BACHELOR
Movement of plastic particles inside a magnetic density separation setup
Rosenkamp, Roy
Award date:
2019
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Movement of plastic particles inside a Magnetic Density
Separation setup
R-1974-B
R.H. Rosenkamp
Eindhoven University of Technology
Department of Applied Physics
Turbulence and Vortex Dynamics Research Group (WDY)
Supervisors:
dr. ir. J.C.H. Zeegers ir. R.A. Dellaert
version 2
Eindhoven, August, 2019
Abstract
Magnetic density separation (MDS) is a technique that can be used to recycle different types of plastics, based on their mass densities. This technique uses ferrofluids and magnets to create a fluid which, with buoyant and magnetic effects combined, has a varying so-called ‘effective density’.
When particles with different mass densities are placed inside these ferrofluids, they will all rise or sink to a certain height, which makes separation more efficient and easy.
In this report, the movement of spherical and ellipsoid plastic particles inside the separation chamber of a MDS-setup is investigated, together with the effects of colliding particles. This movement has been tracked inside water (for comparison) and inside manganese (II) chloride (MnCl2). This is a transparent paramagnetic fluid, which makes it possible to track the particles using Particle Tracking Velocimetry (PTV).
The measurements show that the movement of the plastic particles are in good agreement with numerical simulations. Also, the tracking process of ellipsoid particles has been investigated, resulting in the conclusion that the software that was used has to be modified to track rotating ellipsoids. Finally, a clear relation has been found between a quantified interaction parameter (Lateral center distance (LD)) and the time delay on the movement of a particle, caused by this interaction.
Contents
Contents v
List of Figures vii
List of Tables ix
1 Introduction 1
2 Theory 3
2.1 Spherical particles in water . . . 3
2.2 Magnetic fields and fluids . . . 6
2.2.1 Magnetostatics . . . 6
2.2.2 Magnetic fluids . . . 11
3 Experimental Setup 13 3.1 PTV-setup and changes . . . 13
3.2 Container and release mechanism . . . 14
3.3 Calibration and tracking . . . 15
3.3.1 Calibration . . . 15
3.3.2 Tracking. . . 16
3.4 Measurements. . . 18
3.4.1 Water measurements . . . 18
3.4.2 Measurements in manganese (II) chloride . . . 18
4 Results 19 4.1 Water Measurements . . . 19
4.1.1 Single measurements . . . 19
4.1.2 Collision measurements . . . 21
4.1.3 Ellipsoid particle measurements. . . 24
4.2 Measurements in manganese (II) chloride . . . 26
4.2.1 Single measurements . . . 26
4.2.2 Collision measurements . . . 29
5 Conclusions 33 5.1 Measurements in water. . . 33
5.2 Measurements in manganese (II) chloride . . . 34
Bibliography 35
Appendix 37
A Nomenclature 37
B Measurement graphs 39 B.1 Water Measurements . . . 39 B.2 MnCl2Measurements . . . 45
List of Figures
1.1 A sketch of the MDS-setup. On the left, plastics of different types are inserted.
The plastics are then separated using their mass densities, after which they get post-processed [1]. . . 1 2.1 A graph of the different flow regimes depending on the Galileo number. The regime
left of the first line on the left (G < 165) represents a symmetric wake of the particle.
The other regimes are: steady oblique motion (+), oscillating oblique path with a low frequency (∗), oscillating oblique path with a high frequency (×), periodically zigzagging trajectories (◦) and a three dimensional chaotic path() [2].. . . 6 2.2 An illustration depicting magnetic fields lines around a permanent magnet [3]. . . 7 2.3 A visualization of the difference between ~H, ~M and ~B. Inside the bar magnet, the
field lines are different [4]. . . 7 2.4 A schematic representation of a Halbach array. In this array, permanent magnets,
which are rotated 90 degrees with respect to their neighbour, are placed next to each other to form a strong and weak side [5]. The array on the right is a combination of the two permanent magnets on the left. . . 8 2.5 A simulation of the magnetic field components of H ·dHdz[mA23] above the alternative
Goudsmit magnet. Here, the construction method of the alternative magnet can be seen clearly. From [6]. . . 10 2.6 A graph visualizing the effects of a magnetic field on the magnetization of four
different types of magnetic materials [7]. . . 11 3.1 A typical camera arrangement (top view) of a PTV-setup. The light sources are on
top of the cameras. The lines and numbers determine the calibration positions [6]. 13 3.2 A sketch of the experimental setup that has been used. Camera 3 and 4 were not
used and a piece of cardboard (grey/white) has been placed on these sides of the container during the measurements [6]. . . 15 3.3 A sketch of the used calibration plate. The centres of the white dots are all 10 mm
apart, while the dots have a diameter of 3 mm . . . 16 3.4 Visualization of the drawn ‘lines’ from the cameras. The cameras are positioned on
the x- and y-axes at 0 mm. . . 17 3.5 Visualization of the waist of two polynomials, together with the tracked position of
the particle . . . 18 4.1 Graphs of the height and velocity of the polystyrene particle in water. The results
of numerical simulations were added to the graphs [1]. The green area in the graph represents the expected range of the particle when a simulation is done without a Basset history force added, while the grey area represents the expected range of the particle when this history force is added. The dark green area represents the overlap of the green and grey areas. . . 20
4.2 Graphs of the height and velocity of the santoprene particle in water. The results of numerical simulations were added to the graphs [1]. The green area in the graph represents the expected range of the particle when a simulation is done without a Basset history force added, while the grey area represents the expected range of the particle when this history force is added. The dark green area represents the overlap of the green and grey areas. . . 20 4.3 Graphs of the average Reynolds number Red of the polystyrene and santoprene
particles, derived with equation 2.14 and, instead of the gravitational velocity vg= r
ρp
ρf − 1
gd, the average velocity of the measured particles. . . 21 4.4 A graph of the time delay plotted against the lateral center distance of the collision
experiments using polystyrene and santoprene. Two linear fits were added to the graphs.. . . 23 4.5 A graph of the height of an ellipsoid PVC-U particle in water, plotted against time.
At the end the ellipsoid touches the bottom of the container. . . 24 4.6 Graphs with the average height and velocity of 5 measurements of spherical 5mm
PVC-U particles, together with numerical simulations [1]. . . 27 4.7 Graphs with the average height and velocity of 5 measurements of spherical 5mm
POM particles, together with numerical simulations [1]. . . 28 4.8 A graph of a collision between a 5 mm POM and PVC-U sphere. The dotted line
represents the colliding spheres, while the error bars are the average results of the single measurements of POM and PVC-U spheres, as presented before. . . 29 4.9 A graph showing the point at which the time has been measured to calculate the
time delay.. . . 30 4.10 A plot of the lateral center distance of colliding 5mm spherical PVC-U particles
against the time delay (black), together with a plot of the collision height of these particles against the time delay (red). . . 31 B.1 Graphs of the collision measurements of 5 mm polystyrene and santropene particles
in water.. . . 39 B.2 Graphs of the collision measurements of 5 mm polystyrene and santropene particles
in water.. . . 40 B.3 Graphs of the collision measurements of 5 mm polystyrene and santropene particles
in water.. . . 41 B.4 Graphs of the collision measurements of 5 mm polystyrene and santropene particles
in water.. . . 42 B.5 Graphs of the collision measurements of 5 mm polystyrene and santropene particles
in water.. . . 43 B.6 Graphs of the collision measurements of 5 mm polystyrene and santropene particles
in water.. . . 44 B.7 Graphs of the collision measurements of 5 mm PVC-U and POM particles in MnCl2 45 B.8 Graphs of the collision measurements of 5 mm PVC-U and POM particles in MnCl2 46
List of Tables
4.1 The plastic particles that were used during the water measurements. . . 19
4.2 The plastic particles that were used during the MnCl2 measurements. . . 26
A.1 Nomenclature of the Greek symbols. . . 37
A.2 Nomenclature of the Latin symbols. . . 38
Chapter 1
Introduction
Over the last decades, the use of plastics has been growing vastly. Plastics are a very useful product in all sorts of processes. They have got many advantages, such as:
• There are many types of plastics. Each type has its own properties, which leads to the fact that plastics can be customized to fit any process.
• Plastics can last a long time (varying from 10-1000 years), which makes them cheap to use.
Because of its advantages, plastics are used almost everywhere, which leads to a large plastic waste production. This is a problem that cannot be denied anymore.
In the past, plastic waste has been dumped in oceans, on landfills and other places. This lead to all sorts of issues, regarding sea life as well as the environment people have to live in. This is the reason that more and more plastics are recycled over the past few years.
The most common way to recycle plastic, is to melt all and shape it into new plastics. In theory, in this way all plastic can be reused another time. While this looks very promising, it has one big disadvantage: when melted together, the plastics lose their specific properties, making them not useful anymore for certain processes.
To combat this specific problem, plastics need to be separated down to plastic type before melting. This could be done by hand, but there are many other methods to do this, such as near infrared spectrometry or electrostatic separation. In this report, one of these methods will be investigated: Magnetic Density Separation (MDS).
MDS is a plastic separation method that uses magnets and ‘magnetic fluids’ to separate plastics of different types.
Figure 1.1: A sketch of the MDS-setup. On the left, plastics of different types are inserted. The plastics are then separated using their mass densities, after which they get post-processed [1].
In Figure1.1, a sketch of a MDS-setup is shown. In this sketch, the MDS setup is visualized.
With the use of ferrofluids and the magnets, a liquid with mass density gradient is created inside the MDS-setup. Plastics of different types (which all have different mass densities) will float at different heights, after which they are separated using separation blades.
The MDS-technique can be separated in a couple of steps, such as the flow laminator, the separation chamber and the post-processing. In this report, the movement of the particles inside the (simplified setup of the) separation chamber will be investigated, together with the effects of collisions and other shapes of particles on the separation process.
Chapter 2
Theory
To understand the experiments that will be done to investigate the movement of plastic particles, the theory behind these experiments has to be clarified. In this chapter, the movement of spherical particles will be explained, together with the effects of other shapes of particles, the use of magnets and the effects of a magnetic fluid.
2.1 Spherical particles in water
A moving particle in water is influenced by a couple of forces. In this section, these forces will be explained, together with their influence on the particle.
Gravitational buoyancy force
Every plastic particle experiences a gravitational force, depending on their mass mp and the gravitational acceleration g. When in a fluid, there is another force which is dependent on the gravitational acceleration: the buoyancy force or Archimedes force. This force is an upward force on the particle, opposing the weight of the particle. The magnitude of this force is equal to the gravitational force on the amount of water that would otherwise occupy the volume of the particle.
When these forces are combined in a gravitational buoyancy force, its equation looks like
Fgb = Vp(ρp− ρf)g, (2.1)
with Vp[m3] the volume of the particle, ρpand ρf[mkg3] the mass densities of respectively the particle and the fluid and g[ms2] the gravitational acceleration.
Drag force
When a particle is moving through a fluid, it experiences a drag force. The theory of this force was developed by Sir Isaac Newton [8], from the theories of Galileo Galilei [9]. Just as the drag force in air, this drag force comes from the resistance of the surrounding particles (in this case water) and looks like
FD=ρfACD
2 |v| v, (2.2)
with A[m2] the cross sectional surface area, CD the drag coefficient of the particle and v[ms] the relative velocity vp− vf of the sphere to the fluid.
Added mass force
Next to these two forces, there are a couple of forces that are often left out of the equation. When the particle-fluid density fraction ρρp
f ≈ 1, these forces however become significant.
One of these forces is the added mass force. The added mass force determines the amount of work that needs to be done to change the kinetic energy T of the surrounding fluid [10]. This kinetic energy is given by the following expression:
T = ρf
2 Iv2p, (2.3)
where:
I = Z
V
vf,i
vp vf,i
vp dV, (2.4)
with vf,i(i = 1, 2, 3) the components of the fluid velocity, vp the velocity of the particle and V the entire volume of the fluid. When a sphere, moving with a constant velocity in a fluid, is accelerated, the kinetic energy of the surrounding fluid is likely to rise. The amount of work that needs to be done to change the kinetic energy of the fluid leads to the added mass force Fam. This added mass force looks like
Fam= − 1 vp
dT
dt = −ρfIdvp
dt , (2.5)
out of which it can be concluded that the added mass force depends on the time derivative of the particle velocity.
Basset history force
The second force that is often left out of the equation is the Basset history force. Where the added mass force accounts for a drag term added to the equation, the Basset history force accounts for the viscous effects that take place inside the fluid and is often called the history term. It is called the history term, because it takes into account the entire history of the motion of the particle.
Consider an accelerated infinite flat plate [11]. The equation of motion for the fluid is dvf
dt = ν∂2vf
∂y2 , (2.6)
with ν the kinematic viscosity [ms2] of the fluid. Taking the initial condition vf(0, y) = 0 and the boundary conditions vf(t, 0) = vf,0 and vf(t, ∞) = 0, where vf,0 is the velocity of the plate, the solution to equation2.6is
vf = vf,0erf(η), (2.7)
with η = y
2√ νt or
vf = 2vf,0
√π Z n
0
exp(−λ2)dλ. (2.8)
The local shear stress is given by
τ =
√ρµvf,0
√πt , (2.9)
with µ[mskg] the dynamic viscosity. Now assume that this variation in plate velocity can be broken up into a series of changes. When doing this, the effect on the local shear stress would be
τ =r ρµ π[∆vf,0
√t + ∆vf,1
√t − t1
+ ∆vf,2
√t − t2
+ ...] (2.10)
This series can be rewritten as a sum and (in the limit of ∆t → 0) an integral, which looks like
τ =r ρµ π
Z t 0
du dt0
√t − t0dt0 (2.11)
CHAPTER 2. THEORY
This approach, dividing the variation into steps, can also be applied to the impulsive flow over a sphere at low Reynolds number (which is the case during the experiments in this report). Basset [11] found that the drag force due to this variation was equal to
FBasset,i= 3 2d2√
πρµ Z t
0
d
dt0(ui− vi)
√t − t0 dt0 (2.12)
One-way coupled model
To combine the forces explained above, a one-way coupled model has been used to model the forces acting on a spherical particle inside a fluid. This kind of model has been well studied in the past, which leads to the fact that there are many different solutions to combine the forces explained above. The solution that will be used to model the particle in this report is called the
‘Maxey-Riley equation’, given by equation2.13.
mpdvp
dt = (mp− mf)g + mfDvf Dt −mf
2 (dv dt −Dvf
Dt ) − 3πdµf (vp− vf)
−3 2d2√
πρfµ Z t
t0
dt0
√t − t0(dvp
dt0 −dvf
dt0 ),
(2.13)
with f a correction for the drag force, made by Maxey-Riley [12]. This correction has been made in case the fluid is not in the Stokes regime. As can be seen in equation2.13, vf, the velocity of the fluid is present. During the measurements that will be presented in this report, vf = 0 will be used when computing the numerical modelling results of the movement of the particle, as the particle will often move through a nearly quiescent fluid.
Galileo Number
In this report, the Galileo number will be used to explain why certain particle movements happen during a measurement. The Galileo number is defined as
G = r
ρp ρf − 1
gd3
ν , (2.14)
with d[m] the diameter of the sphere and ν[ms2] the kinematic viscosity of the fluid. This is a dimensionless number which characterizes the behavior of particles falling through fluids. It is equivalent to the the Reynolds number, with the diameter of the spherical particles d as typical length scale and the typical gravitational velocity of vg=
r
ρp
ρf − 1 gd.
When plotting the density fraction ρρp
f against the Galileo number G, the movement of falling spheres can be divided into five regimes, varying from a laminar regime to a three dimensional chaotic path. These regimes can be seen in Figure2.1.
Ellipsoid particles
Next to the movement of spherical particles in water, elliptical particles have been investigated. In the sections above, the rotation of the spherical particle has not been taken into account entirely.
When looking at elliptical particles the rotation of the particles has a much more significant influence on the movement of the particle through the fluid.
In the past, there have been a lot of studies to the motion of ellipsoid particles in fluids [13]
[14]. The measurements that were done for this report were however merely done to test the tracking capabilities of the existing setup. Because of this reason, the rotation and movement of ellipsoid particles will not be discussed in detail.
Figure 2.1: A graph of the different flow regimes depending on the Galileo number. The regime left of the first line on the left (G < 165) represents a symmetric wake of the particle. The other regimes are: steady oblique motion (+), oscillating oblique path with a low frequency (∗), oscillating oblique path with a high frequency (×), periodically zigzagging trajectories (◦) and a three dimensional chaotic path() [2].
2.2 Magnetic fields and fluids
In this section, the theory of the influence of a magnetic field on a polymer particle in a para- magnetic fluid (for example: manganese chloride) is discussed. This influence is an important principle that is used in an MDS setup. To understand the behavior of the particle, first the basics of magnetic fields will be introduced, together with examples of an ideal magnet in a Hal- bach array configuration and the use of magnets in practice. In section 2.2.2, the concept of ferrohydrodynamics will be discussed, and its influence on the dynamics of a spherical particle.
2.2.1 Magnetostatics
The discovery of magnets and magnetic fields dates back to the 6th century BC [15]. At this time philosopher Thales of Miletus, an ancient Greek, discovered the lodestone and its attraction to iron. Ever since then, magnets have been used more and more in electronics as well as other industries.
Magnetic fields
Magnets can be described using magnetic fields. Magnetic fields ~H[Am] are often visualized using magnetic field lines. An example of a magnet and its magnetic field lines can be seen in Figure 2.2. The closer the field lines are together, the stronger the magnetic field on that position. That is why in this figure, the magnetic field is strongest close to the poles.
The magnitude of this magnetic field ~H can be written as H = |H| =q
Hx2+ Hy2+ Hz2, (2.15)
CHAPTER 2. THEORY
Figure 2.2: An illustration depicting magnetic fields lines around a permanent magnet [3].
with Hx, Hy and Hz the components of the magnetic field in the x, y and z-direction. Besides using ~H to describe a magnetic field, often the magnetic flux density ~B[T ] is used. The relation between ~B and ~H is
B = µ~ 0( ~H + ~M ), (2.16)
with µ0[Hm] the magnetic constant and M [mA] the magnetization. In a vacuum, equation 2.16 reduces to
B = µ~ 0H,~ (2.17)
which means that the field lines of both quantities look the same. Inside a material however, the field lines are not the same, which can be explained with the help of equation 2.16. Inside the material, the magnetization ~M is not equal to 0, which leads to different field lines. This difference can be seen in Figure2.3. Here, the field lines of ~H and ~B are sketched, together with the field lines of ~M .
Figure 2.3: A visualization of the difference between ~H, ~M and ~B. Inside the bar magnet, the field lines are different [4].
The magnetization ~M can be described by
M = χ ~~ H, (2.18)
with χ the magnetic susceptibility, a dimensionless quantity. This magnetic susceptibility plays an important role when looking at magnetic fluids, which will be discussed in the next section.
Combining equations2.16and2.18results in
B = µ ~~ H, (2.19) with µ = µrµ0 the permeability and µr = (1 + χ) the relative permeability of the magnetic material. In the case of a permanent magnet, the equation reads
B = µ ~~ H + Br, (2.20)
with Br[T ] the remanent magnetic flux density, the magnetic flux density that remains when removing the external magnetic field from for example a permanent magnet.
Halbach Array
In the MDS-setup, that will be explained in section 3.2, a magnetic field is required that only varies in one direction. In this case, the magnetic field strength has to decrease in the z-direction (the height), while it stays constant in the x- and y-direction. This ensures the reliability of the measurement results.
An ideal way to make or construct a magnetic field with these properties is an infinite Halbach array. This is an array of permanent bar magnets that are placed in such a way, that a magnetic field is created which only changes in the direction perpendicular to the placement of the magnets.
In Figure 2.4, a sketch of such an Halbach array is shown. As can be seen in this figure, permanent bar magnets are placed in such a manner, that on one side of the array, the magnetic fields of the two types of magnets (respectively represented in red and blue) create a ‘strong side’
on top of the array, while the magnetic fields under the array cancel each other out, which is called the ‘weak side’ of the array. This type of Halbach array can be called a practical Halbach array, where the bar magnets are rotated 90 degrees in respect of their neighbour.
Figure 2.4: A schematic representation of a Halbach array. In this array, permanent magnets, which are rotated 90 degrees with respect to their neighbour, are placed next to each other to form a strong and weak side [5]. The array on the right is a combination of the two permanent magnets on the left.
Analytical expression of the magnetic field of an infinite Halbach array
To understand the behavior of particles on top of an infinite Halbach array, an expression of the magnetic field of this array needs to be found. This is easily done numerically, but an analytic expression can be found with ease as well (in contrast to a finite Halbach array, where it is not easily derived).
CHAPTER 2. THEORY
In order to find the field associated with a Halbach array, consider a planar configuration of thickness d, lying in the (x, y)-plane. The magnetization M of this Halbach array can be approximated to
Mx= Mrsin kx, (2.21)
My= 0, (2.22)
Mz= −Mrcos kx, (2.23)
with Mr the remanent magnetization and k = 2πλ[m1] the wave vector and λ = 4p[m] the wavelength, where p is the pole size of the magnets. Using Poisson’s equation [16] [17], the following equation can be found:
∇2φinside= −M0k cos kx, (2.24)
with φinsidethe scalar potential inside the array and M0= µ0Mr, with µ0the vacuum permeability.
Laplace’s equation above and below the array are
∇2φabove= 0 (2.25)
and
∇2φbelow = 0 (2.26)
Equations 2.24, 2.25and2.26can be solved with the following boundary conditions:
φabove= φbelow = 0 (2.27)
φabove= φinside (2.28)
φbelow = φinside (2.29)
at respectively z = ±∞, z = 0 and z = −d.
Because of the fact that the scalar potential φ or the normal flux density must be continuous on the upper and lower surfaces of the array, the following also holds:
∂φabove
∂z =∂φinside
∂z − M0sin kx (2.30)
∂φbelow
∂z =∂φinside
∂z − M0sin kx (2.31)
at respectively z = 0 and z = −d.
Using these boundary conditions, the solution of equations 2.24, 2.25and2.26is:
φabove= −M0
k (1 − e−kd)e−kzcos kx, (2.32)
φinside =M0
k (e−k(d+z)− 1) cos kx, (2.33)
φbelow= 0. (2.34)
This results in a magnetic flux density B above the Halbach array of Bx,above= −∂φ
∂x = −M0(1 − e−kd)e−kzsin kx (2.35) Bz,above= −∂φ
∂z = −M0(1 − e−kd)e−kzcos kx (2.36) As can be seen in equations 2.35 and 2.36, the magnetic flux density of the Halbach array decreases in the z-direction, while it oscillates in the x-direction.
In theory, a Halbach array could be used to construct a perfectly homogeneous magnetic field.
This however can only be done when the array is of infinite length in x- and/or y-direction. This, of course, is not possible in practice. In the MDS-setup, a Halbach array of finite length could be used, but then the effects on the sides of the array should be kept in mind.
Magnet in practice
In the past, Meulenbroek [6] has been building practical Halbach arrays with pole sizes p of 4, 10, 15 and 25 mm. Building a Halbach array with a large pole size (p in Figure2.4), which is needed in the MDS-setup, turned out to be very difficult, because of the fact that working with these strong magnets requires extreme caution. That is why the building of the array used in the setup has been outsourced to an external company. The construction of the array depicted in Figure 2.4 turned out to be expensive. To decrease the cost, an altered construction method has been used, in which a more complex structure of NdFeB blocks has been made. This structure can be seen in Figure 2.5, together with the the magnetic field components of H · dHdz[mA23], which is indicative for the effective mass density above the magnet (this will be discussed in section2.2.2).
This structure will be called the ‘Goudsmit magnet’ from now on.
Figure 2.5: A simulation of the magnetic field components of H · dHdz[Am23] above the alternative Goudsmit magnet. Here, the construction method of the alternative magnet can be seen clearly.
From [6].
As can be seen in Figure 2.5, the magnetic field components are not only varying in the z- direction, but also in the other directions. This is due to the effects coming from the finite length of the array. This ‘hamburger shape’ is however showing promising simulation results in the middle of the magnet. Here, inside the rectangle in Figure2.5, the magnetic field components are almost constant the x- and y-direction. Because of this fact, the measurements on the MDS-setup later on in this report will be executed on top of this part of the ‘Goudsmit magnet’.
CHAPTER 2. THEORY
2.2.2 Magnetic fluids
As described in section 1, to implement a MDS-setup, besides a magnet, a ‘magnetic fluid’ is required. This section will discuss the concept of a ‘magnetic fluid’ and describe the fluid that is used during the measurements. Before discussing this, the different types of magnetism are explained, to gain a full understanding of the properties of magnetism a so-called ferrofluid uses.
Types of Magnetism
Magnetism can be divided into several different types. Four of these types are [18]:
• Diamagnetism
The weakest type of magnetism is called diamagnetism. Diamagnetism appears in all ma- terials and is the tendency to oppose a magnetic field, because of a Lorentz force applied on the electrons circling the nucleus an atom.
• Paramagnetism
Paramagnetism appears in materials with unpaired electrons. It is the type of magnetism that follows the direction of an applied magnetic field.
• Ferromagnetism
Just as a paramagnetic material, a ferromagnetic material has unpaired electrons. In addition to this paramagnetic effect (the magnetic moments are parallel to the applied magnetic field) the magnetic moments also have the tendency to orient themselves parallel to each other.
• Superparamagnetism
Tiny ferromagnetic materials with a radius in the nano-region can behave like a paramag- netic, given the right circumstances. When these materials are within the so called ’super- paramagnetic region’, they act as a paramagnetic with a giant, classical moment.
A visualization of these kinds of magnetism can be seen in Figure 2.6.
Figure 2.6: A graph visualizing the effects of a magnetic field on the magnetization of four different types of magnetic materials [7].
Effects of magnetism
After explaining the types of magnetism, the effects of magnetism during the experiments will be discussed. During the measurements, a paramagnetic salt solution of manganese (II) chloride (MnCl2) has been used. This type of salt has been used, because:
• it is a clear solution, which is an requirements to track particles in it;
• it is relatively cheap to make (in [6], the synthesis of the solution has been explained);
• one of the effects of using this salt solution is the fact that it creates an ‘effective mass density’ in the fluid.
Effective mass density
Because of the paramagnetic properties of MnCl2, the magnetization of the fluid changes linearly with the magnetic field at a given position ( ~M = χ ~H), with χ as a linearization constant. This leads to the fact that the salt solution can be called a ‘magnetic fluid’. When a non-magnetic object is immersed in a magnetic fluid, it has been found that these particles can be levitated inside this magnetic fluid. Because of this reason, there has to be a force on the non-magnetic object. This force is the so-called magnetic body force. After some calculations (see [19] and [6]), this magnetic body force can be written as
Fm= −V µ0M ∇H, (2.37)
with V [m3] the volume of the non-magnetic object, M the average magnetization over the volume of the magnetic fluid and H the magnetic field. This magnetic body force can be combined with the gravitational buoyancy force (equation2.1), resulting in the following equation of the so-called magneto-buoyancy force:
Fmb= Fgb+ Fm,
= V (ρp− ρf)~g − V µ0M ∇H,
= V (ρp− ρef f)~g,
(2.38)
where:
ρef f = ρf−µ0M
g ∇H, (2.39)
with ρef f[mkg3] the effective mass density. This effective density is, thus, not the actual density of the fluid at a position, but is a combination of the fluid density and the magnetic effects. As can be seen in equation2.39, the effective mass density is a function of the magnetic field H and the average magnetization M . Because of this, the effective mass density changes in every point inside the magnetic fluid. In this way, a non-magnetic object rises or sinks to an equilibrium height when placed inside the magnetic fluid, at which point the magnetic buoyancy force is zero.
In MDS-applications, this effective mass density is used to separate plastics of different mass densities, as their equilibrium heights differ. During the experiments in this report, measurements have been done to determine the movement of different kinds of plastics inside a magnetic fluid.
This was needed to find out which properties of an MDS setup are relevant.
Galileo number in MnCl2
In water, the Galileo number is defined as expressed in equation 2.14. In MnCl2, the effective density of the magnetic fluid needs to be taken into account. Here, the Galileo number is defined as:
GMnCl2 = r
ρp
ρef f − 1 gd3
ν , (2.40)
with d[m] the diameter of the particle.
Chapter 3
Experimental Setup
As explained in section1, the experiments that were performed to make this report, were made in a simplified setting. To track the plastic particles inside a liquid, a tracking setup was used. The setup consists of two cameras, the Goudsmit magnet (as explained in section2.2) and a container with a release mechanism. In this section, the so-called particle tracking velocimetry (PTV) camera setup will be explained, together with the working principles of the release mechanism and the measurements in water and manganese (II) chloride.
3.1 PTV-setup and changes
A PTV-setup usually consists of multiple cameras, a light source to illuminate the environment and a computer to analyse the images from the cameras. To obtain a 3D image of the observed space, two cameras are, in theory, enough. However, in practice, researchers usually use three or four cameras to minimize the measurements errors and to measure particles which are (for example) behind each other. A typical camera arrangement of a PTV-setup can be seen in Figure 3.1.
Figure 3.1: A typical camera arrangement (top view) of a PTV-setup. The light sources are on top of the cameras. The lines and numbers determine the calibration positions [6].
While this setup seems to be practical, there are some disadvantages to this arrangement:
1. The light sources need to illuminate the space, to be able to track the particle inside the fluid. This illumination is however often so bright, that it over saturates the cameras on the other side.
2. All kinds of coloured particles are used to observe their behavior in the fluids. When using these particles, objects in the background of the picture can lead to errors during analyzing.
For example: black particles cannot be tracked when they are floating in front of the camera on the other side (which is black as well).
3. Because there are 4 cameras, the amount of data that needs to be processed is doubled, compared to 2 cameras.
Because of these reasons, another arrangement has been used, with two cameras. The two cameras have been placed in the same position as Camera 1 and 2 in Figure 3.1. To make sure that the background does not lead to issues anymore, a piece of paper or cardboard has been placed on the other side (on the container, this will be explained in section3.2). The light source is placed above the camera setup, which eliminates the over saturation of the cameras. In previous research [6], the light source also formed another problem regarding the measurement of the position of the particles. In these experiments, the light source slowly heated the fluid in which the measurements were done, leading to, for example, a slight difference in height inside the magnetic fluid. This of course led to higher measurement errors. In this experiment, a LED light source has been used, which reduces these measurement errors, as a LED light does not radiate as much heat as other light sources.
3.2 Container and release mechanism
To contain the liquid that is used during the measurements, a special container has been made by our technicians. The dimensions of this container are 15x15x15 cm. In Figure 2.5, the rectangle on top of the Goudsmit magnet points out on what position the measurements should take place to find reliable results. As it turns out, this is 15 cm above the Goudsmit magnet.
Inside the container, a release mechanism has been built. In Figure3.2, a sketch of the whole setup can be seen, together with the cameras. The mechanism has been made in such a way, that it can release two particles at the same time. Two pins, attached to a rotating bar, hold the particles back and can be moved away at the same time. In this way, collision experiments can be performed easily, because the particles will also be released exactly above each other. The release point closest to the magnet is placed inside a so-called false bottom, minimizing the space used by the mechanism.
To minimize the measurement error in the experiments, the influence of objects on the back- ground of the camera frames was minimized. This has been done using paper and cardboard, which is placed on the outside of the container. During the measurements, white and black particles have been used. When measuring the white particles, a grey background has been used, while measuring the black particles, a white background was placed on the container. In this way, there is only one background colour and the particle will most of the time show up on the recorded images, as this kind of colours lead to high contrast levels on the black-and-white cameras that have been used.
CHAPTER 3. EXPERIMENTAL SETUP
Figure 3.2: A sketch of the experimental setup that has been used. Camera 3 and 4 were not used and a piece of cardboard (grey/white) has been placed on these sides of the container during the measurements [6].
3.3 Calibration and tracking
To perform measurements with the experimental setup, a set of Matlab scripts has been used.
Some of the scripts were used to calibrate the setup, while others were used to track the particles itself. In this section, the calibration process and the tracking process of the particles will be explained.
3.3.1 Calibration
The PTV-setup records videos of the particles that have to be tracked. In these (black and white) images, the pixels can be used to determine the position of the particle inside the fluid.
This however is not enough to get a ‘real-world’ position of the particle, which is the goal of the tracking process. That is why the setup needs to be calibrated before any measurements can be done.
The calibration process will be executed with the help of a calibration plate. This is a plate, which can be placed inside the container in the setup.
As can be seen in Figure 3.3, there are white dots placed on the plate. These dots are also placed on the other side of the plate and are 10 mm apart from each other. In order to calibrate the space inside the container, the calibration plate is placed on predetermined positions inside the container (see the horizontal and vertical lines in Figure 3.1), for both camera 1 and 2. In this way, a 3D-grid of white dots is made. This 3D-grid of ‘real-world’ coordinates can then be related to the pixel coordinates of the cameras. In this case, the coordinates are related to each other with a polynomial, given by
f (xp, yp, r) = P (r), (3.1)
with xp and yp the pixel coordinates, P the fitted polynomial and r the distance between the
Figure 3.3: A sketch of the used calibration plate. The centres of the white dots are all 10 mm apart, while the dots have a diameter of 3 mm
calibration plate and the camera. There is a unique polynomial for every pixel coordinate (xp, yp), in both cameras. When tracking a particle, a ‘line’ is drawn from both cameras to the particle that is tracked. The particle is estimated to be on the position where these lines are nearest to each other, resulting in a transformation of the position of the particle from the pixel coordinates to the ‘real-world’ coordinates. A visualization of the lines that are drawn can be seen in Figure 3.4.
3.3.2 Tracking
After the calibration is done, the calibration data can be used to track the particles. Just as during calibration, a set of Matlab scripts has been used to track particles. The tracking process of a particle can be divided in several steps. These steps are:
1. A series of images is taken of the moving particle.
2. The part of interest (for example: when the particle is within the measurement area) is cut out of the series of images, which makes the processing time shorter.
3. An average image of all frames is calculated and subtracted from each individual frame. This results in an image with only the particle, which is highlighted.
4. In each frame, a circle is automatically drawn around the particle, with a given radius in pixels. This means that, for each measurement of a different particle, the radius needs to be known beforehand.
5. The pixel coordinates of the centre of the circle in each frame are calculated and saved.
6. After all pixel coordinates are calculated, a track is made of the centres of the circle in each frame.
7. The efficiency of the Matlab scripts is not 100%, which means that there are frames in which no circle is drawn. When this has happened, two different tracks can be glued together and points can be manually added to the track.
8. Finally, the pixel coordinates of the track are transformed to ‘real-world’ coordinates, using the calibration data.
CHAPTER 3. EXPERIMENTAL SETUP
Figure 3.4: Visualization of the drawn ‘lines’ from the cameras. The cameras are positioned on the x- and y-axes at 0 mm.
The measurement error of the tracking of the particles can be derived using the so-called waist of the transformation from pixel coordinates to ‘real-world’ coordinates. During this transformation, the ‘lines’ are drawn from both cameras in the direction of the particle. At a certain position, the distance between these two lines will be minimal. This distance will be used as the measurement error of the tracking, while the centre of this distance is pointed out as the position of the particle.
A visualization of this waist can be seen in Figure3.5.
Figure 3.5: Visualization of the waist of two polynomials, together with the tracked position of the particle
3.4 Measurements
In this section, the measurements in water and the ‘magnetic fluid’ manganese (II) chloride (MnCl2) will be explained.
3.4.1 Water measurements
When conducting the measurements, the container is filled with water to the top and the release mechanism is put into place. In water, the following different measurements were performed:
• Single particle measurements. These measurements were performed using sinking 5 mm Polystyrene (which has a density of 1029 ± 7[mkg3]) and rising 5 mm Santoprene (which has a density of 948 ± 13[mkg3]) particles.
• Collision measurements. These measurements were performed with the same particles as the single measurements, but then simultaneously.
• Single elliptical particle measurements. These measurements were performed using sinking 5- 2.5 mm polyvinyl chloride (PVC-U) (1433±4[mkg3]) and sinking 5-2.5 mm sinking chlorinated polyvinyl chloride (PVC-C) (1582 ± 4)[mkg3]) particles. Goal of these measurements was to review the functionality of the tracking script with particles that are not spherical.
3.4.2 Measurements in manganese (II) chloride
After measuring the movement of the particles in water, the effects of the MnCl2 are measured, when the container is placed on top of the Goudsmit magnet. The following measurements were performed:
• Single particle measurements. These measurements were performed with 5 mm PVC-U (1433 ± 4[mkg3]) and polyoxymethylene (POM) particles, which has a density of 1405 ± 3[mkg3].
• Collision measurements. These measurements were performed with the same particles as the single measurements, but then simultaneously.
Chapter 4
Results
After explaining the theory and the experimental setup, this chapter will give the results of the measurements in water and MnCl2.
4.1 Water Measurements
Using the setup explained in section 3.2, measurements were conducted in water. During these measurements, plastic particles were released from the top and/or the bottom of the setup, while tracking their movement in the water. The following plastic particles were used:
Plastic type Diameter [mm] Mass density [kg · m−3]
Polystyrene 4.89 1029 ± 7
Santoprene 4.73 948 ± 13
PVC-U 5.06 1433 ± 4
Table 4.1: The plastic particles that were used during the water measurements.
Furthermore, the fluid that was used during these experiments (water) had a mass density of 997 [mkg3] and a dynamic viscosity µ of (8.9 ± 0.5) · 10−4[mskg].
4.1.1 Single measurements
During the measurements, the polystyrene and santoprene particles were released from the top and bottom respectively. Because of their mass densities, which can be seen in Table 4.1, the polystyrene particle sinks to the bottom, while the santoprene particle rises to the top of the container.
In Figure 4.1, the measurement results of the tracking of the polystyrene particle are shown.
Three measurements were performed to track the particle. The average height and velocity of the particle, together with the standard deviation, can be seen in this graph. In Figure 4.1(a), the height of the particles is plotted against time, while in Figure 4.1(b), the velocity of the particle is visualized.
The green and grey areas in the graph are results of numerical simulations of the one-way coupled model of the movement of the particle through time. These simulations were done with the equations that were explained in section2.1. In the green area, the results of the simulations are shown when the Basset history force is not added, while in the grey area, the Basset history force is included.
These simulations were all done for a range of mass densities (hence the green/grey areas instead of lines) within the measurement error of the mass density of the particle that was used (see Table4.1). For example, the range that was used in Figure 4.1is 1022-1036 mkg3.
(a) The height of a polystyrene particle, with two numerical simulations
(b) The velocity of a polystyrene particle, with two numerical simulations
Figure 4.1: Graphs of the height and velocity of the polystyrene particle in water. The results of numerical simulations were added to the graphs [1]. The green area in the graph represents the expected range of the particle when a simulation is done without a Basset history force added, while the grey area represents the expected range of the particle when this history force is added.
The dark green area represents the overlap of the green and grey areas.
(a) The height of a santoprene particle, with two numerical simulations
(b) The velocity of a santoprene particle, with two numerical simulations
Figure 4.2: Graphs of the height and velocity of the santoprene particle in water. The results of numerical simulations were added to the graphs [1]. The green area in the graph represents the expected range of the particle when a simulation is done without a Basset history force added, while the grey area represents the expected range of the particle when this history force is added.
The dark green area represents the overlap of the green and grey areas.
In Figure 4.2, the measurement results of the tracking of a santoprene particle are shown.
Just as the measurement of the polystyrene particle, three measurements were performed and the average values can be seen in this figure. The height and velocity of the particle are plotted in respectively Figure4.2(a) and Figure4.2(b). Also, the results of the numerical simulations were added to these graphs.
After these numerical simulations, the same Matlab scripts were used to reversely determine the mass density of the polystyrene and santoprene particles. These values turned out to be 1029 and 953mkg3 respectively. These results show that the measured mass density and simulated density of the polystyrene are similar, while the densities of the santoprene are not (but however still inside the uncertainty range of the measurement). These calculations were done with the model
CHAPTER 4. RESULTS
including the Basset history force. The model without the history force gave values of the mass density far from the measurement. This calculation shows that the model including the history force is the best choice when modelling the movement of the particles in water.
Figure4.1and4.2show that the measurement results all are in the dark green area of the simulation results, which means they are in correspondence with the expectation values of the one-way coupled model. The standard deviations of the measurement results, which were calculated with the average values of the height and velocity of the rising and sinking particle, are also relatively small, except in Figure 4.2(b). Here, the standard deviation of the velocity of the santoprene particles is varying more. This could be caused by several reasons, such as:
• the number of measurements. Only three measurements were performed, which can lead to higher standard deviations.
• the usage of the Savitsky-Golay filter. In Matlab, a Savitsky-Golay filter was used to filter the noisy signals of the velocity profiles of the measurements. This filter predicts the velocity values at each point using the already predicted values. This method could have an effect on the calculated standard deviation.
(a) The Reynolds number of the sinking polystyrene particle, plotted against time.
(b) The Reynolds number of the rising santoprene particle, plotted against time.
Figure 4.3: Graphs of the average Reynolds number Red of the polystyrene and santoprene particles, derived with equation2.14and, instead of the gravitational velocity vg=
r
ρp ρf − 1
gd, the average velocity of the measured particles.
In Figure 4.3, the Reynolds number Red of the measured particles are plotted against time.
As can be seen, when the particles reach their maximum velocity (dv
dt ∼ 0), this number is ∼ 220 for polystyrene and ∼ 260 for santoprene. These values can be compared with the Galileo number that has been discussed in section 2.1. When using equation2.14, and the values mentioned in Table4.1, the Galileo number for the particles turn out to be ∼ 215 and ∼ 253 respectively, which is similar to the measured results.
These Galileo numbers mean that, when looking at Figure2.1 and keeping in mind the ratio ρρp
f
of the materials, both materials can be placed inside the regime of a 3-dimensional chaotic path.
4.1.2 Collision measurements
Next to the single measurements, collision measurements were performed, where polystyrene and santoprene particles were released from respectively the top and the bottom of the container simultaneously. The goal of these measurements was to determine the delay in time of the particles to reach the bottom/top of the container caused by interaction between the two particles. This
interaction can be quantified in the lateral center distance (LD [-]) between the two particles.
This LD can be calculated with:
LD = p(x1− x2)2+ (y1− y2)2
1
2(d1+ d2) , (4.1)
with x1and x2 the x-coordinates of particle 1 and 2, y1and y2the x-coordinates of particle 1 and 2 and d1 and d2the diameters of particle 1 and 2.
With this LD, the interaction between the two particles can be split up in three kinds (as seen from the top view):
• LD = 0. The particles hit each other exactly head-on.
• 0 < LD ≤ 1. The particles hit each other, but not exactly head-on.
• LD > 1. The particles do not hit each other.
To determine the time delay caused by the collision between two particles, the collision meas- urements are compared to the single measurements which were presented in section 4.1.1. The time delay is then calculated using:
tdelay = tcollision− tsingle, (4.2)
with tsingle the time it takes a particle to reach the top/bottom of the container without collision and tcollision this same time, but with a collision.
During the collision experiment, 31 measurements were performed. The XT-graphs of these measurements can be seen in Appendix B.1. Out of these 31 measurements, 21 measurements resulted in a collision (LD ≤ 1). The results of these measurements are plotted in Figure 4.4.
Because of the high Galileo number (∼ 220-260, see section2.1and section4.1.1) of the polystyrene and santoprene spheres in water, it was hard to obtain collisions at first. When this Galileo would have been low (< 165), the movement would have been in the laminar flow regime and the direction of the spheres would have been more predictable, but this was not the case unfortunately.
Before the successful collision measurements could be done, there were some problems with the experimental setup. As can be seen in Figure3.2, there are two horizontal rods keeping the particles in place before the measurement. When the measurement begins, these rods are moved away, to let the particles rise or sink. At first, rods with a diameter of ∼ 5 mm were used for this purpose. This however led to an extra horizontal movement, caused by the wake of the rods.
This resulted in the fact that no collision would be measured. Because of this, new rods with a diameter of ∼ 2 mm were placed on the setup. After replacing the rods, measurements with a LD
≤ 1 were much more common, as can be seen above.
The measurement error of the lateral center distance in Figure4.4is calculated using the waist of the cameras, which has been explained in section3.3.2. After tracking the particles using the steps in this section, the waist of each frame is presented in a graph. During the measurements, this waist (see section 3.3.2) turned out to be around 0.4 mm for most frames. The waist is a 3-dimensional measurement error, so to gain a measurement error in the LD, some calculations need to be done, using equation4.1and the fact that the LD is a 2-dimensional quantity. These calculations resulted in a measurement error of ∼ 0.1 [-].
As can be seen in Figure 4.4, there is a clear relation between the lateral center distance and the time delay the collision causes; a linear trend line can be plotted through the measurement points. A small lateral center distance results in a large time delay. Also, when the particles do not collide, there still is an influence of one particle on the other, as there is a time delay present in all measurements. This relation is of significant importance for the MDS-project. The fact that colliding particles prolong the separation process (through collisions or interaction with the wake of other particles) of the different plastics, has to be taken into account when designing a final MDS-setup.
CHAPTER 4. RESULTS
Figure 4.4: A graph of the time delay plotted against the lateral center distance of the collision experiments using polystyrene and santoprene. Two linear fits were added to the graphs.
Another observation that can be made out of Figure 4.4 is the fact that the particle with a mass density close to that of the fluid (Santoprene, see Table4.1), has a smaller time delay than the other particle most of the times. This is confirmed by 25 out of 31 measurements.
Although these experiments were performed in water, they can give valuable information about collision effects in an MDS-setup. However, to gain a better understanding of these effects, the measurements will have to be executed in manganese (II) chloride as well. These results will be presented in section4.2.
4.1.3 Ellipsoid particle measurements
In the results above, measurements were performed using spherical polystyrene and santoprene particles. Spherical particles were used, because the tracking Matlab script is made to recognize circles in a 2D-image (and thus spherical particles).
In practice however, plastic particles inside the MDS-setup are often made out of all kinds of shapes. Because of this, measurements were performed using ellipsoids, to find out whether the tracking script would still recognize the particles and their tracks inside the fluid.
During these experiments, 10 measurements were performed using 5-2.5 mm PVC-U ellipsoids.
After processing these measurements, it became clear that the Matlab scripts could not handle the ellipsoid shapes very well. Three successful tracks were obtained after processing. One of these tracks can be seen in Figure 4.5. As can be seen in this graph, the height of the particle does not change regularly, but has a slight ‘ripple’ in it. This ripple is probably caused by the rotation of the ellipsoid inside the water during the measurement. Although these ripples are present in the graph, it shows a almost constant average velocity of ∼ 129mms . Out of this can be assumed that the rotation of the ellipsoid doesn’t have a significant effect on its vertical velocity.
No conclusions can however be made about the value of the velocity, as the ellipsoid could not be released with the release mechanism discussed in section 3.2. The mechanism was not able to hold the ellipsoid in place, so the ellipsoid was released from the top of the container instead.
To compare the motion of the ellipsoid particle to the spherical particles discussed above and to research the detailed influence of the rotation of the ellipsoid, experiments with an altered release mechanism should be done in the future.
Figure 4.5: A graph of the height of an ellipsoid PVC-U particle in water, plotted against time.
At the end the ellipsoid touches the bottom of the container.
The rotation is however a cause of problems in the tracking process. When rotating, the 2D- shape in the frames of the cameras changes from frame to frame. Sometimes, it is a circle, while at other times it is an ellipse. This leads to several problems, which are:
• When the ellipsoid rotates, the script does not always recognize the particle, which results in unfinished tracks in the end product.
CHAPTER 4. RESULTS
• When the script does recognize the particle, it often generates an error for the center position of the particle. This leads to greater measurement errors.
Because of these problems, the measurement results of this experiment are not reliable to use in further research. The waist (or measurement error) of these measurement is too big (it went up to values >1, where 0.4 mm is normal).
A solution to these problems is to develop a script that adapts the shape of the particle that has to be found to the input images (so an ellipse when the particle has the shape of an ellipse etc.). Also, the rotation angle of the ellipsoid should be taken in account in this script. In this way, the particle, as well as the center of the particle, can be found in a more reliable way.
4.2 Measurements in manganese (II) chloride
Using the setup explained in section3.2, measurements were conducted in manganese (II) chloride.
During these measurements, plastic particles were released from the top and/or the bottom of the setup, while tracking their movement in the MnCl2. The following plastic particles were used:
Plastic type Diameter [mm] Mass density [kg · m−3]
PVC-U 5.06 1433 ± 4
POM 5.06 1405 ± 3
Table 4.2: The plastic particles that were used during the MnCl2 measurements.
Furthermore, the fluid that was used during these experiments (MnCl2) had a mass density of 1402 ± 0.1[mkg3], a dynamic viscosity µ of 5.54 · 10−3[mskg] and a susceptibility χ of (6.987 ± 0.002) · 10−4[−].
4.2.1 Single measurements
During the measurements, the PVC-U and POM were released from the top and bottom of the experimental setup explained in section3.2. As explained in section2.2.2, the mass density of the MnCl2on top of the Goudsmit magnet changes from around 1500 [mkg3] close to the magnet to the 1402 [mkg3] of the fluid itself. Because of this varying mass density of the fluid, the particles used during the measurements will rise or sink to a equilibrium position, determined by their own mass densities and the local magnetic field.
PVC-U particles
During the first measurements, 5 mm PVC-U particles were released from the top and bottom of the container. 10 measurements were performed, resulting in the graphs in Figure 4.6, with the average height and velocity of the particles during these measurements.
As can be seen in these graphs, the PVC-U particles rise or sink towards an equilibrium position of around 45 mm above the Goudsmit magnet. Also, the particles overshoot this equilibrium position a couple of times before staying at a constant height. A numerical simulation of the movement of the PVC-U particles in MnCl2 has been added to the graphs in Figure 4.6. This is a simulation of the one-way coupled model with the Basset history force added, for a range of mass densities of PVC-U between 1429-1437 [mkg3], which is the range of the mass density of the measured PVC-U with the measurement error.
In Figure4.6, the measurement results of the PVC-U particle are within the range the numerical model predicts. When looked at further detail, there can be seen that there are some places where the practical and numerical results are not the same, especially when looking at the velocity. This could be caused by several reasons, such as:
• the number of measurements. As there were only made 5 measurements for rising particles as well as sinking particles, this could lead to less accurate results.
• effects of the movement of the fluid. The numerical model is based on the assumption that vf is equal to zero, which reduces the complexity of the model. While the experiments are conducted with a nearly quiescent fluid, the movement of the fluid could still play a role in the track of the particle.
POM-particles
After measuring the rising and sinking PVC-U particles, POM particles were measured, to determ- ine their equilibrium position and compare them to the other particles. Because of the fact that the POM particles have got a mass density close to that of the MnCl2 (see Table4.2), only rising
CHAPTER 4. RESULTS
(a) Average height of 5mm PVC-U particles
(b) Average velocity of 5mm PVC-U particles
Figure 4.6: Graphs with the average height and velocity of 5 measurements of spherical 5mm PVC-U particles, together with numerical simulations [1].
POM particles were measured. When placing the POM particles in the top part of the container, they would not come out of the release mechanism. Because the densities of POM and MnCl2
are almost equal, the buoyancy forces are not high enough to release the particle downwards. For future work, a change in release mechanism is required to also measure sinking POM particles in MnCl2.
In Figure4.7, the results of 7 measurements of rising 5 mm spherical POM particles can be seen.
