Cement and Concrete Research

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1 Cement and Conrete Researh 39 (2009) Contents lists available at SieneDiret Cement and Conrete Researh journal homepage: Hydration kinetis modeling of Portland ement onsidering the effets of uring temperature and applied pressure Feng Lin a,b,, Christian Meyer a a Department of Civil Engineering and Engineering Mehanis, Columbia University, New York, NY 10027, USA b Department of Civil Engineering, Tsinghua University, Beijing , China artile info abstrat Artile history: Reeived 7 November 2006 Aepted 28 January 2009 Keywords: Hydration Kinetis Portland ement Modeling Thermodynamis A hydration kinetis model for Portland ement is formulated based on thermodynamis of multiphase porous media. The mehanism of ement hydration is disussed based on literature review. The model is then developed onsidering the effets of hemial omposition and fineness of ement, water ement ratio, uring temperature and applied pressure. The ultimate degree of hydration of Portland ement is also analyzed and a orresponding formula is established. The model is alibrated against the experimental data for eight different Portland ements. Simple relations between the model parameters and ement omposition are obtained and used to predit hydration kinetis. The model is used to reprodue experimental results on hydration kinetis, adiabati temperature rise, and hemial shrinkage of different ement pastes. The omparisons between the model reprodutions and the different experimental results demonstrate the appliability of the proposed model, espeially for ement hydration at elevated temperature and high pressure Elsevier Ltd. All rights reserved. 1. Introdution The hydration of ement and the aompanying phenomena suh as heat generation, strength development and shrinkage are the results of interrelated hemial, physial and mehanial proesses. A thorough understanding of theses proesses is a ritial prerequisite for modeling hydration kinetis of ementitious materials. The advanes of omputer tehnology made it possible for a branh of omputational materials siene to develop rapidly in the past twenty years, i.e., the omputer modeling of hydration and mirostruture development of onrete. Noteworthy studies inlude those of Jennings and Johnson [1], Bentz and Garbozi [2], van Breugel [3], Navi and Pignat [4], and Maekawa et al. [5]. Among these, the NIST model, referred to as CEMHYD3D [6], and the HYMOSTRUC model [3] appear to be the most advaned and widely used ones. These models attempt to simulate ement hydration and mirostruture formation on the elementary level of ement partiles, whih is the most rational way, as long as the hemial, physial and mehanial harateristis of ement hydration are properly taken into aount. However, there exist ertain appliations for whih simpler mathematial models desribing and quantifying hydration kinetis are neessary. In fat, numerous attempts have been undertaken with that objetive and many suh simplified formulations an be found in Corresponding author. Department of Civil Engineering and Engineering Mehanis, Columbia University, New York, NY 10027, USA. address: [email protected] (F. Lin). the literature, e.g., Byfors [7], Knudsen [8], Freiesleben Hansen and Pedersen [9], Basma et al. [10], Nakamura et al. [11], Cervera et al. [12], Shindler and Folliard [13], and Bentz [14]. In ontrast to the aforementioned mirosopi models, most of these simplified models are empirial in nature, based on experimental observations of marosopi phenomena, and they apture the effets of uring temperature, water ement ratio, fineness, partile size distribution and hemial omposition of ement with different degrees of auray. The range of uring temperature in these models is usually small and does not exeed 60 C, and none of them inlude the effet of applied pressure. However, elevated temperatures and high pressures are frequently enountered in oil wells where they must be addressed. In this study, a hydration kinetis model is formulated based on the thermodynamis of multiphase porous media, whih was first proposed by Ulm and Coussy [15]. The hardening ement paste is known to be a multiphase porous material and the thermodynamis theory of [15] is an ideal framework to model hydration kinetis. Cervera et al. [12,16] proposed an equation to desribe hydration kinetis using this theory, but it needs to be elaborated and reformulated to take into aount the experimental observations of ement hydration, espeially uring temperature and applied hydrostati pressure. This paper presents the theoretial formulation of a new model. The model parameters are alibrated and related to the hemial omposition of ement through simple but expliit equations; the model is then used to predit the experimental results of different investigators, inluding those on hydration kinetis, adiabati temperature rise, and hemial shrinkage of ement pastes /$ see front matter 2009 Elsevier Ltd. All rights reserved. doi: /j.emonres

2 256 F. Lin, C. Meyer / Cement and Conrete Researh 39 (2009) Literature survey on hydration of Portland ement Before a mathematial model an be developed that properly onsiders the various influening fators, the known fats of Portland ement hydration should be thoroughly surveyed. A omprehensive study on ement hydration was onduted by van Breugel [3]. However, during the past deades, many new findings, mostly experimental, regarding ement hydration have been reported. Therefore, the most reent findings and their relevane to understand Portland ement hydration will be addressed herein Influene of hemial omposition of ement The omposition of ement is the most important influening fator of all. There exist two forms in whih to desribe the omposition of ement, i.e. the oxide omposition and the hemial omposition. The latter is usually determined using the ompound stoihiometries and the values of the former. The Bogue alulation [17] is the most frequently quoted mathematial proedure among suh indiret methods and thus is used in the present study. Experimental results show that the four major linker phases of ement, viz. C 3 S, C 2 S, C 3 A and C 4 AF have different reation rates with water [18 20]. It is well known that C 3 A reats the fastest, followed by C 3 S and the other two. However, whether the individual onstituent phases hydrate independently from one another or at equal fration rates is still not resolved [3]. Beause of different reation rates of the linker phases and their interations, it is generally aepted that the so-alled degree of hydration of ement is just an overall and approximate measure. Due to the diffiulty of identifying and simulating the ompliated interations, the overall degree of hydration, denoted as α, is still widely used [1 16] and will be employed in this study Influene of water ement ratio w The water ement ratio,, also influenes hydration kinetis. Experimental results have shown that a higher water ement ratio leads to a higher hydration rate after the middle period of hydration, but only has a small effet on the hydration rate in the early stage [21,22]. The water ement ratio also determines the ultimate degree of hydration α u. Theoretially, a water ement ratio of about 0.4 is suffiient for the omplete hydration of ement (i.e. α u =1.0). In other words, a water ement ratio higher than 0.4 will lead to full hydration given enough time. However, ement hydration is retarded at low internal relative humidity, and the theoretial water ement ratio of about 0.4 is not suffiient for full hydration [23]. The hydration produts around the anhydrous ement partiles prevent further hydration if there is insuffiient free water in the maro-pores. From experimental observations, the relationship between the ultimate degree of hydration α u and the water ement ratio an be desribed by a hyperbola with α u 1.0. Mills [24] onduted a series of tests and derived the following equation for α u, α u = 1:031 w 0:194 + w V1:0 ð1þ This equation has been used frequently in hydration kinetis modeling [13]. However, it does not onsider the effets of ement fineness and uring temperature and may underestimate the ultimate degree of hydration in some ases Influene of fineness of ement The fat that fineness of ement influenes the ultimate degree of hydration α u as well as the hydration rate has been observed in experiments and numerial simulations [25,26]. The finer the ement partiles, the higher α u, and the higher the hydration rate. However, Bentz and Haeker [26] also found that at low water ement ratios, the influene of ement fineness on α u diminishes. A finer ement, or ement with a larger surfae area, provides a larger ontat area with water and hene auses a higher hydration rate. Also, at the same degree of hydration, a larger surfae area orresponds to a smaller thikness of hydration produts around the anhydrous ement partiles, whih inreases the ultimate degree of hydration. Apart from fineness, the partile size distribution of ement also influenes the hydration rate [3,8,27,28] Influene of uring temperature The effets of uring temperature on hydration kinetis have been shown to be twofold. On the one hand, the reation rate α inreases with the inrease in temperature [29 31]. On the other hand, the density of hydration produts at higher temperature is higher [32], whih slows down the permeation of free water through the hydration produts. Therefore, during the late period, the hydration rate is lower at elevated temperature and the ultimate degree of hydration may thus also be lower. Based on the experimental results in [32], van Breugel [3] proposed the following equation for the volume ratio between hydration produts and the reated ement at temperature T [K]: Volume of Hydration Produts at T vt ð Þ = Volume h of Reated Cementiat T ð2þ = v 293 exp ðt 293Þ 2 Generally, v aording to Powers and Brownyard [33]. It should be pointed out that the available experimental results regarding the effets of elevated uring temperature on hydration kinetis are still very limited. Very few publiations an be found that onsider uring temperatures higher than 60 C (333 K). Chenevert and Shrestha [34] and Hills et al. [35] are the two studies worth mentioning. An issue related to the effets of uring temperature is the apparent ativation energy E a. Beause ement is a mixture of different hemial omponents rather than a pure material, its ativation energy is merely phenomenologial and an only be referred to as the apparent one. Different and onfliting onlusions regarding the value of the apparent ativation energy have been drawn based on experiments and theoretial analysis. Xiong and van Breugel [3,36] argued that E a is a funtion of hemial omposition of ement, uring temperature and the degree of hydration. Freiesleben Hansen and Pedersen [37], on the other hand, suggested that E a is only a funtion of uring temperature. Shindler [38] proposed an equation for E a that onsiders the hemial omposition and fineness of ement. Among these studies, the work of [38] is found to be most onvining beause of its appliability to a wide range of ements and simpliity Influene of applied pressure The experiments to investigate the influene of applied pressure on hydration kinetis are even sarer than those involving uring temperatures. Roy et al. [39,40] studied strength development of ement pastes at high temperature and high pressure, but did not address hydration kinetis. Bresson et al. [41] onduted hydration tests on C 3 S subjet to hydrostati pressure up to 85 MPa and found that a higher pressure inreases hydration rate. Zhou and Beaudoin [42] also found that applied hydrostati pressure inreases the hydration rate of Portland ement, but has only a negligible effet on the pore struture of hydration produts when different ement pastes at similar degrees of hydration are ompared. The finding of [42] is important sine it implies that the density of the hydration produts is not onsiderably affeted by the applied hydrostati pressure, at least not up to 6.8 MPa, whih is the pressure they used.

3 F. Lin, C. Meyer / Cement and Conrete Researh 39 (2009) Mehanisms and thermo-hemial onepts of ement hydration Based on numerous experimental observations, the mehanism of ement hydration is fairly lear. Aording to [3], it involves three main stages, viz. the early, middle and late periods. A short period of rapid hemial dissolution, termed the pre-indution period, is followed by a dormant stage, whih lasts for several hours. Then ement hydration enters the middle period that an last for 24 to 48 h, in whih ions transport to and from the surfaes of anhydrous ement partiles through gradually growing layers of hydration produts. During the late period, whih is the longest and dominant one, the reation is totally diffusion-ontrolled. The free water in maro-pores permeates through the hydration produts formed around the anhydrous ement ore, making further reations possible. van Breugel [3] laims that ement hydration follows two different mehanisms at different stages. During the early and middle periods, the phase boundary mehanism prevails, while in the late period, the diffusion-ontrolled reation dominates. However, it is very diffiult to define a definite boundary between the two mehanisms. In the present study, it is assumed that the phase boundary mehanism an be regarded as a speial ase of the diffusion-ontrolled reation when the layers of hydration produts around the anhydrous ement ores are very thin. The mehanisms of ement hydration are then unified to be the diffusion-ontrolled one. Cement hydration is interpreted as a hemial reation in whih the free water in the maro-pores ombines as a reatant phase with the anhydrous ement and beomes the hemially bound water as a produt phase, and the dominant mehanism of the reation kinetis is the diffusion of free water through layers of hydration produts, as illustrated in Fig. 1 [15]. The assumption an simplify the model formulation yet still apture the salient feature of ement hydration. Using the thermodynamis of reative porous media and assuming a losed system, the thermodynami inequality equation an be expressed as [15]: σ ij e ij S : T : W + Φ AYB 0 in whih σ ij is the ijth omponent of the stress tensor, ε ij is the ijth omponent of the strain rate tensor, S is the entropy, T is the variation rate of temperature, Ψ is the variation rate of free energy, and Φ A B is the dissipation assoiated with the hemial reation A B. For ement hydration, aording to the aforementioned mehanism, A is the free water in maro-pores and B is the hemially bound water in hydration produts. The hemial dissipation Φ A B an be expressed as: ð3þ the variation rate of the degree of hydration. Using the Arrhenius equation, the hemial affinity A α an be written as: A α = α exp E a ð5þ η α RT in whih η α represents the permeability of the hydration produts around the anhydrous ement, E α is the apparent ativation energy, R is the universal gas onstant, and T is the absolute temperature. The apparent ativation energy E α is used in Eq. (5) beause as pointed out in [3], it is an experimentally obtained quantity that is a net result of the temperature dependeny of several proesses. With Eq. (5), the hydration rate α an be written as: α = A α η α exp E a ð6þ RT Eq. (6) provides an ideal framework for the modeling of hydration kinetis of ement, provided the fators that influene the hemial affinity A α and the permeability η α an be identified and quantified. 3. Model formulation 3.1. Modeling of the influene fators It is obvious that the hemial omposition of ement affets both the hemial affinity A α and the permeability η α, whereas the water ement ratio influenes only the hemial affinity A α, sine it determines the hemial potential of free water in the multiphase system of ement paste to a ertain extent. The following expression may be used to relate the hemial affinity A α to the hemial omposition of ement and water ement ratio [12]: A A α ~k 0 + α ðα kα u αþ ð7þ u in whih means proportional to, k and A 0 are parameters related to the hemial omposition of ement and other fators. A 0 is atually the initial hemial affinity of ement hydration (α=0). Eq. (7) impliitly onsiders the so-alled water shortage effet as pointed out in [3], sine the ultimate degree of hydration α u is affeted by the water ement ratio. From Eq. (6), it an be seen that the effets of A α and η α are demonstrated through their produts. In order to simplify the model, an exponential equation is used to represent the permeability η α,thatis: Φ AYB = A α α 0 ð4þ η α =exp nα ð Þ ð8þ where A α is the affinity of the hemial reation A B, viz. the hemial affinity of ement hydration, and α is the reation rate, i.e. Fig. 1. Diffusion of free water through layers of hydration produts [15]. in whih n is a parameter that depends on uring temperature, degree of hydration, and fineness of ement. η α an be onsidered the normalized permeability, whih means all the model oeffiients are taken into aount by A α. Eq. (8) desribes the gradual derease in permeability of hydration produts with the development of hydration. In fat, the exponential funtion has been frequently used to desribe hydration kinetis [12,13]. Aording to Eqs. (7) and (8), the hemial omposition of ement influenes the three parameters, k, A 0 and n, while the water ement ratio affets only α u. With different values of α u, Eqs. (6) (8) reflet impliitly the effets of water ement ratio on hydration rate. For the sake of simpliity, only the fineness of ement partiles but not the atual partile size distribution is taken into aount in the present model. Sine for most general-purpose Portland ements, the shapes of their partile size distribution urves are similar, this simplifiation does not introdue signifiant errors. The fineness of ement an be represented by the Blaine value with units of m 2 /kg. It influenes the initial reation rate of ement. Therefore, it should

4 258 F. Lin, C. Meyer / Cement and Conrete Researh 39 (2009) appear in the expression for the parameter A 0, whih represents the initial hemial affinity of ement hydration. Hene the following equation is used to express A 0 : A 0 = AV 0 Blaine 350 in whih A 0 is the normalized initial hemial affinity of ement hydration and is only a funtion of the hemial omposition of ement. The Blaine fineness also affets the permeability η α and the ultimate degree of hydration α u. For permeability, it is notied that the finer the ement partiles, the larger the Blaine value, the thinner the layers of the hydration produts, hene the higher the permeability of the hydration produts and the smaller the parameter n (Eq. (8)). Therefore, it is assumed that: n~1 + ð1 αþ 2 ln 350 Blaine ð9þ ð10þ As an be seen in Eq. (10), the effet of Blaine fineness is not onstant throughout the hydration proess. With an inrease in the degree of hydration α, the effet of Blaine fineness dereases. The effets of uring temperature an be onsidered partly by involving one term of the Arrhenius equation, as shown in Eq. (6). However, the densifying effet of elevated uring temperature on the hydration produts should also be onsidered. Sine the density of the hydration produts is losely related to the permeability η α, the parameter n in Eq. (8) should also be a funtion of uring temperature. And sine the hydration rate of ement during the late period dereases at elevated uring temperatures, the parameter n should be an inreasing funtion of the uring temperature T. In this study, the volume ratio equation (Eq. (2)) is used, with the effet of uring temperature T on the parameter n represented as: n~ v 10α vt ð Þ ð11þ in whih α is the degree of hydration, and v(t) and v 293 are as given in Eq. (2). Eqs. (8) and (11) imply that the effet of uring temperature on the permeability η α is a dereasing funtion of the degree of hydration α, whih is onsistent with experimental results [29 31]. The uring temperature T also influenes the ultimate degree of hydration α u, as will be disussed subsequently. The applied hydrostati pressure p is assumed to influene only the hemial affinity A α, as supported by the experimental results in [42]. Aording to the tests onduted by Bresson et al. [41], the hemial affinity A α is set to be: p 0:07 α A α ~ exp 0:02 1 1:5 α 2 +0:4 ð12þ p atm α u α u where p is the applied hydrostati pressure, and p atm is the atmospheri pressure. Eq. (12) shows that the reation rate inreases with the inrease in the applied pressure p. It an be onluded now that the hemial affinity A α is a funtion of the degree of hydration α, the water ement ratio w (through the ultimate degree of hydration α u ), the Blaine fineness (through the initial hemial affinity A 0 and the ultimate degree of hydration α u ), the applied pressure p (through Eq. (12)), and the hemial omposition of ement (through the parameters A 0 and k). And the parameter n that ontrols the permeability of free water through the hydration produts is a funtion of the uring temperature T (through Eq. (11)), the degree of hydration α, the Blaine fineness (through Eq. (10)), and the hemial omposition of ement. Thus, if the ultimate degree of hydration α u is identified, the parameters that remain in the hydration kinetis model will be related to the hemial omposition of ement only Modeling of the ultimate degree of hydration The available experimental and numerially simulated results for α u as a funtion of water ement ratio and fineness of ement [22,24,43 45] were olleted and analyzed. The Blaine fineness of the ements ranges from around 270 m 2 /kg to 420 m 2 /kg and the water ement ratio varies from 0.25 to For some sets of data, in whih the hydration ages were not long enough, small extrapolations were performed based on the trends of the degree-of-hydration vs. time urves to obtain the orresponding ultimate degree of hydration. It must be pointed out that theoretially, the ultimate degree of hydration an never be obtained from experiments, sine very limited reations may still be possible after years. However, the very limited reations at suffiiently late age are of little signifiane for pratial appliation and the final degrees of hydration found in [22,24,43 45] are assumed to be the ultimate ones. From the data analysis, it was found that: 1) The ultimate degree of hydration α u may be expressed as a hyperboli funtion of water ement ratio with α u ) α u annot exeed the theoretial value given by: w α u V 0:4 ð13þ in whih 0.4 is approximately the theoretial water ement ratio neessary for full hydration, whih is the sum of the hemially bound water ratio, about 0.25, and the gel water ratio, about 0.15 [3,33]. 3) α u inreases as the ement partiles get finer, i.e. as the Blaine value inreases. The effet of ement fineness on α u beomes smaller with a derease in water ement ratio [26]. Based on these observations, an equation onsidering both the effets of water ement ratio and fineness of ement is obtained via theoretial analysis and data regression: α u;293 = β 1ðBlaineÞ w β 2 ðblaineþ + w V1:0 ð14þ in whih α u,293 is the ultimate degree of hydration at T=293 K. β 1 and β 2 are both funtions of Blaine fineness and are defined as: β 1 ðblaineþ = β 2 ðblaineþ = 1:0 2:82 +0:38 9:33 Blaine 100 Blaine :78 + 1:656ðBlaine 220Þ ð15þ ð16þ Eqs. (15) and (16) are valid for Blaine 270 m 2 /kg. When Blaineb270 m 2 /kg, β 1 and β 2 are assumed to remain onstant, that is, β 1 β 2 Blaine b 270m 2 = kg Blaine b 270m 2 = kg = β 1 Blaine = 270m 2 = kg = β 2 Blaine = 270m 2 = kg ð17þ ð18þ The appliability of the proposed equation for α u,293 is validated using the experimental results of Mills [24] and Baroghel-Bouny et al. [45], as shown in Fig. 2. The ultimate degree of hydration α u is also related to the uring temperature T. Based on the experimental results of Kjellsen and Detwiler [30] and Esalante-Garia [31] for three different Portland

5 F. Lin, C. Meyer / Cement and Conrete Researh 39 (2009) obtained from experimental results at room temperature and will not vary onsiderably for different values of ativation energy. The hemial affinity A α is expressed as: A α α; w ; Blaine; p A = k 0 + α ðα kα u αþ sðα; pþ ð23þ u in whih the funtion s(α, p) represents the effet of applied hydrostati pressure p (see also Eq. (12)): p 0:07 α sðα; pþ = exp 0:02 1 1:5 α 2 +0:4 p atm α u α u ð24þ The initial hemial affinity A 0 is defined by Eq. (9), and the ultimate degree of hydration α u is given by Eqs. (14) (21). The remaining onstants A 0 and k are funtions of the hemial omposition of ement. The permeability of free water through hydration produts, η α,is expressed by Eq. (8), in whih the parameter n is written as: n = n 0 f 1 ðblaine; αþ f 2 ðt; αþ ð25þ where f 1 ðblaine; α Þ =1+ ð1 αþ 2 ln f 2 ðt; αþ = v 4 10α 293 vt ð Þ 350 Blaine ð26þ ð27þ Fig. 2. Simulated and measured ultimate degree of hydration at T=293 K (a) experimental data from [24]; (b) experimental data from [45]. ements, the following equation is proposed to onsider the effet of uring temperature T: h i α u = α u;293 exp 0:00003ðT 293Þ 2 SGNðT 293Þ ð19þ in whih 1; when T 293K SGNðT 293Þ = 1; when T b 293K ð20þ It should be noted that the equation for α u must also be bounded by the theoretial value. When the water ement ratio is low, α u predited by Eqs. (14) (20) an be higher than the theoretial value of Eq. (13). Therefore, if the alulated α u exeeds that theoretial value, it is assumed that: α u = w 0: Mathematial model of hydration kinetis ð21þ With the above disussion and formulation, a mathematial model of hydration kinetis for Portland ement an now be presented. The rate of hydration α is given by Eq. (6), but normalized, that is, α = A α η α exp E a E exp a ð22þ RT 293R in whih E a is the apparent ativation energy. The normalization in Eq. (22) ensures that the material onstants of the model an be and n 0 is a material onstant that is related to the hemial omposition of ement only. The last material onstant of this model is the apparent ativation energy E α. In this study, it is assumed that E α is onstant for one speifi type of ement. The empirial equation obtained in [38] is used to determine E α, whih is a funtion of the hemial omposition and fineness of ement. 4. Model alibration The proposed model ontains four material onstants, k, A 0, n 0 and the apparent ativation energy E α, all of whih depend on the ement properties only. The first three are funtions of the hemial omposition of ement only, while the apparent ativation energy depends also on the fineness of ement [38]. To alibrate these material onstants, the experimental results on hydration kinetis for eight different Portland ements are used, namely those for Type I, Type II, Type III and Type IV ements of Lerh and Ford [29], and those of Keienburg [25], Danielson [21], Esalante-Garia [31], and Taplin [22]. In Table 1, these ements are identified as #A #H, respetively. The experimental results used over the effets of uring temperature, water ement ratio, and fineness of ement on hydration kinetis of Portland ement, with the mass fration of C 3 S in ement ranging from 0.24 to An equation was proposed in [38] for the apparent ativation energy E α [J/mol], 0:30 0:25 0:35 E a = pc4af ðblaineþ ð29þ p C3 A in whih the Blaine fineness is with units of m 2 /kg. p C3 A and p C4 AF are the Bogue mass frations of C 3 A and C 4 AF, respetively. This equation is used in the present study to obtain the apparent ativation energy if its value was not provided in the experimental data. The Bogue ompositions of the ements investigated are listed in Table 1. The four material onstants are also shown for eah ement. It an be seen from Eq. (22) that for experiments onduted at room

6 260 F. Lin, C. Meyer / Cement and Conrete Researh 39 (2009) Table 1 Cement ompositions and the orresponding material onstants. Cement identifiation # Figure number Bogue omposition Blaine fineness Material onstants C (m 2 /kg) 3 S C 2 S C 3 A C 4 AF E α (J/mol) k (h 1 ) n 0 A 0 (h) #A [29] ,271 a #B [29] ,788 a #C [29] ,955 a #D [29] ,978 a #E [25] #F [21] #G [31] b , #H [22] a Values taken from [38]. b The hemial omposition determined by quantitative X-ray diffration analysis (QXDA) was used for this ement. Calulated value using Eq. (29). temperature, E α is not signifiant for the model. The other three material onstants are obtained by fitting the experimental data. The material onstants are not neessarily the ones that best fit the experimental results, as they have been adjusted to allow for their variations with the hemial ompositions by onduting linear or nonlinear regression analyses with EXCEL. The omparison of the model simulations with the experimental results for the eight ements are shown in Figs. 3 10, respetively. The material onstants k, A 0 and n 0 of Table 1 an be approximately represented by ertain funtions of the Bogue omposition of ement. The normalized initial affinity A 0 an be expressed as a linear funtion of the Bogue mass frations of C 4 AF. The experimental results of Esalante-Garia and Sharp [20] support this relation. It an be seen from their experimental data that the initial reation rate of C 4 AF is muh lower than that of C 3 S, C 2 S and C 3 A, whih have similar initial reation rates. The funtion of A 0 is expressed as: AV 0 = 0:0767p C4 AF +0:0184 ð30þ with p C4 AF being the Bogue mass fration of C 4 AF. The omparison of A 0 alulated by Eq. (30) with the values listed in Table 1 is shown in Fig. 11(a). Fig. 3. Simulated and measured degree of hydration for ement #A (type I) (experimental data from [29]). Fig. 5. Simulated and measured degree of hydration for ement #C (type III) (experimental data from [29]). Fig. 4. Simulated and measured degree of hydration for ement #B (type II) (experimental data from [29]). Fig. 6. Simulated and measured degree of hydration for ement #D (type IV) (experimental data from [29]).

7 F. Lin, C. Meyer / Cement and Conrete Researh 39 (2009) Fig. 7. Simulated and measured degree of hydration for ement #E (experimental data from [25]). Fig. 9. Simulated and measured degree of hydration for ement #G (experimental data from [31]). The onstants n 0 and k an be represented by a linear and a nonlinear funtion of the Bogue mass frations of C 3 S, C 2 S, and C 3 A, respetively: 0:206 0:128 0:161 k =0:56 p C3 S pc2 S pc3 A ð31þ n 0 =10:945p C3 S +11:25p C2 S 4:10p C3 A 0:892 ð32þ The omparisons of k alulated by Eq. (31) and n 0 alulated by Eq. (32) with the values listed in Table 1 are shown in Fig. 11(b) and (), respetively. It an be seen in Fig. 11 that the three model parameters (k, A 0 and n 0 ) orrelate strongly with the hemial omposition of ement. Although only eight ements were used for the data regression analysis, onsidering the fat that the hemial ompositions of these ements are very different, the trends of the funtional relationships appear to be aptured very well. 5. Model verifiation 5.1. Model reprodution of the degree of hydration In order to validate the proposed model, it is used to reprodue hydration kinetis for several ements. First of all, the model apability to predit the degree of hydration of ement under pressure is verified using the experimental results. As mentioned in Setion 3.1, the funtion s(α, p) in Eq. (24) was alibrated using the experimental results for C 3 Sin[41]. The simulated and experimental results are ompared in Fig. 12. In Fig. 13 the model reprodutions of the degree of hydration for an ordinary Portland ement ured under pressure are ompared with the experimental results of [42]. It an be seen that the model is apable of prediting the effet of applied hydrostati pressure on ement hydration. Hydration kinetis of three ements tested by Bentz et al. [46,14] and Hill et al. [35] are simulated using the proposed model. The experimental results of [46] exhibited the effet of uring onditions (saturated or sealed uring); those of [14] showed the effet of water ement ratio; and those of [35] indiated the effet of uring temperature up to 90 C (363 K), but the values of the degree of hydration were not provided. Suh data are obtained by using the maximum heat output equation of Bogue [17], viz. αðþ t Qt ðþ Q max ð33þ The material onstants used in the simulation are obtained through Eqs. (29) (32). Sine the fineness of ement was not provided either, a value of 350 m 2 /kg is used by omparing the Blaine values of similar ements. The model reprodutions are ompared with the experimental results in Figs , whih show reasonably good agreement. Although the degree of hydration at the uring temperature of 90 C [35] is over-estimated to a ertain extent, the overall Fig. 8. Simulated and measured degree of hydration for ement #F (experimental data from [21]). Fig. 10. Simulated and measured degree of hydration for ement #H (experimental data from [22]).

8 262 F. Lin, C. Meyer / Cement and Conrete Researh 39 (2009) Fig. 12. Simulated and measured degree of hydration for C 3 S under pressure (experimental data from [41]). where M e [kg/m 3 ] is the ement ontent by mass in the paste; ρ paste [kg/m 3 ] is the density of ement paste; q paste, q ae and q w are the speifi heats of ement paste, anhydrous ement, and water, respetively; q he [kj/(kg K)] is the fititious speifi heat of the hydrated part of ement, whih an be expressed as: q he =0:0084ðT 273Þ +0:339 ð35þ The temperature rise, ΔT, an be written as: ΔT = Δα Q max M e Δα Q = max ρ paste q paste α q he + ð1 αþq ae + w q w ð36þ where Δα is the inrease in the degree of hydration, and Q max [kj/kg] is the maximum heat of hydration. The values of the speifi heat of ement and water are known to be q ae 0.75 kj/(kg K) and q w 4.18 kj/(kg K), respetively [3]. If onrete or mortar is onerned instead of ement paste, the speifi heat and mass of aggregate should also be inluded in Eqs. (34) and (36). The adiabati tests for the CEM I 52.5 PM CP2 onrete by Bentz et al. [47] are reprodued using the proposed model to verify its apability to predit the adiabati behavior of ementitious materials with different water ement ratios. The material onstants, exept for the apparent ativation energy E α taken diretly from [47], are obtained via Eqs. (30) (32). The results are given in Fig. 17. It an be seen that the model preditions agree well with the experimental results. Fig. 11. Comparisons of the simulations with the regression values (a) Eq. (30) for A 0 ; (b) Eq. (31) for k; () Eq. (32) for n 0. predition auray an be onsidered aeptable in view of the diffiulties of onduting tests at elevated uring temperatures Model reprodution of adiabati temperature rise The exothermi behavior of ement hydration an be simulated by introduing the speifi heat of ement paste, q paste. van Breugel [3] proposed an equation for q paste, whih is a funtion of the degree of hydration α, uring temperature T, and water ement ratio: q paste = M h e α q ρ he + ð1 αþq ae + w i paste q w ð34þ Fig. 13. Reprodued and measured degree of hydration for OPC under pressure (experimental data from [42]).

9 F. Lin, C. Meyer / Cement and Conrete Researh 39 (2009) Fig. 14. Reprodued and measured degree of hydration for CCRL ement 115 under different uring onditions (experimental data from [46]). Fig. 17. Reprodued and measured adiabati temperature hanges for CEM I 52.5 PM CP2 ement with different water ement ratios (experimental data from [47]). Fig. 15. Reprodued and measured degree of hydration for CCRL ement 152 with different water ement ratios (experimental data from [14]). Fig. 18. Reprodued and measured hemial shrinkage for Class H ement at different temperatures and pressures (experimental data from [34]) Model reprodution of hemial shrinkage With the degree of hydration α, the hemial shrinkage of ement paste an be alulated, whih aording to [33], is approximated as: ð v hsh ðαþ = 1 v nþ w n α 0:25 0:25 α v + w = 0:32 + w ð37þ where v hsh (α) is the hemial shrinkage; v n ( 0.75 m 3 /g) is the speifi volume of the hemially bound water, whih means that the hemial shrinkage equals up to 25% of the volume of the hemially bound water; w n is the mass of the hemially bound water; and v ( 0.32 m 3 /g) is the speifi volume of ement. The hemial shrinkage tests by Chenevert and Shrestha [34], Justnes et al. [48,49] and Baroghel-Bouny et al. [45] are reprodued using the proposed model. It should be pointed out that the tests in [34] were onduted at high temperatures and high pressures. The material onstants are again obtained via Eqs. (29) (32). The Fig. 16. Reprodued and measured degree of hydration for OPC at different temperatures (experimental data from [35]). Fig. 19. Reprodued and measured hemial shrinkage for Class G ement with different water ement ratios (experimental data from [48]).

10 264 F. Lin, C. Meyer / Cement and Conrete Researh 39 (2009) Fig. 20. Reprodued and measured hemial shrinkage for P30 k ement with different water ement ratios (experimental data from [49]). predited and experimental results are ompared in Figs , whih show aeptable agreement. The Blaine fineness was not provided in [34], hene a value of 300 m 2 /kg is adopted for the Class H oil well ement used in the tests by referring to the general fineness of this type of ement. Attention is direted to the model apaity of reproduing the behavior of oil well ements (Class G and Class H) under ambient onditions as well as downhole onditions of high temperature and high pressure. It should be noted that in the experiments the ement started to hydrate before the data were reorded. Therefore, values of the original degree of hydration in the model reprodution of the experimental results in [34] are not zero, but set as 0.04 at 38 C (100 F) and 0.12 at 66 C (150 F), respetively. 6. Summary and disussion A hydration kinetis model for Portland ement has been proposed based on the thermo-hemial onepts of Ulm and Coussy [15]. The dominant mehanism of ement hydration is assumed to be the diffusion of free water through the layers of hydration produts. The effets of hemial omposition and fineness of ement, water ement ratio, uring temperature and applied pressure are taken into aount in this model. The influene of the hemial omposition of ement is inorporated in the four material onstants, k, A 0, n 0 and E α. The relationships between these material onstants and the hemial omposition of ement are alibrated. The preditive apabilities of the model are demonstrated for various appliations. The proposed hydration kinetis model is theoretially sound, easy to use and apable of prediting the hydration Fig. 21. Reprodued and measured hemial shrinkage for HS65 ement with different water ement ratios (experimental data from [49]). Fig. 22. Reprodued and measured hemial shrinkage for type I ement with different water ement ratios (experimental data from [45]). development of various ements under different uring onditions. In partiular, the effets of elevated uring temperature and high applied pressure on hydration kinetis of ement an be reprodued. The proposed model an be used for different appliations, suh as the predition of hydration kinetis, adiabati temperature hange and hemial shrinkage of ement paste. Sine only eight ements were used for the data regression analysis, the relationships between the three material onstants (A 0, k and n 0 ) and the hemial omposition of ement may not be optimal. However, the trends of the relationships appear to have been aptured. Aording to the available experimental observations, the initial reation rate of C 4 AF is the lowest, hene the material onstant A 0 should be losely related to its ontent. The values of k and n 0 determine the overall reation rate. The larger the value of k and the smaller the value of n 0, the higher the reation rate. It is well-known that C 3 A and C 3 S reat faster, while C 2 S reats slower, hene the trends as given by Eqs. (31) and (32) are orret. The validity of using one single value of apparent ativation energy E α has been verified, but its dependene on the hemial omposition and fineness of ement deserves further investigation. The present study is foused on ordinary Portland ement without any mineral or hemial admixtures. However, it should be emphasized that this model is fundamentally apable of simulating the behavior of blended ements or ements with admixtures. Of ourse, in this ase, the material onstants would have to involve the effets of the replaed materials or the admixtures, but the model formulation would be similar. The model's apaity of reproduing hydration kinetis of ement under uring onditions of high temperature and high pressure is demonstrated. However, the available experimental results under suh onditions are very limited, therefore, the proposed formulas still need to be verified with more reliable experimental data. Although it was reported in [3,8,27,28] that the partile size distribution of ement influenes hydration kinetis to some extent, this influene is not onsidered in the proposed model. The assumption ould lead to some disrepany in the model simulation but sine the trends of the partile size distribution urves of most generally used Portland ements are quite similar, it is onsidered to be aeptable. The preditive apaity of the model has demonstrated that this assumption does not introdue signifiant errors in this model for a wide variety of Portland ements. It is also noteworthy that different methods to determine the degree of hydration an introdue onsiderable disrepanies among the obtained results. The methods ommonly used to determine the degree of hydration vary from the uses of liberated heat of hydration, amount of hemially bound water, hemial shrinkage, amount of Ca(OH) 2, to the approximations from the speifi surfae,

11 F. Lin, C. Meyer / Cement and Conrete Researh 39 (2009) strength, or dieletri properties of ement. These differenes between the methods to measure the degree of hydration should be borne in mind when modeling hydration kinetis for Portland ement. Aknowledgements The support of this study was provided by Halliburton Energy Servies (HES), whih is ordially appreiated. The enthusiasti help from Dr. Lewis Norman, Dr. Wolfgang Deeg, and many other personnel at Halliburton's Dunan Tehnology Center is gratefully aknowledged. Referenes [1] H.M. Jennings, S.K. Johnson, Simulation of mirostruture development during the hydration of a ement ompound, Journal of the Amerian Cerami Soiety 69 (11) (1986) [2] D.P. Bentz, E.J. Garbozi, A digitized simulation model for mirostrutural development, in: S. 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