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The parameter 0<n<1 specifies the relationship between the impact energy per unit mass and the minimum breakable size dmin and controls the saturation value of dmin for given granulate properties and crusher design.
The parameter k in Eq. (7) controls the shape of the classification function and hence, the shape of the product size distribution. Greater values for k correspond to a higher breakage probability for the large particles. Therefore, k can be expected to decrease with increasing the impact energy. In this work, we assume that k depends directly on the rotor velocity. As the impact energy is proportional to the second power of the rotor velocity, the dependence of k on the impact energy can be written as:
2.2. Breakage function
According to Karra (1982), the breakage distribution function bij represents the fraction of the debris created from breakage of identical parent particles of size dj and passing through a screen with mesh size di. It is assumed that the shape of the size distribution of the debris is independent on the size of the parent particles. As mentioned in a review paper by Kelly and Spottiswood (1990), this is true for almost all of the experimentally studied product size distributions obtained by crushing or grinding of a large number of different minerals.
The breakage distribution function for crushers derived by Whiten and White (1979) reads:
where φdenotes the mass fraction of the fine product; m and l are material coefficients accounting for the shape of the fine and the coarse product size distributions respectively.
The cumulative distribution function derived by Broadbent and Callcott (1956) is actually a normalised Weibull distribution starting from a particle size d=0 and, as already explained, not only provides a better fit to the experimental data but can also be obtained from physical considerations related to brittle fracture.
As for the proportion of the fine product φ, Narayanan and Whiten (1988) found that during impact breakage of single particles, φ increases with increasing the energy intensity while m and l remain virtually unchanged. In the case of impact crushing, this means that φ increases with increasing the rotor velocity. On the other hand, φ should decrease with increasing the feed rate because higher frequency of the particle–particle interactions involves lower average impact energy in the crushing chamber. Taking into account that 0<φ<1 by definition, we can express the above arguments in a mathematical form as follows:
3. Results
The model developed in the previous section has been implemented in an in-house FORTRAN code. It has been validated with experiments performed on a hammer crusher with rotor diameter and width of 0.65 and 0.45 m respectively. The rotor radius is R=0.325 m; the height of the rotor’s impact bars is Hb=0.1 m. The material used is limestone from the region of Tournai, Belgium. The feed has been calibrated by screening and its size ranges from 14 to 20 mm. The maximum particle dimension in the feed is dmax=26 mm. The reference feed rate Q0 and the reference impact energy E0 are taken to be 2 t/h and 300 J/kg respectively. The rest of the parameters in Eq. (8) are identified as follows: c0=1.4, c1=0.12 and n=0.35. The parameters of the breakage function (Eqs. (11) and (12)) m, l and c2 are set to 0.74, 2.6 and 0.55 respectively. The values of the parameters necessary to compute the shape of the classification function are fixed to k0=1.35 and k1=0.1.
The influences of the rotor velocity and the feed rate on the minimum size of the particles that undergo breakage dmin are shown in Fig. 4. It is seen that dmin strongly depends on both the rotor velocity and the feed rate and ranges from 3.8 to 7.8 mm for the given operating conditions.
For comparison, when the model of Whiten and White (1979) is used for simulation of the behaviour of short-head cone crushers, the values for φ, m and l are often fixed to 0.2, 0.5 and 2.5 respectively. A greater value for the fine fraction φ in our case reflects the well-known fact that the product issued from impact crushing contains more fines than that obtained with cone or jaw crushers.