Derivation of equations for a size distribution of spherical particles in non-transparent materials

Abstract

This paper presents a new proposition on how to derive mathematical formulas that describe an unknown Probability Density Function (PDF$_{3}$) of the spherical radii ($r_{3}$) 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 (PDF$_{2}$) the form of which is experimentally obtained on the basis of a set containing two-dimensional radii ($r_{2}$). 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 (PDF$_{1}$). But the form of PDF$_{1}$ 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 PDF$_{1}$ has proportional scopes, the PDF$_{3}$ 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.