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Grain size distribution
Abstract:
The grain radius R distribution function f ( R , t ) with R c ( t ) as critical grain radius is formulated, inspired by the Hillert self-similar solution concept, as product of 1 / R c 4 and of a shape function g ( ρ , t ) as function of the dimension-free radius ρ = R / R c and time t , contrarily to the Hillert self-similar solution concept with time-independent g ( ρ ) . The evolution equations for R c ( t ) as well as for g ( ρ , t ) are derived, guaranteeing that the total volume of grains remains constant. The solution of the resulting integro-differential equations for R c ( t ) and g ( ρ , t ) is performed by standard numerical tools. Remarkable advantages of this semi-analytical concept are: (i) the concept is a deterministic one, (ii) its computational treatment is very efficient and (iii) the shape function g ( ρ , t ) remains localized in a fixed interval of ρ . The shape function g ( ρ , t ) evolves towards the well-known Hillert self-similar distribution, which is demonstrated for two initial shape functions (one of them is triangular). Also a study on “nearly” self-similar distribution functions proposed as useful approximations of experimental data is presented.
