## 2 Materials and Methods  
### Black Soldier Fly Sand and Its Adhesion Module Model  
The BSF sand mixture that was separated in this study was a product formed by the biotransformation of pig manure by black soldier flies, of which the main components were black soldier fly sand, BSFL. To be specific, the black soldier fly sand with the highest content was followed by the BSFL, with a mass ratio of 8 to 10 (Figure 1). The particle size distribution and percentage of the content of BSF sand were obtained using the sieving method. Most of the sand particles of the BSF were similar spheres, and a few were in the form of agglomerates. The particle size distribution was <1.6 mm, 1.6-2.6 mm, 2.6-3.5 mm, and >3.5 mm, respectively. The mass and mass fraction were 148.24 g, 18.53%, 383.68 g, 47.96%, 154.86 g, 19.36%, 113.22 g, and 14.15%, respectively. At the early stage, 50 BSFLlarvae were randomly examined. The cross-section of the BSFL was approximately oval. The BSFL physical parameters achieved the following mean values: a body length of 23.30 mm, standard deviation of 0.70 mm; body width of 4.90 mm, standard deviation of 0.34 mm; and the mass of 0.177 g/piece.  
A certain amount of water exists on the sand particle surface of BSF. The adhesion between the sand particles of the BSF increases due to the action of liquid bridge force. Accordingly, the particles are easily bonded to each other to form a ball, thus leading to the increased particle size of the BSF sand. The round-hole screen and the grid-shaped screen on the conventional tremmel are prone to this phenomenon, thus reducing the screen penetration rate of the tremmel screener. There is adhesion between the sand particles of the BSF due to its moisture content. The particles form secondary and tertiary agglomerated particles during the movement process, thus resulting in the increased particle size. The adhesion process of BSF sand particles is illustrated in Figure 2. Figure 3 presents the established BSF sand particle adhesion nodule model.  
Figure 1: The main components of black soldier fly feces.  
Figure 2: Process of black soldier fly feces adhered to become agglomerations.  
Assuming that the contact angle between particles is zero, the adhesion force \(F\) between the sand particles of the BSF can be calculated as follows:  
\[F=\pi R_{1}^{2}\tau\bigg{(}\frac{1}{R_{2}}-\frac{1}{R_{1}}\bigg{)}+2\pi R_{1}\tau= \pi R_{2}\tau\bigg{(}\frac{R_{1}+R_{2}}{R_{2}}\bigg{)} \tag{1}\]  
\[R_{1}=r(\sec\varepsilon-1) \tag{2}\]  
\[R_{2}=r(\tan\varepsilon-\sec\varepsilon+1) \tag{3}\]  
Based on Equations (1)-(3), this yields:  
\[F=\frac{2\pi r\tau}{1+\tan\frac{\varepsilon}{2}} \tag{4}\]  
where \(\tau\) denotes the surface tension; \(r\) is the radius of the BSF sand particle; and \(\varepsilon\) represents the pincer angle.  
The analysis of Equation (4) indicates that when the BSF sand particles are in adhesive contact, the adhesion force between the particles is correlated with several factors (e.g., the radius of the particles, the water content, and the size of the pincer angle). For \(N\) particles in a space with a certain volume of the trommel screener \(V\), the probability \(P\) that any two particles will collide with each other in time \(t\) is calculated as follows:  
\[P=\frac{4\pi r^{2}tv_{r}N}{V} \tag{5}\]  
where \(N/V\) is the approximate particle concentration of the BSF sand material, and \(v_{r}\) denotes the relative movement speed of the particles.