Fractal Geometry Approach Research Articles

Effect of particle shape produced from wet ball milling on magnetic separation concentrate by fractal dimension

The particle shape in mineral processing is a function of the comminution mechanism. In this study, the effect of the concentrate particle shape, obtained from the wet ball milling and the low-intensity magnetic separator on magnetite ore is investigated by the fractal geometry approach. The particle shape is described by the fractal dimension, calculated by the divider method. The variable parameters of grinding time, loading ratio, and solid weight percentage were studied here. The fractal dimensions of the particles are obtained between 1.12 and 1.42. When the grinding time is increased from 45 to 65 minutes, the fractal dimension and product grades are increased from 1.19 to 1.42 and from 65.61 to 67.70%, respectively. By increasing the loading ratio from 14% to 22%, the fractal dimension and the product’s grades are increased from 1.17 to 1.38 and from 64.80 to 66.68%, respectively. By increasing the solid weight percentage from 35 to 55%, the fractal dimension and product grades are decreased from 1.32 to 1.12 and from 65.51 to 64.65%, respectively. The results show that the particle shape has a significant effect on the performance of the magnetic separator. It is the findings indicate a direct relationship between the particle fractal dimension and the grade of magnetic separation products.

Analysis of Graphene Pythagoras Tree Fractal Antenna with Thin SiO2 Substrate in Terahertz Regime

Abstract The Pythagoras Tree Fractal patch is considered on a SiO2 substrate thickness of 1.5 µm to radiate at four frequencies of 4.975THz, 5.38THz, 6.73THz, and 7.61THz with Voltage Standing Wave Ratio (VSWR) ≤ 2. This is a very creative fractal design of a multiband THz antenna, based on a very thin graphene layer, the study proved a value in terms of high radiation efficiency and high gain, at the level of current research we made a comparison study where the design has evidently huge potential regarding applicability for telecommunication technology in the terahertz regime. The demand for high-performance terahertz (THz) antennas has increased significantly in recent years due to their potential applications in various fields such as medical imaging, security screening, and wireless communications. In this paper, the authors present an analysis of a graphene Pythagoras Tree Fractal (GPTF) antenna with a thin SiO2 substrate for THz regime. The GPTF antenna is designed using a fractal geometry approach, which provides multiple resonant frequencies and enhances the overall radiation efficiency. The thin SiO2 substrate is used to reduce the substrate losses and improve the radiation performance of the antenna. The authors use the Finite-Difference Time-Domain (FDTD) software to simulate the performance of the proposed antenna. The results show that the proposed antenna exhibits high gain, low return loss, and wide bandwidth, making it a promising candidate for THz applications.

Evaluation of the shear strength parameters of a compacted S-RM fill using improved 2-D and 3-D limit equilibrium methods

It is a common construction practice to use mixtures of soils and rock fragments for engineered fills in slopes and embankments, making the assessment of their shear strength properties a fundamental step for determining the stability of such structures. In situ bimtests were performed in an artificial soil-rock mixture (S-RM) slope constructed for an electronic converter station in the Funing County, Yunnan Province, China. Improved 2-D and 3-D limit equilibrium methods, accounting for the difference between the peak pushing horizontal forces applied during the first and the second loading cycle of the test, neglected in previous researches, are proposed for correctly assessing the shear strength parameters of the S-RM. Results show that neglecting the difference between the two peak pushing horizontal forces can produce relatively large errors in the back-calculated strength properties of S-RMs, especially when the cohesive term is low. These errors are particularly significant when the analysis is performed in 3-D condition. The number of sections required in the simplified 2-D approach is also determined based on the relative errors of estimated strength parameters in comparison to those deriving from a 2-D analysis considering a very large number of sections (i.e. 133 sections). The shear mechanical properties of S-RM samples with different volumetric block proportion (VBP) and block strength are subsequently described. S-RMs with higher VBP have more tortuous shear surface, especially when stiff rock blocks are employed. The cohesion of “stiff S-RMs” decreases with the increment of VBP, while the friction angle increases. Conversely, both the strength parameters increase with VBP for the S-RMs containing soft rock blocks (i.e. “soft S-RMs”), even if the cohesion assumes negligible values. The shear behavior of soft S-RMs is different from that of stiff S-RMs when the VBP is the same, indicating that the block strength also may affect the strength of this class of geomaterials. Finally, by adopting a fractal geometry approach, it is demonstrated that there is a large correlation between the degrees of tortuosity of shear surfaces and the friction angles of S-RMs.

Creeping microstructure and fractal permeability model of natural gas hydrate reservoir

The successful pilot testing of depressurization production for natural gas hydrate reservoir was conducted at Shenhu Area in the South China Sea. However, pressure changes due to depressurization production cause shrinkage of creeping pore and throat space for fluid flow, ultimately resulting in permeability damage in this reservoir. To optimize gas recovery for natural gas hydrate reservoir, it is important to understand the substantial variations of the pore structure and physical properties under different pressure conditions. In this study, by integrating computed tomography imaging with water flow experiments on Clay-silt sediment sample, we analyze how pore scale structural parameters (such as average pore and throat radius, pore and throat median radius, maximum pore and throat radius, and fractal dimension) change under different axial stresses by using fractal geometry approach. It is found that there is a negative relationship between axial stress and structural parameters. Meanwhile, with the axial stress increases, the range of pore and throat radius distribution located on the right of center distribution decreases. On the basis of fractal geometry theory, a fractal model is then proposed to explain the effects of axial stress and creeping microstructure on permeability for natural gas hydrate reservoir. Results show that the model provides good match with experimental data when the axial stress is larger, while poor agreements with experimental results at low pressure. This study helps us understand fundamental mechanism for permeability changes during the depressurization of gas hydrate reservoir.

Fractal Dimension as a Diagnostic Tool for Cardiac Diseases

The discovery of fractal geometry has been one of the major developments in mathematics. Fractals, defined as selfsimilar structures, provide a new approach to the understanding of irregular structures. The dimensions of complete fractals can be easily calculated as real numbers with the fractal geometry approach. However, most of the structures in nature do not demonstrate self-similarity fully, and different approaches are needed for dimension calculations. These structures with semi-fractal properties are known as quasi fractals. Fractals derived from time series are called statistical fractals and they are an example for quasi fractals. Various methods have been developed to estimate the dimensions of statistical fractals. In this study, the methods which predict dimension in statistical fractals were investigated and the most suitable method for time series was identified. Fractal dimensions were calculated by using ECGs of 236 individuals, and the data set was divided into four disease groups and a control group of healthy individuals. Various statistical analyses were performed using the fractal dimensions of the graphs computed with MATLAB. Statistical hypothesis tests showed that the differences between the group mean of fractal dimensions are significant. Fractal dimensions of ECGs have the potential to be used as a diagnostic tool in the diagnosis of heart diseases.

Fractal analysis of river newtworks based on remote sensing data: an example from Kelani River Basin, Sri Lanka

River network patterns are important components of basin geomorphology. The geomorphological complexity of river networks has an impact on various natural processes and hazardous events. In order to find those correlations and establish quantitative relationships, a comprehensive textural examination on river network patterns is crucial. In this study, the fractal geometry approach is applied to analyse river network by the means of their geometric distinctiveness. Since river networks are fractal objects formed by repetition of certain natural processes over a long period of time, fractal geometry provides the most sensitive method to analyse their complex branching structure. Five river networks from Kelani river basin (1. Ambalanpiti oya, 2. Gurugoda oya, 3. Pugoda oya, 4. Pusweli oya, 5. Wakoya) are subjected to fractal analysis. Fixed-size box counting algorithm is applied to obtain fractal measures. The recursive algorithm is applied to the same river network twice; allocating weighted-lines for different orders and then allocating non-weighted equal width line for all tributaries, to test the best suitable option to model their geomorphological complexity. Non-plane-filling behaviour of river networks is confirmed by present values which are greater than 1 and lesser than 2. The largest fractal dimension value is for Gurugodaoya tributary, confirming that it has the most complex geomorphologic pattern. Smallest value is from Pugodaoya concluding the least geomorphological complexity. Multifractal spectra f(α) are constructed for each river network and detailed investigation is required (considering lithological features of the basin) to link f(α) to the physical characteristics of river network.

A stress sensitivity model for the permeability of porous media based on bi-dispersed fractal theory

A stress sensitivity model for the permeability of porous media based on bidispersed fractal theory is established, considering the change of the flow path, the fractal geometry approach and the mechanics of porous media. It is noted that the two fractal parameters of the porous media construction perform differently when the stress changes. The tortuosity fractal dimension of solid cluster [Formula: see text] become bigger with an increase of stress. However, the pore fractal dimension of solid cluster [Formula: see text] and capillary bundle [Formula: see text] remains the same with an increase of stress. The definition of normalized permeability is introduced for the analyzation of the impacts of stress sensitivity on permeability. The normalized permeability is related to solid cluster tortuosity dimension, pore fractal dimension, solid cluster maximum diameter, Young’s modulus and Poisson’s ratio. Every parameter has clear physical meaning without the use of empirical constants. Predictions of permeability of the model is accordant with the obtained experimental data. Thus, the proposed model can precisely depict the flow of fluid in porous media under stress.

Design of Fractal Features-Based Partial Discharge Pattern Recognition using Multi Support Vector Machine Method

Partial Discharge (PD) is one of the causes of insulation deteriorisation mode and impacts on the reliability of high voltage equipment. Therefore, PD measurement is used for diagnostic technique of high voltage equipment. Diagnostic output of high voltage equipment contain information about PD type, PD cause, PD location and PD severity. after identification, a proper preventive maintenance pattern can be performed. Therefore PD pattern recognition system is very important on PD diagnostic system to recognize the PD pattern and determine the level of hazard that occurs in specimen object or high voltage equipment‥ In this paper, PD pattern recognition system is designed with fractal geometry approach and support vector machine (SVM) algorithm. The coding and programming of graphical user interface of the application is done. Each PD type and hazard level on various insulating materials (solid, liquid and gas) have the dimensions of the fractal and the lacunarity. The type of PD (void, corona) and its danger level (bad, fair and good) can be identified with the support vector machine (SVM)

Methodology for Fast and Facile Characterisation of Carbon-Based Electrodes Focused on Bioelectrochemical Systems Development and Scale Up.

The development and practical implementation of bioelectrochemical systems (BES) requires an in-depth characterisation of their components. The electrodes, which are critical elements, are usually built from carbon-based materials due to their high specific surface area, biocompatibility and chemical stability. In this study, a simple methodology to electrochemically characterise carbon-based electrodes has been developed, derived from conventional electrochemical analyses. Combined with classical electrochemical theory and the more innovative fractal geometry approach, our method is aimed at comparing and characterising the performance of carbon electrodes through the determination of the electroactive surface and its fractal dimension. Overall, this methodology provides a quick and easy method for the screening of suitable electrode materials to be implemented in BES.

Characterizing Woodiness Carbon Fiber Precursor by Fractal Geometry Approach in Different Curing Conditions

For exploring the variation of the surface morphology of carbon fiber precursor which was obtained from different curing conditions,this paper used the fractal dimension measurement method to analyze morphological characters of carbon fiber precursor surface through the electronic scanning electron microscopy image recognition and computer fractal calculation system. The experimental results show:the surface fractal dimensions of carbon fiber precursor were between 2.480~2.622 by TPSAM method, and the surface fractal dimensions of carbon fiber precursor were between 2.555~2.633 by PCM method, both methods show that the carbon fiber precursor surface is smooth. And the CCM is 2.678~2.755, this method shows that the precursor surface is concave and convex. The fractal dimension value obtained by TPSAM method and PCM method is closer, while the CCM is different. But the trend is almost the same, which shows that using the fractal dimension method to desirable the surface structure of carbon fiber precursor is desirable.

Scaling Analysis of Water Retention Curves: A Multi-fractal Approach

Water retention curve (WRC) is analyzed by means of the fractal geometry approach. Three models accounting for the fractal distribution of either the pore and solid phase of unsaturated porous media have been considered. By using data collected during a field scale internal drainage, we determine the functional relationship between the WRC, and the fractal dimension(s). In particular, it is shown that the fractal scaling of the WRC is feasible provided that a large enough set of measurements at the lowest water contents is available.

Optical assessment of vibrating systems: A fractal geometry approach

Vibrating beams were analyzed by the fractal dimension associated to speckle patterns, obtained by impinging a laser beam onto the surface of the samples and collecting the inelastically scattered light. The results show a significant correlation between the mechanical status of the beams and the corresponding fractal dimension.

Synergetics and fractal dimensions in tribology

The possibility is substantiated of applying a fractal geometry approach to the study of tribological processes, i.e., pre-fracture and fracture (wear) of metallic materials, checking of the critical points in their mechanical state, and prediction of fatigue damageability in contact interaction and friction of rough surfaces. The substantiation is based on quantitative evaluation of the fractal dimension of the deformation relief by the vertical section method using the self-similar ratio of the perimeter to the area of the closed profiles of the fractal. A synergetic approach to the analysis of plastic deformation and fracture with regard to the collective effects and self-organization of dissipative structures and the established hierarchy of structural (scale) levels of deformation gives grounds to search for relationships between the microand macroparameters of plastic deformation and fracture based on the theory of fractals as self-similar defects and fractal dimensions as a characteristic of the dynamic structure of an object.

Investigation of bipolar plate and diffusion media interfacial structure in PEFCs: A fractal geometry approach

The imperfect interfacial contact between the bipolar plate (BP) and the diffusion medium (DM) can have a significant impact on the multi-phase flow and current transport in an operating polymer electrolyte fuel cell (PEFC). The objective of this work is to describe the impact of the BP and DM surface morphologies and the resulting interfacial contact on PEFC performance. In this study, the surface morphology of several BP and DM samples was digitally characterized using optical profilometry (OP). The benchmark surface data were then utilized in a microscopic model developed to simulate the BP|DM interfacial contact under compression. The microscopic model is based on the fractal modeling approach, which provides an accurate representation of the BP|DM interfacial contact by suppressing the resolution dependence of the surface profiles in consideration. Results indicate that the uncompressed surface morphology of mating materials, elasticity of these components, and local compression pressure are the key parameters that influence the BP|DM contact. The model results show that the void space along the BP|DM interface can potentially store a significant amount of liquid water (from 0.85 to 3.5 mg/cm 2), which can result in reduced durability and performance of the PEFC. The model predicts that a 50% drop in the DM surface roughness results in nearly a 40% drop in the BP|DM contact resistance and a 15% drop in the BP|DM interfacial water storage capacity.

Fractal Geometry Approach to Describe Mesostructured Boehmite and Gamma‐Alumina Nanorods

AbstractMesoporous boehmite and gamma‐alumina with nanorod‐like morphology, exhibiting a high surface area and large pore volume, have been prepared by a nonsurfactant‐templating sol–gel technique with a simple precursor system, aluminum isopropoxide, ethanol, 2‐propanol, or 2‐butanol as solvents, and sulfuric or hydrochloric acid as peptizing agents. The effect of the solvent‐type and the peptizing agent were studied, and optimum conditions were determined. A fractal geometry approach was adopted to describe the subtle alterations in the smoothness of the surface as a function of the synthesis conditions. Boehmite and γ‐alumina nanorods were characterized by nitrogen physisorption, wide‐ and small‐angle XRD, SAXS, 27Al MAS NMR spectroscopy, TEM, and TGA–DTA. Both boehmite and gamma‐alumina nanorods have a large surface area, pore volume, and pore size.

On the influence of the surface roughness onto the ultrathin SiO 2/Si structure properties

The surface roughness of the semiconductor substrate substantially influences properties of the whole semiconductor/oxide structure. SiO 2/Si structures were prepared by using low temperature nitric acid oxidation of silicon (NAOS) method and then the whole structure was passivated by the cyanidization procedure. The influence of the surface morphology of the silicon substrate onto the electrical properties of ultrathin NAOS SiO 2 layer was investigated. Surface height function properties were studied by the AFM method and electrical properties were studied by the STM method. The complexity of analyzed surface structure was sensitive to the oxidation and passivation steps. For describing changes in the oxide layer structure, several fractal measures in an analysis of the STM images were used. This fractal geometry approach enables quantifying the fine spatial changes in the tunneling current spectra.

Preparation of self-organized porous tungsten oxide using HFCVD technique

Abstract Hot filament chemical vapour deposition (HFCVD) technique was applied to deposit a porous tungsten oxide film on glass wafers. The tungsten filament was used as a source in a vacuum atmosphere. The porous film was characterized by scanning electron microscopy, energy-dispersive X-ray spectroscopy, X-ray thermodiffraction, nitrogen sorption and small-angle X-ray scattering. From these characterization techniques it was found that porous film presents a clusters-like morphology of WO3−x particles. The particles are arranged on substrate in a way that free spaces are originated, as a 3D network of pores. By increasing temperature, the BET specific surface area of the porous film changes from 38.67 to 34.5 m2 g−1 most likely due to the particles have a tendency to stick together to form aggregates, particularly at high temperature. A fractal geometry approach permits to elucidate the interconnection between the particles and a simple model of the porous structure is proposed.

Fractal Extra-Framework Species in De-aluminated LaY Zeolites and Their Catalytic Activity

Lanthanum-containing Y (LaY) zeolites were prepared by ionic exchange from NaY parent zeolite. The LaY zeolites were de-aluminated by steaming. De-aluminated zeolites presented different Si/Al ratio. The physicochemical properties of these catalysts were characterized by X-ray diffraction, pyridine and xenon adsorption, infrared spectroscopy and 29Si, 27Al, 129Xe, 139La solid-state nuclear magnetic resonance spectroscopy. Furthermore, a fractal geometry approach was adopted to describe the evolution in the texture as a consequence of de-alumination. The catalytic properties of materials were evaluated in the n-hexane cracking reaction. The catalyst with the highest catalytic activity was the zeolite highest de-aluminated (Si/Al ratio of 3.7). Such performance was attributed on the one hand, to active extra-framework aluminum species hosted in the large cavities of zeolites and, on the other hand to redistribution of lanthanum species into the zeolite as a consequence of de-alumination.

Prediction of surface roughness profiles for milled surfaces using an artificial neural network and fractal geometry approach

Abstract Artificial neural networks (ANNs) models were developed for the analysis and prediction of the relationship between the cutting conditions and the corresponding fractal parameters of machined surfaces in face milling operation. These models can help manufacturers to determine the appropriate cutting conditions, in order to achieve specific surface roughness profile geometry, and hence achieve the desired tribological performance (e.g. friction and wear) between the contacting surfaces. The input parameters of the “ANNs” models are the cutting parameters: rotational speed, feed, depth of cut, pre-tool flank wear and vibration level. The output parameters of the model are the corresponding calculated fractal parameters: fractal dimension “D” and vertical scaling parameter “G”. The model consists of three-layered feed-forward back-propagation neural network. ANNs models were utilized successfully for modeling and predicting the fractal parameters “D” and “G” in face milling operations. Moreover, W–M fractal function was integrated with the developed ANNs models in order to generate an artificially fractal predicted profiles at different cutting conditions. The predicted profiles were found statistically similar to the actual measured profiles of test specimens.

Irreversible Adsorption of an Aztec Dye on Fractal Surfaces

Carminic acid was adsorbed into amorphous silicas prepared by the sol−gel process. While cromophore is gradually released in silica when put in contact with water, irreversible adsorption takes place when silica is doped with aluminum. The role of aluminum on effective retention of carminic acid by aluminic silicas is described. A fractal geometry approach was employed to describe changes in silica samples when they are doped with aluminum and when chromophore is incorporated. Inclusion of chromophore provokes aggregates to form interconnected fractal surfaces.