## 3 Results and Discussion
### Prediction of HRR
#### 3.3.2 Determination of \(r\)  
The chemical formula of NC can be written as C\({}_{6}\)H\({}_{7}\)O\({}_{2}\)(OH)\({}_{3}\)(ONO\({}_{2}\))\({}_{y}\), where \(x+y=3\). The customary way to define the composition of NC is to express the nitrogen content as a percentage by weight, i.e., the nitrogen content mentioned before, which is termed as \(n\)[4]. The average number of nitrate groups present in NC can be calculated as:  
\[y=\frac{162n}{1400-45n} \tag{6}\]  
Thus, for NC-I, NC-E, and NC-D samples, their NC substrates can be further specified as C\({}_{6}\)H\({}_{7}\)O\({}_{2}\)(OH)\({}_{0.74}\)(ONO\({}_{2}\))\({}_{2.26}\), C\({}_{6}\)H\({}_{7}\)O\({}_{2}\)(OH)\({}_{0.78}\)(ONO\({}_{2}\))\({}_{2.22}\), C\({}_{6}\)H\({}_{7}\)O\({}_{2}\)(OH)\({}_{0.75}\)(ONO\({}_{2}\))\({}_{2.25}\), respectively. In general, their difference in nitrate group is quite small because of their slight difference in nitrogen content. Assuming all the combustion yields are in the most stable forms, i.e., complete combustion, the chemical reactions for NC and their corresponding humectants and plasticizer can be written as follows:  
for NC-I:  
\[\left\{ \begin{array}{c}\mathrm{C_{6}H_{7}O_{2}(OH)}_{0.74}(\mathrm{ONO_{2} })_{2.26}+5.435\mathrm{O_{2}}\to 6\mathrm{CO_{2}}+3.87\mathrm{H_{2}O}+2.26 \mathrm{NO_{2}}\\ \mathrm{C_{3}H_{8}O}+4.5\mathrm{O_{2}}\to 3\mathrm{CO_{2}}+4\mathrm{H_{2}O} \end{array} \right. \tag{7}\]  
for NC-E:  
\[\left\{ \begin{array}{c}\mathrm{C_{6}H_{7}O_{2}(OH)}_{0.78}(\mathrm{ONO_{2} })_{2.22}+5.445\mathrm{O_{2}}\to 6\mathrm{CO_{2}}+3.89\mathrm{H_{2}O}+2.22 \mathrm{NO_{2}}\\ \mathrm{C_{2}H_{6}O}+3\mathrm{O_{2}}\to 2\mathrm{CO_{2}}+3\mathrm{H_{2}O} \end{array} \right. \tag{8}\]  
for NC-D:  
\[\left\{ \begin{array}{c}\mathrm{C_{6}H_{7}O_{2}(OH)}_{0.75}(\mathrm{ONO_{2} })_{2.25}+5.4375\mathrm{O_{2}}\to 6\mathrm{CO_{2}}+3.875\mathrm{H_{2}O}+2.25 \mathrm{NO_{2}}\\ \mathrm{C_{16}H_{22}O_{4}}+19.5\mathrm{O_{2}}\to 16\mathrm{CO_{2}}+11\mathrm{H_{2}O} \end{array} \right. \tag{9}\]  
According to the definition of \(r\), it can be determined as:  
\[r_{fuel}=4.76\mathrm{s}\frac{M_{air}}{M_{fuel}} \tag{10}\]  
where \(\mathrm{s}\) is the stoichiometric mole ratio of oxygen to fuel, and \(M_{air}\) and \(M_{fuel}\) are the molecular weights of air and fuel, respectively. As the weight percentage of addition agent (denoted as \(p\)) is given in current study, the average \(r\) value for NC-humectant or NC-plasticizer mixtures can be reasonably expressed as:  
\[\left\{ \begin{array}{c}\overline{r_{NC-I}}=pr_{Isopropanol}+(1-p)r_{NC-I} \\ \overline{r_{NC-E}}=pr_{Ethanol}+(1-p)r_{NC-E}\\ \overline{r_{NC-D}}=pr_{DBP}+(1-p)r_{NC-D}\end{array} \right. \tag{11}\]  
Based on the equation above, the average \(r\) values for NC-I, NC-E, and NC-D are estimated as 5.097, 4.709, and 4.217, respectively. Combined with the actual heat of combustion derived by Figure 8, \(H_{c}/r\) can be correspondingly modified as 3251, 3228, and 3201 kJ/kg. Substituting these values in Equation (5), the model based on the flame height will be further modified to give the predicted HRR, as plotted in Figure 9. It indicates that for all three samples, the modified and measured HRRs intertwine with each other for the two tested pool dimensions, implying the modified model would help to predict the HRR more accurately. However, because of the violent oscillation of flames of NC samples shown in Figure 5, the evident fluctuation of HRR derived from flame height can be observed, exhibiting serration patterns. The predicted HRRs in quasi-steady burning stage are in the vicinity of the measured ones for NC-Iand NC-E samples. Specifically, for NC-I, the average values for the predicted values are slightly higher than that for the tested results by about 1.8% and 1.7% for \(L\) = 8 cm and 10 cm configurations, respectively, while that for NC-E are 1.3% and 1.1%. It is interesting to note that the predicted HRR by Equation (5) denotes the total HRR, which is reasonably higher than the measured HRR by ISO 5660.