Particle Fractal Dimension Research Articles

Size distribution and particle shape of iron ore powder based on the fractal theory

The size distribution and particle shape of various iron ores are measured in this paper, then the ballability index and compression strength of green pellet of different iron ores are examined. The particle fractal dimensions of particle size distribution and texture are calculated based on fractal theory. Research shows that the size distribution of iron ores has obvious fractal characters. The fractal dimension of size distribution is significantly related with balling index and specific surface area of particle. Particle shape can be characterised by texture fractal dimension. At the same size level, smaller fractal dimension means smoother particle and smaller compression strength of pellet. The relationship between the texture feature parameters and compressive strength of the pellet is obtained.

Fractal dimensions and escape rates in the two-dimensional Hénon-Heiles potential and its deformation form

From the beginning of the twentieth century the development of nonlinear theory has guided us into a new field of exploration; chaos and fractals are important tools widely used in studying the nonlinear system. Fractal focuses on the description of the physical geometry while Chaos emphasizes the research on the developing process of dynamics coupled with geometry. We have studied the nonlinear behavior of the particles in non-integrable system by means of chaos and fractals.#br#There are fractal structures in a Hénon-Heiles system, by which we can investigate the general escape law of particles in chaotic systems. The motion of particles is traced by dynamical algorithm within the framework of Chin and Chen (2005 Celestial Mechanics and Dynamics Astronomy 91 301). For the first time, we study the escape property of particles in Hénon-Heiles system and make a comparison with it in its hexagon-shaped form. Particles with energy higher than the threshold (1/6) can escape from the Hénon-Heiles system. The self-similar structures are found by calculating the escape time of particles with an energy higher than the threshold for different exit angles. We calculate the escape rate of particles with different energies and make a statistical analysis on the fractal dimensions by ‘box-counting' method. It is that the escape rate and fractal dimensions change with the energy of particles in the Hénon-Heiles system. Moreover, we find that the fractal dimensions are strongly linear with the escape rate.#br#To testify the universality of the conclusion, we calculate the escape rate and fractal dimensions of particles in the hexagon-shaped Hénon-Heiles system. Unlike the motion of particles in the Hénon-Heiles system, the escape of particles is divided into two ranges of energy. In low energy range, the escape rate and fractal dimensions of particles change with energy, which is similar to that in the Hénon-Heiles system. However, the escape rate and fractal dimensions tend to become stable in high energy range. But the general law is still valid–the fractal dimensions are strongly linear with the escape rate.#br#Therefore, the fractal dimensions can be served as a tool to study the escape features of particles in a chaotic system. We can characterize the transport behaviors in a chaotic electronic equipment by investigating the fractal dimensions in the design of mesoscopic devices.

Particle fractal dimension and total phosphorus of soil in a typical watershed of Yangtze River, China

Phosphorous (P) is an essential nutrient in soil which can lead to eutrophication when eroding into rivers or lakes. As a new method to represent soil properties, fractal theory was applied to study soil characteristics and related process. To understand the correlation of total soil P (TSP) and fractal dimension (D) with the effect of soil depth and slope gradient, soil grid samples were collected at 0–10, 10–20 and 20–40 cm depths in Yingwugou watershed, Yangtze River, China. The physico-chemical properties of the samples were analyzed to determine the TSP content and D. Fractal scaling theory was then applied to establish the relationship between D and TSP. With silt as the dominant particle size, the D values of the study area were in the range of 2.42–2.55. On the flat ground, more fine particles (clay and silt) were reserved and on the contrary relatively high content of coarse sand was retained on steep slope. D values decreased with slope and soil depth increased. The average TSP contents of the three layers were 0.65, 0.63 and 0.57 g/kg for 0–10, 10–20 and 20–40 cm, respectively, showing a decreasing tendency with increased soil depth. With increasing slope gradient, the TSP contents decreased first and then increased slowly. The TSP contents were positively related to silt content (P < 0.05) and clay content (P < 0.05) and negatively associated with sand contents (P < 0.01). There was an one-dimensional linear relationship (n = 530, R 2 = 0.16, P < 0.01) between TSP content and D values. The correlation would improve when concrete slope was taken into account. To better understand and control soil erosion, the relationship between D value and soil nutrients deserves more attention.

Effect of particle size distribution on turbidity under various water quality levels during flocculation processes

The dynamic relationship between particle size distribution and water turbidity was characterized using online real-time monitoring devices to explore the impact of particle size distribution on water quality. Particle amount was proportional to water turbidity in the range of 0–40NTU, whereas the correlation was rather poor in the range of 40–100NTU. The number of particles smaller than 5μm (Nd<5) should be strictly monitored and controlled in water treatment works (WTWs) due to their complex behaviors. The fractal dimension of particles (dpf) is well suited to characterize particle size distribution under different water turbidities. The first derivative of dpf was firstly used to explain the relationship between dpf and the percentage of Nd<5, and the turbidity and dpf values could be used together to monitor the temporal evolution of the number of smaller particles. A new block method for analyzing flocculation processes was proposed based on the change in dpf. The particle counting method was more sensitive to the evolution of particle size, which allowed a better understanding of the impact of particle size distribution on water turbidity.

Soil aggregates and fractal features under different land use types in a frequent debris flow area

The stability of soil aggregates and the fractal characteristics of four typical land use types (farmland, grassland, woodland, and bare land) in the Jiangjiagou Ravine (Yunnan, China), a frequent debris flow occurring area, were studied according to the normal mean mass diameter and fractal theory. The present research showed that the stability of the soil aggregates was different for the different land use types. When the soil depth was 0-30 cm, farmland soil formed more aggregates with diameters greater than 0.25 mm, i.e., the farmland soil was more stable than that of the other three land uses. When the soil depth was 30-45 cm, the order of stability of the soil aggregates was woodland > grassland > farmland > bare land. The fractal dimensions had a significant linear positive correlation with the amount of soil particles with diameters of < 0.25 mm, and a significant negative linear correlation with the amount of soil particles with diameters of 0.25-0.5 mm, 0.5-1 mm and 1-2 mm. Smaller fractal dimensions of the soil particles correlated with more stable soil aggregates. The fractal dimensions had a positive linear correlation with the soil bulk density and a negative correlation with the concentration of organic matter. These results showed that soil aggregates can be used as a parameter for characterizing the soil structures and properties. According to these results, the soil particle fractal dimensions could not only objectively characterize the stability of the soil structure but also could be used to indicate soil structure and properties. In addition, these results have great significance for the discussion of the comprehensive evaluation of soil.

A Fractal Approach to Assess the Risks of Nitroamine Explosives

To the best of our knowledge, this work represents the first thermal conductivity theory for fractal energetic particle groups to combine fractal and hot-spot theories. We considered the influence of the fractal dimensions of particles on their thermal conductivity and even on the sensitivity of the explosive. Based on this theory, two types of nitroamine explosives (hexahydro-1,3,5-trinitro-1,3,5-triazine [RDX] and hexanitrohexaazaisowurtzitane [HNIW]) with different sizes, size distributions, and microscale morphologies were prepared using wet milling, solvent/nonsolvent, and ridding methods. The dependence of the explosive sensitivity on the fractal characteristics of the particles was investigated. Specifically, the size distributions and scanning electron microscopy (SEM) images of the samples were used to obtain the fractal dimension (D) and surface fractal dimension (Ds), respectively, by using a least-squares method and fractal image processing software (FIPS). The mechanical sensitivity and thermal stability of the samples were characterized using mechanical sensitivity tests and differential scanning calorimetry (DSC) and were further compared with the previous results upon the investigation about HMX (octahydro-1.3.5.7-tetranitro-1,3,5,7-tetrazocine). The results indicate that the sensitivity of nitroamine explosives largely depends on the fractal dimensions of the particles. Specifically, the sample with a higher D value is more insensitive to impact action, whereas the sample with a higher Ds value is more sensitive to friction action. In addition, the sample with both higher D and Ds values has less heat release and a slower rate of thermal decomposition. All of the above observations can be attributed to the alternation of the formation of hot spots that was controlled by heat mass and thermal conductivity due to the increase of D and Ds values caused by changes in parameters such as fine particle content, specific surface area, porosity content, surface protruding points, and surface roughness. Therefore, the data in these studies were used to develop a thermal conductivity theory for fractal energetic particle groups that could be applied to the prediction of the sensitivity of energetic materials.

Small-scale variability in suspended matter associated with the Connecticut River plume front

[1] We report high spatial resolution observations of optical proxies for suspended particles and dissolved matter measured at the boundary of the Connecticut River plume (CRP) in Long Island Sound (LIS) in April 2000 when river discharge was near the annual maximum. The magnitude of beam attenuation, cp, backscatter, bb, and absorption at short wavelengths indicated higher concentration of suspended particles and colored dissolved matter within the LIS relative to the adjacent CRP. The fractional backscatter from particles indicated relatively higher organic matter fraction within the LIS. An absorption feature centered at 429 nm (aPB), indicated the presence of pigmented heterotrophic bacteria unique to the LIS and the strongest signals were observed at locations closest to shore. The spectral slope of beam attenuation associated with particulate matter (γ) indicated the presence of relatively larger particle assemblages within the LIS. Strong linear relationships between γ and salinity were observed within the energetic CRP boundary region, within 30 m of the front location. Regression residuals indicated a shift to smaller particles and were greatest at the front and decreased with distance toward the plume interior with a length scale similar to previous reports of the kinetic energy dissipation rate. At the same time, the magnitude of cp and bb remained uniform. These results are consistent with the disruption of low fractal dimension particle assemblages due to enhanced turbulence and mixing. The residuals in γ were weakly correlated with salinity and aPB suggesting that aggregate disruption was primarily associated with entrained Long Island Sound water.

Fractal Analysis of Mixing in Zinc-Silica Composite Electrolyte by Particle Image Velocimetry

This paper investigates the mixing effect in zinc-silica composite electrolyte by particle image velocimetry (PIV) and fractal dimension method. Fifteen groups of orthogonal tests including different rotational speed, electric current and height of paddle are carried out to study the mixing homogeneity using fractal dimension method, it is found that the case with paddle height of 1cm, electric current of 0.25 A and rotational speed of 1000 rpm is better than the others, and at last the distribution of silica in composite coating provides a strong evidence for characterizing the mixing homogeneity.

Fractal-statistical analysis of grain-size distributions of debris-flow deposits and its geological implications

Breakage models and particle analyses have been widely used as tools for describing and interpreting various deposits and providing parameters for assessing the particle-size distribution of the deposits. Debris flows can be seen as a two-phase rheological fluid with a clay-fluid composition, and debris-flow deposits comprise mud, silt, sand, and boulders, with grain sizes ranging from less than one μm to more than several meters. As a consequence, according to fractal theory, the particles in debris-flow deposits have self-similarity in geometrical shape and scale invariance in size. In this paper, the fractal dimensions of particles in various debris-flow deposits are calculated and corresponding fractal features are determined based on fractal-statistical theory. The aims of the study are: to provide a quantitative grain parameter that reflects both the grain composition and grain-size distribution in debris-flow deposits; to compare the fractal dimensions of grains in different types of debris-flow deposits and the degree of self-organization of debris flows; as well as to discuss the geological implications of fractal dimensions and fractal features of particles in debris-flow deposits.

Sub-populations and patterns in blast induced fine fragmentation

The understanding of fundamental mechanisms of fragmentation is still being developed. As such the fine fragmentation produced in a blast has been the subject to study. By examining the fines in the context of surface area instead of mass, several patterns and sub-populations have been observed. This paper is a study of fine fragmentation generated in small scale blasts. Each size distribution was transformed from mass into surface area. Each explosive charge was simulated in context of the energy it applied to the sample during breakage. This paper represents a small part of a larger study. The samples blasted in this study produced several aspects of self-similarity in breakage. Samples of the same rock type produced a similar surface area size distribution that was observed at a wide range of applied explosive energies. Each rock type had a characteristic surface area size distribution. Shape characteristics of measured fragments in the √2 sieve series shared similarities across a wide range of size fractions. Also paradoxically, the products of fine fragmentation were found to be multi-component in nature. A change in surface area size distribution was observed at 1 mm, which was found to be statistically significant. The fractal dimension of particles larger than 1 mm were found to be statistically different to particles smaller than 1 mm. Another change in particle character was found at 10 mm fragment size. The generation of fine fragmentation characterisation paradoxically has two fragment size thresholds of note, 10 mm and 1 mm, and is self-similar in nature. It is postulated that fragmentation is not a true fractal but the individual fundamental mechanisms that generate it may well be.

Size and fractal dimension of particles in the River Nakdong

Water samples were collected at seven sites located along the River Nakdong on 30 occasions. Water quality, size and the fractal dimension (dF) of suspended particles were measured. The laser light scattering method was used to obtain the size and dF of suspended particles. The average size of particles in this river ranged from 89 microm and 169 microm, which appears to be relatively coarse compared with other rivers worldwide. The average dF of suspended particles in this study ranged from 1.8 to 1.9. Slight variations in fractal dimension values and other particle characteristics results from various measuring methods available. The correlation analysis showed that DO, TN, NO3 and chlorophyll-a had significant positive relationships with particles size, whereas flow rates and temperature had negative relationships. However, the factors which had positive relationships with particles size showed negative relationships with the dF of suspended particles. Generally, as the size of particles increased, the fractal dimension of particles decreased which indicated that the shape of the larger particles became more irregular relative to that of the smaller ones. To obtain and apply the statistical functional relationship between water quality characteristics, multiple linear regression equations of the size and fractal dimension of particles on explanatory variables such as pH, BOD, TSS, DO, T-N and T-P have been established.

A Fractal Method of Estimating Soil Structure Changes Under Different Vegetations on Ziwuling Mountains of the Loess Plateau, China

Fractal method is a new method to estimate soil structure. It has been shown to be a useful tool in studies related to physical properties of soil as well as erosion and other hydrological processes. Fractal dimension was used to study the soil structure in soil at different stages of vegetative succession on the Ziwuling Mountains. The land use and vegetation types included cultivated land, abandoned land, grassland, two types of shrub land, and three types of forests. The grassland, shrub land, and forested areas represented a continuum in vegetative succession that had occurred naturally, as the land was abandoned in 1862. Disturbed and undisturbed soil samples were collected from ten vegetation types from depths of 0–10, 10–20, and 20–30 cm on the Ziwuling Mountains, at a site with an elevation of about 1500 m. Particle size distribution was determined by the pipette method and aggregate size distribution was determined by wet sieving. The results were used to calculate the particle and aggregate fractal dimension. The results showed that particle and aggregate fractal dimensions varied between vegetation types. There was a positive correlation between the particle fractal dimension and the weight of particles with diameter <0.001 mm, but no relationship between particle fractal dimension and the other particle size classes. Particle fractal dimension was lower in vegetated soils compared to cropland and there was no consistent relationship between fractal dimension and vegetation type. Aggregate fractal dimension was positively correlated with the weight of >0.25 mm aggregates. Aggregate fractal dimension was lower in vegetated soils compared with cropland. In contrast to particle fractal dimension, aggregate fractal dimension described changes in soil structure associated with vegetative succession. The results of this study indicate that aggregate fractal dimension is more effective in describing soil structure and function compared with particle fractal dimension.

A new method for calculating the diameters of partially-sintered nanoparticles and its effect on simulated particle properties

In this paper a new method for the calculation of the diameter of a partially-sintered particle is introduced and shown to be more accurate than the standard dp=6v/a formula. The method was used to determine the particle sintering rates in a flame producing SiO2 nanoparticles. The results of the simulations return a value for the average particle fractal dimension closer to the experimental values than is obtained using the standard formula. The particle ensemble was simulated using an efficient bivariate stochastic particle method, which for the first time, was applied to the non-isothermal environment of a premixed laminar flame. The transition from a bimodal particle size distribution to a unimodal particle size distribution for the silica particles was observed. This paper also contains the results of a test simulation which examines the stochastic algorithm under various simple starting conditions and determines its convergence properties.

Multi-fractal imaging and structural investigation of silica hydrogels and aerogels

In this work, we report on the preparation of TEOS gels by means of the sol–gel method employing different TEOS/ethanol/water ratios at room temperature. Small-angle X-ray scattering (SAXS) measurements provide the first-generation structural parameters, which indicate that particle size and fractal dimension are synthesis and depth dependent. These results are supported by the second- and third-generation aggregates observed using atomic force microscopy (AFM) and scanning electron microscopy (SEM). The fiber-like first-generation clusters aggregate as spherical second-generation particles which, in turn, in the third generation rearrange into complex structures.

비구형 입자의 형상에 따른 단극 확산 하전 특성

Unipolar diffusion charging of non-spherical particles was investigated for various particle shapes. We researched with TiOagglomerates produced by the thermal decomposition of titanium tetraisopropoxide (TTIP) vapor. TTIP was converted into TiO, in the furnace reactor and was subsequently introduced into the sintering furnace. Increasing the temperature in the sintering furnace, aggregates were restructured into higher fractal dimensions. The aggregates were classified according to their mobility using a differential mobility analyzer. The projection area and the mass fractal dimension of particles were measured with an image processing technique performed by using transmission electron microscope (TEM) photograph. The selected aggregates were charged by the indirect photoelectric-charger and the average number of charges per particle was measured by an aerosol electrometer and a condensation particle counter. For the particles of same mobility diameter, our results showed that the particle charge quantity decreases as the sintering temperature increases. This result is understandable because particles with lower fractal dimension have larger capacitance and geometric surface area.

Bivariate Extension of the Quadrature Method of Moments for Modeling Simultaneous Coagulation and Sintering of Particle Populations

We extendthe application of moment methods to multivariate suspended particle population problems—those for which size alone is insufficient to specify the state of a particle in the population. Specifically, a bivariate extension of the quadrature method of moments (QMOM) (R. McGraw, Aerosol Sci. Technol. 27, 255 (1997)) is presented for efficiently modeling the dynamics of a population of inorganic nanoparticles undergoing simultaneous coagulation and particle sintering. Continuum regime calculations are presented for the Koch–Friedlander–Tandon–Rosner model, which includes coagulation by Brownian diffusion (evaluated for particle fractal dimensions, Df, in the range 1.8–3) and simultaneous sintering of the resulting aggregates (P. Tandon and D. E. Rosner, J. Colloid Interface Sci.213, 273 (1999)). For evaluation purposes, and to demonstrate the computational efficiency of the bivariate QMOM, benchmark calculations are carried out using a high-resolution discrete method to evolve the particle distribution function n(ν, a) for short to intermediate times (where ν and a are particle volume and surface area, respectively). Time evolution of a selected set of 36 low-order mixed moments is obtained by integration of the full bivariate distribution and compared with the corresponding moments obtained directly using two different extensions of the QMOM. With the more extensive treatment, errors of less than 1% are obtained over substantial aerosol evolution, while requiring only a few minutes (rather than days) of CPU time. Longer time QMOM simulations lend support to the earlier finding of a self-preserving limit for the dimensionless joint (ν, a) particle distribution function under simultaneous coagulation and sintering (Tandon and Rosner, 1999; D. E. Rosner and S. Yu, AIChE J., 47 (2001)). We demonstrate that, even in the bivariate case, it is possible to use the QMOM to rapidly model the approach to asymptotic behavior, allowing an immediate assessment of when previously established asymptotic results can be applied to dynamical situations of current/future interest.

Monte Carlo Simulation of Particle Aggregation and Simultaneous Restructuring

Ultrafine (“nano”-) particles produced from highly supersaturated vapors or liquids often undergo rapid coagulation and slow interspherule coalescence. Resulting “aggregates” typically contain hundreds of small spherules bound together in tenuous structures characterized by mass fractal dimensions much less than 3. Such aggregates have large and relatively accessible initial surface area but are metastable with respect to more compact configurations, especially in high temperature environments (e.g., flames). Subject to deliberately idealized “uncoupled” rate laws for coagulation and coalescence, we illustrate the power of Monte Carlo simulation methods to obtain the self-preserving joint distribution function (with respect to both particle size and surface area) of populations of coagulating fractal aggregates in the continuum regime, simultaneously undergoing finite-rate restructuring (e.g., via surface-energy-driven viscous flow). Unconditional distributions with respect to either particle volume or area are also obtained from the Monte Carlo simulations. These are conveniently quantified by fitting them to log-normal distributions and we report the sensitivity of the associated spreads to characteristic fusion/coagulation time ratio, χ, and particle fractal dimension,Df, here prespecified. We also calculate and report selected “mixed” moments of the joint pdf with respect to particle volume and surface area needed for engineering calculations of deposition or diffusion-controlled vapor scavenging, as well as the important ratio of actual mean area to that area corresponding to the mean particle volume in the aerosol population. This work sets the stage for tractable simulations of particle dynamics in more complex coagulating systems requiring multi-internal (state-) variables for their more realistic and self-consistent description.

The Fractal Characteristic of Particles in Granular Material Flows and Its Effect on Effective Thermal Conductivity

According to the fact that many pulverized particles possess fractal characteristic, a fractal model for studying fine particles in granular material flows is first proposed. An expression of particles' fractal distribution is derived to describe the relationship between the particle fractal dimensions and particle velocity distribution function. In accordance with this model, the theoretical particle effective thermal conductivity is derived. The analytical results show that for the small Biot–Fourier number, the effective thermal conductivity increases with the square root of the granular temperature. For very large Biot–Fourier number, the effective thermal conductivity linearly increases with the granular temperature. Numerically calculated results show that the thermal conductivity increases with the particle size fractal dimensions and decreases with the particle surface fractal dimensions.

Predicting particle coagulation and sedimentation rates for a pulsed input

Particles provide an important transport mechanism for distributing chemically and biologically reactive elements in the ocean. To aid in predicting the fate of colloidal particles, we update an existing model, developed by Farley and Morel [1986] for predicting particle sedimentation rates in the presence of aggregation. We add some effects of the fractal structure of the particles as well as the hydrodynamic interactions between particles. Changes due to the hydrodynamics have a lesser effect than the fractal dimension. Simulated sedimentation rates can vary by as much as 6 orders of magnitude as the fractal dimension is varied between D=3 and D=2. Following Farley and Morel we develop simple expressions for the sedimentation rate. We make use of special initial conditions and particle input rates as well as a value of 1 for the particle stickiness. Introducing the particle fractal dimension revealed limitations with this class of models, restricting the range of D which we could successfully use.

Boundary Fractal Dimensions from Sections of a Particle Profile

The fractal dimension of particles is commonly evaluated from complete particle boundaries. In this work, a study has been made of the self‐similar nature of complete and incomplete boundary profiles of a range of morphologically different copper powders. Boundary images were captured from SEM micrographs of particle boundaries at a range of magnifications up to nearly 14000X. An algorithm was developed to compute the fractal dimension of boundary segments. This algorithm was tested against the Koch Island fractal, and was found to give excellent estimates of the fractal dimension. For the particle system studied, the boundary fractal was found to be sensitive to magnification with appreciable drops in value at high magnification. This demonstrates that the particles studied did not have true fractal boundaries and the use of fractal theory to study particle surface roughness must be used with caution.