scholarly journals Changes in the two-dimensional and perimeter-based fractal dimensions of kaolinite flocs during flocculation: a simple experimental study

2017 ◽  
Vol 77 (4) ◽  
pp. 861-870 ◽  
Author(s):  
Zhongfan Zhu ◽  
Dingzhi Peng ◽  
Jie Dou
Keyword(s):  
Fractal Dimension ◽  
Steady State ◽  
Surface Area ◽  
Dynamic Equilibrium ◽  
Processing System ◽  
Fractal Dimensions ◽  
Flow Shear ◽  
Steady Level ◽  
Axis Length ◽  
Flocculation Time

Abstract In this study, Couette flow experiments were performed to estimate the temporal evolution of the 2D and perimeter-based fractal dimension values of kaolinite flocs during flocculation. The fractal dimensions were calculated based on the projected surface area, perimeter length and length of the longest axis of the flocs as determined by sampling observation and an image-processing system. The 2D fractal dimension, which relates the longest axis length and projected surface area of flocs, was found to decrease with the flocculation time, corresponding to the production of some porous flocs from the flow shear. This fractal dimension finally reached a steady state, which resulted from a dynamic equilibrium among the floc growth, floc breakage and floc restructuring. The perimeter-based fractal dimension, which characterizes the relationship between the projected surface area and the perimeter of flocs, increases with flocculation time because the flow shear increases the collisions among the primary particles, and some irregular flocs are formed. The perimeter-based fractal dimension reaches a steady level because of the balance among floc aggregation, breakage and restructuring. In addition, a stronger turbulent flow shear makes the steady state of fractal dimensions occur early during flocculation.

Nano/Micro Pore Structure and Fractal Characteristics of Baliancheng Coalfield in Hunchun Basin

2021 ◽  
Vol 21 (1) ◽  
pp. 682-692
Author(s):  
Youzhi Wang ◽  
Cui Mao
Keyword(s):  
Fractal Dimension ◽  
Specific Surface ◽  
Pore Structure ◽  
Surface Area ◽  
Specific Surface Area ◽  
Pore Diameter ◽  
Fractal Dimensions ◽  
Nitrogen Adsorption ◽  
Average Pore ◽  
Coal Reservoir

The pore structure characteristic is an important index to measure and evaluate the storage capacity and fracturing coal reservoir. The coal of Baliancheng coalfield in Hunchun Basin was selected for experiments including low temperature nitrogen adsorption method, Argon Ion milling Scanning Electron Microscopy (Ar-SEM), Nuclear Magnetic Resonance (NMR), X-ray diffraction method, quantitative mineral clay analysis method. The pore structure of coal was quantitatively characterized by means of fractal theory. Meanwhile, the influences of pores fractal dimension were discussed with experiment data. The results show that the organic pores in Baliancheng coalfield are mainly plant tissue pores, interparticle pores and gas pores, and the mineral pores are corrosion pores and clay mineral pores. There are mainly slit pore and wedge-shaped pore in curve I of Low temperature nitrogen adsorption. There are ink pores in curve II with characteristics of a large specific surface area and average pore diameter. The two peaks of NMR T2 spectrum indicate that the adsorption pores are relatively developed and their connectivity is poor. The three peaks show the seepage pores and cracks well developed, which are beneficial to improve the porosity and permeability of coal reservoir. When the pore diameter is 2–100 nm, the fractal dimensions D1 and D2 obtained by nitrogen adsorption experiment. there are positive correlations between water content and specific surface area and surface fractal dimension D1, The fractal dimension D2 was positively and negatively correlated with ash content and average pore diameters respectively. The fractal dimensions DN1 and DN2 were obtained by using the NMR in the range of 0.1 μm˜10 μm. DN1 are positively correlated with specific surface area of adsorption pores. DN2 are positively correlated volume of seepage pores. The fractal dimension DM and dissolution hole fractal dimension Dc were calculated by SEM image method, respectively controlled by clay mineral and feldspar content. There is a remarkable positive correlation between D1 and DN1 and Langmuir volume of coal, so fractal dimension can effectively quantify the adsorption capacity of coal.

FRACTAL PARAMETERS OF SOIL PORE SURFACE AREA UNDER A DEVELOPING CROP

Fractals ◽  
1996 ◽  
Vol 04 (01) ◽  
pp. 97-104 ◽  
Author(s):  
YA. A. PACHEPSKY ◽  
L. P. KORSUNSKAIA ◽  
M. HAJNOS
Keyword(s):  
Fractal Dimension ◽  
Surface Area ◽  
Plant Development ◽  
Fractal Dimensions ◽  
Carbohydrate Oxidation ◽  
Surface Irregularity ◽  
Pore Surface ◽  
Cutoff Values ◽  
Fractal Parameters ◽  
Pore Surface Area

Fractal parameters of soils has become increasingly important in understanding and quantifying transport and adsorption phenomena in soils. It is not known how soil plant development may affect fractal characteristics of soil pores. We estimated pore surface area fractal parameters from mercury porosimetry data on gray forest soil before and during crop development, in samples both containing and not containing soil carbohydrates known to be important structure-forming agents. Two distinct intervals with different fractal dimensions were found in the range of pore radii from 4 nm to 1 μm. This could be attributed to differences in mineral composition of soil particles of different sizes. The interval of the smallest radii had the highest average fractal dimension close to 3. Smaller surface area fractal dimensions corresponding to low surface irregularity were found in the next interval of radii. The plant development affected neither fractal dimensions nor the cutoff values of soil samples. The carbohydrate oxidation caused a significant increase in the fractal dimension in the interval of larger radii, but did not affect fractal dimension in the interval of small radii. The cutoff values decreased after carbohydrate oxidation.

scholarly journals Fractal analysis of bentonite modified with heteropoly acid using nitrogen sorption and mercury intrusion porosimetry

2011 ◽  
Vol 76 (10) ◽  
pp. 1403-1410 ◽  
Author(s):  
Srdjan Petrovic ◽  
Zorica Vukovic ◽  
Tatjana Novakovic ◽  
Zoran Nedic ◽  
Ljiljana Rozic
Keyword(s):  
Fractal Dimension ◽  
Specific Surface ◽  
Surface Area ◽  
Specific Surface Area ◽  
Heteropoly Acid ◽  
Mercury Intrusion Porosimetry ◽  
Fractal Dimensions ◽  
Surface Fractal Dimension ◽  
Mercury Intrusion ◽  
Surface Fractal

Experimental adsorption isotherms were used to evaluate the specific surface area and the surface fractal dimensions of acid-activated bentonite samples modified with a heteropoly acid (HPW). The aim of the investigations was to search for correlations between the specific surface area and the geometric heterogeneity, as characterized by the surface fractal dimension and the content of added acid. In addition, mercury intrusion was employed to evaluate the porous microstructures of these materials. The results from the Frankel-Halsey-Hill method showed that, in the p/p0 region from 0.75 to 0.96, surface fractal dimension increased with increasing content of heteropoly acid. The results from mercury intrusion porosimetry (MIP) data showed the generation of mesoporous structures with important topographical modifications, indicating an increase in the roughness (fractal geometry) of the surface of the solids as a consequence of the modification with the heteropoly acid. By comparison, MIP is preferable for the characterization because of its wide effective probing range.

scholarly journals Relevance of coral geometry in the outcomes of the coral-algal benthic war

2018 ◽  
Author(s):  
Emma E. George ◽  
James Mullinix ◽  
Fanwei Meng ◽  
Barbara Bailey ◽  
Clinton Edwards ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Surface Area ◽  
Essential Element ◽  
Fractal Dimensions ◽  
Environmental Conservation ◽  
Microbial Composition ◽  
Surface Area Ratio ◽  
Surface Fractal Dimension ◽  
Surface Fractal ◽  
The Mean

AbstractCorals have built reefs on the benthos for millennia, becoming an essential element in marine ecosystems. Climate change and human impact, however, are favoring the invasion of non-calcifying benthic algae and reducing coral coverage. Corals rely on energy derived from photosynthesis and heterotrophic feeding, which depends on their surface area, to defend their outer perimeter. But the relation between geometric properties of corals and the outcome of competitive coral-algal interactions is not well known. To address this, 50 coral colonies interacting with algae were sampled in the Caribbean island of Curaçao. 3D and 2D digital models of corals were reconstructed to measure their surface area, perimeter, and polyp sizes. A box counting algorithm was applied to calculate their fractal dimension. The perimeter and surface dimensions were statistically non-fractal, but differences in the mean surface fractal dimension captured relevant features in the structure of corals. The mean fractal dimension and surface area were negatively correlated with the percentage of losing perimeter and positively correlated with the percentage of winning perimeter. The combination of coral perimeter, mean surface fractal dimension, and coral species explained 19% of the variability of losing regions, while the surface area, perimeter, and perimeter-to-surface area ratio explained 27% of the variability of winning regions. Corals with surface fractal dimensions smaller than two and small perimeters displayed the highest percentage of losing perimeter, while corals with large surface areas and low perimeter-to-surface ratios displayed the largest percentage of winning perimeter. This study confirms the importance of fractal surface dimension, surface area, and perimeter of corals in coral-algal interactions. In combination with non-geometrical measurements such as microbial composition, this approach could facilitate environmental conservation and restoration efforts on coral reefs.

scholarly journals Nano-Scale Pore Structure and Its Multi-Fractal Characteristics of Tight Sandstone by N2 Adsorption/Desorption Analyses: A Case Study of Shihezi Formation from the Sulige Gas Filed, Ordos Basin, China

Minerals ◽  
2020 ◽  
Vol 10 (4) ◽  
pp. 377
Author(s):  
Zhelin Wang ◽  
Xuewei Jiang ◽  
Mao Pan ◽  
Yongmin Shi
Keyword(s):  
Fractal Dimension ◽  
Specific Surface ◽  
Pore Structure ◽  
Surface Area ◽  
Mineral Composition ◽  
Fractal Dimensions ◽  
Contribution Rate ◽  
Tight Sandstone ◽  
N2 Adsorption ◽  
Adsorption Desorption

Fractal dimension is a critical parameter to evaluate the heterogeneity of complex pore structure in tight sandstone gas and other low permeability reservoirs. To quantify the fractal dimension of tight sandstone at various pore size classes and evaluate their implications on mineral composition and nano pore structure parameters, we conducted an integrated approach of N2 adsorption/desorption experiment (N2-GA), X-ray diffraction (X-RD), and field emission scanning electron microscopy (FE-SEM) on Sulige tight sandstone reservoirs. By comparing the nine types of fractal dimensions calculated from N2 adsorption data, we put forward the concept of “concentrated” fractal dimensions and “scattered” fractal dimensions (DN2, DN3, DN5, DN7 and DN8) for the first time according to its concentration extent of distribute in different samples. Result shows that mineral composition has a significant influence of a different level on specific surface area (SSA), pore volume (PV), and fractal dimensions (DN), respectively, where the “scattered” fractal dimension is more sensitive to certain specific property of the reservoir, including mineral content and the specific surface area contribution rate (Sr) of type II mesopores (Mesopore-II: 10~50nm). In addition, three type of hysteresis loops were distinguished corresponding to different pore shape combination of N2-GA isotherm curve, which reveals that pore structure heterogeneity is mainly controlled by inkbottle-shaped pores and the volume contribution rate (Vr) of mesopores in this study area. These findings could contribute to a better understanding of the controlling effect of pore heterogeneity on natural gas storage and adsorption.

FRACTAL DIMENSION, PRIMES, AND THE PERSISTENCE OF MEMORY

2003 ◽  
Vol 06 (02) ◽  
pp. 241-249
Author(s):  
JOSEPH L. PE
Keyword(s):  
Number Theory ◽  
Fractal Dimension ◽  
Fractal Dimensions ◽  
Distinguishing Characteristic ◽  
Local Behavior ◽  
Remarkable Similarity ◽  
Recursive Procedures

Many sequences from number theory, such as the primes, are defined by recursive procedures, often leading to complex local behavior, but also to graphical similarity on different scales — a property that can be analyzed by fractal dimension. This paper computes sample fractal dimensions from the graphs of some number-theoretic functions. It argues for the usefulness of empirical fractal dimension as a distinguishing characteristic of the graph. Also, it notes a remarkable similarity between two apparently unrelated sequences: the persistence of a number, and the memory of a prime. This similarity is quantified using fractal dimension.

scholarly journals Supramolecular Fractal Growth of Self-Assembled Fibrillar Networks

Gels ◽  
2021 ◽  
Vol 7 (2) ◽  
pp. 46
Author(s):  
Pedram Nasr ◽  
Hannah Leung ◽  
France-Isabelle Auzanneau ◽  
Michael A. Rogers
Keyword(s):  
Fractal Dimension ◽  
Crystallization Temperature ◽  
Cayley Tree ◽  
Fractal Dimensions ◽  
Self Similarity ◽  
Avrami Model ◽  
Cluster Aggregation ◽  
Box Counting ◽  
Self Assembled ◽  
Limited Diffusion

Complex morphologies, as is the case in self-assembled fibrillar networks (SAFiNs) of 1,3:2,4-Dibenzylidene sorbitol (DBS), are often characterized by their Fractal dimension and not Euclidean. Self-similarity presents for DBS-polyethylene glycol (PEG) SAFiNs in the Cayley Tree branching pattern, similar box-counting fractal dimensions across length scales, and fractals derived from the Avrami model. Irrespective of the crystallization temperature, fractal values corresponded to limited diffusion aggregation and not ballistic particle–cluster aggregation. Additionally, the fractal dimension of the SAFiN was affected more by changes in solvent viscosity (e.g., PEG200 compared to PEG600) than crystallization temperature. Most surprising was the evidence of Cayley branching not only for the radial fibers within the spherulitic but also on the fiber surfaces.

scholarly journals Dynamic characteristics and fractal representations of crack propagation of rock with different fissures under multiple impact loadings

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Bing Sun ◽  
Shun Liu ◽  
Sheng Zeng ◽  
Shanyong Wang ◽  
Shaoping Wang
Keyword(s):  
Fractal Dimension ◽  
Crack Propagation ◽  
Crack Growth ◽  
Impact Test ◽  
Fractal Theory ◽  
Dynamic Change ◽  
Fractal Dimensions ◽  
Peak Stress ◽  
Dynamic Mechanical Properties ◽  
Gravitational Potential Energy

AbstractTo investigate the influence of the fissure morphology on the dynamic mechanical properties of the rock and the crack propagation, a drop hammer impact test device was used to conduct impact failure tests on sandstones with different fissure numbers and fissure dips, simultaneously recorded the crack growth after each impact. The box fractal dimension is used to quantitatively analyze the dynamic change in the sandstone cracks and a fractal model of crack growth over time is established based on fractal theory. The results demonstrate that under impact test conditions of the same mass and different heights, the energy absorbed by sandstone accounts for about 26.7% of the gravitational potential energy. But at the same height and different mass, the energy absorbed by the sandstone accounts for about 68.6% of the total energy. As the fissure dip increases and the number of fissures increases, the dynamic peak stress and dynamic elastic modulus of the fractured sandstone gradually decrease. The fractal dimensions of crack evolution tend to increase with time as a whole and assume as a parabolic. Except for one fissure, 60° and 90° specimens, with the extension of time, the increase rate of fractal dimension is decreasing correspondingly.

LATTICE DYNAMICS OF THE BINARY APERIODIC CHAINS OF ATOMS I: FRACTAL DIMENSION OF PHONON SPECTRA

1995 ◽  
Vol 09 (12) ◽  
pp. 1429-1451 ◽  
Author(s):  
WŁODZIMIERZ SALEJDA
Keyword(s):  
Fractal Dimension ◽  
Lattice Dynamics ◽  
Nearest Neighbor ◽  
Average Distance ◽  
Local Maximum ◽  
Fractal Dimensions ◽  
Dimension Formula ◽  
Phonon Spectra ◽  
Wide Range ◽  
Silver Mean

The microscopic harmonic model of lattice dynamics of the binary chains of atoms is formulated and studied numerically. The dependence of spring constants of the nearest-neighbor (NN) interactions on the average distance between atoms are taken into account. The covering fractal dimensions [Formula: see text] of the Cantor-set-like phonon spec-tra (PS) of generalized Fibonacci and non-Fibonaccian aperiodic chains containing of 16384≤N≤33461 atoms are determined numerically. The dependence of [Formula: see text] on the strength Q of NN interactions and on R=mH/mL, where mH and mL denotes the mass of heavy and light atoms, respectively, are calculated for a wide range of Q and R. In particular we found: (1) The fractal dimension [Formula: see text] of the PS for the so-called goldenmean, silver-mean, bronze-mean, dodecagonal and Severin chain shows a local maximum at increasing magnitude of Q and R>1; (2) At sufficiently large Q we observe power-like diminishing of [Formula: see text] i.e. [Formula: see text], where α=−0.14±0.02 and α=−0.10±0.02 for the above specified chains and so-called octagonal, copper-mean, nickel-mean, Thue-Morse, Rudin-Shapiro chain, respectively.

Fractal dimension based features are useful descriptors of leaf complexity and shape

1999 ◽  
Vol 29 (9) ◽  
pp. 1301-1310 ◽  
Author(s):  
Wojciech Borkowski
Keyword(s):  
Fractal Dimension ◽  
Tree Species ◽  
Fractal Dimensions ◽  
Plant Identification ◽  
Simple Method ◽  
Mass Dimension ◽  
Scale Range ◽  
Computer Identification

An application of fractal dimensions as measures of leaf complexity to morphometric studies and automated plant identification is presented. Detailed algorithms for the calculation of compass dimension and averaged mass dimension together with a simple method of grasping the scale range related variability are given. An analysis of complexity of more than 300 leaves from 10 tree species is reported. Several classical biometric descriptors as well as 16 fractal dimension features were computed on digitized leaf silhouettes. It is demonstrated that properly defined fractal dimension based features may be used to discriminate between species with more than 90% accuracy, especially when used together with other measures. It seems, therefore, that they can be utilized in computer identification systems and for purely taxonomical purposes.

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