Teaching Old Laws New Tricks – Specific Energy Requirements for Breakage Based on Modern Breakage Rate Kernels and a Population Balance Model

Rittinger (1867), Kick (1885) and Bond (1952) all proposed empirical laws to estimate the specific energy requirement for accomplishing a given extent of size reduction expressed in terms of the initial and final particle size (length dimension) and an empirical constant.  In this paper, we show these laws are consistent with a population balance model in which: (a) breakage is first-order in particle concentration, (b) the fragments are self similar, and (c) the rate kernel is power-law in size and linear in specific power input.  One nice result of the model is a relationship for estimating the constants in these older empirical laws.

The breakage rate exponents for particle volume (the power-law) implied by the various energy laws are: Rittinger – 1/3, Kick – 0, Bond – 1/6.   While models using either the Rittinger or Bond exponents yield similarity solutions, the size-independent kernel implied by Kick does not.  Instead, the mathematical attractor for the system is an infinite number of zero-size particles.  Historically, Kick’s law has been applied to crushing of very large particles (1 m-1 mm) [Rhodes (2008)], far from this mathematical attractor. 

Breakage rate kernels based on the theoretical considerations of Vogel & Peukert (2003) and Capece, Bilgili and Dave (2014) are linear in the excess specific power input (power in excess of a minimum value) and have exponents for particle volume of 1/3.  The minimum specific power input is generally small compared to the actual applied power, so the theoretical kernels are consistent with the rate kernel assumed in the population balance model used in this study.  The range of size considered in the experiments of  Vogel-Peukert/Capece-Bilgili-Dave is much smaller (initial sizes ranging 95 micron - 8 mm) than the application range of Kick’s law but is similar to the typical application range of Bond’s law that Rhodes (2008) associates with most industrial grinding operations.  However, the Vogel-Peukert/Capece-Bilgili-Dave exponent of 1/3 is consistent with Rittinger’s law, not Bond’s law.  According to Rhodes (2008), the typical application range of Rittinger’s law is to ultrafine grinding below 30 micron.

Breakage rate exponents for particle volume extracted from 15 literature data sets indicate values in the range 1/4-1.  These bracket the value of 1/3 implied by Rittinger’s law and do not support the values implied by the laws of either Kick or Bond.  The combined implications of the theoretical rate kernels of Vogel-Peukert/Capece-Bilgili-Dave and this literature analysis suggest that Rittinger’s law may be more broadly applicable than has been generally assumed. Furthermore, the literature results suggest that there is a much broader range of “laws” than these three older versions, since the range of rate kernel exponents extracted from data is large.