# Pressure Drop and Particle Settlement of Gas-Solid Two-Phase Flow in a Pipe
## 2 Basic Equations  
Figure 1 shows gas-solid two-phase flow in a pipe with diameter \(D\) and length \(L\). A two-fluid model is used to simulate three-dimensional gas-solid two-phase flow [14]. The particle phase is also considered to be a continuous medium in the two-fluid model, so particle-to-particle interaction has been reflected by the relationship between the stress and strain rates in the second term on the left-hand side of Equation (2). The two phases are regarded as two interacting continuous phases for the model, so the two phases have the same structure of the governing equations. Assuming that the flow field is steady and isothermal, there is no mass exchange between phases, and the particle stress tensor is ignored. Then, the continuity equation, momentum equation, and state equation are:  
\[\
abla\cdot\big{(}\alpha_{g,s}\rho_{g,s}v_{g,s}\big{)} =0, \tag{1}\] \[\
abla\Big{[}\alpha_{g,s}\rho_{g,s}v_{g,s}v_{g,s}+\alpha_{g,s} \Gamma_{g,s}\Big{(}\
abla v_{g,s}+\
abla v_{g,s}^{\intercal}\Big{)}\Big{]} =-\
abla\big{(}\alpha_{g,s}\rho\big{)}+\alpha_{g,s}\rho_{g,s}g+M_{g,s},\] (2) \[p =\rho_{g}RT_{g}, \tag{3}\]  
where subscript \"\(g\), \(s\)\" indicates the gas or solid phase; \(\alpha\) is the phase composition; \(\rho\) is the density; \(v\) is the velocity; \(\Gamma=\rho(v_{l}+v_{l})\) is the diffusion coefficient; \(v_{l}\) and \(v_{l}\) are the molecular and turbulent viscosity coefficients, respectively; \(p\) is the pressure; \(g\) is the gravitational acceleration; \(R\) is the gas constant; \(T\) is the temperature; and \(M\) is the interphase momentum exchange term:  
\[M_{g,s}=K\big{(}v_{g,si}-v_{g,s}\big{)}+p_{g,s}\
abla\alpha_{g,s}, \tag{4}\]  
where subscript \"\(i\)\" indicates the different phases, and \(K\) is the interphase friction coefficient and can be expressed as follows when the particle volume concentration is larger than 0.2:  
\[K=150\frac{\alpha_{s}^{2}}{\alpha_{g}}\frac{\mu}{d_{s}^{2}}+1.75\alpha_{g} \frac{1}{d_{s}}\rho_{g}\big{|}v_{g}-v_{s}\big{|}, \tag{5}\]  
where \(d\) is the particle diameter, and \(\mu\) is the gas viscosity coefficient. When the particle volume concentration is less than 0.2, \(K\) can be expressed based on the aerodynamic force acting on solid particles as follows:  
\[K=\Big{(}C_{D}\alpha_{g}^{-2.65}\Big{)}\bigg{(}\frac{3\alpha_{s}}{2d_{s}} \bigg{)}\frac{1}{2}\alpha_{g}\rho_{g}\big{|}v_{g}-v_{s}\big{|}, \tag{6}\]where the drag coefficient of a single particle \(C_{D}\) and particle's Reynolds number \(Re\) are:  
\[C_{D}=Max\left\{\frac{24}{Re}\Big{(}1+0.15\text{Re}^{0.687}\Big{)},\;0.44\right\},\;Re=\frac{\rho_{g}d_{s}\big{(}\alpha_{g}|v_{g}-v_{s}|\big{)}}{\mu} \tag{7}\]  
In the computation, the turbulent stress in the momentum equation adopts the Buossinesq eddy viscosity model, and the value of the eddy viscosity is determined with the corrected \(k\)-\(\epsilon\) turbulence model.  
The friction between the gas and the pipe wall can be represented by adding a source phase to the gas phase momentum equation of the control body near the pipe wall. Assuming that there is no slip between the flow and the wall, the velocity near the wall is distributed logarithmically, and the wall friction is calculated according to the smooth pipe. The friction between the solid phase and the pipe wall can be calculated by the method in reference [15], i.e., a source phase is added to the momentum equation of the solid phase of the control body near the tube wall.