Soot aggregate sizing in an extended premixed flame by high-resolution two-dimensional multi-angle light scattering (2D-MALS)

Abstract

The spatial distribution of soot aggregate size and morphology within a premixed flat flame (McKenna-type burner and ethyne–air mixture at an equivalence ratio of Φ = 2.7) is characterized by two-dimensional multi-angle light scattering (2D-MALS). A profound investigation of such an extended, radially symmetrical sooting flame with 2D-MALS requires a sophisticated camera calibration to correct for non-linear image scaling and a careful evaluation of the scattering data. Sharp scattering images were acquired in the angular range from 20° to 155° using a rotatable camera system and an automated Scheimpflug adapter. To correct for non-linear variations in horizontal and vertical image magnification occurring at scattering angles differing from perpendicular view, a polynomial-based image transformation algorithm was developed to convert all scattering images into a common coordinate system. Effective radii of gyration and fractal dimensions of soot aggregates were then derived from scattering data by two different approaches. Due to limited amount of angular positions, the classical method based on Guinier and power law analysis shows limitations, as it yields discontinuous results, predominantly in axial direction of the burner. Bayesian analysis was then used for a data fit of the complete structure factor conducting a least square minimization leading to more consistent results. The use of prior knowledge in the Bayesian evaluation allows for improved data fitting and reduced uncertainties in radius of gyration and fractal dimension even for small aggregate sizes.

Appendix

1.1 Derivation of the image calibration function

To find a suitable calibration function with minimum deviations and fast processing speed, we examined polynomial functions for different polynomial degrees n from one to 15. Figure 12 shows the maximum deviations and standard deviations in x- and y-directions for polynomial degrees 1 ≤ n ≤ 15 calculated for all marker points in the region of interest for all 41 angles following the procedure described in Sect. 3.2.

Fig. 12

Maximum deviations (left) and standard deviations (right) for all transformed marker positions in x- and y-directions for polynomial degrees n

For an increasing polynomial degree, the coordinates of the marker points are mapped more precisely according to their set point positions. Both the maximum deviations and the standard deviations for x- and y-directions are decreasing with increasing polynomial degree, whereby the deviations in y-direction are a little lower than in x-direction. A maximum degree of seven seems suitable for both spatial dimensions, as deviations are only slightly decreasing for higher polynomial degrees. Moreover, higher polynomial degrees should be handled with care, as they might lead to oscillations between sampling points and thus wrong mapping. Besides, higher degrees result in more terms and, therefore, in an extended processing time for image transformation. For example, a polynomial degree of 15 needs 816 terms to be solved, while a degree of seven only needs 120 terms and can be processed by hardware around 15 times faster based on MATLAB calculations. For further reduction of terms and improvement of calculation time, we successively reduced the number of maximum allowed polynomial degrees p_max and q_max for the x- and y-variables, respectively, down to a minimum of one, keeping r_max = n for the angular variable. This approach follows the idea of an almost linear behavior in x- and y-imaging. The maximum deviations as well as standard deviations in both x- and y-directions are shown in Fig. 13.

Fig. 13

Maximum deviations (upper row) and standard deviations (lower row) for all transformed marker positions in x-direction (left column) and y-direction (right column) as functions of maximum polynomial degrees p_max and q_max for an overall polynomial degree of n = 7

Both maximum and standard deviation in x-direction are relatively independent of the maximum polynomial degree p_max between two and seven; in y-direction, the maximum deviation obtains minimum values towards lower degrees q_max. The standard deviation in y-direction is very small for any degree q_max. Consequently, we chose a customized polynomial calibration function of degree seven with 39 individual terms in total omitting terms with degree p_max ≥ 3 and q_max ≥ 2 with high computational efficiency.

1.2 Uncertainty in the extinction correction

A profound derivation of R_g,eff and D_f from light scattering in sooting flames requires a correction for extinction caused by laser attenuation and signal trapping. However, the height-specific extinction coefficients used in this work are derived from a simplified approach (see Sect. 3.4 for further details) and thus afflicted with uncertainties, which also affect a reliable determination of R_g,eff and D_f. Figure 14 shows the deviations in R_g,eff and D_f derived from Bayesian inference including prior knowledge along the radial axis at two different HAB with a conservative estimation of the uncertainty of the extinction coefficient α of ± 10%. Besides, R_g,eff and D_f were also derived from scattering without any extinction correction.

Fig. 14

R_g,eff (left) and D_f (right) derived from Bayesian inference including prior knowledge along the radial axis at 10 mm HAB (solid lines) and 20 mm HAB (dashed lines) for the determined extinction coefficient α (black lines) and the regions of uncertainty for each HAB (grey areas). Additional results for a 10% overestimation of α (blue lines) and a 10% underestimation of α (red lines), as well as results determined without extinction correction (thick dark red lines)

Without extinction correction, the values for R_g,eff along radial extent in the ROI (between −20 and 20 mm radial extent and 10 mm and 20 mm HAB) are underestimated by up to 43% or overestimated by up to 28%. At 10 mm HAB, aggregate sizes would exhibit a monotonous decrease in the radial direction, at 20 mm HAB aggregate sizes on the left flame side with R < 0 mm are within the range of uncertainty (grey area), but strongly decrease for R > 0 mm. At 10 mm HAB, the fractal dimensions show no significant deviations outside the region of uncertainty, as they are mainly based on prior knowledge. At 20 mm HAB, however, D_f is overestimated by up to 24% for R < 0 mm.

Assuming a reasonable, yet conservative uncertainty of ± 10% in the extinction coefficient, deviations in R_g,eff and D_f within the ROI are below 5% and thus smaller than the uncertainty obtained from Bayesian analysis of the scattering data.