## 3 Concrete Pumping Particle Flow Simulation  
### The Principle of Particle Flow Simulation Calculation  
During concrete conveying, coarse aggregate particles collide with each other, and the forces involved are influenced by the shape and distribution of the coarse aggregates. This issue is commonly addressed using a combination of finite element and discrete element methods. In concrete conveying simulations, the mortar is treated as a fluid medium, while the aggregate particles are considered to be solid particles flowing within this fluid. The flow behavior of the mortar is calculated using the Eulerian method, while the motion of the solid aggregate particles is analyzed with the Lagrangian method. These two methods are coupled through interaction forces, which include the drag force, buoyancy force, and the pressure generated by the pressure gradient on both sides of the particles. Given the small size of aggregate particles relative to the pumping distance, this paper disregards the pressure generated by the pressure gradient on both sides of the particles. Only the combined effects of drag force and buoyancy are considered. Mortar flow in concrete follows the law of conservation of fluid momentum. The equations of continuity and momentum conservation are used to solve the governing equations, which are expressed as follows:  
\[\frac{\partial}{\partial t}(\alpha_{f}\rho_{s}u_{s})+\
abla\cdot(\alpha_{f} \rho_{s}u_{s})=0 \tag{6}\]  
\[\frac{\partial}{\partial t}(\alpha_{f}\rho_{s}u_{s})+\
abla\cdot(\alpha_{f} \rho_{s}u_{s}u_{s})=-\alpha_{f}\
abla P+\
abla\cdot(\alpha_{f}\mu_{s})+\alpha_ {f}\rho_{s}g-S \tag{7}\]  
where, \(\alpha_{f}\) is the volume fraction of mortar to concrete, \(\rho_{s}\) is the density of mortar, \(u_{s}\) is the velocity of mortar, \(\
abla P\) is the pressure gradient, \(\mu_{s}\) is the dynamical viscosity of mortar, \(S\) is the momentum exchange, and g is the gravitational acceleration. \begin{table}
\begin{tabular}{c c c c} \hline
**Bend Type** & **Type of Conversion** & \
\begin{tabular}{c} **Pipe Diameter/Curvature** \\ **Radius (mm)** \\ \end{tabular} &
\begin{tabular}{c} **Conversion Length** \\ **(m)** \\ \end{tabular} \\ \hline Bent Pipes & everyone & 500 & 12\(\beta\)/90 \\ Bent Pipes & everyone & 1000 & 9\(\beta\)/90 \\ \hline \end{tabular}
\end{table} 
Table 1: Horizontal conversion length of pumped concrete pipes. The motion of concrete aggregate particles includes both translational and rotational components. The state of motion is determined using [PERSON]'s second law, described by the following equation:  
\[m_{i}\frac{du_{i}}{dt}=m_{iS}+F_{p-g}+\sum_{j=1}^{k_{i}}\left(F_{cn,ij}+F_{ct,ij}\right) \tag{8}\]  
\[I_{i}\frac{d\omega_{i}}{dt}=\sum_{j=1}^{k_{i}}T_{ij} \tag{9}\]  
\[F_{p-g}=-\rho_{S}gV+F_{drag} \tag{10}\]  
where, \(m_{i}\) denotes the mass of aggregate particle \(i\), \(u_{i}\) denotes the velocity of particle \(i\), \(\omega_{i}\) denotes the angular velocity of particle \(i\), \(F_{p-g}\) denotes the interaction force between the mortar and the particles, \(k_{i}\) denotes the number of particles in contact with particle \(i\), \(F_{cn,ij}\) and \(F_{ct,ij}\) denote the normal and tangential contact forces of particle \(i\) and particle \(j\), respectively, and \(T_{ij}\) denotes the moment induced by the collision of particles \(i\) and \(j\). \(\rho_{g}\) is the particle density, \(V\) is the particle volume, and \(F_{drag}\) is the resistance force on the particles.