An appraisal of the energy-size reduction relationships for mill scale-up design

In this paper, several well-known energy-size reduction relationships have been analyzed using Kapur’s similarity solution to the integro-differential equation of batch grinding. It has been shown that: (i) relationships proposed in terms of the energy actually utilized for breakage of particles, E, cannot be used to develop a practical energy-size reduction relationship as E is neither measurable nor it is found to be proportional to the measurable net energy input, E_n, (ii) the exponent of the characteristic particle size, $\overline{\mathrm{x}}$, in the relationship between E_n and $\overline{\mathrm{x}}$ is same as the exponent of particle size in the expression for the specific breakage rate function, α, which is not a function of $\overline{\mathrm{x}}$ in the case of ball mills, (iii) α values of 0, 0.5 and 1 cannot be associated with the laws of Kick, Bond and Rittinger, respectively. Based on the results of this analysis, the empirical scale-up models proposed by Bond and Morrell have been modified by replacing the exponents of 80% passing sizes by α. Further investigations have been recommended for resolution of large differences in the energy estimates obtained from these models for materials characterized by grindabilty values less than 1.0 and greater than 3.0?g/rev.