3.4 Measurements
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.
The average height and velocity are plotted against time, together with numerical simulations of the one way coupled model. Because the densities of the POM and the MnCl2are close together, the model with Basset history force added would not converge, so no history force is added to the simulations.
(a) Average height of 5mm POM particles
(b) Average velocity of 5mm POM particles
Figure 4.7: Graphs with the average height and velocity of 5 measurements of spherical 5mm POM particles, together with numerical simulations [1].
As can be seen in Figure 4.7, the measurement results are within the range predicted by the numerical solution. Because of the fact that the POM-particle is at a relatively high height,
CHAPTER 4. RESULTS
the Basset history force is however a non-negligible force, as it has higher influence at heights further away from the magnet (ρρp
f → 1). In Figure4.2, there can be seen that the numerically predicted range including the Basset history force is lower than the range which does not include the history force. As the measurement results of the rising POM-particle are in the lower part of the numerically predicted range in Figure4.7, this could mean that these results are in the range with history force as well. In further research, it could be interesting to find a way to implement the Basset history force in the numerical models in MnCl2, to verify this assumption.
4.2.2 Collision measurements
After the single measurements of PVC-U and POM particles were performed, collision measure-ments have been executed. During the collision measuremeasure-ments in water, the Galileo number of the Polystyrene and Santroprene spheres was around 210-220, making it difficult to obtain colli-sions during the measurements. In MnCl2 however, the Galileo number is lower, making it less difficult to get the particles to collide. In section4.1.2, the results of the collision measurements in water were presented. The conclusion of these measurements was that the interaction between the particles resulted in a time delay. Because of this fact, the hypothesis of this experiment of colliding spheres in MnCl2is that there is also a time delay in reaching the equilibrium position.
During the measurements, 5 mm POM and PVC-U spheres have been released from respect-ively the bottom and top of the container. 10 measurements have been performed; all of them resulted in a collision. In Figure4.8, one of the results of the collision measurements can be seen.
The results of the other measurements are presented in AppendixB.2.
Figure 4.8: A graph of a collision between a 5 mm POM and PVC-U sphere. The dotted line rep-resents the colliding spheres, while the error bars are the average results of the single measurements of POM and PVC-U spheres, as presented before.
In Figure4.8, the dotted line represents the collision measurement, while the red and green error bars (the individual bars cannot easily be seen, as they are close to each other) depict the aver-age results of the single measurements presented in section 4.2.1. As can clearly be seen in the results of the sinking PVC-U spheres, there is a slight time delay between the average height and the height of the colliding sphere. Because this can be seen most clearly at the track of PVC-U (at the track of the POM, the equilibrium position has not been reached yet), this track will be investigated further.
The lateral center distance (LD) of the two colliding spheres at the time of collision has been measured using equation 4.1. Also, the time delay of reaching the equilibrium position between the non-colliding and colliding spheres has been calculated for all 10 measurements using equation 4.2, in the following way:
In Figure 4.9, a plot of the velocity of one of the sinking collided particles can be seen. tcollision
and tnormal(from equation4.2) defined at the point the velocity reaches 0 for the second time (at the arrow).
Figure 4.9: A graph showing the point at which the time has been measured to calculate the time delay.
In Figure 4.10, the results of the collision measurements are shown. The relation between the LD and the time delay cannot be easily seen, such as the measurements in water in Figure4.4.
However, it can be seen that, when the LD is → 0, the time delay is going up, which was expected after measuring the time delay of colliding particles in water. In the same figure, the collision height of the PVC-U and POM particles has been plotted against the time delay. In this case, a clear relation cannot be found either, but roughly, there can be seen that the time delay goes down as the collision height goes up.
During the experiment with colliding particles in MnCl2, 10 collision measurements were done.
As can be seen in Figure4.8, the collision measurements (the striped lines) were only 15 seconds long. This lead to the fact that the time delay could only be measured for the PVC-U particles, as the POM particles were not at an equilibrium position yet.
To get more reliable results from the time delay measurements, longer and more collision
meas-CHAPTER 4. RESULTS
Figure 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).
urements will have to be done in future research. This will probably also give a clearer view on the relation between the LD and the time delay of both particles.
Chapter 5
Conclusions
In this chapter, the conclusions of the measurements will be presented, divided into the conclusions of the measurements in water and manganese (II) chloride.
5.1 Measurements in water
During the experiment, three different kinds of measurements were performed in water:
• Single measurements of 5 mm polystyrene particles.
• Single measurements of 5 mm santoprene particles.
• Collision measurements of polystyrene and santoprene particles.
• Single measurements of ellipsoid 5-2.5 mm PVC-U particles.
After the single measurements were performed, the results were compared to a one-way coupled numerical model, as explained in section2.1. This model included, next to the gravitational buoy-ancy force, the added mass force and the Basset history force.
The single measurements turned out to be consistent with the numerical model, where the model with the Basset history force included turned out to give the best predictions. There were how-ever some complications regarding the relatively large standard deviation. For further research, more measurements (there were only 3 measurements done) should be performed, to increase the reliability of the measurement results.
Also, after calculating the mass density of the particles in reverse, the numerical model with the Basset history force included turned out to give more plausible results. In future research, this model should be used (or further developed) to compare simulations to experiments.
After the collision measurements, a linear fit was made between the lateral center distance (LD) and the time delay caused by the interaction between the two particles. The more head-on the particles collided, the higher the time delay was. Furthermore, there turned out to be a dependence between the mass density of the particle and the time delay. For 25 out of 31 measurements, the particle with a mass density closer to the water had a smaller time delay than the other particle.
This was confirmed by the linear fit, which was lower for the santropene.
Finally, the movement of ellipsoid PVC-U particles was investigated. It turned out to be hard to track the ellipsoid particles, as the particles rotated, leading to errors in the Matlab-scripts that were used. These errors were caused by the constantly changing 2D-image of the ellipsoid on the camera frames.
For further research, a new or modified Matlab-script will have to be made, which takes the changing shape and rotation angle of the ellipsoid into account.
5.2 Measurements in manganese (II) chloride
After the measurements in water, manganese (II) chloride was used to track the movement of plastic particles in a ‘magnetic fluid’. The following measurements were performed:
• Single measurements of rising and sinking 5 mm PVC-U particles.
• Single measurements of rising 5 mm POM particles.
• Collision measurements of PVC-U and POM particles.
Just as the single measurements in water, the single measurements in MnCl2 were compared to a one-way coupled numerical model. The results of the PVC-U particles showed that, most of the time, the results were in accordance with the model. There were some exceptions, which could be caused by the measuring technique of the mass density of the particle, as well as the number of measurements.
Regarding the measurements of the POM particles, the results were completely in accordance with the numerical model. However, the Basset history force was not included, as the model would not converge with mass densities of the particle so close to that of the fluid. To exactly compare the measurement results to the theoretic values of the height and velocity of the POM particle, this model will need to be modified to include the Basset history force as well.
The collision measurements with PVC-U and POM particles showed that, just as during the water measurements, there exists a dependence between the LD and the time delay caused by the interaction between the two particles. However, the dependence was not as clear as the dependence observed during the water measurements. This could be caused by the number of measurements, as well as the length of them.
Bibliography
[1] S. T. H. Kuerten, “Autonomous system analysis of the oscillatory motion of particles in magnetic density separator,” 2019. vii, vii,viii,viii,viii,1,20,27,28
[2] M. Jenny, J. Dusek, and G. BOUCHET, “Instabilities and transition of a sphere falling or ascending freely in a newtonian fluid,” Journal of Fluid Mechanics, vol. 508, pp. 201 – 239, 06 2004. vii,6
[3] Brilliant.org, “Magnetic field lines.” website. vii,7
[4] N. L. Weiss, SME Mineral Processing Handbook, Vol. 2. Society of Mining Engineers of the American Institute of Mining, Metallurgical, and Petroleum Engineers, 1985. vii, 7
[5] J. Kosse, “Design considerations for superconducting magnetic density separation,” Master’s thesis, University of Twente, 2016. vii,8
[6] A. Meulenbroek, “Particle settling dynamics in magnetic fluids for a magnetic density sep-aration application,” Master’s thesis, Eindhoven University of Technology, 2019. vii, vii,vii, 10,12,13,14,15
[7] M. Namdeo, S. Saxena, R. Tankhiwale, M. Bajpai, Y. Mohan, and S. Bajpai, “Magnetic nanoparticles for drug delivery applications,” Journal of nanoscience and nanotechnology, vol. 8, pp. 3247–71, 08 2008. vii,11
[8] I. Newton, I. B. Cohen, and A. Whitman, The mathematical principles of natural philosophy.
University of California press, 1999. 3
[9] G. Galilei, I. Drabkin, and S. Drake, Galileo Galilei On Motion and On Mechanics, Compris-ing De Motu (ca. 1590) Transl. with Introd. and Notes by I.E. Drabkin, and Le Meccaniche (ca. 1600) Transl. with Introd. and Notes by Stillman Drake. Publications in medieval science / the University of Wisconsin, 1960. 3
[10] C. Brennen, “A review of added mass and fluid inertial forces,” 01 1982. 4
[11] C. T. Crowe, J. D. Schwarzkopf, M. Sommerfeld, and Y. Tsuji, Multiphase Flows with Droplets and Particles. CRC Press, aug 2011. 4,5
[12] M. R. Maxey, “Equation of motion for a small rigid sphere in a nonuniform flow,” Physics of Fluids, vol. 26, no. 4, p. 883, 1983. 5
[13] P. H. Mortensen, H. I. Andersson, J. J. J. Gillissen, and B. J. Boersma, “Dynamics of prolate ellipsoidal particles in a turbulent channel flow,” Physics of Fluids, vol. 20, p. 093302, sep 2008. 5
[14] H. Lehle, E. Noruzifar, and M. Oettel, “Ellipsoidal particles at fluid interfaces,” The European Physical Journal E, vol. 26, pp. 151–160, may 2008. 5
[15] Keithley, The Story of Electrical and Magnetical Measurements: From 500 B.C. to the 1940s.
John Wiley and Sons, 1998. 6
[16] J. Masi, “Notes for electromagnetic fields and waves,” 1988. 9 [17] J. Masi, “Notes for partial differential equations,” 1975. 9
[18] J. M. D. T. C. D. Coey, Magnetism and Magnetic Materials. Cambridge University Press, 2010. 11
[19] R. E. ROSENSWEIG, “Fluidmagnetic buoyancy.,” AIAA Journal, vol. 4, pp. 1751–1758, oct 1966. 12
Appendix A
Nomenclature
Symbol Description Dimension Value
η Buoyancy acceleration constant [-]
-λ Wavelength [m]
-µ Dynamic viscosity [mskg]
-µ0 Magnetic constant [Hm]
-µr Relative permeability [-]
-ν Kinematic viscosity [ms2]
-π Ratio of a circle’s circumference to its diameter 3.141592...
ρef f Effective mass density [mkg3]
-ρf Fluid mass density [mkg3]
-ρp Particle mass density [mkg3]
-τ Local shear stress [-]
-χ Magnetic susceptibility [-]
-Table A.1: Nomenclature of the Greek symbols.
Symbol Description Dimension Value
A Surface of a sphere [m2]
-B~ Magnetic flux density [T]
-Br Remanent magnetic flux density [T]
-CD Drag coefficient [-]
-FBasset,i Basset history force [N]
-FD Drag force [N]
-Fgb Gravitational buoyancy force [N]
-g Gravitational acceleration [ms2]
-G Galileo number [-]
-P (~r) Calibration polynomial [-]
-t Time [s]
-t0 Dummy variable integration [-]
-T Kinetic energy [J]
-vf,i Fluid velocity in every point surrounding the particle [ms]
-vp Particle velocity [ms]
-x1 x-position sphere 1 [m]
-x2 x-position sphere 2 [m]
-xp x-position of the particle [m]
-y1 y-position sphere 1 [m]
-y2 y-position sphere 2 [m]
-yp y-position of the particle [m]
-Table A.2: Nomenclature of the Latin symbols.
Appendix B
Measurement graphs
In this Appendix, the graphs of all measurements are shown.
B.1 Water Measurements
(a) (b)
(c) (d)
Figure B.1: Graphs of the collision measurements of 5 mm polystyrene and santropene particles in water.
(a) (b)
(c) (d)
(e) (f)
Figure B.2: Graphs of the collision measurements of 5 mm polystyrene and santropene particles in water.
APPENDIX B. MEASUREMENT GRAPHS
APPENDIX B. MEASUREMENT GRAPHS