CHLORIDE INGRESS PREDICTION - PART 1: ANALYTICAL MODEL FOR TIME DEPENDENT DIFFUSION COEFFICIENT AND SURFACE CONCENTRATION

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1 CHLORIDE INGRESS PREDICTION - PART : ANALYTICAL MODEL FOR TIME DEPENDENT DIFFUSION COEFFICIENT AND SURFACE CONCENTRATION Jens Mejer Frederiksen () and Mette Geiker () () ALECTIA A/S, Denmark () Deartment of Civil Engineering, Technical University of Denmark Abstract Prediction of chloride ingress into concrete is an imortant art of durability design of reinforced concrete structures exosed to chloride containing environment. This aer resents the state-of-the art: an analytical model which describes chloride rofiles in concrete as function of deth and time, and where both the surface chloride concentration and the diffusion coefficient are allowed to vary in time; the Mejlbro-Poulsen model is the general solution to Fick s nd law. The aer also resents conversion formulas for the four decisive but rather abstract arameters to arameters, which makes hysical sense for the design engineer, i.e. the achieved chloride diffusion coefficients at year and years, D and D resectively, and the corresonding achieved chloride concentrations at the exosed concrete surface, C and C. Data from field exosure suorts the assumtion of time deendent surface chloride concentrations and the diffusion coefficients. Model arameters for Portland cement concretes with and without silica fume and fly ash in marine atmosheric and submerged South Scandinavian environment are suggested in a comanion aer based on years field exosure data.. INTRODUCTION Colleardi [] introduced in 97 the so-called error function solution to Fick s nd law in the context of chloride ingress into concrete. However, observations from naturally exosed concrete structures indicate that this simle model gives conservative estimates leading to uneconomic design and maintenance lanning. Poulsen [] showed in 993 how the simle solution suggested in [] also could be alied for a time deendent diffusion coefficient used. The trick was to substitute the instantaneous (or oint wise) time deendent diffusion coefficient with an average (or achieved) diffusion coefficient thereby avoiding mathematical and exerimental comlications. Based on data from natural exosure Maage et al. [3] and Bamforth [4] showed (indeendently) in 995 that the achieved diffusion coefficient was a function of time at the same time we became in Scandinavia aware that Takewaka et al. [5] already in 988 roosed to describe D a (t) as a ower function of time. Since 995 state-of-the-art analytical models for chloride ingress have included a time deendent chloride diffusion coefficient, but a constant surface concentration. However, also the surface concentration varies in time as already in 988 described by Uji et al. [6]. A comanion aer [7] to the resent is based on data from years field 75

2 exosure of concrete anels laced in Träslövsläge on the South Western coast of Sweden. Chloride rofiles for a concrete with a three owder blend (cement, silica fume and fly ash) exosed to two different environments: ATMosheric and SUBmerged are shown in Figure. D a and C sa were obtained by an ordinary least sum of squared deviations regression analysis between the error function solution (eq. ()) and the measured chloride rofiles, similarly to the methodology given in the aendix of NT BUILD 443 [8]. Figures and 3 show calculated arameters vs. time for a grou of concretes (3 concrete tyes, 86 chloride rofiles, cf. [7] for further details). The data suorts the assumtion that both arameters are changing with exosure time. Also, others [8], [9], [] suggest that not only the diffusion coefficient but also the surface concentration of chloride exosed concrete should or may vary in time. Chloride concentration, % mass binder y; Da=38.8 mm²/y ; Csa=.4 %.7 y; Da=8. mm²/y ; Csa=.3 % 5.3 y; Da= 5. mm²/y ; Csa=.6 %.3 y; Da=6.7 mm²/y ; Csa=.3 % ID #-35 ATM (w /b =.33; % FA; 5% SF) 4 6 Deth below exosed surface, mm Chloride concentration, % mass binder y; Da=56. mm²/y ; Csa=.9 %.7 y; Da=9.9 mm²/y ; Csa= 3.7 % 5.3 y; Da=.8 mm²/y ; Csa= 4.9 %.3 y; Da=3.6 mm²/y ; Csa= 5. % ID #-35 SUB (w /b =.33; % FA; 5% SF) 4 6 Deth below exosed surface, mm Figure : Chloride rofiles and their corresonding chloride ingress arameters at different exosure times in the atmoshere zone (left) and the submerged zone (right) of concrete tye -35, cf. Table in [7]. According to Crank [] Fick s nd law can only be solved for certain functions describing the time variance of the surface concentration one of which was suggested by Uji et al. [6]. But, in 996 Mejlbro [] and Poulsen [3] (see also [4]) introduced the so-called comlete solution to Fick s nd law, where both the diffusion coefficient and the surface concentration are allowed to vary in time by any ower function of the tye C s (t) = C t. The more sohisticated Mejlbro-Poulsen Model [], [3] includes four arameters instead of the two arameters resent in the simle Colleardi Model []. This calls for detailed field exosure data in order to obtain trustworthy design arameters. Based on the Mejlbro-Poulsen model and data from a Swedish roject [5] Frederiksen et al. [8] established comosition and exosure deendent design arameters. At that time ( ) only data for u to.5 years existed; the design arameters were later udated based on data u to 5 years [6] and recently years [7] (data from [7] and []). This aer resents the state-of-the art for rediction of chloride ingress in to (a homogeneous and defect free) concrete. The aer also resents conversion formulas for the four decisive but rather abstract arameters to arameters, which makes hysical sense for the design engineer, i.e. the achieved chloride diffusion coefficients at year and years, D and D resectively, and the corresonding achieved chloride concentrations at the exosed 76

3 concrete surface, C and C. A list of symbols is given at the end of the aer; the list covers the notation in both Part and Part [7]. Design arameters for Portland cement concretes with and without silica fume and fly ash in marine atmosheric and submerged South Scandinavian environment are suggested in a comanion aer based on years field exosure data [7]. Achieved diffiusion coefficient mm²/year ATM (4 rofiles) SUB (46 rofiles) Exonential (ATM) Exonential (SUB) Achieved surface concentration %mas binder Exosure time, years Figure : The achieved diffusion coefficient vs. exosure time ATM (4 rofiles) SUB (46 rofiles) Log. (ATM) Log. (SUB) Exosure time, years Figure 3: The achieved surface concentration vs. exosure time. 77

4 . THE MEJLBRO-POULSEN MODEL The Mejbro-Poulsen model is based uon Fick s nd law of diffusion which imlies only one assumtion, namely that the flow of chloride in concrete is roortional to the gradient of the chloride concentration in the medium; i.e. concrete. Fick s nd law of diffusion is a artial differential equation, which can be solved by secifying the material arameters, i.e. the chloride diffusion coefficient, and the initial and boundary conditions. Alying the constraints: C C =, x >, t > t x C(, t) = t, t >, > (constant) () C( x,) =, x > lim C( x, t) =, t > tex x + and using the following ower function for the achieved chloride diffusion coefficient D a (the average over the exosure time) roosed by Maage et al [3]: α tex Da ( t) = Daex, where a () t as well as the following family of functions for the achieved chloride concentration C sa (i.e. the averaged resonse from the concrete to the exosure over the exosure time) []: or: C sa = C i α t ex S + ( t tex ) D aex, where (3) t C sa i = C + S τ, where (4) ex aex S = S ( t D ) and (5) α t tex τ = tex t (6) Mejlbro [] found the comlete solution of Fick s nd law: C( x, t) = Ci + ( C sa Ci ) Ψ ( z), where (7) z =.5x.5x = ( t tex ) Da ( t) τ tex Daex (8) Mejlbro s Ψ functions in eq. (7) are defined as: + ( n) n + ( n) n+ (z) Γ( + ) (.5) (z) Ψ ( z ) = n= (n)! Γ( +.5) n= (n + )! (9) In eq. (9) Γ(y) is the Gamma function defined as: + y Γ( y) = u ex( u) du α for y. () 78

5 The notation used in eq. (9) should be noted: () () () ( n) = ; = ; = ( );... = ( )... ( n + ) () where (n) has n factors. In the case where the chloride concentration of the chloride exosed concrete surface is constant; i.e. = in the general solution (7), the chloride rofile is described by the wellknown error-function solution []:.5x C( x, t) = C sa ( C sa Ci )erf () ( t tex ) Da ( t) If the achieved chloride diffusion coefficient D a of the concrete is not time-deendent, i.e. α =, D a (t) is simly relaced by D a in eq. () and we have the simlest form of the solution to Fick s nd law when diffusion with a constant diffusion coefficient into a halfsace is considered i.e. the well-known error-function solution is a secial case of this more general comlete solution. 3. CONVERSION FORMULAS The four arameters D aex, α, S and govern the chloride rofiles defined by the general solution, eq. (7). D aex and α mainly describe the magnitude and the time-deendency of the transort coefficient, cf. eq. (), while S and (together with D aex and α) describe the magnitude and the time-deendency of the boundary condition, cf. eqs. (3) or (4). Instead of estimating the four decisive arameters α, D aex, S and of the concrete and it s local environment, it is more convenient and makes more hysical sense to estimate the achieved chloride diffusion coefficients D and D at time t = year and t = years resectively, and the corresonding chloride concentrations at the exosed concrete surface C and C [8]. The estimation of these arameters can be based on data from natural exosure, ref. Ste and Ste : Ste. Estimate the achieved chloride diffusion coefficient D and D at exosure time year and years cf. eqs. (-5) in [7]. Ste. Estimate the achieved chloride concentration of the concrete surface C and C at exosure time year and years cf. eqs. (-5) in [7]. From D, D, C and C it is ossible to calculate the corresonding values of α, D aex, S and by a ste wise determination using the following formulae [8]: Ste 3. Calculate: θ = log (3) t ex Ste 4. Calculate: Ste 5. Calculate: θ D D aex = D (4) D log ( C C ) = tex D (5) log tex D 79

6 θ D t ex Ste 6. Calculate: S C = D (6) t ex These formulas are suitable for calculation by means of a sreadsheet and/or a rogrammable ocket calculator. 4. CONCLUSIONS It is suggested to use the Mejlbro-Poulsen model for rediction of chloride ingress in to (homogeneous and defect free) concrete. The model is the comlete solution to Fick s nd law where both the diffusion coefficient and the surface concentration are allowed to vary in time. The assumtion of time deendent surface chloride concentrations and diffusion coefficients is suorted by field observations from years exosure of concrete in South Scandinavian marine environment. NOTATION (resent and the comanion aer [7]) Symbol Unit Descrition and reference to definition in text C(x,t) mass% Chloride concentration of concrete at a deth x beyond the concrete surface at the time t C i mass% Initial chloride content of concrete C s mass% Surface chloride concentration of concrete C sa mass% Achieved surface chloride concentration of concrete, determined by regression analysis of an achieved chloride rofile C mass% Achieved surface chloride concentration (boundary condition) of an achieved chloride rofile after one year of exosure C mass% Achieved surface chloride concentration (boundary condition) of an achieved chloride rofile after years of exosure D m²/s Diffusion coefficient D a m²/s Achieved transort coefficient characterising a chloride rofile after exosure for a non-secified time D m²/s Achieved transort coefficient characterising a chloride rofile after one year of exosure D m²/s Achieved transort coefficient characterising a chloride rofile after years of exosure D aex m²/s Achieved transort coefficient characterising a chloride rofile after exosure at time t ex (a fictive arameter) S mass% Parameter in the Mejlbro-Poulsen Model described the magnitude of the surface chloride concentration. t ex s The time of exosure (to a chlorine environment) x m Distance below the exosed concrete surface α A arameter describing the decrease with time of the achieved chloride diffusion coefficient τ Time arameter - Potential and base of Mejlbro s Λ and Ψ functions x m Deth 8

7 Ψ - Mejlbro s Ψ functions A, B, U, V Regression arameters in the estimation formulas for D, C and α C s mass% Surface chloride concentration of concrete, determined by regression analysis of a chloride rofile from a standard laboratory exosure D ex m²/s Potential transort coefficient characterising a chloride rofile after a standard exosure in laboratory at time t ex eqv (w/c) b The equivalent w/c ratio with resect to the surface boundary conditions C and C eqv (w/c) D The equivalent w/c ratio with resect to the achieved diffusion coefficient after one year of exosure D k α, env Factor encountering the effect of a local environment on α. k C, env Factor encountering the effect of a local environment on the surface boundary conditions C and C k D, env Factor encountering the effect of a local environment on the achieved diffusion coefficient after one year of exosure D k FA Efficiency factor of fly ash for chloride ingress k SF Efficiency factor of silica fume for chloride diffusivity k C, env Factor encountering the effect of time for a local environment on the surface boundary condition C w/b kg/kg Water/binder ratio (water: the total amount of free water in the mixture; binder: Portland cement, fly ash, silica fume) w/c kg/kg Water/cement ratio (water: the total amount of free water in the mixture; cement: Portland cement) y mod % mass Modelled chloride concentration at a certain unsecified environment, deth and time y measured % mass Measured chloride concentration at a certain unsecified environment, deth and time REFERENCES (resent and the comanion aer [7]) [] Colleardi, M.; Marcialis, A.; Turriziani, R.: The kinetics of chloride ions enetration in concrete (in Italian). Il Cemento, Vol. 67, , 97. [] Poulsen, E: On a model of chloride ingress into concrete having time deendent diffusion coefficient. Proceeding if the Nordic Miniseminar in Gothenburg Det. of Building Materials, Chalmers University of Technology, Gothenburg Sweden. Publication P- 93:.993. [3] Maage, M.; Poulsen, E.; Vennesland, Ø.; Carlsen, J.E.: Service life model for concrete structures exosed to marine environment initiation eriod. LIGHTCON Reort No..4, STF7 A948 SIN-TEF, Trondheim, Norway, 995. [4] Bamforth, P.B.: Prediction of the onset of reinforcement corrosion due to chloride ingress. Concrete Accross Borders, International Conference, Odense, 994, [5] Takewaka, K.; Mastumoto, S.: Quality and cover thickness of concrete based on the estimation of chloride enetration in marine environments. American Concrete Institute. Detroit USA. ACI SP 9-7, [6] Uji, Matsuoka & Maruya: Formulation of an Equation for surface Chloride Content due to Permeation of Chloride. Proceedings of the Third International Symosium on 8

8 Corrosion of Reinforcement in Concrete Construction. Elsevier Alied Science. London UK, 99. [7] Frederiksen, J. M. & Geiker, M.R. Chloride ingress rediction Part : Exerimentally based design arameters ibid. [8] Frederiksen, J. M., Nilsson, L.-O., Poulsen, E.; Sandberg, P.; Tang L. & Andersen, A.: HETEK, A system for estimation of chloride ingress into concrete, Theoretical background. The Danish Road Directorate, Reort No. 83, 997. [9] Goltermann, P.: Chloride ingress in concrete structures: extraolation of observations. ACI Materials Journal, March-Aril, 4-9, 3. []Tang, L.; Gulikers, J.: On the mathematics of time-deendent aarent chloride diffusion coefficient in concrete, Cement & Concrete Research, 7. []Crank, J.: The Mathematics of Diffusion, nd ed., Clarendon Press, Bristol UK, 975. []Mejlbro, L.: The Comlete Solution of Fick s Second Law of Diffusion with Timedeendent Diffusion Coefficient and Surface Concentration, Durability of Concrete in Saline Environment. Cementa. Danderyd Sweden.996. [3]Poulsen, E.: Estimation of Chloride Ingress into Concrete and Prediction of Service Lifetime with Reference to Marine RC Structures, Durability of Concrete in Saline Environment. Cementa. Danderyd Sweden.996. [4]Poulsen, E.; Mejlbro, L.: DIFFUSION OF CHLORIDE IN CONCRETE Theory and Alication. Taylor & Francis, London and New York, 6. [5]Sandberg, P., Tang, L. & Andersen, A.: Recurrent Studies of Chloride Ingress in Uncracked Marine Concrete at Various Exosure Times and Elevations, Cement and Concrete Research, Vol. 8, No., , 998. [6]Frederiksen, J. M. & Geiker, M.: On an emirical model for estimation of chloride ingress into concrete, Testing and modelling the Chloride Ingress into Concrete, Proceedings of the nd International RILEM Worksho, Paris France, - Setember. [7]L. Tang: Chloride Ingress in Concrete Exosed to Marine Environment Field Data U to Years' Exosure, SP Reort 3:6, SP Swedish National Testing and Research Institute, Borås, Sweden, 3. [8] NT BUILD 443 Concrete, Hardened: Accelerated Chloride Penetration (htt:// nordicinnovation.net/nordtestfiler/build443.df). [9]Frederiksen, J. M.; Sørensen, H. E.; Andersen, A.; Klinghoffer, O.: HETEK, The effect of w/c ratio on chloride transort into concrete, The Danish Road Directorate, Reort No. 54, 997. []Tang, L.: A collection of chloride and moisture rofiles from the Träslövsläge field site, Publication P-3:3, Det. of Building Technology, Chalmers University of Technology, Gothenburg, Sweden, 3. 8