## 3 Results and Analysis
### Coupling Effects of Pressure and Inclination on Upward Flame Spread  
As the key parameter in flame spread, the FSR is mainly determined by preheating in the preheated zone as \(V_{p}\propto(\vec{q^{\prime\prime}}_{p})^{2}\cdot\delta_{p}/(T_{ig}-T_{ \infty})^{2}\) for thermally thick solid [21]. It means that the FSR is evidently determined by preheating intensity and ignition temperature. According to fundamental theory, the heat balance relationship of preheating in the upward flame spread process could be simplified by the following equation,  
\[\dot{q}^{\prime\prime}_{p}\big{|}_{x_{p}<x_{p}+\delta_{p}}=\dot{q}^{\prime\prime }_{f,rad}+\dot{q}^{\prime\prime}_{f,conv}+\dot{q}^{\prime\prime}_{FPU,cond} \tag{4}\]  
where the conduction heat flux component \(\dot{q}^{\prime\prime}_{FPU,cond}\) is negligible, attributed to the good thermal insulation performance of the FPU. Besides, the proved correlation of \(\dot{q}^{\prime\prime}_{f,rad}\) and \(\dot{q}^{\prime\prime}_{f,conv}\) with pressure are [9],  
\[\left\{ \begin{array}{c}\dot{q}^{\prime\prime}_{f,rad}\sim p\\ \dot{q}^{\prime\prime}_{f,conv}\sim p^{1/2}\end{array} \right. \tag{5}\]  
Considering the scale of the burning area (although larger than that in the downward flame spread for the same FPU board), convection heat flux still tends to be stronger than radiation for both preheated and pyrolysis zones for the board size selected [22]. Consequently, heat flux to the preheated zone could be approximately,  
\[\dot{q}^{\prime\prime}_{p}=\dot{q}^{\prime\prime}_{f,rad}+\dot{q}^{\prime \prime}_{f,conv}\sim p^{\alpha},\ 1/2<\alpha\ll 1 \tag{6}\]  
where \(\alpha\) is the apparent index of pressure effect. As mentioned in Equation (6), \(\alpha\) would be closer to 1/2, yet, the competition of convection and radiation heat fluxes will become tricky under certain inclination conditions, e.g.,  
1. For horizontal (0\({}^{\circ}\)) flame spread, the preheated zone is definitely under convection control by the classical theory described above. Likewise, the preheated zone is almost immersed in flame for vertical (90\({}^{\circ}\)) spread; the convection between hot fire plume and fuel surface should also dominate the preheating mechanism (similar to pyrolysis zone). In these periods, \(\alpha\) is believed to approach 1/2. 2. However, for upward spread with certain inclination, radiation will be enhanced obviously due to stretched flame length and increasing radiation configuration factor owing to flame tilt, which might promote the ratio of radiation in preheating to make \(\alpha\) larger subsequently. On the other hand, influences of inclination angle on preheating in upward spread was deduced by [PERSON] [14] as,  
\[\dot{q}^{\prime\prime}_{p}\propto(\sin\theta)^{2/5} \tag{7}\]  
Combining the pressure effect given in Equation (6), preheating heat flux could be expressed as,  
\[\dot{q}^{\prime\prime}_{p}=C\dot{p}^{\alpha}(\sin\theta)^{2/5} \tag{8}\]where \(C\) is the pre-exponential coefficient by empirical correlation, which is independent of pressure or inclination. Therefore, coupling effects on the FSR could be integrated as follows,  
\[V_{p}\approx(\dot{q^{\prime\prime}}_{p})^{2}\frac{4\delta_{p}}{\pi k\rho_{FPU}c_{ p}(T_{ig}-T_{\infty})^{2}}\sim(\dot{q^{\prime\prime}}_{p})^{2}\cdot\delta_{p} \sim p^{2\alpha}\cdot(\sin\theta)^{4/5}\cdot\delta_{p} \tag{9}\]  
In fact, Equation (9) is proposed to be a theoretical approximation, which is not strict for small inclinations (or with significant deviation since \(\sin\theta\overset{\theta\to 0}{\to}0\)). Fortunately, this mathematical limitation is not necessary to be concerned about under large inclination condition.