The Calculation of the Mean Caliper Diameter of a Body for Use in the Analysis of the Number of Particles Per Unit Volume

Abstract

It is easily shown that the expected number of particle profiles N̄_A per unit area of a plane section of orientation (ø, θ) is given by:

$$ {\bar{N}_{A}}\left( {\phi ,\theta } \right) = {N_{V}}\bar{D}\left( {\phi ,\theta } \right) $$

((1))

in which N_V is the number of particles per unit volume and D̄(ø, θ) is the average projected height of the particles on a line of orientation (ø, θ). If the particles are randomly oriented, then Eq. (1) can be written as

$$ {\bar{N}_{A}} = {N_{V}}\bar{D} $$

((2))

in which D is the projected height averaged over all orientations of a particle; i.e.,

$$ D = \left( {1/4\pi } \right)\int\limits_{0}^{{2\pi }} {} \int\limits_{0}^{\pi } {} \quad D\left( {\phi ,\theta } \right)\sin \phi d\phi d\theta $$

((3))

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Written by special invitation, not presented at Congress