Unconfined Flame Method (Radius-time data-driven)

Chapter 3 Experimental setup and Methodology

3.4 Data Processing

3.4.2 Unconfined Flame Method (Radius-time data-driven)

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𝑆_𝑢 𝑆_𝑢⁰ = (^𝑃

𝑃_𝑖)^𝛼 3.16

The pressure-time data were processed using Eq. 3.1 iteratively to estimate LBV (𝑆_𝑢⁰).

Burned gas mass fraction and its variation with pressure were estimated using Eq. 3.2- 3.13. The pressure data corresponding to an interval 0.25 to 0.5 times that of reduced pressure, 𝑃_𝑟 (Eq. 3.9) was chosen. Power law (Eq. 3.16) was employed in the selected data range, and laminar burning velocity at initial thermodynamic conditions was estimated through extrapolation. The above formulations were implemented in the data range with negligible stretch. For further analysis, the flame stretch was calculated with two different formulations K1[3] and K2[4].

𝐾₁ = ^𝑑𝑃

𝑑𝑡 𝑑𝑥 𝑑𝑃+ ¹

𝑃𝛾𝑢(1−𝑥) 3

2((^𝑃 𝑃𝑖)

1

𝛾𝑢−(1−𝑥))

3.17

𝐾₂ = ²

3(^𝑅𝑣

𝑅_𝑓)

3

(1 + (^{𝑃𝑒−𝑃}

𝛾_𝑢𝑃)) ( ¹

𝑃_𝑒−𝑃_𝑖) (^𝑃𝑖

𝑃)

1 𝛾𝑢𝑑𝑃

𝑑𝑡 3.18

Experimental setup and Methodology

39

or column-wise marching/movement was used to identify the changes in the pixel intensity level. The distribution of pixel intensity levels from the Eastern sector is presented in Figure 3-5. The rate of change in the intensity was coupled with the moving average to identify the edge and nullify discreet points. The post averaged rate of change in intensity is presented in Figure 3-6. The outer edges were identified by specifying the stringent constraints, excluding all other edges in the image.

Figure 3-4 Various stages of a working image

Figure 3-5 Pixel intensity distribution in the Eastern sector of the working image

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Figure 3-6 Rate of change in intensity in the Eastern sector of working image The main challenge was identifying the edges in the ordinal directions as the flame front was at 45° to two cardinal directions. The marching, either row or column-wise, was not possible. Typically image was rotated by 45° to align with cardinal directions. This operation incorporated the interpolation of pixels, leading to distortion of original pixel data. This interpolation-based rotation was avoided by using the matrix-interlacing method.

3.4.2.1.1 Matrix-interlacing method

The pixel intensity distribution of the image was stored in a square matrix. The alternate pixels were separated, similar to the interlacing operation, and reassigned into a new matrix, as shown in Figure 3-7. The two new matrices were referred to as child matrices.

The null points were dropped in the child matrix, as shown in Figure 3-8. New null points were added, and a new matrix was reconstructed, as presented in Figure 3-9. Figure 3-10 is the consolidated figure with all crucial steps of image rotation. The 45° rotated images were obtained without interpolating original pixel intensity data. The image rotation operation is executed on an image and presented in Figure 3-11.

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Figure 3-7 Parent matrix split into two child matrices

Figure 3-8 The removal of null values from the child matrix

Figure 3-9 Reconstructing the child matrix with new null points

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Figure 3-10 A consolidated image capturing the steps of image roatation

Figure 3-11 A test image split into two child images

The rotated images were processed similarly to cardinal sectors. The obtained edge data coordinates were reassigned back to the original configuration. The resultant data was corresponding to the ordinal sectors. All the coordinates were sorted, and duplicates were removed. The final edge list was fitted with a circle equation to compute the radius and its center. The procedure was repeated for all the images to obtain radius-time data. The main advantages of the developed algorithm were 1. The set of operations suitable enough to be implemented on any platform, 2. Minimized dependency on any single parameter, 3. The original pixel intensity distribution remained undisturbed, 4. Unwanted edges except the boundaries were ignored by default in the initial stages. As a result, the developed algorithm can be used for any non-smooth, turbulent, and unstable flame fronts, as shown in Figure 3-12. A word of caution was that edge detection of non-smooth images shown in Figure 3-12 was only for an illustration, and such non-smooth flames were never considered in the estimation of LBV.

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Figure 3-12 Edge detection of non-smooth spherical flames. This illustration is only for demonstrating the ability of the new edge detection algorithm. Non-smooth images were never considered for the estimation of LBV

3.4.2.2 Procedure for the estimation of Unstretched LBV

The measured/simulated radius-time data was processed as shown in the flowchart (Figure 3-13) to obtain unstretched LBV and burned gas Markstein length. Detailed description of each process mentioned in Figure 3-13 is discussed in the following section. Initially, stretched flame speed (Sb) was calculated using Eq. 3.19.

𝑆_𝑏= ^𝑑𝑟𝑓

𝑑𝑡 3.19

where rf was the instantaneous flame radius. Eq. 3.19 was applicable only for thin flames where (𝑟_𝑓⁄ ) ≪ 1 (𝛿𝛿_{𝑓 𝑓} is the flame thickness) without ignition and buoyancy effects. As the spherical flames change curvature during the flame propagation, the stretch rate (K) was estimated using Eq. 3.20.

𝐾 = ²

𝑟_𝑓 𝑑𝑟𝑓

𝑑𝑡 3.20

Then, unstretched flame speed (𝑆_𝑏⁰) at K = 0 was extrapolated using a linear model (Eq.

3.21[54]) and a non-linear model (Eq. 3.22[105]).

𝑆_𝑏= 𝑆_𝑏⁰− 𝐿_𝑏𝐾 3.21

𝑆_𝑏

𝑆_𝑏⁰(1 +^2𝐿𝑏

𝑟𝑓 +^4𝐿𝑏

2

𝑟_𝑓² +^16𝐿𝑏

3

3𝑟_𝑓³ + 𝑂⁴(^𝐿𝑏

𝑟𝑓)) = 1 3.22

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Figure 3-13 Flow chart of data processing in unconfined flame method

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Also, the non-extrapolation model proposed by Xiouris et al. [51] for the estimation of 𝑆_𝑏⁰ as shown in the Eq. 3.23 was also used.

[^{𝑆𝑏−𝑆𝑏}

0 𝐾 ]

𝑆𝑖𝑚 ≅ [^{𝑆𝑏−𝑆𝑏}

0 𝐾 ]

𝐸𝑥𝑝 3.23

The 𝑆_{𝑏,𝑠𝑖𝑚}⁰ was calculated from the freely propagating planar flame model in Ansys Chemkin [106] and instantaneous 𝑆_{𝑏,𝑠𝑖𝑚} and Ksim were estimated using one dimensional, expanding spherical flame model available in Cosilab [107]. For both the simulations, GRI Mech 3.0 [108] was used. From the known information of 𝑆_{𝑏,𝐸𝑥𝑝}and 𝑆_{𝑏,𝑆𝑖𝑚} at each KExp, 𝑆_{𝑏,𝐸𝑥𝑝}⁰ was estimated by following Eq. 3.23. Then, the obtained 𝑆_{𝑏,𝐸𝑥𝑝}⁰ at each KExp

was averaged to find the final value of 𝑆_{𝑏,𝐸𝑥𝑝}⁰ meant for that operating condition. Xiouris et al. [51] mentioned that the choice of chemical kinetic mechanism did not affect the accuracy of Eq. 3.23. Finally, laminar burning velocity (𝑆_𝑢⁰) defined as the relative velocity of the reactants propagating normal to an adiabatic and stretch free planar flame was estimated using Eq. 3.24. Burned gas density (𝜌_𝑏⁰) assumed to be in chemical equilibrium was estimated using equilibrium model in Ansys Chemkin.

𝑆_𝑢⁰ = 𝑆_𝑏⁰(^𝜌𝑏

0

𝜌𝑢) 3.24