Time dependent models of apparent diffusion coefficient and surface chloride for chloride transport in fly ash concrete

Construction and Building Materials 38 (2013) 497–507

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Time dependent models of apparent diffusion coefficient and surface chloride for chloride transport in fly ash concrete Aruz Petcherdchoo ⇑ Department of Civil Engineering, Faculty of Engineering, King Mongkut’s University of Technology North Bangkok, 1518 Pibulsongkram Road, Bangsue, Bangkok 10800, Thailand

h i g h l i g h t s " The inconsistency in diffusion coefficient and surface chloride models is shown. " A consistent chloride transport model for fly ash concrete is developed. " The sensitivity analysis of the developed model coefficients is performed. " Remarks on the diffusion coefficient and surface chloride are shown.

a r t i c l e

i n f o

Article history: Received 16 March 2012 Received in revised form 31 July 2012 Accepted 14 August 2012 Available online 9 October 2012 Keywords: Time dependent models Apparent diffusion coefficient Surface chloride Chloride transport Fly ash concrete

a b s t r a c t This paper presents a Fick-based chloride transport model, which is mathematically consistent with time dependent apparent diffusion coefficient and surface chloride for fly ash concrete. In the paper, the inconsistency in a simple close-formed solution used to predict chloride penetration through fly ash concrete in a previous study of other researchers is pointed out. The inconsistency can be seen by comparing chloride profiles calculated by the simple close-formed solution to those calculated by a finite difference program. The inconsistency is caused by the use of simplified diffusion coefficient and mathematically incompatible surface chloride models. To avoid the inconsistency, a chloride transport model is developed in this study. In developing the model, regression analysis to fit coefficients of the developed model is compared to the regression analysis results obtained from the experiment of the researchers. In the experiment of the researchers, the effect of water to binder ratio and the amount of fly ash replacement in concrete is considered. Furthermore, the chloride transport calculated from the developed model is validated by comparing to other experimental results. Finally, the sensitivity analysis of the model coefficients and some remarks on the developed model are presented. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction For many years, concrete structures have been known as one of the main materials in construction industry. After a period of exposure time, the concrete structures, especially marine structures, inevitably deteriorate. The deterioration of concrete structures can be considered in different terms, for example, concrete deterioration, reinforcement corrosion, or a combination of them. Chloride attack is considered as one of the important factors in these deterioration mechanisms. Whenever the threshold amount of chloride ions at the surface of reinforcement is reached, reinforcement corrosion and concrete cracking may occur resulting in decreasing the bond strength between concrete and reinforcement, and subsequently reducing the flexural or shear strength of the structure. As a result, the corrosion of reinforcement adversely

⇑ Tel.: +66 82 555 2000x8637. E-mail address: [email protected] 0950-0618/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conbuildmat.2012.08.041

affects the safety and serviceability of concrete structures, and hence shortens their service life [1]. To avoid these, there are two main approaches; producing durable concrete [2–4], or applying appropriate maintenance plans [5,6], e.g., preventive maintenance, essential maintenance, or a combination of them. For the first approach, several researchers studied and recommended durable and sustainable materials, such as fly ash [7–9]. It is evident that the use of fly ash increased the chloride binding capacity in concrete. This binding capacity caused reduction of free chloride ions which were found to be related to corrosion of reinforced steel in concrete located in marine environment [9]. For the second approach, the maintenance plans will of course affect the allocation of the reasonable amount of funds in the administration level. According to these two approaches, the development of a model to predict the corrosion initiation time of concrete structures with durable and sustainable materials, e.g., fly ash, and the application time of maintenance is of importance. In many occasions, the corrosion initiation time and the maintenance application time are defined as service life.

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Table 1 Value of b and CS for W/B = 0.65[22]. Mix no.

I65 I65FA15 I65FA25 I65FA35 I65FA50

CS

b 2-yr exp.

3-yr exp.

4-yr exp.

5-yr exp.

2-yr exp.

3-yr exp.

4-yr exp.

5-yr exp.

0.71 0.79 0.81 0.83 0.84

0.71 0.78 0.805 0.83 0.84

0.71 0.78 0.80 0.82 0.83

0.75 0.77 0.78 0.81 0.82

4.0 5.3 5.5 4.8 4.8

5.2 6.5 6.0 5.4 5.4

5.6 6.9 6.3 5.7 5.5

6.2 7.3 7.0 6.3 6.2

In studying the service life of chloride exposed concrete structures, a quantitative assessment is preferable [10]. The diffusion theory based on the Fick’s second law can be used for predicting the chloride penetration through concrete structures. In the study of researchers [11–14], if the surface chloride and the diffusion coefficient were assumed constant, one-dimensional partial differential equation (1-D PDE) of the Fick’s second law could be analytically solved, and a simple close-formed solution can be obtained. However, from many studies [15–19], it was found that both surface chloride and diffusion coefficient were not constant but time dependent. If this is the case, the simple close-formed solution is inapplicable. In the study of Tang and Gulikers [20], the apparent time dependent diffusion coefficient for concrete was proposed so that it could directly be input into the simple close-formed solution. In 2009, Ann et al. [21] compared chloride ingress in concrete by using different forms of time dependent surface chlorides. However, they assumed constant diffusion coefficients in addition to a set of assumed data for study. Subsequently, Chalee et al. [22] proposed a chloride transport model including the time dependent diffusion coefficient and surface chloride, and validated their models with their experimental results. However, their model contained simplified diffusion coefficient and mathematically incompatible surface chloride leading to the inconsistency in predicting chloride transport as indicated in the next section. As a result, in order to bridge the gap, a chloride transport model, which is mathematically consistent with the time dependent surface chloride and diffusion coefficient, is required.

dependent diffusion coefficient D(t), which is based on the simplified model of Magnat and Limbachiya [25], as follows

DðtÞ ¼

 b 1 t

ð3Þ

where b is an empirical coefficient, and t is concrete exposure time (s). Hence, the PDE for the Fick’s second law was written as

@C @ @C ¼ DðtÞ @t @x @x

ð4aÞ

 b 2 1 @ C t @x2

ð4bÞ

@C ¼ @t

in which its close-formed solution was expressed as

2

0

13

6 B x C7 Cðx; tÞ ¼ C S 41  erf @ qffiffiffiffiffiffiffiffiA5 ð1bÞ 2 t1b

ð5Þ

where C(x, t) is the total chloride content (% by weight of binder) at the position x (mm) and the exposure time t (s), CS is the chloride content at the concrete surface (% by weight of binder). By the regression analysis with their experimental data, they computed the values of b and CS for each test, e.g., for W/B = 0.65 as shown in Table 1. By data analysis, they further proposed the equation for the coefficient b and the time dependent surface chloride CS as follows

b ¼ ½0:0015 ðW=BÞ þ 0:0034 ½F þ ½0:175 ðW=BÞ þ 0:840

ð6Þ

2. Statement of problems and observations where W/B and F are the water to binder ratio and the amount of fly ash replacement (%), respectively, and

2.1. Fundamental of chloride transport models The fundamental one-dimensional partial differential equation (1-D PDE) for chloride diffusion through concrete structures [23,24] can be written as

@C @ @C ¼ D @t @x @x

ð1Þ

where C is the chloride content as a function of position x and time t, and D is the chloride diffusion coefficient of concrete. If the initial condition (initial chloride content), boundary condition (surface chloride) and material property (chloride diffusion coefficient) are assumed to be zero, constant CS, and constant D, respectively, the simple closed-form solution for Eq. (1) can be shown as

   x Cðx; tÞ ¼ C S 1  erf pffiffiffiffiffiffi 2 Dt

ð2Þ

where erf() is an error function. 2.2. Time dependency of diffusion coefficient and surface chloride In 2009, Chalee et al. [22] proposed a chloride transport model for fly ash concrete in seawater, and validated the model with their experimental data. For model formulation, they employed the time

C S ðtÞ ¼ ½0:379ðW=BÞ þ 2:064 lnðtÞ þ ½4:078ðW=BÞ þ 1:011

ð7Þ

where CS(t) is the time dependent surface chloride. For comparing the chloride profiles based on the model using Eqs. (5)–(7) to those based on a Crank–Nicolson based finite difference method (FDM), we can approximate Eq. (4a) as

ci;jþ1 ci;j 1 D;jþ1 ðciþ1;jþ1 2ci;jþ1 ci1;jþ1 Þ D;j ðciþ1;j 2ci;j ci1;j Þ þ ¼ 2 Dt ðDxÞ2 ðDxÞ2

!

ð8Þ where cx,t, in a general form, is the chloride content at a mesh point x and time t, and D,t is the diffusion coefficient at time t. In addition, Dx and Dt are the size of the mesh point and the incremental time step, respectively. In computation, they are chosen as 1 mm and 1 week, respectively. It is noted that for the finite difference program, the time dependent diffusion coefficient in Eq. (3) can directly be input. However, for the close-formed solution in Eq. (5), the time dependent diffusion coefficient in Eq. (3) must be integrated over time to have a constant or apparent diffusion coefficient before replacing into Eq. (2) to get Eq. (5). In addition, the time dependent surface chloride in Eq. (7) can directly be input into the finite difference program and the close-formed solution of Eq. (5).

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solved, e.g., concrete structures cyclically repaired by cover replacement which leads to space dependent diffusion coefficient or D(x) [27]. As a result, both the time dependent diffusion coefficient and surface chloride for chloride penetration through fly ash concrete in seawater need to be developed for general application. 3. Development of the chloride transport model 3.1. Time dependent apparent diffusion coefficient According to the study of Tang and Gulikers [20], the time dependent diffusion coefficient can be expressed as

DðtÞ ¼ Dref

In comparison, the water to binder ratio and the amount of fly ash replacement are chosen as 0.55% and 25%, respectively. The chloride profiles for 2-year, 3-year, 4-year, and 5-year exposures are compared in Fig. 1. It is seen that the chloride profiles obtained from the closeformed solution of Chalee et al. [22] are inconsistent with those obtained from the FDM. To find out the cause of the inconsistency, let consider the work of Chalee et al. [22]. It is found that there are three observations in their model formulation: (1) They used simplified time dependent diffusion coefficient model which leads to the inconsistency. For explanation, let consider the work of Tang and Gulikers [20]. They stated on the work of Magnat and Molly [26] who considered D(t) being a little different from Eq. (3) as

 m 1 t

ð10Þ

where Dref is the diffusion coefficient corresponding to the time tref (=28 days), and m is the age factor. They expressed the close-formed solution for the PDE of the Fick’s second law as

Fig. 1. Chloride profiles based on Chalee et al. [22] and the FDM.

DðtÞ ¼ Di

 m t ref t

   x Cðx; tÞ ¼ C S 1  erf pffiffiffi 2 T

ð11Þ

in which

T¼

Z

tþt ex

DðtÞdt

t ex

¼

Dref 1m

" 1m  1m #  m t ref tex tex  t 1þ  t t t

ð12Þ

where tex and t are defined as the age of concrete at the start of exposure and the time after exposure, respectively. By substituting Eq. (12) into Eq. (11) and rearranging, we get

   x Cðx; tÞ ¼ C S 1  erf pffiffiffiffiffiffiffiffi 2 Da t

ð13Þ

in which Da is the apparent chloride diffusion coefficient [20] as

ð9Þ

where Di was defined as the effective diffusion coefficient at the time equal to 1 s. In comparison between Eqs. (3) and (9), it is observed that Di is missing in Eq. (3), or assumed equal to 1 mm2/s. Tang and Gulikers [20] stated the value of 1 mm2/s is far larger than the chloride diffusion coefficient that could be obtained in the diluted bulk solution (2.03  103 mm2/s). This was inapplicable, because the diffusion coefficient in a porous material should be always less than that in the diluted bulk solution. (2) The time dependent surface chloride in Eq. (7) is incompatible with the assumption used to solve the PDE in Eq. (4a). In the other words, the solution in Eq. (5) is for constant surface chloride, hence it is mathematically incompatible to use Eq. (5) to develop the chloride transport model with the time dependent surface chloride. On the contrary, the numerical solutions by the FDM based on Eq. (8) is directly derived from the PDE in Eq. (4a) without assuming the constant surface chloride, therefore the effect of this assumption is not included. (3) The surface chloride according to Eq. (7) is negative in some cases. For instance, if W/B = 0, CS(t) is negative from the beginning of the exposure to 1-year exposure. This phenomenon means that surface chloride ions will move out of concrete, and this is inapplicable in the problem of chloride penetration through concrete. In conclusion, Eqs. (6) and (7) were obtained by the regression analysis of the chloride profiles with the experimental data, but those equations are inapplicable to general application, such as the numerical method by the finite difference method. In many cases, the numerical method is necessary, because some problems are so complicated that close-formed solutions cannot easily be

Da ¼

Dref 1m

" 1m  1m #  m tref tex t ex   : 1þ t t t

ð14Þ

It is noted that if Dref = 1, tex = 0 (or t  tex), and tref = 1, Eq. (14) can be simplified, and Eq. (13) becomes Eq. (5). If we divide Eq. (12) by the time after exposure t, it can be seen that the apparent chloride diffusion coefficient is an average of the time dependent diffusion coefficient over the time after exposure, t, as

Da ¼

T ¼ t

R tþtex t ex

DðtÞdt t

:

ð15Þ

It is noted that if D(t) in Eq. (10) is employed instead of Eq. (3), the problem in the first observation can be solved. 3.2. Time dependent surface chloride The close-formed solution for the PDE of the Fick’s second law is directly related to the time dependent surface chloride, because it is a boundary condition for the PDE. If a constant value of surface chloride is considered, a close-formed solution will be similar to Eq. (2). If the time dependent surface chloride is chosen as a linear function of exposure time, CS = kt (where k is a constant), then the close-formed solution [24] can be shown as

Cðx; tÞ ¼ kt



1þ

      x2 x x x2 erfc pffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffi e4Dt : Dt pDt 2 Dt

ð16Þ

For the surface pffiffi chloride with a square root function of exposure time, C S ¼ k t (where k is a constant), the close-formed solution [24] can be expressed as

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Table 2 Regression parameters for W/B = 0.65 and 0%FA. Parameters

D28 m C0 k

Initial results

Adjusted results

2-yr exp.

3-yr exp.

4-yr exp.

5-yr exp.

2-yr exp.

3-yr exp.

4-yr exp.

5-yr exp.

505.814 0.2 3.999 0.000

406.984 0.2 5.199 0.000

349.334 0.2 5.599 0.001

335.466 0.2 6.199 0.000

746.809 0.2 0.479 2.588

612.53 0.2 0.479 2.829

529.81 0.2 0.479 2.655

515.293 0.2 0.479 2.648

 pffiffiffiffi     pffiffi x p x x2 pffiffiffiffiffiffi erfc pffiffiffiffiffiffi : Cðx; tÞ ¼ k t e4Dt  2 Dt 2 Dt

ð17Þ

In comparison, Eqs. (2), (16), and (17), are different as depending on the form of the surface chloride. This confirms that the close-formed solution is dependent on the selected form of the surface chloride. If the close-formed solution consistent with the surface chloride, such as Eqs. (16), (17), is used, the problem in the second observation can be solved. 3.3. Developed chloride transport model In order to consider the time dependency of the diffusion coefficient and the surface chloride, we start from selecting the surface chloride in form of

pffiffi C S ðtÞ ¼ C 0 þ k t

ð18Þ

where C0 is the initial surface chloride (% by weight of binder) at the exposure time, and k is a constant related to the rate of increase of surface chloride per square root of the exposure time. It is noted that C0 is necessary, because immediately after being exposed to chloride environment, the surface of concrete is surrounded by chloride ions. Hence, the chemistry between cement matrix and chloride ions leads to a certain amount of chloride ions at the surface of concrete [21]. The time dependent term as a square root pffiffi function t is selected rather than a linear function, because the square root function can capture nonlinear surface chloride increase as indicated in the study of Song et al. [28]. It is noted that if the surface chloride in form of Eq. (18) is chosen, the problem in the third observation can be solved. Using the surface chloride in Eq. (18), the close-formed solution for the PDE of the Fick’s second law can be shown as

   x Cðx; tÞ ¼ C 0 erfc pffiffiffiffiffiffiffiffi þ k 2 Da t   pffiffiffiffi    pffiffi  x2 x p x pffiffiffiffiffiffiffiffi erfc pffiffiffiffiffiffiffiffi  t e 4Da t  2 Da t 2 Da t

ð19Þ

for 24 h, all the specimens were removed from the molds and then cured in water for 27 days, then they were transferred to a tidal zone with exposure to wet-dry cycles of seawater in the Gulf of Thailand daily under the temperature ranging between 25 °C and 35 °C. After exposure, the specimens were dry-cored at 2, 3, 4, and 5 years for determining the total chloride content according to ASTM C152 [29]. 4.2. Da and CS in the developed chloride transport model To calculate all the parameters in Eq. (19), let consider the experimental program of Chalee et al. [22]. It is first found that tex = 28 days. And, by selecting tref = 28 days, Eq. (14) becomes

Da ¼

D28 1m

" 1m  1m #  m 28 28 28   1þ 365t 365t 365t

ð20Þ

in which the number of 365 is multiplied for changing the time after exposure from years to days. From Eq. (20), there are two regression parameters, i.e., D28 (mm2/year) and m. In the analysis, m must be constrained between 0 and 1 to ensure that the diffusion coefficient in Eq. (10) is in form of a decay function. Following Thomas and Bamforth [30] and Thomas and Bentz [31], we employ

m ¼ 0:2 þ 0:4ð%FA=50Þ

ð21Þ

where %FA is the amount of fly ash replacement. For CS, Eq. (18) will be used without modification, hence there are two additional regression parameters, i.e., C0 and k. To find all the three regression parameters, the sum of the squared differences between the chloride profile from the regression analysis by Chalee et al. [22] and that by the developed model is minimized by adjusting the regression parameters. The example of the analysis is shown in Table 2. In the initial results obtained by the first regression analysis, it is found that C0 approaches to 4, 5.2, 5.6, and 6.2 at 2-, 3-, 4-, 5-year exposure, respectively, while k is about zero for all the cases. The

where Da is the apparent chloride diffusion coefficient (mm2/year) in Eq. (14), and x and t are the distance from concrete surface (mm), and the time after exposure (years), respectively. It is also noted that Eq. (19) is a combination of Eqs. (2) and (17) [24]. 4. Parameters in the developed chloride transport model The parameters in the developed chloride transport model, Da and CS, are computed by performing the regression analysis to fit with the regression analysis results of Chalee et al. [22]. The method to find these parameters is shown in the following. 4.1. Review of the experimental program In the experiment of Chalee et al. [22], concrete specimens were prepared using class F fly ash to replace Portland cement type I at the 0%, 15%, 25%, 35% and 50% replacement by weight of binder. The W/B ratio was selected as 0.45, 0.55, and 0.65. After casting

Fig. 2. Initial surface chlorides from the initial results and their trend line.

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A. Petcherdchoo / Construction and Building Materials 38 (2013) 497–507

values of C0 are all equal to those obtained from the regression analysis by Chalee et al. [22] as shown in Table 1. On the other hand, the time dependency of the surface chloride vanishes. This occurs because the regression analysis is performed year by year, hence the time dependent effect cannot be imposed. To have the time dependent term, the values of C0 from the initial results are all substituted as CS in Eq. (18) in addition to matching with their

exposure time. After adding a trend line, we get the initial surface chloride, C0, equal to 0.479 as shown in Fig. 2. By setting the initial surface chloride and performing the regression analysis again, we obtained the adjusted results as shown in Table 2. By this method, the regression analysis for

Table 3 Values of D28 from the adjusted results. Mix no.

I45 I45FA15 I45FA25 I45FA35 I45FA50 I55 I55FA15 I55FA25 I55FA35 I55FA50 I65 I65FA15 I65FA25 I65FA35 I65FA50

D28 2-yr exp.

3-yr exp.

4-yr exp.

5-yr exp.

Average (2-yr to 5-yr)

325.16 161.36 199.45 221.42 123.40 405.26 205.92 179.70 220.13 146.39 746.81 278.60 240.49 228.59 253.43

294.21 150.10 165.00 176.00 109.62 405.62 233.35 168.59 201.01 117.86 612.53 267.81 216.14 196.42 225.18

268.81 129.83 154.65 156.80 125.29 347.88 216.94 157.90 205.61 122.78 529.81 234.05 204.87 200.08 232.45

298.76 133.76 150.37 147.34 129.95 311.73 195.78 180.73 249.11 123.86 515.29 244.18 251.07 212.94 252.72

296.73 143.76 167.37 175.39 122.06 367.62 213.00 171.73 218.97 127.72 601.11 256.16 228.14 209.51 240.95

Table 4 Values of C0 from the adjusted results. Mix no.

I45 I45FA15 I45FA25 I45FA35 I45FA50 I55 I55FA15 I55FA25 I55FA35 I55FA50 I65 I65FA15 I65FA25 I65FA35 I65FA50

C0 2-yr exp.

3-yr exp.

4-yr exp.

5-yr exp.

Average (2-yr to 5-yr)

Average (0–50%FA)

0.32 2.35 1.27 1.72 1.55 0.72 1.02 3.37 2.08 1.15 0.48 2.11 2.99 2.32 2.59

0.32 2.35 1.27 1.72 1.55 0.72 1.02 3.37 2.08 1.15 0.48 2.11 2.99 2.32 2.59

0.32 2.35 1.27 1.72 1.55 0.72 1.02 3.37 2.08 1.15 0.48 2.11 2.99 2.32 2.59

0.32 2.35 1.27 1.72 1.55 0.72 1.02 3.37 2.08 1.15 0.48 2.11 2.99 2.32 2.59

0.32 2.35 1.27 1.72 1.55 0.72 1.02 3.37 2.08 1.15 0.48 2.11 2.99 2.32 2.59

1.442

1.668

2.098

Table 5 Values of k from the adjusted results. Mix no.

Fig. 3. Chloride profiles based on Chalee et al. [22] and the developed model.

I45 I45FA15 I45FA25 I45FA35 I45FA50 I55 I55FA15 I55FA25 I55FA35 I55FA50 I65 I65FA15 I65FA25 I65FA35 I65FA50

k 2-yr exp.

3-yr exp.

4-yr exp.

5-yr exp.

Average (2-yr to 5-yr)

Average (0–50%FA)

2.34 1.81 2.46 1.76 1.81 2.42 2.64 1.36 2.01 2.53 2.59 2.36 1.86 1.83 1.64

2.81 1.90 2.30 1.55 1.66 2.40 2.76 1.23 2.06 2.67 2.83 2.65 1.82 1.86 1.70

2.69 1.80 2.31 1.45 1.64 2.23 2.59 1.12 2.05 2.73 2.65 2.50 1.73 1.77 1.52

2.83 1.85 2.44 1.76 1.79 2.46 2.69 1.38 2.02 2.53 2.65 2.42 1.88 1.86 1.69

2.67 1.84 2.38 1.63 1.72 2.38 2.67 1.27 2.04 2.62 2.68 2.48 1.82 1.83 1.64

2.049

2.193

2.091

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A. Petcherdchoo / Construction and Building Materials 38 (2013) 497–507

Fig. 4. Diffusion coefficient at 28 days VS the amount of fly ash (%FA).

concrete containing 0%, 25%, and 50%FA with W/B = 0.65 at 2-, 3-, 4-, 5-year exposures can be performed, and it is found that the chloride profiles calculated by the developed model fit well with those calculated from the regression analysis by Chalee et al. [22] as shown in Fig. 3. By the same method, all the regression parameters in the developed model for concrete containing 0%, 15%, 25%, 35%, and 50%FA with W/B = 0.45, 0.55, and 0.65 at 2-, 3-, 4-, and 5-year exposures can be calculated as shown in Tables 3–5. From Table 3, the diffusion coefficient at 28 days is averaged from 2-year to 5-year exposures and can be plotted with the amount of fly ash replacement (%FA) as shown in Fig. 4. By adding a trend line, we get three linear equations which represent the diffusion coefficient at 28 days for W/B = 0.45, 0.55, and 0.65. From these three equations, the diffusion coefficient at 28 days can be represented in terms of the water to binder ratio and the amount of fly ash replacement as

D28 ¼ 10½1:776þ1:364ðW=BÞ þ ½5:806  18:69ðW=BÞ ½%FA:

ð22Þ

It is noted that D28 is linearly fitted with the amount of fly ash replacement, because during model development, it is found that the chloride profiles with linearly fitted D28 show better fitting with the experimental data of Chalee et al. [22]. The initial surface chloride at the exposure time C0 and the value of k can be averaged over the exposure time and the amount of fly ash replacement as shown in Tables 4 and 5. From the averaged values, we get

pffiffi pffiffi C S ðtÞ ¼ C 0 þ k t ¼ 10½0:814ðW=BÞ0:213 þ 2:11 t

ð23Þ

It should be noted that the value of k is found to be a constant or independent of the water to binder ratio and the amount of fly ash replacement. 5. Validation of the developed chloride transport model 5.1. Comparison to the results of Chalee et al. [22] The chloride transport predicted by the developed model is validated by comparing to that predicted by the simplified model and measured from the experiment of Chalee et al. [22] as shown in Fig. 5 for concrete containing 0%, 25%, and 50% FA with W/B = 0.45, 0.55, and 0.65 at 5-year exposure. It is found that the chloride profiles by the developed model fit well with the experimental data.

Fig. 5. Chloride profiles based on the prediction by the simplified model and the experimental data of Chalee et al. [22], and the developed model.

From Tables 3–5 and Figs. 3 and 5, it is found that the chloride profiles based on the developed model fit well with those based on both the regression analysis results and the experimental data of Chalee et al. [22], respectively. As a result, if Chalee et al. [22] proved that the chloride profiles based on their model fitted well with most of the experimental data, Tables 3–5 and Figs. 3 and 5 can also prove that the chloride profiles based on the developed model fit well with the experimental data via the regression analysis results of Chalee et al. [22].

A. Petcherdchoo / Construction and Building Materials 38 (2013) 497–507

5.2. Comparison to other experimental results The developed model is also validated with other experimental data, i.e., the data of Castro et al. [32], Thomas and Matthews [33], and McPolin et al. [34]. In the study of Castro et al. [32], a chloride profile of normal concrete with W/B of 0.50 under 24- and 45month exposures in a Mexican marine site that varied greatly in humidity and temperature from 60% to 95% and 20 °C to 30 °C, respectively, was published. For Thomas and Matthews [33], an experiment using 0% and 30% FA concrete (W/B of 0.45 and 0.57) exposed for 10 years under an English tidal zone BRE marine site with chloride and sulfate compositions of 18,200 ppm and 2600 ppm, respectively, was chosen for comparison. The major chemical compositions of seawater at that site did not differ greatly from those in the Gulf of Thailand, but other physical factors (such as temperature, humidity, and abrasion–erosion) did. McPolin et al. [34] conducted a laboratory investigation of chloride content in concrete specimens at 24-, 36-, and 48-week exposures

503

in 0.55 M NaCl (3.2% by weight). These specimens were exposed to the solution for 24 h and then removed to dry for 6 days before being immersed again. From the comparison in Fig. 6, it is found that most of the results compared to Castro [32] and Thomas and Matthews [33] fall within 30% margin of error, while those compared to McPolin et al. [34] mostly fall under the line of -30% margin of error for both 0% and 30%FA concrete. This shows that the chloride contents measured in the laboratory by McPolin et al. [34] are overestimated, and they are more overestimated if the amount of fly ash increases. This indicated that the environment in the laboratory test is probably not identical with the actual marine environment. It is also noted that although the exposure times in the study of McPolin et al. [34] are shorter than those of the other papers, some of the chloride contents in their paper are about as high as the chloride contents in the other papers. This is due to the fact that the locations to monitor the chloride contents for the three papers are not the same. 5.3. Comparison to the finite difference method (FDM) The chloride profiles based on the developed model are compared to those based on the numerical solutions computed by the FDM as shown in Fig. 7 for concrete containing 0% and 50% FA with W/B = 0.45 and 0.65 at 2-year, 4-year, and 10-year exposures. The comparison shows that the proposed method for the chloride transport model is applicable to the problem without simplification which causes the inconsistency as shown in Fig. 1 and in the aforementioned observations. 5.4. Sensitivity analysis In developing the chloride transport model, the sensitivity analysis is necessary for observing the effect of each parameter. To perform the analysis, the surface chloride model in Eqs. (23) is written as

pffiffi C S ðtÞ ¼ 10½C1 ðW=BÞC 2  þ C 3 t Fig. 6. Predicted chloride profiles from the developed model VS other experimental chloride profiles.

ð24Þ

where C1–C3 are the sensitivity parameters for the surface chloride.

Fig. 7. Predicted chloride profiles based on the close-formed solutions and the FDM of the developed model.

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Considering the apparent diffusion coefficient in Eq. (20), m and D28 in Eqs. (21) and (22), respectively, are expressed as

D28 ¼ 10½D1 þD2 ðW=BÞ þ ½D3  D4 ðW=BÞ½%FA

m ¼ M 1 þ M 2 ð%FA=50Þ

where M1–M2 and D1–D4 are the sensitivity parameters for m and D28, respectively.

ð25Þ

ð26Þ

Fig. 8. Sensitivity of chloride contents to C1–C3 for concrete with W/B = 0.45–0.65 and 0–50%FA at 20, 60, and 80 mm from surface and 2-, 10-, 20-year exposure.

Fig. 9. Sensitivity of chloride contents to M1–M2 for concrete with W/B = 0.45–0.65 and 0–50%FA at 20, 60, and 80 mm from surface and 2-, 10-, 20-year exposure.

Fig. 10. Sensitivity of chloride contents to D1–D4 for concrete with W/B = 0.45–0.65 and 0–50%FA at 20, 60, and 80 mm from surface and 2-, 7-, 20-year exposure.

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All the nine sensitivity parameters in Eqs. (24)–(26) can be written in a general form as Xi (i = 1, 2, . . . , 9). In the sensitivity analysis, Xi will be disturbed by 10% from the specified values or Xi ± 0.1Xi, except that D1 is disturbed by 9%. If D1 is disturbed by 10%, then D28 will be negative which is inapplicable. The results of the regression analysis are shown in terms of the comparison between the chloride contents at 20, 60, 80 mm from concrete surface and 2-, 7-, 20-year exposure using Eqs. (20)– (23) and those using Eqs. (24)–(26) as shown in Figs. 8–10. It is noted that the exposure time is limited to 20-year, because it is expected that the amount of chloride ions will be changed or removed at least once in the process of maintenance which should occur prior to 20 years. From the analysis, it is found that the chloride contents are not sensitive to C1, C2 and M1, because the difference between the chloride contents calculated using the specified value and those using the specified value with disturbance falls within 10% margin of error. And, they are quite sensitive to C3 and M2, because the difference falls between 10% and 20% margin of error. However, they are very sensitive to D1–D4, because the difference is out of the 20% margin of error in spite of only 9% disturbance of D1 or 10% disturbance of D2–D4. And, they are most sensitive to D1. Owing to this, D1 is selected for further study as shown in Fig. 11. From the chloride profiles in Fig. 11, when the water to binder ratio and/or the amount of fly ash replacement is higher, the chloride profiles are more sensitive to D1. 5.5. Remarks on the diffusion coefficient and surface chloride 5.5.1. Diffusion coefficient (a) Although the chloride profiles based on the close-formed solution of the developed model agree well with those based on the numerical solutions by the FDM as shown in Fig. 7, all the cases show small difference, such as the chloride profiles for concrete containing 50%FA with W/B = 0.65 at 10-year exposure. This difference occurs because the diffusion coefficients used in those two

solutions are not exactly the same. On the other hand, the apparent diffusion coefficient, Da, used in the close-formed solution is the average of the instantaneous time dependent diffusion coefficient over time t, D(t), used in the numerical solution as shown in Eq. (15). To see the difference between them, let consider the ratio of Eq. (14) to Eq. (10) as

Da 1 ¼ DðtÞ 1  m

" 1þ

t ex t

1m 

 1m # tex : t

ð27Þ

If we plot the ratio in Eq. (27) with the time after exposure t as shown in Fig. 12, it can be seen that the ratio is always larger than a unity, or the average value is always larger than the instantaneous one. As a result, the chloride profiles based on the close-formed solution tend to be overestimated than those based on the numerical solution. It is also noted that the ratio nonlinearly increases with time in the early age (prior to 10-year exposure) but becomes almost constant in the later stage. And also, the ratio increases with increasing the amount of fly ash replacement, but decreases with increasing the age of concrete at the start of exposure, tex. (b) The form of the time dependent diffusion coefficient in the program Life-365 [31] is similar to that in the developed model (Eq. (10)), and the ratio between them can be written as

DðtÞLife-365 D28;Life365 ¼ DðtÞDev eloped D28;Dev eloped ¼

10½12:06þ2:4ðW=BÞ  ð365  24  60  60=106 Þ 10½1:776þ1:364ðW=BÞ þ ½5:806  18:69ðW=BÞ ½%FA

: ð28Þ

The factor (365  24  60  60/106) is used for changing the unit of the diffusion coefficient in Life-365 from m2/s to mm2/year. The ratio in Eq. (28) can be plotted with %FA in addition to varying W/B as shown in Fig. 13. From the comparison, it is found that the diffusion coefficient in Life-365 is higher than that in the developed model, and the ratio

Fig. 11. Sensitivity of chloride profiles to D1.

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Fig. 12. Ratio of the apparent diffusion coefficient to time dependent diffusion coefficient.

Fig. 14. Surface chloride by Chalee et al. [22] and the developed model.

sion analysis performed within this period. However, after 5-year exposure, the surface chlorides from the developed model increase faster than those from the simplified model of Chalee et al. [22]. This is the main difference between the square root function and the natural logarithmic function. For this, Song et al. [16] compared different forms of surface chlorides to a set of measured data collected by Weyer et al. [35]. It was found that the natural logarithmic function was the most fitted one (R2 = 0.99), and the square root function shows acceptable fitting results (R2 = 0.90). However, the surface chloride in form of the natural logarithmic function is not used in this study, because its close-formed solution is so complicated to solve that it may not be simple to use in practice. To remedy the problem, another method or technique must be developed, and this is required for further study. 6. Conclusion Fig. 13. Ratio of the diffusion coefficient in Life-365 [31] to that in the developed model.

of them increases with W/B and %FA. When the amount of fly ash replacement increases, the ratio will increase faster if W/B increases. If W/B and %FA are set the same, except using the time dependent diffusion coefficient in Life-365 and the developed model for comparison, the chloride transport using the time dependent diffusion coefficient in Life-365 will be faster leading to shorter time to corrosion initiation, and earlier time of maintenance application due to its higher time dependent diffusion coefficient. Based on Eqs. (10) and (28), it is found that the value of Dref in Life-365 depends on W/B only, while that in the developed model depends on both the amount of fly ash replacement and the W/B. This is due to the assumption in Life-365 to set Dref dependent on W/B only, and the value of m (Eq. (21)) dependent on the amount of fly ash replacement. It should also be observed that m indicates the degree of diffusion coefficient decay with time, hence the degree of diffusion coefficient decay depends on the amount of fly ash replacement. 5.5.2. Surface chloride The surface chloride proposed by Chalee et al. [22] and the developed model for W/B = 0.45, 0.55, and 0.65, can be compared as shown in Fig. 14. It is noted that within 5-year exposure, the surface chlorides from the two models agree well with each others due to the regres-

In this study, the inconsistency due to the use of simplified diffusion coefficient and mathematically incompatible surface chloride models in the simplified model of Chalee et al. [22] is pointed out. Although the simplified model by Chalee et al. [22] can be used to predict the chloride transport in concrete, the models are not general for other application, e.g., numerical analysis by the finite difference method. Hence, a mathematically consistent chloride transport model with the time dependency of apparent diffusion coefficient and surface chloride in fly ash concrete based on the Fick’s second law is developed. The developed model is validated by comparing to experimental results. In developing the model, it is found that 1. From the sensitivity analysis of the model coefficients, the chloride contents are not sensitive to C1, C2, and M1, but quite sensitive to C3 and M2. However, they are very sensitive to D1–D4. In particular, they are most sensitive to D1 leading to being very sensitive to D28. As a result, D28 or Dref must carefully be selected, and cannot be simplified. 2. The apparent diffusion coefficient in the close-formed solution of the developed model is the average over time t of the instantaneous time dependent diffusion coefficient in the numerical solution. The ratio between them is larger than a unity, or the average value is always larger than the instantaneous one. As a result, the chloride profiles based on the close-formed solution tend to be overestimated than those based on the numerical solution. 3. The diffusion coefficient used in Life-365 is higher than that in the developed model, and the ratio of the diffusion coefficient

A. Petcherdchoo / Construction and Building Materials 38 (2013) 497–507

increases with W/B and %FA. By increasing the amount of fly ash replacement from 0% to 50%, the ratio will increase faster if W/B increases. Due to higher diffusion coefficient in Life-365, the chloride transport will be faster leading to shorter time to corrosion initiation and earlier time of maintenance application. 4. Within 5-year exposure, the surface chloride calculated from the simplified model of Chalee et al. [22] and that from the developed model agrees well with each others due to the regression analysis performed within this period. However, after 5-year exposure, the surface chloride from the developed model increases faster than that from the simplified model. This is the main difference between the square root function and the natural logarithmic function, and requires verification by a number of long-term data. 5. It is recommended that a set of consistent long-term data on the diffusion coefficient and the surface chloride should further be collected for making a database such that parameters and coefficients in the model can be updated. In particular, the diffusion coefficient is well known as dependent on a number of factors (e.g., chloride binding capacity, W/B, etc.), which are many times co-related with each others, rather than dependent only on W/B and the amount of fly ash replacement in concrete. The surface chloride also depends on the exposure condition (e.g., temperature, weathering, etc.) rather than W/B only. Furthermore, close-formed solutions for different forms of surface chloride should also be developed. 6. In any real case, the diffusion coefficient and surface chloride are inevitably uncertain due to material variation and exposure conditions. To deal with this uncertainty, the time dependent models of apparent diffusion coefficient and surface chloride in normal and fly ash concrete should be considered as probability based. This topic is recommended for further study.

References [1] Yokota H, Hida K, Sato Y, Sugiyama T, Takewaka K, Ueda T, et al. Recommendation for maintenance and rehabilitation of concrete structures against chloride induced deterioration. In: JCI research committee on Asian concrete model code, recommendation for maintenance and rehabilitation of concrete structures against chloride induced deterioration, Asian concrete model code, level 3 document. Maintenance; 2003. [2] Gjrv OE, Vennesland. Evaluation and control of steel corrosion in offshore structures. In: Scanlon JM, editor. Concrete durability, ACI SP100; 1987. p. 1575–602. [3] Berke NS, Sundberke KM. The effect of admixtures and concrete mix designs on long-term concrete durability in chloride environments. In: Corrosion-89, NACE conference, Texas, US; 1989. p. 386. [4] Kropp J. Chloride in concrete. In: Kropp J, Hilsdorf HK, editors. Performance criteria for concrete durability. E&FN SPON; 1995. p. 138–64. [5] Petcherdchoo A. Maintaining condition and safety of deteriorating bridges by probabilistic models and optimization, PhD. thesis, University of Colorado, Boulder, CO, USA; 2004. [6] Petcherdchoo A, Neves LC, Frangopol DM. Optimizing lifetime condition and reliability of deteriorating structures with emphasis on bridges. J Struct Eng ASCE 2008. [7] Chindaprasirt P, Chotithanorm C, Cao HT, Sirivivatnanon V. Influence of fly ash fineness on the chloride penetration of concrete. Constr Build Mater 2007;21:356–61. [8] Sumranwanich T, Tangtermsirikul S. A model for predicting time-dependent chloride binding capacity of cement-fly ash cementitious system. Mater Struct 2004;37(270):387–96.

507

[9] Cheewaket T, Jaturapitakkul C, Chalee W. Long Term performance of chloride binding capacity in fly ash concrete in a marine environment. Constr Build Mater 2010;24:1352–7. [10] REHABCON. Strategy for maintenance and rehabilitation in concrete structures. CBI, LIT, VUAB, SNRA, BV, NCC, SIKA, IETcc, GEOCISA, BRE, TRL, NCP, WP 2.3 Final report on the evaluation of alternative repair and upgrading options, EC Innovation and SME Programme project no. IPS-2000-0063, Department of Building Materials, LIT, Lund, Sweden; 2004. [11] Kassir MK, Ghosn M. Chloride-induced corrosion of reinforced concrete bridge decks. Cem Concr Res 2002;32:139–43. [12] Hooton RD, Geiker MR, Bentz EC. Effects of curing on chloride ingress and implications on service life. ACI Mater J 2002;99:201–6. [13] Fanahashi M. Predicting corrosion free service life of a concrete structure in a chloride environment. ACI Mater J 1990;87:581–7. [14] West RE, Hime WG. Chloride profiles in salty concrete. Mater Perform 1985;24:29–36. [15] Bamforth PB, Price WF. Factors influencing chloride ingress into marine structures. In: Dhir RK, Jones MR, editors. Proceeding of the international conference, concrete 2000 – economic and durable construction through excellence; 1993, p. 1105–18. [16] Song HW, Pack SW, Moon JS. Durability evaluation of concrete structures exposed to marine environment focusing on a chloride build-up on concrete surface. In: Yokota H, Shimomura T, editors. Proceeding of the International workshop on life cycle management of coastal concrete structures, Nagaoka, Japan; 2006, p. 1–9. [17] Uji K, Matsuoka Y, Maruya T. Formulation of an equation for surface chloride content of concrete duet o permeation of chloride. In: Page CL, Treadaway KWJ, Bamforth PB, editors. Corrosion of reinforcement in concrete. London, UK: SCI; 1990. p. 258–67. [18] Amey SL, Johnson DA, Miltenberger MA, Farzam H. Predicting service life of concrete marine structure: an environment methodology. ACI Struct J 1998;95(2):205–14. [19] Bentz EC, Evans CM, Thomas MDA. Chloride diffusion modeling for marine exposed concrete. In: Page CL, Bamforth PB, Figg JW, editors. Corrosion of reinforcement in concrete construction. Cambridge, UK: SCI; 1996. p. 136–45. [20] Tang L, Gulikers J. On the mathematics of time-dependent apparent chloride diffusion coefficient in concrete. Cem Concr Res 2007;37:589–95. [21] Ann KY, Ahn JH, Ryou JS. The importance of chloride content at the concrete surface in assessing the time to corrosion of steel in concrete structures. Constr Build Mater 2009;23:239–45. [22] Chalee W, Jaturapitakkul C, Chindaprasert P. Predicting the chloride penetration of fly ash concrete in seawater. Mar Struct 2009;22:341–53. [23] Saetta VA, Scotta VR, Vitaliani VR. Analysis of chloride diffusion into partially saturated concrete. ACI Mater J 1993;90(5):441–51. [24] Crank J. The mathematics of diffusion. 2nd ed. Oxford: The Clarendon Press; 1975. [25] Magnat PS, Limbachiya MC. Effect of initial curing on chloride diffusion in concrete repair materials. Cem Concr Res 1999;29:1475–85. [26] Magnat PS, Molly BT. Predicting of long term chloride concentration in concrete. Mater Struct 1994;27:338–46. [27] Song HW, Shim HB, Petcherdchoo A, Park SK. Service life prediction of repaired concrete structures under chloride environment using finite difference method. Cem Concr Compos 2008;31(2):120–7. [28] Song HW, Lee C-H, Ann KY. Factors influencing chloride transport in concrete structures exposed to marine environments. Cem Concr Compos 2008;30:113–21. [29] ASTM. Standard test method for acid-soluble chloride in mortar and concrete, C1152. In: Annual book of ASTM standards. V. 04.01; 1997. [30] Thomas MDA, Bamforth PB. Modeling chloride diffusion in concrete effect of fly ash and slag. Cem Concr Res 1999;29:487–95. [31] Thomas MDA, Bentz EC. Life-365 manual, released with program by Master Builders; 2000. [32] Castro P, De Rincon CT, Pazini EJ. Interpretation of chloride profiles from concrete exposed to tropical marine environment. Cem Concr Res 2001;31:529–37. [33] Thomas MDA, Matthews JD. Performance of Pfa concrete in a marine environment-10-year results. Cem Concr Res 2004;26:5–20. [34] McPolin D, Basheer PAM, Long AE, Grattan KTV, Sun T. Obtaining progressive chloride profiles in cementitious materials. Constr Build Mater 2005;19:666–73. [35] Weyers RE, Fitch MG, Larsen EP, Al-Qadi I, Chamberlin WP, Hoffman PC. Concrete bridge protection and rehabilitation: chemical and physical techniques. Service life estimates, Strategic highway research program, National Research Council, Washington DC; 1994.