dynamics of the two-phase flows in vertical and horizontal ducts separated by 90° elbows. A three-dimensional and transient model has been applied for predictions of volume fractions, pressure, velocities and turbulence properties fields. For the momentum transfer between phases a drag model based on the application of “Churchill” asymptotes techniques was used to obtain a continuous function for all flow regimes. Furthermore, the wall effects over the particulate flow have been investigated. The mathematical model was applied in CFD commercial codes for numerical studies and compared with experimental data obtained by the research group at Lehigh University. The model is solved using the finite-volume method with variables located in a generalized co-ordinate system. The main results present the volume fraction and velocities profiles as a function of the time and position. In addition, the mean velocity and particle concentration of the radial profiles were compared with experimental data in different axial positions of the vertical duct. The results showed a good agreement for the mean particle velocity with the increment of the axial position. Keywords: CFD, gas-particle flow, ducts, elbows, numerical simulation.

WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090041


1

Introduction

In the chemical industry several processes occur in the presence of particulate flows, such as the up flow of catalytic particles in riser reactors in the petroleum industry and particle conveying systems found in the cement industry. From one first point of view the particulate flow in ducts seems to be not so difficult to study in comparison with other geometries where the particulate flow also exists. However, the flow phenomena existent in such flows is not simple. There are a lot of variables that may change the characteristics of the gas-solid flow in ducts, such as particle diameter distribution, particle-wall, particle-particle and particlegas interactions, operational conditions, geometry design and so on. In order to analyze such variables, several research groups are dedicating efforts to obtaining experimental data for the gas-particle flow in different geometries and operational conditions. Tsuji and Morikawa [9] and Tsuji et al. [10] for instance developed a complete analysis of the gas-particle flow in the developed flow region of horizontal and vertical ducts by means of Laser Doppler Velocimeter (LDV). Some years later the research group at Lehigh University (Yilmaz and Levy [12]; Akilli et al [1]; Bilirgen and Levy [2]) analyzed the gas-particle flow by means of a fiber optic probe in the presence of different geometrical configurations and devices. The studied geometry was constituted of a vertical duct connected to the first by a 90° elbow and a second horizontal duct connected with the vertical one with another 90° elbow. In order to understand the particle rope formation, mixture and dispersion phenomena in both sections, the authors evaluated devices for rope dispersion at the beginning of the vertical section just after the horizontal-to-vertical elbow and analyzed the parameters of the geometry and the flow operational conditions as well. As a chronological sequence of works showing the behavior of the twophase flow in ducts after the analysis of the developed and developing regions in horizontal and vertical ducts, as previously reported, Yang and Kuan [11] investigated the behavior of dilute turbulent particle flows inside a curved 90° bend using a 2D Laser Doppler Anemometry (LDA). As a result the authors observed the rope formation phenomena inside the bend. Even so, experimental measurements can bring information about the gasparticle conveying as previously described, although simulations of the twophase flow system are also nowadays required. Basically, two phenomenological approaches are employed in the representation of the two-phase flows: the Eulerian-Lagragian (E-L) and the Eulerian-Eulerian (E-E) models. Several numerical studies showing the two-phase flow behavior in ducts were developed in the literature using the E-L approach, [1, 2, 5, 6, 12], and the E-E approach [4, 7]. It is known that the E-L approach requires a great amount of tracking particles to describe efficiently the two-phase flow in ducts, which increases the computational time. Furthermore, the particle-particle and particle-wall interactions are very sensitive to the attributed restitution coefficient, which makes its application difficult for others flow and geometrical conditions. On the other hand, the E-E approach can be applied in a basic methodology by the adoption of an inviscid model, which does not have viscosity stress for the solid WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


47

characterization. Furthermore, in such a model the solid agglomeration behave as a hypothetic fluid and interact with the gas phase by the drag force model (Decker [4]). In this sense, the intention of this study is to compare the drag force model proposed by Coelho and Massarani [3] based on the application of “Churchill” asymptotes techniques with the model developed by Shiller and Neuman [8], as well as to compare the inviscid model proposed in this study with the model based on the kinetic theory of granular flow (KTGF) for the two-phase flow in a vertical section of the test facility proposed by Yilmaz and Levy [12]. For a better understanding of the flow phenomena, particle velocity and concentration maps are also presented for the measuring sections.

2

Mathematical modeling

The mathematical model applied in this work is based on two-fluid, threedimensional and transient flow approach in Eulerian-Eulerian framework. The main assumptions of the model are:  Phases interpenetrating;  Hypothetical fluid;  The viscous stress and pressure of the particle phase is negligible;  Particles are considered to be uniform spheres of same mean diameter;  The momentum transfers is due only the drag force;  The flow is isothermal. 2.1 Conservation equations The equations of mass conservation for gas and solid phases are expressed by the eqn. (1) and eqn. (2), as follows:  (1) f g  g   .f g  g v g   0 , t  (2) f s  s   .f s  s v s   0 . t Whereas the momentum equations for the gas and solid phases are expressed by eqn (3) and eqn (4), as follows  f g  g v g  . f g g v g v g  f g . Τ ef (3) g  f g  g g  p  Fres , t  (4) f s s v s   .f s s v s v s   f s s g  Fres . t









 

2.2 Constitutive equations The effective tensor on the gas phase ( Τ ef g ) establishes a similar relation to the model for a general newtonian fluid, where the stress is directly proportional to ef the deformation rate ( Τ ef g  2 Dg ) and is giving by the expression: WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)




 T ,

ef Τef g   v g  v g

(5)

  g   .

(6)

ef

where

(t)

In eqn (6) the turbulent viscosity,  ( t ) is obtained from an isotropic turbulence model known as standard k- model: k g2  ( t ) C   g . (7) g There are two additional transport equations, one for the turbulent kinetic energy (k) and another for the rate of dissipation of turbulent kinetic energy ():   ef   (8)  f g  g k g   .f g  g v g k g    f g . k k g   f g G g   g  g  , t   g  ef   fggg  . fgg vgg  fg.  g   fg C1Gg  C2gg . t kg  



 







(9)

The turbulent kinetic energy generation (Gg) for the gas phase is obtained by the tensorial product, which is denominated as double dot product, between the Reynolds tensor ( Τ gt  ) and the velocity gradient for the gas phase (vg), as follows: G g  Τ (gt ) : v g .

(10)

The resistive force between the phases ( Fres ) is modeled by the equation: (11) Fres   g ,s v g  v s  . The interface coefficient g,s can be predicted by dilute flow (fg > 0,8), as: g ,s 

f s g v g  v s 3 . CD 4 dp

(12)

2.2.1 Two correlations for the drag coefficient 2.2.1.1 Coelho and Massarani correlation The drag coefficient (Cd) prediction is made for the studied flows by means of a Coelho and Massarani [3] correlation, valid for all Reynolds number range avoiding possible discontinuities provoked by the flow change, typical of a standard discontinue model where its adopted for each Reynolds number range. The correlation is:

 24 Cd    K1 Re p 

   

0.85

 K2

0.85

 p   and K 2  5.31  4.88p . with K1  0.843. log10   0.065  The particle Reynolds number (Rep) is: g v g  v s d p Re p  . g WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

   

1.18

,

(13)

(14)


49

2.2.1.2 Shiller and Nauman correlation Three regimes are considered:  Stokes Regime: Cd  24 / Re p for 0  Re p  200. 



Viscous Regime: 24 Cd  1  0.15 Re 0p.687 Re p





for

200  Re p  2500.

Inertial Regime:

Cd  0.44 for

2500  Re p .

2.2.2 Kinetic theory of granular material model With the aim to verify the inviscid approach, simulations applied the kinetic theory to predict the stress in the solid phase. So, the solid stress Γs can be written as follow: 2  T  Τ s   ps   s   s    v s    v s   v s   . (15)   3   The solid pressure, ps, and the bulk, s, and shear, µs, viscosities are given in terms of granular temperature, s, the radial distribution function, g0, the coefficient restitution, e, and the particle diameter, i.e., (16) ps   s f s  s 1  2 1  e  g 0 f s  , 0.5

s 

s 

4   f s  s d p g 0 1  e   s  , 3   

0.5 2 10   s d p 0.5  4  s   4 1 1 e g f f  d g 1 e         0 s  s s p 0     . 96 1  e  g 0 f s  5  5  

(17) (18)

The radial distribution function is given by:   f g 0  0.6 1   s   f s ,max  

  

1

1

3

  .  . 

and the granular temperature is obtained from an equation:  s    ps   s  v s . Where the dissipation of fluctuation energy takes the form:  4   0.5   s  3 1  e 2  g 0 f s2  s  s   s     v s  .  dp      

(19)

(20)

(21)

2.3 Boundary conditions The boundary conditions for physical frontiers of the gas-solid flow in ducts are as follows. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

50 Computational Methods in Multiphase Flow V 2.3.1 Inlet It is adopted a uniform and constant gas-solid flow with an initial velocity for both phase (U0=29m/s). All other flow properties are considered uniform. 2.3.2 Outlet Constant pressure with continuity conditions considered for all flow properties. 2.3.3 Wall It is considered null derivates for volumetric fractions (impermeable frontiers), and wall logarithm function for the turbulent properties. The wall condition for the gas velocity for all numerical studies was considered to be no-slip and for the particle velocity, no-slip and slip, depending on the desired simulation condition.

3

Numerical modeling

The numerical simulations where conducted by a CFD commercial package, the CFX 4.4. The pressure-velocity coupling applied was SIMPLEC, and the interpolation scheme of first order was the UPWIND due to the strong convective component of the gas-solid transport. The algorithm was applied with the AMG and ICCG proceeding to solve the discrete algebraic equations. The convergence criteria for all studied cases was 1.10-4 for the Euclidian norm in the mass source. The relaxation factors were used only for the turbulent kinetic energy (k) and for its dissipation rate () in the turbulence model with values equal to 0.7 to guarantee stability and convergence. The interactions were solved for a time step equal to 5.10-4 using one time typical implicit integration.

4

Case study

Fig. 1 presents the geometrical system proposed by Yilmaz and Levy [12] for the analysis of the gas-solid flow and details of the numerical grid used for the simulation process. The system is composed of three different duct sections, two horizontal and one vertical (L1, H and L2), connected among them by 90º elbows with bend ratio equal 1.5. The mass load ratio A/F=1 was used for all numerical simulations. The physical and geometrical properties are showed in Table 1.

5

Results

For the analysis of the gas-solid flow in a vertical duct a sequence of studies was developed. The first study was the investigation of the drag force model Table 1: Physical and Geometrical Properties

Physical and geometrical properties.

g (Kg/m3) g (Kg/m.s) s (Kg/m3) dp (m)

1.225 1.850E-05 1680.000 75E-06

p D (m) L1 and L2 (m) H (m)


1.000 0.154 6.100 3.400


51

proposed by Coelho and Massarani [3] in comparison with the default model implemented in the CFD software developed by Schiller and Neuman [8]. The criteria adopted to define each model are the comparison with the experimental data obtained by Yilmaz and Levy [12] as well as the convergence rate. The results presented in fig. 2 in terms of particle velocity shows that both models are equally far from the experimental data. However the Coelho and Massarani model showed much better convergence stability than the Shiller and Neuman model which hardly reduced the computational time for the mass flow balance convergence. Due to this criteria and not just the comparison with the experimental data the Coelho and Massarani model was chosen to be used in this study for other developments.

Figure 1:

Details of geometry and numerical grid.

Once the drag model is defined the analysis of the inviscid model can be carried out and compared with a model based on the theory of granular materials (KTGF). For both developments the drag force model of Coelho and Massarani [3] has been applied. Fig. 3 shows for L/D = 17 that the inviscid model seems to disperse the particulate phase in the transversal section of the duct much better than the KTGF model and more likely the experimental data of Yilmaz and Levy [12]. In this way, the inviscid model showed to be a relevant tool to be applied for the gas-solid flow modeling due to its simple characteristics when compared with E-L models and some others E-E models in the presence of several functions for the solid pressure and viscosity such as KTGF. The results previously presented considered the no-slip condition at the wall for the particulate phase. However, owing the flow direction change promoted by the 90° elbows an analysis of the wall treatment has been developed and WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

52 Computational Methods in Multiphase Flow V presented in fig. 4. As result it is observed that the wall condition just interfere in the region near the wall. However when compared with the experimental data the slip condition showed to be apparently consistent near the wall. This comparison can just be done in the x/D region near to zero due to the better experimental data acquisition at this region near the wall. Considering that the slip condition at the wall showed a reasonable adjustment in these previous numerical experiments combined with the inviscid proposed model, and considering that the Coelho and Massarani model showed to be faster in the convergence of the mass balance between inlet and outlet, a complete analysis of the two-phase flow in a vertical section is developed. For this analysis snapshots of particle concentration, particle velocity and tangential vectors are showed for L/D = 1, 5, 9 and 17 and presented in fig. 5. As can be seen in fig. 5a and fig. 5b for L/D = 1 the particle flow is concentrated in the outer wall of the duct (x/D = 0 and Cp = 2Kg/m³) due to the tangential forces acting in the particulate flow by the presence of a 90° elbow. At the same instant that the ropes are formed, secondary flows in the order of 5m/s start to act, as showed in fig. 5c, redirecting and dispersing the ropes in direction to the opposite wall (x/D = 1) of the vertical section. Owing to the

Figure 2:

Drag force analysis.

Figure 4:

model

Figure 3:

Inviscid analysis.

Wall treatment condition.


model


L/D = 1

a)

L/D = 5

L/D = 9

53

L/D = 17

Particle Concentration

b) Particle Axial Velocity

c)

Figure 5:

Tangencial Vector Plot

Snapshots in different positions (L/D = 1, 5, 9 and 17) for the vertical measuring section. a) Particle concentration ranging from 0,5 Kg/m³ to 2 Kg/m³; b) particle axial velocity ranging from 16 m/s to 32 m/s; c) tangential vector plot for the particle phase ranging from 0 to 5 m/s.

presence of secondary flows the axial particle velocity is also redirect in direction to the inner wall (x/D = 1) with the increment of the L/D region of the vertical section as seen for L/D = 5 and herewith the particle concentration. After its redirection the secondary flows becomes weak and particle axial velocity and concentration are dispersed along the transversal section with the increment of axial position. These observations are showed in fig. 5a and 5b for L/D regions equal to 9 and 17. Fig. 5c also shows that secondary flows also becomes weak and more homogeneous with the increment of axial position. These observations were also reported by Yilmaz and Levy [12] in their studies. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


6

Conclusions

The CFD tools presented and discussed in this work makes possible to know better the gas-solid flow conditions in a vertical duct and how the secondary flow interact with the axial flow dispersing ropes. It is possible to verify the rope formation in the outer wall formed just after the elbow due to the centrifugal forces and the dispersion toward with the increment of the axial position. Furthermore, the inviscid model showed to be a relevant tool to be applied for the two-phase flow modeling due to its simple characteristics when compared with E-L models and some others E-E models in the presence of several functions for the solid pressure and viscosity such as KTGF.

References [1] Akilli, H., Levy, E. K. & Sahin, B., Gas-solid behavior in a horizontal pipe after a 90º vertical-to-horizontal elbow. Powder Technology, 116, pp. 4352, 2001. [2] Bilirgen, H. & Levy, E. K., Mixing and dispersion of particle ropes in lean phase pneumatic conveying. Powder Technology, 119, pp. 134-152, 2001. [3] Coelho, R. M. L. & Massarani, G. Fluidodinâmica de partículas: ainda sobre correlações em base aos dados experimentais de Pettyjohn e Christiansen. Relatório LSP/COPPE 1/96, 1996. [4] Decker, R. K.; Modelagem e simulação tridimensional transiente do escoamento gás-sólido, Campinas: Faculdade de Engenharia Química da Unicamp, Tese, 2003. [5] Huber, N. & Sommerfeld, M., Modeling and numerical calculation of dilute-phase pneumatic conveying in pipe systems. Powder Technology, 99, pp. 90-101, 1998. [6] Kuan, B., Yang, W. & Schwarz, M. P., Dilute gas-solid two-phase flows in a curved 90° duct bend: CFD simulation with experimental validation. Chemical Engineering Science, 62, pp. 2068-2088, 2007. [7] Mohanarangam, K., Tian, Z. F. & Tu, J. Y., Numerical simulation of turbulent gás-particle flow in a 90° Bend: Eulerian-Eulerian approach. Computers & Chemical Engineering, 32, pp. 561-571, 2008. [8] Shiller, L. & Neuman, A. Z., A drag coefficient correlation, Verein Deutschen Ingenieure Zeitung, 77, pp. 318–320, 1933. [9] Tsuji, Y. & Morikawa, Y., LDV measurements of an air-solid two-phase flow in a horizontal pipe. J. Fluid Mech., 120, pp. 385-409, 1982. [10] Tsuji, Y., Morikawa, Y. & Shiomi, H., LDV measurements of an air-solid two-phase flow in a vertical pipe. J. Fluid Mech., 139, pp. 417-434, 1984. [11] Yang, W. & Kuan, B., Experimental investigation of dilute turbulent particulate flow inside a curved 90° bend. Chemical Engineering Science, 61, pp. 3593-3601, 2006. [12] Yilmaz, A. & Levy, E. K., Formation and dispersion of ropes in pneumatic conveying. Powder Technology, 114, pp. 168-185, 2001.



55

Modelling of solidification of binary fluids with non-linear viscosity models T. Wacławczyk, D. Sternel & M. Schäfer Department of Numerical Methods in Mechanical Engineering, Technische Universität Darmstadt, Germany

Abstract This paper deals with the numerical modelling of multiphase flows with phase transition during solidification of binary alloys. First, verification of the effective viscosity assumption in the regime of moving solid (equiaxed crystals) and liquid (melt) for large solid mass fractions is presented. In order to extend the effective viscosity model to the region of stationary solid (columnar crystals); non-linear dependence of the viscosity on the solid mass fraction and the shear velocity is introduced based on the experimental evidence. The proposed formulation is used in a numerical study of the metal alloy solidification in a rectangular cavity. Keywords: multiphase flow, solidification, mushy zone, phase change.

1 Introduction During solidification of binary fluids, e.g. metal alloy Al–Si, Al–Cu, for a certain range of temperatures and compositions a mushy zone is created (cf. Refs. [1, 2]). The influence of the mushy zone morphology on the flow field can be modelled by two approaches: porous media model, where the Carman-Kozeny relation couples local porosity of the medium with the local liquid fraction, or direct modification of the local fluid viscosity relatively to the local solid fraction, see e.g. Refs. [3, 4], respectively. These two physical models are valid in different regions of the mushy layer: the region of stationary, columnar crystals where the solid velocity is us = 0 and the region of the equiaxed crystals where the velocity of solid is assumed to be equal to the velocity of the melt us = ul , see Fig. 1. The difficulties in the modelling of the mushy zone by a one-field model arise during the transition from the mixture velocity (for fluid and solid) to the interstitial velocity in the porous zone, see Refs. [5, 6]. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090051


Figure 1: Schematic presentation of the mushy zone with two characteristic regions: columnar dendrites us = 0 and advected equiaxed crystals us ≈ ul . The disadvantage of the first method is the necessity of the permeability coefficient estimation by experimental or theoretical investigations. This issue is not straightforward because the mushy zone has a complex morphology dependent on the material and external conditions. A porous media model should be used only in the region where the solid phase is stationary. In the case of the second approach, an assumption about the continuous change of the material properties across the solid, the mushy layer and the liquid is used. A common approach employs a lever rule to approximate density and viscosity in the mushy zone together with a linear dependence of the solid fraction on temperature inherited from the linearised phase change diagram, see Fig. 2 and Refs. [5, 6]. The linear dependence of solid and liquid viscosities based on the lever rule is not appropriate since the solid viscosity μs can not be defined. An alternative for the linear viscosity approximation was given in Ref. [7], however, it does not take into account the dependence of the viscosity on the shear velocity. In this paper, based on the experimental evidence from Ref. [8] and the parametrisation study from Ref. [9], we postulate an alternative non-linear dependence of the viscosity on the solid fraction and the shear velocity. The viscosity values used for partial validation of the model were obtained during measurements in a cylindrical rheometer, see Ref. [8]. In section 3 experimentally obtained data, i.e. viscosities as a function of solid fraction and the shear velocity, were used to confirm the possibility of a modelling of multiphase systems by the effective viscosity model. Numerical simulations presented in this paper were carried out with the commercial software Star-CD where the user coding was used for implementation of WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


57

the physical models. The proposed viscosity data parametrisation is tested in the case of the binary alloy solidification in a rectangular cavity, see Ref. [3].

2 Description of the solidification model A solidification model implemented in the Star-CD (ver. 4.06) commercial software is employed. The description of the model is given in supplementary notes distributed together with the program, for this reason here only a short comparison of the model used and other models presented in the literature, see Refs. [1, 3, 5], is given. The set of the conservation equations that describe the mixture medium is obtained by the volume averaging under additional assumptions: equal solid/liquid velocity us = ul , equal solid/liquid density ρs = ρl = ρ and the mechanical equilibrium ps = pl = p. The set of conservation equations consists of: the momentum, the continuity, the energy and the species mass fraction transport equations: ∂uj 2 ∂ul ∂p ∂ ∂ui ∂ (ρui ) ∂ ρuj ui + μ − μ =− + + δij ∂t ∂xj ∂xi ∂xj ∂xj ∂xi 3 ∂xl (1) + ρref gi 1 − β(T − Tref ) , ∂ ρuj ∂ρ + = 0, ∂t ∂xj ∂p ∂ 2T ∂ (ρh) ∂ ρuj h ∂p ∂ui + + uj = + τij +k 2, ∂t ∂xj ∂t ∂xj ∂xj ∂xj ∂ 2C ∂ (ρC) ∂ ρuj C + =D 2. ∂t ∂xj ∂xj

(2) (3)

(4)

One can notice that the mathematical model in the Star-CD allows for simulation of the material compressibility, therefore, modelling of the shrinkage effects is possible. In the mushy zone the variables and the material properties in the above equations represent quantities obtained by volume averaging (see Ref. [5, 6]). Hence, the velocity ui , the density ρ, the viscosity μ, the enthalpy h, the thermal conductivity k, the species mass fraction C and the species diffusion coefficient D represent mass averaged quantities of the solidified alloy (solid) and the melt (fluid) mixture: φ = Cs φ + Cl φ,

(5)

where Cs = ρs cs /ρ and Cl = ρl cl /ρ are solid and liquid mass fractions, cs and cl denote solid and liquid volume fractions and φ represents the aforementioned variables and the material properties of the mixture. The local thermodynamic equilibrium assumption allows to define the temperature T as the equilibrium WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


a)

b)

Figure 2: The phase change diagram a) typical for binary fluids of α + β composition, b) approximation of the phase diagram currently used in the Star-CD valid for the constant melt composition.

temperature. Therefore the buoyancy effects can be approximated using the Boussinesq assumption, see Eq. (1). The enthalpy in the solid, liquid and mushy zone are calculated as follows, see Fig. 2b:

hs =

hl =

cps T cps Ts

: :

T ≤ Ts , Ts < T < Tl

cpl T + (cps − cpl )Tm + L

:

T ≥ Tl

cpl Tl + (cps − cpl )Tm + L

:

Ts < T < Tl

(6)

,

(7)

where L is the latent heat of fusion, cpl , cps are specific heats of the liquid and solid, respectively, and Tm = (Ts + Tl )/2 where Ts , Tl are solidus and liquidus temperatures. The Star-CD code uses a simplified phase change diagram where the enthalpy is calculated for constant (initial) composition of the binary fluid, see Fig. 2b. This kind of simplification is the source of the main difference between solidification model known in literature and the procedure applied in Star-CD. The equation used for the determination of the liquid volume fraction cl is deduced from the simplified phase change diagram, cf. Fig. 2b: cl =

T − Ts , Tl − Ts

c s = 1 − cl ,

(8)

where solidus Ts and liquidus Tl temperatures are constant and must be supplied by the user, whereas the calculation of the liquid mass fractions from the phase change diagram in Fig. 2a requires information about the local composition C WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


59

since it influences its solidus and liquidus temperatures, see Refs. [5, 10]: Cl =

C − Csα , Clα − Csα

Cl =

ρl cl . ρ

(9)

The approximation in Star-CD is based on the assumption that the composition of the binary fluid remains constant. The main consequence of this simplification is a direct dependence of the liquid volume fraction distribution on the temperature since Ts , Tl are set constant in Eq. (8). Hence, one can expect that according to Eq. (8), the cl distribution must follow isotherms. Thus, modelling of the real shape of the solidification front is largely an approximation. One can also notice another implication of the simplified model. The conservation equation (4) is no more a species mass conservation equation but only a solid and liquid mass conservation equation since the composition of the binary liquid is constant.

3 Verification of the effective viscosity model Cs ≤ 0.45 To verify the hypothesis about the applicability of the effective viscosity model in the case of large solid fractions, experimental data obtained from viscosity measurements of a Al−Si metal alloy carried out in Ref. [8] were used. During the experiment, a cylindrical rheometer was placed in an electrical thermostat allowing to sustain constant temperature around it. Measurements of the torque M on the grooved rod rotating inside the cylinder allowed to calculate value of the tension acting at the surface of the rod, see Eq. (10). The number of the revolutions per minute n was used to calculate the shear velocity γ , cf. Eq. (11). The ratio of the tension τ and the shear velocity γ gives the viscosity of the multiphase fluid, see Eq. (12): 1 1 + δ2 τ= M, (10) 2πLRi2 CL 2δ 2 1 + δ2 π γ = n, (11) δ 2 − 1 30 μ=

τ , γ

(12)

where δ = Ra /Ri is the ratio of the cylinder radius Ra and the rod radius Ri . It is important to notice that Eqs. (10–12) are valid only under the assumption of a linear velocity profile between the external cylinder surface and the rotating rod, i.e. a Couette flow assumption. Since the whole cylindrical rheometer was placed inside of the thermostat, it was possible to assume that the temperature and thus the solid mass fraction are constant during the simulation. This simplification allows to employ only the momentum and continuity equations, see Eqs. (1–2) respectively, where all variables and material properties are defined for the multiphase mixture. The isothermal assumption allow for relatively straightforward simulation of this WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

60 Computational Methods in Multiphase Flow V multiphase flow. The single simulation point in Fig. 4 (left) corresponds to the single viscosity measurement for given shear velocity γ see Eq. (11). Experimental data for two temperatures 594◦C, 605◦C were used, density ρ = 2700 kg/m3 was assumed constant during all simulations. After the convergence criterion for the torque calculated on the grooved rod is obtained (variation of the z-component of the total torque was monitored) the new values of the tension, see Eq. (10) and then viscosity, see Eq. (12), are calculated and compared with experimental findings, see Fig. 4. Obtained results show that when using exact experimental data, the flow in the multiphase system is accurately modelled with the effective viscosity assumption. The difference between the experimental value and the numerical solution = 1 − μcf d /μexp can be defined due to the knowledge about the measured viscosities μexp . One can notice that the value of grows with increasing shear velocity γ , see Fig. 4 (left). In the case of the solid mass fraction Cs ≈ 45%, the error starts to grow from the value = 3.9%, γ ≈ 490 1/s until ≈ 22% for γ ≈ 520 1/s the last computational point in Fig. 4 (left, top). For the solid mass fraction Cs = 33% the error grows from ≈ 3.9%, γ ≈ 500 1/s until the solution with the experimentally obtained viscosities does no more follow the experimental data, i.e., ≈ 44%, γ ≈ 600 1/s. The source of the error variation has a twofold nature. First of all, in the case of large solid fractions, for larger shear velocities γ the non-slip condition at the wall of the cylindrical rheometer and the wall of the grooved rod (rod is grooved to avoid slip effect) can be no more satisfied. Secondly, the important factor that limits the range of the measurements in the cylindrical rheometer is the development of the Taylor instability. This phenomenon occurs for large values of the revolutions per minute

a)

b)

Figure 3: The cylindrical rheometer a) characteristic dimensions of the grooved rod and the cylinder Ra = 13 mm, Ri = 10 mm, b) cross section through the numerical model build from about 8 × 105 CV’s. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


61

V[m/s]

V[m/s]

Figure 4: The viscosity obtained from simulations in the cylindrical rheometer compared with experimental data (left) and velocity magnitude for the last computational point (right) in the case of the two solid mass fractions Cs = 45% (top), Cs = 33% (bottom). In the case of smaller solid mass fraction Cs = 33% vortices developed due to the Taylor instability are visible. n and prevents accurate viscosity measurements since the Couette flow assumption is no more satisfied, see Fig. 4. The accurate prediction of this effect in multiphase systems is difficult, since it is directly connected with the variable viscosity of the mixture fluid. The numerical modelling of the flow in the cylindrical rheometer should be further investigated since it might become a valuable verification tool for the experimental investigations.

4 Extension of the viscosity model for Cs > 0.45 The key problem during modelling of the mushy zone by the effective viscosity model is the extension of this assumption for large solid fractions Cs > 0.45 since in this case viscosity measurements in the rheometer are not possible. In fact, WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

62 Computational Methods in Multiphase Flow V α = 0.999

α = 0.987

α = 0.95

Figure 5: Two parametrisations of the effective viscosity: error function erf and piecewise functional pwf viscosity parametrisations. In the case of the erf parametrisation three different α values were used 0.95, 0.987, 0.999, coefficients β = 1e − 4, γ = 9.81 are the same for all three cases. The second pwf parametrisation uses μl = 1 P a · s, μs = 9e + 4 P a · s and Cs,cr = 0.6. Notice that the relative viscosity erf parametrisation μr = μ/μl is almost independent of the α value when Cs ≤ 0.5. viscosity of the solid μs can be considered only as an auxiliary parameter that does not possess physical meaning. For this reason the commonly used approximation: μ = Cs μs + (1 − Cs )μl ,

(13)

where μl is the liquid viscosity, is not valid since μs can be an arbitrarily large number. Alternatives for this approach are rarely presented in the literature, two examples given in Refs. [4, 9] will be shortly discussed below. In the case of the first parametrisation, the mixture viscosity μ is approximated by a piecewise functional approach, cf. Fig. 5: ⎧ ⎪ μl e4.5Cs : Cs ≤ Cs,cr ⎨ μ (Cs ) = (14) b1 Cs + b2 : Cs,cr + 0.1 > Cs > Cs,cr ⎪ ⎩ (1 − Cs )μl + Cs μs : Cs ≥ Cs,cr + 0.1 where Cs,cr (here after Ref. [4] Cs,cr = 0.6) is the critical solid fraction value that defines the highly viscous zone interpreted as the columnar crystals region, see Fig. 1; b1 and b2 are two constants determined by the solution of the two equation system in point Cs,cr , Cs,cr + 0.1. One needs to notice that Eq. (14) employs also the linear dependence given by equation Eq. (13). The main disadvantage of this approach is its discontinuity, cf. Fig. 5, and its lack of physical justification. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


63

The second effective viscosity parametrisation uses error function to approximate variation of the effective viscosity μ (Cs ): √ −B/α β π Cs 1 + μ (Cs ) = μl 1 − α · erf 2 (1 − Cs )γ

(15)

where α, β, γ are parameters to set, B = 2.5, cf. Ref. [9]. The second approach is non-linear and the error function is known to be a solution of the heat transport equation, when the initial condition is given by the Heaviside function. In the case of solidification in the mushy zone, the aforementioned condition can be interpreted as the jump of the enthalpy caused by the latent heat rejected from the solid to the liquid phase, cf. Fig. 2b. Since the solid fraction variation across the mushy zone is expressed by this function, see Ref. [1], and the effective viscosity depends on the solid fraction, it should also be possible to express it in terms of the error function. 4.1 Dependence on the shear velocity γ To obtain a formula for the effective viscosity μ as a function of the solid fraction Cs and the shear velocity γ , additional normalised variables are introduced: the normalised viscosity μr = μ/μl and the normalised shear velocity γr = γ /γC . The values of μl , μs and γC were set to 0.02 P a · s, 500 P a · s and 1000 1/s, respectively, based on the available experimental data for five temperatures T : 594◦C, 600◦C, 605◦ C, 610◦ C, 615◦C and corresponding solid fractions Cs : 0.45, 0.39, 0.33, 0.25, 0.17, see Fig. 6. First, fitting of the continuous functions to the normalised viscosities μr (γr ) obtained for each solid fraction was carried out, in order to obtain μr (γr , Cs = const.). Afterwards, the obtained functions were used to calculate formula for the B coefficient. The remaining coefficients α = 0.988, β = 1e − 4, γ = 9.66 were set only once and are constant in the whole domain. The new, effective viscosity parametrisation μr (Cs , γr ) is given by the following equations: −B(γr )/α √ β π μr (Cs , γr ) = 1 − α · erf Cs 1 + , 2 (1 − Cs )γ

(16)

B = γ˙ −0.52 + 1.1.

(17)



Figure 6: Comparison of the two effective viscosity parametrisations a) linear viscosity see Eq. (13), b) non-linear viscosity cf. Eqs. (16–17).

In Fig. 6 the proposed viscosity parametrisation is compared with experimental data and the old parametrisation given by Eq. (13). One can notice that the chosen approach allows for relatively accurate approximation of the experimental data unlike the original linear approach given by Eq. (13) that over-predicts viscosity values for Cs ≤ 0.5. Trun = 66s

T ◦C

Trun = 36s

T ◦C 650.6

cl

0.85

0.99

cl

659

654.8 T ◦C

Trun = 22s

0.92 cl

0.85

659 650.3

654.8 0.92

0.99

Figure 7: Comparison of the isotherms (left) and the liquid volume fraction isolines (right) in the case of solidification in the rectangular cavity. The two top figures come from the Ref. [3] at Trun = 66 s, the two figures in the middle present the Star-CD result with the linear viscosity model at Trun = 36 s and the two bottom figures depict the Star-CD solution obtained with the proposed effective viscosity model at Trun = 22 s. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


65

5 Solidification of the metal alloy in the rectangular cavity In order to access the properties of the proposed effective viscosity parametrisation solidification of an Al–Cu alloy (μl = 0.003 Pa · s, Ts =548◦C, Tl =660◦C and ρ = 2525 kg/m3 ) in a rectangular cavity (20 mm × 67 mm) is chosen. A detailed description of the test case is given in Ref. [3]. In order to implement the viscosity model given by Eqs. (16–17) an approximation of the error function by elementary function is used (see Ref. [11]) since erf() is not supplied by the compiler intrinsic functions library. The relative shear velocity γr is approximated by the second invariant of the strain rate tensor that can be considered as the mean shear rate. Initially, the binary alloy has constant temperature Ti =660◦C in the whole cavity. The convective boundary condition q = −hconv (T − Tamb ) (hconv = 1 kW/(m2◦ C), Tamb =20◦C and T is temperature in the domain) causes a drop of the temperature close to the left side of the cavity and initialise solidification process. The obtained results in Fig. 7 are compared for different run times Trun because it was noticed that due to the Courant number restriction a restart is required with a ten times smaller time step ( t = 1e − 5 s). Despite this fact it is possible to compare the main features of the reference solution and the obtained results. From Fig. 7 it is clear that the over prediction of the viscosity by the first model, see also Fig. 6, is responsible for the lack of deformation in the mushy zone visible in the case of the new viscosity parametrisation given by Eqs. (16–17). The temperature distribution and the isotherms obtained with the new parametrisation are closer to the reference solution. As mentioned before, in the case of the solidification model used in Star-CD the front of the solidification always follows isotherms unlike in the reference solution. Moreover, because the first order upwind scheme is the only available discretization for the convective term in Eq. (4), smearing due to the numerical diffusion influences the front of solidification in Fig. 7. Finally, the magnitude of the velocity generated by buoyancy effects is closer to the reference solution in the case of the non-linear viscosity model, however, final quantitative comparison can only be performed when the final result is obtained.

6 Conclusions The paper concerns the binary alloy solidification with the effective viscosity model. It was shown that the effective viscosity assumption can be used in the modelling of the multiphase flow when the solid mass fraction Cs ≤ 0.45 showing good agreement with experimental evidence from cylindrical rheometer. Moreover, the influence of the Taylor instability on the measurements in the cylindrical rheometer was emphasised. The proposed viscosity parametrisation covers more accurately the distribution of the experimental data and gives a realistic solution. Further work on this subject should be devoted to the determination of reliable verification test case based on the experimental evidence. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


References [1] Huppert, H.E. & Worster, M.G., Dynamic solidification of binary melt. Nature, 314, pp. 703–708, 1985. [2] Peppin, L., Aussillous, P., Huppert, H. & Worster, G., Steady state mushy layer experiments and theory. J Fluid Mech, 570, pp. 69–77, 2007. [3] Samanta, D. & Zabaras, N., Numerical study of macrosegregation in aluminium alloys solidifying on uneven surfaces. Int J Heat Mass Transfer, 48, pp. 4541–4556, 2005. [4] Nikrityuk, P.A., Eckert, K. & Grundmann, R., A numerical study of unidirectional solidification of a binary metal alloy under influence of a rotating magnetic field. Int J Heat Mass Transfer, 49, pp. 1501–1515, 2006. [5] Ni, J. & Incropera, F.P., Extension of the continuum model for transport phenomena occuring during metall alloy solidification – i. the conservation equations. Int J Heat Mass Transfer, 38, pp. 1271–1284, 1995. [6] Le Bars, M. & Worster, M.G., Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification. J Fluid Mech, 550, pp. 149–173, 2006. [7] Roplekar, J.K. & Dantzing, J.A., A study of solidification with rotating magnetic field. Int J Cast Met Res, 14, pp. 79–98, 2001. [8] Modigell, M., Pape, L. & Moll, A., Cylindrical rheometer viscosity measurements of the Al − Si alloy. Aachener Verfahrenstechnik, Personal communication, 2008. [9] Costa, A., Viscosity of high crystal content melts: depencence on solid fraction. arXiv:physics, 0510191v1, 2005. [10] Le Bars, M. & Worster, M.G., Solidification of a binary alloy: Finite-element, single-domain simulation and new benchmark solutions. J Comp Phys, 216, pp. 247–253, 2006. [11] Error function, approximation with elementary functions. Wikipedia The Free Encyklopedia.



67

A simulation of the coupled problem of magnetohydrodynamics and a free surface for liquid metals S. Golak & R. Przyłucki Faculty of Materials Science and Metallurgy, Department of Electrotechnology, Silesian University of Technology, Poland

Abstract The simulations of induction melting and stirring of the molten metal have focused on the flows inside the bath, while the meniscus forming on the surface of the bath has been neglected. The fact of ignoring the meniscus results in ignoring a whole range of problems caused by the phenomenon, and in consequence all quantitative analyses of the processes occurring on the metal-gas interface are significantly distorted because of the underestimation of the real free surface of the metal. A tendency to ignore the meniscus in the studies on the processes of induction melting results from the complexity of the problem. In order to correctly simulate the phenomenon we are forced to make allowances for a triple coupling of magnetic field, flow velocity field of the molten metal, and shape of the bath surface. The paper presents the methodology for conducting a simulation in which such allowances are made. Keywords: magnetohydrodynamics, meniscus, induction melting and stirring.

1

Introduction

Today, the processes of induction melting and stirring are commonly used in metallurgy. At the same time constant development of the devices utilised in these processes can be observed. In the case of induction furnaces applied for metal melting the researchers focus on increasing their unit power, with the aim to reduce the charge melting time, which in turn means higher efficiency of the furnace. The increased power of the furnace results in enhanced electrodynamic WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090061

68 Computational Methods in Multiphase Flow V field affecting the already molten metal. Consequently, it is hydrodynamic phenomena that acquire significance in the processes of induction melting. In effect, the research concerning the induction furnace design, which previously concentrated only on the analyses of electromagnetic and temperature fields, now must be extended to the issue of modeling the hydrodynamics of the metal bath. So far it has usually been assumed in the research that the area of the liquid metal is determined after it has been melted. In such a case there is a possibility to run a simulation of the heating process of the liquid metal in two separate stages. The first step is to determine the distribution of electromagnetic forces affecting the metal, which can be done when metal geometry, materials parameters, inductor geometry, and supply parameters are known. With the assumption that the geometry of the liquid metal is fixed the obtained distribution of the field forces is constant, too. The second step includes hydrodynamic calculations for the known geometry of the metal, for the material parameters which can influence hydrodynamics, and for the distribution of forces determined in the first step (Adler and Schwarze [1]). Owing to this most widely used methodology, the problem is boiled down to a single-phase flow, which on the one hand simplifies the calculations but at the same time is far from being realistic. In a real process of metal melting a convex meniscus occurs. It is initially dependent on the resultant of the surface energy of the crucible walls and the surface tension of the liquid metal which is depending on the liquid surface area. However, in case of induction melting the ultimate shape of the meniscus and the degree of the phenomenon are strongly influenced by hydrodynamics inside the metal induced by electromagnetic field. The main difficulty with such simulation of the induction melting process is the fact that the shape of the meniscus is affected (through the hydrodynamic phenomena caused by it) by electromagnetic field, and electromagnetic filed is in turn dependent on the geometry of the liquid metal. When one wants to run a simulation of the melting process taking into account the meniscus, the coupling between the shape of the metal bath and electromagnetic field must not be neglected. In this case the hydrodynamic simulation will concern a two-phase system of liquid metal-atmosphere. Any neglect of the meniscus in induction melting will significantly distort the principal quantitative information concerning the process, that is the power output in the charge. Besides, the information about the real shape of the meniscus is of practical use for this process as it allows a prediction of the potential problems with the barrier protection of metal, oxidation of its surface and wearing off of the crucible lining. A process similar to induction melting is induction stirring, which is often an immediate continuation of the former one. Both processes can also take place simultaneously. The goal of induction stirring is homogenizing of the liquid metal and supporting the processes occurring at the liquid metal-gas interface through providing suitable substrates near the area and transporting the products. An example of such a process might be elimination of impurities from the metal



69

through evaporation, as it takes place in the process of removing lead from copper (Blacha et al. [2]). Also in this process, neglecting the occurrence of the meniscus or imprecise estimation of its shape might distort the obtained results completely. First, it must be remembered that estimating the effectiveness of the metal homogenizing process is based on the determination of metal velocity field, which may be calculated correctly only in case of precisely assigned metal geometry and the distribution of electromagnetic forces. Ignoring the meniscus will determine the velocity field that is far from the real one. Still, this distortion may be seen as relatively small in comparison with the error made when estimating the phenomena occurring at the liquid metal-gas interface. The reason is practically linear dependence of the intensity of these phenomena (e.g. evaporation of lead from copper) on the surface area of contact between metal and gas. An assumption of an unreal shape of the metal surface may totally distort the quantitative measure of the reactions occurring on it. For the above reasons it can be said that precise determination of the meniscus is essential for the accurate calculation of most quantitative measures representing utility values of induction furnaces and stirrers.

2 Methodology The software available on the market enables to run the simulations of the induction melting and stirring processes in which the coupling between electromagnetic field and flow field is taken into account. Examples of such applications packages are Ansys Fluent MHD. However, in case of induction melting and stirring velocities of the liquid metal are so small that their influence on the electromagnetic field can be ignored. That is why this model of coupling between metal hydrodynamics and electromagnetics is of no use in case of the devices discussed in this paper. When analyzing the functioning of electromagnetic furnace and stirrer, the simulation performed should allow for the coupling among electromagnetic field, velocity field induced by it, and the changes in the shape of the liquid metal free surface caused by the metal flow. Unfortunately, to the authors’ knowledge no commercial software available on the market allows such a possibility. Therefore it proved necessary to seek for another solution. The authors of this paper decided to create their own system on the basis of the existing software devoted to hydrodynamic and electromagnetic calculations. The main reason for this decision is a high level of development of these instruments, significant optimisation of their codes and their high efficiency, which are extremely difficult to obtain in one’s own solutions created by a small team. The operation of the induction furnace and stirrer was run in 2D space, which was possible thanks to the fact that the configuration considered here both from electromagnetic and hydrodynamic perspective can be regarded as axially symmetrical. The assumption was that the simulation started with the formation


70 Computational Methods in Multiphase Flow V of the meniscus, whose shape was dependent only on the resultant of the surface energy of the crucible walls and the total surface tension of the liquid metal. The first stage entailed the analysis of hydrostatic field through which the initial geometry of the liquid metal was determined. The meniscus shape was determined by VOF (volume of fluid) method allowing a simulation of behaviour of separate, immiscible phases (Hirt and Nichols [3]). In the study discussed here the interaction occurred between liquid metal phase, gas phase, and the crucible wall. The simulation yielded a set of points defining a curve that describes the shape of the free surface of the metal. During the next stage of the simulation the obtained geometry was used to determine electromagnetic field. The result was the distribution of electromagnetic forces acting on the liquid metal. The meshes applied in the programmes for electromagnetic and hydrodynamic calculations do not overlap. The main reason is different methods of solving the differential equations describing electromagnetic and hydrodynamic fields. In the former the finite element method is widely used, while in the latter the finite volume method is preferred because of greater universality. Additionally, since the electromagnetic and hydrodynamic phenomena are different from each other in nature, the optimal mesh for electromagnetic calculations is not optimal in case of hydrodynamic calculations. For this reason the exchange of data between the programmes for electromagnetic and hydrodynamic calculations was based on a rectangular mesh of density twice as high as the density of the meshes used in electromagnetic and hydrodynamic simulations. The decision concerning such density was taken on the basis of Kotielnikow-Shannon law of signal sampling. The values of the force-field components in the nodes of the mesh used for electromagnetic calculations were recalculated to the nodes of the rectangular mesh by bilinear interpolation. The known distribution of electromagnetic forces made it possible to proceed to the simulation of liquid metal hydrodynamics aimed at the determination of the current distribution of liquid metal velocity and the change in the original shape of the meniscus caused by electromagnetic forces. Certainly, the rectangular mesh containing the distribution of forces had to be transformed into the node mesh applied in hydrodynamic calculations. Once more bilinear interpolation was used. Unfortunately, the change in the shape of the meniscus induced by electromagnetic field causes some changes of the field itself. That is why it was necessary to go back to electromagnetic calculations in order to update the field distribution. The set of points defining the curvature of the free surface are the input data for the electromagnetic calculations programme. The main problem encountered at this point is the rate of the above updating. The update rate of field distribution during the hydrodynamic simulation is closely connected with the communication rate between the two separate programmes and with the need to conduct relatively time-consuming electromagnetic calculations.



71

The hydrodynamic simulation is run in the unsteady mode, meaning that it analyses the change in time. An important element is here to adjust an adequate time step. When VOF method is used to trace the free surface of the metal, a time step below 10-4 second is often required in order to ensure the convergence of solution. Since this is the updating rate of the free surface shape, the optimal solution would be to update the distribution of electromagnetic forces with the same frequency, that is 10000 times a second. However, the exchange of such amount of data between programmes and the necessity to conduct timeconsuming electromagnetic calculations make the whole issue very difficult because the simulation time of the stabilization of free surface shape may even be as long as a few dozen seconds. Total time of simulation would then become excessively time-consuming. The only way out seems to be less frequent updating of the electromagnetic field distribution. In such a case another problem occurs, and namely the decision concerning the moment of calculating the forces. This decision may depend either on the selected time step or the change in the shape of free surface. Because the distribution of electromagnetic field does not depend directly on time but on the actual geometry of the metal, the method based on the monitoring of the surface shape seems to be better justified. For the sake of the simulation a scalar measure was defined which reflects the change in the shape of free surface compared with the surface for which the previous electromagnetic calculations were conducted. Exceeding the threshold value (adjusted experimentally) means the necessity to repeat the calculations. z

ρ=0.00 ρ=0.25 ρ=0.50 ρ=0.75 ρ=1.00

r Figure 1:

Relative radial coordinate.


72 Computational Methods in Multiphase Flow V Before the definition of the above measure is provided first the notion of relative radial coordinate ρ should be introduced. It denotes the ratio of the absolute coordinate r to the crucible radius R ratio at a given height, eqn. (1). Figure 1 shows graphic interpretation of the relative radial coordinator.

ρ=

r R(z )

(1)

where: r, z – radial and axial coordinates of the point R(z) – function representing the dependence of crucible inner radius on radial coordinate On the basis of the measurements are obtained the dependencies of the distance of bath surface from the crucible bottom as function of relative radius for the original surface Ho(ρ) and the current surface Hc(ρ). Knowing these values the measure of the surface change SC can be determined: 1

SC = ∫ H o (ρ ) − H c (ρ ) dρ n

(2)

o

where: ρ - relative radial coordinate Ho(ρ) – dependency of the distance of original metal surface from the bottom in function of relative radial coordinate ρ Hc(ρ) – dependency of the distance of current metal surface from the bottom in function of relative radial coordinate ρ n – empirical coefficient Unfortunately, the experiment proved that the assumption that the distributions of electromagnetic forces between the consecutive updates are constant caused that the threshold value of the shape change measure had to be so low that the updating rate of the force field was only slightly lower than the time step in the hydrodynamic simulation. In any other case the convergence of solutions was not possible to obtain. This situation is easily explained by the physics of electromagnetic field, whose penetration is limited only to small distances from the upper, lower and side surfaces of the liquid metal. That is why the strongest electromagnetic forces occur in the upper and lower areas of the metal, nearby the crucible walls. Even the slightest shifts of the free surface cause significant changes in the distribution of the electromagnetic forces in its vicinity. It proved necessary to apply at least an approximate extrapolation of the field distribution based on the changes in the bath shape between the consecutive updating steps. In the simulation considered a simple mathematical extrapolation was chosen. The process of extrapolation entails a comparison of the free surface curvature for which the electromagnetic calculations were conducted with the current curvature, and on the basis of this a re-scaling of the electromagnetic field determined for the original curvature, eqn. (3). WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


G  H (ρ )  G Fc (ρ , z ) = Fo  ρ , o ⋅ z   H c (ρ ) 

73 (3)

where: G Fc (ρ,z) – extrapolated vector of electromagnetic force at the point with the coordinates (ρ,z) G Fo (ρ,z) – vector of electromagnetic force at the point with the coordinates (ρ,z) calculated from the magnetic field equations Hc(ρ), Ho(ρ) – dependency of the distance of current and original metal surface from the bottom in function of relative radial coordinate ρ The above formula is a heuristic one and it is not derived from any physical equations of electromagnetic field. However, this strictly mathematical spatial transformation observes the rule that the highest values of electromagnetic forces are encountered at the boundaries of the metal charge, and it does not change their directions. Its application resulted in a surprisingly efficient hundredfold reduction of the frequency of electromagnetic calculations, limiting the total simulation time to an acceptable amount. Figure 2 presents a diagram of the simulation methodology discussed here.

HYDRODYNAMIC SIMULATION

ELECTROMAGNETIC SIMULATION

START

STOP

Figure 2:

3

Schema of the method.

Example

An experimental simulation was run as part of the presentation of the method. The stirring process of the liquid aluminium was performed in a cone-shaped crucible in order to prove the high degree of the deformation of the liquid metal surface. Figure 3 presents a diagram of the modeled object. Table 1 presents the materials, supply and geometric parameters of the simulation.


74 Computational Methods in Multiphase Flow V A 176 percent change was recorded in the free surface area of the liquid metal (table 2). It means that if the meniscus is not taken into account in the calculations, a similar degree of error should be expected in case of all quantitative analyses of the phenomena occurring at the metal-gas interface.

coil free surface

liquid aluminium

crucible

` Figure 3: Table 1:

Modeled object.

Parameters of the simulation.

Quantity name Frequency Current of source Resistivity of aluminium Volume of aluminium Density of aluminium Viscosity of aluminium Surface tension Contact angle Height of crucible Bottom radius of crucible Top radius of crucible

Table 2:

Value 3 kHz 2903 A 8.8 10 -8 Ωm -4 3 5.31·10 m 3 2375 kg/m -4 9,5·10 kg/m⋅s 1.0 N/m 120° 0.15 m 0.036 m 0.051 m

Free surface area for different meniscus types.

Meniscus not taken into account natural intensified by electromagnetic field

Free surface area [m2] 0.006617 0.006675 0.017046



b)

a)

Figure 4:

Geometry of the liquid aluminium: a) natural meniscus, b) meniscus intensified by electromagnetic field.

Figure 5:

4

75

Velocity distribution.

Conclusion

The methodology described here is now used by the authors in many simulations concerning the influence of various inductor designs, supply parameters and crucible shapes on the process of induction heating and stirring of metals (Golak [4, 5]). Its main advantage, that is the fact that the deformation of the liquid metal surface is taken into account in calculations, offers new possibilities of a more precise optimisation of the devices for induction heating and stirring of liquid metals. Knowing the shape of the free surface of the liquid metal and being able determine its actual size allow a more precise determination of these utility parameters of the devices that are somehow connected with the surface or generally with the shape of the molten metal. The most important of them are the power emitted in the charge and exchange of the elements between the liquid metal and the surrounding atmosphere. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

76 Computational Methods in Multiphase Flow V Our meta-system for the type of simulation described above consists of two commercial packages, which are (a) programme for electromagnetic calculations Cedrad 2D and (b) programme for hydrodynamic calculations Ansys Fluent. However, the presented methodology of triple coupling of electromagnetic field, flow field and liquid metal geometry supported by mathematical extrapolation of force field may be based on any software created for this kind of calculations.

Acknowledgement This research work was carried out within project No. N508 034 31/1889, financially sponsored by the Polish Ministry of Science and Higher Education.

References [1] Adler K., Schwarze, R.: Numerical Modelling of the Evaporation Process of an Electromagnetically Stirred Copper Melt, FLUENT CFD Forum 2005 Bad Nauheim, 2005 [2] Blacha L., Fornalczyk A., Przyłucki R., Golak S.: Kinetics of the evaporation process of the volatile component in induction stirred melts, 2nd International Conference Simulation and Modelling of Metallurgical Processes in Steelmaking STEELSIM 2007, Graz, Austria, pp. 389-395, 2007 [3] Hirt C. W., B. D. Nichols: Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries. J. Comput. Phys., 39, pp. 201-225, 1981 [4] Golak S., Przyłucki R.: Oxidation of the surface of a liquid metal in the induction furnaces., Acta Metallurgica Slovaca 13, pp. 256-259, 2007 [5] Golak S., Przyłucki R.: The optimization of an inductor position for minimization of a liquid metal free surface, Electrotechnical Review, 11/2008, SIGMA-NOT, pp. 163–164, 2008



77

Ammonia concentration analysis for the steam condenser by combining two phase flow CFD simulation with condensation and process simulation K. Karube1, M. Maekawa1, S. Lo2 & K. Mimura1 1

2

Chiyoda Advanced Solutions Corporation, Yokohama, Japan CD-adapco, UK

Abstract Ammonia corrosion in alumi-brass tubes in steam condensers can be a serious problem. It generally occurs in the high ammonia concentration area. In this case, it is planned to replace the alumi-brass tubes by higher grade material, such as cupronickel. Therefore, it is important to minimize the area to be replaced in order to keep the equipment cost down. It is known that the ammonia concentration is related to the degree of corrosion damage. We propose a hybrid analysis method to obtain ammonia concentration by combining two-phase flow Computational Fluid Dynamics (CFD) analysis and network analysis by a chemical engineering process simulator for the tube bundle. Ammonia concentration obtained by the simulation is therefore used to determine the area to be replaced by the higher grade material tubes. Keywords: condensation, process simulation, CFD, ammonia concentration distribution, network analysis.

1

Introduction

A steam condenser is installed downstream of the steam turbine to recover the exhaust steam. It is known that ammonia corrosion of the alumi-brass tubes in areas of high ammonia concentration often occurs in steam condensers. Ammonia is used in water treatment agents, such as pH adjusters and boiler compounds. Usually, the condenser has thousands of tubes that consist of both higher grade material (such as cupronickel) and lower grade material (such as WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090071

78 Computational Methods in Multiphase Flow V alumi-brass) for reducing equipment cost. The higher grade tube is used in the severe corrosion area. However, ammonia corrosion often occurs in the lower grade tube area when the ammonia concentration is high. It is known that the ammonia concentration is related to the actual corrosion damage. Therefore, it is very important to determine the higher grade tube area quantitatively. High ammonia concentration is caused by steam condensation. Ammonia concentration in the steam is usually very low. However, high ammonia concentration occurs during condensation governed by ammonia-water vaporliquid equilibrium. In order to obtain ammonia concentration, coupled analysis is required for two-phase flow dynamics with condensation and ammonia-water vapor-liquid equilibrium. However, direct coupled analysis is very difficult and needs a lot of calculation time. The Computational Fluid Dynamics (CFD) method was used for simulating two-phase flow with condensation [1, 2]. However, it is very difficult to consider phase equilibrium in CFD. A chemical engineering process simulator can simulate rigorous ammonia-water vapor-liquid equilibrium but cannot simulate complicated two-phase flow patterns in the condenser. In order to solve the above problems, we proposed a hybrid analysis method to simulate ammonia concentration in the steam condenser. In this method, twophase flow CFD analysis was used to obtain the steam-water flow in the condenser and the flow distribution was passed to the chemical engineering process simulator for network analysis. Some researchers have proposed techniques in combining CFD and process simulation [3, 4]. Although these calculation techniques were applied to the stirred tank etc, only gas and liquid flows were considered. Few simulation examples were proposed for two-phase flow with condensation, such as steam condensers. The proposed method was very useful and was used to determine the area to be replaced by the higher grade material tubes to achieve an optimum condenser design while keeping the cost of manufacturing or modification to a minimum.

2

Numerical method

For analyzing ammonia concentration, we proposed a hybrid analysis method to simulate ammonia concentration in the steam condenser. In this method, two-phase flow CFD analysis was used to obtain the mass flow distribution of steam and water in the tube bundle of the condenser and the computed mass flow distribution was passed to the chemical engineering process simulator for network analysis. 2.1 CFD method In the CFD analysis, we tried two methods of analysis. One was a single-phase flow analysis and the other was a two-phase flow analysis. In the single-phase flow analysis only the steam flow was calculated and the steam condensation was treated as mass sink in the mass and momentum equations. In the two-phase flow analysis both the steam and water flows were calculated. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


79

The Navier-Stokes equations were solved for the single-phase flow analysis. The Eulerian multiphase flow model was applied to the two-phase flow analysis [5]. In the Eulerian multiphase flow model, mass and momentum conservation equations were solved for both the gas and liquid phases. Since this analysis assumed isothermal condition, the energy equations were neglected. The gas phase was treated as the continuous phase and the liquid phase as the dispersed phase. The mass and momentum conservation equations used in present CFD method are given by (1) ・ αkρk u k   S mass k ・ αkρk u k u k  

 



αk p αkρk g  ・ αk  k   kt  M k  S p k

(2)

where  and u are the phase fraction, the density, and the velocity, respectively.  and t represent the viscosity stress and the turbulence stress, respectively. In the present study, the standard k -  model is used as the turbulent model to estimate turbulence stress t. p and g are pressure and gravity acceleration, respectively. The subscript k denotes the phase in the two-phase flow model, where k = c represents the continuous phase, k = d represents the dispersed phase. In the single-phase flow model, subscript .k in the above equation is eliminated. And the phase fractions  satisfies the following equation. c  d  1 (3) In the equation (1), Smass represents the interphase mass transfer due to the condensation of steam in the steam condenser, which is considered as a constant (Smass = 1.407 kg m-2 s-1) in the present study, simply. In the equation (2), M, which is eliminated in the single-phase flow model, represents the momentum exchange through the interface or a force per unit volume acting between phases. In the present study, only drag and lift force are considered. Hence, M  MD  ML (4) The drag force MD is estimated using following equation.

FD 

3  d  c C D ud  uc ud  uc  4d

(5)

where d is the diameter of dispersed phase. The drag coefficient CD is estimated using the modified Schiller and Naumann [6] correlation, shown in following equation.



 24 1  0.15 Re 0d.687  C D   Re d 0.44 



; 0  Re d  1000 ; Re d  1000

where the particle Reynolds number is defined as below.


(6)


Re d 

 c ud  uc d c

(7)

Inlet of steam

Tube bundle Area

Outlet of steam

Outlet of condensing liquid

Figure 1: Table 1:

CFD analysis model. Analysis condition summary.

Steam mass flow rate Density of steam Density of condensing liquid

kg/h kg/m3 kg/m3

100,000 0.050 1000.0

The lift force ML is defined as below. M L  C L d  c u d  u c   (  u c ) (8) where CL is the lift coefficient. In these analyses, the general-purpose CFD software Star-CD from CDadapco was used, and steady state and isothermal calculation was conducted. Figure 1 shows the CFD analysis model. Around 7,000 regular fine cells were used in the model. A summary of the analysis condition is provided in Table 1. The steam flow in the steam turbine was assumed as the uniform flow distribution for the longitudinal direction of the tubes. Therefore, the two WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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dimensional axi-symmetry model was applied to the sectional plane of the tube bundle. Inlet was located at the top of model and steam flows towards the outlet at the center of the tube bundle. Condensing liquid flows towards the outlet located at the bottom of the model. Steam was condensed as it moves from outside to inside of the tube bundle. The gray part in Figure 1 shows the tube bundle area. Thousands of tubes contained in the tube bundle area of the steam condenser are simply represented by using porous media model. The pressure drop for the tube bundle was calculated by the following empirical correlation, which is counted as the continuous phase momentum source term Sp c of momentum equation (2) [7].   0.1175 S p k  4  c u c2 Re 0 0.16 0.25  (9) N 1.08 (bT / d 0  1)   where bT, d0, and N are the pitch of tubes, outer-diameter of tube, and tube number per meter, respectively. And Re0 is the Reynolds number based on tube diameter, which is defined as below.

Re 0 

Figure 2:

 c uc d 0 c

(10)

Network model.

2.2 Network analysis method In the network analysis, the tube bundle area was divided into relatively coarse cells shown as Figure 2. Each cell was modeled by the heat exchanger and flash drum module in the chemical engineering process simulator. Non-uniform cells were used (coarse cells outside the tube bundle and fine cells inside) for simplifying the model. A commercial process simulator Pro/2 (Invensys SIMSCI) was used for this analysis. In the case of single-phase flow analysis, only the steam mass flow WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

82 Computational Methods in Multiphase Flow V distribution was transferred to the network analysis. Condensed liquid flow was assumed to flow vertically downwards. In the case of two-phase flow analysis, steam and condensing liquid mass flow distributions were passed to the network analysis. The two-phase flow model was considered to have a higher accuracy than the single-phase flow model. Ammonia concentration in the condensation liquid was calculated by the network analysis. The Non Random Two Liquids (NRTL) activity coefficient model was used for the ammonia-water equilibrium relationship.

3

Results

Figure 3(a) and (b) show the steam velocity distribution in the condenser obtained by the single-phase flow analysis and the two-phase flow analysis, respectively. The calculated steam flow pattern was similar in Figure 3(a) and (b), although the highest velocity was a little bit different. Velocity [m/s] High

Low

Outlet of steam

(a) single-phase flow analysis Figure 3:

Outlet of steam

(b) two-phase flow analysis Steam velocity distribution.

Figure 4(a) and (b) show the distribution of the liquid phase ammonia concentration calculated by the network analysis. According to these results, it was confirmed that the ammonia concentration became higher near the steam outlet. It was considered that steam condensation started immediately as the steam entered into the tube bundle. However, ammonia condensed much later because volatility of the ammonia was much higher than steam. Therefore, WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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ammonia was condensed near the outlet where total steam condensation occurred. As we can see in Figure 4(a) and (b), the ammonia concentrations looked very similar, so it was concluded that the difference of steam velocity between both results did not have any significant effect on the ammonia concentration distribution. Ammonia Concentration ratio [-] High

Low

Outlet of steam

(a) single-phase flow analysis Figure 4:

4

Outlet of steam

(b) two-phase flow analysis

Ammonia concentration distribution in the tube bundle.

Conclusions

We proposed a hybrid analysis method to obtain ammonia concentration by combining two-phase flow analysis and network analysis for the tube bundle of a condenser. In this method, two-phase flow CFD analysis was used to obtain mass flow distribution of steam and water in the tube bundle. The calculated mass flow distribution was passed to the chemical engineering process simulator for network analysis. It was confirmed that there was little differences between single-phase and two-phase flow CFD analysis results. Therefore, single-phase flow analysis was more economical to obtain the flow distribution and made simulation easier without losing accuracy. The proposed method was very useful in condenser design or modification to determine the area where higher grade material tubes should be used to avoid corrosion problems.

References [1] S. S. Gulawania, S. K. Dahikara, J. B. Joshia, M. S. Shahb, C. S. amaPrasadb, D. S. Shuklab, CFD simulation of flow pattern and plume dimensions in submerged condensation and reactive gas jets into a liquid bath, Chemical Engineering Science ,63, pp. 2420 – 2435, 2008 [2] S. S. Gulawania, J. B. Joshia, M. S. Shahb, C.S. RamaPrasadb, D. S. Shuklab, CFD analysis of flowpattern and heat transfer in direct contact



[3] [4] [5] [6] [7]

steam condensation , Chemical Engineering Science ,61, pp. 5204 – 5220, 2006 F. Bezzo, S. Macchietto & C.C.Pantelides, A general methodology for hybrid multizonal/CFD models Part I. Theoretical framework, Computers & Chemical Engineering, 28, pp.501-511, 2004 [4] F. Bezzo, S. Macchietto & C.C.Pantelides, A general framework for the integration of computational fluid dynamics and process simulation, Computers & Chemical Engineering, 24, pp. 653-658,2000 M. Yasuhara & H. Daiguji, Numerical Fluid Dynamics-Basic and Applications-, Tokyo University Inc.: Tokyo, pp. 325-328, 1992 Schiller, L., and Naumann, A., Ŭber die grundlegenden berechungen bei der schwerkraftbereitung. Z. Vereins deutcher Ing., 77(12), pp.318-320,1933 N. Suzuki, Chemical Engineering Standard five edition , Maruzen Inc.: Tokyo, pp. 276-277, 1995



85

An approach to the prediction of wax and asphaltene deposition in a pipeline based on Couette device experimental data D. Eskin, J. Ratulowski, K. Akbarzadeh & T. Lindvig DBR Technology Center, Schlumberger, Canada

Abstract An analysis of similarities of turbulent flows in a pipeline and a Couette device is performed. The transport processes in both cases are determined mainly by the boundary layer structure. A wax deposition model requires a single parameter to be determined from the Couette flow experiments. The asphaltene deposition modeling is more complicated and presented by the model framework only. The effect of the centrifugal force on asphaltene particle transport in a Couette device is investigated numerically. An approach to modeling wax deposition in a pipe based on Couette device experimental results is illustrated by a numerical example. The approaches developed can be straightforwardly applied for the transport pipeline design. Keywords: asphaltene, Couette device, deposition, pipe, precipitation, scaling, turbulent transport, wax.

1

Introduction

Prevention of both wax and asphaltene deposition are important problems of oil transport in pipelines. The mechanisms of wax and asphaltene deposition are different. The major mechanism of wax deposition is a molecular diffusion. Due to intensive heat exchange between the outer pipeline wall and the cold environment (for example, sea water) the oil temperature in a pipeline wall vicinity may fall below the “wax appearance temperature” (WAT). Then, wax crystals precipitate from the fluid phase. On one hand the wax crystal concentration gradient directed towards the wall is highest near the wall where the temperature gradient oriented oppositely is biggest. On the other hand the wax particle concentration gradient causes the opposite gradient of wax WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090081

86 Computational Methods in Multiphase Flow V molecules of the same absolute value. The latter causes the diffusion of wax molecules to the wall leading to the deposit layer formation. The growth of the deposit layer thickness is slowed down by the partial deposit removal caused by the shear flow in the vicinity of the deposit surface. There are a number of papers on wax deposition modeling. Some models describing both the diffusion and the shear removal demonstrated fairly good performance (for example, [1]). The physics of the asphaltene deposition phenomenon is fundamentally different from wax deposition. The asphaltene particles precipitate from oil when due to the friction losses the pressure in the pipeline drops below the “asphaltene precipitation onset pressure”. Precipitated particles grow due to the molecular diffusion and the particle-particle aggregation, and eventually reach the wall forming the deposit layer. The asphaltene particles are usually characterized by a wide size distribution because they are prone to forming agglomerates, sizes of which may reach tens of microns. The major mechanisms of particle transport to the wall are (for example, [2]): 1) turbulent and Brownian diffusion; 2) turbophoresis. When a particle collides with the wall it sticks to the wall if the van der Waals attractive force is sufficient to prevent the particle removal by the shear flow. There are many papers devoted to modeling particle deposition in pipes. Most of them are concentrated on the particle transport to the wall only [2, 3]. There are no physically justified models for asphaltene deposition available in literature. Since modeling both wax and asphaltene deposition is associated with a number of difficulties and uncertainties we employed a Couette Device for imitation of the complex deposition processes. In this device the inner cylinder is rotating while the outer one is immobile. The deposit layer is formed on the outer wall.

2

Hydrodynamic similarity

For providing similarity of the deposition process on the wall in a Couette device to that in a pipe the hydrodynamic conditions in the wall vicinity should be similar. For wax deposition the hydrodynamic similarity provides similarity of the shear removal process. For asphaltene deposition the hydrodynamic similarity guaranties the similarities of both particle transport and the probability of a particle sticking to the wall. Note that in the case of wax deposition the temperature gradient at the wall should also be the same as that at the pipe wall. The majority of transport pipelines operate under turbulent flow conditions. A Couette device imitating such a flow should also be run under similar turbulent conditions. Both pipe and Couette flows are relatively simple shear flows. The flow structure in such geometries can be considered as composed of a boundary layer flow and a core flow. Conventionally, the boundary layer is considered as consisting of a laminar sub-layer, a buffer layer and a turbulent boundary layer [4]. Within the laminar sub-layer the momentum transport is mainly controlled by the molecular viscosity. The thickness of this layer is evaluated as  L  5  [4], WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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where     f u * is the conventional non-dimensional thickness calculated assuming that the Reynolds number, based on the layer thickness and the velocity on its boundary, equals unity; u *   w  f is the friction velocity;  f

is the fluid kinematic viscosity; f is the fluid density;  w is the shear stress at the wall. The velocity distribution within the laminar sub-layer is linear. Within the buffer layer the momentum transport is controlled by both the molecular viscosity and the turbulence. The buffer layer thickness is usually evaluated as  B  b  . Different authors use different values of the constant b. Schlichting and Gersten [4] employed b  65 . The velocity distribution within the buffer layer can be described by either a turbulence model or by a so called wall function. The wall function is a unique normalized velocity distribution valid for a wide range of flow parameters. The wall function for the buffer layer in a Couette device can be found in [4] formulated as u   f y  , where

 







u  u u * ; u is the circumferential flow velocity, y  y  is the normalized coordinate; y is the coordinate determining a position within the boundary layer (y=0 at the wall). This normalized velocity distribution in the buffer layer can be employed for a pipe flow as well. The momentum transport within the turbulent boundary layer is controlled by turbulence only. Maintaining high accuracy it can be assumed that the turbulent boundary layer extends to the channel center (this assumption is valid for both Couette and pipe flows). Then the velocity distribution in a core flow can be calculated by applying the Prandtl mixing length model. The velocity distribution in a pipe can be also accurately described by a power law function (see [4]). As one can see from the above analysis the thicknesses of the laminar sublayer and the buffer layer are functions of the shear stress at the wall,  w , and the

fluid viscosity,  f . Let us assume that the particle transport to the wall is not affected by inertial forces (particles are relatively small). Then the hydrodynamic similarity of two near-wall flows is obtained if the shear stress at the wall,  w , and the wall temperature, determining the fluid kinematic viscosity,  f , are the same. The shear stress at the pipe wall is calculated as [4]: U2  w  f f (1) 8 where f is the Fanning friction factor that is a function of the pipe Reynolds number and the surface roughness and U is the superficial flow velocity. Usually, the maximum roughness of the transport pipe walls is below 50 m . Our calculations show that in the majority of flow regimes the transport pipelines are hydraulically smooth. Moreover, at the initial stage of the deposition process the cavities between asperities forming roughness are filled with deposit material, i.e., after a relatively short time the pipe surface is covered with WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

88 Computational Methods in Multiphase Flow V deposit. A newly formed surface is hydraulically smooth therefore for calculation of the Fanning friction factor the Blausius correlation for a smooth pipe can be employed [4]. Based on the experimental particle velocity distribution in the Couette device buffer layer [4] and the velocity distribution in the core flow obtained on the basis of the Prandtl mixing length approach [4] we derived the analytical expression relating the non-dimensional torque G applied to the Couette device rotor and the Reynolds number Re c :  





Re c G

 ln G    

(2)

where G  T /  f  f2 L is the non-dimensional torque, L is the Couette device height, r0 , R are the inner and outer radii of a Couette device respectively, Re c  r0 R  r0   f is the Couette device Reynolds number, T   w  2R 2 L

is the torque,   r0 R is the Couette device radius ratio,



1  , 2   ln 1 1    



 2 , 1 1   

 is the inner cylinder angular velocity,   0.406 .

Equation (2) is in a good agreement with the experimental data for Re c  13000 [5]. Thus, the rotation speed of a Couette device providing the shear stress at the outer wall that is equal to the stress at the pipe wall (Eq. (1)) is easily calculated by Eq. (2). For asphaltene deposition it is important to consider the mechanism of a particle interaction with the wall. As it was mentioned above the main mechanisms of particle transport to the wall are turbulent and Brownian diffusions, and turbophoresis. Brownian diffusion is the dominating mechanism within the diffusive boundary layer only. The thickness of the diffusive layer can be evaluated by assuming that the Peclet number, calculated on the basis of the diffusive layer thickness and the velocity at the layer boundary, equals unity ( Pe B  u  d  d D B  1 ): L (3) d ~ Sc B where Sc B   f D B is the Schmidt number, D B is the Brownian diffusivity. Since for regular hydrocarbons and asphaltene particles the Schmidt numbers are usually very large (up to 109 ) the diffusive boundary layer is much thinner than the hydrodynamic boundary layer. In the majority of practical cases the Brownian diffusion can be ignored. The turbulent diffusion leads to the particle dispersion over a flow domain and plays an important role in particle transport. The turbulent diffusion decreases with increasing the particle size [2]. The turbophoresis is a WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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phenomenon caused by the sharp decrease in turbulent kinetic energy in the wall vicinity. Due to this phenomenon particles acquire velocities directed to the wall. The turbophoretic effect increases with increasing the particle size [2]. Fortunately, as it will be shown below employing a Couette device for imitation of the deposition in a pipe allows simplified modeling the complicated transport processes.

3

Influence of the centrifugal force on particle transport in a Couette device

The centrifugal force in a Couette flow may lead to particle stratification in the Couette device, i.e., an uneven distribution of particles across the gap. This effect must be understood and quantified to properly imitate the deposition in a pipe by using a Couette device. Let us calculate the particle concentration distribution in a turbulent Couette flow assuming the concentration distribution is steady-state. We will neglect the turbophoresis effect because 1) it reveals itself only in the wall vicinity and 2) it occurs in both pipe and Couette flows. Then the convection-diffusion equation takes the form: dc  cv tr  0 (4) DP dr where D p is the particle diffusivity in a turbulent flow, c  c c m ; c m is the mean particle concentration; v tr is the particle drift velocity in a fluid under action of the centrifugal force. Since the amount of particles deposited is small the total volume of suspended particles in a Couette Device is assumed to be constant. Thus, the boundary condition for Eq. (4) is a volume conservation equation for solids: R

 crdr  0.5R

2

 r02



(5)

r0

Thus, the distribution of the relative concentration c does not depend on c m . Assuming that the particle circumferential velocity equals that of the fluid (the assumption is valid for relatively small particles) the particle drift velocity in radial direction can be evaluated by the Stokes settling equation in a centrifugal force field: u r 2 s  f  d s2 r (6) v tr  18 f where d s is the particle size,  f is the fluid dynamic viscosity, s is the particle density. Equation (6) was derived at ignoring the particle fluctuation velocity due to turbulence. However, it provides reasonable accuracy for relatively small particles, such as asphaltene particles, which do not exceed a few tens of WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

90 Computational Methods in Multiphase Flow V micrometers. These are characterized by a relatively low density ( s  1200 kg m 3 ). Our evaluations show that the particle diffusivity D p in this case is close to the fluid turbulent diffusivity (the eddy diffusivity D t ). The latter is numerically close to the turbulent kinematic viscosity  t because the transport of momentum, mass and heat in a turbulent flow has the same mechanism [4]. Usually, it is assumed that Sc t   t D t is in the range 0.8 – 1.0. Note that the eddy and the turbulent thermal diffusivities are equal to each other. 6

5

Rotation Speed = 2948 rpm Particle Density = 1200 kg/m3 Fluid Density = 800 kg/m3 Fluid Viscosity = 1 cp

Particle Size = 30 microns Particle Size = 50 microns Particle Size = 100 microns

C/Cm

4

3

2

1

0 0.014

Figure 1:

0.016

0.018

0.02 0.022 Radial Coordinate, m

0.024

0.026

0.028

The normalized distributions of the volume concentration of asphaltene particles of different sizes along the radius of a Couette device.

Within the present research we will employ the empirical equation for the eddy diffusivity in the wall vicinity ( y   45 ) that can be found in [6]. The eddy diffusivity in the core flow can be calculated by the Prandtl mixing length model with reasonable accuracy (for example, [4]). In Figure 1 we showed the normalized particle concentration distribution by volume vs. the Couette device radius calculated for the different particle sizes: d s  30, 50 and 100 m . The dimensions of the Couette device were r0  14 mm ,

R  28 mm and L  70 mm . The fluid viscosity was

 f  1 cp , the fluid density,  f  800 kg m 3 . The particle density was WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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s  1200 kg m 3 . The relatively high rotation speed (   2948 rpm ) was selected for calculations. One can see that the concentration distribution is relatively uniform for the 30 m particles, but an increase in the particle size leads to rapid strengthening of particle stratification. Since we expect that not very big asphaltene particles (probably smaller than 20 m ) mainly contribute into the deposition there is a high possibility that the centrifugal stratification can be ignored in a deposition study by a Couette device.

4

Calculation principles of the deposition process

As it will be shown below employing a Couette device for deposition enables us to simplify modeling the complicated mechanisms of the particle transport and the deposit layer formation. 4.1 Wax deposition calculation

Let us characterize wax deposition in a pipeline of a given diameter D at the known superficial velocity U . We employ a hypothesis that in the wall vicinity the precipitated particles are in thermodynamic equilibrium with the fluid. In this case the concentration distribution of precipitated wax particles near the wall is determined by the temperature distribution. Then the wax molecule flux to the wall can be evaluated as [1]:  dT   dc   dc   D m  q w  D m      dr dT   rR   TT R   dr  r R

(7)

where D m is the wax molecular diffusivity; dc dr is the gradient of the wax molecular concentration; dc dT is the rate of the wax molecular concentration change with temperature that can be measured or calculated. Not the whole wax amount transported to the wall by the molecular diffusion will deposit due to shear removal. The removal rate depends on the shear stress at the wall and the rheological properties of the deposit layer. Currently, there is no clear understanding of mechanism of this phenomenon. The shear removal effect can approximately be taken into account by replacing the molecular diffusivity with an effective diffusivity,  , that can be identified from a Couette device experiment. This approach is plausible since it is difficult to separately quantify the oil molecular diffusivity and the shear removal rate. The wax flux contributing to the deposit growth is calculated as:  dc   dc   dT     q d     (8)     dr  r  R  dT  T T R   dr  r  R where  is the effective diffusivity of wax molecules. Note that for this application the Couette device is equipped with a heater incorporated into the inner cylinder, and a cooling jacket mounted over the outer WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

92 Computational Methods in Multiphase Flow V cylinder. This design allows maintaining the temperature of the outer wall constant and below WAT as well as controlling the temperature gradient at the wall to imitate deposition conditions in a pipe. The amount of wax deposited in the Couette device during the time t, at neglecting the wax depletion effect, is calculated as:  dc   dT  M d  q d s 2RL  t      s 2RL  t dT   T T R   dr  r R

(9)

Thus, if the deposit amount in a Couette device is accurately measured, the effective diffusivity  can be straightforwardly calculated from Eq. (9). Let us illustrate how this technique can be applied for calculating the deposition layer thickness in pipe. To determine  we need to run a Couette device experiment. For providing the shear stress at the outer wall equal to that at the pipe wall Eqs. (1) and (2) have to be used to calculate the required inner cylinder rotation speed. For providing the equality of the temperature gradients at equivalent hydrodynamic conditions in the wall vicinity the heat flux through the wall of the Couette device should be equal to that through the pipe wall. The heat balance equation for pipe flow can be written as follows: dTb 4k T  Tw    bw b dx  f c p UD

(10)

where c p is the isobaric heat capacity of a fluid, D is the pipe diameter, k bw is the heat transfer coefficient from the fluid to the pipe wall, Tb is the temperature in the central area of a pipe, Tw is the temperature at the deposit surface (initially, at the wall surface), x is the coordinate along a pipeline. The temperature at the deposit surface is below the wax appearance temperature. The initial temperatures Tb 0 and Tw 0 at x  x 0 are given. For the illustrative purpose only we use a simplified approach. To avoid the routine calculation of the heat transfer through the growing deposit layer we assume that the temperature at the deposit layer surface Tw is constant along a pipe. This assumption to some extent takes into account an observation that the low conductivity of the growing deposit layer prevents the deposit surface from cooling. The same assumption means also that the deposit layer should be relatively thin and we can ignore an increase in the temperature Tw at a fixed pipe cross-section in time due to the deposit layer insulation effect. Then we can also assume that the physical parameters of a fluid remain constant along a pipeline. Thus, the heat transfer coefficient k bw is constant. Then Eq. (10) is solved analytically. After performing a routine math we obtain the distribution of the temperature difference T  Tb  Tw along a pipe as: T  T 0 e



Nu x Re Pr D


(10)


93

where Nu  k bw D  f is the Nusselt number,  f is the fluid heat conductivity, Pr   f a f is the Prandtl number, a f   f c p  f is the fluid thermal diffusivity. The Nusselt number can be evaluated by the empirical equation as follows [7]: Nu  0.027 Re 0.8 Pr 0.33 (11) The temperature gradient at the wall is calculated as: q k T  dT  (12)   bw   bw   f f  dr  r  R where q bw is the heat flux to the wall. Substituting this equation into Eq. (9), we get the following deposit flux to the wall: Nu x

 k  dc   dc  T 0 e Re Pr D q d x        bw    f  dT  TT R   dr  r  R

(13)

The deposit layer thickness is calculated from the deposit volume balance as: q x   t (14) x   d 1  where t is the time,  is the wax deposit layer porosity. 4.1.1 Calculation examples Let us now consider a pipeline with diameter D  0.1778 m and length L p  3200 m in which oil of a certain chemical composition flows at a

superficial velocity U  3.1 m s . The initial oil temperature is set to Tb  350 K and the wall temperature Tw  322 K . The oil dynamic viscosity is assumed to be  f  7  10 3 Pa  s , the density  f  843 kg m 3 , the heat

conductivity  f  0.15 W m  K  , the heat capacity c p  2020 J kg  K  .

The rate of the wax molecular concentration change with temperature at the wall, calculated for the known oil chemical composition by the DBRSolids commercial software, is dc dT T322 K  1.28  10 4 1 K . The wax particle density is s  900 kg m 3 . The deposition experiment in the Couette device (the dimensions were presented above) was conducted at a rotation speed of the inner cylinder set to   3900 rpm . The wall temperature and the temperature gradient were maintained the same as those in the pipeline. The two hour experiment produced M  270 mg of wax deposited on the outer wall. The effective diffusivity, calculated by Eq. (9) was   1.66  10 10 m 2 s that is in line with the literature data on the molecular diffusivity [1]. In Fig.2 we showed the calculated thicknesses of the wax deposit obtained at various flow times: t  10, 50, 100 h . The wax deposit thickness linearly WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

94 Computational Methods in Multiphase Flow V increases with time (Eq. (14)) and sharply decreases along the pipeline due to rapid reduction of the temperature gradient at the wall caused by fluid cooling. Note that the effect of the deposit layer growth rate reduction in time due to the deposit insulation effect is not taken into account by the employed simplified model. 7

t=10 h t=50 h t=100 h

D=0.1778 m U=3.1 m/s Fluid Viscosity = 7 cp Effective Wax Diffusivity=1.66 e-10 m2/s

6

Deposit Thickness, mm

5

4

3

2

1

0 0

500

1000

1500

2000

2500

3000

3500

x, m

Figure 2:

Distribution of the deposit thickness along a pipeline for the different operation times.

4.2 Asphaltene deposition modeling

The forecasting of asphaltene deposition is a more complicated problem than that of wax deposition. In this paper we present the model framework only. Guha [2] suggested a robust convection-diffusion model, according to which the volume flux of particles depositing on the wall is determined as: dc J  D t  D B   cVpy (15) dy where Vpy is the particle drift velocity caused by turbophoresis, that can be calculated for a given particle size [2]. The first right-hand side term determines the particle flux due to Brownian and turbulent diffusion, the second term due to the turbophoresis. Equation (15) requires a boundary condition at the wall that should be set in dependence on the probability of a single particle deposition after a contact with the wall. Note that no clear approach to the boundary condition formulation is available in the literature. The same author [2] showed that the deposition mechanism (diffusion or turbophoresis) is determined by the particle inertia. The particle velocity relaxation time   s d s2 18 f  is employed as the measure of the inertia. It is WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


95

convenient to use the dimensionless particle relaxation time:    u *2  f . Guha [2] compared the deposition rate results obtained by the convection-diffusion model with the experimental data for deposition of aerosol droplets in a pipe. It was assumed in the calculations that all particles reaching the wall deposit on it. The author [2] demonstrated that small particles (    0.1 ) move to the wall mainly due to the diffusion, whereas for large particles (    1 ) turbophoresis dominates. Thus, there is a significant size range where both diffusion and turbophoresis are important. It is should be noted that according to Guha [2] in the medium particle size range ( 0.1     1 ) the calculated results deviate noticeably from the experimental data. An interpretation of the Couette device experimental data for asphaltene deposition is complicated because both the particle size distribution and the deposition mechanism are not known a priori. Let us assume that only particles smaller than a certain (critical) size d cr stick to the wall as a result of a particlewall collision. This assumption is explained by considering the force balance for a particle attached to the wall. A drag force tending to remove the particle from the wall increases with increasing the particle size, while the particle - wall van der Waals attraction force per unit mass is reduced. The flux of “small” particles to the wall can be determined as (see Eq. (15)):  dc small  small   c small Vpy (16) J d   D p  y 5    dy    y 5





where csmall is the volume concentration of “small” particles, i.e. particles small is the mean turbophoretic velocity of smaller than the critical size d cr , Vpy small particles. For the sake of convenience we consider the particle flux at the laminar boundary sub-layer surface ( y   5 ). This flux is practically equal to the flux of particles depositing on the wall under steady-state conditions (the continuity equation for particles within the boundary layer is dJ d dy  0 ). Since the particles are small the particle diffusivity is approximately equal to the eddy diffusivity ( D p  D t ). According to Notter and Sleicher [6] the eddy diffusivity





at the laminar sub-layer surface is: D t y   5  0.104   f . Assuming that the rate of establishing the concentration distribution profile along a Couette device radius is much higher than the rate of changing the concentration of small particles in the Couette device, we obtain an equation that describes the evolution of the mean concentration of small particles in time: dcsmall m t  t   J d 2RL  q small v dt Vc

(17)

is the volumetric rate of generation of small particles, Vc is the where q small v volume of a Couette chamber.


96 Computational Methods in Multiphase Flow V On the basis of Eqs. (17) and (16) we formulate the convection-diffusion equation describing the deposition process:   dc small  dc small  small m   t   c small  q small  c small Vpy v m  0.104 f   dt dy     y  5  2  q small ( t )  c small v m  R 1  2





where c

small

c

small

c small m



y 5 

 2   R 1   2  





(18)



is the relative volume concentration of small

particles,  is the effective deposition velocity, which is constant for given flow parameters. The initial condition for this equation is that the initial volume concentration of small particles is zero: c small m 0   0 . Since the parameter  does not depend on the concentration it can be identified from a constant pressure Couette device experiment where asphaltene particles are pre-generated. The particle generation term in Eq. (18) is zero in 0  c small we obtain the this case. Assuming that the initial condition is c small m m0 analytical solution as:

c small m (t)

2 t  small R (1  2 ) c m0 e

 The deposit mass for the time t is then calculated as: 2t  t   2 2 small  R 12       Mt   s 2RL   c small dt   R ( 1  ) L c 1 e m s m0    0  



(19)

(20)

There are two unknowns in this equation:  and c small m 0 . To exclude the variable csmall we can use the results of two deposition experiments of two m0 different durations ( t1 and t 2 ). Then the effective deposition velocity can be determined from the following equation: 

2 t 1



 M t1  1  e (21)  2 t 2 M t 2   R 1  2  1 e One of the most difficult problems is evaluating the rate of small particle generation. It is expected that this generation rate should change from maximum to zero due to the fluid depletion. The evaluation of this rate is possible if monitor the deposit mass while conducting the deposition experiments at a gradual pressure reduction with the same rate as that observed in the pipeline ( dp dt  U dp dx ). The effective deposition velocity and the rate of small particle generation determined by the Couette device experiment can be straightforwardly employed for predicting the asphaltene deposit thickness in a pipeline. R 1  2



5

97

Conclusions

An analysis of similarities of turbulent flows in a Pipeline and a Couette device has been performed. It has been shown that the transport processes in both cases are determined mainly by the boundary layer structure. The wax deposition model requires a single parameter to be determined from the Couette experiments. The asphaltene deposition modeling is more complicated and only the model framework has been presented here. The effect of the centrifugal force on asphaltene particle transport in a Couette device has been investigated numerically. It has been also demonstrated how experimental data obtained in the Couette device can be used to predict the wax deposition thickness along a pipeline at different production times.

References [1] Akbarzadeh K., Zougari M., Introduction to a Novel Approach for Modeling Wax Deposition in Fluid Flows. 1. Taylor−Couette System Ind. Eng. Chem. Res., 47(3), pp 953–963, 2008. [2] Guha A, Transport and Deposition of Particles in Turbulent and Laminar Flow, Annual Review of Fluid Mechanics, 40, pp. 311-341, 2008. [3] Johansen S.T., The Deposition of Particles on Vertical Walls, International Journal of Multiphase Flow, 17(3), pp. 355-376, 1991. [4] Schlichting H., Gersten K., Boundary-Layer Theory, Springer-Verlag, Berlin, Heidelberg, New York, 2000. [5] Lewis G.S. and Swinney H.L., Velocity Structure Functions, Scaling and Transitions in High-Reynolds-Number Couette-Taylor Flow, Physical Review, 59(5), pp. 5457-5467, 1999 [6] Notter R.H., Sleicher C.A., Eddy Diffusivity in the Turbulent Boundary Layer Near a Wall, Chemical Engineering Science, 26(1), pp. 161-171, 1971. [7] Rohsenow W.M., Hartnett J. and Cho Y.I., Handbook of Heat Transfer, McGraw-Hill, 3-rd ed., 1998.




99

Large amplitude waves in a slug tracking scheme A. De Leebeeck & O. J. Nydal Department of Energy and Process Engineering, Norwegian University of Science and Technology (NTNU), Norway

Abstract Large amplitude roll waves are incorporated into a previously developed slug tracking scheme for two phase gas-liquid pipe flow. The applicability of the tracking scheme to large amplitude waves is demonstrated with a simplified model for the waves. The waves are modelled analogous to slugs on a moving grid with corresponding wave velocities and a pressure variation determined using an orifice type relation. Slugs and waves in the tracking scheme are separated by regions of stratified flow, which are modelled on a stationary grid using the two-fluid model. The computational scheme is described, compared to experimental data on roll waves, and some wave dynamics such as waves developing to slugs and slugs decaying to waves are demonstrated. Keywords: roll waves, tracking, two phase pipe flow, modelling.

1

Introduction

In two phase gas-liquid pipe flow, different flow regimes occur depending on gas and liquid phase velocities, fluid properties, and pipe geometries. Various numerical strategies exist for the different flow regimes in dynamic models. Slug flow, for example, can be treated with unit cell models (Bendiksen et al. [2]), in slug capturing (Bonizzi and Issa [3], Issa and Kempf [7], Renault [12]), or in tracking schemes [Taitel and Barnea [13], Nydal and Banerjee [11]). Although capturing schemes (Issa and Kempf [7], Bonizzi and Issa [3]) can model the initiation of slugs and roll waves, they require the use of fine grids which are computationally expensive and the large computational times are prohibitive for simulation in long pipelines. Tracking schemes, however, use WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090091

100 Computational Methods in Multiphase Flow V orders of magnitude fewer grid points. Tracking schemes can also be suitable for plug simulations (Kjølaas [9]). A combination of capturing and tracking has also been tested by Renault [12]. Large roll waves can have similar scales and behavior to slug flow in that they transport liquid and have a propagating front. A simple model treating waves as moving objects in a similar way as for slugs is therefore desired in the tracking scheme. Similarities between slugs and roll waves such as a propagating front and a sloping tail have been observed experimentally in, for example, Johnson [8]. Pressure variations across wave fronts similar to slugs have also been measured (De Leebeeck et al. [4]). These experiments are used to develop a wave model including the observed pressure variation. The slug tracking scheme of Kjølaas [9] is the starting point for incorporating wave tracking capabilities into a slug tracking scheme. Slug flow is modelled as alternating liquid slugs and bubbles with stratified flow. The two-fluid model is solved on a stationary staggered grid in bubbles, while integral momentum equations are solved in slugs on a moving grid. Before the addition of wave tracking, decaying slugs were replaced immediately with stratified flow. With the addition of wave tracking, slugs can decay into waves, modelled analogous to slugs with integral momentum equations and their own front and tail velocities. The tracking scheme of Hu et al. [6] includes wave tracking with a liquid height profile behind waves and slugs, solving the two-fluid model in combination with modelling the wave front as a hydraulic jump (Hu et al. [6]). Other models for roll waves include, for example, Johnson [8] and Holmås [5] who solve the two-fluid model with modified friction terms in roll waves. Johnson [8] assumes a sequence of repeating “maximum amplitude” waves with a sharp front and includes a unique interfacial friction factor as part of a steady state solution. The model of Holmås [5] model is dynamic and includes increased turbulence at the wave front using a modified Biberg friction model. In our scheme, we simplify waves and slugs as square objects that can be modelled dynamically on a coarse grid. The dynamics of the waves and slugs are determined from mass and momentum balances. The stratified gas regions between waves and slugs are solved with a two fluid model. A large grid gives square shaped bubbles. Slug and wave tails can be reproduced by refining the grid in the bubble region.

2

Description of the model

The wave tracking model builds on a slug tracking scheme (Kjølaas [9]) which is coded in C++ using object oriented programming techniques. Slug flow is represented in one dimension with alternating slug objects that completely fill the pipe and stratified sections including both phases as shown in figure 1A. In stratified sections, the two-fluid model is solved on a stationary staggered grid where phase velocities are determined at section borders while pressure and masses are determined at section centers. Slug sections are modelled as moving



101

objects where liquid phase velocity, slug length, front and tail velocities are determined from mass and momentum balances. Waves, shown in figure 1B, are modelled in a similar way to slugs as moving objects and include a pressure variation due to liquid acceleration at the wave front. In the gas phase, the pressure variation across the wave front is modelled with an orifice type relation. Assuming that the pressure variation is the same in both phases, the phase velocities in the wave can be determined from the momentum balance equations. Front and tail velocities and liquid holdup are also determined in the wave assuming a fixed length as opposed to slugs which have a variable length.

Figure 1:

Schematic of models for A. slug flow and B. wave flow in the tracking scheme. The arrows indicate direction of flow. Dashed lines represent section borders. Gray – liquid phase. White – gas phase.

Gas flow in a slug is modelled using a slip relation, however, in a wave the gas phase flows through a gap between the liquid phase and the upper pipe wall. In this way, gas flow over a large wave can be thought of as similar to gas flow through an orifice, and therefore an orifice type relation is used in the gas momentum balance eqn (1) for waves. The orifice type relation, the second term in eqn (1), replaces the gas wall and interfacial friction terms. Eqn (1) is then used to determine the gas velocity in a wave by relating it to the pressure variation across the wave front.

Mg ∆t

∆U g

  1 − H 2  1 1     ⋅ (U gn+1 − U front ) U gn − U front ρ g 1−  2  2 Cd 1− HR      = (1 − H ) A( PL − PR ) − M g g sin θ + (1 − H ) A

(1)

Using the same pressure variation across a wave front as in eqn (1), the liquid momentum balance eqn (2) is used to determine the liquid phase velocity in a wave. In the liquid phase, the main component giving pressure variation is the WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

102 Computational Methods in Multiphase Flow V acceleration of the liquid at the wave front. There is also a contribution from liquid wall friction and gravity.

Ml ∆U l + Hρ l A(U ln − U front )(U lRn − U ln+1 ) ∆t = HA( PL − PR ) + (− 18 LSl λl ρ l U ln U ln+1 ) − M l g sin θ

(2)

Slugs are modelled as objects with moving boundaries, the front of a slug moves with a front velocity determined from a mass balance across the front while the tail moves with a bubble nose velocity. If the front velocity is greater than the tail velocity, the slug will grow in length, otherwise its length will decrease. Similarly, waves are modelled as moving objects but they have a fixed length of one to two pipe diameters and move with the wave front velocity. The front velocity of a wave is determined from the mass balance across the front, eqn (3), and given in eqn (4). (3) H (U ln +1 − U front ) = H R (U lRn − U front )

U front =

HR H U lRn U ln +1 − H − HR H − HR

(4)

One of the aims of the wave tracking scheme was to have a simplified model, therefore a simple wave tail speed relationship was desired. The wave tail speed is given in eqn (5). U tail = 1.2U l (5) The factor of 1.2 allows for continuous transition between wave and slug flow. When the liquid holdup in a wave approaches unity, the liquid phase velocity in the wave approaches the mixture velocity. The bubble nose velocity or wave tail velocity is commonly related to the mixture velocity by a factor of 1.2. The mass balance equations in a wave or slug are the same, where the change in mass in a given time step is the difference in mass flux in and out. Eqns (6) and (7) are the liquid phase and gas phase mass balance equations respectively. The liquid holdup is given in eqn (8).

∆M l = M ln (U ln+1 − U tail ) / L − M ln (U ln+1 − U front ) / L ∆t ∆M g = (M gn (U gn+1 − U tail ) / L − M gn (U gn+1 − U front ) / L ) ∆t M n+1 H= l ALρ l

(

)

(6) (7) (8)

Since wave fronts are modelled with a fixed length and they move at the wave front velocity, the wave tail velocity only appears in the mass balance eqns (6) and (7). If the front speed is larger than the tail speed the liquid mass in the wave will increase and vice versa. Therefore waves can grow or decay in amplitude.



103

2.1 Transitions and wave insertion Waves can be formed from stratified flow or they can form from decayed slugs. At the transition from stratified flow waves are inserted according to the inviscous Kelvin Helmholtz stability criteria, eqn (9). Neglecting surface tension and viscous effects, stratified flow is stable if (Barnea and Taitel [1], Lin and Hanratty [10]):

 Al Ag  + ρ S  l i ρ g Si

 ( ρ l − ρ g ) g cos θ − (U g − U l ) 2 > 0  

(9)

On the other hand, if a wave is dying, it will be removed when the holdup in the wave approaches the holdup in the stratified section in front of it. A decaying slug will be converted to a wave when its length goes below a user defined minimum, i.e. one or two pipe diameters in length. In the reverse case where a wave grows to a slug, a wave will be converted to a slug if its holdup goes above a user defined maximum, e.g. a holdup of 0.99. Slugs can also form if two waves merge or a slug overtakes a slower moving wave.

3


The roll wave tracking model has been compared with experiments that were conducted in the multiphase flow laboratory at NTNU in a 16 m long, 0.06 m I.D. pipe using air and water at atmospheric pressure (De Leebeeck et al. [4]). The experiments included pipe inclinations from -1 to 3 degrees, Usg from 2 to 11.5 m/s, and Usl from 0.04 to 0.52 m/s where a mixture of waves and slugs occurred. Data for comparison include liquid holdup and pressure time traces, average wave velocities from cross correlation between holdup time traces, and pressure drop. One purpose of doing the experiments was to measure the pressure variation over a wave, as this is an assumption in the tracking model. This was confirmed in the experiments, and led to an estimate of the discharge coefficient in the orifice relation in eqn (1) of Cd = 0.2 to 0.4. The simulations discussed here used a fixed grid size of 2 pipe diameters in wave fronts, a minimum of 20 and maximum of 100 pipe diameters in stratified sections. Pipe length, diameter, fluid properties, and a simulation time of 80 sec were as in the experiments. A discharge coefficient giving the best approximation of experimental wave speed and pressure variation with value Cd = 0.2 was used in the gas momentum equation for waves. The waves were inserted at a similar frequency to the experiments. Figure 2 shows a plot of wave or slug velocity for all experiments and wave velocity from the model plotted against mixture velocity. The largest velocities are associated with slugs which have larger velocities than waves. The experiments with lower velocities contained more waves than slugs. Looking at the data qualitatively, the model gives wave velocities in the same range as the experiments.



Wave or slug velocity (m/s)

7

model experimental

6 5 4 3 2 1 0 0

Figure 2:

2

4

6 8 Mixture velocity (m/s)

10

12

Experimental velocities compared with wave velocities from dynamic wave tracking simulations. For Umix < 4 m/s: mainly slug flow. Umix > 4 m/s: a mixture of waves and slugs.

600

Pressure drop (Pa/m)

500

model experimental

400 300 200 100 0 0

Figure 3:

2

4 6 Mixture velocity (Usl + Usg) (m/s)

8

10

Experimental pressure drop compared with averaged pressure drop from tracking simulations.

In figure 3, the experimental and modelled pressure drops are compared. Although all of the simulations reproduced a pressure variation in waves, some of the pressure drops tended to be low compared to the experimental values. The WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


105

0.5

Pressure (Pa)

0.5

0 6

0 22

23 Time (sec)

A

Figure 4:

Liquid velocity (m/s)

0 Liquid velocity (m/s)

1.0e5

Holdup (-)

1.0e5

1.1e5

1.1e5

1.0e5

Holdup (-)

Pressure (Pa)

1.1e5

Holdup (-)

Pressure (Pa)

magnitude of the pressure drop depends on the number of waves and slugs in the experiment or simulation. An experiment with more slugs will have a larger pressure drop than one with fewer. All the experiments contained a mixture of waves and slugs but some of the simulations, especially downward inclined, reproduced waves which did not grow to slugs resulting in low pressure drops. One advantage of modelling waves in a tracking scheme is that coarse grids can be used allowing for longer pipe systems to be modelled. Using a coarse grid means that waves and slugs are modelled as square objects corresponding to the plots figure 4A, using a maximum stratified section length of 100 pipe diameters. Finer grids can be used, however, allowing for waves with more apparent tails, shown in figure 4B, as occurs in the experimental time traces, for example figure 4C.

6

0.5

0 16

17 Time (sec)

0 32

Time (sec)

33

B

C

Pressure, holdup and liquid velocity time traces where a wave passes at a given location in the pipe. A. A coarse grid with maximum length 100 pipe diameters. B. A fine grid with maximum length 10 pipe diameters. C. Experimental. Usg = 8.0 m/s, Usl = 0.1 m/s, θ = 1 degree.

Figure 5 shows how an experimental holdup time trace compares to a simulated time trace on a fine grid at the same Usg, Usl and pipe inclination. The model time trace in figure 5B shows a mixture of slugs and waves of various sizes occurring in the pipe as well as the shape of the waves. Wave dynamics such as waves growing to slugs, or slugs decaying to waves is inherent in the tracking model. Examples of a wave growing to a slug and a WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


A

Holdup (-)

1 0.5 0 20

25

30

35

40

45 50 Time (sec)

55

60

65

70

25

30

35

40

45 50 Time (sec)

55

60

65

70

B

Holdup (-)

1 0.5 0 20

Figure 5:

Holdup time traces A. experimental and B. simulation using a fine grid with maximum length 10 pipe diameters. Usg = 6.09 m/s, Usl = 0.18 m/s, θ = 2 degrees.

Pressure variation (Pa)

10000

0

Holdup (-)

1

Liquid velocity (m/s)

0 6

0 32

32.2

32.4

32.6 32.8 Time (sec)

A

Figure 6:

33

33.2

33.4

B

Pressure variation, liquid holdup and liquid velocity in a wave vs. time for A. a wave growing to B. a slug. Usg = 6.01 m/s, Usl = 0.2 m/s, horizontal pipe.



A

Figure 7:

B

107

C

Pressure variation, liquid holdup, and liquid velocity in a wave vs. time for A. a wave growing to B. a slug and then decaying to C. a wave again. Usg = 5.87 m/s, Usl = 0.13 m/s, θ = 1 degrees.

wave which becomes a slug and then decays into a wave again are shown in figures 6 and 7 respectively. The pressure variation across the wave, liquid holdup and velocity in the wave object as it moves are plotted against time in both figures. When a wave becomes a slug the pressure variation across it and the liquid velocity increase, and the holdup approaches one. In figure 7, when the slug decays again, pressure variation, liquid velocity and holdup decrease. The time traces are cut off when the wave or slug exits the pipe.

4

Conclusions

A model for large roll waves has been implemented and tested in a slug tracking scheme. The model introduces an orifice type relation for pressure variation across the wave front and a simplified relationship for wave speed in a similar way as for slug flow. Computations have been demonstrated in comparison to experimental data on roll waves in two-phase air-water pipe flow at atmospheric pressure. The model gives a reasonable approximation of wave speed and pressure variations in waves. Looking at pressure drops, modelled pressure drops


108 Computational Methods in Multiphase Flow V are sometimes low compared to experiments due to a difference in the number of waves and slugs in the pipe. The tracking scheme can run with a coarse grid which allows simulation in longer pipes but means that waves and slugs are modelled as square objects without tails. A finer grid allows a more physical representation of waves with tails. The model includes wave dynamics such as a wave growing to a slug or a slug decaying to a wave.

5

A Cd g H

List of Symbols area, m2 discharge coefficient gravity, 9.81 m/s2

liquid holdup internal diameter L length of section, m M mass, kg P pressure, Pa S wetted perimeter, m t time, sec U velocity, m/s Umix mixture velocity, m/s Usg superficial gas velocity, m/s

I.D.

Usl superficial liquid velocity, m/s Greek symbols ∆ change in a given quantity λ friction factor θ angle of pipe inclination, degrees ρ density, kg/m3 Superscripts n current time step n+1 next time step Subscripts front front of a wave or slug g gas phase i interface l liquid phase L left section R right section tail tail of a wave of slug WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


109

Acknowledgement Financial support from Total E&P Norge is gratefully acknowledged by A. De Leebeeck.

References [1] Barnea, D. & Taitel, Y., Kelvin-Helmholtz stability criteria for stratified flow: Viscous versus non-viscous (inviscid) approaches, International Journal of Multiphase flow, 19, pp. 639-649, 1993. [2] Bendiksen, K.H., Malnes, D. & Nydal, O.J., On the modelling of slug flow, Chemical Engineering Communications, 141, pp. 71-102, 1996. [3] Bonizzi, M. & Issa, R.I., A model for simulating gas bubble entrainment in two-phase horizontal slug flow, International Journal of Multiphase flow, 29, pp. 1685-1717, 2003. [4] De Leebeeck, A., Gaarder, A.H. & Nydal, O.J., Experiments on Roll Waves in Air-Water Pipe Flow, 16th Australasian Fluid Mechanics Conference, Gold Coast, Australia, 2007. [5] Holmås, H., Numerical simulation of waves in two phase pipe flow using 1D two-fluid models, Doctoral dissertation, University of Oslo, 2008. [6] Hu, B., Stewart, C., Manfield, P.D., Ujang, P.M., Hale, C.P., Lawrence, C.J. & Hewitt, G.F., A Model for Tracking the Evolution of Slugs and Waves in Straight Pipelines, 6th International Conference on Mulitphase Flow, Leipzig, Germany, 2007. [7] Issa, R.I. & Kempf, M.H.W., Simulation of slug flow in horizontal and nearly horizontal pipes with the two-fluid model, International Journal of Multiphase flow, 29, pp. 69-95, 2003. [8] Johnson, G.W., A Study of Stratified Gas-Liquid Pipe Flow, Doctoral dissertation, University of Oslo, 2005. [9] Kjølaas, J., Plug propagation in multiphase flow, Doctoral thesis, Norwegian University of Science and Technology, 2007. [10] Lin, P.Y. & Hanratty, T.J., Prediction of the initiation of slugs with linear stability theory, International Journal of Multiphase flow, 12, pp. 79-98, 1986. [11] Nydal, O.J. & Banerjee, S., Dynamic slug tracking simulation for gas-liquid flow in pipes, Chemical Engineering Communications, 141-142, pp. 13-39, 1996. [12] Renault, F., A Lagrangian slug capturing scheme for gas-liquid flows in pipes, Doctoral thesis, Norwegian University of Science and Technology, 2007. [13] Taitel, Y. & Barnea D., Effect of gas compressibility on a slug tracking model, Chemical Engineering Science, 53(11), pp. 2089-2097, 1998.




111

Numerical simulation of an intermediate sized bubble rising in a vertical pipe J. Hua1, S. Quan2 & J. Nossen1 1

Department of Process and Fluid Flow Technology, Institute for Energy Technology, Kjeller, Norway 2 Institute of High Performance Computing, Connexis, Singapore

Abstract A Computational Fluid Dynamics (CFD) based front tracking algorithm is applied to investigate the rising behaviour of a single bubble in a vertical pipe with a stagnant or flowing viscous liquid. The ratio of the pipe diameter to the bubble equivalent diameter ( D / Db ) is varied within the range of 1.0∼10.0. The wall effects on the terminal bubble rising speed ( U ) and shape are investigated under various flow conditions, which are characterised by the parameters Archimedes number ( Ar ), Bond number ( Bo ), and bulk liquid flow speed ( U l ). It is found that the terminal bubble rising speed ( U ) relates to the bubble rising speed in an infinite domain ( U ∞ ) and the pipe diameter by the formula (U ∞ − U ) / U ∞ ∝ ( Db / D)α

, where α is an exponent relating to the bubble deformability, and it is found to be in the range of 1.0∼0.7 in this study. In addition, the effects of flowing liquid on the terminal bubble rising speed and shape are also investigated for different sized pipes. It is demonstrated that the bubble rising behaviour is significantly affected by the flowing liquid in the pipe with a small diameter. Moreover, the detailed flow field around the bubble is presented to understand the physics of bubble rising behaviour in a vertical pipe under various flow conditions. Keywords: bubble rising, wall effect, front tracking method, Taylor bubble.

1

Introduction

The dynamics of a gas bubble rising in a vertical pipe filled with a viscous fluid is of great importance in nuclear and process industries, e.g. petroleum, refining, WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090101

112 Computational Methods in Multiphase Flow V bubble columns and boiling flows. The terminal rising speed and shape of a gas bubble in a large domain without wall effects have been studied by many researchers [1–2]. The correlations obtained from such studies have been widely used in the modelling of bubbly flows [3]. However, in reality, most of the bubbly flow problems occur in a container with walls, e.g. a pipe. Unfortunately, the effects from the nearby stationary wall (pipe wall) and moving wall (neighbouring bubble) are usually neglected due to the lack of the knowledge in this aspect. Extensive experimental and numerical studies have been conducted to investigate the rising of a Taylor bubble (elongated bubble compared to pipe diameter) [4], however, studies of intermediate sized bubble rising in a vertical pipe are quite limited [5], and it is of great importance to understand the physics of the wall effects on bubble rising behaviour in pipes. It is believed that the wall plays critical roles in the regime transitions of multiphase pipe flow. For example, the transition between bubbly flow and slug flow is resulted by the changes of the relative sizes of bubbles in the liquid flows [6]. In order to qualitatively justify the wall effect on the bubble rising dynamics, such as the terminal velocity and shape, detailed studies of the wall proximity are necessary. Uno and Kintner [7] experimentally studied the effect of wall proximity on the velocity of a single air bubble rising in a quiescent liquid contained in a cylinder. Four different liquids were used, namely, distilled water, 61% glycerine, diethylene glycol, and a solution of a surface-active agent. On the basis of the experimental results, a single equation was proposed to express a velocity-correction factor in term of the ratio of the bubble diameter to pipe diameter and an empirical constant, i.e. U / U ∞ ∝ (1 − Db / D)α , where Db is the equivalent bubble diameter, D is the pipe diameter, and α is the exponent with a value of 0.765. Numerical simulations of a bubble rising freely in viscous liquid have been performed by a number of researchers. Chen et al. [8] used a modified Volume of Fluid method to simulate a bubble rising in a stationary liquid contained in a closed, vertical cylinder. The effect of density and viscosity ratios on the bubble rising is investigated. Hua and Lou [9] proposed an improved front tracking method to simulate the bubble rising in viscous liquid. The simulation results were compared with the available experimental data. Recently, Mukundakrishnan et al. [5] studied the wall effects on a buoyant gas-bubble rising in a liquid-filled finite cylinder using a front tracking finite difference method coupled with a level contour reconstruction of the front. They made detailed simulations on the flow pattern around the bubble and the bubble shape. They presented preliminary discussions about the wall effects on the terminal bubble rising speed. Since it is of great importance to both the scientific researches and engineering applications, further studies are performed here. In this paper, a Computational Fluid Dynamics based front tracking algorithm is applied to examine the rising behaviour of a single bubble in a vertical pipe with a stagnant or flowing viscous liquid. The ratio of the bubble equivalent diameter to the pipe diameter is varied in the range from 0.1 to 1.0. The simulation is started with the basic cases of single bubble rising in a large domain without wall effects in different regimes, where different terminal bubble WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


113

shapes such as spherical, oblate ellipsoidal, dimpled ellipsoidal, skirted bubble and spherical cap, can be obtained. Then, further simulations are performed to investigate the wall effects by reducing the pipe diameter. The improved front tracking method by Hua and Lou [9] is employed for this study, as this method is extensively validated against a number of experiments on bubble rising in various flow regimes. Our simulation results about the wall effects on the bubble terminal velocity agree reasonably with Uno and Kintner’s [7] correlation when the Archimedes number ( Ar ) and Bond number ( Bo ) are relatively higher. However, we found that the exponent varies with the flow regimes. And a new correlation for bubble terminal rising speed is proposed to take into account the wall effect (U ∞−U ) / U ∞ ∝ ( Db / D)α . The effects of pipe wall on the bubble shape are also qualitatively investigated. The proximity of the cylinder wall tends to elongate the bubble in the pipe axial direction. But in the surface tension dominated flow regime, the bubble shape will remain spherical.

2

Mathematical formulation and computational method

2.1 Physical problem

Ul

ρl , µl

zb

rb

U Db ρb , µb

H

g

z

Ul

D

Figure 1:

r R

Schematic illustration of a gas bubble rising in a vertical pipe with a flowing viscous liquid.

The physical problem of a gas bubble rising in a pipe with a flowing viscous liquid is illustrated in Figure 1. The gas bubble has an equivalent diameter Db = 3 6V π , density ρb , viscosity µ g and rising speed U . The bubble volume is assumed to be V . The pipe has diameter D or radius R and height H . The liquid inside the pipe has density ρl , viscosity µ l , and the bulk speed of liquid flow is assumed to be U l . The flow inside the pipe is assumed to be fully developed laminar flow with a parabolic distribution: u l (r ) = 2.0 ⋅ U l ⋅ [1 − (r / R) 2 ]. . WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

114 Computational Methods in Multiphase Flow V The gravitational acceleration is assumed to be g . Two coordinate systems are employed in the current analysis: a stationary cylindrical system ( r , z ) used for solving the governing equations; and a moving cylindrical system ( rb , z b ) with the origin located on the bubble nose used for post-processing and analysing the flow field around the bubble. 2.2 Mathematical formulation and numerical method In this study, it is reasonable to assume that both the gas and liquid phases are incompressible. The governing equations for the multiphase fluid flow system can be expressed as ∇ ⋅u = 0 , (1) ∂ ( ρ u) + ∇ ⋅ ρ uu = −∇p + ∇ ⋅ [ µ (∇u + ∇u T )] + σ κ f n f δ (x − x f )ds f + ( ρ − ρ l )g , (2) ∂t where u is the fluid velocity, p denotes pressure, ρ and µ stand for the density

∫

and viscosity, respectively, g is the gravitational acceleration, s stands for the arc length measured on the interface, κ f denotes the curvature of the interface, σ is the surface tension coefficient and is assumed to be a constant, n f stands

for the unit normal vector on the interface,

xf

is the position vector on the

interface, and δ (x − x f ) stands for the delta function that is non-zero only when x=xf

.

The governing equations can be further non-dimensionalized using characteristic length (the equivalent bubble diameter Db ) and speed ( gDb ). ∇ ⋅ u* = 0 ,

(3)

∂( ρ u ) 1 + ∇ ⋅ ρ *u *u * = −∇p * + ∇ ⋅ [ µ * (∇u * + ∇ T u * )] Ar ∂τ * 1 + κ *nδ (x * − x * f )ds + ( ρ * − 1)g * Bo * *

(4)

∫ Γ

where,

x* =

κ * = Dbκ

,

x Db

g* =

g g

,

u* =

u gDb

,

τ* =

g t, Db

ρ* =

ρ ρl

,

p* =

p

ρ l gDb

,

µ* =

µ µl

,

. And the Archimedes number ( Ar ) and Bond number ( Bo )

are defined as, Ar =

ρ l g 1 / 2 Db 3 / 2 ρ gD 2 and Bo = l b . µl σ

The Archimedes number ( Ar ) denotes the importance of the buoyancy force over viscous force, and the Bond number ( Bo , also called Eotvos number) represents the relative importance of the buoyancy and surface tension forces. Hence, the problem of bubble rising can be specified by the following flow parameters such as density ratio ( η = ρ g / ρl ), viscosity ratio ( λ = µ g / µl ), WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


115

Archimedes number, Bond number, and the ratio of the bubble diameter to the pipe diameter ( Db / D ), and the bulk speed of the liquid flow inside the pipe ( U l ). Table 1:

Flow conditions for the simulation cases and the predicted terminal bubble shape. D / Db

Simulation No.

Ar

Bo

Ul

1

10

5

0

2

10

50

0

3

50

10

0

4

100

50

0

5

100

100

0

6

10

50

0.2

----

7

10

50

-0.2

----

1

2

4

8

----

The numerical method used in this study is based on the finite volume / front tracking method developed by Hua and Lou [9], which has been extensively validated for simulating single bubble rising freely in a quiescent viscous liquid. In the current numerical simulation, a bubble is released at a short distance above the bottom of the pipe (as shown in Fig. 1), and is initially assumed to be spherical or elliptical (depending upon the pipe diameter) with a dimensionless equivalent diameter of one. The pipe is filled with liquid, and the bubble will be accelerated and move upwards due to buoyancy force. The vertical pipe height is long enough to allow the bubble to reach the steady state. In order to study the effects of the pipe wall proximity on the bubble rising, the pipe diameter is varied within the range of 1.0 ⋅ Db ~ 10.0 ⋅ Db , while other parameters and the fluid properties are kept constant for the simulations. Ar and Bo are varied in order to investigate the wall effects on bubble rising in different flow regimes (spherical, dimpled ellipsoidal, ellipsoidal, spherical cap, skirted bubble) while the density ratio and viscosity ratio are kept constant for all simulations ( ρ b / ρ l = 0.001 ; µ b / µ l = 0.01 ) in this study. The flow conditions for the different simulation cases reported in this paper are listed in Table 1.



3


3.1 Simulation of transient bubble rising in vertical pipe The temporal variations of bubble shape and position while it is rising in the vertical pipes with different diameters are shown in Figure 2. The liquid is initially stagnant. As the pipe diameter is reduced, the bubble inside the pipe is elongated in shape due to the constraints of the pipe wall, and the bubble rising speed is also reduced significantly. To investigate the effect of pipe wall on the bubble rising speed, the detailed flow field around the terminal bubble is shown in Figures 3(a), (b) and (c). Here, the reference system is located on the nose of the rising bubble. It shows that a falling liquid film is formed between the bubble and the pipe wall when the pipe diameter is decreased to the same order as the bubble equivalent diameter. Flow dynamics in the falling liquid film has significant effect on the bubble rising. The velocity profiles along the radial direction crossing the bubble and liquid film inside the pipes with different diameters of 1.0 Db , 1.6 Db and 2.0 Db are shown in Figure 3(d). Here the reference system is located on the stationary pipe. As the pipe diameter becomes smaller, the liquid film thickness decreases, and the length of the liquid film increases with the bubble being elongated. A stable bubble rising speed can be obtained when the buoyancy force acting on the bubble is balanced by the drag force including the viscous shear force from the flow in the falling liquid film. It can be seen from Figure 3(d) that the liquid film thickness decreases as the pipe diameter is reduced, and the maximum downward speed in the falling liquid film increases. A higher velocity gradient is induced in the liquid film, and results in a higher viscous shear stress. Hence, the terminal bubble rising velocity becomes lower as the pipe size decreases. (a)

Figure 2:

(b)

(c)

Bubble rising behaviour in vertical pipes with different diameters (a) D = 1.0 Db ; (b) D = 1.6 Db ; (c) D = 4.0 Db , Bo = 100 , and U l = 0 .



(a)

(b)

117

(c)

(d)

Figure 3:

Detailed flow field around the terminal rising bubble in different sized pipes, (a) D = 1.0 Db ; (b) D = 1.6 Db ; and (c) D = 4.0 Db , and (d) shows the velocity profiles in the bubble and liquid film across the pipe.

3.2 Wall effects on terminal bubble shape and rising speed It is understood that both terminal bubble shape and rising speed can be significantly affected by the relative size of the bubble to the pipe ( Db / D ). Table 1 shows the terminal bubble shapes under different flow conditions. When the pipe diameter is large enough ( D / Db > 4.0 ), the pipe wall has much less effects on the terminal bubble shape for most simulation cases. Significant effect of pipe wall on the terminal bubble shape occurs when the pipe size is small or the bubble size is large with D / Db < 2.0. Generally, when a pipe wall exists in the proximity of a bubble, the bubble will be elongated along the axial direction of the pipe. The bubble rising speed is reduced as a result of the increase of the resistance from the liquid. The variations of bubble shape and terminal speed with the relative size of bubble to pipe behave differently at different flow regimes. In the surface tension dominated regime ( Ar < 5.0 and Bo < 5.0 ), the strong surface tension will always keep the bubble in spherical shape. Even when the pipe diameter is of the same order of magnitude of the bubble, the spherical bubble will not deform significantly as it moves very slowly in the pipe. With WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

118 Computational Methods in Multiphase Flow V slightly increase of both Archimedes number ( Ar > 5.0 ) and Bond number ( 5.0 < Bo < 50.0 ), an oblate ellipsoidal bubble will be observed in liquid without wall effects. When the pipe diameter becomes smaller, the bubble will be elongated and become prolate shaped with spherical cap at both head and tail. With a further increase in the Bond number ( Bo > 50.0 ), the role of surface tension becomes less important in determining the bubble shape, and the balance of inertial and viscous force becomes more important, and the bubble wake starts to affect the shape of the bubble bottom. When the Archimedes number is small ( 5.0 < Ar < 20.0 and Bo > 50.0 ), the bubble bottom becomes dimpled. When the Archimedes number is large ( Ar > 100.0 and Bo > 50.0 ), the strong bubble wake flattens the bubble bottom, creating a spherical cap bubble. Within the intermediate range Archimedes number ( 20.0 < Ar < 100.0 and Bo > 50.0 ), skirted bubbles can be observed. When the bubble is rising in a small pipe ( D / Db ∝ O(1) ) with Bo > 50.0 , the bubble will be elongated, and a falling liquid film will be formed between the bubble and pipe wall. The bubble head normally has a semi-spherical shape, and the bubble bottom may be dimpled, skirted or flattened, depending upon the Archimedes number. (a)

1.0

(b)

Ar=10 Bo=5

0.8

( U∞ - U ) / U ∞

( U∞ - U ) / U ∞

0.8 0.6 0.4

0.2

0.4

0.6

( D b /D )

0.8

0.2

1.0

0.4

1.05

0.6

( D b /D )

0.6

(d)

Ar=100

0.5

Bo=50 ( U∞ - U ) / U ∞

( U∞ - U ) / U∞

0.4

0.0

0.0

0.4 0.3 0.2

0.8

1.0

0.79

0.6 0.5 0.4

Ar=100 Bo=100

0.3 0.2 0.1

0.1

0.0

0.0 0.2

0.4

0.6 ( D b /D )

Figure 4:

Bo=50

0.6

0.2

0.2

(c)

Ar=10

0.8

1.0

0.2

0.4

0.72

0.6

( D b /D )

0.8

1.0

0.71

The effects of pipe wall on the terminal bubble rising speed under different flow regimes.

The wall effects on the bubble rising is investigated in comparison with the corresponding case of bubble rising in an infinite liquid without a wall. Hence, it is reasonable to express the relative change of the bubble terminal velocity ( (U ∞ − U ) / U ∞ ) as a function of the ratio of bubble size to the pipe diameter



119

( Db / D ). Figure 4 shows the correlation between (U ∞ − U ) / U ∞ and Db / D under different flow regimes. It can be concluded that the wall effects on the terminal bubble rising speed follows the correlation (U ∞ − U ) / U ∞ ∝ ( Db / D)α , where α is an exponent depending on the flow regime. It is found that the exponent α is about 0.7~0.8 when Bo > 50.0 . 3.3 Effect of pipe flow on bubble rising Pipes are generally used to transport gas and liquid. Therefore, investigation of the effect of pipe flow on the bubble rising behaviour is of great engineering interest. Figure 5 shows the detailed flow field around the bubble in different sized pipes with liquid flowing upwards ( U l = 0.2 ) or downwards ( U l = −0.2 ). When the pipe diameter is large enough, the terminal bubble shape is not affected by the upward flow or downward flow in the pipe. Figure 6(a) shows the velocity profile along the radial direction crossing the bubble and at the far field away from the bubble (D/Db = 4. 0). From Figure 6(a), it can be concluded that the terminal bubble rising velocity can be estimated by adding the bubble rising speed in stagnant liquid and the pipe flow speed. On the contrary, when the pipe diameter becomes small enough, a liquid film is formed between the bubble and pipe wall. The liquid flow inside the pipe starts to affect the bubble shape and bubble moving speed. As shown in Figure 5, the upward liquid flow enhances the bubble rising speed when D/Db = 2.0, and the dimple at the bubble bottom becomes deeper. On other hand, the downward liquid flow reduces the bubble rising speed significantly and the dimple at the bubble bottom disappears. When the pipe diameter become smaller D/Db = 1.0, the pipe flow affects the bubble shape and movement more significantly. The upward liquid flow helps D/Db = 4. 0

D/Db = 2.0

D/Db = 1.0

Ul = 0.2

Ul = -0.2

Figure 5:

The detailed flow field around the rising bubble in different sized pipes with upward or downward flows. Ar = 10.0 , Bo = 50.0 .


120 Computational Methods in Multiphase Flow V (a)

(b)

Figure 6:

Comparison of velocity profiles crossing the rising bubble and the pipe flow for different sized pipe (a) D/Db = 4. 0 and (b) D/Db = 1.0. Ar = 10.0 , Bo = 50.0 .

the rising of the elongated bubble, creating a high velocity gradient in the pipe and builds a dimple at the bubble bottom. On the contrary, the downward flow pushes the “bullet” shaped bubble downwards. A comparison of the velocity profiles across the bubble under different pipe flow conditions in the small pipe (D/Db = 1.0) is shown in Figure 6(b). It is interesting to find that the velocity profile across the liquid film is independent of the pipe flow conditions for the current study cases. This is maybe due to the high viscosity of the liquid when the Archimedes number is low ( Ar = 10.0 ), and the boundary layer along the pipe wall determines the velocity profile in the liquid film.

4

Conclusion

A Computational Fluid Dynamics (CFD) based front tracking algorithm has been applied to investigate the rising behaviour of a single bubble in a vertical pipe with a stagnant (in the far field) or flowing viscous liquid. It is found that the WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


121

relative size of the bubble and pipe has significant effect on the terminal bubble rising velocity ( U ) and shape. When D / Db > 4.0 , the wall effect on the terminal bubble shape and rising speed can be neglected. A strong wall effect on bubble shape and terminal velocity can be observed when D / Db < 2.0 . In general, a small pipe diameter elongates the bubble in the axial direction and reduces the bubble rising velocity. It is found that the bubble rising velocity can be affected by the pipe diameter as (U ∞ − U ) / U ∞ ∝ ( Db / D)α , where α is an exponent relating to the bubble deformability. The exponent α has a value about 0.7~0.8 when Bo > 50.0 . The liquid flow may also produce significant effects on the bubble moving behaviour in a vertical pipe. It is found that the pipe flow has an effect on the bubble shape and moving pattern only when the pipe diameter is small. In a large pipe, the bubble shape is not significantly affected by the pipe flow, and the bubble moving speed is the sum of the pipe flow speed and its rising speed in the stagnant liquid.

References [1] Clift, R., Grace, J.R., & Weber, M.E., Bubbles, drops, and particles, Academic Press: New York, 1978. [2] Bhaga, D. & Weber, M.E., Bubbles in viscous liquids: shapes, wakes and velocities. J. Fluid Mech., 105, pp. 61–85, 1981. [3] Lucas, D., Krepper, E. & Prasser, H.M., Use of models for lift, wall and turbulent dispersion forces acting on bubbles for poly-disperse flows. Chem. Eng. Sci., 62, pp. 4146-4157, 2007. [4] Nogueria, S., Riethmuller, M.L., Campos, J.B.L.M. & Pinto, A.M.F.R., Flow patterns in the wake of a Taylor bubble rising through vertical columns of stagnant and flowing Newtonian liquids: An experimental study. Chem. Eng. Sci., 61, pp. 7199-7212, 2006. [5] Mukundakrishnan, K., Quan, S.P., Eckmann, D.M. and Ayyaswamy P.S., Numerical study of wall effects on buoyant gas-bubble rise in a liquid-filled finite cylinder. Phys. Rev. E, 76, 36308, 2007. [6] Omebere-Yari, N.K., Azzopardi, B.J., Lucas, D., Beyer M. and Prasser, HM, The characteristics of gas/liquid flow in large risers at high pressures. Int. J. Multiphase Flow, 34, pp. 461-476, 2008. [7] Uno, S. and Kintner, R.C., Effect of wall proximity on the rate of rise of single air bubbles in a quiescent liquid. AIChE J., 2, pp. 420–425, 1956. [8] Chen, L., Garimella, S.V., Reizes, J.A., & Leonardi, E., The development of a bubble rising in a viscous liquid. J. Fluid Mech., 387, pp. 61–96, 1999. [9] Hua, J.S. & Lou, J., Numerical simulation of bubble rising in viscous liquid. J. Comput. Phys., 222, pp. 769–795, 2007.




123

Modelling of stratified two phase flows using an interfacial area density model T. Höhne & C. Vallée Forschungszentrum Dresden-Rossendorf e.V., Dresden, Germany

Abstract Stratified two-phase flow regimes can occur in chemical plants, nuclear reactors and oil pipelines. A relevant problem is the development of wavy stratified flows which can lead to slug generation. The slug flow regime is characterized by an acceleration of the gaseous phase and by the transition of fast liquid slugs, which carry a significant amount of liquid with high kinetic energy. It is potentially hazardous to the structure of the system due to the strong oscillating pressure levels formed behind the liquid slugs as well as the mechanical momentum of the slugs. Because these flow conditions cannot be predicted with the required accuracy and spatial resolution by the one-dimensional system codes, the stratified flows are increasingly modelled with computational fluid dynamics (CFD) codes. In CFD, closure models are required that must be validated. The recent improvements of the multiphase flow modelling in the ANSYS CFX code make it now possible to simulate these mechanisms in detail. In order to validate existing multiphase flow models and to further develop these, measurement data with a high-resolution in time and also in space are needed. For the experimental investigation of co-current air/water flows, the HAWAC (Horizontal Air/Water Channel) was built. The channel allows in particular the study of air/water slug flow under atmospheric pressure. Parallel to the experiments, CFD calculations were carried out. The two-fluid model was applied with a special turbulence damping procedure at the free surface. An Algebraic Interfacial Area Density (AIAD) model on the basis of the implemented mixture model was introduced, which allows the detection of the morphological form of the two phase flow and the corresponding switching via a blending function of each correlation from one object pair to another. As a result this model can distinguish between bubbles, droplets and the free surface. The behaviour of slug generation and propagation at the experimental setup was qualitatively reproduced by the simulation, while local deviations require a continuation of the work. The creation of small instabilities due to pressure surge or an increase of interfacial momentum should be analysed in the future. Furthermore, experiments with pressure and velocity measurements are planned and will allow quantitative comparisons, also at other superficial velocities. Keywords: CFD, stratified flow, slug flow, HAWAC. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090111


1

Introduction

Stratified two phase flows occur in many industrial applications. The effects of the flow on the quantities (such as flow rate, pressure drop and flow regimes) have been always of engineering interest. Wallis and Dobson [1] analysed the onset of slugging in horizontal and near horizontal gas-liquid flows. A prediction of horizontal flow regime transitions in pipes was introduced by Taitel and Dukler [2]. They explained the formation of slug flow by the Kelvin-Helmholtz instability. They also proposed a model for the frequency of slug initiation [3]. The viscous Kelvin-Helmholtz analysis proposed by Lin and Hanratty [4] generally gives better predictions for the onset of slug flow. A general overview of the phenomenological modelling of slug flow was given by Hewitt [5]. Various multidimensional numerical models were developed to simulate stratified flows: Marker and Cell (Harlow and Welch [6]), Lagrangian grid methods and Volume of Fluid method (Hirt and Nichols [7]) and Level set method (Sussman [8]). These methods are in principle able to accurately capture most of the physics of the stratified flows. However, they cannot capture all the morphological formations like small bubbles and droplets if the grid is not reasonable small enough. One of the first attempts to simulate mixed flows was presented by Cerne et al. [9] who coupled the VOF method with a two-fluid model in order to bring together the advantages of the both analytical formulations. A systematic study of numerical simulation of slug flow in horizontal pipes using the two fluid formulation was carried out by Frank [10]. It was shown that the formation of the slug flow regime strongly depends on the wall friction of the liquid phase. In simulations using inlet/outlet boundary conditions it was found, that the formation of slug flow regimes strongly depends on the agitation or perturbation of the inlet boundary conditions. Furthermore Frank showed that the length of the computational domain plays an important role in slug formation. However, the direct comparison between CFD calculations and measurements of the slug generation mechanisms and its propagation in horizontal pipes was not analysed. For the experimental investigation of air/water flows, HAWAC (Horizontal Air/Water Channel) with rectangular cross-section was built at Forschungszentrum Dresden-Rossendorf (FZD). Its inlet device provides defined inlet boundary conditions. The channel allows in particular the study of air/water slug flow under atmospheric pressure. Parallel to the experiments, CFD calculations were carried out [11]. The aim of the numerical simulations presented in this paper is the validation of prediction of the slug flow with newly developed and implemented multiphase flow models in the code ANSYS CFX [12].

2 HAWAC The Horizontal Air/Water Channel (HAWAC) (Fig. 1) is devoted to co-current flow experiments. A special inlet device provides defined inlet boundary conditions by a separate injection of water and air into the test-section. A blade separating the phases can be moved up and down to control the free inlet crossWIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


125

section for each phase. This allows influencing the evolution of the two-phase flow regime. The cross-section of the channel are 100 x 30 mm² (height x width). The test-section is about 8 m long, and therefore the length-to-height ratio L/h is 80. Alternatively, related to the hydraulic diameter, the dimensionless length of the channel is L/Dh = 173. The inlet device (Fig. 1) is designed for a separate injection of water and air into the channel. The air flows through the upper part and the water through the lower part of this device. Because the inlet geometry produces a lot of perturbations in the flow (bends, transition from pipes to rectangular cross-section), four wire mesh filters are mounted in each part of the inlet device. The filters are made of stainless steel wires with a diameter of 0.63 mm and have a mesh size of 1.06 mm. They aim at providing homogenous velocity profiles at the test-section inlet. Moreover, the filters produce a pressure drop that attenuate the effect of the pressure surge created by slug flow on the fluid supply systems. Air and water come in contact at the final edge of a 500 mm long blade that divides both phases downstream of the filter segment. The free inlet cross-section for each phase can be controlled by inclining this blade up and down. In this way, the perturbation caused by the first contact between gas and liquid can be either minimised or, if required, a perturbation can be introduced (e. g. hydraulic jump). Both, filters and inclinable blade, provide well-defined inlet boundary conditions for the CFD model and therefore offer very good validation possibilities. Optical measurements were performed with a high-speed video camera. air outlet air inlet 8m

V G V L

Figure 1:

3

pump

Schematic view of the horizontal channel HAWAC with inlet device for a separate injection of water and air into the test-section.

Free surface modelling

The CFD simulation of free surface flows can be performed using the multi-fluid Euler-Euler modelling approach available in ANSYS CFX. Detailed derivation of the two-fluid model can be found in the book of Ishii and Hibiki [13]. However it requires careful treatment of several aspects of the model: The interfacial area density should satisfy the integral volume balance condition. In case if surface waves are present, their contribution to the interfacial area density should be also taken into account.


126 Computational Methods in Multiphase Flow V The turbulence model should address the damping of turbulence near the free surface. The interphase momentum models should take the surface morphology into account. 3.1 Turbulence damping at the free surface As the goal of the CFD calculation was to induce surface instabilities, which are later generating waves and slugs, the interfacial momentum exchange and also the turbulence parameters had to be modelled correctly. Without any special treatment of the free surface, the high velocity gradients at the free surface, especially in the gaseous phase, generate too high turbulence throughout the twophase flow when using the differential eddy viscosity models like the k-ε or the k-ω model [12]. Therefore, damping of turbulence is necessary in the interfacial area because the normally in industrial applications the mesh is too coarse to resolve the velocity gradient in the gas at the interface. A few empirical models have been suggested, which address the turbulence anisotropy at the free surface, see among others Celik and Rodi [14]. However, no model is applicable for a wide range of flow conditions, and all of them are non-local: they require for example explicit specification of the liquid layer thickness, of the amplitude and period of surface waves, etc. Menter proposed a simple symmetric damping procedure. This procedure provides for the solid wall-like damping of turbulence in both gas and liquid phases. It is based on the standard ω -equation, formulated by Wilcox [16] as follows:

∂ (ρ ⋅ ω) + ∇ ⋅ (ρ ⋅U ⋅ ω) = α ⋅ ρ ⋅ ω ⋅ τ t ⋅ S − β ⋅ ρ ⋅ ω2 + ∇[(µ + σ ω ⋅ µ t ) ⋅ ∇ω] k ∂t

(1)

where α = 0.52 and β = 0.075 are the k-ω model closure coefficients of the generation and the destruction terms in the ω-equation, σω = 0.5 is the inverse of the turbulent Prandtl number for ω, τt is the Reynolds stress tensor, and S is the strain-rate tensor. In order to mimic the turbulence damping near the free surface, Menter [15] introduced a source term in the right hand side of the gas and liquid phase ω-equations. A factor activates a source term only at the free surface, where it cancels the standard ω-destruction term of the ω-equation − ri ⋅ β ⋅ ρ i ⋅ ωi2 and enforces the required high value of ωi and thus the turbulence damping.

(

)

3.2 Algebraic Interfacial Area Density (AIAD) Model Fig. 2 shows different morphologies at slug flow conditions. Separate models are necessary for dispersed particles and separated continuous phases (interfacial drag etc.). Two approaches are possible within the Euler-Euler methodology: Four phases: Bubble/Droplet generation and degassing have to be implemented as sources and sinks. Two phases: Momentum exchange coefficients depend on local morphology. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


Air (conti.)

127

Droplets

Water(conti.) Figure 2:

Bubbles Different morphologies at slug flow conditions.

Four phases: Bubble/Droplet generation and degassing have to be implemented as sources and sinks. Two phases: Momentum exchange coefficients depend on local morphology. For the second approach Yegorov [15] proposed an Algebraic Interfacial Area Density (AIAD) Model. The basic idea of the model is: The interfacial area density allows the detection of the morphological form and the corresponding switching of each correlation from one object pair to another. It provides a law for the interfacial area density and the drag coefficient for full range 0≤rα≤1. The model improves the physical modelling in the asymptotic limits of bubbly and droplet flows. The interfacial area density in the intermediate range is set to the interfacial area density for free surface In an Euler-Euler simulation of horizontal slug flow the air entrainment below the water surface can be caused by the drag force. The magnitude of the force density for the drag is 2 1 (2) | D | = CD A ρ U 2 where CD is the drag coefficient, A the interfacial area density and ρ the density of the continuous phase (if the other phase is a dispersed phase). U is the relative velocity between the two phases. The AIAD model applies three different drag coefficients, CD,B for bubbles, CD,D for the droplets and CD,S for free surface (Fig. 3). Non-drag forces (e.g. lift force and turbulent dispersion force) are neglected. The interfacial area density A also depends on the morphology of the phases. For bubbles it is 6r (3) Aαβ = α

dα

where dB is the bubble diameter and rG is the gas void fraction. For a free surface the interfacial area density is the gradient of the void fraction ∂r (4) AFS = ∇rL = L ∂n For ρ the average density is applied, i.e.

ρ = α G ρG + α B ρ B

(5)

where rL and rG are the liquid and the gas phase density respectively. In the bubbly regime, where aG is low, the average density ρ is close to the liquid phase WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

128 Computational Methods in Multiphase Flow V density ρL. According to the flow regime (bubbly flow, droplet flow or stratified flow with a free surface) the corresponding drag coefficients and interfacial area densities have to be applied (Fig. 5). The simplest switching procedure for the interfacial area density, uses the blending function Fd. Introducing void fraction limits, the weights for flow regimes and length scales for bubbly and droplet flow (dB,dD) are the following: a (r −r ) f B = 1 + e B G B ,limit    f FS = 1 − f B − f D

−1

(6) (7)

Aαβ = f FS Aαβ , FS + f B Aαβ , B + f D Aαβ , D

(8)

CD = f FS CD , FS + f B CD , B + f D CD , D

(9)

Fig. 5 shows different blending functions fB for different VF limits and blending coefficients. For the simulation of slug flow the void fraction limits of rB,limit=0.3 resp. rD,limit=0.3 and blending coefficients of aB=aD=70 are recommended. 3.3 Modelling the free surface drag In simulations of free surface flows eq. (2) does not represent a realistic physical model. It is reasonable to expect that the velocities of both fluids in the vicinity of the interface are rather similar. To achieve this result, a shear stress like a wall shear stress is assumed near the surface from both sides to reduce the velocity differences of both phases (Fig. 3).

Figure 3:

Air velocity near the free surface.

A viscous fluid moving along a “solid” like boundary will incur a shear stress, the no-slip condition, the morphology region “free surface” is the boundary layer, the shear stress is imparted onto the boundary as a result of this loss of velocity

τW = µ

∂u ∂y

y =0


(10)


129

The result is a drag coefficient, which is mainly locally dependent on the velocity gradient and the viscosity of both fluids. ∂u   C D ,S =  µ L ,G , L ,G  ∂y  

a) Fluid domain (channel with inlet blade in horizontal position) Figure 4:

(11)

b) Air volume fraction, initial state (Zoom) [-]

Model and initial conditions of the volume fractions.

3.4 Boundary conditions The HAWAC channel with rectangular cross-section was modelled using ANSYS CFX. The model dimensions are 8000 x 100 x 30 mm³ (length x height x width) (Fig. 4a). The grid consists of 1.2x106 hexahedral elements. A slug flow experiment at a superficial water velocity of 1.0 m/s and a superficial air velocity of 5.0 m/s was chosen for the CFD calculations. In the experiment, the inlet blade was in horizontal position. Accordingly, the inlet blade was modelled (Fig. 4a) and the inlet was divided into two parts: in the lower 50% of the inlet crosssection, water was injected and in the upper 50% air. An initial water level of y0 = 50 mm was assumed for the entire model length (Fig. 4b). In the simulation, both phases have been treated as isothermal and incompressible, at 25°C and at a reference pressure of 1 bar. A hydrostatic pressure was assumed for the liquid phase. Buoyancy effects between the two phases are taken into account by the directed gravity term. At the inlet, the turbulence properties were set using the “Medium intensity and Eddy viscosity ratio” option of the flow solver. This is equivalent to a turbulence intensity of 5% in both phases. The inner surface of the channel walls has been defined as hydraulically smooth with a non-slip boundary condition applied to both gaseous and liquid phases. The channel outlet was modelled with a pressure controlled outlet boundary condition. The parallel transient calculation of 15.0 s of simulation time on 4 processors took 10 CPU days. A high-resolution discretization scheme was used. For time integration, the WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

130 Computational Methods in Multiphase Flow V fully implicit second order backward Euler method was applied with a constant time step of dt = 0.001 s and a maximum of 15 coefficient loops. A convergence in terms of the RMS values of the residuals to be less then 10-4 could be assured most of the time. The implementation of the AIAD model and turbulence damping functions into CFX was done via the command language CCL (CEL, Expressions) and User Fortran Routines.

4

Results: comparison between simulation and experiment

A simulated free surface at the HAWAC channel with small surface instabilities is given in Fig. 6. Fig. 7 shows the resulting Interfacial Area Density variable. The AIAD model uses the following three different drag coefficients: CD,B = 0.44 for bubbles, CD,D = 0.44 for the droplets and CD,S according to eq. 11 for the free surface (see Fig. 8). In the picture sequences (Fig. 9 and 10) a comparison is presented between CFD calculation and experiment: the calculated phase distribution is visualized and comparable camera frames are shown. In both cases, a slug is generated. The sequences show that the qualitative behaviour of the creation and propagation of the slug is similar in the experiment and in the CFD calculation. 1 CoefBubb=70, LimitVFBubb=0.1

0.8

CoefBubb=70, LimitVFBubb=0.3

WeightBubb [-]

CoefBubb=30, LimitVFBubb=0.3

0.6

0.4

0.2

0 0

0.2

0.4 0.6 Air.vfc [-]

0.8

Figure 5: Blending functions fB blending coeff.

1

Figure 6: Air volume fraction [-].

In the CFD calculation, the slug develops, induced by instabilities. The single effects leading to slug flow that can be simulated are: Instabilities and small waves are generated by the interfacial momentum transfer randomly. As a result bigger waves are generated. The waves can have different velocities and can merge together. Bigger waves roll over and can close the channel cross-section. However, a detailed comparison shows quantitative deviations between simulation and measurement. The needed entrance length for slug generation was defined as the length between the inlet and the location nearest the inlet WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


density

Figure 8:

131

Drag coefficient [-].

Figure 7:

Interfacial area Variable [1/m].

Figure 9:

Measured picture sequence at JL = 1.0 m/s and JG = 5.0 m/s with ∆t = 50 ms (depicted part of the channel: 0 to 3.2 m after the inlet).

where a wave closes nearly the entire cross-section. This was observed at about 1.5 m in the experiment and 2.5 m in the calculation. These quantitative differences can be explained with the flow regimes observed at the test-section inlet. In fact, the flow pattern has an important influence on the momentum exchange between gas and liquid, especially at high velocity differences between the phases. Small disturbances of the interface provide a more efficient momentum transfer from the air to the water than in a stratified smooth flow. A high momentum transfer induces a rapid wave growth and therefore slug generation. In this case, in the experiment supercritical flow waves were observed from the inlet of the channel. This means that the boundary conditions chosen for the CFD model do not reproduce all small disturbances



Figure 10:

Calculated picture sequence at JL = 1.0 m/s and JG = 5.0 m/s (depicted part of the channel: 1.4 to 6 m after the inlet).

observed in the experiment. In the end, a quite long channel length is needed before waves appear spontaneously in the simulation. Future work should focus on the proper modelling of the small instabilities observed at the channel inlet.

5

Summary

In the HAWAC test facility, a special inlet device provides well defined as well as variable boundary conditions, which allow very good CFD-code validation possibilities. A picture sequence recorded during slug flow was compared with the equivalent CFD simulation made with the code ANSYS CFX. The two-fluid model was applied with a special turbulence damping procedure at the free surface. An Algebraic Interfacial Area Density (AIAD) model on the basis of the implemented mixture model was introduced and implemented. It improves the physical modelling, detection of the morphological form and the corresponding switching of each correlation is now possible. The behaviour of slug generation and propagation at the experimental setup was reproduced, while deviations require a continuation of the work. Experiments like pressure and velocity measurements are planned and will allow quantitative comparisons, also at other superficial velocities.

Acknowledgements This work is carried out in the frame of a current research project funded by the German Federal Ministry of Economics and Labour, project number 150 1329. Thanks to Yuri Yegorov and Thomas Frank from ANSYS CFX for their fruitful cooperation.



133

References [1] Wallis, G. D., and Dobson, J. E. 1973. Onset of slugging in horizontal stratified air-water flow. Int. J. Multiphase Flow 1, 173-193. [2] Taitel, Y., and Dukler, A. E. 1976. A model for predicting flow regime transitions in horizontal and near horizontal gas-liquid flow. AIChE J. 22, 47-55. [3] Taitel, Y., and Dukler, A.E., 1977. A model for slug frequency during gasliquid flow in horizontal and near horizontal pipes. Int. J. Multiphase Flow, 3, 585-596. [4] Lin, P. Y., and Hanratty, T. J. 1986. Prediction of the initiation of slugs with linear stability theory. Int. J. Multiphase Flow, 12, 79-98. [5] Hewitt, G. F. 2003. Phenomenological modelling of slug flow. Short course modelling and computation of multiphase flows, ETH Zurich, Switzerland. [6] Harlow, F. H., Welch, J. E., 1965. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Physics of Fluids 8 (12), 2182-2189. [7] Hirt, C. W., Nichols, B. D., 1981. Volume of fluid (VOF) method for the dynamics of free boundaries. Journal of Computational Physics 39 (1), 201225. [8] Sussman, M., 1994. A level set approach for computing solutions to incompressible two-phase ow. Journal of Computational Physics 114 (1), 146-159. [9] Cerne, G., Petelin, S., Tiselj, I., 2001. Coupling of the interface tracking and the twofluid models for the simulation of incompressible two-phase flow. Journal of Computational Physics 171 (2), 776-804. [10] Frank, T. 2003. Numerical simulations of multiphase flows using CFX-5. CFX Users conference, Garmisch-Partenkirchen, Germany. [11] Vallée, C., Höhne, T., Prasser, H.-M., Sühnel, T. 2008, Experimental investigation and CFD simulation of horizontal stratified two-phase flow phenomena. NED, Volume 238, Issue 3, March 2008, Pages 637-646 [12] ANSYS CFX, 2008. User Manual. Ansys Inc. [13] Ishii, M., Hibiki, T., 2006. Thermo-fluid Dynamics of Two-phase Flow. Springer-Verlag. [14] Celik, I., and Rodi, W. 1984. A deposition-entrainment model for suspended sediment transport. Report SFB 210/T/6, Strömungstechnische Bemessungsgrundlagen für Bauwerke, University of Karlsruhe, Germany. [15] Yegorov, Y. 2004. Contact condensation in stratified steam-water flow, EVOL-ECORA –D 07. [16] Wilcox, D. C. 1994. Turbulence modelling for CFD. La Cañada, California: DCW Industries Inc.




135

Multi-phase mixture modelling of nucleate boiling applied to engine coolant flows V. Pržulj & M. Shala Ricardo Software, Ricardo UK Limited Shoreham-by-Sea, West Sussex, UK

Abstract The paper reports on the use of the homogeneous multi-phase mixture modelling approach to simulate nucleate boiling in low pressure flows. A variant of the RPI (Rensselaer Polytechnic Institute) mechanistic nucleate boiling model provides closures for the wall thermal conditions and mass transfer rates due to phase change in the bulk flow. The bubble departure diameter at the wall and the bubble bulk diameter are identified as the most influential factors and their original model coefficients are modified. The numerical difficulties due to large density variations associated with phase change are successfully addressed in conjunction with the segregated pressure based SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm. The nucleate boiling model has been compared against published data from two experiments. The computed and measured values for the wall heat flux and vapour volume fractions are in broad agreement. The model capability is also demonstrated for the engine coolant flow where the conjugate heat transfer problem involving complex engine components is solved. Keywords: CFD, nucleate boiling, multi-phase mixture, RPI model, engine cooling.

1 Introduction Nucleate boiling is the exceedingly effective mode of heat transfer from a heated wall to a liquid. In many liquid cooling systems, ranging from nuclear reactors to electronic devices, a change from single-phase convection to nucleate boiling can effectively provide the desired high rates of heat transfer. The high heat transfer rates (thermal loads) also characterise cylinder blocks and heads of internal WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090121

136 Computational Methods in Multiphase Flow V combustion engines. Their cooling is achieved by pumping a water-glycol mixture through a system of connected coolant passages. Whether nucleate boiling is utilised intentionally or not, the design of efficient liquid cooling systems should avoid departure from the nucleate boiling regime, the boiling crisis. This departure, described by the critical (maximum) wall heat flux (CHF) might lead to the hazardous overheating of wall materials. Practical CFD modelling of multi-phase flows requires built-in mechanistic models for interfacial mass, momentum and energy transfer between phases. In the past, the majority of published CFD predictions of boiling flows employed the RPI model of Kurul and Podowski [1] for interfacial transfer closures. It was validated against high pressure flows but later some authors [3, 4, 5] adopted this model for low pressure boiling flows. Model improvements have been proposed by [6, 7], while the model extension to conditions close to CHF was reported in [8]. In the majority of CFD simulations [3–5, 9, 10] the boiling model has been coupled with multi-fluid conservation equations where individual velocity, temperature and other field variables can be solved for each phase. The exception is Bo’s work [11], where the homogeneous multi-phase mixture (single fluid) approach was used in conjunction with the author’s own mass transfer model. In the present study, modelling of nucleate boiling based on the RPI model and using the homogeneous multi-phase mixture approach is used. This choice is motivated by uncertainties in modelling interphase transfers and by computational efficiency of the mixture approach. The objective is to explore this overall approach for modelling low pressure coolant flows in internal combustion engines.

2 Mathematical formulation A homogeneous mixture model is adopted in this work. This implies that relative motion between phases is neglected. The modelling equations describing mass, momentum and energy conservation of the mixture have the same form as the single-phase Reynolds Averaged Navier–Stokes equations. A variant of the conventional k − model, with imposed bounds on the turbulence time scale is used to model turbulence. The near-wall region turbulence is modelled by enhanced wall functions. In the viscous sub-layer they satisfy corresponding wall-limiting expressions and in the fully turbulent region they are identical to the conventional wall functions. Further details regarding the employed k − model and enhanced wall functions can be found in [12]. The transport and thermodynamic properties appearing in the mixture equations depend on the properties of constituent fluid phases k and their volume fractions αk . In case of Nph fluid phases, the mixture properties are calculated as:

Nph

φ=

αk φk = αl φl + αg φg , φ = ρ, µ, λ

(1)

k=1

where φk denotes the property value of the constituent phase; in this work of liquid k = l and vapour k = g. In the above equation, ρ, µ and λ denote the mixture WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


137

density, dynamic viscosity and thermal conductivity,respectively. The mixture specific heat takes a different form:

Nph

Cp =

Xk Cp,k = Xl Cp,l + Xg Cp,g , Xk =

k=1

αk ρk ρ

(2)

where Xk is the phase mass fraction. Individual phase properties are in general dependent on the temperature. The vapour can be treated as an ideal gas so its density is calculated from the equation of state ρg = p/(Rg T ) where p is the pressure and Rg is the gas constant. The volume fractions are governed by their own transport equations: ∂ ∂αk ρk + (αk ρk Uj ) = Γk , k = l, g ∂t ∂xj

(3)

where t is the time, xi are the Cartesian space coordinates and Uj are Favreaveraged velocities. The source term, Γk , represents the phase mass generation rate due to evaporation and condensation. The energy conservation equation is defined in terms of total enthalpy, H = h + 0.5Ui2 + k, with h being the specific enthalpy of mixture and k is the turbulent kinetic energy. In order to calculate temperature from the specific enthalpy and vice versa, the specific enthalpy h and its relationship with temperature T needs to be defined. Considering enthalpies of a liquid phase, hl = Cp,l Tl , and a vapour phase, hg = Cp,l Tsat + ∆hgl + Cp,g (Tg − Tsat ) (Tsat denotes the saturation temperature), and setting T = Tl = Tg , the mixture enthalpy can be defined in terms of temperature as: h = Xl hl + Xg hg = hl,sat + Xg ∆hgl + Cp (T − Tsat ) ,

(4)

where the specific heat of mixture Cp is given by eqn. (2). In the above equation, ∆hgl signifies the latent heat of evaporation: ∆hgl = hg,sat − hl,sat , where hl,sat and hg,sat , are saturated liquid and vapour enthalpies, respectively. The mixture temperature then becomes T = Tsat +

h − (hl,sat + Xg ∆hgl ) Cp

(5)

2.1 Mechanistic boiling model The mechanistic model presented here provides closures for the wall thermal conditions for the energy and the mass transfer rate Γk in the volume fractions eqn. (3).


138 Computational Methods in Multiphase Flow V Thermal conditions for the near-wall heat transfer. Considering the wall heat flux qw , a general approach is to apportion it to the liquid and vapour phases according to their wall volume fractions αl/g,w : qw = αl,w qw,l + αg,w qw,g , αg,w = 1 − αl,w

(6)

In the case of sub-cooled nucleate boiling, vapour bubbles, created at the wall nucleation sites and then departed from the wall, are continuously replaced by fresh liquid (quenching) and by new generation of bubbles. Thus the wall liquid fraction is close to one, as the absence of dry wall regions can be assumed. At high wall heat fluxes, the boiling process can move towards CHF conditions. This means that heat transfer between dry wall regions (αg,w > 0) and vapour has to be considered. A phenomenological function fα,l reported in [8]: fα,l = 1 − 0.5 exp [−20 (αl − αl,cr )] , for αl ≥ αl,cr ,

(7)

fα,l = 0.5 (αl /αl,cr )20αl,cr for αl < αl,cr , αl,cr = 0.2 is used in this work to define the wall liquid fraction αl,w = fα,l in order to model transition from the nucleate boiling to the CHF regime. In the above expression, αl is a value at the near-wall cell. Kurul and Podowski’s RPI model [1] is used to split the heat flux to the liquid into the single-phase convection qc , the wall quenching qq and the evaporation part qe : qw,l = (1 − Ω)qc + Ωqq + qe (8) where Ω is the fraction of the wall area affected by nucleation sites and quenching, see eqn. (11) below. The wall heat flux expression (6) then reads: qw = qw,d + αl,w [Ω (qq − qc ) + qe ] , qw,d = αl,w qc + αg,w qg,w

(9)

where contribution of convection to the liquid and gas phase is described by the wall diffusion flux qw,d . This flux is calculated within the solver using the mixture properties. The liquid convective heat flux qc is calculated in the same way as the wall diffusion flux qw,d except that properties of liquid are used. The evaporative and quenching flux are given by: qe =

3 Dbw 2 π ρg fn Nn ∆hgl , qq = √ fn λl ρl Cp,l tq (Tw − Tl ) 6 π

(10)

where Dbw is the bubble diameter at departure from the wall, fn and Nn are the frequency of nucleation and the number of nucleation sites per unit area, respectively. In the expression for the quenching flux, tq denotes the waiting time elapsed between the departure of a bubble and the nucleation of a new one. It can be estimated as tq = 0.8/fn [1]. The fraction of the wall area affected by nucleation is given as [4]: 2 Cp,l ρl (Tsat − Tl ) Dbw π Nn η , η = 4.8e−Ja/80 , Ja = (11) Ω = min 1, 4 ρg ∆hgl where Ja is the Jacob number. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


139

The closures for the remaining model parameters have been selected based on the assessment of the existing empirical closures reported in literature. For the bubble diameter, the correlation of Tolubinsky, see [5], is modified by introducing an adjustable factor Cbw (Cbw = 0.6 in the original correlation): Dbw = min Cbw × 10−3 e−∆Tsub /45 , 1.4 × 10−3 , ∆Tsub = Tsat − Tl (12) The bubble release frequency and nucleation site density are calculated from Cole’s and Lemmert and Chwala’s correlations (see for example [5]), respectively: fn =

4g (ρl − ρg ) 1.805 , Nn = 210 (Tw − Tsat ) 3ρl Dbw

(13)

Interfacial mass transfer. The mass transfer in the bulk flow due to evaporation Γe or condensation Γc is calculated with the help of empirical correlations for the interface heat transfer coefficient around the representative vapour bubble with diameter Db : Γe/c =

hi,e/c Ai (T − Tsat ) 6αsg (1 − αg ) , Ai = , αsg = min (0.25, αg ) (14) ∆hgl (1 − αsg )Db

where Ai (1/m) is the interfacial area density [1]. For both evaporation and condensation process, the heat transfer coefficient hi is determined from RanzMarshall correlation [3]. The bubble Nusselt number N ub is correlated by Reynolds, Reb , and liquid Prandtl, P rl , numbers: hi =

1 1 N ub λl ρl U b D b µl Cp,l , N ub = 2 + 0.6Reb2 P rl3 , Reb = , P rl = Db µl λl

(15)

The characteristic bubble velocity Ub is taken to be the magnitude of the mixture cell velocity. Then, mass transfer rates in eqn. (3) read as Γg = −Γl = Γe + Γc The bubble mean diameter Db has usually been modelled by either a constant, estimated value [8] or as a linear function of local sub-cooling ∆Tsub = Tsat − Tl , [1, 3, 4] with adjustable reference bubble sizes and corresponding sub-cooling temperatures. As shown by [10], these simple approaches are very deficient when compared against the measured bubble size profiles. To accommodate some observed bubble behaviour such as the coalescence, Wintterle et al. [9] adopted models of Basuki et al. and El Jouhary (see [9]): Db =

6αg πN (αg )

13

−1 αg − 0.015 , N = 1 + exp (N0 − N1 )+N1 (16) 0.004

where N0 = 1.1 · 107 and N1 = 3.0 · 105 are the number densities (1/m3) corresponding to the very low and very high vapour fractions αg . WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


3 Numerical framework The boiling model has been implemented into an advanced, commercial CFD solver, VECTIS-MAX. The governing equations are discretised over general unstructured (polyhedral) grids employing an advanced collocated Finite Volume Method, [12]. The pressure based, segregated SIMPLE-like solution algorithm [2] ensures efficient pressure-velocity-density coupling. Considering the compressible gas flow, in addition to the pressure and velocity corrections, the density corrections are introduced and defined in terms of the pressure corrections as ρ ∝ Cρ p , where Cρ is the density derivative over pressure. In the case of a multi-phase mixture, Cρ can be calculated from the mixture sonic speed cm : ρ αg 1 αl αg . (17) ≈ + Cρ = 2 = ρ cm κRg T ρg ρl c2l ρg Rg T where κ = 1.4 is the isentropic exponent. The above simple formulation for the mixture density derivative over pressure has been effective in handling large density variations associated with phase change.

4 Results and discussion The nucleate boiling model is first validated against published data from two experiments. To assess the model capability for automotive applications, real engine coolant flow simulations are then presented. The initial validation results identified the bubble departure diameter Dbw and the bubble bulk diameter Db as the most influential model parameters. In order to reproduce the experimental trends, it was necessary to adjust their correlations. Therefore, the adjustable factor in eqn. (12) was set to Cbw = 1.2, while the number densities in the expression for Db , eqn. (16), were tuned to the values N0 = 1.6 · 107 and N1 = 3.0 · 106 . Horizontal channel flow. Robinson [13] tried to replicate engine flow conditions in a cylinder head cooling passages by using a rectangular channel (241 mm long, 16 mm wide and 10 mm high) with a heated section (10 × 50 mm2 , positioned 76 mm downstream from the flow inlet) at the bottom wall, see fig. 1. The

Outlet

Inlet

Heated section

Figure 1: Simulated channel geometry and location of the heated section. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


141

heated surface (aluminium alloy) was machine finished and can be considered as smooth. The working fluid was a mixture of 50% water and 50% ethylene glycol by volume. The present results were obtained for the sets of test data with the inlet liquid temperature Tinl = 363 K and with two mass flow rates corresponding to the inlet velocities of 0.25 m/s and 1.0 m/s. Within these sets the pressure values were 1, 2 and 3 bars (the corresponding saturation temperatures are 381 K, 401 K and 415 K, respectively). These pressure values were specified as the outlet boundary conditions. On the heated section, a constant temperature (known from the experiments) was specified. Other walls were considered adiabatic. The calculated wall heat flux values can be compared against the measured ones as shown in fig. 2. Very good agreement has been obtained for the inlet velocity 0.25 m/s, fig. 2 (a). The heat flux values for the high velocity of 1 m/s are significantly under-predicted as shown in fig. 2 (b). This model behaviour can be partly explained by evident under-prediction of the single-phase heat fluxes before the onset of nucleate boiling. Thus the effect of turbulence modelling and wall functions should be anticipated. Another factor is the dependence of the bubble departure size on the local velocity [3] which is not taken into account by eqn. (12).

2

1

2

1.5

Exp, 1 bar Exp, 2 bar Exp, 3 bar Present, 1 bar Present, 2 bar Present, 3 bar

Heat Flux (MW/m )

2

Heat Flux (MW/m )

2

0.5

0 370

380

390 400 410 420 Wall temperature (K)

(a) Inlet velocity: 0.25 m/s

430

440

1.5

1

Exp., 1 bar Exp., 2 bar Exp., 3 bar Present, 1 bar Present, 2 bar Present, 3 bar

0.5

0 370

380


430

440

(b) Inlet velocity: 1.0 m/s

Figure 2: Channel flow, inlet temperature 363 K. Comparison of the predicted wall heat fluxes against measured data from [13].

An example of the computed flow variable fields is depicted in fig. 3. The selected case has the high wall temperature and heat flux which resulted in the large amount of vapour near the heated wall. Consequently, the velocity profiles are distorted when compared to the profiles upstream of the heated section. Upward flow in a vertical annulus. Bae et al. [5] benchmarked their full EulerEuler (two-fluid) multi-phase model with an interfacial area equation using the experiment from the Seoul National University. The sub-cooled boiling of water WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


(a) Temperature, from 363 K (blue) to 414 K (red)

(b) Vapour volume fraction, from 0 (blue) to 0.75 (red)

(c) Velocity vectors, from 0.009 m/s to 0.33 m/s

Figure 3: Channel flow, inlet temperature 363 K, velocity 0.25 m/s. Computed variable fields in the symmetry plane for p = 1 bar, Tw = 418 K.

in a vertical upward flow through a concentric annulus was investigated under constant wall heat flux conditions along the heated inner tube section. The total length of the annulus was 3.06 m. The inner tube consisted of the inflow unheated, central heated and outflow unheated section, each 0.5, 1.87 and 0.69 m long, respectively. The inner and outer annulus radii were Rin = 9.5 and Rout = 20 mm. The radial distribution of the vapour fraction was measured at three axial positions along the heated section. For the present validation, a test case with the following conditions has been selected: the mass flow rate 342.207 kg/(m2 s) (applied as the outlet boundary condition), the wall heat flux at the heated section qw = 212.706 kW/m2 , the pressure 1.21 bar (applied as the inlet boundary condition) and the inlet sub-cooling Tsat − Tinl = 21.695 K. Steady-state simulations are carried out for quarter of the annulus. Apart from the heated inner section, all other walls are assumed adiabatic. As the heat flux is prescribed at the heated section, the wall temperature is calculated with an iterative procedure which ensures that the heat flux eqn. (9), expressed as qw = F (Tw ), is satisfied. The predictions of the radial profile of the vapour fraction at the exit of the heated section are depicted in fig. 4(a). The axial profile of the average vapour fraction is shown in fig. 4(b). The present profiles can be compared with the measured data as well as with two-fluid model predictions from [5]. Two present profiles are presented: the first with the original densities numbers in eqn. (16) for the bubble bulk diameter and the second with modified values of these numbers. The modified densities numbers have improved predictions of the average axial profile with reference to the measured data. Considering the local radial predictions, the present approach over-predicts the amount of vapour near the wall and under-predicts it in the bulk flow. Calculated temperature and vapour fraction fields as obtained with adjusted density numbers are plotted in fig. 5 for the whole solution domain. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

143


Experiment Ref. [Bae et al.] Present, N0 1.1e 7, N1 3e 5 Present, N0 1.6e 7, N1 3e 6

0.5 0.4

Experiment Ref. [Bae et. al.] Present, N0 1.1e 7, N1 3e 5 Present, N0 1.6e 7, N1 3e 6

0.12 Vapour fraction αg

Volume fraction αg

0.6

0.3 0.2 0.1

0.08 0.04 0

0 0

0.2

0.4 0.6 (R-Rin)/(Rout-Rin))

(a) Local radial profile.

0.8

1

0

0.5

1 1.5 Axial length (m)

2

(b) Average axial profile.

Figure 4: Annular flow. Computed and measured vapour volume fraction profiles. (Zero value on the x-axes in (b) denotes beginning of the heated section.)

(a) Temperature, from 356 K (blue) to 376.3 K (red).

(b) Vapour fraction, from 0 (blue) to 0.69 (red).

Figure 5: Vertical annulus (flow direction from the right to the left). Predicted distribution of temperature and vapour fraction.

In the first half of the heated section the wall temperature experiences an oscillatory pattern which corresponds to the vapour fraction pattern. Engine coolant flow. The CFD solution domain needs to include the complete engine assembly consisting of typical components such as the cylinder head, head gasket, engine block, cylinder liners and coolant jacket. These components (except liners) are shown in fig. 6 for the selected engine type. The coolant jacket represents the fluid domain while other components define separate solid material domains, each with their own thermal properties. To make the numerical approach practical, automatic meshing of the solution domain is required because of the very complex geometry. For the present simulation, the Cartesian cut-cell grid, having around 3.3 million cells, has been generated by VECTIS–MAX mesher. This mesher delivers fast meshing without resorting to the boundary surface grid generation. As part of the steady Conjugate Heat Transfer (CHT) simulation, the mass, momentum, turbulence and volume fraction equations are solved for the fluid domain (coolant flow). The energy equation is solved over a global solution domain WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


Figure 6: Engine cooling. Temperature distribution at surfaces of fluid (coolant jacket) and solid (cylinder head, gasket and cylinder block) materials participating in the CHT simulation with boiling model.

(containing all participating fluid and solid domains) in a fully implicit and conservative manner. For this, conformal numerical grids at fluid-solid interfaces are provided. At the coolant inlet the mass flow rate (2.8 kg/s) and temperature (363 K) are specified; at the outlet fixed static pressure (1.01bar) is maintained. The external heat transfer coefficient and temperature describe heat transfer at the solid surfaces exposed to the environment. At the gas side cylinder surfaces the heat flux distribution is prescribed. Fig. 6 presents temperature field at the surfaces of each material domain as calculated with the boiling model. These surfaces are either fluid/solid or solid/solid interfaces or external boundaries. The effect of the boiling model on the temperature levels is clearly demonstrated in fig. 7. Here, one can compare temperature on the cylinder head as predicted with and without the nucleate boiling model. Inclusion of the boiling model reduces temperature levels. In this case, the maximum temperature without the boiling model is 576.8 K, while the maximum temperature with the boiling model reaches 550.5 K. The maximum temperature of coolant is also reduced from 466.6 to 425.2 K. As expected, the temperature reduction indicates presence of a certain amount of vapour. This amount can be quite large as shown in fig. 8. The figure depicts the coolant passage region in which the near-wall vapour fraction has the high value, around 0.3. The spot with the maximum value of 0.61 is not shown in the above figure. Identification of such local spots where departure from WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


145

Figure 7: Engine cooling. Comparison of the surface temperature on the cylinder head as computed with (left) and without (right) boiling model, values from 352 K (blue) to 576.8 K (red).

Figure 8: Engine cooling. Near-wall vapour fraction distribution in the coolant passage region with the large amount of vapour, range: from 0 (blue) to 0.3 (red).

nucleate boiling might lead to overheating of the wall materials is useful for design of engine cooling systems.

5 Conclusion The capability of the multi-phase mixture modelling approach, supplemented with the mechanistic nucleate boiling model, has been assessed for CFD predictions of low pressure coolant flows in internal combustion engines. The nucleate boiling model is based on the popular RPI model. Among a number of modelling parameters, the bubble departure diameter at the wall and the bubble bulk diameter are the most influential and required adjustments. In comparison to the original RPI model, the wall heat flux partitioning is modified to account for the possible heat transfer from the wall to the gas phase. Two boiling flow experiments are used to benchmark the model. The computed and measured values for the wall heat flux and vapour volume fraction are found to be in broad agreement. The CHT simulation of an engine cooling system has shown reasonable model behaviour. Bearing in mind uncertainties in the modelling of nucleate boiling, the present approach can be seen as a good compromise in terms of the accuracy and computational efficiency. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


References [1] Kurul, N. & Podowski, M., Multidimensional effects in forced convection subcooled boiling. Proceedings of the 9th International Heat Transfer Conference, Jerusalem, Israel, 1-BO-04, pp. 21–26, 1990. [2] Ferziger, J. & Peric, M., Computational Methods for Fluid Dynamics. Springer: Berlin, 1997. [3] Koncar, B. & Mavko, B., CFD simulation of subcooled flow boiling at low pressure. Intl. Conf. Nuclear Energy in Central Europe 2001, Portoroz, Slovenia, pp. 208.1–208.8, 2001. [4] S. Narumanchi, D.B., A. Troshko & Hassani, V., Numerical simulations of nucleate boiling in impinging jets: Application in power electronics cooling. Int J Heat and Mass Transfer, 51, pp. 1–12, 2008. [5] B-U. Bae, H.Y.Y., Euh, D.J., Song, C.H. & Park, G.C., Computational analysis of a subcooled boiling flow with a one-group interfacial area transport equation. J Nuclear Science and Technology, 45(4), pp. 341-351, 2008. [6] N Basu, G.W. & Dhir, V.K., Wall heat flux partitioning during subcooled flow boiling: Part 1 - Model development. ASME J Heat Transfer, 127, pp. 131140, 2005. [7] G. Yeoh, M.H. & Tu, Y., Improved wall heat partition for subcooled boiling flows. 6th Int. Conf. Multiphase Flow ICMF 2007, Leipzig, Germany, 2007. [8] Koncar, B. & Mavko, B., Simulation of boiling flow experiments close to CHF with the Neptune-CFD code. Science and Technology of Nuclear Installations, 2008, p. 8, 2008. [9] T. Wintterle, Y.E., E. Laurien & Menter, F., Numerical simulation of a subcooled boiling flow in a nuclear reactor bundle geometry. 11th Workshop on Two-Phase Flow Predictions, Merseburg, Germany, 2005. [10] Yeoh, G. & Tu, J., Two-fluid and population balance models for subcooled boiling flow. Applied Mathematical Modelling, 30, pp. 1370-1391, 2006. [11] Bo, T., CFD homogeneous mixing flow modelling to simulate subcooled nucleate boiling flow. SAE International, 2004-01-1512, 2004. [12] Przulj, V., Birkby, P. & Mason, P., Finite volume method for conjugate heat transfer in complex geometries using cartesian cut-cell grids. CHT-08, Marrakech, Morocco, 2008. [13] Robinson, K., IC Engine Coolant Heat Transfer Studies. PhD thesis, University of Bath, Department of Mechanical Engineering, 2001.



147

On the application of Mesoscopic Eulerian Formalism to modulation of turbulence by solid phase Z. Zeren1,2 & B. Bédat1,2

1 Institut 2 CNRS,

de Mécanique des Fluides de Toulouse, France France

Abstract Recently developed method, Mesoscopic Eulerian Formalism, is searched for its extension to the gas-solid flows where the carrier phase is modified by the solid phase. The possibility is shown to be existing by the introduction of two classes of particles with all the same properties except their initial positions. Classes are distributed homogeneously in space and only one of them is two-way coupled with the flow. The others are with ghost particles (particles with one-way coupling). With increased number of ghost particles, the field of source terms’ of classes become similar letting the fluid realization become the same for each class. Then the conditional one-particle probability density function is definable. Keywords: Mesoscopic Eulerian Formalism, two-way coupling, particle and fluid realization, initial particle conditions.

1 Introduction Spatial distribution of inertial particles in a turbulent flow is very important in understanding different phenomena occurring in gas-solid flows such as particleparticle interactions, interactions fluid-particle, etc. For example, it is shown by Sundaram and Collins [1] that particle spatial distributions can cause significant changes in the collision rates of particles. On the other hand, particle distribution can also have specific effect on the modulation of fluid turbulence, Elgobashi and Truesdell [2]. In this regard, Fevrier et al. [3] have proposed the Mesoscopic Eulerian Formalism (MEF) to have comprehensive understanding on the distribution of finite WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090131

148 Computational Methods in Multiphase Flow V inertia particles in a turbulent flow. The particle spatial velocity correlations are assumed to be induced only via the interactions with the fluid. Specifically, hydrodynamic interactions and inter-particle collisions are assumed not to induce any spatial correlation between the particles. This considers the fact that dispersed phase statistical quantities do not depend on the initial conditions of particles having chaotic motion. The statistical measures are developed then via the definition (1) of one-particle probability density function, fp (cp , x, t, Hf ), conditioned on the single fluid realization. The extension of the method to the gas-solid flows with inter-particle collisions is proposed by Fevrier et al. [3] for dilute regimes. However, implementation in turbulence modulation regimes is not that direct due to the fact that the single fluid realization, Hf , is not obvious to be definable. In this paper, it is shown that Mesoscopic Eulerian Formalism is applicable through the introduction of two classes of particles into a turbulent flow where only one of the classes is coupled with the flow (two-way coupling) and the other is assumed to be ghost particles (particles with one-way coupling). Particles are point sources tracked in the Lagrangian frame of reference whereas the fluid is solved in the Eulerian grid. Particle Source In Cell approximation (PSIC), initialized by Crowe et al. [4], is shown to be not to cause any significant non-physical oscillations on the fluid velocity. Discussion will continue with describing the numerical scheme and the application to two-way coupling will be discussed.

2 Governing equations and numerical configuration The studied configuration is a cubical domain with a volume of L3b = (2π)3 . The domain is discretized with 1283 grid points with periodic boundary conditions for the both phases. Homogeneous isotropic turbulence is generated and kept stationary using a stochastic forcing scheme [5, 6]. The code used is finite difference code with 6th order spectral-like scheme in space and 3rd order Runge-Kutta scheme in time. The code is parallelized with MPI (Message Passing Interface) library. Governing equations of the fluid including the effect of particles are written as: ∂ρ ∂ρui + =0 ∂x ∂xi

(1)

∂p ∂τij ∂ui ∂ρui + ρuj =− + + Πi + fi ∂t ∂xj ∂xi ∂xj

(2)

Equations (1) and (2) are respectively for the conservation of mass and momentum. While solving the Navier-Stokes equations in the Eulerian frame, particles are tracked individually in the Lagrangian frame. The effect of the particles on the fluid is taken into account through the term, Πi , on the right hand side of the momentum equation. Without taking into account the effect of gravity, this term in the context of point-source approximation is written as: WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


Πi = −

Np

(n)

(n)

(n)

Fp,i (xp,i )δ(x − xp,i )

149

(3)

n=1 (n)

where Fp,i is the force applied by the fluid to the nth particle in i direction defined as: (n)

Fp,i = −

(n)

mp

(n) τp

(n)

(n)

(up,i − uf @p,i )

(4)

where uf @p is the fluid velocity at the position of particle and τp is the particle relaxation time defined as: τp(n) =

ρp d2p 18µf fD

(5)

fD = (1 + 0.15Re0.687 ) is the correction to take into account the effect of parp ticle Reynolds number, Rep . Fluid velocity at the position of particle, uf @p , is calculated by 3rd order Lagrangian interpolation scheme. The term fi in the equation 2 is the stochastic forcing term which keeps the turbulence stationary.

3 Initial conditions of particle phase Characteristics of turbulence and particles can be found in table 1 and table 2. TE is the Eulerian time scale calculated by the Eulerian one-point autocorrelation func tion, Te is eddy turnover time defined as Lf /u where Lf is the longitudinal large scales length and u is the characteristic velocity of turbulence. Lg is the transversal large scales. TLf is Lagrangian time scale of large structures calculated by the Lagrangian one-point autocorrelation function. Based on these values, turbulent Reynolds number is defined as ReL = u Lf /νf and Reynolds number based on the Taylor scales is defined as Reλ = u λg /νf . The crucial condition for MEF to be applicable is that the particle spatial correlations are induced only by the interaction with the fluid. As mentioned in introduction, this dictates the fact that the hydrodynamic interactions and interparticle collisions are assumed not to induce any spatial correlation. To this end, small deviations in the initial conditions are quickly magnified, after several relaxation time, particle statistics become independent of the initial conditions and controlled by the interactions with the fluid. Under these assumptions, two populations of particles are introduced into the same fluid realization with the same mesoscopic fields except their positions. The characteristics of particles and of turbulence are defined in table 1 and table 2. Note that in the table, total number of particles is given. Each populations is then with 2 particles per grid in DN S1283 application. These two populations are time-stepped with the fluid realization without two-way coupling so that they settle to an equilibrium with the turbulent field. Their statistical quantities, particle WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

150 Computational Methods in Multiphase Flow V Table 1: Turbulence characteristics without two-way coupling. TE /Te

TLf /TE

Lf /Lb

Lf /Lg

ηf

ReL

Reλ

6.2816 1.0267

TE

0.8275

0.0864

1.9797

0.0177

96

49

Table 2: Particle characteristics. ρp /ρf

dp /ηf

Nptotal

12000 0.0751 4x1283

Table 3: Fluid and particle statistics. Population

qp2

qf2

qf p

qf2 @p

ClassA ClassB

.4544E-02 .4546E-02

.1176E-01 .1176E-01

.8883E-02 .8883E-02

.1141E-01 .1141E-01

Outlet

Inlet

Heated section

Figure 1: Spatial distribution of particles, T + /τp = 0. ClassA on the left and ClassB on the right. kinetic energy qp2 , turbulent kinetic energy qf2 , fluid-particle covariance qf p , turbulent kinetic energy seen by the particles qf2 @p , in stationarity are shown in table 3 and they’re exactly the same. This final field obtained is referred as T + /τp = 0 where T + is the nondimensional time of the simulations. These populations will be useful not only for the verification of the PSIC approximation but also for the test that is performed to validate the MEF’s application to two-way coupling. Initial distributions of the particles are shown in fig. 1. It is clear that the both classes are distribution homogeneously in space. To be more quantitative on the distributions of the both classes, the normalized spatial distribution function, P (C), which is possible number of particles, C, in an elementary volume and longitudinal spatial correlations of particles, Rpp (r) =< up (x)up (x + r) >, are plotted. Shown in fig. 2, both populations have exactly the same correlation and distribution curves validating the equality of the mesoscopic WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


1

Exp, 1 bar Exp, 2 bar Exp, 3 bar Present, 1 bar Present, 2 bar Present, 3 bar

2

Heat Flux (MW m )

2

Heat Flux (MW m )

2

1.5

0.5

0 370

151

380


(a) Inlet velocity: 0.25 m/s

430

440

1.5

1

Exp., 1 bar Exp., 2 bar Exp., 3 bar Present, 1 bar Present, 2 bar Present, 3 bar

0.5

0 370

380


430

440

(b) Inlet velocity: 1.0 m/s

Figure 2: Spatial correlations, Rpp (r) normalized by particle phase kinetic energy, qp2 , and distribution functions of particle classes, P (C), at T + /τp = 0.

(a) Temperature, from 363 K (blue) to 414 K (red)

(b) Vapour volume fraction, from 0 (blue) to 0.75 (red)

(c) Velocity vectors, from 0.009 m/s to 0.33 m/s

Figure 3: The error of the PSIC approximation, qfpp is particle-fluid covariance for the particles effecting the flow and qfg p is covariance for the ghost particles.

fields of both classes.

4 Validity of point source approximation Once being sure of the initial conditions of the particle classes, a numerical study has been performed to verify the point source approximation (PSIC). It is been shown by Eaton [7] how the point source approximation fails with increase in the particle radius. In this paper, particles smaller than the Kolmogorov scale are used (see table 2) so the effect of wake production is negligible. Then all the rest is the validation of the approximation. Simple test is then applied to two-classes of particles introduced into the same fluid realization. One of them is coupled with the fluid and the other is chosen as the ghost particles (particles with one-way coupling) (see Vermorel et al. [8]). If the approximation is to be valid, then the statistical values of each class should be similar with a negligible difference between each other so that they see the same fluid field. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

152 Computational Methods in Multiphase Flow V Experiment Ref. [Bae et al.] Present, N0 1.1e 7, N1 3e 5 Present, N0 1.6e 7, N1 3e 6

0.5 0.4

Experiment Ref. [Bae et. al.] Present, N0 1.1e 7, N1 3e 5 Present, N0 1.6e 7, N1 3e 6

0.12 Vapour fraction αg

Volume fraction αg

0.6

0.3 0.2 0.1

0.08 0.04 0

0 0

0.2

0.4 0.6 (R-Rin) (Rout-Rin))

(a) Local radial profile.

0.8

1

0

0.5

1 1.5 Axial length (m)

2

(b) Average axial profile.

Figure 4: Parameter D, the difference between the fluid realizations and statistics of the two classes, ClassA and ClassB. As seen in fig. 3, the difference of the fluid-particle covariance, qf p , between the two populations is less than 1% which is rather acceptable for the application.

5 Application of MEF to two-way coupling As explained in introduction, for the MEF’s application to the flows where the twoway coupling cannot be ignored, the definition of the probability density function, (1) fp (cp , x, t), is not direct forward preventing the definitions of statistical quantities conditioned on a single realization of fluid turbulence. Initially, the idea was to explore the effect of different initial conditions of particles on the same fluid field, Hf . Performing number of two-way coupled simulations with particles having different initial positions, it was observed how the turbulence evolves with a response to different initial conditions. Specifically, it was curious to find a time range where the fluid turbulence does not differ between different fluid realizations enormously. To compare the fluid fields in these different simulations, the proper method is the utilization of the normalized parameter D: D = (< (uA − uB )2 > /uArms uBrms )1/2

(6)

where uA is the fluid velocity with response to the ClassA and uB is the one to ClassB where < . > denotes volume averaging. Two simulations are performed with 2 classes of particles (see table 2). The particle initial conditions are as explained in section 3. Fig. 4 shows the behavior of D and the particle statistics in time. As seen in the figure, the flow fields differentiate more than 10% and significantly the difference increase in time whereas the statistical quantities are the same for each class (the figure on the right hand side). This difference in instantaneous fluid field is due to the small deviations in the feedback of particles which are quickly amplified in time by the non-linear chaotic nature of turbulence. The conclusion is then the turbulent field does not allow a WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


153

(a) Temperature, from 356 K (blue) to 376.3 K (red).

(b) Vapour fraction, from 0 (blue) to 0.69 (red).

Figure 5: Spatial correlations, Rpp (r) normalized by particle phase kinetic energy, qp2 , and distribution functions of particle classes, P (C), at T + /τp = 8. fluid realization rest the same at least for a small time-range when it is coupled with the particle phase. To come over the non-linear nature of turbulence, the simulation resumed in the first test is considered with only one class being active (two-way coupled to the fluid) and the other class is considered as ghost particles (particles with one-way coupling). In the same fluid realization, difference between the source terms of classes are then expected to be dependent on the number of particles, at least for a time range. Two simulations are performed with the configuration where ClassA and ClassB are the ones active in respective simulations to see the interchangability of the active class. As might be guessed, different classes correspond to different initial conditions and the fundamentality here is then the ghost particles’ utility which is to increase the precision of the computation of the mesoscopic quantities (1) using the p.d.f. fp . As seen in fig. 5, at the end of the simulation, spatial distributions and correlations of the two classes rest the same to each other, shown only for one simulation. This is to say that even with two-way coupling, particle field keeps the mesoscopic field, as in the initial conditions, the same for both classes. However, the effect on the fluid field is not the same when the active class is A or B. Turbulent field responding to ClassA and ClassB in respective simulations is shown in fig. 6. It is clear that the topology of the flow stays the same. However, analyzing closer the field, the dashed-line regions are shown in fig. 7. The dots in the fields shows the approximative centers of the vortexes and as seen, they have slight deviations between each other. The source terms of active and non-active classes in one of the simulations are shown in fig. 8. As seen in the figure, the difference quantified by the parameter D in time stays constant for a time range more than one particle relaxation time. The initial peak on the graph is due to the transition of both the particles and fluid field to arrive at a new equilibrium, it is to be reminded that turbulence is forced by the scheme of Eswaran and Pope [5]. In the stationary period, constant difference seems like promising and more strictly, it has been found out that the difference depends on the number of ghost particles, not shown here. As the number of particles in each class, one-way or two-way, increases the parameter D decreases. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


Figure 6: Fluid velocity vectors, T + /τp = 8, on the left hand side, ClassA is active, on the right hand side, ClassB is active in seperate realizations.

Figure 7: Fluid velocity vector zoom field, the dashed-line regions in fig. 6. So there is a statistical relation between the number of particles and the difference of the source terms among the classes. This sounds a bit like the statistical confirmation of PSIC method. Using high number of particles lets the changing



155

Figure 8: Source term Πx , eq. (3), of particles in x direction at T + /τp = 8, of ClassA (active) on the left and of ClassB (non-active) on the right in the same fluid realization.

Figure 9: Parameter D, difference between the source terms of particles in ClassA and ClassB for the two realizations of fluid.

the active class to any other classes to generate the same fluid field. The definition (1) of the conditional probability density function, fp (cp , x, t, Hf ), is then possible.

6 Conclusion and perspectives The Mesoscopic Eulerian Formalism is shown to be applicable to the flows where the carrier phase is modified by the presence of solid particle phase. From statistical point of view, increasing the number of particles reduces the error of the PSIC approximation where non-physical oscillations are not obtained and also high number of particles lets the definition of a single fluid realization, Hf , in which large number of particle realizations, Hp , is imaginable. Study can be extended to different Stokes numbers to see the difference when there is no concentration of particles. Also classes with different numbers of particles should be very informative on the effects of initial conditions. With the definition of the probability WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

156 Computational Methods in Multiphase Flow V density function of particles conditioned on a single fluid realization, mesoscopic field values become measurable.

Acknowledgements This research project has been supported by a Marie Curie Early Stage Research Training Fellowship of the European Community Sixth Framework Program under contract number MEST-CT-2005-020426’.

References [1] Sundaram, S. & Collins, L.R., Collision statistics in an isotropic particle-laden turbulent suspension. part 1. direct numerical simulations. Journal of Fluid Mechanics, 335, pp. 75–109, 1997. [2] Elghobashi, S. & Truesdell, G.C., On the two-way interaction between homogeneous turbulence and dispersed solid particles. part 1: Turbulence modification. Phys Fluids A, 5(7), pp. 1790–1801, 1993. [3] Fevrier, P., Simonin, O. & Squires, K.D., Partitioning of particle velocities in gas-solid turbulent flows into a continuous field and a spatially uncorrelated random distribution: theoretical formalism and numerical study. Journal of Fluid Mechanics, 533, pp. 1–46, 2005. [4] Crowe, C., Sharm, M. & Stock, D., The particle source in cell (psi-cell) model for gas-droplet flows. J Fluids Engineering, 99, pp. 325–332, 1977. [5] Eswaran, V. & Pope, S., An examination of forcing in direct numerical simulations of turbulence. Computers and Fluids, 16, pp. 257–278, 1988. [6] Zeren, Z. & Bédat, B., Spectral and physical forcing of turbulence. Proceedings of the iTi International Conference on Turbulence, Bertinoro, Italy, 2008. [7] Eaton, J.K., Two-way coupled turbulence simulations of gas-particle flows using point particle tracking. International Journal of Multiphase Flow, In Press, Accepted Manuscript. [8] Vermorel, O., Bédat, B., Simonin, O. & Poinsot, T., Numerical study and modelling of turbulence modulation in a particle laden slab flow. Journal of Turbulence, 4, p. 25, 2003.



157

VOF-based simulation of conjugate mass transfer from freely moving fluid particles A. Alke1 , D. Bothe1 , M. Kroeger1 & H.-J.Warnecke2 1 Center

of Smart Interfaces, Technical University Darmstadt, Germany of Chemical Engineering, University Paderborn, Germany

2 Department

Abstract In this paper two variants of a VOF-based approach for the numerical simulation of the molar mass transport of a diluted species in two-phase flows with deformable interfaces are introduced and compared. The variants differ in the manner of the computation of the mass transfer flux across the interface. The method assumes local thermodynamical equilibrium at the interface and enables the simulation of conjugated mass transfer problems across deformable interfaces, where the mass transport resistance lies in both phases. The considered model also allows for arbitrary distribution coefficients. First numerical simulations show the potential and the present limits of this method. Keywords: Direct Numerical Simulation, two-phase flow, Volume of Fluid, conjugate mass transfer, two scalar approach.

1 Introduction In Process Engineering, mass transfer operations based on dispersed two-phase flows are frequently applied. Typical examples are gas purification by bubbling of a gas through a liquid, oxygenation of aqueous systems in biological processes, and solvent extraction as a thermal separation process. Besides the departure from the phase equilibrium, the mass transfer depends mainly on the characteristics of the dispersed two-phase flow, i.e. on the particle size and shape, slip velocities, internal circulation, swarm behaviour etc., and on the species diffusivities. Besides experimental studies, Direct Numerical Simulations of single fluid particles, which become more and more feasible due to the ongoing increase in computational power, can be very useful since they can provide local data which usually cannot be accessed by experiments. Contrary to heat transfer, in mass transfer probWIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090141

158 Computational Methods in Multiphase Flow V lems the transported scalar - the molar concentration c, say- is not continuous at the interface. To handle the interfacial jump discontinuity numerically is a challenging task. Volume of Fluid (VOF)-based simulations of mass transfer across deforming interfaces have been reported in [1] and in [2, 3]. In the latter papers, transfer of oxygen from air bubbles rising in water or aqueous solutions has been simulated, taking into account the realistic jump discontinuity of the oxygen profiles at the interface. Darmana et al. [4] performed 3D simulations of mass transfer at rising fluid particles for Sc = 1 using the Front Tracking method. There, the transport resistance inside the fluid particle is neglected, i.e. a constant concentration value inside the bubble is assumed. Radl et al. [5] performed 2D simulations of deformable bubbles and bubble swarms with mass transfer in non-Newtonian liquids using a semi-Lagrangian advection scheme. To prevent stability problems, a reduced density ratio between gas and liquid is used there. Recently, first papers on numerical simulation of reactive mass transfer appeared. In [6, 7], the impact of single bubble wake dynamics on the reaction-enhanced mass transfer and on the yield and selectivity of the cyclohexane oxidation reaction is studied numerically for fixed shapes in 2D. In [8], reactive mass transfer at deformable interfaces is examined using a 2D Front Tracking/Front Capturing hybrid method. In [9], a Level Set based method is used to simulate mass transfer across the interface of a moving deformable droplet. This method is extended to reactive mass transfer in [10], where an instantaneous chemical reaction occurs inside a moving droplet which leads to a quasi-stationary problem for the mass transfer. In [11], 2D simulations are performed using a Front-Tracking method to investigate the effect of different Hatta and Schmidt numbers on the catalytic hydrogenation of nitroarenes for single bubbles and bubble clusters. The focus of the present work is on a VOF-based method having the potential to be used for Direct Numerical Simulations (DNS) of mass transport in two-phase flows with deformable interfaces, including droplets, bubbles, falling films etc. For the computation of the mass transfer across the interface, two variants are employed. In the numerical study presented here, single fluid particles rising in a Newtonian fluid with mass transfer from the fluid particle to the surrounding liquid are considered.

2 The governing equations In the following, we consider a fluid-particle (domain Ωd (t)) which is immersed in a liquid (domain Ωc (t)). The deformable interface between the two phases is presented as a surface of zero thickness and is denoted by Σ(t). The transfer component k has a constant initial concentration ck (t0 ) > 0 inside the fluid particle and a zero concentration inside the surrounding liquid. Furthermore, the following assumptions are imposed: • dilute two-phase system, • chemically inert non surface active transfer component, • local thermodynamical equilibrium at the interface, • no phase change, WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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• isothermal conditions, • incompressible bulk phases. The present paper employs a continuum mechanical model in which the governing equations are based on the conservation of mass, momentum, and (molar) mass of the transfer component. Inside the phases the transport of species k is governed by the local balance equation ∂t ck + ∇ · (uck + jk ) = Rk

in

Ωc (t) ∪ Ωd (t).

(1)

Here ck is the volume specific molar concentration of the dissolved species k, jk is the area specific diffusive (molecular) flux density, and Rk is the overall reaction rate accounting for all chemical reactions in which species k is involved. In the following, chemical reactions are not more considered, i.e. Rk = 0. For the diffusive flux density, a suitable constitutive equation is required. Here, only diluted systems are considered. In this case the molecular transport inside the bulk phases can be described by Fick’s law, i.e. jk = −Dk ∇ck ,

(2)

with diffusion coefficient Dk . The concentration ck and with it the flux jk as well as the velocity u are local and time dependent quantities. To solve the parabolic partial differential equations (1) inside the bulk phases, initial and suitable boundary conditions are required. The solutions inside the phases are not independent at the interface and two jump conditions are required. The first one is a transmission condition and comes from the interfacial balance. Since only non surface active transfer components are considered, the normal component of the diffusive fluxes are equal at the interface, i.e. (3) [jk ] · nΣ = 0 with the jump notation [φ] (xΣ ) = lim (φ(xΣ + hnΣ ) − φ(xΣ − hnΣ )) . h→+0

(4)

For the second interfacial condition, local thermodynamical equilibrium is assumed, i.e. the chemical potential µk of component k is continuous at the interface: (5) [µk ] = 0, with µk (T, p) = µ0k (T, p0 ) + RT ln ak .

(6)

The first term in (6) is the chemical potential of component k in a pure system (consisting only of component k) at temperature T and standard pressure p0 , whereas the second term, containing the activity ak of species k in a multicomponent fluid, accounts for mixing effects. For liquid systems, the activity is proportional to the WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

160 Computational Methods in Multiphase Flow V concentration and for gas mixtures it is proportional to the partial pressure. In case of a diluted system, the thermodynamical equilibrium condition (5) reduces to cdk|Σ = cck|Σ /mk ,

(7)

with the distribution coefficient mk = mk (p, T ) > 0, where cdk|Σ and cck|Σ are the one-sided limits of the concentrations at the interface in the dispersed and continuous bulk phase, respectively. Since we consider only isothermal flows with small pressure gradients, the distribution coefficient mk is assumed to be constant. The second condition (5) is only an approximation since the deviation from the local thermodynamical equilibrium is the driving force of the mass transfer. However, this deviation is very small. Therefore, the equilibrium assumption is commonly accepted. The underlying velocity field is governed by the two-phase Navier-Stokes equations expressing conservation of mass and momentum. Assuming continuity of the velocity at the interface, a one-field formulation is possible in which the interfacial momentum jump conditions act as source terms in the momentum equations. For viscous (Newtonian) fluids of constant density and constant surface tension, the governing equations read as ∇ · u = 0, (8) and ρ ∂t u + ρ (u · ∇) u = −∇p + η∆u + ρg + σκnΣ δΣ ,

(9)

where the momentum jump conditions are incorporated via the interfacial Delta distribution δΣ . In this interfacial source term, κ = −∇ · nΣ denotes the curvature (more precisely, the sum of the principal curvatures). In (9) the material properties ρ and η refer to the phase dependent values which are given as ρ = f ρd + (1 − f )ρc

(10)

and η = f η d + (1 − f )η c ,

(11)

where f is the phase indicator function of the phase domain Ω (t). d

3 Numerical method For complex flow situations such as a freely moving fluid particle with a deformable interface, the mathematical model described in the previous section cannot be solved analytically but has to be treated numerically. There are several requirements for an appropriate numerical method. One challenge is the capturing of the interfacial concentration jump of the transfer component at the interface according to (7). Since a Lagrangian fluid particle cannot cross the interface (Lagrangian theorem), the species transport across the interface is purely diffusive. Therefore, the convective transport of the discontinuity shall not lead to an artificial mass transfer WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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across the interface. Furthermore, the continuous-side concentration gradient normal to the interface may be very steep depending on the particle Reynolds number, Re, and the Schmidt number, Sc, of the continuous phase. For practically relevant Sc numbers around 500, say, the concentration boundary layer cannot be resolved without specific computational techniques. The numerical scheme presented here is based on the VOF method [12] using the one-field formulation of the NavierStokes equations (9). In comparison with other free surface simulation methods, the VOF method can handle massive deformations and even topology changes as they can appear in case of large bubbles. Furthermore, the VOF method conserves the phase volume, which is an important issue if chemical reactions shall be accounted for. The phase indicator f is obtained from its initial distribution by solving the advection equation ∂t f + u · ∇f = 0.

(12)

In the Finite Volume context, f corresponds to the volume fraction of phase Ωd inside a computational cell V . The employed Finite Volume based VOF-solver, Free Surface 3D (FS3D) developed by Rieber [13], applies a directional as well as a kind of operating splitting. To avoid systematic errors and unsymmetries, the sequence of processed directions in the splitting scheme is altered in each time step. That is, the convective terms of all transport equations are computed firstly. Than, the forces for the momentum equations are computed and impressed before the diffusive transport terms are calculated. For the volumetric surface tension force the conservative continuum surface stress (CSS)-model of Lafaurie et al. [14] is used. The numerical solution of the discrete version of (12) is based on a geometrical based flux calculation. Application of the so called piecewise linear (or planar in case of 3D) interface calculation (PLIC) scheme for the outgoing phase volume fluxes in interfacial cells prevents interface smearing. 3.1 Transport of molar species mass For the computation of the transport of a transfer species k, the concentration is represented by two separate scalar variables according to φdk (x, t)

=

and

φck (x, t) =

ck (x, t) for x ∈ Ωd (t) 0 for x ∈ Ωc (t)

(13)

0 for x ∈ Ωd (t) ck (x, t) for x ∈ Ωc (t).

(14)

This allows for capturing the different one-sided limits of the concentrations at the interface. In the discrete Finite Volume scheme, these scalars are related to the cell WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

162 Computational Methods in Multiphase Flow V volume V , i.e. in interface-containing cells the cell centered values are given as φdk (t) = and φck (t)

1 |V |

1 = |V |

V ∩Ωd (t)

ck dV

(15)

ck dV.

(16)

V ∩Ωc (t)

3.1.1 Convection The new variables are similar to the VOF-variable f in that these quantities are all nonnegative. The only difference is that φdk and φck can take arbitrary positive values whereas f is always less or equal to one. Therefore, the convective transport of φdk and φck is treated analogously to that of the convective f transport, using the PLIC algorithm for the outgoing flow in interfacial cells to prevent an (artificial) convective mass transfer and a flux limiter scheme inside the bulk phases to minimize numerical diffusion. 3.1.2 Mass transfer across the interface After the computation of the convective transport of species mass (and also of the other quantities like phase volume and momentum) mass transfer across the interface is calculated. Within the two scalar approach, mass transfer is accounted for by source terms. Inside a computational cell Vi containing a part of the interface, the transferred volume specific molar mass is substracted from and added to the corresponding values of φdk,i and φck,i . In this notation, index i stands as a short form for (i, j, k) which is the full index of a grid cell in a 3D Cartesian mesh. For brevity the characters j and k are omitted. In the following, the calculation of onedimensional fluxes are explained only for the x-direction where index i + 1 has the meaning of (i + 1, j, k). For the calculation of mass transfer, where the two jump conditions (3) and (7) have to be accounted for, we employ two different variants. Variant I: Equilibration of interfacial cells In the first variant, we assume that inside a computational cell Vi , containing a part of the interface, the transfer component k is ideal mixed in the separate phases. After the convective transport both variables φck,i and φdk,i have certain values which are assumed to be constant within the respective phases lying in the considered interfacial cell. However, the ratio φdk,i (1 − fi )/(φck,i fi ) does not corresponds to mk , in general. Therefore, the characteristic of this equilibrium approach is the conservative redistribution of the (molar) species mass according to c,eq φdk,i |Vi | + φck,i |Vi | = φd,eq k,i |Vi | + φk,i |Vi | ,


(17)


163

c,eq where the values of the variables φd,eq k,i and φk,i fulfil the thermodynamical condition (7). In this variant, the first transmission condition (3) is not explicitly accounted for. But this condition expresses the local (molar) mass balance at the interface which is inherently fulfilled by this approach.

Variant II: Computation of the one-sided concentration gradient The second variant is based on the computation of the concentration gradient adjacent to the interface at the continuous side. Within an interfacial cell Vi the total (molar) mass flux of species k normal to and across the interface with interfacial area |AΣi | is given by jΣi ,k · nΣi |AΣi | = (jΣi ,k,x nΣi ,x + jΣi ,k,y nΣi ,y + jΣi ,k,z nΣi ,z ) |AΣi | . (18) Here, only directions are accounted for in which the neighbour cell is completely filled with the continuous phase. To calculate the one-dimensional diffusive flux density jΣi ,k,x (where Σi denotes the interface in cell Vi , k the component index, and x the direction) it is assumed that within the interfacial cell the dispersed phase is well mixed. Therefore, the cell centered value of φdk,i is taken as the concentration value cdk|Σ ,i adjacent to the interface. Then, depending on whether cell Vi+1 or cell Vi−1 lies completely in the continuous phase, the one-dimensional flux density is computed as jΣi ,k,x = Dkc

φdk,i /(fi mk ) − φck,i+1 /(1 − fi ) xi+1 − xi

(19)

jΣi ,k,x = Dkc

φck,i+1 /(1 − f1 ) − φdk,i /(fi mk ) , xi+1 − xi

(20)

or

respectively. The local interfacial area in an interfacial cell Vi is calculated from the cell centered value of the gradient of the VOF-variable f according to |AΣ,i | = ∇f i |Vi | .

(21)

After calculating the cell specific total molar mass flux normal to the interface (18), the values for φdk,i and φck,i are updated according to φdk,i = φdk,i − qk,i and φck,i = φck,i + qk,i

(22)

respectively, where the source term is given by qk,i =

jΣi ,k · nΣi |AΣi | ∆t . |Vi |


(23)

164 Computational Methods in Multiphase Flow V Table 1: Physical parameters. phase

dispersed continuous

density

dynamical viscosity

diffusion coefficient

Schmidt number

ρ in kg/m3

η in mP as

D in m2 /s

Sc = ν/D

1.2 1000

18 · 10 10

−3

−6

5 · 10 10−6

3 10

3.1.3 Diffusive transport inside the bulk phases Finally, diffusive transport in the bulk phases is computed. Here, especially in computational cells with a very small f value it might occur that too much species mass leaves the cell during time interval ∆t. The directional splitting scheme allows for a limitation of the diffusive fluxes which are calculated, using the forward differencing scheme according to c/d ck,i − ck,i+1 |Ai,i+1 | , N˙ k,i,i+1 = Dk xi+1 − ci

(24)

where Ai,i+1 is the cell face which connects the cells Vi and Vi+1 . This onen+1 dimensional flux is limited by the equilibrium criteria cn+1 if cni+1 ≥ i+1 ≥ ci n ci and vice versa. Furthermore, diffusive fluxes across cell faces connecting two interfacial cells are also accounted for.

4 Simulation results and discussion To compare the two variants, 2D numerical simulations (i.e. with translational symmetry) of an air bubble rising in a Newtonian liquid have been performed. The area equivalent diameter of the bubble is 3 mm, the liquid has a dynamical viscosity of 10 times higher than that of water and a density of 1000 kg/m3 . To keep the time step sufficiently high, a reduced surface tension of 36 mN/m has been used. The physical parameters are given in Table 1. The computational domain of 1.2 cm × 2.4 cm is resolved by three different computational grids; cf. Table 2. Figure 1 shows the increase of the molar mass of the transfer component within the continuous phase with time. For all resolution cases the mass transfer calculated with the equilibrium approach is higher than that calculated with the gradient approach. But the results of both variants get closer together with higher resolution. This indicates insufficient resolution of the thin concentration boundary √ layer; recall that the latter has a thickness proportional to 1/ Re Sc. Obviously, with the gradient approach the mass transfer is underestimated. The reason for this may be an inaccurate approximation of the interfacial area. The total interfacial area of a fluid particle is given by all cells ∇f i |Vi |. But not only interfacial WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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Table 2: Used numerical grids. Case

number of grid cells

cell width in µm

cells per diameter

A B

256 x 512 512 x 1024

46.9 23.4

64 128

C

1024 x 2048

11.7

256

Figure 1: Molar mass of transfer species in continuous phase related to initial molar mass in fluid particle versus time.

cells have ∇f greater than zero but also the neighboring cells. Therefore, equation (21) yields a too low value for the local interfacial area inside a computational cell. Otherwise, the equilibrium approach inherently overestimates the mass transfer since the (local) thermodynamical equilibrium is only valid adjacent to the interface. The larger the normal distance from the interface the lower is the concentration. The assumption of ideally mixed interface cells can be interpreted as an infinitely fast molecular transport at the interface leading to a too large mass transfer. Therefore, the equilibrium approach yields an upper bound for the mass transfer. However, from Figure 1 it can be seen that grid independence is already reached. The different mass transfer rates are also noticeable in the concentraWIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

166 Computational Methods in Multiphase Flow V tion profiles. Figure 2 shows the concentration distribution within the continuous phase when stationary hydrodynamical conditions are reached, obtained with the two variants (left: equilibrium and right: gradient variant) at the highest resolution (case C). In both cases, species is mainly present in the wake of the fluid parti-

Figure 2: Concentration distribution of transfer component in the continuous phase yielded at highest resolution (Case C): mass transfer calculated with equilibrium variant (left) and with the gradient variant (right). cle. However, with the equilibrium approach the region of high concentrations at the stagnation points is more pronounced than those obtained with the gradient approach.

5 Conclusions and outlook A new VOF-based two scalar approach for simulating the transport of chemical species within a two-phase flow is introduced. The method allows for the simulation of conjugate mass transfer problems across deformable interfaces with an arbitrary distribution coefficient. The treatment of the convective transport is analogous to that of the VOF-variable f , using a geometrical flux calculation. This procedure avoids artificial mass transfer due to convection. First numerical results in 2D at a moderate Schmidt number of 10 are performed. The results show that the mass transfer rate obtained by equilibration of the interfacial cells are always higher as those obtained by using the one-sided limit of the concentration gradient in the continuous phase. However, with finer resolution the results of both variants get closer together. For the equilibrium variant, grid independency is reached. However, the finest resolution presented here will not be sufficient for WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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higher Schmidt numbers and, moreover, is not suitable for 3D simulations. The mass transfer rate obtained with the gradient variant at the highest resolution is still lower than the ”true” rate. The reason for this may lie in the underestimation of the interfacial area. Therefore, further steps are the use of a more accurate interfacial area calculation scheme, the development of a subgrid model for the concentration profile at the interface, use of a moving grid technique to reduce the computational domain as well as a local grid refinement around the bubble.

Acknowledgement We gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG) within the DFG-project ”Reactive mass transfer from rising gas bubbles” (PAK-119).

References [1] Davidson, M.R. & Rudmann, M.J., Volume-of-fluid calculation of heat or mass transfer across deforming interfaces in two-fluid flow. Numerical Heat Transfer, B41, pp. 291–308, 2002. [2] Bothe, D., Koebe, M., Wielage, K. & Warnecke, H.J., VOF simulations of mass transfer from single bubbles and bubble chains rising in aqueous solutions. in Proc. 2003 ASME Joint U.S.-European Fluids Eng. Conf., 2003. [3] Bothe, D., Koebe, M., Wielage, K., Prüss, J. & Warnecke, H.J., Direct Numerical Simulation of Mass Transfer Between Rising Gas Bubbles and Water. Springer Verlag, Berlin, Heidelberg, New York, 2003. [4] Darmana, D., Deen, N.G. & Kuipers, J.A.M., Detailed 3d modelling of mass transfer processes in two-phase flows with dynamic interfaces. Chemical Engineering and Technology, 2006. [5] Radl, S., Tryggvason, G. & Khinast, J.G., Flow and mass transfer of fully resolved bubbles in non-newtonian fluids. AIChE Journal, 53, pp. 1861– 1878, 2007. [6] Khinast, J.G., Impact of 2-d bubble dynamics on the selectivity of fast gas liquid reactions. AIChE Journal, 47, pp. 2304–2319, 2001. [7] Khinast, J.G., Koynov, A. & Leib, T.M., Reactive mass transfer at gas-liquid interfaces: Impact of micro scale fluid dynamics on yield and selectivity of liquid phase cyclohexane oxidation. Chemical Engineering Science, 58, pp. 3961–3971, 2003. [8] Koynov, A., Tryggvason, G. & Khinast, J.G., Mass transfer and chemical reactions at dynamic interfaces. AIChE Journal, 51, pp. 2786–2800, 2005. [9] Deshpande, K.B. & Zimmermann, W.B., Simulation of interfacial mass transfer by droplet dynamics using the level-set method. Chemical Engineering Science, 61, pp. 6486–6498, 2006. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

168 Computational Methods in Multiphase Flow V [10] Deshpande, K.B. & Zimmermann, W.B., Simulations of mass transfer limited reaction in a moving droplet to study transport limited characteristics. Chemical Engineering Science, 61, pp. 6424–6441, 2006. [11] Radl, S., Koynov, A., Tryggvason, G. & Khinast, J.G., DNS-based prediction of the selectivity of fast multiphase reactions: Hydrogenations of nitroarenes. Chemical Engineering Science, 63, pp. 3279–3291, 2008. [12] Hirt, C.W. & Nichols, B.D., Volume of fluid (vof) method for the dynamics of free boundaries. Journal of Computational Physics, 39, pp. 201–225, 1981. [13] Rieber, M., Numerische Modellierung der Dynamik freier Grenzflächen in Zweiphasenströmungen. Ph.D. thesis, ITLR Universität Stuttgart, 2004. [14] Lafaurie, B., Scardovelli, C.N., Scardovelli, R., Zaleski, S. & G.Zanetti, Modelling merging and fragmentation in multiphase flows with surfer. Journal of Computational Physics, 113, pp. 134–147, 1994.



169

Computational fluid dynamic as a feature to understand the heat and mass transfer in a vacuum tower K. Ropelato1, A. V. Castro2, W. O. Geraldelli2 & M. Mori1 1

School of Chemical Engineering, State University of Campinas (UNICAMP), Campinas-SP, Brazil 2 CENPES – PETROBRAS Research and Development Center, Cidade Universitaria, Rio de Janeiro, Brazil

Abstract The understanding of fluid dynamic phenomena in industrial equipment is extremely important for new projects and their optimization. Distillation columns have been used for centuries. Since then many advances have been achieved. The present study shows a methodology to study the heat and mass transfer in empty sections of distillation columns considering the EulerianLagrangian approach. A multiphase model is applied to the vacuum tower, with the vapor-liquid flow being modeled with a Eulerian-Lagrangian approach. The Computational Fluid Dynamic (CFD) technique is adopted as the tool to predict vacuum tower dynamics. The CFD results were validated with real operation behaviour. An ideal multicomponent equilibrium model is adopted to represent the thermodynamics in the heat and mass transfer processes. The characteristic time scales are used in the classification and the understanding of the dominant mechanisms in gas-liquid suspensions. This methodology is general, and therefore can be applicable to any turbulent gas-liquid flow. A discussion on the validity of the models is given, as well as an application to distillation vacuum towers. Keywords: CFD, thermodynamic equilibrium, multiphase, Lagrangian, spray.

1

Introduction

The heat and mass transfer takes place in a distillation column by vapor and liquid contact. In the vacuum towers wash zones, this contact is promoted by WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090151

170 Computational Methods in Multiphase Flow V spray nozzle distributors. This sort of device is widely used in distillation and absorption columns to provide a uniform liquid distribution. One of the challenges in this column section is to reduce the space available for heat and mass transfer, without decreasing process efficiency. The reduction of space between liquid sprays and vapor inlet can bring a couple of advantages: • minimize column height; • less maintenance • lower pressure drop This potential height reduction was observed in experimental measures, indicated by thermal equilibrium near the vapor entrance. The literature presents many studies considering spray simulations, each one having its own modeling approach as presented by Mugele and Evans [1], Meyer et al. [2], Trompiz and Fair [3] and Beck and Watkins [4], where the heat and mass transfer is modeled considering an Eulerian-Eulerian approach. Authors as Lefebvre [5] and Guet et al. [6] argue that an accurate knowledge of drop size distribution as a function of the conditions of the system is an important prerequisite for fundamental analysis of the heat and mass transfer. Kim [7] and Bozorgi [8] considered a Eulerian-Lagrangian model to represent droplet evaporation. Kim considered phase equilibrium using the fugacities of the liquid and gas phases assuming real gas behavior, and its importance in the calculation of the evaporation of a droplet or spray at high pressures. Bozorgi studied variables effects, such as liquid film, total projected surface area of the droplets and velocity profile of the droplets on the performance of the spray scrubber in the aerosol removing process. A Lagrangian model has some advantages in comparison to the Eulerian model, such as: • easy modeling of droplets diameter distribution; • spatial discretization of spray nozzles (computational mesh) is not required since it is represented by points within the domain In this study a 4-m diameter vacuum tower was considered with a liquid distributor having 41 spray nozzles. The vapor-liquid thermodynamic equilibrium is modeled with a “gamma-phi” approach. The model is capable of representing the evaporation and condensation of the droplets. Thus, droplets diameter is variable as a function of system properties. The thermodynamic equilibrium was modeled via FORTRAN subroutines coupled to the commercial CFD code, ANSYS-CFX 11. The main advantage of CFD technique is to allow the user to evaluate conceptual changes in equipment in short-term simulations (compared to experimental measurements) at low computational cost. Based on these statements, the main target of this work is to gather information on vacuum tower fluid dynamic using an Eulerian-Lagrangian model using Computational Fluid Dynamics (CFD) techniques that represent a powerful tool for better understanding of physical phenomena involved in vacuum towers.



2

171

Mathematical modelling

The present work applies simplified multicomponent Fick's Law and gamma-phi approach for liquid-vapor equilibrium in a Eulerian-Lagrangean multiphase flow model together with multicomponent heat and mass transfer. This model was applied successfully before by Paladino et al. [9] and Ropelato et al. [10]. The simplification proposed here is the independence of diffusive mass flux of one component from other components. This implies that each mass flux can be calculated without other fluxes information. In the other hand, thermodynamic equilibrium takes the all components into account. Thus, it is calculated rigorously. With the liquid-vapor equilibrium considering a “gamma-phi” approach, an equation of state is applied to calculate fugacity coefficient (which provides a measure of non-ideality of vapor phase), while an excess Gibbs free energy model is applied to predict the behavior and non-idealities of liquid phase by calculation of an activity coefficient. The considered equilibrium is a traditional approach and can be applied to many different mixtures cases. It is adequate for systems at low or moderate pressure, as the vacuum tower, for example. This model cannot be applied in systems at high pressures. This limitation can be explained because the excess Gibbs free energy models are determined at low pressures. Moreover, the “gamma-phi” equilibrium model is one of the most used equilibrium approach in Petroleum Industry. The conservation equations calculated in CFD code are presented below for the Eulerian phase. The mass conservation of the continuous (Eulerian) phase is: N

P ∂ (ρ C ) + ∇ ⋅ (ρ C v C ) = ΓCD ∂t D =1

∑

(1) The ΓCD represents each component mass flux through the interface. The subscripts C and D are the continuous and dispersed phase respectively. The momentum equation is: ∂ (ρ C v C ) + ∇.(ρ C v C v C ) = −∇p + ∇.T + M C + ρ c g (2) ∂t The Mc represents the interfacial momentum transfer terms, turbulent stress tensors and T is the viscous stress tensor. For Newtonian fluid, viscous stress can be expressed via Stokes equation:

[

T = µ C ∇v C + (∇v C )T

]

(3)

Where µC is the fluid dynamic viscosity. The conservation equation for component “i” in the continuous phase is: ∂ (ρC yiC ) + ∇ ⋅ (v C yiC ) − ∇ ⋅ (ρC DiC (∇yiC ) ) = Γi, mC (4) ∂t The term ρCDiC represents the mass diffusion coefficient, DiC represents the kinematic diffusivity and Γi,mC the source term due to mass transfer to/from dispersed phase. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

172 Computational Methods in Multiphase Flow V The energy equation can be represented as follows: ∂ (ρ C h C ) + ∇ ⋅ (ρC v C h C ) − ∇ ⋅ (λ C∇TC ) = ∂t Ncomp

∑Γ

i , mC h C

+ Q C + Sc

(5)

i =1

where hC is the enthalpy, TC is the temperature and λC is the convective heat transfer from/to liquid phase. Evaporation and condensation rates in the Lagrangian model will be indicated by the droplet mass conservation equation: dm D = Γi,mC = −πd p ρDSh (w L K e − w V ) dt

(6)

The dp is the droplet diameter; D represents the mass diffusivity term; Sh, the droplet Sherwood number; Ke, equilibrium ratio; and wL and wV, the component of mass fraction at the interface of the dispersed and the continuous phases, respectively. The simulation considered a two-way coupling. The coupling concept is very important in a multiphase flow. If the flow of one phase affects the other while there is no reverse effect, the flow is said to be one-way coupled. If there is a mutual effect between the flow of both phases, the flow is two-way coupled. Therefore, the effects of the presence of droplets on the turbulent motion of the continuous phase can be observed. A schematic diagram of coupling is shown in fig 1. The carrier phase is described by density, temperature, pressure and velocity field. The droplet phase is described by concentration, size, temperature and velocity field. Coupling can take place through mass, momentum and energy transfer between phases. Mass coupling is the addition of mass through evaporation or the removal of mass from the carrier phase stream by condensation. Momentum coupling is the result of the drag force on the dispersed and continuous phases. Momentum coupling can also occur with momentum addition or depletion due to mass transfer. Energy coupling occurs through heat transfer between phases.

Figure 1:

Schematic diagram of coupling effects.



173

2.1 Characteristics time scales In the formulation of the transport equations, several characteristic time scales can be defined. These time scales are of fundamental importance in the classification and understanding of the dominant mechanisms in suspensions. (Elgobashi [11]; Peirano and Leckner [12]). Time scales are fundamental to understand the vapor and liquid behavior, and also the effects of phase interaction. The characteristic time scale of the large eddies in the continuous phase ( τ ct ) is defined in accordance with the k-ε model as eqn (7) τ ct = 0.09

k ε

(7)

Where k is the turbulent kinetic energy and ε its dissipation. The Kolmogorov scale dissipative ( τ k ) is the characteristic time scale of the smallest scales. υ  τk =  c   ε 

0.5

(8)

The time of interaction between particle motion and continuous phase fluctuations is defined by eqn (9) τ ct t (9) τ cd = 2 3 Vr 1 + 0.45 2 k t τ cd is the Lagrangian integral time scale seen by the particles (computed along the trajectory of the particle). This means that the time spent by a virtual fluid particle in an eddy is not the same as the time experienced by a liquid particle, due to the mean relative velocity between the two phases. Eqn (10) represents the droplet relaxation time ( τ dt ). τ dt =

4d pρ d 3ρc C D Vr

(10)

The ratio τ dt / τ k represents the Stokes number in Kolmogorov scale (Stk), which indicates the probability of coalescence in the region of study. The Stokes number is a very important parameter in fluid-particle flows. If Stk1, then the droplets will have essentially no time respond to the fluid velocity changes and the droplets velocity will be little affected in the equipment, Crowe et al. [13].


174 Computational Methods in Multiphase Flow V 2.2 Numerical method and geometry details Element based finite volume method (EbFVM) with unstructured is used by ANSYS CFX solver to solve the set of equations. The pressure-velocity coupling formulation is similar to the Rhie-Chow method where the solution of velocity and pressure is in the same node of element in the numerical mesh. A computational mesh study was conducted to obtain mesh independent results. A mesh with approximately 1.106 nodes was considered good for the simulations. The independency of mesh and particle tracking was analyzed considering all characteristics of the model (multiphase, heat and mass transfer). Fig. 2(a) shows the boundary conditions imposed in this study: red color indicates de vapor inlet; yellow indicates the demister pad; blue color, the spray section; and green color represents the bottom of equipment. Fig 2(b) shows the numerical grid details. The vapor inlet doesn’t present symmetrical characteristics, thus the whole geometry was taken into account. The spray distributor with 42 sprays is showed in fig. 3. Each of it with 200 particle tracks being used. Each spray nozzle was modeled as a full cone with 60º. The droplet diameter distribution used in the simulations was obtained from the spray nozzle provider which was characterized for water. A Petrobras internal correlation was applied considering the oil properties. The model has been generated as three-dimensional (3D), steady-state and the continuous phase (vapor) was model using the k-ε turbulence model. The k-ε

Figure 2:

Vacuum tower. (a) Physical geometry with and boundary conditions; (b) Numerical grid details.



Figure 3:

175

Spray section distributor.

turbulence model is the most prominent turbulence models has been implemented in most general purpose CFD codes and is considered the industry standard model. It has proven to be stable and numerically robust and has a well established regime of predictive capability. For general purpose simulations, the k-ε model offers a good compromise in terms of accuracy and robustness (Wilcox [14]). A high resolution scheme (a bounded second order upwind) was used to model the advection terms of the momentum equation. 2.3 Boundary conditions The vapor flows upwards through the vacuum tower and the light gas oil liquid phase flows downwards counter-courant to the vapor phase. The case analyzed is considered a large vacuum tower with high capacity; the vapor/liquid flow ratio is 0.43.

3 Results and discussions The analysis of time scales was considered in the whole domain in two orthogonal slice planes. The vapor and liquid contact considering the effects of vapor inlet was evaluated by time scales. The characteristic time scale of the large eddies in the continuous phase (τc), fig. 4(a), considered the rate of effects of turbulent kinetic energy and the rate of dissipation of turbulent kinetic energy. Regions of high values of dissipation energy represent the low values of τc. These regions are close to the spray section. The Kolmogorov scales (τk) indicates regions of smallest vortices in the domain, hence these regions show the highest values of turbulent energy dissipation fig. 4(b). Fig. 5 considered the effects of vapor velocities in droplets behavior. Regions closed to the vapor inlet indicates low values of droplet relaxation time ( τ dt ), fig. 5(a), which means that these are regions of low droplets inertia. Thus phases are well mixed. Drag values are high in these regions. The time of interaction between particle motion and continuous phase t ), fig. 5(b). The fluctuations is defined by the Lagrangian integral time scale ( τ cd WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


Figure 4:

Vapor fluid dynamics behavior (a) characteristic time scale of the large eddies in the continuous phase (τc), (b) characteristic time scale of the Kolmogorov scales (τk).

Figure 5:

Droplets flow properties, (a) droplet relaxation time ( τ dt ), t ). (b) Lagrangian integral time scale ( τ cd

red color indicates regions where low values of heat and mass transfer occurs, these regions are close to spray section. The Stokes in Kolmogorov scale is a very important parameter in liquidvapor flows. If Stokes number approaches to zero, the response time of droplets is much less than characteristic time associated with the flow field. Thus the droplets will have enough time to respond to the changes in flow field. In other words, droplets and vapor will be flowing at very close velocities. If the opposite behavior is noticed the droplets velocity is unaffected by vapor phase. Fig. 6 shows the Stokes number in the vacuum tower and the regions with Stokes number close to one represent regions with risk of coalescence.



177

The studied region doesn’t show a strong risk to this phenomenon take place. There are no assumptions in the model with this regard. Problems with large coalescence regions should be treated with a more suitable model.

Figure 6:

Stokes in Kolmogorov scale (Stk).

Fig. 7(a) and (b) represent the temperature and volume fraction profile respectively. As the time scale analyses showed, the main region of heat transfer is close to the bottom of equipment. The thermodynamic equilibrium is achieved at half of the height of the equipment. The CFD temperature and thermodynamic equilibrium results were compared with values obtained in a Petrobras` Vacuum Tower with good agreement.

Figure 7:

4

(a) Temperature profile, (b) Volume fraction profile.

Conclusion

The proposed 3D, steady-state, and multiphase model represented adequately the vacuum tower fluid dynamics. The time scale methodology was presented as a feature for fluid dynamics understanding of a vacuum tower. The continuous phase turbulent behavior and its interaction with the droplets were studied in equipment entrance section. The WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

178 Computational Methods in Multiphase Flow V more relevant regions of heat transfer and thermodynamic equilibrium was analyzed. CFD tool proved to be also a powerful tool to predict the fluid dynamic of heat and mass transfer in vacuum towers, and it can be used with success in chemical process optimization to predict critical points of scientific investigation. A potential height reduction in this case is possible as long as the assumptions taken in this model are suitable for the present case. Future studies are being carried out, to propose some optimization points in the studied vacuum tower.

5 CD dp F g k p S t v Vr x y z

Nomenclature drag coefficient, [kg.m3/s] droplet diameter [m] diffusion, resistive force gravity acceleration, [m/s2] turbulent kinetic energy, [m2/s2] pressure, [Pa] source term time, [s] velocity vector, [m/s] slip velocity, [m/s] Cartesian coordinate, [m] Cartesian coordinate, [m] Cartesian coordinate, [m]

Greek letter Γ interphase mass flux [kg/m2/s] ε rate of dissipation of “k”, [m2/s3] µ dynamic viscosity, [kg/m s] ρ density, [kg/m3] τ characteristic time scale (s) ν kinematic viscosity [m2/s] Subscript c gas phase d liquid phase k Kolmogorov scale x cartesian coordinate y cartesian coordinate z cartesian coordinate φ fluid dynamic property Superscript t turbulent WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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References [1] Mugele, R. A and Evans, H. D. Droplet size distribution in sprays. Ind. Eng. Chem., 43, n 6, pp. 1317 – 1324, June 1951. [2] Meyer, M., Hendou, M., Prevost, M. Simultaneous heat and mass transfer model for spray tower design: application on VOCs removal. Computers Chem. Engng, 19, po. S277 – S282, 1995. [3] Trompiz, C. J., Fair, J. R. Entrainment from spray distributors for packed columns. Ind. Eng. Chem, 39, pp. 1797 – 1808, 2000. [4] Beck, J. C., Watkins, A. P. The droplet number moments approach to spray modelling: The development of heat and mass transfer sub-models. Int. Journal of Heat and Fluid Flow, 24, pp. 242 – 259, 2003. [5] Lefebvre, A. H. Atomization and Sprays. Taylor & Francis. Purdue University, West Lafayette, Indiana, 1989. [6] Guet, S., Ooms, G., Oliemans, R. V. A., Mudde, R. F. Bubble injector effect on the Gas lift efficiency. Fluid Mechanics and Transport Phenomena. 49, pp. 2242-2252, 2003. [7] Kim, H., Sung. The effect of ambient pressure on the evaporating of a single droplet and a spray. Combustion and Flame. 135, pp. 261 – 270. 2003. [8] Bozorgi, Y., Keshavarz, P., Taheri, M., Fathikljahi, J. Simulation of a spray scrubber performance with Eulerian/Lagrangian approach in the aerosol removing process. Journal of Hazardous Materials. 2006 . [9] Paladino, E. E., Ribeiro, D., Reis, M. V., Geraldelli, W. O., Barros, F. C. C. A CFD model for the washing zone in coker fractionators. AIChE 2005 Annual Meeting. [10] Ropelato, K., Rangel, L. P., Marins, E. R., Geraldelli, W. O. A CFD Study Comparing Different Feed Nozzle Arrangement within an Empty Spray Section in a Coker Fractionator. AIChE 2007 Spring Meeting. [11] Elgobashi, S. On predicting particle-laden turbulent flows. Applied Scientific Research , 52, pp. 309-329, 1994. [12] Peirano, E., Leckner, B. Fundamentals of turbulent gas-solid flows applied to circulating fluidized bed combustion. Department of Energy Conversion. Chalmers University of technology Göteborg, 1998. [13] Crowe, C., Sommerfeld, M., Tsuji, Y. Multiphase Flows with Droplets and Particles. CRC Press, pp 17-36, 1998. [14] Wilcox, D.C. Turbulence Modelling for CFD. DCW Industries, pp. 314, 2000.




181

Understanding segregation and mixing effects in a riser using the quadrature method of moments A. Dutta, J. Raeckelboom, G. J. Heynderickx & G. B. Marin Laboratorium voor Chemische Technologie, Ghent University, Belgium

Abstract Segregation and mixing effects of particle diameter distributions are numerically investigated in the riser section of a circulating fluidized bed. A granular kinetic theory based approach, which implements the Eulerian quadrature-based moment method to describe the particle phase, is coupled to an Eulerian multi-fluid solver through a population balance model. The gas-solid multiphase flow is two-dimensional, transient and isothermal. The particle distributions fed from each side inlet of the riser have different variances, but the same mean diameter. The core-annular regime used as a numerical benchmark for riser flows is well predicted. A comparison in the homogeneity of particle mixing is made from the lower-order moments of the particle distribution obtained at various positions and at different axial lengths along the riser. It is seen that the relative standard deviation of the particle distribution varies spatially, indicating dynamic mixing inside the riser. Keywords: mixing, riser, multiphase, quadrature method of moments.

1

Introduction

Multiphase flows involving solid particles in contact with a carrier gas in circulating fluidized beds (CFB) are ubiquitous in most chemical, petrochemical and pharmaceutical industries. Gas-fluidized circulating beds are commonly used in coal combustion, the catalytic cracking of crude oil, pharmaceutical granulation, etc. Quantitative understanding of the hydrodynamics of fluidization is needed for the design and scale-up of these processes. In these processes, the knowledge of particle segregation and mixing is limited. Nevertheless, there are applications in which it is important to avoid particle segregation, e.g. in WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090161

182 Computational Methods in Multiphase Flow V catalytic fluidized bed reactors, whereas in other applications it is necessary to achieve complete segregation or mixing, e.g. in fluidized bed combustors [1]. Usually, segregation is referred to as the spatial (both axial and radial) distribution of particles of different size and/or density. Dependence on the size and/or density differences is linked to operating conditions [2]. In a CFB riser, the solids size and/or density distribution gives rise to radial segregation resulting in a core-annulus structure, in which particles flow downwards along the walls in clusters while strands of particles move upwards, together with the dispersed particles in the centre [3]. The single-particle terminal velocity is often a decisive parameter affecting axial segregation in fluidized beds [4], while coreannulus flow is attributed to the radial segregation of the particles. Karri and Knowlton [5] experimentally found that the particle diameter distribution in the core is smaller than in the annulus due to shear and recirculation/backmixing effects generated from the solids downflow near the wall. Numerical studies using computational fluid dynamics (CFD) on solids segregation/mixing have been reported for binary, ternary [6,7] and even quaternary [8] mixtures of particles of varying size/density. To simulate a realistic particle distribution, the approach followed by most researchers [9,10] is the use of discrete approximations of the continuous size distribution function. However, this sectional approach is not feasible when the distribution is broad as it requires a large number of discrete sizes to accurately represent the size distribution leading to high computational costs even for relatively simple situations. Recently, Fan and Fox [11] have implemented a direct quadrature-based moment method to simulate segregation phenomena for a continuous distribution, thus avoiding the need for discrete approximations. In this method, the particle distribution is represented through a finite number of nodes, commonly referred to as abscissas, in the quadrature method. The evolution of these nodes is tracked through the lower-order moments of the distribution. As such, only a few nodes suffice to represent the entire distribution thus representing an attractive alternative to the traditional discrete approaches [11]. In the present study, the mixing effect of particle diameter distributions in a riser is numerically investigated using the quadrature method of moments (QMOM) in a multi-fluid CFD code, based on an Eulerian-Eulerian approach. The influence of polydispersity on the core-annulus flow profile is discussed. The standard deviation and mean determined through the lower-order moments of the distributions are calculated at various positions and at different axial lengths along the riser. These are used to study the influence of the particle distribution on the overall flow behavior and in particular, the mixing and segregation effects in the riser.

2

Numerical approach

A multi-fluid model based on the Eulerian-Eulerian approach is used. The conservation equations are solved for each phase in the Eulerian frame. The gas phase is considered as the primary phase, whereas the solid phase is considered as secondary or dispersed phase. For the gas phase the mass and momentum conservation equations are Reynolds averaged. The effects of turbulence are WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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taken into account via a dispersed k-ε turbulence model, which is an extension of the single-phase k-ε model [12], adapted for gas–solid interactions. For the solid phase, the transport equations for mass, momentum and granular temperature are obtained via the kinetic theory of granular flow (KTGF) [13]. 2.1 Population balance model (PBM) The general form of a particle population balance conservation equation, as written by Ramkrishna [14] is:   [  s n( x; t )]  .[  s n( x; t )u s ]   s S ( x; t ) (1) t where S ( x; t )  B( x; t )  D( x; t ). In eq. 1, the number of particles, n, is distributed with respect to some  intrinsic parameter, x (say particle size). The variables  s and u s refer to the density and velocity vector of the particles respectively. The source terms B and D represent birth and death rates due to aggregation, breakage, etc. Several approaches for solving these equations are available, the most common being the discrete approach. However, a computationally attractive approach is the method of moments in which the population balance is formulated in terms of the lowerorder moments in closed form [15]. For a homogeneous system, considering the number density function n(x;t) in terms of the particle size length (i.e. x≡L), the kth moment is defined by: 

mk (t )   Lk n( L; t )dL 0

k = 0, 1,…..,2N-1.

(2)

It is important to note that, usually, L is referred to as an internal coordinate in contrast to x and t, which are external coordinates. A number of lower-order

moments are sufficient to represent the particle number density, total surface area and total volume of the particles. The resulting moment transport equation obtained by applying the above moment transform to eq. 1 is:  (  s mk )  .(  s us mk )   s ( B k  D k ). (3) t A major drawback of this method is the need for an exact closure model for the source contributions. An exact closure is only available for constant or simple linear forms of the aggregation kernel and size-independent growth [15]. This constraint is, however, avoided by replacing the exact closure by an approximation of the unclosed terms using an ad hoc Gaussian quadrature formula. 2.1.1 The quadrature approximation (or Gaussian quadrature) In order to solve the transport equations (eq. 3) for the moments, a Gaussian quadrature is used to approximate the integral (eq. 2) by a finite summation of the products of weight,  and abscissa, L [15] as: 

N

mk (t )   Lk n( L; t )dL   i Lki 0

i 1


(4)

184 Computational Methods in Multiphase Flow V Thus a quadrature approximation of order N is defined by its N weights  j and abscissas

Lj

and can be calculated from the first 2N moments

m0 , m1 ..., m2 N 1 by writing the recursive relationship for the polynomials in terms of the moments mk . From a practical point of view, the system is ill-conditioned because the direct solution would require a non-linear search. McGraw [16] recommends the use of Product Difference (PD) algorithm, first described by Gordon [17]. The PD algorithm requires only the moments as inputs and gives the weights and abscissas as output. This algorithm is based on the minimization of the error introduced by replacing the integral in eq. (2) with its quadrature approximation by deriving a tridiagonal matrix of rank N/2 and finding its eigenvalues and eigenvectors.

2.2 Coupling CFD and PBM

The population balance model is coupled to the hydrodynamic model through a Sauter mean diameter, d32 , of the particle distribution given as: d32 

N d N d i

i

3 i 2 i



m3 . m2

(5)

Here, d32 is calculated from the ratio of the moments, m3 and m2 . This value is then replaced with the particle diameter in the drag model and updated with every time step. The use of the Sauter mean diameter causes only a single set of Navier-Stokes equations to be solved. As such, the particle distribution and with it, the moments, will be transported at the same velocity of the solids phase. This implies that particles of different sizes will have the same velocity.

3

System description

The riser set-up used in the present study is similar to the one used in the challenge problem reported by Knowlton et al [18] and explained in detail by Benyahia et al [19,20]. Knowlton et al [18] presented their experimental data obtained from a CFB column riser with FCC particles at the 8th International Fluidization Conference. They measured solids concentration and flux at given locations of the riser and observed core-annulus flow behavior along the riser height. The geometry of the 0.2-m riser with a total height of 14.2 m is similar to the experimental set-up used by Knowlton et al. [18]. The bottom inlet of the riser is a gas distributor yielding a superficial velocity of 5.2 m/s. Gas and solids are fed from two side inlets with a velocity of 0.476 m/s. The total solids volume fraction at the riser inlets is 40%. Although the original geometry of the riser has a single solid inlet [18], the main reason for selecting a two-inlet geometry design is to obtain mixing effects in the riser entrance zone similar to the experimental results [19]. At the inlets, the velocity and volume fraction of both the phases are specified. At the outlet, only the pressure (atmospheric) is specified. At the walls, the gas velocities are set equal to zero by introducing a no-slip boundary condition. The partial-slip boundary condition for the solids at WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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the wall is specified by applying an equation for the tangential velocity of the particles [21] using a specularity coefficient of 0.0001. The value is justified because an increase in the specularity coefficient would result in a reduction of the solids volume fraction at the walls, causing the core-annulus behavior to disappear [22]. Two lognormal distribution functions with a mean average diameter, d ave of 77 μm and a relative standard deviation,  / d ave of 0.2 and 0.5 are implemented at the side-inlets. The corresponding d32 of the inlet feed distribution are 83.3 μm and 120.3 μm respectively. This approach is also a check to ensure that the QMOM model returns the correct non-varying average diameter as the statistical mean. In the simulations, only the first six moments of the lognormal distributions are tracked (i.e. m0 , m1 ,.., m5 ), which implies the use of a 3-node quadrature approximation (i.e. N = 3).

4


A two-dimensional transient model approach incorporating the kinetic theory for the solid particles is used in the commercial code Fluent, version 6.3, to simulate the gas-solids flow in the riser. A transient two-dimensional approach simulates the dynamic behavior of a multiphase flow in a Circulating Fluidized Bed (CFB) within a reasonable period of time [23]. The simulations are continued for 40 s of real time. The time-averaged distribution of the variables is then computed considering the last 30 s of the simulation. Both log-normal (  / d ave = 0.2, 0.5) and monosize ( d  77 μm) particle distributions at the inlet are simulated for comparison with the experimental observations of Knowlton et al [18]. It is noteworthy to mention that the value of 77 μm for the monosize distribution corresponds to the probabilistic mean diameter of the log-normal distribution to ensure a test-similarity. The flow patterns in the riser show a transient behavior of the solids volume fraction and the solids velocity profiles for both log-normal (fig. 1(i)(a) and 1(ii)(a)) and monosize (fig. 1(i)(b) and 1(ii)(b)) distributions. Note that fig. 1 corresponds to the lower and upper section of the riser respectively. In both the cases, the solids volume fraction is higher near the walls where the solids velocity is lower. Clusters are seen to be formed in the top of the riser and to flow downwards along the walls. In the centre of the riser, the solids velocity is considerably higher giving rise to core-annular flow: a dilute core region and a dense region near the walls. The core-annulus profile appears to be more uniform when the solids are log-normally distributed through their moments. This is probably due to the coupling of the gas-solid hydrodynamics with the population balance model through the Sauter mean diameter, d32 , which varies with the moments of the distribution. Figure 2 shows a comparison of the time-averaged solids density distribution (  s  s ) at a height of 3.9 m from the bottom of the riser. The flow in the core is better predicted using a log-normal size distribution than a monosize distribution. The core width agrees reasonably well with the experimental data. However, the solids density at the wall is underpredicted, possibly due to inaccuracies in the applied wall boundary condition [20]. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


(i) Lower section Figure 1:

(ii) Upper section

Comparison of the time-averaged (10-40s) solids volume fraction and velocity profiles for (a) log-normal and (b) monosize distribution, for the (i) lower (0-8 m) and (ii) upper (8-14 m) section of the riser.

Although the density of the solid particles used in the experimental set-up [18] is slightly different from the density of the particles used in the present study, a good qualitative fit for the time-averaged profile of the solids mass flux is obtained in the core of the riser when using a log-normal distribution. Similar to the experimental observations [18], the solids mass flux is found to be maximum at the central core of the riser (see fig. 3), although the solids density is at its lowest value there (see fig. 2). This is attributed with the core-annulus behavior i.e. with a high upward solids (and gas) axial velocity in the core and a low downward solids (and gas) axial velocity in the annulus of the riser. Thus a particle distribution captures a more realistic dynamic flow behavior in a gassolid riser. Figure 4 show a comparison of the time-averaged solids volume fraction profile for the log-normal and monosize distributions respectively. The effect of riser height on solids volume fraction is investigated by studying the profiles at 1m, 3.9 m and 10 m from the bottom of the riser. Although a coreannular flow is observed in both the cases, there is a slight variation in the radial distribution of the solids volume fraction with the height of the riser. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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Figure 2:

Time-averaged solids density distribution, using both log-normal and monosize distribution at 3.9 m from the bottom of the riser, compared with the numerical simulation [19] and experimental data [18].

Figure 3:

Time-averaged solids mass flux distribution using a log-normal distribution at 3.9 m from the bottom of the riser, compared with the numerical simulation [19] and experimental data [18].

The log-normal disribution almost maintains the core-annulus pattern higher up the riser (e.g. at 10 m) whereas this pattern is assymetrically flattened in a monosize distribution. Indeed, the time-averaged profile of solids volume fraction differs remarkably along the height of the riser, as shown in fig. 4(b). A shift in the solids (and gas) axial velocity towards the right wall (see fig. 1(ii)) is indicated by a two-fold decrease in its solids volume fraction compared to the left wall. The difference in the solids flow behaviour in both the cases indicates that the particle distribution does, in fact, influence the gas-solid hydrodynamics in the riser. A similar conclusion can be drawn from the time-averaged axial solids velocity profiles given in fig. 5. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


(a)

(b) Figure 4:

Effect of riser height on the time-averaged solids volume fraction profile in the riser for (a) log-normal and (b) monosize distribution.

(a)

(b) Figure 5:

Effect of riser height on the time-averaged solids velocity profile in the riser for (a) log-normal and (b) monosize distribution.



189

As seen in fig. 5, the peak of the solids axial velocity shifts slowly towards the wall along with the height of the riser, but in different directions. This shift in the peak of the the dilute core zone is slightly more for the monosize distribution (also see fig. 1). The influence of the particle distribution on the hydrodynamic behaviour in the riser can thus be clearly seen through the comparison in fig. 5. At the wall, the solids have a negative axial velocity which indicates the downward movement of clusters along the walls. Remark that the calculated solids axial velocities are higher for the monosize distribution, due to a smaller width of the core. To evaluate mixing of the particle distributions, a user-defined subroutine is written to obtain the values of the moments calculated for various positions and for multiple cross-sectional surfaces at different axial locations along the riser. The mean,  , and standard deviation,  , is then calculated from the following:



m1 m0

(6)

2

m2  m1   (7)  . m0  m0  Applying these formulas give the values of the mean and the standard deviation of the local and global distributions respectively. The standard deviation gives insight on how the particles are distributed over a surface depending on their size. The smaller the standard deviation (i.e. smaller range of different sizes), the more segregated is the particle distribution in the surface. The larger this value (i.e. larger range of different sizes), the more mixed is the distribution. Based on a standard deviation, it is possible to conclude whether two or more surface distributions are equally mixed or not. Through the calculation of the mean of the distribution for a surface, it is possible to determine differences in the sizes of the particles present in these surfaces. The mean, together with the standard deviation, are thus a measure of the radial segregation along the riser height. The results obtained at 1 m, 3.9 m and 10m from the bottom of the riser are listed in table 1 and table 2 respectively at a simulated time of 40 s. Note that the left and right positions in table 1 refer to 5 mm radial distance from the left and the right wall of the riser respectively, while the middle refers to the mid-point of the cross-sectional surface. The mixing and segregation of the lognormal distributions and the effect on the riser hydrodynamics can be observed from the results of table 1 and 2. From table 1, it can be concluded that the radial mixing of particles over a cross section of the riser along the height of the riser is very dynamic. At a height of 1m, the two distributions are mixing in the middle because the standard deviation of the distribution in the middle lies between the values of those at the walls. Higher up the riser i.e. at 3.9 m and further at 10 m from the bottom, it is observed that the distribution in the middle becomes more segregated, meaning that only a fraction of the particles introduced in the riser resides in the middle part of the riser. This also indicates the formation of a core-annular regime higher up the riser.





Estimation of the local particle diameter distribution over a surface for three positions along the riser height.

Height from the bottom of the riser (m)

Left

1.0 3.9 10.0

7.70  10-5 7.70  10-5 7.70  10-5

1.0 3.9 10.0

3.72  10-5 3.52  10-5 3.25  10-5

1.0 3.9 10.0

1.0 1.0 1.0

Table 2:

Radial position Middle Right Average diameter, local 7.70  10-5 7.70  10-5 -5 7.70  10 7.70  10-5 -5 7.70  10 7.70  10-5 Standard deviation,  local 3.41  10-5 3.01  10-5 -5 2.75  10 3.02  10-5 -5 2.96  10 3.01  10-5 Spread ratio,  local /  'local 0.92 0.81 0.78 0.86 0.91 0.93

Estimation of the global particle diameter distribution for different radial surfaces along the riser height.

Parameters of distribution Average diameter, global

Height from the bottom of the riser (m) 1.0 3.9 10.0 7.70  10-5 7.70  10-5 7.70  10-5

Standard deviation,  global

3.37  10-5

2.93  10-5

3.02  10-5

Spread ratio,  global /  'global

1.0

0.87

0.89

Rel. std. deviation,  global / global

0.438

0.38

0.392

The surface-averaged mean and standard deviation of the lognormal distributions are given in table 2 for surfaces at different radial cross-sections along the height of the riser. It can be concluded that the distribution of particles along the riser height show segregation which depends on the hydrodynamic conditions of the riser. The correspondence of the local and the global mean with the mean average diameter of the distributions indicates the robustness of the QMOM model in returning the correct average diameter for multiple distributions with a varying standard deviation. These observations confirm that the gas-solid simulation in a riser using particle distributions show segregation and mixing effects, which is influenced by the overall riser hydrodynamics.

5

Conclusion

The Quadrature-based Method of Moment (QMOM) has been successfully applied to solve the evolution of a continuous particle distribution in a multiWIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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fluid CFD model for flows in a gas-solid riser. The core-annulus profile usually observed in riser flow is well predicted by this model. The simulation results considering a log-normal particle diameter distribution give a better qualitative fit with the experimental data as compared to the fit when using a monosize distribution. In comparison with the results for a monosize distribution, the coreannulus profile is much more uniform and maintained higher up in the riser for the log-normal distribution. It can be said that polydispersity has an influence on the gas-solid hydrodynamics of a riser, especially on the core-annulus flow profile. The radial segregation and mixing for the particle distributions, calculated for various positions and for multiple cross-sectional surfaces at different axial locations along the riser, is evaluated from the local and surfaceaveraged moments of the distribution. It is seen that the radial mixing profile is dynamic as it changes along the height of the riser. Although the distributions are quite mixed in the middle of a cross-sectional surface close to the inlet, it becomes more segregated as it approaches the outlet. The effect of mixing homogeneity due to particles size distribution on the hydrodynamic conditions of a riser is investigated. Spatial segregation of particles can be observed if, according to their size, different velocities of the particles are computed. For this, a modification in the quadrature-based moment method is necessary.

References [1] Das, M., Banerjee, M. & Saha, R.K., Segregation and mixing effects in the riser of a circulating fluidized bed. Powder Technology, 178(3), pp. 179186, 2007. [2] Nakagawa, N., Dingrong Bai, Shibuya, E., Kinoshita H., Takarada, T. & Kato, K., Segregation of particles in binary solids circulating fluidized beds. Journal of Chemical Engineering of Japan, 27(2), pp. 194-198, 1994. [3] Bader, R., Findlay, J. & Knowlton, T., Gas/solid flow patterns in a 30.5-cm diameter circulating fluidized Bed. Circulating Fluidized Bed Technology: II, Basu, P. and Large, J. F. (eds.), Pergamon Press, p. 123, 1988. [4] Hirschberg, B. & Werther, J., Factors affecting solids segregation in circulating fluidized-bed riser. AIChE Journal, 44(1), pp. 25–34, 1998. [5] Karri, S.B.R. & Knowlton, T.M., Flow direction and size segregation of annulus solids in a riser In: L.S. Fan and T.M. Knowlton, Editors, Fluidization IX, Engineering Foundation, New York, pp. 189-194, 1998. [6] Iddir, H., Arastoopour, H. & Hrenya, C.M., Analysis of binary and ternary granular mixtures behavior using the kinetic theory approach. Powder Technology, 151(1-3), pp.117-125, 2005. [7] Huilin, L. & Gidaspow, D., Hydrodynamics of binary fluidization in a riser: CFD simulation using two granular temperatures. Chemical Engineering Science, 58(16), pp. 3777-3792, 2003. [8] Nienow, A.W., Naimer, N.S. & Chiba, T.S., Studies of segregation/mixing in fluidized beds of different size particles. Chemical Engineering Communications 62(1-6), pp. 53-66, 1987.


192 Computational Methods in Multiphase Flow V [9] Mathiesen, V., Solberg, T. & Hjertager, B.H., Predictions of gas/particle flow with an Eulerian model including a realistic particle diameter distribution. Powder Technology 112(1-2) pp. 34-45, 2000. [10] van Wachem, B.G.M., Schouten, J.C., van den Bleek, C.M., Krishna, R. & Sinclair, J.L., CFD modeling of gas-fluidized beds with a bimodal particle mixture. AIChE Journal, 47(6), pp.1292-1302, 2001. [11] Fan, R & Fox, R., Segregation in polydisperse fluidized beds: Validation of a multi-fluid model. Chemical Engineering Science 63(1) pp.272 -285, 2008. [12] Hinze, J., Turbulence: An Introduction to its Mechanism and Theory. McGraw-Hill, New York, 1959. [13] Gidaspow, D., Multiphase Flow and Fluidization, Continuum and Kinetic Theory Descriptions. Academic Press, New York, 1994. [14] Ramkrishna, D., Population balances. Theory and applications to particulate systems in engineering. Academic Press, New York, 2000. [15] Marchisio, D.L., Pikturna, J.T., Fox, R.O., Vigil, R.D. & Barresi, A.A., Quadrature method of moments for population-balance equations. AIChE Journal, 49(5), pp. 1266-1276, 2003. [16] McGraw, R., Description of Aerosol Dynamics by the Quadrature Method of Moments, Aerosol Sci. Tech. 27(2), pp. 255-265,1997. [17] Gordon, R. G., Error Bounds in Equilibrium Statistical Mechanics. J. Math. Phys., 9, pp.655-663, 1968. [18] Knowlton, T., Geldart, D., Matsen, J. & King, D. Comparison of CFB Hydrodynamic Models. PSRI Challenge Problem, 8th International Fluidization Conference, Tour, France, May, 1995. [19] Benyahia, S., Arastoopour, H. & Knowlton, T. M. Two-dimensional transient numerical simulation of solids and gas flow in the riser section of a circulating fluidized bed. Chemical Engineering Communications, 189(4), pp. 510-527, 2002. [20] Benyahia, S., Arastoopour, H., Knowlton, T.M. & Massah, H., Simulation of particles and gas flow behavior in the riser section of a circulating fluidized bed using the kinetic theory approach for the particulate phase. Powder Technology, 112(1-2), pp. 24-33, 2000. [21] Johnson, P.C. & Jackson, R., Frictional-collisional constitutive relations for granular materials, with application to plane shearing. Journal of Fluid Mechanics 176, pp. 67-93, 1987. [22] Benyahia S., Syamlal M. & O’Brien, T.J. Study of the ability of multiphase continuum models to predict core-annulus flow. AIChE Journal, 53(10), pp.2549-2568, 2007. [23] Tsuo, Y. P. & Gidaspow, D., Computation of flow patterns in circulating fluidized beds. AIChE Journal, 36(6), pp. 885-896, 1990.



193

Numerical simulation of heavy oil flows in pipes using the core-annular flow technique K. C. O. Crivelaro1, Y. T. Damacena1, T. H. F. Andrade2, A. G. B. Lima1 & S. R. Farias Neto2 1

Mechanical Engineering Department, Federal University of Campina Grande, Brazil 2 Chemical Engineering Department, Federal University of Campina Grande, Brazil

Abstract The importance of heavy oils in the world market for petroleum has increased very quickly in the last years. The reserves of heavy oils in the world are estimated at 3 trillion barrels, while reserves of light oils have reduced progressively in the last decade. The high oil viscosity creates major problems in the production and transportation of the oil. This situation leads to the high pressure and power required for its flow, overloading and damaging the equipment, increasing the cost of production. Due to the need to develop new alternatives that will make the production and transport of heavy oil economically viable, this work has the objective to study, numerically, the behavior of isothermal multiphase flow (heavy oil and water), type “core flow”, in pipelines, using the software CFX® 3D. The pressure drop was determinated to a core-flow in a pipe with 7 in. diameter, 2.7 Pa.s oil viscosity and water at environment temperature. Results of the pressure, velocity and volume fraction distributions of the phases are presented and analyzed. It was verified that the pressure drop was reduced 58 times when compared to that obtained with oil flow alone in the pipe. Keywords: heavy oil, numerical simulation, CFX®, two-phase flow oil-water.



1

Introduction

The reserves of heavy oils in the world are estimated at 3 trillion barrels, while reserves of light oils have reduced progressively in the last decade. This leads to much economic interest in the reserves of heavy oils and research to make its production economically viable. The heavy oil has a low degree API (between 10º and 22º) and high viscosity (between 100 cp e 10000 cp). Besides, it has a high ratio of carbons/hydrogen, a large amount of residue carbon, asphaltenes, sulfur, nitrogen, heavy metal and paraffin. The major problems in heavy oil production are: high density, which increases the fluid hydrostatic weight, high viscosity, which makes the flow very difficult, resulting in high pressures and therefore high power being required for its flow, increasing the cost of production. In offshore fields, these difficulties become more serious due to the adverse conditions present during production and transportation. Nowadays, the methods currently used for heavy oil production are based on the reduction in the viscosity of the oil within the reservoir and well and, frequently, are associated with a system lift. The core annular flow technique can be called core-flow; different to other techniques, it is based on the lateral injection of relatively small quantities of water into the pipe and is capable of generating an annular pattern of two-phase flow. This annular pattern will be very stable, since the two fluids are immiscible, where the oil is in center of the pipe and a thin layer of water is flowing near the wall surface. This injection of water will lubricate the wall of the pipe, reducing the friction between the wall and the oil along the flow, as reported by Prada and Bannwart [13] and Silva [14]. According to Bensakhria et al. [6], this technique was created by Isaacs and Speed in 1904, patent request nº 759374 in the United States and was the first to mention the ability to transport viscous product through water lubrication. However, only in 1970 was a largescale industrial pipeline built (length of 30 km and diameter of 15 cm) to transport heavy oil by the Shell Company near Bakersflield in California. For more than ten years, a viscous crude oil has been produced at the flow rate of 24000 bbl/d in a water lubricated regime. The interest in heavy oil production employing the core annular flow technique has increased in recent years because of the large reserves of heavy oil accessible. This technique brings attractive results with regard to energy consumption. This fact is related to the reduction in pressure drop during the water/oil flow type core-annular when compared with the transport of oil alone (Andrade [1]). It has been observed in the literature related to the use of this technique to improve the transport of heavy oils using water as lubricant (Andrade [1]; Bai [3]; Bannwart [4]; Bensakhria et al. [6]; Joseph et al. [7]; Ko et al [8]; Oliemans et al. [9]; Ooms and Poesio [10]; Prada and Bannwart [13]). Bannwart [4] proposed a theory to stabilize the pattern annular when two liquids of density and viscosity flow differently in a horizontal pipe. The theory is based on the analysis of the linear momentum equation in a cross section of the pipe leading to account for the effect of interfacial tension. Bensakhria et al. [6] evaluated the radial position of the annular flow and showed that the position WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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depends only on the relationships between the contact perimeter of the wall of the pipe and the fluid that forms the core (oil) and the perimeter of the pipe. This ratio depends on the difference in density between the fluids to transport and the fluid to lubricate, as well as of the amount of injected water. The ideal annular flow or perfect core annular flow (PCAF) corresponds to an exact solution of the concentric fluid flow with a different density and viscosity in a pipe of circular cross section. According to Bensakhria et al. [6], the ideal or perfect annular flow appears to be very rare and can exist only for density matched fluid. Prada and Bannwart [13] also study a simplified solution to perfect annular flow, where two immiscible Newtonian fluids (oil and water) are flowing in a vertical pipe in a concentric configuration with a smooth circular interface. According to Prada and Bannwart [13], core flow lift is a new alternative for heavy oil production, because it significantly increases the well productivity by decreasing of the frictional pressure losses, without the addition of heat and without the use of chemical agents or diluents. Hence, artificial lift technology can be applied in either onshore or offshore fields, horizontal or vertical wells, and its installation in the field is relatively simple. Ooms and Poesio [10] analyzed the annular flow in a stationary regime in a horizontal pipe and proposed a theoretical model based on hydrodynamic lubrication theory. According to this model, a harmonic motion was observed in the annular flow. Indeed, the more viscous fluid (heavy oil) is moved to a wave form in the center of the horizontal pipe; this behavior is known as Wavy Core Annular Flow (WCAF). Ooms et al. [11] theoretically investigated the hydrodynamic counterbalancing of buoyancy force on a core of heavy oil flowing in the pipe, by considering the difference in density between two fluids. During the study it was assumed that the fluid that forms the core consists of a solid center surrounded by a high-viscosity liquid layer. To represent the pressure drop correctly it was necessary to model the effects of the WCAF, such as turbulence and fluctuability. The buoyancy term favors the heavy oil flow, but this is affected by the wave interface in the two-phase flow water/heavy oil (Prada and Bannwart [13]). In this sense the aims this work are to study, by numerical simulation, the isothermal behavior of the two-phase flow (oil and water) type core-flow of ultra-viscous heavy oils in pipe, using the finite-volumes method through the software CFX® 3D. Through these simulations, the pressure, velocity and volume fraction distributions of the phases present will be analyzed.

2

Methodology

2.1 Mathematical modeling The set of equations to be solved by CFX are as follows. Continuity:  t



 f         f    U    S MS 

NP     1


(1)

196 Computational Methods in Multiphase Flow V Momentum equation: 





t

 f    U       f    U 



  U

   f p





  T    f    U   U  



NP        U    U   S M  M   1





(2)

where α and β represent the phases involved (water or oil), f is the volume  fraction, ρ is density, U is the velocity vector, Np is the number of phases involved, p is the pressure and  is viscosity. In addition, the term SMSα describes user specified mass sources, Гαβ is the mass flow rate per unit volume from phase β to phase α, SMα describes momentum sources due to external body forces (buoyancy force and rotational force), Mα describes the interfacial forces acting on phase α due to the presence of other phases (drag force, lift force, wall lubrication force, virtual mass force and interphase turbulent dispersion force),   and the term   U    U  represents momentum transfer induced by  



interphase mass transfer. To simplify the model and the governing equations solution, some consideration were assumed (table 1). The following boundary conditions were used: a) oil: u = v = 0 and w = Uo in z = 0 to  (x,y); u = v = w = 0 to  (x,y,z) / x2 + y2 = r2, where “r” represents the radius of the pipe. Laminar flow regime. b) water: u = v = 0 e w = Uw em z = 0 para  (x,y); u = v = w = 0 para  (x,y,z) / x2 + y2 = r2; Turbulent flow regime. The turbulence model used to water flow was the k-Epsilon model. In this model it is assumed that Reynolds stresses are proportional to the average velocity gradient, with the constant of proportionality characterized by turbulent viscosity (known as the hypothesis of Boussinesq). Table 1:

Considerations adopted for solving the governing equations.

Flow thermal-physical properties Interphase transfer models Buoyancy force Mass convergence criteria Advection scheme Pressure interpolation type Velocity interpolation type

Consideration Two-phase (water/oil), tridimensional, incompressible and isothermal. constant Mixture model Not considered 10-7 kg/s High resolution Trilinear Trilinear



197

The transport equations to turbulent kinetic energy, k, eqn. (3), turbulent dissipation,  , eqn. (4), are:       t   (   f k  ) k        f    U  k     (3)  t  k        f G     



   t    (   f   )       f   U           t    f

 k

  

(4)

C1G  C 2    

where G is the generation of turbulent kinetic energy inside the phase α and

C1 and C2 are empirical constants. In eqn. (5),   is the rate of dissipation of the turbulent kinetic energy of the phase α, eqn. (6), defined by: 3 c  q   l and k α is the turbulent kinetic energy to phase α given by: 2 q k  2 where l is the spatial scale length, q is the scale of velocity and empirical constant calculated by eqn. (7), given by: 2 c   4c In this equation

(5)

(6)

c is an (7)

c is an empirical constant and t corresponds to

turbulent viscosity, defined by eqn. (8) as follows: 2 k  t  c   



(8)

The constant used in the eqns. (3)–(8) are: C1 = 1,44; C 2 = 1,92; c  = 0,09;

 k = 1,0;   = 1,3. 2.2 Numerical solution The software CFX 10 was used to generate the mesh and numerical solution of the physical problem. This software uses the methodology of the finite volumes to solve its equations. The thermal-physical properties of the fluids and inlet velocity are illustrated in table 2.



Water Oil

The thermal-physical properties of the water and oil used in all simulations. Thermal-physical properties Density (kg/m3) Viscosity (Pa.s) 997,00 0,0008899 989,00 2,7

Figure 1:

Inlet velocity (m/s) 1,0 0,8

Numerical mesh used in all simulations.

For the initial condition, we considering that the pipe is full with water, and has a null vector velocity (Vx water = Vy water = Vz water = 0 m/s). 2.3 Numerical mesh All the simulations were developed in the Thermal and Fluids Computational Laboratory, Mechanical Engineering Department, in the Center of Science and Technology of the Federal University of Campina Grande. The development of the mesh was a simplification of a nozzle reported by Prada and Bannwart [12]. The domain of study was created by definition of the points, curves, surfaces and solids and by describing the size and shape (D = 0,1778m e L = 15m) of the nozzle, so we generate the unstructured mesh and after several refinements we obtain a mesh with 105700 elements as illustrated in fig. 1.

3

Numerical results

The numerical results, in the transient state, were obtained for an elapsed time of t = 90 s and time steps of Δt = 0.3 s. In the steady state, we analyze the volume fraction, pressure field and velocity profiles. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


199

Fig. 2 shows the pressure drop in different positions along the length of the pipe (z axis). A decrease of pressure along of the pipe is verified, with the highest pressure in the inlet and the lowest in the outlet of pipe. A pressure drop Δp = 675.23 Pa was found, this being necessary to dislocate the oil and water. No gravity effects were considered. The mass flow rate of the oil and water were 15.86 kg/s and 4.69 kg/s, respectively. Figs. 3 and 4 show the volume fractions profile of the oil in the center. It is observed that the oil flows almost in central region of the pipe; approximately 95% of the center of the pipe is full of oil.

Figure 2:

Figure 3:

Pressure drop of the core flow as a fraction of the radial position for different locations along the pipe.

Volumes fractions of the oil in the core-flow along the pipe.



Volume fractions of oil

1

0.8

0.6

0.4

0.2

0 -0.12

Figure 4:

-0.08

-0.04

0

0.04

Radius (m)

0.08

0.12

Volume fractions of the oil as a function of the radial position in z = 7.5 m.

Volume fractions of the water

1

0.8

0.6

0.4

0.2

0 -0.12

Figure 5:

-0.08

-0.04

0

0.04

Radius (m)

0.08

0.12

Volume fractions of the water as a function of the radial position in z = 7.5 m.

The volume fractions of the water in a steady state in different regions of the pipe are shown in fig. 5. As expected, the water moves near the wall of the pipe in the form of a ring. In addition, in fig. 5, we observe that a small portion of the water moves together with the oil in the center of the pipe (  5% ). This can be justified by the occurrence of dispersion of the water phase in the center of the pipe due to the turbulence level of the flow. The oil velocity profiles along of the pipe are shown in fig. 6. A small region, the so called hydrodynamic entrance length, where the fluid moves with constant velocity profiles along the pipe, is verified.



Figure 6: Table 3:

201

Oil velocity profiles in a core-flow along the pipe.

Comparison of the pressure drop among water and oil single-phase flow and core annular flow. Flow

Core-flow Water single-phase flow Oil single-phase flow

Pressure drop (Pa) 675.23 539.52 39112.18

Reduction factor (Ω) 57,2

The oil velocity profiles in a pattern core-flow stay almost constant in the center of the pipe until the annular region, where a water fraction of 5% can be seen in fig. 7. In the wall, the water velocity profiles were made null by increasing the distance of the wall until the region is found where the oil flows alone, see fig. 7. To evaluate the efficiency of the core-flow technique, we compare the pressure drop in the two-phase flow with the pressure drop obtained in the water and oil single-phase flow. The water in the core flow is always in contact with the internal wall of the pipe and the pressure drop in the two-phase flow should be close to the pressure drop in the water flow alone at the mass flow rate of the mixture (Barbosa [5]). Table 3 shows this pressure drop. It was necessary for Δp  w  20.6 kg/s along the pipe, without = 539.52 Pa to move a water flow rate of m considering the effects of gravity. This pressure drop was lowest at the value obtained in the core-flow (Δp = 675.22 Pa). This demonstrates that through this method, we can move ultra-viscous petroleum with a pressure drop near to the value found in water flow. The pressure drop in the case of the single-phase flow WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


Velocity (m/s)

0.8 0.6

água óleo

0.4 0.2 0 -0.12 -0.08 -0.04

0

0.04

Radius (m)

Figure 7:

0.08

0.12

Velocity profiles of the water and oil in the center of the pipe as a function of the radial position.

of oil is introduced in fig. 7. It was necessary for Δp = 39112.1 Pa to move an oil flow rate of 20.44 kg/s along the pipe. By comparing the Δp of the cases coreflow, a reduction factor of the pressure drop of approximately 58 times was obtained.

4

Conclusions

In this work a numerical study of the phenomenon associated with water-oil twophase flow of the core-flow type was developed. Results of the velocity, pressure and volume fraction of the phases were obtained, analyzed and compared, by showing the efficiency of the core-flow technique in the production and transport of heavy oils. In this study, it can be concluded that Δp = 676.17 Pa was necessary to move the heavy oil in a core-flow, in a pipe of 7 inch diameter and 15.0 m length. The mass flow rate of oil and water were 15.85 kg/s and 4.68 kg/s, respectively. By comparing the core-flow technique with a water singlephase flow, we notice that the two-phase pressure drop (Δp = 676.17 Pa) approaches the pressure drop in water single-phase flow (Δp = 501.5 Pa) to oil viscosity of 2.7 Pas. By comparing the core-flow technique with an oil singlephase flow, the reduction factor of the pressure drop was approaching 58 times. Due to these benefits, we observe great interest in the use of the core-flow technique to solve the problems related to the production and transport of ultraviscous oil in pipes.

References [1] Andradre, T. R. F. Numerical study of heavy oils transport on pipe lubrificated by water. Master thesis in Chemical Engineering – Federal University of Campina Grande, Paraíba, Brazil, 2008. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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[2] ANSYS, CFX-Theory Manual, 2005. [3] Bai, R. Traveling waves in a high viscosity ratio and axisymmetric core annular flow. PhD Thesis, Faculty of Graduate School of the University of Minnesota, Minnesota-USA, 1995. [4] Bannwart, A. C. Modeling aspects of oil–water core annular flows, Journal of Petroleum Science and Engineering, vol., 32, pp. 127– 143, 2001. [5] Barbosa, A. Transient effects in the pressure drop for heavy oil-water core annular flow in metallic pipes. Master thesis in Petroleum Engineering – State University of Campinas, Campinas, 2004 [6] Bensakhria, A.; Peysson, Y. & Antonini, G., Experimental study of the pipeline lubrication for heavy oil transport. Oil & Gas Science and Technology – Rev. IFP, vol. 59, N°. 5, pp. 523-533, 2004. [7] Joseph, D. D., Bai, R., Chen, K. P. & Renardy, Y. Y., Core annular flows. Annual Reviews Fluid Mechanical, vol. 29, pp.65–90 1997. [8] Ko, T.; Choi, H. G.; Bai, R. & Joseph, D.D. Finite element method simulation of turbulent wavy core annular flows using a k-w turbulence model method. International Journal of Multiphase Flow, vol. 28, Nº 7 pp. 1205–1222, 2002. [9] Oliemans, R.V.A.; Ooms, G.; Wu, H.L. & Duijvestijn. A., The core annular oil/water flow turbulent-lubricating-film model and measurements in a 5 cm pipe loop. International Journal of Multiphase Flow, vol. 13, Nº. 1, pp. 23-31, 1987. [10] Ooms, G. & Poesio, P., Stationary core-annular flow through a horizontal Pipe. Physical Review, vol. 68, 2003. [11] Ooms, G.; Vuik, C. & Poesio, P., core-annular flow through a horizontal Pipe: hydrodynamic counterbalancing of buoyancy force on core. Physics of Fluids, vol. 19, Nº 092103 (2007). [12] Prada, J. W. V. & Bannwart, A. C. Pressure drop in vertical core annular flow, XV Brazilian Congress f Mechanical Engineering, Água de Lindóia, São Paulo, 1999. [13] Prada, J. W. V. & Bannwart, A., C., Modeling of vertical core annular flows and application to heavy oil production. Energy for the New Millennium February 14-17, New Orleans, LA. Proceedings of ETCE/OMAE, 2000. [14] Silva, R.C.R. Wettability alteration of internal surfaces of pipelines for used in the transportation heavy oil via core-flow. Master thesis in Science and Petroleum Engineering – State University of Campinas, São Paulo, 2003.




205

Simulation of flow and modelling the residence time distribution in a continuous two impinging liquid-liquid streams reactor using the Monte Carlo Technique M. Sohrabi & E. Rajaie Amirkabir University of Technology, Department of Chemical Engineering, Iran

Abstract In the present study, a stochastic model for the residence time distribution (RTD) in a coaxial counter current liquid-liquid impinging streams system has been developed. Simulations of droplets movements in the impinging spray systems determine the residence time distribution. Droplets dynamics has been formulated from the Boltzmann equation using direct simulation Monte Carlo (DSMC) method. The data predicted from the model has been correlated with the experimental results obtained from a coaxial counter current two impinging liquid-liquid steams apparatus. The reactor consisted of a cylindrical vessel made of Pyrex glass, length 60 cm and internal diameter 14 cm. The apparatus was equipped with two circular plates placed at the two ends of the reactor. Thus the length of the contact compartment could be varied by moving the plates away from or towards each other. Water and toluene were used as two immiscible liquid streams. These were sprayed into the reactor by applying pressurized nitrogen via special ducts, placed behind the feed nozzles. The degree of agreement between the experimental results and those predicted from the model was 85%. Keywords: impinging streams reactor, residence time distribution, Monte Carlo method, Boltzmann relation, Navier-Stokes equation.



1

Introduction

One of the important operations in chemical engineering is the mass transfer between immiscible phases. In an impinging streams apparatus a unique flow behavior is utilized by which the transfer processes in heterogeneous systems is intensified [l]. In such systems two feed streams, flowing parallel or counter currently collides with each other at a zone in which the two streams impinge. Impinging streams systems, first proposed by Elperin [2], have been applied as some suitable devices for enhancing mass and heat transfer processes in flowing gas-liquid, gas-solid and liquid-liquid emulsions. In such apparatus, two droplet-laden gaseous jets flowing in opposite directions are allowed to impinge. Some of the multiphase reactions carried out in two impinging streams reactors include two-phase mono-nitration of toluene [3], chemical absorption of CO2 gas in both sodium hydroxide [4] and mono ethanolamine solutions [1] and isomerization of glucose to fructose by an immobilized enzyme [5]. Impinging streams systems have been successfully applied to other chemical processes such as dissolution and mixing as well as mass and heat transfer operations [6–9]. In the present study a stochastic model for the residence time distribution (RTD) of the aqueous phase within a coaxial counter current two liquid-liquid impinging streams contactor (TISC) has been carried out. In addition a comparison has been made between the experimentally determined RTD data and those predicted from the model.

2

Contactor system

The experimental apparatus used in the present study is shown in Fig. 1. The contactor consists of a cylindrical vessel made of Pyrex glass, length 60 cm and internal diameter 14 cm. The contactor is equipped with two circular plates made of “Teflon” placed at the two ends of the contactor. Thus the length of the contact compartment can be varied by moving the plates away from or towards each other.

Figure 1:

Contactor dimensions (figures are in mm).

The two immiscible liquids (toluene and water) were kept in separate glass containers and fed to the contactor via four identical feed nozzles made of glass, situated at the two ends of the vessel. Liquids were transported to the nozzles WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


207

using metering pumps. The liquid streams were sprayed into the contactor by applying pressurized nitrogen via special ducts, placed behind the feed nozzles. In figure 2, the diagram of a feed nozzle is shown. As it may be observed from this figure, there is no mixing of fluids within the nozzle. The mixing is occurred only at the exit of the system. In each end of the contactor two identical feed nozzles were installed spraying two different liquids towards the impingement zone.

Figure 2:

Diagram of a feed nozzle.

Operating conditions are shown in Table 1. Table 1: Temperature (ºC)

20.1

Operating conditions of the impinging streams contactor. Distance between the two feed nozzles (cm) 25

Flow rate of nitrogen (cm3/min)

Toluene flow rate (cm3/min)

Water flow rate (cm3/min)

1300

320

325

In order to determine the suitable positions and angles for the feed nozzles at which stable jets of liquids would be established, a number of experiments were performed. This experimental set up allows consideration of the effects of changing certain pertinent parameters of the system on the residence time distribution of materials and hence on the extent of the heterogeneous liquid-liquid mixing and reaction.

3


3.1 Measurement of the residence time distribution of aqueous phase To determine the residence time distribution of the aqueous phase within the contactor, the following experiment was performed. Toluene and distilled water were fed to the contactor via spray nozzles. At a time, a change from the water stream to one of water containing a mineral salt (potassium dichromate) with known concentration and colour intensity (Co) was rapidly performed (step input). Samples at the outlet of the contactor were collected, using a circular vessel divided into 24 segments with equal volumes, placed under the exit port WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

208 Computational Methods in Multiphase Flow V and rotated at a pre-set speed applying an electric motor. Successive samples were obtained at equal time intervals by this method. Change in the rotating speed altered the time intervals between the sample collections. The content of each segment was transferred to a separating funnel. The aqueous phase was separated from the organic layer and the concentration (colour intensity) of the former was measured (C), using a UV spectrophotometer. From a plot of C/Co versus time, the RTD data were determined. 3.2 Developing a stochastic model for the residence time distribution of the aqueous phase in TISC 3.2.1 Gas flow simulation Gas flow pattern is required to simulate the droplet movement. By the velocity pattern, the drag force which affects the droplet velocity may be estimated. Complete solution of Navier-Stokes equation for such a system can provide the necessary data, for this pressure, although the approximate solution of this equation (analytical relations) for free jet may be also used. In the present study the Navier-stokes equation has been solved, using the SIMPLE method and a mesh system shown in figure 3.

Figure 3:

Pyramid mesh for the contactor.

The gas stream is assumed to be an incompressible flow (This assumption is valid for the fluid flows having Mach number lower than 0.3). Calculation of pressure contours is the main problem in this kind of fluid flow and the SIMPLE method seems to be a way by which the pressure profile under these conditions may be estimated [10]. The results are shown in figure (4).



Figure 4:

209

Velocity vectors and streamlines.

3.2.2 Droplet simulation An analogy between the droplet collisions in emulsions and the molecular collisions, described in the kinetic theory of gases, enables the application of the Boltzmann equation to the droplets, as first suggested by Pai [11]. In this work, the direct simulation Monte Carlo (DSMC) method, first proposed by Bird [12] for solving the Boltzmann equation in molecular gas dynamics, is used for modeling the droplet interactions in dense sprays. 3.3 Mathematical formulations The Boltzmann equation may be written as, +∞ 4π

(

)

r

∂ ∂ ∂ (1) ( nf ) + V . ( nf ) + F. ( nf ) = ∫ ∫ n2 f * f1* − ff1 V σ d ΩdV1 ∂t ∂r ∂v −∞ 0 In this equation, n is the number density of droplets, f is the velocity probability distribution function of droplet of class having the velocity V, f1 is the velocity distribution function of the droplet of class having the velocityV1, Vr is the velocity of a test droplet in the class of droplets having the velocity V1, f * ,

is the post-collision velocity probability distribution function and f1* is the postcollision velocity probability distribution function of the droplet of class having

the velocity V1, F, is the external force per unit mass and Ω is the angle in the spherical coordinates. The key ideas of the DSMC method are: (a) the uncoupling of droplet motions and collisions during a time step ∆tm i.e. the use of the operatorsplitting technique (b) the simulation of droplet collisions by disregarding droplet position coordinates within spatial coils: and (c) the simulation of fewer droplets than those present in the real flow, while normalizing the collision cross-section so that the collisions rate is not changed. Assumption (a) is valid when ∆tm is smaller than the time between collisions, and larger than that of collision duration, and assumption (b) is valid provided that the cell is so small that the spatial variation of flow variables in the cell is negligible. Assumption WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

210 Computational Methods in Multiphase Flow V (c) may not be necessary for dilute emulsion flow, due to relatively small droplet number densities; in the present work, however, due to the formation of large number of fragments, the implementation of this procedure is required. Under the assumptions listed above the DSMC method for the solution of Boltzmann kinetic equation, describing the flow of gas and droplets emulsions can be formulated as follows. The flow system is divided into equal-volume cells. Simulated droplets are distributed in the system, with their positions, sizes and velocities sampled from the initial distribution function. When a stationary kinetic equation is solved, the initial distribution function is chosen arbitrarily. The droplets population is normalized such that each Kf droplets of identical size in the real system is substituted by a single droplet in the simulation, having the same diameter as these droplets. The collision cross-section for each such simulated droplet is accordingly increased by the factor Kf so as to preserve the true collision rate. Provided that the droplet distribution function at time (n - 1) ∆tm is determined, the distribution function at time n ∆tm maybe calculated as follows. Droplets are allowed to move in the system, without colliding with each other, for a time interval ∆tm , with each droplet’s subsequent position, velocity and diameter are determined from the droplet equation of motion. Within the time interval ∆tm , a droplet may encounter a boundary either an open boundary, through which it leaves the system or a wall, onto which it sticks. Following the collisionless flow, droplets are allowed to collide with each other. The droplet population is discredited by location and size such that the number of droplets of type k in cell m, N m ,k having a volume Vm , , is given as, ∞

δ kl

0

δk

N m,k (t ) = ∫ dr ∫ dv ∫ s dδ f (r , v, δ , t ) Vm

(2)

Where, δ k and

δ kl are the lower and upper diameter limits, respectively, for droplets of type k . The total number of droplets in cell m is, s

s

N m = ∑ N m,k

(3)

k =1

In the stochastic model it has been assumed that collision durations are negligible, so that droplet motion can be described as a free motion, disrupted instantaneously by collisions. For high collision velocity, which is characteristic of impinging streams systems, shorter collision durations may be expected. Another assumption made was that droplets are spherical and the aerodynamic break up due to gas shear during free droplet motion is negligible. Borisov et al. [13] investigated experimentally the break up of single droplets moving in a gas flow. Their results show that droplet break up dose not occur when Weber number < 6. When only fluid drag acts upon droplets during ∆tm , droplet trajectory is calculated by integrating the following ordinary differential equation, d 2r = 0.75CD  ρ g ( ρ pδ )  U − v (U − v ) + g dt 2 WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

(4)


211

where, U, is the gas velocity at position r that is determined according to the procedure presented in gas flow simulation section, g, is the gravity acceleration and CD , is the gas drag coefficient for liquid droplet flowing in a gas and may be determined from the following relation proposed by Hestroni [14].

8(3Θ + 2) /  Re p ( Θ + 1)   for Re p < 2     CD =   0.78 1/ 5  14.9 / Re p + Θ ( 24 / Re ) + 4 / Re p  /(Θ + 1) for 2 < Re p < 500  Θ = µ p / µg Re p = ρ gδ U − v µ g

{

(

)}

(5)

In this model the droplet were considered to be spherical and collide with each other at different speeds. In simulation of processes, using the DSMC, two grid systems are normally considered. The first grid system is used to calculate the averages of flow properties. This grid system is chosen to be fine enough in order to increase the computational accuracy. The grid system is refined until the variations of the flow properties are not substantial (the variations of the flow properties should be less than 2%). The second grid system (Fig. 5) is selected to be extremely fine (the mesh size is equal to 0.2 times of the mean free path of the droplets) so that the collision of droplets could be controlled within each mesh with high accuracy. The grid system chosen in this study consisted of 88 divisions in X direction, 54 divisions in Y direction, 54 divisions in Z direction (totally 256608 meshes) and the total number of model droplets was 3,800,000. Each model droplet consists of K f = 70 real droplets. The size of the mesh is in order of the mean free path of droplets and the time step in the simulation process is chosen to be 0.2 times the collision time [12].

Figure 5:

Contactor mesh for DSMC simulation.

The droplets are distributed in the mesh system according to the normal distribution. The initial velocity of the droplets is chosen based upon the velocity of gas jets in the flow field. Then as time passes the new position of the droplets is designated. The collisions of the model droplets are occurred based on the pattern put forward by Bird [12]. Passage of time is continued until the statistical fluctuations of the flow properties attain minimum values. To implement the DSMC method, spatial cells have to be produced in the first step. The model droplets are placed in the cells with appropriate distribution. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

212 Computational Methods in Multiphase Flow V With regard to local gas velocity, a normal distributed velocity is supposed for droplets. Without any statistical sampling, droplets are moved (NPS times repetition), until the latter attain suitable conditions for statistical sampling. For large domain simulations, this step can be ignored. NPS value definition is dependent on cells conditions; however optimization can not be approached without sufficient experience. At the next time step, according to each droplet velocity and the extended time step, the droplets are moved and so their new coordinates will be attained. If a droplet passes cell boundary, it will be eliminated from the cell, and a new droplet would be entered to the domain. Furthermore, if in this replacement, the droplet collides with a physical surface, it will be reflected from the surface according to the collision of “surface-droplet model”. The index number of cell and sub-cell containing the droplet is defined; and then the probability droplet collision is investigated based on the model and if there is any collision, the velocity and location of the droplet will be modified. After NIS times repetition of movements, indexing, and collision investigations, statistical sampling should be done, the output of the program will be provided after NPT times repetition of sampling. This output is corrected up to NPT times. The expected number of collisions in a cell, during a time interval ∆tm is given as, N c = 0.5 n 2 σ T cr

(6)

The probability P of collision between two simulated droplets over the time interval ∆tm is equal to the ratio of the volume swept out by their total crosssection moving at the relative speed between the droplets to the volume of the cell, i.e., (7) P = FN σ T cr ∆t / vc Maximum efficiency is achieved if the fraction is such that the maximum probability becomes unity. The fraction is given by, Pmax = FN (σ T cr )max ∆t / vc

In the above equation

σT

(8)

is collision cross section and vc is the relative

droplet velocity. The average number of real droplet in the cell is nvc and the average number

of simulated droplets is N = nvc / FN , where, n , is the number density in liquid phase. full set of collisions could be calculated by selecting in turn, all N ( N − 1) / 2 pairs in the cell and by computing the collision with probability P. Pairs are selected from the cell at the time step, and the collision is computed using the σ T cr / (σ t cr ) probability relation given by Bird [12]. max

In the present work two different liquids (water and toluene) have been used. Therefore, the collisions were between similar or unlike droplets so the method has to be slightly modified. A pair of colliding droplets of type p and q is sampled from the possible pairs of these types with the above probability. This WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


213

term has been substituted for σ T cr / (σ t cr ) in all related equations. Simulation max results for these droplets are shown in figures 6-8.

Figure 6:

Droplets velocities in X direction.

Figure 8:

Figure 9:

Figure 7:

Droplets velocities in Y direction.

Droplets velocities in Z direction.

A typical residence time distribution data.


214 Computational Methods in Multiphase Flow V A typical experimental residence time distribution (RTD) curve obtained for the continuous two impinging streams contactor and those predicted applying the Markov chain model and DSMC technique are shown in figure 9.

4

Conclusion

The Direct Simulation Monte Carlo (DSMC) technique was applied to predict the motion of liquid droplets and residence time distribution in a two impinging streams contactor. This method was found to be more accurate and flexible in prediction of RTD data compare to the Markov chain discrete time formulation applied by Sohrabi et al [3,5,15] in some previous studies. The degree of agreement between the data estimated from the present model and those determined experimentally was within 85%. While in case of Markov formulation the degree of agreement was lower than 75%.

References [1] Sohrabi, M., Jamshidi, A.M., Studies on the behaviour and application of the continuous two impinging streams reactors in gas-liquid reactions, J. Chem. Tech. Biotechnol. 69, pp. 415 420, 1997. [2] Tamir, A., Impinging Streams Reactors, Fundamentals and Applications, Elsevier B. V., Amsterdam, The Netherlands, 1994. [3] Sohrabi, M. Kaghazchi, T. & Yazdani, F., Modelling and application of the continuous impinging streams reactors in liquid - liquid heterogeneous reactions, J. Chem. Tech. Biotechnol., 58, pp. 363 370, 1993. [4] Tamir, A., Herskovitz, D., Absorption of CO2 in a new two-impingingstreams absorber, Chem. Eng. Sci. 40, pp. 2149 2160, 1985. [5] Sohrabi, M., Ahmadi Marvast, M., Application of a continuous two impinging streams reactor in solid-liquid enzyme reactions, Ind. Eng. Chem. Res., 39, pp.1903 1910, 2000. [6] Tamir, A., Kirton, Y., Chem. Eng. Comm., 50, pp. 241 252, 1987. [7] Tamir, A., Kirton, Y., Drying Technol., 7, pp. 183 191, 1989. [8] Tamir, A., Luzzatto, K., Mixing of solids in impinging streams reactor, J. Powder Bulk Solids Technol., 9(15), pp. 15 17, 1989. [9] Tamir, A., Chem. Eng. Progress, 85, pp. 53 67, 1989. [10] Patankar, S.V., Numerical Heat Transfer and Fluid Flow, Elsevier, B.V., The Netherlands, pp. 123 144, 1993. [11] Pai, S.I., Sci. Tech. Mech. Appl., 19, pp. 605 621, 1974. [12] Bird, G.A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford University Press, UK, pp. 196 211, 1994. [13] Borisov, A.A., Gelfand, B., Natanzon, M.S. & Kossov, O., J. Eng. Phys. 40, pp. 44 49, 1981. [14] Hetsroni, G., Handbook of Multiphase Systems, Hemisphere Inc., Washington, 1982. [15] Sohrabi, M., Zareikar, B., Modeling of the residence time distribution and application of the continuous two impinging streams reactors in liquidliquid reactions, Chem.Eng.Technol. 28, pp. 61 66, 2005. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

Section 2 Interaction of gas, liquids and solids



217

Vortex study on a hydraulic model of Godar-e-Landar Dam and Hydropower Plant R. Roshan1, H. Sarkardeh2 & A. R. Zarrati3 1 Water Research Institute, Iran 2 Department of Engineering, Tarbiat Moallem University of Sabzevar, Iran 3 Department of Civil Engineering, Amirkabir University of Technology, Iran

Abstract In hydropower plants the kinetic energy of falling water is captured to generate electricity. In this process, the formation of vortices with an air core at the power intake entrances is expected at lower reservoir levels. The entrainment of air and swirl into the power tunnels leads to a reduction of power generation and vibration and damage to the turbine blades. To use the maximum potential of water power at lower reservoir levels when water is scarce, it is necessary to prevent vortex formation. Anti-vortex devices are usually considered as an efficient method for vortex prevention. The Godar-e-Landar Dam and Hydropower Plant is sited on the Karun River in the province of Khuzestan, Iran, with the capacity of 2000 MW. There are four horizontal power intakes where the capacity of each intake is equal to 375 m3/s. The dam is a rock fill type with 170 m height from the foundation. The dam has a gated spillway with an ogee chute and stilling basin. In the present work a physical model was used to study the formation and prevention of air core vortices at power intakes of the dam. Studies showed that vortices form when the reservoir water level decreases from a certain elevation and air enters the power tunnels. The performance of anti-vortex walls was therefore examined to eliminate vortices or reduce their strength and prevent entrainment of air. The anti-vortex walls were constructed on top of each intake to increase the friction stresses within the vortex path. To distinguish the vortex type, light colorful objects were released in the flow. Results of experiments showed Type 4 vortices (which may be a stronger air core Type 6 in prototype) reduced to a weak vortex Type 2 and 1 when an anti-vortex wall was installed. Moreover, the vortices became very unstable. Keywords: hydropower plant, power intake, vortex, anti-vortex wall, vortex prevention, trash rack. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090191


1

Introduction

Power generation is one of the main targets in dam construction. To do so, many hydropower plants have been constructed and put into operation alongside dams all over the world. The formation of vortex at power intakes is an undesirable phenomenon that may cause a number of problems, such as decreasing the efficiency of turbines and their vibration, increasing hydraulic losses at the entrance of power intakes, entraining debris that may block the trash racks, entraining air into the power tunnel and reducing the working life of turbines [1]. The stronger the vortex the greater will be its negative effects on power intake performance. Based on Alden Research Laboratory visual classification, vortices are divided into 6 types (Figure 1).

Coherent surface swirl

Surface dimple coherent swirl at surface

Dye core to intake coherent swirl throughout water column

Figure 1:

Vortex pulling floating trash but not air

Vortex pulling air bubbles to intake

Full air core to intake

Different types of vortices [1].

Vortex Type 1 is observed as a weak rotation of flow at water surface. In vortex Type 2 in addition to water surface rotation a drop is also observed in water surface. In vortex Type 3 the rotation of flow is extended down to the intake itself. In vortex Type 4, debris is dragged into the intake. In vortex Type 5 some air bubbles are entrained from water surface and are transported down to the intake. In the strongest Type 6 vortex, a stable air core is formed in the centre of the vortex and air is entrained into the power tunnel steadily [1]. To prevent formation of an air core vortex, a minimum operating depth, called critical submerged depth ‘hcr’ is recommended for the intakes. Submerged depth is defined as the distance between water surface and the axis of the intake (Figure 2). Many researchers have tried to find a relationship for hcr based on flow parameters such as the intake Froude number defined as:

Fr 

V where V is intake velocity, g is gravitational acceleration and D is gD

the power tunnel diameter (Berge 1966, Gordon 1970, Reddy and Pickford 1972, WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


219

Amphlett 1976, Chang 1977) [2–4, 11, 12]. However, since many factors such as the reservoir geometry affect the strength of vortices these relationships are not very accurate and are valid within their data limits and experimental setups. Therefore in design of power intakes, especially for large dams, physical model study is required.

S

Figure 2:

Formation of an air core vortex at the intake.

Critical submerged depth can be determined in a physical model. It should be noticed that since the operating level of the power intake should not be reduced below the critical submerged depth, the volume of water in this region can not be used for power generation. Therefore increasing the submerged depth of the intake for prevention of vortex formation may not always be an economical solution. On the other hand, construction of deeper intakes may be more expensive. A strategy for preventing vortex formation or in another word decreasing the critical submerged depth is to employ anti-vortex devices. Considering the factors which affect vortex strength, it can be concluded that vortex formation can be prevented or its strength can be reduced if there is a disturbance and an increase in flow resistance along the vortex path. Knauss [1] introduced various anti-vortex devices for intakes. These devices include: i) Vertical walls on top of the intake which induce friction on vortices and reduce their strength, ii) half cylinder walls in front of the intakes which cuts the vortex path and cause additional resistance to flow rotation iii) floating plates at reservoir water surface, which prevents vortex rotation due to friction. Since this device is floating on water surface it can adjust itself above the power intake at any water surface elevation, and iv) horizontal plates installed on top of the intakes This device though is fixed above the intake, cuts the vortex path and prevents its rotation. (Figure 3). In the present work formation of vortices was studied on power intakes of Godar-e-Lander Dam. To reduce the minimum operating water level of the reservoir, performance of anti-vortex walls were examined on the intakes. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


Intake

Figure 3:

Vertical Wall

Half Cylender wall

Floating Perforated Plate

Horizontal perforated Plate

Few types of anti-vortex devices used at horizontal intakes (courtesy of Water Research Institute, Tehran, Iran).

In additional experiments the effect of trash rack and spillway operation simultaneously with the power intakes on type of vortices was studied at different reservoir levels.

2

Experimental setup

Godar-e-Lander is rock fill dam 170 m high. Four power tunnels with 10 m diameter were constructed to supply the 8 power units (Figure 4). Design discharge of each intake was 375 m3/s. Elevation of intake axis was 340.98 m above the sea level (masl). Maximum and minimum reservoir operation level was designed at 372 masl and 363 masl respectively. As a part of design, a 1:66.67 scale model of the dam was constructed in Water Research Institute Laboratory in Tehran. The hydraulic model of Godar-eLander Dam included: dam body, the reservoir, spillways, power intakes and a part of their downstream tunnels (Figure 5). To include the effect of reservoir geometry on inlet flow pattern, an extension of 933 m of the reservoir upstream of the dam body and width of 667 m along the dam axis in prototype scale was constructed in the model. To be able to see the flow pattern at power intakes the reservoir walls and power intakes and tunnels were made from clear Perspex. Standard orifice plates installed downstream of each power tunnel in the model were used for discharge measurement. A point gauge with 0.1 mm WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


221

accurate was also used to measure elevation of water surface in the reservoir. To distinguish the vortex type in each test, light colorful objects were released in the flow. This model study was used to collect hydraulic design data for the anti-vortex wall. The hydraulic information obtained from the model study included: type of vortices, vortex stability in various alternatives of anti-vortex walls.

Figure 4:

Figure 5:

Plan view of power intakes and spillways.

1:66.67 scale model of the Godar-e-Lander Dam.



3

Model results

Model studies showed that with operation of the intakes, at reservoir level of 367.5 masl that is 4.5 m above the designed minimum operation water level, stable vortex Type 4 formed. These tests were repeated with installation of trash rack on intakes and different combinations of intakes operation with similar results (Table 1). It was also observed that operation of the spillways simultaneously with the power intakes reduces the vortices strength by inducing turbulence in the reservoir. Additional tests showed that, since the vortex location was closer to the intake head wall at elevations below 365.5 masl, the vortex type reduced one level (Table 1 and Figure 6) [5,6].

Figure 6:

Side and front view of dam, power intake location and anti-vortex wall.

Owing to viscous effects in small scale models, strengths of vortices may be underestimated and therefore stronger air core vortices may form in the prototype. Considering negative consequences of vortex formation and air entrainment at power intakes, design of an anti-vortex wall was proposed to enable the dam operators to use the power units until the designed minimum water level of 363 masl. In hydropower tunnels air should not be drawn into the intakes. This corresponds to vortex Type 5 and 6. To evaluate the performance of the antivortex wall and considering the scale effects it was decided not to have vortices stronger than Type 2 in the model. This is especially acceptable since the cost of constructing anti-vortex walls is not high. A wall was installed in the model at elevation 365.5 masl above each intake on top of the intake head wall and with the same slope (Figure 6). The top elevation of this anti-vortex wall was selected 368.5 masl which is the height above it no vortex was observed (see Table 1). Different combinations for operation of intakes were tested at 3 reservoir surface elevations of 363 masl, 367.5 masl and 372 masl and 2 discharges for spillway operation and formation of vortices was monitored. The results of these tests showed that the strongest vortex with the presence of the anti-vortex wall was Type 2 at elevation 367.5 masl. The vortices with the presence of wall were also very unstable (Table 1). These results showed the successful performance of the anti-vortex wall. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


Table 1:

223

Summary of results on vortex formation and the performance of an anti-vortex wall at different conditions.

Reservoir Level (masl)

Discharge of Spillway (m3/s)

Trash rack

Number of Active Intakes

AntiVortex Wall

The more Strength Type of Vortices

Stability

372

Closed

No Trash rack

1,2

No wall

No Vortex

No Vortex

367.5

Closed

No Trash rack

1,2

No wall

4

Stable

363

Closed

No Trash rack

1,2

No wall

3

Stable

367.5

2500

No Trash rack

1

No wall

2

Stable

367.5

2500

No Trash rack

1,2

No wall

2

Stable

367.5

Closed

No Trash rack

1,2,3,4

No wall

3

Stable

367.5

2500

No Trash rack

1,2,3,4

No wall

2

Unstable

367.5

5000

No Trash rack

1,2,3,4

No wall

No Vortex

No Vortex

372

Closed

Trash rack

1,2

No wall

No Vortex

No Vortex

367.5

Closed

Trash rack

1,2

No wall

4

Stable

363

Closed

Trash rack

1,2

No wall

3

Stable

367.5

2500

Trash rack

1

No wall

2

Stable

367.5

2500

Trash rack

1,2

No wall

2

Stable

367.5

Closed

Trash rack

1,2,3,4

No wall

3

Stable

367.5

2500

Trash rack

1,2,3,4

No wall

2

Unstable

367.5

5000

Trash rack

1,2,3,4

No wall

No Vortex

No Vortex

372

Closed

Trash rack

1,2

yes

No Vortex

No Vortex



4

Continued.

Reservoir Level (masl)

Discharge of Spillway (m3/s)

Trash rack

Number of Active Intakes

AntiVortex Wall

The more Strength Type of Vortices

Stability

367.5

Closed

Trash rack

1,2

yes

2

Unstable

363

Closed

Trash rack

1,2

yes

1

Unstable

367.5

2500

Trash rack

1

yes

No Vortex

No Vortex

367.5

2500

Trash rack

1,2

yes

No Vortex

No Vortex

367.5

Closed

Trash rack

1,2,3,4

yes

2

Unstable

367.5

2500

Trash rack

1,2,3,4

yes

1

Unstable

367.5

5000

Trash rack

1,2,3,4

yes

No Vortex

No Vortex

Summary and conclusions

Formation of vortices at power intakes is an undesirable phenomenon, which cause number of problems. Stronger vortices have more negative effects on performance of a hydropower plant. Godar-e-Lander Dam and Hydropower Plant is placed in west of Iran. It is a rock fill dam with 170 m height from the foundation and a hydropower plant with the capacity of 2000 MW. In physical model study it was found that vortices of Type 4 were formed at the intakes at depths more than the minimum designed reservoir water level. Owing to scale effects stronger air core vortices may form in the prototype. Experiments also showed that operation of spillways has a great effect on reducing or eliminating vortices. It was also observed that trash rack effect on reducing type of vortices was negligible. To reduce the strength of vortices anti vortex walls were installed on top of each intake head wall and with the same slope. In fact the intake head wall was extended an extra 3 m. Results of experiments showed that type of vortices reduced from Type 4 to Types 2 and 1 when anti-vortex walls were installed. Moreover vortices became very unstable when anti-vortex walls were present. The anti-vortex wall was then recommended as a cheap method to prevent vortex formation in the prototype.

Acknowledgement The authors would like to thank Water Research Institute, Tehran, Iran for their permission in using the model data. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


225

References [1] Knauss, J, “Swirling Flow Problems at intakes”, Balkema, 1987. [2] Gordon, J.L, “Vortices at Intakes”, Water Power, 1970. [3] Reddy, Y.R. & Pickford, J.A., “Vortices at Intake in Conventional Sumps”, Water Power, No.3, 1972. [4] Amphlet, M.B, ”Air Entraining Vortices at Horizontal Intake”, HRC Wallingford Rep., No .OD/7,1976 [5] Sarkardeh, H, Safavi, K & Karaminejad, R “Experimental study of vertical anti-vortex wall at power intakes of Karun III dam”, 2nd IJREW on Hydraulic Structures, Italy, 2008. [6] Sarkardeh, H, Zarrati, A.R & Roshan, R “Effect of intake head wall and trash rack on type and strength of vortices”, Journal of Hydraulic Research, Submitted for publication, 2008. [7] Lugt, H.J, “Vortex Flow in Nature and Technology”, John Wiley & Sons, 1983. [8] Hecker, G.E, “Model-Prototype Comparison of Free Surface Vortices”, Journal of Hydraulic Engineering, Vol.107, No.10, pp 1243-1259, 1981. [9] Anwar, H.O., Weller, J.A & Amphlett M.B, “Similarity of Free Vortex at Horizontal Intake”, Journal of Hydraulic Research,No.2,pp 95-105, 1978 [10] Jain, A.K, Raju, K.G.R & Grade, R.J, “Vortex Formation at Vertical Pipe Intake”, Journal of Hydraulic Engineering, Vol.100, No.10, pp 1427-1445, 1987. [11] Berge, J.P. (1966). A Study of Vortex Formation and Other Abnormal Flow in a Tank with and without a Free Surface. La Houille Blanche, No.1. [12] Chang, E. (1977). Review of Literature on Drain Vortices in Cylindrical Tanks. BHRA Report, TN.1342, March.




227

Minimum fluidization velocity, bubble behaviour and pressure drop in fluidized beds with a range of particle sizes B. M. Halvorsen1,2 & B. Arvoh1 1

Institute of Process, Energy and Environmental Technology, Telemark University College, Norway 2 Telemark Technological R&D Centre (Tel-Tek), Norway

Abstract Fluidized beds are used in the production of pure silicon for solar cells. The particles are fully consumed during the reaction and the particles in the reactor have a large range of diameters. When the range of particle sizes is wide, the particles have a tendency to segregate. A series of experiments are performed to study the particle segregation and the influence of particle segregation on the bubble formation and flow behaviour. Experiments are performed in a two dimensional bed. The minimum fluidization velocity and the pressure drop have been measured. Spherical glass particles with different ranges of particle sizes are used in the experiments. Superficial gas velocities well above the minimum fluidization velocities are used in the study of segregation and bubble formation. Corresponding simulations are performed by using the commercial CFD code Fluent 6.3. The computational results are compared to the experimental data and the discrepancies are discussed. Keywords: fluidized bed, minimum fluidization velocity, particle size, pressure drop, particle segregation, fluent.

1

Introduction

Fluidized beds are widely used in industrial operations due to their large contact area between phases, which enhances chemical reactions, heat transfer and mass transfer. The efficiency of a fluidized bed reactor is highly dependent on the flow conditions which also control the mixing of the bed. Particle sizes, range of WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090201

228 Computational Methods in Multiphase Flow V particle sizes, particle size distribution and superficial gas velocity are influencing on the flow behaviour and mixing. Fluidized bed reactors are used in production of silicon for solar cells. The gas-particle reaction is a continuous process where the particles are fully consumed during the reaction and the particles in the reactor may therefore have particle diameters that range from 0 to about 500 µm. In a reactor like this, the temperature becomes very high and it is therefore extremely important to keep the particles fluidized and well mixed. The aim of this work is to study how the range of particle sizes and the particle size distribution influence on the flow behaviour in fluidized bed. When the powders have a large range of particle sizes, the chance of segregation is significant. Segregation and low degree of mixing may give very high temperatures in parts of the bed, and the consequence may be melting of the reactor. Knowledge about mixing and segregation are therefore essential.

2

Fluidized bed dynamics

2.1 Characterization of particles Computational studies have been performed on a two dimensional fluidized bed. Spherical particles with a mean diameter of 154 µm, 488 µm and 960 µm and a density of 2485 kg/m3 are used. The behaviour of particles in fluidized beds depends on a combination of the particle size and density. Geldart fluidization diagram [1], shown in Figure 1, is used to identify characteristics associated with fluidization of powders. The dots represent the powders used in this study. The powder with mean diameter of 154 µm is close to group A particles, whereas the powder with mean diameter of 960 µm is close to group D particles. The fluidization properties for these groups of particles differ significantly from each other.

Figure 1:

Geldart classification of particles [1].



229

Particles characterized in group A are easily fluidized and the bed expands considerably before bubbles appear. This is due to inter-particle forces that are present in group A powders [2]. For group B particles the inter-particle forces are negligible and bubbles are formed as the gas velocity reaches the minimum fluidization velocity. The bed expansion is small compared to group A particles. Geldart group D describes large and/or dense particle powders. These powders need a large amount of gas to get fluidized, and bubbles may occur if the gas velocity is kept close to the minimum fluidization velocity. Group D powders give low degree of solid mixing and gas back-mixing compared to group A and B powders [3]. 2.2 Model Computational studies have been performed on a two dimensional fluidized bed. The computational work is performed by using the commercial CFD code Fluent 6.3. The model is based on an Eulerian description of the gas and the particle phases. The combinations of models that are used in this work are presented in Table 1: Table 1:

Models used in Fluent.

Property Drag Granular viscosity Frictional viscosity Frictional pressure Solid pressure

Model Syamlal & O’Brien Syamlal & O’Brien Schaeffer Based-ktgf Ma-ahmadi

In a bubbling fluidized bed the concentration of particles varies from very low to very high. In dilute regions, the kinetic of the particles will dominate the solids viscosity, and the solid pressure will be close to zero. In regions with higher concentration of particles, the collisions between particles will dominate the solids viscosity, and the solid pressure will increase. At very high concentration of particles, the frictional stresses dominate the solid viscosity. The drag describes the momentum exchange between phases and is expressed by the drag coefficient in the momentum equation. The Syamlal & O’Brien drag model is expressed by [4]: Φ sg = CD

G G 3ε sε g ρ g U g − U s

(1)

2

4vr d s

where εg and εs are the gas and solid fractions, ρg is the gas density, Ug and Us are the gas and solid velocities and ds is the particle diameter. The terminal velocity correlation for the solid phase, vr, is a function of void fraction and Reynolds number. An empirical equation for vr is developed by Garside and Al Dibuouni [5]. The drag factor is proposed by Dalla Valle [6] and is expressed by:


230 Computational Methods in Multiphase Flow V  4.8 CD =  0.63 +  Re s / vr 

   

2

(2)

The minimum fluidization velocity can be developed from the buoyantequals-drag balance: (3) (1 − ε )(ρ − ρ )g = Φ sg (u − u ) g

s

εg

g

g

s

where the drag coefficient is developed by Syamlal & O’Brien. The equation for minimum fluidization velocity is [7]: ε mf 2 (1 − ε mf )(ρ s − ρ g )g (4) U = mf

Φ sg

Multiple particle phases are used in the simulations to account for the particle size distribution. The model, Syamlal-O’Brien-symmetric, is used to express the particle-particle momentum exchange [8]. The radial distribution function included in the Syamlal-O’Brien-symmetric equation is expressed by Ma and Ahmadi [9].

3

Results

Flow behaviour in a two dimensional bed with a uniform air distribution is studied experimentally and computationally. The purpose of this work is to study minimum fluidization velocity, particle segregation, bubble activity and pressure drop on a simple well-defined model. The experimental results are used to validate the CFD model that is used in the simulations. 3.1 Experimental set-up A lab-scale fluidized bed with a uniform air distribution is constructed. The width, depth and height of the bed are 0.20 m, 0.025 m and 0.80 m respectively. The pressure is measured at five positions in the bed. A digital camera is used to record the bubble behaviour and particle segregation. The experimental set-up is shown in Figure 2. Spherical glass particles with density 2485 kg/m3 and different particle diameters are used in the experiments. The mean particle size and the range of particle sizes are presented in Table 2. Table 2: Experiment no. 1 2 3 4 5 6

Particle diameters.

Particle range [µm] 100-200 400-600 750-1000 (100-200)+( 400-600) (100-200)+( 750-1000) ( 400-600) +( 750-1000)


Mean diameter [µm] 153 488 960 320 556 724


231

For all the powders and mixture of powders, the minimum fluidization velocities are observed. Experimental studies of bubble behaviour and particle segregation are performed with velocities well above the minimum fluidization velocity. The pressure is measured as a function of time. Photos are taken initially and after a few minutes. The experiments have been run for 2 to 10 minutes.

Figure 2:

Experimental set-up.

3.2 Computational set-up The simulations are performed with particles with diameters equal to the mean diameters of the glass powders used in the experiments. The data are given in Table 3. Two particle sizes are used to simulate the mixtures of two powders with different mean particle size. Table 3: Simulation no. 1 2 3 4 5

Particle diameters.

1 particle phase Mean diameter [µm] 153 488 960

2 particle phase Mean diameter [µm]

153(50%)+488(50%) 153(50%)+960(50%)

The simulations are run with the same velocities and initial bed heights as in the experiments. Two-dimensional Cartesian co-ordinate system is used to WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

232 Computational Methods in Multiphase Flow V describe the geometry. The grid is uniform in both horizontal and vertical direction, and the grid size is 0.5x0.5 mm. The models used are given in Table 1. The simulations have been run for 10 to 30 s. 3.3 Results The measured and the theoretical minimum fluidization velocities are presented in Figure 3. The theoretical fluidization velocities are calculated from eqn (7) using the Syamlal and O’Brien drag model. The minimum fluidization velocity is influenced of the void fraction in the bed. The theoretical minimum fluidization velocities are calculated for a range of void fractions, from 0.37 to 0.44. The deviation between the theoretical and the experimental fluidization velocities increases with increasing void fraction. The experimental minimum fluidization velocity is about equal to the theoretical for the small particles (100-200 µm). For the larger particles the experimental minimum fluidization velocities are significantly lower than the theoretical. In the mixtures, the experimental minimum fluidization velocities are about equal to the minimum fluidization velocity for the smallest particles in the mixture. The experiments with small particles, mixture of 50% small and 50% medium particles and mixture of 50% small and 50% large particles give minimum fluidization velocities of 0.033, 0.033 and 0.04 m/s respectively. This indicates that the theoretical minimum fluidization velocity cannot be based only on the mean particle sizes. In a bed with a large range of particle sizes, the smaller particles will fill the void between the larger particles, and the bulk density can be rather high compared to a bed with one sized particles. This will highly influence on the fluidization condition, bubble formation and bubble behaviour in the bed.

Minimum fluidization velocity [m/s]

0.6

0.5 Ex. Small+mix small 0.4

Ex. Medium + mix medium Ex. Large Calc., void=0.37

0.3

Calc., void=0.38 Calc., void=0.40 0.2

Calc., void=0.42 Calc., void=0.44

0.1

0 0

200

400

600

800

1000

1200

Mean particle diameter [µm]

Figure 3:

Experimental and calculated minimum fluidization velocities.

Experiments are performed with superficial gas velocities well above the minimum fluidization velocities for the different powders and mixture of WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


233

powders. Figure 4 and 5 show the results from experiment and simulation with the smallest particles. The mean particle diameter is 154 µm and the superficial velocity is 0.1 m/s. The superficial velocity is well above the minimum fluidization velocity. The simulations are performed with one particle phase. The computational results agree well with the experimental results, both with respect to bubble behaviour and pressure drop. The pressure drop over the bed is about 50 mbar.

t=2 s Figure 4:

t=10 s

Initially

t=2 min

Bubble behaviour. Small particles. Superficial velocity 0.1 m/s.

80 Pressure gauge [mbar]

70 60 50 Simulation

40

Experiment

30 20 10 0 0

0.1

0.2

0.3

0.4

0.5

0.6

Height above gas distributor [m]

Figure 5:

Pressure as a function of height above the gas distributor, small particles, superficial gas velocity 0.1 m/s.

The mixture of 50% small and 50% medium particles gives a mean diameter of 320 µm. The theoretical minimum fluidization velocity is 0.10-0.17, depending on the void fraction. The measured minimum fluidization velocity is 0.033. Superficial velocity 0.1 m/s is used in experiment and simulation. The simulation is performed with two particle phases. The computational and experimental flow behaviour is presented in Figure 6. The experimental results WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

234 Computational Methods in Multiphase Flow V show very clearly the segregation of particles. The bubble formation occurs above the layer of large particles in the lower part of the bed. The tendency of particle segregation is also observed in the simulations, where high concentration of large particles is located close to the gas distributor. Figure 7 shows the comparison of the experimental and computational pressure drop for the mixture. The experiment gives higher pressure than the simulations in the upper part of the bed. This may be due to a higher bed expansion in the experiment. In the experiments, the pressure is not measured at positions below 0.185 m above the distributor.

t=3 s Figure 6:

t=25 s

Initially

t=3 min

Bubble behaviour. Mixture of small and medium particles. Superficial velocity 0.1 m/s.

Pressure gauge [mbar]

60 50 40 Simulation

30

Experiment

20 10 0 0

0.1

0.2

0.3

0.4

0.5

0.6

Height above the gas distributor [m]

Figure 7:

Pressure as a function of height, 50% small and 50% medium particles.

Figure 8 shows results from experiments and simulations performed with a mixture of 50% small and 50% large particles. The mean particle diameter of the mixture is 556 µm and the superficial velocity used is 0.1 m/s. The theoretical WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


235

minimum fluidization velocities range from 0.21 to 0.34 m/s for void fraction 0.37 to 0.44 respectively. The minimum fluidization velocity observed in the experiments is 0.04 m/s. The simulations are performed with two particle phases with diameter 153 µm and 960 µm. Bubbles are formed in the simulations although the superficial gas velocity is well below the theoretical minimum fluidization velocity for the mixture. Segregation of particles is very clear both in the simulation and experiment. The pressure as a function of bed height is presented in Figure 9. The computational pressure agrees well with the experimental pressure measurement.

t=5 s Figure 8:

t=18 s

Initially

t=2 min

Bubble behaviour. Small and large particles. Superficial velocity 0.1 m/s.

Pressure gauge [mbar]

60 50 40 Simulation

30

Experiment

20 10 0 0

Figure 9:

0.1

0.2 0.3 0.4 Height above gas distributor [m]

0.5

0.6

Pressure as a function of height, 50% small and 50% large particles.

Experiments and simulations have been performed with medium and large particles. The superficial velocity is 0.2 m/s for medium particles, 0.33 m/s for WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

236 Computational Methods in Multiphase Flow V large particles. The velocities are well above the experimental minimum fluidization velocities, but below the theoretical minimum fluidization velocities. The experiments give bubbles, whereas the simulations give flow through a fixed bed. The simulations are performed with one particle phase which means that all the particles in the simulation have the same diameter. This gives a rather high void fraction and therefore a higher gas flow is required to get the particles fluidized. For these cases the fluidization does not occur at current velocities in the simulations. Figure 10 and 11 show the flow behaviour in the bed with medium and large particles respectively.

t=10 s Figure 10:

t=2 min

Bubble behaviour. Medium particles, superficial gas velocity 0.2 m/s.

t=10 s Figure 11:

Initially

Initially

t=2 min

Flow behaviour. Large particles, superficial gas velocity 0.33 m/s.



4

237

Conclusion

Flow behaviour in a two dimensional bed with a uniform air distribution is studied experimentally and computationally. The experiments are performed with spherical glass particles with different particle sizes and range of particle sizes. The minimum fluidization velocity is measured for all the different powders. The pressure is measured at different heights of the bed. The bubble behaviour is observed by using a digital camera. Corresponding simulations are performed by using the commercial CFD code Fluent 6.3. The model is based on a multi-fluid Eulerian description of the phases. The measured minimum fluidization velocities are compared to theoretical minimum fluidization velocities. The theoretical minimum fluidization velocities are developed from the buoyant-equals-drag balance by using Syamlal & O’Brien drag coefficient. The theoretical minimum fluidization velocity is significantly higher than the observed experimental velocity except for the smallest particles. This may be due to the particle size distribution and range of particle sizes in the experimental powder which influence on the void fraction and fluidization properties. The theoretical minimum fluidization velocity is based on a mean particle diameter. Powders with mono-sized particles have a significantly higher void fraction than powders with a particle size distribution. 50%-50% mixtures of two powders has a about the same minimum fluidization velocity as the smallest particles in the mixture. Comparison of computational and experimental bubble behaviour is performed, and the results show good agreement according to bubble formation, pressure drop and particle segregation for the small and mixtures of small and larger particles. Simulations with medium and large particles give no fluidization, whereas fluidization and bubbles are obtained in the corresponding experiments. This indicates that more than one particle phase is needed in the simulation to account for the particle size distribution in the powder.

References [1] Geldart, D., Types of Gas Fluidization, Powder Technology, 7, pp. 285-295, 1973. [2] Geldart, D., Gas Fluidization Technology, John Wiley & Sons Ltd., 1986. [3] Kunii, D., Levenspiel, O., Fluidization Engineering, Second Edition, Butterworth-Heinemann, Newton, US, 1991. [4] Syamlal, M., O’Brien, T.J., A Generalized Drag Correlation for Multiparticle Systems, Morgantown Energy Technology Center, 1987. [5] Garside, J. & Al Dibuouni, M.R., Velocity-Voidage Relationships for Fluidization and Sedimentation, I&EC Process Des. Dev., 16, pp. 206-214, 1977 [6] Dalla Valle, J.M., Micromeritics, Pitman, London, 1948. [7] Gidaspow, D., Multiphase Flow and Fluidization, Academic Press, Boston, 1994.


238 Computational Methods in Multiphase Flow V [8] Syamlal, M., The Particle-Particle Drag Term in a Multiparticle Model of Fluidization., National Technical Information Service, Springfield, VA, 1987. [9] Ma, D., Ahmadi, G., Thermodynamical Formulation for Dispersed Multiphase Turbulent Flows, Int. J. Multiphase Flow, 16, pp. 323-351, 1990.


Section 3 Turbulent flow



241

Mathematical modelling on particle diffusion in fluidised beds and dense turbulent two-phase flows R. Groll Center of Applied Space Technology and Microgravity, Germany

Abstract Volume-fraction weighted and Reynolds averaged momentum transport equations are solved in an Euler/Euler approach to numerically simulate the turbulent, dispersed two-phase flow in a two-dimensional channel and a three-dimensional conic diffuser flow. Particular attention is given to the modelling of turbulent diffusion and particle wall interaction, assuming local equilibrium, but introducing individual terms for particle/fluid drag interaction, particle collisions and trajectory crossings. These influences have been quantified in terms of partial viscosities, a restitution power and a turbulence structure parameter. Boussinesq approximations have been used for each phase and the formulation of their interaction was provided in the framework of the eddy-viscosity modelling concept. Keywords: two-phase flow, particle diffusion, particle collision, fluidised bed.

1 Introduction The momentum transport equation includes a turbulent diffusion term, which characterises motions that are not resolved by the convective term. This turbulent diffusion depends on the turbulent kinetic energy and the turbulence characterising Eulerian time scale. Based on ”Csanadys Approximation” [3], the time scale of the dispersed phase is coupled with the turbulent time scale of the continuous phase. This time scale quantifies the diffusion intensity [5] and is influenced by the drag interaction of the particles with the viscous gas phase and inter-particle collisions [4, 9]. The fluctuation of a filtered variable φ is written: {φ}k = φ− < φ >k . WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090211

(1)

242 Computational Methods in Multiphase Flow V The index k stands for C (continuous phase) or D (dispersed phase). The following negation notation is used here, i.e. C¯ = D, D¯ = C. The drag force relaxation time scale τpk depends on the particle Reynolds number, based on the velocity difference between the continuous and the dispersed phase [12], see e.g. Crowe et al. (1998) [2]. The equation terms are described with the help of an averaging operator < . >k [11]:

< uki >l =

α l uki ; α¯ l

uki =< uki >l +{uki }l .

(2)

Summarising the different diffusion generating components to a combined particle-turbulence shear stress model the diffusion of dispersed particles is predicted very well, especially inside turbulent shear layers.

2 Transport equations Modelling the diffusion character of particle flows transported in a viscous carrier phase the diffusion is not dominated by the eddy dissipation as it is in turbulent shear flows in a continuous gas phase. This dispersed phase diffusion is characterised by unsteady drag influences and crossing trajectory characteristics defined by the correlation of velocities of the continuous phase and the dispersed phase itself. The Reynolds averaged, volume-fraction weighted momentum transport equations for the continuous and the dispersed phase read:

∂ k k ∂ (ρ α¯ < uki >k ) + (ρ k α¯ k < uki >k < ukj >k ) ∂t ∂xj = −α¯

∂σijk ∂pC α¯ k ¯ ¯ + α¯ + ρ k k (< uki >k − < uki >k ) ∂xi ∂xj τp

+ ρ k α¯ k gi −

∂ (ρ k α¯ k < {uki }k {ukj }k >k ) ∂xj

(3)

Collision and crossing trajectory terms were approximated by Grad [4] and Csanady [3]. The pressure gradient of the continuous phase is equivalent to the lift force. And the sum would be zero in a hydrostatic case. σijk defines the viscous stresses inside the phase k. Because of the solid character of the dispersed phase σijD is zero. The underlying turbulence model for both phases is based on the Boussinesq analogy, employing eddy viscosity as the model quantity, whose formulation was provided in the framework of the standard k − modelling concept. k C defines the turbulent kinetic energy of the continuous phase. The corresponding variable of the dispersed phase k D describes the particle velocity variance at a given position. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

243


∂ ∂ k k k (ρ α¯ k ) + (ρ k α¯ k k k < ukj >k ) ∂t ∂xj k ∂ < uki >k ∂ ∂k k k k k νt α¯ μ + ρ k − ρ k α¯ k < {uki }k {ukj }k >k = ∂xj ∂xj σk ∂xj with σkC = 1, 0 and σkD = 2.5

(4)

The transport equations for the turbulent kinetic energy k k and its dissipation rate tk differ from those for a single-phase flow by several additional production terms and the modified dissipation. This total turbulent kinetic energy loss αk in the present model arises from relative drag and particle collision processes. ∂ k k k ∂ k k k ρ α¯ t + ρ α¯ t < ukj >k ∂t ∂xj k ∂ k ∂ < uki >k ∂ k t k k k νt = α¯ μ + ρ k − C1 tk ρ k α¯ k < {uki }k {ukj }k >k ∂xj σ ∂xj k ∂xj k k k k α¯ − C2 ρ k α¯ k t k α + C3 tk ρ k k q k − 2k k k k τp with σC = 1.3, C1 = 1.44, C2 = 1.92, C3 = 1.2

(5)

Because of the different phase velocities inside the drag relation term a general formulation is needed for the complimentary index of k. The velocity covariance ¯ ¯ q k =< {uki }k {uki }k >k of He and Simonin (1993) [5] represents the trace of the velocity vector correlation tensor of both phases: ∂ k k k ∂ k k k ρ α¯ q + ρ α¯ q < ukj >k ∂t ∂xj ∂ ∂q k α¯ k ¯ ρ k α¯ k ναk + ρ k k 2Z k k k + 2k k − 1 + Z k q k = ∂xj ∂xj τp ¯

− ρ k α¯ k < {uki }k {uj }k >k with Z k =

ρ k α¯ k ρ k¯ α¯ k¯

¯

∂ < uk > k ∂ < uk > k ¯ ¯ ¯ ¯ − ρ k α¯ k < {uki }k {ukj }k >k ∂xj ∂xj

,

(6)

which completes the present three-equation model for each of both phases. Velocity correlations, representing the turbulent momentum diffusion, are modelled by the following Boussinesq approximations: − < {uki }k {ukj }k >k 2 = − k k δij + νtk 3

∂ < ukj >k ∂ < uki >k 2 ∂ < ukl >k + − δij ∂xj ∂xi 3 ∂xl


(7)

244 Computational Methods in Multiphase Flow V ¯ ¯

− < {uki }k {ukj }k >k 1 = − q k δij + ναk 3

∂ < ukj >k ∂ < uki >k 2 ∂ < ukl >k + − δij . ∂xj ∂xi 3 ∂xl

(8)

Modelling the viscous continuous phase the eddy viscosity is modelled based on eddy dissipation relating components. Because of the phase interaction influence on the turbulent diffusion of both phases an additional transport equation for the velocity covariance has to be defined. The diffusion coefficients of momentum and turbulent transport equations are given by the characteristic diffusion time scale ταk : 2 νtk = ταk · k k 3

and

2 ναk = ταk · q k . 3

(9)

To close the present formulation of particle and carrier gas phase motion this time scale has to be modelled. Based on this new kind of particle diffusion modelling also equilibrium-turbulence boundary conditions are modified by the influences of inter-particle collisions and phase-interactions of dispersed and continuous phase.

3 Diffusion modelling Based on this model of momentum diffusion, which depends on the velocity gradients of the diffusing phase, the characteristic diffusion time scales have to be defined by the velocity correlation and its associated loss rate αk . Local equilibrium describes the equivalence of production and loss of turbulent kinetic energy. Assuming ∂/∂x1 ≈ 0 yields an expression (i = 1, j = 2) for the nondiagonal element of the Reynolds stress tensor. αk = − < {uk1 }k {uk2 }k >k

⇒

−
k > = αk k

2

∂ < uk1 >k ∂x2

(10)

∂ < uk1 >k ∂x2

With the definition of the turbulence structure parameter Cαk of the phase k Cαk = −

⇒

νtk

uk1

∂< ∂x2

>k

< {uk1 }k {uk2 }k >k 2 k 3k

= − < {uk1 }k {uk2 }k >k =

(11)

2 k k 3 Cα k αk

2 ∂ < uk1 >k ∂x2

the turbulent viscosity is calculated using the turbulent kinetic energy k k , its loss rate and the turbulence structure parameter. Based on the diffusion definition (eq.9) WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


245

Table 1: Restitution power and turbulence loss components. γ

model

β

Jones et al.

d

Csanady

p

Wallis

c

Jenkins et al.

πγC

γC

2 β C 3 β

tC

0.

0.

2 2k C −q C 3 τpC

2k C −q C τpC

0.

0.

2 kD 3 ταC

πγD

γD

0.

0.

1 + CβC (ξrC )2 2 2k D −q D 3 τpD σcD 2 k k τpD 3

0. 2k D −q D τpD

kk τcD

1 − (ecD )2

the turbulent time scale is also defined by these values: 2 νtk = − ταk k k 3

⇒

ταk =

νtk 2 k 3k

=

2 k 2 k k Cα 3 αk

(12)

With the definition of the restitution power παk , the turbulent viscosity is determined by the turbulent kinetic energy and the restitution power. παk =

kk

2 = 3 ταk

αk 2 Cαk

⇒ − < {uk1 }k {uk2 }k >k =

2 k 3k παk

2 ∂ < uk1 >k ∂x2

(13)

The power παk describes the restitution of turbulent shear forces based on the dissipation and structure of turbulence and reduces the turbulent diffusion. This restitution power consists of the partial powers defined by four different effects (see Table 1), which are described in the following subsections with the indices β, p, c and d. The total turbulence loss rate αk is given by the sum of individual loss rates (γk , see Table 1): αk =

γk = βk + dk + pk + ck

(14)

γ

The different diffusion rates and turbulence loss rates are induced by the viscous turbulent shear stress (πβk , βk : Jones et al. [7]), crossing trajectory effects (πdk , dk : Csanady [3]), drag forces (πpk , pk : Schiller and Naumann [12]) and collision terms (πck , ck : Jenkins and Richman [9]). Adding together these influences, the new restitution power term of the turbulent diffusion is modelled. παk =

πγk = πβk + πdk + πpk + πck

γ WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

(15)


1 Eul./Eul. [2] UD D Eul./Eul. [1] U D Eul./Lag. U D Exp. U

y/h

0.6

0.8 0.6 y/h

0.8

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6 D

D

Eul./Eul. [2] u mag D Eul./Eul. [1] u mag Eul./Lag. uDmag Exp. uDmag

0.8

0

1

0

0.05

C

U /U

0.1 u

0

0.15

0.2

0.25

D

C mag/U 0

Figure 1: Normalised stream-wise particle mean velocity U D and normalised standard deviation uD mag of the particle velocity magnitude in the fullydeveloped, particle-laden channel flow. Simulating the general restitution power παk and energy loss αk the resulting turbulence structure parameter Cαk is computed in the following way: παk

2 kk αk = = 2 3 ταk Cαk

⇒

Cαk

=

αk παk

(16)

The turbulent time scale ταk depends on the sum of all diffusion rates: ταk =

2 k 3k

πβk + πdk + πpk + πck

(17)

This way of calculation yields a deterministic method to compute the turbulent time scale, which is needed for the calculation of the general turbulent viscosity of both phases (eq.12).

4 Computational results and discussion This model was validated using experimental data of 70 μm copper particles in a fully-developed channel flow ([10]; experimental results). The results using the present model (Eul./Eul. [2]) were also compared with the results obtained by an Euler/Lagrange scheme (Eul./Eul.) [6, 8] and a well-known Euler/Eulerian diffusion approach (Eul./Eul. [1]) [5, 11]. Gravity acts in the positive x-axis direction. The channel flow Reynolds number, based on channel height (2 h = 40 mm) and single phase channel centreline velocity (U0C = 10.5 m/s) is Re2h = 27600. The flow is regarded as fully developed after 125 channel heights and at this position it is assumed that the particle velocity and particle turbulence has reached an asymptotic state. The copper particles have a density of ρ D = 8800 kg/m3 and a diameter of Dp = 70 μm. The inlet mass loading of particles is Z0D = 10% and the parameter of WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


1

D C

0.8

D C

0.8

y/h

0.6

y/h

0.6

247

0.4

0.4

0.2

0.2

0

0

1000 2000 3000 4000 5000 6000 Restitution Power [m2/s3]

0

0

0.1

0.2

0.3

0.4

0.5

Turbulence Structure Parameter

Figure 2: Turbulent kinetic energy loss rate αk and turbulence restitution power παk of the dispersed phase (D) and the continuous phase (C). elasticity is ecD = 0.90. The results obtained using the present method were also compared with the computational results obtained by an Euler/Lagrange scheme [8]. Comparing the present model (Eul./Eul. [2]) with a standard particle diffusion model (Eul./Eul. [1]), the velocity and the standard deviation of the velocity magnitude of the present model agree better with the Euler/Lagrangian results. Because of the assumed isotropy of the present model, the standard deviation of the stream-wise particle velocity does not agree as well with the predicted standard deviation of the particle velocity magnitude. The characteristic model values describing diffusion: the restitution power παk and the turbulence structure parameter Cαk are shown in Fig. 2. The dissipation loss of the continuous phase and the restitution power of the dispersed phase dominate over the respective values of the other phase. As expected the turbulence structure parameter of the

dispersed phase CD decreases near the wall, because of the decreasing ratio αD /παD against the nearly constant turbulence structure parameter of the continuous phase CαC . Comparing the positions of these restitution maxima of the momentum diffusion with local minima of the turbulence structure parameter CαD (Fig. 2, right) near the wall the difference between the different kinds of diffusion, with and without viscous turbulence dissipation, in dispersed and continuous phase are shown. Validating this new particle diffusion model a test case of Bohnet and Triesch [1] has been chosen. Simulating a fluidised bed with a particle loading Z = 1 inside a rising vertical diffuser flow. Glass particles are used approaching the dispersed phase. The half-cone angle of the conic diffuser is 6◦ and the outlet/inlet diameter ratio is D2 /D1 = 1.45. The particle laden flow enters the diffuser part of the channel after a distance, which is long enough to develop a fully turbulent flow. Analysing the velocity profiles (Fig. 3) local loading, gas and particle velocities are shown at entrance and exit of the diffuser. The gas phase inlet-boundary condition is a velocity block profile of U0 = 26.0 m/s. The symmetry line is at x = 0. Inside the diffuser the gas velocity decreases and is overtaken by the particles. The particle velocity decreases during the relaxation process upside the WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


Figure 3: Normalised radial profiles of axial gas velocity, particle velocity and loading in comparison with experimental results.

diffuser. The particle loading has a similar maximum at the exit of the diffuser. Also this maximum decrease during the homogenising diffusion process upside the diffuser. The numerical data are compared with experimental data of Bohnet and Triesch [1].

5 Conclusions Using this kind of diffusion blending, simulations are able to give better results for turbulent wall layers of the dispersed phase, including their turbulence production. The prediction of turbulent particle diffusion is limited by the quality of modelling of the momentum diffusion and the turbulence production of the dispersed phase. Compared to classical equilibrium models, which solve an additional differential equation for the energy loss, coming up to the dissipation rate in viscous systems, the energy loss of the dispersed phase is given by algebraic equations. The turbulence structure parameter remains nearly constant in the dissipative systems examined here. So the ratio of restitution and dissipation power of the involved sub-models keeps also nearly constant. For the not viscous, dispersed phase the turbulence structure parameter decreases corresponding to the high restitution and the locally low momentum diffusion inside the wall layer. This characteristic behaviour is based on the additional restitution power without the corresponding loss rate induced by crossing trajectories effects. Basically the added modelling of the turbulence structure parameter influences the production of particle velocity variance near the wall in that kind that the gradients of the stream-wise particle velocity agree with the measurement data.

References [1] Bohnet A. and Triesch O., Influence of particles on fluid turbulence, Chem. Eng. Techn. 26:1254ff., 2003. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


249

[2] Crowe C.T., Sommerfeld M., and Tsuji Y., Multiphase flows with droplets and particles. CRC Press LLC, Florida, 1998. [3] Csanady G.T., Turbulent diffusion of heavy particles in the atmosphere. J. Atm. Sc. 20:201ff., 1963. [4] Grad H., On the kinetic theory on rarefied gases. Communications on Pure and Applied Mathematics 2(4):331ff., 1949. [5] He J., and Simonin O., Non-equilibrium prediction of the particle-phase stress tensor in vertical pneumatic conveying. ASME-FED : Gas-Solid Flows 166:253, 1993. [6] Huber N., and Sommerfeld M., Characterization of the cross-particle concentration distribution in pneumatic conveying systems. Powder Techn. 79:191ff., 1994. [7] Jones W. P., and B. E. Launder, The prediction of laminarization with a twoequation model of turbulence. Int. J. Heat and Mass Transf. 15:301ff., 1972. [8] Kohnen G., ber den Einfluss der Phasenwelchselwirkungen bei turbulenten Zweiphasenstrmungen und deren numerische Erfassung in der Euler/Lagrange Betrachtungsweise (tranl. On the influence of phase interactions of turbulent two-phase flows and their numerical description in the Euler/Lagrangian approach), Shaker Verlag, Aachen, Germany, 1997. [9] Jenkins J. T., M. W. Richman, Grads 13-moment-system for a dense gas of inelastic spheres. Arch. Ration. Mech. Anal. 87:355ff., 1985. [10] Kulick J. D., Fessler J.R., Eaton J.K., Particle response and turbulence modification in fully developed channel flow. J. Fluid Mech. 277:109ff., 1994. [11] Politis S. 1989. Prediction of two-phase solid-liquid turbulent flow in stirred vessels, PhD Thesis, Imperial College London, 1989. ¨ [12] Schiller L., A. Naumann, Uber die grundlegenden Berechnungen bei der Schwerkraftaufbereitung (transl.: On the basic calculations in gravity conditioning), VDI-Journal 77:318ff., 1933.




251

A numerical study of the scale effects affecting the evolution and sediment entrainment capacity of a gravity current, propagating over a loose bed containing large-scale roughness elements T. Tokyay & G. Constantinescu Department of Civil and Environmental Engineering, The University of Iowa, USA

Abstract When compositional gravity currents propagate over a loose bed, they can entrain, carry, and deposit large quantities of sediment over considerable distances from the entrainment location. The capacity of compositional gravity currents to entrain sediment of various sizes from a loose bed containing largescale roughness elements in the form of two-dimensional (2D) ribs is investigated using high-resolution Large Eddy Simulation (LES). The compositional gravity current is produced by the instantaneous release of a large volume of heavier lock fluid in a straight open channel. The present study focuses on the influence of the Reynolds number and the presence of 2D ribs on the evolution of the lock-exchange gravity current and on the distributions of the bed friction velocity. At the higher Reynolds number considered in the present investigation, the structure of the gravity current is closer to the inviscid state often assumed in simplified models. Keywords: gravity currents, sediment entrainment, Large Eddy Simulation, large-scale roughness.

1

Introduction

Gravity currents are mainly horizontal flows moving under the influence of gravity and generated by predominantly horizontal density differences within a WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090221

252 Computational Methods in Multiphase Flow V fluid or between two fluids. Predicting and understanding the evolution of gravity currents is of considerable interest to many applications in geophysics, in particular due to their impact on the environment. In the case in which they propagate over a loose bed (e.g., in rivers, lakes or oceans), compositional gravity currents can entrain substantial amounts of sediment and induce the formation of a turbidity gravity current. The turbulent motions within the body of the current detach particles from the bed and entrain them into the channel to form a turbidity current. Additionally, in most cases the loose bed over which the gravity current propagates is not smooth and flat. The presence of large-scale roughness elements can significantly modify the capacity of the compositional gravity current to entrain sediment with respect to the case in which the bed is flat. For example, erosion by gravity currents is one of the main causes for the formation of submarine canyons. Detailed measurements of the velocity and density fields within the gravity current are seldom available from experimental studies. In many applications involving turbidity currents or compositional currents propagating over loose beds, information on the spatial and temporal distributions of the bed friction velocity are essential to determine the amount of sediment entrained and carried by the current. High-resolution three-dimensional numerical simulations using eddy-resolving techniques like Direct Numerical Simulation (e.g., see Hartel et al., [3]) and Large Eddy Simulation (e.g., Ooi et al., [6]) have the advantage that can predict the spatial and temporal evolution of these variables. In particular, LES allows performing simulations at Reynolds numbers that are closer to the range typically encountered in practical applications. In many applications in river and ocean engineering, the Reynolds numbers are very high and the structure of the gravity current approaches the inviscid limit. Laboratory studies are typically conducted at much lower Reynolds numbers.

Figure 1:

Sketch of a lock exchange flow in a channel after the gate was removed with associated front velocities of the heavier (Uf) and lighter (Ufl) currents.

In this work we investigate the effect of the Reynolds number on the evolution and capacity to entrain sediment of a lock-exchange compositional Boussinesq gravity current propagating over a flat bed containing an array of 2D ribs. We are also studying how the capacity of the gravity current to entrain sediment changes with respect to the case when the gravity current propagates over a flat bed. As a result of the density differences between the lock fluid and the surrounding fluid and of the hydrostatic pressure differences, the heavy fluid starts spreading underneath the lighter fluid, as shown in Figure 1. In the set up WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


253

considered in this study, the gravity current containing lock fluid propagates in a rectangular horizontal plane channel. We consider the case in which the initial volume of the release (heavier lock fluid) is high (x0>>H) and occupies the whole depth of the channel (full-release case). The channel is long enough to avoid interactions of the gravity current with the end walls during the simulation time. Thus, this paper considers only the evolution of a gravity current during the slumping phase. The wavelength of the square ribs is 3H and their height is 0.15H (see Figure 2), where H is the channel depth. The changes in the structure of the gravity current and its capacity to entrain sediment are investigated between Re=48,000 (Re=ubH/ν, where ν is the molecular viscosity, ub = g ' H is the buoyancy velocity and g’ is the reduced gravity), in the range

at which most laboratory experiments are conducted, and Re=106, closer to field conditions.

Figure 2:

2

Sketch showing dimensions and spacing of the 2D ribs on the bottom surface.

Numerical model

A finite-volume LES code is used to solve the governing equations on nonuniform Cartesian meshes. A semi-implicit iterative method that employs a staggered conservative space-time discretization is used to advance the equations in time while ensuring second order accuracy in both space and time. A Poisson equation is solved for the pressure using multigrid. The algorithm discretely conserves energy, which allows obtaining solutions at high Reynolds numbers without artificial damping. A dynamic Smagorinsky model is used to estimate the subgrid-scale viscosity and diffusivity. All operators are discretized using central discretizations, except the convective term in the advection-diffusion equation solved for the concentration for which the QUICK scheme is used. Detailed validation of the code for 3-D LES simulations of cavity flows with or without an incoming turbulent flow is described in Chang et al. [1]. The same code was successfully used by Chang et al. [2] to predict ejection of buoyant pollutants from bottom channel cavities and by Ooi et al. [6] to predict the evolution of intrusive gravity currents. The density difference between the lock fluid and the ambient fluid is small enough to use the Boussinesq approximation. The Navier-Stokes equations and the advection-diffusion equation for the concentration are made dimensionless using the channel depth, H, and the buoyancy velocity, ub. The time scale used in the discussion of the results is t0=H/ub. The non-dimensional concentration is defined as C = (C − Cmin ) /(Cmax − Cmin ) , where Cmax , Cmin represent the WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

254 Computational Methods in Multiphase Flow V maximum (lock fluid) and minimum (ambient fluid) concentrations in the domain and C is the dimensional concentration. The lock gate is positioned in the middle of the computational domain (x/H=0.0). The top and bottom surfaces are simulated as no-slip smooth walls. The flow is assumed to be periodic in the spanwise direction (z). A zero normal gradient boundary condition is assumed for the concentration at the top, bottom and at the two end boundaries. All simulations discussed in this paper were conducted with a value of the viscous Schmidt number of 600 corresponding to saline water. No assumptions are needed on the value of the turbulent Schmidt number as the dynamic procedure (Chang et al. [1]) directly estimates the value of the subgrid-scale diffusivity based on the resolved velocity and concentration fields. The flow field was initialized with the fluid at rest. The length of the computational domain in the streamwise and spanwise directions was 40H and H, respectively. The mesh contained over 40 million cells and the mesh spacing in the wall normal direction was sufficiently small to resolve the viscous boundary layer (no wall functions were used) in all the simulations. The time step was 0.001t0. To maintain the anti-symmetry of the forward and backward propagating currents containing lock and ambient fluid, respectively, the roughness elements were placed on the bottom surface in the region with x/H>0 and on the top surface in the region with x/H20to) the interfacial billows shed in the dissipative wake region are not strong enough to amplify significantly the bed friction velocity values beneath them. This is because the WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

258 Computational Methods in Multiphase Flow V cores of the Kelvin Helmholtz billows are strongly disturbed in the formation region, in part due to their interactions with the lobes and the clefts present at the front of the current. This explains why the distribution of the bed friction velocity is relatively uniform behind the front region during the later stages of the slumping phase for a gravity current propagating over a smooth surface. The distributions of uτ/ub are qualitatively similar for the Flat_LR simulation but, as expected, the mean value of uτ/ub is larger (~0.028).

Figure 6:

Distribution of the non-dimensional friction velocity, uτc /ub, on the bottom wall (x/H>0) at t=32t0 in the Flat_HR (top) and Ribs_HR (bottom) simulations. Also shown are the corresponding concentration contours (aspect ratio is 1:2) in an x-y section.

In the Ribs_HR simulation results, the distribution of uτ/ub is strongly modulated by the presence of the roughness elements. The bed-friction velocity is strongly amplified downstream of each of the ribs, in the region where the flow inside the gravity current plunges downwards in the form of a jet-like flow and reaches the flat part of the bed surface. After the front passes several ribs, the largest amplification does not always occur downstream of the rib situated the closest to the front, as was the case in the initial stages of the slumping phase. This is because as a result of the interaction between the front and the ribs, the mixing in the head region is very high (see concentration contours for Ribs_HR simulation in Figure 6) and the mean concentration in the head region is smaller than the one in the upstream part of the gravity current (e.g., around the second rib behind the front in Figure 6). The lowest values of the bed-friction velocity are observed in the region situated upstream of each rib, where the heavier gravity current fluid decelerates and increases its potential energy such that it can be convected over the top of the rib. Figures 7 and 8 allow a direct comparison of the spanwise averaged bed friction velocity among the four simulations in the later stages of the slumping phase (t=32to). The streamwise distributions of the bed friction velocity are qualitatively similar in the low and high Reynolds number simulations. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


259

However, important differences are observed in the distribution of this variable between the simulations with a flat bed and the simulations with a bed containing ribs. Consistent with the results in Figure 6, the presence of the ribs induces large-scale variations in the streamwise distribution of the spanwise-averaged bed friction velocity. The peak values present in the region where the current reattaches to the flat part of the bed in the simulation containing the ribs are larger than the values observed in the same region in the simulation with a flat bed conducted at the same Reynolds number. For example, the nondimensional peak values of the bed friction velocity behind the head region are around 0.02 in the Ribs_HR simulation, while the corresponding values in the Flat_HR simulation are around 0.015 (see Figure 7). The differences are even larger in the simulations conducted at Re=48,000. The peak values in the Ribs_LR simulation are close to 0.045 while the corresponding values in the Flat_LR simulation are around 0.03 (see Figure 8).

Figure 7:

Streamwise variation of the spanwise-averaged friction velocity on the bottom wall (x/H>0) at t=32t0 in the Flat_HR (dashed line) and Ribs_HR (solid line) simulations.

Figure 8:

Streamwise variation of the spanwise-averaged friction velocity on the bottom wall (x/H>0) at t=32t0 in the Flat_LR (dashed line) and Ribs_LR (solid line) simulations.

Based on the results in Figure 7, the mean value of the spanwise-averaged bed friction velocity in between x/H=0.0 and the front is 0.011 in the Ribs_HR simulation and 0.013 in the Flat_HR simulation. This means that a gravity current propagating over a flat bed in a high Reynolds number (~106) lockexchange flow will have a slightly higher capacity to entrain fine sediment compared to one propagating over a bed containing large-scale roughness elements in the form of 2D ribs. Same trend is observed in the simulations conducted at Re=48,000 where the relative difference between the streamwiseaveraged values in between x/H=0.0 and the front is similar (0.026 in the Flat_LR simulation and 0.021 in the Ribs_LR simulation). Of course, at both Reynolds numbers the capacity of the gravity current to entrain coarser sediment is expected to be higher in the case in which the gravity current propagates over an array of 2D ribs mainly due to the strong amplification of the bed friction WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

260 Computational Methods in Multiphase Flow V velocity in the region where the jet-like flow of heavier fluid reattaches to the bed. While in the flat bed case one expects the capacity of the current to entrain sediment will peak behind the front which is propagating with constant velocity, in the case in which ribs are present the capacity of the current to entrain sediment will be the highest downstream of the first 3-4 ribs behind the front. To quantitatively compare the capacity of the gravity current to entrain sediment particles of a certain size in the four simulations, the flux of the sediment entrained from the bed per unit time and unit width is calculated for a certain sediment size and Shields critical shear velocity, uτc, using: 1 (1) F( t ) = ∫ P dA W A where W (=H) is the width of the channel, A is the bed area and P is the local pick-up rate for the sediment. In the present work the expression proposed by van Rijn (1984) was used to estimate P when uτ>uτc.  u 2 − uτ2c P = 0.00033 τ  u2 τc 

   

1.5

( s − 1 ) 0.6 g 0.6 d 0.8

ν 0.2

(2)

The pick-up rate P is expressed in units of volumetric flux per unit area per unit time. If at a certain location uτ0) for particles with a threshold value for entrainment of uτc/ub=0.03 (top) and uτc/ub=0.01 (bottom) in the Flat_LR (circles) and Ribs_LR (squares) simulations.

Assuming a length scale h=1.7m corresponding to ∆ρ/ρ0=0.02, the buoyancy velocity in the Re=106 simulations is ub=0.58m/s and t0~3s, if the ambient fluid WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


261

is water. The flux F(t) was plotted in Figure 9 for the simulations conducted at Re=106 for four sediment sizes (d=14, 30, 50 and 100 µm). Using Shields diagram, the critical bed friction velocity uτc for sediment entrainment is 0.006, 0.008, 0.01, 0.012 m/s for the four sediment sizes with d=14, 30, 50 and 100 µm. Past the initial stages of the slumping phase (t>40s during which the front of the gravity current passes the first rib in the Ribs_HR simulation) the capacity of the gravity current propagating over ribs to entrain particles with d40s is three to five times larger for the gravity current propagating over ribs. For d>50 µm (Figures 9c and 9d) the gravity current propagating over a flat bed cannot entrain sediment past the initial stages of the slumping phase as uτ1 means that the oil travels faster than water in the pipe while S 10 3

kg m3

(7)

Particle inertia occurred because the particle Reynolds number ReP is greater than 10−1 [27], as shown following: Re p =

ρ c ⋅ γ ⋅ a 2 1000 ⋅ 50 ⋅ 0.05 2 = = 12.5 ⋅ 10 −1 > 10 −1 ηc 10 −3

(8)

It is possible to estimate the influence of settling through the calculation of the experimental time texp required for a single sphere to migrate over a length l (l is equal to H, the gap height) as shown following: t exp =

ηc ⋅ l 9 9 10 −3 ⋅ 1 ⋅ = ⋅ = 0.115 sec 2 ∆ρ ⋅ g ⋅ a 2 2 1600 ⋅ 9.81 ⋅ 0.05 2

(9)

where ∆ρ = |ρd−ρc|, g is the acceleration of gravity and a is the radius of the particle with maximum size in the mixtures [27]. In the following the flow curves for shear rate more than the critical shear rate have been shown.


312 Computational Methods in Multiphase Flow V 4.2 Experimental results and model fitting In fig. 6 the experimental data and the theoretical flow curves in a semi logarithmic scale diagram for each material analyzed, at different solid volumetric concentration Cv, have been reported: the solid line represents the theoretical model function and the points indicate the experimental data. The best fitting model of the experimental data is the theoretical model of Bingham. The model fitting parameters have been reported in Table 3. First of all, it is noted that, after the critical shear rate, all the debris flow mixtures investigated behave like a shear-thinning yield stress fluids: shear stress increases with the increase of shear rate and the viscosity decreases. The influence of the solid volumetric concentration Cv on the rheological parameter of debris flow material mixtures tested has been evaluated. Proportionally higher values of shear stress and viscosity with increasing of the solid volumetric concentration have been noted. Moreover, on equal solid volumetric concentration, the yield stress and the viscosity of material B are higher than the yield stress and the viscosity of materials A and C. The rheological parameters of material C are the lowest according to the volcanic particle nature. Table 3: Cv (%) 0.20 0.30 0.40 0.20 0.30 0.20 0.30 0.40

Material A B C

Bingham model parameters. τy (Pa) 0.732 0.927 1.337 3.792 16.88 0.047 0.181 0.607

ηB (Pa · s) 0.0037 0.0059 0.0233 0.0145 0.0333 0.0033 0.0042 0.0074

R2 0.998 0.999 0.999 0.991 0.969 0.999 0.998 0.999

In fig. 7 the yield stress τy (obtained by the results of the vane rotor) versus the solid volumetric concentration Cv has been reported: the yield stress τy exponentially increase with the increase of solid volumetric concentration Cv. According to some previous study [4–6, 10, 12, 13, 17], the following relation could be used:

τ y = α ⋅ e β ⋅Cv

(10)

where α and β are fitting parameters. Their values have been reported in table 4. Table 4: Material A B C

Fitting parameters α and β. α 0,2464 0,1333 0,0089


β 0,1215 0,1909 0,1066


313

Shear stress (Pa)

a) 100

10

1

Cv 0.20 Cv 0.30 Cv 0.40

0,1 50

200

350 500 Shear rate (1/s)

650

800

Shear stress (Pa)

b) 100

10

1 Cv 0.20 Cv 0.30 0,1 50

200

350 500 Shear rate (1/s)

650

800

Shear stress (Pa)

c) 100

10

1 Cv 0.20 Cv 0.30 Cv 0.40

0,1 50

200

350

500

650

800

Shear rate (1/s)

Figure 6:

Experimental data and theoretical flow curves at different volumetric concentration Cv : a) material A; b) material B; c) material C.



Yield stress (Pa)

100 Material A Material B Material C

10

1

0,1

0,01 10

Figure 7:

5

20 30 40 Solid volumetric concentration (%)

100

Yield stress τy versus solid volumetric concentration Cv.

Conclusion

In order to evaluate the rheological behaviour of natural debris flow material, laboratory tests involving soils taken from the source area of three debris flows occurred in Campania region (southern Italy) have been carried out. Mixtures with varying concentration of fine sediment with maximum diameter less than 0.1 mm and distilled water have been prepared. The debris flow mixtures have been investigated in a standard rheometer with two different geometries, the parallel plates system and the vane rotor system. The vane geometry seems to be an appropriate rheometrical tool for quantitative evaluation of the rheological behaviour of debris flow materials (but using the parallel plates system is also possible to give some simple qualitative ideas about debris flow mixtures). The comparison between two geometry configurations allows checking the range of shear rate where there are not disturbing effects and misleading evaluations. In this range of shear rate all the debris flow mixtures tested behave like a shear thinning yield stress fluid: shear stress increases with the increase of shear rate and viscosity decreases when increasing shear rate. These experimental data have been fitted with the theoretical Bingham model. The increase in solid volumetric concentration produces an increase of shear stress and viscosity. The yield stress dependent exponentially on the solid volumetric concentration. The experimental results presented have to be preliminary, because they have been carried out only on particles less than 0,1 mm. For determining the rheological behaviour of gravel-sand mixtures, new apparatus have been set up.

References [1]

Costa, J.E. & Williams, G.P., Debris flow dynamics (videotape), US Geological Survey Open file 84 – 606, 22min, 1984. www. pubs.usgs.gov/of/1984/ofr84-606/ WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


[2] [3] [4]

[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

315

Johnson, A.M., Debris flow. Topics in Slope Instability, eds. D. Brunsden and D.B. Prior, Wiley, New York, pp. 257-361, 1984. Coussot, P. & Piau, J.M., On the behaviour of fine mud suspensions. Rheologica Acta, 33, pp. 175-184, 1994. O’Brien, J.S. & Julien, P.Y., Physical properties and mechanics of hyperconcentrated sediment flow. Proc. of the Specialty Conference on Delineation of landslide, flash flood and debris flow hazard in Utah, Utah Water Research Laboratory, General Series, UWRL/G-85/03, pp. 260278., 1984. Phillips, C.J. & Davies, T.R.H., Determining rheological parameters of debris flow material. Geomorphology, 4, pp. 573-587, 1991. Major, J.J. & Pierson, T.C., Debris flow rheology: experimental analysis of fine – grained slurries. Water Resources Research, 28 (3), pp. 841-857, 1992. Savage, S.B. & Hutter, K., The motion of a finite mass of granular material down a rough incline. Journal of Fluid Mechanics, 199, pp. 177215, 1989. Iverson, R.M., The physic of debris flow. Reviews of Geophysics, 35, pp. 245-296, 1997. Takahashi, T., Debris flow Mechanics, Prediction and Countermeasures, Taylor and Francis Group: London, pp. 35-38, 2007. Coussot, P., Mudflow Rheology and Dynamics, IAHR Monograph Series, A.A. Balkema: Rotterdam, pp. 252, 1997. Bagnold, R.A., Experiment on a gravity-free dispersion of large solid sphere in a Newtonian fluid under shear. Proc. of The Royal Society London, Series A, pp. 49-63, 1954. O’Brien, J.S. & Julien, P.Y., Laboratory analysis on mudflow properties. Journal of Hydraulic Engineering, 144, pp. 877-887, 1988. Coussot, P. & Piau, J.M., A large-scale field cylinder rheometer for the study of the rheology of natural coarse suspensions. Journal of Rheology, 39 (1), pp. 105-123, 1995. Contreras, S.M. & Davies, T.H.R., Coarse-Grained Debris Flows, Hysteresis and Time-Dependent Rheology. Journal of Hydraulic Engineering, 126, pp. 938-941, 2000. Ancey, C. & Jorrot, H., Yield stress for particle suspensions within a clay dispersion. Journal of Rheology, 45 (2), pp. 297-319, 2001. Schatzmann, M., Rheometry of large particle fluids and debris flows, PhD Dissertation No 16093, ETH, Zürich, Switzerland, pp. 192, 2005. Kaitna, R., Rickenmann, D. & Schatzmann, M., Experimental study on rheological behaviour of debris flow material, Acta Geotechnica, 2, pp. 71-85, 2007. Nguyen, Q.D. & Boger, D.V., Measuring the flow properties of yield stress fluids, Annual Review of Fluid Mechanics, 24, pp. 47-88, 1992. Chhabra, R.P. & Richardson, J.F., Non-Newtonian Flow in the Process Industries. Butterworth-Heinemann. Oxford, pp 436, 1999.


316 Computational Methods in Multiphase Flow V [20] [21] [22] [23] [24] [25] [26] [27]

Scotto di Santolo, A., Analisi geotecnica dei fenomeni franosi nelle coltri piroclastiche della provincia di Napoli, PhD thesis, University of Naples “Federico II” and Rome “La Sapienza”, 2000. Ruopolo, S., Analisi dei fenomeni franosi nella coltre piroclastica non satura del cratere degli Astroni, Graduate thesis, Department of Geotechnical Engineering, University of Naples “Federico II”, 2006. Papa, R., Indagine sperimentale di una copertura piroclastica di un versante della Campania, PhD thesis, University of Naples “Federico II”, 2007. Alderman, N.J., Meeten, G.H. & Sherwood, J.D., Vane rheometry of bentonite gels, Journal of Non-Newtonian Fluids Mechanics, 39, pp. 291310, 1991. Nguyen, Q.D. & Boger, D.V., Direct yield stress measurement with the vane method, Journal of Rheology, 29, pp. 335-347, 1985. Pignon, F., Magnin, A. & Piau, J.M., Thixotropic colloidal suspension and flow curve with a minimum: identification of flow regimes and rheometric consequence, Journal of rheology, 40, pp. 573-587, 1996 Larson, R.G., The Structure and Rheology of Complex Fluids. Oxford Univ. Press, New York, 1999. Macosko, C.W., Rheology. Principles, Measurements and Applications. Wiley-VCH, Inc. pp 550.1994



317

Experimental and numerical investigation of mixed flow in a gallery S. Erpicum1, F. Kerger1,2, P. Archambeau1, B. J. Dewals1,2 & M. Pirotton1 1

Research Unit of Hydrology, Applied Hydrodynamics and Hydraulic Constructions – HACH, ArGEnCo Department, Liege University, Belgium 2 Belgian Fund for Scientific Research – FRS-FNRS, Belgium

Abstract Experimental investigations on a physical model of a gallery performed in the Laboratory of Structures Hydraulics at the University of Liege are presented. The study focuses on the influence of the gallery aeration rate on the mixed flow pattern and the pressure distribution. In particular, the effect of air vents on the flow patterns and the release capacity of the gallery are assessed. An unusual mechanism leading to a two-phase instability is also pointed out when the aeration rate is not sufficient. Experimentations are completed with numerical computations performed with an original 1D model developed by the authors to give new insight into the mechanisms involved. Keywords: hydraulics, mixed flows, civil engineering, aeration.

1

Introduction

Mixed flows, characterized by the simultaneous occurrence of free-surface and pressurized flows, are frequently encountered in rivers networks, sewer systems, storm-water storage pipes, flushing galleries… As a matter of fact, some hydraulic structures are designed to combine free-surface and pressurized sections (e.g. water intakes). Dynamic pipe filling bores may occur in hydraulic structures designed only to convey free-surface flow. During such a transition, highly transient phenomena appears and may cause structural damages to the systems [1], generate geysers through vertical shafts [2] and engender flooding. What is more, air/water interactions may arise in such structure (particularly at the transition bore [3]) and thoroughly alter the flow regime. Sometimes the WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090271

318 Computational Methods in Multiphase Flow V interactions produce beneficial effects. However, more often than not, the effects are not beneficial and the remedial actions are expensive. For instance, the presence of air in pipelines can severally affect the water carrying capacity of the line, increase head losses, produce undesirable pressure rise, and even induce water hammer or blowbacks by air evacuation [4]. The effect of air in such situations depends on the location and amount of the un-dissolved air as well as the configuration of the hydraulic structure. In addition, the prediction of such flows by means of a numerical scheme remains challenging by the lack of suited mathematical model and closure relations [5]. Traditionally, the common design of hydraulic structures was conducted assuming either stratified or pressurized single phase flow. The need for additional flexibility in the hydraulic electricity production on the one hand and more rigorous restriction due to environmental and political demands on the other hand lead to an increasing demand for further research on air/water interaction for civil engineering applications. As pointed out in [6], multiphase flow in hydraulic structures is a topic of recent interest. In fact, the only comprehensive description of the relevant flow patterns could be found in approaches originating from chemical and process engineering. As the typical internal diameter of the conduits in chemical engineering is at least one order of magnitude smaller than in hydraulic engineering, the application of these concepts to hydraulic works remains challenging. In this paper, the results of the experimental study of stationary mixed flows taking place in a gallery are presented in detail. Experimental investigations have been carried out on a model of a gallery in the Laboratory of Structures Hydraulics (HACH) of the Liege University. The model includes a Plexiglas pipe linking two tanks. The topography of the upstream and downstream tanks has been built regarding realistic in-situ conditions. The aim of the experimental study was to: 1. Determine the expected flow discharge and the influence of the aeration rate on the flow discharge. 2. Identify hydrodynamic characteristics of the flows appearing in the gallery and appearance conditions for each pattern. In particular, the experimental apparatus is clearly described in this paper as well as the measurement system and the investigation method. Finally, numerical simulations have been performed with an original 1D model for mixed flows, developed by Kerger et al. [5]. The model is implemented in the 1D module of the software package WOLF. WOLF is a finite volume flow simulation modelling system developed within the Laboratory of Hydrology, Applied Hydrodynamics and Hydraulic Constructions (HACH) at the University of Liege. Stationary numerical results give new insight into the mechanisms regulating flows in the gallery.

2

Experimental set up

2.1 Physical model The experimental facilities are made of two tanks, an upstream and a downstream one, linked by a circular gallery 5 m long with a .14 m diameter. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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The natural topography of a mountain river bed is represented in both tanks, as the gallery bypasses a river meander. The gallery inlet and outlet are located in the right bank of the river, at the level of the river bottom, and the constant gallery slope is 6.96%. The inlet structures are profiled to decrease the head losses at the gallery entry and a .17 m long square to circle transition links the inlet to the gallery. The gallery is not straight but counts for an upstream bend of 27.68 degrees with a curvature radius of 1.81 m and a sharp angle of 2.5 degrees at its middle (figure 1). The outlet is rectangular with a radial gate .12 m high and .12 m wide to control the flows through the gallery. A .17 m long circle to square transition links the gallery to the outlet. The tanks are made of steel; the tanks topography has been build with concrete blocks and mortar painted with latex. The gallery is in transparent Plexiglas and the inlet and outlet are made of aluminium and PVC. The roughness height of the gallery has been estimated to be 2.10-5 m. In a second part of the study, three air vents with a .02 m diameter have been added upstream of the gallery, on the top of the circular cross section, at the beginning of the constant slope. The purpose of these vents is to feed the gallery with air to prevent the formation of low pressures within the flowing water. Air vents have been designed according to the recommendations in [7]. Alimentation basin

Natural river bed

Air vents Inlet

Circular pipe

Upstream tank

Downstream tank

Upstream tank Air vents Outlet

Natural river bed

Natural river bed Downstream tank Inlet Alimentation basin

Radial gate Circular pipe Outlet

Figure 1:

Natural river bed

Sketch of the experimental device.

2.2 Water alimentation and boundary conditions The water feeding system is a closed circuit with a pump taking water from a 400 m³ underground reservoir to inject it through under pressure galleries in an alimentation basin in the upstream tank. The alimentation basin is separated from WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

320 Computational Methods in Multiphase Flow V the upstream tank by a permeable screen to make uniform the velocity fields entering the physical model. Downstream of the model, the water is collected in a free-surface channel to go back to the underground reservoir. The discharge in the upstream tank is the upstream boundary condition. The head level upstream of the gallery regulates naturally regarding the gate opening rate and the system release capacity. Downstream of the physical model, the natural topography is very steep so no specific boundary condition is needed (supercritical flow). b. Downstream tank

a. General view of the physical model

c. Upstream tank

e. Inlet structure in the upstream tank

d. Gallery

Figure 2:

Details of the physical model.

2.3 Measurement system The model is equipped with the following measurement system:  The upstream discharge is measured with an electromagnetic dischargemeter (accuracy of  l l/s) on the pumping system;  The water level in the upstream tank is measured using a limnimeter (accuracy of 0.1 mm) and a Pitot tube (accuracy of  0.1 mm);  9 Pitot tubes are regularly distributed along the gallery to measure the pressure head in the gallery (accuracy of  0.1 mm);  14 graduated scales are fixed on the gallery perimeter to measure the water level for stratified flows.

3

Numerical model

In the last decades, large literature [8, 9] has been dedicated to a new approach in experimental hydraulics consisting in coupling experimental investigations with CFD computations. Comparison of the experimental data with numerical results



321

from a modelling system is shown to give new insight into the flow conditions and to significantly reduce the cost of the physical models. The numerical model used in this paper is an original 1D model for mixed flows developed by Kerger et al. [5]. It is implemented in the one-dimensional module of the modelling system WOLF. WOLF is finite volume scheme developed within the Laboratory of Hydrology, Applied Hydrodynamics and Hydraulic Constructions (HACH) of the University of Liege [10]. WOLF 1D is based on a set of hyperbolic Partial Differential Equations (PDE), usually called Saint-Venant equations [11], describing one-dimensional unsteady open channel flow. The Saint-Venant equations are derived from cross section integration of the Navier-Stokes equation: 0  Q     A        2 (1)  Z    t  Q  x  Q   gA   Sf    A    x  where A[m²] is the cross section, Q[m³/s] is the flow discharge, g[m²/s] is the gravity, Z[m] is the free surface elevation, Sf[-] is the friction slope resulting from the resistance law. Friction slope SF may be computed using the DarcyWeisbach relation and the Colebrook relation for the friction factor:  k fu 2 1 2.51  (2) SF  with  2 log      f 2 Dh  3.7 Dh Re f  with Dh[m] the hydraulic diameter of the cross-section, k[m] the roughness height, u[m] the water velocity and Re[-] the Reynolds Number. Pressurized flows are commonly described through the Allievi equations [12]. According to the Preissmann slot model [13], pressurized flow can equally be calculated through the Saint-Venant equations by adding a conceptual slot on the top of a closed pipe [5]. When the water level is above the cross section maximum level, it provides a conceptual free surface flow, for which the gravity wave speed is c  g  T f (Tf is the slot width). Physically, the slot accounts naturally for the water compressibility and the section dilatation under a variation of pressure. In order to simulate pressurized flows with a piezometric head below the top of the pipe section, an original concept, called negative Preissmann slot [5], has been developed.

4

Results

Investigations focused mainly on stationary flows and aimed at determining the flow discharge as a function of the upstream pressure head and downstream gate opening. In this case, strong air/water interactions alter the flow behaviour. In particular, the flow discharge through the gallery is strongly influenced by air/water interaction and depends of the aeration rate. In this section, the influence of both the upstream pressure head and the aeration conditions on the release capacity of the gallery is analyzed.


322 Computational Methods in Multiphase Flow V 4.1 Without air vents Figure 4 shows the experimental relation between the flow discharge through the gallery and the upstream pressure head (zero level is set at the upstream reservoir bottom level) without air vents. Various two-phase flow patterns are observed according to the flow discharge. 5 areas corresponding to the 5 flow patterns that are usually mentioned in literature [14] can be defined. Details of each flow pattern are provided on figure 3: 1. Pure water fully free surface flow (or smooth stratified flow) is observed for pressure head below 30 cm (figure 3.a). It is characterized by both phases, air and water, flowing separately by gravity. A smooth interface between phases appears only if both phases flow with almost the same velocity. 2. Wavy stratified flow is observed for pressure head between 30 cm and 40 cm (figure 3.b). This flow does not differ much from smooth stratified flow in terms of hydro-mechanic characteristics. Surface ripples and waves are building up and create a rather rough interface. The volume of airflow, entering by inlet as dissolved air in water and through a vertical vortex appearing at the gallery intake, is equal to the air volume insufflated into the flow by self-aeration plus the air volume flowing above the water surface as a result of the air-water shear forces. 3. Intermittent flow is observed for pressure head between 40 cm and 55 cm. This latter category includes slug flow (figure 3.c), where waves touch the top of the tube and form a liquid slug which passes rapidly along the gallery; as well as plug flow (figure 3.d), in which there are large bubbles flowing near the top of the tube. As the transition from plug to slug flow is gradual and not very sharp, slug and plug flow together are often simply referred to as intermittent flow patterns with no further specification. This flow pattern is not strictly speaking a steady flow as it is characterized by a time periodic oscillation between a free surface flow and slug flow, according to the aeration rate and the amplitude of the waves. This leads to significant fluctuations in the measured data. 4. Bubbly flow, characterized by the entrainment of small bubbles dispersed in the liquid continuum, is observed for pressure head between 55 cm and 70 cm (figure 3.e). The larger bubbles (but smaller than the resulting pockets during plug flow) propagate below the conduit ceiling due to buoyancy. Smaller bubbles, primarily transported by liquid turbulence, may be detected dispersedly over the whole cross-section. Bubbles appear where the pressure of the liquid falls below the atmospheric pressure. 5. A pure water pressurized flow is observed for pressure head above 70 cm (figure 3.f). Two curves computed with WOLF are represented on the graph of figure 4. The dotted line is computed assuming that a free surface appears if the water height is below the cross section top (air phase above the free surface is at atmospheric pressure – high aeration rate). The continuous line is computed by activating the negative Preissmann slot (sub-atmospheric pressurized flow – low aeration rate). WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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a. Smooth Stratified flow

W a te r Flow

b. Wavy Stratified flow Wa te r Fl o w

c. Intermittent – Slug flow Wa te r Fl o w

d. Intermittent – Plug flow

Wa te r Fl o w

e.Bubbly flow Wa te r Fl o w

f. Pressurized flow Wa te r Fl o w

Figure 3:

Flow patterns visualization.

Numerical results are in good accordance with experimental data for smooth stratified flows and fully pressurized flows. Bubbly and intermittent flows show a similar behaviour to the sub-atmospheric pressurized flows. This point underlines the aeration rate in the gallery is too small to create a free surface flow.



Upstream pressure head [cm]

90 80

Pure water fully pressurized flow

70

Bubbly flow

60 50

Intermittent flow: Plug/Slug flow

40

Oscillations

Wavy stratified flow

30

Smooth stratified flow

20

Experimental Numerical (Atmospheric pressure)

10

Numerical (Sub‐Atmospheric pressure) Upstream reservoir bottom level

0 5

Figure 4:

15

25

35

Flow discharge [l/s]

45

55

Experimental relation (upstream pressure head-flow discharge) and flow patterns observed.

4.2 Periodic instabilities A periodic instability between two unstable steady flow regimes occurs in the area denoted by wavy stratified flows (figure 4). The instability induces large period (10 to 60 seconds) oscillations of the water level in the upstream reservoir. The amplitude of the oscillations can reach 2 cm. In addition, pressure oscillations, whose amplitude can reach 4 cm, are observed all along the gallery. The inception of the instability is intimately linked with the aeration rate of the gallery, and in particular the amount of air entrained through the vertical vortex appearing at the water intake. Indeed, the minimum of the oscillation corresponds to a highly aerated stratified flow below an air phase at subatmospheric pressure. The distribution of pressure along the gallery in this case is given in figure 5.b. As pointed in figure 4, this flow pattern gives the minimum water carrying capacity for a given upstream head. If the upstream reservoir is supplied with a constant flow discharge, the water level in the reservoir increases. As the water level arises in the reservoir, the amount of air entrained through the vortex decreases [15]. Decrease of the air void fraction in the flow generates the formation of Kelvin-Helmholtz instabilities characteristic of intermittent flows (pressurized flow pattern). The maximum of the oscillations corresponds to this poorly aerated intermittent flow. The distribution of pressure along the gallery in this case is given in figure 5.a. As pointed in figure 4, this flow pattern gives the maximum water carrying capacity for a given upstream head. Then, the water level in the reservoir decreases because the flow discharge in the gallery is higher than the flow discharge supplying the reservoir. The cycle of the instability is then explained. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

Computational Methods in Multiphase Flow V a. Pressure profile for a discharge of 38.4 l/s : Sub-atmospheric pressurized flow

0,5

Topography

0,4

Pipe crown

Numerical Pressure head/Free surface level Experimental total head

Altitude (cm)

0,3

Numerical total head

0,2 0,1 0,0

-0,1 -0,2 -0,3

Experimental Pressure head/Free surfacelLevel

Abscissa(m)

-0,4

b. Pressure profile for a discharge of 38.4 l/s : Free surface flow

0,6

Topography

0,5 0,4

Numerical total head Numerical Pressure head/Free surface level Experimental total head

Altitude (cm)

Pipe crown


Figure 5:

325

0,3 0,2 0,1 0,0

-0,1 -0,2 -0,3

Abscissa(m)

-0,4

Computed total head and pressure head distribution for an intermittent flow sub-atmospheric pressurized flow and freesurface flow computation.


90 80


70

Bubbly flow

60


50


40 30


Experimental Experimental - Air vents added Numerical (Atmospheric pressure) Numerical (Sub-Atmospheric pressure) Upstream reservoir bottom level

20 10 0 5

Figure 6:

15

25

35


45

55

Effects of the addition of air vents on the experimental relation head/discharge and the flow pattern observed.


326 Computational Methods in Multiphase Flow V 4.3 Air vents effect In this section, the effect of the 3 air vents added upstream of the gallery, on the top of the circular cross section, at the beginning of the constant slope is assessed. The purpose of these vents is to enable air to enter the gallery to prevent the formation of low pressures within the flowing water (figure 5.a). The experimental relations between the flow discharge through the gallery and the upstream pressure head is shown on figure 6. The graph compares the new relation with the relation without air vent. Addition of air vents does not affect pure water flow, as expected. On the opposite, a diminution of the water carrying capacity of the gallery is observed for multiphase flow patterns. Area of observation of each two-phase flow pattern is affected as well. The ranges of flow discharge corresponding to bubbly flows and corresponding to intermittent flows are narrowed to the benefit of the wavy stratified flow area.

5

Conclusions

Experimental investigations on a physical model of a gallery have been presented. They were completed with CFD computations to give more insight into the mechanisms involved. Flow patterns description and mathematical models originating from chemical/process engineering served as a referential basis. The key issue of the present work is to show concepts develop for small size pipe can be adapted for conduit sizes significantly larger. Flow pattern identification has been performed for various aeration rates. In particular, the effect of the aeration rate and the flow pattern over the water carrying capacity has been assessed. Particular insight has been given on the nature of the transition between free surface and pressurized flows. Finally, a two-phase instability has been outlined. The fundamental concepts introduced pave the way for further research. Experimental research is required to derive air entrainment predictive relation and flow pattern maps more specific to hydraulic structures.

Acknowledgement A part of the experimental results is the property of EDF-CIH. The HACH gratefully acknowledge EDF-CIH for the authorization to publish those results.

References [1] Zhou, F., F.E. Hicks, and P.M. Steffler, Transient Flow in a Rapidly Filling Horizontal Pipe Containing Trapped Air, Journal of Hydraulic Engineering, 128(6): p. 625-634, 2002 [2] Guo, Q. and C. Song, Dropshaft Hydrodynamics under Transient Conditions, Journal of hydraulic Engineering, 117(8): p. 1042-1055, 1991 WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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[3] Vasconcelos, J. and S. Wright, Experimental Investigation of Surges in a Stormwater Storage Tunnel, Journal of hydraulic Engineering, 131(10): p. 853-861, 2005 [4] Estrada, O.P., Investigation on the Effects of Entrained Air in Pipelines, in Eigenverlag des Instituts für Wasserbau der Universität Stuttgart. 2007, Universität Stuttgart. p. 200. [5] Kerger, F., S. Erpicum, P. Archambeau, B.J. Dewals, and M. Pirotton. Numerical Simulation of 1D Mixed Flow with Air/Water Interaction in Multiphase Flow New Forest, 2008 [6] Keller, U., in Versuchanstalt für Wasserbau, Hydrologie und Glaziologie der Eidgenössichen. 2006, ETH Zürich: Zürich. p. 250. [7] Falvey, H.T., Air-Water Flow in Hydraulic Structures. Engineering Monogaph. Vol. 41: United States Department of the Interior. 1980 [8] Dewals, B.J., S. Andre, M. Pirotton, and A. Schleiss. Quasi 2D-numerical model of aerated flow over stepped chutes. in 30th IAHR Congress, Greece, 2003 [9] Erpicum, S., P. Archambeau, B.J. Dewals, S. Detrembleur, A. Lejeune, and M. Pirotton. Interactions between Numerical and Physical Modelling for the design and Optimization of Hydraulic Structures - Example of a Large Hydroelectric Complex. in International Symposium on Hydraulic Structures, XXII Congresso Latinoamericano de Hidrahulica, Ciudd Guayana, Venezuela, 2006 [10] Dewals, B.J., S. Erpicum, P. Archambeau, S. Detrembleur, and M. Pirotton, Depth-Integrated Flow Modelling Taking into Account Bottom Curvature, Journal of Hydraulic Research, 44(6): p. 787-795, 2006 [11] Cunge, J.A., F.M. Holly, and A. Verwey, Practical Aspects of Computational River Hydraulics. [Monographs and surveys in water resources engineering], 3. Boston: Pitman Advanced Pub. Program. 1980 [12] Wylie, E.B. and V.L. Streeter, Fluid transients. Première ed, ed. M.-H. Inc., 385, 1978 [13] Preismann, A. Propagation des intumescences dans les canaux et rivieres. in First Congress of the French Association for Computation, Grenoble, France, 1961 [14] Wallis, G.B., One-dimensional Two-phase Flow, ed. M.-H.B. Company. 410, 1969 [15] Quick, C.M., Efficiency of Air-Entraining Vortex Formation at Water Intake, Journal of the Hydraulics Division, Proceedings of the American Society of Civil Engineers, 96(7): p. 1403-1416, 1970




329

Sediment transport via dam-break flows over sloping erodible beds M. Emmett & T. B. Moodie University of Alberta, Canada

Abstract When a semi-infinite body of homogeneous fluid initially at rest behind a vertical retaining wall is suddenly released by the removal of the barrier, the resulting flow over a horizontal or sloping bed is referred to as a dam-break flow. When bed resistance is neglected the exact solution, in the case of a stable horizontal bed, may be obtained on the basis of shallow-water theory via the method of characteristics and the results are well known. Discrepancies between these shallow-water based solutions and experiments have been partially accounted for by the introduction of flow resistance in the form of basal friction. This added friction significantly modifies the wave speed and flow profile near the head of the wave so that the simple exact solutions no longer apply. Various asymptotic or numerical approaches must be implemented to solve these frictionally modified depth-averaged shallow-water equations. When the bed is no longer stable so that solid particles may be exchanged between the bed and the fluid, the dynamics of the flow become highly complex as the buoyancy forces vary in space and time according to the competing rates of erosion and deposition. It is our intention here to study dam-break flows over erodible sloping beds as agents of sediment transport, taking into account basal friction as well as the effects of particle concentrations on flow dynamics including both erosion and deposition. We shall consider shallow flows over initially dry beds and investigate the effects of changes in the depositional and erosional models employed, in the nature of the drag acting on the flow, and in the slope of the bed. These models include effects hitherto neglected in previous studies and offer insights into the transport of sediment in the worst case scenario of the complete and instantaneous collapse of a dam. Keywords: gravity current, dam-break flow, sediment transport, dilute sediment.



1 Introduction Dam-break flows, which are represented by the sudden release of fluid contained in a semi-infinite reservoir behind a vertical barrier, are of both practical and fundamental importance in fluid mechanics, engineering, and geomorphology. They have played a crucial role in underpinning simple models for a number of natural and catastrophic events, such as break-out floods from the failure of end moraine dams and various sheet flow events, as well as the formative stages of lahars or debris flows [1–3]. Although in practice the release of water upon collapse of the retaining barrier will often be more gradual than that in the idealised mathematical models, one can view these models as providing the worst case scenario for these events [3, 4]. The earliest work on dam-break flows considered single phase, low aspect ratio, frictionless flows in rectangular geometry taking the shallow-water equations as the governing model equations. With the bed below the dam assumed horizontal and dry, the solution for the flow is a centred simple wave that was first developed by Ritter [5]. If the initial depth of water behind the vertical dam is h0√ , the front of the flow advances as a wave over the dry bed with constant speed 2 gh0 , while the √ reduction of depth spreads back from the initial position of the dam with speed gh0 , where g is the acceleration due to gravity. In the disturbed region between the two extremes of depth, the velocity u and the depth of the flow h are given by 2 x + gh0 , and (1) u= 3 t 1 x 2 gh0 − , (2) gh = 3 t where x measures distance downstream of the original position of the dam and t measures the time elapsed since its collapse. Although these solutions do provide a reasonably good match to the experimental observations when the transients associated with the initial release have died down there are still important properties of the flow that are not captured by the classical shallow-water model used in the construction of the solutions displayed in (1) and (2). It has been observed in particular [3, 6] that for the dam-break flow experiments the water near the tip piles up and the front speed is appreciably less than that predicted by the simple theory. In order to account for this blunting of the tip and the slowing down of the front several authors [4, 7, 8] have postulated that near the tip, where the depth of flow drops to zero, frictional resistance and the resulting turbulence dominate the flow. To account for this basal friction a Chézy resistance term is added to the momentum equation [4, 8]. Various asymptotic procedures were employed [4, 7, 8] to determine the influence of this resistance and it was found that its inclusion brought theory and experiment into closer accord. Since we are interested in developing and exploiting models for sediment transport that employ dambreak flows on down-sloping topography as paradigms for certain geological and engineering processes we shall extend the model beyond what is discussed in this WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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paragraph while appreciating the significant gains achieved through the addition of the resistance term. In fact, basal friction with realistic parameterisations for geological applications appears to have a much greater influence on flow dynamics than does the presence of particles in suspension [9]. Recent studies [1, 2] have employed dam-break flows as agents of sediment transport. In [2] the authors explore dam-break flows over beds that consist of fine sediment that can be entrained into the water column and transported in suspension. The sediment transport was passive in that the suspended particles did not influence the flow dynamics which could then be totally specified employing the simple exact solution of the shallow-water equations for both a dry bed [5] and a bed with ‘tail water’. Recent studies by the current authors [9, 10] have shown that under the assumption of dilute suspensions employed in [2] the suspended particles will play a relatively minor role in modifying the flow dynamics so that passivity of sediment transport does not produce large errors. The omissions of basal friction and bed topography however, as was the case in [2], were shown to have a more profound negative influence on the accuracy of the results [10]. In the present work we shall employ dam-break flows over erodible beds as agents of sediment transport. The inclusion of a velocity dependent basal friction as well as bottom topography and non-passive particle transport adds several important mechanisms that were absent from previous studies. We shall assume that our flows are shallow so that the pressure remains hydrostatic throughout the flow regime [2, 8–13]. We shall also assume that the particle concentration in the flow remains sufficiently low so that we may treat the particles as being isolated and employ a Boussinesq approximation whereby the particles appear in the momentum equations only in the buoyancy terms. These assumptions put definite constraints on the range of particle volume fractions φ(x, t) in our well-mixed suspensions. When erosion exceeds deposition so that particle concentrations are increasing we shall assume that our model calculations are valid up until φ ≈ 0.05 [9, 10]. Although the bottom boundary shear stress could be calculated from the full governing equations we shall adopt the common and much simpler approach of introducing a Chézy drag coefficient CD which when viscous effects are small (large Reynolds number flows) gives the boundary shear stress as τb = CD ρf u2 , where ρf is the density of the fluid and u a depth averaged horizontal velocity [4,8–10]. The Chézy drag coefficient is dimensionless and usually falls in the range 0.01-0.001 for most environmental flows [8]. With all of our additions to the model of Pritchard and Hogg [2] (basal drag, bottom topography, and particle modified flow dynamics) the simple shallow-water based solutions [5] will no longer apply and the approach adopted in [2] is not available so that an alternative approach will have to be adopted.

2 Model development We consider the two-dimensional flow resulting from the sudden release of water initially held at rest behind a plane vertical retaining wall of height h0 . The bed below the dam which is initially located at x = 0, is gently varying and specified WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

332 Computational Methods in Multiphase Flow V z

ρ(x, t) = φ(x, t)ρp + (1 − φ(x, t))ρf reservoir

u(x, t) h(x, t)

x b(x, t)

Figure 1: Schematic for the dam-break flow on a dry sloping and erodible bed. by the topography z = b(x, t). It is assumed that initially there is no water below the dam and that the bed is comprised of fine or cohesive material which, once a threshold shear stress is exceeded, is entrained into the water column and transported in suspension to possibly be deposited downstream. With our assumption of dilute suspensions the erosion or deposition depth is small relative to the depth of the current so that the position of the bottom boundary may be kept fixed in most of our calculations. Although in the early stages of such dam-break flows the stream-wise and vertical scales of the motion will be comparable, there soon comes a time when stream-wise scales dominate the vertical ones and the flow may be considered to be a shallow flow with negligible vertical accelerations and a hydrostatic pressure distribution can be adopted [2, 8]. We shall assume that once particles have been suspended into the water column they are always vertically well-mixed by the turbulence of the flow [2,11,12,14] so that the volume fraction of particles in suspension φ is a function of the horizontal coordinate x and the time t elapsed since collapse of the dam. We shall always assume that the water initially behind the dam is particle free. The bulk density ρ of the suspension is then given by ρ(φ) = ρp φ + (1 − φ)ρf ,

(3)

where ρp (> ρf ) is the particle density. The setting under which the model is developed is summarised by the schematic presented in Figure 1. The depth-averaged continuity and momentum equations for the system under the Boussinesq approximation (φ 1) and the hydrostatic assumption [10] are ∂ ∂h + (hu) = 0, and ∂t ∂x ∂ ρgh ∂b 1 g ∂ 2 2 βhu + − CD u2 , ρh = − (hu) + ∂t ∂x 2 ρf ρf ∂x

(4) (5)

where β is the Boussinesq coefficient (shape factor) [8]. The magnitude of β ≥ 1 corresponds to the amount of shear present in the horizontal velocity field and may depend on such factors as the Reynolds number or the boundary roughness [8]. The WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


333

particle mass conservation and bed evolution equations [9, 10, 12] are ∂ ∂ ρp φh + ρp φhu = qe − qd , ∂t ∂x

and ρp

∂b = −qe + qd . ∂t

(6)

where qe is the mass erosion flux and qd is the mass deposition flux. Observations have shown that qe and qd are functions of both the fluid velocity u and the volume fraction of particles in suspension φ [15]. We adopt the usual expression for mass deposition rate, that is, qd = ρp φvs , where vs is the Stokes settling velocity [2,9–14]. With no particles in suspension behind the dam we need only consider deposition when u > uc , where uc is some critical velocity below which particles are not entrained into the fluid column. We adopt the expression 2 n for |u| ≥ uc , ρp ve uu2 − 1 c qe (u) = (7) 0 for |u| < uc for the erosion rate qe , where ve is a sediment entrainment rate. This model is used to describe the erosion of sediment from a cohesive bed or from a bed of fine cohesionless material where some critical shear stress must be exceeded in order to entrain particles from the bed into the water column [2, 9, 10, 15–17]. We make all equations non-dimensional using the non-dimensionalisation and scaling scheme presented in [10]. Of particular importance is the typical volume fraction scale φ0 and velocity scale U . The velocity scale U is the familiar √

g h0 wherein

g is a ‘modified gravity’ which is given by (γ φ0 + 1)g where γ = (ρp −ρf )/ρf . Rendering all equations non-dimensional gives ∂ ∂ h+ (hu) = 0, ∂t ∂x ∂ 1 ∂ (hu) + βhu2 + h2 1 − + φ = −h 1 + φ bx − CD u2 , ∂t ∂x 2

qd = ud φ,

∂ ∂ (φh) + (φhu) = qe − qd , ∂t ∂x ∂b = φ0 q d − q e , ∂t n ⎧ 2 ⎪ ⎨u u − 1 for |u| ≥ uc , e and qe (u) = u2c ⎪ ⎩ 0 for |u| < uc

(8) (9) (10) (11)

(12)

where we have introduced the non-dimensional parameter = 1 − 1/(γ φ0 + 1).

3 Analysis of the model In this section we highlight various aspects of dam-break flows over erodible beds based upon our model equations. The parameter values used in the simulations are typical for flows of a geological scale with h0 = 20m [11]. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

334 Computational Methods in Multiphase Flow V (a) h(x, t) vs x

h(x, t)

1

0.5

0 −300

−200

−100

0

x

100

200

300

400

200

300

400

(b) u(x, t) vs x

u(x, t)

2

1

0 −300

−200

−100

0

x

100

Figure 2: Solutions of (a) height and (b) velocity of the particle free modified dambreak flow over a flat bottom with basal drag at t = 300. Solid lines show numerical solutions with β = 1.0, 1.1, and 1.2 (from left to right, at the front). Dashed line shows the Ritter solution. Parameter value used is CD = 0.001.

We begin by presenting an exploration of the effects of those mechanisms (velocity shear, bed slope, and drag) that do not necessarily involve suspended particles. To do so we examine solutions of the initial value problem corresponding to the dam-break flow for the non-dimensional pair of coupled equations (8) and (9) with ≡ 0. Subsequently, we present an overview of the effects of suspended particles by allowing the flows to entrain particles through bed erosion but fixing the bed so that b(x, t) is in fact independent of t. A more detailed study of these scenarios is presented in [10]. Finally, we present a brief note regarding the evolution of the bed topography when the bed is eroded according to (11). 3.1 Modified dam-break flows with drag In this subsection we present solutions for modified dam-break flows with basal drag in the absence of bed topography or particles. That is, following the approach employed in [2, 6–9, 18] we retain the basal drag term in order to bring our model calculations into closer accord with the experimental results of [7] wherein dambreak flows over a horizontal bed were examined. In Figure 2 we have plotted both the height and velocity profiles, respectively, for both the classic solution [5] and the numerical solution including basal drag. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


335

We note immediately that the presence of drag has significantly altered the shape of the depth profile in the immediate vicinity of the leading edge as well as the velocity structure of the flow in this region. We also note that, in agreement with the experimental results [7] and the theoretical approach adopted in [4], the velocity is nearly uniform in the blunt snout. That is, in the deformed tip basal drag retards the flow so that the velocity profile is approximately horizontal there. The assumption that the velocity in the tip depends only on time was crucial to the theoretical development in [4] and appears to be confirmed by our numerical work. Furthermore, we note that the effect of vertical shear (β > 1) in the horizontal velocity profile is most dramatic in the immediate vicinity of the leading edge where the depth, and hence momentum, of the flow is small allowing the effect of vertical shear in the horizontal velocity to be accentuated. 3.2 Modified dam-break flows with drag over a linear slope In this subsection we will examine modified dam-break flows over sloping beds in order to isolate the effects of the interplay between the bottom slope and basal drag. We shall take the bottom topography to be specified by b(x, t) = −sx (x)

(13)

where s is a small non-dimensional parameter and is the Heaviside step function. Employing this simple linear form for the bottom topography allows us to appeal to our intuition while interpreting both theoretical and numerical results. Furthermore, the stream-wise gradient of a linear slope is constant, affording us the opportunity to perform an asymptotic expansion over the bed slope s, which was presented in [10]. In Figure 3 we have plotted the numerical solutions for both the height and velocity profiles of a particle free dam-break flow over a sloping bottom with drag for β = 1. We note that both the height and velocity profiles are nearly horizontal in the bulk of the flow over the linearly sloping bed. In the presence of a sloping bottom the blunt snout in the height profile has become more abrupt and falls steeply to zero at the front. As demonstrated in previous sections, the effect of drag is to retard the front and create a blunt snout, while the effect of a sloping bottom is to draw out the fluid, reducing its height in the bulk of the flow over the sloping bed, and slightly increasing its height directly behind the front. Furthermore, in contrast to the drag-free case, the presence of a sloping bottom has a significant effect on the front position of the flow. As one may expect, the front position is greater for flows over steeper beds. 3.3 Modified dam-break flows with drag and sediment In this subsection we will examine modified dam-break flows as agents of sediment transport over flat erodible beds while allowing the particle volume fraction to change through the mechanisms of particle advection, deposition, and entrainment through bed erosion. In our previous work [9] we demonstrated the role played WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

336 Computational Methods in Multiphase Flow V (a) h(x, t) vs x

h(x, t)

1

0.5

0 −100

0

x

100

200

100

200

(b) u(x, t) vs x

u(x, t)

2

1

0 −100

0

x

Figure 3: Solutions of (a) height and (b) horizontal velocity for particle free dambreak flows over linear slopes with drag. Solid lines show numerical solutions at t = 100 with slope s = 0, 0.001, and 0.01 (from left to right, at the front). Dashed lines show the Ritter solutions. Parameter values used are CD = 0.01 and β = 1.

by the critical bed velocity uc in the long term competition between erosion and deposition and displayed the strong effect of basal drag on the ultimate outcome of this competition. Furthermore, we observed that the inclusion of particles did not have a significant effect on the height or velocity profiles of the flows. Generally, flows with particles were slightly faster compared to analogous particle free flows, but only by a few percent at most. The maximum rate of sediment entrainment occurred at the front and was nearly uniform within the snout. Peaks in the volume fraction of sediment profiles were observed directly behind the front where the height of the fluid decreased sharply to zero. These peaks in the volume fraction of sediment profiles where highest for short post-release times since the velocity was also highest for short post-release times, and decreased with time. In Figure 4 we have plotted a typical numerical solution of the full model equations with drag, particle deposition, and bed erosion over a flat bed for a parameter configuration in which the drag coefficient CD and critical bed velocity uc were small enough so that particles were entrained by the snout for the duration of the flow. The interplay between basal drag and the critical bed velocity was further demonstrated in our recent work [9] by examining the particle flux at a fixed WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

Computational Methods in Multiphase Flow V φ(x, t) vs x

1

φ

337

t = 60

0.5

t = 600

0 0

100

200

x

300

400

500

Figure 4: Solutions (volume fraction) of the dam-break flow over a flat bed with basal drag, particle deposition, and particle entrainment at various times t = 60, 120, . . . , 540, 600. Parameter values used are CD = 0.001, φ0 = 0.01, γ = 2.5, ud = 0.005, ue = 0.0015, n = 1.2, uc = 0.5, and β = 1.

station. Flows with high basal drag did not continue to erode the bed and entrain particles once the front had passed a given station. Flows with a critical bed velocity above 2/3 were also dominated by deposition since the long term Ritter velocity is less than 2/3 regardless of the presence of basal drag. Flows with low drag and a critical bed velocity below 2/3 continued to erode the bed and advect entrained particles downstream. As time progressed, the horizontal velocity at the station approached its Ritter solution and the particle flux due to erosion approached a steady value while the particle flux due to deposition increased with increasing volume fraction until an equilibrium between erosion and deposition was reached. 3.4 Modified dam-break flows with drag and sediment over a linear slope In this subsection we will examine modified dam-break flows as agents of sediment transport over sloping erodible beds. In our previous work [10] we observed that, as in the previous case with flat beds, the inclusion of particles did not have a significant effect on the height or velocity profiles of the flows. Particles entrained into the flow maintained their relative position within the flow which resulted in a nearly linear volume fraction profile that increased in the downstream direction. Furthermore, the maximum attained by the volume fraction continued to increase for all post-release times, in contrast to the flat bed case. The peak in the volume fraction occurred directly behind the front, and was primarily due to advection coupled with the nearly horizontal velocity profile over the sloping bed. In Figure 5 we have plotted the horizontal velocity u and volume fraction of sediment φ for a typical numerical solution to the full model equations with basal drag, particle deposition, bed erosion, and bed slope. These plots elucidate the relationship between the velocity and volume fraction of sediment entrained by the flow. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

338 Computational Methods in Multiphase Flow V (a) u(x, t) vs x

2

t = 100

t = 20 u1

0 −100

0

x

100

(b) φ(x, t) vs x

0.5 0.4

t = 100

0.3 φ

0.2 t = 20

0.1 0 −0.1 −100

0

x

100

Figure 5: Solutions (a) horizontal velocity and (b) volume fraction of the dambreak flow with basal drag, particle deposition, and bed erosion at various times t = 20, 40, . . . , 80, 100. Parameter values used are CD = 0.001, φ0 = 0.01, γ = 2.5, ud = 0.005, ue = 0.0015, n = 1.2, uc = 0.5, β = 1, and s = 0.001.

The linear nature of the volume fraction profile shown in Figure 5 is in contrast to the flat case (see Figure 4) in which the volume fraction was highest in the snout and decayed in a non-linear fashion in the upstream direction [9]. In the sloping case, the peak in the volume fraction is primarily due to advection coupled with the nearly horizontal velocity profile, and the peak increases for all post-release times. 3.5 Modified dam-break flows with drag and sediment over a variable bed In this subsection we will examine modified dam-break flows as agents of sediment transport over initially flat erodible beds. We allow the bed topography to change with time through the mechanism of erosion according to (11). In Figure 6 we have plotted the bed topography b(x, t) and horizontal velocity u of the numerical solution to the full model equations with basal drag, particle deposition, and bed erosion. These plots show the scour pit resulting from bed erosion. We note that the pit is deepest slightly downstream from the original position of the dam. There is peak in the horizontal velocity directly above the scour pit since the fluid gains momentum as it falls into the pit. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


339

(a) b(x, t) vs x

0.01 0 b(x, t)

−0.01 −0.02 −0.03 −0.04 −0.05

−100

0

x

100

200

100

200

(b) u(x, t) vs x

u(x, t)

2

1

0 −100

0

x

Figure 6: Solutions of (a) bed height and (b) horizontal velocity for particle laden dam-break flows over variable beds with drag at time t = 120. Solid lines show numerical solutions with CD = 0.001, φ0 = 0.01, γ = 2.5, ud = 0.005, ue = 0.0015, n = 1.2, uc = 0.5, β = 1, and s = 0.001. Dashed lines show the Ritter solutions.

Further exploration of the complex interplay between the fluid, sediment, and bed dynamics will be carried out in subsequent research.

4 Discussion We have developed a model to describe the transport of dilute sediment under dam-break flows over sloping beds with basal drag and presented numerical solutions to this model in order to investigate the influence of various model parameters. The model developed is an extension to previous models, includes a velocity dependent basal drag force, incorporates the effects of a spatially dependent bed topography, and allows the variable concentration of suspended particles, through the mechanisms of deposition and erosion, to influence the flow dynamics. The model is unique from existing models which do not couple the flow and sediment dynamics and do not include basal drag [2] or bed topography [9]. The numerical results show that this coupling is especially important for the sediment dynamics and should not be ignored when the understanding of sediment processes is vital to a given study. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


References [1] Capart, H. & Young, D., Formation of a jump by the dam-break wave over a granular bed. J Fluid Mech, 372, pp. 165–187, 1998. [2] Pritchard, D. & Hogg, A., On sediment transport under dam-break flow. J. Fluid Mech, 473, pp. 265–274, 2002. [3] Stansby, P., Chegini, A. & Barnes, T., The initial stages of dam-break flow. J. Fluid Mech, 374, pp. 407–424, 1998. [4] Whitham, G., The effects of hydraulic resistance in the dambreak problem. Proc R Soc London, Ser A, 227, pp. 399–407, 1955. [5] Ritter, A., Die fortpflanzung der wasserwellen. Z Verein Deutsch Ing, 36, pp. 947–954, 1892. [6] Dressler, R., Hydraulic resistance effect upon the dam-break functions. J. Res Nat Bur Stand, 49, pp. 217–225, 1952. [7] Dressler, R., Comparison of theories and experiments for the hydraulic dambreak wave. Proceedings of the Commission des Eaux de Surface at the Assemble Gnrale de Rome 1954, International Association of Hydrological Sciences, volume 38, pp. 319–328, 1954. [8] Hogg, A. & Pritchard, D., The effects of hydraulic resistance on dam-break and other shallow inertial flows. J Fluid Mech, 501, pp. 179–212, 2004. [9] Emmett, M. & Moodie, T., Dam-break flows with resistance as agents of sediment transport. Phys Fluids, 20(8), p. 086603, 2008. [10] Emmett, M. & Moodie, T., Sediment transport via dam-break flows over sloping erodible beds. Stud Appl Math, 2009. Submitted. [11] Bonnecaze, R., Huppert, H. & Lister, J., Particle-driven gravity currents. J. Fluid Mech, 250, pp. 339–369, 1993. [12] Moodie, T., Pascal, J. & Swaters, G., Sediment transport and deposition from a two-layer fluid model of gravity currents on sloping bottoms. Stud Appl Math, 100, pp. 215–244, 1998. [13] Moodie, T. & Pascal, J., Axisymmetric particle-bearing gravity flows on sloping bottoms. Can Appl Math Q, 7, pp. 17–47, 1999. [14] Moodie, T. & Pascal, J., Non-hydraulic effects in particle-driven gravity currents in deep surroundings. Stud Appl Math, 107, pp. 217–251, 2001. [15] Teisson, C., Ockenden, M., Hir, P.L., Kranenburg, C. & Hamm, L., Cohesive sediment transport processes. Coastal Eng, 21, pp. 129–162, 1993. [16] Kerswell, R.R., Dam break with Coulomb friction: a model for granular slumping? Phys Fluids, 17(5), pp. 057101, 16, 2005. [17] Blanchette, F., Strauss, M., Meiburg, E., Kneller, B. & Glinsky, M., Highresolution numerical simulations of resuspending gravity currents: Conditions for self-sustainment. J Geophys Res, 110, p. C12022, 2005. [18] Whitham, G., Linear and Nonlinear Waves. Wiley, New York, 1974.


Section 5 Bubble and drop dynamics



343

Hydrodynamic drag and velocity of micro-bubbles in dilute paper machine suspensions A. Haapala1, M. Honkanen2, H. Liimatainen1, T. Stoor1 & J. Niinimäki1 1

University of Oulu, Fibre and Particle Engineering Laboratory, Finland Tampere University of Technology, Energy and Process Engineering, Finland

2

Abstract This paper studies hydrodynamic drag forces acting on freely rising microbubbles in dilute paper machine suspensions under turbulent flow conditions. Dissolved, colloidal and numerous solid materials i.e. process chemicals, wood extractives, fillers and wood fibre fractions present in these suspensions disturb the rise of micro-bubbles increasing their drag. The aim of this study is to characterise the terminal velocities and drag coefficients of the bubbles as a function of their Reynolds number in several paper machine circulation waters, i.e. white waters, and in some model suspensions. Characterisation is performed experimentally with a high-speed CMOS camera and a submersed back-light illumination in a pressurised bubble column. Image sequences of bubbly flow are analysed with automatic image processing algorithms that measure not only the bubble size and velocity, but also the velocity of the fluid surrounding bubbles, revealing the initial slip velocity of each bubble. Bubbles are tracked in time to provide time series data for every bubble that passes the focal plane of the imaging system. Results show how some suspension properties – concentration, apparent viscosity and surface tension – affect the motion of micro-bubbles. Results also show the changes in micro-bubble formation with pressure drop and differences of bubble size distributions in a variety of suspensions and solutions. Finally, a mathematical model describing the bubble rise velocities and drag coefficients with respect to the bubble Reynolds number is developed for the investigated white waters. Keywords: bubble motion, drag coefficient, drag force, white water, papermaking, high-speed imaging, high-speed imaging, bubble sizing. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090291


1

Introduction

Bubbly gasses and their interaction with other substances in pulp suspensions may cause substantial problems in papermaking processes. Most of them relate to stock filtration, dewatering, sheet formation or pumping but also process measurements. For example, consistency can be affected as recently presented by Stoor [1] and Helle et al [2]. Deaeration in papermaking usually takes place in a pressurised tank or a special gas removing pump, while passive methods rely solely on bubble rise and evacuation from suspension. Today, different solutions for gas removal are developed using mathematical models and computational fluid dynamics (CFD) tools that describe the phenomenon of bubble motion. These models in turn require experimental knowledge on the kinetics and hydrodynamics behind the free gas bubble rise in process suspensions. The rise of free gas bubbles in papermaking suspensions is hindered by apparent viscosity and solids, while surface tension of the liquid phase affects the size distribution of formed bubbles [3–8]. This is also typical for fibre-free filtration waters drained from sheet former, commonly referred as white waters. Suspended and dissolved solids are the main components contributing to bubble drag in white waters i.e. fibre fines fractions and inorganic particles such as clay fillers [7, 8]. Their composition and overall chemical state varies from one paper machine to another, whilst process waters slowly accumulate dissolved and colloidal materials from raw materials, process chemicals and fresh water. While the effect of surfactants on the drag coefficient of rising bubbles was shown previously in the work of Okazaki [6] in 1962, so far research on experimental values for bubble drag and rise velocity in complex papermaking white waters has not been published. Image-based measurements have become a powerful tool to determine the gas bubble size, velocity and the velocity of suspension surrounding the gas bubble [8, 9]. Robust image analysis algorithms [10, 11] can automatically recognise individual, in-focus bubbles from image sequences of complex multiphase suspensions, where bubbles commonly overlap. Optical measurements are, however, limited to dilute multiphase flows, because they need undisturbed optical access to the measurement volume. Dispersed phase particles and bubbles in these suspensions scatter the incident light and damp the light intensity, which restricts the penetration of light through the suspension. In addition, they scatter light on the optical path between the illumination source and camera causing image distortions. Opaque multiphase suspensions, such as white waters, are often testing the limits of digital imaging methods. However, an underwater measurement probe [13] or illumination also provides a way to visualise these opaque multiphase suspensions inside large vessels. In this study, the bubbly multiphase flows are visualised close to the column wall with submerged backlight illumination optics. The aim of the study is to investigate the effect of white water properties on size distribution and motion of micro-bubbles in white waters in order to produce consistent experimental data for the validation of CFD models that describe the deaeration phenomenon of these suspensions. Micro-bubbles are generated in WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


345

characterised white waters and model waters of varying consistencies, apparent viscosities and surface tension by dissolving air under pressure and causing a sudden pressure drop. Generated micro-bubbles are then visualised using a highspeed CMOS camera and submerged diode laser back-light. Bubble size distributions, rise velocities of bubbles and bubble drag coefficients (CD) are defined for each white water as a function of bubble Reynolds number.

2

Materials and methods

2.1 Studied suspensions The examined white waters (WWs) originate from 8 paper machines that utilise various raw materials and produce a variety of paper grades: mechanical pulp to produce newsprint and magazine grades while eucalyptus, pine or similar kraft pulp to produce different fine paper grades. Thus fines and fillers content in white waters varied accordingly for each machine. Analyses on waters were made according to the following standards and methods; pH (SFS 3021), conductivity (SFS-EN 27888) and dry matter content (SFS-EN 20638). Surface tension was measured with a Krüss K8600 (du Noüy’s ring method) and apparent viscosities with a Haake 501 viscometer. In addition, apparent mean particle size of white water solids were determined using a multiwavelength particle size analyser (Beckman Coulter LS 13 320). Physicochemical properties of tested white waters are summarised in table 1. Table 1: pH (20°C)

Properties of white waters.

WW 1

WW 2

WW 3

WW 4

WW 5

WW 6

WW 7

WW 8

7.9

4.0

6.9

7.7

7.7

7.7

7.7

7.6

988 989 988 987 991 991 995 989 Density (40°C) [g/L] Conductivity 1.5 2.0 1.0 0.9 1.0 0.9 1.1 1.4 (20°C) [µS/cm] Dry Matter 0.58 0.49 0.50 0.25 0.22 0.22 0.25 0.21 Content [%] Surface tension 53 51 48 59 58 53 66 61 (20°C) [mN/m] Apparent viscosity 1.7 1.8 1.8 1.4 1.3 1.2 1.4 1.5 (40°C) [mPas] Mean particle size, (from 10.

Figure 6:

Bubble drag coefficients and mean bubble velocities according to the micro-bubble Reynolds number and size for white waters.



353

The non-linearity of certain drag and micro-bubble velocity measurements can be attributed to a small population of detected bubbles in each size class and to the attachment of some, especially larger micro-bubbles on to pulp fragments. This was mostly seen on white waters 2, 3 and 4 in fig. 6. Based on experiences with different suspensions, we can conclude that the differences in solid contents and deviations in apparent viscosity or suspended particle sizes are so minor that they can be approximated with a single model. However, the rise velocity of micro-bubbles appears to be the weakest in suspensions WW1-WW3 and WW5 that mostly contain materials originating from mechanical pulp, e.g. wood based extractives, colloidal pitch, etc. Their appearance is not seen in present analyses of water properties and would thus make ideal continuation for this line of study.

4

Conclusions

A digital imaging method was used to present the degree on which apparent viscosity, surface tension and solids consistency affect micro-bubble formation, drag and rise in dilute suspensions. Similarity of micro-bubble formation and uniformity of drag coefficient development in white waters was shown with data needed to devise a novel model describing drag forces affecting micro-bubble rise in white waters. Effects of contaminants in process water on dynamics of micro-bubbles were shown to greatly affect the rate of bubble evacuation and thus the kinetics of suspension deaeration process. Results revealed that contamination decreases bubble size and reduces the micro-bubble concentration. The results can be explained with the changes in surface tension and viscosity of the suspension. In white waters, the drag co-efficient of microbubbles is higher than in pure water when ReB < 10 and lower when ReB > 10.

Acknowledgements Authors would like to thank Tekes (The Finnish Funding Agency for Technology and Innovation), PaPSaT graduate school and Academy of Finland for financially supporting this research.

References [1] Stoor, T. Air in pulp and papermaking processes, Oulu Univ. press, pp. 66, 2006. [2] Helle, T-M. Qualitative and quantitative effects of gas content on papermaking. Paper & Timber, 82(7), pp. 457-463, 2000. [3] Heindel, T.J. Bubble size in cocurrent fiber slurry. Industrial and Engineering Chemistry Research, 41, pp. 632-641, 2002. [4] Reese, J., Jiang, P. & Fan, L-S. Bubble characteristics in three-phase systems used for pulp and paper processing. Chemical Engineering Science, 51(10), pp. 2501-2510, 1996.


354 Computational Methods in Multiphase Flow V [5] Margaritis, A., te Bokkel, D.W. & Karamanev, D.G. Bubble rise velocities and drag coefficients in non-Newtonian polysaccharide solutions. Biotechnology and Bioengineering, 64(3), pp. 257-266, 1999. [6] Okazaki, S. The Velocity of Air Bubble Ascending in Aqueous Solution of Surface Active Substance and Inorganic Electrolyte. Colloid & Polymer Science, 185, pp. 154-157, 1962. [7] Hubbe, M.A. Water and papermaking 2, white water components. Paper Technology, 48(2), pp. 31-40, 2007. [8] Garver, T.M., Xie, T.B. & Kenneth H. Variation of white water composition in a TMP and DIP newsprint paper machine. Tappi Journal, 80(8), pp.163-173, 1997. [9] Lindken, R. & Merzkirch, W. A novel PIV technique for measurements in multiphase flows and its application to two-phase bubbly flows. Experiments in Fluids, 33, pp. 814-825, 2002. [10] Honkanen, M., Saarenrinne, P., Stoor, T. & Niinimäki, J. Recognition of highly overlapping ellipse-like bubble images. Measurement Science and Technology, 16, pp. 1760-1770, 2005. [11] Honkanen, M. & Marjanen, K. Analysis of the overlapping images of irregularly-shaped particles, bubbles and droplets. Proc. of Int. Conf. on Multiphase Flow, Leibzig, Germany, paper 559, 2007. [12] Honkanen, M. Direct optical measurement of fluid dynamics and dispersed phase morphology in multiphase flows. Univ. print, Tampere, pp. 80, 2006. [13] Honkanen, M., Eloranta, H. & Saarenrinne, P. Submersible, planar shadow image velocimetry system for online, in-situ analysis of multiphase flows in industrial processes. Proc. of the 13th Int. Symposium on Flow Visualization, Nice, France, paper 222, 2008. [14] Kamgar-Parsi, B. & Netanyahu, N.S. A nonparametric method for fitting a straight line to a noisy image, IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(9), pp. 998-1001, 1989. [15] Ouellette, N.T., Xu, H. & Bodenschatz, E. A quantitative study of threedimensional Lagrangian particle tracking algorithms. Experiments in Fluids, 40, pp. 301-313, 2006. [16] Frank, X., Li, H. & Funfschilling, D. An analytical approach to the rise velocity of periodic bubble trains in non-Newtonian fluids. European Physics Journal E, 16, pp. 29-35, 2005. [17] Hadamard, J. Mouvement permanent lent d’une sphère liquide et visqueuse dans un liquide visqueux. Comptes Rendus Mathématique. Académie des Sciences, Paris, 152, pp. 1735-1738, 1911. [18] Moore, D.W. The boundary layer on a spherical gas bubble. Journal of Fluid Mechanics, 16, pp. 161-176, 1963. [19] Mei, R., Klaussner, J.F. & Lawrence, C.J. A note on the history force on a spherical bubble at finite Reynolds number. Physics of Fluids, 6, pp. 418420, 1994. [20] Haapala, A., Stoor, T., Liimatainen, H., Nelo, M. & Niinimäki, J. Passive white water deaeration efficiency in open channel flow. 62nd Appita Ann. Conf., Rotorua, NZ, 2008. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


355

Effects of physical properties on the behaviour of Taylor bubbles V. Hernández-Pérez, L. A. Abdulkareem & B. J. Azzopardi Department of Chemical and Environmental Engineering, Faculty of Engineering, University of Nottingham, UK

Abstract Gas-liquid flow in vertical pipes, which is important in oil/gas wells and the risers from sea bed completions to FPSOs, was investigated to determine the effects of physical properties on the characteristics of the mixture such as void fraction, structure frequency and velocity as well as the shape of 3D structures. These are difficult to visualise using conventional optical techniques because even if the pipe wall is transparent, near-wall bubbles would mask the flow deep in the pipe. Therefore, more sophisticated methods are required. Two advanced wire mesh sensors (WMS) were used and the two-phase mixtures employed were air-water and air-silicone oil. The effect of fluid properties is accounted for in terms of the Morton number. It was found that the flow pattern is affected by the fluid properties as the results revealed that contrary to what is commonly assumed when modelling pipe flow, the flow is not symmetric, with a lot of distortion, which is even higher for the air-silicone oil mixture. Keywords: gas/liquid, vertical, wire mesh sensor, slug flow, void fraction, structure velocity.

1

Introduction

One of the most common structures found in gas-liquid flows is the Taylor bubble; it is related directly to the slug flow pattern in upward vertical flow, for instance in oil/gas applications. This bubble occupies the greater part of the pipe cross section. Liquid between the Taylor bubble and the pipe wall flows around this bubble as a thin film. Traditionally the Taylor bubble has been stereotyped as a bullet-shape bubble and most hydrodynamic models of pressure drop for WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090301

356 Computational Methods in Multiphase Flow V slug flow take on this shape as suitable, but in reality the behaviour of Taylor bubbles is rather complex and they can adopt different shapes affected by different parameters, such as pipe diameter and fluid properties. Several authors have reported in the literature a lack of slug flow in relatively large pipe diameters. The behaviour of bubbles in turn can affect parameters such as pressure drop and heat transfer coefficients. Of particular importance can be the role that fluid properties play in oil/gas production, where they can vary from one well to another. The simple change of fluid from one mixture to another can result in the variation of several important properties such as viscosity, density and surface tension. In fact, the viscosities measured for different heavy oils can vary by orders of magnitude. However in the literature, there is a lack of information regarding the effect of fluid properties on the behaviour of Taylor bubbles, particularly at high Reynolds numbers. Most of the work using different gasliquid mixtures has been directed towards the measurement of average liquid holdup and pressure drop and flow pattern transition, for instance Weisman et al. [1] as well as Nädler and Mewes [2], have studied effect of fluid properties on flow patterns in horizontal two-phase flow. For vertical flow, not surprisingly, most of the extensive work reported has been regarding the motion of single Taylor bubbles using air-water mixtures, as it is the basis for the study of more complicated flows. One of the first investigations on these large bubbles was carried out by Davies and Taylor [3], who carried out viscous potential flow analysis of cap bubbles and found that the terminal rise velocity was simply related to the curvature radius of the cap bubble. However due to disturbances induced by the bubbles in the liquid, it is evident that the Taylor bubbles in a train of bubbles will behave different from a single bubble in static liquid. In general, the theoretical approach has been limited to the low Reynolds number regime and several studies have been published over the years, for instance Moore [4] carried out a study of a gas bubble in a viscous liquid and more recently Tomiyama et al. [5] studied the terminal velocity of a single bubble rising through an infinite stagnant liquid in a surface tension force dominant regime theoretically and experimentally. In many cases investigators have followed the experimental approach in order to tackle the behaviour of Taylor bubbles in more attention demanding problems. Nicklin et al. [6] established a correlation for translational velocity of Taylor bubbles in moving liquid while White and Beardmore [7] used dimensionless groups to account for the combined effect of several variables, including fluid properties. Later developments in instrumentation have allowed researchers to look further at the two-phase flow; Van Hout et al. [8] measured the translational velocity of elongated bubbles in continuous slug flow, Hassan et al. [9] studied two-phase flow field and 3D structures in bubbly flow using particle image velocimetry (PIV). However, no effect of physical properties has been reported. Not until recently have a few studies been presented on numerical simulation of Taylor motion, usually they are limited by the sort of assumptions that need to be made, such as flow symmetry and stagnant liquid. For example Clarke and Issa [10] modelled the motion of a periodic train of Taylor bubbles in vertical flow by WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


357

imposing cyclic conditions at the inlet and outlet of the slug unit based on the assumption that the flow pattern repeats itself over consecutive slug units. Taha and Cui [11] among others have highlighted the use of dimensionless groups in the study of Taylor bubbles along with the numerical approach, as correlations of experimental data are generally developed in terms of dimensionless groups rather than in terms of the separate dimensional variables in the interests of compactness and in the hope of greater generality. The shape of the bubbles can change with the local hydrodynamic conditions, adding new degrees of freedom to an already complex problem. Knowledge of the Taylor bubble behaviour in moving liquid is fundamental to our understanding of multiphase flow, particularly when fluid properties are varied. In this work, a study of the effect of liquid properties on Taylor bubble behaviour is performed based on experimental data obtained at high Reynolds numbers. Particular emphasis is put in the extraction of 3D structures, with the use of advanced instrumentation known as wire mesh sensor, looking forward to use this information for validation of computational models for determination of gas-liquid interface.

2

Experimental facility

Two advanced wire mesh sensors (WMS), developed at Forschungszentrum Rossendorf-Dresden, have been used in a two-phase flow facility at Nottingham. The basic part of the facility has been frequently described elsewhere, for example Hernández-Pérez et al. [12] and Azzopardi et al. [13], and for the sake of simplicity details are omitted here. The sensors are described in detail by Prasser at al. [14] and Da Silva et al. [15] respectively. Both of them have a grid of 24 × 24 measurement points evenly distributed across the pipe cross-section given by a 24 × 24 wire configuration in two planes. The first is based on conductivity measurements and is suitable for water; the second is based on capacitance measurements and works with non-conductive materials such as oil. Figure 1 shows a picture of the wire mesh sensor. The two-phase mixtures employed were air-water and air-silicone oil. The latter liquid has a surface tension about one third that of water and a viscosity of ~5x water. The physical properties of the fluids used are given in Table 1. Table 1:

Properties of the fluids.

Fluid

Density (kg/m3)

Viscosity (kg/ms)

Air

1.224

0.000018

Water

1000

0.001

Silicone oil

900

0.005


Surface tension (N/m) 0.072 0.02


Figure 1:

Wire-mesh sensor (2 × 24 electrode wires).

The data were gathered in two campaigns: In the first campaign the air-water mixture was used with the conductivity wire mesh sensor and a pair of capacitance probes in order to obtain structure velocity, as reported by Hernández-Pérez et al. [12]. In the second campaign air-silicone oil was employed with the capacitance wire mesh sensor as well as Electrical capacitance tomography (ECT), as described by Abdulkareem et al. [16]. The test pipe is 6 m long and 67 mm diameter. In both campaigns, the wire mesh sensor was located at 4.92 m from the mixing or inlet section and we focus on vertical flow. Conditions studied were superficial velocities: for air ranging from 0.05 to 4.7 m/s and for liquid from 0.0 m/s to 0.7 m/s for both water and silicone oil. The data were taken at a data acquisition frequency of 1000 Hz over an interval of 40 s for the wire mesh sensors and 200 Hz for both capacitance probes and ECT. In addition, high speed video system was used in order to obtain real images of the flow regimes under different conditions.

3


The behaviour of the Taylor bubble is described by means of several parameters such as time series of cross-sectional area averaged void fraction. Further analysis of these time series will allow flow patterns and structure frequencies to be extracted and compared. Finally the full 3D structures will be presented. Applying dimensional analysis, the effect of the properties can be put in terms of dimensionless numbers. For gas bubbles rising in liquids, the viscosity ratio and density ratio tend to be very small and therefore it is generally sufficient to consider only three dimensionless groups, namely the Morton number (M), the Eotvos Number (Eo) and the Froude number (Fr) as identified by White and Beardmore [7]. M is also called the properties number and together with the Eötvös number, is useful to characterize the shape of bubbles. These numbers are defined respectively as:



359

M  g L4 ( ρL  ρG )  L2  3

(1)

Eo  g (  L   G ) D / 

(2)

Fr  U TB / gD(  L  G ) /  L

(3)

2

and Here g is the gravity acceleration, µ is the viscosity, ρ is the density, σ is the surface tension, D is the pipe diameter and UTB is the velocity of a Taylor bubble rising in motionless liquid. The subscripts L and G refer to liquid and gas respectively. Therefore for air-water M = 2.64 10-11 and Eo=610.9 whereas for air–silicone oil M = 1.36 10-9 and Eo=1970. According to Bhaga and Weber [17], since M for both of the two-phase mixtures employed in this work are lower than 4 10-3, the bubble behaviour is expected to be a function of both Morton and Reynolds numbers. It can be observe from eqn (1) that variations in M are mainly due to the factor, µ4, since ρ and σ do not vary much from water to silicone oil. Water is usually considered as a low M number fluid. 3.1 Time series of void fraction The time series of cross sectional average void fraction show in a simple way the occurrence of structures as the gas-liquid mixture flows along the pipe. It also constitutes the raw data for application of statistical analysis to judge the flow behaviour. Figure 2 shows a typical run in which slug flow is present, the appearance of Taylor bubbles can be identified as high void fraction intervals whereas the low void fraction intervals correspond to the liquid slugs. The irregular variation of the void fraction reveals the transient nature of the Taylor bubbles. In addition, the liquid film variation can be observed to have an irregular shape. void fraction

1 0.8 0.6 0.4

0

Figure 2:

0.2

0.4

0.6

0.8

1 1.2 time (s)

1.4

1.6

1.8

2

Typical time series of cross-section average void fraction.

3.2 Flow patterns and structure frequency In considering Taylor bubbles, it is important to know the flow conditions under which these particular structures occur. Statistical analysis of the time series data can provide an insight into flow pattern identification. In this sense, a widely used and accepted method is Probability Density Function (PDF). It is also important to characterise the structures in terms of their occurrence frequency, WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


1

1

0.8

0.8 Void fraction

Void fraction

which was found from the time series utilizing the Power Spectral Density (PSD) technique. Figure 3 shows the time series, the PDF and the PSD for both two-phase mixtures at a particular flow condition. In this figure a time interval of 10 s is used to plot the time series however for processing the PDF and the PSD, the data gathered in 40 s have been considered. It is apparent from both the time series and PDF plots that the fluctuation amplitude of void fraction is bigger for the case of air-water mixture. Also, the double peak shape of the PDF plot, characteristic of slug flow is not well defined; indeed the flow is greatly distorted as illustrated by the 3D structures in section 3.3.

0.6 0.4 0.2

0.4 0.2

0

5 Time (s)

0

10

4

4

3

3

PDF (%)

PDF (%)

0

0.6

2 1 0

0

0.5

0

x 10

2

x 10

1.5

PSD

PSD

1

5

3 2 1

1 0.5

0

1

2

3

Frequency (Hz)

Air-water Figure 3:

0.5

Void Fraction

5

0

10

1

Void Fraction 4

5 Time (s)

2

0

1

0

4

0

0

1

2

3

4

Frequency (Hz)

air-silicone oil

Effect of fluid properties on flow pattern transition, liquid superficial velocity 0.2 m/s and gas superficial velocity 0.94 m/s.

Due to the random distortion and fluctuations of Taylor bubbles, in practice, it is usually more convenient to study its behaviour in terms of average parameters such as the frequency. Looking at Figure 3, it can be observed that the frequency of Taylor bubbles is also affected by the change in physical properties of the liquid phase, for a particular condition the structure frequency increases when the viscosity is increased as the dominant frequency predicted with PSD is higher for the case of silicone oil liquid. Also, the liquid film in the Taylor bubble is in



361

general thinner for the case of the air-water mixture. This is in agreement with Goldsmith and Mason [18] who found a similar behaviour. Figure 4 shows that, the change of liquid from water to silicone oil, which means an increase in the Morton number, keeping the liquid superficial velocity constant produces a displacement of the bubbly-cap bubble boundary towards a higher gas superficial velocity. This behaviour was observed to be evident at higher liquid superficial velocities (0.7 m/s). This is similar to the findings of Weisman et al. [1] in terms of the flow pattern boundaries being affected by the physical properties of the fluids and can be thought of as an indication of a longer development distance required by a more viscous flow. Most of the PDFs in Figure 4 show a shape that corresponds to the cap bubble flow pattern. Usg=0.05 m/s

10

0.15 m/s

0.28 m/s

0.47 m/s

0.94 m/s

1.4 m/s

Bubbly flow

Cap bubble

Cap bubble

Cap bubble

Churn flow

Cap bubble

Taylor bubble

Taylor bubble

8

Bubbly flow

6 4

PDF (%)

2 0

10

10

Cap bubble

5

0

0

0.5

Cap bubble

1

0

0.5

Cap bubble

1

0

0.5

5

1

0

0

0.5

1

0

0.5

1

0

0.5

1

Void Fraction

Figure 4:

Probability density function of the time series of the cross-section average void fraction at liquid superficial velocity 0.7 m/s. Lower row air-water, upper row air-silicone oil.

3.3 Shape of 3D structures The detection of the sharp gas-liquid interfaces plays an important role in identification of flow patterns. The wire mesh sensors employed in this study are capable of providing the Taylor bubbles shapes by creating a 3D reconstruction of the flow as shown in Figure 5, where the Taylor bubbles have been brought out for two cases. A time interval of 2 s has been used in both figures 5 and 6. The main features observed in Figure 5 are asymmetry and distortion of the Taylor bubbles, the distortion of bubbles appears to be higher in the air-silicone oil mixture, which can be due to the lower surface tension of the silicone oil, as the surface tension acts as the force that restrains deformation, despite its higher viscosity, as the main effect of viscosity on the displacement of a bubble is the production of a drag force that tends to slow down its relative motion in the surrounding fluid. From the 3D representation in Figure 5 it is clear that the physical properties greatly influence the shape of the structures. For the conditions used in this work the typical Taylor bubble shape is rarely observed for the case of silicone oil.



air-water Figure 5:

air-silicone oil

Comparison of Taylor bubble shapes in 3D. Superficial velocities: liquid 0.7 and gas 1.4 m/s.

Both the increase of viscosity and the decrease of surface tension in silicone oil with respect to water contribute to the increase of the Morton number. Therefore the more deformation observed for the air-silicone oil is in agreement with Duineveld [19], who observed that for water, low Morton number, the bubbles rising have relatively low deformation. The deformation of a Taylor bubble is also related to the stresses generated from the translation movement; as a result, as the mixture velocity increases, it can be observed in Figure 6 that the Taylor bubble is totally broken when the gas superficial velocity reaches 0.94 m/s. This phase interaction mechanism might be the reason why, as reported by Azzopardi et al. [13], liquid structures inside the gas core of the Taylor bubble (Wisps) have been found to exist in the churn flow regime. The deformation due to the high mixture velocity is also related to the high level of turbulence intensity in the flow. The flow conditions employed in the present work, involve large Reynolds numbers. Indeed the classical bullet shape Taylor bubble is rarely observed at the conditions of liquid superficial velocity 0.7 m/s for both water and silicone oil. It is generally accepted that turbulence enhances bubble breakup, however currently turbulence remains an



363

unsolved problem although different turbulence modelling approaches have emerged. Bubble distortion increases not only with the Morton number but also with Eotvos number, which involves the pipe diameter. This is congruent with the fact that Taylor bubble distortion is proportional to pipe diameter, as it has been reported in the literature.

Usg= 0.15 Figure 6:

0.28

0.47

0.94 m/s

Effect of mixture velocity on Taylor bubble shape, air-water at liquid superficial velocity 0.7 m/s and different gas superficial velocities.

3.4 Translational velocity The translational velocity, which is an essential parameter to characterise Taylor bubbles, has been obtained by cross correlating a pair of signals delivered by the capacitance probes for the case of air-water and ECT for air-silicone oil. Some of these results are plotted in Figure 7. They show a higher translational velocity for the case of the silicone oil. The higher translational velocity is related to the higher frequency shown in the PSD plot in Figure 3 and the higher overall liquid holdup in the pipe for the air-silicone oil. This finding is in agreement with Van Hout et al. [8], who found that the drift velocity for continuous slug flow is enhanced by the dispersed bubbles in the liquid slug body. Similarly, Hills and Darton [20] found considerable enhancement in the velocity of large bubbles in bubble swarms. Based on observations of Hills [21] who found that when a cap bubble rises in a swarm of small bubbles, the small bubbles never coalesce with the nose of the cap bubble, they suggested that the enhancement is due to the shape change in the cap bubble. However, Hills’ arrangement consisted of a twodimensional bubble column of width so much larger than the cap bubble. Therefore the small bubbles in most cases manage to escape from the cap bubble driven by the liquid velocity field. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


Figure 7:

4

Effect of physical properties on translational velocity. Liquid superficial velocity 0.2 m/s.

Conclusion

A study of the effect of the change of fluid properties on Taylor bubble behaviour has been carried out with the use of advanced instrumentation, and a clear effect has been observed. For a higher Morton number, which corresponds to the silicone oil, there is more distortion. Other flow features such as the liquid film and structure frequency are also affected. This comparison shows a remarkable effect of the physical properties on the flow pattern. These results can be used to validate the numerical modelling of Taylor bubbles, which is increasingly gaining popularity, as computational fluid dynamics codes are becoming more widespread.

Acknowledgements This work has been undertaken within the Joint Project on Transient Multiphase Flows and Flow Assurance. The Authors wish to acknowledge the contributions made to this project by the UK Engineering and Physical Sciences Research Council (EPSRC) and the following: - Advantica; BP Exploration; CD-adapco; Chevron; ConocoPhillips; ENI; ExxonMobil; FEESA; IFP; Institutt for Energiteknikk; PDVSA (INTERVEP); Petrobras; PETRONAS; Scandpower PT; Shell; SINTEF; StatoilHydro and TOTAL. The Authors wish to express their sincere gratitude for this support. V. Hernández-Pérez was supported by Mexican Council for Science and Technology (CONACyT) with a PhD scholarship. L. Abdulkareem would like to thank The Kurdish Government for supporting his PhD study.



365

References [1] Weisman, J., Duncan, D., Gibson, J. and Crawford, T., Effect of fluid properties and pipe diameter on two-phase flow pattern in horizontal lines. Int. J. Multiphase Flow Vol. 5, pp. 437–462, 1979. [2] Nädler, M. & Mewes, D., Effects of the liquid viscosity on the phase distributions in horizontal gas-liquid slug flow, Int. J. Multiphase Flow, Vol. 21, pp. 253-266, 1995. [3] Davies, R.M. & Taylor, G.I., The mechanics of large bubbles rising through liquids in tubes. Proc. R. Soc. Lond. Vol. A200, pp. 375–390, 1950. [4] Moore, D.W., The rise of a gas bubble in a viscous liquid. J. Fluid Mech. Vol. 6, pp 113-130, 1959. [5] Tomiyama, A., Celata, G.P., Hosokawa, S. & Yoshida S., Terminal velocity of single bubbles in surface tension force dominant regime, Int. J. Multiphase Flow Vol. 28, pp. 1497-1519, 2002. [6] Nicklin, O.J., Wilkes, J.O., & Davison, J. F., Two-phase flow in vertical tubes, Trans. I. Chem. Engrs. Vol. 40, pp. 61-68, 1962. [7] White, E.R. & Beardmore, R.H., The velocity of rise of single cylindrical air bubbles through liquids in vertical tubes, Chem. Eng. Sci. Vol. 17, pp. 351-361, 1962. [8] Van Hout, R., Barnea, D., & Shemer, L., Translational velocities of elongated bubbles in continuous slug flow, Int. J. Multiphase Flow Vol. 28, pp. 1333-1350, 2002. [9] Hassan, Y.A., Schmidl, W. & Ortiz-Villafuerte, J., Investigation of threedimensional two-phase flow structure in a bubbly pipe flow, Meas. Sci. Technol. Vol. 9, pp. 309–326, 1998. [10] Clarke, A. & Issa, R. I., A numerical model of slug flow in vertical tubes”, Computers & Fluids, Vol. 26, pp. 395-415, 1997. [11] Taha, T. & Cui, Z.F., CFD modelling of slug flow in vertical tubes, Chem. Eng. Sci. Vol. 61, pp. 676-687, 2006. [12] Hernández-Pérez, V., Azzopardi, B.J. & Morvan, H., Slug flow in inclined pipes. 6th Int. Conf. Multiphase Flow, Leipzig, Germany 5 to 9 July 2007. [13] Azzopardi, B.J., Hernández-Pérez, V., Kaji, R., Da Silva M.J., Beyer, M., & Hampel, U., Wire mesh sensor studies in a vertical pipe, HEAT 2008, Fifth International Conference on Transport Phenomena in Multiphase Systems, Bialystok, Poland, 2008. [14] Prasser, H.-M., Bottger, A., & Zschau J., A new electrode-mesh tomograph for gas-liquid flows, Flow Meas. Instr. Vol. 9, pp. 111-119, 1998. [15] Da Silva, M.J., Schleicher E. & Hampel, U., Capacitance wire-mesh sensor for fast measurement of phase fraction distributions. Meas. Sci. Tech. Vol. 18, pp. 2245-2251, 2007. [16] Abdulkareem, L.A., Hernández-Pérez, V., Azzopardi, B.J., Sharaf, S., Thiele, S. &Da Silva, M., Comparison of different tools to study gas-liquid flow. ExHFT-7, 28 June – 03 July 2009, Krakow, Poland, (Submitted) [17] Bhaga, D. & Weber, M. E., Bubbles in viscous liquids: shapes, wakes and velocities. J. Fluid Mech. 105, pp 61-85, 1981. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

366 Computational Methods in Multiphase Flow V [18] Goldsmith, H. L. & Mason, S. G., The movement of single large bubbles in closed vertical tubes, J. Fluid Mech. Vol. 14, pp 42-58, 1962., [19] Duineveld, P.C., The rise velocity and shape of bubbles in pure water at high Reynolds number, J. Fluid Mech. Vol. 292, pp. 325-332, 1995. [20] Hills, J.H. & Darton, R.C., The rising velocity of a large bubble in a bubble swarm, Trans. I. Chem. Engrs. Vol. 54, pp. 258-264, 1976. [21] Hills, J.H., The rise of a large bubble through a swarm of smaller ones, Trans. I. Chem. Engrs. Vol. 53, pp. 224-233, 1975.



367

Numerical simulation of one-dimensional mixed flow with air/water interaction F. Kerger1,2, S. Erpicum1, P. Archambeau1, B. J. Dewals1,2 & M. Pirotton1 1

Laboratory of Hydrology, Applied Hydrodynamics and Hydraulic Constructions (HACH), Liège University, Belgium 2 Belgian Fund for Scientific Research F.R.S-FNRS, Belgium

Abstract An original one-dimensional unified numerical model dealing with aerated mixed flow, characterized by a simultaneous occurrence of free-surface and pressurized flow, is derived and applied to the case of a gallery. The mathematical model is based on a new integration of the Homogeneous Equilibrium Model (HEM) over the cross section of a free-surface flow and consists of a simple set of equations analogous to the Saint-Venant equations. In addition, both free-surface and pressurized flows are mathematically modeled by means of the free-surface set of equations (Preissmann slot model). The original concept of the negative Preissmann slot is proposed to simulate sub-atmospheric pressure. This model is shown to be particularly well suited for the simulation of bubbly and intermittent flows. Keywords: hydraulics, Preissmann slot, homogeneous equilibrium model.

1

Introduction

Mixed flows, characterized by the simultaneous occurrence of free-surface and pressurized flows, are frequently encountered in river networks, sewer systems, storm-water storage pipes, flushing galleries, bottom outlets,… As a matter of fact, some hydraulic structures are designed to combine free-surface and pressurized sections (e.g. water intakes). In addition, dynamic pipe filling bores may occur in hydraulic structures designed only for conveying free-surface flow under an extreme water inflow or upon starting a pump. During such a transition, highly transient phenomena appear and may cause structural damages to the WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090311

368 Computational Methods in Multiphase Flow V system [1], generate geysers through vertical shafts [2], engender flooding,… What is more, air/water interactions may arise, particularly at the transition bore [3], and thoroughly alter the flow regime and its characteristics. On account of the range of applications affected by mixed flows, a good prediction of mixed flow features is an industrial necessity. Numerical simulation of mixed flow remains, however, challenging for two main reasons. Dissimilarity in the pressure term arises between the classical sets of equations describing free-surface and pressurized flows. Air/water interaction has to be taken into account through a two-phase flow model. Different mathematical approaches to describe mixed flows have been developed to date. Firstly, the so-called shock-tracking approach consists of solving separately free-surface and pressurized flows through different sets of equations [4, 5]. Secondly, the Rigid Water Column Approach [6] treats each phase (air/water) separately on the basis of a specific set of equations. The latter approach succeeds in simulating complex configurations of the transition. However, using this method for practical application is not possible because of the complexity and specificity of the algorithm. Thirdly, the so-called shockcapturing approach is a family of methods that computes pressurized and free-surface flows by using a single set of equations [7–9]. In this paper, a shock-capturing approach is used, based on the widespread model of the Preissmann slot [7]. Free-surface flow and pressurized flow are in this way equally solved through a free surface set of equations. An original concept developed by the authors, the negative Preissmann slot, extends the Preissmann slot model to simulate sub-atmospheric pressurized flows. Computing air-water interaction requires using a two-phase flow model. On the one hand, to the authors’ knowledge, no mixed flow model takes into account the effect of entrained air in the water flow. Only the air phase pressurization is usually modeled, as in the Rigid Water Column [6] and in the shock-capturing model of Vasconcelos et al. [8]. On the other hand, the usual multiphase flow investigations focus mainly on fully pressurized flow in small diameter pipes for chemical and mechanical engineering applications. There have been only a few attempts, often based on a transport equation [10], to simulate air entrainment in large hydraulic structures. Consequently, the current research aims at applying the classical model for multiphase flow to civil engineering applications. In this paper, an Homogeneous Equilibrium Model (HEM) coupled with the Preissmann slot model is derived by using the time and area averaging methods proposed by Ishii and Hibiki [11]. These developments have been carefully implemented in the one-dimensional module of the software package WOLF. WOLF is finite volume flow simulation modeling system developed within the Laboratory of Hydrology, Applied Hydrodynamics and Hydraulic Constructions (HACH) of the University of Liege. Application to this new model to the case of flows in a gallery is presented in this paper. Experimental results from a physical model build in the Laboratory of Structures Hydraulics of the University of Liege are used for comparison with numerical results. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


2

369

Mathematical model

2.1 Homogeneous equilibrium model for free-surface flow Homogenous flow theory provides the simplest technique for analyzing multiphase flows. Using suitable averaged properties, the fluid is treated as a pseudo-fluid that obeys the usual equations of a single-component flow. Under the assumption of equality between air and water properties, one obtains the Homogeneous Equilibrium Model (HEM) as demonstrated by Ishii and Hibiki [11]. This assumption is particularly suited for dispersed bubbly flow. The model is commonly used for the simulation of heat exchangers [12, 13], two-phase flow in ducts [14],… 2.1.1 3D Time-averaged governing equations HEM may be considered as a simplification of the Drift Flux Model [11] if the drift or diffusion of mass is negligible regarding the continuity equation for the gas phase. Thus, HEM 3D model is derived through the time averaging of the Local Instant Formulation for multiphase flow, the introduction of suitable mixture variables and the assumption of equilibrium between phases. For further details, we refer the interested reader to the classical book of Ishii and Hibiki [11]. The resulting equations write:  m  continuity equation   t    m v m   0    g    g v m   g  diffusion equation   (1)  t  m v m    m v m v m   p m    m  T   m g  M m   t    momentum equation  

where ρm[kg/m³] is the mixture density, vm[ms-1] is the mixture velocity vector (under the assumption of velocity equilibrium, vm= vwater= vair), g[-] is the air void fraction, g[s-1] the phase change volume generation, pm[Nm-2] is the mixture pressure, m[Nm-2] and T[Nm-2] are the viscous and turbulent stress tensors, g[ms-2] is the gravity and Mm[kgs-2m-2] is the interfacial momentum source. Closure of eqn (1) requires the definition of the mixture variables and a constitutive equation. Air and water are supposed to be incompressible Newtonian fluid, and the mixture properties write then: m  g g  1   g  w  1  g  w (2) T m   g g  1  g   w  .v   .v  At this point, no assumption is needed for the constitutive equations of the turbulent stress T, the phase change volume generation g, the pressure distribution pm and the mixture momentum source Mm.






370 Computational Methods in Multiphase Flow V 2.1.2 Area average of three-dimensional homogeneous equilibrium model 1D HEM equations are obtained by area-averaging eqns (1) and (2) over the cross section of the flow. The originality of the following development is to consider a free-surface flow. It is indeed shown in section 3 how we use the freesurface set of equations to simulate pressurized flow as well. The whole process of integration is beyond the scope of this paper. It is performed by analogy to the integration of the Saint-Venant equation for pure water flow, as exposed in [15]. The flow is assumed to flow in the x-direction. Successive integration over the flow width (y-abscissa) and the flow depth (z-abscissa) results in a set of conservative hyperbolic partial differential equations that describe the flow below a free-surface. A non-dimensional analysis of eqn (1) leads to the following expression for the mechanical constitutive equation: z

p(x, z, t) 

 

m





g cos dz  1   w g  z  h b 

 hb

(3)

where hb is the distance between the z-abscissa and the bottom of the flow area. Eqn (3) simply states that the pressure distribution over the flow area is hydrostatic. 1D HEM for free-surface flow writes at the end of the day:            u   m   g       1        (4)  1   um  0       x t       2    Z   1   um    SF   gp w  1   u m    g 1       x      x 

   

where p w 

   

hs

 h

s

 

 z  l  z  dz , [m²] is the flow cross-section area, um is the

 hb

flow velocity, Z[m] is the free surface elevation, SF[-] is the friction slope (resulting from the integration of the viscous, turbulent shear stress and the interfacial momentum source). The area-average of a given function f is defined as:

1 f  x, t   

h f lg



f  x, y, z, t  dydz

(5)

 hb ld

As pointed in [13], two-phase friction slope SF may be computed using the Darcy-Weisbach relation. The Colebrook relation for the friction factor still holds if a suitable “virtual viscosity” for the mixture is computed:  k fu 2 1 2.51   2 log   S F  m with    f 2 Dh  3.7 Dh Re f  (6) 1    wum Dh Re  with m  1     w m with Dh[m] the hydraulic diameter of the cross-section, k[m] is the roughness height, Re[-] the Reynolds Number and µw[kgm-1s-1] the dynamic viscosity. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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To close the partial differential system given by eqn (4), we still need to give an expression for the phase change volume generation g. Literature is abundant for empirical relations. To keep the generality of the model, a very fundamental relation given in [10] for air entrainment in free-surface flow is used:



 g  m    eq



(7)

where  and g are constants calibrated with experimental results. The onset of air entrainment is controlled by the parameter m=1 or m=0. Discretization of eqns (4)-(7) is performed by means of a finite volume scheme with an original flux vector splitting [16]. Time discretization is based on an explicit Runge-Kutta scheme to enhance the convergence in steady state applications. 2.2 Preissmann slot model

Pressurized flows are commonly described through the Allievi equations [17]. According to the Preissmann slot model [7], pressurized flow can be equally calculated through the free-surface equations by adding a conceptual slot at the top of a closed pipe (Figure 1b). When the water elevation is above the pipe crown, it provides a conceptual free-surface flow, of which the gravity wavespeed is given by c  g  Ts (Ts is the slot width). Strictly speaking, the pressure wave celerity of a flow in a full pipe, referred by a[m/s], depends on the properties of the fluid, the pipe, and its means of support. In first approximation, its value is not dependant of the pressure value and may be computed on the basis of solid mechanics relations [17]. It is then easy to choose a slot width Ts which equalizes the gravity wavespeed c to the water hammer wavespeed a: g dp (8) Ts  with a²  A a² d  A  From a hydraulic point of view, all the relevant information is summarized in the relation linking the water height and the flow area (H-A). A specific relation corresponds to each geometry of the cross section (Figure 1a). Adding the Preissmann slot leads to linearly extend the relation beyond the pipe crown head. In order to simulate pressurized flows with a piezometric head below the pipe crown, the authors propose a new concept, called negative Preissmann slot. It consists in extending the Preissmann straight line for water height below the pipe crown (Figure 1c). Two values of the flow area correspond to each water level below the pipe crown: one for the free surface flow and one for the pressurized flow. The choice between the two relations is done according to the local aeration conditions. For steady flow applications, the choice of the slot width may be arbitrary. On the one hand, the wave celerity does not affect the steady state of a flow. On the other hand, explicit numerical schemes are characterized by a time step t that is limited by a CFL condition of the form x (9) NbC  1 with NbC  max  u m  c  * t WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


Water level

c. Sub-atmospheric pressurized flow

Ts  gA

a2

Slot accounts for the radial flow decrease and the water dilatation

Ts  gA

0.50

0.25

0.00 0.00

0.50

1.00

1.50

2.00

1.00

0.75

0.50

0.25

0.00 0.00

Non-dimensional water height (-)

Figure 1:

0.50

1.00

1.50

2.00


1.25

a2

Sub-atmospheric Pressurized flow pressurized flow

1.00

0.75

pipe crown height

0.75

Pressurized flow

pipe crown height

1.00

Free-surface flow

Non-dimensional flow area (-)

1.25

Free-surface flow

pipe crown height


1.25

Relation flow area/water height

b. Pressurized flow Slot accounts for the radial flow increase and the water compressibility


Preissmann slot model (incl. negative slot)

a. Free surface flow

0.50

0.25

0.00 0.00

0.50

1.00

Choice based on aeration conditions

1.50

2.00


The Preissmann slot method under different flow conditions.

Alimentation basin

Natural river bed

a. Sketch of the experimental model

b. View of the physical model

Inlet

Upstream tank

Circular pipe Downstream tank

Upstream tank Outlet

Natural river bed

Natural river bed Downstream tank Inlet

Radial gate

Alimentation basin

Circular pipe Outlet

Figure 2:

Natural river bed

Description of the experimental setup.

It seems then reasonable to impose a wider slot than the width calculated with eqn (4) in order to decrease the computation time.

3

Steady flow application

This section outlines the application of the one-dimensional HEM solver for simulating stationary mixed flows taking place in a gallery. Numerical results are compared with experimental results provided by experimental investigations carried out in the Laboratory of Structures Hydraulics (HACH) of the University of Liege. The model (figure 2) includes a Plexiglas circular pipe linking two tanks. Topography of the upstream and downstream tanks has been built WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


373

regarding realistic in-situ natural conditions. The gallery inlet and outlet structures are also represented. Experimental apparatus, measurement systems and results are described in details in [18]. 3.1 Experimental investigations

Investigations focus mainly on stationary flows and aims at determining the flow discharge through the gallery as a function of the upstream pressure head. Strong air/water interactions may alter the flow behavior. In particular, the flow discharge through the gallery is strongly influenced by air/water interaction, and consequently depends of the aeration rate as well. Various two-phase flow patterns are observed according to the flow discharge through the gallery. Figure 4 shows the experimental relation between the flow discharge and the upstream pressure head (zero level is set at the upstream reservoir bottom level). The curve defines 5 areas corresponding to the 5 flow patterns (Figure 3) traditionally mentioned in the literature [13]: 1. A pure water fully free surface flow or smooth stratified flow. 2. A wavy stratified flow. 3. An intermittent flow that includes slug flow as well as plug flow. 4. A bubbly flow. 5. A pure water pressurized flow.


90 80

Water Flow

Smooth Stratified flow


70

Bubbly flow

60 50


Water Flow

Wavy Stratified flow

Water Flow

Slug flow Intermittent

Water Flow

Plug flow Intermittent

Water Flow

Bubbly flow

Wate r Flow

Pressurized flow

40


30 20 10


0

Upstream reservoir bottom level 5

Figure 3:

15

25

35


45

55

Experimental relation (upstream pressure head-flow discharge).

3.2 Pure water simulation

Simulations are performed under the assumption of a pure water flow (void fraction is equal to zero). They use a discretization step x=3.33cm, a CFL number limited to 0.5 and a roughness height k = 2.10-5m. The flow discharge varies between 5l/s and 55l/s. Results provide new insight into the flow behavior. A first head/discharge relation (dotted line in figure 4) is computed with the HEM model and assuming a free surface in each mesh if the water height is below the pipe crown (air phase above the free surface is at atmospheric pressure). The second head/discharge relation (continuous line) is computed by activating the negative Preissmann slot (sub-atmospheric pressurized flow). WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)



90 80


70

Bubbly flow

60 50


40


30


20


10

Numerical (Sub‐Atmospheric pressure) Upstream reservoir bottom level

0 5

15

25

35


45

55

Computed flow discharge relation for pure water simulations.

Figure 4:

0,3

Topography

a. Pressure profile for a discharge of 9.5 l/s

0,2


Altitude (cm)

Pipe crown

0,1 0,0

-0,1 -0,2 -0,3

Experimental Pressure head/Free surface level

-0,4

1,0

Topography

Abscissa (m)

b. Pressure profile for a discharge of 48.4 l/s

0,8


Altitude (cm)

Pipe crown

0,6 0,4 0,2 0,0 -0,2


Figure 5:

-0,4

Abscissa(m)

Computed total head and pressure head distribution for a smooth stratified flow and a pressurized flow.

Numerical results are in good accordance with experimental data for smooth stratified flows and fully pressurized flows. Bubbly and intermittent flows show a similar behavior two the sub-atmospheric pressurized flows. A periodic instability between to unstable steady flow regimes occurs in the area of wavy stratified flows. The instability induces large period (10s to 60s) oscillation of the water level in the upstream reservoir. The amplitude of the oscillation reaches 2cm. Consequently, very large pressure oscillations (up to 4cm) are observed in the gallery. For further details over this regime, we refer the interested reader to the paper of Erpicum et al. [18]. Experimental and numerical data for the distribution of the total head and the pressure head (water level for free surface flow) along the gallery length are given in figure 5 for a smooth stratified flow (discharge of 9.5l/s) and a fully WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


375

pressurized flow (discharge of 48.4l/s). In the latter case, results are in full agreement. In the former case, a slight discrepancy is observed in the total head curve. It results from the effect of the air phase flowing above the free surface that is not taken into account in the computation. 0,5

Topography

0,4

Pipe crown


0,3

Altitude (cm)


a. Pressure profile for a discharge of 38.4 l/s : Sub-atmospheric pressurized flow

0,2 0,1 0,0 -0,1 -0,2 -0,3


0,6

Topography

0,5


Figure 6:

Altitude (cm)


b. Pressure profile for a discharge of 38.4 l/s : Free surface flow

0,4

Pipe crown Numerical total head

Abscissa(m)

-0,4

0,3 0,2 0,1 0,0 -0,1 -0,2 -0,3 -0,4

Abscissa(m)

Computed total head and pressure head distribution for an intermittent flow sub-atmospheric pressurized flow and free-surface flow computation.

A comparison of the results given by the computation for an intermittent flow of 38.4l/s discharge is shown in figure 6. Pressure distribution along the gallery is computed in figure 6b under the assumption of a free surface flow. Large discrepancies of the results are observed. The upstream pressure head is overestimated. In figure 6a, activation of the negative Preissmann slot gives the curve corresponding to a pressurized flow. We consequently identify a large area of sub-atmospheric pressure in the upstream part of the pipe. Results are now in better accordance and it has been concluded that the aeration rate of the pipe is not sufficient to induce the apparition of a free surface flow. However, some differences still remain. 3.3 Air-water mixture simulation

Application of the Homogeneous Equilibrium Model enables to overcome the results discrepancy observed in section 3.2 for bubbly and intermittent flows (figure 4 and 6). The effect of the entrained air on the water flow is accurately computed by using the eqn (7) for the phase change volume generation g. The parameter  is set at 25 and g is calibrated according to the flow pattern observed. For bubbly flows, as bubbles arise from the air dissolved in water, WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

376 Computational Methods in Multiphase Flow V equilibrium void fraction is chosen between 0.5% and 2%. For intermittent flows, an additional air supply is provided through a vertical vortex appearing at the water intake. Equilibrium void fraction is then chosen between 2% and 4.5%. Figure 7a shows a comparison between experimental and numerical data for the relation between the flow discharge through the gallery and the upstream pressure head. Taking into account air/water interaction in the computation obviously gives more accurate results for bubbly and intermittent flows. The void fraction relation corresponding to this new relation is given in Figure 7b. A comparison between experimental data and numerical results computed with the HEM scheme is drawn on figure 8. Computation is performed with a flow discharge of 38.4l/s and a void fraction of 4.5%. Results are in full agreement.

Air Void Fraction [%]


50


40


30


20


10

Upstream reservoir bottom level 5

15

25

4 3 2 1

Numerical (Sub‐Atmospheric pressure and void fraction variable)

0

5


60

Bubbly flow

Bubbly flow


6


70


7

80

Intermittent flow

90

0 35


45

a. Head/Discharge relation computed with variable void fraction

55

5

15

25

35


45

55

b. Void fraction distribution in bubbly and intermittent flows

Results of air-water mixture simulation.

Figure 7: 0.5

Topography

0.4

Pipe crown


Altitude (cm)


0.3 0.2 0.1 0.0

-0.1 -0.2 -0.3


Figure 8:

4

-0.4

Abscissa(m)

Computed total head and pressure head distribution for a bubbly (flow discharge of 38.4 l/s and void fraction of 4.5%).

Conclusion

The original mathematical model derived in this paper is a first step towards a completely unified model for the simulation of highly transient mixed flow in multi-scale hydraulic structures. Thanks to the Preissmann slot method, both free-surface and pressurized flow are calculated through the free-surface set of equation by adding a narrow slot at the top of the pressurized sections. In WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


377

addition, an original negative Preissmann slot has been added to simulate subatmospheric pressure. Area-integration of the Homogeneous Equilibrium Model (HEM) over the cross section give a simple set of equations, analogous to the Saint-Venant equations, for analyzing air-water flows. This assumption has been shown to be particularly well-suited for the simulation of bubbly and intermittent flows. The fundamental concepts introduced in the previous pages pave the way for further research. Experimental research is required to develop appropriate source terms as phase change volume generation and interfacial momentum source. Development of a stratified air/water model would give us insight into wavy stratified flows. All results should be then easily extended to multidimensional problems.

References [1] Zhou, F., F.E. Hicks, and P.M. Steffler, Transient Flow in a Rapidly Filling Horizontal Pipe Containing Trapped Air, Journal of Hydraulic Engineering, 128(6): p. 625-634, 2002 [2] Guo, Q. and C. Song, Dropshaft Hydrodynamics under Transient Conditions, Journal of hydraulic Engineering, 117(8): p. 1042-1055, 1991 [3] Vasconcelos, J. and S. Wright, Experimental Investigation of Surges in a Stormwater Storage Tunnel, Journal of hydraulic Engineering, 131(10): p. 853-861, 2005 [4] Cardle, J. and C. Song, Mathematical Modeling of Unsteady Flow in Storm Sewers, International Journal of Engineering Fluid Mechanics, 1(4): p. 495-518, 1988 [5] Politano, M., A.J. Odgaard, and W. Klecan, Numerical Evaluation of Hydraulic Transients in a Combined Sewer Overflow Tunnel System, Journal of Hydraulic Research, 133(10): p. 1103-1110, 2007 [6] Li, J. and A. McCorquodale, Modeling Mixed Flow in Storm Sewers, Journal of hydraulic Engineering, 125(11): p. 1170-1180, 1999 [7] Preissmann, A. Propagation des intumescences dans les canaux et rivieres. in First Congress of the French Association for Computation, Grenoble, France, 1961 [8] Vasconcelos, J., S. Wright, and P.L. Roe, Improved Simulation of Flow Regime Transition in Sewers : The Two-Component Pressure Approach, Journal of Hydraulic Engineering, 132(6): p. 553-562, 2006 [9] Bourdarias, C. and S. Gerbi, A Finite Volume Scheme for a Model Coupling Free Surface and Pressurized Flows in Pipes, Journal of Computational and Applied Mathematics, 209: p. 109-131, 2007 [10] Dewals, B.J., S. Andre, M. Pirotton, and A. Schleiss. Quasi 2D-numerical model of aerated flow over stepped chutes. in 30th IAHR Congress, Greece, 2003 [11] Ishii, M. and T. Hibiki, Thermo-fluid dynamics of two-phase flow. First ed, ed. U. Springer Science: Springer Science, USA. 430, 2006


378 Computational Methods in Multiphase Flow V [12] Clerc, S., Numerical Simulation of the Homogeneous Equilibrium Model for Two-phase Flows, Journal of Computational Physics, 161: p. 354-375, 2000 [13] Wallis, G.B., One-dimensional Two-phase Flow, ed. M.-H.B. Company. 410, 1969 [14] Guinot, V., Godunov-type Schemes: An introduction for engineers, ed. E. Science. Amsterdam. 480, 2003 [15] Cunge, J.A., F.M. Holly, and A. Verwey, Practical Aspects of Computational River Hydraulics. [Monographs and surveys in water resources engineering], 3. Boston: Pitman Advanced Pub. Program. 1980 [16] Dewals, B.J., S. Erpicum, P. Archambeau, S. Detrembleur, and M. Pirotton, Depth-Integrated Flow Modeling Taking into Account Bottom Curvature, Journal of Hydraulic Research, 44(6): p. 787-795, 2006 [17] Wylie, E.B. and V.L. Streeter, Fluid transients. Première ed, ed. M.-H. Inc., 385, 1978 [18] Erpicum, S., F. Kerger, P. Archambeau, B.J. Dewals, and M. Pirotton. Experimental and numerical investigation of mixed flow in the flushing gallery of a dam. in Multiphase Flow New Forest, 2008



379

Geometry effects on the interaction of two equal-sized drops in simple shear flow at finite Reynolds numbers S. Mortazavi & M. Bayareh Department of Mechanical Engineering, Isfahan University of Technology, Iran

Abstract The effect of geometry on the interaction of two equal-sized drops in shear flow is presented. The full Navier-Stokes equations are solved by a finite difference/front tracking method. The interaction of drops was studied at finite Reynolds numbers for viscosity ratio (λ) of one. The distance between drop centres along the velocity gradient direction (z) was measured as a function of time. The interaction of two drops contains approach, collision, and separation. Based on experimental data, we simulated different geometries by changing the offset and size of drops. It was found that ∆z increases after collision and reaches to a new steady-state value after separation. The values of ∆z, during three stages of interaction, increases with the increasing initial offset. To investigate the drop shape evolution, we calculated the deformation and the orientation angle formed by the drop major axis and horizontal direction. The deformation of the drops is maximum when the drops are pressed against each other and minimum when they are drawn a part. Our results show that the time of approaching of drops at low initial offset is greater than the other ones, but the maximum deformation is the same for equal drop sizes. The deformation decreases with the decreasing size of drops. As the initial offset increases, the drops rotate more quickly and the available contact time for film drainage decreases. We found that the trajectories of drops in the approaching stage are different owing to the different initial offsets. However, after the drops come into contact, it can be seen that they follow the same trajectories, similar to experimental results. Keywords: two-phase flow, front tracking, Reynolds number, Weber number, capillary number, offset, drainage time.



1

Introduction

Immiscible polymer blending plays a fundamental role in determining material properties of industrial interest. It is important to understand and control the size and size distribution of the dispersed drops because the properties of the blend depend on them. The final size distribution is determined by a balance between flow-induced break up and coalescence. While break-up involves a single drop and is not very affected by the presence of neighbouring drops in a blend (Leowenberg and Hinch [1], coalescence is the result of many-particle interaction process. The majority of numerical simulations are based on the interaction of two deformable drops in a shear flow, the drainage of the thin film between two colliding drops and the problems of coalescence of two deformable drops. Wang et al. [2] computed the coalescence of two undeformed spherical drops. Leowenberg and Hinch [3] presented the numerical simulations of the interaction between deformable drops based on boundary integral calculations. They showed that if capillary number is much smaller than one, the tendency for coalescence is greatest when drops are pressed against each other by the shear flow. Viscosity ratio effects on film drainage between interacting drops were studied by Bazhlekov et al. [4]. Cristini et al. [5] simulated the drop break-up and coalescence by an adaptive mesh algorithm. Effects of inertia on the rheology of a dilute emulsion of drops in shear flow are investigated by Zhao [6] using direct numerical simulation. The drop shape and flow are computed by solving the Navier-Stokes equations in two phases using front tracking method. On the other hand, most of experimental works are based on blending studies that analyses the drop size distribution of a blend or a concentrated emulsion. The collision of two equal-sized drops immersed in an immiscible liquid phase undergoing a shear flow in a parallel apparatus was investigated by Guido and Simeone [7] over a range of capillary numbers. Trajectories of a pair drops and their deformations were presented. The coalescence efficiency of two drops in a simple shear flow was also investigated by Mousa et al. [8]. The effect of viscosity ratio on the flow-induced coalescence of two equal-sized drops with clean interfaces was investigated by Yoon et al. [9]. Their studies showed that when the viscosity ratio is greater than O(0.1), the critical capillary number decreases with increasing offset only for the smallest offset. Zhao [10] investigated the drop break up in dilute Newtonian emulsions in simple shear flow by using high-speed microscopy over a wide range of viscosity ratio, focusing on high capillary number. He showed the final drop size distribution intimately links to the drop break up mechanism, which depends on viscosity ratio and capillary number. In this article, we present numerical simulation data describing the motion of a pair of drops under simple shear flow at finite Reynolds numbers. We consider the special case of drops with the same viscosity as the continues-phase fluid. Formulation and numerical method are described in §2, results are presented in §3, and concluding remarks are given in §4.



2

381

Formulation and numerical method

2.1 Formulation The governing equations for the motion of unsteady, viscous, incompressible, immiscible two-fluid systems are the Navier-Stokes equations in conservative form: ∂ρu (1) + ∇ . ρ uu = −∇ P + ∇ . µ ( ∇ u + ∇ u T ) + σ κ n δ β ( x − X ) d .

∫

∂t

S

Here u is the fluid velocity, p is the pressure, ρ is the fluid density, µ is the fluid viscosity, σ is the surface tension coefficient, g is the acceleration due to gravity. δ β is a two- or three-dimensional delta function (for β=2 and β=3) respectively. κ is the curvature for two-dimensional flows and twice the mean curvature for three-dimensional flows. n is a unit vector normal to the drop surface pointing outside of the drop. x is the position in Eulerian coordinate and X is the position of front in Lagrangian coordinate. Both of immiscible fluids are taken to be incompressible, so the divergence of velocity field is zero: (2) ∇ .u = 0 . Equations of state for the density and the viscosity are: (3) Dρ Dµ = 0, Dt

Dt

= 0.

Continuity of stresses at the fluid boundary shows that the normal stresses are balanced by surface tension. The force due to surface tension is (4) ∆ F = σ kn . Three governing non-dimensional numbers of the flow are the Reynolds number (bulk and particle Reynolds numbers), the Weber number and the capillary number. Only two of these non-dimensional numbers are independent (one Reynolds number and the Weber or capillary number): (5) ρ 0 GH 2 ρ 0 GR 2 ρ G R µ GR Re b =

µ0

,

Re P =

µ0

,

2

We =

0

σ

3

,

Ca =

0

σ

.

Here ρ 0 is the density of ambient fluid, µ 0 is the viscosity of the ambient fluid, R is the initial radius of the drop, H is the width of the channel and G is the shear rate. The shear rate is u − ub (6) G = t . H

where ut and ub are the velocity of top and bottom walls, respectively. It is usual to define a scalar measure of the drop deformation (the Taylor deformation) by: l−b (7) . D = l+b

where l, b are the major and minor semi-axes of the drop (defined by the largest and smallest distances of the surface from the centre). In addition, the collision or film drainage time is the time between the points where the centre-to-centre distance is equal to one undeformed drop diameter to the instant of coalescence. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

382 Computational Methods in Multiphase Flow V 2.2 Numerical method Various methods have been used to simulate the two-phase flows. These methods include the Marker-And-Cell (MAC) method, the Volume-Of-Fluid (VOF) method, and the level set method. In general, the interface representation can be explicit (moving mesh) or implicit (fixed mesh) or a combination of both. The front-tracking method is combination of fixed and moving mesh method. Although an interface grid tracks the interface, the flow is solved on a fixed grid. The interface conditions are satisfied by smoothing the interface discontinuities and interpolating interface forces from the interface grid to the fixed grid. In this method, the governing equations are solved separately for each fluid. Front capturing has two difficulties. The first is a sharp boundary between the fluids and the second is accurate computation of surface tension. Different attempts have been made in overcoming these problems. For the simulations presented here, the method developed by Unverdi and Tryggvason [11] is used. They simulated the motion of buoyant bubbles in a periodic domain. Eqns (1), (2), and (3) are solved in a rectangular threedimensional domain with a finite difference method. The spatial differentiation is calculated by a second order finite difference scheme on a staggered Eulerian grid. We use an explicit second-order time integration method. Combining the incompressibility condition and momentum equations results in a non-separable elliptic equation for the pressure. Due to the similarity in density between the drop and the ambient fluid, a quick poisson solver solves the pressure equation. The force due to surface tension on each element of front is (8) δ F = σκ nd . σ

∫

S

∆S

In three-dimensional flow, the average surface curvature is (9)

κn = (n × ∇ ) × n.

Then, the force on each element surface is δ Fσ = σ ∫ κ nd A = σ σA

∫ ( n × ∇ ) × nd

σA

A

= σ ∫ t × nd S .

(10)

S

The integration is over the boundary of each element representing the front. t and n are the tangent and the normal vector to each element, respectively.

3

Results

The reference system to describe the results is shown in fig. 1. According to experiments of Yoon et al. [9], initial offset is defined as the shortest distance from the centre of the drop to the inflow axis (∆) divided by drop radius R. The centre-to-centre distance between drops is 4R as shown in fig. 1. The coordinate axes are oriented as follows: the x-axis is parallel to flow direction, the y-axis is parallel to the vorticity direction, and the z-axis is parallel to the velocity gradient. The relative trajectory of the two drops will be expressed in terms of the differences ∆ z = z 2 − z 1 and ∆ x = x 2 − x1 , where x i and z i are the centre-ofWIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


383

mass coordinates of the ith drop. The difference ∆y between the y-coordinates of the two drop centres is zero. In all plots, ∆x and ∆z will be made dimensionless by the radius R of the undeformed drops as the characteristic length. We will compare our results with experimental results of Guido and Simeone [7] and Yoon et al. [9] and numerical results of Loewenberg and Hinch [3]. Sequences (1-4) show the interactions between two drops in simple shear flow (fig. 2). Initially, each drop has the steady shape under same flow conditions.

z y d

4R

x '

Į

Figure 1:

Schematic of the relative trajectory between a pair of deformable interacting drops in shear flow (Offset = ∆ / R).

1

Figure 2:

2

3

4

Sequences (1-4) showing the interaction between two drops in simple shear flow with Ca = 0.13, Offset = 0.512, λ = 1.

In fig. 3, ∆z is plotted as a function of ∆x during approach, collision and separation between two drops. The data correspond to sequence depicted in fig. 2. It can be seen that ∆z starts increasing after the drops come into apparent contact (∆x ~ -2R), reaches a maximum value, and, after separation, reaches a new steady-state value. The final value of ∆z (which is 1.4 for offset = 0. 2, 1.52 for offset = 0.512, and 1.72 for offset = 0.8) is greater than the value before collision. In the other words, if the drops were made to collide again by reversing the flow direction, ∆z increased further. So, the effect was irreversible, and repeated collisions lead to increasing values of ∆z until drop interaction became negligible (Guido and WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

384 Computational Methods in Multiphase Flow V Simeone [7]). This is also an agreement with the numerical simulations of Loewenberg and Hinch [3]. Experimental results of Guido and Simeone [7] was based on λ = 1.4 and numerical results of Loewenberg and Hinch [3] were presented for viscosity ratio of one. 2.2 2 1.8 1.6

Z/R

1.4

1.2 1 0.8

Offset = 0.2 Offset = 0.512 Offset = 0.8 Expriment (Guido & Simeone)

0.6 0.4 0.2

Figure 3:

0

∆X / R

5

10

Cross-flow separation (velocity gradient direction) versus ∆x / R between interacting drops with Ca = 0.13, λ = 1 (present work) and λ = 1.4 (experiment), and different offsets.

The deformation parameter is shown in fig. 4 as a function of dimensionless time with viscosity ratio of one and Ca = 0.3. Based on experimental observation of Guido and Simeone [7], deformation of two drops is the same. Deformation slightly increases, and then reaches a maximum, a minimum, a second maximum, and eventually reaches a steady state value from before the collision. Numerical simulations of Loewenberg and Hinch [3] show no difference before and after collision at low-Reynolds numbers. Comparison between the results shows that the time of approaching of drops at low initial offset is greater (for current simulations). Following Allan and Mason [12] and Guido and Simeone [7], the minimum value of deformation is lower than the steady-state value before (or after) collision. This can be explained as the result of two processes: (i) relaxation of drop shape once they leave the compressional axis and (ii) action of the surrounding fluid on the drops. At fixed initial offsets, as the size of drops increases, the deformation increases as shown in fig. 5. We see that the approach-collision-separation times of drops are different owing to the different initial sizes. The film drainage time increases with increasing size of drops. The dimensionless drainage time is 5.21 for D / H = 0.3 and 5.86 for D / H = 0.36. Fig. 6 shows the trajectories of the drops for different initial offsets with the same capillary number for a viscosity ratio of 1. As the initial offset increases, WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


385

the drops rotate more quickly and the available time for film drainage decreases as shown in fig. 6. Therefore, we should expect that the critical capillary number for coalescence will decrease with increasing offset. This was found in studies of Yang et al. [13], where the viscosity ratio was 0.096. 0.55 0.5 0.45

Deformation

0.4 0.35 0.3 0.25 0.2 0.15 0.1

Offset = 0.12 Offset = 0.2 Numerical simulations (Loewenberg & Hinch)

0.05 0

Figure 4:

10

20

t*

The deformation parameter as a function of dimensionless time between interacting drops with Ca = 0. 3, λ = 1 and different offsets. 0.2 0.18 0.16

Deformation

0.14 0.12 0.1 0.08 0.06 D / H = 0.3 D / H = 0.36

0.04 0.02 0

Figure 5:

5

t*

10

15

The deformation parameter as a function of dimensionless time between interacting drops with Ca = 0. 075, λ = 1 and different size of drops.

In fig. 7 we see that the approaching parts of the trajectories are different from each other owing to different initial offsets. However, after the drops come into WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

386 Computational Methods in Multiphase Flow V contact, it can be seen that they follow the same trajectories of separation distance versus orientation angle (α) formed by the drop major axis and horizontal direction. This is an agreement with the experimental results of Yoon et al. (2005). They said that they have no explanation for this result. 2.5

Distance, d / 2R

2

1.5

1

0.5

0

Offset = 0.08 Offset = 0.12 Expriment (Yosang Yoon et al.), Offset = 0.08 Expriment (Yosang Yoon et al.), Offset = 0.12

0

5

10

15

20

t*

Figure 6:

Trajectories of drops for different initial offsets, separation distance versus dimensionless time, with Ca = 0.00481, λ = 1 (present work), and λ = 1.2 (experiment). 2.5

Distance, d / 2R

2

1.5

1

0.5

0

Offset = 0.08 Offset = 0.12 Expriment (Yosang Yoon et al.), Offset = 0.08 Expriment (Yosang Yoon et al.), Offset = 0.12

0

25

50

75

100

Angle (deg.)

Figure 7:

Trajectories of drops for different initial offsets, separation distance versus orientation angle (α), with Ca = 0.00481, λ = 1 (present work), and λ = 1.2 (experiment).



4

387

Conclusion remarks

The effects of geometry include initial offset and size of drops on the interaction of two equal-sized drops in simple shear flow has been presented using finite difference/front tracking method. Simulations were studied at finite Reynolds numbers for viscosity ratio of one. The deformation, relative trajectories, and film drainage time were examined by changing the initial offset and size of drops. We changed offset and size of drops, Based on experimental data. It was found that ∆z increases after collision and reaches to a new steady-state value after separation. The values of ∆z, during the interaction, increases with the increasing initial offset. Our results showed that the time of approaching of drops at low initial offset is greater than the other ones, but the maximum of deformation is the same for equal drop sizes. The deformation decreases with the decreasing size of drops. As the initial offset increases, that time for film drainage decreases. Also, the approaching parts of the trajectories are different from each other owing to the different initial offsets. However, after the drops come into contact, it can be seen that they follow the same trajectories of separation distance versus orientation angle formed by the drop major axis and horizontal direction, similar to experimental results.

References [1] Loewenberg, M. & Hinch, E.J., Numerical simulation of a concentrated emulsion in shear flow. J. Fluid Mech., 321, pp. 395-419, 1996. [2] Wang, H., Zinchenko, A.Z. & Davis, R.H., The collision rate of small drops in linear flow-fields. J. Fluid Mech., 265, pp. 161-188, 1994. [3] Loewenberg, M. & Hinch, E.J., Collision of two deformable drops in shear flow. J. Fluid Mech., 338, pp. 299-315, 1997. [4] Bazhiekov, I.B., Chesters, A.K. & Van De Vosse, F.N., The effect of the dispersed to continuous-phase viscosity ratio on film drainage between interacting drops. Intl J. Multiphase Flow, 26, pp. 445-466, 2000. [5] Cristini, V., Blawzdziewicz, J. & Loewenberg, M., An adaptive mesh algorithm for evolving surface: simulation of drop break-up and coalescence. J. Comput. Phys. 168, pp. 445-463, 2001. [6] Zhao, X., Effects of inertia on the rheology of a dilute emulsion of drops in shear flow. J. Rheology, 49, pp. 1377-1394, 2005. [7] Guido, S. & Simeone, M., Binary collision of drops in simple shear flow by computer-assisted video optical microscopy. J. Fluid Mech., 357, pp. 1-20, 1998. [8] Moua, H., Agterof, W. & Mellema, J., Experimental investigation of the orthogenetic coalescence efficiency of droplets in simple shear flow. J. Colloid Interface Sci., 240, pp. 340-348, 2001. [9] Yoon, Y., Borrell, M., Park, C.C. & Leal L.G., Viscosity ratio effects on the coalescence of two equal-sized drops in a two-dimensional linear flow. J. Fluid Mech., 525, pp. 355-379, 2005. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

388 Computational Methods in Multiphase Flow V [10] Zhao, X., Drop break up in dilute Newtonian emulsions in simple shear flow: new drop break up mechanism. J. Rheology, 51, pp. 367-192, 2007. [11] Unverdi, S.O. & Tryggvason, G., Computations of multi-fluid flows. Phys. Fluids, D60, pp. 70-83, 1992. [12] Allan, R.S. & Mason, S.G., Particle motion in sheared suspensions. XIV. Coalescence of liquid drops in electric and shear fields. J. Colloid Interface Sci., 17, pp. 383-408, 1962. [13] Yang, H., Park, C.C, Hu, Y.T. & Leal, L.G., The coalescence of two equalsized drops in a two-dimensional linear flow. Phys. Fluids, 13, pp. 10871106, 2001.


Section 6 Flow in porous media



391

Modelling the tide effects in groundwater J. Mls Faculty of Science, Charles University, Czech Republic

Abstract It is well known that in aquifers in contact with the ocean, sinusoidal fluctuations of groundwater level occur in response to tides. Similar semidiurnal fluctuations have been observed in the piezometric head of confined aquifers without any contact with the ocean. These fluctuations are of interest to hydrogeologists and geophysicists as they indicate that the observed wells are sensitive indicators of the surrounding material deformations either by changes in the surface loading or by applied strain. The aim of the contribution is to model the pressure-head fluctuations in a leaky confined aquifer as a result of periodic changes of the load originating in the tidal changes of gravity acceleration. The Biot approach was used to derive the governing equations. The aquifer characteristics and the imposed boundary conditions were determined according to the known geology of the particular site where the water level fluctuations were observed. The problem was solved numerically and the obtained results were compared with the measured values. Keywords: groundwater tides, water level fluctuations, tide potential, tide acceleration, aquifer, solid phase compressibility, Biot’s approach.

1 Introduction According to Rojstaczer and Agnew [5], high porosity aquifers show high sensitivity to atmospheric loading while high sensitivity to the applied areal strains occurs for low porosity aquifers, and both increase with decreasing compressibility of the solid matrix. Involving an inertia term, Bodvarsson [2] generalized the Darcy law and gave a theory of strain-induced pressure fluctuations in a well-aquifer system. In his paper, he emphasized the necessity of the wells being connected with a sufficiently confined aquifer. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090331

392 Computational Methods in Multiphase Flow V Response of the water level in a well to Earth tides and atmospheric loading under unconfined conditions was studied by Rojstaczer and Riley [6]. The authors define physical properties of aquifers to exhibit a significant effect of water level fluctuation: high specific storage, high hydraulic conductivity, and low specific yield. Boreholes where measurable values of tide oscillation have been found indicate locations where the solid-phase tension of the geological layer does not match the tide forces. At such places, the pressure of the liquid phase takes over a part of the solid-phase tension. During hydrogeological investigation of the Police basin, situated in northeastern Bohemia, oscillations of water table were found in several boreholes (Krásný et al. [3]). Subsequent more detailed research checked this phenomenon with the effects of atmospheric pressure and groundwater storage due to recharge/discharge processes. The relation was confirmed in all but one of the cases. Water table of the borehole V-34 oscillated in its own way regardless of atmospheric pressure and the pumping process in its neighbourhood. According to the frequency of the water table changes, Krásný et al. [3] came to the conclusion that the observed phenomenon were tidal oscillations of groundwater hydraulic head. The knowledge of the geology of the investigated region and the knowledge of hydraulic characteristics of the aquifer and surrounding layers make it possible to formulate a simple model of the two phase system and numerically simulate the supposed process. To make this, the Biot [1] approach has been applied connecting the solid-phase elastic deformation with the pressure and flow of the liquid phase within the pore system.

Figure 1: Water level fluctuations indicated in boreholes VS-20 and V-34. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


393

Figure 2: Position of boreholes VS-20 and V-34 and of the faults.

2 Geological conditions Figure 1 shows a water level fluctuation as it was recorded in two boreholes, VS-20 and V-34, in the Police basin. Initially, it was explained by pumping from a water supply borehole lying about 1,5 km north of the measured boreholes. It was later realised that the borehole V-34 was separated from the pumping site by the Skalsk fault. Subsequent test confirmed the assumption that the fault built an impervious obstacle which did not allow for any propagation of hydraulic head changes. The borehole V-34 is situated in a narrow strip bounded by two faults, Skalsk fault and a shorter parallel one, see Figure 2. The borehole is open to a confined aquifer, the part of which, situated between the planes of the faults, is affected by tension originated in tide forces. Hydrogeology of the region, the Police basin, and hydraulic characteristics of particular layers are known from the study published by Krsn et al. [3]. This makes it possible to define the conceptual model of the investigated process: the domain, its parameters and the boundary conditions. The studied domain is the aquifer between the faults. The problem is supposed to be one dimensional defined in horizontal direction normal to the plane of the faults. The coordinate is oriented in the northwest direction and denoted x. The domain WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


Figure 3: North part of the sectional view AA . is then = (0, L), L being the distance of the faults. In Figure 3, A2 denotes the aquifer, A1 /A2 denotes the underlying layer and A/C denotes the overlying layer. The underlying layer is supposed to be an aquiclude and the overlying layer is supposed to be an aquitard. Hydraulic conductivities of these layers, specific storativity of the aquifer and pressure head in the aquifer lying above the layer A/C are known from (Krsn et al. [3]). The region north of the Skalsk fault is elevated about 100 m above the southern part of the basin. Hence, it is supposed that the part of the massif situated south of the domain , having smaller load, is easier deformable than the northern one. Consequently, it is supposed that the northern part of the basin is rigid and the modelled part of the aquifer is pushed against the plane of the fault.

3 Governing equations Part of the tension arising within geological layers originates in tide acceleration acting upon the mass of the layers. In most cases, the solid phase, due to its toughness, holds most of the tension. There are, however, places where the tide forces affect the liquid phase within the pore space in such a way that the corresponding pressure-head fluctuation is measurable. Biot [1] published his theory of consolidation which he considered as a process of squeezing water out of a deformable solid-phase body. The Biot theory connects laws of elastic deformation with laws of Darcian flow of fluid in porous media. In this way, it makes it possible to relate the solid-phase tension with liquid-phase pressure and to formulate governing equations of the investigate process. The initial Biot’s assumptions are ∂uj 1 σ 2p ∂ui ϑ δi,j + δi,j , + = τi,j − ∂xj ∂xi μ μ(1 + σ ) 3H and θ = θ0 +

p ϑ + . 3H R


i, j = 1, 2, 3,

(1)

(2)


395

Hence, taking into account the equilibrium conditions of a stress field ∂τi,j = 0, ∂xj

i = 1, 2, 3,

(3)

the resulting equations of the Biot theory are μ ui +

μ ∂p ∂

−α = 0, 1 − 2 σ ∂xi ∂xi

and ∂

α + ∂t

α 1 − R H

i = 1, 2, 3,

k ∂p = p, ∂t η

(4)

(5)

where x1 , x2 , x3 are the space coordinates, t is time, u = (u1 , u2 , u3 ) is the solidphase displacement vector, τi,j is the tensor of the solid-phase stress, p is the liquid-phase pressure, θ is the water content, δi,j is the Kronecker tensor, μ is the shear modulus of the solid phase, σ is the Poisson ratio, H and R are Biot’s coefficients: 1/H is a measure of compressibility of the solid phase for a change in water pressure and 1/R is a measure of the change in water content for a change in water pressure, k is the permeability of the aquifer, η is the liquid-phase dynamic viscosity, ϑ = τj,j ,

(6)

=

∂uj , ∂xj

(7)

α=

2(1 + σ )μ . 3(1 − 2 σ )H

(8)

and

4 The solved problem According to the assumptions of our model, Equations (4) and (5) are rewritten in the one-dimensional form as 2 μ(1 − σ ) ∂p u − α = 0, 1−2σ ∂x and ∂ 2u α + ∂t∂x

α 1 − R H

k ∂p = p. ∂t η

(9)

(10)

One-dimensional form of Equation (1) is τ=

2 μ(1 − σ ) ∂u − α p. 1 − 2 σ ∂x


(11)

396 Computational Methods in Multiphase Flow V Now, every triplet of functions {u(x, t), p(x, t), f (t)} satisfying equation 2 μ(1 − σ ) ∂u −αp =f 1 − 2 σ ∂x

(12)

satisfies also Equation (9). From Equations (11) and (12), it follows that the function f represents solid-phase stress and that it can be determined provided function τ is known as a function of time at a single value of coordinate x. Consequently, denoting T (t) the tension at a boundary point, it follows from (12) α(1 − 2 σ ) ∂p 1−2σ ∂ 2u = + T (t). ∂t∂x 2 μ(1 − σ ) ∂t 2 μ(1 − σ )

(13)

The last equation makes it possible to exclude the unknown function u from Equation (10) and to solve the resulting second order parabolic equation with the liquid-phase pressure as the unknown function. According to the aquifer and its neighbouring layers, the solved equation is either linear or nonlinear. Solving the problem for a leaky confined aquifer and horizontally oriented domain , it is necessary to introduce a source term and to deal with the continuity equation of the liquid phase in the form ∂θ + div v = q, ∂t

(14)

where v is vector of the liquid-phase flux density and q is the source term, i.e. the volume of water arising in unit volume of the aquifer per unit time. As the hydraulic head of the overlying aquifer as well as the height and the permeability of the aquitard are known, the source term can be determined as a function of the liquid-phase pressure: k pˆ − p q= ρw g + , (15) η Z where ρw is the liquid-phase density, g is the Earth’s gravity acceleration, pˆ is the liquid-phase pressure at the top of the aquitard and Z is height of the aquitard. Biot [1] discussed consolidation of saturated soils as a special case of his general equations. He comes to the conclusion that following equation holds for saturated soils: HR → ∞. (16) α = 1 and H −αR Hence, the diffusivity coefficient of the liquid-phase pressure is c=

2 μ k (1 − σ ) . η (1 − 2 σ )

(17)

Concerning the boundary conditions, it is supposed that the top of the aquifer is the only part of its boundary that allows for discharge or recharge. Consequently, the boundary ∂ is impervious and the Neumann boundary condition is imposed in the form ∂p (x, t) = 0, t > 0, x ∈ ∂. (18) ∂x WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


397

Figure 4: Comparison of observed water level fluctuations with results of a simulation.

5 Results and discussion The tension T is not known in our case. On the other hand, the amplitude of the detected oscillations can be estimated. Hence, function T can be determined by means of the inverse formulation of the problem. Assuming that the tension T (t) originates in tide acceleration, it can be expressed as (19) T (t) = MN γ (t), where MN is a constant and γ is the projection of the tide acceleration vector into axis x (the normal to the Skalský fault). Choosing a value of MN , the forward problem can be solved as described above. Figure 4 presents results of such a simulation. In this study, the tide acceleration is supposed to be determined by Sun and Moon, the effect of other planets being neglected. Function γ was calculated from known coordinates of tide acceleration vector. The code SPZ SM 01 (Ondovˇcin [4]), returning values of tide potential and tide-acceleration vector to given time and position on Earth, was used for this purpose. It can be seen from Figure 4 that in this case, the value of MN was overestimated. On the other hand, the figure shows the most important result of the simulation: there is clear correspondence between the simulated tide effects and the observed data. Consequently, the results confirm the previous conclusion made by Krásný et al. [3]. The oscillations of the computed curve are determined by function γ and the value of MN affects just the amplitude. Hence, it is a simple problem to iterate for its proper value. The value MN = 2 × 108 kg/m2 was found as the best match of the observed amplitude. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


Acknowledgements This paper is based upon work supported by the Grant Agency of the Czech Republic under grant No. 205/07/1311 and by the Ministry of Education of the Czech Republic under grant No. MSM 0021620855. The author is also indebted to the Journal of Geological Sciences for the permission to use Figures 1, 2, and 3.

References [1] Biot, M. A., 1941, General theory of three-dimensional consolidation, Journal of Applied Physics, 12, 155–164 [2] Bodvarsson, G., 1970, Confined fluids as strain meters, Journal of Geophysical Research, 75, 14, 2711–2718 [3] Krsn, J., Buchtele, J., ech, S., Hrkal, Z., Jake, P., Kobr, M., Mls, J., Łantrek, J., Łilar, J., Valeka, J., 2002, Hydrogeology of the Police basin: Optimisation of groundwater development and protection, Journal of Geological Sciences, 22, 5–100 (in Czech) [4] Ondovˇcin, T., 2007, Description of the code SPZ SM 01, UK, Prague, (unpublished) [5] Rojstaczer, S. and Agnew, D. C., 1989, The influence of formation material properties on the response of water levels in Wells to Earth tides and atmospheric loading, Journal of Geophysical Research, 94, B9, 12, 403–12, 411 [6] Rojstaczer, S. and Riley, F. S., 1990, Response of the water level in a well to Earth tides and atmospheric loading under unconfined conditions, Water Resources Research, 26, 1803–1817 [7] van der Kamp, G., and Gale, J. E., 1983, Theory of Erth tide and barometric effects in porous formations with compressible grains, Water Resources Research, 19, 538–544



399

Modelling of diffusion in porous structures E. du Plessis & S. Woudberg Applied Mathematics, Department of Mathematical Sciences, Stellenbosch University, South Africa

Abstract An existing pore-scale model is used to predict the effective diffusivity of staggered two-dimensional rectangular unconsolidated arrays through the use of a Representative Unit Cell concept. A tri-diagonal matrix algorithm is used to solve the diffusive flux field and to compute the effective diffusion coefficient for concentration gradients of staggered arrays. The numerical results and analytical model are compared critically with theoretical and numerical studies, as well as experimental data reported in literature. The good correlations obtained for the effective diffusivity provide confidence in both the computational and analytical work. Keywords: diffusion, porous media, pore-scale, effective diffusion coefficient, modelling, microstructure, fibres.

1 Introduction The study of molecular diffusion forms an important cornerstone in the analysis of more involved multiphase processes, such as dispersion and combustion in porous media, which form part of many processes in the coal industry, e.g. the recovery of methane from coal beds (Kim et al. [1]) and similar industries. Natural substances, such as coal, vary considerably in structural morphology. The microstructure is seldom simple and most often present a mixture of different types of porous environments. The flow processes thus often consist of a mixture of convective flow in macropores and molecular diffusion in micropores. Following up on considerable success with a pore-scale model on the modelling and subsequent prediction of drag during convection in porous media, our next goal is the modelling of diffusion making use of the same geometrical and modelling practices. To this end, descriptions of diffusion in two-dimensional rectangular arrays of non-staggered WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090341

400 Computational Methods in Multiphase Flow V and of fully staggered solid rectangles are needed to formalize the theory and to compare it with published experimental, computational and theoretical models.

2 Diffusion equation Fick’s second law of diffusion or simply the diffusion equation is given by ∂ρA + ∇· D∇ρA = 0, ∂t

(1)

where D is the mass diffusivity or diffusion coefficient for component A diffusing through component B and ρA is the mass concentration of species A. The diffusivity D is a property of a specific system, dependent upon the system’s temperature, pressure and composition (Welty et al. [2]). Eqn. (1) applies to a stationary, incompressible fluid without chemical production. For a steady incompressible fluid, constant diffusion coefficient, no chemical production and no fluid motion the Laplace equation for mass transport is obtained (Welty et al. [2]), i.e. ∇ 2 ρA = 0.

(2)

3 Effective diffusion coefficients of porous media Equation (2) is a point equation which governs the transport of mass of the chemical species considered. When regarding diffusion in porous media, eqn (2) needs to be solved at every point within the porous medium which leads to enhanced mathematical complexity. The characterization of diffusion in porous media therefore relies on the use of macroscopic dependent variables (Sáez et al. [3]). Eqn (2) thus needs to be volume averaged over a representative portion of the porous domain according to the volume averaging theory (e.g. Whitaker [4]). Boundary conditions also need to be imposed together with conditions at the fluidsolid interfaces (Sáez et al. [3]). For the present work no mass transfer between the phases will be considered. Volume averaging of eqn (1) leads to the following macroscopic diffusion equation: ∂ρA f = Deff : ∇∇ρA f , ∂t

(3)

where Deff is the effective diffusion coefficient which is only a function of the geometric structure of the porous medium at the pore level (Sáez et al. [3]) and may thus be expressed as a function of the porosity alone, i.e. Deff = f (). D

(4)

In the next section a geometric pore-scale model will be introduced which will be used to predict the effective diffusivity analytically. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


401

n

Uo ds Uf

d

Us

L

Figure 1: RUC model for fibre beds.

4 Geometric pore-scale model for fibre beds The geometric pore-scale model for fibre beds was originally introduced by Du Plessis [5]. It is based on a rectangular Representative Unit Cell, abbreviated RUC, which is defined as the smallest rectangular control volume, Uo , into which the average geometrical properties of the porous medium may be embedded. The RUC model for fibre beds is shown in Fig. 1. Only cross-flow – flow perpendicular to the prism axis – will be considered, as indicated by the streamwise direction n in Fig. 1. Cross-flow through prismatic porous media can therefore be approximated as two-dimensional flow. Two arrays are considered, namely a fully staggered array in which maximum staggering occurs in the streamwise direction and a non-staggered array in which no staggering occurs in the streamwise direction. The fluid filled volume within the RUC is denoted by Uf and Us denotes the volume of the solid phase. The RUC model is assumed to be homogeneous and isotropic with respect to the average geometric properties of the fibre bed. The porosity of the RUC model is defined as =

Uf . Uo

(5)

4.1 Volume partitioning The total fluid filled volume Uf within the RUC may be expressed as U f = U + U⊥ + Ug + Ut ,

(6)

where the streamwise volume U is the total fluid volume parallel to the streamwise direction, the transverse volume U⊥ is the total fluid volume perpendicular to the streamwise direction, in the stagnant volume, Ug , the fluid remains stationary and in the transverse fluid volume, Ut , no wall friction occurs due to the absence of adjacent solid surfaces. From eqn (5) and quantification of the respective fluid volumes, the following relationship between the linear dimensions of the RUC WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

402 Computational Methods in Multiphase Flow V model for fibre beds is obtained in terms of the porosity (Du Plessis [5]): √ ds = d 1 − .

(7)

Diedericks and Du Plessis [6] defined the geometrical tortuosity as the ratio of the streamwise displacement d to the total path length de of the fluid traversing through the constant cross-sectional area Ap in the RUC. The tortuosity may also be expressed as a volumetric ratio, but with stagnant volumes excluded, yielding [7] U + Ut + ξ U⊥ U + Ut + ξ U⊥ de = = , (8) χ= d A p d U + Ut where the coefficient ξ was introduced to account for the reduction in the tortuosity due to the splitting of the streamtube into two equal but directionally opposite transverse parts in a fully staggered array. It thus follows that ξ = 1/2 for the fully staggered array and ξ = 0 for the non-staggered array. Quantification of the tortuosity leads to the following expression for the fully staggered array χ =1+

1√ 1 − e, 2

(9)

and for the non-staggered array χ = 1. In order to account for the effect of stagnant regions a geometric factor ψ was introduced and defined as (Lloyd et al. [7]) ψ≡

U + Ut + U⊥ + Ug Uf Uf = = . U + Ut A p d U + Ut

(10)

The geometric factor expressed as a function of the porosity alone yields ψ=

1+

. √ 1−e

(11)

4.2 Effective diffusivity predicted by the RUC model According to Kim et al. [1] it is common for isotropic processes to express the effective diffusivity Deff over the diffusivity D as Deff = . D χ

(12)

Making use of eqn (9) for the tortuosity obtained by the RUC model, yields Deff = , √ 1 D 1+ 2 1−e

(13)

for the fully staggered array. An alternative expression for the effective diffusion coefficient can be obtained by using the geometric factor supplied by the RUC WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


403

model, which takes into account stagnant regions, instead of the tortuosity, i.e. √ Deff = 1 + 1 − e. D

(14)

Equation (14) applies to both the staggered and non-staggered arrays. Weissberg [8] made use of a variational approach which they applied to a bed of spheres to obtain the following expression for the effective diffusivity: −1 Deff 1 = 1 − ln . D 2

(15)

Maxwell (Kim et al. [1]) analyzed a dilute suspension of spheres analytically and obtained the following expression for the effective diffusivity: −1 Deff 1 = 1 + (1 − ) . D 2

(16)

The micropore-macropore model of Wakao and Smith [9] is given by Deff = 2. D

(17)

Based on their experimental data Kim et al. [1] proposed the following empirical equation for the effective diffusivity: Deff = 1.4 . D

(18)

5 Series-parallel formulae A popular concept to model diffusion in arbitrary composites is to mix two formulae that applies to composites made up of laminates of different diffusivities, namely the series and the parallel formula (Crank [10]). The assumptions for these models are that each laminate has uniform diffusive properties and the diffusive transport is uni-directional. It will be assumed that the solids are impenetrable, i.e. Ds = 0 (Crank [10]). 5.1 Series-parallel (SP) model In the SP model (Crank [10]) the composite is split into thin cross-stream strips, after which an effective diffusion coefficient is calculated for each strip using the parallel formula and, finally, by summing the strips in series an effective diffusion coefficient is obtained for the composite as a whole. Thus, in the SP-model the parallel formula is first applied, followed by the series formula. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


Diffusion

Figure 2: Overlapping fully streamwisely staggered array.

5.2 Parallel-series (PS) model In the PS model (Crank [10]) the composite is split into thin strips parallel to the streamwise direction, after which an effective diffusion coefficient is calculated for each strip using the series formula and, finally, by summing the strips in parallel an effective diffusion coefficient is obtained for the composite as a whole. Thus, in the PS-model the series formula is first applied, followed by the parallel formula.

6 Effective diffusion coefficients of ordered arrays For the present work only the overlapping fully streamwisely staggered array will be discussed. A regular array, a non-overlapping fully staggered array and an overlapping fully transversally staggered array have also been studied but will not be included in this work. In a regular array no staggering and no overlap of the solids occur in any of the two principle directions. In the non-overlapping fully staggered array the solid rectangles are staggered in both principle directions but with no overlap of the solids. In the overlapping fully streamwisely staggered array staggering and overlapping of the solids occur only in the streamwise direction. A schematic representation of an overlapping fully streamwisely staggered array is shown in Fig. 2. The unit cell is indicated by the dashed lines and also shown in Fig. 3. Both the SP and the PS models were applied to the unit cell. The results of the overlapping and non-overlapping streamwisely staggered arrays are presented in Table 1. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


405

Figure 3: Unit cell of streamwise fully staggered array with overlaps.

Table 1: Diffusion coefficients based on the SP- and PS-models. Non-overlapping fully staggered array: SP model:

PS model:

Overlapping fully streamwisely staggered array: SP model:

PS model:

−1 ds ds⊥ DSP = 1+ D d (2d⊥ − ds⊥ ) d⊥ − ds⊥ DP S = D d⊥ −1 ds ds⊥ DSP = 1+ D d (2d⊥ − ds⊥ ) DP S =0 D

7 Weighted average of the SP and PS models The following weighted average is suggested by Bell and Crank [11] to predict the effective diffusivity: Deff = θ DSP + (1 − θ )DPS , (19) where DSP and DPS are the estimates produced by the SP and PS models and 1 1 ds + 0.4 (d⊥ − ds⊥ ) . (20) θ = 0.56 − 0.5 2 2 WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

406 Computational Methods in Multiphase Flow V 1 0.9 0.8

DSP and DPS models

Weighted average Numerical data

0.7

eff

D /D

0.6

SP curve

0.5 0.4

PS curve

0.3 0.2 0.1

0 0

20

80

60

40

100

100h=100×(d −1/2ds ) ⊥

⊥

Figure 4: Effective diffusion coefficient as a function of the transverse pore width h for σ = 0.2.

For the present analysis let the streamwise solid dimension be denoted by σ = and the transverse pore width be denoted by h = 12 (d⊥ − ds⊥ ).

1 2 ds

8 Numerical computations A tri-diagonal matrix algorithm was used to solve the diffusive flux field and to compute the concentration gradients from the discritized form of eqn (2). The concentration values were used to calculate the effective diffusion coefficient. The ratio of Deff /D was calculated as the ratio of the sum of the fluxes of all the cells within the porous matrix over the ratio of the total flux when no solids are present. 8.1 Fully staggered array of rectangles Fig. 4 shows the effective diffusion coefficient as a function of the transverse pore width h for σ = 0.2. In RUC notation it follows that for this specific case d = d⊥ and 12 ds = 0.2, whilst 12 (d⊥ − ds⊥ ) is allowed to vary. For 0 ≤ h ≤ 0.5 the SP and PS models for the overlapping fully streamwisely staggered array were used and for 0.5 ≤ h ≤ 1 the SP and PS models for the non-overlapping fully staggered array were used. The predictions by both the SP and PS models are shown in Fig. 4 together with the weighted average of the two models. The agreement between the weighted average model and the numerical data is satisfactory. Fig. 4 is similar to Fig. 9(a) of Bell and Crank [11]. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


407

1 RUC model (eqn. (14)) RUC model (eqn. (13)) Weissberg [8] Maxwell (Kim et al [1]) Wakao and Smith [9] Emp eqn. (Kim et al [1]) Exp. data (Currie [12]) Exp. data (Kim et al [1]) Exp. data (Hoogschagen [13]) Num. data (present work) Num. data (Kim et al [1])

0.9 0.8 0.7

eff

D /D

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6

Porosity, ∈

0.8

1

Figure 5: Effective diffusion coefficient as a function of porosity for a fully staggered array of squares.

8.2 Fully staggered array of squares The effective diffusion coefficient for the fully staggered isotropic array was computed by setting 2 d = d⊥ , because if a square cell with solid squares is used, then the effective diffusion coefficient cannot be computed for porosities less than 0.5 because the solid squares will begin to overlap. In the way Bell and Crank [11] formulated their cells it is most probable that they encountered this restriction for further analysis. The effective diffusion coefficients predicted by the SP and PS models for square arrays can be obtained from table 1 as a function of the porosity by setting d = d⊥ = d and ds = ds⊥ = ds . Fig. 5 compares the numerical data to the predictions for the effective diffusivity obtained by the RUC model and with experimental and numerical data as well as several analytical models from literature. The model of Wakao and Smith [9] underpredicts the diffusivities obtained from the experimental data whilst both RUC models are in good agreement with the experimental data as well as all the other analytical models. The experimental data of Currie [12] and Hoogschagen [13] were obtained from beds of spheres. The RUC model involving the tortuosity proves to be more accurate at porosities greater than 0.8 than the RUC model based on the geometric factor. Nevertheless, both RUC models as well as the numerical results obtained within this work are in good agreement with the experimental and numerical data from literature as well as with the other analytical models from literature. Kim et al. [1] states that simple two-dimensional models can be used to predict the transport properties of isotropic systems which is confirmed by the present work. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


9 Conclusions A pore-scale model of simple rectangular geometry is used to predict the effective diffusion coefficient for isotropic systems. The present results show that the effective diffusivity can be accurately predicted in terms of only the porosity and the particle geometry. Numerical computations were performed to compute the effective diffusion coefficient for staggered arrays of solid rectangles. Both the analytical RUC model and the numerical computations are in good agreement with data and predictive models from literature. The successful interim results pave the way for a more advanced study towards analysis of coupled convectivediffusive processes in double porosity media which is important in the study of coal processing.

References [1] Kim, J., Ochoa, J.A. & Whitaker, S., Diffusion in Anisotropic Porous Media. Transport in Porous Media, 2, pp. 327–356, 1987. [2] Welty, J.R., Wicks, C.E. & Wilson, R.E., Fundamentals of Momentum, Heat and Mass Transfer. John Wiley and Sons, 1969. [3] Sáez, A.E., Perfetti, J.C. & Rusinek, I., Prediction of Effective Diffusivities in Porous Media using Spatially Periodic Models. Transport in Porous Media, 6, pp. 143–157, 1991. [4] Whitaker, S., The Method of Volume Averaging. Kluwer Academic Publishers, 1999. [5] Du Plessis, J.P., Saturated crossflow through a two-dimensional porous medium. Advanced Water Resources, 14(3), pp. 131–137, 1991. [6] Diedericks, G.P.J. & Du Plessis, J.P., On tortuosity and areosity tensors for porous media. Transport in Porous Media, 20, pp. 265–279, 1995. [7] Lloyd, C.A., Du Plessis, J.P. & Halvorsen, B.M., On Closure Modelling of Volume Averaged Equations for Flow Through Two-Dimensional Arrays of Squares. Proceedings of the Fifth International Conference on Advances in Fluid Mechanics, March 2004, Lisbon, Portugal, pp. 85–93, 2004. [8] Weissberg, H.L., Effective Diffusion Coefficient in Porous Media. Journal of Applied Physics, 34(9), pp. 2636–2639, 1963. [9] Wakao, N. & Smith, J.M., Diffusion in catalyst pellets. Chemical Engineering Science, 17, pp. 825–834, 1962. [10] Crank, J., The Mathematics of Diffusion. Clarendon Press, Oxford, 1975. [11] Bell, G.E. & Crank, J., Influence of imbedded particles on steady-state diffusion. J Chem Soc, Farabay Trans 2, 70, pp. 1259–1273, 1974. [12] Currie, J.A., Gaseous diffusion in porous media part 1. - a non-steady state method. British Journal of Applied Physics, 11, pp. 314–324, 1960. [13] Hoogschagen, J., Diffusion in Porous Catalysts and Adsorbents. Industrial and Engineering Chemistry, 47, pp. 906–913, 1955.



409

Measurement and prediction for air flow drag in different packing materials C. Rautenbach1 , B. M. Halvorsen2, 3 , E. du Plessis1, S. Woudberg1 & J. P. du Plessis1 1

Department of Mathematical Sciences, Applied Mathematics Division, Stellenbosch University, South Africa 2 Institute for Process, Energy and Environmental Technology, Telemark University College, Norway 3 Telemark Technological R&D Centre (Tel-Tek), Norway

Abstract Packed bed reactors are widely used in industry to improve the total contact area between two substances in a multiphase process. In some cases, like for the packing elements of some CO2 absorption towers, the packing material can be of such geometric nature that during discharge through them different flow conditions can be present in different parts of the packing. This renders prediction of pressure drops quite difficult. This paper concerns experimental and modelling activities to improve predictive equations for pressure drops over a packed bed of Raschig rings. It is shown that the application of some corrective measures can dramatically improve the correlation between theory and experiment, but that more research is needed in this field, regarding both carefully controlled experiments and mathematical modelling. Keywords: pressure drop, porous media, packing materials, Raschig rings, drag models.

1 Introduction In the chemical engineering industry packed bed reactors are widely used to improve the total contact area between two substances in a multiphase process. The process typically involves forced convection of liquid or gas through either structured or dumped solid packings. Applications of such multiphase processes include mass WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090351


(a)

(b)

Figure 1: Typical examples of random packings. (a) Metallic Raschig rings and (b) glass Raschig rings.

transfer to catalyst particles forming the packed bed and the adsorption of gases or liquids on the solid packing. For reactor design the drag laws are needed to predict the pressure drops over such reactors. In many cases the packing material can be of such geometric nature that during discharge through them different flow conditions can be present in different parts of the packing. This renders prediction of pressure drops quite difficult. This paper concerns experimental and modelling activities to improve predictive equations for pressure drops over a packed bed of Raschig rings. An experimental study on the determination of air flow pressure drops over different packing materials was carried out at the Telemark University College in Porsgrunn, Norway. The packed bed consisted of a cylindrical column of diameter 0.072m and height 1.5m, filled with different packing materials. Air was pumped vertically upwards through a porous distributor to allow for a uniform inlet pressure. Resulting pressure values were measured at regular height intervals within the bed. Due to the geometric nature of a Raschig ring packing the wall effects, namely the combined effects of extra wall shear stress due to the column surface and channelling due to packing alignment adjacent to a solid column surface, were assumed to be negligible. Several mathematical drag models exist for packed beds of granular particles and an important question arises as to whether they can be generalized in a scientific manner to enhance the accuracy of predicting the drag for different kinds of packing materials. Problems with the frequently used Ergun equation, which is based on a tubular model for flow between granules and then being empirically adjusted, will be discussed. Some theoretical models that improve on the Ergun equation and their correlation with experimental work will be discussed. It is shown that a particular pore-scale model, that allows for different geometries and porosities, is superior to the Ergun equation in its predictions. Also important in the advanced models is the fact that it could take into account anomalies such as dead zones where no fluid transport is present and surfaces that do neither contribute to shear stress nor to interstitial form drag. The overall conclusion is that proper modelling of the dynamical situation present in the packing WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


411

3000 glass Raschig ring data 2500

2000

1500 Experimental error

1000

500

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Figure 2: Experimental results for glass Raschig rings.

can provide drag models that can be used with confidence in a variety of packed bed applications. There is a wide range of different packing materials available. The packing material used varies from application to application. Factors that need to be considered include the pressure drop produced by the packed bed, chemical stability of the packing and size of the packing, to name but a few. Porous media created by using the packing materials illustrated in Figure 1 are called random dumped packings, as they are randomly placed into the container. Raschig rings and small glass spheres were provided by the TUC in Porsgrunn Norway and were used to produce the experimental results presented in this study. An example of data retrieved during the experiments conducted at the TUC is given in Figure 2. As indicated in Figure 2, there is a data point that does not follow the trend of the rest of the data. This is assumed to be caused by experimental error and is ignored for the remainder of the analysis.

2 Existing models used to predict flow behavior through porous packed beds In this section some existing available models are discussed. For convenience of comparison all models will be rewritten in terms of a F , defined by ∆p = µqF, L WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

(1)

412 Computational Methods in Multiphase Flow V with ∆p the measured pressure drop, corrected for gravity, and L the bed height. The Reynolds number, Rep , is defined as: ReDp ≡

ρqDp , µ

(2)

for use in the present study. 2.1 Ergun equation The Ergun equation [1], empirically based on results obtained from experimental packed beds of identical spheres, is given in the present notation, with Dp the particle diameter, as: F Dp2 =

150(1 − )2 1.75(1 − ) + · ReDp . 3 3

(3)

The empirically based constants compensate for the assumptions made in the capillary model. One assumption is that the porous medium is statistically uniform so that there is no channelling. Of course this is a crude assumption as channelling is common place in practical applications. Another more practical assumption is that the column diameter is large in comparison to the particle dimensions. The Ergun equation [1] also assumes a uniform particle size. As more data on irregular particles becomes available, the modelling can be improved to represent a wider spectrum of packed bed geometries. 2.2 RUC model An RUC drag model, [2], is used in this work to give a possible prediction of single phase flow through the packing elements of a CO2 absorption tower. This universal model can be applied to different types of porous media and for this work our interest is in the granular and foam versions. 2.2.1 Granular RUC model The granular RUC model is model aims to approximate porous media such as sand, consisting of small granular particles. The expression for the drag factor for granular porous media is expressed as:

F Dp2 =

25.4(1 − )4/3 cd (1 − ) 2 + 2 ReDp . (4) 1/3 2/3 (1 − (1 − ) ) 1 − (1 − ) 2 1 − (1 − )2/3

The form drag coefficient, cd , should typically be determined either numerically or empirically and is frequently assigned the value 1.9. The pressure drop can then be determined via the drag factor F from equation (4). WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


413

2.2.2 Foam model The RUC foam model was developed to accurately predict flow behavior through foamlike porous media [3]. A typical example is a spongelike metallic foams. The two variations that exist in the foam model are the doubly staggered model and the singly staggered model, the latter yielding a smaller pressure drop for the same discharge. The drag factor is given as: F d2 =

24ψ 2 (ψ − 1) cd ψ 2 (ψ − 1) ρqd · , + 2 2 2 (3 − ψ) µ

(5)

in the case of the doubly staggered model and as: F d2 =

36ψ 2 (ψ − 1) cd ψ 2 (ψ − 1) ρqd · , + 2 3 (3 − ψ) µ

(6)

for the singly staggered model. An expression for the geometric factor, ψ, as given in equations (5) and (6), is given as:

1 4π −1 + cos (2 − 1) , ψ = 2 + 2cos 3 3

(7)

for foamlike media. The micro-scale parameter d is given by the length of a cube that would produce N cubes in the total packing volume. The total number of particles in a fixed bed is represented by N . 2.3 The Sonntag correction A small change in the porosity has a large impact on the pressure drop and thus this effect can have a large influence on the pressure drop, predicted by the models. Sonntag [4] introduced the influence of the fraction m of the inner volume Vi of a Raschig ring that is stagnant and does not contribute to the shear stress nor to the interstitial form drag. After experimental correlations he stated that only approximately 20% of the inner volume of the ring is available for flow, i.e. m = 0.2. The effect of the decrease in the volume available for flow is a decrease in the effective porosity. 2.4 Nemec’s equation Nemec [4] applied the Sonntag correction to the Ergun equation. In Nemec’s work the experimentally determined values for the tortuosity, χ, and the friction factor, f , are kept the same as stated by Ergun. The reasoning is that when Sonntag derived his 20% criterion he used the original Ergun equation. Thus if adapted values for the tortuosity, χ, and the friction factor, f , are used, say for a bed consisting WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

414 Computational Methods in Multiphase Flow V of equivalent solid cylinders, the value of Sonntag’s correction would change. The equation put forward by Nemec can be written as: 150(1 − )2 3 de (Sf c + mSi ) 2 F de = × 3 3 6Vp (1 − (1 − ) (Vf c − mVi ) /Vp ) 1.75de ρq(1 − ) 3 + 3 µ3 (1 − (1 − ) (Vf c − mVi ) /Vp ) 2 de (Sf c + mSi ) × , (8) 6Vp with Vi the volume of the inner void cylinder, Vp the particle volume and Vf c the volume of a hypothetical full cylinder with the same outer dimensions. In equation (8) the surfaces are indicated by an S and the subscripts have the same meaning as previously mentioned volumes. The fraction of the inner void of each ring available for flow is denoted by m and can be taken as 20% according to Sonntag [4]. The equivalent particle diameter is defined as 6Vp /Sp and is denoted by de in equation (8). 2.5 Mackowiac’s equation Using experimental results from Raschig ring packings, Mackowiak [5] arrived at the following drag equation for perforated Raschig ring packings when he investigated the influence of the fluid-solid interface on the drag: 2 1 725.6 Fv 1− . (9) + 3.203 (1 − ϕ) F = 3 µq Rev dp K Here the form factor ϕ becomes zero for non-perforated packings, like Raschig rings. According to Mackowiak the value for the resistance coefficient, ψ0 , has been determined experimentally and has been found to be given by: 725.6 ψ0 = + 3.203 , (10) Rev with Rev the modified Reynolds number and is defined as: Rev =

qdp K. (1 − )µ

(11)

3 Comparative results The comparisons of the models to the experimental data for glass Raschig rings are given in Figure 3. The models were compared to the data and to each other, for metallic Raschig rings in Figure 4 (a). The metallic rings produced a higher WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


415

3000 Data Nemec eqn. (5.1.37) Doubly staggered foam eqn. (2.2.12) Singly staggered foam eqn. (2.2.13) Granular model eqn. (2.2.9) Ergun eqn. (2.3.4) Mackowiak eqn. (5.1.48)

2500

∆ p/L [Pa/m]

2000

1500

1000

500

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

q [m/s]

Figure 3: Comparison of model with the data for glass Raschig rings (refer to Figure 1(a)).

porosity than the glass ring bed. The result is that the effect of the wall on the over all pressure drop is lower with the metallic rings. This is also slightly evident from Figures 3 and 4 (a). It is suspected that wall effects are the cause of the difference in curvature of the models and the actual data. Confirmation of this suspicion is that when pressure drop measurements through small spherical particles were collected, there were no curvature discrepancy. In Figure 4 (b) the comparison of the Granular RUC model to the data acquired with flow through non-uniform spherical particle is given. Thus the difference in curvature could not be caused by incorrect data processing or incorrect experimental methods.

4 Adaptations 4.1 Shape-factor In the case of irregular shaped packings, shape factors can be used to determine the equivalent diameter of a sphere with the same volume as the element or particle (nominal diameter). The sphericity, φs , of an element is the ratio of the surface of the equivalent sphere to the actual surface area of the element. In Figure 5 the effect of a shape factor (sphericity) is investigated. It is evident that the sphericity marks a significant improvement on the RUC granular model and Ergun equation’s accuracy. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

416 Computational Methods in Multiphase Flow V 1200 Data Nemec eqn. (8) Doubly staggered foam eqn. (5) Singly staggered foam eqn. (6) Granular model eqn. (4) Ergun eqn. (3) Mackowiak eqn. (9)

1000

∆ p/L [Pa/m]

800

600

400

200

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

q [m/s]

(a) 4

6

x 10

5

Granular model eqn. (2.15), with a particle diameter of 0.0001m Granular model eqn. (2.15), with a particle diameter of 0.0002m Granular model eqn. (2.15), with a particle diameter of 0.0003m data

∆ p/L [Pa/m]

4

3

2

1

0 0.004

0.006

0.008

0.01

0.012 0.014 q [m/s]

0.016

0.018

0.02

0.022

(b)

Figure 4: (a) Comparison of model with the data for metallic Raschig rings Figure 1 (b). (b) Comparison of the RUC granular model [2] with experimental data given a range of particle sizes. 100 − 200µm powder was used to acquire the data.



417

3000 Data Nemec eqn. (8) Doubly staggered foam eqn. (5) Singly staggered foam eqn. (6) Granular model eqn. (4) Ergun eqn. (3) Mackowiak eqn. (9)

2500

∆ p/L [Pa/m]

2000

1500

1000

500

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

q [m/s]

Figure 5: Investigation of the effect of the sphericity on the Ergun equation and the RUC granular model.

6000 Data Nemec eqn. (8) Doubly staggered foam eqn. (5) Singly staggered foam eqn. (6) Granular model eqn. (4) Ergun eqn. (3) Mackowiak eqn. (9)

5000

∆ p/L [Pa/m]

4000

3000

2000

1000

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

q [m/s]

Figure 6: (a) Comparison of models with the Sonntag correction equal to 20% (with glass Raschig rings). (b) Pressure gradients of the different models and the data.


418 Computational Methods in Multiphase Flow V 4.2 The Sonntag correction This stagnant region within the rings can also be the reason for the under prediction of the Ergun equation. In Figure 6 the effect of the Sonntag correction is given. Thus using the sphericity and the Sonntag correction the best correlation to the experimental data is obtained.

5 Conclusion The major contribution of this work was the generation of a set of data by means of experiments and analyses of possible predictive models. They produced satisfactory correlations to data and thus provide confidence in the capability of mathematical models to predict experimental trends for various fixed bed reactors. The initial aim of using the spherical particle powders in the experimental part of this study was to check how experimental results would compare with the well known Ergun equation. As the Ergun equation was adapted empirically, based on the data obtained for flow through uniform spheres, it is expected to describe the flow through non-uniform spheres to some extent. The reason for such an assessment was because of the non-linearity of the data acquired with Raschig rings. The curvature differed from what the models predicted and the suspicion arose that the data may have been processed incorrectly. Thus, after the powders produced data that had the same behavior as the models, especially the Ergun equation, it can be concluded that the problem with the Raschig rings was not the result of incorrect data processing. The main cause is expected to be wall effects, due to the column to ring size ratio being small, only about 10. If this ratio is much larger, i.e. if dc /dp >> 10, the effect of the wall can be ignored [5]. With the incorporation of the Sonntag correction all the models perform better. On the modelling side the Sonntag correction was applied with great success to existing empirical and pore-scale models. Using any of the models discussed in this work without Sonntag’s correction will result in a under-prediction of the pressure drop for Raschig ring experiments. The deviations in the models from the data could be attributed to a variety of different effects. Almost none of these effects could be pin-pointed satisfactorily in this work due to time constraints. It is thus advisable to verify which effects predominated in both the Raschig ring and powder beds. The next logical step would then also be to find ways in which to combat the effects or to adapt the models to take these effects into account.

References [1] Ergun, S., Fluid flow through packed columns. Chemical Engineering Progress, 48, pp. 89–94, 1952. [2] du Plessis, J.P. & Woudberg, S., Pore-scale derivation of the ergun equation to enhance its adaptability and generalization. Chem Eng Sci, 63, pp. 2676–2586, 2008. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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[3] Crosnier, S., du Plessis, J.P., Riva, R. & Legrand, J., Modeling of gas flow through isotropic metallic foams. Journal of Porous Media, 9, pp. 35–54, 2006. [4] Nemec, D. & Levec, J., Flow through packed bed reactors: 1. single-phase flow. Chemical Engineering Science, 60, pp. 6947–6957, 2005. [5] Mackowiak, J., Extended channel model for prediction of the pressure drop in single-phase flow in packed columns. Chemical Engineering Research and Design, 2008. [6] Mcdonald, I.F., El Sayed, M.S., Mow, K. & Dullien, F.A.L., Flow through porous media - the ergun equation revisited. Ind Eng Chem Fundam, 18, pp. 199–208, 1979.




421

CFD simulation with multiphase flows in porous media and open mineral storage pile S. Torno, J. Toraño, I. Diego, M. Menéndez, M. Gent & J. Velasco School of Mines, University of Oviedo, Spain

Abstract In open storage piles in bulk solids port terminals, power stations and cement factories, not only the solid and porous barrier behaviour in front of the pile is important, but also the effect that porous and solid barriers produce when they are behind the pile. Considering the dust propagation behind the pile, the solid barriers are more effective than the porous ones. But, the effect of the porous barriers on the wind velocity distribution, mainly in zones between the barriers and the piles should be taken into account, regarding the total dust emission to the atmosphere and its propagation. In these studies, wind and dust concentration measurements in laboratory and field were carried out. A hot-wire anemometer (Velocicalc Plus (TSI)) was used in the wind study and two dust collectors (E-Sampler, Met One Instruments, Inc., Oregon, USA) were used to measure the dust concentration (Total Particle like PM10); a meteorological station (E-Sampler, Met One Instruments, Inc., Oregon, USA) attached to a PM10 collector was also employed. All these measurements were used to adjust the 3D CFD computational model (Ansys CFX 10.0): In the wind case through a k-epsilon turbulence model and the dust case by Lagrangian method. These adjusted models allow us to carry out several simulations combining the effect of solid and porous fences in front of and behind the pile, as well as pile shape modifications and behaviour analysis according to the dust emission from several special configurations and its relation to the wind gust preferential directions. Keywords: particle emission, Computational Fluid Dynamics (CFD), open storage pile, air pollution.



1

Introduction

The movement of minerals in open storage piles produces particle emissions to the atmosphere, which were studied to minimize their effect in the area more or less near the pile. In previous research [1], the use of solid barriers in storage pile protection which reduces dust emission to 66% in the worst environmental and industrial conditions, was shown. Nevertheless, the use of this solid barrier produces a high velocity vortex between the barrier and the pile, causing a greater dust emission to the windward side of the pile, [2–4]. The authors, based on their own experience and other research like [5] and [6], changed a solid barrier for a 30% porosity barrier, proving that the dust emission is reduced by 78%. The 30% porosity is obtained from research carried out by [7], who have determined from the “Particle Tracking Velocimitry” that a porosity of 30% is the more effective. Once the barrier effect (solid and porous ) in front of the open storage pile is studied, the need of studying the barrier effect behind the pile arises, thus the pile is protected against the wind with a barrier in front of it and the emitted dust is collected with a barrier behind the pile. In this paper, the barrier effectiveness, solid and porous, in each of the studied position, is shown. The 3D Computational Fluid Dynamics or CFD code Ansys CFX 10.0 is used to carry out the simulations, through the K-epsilon turbulence model [8], which was adjusted by the measurement campaigns carried out in the Mine of Carbonar S.A. situated in Asturias (North of Spain).

2 Experimental description 2.1 Introduction The scale tests were carried out in the installations of Carbonar, which are shown in fig. 1. The equipment consisted of a metalic cone of 52cm in height (h) and 69cm in diameter, a metalic ventilation pipe of 3m in length and 300mm in diameter, which is connected to a 12kw fan, which generates the wind flow. Two barriers are added to these elements, which are placed in front of and behind the cone. 2.2 Barrier in front of the cone In this case, the barrier has 52cm in height and 1.78m in length such as it was explained in [1] for the solid barrier, and it is placed at 25cm from both the pipe and the cone. In fig. 2 distribution of the elements used in the scale tests for the porous barrier, is shown. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


Figure 1:

423

Zone where the equipment was installed.

2.3 Barriers behind the cone. In this case the barrier is placed at 3.5m behind the cone and it has 1m in height and 2.04m in length. In fig. 3 the distribution of the elements for measurements in field with the dust collectors at 2.5m behind the barrier, is shown. 2.4 Velocity and concentration measurements. Flow velocity measurements were carried out with a hot-wire anemometer (Velocicalc Plus (TSI)), which was also used by other authors like [9] and [10], in 73 measurement points distributed in the cone zone and the barrier. Two dust collector (E-Sampler, Met One Instruments, Inc) measuring total particles and PM10 ( φL  ρ L , µ=

(7)

ρ − ρG ( µ L − µG ) + µG , ρ L − ρG

(9)

(10)

where φ L and φG are respectively the maximum and minimum order parameter for marking bulk liquid and gas; ρ L and ρ G are respectively the density of liquid and gas phases; µ L and µG are respectively the dynamic viscosity of liquid and gas phases. In Eq. (10), a simple linear function is applied to approximate the density within the interface; this enables the present method to obtain fα( eq ) (x, t ) and gα( eq ) (x, t ) in a simple form and thereby improve computation efficiency. For example, the calculations of second-order tensor G ( ρ ) , first partial derivative of ρ , etc., can be avoided in the present model, but they have to be calculated in the models such as [21]. To enable the method to treat two-phase fluids interacting with confined solid surfaces with wetting boundary potentials, for the current isothermal system, a simple form of representation of the free energy density ψ (φ ) , as suggested in [23], rather than the van der Waals free energy used in the traditional model, is applied in the present simulation, namely, ψ (φ ) = β (φ − φG ) 2 (φ − φL ) 2 + µbφ − pb ; (11) where β is a constant relating to interfacial thickness; µb and pb are the bulk chemical potential and bulk pressure, respectively. By substitution of Eq. (11), Eq. (7) becomes p0 = β (φ − φ L )(φ − φG )(3φ 2 − φφL − φφG − φLφG ) + pb .

(12)

In a plane interface under an equilibrium condition, the density profile across the interface is on equilibrium and can be represented as [22]

φ L + φG

φ L − φG

 2ξ  tanh  ; (13) D where ξ is the coordinate normal to the interface; the interface thickness D is given by 4 k . (14) D= φ L − φG 2β

φ (ξ ) =

2

+

2



499

The fluid-fluid (liquid-gas) surface tension force σ is expressed as [24] (φ − φ ) 3 σ= L G 2kβ . (15) 6 2.2 Correction for pressure It should be pointed out that the predicted velocity u * is not divergence free. To obtain the velocity field which satisfies the continuity equation ( ∇ ⋅ u = 0 ), u * is corrected by following equations: ∇p u − u* = − , (16)

ρ

 ∇p   ; ∇ ⋅ u* = ∇ ⋅  (17)  ρ  where p is the pressure of the two-phase fluid. Eq. (17) can be approximated by the LBM framework equation: ω 1 1 (18) ∇ ⋅u*; hα (x + e α , n + 1) = hα (x, n) − [ hα (x, n) − ω α p (x, n)] − α 3 ρ τ where, n is the number of iterations and τ = 0.5 + 1 / ρ is the relaxation time. The pressure at step n + 1 is given by p(x, n + 1) = ∑ hα (x, n + 1) . (19) α

The convergent pressure p is determined when ∀x ∈ V , | p( x, n + 1) − p (x, n + 1) |< ε ; (20) where V denotes the whole computational domain. Substituting the newly obtained pressure p into and solving Eq. (16) gives the corrected u , the velocity field.

2.3 Boundary treatment Applying the present LBM model, no-slip boundary conditions can be implemented by simply specifying the zero velocity on the solid boundaries, i.e. the boundary velocities u and u * in Eqs. (4-5) and (16-18) are given by u w = 0 , u*w = 0 . (21) As there is always a thin liquid layer in the vicinity of the solid boundary surface due to the intermolecular forces between the liquid and solid substrate [25, 26], it is assumed that a thin liquid occupies one layer of the lattice spacing, the order parameter on the boundary used in Eqs. (4-5) is then determined by φw = φ L . (22) In the present simulation, the finite-difference of the order parameter on the boundary are given by  ∂φ 1  ∂φ ∂φ ∂ 2φ ; (23) − +4 ≈ − 3 2 ∂ζ ζ =1 ∂ζ ζ =2  ∂ζ ζ =0 2  ∂ζ ζ =0 WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

500 Computational Methods in Multiphase Flow V where, ζ is the direction perpendicular to the wall. In this scheme, the first term on the right hand side of Eq. (23) is determined by a right-handed finitedifference; the second term is calculated by a standard centred finite-difference formula. Finally, it is found empirically that the best choice for the third term is a left-handed finite-difference formula taken back into the wall, namely, ∂φ ∂ζ

3

≈ z =2

(

)

1 3φ ζ =2 − 4φ ζ =1 + φ ζ =0 . 2

(24)


The motion of air bubbles surrounded by water flow in a three dimensional rectangular microchannel is considered. The gravitational force is taken into account by adding the term −3ω α e α z (1 − ρ G / ρ ) g to the right hand side of Eq. (3), where g is the dimensionless gravitational acceleration. Fig. 3 shows the computational domain and initial and boundary conditions of the modelling. Initially, two air bubbles with same diameter d = 200µm are ~ placed 300µm apart in water inside the channel of the length L x = 1200 µm , the ~ ~ width and the height L y = L z = 300 µm . The channel has an inlet boundary on the left hand side of the channel and a free outflow boundary on the right hand side of the channel. The other four sides of the channel are no-slip solid walls. Naturally, the densities of two fluids are set at ρ~L = 1000kg / m 3 and ρ~ = 1.0 kg / m 3 (making the density ratio to be 1000); meanwhile the dynamic G

viscosities of them are µ~L = 1 × 10 −3 kg / ms , µ~G = 2.1 × 10 −5 kg / ms , respectively. The initial surface tension between water and air is of σ~ = 1 × 10 −3 kg / s 2 and the gravitational acceleration is at g~ = 9.8m / s 2 . To relate the physical parameters with simulation parameters, a length scale of L0 = 1 × 10 −5 m , time scale of T0 = 1 × 10 −7 s and mass scale of M 0 = 1 × 10 −14 kg are applied; these lead to the dimensionless parameters: ρ L = 100 ; ρ G = 0.1 ;

µ L = 0.1 ; µ G = 2.1 × 10 −3 ; φL = 0.4 ; φG = 0.1 ; k = 0.05 ; and g = 9.8 × 10 −9 , respectively. Unless otherwise specified, the following simulations are within a computational domain occupied by 120 × 30 × 30 cubic lattices; ε in Eq. (20) is set at ε = 1×10−6 . The velocity distribution at the inlet boundary is specified as, u x (0, y, z ) = 16U ( L y − y )( L z − z ) yz /( L y L z ) 2 ;  u y (0, y, z ) = 0;  u z (0, y, z ) = 0; WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

(25)


Figure 2:

501

Computational domain and initial/boundary conditions.

where, U is the maximum value of u x (0, y, z ) . Thus, the Reynolds number is defined as ρ UL (26) Re = L z .

µL

The bubbles flow and behaviour in the microchannel at Re=100 is first simulated. The evolution with time of bubbles shapes and the behaviour of interactions are shown in Fig. 4. It can be seen clearly that the bubbles move in x-direction by the thrust force of surrounding water flow and meanwhile go up in y-direction due to the effect of buoyancy force; and with time marching, the two bubbles coalesce into a lager one. To focus only on the shape evolution of the left bubble at the early stage, it is found that the lower part of the bubble moves more quickly in x-direction than the upper part, which is caused mainly by the effects of velocity boundary layer near the solid wall of the channel.

1ms 0.5ms

y

z

x 2ms 1.5ms

Figure 3:

Evolution with time of bubble shapes and behaviour at Re=100.

The velocity fields are obtained through the numerical modelling. For ~ example, at t = 2 × 10 −3 s , the velocity distribution at different cross sections of WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

502 Computational Methods in Multiphase Flow V y-z plane in Fig. 4 such as x = Lx / 2 , 2 Lx / 3 and 3Lx / 4 , respectively, is shown in Fig. 5; where the solid line, the constant density line, indicates the interface between the two phases. As both pressure and velocity distributions across the interface are normally excellent indicators of numerical stability for the LBM calculations [27], Figs. 5 and 6 have actually shown that the present LBM can be used to obtain reasonable and stable velocity fields. Indeed, similar to the conventional CFD, the numerical instabilities of the LBM for two-phase flow of large density ratios are mainly caused by spurious velocities and/or the large oscillation of the pressure distribution across the phase interface. However, in the present method, the velocity and pressure are both corrected by solving an additional Poisson equation after each collision-stream step. Such corrections are able to ensure the velocity to satisfy the continuity equation and smooth pressure distributions even across the interface, so that to ensure the numerical stability.

A

Buoyancy forces drive the coalescent bubble upward

B

C

0

2ms

High pressure region

A

Figure 4:

x

B

C

~ Velocity field at different cross section, t = 2ms , Re=100: (A: x = L x / 2 , B: x = 2 L x / 3 , C: x = 3L x / 4 ).

Figs. 6(a) and (b) show the velocity vector and the vorticity contours, ~ respectively, at t = 2 × 10 −3 s , and at y = L y / 2 on x-z plane. It can be seen that the local distribution in coherent structures is evident; the shape of the coalescent bubbles is a result of the interaction between the fields of velocity and density concentration, and this is mainly affected by the effects of buoyancy force of the bubbles [28]. Fig. 7 shows the evolution of the bubble shapes at a low Reynolds number (Re=50). The results show an interesting evolution of the bubbles flow and WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


503

z 2ms

y x

(a) (b) vortices centres

Figure 5:

Velocity vector and vorticity contours of coalescent bubble at ~ y = L y / 2 on x-z plane for t = 2ms , Re = 100.

0.5ms

1ms

1.5ms

2ms

2.3ms

2.4ms

2.5ms

2.6ms

2.7ms

Figure 6:

Time evolution of bubble shapes at Re=50.

coalescence. At the early stage of the flow, the evolution of the bubbles is quite similar to that at Re=100, two bubbles coalesce at 1.5 ~ 2 .0 ms; however, at the later stage, the newly coalescent bubble re-breaks up into two bubbles; this shows probably a typical phenomenon of the flow at low Reynolds number. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

504 Computational Methods in Multiphase Flow V Obviously, such separation is caused by the strong effect of shear boundary layer near the upstream boundary. As at low Reynolds number, the buoyancy effect is more evident; which forces the bubbles migrating to the upper boundary of the channel and meanwhile results in the displacements in x-direction smaller and forms a relative returning flow in the vicinity of the bubbles interface as the sheer stress tending to different directions.

4

Conclusions

In this paper, a newly modified LBM model is developed to simulate bubbles flow in a rectangular microchannel. In the current model, a simple linear function of order parameter is applied to approximate the density within the interface of two fluids; meanwhile, a new form of the free energy density (rather than the van der Waals free energy) is used to enable the model treat confined surface and wetting boundary conditions. Based on the developed LBM model, the evolution of two isothermal air bubbles move through a water-filled microchannel are investigated numerically. Both the bubble shapes and the velocity files are imported to analyze the bubble-water and bubble-bubble interactions. The effect of Reynolds number on the flow is also examined. It is found that two bubbles can finally coalesce into one larger bubble at the relatively high Reynolds number; however, under the lower Reynolds number (e.g. Re=50), the newly coalesced bubble can be separated again by the stronger shear flow upstream.

Acknowledgement This work is supported by the EPSRC under grant EP/D500125/1.

References [1] Mishima, K., and Hibiki, T., 1996, Some characteristics of air-water twophase flow in small diameter vertical tubes, Int. J. Multiph. Flow, Vol. 22, pp 703-712. [2] Coleman, J. W., and Garimella, S., 1999, Characterization of two-phase flow patterns in small diameter round and rectangular tubes, Int. J. Heat Mass Transf., Vol. 42, pp 2869-2881. [3] Triplett, K. A., Ghiaasiaan, S. M., Abdel-Khalik, S. I., and Sadowski, D. L., 1999, Gas-liquid two-phase flow in microchannels - Part I: two-phase flow patterns, Int. J. Multiph. Flow, Vol. 25, pp 377-394. [4] Chen, W. L., Twu, M. C., and Pan, C., 2002, Gas-liquid two-phase flow in micro-channels, Int. J. Multiph. Flow, Vol. 28, pp 1235-1247. [5] Kawahara, A., Chung, P. M. Y., and Kawaji, M., 2002, Investigation of two-phase flow pattern, void fraction and pressure drop in a microchannel, Int. J. Multiph. Flow, Vol. 28, pp 1411-1435.



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[6] Chung, P. M. Y., Kawaji, M., Kawahara, A., and Shibata, Y., 2004, Twophase flow through square and circular microchannels - Effects of channel geometry, J. Fluids Eng., Trans. ASME, Vol. 126, pp 546-552. [7] Cubaud, T., and Ho, C. M., 2004, Transport of bubbles in square microchannels, Phys. Fluids, Vol. 16, pp 4575-4585. [8] Xiong, R. Q., and Chung, J. N., 2007, An experimental study of the size effect on adiabatic gas-liquid two-phase flow patterns and void fraction in microchannels, Phys. Fluids, Vol. 19, pp 1/033301-8/ 033301. [9] Qu, W. L., Yoon, S. M., and Mudawar, I., 2004, Two-phase flow and heat transfer in rectangular micro-channels, J. Electron. Packag., Trans. ASME, Vol. 126, pp 288-300. [10] Sussman, M., P. Smereka, S. Osher, A level set approach for computing solutions to incompressible two-phase flow, Journal of Computational Physics 114 (1994) 146-159. [11] C.Y. Ji, Y.Y. Yan, 2008. A numerical study of bubbly flow in a rectangular microchannel. Proc. 6th Int. ASME Conf. on Nano-, Micro- and Minichannels, ICNMM2008-62117, June 23-25, 2008, Darmstadt. [12] Y.Q. Zu, Y.Y. Yan, 2008, A numerical study of quasi-nucleate boiling in mini- and micro channels, ICNMM2008-62112, June 23-25, Darmstadt. [13] Y.Q. Zu, S. Gedupudi, Y.Y. Yan, T.G. Karayiannis, D.B.R. Kenning, 2009, Numerical simulation and experimental observations of confined bubble growth during flow boiling in a mini-micro channel with a rectangular cross section of high aspect ratio, ICNMM2009-82118, Puhang, South Korea. [14] Chen, Shiyi, and Doolen, Gary D., 1998, Lattice Boltzmann method for fluid flows, Ann. Rev. Fluid Mech., Vol. 30, pp 329-364. [15] Succi, S., 2001, The lattice Boltzmann equation for fluid dynamics and beyond, Clarendon Press, Oxford. [16] Gunstensen, A. K., Rothman, D. H., Zaleski, S., and Zanetti, G., 1991, Lattice Boltzmann Model of Immiscible Fluids, Phys. Rev. A, Vol. 43, pp 4320-4327. [17] Shan, X. W., and Chen, H. D., 1993, Lattice Boltzmann Model for Simulating Flows with Multiple Phases and Components, Phys. Rev. E, Vol. 47, pp 1815-1819. [18] Swift, M. R., Orlandini, E., Osborn, W. R., and Yeomans, J. M., 1996, Lattice Boltzmann simulations of liquid-gas and binary fluid systems, Phys. Rev. E, Vol. 54, pp 5041-5052. [19] Swift, M. R., Osborn, W. R., and Yeomans, J. M., 1995, Lattice Boltzmann Simulation of Nonideal Fluids, Phys. Rev. Lett., Vol. 75, pp 830-833. [20] He, X. Y., Chen, S. Y., and Zhang, R. Y., 1999, A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh-Taylor instability, J. Comput. Phys., Vol. 152, pp 642-663. [21] Inamuro, T., Ogata, T., Tajima, S., and Konishi, N., 2004, A lattice Boltzmann method for incompressible two-phase flows with large density differences, J. Comput. Phys., Vol. 198, pp 628-644. WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

506 Computational Methods in Multiphase Flow V [22] Yan, Y.Y., Zu, Y.Q., 2007, A lattice Boltzmann method for incompressible two-phase flows on partial wetting surface with large density ratio. Journal of Computational Physics, 227(1), 763–775. [23] Jamet, D., Lebaigue, O., Coutris, N., and Delhaye, J. M., 2001, The second gradient theory: a tool for the direct numerical simulation of liquid-vapor flows with phase-change, Nucl. Eng. Des., Vol. 204, pp 155-166. [24] Rowlinson, J.S., and Widom, B., 1989, Molecular Theory of Capillarity, Clarendon, Oxford. [25] Wayner, P.C., 1999, Intermolecular forces in phase-change heat transfer: 1998 Kern Award Review, AIChE Journal, 45(10), 2055-2068. [26] Ji, C., Yan, Y.Y., 2008, A molecular dynamics simulation of liquid-vapoursolid system near triple-phase contact line of flow boiling in a microchannel. Applied Thermal Engineering, 28(2-3), 195-202. [27] Lee, T., and Lin, C. L., 2005, A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio, J. Comput. Phys., Vol. 206, pp 16-47. [28] Mazzitelli, I. M., Lohse, D., and Toschi, F., 2003, The effect of microbubbles on developed turbulence, Phys. Fluids, Vol. 15, pp L5-L8.



507

Experimental investigation of a 2d impinging jet on a liquid surface R. Berger1,2, S. Depardon1, P. Rambaud2 & J. M. Buchlin2 1

Division of Research and Automotive Innovation, PSA Peugeot Citroën, France 2 Environmental and Applied Fluid Dynamics Department, von Karman Institute, Belgium

Abstract An experimental investigation of impinging 2d planar air jet on a water surface is performed by means of Particle Image Velocimetry (PIV). The difficulty encountered to measure the two dimensional velocity and turbulence fields close to the interface requires a special treatment of the PIV data obtained in the air and water sides. The resulting technique enables the localisation of the moving interface, the use of the inter-correlation and the calculation of statistics in both phases. The results show that the proposed PIV interface detection technique is in good agreement with the LeDaR detection technique developed at the von Karman Institute. The analysis of the air and water velocity field close to the interface highlights the topological differences between dimpling and incipient splashing configurations. The results obtained in this study will serve to model later turbulence transfer in impinging jet configuration. Keywords: turbulence, 2d impinging jet, interface, PIV – LeDaR.

1

Introduction

Gas jets impinging a liquid surface are encountered in a broad swathe of engineering processes related to automotive, metallurgical, chemical industries and safety systems. The gas jet impingement causes a depression at the liquid surface. The development of the lateral gas flow from the impact point induces water motion and wave formation. However, the physical interaction between a gas (air) and a liquid (water) involves a number of complex phenomena occurring in WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090431

508 Computational Methods in Multiphase Flow V both phases at the vicinity of the interface. The momentum transfer from the air to the water produces waves on the surface of the liquid and involves underwater flow and turbulence. The study of the flow structure above and below the interface is a very challenging task, especially in the presence of waves. The airflow structure above water surface has been studied for 50 years but the contribution of the near surface turbulence to momentum exchange between the two fluids is not well quantified and its modelling often in default. This lack of knowledge comes mainly from the difficulties to measure with accuracy physical quantities in the near interface region. In carrying a study of wind-wave flumes in oceanography, Shaikh and Siddiqui [1] show that PIV is a reliable optical nonintrusive measurement technique for investigating interface problem. However, it is also known that PIV measurements are very sensitive to the quality of the image [2], so that the moving interface needs to be accurately located. Another limitation in the previous near interface PIV studies is the lack of simultaneous measurements. Indeed, lots of measurements and analysis are done either in the liquid phase [3, 4] or in the gas phase [1], but very none are addressed to simultaneous measurements in both phases. The present study is focused on the development of a PIV technique to be applied both in air and water above and below an interface in the case of a planar 2d impinging jet. It is a pioneer study, which reports an analysis of the structure of the flow in both phases in the case of an impinging jet configuration and which yields a better understanding of the related mechanisms.

2

Experimental setup

The experiments are conducted on a dedicated setup designed and built at the von Karman Institute for fluid dynamics in Rhode-Saint-Genèse, Belgium (Figure 1). The facility is composed of a 1m long, 30cm wide and 15cm high transparent Plexiglas tank that allows optical measurements. In this tank, a removable double floor is used to set the water depth around 15mm and the distance with the nozzle outlet close to 30 mm. The 2d planar jet is created by a nozzle with an outlet slot of 5mm. The nozzle is composed of a “stagnation” chamber and a convergent section. The nozzle is fed a 10 bars compressed air system through a pressure regulator that sets the flow rate, which is determined from the measurement of pressure inside the stagnation chamber. The turbulence in the jet potential core is quantified by means of a hot wire and evaluated at 4%. The jet Reynolds number, Re, is based on the nozzle slit. In the present study the Re value of 2000 and 2700 are considered. They correspond to dimpling jet and incipient splashing jet, respectively, as defined by Molloy [5] (see Figure 2). PIV [2], with parameters summarised in Figure 3, is used to measure 2d instantaneous velocity fields in a plane perpendicular to the water surface along the centre line of the tank. The measurement chain is made of a 250 mJ two cavities pulsed Nd-Yag laser and a PCO 1280 x 864 pixels CCD camera with a 35mm objective and their related acquisition system. The camera and the laser are synchronised at 3 Hz by means of a timing hub. The laser sheet is created and WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


Figure 1:

Dimpling

Figure 2:

509

Schematic of the setup.

Splashing

Penetrating

Impinging jet configurations – Molloy [5].

focused in the measurement plane with the use of a spherical and a cylindrical lens, which guide it to the measurement region by the side of the tank for the water PIV and by the bottom of the tank for the air PIV as shown in Figure 4. The instantaneous character of the PIV measurements allows the computation of detailed statistics of the flow field in terms of mean and fluctuating quantities.


510 Computational Methods in Multiphase Flow V Air phase PIV oil vapour

Water Phase PIV Vestosint 10mg/L Water Water – fluoreseine mixture ~1 µm ~5 µm ~35 mm x 25 mm ~0.04 ms ~4ms WIDIM [6] 96 x 96 2 75% 1 vector / ~0.17 mm

seeding particle Liquid Particle diameter field of view size separation time Processing software initial window size refinement step Overlapping final resolution

Figure 3:

PIV parameters.

Water phase PIV

Air phase PIV Cylindrical lens

Spherical lens

Prism

Figure 4:

Laser arrangement overview.

In addition to PIV measurements, the LeDaR interface tracking technique developed at the VKI [7] is also applied to validate by comparison the interface detection from the PIV images. The LeDaR database is composed by 3000 images sampled at 2.5 kHz rate by means of an 800 x 600 pixels high speed camera. The measurements allow also the detailed statistics of the interface deformation (mean and fluctuations), the dynamic analysis of the data.

3

Interface detection

In order to use WIDIM PIV algorithm developed at the VKI, a special treatment is used for each phase. The objective of these treatments is to provide images only in the medium (air or water) of interest. For both phases, the PIV images are WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


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used to detect the position of the interface. Once the interface is detected, a simple filtering is applied to blank the part of image that is not in the investigated phase. However, the procedure to determine the interface position is different depending on whether one uses air or water PIV data. 3.1 LeDaR interface detection method Level Detection and Recording (LeDaR) algorithm is a reference measurement method for interface detection developed at the VKI. The principle and the performance of this technique are described by Planquart [8] and Bouchez [9]. The algorithm used is an improved version developed by Toth [10] for grey level sensor camera. The enhanced version used is based on the Maximum Forward Step Filter method that has already been shown [10] as the most robust detection method among the others available in LeDaR. An example of LeDaR interface detection is shown in Figure 5. Based on the accuracy of the algorithm and the spatial resolution, the resulting uncertainty of the interface location is estimated to be within ±70 µm.

a

b Figure 5:

Example of LeDaR application: a – raw image, b – enhanced images with resulting interface (white line).

3.2 Water phase interface detection From the water PIV raw images, the technique is relatively complex because the interface doesn’t appear clearly due to the “Mirage effect” (Dias [11, Nogueira [12]). Therefore, an image texture separation has been developed and used (Figure 6). The different steps of the detection procedure are successively: • Grey level and Sobel filtering of the raw image to highlight the particle in the images. The use of the fluorescein is justified at this step for a better contrast and an efficient filtering of the image background and the “Mirage effect” • Texture segmentation and closure. This step appears as the most critical because highly depend on the seeding density. It is required to be homogeneous between all the images for an automated process. The result is a binary image where the value “0” corresponds to the air phase and the value “1” to the water phase. • Localisation of the binary jump for each pixels column, defined as the interface location WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

512 Computational Methods in Multiphase Flow V • •

Spatial smoothing of the detected interface. Blanking of the above interface part in order to obtain the final images used for the cross-correlation process. The process is applied to a set of 1000 images. The results show that small deviations of the interface appears due to strong light reflections on the interface by floating particles or low seeding density zones within less than 10% of the images. The difference between the computed interface and the real interface is manually checked on a set of 20 images randomly selected and for each one the maximal deviation recorded. The resulting uncertainty of the interface location is estimated to be within ±180 µm. This ambiguity is mainly due to the closure and segmentation parameters, to the seeding density and the spatial resolution of the images. Raw images

Sobel filter + Grey level filter

Segmentation + closure

Spatial Smoothing

Interface location + blanking

Figure 6:

Water side PIV interface detection method illustration.

3.3 Air phase interface detection The detection scheme from the air phase PIV is completely different. Indeed to avoid light saturation and to provide useful pictures for the interface detection, the setting respects the schematic drawn in Figure 7. The way of detecting the interface in the air phase PIV image relies on the assumption that the deformed surface assumption has a 2d character. Instead of detecting the interface in its real position (A in Figure 7), the interface is recorded as the image of the laser reflection footprint (B in Figure 7) on the wall of the double floor through the deformed interface (B’ in Figure 7). This point is recorded as point b on the camera picture. In addition we assume that the line of WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


513

high intensity on the picture is the real interface deformation with a small shift that is very easy to determine manually. So applying the LeDaR algorithm [7] on the raw image and using a shift parameter we are able to accurately determine the position of the real interface (Figure 8). Applying the same accuracy evaluation than for the water phase, the interface is found to be localised within a tolerance of ±120 µm. The determination from the air side is more accurate than from the water side thanks to the accuracy of the LeDaR algorithm.

Figure 7:

Detail of Air side PIV image.

a

c

b

d LeDaR Detected

Figure 8:

Real Interface

Air phase PIV based interface detection illustration: a – raw images, b – raw images with highlighted interface, c – blanked image, d – image with velocity vectors.



4


4.1 Interface tracking scheme evaluation The PIV based interface localisation gives statistical information on the position of the interface that can be compared to the data obtained with a specific interface tracking measurement technique. The previous data extracted from the PIV are confronted with data obtained by the LeDaR technique. Figures 9 and 10 are typical examples. Figure 9 shows that good agreement is obtained between the interface tracking methods for the dimpling configuration of the impinging jet. The trough (0<X/b 4.5). However, the water PIV based interface position is underestimated in the region of the “lip” (2.5 < X/b < 4.5). This inconsistency can be explained by lighting and seeding artefact. To obtain images respecting the needs of our interface detection scheme, the laser sheet is carried by the side of the tank. The consequence is the missing part upside the one detected is present in a shadow and low seeded region. The resolution of this problem is still under investigation. The consequence is a loss of PIV data in this region even though the technique gives satisfactory results in overall. 4.2 PIV qualitative analysis PIV provides a series of snapshots of the velocity field at fixed time interval. In a fixed Eulerian coordinate system, a time series of the velocity data at any spatial location between the top and the trough of the waves cannot be extracted. This limitation is mainly due to the fact that a given spatial location can either lie in the water or in the air. Hsu and Hsu [13] already show that time-averaged quantities do not exist in a useful form at a fixed position in the crest-trough region of a wavy interface. Two ways of computing the quantities are given: • Conditional averaging. This method uses fixed spatial location and takes in account only the data that are in the investigated phase. • Wave-following Eulerian averaging: in order to obtain time-averaged quantities especially for high amplitudes waves, Hsu and Hsu [13] as Shaikh and Siddiqui [1] use another system they show as more accurate and meaningful. They transformed the fixed Eulerian coordinate system into a wave-following Eulerian system. The transformation imposes that the new vertical origin of the coordinate system is located on the interface for each instantaneous field. The quantities are then arranged with reference to the water surface using a new vertical coordinate system ζ, such that ζ=0 is the location of the interface in the each field and the positive ζ-axis is pointing upwards. Since the first method could be sufficiently accurate for a smooth interface as observed in the dimpling condition, the issue could be more chaotic in the incipient splashing configuration where the surface becomes strongly wavy. That is why all the present study statistics are computed using the wave following coordinate system. Figure 11 shows the averaged velocity field on both phases for the two jet configurations. We can clearly distinguish the typical behaviour of each configuration with the lip presence (X/b~3). For both flow configurations the bulk rotating motion in the water phase, created by the water surface dragging [14] is captured. Compared to the dimpling configuration, the centre of the rotating motion in the incipient splashing configuration is located outside the field of view, because of higher velocity and wider water depression at the impingement. For both cases it is worthwhile focussing on the region close to the interface. The measurement resolution is not sufficient to check the velocity continuity WIT Transactions on Engineering Sciences, Vol 63, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

516 Computational Methods in Multiphase Flow V across the interface due to the one two order of magnitude difference between the velocities in both phases. Indeed, to be able to confirm continuity, we need to access the viscous sub-layer velocity in each phase as shown by Davis [15], who argues that the viscosity has a significant influence on the momentum transfer across the interface. That very small layer has not been resolved in the present study.

I

Figure 11:

Water and air sides averaged velocity magnitude fields and streamlines – X/b>0: splashing configuration – X/b

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