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In this study, flow structures and mixing performance in a blade-free planetary mixer, which combines rotation and revolution motions inside a cylindrical vessel, are numerically investigated. Flow fields in the mixer vessel are simulated in a single rotating reference frame with various revolution speeds and a fixed rotation speed. The mixing process is investigated by a Lagrangian particle tracking method and the mixing performance is evaluated based on particle concentration. The results of the numerical simulations show that a vortical flow with an axis inclined with respect to the rotation axis of the vessel is generated by the combined influence of the rotation and revolution motions. The flow structure and vortical flow intensity vary as a function of the precession rate, which is the ratio of the revolution speed to rotation speed. The mixing performance of the blade-free planetary mixer is found to be maximum at aspecific precession rate.

Liquid blending and solid-liquid mixing are important operations in the production processes of various industries such as chemical, pharmaceutical, and food. A typical method for the blending and mixing is agitation of a fluid in a vessel with impellers. Various techniques, such as visualization of mixing process [

A blade-free planetary mixer is a new technology for mixing liquids without impellers and agitators. This type of mixer can mix highly viscous fluids with less contamination as the flow in the vessel is induced by precession that combines rotation and revolution of the vessel. The fluid mechanics in a precessing cylinder, which is the basic principle of the blade-free planetary mixer, has been studied in connection with the instability of the fluid motion [

Two approaches to evaluate mixing performance with CFD have been reported in the literature: Eulerian-Lagrangian method [

The objective of this study is to investigate the laminar mixing performance by numerical simulation in a blade-free planetary mixer, which is related to flow structures induced by the combined influence of rotation and revolution. The influence of the precession rate on the flow structure and mixing performance was also investigated.

Flow structures in a blade-freeplanetary mixer vessel were investigated by three-dimensional numerical simulations [

∂ u ∂ t + u ⋅ ∇ u = − 1 ρ ∇ p + ν ∇ 2 u − 2 Ω × u − Ω × Ω × r (1)

∇ ⋅ u = 0 (2)

where t, u, p, ρ, and ν denote time, fluid velocity, pressure, fluid density, and viscosity, respectively. Note that the fluid velocity and position vectors are defined as the relative velocity and position vectors in the rotating frame. The third term on the right-hand side of Equation (1) is the Coriolis force, where Ω is the revolution speed of the rotating frame. The fourth term is the centrifugal force, where r represents the position vector from the revolution axis.

_{rev} = 83 mm at the bottom center of the vessel. The configuration of the mixer was determined based on an actual machine designed for the purpose of experimental investigation. The rotation speed of the vessel was set to n_{rot} = 60 rpm, and the fluid was silicone oil with kinematic viscosity of ν = 1.5 × 10^{−5} m^{2}/s to keep the flow regime laminar in the range of the computational conditions. Therefore, the Reynolds number was Re ( = 1 / 4 ⋅ ω r o t ⋅ d 2 / v ) = 605 , where ω_{rot} is the angular velocity. Note that the present Reynolds number is smaller than the critical Reynolds number of 1 × 10^{4} in a stirred cylindrical vessel with weak precession [_{rev} = 0, 15, 30, 45, and 60 rpm, and the precession rates in the simulation were Γ (= n_{rev}/n_{rot}) = 0, 0.25, 0.5, 0.75, and 1. The rotation and revolution directions were opposite.

The computational domain was in the cylindrical vessel which was filled with the fluid. At the wall boundaries of the vessel, the rotation speed was applied to define the rotational motion of the vessel. The governing equations were discretized using the finite volume method with the quadratic upstream interpolation for convective kinematics (QUICK) scheme for the convective term and with the central differencing scheme for the other terms. The pressure implicit with splitting of

operator (PISO) algorithm was used to solve the governing equations. Independence of the grid convergence was evaluated using fine, medium, and coarse grids with 6.5 × 10^{5}, 3.8 × 10^{5}, and 4.8 × 10^{4} cells, respectively. Root-mean-square errors for the velocity fields based on the fine grid were 0.1% and 1.8% for the medium grid and the coarse grid, respectively. Consequently, the medium grid shown in

In this study, the mixing performance of the blade-free planetary mixer was evaluated by the Eulerian-Lagrangian method. The degree of mixing can be estimated from the number density distribution of particles by tracking a sufficient number of the particles in the vessel. Particle trajectories were simulated with the Lagrangian particle tracking method as passive particles after flow in the vessel reached the steady state. In this study, for experimental validation, it was assumed that solid particles gravitated to the bottom before mixing and were transported throughout the vessel by mixing. The particles were spherical with 10 μm diameter, and the particle mass density was equal to the mass density of the fluid in order to evaluate the mixing performance of the flow field itself. Collisions between particles were ignored.

The mixing performance was evaluated by the mixing index, which was calculated based on the particle number density in the vessel [_{i} was defined as the ratio of the particle number density in the ith sub-domain to the maximum particle number density in all sub-domains. The standard deviation of the number density σ over all sub-domains is defined as follows:

σ = 1 N ∑ i = 1 N ( ρ i − ρ m ) 2 (3)

where, ρ_{m} represents the average particle number density in the entire vessel. The mixing index I_{m} is given by the following equation:

I m = 1 − 2 σ (4)

In this study, the number of sub-domains was N = 6000 and the number of supplied particles was 6000 per second.

Velocity fields in a blade-free planetary mixer vessel were computed to understand the mixing mechanism. _{rot} = 60 rpm and precession rates Γ (= n_{rev}/n_{rot}) =0 (a), 0.25 (b), 0.5 (c), and 1 (d). The results for Γ = 0.75 are omitted because they are similar to the results for Γ = 1. Streamlines (left) are shown to analyze the three-dimensional flow structures, and velocity magnitude contours of the in-plane velocity components are shown

in the x-z plane (middle) and in the y-z plane (right). The velocity component was normalized by the wall velocity U_{0} (= 1 2 d ω rot ). At this Reynolds number, a steady flow was observed after a certain transient state. The results from panel (a) show circular streamlines parallel to the bottom and no in-plane velocity component due to the two-dimensional rotating flow. In the precession cases, the streamlines show the generation of vortical flows with axes inclined with respect to the spin axis of the vessel. The velocity fields for the lower precession rate of Γ = 0.25 (b) show strong rotational flow in the vertical cross-section, but the flow structure was nearly two-dimensional. The results for Γ = 0.5 (c) show a spiral axis of the rotating flow and a complex vortical flow structure. Non-axisymmetric rotating flow was observed in the x-z plane, and twin corner vortices occurred near the upper left and lower right corners in the y-z plane. As the precession rate increases further, the flow structure for Γ = 1 (d) became more complex. However, the low-speed region extended around the vortical flow axis. These characteristics of the vortical flow structure with the spiral and inclined axis in the blade-free planetary mixer are similar to the results from others presented in the literature [

The mixing performance was investigated by tracking tracer particles in the vessel and evaluating the particle number density.

To quantify the mixing performance, the mixing index was evaluated from the particle number density at different precession rates.

The flow structures and the mixing performance of a blade-free planetary mixer were investigated by numerical simulations. The flow fields in the mixer vessel were determined by solving the Navier-Stokes equation considering the influence of the Coriolis and centrifugal forces in a rotating reference frame. The mixing process was numerically visualized by a particle tracking method, and the mixing index was evaluated from the concentration of particles. In the numerical results, a vortical structure with an axis inclined with respect to the rotation axis was induced by the combined influence of rotation and revolution at lower precession rates. This vortical flow generated a rotating flow in the vertical cross-section of the vessel. As the precession rate increases, vortical flows showed complex three-dimensional flow structures with a spiral rotation axis. It is found from the present numerical simulation that the laminar mixing performance of the blade-free planetary mixer becomes maximum around the precession rate 0.5, which is contributed by the occurrence of rotating flow with an inclination angle to the spin axis and the generation of highly magnified flow around the axis caused by the precession. However, too strong precession prevents the formation of the large-scale vortical flow.

The authors express thanks to Mitsuboshi Co. Ltd. for the help and suggestions during the course of this study.

The authors declare no conflicts of interest regarding the publication of this paper.

Yamagata, T. and Fujisawa, N. (2019) Effect of Rotation and Revolution on Performance of Blade-Free Planetary Mixer. Journal of Flow Control, Measurement & Visualization, 7, 1-10. https://doi.org/10.4236/jfcmv.2019.71001

d vessel diameter [mm]

h vessel height [mm]

I_{m} mixing index

N number of sub-domains

n_{rev} revolution speed [rpm]

n_{rot} rotation speed [rpm]

p pressure [Pa]

Re Reynolds number

R_{rev} revolution radius [mm]

r position vector from a revolution axis

t time [s]

U_{0 } wall velocity [m/s]

u velocity vector [m/s]

u, v, w velocity components [m/s]

x, y, z Cartesian coordinates

Γ precession rate (=n_{rev}/n_{rot})

ν kinematic viscosity [m^{2}/s]

ρ fluid density [kg/m^{3}]

ρ_{i} particle number density in the ith sub-domain

ρ_{m} averaged number density

σ standard deviation of number density

Ω_{ } revolution rate [rad/s]

ω_{rot}_{ } rotation rate [rad/s]