## 2 Materials and Methods
### Physical Modeling of Bulk Transfer Pipelines
#### 2.1.2 Mathematical Modeling of Pressure Drop in Pipeline  
As shown in Figure 3, the dilute-phase transfer method has a typical system layout and generally constitutes a highly reliable system. In this method, the pressure loss performance of the BTS is composed of several main characteristic components. Each pressure loss component can be further classified into the pressure drop for only air (\(\Delta P_{L}\)), pressure drop due to acceleration (\(\Delta P\)), additional pressure loss due to the presence of solids (\(\Delta P_{Z}\)), lift pressure loss (\(\Delta P_{G}\)), and bend pressure loss (\(\Delta P_{B}\)) [25]. Additional losses include the pressure loss due to system wear and the step loss at pipe connections. The sum of these pressure loss components specifies the gas pressure required during the design phase. The parameters used in the equations are represented in the SI unit system. Pressure Drop for Only Air (\(\Delta P_{L}\))  
Equation (1) represents the horizontal head loss. The pressure loss in the pipe is generalized by [PERSON]'s equation [25]  
\[\Delta P_{L}=\lambda_{L}\frac{\rho}{2}V^{2}\frac{L}{d}, \tag{1}\]  
where \(\lambda_{L}\) is the friction factor, \(\rho\left(\text{kg}^{3}/\text{m}\right)\) is the density of the conveying gas, \(V(\text{m}/\text{s})\) is the average speed of the conveying gas, \(L(\text{m})\) is the length of the pipe, and \(d(\text{m})\) is the pipe diameter. Pressure Drop Due to Acceleration (\(\Delta P_{A}\))  
Solids are deposited in bunkers over a feeder. The flow of the feeder is the main form of transport according to various flow ratios and pressures. The solids accumulated in the bunkers at atmospheric pressure are immobile and are moved by the flow gas. Rapid changes in momentum cause high pressure losses. The length of the horizontal pipe is sufficient to allow the particles to accelerate from a  
Figure 3: Conceptual diagram and pressure loss configuration of BTS. stationary state to an average feed rate. The pressure loss due to acceleration is associated with the acceleration zone [25]. \[ \begin{array}{c}\mathrm{c}=v(1-0.65d^{0.92}\rho_{P}^{0.5}\rho^{-0.2}D^{-0.54}),\\ \Delta P_{A}=\upnu_{1}\rho_{1}c,\end{array}  \tag{2}\]  
where \(c(\mathrm{m/s})\) is the particle velocity, \(d\) is the instantaneous total drag coefficient, \(\rho p\left(\mathrm{kg^{3}/m}\right)\) is the bulk density, \(\rho\left(\mathrm{kg^{3}/m}\right)\) is the carrier gas density, and \(D(\mathrm{m})\) is the pipe diameter. Additional Pressure Drop Due to Presence of Solids (\(\Delta P_{Z}\))  
When the bulk is being transferred, the bulk itself has an additional pressure loss. The additional pressure drop due to the presence of the solids is given as [25]  
\[\Delta P_{Z}=\mu\lambda_{Z}\frac{\rho}{2}v^{2}\frac{\Delta L}{D}, \tag{3}\]  
where \(\mu\) is the viscosity coefficient, \(\lambda_{Z}\) is an additional pressure drop factor, \(\rho\left(\mathrm{kg^{3}/m}\right)\) is the density of the carrier gas, \(v\left(\mathrm{m^{2}/s}\right)\) is the kinematic viscosity, \(L(\mathrm{m})\) is the pipe length, and \(D(\mathrm{m})\) is the pipe diameter. Pressure Drop Due to Gravity (\(\Delta P_{G}\))  
Unlike in the horizontal pipe, the particle flow in the vertical pipe takes into account the additional particle size and velocity and the pressure loss due to gravity acting on the pipe length [25]. \[\Delta P_{G}=\rho^{*}g\Delta Z,\ \rho^{*}=\frac{\mu\rho}{c/v}, \tag{4}\]  
where \(\rho^{*}\left(\mathrm{kg^{3}/m}\right)\) is the apparent bulk density and conveying density, \(g\left(\mathrm{m/s^{2}}\right)\) is the gravitational acceleration, \(\Delta Z(\mathrm{m})\) is the height difference of the vertical pipe, \(\mu\) is the viscosity, \(\rho\left(\mathrm{kg^{3}/m}\right)\) is the air density, \(c(\mathrm{m/s})\) is the particle velocity, and \(v\left(\mathrm{m^{2}/s}\right)\) is the kinematic viscosity. Bend Pressure Loss (\(\Delta P_{B}\))  
The suspension velocity will be considerably reduced if by the bend angle of the pipe is too large.