# Experimental Comparison of Diesel and Crude Rapeseed Oil Combustion in a Swirl Burner
## 2 Materials and Methods
### The Continuous Wavelet Transform  
The temporal pressure signal of Case 1 is shown in Figure 4a, where blowout occurs at 1 s. To gather spectral data, Fourier analysis is commonly used. Figure 4b shows the spectrogram of the above signal that consists of \(2^{14}\) samples at 12 kHz sampling frequency, i.e., 1.37 s. The previously mentioned FFT is excellent for steady operation, however, this transient state cannot be characterized well by this technique. Hence, the signal is cut into shorter parts, and FFT is performed on all of them, which is the Short-Time Fourier transform (STFT). The best resolution was achieved at a window size of 512 samples and 50% overlapping, which is still not ideal for identifying the governing dynamics that precede flame blowout. The continuous wavelet transform (CWT) of the raw pressure signal is shown in Figure 4c by using a Morse wavelet (time-bandwidth product parameter: 60 and symmetry parameter: 3). The white dashed line sets a boundary for the cone of influence under which the edge effect might excessively bias the result, i.e., this region should be omitted from the evaluation. While the STFT is characterized by a homogeneous time-to-frequency window aspect ratio, the CWT provides poor time and fine frequency resolution at low frequencies, which ends up with the opposite in the high-frequency regime. The time-to-frequency aspect ratio continuously decreases by a factor of 2 in each step due to the high-pass filtering nature of the CWT [57]. Since frequencies well below 100 Hz are typically out of interest [58], the poor time resolution at lower frequencies does not affect the analysis. At a few hundred Hz, the temporal resolution reveals the amplitude fluctuations in the time domain, while the frequency resolution is ~20 Hz. Since combustion is a broadband phenomenon [59], accurate resolution in the frequency domain is not mandatory for control logic. The high temporal resolution also contributes to the understanding of the evolution of the frequency peaks. This phenomenon is facilitated by the variation in the speed of sound through the heat release oscillation. Four frequencies are emphasized in the present study, following the representative results of Figure 4. They are \(f_{1}=230\) Hz,\(f_{2}=500\) Hz,\(f_{3}=1600\) Hz, and \(f_{4}=3200\) Hz. The first two are associated with the recirculation zones in V-shaped flames, while the latter two dominate the straight flames. The peak near 200 Hz is typically in the focus of blowout and flame transfer function investigations of V-shaped flames [60; 61; 62]. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(p_{g}\) [Bar] & \(SMD\) [\(\mu\)m] & **We [–]** & \(AFR\) [–] \\ \hline
0.3 & 27.1 & 779 & 0.778 \\
0.45 & 22.0 & 1121 & 0.948 \\
0.6 & 19.3 & 1438 & 1.09 \\
0.8 & 17.1 & 1830 & 1.25 \\
1.1 & 15.1 & 2363 & 1.45 \\
1.6 & 13.3 & 3143 & 1.73 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Estimated properties of diesel oil atomization at various \(p_{g}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(p_{g}\) [Bar] & \(SMD\) [\(\mu\)m] & **We [–]** & **AFR [–]** \\ \hline
0.3 & 22.8 & 908 & 0.681 \\
0.45 & 18.0 & 1307 & 0.829 \\
0.6 & 15.5 & 1678 & 0.953 \\
0.8 & 13.5 & 2135 & 1.09 \\
1.1 & 11.7 & 2757 & 1.27 \\
1.6 & 10.1 & 3666 & 1.51 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Estimated properties of crude rapeseed oil atomization at various \(p_{g}\). To address the uneven behavior of the temporal series of CWT magnitudes in the scalogram, it is a possibility to filter the results to a single frequency, then perform a further transform on it. This procedure in FFT is well-known in the field of bearing diagnostics to find nonlinear signal sources, e.g., cracks in the rings or a damaged ball. This step removes the harmonic components, hence the non-periodic terms will remain. The scalogram of the CWT coefficients at \(f_{1}\) is shown in Figure 5, showing no apparent deterministic behavior. The effective frequency axis is practically limited to 115 Hz, according to the Nyquist-Shannon sampling theorem.