Numerical simulation of particulate flow with liquid bridge between particles (simulation of centrifugal tumbling granulator)

doi:10.1016/S0032-5910(99)00226-0

Powder Technology

Volume 109, Issues 1–3, 3 April 2000, Pages 49-57

https://doi.org/10.1016/S0032-5910(99)00226-0 Get rights and content

Abstract

It is well known that a small amount of liquid has significant effects on the motion of particles. A numerical simulation of mono-sized spherical particle flows with a small amount of water in a centrifugal tumbling granulator was made by using discrete element method (DEM). A model for the inter-particle force due to a liquid bridge was applied to the present work because granulating processes are usually performed by adding binding liquid. In the present simulation, this behavior was predicted when bridges are formed within the layer. Once the locations of bridge were determined, the water was assumed to distribute uniformly among all locations. In addition to the simulation, measurements were made to validate the simulation results. Particle velocities on the surface of a moving particle bed was measured with a particle tracking velocimetry (PTV) technique. The calculated variation of particle velocities against the water content agreed well with the experiments.

Introduction

The discrete element method (DEM) simulation has become a popular method of predicting particulate flows. Most previous works have dealt with comparatively simple cases where various complicated factors inherent to fine particles are neglected. The adhesion force due to liquid bridges between particles is one of those factors which has long been attracting attention in the field of powder technology. Theories of the liquid bridge between two particles have been developed by several researchers 1, 2. The arrangements of every particle are known in the DEM calculation. Thus, it is not difficult to combine the theoretical models of a liquid bridge with DEM calculations. The present authors [11]attempted to develop a method considering the effects of the liquid bridge in the DEM simulation. Results of a rigorous theory of the adhesion force were used in the formulation of contact forces between particles, where the adhesion force acting on each particle pair was taken into account.

Thornton et al. [3]and Lian et al. [4]also carried out the DEM simulation of agglomerate collisions taking account of the liquid bridge. In their study, a specific volume of liquid was assigned to all particle–particle gaps where the liquid bridges are able to form. In an actual granulator, however, the given condition is the total amount of water added to the particle layer. In the present work, we assume that the water is distributed uniformly in all particle–particle gaps where the liquid bridges are able to form. To accomplish this, we also developed the algorithm to treat the liquid bridge based on the volume of water added in the system.

The method was applied to the flow in a centrifugal tumbling granulator. The granulating process is made by adding a binding liquid. It is well known that even a small amount of liquid has a significant effect on the particulate flow. The present numerical simulations revealed that the particle motion was retarded by the liquid as was expected. Since it is difficult to measure the particle velocities inside the layer, we tried to measure the particle velocities on the surface of a moving particle bed for quantitative comparison between simulations and experiments. The experimental method was based on a PTV (particle tracking velocimetry) technique. It was confirmed that calculated particle velocities compared well with measurements.

Section snippets

Equipment

Fig. 1 shows a schematic diagram of a granulator to which the numerical simulation was applied. The granulator consists of a rotor, 360 mm in diameter and a stator, 376 mm in diameter. The standard content of particles in the present granulator is 3 kg. The granulator of this size is used for small-scale experiments or production in pharmaceutical companies. Particles form an annular layer and show a spiral motion as shown in Fig. 1. In practice, the granulation is progressed by adding the

Model of particle–water system

The particle–water systems show several patterns depending on the amount of water. In the case of a small volume fraction of water, the liquid bridges formed in the gap of particles are isolated from each other as shown in Fig. 2. As the amount of water increases, the bridges become combined. A further increase of water leads to the formation of a slurry. Granulating processes that produce agglomerate particles from fine particles are generally operated under the condition of a small amount of

Adhesion force of liquid bridge

When the liquid bridge is formed between two particles, the shape of the liquid bridge and the capillary force are calculated from the Laplace's equation [8]. The Laplace's equation, however, cannot be solved analytically. Before above exact study, Fisher [1]had supposed that the shapes of liquid bridge approximate to an arc of a circle as shown in Fig. 4. This approximation gives the adhesion force f_L as $f_{L} =πr^{2}_{2} γ 1 r_{1} − 1 r_{2} +2πr_{2} γ$ where γ is a surface tension of liquid, and r₁,r₂ are the radii of

Results of calculation

The conditions of the simulation are shown in Table 1. The present simulation dealt with mono-sized sphere particles. Actual physical properties of grass beads were used except the Young's modulus. For the calculation to be fast using longer time steps, a smaller Young's modulus than the actual one was used [6]. Periodic boundaries were applied for the region of θ_b∼−θ_b as shown in Fig. 1 in order to reduce the number of particles treated. The number of particles 8880 is equivalent to 3 kg of

Experiment

The particle velocity on the surface of the particle layer was measured [10]by using a PTV technique. The apparatus used for the measurement is shown schematically in Fig. 8. The particle flow was recorded by using a high-speed VTR system with speed of 250 images per second. Many images were extracted from VTR and those images were transmitted to PC consecutively. Particle positions were measured from each image. Brown grass beads (1.92 mm, 2.52×10³ kg/m³) were used as test particles. As tracer

Comparison of experimental and calculated results

Fig. 11(a),(b) shows the comparison between calculation and measurement about the particle velocity profile. The calculated results are shown by lines and experimental ones by points. These lines were obtained by averaging data over 2.0–3.0 s when the particulate flow became stationary.

The circumferential components of particle velocity are shown in Fig. 11(a), where it is found that experimental and calculated results show a good agreement. The velocities decrease along r/R in common.

Conclusion

Numerical simulations of a three-dimensional particulate flow with a small amount of water were performed by using the DEM taking account of the adhesion force due to water. The adhesion force acting on two particles is formulated by using a theory of the liquid bridge.

The present method was applied to the flow in a centrifugal tumbling granulator. The simulation indicated that the adhesion force due to liquid bridges largely affects particulate flow, particularly the circulating flow in the

Nomenclature

f_ij	Forces acting between particles i and j
f_Cij	Contact force between particles i and j
f_Lij	Adhesion force of liquid bridge between particles i and j
F_L	Dimensionless adhesion force
g	Gravitational acceleration vector
h,H	Length of the gap between particles
I_i	Moment of inertia of the particle i
m_i	Mass of the particle i
N_p	Number of particles
N_s	Number of gaps smaller than H_max
r_i	Position vector of the particle i
r₀	Particle radius
r₁,r₂, R₁, R₂	Radii of curvatures of the liquid bridge surface
T_ij	Torque caused by

References (11)

G Lian et al.
J. Colloid Interface Sci.
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R.A Fisher
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C. Thornton, G. Lian, M.J. Adams, Proc. 2nd Int. Conf. Discrete Element Methods, 1993, pp....
G. Lian, C. Thornton, M.J. Adams, Proc. 2nd Int. Conf. Micromechanics of Granular Media, 1993, pp....
H Tanaka
Yakuzaigaku
(1972)

There are more references available in the full text version of this article.

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