Effects of fractality on the accessible surface area values of zeolite adsorbents

The effects of the molecular sizes of adsorbates on the accessible surface area values of the zeolites 13X, 5A, silicalite and NaY as well as the modified forms of NaY are investigated by taking into account the concept of fractality. For this aim, a relationship developed by combining the Pfeifer–Avnir and the surface area equations, which relates the surface areas of adsorbents to the molar volumes and the cross-sectional areas of adsorbates is utilized. The expected relationship, signifying that the accessible surface areas of adsorbents having fractal dimensions above and below 2, increase and decrease, respectively, with decreasing size of the adsorbate, is quantified in this study for the above zeolite adsorbents. Modifying the properties of adsorbents by using various treatment methods is seen to be potentially useful for enhancing the performances of processes involving adsorption, e.g. adsorption heat pump applications. Since the treatments employed may change the fractal dimension of the original sample, adsorbates having proper sizes should be used with the modified forms in order to achieve a good result. The modified hydrogen form of NaY provides the opportunity to increase the efficiency of the adsorption heat pumps by almost 40% with respect to the utilization of the original sample when methanol is used as the adsorbate.     

Simulated fractal permeability for porous membranes

Fractal permeability model for bi-dispersed porous media is developed based on the fractal characteristics of pores in the membrane. The fractal permeability model is found to be a function of the tortuosity fractal dimension, pore area fractal dimension, sizes of particles and clusters, micro-porosity inside clusters, and the effective porosity of a medium. The pore area fractal dimension and the tortuosity fractal dimension of the porous membranes are determined by the box counting method. To verify the validity of the model, the predicted permeability were compared with the experimental data utilizing H₂ gas permeating through porous Pd-alumina, silicalite-1 and B-ZSM-5, and O₂ across perovskite-alumina membranes form the past effort.     

ATTRACTORS FOR DISCRETIZATION OF GINZBURG-LANDAU-BBM EQUATIONS

1. IntroductionIn this paper) we consider the following periodic initial value problem for the system ofGinzburg-Landau equation coupled with BBM equationwhere e(x,t) is a complex function, n(x, t) is a real scalar function, at a, 5, 7, al, a2, FI, adZare real constants, and gi (x), g200 are given real functions.This problem describes the nonlinear interactions between Langmuir wave and ion acousticwave in plasma physics, e(x, t) denotes electric field, n(x, t) the perturbation of density (…     

Construction of fractal surfaces by recurrent fractal interpolation curves

A method to construct fractal surfaces by recurrent fractal curves is provided. First we construct fractal interpolation curves using a recurrent iterated functions system (RIFS) with function scaling factors and estimate their box-counting dimension. Then we present a method of construction of wider class of fractal surfaces by fractal curves and Lipschitz functions and calculate the box-counting dimension of the constructed surfaces. Finally, we combine both methods to have more flexible constructions of fractal surfaces.     

Box-counting dimensions of fractal interpolation surfaces derived from fractal interpolation functions

A construction method of Fractal Interpolation Surfaces on a rectangular domain with arbitrary interpolation nodes is introduced. The variation properties of the binary functions corresponding to this type of fractal interpolation surfaces are discussed. Based on the relationship between Box-counting dimension and variation, some results about Box-counting dimension of the fractal interpolation surfaces are given.     

The Minkowski dimension of the bivariate fractal interpolation surfaces

We present a new construction of continuous bivariate fractal interpolation surface for every set of data. Furthermore, we generalize this construction to higher dimensions. Exact values for the Minkowski dimension of the bivariate fractal interpolation surfaces are obtained.     

Multifractal scaling of crack images from pyroligneous acid dried sludge

Tannery effluent (sludge, wastewater) is treated by natural way. The waste sludge has been taken for two treatment process. The alkali chemicals are neutralized by pyroligneous acid which is obtained by pyrolysis process of wood. This sludge is sent out for drying. The dried sludge contains some crack pattern formation. Photographs were used to record two sludge cracking surfaces. Experiment has been performed to study the fracture pattern formation in thin film sludge. We studied changes of crack surface of a sludge by image analysis and also assessed applicability of fractal scaling to crack surfaces. The calculated crack surface dimension shows that the fracture surface exhibit fractal structure. Image size was 256 × 256 pixels. MFA (multifractal analysis) was carried out to the method of moments, i.e., the probability distribution was estimated for moments ranging from ?150 < q < 150 and the generalized dimension were calculated from the log/log slope of the probability distribution for the respective moments over box sizes. Generalized dimension D(q) were attained for this box size range, which are capable of characterizing heterogeneous spatial crack structure. Multifractal spectra analyzed two fracture surface of the image and results were indicated that the width of spectra increases due to pyroligneous acid. Multifractal method is sensitive enough to measure the fracture distribution and can be seen as a different approach for changing research of crack images of manure sludge drying.     

粉末注射成形坯孔隙度的分形模型

粉末注射成形坯是一种具有分形特性的典型的多孔介质,借助于多孔介质孔隙结构的分形理论,对粉末注射成形坯孔隙率的分形模型进行推导。首先分析了粉末注射成形坯孔隙结构的双重分形特性,介绍了粉末注射成形坯孔隙分布分形维数和孔隙迂曲分形维数,然后推导出粉末注射成形坯孔隙度的分形模型。     

Construction of recurrent bivariate fractal interpolation surfaces and computation of their box-counting dimension

Recurrent bivariate fractal interpolation surfaces (RBFISs) generalise the notion of affine fractal interpolation surfaces (FISs) in that the iterated system of transformations used to construct such a surface is non-affine. The resulting limit surface is therefore no longer self-affine nor self-similar. Exact values for the box-counting dimension of the RBFISs are obtained. Finally, a methodology to approximate any natural surface using RBFISs is outlined.     

Closed contour fractal dimension estimation by the Fourier transform

This work proposes a novel technique for the numerical calculus of the fractal dimension of fractal objects which can be represented as a closed contour. The proposed method maps the fractal contour onto a complex signal and calculates its fractal dimension using the Fourier transform. The Fourier power spectrum is obtained and an exponential relation is verified between the power and the frequency. From the parameter (exponent) of the relation, is obtained the fractal dimension. The method is compared to other classical fractal dimension estimation methods in the literature, e.g., Bouligand–Minkowski, box-counting and classical Fourier. The comparison is achieved by the calculus of the fractal dimension of fractal contours whose dimensions are well-known analytically. The results showed the high precision and robustness of the proposed technique.     

岩石裂隙网络的分形特征

分形特征与分形维数广泛应用于岩石裂隙网络的量化,及与工程参数的关系模型建立.然而,严格的分形维数的极限定义形式难以直接应用,工程应用中多用近似分形维数值代替,近似的结果在建立量化关系模型时会产生蝴蝶效应,在量化及预测过程中产生巨大偏差.本文回顾了分形研究一系列的发展过程,并基于最新的分形定义提出了一种新的分形维数计算方法.通过对于十个岩石裂隙网络分形维数的计算,证明该方法能够准确有效的计算出图形的复杂度,避免了以往计算分形维数所产生的问题.     

Attractors and Dimensions for Discretizations of a NLS Equation with a Non-local Nonlinear Term

In this paper we consider a semi-dicretized nonlinear Schrödinger (NLS) equation with local integral nonlinearity. It is proved that for each mesh size, there exist attractors for the discretized system. The bounds for the Hausdorff and fractal dimensions of the discrete attractors are obtained, and the various bounds are independent of the mesh sizes. Furthermore, numerical experiments are given and many interesting phenomena are observed such as limit cycles, chaotic attractors and a so-called crisis of the chaotic attractors.     

粗糙面分形计算理论研究进展

为提出一种工程上适用可靠的粗糙面分形维数计算方法,在分形曲线的维数计算方法(码尺法,盒维法)基础上,先后提出了星积分形曲面的维数计算方法、三角形棱柱表面积法、投影覆盖法、立方体覆盖法、改进的立方体覆盖法、分形的增变量描述法等曲面分形维数理论.鉴于上述方法的共有缺陷——获取三维坐标的激光表面仪器的扫描尺度限制,研究者提出了粗糙面图像维数计算理论,包括二值化图像维数、灰度图像维数、RGB图像维数计算理论.最后,本文展望了分形维数计算理论领域内亟待解决的三大问题.     

Dimension analysis on a class of Bush type fractal surfaces

Abstract. In this paper,the authors construct a class of fractal surfaces,Bush type surfaces,based on the Bush type functions. The Box dimension,Packing dimension and Hausdorff dimen-sion of such surfaces are investigated.     

On 2D generalization of Higuchi’s fractal dimension

We propose a new numerical method for calculating 2D fractal dimension (DF) of a surface. This method represents a generalization of Higuchi’s method for calculating fractal dimension of a planar curve. Using a family of Weierstrass–Mandelbrot functions, we construct Weierstrass–Mandelbrot surfaces in order to test exactness of our new numerical method. The 2D fractal analysis method was applied to the set of histological images collected during direct shoot organogenesis from leaf explants. The efficiency of the proposed method in differentiating phases of organogenesis is proved.     

离散的Burgers-Ginzburg-Landau方程组的吸引子

文章通过对空间变量的有限差分方法离散了具有周期边值的Ｂｕｒｇｅｒｓ Ｇｉｎｚｂｕｒｇ Ｌａｎｄａｕ方程组．研究了这个离散方程组初值问题解的适定性．证明了当差分网格足够大时离散方程组存在吸引子，并得到了吸引子的Ｈａｕｓｄｏｒｆｆ维数和分形维数的上界估计．这个上界不会随着网格的加细而无限增大，因此数值分析离散的有限维系统的吸引子可以近似探讨原无限维系统的吸引子．     

Temporal variation of velocity components in a turbulent open channel flow: Identification of fractal dimensions

Fractals are objects which have similar appearances when viewed at different scales. Such objects have details at arbitrarily small scales, making them too complex to be represented by Euclidian space; hence, they are assigned a non-integer dimension. Some natural phenomena have been modeled as fractals with success; examples include geologic deposits, topographic surfaces and seismic activities. In particular, time series have been represented as a curve with fractal dimensions between one and two. There are different ways to define fractal dimension, most being equivalent in the continuous domain. However, when applied in practice to discrete data sets, different ways lead to different results. In this study, three methods for estimating fractal dimension are described and two standard algorithms, Hurst’s rescaled range analysis and box-counting method (BC), are compared with the recently introduced variation method (VM). It was confirmed that the last method offers a superior efficiency and accuracy, and hence may be recommended for fractal dimension calculations for time series data. All methods were applied to the measured temporal variation of velocity components in turbulent flows in an open channel in Shiraz University laboratory. The analyses were applied to 2500 measurements at different Reynold’s numbers and it was concluded that a certain degree of randomness may be associated with the velocity in all directions which is a unique character of the flow independent of the Reynold’s number. Results also suggest that the rigid lateral confinement of flow to the fixed channel width allows for designation of a more-or-less constant fractal dimension for the spanwise velocity component. On the contrary, in vertical and streamwise directions more freedom of movements for fluid particles sets more room for variation in fractal dimension at different Reynold’s numbers.     

Fractal analysis of time series in oil and gas production

The peculiarities of fractal characteristics’ calculations for time series are described in this article. An algorithm for calculation of fractal dimension is suggested. It has been proved that the suggested method possesses high accuracy and the rapidity of convergence on the limited number of measurements compared to the methods of covering.The criteria of early diagnosis for changes in the condition of hydrodynamic processes, which do not vary by fractal dimension, have been recommended.The presented method is applicable for practical engineering calculations with self-affine, chaotic data, usually with relatively limited number of measurements. It is quite a simple method for calculation of fractal dimension, algorithm can be easily realized and it should be useful for engineers.The applicability of the proposed algorithm for fractal dimension calculation and early diagnosis criteria of qualitative changes in the behaviour of various dynamic systems has been tested both on simulated as well as practical examples of oil and gas production.     

Strength vs. toughness optimization of microstructured composites

Aim of this paper is to present a new fractal approach linking the macroscopic mechanical properties of micro- and nano-structured materials with the main parameters: composition, grain size and structural dimension, as well as contiguity and mean free path. Assuming the key role played by the interfaces, the proposed fractal energy approach unifies the influences of all the above parameters, through the introduction of a fractal structural parameter (FSP), which represents an extension of the Gurland’s structural parameter. This modeling approach is assessed through an extensive comparison with experimental data on poly crystalline diamond (PCD) and WC–Co alloys. The results clearly show that the theoretical fractal predictions are in a fairly good agreement with the experiments on both hardness and toughness. This new synthetic parameter is thus proposed to investigate, design and optimize new micro- and nano-grained materials. Eventually, FSP-based optimization maps are developed, that allow to design new materials with high hardness and toughness.