Incorporating truncated fractal distribution into Maxwell model to quantify thermal conductivity and its uncertainty in heterogeneous multiparticle systems

An approach is developed to examine the mean and uncertainty of thermal conductivity of a heterogeneous multiparticle system,where the particle concentration or void fraction is treated as a truncated fractal distribution.The truncated fractal distribution is then integrated into the Maxwell model,which is equivalent to a cell model in which the multiparticle system is conceptualized as a spherical fluid cell that envelopes a solid particle.The developed mean thermal conductivity is compared with four experimental data sets of liquid-saturated media from the literature.The effect of fractal characteristics is quantified and discussed.Incorporating particle concentration or void fraction truncated fractal distribution can better capture scatters in the experimental results.The thermal conductivity and its standard deviation decrease with increasing fractal dimensions.When the void fraction is truncated fractal,the uncertainty increases mostly in the low mean void fraction range and drops more quickly with the increasing mean void fraction compared to the case where the particle concentration is truncated fractal.In a typical case of multiparticle system when the solid particles are more conductive than the fluid,the faster increase rate of standard deviation with the ratio of solid over fluid conductivities occurs when the mean void fraction is smaller.

Fractal characteristics of cracks and fragments generated in unloading rockburst tests

True triaxial rockburst experiments with four different unloading rates were performed on four prism specimens of granite sampled from Beishan, China. The damage evolution in the rockburst test was investigated from two aspects including fracture surface crack and fragment characteristics. The scanning electron microscopy was used to observe the micro crack information on fragment surface. Combing binarization and box counting dimensions, the fractal dimensions of cracks were obtained. Meanwhile,the fragments were collected and a sieving experiment was conducted. We weighed the fragments qualities, counted the amount of fragments and measured the fragments length, width and thickness.Utilizing four methods to calculate damage fractal dimensions of fragments, the trend of fractal value changing with unloading rates can be roughly described. It can be concluded from these experiments that the fractal dimension either for crack or for fragment holds a decreasing trend with the decreasing unloading rate, indicating a reduction of damage level.

Effect of Fiber Properties on Nonwovens' Pore Structures with Fractal Geometry Analysis

Nonwovens＇ pore structures are very important to their mechanical and physical properties. And the pore structures are influenced by the fiber properties and fibers arrangement in web. In this paper, the fractal geometry, in combination with computer image anaysis, is used to express the irregularity of pore size distribution in nonwovens, and the effect of fiber properties on fractal dimension of pore size distribution is discussed by using simulated images which are composed of nonlinear staple fibers. The results show that the fiber properties, such as crimp, diameter, angular distribution, and especially the number of fibers prominently influence the pore structure.

Modeling porous structure of oil-pressboard interface and its effect on electric field distribution

The oil-pressboard insulation is a typical composite insulation system widely used in the design and manufactory of large power apparatus. The implement of oil-pressboard insulation may lead to surface electrification and discharge at the interface under certain condition. It is of significant importance to take an insight into the phenomenon occurring at the interface. Through experiment, the pressboard is found as a porous material. The interface changes abruptly from bulk pressboard to the bulk oil as a result of the porous structure. A new model is proposed which divides the interface into bulk oil region, transition region, and bulk pressboard region. The width of the transition region is decided according to the microtome figure. The effective permittivity of the transition region is calculated using a new model based on fractal theory. The model is validated and compared with previous calculation model. The effect of the existence of transition region on the electric field distribution is discussed.

Micro-and macro-fractures of coarse granite under true-triaxial unloading conditions

We studied the mechanical behavior of rock under different boundary conditions, based on the fractal characteristics of fractures in terms of microscopic and macroscopic investigations. Three rectangular granite specimens of similar dimensions were tested by a triaxial rock testing machine under uniaxial compression （UC）, confined compression （CC） and true-triaxial unloading conditions （RB） under rock burst boundary conditions. The failure processes of these specimens were investigated via examinations of their fracture behavior on a macro-scale by laser profilometers and on a micro-scale by a scanning electron microscopic （SEM） imaging technique. The SEM images, showing the spailing features of RB frag- ments, are compared with the grain dislocations under UC and CC conditions. Based on a variogram method, two fractal parameters, i.e., fractal dimensions （Dr^d） and the scale dependent fractal parameter Kv, were induced to present the surface roughness of scanning profiles in all directions. The fitted ellipses of Dr^d distribution show that RB surface has the smallest eccentricity, followed by the CC surface, while the UC surface had the largest eccentricity. As a result of this assessment, we conclude that rocks are affected by shear traction in an intermediate stress direction, which will cause fractures generated during rock bursts to twist rather than to tilt as shown in the uniaxial compression and the confined compres- sion tests.

Comments on "Reexamination of Correlations for Nucleate Site Distribution on Boiling Surface by Fractal Theory

In recently published paper by Yang Chunxin[1], I reexamined the paper. On page 128, the paper 'pointed out that the size and spatial distribution density of nucleation sites presented on real boiling surface can be described by the normalized fractal distribution function, and the physical meaning of parameters involved in some experimental correlations proposed by early investigations are identified according to fractal distribution'. However, the definition on fractal dimension given by Yang[1] is highly questionable, and the results obtained by Yang are contradictory to the basic fractal theory. Here are my comments:

Collision Statistics of Driven Polydisperse Granular Gases

We present a dynamical model of two-dimensional polydisperse granular gases with fractal size distribution, in which the disks are subject to inelastic mutual collisions and driven by standard white noise. The inhomogeneity of the disk size distribution can be measured by a fractal dimension df. By Monte Carlo simulations, we have mainly investigated the effect of the inhomogeneity on the statistical properties of the system in the same inelasticity case. Some novel results are found that the average energy of the system decays exponentially with a tendency to achieve a stable asymptotic value, and the system finally reaches a nonequilibrium steady state after a long evolution time. Furthermore, the inhomogeneity has great influence on the steady-state statistical properties. With the increase of the fractal dimension df, the distributions of path lengths and free times between collisions deviate more obviously from expected theoretical forms for elastic spheres and have an overpopulation of short distances and time bins. The collision rate increases with df, but it is independent of time. Meanwhile, the velocity distribution deviates more strongly from the Gaussian one, but does not demonstrate any apparent universal behavior.

SIGNATURES OF UNIVERSAL CHARACTERISTICS OF FRACTAL FLUCTUATIONS IN GLOBAL MEAN MONTHLY TEMPERATURE ANOMALIES

This paper proposes a general systems theory for fractals visualising the emergence of successively larger scale fluctuations resulting from the space-time integration of enclosed smaller scale fluctuations. Global gridded time series data sets of monthly mean temperatures for the period 1880- 2007/2008 are analysed to show that data sets and corresponding power spectra exhibit distributions close to the model predicted inverse power law distribution. The model predicted and observed universal spectrum for interannual variability rules out linear secular trends in global monthly mean temperatures. Global warming results in intensification of fluctuations of all scales and manifested immediately in high frequency fluctuations.