To compute any physical quantity for a random particle, the mathematical shape of the particle must be known. The mathematics for regular particles like spheres and ellipsoids is simple. Mathematically characterising the shape of random particles with realistic shapes had not been done previously. However, since around the year 2002, a method that combines X-ray computed tomography and spherical harmonic analysis has been developed to give analytical, differentiable mathematical functions for the three-dimensional shape of star-shape particles, which are a broad class of particles covering most industrial particles of interest, ranging from micrometre scale to millimetre scale particles.
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