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2. Model development
2.1. Impact energy per unit mass
Consider the impact between a single particle and a crushing bar attached to the rotor of a hammer crusher. Given that the rotor mass is much greater than the mass of a single particle in the feed, and that before impact the linear velocity of the crushing bar is much more important than the particle velocity, the kinetic energy associated with a single particle is negligible compared with that of the rotor. Considering the conservation of linear momentum of the system particle-crushing bar before and after impact, Attou et al. (1999) derived the following expression for the impact energy per unit mass:
In vertical-axis crushers, the particles are fed to a horizontal turning table (rotor) with radially oriented guides and are projected towards the crusher’s walls by the centrifugal forces. Unlike in hammer crushers, here most of the fragmentation takes place at the crusher’s walls rather than at the rotor’s periphery. With the assumption that the particle energy does not change during its flight from the rotor periphery to the crushing walls, i.e., the particle–particle interactions are neglected in a first approximation, Nikolov and Lucion (2002) derived the following expression for the impact energy per unit mass:
It is interesting to note that for the same rotor radius, the impact energy per unit mass provided by hammer crushers is lower than that provided by vertical-axis crushers. This could explain the fact that vertical-axis crushers produce more fines and perform better when finer granulate must be reduced in size, which is most probably due to the higher level of impact energy reached in these machines.
where E (J/kg) is the average impact energy per unit mass, Q (t/h) is the feed rate; a and s depend on the specific crusher design and size; k and n are related to the material properties of the granulate. In Eq. (6) the breakage probability for particles with size of several dozens of microns is not exactly zero at given feed rate and rotor velocity, which is in contradiction with the experimental evidence. Moreover, Eq. (6) is an almost step-like function of the particle size when applied to typical size distributions. Actually, Eq. (6) has been fitted to experimental data for breakage of identical particles with different impact energies while the inverse problem (breakage of particles with different size with the same impact energy) has not been investigated. According to Eq. (6), a very limited number of particles should have breakage probabilities different from one or zero, which is unrealistic.
In order to deal with these problems, Nikolov and Lucion (2002) recently proposed a new classification function in the form:
As for the influence of the feed rate on dmin, an increase in the feed rate, keeping the feed size unchanged, results in higher frequency of the particle–particle collisions. As each collision dissipates energy, the average impact energy will decrease at higher feed rates and that would result in a coarser product, i.e., a greater value for dmin. Taking into account these considerations, we can write:
where Q and E are the feed rate and the average impact energy per unit mass respectively; Q0 and E0 are reference feed rate and reference impact energy respectively; dmax (mm) is the maximum particle dimension in the feed; n is a material parameter; c0 is a rate constant and c1 accounts for the intensity of the particle–particle interactions.
The dependence of dmin on the feed rate is modelled with a logarithmic function because of the fact that relatively important variations in the feed rate do not change significantly the product size distribution obtained with impact crushers. In addition, at reference feed rate Q=Q0, ln(Q0/Q)=0 so that all other model parameters can be identified independently on c1. It is noted that c1 should depend, among the other factors, on the volume enclosed between the rotor and the crusher’s internal walls. A larger volume of the fragmentation chamber leads to a lower volume fraction of solids for the same feed rate and particle size and therefore, the particle–particle interactions are less frequent.