Numerical simulation of the settling behaviour of particles in thixotropic fluids
M. M. Gumulya, R. R. Horsley, and V. Pareek
Citation: Physics of Fluids 26, 023102 (2014); doi: 10.1063/1.4866320
View online: http://dx.doi.org/10.1063/1.4866320
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PHYSICS OF FLUIDS 26, 023102 (2014)
Numerical simulation of the settling behaviour of particles
in thixotropic fluids
M. M. Gumulya, R. R. Horsley,
a)
and V. Pareek
Department of Chemical Engineering, Curtin University, U1987, Perth, WA 6845, Australia
(Received 16 April 2013; accepted 7 February 2014; published online 25 February 2014)
A numerical study on the settling behaviour of particles in shear–thinning thixotropic
fluids has been conducted. The numerical scheme was based on the volume of fluid
model, with the solid particle being likened to a fluid with very high viscosity.
The validity of this model was confirmed through comparisons of the flow field
surrounding a sphere settling in a Newtonian fluid with the analytical results of
Stokes. The rheology model for the fluid was time–dependent, utilising a scalar
parameter that represents the integrity of a “structural network,” which determines its
shear thinning and thixotropic characteristics. The results of this study show that the
flow field surrounding the settling sphere is highly localised, with distinct regions of
disturbed/undisturbed fluids. The extension of these regions depends on the relaxation
time of the fluid, as well as its shear thinning characteristics, and reflects the drag
force experienced by the sphere. As the sphere settles, a region of sheared fluid that
has significantly lower values of viscosity is formed above the sphere. This region
slowly recovers in structure in time. As a result, a sphere that falls in a partially
recovered domain (e.g., due to the shearing motion of an earlier sphere) tends to
attain a greater velocity than the terminal velocity value. This was found to be true
even in cases where the “resting time” of the fluid was nearly twice the relaxation time
of the fluid. The results of this study could provide a framework for future analysis
on the time–dependent settling behaviour of particles in thixotropic shear–thinning
fluids.
C
2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4866320]
I. INTRODUCTION
Suspensions of fine mineral particles that are encountered in the mineral processing indus-
try generally possess non–Newtonian flow characteristics, usually in the form of viscoplasticity,
thixotropy, and/or shear thinning flow behaviour, or various combinations of these characteristics.
As can be expected, the settling behaviour of particles suspended in slurries and suspensions is
greatly dependent on the rheology of the suspending fluids (Ref. 1). In turn, the design and per-
formance of many slurry based units of operations, most notably the tertiary grinding circuits, are
highly influenced by the tendency of suspended particles to settle and fall through the suspension.
Therefore, to improve the efficiency of these units of operations, a greater understanding of the
movement of particles through suspensions and slurries needs to be obtained.
The settling behaviour of particles in viscous fluids has been examined extensively by Gumulya
et al.,
2, 3
using solutions of polyacrylamide in water. These solutions were identified as highly
shear thinning with thixotropic characteristics. Under short timescales, the existence of a critical
value of shear stress that has to be exceeded for the fluid to start to flow (generally termed as
yield stress) is apparent, indicating that the fluids can be represented by a simple viscoplastic fluid
model. However, in cases where longer time frames are involved (e.g., in experiments involving the
settling of consecutive particles with a finite time lapse between them (Refs. 2 and 3)), it was found
that both the thixotropic and shear thinning characteristics of the materials need to be considered.
a)
Deceased.
1070-6631/2014/26(2)/023102/16/$30.00
C
2014 AIP Publishing LLC26, 023102-1
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023102-2 Gumulya, Horsley, and Pareek Phys. Fluids 26, 023102 (2014)
Gumulya et al.
2, 3
observed that a sphere that is released into a fluid medium that has recently been
subjected to shear tends to possess a settling velocity that is significantly higher than a sphere that
is settling through a fluid medium that has been rested for some time. This finding suggests that the
settling motion of a sphere through the fluid introduces changes in the structure of the fluid (it was
hypothesised that this network results from hydrogen bonding between polyacrylamide and water
molecules in the fluid), which in turn requires some “resting” time for its full recovery. It was thus
concluded that the analysis of the settling behaviour of particles through complex fluids requires the
shear induced changes in the internal structures of the fluids to be characterised.
While various studies have been conducted on the characterisation of the flow field surrounding
particles settling in yield stress fluids (Refs. 4–8), such studies on shear thinning, thixotropic
fluids are still very scarce, despite the fact that some yield stress fluids are known to possess a
combination of all these characteristics. Gueslin et al.
9, 10
and Putz et al.
11
have conducted flow–
visualisation experiments on the flow surrounding spheres settling in thixotropic yield stress fluids
with viscoelastic properties. Yu et al.
12
have presented a numerical study on the sedimentation and
aggregation of particles in shear–thinning thixotropic fluids, and concluded that the aggregation
of two particles settling one above the other is caused by the time–dependent and shear thinning
characteristics of the fluid. In addition, it was also observed that the aggregation of randomly
distributed particles in such fluids into stable clusters or columns, which is a phenomenon that has
been observed in some non–Newtonian fluids (Refs. 13 and 14), requires the suspending fluid to
exhibit a degree of elasticity. The dynamics of shear thinning, thixotropic fluids in other systems
have been analysed by Derksen,
15
Potanin,
16
and Møller et al.
17
In order to characterise the dependency of the viscous parameters of a fluid on its internal
structural network, Gumulya et al.
3
utilised a fluid model that is similar to the model of thixotropy
proposed by Møller et al.
17
The fluid model assumes that the fluid is completely non–elastic.
Furthermore, it does not include a constant yield stress value. However, through a series of numerical
studies, it was found that the transient response predicted by the model towards a variety of changes
in shear conditions (under short time scales) tends to resemble the observed rheological response of
typical viscoplastic fluids (Ref. 18).
In this paper, a numerical study of the flow field of a thixotropic, shear thinning fluid surrounding
a settling spherical particle will be presented. The utilised numerical method is the Volume–of–Fluid
(VOF) method, where the solid particle is likened to a fluid with very high viscosity. While this
method is more commonly applied to immiscible fluids with deformable interfaces, the use of VOF
to model the movement of a non-deformable material in a fluid domain has previously been examined
by Chen et al.
19
(in yield–stress fluids), as well as Gumulya
18
(in Newtonian fluids of Re ≤ 4.2).
The apparent agreement shown by Gumulya
18
in the case of a solid particle settling in Newtonian
fluids with published experimental data suggests that this method can be used to analyse the flow
fields surrounding a particle settling in a fluid domain, without prior knowledge of the drag force
experienced by the particle. Further discussion on the validation of this method will be presented
in Sec. II B, and the use of this method to analyse the highly coupled and non–linear equations
surrounding the settling of a spherical particle in time–dependent non–Newtonian fluids will be
examined in this paper. In Sec. II C, the details of the rheology model and its implementation in the
VOF framework will be presented. Furthermore, the parameters required for the modelling of the
solid particle as a highly viscous fluid will also be discussed (Sec. II D). Section III presents the
resulting flow fields and drag force, along with comparisons to relevant experimental data.
II. NUMERICAL MODEL
A. Mathematical formulation and numerical method
Consider an initially stationary metal sphere in a cylinder filled with fluid. The metal sphere
exerts a downward shear force on the fluid due to gravitational effects, inducing changes in viscosity
and flow in the fluid medium. If the metal sphere is likened to a fluid with very high viscosity (see
Sec. II B), this flow problem can be considered to be a dual–phase problem, with the two phases
possessing vastly different values of density and viscosity.
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023102-3 Gumulya, Horsley, and Pareek Phys. Fluids 26, 023102 (2014)
Navier–Stokes equations are applied over the whole domain,
∇·U = 0, (1)
∂
∂t
(
ρU
)
+∇·
(
ρUU
)
=−∇p +∇·τ + ρg, (2)
where U is the velocity vector, ρ is the density, p is the pressure, and g is the gravitational acceleration.
The stress tensor τ is given by
τ = μ(˙γ )
˙
γ , (3)
where ˙γ is the second invariant of the rate-of-deformation tensor (
˙
γ ). At each control volume, the
density and viscosity (μ)are
ρ =
α
q
ρ
q
, (4)
μ =
α
q
μ
q
, (5)
where α
q
is the volumetric fraction of the qth phase. In the current study, a value of unity for α
indicates that the cell is full of the fluid phase, whereas α = 0 indicates that the cell is occupied by
the “metal” phase.
In addition, the tendency of the particle to distort from its spherical shape was minimised by
the incorporation of a surface tension parameter between the two phases, implemented through the
continuum surface force (CSF) model of Brackbill et al. ,
20
F =
ρσ∇α
0.5
(
ρ
1
+ ρ
2
)
∇.
∇α
|
∇α
|
, (6)
where σ is the surface tension coefficient between the “metal” and fluid phases. The magnitude
of this coefficient in relation to the viscous and inertial forces in the two phases will be discussed
further in Sec. II C.
The flow problem described above was spatially discretised on a two–dimensional axisymmetric
numerical grid, which was divided uniformly such that there were 40 control volumes across the
diameter of the sphere. An implicit differencing scheme was employed for the temporal discretisation
of all the governing equations, whereas the convection terms were discretised using the standard
central differencing scheme. The coupled method was adopted for the velocity-pressure coupling
in conjunction with the PRESTO (Pressure Staggering Option) scheme for the interpolation for the
pressure field in the momentum equation. The resulting equation was implemented into the ANSYS
FLUENT 13.0 through a series of user-defined functions (UDFs). Throughout the calculation process,
the motion of the sphere was tracked through the Piecewise-Linear Interface Calculation (PLIC)
interface–tracking scheme suggested by Youngs.
21
The time step used throughout the calculation
process was set such that the global Courant number is less than 0.1.
B. Model validation
The validity of the proposed numerical method was examined through comparisons of the
predicted drag force experienced by spheres (ρ
s
= 7638 kg/m
3
;d= 6.25 mm) settling in Newtonian
fluids with the representative Stokes drag value. The viscosity of the Newtonian fluid ranges from
0.4 to 5 Pa s, and the fictional surface tension parameter between the two phases was set to 3.5 N/m.
The resulting Capillary number
Ca = μ
f
V
t
/σ
for these simulations ranges from 0.027 to 0.033.
The development of spurious/parasitic currents in the interface of the two phases, which have been
known to occur in CSF-based models with considerable imbalance in viscous and surface tension
forces (Ref. 22), is therefore expected to be minimal.
The effect of the fictional viscosity parameter for the “solid” material (μ
s
) on the development
of the flow field around the settling sphere was first examined, based on a fluid viscosity value,
μ
f
, of 5 Pa s (Figure 1). The settling velocity of the sphere in this case can be seen to be highly
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023102-4 Gumulya, Horsley, and Pareek Phys. Fluids 26, 023102 (2014)
FIG. 1. The development of settling velocity of the “solid” sphere, using several different ratios of μ
s
/μ
f
. The viscosity of
the “fluid” phase is constant (i.e., Newtonian) at 5 Pa s.
dependent on the μ
s
value. At higher values of μ
s
/μ
f
, the development of the velocity field around
the sphere tends to be sluggish, as reflected by the lower settling velocity of the sphere at μ
s
/μ
f
= 250. With lower μ
s
/μ
f
values, the development of the velocity field around the sphere was found
to be considerably more rapid. However, a velocity overshoot was found in the case where μ
s
/μ
f
= 2.5, indicating that the low value of μ
s
tends to cause considerable diffusion within the “solid”
phase, leading to the formation of a secondary flow within the solid sphere.
The formation of the secondary flow within the “solid” sphere can be inspected further in
Figure 2, where the development of the velocity field within the “solid” sphere has been presented.
In this figure, it can be seen that while the case with μ
s
/μ
f
= 250 presents minimal distribution of
velocity within the “solid” phase, the opposite is true with the case where μ
s
/μ
f
= 2.5. In the latter
case, the diffusion of momentum within the “solid” phase can be seen to cause the velocity distribution
within the sphere to be in excess of 20%, with the highest velocity being at the centre of the sphere.
The selection of the fictional solid viscosity parameter therefore requires careful consideration of
the velocity and flow field development within the two phases. Within the parameters of this study,
a μ
s
/μ
f
ratio of 25 has been found to present a satisfactory balance of rapid flow field development
and minimal secondary flow development within the solid phase.
FIG. 2. Magnitude of the velocity field within the “solid” sphere, in a Newtonian fluid (μ
f
= 5.0 Pa s). The velocity parameter
has been normalised against the average settling velocity of the sphere, and the x axis represents the distance from the centre
of the sphere in the radial direction, normalised against the radius of the sphere.
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