## 3 Results and Discussion
### Prediction of HRR  
In general, flame height is closely relevant to the air entrainment, i.e., the flame should extend to a certain height where the total amount of air engulfed along this height is sufficient to fulfill the combustion process [30]. Therefore, as expounded in Section 3.1, the flame height should be intrinsically related to the pyrolysis rate of the combustible. Conventionally, their relationship is summarized as:  
\[z_{f}=0.235\dot{Q}^{2/5}-1.02D \tag{1}\]  
where \(D\) is the hydraulic diameter of the burner equivalent to \(\frac{2}{\sqrt{\pi}}L\)[34]. Here, As the flame height is concretely plotted in Figure 6, Equation (1) can be used to solve for \(\dot{Q}\), i.e.,:  
\[\dot{Q}=\left(\frac{z_{f}+1.02D}{0.235}\right)^{5/2} \tag{2}\]  
The calculated HRRs together with corresponding HRRs measured by the cone calorimeter are plotted in Figure 7. It clearly shows that the prediction model derived from Equation (1) can well predict the general variation trend of HRR for the three NC samples, while the HRR values tend to be underestimated, especially when the HRRs are relatively larger. The average values in quasi-steady burning stages predicted by Equation (2) for NC-I are 18.8% and 18.6% lower than the measured HRR values for sides of 8 cm and 10 cm, respectively, while that for NC-E are 16.1% and 13.0%. Figure 7c further demonstrates the feasibility of Equation (2) in predicting the unsteady combustion process. However, the accuracy of the prediction model needs to be improved in consideration of the discrepancy shown in Figure 7.  
Figure 6: Flame heights for all the configurations.  
Figure 7: _Cont._In effect, the coefficient 0.235 in Equation (1) is a typical empirical value applicable for a larger number of gaseous and liquid fuels under normal atmospheric conditions. Its feasibility for NC samples needs to be further clarified. Equation (1) is originally expressed as the following form [40]:  
\[\frac{z_{f}}{D}=-1.02+15.6N^{\frac{1}{5}} \tag{3}\]  
with the dimensionless parameter \(N\) defined as:  
\[N=\left[\frac{c_{p}T_{\infty}}{g\rho_{\infty}^{2}(H_{c}/r)^{3}}\right]\frac{ \dot{Q}^{2}}{D^{5}} \tag{4}\]  
Apart from the environmental parameters, i.e., the specific heat of air \(c_{p}\), ambient temperature \(T_{\infty}\), gravitational acceleration \(g\) and air density \(\rho_{\infty}\), the actual heat of combustion \(H_{c}\) and mass stoichiometric ratio of air to volatiles \(r\) substantially depend on the physical properties of combustibles. Based on Equations (3) and (4), the HRR can be recalculated as:  
\[\dot{Q}=A\Big{(}z_{f}+1.02D\Big{)}^{5/2}(H_{c}/r)^{3/2} \tag{5}\]  
where \(A=\Big{(}\frac{8\rho_{\infty}^{2}}{15.6^{c_{p}}T_{\infty}}\Big{)}^{1/2}\) is constant for the fixed ambient environment. When \(H_{c}/r\) remains within the range of 2900-3200 kJ/kg, the coefficient 0.235 might be adequate for the rough estimation of HRR [36; 37]. However, to precisely predict the HRRs, \(H_{c}/r\) has to be reconsidered for current NC samples due to its 3/2 power law relationship with HRR.  
Figure 7: Comparisons of the predicted and measured HRRs: (**a**) NC-I; (**b**) NC-E; and (**c**) NC-D.