# Assessment of Lightweight Concrete Thermal Properties at Elevated Temperatures
## 4 Proposed Thermal Properties Relationships
### Specific Heat Equations  
Data generated from the specific heat measurements are used to develop a specific heat relationship for LWC. This thermal property is expressed in the form of an empirical relationship in the temperature range of 20-1000 \({}^{\circ}\)C, as shown in Figure 10a. The accuracy of the statistical model is represented by the coefficient of determination, R\({}^{2}\), whose value lies between 0.9 and 0.97.  
Specific heat is mainly influenced by the temperature range. To capture this trend, several equations are proposed. These relationships are presented in Equation (11) through Equation (19) in J/kg\({}^{\circ}\)C.  
\[c_{P}(T) =950 \tag{11}\] \[c_{P}(T) =0.0018\Big{[}\frac{I}{\text{kg}^{\circ}\text{C}}\Big{]}\times T^ {3}-0.9352\Big{[}\frac{I}{\text{kg}^{\circ}\text{C}}\Big{]}\times T^{2}+157.16 \Big{[}\frac{I}{\text{kg}^{\circ}\text{C}}\Big{]}\times T-7025.4\big{[}\frac{I} {\text{kg}^{\circ}\text{C}}\Big{]}\ \big{(}R^{2}=0.96\big{)} 100<T\leq 200\ ^{\circ}\text{C}\] (12) \[c_{P}(T) =1250\] \[c_{P}(T) =-0.0069\Big{[}\frac{I}{\text{kg}^{\circ}\text{C}}\Big{]}\times T^ {3}+9.8164\Big{[}\frac{I}{\text{kg}^{\circ}\text{C}}\Big{]}\times T^{2}-4590.3 \Big{[}\frac{I}{\text{kg}^{\circ}\text{C}}\Big{]}\times T+709,200\big{[}\frac{I }{\text{kg}^{\circ}\text{C}}\Big{]}\ \big{(}R^{2}=0.97\big{)} 430<T\leq 450\ ^{\circ}\text{C}\] (14) \[c_{P}(T) =1.3695\Big{[}\frac{I}{\text{kg}^{\circ}\text{C}}\Big{]}\times T^ {2}-1341.5\Big{[}\frac{I}{\text{kg}^{\circ}\text{C}}\Big{]}\times T+329,744 \big{[}\frac{I}{\text{kg}^{\circ}\text{C}}\Big{]}\ \big{(}R^{2}=0.97\big{)} 450<T\leq 500\ ^{\circ}\text{C}\] (15) \[c_{P}(T) =1250\]  
Figure 10: Specific heat: (**a**) comparison of the experimental and proposed specific heat equations; (**b**) comparison of proposed equations, simplified equation, and Eurocodes 2 and 4 values.  
\[c_{p}(T) =0.5463\cdot e^{0.0117\times T}\left(R^{2}=0.95\right) 650<T\leq 850\ ^{\circ}\mathrm{C} \tag{17}\] \[c_{p}(T) =-208.51\left[\frac{I}{\mathrm{kg}^{\circ}\mathrm{C}^{2}}\right] \times T+185,723[\frac{I}{\mathrm{kg}^{\circ}\mathrm{C}}]\ \left(R^{2}=0.91\right) 850<T<900\ ^{\circ}\mathrm{C}\] (18) \[c_{p}(T) =850 900\leq T\leq 1000\ ^{\circ}\mathrm{C} \tag{19}\]  
The current Eurocodes 2 and 4 for evaluating the fire resistance of concrete elements assume that specific heat for normal concrete can be simplified with an equation as a function of temperature. However, for LWC, this assumption has numerous drawbacks. For LWC, Eurocodes 2 and 4 propose a constant value (840 J/kg\({}^{\circ}\)C). In this research work, to fit a specific heat equation to LWC, the following equation is presented for the temperature range of 20-700 \({}^{\circ}\)C:  
\[c_{p}(T)=3\cdot 10^{-6}\left[\frac{J}{\mathrm{kg}^{\circ}\mathrm{C}^{4}} \right]\times T^{3}-0.005\left[\frac{J}{\mathrm{kg}^{\circ}\mathrm{C}^{3}} \right]\times T^{2}+2.7\left[\frac{J}{\mathrm{kg}^{\circ}\mathrm{C}^{2}} \right]\times T+786[\frac{J}{\mathrm{kg}^{\circ}\mathrm{C}}] \tag{20}\]  
where \(c_{p}(T)\) is the specific heat of LWC as a function of temperature and \(T\) is the temperature in \({}^{\circ}\)C.  
Figure 9(b) shows the comparison between the fitted equation, the proposed equation, and the Eurocodes. The Eurocodes underestimate the specific heat of LWC. However, the trendline for proposed values from the current study is close to the line depicting results without taking into account the peaks caused by the vaporization of free water, the dissociation of calcium hydroxide, and the dissociation or decomposition of expanded clay.