# Experimental Study on Horizontal Pipeline Transportation Characteristics of Coarse Particle
## 2 Methodology  
### Theory of Solid-Liquid Two Phase Flow Model  
In the realm of dredging engineering, the physical properties of the solid-liquid two-phase flow, which results from the mixture of coarse dredging materials and water, significantly influence the dynamics of the two-phase flow [16]. These properties are determined collectively by both the solid and liquid components. The characteristics pertinent to the mixture and its hydraulic transportation can be described by the following formula:  
\[\gamma_{m}=\frac{G_{m}}{\varnothing_{m}} \tag{1}\]  
where \(\gamma_{m}\) represents the bulk density of slurry; \(G_{m}\) represents the total weight of the slurry; \(\varnothing_{m}\) represents the total volume of slurry. This physical quantity reflects, to some extent, the concentration of the solid-liquid two-phase flow. The settling velocity of solid particles in a liquid is a critical physical parameter that characterizes the behavior of two-phase flow. This velocity significantly influences the formation and evolution of flow patterns within the pipeline. The settling velocity is primarily determined by factors such as the density of the particles, their diameter, the viscosity and density of the liquid, and the local gravitational acceleration. According to [PERSON] law, when the settling velocity of particles is low and the relative velocity between the particles and the liquid is minimal, the settling velocity of the particles can be calculated using the following formula [17]:  
\[\omega=\sqrt{\frac{4}{3}gd\left(\frac{\rho_{s}-\rho}{\rho}\right)/C_{D}} \tag{2}\]  
where \(\omega\) represent the settling velocity of particles, \(g\) represent the gravitational acceleration, \(\rho_{s}\) represent the density of particles, \(\rho\) represent the density of liquids, \(d\) represent the diameter of particles, and \(C_{D}\) represent the drag coefficient. The spatial volume concentration \(C_{\text{\tiny{gs}}}\) is the volume occupied by the solid \(\varnothing_{s}\) divided by the total mixture volume \(\varnothing_{m}\) of the pipeline section. \[C_{\text{\tiny{gs}}}=\frac{\varnothing_{s}}{\varnothing_{m}} \tag{3}\]  
The transport volume concentration \(C_{\text{\tiny{gt}}}\) is the solids volume flow rate \(Q_{s}\) divided by the total mixture volume flow rate \(Q_{m}\). \[C_{\text{\tiny{gt}}}=\frac{Q_{s}}{Q_{m}} \tag{4}\]For a certain control volume, if a certain linear velocity \(V_{ls}\) is given and the volumetric spatial concentration \(C_{\text{\tiny{IS}}}\) and slip velocity \(V_{sl}\) are known, the volumetric transport concentration \(C_{vt}\) can be determined by the following equation:  
\[C_{vt}=\left(1-\frac{V_{sl}}{V_{ls}}\right)\cdot C_{\text{\tiny{IS}}}=(1-\xi) \cdot C_{\text{\tiny{IS}}} \tag{5}\]  
In the field of pipeline hydraulic transportation, there is a difference in velocity between liquid and solid phases. Slip velocity is the difference in velocity between one phase and another, while linear velocity is the flow velocity inside the fluid pipeline. \[V_{\text{sl}}=V_{l}-V_{s}=V_{ls}-V_{s}=V_{ls}\cdot\left(1-\frac{V_{s}}{V_{ls}} \right)=V_{ls}\cdot(1-\xi) \tag{6}\]  
where \(V_{sl}\) represents the slip velocity; \(V_{l}\) represents the liquid phase velocity; \(V_{s}\) represents the solid velocity; \(V_{sl}\) represents the linear velocity of the fluid inside the pipeline; \(\xi\) represents the slip ratio, which is the ratio of the slip velocity \(V_{sl}\) to the linear velocity \(V_{ls}\). The magnitude of the slip ratio is more related to the development of flow patterns between flows in the pipe [18]. Generally speaking, when the slip ratio is close to 1, it means that the velocities of the two phases are close to each other, the relative motion is small, and the closer the flow pattern in the pipeline is to the homogeneous suspension flow; when the slip ratio is less than 1, it means that the velocity of the gas phase or the liquid phase is greater, the relative motion is more obvious, and the closer the flow pattern in the pipeline is to the sedimentary bed type.