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STOCHASTIC DISPERSION OF FIBERS IN TURBULENT FLOW USING EULERIAN-LAGRANGIAN METHOD

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STOCHASTIC DISPERSION OF FIBERS IN TURBULENT FLOW USING EULERIAN-LAGRANGIAN METHOD

Understanding the motion of elongated particles is important in a wide range of areas, including inhalation toxicology, targeted drug delivery, paper industries and food processing. For example, during inhalation elongated fibers tend to align with mean flow and may penetrate to deep lung regions. The respiratory deposition of fibers are hazardous to human health and increase the occurrence of lung cancer. On the other hand, the alignment of fibers with mean flow indicates that for certain applications an improved drug delivery performance can be achieved using such fiber like carriers.

The dynamics of elongated fibers is complex because of the anisotropic shape effect, meaning that combined rotational and translation movement of the fibers must be considered. In this study dispersion and deposition of non-spherical rigid particles in various laminar and turbulent pipe and duct flows were investigated numerically. For numerical simulation the available CFD codes are limited to creeping flow and they are not able to handle non-creeping flows as well as employing accurate stochastic modeling for turbulence fluctuations. In order to extend numerical simulations to the more complex geometries, user defined subroutines are developed and coupled to the Fluent discrete phase model (DPM) module. These user defined subroutines are proposed to solve for coupled translation and rotational equations of motion for ellipsoidal fibers using the simulated flow field from Fluent package. Simulations are performed for dispersion and deposition of fibers in several test cases including laminar and turbulent channel flows as well as a model of human nasal cavity. For the particle equations of motion of laminar flows, new non-creeping flow (finite Re_p) correlations were applied for the hydrodynamic forces and torques as well as creeping flow formulations. Satisfactory results were obtained for dispersion and deposition of non-spherical particles in various test cases. A new empirical model for estimating fiber deposition efficiency for finite particle Reynolds numbers, Re_p<10 was developed. The simulation results were compared with earlier theoretical and numerical studies, as well as the new empirical equation, and good agreement was found. Particular attention was given to the effect of fiber size and aspect ratio on the dispersion and deposition of particles under various flow conditions for fully developed laminar pipe flow. For these simulations particle size is changed from 1μm to 20μm, aspect ratio from 2 to 50 and flow Reynolds number varied from 10 to 400. Using the results of numerical simulations, it is concluded that the new non-creeping flow formulation for hydrodynamic forces and torques are able to properly predict the dispersion and deposition of non-spherical particles in both creeping flow and non-creeping flow regimes. In addition, good agreement is observed between the predicted deposition velocity for turbulent pipe and duct flows with the reported experimental data in literature.

For the case of turbulent flows, the role of fluctuating velocity and velocity gradients are incorporated using proper stochastic modeling. For these analysis, flow Reynolds number is varied from 17000 to 87000 and the stochastic modeling is based on modified Langevin equations. Results for dispersion and deposition of particles for several turbulent flows, i.e. isotropic homogeneous turbulence, non-homogenous turbulence were presented and discussed. The simulation results were compared with earlier numerical and experimental studies for flow statistics and deposition velocity and good agreements were observed. It was shown that using appropriate stochastic models lead to satisfactory evaluation of dispersion and deposition of fibers in the turbulent flows. Results also highlighted the importance of proper turbulence fluctuations modeling in the deposition of fibers in turbulent flows. Good agreement is observed in the prediction of fiber deposition in the case of human nasal cavity in comparison with the previous experimental and numerical results for both laminar and turbulent flows.

Keywords: Ellipsoidal fibers, stochastic modeling, fluctuating velocity gradient, fiber deposition, Eulerian-Lagrangian method, laminar flow, turbulent flow, creeping flow formulation, non-creeping flow formulation

دسته: زبان خارجه, زبان خارجه برچسب: creeping flow formulation, Eulerian-Lagrangian method, fiber deposition, fluctuating velocity gradient, Keywords: Ellipsoidal fibers, laminar flow, non-creeping flow formulation, stochastic modeling, turbulent flow

توضیحات

Table of contents

Abstract III

Nomenclature. XIII

Chapter 1 .Introduction. 1

1.1 Introduction. 2

1.2 Literature survey. 4

1.3 Motivation. 8

1.4 Objectives. 11

1.5 Innovative features. 13

1.5.1 Flow model 13

1.5.2 Particle model 14

1.5.3 Deposition of fibers in the model of human nasal cavity. 14

Chapter 2 .Theoretical Background. 15

2.1 Different types of multiphase flow.. 16

2.1.1 Eulerian-Lagrangian method. 16

2.1.2 Eulerian-Eulerian method. 17

2.1.3 PDF method. 18

2.2 Interaction effects. 18

2.3 Governing equations. 22

2.3.1 Creeping flow formulations for hydrodynamic forces and torques. 25

2.3.1.1 Hydrodynamic force. 25

2.3.1.2 Hydrodynamic torques. 26

2.3.2 New formulations for hydrodynamic forces and torques. 27

2.3.2.1 Hydrodynamic force. 28

2.3.2.2 Hydrodynamic torques. 29

Chapter 3 .Stochastic modeling of particle dispersed flow.. 32

3.1 Stochastic processes. 34

3.2 Stochastic models for particle dispersion. 37

3.2.1 Stochastic models for velocity. 37

3.2.2 Near wall correction for fluctuating velocities. 43

3.2.3 Lagrangian time scale and RMS fluctuating velocities. 45

3.2.4 Stochastic model for velocity gradient tensor 46

Chapter 4. Results. 50

4.1 Laminar flow.. 52

4.1.1 Case1: Comparison of fiber trajectory in fully developed laminar pipe flow 52

4.1.2 Case2: Comparison between time evolutions of fiber direction cosine. 55

4.1.3 Validation using non-creeping expressions for hydrodynamic forces and torques 56

4.1.4 Case3: Comparison between fiber velocities in a laminar pipe flow.. 60

4.1.5 Single fiber motion. 61

4.1.6 Dispersion and deposition of fibers. 67

4.2 Turbulent flow.. 77

4.2.1 Synthesizing isotropic-homogeneous turbulence. 77

4.2.1.1 Time evolution of moments of fiber displacement in an isotropic homogeneous field. 79

4.2.1.2 Time evolution of moments of fiber displacement in a fully developed turbulent flow field. 81

4.2.2 Deposition of fibers in fully developed turbulent pipe and duct flows. 83

4.2.2.1 Effect of lift force on deposition of fibers in fully developed turbulent pipe and duct flows. 91

4.3 Results from the developed UDF in Fluent 92

4.3.1 Laminar flow.. 93

4.3.1.1 Trajectory of single fiber in a laminar pipe flow.. 94

4.3.1.2 Deposition of fibers in a laminar pipe flow.. 95

4.3.1.3 laminar deposition of fibers in human nasal cavity. 97

4.3.2 Turbulent flow.. 100

4.3.2.1 Deposition of fibers in a turbulent channel flow.. 100

4.3.2.2 Turbulent deposition of fibers in human nasal cavity. 102

4.3.2.3 Effect of fluctuating velocity gradient 107

Chapter 5. Summary, Conclusions and Recommendations. 108

5.1 Summary and conclusions. 109

5.1.1 Laminar flow.. 109

5.1.2 Turbulent flow.. 110

5.2 Suggestions for further works. 112

References. 113

Appendix A.. 122

Appendix B.. 123

Appendix C.. 124

Appendix D.. 127

List of figures

Figure 1.1 Sample of non-spherical particles, Damstedt et al. (2007) 2

Figure 1.2 Typical shape of fibers. 3

Figure 1.3 HRT and various deposition mechanisms for fibers, Sturm and Hofmann (2007) 4

Figure1.4. Results for spherical particle deposition in a fully developed pipe flow from the implementation of DRW model in Ansys Fluent in comparison with experimental data. 9

Figure1.5. Results from CRW model, Dehbi (2011) 11

Figure 2.1. Different interaction regimes between particles, : particle phase volume fraction, particle response time, particle time constant, turn over time of a large eddy (Elgobashi 1994) 20

Figure 2.2. Dilute-dense flow for air at standard conditions, (Sommerfeld 1994. 21

Figure 2.3. Fiber geometry and different coordinate systems. 23

Figure 2.4. Geometry of an ellipsoidal fiber. 25

Figure 2.5. Angle of incidence in the fiber frame. 29

Figure 3.1. a) Solution of ordinary differential equation, b) Solution of stochastic differential equation, Pope (2000) 35

Figure 3.2. Schematic of DRW model 39

Figure 3.3. Results for particle deposition from implementation of DRW model with near wall correction in Ansys Fluent, compared against those from DRW model without near wall correction and experimental data. 40

Figure 4.1. Comparison of predicted fiber trajectory projections with the earlier study of Hogberg et al. (2008), a) Projection of fiber trajectory on XY plane, b) Projection of fiber trajectory on YZ plane, c) Comparison between location of the fiber center of mass in XY plane, d) Geometry of pipe and reference coordinate system.. 54

Figure 4.2. Comparison of predicted time evolution of fiber direction cosine cos(Y, ) with the earlier studied of Tian et al. (2012) and Feng and Kleinstreuer (2013) 55

Figure 4.3. Comparison of drag coefficients for an ellipsoid with β=2 for a range of Reynolds numbers, =0°. 57

Figure 4.4. Comparison of drag coefficients for an ellipsoid with β=2 for a range of Reynolds numbers, =90°. 58

Figure 4.5. Comparison of drag coefficients for an ellipsoid with β=5 for a range of Reynolds numbers, =0°. 58

Figure 4.6. Comparison of drag coefficients for an ellipsoid with β=5 for a range of Reynolds numbers, =90°. 59

Figure 4.7. Comparison of rotational torque coefficients for an ellipsoid with β = 2 for various rotational Reynolds numbers. 59

Figure. 4.8. Comparison of rotational torque coefficients for an ellipsoid with β = 5 for various rotational Reynolds numbers. 60

Figure 4.9. Comparison of fiber velocity in the present study with results from Feng and Kleinstreuer (2013) 61

Figure 4.10. a) Projection of an ellipsoid’s trajectories and orientations on the X-Z plane during its motion in a pipe. b) Trajectories of ellipsoid mass center in the X-Y plane. c) Time evolution of the first quaternion ε_1.Here a = 10 μm, = 2 in a pipe with R = 2 mm, = 0.012 m/s, 0.03 < Re_p< 0.044, a = 10 μm and = 2. Fiber initial orientation: = -45°, = 45°, = 0. Red: Creeping flow correlation. Green: Zastawny et al. (2012) correlation. Blue: Holzer and Sommerfeld (2009) correlation. 64

Figure 4.11. a) Projection of an ellipsoid’s trajectories and orientations on the X-Z plane during its motion in a pipe. b) Trajectories of ellipsoid mass center in the X-Y plane. c) Time evolution of the first quaternion ε_1.Here a = 10 μm, = 2 in a pipe with R = 2 mm, = 0.012 m/s, 0.031 < Re_p< 0.045. Fiber initial orientation: = 90°, = 30°, = 0. Red: Creeping flow correlation. Green: Zastawny et al. (2012) correlation. Blue: Holzer and Sommerfeld (2009) correlation. 65

Figure 4.12. a) Projections of an ellipsoid’s trajectories and orientations on the X-Z plane during its motion in a pipe. b) Trajectories of ellipsoid mass center in the X-Y plane. c) Time evolution of the first quaternion ε_1. Here a = 10 μm, = 2 in a pipe with R = 2 mm, = 0.06 m/s, 0.03 < Re_p< 0.22. Fiber initial orientation: = 90°, = 90°, = 90. Red: Creeping flow correlation. Green: Zastawny et al. (2012) correlation. Blue: Holzer and Sommerfeld (2009) correlation. 67

Figure 4.13. Comparison of the predicted deposition efficiency of fibers in a fully developed laminar pipe flow with the study of Tian et al. (2012). Q = 0.5 L/min, 69

d = 4.2 mm, L = 70 cm. 69

Figure 4.14. Comparison of the predicted deposition efficiency of fibers in a fully developed laminar pipe flow with the study of Tian et al. (2012). Q = 1.6 L/min, 70

d = 4.2 mm, L = 70 cm. 70

Figure 4.15. Comparison of predicted deposition efficiency of fibers with correlations of Pich (1972) and Yeh et al. (1979) for a = 10 µm, β = 2, U = 0.1 m/s. λ is varied by decreasing fluid viscosity. 74

Figure 4.16. Comparison of predicted deposition efficiency of fibers with correlations of Pich (1972) and Yeh et al. (1979), for a = 10 µm, β = 2, λ is varied by decreasing average flow velocity. 76

Figure 4.17. Comparison of predicted deposition efficiency of fibers with correlations of Pich (1972) and Yeh et al. (1979), for a = 10 µm, β = 5, and λ is varied by decreasing average flow velocity. 77

Figure 4.18. Comparison of fiber mean-square translational displacements in an isotropic homogeneous turbulent flow for, NP=1000, a=2μm, b=c=50μm, , initial location, , , . 80

Figure 4.19. Comparison of first and second moments of translational displacement in a fully developed pipe flow for a fiber with a=2μm, b=c=50μm released from initial location of , , for , R=0.025m. 82

Figure 4.20. Fiber deposition scheme on a smooth surface. 84

Figure 4.21. Dispersion pattern of fibers in turbulent pipe flow for, Red, Re=30000, NP=1000, u*=0.6, a=1μm, dt=10^-5s, β=20, , Dispersion pattern of fibers if moving with mean flow, Black. 86

Figure 4.22. Comparison of the simulated deposition of glass fibers with different length in a turbulent pipe flow for Re=30000, u*=0.6, a=1.9μm, dt=10^-6s, and with initial random orientation with the experimental data of Shapiro and Goldenberg (1993). 87

Figure 4.23. Comparison of floor deposition velocity of glass fibers in a turbulent pipe flow, a=1.5μm, initial random orientation. 89

Figure 4.24. Comparison of floor deposition of glass fibers with different length in a turbulent duct flow for Re=17000, a=2.5μm, dt=10^-6s, initial random orientation. 91

Figure 4.25. Comparison between projection trajectories of a fiber in a laminar pipe flow. Red, developed UDF and Blue, developed Code. 94

Figure 4.26. Comparison between trajectories of a fiber center of mass in a laminar pipe flow. 95

Figure 4.27. Comparison between deposition efficiency of fibers in a laminar pipe flow,( Δt=5×10^-6s, L/R=3, a=5μm, β=5, ) 96

Figure 4.28. Comparison between deposition efficiency of fibers in a laminar pipe flow, (Δt=5×10^-6s, L/R=3, a=5μm, β=10, ) 96

Figure 4.29. Lateral view of studied nasal cavity. (All dimensions are in cm) 97

Figure 4.30. Comparison between projections of trajectory of a single fiber in the model of human nasal cavity. 99

Figure 4.31. Comparison between present deposition efficiency of fibers in the model of human nasal cavity with previous study. 100

Figure 4.32. Geometry of channel flow in the study of Kvasnak and Ahmadi (1995) 101

Figure 4.33. Comparison of deposition efficiency of fibers with experimental study of Kvasnak and Ahmadi (1995) 102

Figure 4.34. Typical deposition pattern of fibers on the bottom wall in the turbulent channel flow similar to study of Kvasnak and Ahmadi (1995), Re=17000. 103

Figure 4.35. Comparison between pressure drop in the nasal cavity for various breathing conditions. 104

Figure 4.36. Comparison between turbulent deposition efficiency of fibers in the model of human nasal cavity with previous studies. 106

Figure 4.37. Comparison between laminar and turbulent deposition efficiency of fibers in the model of human nasal cavity. 106

Figure 4.38. Comparison between deposition efficiency of fibers in the model of human nasal cavity with and without fluctuating velocity gradient 107

List of tables

Table 2.1 Constants in correlations of Zastawny et al. (2012) for hydrodynamic force and torque coefficients………………………………………………………………………………. 31

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