Table Of Content

Diffusion of Chloride in Concrete Modern concrete technology series A series of books presenting the state-of-the-art in concrete technology Series Editors Arnon Bentur Sidney Mindess National Building Research Institute Department of Civil Engineering Faculty of Civil and Environmental University of British Columbia Engineering 6250 Applied Science Lane Technion-IsraelInstituteofTechnology Vancouver, B.C. V6T 1Z4 Technion City Canada Haifa 32 000 Israel 1. Fibre Reinforced Cementitious Composites A. Bentur and S. Mindess 2. Concrete in the Marine Environment P.K. Mehta 3. Concrete in Hot Environments I. Soroka 4. Durability of Concrete in Cold Climates M. Pigeon and R. Pleau 5. High Performance Concrete P.C. A¨ıtcin 6. Steel Corrosion in Concrete A. Bentur, S. Diamond and N. Berke 7. Optimization Methods for Material Design of Cement-based Composites Edited by A. Brandt 8. Special Inorganic Cements I. Odler 9. Concrete Mixture Proportioning F. de Larrard 10. Sulfate Attack on Concrete J. Skalny, J. Marchand and I. Odler 11. Fundamentals of Durable Reinforced Concrete M.G. Richardson 12. Pore Structure of Cement-Based Materials: Testing, Interpretation and Requirements K.K. Aligizaki 13. Aggregates in Concrete M.G. Alexander and S. Mindess 14. Diffusion of Chloride in Concrete L. Mejlbro and E. Poulsen Diffusion of Chloride in Concrete Theory and Application Ervin Poulsen Professor Emeritus of Civil Engineering Technical University of Denmark Leif Mejlbro Professor Emeritus of Mathematics Technical University of Denmark Firstpublished2006 byTaylor&Francis 2ParkSquare,MiltonPark,Abingdon,OxonOx144RN SimultaneouslypublishedintheUSAandCanada byTaylor&FrancisInc, 270MadisonAve,NewYork,NY10016,USA Taylor & Francis is an imprint of the Taylor & Francis Group This edition published in the Taylor & Francis e-Library, 2010. To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk. (cid:2)c 2006ErvinPoulsenandLeifMejlbro Allrightsreserved. Nopartofthisbookmaybereprinted orreproducedorutilisedinanyformorbyanyelectronic, mechanical,orothermeans,nowknownorhereafterinvented, includingphotocopyingandrecording,orinanyinformation storageorretrievalsystem,withoutpermissioninwritingfrom thepublishers. Thepublishermakesnorepresentation,expressorimplied,with regardtotheaccuracyoftheinformationcontainedinthis bookandcannotacceptanylegalresponsibilityorliabilityfor anyeffortsoromissionsthatmaybemade. British Library Cataloguing in Publication Data Acataloguerecordforthisbookisavailablefromthe BritishLibrary Library of Congress Cataloging in Publication Data Mejlbro,Leif. Diffusionofchlorideinconcrete:theoryandapplication/Leif Mejlbro,ErvinPoulsen. p. cm. —(Modernconcretetechnology;14) Includesbibliographicalreferencesandindex. ISBN0–419–25300–9 1.Concrete—Deterioration. 2.Reinforced concrete—Deterioration. 3.Concrete—Effectofsalton. I.Poulsen,Ervin. II.Title. III.Series. TA440.M3942005 620.1(cid:2)36(cid:2)2—dc22 2005050889 ISBN 0-203-96371-7 Master e-book ISBN ISBN10:0–419–25300–9 (Print Edition) ISBN13: 9–78–0–419–25300–6 Contents List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xvii List of Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxv 1 Introduction and Reader’s Guide 1 1.1 The process of chloride ingress . . . . . . . . . . . . . . . . . . 2 1.1.1 Types of chloride transport in concrete. . . . . . . . . . 2 1.1.2 Equations of diffusion . . . . . . . . . . . . . . . . . . . 3 Fick’s second general law of diffusion . . . . . . . . . . . . . 4 Fick’s first law of diffusion . . . . . . . . . . . . . . . . . . . 5 The differential equation of diffusion . . . . . . . . . . . . . . 5 Effect of cracks on chloride ingress . . . . . . . . . . . . . . . 7 1.1.3 Initial and boundary conditions . . . . . . . . . . . . . . 7 The Collepardi model . . . . . . . . . . . . . . . . . . . . . . 7 The LIGHTCON model . . . . . . . . . . . . . . . . . . . . . 8 The HETEK model . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.4 Chloride binding . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Chloride-laden environments . . . . . . . . . . . . . . . . . . . 9 1.2.1 Sources of chloride . . . . . . . . . . . . . . . . . . . . . 10 Seawater . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Chloride containing de-icing salts. . . . . . . . . . . . . . . . 10 PVC fire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Industrial processes . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.2 Marine environments . . . . . . . . . . . . . . . . . . . . 11 Governing parameters of marine structures . . . . . . . . . . 11 Concrete submerged in seawater versus marine atmosphere . 13 1.2.3 Road environments . . . . . . . . . . . . . . . . . . . . . 15 Parameters of the road environment . . . . . . . . . . . . . . 15 Road traffic zones . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3 Corrosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.3.1 Corrosion and its consequences . . . . . . . . . . . . . . 21 Anodes, cathodes and incipient anodes . . . . . . . . . . . . 21 vi CONTENTS Corrosion current and corrosion rate . . . . . . . . . . . . . . 23 1.3.2 Threshold value of chloride in concrete . . . . . . . . . . 24 1.3.3 Corrosion inhibitors . . . . . . . . . . . . . . . . . . . . 25 Effects of inhibitors . . . . . . . . . . . . . . . . . . . . . . . 25 Corrosion tests with migrating inhibitors . . . . . . . . . . . 28 Field tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.3.4 Initiation period of time and service lifetime. . . . . . . 29 1.3.5 Corrosion multi-probes. . . . . . . . . . . . . . . . . . . 29 1.3.6 Design against corrosion . . . . . . . . . . . . . . . . . . 30 Class of environment and method of safety . . . . . . . . . . 31 Composition of the concrete . . . . . . . . . . . . . . . . . . 31 Structural design and design of rebar cover . . . . . . . . . . 32 1.4 Test methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.4.1 Analysis of chloride content of concrete . . . . . . . . . 32 Sources of uncertainties . . . . . . . . . . . . . . . . . . . . . 33 Exposure conditions in field . . . . . . . . . . . . . . . . . . . 34 Laboratory exposure conditions . . . . . . . . . . . . . . . . 34 Preparation of powder samples for analysis . . . . . . . . . . 36 Analysis of chloride content . . . . . . . . . . . . . . . . . . . 37 1.4.2 Chloride profiles . . . . . . . . . . . . . . . . . . . . . . 38 Chloride profile of concrete exposed to seawater . . . . . . . 39 1.4.3 Determination of chloride parameters . . . . . . . . . . 41 Chloride profiles for diffusion . . . . . . . . . . . . . . . . . . 43 Deviations from the ideal shape . . . . . . . . . . . . . . . . 44 Othermethodsforthedeterminationofthechlorideparameters 45 1.4.4 Characteristic value of observations. . . . . . . . . . . . 46 1.4.5 Examination of concrete . . . . . . . . . . . . . . . . . . 48 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 1.5 Maintenance and renovation of RC structures . . . . . . . . . . 49 1.5.1 Repair of corrosion with corrosion inhibitors. . . . . . . 49 Choice of corrosion inhibitor for repair of RC structures . . . 49 Application of corrosion inhibitors in concrete repairs . . . . 50 1.5.2 Electro-chemical chloride removal . . . . . . . . . . . . . 50 1.5.3 Cathodic protection . . . . . . . . . . . . . . . . . . . . 52 1.5.4 Surface protection . . . . . . . . . . . . . . . . . . . . . 53 1.6 Design of chloride exposed RC structures . . . . . . . . . . . . 53 1.6.1 Service life . . . . . . . . . . . . . . . . . . . . . . . . . 53 1.6.2 Methods of design . . . . . . . . . . . . . . . . . . . . . 53 Deterministic design . . . . . . . . . . . . . . . . . . . . . . . 54 Stochastic design . . . . . . . . . . . . . . . . . . . . . . . . . 54 CONTENTS vii 2 Constant Chloride Diffusivity 55 2.1 Parameters of the chloride profile . . . . . . . . . . . . . . . . . 55 2.1.1 Chloride profile . . . . . . . . . . . . . . . . . . . . . . . 56 Equation of a chloride profile . . . . . . . . . . . . . . . . . . 57 The ‘first year chloride ingress’ . . . . . . . . . . . . . . . . . 58 Flux of chlorides . . . . . . . . . . . . . . . . . . . . . . . . . 59 Intensity of penetrating chloride . . . . . . . . . . . . . . . . 60 Diffusion rate of chloride . . . . . . . . . . . . . . . . . . . . 60 Summary of 2.1.1 . . . . . . . . . . . . . . . . . . . . . . . . 61 2.1.2 Types of chloride profiles . . . . . . . . . . . . . . . . . 65 Achieved chloride profile . . . . . . . . . . . . . . . . . . . . 65 Potential chloride profile . . . . . . . . . . . . . . . . . . . . 65 Deviations from the ideal shape . . . . . . . . . . . . . . . . 66 2.1.3 Chloride parameters determined by approximation . . . 67 Test of heterogeneity. . . . . . . . . . . . . . . . . . . . . . . 67 Method of surface tangent . . . . . . . . . . . . . . . . . . . 68 Three sets of observation . . . . . . . . . . . . . . . . . . . . 70 2.1.4 Chloride parameters by regression analysis . . . . . . . 74 Non-linear curve-fitting . . . . . . . . . . . . . . . . . . . . . 75 2.2 Chloride ingress into prismatic specimens . . . . . . . . . . . . 76 2.2.1 Chloride ingress into walls from opposite sides . . . . . 76 Fick’s second law of diffusion . . . . . . . . . . . . . . . . . . 77 Chloride profiles . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.2.2 Chloride ingress into specimens having square cross- sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Fick’s second law of diffusion . . . . . . . . . . . . . . . . . . 80 Chloride profiles . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.3 Chloride ingress from de-icing salt . . . . . . . . . . . . . . . . 83 2.3.1 Chloride content of concrete surface . . . . . . . . . . . 84 Fick’s second law of diffusion . . . . . . . . . . . . . . . . . . 85 Chloride profiles of de-iced concrete . . . . . . . . . . . . . . 86 2.4 Old marine RC structures . . . . . . . . . . . . . . . . . . . . . 89 2.4.1 Fick’s second law for constant chloride diffusivity . . . . 90 2.4.2 Chloride ingress into old concrete . . . . . . . . . . . . . 90 2.4.3 Initiation period . . . . . . . . . . . . . . . . . . . . . . 91 Determination by a chloride profile . . . . . . . . . . . . . . . 91 2.4.4 Corrosion domain . . . . . . . . . . . . . . . . . . . . . 92 (t,C)-diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 92 (t,D)-diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.4.5 Service lifetime . . . . . . . . . . . . . . . . . . . . . . . 96 2.4.6 Corrosion multi-probe . . . . . . . . . . . . . . . . . . . 96 Observations from a corrosion multi-probe . . . . . . . . . . 97 Rough estimates by chloride indicators . . . . . . . . . . . . 98 2.4.7 Probabilistic analysis. . . . . . . . . . . . . . . . . . . . 99 Introduction to the reliability index . . . . . . . . . . . . . . 99 viii CONTENTS The multi-dimensional problem . . . . . . . . . . . . . . . . . 101 Geometrical meaning of the reliability index . . . . . . . . . 103 Probability of corrosion . . . . . . . . . . . . . . . . . . . . . 104 Characteristic initiation period of time . . . . . . . . . . . . 107 Determination of the characteristic value by means of a table 109 3 Error Function and Related Functions 113 3.1 The gamma function . . . . . . . . . . . . . . . . . . . . . . . . 114 3.1.1 Definition and extensions . . . . . . . . . . . . . . . . . 114 3.1.2 Special values . . . . . . . . . . . . . . . . . . . . . . . . 116 3.1.3 Important formulæ . . . . . . . . . . . . . . . . . . . . . 117 The reflection formula . . . . . . . . . . . . . . . . . . . . . . 117 The duplication formula . . . . . . . . . . . . . . . . . . . . . 117 Gauß’s multiplication formula . . . . . . . . . . . . . . . . . 117 3.1.4 Pochhammer’s symbol and related symbols . . . . . . . 117 3.1.5 Approximation formulæ . . . . . . . . . . . . . . . . . . 118 Stirling’s formula . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.2 The error function and related functions . . . . . . . . . . . . . 119 3.2.1 Special values of erfc(u) . . . . . . . . . . . . . . . . . . 120 3.2.2 Connection with Fick’s second law . . . . . . . . . . . . 121 3.2.3 Series expansions of erfc(u) . . . . . . . . . . . . . . . . 122 Variant of Taylor’s formula . . . . . . . . . . . . . . . . . . . 123 3.2.4 Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 124 3.2.5 Approximations. . . . . . . . . . . . . . . . . . . . . . . 126 3.2.6 Inverse of the complementary error function . . . . . . . 127 3.2.7 Repeated integrals of erfc(u) . . . . . . . . . . . . . . . 129 3.2.8 Extension of Fick’s second law . . . . . . . . . . . . . . 133 3.2.9 Series expansions for inerfc(u) . . . . . . . . . . . . . . 135 3.3 The Ψp(u) functions . . . . . . . . . . . . . . . . . . . . . . . . 137 3.3.1 Definition and main theorem . . . . . . . . . . . . . . . 138 3.3.2 Fick’s second law for Ψp(u) . . . . . . . . . . . . . . . . 140 3.3.3 Ho¨lder’s inequality and related results . . . . . . . . . . 145 3.3.4 Differentiation and Taylor expansion of Ψp(u) . . . . . 149 3.3.5 Series expansion of Ψp(u) . . . . . . . . . . . . . . . . . 152 3.3.6 Some estimates . . . . . . . . . . . . . . . . . . . . . . . 157 3.3.7 Polynomial approximation of Ψp(u) . . . . . . . . . . . 160 3.3.8 Generalized repeated integrals of Ψp(u) . . . . . . . . . 166 3.3.9 Connection with hypergeometric functions(cid:2) . . . . . . . 168 3.3.10 The Λp(u) functions . . . . . . . . . . . . . . . . . . . . 169 3.4 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 3.4.1 Bessel functions. . . . . . . . . . . . . . . . . . . . . . . 174 3.4.2 Recurrence formulæ . . . . . . . . . . . . . . . . . . . . 177 3.4.3 Zeros of Jν(r) and Jν(r)Yν(λr)−Jν(λr)Yν(r) . . . . . . 177 3.4.4 Bessel functions for ν =n+ 12, n∈N0 . . . . . . . . . . 180 3.5 Other useful functions . . . . . . . . . . . . . . . . . . . . . . . 182 CONTENTS ix 3.5.1 The function H(ξ,τ) . . . . . . . . . . . . . . . . . . . . 182 3.5.2 The function H1(ξ,τ) . . . . . . . . . . . . . . . . . . . 184 4 Fick’s Second Law for Constant Diffusion Coefficient 185 4.1 The general initial/boundary value problem . . . . . . . . . . . 187 4.2 Eigenfunction expansions . . . . . . . . . . . . . . . . . . . . . 190 4.2.1 Helmholtz’s equation . . . . . . . . . . . . . . . . . . . . 191 4.2.2 Change of coordinates in the operator (cid:3)2 . . . . . . . . 192 4.2.3 Separation of the variables . . . . . . . . . . . . . . . . 192 4.2.4 Eigenfunction expansions . . . . . . . . . . . . . . . . . 197 4.2.5 Method of solution of Fick’s second law . . . . . . . . . 203 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 4.3 A catalogue of solutions of Fick’s 2nd law . . . . . . . . . . . . 214 4.3.1 Half-infinite interval, 1 dimension . . . . . . . . . . . . . 214 4.3.2 Bounded interval, 1 dimension . . . . . . . . . . . . . . 215 4.3.3 Bounded two-dimensional interval . . . . . . . . . . . . 217 4.3.4 Bounded three-dimensional interval. . . . . . . . . . . . 219 4.3.5 Circle in the plane . . . . . . . . . . . . . . . . . . . . . 221 4.3.6 Sector of a circle in the plane . . . . . . . . . . . . . . . 222 4.3.7 Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 4.3.8 Sector of an annulus . . . . . . . . . . . . . . . . . . . . 226 4.3.9 Finite cylindrical column . . . . . . . . . . . . . . . . . 228 4.3.10 Sector of a finite cylindrical column . . . . . . . . . . . 230 4.3.11 Finite pipe . . . . . . . . . . . . . . . . . . . . . . . . . 233 4.3.12 Sector of a finite pipe . . . . . . . . . . . . . . . . . . . 238 5 Time-Dependent Chloride Diffusivity 241 5.1 Constant surface chloride content . . . . . . . . . . . . . . . . . 242 5.1.1 LIGHTCON model of chloride ingress . . . . . . . . . . 242 Problem of estimating the chloride ingress into concrete . . . 242 Assumptions of the LIGHTCON model . . . . . . . . . . . . 243 Mass balance of chloride in an element volume of concrete. . 243 Achieved chloride diffusion coefficient . . . . . . . . . . . . . 244 Boundary condition . . . . . . . . . . . . . . . . . . . . . . . 246 Graphs of chloride profiles. . . . . . . . . . . . . . . . . . . . 248 5.1.2 Chloride ingress into concrete . . . . . . . . . . . . . . . 252 5.1.3 Initiation period of time . . . . . . . . . . . . . . . . . . 253 Determination by a chloride profile . . . . . . . . . . . . . . . 255 5.1.4 Corrosion domain . . . . . . . . . . . . . . . . . . . . . 257 (t,D)-diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 257 5.1.5 Corrosion multiprobe . . . . . . . . . . . . . . . . . . . 261 Observations from a corrosion multiprobe . . . . . . . . . . . 261 5.1.6 Chloride ingress into prismatic RC components . . . . . 263 Chloride ingress into walls from opposite sides . . . . . . . . 263 5.1.7 Chloride ingress from de-icing salt . . . . . . . . . . . . 267

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