PIV EXPERIMENTS ON THE FLOW                              INDUCED BY A SPHERE                         SEDIMENTING TOWARDS A...
AbstractThe motion induced by gravity of solid spheres in a vessel lled with uid has beeninvestigated experimentally at Re...
List of symbolsRoman symbolsSymbol        Quantity                                                Dimensiona              ...
n               unit vector                                           ;m               sphere mass                        ...
Greek symbolsSymbol      Quantity                                                              Dimension            angle ...
vi
ContentsAbstract                                                                                                          ...
3.5 Image acquisition . . . . . . . . . . . . . . . .       .   .   .   .   .   .   .   .   .   .   .   .   .   16  3.6 An...
6.2.5 Sphere . . . . . . . . .     .   .    .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   . ...
Chapter 1Introduction1.1 General backgroundIn many industrial processes, two-phase ows play an important role. A commonkin...
2                                                              Chapter 1. Introduction    Figure 1.1:   Simpli cations mad...
1.4. Outline of this thesis                                                          3The second objective is to look into...
Chapter 2Motion of a single particle in a uid2.1 IntroductionConsider a spherical particle hanging at rest in a rectangula...
6                                   Chapter 2. Motion of a single particle in a uidequation of motion for the particle wil...
2.4. Steady particle/ uid motion                                                       7 eld forces which act on the parti...
8                                   Chapter 2. Motion of a single particle in a uid2.5 Unsteady particle/ uid motionWhen u...
2.6. Simpli ed equation of motion                                                    9called the added-mass term, the thir...
10                                   Chapter 2. Motion of a single particle in a uidThe value for t can be experimentally ...
2.8. Wall impact                                                                   11     if the ow is Stokes ow, asymmetr...
12                                  Chapter 2. Motion of a single particle in a uidand Hunt, 1999]. The model predicts tha...
Chapter 3PIV TheoryOne of the objectives of this research is to measure the ow eld around a spherefalling in a liquid and ...
14                                                           Chapter 3. PIV Theory              : Schematic representation...
3.4. Laser sheet                                                                     15      The particle diameter must be...
16                                                              Chapter 3. PIV Theory                    (a)              ...
3.6. Analysis of images                                                                     17   Figure 3.3   : Analysis o...
18                                                          Chapter 3. PIV Theory              : RMS measurement error as ...
3.7. Spurious vectors                                                              19Monte-Carlo simulations and the solid...
20                                                              Chapter 3. PIV Theory3.8 Estimation of vorticityAfter velo...
Chapter 4Experimental set-up for spheretrajectory measurementsIn this chapter, rst a global description is given of the me...
22             Chapter 4. Experimental set-up for sphere trajectory measurementsis placed between the spotlight and the ve...
4.2. Detailed description of some components                                               23                             ...
24                Chapter 4. Experimental set-up for sphere trajectory measurements                         (a)           ...
4.3. Image processing                                                              25used for the highest pixel value. In ...
: Example of a picture of a ruler used for calibration of trajectory mea-Figure 4.4             surements.
Chapter 5Results and discussion oftrajectory measurementsIn this chapter, results of sphere trajectory measurements will b...
28                  Chapter 5. Results and discussion of trajectory measurements              Table 5.1: Accuracies of mea...
5.1. Accuracy and reproducibility                                                             29                14        ...
30                           Chapter 5. Results and discussion of trajectory measurements                 Table 5.2 : Acce...
5.2. Trajectories: Accelerating motion to steady-state                                                     31             ...
32                      Chapter 5. Results and discussion of trajectory measurements                         : Order of ma...
5.3. Trajectories: Wall impact                                                               33                  5        ...
34                       Chapter 5. Results and discussion of trajectory measurements               0.14                  ...
5.4. Critical Stokes number measurements                                                        35                  1     ...
36                         Chapter 5. Results and discussion of trajectory measurements           0.00014           0.0001...
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL
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PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL

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The motion induced by gravity of solid spheres in a vessel filled with fluid has been investigated experimentally at Reynolds numbers in the range from 1-74 and Stokes numbers ranging from 0.2-17. Trajectories of the spheres have been measured with a focus on start-up behavior, and on impact with a horizontal wall. Two models have been investigated. The first describes the accelerating motion of the sphere. The second model predicts the distance from the wall at which the sphere starts decelerating.

The flow in the vicinity of the sphere was measured by means of PIV. The time scales and flow structures strongly depend on the Reynolds number. Measurements performed are in good agreement with simulations performed at the Kramers Laboratorium.

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PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL

  1. 1. PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALL C.H. Nieuwstad Masters ThesisDelft University of Technology Delft, January 2001Faculty of Applied SciencesDepartment of Applied PhysicsKramers Laboratorium voorFysische TechnologiePrins Bernhardlaan 62628 BW DelftProfessor: prof. dr. ir. H.E.A. van den AkkerSupervisors: ir. A. ten Cate dr. ir. J.J. Derksen
  2. 2. AbstractThe motion induced by gravity of solid spheres in a vessel lled with uid has beeninvestigated experimentally at Reynolds numbers in the range from 1-74 and Stokesnumbers ranging from 0.2-17. Trajectories of the spheres have been measured witha focus on start-up behavior, and on impact with a horizontal wall. Two modelshave been investigated. The rst describes the accelerating motion of the sphere.The second model predicts the distance from the wall at which the sphere startsdecelerating.The ow in the vicinity of the sphere was measured by means of PIV. The timescales and ow structures strongly depend on the Reynolds number. Measurementsperformed are in good agreement with simulations performed at the Kramers Lab-oratorium. i
  3. 3. List of symbolsRoman symbolsSymbol Quantity Dimensiona t parameter ;Ap particle surface area m2AT element of a matrix ;Ac acceleration modulus ;b t parameter ;BT element of a matrix ;C0 proportionality constant ;CA added-mass coe cient ;CD drag coe cient ;CH history coe cient ;CT element of a matrix ;ds seeding particle diameter md diameter of particle image pix:D particle diameter mDapp lens aperture mDT element of a matrix ;f focal length of camera lens mf1 focal length of cylindrical lens mf2 focal length of convex lens mfc image acquiring frequency HzFvisc viscous force of resistance acting on sphere NFarchimedes Archimedes force NFgravity gravity force NFD , Fdrag drag force NFD Basset Basset force Ng gravitational acceleration m=s2h distance between bottom apex sphere and bottom vessel mh non-dimensional distance ;hc critical distance m(i j ) indices of interrogation area ;Ii j discrete image intensity ;I mean image intensity ; iii
  4. 4. n unit vector ;m sphere mass ; m0 n0 ] integer displacement of the displacement-corr. peak (pix: pix:)M image magni cation ;MD displacement modulus ;ni refractive index at point of interest ;nr refractive index at reference point ;nr 1 refractive index of medium ;nr 2 refractive index of medium ;N image diameter in pixels pix:p pressure PaP laser power Wqi complex beam parameter at point of interest 1=mqr complex beam parameter at reference point 1=m ^RII r s] estimator for cross-correlation between two IAs ;(rD sD ) in-plane particle image displacement (pix: pix:)R;1 R0 R1 Heights of highest peak (0) and neighbours (+1,-1) ;Rep instantaneous Reynolds number of the particle ;Ri radius of wave front at point of interest mRr radius of wave front at reference point mRes Reynolds number based on vs ;Stc critical Stokes number ;Sts Stokes number based on vs ;t time st0 release moment of the particle stdiff di usion time scale stadv advection time scale stillumination illumination time sT non-dimensional time ;Ug gravitationally induced velocity m=sv0 initial particle velocity m=svf uid velocity m=svr relative velocity m=svp instantaneous particle velocity m=svs calculated particle velocity m=svs+ maximum measured particle velocity m=sv non-dimensional particle velocity ;Vp particle volume m3V measured photo multiplier voltage VVmax maximum measured photo multiplier voltage V(x y) coordinate of eld of view (m m)X non-dimensional distance travelled by particle ;z distance from laser exit mz0 image distance mZ0 object distance m iv
  5. 5. Greek symbolsSymbol Quantity Dimension angle rad Z focal depth m diameter of a pixel pix: t time between two consecutive images acquired s x distance between top left of light sheet and right side of particle m s distance between origin of cross-corr. plane and cross-corr. peak pix: z distance between light sheet and centre of mass of particle m( X Y) distance between mesh points of PIV data (m m)( xv yv ) vortex position relative to sphere centre (m m) d beam divergence at laser exit rad( m n) fractional displacement (pix: pix:) G Gaussian fractional displacement estimator pix: particle/ uid density ratio ; correction factor ; wavelength of light m f uid viscosity Pa s!0 minimal beam spot size m!i beam radius at point of interest m!r beam radius at reference point m(!Z )i j out-of-plane component of the vorticity 1=s f uid density kg=m3 p particle density kg=m3 s seeding particle density kg=m3 shear stress Pa p particle relaxation time s particle diameter to vessel width ratio ; distance travelled by particle mMiscellaneousSymbol QuantityCCD Charge Coupled DeviceFOV Field of ViewIA Interrogation AreaPIV Particle Image VelocimetryT Transpose v
  6. 6. vi
  7. 7. ContentsAbstract iList of symbols iiiContents vii1 Introduction 1 1.1 General background . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Micro-hydrodynamics in crystallisation . . . . . . . . . . . . . . . . . 1 1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Motion of a single particle in a uid 5 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Equation of motion of the uid . . . . . . . . . . . . . . . . . . . . . 6 2.3 Equation of motion of the particle . . . . . . . . . . . . . . . . . . . 6 2.4 Steady particle/ uid motion . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 Unsteady particle/ uid motion . . . . . . . . . . . . . . . . . . . . . 8 2.6 Simpli ed equation of motion . . . . . . . . . . . . . . . . . . . . . . 9 2.7 The wall e ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.8 Wall impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 PIV Theory 13 3.1 Principle of Particle Image Velocimetry . . . . . . . . . . . . . . . . 13 3.2 Optical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 Seeding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.4 Laser sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 vii
  8. 8. 3.5 Image acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.6 Analysis of images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.6.1 Analysis of displacement . . . . . . . . . . . . . . . . . . . . . 17 3.6.2 Calculation of the cross-correlation . . . . . . . . . . . . . . . 17 3.6.3 Estimation of fractional displacement . . . . . . . . . . . . . 18 3.6.4 Optimization of correlation . . . . . . . . . . . . . . . . . . . 19 3.7 Spurious vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.8 Estimation of vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Experimental set-up for sphere trajectory measurements 21 4.1 Measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Detailed description of some components . . . . . . . . . . . . . . . . 22 4.2.1 Rectangular vessel . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2.2 Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2.3 Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2.4 Release mechanism . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2.5 Digital camera . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.3 Image processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.3.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Results and discussion of trajectory measurements 27 5.1 Accuracy and reproducibility . . . . . . . . . . . . . . . . . . . . . . 27 5.2 Trajectories: Accelerating motion to steady-state . . . . . . . . . . . 30 5.3 Trajectories: Wall impact . . . . . . . . . . . . . . . . . . . . . . . . 32 5.4 Critical Stokes number measurements . . . . . . . . . . . . . . . . . 35 5.5 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 PIV experimental set-up and measurement method 39 6.1 Measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.2 Detailed description of some components . . . . . . . . . . . . . . . . 41 6.2.1 Rectangular vessel . . . . . . . . . . . . . . . . . . . . . . . . 41 6.2.2 Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.2.3 Seeding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.2.4 Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 viii
  9. 9. 6.2.5 Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.2.6 Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.2.7 Sheet formation optics . . . . . . . . . . . . . . . . . . . . . . 43 6.3 Image processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.3.1 Sphere position . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.3.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.4 PIV processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 Results and discussion of PIV measurements 47 7.1 De nition of Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 7.3.1 Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 7.3.2 Flow pattern of a sphere . . . . . . . . . . . . . . . . . . . . . 56 7.3.3 Flow eld shortly before wall impact . . . . . . . . . . . . . . 56 7.3.4 Comparison of the ow eld in the middle and bottom of the vessel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 7.3.5 Development of the ow eld of the accelerating sphere . . . 58 7.3.6 Vorticity at a height of 0.045 D above the wall . . . . . . . . 60 7.3.7 Time series of uid velocity in a monitor point . . . . . . . . 60 7.3.8 Time series of vortex position . . . . . . . . . . . . . . . . . . 62 7.4 Comparison with simulations performed at this laboratory . . . . . . 638 Conclusions and recommendations 69 8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 8.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70A A solution for the equation of motion of the sphere 71B ABCD-law for Gaussian beams 75Bibliography 79 ix
  10. 10. Chapter 1Introduction1.1 General backgroundIn many industrial processes, two-phase ows play an important role. A commonkind of a two-phase ow is a particulate ow. This is a ow in which the dispersedphase is a solid in the form of particles and the continuous phase is a gas or a liquid.Such ows can be found in the elds of mechanical, petroleum, mining, chemical andnuclear engineering. Other examples of such ows can be found in natural processeslike snow avalanches, erosion, sand storms, dune formation, etc.Despite the prominence of this type of ows, its behaviour is not well understood.A large part of the industrial applications are designed empirically. Due to theirprevalence, even small improvements in the performance of these technologies couldhave an important economic impact.1.2 Micro-hydrodynamics in crystallisationThis research contributes to a larger project to improve the microscopic modelling ofhydrodynamics in industrial crystallisers. In these crystallisers, particulate ows arevery important, because the slurry in the crystalliser consists of solid particles, thecrystals, dispersed in a liquid. To get a better insight in the process hydrodynamics, ow simulations of the crystalliser as a whole using the lattice-Boltzmann methodhave been performed, for more details see Ten Cate et al., 2000]. These simulationsonly give information about the uid ow on length scales much larger than thecrystal size. However, as an important parameter in crystallisation is the desiredcrystal size, which is in uenced by the rate at which crystal-crystal collisions occur,it is also important to obtain information about the processes that occur on thesmall length scales. Microscopic modelling deals with the modelling of these smallscale phenomena in the crystalliser.As the motion of single particles through a crystalliser is too complex to simulatenumerically, a simpli ed approach is used to get insight in the hydrodynamics ofsingle crystals. Approximately 100 spheres are put in a box in which turbulent ow 1
  11. 11. 2 Chapter 1. Introduction Figure 1.1: Simpli cations made for validating numerical simulations of a crys- talliser.is simulated. These spheres have Reynolds numbers ranging from 10 to 60. The ow conditions in the box are comparable to the ow conditions in the crystalliser.More information about those simulations can be found in Ten Cate et al., 2000].However, the question arises if the lattice-Boltzmann method is capable of accuratelysimulating the movement of spheres in a liquid. To check this, a model system isde ned: a simulation of a single settling sphere in a box, again at Reynolds numbersranging from 10 to 60 is done. Note that this is the most simple case of particulate ow. This simulation of a single sphere settling under gravity and colliding ontothe bottom wall of a box is considered a good, rst test case for the many-particlebox simulations, see Ten Cate et al., 2000]. This simple case has to be veri ed withexperiments. As a summary, all simpli cations made are shown in the left part of gure 1.1.1.3 ObjectivesThe main objective of this research is to set up a series of experiments to obtain datathat can be used for validating the single sphere simulation mentioned above. To dothis, experimental set-ups are built with a box and a sphere having sizes that makeit possible to use exactly the same geometry for simulations. This makes it possibleto compare simulations and experiments directly, on a 1-on-1 scale, as illustrated in gure 1.1. For validation, measurements have been performed at Reynolds numbersranging from 1 to 100, of two parameters: The trajectory of the sphere. The ow eld in the uid induced by the motion of the sphere. This ow eld is measured using Particle Image Velocimetry (PIV).It is desirable to use a particle and a uid such that the particle to uid densityratio is comparable to 1.4 (a typical value for crystals in crystallisers).
  12. 12. 1.4. Outline of this thesis 3The second objective is to look into two models for the motion of the sphere. The rst model describes the accelerating motion of the particle, the second predicts thedistance from the bottom of the vessel at which it starts to decelerate before it hitsthe wall.1.4 Outline of this thesisIn chapter 2 of this report the motion of a single particle in a uid will be discussed.Theory about Particle Image Velocimetry needed for performing the experiments isgiven in chapter 3. In chapter 4 the experimental set-up used for trajectory measure-ments is presented. Results of trajectory measurements are discussed in chapter 5.In chapter 6 the used PIV measurement set-up is explained. The results of PIVmeasurements performed are given in chapter 7. Finally, in chapter 8 conclusionsand recommendations concerning the research are given.
  13. 13. Chapter 2Motion of a single particle in a uid2.1 IntroductionConsider a spherical particle hanging at rest in a rectangular vessel lled with liquid,see gure 2.1. If the density of the particle is larger than the density of the liquid,the particle starts falling down when it is released. Forces acting on the particle arethe gravity force, Archimedes force and the drag force. These forces are illustratedin gure 2.1. The trajectory the particle travels can be divided into several parts.First, the forces acting on the particle will make it accelerate until a maximumvelocity is reached. After this, the particle will (quickly) decelerate and come to reston the bottom of the vessel, or it will bounce back from the bottom if its energy islarge enough.In the next sections, a more detailed treatment of the foregoing phenomena willbe given. First, the equation of motion of the uid is given. Second, the equationof motion of the particle in an in nite medium will be given. Third, steady stateparticle motion and unsteady particle motion will be discussed. Fourth, a simpli ed Figure 2.1 : Particle in a rectangular vessel lled with liquid. 5
  14. 14. 6 Chapter 2. Motion of a single particle in a uidequation of motion for the particle will be presented. Fifth, in uences from theside-walls of the box will be discussed. Finally, criteria will be given, for bouncingback from the bottom wall of the vessel.2.2 Equation of motion of the uidThe liquid in the vessel is assumed to be Newtonian. The reason for this, is that theequations derived in the next paragraphs apply to Newtonian liquids. During thisresearch project, glycerol-water-solutions and silicone oil are used. These liquids areNewtonian uids. Newtonian liquids follow the law of Newton, which states: = f (rvf + rvT ) f (2.1)where is the shear stress, f is the dynamic viscosity of the uid and vf is thevelocity of the uid. According to this equation, the shear stress is proportional tothe strain rate 1 (rv f + rvT ). Further, it is assumed, that the uid is homogeneous 2 fand incompressible. As a consequence the density of the uid is constant. The set ofequations, that describe the ow of the uid, are the continuity and Navier-Stokesequations, and in this case they simplify to: r vf = 0 (2.2)and @vf @t + (v f r)vf = ; 1 rp + f r2 v f + g (2.3) f fwhere f is the density of the uid, p is the pressure and g is the gravitationalacceleration.2.3 Equation of motion of the particleIn this project, as well as in most other studies on related subjects, the geometricaldimensions and the density of the particle are supposed to remain constant. Theparticles are rigid spheres and have a smooth surface. The equation of motion forthe particle can be obtained from a momentum balance following Newtons secondlaw: d(vp) Z Z Z Z Z Z Z p Vp dt = ; pndAp + ndAp + g( p ; f )dVp (2.4) A p Ap Vpwhere Vp is the sphere volume, Ap is the sphere surface, vp is the sphere velocity, p is the sphere density and n is the unit vector perpendicular to the surface of thesphere. The rst term on the right-hand side of this equation is the local pressure ofthe uid. The second term is the force of friction, which acts on the surface and isdue to viscosity. Often these terms are combined to give one term, which is referredto as the drag force. In principle, the volume integral term comprises all external
  15. 15. 2.4. Steady particle/ uid motion 7 eld forces which act on the particle. In general, however, the external force eld isgiven by the gravitational eld only. Taking into account all assumptions mentionedabove, this equation can be simpli ed to: dv Vp p dtp = Vp( p ; f )g + F D (2.5)where F D is the drag force exerted on the particle. This drag force takes into accountboth the pressure and the shear e ect.2.4 Steady particle/ uid motionThe motion of the particle is called steady-state if the velocities of both the particleand the uid are constant. If the particle velocity is di erent from the uid velocity,the force exerted by the uid on the particle depends on the relative velocity (orslip-velocity), vr = v f ; vp . As mentioned in section 2.3, this force is called thedrag force. Unfortunately, an expression for the drag force can not be derivedanalytically, even for a simple shape such as a sphere, except for extremely lowReynolds numbers. This is due to the fact, that exact solutions for the ow eldare not available, except for the creeping ow region. Therefore, it proves usefulto introduce an experimentally determined drag coe cient CD , valid for a single,smooth, non-rotating sphere in a steady, incompressible and unbounded ow eld,in the expression of the drag-force. This coe cient is de ned as the ratio of thedrag force over the dynamic head of the ow and the frontal area of the particle.For spherical particles, the de nition is: 1 1 FD = CD 4 D2 2 f (vf ; vp )2 (2.6)where D is the particle diameter. Since the drag may be positive or negative, de-pending on whether the uid velocity is greater or smaller than the particle velocity,the square of the relative velocity is usually replaced by (vf ; vp)jvf ; vp j. The forceis then always obtained with the correct sign. The drag coe cient is a function ofthe particle Reynolds number, Rep, which is de ned as: Rep = f Djvf ; vpj (2.7) fResults from measurements covering a wide range of experimental conditions can befound in the literature. From these measurements correlations for CD as a functionof Reynolds number have been determined by tting equations to measured data.They can be found in most texts on uid mechanics, for example Janssen andWarmoeskerken, 1991]. The correlation for the drag coe cient used in this researchis the expression which has been proposed by Abraham, 1970]: !2 CD = 9:24 2 9:062 + 1 06 Re1= (2.8) pwhich is in good agreement with experiments (it has an average relative deviationof 7.6% and maximum relative deviations of +13 and -14%) for 0 Rep 5000 Abraham, 1970].
  16. 16. 8 Chapter 2. Motion of a single particle in a uid2.5 Unsteady particle/ uid motionWhen unsteady situations are considered, the problem grows more complex. Adimensional analysis, described by Clift, 1978], gives insight into which parametersmay in uence the interaction between the uid and the particle. In general, the(overall) drag coe cient of particles in unsteady motion is considered to depend onthe following non-dimensional groups. First, the the instantaneous Reynolds numbershould be mentioned, de ned in equation (2.7). Second, the displacement modulusMD , de ned as the ratio of the displacement of the particle since it started to moveand particle diameter, is an in uencing parameter. The overall drag coe cientfurthermore depends on the solid/ uid density ratio , de ned as the ratio of particledensity and uid density. Last but not least, the acceleration modulus, Ac, is usedas a non-dimensional parameter. In general, this acceleration modulus is de ned by: (f p2 Ac = dvv ; vv) )=D (2.9) ( f ; p =dtThis number is a measure of the ability of the ow eld around the particle toadjust to changing relative velocity. It stands for the ratio between forces due to 2convective acceleration ( f vp @vp =@x f vp =D) and forces due to local acceleration( f @vp =@t f dvp =dt). If the forces due to convective acceleration are small, com-pared to the forces due to local acceleration, the acceleration modulus will be low.Below a certain value of Ac the contribution of the convective acceleration may bedisregarded. For the situation of a particle moving in a liquid, the density ratio isrelatively low, i.e. up to a value of about 10. In that case, the unsteady motion canbe characterised by a rapid change of Reynolds number with distance. The principalconsideration in unsteady motion of this kind is the development of the ow patternand associated entrainment of the uid. Since the instantaneous ow pattern mayhave little resemblance with the fully-developed ow, the instantaneous drag maydi er radically from and be even almost unrelated to the steady motion drag.For the drag in creeping ow, the convective acceleration term in the Navier Stokesequation is negligible. Although this results in a considerable simpli cation of theproblem, solution is still complicated and requires a thorough mathematical analy-sis. For details, see Talman, 1994] or Althuisius, 1997].Equation of Odar and HamiltonFor the drag at Reynolds numbers higher than that of creeping ow, no theoret-ical expression is available. Odar and Hamilton, 1964] suggested an expression forthe drag force. By making use of empirical coe cients they proposed the followingequation: Zt 1 D2 1 jv jv + C 1 D3 dvr + C 1 D2 p FD = CD 4 2 f r r A 6 dvr =d d p f dt H4 f f 0 t; (2.10)where CD is the steady drag coe cient, CA is the (empirical) added-mass coe cientand CH is the (empirical) history coe cient. The second term on the right is usually
  17. 17. 2.6. Simpli ed equation of motion 9called the added-mass term, the third the Basset force or history term. Based onmeasurements with a sphere oscillating in a viscous liquid, Odar, 1966] gave thefollowing empirical formulas for these coe cients: CA = 1:05 ; Ac0:+ 0:12 2 066 (2.11) CH = 2:88 + (Ac :+ 1)3 3 12 (2.12)Note that is not included in these expressions. While these expressions were de-rived from experiments with spheres in harmonic motion, Odar, 1966] found, againexperimentally, that the above expressions were equally applicable to the situationof a sphere falling in a liquid. However, the opposite has been demonstrated in laterexperiments by other researchers, who have found several empirical relationshipsfor di erent experimental conditions, see Talman, 1994]. From this the conclusionmust be drawn that, unfortunately, the correlations for CA and CH are not univer-sally valid, so that each problem requires its own expression, or perhaps di erentparameters are needed to describe the problem correctly.2.6 Simpli ed equation of motionThe problem with equation (2.10) is that the integral makes it di cult to get ananalytical solution for the equation of motion. To get an analytical solution, Ferreiraand Chhabra, 1998] entirely neglect the history term and consequently put CH =0. They also put CA = 1=2, which is the value for the added-mass coe cientfor irrotational ow. According to Ferreira, this approximation is consistent withprevious studies in this eld. The equation of motion becomes: m dvp = mg 1 ; f ; 1 D2 CD f vp ; 12 D3 f dvp dt 8 2 1 dt (2.13) pThis equation can be solved analytically. The result can be found in Appendix A.The signi cance of the Basset-force diminishes with increasing density of the spherein comparison with the uid, i.e. p f . An order of magnitude estimation forthis force can be made from equation (2.10) by assuming dv constant: dt r FD Basset CH 41 D2 p dvr Z t p d (2.14) f fd 0 t;Now an estimation for CH has to be made. For unsteady, rectilinear movement ofa sphere at low Reynolds numbers, CH = 6. Odar and Hamilton, 1964] determinedCH for a sphere in oscillatory motion at high Reynolds numbers. They found, thatCH = 6 is a maximum. Dohmen, 1968] found values of CH around 1 for a fallingsphere. Here a value of CH = 1 is used. After integration, equation (2.14) becomes: p FD Basset 4 D2 p f f dvr 2 t 1 d (2.15)
  18. 18. 10 Chapter 2. Motion of a single particle in a uidThe value for t can be experimentally estimated as the time it takes for the sphereto reach steady state falling velocity vs . The factor dv can than be estimated by dt r s;calculating vt;tv0 = vt if the particle starts from rest at t = 0. s 02.7 The wall e ectThe theory for steady and unsteady motion of a sphere in a uid as discussed inthe foregoing sections, was derived for an in nite medium. Since in the experimentsthe uid is bounded by the walls of the vessel, it is necessary to investigate possiblewall e ects. In general it may be said, that if the diameter of the sphere becomesappreciable with respect to the diameter of the vessel in which it is settling, the wallsof the vessel will exert an additional retarding e ect, resulting in a lower steady stateparticle velocity than would be the case in an in nite medium. It is brought aboutby the upwards counter ux of uid which balances the downward ux of the solidand that of the entrained down ow Di Felice, 1996]. The smaller the area availablefor the counter ux, i.e. the smaller the the vessel cross sectional area comparedto the particle size, the more important the phenomenon is. The magnitude ofthis e ect also depends on the Reynolds number. In case of creeping ow the walle ects are larger than for the situation of intermediate or high Reynolds number ow around the sphere. For intermediate ow (Rep roughly 1-100) around a singlesettling sphere at steady state falling velocity, Di Felice, 1996], found the followingempirical correlation for a sphere falling in a cylindrical tube: vs+ 1; vs corr = 1 ; 0:33 (2.16) = 3:3 + 0::085Res 1 + 0 1Re (2.17) swhere vs+ is the terminal velocity corrected for wall e ects, vs is the terminal settlingvelocity in an in nite medium, is the particle to the tube diameter ratio and Resis the Reynolds number based on the terminal velocity in an in nite medium. Ithas to be noted however, that this correlation is not very accurate, especially forsmall values of , because reported experimental evidence for intermediate Reynoldsnumber ow are scarce. The equation mentioned above is derived for a tube, butit is also valid for a rectangular vessel, if for the particle diameter to vessel widthratio is used Althuisius, 1997].2.8 Wall impactSeveral things happen when a sphere approaches the bottom of a square vessel: there is a change in ow- eld, and consequently, a change of drag on the sphere. uid is pushed out of the gap between the sphere and the vessel. added mass of the sphere is changed, due to the approach of the wall.
  19. 19. 2.8. Wall impact 11 if the ow is Stokes ow, asymmetry of the ow is developed.When the sphere approaches the wall, it decelerates. This phenomenon has beeninvestigated by Gondret et al., 1999]. He studied rigid spheres approaching a wallmoving in a viscous uid. In this case its kinetic energy is dissipated by viscousforces as it approaches the wall. For Stokes ow, Brenner, 1961] has calculatedthe viscous force of resistance Fvisc acting on the sphere moving at the velocity vpinto a uid of viscosity f at a distance h between its bottom apex and the wall,Fvisc = 3 f Dvp , where = (h=D) is a correction to Stokes law given by anin nite series. There is no correction when the sphere is far away from the wall, so ! 1 when h=D ! 1. When the distance gets in nitely small, ! D=h whenh=D ! 0. The rate of approach is asymptotically slow and the sphere does notrebound. The approach of the wall of the sphere in Stokes ow has been testedexperimentally by Gondret et al., 1999] and the results are in good agreement withBrenners theory. This makes it possible to calculate the trajectory of the sphere,and to predict at which height hc the sphere starts feeling the wall, i.e. when itstarts decelerating. For larger Reynolds numbers, uid inertia needs to be consideredas well. A quick estimate of the magnitude of hc can be made by comparing thetime of di usion for the momentum on the length scale h, tdiff = h2 f = f and thetime for the particle advection on the same length scale, tadv = h=vs+ . A criticaldistance hc can be de ned as the height at which these two times are equal: hc = 1 (2.18) D Res+Where Res+ is the Reynolds number of the particle based on vs+. Hence the distancefrom the wall at which the sphere starts decelerating, is smaller when Res+ is larger.For the intermediate ow regime, the region between Stokes and turbulent ow,rebound may occur. This depends on the part of the particle kinetic energy thatwill be dissipated in the uid and by deformation of both the particle and the vesselwall. If the elastic deformation of the sphere and the vessel wall is large enough,rebound occurs. The relevant parameter for the bouncing transition is the Stokesnumber: St = 9 pDvs+ 1 (2.19) fThis dimensionless number characterises particle inertia relative to viscous forces.The critical Stokes number Stc , which de nes the non-bouncing to bouncing tran-sition, was found to be around 20 by experiments with a steel sphere falling in aliquid towards a glass wall Gondret et al., 1999].A correlation for Stc derived from experiments with di erent particles in water isreported by Zenit and Hunt, 1999]: ; Stc = eC 2 f (2.20) 0 pwhere C0 is a proportionality constant. Values of C0 used are 0.05, 0.03 and 0.02.Comparisons between this model and experiments show qualitative agreement Zenit
  20. 20. 12 Chapter 2. Motion of a single particle in a uidand Hunt, 1999]. The model predicts that when the density ratio increases, thecritical Stokes number decreases. Hence, as the particle is lighter, the viscous forcesbecome more dominant and therefore the critical Stokes number increases.
  21. 21. Chapter 3PIV TheoryOne of the objectives of this research is to measure the ow eld around a spherefalling in a liquid and colliding with a wall using Particle Image Velocimetry (PIV).In this chapter, theory needed for designing and operating a PIV experimental set-upis given. Also the data analysis procedure is explained.3.1 Principle of Particle Image VelocimetryPIV is a technique to measure the velocity eld in a uid ow. A plane is illuminatedwith a sheet of laser light. The ow is seeded with small tracer particles, thatscatter light when they are in the laser sheet. Pictures are taken from consecutiveilluminations. From these pictures, the instantaneous velocity distribution of the ow eld is reconstructed by measuring the distance travelled by groups of particlesbetween two consecutive illuminations. The pictures taken are divided in smallareas. Using an interrogation method, the distance travelled by the particles inthese areas can be determined. When this distance is divided by the time betweenthe two consecutive illuminations, the velocity of the uid is known. This procedurewill be explained in the following sections by discussing the following topics: Optical system Seeding Laser sheet Image acquisition Analysis of images Spurious vectorsIn these sections also some choices that are made with respect to the laser sheet andthe way of acquiring and processing images are explained. 13
  22. 22. 14 Chapter 3. PIV Theory : Schematic representation of a PIV imaging setup. Taken from West- Figure 3.1 erweel, 1993].3.2 Optical systemIt is important to obtain pictures with clear and sharp particle images. A schematicrepresentation of the light sheet and optical system for imaging tracer particles in aplanar cross section of a ow is shown in gure 3.1. It is assumed, that the systemconsists of an abberation-free, thin circular lens with a focal length f and a diameterDapp . The object distance Z0 and image distance z0 satisfy the geometrical lens law: 1 +1 =1 (3.1) Z0 z0 fThe image magni cation is given by: z M = Z0 (3.2) 0The particles in the object plane are illuminated with a thin light sheet from acoherent light source, for example a laser with wavelength . It is important thatall observed particles are in focus. This condition is satis ed if the thickness of thelight sheet is smaller than the the object focal depth Z of the imaging lens, as givenby Adrian, 1991]: 1 2 f2 Z =4 1+ M 2 Dapp (3.3)3.3 SeedingThe ow has to be seeded with particles, that have to obey four criteria: The particles must be neutrally buoyant. If they are not, they will be dis- placed with a gravitationally induced sedimentation velocity Ug = d2 ( 18; ) g s s f (calculated for very low Reynolds numbers Ra el et al., 1998]). f
  23. 23. 3.4. Laser sheet 15 The particle diameter must be small compared to the length scales that are to be investigated. To follow the uid motion correctly, the particle relaxation time p = d2 18 s s has to be small Ra el et al., 1998]. This implies that the particle diameter f must be small. The light scattered by the particles has to be as intense as possible. For that reason, the di erence between the refractive index of the particles and the uid has to be as large as possible. Because the intensity of the light scattered by particles between 1 and 100 m linearly increases with increasing particle volume, see Adrian, 1991], the size of the particles must not be too small.3.4 Laser sheetFor illumination of the particles, a laser sheet is used. A laser makes it possibleto get a high light intensity in a small volume, which ensures a detection of theparticles. Di erent ways of making the sheet are: Sweeping sheet Continuous sheet Pulsating sheetA large sweeping sheet can be made using a rotating polygon mirror.A small continuous sheet can be made using one or more lenses. The essentialelement for generating a continuous light sheet is a cylindrical lens. When using alaser with a su ciently small beam diameter and divergence, for example an Argon-ion laser, one cylindrical lens can be su cient to generate a light sheet of appropriateshape. However, often additional spherical lenses are used.For pulsating sheets, often NdYAG lasers are used. This type of lasers is able togenerate a pulsating sheet with high energy and frequency. However, it is alsopossible to make a pulsating sheet using a continuous (for example Argon-ion) laser.To do this, in front of the laser-exit a chopper has to be installed, which chopsthe laser beam periodically. A disadvantage of this method is the loss of e ectivelaser-power when the chopper is closed.For the experiments performed in this research, a small sheet is su cient. Evaluationof the data is done using multi-frame/single exposure PIV (see section 3.5). Forthese reasons, a continuous light sheet has been chosen. Another advantage of usinga continuous sheet is that it is possible to use all laser power available.
  24. 24. 16 Chapter 3. PIV Theory (a) (b) : PIV recording methods. (a) single frame/multi-exposure PIV. (b) multi- Figure 3.2 frame/single exposure PIV. Taken from Ra el et al., 1998]3.5 Image acquisitionWhen discussing image acquisition, two topics are important: the recording methodand the camera. A choice has to be made for a recording method. There are twodistinctive fundamentally di erent recording methods: single frame/multi-exposurePIV and multi-frame/single exposure PIV. A schematic picture of these two meth-ods is shown in gure 3.2. The principal distinction is that without additional e ortthe rst method does not retain information on the temporal order of the illumi-nation pulses, giving rise to a directional ambiguity in the recovered displacementvector. In contrast, multi-frame/single exposure PIV recording inherently preservesthe temporal order of the particle images and hence is the method of choice if afast camera is available. Also in terms of evaluation, this method is much easier tohandle. For these reasons, the choice is made to use multi-frame/single exposurePIV in this research.A camera is used for registration of the illuminated particles in the light sheet.Both a digital CCD camera and a photographic camera can be used. As analysisof the pictures using a computer requires digital pictures, a big advantage of usinga CCD camera is digital registration. Another advantage of CCD compared tophotographic cameras is the high frequency at which images can be taken. However,this frequency is inversely proportional to the spatial resolution of the camera (e.g.,when the number of pixels of the CCD array is increased by a factor 4, the maximumcapturing frequency is 4 times lower). Most conventional CCD cameras have lowresolution compared to photographic lm, for example 512x512 pixels comparedto 3000x4500 lines. A disadvantage of CCD cameras compared to photographiccameras is the amount of light needed. CCD cameras generally need more scatteredlight for imaging of tracer particles. As a consequence, more laser power is requiredwhen using a CCD camera. In general, a high recording frequency is needed whenusing the multi-frame/single exposure PIV recording method. For this reason, aCCD camera is used in this research.
  25. 25. 3.6. Analysis of images 17 Figure 3.3 : Analysis of double frame/single exposure recordings: the digital cross- correlation method. Taken from Ra el et al., 1998].3.6 Analysis of images3.6.1 Analysis of displacementTo analyse digital PIV images, an interrogation method is used. The images aredivided in small areas, the interrogation areas (IAs), as shown in gure 3.3. In thisresearch, the multi-frame/single exposure PIV recording method is used. In thiscase, analysis of images is performed by cross-correlation. The cross-correlation be-tween two interrogation windows out of two consecutive recordings is calculated. Thevector s between the origin of the cross-correlation plane and the cross-correlationpeak resulting from this calculation is a measure for the average displacement ofthe particles in the IA. As the time t between two consecutive frames and themagni cation factor of the images are known from the measurement, the velocityvf of the uid corresponding to this IA can easily be calculated: vf = M s t (3.4)3.6.2 Calculation of the cross-correlationConsider an image of N N pixels. Two coordinates i j determine the place of apixel with intensity Ii j . The spatial average of this region is de ned as: N N 1 XXI I = N2 (3.5) ij i=1 j =1This spatial average is used in calculating the image cross-correlation for two IAs,Ii0 j and Ii00j for a shift over r,s], see Westerweel, 1993]: N ;jrj N ;jsj X X 0 ^ 1 RII r s] = N 2 (Ii j ; I )(Ii00+r j +s ; I ) (3.6) i jwith: N ;jrj(P X N ;r for r 0 = PiN=1 (3.7) i i=1;r for r < 0
  26. 26. 18 Chapter 3. PIV Theory : RMS measurement error as a function of displacement. Figure 3.4 Taken from Westerweel et al, 1997].The image mean is calculated using equation 3.5 for the entire image, while the cross-correlation is estimated at each interrogation position from the pixels in the IA. Inpractise, this is done using discrete Fourier transformation. For more informationabout this method, see Westerweel, 1993].3.6.3 Estimation of fractional displacementThe particle image displacement is given by the location of the cross-correlation peakwith respect to the origin. Provided there are su cient matching particle image ^pairs, the cross-correlation peak is de ned as the maximum of RII r s], located atm0 n0 ]. The actual displacement (rD sD ) can be written as: rD = (m0 + m ) and sD = (n0 + n) (3.8)where m0 and n0 are de ned as the nearest integer displacements, and m and n asthe fractional displacements, which both have a value between -0.5 and 0.5 pixels. is the size of a pixel. The fractional displacement can be estimated using athree-point estimator. This is a function that estimates the position of a peak by tting a function (e.g. a Gaussian function or a parabola) to the cross-correlation ^function. It is assumed that RII r s] is circularly symmetric and separable in r ands and that m and n are statistically orthogonal: covf ^ ^ g = 0. This makes it m npossible to deal with the fractional displacement in the two directions as if they areindependent. The Gaussian peak t estimator gives the best result for the fractionaldisplacement Westerweel, 1993]. This estimator is given by: 1 ; ln ^G = 2(ln Rln R;2 ln R R+1 R ) (3.9) ;1 ; 0 + ln +1where ^G is the Gaussian fractional displacement estimator, R0 is the highest valueof the peak and R;1 and R+1 are the values adjacent to the highest value. Thedisplacement estimation error is a function of the displacement Westerweel, 1997].This can be seen in gure 3.4. In this gure the RMS measurement error for thedisplacement in pixel units for a 32 32 pixel IA in cross-correlation is given, forparticle images with a diameter d of 2 and 4 pixels. The dots are the results from
  27. 27. 3.7. Spurious vectors 19Monte-Carlo simulations and the solid lines are analytical results Westerweel, 1993].From this gure the measurement error is approximately constant if the displacementdue to the ow is larger than 0.5 pix, but for displacements smaller than 0.5 pix, theerror is linear with displacement. Therefore, the absolute measurement accuracy forparticle images with a diameter of 4 pix is approximately 0.1 pix for displacementslarger than 0.5 pix and the relative measurement accuracy is approximately 17% fordisplacements smaller than 0.5 pix.3.6.4 Optimization of correlationTo get optimal results from the PIV analysis, Keane and Adrian, 1990] carriedout Monte-Carlo simulations to determine the requirements for the experimentalparameters to yield optimal performance of the PIV analysis. They recommend thefollowing criteria: The number of particles per interrogation area should be at least 15 The particle-image displacement in the direction perpendicular to the light sheet ("out-of-plane" displacement) should be less than 1/4 of the thickness of the light sheet The in-plane displacement of the particle images should be about or less than 1/4 of the size of the interrogation area The velocity di erence over the interrogation area should be at most 5% of the mean velocity3.7 Spurious vectorsIn PIV one often nds that measurement results contain a number of "spurious"vectors. These vectors deviate un-physically in magnitude and direction from nearby"valid" vectors. In general these vectors originate from IAs containing insu cientparticles. In practise, the number of spurious vectors in a PIV data set is relativelylow (typically less tan 5% for data with good signal to noise ratio). However, theiroccurrence is more or less inevitable: even in carefully designed experiments thereis a probability that an interrogation yields a spurious vector.Spurious vectors can be detected using the local median test Westerweel, 1994]. Inthis test, the median of the length of all the neighbouring (max. 8) vectors is usedto determine the residual of every displacement vector. From a histogram of theseresiduals the cut-o residual is determined. All vectors having a larger residual thanthe cut-o residual are removed. After this removal, the resulting empty spaces canbe lled by vectors determined from the local mean displacement of the neighbouringvectors. In this research, Cleanvec Solof and Meinhart, 1999] is used for both theremovals and the replacements.
  28. 28. 20 Chapter 3. PIV Theory3.8 Estimation of vorticityAfter velocity vector elds have been obtained and spurious vectors have been re-placed, the out-of-plane component of the vorticity of the ow eld can be calculated.In this research, the vorticity estimate for point (i j ) is calculated using a circulationestimate around the neighbouring eight points Ra el et al., 1998]: (!Z )i j = 4 ;i j Y X (3.10) 1 ;i j = 2 X (vf x )i;1 j ;1 + 2(vf x )i j ;1 + (vf x )i+1 j ;1 ] 1 + 2 Y (vf y )i+1 j ;1 + 2(vf y )i+1 j + (vf y )i+1 j +1 ] (3.11) ; 1 X (vf x )i+1 j+1 + 2(vf x )i j+1 + (vf x)i;1 j+1] 2 ; 1 Y (vf y )i;1 j+1 + 2(vf y )i;1 j + (vf y )i;1 j;1] 2where X and Y are the distances between the mesh points in a PIV data set inthe x- and y-direction.
  29. 29. Chapter 4Experimental set-up for spheretrajectory measurementsIn this chapter, rst a global description is given of the measurement setup forsphere trajectories. Second, a more elaborate description of some parts of the setupis given. Third, image processing needed for extracting place-time graphs of thesphere from raw measurement data will be explained.4.1 Measurement setupIn gure 4.1 a schematic representation of the measurement setup is shown. Themain part of this setup is a rectangular vessel. On the bottom of the vessel lies a 19mm thick glass plate. The vessel is lled with a uid, in this case a glycerol-watermixture of a certain viscosity and density. In the top of the vessel a glass sphereis hanging at a release mechanism. The sphere is positioned in the centre of thevessel. The distance between the top of the sphere and the top of the glass plate is130 mm. This sphere is illuminated by a 750 W spotlight. A dimming glass plate (5) (7) (8) (1) (4) (6) (3) (2) Figure 4.1 : Schematic picture of the experimental set-up used for trajectory mea- surements: (1) Rectangular vessel, (2) Glass plate, (3) Fluid, (4) Glass sphere, (5) Release mechanism, (6) Spotlight, (7) Dimming glass plate, (8) Camera. 21
  30. 30. 22 Chapter 4. Experimental set-up for sphere trajectory measurementsis placed between the spotlight and the vessel to make the light of this lamp morehomogeneous. In front of the vessel, a digital camera is placed to take pictures ofthe dark sphere against a white background.A measurement is made by starting capturing images with the digital camera. Start-ing the camera triggers the release mechanism the sphere is released and will startfalling down until it reaches the bottom of the vessel and comes to rest. During thesettling, images of the falling sphere are taken at high frequency. When the mea-surement is nished, a lm that can be saved to computer disk has been obtained.Using a C-routine in SCIL-image, an image processing package Scilimage, 1998], aplace-time data le of the sphere can be extracted from the lm. Thus, informationof the trajectory of the sphere as a function of time is obtained. In the next section,more will be said about: Rectangular vessel Spheres Fluid Release mechanism Digital camera4.2 Detailed description of some components4.2.1 Rectangular vesselThe experiments are performed in a transparent perspex rectangular vessel. Theinner dimensions are: d w h = 134 134 185 mm. It is lled with liquid to aheight of 165 mm. At the bottom lies a glass plate of d w h = 120 120 19mm. (This plate has been inserted because the initial idea was to do bouncingexperiments with the sphere. However, it turned out that at interesting Reynoldsnumbers no rebound occurs.)4.2.2 SpheresThe spheres used in this research are non-transparent black glass beads from Brooks ow-meters. They are used in ow-meters, which implies that they may be consid-ered as perfectly spherical. Spheres of four di erent diameters are used. The diame-ter was determined experimentally using a slide-rule: 9.53, 6.35, 5.55 and 3.18 mm.The mass of each sphere was measured using a substitution balance (type MettlerB5). Dividing by the volume of the sphere results in the density p of the spheres: p = (2:55 0:02) 103 kg m;3
  31. 31. 4.2. Detailed description of some components 23 (7) (8) (3) (5) (6) (2) (4) (1) (9) Figure 4.2 : Schematic picture of the release mechanism: (1) Small opening of Pas- teur pipette, (2) Large opening of Pasteur pipette, (3) To vacuum net, (4) Vessel to capture uid leaking around the sphere, (5) Electronic valve, (6) Surroundings, (7) Valve, (8) Pressure gauge, (9) Sphere.4.2.3 FluidThe uids used in this experiment are glycerol-water mixtures. They behave as New-tonian liquids. The more glycerol is contained in the mixture, the larger the dynamicviscosity is. A wide range of viscosities can be created. For example: a solution of 5weight-% glycerol has a viscosity of 1.143 mPas at 293 K and 100 weight-% glycerolhas a viscosity 1499 mPas at 293 K, see Janssen and Warmoeskerken, 1991]. Anadvantage of using glycerol-water solutions, is that they are cheap. A disadvantageis the temperature sensitivity of the viscosity. For example: a 80 weight-% glycerol-water solution has a viscosity of 62.0 mPas at 293 K, but a viscosity of 45.86 mPasat 298 K. This is a decrease of 3.2 mPas/K. This decrease can be a problem whendoing experiments throughout the day: the viscosity of the uid changes.At the end of each series of measurements, the viscosity of the uid is measured,to decrease the in uence of temperature change. This is done using a ContravesRheomat 115 rotational viscosimeter. The density of the uid is determined usinga 100 ml ask and a Mettler balance.4.2.4 Release mechanismA schematic picture of the release mechanism is given in gure 4.2. The main part ofthe release mechanism is a Pasteur pipette (length 23 cm). The large opening of thepipette (diameter 5 mm) is both connected to the vacuum net of the laboratoryand an electronic valve which stands directly in contact with the surroundings. Thesphere is hanging at the small (diameter 1 mm) opening of the pipette, sucked bythe vacuum. A valve is inserted between the pipette and the vacuum net to controlthe pressure drop over the sphere. A pressure gauge is put between the pipette andthe vacuum net, to get an idea of the pressure-drop. When the electronic valve isopened by the trigger signal, the vacuum vanishes. The sphere is released.
  32. 32. 24 Chapter 4. Experimental set-up for sphere trajectory measurements (a) (b) (c) : Steps to determine the centre of mass of a sphere from raw data. (a) Raw Figure 4.3 image after contrast-stretch (by eye, the raw image without contrast- stretch looks the same), (b) Image a after background subtraction, in- verting and another contrast-stretch. (c) Image b after thresholding.4.2.5 Digital cameraDuring this experiment two CCD (Charge Coupled Device) cameras are used. The rst one is a Dalsa CA-D6-0512W-BCEW camera of 512 512 pixels and a maximumcapturing frequency (f c)max =250 Hz, the second a Dalsa CA-D6-0256W-ECEWcamera of 256 256 pixels with (f c)max =1000 Hz. The pixel-size is 10 10 mfor both cameras. The camera used is connected to a computer running a data-acquisition program. With this program it is possible to adjust camera settings, likerecording frequency and number of frames to acquire.For trajectory measurements two lenses are used. The accelerating and steady statemotion is measured with a magni cation factor M=0.038 using a lens with a focallength of 16 mm and a minimum f-number (ratio of focal length and aperture diam-eter or the opening of the diaphragm of the lens) of 1.4, trajectories close to the wallwith M=0.14 using a Nikon lens of focal length of 55 mm and a minimum f-numberof 2.8.4.3 Image processingIn this research, the image processing package SCIL-Image is used to nd the centreof mass of the sphere on recorded lms. The images consist of a number of pointsin a two-dimensional plane. Each pixel is assigned a grey-value according to theintensity of the on-falling light. This grey-value is in the range of 0 (black) and 255(white). In this section, the steps needed to nd the centre of mass are explained.An example of an image is shown in gure 4.3a. This picture is a contrast-stretchedversion of the raw measurement data. Contrast-stretching is performed to obtain anoptimum contrast between objects in an image and its background. The entire rangeof existing grey-values in the new image has been rescaled to the maximum possiblevalues 0-255, while all other pixels have been multiplied using the same factor as
  33. 33. 4.3. Image processing 25used for the highest pixel value. In the recordings resulting from the experimentsthe particle is dark and the background is light. Contrast stretching has assigned avalue of 255 to the background.It is desirable to remove objects other than the sphere, for example the vessel wall,from the image before processing. This is done using background subtraction. Arecorded background image is subtracted from the image. To get an image of a whitesphere on a black background, the image is inverted. Finally a second contrast-stretch is performed on the image. This results in an image of a white sphere on adark background, as shown in gure 4.3b.The next step is to divide the image into a number of objects, containing the sphere,and a background. This is done using image segmentation. One of the most simpleways to perform this segmentation is thresholding. Pixels with grey-values equalto or higher than the adjusted threshold value are made object pixels (value 1)and pixels with grey-values below the threshold values are made background values(value 0). The result is a binary image shown in gure 4.3c: the pixels are either 0 or1. In general, the main problem using this type of segmentation is to nd a suitablethreshold value. For this purpose, a histogram of grey-values may be constructed.This shows that the background of the images have pixel values that do not exceedthe value of 5. Therefore, the threshold value has been chosen just above this value,at a value of 10. This way a clear distinction between object and background hasbeen obtained.After this the objects in the image have to be labelled: every object is given anumber. The sphere is selected and the x- and y-coordinate of the centre of massare determined. Using a C-routine, the whole lm is processed automatically andthe result, the trajectory the sphere, is written to a data- le.4.3.1 CalibrationThe last thing that requires attention is that the place coordinates of the spherestored in the data le are in pixels. By calibration of the recordings the absolutevalues of the coordinates can be calculated. This calibration is performed by placinga ruler at the place where the sphere is falling. By taking an image ( gure 4.4), thearea to which a pixels corresponds (m m) is determined.To check if there is image distortion, the distance between the 11 and 12, 5 and 6and the 0 and 1 cm mark on the ruler was determined. These are 39, 39 and 40pixels, respectively. This indicates that image distortion can be neglected: there isno large di erence between the area to which a pixel corresponds in the top, middleand bottom of the eld of view of the camera.
  34. 34. : Example of a picture of a ruler used for calibration of trajectory mea-Figure 4.4 surements.
  35. 35. Chapter 5Results and discussion oftrajectory measurementsIn this chapter, results of sphere trajectory measurements will be presented. First,the accuracy and reproducibility of the used measurement method is discussed. Sec-ond, results of measured trajectories are shown and derived quantities, for examplevelocity, will be calculated. Third, some nal remarks will be made.5.1 Accuracy and reproducibilityThe accuracy and reproducibility of the measurements is discussed in this section.First, spatial and grey-value discretization are discussed. Second, the positioningof the camera is explained. Third, attention is given to the release moment of thesphere. Fourth, errors in measured and calculated quantities are given. Fifth, thereproducibility of the measurements is discussed.Every recorded image consists of a limited number of pixels (512 512 or 256 256).The diameter of the sphere calculated from the recordings should be equal to thereal diameter. For example, in a measurement the diameter for the 6.35 mm spherewas 24 or 25 pixels. The area to which a pixel corresponds was 0:26 0:26 mm2 ,resulting in sphere diameters of 6.24 and 6.50 mm. Because of the good resemblancebetween measured and real sphere diameters, the size of the error due to spatialdiscretization is estimated 1 pixel. For the measurements of acceleration and steady-state trajectories this results in errors relative to sphere diameter of 3%, 5%, 5%and 9% for sphere diameters of 9.53, 6.35, 5.55 and 3.18 mm, respectively. For theexperimental conditions of the wall impact measurements, this error is 3%.In section 4.3 it has been described how on recorded lms the place of the sphereis determined. It is reasonable that a change of threshold level (between certainlevels) for a spherical particle on a dark background should have no in uence on thedetermined centre of gravity of the particle. For this reason, some recordings havealso been processed using a threshold value of 50 instead of 10. There was observedno di erence between the position of the centre of gravity. Since the error in the 27
  36. 36. 28 Chapter 5. Results and discussion of trajectory measurements Table 5.1: Accuracies of measured and calculated quantities. quantity accuracy f 1% f 1% vs 4% vs+ 1% Res 4%experimentally determined diameter of the sphere is of the order of 1 pixel, the errorin the determined centre of gravity of the sphere will be less than 1 pixel.If the camera is not positioned exactly horizontally, deviations will occur in thecoordinates of the particle position. A spirit-level was placed on top of the camera,to position it horizontally. To align the camera more accurate, a plummet was hungin the vessel. The camera was tilted until the plummet was imaged exactly on onevertical line of pixels. This means that the camera is aligned with the direction ofgravitational acceleration.Due to variations in the measurement process the sphere was not always releasedat the same time. On average, the sphere was released 4 frames after the camerastarted acquiring frames, with a deviation of 2 frames. This di erence in releasemoment can be compensated for by applying a small time-shift to the resultingplace-time graph.Inaccuracies of some measured and calculated quantities are given in table 5.1. Theinaccuracy of f is mainly due to the inaccuracy of using the ask. In f theinaccuracy is due to the measurement error of the viscosimeter. In vs it is due tothe error in the used correlation for the drag curve. The inaccuracy in vs+ mainlyoriginates from performing linear regression to calculate this velocity from the data.In Res the main in uence is again the inaccuracy of the used drag curve to calculatevs .The acceleration to steady-state measurements were reproduced four times. Theapproach of wall measurements were reproduced two times. The results for Res =74:2 and Res = 1:5 are shown in gures 5.1 and 5.2. Plots have been made of h/Dagainst t, where h is the distance between the bottom apex of the sphere and thebottom of the vessel and t is time. In gure 5.1 all trajectories overlap. In gure 5.2there is also little di erence in trajectory. This indicates that these experimentswere reproducible. All other cases measured were also reproducible, because thetrajectories overlapped as well.
  37. 37. 5.1. Accuracy and reproducibility 29 14 12 10 8 h/D (-) 6 rst measurement 4 second measurement third measurement 2 fourth measurement fth measurement 0 0 0.1 0.2 0.3 0.4 0.5 t (s) Figure 5.1: Accelerating motion of sphere until steady-state for Res =74.2. 5 4.5 rst measurement second measurement 4 third measurement 3.5 3 h/D (-) 2.5 2 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 t (s) : Approach of bottom of vessel for Res =1.5 Figure 5.2
  38. 38. 30 Chapter 5. Results and discussion of trajectory measurements Table 5.2 : Accelerating motion and steady-state: de nition of cases. D f f vs Res (mm) (mPas) (kg=m3 ) (cm=s) (-) 3.18 50.3 1205 8.8 6.7 5.55 50.3 1205 18.7 24.8 6.35 52.0 1205 21.6 31.8 9.53 52.0 1205 33.6 74.2 40 Re = 6:7 35 s Re = 24:8 s 30 Re = 31:8 s Re = 74:2 s 25 h/D (-) 20 15 10 5 0 0 5 10 15 20 25 30 35 40 45 tv =D (-) s Figure 5.3: Accelerating motion of a sphere until steady-state for 4 di erent Reynolds numbers.5.2 Trajectories: Accelerating motion to steady-stateThe accelerating motion of a sphere to steady-state has been measured for four dif-ferent Reynolds numbers. Res was varied by changing sphere diameter and keeping uid viscosity and density constant. The de nition of cases measured is given intable 5.2. In gure 5.3 plots have been made of h/D against tvs=D. By scaling thisway, the trajectories get a slope of approximately -1 at steady-state settling velocity.Before the results are discussed further, it has to be checked if the measured steady-state velocities make sense. This is done by comparing the measured maximumfalling velocities vs+ in the vessel to the calculated theoretical steady-state velocitiesvs in an in nite medium. The results are shown in table 5.3. The highest measuredvelocity in the vessel is smaller than the theoretical value. This is to be expecteddue to the wall e ect. The ratio vs+ =vs is a measure for the wall e ect as de nedin section 2.7. However, care has to be taken when interpreting vs . It has beencalculated using the empirically determined drag coe cient CD . The drag curve
  39. 39. 5.2. Trajectories: Accelerating motion to steady-state 31 Table 5.3 : Check of particle settling velocity. Res vs+ vs v+ s v+ s v s v corr s (-) (cm/s) (cm/s) (-) (-) 6.7 8.2 8.8 0.93 0.96 24.8 16.6 18.7 0.89 0.96 31.8 18.8 21.6 0.87 0.95 74.2 29.4 33.6 0.88 0.94 12.5 12 11.5 11 h (cm) 10.5 10 9.5 9 model of Ferreira et al. measurement 8.5 0 0.05 0.1 0.15 0.2 0.25 t (s) Figure 5.4: Accelerating motion of sphere for Res =31.8. Together with model of Ferreira et al.used in this thesis, see section 2.4, has an average inaccuracy of 7.6%. This meansthat the average inaccuracy in vs is around 4%.The calculated ratio of vs+ =vs can be compared with this ratio predicted by the walle ect correlation of Di Felice as mentioned in section 2.7. This correlation has beencalculated and the result is shown in the last column of table 5.3. The correlationgives a higher value for (vs+ =vs )corr than the measured value. If the inaccuracyof the drag-curve is taken into account upper boundaries for vs+ =vs are 0.96, 0.92,0.90 and 0.91 for increasing Reynolds number, respectively. Now for the Res=6.7measurement the predicted value is the same as the measured result.It is clear that the correlation proposed by Di Felice underestimates the wall e ectfor the cases measured. This is possibly due to the fact that the correlation is notvery accurate, especially for small values of particle to vessel width ratio.Ferreira modelIn gure 5.4, a measurement result of the accelerating motion of a sphere together
  40. 40. 32 Chapter 5. Results and discussion of trajectory measurements : Order of magnitude estimation of the Basset force. Table 5.4 Res t FD Basset F F D Basset ( N) D stationary (-) (s) (-) 6.7 0.3 10 0.05 24.8 0.25 70 0.06 31.8 0.25 106 0.06 74.2 0.2 415 0.07 : Approach of bottom of vessel: De nition of cases. Particle used: Table 5.5 D=3.175 mm. f f vs Res (mPas) (kg=m3 ) (cm=s) (-) 122.3 1218 4.6 1.5 76.0 1219 6.7 3.4 54.6 1215 8.4 5.9 41.7 1215 10.0 9.1 26.1 1198 13.0 18.8with the model of Ferreira as described in section 2.6 is shown. The Ferreira modelis in qualitative agreement with measurements: it predicts approximately the sameshape of particle trajectory as measured. However, it predicts higher sphere veloc-ities. Several reasons exist for this. First, the wall e ect is not included in thismodel. Second, the added-mass-coe cient used is 0.5 for irrotational ow. Third:the Basset-force is not included in this model. An estimation of this force can bemade using equation 2.15. The results of these calculations are given in table 5.4.It is clear that it is not allowed to neglect this force under these circumstances: ithas a magnitude of around 6% of the steady-state drag force acting on the particlein an in nite medium. Reasons for this are already mentioned in section 2.6. TheBasset force diminishes with increasing particle to uid density ratio. In this casethis ratio is low (around 2). The Basset force is also proportional to the squareroot of the viscosity of the uid. In this research uids with relatively high viscos-ity have been used. If the experiments would have been performed in water, theBasset-force would have been smaller and the Ferreira model would have been inbetter agreement with the measured trajectory.5.3 Trajectories: Wall impactFor 5 di erent Reynolds numbers trajectories of a sphere approaching the bottom ofthe vessel have been measured. The de nition of cases measured is given in table 5.5.Res was varied by varying the uid viscosity. In gure 5.2 it can be seen that the
  41. 41. 5.3. Trajectories: Wall impact 33 5 4.5 Re =1.5 s Re =3.4 4 s Re =5.9 s 3.5 Re =9.1 s Re =18.8 3 s h/D (-) 2.5 2 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 t (s) Figure 5.5: Approach of bottom of vessel for ve di erent Reynolds numbers.sphere decelerates as it approaches the wall. The question arises how the approachto the bottom wall is in uenced by Reynolds number. For this reason, a plot fordi erent Reynolds numbers has been made in gure 5.5. Clearly, the sphere startsto decelerate for larger h=D when Res+ gets smaller. This is in agreement with themodel of Gondret et al., 1999] which has been described in section 2.8.To get another view of the approach of the sphere to the bottom wall, the trajec-tories have been di erentiated. For the two lowest Reynolds numbers this has beendone using a least squares method Ra el et al., 1998]. This is equivalent to tting astraight line through 5 data points and taking the slope of this line as the derivative.The other trajectories are di erentiated using an ordinary central di erence scheme.In gure 5.6 plots were made of vp against h/D. This graph gives a di erent interpre-tation. If the slope of the curves in the graph is not zero, the particle is decelerating.This can clearly be seen for the Res =1.5 measurement: the plot is already slightlycurved for h/D=4, indicating that the particle is already decelerating at this height.The Res =18.8 measurement however shows a at plot until approximately h/D=1:the particle starts to decelerate at a much smaller value of h=D.It is interesting to determine the force acting on the sphere as a function of height.For this the acceleration of the sphere has to be determined form the data. However,the data is not accurate enough to calculate the second derivative.To get an indication of the size of this force, a function is tted. To do this, a non-dimensional velocity v is de ned as vp =vs and h is de ned as h=D. The followingfunction was used: ah v (h ) = 1 + bh (5.1)The coe cients a and b were determined using non-linear regression. The resultsare given in table 5.6. As an example, the result for Res =5.9 is shown in gure 5.7.
  42. 42. 34 Chapter 5. Results and discussion of trajectory measurements 0.14 Re = 18.8 s 0.12 Re = 9.1 s Re = 5.9 s 0.1 Re = 3.4 s Re = 1.5 s 0.08 v (m/s) 0.06 p 0.04 0.02 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 h/D (-) Figure 5.6: Velocity of the sphere as a function of h/D for di erent Res . Table 5.6: Result for t coe cients. Res a inaccuracy a b inaccuracy b (-) (-) (-) (-) (-) 1.5 2.81 0.05 3.06 0.06 3.4 3.12 0.09 3.12 0.12 5.9 10.0 0.6 10.5 0.7 9.1 18.6 0.9 18.7 0.9 18.8 49.2 5.5 50.5 5.8
  43. 43. 5.4. Critical Stokes number measurements 35 1 0.9 0.8 0.7 v /v (-) s 0.6 p 0.5 0.4 data t 0.3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 h/D (-) Figure 5.7: Fit of function to Res =5.9 data.The t looks reasonable. It shows the appropriate limit-behaviour. From this ttedfunction, the qualitative shape of the force acting on the particle as a function ofheight can be calculated by di erentiation using the chain rule ( dv = dh dh ): dt dv dt p p 2 d F (h ) = m dvp = m vs dh ( 1 v (h ))2 dt D 2 (5.2)The result of this calculation for Res = 5:9 is given in gure 5.8. The result is inagreement with what can be expected. When the sphere is far away from the wall, itis falling at steady-state velocity. There is no net force acting on the particle. Whenit starts approaching the bottom of the vessel, the force increases until it reaches amaximum value. Finally the force decreases to zero when the particle is at rest.5.4 Critical Stokes number measurementsAn attempt has been made to measure the critical Stokes number of a glass spherecolliding on the glass bottom of the vessel. By varying uid viscosity the Stokesnumber of the particle was changed. To judge whether or not a rebound occurredthe lms resulting from the measurement were studied by eye. The results are shownin table 5.7. It follows that Sts, and thus the critical Stokes number based on thevelocity of the sphere in an in nite medium lies between 9.75 and 13.4. This resultis not very accurate. However, it is in agreement with the value of 20 resultingfrom measurements performed by Gondret et al., 1999], because these measure-ments were also not very accurate and a steel instead of a glass sphere has beenused. The correlation derived by Zenit and Hunt, 1999] in section 2.8 gives valuesStc =8.1, 22.6 and 50.83 for the three di erent t parameter values mentioned. Asthe measured values are in this range, they are not contradicting this correlation. Todo more accurate measurements of Stc, a more sophisticated measurement method
  44. 44. 36 Chapter 5. Results and discussion of trajectory measurements 0.00014 0.00012 0.0001 8e-05 F (N) 6e-05 4e-05 2e-05 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 h/D (-) Figure 5.8: Force acting on the particle as a function of height. Derived by calcu- lating the derivative of the t of function to Res =5.9 data. Table 5.7: Measurement results for determination of critical St-number. Particle used: D=6.35 mm. f f vs Res Sts Rebound? (mPas) (kg=m3 ) (cm=s) (-) (-) 43.3 1200 23.5 41.3 9.75 Certainly not 38.0 1197 24.8 49.7 11.8 Do not know 37.0 1195 25.1 51.4 12.2 Do not know 34.7 1192 25.8 56.3 13.4 Certainly 25.3 1189 28.9 86.1 20.5 Certainly

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