Single Particle Breakage Tests

Grinding in a ball mill involves a complex interaction between material effects, stressing conditions and environmental effects inside the mill which, in overall, determines the product size distribution and product quality. In this aspect, single particle breakage tests provide insight to understand breakage process in microscale event basis. Single particle breakage tests are classified with respect to the mode of loading: Single particle will be broken either by impact or compression or shearing.

The following can be estimated from single particle breakage data:

-Functional relationship between specific impact energy and product size distribution (Napier-Munn et al., 1996)

-Specific fracture energy of a single particle (J/g) and specific fracture energy or fracture strength distribution of a given material (Bourgeois et al., 1992; Tavares and King, 1998; Tavares, 2007)

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-Breakage probability of particles as a function of stressing energy or specific stressing energy applied (Aman et al., 2010; Krogh, 1980; Tuzcu et al., 2011) -Effect of particle size, shape, material physical properties and modes of loading on particle breakage characteristics (Tavares, 2007)

2.2.2.1 Drop Weight Testing

One of the most commonly used single-particle impact testing method is the drop weight testing. It provides extended input energy range, shorter test duration, extended particle size range and possibility to conduct particle-bed breakage studies (Napier-Munn et al., 1996). However, the experimental procedure becomes tedious as the feed size decreases such that a large number of particles should be broken to get a sufficient weight of sample to be analyzed for size distribution.

As illustrated in Figure 2.4, the test consists of stressing each particle placed on an anvil by dropping a steel weight from a certain height. The weight of the drop head and drop height can be adjusted with respect to the energy input, where the weight of drop head can be up to 50 kg, and the standard range of drop height is between 0.05 and 1 m. The specific impact energy applied to an average weight of the particle in a given set of particles is the potential energy of drop head with respect to surface of the anvil, assuming that frictional losses occurred during the falling motion of drop head are negligible. Then, the specific impact energy applied to a given set of particles (Eis) can be calculated as:

E_is= 0.0272 M (h₀-h_f) m̅ (15)

where E_is is the specific impact energy applied (kWh/t), M is the mass of the drop weight (kg), h0 is the initial drop height (cm) measured from the surface of anvil to the bottom of the drop head and m̅ is the average weight of a particle in the set of particles tested (g). Also, it may be required to subtract an average offset height (h_f)

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from “h₀” term in Equation 15 for a more precise estimation of specific impact energy although this term is relatively small with respect to initial drop height. In this case, offset height is defined as the height (cm) between bottom of the drop weight and surface of the anvil after impacting the particle. It should be noted that average offset height can only be calculated after breaking all particles in a given test. Thus, precise estimation of E_is is possible at the end of the experiment.

It has been observed that drop weight might rebound at high impact energies. This rebound energy is not directly measured, yet it is known to be small relative to the input energy. This might be eliminated by using different combination of drop weight and drop height that gives the same input energy.

Figure 2.4. Schematics of a drop weight tester

The key concept in the drop weight test is to estimate product size distribution as a function of specific impact energy. In order to model this breakage function, a set of cubic spline curves is employed to describe the product size distribution obtained by breakage of a set of particles in narrow size intervals at various specific impact energies. These curves are referred to as one-parameter family curves

(Napier-17 distribution at a given t10value. Therefore, the product size distribution and impact breakage distribution function can be estimated by calculating t10 from a given specific impact energy. Moreover, the relationship between specific impact energy and t10 for each narrow size fraction might be fitted to the following functional form (Napier-Munn et al., 1996):

t10 = A 1-e p(-b E_is) (16)

where A and b are the impact breakage parameters to be fitted.

Figure 2.5. One-parameter family curves

In addition to functional relationship between product size distributions (or impact breakage distribution functions) and various specific energy levels, breakage

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probability and specific fracture energy distribution might be estimated in drop-weight testing. These two concepts are analogous to specific breakage rate in ball milling. Considering a sample of a given size, the specific fracture energy distribution is obtained by measuring the primary specific fracture energy (the energy per unit mass required up to the first instant of failure) of each particle in the sample, then calculating the cumulative probability distribution of specific fracture energy. The breakage probability is calculated by measuring the cumulative percentage of particles broken either in mass basis or number basis for a given specific impact energy. Breakage probability could be easily estimated in conventional drop weight testers, while primary fracture energy is measured through a specialized drop weight device called UFLC (Ultra Fast Load Cell).

Determination of specific fracture energy of a particle is beyond the scope of this work, and the details to estimate specific fracture energy and specific fracture energy distribution in UFLC are given in the literature (Bourgeois et al., 1992;

Bourgeois, 1993; Tavares and King, 1998; Tavares, 2007).