## 3 Prediction of the Pool Shape
### Empirical Formulae Predicting the Spreading Extent of Spilled Pools on Inclined Surfaces  
Empirical formulae for predicting the pool shape that forms upon the spillage of JP-5 on an inclined steel plate were developed based on the experimental results. The pool shape was predicted under different inclination angles, spill rates, and spillage duration. To derive the formulae for predicting these variables, the effect of each variable on the pool shape was evaluated. The correlation between the pool length, substrate inclination, and discharge rate is shown in Figure 7. This relationship was used to formulate the effects of the substrate inclination and discharge rate on the pool length for each time interval, which can be found in Figure 6. Three coefficients, \(X_{1}\), \(X_{2}\), and \(X_{3}\), were derived using the experimental results. \(X_{1}\) and \(X_{2}\) represent the coefficients of the inclination angle and spill rate terms, respectively. \(X_{3}\) represents the effect of time without the influence of the inclination angle and spill rate. These coefficients were fitted with respect to the spillage duration. The length \(L\) (m), breadth \(B\) (m), and area \(A\) (m\({}^{2}\)) of the spilled pool can be empirically predicted using the inclination angle \(\theta\) (\({}^{\circ}\)), spill rate \(\dot{V}\) (mL/min), and spillage duration \(t\) (s):  
\[L\left(\theta,\dot{V},t\right)=X_{1}\theta+X_{2}\dot{V}+X_{3} \tag{1}\]  
where \(X_{1}=8.557t\), \(X_{2}=0.0914t\), \(X_{3}=-10.48t\). \[B\left(\theta,\dot{V},t\right)=Y_{1}\theta+Y_{2}\dot{V}+Y_{3} \tag{2}\]  
where \(Y_{1}=-0.4110t\), \(Y_{2}=-0.4184e^{-0.1569t}+0.4133\), \(Y_{3}=6.183t\). \[A\left(\theta,\dot{V},t\right)=Z_{1}\theta+Z_{2}\dot{V}+Z_{3} \tag{3}\]  
where \(Z_{1}=634.0t\), \(Z_{2}=24.79t\), \(Z_{3}=-2576t\). The HRR of a pool fire can be predicted using an equation which correlates the HRR with the pool dimensions, thermochemical constants, and empirical constants of liquid fuels [30]. The formula for pool area prediction is substituted into this equation to estimate the HRR on inclined surfaces with improved accuracy. The following equation is the proposed equation for HRR prediction for inclined conditions:  
\[\dot{q}=\Delta l_{c}\dot{m}_{\infty}^{\prime\prime}\left(1-e^{-k\beta D} \right)\times\left(Z_{1}\theta+Z_{2}\dot{V}+Z_{3}\right) \tag{4}\]  
where \(\dot{q}\), \(\Delta l_{c}\), \(\dot{m}_{\infty}^{\prime\prime}\), \(k\beta\), \(D\), and \(A\) represent the HRR, heat of combustion, mass burning rate per unit area, the product of the extinction absorption coefficient \(k\) and the beam-length corrector \(\beta\), equivalent pool diameter, and surface area of the pool fire, respectively. The range of the substrate inclination angle and discharge rate at which the formulae can be reliably used is limited by the experimental scenarios, as the formulae above were developed based on the experimental results. The pool lengths obtained using the proposed empirical formula were compared with the experimentally measured data in Figure 8a. The predicted and measured values at the final state (\(t_{m}/t_{d}=1\)) are shown. As no study listed in Table 1 predicted the pool length and breadth as defined in this study, no inter-study comparisons are provided. The pool length predicted using the proposed formula agreed well with the experimental values, and the maximum error was within 10% when excluding predictions with low area values. The mean and coefficient of variance (COV) were 1.048 and 0.140, respectively. This result was expected, as the proposed formulae were developed using the experimental results. Figure 8b compares the predicted and experimentally measured pool breadths. The predicted values were obtained using the proposed formula. The accuracy of the proposed formula in predicting the breadth was relatively low compared with the results shown in Figure 8a, attributable to the nonlinearity of the pool breadth with time displayed in Figure 6b.