Fractal Dimension Mathematical Modeling And Optical Property Calculation Of Soot Particles

Soot is mainly produced by fossil fuel and biomass burning,which has a fractal-like aggregates structure.Soot particles strongly absorb sunlight,altering the radiative forcing balance of the atmosphere and affecting the climate change.The optical properties and radiative forcing of soot particles vary,when their morphologies and mixing states change during aging process in the atmosphere.At present,people mainly study the morphology of soot through laboratory simulation and combustion experiments,but there are few studies focus on atmospheric soot and their morphologies.In this study,we induced a parameter,fractal dimension,to demonstrate the complex morphology of soot.We systematically compared the advantages and disadvantages of two-dimensional method boxing count and sliding box counting,and the three-dimensional method scaling law.Besides,we collected the soot particles directly emitted from vehicles,and we collected soot particles at Kaiyuan tunnel,Jinan City,mountaintop of Mt.Tai,and background site Changdao.Then we analyzed their morphologies and mixing states with transmission electronic microscope(TEM).Afterwards,we calculated the optical properties of soot particles with various fractal dimensions and mixing states using discrete dipole approximation(DDA)and Mie theory.We found that the maximum box size of box counting method should be limited in 18-30,and the step size should be larger than one.The maximum box size of sliding box counting method should be larger than 21,and this method was most stable when the maximum box size was 40.The fractal dimension in three-dimensional space more accurately described the morphology of soot.We used box counting,sliding box counting,and scaling law method to calculate the fractal dimension of soot particles directly emitted from vehicle exhaust.The calculation results of box counting were similar to that of sliding box dimension for these two-dimensional calculation methods.However,the results of box counting and sliding box counting were too large.The scaling law of the three-dimensional calculation method can obtain the accurate fractal dimension of bulk soot particles in three-dimensional space with high efficiency.According to the morphology and mixture state of soot particles observed in TEM image,we divided the soot particles of four sampling points in North China into three types:bare-like,partly coated and embedded.The bare-like type dominated the soot particles collected at the tunnel site,while the other sampling points are mainly of the partly coated and embedded type.Different types of soot particles had fractal dimensions ranging from 1.80 to 2.16,and the order of magnitude is bare-like<partly coated<embedded.From the tunnel site to the background site,the average of the fractal dimensions of the four sampling sites varies from 1.8 to 2.0.The fractal dimensions reasonably characterized the shape of soot aggregates,and their changes reflected the soot aging process.Compared to the fractal dimensions of soot particles reported in other studies,we found that the Df = 1.8 used in previous optical models mainly represented freshly emitted soot particles,rather than the actual morphological structure of atmospheric soot particles including aged and unaged ones..The Df of soot particles in the polluted air of the North China Plain obtained in this study was 1.91.This value can represent the fractal dimension of soot particles in general dry polluted air for optical calculation.We constructed models similar to soot particles in the atmosphere,and used the discrete dipole approximation(DDA)to calculate the optical properties of soot particles.We found that the absorption cross section of the externally mixed black carbon tended to decrease with the increase of the fractal dimension,while the scattering cross section increased with the increase of the fractal dimension.The scattering cross section of the partly coated soot particles increased with the increase of the volume of the sulfate coating,and the shape of the sulfate coating affected the scattering cross section.The spherical sulfate coating had an underestimation of the absorption enhancement(Eabs),and the shape of the sulfate coating had a great influence on the absorption enhancement of the soot particles.The absorption enhancement(Eabs)of all embedded soot particles showed a tendency to increase first and then remained stable at Dp/Dc = 2.3.The Core-shell model overestimated the absorption enhancement(Eabs)of soot particles.This study showed that formation of soot particles in the atmosphere,mixing state,and structural model helped to improve the accuracy of the calculation of the optical properties of black carbon,which is essential for reducing the uncertainty of black carbon radiative forcing simulation.