(388h) Three-Dimensional Simulation of a Solid-Liquid Flow by a Lagrangian-Lagrangian Approach

Three-dimensional simulation of a
solid-liquid flow by a Lagrangian-Lagrangian approach

Xiaosong
Sun, Mikio Sakai, Yoshinori Yamada

School
of Engineering, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo
113-8656, Japan

  1      
Introduction

Solid-liquid flows are widely encountered
in various industrial processes like dispersion and classification. Accurate
numerical simulations would be helpful both for the design and investigation of
operational conditions in these processes. For these simulations, modeling of
the free surface and the solid-liquid coupling is important. Because the
Lagrangian approaches do not need complicated techniques like adaptive mesh
required by Eulerian approaches, they may handle these problems effectively. As
far as the Lagrangian approaches for fluids are concerned, smoothed particle
hydrodynamics (SPH) (Monaghan, 1988) and moving particle semi-implicit (MPS)
method (Koshizuka & Oka, 1996) have been thoroughly studied. These
Lagrangian methods have advantages in that they are not affected by the
numerical diffusion and can simulate a large deformation of the free surface
flows relatively easily.

Two types of approaches have been used to
simulate solid-liquid flows. One is direct numerical simulation (DNS) and the
other is the local volume average technique (Anderson & Jackson, 1967). For
the DNS-based method, the disadvantage is that too fine a discretization is
required for the precise computation. On the other hand, although experimental
equations are needed, the local volume average technique can reduce the
computational load significantly for practical systems.

In the present study, we developed a new
approach with the local volume average technique namely the DEM-SPH method, by
coupling the discrete element method (DEM) (Cundall & Strack, 1979) with
the SPH method. The DEM is also a Lagrangian method which has been widely
applied in powder systems. With the DEM-SPH method we perform three-dimensional
simulations of solid-liquid flows inside a cylindrical tank and conduct
validation experiments. The angles of repose and solid particle distributions
obtained from simulations and experiments were compared, where a good agreement
was shown. Consequently, the DEM-SPH method can accurately simulate a 3D
solid-liquid flow.

  2      
Numerical methodology

  2.1     
The liquid-solid interaction model

On simulating a multi-phase flow, the
interaction model must be constituted properly to describe momentum exchanges
between phases. The void fraction (the volume fraction of fluid phase in the
context of solid-liquid flows) determines the liquid-solid interaction force
(the drag force) as

where ε denotes the void
fraction. For the value of β, a combination of (Ergun, 1952) and (Wen
& Yu, 1966) is employed and a void fraction of 0.8 has been adopted as the
boundary between these two regimes.

  2.2     
Solid phase

In this study, the forces of drag, pressure
gradient, gravitation and contact were taken into consideration. The contact
force acting on a solid particle is estimated by the DEM, where the
inter-particle forces are determined using a Voigt model as shown in Figure 1.

Figure 1
Discrete element method (DEM)

  2.3     
Liquid phase

In the SPH method, the continuous material
is divided into assemblies of particles. These SPH particles carry physical
quantities and move according to the Lagrangian material velocity. The SPH
particles are totally mess-free, which allowed more flexibility in modeling
fluid motions. An SPH particle interacts with its vicinity according to a
kernel function W(r, h), where r
is the relative position vector and h defines the influence domain of
the kernel estimation. In this study we deployed the commonly used cubic spline
kernel proposed by (Monaghan & Lattanzio, 1985).

Incorporating the void fraction and drag
force, the momentum conservation equation in SPH formulation can be written as

where σ is the
stress tensor and Π
is an artificial viscosity term. To solve the above
equation, the "equation of state" for liquid is used to estimate the pressure
under a weak compressibility assumption (Monaghan, 1994),

where c is the speed of sound in the
fluid, ρ₀ is the reference density, and γ is set
to 7.0.

  3      
Numerical example

The DEM-SPH method was applied to the
solid-liquid flow in a cylindrical tank. The results were herein compared with
validation experiments.

  3.1     
Calculation conditions

Figure 2 shows a schematic diagram of the
domain of interest. It was a cylinder with inner diameter and depth of 100 mm.
The water level was set to be half the height of the tank. The tank was rotated
at 102 rpm. The properties of the solid and liquid phases are given in Table 1.

Two different calculations were performed
in the current study. The number of solid particles simulated was consistent
with the bed height in the validation experiments, and was set to be 7755 in
Case 1 and 5304 in Case 2.

Figure 2 Schematic diagram of the cylindrical tank

Table 1 Physical
properties and computational parameters

  3.2     
Validation test

The experimental device is of the same
dimension as the simulated system. Glass beads with an average diameter of 3.0 mm
were used as solid particles. The fluid was water. Photographs of the rotating
cylinder were taken to obtain the results. Figure 3 shows the initial
configuration of Case 1.

Figure 3 Initial configurations of the experiment
and simulation (Case 1)

  4      
Results and discussion

In this study, we concentrate on the
behavior at a quasi-steady state. A set of snapshots from Case 1 is shown in
Figure 4 to help understand transient changes in the rotating tank.

Figure 4 Transient behaviors of the simulation (Case
1)

The results for Case 1 are described below.
Representative snapshots were shown in Figure 5. The simulation results were in
agreement with the experimental results: the bed slope was bilinear (as shown
by red lines in Figure 5); the angle of repose at the upper side, the solid bed
height and width were of similar values, as shown in Table 2. The close
correspondence shows the accuracy of the DEM-SPH method.

Figure 5 Comparison of the experimental and
simulation results (Case 1)

Table 2
Comparison between the simulation and experimental results (Case 1)

The results of Case 2 at quasi-steady state
are shown in Figure 6. Again, a bilinear shape of the bed slope as found for both
the experiment and simulation was observed. Numerical comparisons of the angles
of repose, the solid bed height and width are measured and presented in Table
3. Note that there is a larger discrepancy between simulated and experimental
results in this case than in Case 1. However, these results are still close and
reasonable.

Figure 6 Comparison of the experimental and
simulation results (Case 2)

Table 3
Comparison between the simulation and experimental results (Case 2)

These comparisons showed good agreements in
macroscopic respects. This implies that the DEM-SPH method has successfully
modeled free surfaces of fluids, solid-fluid coupling and solid-solid
interaction.

However, two factors may have contributed
to the observed differences between the simulations and experiments. Firstly,
in modeling the drag force, it was assumed that the solid particles would be
distributed evenly over the solid-liquid boundary. Deviations from this
assumption would perturb estimating the drag force. Secondly, some interaction
effects such as lubrication were not taken into consideration in the present
model.

  5      
Conclusions

The DEM-SPH method was developed to perform
a three-dimensional simulation of a solid-liquid flow involving free surfaces.
It was validated with the two phase flow in a rotating cylindrical tank. Hence,
we have shown that the DEM-SPH method is effective in simulating solid-liquid
flows involving free surfaces. Experiments using a Particle Image Velocimetry
are planned in the near future.

  Acknowledgement

This study was financially supported by a
Grant (22760579) from the Ministry of Education, Culture, Sports, Science and
Technology (MEXT), Japan.

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