# 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  
The construction of the coal mass distribution equation and the dynamic coal flow mathematical model is a key foundation for accurately describing the coal transportation process in scraper conveyors. Based on the law of conservation of mass, this model comprehensively considers important characteristics such as accumulation, dispersion, and continuity of coal flow, realistically reflecting the spatiotemporal dynamics of coal flow during mining and transportation. This provides a solid theoretical basis for subsequent energy consumption analysis and optimization. Similar modeling approaches have been widely applied in bulk material transport to characterize mass transfer and transient flow behaviors [24]. The following sections will detail the formulation, derivation, and mathematical representation of the dynamic coal flow model. 1. Derivation of the coal flow distribution equation According to the law of mass conservation, the change in coal volume within a given region over time equals the difference between the inflow and outflow coal flow rates, as expressed in Equation (1): \[\frac{\partial M(x,t)}{\partial t}=Q_{in}(x,t)-Q_{out}(x,t)\] (1) where \(M(x,t)\) represents the coal volume distribution at position \(x\) and time \(t\), and \(Q_{in}(x,t)\) and \(Q_{out}(x,t)\) represent the inflow and outflow coal flow rates, respectively. \(Q(x,t)\) is defined as the coal flow rate at position \(x\), as shown in Equation (2):  
\[Q_{in}(x,t)=Q(x,t)\,\ Q_{out}(x,t)=Q(x+\Delta x,t) \tag{2}\]  
Thus, the equation can be rewritten as given in Equation (3):  
\[\frac{\partial M(x,t)}{\partial t}=Q(x,t)-Q(x+\Delta x,t) \tag{3}\]  
As \(\Delta x\) approaches an infinitesimally small value, Taylor expansion is applied to approximate \(Q(x+\Delta x,t)\) linearly, as given in Equation (4):  
\[Q(x+\Delta x,t)\approx Q(x,t)+\frac{\partial Q(x,t)}{\partial x}\Delta x \tag{4}\]  
which simplifies to:  
\[\frac{\partial M(x,t)}{\partial t}+\frac{\partial Q(x,t)}{\partial x}=0 \tag{5}\]  
During coal transportation, the distribution of coal volume varies with both space and time. To develop a mathematical model, coal is treated as a continuous medium on a macroscopic scale, mass conservation is assumed with no external coal input or loss, and the flow is considered one-dimensional along the conveyor, neglecting variations in other directions. 2. Steady-state coal load equation According to the coal mass conservation equation, assuming the conveying velocity \(v(x,t)\) and the coal flow rate \(Q(x,t)\) are locally steady and can be treated as constants, the mass conservation equation can be simplified as follows:\[\frac{\partial M(x,t)}{\partial t}+v\frac{\partial M(x,t)}{\partial x}=Q(x,t) \tag{6}\]  
Under steady-state conditions, the coal load \(M\)(\(x\),\(t\)) reaches equilibrium over time; thus:  
\[\frac{\partial M(x)}{\partial x}=0 \tag{7}\]  
The equation can be further simplified to:  
\[v\frac{\partial M(x)}{\partial x}=Q(x) \tag{8}\]  
By integrating this equation, the coal load at any position on the scraper conveyor can be expressed as follows:  
\[M(x)=M_{0}+\frac{1}{v}\int\limits_{0}^{x}Q(x^{{}^{\prime}})dx^{{}^{\prime}} \tag{9}\]  
where \(M_{0}\) represents the initial coal load. To better describe the coal accumulation effect, an accumulation factor \(\alpha\) is introduced. It is defined as the rate of change in coal load with respect to spatial position \(x\), and is given by:  
\[\alpha=\frac{1}{v} \tag{10}\]  
A dispersion factor \(\beta\) is used to describe material losses during coal transportation due to friction, collisions, and other factors. It represents the attenuation of coal flow per unit distance and ranges from 0 to 1, reflecting the loss ratio of coal flow during transport.