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Analysis of the Behavior of Solid-liquid Systems Based on the Shape, Size Distribuition
and Density of the Solid Particles
F. O. Arouca, C. G. Azevedo, M. H. A. Oliveira, J. J. R. Damasceno.
Federal University of Uberlândia – School of Chemical Engineering
Av. João Naves de Ávila, 2160, block K, Campus Santa Mônica.
Tel. (55 34) 3239 – 4292, fax. (55 34) 3239 – 4249, e-mail: [email protected]
Zip Code 38408-100, Uberlândia – MG – Brazil.
Keywords: Solid-liquid systems, particle shape, batch sedimentation.
Abstract: The dynamic analysis of behavior of solid particles in porous media such as
settling processes are important for the dimensioning increasingly precise of pieces of
equipment that promote the solid-liquid separation. Several factors can influence in dynamics
of the fall of solid particles into a fluid medium; among them, the shape, distribution of sizes,
and particle density. The main objective of this work is to analyze the behavior of solid-solid
system based on the shape, size distribution and density of solid particles. The initial settling
velocity in batch settling tests and the accommodation of particles in the sediment formed are
evaluated for different materials. The gamma-ray attenuation technique was used in the
experimental tests. Comparison of results obtained allowed evaluating in an exploratory
research the significance of variables involved.
1 – Introduction
The great industrial usage of equipment that promotes the solid-solid separation is
responsible for an increasing interest in the development of studies on the variables involved
in the process. Dynamics of particle settling and the sediment compressibility through
accommodation of particles are particular characteristics of the type of solid used.
Separation of solid and liquid phases by the settling phenomenon basically occurs by
difference of density of constituent materials and by the action of gravitational force.
However, when results of tests with different solid materials subjected to the same
experimental conditions are compared, it is noted that the behavior of suspensions is
depending of the material used. A small, homogeneous spherical particle displacing under the
action of gravity into an infinite stagnant liquid, without a wall effect, theoretically undergoes
a minor resistance to fall compared with a particle of uneven shape or low sphericity. Fig. 1
illustrates the stream lines of laminar discharges around several types of solid particles.
Naturally, resistance of fluids to flow around particles of irregular shapes is higher than
around more spherical particles.
Fig. 1 -
Fluid flow around particles of different shapes.
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The accommodation of particles in the sediment formed from the batch settling tests is
a function of the particle shape and its size distribution of the sample. The consolidation of
irregular-shaped particles in the sediment certainly takes place in a more complex manner and
the final accommodation defines the system compressibility that it is a property of each solidliquid system. Fig. 2 shows examples of accommodation of irregular and regular particles in
sediments.
A
B
Fig. 2 - Consolidation of irregular and regular shape particles in the sediment.
The main objective of this work is to analyze the sedimentation kinetic of solid-liquid
systems based on the shape, size distribution, and density of solid particles. The initial
velocity of settling in batch settling tests at the free settling region, a region whose tension
effect on solids is negligible, is used for evaluation of fall dynamics of solid particles with
different shapes. The local concentration of solids is measured in static tests in tube by using
the technique of gamma-ray attenuation technique, thus allowing the study of material
compressibility.
2 - Theoretical Fundamentals
2.1 - Gamma-Ray Attenuation
The variation of intensity of a collimated monoenergetic gamma rays beam, when
these go through any physical medium is determined by the Lambert equation presented by
Gardner and Ely Jr. [1]:
I (E ) = I 0 exp(- σ (E )ρy )
(1)
where I and I0 are, respectively, the relative number of monochromatic gamma-ray pulses
before and after crossing a physical medium with thickeness equal to y and mass attenuation
coefficient σ.
By considering that the physical medium is a mixture and using as reference the
attenuation in test tube that contains pure water, Eq. 1 can be rewritten as:
I 
ln 0  = (σ s ρ s − σ l ρ l )ε s y
(2)
I 
which shows that the logarithm of gamma-ray attenuation of a monoenergetic beam is a linear
function of the volumetric concentration of solid, that is:
I 
ln 0  = Bε s
(3)
I 
So, a curve of calibration for the analytical system can be established, for each solidliquid system, measuring the attenuation of gamma rays produced by known concentrations of
the suspension, so as to determine the constant of calibration B. With the value B, the local
concentration of any suspension of the same media, dispersed and dispersant, can be obtained
through the simple measure of gamma-ray attenuation produced by it.
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2.2 – Equation of Motion for the solid-liquid separation
A solid-liquid mixture is subjected to tests in a test tube. The equation of motion for
the solid phase can be expressed by Eq.4 [2]:
µε l  ql q s 
dPs
 −  + ( ρ s − ρ l )ε s g
=
dz k (ε l )  ε l ε s 
(4)
where P is the pressure in solids, µ is the viscosity of the liquid, k is the permeability of the
porous medium, ρ is the density, ε is the local volumetric concentration and q is the surface
velocity. The indicators s and 1 are relative to the solid and liquid, respectively.
Taking as reference the positive sense of the axis z from the top to the base of the
sediment, at the end of a test in tube and considering the stage of the phenomenon in which
the height of the sediment does not vary anymore with the time, the constitutive equations for
the pressure in solids and permeability of the medium can be determined from Eq.4. Taking
into account that in the stationary state the surface velocities of both constituents, liquid and
solid, are null and distribution of concentrations in the sediment is known, the pressure in
solids can be determined by considering the vertical position from Eq. 5:
Ps = ( ρ s − ρ l )g ∫ ε s ( z )dz
L
0
(5)
where L is the height of the sediment measured from the top. So, when the distribution of
volumetric fractions of the solid is known it is possible to determine the distribution of
pressures of that component.
3 – Materials and Methods
For carrying out the experiments, kaolin aqueous suspensions, calcium carbonate, and
glass microspheres were used with densities equal to (2.577±0.001) g/cm³, (2.69±0.01) g/cm³
and (2.50±-0.01) g/cm³, respectively.
Shapes of particles from solids kaolin, calcium carbonate and glass microspheres were
visualized by using a Neophot 2I light microscope and are presented in Fig. 3. Glass spheres
are produced by Potters Industrial Ltda and their solid particles present minimum sphericity
of 80%.
Fig. 3 – Microscopy of the particles: kaolin(A), calcium carbonate(B) and glass spheres(C).
By observing Fig. 3, it can be found that kaolin solid particles and calcium carbonate
have very irregular shapes unlike glass spheres that present greater sphericity.
Granulometric distributions of materials were obtained through the Laser
Diffractometry with a Malvern Mastersizer Microplus. Fig. 4 shows the curves representing
the granulometric distributions of solids. It is noted from Fig. 4 that calcium carbonate
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particles (d50=3µm) are, in general, smaller than the other solids, kaolin (d50=16.8µm) and
glass microspheres (d50=37.2µm).
100
Y - cumulative mass fraction [%]
100
100
80
80
80
60
60
60
40
40
40
20
20
20
Kaolin
0
0
50
100
150
Diameter [µm]
200
Glass spheres
Calcium carbonate
250
0
0
2
4
6
8
10
12
14
Diameter [µm]
16
0
0
10
20
30
40
50
60
70
80
Diameter [µm]
Fig. 4 – Particle size distribuition of the solids: kaolin, calcium carbonate and glass spheres.
The experimental system used in the determination of solids concentration was
basically composed of a source of gamma rays, collimators, equipment for detection of
radiation, a test tube in which the suspension of the solid under study was placed, and a
device for promoting the vertical displacement of the test tube, which made it possible the
study of the sedimentation process in several positions (Fig.5).
Fig. 5 – Experimental system.
For carrying out the experiments the mass of material was weighed in such a way
that an aqueous suspension with volumetric concentration previously known were
produced. For the determination of the initial velocity of particle settling, aqueous
suspensions with solids concentrations of 4% by volume were used. The methodology
employed is based on the monitoring of the upper interface in batch settling tests as a time
function for the free settling region whose tension effect in solids is negligible. In such a
region, solid particles decant freely with a constant velocity and equal to the initial velocity of
settling [3]. The sloping of the linear part of the graph z versus t is equivalent to the initial
velocity of solids settling.
In tests of batch settling for determination of concentration distributions in sediments
and tensions in solids, aqueous concentrations with solids concentrations of 8% were used by
volume. Half liter of distilled water was transferred to the test container from a
volumetric balloon. The weighed solid was then carefully poured onto the liquid. The
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suspension formed was vigorously homogenized so as to minimize the initial gradients
of concentrations. Following the homogenization procedure, the sealing cap of the
system was carefully placed. The system remained at rest for a period of 48 hours so
that the process of decantation were completed. From this point on, measures of gammaray attenuation were taken in various vertical positions inside the sediment, beginning at
the base of the test tube up to the top of the sediment. Measures of concentration
distributions in sediments enabled the comparative analysis of particle accommodation among
solids and tensions in solids were determined through Eq. 5. Three replicates were carried out
for each solid in order to verify the reproducibility in each test.
4 – Results and Discussion
z [cm]
Batch settling tests were conducted from aqueous suspensions of the same initial
volumetric concentration (εs0= 4%) for each solid under study. Initial volumes of porous
media being identical for all solids, the fall dynamics of particles was analyzed at the region
of free settling in which solid particles go along their way freely without the effect of tension
in solids. Monitoring of displacement of the upper interface as a time function in each test
allowed the determination of initial fall velocities of particles by the inclination of straight
lines presented in Fig. 6.
26
24
22
20
18
16
14
12
10
8
6
4
2
kaolin (Replicate 1)
kaolin (Replicate 2)
kaolin (Replicate 3)
Calcium Carbonate (Replicate 1)
Calcium Carbonate (Replicate 2)
Calcium Carbonate (Replicate 3)
Glass spheres (Replicate 1)
Glass spheres (Replicate 2)
Glass spheres (Replicate 3)
0
400
800
1200
1600
2000
2400
t [s]
Fig. 6 - Monitoring of upper interfaces in tests of batch settling.
The initial velocities of settling measured for each solid were vs=(-18.9 ± 0.7)⋅10-3cm/s
for kaolin aqueous suspensions, vs=(-3.4 ± 0.3)⋅10-3cm/ for calcium carbonate and vs=(-102 ±
9)⋅10-3cm/s for glass microspheres. It is observed that the settling of glass microspheres takes
place much more rapidly than the other solids.
Measures of concentrations distributions in sediments formed from aqueous
suspensions of initial concentrations of 8% by volume allowed the analysis of particle
accommodations presented in diagrams of Fig. 7.
Volumetric concentration [-]
0.30
A
0.25
0.60
B
0.20
0.16
0.55
0.12
0.50
0.08
0.45
C
0.20
0.15
0.10
Replicate 1
Replicate 2
Replicate 3
0.00
0
2
4
Replicate 1
Replicate 2
Replicate 3
0.04
6
8
z [cm]
10
12
14
0
2
4
0.40
6
8
z [cm]
10
12
14
16
0.35
0.0
Replicate 1
Replicate 2
Replicate 3
0.5
1.0
1.5
2.0
2.5
3.0
3.5
z [cm]
Fig. 7 - Concentrations distributions: kaolin (A), calcium carbonate (B) e glass spheres (C).
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Pressures in solids determined through Eq. 5 are presented in Fig. 8.
400
300
300
300
200
100
Replicate 1
Replicate 2
Replicate 3
350
]
Replicate 1
Replicate 2
Replicate 3
[ g
Pressure on solids [Kg/ms2]
500
Replicate 1
Replicate 2
Replicate 3
250
250
200
200
150
150
100
100
A
50
0
0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28
Volumetric concentration [-]
0
50
B
0.10
0.11
0.12
0.13
0.14
0.15
Volumetric concentration [-]
0.16
0
0.42
C
0.45
0.48
0.51
0.54
0.57
0.60
Volumetric concentration [-]
Fig. 8 – Pressure in solids: kaolin (A), calcium carbonate (B) e glass spheres (C).
Through a general analysis based on the results obtained in this work, it was observed
that significant differences exist in dynamic behaviors and in the accommodation of solid
materials in an aqueous medium and such differences are not restricted only to density
differences, but possibly as well to shapes and size distributions.
Glass microspheres, despite possessing lower density with regard to the other solids,
presented settling velocity about 30 times higher than particles of calcium carbonate that, for
their turn, presented a greater density and smaller diameters among the materials. Size
distributions certainly are significant since larger particles possess more mass and hence settle
more rapidly.
Fig. 7 shows that sediments formed by more spherical particles, in this case, glass
microspheres, settle and produce more compacted sediments, solids concentrations of 56% by
volume at the base, while kaolin presented 26% and calcium carbonate 16%. However, Fig. 8
shows that pressures in kaolin solids were superior to the other solids, certainly due to the
sediment to be taller.
5 - Conclusions
Effects related to processes of sample dispersion and the shape of materials cause the
analysis of the process dynamic to be more complex than it may seem at first sight. Based on
the exploratory studies carried out in this work, it follows that more spherical particles settle
more rapidly than irregular-shaped particles and that the dynamic behavior must not restrict
only densities of solids. The particle consolidations in sediments are certainly functions of the
shape and size distributions of solids. Particles possessing irregular shapes produce less
compacted sediments.
6 – References
[1] R.P. Gardner, R.L. Ely Jr., Reinhold Publishing Corporation (1967).
[2] J.J.R. Damasceno, Doctoral Thesis (1992), p.65-68.
[3] G.J. Kynch, Trans. Faraday Soc. (1952), p.166-176.
7 – Acknowledgment
The authors are grateful to CAPES for the financial support given to this research.
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