Experimental and numerical investigation of flow field in a static ...

Experimental and numerical investigation of flow field in a static mixer at low Reynolds number regime

H. Rajaei Number : WET

Graduation Thesis Professor : Pof. Dr. Ir. A.A. van Steenhoven Supervisor: Dr. Ir. M.F.M. Speetjens Advisor : ¨ Baskan MSc. O.

Graduation committee: Pof. Dr. Ir. A.A. van Steenhoven Dr. Ir. M.F.M. Speetjens Prof. Dr. H.J.H. Clercx Dr. Ir. H.C. de Lange

Eindhoven University of Technology Mechanical Engineering Department Thermo Fluids Engineering Division Energy Technology Section

Eindhoven, August 2013

EINDHOVEN UNIVERSIRT OF TECHNOLOGY

Abstract Faculty of Mechanical Engineering Thermo Fluids Engineering Division Master of Science in Mechanical Engineering Experimental and numerical investigation of flow field in a static mixer at low Reynolds number regime by Hadi Rajaei

Static mixers are widely used in chemical process industries. In the field of mixing, laminar mixing is important for a wide range of industrial processes, encompassing mixing of viscous fluids (e.g. food industry, polymers) and micro-fluidics. In the present study, a representative industrial static mixer, the so-called Q-type mixer is the subject of the investigation. Q-type mixer is an industrial inline mixer which is composed of a circular tube and internal elements. The strategy is a combined numerical-experimental investigation. The experimental studies are performed for Re = 3.4. Three-Dimensional Particle Tracking Velocimetry (3DPTV) is utilized for the experiments. For the processing of the images, an algorithm developed at ETH Z¨ urich is used. The periodic flow field characteristics are investigated via implementation of several mixing elements into the experimental facility. The numerical simulations are performed using COMSOL Multiphysics 4.2 software. The numerical studies are performed for five different Reynolds numbers; 3.4, 6.8, 34.2, 68.4, and 100. The effects of the variation of Reynolds number on the velocity field and streamlines are investigated. Furthermore, the numerical concentration studies are performed for different Reynolds numbers to enhance our knowledge on the mixing performance of the subject static mixer. The numerical predictions are validated via experimental results. It shows a good agreement between numerical and experimental studies. Finally, the effects of the inlet conditions on the periodicity are investigated via prescribing three different inlet conditions.

Acknowledgements I would like to express my deepest appreciation to all those who provided me the possibility to complete this thesis. First of all, a very special gratitude I give to my supervisor Michel Speetjens for his great supervision. Furthermore, I would also like to acknowledge with much appreciation the crucial role of Prof. Herman Clercx for the useful comments, remarks and engagement through the learning process of this master thesis. A special thanks goes to my ¨ advisor, Ozge Baskan, whose contribution in stimulating suggestions and encouragement, helped me to coordinate my project amd in writing this report. I would like to appreciate my graduation professor Anton van Steenhoven for his advices during the various meetings. I would like to thank Ad Holten for his help and advices on the PTV measurements. Also many thanks to Henri Vliegen, Geert-Jan van Hoek, Jan Hasker, Paul Bloemen, and Frank Seegers of the TFE technical staff. Furthermore, I would like to thank the members in my graduation committee. I would also like to thank my parents for their endless love and support. Finally, I would like to thank my friends.

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Contents Abstract

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Acknowledgements

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List of Figures

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List of Tables

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1 Introduction 1.1 Scope and motivation 1.2 Previous study . . . . 1.3 Problem definition . . 1.4 Outline of the thesis .

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1 1 2 4 4

2 Theory and methods 2.1 Flow model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Pipe-entrance conditions . . . . . . . . . . . . . . . . . . . . . . 2.3 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Particle Tracking Velocimetry . . . . . . . . . . . . . . . 2.3.1.1 Performance issues . . . . . . . . . . . . . . . . 2.3.2 Interpolation of 3DPTV data . . . . . . . . . . . . . . . 2.3.2.1 Validation of the interpolation on regular grid 2.4 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Mesh validation . . . . . . . . . . . . . . . . . . . . . . .

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3 Further Development of The Experimental Set-up 3.1 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . 3.2 Matching Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Lift force, density effects and response time of particle 3.3 Opacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Distribution of the particles . . . . . . . . . . . . . . . . . . . 3.5 Fluctuations at low Reynolds numbers . . . . . . . . . . . . . 3.6 Internal elements . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Modification in calibration . . . . . . . . . . . . . . . . . . . . iii

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Contents 4 Numerical simulation of 3D flow field 4.1 One Period . . . . . . . . . . . . . . . . . 4.1.1 Velocity Field . . . . . . . . . . . . 4.1.2 Streamlines . . . . . . . . . . . . . 4.1.3 Concentration field . . . . . . . . . 4.2 Periodic flow . . . . . . . . . . . . . . . . 4.2.1 Representative cross section . . . . 4.2.2 On random points along the mixer 4.2.3 Periodic flow field . . . . . . . . .

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5 Experimental analysis of general flow properties 5.1 Flow physics . . . . . . . . . . . . . . . . . . . . . 5.1.1 One period . . . . . . . . . . . . . . . . . . 5.1.1.1 Velocity field . . . . . . . . . . . . 5.1.1.2 Streamlines . . . . . . . . . . . . . 5.1.2 Periodic flow . . . . . . . . . . . . . . . . . 5.2 Velocity field error analysis . . . . . . . . . . . . . 5.2.1 Random error . . . . . . . . . . . . . . . . . 5.2.2 Systematic error . . . . . . . . . . . . . . . 5.3 Streamlines error analysis . . . . . . . . . . . . . . 6 Comparative numerical-experimental analysis of 6.1 Fully developed flow . . . . . . . . . . . . . . . . 6.2 63% of the inlet being blocked . . . . . . . . . . . 6.2.1 Numerical results . . . . . . . . . . . . . . 6.2.2 Experimental results . . . . . . . . . . . . 6.3 A quarter of the inlet being blocked . . . . . . . 6.3.1 Numerical results . . . . . . . . . . . . . . 6.4 Effects of the inlet condition on periodicity . . .

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25 26 26 30 32 33 34 36 37

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41 41 41 41 44 46 48 49 51 53

inlet conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Conclusions and recommendations 65 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

A Setting valve

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B Flowmeter

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C Working principle of 3D PTV C.1 Performance issues . . . . . . . . . C.2 Epipolar line intersection technique C.3 Two views for particle tracking . . C.3.1 ETH code . . . . . . . . . .

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70 70 72 72 73

D Higher order elements

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E General improvements of the experimental setup

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Contents Bibliography

v 79

List of Figures 1.1 1.2

1.3 1.4 1.5 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

Mixing mechanism of static mixers[1]. . . . . . . . . . . . . . . . . . . . . . . . .

1

Elements of different commercial static mixers ; Kenics (top-left), low pressure drop (top-right), HEV (middle-left), inline series 45 (middle-right), Custody transfer mixer (bottom-left), SMR (bottom-right)[32]. . . . . . . . . . . . . . . . . . . . . . . . . . . Blade geometry of Q-type mixer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measured (left) vs. simulated (right) 3D velocity field. . . . . . . . . . . . . . . . . . . Measured (blue) vs. simulated (red) streamlines for the whole domain. . . . . . . . . .

2 3 3 3

Blades geometry and frame of reference. . . . . . . . . . . . . . . . . . . . . . . . . . The entrance length in a pipe flow [53]. . . . . . . . . . . . . . . . . . . . . . . .

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Epipolar geometry in two views for different refractive indices geometry (left) and principles of epipolar line method for four cameras (right) [19]. . . . . . . . . . . . . . . . Experimental Poiseuille profile -dots- and the theoretical Poiseuille profile-line-(left) (left) Coordinate axes (middle) Calibration body (right). . . . . . . . . . . . . . . . . . . .

refraction (left), critical angle (middle), reflection (right) [28]. . . . . . . . . . . . Arbitarary surface for interpolation test (left) and the error of the interpolation (right). Blade geometry of Q type mixer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Difference in velocities for two different mesh numbers for inlet(top-lef) middle(top-right) and outlet(bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Difference in concentration for two different mesh numbers for outlet (left) and 60 [mm] after outlet (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1 3.2

Schematic view of the setup.

3.3 3.4

Opaque silicon oil . . . . . . . . . . . . . . . . . . . . . . The position of the feeding tube and top reservoir before ment (lright). . . . . . . . . . . . . . . . . . . . . . . . . Particles which are trapped inside the water droplets. .

3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14

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Density effects (left) lift force (middle), and response time (right) of the particles in silicone oil and water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Absolute transverse velocity contours ux (left) and uz (right)- the axial velocity is 12.3 [ mm s ]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ux (top) and uz (bottom) before (left) and after (right) changing the setting valve. . . .

Deterioration of the tracking when 3D printed are used

. . . . . . . . . . . The image of the old blades (left) and 3D printed blades (right). . . . . . . . . . Experimental Poiseuille profile -dots- and the theoretical Poiseuille profile-line. . . Real calibration image (left) and artificial one (right). . . . . . . . . . . . . . .

. . . . Deviation in position due to curve tube in ideal case (left) and current case (right). . The error in z (left) and x (right) directions for current case. . . . . . . . . . . . . . Error after modification in calibration in z(left) and x(right) directions. . . . . . . . . vi

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8 9 10 11 12 13 13 14 16 17 18 18 19 20 20 21 21 22 22 23 24

List of Figures 3.15 Experimental Poiseuille profile (dots) and the theoretical Poiseuille profile(line) after modification in calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 4.2

4.3

4.4 4.5

4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30

Blade geometry of Q type mixer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-sectional velocity field in the upstream region at y = +20[mm], ux , uz (transverse components), and uy (axial component) -left to right- for Re = 3.42 (top) and Re = 34.2 (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-sectional velocity field in the mixing region at y = −28[mm], ux , uz (transverse components), and uy (axial component) -left to right- for Re = 3.42 (top) and Re = 34.2 (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The velocity field inside the first element. . . . . . . . . . . . . . . . . . . . . . . . . Cross-sectional velocity field at the outlet at y = −112[mm], ux , uz (transverse components), and uy (axial component) -left to right- for Re = 3.42 (top) and Re = 34.2 (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-sectional velocity field at y = −152[mm], ux , uz (transverse components), and uy (axial component) -left to right- for Re = 3.42 (top) and Re = 34.2 (bottom). . . . . Transverse velocity component for the entire domain for Re = 3.4, 34.2, and 100. . . . . Formation of the vortical structure at the outlet of the mixing element at Re =100 . . . Numerical velocity field for Re = 3.4 (left), and 100 (right). . . . . . . . . . . . . . . . Simulated streamlines for the entire domain for Re = 3.4(left) and 34.2 (right). . . . . . Simulated streamlines for the inlet region for Re =3.4 (left) and 34.2 (right). . . . . . . Cutting and stacking of the flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Streamlines in the mixing region (left) and at the outlet (right). . . . . . . . . . . . . . Streamlines at the outlet for Re =3.4(top), 34.2(middle), and 100 (bottom). . . . . . . Concentration field at y = 0, −28, −56, −85, and − 112[mm] (left to right) for Re = 3.4. Concentration field within the first element at y = −12[mm] (top) and y = −48[mm] (bottom) for Re = 3.4, 34.2, and 100 (left to right). . . . . . . . . . . . . . . . . . . . Concentration field within the second element at y = −64[mm] for Re = 3.4, 34.2, and 100 (left to right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concentration field at the outlet, y = −112[mm] (top) and at 40[mm] after the outlet, y = −152[mm] (bottom) for Re = 3.4, 6.8, 34.2, 68.4, and 100 (left to right). . . . . . . Simulation for four elements for Re = 3.4 . . . . . . . . . . . . . . . . . . . . . . . . Five cross sections which the data are analyzed on. . . . . . . . . . . . . . . . . . . . Velocity field on cross section 1 for second period (left) and third period (middle) and the difference between them (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-sectional periodic velocity field in the middle of the first element, ux , uz , (transverse components) and uz (axial component) -left to right- for Re = 34.2. . . . . . . . . . . Cross-sectional periodic velocity in the middle of the second element, ux , uz , (transverse components) and uz (axial component) -left to right- for Re = 34.2. . . . . . . . . . . . The velocity field for Re =3.4(left) and 34.2 (right). . . . . . . . . . . . . . . . . . . The transverse velocity field for Re =3.4(left) and 34.2 (right). . . . . . . . . . . . . . Simulated streamlines for Re = 3.4 (left)and 3.42(right). . . . . . . . . . . . . . . . . Periodic simulated streamlines for Re = 34.2. . . . . . . . . . . . . . . . . . . . . . . Concentration field for Re = 3.4 (top), and 34.2 (bottom). . . . . . . . . . . . . . . . . Concentration contours for the inlet of the second, third, fourth, and fifth period (left to right) for Re = 56. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The concentration contours at the inlet of second to ninth period respectively. . . . . .

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28 29 29 30 30 31 31 31 32 33 33 34 34 35 35 35 36 37 38 38 38 39 39 40 40 40

List of Figures 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22 5.23 5.24 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

Cross-sectional velocity field in the inlet region, 20[mm] before the onset of the mixing elements, for ux , uz (transverse components), and uy (axial component) -left to right. . . Cross-sectional velocity field in the mixing region at the middle of the first element for ux , uz (transverse components), and uy (axial component) -left to right. . . . . . . . . . Numerical (left) and confined color experimental (right) contours for z-component. . . . Cross-sectional velocity field at the middle of the period for ux , uz (transverse components), and uy (axial component) -left to right. . . . . . . . . . . . . . . . . . . . . . . Cross-sectional velocity field at the outlet for ux , uz (transverse components), and uy (axial component) -left to right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Confined color experimental velocity contour for z direction at the outlet . . . . . . . . Cross-sectional velocity field at y = −152[mm], ux , uz (transverse components), and uy (axial component) -left to right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity field (left) and transverse velocity field (right) for the whole domain. . . . . . . Simulated streamlines for the entire domain (left) and the streamlines for the inlet of the mixer (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental particle trajectories, front view (left) and rear view (right). . . . . . . . Numerical (red) and experimental (blue) streamlines in the mixing region (left) and at the outlet (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental velocity field (red) versus numerical velocity field (blue) for second period. Experimental velocity contour(left) and corresponding numerical velocity contour for one period (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross sectional velocity field at the middle of the first element of the second period for ux , uz (transverse components), and uy (axial component) -left to right. . . . . . . . . . Experimental(blue) and numerical(red) periodic streamlines. . . . . . . . . . . . . . . Inlet velocity contours for two consecutive experiments. . . . . . . . . . . . . . . . . . Inlet velocity contour after the rejection of the bad data. . . . . . . . . . . . . . . . . Experimental velocity contours (left) versus simulated velocity contours (right) for inlet, middle and outlet of the mixer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differences between experimental and numerical flow field and their corresponding histograms for inlet/middle/ outlet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Histogram for comparison between direct 3DPTV points and numerical studies for ux , uy , uz , U. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental (blue) and numerical (red) velocity vectors for three main regions. . . . . Number of the tracks (y) versus the track lengths (x). . . . . . . . . . . . . . . . . . . Experimental particle trajectories, front view (left) and rear view (right). . . . . . . . Full trajectories from different point of views. . . . . . . . . . . . . . . . . . . . . . . Five cross sections which the data are analyzed on. . . . . . . . . . . . . . . . . . . . Simulated flow field for fully developed flow as inlet condition, yz plane. . . . . . . . . Implementation of a square at the inlet for disturbing the flow. . . . . . . . . . . . . . Simulated velocity field for 1, 2, 3, and 4 for the second (top) and the third periods (bottom) when 63% of the inlet is blocked. . . . . . . . . . . . . . . . . . . . . . . . . Simulated streamlines for the second (blue) and the third (red) periods. . . . . . . . . . Simulated flow field when 63% of the inlet is blocked. . . . . . . . . . . . . . . . . . . Experimental (blue) and numerical (red) velocity vectors within the second period. . . . Numerical (left) and experimental (right) velocity field in the second period for the middle of the second element (top) and the middle of period (bottom). . . . . . . . . . . . . . A quarter of the inlet is blocked (black). . . . . . . . . . . . . . . . . . . . . . . . . .

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42 42 43 43 43 44 44 45 45 46 46 47 48 48 48 49 50 52 53 and

54 55 55 56 56 58 58 58 59 60 60 61 61 62

List of Figures 6.10 Simulated velocity field for 1, 2, 3, and 4 for second (top) and third periods (bottom) when a quarter of the inlet is blocked. . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11 Simulated flow field for fully developed flow (top) a quarter of the inlet is blocked (middle) and 63% of the inlet is blocked (bottom) . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 Simulated flow field for fully developed flow (top) a quarter of the inlet is blocked (middle) and 63% of the inlet is blocked (bottom). . . . . . . . . . . . . . . . . . . . . . . . .

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63 64 64

A.1 Globe valve(left) and ball valve (right) . . . . . . . . . . . . . . . . . . . . . . . . . .

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B.1 Flowmeters characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Flowmeters characteristics (continued) [23] . . . . . . . . . . . . . . . . . . . . .

68 69

C.1 Spinning particle in the boundary layer [19]. . . . . . . . . . . . . . . . . . . . . . C.2 (left) The two cameras are indicated by their centers C and C 0 and image planes. The

71

camera centers, 3-space point X, and its images x and x0 lie in a common plane π. (right) An image point x back-projects to a ray in 3-space defined by the first camera center, C, and x. This ray is imaged as a line l0 in the second view. The 3-space point X which projects to x must lie on this ray, so the image of X in the second view must lie on l0 [52]. 72

C.3 3D PTV configuration using 4 cameras and epipolar intersection technique [6] . . 73 C.4 image space based tracking technique (left) object space based tracking technique (right) 74 C.5 Processing scheme for tracking the particles [6] . . . . . . . . . . . . . . . . . . . 75 D.1 Taylor-Hood element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.2 Simulated concentration field with different mesh numbers; 6,500,000 linear elements(left) 650,000 linear elements(middle) and 650,000 quadratic elements (right). . . . . . . . . .

76

E.1 The transparent overflow tube section and transparent main tube. . . . . . . . . . . . .

78

77

List of Tables 2.1

The statistics of the velocity analysis for inlet, middle and outlet (the numbers are in percentage). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The statistics of the concentration analysis for outlet and 60[mm] afterthe outlet (the numbers are in percentage). . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

3.1

Working fluid properties[22, 24, 25, 50, 55]. . . . . . . . . . . . . . . . . . . . . .

16

4.1

The statistics of the cross section analysis for two Reynolds numbers (the numbers are in percentage). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The statistics of the analysis on the random points along the mixer (the numbers are in percentage). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

4.2 5.1 5.2 5.3 5.4 6.1 6.2 6.3 6.4 6.5 6.6

The statistics of the analysis on the random points along the mixer (the numbers are in percentage). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The limit value of the Q test [47]. . . . . . . . . . . . . . . . . . . . . . . . . . . . The statistics of the analysis on representative cross sections (the numbers are in percentage). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The statistics of the analysis directly on 3D-PTV data points (the numbers are in percentage). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The statistics of the analysis on five cross sections when 63% of the inlet area is blocked (the numbers are in percentage). . . . . . . . . . . . . . . . . . . . . . . . The statistics of the analysis on 25,000 numerical points when 63% of the inlet is blocked (the numbers are in percentage). . . . . . . . . . . . . . . . . . . . . . The statistics of the analysis on 18,000 experimental points along the second period when 63% of the inlet area is blocked (the numbers are in percentage). . . The statistics of the analysis on five cross sections when a quarter of the inlet area is blocked (the numbers are in percentage). . . . . . . . . . . . . . . . . . . The statistics of the analysis on 25,000 numerical points when a quarter of the inlet is blocked (the numbers are in percentage). . . . . . . . . . . . . . . . . . . The statistics of the analysis on second period when a quarter of the inlet area is blocked and a fully developed prescribed at the inlet (the numbers are in percentage). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

13

36 37 47 50 51 53 59 60 62 62 63

64

Chapter 1

Introduction 1.1

Scope and motivation

Mixing process is one of the most significant processes in the industry. Mixers are categorized according to whether they have moving parts or not. Based on this categorization, there are two distinct type of mixers; static mixers and dynamic mixers. The static mixers, also known as motionless mixers, have potential to be an attractive alternative to the conventional agitation due to the fact that they typically have lower energy consumptions and reduced maintenance requirements because they have no moving parts. Good performance can be achieved with lower cost compared to dynamic mixers. The static mixers were not utilized in industrial process until the 1970s. However, the patent is much older. There are more than 2000 US patents and 8000 literature articles that explain static mixers and their applications[2]. The working principle of the mixers is based on the stretching, cutting and stacking which is illustrated in Figure 1.1[1]. In the field of mixing, laminar mixers are an important class of mixers. Laminar flow concerns two important practical categories: (i) viscous fluids (e.g. food industry, polymers), and (ii) micro-fluidics. In recent years, many theoretical and experimental studies have been carried out for laminar mixing and have provided much beneficial information on how laminar mixing can be enhanced. It has been found that due to the dominating laminar flow, these mixers relies mainly on molecular diffusion and chaotic advection [32]. Figure 1.2 shows the elements of different commercial static mixers.

Figure 1.1: Mixing mechanism of static mixers[1].

1

Chapter 1. Introduction

2

Figure 1.2: Elements of different commercial static mixers ; Kenics (top-left), low pressure drop (top-right), HEV (middle-left), inline series 45 (middle-right), Custody transfer mixer (bottom-left), SMR (bottom-right)[32].

1.2

Previous study

In 2011, Jilisen built a setup to investigate the 3D-flow field in the static mixers utilizing 3DPTV [19]. The subject of his investigation was a representative industrial static mixer, so-called Q-type mixer, see Figure 1.3. He performed the experiments for Reynolds number of 502. However, measurements at Re < 500 were not reliable because distilled water was working fluid and the oscillations became dominant due to the fluctuations in the setting valve at very low flow rates. To validate the experiments, he did numerical study in parallel to experiments. Figure 1.4 shows the experimental and numerical velocity field inside the mixer. The inlet of the mixer is located at z = 0. Since mixing flows exhibited (locally) chaotic advection, implying that the particle trajectories are extremely sensitive to initial particle positions [19, 26], comparison of individual trajectories was not useful. Therefore, Jilisen examined the trajectories in term of coherent structures. Figure 1.5 shows the experimental (blue) and simulated streamlines (red). It shows that the 3D structures which were induced by the mixing element are well captured by the experimental results. The swirling exit flow is also apparent in the experimental results. He showed that the deviation between numerical and experimental results for 3D-velocity field lies between 10 − 15%[26]. The aim of Jilisen’s project was to build a very first version of an experimental setup which enables us to conduct 3D flow measurements. Since there are lots of processes which static mixers are working at very low Reynolds number, in the present study we are mainly interested in the flow characteristics at very low Reynolds number regime. However with the previous


3

experimental setup, it was not possible to reach Reynolds numbers lower than 500. Moreover, the periodic flow field characteristics and the effects of the inlet condition on periodicity are two important questions which have not been considered by Jilisen.

Figure 1.3: Blade geometry of Q-type mixer.

Figure 1.4: Measured (left) vs. simulated (right) 3D velocity field.

Figure 1.5: Measured (blue) vs. simulated (red) streamlines for the whole domain.


1.3

4

Problem definition

The goals for this thesis are • Adjustment of the laboratory set-up for 3D-PTV measurements at low Reynolds number regime and improvement upon the accuracy of the measurements. • Comparative analysis of the particle trajectories and the flow field in a static mixer. • Investigation of the periodic flow field characteristics of a real static mixer via implementation of several elements into the system. • Examining the effects of the inlet conditions on periodicity by using different inlet conditions. • Comparison of the experimental results with the numerical predictions.

1.4

Outline of the thesis

In this thesis the flow field of a static mixer is investigated. Experimental validation of the numerical simulations has been performed. Chapter 2 begins with the discussion of the governing equations. Working principles of PTV is covered in this chapter. At the end of the chapter, the numerical method is given. Chapter 3 starts with a brief introduction to the experimental setup. Then, the modifications that have been implemented in the setup are discussed. The numerical results for the flow field and the concentration field are given in Chapter 4. Furthermore, the physical aspects of the flow are discussed in this chapter. Chapter 5 revolves around the experimental results. It concentrates on the experimental investigation of the relevant physical aspects discussed in Chapter 4 and the error analysis. Chapter 6 involves the periodic flow measurements. The effects of inlet conditions on the periodicity are discussed in this chapter. Last chapter is dedicated to the conclusions and recommendations.

Chapter 2

Theory and methods In the present study, a representative industrial static mixer, the so-called Q-type mixer is the subject of the investigation. Q-type mixer is an industrial inline mixer which is composed of a circular tube and internal elements. The internal element consists of chevron-shaped central plates of width D = 56[mm] and length L = 56[mm]. Four perpendicular elliptical segments are extending to the outer cylindrical wall. Figure 2.1 shows the blade geometry. The 3D frame of reference is defined in such a way that y direction is along the static mixer (axial direction) and x and z directions are transverse, see Figure 2.1 for the frame of reference.

Figure 2.1: Blades geometry and frame of reference.

2.1

Flow model

Conservation laws for mass and momentum are the governing equations for description of the three dimensional flow field. The flow in the subject static mixer in our experiments is steady state. No-slip condition is enforced at the walls. The outlet condition is the constant pressure. For the inlet, Poiseuille profile is prescribed. For the steady state and incompressible flow conditions, the flow is governed by the continuity equation (mass conservation) and Navier-Stokes

5

Chapter 2. Theory and methods

6

(momentum conservation) equation[7]; ∇·u=0

(2.1)

ρu · ∇u = −∇p + µ∇2 u

(2.2)

where u the velocity, ρ the density, µ the dynamic viscosity, and p the pressure. Non-dimensional model shows the important non-dimensional numbers. Using the scaling parameter, it is possible to find the non-dimensional momentum equation. u∗ = Uu , ∇∗ = L∇, p∗ = ρUp 2 For constant density, it is possible to multiply both sides by ρUL 2 which gives: 1 ∗2 ∗ ∇ u Re where Re is the Reynolds number which is expressed by the following equation: u∗ .∇∗ u∗ = −∇∗ p∗ +

(2.3)

ρU L µ

(2.4)

Re =

where U is the velocity magnitude and L is the characteristic length. Apart from Re and Sr numbers, there are two important geometrical non-dimensional numbers. The first one is the L aspect ratio which is defined as Λ = D . In the current case, Λ = 1. The second important geometrical parameter is the number of the elements, N . Based on the definition of the Reynolds number, Equation 2.4, L is the characteristic length which for an open tube is considered to be the diameter of the tube. In an open tube, the flow is laminar when Re < 2300, and for Re > 2300 the flow is turbulent. Due to the existence of the mixer blades, the flow accelerates in some regions and it reaches high velocities. However, even in high velocity regions the flow is far from turbulent state. The working medium is considered kg L to be silicone oil (ρ = 970[ m 3 ] and µ = 100[mP a.s]). The flow rate of 56[ h ] resulting in a mean velocity of u = 6.3[ mm s ] corresponding to Re = 3.4 is prescribed at the inlet. Calculations show that the maximum local Reynolds number is less than 15 which indicates a pure laminar flow field. The local Reynolds number is defined by Equation 2.4, where U is the local velocity and L is the diameter of the tube. Furthermore, it was shown that the Kenics mixer has a threshold to transition at Re = 1000 [33]. Since the subject geometry is in close resemblance to Kenics mixer, it can be inferred that the flow is in deep laminar regime. Besides Navier-Stokes and mass conservation equation which govern the flow field, there are two other important equations in the current study. The motion of passive tracers is governed by the three-dimensional kinematic equation dx = u(x, t), x(0) = x0 dt

(2.5)

which is the main equation for the Lagrangian description of mixing. It describes the temporal evolution of the positions x of tracers released at the position x0 in the flow field u(x, t). The general solution to the equation is x(t) = Φt (x0 ). The general solution uniquely determines the position x at time t for given initial tracer position x0 . For steady state cases, the solution is u = u(x). It shows that the tracer paths coincide with the streamlines of the flow [49]. For concentration field, the advection-diffusion equation is the governing equation. u · ∇(c) = ∇ · (α∇c) + R

(2.6)

where c the concentration of the species, α the diffusion coefficient, R sources or sinks of the quantity c, and u is the velocity vector. The first term accounts for the convective transport due


7

to velocity field, u. On the right-hand side, the first term describes the diffusion transport, and finally the second term on the right-hand side represents a source or sink term. The important non-dimensional number in advection-diffusion equation is the Peclet number. It is defined to be the ratio of the rate of advection by the flow to the rate of diffusion of a scalar quantity. Peclet number is defined as follow; UL Pe = (2.7) α The Peclet number is the product of the Reynolds number and the Schmidt number. Schmidt number is defined as the viscous diffusion rate over molecular diffusion rate. Sc =

2.2

µ ρα

(2.8)

Pipe-entrance conditions

For comparison of measured velocity profiles and analytical solutions at the inlet and outlet (i.e. swirl), knowledge about pipe entrance conditions is necessary. Suppose that there is a plug flow at the entrance of the pipe. At the entrance, there is no boundary layer, but exactly after the flow enters the pipe, the boundary layer starts to grow. It takes some time for boundary layer to grow and fill the pipe completely, see Figure 2.2. The distance from the entrance to the position in the pipe where the boundary layer is completely developed (Poiseuille flow) is called the entrance length, Le . The entrance length for laminar flow (ReD < 2300 for tube without obstruction) is given by the following formula [17] Le = 0.058ReD D

(2.9)

where ReD is the Reynolds number and D is the hydraulic diameter of the tube.

Figure 2.2: The entrance length in a pipe flow [53].

2.3 2.3.1

Experimental Methods Particle Tracking Velocimetry

PTV is based on Lagrangian viewpoint, it tries to reconstruct the trajectories of the individual particles in the flow field. In order to visualize the flow field, small, reflecting, neutrally buoyant tracer particles are seeded in the flow field. Sufficient illumination is required for visualization which is supplied by a light source. Multiple synchronized cameras are recording the particles positions in the flow field. For 2D PTV, one camera is sufficient. However, in 3D PTV case at least two synchronized cameras are needed because in 3D case the depth of the field is also unknown and one camera is unable to determine the depth of the field. This is the reason that


8

human has two eyes. Consecutive images are used to track individual particles. When two cameras view a 3D scene from two distinct positions, there are a number of geometric relations between the 3D points and their projections onto the 2D images that lead to constraints between the image points. The epipolar geometry is used to make correspondence automatically. The technique has been tested and successfully applied for PTV measurements[8–10]. Figure2.3 shows the projection of the point X on image 1 is x1, then the epipolar line can be calculated on image 2 which the corresponding point, x2 has to be found.

Figure 2.3: Epipolar geometry in two views for different refractive indices geometry (left) and principles of epipolar line method for four cameras (right) [19].

Like all other PTV systems, a software is needed to track the particles. In the present study, an algorithm which is developed and implemented at ETH, Zurich is used. This code uses a combination of the image space based tracking technique and object space based tracking technique. The software is freely available for non-commercial use [51]. See Appendix C for detailed discussion on ETH code and general PTV working principle. Like all other measurement devices, PTV requires to be calibrated. The ETH code enables us to use either a 3D calibration body or a 2D translating plate. In case of 2D plate, the plate should be placed in multi-positions to get a 3D grid. However, inaccuracy in positioning the plate leads to errors in calibration. Hence, 3D calibration body has been chosen. The calibration body is shown in Figure 2.4-right. It can be inferred that the grid close to the points on the calibration body is accurate, however, it suffers from inaccuracy far from the points of the calibration body due to the fact that there are extrapolations to generate the grid in these regions. Figure 3.10-left shows a view of the calibration body and cameras. The coordinate axes is defined in such a way that y is pointed upward, x is pointed to the right and z is coming out of the plane. Cameras are located at the z positive region. The cameras are oriented in such a way to have maximum field of view. The rotation angles are obtained during the calibration procedure. To get the maximum observation area, the cameras are rotated sideways, so the rotation around z axes is 90◦ .


9

Figure 2.4: Experimental Poiseuille profile -dots- and the theoretical Poiseuille profile-line(left) (left) Coordinate axes (middle) Calibration body (right).

2.3.1.1

Performance issues

Two important source of the disturbances in optical measurements are reflection and refraction. In fact these two disturbances are the primary challenge for the optical measurement system. Up to now, the 3D-PTV flow measurements are limited to measurements in simple domains: rectangular tanks[34–40]; the flow field inside the cylindrical object is viewed through its flat end walls or indirectly through a rectangular view box which surrounds the cylindrical tube[41]. However, the flow inside a static mixer has much more complexity due to the cylindrical boundary and presence of the internal geometry. This complex domain results in two abovementioned optical disturbances which dramatically deteriorate the tracking performance. Jilisen solved these problems [19]. Reflection : When a ray of light falls onto a plate, a portion of it is reflected. Fresnel equations describe the fraction of incidence energy transmitted or reflected at a plate surface and the fraction depends on the change in the refractive indices, the incidence angle and the polarization of the incidence angle [20]. In some areas, the intensity of the reflected light is higher than that of the tracer particles. This reflected light is displayed in the image as a bright dot/zone which completely deteriorates the performance of the tracking of particles. Moreover, when the incident angle becomes larger than a critical value, all the light is reflected, see Figure 2.5. With curved surfaces, the whole situation is exacerbated. Curved surfaces deliver more internal and external reflection [19]. Therefore, the reflection plays an important role in PTV performance and must be taken into account for designing the PTV setup. Using the optical filters and the fluorescent tracer particles can resolve the reflection problem [19]. Refraction : As can be seen in Figure2.5 (left), a ray of light passes through different media refracts at the interface. The refractive angle of two media is determined by using Snells law [20]. n1 × sin(θ1 ) = n2 × sin(θ2 )

(2.10)

where θ1 and θ2 are the incidence and refracted angle respectively, n1 and n2 are refractive indices of the two media. Refraction due to the flat walls can easily be managed by the PTV image


10

Figure 2.5: refraction (left), critical angle (middle), reflection (right) [28].

processing code (ETH code). However, refractions by the cylindrical tube are not considered in the ETH code. Placement of the test section inside a cubical view box filled with working fluid, decreases the internal refraction to acceptable levels[19, 26]. In order to decrease the refraction even more, it is possible to use refractive index matching technique[21, 22]. In this method, the refractive index of liquid is match to that of the walls and test section. In Chapter 3 we will see that the refractive index of the new working fluid is better matched with the refractive index of the walls. Above all, modifications in ETH code has been implemented to make it possible for ETH code to handle the cylindrical objects (The modeling of the set-up is performed by Ad Holten). Hence, there will be no more errors due to the curved object. It is worth mentioning that the refractions due to the internal geometry are calculated in [26] which shows that they are negligible.

2.3.2

Interpolation of 3DPTV data

In the current setup, the tracer particles are distributed in the bottom reservoir. In fact, setting the initial position of a tracer particle is beyond our control. Hence, the density of the velocity vectors is not uniform on a cross section. In order to plot the velocity contours in different cross sections, a regular grid (100 × 100) has been introduced. Furthermore, interpolation on a regular grid enable us to compare data from different experimental runs. Interpolation has been executed using built-in MATLAB cubic interpolation scheme.

2.3.2.1

Validation of the interpolation on regular grid

A test has been done in MATLAB to validate the regular grid for interpolation. An arbitrary analytical function has been defined in such a way that its shape is similar to the velocity contours at the inlet of the mixer. The following equation describes the surface:

2 −0.01(y+10)2 )

z(x, y) = 20 × e(−0.01(x+10)

2 −0.01(y−10)2 )

+ 20 × e(−0.01(x−10)

Figure 2.6-left shows the defined equation which consists of two maxima. x and y vary between −28 and 28. In order to examine the interpolation error, the exact values of the z function on one sample PTV data are calculated. Then, the data are interpolated on the regular grid. Therefore, the interpolation error can be estimated using the difference between exact values and interpolated ones, which shows a σ = 0.167 (standard deviation). By knowing the maximum value of the function which is 20, the relative error is calculated to be 0.8%. Compared to the


11

PTV error (we will see that the PTV error is around 6%, Sec. 5.2.2), the interpolation error is negligible. Figure 2.6-right shows the subtraction of the exact values and interpolated ones on the regular grid.

Figure 2.6: Arbitarary surface for interpolation test (left) and the error of the interpolation (right).

2.4

Numerical methods

In the field of fluid dynamics, there are different software packages which enable researchers to investigate the flow field. In the present study, the analysis of the flow is performed by COMSOL Multiphysics 4.2 software. COMSOL uses finite element methods for solving the governing equations. Besides the numerical study on the flow field, concentration study has been done. This concentration study can be used as a reference for future studies. COMSOL can solve various physics; electrical, mechanical, chemical, fluids ,and etc. In the present studies, the Laminar Flow (spf) was selected. The model was defined to be stationary. The solver was set on generalized minimal residual method (GMRES) with preconditioning. The convergence criterion was set to a residual of 10−7 for velocity field. COMSOL uses a Newton-type iterative method to solve the nonlinear terms. The momentum equation is convection-diffusion equation. Such equations are unstable if discretized using Galerkin finite element method. There are three types of stabilization methods available for Navier-Stokes equation in COMSOL; streamline diffusion, crosswind diffusion, and isotropic diffusion. Streamline diffusion and crosswind diffusion are consistent stabilization methods and isotropic diffusion is an inconsistent stabilization method. In the present study, streamline diffusion and crosswind diffusion are used for stabilization. For the concentration field, Transport of Dilute Species (chds) has been selected for the physics. The model was defined to be stationary. Since for the concentration study, the velocity field is required, two different steps are defined for solution. In the first step, COMSOL solves the governing equation of the flow field. Then, in the second step, COMSOL uses velocity field which is the consequence of the previous step, to solve the concentration filed.

2.4.1

Geometry

The geometry has been generated in COMSOL itself, using its geometry tools. In order to mimic the experimental conditions, the thickness of the surfaces are set to 1[mm]. The mixer geometry is located inside a circular tube. The diameter of the tube is set as 56[mm]. For one period


12

study, the length of the tube is equal to 212[mm]. The length of the period itself is 112[mm]. An entrance length of 40[mm] is introduced. In order to decrease the effects of the outlet boundary conditions, an outlet tube with a length of 60[mm] is defined. Figure 2.7 shows the blade geometry and the coordinate axes. The inlet of the tube is located at y = +40[mm]. The inlet of the first element is at y = 0. In this context a period refers to the entire blade geometry, see Figure 2.7, and one element refers to half of a period. In other words, each period consists of two elements.

Figure 2.7: Blade geometry of Q type mixer.

2.4.2

Mesh

COMSOL is able to generate the geometry and mesh itself. Tetrahedral elements are selected for meshing the geometry. For this study, linear elements are chosen for the flow field and quadratic elements for the concentration simulations. Quadratic elements shows much better performance for the concentration field, see Appendix D. To get the best results for the concentration field, the mesh size is set to the finest that can be calculated on the Mechanical Engineering computer cluster. The maximum element size is set to 1.8[mm]. The field consists of 1,900,000 elements. The grid can be validated for larger mesh sizes for velocity field, but for concentration field, the error increases with increasing the mesh size. For four periods simulation, the number of elements is the same as one period, 1,900,000, but the mesh is coarser. The maximum mesh size is set to 2.4[mm].

2.4.3

Mesh validation

In order to validate the mesh, simulation for different mesh sizes has been performed. The mesh is generated in COMSOL. The first mesh consists of 1.200,000 elements and the second mesh consists of 1,900,000 elements. The velocity at different cross sections are compared. The deviation is quantified via ≡ (us1 −us2 )/us,max where us1 and us2 are the local velocities in the cross sections and us,max is the maximum velocity. Figure 2.8 shows the differences in velocities for three different cross sections. The statistics are given in Table 2.1. The error occurs close to the surfaces, where the velocity is zero. There is a steep velocity gradient close to the surfaces, which leads to interpolation error in these regions.

Chapter 2. Theory and methods Velocity component inlet middle outlet

13 σ (standard deviation) 0.6 1.1 0.4

µ (mean) 0.08 0.10 0.05

M (median) 0.005 0.023 0.003

Table 2.1: The statistics of the velocity analysis for inlet, middle and outlet (the numbers are in percentage).

Figure 2.8:

Difference in velocities for two different mesh numbers for inlet(top-lef) middle(top-right) and outlet(bottom).

The same analysis has been performed for the concentration field. The deviation is quantified via ≡ (cs1 − cs2 )/diff(cs ) where diff(cs ) is the concentration difference. The analysis has been performed for two cross sections ; outlet of the geometry and 60[mm] after the outlet. Figure 2.9 shows the difference for two different mesh numbers. The statistics are given in Table 2.2 for outlet/60 [mm] after outlet. The most of the error occurs where the concentration gradient is high. In overall, 1,900,000 quadratic elements yields acceptable results. Velocity component inlet middle

σ (standard deviation) 0.00228 0.9917

µ (mean) 0.00028 0.0041

M (median) 0 0.0048

Table 2.2: The statistics of the concentration analysis for outlet and 60[mm] afterthe outlet (the numbers are in percentage).

Figure 2.9: Difference in concentration for two different mesh numbers for outlet (left) and 60 [mm] after outlet (right).

Chapter 3

Further Development of The Experimental Set-up 3.1

Experimental set-up

Pressure gradient is the driving force in the real static mixers in industries. Static mixers consist of mixing elements placed inside a pipe or channel. This condition is imitated by positioning the mixing element inside an optically accessible test section. Figure 3.1 shows a schematic view of the setup. The system is a closed system. The working fluid is pumped from bottom reservoir (K) to top reservoir (A). In order to keep the working fluid level in the top reservoir, and consequently the pressure head, constant, a smaller bucket (B) is positioned inside the A. The B contains 5L of the working fluid. Reservoir B is fed by pump continuously. When the reservoir B overflows, it indicates that the working fluid level - consequently, the pressure head - is constant. The working fluid flowing out of the reservoir B returns back to the bottom reservoir through the over flow tube (M). The main stream of the working fluid passes through a flow straightener (C), test section (F) and flowmeter (I) and then it flows into the bottom reservoir.

Figure 3.1: Schematic view of the setup.

14

Chapter 3. Further development of the experimental set-up

15

The inner diameter of the tube is the same over the entire line. In order to be able to insert elements into the test section, three couplings are introduced (D). Tube E ensures that the flow is fully developed at the inlet of the static mixer. Entrance length is dependent on Reynolds number. In the present study, since the Reynolds number is quite small, the entrance length is small too. Based on Equation 2.9, the entrance length is obtained to be 11.4[mm]. Two arrays of light-emitting diodes (LEDs) (G) are designed for the illumination of the seeding particles. Test section is comprised of a rectangular box (view box) and a cylindrical tube. The cylindrical tube is placed in the upright position at the center of the rectangular box. The rectangular box is placed to reduce the refraction due to the cylindrical tube. The rectangular box is filled with the same liquid as the working fluid. Four synchronized CCD cameras (MegaPlus ES2020, 1600 pixels-50[mm] lenses) are recording the flow field. The maximum frequency of the cameras is 30[Hz] which leads to a data transfer of 230[ MsB ]. The working fluid passes through setting valve (H) and a rotameter. The factory calibration of the rotameter is only suited for water as working fluid. Here, the rotameter is calibrated by using the outcome of the PTV image processing for Poiseuille profile. The working fluid is pumped from bottom reservoir to top reservoir by a barrel pump (J). Previous pump was a typical centrifugal pump which is designed for transportation of the low viscosity liquids. Barrel pumps which can work with highly viscous fluids are operating on progressive cavity (screw) pumping principle. Further information about the design of the experimental setup is available in [19].

3.2

Matching Fluid

Since, measurements at low Reynolds numbers are desired, the working fluid has been changed. The working fluid used to be water. It was not possible to reach the Reynolds number less than 500 with water [19]. The selection of the new working liquid was done by the following criterion; not flammable at low temperature, high flashpoint temperature, non-toxic, appropriate density, appropriate viscosity, availability, transparency, high boiling temperature, price, and appropriate refractive index. Table3.1 shows characteristics of the various liquids. As can be seen from the table, silicone oil has a wide range of viscosities. It also satisfies all other parameters for a suitable liquid. Its refractive index is also closer to that of glass compared to water. It implies that the error due to the cylindrical object decreases when the working liquid is silicone oil.

3.2.1

Lift force, density effects and response time of particle

The tracer particles are polymethylmethacrylate (PMMA) particles which show a proper fluoreskg cence. The density and diameter of the tracer particles are 1190[ m 3 ] and 20-50[µm], respectively. Lift force, density effects and response time of the particles are three important factors in selection of the working liquid. For detailed discussion and formula, please refer to Appendix C. The error due to the lift force, the density effects and the response time must have a reasonable value. Otherwise, the whole particle tracking measurements is in vain. Although the density of the silicone oil is smaller than water, the density effects in the silicone oil is minute compared to water. It is mainly due to the high viscosity of the silicone oil. As can be seen from figure 3.2-top-left, vertical velocity due to the density effects in case of silicone oil is much smaller than that of water. Therefore, using silicon oil decreases the density effects.

Chapter 3. Further development of the experimental set-up Fluid Water Soybean oil Turpentine Silicon oil mixture Olive oil Mineral oil phosphorus trichloride tin tetrachloride germanium tetrachloride antimony pentachloride Kerosene Tung oil Castor oil Corn oil Canola oil Sunflower oil

n 1.33 1.47 1.47 1.40 1.47 1.48 1.51 1.50 1.46 1.59 1.45 1.51 1.48 1.47 1.47 1.47

Density 1000 930 870 970 920 850 1570 2226 1880 2340 820 930 960 925 920 918

16

Viscosity (mPa.s) @ 20C 1 69 1.4 0.006-1000000 84 10 20.5 986 65 57 (@ 25C) 49 (@25C)

Table 3.1: Working fluid properties[22, 24, 25, 50, 55]. −8

4

x 10

0.18 Water Silicone oil

0

−50

Water Silicone oil

0.16 0.14

−100

−150

Response time (ms)

3

Lift force (N)

Vertical velocity ( µm s )

3.5

Water Silicone oil

2.5 2 1.5

−200

0.1 0.08 0.06 0.04

1 −250

0.12

0.02

0.5

0 −300 20

0 25

30

35

40

Particle diameter(µm)

45

50

0

100

200

300

400

Reynolds Number

500

600

20

25

30

35

40

45

Particle diameter(µm)

Figure 3.2: Density effects (left) lift force (middle), and response time (right) of the particles in silicone oil and water.

Figure 3.2-top-right shows that the lift force. Lift force depends on Reynolds number. In general, the lift force in silicone oil is higher than water for a specific Reynolds number. However, since the minimum possible Reynolds number with water is 500, the lift force in silicone oil is compared with water at Reynolds number of 500. The lift force in silicone oil for Reynolds number of 3.45 (the Reynolds number which the experiments are performed) is calculated to be 1.6 × 10−8 [N ] which gives an acceleration of 1.5 × 10−7 [ sm2 ] to the particles. Figure 3.2-bottom shows the response time for the seeding particles. The figure shows that the response time for the tracer particles in silicone oil is much less compared to water (water of O(0.1[ms]) and for oils O(10−3 [ms])). Considering all these effects and other properties of silicon oil like, little change in physical properties over a wide temperature range compare to other oils, low toxicity, non-aggressive, non-corrosive, wide range of viscosity available, no contamination, low flammability and antifoam, make it a suitable candidate for PTV measurements.

50


3.3

17

Opacity

Unlike water, silicon oil is extremely sensitive to dirt and air bubbles. Small amount of water makes the entire silicone oil opaque. Figure 3.3 shows the silicone oil when it becomes opaque. The second main cause of opacity is the air bubbles. Since, the silicone oil has higher viscosity than water, it is more difficult to remove the air bubbles. In order to minimize the air bubbles, the position of the feeding tube has been changed. As described at the beginning of this chapter, the silicone oil is pumped from the bottom reservoir to the top reservoir. Figure 3.4 shows the position of the inlet tube and the bucket. The inlet tube and the main tube used to be concentric. Therefore, at the beginning of the experiment, which the main tube is empty, the silicone oil will fall all the way to the bottom reservoir. This fall causes more air entrainment. In order to avoid falling of the silicone oil, the position of the inlet tube has been changed. Figure 3.4 shows the new position of the inlet tube. After the adjustment, the silicone oil tends to flow over the boundaries, which leads to less air bubbles generation.

Figure 3.3: Opaque silicon oil

3.4

Distribution of the particles

The polymethylmethacrylate (PMMA) particles are the best option for proper visualization. They have a reasonable size which occupy one pixel at least. Furthermore, they are good fluorescence. The particles diffuse into the water easily and they distribute in the water uniformly. However, in the silicone oil, they behave differently. The manufacturer makes a suspension of the particles in the water. As it is expected, the water is not dissolvable into the oil, so the water molecules have a tendency to stick together and make droplets. Figure 3.5 shows the


18

Figure 3.4: The position of the feeding tube and top reservoir before (left) and after adjustment (lright).

water droplets with tracer particles in the silicone oil with arrows. In order to deal with this problem, the particles and water should be separated. The easiest way for separation is to let the water evaporate. Since, a tiny amount of the particles gives the appropriate particle density, there is no need to heat up the suspension for evaporation.

Figure 3.5: Particles which are trapped inside the water droplets.


3.5

19

Fluctuations at low Reynolds numbers

Jilisen performed the experiments for umean = 9[ mm s ] and higher. He mentioned that due to some fluctuations at low mean velocities, he didn’t perform experiments at Re < 500[19]. Based on the description of the fully developed flow, the transverse velocity components should be zero. However, as Figure 3.6 displays, ux and uz suffer from fluctuations.

Figure 3.6: Absolute transverse velocity contours ux (left) and uz (right)- the axial velocity is 12.3 [ mm s ].

Apart from the abovementioned fluctuations in ux and uz , the flow rate was unstable. There was a decrease in flow rate over the time. It was found that the source of this decrease was the setting valve. The setting valve is responsible for adjusting the flow rate. Changing the valve had considerable effect on fluctuations too. Figure 3.7 shows the transverse velocity components for the entire measuring volume before (left) and after changing the valve (right). As can be seen from the graph, the fluctuations decreased after changing the valve. The maximum error in ux used to be 15%, but after the substitution it becomes 5%. The maximum error in uz decreased from 20% to 8%. Therefore, the defective valve was one of the sources of the fluctuations at low velocities in Jilisen’s experiments. See Appendix A for discussion on the setting valve.

3.6

Internal elements

There are some drawbacks to the old internal elements. First of all, the accuracy of the old blades is not perfect. The second problem is related to the connection of two elements. It used to glue them toghether for connection of the two parts. For two elements, it is possible, however, for higher number of elements the structure is not rigid enough. The third disadvantage of the old internal elements is a rod at the downstream region to fixate the blades. Therefore, a new construction method, 3D printing, has been investigated and tried. The product of the 3D printers has much more accuracy compared to the old fashion ways. Furthermore, It is possible to construct different elements regardless of the complexity of their geometry. Above all, it is possible to make a male/female mating surfaces which makes a strong connection for two parts.


20

Figure 3.7: ux (top) and uz (bottom) before (left) and after (right) changing the setting valve.

However, making the outcome of the printers transparent is a tedious job. Mixer geometry with 3D printing technology has been constructed and trialed. Unfortunately, the blades reflect the light at almost the same wavelength which the fluorescent particles do. So, the blades are visible in the images, see Figure 3.9. Furthermore, the edges of the blades are not clear enough. There is light distortion on the edge of the blades. Elimination of the static background has been tested, but the tracking results is not yet satisfying. Figure 3.8 shows that tracking of the particles is completely deteriorated by the edges of the new blades. Moreover, The blades are not transparent enough, so most of the particles which travel through the back of the blades are not captured by the cameras.

Figure 3.8: Deterioration of the tracking when 3D printed are used

Since it is not possible to use the 3D-printed blades, the new blades are produced in the Mechanical Engineering workshop. There are two differences between the new blades and old ones. The previous blades were glued together, however, for the new blades a female/male connection is introduced. Furthermore, In the new configuration, the rod is eliminated because the blades are fitted the cylindrical wall, so they don’t move.


21

Figure 3.9: The image of the old blades (left) and 3D printed blades (right).

3.7

Modification in calibration

The ETH code cannot handle the curved cylinder and it simply ignores it. Consequently, there is always a systematic error which is independent from the accuracy of calibration body and calibration procedure. This error can be seen in experimental results too. In order to take a closer look at the error in experimental results, the flow field in an empty tube has been investigated. The fully developed flow in the tube is known as Poiseuille flow, see Figure 3.10. The blue dots are the experimental results and the blue line is the theoretical Poiseuille profile. Blue dots and the line must coincide in ideal case, however, there are differences between experimental results and theoretical Poiseuille profile. The differences are higher closer to the cameras (cameras are located in the positive r region). As mentioned previously, the ETH code cannot handle the curved object and it simply ignores

Figure 3.10: Experimental Poiseuille profile -dots- and the theoretical Poiseuille profile-line.

it. A model of the setup was developed in the C programming language based on configuration of the cameras and Snell’s law. The program was developed in Applied Physics Department by Ad Holten. In the modeling, the procedure of the image processing (ETH code) was reversed. The working principles of the ETH code and the model are given as follows; ETH code: ETH code generates a grid based on the calibration images. Based on the generated grid, ETH code figures out the position of the particles in the space. Four cameras capture the flow field inside the cylinder. By processing these four images and using the generated grid, ETH code determines the position of the particles in space. Model: The first essential step in the modeling of the setup is the generation of the artificial


22

calibration images. By knowing the configuration of the cameras, and using Snell’s law it is possible to make artificial calibration images. Figure 3.11 displays the artificial and real calibration images for the first camera. The two images are almost the same which shows accurate modeling of the setup. The second step is the generation of the images of the particles in space.

Figure 3.11: Real calibration image (left) and artificial one (right).

In order to do that, a set of 5,000 particles are scattered into the space. The exact positions of all these 5,000 particles are known. Using the artificial calibration images, it is possible to define how each camera sees the particles. The C script produces artificial images of the particles for four cameras. The artificial images were processed via ETH code and the outcome of the ETH code gives the position of the particles. The exact position of the particles are known at the beginning. The outcome of the ETH code provides the position of the particles when the curved cylinder is neglected. The comparison of the exact positions and the outcome of the ETH code, implies the error due to the curved tube. Figure 3.12 shows the exact position (red circle) versus the processed positions (blue dots) for ideal case (left) and current case (right). The difference between the ideal case and the current case is that in the ideal case the artificial calibration images are used for the determination of the particles positions. However, in the current case, the real calibration images are used. The number of detected particles are much less in the current case compared to the ideal case due to the small differences between real and artificial calibration images.

Figure 3.12: Deviation in position due to curve tube in ideal case (left) and current case (right).


23

Figure 3.13 shows the difference between the exact value and the outcome of the ETH code in z and x directions. The difference (error) in z direction is much higher than the difference in x direction, as it was expected. Note that the cameras are located at positive z region, see Figure 3.10. It is definitely more difficult for cameras to distinguish the displacement in depth of the tube (z direction) rather than across the tube (x direction). Figure 3.13-left shows that the error is higher at positive z region where the cameras are located. This conclusion can be drawn from the experimental results, Figure 3.10, as well in which the deformation occurs at positive r region where the cameras are located.

Figure 3.13: The error in z (left) and x (right) directions for current case.

The following coefficients are introduced to compensate the error due to the curved tube.

For z > 0 , z = z(1 − 1.8 × R1 ) For z < 0 , z = z(1 − 1.0 × R1 ) For x > 0 , x = z(1 + 0.09 × R1 ) For x < 0 , x = x(1 + 0.3 × R1 ) where R is the radius of the tube (R=28[mm]). The coefficients are somehow arbitrary which were obtained via trial and error. It is out of the outline of this thesis to implement the new calibration coefficients in the ETH code, however, it is possible to modify the calibration when analyzing the results in MATLAB. Figure 3.14 shows the error in Z and X directions after the implementation of these coefficients. It shows a tremendous improvement specially in z direction. There is a peak in z = 0[mm] and x = −20[mm] which is a random error due to mismatching of a particle. The error in Y direction is less than 0.3[mm] which is smaller compared to the errors in other directions, so it is neglected. The experimental results proved the improvement in calibration as well. Figure 3.15 shows that after modifying the calibration, the experimental data are better fitted with the theoretical Poiseuille profile. The blue dots are the experimental data points and the line is the theoretical Poiseuille profile. The deviation is quantified via ≡ (uth − ue )/uth,max where uth is the theoretical velocity, ue is the corresponding experimental velocity and uth,max is the maximum theoretical velocity. The statistics read µ = 2.42%(mean) σ = 2.68%(standard deviation) before implementation of the modification. Mean and standard deviation become µ = 1.77%(mean) σ = 2.28%(standard deviation) after modification in calibration. It shows almost 15% improvement.


24

Figure 3.14: Error after modification in calibration in z(left) and x(right) directions.

It is worth mentioning that implementation of the modification coefficients in the ETH code is out of the scope of this thesis. However, the modifications are implemented in the MATLAB code for post processing of the ETH code data. Hence, the ETH code still cannot handle the curved interfaces, but it is possible to include the effect of curved cylinder in the post processing of the data.

Figure 3.15: Experimental Poiseuille profile (dots) and the theoretical Poiseuille profile(line) after modification in calibration.

In the next chapter, we will see that the error of the experimental results will be around 6%. Jilisen observed an error of 10 − 15% [26]. It shows a good improvement compared to the results of Jilisen.

Chapter 4

Numerical simulation of 3D flow field The performance of the subject static mixer is evaluated via experimental 3D-PTV measurements and numerical predictions. The results are concentrating on three properties relevant to mixing ; (i)the velocity field, (ii) the coherent structures that form in the 3D streamlines pattern and (iii) the concentration field (only simulations). Figure 4.1 shows the blade geometry and the coordinate axes. The inlet of the tube is located at y = +40[mm]. The inlet of the first element is at y = 0. In this context a period refers to the entire blade geometry, see Figure 4.1, and one element refers to half of a period. In other words, each period consists of two elements. The 3D frame of reference is defined in such a way that y direction is along the static mixer (axial direction) and x and z directions are transverse.

Figure 4.1: Blade geometry of Q type mixer.

The boundary condition at the inlet is Poiseuille profile with mean velocity of umean = 6.3[ mm s ] kg L which corresponds to flow rate of 56[ h ]. The density of the silicon oil is considered to be 970[ m3 ]. No-slip condition is assumed for all walls. At the outlet, constant pressure condition is imposed. The constant pressure gauge of p = 0[pa] is applied for the outlet. In order to vary the Reynolds number, the dynamic viscosity has been changed. Numerical simulations has been performed for Re = 3.4, 6.8, 34.2, 68.4, and 100 which correspond to the dynamic viscosity, µ, of 100, 50, 10, 5, and 3.4[mpa.s]. For the concentration study, c=0 and c=5 are prescribed in one half of the inlet. The diffusion 25

Chapter 4. Numerical simulation of 3D flow field

26

coefficient is assumed to be 5e − 8[ m2 s ] which corresponds to Peclet number of 70560. The diffusion coefficient is assumed small to see the mixing occurs by advection.

4.1 4.1.1

One Period Velocity Field

The velocity contours for four cross sections along the mixer have been analyzed for five different Reynolds numbers; 3.4, 6.8, 34.2, 68.4, and 100. The velocity contours for 20[mm] before onset of the mixing element are depicted in Figure 4.2 (y = +20[mm]) for two different Reynolds numbers; 3.4, and 34.2. Poiseuille profile is prescribed at the inlet of the tube, so, far from the inlet of the mixer geometry the velocity profile is still Poiseuille profile which shows a circular profile. However, closer to the mixing geometry, the circular pattern is slightly deformed, see Figure 4.2-right. The deformation is due to the effects of the mixing elements. The transverse velocity components, ux and uz , are small in magnitude compared to the axial velocity component, uy (O(7%) of the axial velocity) which shows that the circulations are small in the upstream region. Comparing the different velocity contours for different Reynolds numbers at upstream region reveals that the transverse velocity components become smaller at higher Reynolds numbers. In other words, the effects of the mixing geometry on the upstream flow decreases as the Reynolds number increases. It is worth mentioning that the differences in the upstream region for different Reynolds numbers are small.

Figure 4.2: Cross-sectional velocity field in the upstream region at y = +20[mm], ux , uz (transverse components), and uy (axial component) -left to right- for Re = 3.42 (top) and Re = 34.2 (bottom).


27

The cross-sectional profile in the middle of the first element, shows that the transverse velocity components become stronger compared to upstream region, see Figure 4.3. It indicates that there are more circulations in this region. Looking at ux it can be seen that two distinct regions are separated by the mixing element; top region (red) and bottom region (blue). The materials in the red region are travelling rightward and the blue region materials are moving leftward. Figure 4.4 shows the velocity field within the first element. The velocity field shows the materials are traveling rightward at one side of the element. Looking at uz it can be inferred that in each side of the mixing element, there are two different regions. One stream is going upward (red) and the other one is going downward (blue). These opposite movements show that the materials undergo shear stress and consequently, stretching in each side of the element. For higher Reynolds numbers, the flow shows the same behavior, see Figure 4.3-bottom. The maximum value of ux is almost constant for different Reynolds numbers, however, uz shows stronger dependency on Reynolds number. Increasing the Reynolds number results in a growth in uz which indicates the existence of the stronger shear stress and stretching in this region at higher Reynolds numbers.

Figure 4.3: Cross-sectional velocity field in the mixing region at y = −28[mm], ux , uz (transverse components), and uy (axial component) -left to right- for Re = 3.42 (top) and Re = 34.2 (bottom).

The velocity components at the outlet are depicted in Figure 4.5-top for Re = 3.42. The xcomponent velocity, ux , is quite considerable in magnitude. However, it is not covering all the cross sectional area. It shows locally transversal motion. In contrast, uz covers most of the cross section, but its magnitude is not large compared to axial velocity component (O(15%)). uz shows quite a strong dependency on the Reynolds number. The maximum value of uz inmm creases from 3[ mm s ] to 11[ s ] for Re = 3.4 to Re = 100. In contrast, ux doesn’t show such a mm strong dependency on the Reynolds number. It increases from 8[ mm s ] to 10[ s ] for Re = 3.4 to Re = 100. In general, the transverse velocity components become larger with increasing Reynolds number. Furthermore, the x-component of the velocity covers a larger cross-sectional area at higher Reynolds numbers. Therefore, this conclusion can be drawn that the swirling


28

Figure 4.4: The velocity field inside the first element.

exit flow is stronger at higher Reynolds numbers.

Figure 4.5: Cross-sectional velocity field at the outlet at y = −112[mm], ux , uz (transverse components), and uy (axial component) -left to right- for Re = 3.42 (top) and Re = 34.2 (bottom).

Figure 4.6-top shows the velocity contours at 40[mm] after the outlet of the mixer, y = −152[mm], for Re = 3.4. Once the flow leaves the mixer geometry, it tends to become a fully developed flow (Poiseuille profile). At 40[mm] after the outlet, the transverse components become negligible (O(2%) of the axial velocity) and the flow shows a fully developed flow behavior. It can be inferred that the flow damps almost all the transverse components within 40[mm]. Note that the entrance length of the tube is 11.4[mm] for Re = 3.4. At the outlet of the mixer it takes longer distance to develop into Poiseuille profile due to the disturbances in the flow field at this region. Figure 4.6-bottom shows the velocity contours at the same cross section for Re = 34.5. As mentioned before, once the flow leaves the mixer geometry, it tends to become a fully developed flow


29

(Poiseuille profile). Since at higher Reynolds numbers, the transverse velocity components are larger in magnitude it takes longer distance to become fully developed. The velocity contours show that the flow is not fully developed 40[mm] after the outlet. In fact, the flow doesn’t show the fully developed behavior even at the end of the outlet tube (the length of the outlet tube is 60[mm]). The entrance length at Re = 34.2 is 114[mm].

Figure 4.6: Cross-sectional velocity field at y = −152[mm], ux , uz (transverse components), and uy (axial component) -left to right- for Re = 3.42 (top) and Re = 34.2 (bottom).

Figure 4.7 shows the transverse velocity field for Re = 3.45, 34,5 and 100. As can be seen from the graphs, the transverse velocity field is stronger for higher Reynolds numbers specially at the outlet. For Re = 3.45, the transverse velocity components are small at the outlet of the tube which is an indication of weak circulations. However for higher Reynolds numbers, the transverse velocity components are larger at the outlet of the mixer. The transverse velocity field allows the distinction of the vortical structure. Figure 4.8 shows the formation of a vortical structure at the outlet for Re = 100 whereas Figure 4.9 shows the velocity field for Re = 3.4 and 100.

Figure 4.7: Transverse velocity component for the entire domain for Re = 3.4, 34.2, and 100.


30

Figure 4.8: Formation of the vortical structure at the outlet of the mixing element at Re =100 .

Figure 4.9: Numerical velocity field for Re = 3.4 (left), and 100 (right).

4.1.2

Streamlines

Figure 4.10 shows the numerical streamlines in the static mixer for two different Reynolds numbers; Re =3.45, and 34.5. Comparison between two graphs shows that the flow structure for both cases are almost the same. However, there are distinct differences. Detailed discussion on the flow structure is given as follows; Figure 4.11 shows the streamlines at the inlet of the mixing geometry. It shows that the streamlines are deflected before entering the mixing element. The same effect can be tracked from the velocity contours too, where the upstream velocity contour was deformed slightly, see Figure 4.2. Soon after entering the mixer, the streamlines follow the blade geometry. The streamlines deflect which promotes the stretching of the materials. In the middle of the mixing element, the streamlines undergo a lateral dispersion due to bifurcation of the fluid stream by the downstream element, see Figure 4.13-right. It can be seen that the flow in one side of the first element is split into two parts and each part is going to one side of the second element. This is where cutting take place. The flow in one side of the second element is made of two streams each from one side


31

Figure 4.10: Simulated streamlines for the entire domain for Re = 3.4(left) and 34.2 (right).

Figure 4.11: Simulated streamlines for the inlet region for Re =3.4 (left) and 34.2 (right).

of the first element, see Figure 4.12. This shows the stacking in the current mixing configuration. By stacking we mean that the two streams from two different sides of the first element are joining together in the second element. This is actually the basic principles of the current configuration.

Figure 4.12: Cutting and stacking of the flow.

Figure 4.13-left shows the streamlines for two different Reynolds numbers; 3.4, and 34.2. The figure shows that the streamlines initial positions of which are close to the cylindrical wall don’t bifurcate for Reynolds number of 3.4 (blue lines). However, for higher Reynolds number, 34.2, (red lines) the bifurcation occurs. Regardless of the initial positions of the streamlines, the bifurcation for higher Reynolds numbers is more severe.


32

Figure 4.13: Streamlines in the mixing region (left) and at the outlet (right).

At the outlet, the flow enters an open tube which means that the flow starts to develop into the Poiseuille profile. Therefore, the flow has a tendency to damp the transverse velocity components. It can be seen that soon after the outlet, the streamlines start to travel through a straight line, and the swirling motion vanishes for Re =3.4. It indicates that there is no further considerable mixing when the flow leaves the elements at this Reynolds number. However, for higher Reynolds numbers, the streamlines show more swirling movement. Figure 4.14 shows the fluid stream passing one side of the downstream mixing element for three different Reynolds numbers ; 3.4, 34.2, and 100. It can be seen that at higher Reynolds numbers, a vortical structure which span most of the cross section appears.

4.1.3

Concentration field

The concentration field simulation will be used as a reference for future experimental measurements on concentration. Figure 4.15 illustrates the evolution of the concentration field for Re = 3.4. It shows how two different materials are mixing along the mixer. Three steps of a static mixer (stretching, cutting, and stacking) can be detectable in the concentration field. Stretching of an interface between blue and a red fluid is caused by the simple shear within the element. Cutting and stacking is cleverly realized by placing the next element under 90◦ . Figure 4.16 shows the concentration fields for two cross sections within the first element for three different Reynolds numbers; 3.4, 34.4, and 100. The top graphs show the concentration fields of a cross section which is located at y = −12[mm] and the bottom one is located at y = −48[mm]. As can be seen, as the materials are passing along the first element, they stretch. Comparison between two cross sections, clearly shows that the materials in blue and red region are stretching. For higher Reynolds numbers, the same behavior was observed. Figure 4.17 shows the concentration field within the second element at y = −64[mm]. It shows how the concentration field is developing. Figure 4.18-top shows the outlet concentration contour for five different Reynolds numbers; 3.4, 6.8, 34.2, 68.4, and 100. Figure 4.18-bottom shows the concentration field at 40[mm] after the outlet for the corresponding Reynolds numbers. Comparison between top graphs and bottom graphs shows that the concentration field doesn’t undergo considerable changes after the mixing


33

Figure 4.14: Streamlines at the outlet for Re =3.4(top), 34.2(middle), and 100 (bottom).

Figure 4.15: Concentration field at y = 0, −28, −56, −85, and − 112[mm] (left to right) for Re = 3.4.

element for low Reynolds number. This is consistent with our findings on weak swirling exit flow based on velocity contours and streamlines at low Reynolds numbers. Swirling exit flow at Re = 100 is such strong that it completely changes the pattern of the concentration at the outlet of the mixing element. Furthermore, it can be seen from the graphs that increasing the Reynolds number results in a more efficient mixing. The materials in the red region at the bottom side of the cylinder have a tendency to move towards the top side of the cylinder with increasing Reynolds number.

4.2

Periodic flow

The simulations for four periods have been performed for two different Reynolds numbers; 3.45, and 34.2. Numerical results show that the flow becomes periodic soon after the entrance for both Reynolds numbers. Figure 4.19 shows the velocity profile along the mixer for Re = 3.45.


34

Figure 4.16: Concentration field within the first element at y = −12[mm] (top) and y = −48[mm] (bottom) for Re = 3.4, 34.2, and 100 (left to right).

Figure 4.17: Concentration field within the second element at y = −64[mm] for Re = 3.4, 34.2, and 100 (left to right).

As can be seen, the flow becomes periodic within the first element and the flow is completely periodic in the second element, qualitatively. To examine the periodicity quantitatively, two approaches are chosen; (i) the velocity fields on five representative cross sections are examined for second and third periods (ii) the comparison between the velocity field in the second and the third periods has been performed.

4.2.1

Representative cross section

In order to examine the periodic flow field, the velocity fields on five cross sections in the second period are compared with the corresponding cross sections in third period. Figure 4.20 shows the cross sections that are selected in the second period. The corresponding cross sections in the third period are not depicted in the figure. The data are interpolated on a regular grid (100 × 100) using the built-in MATLAB interpolation scheme. The statistics for analysis on cross sections are given in Table 4.1. The numbering of the cross sections is based on the Figure 4.20. For example, (1) means the difference in velocity field on cross section (1) in the second period and its corresponding cross section in third period. Figure


35

Figure 4.18: Concentration field at the outlet, y = −112[mm] (top) and at 40[mm] after the outlet, y = −152[mm] (bottom) for Re = 3.4, 6.8, 34.2, 68.4, and 100 (left to right).

Figure 4.19: Simulation for four elements for Re = 3.4

4.21-right displays (1) for Re = 3.42. As can be seen from the graph, the error occurs close to the surfaces where the velocity gradient is high due to the fact that the velocity is zero on the surfaces. Steep gradients are the main source of the error in the interpolation. Figure 4.21-left and middle show the velocity contours on the cross section (1) for second and third periods, respectively, for Re = 3.42.

Figure 4.20: Five cross sections which the data are analyzed on.

Chapter 4. Numerical simulation of 3D flow field Reynolds number 3.4

34.2

Cross section 1 2 3 4 5 1 2 3 4 5

σ (standard deviation) 0.18 0.04 0.16 0.05 0.12 0.27 0.03 0.23 0.06 0.33

36 µ (mean) 0.075 0.002 0.07 0.002 0.012 0.067 0.0054 0.04 0.004 0.05

M (median) 0.026 0.004 0.007 0.004 0.002 0.002 0.003 0.009 0.03 0.002

Table 4.1: The statistics of the cross section analysis for two Reynolds numbers (the numbers are in percentage).

Figure 4.21: Velocity field on cross section 1 for second period (left) and third period (middle) and the difference between them (right).

4.2.2

On random points along the mixer

In order to examine the flow field inside a period, more than 25,000 points within the second period have been chosen. The velocity components (ux , uy , uz ) and magnitude (U ) on these points are compared with the corresponding points in the third period. Since the generated mesh in the second and the third periods are not exactly the same, interpolation for the third period is essential. The statistics are given in Table 4.2 for different velocity components and the velocity magnitude. The error relates to the interpolation and the steep gradients close to the surfaces. The error here, is higher than the error when the cross sections are examined. In COMSOL, it is possible to export the data with different resolutions. For the cross sectional analysis, the highest resolution has been selected for exporting the data due to the fact that the number of the points on a cross section is not too high for calculations. However, if the whole domain is considered, the number of the data points with high resolution is huge. The interpolation time in MATLAB is highly dependent on the number of the data points. Therefore, to make the interpolation time reasonable, a lower resolution has been chosen for exporting the data points. Lower resolution data is the main cause of the higher error. Based on the numerical studies, it can be concluded that the flow becomes periodic within the first period for both Re = 3.4 and 34.2.

Chapter 4. Numerical simulation of 3D flow field Reynolds number 3.4

34.2

Velocity component ux uy uz U ux uy uz U

σ (standard deviation) 1.42 2.83 1.63 2.43 1.13 2.06 1.11 1.92

37 µ (mean) 0.016 0.50 0.007 0.533 0.006 0.370 0.004 0.433

M (median) 0 0 0 0.03 0 0.117 0 0

Table 4.2: The statistics of the analysis on the random points along the mixer (the numbers are in percentage).

4.2.3

Periodic flow field

Figure 4.22 shows the velocity contours for 28[mm] after the inlet of the second period for Re = 34.2. Note that the flow is periodic in the second period, so these velocity contours will repeat themselves in the consecutive periods. Comparison between periodic flow and the flow for one period shows that they exhibit almost similar flow behavior. However, the maximum value for transverse velocity components are larger in periodic flow. It can be an indication of stronger shear stress and as a result, stronger stretching.

Figure 4.22: Cross-sectional periodic velocity field in the middle of the first element, ux , uz , (transverse components) and uz (axial component) -left to right- for Re = 34.2.

The velocity field at the middle of the second element is depicted in Figure 4.23. It can be seen that the velocity field shows the same behavior like middle of the first element. The stretching of the materials is obvious in the second element as well. At the outlet of the fourth period, the flow shows the same behavior as in the outlet of the one period case. It indicates that regardless of the number of the periods, the flow always shows the same behavior at the outlet of the mixer (This conclusion is drawn only for Re = 3.4 and 34.2). This conclusion is in agreement with the previous statement that the flow becomes periodic within the first element. Figure 4.24 shows the periodic velocity field for Re =3.4 and 34.2 whereas the transverse velocity field is depicted in Figure 4.25. The vortical structure at the outlet of each period is


Figure 4.23: Cross-sectional periodic velocity in the middle of the second element, ux , uz , (transverse components) and uz (axial component) -left to right- for Re = 34.2.

apparent in the figures.

Figure 4.24: The velocity field for Re =3.4(left) and 34.2 (right).

Figure 4.25: The transverse velocity field for Re =3.4(left) and 34.2 (right).

38


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Figure 4.26 shows the streamlines inside the mixing elements for four periods. All the streamlines are passing through one side of the first element. It shows that in both cases, the streamlines are spanned over the entire cross section at the outlet. It is an indication of efficient mixing.

Figure 4.26: Simulated streamlines for Re = 3.4 (left)and 3.42(right).

Figure 4.27 shows the periodic streamlines for Re =34.2. The streamlines at the upper side of the first element are colored blue and the streamlines at the bottom side are colored red. The figure clearly shows that the flow at each side of the first element bifurcates. Moreover, it shows that the flow at each side of the second element is made of two distinct streams, each of which comes from one side of the first element. The same behavior is observed for the flow in one period case.

Figure 4.27: Periodic simulated streamlines for Re = 34.2.

Figure 4.28 shows the concentration field for Re =3.4 (top) and 34.2(bottom). The figure shows how the concentration field develops along the mixer. Moreover, it can be inferred that the mixing at higher Reynolds numbers is more efficient. In mixers, self-similar pattern formation is an indication of periodicity. In other words, once the flow becomes periodic, the concentration field starts to repeat itself. In order to capture these self-similar patterns, the simulations have been performed for ten periods for Re = 56[42]. Figure 4.29 shows the concentration contours along the mixer. If the color range for each cross section is set to the maximum and the minimum value of the same cross section, the self-similar patterns become visible. Figure 4.30 shows the concentration field along the mixer. It is shown


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Figure 4.28: Concentration field for Re = 3.4 (top), and 34.2 (bottom).

that the flow field becomes periodic within the first element, however, the concentration field has not yet reached the state of self-similar decay. It is revealed that the concentration field becomes periodic after passing through two periods (four elements), see Figure 4.30.

Figure 4.29: Concentration contours for the inlet of the second, third, fourth, and fifth period (left to right) for Re = 56.

Figure 4.30: The concentration contours at the inlet of second to ninth period respectively.

Chapter 5

Experimental analysis of general flow properties This chapter starts with the flow physics based on the velocity field and streamlines for one period and periodic flow. The experimental error analysis of the velocity field is given afterwards. The general tracking performance and the error analysis on streamlines are given at the end of this chapter.

5.1

Flow physics

The performance of the subject static mixer is evaluated via experimental 3D-PTV measurements. The experimental results are concentrating on two properties relevant to mixing ; (i) the velocity field, (ii) 3D streamlines pattern. L The experiments have been performed for umean = 6.3[ mm s ] (flow rate = 56[ h ]) which corresponds to Re = 3.42.

5.1.1

One period

The experimental velocity field and streamlines for one period are discussed in this section. The experimental periodic velocity field will be considered in the next section.

5.1.1.1

Velocity field

The experimental velocity contours for five cross sections along the mixer have been analyzed. Like numerical studies, the flow is fully developed far from the inlet of the static mixer. Note that based on Equation 2.9 the entrance length is calculated to be 11.4[mm], which shows that the flow becomes fully developed within a very short distance. The experimental velocity contours for 20[mm] before the onset of the mixing element are depicted in Figure 5.1. The data show that the circular pattern is slightly deformed, see Figure 5.1-right. This behavior has been observed in the numerical simulations as well. The deformation is due to the effects of the mixing elements. The transverse velocity components, ux and uz , are much smaller than the axial velocity component, uy (O(7%) of the axial velocity) which indicates that there are small 41

Chapter 5. Experimental analysis of general flow properties

42

circulations at the upstream region. The same behavior is observed in the numerical studies. Comparably poor definition of the transverse components, ux and uz is related to their small magnitude relative to the axial flow. This makes the former more susceptible to noise. The situation is worse for uz because the cameras are located in the positive z region (see Chapter 3 for coordinate axes). It is harder for cameras to determine the displacement of the particles in the depth of filed of view.

Figure 5.1: Cross-sectional velocity field in the inlet region, 20[mm] before the onset of the mixing elements, for ux , uz (transverse components), and uy (axial component) -left to right.

The cross-sectional profile in the middle of the first element is depicted in Figure 5.2. Looking into the transvers velocity components, it can be inferred that there are more circulations in this region compared to upstream region. ux shows that the mixing element splits the flow field into two distinct regions; top region (red) and bottom region (blue). Numerical results showed that uz consists of four regions see Figure 5.3-left. In the experimental results, two of the regions at the top are clearly distinguishable. However, the two regions at the bottom are comparably vague. It is due to the mixing element which obstructs the cameras view to some extent. If the color range is confined between -2.1 and 2.1 (the limits are based on the numerical simulations), four regions become more clear in experimental results, see Figure 5.3-right. This is basically due to some outliers in the experimental results which completely change the color range.

Figure 5.2: Cross-sectional velocity field in the mixing region at the middle of the first element for ux , uz (transverse components), and uy (axial component) -left to right.

Figure 5.4 shows the experimental velocity contours at the middle of the mixer. ux consists of two small but strong regions. The same behavior can be tracked in the numerical simulations. uy shows four peaks, however, the interfaces of the peaks are not distinguishable. As previously


43

Figure 5.3: Numerical (left) and confined color experimental (right) contours for z-component.

mentioned, this is due to the existence of four surfaces in the middle of the period.

Figure 5.4: Cross-sectional velocity field at the middle of the period for ux , uz (transverse components), and uy (axial component) -left to right.

The velocity components at the outlet are depicted in Figure 5.5. The x-component velocity, ux , is quite considerable in magnitude. However, it is not covering all the cross sectional area. It shows locally transversal motion. The same behavior has been seen in the numerical simulations. uz velocity field is not clear enough. As mentioned before, this is due to the some random outliers. However, if the color range is confined between -3 and +3 (based on numerical results), four regions in uz appear, see Figure 5.6.

Figure 5.5: Cross-sectional velocity field at the outlet for ux , uz (transverse components), and uy (axial component) -left to right.

The velocity field for 40[mm] after the outlet has been depicted in Figure 5.7. The figure shows that the flow is fully developed. The same result was obtained by analyzing the numerical


44

Figure 5.6: Confined color experimental velocity contour for z direction at the outlet

simulations. There are still some fluctuations in x and z direction, however, these fluctuations are unavoidable. The measurements for Poiseuille profile are discussed in Appendix B. The errors here are a bit higher than the errors obtained in the Appendix B because the tracking parameters in the ETH code are set more flexible in tracking with the internal geometry. Note that it is possible to set the maximum allowable displacement of the particles for each direction in the ETH code. For fully developed flow, the parameters in x and z directions are set comparably restricted, however, for measurements with internal element, the displacement range in x and z should be flexible. Due to this flexible range, the code sometimes establishes link between two different particles, but near each other, in two consecutive images. It indicates that the number of wrong established links increases, which results in more error. This is a main reason for higher fluctuations compared to Sec. 3.5. Figure 5.8 shows the velocity field for the whole domain(left) and transverse velocity field (right). It shows the same flow behavior as in the numerical simulations. The swirling exit flow and disturbances at the upstream region are detectable in the transverse velocity field.

Figure 5.7: Cross-sectional velocity field at y = −152[mm], ux , uz (transverse components), and uy (axial component) -left to right.

5.1.1.2

Streamlines

Figure 5.9-right shows the experimental (blue) and numerical (red) streamlines in the static mixer for Re =3.42. The figure shows a good agreement between the experimental and the numerical streamlines. Figure 5.9-right displays the streamlines at the inlet of the mixing geometry. It can be inferred that the streamlines are affected by the mixer geometry at the upstream. The streamlines are deflected before entering the mixing element. The same effect can be tracked from the velocity


45

Figure 5.8: Velocity field (left) and transverse velocity field (right) for the whole domain.

contours too, where the upstream velocity contour is deformed slightly, see Figure 5.1.

Figure 5.9: Simulated streamlines for the entire domain (left) and the streamlines for the inlet of the mixer (right).

As discussed in the numerical analysis, soon after entering the mixer, the streamlines follow the blade geometry. In the middle of the mixing element, the streamlines undergo a lateral dispersion due to bifurcation of the fluid stream by the downstream element. The bifurcation of the flow within the first element can be tracked in the experimental results as well. Figure 5.10 shows the streamlines in the mixing region. The bifurcation of the flow is detectable at the front side of the blades, however, the cameras cannot capture the bifurcation at the rear side of the blade. At the outlet, see Figure 5.11-right, the flow enters an open tube which means that the flow starts to develop into the Poiseuille profile. Therefore, the flow has a tendency to damp the transverse velocity components. It can be seen that soon after the outlet, the streamlines are becoming straight lines. It is an indication of the weak swirling exit flow. The same behavior is observed in numerical simulations.


46

Figure 5.10: Experimental particle trajectories, front view (left) and rear view (right).

Figure 5.11: Numerical (red) and experimental (blue) streamlines in the mixing region (left) and at the outlet (right).

5.1.2

Periodic flow

In order to investigate the periodic flow, three periods (consists of 6 elements) were built in the mechanical engineering workshop. A new female/male connection has been introduced to make the long mixer blades rigid. The velocity field for the second period is measured while the inlet condition is a fully developed flow (Poiseuille profile). In practice, it is not possible to compare the experimental velocity field of the second period with that of third period due to the fact that the experimental velocity field is not dense enough, so the interpolation of the experimental velocity field is risky. Therefore, the experimental results are compared with the numerical simulations. Since, it was shown that the numerical flow field is periodic in the second field of the case with Poiseuille profile as an inlet condition, the data on the second period is considered as periodic benchmark and all the experimental data are compared with this benchmark. More than 18,000 points in the second period are selected. The velocity components and magnitude are compared for these points. The statistics are given in Table 5.1. The errors are almost the same as the error for velocity field of one period, see Table 5.4. It indicates that the flow becomes periodic within the first period when the inlet flow is a fully developed flow (Poiseuille profile). The same conclusion is drawn from the numerical simulations.

Chapter 5. Experimental analysis of general flow properties Velocity component ux uy uz U

σ (standard deviation) 4.56 7.55 6.82 6.29

47 µ (mean) 0.15 0.55 0.49 0.46

M (median) 0.06 0.12 0.11 0.10

Table 5.1: The statistics of the analysis on the random points along the mixer (the numbers are in percentage).

Figure 5.12 shows the experimental velocity vectors (blue) versus the corresponding simulated velocity vectors for the second period. Figure 5.13 shows the experimental and numerical ve-

Figure 5.12: Experimental velocity field (red) versus numerical velocity field (blue) for second period.

locity contour, xy plane. Nine consecutive experimental runs are performed in order to be able to eliminate bad data using Q test method. The data are interpolated on a regular grid of 100 × 200. Figure 5.13 is the result of 27 experiments. It shows a good agreement with the numerical simulations qualitatively. Figure 5.14 shows the experimental velocity contours at the middle of the first element of the second period. The velocity contours are almost similar to the contours in Figure 5.2. They are not exactly the same because there are always errors in the experimental measurements. The similarity indicates that the flow becomes periodic within the first element. The flow at the middle of the mixing period shows the same behavior as in Figure 5.4. Figure 5.15 shows the periodic experimental and the numerical streamlines. As can be seen from the graph, the experimental and numerical streamlines show the same behavior. Deflection and bifurcation of the flow are apparent in the experimental and numerical streamlines.


48

Figure 5.13: Experimental velocity contour(left) and corresponding numerical velocity contour for one period (right).

Figure 5.14: Cross sectional velocity field at the middle of the first element of the second period for ux , uz (transverse components), and uy (axial component) -left to right.

Figure 5.15: Experimental(blue) and numerical(red) periodic streamlines.

5.2

Velocity field error analysis

Errors are normally classified in two categories: systematic errors, and random errors. Random errors in experimental measurements are caused by unknown reasons. Random errors are positive and negative fluctuations. The causes of random errors are not known, so they cannot be


49

excluded. Systematic errors are due to identified causes and can, in principle, be eliminated. Systematic errors are biases in measurement like error in flow meter and calibration body.

5.2.1

Random error

Random errors are positive and negative fluctuations. Almost in all experimental data, there are some outliers which are definite errors. Measurements with internal geometry has more complexity. The geometry acts as an obstacle, so some particles are blocked by the geometry and the cameras cannot capture them easily. Furthermore, as we discuss it previously, the tracking parameters in the ETH code are set more flexible to be able to track the curved movements which leads to higher number of wrong established links. Figure 5.16 shows the velocity profile at the inlet of the mixer for two consecutive experiments. The outliers can be detected easily. The only way that allows to reject a data point is to use a statistically valid method.

Figure 5.16: Inlet velocity contours for two consecutive experiments.

Dixon’s Q Test : There are couple of statistical methods for rejection of a result like Peirce’s criterion, Dixon testQ test, Grubbs test and etc [43–47]. In the present study, Q test has been implemented[47, 48]. It is the simplest statistical test for rejection of a data point and the outcome of the test for the present studies is acceptable. Consider that there are experimental data from different experimental runs. It is possible to define Q as follows:

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50

If Q value is greater than the value listed in table 5.2 the questioned data point can be rejected. The confidence limit plays a role in the Q test as well. The confidence limit is defined as follows;

λ=

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× t1− α2 ,N −1

where s is the sample standard deviation, N is the sample size, α is the desired significance level and t1− α2 ,N −1 is the 100(1 − α2 ) percentile of the t distribution with N − 1 degree of freedom. The critical values of student’s t distribution with different degree of freedom are available in tables in statistics books. Although the choice of confidence coefficient is somewhat arbitrary, in practice 90%, 95%, and 99% intervals are often used, with 95% being the most commonly used. Number of values Q90% Q95% Q99%

3 0.941 0.970 0.994

4 0.765 0.829 0.926

5 0.642 0.710 0.821

6 0.560 0.625 0.740

7 0.507 0.568 0.680

8 0.468 0.526 0.634

9 0.437 0.493 0.598

10 0.412 0.466 0.568

Table 5.2: The limit value of the Q test [47].

It is worth mentioning that the data on cross sections are scattered. Therefore, in each experiment, the location of the particles, hence the data points are different. In order to have comparable results, the data points are interpolated on a regular grid (100 × 100).

Figure 5.17: Inlet velocity contour after the rejection of the bad data.

Figure 5.17 shows the outcome of the Q-test for 9 consecutive experiments. The improvement is considerable and almost all the bad data are rejected. The number of rejected points is less than 1% of the data points. In principle, it is possible to implement the Q test for the whole domain. However, in practice, it requires numerous number of interpolation which takes a lot of time. Therefore, the test has been implemented on some representative cross sections only.


5.2.2

51

Systematic error

Systematic errors are due to identified causes and can, in principle, be eliminated. In case of systematic error, the measured values are consistently too high or consistently too low. In the current experimental system, the error due to the curved cylinder was one of the most prominent systematic errors. Error in the calibration of the flow meter is another source of systematic errors. Quantitative studies on the accuracy of the data is carried out in two ways; on representative cross sections and directly in the 3D-PTV data points. Representative cross sections: The numerical and experimental velocity contours are depicted for three representative cross sections; inlet, middle and outlet. Figure 5.18-left displays the experimental velocity contours for abovementioned cross sections. Figure 5.18-right shows the corresponding numerical contours. The graphs show a good agreement between numerical and experimental results. Numerical simulations reveals that the flow inside the ,mixing geometry is symmetric. However, looking closer to the experimental velocity contours, it reveals that the velocity profile suffers from asymmetry. For instance, the velocity contour at the inlet has two maxima, but the maximum which is located in the negative x region is stronger than the other one. Since the position of the stronger peak has changed with rotation of the blades, this conclusion has been drawn that this asymmetry relates to the imperfections in geometry. Quantitative analysis of the experimental results has been performed on five various cross sections; inlet, middle of the first element, middle of the period, middle of the second element and outlet. Figure5.19 displays the subtraction of the experimental results from the simulated one for inlet, middle of the period, and outlet. The deviation is quantified via ≡ (us − ue )/us,max where us is the velocity from the simulations, ue is the corresponding experimental velocity and us,max is the maximum numerical velocity on the cross section. The associated statistics for inlet/middle of the first element/middle of the period/middle of the second element/outlet are given in Table 5.3. The error in the middle of the mixing element is higher compared to the remaining parts of the mixer due to the fact that a male/female connection is used to connect two elements. This connection blocks the view. Besides, in the experimental measurements, there are no measurements close to the surfaces due to the particles tendency to move away the walls. Lack of data points close to the surfaces indicates that the velocity fields in these regions are the results of the interpolation. Since, there are four surfaces in the middle of the period, it is expected that the error is higher in this region. Cross section inlet middle of the first element middle of the period middle of the first element outlet

σ (standard deviation) 8 5 17 7 6

µ (mean) 3 1 5 3 3

Table 5.3: The statistics of the analysis on representative cross sections (the numbers are in percentage).


52

Figure 5.18: Experimental velocity contours (left) versus simulated velocity contours (right) for inlet, middle and outlet of the mixer.

Figure 5.19 displays the difference between experimental and numerical velocity contours for inlet, middle and outlet. The corresponding histograms are depicted as well. 3D-PTV data points: A quantitative comparison between experiments and simulated velocity field is carried out directly on 3D-PTV data points as well. More than 17,400 points are selected and the comparison has been performed for velocity components (ux , uy , uz ) and magnitude (U ). The deviations of the data are quantified via i ≡ |is −ie |/|is,max | where i can be x, y, z or U . The associated statistics are given in Table 5.4. Figure 5.20 shows the histogram of the data. The error is between 6 to 7% which implies a good agreement between numerical and experimental results. Note that the error here is less than the error on representative cross sections. It is due to the fact that in this case the numerical data are interpolated on the experimental data points and the exact


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µ (mean) 0.40 0.74 0.71 0.66

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Table 5.4: The statistics of the analysis directly on 3D-PTV data points (the numbers are in percentage).

Figure 5.21 shows the experimental velocity vector (blue) versus simulated velocity vectors (red) in the inlet, mixing and outlet region. The following items play an important role in the error; mixing geometry is not perfect, the exact location of the mixing geometry in the tube is unknown, there are random errors, and there is an error in the flow meter. They do not contribute in the error equally, however it is not possible to give a firm statement which one is the dominating one.

5.3

Streamlines error analysis

The trajectory of individual particles are recorded within the cameras field of the view. The flow field is partitioned in three regions; (i) inlet region including the upstream section (ii) mixing


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region (iii) outlet region consisting of downstream section. Images are recorded at maximum recording frequency, 30 frames per second, and at an exposure time of 5 [ms]. For each experiment, the cameras capture the images of the flow field for 45[s] yielding 1392 images per camera which is equal to 10[GB] of data in size. On average, 270 particles are detected on each set of the images. 210 particles are detected on all four cameras (consistent quadruplets) which can be used for tracking. This indicates that almost 77% of the particles are capable of being tracked. The tracks which has a length less than 50 steps, are ignored. Figure 5.22 displays the distribution of the tracks lengths (Lp ) against the number of the tracks (Np ) with a given length of Lp . Note that the y axis is in logarithmic scale. The graph reveals that the relation between number of tracks and track length is exponential decay. The equation which describes (dashed line) the exponential decay is log(Np ) = −0.0042 × Lp + 3.4918. The mean of the track lengths is 138 steps. The particles which either have been tracked for more than 1000 steps or less than 50 steps are not considered in the graph. The particles, which are tracked for more than 1000 steps, are either stuck to the internal element or moving very slowly due to closeness to the walls, which in both situations they occurs randomly. Note that the performance of the PTV depends on the internal element as well. Figure 5.23 depicts the trajectories for a typical run. A quick look at the Figure 5.23 shows that the internal element hinders the cameras view to some extent, especially at the region where two elements are connected. In Figure 5.23-right, which shows the trajectories at the rear side of the element, some tracks are interrupted. However, in Figure 5.23-left, which shows the trajectories at the


55

Figure 5.21: Experimental (blue) and numerical (red) velocity vectors for three main regions.

Figure 5.22: Number of the tracks (y) versus the track lengths (x).

front side of the element, most of the tracks are complete. Figure 5.24 shows full trajectories. The blue line is the experimental trajectory and the red one is the simulated one. The numerical trajectories are generated by COMSOL itself. Although the experimental and numerical trajectories have the same trends, there are still differences between them. One of the causes of the differences is the fact that the experimental trajectories are the


56

Figure 5.23: Experimental particle trajectories, front view (left) and rear view (right).

results of three experiments. As mentioned previously, the flow field is partitioned into three regions. An error of 2% is introduced for connecting the three regions streamlines. Furthermore, the imperfection in the mixer geometry, minor changes in the flow rate for each experiments, approximate positioning of the mixing geometry are the other sources of the differences. The last but not the least reason for the differences is the fact that in the numerical simulations, the particles are imaginary ones. It is implying that the density effects, lift forces and response time are ignored. The numerical results are streamlines rather than real particle trajectories. 50

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Chapter 6

Comparative numerical-experimental analysis of the effect of inlet conditions Periodic flow occurs when the physical geometry of interest and the expected pattern of the flow solution have a periodically repeating nature. The static mixers with more than one period are usually used in the industries. It is important to know where the flow becomes periodic in a static mixer because as soon as the flow becomes periodic, the flow field in the consecutive elements is known. In this chapter, the effect of the inlet conditions on periodicity for Re = 3.4 is discussed. The strategy is a combined numerical-experimental method. Three different inlet conditions are taken into account; fully developed flow, 63% of the inlet being blocked, and a quarter of the inlet being blocked. The last one is analyzed only by numerical simulations.

6.1

Fully developed flow

The numerical periodic flow is discussed in Chapter 4, Sec.4.2. When the flow becomes periodic, it means that the velocity field repeats itself in the consecutive periods. In order to investigate whether the flow is periodic or not, two approaches are adopted; (i) the comparison of velocity field on five representative cross sections in the second and the third periods (ii) and comparison between the velocity fields on random data points in the second and the third periods. More details are available in Sec.4.2. Figure 6.1 shows the cross sections which the analysis has been performed on. The abovementioned methods are used to examine the periodicity in numerical simulations, however, for the experimental results analysis the methods are different. The experimental results are compared with numerical periodic flow filed. In practice, it is not possible to compare the experimental results of two consecutive periods because the experimental velocity field is not dense enough. The numerical periodic flow field for Poiseuille profile as inlet condition has been discussed in Chapter 4, Sec. 4.2. The corresponding experimental analysis has been discussed in Chapter 5, Sec. 5.1.2. The statistics are given in Table 4.1, 4.2, and 5.1. 57

Chapter 6. Comparative numerical-experimental analysis of the effect of inlet conditions

58

Figure 6.1: Five cross sections which the data are analyzed on.

Based on the numerical and experimental studies it is shown that the flow becomes periodic within the first period when the inlet condition is Poiseuille profile, see Sec.4.2 and Sec. 5.1.2. Figure 6.2 shows the velocity field, yz, plane. It can be seen from the graph that the flow becomes periodic within the first period. Therefore, the simulated velocity field in the second period can be used as a benchmark of the periodic flow field.

Figure 6.2: Simulated flow field for fully developed flow as inlet condition, yz plane.

6.2

63% of the inlet being blocked

Changing the inlet condition is achieved by using an arbitrary blockage at the inlet, see Figure 6.3. The blockage covers 63% of the inlet surface. It is a square inside a circle. The diameter of the square is equal to the diameter of the circle. The blockage is introduced exactly at the inlet of the mixing geometry. This situation is mimicked in the experimental facility by gluing a square surface on the mixing geometry.

Figure 6.3: Implementation of a square at the inlet for disturbing the flow.

The numerical and experimental results are discussed in the following sections;


6.2.1

59

Numerical results

The obstruction which is shown in Figure 6.3 is introduced in the numerical simulations at the inlet of the mixing geometry. In order to analyze the periodicity, the same approaches as previous section have been adopted. The flow is analyzed on the five representative cross sections. Furthermore, the flow field inside the second period is compared with the flow field in the third period on more than 25,000 points. Figure 6.4 shows four cross sectional velocity contours in the second (top) and the third periods (bottom). The statistics of the analysis are given in Table 6.1. The differences between two cross sections are small which indicates that the velocity contours are repeating themselves.

Figure 6.4: Simulated velocity field for 1, 2, 3, and 4 for the second (top) and the third periods (bottom) when 63% of the inlet is blocked.

Cross section 1 2 3 4 5

σ (standard deviation) 0.20 0.03 0.16 0.05 0.20

µ (mean) 0.043 0.002 0.084 0.002 0.024

M (median) 0.002 0.003 0.047 0.004 0.012

Table 6.1: The statistics of the analysis on five cross sections when 63% of the inlet area is blocked (the numbers are in percentage).

Apart from the analysis on cross sections, the flow field inside the second period is compared with the third period. More than 25,000 points within the second period have been chosen. The velocity components (ux , uy , uz ) and magnitude (U ) on these points are compared with the corresponding points in the third period. Since the generated mesh in the second and third periods are not exactly the same, interpolation for third period is essential. The statistics are given in Table 6.2

Chapter 6. Comparative numerical-experimental analysis of the effect of inlet conditions Velocity component ux uy uz U


µ (mean) 0.004 0.39 0.016 0.46

60

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Table 6.2: The statistics of the analysis on 25,000 numerical points when 63% of the inlet is blocked (the numbers are in percentage).

Figure 6.5 shows the streamlines in the second (blue) and third (red) periods. As can be seen from the graph, It shows that the streamlines are perfectly matched.

Figure 6.5: Simulated streamlines for the second (blue) and the third (red) periods.

Figure 6.6: Simulated flow field when 63% of the inlet is blocked.

Based on the statistics and figures, this conclusion can be drawn that the flow becomes periodic within the first period even when 63% of the inlet area is obstructed from numerical point of view.

6.2.2

Experimental results

In order to analyze the periodicity from experimental point of view, more than 18,000 points in the second period are selected. As discussed before, the experimental results are compared with numerical simulations. Like previous sections, the velocity components (ux , uy , uz ) and


61

magnitude (U ) are compared for these points. The statistics are given in Table 6.3. The errors are the same as the error of PTV itself, see Table 5.4. It indicates that from the experimental point of view the flow becomes periodic within the first period even when 63% of the inlet area is blocked. Figure 6.7 shows the experimental (blue) and numerical (red) velocity vectors in the second period. It shows that the experimental velocity field is matched with the numerical periodic velocity field. Figure 6.8 shows the numerical and experimental velocity contours for two cross sections within the second period. It shows a good agreement between numerical and experimental results. However, as discussed previously, the error of the velocity contours at the middle of a period is high due to the existence of four surfaces.

Figure 6.7: Experimental (blue) and numerical (red) velocity vectors within the second period.

Figure 6.8: Numerical (left) and experimental (right) velocity field in the second period for the middle of the second element (top) and the middle of period (bottom).

Chapter 6. Comparative numerical-experimental analysis of the effect of inlet conditions Velocity component ux uy uz U


µ (mean) 0.15 1.28 1.15 1.06

62

M (median) 0.053 1.22 1.09 1.01

Table 6.3: The statistics of the analysis on 18,000 experimental points along the second period when 63% of the inlet area is blocked (the numbers are in percentage).

6.3

A quarter of the inlet being blocked

In order to examine the different inlet conditions, a second kind of obstruction has been introduced. A quarter of the inlet has been blocked, see Figure 6.9. This inlet condition has been examined only by numerical simulations.

Figure 6.9: A quarter of the inlet is blocked (black).

6.3.1

Numerical results

The obstruction which is shown in Figure 6.9 is introduced in the numerical simulations. In order to analyze the periodicity, the same approaches as previous sections have been adopted. The flow is analyzed on the five cross sections and on random points along the mixer. The statistics for five cross sections are given in Table 6.4. Figure 6.10 shows the velocity contours for 1, 2, 3, and 4 in the second and the third periods. Cross sectiont 1 2 3 4 5

σ (standard deviation) 0.14 0.05 0.20 0.03 0.13

µ (mean) 0.053 0.009 0.004 0 0.08*

M (median) 0.063 0.007 0.025 0 0.066

Table 6.4: The statistics of the analysis on five cross sections when a quarter of the inlet area is blocked (the numbers are in percentage).

In order to examine the flow field inside a period, more than 25,000 points within the second period have been chosen. The velocity components (ux , uy , uz ) and magnitude (U ) on these points are compared with the corresponding points in the third period. Table 6.5 shows the


63

Figure 6.10: Simulated velocity field for 1, 2, 3, and 4 for second (top) and third periods (bottom) when a quarter of the inlet is blocked.

statistics of this analysis. Velocity component ux uy uz U


µ (mean) 0.011 0.353 0.015 0.408

M (median) 0 0.104 0 0.141

Table 6.5: The statistics of the analysis on 25,000 numerical points when a quarter of the inlet is blocked (the numbers are in percentage).

Figure 6.11 shows the streamlines for second (blue) and third period (red). It can be seen that the streamlines coincide which is an indication of the periodic flow field. Figure 6.12 shows the velocity contour, yz palne, along the mixer when a quarter of the inlet is blocked. It can be seen that the flow becomes periodic within the first element. Based on statistics and figures, this conclusion can be drawn that the flow becomes periodic within the first period even when 25% of the inlet is obstructed.

6.4

Effects of the inlet condition on periodicity

Three different inlet conditions have been analyzed. Looking into the numerical and experimental results for various inlet conditions shows that any disturbances at the inlet will be vanished within the first element. Moreover, the periodic flow patterns in all three cases are the same. For intense, the simulated velocity field of the second period when a quarter of the inlet is blocked is compared with the simulated velocity field when the inlet is a Poiseuille profile. The statistics are given in Table 6.6 which show that the flow patterns are the same.


64

Figure 6.11: Simulated flow field for fully developed flow (top) a quarter of the inlet is blocked (middle) and 63% of the inlet is blocked (bottom)

Figure 6.12: Simulated flow field for fully developed flow (top) a quarter of the inlet is blocked (middle) and 63% of the inlet is blocked (bottom).

In practice, this is an important conclusion because it is indicating that the static mixers can be installed anywhere inside the tube i.e. the upstream condition is not affecting the periodicity at low Reynolds numbers. Velocity component ux uy uz U


µ (mean) 0.006 0.036 0.003 0.042

M (median) 0 0.076 0 0.110

Table 6.6: The statistics of the analysis on second period when a quarter of the inlet area is blocked and a fully developed prescribed at the inlet (the numbers are in percentage).

Chapter 7

Conclusions and recommendations 7.1

Conclusions

In this thesis a pressure driven flow through a Q type static mixer element has been studied using numerical simulations and 3D-PTV experiments. The main objective of this thesis is to investigate the flow properties of a mixing flow at very low Reynolds number regime. The strategy is a combined numerical-experimental method. A summary of the main conclusions is presented here. The new version of the laboratory setup is capable of working with highly viscous liquids. It turned out that the best option for new working fluid is silicone oil. However, it is extremely sensitive to contamination, so small amount of contamination makes the silicone oil completely opaque. It was found that a defective valve was the source of fluctuations at low velocities in Jilisen experiments. The experimental images are processed via ETH code, however, it cannot handle curved interfaces. A modification in the calibration has been performed to include the effects of curved cylinder in the experimental results (Modeling of the setup has been performed by Ad Holten). Due to modification in calibration and decreasing the fluctuations via changing the defective valve, the error decreased from 10-15% to 6-7%. The real 3D experimental flow field has been investigated for a real static mixer at Re = 3.4. The experimental results show a good agreement with numerical predictions. The bifurcation of the flow inside the mixing element and the swirling exit flow are conspicuous in the experimental results. It is shown that the swirling exit flow is quite weak for Re = 3.4 compared to higher Reynolds numbers. Furthermore, experimental periodic flow field has been examined via implementation of several elements into the set-up. The corresponding numerical studies have been performed for five different Re; 3.4, 6.8, 34.2, 68.4, and 100. The effects of the Reynolds number on the flow field and streamlines have been investigated. It has been shown that the bifurcation and swirling exit flow are stronger at higher Reynolds numbers. Furthermore, the effects of Reynolds number on periodic flow field were discussed. Apart from velocity field and streamlines investigation, the numerical study on concentration field has been performed as a reference for future studies. At low Re, it is revealed that the downstream flow doesn’t considerably contribute to the mixing. As the flow leaves the mixing geometry, there will be no more considerable mixing. In contrast, at higher Re, the downstream flow contribute to the mixing. 65

Chapter 7. Conclusions and recommendations

66

At the end, the effects of the inlet condition on periodicity have been investigated for Re = 3.4. Three different inlet conditions were explored via implementation of different blockages at the inlet. It is shown that regardless of the inlet condition, the flow becomes periodic within the first element for Re = 3.4. The same results are achieved in the experiments.

7.2

Recommendations

Since the flow domain is partitioned into three regions, achieving full trajectories over the entire domain introduces an error. The reason that the domain is partitioned in three regions is the limited observation field of the cameras. By using higher resolution cameras, it is possible to move the cameras further away the test section. Therefore, the observation field of view can be larger. As a result, full trajectories can be obtained via using higher resolution cameras together with construction of longer test section. Mixing in the current geometry is not yet investigated in detail. Achieving the experimental Poincare section is a follow-up study to investigate the mixing performance. The first step for achieving the Poincare section is finding the full trajectories over the entire period. Furthermore, the set-up facility allows to place different type of mixing elements in the test section. Therefore, the mixing performance of other mixing geometries can be investigated. To this end, the velocity field is known, however, the experimental concentration field is still unknown. Experimental concentration studies are follow-up to this thesis to enhance knowledge and improvement of mixing. ETH code cannot handle the curved interfaces. Furthermore, a new method (using polynomial for particle mapping) for link establishment is in progress. It is recommended to validate the new version of the ETH code when it is available. In this thesis numerical studies on the concentration field focused on the pure convection. The effects of Peclet number on concentration field can be a follow-up study for the numerical simulations.

Appendix A

Setting valve The old setting valve was a typical Globe valve. Figure A.1-left shows the schematic view of the Globe valve. The washer in the Globe valve was suspected to be the defective part. Since the working liquid used to be distilled water which is more aggressive compared to normal water and the washer is continuously in contact with working liquid, it decays over the time. The valve has been substituted with a ball valve. A ball valve is a valve with a spherical disc, the part of the valve which controls the flow through it. The sphere has a hole, or port, through the middle so that when the port is in line with both ends of the valve, flow will occur. When the valve is closed, the hole is perpendicular to the ends of the valve, and flow is blocked. The advantage of the ball valve over Globe valve in current case is that ball valve doesn’t have washer inside.

Figure A.1: Globe valve(left) and ball valve (right)

67

Appendix B

Flowmeter The experimental setup can be improved by installing an accurate real-time flow rate measuring device. Table B.1 - B.2 show the different flow meters characterestics.

Figure B.1: Flowmeters characteristics

68

Appendix B. Flowmeter

69

Figure B.2: Flowmeters characteristics (continued) [23]

The last column of the Table B.1, indicates the minimum Reynolds number which the flowmeter can be used. Knowing that the experiments are going to perform at low Reynolds numbers, three real-time flowmeters fulfill the minimum Reynolds number requirement; Coriolis Mass, Positive displacement, and Magnetic flowmeter. Magnetic flowmeters can be applied only for conductive fluids. Since silicone oil is not a conductive fluid, the magnetic flowmeter is out of table. Coriolis mass flowmeters are really expensive and they have high pressure drop, so it is not reasonable to use them. The last option is the positive displacement flowmeter. However, there are still problems with these kind of flowmeters. First of all, positive displacement flowmeters have high pressure drops. Knowing that the setup is 2.5[m] high, the maximum pressure head is calculated to be 23 [kPa] approximately. If the pressure drop in the tubes and over the blade are taken into account, the maximum allowable pressure drop in the flowmeter shouldn’t exceed 15 [kPa]. This number is a really low pressure drop for positive displacement flowmeters. The situation exacerbated when the viscosity goes up. Above the pressure drop, the positive displacement flowmeters are suitable only for clean fluid. In our experiments, there are some small tracer particles in the fluid and these particles will clog the flowmeter. In conclusion, there is no appropriate real-time flowmeter for our case of study. One remedy is to use another pump at the top of the setup and push the fluid into the tubes. Therefore, there is more pressure head and it will be possible to use Coriolis mass flowmeter. However, it is quite expensive.

Appendix C

Working principle of 3D PTV C.1

Performance issues

PTV like all other experimental measurement devices has some performance issues which should be considered for yielding reliable results. Seeding density: the seeding density should not be too high. High seeding density deteriorates the performance of matching algorithm by increasing the likelihood of the mismatched particles. Very high seeding density ruins the whole particle tracking because epipolar line intersection technique is unable to cope with the ambiguities [5]. Spurious trajectories: there are particles other than seeding particles in the flow field like air bubbles, contaminants and particles stick to the walls. These redundant particles may be taken into account and causes mismatching. To minimized the redundant particles, sufficient cleaning of the walls is needed. Furthermore, the purified liquid should be used to prevent this particles [5]. Sampling frequency (Temporal resolution) : sampling frequency plays an important role to get reliable matching. At low sampling frequencies, particles may travel quite a large distance over one time step. This deteriorates the matching quality and increases the likelihood of the mismatch of particles. On the other hand, at high sampling frequencies, some particles may have a displacement of the order of single pixel or even sub-pixel level. Therefore, the exact position of the particles will be unknown due to insufficient camera resolution [5]. Particle inertia : when the fluid velocity changes, particles require a certain time to acquire the fluid velocity. In order to describe this in formula, it is possible to use the Newtons law. Assume that the particles are spheres, the fluid is incompressible and has a constant viscosity, and the particle concentration is low enough for particle interaction to be negligible. The total viscous friction force comprises three contributions; Stokes force, added-mass force, and Basset force . The added-mass and Basset force are neglected, so by using the Newtons law, we have dUd 4 = 6πµR(Ud − U ) ma = W → πR3 3 dt 70

(C.1)

Appendix C. Working principle of 3D PTV

71

with Ud the particle velocity and U the flow, µ the dynamic viscosity and R the radius of the particles. Inserting the initial condition (Ud = 0 at t = 0) and by substitution of 2R = Dd this yields t Ud = U (1 − exp(− ) (C.2) τ where τ is the relaxation time and is defined as τ=

Dd2 ρd 18µ

(C.3)

Small τ is desired because it implies that the particles have a fast response time to the changes in the velocity [5]. Particle density : it is obvious that the difference in densities leads to sinking or rising of the tracer particles in the flow field and thus, the particles do not follow the stream line. In ideal case, the tracer particle density and the fluid density should be the same. If the densities differ, two distinct forces act on the tracer particles; buoyancy force and viscous force. The vertical motion of particles is determined by a force balance between these two acting forces. The assumptions are the same as assumptions were made for particle inertia [5]. 4 6πµR(Vd − V ) + πR3 (ρd − rho)g = 0 3

(C.4)

where Vd the vertical particle velocity and V the vertical flow, ρd and ρ the particle and the fluid density, respectively, µ the dynamic viscosity and R the radius of the particles. Particle lift force in boundary layers : Since there is a velocity gradient in the boundary layer, the lower side of the particles travel with a lower speed compared to the upper side. This difference in velocities leads to spinning of the particle. Figure C.1 demonstrates this effect. Spinning particles experience a lift force which is given by Kutta-Joukowski lift theorem [18], F = −ρU γ

(C.5)

where U is the flow velocity and γ is the circulation. After some calculations [5] it reveals that the lift force for 3D situation is given by, s 1 dU F = ρπDd2 U (C.6) 2 dy In the boundary layer dU dy > 0 and this implies a lift force away the solid walls. This is one of the reasons why visualization close to the walls is difficult. To decrease the lift effects, the particles must be small enough.

Figure C.1: Spinning particle in the boundary layer [19].


C.2

72

Epipolar line intersection technique

When two cameras view a 3D scene from two distinct positions, there are a number of geometric relations between the 3D points and their projections onto the 2D images that lead to constraints between the image points. The epipolar geometry is used to make correspondence automatically. The technique has been tested and successfully applied for PTV measurements[8–10]. Figure C.2 displays two cameras focusing on point X. C and C 0 are the pinhole of the cameras, whereas x and x0 are the projection of the point X on the image plane. As can be seen from the following figure, image points, point X and camera centers are coplanar. Hence, it is possible to define an arbitrary plane, π, which passes through these points. Supposing now that we only know the 2D projection of point X on first camera image x, it may be asked how the corresponding point x0 is constrained. The epipolar plane, π, is defined by the baseline-the line joining the camera centers- and the ray defined by x. By defining the epipolar plane, the search area is limited to a line, l0 , which is called the epipolar line. Note that in general case the epipolar line will be bent because of lens distortion. In 3D PTV, more than two cameras are usually utilized because it is highly likely to find more

Figure C.2: (left) The two cameras are indicated by their centers C and C 0 and image planes. The camera centers, 3-space point X, and its images x and x0 lie in a common plane π. (right) An image point x back-projects to a ray in 3-space defined by the first camera center, C, and x. This ray is imaged as a line l0 in the second view. The 3-space point X which projects to x must lie on this ray, so the image of X in the second view must lie on l0 [52].

than one candidate on the epipolar line. The situation is exacerbated when the number of dye particles is high. In case of using three cameras, the searching area is limited to the intersection of the epipolar line of the second camera and the epipolar line of the third camera. The intersection of two lines is a point, so the searching area is much more confined. The method itself can be used by any arbitrary number of cameras. However, four cameras is the optimum choice and the use of more than four cameras is not necessary, practical and cost efficient [11]. Figure C.3 shows the epipolar lines in four cameras.

C.3

Two views for particle tracking

Image space based tracking technique In this technique, consecutive images of a camera is analyzed and particles trajectories are determined in 2D. Then, the image of different cameras are combined to establish the spatial correspondence between the 2D trajectories [6, 12–14].


73

Figure C.3: 3D PTV configuration using 4 cameras and epipolar intersection technique [6]

Object space based tracking technique This tracking technique determines the particle positions for each time step in 3D space. Then tracking in 3D space starts. Determination of particles positions in 3D space is performed by epipolar line intersection technique. The main drawback of this technique is that there are usually some ambiguities which cannot be solved by epipolar line technique. Hence in 3D space there are some missing particles which harden the tracking process [6]. Details of the tracking procedure can be found in [15, 16]. Figure C.4 displays the fundamental of these techniques.

C.3.1

ETH code

Like all other PTV systems, a software is needed to track the particles. In the present study, an algorithm which is developed and implemented at ETH, Zurich is used. This code uses a combination of the image space based tracking technique and object space based tracking technique. Figure C.5 shows the algorithm of the tracking software. The software is freely available for non-commercial use. For detailed information about the code and the algorithm please refer to [6].


Figure C.4: image space based tracking technique (left) object space based tracking technique (right)

74


Figure C.5: Processing scheme for tracking the particles [6]

75

Appendix D

Higher order elements Taylor-Hood family are very suitable for the standard Galerkin methods. However with respect to special methods such as Penalty function, the discontinuous pressure elements are most favorable [27]. Taylor-Hood elements are characterized by the fact that the pressure is continuous. A 2D triangular Taylor-Hood element uses 6 nodal points for velocity and 3 points for pressure, see Figure D.1. In this element the velocity is approximated by a quadratic polynomial and the pressure by a linear polynomial. The accuracy of velocity and pressure is O(h3 )and O(h2 ), respectively, where h is the characteristic length. The basis functions for u and p for 2D calculations are (for detailed discussion refer to [27]-p 51-53 and [28]). uei =

6 X

− uij ϕj (→ x)

(D.1)

j=1

pe =

3 X

pj Ψj

(D.2)

j=1

Figure D.1: Taylor-Hood element

Quadratic elements showed much better performance on the concentration field. They can capture the steep gradients in concentration field, however, the linear elements have low efficiency in solving the concentration field. Figure D.2 shows the concentration field for linear and quadratic elements. Figure D.2-left consists of 6,500,000 linear elements, the middle one consists of 650,000 linear elements and the right one consists of 650,000 quadratic elements. It can be seen that steep gradients are much better captured by the quadratic elements. 76

Appendix D. Higher order elements

Figure D.2: Simulated concentration field with different mesh numbers; 6,500,000 linear elements(left) 650,000 linear elements(middle) and 650,000 quadratic elements (right).

77

Appendix E

General improvements of the experimental setup • The tube which connects the top reservoir to the test section used to be a PVC tube. The new tube is a transparent tube, so the flow inside the tube is visible. • The objective of the over flow tube is to ensure us that the small bucket is filled with working medium. The overflow tube used to be a PVC tube. As a results, it was not possible to see inside the overflow tube. A transparent section is added to the overflow tube. • The small bucket in the top reservoir was changed from 20L to 5L which increases the life time of the pump.

Figure E.1: The transparent overflow tube section and transparent main tube.

78

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