### 1. Introduction

### 2. Numerical Formulation and Conditions

### 2.1 Governing Equations and Numerical Modeling

*α*is the volume fraction,

*ρ*is the density,

*t*is time,

**is the velocity vector,**

*u**p*is the pressure,

**is the stress,**

*τ*

*τ**is the turbulent stress,*

^{t}**is the gravity acceleration,**

*g***is the momentum transfer term between the different phases,**

*M*

*F*_{i}

*is the internal forces, and the subscript*

_{nt}*i*indicates the type of fluid.

*α*is the volume fraction for solid phase, which was obtained as 0.2 through an experiment.

_{p}*μ*is the viscosity of the liquid phase,

_{c}*ρ*is the density of liquid,

_{c}*l*is the average diameter of particles,

*α*is the volume fraction for liquid, and

_{c}*u*is the relative velocity between adjacent phases. The drag coefficient

_{r}*C*

*is corrected by a correlation equation from Schiller and Naumann and is written as Eq. (4): where the Reynolds number for the dispersed phase can be expressed in Eq. (5):*

_{D}*u*is the minimum fluidization velocity, and

_{mf}*C*

*is the drag coefficient of the solid particle group, which is presented as Eq. (7):*

_{D}*is the drag coefficient of a single solid particle and is given in Eq. (8): where*

_{Ds}*Re*is the Reynolds number of a single solid particle in Eq. (9), and

_{s}*V*is the ratio of the terminal velocity of the single particle in Eq. (10) and the total particles: where

_{r}*μ*is the viscosity coefficient of the solid phase, and

_{p}*ρ*is the density of the solid phase.

_{p}*u*is the velocity of fluid, and

*C*is the lift coefficient. In the case of the lift coefficient, the default value is set to 0.25, but in this study, a value of 0.1 was used because it is suitable for small-sized particles according to Ekambara et al. (2009).

_{L}*σ*and

_{p}*σ*are the turbulent Prandtl numbers of the solid and fluid, respectively, and both are set to a value of 1. Additionally,

_{s}*ν*and

_{p}^{t}*ν*represent the kinematic viscosity of the solid and fluid due to turbulence, respectively. Inside the solid phase, a pressure force between the solid phases acts when the distribution of the solid reaches the maximum distribution. In order to consider the pressure of the solid, in this study, the granular pressure model was used, as represented by Eq. (15): where

_{s}^{t}*P*is the solid pressure,

_{p}*μ*is the effective granular viscosity,

_{p}*I*is the isotropic tensor, and

*ξ*is the bulk viscosity.

_{p}*g*

_{0}of Ding and Gidaspow (1990) if the particle distribution is lower than the maximum distribution criterion as Eq. (16): where

*α*is the particle volume fraction, and

_{p}*α*,

_{p}_{max}is the maximum particle volume fraction.

*g*is used to calculate the granular temperature

_{o}*θ*, which determines the effective granular viscosity. The effective granular viscosity is composed of the collisional (

_{p}*C*) and kinetic (

*K*) contributions (Gidaspow, 1994) as given in Eqs. (17)∼(19):

*ϕ*is given as 25° from Schaeffer (1987). In addition, the maximum distribution criterion for rigid spherical solid particles is applied based on the representative value of 0.624 obtained from an experiment.

*ϕ*is the angle of internal friction, and

*I*

_{2}

*is the second invariant of deviator of is the strain rate tensor.*

_{D}### 2.2 Geometric Shape and Ggrid System

^{3}and viscosity of 0.0045 Pa·s was used as the working fluid. The Reynolds number was calculated based on the maximum diameter of the impeller as about 10

^{6}.

### 2.3 Boundary Conditions

### 3. Simulation Results

### 3.1 Verification of Single-phase Swirling Flow Around Agitator

*k*−

*∊*(SKE), realizable

*k*−

*∊*(RKE), and

*k*−

*ω*models (KW) models. The initial conditions for turbulence in the flow field for all subsequent cases were turbulence intensity (TI) = 0.01 and turbulent viscosity ratio (TVR) = 10. Fig. 6 shows the time-averaged profiles of the simulated axial and tangential velocities compared to the experiments. Similarly, in both the simulation and experiment, the magnitude of the velocity tends to increase toward the end of the blade with a radius of 0.072 m and then gradually decreases. However, it was found that the tendency of the velocity profile is slightly different between the simulation results depending on turbulence models employed in the simulation. It seems that all of the turbulence models cannot predict accurately the peak value of a tangential velocity.

### 3.2 Verification of Liquid-solid Multi-phase Flow Around Agitator

^{3}by adding NaCl. Spherical glass beads (d = 2.85–3.30 mm) with a density of 2,485 kg/m

^{3}were used as the solid. Initially, solid particles are evenly distributed in the space and mostly settle down from their own weight for 3 seconds. After 3 seconds, a swirling flow is generated by the rotational motion of the impeller in the agitator, and then the particles slowly start to suspend.