## 1 Introduction  
Comminution and especially grinding is one of the most important operations of the materials processing industry. For many decades, it has been the subject of intensive research in several industrial sectors including mineral processing [1, 2], metallurgy [3], cement, chemicals and ceramics production [4, 5]. In addition, interest has been paid in recent years on the production of ultra-fine materials due to their improved physical and mechanical properties [6]. It is also known that communication is an energy intensive process and has low efficiency, due to the fact that most energy consumed is absorbed by the device and only a small share is used for size reduction [7]. Considering these factors, it is deduced that improvement of process efficiency will definitely have a large impact on the operating cost of the mill, the environment and the conservation of resources. Scientific research, aiming to improve grinding efficiency, was initiated with the empirical relationships of [PERSON], [PERSON], [PERSON] as well as [PERSON] and [PERSON], which described the energy-particle size relationship [8]. Later, theories which incorporated the concept of particle size distribution were developed [9, 10], while recently research focused on the use of the mechanistic approach which is based on the recognition of physical events that take place during grinding [11, 12, 13]. The mechanistic approach includes the matrix model, which considers grinding as a series of breakage events and the population balance model, which considers grinding as a continuous process. The population balance model considers first order kinetics and uses two functions, namely the breakage rate \(S_{i}\) and the breakage function \(b_{i,j}\)[14; 15]. These functions generate the fundamental size-mass balance equation which is applied for fully mixed batch grinding operations. It is mentioned that many researchers have underlined the advantages of these functions [16; 17], while the scale-up from laboratory to industrial mills has also been discussed in a number of studies [18; 19; 20; 21]. The variation of breakage rate \(S_{i}\) (\(\min^{-1}\)) as a function of particle size \(x_{i}\) (mm) has been considered by [PERSON] [15] and is expressed with Equation (1), which is accepted by many researchers [22; 23; 24]. \[S_{i}=\alpha_{T}\cdot\left(\frac{x_{i}}{x_{0}}\right)^{\alpha}\cdot Q_{i} \tag{1}\]  
where, \(x_{i}\) (mm) is the upper size of class \(i\), \(x_{0}\) is the standard size (1 mm), \(\alpha_{T}\) (parameter that depends on milling conditions) is the breakage rate for size \(x_{i}=1\) mm and \(\alpha\) is a characteristic parameter depending on material properties. \(Q_{i}\) is a correction factor, which is 1 for small particles (normal breakage) and smaller than one for large particles that need to be nipped and fractured by the grinding media. In the abnormal breakage region, the material behaves as it consists of a soft and a harder fraction [16]. \(Q_{i}\) is calculated from Equation (2) [15],  
\[Q_{i}=\left(\frac{1}{1+\left(x_{i}/\mu\right)^{\Lambda}}\right) \tag{2}\]  
where, \(\mu\) is a parameter that depends on milling conditions and \(\Lambda\) is a positive number that depends on material type and shows how rapidly the breakage rate decreases as size increases (\(\Lambda\geq 0\)). In addition, the particle size \(x_{m}\) for which the breakage rate reaches its maximum value is related with parameter \(\mu\) through the following equation [25; 26],  
\[x_{m}=\mu\cdot\left(\frac{\alpha}{\Lambda-\alpha}\right)^{1/\Lambda},\ \ \Lambda>\alpha \tag{3}\]  
Equation (3) shows that \(x_{m}\) is directly proportional to \(\mu\) since for the same material \(\alpha\) and \(\Lambda\) are constants. The physical and mechanical properties of a material play a crucial role in the design of a mineral processing circuit. The reliable determination of these properties enables the prediction of milling results and in this case no extensive milling tests that require large amounts of time and material are needed [27].