## 2 Materials and Methods
### DEM Simulation of Separation Processes
#### 2.4.2 Establishment of the Adhesion Model  
The adhesion between raisins and foreign object, combined with sugar approaching vitrification after quick-freezing, creates a raisin-sugar-foreign object structural configuration. Separation mechanisms involve three potential pathways: fracture of the sugar layer under external force, interfacial detachment due to weak adhesion at sugar-surface interfaces, or surface delamination caused by strong adhesion under mechanical stress. Experimental validation through rolling extrusion demonstrated reduced residue and damage rates, justifying the exclusion of the third mechanism in simulations. Consequently, the Hertz-Mindlin with bonding contact model was implemented to simulate adhesive interactions between raisins and foreign object, as depicted in Figure 17.  
Figure 16: Distinct material models. The discrete element model is used for EDEM simulation, while the discrete element adhesion model is used for EDEM-Fluent simulation.  
The simulation employing the Hertz-Mindlin with bonding contact model necessitates the determination of parameters including inter-particle contact radius (\(R\)), normal stiffness per unit area (\(K_{n}\)), shear stiffness per unit area (\(K_{t}\)), critical normal stress (\(\sigma_{c}\)), and critical shear stress (\(\tau_{c}\)) [25]. In the raisin-sugar-foreign object system where sugar acts as the bonded component, the sugar-related parameters are challenging to measure experimentally. Consequently, this study adopts theoretically calculated values as model parameters. According to [PERSON] contact theory, the contact radius formula is presented in Equations (11)-(13).  
\[R=\sqrt[3]{\frac{3F_{n}R_{eff}}{4E^{*}}} \tag{11}\]  
\[\frac{1}{R_{eff}}=\frac{1}{R_{1}}+\frac{1}{R_{2}} \tag{12}\]  
\[\frac{1}{E^{*}}=\frac{1-{v_{1}}^{2}}{E_{1}}+\frac{1-{v_{2}}^{2}}{E_{2}} \tag{13}\]  
In the equation, \(F_{n}\) denotes the average tensile force at separation, \(R_{eff}\) represents the effective radius of curvature, \(E^{*}\) indicates the equivalent elastic modulus, \(R_{1}\) corresponds to the radius of curvature at the raisin contact point, \(R_{2}\) signifies the radius of curvature at the foreign object contact point, \(v_{1}\) is the Poisson's ratio of the raisin, \(v_{2}\) denotes the Poisson's ratio of the foreign object, \(E_{1}\) represents the elastic modulus of the raisin, and \(E_{2}\) corresponds to the elastic modulus of the foreign object. The normal stiffness per unit area \(K_{n}\) is calculated as shown in Equation (14).  
\[K_{n}=\frac{4}{3}E^{*}\sqrt{R} \tag{14}\]  
The calculation of the tangential stiffness per unit area (\(K_{t}\)) is presented in Equations (15) and (16).  
\[K_{t}=\frac{8G^{*}\sqrt{R}}{2-v} \tag{15}\]  
\[\frac{1}{G^{*}}=\frac{2(1+v_{1})}{E_{1}}+\frac{2(1+v_{2})}{E_{2}} \tag{16}\]  
In the formula, \(G^{*}\) represents the equivalent shear modulus. The critical normal stress \(\sigma_{c}\) and the critical tangential stress \(\tau_{c}\) are calculated as shown in Equations (17) and (18).  
\[\sigma_{c}=\frac{F_{n}}{\pi R^{2}} \tag{17}\]  
\[\tau_{c}=\frac{\sigma_{c}}{\sqrt{3}} \tag{18}\]  
Figure 17: Bonding contact model.