## 3 Theoretical Models and Results
### Product Quality Model Considering Liner Wear  
In mineral engineering, particle size and shape are two key factors that reflect the quality of product [22]. In previous research, the process of the material flowing through the crushing chamber could be modeled as a series of successive crushing events [20]. On the basis of this research and the wear model, equations were formulated for PSD as the liner wear. Size distribution of the final product can be described by Equations (8) and (9), as shown in:  
\[P(t)=\prod\limits_{i=1}^{K_{0}}\left[B_{i}(t)S_{i}(t)+[1-S_{i}(t)]P_{(1)}(t) \right., \tag{8}\]  
\[\epsilon_{i}(t)=\frac{s_{i}}{b_{i}+\Delta\omega_{i}\left(t\right)}, \tag{9}\]  
where \(P(t)\) is the product size distribution of \(t\) time, \(K_{0}\) is the total number of all crushing zones, \(S_{i}(t)\) is the selection function and \(B_{i}(t)\) is the breakage function. \(S_{i}(t)\) and \(B_{i}(t)\) can be respectively obtained by relevant compression experiments. \(\epsilon_{i}(t)\) is the compression ratio in the crushing zone \(i\), which describes how much the rock material is compressed in the crushing zone \(i\) over time. Piston and die tests were performed to obtain selection function and breakage function [12; 23], which were determined by the compression ratio. The compression ratio in crushing zone could be calculated with parameters of the cone crusher structure and wear model. The basalt was used to study the selection function \(S_{i}\) and breakage function \(B_{i}\). The selection function model and breakage function model was developed, as shown in Equations (10) to (12):\[S_{i}(\varepsilon(t)_{i},\sigma_{i}) = 15.3917(\varepsilon(t)_{i})^{2}{\sigma_{i}}^{2}-4.1471(\varepsilon(t )_{i})^{2}{\sigma_{i}}-3.5993(\varepsilon(t)_{i})^{2}-16.2863(\varepsilon(t)_{i }){\sigma_{i}}^{2}\ \,\] \[+4.3881(\varepsilon\ (t)_{i}){\sigma_{i}}+3.8085(\varepsilon(t)_{i })+0.1613{\sigma_{i}}^{2}-0.0435\sigma_{i}-0.0377\] \[B_{i}(x_{N},\varepsilon(t)_{i}) = (1-(0.09545+0.4137\varepsilon(t)_{i}))x_{Ni}^{33.62+4.635 \varepsilon(t)_{i}}\ \,\] \[+(0.09545+0.4137((\varepsilon(t)_{i})))x_{Ni} \tag{11}\] \[x_{N} = \frac{\log_{2}(x/x_{\min})}{\log_{2}(x_{0}/x_{\min})}, \tag{12}\]  
where \(x_{\min}\) is the minimum particle diameter of the feed material, \(x_{0}\) is the maximum particle diameter of the feed material, \(x\) is the particle diameter needed to solve the relevant weight percentage undersize passing, and the units of these variables are mm. The model for particle shape prediction was firstly described by [PERSON] and [PERSON] [24]. [PERSON] and [PERSON] [25; 26] developed a model able to directly calculate the mass percentage of flakiness in the product. The particle shape is also affected by liner wear. Thus, a model of particle shape is developed considering liner wear as shown in Equations (13) and (14):  
\[\text{FI}_{\text{TOTAL}}=\sum_{j=1}^{n}P_{j}\text{FI}_{j}\left(\overline{F}, \text{CSS}\left(t\right),\overline{d}_{j}\right), \tag{13}\]  
\[\text{CSS}\left(t\right)=\text{CSS}+\Delta\omega_{p}(t) \tag{14}\]  
In this constitutive equation \(P_{j}\) is a component of the vector which represents the proportion of particles in the size range \(j\) compared to the total product, and \(n\) is the total number of size classes. \(\overline{d}\) is the average size of the particles in the size range \(j\), and \(\overline{F}\) is the average particle size of the feed, \(\text{FI}_{\text{TOTAL}}\) is just the percentage of flakiness in the total product. Based on the cone crusher design, the maximum compression ratio in each crushing zone was calculated and the compression ratio and size distribution index entered into the pressure model, thus, the pressure distribution in the first crushing zone could be achieved. Then, the pressure distribution in each crushing zone and working time of crusher was entered into the wear model, and the liner wear of crusher was emulated.