# Hydration, Setting and Crack-Resistance Properties of Activated HVFACs at Early Stages
## 4 Models and Discussion  
### Existing Method for Predicting the Setting Time of FACs: Brooks Model  
Based on a theoretical initial spacing between the particles of un-hydrated cementitious material and the rate of growth of the hydration products, [PERSON] [37] proposed a model for predicting the initial setting time (\(T_{I}\)) of concrete with and without fly ash, as shown in Equation (1). In this model, the following factors have been considered to quantify: the fineness and specific gravity of cementitious material, water/binder ratio, temperature and the chemical composition of the blended cement. Meanwhile, according to this model, the final setting times (\(T_{F}\)) of FACs are linearly related to their initial times with a low error coefficient that can be given by Equation (2). Initial time:  
\[T_{I}=R_{IS}R_{KT}\frac{3}{k\gamma_{bc}S_{bc}}\left[1.613\,\left(1+\gamma_{bc} \frac{w}{c}\right)^{\frac{1}{3}}-1.0\right] \tag{1}\]  
where:  
\(R_{IS}\): relative initial setting time;  
\(R_{KT}\): relative rate coefficient taking into account the effect of temperature on setting;  
\(k\): a rate coefficient that represents the rate of growth of the initial hydration;  
\(S_{bc}\): the specific surface area of the blended cement;  
\(1/\gamma_{bc}\): the volume of blended cementitious materials per g;  
\(w/c\): water/cement ratio;  
Final time:  
\[T_{F}=1.35\,T_{I} \tag{2}\]  
Figure 13: Microstructures vs. properties of FACs. In Equation (1), a combined specific gravity of the FA-PC system should be used as the form of the volume of blended cementitious material per g (\(1/\gamma_{bc}\)), which is calculated as Equation (3). It should be noted, according to the selected FA and cement, that the volume ratio of FA in this equation is 0.8-times its mass ratio. \[\frac{1}{\gamma_{bc}}=\frac{1}{100}\left[\frac{100-R}{\gamma_{c}}+\frac{R}{ \gamma_{ma}}\right] \tag{3}\]  
where \(R\) is the mass percentage of the mineral admixture replacing cement. \(\gamma_{ma}\) is the specific gravity of the mineral admixture. For this model, the specific surface area of blended cement \(S_{bc}\) will be considered as the ratio of FA (\(R\)) and the total surface area of cement (\(S_{c}\)) in concrete, as given in Equation (4). In this paper, the specific surface of mineral admixture (FA) \(S_{ma}\) can be obtained from previous studies [4; 9], which is related to the fineness of the mineral admixture given as the percentage retained on a 45-\(\upmu\)m sieve which is given as,  
\[S_{bc}=\frac{1}{100}\left[S_{c}\left(100-R\right)+S_{ma}R\right] \tag{4}\]  
\[S_{ma}=-425+9.5\left(100-P_{\upmu 45}\right) \tag{5}\]  
Besides, [PERSON] model considered that the setting rate coefficient can be presented as a consistent trend when the chemical composition of blended cement was described as a blend oxide ratio, CaO/(SiO\({}_{2}\) + Al\({}_{2}\)O\({}_{3}\) + Fe\({}_{2}\)O\({}_{3}\)), which is shown as Equation (6). In this study, little lime (CaO) (10% of cement mass) is considered in the calculation of this oxide ratio, as well. \[k\times 10^{4}=0.192\times\left(\frac{CaO}{SiO+Al_{2}O_{3}+Fe_{2}O_{3}}\right) \tag{6}\]  
At the same time, the effects of temperature (\(T\)) and the dosages of a highly-effective water-reducing admixture (\(D\)) on the hydration heat of FAC have been considered in the model, via using two factors, \(R_{KT}\) and \(R_{IS}\), shown in Equation (1), given in Equations (7) and (8), respectively. \[R_{KT}=0.0333\ (T+10) \tag{7}\]  
\[R_{IS}=1+0.038D^{1.73} \tag{8}\]  
In this study, the specific gravity of Portland cement and FA are taken as 3.14 and 2.42, respectively. The fineness of cement and the temperature of concrete construction are taken as 350 m\({}^{2}\)/kg and 26 \({}^{\circ}\)C, respectively. Because the effect of AC on setting had not been taken into account in [PERSON] model, the predicted setting times with the increasing FA will overestimate the corresponding experimental setting times in theory.