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On the Dispersion of Solid Particles in a Liquid Agitated by a Bubble Swarm

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Abstract

This article deals with the dispersion of solid particles in a liquid agitated by a homogeneous swarm of bubbles. The scale of interest lies between the plant scale (of the order of the tank) and the microscale (less than the bubble diameter). The strategy consists in simulating both the two-phase flow of deforming bubbles and the motion of solid particles. The evolution of the spatial distribution of particles together with the encounter and entrainment phenomena is studied as a function of the void fraction and the relative size and mass of particles. The influence of the shape of the bubble and of the model of forces that govern the motion of particles is also considered.

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Abbreviations

ρ :

local density (kg/m3)

μ :

local dynamic viscosity (kg/m/s)

C M :

added mass coefficient (here, C M = 1/2)

D :

bubble diameter (m)

d p :

particle diameter (m)

E coll :

encounter efficiency

E entr :

entrainment efficiency

g :

gravitational acceleration (m/s2)

H :

height of the computational domain (m)

m f :

mass of the liquid of same volume as the particle (kg)

m p :

mass of the particle (kg)

n :

unit vector normal to the interface

P :

pressure (Pa)

U :

local velocity (at the particle location for Eq. [3], m/s)

U l :

local velocity in the liquid (m/s)

V :

particle velocity (m/s)

V T :

bubble terminal velocity (m/s)

μ g :

dynamic viscosity of the gas (kg/m/s)

μ l :

dynamic viscosity of the liquid (kg/m/s)

ρ g :

density of the gas (kg/m3)

ρ l :

density of the liquid (kg/m3)

ρ p :

density of the particles (kg/m3)

σ :

surface tension between the gas and the liquid (N/m)

δ I :

surface delta function

α = 2D 3/3H 3 :

volume fraction

Bo = ρ l gD 2/σ :

Bond number

Mo = gμ 4 l /ρ l σ 3 :

Morton number

U p  = (ρ l – ρ p )gd 2 p /18μ l :

Stokes sedimentation velocity of a solid particle

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Author information

Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL, 32611, USA

    Thomas Bonometti (Postdoctoral Associate)

  2. Institut de Mécanique des Fluides de Toulouse, UMR CNRS/INPT/UPS, 5502, Allée Camille Soula, 31400, Toulouse, France

    Jacques Magnaudet (Research Director)

  3. Process Engineering Department, Arcelor Research, Voie Romaine, BP30320, 57283, Maizières-lès-Metz Cedex, France

    Pascal Gardin (Technical Manager)

Authors
  1. Thomas Bonometti
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  2. Jacques Magnaudet
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  3. Pascal Gardin
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Corresponding author

Correspondence to Thomas Bonometti.

Additional information

Manuscript submitted January 11, 2007.

Appendix

Appendix

Influence of added-mass and shear-induced lift forces

An important issue is to find out how the force models on the right side of Eq. [3] influence the results involving the particle motion and the efficiency statistics. Some aspects of the role played by the gravity, drag, and added-mass forces were already presented in Section III–A. However, we can still question the influence of the added-mass effects on the migration phenomenon. Figure 18 compares the particles distribution when added-mass effects are taken into account. When the added-mass force is switched off, some particles remain in the bubble wake (Figure 18(a)). In contrast, when added-mass effects are taken into account, no particle remains in the bubble wake (Figure 18(b)). Therefore, it turns out that added-mass effects are part of the origin of the migration phenomenon. We could also wonder whether including the shear-induced lift force would have changed the results. To check this point, we performed a computation in which the lift force was included (Figure 18(c)). Following McLaughlin’s theoretical result[34] valid for particles moving at a low but finite Reynolds number in a pure shear flow, an extra term F L should be added to the right side of Eq. [3], namely

$$ F_{L} = \,{\text{ }}m_{f} C_{L} {\left( {U - V} \right)}{\text{ }} \times {\text{ rot}}{\left( {U - V} \right)} $$
(6a)

with

$$ C_{L} = \frac{9} {{\pi ^{2} }}\frac{{2.255}} {{{\sqrt {\text{Re} _{p} \,{\text{Sr}}} }}}{\left( {1 + 0.2\frac{{\text{Re} _{p} }} {{{\text{Sr}}}}} \right)}^{{ - 3/2}} {\text{with Sr}} = \frac{{2R\,\,{\left| {\,{\text{rot}}{\left( {U - V} \right)}\,} \right|}}} {{{\left| {U - V} \right|}}} $$
(6b)
Fig. 18
figure 18

Influence of added-mass and shear-induced lift forces on the particle distribution, (a) without both added-mass and shear-induced lift and (b) with added-mass included but without shear-induced lift; and (c) with both added-mass and shear-induced lift (tV T /D = 28.5; α = 17 pct; ρ p = 0.36ρ l ; and Re p = 5)

Full size image

Strictly speaking, this model for the shear-induced lift force is only valid under the assumptions Re p << 1 and (Sr/Re)1/2 << 1. However, extensive direct numerical simulations[35] revealed that it provides a reasonable estimate of the shear-induced lift force up to Re p  ≈ 5. Present computations indicate that the particle distribution is very similar whether the shear-induced lift force model is included or not (Figure 18). Figure 19 compares the evolution of the averaged forces acting on the particles, in both cases. Clearly, the presence of the shear-induced lift force only affects the added-mass force at long times and leaves the other contributions unchanged. Note that the intensity of the shear-induced lift force is one order of magnitude smaller than that of the other forces. Therefore, we can conclude that shear-induced lift effects do not play a significant role in the situation considered in this article.

Fig. 19
figure 19

Evolution of the averaged forces acting on particles (α = 17 pct; ρ p = 0.36ρ l ; and Re p = 5): (a) without shear-induced lift and (b) including the shear-induced lift model of Eqs. [5a] and [5b]. (·····) Added-mass force; (—) drag; (- - - - - -) Archimedes force; and (-·-·-·-·-·-) shear-induced lift. All forces are normalized by the weight of the particle

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Bonometti, T., Magnaudet, J. & Gardin, P. On the Dispersion of Solid Particles in a Liquid Agitated by a Bubble Swarm. Metall Mater Trans B 38, 739–750 (2007). https://doi.org/10.1007/s11663-007-9085-7

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