# Dynamic Coal Flow-Based Energy Consumption Optimization of Scraper Conveyor
## 2 Development of the Dynamic Coal Flow Model and EDEM Simulation
### Construction of Dynamic Coal Flow Model
Assuming that coal flow follows an exponential decay pattern during conveying, and that the dissipation rate per unit distance is \(k\), the coal load \(M\)(\(d\)) after traveling a distance \(d\) can be expressed as follows:  
\[Q_{in}(x,t)=Q(x,t)\,\ Q_{out}(x,t)=Q(x+\Delta x,t) \tag{2}\]0  
where \(k\) is the dissipation coefficient related to friction, collisions, and other losses, indicating the coal loss rate per unit distance. Based on the above, the dispersion factor \(\beta\) is defined as the proportion of coal loss per unit distance:  
\[Q_{in}(x,t)=Q(x,t)\,\ Q_{out}(x,t)=Q(x+\Delta x,t) \tag{2}\]1  
Under steady-state conditions, the coal load on the scraper conveyor depends not only on the coal flow rate \(Q(x)\), but also on the conveying speed \(v\), the accumulation factor \(\alpha\), and the dispersion factor \(\beta\). The final expression for calculating the coal load at any point along the conveyor is:  
\[Q_{in}(x,t)=Q(x,t)\,\ Q_{out}(x,t)=Q(x+\Delta x,t) \tag{2}\]2  
3. Modification of the dynamic coal flow model  
Building upon the coal distribution equation and coal load equation, the relationship between coal flow rate \(Q\)(\(x\),\(t\)) and coal mass \(M\)(\(x\),\(t\)) is further explored. It is assumed that the coal flow rate is proportional to the inlet coal mass, given by:  
\[Q_{in}(x,t)=Q(x,t)\,\ Q_{out}(x,t)=Q(x+\Delta x,t) \tag{2}\]3where \(v(x,t)\) is the coal flow velocity at position \(x\) and time \(t\), in meters per second (m/s). Substituting this expression into the coal distribution equation yields the fundamental equation of dynamic coal flow, as shown in Equation (15):  
\[Q_{in}(x,t)=Q(x,t)\,\ Q_{out}(x,t)=Q(x+\Delta x,t) \tag{2}\]4  
This equation serves as the basic mathematical model for dynamic coal flow, accurately describing the spatial-temporal distribution of coal. However, in addition to mass conservation, the dynamic model must also incorporate the effects of accumulation and dissipation. The accumulation effect reflects the localized build-up of coal in certain regions, often occurring when the conveyor speed is low or the coal inflow rate is high. To represent this, an accumulation factor \(\alpha(x,t)\) is introduced, indicating the accumulation rate of coal flow. The modified coal distribution equation becomes:  
\[Q_{in}(x,t)=Q(x,t)\,\ Q_{out}(x,t)=Q(x+\Delta x,t) \tag{2}\]5  
The dissipation effect describes the gradual reduction in coal flow due to friction, gravity, and other factors during transportation. This is modeled by introducing a dissipation factor \(\beta(x,t)\), representing the coal loss rate:  
\[Q_{in}(x,t)=Q(x,t)\,\ Q_{out}(x,t)=Q(x+\Delta x,t) \tag{2}\]6  
where \(\beta(x,t)\) is influenced by factors such as coal particle size, conveying speed, and friction coefficient. By incorporating the accumulation factor \(\alpha(x,t)\) and dissipation factor \(\beta(x,t)\), the dynamic coal flow equation is further revised as follows:  
\[Q_{in}(x,t)=Q(x,t)\,\ Q_{out}(x,t)=Q(x+\Delta x,t) \tag{2}\]7  
This equation indicates that the inlet coal flow rate is a key parameter in the dynamic coal flow model, directly determining the evolution and distribution of coal over time and space. The inlet rate is mainly affected by the cutting process of the shearer drum, which governs the volume of coal entering the scraper conveyor per unit time. As shown in Figure 2, to clearly describe the coal flow formation during drum cutting, a differential surface element is defined in polar coordinates. Let the radius from the drum center to the coal flow be \(R_{i}\), with small angular increment \(d\theta\) and radial increment \(dR_{i}\).