# Analysis of Flame Evolution Generated from Methyl Laurate Droplet Using Deep Learning
## 3 Results and Discussion
### Flame Volume and Surface  
Flame volume is a critical parameter in characterizing combustion, particularly in studying the evolution of flames generated from droplet combustion. Measuring flame volume provides insights into the intensity and energy distribution released during the combustion process, which is essential for understanding heat and mass transfer mechanisms in combustion environments. One of the commonly used approaches is the frustum-based geometric method, which assumes that the flame has a truncated cone shape with varying radii along its length. This method has been widely applied in combustion studies, including the jet fire model, where the flame is idealized as a frustum of a cone to predict geometric characteristics and volume, as well as to model thermal radiation hazards in natural gas jet fires [68]. Furthermore, 3D point cloud techniques have been employed to reconstruct flame volume based on spatial surface data, enabling more accurate volume estimation under various combustion conditions [69]. In this study, the flame volume generated by the combustion of methyl laurate droplets is analyzed using the frustum-based geometric method. This approach integrates machine vision and machine learning techniques to reconstruct flame structures with high accuracy. The use of artificial neural networks has been also explored to extract features for flame structure recognition during lean blowout [70], and other researchers have employed a residual network with attention mechanisms to analyze self-luminous flame images in supersonic combustion environments [71]. The surface area of the flame can be calculated using a geometric frustum-based approach, which models the flame as a series of truncated cones formed between adjacent radii along the centerline. The total surface area is computed as the sum of the lateral surface areas of these frustums. The lateral surface area of each frustum is given as follows:  
\[A_{i}=\pi(r_{i}+r_{i+1})l_{i} \tag{4}\]where \(r_{i}\) and \(r_{i+1}\) represent the radii at two consecutive points along the flame centerline, and \(l_{i}\) is the distance between these points, calculated as follows:  
\[l_{i}=\sqrt{\left(y_{i+1}-y_{i}\right)^{2}+\left(x_{i+1}-x_{i}\right)^{2}} \tag{5}\]  
The total surface area \(A\) is then expressed as follows:  
\[A=\sum_{i=1}^{n-1}A_{i} \tag{6}\]  
The flame volume is computed using the frustum-based approach, particularly when the flame radius varies significantly along the centerline. The equations utilize a Cartesian coordinate system to represent the positions of points along the flame centerline, where \(y_{i}\) and \(y_{i+1}\) denote vertical coordinates and \(x_{i}\) and \(x_{i+1}\) represent horizontal coordinates. The distance between consecutive points (\(l_{i}\)) is calculated using the Pythagorean theorem in 2D space. In this method, the total volume is approximated as the sum of frustums formed by small segments along the flame centerline. The volume is given by the following equation:  
\[V=\sum_{i=1}^{n-1}\frac{1}{3}\pi\Big{(}r_{i}^{2}+r_{i}r_{i+1}+r_{i+1}^{2} \Big{)}\Delta\text{y} \tag{7}\]  
where \(r_{i}\) and \(r_{i+1}\) are radii at points \(i\) and \(i+1\), respectively; \(\Delta\)y is the distance between these points along the centerline; and \(n\) denotes the total number of points dividing the flame centerline for volume approximation. This approach provides volume estimation when the radius exhibits significant variation along the flame length. By employing the above method, the overall geometry of the flame can be reconstructed, providing a three-dimensional representation of the flame structure over time.