scholarly journals Probability density function of steady state concentration in two-dimensional heterogeneous porous media

2011 ◽  
Vol 47 (11) ◽  
Author(s):  
Olaf A. Cirpka ◽  
Felipe P. J. de Barros ◽  
Gabriele Chiogna ◽  
Wolfgang Nowak
Keyword(s):  
Porous Media ◽  
Steady State ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Heterogeneous Porous Media ◽  
Two Dimensional ◽  
Steady State Concentration ◽  
State Concentration

scholarly journals Finsler Geometry for Two-Parameter Weibull Distribution Function

2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
Emrah Dokur ◽  
Salim Ceyhan ◽  
Mehmet Kurban
Keyword(s):  
Probability Density Function ◽  
Weibull Distribution ◽  
Probability Density ◽  
Density Function ◽  
Finsler Space ◽  
Finsler Geometry ◽  
Cumulative Probability ◽  
Two Dimensional ◽  
Energy Potential ◽  
Two Parameter

To construct the geometry in nonflat spaces in order to understand nature has great importance in terms of applied science. Finsler geometry allows accurate modeling and describing ability for asymmetric structures in this application area. In this paper, two-dimensional Finsler space metric function is obtained for Weibull distribution which is used in many applications in this area such as wind speed modeling. The metric definition for two-parameter Weibull probability density function which has shape (k) and scale (c) parameters in two-dimensional Finsler space is realized using a different approach by Finsler geometry. In addition, new probability and cumulative probability density functions based on Finsler geometry are proposed which can be used in many real world applications. For future studies, it is aimed at proposing more accurate models by using this novel approach than the models which have two-parameter Weibull probability density function, especially used for determination of wind energy potential of a region.

scholarly journals Derivation of Equations for a Size Distribution of Spherical Particles in Non-Transparent Materials

2021 ◽  
Vol 5 (4) ◽  
pp. 53-60
Author(s):  
Daniel Gurgul ◽  
Andriy Burbelko ◽  
Tomasz Wiktor
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Size Distribution ◽  
Density Function ◽  
Cross Section ◽  
Three Dimensional ◽  
Spherical Particles ◽  
Two Dimensional ◽  
Mathematical Formulas ◽  
Transparent Materials

This paper presents a new proposition on how to derive mathematical formulas that describe an unknown Probability Density Function (PDF3) of the spherical radii (r3) of particles randomly placed in non-transparent materials. We have presented two attempts here, both of which are based on data collected from a random planar cross-section passed through space containing three-dimensional nodules. The first attempt uses a Probability Density Function (PDF2) the form of which is experimentally obtained on the basis of a set containing two-dimensional radii (r2). These radii are produced by an intersection of the space by a random plane. In turn, the second solution also uses an experimentally obtained Probability Density Function (PDF1). But the form of PDF1 has been created on the basis of a set containing chord lengths collected from a cross-section.The most important finding presented in this paper is the conclusion that if the PDF1 has proportional scopes, the PDF3 must have a constant value in these scopes. This fact allows stating that there are no nodules in the sample space that have particular radii belonging to the proportional ranges the PDF1.

scholarly journals Probability density function approach for modelling multi-phase flow with ganglia in porous media

2011 ◽  
Vol 688 ◽  
pp. 219-257 ◽  
Author(s):  
Manav Tyagi ◽  
Patrick Jenny
Keyword(s):  
Porous Media ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Transport Equations ◽  
Kolmogorov Equation ◽  
Phase Flow ◽  
Correlation Times ◽  
Multi Phase Flow ◽  
Multi Phase

AbstractA probabilistic approach to model macroscopic behaviour of non-wetting-phase ganglia or blobs in multi-phase flow through porous media is proposed. The key idea is to consider a set of stochastic Markov processes that can mimic the microscopic multi-phase dynamics. These processes are characterized by equilibrium probability density functions (PDFs) and correlation times, which can be obtained from micro-scale simulation studies or experiments. A Lagrangian viewpoint is adopted, where stochastic particles represent infinitesimal fluid elements and evolve in the physical and probability space. Ganglion mobilization and trapping are modelled by a two-state jump process with transition probabilities given as functions of ganglion size. Coalescence and breakup of ganglia influence the ganglion size distribution, which is modelled by a Langevin type equation. The joint probability density function (JPDF) of the chosen stochastic variables is governed by a high-dimensional Chapman–Kolmogorov equation. This equation can be used to derive moment (e.g. saturation, mean mobility etc.) transport equations, which in general do not form a closed system. However, in some special cases, which arise in the limit of one time scale being smaller or larger than the others, a closed set of moment transport equations can be obtained. For slowly varying and quasi-uniform flows, the saturation transport equation appears in closed form with the mean mobility fully determined, if the equilibrium PDFs are known. Furthermore, it is shown how statistical parameters such as mobilization and trapping rates and equilibrium PDFs can be obtained from the birth–death type approach, in which ganglia breakup and coalescence are explicitly considered. A two-equation transport model (one equation for the total saturation and one for the trapped saturation) is obtained in the limit of very fast coalescence and breakup processes. This model is employed to mimic hysteresis in relative permeability–saturation curves; a well known phenomenon observed in the successive processes of imbibition and drainage. For the general case, the JPDF-equation is solved using the stochastic particle method, which was proposed in our previous paper (Tyagi et al. J. Comput. Phys. 227, 2008, 6696–6714). Several one- and two-dimensional numerical simulation results are presented to show the influence of correlation times on the averaged macroscopic flow behaviour.

scholarly journals A Novel Limit Distribution for the Analysis of Randomly Mistuned Bladed Disks

1996 ◽  
Author(s):  
Marc P. Mignolet ◽  
Chung-Chih Lin
Keyword(s):  
Steady State ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Perturbation Technique ◽  
Practical Case ◽  
Numerical Examples ◽  
Critical Effect ◽  
Form Model ◽  
Steady State Amplitude

A recently introduced perturbation technique is employed to derive a novel closed form model for the probability density function of the resonant and near-resonant, steady state amplitude of blade response in randomly mistuned disks. In its most general form, this model is shown to involve six parameters but, in the important practical case of a pure stiffness (or frequency) mistuning, only three parameters are usually sufficient to completely specify this distribution. A series of numerical examples are presented that demonstrate the extreme reliability of this three-parameter model in accurately predicting the entire probability density function of the amplitude of response, and in particular the large amplitude tail of this distribution which is the most critical effect of mistuning.

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