# Study on Hydration Heat Release Model and Its Influence Coefficient of Addition Concrete
## 2 Concrete Hydration Exothermic Model and Its Finite Element Analysis Method
### Introduction to Exothermic Modeling of Concrete Hydration  
Currently, three types of exothermic concrete hydration models are commonly used in engineering research: mono-exponential, hyperbolic, and composite exponential models. In this section, these three exothermic hydration models are briefly introduced, and the related calculation formulas and parameters are provided to compare and select the most suitable calculation model. 1. Mono-exponential model  
The mono-exponential model was proposed by the U.S. Bureau of Reclamation, and use of the exponential function can be more accurate and better suits the requirements of the adiabatic temperature-rise model in engineering. However, the formula has only one variable parameter; therefore, its accuracy may be reduced when calculating the heat of hydration of different types of concrete. The formula is expressed as follows:  
\[Q(t)=Q_{0}(1-e^{mt}) \tag{1}\]  
where \(Q(t)\) is the total amount of heat of hydration released when the age of the concrete is \(t\) (kJ/kg); \(Q_{0}\) is the maximum value of concrete hydration heat release; \(t\) is the age of the concrete; and \(m\) is the specific calculation parameter, which is specific for the type of cement. 2. Hyperbolic model  
The hyperbolic model proposed by [PERSON] [13] introduces the concept of half-age when the concrete hydration exotherm is half so that the model is closer to reality. However, before applying the model for any specific application, it is necessary to know in advance the age the concrete will be when it reaches half of the total amount of exotherm. Thus, there are some limitations to the application of this formula. The specific model calculation formula is expressed as follows:  
\[Q(t)=\frac{Q_{0}t}{n+t} \tag{2}\]  
where \(n\) is the age at which half of the total exothermic heat is reached; corresponding to the setting of \(n\), the values of other parameters are determined according to the following expressions: when \(t=0\), \(Q(t)=0\); when \(t=n\), \(Q(t)=Q_{0}/2\); and when \(t=\infty\), \(Q(t)=Q_{0}\). 3. Composite exponential model  
The composite exponential model was proposed by [PERSON] [14], an academic at the Academy of Engineering, China. The formula is optimized for the single exponential model and two parameters are introduced to the age \(t\) for calculation, which greatly improves the accuracy of the model and can be applied to various types of high-performance concrete. The specific model calculation formula is expressed as follows:  
\[Q(t)=Q_{0}\Big{(}1-e^{-at^{b}}\Big{)} \tag{3}\]  
where \(a\) and \(b\) are the model influence coefficients related to the type of cementitious material involved in the hydration reaction. The specific values are listed in Table 1.  
4. Selection of hydration exothermic models  
From the foregoing discussion on the three hydration exothermic models, it is evident that the composite exponential model is the best hydration exothermic model with the best calculation accuracy and applicability. Therefore, the subsequent calculations are based on the composite exponential model, which has three main parameters, i.e., \(Q_{0}\), \(a\), and \(b\). Among them, the availability of information on the influence of coefficients \(a\) and \(b\) is relatively low, especially for different multi-mixed additions of concrete, and there is a lack of relevant experimental support data as well. For the \(Q_{0}\) of different additions, the value \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Cement Varieties** & \(Q_{0}\) & \(a\) & \(b\) \\ \hline CEM I 42.5 CHN & 330 & 0.69 & 0.56 \\ CEM I 52.5 CHN & 350 & 0.36 & 0.74 \\ CEM II/A 42.5 CHN & 270 & 0.79 & 0.70 \\ CEM II/A 52.5 CHN & 285 & 0.29 & 0.76 \\ \hline \hline \end{tabular} Note: The values in the table refer to research by [PERSON] [14].  
\end{table} 
Table 1: Composite exponential model coefficients corresponding to different cement grades.