Mathematical Model for the Particle Size Distribution of a Kieselguhr Filter Granulation

Diatomaceous Earth, also known as diatomite, diatomaceous silica, or kieselguhr is the material of natural origin which is
composed of the skeletons of diatoms, single-celled algae, which are accumulated in huge beds. Some diatomite’s are industrially
processed and used for different applications most frequently as filter aid. Particle size distribution of any filter granulation,
composed of these materials, has to be appropriately prepared. A mathematical model of particle size distribution of a kieselguhr
granulation is presented in this paper and experimentally verified. Particle sizes of a kieselguhr granulation are measured by the
microscopic morphometric method, ordinarily used in the experimental microbiology, and the empirical probability density function
(pdf) of particle sizes is formulated. Experimental pdf data are fitted according to the exponential function, which represents a
solution of a developed model of differential equation. The fitting procedure followed the iterative algorithm of LevenbergMarquardt to adjust the parameter values. This way, an experimentally determined pdf can be approximated with an analytic
function with a small number of constants in the adopted model of fit function. This approach facilitates further modeling and
analysis of filtering mediums. Based on the particle sizes, pdf’s of different kieselguhr, and perlite granulations, it is possible to
determine an optimal medium mixture (containing different filtration granulations), depending of the filtration requirements