# Fractal Dimensions of Biomass Burning Aerosols from TEM Images Using the Box-Grid and Nested Squares Methods

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^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

_{g}, as follows [24,25,26,27,28,29]:

_{f}is the three-dimensional fractal dimension, k

_{o}is the fractal pre-factor, a is the radius of the monomer, k is a pre-factor, A

_{a}is the projected area of the aggregate, A

_{p}is the projected area of the monomer, and α is the overlap parameter (the ratio of the monomer diameter to the distance between the centers of two touching monomers) [5,30,31,32,33,34,35,36]. However, calculating the overlap parameter, α, and the pre-factor, k, is difficult due to the level of image analysis required.

_{f}using Equation (1). Empirical relationships to determine 3D fractal dimensions from 2D results have also been developed [25,26,28,39,44].

_{p}to the mass fractal dimension D

_{f}(3-d) using the empirical relationship [31,44]:

## 2. Materials and Methods

_{2}as the change in concentration of carbon monoxide and carbon dioxide, respectively. An MCE greater than 0.95 represents flaming-dominated combustion and an MCE less than 0.92 represents smoldering-dominated combustion, with a transition region between [47].

#### 2.1. Nested Square Method (NSM)

#### 2.2. Box-Grid Method (BGM)

## 3. Results

## 4. Discussions

#### 4.1. Comparing Fractal Dimensions with the Literature Values

_{p}from 1.47 to 1.94, which were within the range of values reported in the literature [5,31,32,42,54]. Previous measurements involved both laboratory and field measurements of biomass burning aerosols from the combustion of mixed wildland fuels from North and South America and roadside urban samples from diesel emissions for both fresh and aged samples [55,56]. The range of D

_{f}determined computationally and experimentally for soot, silica, and black carbon lay in the range 1.6–1.9 [30], which was confirmed experimentally using image analysis and light scattering predictions [25,57]. Analysis of field studies by McDonald and Biswas [54] using the NSM yielded D

_{p}between 1.39 and 1.89, while their BGM analysis yielded D

_{p}between 1.66 and 1.83. In some studies, changes in morphology with fuel type have been observed, similar to ours [58]. Using the same methodology Katrinak and Rez [42] found D

_{p}values ranging between 1.35 and 1.89 for urban aerosols. The calculation by Dye et al. for urban roadside emissions was D

_{f}= 1.56 and 1.57 [59]. Using NSM, Samson found D

_{p}values from 1.75 to 1.95 [60] for Acetylene soot. For diesel soot aggregates, Wentzel [56] found D

_{p}values that were generally lower and more fractal, ranging between 1.44 and 1.55.

_{f}was in the range of 1.67–1.83 and D

_{p}was between 1.68 and1.74, which is similar to our results. In the field measurement studies of images of both ambient and denuded aerosols, China et al. [32] calculated the D

_{f}using Equation (1) and found 1.85 ± 0.05 for ambient and 1.53 ± 0.07 for denuded samples. Clearly, D

_{f}reflects the history of fractals and is controlled by combustion conditions and aging processes. In a study at urban, mountaintop, and background sites in China, the D

_{f}for fresh particles remained at a consistent value of 1.82 at all sites [41]. Most current studies put D

_{f}= 1.8 to simulate the structures of soot particles for optical properties calculations [35,61,62,63], but this value does not represent aged particles. A new method named soot parameters (SP) that uses Equation (1) with the scaling law and image recognition technology to automatically determine D

_{f}was used to show differences in D

_{f}values for soot particles from cars (1.66 ± 0.17), BB aerosols (1.75 ± 0.18), and coal burning aerosols (1.76 ± 0.18) [33,64].

_{fm}which is the 3D fractal dimension for fresh aerosol emitted during flaming-dominated emissions was 2.26 ± 0.05 and commensurate with previously studied externally mixed BC and diesel exhaust particles. The 3D fractal dimensions for biomass burning aerosols were in the range 1.67–1.83 determined using Equation (1) from TEM images in one study or 1.85 in another [41]. These values are within the range of our values obtained by using the empirical Equation (4) to convert the 2D fractal dimensions calculated from NSM and BGM. However, the values were much larger when using empirical Equation (3) to the point of being unreasonable. Since these values were also larger than what we experimentally determined for the flaming-dominated combustion of similar fuels and other values in the literature, Equation (3) is not appropriate for flaming-dominated BB aerosols.

#### 4.2. BGM Considerations

_{f}of both individual and ensemble aggregates. They compared both the NSM and BGM methods with the ensemble method (EM) calculation of fractal dimension based on Equation (1) and determined that EM was the only method that could be used to reliably determine D

_{f}from 2D images. They hypothesized that the errors in the value of D

_{f}by NSM and BGM were due to the “non-self-similar” property of aerosol aggregates and their 2D images. The repeating unit of a fractal object should appear similar under any magnification, but this assumption of “self-similarity” breaks as the length scale approaches the monomer size in most cases.

#### 4.3. NSM Considerations

_{f}[70].

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Plot of 3D fractal dimensions derived from Equations (3) and (4) as a function of MCE. The MCE values for BGM fractal dimensions were offset by 0.001 to allow for better viewing. Note the different scales for each equation.

**Figure 6.**A graph produced by using the nested square method. The slope in this graph is 1.6 which corresponds to the fractal dimension of the measured aerosol.

**Table 1.**The average 2D fractal dimensions calculated using BGM and NSM and the 3D fractal dimensions derived using Equations (3) and (4). (Errors are ± 1σ.).

Nested Square Method | Box-Grid Method | |||||
---|---|---|---|---|---|---|

Sample | D_{p} | D_{f} (Equation (3)) | D_{f} (Equation (4)) | D_{p} | D_{f} (Equation (3)) | D_{f} (Equation (4)) |

Eucalyptus 1 (MCE = 0.96) | 1.73 ± 0.07 | 2.43 ± 0.13 | 1.82 ± 0.07 | 1.72 ± 0.07 | 2.44 ± 0.11 | 1.81 ± 0.06 |

Eucalyptus 2 (MCE = 0.99) | 1.74 ± 0.07 | 2.41 ± 0.12 | 1.83 ± 0.07 | 1.73 ± 0.06 | 2.43 ± 0.10 | 1.82 ± 0.06 |

Olive (MCE = 0.99) | 1.63 ± 0.15 | 2.61 ± 0.27 | 1.74 ± 0.11 | 1.6 ± 0.11 | 2.66 ± 0.19 | 1.72 ± 0.08 |

Pine (MCE = 0.99) | 1.59 ± 0.07 | 2.68 ± 0.07 | 1.70 ± 0.05 | 1.58 ± 0.07 | 2.69 ± 0.12 | 1.70 ± 0.04 |

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Honablew, T.; Fiddler, M.N.; Pokhrel, R.P.; Bililign, S.
Fractal Dimensions of Biomass Burning Aerosols from TEM Images Using the Box-Grid and Nested Squares Methods. *Atmosphere* **2023**, *14*, 221.
https://doi.org/10.3390/atmos14020221

**AMA Style**

Honablew T, Fiddler MN, Pokhrel RP, Bililign S.
Fractal Dimensions of Biomass Burning Aerosols from TEM Images Using the Box-Grid and Nested Squares Methods. *Atmosphere*. 2023; 14(2):221.
https://doi.org/10.3390/atmos14020221

**Chicago/Turabian Style**

Honablew, Timothy, Marc N. Fiddler, Rudra P. Pokhrel, and Solomon Bililign.
2023. "Fractal Dimensions of Biomass Burning Aerosols from TEM Images Using the Box-Grid and Nested Squares Methods" *Atmosphere* 14, no. 2: 221.
https://doi.org/10.3390/atmos14020221