On the Shape Factor of Particles by ‘Curve-Fitting Method’

Abstract

A minimum circle circumscribing the projection of a particle is devided into four parts by defining the circumcenter to the origin of a Cartesian coordinate system.
The shape index of the projection α; is obtained by fitting the contour of each parts of the projection to a mathematical equation y=bⁿ/aⁿ(aⁿ-xⁿ)^1/n.
The degree of ratio to circle, φ_p defined as the ratio of an area of a figure by curve-fitting to that of a minimum circumscribed circle of the projection is given as follows; φ_p=a(α₁b₁+α₂b₂+α₃b₃+α₄b₄)/(π/4)d₀², where d₀ is the diameter of the minimum circumference and is related to the major axis of the projection by a_n=a=d₀/2, and b is the minor axis of the projection.
This type of analysis for a shape factor of projection is named as‘Curve-fitting method’.
The ratio to sphere, Ψ_t=φ_t√φ_t' which is determined by the ratio of true area of the projection of a model particle to the area of a circumscribed circle for the same projection, coincides well with that determined by the‘Curve-fitting method’.
By using various kinds of model particles, the comparisons among the Wadells' degree of true sphericity, the space filling factor and the degree of ratio to sphere are carried out.
The results clearly demonstrate that the value of Ψ_p determined by the‘Curve-fitting method’ falls in between the other two quantities and that the degree of ratio to sphere for these model particles calculated from the ratio of the true area of the projection agrees fairly well with the ideal value.