**2. Materials and methods**

Quartz sand used in laboratory-scale grinding experiments was obtained from a private mining company located in Şile, on the Black Sea coast of İstanbul. Chemical analysis values of quartz sand are given in **Table 1**.

In this study, specific rate of breakage values of Şile region quartz sand in three different mono-size intervals were determined. For this purpose, Şile region quartz sand was prepared in mono-size intervals of �0.090 + 0.075, �0.075 + 0.063, �0.063 + 0.053 mm according to the <sup>4</sup> √2 sieve series. In order to determine the specific rate of breakage values of quartz sand, a 150x150 mm (diameter x length) stainless steel ball mill was used as the grinding medium. The diameters of the grinding balls in this grinding medium were chosen as 6.35, 7.94, 9.52, 12.70, and 19.05 mm. In order to determine the specific rate of breakage values of quartz sand in three mono-size intervals, it was ground in batches at certain time periods (1, 2, 4, 8, 16, 32, and 64 minutes). After each grinding period, all the powder in the mill was discharged, and representative samples were taken. A laser diffraction device was used to measure the particle sizes of the representative samples belonging to the grinding periods. Based on each time period of grinding, semi-logarithmic graphs of the material fractions staying in the high points of the particles'size limits were drawn in contact with the grinding periods. The first-order zone breakage is represented by the zone in which this graph decreases linearly. The slope of the line in the first-order breakage zone gives us the specific rate of breakage of the material in that particle size range. The formula for the specific rate of breakage (*Si*) is given in Eq. 1.

$$\mathbf{S}\_{i} = \mathcal{a}(\mathbf{x}\_{i}/\mathbf{1}\text{ mm}\ )^{a}\mathbf{Q}\_{i} \tag{1}$$

The symbol "*a*" given in Eq. 1 is the model parameter. This parameter depends on the mill conditions. "*α*" value is a positive number, normally in the range 0.5 to 1.5, which is characteristic of the material properties. "*xi*" symbolizes the upper dimension (mm) in the fraction *i*. *Qi* is the correction factor and is taken as 1 for smaller sizes. The experimentally established values of *Qi* can be seen in Eq. 2 [21].

$$Q\_i = \frac{1}{1 + (\boldsymbol{\chi}\_i/\mu)^{\boldsymbol{\Lambda}}} \quad \boldsymbol{\Lambda} \ge \mathbf{0} \tag{2}$$


**Table 1.**

*Chemical composition of quartz sand, mass-%.*

*The Effects of Mill Conditions on Breakage Parameters of Quartz Sand in the District… DOI: http://dx.doi.org/10.5772/intechopen.102554*

Eq. 2 refers to the fact that "*μ* is the particle size at which the correction factor 1/2 and *Λ* a positive number which is an index of how rapidly the rates of breakage fall as size increases (the higher the value of *Λ*, the more rapidly the values decrease)" [21].

In laboratory grinding studies, the rotational speed of the ball mill was chosen to be 70% of the critical speed value of the ball mill. The critical speed of the ball mill was calculated using Eq. 3. Amounts of ball and material to be fed to the mill with Eqs. 4–6 respectively and the mill's interstitial filling rates were found.

$$\text{Critical speed } (N\_c) = \frac{42.3}{\sqrt{(D-d)}} \tag{3}$$

In Eq. 2, *D* represents the internal mill diameter (m) and *d* represents the maximum ball diameter (m) [21].

$$J = \frac{\text{Mass of balls/Ball density}}{\text{Mill volume}} \ast \left(\frac{1}{0.6}\right) \tag{4}$$

$$f\_c = \frac{\text{Mass of powder/Power density}}{\text{Mill volume}} \ast \left(\frac{1}{0.6}\right) \tag{5}$$

$$U = \frac{f\_c}{0.4 \ast f} \tag{6}$$

The properties of the ball mill, experimental conditions, alloy steel balls, and quartz sand used in the laboratory grinding process are given in **Table 2**.

Shoji et al. (1982) found a simple relationship between powder filling and ball load in the mill [23]. It is seen in Eq. 5. In Eq. 6, the net mill power (*mp*) as a function of ball load was fitted by the empirical function. Eq. 7, which is the combination of Eqs. 5 and 6, was used to calculate the specific grinding energy as a function of ball filling. Combining Eqs. 5 and 6 gives the result shown in **Figure 1** [21].



**Table 2.**

*Ball mill characteristics and test conditions for grinding of quartz sand.*

**Figure 1.** *Relative specific grinding energy as a function of ball filling: dry grinding in a laboratory mill [18].*

where *c* is 1.32 and 1.2 for wet and dry grinding respectively.

$$m\_p \propto \frac{1 - 0.937f}{1 + 5.95f^5}, \quad 0.2 \le f \le 0.6\tag{8}$$

$$\text{Specific grinding energy} \propto \left\{ \frac{1 - 0.9375f}{1 + 5.95f^5} \right\} / \left\{ \frac{Ue^{-1.2U}}{1 + 6.6f^{2.3}} \right\} \tag{9}$$

Austin et al. (1984) express the connection between specific grinding energy and ball load as follows: "Although the capacity of a laboratory mill is a maximum at 40 to 45% ball load, the relative specific grinding energy *mp/SW* is a minimum at about 15 to 20% ball load. In practice, ball loads less than 25% are not normally used because low ball loads can give excessive liner wear. In addition, mill capacity is clearly lower for lower ball loads [21]." Dependent upon Eq. 7 and **Figure 1**, the values of specific grinding energy changes based on the ball loads *J* and powder-ball loading ratio *U*. As a result, in this study the specific grinding energy values are obtained from Eq. 7 for 0.35 ball filling ratio was calculated as 3.38.
