Fractal Model Research Articles

Fractal model of thermal contact conductance of rough surfaces based on axisymmetric cosinusoidal asperity

PurposeThe purpose of this study is to propose a fractal model of thermal contact conductance of rough surfaces based on axisymmetric cosinusoidal asperity.Design/methodology/approachThe effects of contact load, fractal dimension, fractal roughness and friction coefficient on the thermal contact conductance of rough surfaces were investigated in this study.FindingsThe findings suggest that as the contact load increases, the thermal contact conductance of rough surfaces also increases. In addition, an increase in the fractal dimension corresponds to an increase in the thermal contact conductance. Conversely, an increase in fractal roughness leads to a decrease in thermal contact conductance. The smaller the friction coefficient, the lower the thermal contact conductance of the rough surface. In practical engineering applications, it is possible to achieve the desired thermal contact conductance of rough surfaces by selecting surfaces with appropriate roughness.Originality/valueA fractal model of thermal contact conductance of rough surfaces based on axisymmetric cosinusoidal asperity was established in this study. The findings of this study offer a theoretical foundation for investigating the thermal contact conductance of rough surfaces.

The Spectrum of Low-pTJ/ψ in Heavy-Ion Collisions in a Statistical Two-Body Fractal Model.

We establish a statistical two-body fractal (STF) model to study the spectrum of J/ψ. J/ψ serves as a reliable probe in heavy-ion collisions. The distribution of J/ψ in hadron gas is influenced by flow, quantum and strong interaction effects. Previous models have predominantly focused on one or two of these effects while neglecting the others, resulting in the inclusion of unconsidered effects in the fitted parameters. Here, we study the issue from a new point of view by analyzing the fact that all three effects induce a self-similarity structure, involving a J/ψ-π two-meson state and a J/ψ, π two-quark state, respectively. We introduce modification factor qTBS and q2 into the probability and entropy of charmonium. qTBS denotes the modification of self-similarity on J/ψ, q2 denotes that of self-similarity and strong interaction between c and c¯ on quarks. By solving the probability and entropy equations, we derive the values of qTBS and q2 at various collision energies and centralities. Substituting the value of qTBS into distribution function, we successfully obtain the transverse momentum spectrum of low-pT J/ψ, which demonstrates good agreement with experimental data. The STF model can be employed to investigate other mesons and resonance states.

Discrete element modelling of the size-dependent mechanical behavior of Al2O3 ceramic under quasi-static uniaxial compression

The size-dependent mechanical behavior of Al2O3 ceramic under quasit-static uniaxial compression was simulated via two-dimensional discrete element method. The numerical sample of Al2O3 ceramic was established by adopting the linear parallel bond contact model, and the inherent defects was further generated by randomly removing a subset of elementary discs. The uniaxial compressive strength of Al2O3 ceramic samples with different sizes follows the Weibull distribution. With an increase in sample size, the magnitude of the uniaxial compressive strength decreases, while its variability increases. Additionally, with sample size increases, both the Young's modulus and Poisson's ratio decrease. Meanwhile, the variability in Young's modulus and Poisson's ratio is more pronounced in larger samples than in smaller ones. The progressive fracture process of Al2O3 ceramic sample can be categorized into four typical stages, namely linear-elastic stage, crack initiation stage, crack propagation stage and complete failure stage. The final fragment size distributions of Al2O3 ceramic samples of different sizes can be accurately described by the fractal model. Notably, smaller samples tend to exhibit a greater breakage extent compared to larger samples. Finally, a size-dependent statistical equation of uniaxial compressive strength of Al2O3 ceramic is proposed based on the numerical results and the inverse function of the Weibull distribution, which can provide a theoretical basis for the design and production of ceramic materials.

Stress-Dependent Fractal Model for Predicting the Relative Permeability of Rocks Based on Equivalent Capillaries

Stress-Dependent Fractal Model for Predicting the Relative Permeability of Rocks Based on Equivalent Capillaries

A SCALING LAW RELATING THE RATE OF DESTRUCTION OF A SOLID TUMOR AND THE FRACTAL DIMENSION OF ITS BOUNDARY

In this paper, we investigate the scaling law relating the size of the boundary of a solid tumor and the rate at which it is lysed by a cell population of non-infiltrating cytotoxic lymphocytes. We do it in the context of enzyme kinetics through geometrical, analytical and numerical arguments. Following the Koch island fractal model, a scale-dependent function that describes the constant rate of the decay process and the fractal dimension is obtained. Then, in silico experiments are accomplished by means of a stochastic hybrid cellular automaton model. This model is used to grow several tumors with varying morphology and to test the power decay law when the cell-mediated immune response is effective, confirming its validity.

New Model for Coal Gas Diffusion Based on the Fractal Tree-Like Bifurcation Network Structure.

To investigate the nonstationary diffusion and transport law of gas in coal, an innovative fractal diffusion model based on fractal theory and a treelike bifurcation network under different diffusion modes were built. In addition, the quantitative relationships among the diffusion coefficient, temperature, pressure, and structural parameters were determined. The model considers the effect of pore tortuosity and connectivity on gas diffusion, which renders it more realistic than the previously presented single-pore diffusion models. Moreover, each parameter in the model has a definite physical meaning and does not contain any empirical constants. The applicability of the new model was verified by experimental data and other model. Subsequently, a sensitivity analysis of the gas diffusion coefficient was performed to study the effect of the microstructural parameters on gas diffusion. Finally, the dispersion degree of the diffusion coefficients at different temperatures and pressures was analyzed to determine the main influencing factors of gas diffusion at different diffusion stages in coal and study their evolution. The results show that the gas diffusion state is more sensitive to pressure variations. The diffusion behavior of gases in coal-based porous media is more influenced by the temperature and pressure at the beginning of the diffusion process.

Numerical simulation for electric field intensity and jet space of the quadratic spiral spinneret with auxiliary electrode

To produce a high quality electrostatic spinneret with low voltage requirements, low energy consumption, and narrow fiber diameter distribution, the mechanical model of the spinning unit of the fractal spiral spinneret was established by using the quadratic fractal spiral parametric equation. The established fractal spiral electrostatic spinning model was imported into COMSOL Multiphysics finite element analysis software, mesh division was performed on the model, multiple spinning units were combined to form the array spinneret, and the optimal parameters of the fractal structure were optimized. However, there are still two sides of the field intensity other than the middle field intensity of the high situation. In order to equalize the field intensity, we used the spinneret properties of the same disc auxiliary electrode and quadratic fractal spiral spinneret in a linear arrangement, and the auxiliary electrode and spinneret distance and spinneret radius of the two important parameters to optimize the simulation, resulting in obtaining a more uniform field intensity distribution of the quadratic fractal spiral spinneret electrospinning equipment. Based on the calculation of the fractional dimension of the Von Koch curve, the fractional dimension of the quadratic fractal spinneret was 1.77. The critical field intensity of the system was calculated to be 2.13 × 106 V/m, and the jet space was 1.22–3.13 mm. The optimized quadratic fractal spinneret model with auxiliary electrode faced the receiving plate in the range of 60° from the left to the right of the spinning sites, –5 to 5 sites, which may emit the spinning jet.

FRACTAL MODEL FOR EFFECTIVE THERMAL CONDUCTIVITY OF COMPOSITE MATERIALS EMBEDDED WITH A DAMAGED TREE-LIKE BIFURCATION NETWORK

Scientists worldwide have always been interested in the study of networks that resemble trees and porous materials. Therefore, based on fractal theory, this paper systematically studies the heat transfer problem of the damaged tree-like networks under different saturations in multi-medium composite materials and derives their dimensionless thermal conductivity (DTC). The research has shown that the dimensionless thermal conductivity (DTC) decreases with an increase in damaged channels. The saturation significantly impacts the heat transfer of the damaged tree networks, and it exhibits linear changes with the ratio of the gas phase, liquid phase, and solid phase. Additionally, it can be observed that the fractal dimension of length distribution and diameter distribution, bifurcation numbers, bifurcation series, porosity, and other parameters in the networks are related to thermal conductivity. The comparison between the data derived from this model and experimental data shows that the proposed model can effectively deepen our understanding of the heat transfer mechanism of the damaged networks in composite materials under different saturation levels. Additionally, the model in this paper does not have an empirical constant, which avoids the influence of potential factors.

Analysis of the path to enhance the quality development of rural tourism with the help of rural revitalization strategy in the context of the Internet

Abstract This paper first determines the research method of rural tourism and the selection of research objects and carries out the corresponding pre-processing operation for the data. Secondly, the spatial distribution characteristics of key villages of rural tourism are analyzed by using the nearest neighbor index, standard deviation ellipse, geographic concentration index and geographic detector so as to understand the development of rural tourism in the five regions of east, west, south, north and central. The spatial clustering structure of key villages in rural tourism is calculated again using the fractal model to explore grid aggregation and correlation of high-quality development in rural tourism. From the spatial distribution characteristics, the nearest neighbor index of the southern region is 0.838, showing an aggregation-random distribution pattern, and the standard deviation ellipse of the southern region is 51.67°. From the spatial clustering structure, the aggregation dimension of the southern region is 0.485, and the correlation dimension of the four regions of east, west, south, north and south is relatively close to that of highway accessibility, for example. Thus, rural tourism in the context of the Internet needs to further strengthen infrastructure construction, enhance the means of training high-quality personnel, build diversified marketing channels, and realize the rural revitalization strategy to boost rural tourism.

Models for Simulation of Fractal-like Particle Clusters with Prescribed Fractal Dimension

This review article delves into the growing recognition of fractal structures in mesoscale phenomena. The article highlights the significance of realistic fractal-like aggregate models and efficient modeling codes for comparing data from diverse experimental findings and computational techniques. Specifically, the article discusses the current state of fractal aggregate modeling, with a focus on particle clusters that possess adjustable fractal dimensions (Df). The study emphasizes the suitability of different models for various Df–intervals, taking into account factors such as particle size, fractal prefactor, the polydispersity of structural units, and interaction potential. Through an analysis of existing models, this review aims to identify key similarities and differences and offer insights into future developments in colloidal science and related fields.

A study of gas transport mechanisms in shale's confined nanopores: Examining irregularity, adsorption effects, and stresses

Many difficulties and challenges have been encountered during the exploration and development of shale gas, among which high flexibility of the reservoir structure and low permeability have been the most notable problems that have restricted the efficient development of shale gas. In this paper, we have developed a fractal apparent permeability model for shale based on fractal theory that has taken into account the confinement effects. Also considering the effect of pore deformation on porosity, the defining equation of pore size under the combined effect of multiple factors is obtained, which, in turn, leads to the defining equation of dynamic fractal dimension. Due to the significant confinement effect due to the development of nanopores in shale reservoirs, the Peng–Robinson equation of state is modified using the adsorption effect, and the influence of the confinement effect on the critical properties and each permeability parameter is considered. Based on this, a shale fractal apparent permeability model coupled with slip flow, Knudsen diffusion, and surface diffusion was developed, and the model was validated with experimental data. The results revealed that the developed model was in relatively better agreement with the measured data. Furthermore, the confinement effect performed a positive role in shale's apparent permeability, with the calculated values of model permeability that considered the confinement effect was greater than the calculated values of model permeability, without the confinement effects being considered.

Fractal theory and dynamic contact angle-based imbibition model for two-phase flow in porous media

During the development of tight oil reservoirs, there are significant occurrences of spontaneous imbibition. Understanding the spontaneous imbibition behavior at the core scale of tight sandstone holds significant importance in improving the recovery rate. This study presents a novel mathematical model for characterizing the spontaneous imbibition phenomenon in tight porous media, drawing upon the fractal theory and the dynamic contact angle in capillary bundles. The proposed model has been verified by the results of core imbibition experiments in the literature. Furthermore, we conducted spontaneous imbibition simulation studies using core structures of different pore types extracted from real tight reservoirs to validate the applicability of the new mathematical model. Comparative analysis shows that the derived mathematical approach fits well with the simulation results, but the heterogeneity of the pore space can lead to certain errors between the model and the simulation results. The influencing factors analysis suggests that the higher the porosity, the higher the final recovery rate, whereas an increase in pore fractal dimension has little effect on the final recovery rate.

Research on Effective Thermal Conductivity in Porous Media Embedded with Randomly Distributed Damaged Tree-like Bifurcation Networks

Due to the complexity of the microstructure of porous media, it is of great significance to explore the heat transport mechanism in porous media in many engineering applications. In this study, an expression for effective thermal conductivity (ETC) of porous media embedded with randomly distributed damaged tree-like bifurcation networks is derived based on the theory of thermodynamics and fractal features of tree-like bifurcation networks. We investigate the effect of heat conduction and heat convection in porous media embedded with randomly distributed damaged tree-like bifurcation networks on the ETC of the porous media. It is found that our fractal model has good consistency with the existing available experimental data. In addition, the influence of the microstructural parameters of the model on heat transfer in the porous media is analyzed in detail. The research results can provide significant theoretical guidance for the development and design of heat transfer systems.

A Fractal Prediction Method for Contact Stiffness of Helical Gear Considering Asperity Lateral Contact and Interaction

The normal contact stiffness (NCS) on rough surfaces has a significant impact on the dynamic characteristics of helical gear. Aiming at the problem of inaccurate calculation of the NCS model under the traditional Hertz theory of smooth surfaces, a fractal prediction model of helical gear contact stiffness considering asperity lateral contact and interaction between asperities is proposed in this paper. The variation formula of asperity and the correction coefficient of a tooth contact surface under asperity lateral contact and interaction are derived, and the influence of micro-elements on normal load and NCS is qualitatively analyzed. The results show that the NCS of considering the interaction and lateral contact of asperity is closer to the experimental results; the contact surface correction coefficient increases with the increase of curvature radius and load. The NCS of a tooth surface increases with the increase in fractal dimension D or the decrease in roughness amplitude G. The influence of asperity lateral contact and interaction decreases with the increase in D and the decrease in G. The NCS of the helical gear decreases under the lateral contact and interaction of the asperity, which is critical for exact estimation of the NCS of contact surfaces in gear.

Spherical seepage model of Bingham fluid in rough and low-permeability porous media

Based on the microstructure of porous media that exhibits statistical self-similarity fractal features, this paper investigates the radial flow characteristics of non-Newtonian fluids within rough porous media. The analytical equation of permeability and starting pressure gradient of Bingham fluid in low permeability rough porous media are established. It is found that the relative roughness is inversely proportional to the permeability and proportional to the starting pressure gradient. In addition, it is also found that the permeability of low permeability porous media decreases spherically with the increase of radial distance and curvature fractal dimension, and increases with the increase of pore area fractal dimension and porosity. Furthermore, the staring pressure gradient is directly proportional to the radial distance, yield stress and curvature fractal dimension. By comparing the model in this paper with the existing experimental data, the correctness and rationality of the spherical seepage fractal model are effectively verified.

Fractal modeling of oil and gas geochemical data in the Taiwan Strait Basin

This study investigated the fractal characteristics of oil and gas indicators in the Taiwan Strait Basin using multifractal methods. The results revealed that these indicators can be categorized into five groups through cluster analysis, principal component analysis, and multifractal analysis. Acidolysis hydrocarbon indicators exhibited strong multifractal characteristics, while other indicators showed weak multifractal characteristics. The high values of Polycyclic aromatic hydrocarbons (F) indicators were mainly found in the southwest direction of the study area, whereas the high values of total aromatic hydrocarbons and their derivatives (UV) indicators were distributed in the eastern part. Erosion carbonate (ΔC) indicators were predominantly located in the western part, while the distribution of acid extractable hydrocarbons (A) and pyrolytic hydrocarbons (P) indicators was sporadic. The distribution of Erosion carbonate (ΔC) indicators in the Jinjiang Depression resembled that in the Jiulongjiang Depression, but the distribution of other indicators varied. The S-A method was employed to identify oil and gas anomaly target areas and predict and evaluate favorable prospect zones in the basin. This study provides a theoretical foundation for future oil and gas exploration and evaluation in the Taiwan Strait.

Local Fuzzy Fractional Partial Differential Equations in the Realm of Fractal Calculus with Local Fractional Derivatives

In this study, local fuzzy fractional partial differential equations (LFFPDEs) are considered using a hybrid local fuzzy fractional approach. Fractal model behavior can be represented using fuzzy partial differential equations (PDEs) with local fractional derivatives. The current methods are hybrids of the local fuzzy fractional integral transform and the local fuzzy fractional homotopy perturbation method (LFFHPM), the local fuzzy fractional Sumudu decomposition method (LFFSDM) in the sense of local fuzzy fractional derivatives, and the local fuzzy fractional Sumudu variational iteration method (LFFSVIM); these are applied when solving LFFPDEs. The working procedure shows how effective solutions for specific LFFPDEs can be obtained using the applied approaches. Moreover, we present a comparison of the local fuzzy fractional Laplace variational iteration method (LFFLIM), the local fuzzy fractional series expansion method (LFFSEM), the local fuzzy fractional variation iteration method (LFFVIM), and the local fuzzy fractional Adomian decomposition method (LFFADM), which are applied to obtain fuzzy fractional diffusion and wave equations on Cantor sets. To demonstrate the effectiveness of the used techniques, some examples are given. The results demonstrate the major advantages of the approaches, which are equally efficient and simple to use in order to solve fuzzy differential equations with local fractional derivatives.

Prediction of relative permeability from capillary pressure based on the fractal capillary bundle model

The accurate prediction of relative permeability is one of the most important issues in the study of multiphase flow in porous media and is involved in many fields, such as geothermal energy development and petroleum engineering. Traditional models such as the Purcell model and Burdine model for predicting relative permeability from capillary pressure usually assume that the wetting phase (WP) flows in small pores and that the nonwetting phase (NWP) flows in large pores, which is a limiting assumption, such that these models cannot describe the process of WP being displaced by NWP. In this study, we propose a generalized theoretical model for calculating relative permeability based on the fluid distribution characteristics and the fractal capillary bundle model. The relative location α of the interface between NWP and WP is introduced to characterize the transport of the two-phase interface in porous media. The developed model was validated with relative permeability experimental data of gas–water/oil and compared with the Li model and the Brooks–Corey model. The comparison results show that the new model provides an improved match to experimental data than the Li model and the Brooks–Corey model. The sensitivity analysis results of the model show that tortuosity and pore size distribution play a key role in predicting relative permeability from the capillary pressure curve. The relative permeability of wetting phase decreases significantly with increasing fractal dimension of tortuosity and fractal dimension of pore size distribution. The effect of parameter α, which reflects fluid transport characteristics, on relative permeability of wetting phase is more pronounced in porous media with complex structures and low/ultralow permeability.

Geochemical prospectivity mapping using compositional balance analysis and multifractal modeling: A case study in the Jinshuikou area, Qinghai, China

To effectively map the prospecting targets related to deposit-type sought, local indicators of spatial association (LISA), compositional balance analysis (CoBA), and multifractal modeling were applied to process stream sediment geochemical data that are collected from the Jinshuikou area, Qinghai province, China. In this study, 14 elements, including Zn, Cu, Ni, Mo, Co, V, Ti, Sb, Ag, Pb, Sn, Hg, Cd, and W, were analyzed in 493 samples. The result of LISA shows that the ore-forming elements Zn and Cu have a close spatial autocorrelation, which supports the construction of balance b1 related to the Zn-Cu mineralization assemblage. A traditional contrast additive method was applied to compare with balance b1 for geochemical anomaly identification. The results clearly indicate that balance b1 significantly assisted with not only reduction of geochemical data internal imbalance caused by extreme value, but also intensification of Zn-Cu assemblage geochemical anomalies associated with Zn polymetallic mineralization. The high anomalies of the two balances, b5 (Co-V-Ti) and b8 (Sn-Pb-Ag-Sb), reflect well the distribution of basic and intermediate-acid rocks in the area, respectively. The geochemical distribution pattern of the balance b1 can be depicted by multifractal parameters such as singular index α, as well as asymmetry index R, which are useful for describing the spatial heterogeneity of the balance b1. The singularity indices of balance b1 were classified by the concentration-area (C-A) fractal model to produce the high potential areas of the Zn-Cu assemblage. The high potential areas were obviously correlated with all Zn polymetallic occurrences. So, combined with the metallogenic geological characteristics of the study area, five prospective prediction targets were mapped. The anomaly verification demonstrates that the Zn concentration has reached the lowest industrial grade and cut-off grade corresponding to the target area I-1 and I-2, respectively, and Cu also has certain metallogenic potential. There is good potential for Zn-Cu mineralization as represented by this integrated method.

Thermal conductivity of porous building materials: An exploration of new challenges in fractal modelling solutions

The improvement in the insulation material performance is one of the recent crucial problems. The energy consumption in the construction and buildings field has a significant impact on the society and the environment. For these reasons, researchers have focused on studying their thermal behaviour in order to improve fabrication methods and material design structures. In this sense, a great contribution has been offered by the modeling procedures. A remarkable attention has been dedicated to the application of fractal geometry which seems to be a promising method to replicate the porous structures as well as to predict the effective thermal conductivity. In this paper, a review of different modeling procedures is presented, comparing both traditional and fractal theory-based approaches. Fractal models demonstrate high reliability in reproducing experimental data under various conditions, including dry and moist systems. This is further enhanced by the application of recursive formulas, which streamline calculations even for complex porous microstructures. The choice between one model and another depends on the specific characteristics of the materials under study. In all cases, the versatility of the analytical procedures enables one to achieve a remarkable agreement with experimental data.