Predicting Method for Time-Dependent Concrete Carbonation Depth (I): Traditional Model and Its Error Distribution Zhijian Shu School of Engineering, Lishui University, Lishui, China Wei Wang* School of Engineering, Shaoxing University, Shaoxing, China *Corresponding author, e-mail: [email protected] Chunyang Cheng School of Engineering, Shaoxing University, Shaoxing, China Xiaocui Tang School of Engineering, Lishui University, Lishui, China ABSTRACT Carbonation of concrete is one important subject in civil engineering because it is harmful to compressive strength and durability of corresponding structures such as subways, buildings and bridges. It is of great importance to properly understand its mechanism and developing process. In this paper, main influencing factors of concrete carbonation are discussed. Traditional mathematical models to predict time-dependent carbonation depth based on Fick's first law of diffusion are classified and generalized. Three sets investigated data are used to verify the applicability and accuracy of the generalized k-0.5 model and its fitted error distribution. Finally, common shortcoming and its essential reason of the k-0.5 model are proclaimed. KEYWORDS: carbonation depth, concrete, mathematical model, error distribution INTRODUCTION Carbonation of concrete indicates the physical-chemical action process between concrete stone and carbon dioxide from atmosphere producing calcium carbonate (Das, et al. 2012; He and Jia 2011). This action process is often dominated by three aspects. The first aspect is concrete materials including cement/water ratio, kind and mass of cement, fine and coarse aggregates and their gradations, and admixture (Nishi 1962; Zhang and Jiang 1990; Zhu 1992; Bouzoubaâ, et al. 2010; Mohamed, et al. 2013; Gao, et al. 2013). The second aspect is environment condition such as surrounding temperature and humidity, carbon dioxide concentration (PaPadakis 1962; He and - 2297 - Vol. 18 [2013], Bund. L 2298 Jia 2011). The third aspect is construction technology of concrete structures including curing method and curing time (He and Jia 2011; Liang, et al. 2013). Concrete carbonation can be simply expressed as: Ca OH +CO CaCO +H O (1) 2 232 Although the produced calcium carbonate is harder than calcium hydroxide which seemly presents bigger compressive strength to concrete, this carbonation action reduces high alkaline environment of the concrete which weakens environment proof and brings potential dangerous to steel bar in concrete. As a result, carbonation often gives rise to steel bar rust of reinforced concrete, which may dramatically reduce the bearing capacity of reinforced concrete pile foundations and structures such as subways, buildings and bridges (Du and Liang 2005; Hussain 2011). Consequently, it may weaken their durability and shorten their service life. So it is an interesting must for engineer to establish mathematical model or empirical formula to well describe concrete carbonation developing process and properly predict carbonation depth to achieve good project design. Lot’s of effort has been devoted to this subject in terms of theory analysis, laboratory test and field investigation, and several mathematical models have been put forward to predict time-dependent carbonation depth of concrete (Sun 2006; Jonathan, et al. 2013; Marques, et al. 2013). However, these models are often associated with big fitting error. The object of this paper is to discuss mechanism and main influencing factor of concrete carbonation in order to find good prediction model. The result of this study is helpful to reasonable design and health evaluation of existed or to be built concrete structures. TRADITIONAL PREDICTING MODELS Basic Model based on Fick's First Law of Diffusion The classic model for concrete carbonation is deduced from Fick's first law of diffusion: dC JD c (2) dX where, J is diffusion flux of carbon dioxide in concrete porous; D is effective diffusion coefficient of carbon dioxide in concrete; Cc is concentration of carbon dioxide in concrete; and X is the normal distance from concrete surface. Considering boundary conditions (Xiao, et al. 2010; He and Jia 2011), following equation can be obtained to describe relation between time and carbonation depth: 2Dc x 0 t 0.5 (3) m0 where, x is carbonation depth; t is carbonation time; c0 is the carbon dioxide concentration at the surface of concrete; m0 is the quality of carbon dioxide that absorbed by unit volume of concrete. Vol. 18 [2013], Bund. L 2299 Prediction Models based on Water and Cement Mass Many laboratory tests show that water and cement masses in unit volume of concrete and water/cement ratio play an important role in carbonation behavior of concrete. Several semi- empirical models were presented based on Eq. (3) and investigated data. Nishi (1962) proposed one carbonation time-depth model: g xt 0.5 K w 2 KWCWCWCw 0.3 11.5 3 / / / 0.25 / 0.6 (4) 2 KWCw 7.2 / 4.6 / 1.76 WC / 0.6 where g is one undetermined parameter rest on aggregate, cement, and admixture; Kw is the other parameter dependent on water/cement ratio, W/C; W and C are the masses of water and cement in unit volume of concrete, respectively. Taking meteorology condition into account, Zhu (1992) presented following model: 0.5 x 12312.1WC / 3.2 t (5) where γ1 is cement kind factor ranged from 0.5 for common Portland cement to 1.0 for slag cement; γ2 is fly-ahs factor; γ2 is meteorology condiction factor varied from 0.5 in wet area to 1.2 in arid area. Zhang and Jiang (1990) proposed one formula considering cement/water ratio and cemment mass after labortory testing of nature carbonation and artifical carbonation: CW0.5 x 0.74 0.137 0.163 t (6) 100 C After discussing the effect of replacing ratio of fly ash, ambient relative humidity, temperature, curing age and W/C on carbonation of concrete made with low-volume fly ash, He and Jia (2011) proposed one composite semi-empirical formula to predict carbonation depth: 81.78kf ( RH , T , w / b ) Vco x 2 t 0.5 (7) 0.31WFASiOCe (1 4.48 %2 %) In Eq. (7), WCe has same meaning to W in above equation denoting cement mass, and meanings of other parameters can be found in literature (He and Jia 2011). Prediction Models based on Compressive Strength Considering curing method, working environment and cement kind, one compressive strength expressed model is established by Niu to calculate depth of concrete: Vol. 18 [2013], Bund. L 2300 57.94 x 2.56KKKKKT40 1 RHRH m 0.76 t.5 (8) mc f co2 p s c fcu This model has nine undetermined parameters and their detailed meanings can be found in literature (Xiao, et al. 2010). To this point, Smolczyk obtained another strength-dependent model (Du and Liang 2005): 11 x 79.06t 0.5 (9) FFcs where Fc denotes compressive strength of concrete, and FS denotes its ultimate strength assuming no carbonation. Generalized Mathematical Behavior Although above mentioned models have different parameters, they have one same behavior that concrete depth is power function of time with exponent 0.5. From mathematical point, those can be generalized as a simple form with one undetermined parameter, k: x kt 0.5 (10) According to parameter k and exponent 0.5, we call Eq. (10) as the k-0.5 model. Eq. (10) can be rewritten as: x2 Kt (11) Eq. (11) demonstrates that, in above mentioned models, square of carbonation depth is linear to carbonation time with slope K. MODEL APPLICATION AND FITTED ERROR ANALYSIS Model Application In order to verify the applicability and accuracy of traditional models, comparison with laboratory tests should be conducted. In this paper, six samples carbonation depth-time data of there different concrete sets are used to discuss the applicability and accuracy of Eq. (10) and Eq. (11). Investigated and fitted data, fitted error distributions, square of carbonation depth and time data are shown as in Fig. 1, Fig. 2, table 1 and table 2. Investigated data of Fig. 1 (set 1) is cited from Sun’s literature (Sun 2006) which conducted carbonation tests on different compressive strength concrete, and the investigated data in this figure is for concrete with C10 and C30 compressive strength. Vol. 18 [2013], Bund. L 2301 4 Investigated C30 3 Eq. 10 fitted C30 m 2 Depth / m / Depth 1 0 0 50 100 150 200 250 300 350 400 Time / d (a) Depth-time data of C30 concrete 6 Investigated C10 5 Eq. 10 fitted C10 4 m 3 2 Depth / m / Depth 1 0 0 50 100 150 200 250 300 350 400 Time / d (b) Depth-time data of C10 concrete Vol. 18 [2013], Bund. L 2302 24 C30 16 C10 8 0 0 50 100 150 200 250 300 350 400 -8 Time / d Relative error / % Relative -16 -24 -32 (c) Fitted error distribution 30 C30 25 C10 2 20 / mm 2 15 Depth 10 5 0 0 50 100 150 200 250 300 350 400 Time / d (d) Depth2-time data Figure 1: Fitted and investigated results of set 1 carbonation data Investigated data of Fig. 2 (set 2) is quoted from Du’s literature (Du and Liang 2005) which conducted carbonation tests on concrete with composite admixture including fly ash and slag. Admixture of sample A1 is 18 % fly ash and 37% slag, and that of A2 is 15% fly ash and 25% slag. Vol. 18 [2013], Bund. L 2303 30 Investigated A1 25 Eq. 10 fitted A1 20 m 15 10 Depth / m / Depth 5 0 0 20406080100 Time / d (a) Depth-time data of sample A1 30 Investigated A2 25 Eq. 10 fitted A2 20 m 15 10 Depth / m / Depth 5 0 0 20406080100 Time / d (b) Depth-time data of sample A2 Vol. 18 [2013], Bund.