Chord Length Distribution Function Research Articles

Orientation-Dependent Chord Length Distribution in a Convex Quadrilateral

This work contributes to the research devoted to the recognition of a convex body by probabilistic characteristics of its lower-dimensional sections. In this paper, for any convex quadrilateral, five orientation-dependent characteristics are introduced and explicitly evaluated per direction. In terms of these characteristics, simple explicit representations of the orientation-dependent chord length distribution function and the covariogram are obtained not only for an arbitrary convex quadrilateral but also for any right prism based on it.

Small angle X-ray scattering and positron annihilation spectroscopy of polyethyleneimine functionalized ordered mesoporous silica SBA-15 microstructure

Owing to its advantages of high specific surface area, large pore volume, adjustable pore size, good thermal stability and relatively low cost, SBA-15 has a wide range of application prospects in adsorption, separation, catalysis, nanomaterials and other fields. And the use of organic functional groups to modify SBA-15 has become one of the hot spots of research on materials, but the introduction of organic functional groups will inevitably affect the pore structure of material, affecting its performance. Therefore, how to more comprehensively characterize the pore structure of material has received much attention. In this work, small angle X-ray scattering (SAXS) technique is used to characterize the pore structure of PEI/SBA-15 mesoporous molecular sieve. The pore structure and periodicity information of PEI/SBA-15 are obtained by using correlation function and string length distribution theory, and compared with those obtained by positron annihilation lifetime spectroscopy (PALS) technique. The results show that the periodic structure of PEI/SBA-15 mesoporous molecular sieve does not change significantly with the increase of PEI mass percent, and the pore size of PEI/SBA-15 mesoporous molecular sieve only decreases from 8.3 nm to 7.6 nm by the chord length distribution function. Two long-life components<i>, τ</i><sub>3</sub> and <i>τ</i><sub>4</sub>, are obtained by PALS, and <i>τ</i><sub>3</sub> reflects the random pores structure in SBA-15 matrix, while <i>τ</i><sub>4</sub> denotes the size of SBA-15 hexagonal pores. Compared with the results of SAXS, the mesoporous pore size obtained by PALS technique shows the same trend. By combining SAXS technique and PALS technique, the evolution of material microstructure can be revealed in more depth, thus providing a unique method for studying the structural characterization of functional nanocomposites in the future.

Covariogram and orientation-dependent chord lenght distribution function for oblique prism

In the present paper we obtain explicit form of covariogram and oriented-dependent chord length distribution function for oblique prism when we know the covariogram of base. Additional we calculate the covariogram and oriented-dependent chord length distribution function in the case if the base is any trapezoid.

Additive manufacturing: A machine learning model of process-structure-property linkages for machining behavior of Ti-6Al-4V

Prior studies in metal additive manufacturing (AM) of parts have shown that various AM methods and post-AM heat treatment result in distinctly different microstructure and machining behavior when compared with conventionally manufactured parts. There is a crucial knowledge gap in understanding this process-structure-property (PSP) linkage and its relationship to material behavior. In this study, the machinability of metallic Ti-6Al-4V AM parts was investigated to better understand this unique PSP linkage through a novel data science-based approach, specifically by developing and validating a new machine learning (ML) model for material characterization and material property, that is, machining behavior. Heterogeneous material structures of Ti-6Al-4V AM samples fabricated through laser powder bed fusion and electron beam powder bed fusion in two different build orientations and post-AM heat treatments were quantitatively characterized using scanning electron microscopy, electron backscattered diffraction, and residual stress measured through X-ray diffraction. The reduced dimensional representation of material characterization data through chord length distribution (CLD) functions, 2-point correlation functions, and principal component analysis was found to be accurate in quantifying the complexities of Ti-6Al-4V AM structures. Specific cutting energy was the response variable for the Taguchi-based experimentation using force dynamometer. A low-dimensional S-P linkage model was established to correlate material structures of metallic AM and machining properties through this novel ML model. It was found that the prediction accuracy of this new PSP linkage is extremely high (>99%, statistically significant at 95% confidence interval). Findings from this study can be seamlessly integrated with P-S models to identify AM processing conditions that will lead to desired material behaviors, such as machining behavior (this study), fatigue behavior, and corrosion resistance.

Chord length distribution and the distance between two random points in a convex body in Rn

In this article for n-dimensional convex body D the relation between the chord length distribution function and the distribution function of the distance between two random points in D was found. Also the relation between their moments was found.

Upward two-phase flow patterns in vertical circular pipe under rolling condition

The experiments on the air-water two-phase flow patterns in a 24 mm diameter vertical circular pipe under both static and rolling conditions were conducted. The superficial velocities were in the range of 0.05–4.07 m/s and 0.02–20.55 m/s for water and air, respectively. Five typical flow pattern morphologies resulting from a rolling motion were observed: dispersed bubbles; bubbles; slug; churn; and annular flow. Flow pattern transitions were identified by analysis of visual images and a bubble chord-length probability distribution function (PDF). The static experimental data was used to validate existing flow pattern transition models that were then modified to fit rolling conditions by considering the additional forces caused by rolling motion. These models also agreed well with the experimental data. Investigation of the influence of the rolling motion on flow pattern transition showed that, as the rolling amplitude increases, the formation of dispersed bubble flow requires a higher superficial liquid velocity. It was also found that, the greater the rolling amplitude, the larger region of churn. The rolling period, however, has little impact on flow pattern transitions.

The chord-length distribution of a polyhedron.

The chord-length distribution function [γ''(r)] of any bounded polyhedron has a closed analytic expression which changes in the different subdomains of the r range. In each of these, the γ''(r) expression only involves, as transcendental contributions, inverse trigonometric functions of argument equal to R[r, Δ1], Δ1 being the square root of a second-degree r polynomial and R[x, y] a rational function. As r approaches δ, one of the two end points of an r subdomain, the derivative of γ''(r) can only show singularities of the forms |r - δ|-n and |r - δ|-m+1/2, with n and m appropriate positive integers. Finally, the explicit analytic expressions of the primitives are also reported.

Super-resolution of real-world rock microcomputed tomography images using cycle-consistent generative adversarial networks.

Digital rock imaging plays an important role in studying the microstructure and macroscopic properties of rocks, where microcomputed tomography (MCT) is widely used. Due to the inherent limitations of MCT, a balance should be made between the field of view (FOV) and resolution of rock MCT images-a large FOV at low resolution (LR) or a small FOV at high resolution (HR). However, large FOV and HR are both expected for reliable analysis results in practice. Super-resolution (SR) is an effective solution to break through the mutual restriction between the FOV and resolution of rock MCT images, for it can reconstruct an HR image from a LR observation. Most of the existing SR methods cannot produce satisfactory HR results on real-world rock MCT images. One of the main reasons for this is that paired images are usually needed to learn the relationship between LR and HR rock images. However, it is challenging to collect such a dataset in a real scenario. Meanwhile, the simulated datasets may be unable to accurately reflect the model in actual applications. To address these problems, we propose a cycle-consistent generative adversarial network (CycleGAN)-based SR approach for real-world rock MCT images, namely, SRCycleGAN. In the off-line training phase, a set of unpaired rock MCT images is used to train the proposed SRCycleGAN, which can model the mapping between rock MCT images at different resolutions. In the on-line testing phase, the resolution of the LR input is enhanced via the learned mapping by SRCycleGAN. Experimental results show that the proposed SRCycleGAN can greatly improve the quality of simulated and real-world rock MCT images. The HR images reconstructed by SRCycleGAN show good agreement with the targets in terms of both the visual quality and the statistical parameters, including the porosity, the local porosity distribution, the two-point correlation function, the lineal-path function, the two-point cluster function, the chord-length distribution function, and the pore size distribution. Large FOV and HR rock MCT images can be obtained with the help of SRCycleGAN. Hence, this work makes it possible to generate HR rock MCT images that exceed the limitations of imaging systems on FOV and resolution.

Characterization of the 3D microstructure of Ibuprofen tablets by means of synchrotron tomography.

A new methodology to segment the three-dimensional (3D) internal structure of Ibuprofen tablets from synchrotron tomography is presented, introducing a physically coherent trinarization for greyscale images of Ibuprofen tablets consisting of three phases: microcrystalline cellulose, Ibuprofen and pores. For this purpose, a hybrid approach is developed combining a trinarization by means of statistical learning with a trinarization based on a watershed algorithm. This hybrid approach allows us to compute microstructure characteristics of tablets using methods of statistical image analysis. A comparison with experimental results shows that there is a significant amount of pores which is below the resolution limit. At the same time, results from image analysis let us conjecture that these pores constitute the great majority of the surface between pores and solid. Furthermore, we compute microstructure characteristics, which are experimentally not accessible such as local percolation probabilities and chord length distribution functions. Both characteristics are meaningful in order to quantify the influence of tablet compaction on its microstructure. The presented approach can be used to get better insight into the relationship between production parameters and microstructure characteristics based on 3D image data of Ibuprofen tablets manufactured under different conditions and elucidate key effects on the strength and solubility kinetics of the final formulation. LAY DESCRIPTION: A typical formulation of uniaxial compacted Ibuprofen tablets consist of a mixture of an excipient (microcrystalline cellulose) with an active ingredient (a ground fraction of Ibuprofen). The final mechanical strength of the tablet as well as the release kinetics are strongly influenced by the underlying microstructure, i.e. the spatial arrangement of the microcrystalline cellulose and Ibuprofen within the tablet. In order to optimize the performance of the tablet, it is important to investigate the relationship between its microstructure and the corresponding production parameters. For this purpose, 3D imaging is a powerful tool as it allows computing microstructural properties such as the internal arrangement, interconnectivity and pore location and distribution, characteristics that cannot be computed by experimental characterization techniques. In the present study, a new algorithm for an accurate trinarization of 3D image data obtained by synchrotron tomography is presented. Trinarization means that we reconstruct microcrystalline cellulose, Ibuprofen and pores on the basis of the 3D images, where one can only observe different greyscale values, but not the different constituents themselves. For this purpose, a hybrid approach combining a trinarization by means of artificial intelligence with a trinarization based on a geometrically motivated algorithm is developed. This hybrid approach allows to compute microstructure characteristics of tablets using image analysis. A comparison with experimental results shows that there is a significant amount of pores below the resolution limit. At the same time results from image analysis lead to the conjecture that these pores constitute the major part of the surface between pores and solid. Moreover, characteristics are computed by image analysis, which are meaningful in order to quantify the influence of tablet compaction parameters on its microstructure. The presented novel approach can be used to elucidate the relationship between production parameters and microstructure characteristics based on 3D image data of Ibuprofen tablets manufactured under different mixing, loading and processingconditions.

The chord length distribution function of a non-convex hexagon

Abstract In this paper we obtain the chord length distribution function of a non-convex equilateral hexagon and then derive the associated density function. Finally, we calculate the expected value of the chord length.

Orientation-dependent chord length distribution as a function of maximal chord

In [3] and [4] have been proved that covariogramand orientation-dependent chord length distribution function of a triangle and an ellipse depends on maximal chord tmax(u) in direction u. In this paper a necessary condition for orientation-dependent chord length distribution function as a function of maximal chord is obtained. A class of parallelograms is constructed for which this necessary condition is violated.

Orientation-dependent distribution of the length of a random segment and covariogram

The paper considers a random segment L(ω) in Rn with fixed direction and length that intersects a given bounded convex body D ⊂ Rn. We obtain a relationship between the covariogram of D and the distribution of the random variable |L| — the length of L ∩ D. A relationship between the distribution of |L| and the orientation-dependent chord length distribution is also obtained. As a consequence, we obtain a relationship between the chord length distribution function and the distribution of the random variable |L|.

Covariogram of a cylinder

In this paper we establish relationships between the covariogram and the orientation-dependent chord length distribution function of a cylinder and those of its base. Also, we obtain explicit expressions for the covariogram and the orientation-dependent chord length distribution function of a cylinder with cyclic, elliptical and triangular bases.

Covariogram of a parallelogram

In this paper we obtain the explicit forms of the covariogram and the orientation-dependent chord length distribution function for any parallelogram. The explicit form of the chord length distribution function for a parallelogram is also obtained.

English

The paper continues the investigations in (1). We consider the problem of recognition of convex bodies in n-dimensional Euclidean space by k-flats sections, in particular for n=3 by linear or planar sections. This problem is equivalent to recognition of convex bodies by orientation-dependent chord length distributions (for linear sections) and orientation-dependent area distributions (for planar sections). The main goal of the article is to enlarge the class of bodies for which the form of the orientation dependent chord length distribution function and the cross- section area distribution function are known.

Analytical Dancoff factor evaluations for reactor designs loaded with TRISO particle fuel

A new mathematical model, the dual-sphere model, is proposed to analytically evaluate Dancoff factors of TRISO fuel kernels based on the chord method. The accurate evaluation of fuel kernel Dancoff factors is needed when one analyzes nuclear reactors loaded with TRISO particle fuel. In these reactor designs, fuel kernels are randomly distributed and shield each other, causing a shadowing effect. The Dancoff factor is a quantitative measure of this effect and is determined by the spatial distribution of fuel kernels. A TRISO fuel particle usually consists of four layers that form a coating region outside the fuel kernel. When fuel particles are loaded in the reactor, the spatial distribution of fuel kernels can be affected by the thickness of the coating region. Therefore, the coating region should be taken into account in the calculation of Dancoff factors. However, the previous model, the single-sphere model, assumes no coating regions in the Dancoff factor predictions. To address this model deficiency, the dual-sphere model is proposed by deriving a new chord length distribution function between two fuel kernels that explicitly accounts for coating regions. The new model is employed to derive analytical solutions of infinite medium, intra-fuel pebble and intra-fuel compact/pin Dancoff factors over a wide range of volume packing fractions of TRISO fuel particles, varying from 2% to 60%. Comparisons are made with the predictions from the single-sphere model and reference Monte Carlo simulations. A significant improvement of the accuracy, over the ranges of packing fractions under current investigation, has been observed. The new model provides a reliable numerical evaluation of Dancoff factors for TRISO particle fuel and can be used as a routine method for the neutronic design and analysis of reactors loaded with TRISO particle fuel.

Recognition of triangles by covariogram

The paper considers the problem of recognition of triangles by orientation-dependent chord length distribution. The following results are obtained: 1. The explicit form of the covariogram and orientation-dependent chord length distribution function for a triangle. 2. The explicit form for the chord length distribution function for a triangle. 3. The length of the maximal chord of a triangle is continuous function on direction u ∈ S1 (S1 is the space of all directions in the plane). 4. If we have orientation-dependent chord length distribution function for an everywhere dense set of S1, then we can uniquely recognize the triangle with respect to reflections and translations. 5. For any finite subset A ⊂ S1, there are two non-congruent triangles with the same values of orientationdependent chord length distribution functions on A.

Linking pore diffusivity with macropore structure of zeolite adsorbents. Part II: simulation of pore diffusion and mercury intrusion in stochastically reconstructed zeolite adsorbents

In the present study we complete the evaluation of three dimensional digitized reconstructions of a binderless zeolite adsorbent with improved mass transfer rates, by performing simulations of pore diffusion and Hg-intrusion porosimetry in these structures. It is seen that an excellent agreement with the experimental diffusivity is achieved (relative error of 1.2 %) for a pore structure that matches, besides low order correlations, chord length distribution functions that account for higher order correlations. Furthermore, simulations on a variety of reconstructed samples indicate that matching chord length distribution functions is a necessary (though probably not sufficient) condition for accurate structural representation. The average tortuosity factor is 2.68 and is nearly constant over a broad spectrum of pressures, when properly normalized. Hg-intrusion porosimetry simulations, performed with a pure morphology method, show a good agreement with the experimental curve for normalized cumulative intrusion volumes in the range of 50–88 %, but cannot make a distinction between structures with differences in higher order correlations. It is believed that SEM micrographs, properly obtained to represent realistic 2D sections of the material, contain sufficient structural information that can distinguish among pore structures with different mass transfer rates, when combined with stochastic reconstruction methods. Evidently, the direct link between these structural parameters and pore diffusivity will provide the necessary route to improve the mass transfer rate of porous adsorbents.

Linking pore diffusivity with macropore structure of zeolite adsorbents. Part I: three dimensional structural representation combining scanning electron microscopy with stochastic reconstruction methods

In the present study we have employed advanced experimental and computational methods to characterize the structure of a binderless zeolite adsorbent with improved mass transfer characteristics. Hence, we have used standard and hybrid simulated annealing (SA) methods to stochastically reconstruct in three dimensions the adsorbent structure, by matching low order correlations, namely porosity and two point correlation function. These correlations are measured on two dimensional images obtained by Back-Scattered SEM micrographs taken on different samples of the adsorbent. In the standard SA method we have started from a purely random structure, while in the hybrid method we have started from a uniformly consolidated random sphere pack made by the Lubachevsky–Stillinger algorithm. It is found that the hybrid method preserves, besides low order correlations, pore and mass chord length distribution functions, which contain information on higher order correlations, while the standard SA method matches only low order correlations. This is because in the hybrid process we have initiated the SA algorithm starting from a structure that contains structural information of the sintered zeolite powder, in the form of a consolidated random sphere pack with the porosity of the final structure. Evidently, diffusion studies will enable us to further evaluate the structures developed by each method, as will be explored in a subsequent study.

Porous media generated by using an immiscible lattice-gas model

To get a right structure–property relationship in porous media it is necessary to have an appropriate digital representation of the microstructure. In this paper a methodology based on a phase ordering dynamics, which it is possible to obtain digital representations of a great variety of isotropic and anisotropic microstructures, is presented. The ordering dynamics, guided by an immiscible lattice gas in a D3Q19 geometry, was characterized by using the structure factor. It was found that, for long times and high particle densities, the obtained patterns are similar to those observed in ordering processes under shear effects. The materials were characterized by means of their two-points correlation and chord length distribution functions. As an application of the model an approximated digital representation of the cubic section of a ceramic material is presented.