Open AccessDerivation of Equations for a Size Distribution of Spherical Particles in Non-Transparent Materials
Author(s) -
D. Gurgul,
A. Burbelko,
Tomasz Wiktor
Publication year - 2021
Publication title -
journal of casting and materials engineering
Language(s) - English
Resource type - Journals
ISSN - 2543-9901
DOI - 10.7494/jcme.2021.5.4.53
Subject(s) - mathematics , intersection (aeronautics) , basis (linear algebra) , chord (peer to peer) , probability density function , space (punctuation) , function (biology) , plane (geometry) , mathematical analysis , planar , geometry , statistics , computer science , distributed computing , computer graphics (images) , evolutionary biology , engineering , biology , aerospace engineering , operating system
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.