Electrochemical Process Simulation III

Electrochemical Process Simulation III

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THIRD INTERNATIONAL CONFERENCE ON SIMULATION OF ELECTROCHEMICAL PROCESSES

ELECTROCOR 2009 CONFERENCE CHAIRMAN C. A. Brebbia Wessex Institute of Technology, UK

INTERNATIONAL SCIENTIFIC ADVISORY COMMITTEE K. Amaya S.G.R. Brown J. Deconinck R. Kelly E. Lemieux P. Mandin A. Peratta M. Pocock A. Powell D.A. Shifler A. Taleb

Organised by Wessex Institute of Technology, UK 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

Electrochemical Process Simulation III

Editors C. A. Brebbia Wessex Institute of Technology, UK R. A. Adey Wessex Institute of Technology, UK

Editors: C. A. Brebbia Wessex Institute of Technology, UK R. A. Adey 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: 978-1-84564-192-4 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

This book contains papers presented at the third international conference on the Simulation of Electrochemical Processes held in Bologna, Italy in 2009. The meeting was organised by the Wessex Institute of Technology. The conference objective was to bring together researchers, engineers and scientist to present and discuss the state of the art in the computer simulation of electrochemical processes and its application in the areas of corrosion, corrosion related fracture and fatigue and coating and deposition processes. The theme of the conference was to encourage papers describing the development of computational models and their application in practice, including the comparison with experimental measurements and case studies. The editors are indebted to the members of the International Scientific Advisory Committee for their help in reviewing the abstracts and papers. The Editors Bologna, 2009

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Contents Primary to ternary current distribution at a vertical electrode during two-phase electrolysis Ph. Mandin, H. Roustan, J. B. Le Graverend & R. Wüthrich .............................. 1 Hydrogen production by the Westinghouse cycle: modelling and optimization of the two-phase electrolysis cell S. Charton, P. Rivalier, D. Ode, J. Morandini & J. P. Caire ............................ 11 A hydraulic model to simulate the hydrodynamics of a fluorine electrolyser J. P. Caire, G. Espinasse, M. Dupoizat & M. Peyrard ...................................... 23 Computational modelling of cathodic protection systems for pipelines in multi-layer soil A. B. Peratta, J. M. W. Baynham & R. A. Adey ................................................. 35 Numerical modelling of cathodic protection systems for deep well casings A. B. Peratta, J. M. W. Baynham & R. A. Adey ................................................. 47 Functional relationship between cathodic protection current/potential and duration of system deployment in desert conditions A. Muharemovic, I. Turkovic & S. Bisanovic..................................................... 59 Optimization of a ship’s ICCP system to minimize electrical and magnetic signature by mathematical simulation S. Xing, J. Wu & Y. Yan ..................................................................................... 69 Numerical analysis assisted monitoring method for the coating condition on a ballast tank wall K. Amaya, A. Nakayama & N. Yamamoto ......................................................... 79

The influence of coating damage on the ICCP cathodic protection effect J. Wu, S. Xing & F. Yun ..................................................................................... 89 Field-based prediction of localized anodic dissolution events taking place on ZnAl alloy coatings in the presence of 5% NaCl solution S. G. R. Brown & N. C. Barnard ....................................................................... 97 Eulerian-Lagrangian model for gas-evolving processes based on supersaturation H. Van Parys, S. Van Damme, P. Maciel, T. Nierhaus, F. Tomasoni, A. Hubin, H. Deconinck & J. Deconinck ................................... 109 Simulation of gas pipeline leakage using the characteristics method E. Nourollahi ................................................................................................... 119 Two-dimensional numerical modelling of hydrogen diffusion assisted by stress and strain J. Toribio, D. Vergara, M. Lorenzo & V. Kharin ............................................ 131 Mathematical modelling of electrochemical reactions in aluminium reduction cells R. N. Kuzmin, O. G. Provorova, N. P. Savenkova & A. V. Shobukhov........................................................................................... 141 Atomistic simulation of the nano-structural evolution of Raney-type catalysts from spray-atomized NiAl precursor alloys during leaching with NaOH solution N. C. Barnard, S. G. R. Brown, F. Devred, B. E. Nieuwenhuys & J. W. Bakker................................................................................................. 151 Corrosion of mild steel and 316L austenitic stainless steel with different surface roughness in sodium chloride saline solutions L. Abosrra, A. F. Ashour, S. C. Mitchell & M. Youseffi................................... 161 Recovering current density from data on electric potential J. Irša, A. N. Galybin & A. Peratta.................................................................. 173 Author Index .................................................................................................. 185

Simulation of Electrochemical Processes III

1

Primary to ternary current distribution at a vertical electrode during two-phase electrolysis Ph. Mandin1, H. Roustan2, J. B. Le Graverend1,3 & R. Wüthrich3 1

LECIME UMR 7575 CNRS-ENSCP-Paris6, ENSCP, France Alcan – Centre de Recherche de Voreppe, France 3 Department of Mechanical & Industrial Engineering, Concordia University, Canada 2

Abstract During two-phase electrolysis for hydrogen or fluorine industrial production, there are bubbles that are created at vertical electrodes, which imply quite important electrical properties and electrochemical processes disturbance. Bubbles are motion sources for the electrolysis cell flow, and then hydrodynamic properties are strongly coupled with species transport and electrical performances. The presence of the bubbles modifies these global and local properties: the electrolysis cell and the primary to ternary current density distribution are modified. This disturbance leads to the modification of the local current density. The goal of this proposition is to present the electrochemical engineering study and modelling of two-phase electrolysis properties at electrode vicinity. This work is due to the necessity for a better knowledge of the actual interface condition during electrolysis, for example to have a better process optimisation or electrode consumption prevention. Keywords: two-phase electrolysis, modelling, electrochemical engineering.

1

Introduction

Gas release and induced fluid flow at electrodes are characteristic for several electrochemical processes, such as hydrogen or fluorine production. The twophase phenomena in gas evolving electrodes are in general neglected because of the major difficulty to be correctly taken into account. Nevertheless, nowadays, with the increasing interest in hydrogen production and clean, sustainable WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/ECOR090011

2 Simulation of Electrochemical Processes III fluorine production for the nuclear industry, these processes have to be revisited with modern experimental and numerical tools for further optimization. Many authors [1–20] have worked on this topic and some interesting experimental observations and numerical modelling are now available for engineering optimization and process intensification. Because of gas evolving in the gravity field, the industrial two-phase electrolysis processes generally use vertical electrodes to promote bubble detachment and avoid gas accumulation. This is the reason why the present work focuses upon vertical electrodes.

2

Experimental set-up

The electrochemical reactors for continuous production are opened. A pump ensures circulation through the cell and the mass flow rate is controlled. The electrodes, anode and cathode have a large surface to ensure massive production and large current intensity is applied. The average current density is from 0.1 to 1 A cm-2. The produced bubbles are accumulated at the top of the cell. The laboratory study of these processes has lead us to define an adapted geometry for the closed electrochemical cell presented in figure 1.

Figure 1:

Electrochemical cell with gas evolving at electrodes.

In accordance with in situ configuration and geometry, two vertical plane electrodes separated with distance d=4.2 cm have been considered. The height is H=10 cm whereas the width is l=2 cm. The anode is made with the usual nickel (on the left in figure 1) and the cathode is made with the usual copper (on the right in figure 1). An Autolab PGSTAT 302 N is used in combination with a 20 A current booster to ensure average current density from 0 to 1 A cm-2 on the 20 cm2 electro active surface configuration studied. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

Simulation of Electrochemical Processes III

3

The classical alkaline water electrolysis has been chosen as a representative two-phase electrolysis process: 2 H2O + 2 e-  H2 + 2 OH-aq 2 OH-aq  0.5 O2 + H2O + 2 e-

E° = -0.83 V/ENH (R1) E° = 0.40V/ENH (R2)

Various alkaline liquid electrolytes have been considered in the present study: NaOH and KOH with various molar concentrations.

3

Results and discussion

The experiments have been performed under imposed current conditions. The cell potential evolution with time was then measured for various imposed current values. After a time defined with induced hydrodynamic conditions, a steady potential is measured at the electrochemical cell. Table 1 gives the associated measured potential for different values for the imposed current, the electrolyte concentrations and nature. As can be seen, it is easier to obey a given imposed current with NaOH than with KOH for concentration 0.1 M. However, at 1 M, the voltage to apply to impose the current is smaller with KOH. The explanation is difficult to give and needs further modelling effort to take into account multi-physico-chemical coupled phenomena. These measurements are really important for electrochemical engineering. The smaller the voltage is, the cheaper the hydrogen production. The hydrogen volume flow rate is directly related to the current according with the Faraday law: QH2 = (Iimp/n.F)*(R.Tamb/Pamb) (1) with QH2 the volume flow rate in m3 s-1, n=2 the transferred electron number in the electrochemical reactions, F=96500 C mol-1 the Faraday constant, R=8.314 J mol-1 K-1 the ideal gas constant, and Tamb =300 K and Pamb =1.013 105 Pa the ambient temperature and pressure, respectively. Table 1:

Average voltage U (V) at the electrochemical cell for various conditions of imposed current Iimp, electrolyte ions and concentration.

Voltage U Iimp (A)

KOH 0.1 M

QH2 mm3/s

NaOH 1M

0.1 M

1M

0.05

2.6 V*

2.0 V

2.52 V

2.03 V

6.4

0.1

3.4 V

2.13 V

3.2 V

2.17 V

12.8

0.2

4.6 V

2.33 V

4.4 V

2.45 V

25.5

0.5

8.2 V*

2.83 V

7.9 V

3.05 V

63.8

WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

4 Simulation of Electrochemical Processes III

8 7

NaOH 0.1 M 0.05 A NaOH 0.1 M 0.1 A NaOH 0.1 M 0.2 A NaOH 0.1 M 0.5 A

U (Volts)

6 5 4 3 2 1 0

1

2

3

4

5

6

7

8

9

10

t (sec)

Figure 2:

Evolution of the cell potential U (V) with time for NaOH concentration 0.1 M, for various imposed current Iimp values. y

Iimp

U

Figure 3:

jx(y)

Vertical gas-evolving electrode: left is the geometric configuration and right is a current distribution example (y: electrode height; jx(y): local current density).

Figure 2 shows the evolution necessary to impose potential at the electrochemical cell to obtain a given current for various NaOH concentration values. As has been observed in table 1, the more concentrated the electrolyte is, the easier and cheaper the hydrogen production. The goal of the study is now to understand and then model the reasons for such an evolution. This is the first step for optimization and process intensification. The numerical modelling is now presented.

WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

Simulation of Electrochemical Processes III

5

Electrolyte Hydrodynamics

Velocity field V

Navier-Stokes relations

Void fraction ε

Newton law for discrete phase

Electrical conductivity κ

Current density field j

Bruggeman relation: κ = κ 0 (1-ε) 3/2 (2) Current balance : div j =0 (3)

Current density distribution

Bubble mass flow rate q

Figure 4:

4

Faraday relation: q = M jn / (n F) (4)

Calculation flow-sheet for the coupling effect in electrochemical cell due to the presence of bubble release.

the

Numerical modelling

A primary current distribution at the electrodes is first developed. The schematic mathematical configuration is presented in figure 3. Figure 4 shows the simplified calculation algorithm for primary current distribution at the working electrode (hydrogen production). A classical CFD software is used to solve the Navier-Stokes equations under laminar hypothesis with finite volume methods. Then, for a given injected gas mass flow rate with constant spherical shape and diameter (mono disperse bubbles), each particle’s trajectory is calculated according to local conditions. The bubble friction is associated to a liquid electrolyte local motion source term, which ensures strong coupling between the continuous liquid phase and the discrete bubbles phase. According to the bubble residence time, an average void fraction ε (-) is calculated and then the local electrical conductivity κ (S m-1), the value of which is smaller than the pure liquid value κ 0 because of the insulating character of the gas, according to Bruggeman’s law (2). Then, the local current density j (A m-2) balance equation (3) can be calculated and yields to a smaller current density value where the bubble concentration is large, at the top of the electrodes. According to Faraday’s law (4), the smaller the current density j, the smaller the local gas mass flow rate q (kg s-1 m-2). WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

6 Simulation of Electrochemical Processes III The Navier-Stokes calculation leads to a proposition of hydrodynamic profile, but the resulting current is not in good accord with the experimental results shown in table 1 and figure 2. The main reason is the great difficulty in correctly modelling the hydrodynamic properties. Then, to validate the hydrodynamic model itself, it has been decided to perform some experimental local field velocity experiments using a PIV laser plane. The bubbles are supposed to perfectly follow the local flow direction and magnitude because they are small enough. A fast CCD camera is used, which registers 170 picture couples at a frequency of 15Hz (66.67 ms). Each couple of picture is measured with a delay of 8 ms. Then, the post-processing of the picture couple allows the definition of a local velocity vector according to the bubble’s displacements during the 8 ms between the two post-treated pictures. The experimental set-up is presented in figure 5.

Figure 5:

PIV measurements experimental set-up.

60

60

0

0.0 6

4

0

position(2)

0 0.02

0

0.12

0.02

0.04

0.0 2 0.0 4 2 0.0

0

0.0

2 0.0

0

position(2)

0.02

0

0.02

0.04

6

0

0.04

0

10

2

0

20

0.0

0.1 0.08

0

30

Vy 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02

0.02

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

0

0.06

0.0

0

0

10

40 0

0

20

50

0

30

Vy 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02

4

0

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

0.0

0.06

40

0.1 2 0.0

0.06

50

-10

-10

0

0

-40

Figure 6:

-20

0

position

20

40

-40

-20

0

position

20

40

Unsteady vertical velocity component contours and stream function at I=1A, after 220 s electrolysis (left) and 220s + 80 ms (right).

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Simulation of Electrochemical Processes III

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For each electrolysis condition, four times 170 couple of pictures (8 ms time difference) have been measured at 15Hz (66.7 ms) during the unsteady and steady regime. The post-processing of the results is in progress, but leads to interesting and difficult to understand results and consequences. The electrodes with bubbles evolving seem to lead in the confined electrochemical cell to a hydrodynamic “puffing phenomenon”, clearly three dimensional and unsteady in the cell. Then all the coupled phenomena, transports and reactions are affected and should actually be three dimensional and unsteady. The primary current distribution modelling described is to be completed with an accurate homogeneous and surface realistic chemistry modelling. This is called the ternary current distribution modelling. This modelling needs the knowledge of transport kinetic properties (diffusion coefficients), speciation and reactions, which occur in the cell, with the associated thermodynamic and/or kinetic reactive properties. An example of local speciation and chemistry is given in figure 7. C (mol m-3)

Distance from anode (m) Figure 7:

5

Speciation profiles in the anode vicinity in the case of aluminium production according to the Hall-Heroult process [12].

Conclusion and interest in zero gravity experiments

As shown previously, the hydrodynamic properties are really difficult to model: both electrodes are naturally induced motion sources with one or two phase flows; this flow can be from laminar to turbulent and really confined. This is the reason why many years researchers [7, 8] have tried to avoid this difficulty with zero gravity experiments. The Japanese team has performed its experiments in a drop tower, whereas we plane to perform zero gravity in an Airbus A300-0G aircraft. This is possible if the plane follows what is called a parabolic trajectory (see figure 8 left). Various electrolysis experiments are performed for different liquid electrolyte and electrode conditions and each time the electrical and video properties are measured. At the beginning of a parabole the gravity level is 1G (normal); first the plane is accelerated at 2G and after 20 seconds, the engines are turned off and the zero gravity experiment begins at 22 seconds. After this the WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

8 Simulation of Electrochemical Processes III engines are turned on and the plane accelerates at 2G for 20 seconds to stabilize and access the normal horizontal trajectory. At 1G and 2G the bubbles are produced and evolve vertically; at zero gravity their motion is “frozen” and they stop almost immediately. They have no motion at 22 seconds (fig. 8 right). The experimental measurement treatments are in progress. These experiments then allow the study of the gas, bath and electrode material in perfect absence of any convective transport during this time, which is really interesting for the twophase electrolysis modelling effort in progress.

Figure 8:

Description of the parabolic trajectory (left) and presentation of the bubbles behaviour for 1G, 2G and 0G (right).

Acknowledgements The authors are grateful to Alcan, CEA and AREVA and the French project AMELHYFLAM (ANR 2007) for their financial support. (AMELHYFLAM is the French acronym for the programme to improve Hydrogen, Fluorine and Alumina industrial production processes by coupled modelling of two-phase and electrochemical phenomena).

References [1] [2] [3] [4]

F. Jomard, J.P. Feraud, J.P. Caire, International Journal of Hydrogen Energy 33 (2008) 1142-1152 H. Kellogg, Anode Effect in Aqueous Electrolysis, Journal of the Electrochemical Society, vol 97 N°4 (1950) Kazuhisa Azumi, Tadahiko Mizuno, Tadashi Akimoto, and Tadayoshi Ohmori, Light Emission from Pt during High-Voltage Cathodic Polarization J. Electrochem. Soc. 146 (1999) 3374 L.J.J. Janssen, J.G Hoogland, The effect of electrolytically evolved gas bubbles on the thickness of the diffusion layer, Electrochimica Acta, Vol. 15 ( 1970) 1020 WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

Simulation of Electrochemical Processes III

[5]

[6] [7]

[8] [9] [10] [11] [12] [13]

[14]

[15]

[16] [17]

9

L.J.J Janssen, C.W.M.P Sillen, E. Barendrecht et S.J.D Van Stralen, Bubble behaviour during oxygen and hydrogen evolution at transparent electrodes in KOH solution, Electrochimica Acta, Vol. 29, No 5 (1984) pp633-642 S.M. Korobeinikov, A.V. Melekhov, Yu. N. Sinikh and Yu. G. Soloveichik, Effect of strong electric field on the behaviour of bubbles in water, in High Temperature, Vol. 39, No 3 (2001) pp368-372 H. Matsushima, T. Nishida, Y. Konishi, Y. Fukunaka, Y. Ito, K. Kuribayashi, Water electrolysis under microgravity: Part I. Experimental technique, Electrochimica Acta, Volume 48, Issue 28 (2003) pp41194125 H. Matsushima, Y. Fukunaka, K. Kuribayashi, Water electrolysis under microgravity: Part II. Description of gas bubble evolution phenomena, Electrochimica Acta, Volume 51, Issue 20 (2006) pp4190-4198 H. Vogt, Ö. Aras, R. J. Balzer, The limits of the analogy between boiling and gas evolution at electrodes, International Journal of Heat and Mass Transfer, Volume 47, Issue 4 (2004) pp787-795 H. Vogt, R.J. Balzer, The bubble coverage of gas-evolving electrodes in stagnant electrolytes, Electrochimica Acta, Volume 50, Issue 10 (2005) pp2073-2079 J. Eigeldinger, H. Vogt, The bubble coverage of gas-evolving electrodes in a flowing electrolyte, Electrochimica Acta, Volume 45, Issue 27 (2000) pp4449-4456 Ph. Mandin, J. Hamburger, S. Bessou, G. Picard, Calculation of the current density distribution at vertical gas-evolving electrodes, Electrochimica Acta, Volume 51, Issue 6 (2005) pp1140-1156 R Wüthrich, L.A. Hof, A. Lal, K. Fujisaki, H. Bleuler, Ph. Mandin, G. Picard, Physical principles and Miniaturization of Spark Assisted Chemical Engraving (SACE), J of Micromech. Microeng., 15 (2005) 268275 Ph. Mandin, H. Roustan, R. Wüthrich, J. Hamburger & G. Picard, Twophase electrolysis process modelling: from the bubble to the electrochemical cell scale Transactions on Engineering Sciences, 2007 WIT Press, Simulation of Electrochemical Processes II, p73 Ph. Mandin, A. Ait Aissa, H. Roustan, J. Hamburger, G. Picard, Twophase electrolysis process: from the bubble to the electrochemical cell properties, Chemical Engineering and Processing: Process intensification, 47 (2008) pp1926-1932 C. Gabrielli, F. Huet, R.P. Nogueira, Fluctuations of concentration overpotential generated at gas-evolving electrodes, Electrochimica Acta, Volume 50, Issue 18 ( 2005) pp3726-3736 Allen J. Bard and Larry R. Faulkner, Electrochemical MethodsFundamentals and Applications, John Wiley & Sons, 2nd. Edition, 2001, 243.

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10 Simulation of Electrochemical Processes III [18] [19] [20]

F. Lapicque, Electrochemical Engineering: An Overview of its Contributions and Promising Features, Chemical Engineering Research and Design, Volume 82, Issue 12 (2004) pp1571-1574 I. Zaytsev, G. Aseyev: Properties of Aqueous Solution of Electrolytes, CRC Press, Boca Raton, Ann Arbor, London, Tokyo (1992) R. Wüthrich, Ch. Comninellis, H. Bleuler: Bubble evolution on a vertical electrode under extreme current densities, Electrochimica Acta 50 (2005) pp5242-5246

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Hydrogen production by the Westinghouse cycle: modelling and optimization of the two-phase electrolysis cell S. Charton1, P. Rivalier1, D. Ode1, J. Morandini2 & J. P. Caire3 1

Commissariat à l’Énergie Atomique, Marcoule – DTEG/SGCS/LGCI, France 2 Astek Rhône-Alpes, France 3 LEPMI, Grenoble INP, France

Abstract Hydrogen is currently viewed as a promising energy carrier for transportation applications. In this context, mass production of hydrogen is a major issue for the coming decades. Among the viable production processes, the hybrid sulphur cycle, or Westinghouse cycle, is studied by the Nuclear Energy Division of the French CEA. In this work, the influence of H2 bubbles on the current distribution within the electrolyser is studied. Turbulent two-phase flow simulations are performed with Ansys-Fluent CFD code in the 3D calculation domain using an Euler-Euler model. The electrokinetic problem is solved in the same domain by a finite element code, Flux-Expert, which is able to compute the secondary current distribution by means of specific interfacial elements. Parameters including the cell orientation (vertical or horizontal electrode), flow regime and bubble size are investigated, and the current model development status and needs are discussed. Keywords: two-phase flow, numerical model, hydrogen, fluent, flux-expert.

1

General context

Hydrogen is a unique zero carbon content energy vector. Its conversion into power and electricity within fuel cells would allow various applications, especially for transportation. Together with fuel cell development and hydrogen storage, hydrogen production is a major issue for the coming decades. The obvious inexhaustible feedstock for hydrogen production is water. Among the WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/ECOR090021

12 Simulation of Electrochemical Processes III potential thermochemical cycles able to split water into hydrogen and oxygen, the Westinghouse process appears particularly attractive for its low theoretical energy requirement (in association with a solar or nuclear high temperature source) and its minimal reagent inventory. The hybrid sulphur cycle (HyS) was developed and patented in the 1970s by the Westinghouse Electric Corporation and, since 2000, has been the subject of renewed interest, especially in the USA and in France. It consists of two steps: an electrochemical step based on sulphur dioxide (SO2) electro-oxidation into H2SO4 (E0 = 0,17VSHE under normal conditions), accompanied by hydrogen gas (H2) production at the cathode, and a thermochemical step in which sulphuric acid is decomposed at high temperature to recover SO2, which is recycled to the electrolyser. Among the phenomena and parameters that must be controlled and optimized in order to meet the requirements of hydrogen mass production, this paper focuses on the issues and modelling of two-phase forced convection flow through the electrochemical cell.

2

Overview of electrolyser modelling approaches

Numerical models for electrochemical process performance assessment or dimensioning generally assume uniform properties or one-dimensional property variations. For example, plug flow with axial dispersion is usually assumed within filter-press electrolysers [1], whereas a Darcy flow model is commonly used within the gas diffusion layer of PEM electrolysers and fuel cells [2]. Industrial electrochemical cells, for which the local current distribution must be precisely determined, mainly involve multidimensional approaches. The electrokinetic problem is then solved using specific finite element codes such as COMSOL or Flux-Expert. Conversely, when hydrodynamic phenomena are predominant or limiting with regard to electrochemical reactions, the flow field is modelled by CDF and the electrokinetic problem is simplified. Focusing on gas-producing electrolysers, Mandin et al. [3] have adopted a Lagrangian approach using Fluent while Agranat et al. [4] have chosen a true two-phase approach using the Euler-Euler algorithm available in PHOENICS. The same model was also implemented by Mat et al. [5], who solved the ionic species transport as well in order to derive the tertiary current density distribution within the simulation domain. In ref. [3] the electrical potential is calculated as a user-defined scalar undergoing a diffusion-like transport equation using Dirichlet boundary conditions at electrodes to obtain a primary current distribution. Chemical species transport is modelled neither in ref. [4], where the current density is assumed to be constant at the electrode surfaces, nor in ref. [5]. The simulation domain is generally two-dimensional and often reduced to single fluid zone. Bubbles are assumed to behave as rigid spheres of constant and uniform size. A few models combine a refined description of both the electrokinetic and hydrodynamic processes. Multiphysics problems are solved by coupling specific codes, either indirectly [6] or by enabling data exchange during the iterative WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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process by means of user-defined library routines. The latter formalism was employed by Jomard et al. [7] who coupled Flux-Expert and Fluent to account for the effect of H2 bubble release on the overall performance of the Westinghouse electrolysis process. The same methodology is employed here. Although gas-evolving electrodes are frequently used in industrial processes, the behaviour of electrogenerated bubbles remains difficult to predict and model. In industrial reactors, large gas release may however strongly influence the electrolyser performance due to hydrodynamic modifications [8] and decreases electrolyte conductivity. These phenomena are examined here, where the process sensitivity to either operating conditions or unknown parameters, such as H2 bubble size and departure angle, is investigated. More fundamental aspects, such as bubble nucleation, growth and departure, are currently being studied within the scope of the AMELHYFLAM (AMELHYFLAM is the French acronym for Hydrogen, Fluorine and Alumina industrial production processes improvement by coupled modelling of biphasic and electrochemical phenomena) project, supported by the National Research Agency (ANR) and dedicated to this theme.

3

Model description

Although the Fluent user-defined functions have been modified in order to optimize data collection and source allocation in the two-phase domain, both the modelling strategy and the simulation domain are similar to the ones developed by Jomard et al. [7]. Hence, only the relevant information and new results are reported here. The exchange data formalism between the two codes is therefore not described. H2SO4 H2SO4 + SO2

H2

    

z

+

x y

X = 0.04m

H2SO4 Y = 0.013m

Figure 1:

Z = 0.16m

H2SO4 + SO2

Filter-press electrolyser configuration ( cathode,  hydrogen release zone,  catholyte,  membrane,  anolyte,  anode).

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14 Simulation of Electrochemical Processes III 3.1 Description of the electrochemical cell The three-dimensional simulation domain is described Figure 1 for a twocompartment filter-press electrolyser. The cathode and anode compartments are separated by a rigid impermeable membrane, to prevent SO2 crossover. The conductivity of a CMX Neocepta® membrane is assumed for simulation. Both Pt electrodes are flat and parallel. Counter-current fluid flow is maintained under steady-state conditions in the plane-parallel compartments. The dimensions are those of the FM01-LC model, manufactured by ICI Chemical & Polymer Company, and used in the Westinghouse pilot test facility implemented in the Marcoule Laboratory [9]. 3.2 Simplifying assumptions Fluid flow and local mass transport effects in the FM01-LC cell has been investigated by Brown et al. [10] in the case of cupric ions reduction to copper using copper-printed segmented electrodes in the longitudinal or transverse directions. Experiments were carried out under laminar inflow conditions, in the range 212 < Re < 855, in a single compartment cell. According to their experimental results, the authors concluded that it is essential to use a turbulent promoter in order to avoid large variations in the current distribution both along and across the flow direction, due to insufficient radial mixing and to the inlet distributor pattern. Moreover, tracer experiments performed by Trinidad et al. [1] in the range 900 < Re < 1900 exhibits plug flow with axial dispersion behaviour and no evidence of dead zone or fluid channelling when a turbulent promoter is used. In the process under study here, where SO2 oxidation is the limiting half reaction, the electrolyte flow rates are controlled by volumetric pumps to ensure forced convection. Moreover, both the anode and cathode compartments are provided with a plastic mesh turbulence promoter. The flow is therefore assumed fully turbulent and a uniform velocity profile is assumed at the inlet. However, for simplification, these devices are not represented in the simulation domain. Although the turbulence promoter should actually influence the bubble population, no reference has been found on its effects. Two additional simplifying assumptions result from the pilot facility operating conditions. First, a high recirculation rate is imposed in both compartments, leading to a low SO2 conversion rate per pass, typically 5%, as given by the mass balance calculation described in [9]. The anolyte composition, CSO2 = 0.24 mol/l, is therefore assumed to remain constant along the cell, in order to achieve nearly uniform current distribution along the electrode. Furthermore, the catholyte loop is provided with an efficient phase separation device. The gaseous H2 volume fraction can therefore be neglected at the cell inlet. 3.3 Governing equations Current density, eqn. (1), is calculated by the finite-element code Flux-Expert solving the Laplace equation for potential, eqn. (2) throughout the computational domain. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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  (1) j     (2)      0 where  is the electrical potential in V and  the electrical conductivity in S.m-1. The SO2 oxidation process is kinetically limited, inducing a non-negligible anodic overvoltage, given by a Tafel electrokinetic law [11]. In order to account for the secondary potential distribution prevailing in the domain, potential jumps are calculated at the anode/fluid interface thanks to the interfacial-type elements provided within the solver. These zero-width finite elements allow electrical potential discontinuities to be managed, as described in ref. [7]. The same formalism is used at the cathode/fluid interface as well to model the proton reduction overvoltage. The fluid electrical conductivity  is assumed to be linearly dependent on temperature in all fluid zones and in the membrane as well. The linear law coefficients were fitted on impedance spectroscopy measurements performed over a wide range of temperatures and H2SO4 concentrations. The catholyte conductivity is also dependent on the gas volume fraction 2 following the Bruggeman relation: (3)    1   1.5



T , 2 



T ,0

2

The materials conductivity values à 50°C are given in Table 1. Table 1:

Materials conductivity at 50°C. 30%wt H2SO4 98,5

CMX 17,5

-1

 (S.m )

50%wt H2SO4 78,5

Both the temperature field and gas volume fraction distribution are computed by Fluent during the hydrodynamic iteration process. The steady-state fluid mechanics problem is solved using the Fluent EulerEuler multiphase model in the fluid domains. Mass, momentum and energy balances, the general forms of which are given by eqn. (4), (5), and (6), are solved for both the liquid and the gas phases. In solid zones the energy equation reduces to the simple heat conduction problem with heat source. By convention, i=1 designates the H2SO4 continuous liquid phase whereas H2 bubbles constitute the dispersed phase (i=2).     i  i ui   S i

 

(4)

 i  i u i . ui   i P     i   i  i g   K ji u j  ui 











 





(5)

j

     i C Pi  i uiTi    eff i Ti     i .u i  S Qi 

 Q ji

(6)

j

The source term Si in eqn (4) is zero in the overall domain but in the H2 release zone. The source term SQi in eqn (6) stands for the heat dissipation rate inherent WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

16 Simulation of Electrochemical Processes III in the electrokinetic processes. The last terms on the right-hand side of eqn. (5) and eqn. (6) represent the interphase interactions, where Kji is computed according to the Schiller and Naumann model for the drag coefficient, as in ref. [5], and Qji using the default FLUENT assumptions. The renormalized k- model of FLUENT is used to compute the Reynolds stress tensors i and the effective properties for each phase. A k- turbulence model is also used in [3] and [8]. A less CPU-consuming model based on mixture-lengths is used in [4]. 3.4 Interphase and source terms In the Westinghouse electrochemical step, hydrogen is released at the cathode interface. In the numerical study, bubble generation is assumed to be localized in the first row of fluid cells neighbouring the electrode. In this special zone analogue to a boundary layer, the rate of gas production is assumed equal to the rate of the reduction process. It is modelled by the source term S2 of the dispersed phase mass-balance equation, assuming a 100% Faradic yield. (7) M H2   S2  j  n int   S1 2eF where MH2 is the gas molar mass (2 10-3 kg·mol-1), F = 96485 C.mol-1 is the Faraday constant and e = 10-4 m the width of the H2 release zone. The interfacial current density is calculated by Flux-Expert. The assumption S1 = -S2 was first made by analogy with boiling, although no significant difference in the simulation results was observed by assuming S1 = 0. Also provided by the finite element solver is the source term SQ in the energy balance, eqn. (6). Except within the H2 release zone, the volumetric heat flux corresponds to heat losses by Joule effect in the conducting materials. In zone , the heat dissipated by the irreversible interfacial processes is computed instead.   (8) SQ    j  n y  e

 

 

3.5 Boundary conditions Material properties are assumed to be constant in the CFD simulation. Data representative of the operating conditions frequently described in the literature [9,11] are used: the cell is operated under atmospheric pressure conditions with sulphuric acid solutions at 30 wt% in the cathode compartment and 50 wt% in the anode compartment, as recommended by the Westinghouse cycle efficiency evaluations. The desired specific flow-rate is imposed at the fluid inlet, where the temperature is maintained at 323 K. Adiabatic conditions prevail at the domain boundaries. A monodisperse population of spherical H2 bubbles is assumed. The bubble diameter was not measured and was arbitrarily set. The gas flow rate was computed from the current density, but regarding bubble impulsion we had a degree of freedom. Thus a constant departure angle θ is imposed to help compute the initial velocity components in the three directions of space (fig. 2). We WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

Simulation of Electrochemical Processes III

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assume therefore that bubbles are ejected perpendicularly (y-direction) and that they are deviated from an angle θ in the flow direction by the convective flow. Since bubbles are not allowed to evolve along the surface (i.e. in the xz-plane), the x-component of the departure velocity is zero. Regarding the electrokinetic problem, an imposed current supply is assumed at the anode corresponding to a uniform current density of 2000 A.m-2. The cathode potential is set at 0 V. CFD simulations were performed in single precision while the electrokinetic problem was solved in double precision. Double precision is required since current density is post-processed from the potential distribution and careful meshing is necessary as shown by Caire and Chifflet [12].

4

Parameter study

From a purely electrokinetic point of view (i.e. for single-phase flow), the calculated cell voltage equal to 0.732 V is mainly attributed to SO2 oxidation kinetics which is responsible for the anodic overvoltage (fig. 3). catholyte

anolyte

catholyte

gz

 vy

x

x

y

vbubble

z

z

x

anolyte

gy

gx

z

Figure 2:

catholyte

anolyte

vz

y

y

Schematic representation of the cell orientations (left) and bubble departure from the electrode surface (right). 0,800 anoodic overpotential

0,700

Voltage (V)

0,600 0,500 ohmic drop membrane

0,400 0,300 cathodic overpotential

0,200 0,100 0,000 0

0,002

0,004

0,006

0,008

0,01

0,012

y coordinate (m)

Figure 3:

Cell potential distribution for single-phase flow (x = 0.02, z = 0.08).

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18 Simulation of Electrochemical Processes III The model was used to study the process sensitivity to the major unknown parameters: the H2 bubble size (50−200 µm) and their departure angle (30−60°). The effect of cell orientation and electrolyte flow rate in the range of the pilot facility operating conditions (75−200 L.h-1) was also investigated. Process sensitivity is measured in term of cell voltage and gas fraction maximum and average values. Gaseous phase spatial distribution is also examined. Major simulation results are summarized in table 2. Three kinds of cell orientation were studied in reference with gravity (fig. 2): i) vertical gas-evolving electrode, like in prior work [3–5,7], ii) gravity vector oriented downward in x-direction, which is the configuration of the pilot facility, and iii) horizontal cathode facing upward. They are referred to as gz, gx and gy respectively. For each case the gas fraction distribution in yz-plane within the cathode compartment is shown in fig. 4. When the flow direction is parallel to gravity (gz), a gas curtain remains confined near the electrode. Bubbles progressively spread within the liquid (in the y-direction) with increasing z. The maximum penetration depth is closed to 20% of the gap width. Additional simulations indicate that increasing flow rate confines the bubbles at the neighbouring electrode, but decreases the overall gas amount since bubbles are conveyed by the liquid flow. The same effect is, to a lesser extent, achieved by increasing the departure angle (Table 2). In the second case (gx), the plume is still present but a second bubble layer develops in the y-direction, expanding in the x-direction according to the gravity field. The gas fraction is therefore increasing, outside the bubble curtain, with increasing distance from the electrode. Gas fraction and voltage distribution contours in the catholyte compartment are illustrated fig. 5. In the gz case (left), the gas fraction increases in the flow direction (z), whereas in the gx case (right) H2 is trapped at the top of the cell, under the effect of gravity. In this configuration, a slight asymmetry can be observed in the liquid flow direction. The voltage distribution reveals the corresponding local variations of the fluid conductivity. In the case of the horizontal cathode facing upward (gy), the plume along the flow direction vanishes and gas accumulates just below the rigid membrane, where the simulation predicts a phase segregation (2  1). Convergence 2

2

2

0.30 0.20 alpha 0.10 0.00 1,5 YY(m) 3,5

1.00

0.20

0,35

1,0

0,2

0,30 0,25

0,20

0,1

0,10 0,05

0,1 10

0,05

0,0035 0,0045 0,0055 0,0065

5,5

y (mm)

Figure 4:

alpha

0,15

0,00

0,0015 0,0025

0.50

0.10 alpha

0,15

0,00

0,00

Z(m) 0,10

z (m)

0,15

0.00 1,5Y (m) 3,5 0,0

0,0015 0,0025

0,10 0,05

0,0035 0,0045 0,0055 0,0065

5,5

0,00

0,00

Z (m) 0,10

z (m)

y (mm)

0,5

0.00 1,5

0,15

0,0

0,10

0,0015 , 0,0025 0 0025

0,05 0,0035 0,0045 0,0055 0,0065

Y (m) 3,5

5,5

0,00

0,00

Z (m) 0,10

z (m)

y (mm)

Gas distribution in the cathode compartment for gz at x = 0.02 (left), gx at x = 0.03 (middle) and gy at x = 0.01 (right) for  =30°, dbubble =100 µm, flow rate 75 L.h-1.

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Simulation of Electrochemical Processes III H2 (Fluent) gz

0,770

0,060

0,768

0,050

0,766

Gx orientation Gz orientation

0,764

0,030

0,762

0,020

0,760

0,010

0,758 0,756

0,000 50

100

d_bubbles (µm)

Figure 6:

V (Flux-Expert)

Surface plots of gas volume fraction 2 and voltage  in the cathodic compartment for dbubbles =100µm,  30°, flow-rate 75 L· h-1 in the gz (near the cathode) and gx (near the membrane) configurations.

0,070

0,040

gx

200

flow Cell voltage (V)

Gas fraction (average)

Figure 5:

H2 (Fluent)

V (Flux-Expert)

19

x z

Bubble trajectory

Left: gas holdup (circles) and cell voltage (triangles) evolutions with bubble size ( 30°, flow-rate 75 L·h-1) in the gx (top) or gz (bottom) orientation. Right: illustration of the bubbles trajectories in the gx configuration for small (top), medium (middle) and large (bottom) bubbles.

troubles were encountered during the resolution process since the dispersed phase became predominant. The same difficulties were encountered for the same reason when simulating the gx - dbubble = 200 µm configuration. The variation of the average gas fraction and the cell voltage with the bubble size is depicted in the left part of fig. 6 for the gz and gx configurations. In the gz configuration, the gas holdup (and correlatively the cell voltage) decreases with the bubble diameter, according to the bubbles rising velocity evolution.

WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

20 Simulation of Electrochemical Processes III In the gx configuration, the bubble free rising direction is perpendicular to the flow direction. Increasing the bubble size is first leading to an increase of the gas holdup. Indeed, the bigger bubbles trajectories are more deviating from the flow trajectory, thus increasing their residence time in the cathode compartment (fig. 6, right). If the bubble diameter is further increased, the reverse phenomenon is observed. In this case, bubbles are rapidly rising from the cathode to the top of the, finite-width, electrolyser, from which they are conveyed straightforwardly toward the cell outlet by the convective flow, with a reduced residence time. Table 2: monophasic gx gx gx gx gx gy gz gz gz gz gz

5

dbubbles / 50 µm 100 µm 200 µm 100 µm 100 µm 100 µm 50 µm 100 µm 200 µm 100 µm 100 µm

Parameter study assumptions and results. Flow rate 75 L·h-1 75 L·h-1 75 L·h-1 75 L·h-1 75 L·h-1 200 L·h-1 75 L·h-1 75 L·h-1 75 L·h-1 75 L·h-1 75 L·h-1 200 L·h-1



2,max

2,mean

/° 30° 30° 30° 60° 30° 30° 30° 30° 30° 60° 30°

/ 0.458 0.726 0.936 0.774 0.258 0.996 0.386 0.367 0.350 0.357 0.295

/ 0.057 0.062 0.046 0.064 0.018 0.044 0.021 0.019 0.018 0.015 0.014

 0.732 V 0.767 V 0.769 V 0.766 V 0.768 V 0.757 V 0.773 V 0.761 V 0.761 V 0.761 V 0.760 V 0.757 V

Conclusion

A parameter study of the two-phase filter-press electrolyser was carried out in the scope of optimizing the Westinghouse electrochemical step. The process sensitivity to biphasic phenomena was qualitatively assessed using a multiphysics model. As compared with the “monophasic” case (i.e. the purely electrokinetic problem solved by Flux-Expert, without CFD coupling) and in the range of conditions explored, hydrogen bubbles are responsible for an additional overvoltage ranging from 25 to 40 mV. Among the investigated parameters, the electrolyte flow rate appeared to have the greatest influence on the cell voltage, irrespective of the cell orientation. The model has demonstrated its versatility, allowing a detailed description of the various phenomena involved, but failed in predicting nearly segregated configurations for which numerical difficulties occurred. As the gas fraction approaches 1, the electrolyte conductivity tends towards 0. Moreover, from a fluid mechanics standpoint, the validity of the Euler-Euler model is questionable and a Volume of Fluid approach should be preferred. Available experimental data, measured under fixed cell voltage conditions, exhibit unexpectedly low current densities. Indeed, although an ideal impermeable membrane was considered for the simulation, non-negligible SO2 crossover occurred during the experiments, due to the poor selectivity of the available membrane. Therefore, an appreciable fraction of the expected current WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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density was consumed by side reactions induced by the sulphur species. The occurring of the latter side reactions, not taken into account in the model, prevents us from validating the two-phase model assumptions, unless a more efficient separator is used. A detailed description of species transport, membrane exchanges and electrochemical processes is required as well to reach a sufficient degree of understanding of the electrolytic cell functioning, and therefore to complete the design and optimization of an industrial-scale process. Such a phenomenological model is currently being developed. Experiments have also been designed to improve the SO2 oxidation overvoltage model, which will be described in a future paper. A phenomenological description of H2 bubbles evolving at industrial electrodes is currently in progress within the scope of the AMELHYFLAM project, and the qualitative phenomena highlighted by this simulation study will be helpful in designing the future cell instrumentation.

Acknowledgements The authors are grateful to the ANR project AMELHYFLAM for financial support and to the LECNA team of the CEA Saclay laboratory for the supplied electrochemical properties.

References [1] [2] [3] [4]

[5] [6] [7]

Trinidad P, Ponce de León C. & Walsh F.C., The application of flow dispersion models to FM01-LC laboratory filter-press reactor. Electrochemica Acta, 52, pp. 604-613, 2006. Duerr M., Gair S., Cruden A. & McDonald J., Dynamic electrochemical model of an alkaline fuel cell stack. Proc. of 16th World Hydrogen Energy Conference, June 13-16, Lyon (France), 2006. Mandin P., Hamburger J., Bessou S. & Picard G., Modelling and calculation of the current density distribution evolution at vertical gasevolving electrodes. Electrochemica Acta, 51, pp.1140-1156, 2005. Agranat V., Zhubrin S., Maria A., Hinatsu J., Stemp M. & Kawaji M., CFD modelling of gas-liquid flow and heat transfer in a high pressure water electrolyser system. Proc. of FEDSM, July 17-20, Miami (Florida) 2006. Mat M., Aldas K. & Ilegbusi O., A two-phase model for hydrogen evolution in an electrochemical cell. Int. Jal of H2 Energy, 29, pp. 10151023, 2004. Roustan H., Caire J.P., Nicolas F. & Pham P., Modelling coupled transfers in an industrial fluorine electrolyser. Jal of Applied Electrochemistry, 28, pp. 237-243, 1998. Jomard J, Feraud J.P., Morandini J., Du Terrail Couvat Y. & Caire J.P., Hydrogen filter press electrolyser modelled by coupling Fluent and Flux Expert codes. Jal of Applied Electrochemistry, 38, pp. 297-308, 2008. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

22 Simulation of Electrochemical Processes III [8] [9]

[10] [11] [12]

Espinasse G., Peyrard M., Nicolas F. & Caire J.P., Effect of hydrodynamics on Faradaic current efficiency in a fluorine electrolyser, Jal of Applied Electrochemistry, 37, pp. 77–85, 2007. Rivalier P., Charton S., Ode D., Duhamet J., Boisset L., Pabion J.L., Gandi F. & Croze J.P., Design study of a pilot test plant for hydrogen production by a hybrid thermochemical process, Proc. of the16th Int. Conf. on Nuclear Engineering, May 11-15, Orlando (Florida), 2008. Brown C.J., Pletcher D., Walsh F.C., Hammond J.K. & Robinson D., Local mass transport effects in the FM01 laboratory electrolyser. Jal of Applied Electrochemistry, 22, pp. 613-619, 1992. Appleby A.J. & Pinchon B., The mechanisms of the electrochemical oxidation of sulphur dioxide in sulphuric acid solutions. Jal of Electroanalytical Chemistry, 95(1), pp. 59-71, 1979. Caire J.P. & Chifflet H., Meshing noise effect in design of experiments using computer experiments, Environmetrics, 13(1), pp. 1-8, 2002.

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A hydraulic model to simulate the hydrodynamics of a fluorine electrolyser J. P. Caire1, G. Espinasse2, M. Dupoizat3 & M. Peyrard4 1

LEPMI, Grenoble INP, France BERTIN Technologies, France 3 AREVA Business Unit Chimie, Secteur Mines Chimie Enrichissement, France 4 ASTEK Rhône-Alpes, France 2

Abstract Fluorine electrolysis is characterized by very large overpotentials and bubble effects that are not yet fully understood. The two-phase free flows appearing in the fluorine reactor are complex and attributable mainly to hydrogen bubbles evolving at the cathode. However, large fluorine bubbles gliding along the anode help to drag the electrolyte up along the anode and in doing so also take part to the two-phase movement. The fluorine electrolyser has been modelled in the past but there has been no means of comparing hydraulic computations with measurements in such an aggressive environment. A hydraulic model is presented here to test the ability of the Estet-Astrid finite volume code to model the fluorine reactor. The two-phase free flow was modelled using an Euler-Euler scheme assuming bubbles of uniform diameter. Laser Particle Image Velocimetry was used to measure both gas and liquid velocities. This paper presents the experimental study and the model made to obtain the plume shape. Numerical and experimental results are compared and some discrepancies are explained. Improvements are suggested for future modelling Keywords: hydrogen, fluorine, electrolysis, two-phase flow, hydraulic model, free convection, Estet-Astrid.

1

Fluorine electrolysis

Fluorine electrolysis is characterized by very large overpotentials and bubble effects [1, 2]. The fluorine electrolyser has been extensively described in [3]. The WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/ECOR090031

24 Simulation of Electrochemical Processes III characteristics of fluorine bubble release are not yet completely understood [1, 4]. The two-phase free flow is mainly due to hydrogen bubbles evolving at the cathode, but large fluorine bubbles also take part in the two-phase movement, as seen in fig. 1a. H. Roustan [1] has shown that to some extent the large gliding fluorine bubbles drag the electrolyte up along the anode. A numerical model was gradually developed in France to account for all the strongly linked phenomena involved in fluorine electrolysis [2–6]. The final numerical model combined two commercial codes, Flux-Expert (FE) and Estet-Astrid (EA), both distributed by Astek [7, 8]. Estet-Astrid is a CFD finite volume code for computing the free convection of two-phase flows. In the present study it was used with an EulerEuler model. The Estet-Astrid code took into account the forces acting on individual bubbles, namely buoyancy, drag force, lift force and added mass. Thanks to EA, Espinasse et al. estimated the fraction of the hydrogen plume that recombines with fluorine and thereby decreases Faraday current efficiency in an industrial fluorine electrolyser [6].

Figure 1:

a) Fluorine and hydrogen bubbles. b) Equivalent scheme of hydrogen plume and fluorine bubbles.

Since the plume of hydrogen bubbles is so important in a fluorine electrolyser, experimental work was carried out to validate the accuracy of plume prediction by EA. KF-2HF molten salt electrolyte is so aggressive that it makes laser Particle Image Velocimetry (PIV) measurements difficult in a large fluorine cell. We therefore designed a hydraulic set-up using water (see fig. 3) in order to simulate the two-phase hydrodynamics of the fluorine electrolyser. We present the specific experimental study performed to determine the shape of the plume and the EA computations. The experimental and CFD results are then compared. Hydrogen bubbles generated in fluorine electrolysis are very small (diameter close to 250 µm). Fluorine behaves in an unusual way: fluorine bubbles slip along the anode instead of detaching from the electrode in the same way as hydrogen bubbles. Fig. 1b) shows a schematic representation of the plume of hydrogen and the bubbles of fluorine that slip along the electrode, partially coated with CFx [4]. The hydrodynamics of a fluorine electrolyser are completely atypical. Roustan [1] showed that free convection of KF-2HF electrolyte was induced by both the hydrogen plume and the slipping film of fluorine, as shown in fig. 2. Electrolyte velocity was measured in the vicinity of the anode by PIV thanks to small alumina beads placed in a laboratory cell. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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Fig. 2 shows that velocity increases with height in an almost linear way along the cylindrical anode at low current density. With an increase in current density, i.e. in the fluorine flow rate on the anode, the velocity increases up to a plateau [1].

Figure 2:

Fluid velocity in vicinity of a cylindrical anode in a laboratory fluorine cell.

This drag effect was interpreted as follows: the almost continuous film of fluorine bubbles adhering to the anode slips upwards increasingly quickly and moves the electrolyte in its vicinity. Because of the large gas cover rate on the anode, the local current density is extremely high and the bubbles interact with the electrolyte, according to us, when small liquid jets exchange mass between the electrode and the bulk in a way similar to that described in Higbie’s penetration model [9]. Thus, the film of fluorine bubbles acts as a mobile wall that moves the liquid roughly at its speed (fig. 2). This idea of a moving wall was used in the hydraulic model. The hydrogen bubbles evolving at the cathode are driven by the molten salt and in the absence of a separator between the anode and cathode compartments (no diaphragm is resistant enough for use in a fluorine electrolyser) they can come in contact with fluorine and recombine spontaneously into hydrofluoric acid. Espinasse et al. [6] showed that when the intensity of the cell increases, Faraday efficiency decreases because of this recombination. It is thus particularly important to accurately simulate the hydrodynamics of the fluorine cell so as to predict its operation correctly. KF-2HF molten salt at 95 °C is very corrosive and extremely delicate to handle. It was thus decided to build a hydraulic model using water and nitrogen bubbles, which is easier to study and presents fewer hazards, the final objective being to validate the EA numerical model used previously for the fluorine cell.

2

Hydraulic model

The 60x60x10 cm scale model made out of Plexiglas is shown in fig. 3. Nitrogen flowing through a gas sparger fitted with a circular sintered-glass filter created a fairly realistic plume of nitrogen bubbles, probably similar to the hydrogen WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

26 Simulation of Electrochemical Processes III

Figure 3:

Scheme of the hydraulic model with the 3 axis directions used in this study.

plume in the fluorine cell. The behaviour of the fluorine in the linear part of fig. 2 was simulated by a notched belt moving at constant speed which sweeps water along in the tank just as fluorine drags the electrolyte in the fluorine cell. The belt was covered with a partially opened casing to simulate an equivalent anode; the flow was therefore channelled only in the vertical direction. The belt casing and its associated electric motor could be moved along the x-axis to visualize the influence of the spacing between the two pseudo-electrodes. The model is not very thick for two reasons: i) PIV cannot be used to measure velocity along the y-axis, ii) free convection movements in the vertical plane are favoured, though Fig. 6 shows that some small parasitic movements appear in the y-direction. Thus it can be estimated that the fluorine film creeping along the anode surface is conveniently represented by the moving belt.

3 Measurements Particle Image Velocimetry (PIV) was used to determine the velocity vectors in the tank. The method is based on the image processing of a laser light sheet created in the vertical z-x plane which allows measurement of the displacement of 14 µm metallised hollow glass beads carried by the flow. Nitrogen flow was injected between 0 and 100 l/h through a circular sintered-glass filter. Ethylene glycol was added to modify the viscosity of the fluid from 1 to 3.4 mPa.s and decrease the diameter of the nitrogen bubbles from 5 to 1 mm, value close to the estimated hydrogen bubbles diameter in the fluorine reactor [13]. The uniform linear speed of the belt could be varied between 0 and 0.3 m/s to simulate in some extent the fluorine behaviour. 3.1 Study of bubble plume alone Since the PIV system could retrieve only one signal at a time for each phase it was not possible to obtain simultaneous measurements of both liquid and WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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nitrogen bubble velocities in the plume. Thus, the velocity vectors were obtained successively through the left window of fig. 4 for the nitrogen bubbles in the plume and through the right window for the liquid.

Figure 4:

The two windows used for measuring gas and liquid velocity vectors in z-x plane.

Figure 5:

Plume of nitrogen bubbles and schematic movements of fluids in the y-z and z-x directions (bubble diameter 5 mm – gas flow rate 50 l/h).

Fig. 5 shows the bubble plume in front view (y-z plane) and profile (z-x plane). In both cases the photo is accompanied by a drawing schematizing the observed flows. For a constant gas input, the total gas/liquid contact surface is greater for small bubbles than for larger bubbles and the gas-liquid interactions are also greater. It was observed that the smaller the bubbles, the greater the liquid velocity, the more agitated and more turbulent the bubble plume, and the more spread out the plume at the air/liquid interface. 3.2 Study of the notched belt alone The speed of the notched belt could be changed to visualize its convective effect on the liquid initially at rest. Fig. 6 shows a photo of the liquid, with the moving WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

28 Simulation of Electrochemical Processes III belt seen at left. The velocity vectors computed from PIV measurements were superimposed on the photo. The picture is of the upper part of the tank taken when the movement of the liquid was instigated by the belt alone.

Figure 6:

Figure 7:

Photo of liquid circulated by the notched with the reference points A, B, C.

PIV velocity of liquid circulated by the notched belt alone.

Fig. 7 presents the liquid velocities measured at A, B, C (see fig. 6) when the belt alone is actuated. A laminar/turbulent transition appears at a belt speed of 0.1 m/s. For speeds higher than 0.1 m/s, the flow becomes turbulent, and the liquid also moves in the third dimension i.e. along y-axis, as seen in fig. 5-left. In turbulent conditions, water is no longer dragged at the belt speed, and the slip velocity becomes significant. A Reynolds number could be assigned to the belt: Reb =

LV ν

(1)

where L is the belt width in m, V the velocity of the belt in m/s and ν the kinematic viscosity in m²/s. With this definition, the laminar-turbulent transition was observed at a Reb value of 2000, i.e. for V equal to 0.1m/s. In laminar conditions, the maximum velocities observed in the liquid (see fig. 7) are very WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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close to the belt speed. At large distances from the belt and particularly at points A, B, and C the influence of the belt is less marked and the liquid velocities are lower. In turbulent conditions, maximum velocity is lower since belt-liquid coupling is lessened and the slip velocity becomes significant. Similar behaviour was observed for a fluorine film in a small laboratory cell when current density was increased i.e. when the velocity of the fluorine film on the anode was increased [1]. This confirms that the moving belt is a convenient method for simulating the fluorine film.

4

3D Numerical modelling

The Estet-Astrid commercial CFD code was used to simulate the hydraulic model. This finite volume code was used with a two-phase Euler-Euler K-ε turbulence model [6]. In this case the code solved a mass balance equation and a momentum balance equation both for the liquid and the gas. In this model, the liquid is regarded as the continuous phase, while the gas constitutes the minority phase, i.e. the dispersed phase. Compared with a Euler-Lagrange two-phase model, strong coupling between phases is easily treated and transients are more easily computed. The model did not take into account bubble diameter distribution (Log-Normal distributions are often observed in this case). However, Antal et al. [10] have shown that distribution of bubble diameters is the most significant parameter for a good description of plumes. The bubbles are regarded as rigid spheres. The two-phase model took into account as in [6] the following forces exerted on bubbles: 1 – gravity, 2 – buoyancy (Archimedes’ force), 3 – the force exerted by the flow on the bubble which results from two terms: the pressure effect, strongly related to the existence of a wake for poorly shaped particles; and the effect of the viscous term that dominates in well shaped bodies, which is the case for a spherical gas bubble. These effects can be split into two forces: a drag force in the direction of incidental flow and a lift force in the direction perpendicular to incidental flow, 4 – added mass, i.e. the force on the liquid related to the particle acceleration. It can be observed that the bubble diameter comes into play in all these forces at power 2 or 3, making the computation very sensitive to this parameter. There are many other forces which could be considered (compressive force due to the liquid, surface tension forces, wake-related forces exerted by neighbouring bubbles, electrostatic repulsion forces between bubbles, etc.). However, these forces were not taken into account in this numerical model. The physical data used for the two phases are shown in Table 1. Table 1:

Physical properties of the continuous and disperse phases.

Pressure Density Kinematic viscosity Inlet mass flow rate Bubbles diameter

H2O 101300 Pa 998 kg m-3 10-6 m2 s-1 0,0124 kg m-2 s-1

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N2 101300 Pa 1.12 kg m-3 2.18.10-5 m2 s-1 0.001-0.005 m

30 Simulation of Electrochemical Processes III 4.1

Initial conditions

Initially the water was assumed to be at rest and the gas fraction null at any point of the system. At initial time, the gas (made up of bubbles of diameter d) was introduced at a constant mass flow rate with a given speed. The problem was thus time-dependent, and therefore solved in non-stationary conditions until stationary conditions were reached. The solution then perfectly described the transient evolution of the gas plume.

Figure 8: 4.2

Boundary conditions and curvilinear structured meshing.

Boundary conditions

Fig. 8 describes the boundary conditions used here for a plume of nitrogen composed of 1 millimetre-diameter bubbles. At gas output, the free surface of the water was assumed to be constantly at rest and the normal liquid velocity vectors null: ∂V = 0 ∂n

(2)

On this symmetry plane, the boundary condition for pressure is as follows: ∂ 2P =0 ∂τ∂n

(3)

This classical condition of EA [8] describes the ideal free liquid surface and let the gas pass freely through it. At walls of the tank the velocities were assumed to be null for the two phases. At gas input, a uniform velocity profile was imposed for the dispersed phase. It was assumed here that the input velocity vectors were tilted at 45° upwards from the horizontal in order to obtain realistic hydraulic WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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conditions. This arbitrary condition was required since convergence problems appeared when the bubbles were injected perpendicularly to the sintered-glass filter. 4.3 Grid mesh Fig. 8 shows the structured curvilinear grid used in these calculations. The grid related to the hydraulic model is made of 18 planes in x-direction, 59 in ydirection, 45 in z-direction and 47790 nodes. No special meshing law was required at tank wall since a no slip condition was imposed in EA [8]. To describe the formation of plume which fully develops in 2 seconds it was necessary to use a time step close to 10-5 second, requiring a computing time of approximately 70 hours on an INTEL bi Xeon 2.5 GHz PC. Fig. 9 presents the velocities calculated by EA for gas bubbles and liquid along AB. The bubble velocity and liquid velocity increase with height. Both velocities have a similar profile on AB, but the bubble velocity is much higher than that of the liquid. The difference, close to 0.38 m/s, is attributed to the slip velocity between bubbles and liquid. This value is much more realistic than the 1.3 m/s free rising velocity of bubbles calculated from the Stokes formula.

5

Comparison of experimental and numerical results

The comparison between measurements and numerical results of fig. 10 shows some disparities. Three zones can be distinguished. In zone 1, close to the sparger at the plume bottom, the diameter of the bubbles is gauged at less than 1 mm. Along the plume, the diameter of the rising bubbles increases, probably by coalescence. Since all the forces acting on the bubbles depend on their diameter, the numerical model overestimates the size of the bubbles in the bottom of the plume. In the second zone, computed and measured velocities are practically identical. It is thought that the bubble diameters have their nominal value in this zone.

Figure 9:

Velocity computed on path AB situated at 2 cm from the left wall.

The third zone is at the top of tank where the plume spreads out. The numerical model again overestimates the gas velocity, probably because the WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

32 Simulation of Electrochemical Processes III effect of the interface on escaping bubbles was not taken into account in this model. Closer examination of the bubble plume in fig. 11-left reveals that the bubble plume widens along the air-water interface whereas the computed plume seen in fig. 11-right is very narrow in this area.

Figure 10:

Comparison between numerical model and measurements for gas velocity on AB.

The difference between model and calculation undoubtedly comes from the effect of the free interface, which was not taken into account here. This model also lacks the interactions between bubbles: in reality, the rising bubbles collide with each other and then lose kinetic energy. The numerous collisions between quasi-rigid bubbles probably explain the widening of the plume seen in fig. 11. Moreover, the numerical model did not take into account the interactions that might exist between the bubbles in the plume and those at the sparger outlet. Nor did the model take into account bubble diameter distribution. This poor description of a plume is not related to EA.

Figure 11:

Photo of nitrogen plume and computed volume fraction of gas.

None of the usual commercial codes satisfactorily describes this type of plume, as underlined by Antal et al. [10] when comparing the three commercial codes CFX, FLUENT and NPHASE with the same plume benchmark. The main WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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issues are probably related to the lack of interactions between bubbles and their environment (other bubbles, wakes of other bubbles, walls, air-liquid interface). Antal et al. [10] have also pointed out that it is not realistic to assume uniform diameters since diameter distributions of bubbles usually follow a Log-Normal law [13]. These problems are encountered in many fields [11], and particularly in the modelling of electrochemical reactors [11–12]. It is thus necessary to take account of these observations in order to improve models of two-phase fluids and obtain more realistic bubble plumes.

6

Conclusions

A hydraulic model giving a vertical two-phase flow similar to that of a fluorine cell was designed and investigated. Experimental results confirmed the similarities in behaviour between the hydraulic model and the fluorine cell. The comparison of numerical and PIV results for the hydraulic model showed that gas and liquid velocity vectors were realistic and of the same order of magnitude. The results were in good agreement for the nominal diameter of the bubbles and discrepancies occurred when coalescence was visible. The belt used as a moving wall behaved in a similar way to the gliding film of fluorine bubbles on an anode. However, the discrepancies observed - particularly for the gas plume reflected the lack of interactions between bubbles in this model. Moreover, the model required a long computing time due to the very small time step necessary to properly describe the transient flows in the tank and attain the stationary state. Further modelling work is in progress within the framework of the AMELHYFLAM project to improve the hydrodynamics of fluorine reactors.

Acknowledgements The authors are grateful to AREVA and the French project AMELHYFLAM (ANR 2007) for their financial support.

References [1] [2] [3] [4] [5]

Roustan H., Modélisation des transferts couplés de charge et de chaleur dans un électrolyseur industriel de production de fluor, INPG thesis, Grenoble, 1998. Roustan H., Caire J.P., Nicolas F. & Pham P., Modelling coupled transfers in an industrial fluorine electrolyser. Jal of Applied Electrochemistry, 28, pp. 237-243, 1998. Nicolas F., Techniques de l’Ingénieur, J 6 325, Editions T.I., Paris. Groult H., Devilliers D., Lantelme F. & Caire J.P., Combel M., Nicolas F., Origin of the anodic overvoltage observed during fluorine evolution in KF-2HF, J. of Elelectrochem. Soc., 149, E485-E492, 2002. Roustan H., Caire J.P., Nicolas F. & Pham P., in J.W. Van Zee, T.F. Fuller, P.C. Foller and F. Hine (Eds), ‘Advances in Mathematical WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

34 Simulation of Electrochemical Processes III

[6] [7] [8] [9] [10]

[11] [12] [13]

Modeling and Simulation of Electrochemical Processes’, Electrochem. Soc. Proc. USA 98-10, 202, 1998. Espinasse G., Peyrard M., Nicolas F., Caire J.P., Effect of hydrodynamics on Faradaic current efficiency in a fluorine electrolyser, Jal of Applied Electrochemistry 37, 77–85, 2007. Flux-Expert, Guide de l'Utilisateur, ASTEK, Paris, 2001. Estet-Astrid, Guide de l'Utilisateur, ASTEK, Paris, 2001. Higbie R., The rate of absorption of pure gas into a still liquid during short periods of exposure. Transactions of A.I.Ch.E. 31 365, 1935. Antal, S.P., Ettorre, S.M., Kunz, R.F., Podowski, M.Z., Development of a next generation computer code for the prediction of multicomponent multiphase flows. In: proceeding of the International Meeting on Trends in Numerical and Physical Modeling for Industrial Multiphase Flow, Cargese, France, 2000. Mandin P., Hamburger J., Bessou S. & Picard G., Modelling and calculation of the current density distribution evolution at vertical gasevolving electrodes, Electrochemica Acta, 51, pp.1140-1156, 2005. Jomard J., Feraud J.P., Morandini J., Du Terrail Couvat Y. & Caire J.P., Hydrogen filter press electrolyser modelled by coupling Fluent and Flux Expert codes, Jal of Applied Electrochemistry, 38, pp. 297-308, 2008. Caire J.P., unpublished results.

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Computational modelling of cathodic protection systems for pipelines in multi-layer soil A. B. Peratta, J. M. W. Baynham & R. A. Adey CM BEASY Ltd, UK

Abstract Computational modelling of cathodic protection (CP) systems involving thin multi-layer media represents a real challenge in terms of accuracy and efficiency required in the numerical calculation. In the case of CP for transmission pipelines, these long metallic structures are usually buried a distance H (approximately a metre or so) below ground level and extend horizontally typically more than ten thousand times H. A number of impressed current anode beds are distributed along the pipeline, providing protection against corrosion of the structure. In addition, the vertically stratified nature of the soil needs to be considered in the model, in order to obtain more accurate representation of the environment. This is particularly relevant when considering the effect of different types of rocks, soil porosity, or water saturation, at different depths. This type of scenario requires three dimensional modelling involving a thin multi layered electrolyte, with a typical aspect ratio (lateral extension to thickness) of the order 1E4 to 1E6. The paper presents an efficient and accurate computational approach based on the Boundary Element Method for simulating the level of protection against corrosion of the pipeline as well as current densities and electric potential in different points of the soil. The resulting modelling approach is then applied to assessing real case scenarios. The simulation approach considers the non linear electrode kinetics on the metal surfaces in the form of polarisation data and also the internal resistance of the pipeline and other electrical connections involved in the CP system. Example applications are presented showing how the model can be used to predict the “signatures” associated with different defect types in the pipe coating. Keywords: cathodic protection, multi-layer, Boundary Element Method, transmission pipelines. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/ECOR090041

36 Simulation of Electrochemical Processes III

1

Introduction

One of the most effective ways to control corrosion in metallic structures embedded in an electrolyte, such as the case of underground or undersea transmission pipelines, is by means of a combination of coating and impressed current cathodic protection (ICCP) systems. Field surveys such as the Direct Current Voltage Gradient (DCVG), and monitoring systems are aimed at different aspects of the assessment of the corrosion control system. Field measurements require skilled technicians, are often expensive and difficult to obtain. In addition, different sections of the pipeline are not always accessible for the surveyors, and in most cases the noise in the potential measurements can mask developing defects in the pipeline coating. Moreover, the interpretation of field data and its correlation to the performance level of the corrosion control system is not always straightforward. The direct computational modelling of CP systems offers a variety of tools for processing the information collected from the field and complementing it, allowing better interpretation of field measurements, as well as enabling the correlation against the CP design parameters and their impact on the observable magnitudes. By direct CP simulation we mean the process of predicting field results such as: • ON and OFF potentials, • potential gradients and currents fields in the soil, as well as • over potential and current density along the pipeline, Provided that the ICCP design parameters such as for example: • geometrical arrangement and material properties of pipelines and anodes, • electrical resistivity distribution of the soil, • type and location of the electrical connections between rectifiers, anodes and structure • Coating breakdown factor along the pipeline are given as input data. The direct simulation of ICCP systems by computational modelling has achieved in the past few years quite a mature state producing reliable results for industrial environments. This type of simulation is particularly useful at the design stage for the analysis of “what-if” case scenarios including the problems of interference with foreign CP systems or metallic structures. The purpose of this work is to present an efficient 3D direct simulation tool based on the Boundary Element Method, applied to two case scenarios involving transmission pipelines. The paper is focused on the effects in the potential measurements introduced by variable breakdown factors associated with the coating. This document is organised in the following way. Section 2 describes modelling approach and provides pointers to the relevant literature involved in the method. Section 3 introduces a simple case for study where the effects of localised damages in the coating along the pipeline can be detected. Section 4 WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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shows a more realistic case scenario involving three parallel pipelines running along the same corridor. Finally Section 5 establishes the conclusions.

2

Direct modelling

The direct modelling consists of obtaining results of electric potential and current density any point in the electrolyte and at surfaces of electrodes. Under most common situations, this requires solving the steady state charge conservation equation in the electrolyte in 3D space given by: ∇⋅ j= 0 , x∈Ω (1) where j = −σ (x) ∇u (x) (2) represents current density, σ is the electrolyte conductivity, u(x) is the potential field, ∇ is the 3D Laplace operator, and Ω represents the integration domain (electrolyte). Eqs. (1)-(2) can be solved together with the corresponding boundary conditions which are usually prescribed by imposing polarisation curves at the electrode surfaces, isolating conditions at ground level, and/or fixed potentials at any known equipotential surfaces in the electrolyte (if any). The Boundary Element Method (BEM) [1] has been widely used to solve Laplacian equations and in particular simulate cathodic protection systems for underground and offshore structures [2–4]. The most significant advantages of the method are first that the formulation is based on the fundamental solution of the leading partial differential operator in the governing equation, and second that it requires only mesh discretisation on the boundaries of the problem. The former aspect confers high accuracy, while the latter substantially simplifies the pre-processing stage of the model, since volume discretisation is not needed. The forward modelling of long transmission pipelines involves considering the soil as a thin film electrolyte (see Figure 2), since the pipeline span (L) is much bigger than the soil depth relevant for the modelling (h). In addition, the soil is generally stratified in one or many layers along the vertical (z) direction. This thin film stratified integration domain is very difficult to solve with standard modelling techniques such as FEM or BEM. Therefore, in order to be able to solve this type of integration domains without the need for extraordinarily high computational resources, a “multi-layer” BEM has been developed (MLBEM). The idea behind the ML-BEM is that the stratified nature of the medium is packaged into the corresponding Green’s function. In other words, the BEM is applied in the same way as in the case of the homogeneous electrolyte, except that the Green’s function for the homogeneous Laplace equation given by 1 / (4π r ) is replaced by the multi-layer Green’s function given by:

G (x i , x j , m, n) =

1 4πσ m

N exp

α ijml

k =1

x i − x j + g ij

∑

WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

(3)

38 Simulation of Electrochemical Processes III where x denotes the 3D coordinates, the sub indices i and j stand for the source and field point, respectively; m and l indicate the layer of the source and field points, respectively; α ijml is a weight coefficient and g ijml denotes a displacement vector. The Green’s function written in this way can be regarded as the one produced by a weighted method of images. The calculation of the weight and displacement vectors goes beyond the scope of this paper and can be derived from references [5] and references therein. Finally, the Green’s function (3) replaces the 1/r kernel used for homogeneous regions, and the same BEM strategy can be employed. Layered non-homogeneous soil CP and Pipeline Network

Typical pipeline network configuration In particular, the condition h<< L can be efficiently handled by the software.

Figure 1:

The soil (electrolyte) considered as thin film stratified media.

3 Case study 1 Figure 2 shows a 16km length of pipeline, composed of 5 electrically connected subsections. At the connection joints between subsections there are four rectifiers delivering constant current to 4 anodes A1 to A4. The table on the right hand side of the figure shows the xy coordinates of the anode in the ground. In the figure, Dz represents the vertical dimension of the anode, which extends down from the ground surface. Details of the pipeline are shown in Table 1: Pipeline specifications. The total current delivered by the anodes is 40 Amps (10 A each).

Figure 2:

Pipeline layout and relevant anode parameters (right).

The soil model consists of two layers, with depths and conductivity as shown in Table 2. The top of the first layer and the bottom of the second layer are insulating. The system of coordinates is such that the pipeline is oriented in x direction from x=0 to x=16000m; the z direction indicated depth (z=0 represents WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

Simulation of Electrochemical Processes III

Table 1:

Pipeline specifications.

Table 2: Layer 1 2

39

Zmin to Zmax 0 to -30m -30m to -100m

Soil properties. Conductivity [S/m] 0.02 0.007

ground level, and z=4m is the centreline of the pipeline). Note that models with this type of boundary conditions are not easily solved with standard techniques such as half space, or method of images [6]. Figure 3 shows the over potential along the pipeline (which is almost the same as the OFF potential calculated above the pipeline at ground level) for different conditions of the pipeline. The red curve corresponds to a uniform breakdown factor BF0 = 1 (100%, bare steel) for the whole pipeline. The green curve corresponds to the case of a uniformly partially damaged coating with BF0 = 0.02 (2%). The black dotted curve corresponds to undamaged uniform coating characterised by uniform BF0 = 0.01 (1%). The Breakdown Factor (BF) is a convenient method for representing general degradation of the pipeline coating. A 10% BF implies that the coating has degraded to the point where an area equivalent to 10% of the surface area is bare metal.

Figure 3:

Comparison between different coating damages in the over potential distribution along the pipeline.

The solid black curve shows the pattern of results produced by 3 localised coating defects at 6000, 8000 and 1000m (with local breakdown factor BF equal WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

40 Simulation of Electrochemical Processes III to 0.02 (2%), 1 (100%) and 1 (100%), respectively) on the pipeline which has an otherwise uniform breakdown factor of 1%. The length of each defect along the pipeline is 2m. It can be observed that the effect of the local damage on the over potential is very localised, and that there is no observable general shift of overpotential. This is because in this case the change of current density is small at regions away from the damage. The pattern observed for the case of 19 localised defects is shown in Figure 4. The distribution of defects is shown in Figure 5 in terms of the breakdown factor in function of length along the pipeline.

Figure 4:

“Signature”Over potential along the pipeline produced by 19 defects distributed as shown in Figure 5.

Figure 5:

BF along pipeline corresponding to the pattern shown in Figure 4.

The method of using breakdown factors in the model to represent local breakdown in the quality of the coating is adequate when the area of damage is relatively large (i.e. larger than a few metres) and the model is capable of predicting what the potential and gradient signatures generated near the pipeline are. This provides a methodology of characterising the size and extent of the defect from its signature. However for more distinct local holidays in the coating a different type of model is required as it is necessary to model the variation of the field around the pipe as discussed in the next section. The simulation tool is also capable of modelling in more detail the effect of localised damage, and the pattern/signature of potential and potential gradient at WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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Figure 6:

41

Detailed representation of a local defect.

ground level produced by small defects in the coating. The rest of this sections shows the simulation results applied to the case of a 0.2 x 0.2 m zone of damaged coating at x = 10000 m. The situation is illustrated in Figure 6 where a three dimensional model of the pipeline is shown. This model is different to the previous in that the elements consider the variation of the coating quality around the pipe. Figure 7 shows the over potential near the holiday. The rest of the pipeline (total length 16 km) is also included in the model and it is considered to have uniform coating breakdown factor of 0.1%. In this case each ICCP anode injects 2A to the system, and an increased number of elements are used in the region that contains holiday.

Figure 7:

Overpotential near a lateral holiday.

The on-potential at ground level with respect to the pipeline can be observed in Figure 8. Figure 9 represents the surface distribution of transversal current density at ground level when the CP system is switched on. The observation plane (at ground level) covers a region of 30 x 15 m centred at x =10000m. It is interesting to compare the different types of pattern produced by the different types and location of defects in the pipeline. In particular, the contrast of voltage and voltage gradient profiles with respect to the case of a healthy pipeline (no holiday at all) is shown in Figures 10 and 11. Figure 10 on the left shows a comparison of longitudinal voltage gradients at ground (z=0) level at y =0.5m (The pipeline is in the plane y = 0, at depth z = WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

42 Simulation of Electrochemical Processes III 4m). The figure compares the case of a uniformly coated pipeline (no defect) and the same pipeline with the holiday (defect). Figure 10 on the right shows the same comparison but with the transversal voltage gradient at different y positions.

Figure 8:

Soil-to-pipe potential at ground level and electrolyte potential on the pipeline.

Figure 9:

Surface distribution of Y component of current density at ground level.

Figure 10:

Voltage gradient in X (left) and Y (right) direction at ground level.

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Figure 11 compares the potential profiles for the pipeline with and without holiday. The picture on the left shows the soil-to-pipe ON-potential at ground level above the pipeline (i.e. y = 0, z = 0). The corresponding over potential profile of each pipeline is shown on the right (z = 4m, y = 0.5m). The results demonstrate how the pattern of electrical signatures observed at ground level due to the presence of small defects on the coatings can be determined in addition to the attenuation of the potential and potential gradients as the observation point moves farther away from the coating.

Figure 11:

(Left) Soil-to-pipe potential. (Right) Over potential on pipeline near the defect.

Figure 12:

Polarisation curve for the metallic structure.

4 Case study 2 The model shown in Figure 13 considers three parallel pipelines of approximately 70km in length sharing the same CP system which consists of 11 ICCP deep anodic beds delivering 10A each. The distance between pipelines is 10m in the local normal direction. The model considers two soil layers (0.02 S/m from ground level to -35m, and 0.005 from -35m to -100m). Figure 14 (left) shows the over potential of the three pipelines along their developed length. In this case, the difference between over potential and OFF potential is negligible. The ohmic drop in the return path due to finite conductivity of the metal is not negligible in large distances, and usually affects the CP parameters. In this example, the voltage difference at different points of the structure is approximately 300 mV, as observed in Figure 14 (right). The maximum WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

44 Simulation of Electrochemical Processes III Pipelines

Anodic bed

Rectifier

Anodic bed

Soil layers

Figure 13:

Schematic representation of case study 2. (Left) Plan view, the dots indicate the location of the ICCP anodes. (Middle and right) Detail of electrical connections at each between anode and the pipelines at each connection point. Table 3:

Pipeline Name Type Pipeline length (m) Ext.diameter (inch) Wall thickness (Inch) Metal resistivity (Ohm m) Breakdown factor

Name A1 A2 A3 A4 A5 A6

Figure 14:

X 266543.4 269839.0 270400.0 274181.0 278114.0 281253.0

Relevant pipeline properties for case study 2.

L De t rho BF

A NPS6 SCH80s 68 442 m 6.625” 0.432“ 1.74E-7 Uniform=50%

B NPS16 SCH 68 442 m 16” 0.656” 1.74E-7 Uniform 3%

Table 4:

Anodes location.

Y 1795525.1 1797394.0 1803073.0 1810219.0 1815776.0 1824888.0

A7 A8 A9 A10 A11

Name

X 285301.0 287517.0 286274.0 287682.0 286435.0

C NPS10 SCH80 68442 m 10.750” 0.594” 1.74E-7 Uniform 100%

Y 1829520.0 1833957.0 1840641.0 1848656.0 1853240.0

(Left) Over potential along the three pipelines. (Right) Metal voltage potential along the pipelines.

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Figure 15:

45

Over potential along pipelines with different coating qualities of pipeline C. (Top) Variable breakdown factor estimated from Close Interval Survey (CIS) measurements. (Bottom) Pipeline C with 100% BF.

variations are registered in pipeline A due to its smaller cross sectional area which introduces a higher resistance per unit length. The simulation has been repeated considering different qualities of the coating for pipeline C. The effects of the quality of the coating of pipeline C on the over potential profile along the neighbouring pipelines can be observed in Figure 15. The top figure shows the case of variable breakdown factor estimated from a CIS survey, in this case, the BF ranges from 0 to approximately 35%. The bottom figure represents the case of pipeline C with no coating at all. The results obtained demonstrate a number of features. First, the feasibility of the modelling approaches for estimating the quality of the coating using data from survey data which is generally spiky and noisy. Second, they reveal how the quality of the coating of one of the pipelines can affect the level of protection in the neighbouring structures.

5 Conclusions A BEM based modelling tool has been developed for the direct simulation of CP systems involving pipelines in multi-layer media. The formulation can represent several layers of geological regions in the soil and does not require the mesh discretisation of the interface between layers. As a consequence, the computational cost of the simulation does not increase with the number of layers.

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46 Simulation of Electrochemical Processes III Two case scenarios were presented in order to illustrate some of the applications of the tool. The first one provides comprehensive results on the impact of the distribution of localised damage to the coating on the over potential from a global point of view. In addition, the effects of a localised coating damage on the local patterns of voltage gradient, soil-to-pipe potential at ground level have been obtained, including both localised degradation and holidays in the coating. In the second case, the interference between three 70km pipelines running parallel along the same corridor has been modelled. The results obtained demonstrate the models ability to represent the case where the damage in the coating of one of the pipelines affects the level of protection of the others. The modelling tool has been used to investigate cases of interference due to varying quality of coating breakdown factor. Once calibrated, the simulation becomes a powerful tool for analysing different types of scenarios, to conduct optimisations on existing CP systems, to provide better interpretation of survey data, and to improve the survey techniques themselves. In particular the modelling approach can be applied to interpret survey data in regions where access is difficult by using adjacent measurements in conjunction with the model to simulate the data required.

References [1] C.A. Brebbia, J.C.F. Telles and L.C. Wrobel. Boundary Element Techniques – Theory and Application in Engineering. Springer Verlag Berlin, Heidelberg NY, Tokyo. 1984. [2] D. P. Riemer and M. E. Orazem, “Modelling Coating Flaws with NonLinear Polarization Curves for Long Pipelines,” in Corrosion and Cathodic Protection Modelling and Simulation, Volume 12 of Advances in Boundary Elements, R. A. Adey, editor, WIT press, Southampton, 2005, 225-259. [3] D. P. Riemer and M. E. Orazem, “Application of Boundary Element Models to Predict the Effectiveness of Coupons for Accessing Cathodic Protection of Buried Structures,” Corrosion, 56 (2000) 794-800. [4] R.A. Adey, J. Baynham. Design and Optimization of Cathodic Protection Systems using Computer Simulation. CORROSION 2000, Paper \ 723. Houston, Texas. NACE International, 2000. [5] Andres B Peratta, John M W Baynham, and Robert A. Adey. A Computational Approach for Assessing Coating Performance in Cathodically Protected Transmission Pipelines. CORROSION 2009, Paper 6595 Atlanta, Georgia. NACE International 2009. [6] J.D. Jackson. Classical Electrodynamics. 2nd ed. John Wiley & Sons. NY, Chichester, Brisbane, Toronto, Singapore. 1975

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Numerical modelling of cathodic protection systems for deep well casings A. B. Peratta, J. M. W. Baynham & R. A. Adey CM BEASY Ltd, UK

Abstract This work is focused on the 3D simulation of cathodic protection (CP) systems for metallic deep well casings immersed in stratified soil. The soil is considered as a multi-layered electrolyte with electric conductivity, which varies from layer to layer. Given a CP system, the aim of the simulation is to predict the level of protection (including the normal current entering the structure) against corrosion along the well casing. It is common practice to fill the annular space between soil and metallic structure with cement along the whole depth. The paper presents a modelling approach which incorporates this feature. In addition, a unique CP system may be used to protect more than one well. In this case, the simulation is used to analyse problems of interference and load unbalance to the electrical resistance in the power lines and flow lines. The modelling approach is aimed at simplifying the model design, especially for dealing with multi-layered electrolyte, and improving the accuracy over existing traditional techniques, which entail a detailed and time consuming representation in the model of the interfaces between different layers of the electrolyte. The effects of different scenarios of electrolytes on the CP system are considered. Keywords: cathodic protection, Boundary Element Method, deep well casings.

1

Introduction

The predictive simulation of cathodic protection (CP) systems applied to well casings provides a number of advantages at the design, monitoring and optimization stages of the project. Accurate modelling capabilities help to gain better understanding of the relationship between different CP variables, the infrastructure to be protected and its surrounding environment.

WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/ECOR090051

48 Simulation of Electrochemical Processes III An effective method to monitor corrosion in buried structures is to measure the pipe/soil potential along its span. With horizontal pipelines this is technically feasible; however the access to data on the local CP performance in well casings is very limited and can only be obtained using expensive logging tools. Deep well casing depths vary usually from a few hundred meters to a few kilometres. Along its depth the casing may penetrate different geological regions with varying physical properties including water content, porosity, permeability, and salinity among others. Therefore, from the modelling point of view, the casing can be considered as exposed to a stratified (multi-layered) electrolyte of varying electrical conductivity. Moreover, from layer to layer the polarization properties of the metallic structure may vary in addition to the characteristic conductivity, due to the corrosivity of the medium. The electrical conductivity of the soil and its distribution are key parameters, which determine how current flows between the anode and the metallic structure, and therefore strongly influence the levels of corrosion protection along the casing. The modelling of CP systems involves predicting the current and potential fields at any point in the electrolyte and at surfaces of electrodes. Under most common situations, this requires solving the steady state charge conservation equation in the electrolyte in 3D space given by: ∇ ⋅ j = 0 , x ∈ Ω where j = −σ (x) ∇Ve (x) represents current density, σ is the electrolyte conductivity, Ve is the potential in the electrolyte measured against remote earth, and Ω represents the integration domain (electrolyte). The Boundary Element Method (BEM) [1] has been widely used to solve Laplacian equations and in particular simulate cathodic protection systems for underground and offshore structures [2–6]. The most significant advantages of the method are first that the formulation is based on the fundamental solution of the leading partial differential operator in the governing equation, and second that it requires only mesh discretisation on the boundaries of the problem. The former aspect confers high accuracy, while the latter substantially simplifies the pre-processing stage of the model, since volume discretisation is not needed. The standard BEM is traditionally aimed at solving homogeneous electrolytes. In case of non-homogeneous conditions it is common practice to combine BEM with domain decomposition or multi-region (MR) technique. In this way the electrolyte, considered as piecewise homogeneous, is represented as a collection of sub-regions, each one of them with homogeneous conductivity. Then, neighbouring regions are connected to each other by prescribing continuity of potential and normal current density through their common interface [3]. In this paper a new technique for computing accurately and efficiently 3D problems involving deep well casings immersed in multilayered soil is used and compared against the traditional MR. The goal of the Multi Layer (ML) approach is to avoid including in the model the interfaces between different layers, by employing a fundamental solution specifically designed for multilayer materials. The major advantages over the more traditional MR approach are mainly the reduction in the engineering time to prepare a model, since interfaces do not need to be included in the model. Also the solving time is minimised, since only the degrees of freedom representing the well casing and the anodes WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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need to be considered. An important consequence of this feature is that the number of layers will not significantly affect the computational cost of the calculation. Moreover, a common limitation in BEM is that the distance between two elements of the mesh must not be too small in comparison to the characteristic length of the largest element, otherwise accuracy and stability of the solution is compromised. This translates into a practical limitation in modelling thin layers of electrolyte, or models in which the thickness of the layer is small in comparison with its lateral extension. However, the ML approach does not suffer from this limitation, and therefore allows the end user to include thin layers without major problems of accuracy or high computational cost. The paper is organised as follows: Section 2 provides briefly some background information for the ML. Section 3 presents a comparison between MR and ML results in a particular case scenario. Section 4 demonstrates the modelling capabilities of subsurface thin layers. This type of modelling is difficult and time consuming with the standard MR approach, but rather straightforward to conduct with the ML. Section 5 presents a method for modelling the cement in the annular space between the outer tube and soil. Section 6 presents the case of a single anodic bed protecting two distant casings. Overall conclusions are given in Section 7.

2

Multi layer modelling approach

The modelling of CP systems protecting well casings is intrinsically three dimensional (3D). Hence 3D BEM kernels are required. The ML approach incorporates the stratified nature of the medium into the corresponding Green’s function. Then BEM is applied in the same way as in the case of the homogeneous electrolyte, except that the Green’s function for the homogeneous Laplace equation given by 1 / (4π r ) is replaced by the multi-layer Green’s function given by:

G (xi , x j , m, n) =

1 4πσ m

N exp

α ijml

k =1

xi − x j + g ij

∑

(1)

where x denotes the 3D coordinates, the sub indices i and j stand for the source and field point, respectively; m and l indicate the layer of the source and field points, respectively; α ijml is a weight coefficient and g ijml denotes a displacement vector. The calculation of the weight and displacement vectors goes beyond the scope of this paper and can be derived from ref [2, 3, 7–9, 11] and references therein. Finally, the Green’s function (5) replaces the 1/r kernel used for homogeneous regions, and employs any standard BEM strategy. In this case, collocation direct BEM has been adopted.

3 Case study: comparison between MR and ML A number of comparison tests and validations of the ML have been performed using as reference the MR technique. This section introduces a conceptual model WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

50 Simulation of Electrochemical Processes III used to compare the results and calculation performance of both methods. The conceptual model consists of a single well casing and one ICCP anodic bed in stratified soil consisting of 7 layers. The well is 1750m deep and consists of 4 sections of different diameter. The specification of the main metallic tube relevant to the simulation is shown in Table 1. Table 1: Section

Soil layer

1 2 3 4

1 2 2,3,4 4,5,6,7

Main tube specifications. Zmin m 0 300 500 1200

Zmax m 300 500 800 1750

Dext m 0.35 0.25 0.175 0.15

t m 0.012 0.012 0.012 0.012

R Ohm/m 1.37E-05 1.94E-05 2.83E-05 3.34E-05

Figure 2 (right) illustrates the conceptual model. The numbers on the left hand side indicate the conductivity of each layer, while the scale on the right hand side indicates the z coordinate of each interface. The dot circle labelled as “O” in the figure represents the origin of the XYZ system of coordinates assumed in this work. The anode bed, represented by a 30cm diameter by 8 m long cylinder, is located 75 m apart from the well in x direction. Casing Conductivity [S/m]

Anode 75 O

0

Ground level

σ1 = 0.1 300

σ2 = 0.5 800

σ3 = 1.0

900

σ4 = 0.5 1300 σ5 = 0.67

1500

σ6 = 0.001

1600

σ7 = 0.02 2500 z

Figure 1:

Conceptual model of MR (left). ML model (right).

The size of the linear system of equations that needs to be solved in this case would be N DOF × N DOF = 3339× 3339 , where NDOF is the number of degrees of freedom. The top of the anode is at 30m deep and the total current injected into the CP system is IA =10 A. The scenario posed for this case study is similar to the one presented in earlier works apart from minor variations. The discretisation mesh of the MR model is shown in Fig 3. The interfaces between layers need to be discretised and incorporated in the model. On the other hand, the ML requires the discretisation of the casing and anode only, thus avoiding elements on the interfaces and bounding box. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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Figure 2:

51

Mesh discretisation for the MR model.

In this example the casing was discretised with 177 tube elements and the anode with 10 elements. Hence, the total number of degrees of freedom is 187, and the corresponding linear system of equations is of size 187 x 187. The ML model is shown in Fig.1 (right). A comparison between MR and ML for the over potential along the structures is shown in Fig 4. More detailed results of the comparison can be found in ref [3].

Figure 3:

Over potential along depth. Comparison between ML (thick red) and MR (thin blue).

4 Effect of thin layers One of the major advantages of the ML is that it is not limited on the number of layers, horizontal extension or thickness of each layer. In order to illustrate this WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

52 Simulation of Electrochemical Processes III feature and to provide better understanding of the effects of the conductivity of thin layers on the over potential distribution on the whole structure, an uppermost subsurface layer 30 m thick with variable conductivity has been added to the model. This type of study is more difficult to conduct with the MR approach in view of the fine discretisation needed for the interface of the thin layer. The over potential in function of depth for different sub-surface conductivities in layer 0 is shown on the left of Figure 4. It can be seen that the conductivity of the soil in the first layer does not substantially affect the over potential anywhere else apart from the first 30 m in contact with that layer. Each curve corresponds to a different soil conductivity in layer 0 which extends from ground level to 30m. The depth is represented in logarithmic scale, since the main changes of over potential occur in the first few soil layers starting from ground level. Figure 4 on the right shows the average over potential in the region 0 < z < 30m of the casing for different electrolyte conductivity of the first layer. Lower conductivity in the subsurface layer translates into more negative over potentials.

Figure 4:

Over potential along the well casing in the 8-layer soil model.

5 Cement layer This section considers the influence of a cement layer surrounding the casing from the electrical conductivity point of view, i.e. any other effect apart from the ohmic drop has not been considered. Figure 5 (left) shows an arbitrary cross section of the well under consideration. The current density in the concrete is considered to be oriented in radial direction. Under these circumstances, the concrete is introduced into the model as a passive media which introduces an additional ohmic-type voltage drop in the electrolyte. The idea is to derive the equivalent resistance offered by the concrete and use it to adjust the original polarization curve of the steel, as illustrated in Figure 5 (right). The original polarization curve of the steel shown in red is corrected with a linear ohmic drop (dashed line). The resulting polarization curve used in the model is represented with a thick black continuous WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

Simulation of Electrochemical Processes III

53

line. Assuming that the current density vector has only radial component, its magnitude at the internal radius r2 varies linearly with the voltage drop across the cement according to: j (r2 ) = k c r2 log(r1 r2 )∆φ , where ∆φ = φ ( r1) − φ ( r 2) is the potential difference across the cement, and kc is the cement electrical conductivity. For practical convenience, we define the surface distribution of ohmic resistance (ρs ) as ρ s = (r2 k c ) log(r1 r2 ) . Then the over potential (V) of the equivalent polarization contemplating the IR drop across the concrete is corrected according to: V = V − ρ s j . In order to investigate the effect of the concrete surrounding the metallic case, the same model of well casing described above was considered in the following scenarios named “BARE”, “WET”, “DRY” and “MIX”. • • • •

“BARE”: This scenario considers the case of bare steel (no concrete at all). “WET”: The metallic case is inside a cement cylinder of external diameter 0.9m. The concrete is considered highly porous and water has infiltrated. “DRY”: Same as wet, but with lower conductivity. “MIX1”: This scenario assumes that the pipe sections 1,2, and 3 are inside a cylindrical column of dry cement, while section 4 (1200< z < 1750m) is considered to be damaged and therefore with the conditions of the wet case. CURRENT DENSITY

CONCRETE

r1

r2

Figure 5:

OVERPOTENTIAL

∆φ

STEEL

∆φ

Cross section of the well casing (left). Adjustment of the original polarization curve due to the ohmic drop in the cement (right).

The two types of concrete considered are shown in table 2. Table 2: Type of concrete Wet Dry

Cement conductivity. Electric Conductivity 0.025 S/m 0.004 S/m

The details of each case scenario are shown in table 3. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

54 Simulation of Electrochemical Processes III Table 3: Sec tion 1 2 3 4

Span Zmin m 0 300 500 1200

Zmax m 300 500 800 1750

r1

r2

h =r1-r2

m 0.45 0.45 0.45 0.45

m 0.175 0.125 0.0875 0.075

m 0.05 0.1 0.1375 0.15

Figure 6:

Figure 7:

Case scenarios. ρs WET Ohm m2 2.87 2.78 2.49 2.33

DRY Ohm m2 17.95 17.38 15.56 14.59

MIX1 Ohm m2 17.95 17.38 15.56 2.33

Polarisation curves.

Normal current density (top left), over potential (top right), metal voltage (bottom left) and axial current density (bottom right) in function of depth for “bare”, “wet”, “dry” and “mix” scenarios.

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The external diameter of the cement lining is kept constant and equal to 0.45 m, hence the thickness of the cement wall (h) varies along depth. As a consequence the resulting polarization curve is modified with different IR drop resistances depending on the sector of the well considered. Fig 7 shows a comparison of normal current density, over potential, metal voltage and axial current density for the different scenarios. These results indicate how the model can be used to investigate different scenarios and conduct “what if” studies. For example, the ohmic drop across

6 Multiple wells The physical situation of this case scenario is shown in Fig 8. The model corresponds to the bare steel case and includes the IR drop in the connections to the power supply but not those in the associated flow lines. The two wells are protected with a single ICCP anode. R1

R2 I

ICCP anode

WC2

WC1

Soil

75m

Figure 8:

425m

Two well casings connected to a unique ICCP anode.

The anode is 75m apart from WC1 and 425 m apart from WC2. The electrical resistance of the power supply cable is represented with the resistors R1 and R2 of 0.4 and 1.44 Ohm respectively. Therefore the unbalance in the currents flowing to the two wells is not only due to the different resistance in the return path circuit but also due to the ohmic IR drop in the soil. Figure 9 shows the results of normal current density, over potential, metal voltage and axial current flowing along each well casing. As expected, the normal current density on the well casing more distant from the anode (WC2) is significantly lower in comparison to the one closer (WC1). In addition, the over potential on the former is in average approximately 100 mV lower than on the latter, except in the less conductive soil layer at depth between 1500 and 1600 m where the difference in over potential is negligible. In this case none of the well casings show anodic behaviour but with different conditions this may occur and would be predicted by the model. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

56 Simulation of Electrochemical Processes III

Figure 9:

Comparison of normal current density, overpotential, metal voltage and total axial current for the two well casings connected to the ICCP anode.

7 Conclusions A new approach for the numerical modelling of CP systems applied to well casing structures in vertically stratified soil has been developed. A significant advantage over the traditional multi-region technique is the reduction of the number of degrees of freedom. In addition the modelling preparation is simplified, as interfaces between soil layers do not need to be modelled at all. The multilayer (ML) approach can easily represent thin layers of electrolyte without additional mesh discretisation thus substantially reducing the work of the user. A feature of the ML is that the computational cost of the calculation does not increase with the number of layers. The results obtained with the ML are in good agreement with the traditional multi-domain approach. The most important benefit of the ML approach is the simplicity in its application to well casing applications as the data can be prepared very easily and the calculations made within a few minutes. Consequently sensitivity studies on proposed new designs can be quickly completed. The model can used to optimise the use of measurement techniques such as surface E-LogI measurements and CPET logs to improve the cost effectiveness and reliability of CP designs. An approach for incorporating the cement that fills the annular space between the metal and the soil has been proposed and the influence of different qualities WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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of the cement on the CP performance has been quantified. In cases where there is doubt about the quality of the cement in a particular layer the model can be used to assess its impact and identify effective corrosion control options. The model has been applied to multiple well situations where there is the possibility of interference between the wells. Results have been presented where not only the ohmic drop in the electrolyte is considered but also the IR drop in the power supply cables, flow lines and other associated equipment and structures is modelled.

References [1] [2]

[3]

[4]

[5] [6] [7] [8] [9]

[10] [11]

C.A. Brebbia, J.C.F. Telles and L.C. Wrobel. Boundary Element Techniques – Theory and Application in Engineering. Springer Verlag Berlin, Heidelberg NY, Tokyo. 1984. Andres B Peratta, John M W Baynham, and Robert A. Adey . A Computational Approach for Assessing Coating Performance in Cathodically Protected Transmission Pipelines. CORROSION 2009, Paper 6595 Atlanta, Georgia. NACE International 2009. Andres B Peratta, John M W Baynham, and Robert A. Adey . Advances In Cathodic Protection Modelling of Deep Well Casings In Multi-Layered Media. CORROSION 2009, Paper 6555 Atlanta, Georgia. NACE International 2009. D. P. Riemer and M. E. Orazem, “Modelling Coating Flaws with NonLinear Polarization Curves for Long Pipelines,” in Corrosion and Cathodic Protection Modelling and Simulation, Volume 12 of Advances in Boundary Elements, R. A. Adey, editor, WIT press, Southampton, 2005, 225-259. D. P. Riemer and M. E. Orazem, “Application of Boundary Element Models to Predict the Effectiveness of Coupons for Accessing Cathodic Protection of Buried Structures,” Corrosion, 56 (2000) 794-800. R.A. Adey, J. Baynham. Design and optimization of cathodic protection systems using computer simulation. CORROSION 2000, Paper \ 723. Houston, Texas. NACE International, 2000. V. Seremet. Handbook of Green’s Functions and Matrices. WIT Press Southampton, Boston. 2003 H. Ymeri, B. Nauwelaers and K. Maex. Computation of conductance and capacitance for IC interconnects on a General Lossy Multilayer substrate. Active and Passive Elec. Comp. Vol 24. pp 87-114, 2001 T. Smedes, N.P. van der Mejis and A.J. van Gendered. Boundary Element methods for 3D capacitance and substrate resistance calculations in inhomogeneous media in a VLSI layout verification package. Advances in Engineering Software 20. pp 19-27. 1994. M. Roche, J. Vittonato and M. Jebara. “Cathodic Protection Modelling of Deep Wells Casing by 3D software simulation: Comparison with E-Logl and CPET Data”. Paper 08273. NACE Corrosion 2008. J.D. Jackson. Classical Electrodynamics. 2nd ed. John Wiley & Sons. NY, Chichester, Brisbane, Toronto, Singapore. 1975 WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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Functional relationship between cathodic protection current/potential and duration of system deployment in desert conditions A. Muharemovic, I. Turkovic & S. Bisanovic University of Sarajevo, Department of Electrical Engineering, Bosnia and Herzegovina

Abstract This paper analyzes the attenuation of cathodic protection (CP) current/potential in pre-stressed concrete cylinder pipes while using galvanic anodes during system employment. The focus in this paper is on the functional relationship between CP current/potential and the duration of system employment in desert conditions. The purpose of this paper is to reach an approximate formula for the functional relationship current/potential following a deployment of a CP system. Consequently, costly field measurements of said PCCP current/potential can be avoided, thus rendering a more efficient operation of the CP system. These measurements have been taken in desert conditions in sections with very high and low soil resistivity during 22 months. Field measurements confirm that there is a substantial reduction of CP current magnitude following initial CP system deployment. Field data confirm the existence of the correlation between CP current and the time of system exploitation, until the point where full system polarization occurs. For both values of soil resistivity, based on measured current values over time, the paper presents a regression model that is a determined twoterm exponential equation. Keywords: cathodic protection, correlation model, cylinder pipes, protection current, measurements, desert conditions.

1

Introduction

In this paper we analyze the attenuation of cathodic protection (CP) current/potential in conditions of high and low soil resistivity that are typical for WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/ECOR090061

60 Simulation of Electrochemical Processes III desert oasis environments. Our study focuses on cases where CP has been applied to pre-stressed concrete cylinder pipes (PCCP) while using galvanic anodes. We present a functional dependency between CP current/potential and the duration of system deployment for cases of high and low soil resistivity. We conclude that total polarization occurs after approximately six months of system deployment, which is the time needed for polarization of bare steel structures in the electrolyte with low soil resistivity. The results have both technical and commercial value, since they make long-term field measurements no longer necessary. The derived dependency between CP current/potential and the duration of system deployment enables a more accurate computation of current/potential parameters in CP systems. The attenuation of the protection current, until the point of full PCCP polarization in the CP system, is caused by [1–5]:  polarization of the PCCP;  partial wear of the cast zinc anode as a result of CP system operation;  increase of anode resistance RA during system operation. Purpose of this paper is reaching an approximate formula for the polarization current/potential following a deployment of a CP system. Consequently, costly field measurements of said PCCP current/potential can be avoided, thus rendering a more efficient operation of the CP system.

2

Experimental basis and functional correlation

A computerized statistical method that calculates current/potential as functional correlation of duration based on measurement data for CP system is introduced. The method uses the regression and correlation analysis of measurements of current and potentials of the piping network in desert environment. This approach ensures during the time installation of more CP capacity with distributed anodes around the piping network and examination of the protection potentials without need for new expensive measurements. This procedure is recommended for the improvement of the existing and new CP system. Based on this approach, developed program determines the relationship between two or more variables from a group of known values from such variables using regression and correlation analysis. The main objective of the program is, with a set of data and based in a curve as a model, to use the regression analysis to obtain the coefficients of the curve to fit the best correlation (minimal standard deviation) between the mathematical model and the set of known data. 2.2 The measured values and functional relationship for high soil resistivity For high soil resistivity we have measured values of the grounded current over time as presented in Table 1. Measured values are given for several points. Also, in Table 2 are given measured values changes of polarization potential over time. Graphical illustration of the change in current over time is presented in Figure 1.

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Simulation of Electrochemical Processes III

Table 1:

61

The change in the grounded current over time [6,7].

section

st 224+240

st 224+850

st 225+420

st 226+020

st 228+950

t [days]

I [A]

I [A]

I [A]

I [A]

I [A]

10

1,36

1,26

1,08

1,06

1,08

70

1,23

1,06

1,00

1,00

1,00

160 250

1,18 1,06

0,96 0,84

0,94 0,86

0,94 0,82

0,96 0,82

340

0,95

0,70

0,78

0,70

0,68

410

0,84

0,54

0,68

0,62

0,60

530

0,74

0,44

0,58

0,54

0,52

620

0,70

0,40

0,52

0,46

0,48

710

0,68

0,40

0,50

0,45

0,47

 [m]

860

875

1030

1170

970

1,5 1,3

st 224+240

st 224+850

st 226+020

st 228+950

st 225+420

current [A]

1,1 0,9 0,7 0,5 0,3 10

70

160

250

340

410

530

620

710

time of energization [days]

Figure 1:

The change in current over time.

On the basis of the measured values of current over time, presented in Table 1, a sample correlation of the results was conducted so as to obtain the functional relationship of the current change over time. To express correlation, we select an exponential regression line: WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

62 Simulation of Electrochemical Processes III Table 2:

Changes of polarization potential over time [6,7].

section

st 224+240

st 224+850

st 225+420

st 226+020

st 228+950

Vnatural

-23 [mV]

-47 [mV]

-32 [mV]

-42 [mV]

-15 [mV]

t [days]

– V [mV]

– V [mV]

– V [mV]

– V [mV]

– V [mV]

10

429

302

380

228

612

70

442

295

365

242

618

160

491

332

375

298

602

250

430

265

341

284

476

340

328

196

286

268

370

 [m]

860

875

1030

1170

970

Figure 2:

Regression curve for st 224+240. I  I 01e at  I 02ebt

An example of the regression curve for st 224+240 ( = 860 [m]) is given in Figure 2. For this section, the functional relationship is: I  1, 374972e 0,001150t  0, 000124e 0,008793t

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In the case of nonlinear regression, following formula is used to determine the correlation coefficient: sy2

k  1

y2

 1

 ym  yr 2  ym  y 2

where: 

ym – the measured values of the current at given points;



yr – the values of current at given points, calculated with regression curve;



y 

1 n yim – the sample mean of the measured values of current n i 1



at given points. Values of the correlation coefficient k that are close to 1 indicate that the selected regression curve is in close proximity to the measured data. In that case, we say that there is high stochastic nonlinear relationship between the varying values. Another factor that can determine the quality of the stochastic correlation of results is the standard deviation s of the random error component of the measurement results. This is calculated as follows: s

1 n yr  y 2 n  1 i 1



The values of the standard deviation that are close to 0 indicate that the chosen model has a small random error component and that the regression was done properly. In case st 224+240, these factors are k = 0,995124 and Table 3: section st 224+240 st 224+850 st 225+420 st 226+020 st 228+950

Coefficients for the correlation function for given sections.

I 01

I 02

a

b

k

s

1,374972 0,000124

0,001150

0,008793 0,995124 0,024634

1,193506 0,075234

0,002043

0,000042 0,991874 0,040072

1,399827 -0,306529 0,000744

0,000194 0,995527 0,020199

1,165009 -0,107443 0,001445 -0,015353 0,996517 0,019431 1,121880 0,000002

0,001416

0,014730 0,991647 0,030585

WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

64 Simulation of Electrochemical Processes III s = 0,024634, which suggests that the functional relationship between protection current and time of polarization is almost complete. Table 3 gives the coefficients for the correlation function for specific sections. Table 4:

The change in the grounded current over time [6,7].

section

st 261+660

st 262+270

st 262+870

st 263+340

st 263+960

t [days]

I [A]

I [A]

I [A]

I [A]

I [A]

10

4,18

4,72

3,82

4,00

3,98

70

2,50

3,10

2,30

2,55

2,10

160

1,75

2,44

1,78

1,95

1,55

250

1,40

2,20

1,50

1,65

1,35

340

1,14

2,00

1,25

1,44

1,18

410

1,00

1,85

1,12

1,30

1,02

530

0,92

1,75

1,11

1,22

0,91

620

0,90

1,73

1,10

1,21

0,85

710

0,89

1,72

1,10

1,20

0,84

 [m]

35

30

25

35

40

5,3 4,8

st 261+660

st 262+270

st 263+340

st 263+960

st 262+870

4,3

current [A]

3,8 3,3 2,8 2,3 1,8 1,3 0,8 0,3 10

70

160

250

340

410

530

time of energization [days]

Figure 3:

The change in current over time.

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620

710

Simulation of Electrochemical Processes III

Table 5:

65

Changes of polarization potential over time [6,7].

section

st 261+660

st 262+270

st 262+870

st 263+340

st 263+960

Vnatural

-8 [mV]

-10 [mV]

-13 [mV]

-7 [mV]

-10 [mV]

t [days]

– V [mV]

– V [mV]

– V [mV]

– V [mV]

– V [mV]

0

560

615

624

540

552

20

570

585

565

529

523

80

480

545

435

454

440

250

387

430

352

389

370

340

327

354

283

314

306

 [m]

35

30

25

35

40

The contribution of the second factor of regression equation during the first year of operation of the CP system does not surpass the value of 2% of total protection current. Thus, we can ignore the second factor in the regression equation during the first year of operation, without any significant effect on the accuracy of results. If we analyze the data in Table 3 we see that the I02 and b coefficients in regression function are becoming more significant when the values of protection potential fall below the allowed protection level. This coefficient becomes important only when we have low resistivity of surrounding soil or a prolonged period of operation of the CP system. This computerized method can be run on a personal computer and can provide timely design support and aids the professional designer in predicting, with a greater degree of certainty, the performance of CP systems. 2.2 The measured values and functional relationship for low soil resistivity For low soil resistivity we have measured values of the grounded current over time as presented in Table 4. Measured values are given for several points. Also, in Table 5 are given measured values changes of polarization potential over time. Graphical illustration of the change in current over time is presented in Figure 3. Table 6 gives the coefficients of correlation function for specific sections. If we analyze the data in Table 6, we can see that the I01 and a coefficients in regression function are becoming more significant when the values of protection potential fall below the allowed protection level. This coefficient becomes important only when we have a low soil resistivity or a prolonged period of operation of the CP system. On the basis of the measured values of current over time, presented in Table 4, a sample correlation of the results was conducted so as to obtain the functional relationship of the current change over time. To express correlation, we select an exponential regression line: I  I 01e at  I 02ebt WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

66 Simulation of Electrochemical Processes III Table 6: section st 261+660 st 262+270 st 262+870 st 263+340 st 263+960

Coefficients for the correlation function for given sections.

I 01

I 02

a

b

k

s

2,973364 1,614316

0,014779 -0,000966 0,998193 0,065652

2,748315 2,417513

0,017469 -0,000544 0,998371 0,056016

2,555295 1,648338

0,016530 -0,000682 0,995738 0,083149

2,515544 1,841271

0,015385 -0,000693 0,997693 0,062871

1,777417 2,875912

0,001187 -0,025777 0,999124 0,042028

The conclusions we have made about changes in protection current values can be used effectively in the periodical recording of CP parameters in similar facilities, which cannot be considered as being isolated by means of a proper passive isolation. This is a simpler and more efficient way to reach valuable and realistic conclusions, especially in difficult climates, about:    

a realistic calculation of CP system lifetimes (seeing as the protection current decreases, as does the driving voltage), a realistic calculation of anode grounded configuration from the viewpoint of anode/solution resistance, considering that there is a decrease of total protection current, an estimation of the length of protective zone (a decrease of protection current and PCCP potential), the elimination of costly and long measurements, CIPS recording, as the values of protection current can be determinate at any moment during the time of exploitation of the CP system

Naturally, the defining of approximate analytical equations to express the functional relationship of changes in potential and in protection current values during the time of exploitation, are followed by appropriate recordings on the concrete object installed in desert ambient [6–9].

3

Conclusion

Extensive and cost – intensive field measurements confirm that there is a substantial reduction of CP current magnitude following initial CP system deployment. Thus, there exists a need for repetitive field measurements in order to establish and confirm the scope of CP protection for the object of interest [10,11]. Field data shown in this paper confirm the existence of the correlation between CP current and duration of system deployment, until the point where full system polarization occurs. Specifically, the case of extreme desert conditions necessitates a high soil resistivity, where the CP current decays WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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67

exponentially with the duration of deployment. The same conclusion can be applied in case low soil resistivity, but with other functional relationship. This paper provides a set of parameters which enables a computation of CP current at any point of system deployment thus enabling CP system architects to estimate the CP current without making expensive filed measurements.

References [1] A. Muharemovic, Electric power system and environment, Sarajevo, 1996. [2] J. Morgan, Cathodic protection, Second edition, published by National Association of Corrosion Engineers, Houston, Texas, 1993. [3] W. Baeckmann, W. Schwenk & W. Prinz, Handbook of cathodic corrosion protection, Gulf Publishing Company, Houston, Texas, 1997. [4] A. W. Peabody, Control of pipeline corrosion, Second edition, published by National Association of Corrosion Engineers, Houston, Texas, 2001. [5] L. Lazzari & P. Pedeferri, Cathodic protection, Polipress, Milano, 2006. [6] Energoinvest Sarajevo, External cathodic protection system – scope of work, 19051-S-395-10-MC-0028-00, Sarajevo, 2006. [7] Energoinvest Sarajevo, Potential survey procedure, 19051-S-395-10-MC0028-00, Sarajevo, 2006. [8] Energoinvest Sarajevo, External cathodic protection project for PCCP, Operation & maintenance period final report, 19051-S-395-10-MC-002800, 2006. [9] A. Muharemovic, I. Turkovic & A. Kamenica: Basic specifics for the assessment of Zinc anode strings for cathodic protection, 13th International expert meeting Power Engineering, Maribor, 2004. [10] NACE Standard RP 0187, Design considerations for corrosion control of reinforcing steel in concrete, NACE Int., Houston, Texas, 2000. [11] NACE Standard RP 0169, Control of external corrosion on underground or submerged metallic piping systems, NACE Int., Houston, Texas, 1996.

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Optimization of a ship’s ICCP system to minimize electrical and magnetic signature by mathematical simulation S. Xing1,2, J. Wu1,2 & Y. Yan1,2 1 2

State Key Laboratory for Marine Corrosion and Protection, P.R. China Luoyang Ship Material Research Institute, P.R. China

Abstract In this paper, the protective effect and distribution of the magnetic field of cathodic protection (CP) systems designed by experience were calculated by the boundary element method (BEM), and the calculation results had agreement with the measured results when the ship berthed at port, which suggested that calculation results are accurate. The performance of the CP system was optimized by moving the position of the anodes, as well as changing the output current of the anodes. The objective was to design the system to minimize the electric and magnetic field and provide adequate protection. Calculation results show that when the anodes were symmetrically installed at the 130th frame and the 232nd frame, the corrosion related magnetic field and total output current were reduced by about 40% and 8.6% respectively. Keywords: cathodic protection, mathematical simulation, boundary element method, electric and magnetic field.

1

Introduction

Coatings combined with impressed current cathodic protection (ICCP) are the most common means for the protection of shipboard. They interact with each other to protect shipboard. Coatings provide primary corrosion protection by isolating the hull metal from the seawater, while ICCP provides secondary corrosion protection in those areas where the paint is damaged or degraded. The protective effect is directly related to the ICCP configurations; an incorrectly designed ICCP system would not only influence the protective effect but also WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/ECOR090071

70 Simulation of Electrochemical Processes III influence the distribution of Underwater Electrical Potential (UEP) and corrosion related magnetic (CRM), which are known as the signature of the ship. Electric fields and coincident magnetic fields arise around a ship due to the current flow from the ICCP system. UEP and CRM signatures exist even in the absence of a cathodic protection (CP) system. They are caused by the galvanic potential differences between the metallic structures in contact with the seawater. For example, the hull and propeller provide sufficient driving potential to create an electric field. These electromagnetic fields take the form of steady electric and steady magnetic fields arising from the steady flow of current around the hull of the vessel. Modern mines can detect these fields and use them to detect and classify passing ships. Researches [1–4] suggest that the design and optimization of the ICCP system is an effective means of preserving the integrity of a vessel and controlling the signatures. The design of an optimized ICCP system requires accurate placement of the anodes and reference electrodes on the ship’s hull. Many shipboard ICCP systems currently in use were designed using relationships and engineering judgment. However, the protection effect of the CP system designed by experience cannot be known and sometimes it might lead to under or over protection and a high electric signature. Today, computer simulation techniques based on BEM, such as BEASY, have enabled the electric and magnetic field generated by the galvanic interaction of the ship’s metallic structure and ICCP system to be predicted. Thus, it provides a tool to predict changes in the protection level of the ship and the electric field in the seawater caused by the ICCP system In this paper, BEASY was used to predict and optimize the distribution of potential electromagnetic field signature.

2

Experimental

2.1 Model A vessel with a length of 125 meters was investigated. The ship had 250 frames with an interval of 0.5m and the frames were assigned 0 to 250 from bow to stern. The geometry of interest in the boundary element model was the wetted surface of the hull and major appendages. The ship has two propellers and two rudders, and the propellers were made of nickel-aluminium-bronze alloy (NAB) and modelled as solid disks with a surface area equivalent to the real propellers. The shaft is made of carbon steel and the propellers and shafts were assumed to be uncoated because of turbulence engendered by propeller movement. The ship’s hull and rudders are also made of carbon steel, which were coated to prevent corrosion. The ICCP system evaluated included four anodes and a centre controlled power supply. The half of ship BEM model was shown in Figure 1. 2.2 Boundary condition The cathodic polarization curves of uncoated and coated carbon steel were shown in figure 2(a) and the polarization curve of NAB was shown in figure WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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71

z y

x Figure 1:

BEM model of the ship.

2(b), which was set as the boundary condition of the ship model. During calculation, the polarization curve of coated steel was set as the boundary condition of the ship’s hull and rudder, the polarization curve of uncoated steel was set as the boundary condition of the shafts and the polarization curve of NAB was set as the boundary condition of the propellers. A box of 2500 meters was created as the infinite boundary where the current density was zero. 2.3 ICCP optimization The optimization process required the problem to be posed in the formed of an objective function, designed variables and constraints. The least squares of the potential on the element of the surfaces with respect to the potential target value of the surfaces were used as an objective function to smooth the potential [5, 6]. ns

ne

obj   (ij  t arg et ,i ) 2

(1)

i 1 j 1

Subject to the following constraints on the surface of the cathode [3]

gi  gj 

i   max,i  min,i

 min, j   j  min, j

 0 on C

i=1， 2， …， n

(2)

 0 on C

j=1， 2， …， n

(3)

where ns is the number of surfaces ne is the number of elements per surface φtarget is the target potential per surface max, j and min, j are the max and min protection potential respectively. 2.4 Magnetic fields The CRM signature generated by the currents following in the sea water could be calculated by solving the vector potential [1]:

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72 Simulation of Electrochemical Processes III

 2 A   J

(4)

where

A = vector potential J = vector of current density components The magnetic field is then simply B = curl A.

3

Results and discussion

3.1 The performance of initial ICCP system on shipboard On the vessel single zone four anodes ICCP system was used to protect the hull and appendages. The anodes were symmetrically installed at 142nd frame and 242nd frame and the initial input current was 8A and 10 A respectively. SSE reference electrode was installed at 186th frame. When the reference electrode read of -850mV (vs. SSE), the potential distribution on the hull was shown in Figure 3. The calculation results indicated that the hull, propellers and rudders were under protect, the range in potential of hull was from -846mV to -1000mV. The current output of anode at 142nd frame and 242nd frame was 5.74A and 7.40A respectively. At the same time, the practical ship protective potential of 37th, 72nd, 128th, 168th and 228th frame was measured when the ship berthed at port by Cu/CuSO4 reference electrode. Compared with the measured results, the calculation results showed the same trend as the practical measured data and were near the same values as the measured results, as shown in figure 4. That is to say the calculation results are accurate. The CRM field distribution on the sea level was shown in figure 5, where x axis was parallel with the keel of ship and y axis was vertical to the keel. The keel of the ship was at the position of y=0 m. The calculation results indicated that the most strength magnetic field position was the region around 142nd frame where anodes were installed and then was the position of stern around 240th frame, while the normal of the magnetic field at the two positions was inverted. In figure 5, figure 7 and figure 9, the same color denoted the same strength of the magnetic field and ever color represented the same gradient of magnetic field. -3.4

-4.0

-3.6

-4.5 -5.0

-3.8

-5.5

coated steel uncoated steel

2

lgi(A/cm )

-4.0

2

logi(A/cm )

-6.0 -6.5 -7.0 -7.5

-4.2 -4.4 -4.6

-8.0

-4.8

-8.5

-5.0

-9.0 -9.5 -1100

-1000

-900

-800

-700

Potential/mV (vs SSE)

(a) carbon steel Figure 2:

-600

-500

-5.2 -1100

-1000

-900

-800

-700

Potential/mV (vs SSE)

(b) NAB Boundary condition.

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-600

-500

-400

73

Simulation of Electrochemical Processes III

vs. SSE Figure 3:

Potential plot of vessel protected by the initial ICCP system. -0.80

measurement results calculation results

Potential/V (vs SSE)

-0.82

-0.84

-0.86

-0.88

-0.90

0

50

100

150

200

250

frame number

Figure 4:

Potential vs. frame.

0

20

Y/m

0

-20

Keel of the ship -40 0

50

Figure 5:

100

150

Frame number

200

CRM field generated by the initial ICCP system.

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250

Increase Inverse increase

nT 40

74 Simulation of Electrochemical Processes III 3.2 ICCP system optimization 3.2.1 Anodes position optimization Based on experientially designed ICCP system, one group of anodes was set at 232th because the propellers consume more current. The other group of anodes was initially set at 115th frame, the candidate positions were set at 115th, 120th, 130th, 135th, 142nd and 150th frame. By comparing the potential and magnetic field distribution of different anodes setting ICCP system, the best ICCP system was sought, which anodes were installed at 130th frame and 232nd frame. When the initial input current of frame1 and frame2 was 5.7A and 7.4A respectively and the reference electrode read of -850mV vs. SSE, the potential distribution of ship hull was shown in figure 6, which ranged from -821mV to -1000mV. The current output of anode at 130nd frame and 232nd frame was 5.32A and 7.51A respectively. After optimization, the CRM field distribution on sea level was shown in figure 7. In figure 7, we can know that the most strength magnetic field position changed to 130th frame, the relative strong magnetic field position changed to 160th frame. Compared with the CRM distribution of initial ICCP system, by moving the position of anode, although the CRM field strength increased at stern, the magnetic field around ship was improved as shown in figure 8, which show the magnetic field distribution at y=0 from bow to stern. The magnetic signature increase between 150th frame and 200th frame may be related to the distance increase between the two group anodes. The distance increase resulted in the current density increase between them, which led to the magnetic signature increased. By anode position optimization, the magnetic field strength was reduced about 30%. 3.2.2 Anodes output current optimization Once the position of anode was fixed, the output current of anodes optimization was carried out. The purpose of current optimized was to minimize the corrosion

vs. SSE Figure 6:

Potential plot of the vessel protected by the optimized anodes position ICCP system.

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Simulation of Electrochemical Processes III

nT Keel of the ship

20

Increase Inverse increase

40

Y/m

0

0

-20 -40 0

Figure 7:

50

100

150

Frame number

200

250

CRM field distribution after anodes position optimization.

0

Figure 8:

CRM field distribution at y=0m.

related electromagnetic signature while the hull obtained sufficient CP effect. The initial input current of frame1 and frame2 was 5.3A and 7.5A respectively. After current optimization, the potential distribution on the hull was shown in figure 9. Compared with figure 3 and figure 5, the protective potential obviously changed more positive after current optimization, but the protective potential lower than -800mV, while the current output of anodes at 130th frame and 232nd minimized to 5.08A and 6.93A respectively. Thus by anode position and output current optimization, the output current was reduced 8.6%. The magnetic field was further reduced after current optimization as shown in figure 10. Compared with the magnetic field generated by initial ICCP system, the strength of magnetic was reduced about 40%. Like the magnetic field of anode position optimized ICCP system, the magnetic field around the position where anode was installed was most strong. In order to more intuitively exhibit the magnetic field distribution of initial and optimized ICCP systems, the WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

76 Simulation of Electrochemical Processes III

vs. SSE Figure 9:

Potential distribution after anodes position and output current optimization. nT Increase

40

Keel of the ship 20

Inverse increase

Y/m

0

0

-20 -40 0

Figure 10:

50

100

150

Frame number

200

250

CRM field distribution after anodes position and current optimization.

magnetic signature at y=0 was shown in figure 8. In this figure, we can distinctly know that the magnetic field around ship was step by step reduced by position and current of anodes optimization.

4

Conclusions

The BEM can reasonably predict and optimize the potential distribution and minimize magnetic signature of vessel. Anode position optimization is an effective approach to control CRM signature, for the ship we studied, when the anodes were symmetrically installed at 130th frame and 232nd frame, the signature was least. The magnetic signature was reduced about twenty-five percent by the anode position and output current optimization.

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References [1] E. Santana. Diaz, R. Adey, J. Baynham et.al, Optimisation of ICCP systems to minimise electric signatures, www.beasy.com/publications [2] R. Adey and J. Baynham, Predicting corrosion related electrical and magnetic fields using BEM, www.beasy.com/publications [3] E. Santana Diaz and R. Adey, Optimisation of the performance of an ICCP system by changing current supplied and position of the anode, Boundary Elements XXIV, pp. 1-16.2003 [4] E. S. Diaz, A Complete Underwater Electric and Magnetic Signature Scenario Using Computational Modelling, www.beasy.com/publications [5] E. Santana Diza and R. Adey, Optimising the location of anodes in cathodic protection systems to smooth potential distribution, Advances in Engineering Software, 36, pp. 592, 2005 [6] E. Santana Diza and R. Adey, Predicting the coating condition on ships using ICCP system data, www.beasy.com/publications

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Numerical analysis assisted monitoring method for the coating condition on a ballast tank wall K. Amaya1 , A. Nakayama1 & N. Yamamoto2 1 Graduate School of

Information Science and Engineering Tokyo Institute of Technology, Japan 2 Research Institute Nippon Kaiji Kyokai, Japan

Abstract We developed the numerical analysis assisted monitoring method for the coating condition on a ballast tank wall. We proposed that the coating condition is evaluated with the surface resistance. We developed the identification method to obtain the whole surface resistance from the differential potential induced by the impressed current from an optional anode inside a tank. We introduced differential potential measurement and inverse analysis to obtain the surface resistance representing the coating condition. The potential measurement and quantitative evaluation were conducted in the actual ship. The verification was performed and there was the good agreement between the proposed method and the preliminary visual inspection. Keywords: corrosion analysis, numerical analysis, surface resistance, inverse problem, coating condition, ballast tank.

1 Introduction In order to prevent corrosion loss, it is very important to inspect the coating condition inside the ballast tank of ship, such as oil tankers, LNG ships and cargo ships. The current inspection standard defines that the ratio of paint defect surface area in the total surface is visually inspected every two or three years periodically. However, the current visual inspection has some problems, such as the dangerous and dark environment for inspectors, the inaccurate evaluation result depending on the inspector’s skill and the time consuming and heavy labor required in order to inspect all the hundreds of tank compartments. In order to overcome these issues, WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/ECOR090081

80 Simulation of Electrochemical Processes III a quantitative, safe, economic and efficient monitoring method for the coating condition is extensively expected. A ballast tank filled with seawater is easily corroded. Corrosion protection by the paint on the metal surface inside the tank, which improves the insulation for the corrosion current, is conducted. The paint has problems with age-related degradation and incipient failure. To protect from the corrosion caused by these problems, plural sacrificial anodes are usually installed in the tank. When seawater is loaded in the tank, the surface of the inside tank becomes cathode and the protective potential works, because of the anode effects. The worse the coating condition becomes, the worse the insulation of the paint becomes and the lower the surface resistance becomes. Therefore, there is the possibility that the coating condition can be evaluated with the monitoring of the surface resistance. On the other hand, the potential measurements are sometimes conducted for the evaluation of the cathodic protection. However because the environmental factors, such as temperature, affect the measured potential and the potential is determined by the number of anodes and the coating condition, the coating condition could not be evaluated directly by the potential measurement. In this paper, we developed the numerical analysis assisted monitoring method for coating condition on ballast tank wall. We proposed the coating condition is evaluated with the surface resistance. We developed the identification method to obtain the whole surface resistance from the differential potential induced by the impressed current with an optional anode. We introduced differential potential measurement and inverse analysis to obtain the surface resistance representing the coating condition. The potential measurement and quantitative evaluation were conducted in the actual ship. The verification was performed.

2 Proposed method 2.1 Coating condition and surface resistance The relationship between potential φ[V] and current density i[A/m2] in an electrolyte near the metal surface represents the function called polarization curve as shown in Eqn. (1). The function is normally non-linear and is determined experimentally. The polarization curve depends on kinds of metal, electrolyte and environmental conditions such as temperature, pH, and concentration of oxygen. In this paper, it is noted that the potential φ at a certain location in electrolyte is defined with referring to the metal and has the inverse sign of the potential E employed in electro-chemistry in which the potential is defined to a reference electrode such as SCE or Ag/AgCl. φ = −fm (i)

(1)

Let us consider the small change of Eqn. (1). δφ  −

dfm δi = −R · δi di

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(2)

Simulation of Electrochemical Processes III

R≡

dfm di

81 (3)

The surface resistance R[m2 ] makes the relationship between the potential change and the current density change. When the resistance is big, it indicates high insulation. Therefore, we propose that the coating condition can be indicated by the surface resistance R. In the case that the non-coating surface resistance is R0 , the ratio of the paint defect area against total surface area can be evaluated with the following [6] (Eqn. (4)): α=

R0 R

(4)

2.2 Summary and procedure of proposed monitoring method Let us describe the proposed monitoring method in this section. An optional anode and an Ag/AgCl reference electrode for potential measurement are placed in the seawater filled in a tank to be inspected as shown in Figure 1. The potential changes at several location in the tank with reference electrode are measured in the two cases. The first case is that the prescribed current is impressed with an optional zinc anode and the second case is that no current is impressed. Each case is represented with subscript ”ON” and ”OFF” respectively ¯ in the following. The differential potential δ φ(= φ¯ ON − φ¯OFF ) is calculated from the results. Potential in any point is calculated with various R using a numerical analysis. It is assumed that the surface resistance R is constant in this numerical model. Let us consider the residual of the measured differential potential δ φ¯ and the numerically

Optional anode (Zn)

A V

Anode (Zn)

Seawater

Reference electrode

SS wall

Figure 1: Simple measurement model in ballast tank. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

82 Simulation of Electrochemical Processes III calculated δφ(R). (Eqn. (5)) F (R) =

N  {δ φ¯ j − δφj (R)}2

(5)

j

Where N is the number of the combinations of the potential measurement points and the anode locations. Obtaining the surface resistance R is to search R which minimizes Eqn. (5). The general optimization method, such as Brent’s method [5], can be applied for searching R. 2.3 Analysis method for differential potential In this section, numerical analysis method to calculate the potential φ(R) or the differential potential δφ(R) for the assumed surface resistance R is described. First, let us consider the mathematical model of the electrical field in the tank. It is assumed that the surface of electrolyte domain  is surrounded by (= d + n + m ), where m is the metal surface, and the potential and current density are prescribed on d and n respectively. The potential φ in 3-D homogeneous electrolyte domain  satisfies the Laplace’s equation: ∇ 2φ = 0

in 

(6)

The general boundary conditions are given with φ = φ0   ∂φ = i0 i ≡κ ∂n

on d

(7)

on n

(8)

−φ = fm (i)

on m

(9)

where φ0 and i0 are the prescribed values of potential φ and current density i, respectively, ∂/∂n is the outward normal derivative, and κ is the conductivity of the electrolyte. The actual boundary condition for numerical analysis is described in Table 1 in section 3.3. The potential φON and φOFF correspond to the case (ON) with the optional anode and the case (OFF) without it respectively. δφ = φON − φOFF

(10)

Since the two potentials φON and φOFF satisfy the Laplace equations, the differential potential δφ also satisfies the equation. ∇ 2 (δφ) = 0

in 

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(11)

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83

Eqn. (11) implies that in case there are two boundary conditions for one model, the following three conditions are applied instead of Eqn. (7) to (9). δφ = φON 0 − φOFF 0 on d

(12)

δi = iON 0 − iOFF 0 on n dfm δi on m −δφ = fm (iON ) − fm (iOFF )  di

(13) (14)

In case that δi in Eqn. (14) is small, the approximation becomes better in terms of Taylor expansion. Also, the constant term of polarization curve in Eqn. (1) representing open circuit corrosion potential can be eliminated and it only depends on the surface resistance dfdim = R. Eqn. (1) becomes Eqn. (15) δφ = −R · δi

(15)

The constant term depends on the environmental conditions such as temperature, pH, concentration of oxygen and the reference electrode offset. But the differential method without the term has advantages on them, in case that the same reference electrodes are used in the short-time measurement. This formulation easily eliminates the effect of open circuit corrosion potential and reference electrode offset. If the potential or current density are constant in two boundary conditions, the differential boundary conditions are zero according to Eqn. (12) or Eqn. (13). A boundary element method is employed for the numerical analysis to solve Eqn. (11) with Eqn. (12) to Eqn. (14). The formulation of the boundary element method is the same as the standard boundary element method with Eqn. (6) to Eqn. (9).

3 Verification of differential method 3.1 Experiment summary Potential measurement was conducted in a ballast tank of the 23 year-old LNG ship. The LNG ship has ten ballast tanks and each ballast tank partitioned to approximately ten compartments with bulkheads transversely and vertically. One of the compartments is measured by the present method. Five years have passed since the paint-coating of the ballast tank was repaired. The measured compartment is a rectangular parallelepiped (L 3.75 m × W 4 m × H 12.8 m) filled with seawater up to 7.6 m in height. There were eight zinc anode electrodes (L 0.1 m × H 0.1 m × W0.5 m) in the seawater within the compartment as shown in Figure 2. The measurements were conducted with the optional anode (φ15 mm×L150 mm) and both (ON) and (OFF) were measured. The potential along the center line was measured with an Ag/AgCl reference electrode every 0.2 meters back and forth between top and the bottom of the seawater. The current from the optional zinc anode was measured. HIOKI 8422-50 logger and HIOKI 3257-50 DMM were used for the potential and current measurement respectively. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

84 Simulation of Electrochemical Processes III Center line

Depth 5.2 m

(measuring line)

1.6 m Optional Anode

Anode x 8

7.6 m filled with seawater

Hole to the lower compartment Depth 12.8 m 3.75 m

4m

Figure 2: Ballast tank compartment with seawater. 0.902 0.900

1ON

0.898

1OFF 2ON 2OFF 3ON 3OFF

Potential[V]

0.896

(ON) with optional anode

0.894 0.892 0.890 0.888 0.886

(OFF) without optional anode

0.884 0.882 0.880

5

6

7

8

9

10

11

12

13

14

Depth[m]

Figure 3: Potential distributions (optional anode depth 1.6m from seawater surface).

3.2 Measurement results One set measurement takes about fifteen minutes. The optional anode was located at the depth of 1.6m from the seawater surface in the first three sets (Fig. 3). The electrode measuring the potential was attached to the cable on edge. The depth of Figure 3 is actually the cable length from the ballast tank ceiling and the depth more than 12.8 meters shows that the electrode reached the bottom. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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85

Node: 3619 Element: 5670

Figure 4: Mesh for compartment. Seawater

2 surface

1 Optional Anode

4

5

SS walls

Anode x 8

7.6 m

3 Access hole to the lower comartment

4m

3.75 m

Figure 5: Ballast tank compartment with seawater. 3.3 Numerical analysis and surface resistance identification In order to verify the proposed differential method, the differential potential distribution within the tank are compared between the numerical result and the experimental result. Firstly, the differential potential distribution within the tank shown in Figure 2 was measured. The experimental condition is set to be the same in the former section. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

86 Simulation of Electrochemical Processes III

Table 1: Boundary conditions of differential model for the compartment. No.

Boundary condition

Place

1  2  3 

δI =30 [mAmps]

Optional anode side (ON)

δi=0

Seawater surface

δφ=0.005 [V]

Access hole to the lower

4 

δφ = −RSS · δi

compartment SS (walls) RSS =1000, 500, 200, 100, 50, 20, 10, 5 or 2 [m2] 5 

δφ = −0.16δi

Zn (anodes)

Secondly, the tank is modeled as shown in Figure 5 for the numerical analysis. The boundary element mesh in Figure 4 was used and the elements are all constant triangle elements. The number of the elements and nodes are 5670 and 3619, respectively. The boundary condition is applied as shown in Table 1. The surface resistance of the anodes is set to be 0.16 [m2 ]. The total current amount of the optional anode is set to be 30mA and the access hole boundary condition is set to be δφ=0.005 [V] from the experiment. The differential potential distributions are calculated for nine cases whose surface resistances RSS are varied. The numerical results for nine cases of the surface resistances are shown in Figure 6. It shows that the whole potential is high when the surface resistance is high. The maximum voltage of the sidelobe of the optional anode is 0.005 [V]. Thirdly, the surface resistance was estimated by minimizing the residual function Eqn. (16). The residual function consists of three residuals between measurement and calculated differential potential at the depth of seven meters. The minimization was performed with the brute-force search. F (RSS ) =

 {δ φ¯ j − δφ(RSS )}2

(16)

j

The both numerical and experimental potential distributions are shown in Figure 7 and they show good agreement. It is noted that the good agreement can be achieved even if the open circuit corrosion potential is unknown, polarization curve is non-linear and unknown and there are the offsets of reference electrodes. The paint defect ratio is lower than 0.2% under the assumption of the surface resistance of bare SS R0 = 1[m2]. (See Eqn. 4) This result agrees with the preliminary visual inspection. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

Simulation of Electrochemical Processes III

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0.018 0.016 R=1000 R=500 R=200 R=100 R=50 R=20 R=10 R=5 R=2

0.014

Voltage [V]

0.012 0.010 0.008 0.006 0.004 0.002 0

5

6

7

8

9 10 Depth [m]

11

12

13

Figure 6: Voltage distributions with various surface resistances. 0.018 Meas. 1

0.016

Meas. 2

Voltage [V]

0.014

Meas. 3

0.012

Analysis

0.010 0.008 0.006 0.004 0.002 0 5

6

7

8

9 10 Depth [m]

11

12

13

14

Figure 7: Voltage distributions (depth 1.6m from seawater surface, SS surface resistance: 500 [m2 ]).

4 Conclusion We developed the quantitative monitoring method for coating condition inside a ballast tank. We proposed the coating condition is evaluated as the surface resistance. We developed the identification method to obtain the whole surface resistance from the differential potential induced by the impressed current from an optional anode inside a tank. We introduced differential potential measurement and inverse analysis to obtain the surface resistance representing the coating condition. The potential measurement and quantitative evaluation were conducted in the actual ship. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

88 Simulation of Electrochemical Processes III The both numerical and experimental potential distributions agrees well with the differential method. The good agreement can be achieved even if the open circuit corrosion potential is unknown, polarization curve is non-linear and unknown and there are the offsets of reference electrodes. The paint defect ratio calculated from the differential method agrees with the preliminary visual inspection.

References [1] Brebbia, C. A., 1978. “The boundary element method for engineering”, Pentech Press, London, pp. 1–178. [2] Amaya, K. and Aoki, S., 2003. “Effective boundary element methods in corrosion analysis”, Engineering Analysis with Boundary Elements, 27, pp. 507–519. [3] Aoki, S., Amaya, K., Nakayama, A., and Nishikawa, A.C1998. “Elimination of error from nonuniform current distribution in polarization measurement by boundary element inverse analysis”, CCorrosion, 54(4), pp. 259–264. [4] Aoki, S., Amaya, K., Urago, M., and Nakayama, A. 2004. “Fast multipole boundary element analysis of corrosion problems”, CMES-Computer Modeling in Engineering & Sciences, 6(2), pp. 123–131. [5] Atkinson, 1989. An Introduction to Numerical Analysis (2nd ed.), Section 2.8. John Wiley and Sons. [6] Purcar, M., Deconinck, J., Van den Bosshe, B., Bortels, L., and Stehouwer, P.J., 2005. “Numerical 3D BEM simulation of a CP system for a buried tank influenced by a steel reinforced concrete foundation”, Simulation of Electrochemical Processes, WIT Transaction on Engineering Sciences, 48, pp. 47–56. [7] DeGiorgi, V.G. and Hogan, E.A., 2005. “Experimental vs. computational system analysis”, WIT Transaction on Engineering Sciences, 48, pp. 37–46. [8] W.H.Press, 1992. “Numerical recipes in C”, Cambridge University Press.

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The influence of coating damage on the ICCP cathodic protection effect J. Wu1,2, S. Xing1,2 & F. Yun1,2 1 2

State Key Laboratory for Marine Corrosion and Protection, P.R. China Luoyang Ship Material Research Institute, P.R. China

Abstract A one/hundredth scale hull model in linear scaled resistivity was used to simulate the cathodic protection potential distribution of a hull with different coating damage extent and position. The potential distribution of the hull was simultaneously measured by a PXI modular instrument and program written by Labview 7.1. At the same time, the potential and current density of the vessel with damage of 2%, 4% and 6% were calculated by the boundary element method (BEM). The results obtained from the physical scale model (PSM) method had agreement with the calculation results. The impressed current cathodic protection (ICCP) protection effect and its induced electric signature were simultaneously influenced by the coating damage extent and position. The damage extent mainly influenced the ICCP output current and the x axis’ electric signature; the damage position mainly influenced the electric signature of the y axis. Calculation results show that the electric signature increased about 35.7% due to coating damage of 6%. Keywords: cathodic protection, physical scale model, computer simulation, electric signature.

1

Introduction

Corrosion damage is a major factor in ship maintenance and availability. Coatings combined with the impressed current cathodic protection (ICCP) system are the most common means for shipboard corrosion control. They interact with each other to protect the shipboard. Coatings provide primary corrosion protection by isolating the hull metal from seawater, while ICCP systems protect the hull by applying an external source of current to the ship where the paint is damaged or degraded. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/ECOR090091

90 Simulation of Electrochemical Processes III The cathodic protective effect is related to the ICCP designed and coating state. Before the 1980s, the effect of ICCP systems designed by experience could not be exactly known. In the past decade, considerable effort has been spent in the development of rapid estimate and design methodologies that have a scientific rather than designer expertise basis. Two such methodologies are the physical scale model (PSM) and computational modelling using boundary element techniques [1]. The PSM technique has been developed to assess the performance of the ICCP systems installed on shipboards since the 1980s. It has been reported that directly proportional dilution of the seawater results in similar potential with reference to miniature electrodes placed along the side of the model as the actual potential on a full-size ship [2]. Apart from evaluation, this technique, which has been used to design ICCP systems installed on US Navy ships, could help in examining the relationships between the coating state and the ICCP protection affect, and optimize the initial position of anodes and reference electrodes. However, the PSM is only physical scale not chemical scale, so it is impossible to overcome the change in polarization behaviour in a scaled resistivity electrolyte [3]. In recent years, computer simulation technology has been developed for calculating the potential and current density distribution in electrochemical systems. Computer simulation techniques can rapidly obtain the potential and current density distribution on the hull with different coating states to estimate the impact of coating damage extent and position on the ICCP system. A primary limitation of computer simulation is the need for accurate and appropriate polarization response data [4], while the PSM does not require polarization response data. The concept of a unified approach between computational simulation and the PSM is very attractive, since it uses the strengths of each method to compensate for the weaknesses of the other [4]. The unified approach uses validated computational tools to examine hull geometry, boundary conditions and other parameters to define a relatively small set of experiments that can be completed in a time and cost efficient manner. According to experience, the coating state plays an important role in the distribution of cathodic protective potential and the electric signature, but there were few reports about how the coating damage influences the potential signature distribution. In order to control the signatures and to preserve the integrity of a vessel, it is essential to predict the impact of the extent of the coating damage and the position of the electric fields and potential distribution. In this paper, PSM and boundary element method (BEM) technologies (BEASY10.0) were used to study the influence of the coating damage on the protective potential electromagnetic signature.

2

Experimental

2.1 Physical scale model A vessel with a length of 125 meters was investigated. The ship had 250 frames with an interval of 0.5m and the frames were assigned 0 to 250 from bow to WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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stern. The scale model of the ship was a one/hundredth scale hull model with the same materials as the ship and the model was immersed in seawater diluted to the same scale ratio. The scale model was shown in Figure 1. The definition of the coating damaged position and area shown in Table 1 was based on observations of the ships in dry-dock after a period of service time. Throughout, the tests were performed at room temperature and potential measurements were made with eight Ag/AgCl reference electrodes fitted with Luggin capillaries. e n i l r e t a w

Figure 1: Table 1: Coating state Damage position

The physical scale model of vessel.

Shipboard coating damage extent and position. 2% damage th

50 frame (symmetrical)

4% damage th

th

120 and 200 frame (symmetrical)

6% damage 50th, 120th and 200th frame (symmetrical)

2.2 Measurement system The protective potential of different positions on the vessel was simultaneously gathered by National Instruments and the Labview 7.1 program. 2.3 Potential measurements The scale model was immersed in a sea water holding tank with the dimensions of 1.8m×0.6m×0.9m and polarized to -850mV for two days for the formation of calcium and magnesium deposition, and then the model was moved into diluted sea water with the conductance of 379μS/cm. The potential of the 10th, 30th, 50th, 70th, 90th, 110th, 130th, 150th, 170th, 190th, 210th, 230th and 250th were measured when the reference electrode potential was set at -850mV. 2.4 Computer simulation The ship has two propellers and two rudders. The propellers were made of nickel-aluminium-bronze alloy (NAB) and modelled as solid disks with a surface area equivalent to the real propellers. The shaft is made of carbon steel and the propellers and shafts were assumed to be uncoated because of turbulence engendered by propeller movement. The ship’s hull and rudders are also made of carbon steel, which were coated to prevent corrosion. The ICCP system evaluated included four anodes and a centre controlled power supply. The half of ship BEM model was shown in figure 2. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

92 Simulation of Electrochemical Processes III z y

x

Figure 2:

BEM model of ship. -400

-500

coated steel uncoated steel

-600

-500

-600

-700 -800

E(mV)

Potential(mV)

-700

-800

-900

-900 -1000

-1000 -1100 -10

-9

-8

-7

-6

-5

-4

-1100 -5.2

-5.0

-4.8

-4.6

-4.4

2

-4.0

-3.8

-3.6

-3.4

lgi(A/cm )

(a) carbon steel Figure 3:

-4.2 2

logi(A/cm )

(b) NAB Boundary condition.

The cathodic polarization curves of uncoated and coated carbon steel were shown in figure 3(a) and the polarization curve of NAB was shown in figure 3(b), which was set as the boundary conditions of the ship model.

3

Results and discussion

3.1 Potential simulation Former studies indicated that the ICCP system performed best when the anodes were symmetrically installed at the 130th frame and the 232nd frame and the reference electrode was installed at the 186th frame. The hull of the ship was under sufficient protection, as shown in figure 4, when the coating was intact. The PSM measured data and computational results had good agreement with each other, except the anode zones. The difference between the PSM and the BEM at the anode zones was most likely due to the change in polarization of the materials in the diluted seawater, because the PSM supposes that dilution of the electrolyte does not significantly change the polarization behaviour of the materials [5].

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When the coating was symmetrically damaged at the 50th frame and the damaged area was about 2% of the whole underwater area, the potential distribution measured by the PSM experiment and calculated by computer simulation was shown in figure 5. According to the PSM experimental results and calculation data, the ICCP system can still provide enough cathodic protection to the hull with the potential ranged from -800mV to -1100 mV when the reference electrode reads -850mV. Because the coating damage and damage positions are far away from the anodes, the potential at coating damage positions changed more positively in comparison with elsewhere. Even though the coating damage area increased, it had little influence on the potential distribution, as shown in figure 6. The hull was under protection and there was no potential wave in the coating damage regions. According to ohm law, the resistance is directly proportional to length, so if the coating damage position was neighbouring the anode, the potential of the uncoated steel could be easily polarized to the designed level, even though uncoated steel consumed more current than coated steel. Similarly, the coating was symmetrically damaged at the 50th frame, the 120th frame and the 200th frame, respectively, and the damaged area was in the same place, the potential variation at the 50th frame was the biggest of the coating damaged regions as shown in figure 7. So, for the ICCP system, when the coating damage extent was less than 6%, the protective potential was mainly influenced by the coating damage position not the damaged area. -800

PSM measure results calculation results

-820 -840 -860

Potential/mV

-880 -900 -920 -940 -960 -980 -1000 -1020 -1040 -1060

0

50

100

150

200

250

Frame number

Figure 4:

The potential (mV) distribution of the ship’s hull when the coating is perfect.

-780

PSM measure results calculation results

-800 -820 -840

Potential/mV

-860 -880 -900 -920 -940 -960 -980 -1000 -1020 -1040 -1060 -1080 -1100

0

50

100

150

200

250

Frame number

Figure 5:

The potential (mV) distribution of the ship’s hull when the coating is 2% damaged.

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94 Simulation of Electrochemical Processes III -800

PSM measure results calculation results

-820 -840 -860

Potential/mV

-880 -900 -920 -940 -960 -980 -1000 -1020 -1040 -1060 -1080

0

50

100

150

200

250

Frame number

Figure 6:

The potential (mV) distribution of the ship’s hull when the coating is 4% damaged.

-780

PSM measure results calculation results

-800 -820 -840 -860

Potential/mV

-880 -900 -920 -940 -960 -980 -1000 -1020 -1040 -1060 -1080 -1100 -1120

0

50

100

150

200

250

Frame number

Figure 7:

The potential (mV) distribution of the ship’s hull when the coating is 6% damaged.

3.2 Electromagnetic signature simulation A line of sampling points was defined at a depth of 10m beneath the sea level, and the vertical distance between simulation points and keel was about 10m. The solution at the sampling points along this line is used to compute the electric signature. The underwater potential was shown in figure 8 and the current density of the x, y and z axes was shown in figure 9. According to the calculation results, the potential gradient near the anodes increased when the coating damage increased and the electric signature increased about 35.7% when the coating was 6% damaged, compared with perfect coating. Because the coating damage positions are far away from the anodes, the coating damage of the 50th frame resulted in the potential gradient increasing in the bow region. It is well known that the polarization current of uncoated steel is obviously larger than that of coated steel. So if the coating of the hull was damaged in some regions, the ICCP system must increase the output current to maintain the hull’s potential under critical potential, and then the total output current increases with the damage extent increase. However, the current density of the x, y and z axes distribution at the same time was influenced by the damage position. From figure 9 we know that the x axis’ current density was mainly influenced by the damage extent, the current density of the y axis was mainly influenced by the damage position and the z axis’ current density was influenced by both damage extent

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-845

perfact coating 2% damage 4% damage 6% damage

-850 -855

Potential/mV

-860 -865 -870 -875 -880 -885 -890 -895 -900

Figure 8:

0

50

150

Frame number

200

250

Underwater potential distribution along the keel.

(a) x axis Figure 9:

100

(b) y axis

(c) z axis

Current density distribution along the keel.

and position. That is to say, the coating damage extent may result in the electric signature increasing along the vessel and the bow’s coating damage may induce the electric signature to increase in the vertical orientation.

4

Conclusions

According to the PSM and computer simulation studies, the ICCP protection effect and its induced electric signature were simultaneously influenced by the coating damage extent and position. The potential gradient and electric signature increased with the increase of the coating damage area and the electric signature increased by about 35.7% when the coating damage area extended to 6%.The damage extent mainly influenced the ICCP output current and x axis’ electric signature and the damage position mainly influenced the electric signature of the y axis.

References [1] DeGiorgi V.G, Thomas E.D, Lucas K.E and Kee A. A combined Design Methodology for Impressed Current Cathodic Protection Systems. Boundary Element Technology XI, Computational Mechanics Pub. 1996: 335-345 WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

96 Simulation of Electrochemical Processes III [2] R. W. Ditchfield, J. N. McGrath and D. J. Tighe-Ford. Theoretical validation of the physical scale modelling of the electrical potential characteristics of marine impressed current cathodic protection. J. Applied Electrochemistry. 1995.25:54-60 [3] I. Gurrappa. Physical and Computer Modelling for Ship’s Impressed-current Cathodic-protection Systems. [4] V.G. DeGiorgi, S.A. Wimmer, E. Hogan and K.E. Lucas. Modelling the experimental environment for shipboard ICCP systems. Boundary Elements XXIV: 440 [5] N Gage and P Mart. Further Cathodic Protection Studies of Naval Ships[J].CAP.2002:1-13

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Field-based prediction of localized anodic dissolution events taking place on ZnAl alloy coatings in the presence of 5% NaCl solution S. G. R. Brown & N. C. Barnard Materials Research Centre, School of Engineering, Swansea University, UK

Abstract An extension to a previously-developed numerical model is presented that predicts the sites of localized anodic dissolution on exposed surfaces of ZnAl alloy coating; often employed in the protection of steel-strip substrate material. Long-established thermodynamic concepts and galvanic-coupling mechanisms are used to identify sites susceptible to local metallic dissolution. Estimation of metal loss over time is linked to the electrode potentials predicted at the exposed alloy surface. Both alloy composition and the concentration of multiple species in the NaCl solution are considered in determining these electrode potentials. The temporal evolution of the Zn-Al-Fe system – represented using a cellular automaton framework – is predicted via field-based calculations on steady-state voltage and non-steady state diffusion/migration fields. Concentration perturbations in the electrolyte are captured and resultant potential fields are generated using a straight-forward finite difference technique on an irregularstructured computational grid. The influence of the microstructural features in the ZnAl alloy coatings is assessed in terms of metal loss, current density fields and pitted-depth. Simulated results are validated at the meso-scale in comparisons made to experimental observations in accelerated testing of these alloys in concentrated aqueous NaCl solutions. Simulations have been performed that quantitatively assess the localized corrosion at the surface of ZnAl alloy coatings and cut-edge scenarios where Zn, Al and Fe are exposed. The model is extended to predict the effect of altering the processing conditions, i.e. cooling rate and coating thickness. Keywords: Zn-Al coatings, galvanic corrosion, de-alloying, finite difference.

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98 Simulation of Electrochemical Processes III

1

Introduction

In an effort to protect steel from the deleterious effects of corrosion it has been a long-standing practice to use zinc and zinc-bearing alloys as coatings. The use of Galfan-type coatings to protect sheet is well-established and forms the substrate for much of the organically coated cladding and guttering of modern buildings. Galfan-type coatings are typically applied via hot-dip routes and the coating thickness, alloy composition and bath chemistry are the focus of much attention [5–7]. A further attempt to model the micro-scale corrosion effects experienced by ZnAl alloy coatings applied to a 0.47mm thickness mild steel substrate is presented here. Whilst the performance of the zinc-aluminium coatings themselves is of interest, the performance of these coating when both coating and steel substrate is exposed, i.e. at a cut-edge, is of primary concern since the coating surface is protected by an organic coating layer. During cooling of the coating from the ZnAl hot-dip bath, at ~4.5wt.% Al, primary zinc dendrites solidify within a predominantly lamellar ZnAl (~5wt.%Al) eutectic. During subsequent accelerated aqueous corrosion testing it is the zinc from the dendritic phase that undergoes anodic dissolution via equation 1. 2 Zn (s)  Zn (aq)  2e  (1) In the case of the surface exposure of the metallic coating layer it is eutectic phase that acts exclusively as the cathodic surface, whereas in the case of cutedge exposure the coupled steel acts cathodically with respect to zinc. In aerated solutions at near-neutral pH, these represent the sites for the supporting oxygen reduction reaction, equation 2.  O 2  2H 2 O  4e   4OH (aq) (2) During the processes described by equations 1 and 2, dissolution from the coating layer gives rise to the deposition of a hydroxide layer; impeding the cathodic oxygen reduction reaction. [8] This reaction involves the rapid hydrolysis of water via equations 3 and 4. Zn 2   H 2 O  ZnOH   H  (3) ZnOH   H 2 O  Zn(OH) 2  H  (4) In order to predict the localized degradation experienced by these steel coatings in the surface and cut-edge situation equations 1-4 need to be adequately described by a model. Here, the evolution and movement of anions and cations, in addition to the effect of electrical potential on the kinetics of these processes is included. The model described here has been developed from a metallurgical standpoint, rather than an electrochemical one and originated as a cellular automata (CA) finite difference model. This approach dealt with the evolution of a representative concentration and electrical potential throughout the electrolyte only [1]. The model was able to predict morphological features such as localized corrosion pits and capping but was limited to qualitative simulation. However the CA method has found alternative applications in the growth of corrosion pits WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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where use of CA and Monte Carlo (MC) has been studied [9], including an associated probability for re-passivation and film formation. The CA approach has also been employed in simulating the simple corrosion process taking place at an electrode [10] and most recently in simulating the initiation and growth of multiple pits [11]. Multiple concentration fields are used here in an attempt to capture the dominant effects brought about by equations 1 and 2 on the spatially discrete anodic and cathodic areas formed during exposure to 5% NaCl solution. Multiion electrolyte simulation has also been documented [12] using a nested radial basis function (RBF) approach to predict concentration profiles around a rotating disk. Here, the evolution of [Mn+], [O2], [OH-] and [H+] fields around a planar interface is predicted, governed by corrosion rates determined using data obtained from the rotating disc technique. Previously [3], quantitative predictions regarding the corrosion experienced at the cut-edge with respect to the cooling rate applied to the Galfan-type coating have been shown to be in agreement with experimental findings [5]. However, the trend in corrosion rate at the surface of the ZnAl coating layer was inconclusive. Refinement of the current model has resulted in the observed trend being replicated in simulations and an overview of the results is given here. As an extension to the work, modelling the cut-edge corrosion behaviour has been extended to predict the corrosion rate experienced by steel that has been coated at different weights in slightly differing compositions [7]. The result of these alterations in processing conditions is a change in primary phase fraction (  Zn) and dendrite morphology. The experimental work carried out by Elvins et al. (2005) [5] and Penney et al. (2008) [7] have been performed using in situ corrosion studies using the scanning vibrating electrode technique (SVET). [13] The SVET measures the voltage drop over an amplitude of ~30µm giving a measure of the vertical current density, Jy (Am-2), 100µm above a corroding surface. An estimation of the material loss experience can then be made from the current density recorded at discrete time intervals.

2

The model

The algorithm used in the model is described in more detail below. There are effectively four main sections to the algorithm: 1. Set up: Geometry and initial values on the computational mesh. 2. Determination of electrical potential throughout the solid electrolyte. 3. Dissolution/deposition. 4. Diffusion/migration of multiple species in electrolyte. 2.1 Initialization The model uses an irregular structured cell-centred finite difference computational mesh. Grid spacings are typically 1-5µm for cells in the coating layer and rise to 10-25µm in steel cells close to the centre of the substrate and WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

100 Simulation of Electrochemical Processes III electrolyte cells far from the solid-liquid interface. The microstructure of the coating layer is entirely synthetic; however the geometry of the dendritic phase and volume fraction of this primary phase is based upon metallographic observation. Once a computational mesh has been set up all variables are initialized and the simulation begins. Each site on the mesh is described by several variables. For electrolyte sites and partially corroded sites these include concentration, Cn, (individual fields for Mn+, dissolved O2, OH-, and H+), volume fraction of corrosion product, p, volume fraction of solid material in a cell (to allow cells to corrode gradually), frac, and electrical potential, ø. Un-corroded solid cells are defined by electrical potential, ø, and whether the site is primary Zn, eutectic, inert (polymeric material) or steel substrate. In all cases below the side faces of the computational mesh possess an insulation boundary condition. 2.2 Electrical potential The first stage in calculating the electrical potential field, ø, is to subdivide all elements into 4 groups; electrolyte only elements, electrolyte interface elements, solid interface elements and solid only elements. Interface elements are any elements where electrolyte and solid material are in contact, they may be solid elements in contact with electrolyte or vice versa. They will also include all those elements that are partially corroded and contain both solid and electrolyte components (i.e. 0 < frac < 1).The electrode potential, ø0 (V) at all electrolyte interface elements is calculated (weighted via areas in contact for any given element) using equation 5. RT  [Ox]   ln 0  E0  (5) nF  [ Red ]  Where E0 is the standard electrode potential (vs. standard hydrogen electrode) in volts, R is the molar gas constant (J.K-1.mol-1), n is the electronic charge, T is temperature (K) and [Ox] and [Red] are the concentrations of the oxidized and reduced form of the species (mol dm-3). F is Faraday’s constant (C mol-1). These calculated electrolyte interface electrical potentials are then used as Dirichlet boundary conditions to determine the electrical potential field throughout the solid elements (solid interface elements and solid only elements) via the Laplace equation. For solid interface elements (which may be partially electrolyte) the electrical conductivity (Sm-1) is calculated via

 *   solid  frac  electrolyte 1 frac

(6)

A precursor step to solving the electrical potentials within the electrolyte is carried out, whereby electro-neutrality is enforced at the solid-electrolyte interface via the adjustment of all solid-containing interface cells. These potentials are held as Dirichlet boundary conditions and the final electrical potential field throughout the electrolyte is calculated using ( electrolyte )  0 (7)

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2.3 Dissolution/deposition The amount of dissolution that may occur at interface elements containing solid over the current time increment, Δt (s), is calculated via equation 8 and the total is defined as idissolution. nF (1   )nF   (8) i  i0 exp   exp  RT RT   Where i0 is the exchange current density (Am-2), α is the transfer coefficient and η is the overpotential (V); (ø0-E0) predicted using equation 5. Parameters necessary to determine the corrosion rate of these cells are derived from rotating disk experimentation carried out in 5% NaCl. Polarization curves obtained for pure zinc and commercially pure aluminium are shown in figure 1 below. In the case of dissolution of the eutectic phase, two rates are calculated for zinc and aluminium and a harmonic mean is applied to describe the rate of degradation, weighted by volume fractions present in the eutectic phase. However, the amount of corrosion that will occur will depend on the maximum cathodic support that is available, controlled by the diffusion of oxygen to the surface. The availability of oxygen during the current time step, Δt, is determined by the availability of oxygen within one diffusion length (m), of the interface. This total is defined as icapacity. If icapacity< idissolution then the amount of corrosion to occur during the current time increment is scaled down uniformly for each element so that icapacity= idissolution. Each corroding interface site will now have its amount of solid material, frac, reduced. Concentration contributions of Mn+, O2 and OH- from each solid interface cell are distributed geometrically to adjacent electrolyte interface cells.

Figure 1:

Polarization curves in 0.86M NaCl of a zinc and aluminium electrode.

It is assumed that 5-10% of the volume of material that is lost during dissolution over the time step is re-deposited at the surface of the material as corrosion product. Electrolyte interface sites are selected for corrosion product deposition according to voltage and concentration tolerances and here further ionic concentration contributions arise in Mn+ and H+. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

102 Simulation of Electrochemical Processes III 2.4 Diffusion Fickian diffusion (not including interaction terms but including migration effects where appropriate) is carried out for four species within the electrolyte, [Mn+], [O2], [OH-] and [H+], according to the standard equation ci  Di ci   z i F(u i ci  ) i  1, 4 (9) t Where Di is the diffusion coefficient (m2s-1) in 0.86M NaCl solution at 298K, Ci is the molar concentration (mol dm-3) and u is the ionic mobility (m2s-1V-1) of species i. Side faces of the computational mesh are insulated whereas the top and bottom faces are both fixed at the initial concentration value. For corroding cells at the interface that contain corrosion products an effective diffusion coefficient, Deff, must be calculated. In the absence of accurate data this is done using a quadratic form [3] so that the effect of corrosion product at the surface is to increasingly prevent diffusion occurring through that cell. Finally, once the diffusion step is completed for all four species, both [OH-] and [H+] are adjusted to enforce the condition [OH-].[H+]=10-14 at all elements containing electrolyte.

3

Results and discussion

During previous investigations the predicted trend at the cut-edge of Zn – 4.5wt.% Al coating layers that have been cooled at different rates replicated that from SVET experimentation [7]. However, the predicted trend reported in [3] did not match that observed. To this end, changes have been made to the solid model that not only better represent the microstructures of the coatings, but also surface depressions around eutectic grain boundaries have been better represented. Again, the cooling rates described here relate to the power output of the coolers that are used after the hot-dip bath and so are expressed as a % of the total capability rather than an absolute cooling rate usually associated with microstructural solidification, 55%, 80% and 100%. All cooling rates simulated correspond to 25µm coating thickness and primary-Zn fraction of 20%. A further examination of the extended model was performed in the cut-edge situation. This involved simulating the localized effect brought about by galvanic corrosion when the weight of the Galfan-type coating applied is altered. Changes in the coating weight and associated changes in the coating composition modelled are outlined in table 1 below. Table 1:

Properties of Galfan-type coatings applied at different weights [7].

Coating Weight (gm-2) 150 255 300

Coating thickness (µm) Al content (%) 16.5 21.5 39.0

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4.5 4.8 4.9

β-Zn (%) 32.6 22.0 16.8

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3.1 Anodic dissolution The anodic dissolution modelled using equations 1 and 8 can be seen to give rise to perturbations in metal cation concentrations from zinc-rich phases present in both surface and cut-edge cases shown in figure 2 below. Despite the fact that the eutectic material is permitted to act anodically and cathodically in the case of the cut-edge corrosive case (figure 2(a)), perturbations predominantly arise from regions occupied by zinc dendrites owing to the weighting applied in equation 8 bringing the dissolution rate of the ZnAl eutectic close to that of pure aluminium.

Figure 2:

Mn+ [M] concentration fields predicted around a solid model of 150gm-2 alloy coating after 9.6 hours exposure. (a) Iso-contours with bulk electrolyte removed, and (b) 2D section of ZnAl coating.

The distribution of cation concentration across the cut-edge is in agreement with studies of a Fe/Zn galvanic couple [13]. Diffusional effects are in evidence in figure 2(b) where the highest iso-contour of cation concentration can be seen to exist within a pit formed by dissolution from a much smaller dendrite. A reduced zinc loss can be seen to be offset by the more tortuous route from the pit, impeded further still by closer proximity of cells containing corrosion product. 3.2 Cathodic oxygen reduction The result of the supporting cathodic reaction, equation 2, on the dissolved oxygen field can be seen for cut-edge case and surface corrosion in figure 3 (a) and (b) respectively. The lowest values of oxygen can be seen in both to be lowest at sites where entire solid removal has taken place, i.e. the longest diffusion path. This has been distinguished in figure 3(b) at A, whereas at B it is the reduction of oxygen that has been the cause of local decrease in oxygen. It can be seen that the cathodic reaction creates a more diffuse field at the cut-edge owing to the presence of the steel substrate. It is only at these sites that the overpotential, η, in equation 8 will be affected as cathodic and anodic processes are identified on a metallurgical basis. 3.3 Cooling rate vs. surface corrosion In a previous attempt to model the surface degradation of Galfan-type coatings it was assumed, as is the case in cut-edge simulations, that the morphology of the WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

104 Simulation of Electrochemical Processes III

(a)

(b)

Figure 3:

Dissolved oxygen (mol dm-3) in 5% NaCl solution after (a) 12 hours contact with 255gm-2 alloy coating at cut-edge, and (b) 19 hours in contact with surface of 80% cooling output.

Figure 4:

Planes showing anodic (light) to cathodic electrical potentials from the solid electrolyte interface to 150µm above; current density vectors also shown.

dendritic phase was the dominant factor in the corrosion rate. However, the trend predicted did not correspond to that in experimentation. It is proposed however, the influence of the depressions around the eutectic cells present dictates the localized corrosion effects that ensue. The cell size has been observed to decrease with an increase in cooling rate [5], giving the greatest opportunity for the dendrites to be present at the surface for the faster cooled coating layers. Figure 4 shows the solid model including the steel substrate at A, primary zinc WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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and eutectic cell depressions with slices showing the electrical potentials into the electrolyte at 30μm intervals. The effect of the surface depressions is evident, where the incidents of anodic attack do not, as was the case in [3], occur evenly over the surface but at positions along depressions. The anodic potentials still correspond to those sites where dendrites make it to the surface. Interestingly, from figure 4 the anodic effects present at the points indicated at B do not consistently impact the voltage field in the electrolyte at regions remote from the corroding surface (where an SVET would operate). The authors are thus reluctant to express the corrosion rate as a function of anode lifetime or dendrite removal predicted at 100μm as in much experimental reporting. Figure 5 shows the predicted vertical current densities 100μm above the solid model. It can be seen from figure 4 that this is a degraded representation but nevertheless a source of validation. A coupled approach to SVET experimentation and numerical prediction also considers this [15]. Figure 5 shows a marked decrease in the frequency of anodic instances from [3] and it is apparent that the reduction in eutectic size corresponds to an increase in corrosion rate. The intensity of the anodic events is now more consistent within simulations and observation. In addition to the increased corrosion rate, decreasing the eutectic cell size coincides with an increase in maximum pit depth: rising from 14.6μm to 19.2μm over a 24 hour period – a clear indication of the relative barrier properties. 0.45

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Vertical current density maps 100µm above ZnAl coating after 12 hours exposure: white -3.5 Am-2 to black +3.5 Am-2. (a) 55%, (b) 80% and (c) 100% cooler output power (length scale mm).

3.4 Coating weight vs. cut-edge corrosion In simulating the cut-edge corrosion at differing coating weights, coating thickness, dendrite morphology and volume fraction of the primary phase will all have a significant influence. Figure 6(a) shows the material loss during the early stages of accelerated corrosion testing can be almost indistinguishable. After 2 hours simulated exposure to 5% NaCl solution differences in the coating layers can be seen to take effect. It is the 150gm-2 coating weight that exhibits the greatest acceleration in mass loss rate. As the steel substrate thickness in all cases is equivalent, the greatest anodic activity would be expected at this coating weight since the cathode to anode area is greatest in this case. This is further WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

106 Simulation of Electrochemical Processes III compounded by the increase in primary phase fraction. In figure 6(a) at certain exposure times the degree of corrosion of the 255gm-2 all can be seen to be equivalent to both 150gm-2 and 300gm-2 coating weights. This would seem to suggest that at the initial solid-electrolyte interface, the dendrite morphology plays a less significant role than volume fraction and coating thickness, but dendrite mass (smaller in lesser coating weights) becomes more significant later on exacerbating the situation in the 150gm-2 coating later in the simulation. Also evident in figure 6(a) is the adjustment of the time step ensuring that no solid fraction greater than that present at the interface can transform in 1 time step. Figure 6(b) shows the total material removed from the solid model over a simulated 24 hour period. It can be seen that the greatest material loss is experienced by the 150gm-2 coating, and decreases as the coating weight is increased. This is in good agreement with experimentation, although as can be seen in figure 6(b) the simulated mass loss is an underestimation, however the experimental findings themselves are calculated from current density maps [7].

(a) Figure 6:

(b) Material loss from the exposed steel cut-edge coated at different weights of ZnAl: (a) cumulative loss over 12 hours, and (b) mass loss during 24 hours exposure compared with SVET testing [7].

Also in agreement is the changing trend as, unlike in cooling rate a linear trend is observed, the decrease in mass loss as coating weight is increased is not linear and the difference between 255gm-2 and 300gm-2 is to a much lesser extent than is the case between 150gm-2 and 255gm-2. In terms of validating the model the most accessible data is that from SVET testing. Figure 7 shows the vertical current density 100μm above the initial solidelectrolyte interface after 12 hours simulated exposure. The form of the current density maps compare favourably to those reported [7]. The intensity of anodic events corresponding to the coating layer can be seen to decrease as the coating weight increases and it would be expected that heavier coating weights would WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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experience lower delamination of applied organic coating layers. Curiously, the emergence of one coating layer offering the bulk of cathodic protection to the steel at heavier coating weights is also predicted by the model, which cannot be attributed to processing conditions since the coating layers are formed using the same parameters. 1

1.1

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Vertical current density maps, Jy (Am-2) 100µm above ZnAl coated steel after 12 hours exposure; (a) 150gm-2, (b) 255gm-2 and (c) 300gm-2 (length scale mm).

Conclusions

Extensions to a previously described algorithm have resulted in the capability for the prediction of corrosion at both the surface and cut-edge of steels coated with Galfan-type zinc-aluminium coatings. Using well-established electrochemical relationships and straight-forward finite difference techniques the relationship between corrosion performance and coating weight of ZnAl coatings at an exposed cut-edge has been successfully simulated.

References [1] Barnard N.C., Brown S.G.R. & McMurray H.N., in Proc. of 1st International Conference on Simulation of Electrochemical Processes, 2005. Cadiz, Spain: WIT Press. [2] Brown S.G.R. & Barnard N.C., Corrosion Science. 48 (2006), 2291. [3] Brown S.G.R. & Barnard N.C., in Proc. of 2nd International Conference on Simulation of Electrochemical Processes, 2007, Myrtle Beach, USA: WIT Press. [4] Barnard N.C. & Brown S.G.R., Corrosion Science, 50 (2008), 2846. [5] Elvins J., Spittle J.A., & Worsley D.A., 47 (2005), 2740.

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108 Simulation of Electrochemical Processes III [6] Elvins J., Spittle J.A., & Worsley D.A., in Proc. 7th International Conference on Zinc and Zinc Alloy Coated Steel Sheet. 2007. Osaka, Japan: Iron and Steel Institute of Japan. [7] Penney D.J., Sullivan J.H., & Worsley D.A., Corrosion Science, 49 (2007), 1321. [8] Worsley D.A., McMurray H.N., & Belghazi A., Chemical Communications, 24 (1997), 2369. [9] Malki B. & Baroux B., Corrosion Science, 47 (2005), 171. [10] Córdoba-Torres P., et al., Electrochimica Acta, 46 (2001), 2975. [11] Pidaparti R.M., Fong L., and Palakal M.J., Computational Materials Science, 41 (2008), 255. [12] La Rocca A. & Power H., International Journal for Numerical Methods in Engineering, 64 (2005), 1699. [13] Souto R.M., et al., Corrosion Science, 49 (2007), 4568. [14] Tada E., et al., Electrochimica Acta, 49 (2004), 2279. [15] Thébault F., et al., Electrochimica Acta, 53 (2008), 5226. [16] Tada E., et al., Electrochimica Acta, 49 (2004), 2279.

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Eulerian-Lagrangian model for gas-evolving processes based on supersaturation H. Van Parys1, S. Van Damme1, P. Maciel1, T. Nierhaus2, F. Tomasoni3, A. Hubin1, H. Deconinck2 & J. Deconinck1 1

Vrije Universiteit Brussel, Research Group of Electrochemical and Surface Engineering, Belgium 2 von Karman Institute for Fluid Dynamic, Department of Aeronautics and Aerospace, Belgium 3 von Karman Institute for Fluid Dynamic, Department of Environmental and Applied Fluid Dynamics, Belgium

Abstract Predicting the macro-scale behaviour of gas bubbles and understanding the coupling between bubbles and fluid flow is essential if one wants to describe the local concentration of all electrolyte components and the current density distribution at an electrode surface. The presented two-phase model uses an Eulerian-Lagrangian approach to simulate the hydrodynamic phenomena occurring in the bubble-laden flow. The model describes the continuous electrolyte phase via the incompressible Navier-Stokes equations, while the trajectories of the gas bubbles are tracked sequentially in space and time. The electrochemical behaviour of the reactor is described using the Multi-Ion Transport and Reaction Model (MITReM), which considers for all relevant species the effect of convection, diffusion and migration and homogeneous reactions. For the electrode reactions, Butler-Volmer kinetics are used. The MITReM provides the concentration of dissolved gas at the electrode and in the electrolyte. At predefined nuclei gas bubbles are formed proportional to the local supersaturation. After a certain time the spherical bubbles are large enough to become detached bubbles. While the bubbles are growing on the electrode surface, they block the surface and therefore also influence the current density distribution. The first steps in view of the validation of the proposed two-phase model against experiments are reported. Keywords: gas evolving electrode, supersaturation, bubble nucleation and growth, hydrogen evolution.

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110 Simulation of Electrochemical Processes III

1

Introduction

Many electrochemical applications, such as the electroplating of chrome, zinc or nickel, suffer from unwanted side-effects related to gas production on electrodes. In other processes gas formation is the only aim. In all cases the formation of gas bubbles has an effect on the mass transport of the electrochemically active species in the process [1]. The gas bubbles sticking at the nucleation sites block the electrode and decrease the active surface [2–6], while the bubbles that are detached into the electrolyte solution change the conductivity of the electrolyte [7–9]. Furthermore, momentum exchange occurs between the bubbles and the surrounding electrolyte, affecting the motion of the electrolyte [10]. The mechanism of bubble formation by nucleation requires supersaturation of the dissolved gas [11–13] and a nucleus radius greater than the critical [7]. The main sources of heterogeneous nucleation are usually surface irregularities capable of containing entrapped gas, e.g. pits and scratches. The bubbles typically develop over the electrode surface, grow in size until they reach a break-off diameter and subsequently detach into the electrolyte. After detachment, some residual gas remains at the nucleation site and another bubble will form at the same place [2,13,14]. In most two-phase flow simulations [15–19], it is assumed that bubbles detach with a constant diameter, although from experiments [20,21] it is know that electrochemically formed bubbles show a size distribution. Present models are based on primary and secondary current density distributions, with the gas production based on an empirical correlation with current density [15–19]. These models don't solve for concentrations. Therefore they cannot describe properly the bubble formation mechanism. We propose a new macroscopic model based on supersaturation, providing a physically relevant link between the dissolved gas production and the bubble formation mechanism.

2

Eulerian fluid flow and ion transport model

For the simulation of gas-evolving electrochemical processes, a new approach is proposed that combines numerical models for electrolyte flow, ion transport and gas evolution. The mass and momentum conservation of the electrolyte flow is modelled by the incompressible Navier-Stokes equations, solving for the fluid velocity u and the pressure p:  u  0 , (1) u 1  (u )u  p  2u  0 . (2) t  In equation (2), ρ is the electrolyte density and ν is the viscosity. Based on the Multi-Ion Transport and Reaction Model (MITReM) [22,23], we can state a balance equation for each species in the system:

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R ci (3)    N i   sir vr t r 1 with ci the concentration of species i. The source term on the right hand side of equation (3) comes from the homogeneous reactions, where vr is the rate of reaction r and sir is the stoichiometric coefficient of species i in that reaction. The flux Ni is given by convection, diffusion and migration as follows: z FD Ni  ci u  Di ci  i i ci U (4) RT with Di the diffusion coefficient, zi the charge, F Faraday’s constant, R the universal gas constant, T the temperature and U the potential. The flux perpendicular to an electrode is zero or given by the heterogeneous reactions E

N i  1n   sie ve .

(5)

e 1

For these reactions, the Butler-Volmer kinetics are used [23].

3

Supersaturation model

In this model, bubbles are formed by consuming dissolved gas, with supersaturation as the driving mechanism for bubble nucleation and growth. Supersaturation is given by c-csat, where csat is the saturation concentration of a species [10]. In a first approach, the bubble growth rate is considered as a single heterogeneous reaction taking place continuously with a linear rate law: c  csat  0 (6) v  k  c  csat  if with k the rate constant. As it is known that gas bubbles are formed at nuclei, the produced gas volume needs to be attributed to bubbles at discrete nucleation sites. Here it is assumed that the nuclei are located at predefined places. Additional fundamental research is needed to justify or correct this rather simple assumption. In practice many parameters are influencing the number of nuclei e.g. surface roughness, surface tension, contact angle, current density, etc. A predefined bubble diameter at which detachment will occur is assigned to every nucleus. Actually, this is done randomly in such a way that the measured lognormal bubble diameter distribution is reproduced. As in practice, also the bubble diameter depends on flow velocity, current density, roughness, etc., this bubble size distribution is measured under the same working conditions as the simulation. It is clear that this approach can be improved by incorporation of more details. Finally, to every nucleus a given surface is assigned from where the gas is feeding the growing bubble. In this way the local gas production rate can be integrated over time and space. So, the amount of produced gas is accumulated in a growing spherical bubble having a zero contact angle with the electrode. The projection of the actual bubble size on the electrode surface is assumed to cover the surface such that the local active electrode surface is reduced resulting in a changed local current density. All this is calculated in an Eulerian way. Once the bubble diameter has reached its assigned diameter, the bubble is released and transferred to the Lagrangian bubble tracker and a new WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

112 Simulation of Electrochemical Processes III bubble diameter is randomly assigned to that nucleus. Also here much better and more refined models can be conceived.

4

Lagrangian bubble tracking model

Every detached bubble enters the electrolyte and a Lagrangian tracking procedure is used to update the velocities and positions of all dispersed gas bubbles in the electrolyte at each time step of the Navier-Stokes solver. From Newton’s second law, an equation of motion can be obtained for every bubble, based on the formulation stated in [24]. Together with the relation between the particle’s position and velocity, a set of two ordinary differential equations in three space dimensions can be formed in order to update the bubble trajectory x (7) v t v 1 Du (8)  (u  v )  3  2CL ((u  v )  ω)  2g, t  b Dt where v is the bubble velocity, u is the fluid velocity and ω the fluid vorticity at the bubble position. The derivative  t follows the moving bubbles in time, while D/Dt is the total acceleration of the carrier fluid as seen by the bubble, evaluated at the bubble position x. The terms of the right hand side of equation (8) represent the forces acting on the bubble. From left to right, these are the drag force, the added mass force, the Saffman lift force (CL is the lift coefficient) and the force due to gravity g, i.e. the buoyancy force. The drag force represents an inter-phase momentum transfer and reduces the velocity difference between the phases, depending on the bubble response time  b for Stokes flow:

d2 , (9) 36 where d is the bubble diameter. If the bubble accelerates relative to the carrier fluid, it will immediately experience a deceleration due to its inertia to the other phase. This effect is known as the added mass force and represents the amount of volume of the fluid displaced by the bubble in its relative motion. The Saffman lift force represents the lift on the bubble induced by a shear velocity, where CL is the lift coefficient. In case of a small spherical bubble, CL takes a value of 0.53, as proposed in [25]. The bubbles occurring in the present electrochemical system are small enough to fulfil the criterion of Stokes flow. Thus, turbulent wakes arising behind bubbles can be neglected [26]. Furthermore, we consider all bubbles in the present simulations as small, non-deformable and rigid spheres. This hypothesis holds for bubbles of low Eötvös numbers Eo:  d 2 g . (10) Eo 

b 



Here,  denotes the surface tension of the bubble and Δρ the difference in density of the two phases. The sphericity assumption of bubbles is valid when

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both Eo and the bubble Reynolds number Red  v d  are of the order of 10 or lower. For hydrogen bubbles of sizes below 300μm, this restriction is valid and thus sphericity is assured [27]. Equations (7) and (8) form a system of six ordinary differential equations in three space dimensions for each individual bubble. This system is integrated in time using a Crank-Nicholson scheme, which provides second order accuracy. Sequential tracking of all bubbles in the system is performed at each time step of the Navier-Stokes solver.

5

The electrochemical system

For the electrolyte solution a 1M Na2SO4 solution is taken. The pH is adjusted to pH 2.5 by adding concentrated H2SO4. The diffusivities and bulk concentrations of all species are given in table 1. Dissolved hydrogen formation occurs on the cathode and dissolved oxygen on the anode according to reactions (7) and (8) respectively: (11) 2 H   2e  H 2(l ) 2 H 2O  O2( l )  4 H   4e  The electrode reactions kinetic parameters are indicated in table 2.

Table 1:

Electrolyte species diffusivity and bulk concentrations [23,28,29]. Species

D [10-9 m2 s-1]

Cbulk [mol m-3]

Na+

1.334

2000.

1.065

1000.

1.330

3.16

9.312

3.16

5.260

3.16 10-9

H2O

2.300

55000.

H2(l)

4.870

0.

SO4

2-

HSO4

-

H+ OH

Table 2:

(12)

-

Electrode reactions kinetic parameters deduced from polarization curves on a rotating disc electrode. E0 [V]

koxi [m s-1]

kred [m s-1]

αoxi [1]

αred [1]

5.700 10-13

-

0.990

-

0.045

-

2 H   2e   H 2(l ) 0.000

-

2 H 2O  O2( l )  4 H   4e  1.230

5.312 10-11

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114 Simulation of Electrochemical Processes III For simplicity and without any loss of generality oxygen gas evolution is not considered and therefore the dissolved oxygen was excluded from the model. The transfer from dissolved hydrogen to gaseous hydrogen is as follows (13) H 2(l )  H 2( g ) −6 -1 with kinetics according to eq. (6), a rate constant k=10 m s and the solubility of H2 in water csat = 0.00075 mol.m-3. The hydrogen gas evolution reaction is considered along the whole lower boundary (including insulators).

6

Results

It has been shown [30] that the kinetic constant k of reaction (6) plays an important role. For high reaction rates, the local gas evolution is almost proportional to the local current density, whereas for lower reaction rates one can have a substantial difference between the local bubble formation rate and the local current density. Indeed much dissolved gas enters the solution by diffusion and convection. It is worth mentioning that in this situation gas evolution can also occur on nuclei that do not belong to the cathode. In view of evaluation of the simulation approach, a parallel flow reactor was build with the following dimensions: length 1 m, width 10 cm and height 1 cm. The inlet is made large enough to assure steady flow along all electrodes. The electrodes are 10 cm wide, 5 cm long and configured such that the cathode is fully optically accessible (figure 1). This involves that they are not aligned and that the current density distribution is intentionally non-uniform.

Figure 1:

The reactor with unaligned electrode configuration.

As the upstream anode edge is situated 1.5 cm from the cathode’s downstream edge, bubbles formed at the anode are neglected. A typical current density distribution along the cathode is given in figure 2. The small peak on the right hand side of the curve is attributed to the proton reduction to H2 (see eq. (11)). However, the H+-ions become rapidly depleted and as a consequence, the pH rises in the immediate vicinity of the electrode surface making the reduction from H2O molecules thermodynamically more favourable (see eq. (12)). The bubble size distribution was also measured at a cell potential of 3.3V and is given in figure 3. To that purpose backlighting or shadowgraphy, being an in situ and non-intrusive optical measurement technique, was used. In backlighting, the bubble is illuminated by a diffuse light source from one side and its shadow is imaged with a high speed camera. After calibration the WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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Current density (A/m²)

1000 800 600 400 200 0 0.00

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Developed length cathode (m)

Figure 2:

Current density distribution along the cathode for an applied total voltage of 3.3V.

Figure 3:

Hydrogen bubble size distribution for an applied total voltage of 3.3V and a mean flow of 0.2 m/s.

projected shadow image is a measure for the bubble diameter. A lognormal distribution is fitted to the bubble diameter distribution. A mean diameter of 170μm with a standard deviation of 64μm is obtained. As explained above, this bubble distribution is used as input data for the simulations. In order to avoid calculating the whole transient process, steady state flow and MITReM conditions are calculated first. From this situation on the simulation of time dependent bubble evolution is started. A two-way interaction between bubbles and flow is considered. This means that the combined effect of the influence of the fluid flow on the bubble trajectories and the effect of the bubble movement on the fluid flow is taken into account. In figure 4 simulated situations at several time steps are shown. The mean flow is 0.2 m/s. The cathode is on the right.

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116 Simulation of Electrochemical Processes III

a) t=1s

b) t=2s

c) t=3s

d) t=4s Figure 4:

Side view of simulated hydrogen bubble dispersion for different time steps.

Figure 5:

Isometric view of simulated hydrogen bubble dispersion for t=2s. The mean flow is 0.2 m/s.

In figure 5 an isometric view is given for t=2s. In this figure bubbles on the surface are coloured in dark grey whereas bubbles in the flow are marked in medium grey. It is clearly observed that more bubbles are formed at the cathode edge where the current density is larger and that larger bubbles are rising faster. Unfortunately these simulated bubble dispersions cannot be directly compared with measurements as in reality always a transient current density phenomenon takes place when the voltage is applied. Also steady state calculations, performed in different applied flow and potential conditions, need still to be compared in full detail with measurements performed in the same conditions. It is believed however that it is already proven that the concepts work and will open new possibilities for bubble simulations in electrochemical reactors. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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117

Conclusions

Models that directly relate the gas bubble flux to the current density implicitly assume that the nucleation and growth of bubbles is very fast. This assumption is not needed in the present modelling approach as gas evolution is related to supersaturation instead of current density. Although so far still rather drastic simplifications have been introduced, it is clear that the presented method offers new ways to model electrochemical gas evolution, at least at moderate gas fractions where bubbles are dispersed in the solution. Future work will be directed towards a more detailed description of nuclei and bubble formation, bubble size determination, all influenced by local quantities such as surface roughness, charge distribution, surface tension (depending on local concentrations), concentration of dissolved gas.

Acknowledgement The authors thank the Flemish Institute for support of Scientific-Technological Research in Industry (IWT SBO contract number 040092, project acronym: MuTEch and IWT post-doc research mandate).

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

J.E. Funk, J. Electrochem. Soc. 116 (1969) 48. H. Vogt, Electrochim. Acta 25 (1980) 527. P.J. Sides, C.W. Tobias, J. Electrochem. Soc. 127 (1980) 288. M. Krenz, L. Müller, A. Pomp, Electrochim. Acta 31 (1986) 723. J. Dukovic, C.W. Tobias, J. Electrochem. Soc. 134 (1987) 331. H. Vogt, R.J. Balzer, Electrochim. Acta 50 (2005) 2073. H. Vogt in: J.O'M. Bockris (Ed.), B. E. Conway (Ed.), E. Yeager (Ed.), Comprehensive Treatise of Electrochemistry, vol. 6, Plenum Press, London, 1983, Ch.7. P. Byrne, P. Bosander, O. Parhammar, E. Fontes, J. Appl. Electrochem. 30 (2000) 1361. G. Kreysa, M. Kuhn, J. Appl. Electrochem. 15 (1985) 517. E. Delnoij, J.A.M. Kuipers, W.P.M. Van Swaaij, Chem. Eng. Sci. 54 (1999) 2217. D. Kashchiev, A. Firoozabadi, J. Chem. Phys. 98 (1993). J. Eigeldinger, H. Vogt, Electrochim. Acta 45 (2000) 4449. S. Lubetkin, Langmuir (2003) 2575. S. Lubetkin, Electrochim. Acta 48 (2002) 357. P. Mandin, J. Hamburger, S. Bessou, G. Picard, Electrochim. Acta 51 (2005) 1140. R. Wüthrich, C. Comninellis, H. Bleuler, Electrochim. Acta 50 (2005) 5242. M.D. Mat, K. Aldas, Int. J. Hydrogen Energy 30 (2005) 411. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

118 Simulation of Electrochemical Processes III [18] K. Aldas, Applied Mathematics and Computation 154 (2004) 507. [19] G. H. Kelsall, G. Li in: R. Woods (Ed.), F.M. Doyle (Ed.), Int. Symp. on Electrochemistry in Mineral and Metal Processing V, Electrochem. Soc., N.J., (2000) 303. [20] S. Dehaeck, Ph.D. Thesis, Ghent University, Belgium [21] P. Boissoneau and P. Byrne, J. Appl. Electrochem. 30 (2000) 767 [22] L. Bortels, J. Deconinck, B. van den Bossche, J. Electroanal. Chem. 404 (1996) 15. [23] J. Newman, Electrochemical Systems, 3rd ed., Prentice-Hall, Englewood Cliffs, NJ, 2004. [24] M.R. Maxey and J.J. Riley, Phys. Fluids 26 (1983) 883 [25] T.R. Auton, Ph.D. Thesis, University of Cambridge, United Kingdom (1983) [26] A.W.G. de Vries, A. Biesheuvel and L. van Wijngaarden, Int. J. Multiphase Flow 28 (2002) 1823 [27] R. Clift, J.R. Grace and M.E. Weber, in Bubbles, drops and particles, 2nd Ed, Academic Press, New York, United States (2005) [28] B.E. Conway, Electrochemical Data, Elsevier, Amsterdam, 1952. [29] V. P. Arkhipov, M. I. Emel'yanov, F. M. Samigullin, N. K. Gaisin, J. Struct. Chem. 19 (1979) 709. [30] P. Maciel, T. Nierhaus, S. Van Damme, H. Van Parys, J. Deconinck, A. Hubin, Electrochemistry Communications 11 (2009) 875

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Simulation of gas pipeline leakage using the characteristics method E. Nourollahi NIGC (National Iranian Gas Company), Department of Mechanical Engineering, Ferdowsi University, Iran

Abstract The modeling of gas leakage for understanding the process of pressure reduction and gas leakage rate of a hole is accomplished by zero-dimensional models of the tank and pipe. These models of the tank and pipe are suitable for simulation when the hole is very small or where there is a complete fracture. However, use of these two methods or a combination of them is suitable rather when the pipelines are very long. In zero-dimensional models the effects of complex boundaries are also ignored. In this study, the pipeline leakage is simulated by use of the one dimensional characteristics method. This model is perfect compared with both pipe or tank models, or a combination of them, and we can expand the simulation of gas pipeline leakage to the short pipelines and variable boundary conditions. This is important in order to calculate the volume of gas discharged from the pipelines when an accident occurs. Keywords: leakage, gas pipeline, characteristics method.

1

Introduction

Natural gas is considered as a clean source of energy worldwide. One of the usual problems is finding methods for prevention of wasting natural gas during transportation and distribution. The method that is presented in this paper can result in somewhat subtle calculations when a hole is creating in the pipeline surface and consequently has an important role in the estimate of the received losses. Pipe surface leakage or pipe section dismissal can be due to various reasons, such as corrosion, earthquake or mechanical stroke, which may be implemented in the pipe surface, and also overload compressors. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/ECOR090121

120 Simulation of Electrochemical Processes III After the leakage has been created, the flat expansion pressure waves are propagated in two converse sides. These waves have sonic speed and after clashing to the upstream and downstream boundaries, return to the form of compression or expansion wave depending on the edge type (Fig. 1). In the leak location, depending whether the ratio of pressure to ambient pressure is more or less than the CPR quantity, the equation of which is showed in equation (1), the flow will be sonic and ultrasonic or subsonic respectively. k

P  2  k 1 (1) CPR  out    P1cr  k  1  In the above equation, P1cr is the critical pressure of point 1 and k is the thermal capacity ratio. Therefore, the problem physics change to one of the two stated cases below. If the flow is sonic and ultra sonic, the sonic reporter wave does not leak from out of the pipe to enter the pipe. Hence, the changes of the flow field are accomplished due to the flat pressure waves and the real boundary conditions on the start and end of the pipe. The mass flow outlet of the hole also depends only on the stagnation pressure in the leak location and on the area of the hole and is not related to the form of the orifice cross section. However, if the flow in the leak location is subsonic, the hole could be the source of the production of a compression or expansion sonic wave and the effect on the flow field. In this situation, the pipe and leakage spot act as a T junction and the mass outflow of the leak location depends on the ambient pressure and a coefficient known as the empirical discharge coefficient in addition to the stagnation pressure and the orifice area. The pressure with respect to the ambient pressure in the gas transition pipelines are good, the ratio is more than CPR and the gas flow is also provided from the entering sources, such as the permanent gas refinery, hence in this paper the problem physics is considered to be like the state in eqn. (1).

Figure 1:

Leakage display and the waves that are produced.

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According to what has been stated about problem physics (which means that changes in the solution field only occur by the flat pressure wave), the solution of gas flow in a leaking pipe can be suitable for use of the one dimensional numerical method. Therefore, in this paper is the one dimensional characteristics method is used for modeling the compressible gas flow in the pipe and for calculation of the leakage flow. The assumption is that the inner flow of the pipe is homentropic; hence the solution of the mass and momentum conservation addition to the constant entropy flow assumption are sufficient for flow simulation. In continuance the characteristics method is introduced, then the necessary correction is stated for considering the leakage effect on the flow field in order to determine the leakage flow and the pressure distribution in the pipe line.

2

Characteristics method

Homentropic flow (a flow that has the same entropy level in the whole field) is a special case of the general flow problems and its solution is fairly straightforward compared with the more complicated general flow problems. In this part the method of characteristics is introduced to solve this problem [1, 2]. The equations of mass and momentum conservation, without considering friction, are the forms below. The continuity equation is:

 ( u )   x t

(2)

The momentum equation is:

Du P  x Dt In these equations  is gas density and u is the speed of the gas. With attention to the definition of the speed of the sound a by: p a2  ( )s  

(3)

(4)

For an ideal gas:

a2 

kp



(5)

By using of the relationship between the sonic speed and the pressure in an ideal gas, these equations are changed to the below forms after some steps of rewriting of the mass and momentum conservation equations:

{

a k  1 u a u  (u  a ) }  {  (u  a) }  0 t x 2 t x

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122 Simulation of Electrochemical Processes III

{

a k  1 u a u  (u  a ) }  {  (u  a ) }  0 t x 2 t x

(7)

Equations (6) and (7) form a set of quasi-linear hyperbolic partial differential equations. A solution of the form:

a  a ( x, t )

u  u ( x, t ) is required. In order to obtain a characteristic solution it is assumed that a and u are uniquely related by this expression:

c  c(u, a )

Then, the solution to eqns. (6) and (7) will be of the form:

c  c ( x, t )

The solution may be represented by the curved surface bounded by, say, the edges PQRS as shown in Fig. 2. According to the definition, if in a special point on the surface of c  c( x, t ) for a reviewing special curve from that point, the slope of the projected curve on the x-t plane is equal with the quantity of the curve of that point, the passing direction of that point is known as the characteristic direction. We have in mathematical expression:

Figure 2:

Graphical interpretation of the characteristics method. (a) Threedimensional surface defining c  c( x, t ) . (b) Projection of line on the characteristics surface to the plane at c  0.

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C= (dx / dt ) (8) By using this complete derivative definition, the sonic speed and particle speed parameters are determined with respect to the time of a characteristic length, such as the below:

a a  da  c    x  dt char t u u  du  c    x  dt char t

(9) (10)

Therefore if c  c(u, a ) is defined as the form of

c1  u  a c2  u  a

, eqns. (6) and

(7) in length of two characteristics c1 and c2 are rewritten like this:

k  1  du   da       0 2  dt  c1  dt  c1 k  1  du   da       0 2  dt  c 2  dt  c 2

(11)

Consequently a total solution is always calculable by use of the below characteristic equations and also by use of the numerical or graphic methods.

dx ua dt da k 1  du 2

(12)

Hence, the solution of these momentum and mass conservation equations in any time step should be determined two characteristics: c1  u  a and c2  u  a 2.1 The numerical solution method At first, the non-dimensional parameters of A and U are defined as below in the characteristics method:

U

u a ref

In the above equation

;

A

a a ref

(13)

aref is the sonic speed at the start point. Then, the

Reimann non-dimensional characteristics are defined as follows:

  A

k 1 U 2

;

  A

k 1 U 2

(14)

Therefore, determination of the two parameters of  ,  in any point of the solution field is going to bring about the speed and pressure at that point. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

124 Simulation of Electrochemical Processes III An explicit equation between Reimann variables in the inner points of the solution field is presented below, which is for each step:

t b  in1  a  in1  in1   in (15) x t  in 1   in b in1  a in1  in1   in  (16) x For determination of the relation between the Reimann variables in boundary points, it should be known that the boundary condition means the equation of state, which is determining for the relation between a and u . Therefore, various forms of boundary conditions can be introduced for numerical solution of the problem. These boundary conditions can be open start or open end, close start or close end, the start contact to an entering valve with the special opening percentage, the end contact to an exit nozzle with special opening percentage, the related end to a valve that has definite pressure drop, the start contact to a reservoir, the end without the changes with respect to the location, the close start and close end or the other boundary conditions. By distinction of the state equation in the boundary, a mono-equation is created between Reimann variables  ,  and so always in any boundary, one of these variables is known and the other one is unknown, then the unknown Reimann variable can be calculated, so the effect of the boundary transfers to the solution field is obtained.

 in 1   in







2.2 The implementation of the leakage effect For implementation of the leakage effect on the flow field, the mesh is chosen in a way that the hole location would be stated between two nodes, a in Figure 3.

Figure 3:

Location of the hole between two nodes.

When the hole is created in the pipe surface, as is said, the pressure ratio to the ambient pressure below the hole, which is in the pipe, is more than the CPR in the later time steps. Therefore, the flow is checked in the hole location and the outflow of the leak location is calculated by the equation below [3]: k 1

M  2  k 1 .k . Q  Aor Pl .  ZRTl  k  1 

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125

Aor is the hole (orifice) area, M is the molecular weight, Z is the compressibility factor, R is the constant of gas and T1 is the In the above equation,

temperature of gas in below the hole. The leakage point in any time step acts as a boundary and two expansion waves, depending on the direction of the flow in the pipe, would reach the a and b points with little time difference and create the same change in the nondimensional speed of U , as in the form below:

Ua  Ua 

Q Ua m a

Q Ub Ub  Ub  m b Therefore, two known characteristics, such as

(18)

a

and

 b , according to the

definition of the positive direction (towards boundary) for the particle speed are corrected like this:

a 

2  Aa  (k  1)  U a  2

2  Ab  (k  1) U b  b  Then the unknown parameters

b

2

and

a

(19)

are calculated as in the form below:

 a  a  (k  1)  U a b   b  (k  1)  U b

(20)

Therefore, the state of two points a and b in any time step, considering the corrected leakage effect and hence by notice to the equations that govern the problem, being hyperbolic types of equations, during the time of the leakage, the effect is transferred permanently as a third boundary addition to the upstream and downstream boundaries to the solution field.

3

Results of numerical solution

In this part, the results of the simulation of the leakage are presented for a pipe which of 250 meter in length and with a hole (orifice) of 1 cm2 area on the surface, by using a grid system with 100 nodes. It is also considered that the hole is in the middle of the pipe length and it is assumed that the initial gas pressure and initial gas speed are 30 bar and 41 ft/s, respectively [4]. It is considered that the upstream boundary condition is the reservoir with constant pressure and the downstream boundary condition is stated with three forms:  The boundary with no changes with respect to the location.  The valve with constant coefficient of pressure drop.  Close end. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

126 Simulation of Electrochemical Processes III The first situation is when the pipe length is very long compared to the hole, where gas pipelines are like this in most of the cases. The second situation is when the leakage occurred before a valve with the constant pressure drop in a part of the pipe. The third situation is when the pipe end is closed (e.g. when the exit valve is closed completely). In continuing, the respective results and discussions for the pressure changes at the beginning (because of the propagation of the expansion wave from the leak location) and changes of mass flux by time are presented.

4

Conclusions

The related diagrams are shown as mentioned before, for the entering boundary of the constant pressure reservoir and three downstream boundaries, which are described here. Figures 4 and 5 are for the non-gradient boundary (with respect to the location), Figures 6 and 7 are for the downstream boundary related to the valve with the constant pressure drop and Figures 8 and 9 are for the close downstream boundary. By studying these situations the below conclusions can be inferred. According to Figures 4(a), 6(a) and 8(a), we observe that the pressure changes on the primary times and the reaching time to the hole; it can be seen that after passing the hole, pressure returns to nearly the first state. We have the pressure increase only in the state of the close downstream boundary condition, due to the constant entering flow. It is also observed that two weak and strong waves are propagated depending on the flow direction. Flow direction in this study is assumed from left to right, then the expansion wave, which moves from the center to the right, is stronger. By attention to Figures 4(b), 6(b) and 8(b), it is seen that the exit mass flux from the leakage location have an intense fluctuant behavior. The boundary conditions in upstream and downstream also affect the time average of the exit mass flux and the amplitude and frequency of the fluctuations. The average at the first type of boundary condition is about 118 kg/s, at the second state it is about 110 kg/s after 70 seconds and at the third boundary state this average reaches 350 kg/s. In the Figures 5(a), 7(a) and 9(a) the quantity of the exit mass flux changes are shown by increasing the hole area and for various lengths of the pipe. It is clear that by increasing the hole area, the quantity of exit mass flux is increased for all the lengths and this subject is true for all three states. By attention to these diagrams, it is also observable that the gradient of the diagram is zero on the long lengths gradually. In other words, the increase of the hole area does not have any effect on the exit mass flux, which is more obvious in figure 5(a). In Figures 5(b), 7(b) and 9(b), the exit mass flux changes are shown by the pressure and for various lengths of pipes. It is seen that the exit flux quantity is increased for all lengths by increasing the pressure of the pipeline, but the manner of the increase with respect to the various lengths depends on the boundary conditions. So, at the first boundary condition (Figure 5(b)) the increase of the pipe length does not have any effect and the gradient of all the WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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(a) Figure 4:

(b)

State (1) of the boundary conditions. (a) Pressure changes by increasing the pipe length at primary times. (b) Changes to the exit mass flux by time.

(a) Figure 5:

127

(b)

State (1) of the boundary conditions. (a) Changes to the exit mass flux by increasing the hole area and pipe length. (b) Changes to the exit mass flux by increasing the pipe pressure and pipe length.

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128 Simulation of Electrochemical Processes III

(a) Figure 6:

State (2) of the boundary conditions. (a) Pressure changes by increasing the pipe length at primary times. (b) Changes to the exit mass flux by time.

(a) Figure 7:

(b)

(b)

State (2) of the boundary conditions. (a) Changes to the exit mass flux by increasing the hole area and pipe length. (b) Changes to the exit mass flux by increasing the pipe pressure and pipe length.

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(a) Figure 8:

(b)

State (3) of the boundary conditions. (a) Pressure changes by increasing the pipe length at primary times. (b) Changes to the exit mass flux by time.

(a) Figure 9:

129

(b)

State (2) of the boundary conditions. (a) Changes to the exit mass flux by increasing the hole area and pipe length. (b) Changes to the exit mass flux by increasing the pipe pressure and pipe length.

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130 Simulation of Electrochemical Processes III diagrams are rather equal. However, in the second and third states (Figure 7(b) and 9(b)) the quantity of the exit mass flux is raised by increasing the pipe length.

References [1] The Thermodynamics and Gas Dynamics of Internal Combustion Engines, Volume 1, Rowland S. Benson, Edited by J.H. Horlock and D.E. Winterbone, Clarendon Press Oxford 1982. [2] Nonsteady, One-Dimensional, Internal, Compressible Flows, JOHN A.C. KENTFIELD, Oxford University Press 1993 [3] Evaluation of gas release rate through holes in pipelines, Dong Yuhua, Gao Hilin, Zhou Jing, Feng Yaorong,2002 [4] Pipeline Design & Construction: A Practical Approach, M. Mohitpour, H. Golshan, A. Murray, TJ930. M57 2003.

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Two-dimensional numerical modelling of hydrogen diffusion assisted by stress and strain J. Toribio, D. Vergara, M. Lorenzo & V. Kharin Department of Materials Engineering, University of Salamanca, Spain

Abstract This work is based on previous research on the one-dimensional (1D) analysis of the hydrogen diffusion process, and proposes a numerical approach to simulate the phenomenon in two-dimensional (2D) situations, e.g., near notches. The weighted residual method was used to solve numerically the differential equations set out when the geometry was discretized through the application of the finite element technique. This developed procedure can be a suitable practical tool to analyze hydrogen embrittlement phenomena in structural materials. Keywords: numerical modelling, weighted residual method, hydrogen diffusion, axisymmetric notch.

1

Introduction

The influence of hydrogen on fracture depends on hydrogen concentration, C, in the sites where localized material damage might occur. The accumulation of hydrogen in these zones proceeds by diffusion from external or internal sources, i.e., local fracture event takes place when and where hydrogen concentration reaches some critical value, Ccr, which is conditioned by the stress-strain state in material [1]. This is expressed by the following equation Ccr = Ccr (i, i) (1) that reflects the influence, in a general case, of the stress-strain state through its invariants, represented by the principal components of stresses and strains i and i (i = 1, 2, 3), respectively. Hydrogen diffusion within metals is also known to be governed by the stressstrain state therein. Roughly, it may be considered that hydrogen diffuses in metals obeying a Fick type diffusion law including additional terms to account for the effect of the stress-strain state. Concerning the role of stress, this is WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/ECOR090131

132 Simulation of Electrochemical Processes III commonly associated with its hydrostatic component . According to this, hydrogen diffuses not only to the points of minimum concentration (driven by its gradient), but also to the sites of maximum hydrostatic stress (driven by its gradient). The diffusion process may be also conditioned by the plastic strain. Following this way, i.e., considering the diffusion assisted by stress and strain as a responsible transport mode, it is possible to evaluate the amount of hydrogen accumulated in metal, and this way to determine the locations where fracture initiation process might take place in hydrogen embrittlement phenomena. In order to improve the understanding of the diffusion phenomena inside materials, and particularly the effects that stresses exert on diffusion, it is useful to reveal the time evolution of hydrogen concentration in the relevant sites concerning particular geometries of the studied test-pieces or components, especially in the more representative locations therein. To this end, numerous analyses have focused on various aspects of hydrogen diffusion in metals affected by mechanical factors, such as stress or strain. Some analytical closed-form solutions as well as numerical approaches have been developed under certain more or less restraining assumptions or simplifications. Concerning these latter, several previous studies were limited to considerations with one spatial dimension of the sole diffusion depth and straight-line diffusion flux (1D situations), but with an advantage of taking explicitly into account a lot of complicating factors which could arise during nonsteady-state elastoplastic loading histories. Several analyses dealt with two-dimensional (2D) situations, when diffusion was disturbed by geometrical or stress-state inhomogeneities and proceeded along curvilinear trajectories, e.g. near notches or cracks. However, these studies have been notably less extensive so far, in particular as regards the nonsteady-state stress-strain fields, e.g., slow strain-rate test conditions or cyclic loading. Thus the present paper aims to give a preliminary depiction of some advances towards the modelling of 2D stress-strain assisted hydrogen diffusion under transient loading conditions.

2

Problem statement

To save the space, further development deals solely with stress assisted diffusion since its generalization for the case of stress-assisted diffusion is straightforward. According to previous studies [2], it is assumed that hydrogen diffusion through material proceeds toward the sites where the lowest concentration or the higher hydrostatic stresses occur. The combination of these factors results in an equation for the stress-assisted diffusion flux of hydrogen which is:  V J  D C  D  C  with   H (2) RT where D is the hydrogen diffusion coefficient, VH is the partial molar volume of hydrogen in metal, R the universal gas constant and T the absolute temperature. Following the standard way, using the matter conservation law [3]:  dC  divJ (3) dt WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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together with the expression (2) for the flux, the equation of diffusion in terms of the sole concentration can be obtained dC   ( D  C  D  C  ) (4) dt In this formulation, neither stress state, nor temperature (and thus, temperature dependent material characteristics, such as D or ) are required to be stationary, but can be time dependent. To simplify this preliminary study, diffusion coefficient, D, as well as temperature is considered to be spatially uniform, i.e., their gradients are zero, although this is not an essential restriction. This leads to the equation of stress-assisted hydrogen diffusion in terms of concentration: dC  D ( 2 C   C    C  2 ) (5) dt where the coefficients may be time dependent. The geometry of interest, selected here as an example to develop the analysis methodology, is sketched in fig. 1, which shows how the three-dimensional testpiece geometry can be analyzed as a two-dimensional region (shaded figure) due to its axial symmetry.

Figure 1:

Figure 2:

Diagram of analyzed geometry.

Boundary conditions applied to an axisymmetric geometry.

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134 Simulation of Electrochemical Processes III The boundary conditions, both the mechanical and those for hydrogen entry into metal, are depicted in fig. 2. There an arbitrary axial load is applied over the boundary SL, a definite environmentally controlled equilibrium concentration of hydrogen, Ceq, is imposed over the boundary Seq exposed to the hydrogenating environment, and  on both symmetry axes denoted Sf, the null values of the hydrogen flux, J , and of the normal component of displacement are imposed. The following expressions represent explicitly the mentioned boundary conditions for diffusion:   J n S  J S  0 (6) f

f

C S  C eq (7) eq  where n is the external unit normal vector to the respective surface. For convenience in further numerical implementation of the diffusion boundaryvalue problem (5)-(7), the equilibrium equation (7) at the side of hydrogen entry from the environment is substituted by the next mass-exchange condition   J  n S  J  S   (C  C eq ) (8) eq

eq

where  is the mass-exchange rate coefficient which controls the velocity of approaching the equilibrium of hydrogen between environment and the entry surface layer Seq of a testpiece. To represent adequately the equilibrium entry condition (7) by means of relation (8), this rate coefficient must be chosen arbitrarily, but large enough to ensure practically instantaneous (with respect to the characteristic time scale for diffusion) attainment of the equilibrium given by equality (7). The adequacy of a choice can be easily confirmed a posteriory. Finally, to finish the diffusion problem statement, hydrogen accumulation in the initially hydrogen-free sample may be considered, so that the zero initial condition C t 0  0

(9)

will be placed throughout a whole testpiece of interest.

3

Numerical approach

Obviously, quantitative modelling of stress-assisted hydrogen diffusion requires the stress field in a testpiece of interest to be known. Even for rather simple cases, such as a notched bar being considered here, neither the exact solutions nor the closed form ones are usually available. Thus, one must count on some sort of the numerical solution of the mechanical portion of the coupled problem of the stress-assisted diffusion. The finite element method (FEM) approach, well-developed for both linear and nonlinear analyses of deformable solid mechanics, is a right choice to perform the stress analysis as a prerequisite for diffusion calculations. In due course, simulation of diffusion which accompanies mechanical loading of a sample also requires numerical treatment. To this end, expansion of the WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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FEM approach based on the same framework of the spatial finite elements used in the stress analysis is a natural choice for numerical modelling of diffusion. Some advanced general purpose finite-element codes, well adapted for stress analysis in particular, e.g. ABAQUS or MSC.MARC, have certain capabilities to simulate the stress-assisted diffusion, too. Unfortunately, they still are limited in some rather important aspects. As regards ABAQUS, this allows to perform simulations of the stress-assisted diffusion governed by equation (5) "over" the data of an accomplished solution of a geometrically and physically nonlinear stress-strain analysis, i.e., for the stationary stress field at the end of some preliminary loading trajectory, but not for the case of simultaneous transient loading and hydrogenation. Another one, the MSC.MARC code with certain user subroutines may be employed to simulate the transient stress-assisted diffusion as far as corresponding transport equation (5) has mathematically the same form as the equation of convective heat transfer implemented in this software, although, not for the geometrically nonlinear (large deformation) cases. Besides, it has another rather strong shortcoming in that it requires one to define the values of stress gradient at the finite element nodes, which is accompanied by the accuracy loss in the displacement-based FEM procedures. Indeed, there the stresses per se are derived with an inevitable loss of accuracy from the displacement gradients, so that the best approximation of stresses is achieved at the element integration points, and calculation of the second derivatives of displacements must worsen the analysis accuracy substantially. With this in mind, it seems to be a reasonable compromise to consider a FEM implementation of the modelling of stress-assisted diffusion over the previously (or simultaneously) performed stress analysis taking the nodal values of stresses, obtained with a post-processing technique, as the entry data for diffusion, i.e., constructing a finite-element approximation of the stress field with the aid of the same finite-element shape functions used in the mechanical analysis to approximate the displacement fields. Following this way, the standard weighted residuals procedure together with finite element approximation of both fields of the hydrostatic stress  and the hydrogen concentration C [6] may be adopted to develop corresponding procedure for diffusion modelling coupled with the stress analysis. Proceeding with the standard weighted residuals approach [6] to find an approximated solution of the diffusion boundary-value problem (2)-(6) and (8), the approximation of concentration is represented in terms of a linear combination of the spatial trial functions generated over a certain sort of finite elements which discretise the solid under consideration, Ni(x), where x stands for the instantaneous coordinates of material points over the volume of considered region V occupied by a testpiece, so that C(x,t) =  Ci(t)Ni(x), i = 1, 2, …, n (10) where Ci is the nodal values of concentration, and the sum is taken over all the nodes of the finite element discretisation. Then, the best approximation of the considered boundary-value problem will be obtained with the nodal concentrations Ci which nullify all the residuals, which correspond to both the WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

136 Simulation of Electrochemical Processes III differential equation (5) and the boundary conditions (6) and (8), with the weights Wi(x), so that this yields the system of equations as follows  dV   Wi  D C - D  C  dV    Wi C V

V





 Wi J  J   0

Sf Seq

  i = 1, 2,…, n (11) where  represents the inner product. Adopting the Galerkin method as a particular form of weighted residuals, i.e., considering the weights Wi to be the same as the trial functions Ni, after standard transformations of integrals in the relation (11), the next system of the ordinary differential equations with respect to nodal concentrations Ci(t) may be derived:  

 C j  Ni N j dV  C j  D  N j   Ni   D   j

V

V



N j   Ni dV

(12)    Ni J  dS  C j  Ni N jdS  Ceq  Ni dS  0 Sf Seq Seq  or in the matrix form,  KCF MC (13) where the dot represents time derivative and the boldface layout is used to denote matrices (vectors). Within the standard framework of development of the finite element procedures, considering the region V subdivided with a set of finite elements Ve, that is V = Ve, corresponding global matrices which appear in equation (13) are the result of the assembling of respective element matrices defined as follows me 

 N i N j dV

Ve

ke 

 D N i  N j  D   N i  N j

Ve

dV    N i N j dS

f e    N i J  dS   C eq Sf

(14) (15)

Seq

 N i dS

(16)

Seq

where trial functions Ni acquire now the meaning of the corresponding element shape functions. In these equations the stress-field is supposed to be known as a certain finite element approximation with the use of the same trial functions (or element shape functions) of the form similar to the one employed for the concentration (10), i.e. (x,t) = i(t)Ni(x), i= 1, 2, …, n (17) where i(t) are the known nodal stress values over a prescribed loading history, which must be obtained on the phase of purely mechanical analysis. This latter may be performed either simultaneously with or previously to the diffusion calculations, as far as hydrogen diffusion is not supposed to affect the stressstrain state evolution in a solid. Now, having reduced the diffusion boundaryproblem to the system of first-order differential equations (12) or (13), the WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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solution of this latter along the t-axis may be undertaken with the aid of the general easily programmable time-marching numerical scheme proposed for diffusion-type equations elsewhere [7]. Limiting to the first-order approximation of the unknowns within every single time interval, the nodal concentration values Cm–1 and Cm at the start and the end, respectively, of the m-th time interval [tm–1,tm] are related as follows C m  C m 1 M   t K  / t  K C m 1  F (18) where t = tm – tm–1, and the constant  must be chosen in a manner assuring the stability of this time-marching scheme. Obviously, for the first time interval (at m = 1) the array C0 must be determined according to the prescribed initial conditions of the problem. Then subsequent values of Cm are to be found from (18) solving corresponding linear algebraic system by any suitable means. In particular, in a symbolic form the matrix equation (18) renders the next solution C m  C m 1  M   t K 1 F  K C m 1 t

(19)

which invites one to employ suitable algorithms of matrix inversion on this route. Described procedure of time integration was proven to be unconditionally stable when [0.5, 1]. This way, the numerical approach to the modelling of the stress-assisted diffusion is determined in general terms. Its further transformation into a working practical code follows established FEM procedures of element formulation (i.e., the choice of appropriate element geometry, its shape functions, derivation of respective element matrices, which are involved in equations (13), the use of numerical integration, etc). Since diffusion modelling under consideration is planned as a supplementary one to a stress-strain analysis to be performed with the use of a certain general purpose FEM code, it seems naturally to use the same spatial element formulations for both mechanical and diffusion phases whenever there appeared no specific reasons to change the element type. Concerning described numerical approach, some final comments are worthy to be made. First, it is known [8] that strong accuracy deterioration may occur when Galerkin method is applied to the transport equation (5), which is a kind of a convection-dominated problem, and a mesh-related parameter called the Peclet number increases too much. In such cases Petrov-Galerkin methods are considered to be a better choice. Fortunately, this complication has never been approached in performed simulations with the magnitudes of governing material parameters associated with common metal-hydrogen systems, considered geometries and loadings, as well as reasonable finite element meshes. Next, whenever diffusion is considered to proceed simultaneously along with a non-steady state loading history, such as if slow strain rate tests were simulated, the stress-field is obviously time dependent, and so, the stress dependent element matrices do, too. Besides, when large geometry changes occur, the deformed distances become the diffusion paths of interest, so that coordinates x must be continuously updated with deformation displacements, and thus, they also become time dependent. As a result, all the element matrices in equations (13) must be updated throughout the simulation histories, i.e., they WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

138 Simulation of Electrochemical Processes III must be recalculated on every time step of diffusion modelling. This makes the full-scale calculations extremely time consuming. To diminish this, some reasonable model reductions might be not only advisable, but necessary.

4

Improvement of the workability of the developed modelling scheme

To proceed with simulations of stress-assisted diffusion with rather modest computational facilities available, it turned out to be indeed necessary to reduce the FEM-problem size. Among two possible approaches, i.e., coarsening of the mesh of the modelled "full-scale" specimen or shrinking the domain of diffusion simulation focusing on the locations of prospective hydrogen assisted fracture initiation near the notch, the second one seems to be preferable. The relevant data about stress fields may be transferred to this domain from the full scale mechanical analyses, performing their interpolation for the finite element mesh for diffusion, if convenient. To succeed on this way, in our particular application case of the notched tensile specimens, one may take an advantage of that the notch can act as a localized disturbance of the uniforms stress field in a smooth tensile specimen, depending on notch parameters of depths and width, as it is displayed by the data of the hydrostatic stress distribution in fig. 3.

Figure 3:

Hydrostatic stress distribution in a notched bar under traction along the horizontal axis indicating homogenization of the stressstate away from the notch (obtained with the finite element code MSC.MARC).

As well, a notch may do the same with regard to the diffusion from the points of view of the geometry and the stress effects on the transport phenomenon, if compared with the stress-unassisted diffusion in a smooth cylinder. In particular, the range of the disturbing effect of a notch on stress in assisted transport phenomena in solids can be estimated from fig. 4, where vanishing of the notch effect corresponds to fairly radial flow trajectories, or concentration contour bands parallel to the cylinder surface, the same as it occurs in smooth bars. Thus, it follows that at some reasonable distance from the notch, these effects on hydrogen diffusion (of notch geometry and non-uniform stresses) vanish and diffusion becomes stress-notch unaffected. According to this, a reduced WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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geometry can be considered, which includes the region of stress-assisted and notch affected transport. This zone of interest may be defined from analyses of the FEM-solutions of the problems of mechanics about the stress-strain state in notched bars and stress-strain unassisted transport, which may be obtained, e.g., with the aid of whichever available FEM-code, such as examples displayed in figs. 3 and 4. To this end it is only necessary to substitute the rest of the specimen (the "remote" portion) by corresponding boundary conditions derived, e.g., from the available closed-form solution of the transport problem for smooth homogeneous cylinder, which may be found elsewhere [9]. As an example, the reduced size domain to calculate the stress affected distribution of diffusible hydrogen in a particular tensile notched specimen, the same as considered above, is verified as shown in fig. 5, where corresponding boundary conditions are indicated. There a mesh of linear triangular elements is employed, although, this is not a matter of particular essence with respect to the presented approach.

Figure 4:

Diffusing specie distribution in a bar in the course of stress-strainunassisted diffusion.

Figure 5:

Reduced size domain and boundary conditions for simulations of stress assisted diffusion.

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140 Simulation of Electrochemical Processes III

5

Conclusions

The numerical approach presented in this paper allows one to calculate the distribution of hydrogen in stressed solids with limited expenditure of computer resources. Its generalization for the case of stress and strain assisted diffusion is straightforward. This model is considered to be useful to improve the knowledge of the role played by the factor of hydrogen accumulation in prospective rupture sites by stress-assisted diffusion, one of the key items in hydrogen embrittlement, a very dangerous phenomenon that frequently accompanies structural metals and alloys in service. Proposed computational model seems to be a promising tool as an aid to develop the life-prediction analyses for metallic components and structures subjected to any king hydrogen embrittlement in service.

Acknowledgements The authors wish to acknowledge the financial support provided by the following Spanish Institutions: Ministry for Science and Technology (MCYT; Grant MAT2002-01831), Ministry for Education and Science (MEC; Grant BIA200508965), Ministry for Science and Innovation (MCINN; Grant BIA2008-06810), Junta de Castilla y León (JCyL; Grants SA067A05, SA111A07 and SA039A08).

References [1] Toribio, J. & Kharin, V., Evaluation of hydrogen assisted cracking: the meaning and significance of the fracture mechanics approach. Nuclear Engineering and Design, 182, pp. 149-163, 1998. [2] Toribio, J. & Kharin, V., A hydrogen diffusion model for applications in fusion nuclear technology. Fusion Engineering and Design, 51-52, pp. 213218, 2000. [3] Shewmon, P., Diffusion in solids, TMS, 1989. [4] Sofronis, P. & McMeecking, R.M., Numerical analysis of hydrogen transport near a blunting crack tip. Journal of the Mechanics and Physics of Solids, 37, pp. 317-350, 1989. [5] Vázquez, M. & López, E., El método de los elementos finitos aplicado al análisis estructural, Editorial Noela, 2001. [6] Zienkiewicz, O.C. & Morgan, K., Finite elements and approximation, Wiley-Interscience Publication, 1983. [7] Zienkiewicz, O.C., Wood, W.L., Hine, N.W. & Taylor, R.L., A unified set of single step algorithms. Part 1: General formulation and applications. International Journal for Numerical Methods in Engineering, 20, pp. 15921552, 1984. [8] Zienkiewicz, O.C. & Taylor, R. L., The finite element method: Solids and fluids mechanics, dynamics and non-linearity, McGraw-Hill Book Company, 1991. [9] Carslaw, H.S. & Jaegger, J.C., Conduction of heat in solids, Oxford Clarendon Press, 1959. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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Mathematical modelling of electrochemical reactions in aluminium reduction cells R. N. Kuzmin1, O. G. Provorova2, N. P. Savenkova1 & A. V. Shobukhov1 1 2

Lomonosov Moscow State University, Russia Siberian Federal University, Russia

Abstract This paper reports the mathematical modelling of electrochemical processes in the Søderberg aluminium electrolysis cell. We consider anode shape changes, variations of the potential distribution and formation of a gaseous layer under the anode surface. Evolution of the reactant concentrations is described by the system of diffusion-convection equations while the elliptic equation is solved for the Galvani potential. We compare its distribution with the CO2 density and discuss the advantages of the finite volume method and the marker-and-cell approach for mathematical modelling of electrochemical reactions. Keywords: aluminium reduction, electrochemical reaction, gaseous layer, mathematical modelling, finite volume method.

1

Introduction

In aluminium reduction cells the carbon anode is placed in the upper part of the bath parallel to the liquid aluminium layer at the bottom, which acts as the cathode. The electrolyte between these electrodes consists of cryolite melt Na3AlF6 and alumina Al2O3 dissolved in it; it may also contain such admixtures as AlF3, CaF2 and others. Dissolving of alumina in the bulk of the cell may be described as follows: Na3AlF6 + Al2O3 ↔ 3Na+ + 3AlOF2 -. (1) Being driven by the electric field, the diffusion and the force of gravity, the ions AlOF2- reach the electrodes. There they donate and accept the electric charges. The cathode process is: 3AlOF2- + 6e = 2Al +6F- + AlF4-, (2) WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/ECOR090141

142 Simulation of Electrochemical Processes III while the anode process looks as: 3AlOF2- - 6e + 3/2 C = 3/2 CO2 + 3Al3+ + 6F-, (3) where e equals the charge of a single electron. Aluminium is deposited at the cathode; simultaneously the CO2 layer is being formed along the lower horizontal side of the carbon anode, which is consumed in this reaction. Unlike the conducting liquid metal layer at the bottom, the side walls of the cell are covered with ledge, which is a good insulator. As a result, most of the electric current goes between the cathode and the horizontal part of the anode. It should be emphasised that the conductivity of an electrolyte strongly depends of its composition. In particular, the increase of the CO2 concentration immediately under the anode surface causes an increase of ohmic resistance in this area and leads to the surge of the cell voltage caused by the formation of an insulating layer and to the consequent breakdown of this layer. This phenomenon is known as the anode effect. Many investigators have carried out mathematical modelling of aluminium electrolysis. In [1–2], for example, the authors dealt mainly with the magnetohydrodynamic (MHD) instability problem and therefore studied the electrolyser behaviour during approximately the first minute after switching on; for this reason they could completely ignore electrochemical reactions. Simulation of the electric field in different parts of the cell and the anode current density distribution was performed using the finite difference, finite elements and the finite volume methods [3–8]. The current distribution in the electrolyte and the variation of the anode shape with time for periods of several days were considered in [4–7]. Never the less, numerical investigation of the simultaneous variations of electrical current and ion concentrations inside the cell have never been performed. In this paper we combine the approach of [6], which consists in solving the equations for the electric fields in the anode, cathode and the electrolyte under steady state conditions, with our own approximation of the electrochemical reaction and the transport of reactants. We solve a 2D problem for the Laplace equation coupled with a system of the convection-diffusion equations through use of the boundary conditions. Therefore our problem becomes non-stationary. We study the time period of about one hour and observe the formation of the CO2 boundary layer and the variation of the Galvani potential caused by it.

2

Theory

We consider a 2D vertical cross section of an aluminium reduction cell and calculate the electrical field and the concentrations of the reactants in the cell. The temperature is supposed to be constant and equal to 960ºC; the influence of the magnetic field on the electrical field and concentrations is ignored. We also do not simulate the formation of the gas bubbles, but assume that the spreading of CO2 in the sub-anode layer is diffusive. We denote the Galvani potential of the electrical field as φ and the concentrations of AlOF2- and CO2 as c1 and c2 respectively. Initially the anode is rectangular and the concentration of CO2 is zero in the whole cell. According to WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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[6], after the voltage is applied to the electrodes, the potential φC at the cathode surface is given by C  c  jC , (4) where jC is the cathode current density and the coefficient c=8·10-2 Ohm·cm2, and the anode surface potential φA is given by  A  U  Erev  a  b  lg( j A ) , (5) where jA=0.75A·cm-2 is the averaged anode current density; a=0.50 V and b=0.25V decade-1 are the Tafel coefficients of the reaction (3) valid for the electrolyte that contains 81 wt% Na3AlF6, 3 wt% Al2O3, 11 wt% AlF3, 5 wt% CaF2 at 960˚C [6]; Ucell is the cell potential and Erev=1.23V is the reversible electrode potential. The distribution of the electrical field in the cell is described by div ( (c2 )  grad  )  0 . (6) where σ(c2) is the electrolyte conductivity. We assume that it depends on the concentration of CO2 in the following way:  (c2 )   0  ( 1   0 )  (1  exp(  c2 )), (7) Here σ0=2.0·100 (Ohm·cm)-1 and σ1=1.0·10-6 (Ohm·cm)-1 are the specific conductivities of pure electrolyte and pure CO2 at 960˚C; α=3·10-1 is an empirical parameter. We suppose that the side walls of the cell are electrical insulators as well as the upper electrolyte surface around the anode, because all these surfaces are covered with insulating ledge. This supposition and the eqns (4)-(5) give us the full set of boundary conditions for φ:

   C at the cathode surface;    A at the anode surface;   0 at the side walls and at the upper surface. 

(8)

where ν is the normal vector to the boundary. The electrode processes (2) and (3) run in accordance with the Faraday law; therefore the concentrations c1 and c2 satisfy the following equations at the cathode and anode surfaces:

D1  D1 

j c1 c2  C ;  0;  nC F 

j c1 c 1 j  A ; D2  2   A .  n A F  2 n A F

(9) (10)

where nC=3 and nA=4 are the numbers of electrons participating in the cathode and anode processes; D1 = D2 ≈ 1.0·10-5 cm2s-1 are the diffusion coefficients [9][10]; ν is the normal vector to the electrode surface and F is the Faraday's constant. Inside the cell concentrations satisfy the convection–diffusion equations [11–13]:

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144 Simulation of Electrochemical Processes III

c1  div ( D1  grad c1  m  c1  grad  ), t c2  div ( D2  grad c2 ), t

(11) (12)

where m=-1 is the electric charge of the AlOF2- ion measured in the electron charge units. The reactants do not permeate through the side walls of the cell; therefore the boundary conditions for c1 and c2 at them look as follows:

c1 c2   0,  

(13)

where ν is the normal vector to the walls. Finally we should take into account the changes of the electrodes [6] during the electrochemical reaction. From Faraday's law it follows that the rate of the cathode vertical shift is

uC 

M1  jC , nC F1

(14)

where M1 and ρ1 are the molar mass and the density of aluminium, while the rate of the anode shift in the direction orthogonal to its surface is

uA 

M2  jA , nA F 2

(15)

where M2 and ρ2 are the molar mass and the density of carbon. It should be mentioned that the cathode surface remains horizontal, because it is the surface of the liquid aluminium pool and the normal vector to it remains vertical, while the anode surface changes its shape, and the direction of its normal vector varies. We study rather small periods of time, around one hour, so the anode consumption is practically negligible, though we take it into account as well. Yet, unlike [4–7], we do not look for the steady state anode shape.

3

Finite volume approximation

We use the finite volume method for numerical approximation of the eqns (4)(14). The whole area is treated as a set of rectangular blocks – see Fig.1

Figure 1:

Block of the grid.

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The blocks, or elementary volumes, form the grid. The blocks are numbered with two indexes, i and j, in the horizontal and vertical direction respectively. The horizontal axes is denoted as x and the vertical one - as z. The sizes of the block along these axes are denoted as Δx and Δz. The values of φ, c1, c2 and σ are located in the centre of the block and are supposed to be constant in it. The current density vector j has the horizontal and vertical components: j=(jx, jz); these components are assigned to the centres of the block sides; see Fig.1. Ohm's law in vector notation gives us the relation between φ and j:

j x  

 ; x

j z  

 . z

(16)

The eqns (15) are approximated on the grid in the following way:

jx  

 i 1, j   i , j i 1, j  i , j

;  2 x   i , j i , j 1  i , j  . jz   i , j 1  2 z

(17)

Similar approximations are done for jx- and jz-. The eqn (6) may be rewritten as an integral of the current density normal to the block boundary:

 ( j, )dS  0,

(18)

S

where S is the boundary of the block and ν is its normal vector, and therefore is approximated as

( jx  jx )  z  ( jz  jz )  x  0.

(19)

The eqns (11)-(12) are treated in the same manner: given the concentration c (where c stands for c1 or c2) in the centre of the block, we approximate the concentration gradient h=(∂c/∂x, ∂c /∂z) at the centres of the block sides:

hx 

ci 1, j  ci , j x

; hz 

ci , j 1  ci , j z

.

(20)

Again similar approximations are done for hx- and hz-. Then the divergences in eqns (11-12) are treated as the integrals of the normal components of their arguments over the block boundary as it was done with the eqns (18-19). Finally for approximation of the time derivatives in the left-hand side of the eqns (11-12) we apply the symmetrical difference scheme. We replace the time derivatives with appropriate finite differences:

c ci , j (t  t )  ci , j (t )  t t and equate them to half-sums of the right hand side approximations on the next and current time layers. The obtained system of algebraic equations is solved together with (18) using Seidel’s iterative method. This scheme approximates the original equations with the second order of accuracy with respect to t, x, z and is WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

146 Simulation of Electrochemical Processes III unconditionally stable [14]. Besides it guarantees the conservation of current and matter: no artificial sources or drains of electric charges and reagents may appear during computation [6, 14]. All elementary volumes are divided into three types: A (anode), B (electrolyte) and C (cathode) blocks (see Fig.2). Eqns (18-19) may be written only for blocks B, because the electrodes (that is, blocks A and C) are equipotential and contain only carbon and aluminium, while blocks B contain a mixture of ions and molecules, AlOF2- and CO2 in particular.

Figure 2:

Scheme of the electrolyser.

Those blocks of types A and C that are situated at the electrolyte-electrode border are used for approximating the boundary conditions given by eqns (8) and (16). The electrode surfaces are determined in the following way [15]: we introduce special markers (imaginary particles) that are originally placed in the centres of those blocks of type B that are adjacent to the blocks of types A and C. The coordinates of these markers are found from eqns (14) and (15) so they delimitate the border. As soon as any of these particles leaves the B-type block and enters the A-type or C-type one, the type of the entered block is changed to B. If the particle leaves the B-type block for another B-type one (as it happens at the cathode surface), the empty block is assigned the type of the adjacent non-Btype block (in the considered problem it's always C).

4

Results and discussion

We considered the aluminium reduction cell section one meter wide and halfmeter high (Lx=Lz=50 cm at Fig.2). Initially the depth of the cathode aluminium pool is 5 cm; horizontal and vertical sizes of the carbon anode are 70cm and 30cm respectively; the gap between the anode and the side walls and the distance between the electrodes equals 15 cm. We use the rectangular grid containing 100 WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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blocks vertically and 200 blocks horizontally; it gives us Δx=Δz=5·10-1 cm. The step Δt is taken equal to 1 second and 104 time layers are computed in a single series, which gives us nearly 3 hours of the physical time. In Figure 3 we demonstrate the equipotential lines of φ with the step 0.4V for t=1 sec. The distances between these lines are practically equal. It proves that the conductivity of electrolyte at this moment doesn’t vary in the whole volume of the cell. The boundary conditions for φ are given by eqn (8); it imposes ∂φ/∂ν =0 along the walls and the upper anode boundary. However, the distribution of the potential is different for t = 1 hour. The same equipotential lines for t = 1 hour are shown in Fig.4. It is easy to see that now the greatest voltage drop occurs in a narrow region directly below the anode.

Figure 3:

Equipotential lines in the electrolyte (units in V) for t=1 sec.

Figure 4:

Equipotential lines in the electrolyte (units in V) for t=1 hour.

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Figure 5:

Contours of the CO2 concentration for t=1 hour. The levels: 1 – 1·10-6 mol/cm3, 2 – 1·10-5 mol/cm3, 3 – 5·10-5 mol/cm3.

The anode potential also starts growing. These results are easily explained by considering the concentration of CO2 in the bath. Initially there is no CO2 in the cell. It appears as a result of the anode process (3) and forms a layer with low specific conductivity, as it is shown in Fig.5. Comparing Fig. 4 and 5, we see that the voltage drop region coincides with the zone of high CO2 concentration. The change of the anode shape is rather small and may be observed mainly in the corners of the electrode. It agrees with previous experimental and numerical results [6–8]; it takes 6 to 8 days to reach the constant anode shape. Never the less, the amount of carbon, consumed during the first hour, is enough to form a layer of CO2, sufficient for transform the electric current distribution inside the cell.

5

Conclusions

It follows from the above discussion and numerical results that even a simple convective-diffusive model of concentration behaviour mechanism gives realistic results and yields a satisfactory description of the formation of the gaseous layer under the anode surface. The model may be improved by adding the electrolyte circulation and electromagnetic forces; yet we hope that it will not change the main conclusions. The finite volume method proves to be a flexible and sufficiently accurate numerical technique for solving both the equations for the Galvani potential and the reactant concentrations. The marker-and-cell approach makes it possible to outline the electrode surfaces easily.

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References [1] Munger, D., Vincent, A., A level set approach to simulate magneto hydrodynamic instabilities in aluminium reduction cells. Journal of Computational Physics, 217, pp. 295-311, 2006. [2] Safa, Y., Flueck, M., Rappaz, J., Numerical simulation of thermal problems coupled with magneto hydrodynamic effects in aluminium cell. Applied Mathematical Modelling, 33, pp. 1479-1492, 2009. [3] Piotrowski, A., Pietrzyk, S., Metall. & Foundry Eng., 16, pp.315-337, 1990. [4] Kuang Z., Thonstad, J., Current distribution in aluminium electrolysis cells with Søderberg anodes. Part I: Experimental study and estimate of anode consumption. Journal of Applied Electrochemistry, 26, pp. 481-486, 1996. [5] Zoric, J., Roušar, I., Kuang Z., Thonstad, J., Current distribution in aluminium electrolysis cells with Søderberg anodes. Part II: Mathematical modelling. Journal of Applied Electrochemistry, 26, pp. 795-802, 1996. [6] Zoric, J., Roušar, I., Thonstad, J., Mathematical modelling of industrial aluminium cells with prebaked anodes. Part I: Current distribution and anode shape. Journal of Applied Electrochemistry, 27, pp. 916-927, 1997. [7] Zoric, J., Roušar, I., Thonstad, J., Mathematical modelling of industrial aluminium cells with prebaked anodes. Part II: Current distribution and influence of sideledge. Journal of Applied Electrochemistry, 27, pp. 928-938, 1997. [8] Provorova O.G., Kuzmin R.N., Savenkova N.P., Shobukhov A.V., Mathematical modelling of aluminium electrolysis. Proc. of the 9th Int. Conf. on Boundary Value Problems and Mathematical Modelling, Novokuznetsk, 1, pp.103-106, 2008. [9] Sterten, Ǻ., Solli, P.A., Cathodic process and cyclic redox reactions in aluminium electrolysis cells. Journal of Applied Electrochemistry, 25, pp. 809816, 1995. [10] Sterten, Ǻ., Solli, P.A., An electrochemical current efficiency model for aluminium electrolysis cells. Journal of Applied Electrochemistry, 26, pp. 187-193, 1996. [11] Powell, A.C., Shibuta, Y., Guuer, J., Becker, C.A., Modelling Electrochemistry in Metallurgical Processes. The Journal of the Minerals, Metals & Materials Society, 59(4), pp. 50-58, 2007. [12] Grafov, B.M., Martemjanov, S.A., Nekrasov, L.N., The Turbulent Diffusion Layer in Electrochemical Systems, Nauka: Moscow, pp. 6-22, 1990. [13] Damaskin, B.B., Petrii, O.A., Tsirlina, G.A., Electrochemistry, Khimiya: Moscow, pp. 417-422, 2006. [14] Samarskii, A.A., Goolin A.V., Numerical Methods in Mathematical Physics, Nauchnii Mir: Moscow, pp.271-274, 2000. [15] Harlow, F.H., Welch, J.E., Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface. Physics of Fluids, 8, pp.2182-2189, 1965.

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Atomistic simulation of the nano-structural evolution of Raney-type catalysts from spray-atomized NiAl precursor alloys during leaching with NaOH solution N. C. Barnard1, S. G. R. Brown1, F. Devred2, B. E. Nieuwenhuys2 & J. W. Bakker2 1

Materials Research Centre, School of Engineering, Swansea University, UK 2 Leiden Institute of Chemistry, Leiden University, The Netherlands

Abstract A model for the nano-structural evolution of Raney-type nickel catalysts (widely used in hydrogenation reactions) from the constituent intermetallic phases present in nickel-aluminium precursor alloys is presented here. Nano-porous nickel catalysts are prepared via a caustic leaching process where the NiAl alloy powder (typically 50-50 at.%) is immersed in concentrated NaOH solution in order to leach away the aluminium present to leave a highly-porous nickel catalyst (often referred to as spongy nickel). The temporal evolution of the nickel catalyst is described by a kinetic Monte Carlo (kMC) method, which captures two dominant processes taking place on leaching with NaOH solution: rapid aluminium dissolution from the crystalline intermetallic phases and surface adatom diffusion of nickel and aluminium. Structural changes from NiAl3 and Ni2Al3 phases present in the starting alloy are considered on an event-by-event basis. Individual crystallite size, pore diameter and effective surface area are predicted. Finally, a Metropolis Monte Carlo method is used to investigate the surface nature of the as-leached catalyst and compared to concurrent experimental investigations. Keywords: de-alloying, leaching, Raney-Ni catalysts, Monte Carlo simulation.

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1

Introduction

Raney-type nickel catalysts are typically prepared by leaching aluminium from a Ni-Al alloy using a concentrated sodium hydroxide solution [1–3]. This process of activation critically affects the structure and properties of Raney-type nickel catalysts. The initial structure and composition of the starting alloy also influence the performance of the final catalyst [4–7]. In this paper, numerical modelling is compared to experimental measurements in an attempt to simulate both the 3D morphology of as-leached Raney-Ni catalyst material and investigate the nature of the exposed catalyst surfaces. Firstly, a kinetic Monte Carlo (kMC) [8] for the nano-structural evolution of so-called spongy nickel from the constituent intermetallic phases present in nickel-aluminium precursor alloys is described. Experimental data concerning nano-porous nickel catalyst powder used in this paper are derived from leached NiAl alloy powder produced via a spray-atomization route rather than the conventional cast-and-crushed route. The temporal evolution of the nickel catalyst during de-alloying described by the kMC model captures two dominant processes taking place during leaching with NaOH solution (i) de-alloying via rapid aluminium dissolution from the crystalline intermetallic phases and (ii) surface adatom diffusion of nickel and aluminium. Structural changes from NiAl3 and Ni2Al3 phases present in the starting alloy are considered on an event-by-event basis. Individual crystallite size, pore diameter and effective surface area are predicted and compared to concurrent experimental investigations [9]. In its role as a catalyst, the surface condition of the as-leached nickel nanostructure is highly important. In particular, the sites occupied on the surface of the catalyst by any residual aluminium atoms are of interest. In order to investigate the surface configuration of as-leached surfaces a Metropolis Monte Carlo (MMC) model is subsequently applied. As a starting point the MMC model exploits the nano-porous structure already predicted by the kMC approach. The MMC model aims to simulate the appearance of Al-rich surface configurations that are routinely observed in experiment.

2

Numerical models

2.1 Kinetic Monte Carlo (kMC) method The Monte Carlo model is based on a technique originally used to model the leaching of silver from a Au – 50 at.% Ag alloy, examining the evolution of nano-porosity [10]. For NiAl alloys two key processes are assumed to be taking place during leaching, namely adatom diffusion (Ni or Al) and Al dissolution. These processes are assumed to occur at known rates that serve as the inputs for the model. Both diffusion and dissolution of adatoms are assumed to proceed at a rate described by the following Arrhenius relationships respectively:  nE  (1) k ndiff  vdiff exp  b  k T b   WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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 nE  (2) kndiss  vdiss exp  b   kbT  where kndiff is the rate constant for diffusion of an adatom (Ni or Al), kndiss is the rate constant for dissolution of an adatom (Al only), νdiff and νdiss are frequency factors (1×1013s-1 and 1×1017s-1 respectively), n is the number of bonds, kb is the Boltzmann constant and T is absolute temperature (K). The value of 1×1017s-1 is at the higher end of reported frequency factors [11]. The bond energies are taken as Eb = 0.165, 0.1875 and 0.069 eV for Ni-Ni, Ni-Al and Al-Al bonds respectively. These values are based on published pair potentials [12] and calculated over the number of nearest neighbour atoms [13]. In the present model it is assumed that the crystal is without any type of lattice defect and the role of any reaction products is not included. Furthermore, it should be noted that the model does not include recently proposed fragmentation mechanisms that may occur during leaching [14]. The surface diffusion of both elements during leaching means that the final nano-metric spongy nickel structure is not ‘revealed’ but ‘constructed’ during leaching by diffusion of Ni adatoms. Equations 1 and 2 ensure that atoms with fewer bonds are more likely to dissolve or diffuse. Adatoms either already present in, or diffusing to, lower energy sites (i.e. sites of higher coordination number) will tend to become less likely to diffuse further. In this way Ni adatoms tend to cluster together at surfaces that are increasingly exposed during Al dissolution, thus building up a nano-scopic structure. Knowledge of the rates of all permitted transitions allows the time increment for any iteration of the model to be calculated. This has been done using the well-known Bortz-KalosLiebowitz (BKL) algorithm described before [8]. To carry out a simulation a 3D computational domain is created containing a total of 2003 Ni and Al atoms. Initially a cubic region of material is created, all faces of which are considered to be exposed to concentrated NaOH solution. There are three phases usually encountered in the precursor NiAl alloy powders: (i) NiAl3 phase, which is represented by an L12 structure with an ABCABC stacking sequence, (ii) Ni2Al3 phase, the structure of which is a five-layer closepacked unit cell with stacking ABCBCA, (iii) Al-rich eutectic phase which is assumed to leach away very quickly and is not dealt with in the model. kMC simulations of the NiAl3 and Ni2Al3 phases are used here. Once the BKL algorithm is applied Al atoms are progressively removed (at a rate that decreases with time) (8). Simultaneously, adatom diffusion of Ni results leads to the development of a nano-porous structure. Figure 1 shows the final predicted as-leached structure of the NiAl3 precursor material after 3 hours of leaching at 353K showing the nano-porous structure that has developed. Figure 2 shows the results of a simulation for Ni2Al3 precursor material leached under the same conditions, a nano-porous structure has developed but is much less open than the structure from the NiAl3 precursor material. These two predicted morphologies are used as the starting point for MMC calculations. In both figures the nano-porous structure has surfaces which are almost exclusively exposed (100) or exposed (111) planes (coordination WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

154 Simulation of Electrochemical Processes III numbers of 8 and 9 respectively). Atoms with coordination numbers of 7 are also common along the ridges at the edges of these planes. 2.2 Metropolis Monte Carlo (MMC) method In order to address the experimental observations that leached Raney-Ni catalytic powders containing <10% Al, can nevertheless possess 30-50% metallic Al at the surface a Metropolis Monte Carlo modelling approach has been adopted. The MMC algorithm differs from the kinetic Monte Carlo method in that it attempts to determine the minimum energy for a system, but without the specific reference to time. The kMC technique has been used to simulate the time evolution of nano-porous structures during leaching. In contrast, the MMC technique is used here to determine the lowest energy configuration for these computed structures in terms of optimum interatomic spacings and distribution of atomic species. The general MMC algorithm is described widely in the literature, e.g. [15]. Here it is used in canonical ensemble mode. The number of atoms, N, temperature (300K) and pressure are fixed. The number of each kind of atom (Ni or Al) is also fixed. Two types of trial are performed. (1) Random displacement of each atom in the computational domain from its current position. The magnitude of this displacement is of the order 0.003Å. Once each atom has been displaced the decision on acceptance of the new configuration is based on the standard Metropolis method: Pnew   U  (3)  exp  Pold  kT  Where kT is the Boltzmann factor and U is the potential energy difference. If Pnew/Pold>1 the new configuration is accepted. If not it is still accepted with a probability of Pnew/Pold. (2) An atom is exchanged with another atom selected at random. The decision on acceptance of the change is also according to equation 3 above. After all atoms in the simulation have undergone several of these steps the lattice parameter of one box direction, [001] [010] or [100], is selected at random and altered. Random changes in lattice parameter of up to 1Å are used. The relative probability of acceptance of such an alteration is given by: Pnew   (U  PV  NkT ln V )  (4)  exp  Pold kT   Here V is the volume of the box and P is pressure (assumed to be zero). These trials are repeated until the system effectively reaches steady state. While several data sets have been made available to model NiAl alloys we choose the tightbinding (TB) second-moment approximation (SMA) approach adopted by Papanicolaou et al. [16]. This TB-SMA approach defines the total energy of the system (U in equation 3) as 12 N     rij    rij     2 exp  2q    1   (5) U     A exp  p    1     i 1  j  i  r0   j i  r0       WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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Here the first term is a pair-potential repulsive term and the second term corresponds to the band-structure term. N is the total number of atoms, rij is the distance between atoms i and j of the species  and , respectively, ( and  stand for Al and Ni) and the sum j is extended up to fifth neighbours. Numerical values of the constants A, ξ, p, q and r0 are given in [16]. MMC simulations of this type have been carried out in the past to investigate the surface configurations of isolated nano-clusters of NixAly alloys [17]. In summary, a kMC model is used to generate a nano-porous structure, hopefully representative of real as-leached spongy nickel. The predicted 3D arrangement is then used as a reference structure on which different proportions of Ni and Al atoms can be placed. While maintaining the numbers of Al and Ni atoms constant, the MMC algorithm repeatedly swaps Al and Ni atoms whilst also adjusting the overall lattice parameters to compute the lowest energy configuration for the given atomic arrangement and relative proportions of Al and Ni atoms. No further diffusion or leaching occurs during the MMC simulation. The MMC model is thus used in a similar manner to published work on isolated nano-clusters of different compositions [17]. The difference here is that the MMC simulation will be used on the non-spheroidal clusters predicted from kMC.

3

Results and discussion

In order to compare simulation results to experimental measurements data related to as-leached catalyst morphology as well as data related to surface composition is required. For morphological comparison a TEM bright field image is used. Prior to the structural characterization the leached samples were slowly passivated with oxygen. The image was taken with a Philips CM 20, equipped with a LaB6 cathode and operated at 200 kV accelerating voltage. This information is combined with Brunauer, Emmett and Teller method (BET) measurements providing surface area data. For surface composition information X-ray photoelectron spectroscopy (XPS) data is used. Figure 3 shows a TEM image of an as-leached particle where two distinct regions can be seen. To the right is a relatively open structure that has formed from NiAl3 phase. To the left is a denser region formed from Ni2Al3 phase. In Figure 3 the box region to the left of A is the computed kMC nano-porous structure from figure 2 overlaid onto the TEM image at the corresponding magnification. The box region to the left of B is the computed kMC nano-porous structure from figure 1 again at the corresponding magnification. A visual comparison indicates that the predicted nano-porous structures are similar to those observed in a real particle. Before considering surface area a note of warning must be sounded. In precursor alloys containing Ni2Al3 phase it has been reported that the Ni2Al3 phase does not leach properly and is still present in the final catalyst [18–20]. This would therefore confuse interpretation of results. Consequently, in this paper we compare to BET measurements made on catalyst powder made from a Ni-Al 82.5 at.% precursor material (with a particle size in the range 106-150 μm). This alloy does not contain any Ni2Al3 phase and thus can safely be WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

156 Simulation of Electrochemical Processes III assumed to have properly leached. Table 1 shows both bulk and surface compositions for this material as well as the surface area per weight from BET. The structures predicted from the kMC model (Figures 1 and 2) have estimated surface areas of 49.9 and 49.8 m2g-1 respectively, showing that despite its simplicity the model is capable of providing results at the correct length scale.

Figure 1:

3D predicted nano-porous structure of leached NiAl3 precursor material. The box at the top left shows a magnified view. Region A is an exposed (111) plane and just to the right of B is an exposed (100) plane. Dark atoms are Ni and light atoms are Al. Edge length of the box is 50 nm.

Table 1:

Measured bulk and surface data for an as-leached catalyst powder manufactured from spray-atomized Ni-Al 82.5 at.% precursor alloy.

Catalysts Ni-Al 82.5at.% (106-150 μm)

Ni (at %) Bulk 95 ± 1

Al (at %) Bulk 5±1

Ni (at %) XPS 68 ± 1

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Al (at %) XPS 32 ± 1

BET (m2g-1) 48 ± 2

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Table 1 shows a very interesting result. Although the powder has a bulk Al content of 5 ± 1 at.% this has risen to 32 ± 1 at.% Al at the surface. To simulate this effect the MMC model was run on an isolated nano-cluster 54 atoms in diameter (which equates to a sphere with approximately 50 m2g-1). The nanocluster was set up as an FCC Ni structure with 5 at.% Al present distributed randomly within the particle. Figure 4 shows the nano-cluster once the MMC model has effectively converged to the lowest energy configuration. The final predicted lattice parameter from the model is 0.343 nm with a predicted cohesive energy per atom of 4.38 eV, in good agreement with previous published results [16]. The final surface composition of the nano-cluster is 29.5 at.% Al where importantly there are no Al-Al bonds present.

Figure 2:

3D predicted nano-porous structure of leached Ni2Al3 precursor material. Dark atoms are Ni and light atoms are Al. Edge length of the box is 50 nm.

This predicted surface composition is thus in very good agreement with the BET measurement. Interestingly, the nano-cluster actually contains enough Al atoms to provide a 69 at.% surface coverage of Al. However, this would entail Al-Al bonding which is not energetically favourable [17]. This last result provides the possibility of estimating the surface configuration of predicted leached structures. Segregation of Al to the surface is energetically favourable, provided there are no Al-Al bonds. In Figure 1 we have a structure which is 95 at.% Ni. As many as possible of the Al atoms have been placed at surface sites ensuring that no Al-Al bonds are present. As a result, the predicted structure in Figure 1 has a surface area per gram of 49.9 m2g-1, a bulk Al composition of 5 at.% and a surface Al composition of 28.9 at.% all in very close agreement with the experimental data in Table 1. As an aside, the measured value [21] of surface aluminium content for an exposed (111) plane is 25 at.% (Ni90Al10 at 1100K). A theoretical maximum of WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

158 Simulation of Electrochemical Processes III surface aluminium content for an exposed (100) plane, precluding any Al-Al bonds, is 50 at.%. Since a structure predominantly comprising exposed (111) and (100) planes is predicted, it might therefore be expected that as-leached RaneyNi powders with residual Al would possess surface concentrations of Al in the range 25-50 at.% which is what is routinely observed.

Figure 3:

4

TEM bright field image of leached NiAl alloy where two distinct regions can be seen. To the right is a relatively open structure that has formed from NiAl3 phase. To the left is a denser region formed from Ni2Al3 phase. (TEM image courtesy of Dr. Ute Hörmann, Ulm University, Germany.)

Conclusions   

A kinetic Monte Carlo model has been shown to be capable of simulating as-leached nano-porous structures that develop during leaching of NiAl alloys. As shown by previous workers the Metropolis Monte Carlo calculations indicate that segregation of any residual Al to the surface is energetically favourable provided that no Al-Al bonds form. Simulated surface area per gram of catalyst and simulated surface concentration of Al are both in good quantitative agreement with measurements carried out on a leached NiAl precursor alloy that did not contain Ni2Al3 phase.

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Figure 4:



159

Isolated NiAl nano-cluster 54 atoms in diameter (equating to a sphere approximately 50 m2g-1) with an FCC structure, bulk composition 5 at.% Al. Dark atoms are Ni and light atoms are Al. The final surface composition computed via the MMC model is 29.5 at.% Al. NB there are no Al-Al bonds present. The predicted surface condition of leached NiAl material is one with predominantly (111) and (100) exposed planes where residual Al is present with no Al-Al bonds.

Acknowledgements The authors would like to thank Dr Ute Hörman of Ulm University in Germany for kindly providing the TEM image. This work has been carried out as part of the Intermetallic Materials Processing in Relation to Earth and Space Solidification (IMPRESS) European integrated project (contract no. NM3-CT2004-500635) coordinated by the European Space Agency (ESA) [22].

References [1] M. Raney, US Patent 1.563.787 (1925). [2] P. Fouilloux, G.A. Martin, A.J. Renouprez, B. Moraweck, B. Imelik and M. Prettre, J. Catal, 25 (1972) 212. [3] A.J Smith and D.L. Trimm, Annu. Rev. Mater. Res. 35 (2005) 127. [4] A.B. Fasman, V.F. Timofeeva, V.N. Rechkin and Y.F. Klyuchnikov, Kyn. and Khem. 13 (1971) 1513. [5] J. Freel, W.J.M. Pieters and R.B. Anderson, J. Catal. 14 (1969) 247. [6] S. Sane, J.M. Bonnier, J.P. Damon and J. Masson, Appl. Catal. 9 (1984) 69.

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160 Simulation of Electrochemical Processes III [7] M.S. Wainwright, Handbook of Heterogeneous Catalysis, Vol 1, G. Ertl, H. Knozinger and J. Weitkamp (Eds), VCH Publ., New York (1997), ISBN 3527-29212-8. [8] N.C. Barnard and S.G.R. Brown, Simulation of Electrochemical Process II, V.G. DeGiorgi, C.A. Brebbia and R.A. Adey (Eds), WIT Press, Southampton, UK (2007), ISBN 978-1-84564-071-2, 53. [9] F. Devred, A.H. Gieske, N. Adkins, J.W. Bakker, B.E. Nieuwenhuys, Applied Catalysis A, General 356 (2009) 154. [10] J. Erlebacher, M. J. Aziz, A. Karma, N. Dimitrov and K. Sieradzki, Nature, (2001) 410: 450. [11] D. D. MacNeil and J. R. Dahn, J. Phys. Chem., A (2001), 105, 4430. [12] J. Mei, Bernard R. Cooper and S. P. Lim, Phys. Rev. B, 54, 1, (1996), 178. [13] M. I. Baskes and C. F. Melius, Phys. Rev. B, 20, 8, (1979), 3197. [14] R. Wang, H. Chen, Z. Lu, S. Qiu and T. Ko, J. Mat. Sci. 43 (2008) 5712. [15] M. P. Allen and D. Tildesley, Computer Simulation of Liquids, Clarendon, Oxford, 1987. [16] N. I. Papanicolaou, H. Chamati, G. A. Evangelakis and D. A. Papaconstantopoulos, Comp. Mat. Sci., 27 (2003) 191. [17] E. E. Zhurkin and M. Hou, J. Phys. Condens. Matter, 12 (2000) 6735. [18] H. Lei, Z. Song, D. Tan, X. Bao, X. Mu, B. Zong and E. Min, Appl. Cat. A: General 214 (2001) 69. [19] M. L. Bakker, D. J. Young and M.S. Wainwright, J. Mat. Sci. 23, (1988) 3921. [20] R. Wang, Z. Lu and T. Ko, J. Mat. Sci. 36 (2001) 5649. [21] T. Schulthess and R. Monnier, Phys. Rev. B, 50, 24, (1994), 18564. [22] D.J. Jarvis and D. Voss, Mat. Sci. Eng. A, 413-414, (2005), 583.

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Corrosion of mild steel and 316L austenitic stainless steel with different surface roughness in sodium chloride saline solutions L. Abosrra, A. F. Ashour, S. C. Mitchell & M. Youseffi School of Engineering, University of Bradford, UK

Abstract The corrosion behaviour of mild steel and 316L austenitic stainless steel was investigated in saline solution containing 1 and 3%NaCl. Specimens with surface roughness of 200, 600 grit emery paper and 1μm diamond paste were investigated. The anodic polarization measurement technique was performed at a scan rate of 1mV/s for a fixed period of 1 hour. The experimental results revealed that chloride ions have a significant effect on the corrosion behaviour of both steels as expected. As the surface roughness of 316L stainless steel increased, the breakdown potential (Ebreak), the free corrosion potential (Ecorr) and the width of passivity decreased, hence the corrosion rate increased. However, in the case of mild steel specimens, improving surface finish lead to shifts in the corrosion potential to more noble states and increased the corrosion rate. Metallographic examination of corroded specimens after electrochemical corrosion tests confirmed that the breakdown of the passive region was due to pitting corrosion. Keywords: mild steel, 316L SS, anodic polarization, corrosion, surface roughness, saline solution.

1

Introduction

The study of corrosion properties of mild steel and stainless steel in aqueous solutions has received a great general attention. Mild steel is the most common structural material and is used in a wide range of environments. It is well known that when mild steel corrodes, anodic and cathodic areas develop over the corroded surface. Conventionally, these pits are known to change in shape and move across the surface, resulting in early corrosion that is approximately WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/ECOR090161

162 Simulation of Electrochemical Processes III uniform [1]. However, when mild steel is exposed to saline solutions and a marine environment this type of corrosion is not observed. Anodic regions and micro-pits develop very quickly after early exposure and the further presence of shallow broad pits occurs [2, 3]. When a high corrosion resistance is required, stainless steel is recommended. Type 316L stainless steel is austenitic and has been used in the chemical and petrochemical industries and offshore structures for many decades. The excellent corrosion resistance of this stainless steel is attributed to the formation of a stable passive oxide layer, but nevertheless stainless steel is susceptible to localized corrosion by chloride ions [4]. In addition to choosing the right stainless steel grade for good corrosion resistance, it is equally important to specify the right surface condition of the materials used in many applications [5]. The surface condition affects the corrosion resistance to a certain extent, which implies that it is possible to meet certain requirements by specifying the proper finish rather than upgrading the chosen alloy. The effect of surface condition on corrosion resistance of 301, 304L and duplex stainless steel has been documented elsewhere [6–8]. Since the pitting potential actually defines a minimum condition under which pits can become stable, the aim is to lower the pitting potential. Metastable pitting occurs throughout the passive region of stainless steel and the potential at which the transition to stable pit growth occurs is an important parameter describing the stability of the metal [6, 9, 10]. In this study the effect of surface roughness on the corrosion resistance of mild steel and 316L austenitic stainless steel in the presence of different chloride concentrations was investigated using the electrochemical technique. Corrosion parameters, such as the breakdown potential, the free corrosion potential, and the corrosion rate, were determined.

2

Experimental methods

2.1 Sample preparation The chemical compositions are given in weight % for mild steel as: 0.84 Mn, 0.53 Cu, 0.24 Si, and 0.19 C and for 316L stainless steel as: 18Cr, 14 Ni, 2.0 Mn, 0.75 Si, 0.03 C and 0.27 Mo. Corrosion tests were performed on cylindrical specimens with 9.5mm diameter and 12.8mm length with an exposed area of 4.5cm². Specimen surfaces roughness was generated by wet grinding using silicon carbide papers of 200 and 600 grits and the polished surface was generated using 1μm diamond paste. 2.2 Electrochemical measurements Electrochemical tests were carried out in saline solutions containing 1 and 3% NaCl using the standard electrochemical corrosion cell containing three electrodes: the saturated calomel electrode (SCE), the counter electrode (graphite) and the working electrode (the sample). All electrodes were immersed in a suitable glass vessel containing 600ml of the saline electrolyte. The anodic and cathodic polarization curves were measured after 1hr of working electrode WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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immersion at ambient temperature (23± 2°C) in the investigated environment. Subsequently, the anodic polarization curves for both mild steel and 316L stainless steel were measured from potential values below and above the free corrosion potentials of  250 mV E corr and  800 mV, respectively, with a scan rate of 1mV/sec. The experiments were repeated 3-4 times for all specimens in all solutions. Corrosion rates were calculated for steel immersed in aqueous solutions using eqn. (1) [11].

CR 

0.13I Corr W DA

(1)

where I Corr  corrosion current density ( A / cm ) , W = Atomic weight of 2

3

steel, D = Density of steel in gm / cm , and A is the exposed surface area of 2

the steel in cm .

3

Results

3.1 Electrochemical data analysis 3.1.1 Mild steel – potentiodynamic results Figure 1 gives the potentiodynamic polarization curves of mild steel with surface roughness of 200 and 600 grit papers as well as 1μm diamond paste polished surface tested in 1% NaCl solution. The corrosion behaviour was affected by the degree of surface roughness. An increase in roughness from polishing (like mirror) to 600 and 200 grit surface roughness shifted the free corrosion potential to more active values as shown in Figures 1 and 3. The corrosion rate values were not expressing the same trend recorded for the free corrosion potential. A specimen of diamond polished surface showed a higher corrosion rate than the 600 and 200 grit surface finished specimens as shown in Figure 4. Figure 2 shows the corrosion behaviour of 1μm diamond polished (like mirror), 600 and 200 grit surface roughness mild steel specimens in 3% NaCl saline solution. Corrosion resistance was drastically reduced with increasing sodium chloride concentration up to 3%. Figure 3 indicated that as the surface became rougher the free corrosion potential moved to a more active state. Corrosion rate measurements showed contradictory results and the conventional trend is that increasing the surface roughness decreases corrosion resistance. The specimens with diamond polished surfaces had the highest corrosion rate as illustrated in Figure 4. 3.1.2 Stainless steel (316L) – potentiodynamic results Potentiodynamic polarization curves and the retrieved corrosion data for 316L stainless steel in 1% and 3% NaCl are shown in Figures 5–9. The polarization curves shown in Figures 5 and 6 indicated that the tendency of the stainless steel WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

164 Simulation of Electrochemical Processes III

grit 200

grit 600

polished

-200

Potential (mV) vs SCE

-300 -400 -500

-600 -700 -800 -900 -2

-1

0

1

2

3

4

5

Log current density (μA/cm²)

Figure 1:

Polarization curves of mild steel at different surface roughness in 1% NaCl saline solution.

grit 200

grit 600

polis hed

200 100 Potential (mV) vsSCE

0 -100 -200 -300 -400 -500 -600 -700 -800 -900 -2

-1

0

1

2

3

4

Log current density (μA/cm²)

Figure 2:

Polarization curves of mild steel at different surface roughness in 3% NaCl saline solution.

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-540

corrosion potential (mV) vs SCE

-550 -560 -570 -580 1% NaCl

-590

3% NaCl

-600 -610 -620 -630 -640 grit 200

grit 600

polished

surface roughness

Figure 3:

Corrosion potential vs. surface roughness in 1% and 3% NaCl saline solution. 12

corrosion rate (mpy)

10

8 1% NaCl

6

3% NaCl

4

2

0 grit 200

grit 600

polished

surface roughness

Figure 4:

Corrosion rate vs. surface roughness in 1% and 3% NaCl saline solution.

was to undergo oxidation and passivation, followed by breakdown, i.e. a typical characteristic. The polarization curve of specimens with surface finishes of 1µm diamond polish, 600 and 200 grit papers tested in 1% NaCl saline solution is shown in Figure 5. The passivity and hence the breakdown potential was affected by the surface roughness. Specimen with less roughness, i.e. 1µm diamond finish surface, had the highest breakdown potential of +317 mV, followed by 600 and 200 grit surface roughness specimens. The passivity of rougher surfaces of 200 and 600 grit failed at less noble potentials of +286 mV and +296 mV, WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

166 Simulation of Electrochemical Processes III respectively. Figure 6 shows the corrosion behaviour of specimens tested in 3% NaCl. The polarization curves of the investigated specimens showed the conventional trend of the effect of chlorides and surface roughness. As the chloride concentration and the surface roughness increased, the breakdown potential values decreased and the free corrosion potential moved in a more grit 200

grit 600

polished

600 500 Potential (mV) vs SCE

400 300 200 100 0 -100 -200 -300 -400 -4

-2

0

2

4

6

Log current density (μA/cm²)

Figure 5:

Polarization curves for different surface roughness of 316L stainless steel in 1% NaCl saline solution. grit 200

grit 600

polis hed

500 400 Potential (mV) vs SCE

300 200 100 0 -100 -200 -300 -400 -500 -4

-2

0

2

4

6

Log current density (μA/cm²)

Figure 6:

Polarization curves for different surface roughness of 316L stainless steel in 3% NaCl saline solution.

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active direction (less stable). The free corrosion potentials, breakdown potentials and corrosion rates in 1 and 3% NaCl were recorded and displayed in Figures 7–9. Figure 7 shows that for the same surface finish specimen, the free corrosion potential increased to a more noble state as the surface roughness decreased. The chloride concentration had a significant effect and reduced the free corrosion potential as the NaCl content increased from 1 to 3%. Figures 8 and 9 show the breakdown potential and the corrosion rate results, respectively. 3.2 Morphology results Corroded specimens after testing were investigated at low and high magnifications for microscopy studies. Examples of mild steel specimen corrosion morphology are shown in Figure 10. Figure 10(a) shows the corrosion pits on the mild steel specimen of diamond polished surface tested in 3% NaCl. This figure shows clearly the severity of the corrosion damage and this confirms and justifies the reason for the highest corrosion rate values recorded for this specimen. Corrosion on 200 grit paper surface roughness tested in 3% NaCl shown in Figure 10(b) clearly reveals a cluster of less deep pitting corrosion, as indicated by the white arrows. Corrosion morphology on 316L stainless steel specimens was investigated using the scanning electron microscope (SEM). The results are shown in Figure 11. Figure 11(a) shows several open corrosion pits on the surface of the 200 grit specimen. The pitting corrosion morphology transformed to another type, called lacelike corrosion pits, as the surface became smoother as can be seen in Figure 11(b). These types of corrosion pits are due to

corrosion potential (mV) vs SCE

0

-50

-100 1% NaCl 3% NaCl

-150

-200

-250

-300 grit 200

grit 600

polished

surface roughness

Figure 7:

Corrosion potential of 316L stainless steel with various surface roughness in 1% and 3% NaCl saline solution.

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168 Simulation of Electrochemical Processes III

350

breakdown potential (mV)

300

250

200

1% Nacl 3% Nacl 150

100

50

0 grit 200

Figure 8:

grit 600

surface roughness

polished

Breakdown potential of 316L stainless steel with various surface roughness in 1% and 3% NaCl saline solution.

0.3

corrosion rate (mpy)

0.25

0.2 1% NaCl

0.15

3% NaCl

0.1

0.05

0 grit 200

grit 600

polished

surface roughness

Figure 9:

Corrosion rate of 316L stainless steel with various surface roughness in 1% and 3% NaCl saline solution.

the collapse of passive film during pit propagation. The specimen of polished surface, Figure 11(c), showed a high number of lacelike corrosion pits. The corrosion morphology shows that the lacelike corrosion is propagated by the mechanism of diffusion until the final collapse, where the pit became stable. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

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Figure 10:

Optical photographs (X400) of corrosion pits on mild steel surfaces: (a) polished grit and (b) 200 grit in 3% NaCl saline solution.

Figure 11:

SEM images (X300): (a) 200, (b) 600 grit surface roughness specimens and (c) diamond polished specimen, respectively.

4

Discussion

Mild steel initially showed some weak passivation range and, due to the presence of chloride on the vicinity of the steel surface, the passivity was easily destroyed. At the end of each experiment on mild steel samples, localized corrosion and red rust was visually observed, indicating corrosion attack and products of various kinds due to the presence of aggressive chloride ions, which are a powerful oxidizing agent and combine rapidly with the metal forming metal chloride. Polarization curves revealed weak passivation when sodium chloride increased from 1 to 3%. This is due to the absorption of water molecules and chloride ions on the steel surface causing breakdown of the oxide film. Corrosion mechanisms retrieved from these results are in agreement with the interpretation provided by others [12, 13]. Increasing the surface roughness shifted the corrosion potential in a more noble direction; however the corrosion current density decreased. This was attributed to a weak passive film which formed on the metal surface, causing the early breakdown of passivity with more aggressive chloride ions. The interesting result obtained in this work was that mild steel with smooth polished surfaces showed the highest corrosion rate compared with 600 and 200 grit surface finished specimens. This can be attributed to the high rate of corrosion propagation after initiation. Pitting corrosion is controlled by the diffusion process and in this case once the pitting started, it propagated at a fast rate due to the continuous diffusion process and the formation of acid media at the bottom of the pits. The corrosion resistance of stainless steel is attributed to the presence of chromium forming an oxide film and protecting the steel from further WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

170 Simulation of Electrochemical Processes III corrosion in the passive region. However, the presence of certain ionic species, such as chloride ions, in the electrolyte can be very aggressive towards the corrosion behaviour. In this study the anodic polarization curves in various chloride concentrations with different surface roughness clearly revealed that the free corrosion potentials and breakdown potential dropped from more noble values to less noble values as sodium chloride increased from 1 to 3%. This is because chloride ions lower the activation energy required for corrosion reaction to occur. This interpretation is in agreement with the work provided by others [14–17]. The chloride formation did not stop the oxide film formation but reduced the passive region as seen in the polarization curves in Figures 5 and 6. There are other reasons for more aggressive behaviour, reflected in the reduction in the passive region in saline solution, which can be due to chloride ions with higher charge density and higher capacity to form soluble species. By entering into the lattice film, the chloride introduces lattice defects, which reduce the resistance of the oxide film to corrosion as seen from the corrosion rate values in Figure 9. The role of surface roughness on the breakdown potential was confirmed as smoother surfaces had a higher breakdown potential. The potentiodynamic results in Figures 5 and 6 showed that the width of the passive region is highly dependent on the surface condition. When the surface finish varied from smooth to rough, the passive film terminated at a lower breakdown potential and the passive region became shorter due to the effect of surface topography, which enhanced the presence of more aggressive corrosion media inside such rough surfaces. The pitting potential is more sensitive to surface roughness changes and the relatively large increase in pitting potential by several tenth mV from the roughest to smoothest surface, suggested that both the nucleation and propagation of metastable pits depends on the steel surface [6]. This implies that for the metastable pit or pits to grow on a smoother surface is more difficult than on a rougher surface and the presence of chloride ions in the solution results in the destruction of the passive film as observed elsewhere [9, 10]. The mechanism of pit growth is autocatalytic by anodic dissolution of the steel, which leads to the introduction of passive metal ions in solution that cause migration of chloride ions. In turn, metal chloride reacts with water causing the pH to decrease where the cathodic reaction is on the surface near the pit mouth. The results in this study confirmed that when the surface is smooth, the pit will survive more due to the formation of lace cover on the pit mouth maintaining the diffusion process. The corrosion morphology for the diamond surface finish behaved in this manner and showed pitting corrosion with lacelike pits as seen in Figure 11.

5

Conclusions 

Corrosion rate measurements of mild steel specimens showed contradictory results with the conventional trend of corrosion reduction as surface roughness increased. The specimens with diamond polished surfaces had the highest corrosion rate compared with 600 and 200 grit surface finished specimens.

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



171

The free corrosion potential, breakdown potential and corrosion rate of 316L stainless steel are chloride concentration dependent. As the chloride concentration increased, the free corrosion potential and the breakdown potential became less noble. The corrosion rate increased as the chloride concentration increased as expected. Reduction in the corrosion resistance of the 316L stainless steel is attributed to the destruction of the passive film in the presence of chloride. The roughest surface of 200 grit showed early passivity breakdown at the highest rate of corrosion and lower breakdown potential compared with the rest of the surfaces.

References [1] [2] [3]

[4] [5] [6] [7] [8] [9] [10] [11]

Uhlig, H. and Revie, R., Corrosion and corrosion control, 3rd edition, John Wiley and Sons: New York, 1985. Forgeson, B., Southwell, C. & Alexander, A., Corrosion of metals in tropical environments, part 3-underwater corrosion of ten structural steels, Corrosion, 16, pp 105-114, 1960. Blekkenhorst, F., Ferrari, G., Wekken, C. & IJsseling, F., Development of high strength low alloy steels for marine applications, Part 1: results of long term exposure tests on commercially available and experimental steels, British corrosion journal, 21, pp 163–176, 1986. Congmin, X., Zhang, Y. & Cheng G., Pitting corrosion behaviour of 316L stainless steel in the media of sulphate- reducing and iron – oxidizing bacteria, Materials characterization, 245, pp 245-256, 2007. Kold, J., Anne, R. & Moeller B, Influence of various surface conditions on pitting corrosion resistance of stainless steel tubes type EN 1.4404, presented at NACE corrosion, paper 06095, San Diego, 2006. Sasaki, K. and Burstein K.T., The generation of surface roughness during slurry erosion- corrosion and its effect on the pitting potential, corrosion science, 38, pp 2111-2120 1996. Moayed, M.H., Laycock, N.J. & Newman, R.C., Dependence of the critical pitting temperature on surface roughness, corrosion science, 45, pp 1203-1216, 2003. Elhoud, A., Renton, N.C. & Deans, W.F., Effect of surface roughness on pitting corrosion of 25 Cr duplex stainless steel in chloride solution. The 9th Libyan corrosion conference, Tripoli-Libya, 2007. Hong, T. & Nagumo M., Effect of surface roughness on early stages of pitting corrosion of type 301 satinless steel, corrosion science, 39, pp 1665- 1672,1997. Cruz, R.P., Nishikata A., & Tsuru T., Pitting corrosion mechanism of stainless steel under wet- dry exposure in chloride containing environments, corrosion science, 40, pp 125-139, 1998. Trethewey, K.T. & Chamberlain, J., Corrosion for science and engineering, 2nd edition, UK, 1995.

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172 Simulation of Electrochemical Processes III [12] [13]

[14] [15] [16] [17]

Jeffrey, R. and Melches, R., The changing topography of corroding mild steel surface in sew water, Corrosion science, 49, pp 2270-2288, 2007. Kim, DK., Muralidharan, S., Ha, T.H., Bae, J.H., Ha, Y.C., Lee, H.G. & Scandebury, JD., electrochemical studies on the alternating current corrosion of mild steel under cathodic protection in marine environments, Electrochemical Acta, 51, pp 5259-5267, 2006. AL-Fozan, S. & Umalik, A., Pitting behaviour of type 316L stainless steel in Arabian Gulf seawater technical report no. SWCC (RDC), 1992. Abd El- Meguid, E., Mohmoud, N., and Gouda, A., Pitting corrosion behaviour of 316L in chloride containing solution, British corrosion journal, 33, pp 42-48, 1998. Chuan, M. and Tseng, W., Environmentally assisted cracking behaviour of single and dual phase stainless steel in hot chloride solution, Materials chemistry and physics, 84, pp 162-170, 2004. Refaey, S.A., Taha F. & Abd El- Malak, AM, Corrosion and inhibition of 316L stainless steel in neutral medium, International Electrochem Society, 1, pp 80-91, 2006.

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Recovering current density from data on electric potential J. Irša, A. N. Galybin & A. Peratta Wessex Institute of Technology, Southampton, UK

Abstract In this paper, an inverse problem for galvanic corrosion in two-dimensional Laplace’s equation was studied. The considered problem deals with experimental measurements on electric potential, where due to lack of data, numerical integration is impossible. The problem is reduced to the determination of unknown complex coefficients of approximating functions, which are related to the known potential and unknown current density. By employing continuity of those functions along subdomain interfaces and using condition equations for known data leads to over-determined system of linear algebraic equations which are subjected to experimental errors. Reconstruction of current density is unique. The reconstruction contains one free additive parameter which does not affect current density. The method is useful in situations where limited data on electric potential are provided. Keywords: current density, potential measurements, reconstruction, complex variables.

1

Introduction

The evaluation of field of current density is essential in problems of galvanic corrosion. In many cases the direct measurement of current density is not feasible, while the electric potential can be obtained from experimental measurements. This is particularly true in case of cathodic protection systems in general, where many surveying techniques (for example DCVG and CIS for underground structures) rely in potential measurements at different points at the electrolyte in order to identify the current distribution along the metallic structures.

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174 Simulation of Electrochemical Processes III In these cases, it is common that the region of interest for collecting data is not accessible to the operator and in order to exploit the available measurements some indirect method is required. There are many studies that imply numerical methods for the forward modelling of galvanic corrosion problem. These techniques are based mainly on boundary value problems (BVP) formulations in order to obtain or verify results, such as finite element method (FEM), finite difference method (FDM) or boundary element method (BEM). These methods are successfully used and showed to be very accurate to solve BVPs. Some of them are also implemented in commercial software. Situations may arise, where the boundary conditions are unknown and only some experimental data in certain locations are known. In this case, the problem is defined as an inverse one. This situation often occurs in many branches of science and mathematics where only the values of some model parameters can be obtained from observed data or measured data. Data on electric potential can be obtained in galvanic corrosion as a set of discrete data with one free parameter due to measuring potential differences. This situation, where measurements on electric potential can be provided as a set of discrete data within simply connected domain Ω imposes the problem to be inverse. Theory of complex variables [1, 2] is used in order to connect current density and electric potential with one holomorphic function. The real part of it represents electric potential and the derivative is related with current density. The domain is divided into smaller subdomains where the holomorphic function is approximated with quadratic function and its derivative with piece wise linear function. These approximating functions obey continuity across subdomain interfaces. Similar method was used in [3] with heat flow and in [4, 5] with stress trajectories. The purpose of this work is to develop efficient method for identification of current densities within a domain where some data on electric potential are provided.

2

Galvanic current density distribution in electrolyte

Current density is a measure of the density of flow of a conserved charge. The equation governing the distribution of potential and current flow in electrolyte can be derived from the continuity equation, charge conservation. The divergence of the current density is equal to the negative rate of change of the charge density [6]:

∇⋅I = −

∂q ∂t

(1)

∂q For steady state current distribution, ∂t = 0 in uniform isotropic electrolyte, where the conductivity of electrolyte is constant, the electric potential satisfies Laplace’s equation:

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∆φ = 0

175 (2)

The current density can be defined at any point as: I j = −k

∂φ ∂x j

(3)

The fact that electric potential φ ( x, y ) is harmonic, which comes from (2) allows to of use theory of complex variables, for 2D potential problems. Introducing harmonic conjugate functionψ ( x, y ) , where both of them satisfy CauchyRiemann equations (CR), one has holomorphic function: W ( z ) = φ ( x , y ) + iψ ( x , y )

(4)

where the real part is electric potential and the imaginary part is function harmonic conjugated, related with fluxes. φ ( x, y ) = const. are equipotentials and ψ ( x, y ) = const. flux lines. Using CR and transformation rules in calculus, one can see that the derivative of holomorphic function W is related to the current density as follows: H ( z) ≡

∂W ∂φ ∂φ = −i ∂z ∂x ∂y

(5)

Comparison of (3) and (5) shows that derivative of holomorphic function, which has real part equal to electric potential is related to the conjugated current density ~ ~ I in the form H ( z ) = I . −k

3

Problem description

The problem is to recover the current density, based on experimental data on electric potential. Let us consider a domain Ω, to be a simply connected domain or a subdomain of a bigger domain which is not necessarily simply connected and bounded. Let potential measurements be known at some discrete points (j=1,…,N), located within Ω. The general problem is formulated as follows: given the discrete data on electric potential φ j = 1...N find field of current density within the domain Ω. There are no other restrictions, such as data type or boundary conditions. The data can be redistributed uniformly or randomly. 3.1 On uniqueness of reconstructed solution The problem deals with reconstruction of a holomorphic function, where the real and imaginary parts are harmonic and satisfy CR. Therefore having one function known, the second one can be easily obtained by integration of CR. Indeed, assume, for instance, that the real part of the holomorphic function is known analytically φ ( x, y ) . CR allows finding the former by integration: WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

176 Simulation of Electrochemical Processes III ψ ( x, y ) =

∫

∂φ ( x, y ) dy + p ∂x

(6)

This formula indicates that the reconstruction can be achieved but it will always have one undetermined parameter as a free parameter. In this particular case is the free parameter an additive constant in the imaginary partψ ( x, y ) of holomorphic function W (z ) . After performing derivative of W (z ) , the free parameter vanishes. It is therefore obvious that from electric potential one can recover the current density without any free parameter. In the case when data are known at discrete points, reconstruction, in general, is non-unique and can have any finite number of free parameters, which is evident from the following. Let an exact solution f ( z ) = u ( x, y ) + iv( x, y ) of the problem be found by any method. It is obvious, that the modulus must be real; therefore it satisfies the following:

[

Im f ( z j )e

− iω ( z j )

] = 0, j = 1...N

(7)

where ω is argument of f(z). Let us consider a function P(z) in the form: m

P( z ) =

∏C

j (z

−zj)

mj

j =1

where C

j

(8)

are arbitrary complex parameters, m j are arbitrary positive integers

and z j are the data points. It is evident from (8) that the sum, f(z)+P(z), satisfies the condition (7) and therefore it is also the solution of the problem. In order to have sense the solution has to be sought in certain class of functions. Hereafter the solution is sought as a piecewise linear holomorphic functions. It will be shown that this set of functions allows reconstructing the current density with high accuracy.

4

Mathematical model

Similar mathematical formulation has been described in [3], where the required function was related to the modulus of holomorphic function. In this case is the required current density related directly with the derivative of holomorphic function W ( z ) . The domain Ω is divided into n smaller subdomains (elements) of an arbitrary shape, as shown in Figure 1. In order to achieve piecewise linear function of current density, the function W(z) must be approximated by quadratic function. Both these approximations obey continuity along the interfaces (at certain collocation points located on the interfaces).

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Figure 1:

177

Discretization of the domain with approximating functions.

For function H(z), related to current density is approximating function required to be piece-wise linear function:

χ H( m) ( z ) = a ( m ) + b ( m) z, m = 1… n

(9) for function W(z), which has real part electric potential is therefore the approximating function the integration of H(z): 1 2

χ W( m) ( z ) = c ( m ) + a ( m ) z + b ( m) z 2 , m = 1… n

(10)

Here a ( m) = a 0( m) + ia1( m) , b ( m ) = b0( m ) + ib1( m) , c ( m ) = c 0( m ) + ic1( m ) are 3 unknown complex constants; therefore 6 real constants are unknown in every element. The method is not restricted to these two particular approximating functions, however it will be demonstrated that the use of these two particular functions provides results with very high accuracy. 4.1 Discretization of the domain In order to obey the continuity of the above mentioned functions, a grid was used consisting of square elements with collocation points (CP) on its interfaces. The CP are distributed regularly. The minimum number of nodes strictly depends on the number of unknowns in the approximating functions as well as the element type and number of known data. Should the system consist only from the continuity equations and no data imposed on electric potential, one can find the number of required collocation points as follows. Given that, the square elements are placed in the regular grid with J elements along x-axis and K elements along y-axis and 6 unknown real coefficients in every element, one finds the total number of unknowns N unkn as: N unkn = 6 N elem = 6 JK

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(11)

178 Simulation of Electrochemical Processes III The required number of collocation points depends on the number of element interfaces, which is N int = ( J − 1) K + ( K − 1) J . Using the same number of collocation points per each interface, keeping in mind that separation of complex equations into real and imaginary parts presents two real equations at every collocation point, the required number of collocation points per interface is: N cp ≥

N unkn 6 JK 3 JK = = 2 int 2[( J − 1) K + ( K − 1) J ] 2 JK − J − K

(12)

For the domain, where J or K is equal to 1, the number of collocation points per interface is N cp ≥ 3 . In general case the number of collocation points per interface N cp > 1 , therefore N cp = 2 is used further on. Given dense data on electric potential, one could reduce the number of CP with respect to unknowns and total number of equations. Other types of mesh could be used, such as triangular or polygonal with the straight or curvilinear sides. The square grid is convenient from computational point of view because the distances between collocation points are regular, which has good influence on the structure of the matrix of the system. 4.2 Equations of continuity and condition equations The problem consists of continuity equations and two types of condition equations. For the k-th collocation point, lying on the interface between the elements numbered m and m+1, the equation of continuity for W(z) is following:

χ W( m) ( z k ) − χ W( m +l ) ( z k ) = 0, m = 1… n; k = 1… N CP

(13)

The second equation of continuity for H(z) is expressed following:

χ H( m) ( z k ) − χ H( m +l ) ( z k ) = 0, m = 1… n; k = 1… N CP

(14)

The condition equation containing data on electric potential φ comes from W(z), where the real part of it is equal to the electric potential, therefore:

(

)

Re χ W( m) ( z j ) = φ j ,

j = 1… N

(15)

this condition equation is satisfied in elements, where the data occurs. It is obvious that from the collocation method, one can find the differences between interfering elements coefficients only. In the condition equation (15) are presented all coefficients except the imaginary part of c, it stays undetermined and from the continuity equations can not be obtained uniquely. This coefficient is the free additive parameter discussed in section 3.1. This would decrease the rank of matrix and would require regularisation procedures. In order to avoid these procedures which contain additional parameters as regularization parameters etc., second condition equation is used. The second condition WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

Simulation of Electrochemical Processes III

179

equation is due to free parameter c1 , it provides the average of all constants c1 across whole domain zero. The condition is following:

∑ Im(c ) = 0 N

(i )

(16)

i =1

The system of linear algebraic equations (SLAE) is formed by equations (13)(16). 4.3 Solution of the linear system By extracting the real and imaginary parts of all equations (equations of continuity, condition equations) the real SLAE can be obtained and rewritten in a matrix form: Ax = b mxn

(17)

m

, b ∈ R and m>n. x is the vector of the unknown real where A∈ R coefficients, of length n. The vector x consists of real and imaginary parts of unknown complex coefficients. Vector b is known exactly, while, A the matrix of the SLAE depends on CP, type and size of element. The matrix A is not a square, the system is over-determined and therefore the left-hand side Ax does not exactly equal to b and thus the system is inconsistent. However an approximate solution, x*, can be found by means of the least squares methods [5] that minimise the residual: Ax * −b

where ...

2

2

≤ε

(18)

stands for the L2 norm. If the arising system is well-posed and not

large, then the inversion of the matrix does not meet any difficulties and solution takes the form [7]: x* = ( A T A) −1 A T b

(19) The condition number (CN) is used in the numerical examples to control wellposedness of the SLAE. In the case of high CN, the methods for ill-posed problems should be used [8].

5

Numerical analysis

For the purpose of testing the method, synthetic function was chosen in the form: i i   W ( z ) = 54 z −  − 30 z −  3 3  

3

(20)

Taking the real part and derivative of (20) one has the electric potential and current density for ideal function, which are used as given data (electric potential) and for comparison with numerically obtained result (current density). WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

180 Simulation of Electrochemical Processes III Conductivity of electrolyte presents multiplicative parameter to the current density, which was chosen as k=1. Different types and structure of data were tested (ideal data and data subjected to errors). 5.1 Ideal data This example uses data obtained from (20). It was used 81 data uniformly redistributed within the domain and 25 elements (Figure 2).

Figure 2:

Figure 3:

Discretized domain with given data on electric potential.

Ideal electric potential – left and reconstructed electric potential – right.

Reconstructed electric potential and current is shown in Figures 3 and 4. This configuration revealed accuracy with average error 4.7% by using 25 elements only. From Figure 5 can be seen the improvement with refinement in residual, maximum and average errors as well as behaviour of condition number (CN).

WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

Simulation of Electrochemical Processes III

Figure 4:

181

Ideal current density – left and recovered current density – right. Refinement results

10000

Average error in [%] Maximum error in [%] Condition number r/Equations

1000 100 10 1 0

50

100

150

200

250

0.1 0.01 number of elements

Figure 5:

Figure 6:

Refinement results for 81 ideal data.

Ideal electric potential – left and reconstructed electric potential from data subjected to errors – right.

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182 Simulation of Electrochemical Processes III 5.2 Data subjected to errors Random error was introduced to the potential data to simulate experimental errors. It was used error ±10% on 81 uniformly redistributed data and 49 elements. The results for electric potential and current density are shown in Figures 6 and 7. The Average error was 2.9%. Condition number was 571 and r/Eq. was 0.2943.

Figure 7:

Ideal current density – left and recovered current density from data subjected to errors – right.

5.3 Random data subjected to errors The measurements are usually randomly redistributed. Random data subjected to errors ±10% were used. It was used 81 data and 49 elements (Figure 8.). The results are shown in Figure 9. Average error was 3.8%. CN was 754 and r/Eq.=0.2672.

Figure 8:

81 random data subjected to errors and discretization.

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Simulation of Electrochemical Processes III

Figure 9:

6

183

Reconstructed electric potential – left and recovered current density – right.

Conclusion

The recovering of current density from data on electric potential, satisfying Laplace’s equation was studied. In experiments, it is difficult or expensive to obtain many measurements and therefore numerical integration cannot be performed. The recovered results revealed high accuracy with synthetic ideal function, as for ideal data, so does for data subjected to high errors. The method uses complex variable theory where one can obtain holomorphic function, related to the electric potential and its derivative related with the current density.

Acknowledgement The authors are grateful to EPSRC for the financial support of this work through the Research Grant EP/E032494/1.

References [1] M. Rahman, Complex Variables and Transform Calculus, Computational Mechanics Publications, Southampton, 1997. [2] Y.K. Kwok, Applied complex variables, Cambridge University Press, Cambridge, 2002. [3] J. Irsa, A.N. Galybin, Heat flux reconstruction in grinding process from temperature data, Computational Methods and Experimental Measurements XIV, WIT Press, Southampton, 2009. [4] A.N. Galybin and Sh.A. Mukhamediev, Determination of elastic stresses from discrete data on stress orientations, IJSS, 41 (18-19), 2004, 51255142. [5] Sh.A. Mukhamediev, A.N. Galybin and B.H.G. Brady, Determination of stress fields in elastic lithosphere by methods based on stress orientations, INT J ROCK MECH MIN,43 (1), 2006, 66-88. WIT Transactions on Engineering Sciences, Vol 65, © 2009 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

184 Simulation of Electrochemical Processes III [6] J.X. Jia, G. Song, A. Atrens, D.St John, J. Baynham and G. Chandler, Evaluation of the BEASY program using linear and piecewise linear approaches for the boundary conditions, Materials and Corrosion, Vol. 55, Is. 11, 845-852. [7] G.H. Golub, C.F. Van Loan, Matrix Computations, London: The John Hopkins Press Ltd., 1996. [8] A.N. Tikhonov, V.Y. Arsenin, Solution of Ill-posed Problems, Winston, Wiley, New York, 1977.

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185

Author Index Abosrra L. ................................ 161 Adey R. A. ........................... 35, 47 Amaya K. ................................... 79 Ashour A. F. ............................ 161

Nakayama A. ............................. 79 Nierhaus T. .............................. 109 Nieuwenhuys B. E. .................. 151 Nourollahi E. ........................... 119

Bakker J. W. ............................ 151 Barnard N. C. ..................... 97, 151 Baynham J. M. W. ............... 35, 47 Bisanovic S. ............................... 59 Brown S. G. R.................... 97, 151

Ode D......................................... 11

Caire J. P. ............................. 11, 23 Charton S. .................................. 11

Peratta A. B. ........................ 35, 47 Peratta A. ................................. 173 Peyrard M. ................................. 23 Provorova O. G. ....................... 141 Rivalier P. .................................. 11 Roustan H. ................................... 1

Deconinck H. ........................... 109 Deconinck J. ............................ 109 Devred F. ................................. 151 Dupoizat M. ............................... 23

Savenkova N. P........................ 141 Shobukhov A. V. ..................... 141

Espinasse G................................ 23

Tomasoni F. ............................. 109 Toribio J................................... 131 Turkovic I. ................................. 59

Galybin A. N............................ 173 Hubin A. .................................. 109

Van Damme S.......................... 109 Van Parys H. ............................ 109 Vergara D. ............................... 131

Irša J......................................... 173 Kharin V. ................................. 131 Kuzmin R. N. ........................... 141

Wu J. .................................... 69, 89 Wüthrich R. ................................. 1 Xing S. ................................. 69, 89

Le Graverend J. B. ....................... 1 Lorenzo M. .............................. 131 Maciel P. .................................. 109 Mandin Ph.................................... 1 Mitchell S. C. ........................... 161 Morandini J. ............................... 11 Muharemovic A. ........................ 59

Yamamoto N. ............................ 79 Yan Y......................................... 69 Youseffi M............................... 161 Yun F. ........................................ 89

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