Fractal relationship between energy and particle size during multistage fragmentation of magnetite ore

To study the theoretical relationship between the ore fragmentation energy and the particle size, the multistage crushing test model of magnetite was constructed with the fractal geometry to simulate the general process of magnetite crushing from the initial particle size to the target particle size, and the theoretical relationship between the fragmentation energy and fragment size and fractal dimension was established. Results show that the multistage crushing of magnetite has fractal characteristics, the rate of particle size reduction in the multistage crushing process decreases, and the fractal dimension increases gradually to be a stable value. The cumulative unit fragmentation energy is affected by the fragment size and fractal dimension, which is defined as the ratio of the cumulative fragmentation energy to ore mass. With the increase in the median particle size of the fragments (d50), the cumulative unit fragmentation energy decreases in the form of exponential function. With the increase in the fractal dimension of the fragment size, the logarithm of the cumulative unit fragmentation energy increases linearly. The smaller the fragment size is, the faster the cumulative unit fragmentation energy changed with the fractal dimension. The larger the fractal dimension of the fragments is, the more significant change in the cumulative unit fragmentation energy with the fragment size is. The mathematical relationship between cumulative unit fragmentation energy, fragment particle size, and fractal dimension is determined.