Predicting Method for Time-Dependent Concrete Carbonation Depth (II): Improved Empirical Simulating Formula

Wei Wang* School of Engineering, Shaoxing University, Shaoxing, China *corresponding author，email: [email protected]

Zhijian Shu School of Engineering, Lishui University, Lishui, China

Qiang Zheng School of Engineering, Shaoxing University, Shaoxing, China

Chunyang Cheng School of Engineering, Shaoxing University, Shaoxing, China

Weimin Jin School of Engineering, Shaoxing University, Shaoxing, China

ABSTRACT Empirical simulating formula for time-dependent carbonation depth of concrete are studied based on k-0.5 model and k-m model. Advantage of the k-m model is discussed which can reduce even eliminate the X-shape distributed simulating error of the traditional k-0.5 model. According to the basic k-m model, three improved four-parameter empirical formulae are presented to simulate time-dependent carbonation depth of different kind concrete. First of all, empirical k-m-CS formula is put forward to describe carbonation behaviour of concrete with different compressive strength. Then, empirical k-m-CC formula is proposed to simulate this behaviour of concrete subjected to various carbon dioxide concentrations. Lastly, empirical k- m-RF formula is established to predict the behaviour of concrete with different replacing ratio of fly ash. Approaches to determine the parameters in the empirical formulae are suggested in detail. Simulating error of presented empirical formula is much smaller than that of the k-0.5 model based formula. Finally, good agreements have been found between investigated carbonation data and improved empirical formulae simulations. KEYWORDS: carbonation, concrete, time, fitting model, simulating formula

INTRODUCTION Carbonation of concrete is a time-dependent process which often continues for a long time associated with gradually increased carbonation depth. It is a complex physical-chemical action

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Vol. 18[2013], Bund. L 2310

(Das, et al. 2012; He and Jia 2011). This carbonation action has to with crude materials concrete (Nishi 1962; Zhang and Jiang 1990; Zhu 1992; Bouzoubaâ, et al. 2010; Gao, et al. 2013), working environment (PaPadakis 1962; He and Jia 2011), and construction technology (Liang, et al. 2013). This action usually damages high alkaline environment inner concrete and develops rust risk to steel bar set in the concrete. Consequently, it produces a vast potential threat to bearing capacity reduction, deformation increase and durability decrease of reinforced concrete structures such as buildings, pile foundations, bridges and dams (Du and Liang 2005; Hussain et al. 2011). As a result, service life shortening of structure is likely to occur. One of the best ways to prevent this negative and dangerous happening is to rationally describe and simulate the time-dependent carbonation and evaluate working ability of concrete structures by taking engineering method. Many researchers and engineers have devoted their efforts to this study, and several models have been proposed (Sun 2006; Jonathan, et al. 2013; Marques, et al. 2013; Mohamed, et al. 2013). Among these models, the k-0.5 model is widely used due to its simple and concise expression. Nevertheless，this model often develops big X- shape fitted error which cannot satisfy with practical engineering demand (Shu, et al. 2013). This paper is set about to present one more precise fitting model of single depth-time carbonation process and to establish improved empirical formulae to simulate this process of concrete considering compressive strength, various carbon dioxide concentration and varied replacing ratio of fly ash.

BASIC k-0.5 MODEL AND k-m MODEL

The k-0.5 Model Although several models have been proposed based on Fick's first law of diffusion (Zhu 1992; Xiao, et al. 2010; He and Jia 2011; Gao, et al. 2013), they can be generalized as k-0.5 model with slope k and exponent 0.5 and be expressed as:

x  kt 0.5 (1) where x is carbonation depth of concrete, and k is one undetermined parameter. Basic behavior of this model is shown as Fig.1.

16 k=1.4 k=1.3

k=1.0 k=0.7 12

8 Depth / mm / Depth 4

0 0 20406080100120 Time / d Figure 1: Mathematical property of the k-0.5 model Vol. 18[2013], Bund. L 2311

Many literatures show that Eq. (1), namely the k-0.5 model, has no enough ability to well fit investigated time-depth behavior of concrete, and it often produces X-shape distributed big fitting errors which limits the application of the k-0.5 model to practical engineering(Shu, et al. 2013). In fact, in the k-0.5 model, there is only single parameter k which can be just looked as one coefficient to function x=t0.5. That is to say, this model is limited to overall amplify with k>1 or reduce with k<1 the locus and shape of function x=t0.5, shown as in Fig. 1. Whatever the k will be, the k-0.5 model inevitably cannot overcome the X-shape distributed fitting error.

The k-m Model In order to cope with the fitting error of the single-parameter k-0.5 model, one two-parameter k-m model is presented, and it can be described as:

x  kt m (2) where x is carbonation depth of concrete, and k and m are two undetermined parameters. It can be proved that, Eq. (2), namely the k-m model, has following mathematical properties: (1) Not like in the k-0.5 model that square of carbonation depth is linear to carbonation time, this relationship in the k-m model can be deduced as:

222m x  kt (3) (2) It can describe overall amplify or reduce the locus and shape of function x=tm like the k- 0.5 model. If k=0.5, this behavior can be indicated as shown in Fig. 1. (3) If k is fixed as a constant, the k-m model is an increase function with parameter m, shown as in Fig. 2. In this figure, k is designed as a constant 1.3 and m is designed as 0.2, 0.4 and 0.6. It is evident that different values of m produce different depth-time carbonation behaviors. However, value of the k-0.5 model cannot change in such case.

24 k=1.3, m=0.6 k=1.3, m=0.4 20 k=1.3, m=0.2 16

Depth / mm 8

0 0 20406080100120 Time / d Figure 2: Values of the k-m model with different m

(4) With various values of k and m, curves of the k-m model may cross as X-shape, shown as in Fig. 3. In this figure, (k, m) are planned as (0.7, 0.5), (1.5, 0.3) and (1.3, 0.6), and their corresponding curves cross each other obviously. This property enables the k-m model to Vol. 18[2013], Bund. L 2312 overcome the X-shape fitting error resulted from the k-0.5 model and to fit the tested time- dependent carbonation depth with more accuracy.

8 k=0.7, m=0.5 k=1.5, m=0.3 k=1.3, m=0.6 6

4 Depth / mm / Depth 2

0 0 20 40 60 80 100 120 Time / d Figure 3: Crossed X-shape curves of t the k-m model

Investigated Data Fittings In order to verify the accuracy and applicability of the k-m model, two typical sets investigated data are selected for fitting samples. Case 1. Tests on concrete with different compressive strength has been designed and conducted by Sun (2006). Taking the carbonation depth-time data of C30 concrete as example, investigated data and fitted results of the k-0.5 model with k=0.166 and of the k-m model with (k, m)= (0.329, 0.369) are plotted in Fig. 4. Depth-time curves of this case in Fig. 4-(a) show that the fitting curve using the k-m model is much closer to the investigated than that using the k-0.5 model. On the other hand, in Fig. 4-(b), the k-m model produces better agreement with the investigated nonlinear depth2-time curve than that of the k-0.5 model assuming linear depth2-time curve.

4 Investigated data 35 Investigated data k-0.5 model 30 k-0.5 model 3 k-m model k-m model 25 2

2 mm / 2 15

Depth 10 Depth / mm / Depth 1 5

0 0 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 Time / d Time / d

(a) depth-time curves (b) depth2-time curves Figure 4: Investigated and fitted carbonation data of C30 concrete

Case 2. In order to study the carbonation behavior and elevate the service life of subway concrete structures in Nanjing city, China, laboratory tests are conducted by Du (2006). In this test, slag and fly ash are designed as admixtures adding into cement with different replace ratio in Vol. 18[2013], Bund. L 2313 mass. The prepared 10cm*10cm*10cm cube concrete samples are cured under standard curing condition for 28d. Then, these samples are put into carbonation box with (20±3)% concentration of carbon dioxide, (70±5)% relative humidity and (20±3)Ԩ temperature until reaching designed carbonation time as 3, 7, 14, 28, 60 and 90d. Taking depth-time data of concrete sample with replace ratio of 18% fly ash and 28% slag as example, denoted as F-S concrete, tested data and fitted results of the k-0.5 model with k=2.838 and of the k-m model with (k, m) =(4.970, 0.357) are listed in table 1.

Table 1: Tested and fitted carbonation depth of F-S concrete k-m model k-0.5 model Time/d Tested depth /mm fitted depth/ mm error/% fitted depth/ mm error/%

3 7.30 7.35 0.74 4.92 -32.66 7 8.40 9.95 18.43 7.51 -10.60 14 13.50 12.74 -5.64 10.62 -21.34 28 17.50 16.31 -6.80 15.02 -14.18 60 21.40 21.40 0.02 21.99 2.73 90 24.20 24.73 2.21 26.93 11.26

Table 1 shows following key benefits of the k-m model. (1) Both the k-m model and the k-0.5 model can describe the general developing tendency of investigated carbonation data. (2) The k-m model produces much smaller fitted error than the k-0.5 model. The biggest absolute value of this error is reduced from 32.66 to 18.43; at the same time, the smallest value of it is reduced from 2.73 to 0.02. (3) The k-m model’s fitted error is positive at two end parts and negative at middle part of the time-dependent carbonation process; however, that of the k-0.5 model is still only changed from negative to positive like discussed in Shu’s literature (Shu, et al. 2013). Case1 and case 2 demonstrate the advantages of the k-m model by using figure curves and table data. However, this is just fitting stage to single sample data. For engineering design and management, it is more necessary to simulate or predict the whole time-dependent carbonation behavior based on limited and typical laboratory tests instead of carrying out lots of time-wasting and cost-wasting experiments. For well simulating the carbonation behavior of concrete, it is a key task to express the (k, m) value by test conditions such as concrete compressive strength, concentration of carbon dioxide and replace ratio of admixture. Object of further study is to establish empirical formula based on such test conditions. Vol. 18[2013], Bund. L 2314

IMPROVED EMPIRICAL FORMULA FOR CARBONATION DEPTH

Empirical k-m-CS Formula In case 1 of above section, Sun (2006) presented detailed depth-time carbonation data of concrete with different compressive strength, shown as in table 2. In order to prove good behavior of the k-m model, five series data are chosen to empirically determine the value of parameter (k, m) and to demonstrate the k-m model’s simulating ability. Table 2: Tested carbonation depth of concrete with different compressive strength (Sun 2006)

Time/d C5 C10 C30 C50 C60

14 1.75 1.50 0.81 0.50 0.40 28 2.30 1.70 1.05 0.65 0.55 60 3.50 2.72 1.47 0.85 0.67 90 4.67 3.67 1.83 1.10 0.96 180 5.88 4.15 2.37 1.25 1.08 365 6.50 5.32 2.80 1.67 1.45

After fitting these five series data by using the k-m model, Eq. (2), values of parameter (k, m) can be obtained and plotted in Fig. 5 against compressive strength, denoted by f. Thus, empirical formula between (k, m) and compressive strength can be written as:

 ke 0.7768 0.0272 f  (3) mf0.0001  0.3757

1.0 0.6

0.8 0.5 m = - 0.0001f + 0.3757 k = 0.7768e-0.0272f 0.6 k m 0.4 0.4 Parameter

Parameter 0.3 0.2

0.0 0.2 0 10203040506070 0 10203040506070 Compressive strength f / MPa Compressive strength f / MPa Figure 5: Fitted (k, m)-f relations Commonly, Eq. (3) can be generalized as:

 kFe Ff2  1 (4) mFfF 34 Vol. 18[2013], Bund. L 2315

Putting Eq. (4) into Eq. (2), we can deduce an empirical formula for simulating or predicting the time-relative carbonation depth of concrete with different compressive strength, and it can be called as k-m-CS (CS, Compressive Strength ) formula with expression as:

Ff2 Ff34 F xFet 1 (5) where F1, F2, F3 and F4 are four parameters which can be easily determined by method like Fig. 3 and Eq. 5. Placing Eq. (3) into Eq. (5), simulated results comparing with investigated data can be both obtained shown as in Fig. 6. Moreover, distribution of simulated error is schemed, shown as in Fig. 7. These two figures indicate that the empirical k-m-CS formula can simulate investigated data with good accuracy.

Simulated, f=5 8 Investigated, f=5 Simulated, f=10 Investigated, f=10 7 Simulated, f=30 Investigated, f=30 Simulated, f=50 6 Investigated, f=50 Simulated, f=60 Simulated, f=60 5

Depth / mm / Depth 3

0 0 100 200 300 400 Time / d

Figure 6: Simulated and investigated carbonation process of the k-m-CS formula

24 f=5 f=10

f=30 f=50 18 f=60 12

m 6

Depth / m / Depth 0 0 100 200 300 400 -6

-12

-18 Time / d Figure 7: Simulated error distribution of the k-m-CS formula Vol. 18[2013], Bund. L 2316

Empirical k-m-CC Formula In order to estimate the service life of thin-walled structures of one long-span flume in Hebei Province section of South-to-North Water Transfer Project in China, accelerated carbonation tests by using bigger 10%, 20% and 27% concentrations of carbon dioxide are conducted (Xie 2005). The concrete sample is 10cm*10cn*30cm prismoid, and its mixing proportion is listed in table 3 in which NH3G is water reducing agent and DH9 is air entraining agent. After reaching carbonation time t=(7, 14, 21, 28, 60)d, carbonation depth of different concentration (denoted by c) are investigated as: xc=10%=(4.86, 7.30, 7.86, 8.40, 11.75)mm, x c=20%=(6.80, 10.22, 11.12, 11.85, 15.68) mm, x c=27%= (7.46, 11.23, 12.27, 13.33, 18.07)mm. Table 3: Mixing proportion of tested concrete sample

Cement material/kg sand/kg gravel/kg water/kg NH3G/kg DH9/g 425 650 1155 170 4.25 25.5

Like the Fig. 5 and Eq. (3), Fig. 8 can be plotted and empirical Eq. (6) using the k-m model can be deduced as:

4.0 0.6

3.0 0.5 m = 0.030 c + 0.388 k m 2.0 0.4 k = 6.489 c 0.431 Parameter Parameter 1.0 Parameter 0.3

0.0 0.2 0% 5% 10% 15% 20% 25% 30% 35% 0% 5% 10% 15% 20% 25% 30% 35% Concentration of carbon dioxide c Concentration of carbon dioxide c Figure 8: Fitted (k, m)-f relations

 kc 6.489 0.431  (6) mc0.030 0.388 Consequently, generalized relations between (k, m) and c can be formatted as:

 kCc C2  1 (7) mCcC 34 Finally, empirical formula for depth-time carbonation behavior under different carbon dioxide environment can be obtained by putting Eq. (7) into Eq. (2), and it can be entitled as the k-m-CC (CC, Concentrations of Carbon dioxide) formula with expression as:

C2 Cc34 C xCct 1 (8) where C1, C2, C3 and C4 are four parameters. Vol. 18[2013], Bund. L 2317

Placing Eq. (6) into Eq. (8), the empirical k-m-CC formula simulating results of x c=10%, x c=20% and x c=27% and its error distribution are drawn in Fig. 9 and Fig. 10. These two figures point out that the empirical k-m-CC formula can well agree with investigated data.

24 Simulated, c=27% Investigated, c=27% Simulated, c=20%" 20 Investigated, c=20% Simulated, c=10%" 16 Investigated, c=10%

12 Depth / mm 8

0 0 20406080 Time / d

Figure 9: Simulated and investigated carbonation process of the k-m-CC formula

m 0 0 10203040506070 -3 Depth / m / Depth Time / d c=10% -6 c=27%"

-9 c=20%"

-12

Figure 10: Simulated error distribution of the k-m-CC formula

Empirical k-m-RF Formula Fly ash is one of the most widely used industrial by-product mixed into concrete replacing some cement because it can save cement amount, reuse waste material, reduce hydration heat and improve mechanical behavior of concrete (Du and Liang 2005; Shu, et al. 2013). It is reported that fly ash often makes big difference to concrete carbonation behavior. He and Jia (2011) discussed the effect of replacing ratio of fly ash, ambient relative humidity, temperature, curing Vol. 18[2013], Bund. L 2318 age and W/C on carbonation of concrete. This process was accelerated in artificial carbonation laboratory by controlling 20% concentration of CO2, 20Ԩ-60Ԩ temperature and 30%-70% relative humidity. Carbonation time is designed as t=(3, 7, 14, 28, 42)d, carbonation depth of different fly ash replacing ratio (r) are presented as: xr=0%=(8.62, 13.19, 18.62, 21.25, 24.20)mm, xr=10%=(9.21, 14.09, 19.89, 22.04, 26.40) mm, xr=20%= (10.27, 15.71, 22.18, 25.27, 29.11)mm and xr=25%=(10.76, 16.46, 23.24, 26.77, 30.24) mm. By fitting above tested data using Eq. (2), empirical formula between (k, m) and the replacing ratio (r) including its generalized form can be put forward as:

 kr(6.714 6.550)  (9) mr0.200  0.358

 kRrR12  (10) mRrR 34 So, empirical formula considering fly ash replacing ratio can be obtained by putting Eq. (9) into Eq. (2), and it can be named as k-m-RF (RF, Replacing ratio of Fly ash) formula with following form:

Rr34 R xRrRt()12 (11) where R1, R2, R3 and R4 are four parameters. Substituting Eq. (9) into Eq. (11) gives rise to the simulating using the k-m-RF formula. On the other hand, the k-0.5 model is also used to fit these four series data resulting empirical relation show as in Fig. 11. Then, the empirical k-0.5-RF formula can be determined as:

6.0

4.0 0.921r K K = 4.098e

2.0 Parameter Parameter

0.0 0% 5% 10% 15% 20% 25% 30% Replace ration of fly ash r

Figure 11: Fitted (k, 0.5)-r relations

x kt0.5 4.098 e0.921r t 0.5 (12) Vol. 18[2013], Bund. L 2319

Simulating results of the k-m-RF formula and the k-0.5-RF formula and their simulated errors are all listed in table 4 to table 7. These four tables show that the k-m-RF formula matches better with tested data than the k-0.5-RF formula with reduced error.

Table 4: Tested and simulated carbonation depth with r=0% k-m-RF formula k-0.5-RF formula Tested depth Time/d /mm simulated depth error simulated depth error / mm /% /mm /% 3 8.62 9.70 12.56 7.10 -17.66 7 13.19 13.14 -0.39 10.84 -17.81 14 18.62 16.83 -9.59 15.33 -17.66 28 21.25 21.57 1.52 21.68 2.04 42 24.20 24.94 3.05 26.56 9.74

Table 5: Tested and simulated carbonation depth with r=10% k-m-RF formula k-0.5-RF formula Tested depth Time/d /mm simulated depth error simulated depth error / mm /% /mm /% 3 9.21 10.67 15.90 7.78 -15.50 7 14.09 14.43 2.40 11.89 -15.63 14 19.89 18.46 -7.18 16.81 -15.48 28 22.04 23.63 7.19 23.77 7.87 42 26.40 27.29 3.37 29.12 10.29

Table 6: Tested and simulated carbonation depth with r=20% k-m-RF formula k-0.5-RF formula Tested depth Time/d /mm simulated depth error simulated depth error / mm /% /mm /% 3 10.27 11.64 13.35 8.53 -16.92 7 15.71 15.71 -0.01 13.03 -17.03 14 22.18 20.07 -9.50 18.43 -16.89 28 25.27 25.65 1.50 26.07 3.16 42 29.11 29.61 1.70 31.93 9.68

Vol. 18[2013], Bund. L 2320

Table 7: Tested and simulated carbonation depth with r=25%

Tested depth k-m-RF formula k-0.5-RF formula Time/d /mm simulated depth error simulated depth error /mm /% /mm /% 3 10.76 12.12 12.67 8.93 -16.96 7 16.46 16.35 -0.70 13.65 -17.08 14 23.24 20.87 -10.19 19.30 -16.95 28 26.77 26.65 -0.44 27.30 1.97 42 30.24 30.75 1.69 33.43 10.55

CONCLUSIONS AND DISCUSSIONS Empirical mathematical method for time-relative concrete carbonation depth is dissected in details based on the k-m model, and main work and conclusions are as follows. (1) The k-0.5 model containing only single parameter has no enough ability to properly describe carbonation behavior, because this model is limited to overall amplify or reduce the locus and shape of function x=t0.5, which inevitably cannot overcome the X-shape distributed fitting error. (2) With various values of k and m, the k-m model can conquer the X-shape fitting error of the k-0.5 model and fit tested data with reduced error which is proved by two cases. (3) Empirical k-m-CS formula, k-m-CC formula and k-m-RF formula are deduced based on the k-m model, which are suited to concrete with different compressive strengths, various carbon dioxide concentrations and varied replacing ratios of fly ash, respectively. (4) Methods to determine the parameters in the empirical formulae are presented, and good agreements with reduced error have been found between tested data and the empirical formulae simulating results. It is worth emphasizing that this study just discusses the empirical formula for fitting or simulating the depth-time carbonation behavior of concrete from the point of mathematical angle. In fact, this behavior is affected by multiple factors. It is a prime and interesting task to investigate the carbonation process from the point of physical and chemical interactions under complex environment working conditions among concrete materials sun as cement material, fine aggregate and coarse aggregate in the future study.

ACKNOWLEDGMENTS The authors thank the reviewers who gave a through and careful reading to the original manuscript. Their comments are greatly appreciated and have help to improve the quality of this paper. The authors would like to acknowledge the financial supports from Science Technology Department of Zhejiang Province (NO. 2012C21009, NO. 2012C23063), Ministry of Housing and Urban-Rural Development of China (NO. 2012-k3-4), and Science Technology Bureau of Shaoxing City (NO. 2012B70022). Vol. 18[2013], Bund. L 2321

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