#### 4.2. Determination of Breakage Parameters by Experimental Methods

The specific rate of breakage can be easily calculated experimentally. A sample having a single size interval was ground for a pre-determined grinding time, t. The fraction remaining in the original size interval, w(t), was measured with sieve shaker (Ro-tap). By repeating these tests several times, a graph of the weight fraction less than size against the grinding time was plotted on a semi-log scale (first-order plot), and the specific rate of breakage was determined directly from the slope.

A first-order-plot of 16 × 20 mesh samples is shown in

Figure 4 to identify the specific rate of breakage

S of the unheated raw limestone (hereinafter referred to as “(

O)”), the brown sample group after heat treatment (“(

B)”) and the white sample group after heat treatment (“(

W)”). (

O) and (

B) show a typical linear decrease in the fraction remaining in the largest size interval, indicating ideal first-order breakage. On the other hand, (

W) shows a straight decrease for up to 4 min, but afterwards more than 98% of the particles were pulverized, resulting in a large amount of fine powder, showing first-order breakage. The

S values of samples (

O), (

B) and (

W) were 0.35, 0.54 and 0.98, respectively, which means that 29%, 42%, and 56% of the largest size particles are broken down within 1 min of grinding (1 – 1/

e_{s}).

The primary breakage distribution is usually measured experimentally using the BII method [

11]: milling is performed for a small degree of grinding in which secondary breakage does not occur. Based on previous literature, the BII method gives a reasonable result: up to approximately 30% of the particles in the largest size interval are broken. Therefore, the cumulative size distribution of the ground product for 30 s of grinding time is shown in

Figure 5, and the parameters calculated using the BII method are shown in

Table 4. In this study, 17%, 25% and 44% of the particles in the largest size interval were broken at 30 s of grinding for (

O), (

B) and (

W), respectively. As shown in

Figure 4 and

Table 3, the

γ values of (

O) and (

B), which have a large influence on the composition of the fragment after breakage, are 0.6 and 0.7, respectively. On the other hand, in the case of (

W), a

γ value of 0.3 means that more fine particles can be generated.

#### 4.3. Determination of Breakage Parameters by Back-Calculation Methods

The breakage parameters can be experimentally determined, but the amount of experimental error and the difficulty in estimating the overall breakage parameters for various grinding times are relevant issues here [

14]. In particular, it may be difficult to apply experimental methods when the breakage does not follow first-order kinetics. Therefore, the optimum parameters were evaluated using a back-calculation method based on nonlinear programming.

Nonlinear programming is an approach to solving optimization problems where the objective function is nonlinear. In general, the goal of nonlinear programming is to minimize any objective function within certain constraints. Equation (4) shows that there are equations and inequalities in the constraint conditions, generalized using

m inequalities

g_{i}(

x) and

n equations

h_{j}(

x) [

12].

Among the many ways to solve nonlinear programming, the simplex method, which does not use a differential, was used in this study because it is difficult to obtain the form of the differential forms of specific rate of breakage and primary breakage distribution, and the time required for the overall computation is very small. Applying the simplex method to this study, the objective function becomes the SSQ, which is the sum of squares of the difference between the experimental values and the calculated values as shown in Equation (5) [

11]. The constraint is that the grinding constants are all positive, and the breakage parameters

A and

α are related to the specific rate of breakage, and

γ,

Φ,

β are related to the primary breakage distribution.

Here, w_{i} is the weight fraction of original material of size i and p_{i} is the weight fraction less than size i of the product.

Table 5 shows the results for the five breakage parameters of the three sample groups obtained using back-calculation. In this case, using the values for

A and

α in Equation (1), the

S values of 12 mesh (1.7 mm) were 0.34, 0.54 and 1.05 for (

O), (

B), (

W)

, respectively. This result is very similar to the experimental values, so it is judged that the constants calculated by the back-calculation method are reasonable. A similar tendency was observed to the values obtained by experimental methods in the case of the primary breakage distribution, but the differences between the experimental and back-calculation method increased in the order from (

O) to (

B) to (

W). The determination of the breakage parameters by experimental methods using the BII method is inadequate because of the fast breakage in a short grinding time, as in this case.

The experimental and simulation (calculated based on parameters obtained by back-calculation) product size distributions are compared in

Figure 6,

Figure 7 and

Figure 8 for samples (

O), (

B) and (

W)

, respectively, for various grinding times. The breakage parameters determined by back-calculation were reasonably accurate when the appropriate parameters were used. The prediction values of the size distribution agree well with the actual measurements over the entire size range, and the results reflect that the back-calculated values of the two breakage functions (specific rate of breakage and primary breakage distribution) are valid.

Figure 9 shows the variation of the median particle size,

d50 (50 wt % passing size) with grinding time for the three samples. As shown in the graph, compared with the original sample (

O), the decrease in the median size of the samples (

B) and (

W) after the heat treatment was remarkable. For example, in the case of sample (

W)

, the grinding time for reaching a median size of 100 μm was reduced from 17 min (in the case of (

O)) to 4 min. Therefore, via appropriate heat treatment and optical sorting processes, the energy consumption can be greatly reduced by shortening the grinding time. By understanding the grinding characteristics of heat-treated materials based on kinetic models, the efficiency can be significantly improved, which is presently the biggest concern in the limestone industry.