Please note this is a comparison between Version 3 by Jason Zhu and Version 2 by Jason Zhu.

The temperature field and chemiluminescence measurements of axisymmetric flame are obtained simultaneously in only one image. Digital Laser Speckle Displacement measures temperature fields, and direct image flame determines chemiluminescence values. Applying the Abel transform of axisymmetric objects for volume visualization requires smooth intensity profiles. Due to the nature of the experimental setup, direct image flame is corrupted with speckle noise and a crosstalk effect. These undesirable effects deteriorate the measurement results. Then, experimental data need crosstalk correction and speckle noise reduction to improve the measurements.

- chemiluminescence
- optics
- fluid flow
- premixed flames

Studying a combustion process is a very complex task ^{[1]}. However, the characterization of a flame is essential to improve the combustion process’s efficiency and reduce pollutant emissions. Measuring several variables in a flame is used to monitor the efficiency of combustion processes. Generally, some measured variables are flame temperature, soot concentration, radical intensities, flame size, luminosity, and flickering ^{[2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]}. Full-field optical techniques are usually used to make this kind of measurement because they give a global description of the process and do not disturb the sample. On the other hand, simultaneously measuring several variables is preferable because this avoids the lack of correlation between them ^{[9][16][17][18][19]}. However, a complex optical setup is usually employed for measurement in such cases, making the task costly and challenging ^{[16][18]}.

The flame equivalence ratio (𝛷) is an adimensional variable used to determine if a fuel-oxidizer mixture is rich, lean, or stoichiometric ^{[1]}. When 𝛷>1, it denotes a fuel-rich mixing process; on the other hand, 𝛷<1 is related to fuel-lean mixtures, and the last 𝛷=1 in the ratio relationship is the value for stoichiometric mixing. This relation is intrinsically linked to determining a system’s performance. This study, as in others, corroborates that the maximum temperature value is obtained at a value of 𝛷=1.05 for LPG fuel. Other variables that can be connected directly to the flame equivalence ratio are the temperature and radical intensities (chemiluminescence) ^{[10][11][12][13][14][15][16][17][18][19]}. In a combustion process, some released species are OH, CH, CH_{2}O, and C_{2,} which have a specific emission spectrum ^{[10][11][12][13][14][15][16][17][18][19]}. In these combustion flames, the presence of radicals C∗2 and CH∗ are inherent in the reaction zone. These two chemical species are most abundant within the flame. Furthermore, these radicals emit in the visible region of the electromagnetic spectrum, centered at wavelengths 430 nm and 525 nm, respectively ^{[14]}. For this reason, digital color cameras play an essential role in chemiluminescence measurements ^{[11][12][13][14][15][16][17][18]}. In this analysis, each color channel contributes specific information about the object under study. However, most digital color cameras suffer from the crosstalk effect; this occurs when sensors have overlapping sensibilities, contaminating the recorded data and needing correction ^{[15][17]}.

There are few works in which chemiluminescence and flame temperature are measured ^{[3][16][17][18]}. In one of the works, the optical system is simple; they only use a camera, and the temperature is measured using two-color pyrometry. However, temperature measurement is applicable only for sooty flames ^{[3]}. In other research, two-color pyrometry is also used to determine the flame’s temperature; however, the optical arrangement is complex ^{[16][18]}. In ^{[17]}, the optical configuration is simple to implement, and Digital Laser Speckle Displacement (DLSD) is used to determine the temperature fields. However, the chemiluminescence measurements are contaminated with speckle noise, making determining these values difficult.

$$\rho \left(x,z\left(\right)=2{\int}_{x{\mathrm{\infty}{}_{}^{}\frac{I\left(r,z\left(\right)rdr\frac{\sqrt{{r\sqrt{{2\sqrt{{}^{}-{x\sqrt{{2\sqrt{{}^{}}\frac{}{}}^{}}}^{}}}^{}}}^{}}}{}\right)}{}}_{}^{}}^{}\right)$$

$$I\left(r,z\left(\right)=\frac{1\frac{\pi \frac{}{}{\int}_{r{\mathrm{\infty}{}_{}^{}\frac{\partial \rho \left(x,z\left(\right)\frac{\partial x\frac{}{}\frac{dx\frac{\sqrt{{x\sqrt{{2\sqrt{{}^{}-{r\sqrt{{2\sqrt{{}^{}}\frac{}{}}^{}}}^{}}}^{}}}^{}}}{}}{}}{}\right)}{}}_{}^{}}^{}}{}}{}\right)$$

$${\rho {(x,z){{S{rgb{}_{}{}_{}^{}={{\rho {(x,z){{O{rgb{}_{}{-\rho {(x,z){{R{rgb{}_{}{}_{}^{}}_{}}_{}}_{}^{}}_{}^{}}_{}^{}}_{}}_{}}_{}^{}}_{}^{}}_{}^{}}_{}^{}}_{}}_{}}_{}^{}}_{}^{}}_{}^{}$$

$${\rho {(x,z){{S{c{}_{}{}_{}^{}\left\{\begin{array}{c}N{\rho {(x,z){{S{c{}_{}{}_{}^{}\le t\begin{array}{c}\\ F{\rho {(x,z){{S{c{}_{}{}_{}^{}t\begin{array}{c}\end{array}\left\{\right.}_{}}_{}}_{}^{}}_{}^{}}_{}^{}\end{array}}_{}}_{}}_{}^{}}_{}^{}}_{}^{}\end{array}\right.}_{}}_{}}_{}^{}}_{}^{}}_{}^{}$$

$${\nabla {2{}^{}{\rho {(i,j){n{}_{}^{}=\frac{{\rho {i+1,j{n{}_{}^{}+{\rho {i-1,j{n{}_{}^{}+{\rho {i,j+1{n{}_{}^{}{+\rho {i,j+1{n{}_{}^{}-4{\rho {i,j{n{}_{}^{}\frac{{h{2{}^{}\frac{}{}}^{}}^{}}{}}_{}^{}}_{}^{}}_{}^{}}_{}^{}}_{}^{}}_{}^{}}_{}^{}}_{}^{}}_{}^{}}_{}^{}}_{}^{}}_{}^{}}_{}^{}}_{}^{}}_{}^{}}{}}_{}^{}}_{}^{}}_{}^{}}^{}}^{}$$

$${\rho {i,j{n+1{}_{}^{}={\rho {i,j{n{}_{}^{}-\omega {\nabla {2{}^{}{\rho {\left(i,j\left(\right){n{}_{}^{}}_{}^{}\right)}_{}^{}}_{}^{}}^{}}^{}}_{}^{}}_{}^{}}_{}^{}}_{}^{}}_{}^{}}_{}^{}$$

$$\rho \left(x,z\left(\right)=\sum _{i=1\underset{}{\overset{}{k\underset{}{\overset{}{}}{w{i{}_{}{f{i{}_{}(x,z)}_{}}_{}}_{}}_{}}}}^{}\right)$$

$$\left(\begin{array}{c}\begin{array}{c}{f\begin{array}{c}\begin{array}{c}{1\begin{array}{c}\begin{array}{c}{}_{}({x\begin{array}{c}\begin{array}{c}{1\begin{array}{c}\begin{array}{c}{}_{},z)\begin{array}{c}{f\begin{array}{c}\begin{array}{c}{2\begin{array}{c}\begin{array}{c}{}_{}({x\begin{array}{c}\begin{array}{c}{1\begin{array}{c}\begin{array}{c}{}_{},z)\begin{array}{c}\begin{array}{c}{f\begin{array}{c}\begin{array}{c}{1\begin{array}{c}\begin{array}{c}{}_{}({x\begin{array}{c}\begin{array}{c}{2\begin{array}{c}\begin{array}{c}{}_{},z)\begin{array}{c}{f\begin{array}{c}\begin{array}{c}{2\begin{array}{c}\begin{array}{c}{}_{}({x\begin{array}{c}\begin{array}{c}{2\begin{array}{c}\begin{array}{c}{}_{},z)\begin{array}{c}\begin{array}{cc}& \begin{array}{c}\begin{array}{c}\dots \begin{array}{c}{f\begin{array}{c}\begin{array}{c}\begin{array}{c}{n\begin{array}{c}\begin{array}{c}\begin{array}{c}{}_{}({x\begin{array}{c}\begin{array}{c}\begin{array}{c}{1\begin{array}{c}\begin{array}{c}\begin{array}{c}{}_{}\begin{array}{c}\begin{array}{c},z)\begin{array}{c}\begin{array}{c}\begin{array}{c}\dots \begin{array}{c}{f\begin{array}{c}\begin{array}{c}\begin{array}{c}{n\begin{array}{c}\begin{array}{c}\begin{array}{c}{}_{}({x\begin{array}{c}\begin{array}{}\end{array}\end{array}}_{}\end{array}\end{array}\end{array}}_{}\end{array}\end{array}\end{array}}_{}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}}_{}\end{array}\end{array}\end{array}}_{}\end{array}\end{array}\end{array}}_{}\end{array}\end{array}\end{array}}_{}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}}_{}\end{array}\end{array}}_{}\end{array}\end{array}}_{}\end{array}\end{array}}_{}\end{array}\end{array}\end{array}}_{}\end{array}\end{array}}_{}\end{array}\end{array}}_{}\end{array}\end{array}}_{}\end{array}\end{array}\end{array}\end{array}}_{}\end{array}\end{array}}_{}\end{array}\end{array}}_{}\end{array}\end{array}}_{}\end{array}\end{array}\end{array}}_{}\end{array}\end{array}}_{}\end{array}\end{array}}_{}\end{array}\end{array}}_{}\end{array}\end{array}\right)$$

$$\stackrel{}{{I{i{}_{}\stackrel{}{^}\left(r,z\left(\right)=\sum _{i=0\underset{}{\overset{}{k=2\underset{}{\overset{}{}}{A{i,j{}_{}{I{j{}_{}}_{}}_{}}_{}}_{}}}}^{}\right)}_{}}_{}}$$

𝐼𝑟̂=𝐴𝑟0𝐼0+𝐴𝑟1𝐼1+𝐴𝑟2𝐼2 $$$$
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𝐼𝑔̂=𝐴𝑔0𝐼0+𝐴𝑔1𝐼1+𝐴𝑔2𝐼2 $$$$
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𝐼𝑏̂=𝐴𝑏0𝐼0+𝐴𝑏1𝐼1+𝐴𝑏2𝐼2 $$$$
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𝐼𝑔̂≈𝐴𝑔𝐺𝐼𝐺+𝐴𝑔𝐵𝐼𝐵 $$$$
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≈
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𝐼𝑏̂≈𝐴𝑏𝐺𝐼𝐺+𝐴𝑏𝐵𝐼𝐵 $$$$
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$${I{G{}_{}=\frac{{A{bB{}_{}\stackrel{}{{I{g{}_{}\stackrel{}{^}-{A{gG{}_{}\stackrel{}{{I{b{}_{}\stackrel{}{^}\frac{{A{gG{}_{}{A{bB{}_{}-{A{gB{}_{}{A{bG{}_{}\frac{}{}}_{}}_{}}_{}}_{}}_{}}_{}}_{}}_{}}{}}_{}}_{}}}_{}}_{}}_{}}_{}}}_{}}_{}}{}}_{}}_{}$$

$${I{B{}_{}=\frac{{A{gBG{}_{}\stackrel{}{{I{b{}_{}\stackrel{}{^}-{A{bG{}_{}\stackrel{}{{I{g{}_{}\stackrel{}{^}\frac{{A{gG{}_{}{A{bB{}_{}-{A{gB{}_{}{A{bG{}_{}\frac{}{}}_{}}_{}}_{}}_{}}_{}}_{}}_{}}_{}}{}}_{}}_{}}}_{}}_{}}_{}}_{}}}_{}}_{}}{}}_{}}_{}$$

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