Study on Fractal Mathematical Models of Pulverizing Theory for Ore

• The concept of fractal size, which is corresponding to the distance r, is proposed.
• The volume, surface and line fractal dimensions are defined in an unified way.
• The relationships of 3 fractal dimensions are obtained and verified experimentally.
• The fractional mathematical models reflect the hierarchy of the self-similar.
• Three new algorithms are developed to measure and calculate 3 fractal dimensions.

Fractal dimension is an important parameter to characterize size distribution and shape features of particles. However, it still remains unclear whether the fractal dimensions measured with different methods for the same particles will have the identical results, or whether the similar quantitative relationship exists between D3 and D2 or D3 and D1 as stated in the product–sum theorem. Silicon dioxide and potassium feldspar were selected for PDS measurements and 2DSPI extractions in this paper. By introducing the concept of fractal size, D3, D2 and D1 are defined in a unified way, and their relationships are obtained under the defined assumptions including the product–sum theorem. The two fractional mathematical models and the three new algorithms including the power spectrum method presented in this paper provide the solutions to the problems encountered in the measurement of fractal dimensions. The Gauss–Newton iterative method of R–R distribution is superior to the existing fitted method. The four relationships of the fractal dimensions have been verified both experimentally and by the computational models. The results show that the relationships among D3, D2 and D1 are rational and valid.

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