Numerical and Experimental Study of Deposition of Polystyrene Particles in Multiphase Pipe Flows. by Fasil Ayelegn TASSEW. supervisor Alex C.

Transcription

1 UNIVERSITY OF BERGEN Numerical and Experimental Study of Deposition of Polystyrene Particles in Multiphase Pipe Flows by Fasil Ayelegn TASSEW supervisor Alex C. HOFFMANN A thesis submitted in partial fulfilment for the degree of Master of science in process technology Faculty of Mathematics and Natural Sciences Institute for physics and technology June 2015

2 Declaration of Authorship I, Fasil Ayelegn Tassew, declare that this thesis titled, Numerical and Experimental Study of Deposition of Polystyrene Particles in Multiphase Pipe Flows and the work presented in it are my own. I confirm that: This work was done wholly while in candidature for a master s degree at the University of Bergen. Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. Where I have consulted the published work of others, this is always clearly attributed. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. I have acknowledged all main sources of help. Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself. Signed: Date: i

3 UNIVERSITY OF BERGEN Abstract Faculty of Mathematics and Natural Sciences Institute for physics and technology Master of science by Fasil Ayelegn TASSEW Solid particle deposition on surfaces occur in various industries. Although particle deposition may be beneficial in some industries where processes such as spray coating or filtration is essential, for many other industries particle deposition is seen as a problem and poses a challenge. Deposited particles often block process equipment, pipelines etc. In order to alleviate this problem it is important to understand why and how particles deposit and what factors influence the deposition behaviour of particles. Years of research have been dedicated to accumulate knowledge about particle deposition. In this thesis the deposition of spherical Polystyrene particles with a diameter of 100µm was studied in a flow cell containing an obstruction. The influence of the flow and the particle properties such as the Reynolds number, the work of cohesion/adhesion, the Adhesion parameter and the Tabor parameter (determines the particle stiffness) were investigated by numerical simulations using a commercial computational fluid dynamics software called STAR-CCM+ as well as laboratory experiments. The discrete element model coupled with the Lagrangian multiphase model was used to simulate the effects of variations of the Reynolds number, the Adhesion parameter and the Tabor parameter and the results were analysed and discussed. Laboratory experiments were also carried out by varying the Reynolds number values to validate the results from the simulations. The results from the simulations were found to be in agreement with the results from the laboratory experiments. Moreover, a literature review was carried out to validate the findings and they were found to be in good agreement with the simulation and laboratory experiment observations in this study.

4 Acknowledgements First and foremost I would like to thank my academic supervisor Professor Alex Christian Hoffmann for his invaluable support and supervision throughout this study. I would also like to thank Maryam Ghaffari who contributed and provided suggestions during meetings, laboratory experiments and simulations. Professor Tanja Barth and Marit Bøe Vaage helped me with providing distilled water and other apparatuses for the experimental work and I am very thankful. I also would like to extend my thanks to Professor Pawel Kosinski and associate professor Boris Balakin for facilitating simulation software training, installation and updates. I would like to thank my friends at multiphase group who have been very friendly and supportive: Arman Salimi, Ingeborg Elin Kvamme, Jaime Luis Suarez, Yuchen Xie and Kari Halland from safety group. I would like to thank Steve R. Gunn and Sunil Patel for providing the LaTex script that greatly helped me prepare this document[1]. Special thanks to my family who have been understanding and supportive: Ayelegn Tassew, Yeshi Teshager, Frehiwot Ayelegn, Tanawork Ayelegn, Samuel Ayelegn and Kidus Ayelegn I miss you every day. Finally, I would like to express my gratitude to Sunniva Lode Roscoe who have been supportive and helpful. iii

5 Contents Declaration of Authorship i Abstract ii Acknowledgements iii List of Figures List of Tables Abbreviations Symbols vii ix x xi 1 Theory Background Objectives Multiphase flow modelling and particle deposition Description of contact models Particle-Particle interactions Hard sphere model Soft sphere model Particle-wall interactions Particle aggregation and deposition Origin of adhesive forces Factors affecting particle deposition Mechanisms of particle deposition Numerical methods Computational fluid dynamics Explicit and implicit methods Numerical stability and convergence iv

6 Contents Multiphase flow simulation Continuous phase equations Single particle equations Dispersed phase equations Properties of the dispersed phase Multiphase flow simulation in STAR-CCM Experimental methods Numerical experiments Dimensional analysis Numerical experiment set-up Numerical experiment variables Laboratory experiments Experimental set-up Laboratory equipment Polystyrene particles Re-experiments Ensuring correct fluid velocity Results and discussion Numerical experiments Results from Re-experiments Results from low cohesivity experiments Results from high cohesivity experiments Adhesion parameter results Tabor parameter results Laboratory experiments Conclusions 79 6 Recommendations Simulation capability Post-processing Laboratory experiments A Dimensional analysis 84 B Grid points at the boundary: Polynomial approach 87 C Post-Processing particle tracks 89 D Extended hard sphere model equations 91 D.1 Solution manual for the extended hard sphere model particle-wall collision equations v

7 Contents D.1.1 Case-I D.1.2 Case-II D.1.3 Case-III Bibliography 94 vi

8 List of Figures 1.1 Hard sphere collision of particles Geometry of hard sphere collision Soft sphere model analogy Particle-wall collision A Particle-wall collision B Comparison of standard and extended hard sphere models Formation of liquid bridge Particle-wall collision Discrete grid points Illustration of volume averaging procedure Moving control surface enclosing a particle Flow cell geometry dimensions Flow cell geometry regions Flow cell geometry after mesh operation Diagram of the experimental set up Experimental set-up The effect of Reynolds number on deposition (low cohesivity) Velocity profile of the particle tracks Front view of particle deposition (low cohesion) Bottom view of particle deposition (low cohesion) The effect of Reynolds number on deposition efficiency (high cohesivity) Front view of particle deposition (high cohesivity) Bottom view of particle deposition (high cohesivity) The effect of adhesion parameter on deposition efficiency (Re=333) The effect of adhesion parameter on deposition efficiency (Re=1733) The effect of adhesion parameter on deposition efficiency (particle slip velocity) Front view of particle deposition (adhesion parameter) Bottom view of particle deposition (adhesion parameter) Front view of particle deposition (tabor parameter) Bottom view of particle deposition (tabor parameter) vii

9 List of Figures 4.15 Flow cell locations of interest Particle deposition in Section A Particle deposition in Section B Particle deposition in Section C B.1 Boundary grid points[5] viii

10 List of Tables 2.1 Summary of single particle equations Experimental values for Re simulations Experimental values for Adhesion parameter simulations Experimental values for Tabor parameter simulations Description of physics model particle-particle interaction model particle injector settings Velocity and Volumetric flow rate values for Re simulations Threshold velocity values High cohesivity threshold values and deposition efficiency The effect of Adhesion parameter on deposition efficiency Tabor parameter results ix

11 Abbreviations CFD DE DEM DMT DPM JKR LES MD MYD PDE RANS RTT Computational Fluid Dynamics Deposition Efficiency Discrete Element Method Derjaguin Muller Toporov Discrete Parcel Method Johnson Kandall Roberts Large Eddy Simulation Maugis Dugdale Muller Yushchenko Derjaguin Partial Differential Equation Reynolds Averaged Navier Stokes Reynolds Transport Theorem x

12 Symbols Symbol Name Unit a/a H /a DMT /a JKR Contact radius m a 1 Liquid curvature radius m a 2 Liquid bridge radius m A p Projected area m 2 Ad Adhesion parameter Dimensionless A interface Area of interface m 2 A vdw Hamaker constant J B disp Dispersion energy coefficient Dimensionless c d Specific heat of dispersed phase J/kg K C D Drag coefficient Dimensionless C fs Static friction coefficient Dimensionless C vm Virtual mass coefficient Dimensionless d Surface separation distance m D Particle diameter m D 0 Interfacial contact separation m D ch Characteristic channel diameter m e Restitution coefficient Dimensionless E eq Equivalent Young s modulus Pa f Body force N f c Collision frequency s 1 f u User defined body force per unit volume N m 3 F adh Force of adhesion N xi

13 Symbols F contact Contact force between spheres N F coh Force of cohesion N F d Particle drag force N F g Gravitational force N F Liquid bridge Liquid bridge force N F n Normal component of contact force N F p Particle pressure gradient force N F s Particle surface force N F t Tangential component of contact force N F vdw Van Der Waals force of attraction N F vm Particle virtual mass force N F u User defined body force N g Gravitational constant m s 2 G Total Gibb s free energy KJ mol 1 G b Gibb s free energy with bulk properties KJ mol 1 G eq Equivalent shear modulus Pa G s Gibb s free surface energy KJ mol 1 I Moment of inertia Ns J Impulse force Ns k c Thermal conductivity of the continuous phase W/mK K n /K t Normal/tangential spring stiffness coefficients N m 1 l Characteristic dimension dispersed/continuous phase m l sd Length of interparticle spacing m L Characteristic dimension of physical system m m/m/m p Mass of particle kg M eq Equivalent mass of particle kg n Number density of particles m 3 N 1 /N 2 Number of molecules per unit volume m 3 P Pressure Pa Q Volumetric flow rate m 3 s 1 r Distance between isolated molecules m xii

14 Symbols r m Meniscus radius m R Particle radius m Re Reynolds number Dimensionless R eq Equivalent radius of particle m R Cmin Minimum contact radius m R min Minimum sphere radius m St v Stokes number for particle velocity Dimensionless St mass Stokes number for particle mass Dimensionless St T Stokes number for particle energy Dimensionless t Time s T Temperature K U(r) Dispersion energy J v r Relative velocity between particles m s 1 V Particle velocity m s 1 V ave Characteristic dimension of averaging volume m 3 V c Volume of continuous phase m 3 V d Volume of dispersed phase m 3 V p Volume of a particle m 3 V Rayleigh Velocity of Rayleigh wave m s 1 V s Particle slip velocity m s 1 V threshold Particle threshold velocity m s 1 W adh Work of adhesion J m 2 W coh Work of cohesion J m 2 Pstatic Static pressure gradient Pa m 1 α Thermal diffusivity m 2 s 1 α c Continuous phase volume fraction Dimensionless α d Dispersed phase volume fraction Dimensionless γ Surface energy J m 2 δ n /δ t Normal/tangential deformation distances m ɛ Interatomic spacing m η Damping coefficient Dimensionless xiii

15 Symbols η n /η t Normal/tangential damping coefficients Dimensionless θ Contact angle Degree µ c Fluid dynamic viscosity Pa s 1 µ tabor Tabor parameter Dimensionless µ c tabor Modified Tabor parameter Dimensionless ν Poisson s ratio Dimensionless ρ f Fluid density kg m 3 ρ p Particle density kg m 3 σ Capillary force N m 1 τ 1 DEM time step s τ c Time between collisions s τ F Time characteristic of flow field s τ M Characteristic mass transfer time s τ T Thermal response time s τ V Momentum response time s υ Fluid velocity m s 1 υ 0 Fluid inlet velocity m s 1 ω Angular momentum s 1 xiv

16 Dedicated to Ayelegn Tassew, Yeshi Teshager, Frehiwot Ayelegn, Tanawork Ayelegn, Samuel Ayelegn and Kidus Ayelegn xv

17

18 Chapter 1 Theory This chapter lays out the theoretical background for particle deposition, reviews contact models that describe contact between solid bodies, discusses the origins of adhesive forces as well as factors that affect particle deposition. 1.1 Background Deposition of solid particles is a common problem in multiphase flow transport. Various industries that require pipe transportation of multiphase systems or systems that involve solid particle contact with a surface often had to deal with deposition of solid matter on surfaces at a big maintenance costs. In oil and gas industries deposition of wax on inner subsea pipeline surfaces cause gradual decrease in flow rate and ultimately lead to complete blockage of the pipe unless the deposit is removed periodically[6]. Adhesion of powder particles on to solid surfaces is a challenge in pharmaceutical and food industries[7][8]. The problem of particle adhesion/deposition on to pipe surfaces also extends its effect on human health. Deposition of particulate matter in human lung[9], trachea[10] and blood vessels[11] have been known to cause significant health risks. Understanding how particles adhere on to solid surfaces and what factors influence their deposition behaviour as well as understanding how these factors affect the extent of deposition is important in preventing or reducing solid matter build-up. 2

19 Chapter 1. Theory Objectives At the beginning of August 2014 several thesis objectives were being considered. After consulting with my academic supervisor and reviewing available literature the following topics were chosen as the thesis objectives: To understand how the deposition of Polystyrene particles is affected by the variation in the Reynolds number within the laminar flow regime in an obstructed flow cell. To assess the influence of cohesivity of the particle (W coh and W adh ) on the deposition of Polystyrene particles. To understand how the Adhesion parameter affects the deposition of Polystyrene particles in an obstructed flow cell. To understand how the Tabor parameter, Young s modulus and Poisson s ratio affect the deposition of Polystyrene particles. To establish the capability of the DEM-Lagrangian multiphase model in simulating Polystyrene particle deposition in an obstructed flow cell. To compare and contrast the results from the numerical simulations with the results from the laboratory experiments. 3

20 Chapter 1. Theory Multiphase flow modelling and particle deposition Development of powerful computational fluid dynamics(cfd) tools that simulate particle (dispersed phase) and fluid (continuous phase) flows as well as particlefluid, particle-particle and particle-boundary interactions provided an attractive approach to study particle deposition. There are various commercial CFD software that are used to simulate particle-particle and particle-fluid interactions in multiphase flows. In this thesis STAR-CCM+, one of these commercial CFD software, was used to simulate particle deposition in multiphase pipe flows. STAR-CCM+ software provide the discrete element model (DEM) and the Lagrangian multiphase models which were extensively used to simulate the deposition of the Polystyrene particles. Description of both DEM and Lagrangian multiphase models are given in Chapter two Section Description of contact models The contact of solid bodies have been studied for several decades and various theories have emerged over the years. In his pioneering work Hertz described the contact between spherical elastic solid bodies. His work showed that the radius of the circle of contact between the solid bodies depends on the pressure (P), the spherical radius(r) and the elastic properties of the bodies involved in the contact(e eq )[12]. E eq = 4 3 a 3 H = P R E eq (1.1) (1 υ υ2 ) 2 E 1 E 2 (1.2) Where, υ 1 and υ 2 are the Poisson s ratios for the solid bodies and E 1 and E 2 are the Young s moduli. The Hertz model works well for contacts where adhesion doesn t play a significant roll. However, when the adhesive force become significant, such as in cases of contacts that involve small, soft particles with high adhesive/cohesive nature, the Hertzian model becomes less effective. Several other models have 4

21 Chapter 1. Theory been proposed to accommodate the adhesivity of solid bodies in contact mechanics models. One such proposal is the DMT model proposed by Derjaguin, Muller and Toporove[13]. In their model they added an extra load term into the Hertz equation for the contact radius to account for the adhesive effect. a 3 DMT = (P + 2π γr)r E eq (1.3) Where γ is the work of adhesion and the extra load term is P DMT = 2π γr Johnson, Kendall and Roberts in their JKR theory studied the contact between a rigid solid surface and an elastic half-space (a half space is a segment of an n- dimensional space that remains when a segment on the side of an (n-1) dimensional hyperplane is removed[14]) and came up with a model for the contact radius (a JKR ) a 3 JKR = R [P + 3π γ + ( 6π γrp + (3π γr) 2) ] 1 2 E eq (1.4) Others such as the MD(Maugis and Dugdale) and MYD (Muller, Yushchenko and Derjaguin) models are also proposed. Each with their own advantages and disadvantages. Although, they are not discussed in this thesis, an interested reader may refer to the following literature[12][15] Particle-Particle interactions In dilute particle flows particle-particle interactions are rare. As a result, the effect of the interactions on the flow behaviour can safely be ignored. However, if the number density (the number of particles per unit volume) of the particles is increased enough, the flow can be considered as a dense flow and the frequency of particle-particle interactions and subsequent loss of kinetic energy can not be ignored. In such cases particle-particle interactions have to be studied and accounted for. Several models have been proposed to model particle-particle as well as particle-wall collisions for different particle inertias and continuous phase flow conditions[16]. For example, in 1956 Saffman and Turner[17] proposed a model for the collision of droplets of equal sizes in turbulent flows. Later in 1975 Abrahamson[18] proposed a model for the rate of particle collisions in a high intensity turbulent flows. Other examples of particle collision models such as 5

22 Chapter 1. Theory Veeramani et al.[19], who studied collision between two non-brownian particles in multiphase flows and suggested a model based on stereomechanical impact model, were also attempted. However, the two most common models that are used to study particle-particle interactions are the hard sphere and soft sphere models. A review of each model is presented below Hard sphere model The hard sphere model is based on the assumptions that particles are rigid spheres with no deformation during collision or contact, the Coulomb s friction law dictates the friction on sliding particles and once a particle stops sliding no further sliding occurs. Figure 1.1: Hard sphere collision of particles. Where ω 1 and ω 2 are angular momentums and V 1 and V 2 are particle velocities of particles 1 and 2 respectively The hard sphere model is based on integrated form of the Newtonian equations of motion for the colliding particles. The collision is not resolved in time, which means the particles translational and rotational velocities after the collision are determined by the integration of the conservation law. However, this restricts the hard sphere model to be applicable only for a binary collision at a time. The equations that govern the hard sphere collision of particles are based on impulsive forces[3]. 6

23 Chapter 1. Theory m 1 = (V 1 V (0) 1 ) = J (1.5) m 2 = (V 2 V (0) 2 ) = J (1.6) I 1 = (ω ω 1 ω (0) 1 ) = r 1 n J (1.7) I 2 = (ω ω 2 ω (0) 2 ) = r 2 n J (1.8) Where, n is the unit normal vector, J is the impulse force on the particles, I is the moment of inertia, r is the particle radius, m is the particle mass and superscript (0) indicates values before the collision. Extended hard sphere model Kosinski and Hoffmann[2] provided a model to account for the cohesivity of the particles during particle-particle collision by directly incorporating a cohesive impulse into the impulse based hard sphere collision model. Plane of collision Particle 2 J t n J t J n Particle 1 Figure 1.2: Geometry of hard sphere collision[2] Since J, the impulse that is acting on the particles during the collision, represents particle repulsion due to elastic deformation then incorporating a factor for the 7

24 Chapter 1. Theory particle cohesivity would account for particle attraction during the collision. In the standard hard sphere model, the normal component of the impulse(j ns ) is given as: J ns = m 1m 2 m 1 + m 2 (1 + e)(n.g (0) ) (1.9) J t = fj ns (1.10) Where, J t, f, e, m 1, m 2, n and G (0) are the tangential impulse component, friction coefficient, restitution coefficient, mass of particle 1, mass of particle 2, unit vector from particle 1 to particle 2 and relative velocity before the collision. Kosinski and Hoffman argued that the above equation should include a cohesive impulse component. The new normal component of the impulse(j n ) would then be: J n = J ns + J n,c (1.11) J t = f(j ns J n,c ) (1.12) Where, J n,c is the cohesive impulse that accounts for particle attraction due to the cohesivity of the particle during the collision. Consequently, equations were derived for the particles post-collision velocities. For particles that slide throughout the collision the requirement and post collision velocities are: Requirement n.g (0) < m 1 + m 2 m 1 m 2 J n,c ct 1 + e + 2 G(0) 7f(1 + e) (1.13) V 1 = V (0) 1 + J n,c m 1 (n ft) m 2 m 1 + m 2 (1 + e)n.g (0) (n + ft) (1.14) V 2 = V (0) 2 + J n,c m 1 (n ft) + (1 + e)n.g (0) (n + ft) (1.15) m 2 m 1 + m 2 ω 1 = ω (0) (n t)f 2r 1 [ J n,c m ] 2 (1 + e)n.g (0) m 1 m 1 + m 2 (1.16) 8

25 Chapter 1. Theory ω 2 = ω (0) (n t)f 2r 2 [ J n,c m ] 1 (1 + e)n.g (0) m 2 m 1 + m 2 (1.17) Similarly equations for particles that stop sliding during the collision were also formulated. The requirement for this condition is the same as the requirement in the standard collision model. The equations for post-collision velocities are given below: Requirement J t = 2m 1m 2 7(m 1 + m 2 ) G(0) ct (1.18) [( ) ] V 1 = V (0) 1 + Jn,c m 1 m 2 m 1 +m 2 (1 + e)n.g (0) n 2m 2 7(m 1 +m 2 ) G(0) ct.t (1.19) [( ) ] V 2 = V (0) 2 + Jn,c m 2 m 1 m 1 +m 2 (1 + e)n.g (0) n 2m 1 7(m 1 +m 2 ) G(0) ct.t (1.20) ω 1 = ω (0) 1 5 (n + t) G (0) 7r 1 ct m 2 m 1 + m 2 (1.21) ω 2 = ω (0) 2 5 (n + t) G (0) 7r 2 ct m 1 m 1 + m 2 (1.22) Soft sphere model Unlike the hard sphere model, the soft sphere model is based on differential form of the Newtonian equation of motion of the particles. The collisions are resolved in time, hence, the collision of more than two particles at a time can be studied [20] The soft sphere model takes particle deformation into consideration. When particles collide they exert energy on each other, this makes them to be momentarily deformed and lose energy. Due to the loss of energy, the coefficient of restitution is less than one. The coefficient of restitution is the ratio of pre-collisional and post-collisional velocities. The energy loss can be modelled using a dash-pot or viscous damper. The equation for a damped oscillator is given as follows: m d2 x dt + η dx + kx = 0 (1.23) 2 dt 9

26 Chapter 1. Theory Where the first term is mass times acceleration, the second term is the damping term and the third term is the spring term. The constants η and k are the damping coefficient and the stiffness of spring, respectively. k m η x x Figure 1.3: Soft sphere model analogy using spring-damper system[3] Particle-wall interactions There are two types of particle-wall interactions. The first one is hydrodynamic interactions. These interaction arise due to the close proximity of particles to a wall such as a pipe surface. An example of this is the Saffman lift force where velocity gradient prevents the particles from contacting the wall. The second category is the fluid force interactions with a particle. When a particle approaches a wall, pressure increases and prevents the particle from contacting the wall. If the particle has a large inertial force, contact with the wall takes place regardless of the hydrodynamic and fluid forces. A particle that has collided with a wall has two likely outcomes, either it bounces back off the wall with a loss of some of its kinetic energy or it remains stuck on the wall if the adhesive/cohesive forces are dominant and can overcome the particles inertial force[3]. 10

27 Chapter 1. Theory Momentum and energy exchange during particle-wall interaction The hard sphere model describes particle-wall collision using the impulse equations, however, the impulse equations alone do not establish the relationship between pre-collision and post-collision velocities of the particle. To achieve this, the coefficient of restitution and the coefficient of friction must be used. There are different ways to define the coefficient of restitution. Crowe et.al [3] provided some of these definitions. e = V (2) V (0) (1.24) e x = V x (2) V x (0) e = V y (2) V y (0) = e y = V y (2) V y (0) e = J y (2) J y (1) (1.25) (1.26) (1.27) V x (0) V (2) =(V (2) X ),V (2) Y,V (2) Z ) ω (2) V x (0) V y (0) V (0) V (2) V y (2) V (0) =(V (0) X ),V (0) Y,V (0) Z ) ω (0) wall Figure 1.4: Particle-wall collision A.[3] Figure 1.5: Particle-wall collision B[3] If the coefficient of restitution and the coefficient of friction are known for a spherical particle colliding with a flat wall, then the impulse equations can be used to solve the post-collision translational and angular velocities. 11

28 Chapter 1. Theory During particle-wall collision a particle passes through a compression and recovery periods as well as brief slide across the wall surface. The post-collision velocities are dictated by how long the particle slides and at what period the particle stopped sliding. Generally, there are three cases, the first one is when the particle stops sliding in the compression period, the second one is when the particle stops sliding in the recovery period and the last case is when the particle slides in both the compression and recovery periods. Kosinski and Hoffmann[21][4] provided an extension of the hard sphere model to account for the adhesive force during particle-wall collisions. They introduced an impulse term in to the impulse equations of the standard hard sphere model. Both the extended and standard models consider the three cases. Y X J ys J J ys J y J J x J y,t J x Standard hard sphere model Extended hard sphere model Figure 1.6: Comparison of standard and extended hard sphere models[4] In the first case the particle stops sliding during the compression period, in the second case the particle stops sliding in the recovery period and the last case considers a particle that continues to slide throughout the collision period. All the impulse terms in the standard model that are acting on the particle as a push-off force were given an extra impulse term in the extended model to account for the force acting on the wall surface due to adhesive force. The extra impulse force acts in the Y-direction (i.e J t = (0, J y,t, 0)). In the case where particle stops sliding in the compression period, the modified equations are: J (s) x = m(v (s) x V (0) x ) (1.28) 12

29 Chapter 1. Theory J (s) y s + J (s) y,t = m(v (s) y V (0) y ) (1.29) J (s) z = m(v (s) z V (s) z ) (1.30) Where, the superscripts (s) and (0) indicate the sliding and the pre-collision periods. For a particle that stops sliding in the recovery period, the equations are: J x (r) = m(v x (1) V x (s) ) (1.31) J (r) y s + J (r) y,t = m(v y (1) V y (s) ) (1.32) J z (r) = m(v z (1) V z (s) ) (1.33) Where, the superscripts (r) and (1) indicate the recovery and the compression periods. For a particle that continues to slide throughout the whole collision period, the equations are: J (2) x = m(v (2) x V (1) x ) (1.34) J (2) y s + J (2) y,t = m(v (2) y V (1) y ) (1.35) J (2) z = m(v (2) z V (1) z ) (1.36) Since the standard model doesn t contain the impulse in the Y-direction for the equations of the particle rotation, both the extended and the standard models have the same equations for the particle rotation. They are given as: aj (s) z = I(ω (s) x ω (0) x ) (1.37) 13

30 Chapter 1. Theory 0 = I(ω (s) y ω (0) y ) (1.38) aj (s) x = I(ω (s) z ω (0) z ) (1.39) aj z (r) = I(ω x (1) ω x (s) ) (1.40) 0 = I(ω (1) y ω (s) y ) (1.41) aj x (r) = I(ω z (1) ω z (s) ) (1.42) aj (2) z = I(ω (2) x ω (1) x ) (1.43) 0 = I(ω (2) y ω (1) y ) (1.44) aj (2) x = I(ω (2) z ω (1) z ) (1.45) Where, I( ω = r J), r=(0,-a,0) and a is the particle radius. In addition the surface velocities of the particle at the point of contact are given as: (V (s) x + aω z (s) )i + (V z (s) aω x (s) )k = 0 (1.46) (V (1) x + aω (1) z )i + (V (1) y )j + (V (1) z aω (1) x )k = 0 (1.47) (V (2) x + aω (2) z )i + (V (2) z aω (2) x )k = 0 (1.48) The relationships between the impulses are given as: 14

31 Chapter 1. Theory J (s) x = ε x f(j (s) y s J (s) y,t ) (1.49) J (s) z = ε z f(j (s) y s J (s) y,t ) (1.50) ε 2 x + ε 2 z = 1 (1.51) J y (2) = (J y (2) s + J (2) y,t ) = e m (J y (s) s + J y (r) s + (J (2) y,t ) (1.52) Where, ε x and ε z are direction cosines for the sliding particle, J (r) y s is the impulse during the remainder of the recovery period and e m is the equivalent restitution coefficient in the extended hard sphere model. Kosinski and Hoffmann[4] also derived estimations of the extra impulse terms, namely the impulse during compression period(j (1) y,t) and the impulse during the recovery period (J (2) y,t). In addition they set out a particle deposition requirement: J (1) y,t = m ( 2F y,t m (D D 1) + (V (0) y ) 2) (1.53) J (2) y,t = m ( 2F y,t m (D 1 D c ) + V 2 2 V 2 ) (1.54) F y,t = aa vdw 6D c D 1 (1.55) V 1 = 2Fy,t m (D c D 1 ) + (V (0) y ) 2 (1.56) V 2 = V 1 e m (1.57) Where, A vdw is the Hamaker constant, D is the surface separation distance, D 1 is surface separation distance just before the collision where the attractive interaction is still negligible, D c is the surface separation distance at the end of the compression 15

32 Chapter 1. Theory period, V 1 is the velocity of the particle at impact and V 2 is the velocity of the particle after the impact. The criteria for the deposition of the particle is given as: 2F y,t (D 1 D c ) m + V (1.58) A solution manual for each of the above cases is provided in Appendix D Particle aggregation and deposition Particles dispersed in a continuous phase often exhibit aggregation and deposition behaviours. Aggregation of particles occur when individual particles associate to form clusters whereas deposition occur when particles are transported to a surface where they become attached. Aggregation and deposition processes have several similarities and each can be considered as an extreme form of the other (eg. deposition can be considered as heteroaggregation where particles of different type form aggregates). Both processes involve the transport and attachment steps. Various mechanisms of particle transport, hydrodynamic and electrical forces play part in determining the fate of the aggregated clusters and deposits [22]. Since the topic of this thesis is about particle deposition, detailed discussions were limited to the mechanism of particle-wall interactions and particle deposition only. Aspects related to particle aggregation and particle-particle interactions were only discussed briefly. (A previous masters student wrote a masters thesis on the subject of cohesive particle agglomeration[23]) Origin of adhesive forces Adhesion between particles and surfaces is thought to be a resultant force from several contributing forces such as the electrostatic, capillary and the van der Waals forces. Most of these contributing forces are active in short distance. The extent of contribution of each force to the overall adhesion force depends on the environmental and the experimental conditions as well as physicochemical properties of the involved particles and surfaces[7]. 16

33 Chapter 1. Theory Salazar-Banda et al.[8] showed that the adherence of dry and inert particles onto solid surfaces is dominated by the van der Waals force. However, the adhesion of wet particles is dominated by the capillary force due to liquid bridge[3]. van der Waals force The van der Waals force is a short range attractive force between macroscopic surfaces due to intermolecular interactions. The pairwise summation of all intermolecular interactions gives the van der Waals force [22]. Hamaker(1937)[24] studied the van der Waals interactions between two spherical particles as a function of separation distance between the particles and the diameter of the particles and It was found out that van der Waals force depends on the geometry and the molecular property of the interacting bodies. In general, the van der Waals interaction for any given two bodies such as two spheres, a sphere and a flat plate or two flat plates can be expressed as a product of A vdw, the Hamaker constant and H, a factor that is dependent on the geometry and dimension of the interacting bodies. For example, for two identical spherical particles H is, -D/24d, where D is the diameter of the sphere and d is the separation distance between the spheres. Capillary force When interacting bodies are in a liquid medium, the capillary force comes into significance. The role of the capillary force in particle adhesion/aggregation is that when the contacting bodies are approaching each other, there will be a build-up of pressure that acts as a buffer and prevents the contact. This can be countered if the contacting bodies have a high enough momentum and become close enough with each other for the capillary force to be active. ( 1 F liquid bridge = πa 2 2σ + 1 ) + 2πa 2 σcosθ (1.59) a 1 a 2 The above equation is used to quantify the force due to the liquid bridge and the capillary force and it is called the Younge-Laplace equation[3]. In the equation, 17

34 Chapter 1. Theory a 1 is the curvature radius of the bridge, a 2 is the radius of the liquid bridge, σ is the capillary force and θ is the contact angle. a 1 a 2 d Figure 1.7: Formation of liquid bridge Electrostatic force Since the van der Waals interaction between two solid surfaces in a liquid medium is always attractive[25], one would expect the particles to immediately aggregate or deposit. However, this is not always the case and the reason is because of the presence of the electrostatic repulsive forces. There are various mechanisms in which the electrostatic charges form on the surfaces of interacting bodies in liquid medium but apart from DVLO theory, which will be discussed in the next section, it is beyond the scope of this thesis to go further into electrostatic charge formation. However, interesting read can be found on this subject in books such as Intermolecular and surface forces by Jacob N. Israelachvili and An introduction to interfaces & colloids: a bridge to nanoscience by John C. Berg Factors affecting particle deposition There are several factors that affect particle deposition on a surface. The rate and the magnitude of deposition of particles is heavily influenced by the electrical properties of both the particles and the continuous phase, the flow properties, 18

35 Chapter 1. Theory the surface properties, the particle properties and the physical properties. Brief overviews of the above properties are given below. Electrical properties The electrical interaction between particles influence the particle stability, aggregation and deposition [22]. Most particles in aqueous solutions are electrically charged but the distribution of ions around the particles are in such a way that give rise to an electrical double layer. This means the surface charge present on charged particles are balanced by oppositely charged counter ions. There are different models that describe the electrostatic interactions between charged particles and how that affect the deposition and aggregation behaviour of particles. One such theory is the classical DLVO theory. The DLVO theory explains the stability of dispersed particles as a product of a balance between the attractive van der Waals force and the repulsive electrostatic Coulomb force interactions. Moreover, It has been shown that the electrical double layer interactions play a significant role in both the kinetics of particle deposition and the structure of deposited particles[26] [27]. The van der Waals force between two particles is always an attractive force and its magnitude depends on the size of the particles involved and their shape. Hamaker(1937) and De Boer (1936) suggested that the Van Der Waals interaction between two macroscopic objects is the summation of energies acting between all molecules in one macroscopic object with those in the other object.[28] For two identical spherical particles of radius R that are separated by a distance of d, the van der Waals attractive force, F vdw, as a function of separation distance between the particles is given as: F vdw = A vdw R/12d (1.60) As mentioned earlier, the interaction between macroscopic objects is a product of the Hamaker constant and a function derived from the geometries of the objects involved in the interaction. Hamaker constant is dependent on the nature of the interacting materials and it is defined as: [29] 19

36 Chapter 1. Theory A vdw(12) = π 2 N 1 N 2 B 12 (1.61) Where, N 1 and N 2 are the number of molecules per unit volume in material 1 and 2, respectively. B 12 is a coefficient in an expression for the dispersion energy, U(r), between isolated pairs of molecules 1 and 2 by a distance of r. Generally, B 12 may be simply referred to as B disp and it represents sum of the dipole-dipole, dipole-induced dipole (debye) and the london dispersion interactions[28]. U(r) = B 12 r 6 (1.62) Flow properties The manner and extent in which particles aggregate with each other or deposit on a surface depends on the flow characteristics to which the particles are subjected. There are several different flow characteristics that influence particle aggregation and deposition. The most important flow characteristics are discussed below. Reynolds number The Reynolds number, Re, is defined as the ratio of inertial and viscous forces of the fluid flow. It is an important parameter in determining the regime of a flow(i.e whether the fluid flow is laminar or turbulent). It is a dimensionless number and is usually defined as: Re = ρ fd ch υ µ c (1.63) Where ρ f is the density of the fluid, D ch is the characteristic length of a channel in which the fluid is flowing (eg. Diameter in pipe flows), υ is the velocity of the fluid and µ c is the viscosity of the fluid. The effect of Reynolds number on particle deposition and aggregation characteristics have been studied extensively. Adams et al. [30] studied the effect of Reynolds number on the deposition and dispersion of spherical particles in turbulent square duct flows using LES and RANS methods and found out that generally the rate of the deposition increases with the Reynolds number. 20

37 Chapter 1. Theory Stokes number The Stokes number is a ratio of the particles response time to that of the fluid s response time. It is a dimensionless parameter and it can be defined in relation to the particles mass, velocity and energy. For example, the Stokes number related to the particle velocity is defined as follows[3]: St v = τ v τ F = ρ dd 2 /18µ c D ch /υ (1.64) In the above equation, τ v and τ F are the particles velocity response time and the time characteristic of the flow field, D ch is length or diameter in which the fluid is passing through and υ is the flow velocity. The Stokes number can also be defined for the particles mass and energy as follows. St mass = τ m τ F = τ mυ D ch (1.65) St T = τ T τ F = τ T υ D ch (1.66) where τ m is the particles mass transfer response time and τ T is the particles thermal response time. When the Stokes number is much less than one, the particles response time is small, hence any change in the flow field would not affect the particles properties significantly. At a small response time the particle takes a very short time to adjust to the change in the flow field. On the other hand, a high Stokes number means the response time is high, hence the particle would take a long time to adjust to the change in the flow field. As a result, the particle properties would be affected by changes in the flow field. This will have implications when there is a need to consider coupling between the continuous and dispersed phases. In multiphase flows phase coupling describes how the continuous and dispersed phases affect one another. The effect can be one-way coupling where only the dispersed phase is affected by the continuous phase but not the other way around or it can be two-way coupling, where both phases are affected by one another. Flow properties of the continuous and dispersed phases such as velocity, temperature, particle size, density etc can be affected by phase coupling[3]. 21

38 Chapter 1. Theory Adhesion parameter The Adhesion parameter, Ad, is a dimensionless parameter that is often used in studies involving adhesive particle flows. The Adhesion parameter is defined as the ratio of adhesive surface energy and the particle inertia[31]. Ad = 2γ ρ p V 2 D (1.67) Where, γ is the adhesive surface energy, ρ p is the particle density, V is the particle velocity and D is the particle diameter. The adhesive surface energy can be related to Hamaker constant as follows:[25]. γ = A vdw 24πD 2 0 (1.68) In the above equation D 0 is the interfacial contact separation. Due to ambiguities of what value to use for D 0, Israelachvili (2011) suggested that a universal value that is less than interatomic center-to-center distances be used and this yields values of adhesive surface energies that are in good agreement with experimental values. For large values of the Adhesion parameter particles colliding with each other or with the wall tend to stick, forming particle agglomerates and deposits. The adhesive force facilitates formation of agglomerates and deposits by reducing the rebound velocity. Tabor parameter The Tabor parameter, µ tabor, is a dimensionless number often used in particle adhesion/cohesion studies. It is defined as[12]. µ tabor = ( ) RW 2 1/3 adh (1.69) Eeqɛ 2 3 E eq = 1 1 ν 2 1 E ν2 2 E 2 (1.70) 22

39 Chapter 1. Theory Where: E eq is the equivalent Young s modulus, E 1 and E 2 are Young s moduli for the contacting particles and surface material, respectively ν 1 and ν 2 are Poissons coefficients for the particle and surface material respectively, ɛ is the interatomic spacing (equilibrium spacing in Lennard-Jones potential), R is the equivalent radius and W adh is the work of adhesion. µ tabor is introduced by Tabor et al. [32][33]. It is understood to be the ratio of elastic deformation due to adhesion and the effective range of surface forces[12]. It is closely associated with adhesive contact models such as JKR and DMT models. More rigid solid particles with small radius tend to have low µ tabor values (µ tabor < 0.1) in such cases DMT model provides a better alternative [33]. In the other extreme, soft solids with large radius have high µ tabor values (µ tabor > 5) and they are better suited for the JKR model. Description of the JKR contact model is presented in Section For the adhesion of particles in the presence of liquids a modified Tabor parameter can be used as suggested by Xu et al.,2007 [34]. In such cases, the equilibrium separation distance in Lennard-Jones potential value, ɛ, should be replaced with the mean radius of the meniscus (r m ). ( ) RW µ c 2 1/3 tabor = adh (1.71) E 2 eqr 3 m Surface properties Properties of the wall surface such as surface energy, surface roughness and contact area play a role in the mechanism for, and extent of, particle deposition. Due to asymmetry of intermolecular force field near interfaces, the density of molecules near an interface is different from that of the bulk density. Hence, real interfaces always have a concentration gradient[35].the asymmetry of intermolecular forces gives rise to excess energy at the interface. This excess energy is defined in terms of Gibb s free surface energy: G s = G Gb A interface (1.72) 23

40 Chapter 1. Theory Where G is total Gibbs free energy, G b is Gibbs free energy if the interface properties were the same as that of the bulk and A interface is area of the interface. For practical purposes Gibbs free surface energy, G s, is assumed to be equal to surface energy, γ G s = γ = ( ) G (1.73) A p,v,n i Surface energy, γ, is related to work of cohesion and work of adhesion. For phases A and B, Work of cohesion, W coh, and work of adhesion, W adh, are defined as: W coh = 2γ A (1.74) W adh = γ A + γ B γ AB (1.75) STAR-CCM+ software provides a model for work of cohesion in DEM simulations that can be defined both for particle-particle as well as particle-wall multiphase interactions. Since surface energy and work of cohesion have units of energy per unit interfacial area, it is then imperative to consider what the interfacial area is and what role it has in particle deposition. In ideal cases where there are only perfectly smooth surfaces, it is easy to estimate the interfacial area, in practice surfaces have some degree of roughness. Tabor et al. (1977) performed an experiment to study the effect of surface roughness on adhesion and they observed that adhesion decreases with increasing surface roughness[32]. In addition, they concluded that higher roughness prevents effective adhesion between surfaces by forcing the surfaces apart. Particle properties Properties of the dispersed particles have various effects on their agglomeration and deposition behaviours. Particle size and shape have been known to have influences on the contact area and the surface energy and ultimately on the magnitude of adhesive/cohesive forces[36]. 24

41 Chapter 1. Theory Ermis et al. [37] studied the effect of size and shape of salt and glass particles on particle adhesion strength to a flat surface. They observed that the adhesive force between the particles and the surface increased with the increase of the particle size. However, when the particle size was increased even further the adhesive force weakened. They ascribed this to the increase in the impact force, which is directly related to the detachment force, with the increasing particle size. Overall, they observed the strongest adhesive force in medium sized particles. They also studied the effect of the particle shape on adhesion by using particles of different geometries such as sphere, cube, tetrahedron and octahedron. They observed that spherical particles show the lowest contact area whereas cubical particles show the highest Mechanisms of particle deposition The Hertz contact model doesn t account for the adhesive force during particle contact, however presence of the adhesive force during contact results in a small but observable increase in the contact area[38]. The JKR theory takes the effect of adhesion force on the elastic deformation into consideration [39]. Most of present understanding of particle adhesion stems from the JKR theory[40]. It is composed of three distinct terms; mechanical energy, elastic energy and surface energy terms. Assume an elastic particle thrown to a flat wall, since the particle is elastic the collision with the wall results in a temporary deformation. The extent of the deformation is dependent on how fast the particle is thrown, how elastic the particle is and how strongly the wall surface attracts the particle. The deformation can be measured as a contact radius. The contact radius in the JKR theory can be given in terms of W adh as follows: a 3 JKR = D 2E eq P + 3 ( ) 2 2 w 3πWadh D adhπd + 3πW adh DP + (1.76) 2 If the surface energy is not accounted for the JKR model will be reduced to the Hertzian contact model. Since the surface energy is zero then, W adh = 0, as a result the contact radius becomes: a 3 JKR = a 3 H = DP 2E eq (1.77) 25

42 Chapter 1. Theory There are also other contact models that take the contribution of adhesive forces into account such as the DMT model, the MD model and the MYD model. The DMT model, unlike the JKR model, considers a long range surface force, whereas the MD model uses a square-well potential for the model. The MYD model, on the other hand, uses both short and long range surface forces for the model[12]. D a Surface energy term wall Figure 1.8: Particle-wall collision In STAR-CCM+ three types of DEM contact models are provided to model DEM phase interactions. They are Hertz-Mindlin, Linear spring and Walton Braun models. The Hertz-Mindlin model is the standard contact model in STAR-CCM+ particle contact simulations. It is a variant of the spring-damper model and it assumes particles as elastic, perfectly smooth and with a small contact area during contact [41]. The contact force between two spheres is given as a sum of the normal and tangential components [42]. F contact = F n + F t = ( K n δ n η n V n ) + K nδ n C fs δ t δ t (1.78) 26

43 Chapter 1. Theory The tangential term in the above equation becomes K t δ t η t V t if K t δ t is less than K n δ n C fs. Moreover, the normal and tangential spring stiffness terms, K n and K t as well as the normal and tangential damping terms, η n and η t are calculated as: K n = 4 3 E eq δn R eq (1.79) K t = 8G eq δt R eq (1.80) η n = 5K n M eq η (1.81) η t = 5K t M eq η (1.82) where R eq is the equivalent radius, M eq is the equivalent particle mass, E eq is the equivalent Young s modulus, G eq is the equivalent shear modulus,c fs is the static friction coefficient, δ n and δ t are overlaps in the normal and tangential directions at the contact point and η is the damping coefficient. Subscripts A and B denote particle A and particle B. The equivalent values are calculated as follows: 1 R eq = (1.83) R A R B 1 M eq = (1.84) M A M B G eq = 1 2(2 ν A )(1 + ν A ) + 2(2 ν B)(1 + ν B ) E A E B (1.85) The above model is formulated for contact of two spheres(particle-particle interaction) but it can be applied for a contact between a sphere and a wall (particle-wall 27

44 Chapter 1. Theory interaction). In a case where particle-wall interaction is needed the formulas to be used are the same as above except that the radius and the mass of the wall are assumed to be infinite. This results the equivalent radius and the equivalent mass to be equal to the radius of the particle and the mass of the particle, respectively[42]. Adhesion/cohesion modelling in STAR-CCM+ is provided for the JKR and DMT models. In both cases the force of cohesion/adhesion is calculated as: F coh/adh = R Cmin πw coh/adh F (1.86) Where R Cmin is the minimum contact radius and F is a factor of 1.5 for the JKR model and 2 for the DMT model. 28

45

46 Chapter 2 Numerical methods This chapter is devoted to the discussion of numerical methods used in multiphase flow simulations. The basic principles of computational fluid dynamics as well as the techniques used to simulate fluid flows in general and multiphase flows in particular are discussed. Common techniques used in computational fluid dynamics such as discretization, finite difference method, implicit and explicit approaches are summarized. Brief reviews of the continuous and dispersed phase equations are also given from the viewpoint of multiphase flow simulation. 2.1 Computational fluid dynamics Traditionally, fluid dynamics problems have been solved using combinations of theoretical and experimental approaches. However, the development of modern computers and algorithms allowed inclusion of a third approach into solving fluid dynamics problems, that is a computational approach[5]. The CFD approach lies in between the theoretical and experimental approaches and it simplifies problems and reduces costs associated with experiments. Fluid flows are governed by three fundamental principles: the conservation of mass, the conservation of momentum and the conservation of energy. These principles can be expressed in mathematical equations and they are generally referred to as the governing equations. An equation derived by applying the mass conservation principle is called the continuity equation. Similarly the momentum and 30

47 Chapter 2. Numerical methods energy equations are derived by applying the principles of momentum and energy conservation on a fluid system, respectively. The governing equations can be derived in different ways and they can have different mathematical expressions depending on the type of flow system involved. For example, the governing equations for the continuity and momentum equations (only in x direction) for an unsteady, three-dimensional, compressible, viscous flows in a conservation form are given below [5]: ρ t + (ρv ) = 0 (2.1) (ρu) t + (ρuv ) = P x + τ xx x + τ yx y + τ zx z + ρf x (2.2) τ xx = λ(.v ) + 2µ u x [ v τ yx = µ x + u ] y [ u τ zx = µ z + w ] x (2.3) (2.4) (2.5) λ = 2 3 µ (2.6) Where ρ is density, u,v and w are velocity components in the x, y and z directions, V is velocity, P is pressure, f x is body force per unit mass acting on the fluid in the x-direction, τ xx, τ yx and τ zx are shear stress on different sides of the fluid element and λ is the second viscosity coefficient. The governing equations of fluid dynamics are a collection of partial differential and integral equations that must be solved to get any meaningful result. However, the process of solving these equations is not straightforward. There are several ways in which the equations can be solved. In CFD, partial differential equations and integral components of the governing equations are replaced with an equivalent and approximated discrete algebraic equations through a process called discretization. Discretization is a process in which a closed form mathematical equations such as partial differential equations (PDE) are approximated into a discrete and finite values[5]. Closed form equations such as partial differential equations that govern flow fields have dependent variables that continuously vary throughout the domain. 31

48 Chapter 2. Numerical methods The purpose of the discretization process is to replace these continuously varying variables with an approximate and finite values at specified grid points in the domain. There are different methods of discretization such as the finite difference, finite volume and finite element methods. An example of how a discretization is carried out is given below using finite difference method[5]. In the finite difference method a numerical grid is defined for a given geometric domain[43] as shown below and the finite values for each grid point is calculated using an approximation of the Tyler s serious expansion or other alternatives such as polynomial fitting. Y X i,j+1 P 2 Y i-2,j i-1,j i,j i+1,j i+2,j P 1 i,j-1 X Figure 2.1: Discrete grid points[5] If one has u i,j at point p 1, then according to the Tyler s polynomial expansion, at point p 2, u i+1,j is given as: u i+1,j = u i,j + ( ) u x + x i,j ( ) 2 u x 2 i,j ( x) 2 2 ( ) 3 u + x 3 i,j ( x) (2.7) Getting the exact value of u i+1,j, requires inclusion of an infinite number of derivatives. However, a reasonable accuracy can be achieved by using only the first few derivatives and truncating the rest. This, of course, introduces a truncation error. For applications where more accurate results are required, one can include the second and third even the fourth derivatives to obtain finite values at each grid 32

49 Chapter 2. Numerical methods points and then truncate the rest of the derivative terms. But, this will increase the computational time and cost dramatically. If the above equation is rearranged one can have the following expression: ( ) u = x i,j u i+1,j u i,j } x {{} finite difference representation ( ) 2 u ( ) ( x) 2 3 u ( x) 3... x 2 i,j 2 x 3 i,j 6 }{{} truncation error (2.8) The above equation is a first order forward difference equation with respect to x. This is because in order to solve the term, ( u ), the algorithm uses the value x i,j of u at a grid point one step forward (i + 1, j) from the starting grid point, (i, j). The term can also be solved using the first order rearward difference and the first order central difference methods. In such cases, the algorithm uses the value of u at a grid point one step backward at (i, j 1) for the rearward difference method and uses two grid points (one grid point forward and one grid point backward(i + 1, j and i, j 1)) for the central difference method. A similar approach can be used to evaluate second order partial differential equations such as ( 2 u) and x 2 ( 2 u ) that are often found in the Navier-Stokes equations. Once the partial x y differential equations are replaced with the finite difference representations, the resulting equation is called a difference equation[5]. Second order central second difference equation with respect to x and second order central mixed difference equation with respect to x and y without the truncation errors are given below[5]. ( ) 2 u x 2 i,j = u i+1,j 2u i,j + u i 1,j ( x) 2 (2.9) ( ) 2 u = u i+1,j+1 + u i 1,j 1 u i 1,j+1 u i+1,j 1 x y i,j 4 x y (2.10) At the boundaries of the grid points the value of u or any other variable can be determined using different methods. One of such methods is the polynomial approach[5]. An example of a polynomial evaluation of u at the boundary of the grid points is shown in Appendix B [5]. 33

50 Chapter 2. Numerical methods The variables in the governing fluid dynamics equations are calculated based on the grid points that are discretized at specified locations. This is known as space marching. However, governing equations also require time marching, which is a discretization where an unknown value of a variable at time step n + 1 is solved from known values at time step n[5]. The solutions obtained using time discretization account for the transient effects of the fluid flow[44] Explicit and implicit methods Algorithms that are used to solve the governing equations of fluid dynamics use two different methods to achieve the solution. These are, the explicit and the implicit methods. In the explicit method, the value of an unknown variable is calculated exclusively from a single equation where the values of all the other variables are known. John D. Anderson[5] illustrated the difference between the two methods using the following one dimensional heat conduction equation. T t = α 2 T x 2 (2.11) first order forward difference: T t = T n+1 i Ti n t ( central second difference: 2 T T n x = α i+1 2Ti n + T n 2 ( x) 2 i 1 ) (2.12) (2.13) Where n is the time step, α is a constant and T is temperature. Rearranging the above equation gives: T n+1 i = tα ( ) Ti+1 n 2Ti n + Ti 1 n + T n ( x) 2 i (2.14) In the above equation there is only one unknown variable, T n+1 i, and the equation can readily be solved. An alternative method of solving equation 2.11 is to use the implicit method. The Crank-Nicolson implicit method for example uses the average values of variables T i+1, T i and T i 1 at time steps n and n + 1 to solve for T n+1 i. 34

51 Chapter 2. Numerical methods T n+1 i t T n i = α 1 2 n+1 (Ti+1 + T i+1) n + 1 n+1 ( 2T 2 i 2Ti n ) + 1 n+1 (T 2 i 1 + T i 1) n (2.15) ( x) 2 In the above equation new unknowns are introduced, namely T n+1 i+1, T n+1 i and. As a result, the equation cannot be solved on its own. In order to solve T n+1 i 1 it, the values for all equations at all grid points must be solved concurrently. The implicit method has an advantage of maintaining the stability of the solution even at a relatively large t values but it is also more difficult to program and it is computationally more demanding than the explicit method Numerical stability and convergence The representation of a partial differential equation with a difference equation introduces a truncation error. In addition, a round-off error can become significant. These errors will propagate and amplify as the calculation is performed while marching in space and time. The amplified errors might become too big and ultimately makes the numerical method unstable. Due to this, there are restrictions placed on the values of t and x in order to keep the numerical method stable[5]. When the numerical calculation progresses from a time step n to n + 1 the error, ɛ i, should shrink or stay unchanged for the solution to be stable. ɛn+1 i ɛ n i 1 (2.16) For example, the application of the above requirement on the one dimensional heat conduction equation (Von Neumann method) reveals that the stability criteria should be as follows[5]: α t ( x) (2.17) In order to get an accurate numerical solution, a numerical algorithm usually involves an iterative process, where the value of a variable is calculated repetitively while the accuracy of the value become progressively better. After a number of repetitions the difference between values from any two consecutive calculations become close to zero (this difference value is referred to as, residual). When the 35

52 Chapter 2. Numerical methods residual becomes lower than a pre-defined value the numerical solution is said to be convergent[45] Multiphase flow simulation Multiphase flows involve mixtures of macroscopically distinct phases known as dispersed phases and continuous phases. Flow of gases in liquids and solid particles in liquids are examples of such flows[42][46]. These phases have clearly defined interfaces and they often interact with each other as well as with the boundary of the flow system (nature of the particle-particle and particle-wall interactions are described in the first chapter). These interactions are often sources of complexities and due to this it is difficult to use experimental and analytical approaches to investigate multiphase flows[47]. Numerical simulations give an attractive alternative to investigate the behaviour of multiphase flows. However, numerical simulation of multiphase flows come with complexity of their own. An accurate and detailed description of multiphase flows are often heavy on computational time and cost. In addition, industrial applications, that often consist of trillions of particles, are restricted by computational capability[3] Continuous phase equations The presence of dispersed phase particles in the continuous phase complicates the numerical solution. Hence, for practical applications the continuous phase equations are based on averaging procedures where the average property of a flow over time, volume/space or ensemble is used to formulate the continuous phase equation[48]. A detailed description of each averaging procedure is presented by Crowe et al. [3] but here, only the volume averaging procedure is discussed. The Volume averaging procedure is the most commonly used method since computational models are themselves based on averaged discretized cells of a flow domain. If B is a flow variable of the continuous phase such as fluid density or velocity and if V ave is the averaging volume, then the volume average of B in the continuous phase is defined as: B = 1 BdV (2.18) V ave V c 36

53 Chapter 2. Numerical methods V d V c l L Figure 2.2: Illustration of volume averaging procedure. Where B is the volume averaged property of the continuous phase and V c is the volume of the continuous phase. In order to apply the volume averaging technique, the characteristic dimension of the averaging volume(v ave ) must be much smaller than the characteristic dimension of the physical system over which the flow variables change significantly (L), and much larger than the characteristic dimension of the continuous phase (usually the average inter-particle spacing)(l)[49]. This requirement is needed to ensure that microscopic variations in the flow variables (B) are levelled off and the volume averaged properties are continuous. When the averaging volume is too small, microscopic fluctuations of the flow variables affect the averaged properties significantly. On the other hand, when the volume is too large the averaged properties will be affected by macroscopic variations of the variables and the dimensions of the physical system. At the right averaging volume, the average properties become independent of the average volume. The local volume average of Equation 2.18 is given as: B = V c 1 BdV (2.19) V ave V c V c V c V ave = α c (2.20) B = α c B (2.21) 37

54 Chapter 2. Numerical methods Where, B is the local volume average of the flow variable, α c, is the volume fraction. When the average properties of the flow become independent of the averaging volume, the local and global average properties become equal. B = B (2.22) The mass, momentum and energy conservation equations of the continuous phase can then be formulated based on the above averaging technique. The conservation equations can be formulated as quasi-one-dimensional form or three-dimensional form Single particle equations The basis for the formulation of single particle equations is the application of the Reynolds transport theorem (RTT) on a control volume. A control volume is an arbitrary section of the flow field in which the conservation laws such as mass conservation are applied. The resulting equations of the control volume are then related to the whole flow system using the Reynolds transport theorem. For a control volume in a flow field with a defined control surface at its boundary, the Reynolds transport theorem states that the rate of change of an extensive property of the system is equal to the time rate of change of the extensive property in the control volume and the net rate of flux of the extensive property across the control surface. In other words, the total change in the property of a system in the control volume is the sum of the property of the control volume at time t and the difference in the property that flows in to the control volume and the property that flows out of the control volume across the control surface. Mathematically, the Reynolds transport theorem is given as: db sys dt = cv (ρβ) t dv + cs ρβu i n i ds (2.23) Where, u i is the velocity of the fluid at the control surface with respect to the coordinate reference frame, n i is a unit vector normal to the control surface, ρ is the fluid density, S is the surface area of the control surface, B sys is the extensive property of the system and β is the corresponding intensive property of the system. The subscripts cv and cs refer to control volume and control surface, 38

55 Chapter 2. Numerical methods respectively. The Reynolds transport theorem equation can now be applied to formulate the single particle equations and the dispersed phase equations in general. dm For example, the mass conservation principle states that, = 0. In this case dt the extensive property is mass and since the intensive property of mass is one (i.e M/M), application of the Reynolds transport theorem on the mass conservation principle gives: dm dt = cv ρ d dt dv + dm dt = cs cs ρ s w i n i ds = 0 (2.24) ρ s w i n i ds (2.25) If the efflux velocity(w i ) and the fluid density are uniform across the control surface, then the above equation can be written as: dm dt = ρ sws d (2.26) Where ρ d is the particle density, ρ s is the density of the fluid at the control surface, S d is the surface area of the particle, w i is the velocity across the control surface with respect to the control surface and w is the magnitude of efflux velocity vector. The above strategy can be applied on the momentum as well as the energy conservation principles to formulate their respective equations. x 3 r 1 w 1 v 1 Control surface x 2 x 1 Figure 2.3: Moving control surface enclosing a particle[3] 39

56 Chapter 2. Numerical methods In the momentum conservation, the Newton s second law of motion states that: F i = d(mu i) dt (2.27) The extensive variable is linear momentum (MU i ) and the intensive variable is velocity (U i ). principle, it gives: When the Reynolds transports theorem is applied on the above F i = d dt = cv ρ d U i dv + cs ρ s U i,s w i n i ds (2.28) Where, U i,s is the velocity of the fluid at the control surface with respect to the inertial reference frame. Similarly from the moment of momentum conservation equation: T i = d(mh i) dt (2.29) Where, T i is the applied torque vector and H i is the moment of momentum per unit mass. In this case the extensive property is the moment of momentum (MH i ) and it s intensive counterpart is the moment of momentum per unit mass(h i ). Hence, applying the Reynolds transport theorem gives: T i = d dt cv ρ d h i dv + cs ρ s h i w i n i ds (2.30) The Reynolds transport theorem can also be applied on the first law of thermodynamics (principle of energy conservation) in a control volume. The first law of thermodynamics states that: de dt = Q W (2.31) E = M(i + U iu i ) + Sσ = Me + Sσ (2.32) 2 Where, E is the sum of internal, external and surface energies, i is internal energy per unit mass, U i is velocity, Q is the rate of heat transfer and W is the rate of work done. When the Reynolds transport theorem is applied on the above equation it 40

57 Chapter 2. Numerical methods gives: dme dt = d ρ d e dv + dt cv cs ρ s ew i n i ds (2.33) Q W = d dt cv ρ d e dv + cs ρ s e s w i n i ds + d(sσ) dt (2.34) Detailed derivations for linear momentum, moment of momentum and energy equations for a dispersed phase particle can be found in Crowe et al.[3] and Jakobsen et al.[49]. However, the results from the derivations are summarized in the following table. Conservation principle Equation Single particle equation Mass conservation Momentum conservation F i = d(mu i) dt Angular momentum conservation T i = d(mh i) dt Energy conservation dm dt = 0 dm dt = ρ sωs d de dt = Q W F i = m dv dt + cs ρ(rn + w)w.n ds T i = I dω dt Table 2.1: Summary of single particle equations mc d dt d dt = Q + m(h cs h d + ω ω 2 ) Dispersed phase equations The dispersed phase equations are classified into two groups.the Lagrangian and Eulerian approaches. The Lagrangian approach is further classified into two groups as the discrete element method(dem) and the discrete parcel methods(dpm). The difference between the DEM and the DPM methods is that DEM simulates the flow properties of every single dispersed phase particle whereas in DPM the smallest unit of simulation is a parcel of the dispersed phase particles. In both DEM and DPM, the dispersed phase particles are assumed as a separate entities from the continuous phase. Here, an important distinction between the Lagrangian and Eulerian approaches arise. In Eulerian approach the dispersed phase particles are assumed to posses properties of the continuous phase. Due to this the Eulerian approach is also known as the two fluid model. The choice of what approach to use is dependent on whether the fluid flow is dilute or dense with the dispersed phase particles. If the flow is dilute, the dispersed 41

58 Chapter 2. Numerical methods phase particles are sparsely present and they can t be considered continuous. As a result the Eulerian approach can t be used. The only choice will then be the Lagrangian approach. In dense flows where the dispersed particles are abundant, contact between the dispersed particles will also be abundant. The dispersed phase starts to show behaviours of continuity, the abundance of particle-particle contact would enable information to travel in all directions[3]. In this case the the dispersed phase can be assumed to be a continuous phase and one can use the Eulerian approach. Crowe et al. presented a dimensionless number to determine whether a dispersed phase flow is dilute or dense. It is the ratio of the momentum response time(τ v ) and the average time between collisions(τ c ). It is given as: Where, f c τ v τ c = τ v 1/f c = ρd 2 18µ c 1 nπd 2 v r = nπρd 4 v r 18µ c (2.35) is the collision frequency, n is the number density of the particles, v r is the relative velocity of one particle to other particles. If the value of the dimensionless number is less than one, then the flow is dilute. If it is more than one the flow is dense Properties of the dispersed phase Multiphase flows involve a continuous phase and dispersed phase particles. One of the main difference between the two phases is that unlike the continuous phase, the dispersed phase particles are not in continuum. The properties of the dispersed phase particles affect the overall property of the flow. In this section some of the most important properties of the dispersed phase are discussed. Particle spacing The average distance between the dispersed particles is an important property of the dispersed phase. It s importance lies in determining whether a particle should be considered isolated from other particles or not. An isolated particle exerts little 42

59 Chapter 2. Numerical methods or no influence on other particles. The ratio of the average interparticle distance and the particle diameter is used to quantify particle spacing. If two particles are at the centres of two adjacent cubes of length, l sd then the ratio of the interparticle spacing and the particle diameter is given as: [ ] 1 l sd π D = 3 6α d (2.36) Where, α d is the volume fraction of the dispersed phase. If the value of l sd D exceeds a certain limit, particles of the dispersed phase are considered isolated. The limit depends on the nature of the continuous and dispersed phases. For example, for systems involving gas-particle flows an l sd value of 10 or above indicates particle D isolation[3]. Particle response times How fast particles respond to changes in the flow characteristics give an important information about the overall flow. Two important response times are usually encountered. These are the momentum and the thermal response times. The momentum response time is the time it takes for the dispersed phase particle to respond to changes in the fluid velocity. Whereas the thermal response time is the time it takes for the dispersed phase particles to respond to changes in the fluid temperature. τ V = ρ dd 2 18µ c (2.37) τ T = ρ dc d D 2 12k c (2.38) Where, τ V is the momentum response time, τ T is the thermal response time, c d is the specific heat of the particle material and k c is the thermal conductivity of the continuous phase. Another property that determine the interaction between the continuous and dispersed phases is the the Stokes number. The Stokes number is a dimensionless quantity that is usually defined as a ratio of time characteristic 43

60 Chapter 2. Numerical methods of the dispersed phase to that of the continuous phase. Some examples of Stokes number are given in Chapter 1 Section How the dispersed particles are packed in the flow is also an important property. It is used to classify flows as dense or dilute flows. In dense flows the particles are densely packed and particle-particle collisions are frequent. As discussed in section , this affects the flow behaviour in a fundamental way in part because when particles are highly densely packed the limit of dispersion will be reached and the particles can be considered to be in a continuum. The continuous and dispersed phases also exert influences on each other. As discussed in Chapter 1 Section 1.2.5, these influences are called phase coupling. In one-way coupling the continuous phase influences properties of the dispersed phase but not the other way around. However in two-way coupling the dispersed phase also influences the continuous phase properties. When the two phases are in a dynamic and thermal equilibrium the flow can be considered as a single phase flow Multiphase flow simulation in STAR-CCM+ In general, multiphase flows can be classified into two broad categories as dispersed flow and stratified or separated flows[42][50]. Dispersed flows are flows where the flow consists of distinct particles such as bubbles and solid particles along with the continuous phase, whereas separated flows are flows where the flow is consisted of multiple continuous phases separated by interfaces. There are six models for the simulation of multiphase flows in STAR-CCM+. They are: Lagrangian multiphase Discrete element method Multiphase segregated flow Dispersed multiphase model Fluid film Volume of fluid 44

61 Chapter 2. Numerical methods Since the later four models are not the focus of this paper, only the description of the Lagrangian multiphase and the discrete element method models are presented. Lagrangian multiphase model In Lagrangian multiphase model the basic unit for calculation is a parcel. A parcel is composed of elements of the dispersed phase. The model selects a statistically representative number of parcels instead of considering a large number of dispersed phase elements. This enables faster simulation. For each parcel, the Lagrangian multiphase model solves the equation of motion[42]. The Lagrangian multiphase model is provided in STAR-CCM+ software and it is suited for flows where particle-wall interactions are important. The Lagrangian multiphase model is able to model several properties of the dispersed phase and record the state of the parcels as a separate track file from which post processing of the results could be carried out. A detailed formulation of the model is found in the user guide manual for the simulation software[42]. Discrete element model The discrete element method is an extension of the Lagrangian multiphase model. It is used for the simulation of several interacting, usually solid particles. The DEM method is suited for flows where particles are densely packed, collide frequently or when flow behaviour is dependent on particle size and shape as well as contact mechanics. Hence, the DEM is a good alternative to study the effect of particle-particle and particle-wall interactions. However, the DEM requires a small integration time steps for an adequate resolution of particle-particle and particle-wall contact properties, this makes the DEM simulations time and cost intensive[51]. DEM formulation The momentum balance for the material particle is generally given as the sum of particles surface and body forces: m p dv p dt = (F s) + (f) = (F d + F p + F vm ) + (F g + F u ) (2.39) 45

62 Chapter 2. Numerical methods Where, F s,f,f d,f p,f vm, F g, and F u are the particles surface, body, drag, pressure gradient, virtual mass, gravity and user defined body forces respectively. F d = 1 2 C DA p V s. V s (2.40) F p = V p Pstatic (2.41) ( DV F vm = C vm ρv p Dt DV ) p Dt (2.42) F g = m p g (2.43) F u = V p f u (2.44) Where C D is the particle drag coefficient, V s is the particle slip velocity, A p is the particle projected area, V p is the particle volume, Pstatic is the gradient of static pressure in the continuous phase and C vm is the virtual mass coefficient. In DEM an extra body force is introduced to account for the particle-particle and the particle-wall contacts. Hence, the body force in DEM becomes the sum of gravity force, user defined body force and contact force: f = F g + F u + F contact (2.45) DEM time step The maximum time-step that is allowed for a DEM particle is limited due to the assumption that the force acting on a particle is only affected by the particle s immediate neighbours during a single time-step. Hence, the time-step is limited by the time it takes for the Rayleigh wave to propagate across the surface of the sphere to the opposite pole and the minimal sphere radius (R min )[42]. R min τ 1 = π (2.46) V Rayleigh 46

63

64 Chapter 3 Experimental methods This chapter provides descriptions of the experimental techniques used for the numerical simulations and the laboratory experiments employed in this thesis. The experimental set-ups are discussed and the values for the selected experimental variables are given along with the justifications for the selections. 3.1 Numerical experiments Several numerical experiments were carried out during the course of this thesis. In order to accomplish the best possible accuracy for the simulations while keeping simulation time as shorter as possible, some considerations were implemented. For example, the number of injected particles was chosen to be 1000 after trial simulations showed that it was possible to assess particle deposition with a reasonably shorter time (about 12 hrs) than the time it would take if higher number of injected particles were used. Other considerations were made so that the simulation parameters are not over complicated and over specified hence, parameters such as coefficient of restitution and coefficient of friction as well as some of the solver properties were left at default values. To identify parameters that affect particle deposition, dimensional analysis and literature review were carried out. In the following sections the experimental parameters and their values as well as the justifications for their choice will be presented. 48

65 3.1.1 Dimensional analysis Chapter 3. Experimental methods Dimensional analysis was carried out to identify the most important variables that influence particle deposition. Ultimately it was decided to use variables such as the Reynolds number, the Adhesion parameter and the Tabor parameter as the variables to investigate particle deposition. However the dimensional analysis gave an interesting insight into other variables that may also influence particle deposition. The dimensional analysis can be seen in Appendix A Numerical experiment set-up Numerical experiments were carried out using a commercial CFD software STAR- CCM model. The numerical experiments involve the following steps: Geometric preparations, defining surface and boundary types, meshing, defining the physics models and interactions between the boundary, the fluid and the particles, set-up and define the injector settings, set-up the numerical solver settings. After these steps were completed the simulations were run until converging solutions were obtained. Most of the simulations took about 12 hours to complete. The simulation time depends on factors discussed in chapter 2 Section After solutions were obtained, post-processing of the solutions were performed. The track files method provided in the STAR-CCM+ software were deemed to be enough to extract data about the particle deposition Numerical experiment variables The variables for the numerical experiments were the Reynolds number, the Adhesion parameter and the Tabor parameter. All of the experiments were carried out within the laminar flow regime(below Re value of 2300). The experimental parameters for each variable are presented in the following tables: 49

66 Chapter 3. Experimental methods Experiment name Re333 Re800 Re1266 Re1733 Re2200 Viscosity of the fluid(pa.s) Density of the fluid(kg/m 3 ) Diameter of flow cell pipe(m) Velocity of the fluid(m/s) Reynolds number Table 3.1: Experimental values for Re simulations Experiment name Ad1 Ad2 Ad3 Ad4 Density of the particle(kg/m 3 ) Particle diameter(m) Particle velocity(m/s) Work of cohesion(j/m 2 ) Adhesion parameter Table 3.2: Experimental values for Adhesion parameter simulations Experiment name Tab1 Tab2 Tab3 Tab4 Young s modulus(pa) Poisson s ratio Separation distance (m) Work of cohesion (J/m 2 ) Particle radious (m) Tabor parameter Table 3.3: Experimental values for Tabor parameter simulations Geometry, surface preparation and boundary types This thesis focuses on study of particle deposition on a pipe containing an obstruction. The shape and dimensions of the pipe is given in Figure 3.1 and 3.2: This geometry was made using STAR-CCM+ 3D CAD design tool. After the geometry was made, the inlet, the outlet and the wall of the pipe were assigned a specific region and each region was assigned an appropriate boundary type. For the inlet the boundary type was velocity inlet and for the outlet it was flow split outlet boundary type. The wall region was assigned wall boundary type. 50

67 Chapter 3. Experimental methods 20 cm 1 cm 0.7 cm 1.4 cm 9 cm 9.6 cm Figure 3.1: Flow cell geometry dimensions Pipe inlet Wall Wall Pipe outlet Figure 3.2: Flow cell geometry regions Meshing Meshing is a step in which the designed geometry is discretized to represent the computational domain [42] in which the physics solver finds a solution. In this thesis the polyhedral mesher with a base size of was selected along with the surface remesher and the surface wrapper. The Polyhedral mesher generates volume meshes of polyhedral shapes that have the advantage of being easy and efficient to build as compared to other mesh types (such as tetrahedral meshes). The Surface remesher was used to refine and improve the geometry surfaces. The Surface wrapper was used to remove any problem that may arise due to the complexity of the geometry, intersection of parts or sharp edges. 51

68 Chapter 3. Experimental methods Figure 3.3: flow cell geometry after mesh operation Physics models The physics models provides settings, variables and constants that will be used for the calculation of the numerical solutions within the computational domain. In this thesis the aim is to study the deposition of Polystyrene particles using the DEM and The Lagrangian multiphase models. In general, the following models were used: Setting Equation of state Time Space Gradient metrics Flow Viscous regime Multi-phase interaction Optional Model Constant density Implicit unsteady 3D Gradients Segregated flow Laminar Multiphase interaction Gravity, passive scalar, DEM, Lagrangian multiphase Table 3.4: Description of physics model The Lagrangian multiphase model provides a model that helps define the Lagrangian phase. In this paper, the Lagrangian phase consists of solid Polystyrene particles whose properties and interactions are defined as constant density, DEM particle type, drag force, residence time, solid particle, spherical particle and track file. The multiphase interaction model lets the interaction between phases to be 52

69 Chapter 3. Experimental methods defined. Polystyrene-Polystyrene and Polystyrene-wall interactions were defined using the physics model. The values for each interactions are listed below: DEM phase interactions First phase Polystyrene Second phase Polystyrene Static frictional coefficient 0.61 Hertz Mindlin Normal restitution coefficient 0.6 Tangential restitution coefficient 0.6 Linear cohesion Work of cohesion(j/m 2 ) Factor 1.5 Rolling resistance method Coefficient of rolling resistance Table 3.5: particle-particle interaction model Particle injector The particle injectors provides a way to introduce the Lagrangian particles into the computational domain. There are different types of particle injectors in STAR- CCM+. The Part injector, the Point injector and the Surface injector are some examples. However, the Random injector was selected because it is best suited for DEM simulations and it represents the random fashion in which particles are distributed in the domain better than the other injectors. The conditions and values of the Random injector are given as follows: condition setting value Particle amount specification number of particles 1000 Particle size specification diameter m Particle velocity specification absolute various Table 3.6: particle injector settings 53

70 Chapter 3. Experimental methods Solver settings and stopping criteria Unlike the steady state solvers, the implicit unsteady solver offers information about the particles at each specified time step. For this reason the implicit unsteady solver is favoured over the steady solver. The implicit unsteady solver requires a time step specification and a time step of 0.01s was selected for all simulations. Other properties of the solver were left at default values. When the Lagrangian multiphase model is selected for the physics model, as is the case in the thesis, a separate solver called Lagrangian multiphase implicit unsteady solver is also activated. This solver offers controls for the maximum substeps and verbosity (level of detail of information required) among other things. All of these properties were again left at the default values or settings. The stopping criteria is used to tell the solver how long the computation runs and how and when the solver should stop it. Once the stopping criterion are specified, they are evaluated after each iteration or time-step. When the criterion are met the simulation ends. For the unsteady solvers there are three important stopping criterion that must be specified [42]. They are the maximum inner iterations, the maximum physical time, and the maximum steps. Generally, the choice of the stopping criteria depends on how much detailed information is required, how much computational time and cost is available and other related aspects. In this study, the maximum inner iteration was set to 2, the maximum physical time was 2 s and the maximum steps were 10, Laboratory experiments Laboratory experiments were carried out with the objective of comparing results with the results suggested by the low cohesion Re-simulations. In order to accomplish this, a laboratory experiment was designed. In the next sections, the laboratory set-up, the experimental procedure, the equipments and the apparatus used for the experiments are discussed. 54

71 3.2.1 Experimental set-up Chapter 3. Experimental methods The experimental observations were carried out based on the following procedure: Two litres of distilled water was added into tank-1 (see Figure 3.4) followed by two drops of concentrated polystyrene particle suspension (this corresponds to nearly a thousand polystyrene particles). The tank was continuously stirred using a magnetic stirrer to ensure uniform particle distribution. Contents from tank-1 were then allowed to pass through connecting tubes into the flow cell. A microscope connected to a computer was mounted in such a way to monitor particle deposition behaviour on a particular sections of the flow cell. The microscope was used to take pictures every 5 minutes. The pictures were then transferred to a computer where a software was used to analyse the extent of deposition of particles. The undeposited particles and the fluid were then allowed to pass to tank-2 from which a pump transported the contents back into tank-1. The recirculation process was continued for 15 minutes. The velocity of carrier fluid was controlled by using the pump and the inlet valve at tank-1. The experiments were carried out at various Reynolds number values. The results were then compared with results from the numerical experiments. Figure 3.4: Diagram of the experimental set up 55

72 Chapter 3. Experimental methods Laboratory equipment The experimental set-up involved the use of the following laboratory apparatus and equipment: Two reservoir tanks Connecting tubes A light source A magnetic stirrer Polystyrene particles KnF lab liquiport pump A flow cell with an obstruction A Nikon SMZ800 microscope with Q-imaging camera and software Figure 3.5: Experimental set-up Polystyrene particles Dark red micro-particles made from polystyrene were purchased from SIGMA- ALDRICH. The particles were spherical with a diameter of 100µm. The Polystyrene particles were in a 10 ml aqueous suspension form with a solid content of 5 wt%. 56

73 3.2.4 Re-experiments Chapter 3. Experimental methods Laboratory experiments were carried out for different values of the Reynolds number. All of which were within the laminar flow regime. The value of the Reynolds number for each experiment were controlled by using the velocity of the fluid as a variable. For polystyrene particles, the diameter and the density are known and constant(10 4 m and 1050kg/m 3, respectively), in addition, the continuous phase (water) has a viscosity of Pa.s at 20 C, this leaves the fluid velocity as the only variable that can be used for the Reynolds number variation. Re = ρυd µ c (3.1) For example for the Reynolds number value of 1000, the corresponding fluid velocity will be: υ = Reµ c ρd = (1000)(0.001 P a.s) (1050 kg/m 3 )(10 4 m) = 9.55 m/s (3.2) Ensuring correct fluid velocity In order to ensure a precise fluid velocity, a pump and a value located at the outlet of tank-1 were used in combination. The pump used in the experiments, KnF lab liquiport pump, has a volumetric flow rate capability ranging from 0.2 lt/min to 3.0 lt/min. Based on a starting 2 lt volume of water in tank-1 and flow cell diameter of 0.01 m, the volumetric flow rate required to ensure m/s flow velocity (corresponding to Re of 333) will be: Q 333 = υ 333.D 2 flowcell = (0.0333m/s)(0.001m 2 ) = m/s = 0.2lt/min (3.3) Once the flow rate for a given velocity of the fluid was known, the next step was to turn the pump to that flow rate value, then the outlet value on tank-1 was adjusted so that the volume of water in the tank remains constant. The recirculation of the water was continued with out adding the polystyrene particles into the tank for 57

74 Chapter 3. Experimental methods up to 10 minutes until the volume of water in tank-1 was completely stabilized. When there was no more variation of the water volume in tank-1, the Polystyrene particles were added into tank-1. Images of the particle deposition were then taken at 5, 10 and 15 minutes after the particles were added. This procedure was repeated for different Reynolds number values. Re Q(lt/min) υ(m/s) Table 3.7: Velocity and Volumetric flow rate values for Re simulations 58

75

76 Chapter 4 Results and discussion This chapter presents the findings from the numerical and the laboratory experiments. Comparisons of the simulations with the laboratory experiments as well as with the results from published literature are discussed. 4.1 Numerical experiments Simulation of deposition of particles using the DEM-Lagrangian multiphase model was performed. The Reynolds number, the Adhesion parameter and the Tabor parameter were varied to understand how they affect the deposition. In the next sections the results are presented and discussed in detail Results from Re-experiments The effect of the Reynolds number on the particle deposition was investigated using two sets of numerical experiments. Each set consisted of five experiments for the Reynolds number values of Re=333, Re=800, Re=1266, Re=1733 and Re=2200. All the other parameters and model specifications were kept constant. The first set of experiments were assigned a low cohesivity (W coh = J/m 2 ), while the second set of experiments were assigned a high cohesivity (W coh = 0.814J/m 2 ) 60

77 Chapter 4. Results and discussion Results from low cohesivity experiments The amount of deposition of Polystyrene particles was determined based on the particle velocity near the wall surfaces. A threshold derived part was created using STAR-CCM+ and particles with velocities below the threshold velocity were recorded as deposited particles. The threshold velocity of the particles depends on the inlet velocity of the fluid and was defined in such a way that made comparisons of particle depositions from different Reynolds number experiments possible. Initial post-processing of the numerical experiments showed that using a constant threshold velocity for all Reynolds number experiments led to a false increase in the deposition of particles. This effect was specifically observed for Re=333 and Re=800 experiments. Lower inlet fluid velocities were found to increase this effect. Hence, it was deemed necessary to define threshold velocities in comparison with the inlet fluid velocities of each numerical experiment. Threshold velocities calculated based on this definition are shown below: V threshold = υ (4.1) Where, V threshold is particle threshold velocity and υ 0 is fluid inlet velocity. Re υ 0 (m/s) V threshold (m/s) Table 4.1: Threshold velocity values Deposition efficiency (DE) was then defined for the flow and plotted against the Reynolds number. Deposition efficiency was defined as the ratio of the number of particles deposited and the number of particles that were injected into the simulation domain. In all the numerical experiments the number of Lagrangian phase particles (Polystyrene) to be injected into the domain was specified to be However, after each simulation the total number of particles in the domain was found to be slightly less 61

78 Chapter 4. Results and discussion than The reason for this is, in Random injectors the number of injected particles depend on the number of seeds and the injection cycle limit, both of which can be specified to keep the total number of injected particles within an acceptable range. The number of seeds is defined as the number of particles that are injected at the same time. Generally, it s value is set at a tenth of the total number of particles that can be packed inside the volume of the injector region. The injection cycle limit controls the number of times the injector injects new particles[42]. Both the number of seeds and the injection cycle limit introduce a small degree of uncertainty to the injector and this leads to discrepancies between the specified number of particles to be injected and the actual number of particles in the domain[23] [52]. Nevertheless, it was found necessary to use the total number of particles in the domain after the simulations were completed as the number of injected particles for the calculation of deposition efficiencies(de). DE = ( ) Deposited particles 100% (4.2) Injected particles Numerical experiments were carried out to study the effect of Reynolds number variation on the deposition efficiency of Polystyrene particles. Reynolds number values ranging from 333 to 2200 were used. Deposition efficiency of each Reynolds number experiment was then calculated. Overall, it was found that deposition efficiency of Polystyrene particles increased with increasing Reynolds number. A sharp increase was observed in the lower range of Re-values. An increase from Re=333 to Re=800 resulted an increase in deposition efficiency from 13% to 35.9%. The increase became less sharp at higher Re-values. An increase in Reynolds number from 1266 to 2200 resulted an increase from 62.5% to 68.3% in deposition efficiency. In addition to deposition efficiency, the location in which particles deposited were visually investigated. It was found that more particles were deposited after the obstructed section of the flow cell (for nomenclature of the flow cell sections, please refer to Figure 4.15). The particles also deposited on the edges of the obstructed section. However, no significant deposition was observed on the surface the obstruction itself. It was also found that the location of particle deposition did not vary significantly from one Reynolds number to the other. 62

79 Chapter 4. Results and discussion DE Reynolds number Figure 4.1: The effect of Reynolds number on deposition efficiency (low cohesivity, W coh = J/m 2 ) Figure 4.2: Velocity profile of the particle tracks (Re=333) 63

80 Chapter 4. Results and discussion The type of velocity profile for particle tracks shown in Figure 4.2 can be converted into a more realistic representation of the particle flow and deposition. STAR- CCM+ provides a tool to convert particle tracks into animation and record them. All particle tracks from experimental simulations were converted into animation for a better visual analysis. The procedure to convert particle tracks can be found in Appendix C. Flow direction Re=333 Re=800 Re=1266 Re=1733 Re=2200 Figure 4.3: Front view of particle deposition (low cohesion simulation, blue dots represent deposited particles) Results from high cohesivity experiments In order to investigate and compare how Reynolds number variation affects particle deposition at higher cohesivity as compared to the low cohesivity experiments discussed above, numerical experiments were carried out for Reynolds number values ranging from 333 to 2200 with a high particle cohesivity. The parameters and procedures in the simulations were the same as the low cohesivity experiment counterparts. However, in the high cohesivity experiments the work of cohesion for each experiment was assigned to be 0.814J/m 2. The same method was also used to calculate deposition efficiency. The threshold velocity for high cohesivity experiments was defined in a slightly different way. Initially, the definition used in the low cohesivity experiments was 64

81 Chapter 4. Results and discussion Flow direction Re=333 Re=800 Re=1266 Re=1733 Re=2200 Figure 4.4: Bottom view of particle deposition (low cohesion simulation) used but the results were found to be unrealistic. Hence, a new definition of threshold velocity had to used. In the new definition the velocity of the particles still had to be reasonably near zero so that particles with velocities below the threshold velocity could safely be considered deposited while obtaining a realistic result. Several trial simulations and threshold values were used to obtain more accurate results and ultimately the following definition of threshold velocity was chosen. V threshold = υ (4.3) Re V threshold (m/s) Deposition efficiency(%) Table 4.2: High cohesivity threshold values and deposition efficiency 65

82 Chapter 4. Results and discussion Results from the high cohesivity experiments indicated that, overall, as the Reynolds number increased deposition efficiency also increased. This trend was found both in the low and high cohesivity experiments. Another similar trend observed was the fact that increase in the deposition efficiency was more pronounced when the Reynolds number was increased from Re=333 to Re=1266 than from Re=1266 to Re=2200. Adams et al.[53] studied the effect of Reynolds number on the deposition of spherical particles of different sizes using LES-RANS approach coupled with Lagrangian particle tracking method and found out that overall, increase in Reynolds number led to increase in particle deposition. However, their study also concluded that the increase in the deposition was more pronounced for particles with larger diameter (500µm) than particles with diameters from 50µm-100µm. Similarly, Afkhami et al.[54] studied the Reynolds number effect on particle agglomeration in turbulent channel flows using LES-DEM method. They found out that increased Reynolds number led to increased agglomeration and observed that particle agglomeration often happened near channel walls DE Reynolds number Figure 4.5: The effect of Reynolds number on deposition efficiency (high cohesivity, W coh = J/m 2 ) 66

83 Chapter 4. Results and discussion Flow direction Re=333 Re=800 Re=1266 Re=1733 Re=2200 Figure 4.6: Front view of particle deposition (high cohesivity) Flow direction Re=333 Re=800 Re=1266 Re=1733 Re=2200 Figure 4.7: Bottom view of particle deposition (high cohesivity) 67

84 Chapter 4. Results and discussion Adhesion parameter results Four numerical experiments were carried out to study the effect of the Adhesion parameter on the particle deposition. Adhesion parameter values ranging from 1722 to were used for the experiments. In each experiment the fluid velocity was set to m/s (Re=333) and the threshold velocity was set to m/s. Deposition efficiency was then calculated and the following results were obtained. Ad υ 0 (m/s) V threshold (m/s) Deposition efficiency(%) Ad Ad Ad Ad Table 4.3: The effect of Adhesion parameter on deposition efficiency DE Adhesion parameter (x10 4 ) Figure 4.8: The effect of Adhesion parameter on deposition efficiency (Re=333) The results showed that as Adhesion parameter increased the deposition efficiency also increased. From Ad1 to Ad2 there was a sharp increase in deposition efficiency, however the increase from Ad2 to Ad3 was minimal. At the highest Adhesion parameter (i.e Ad4) the deposition efficiency decreased sharply. The decrease was not expected since, theoretically, increase in Adhesion parameter should also lead 68

85 Chapter 4. Results and discussion to increase in deposition efficiency. Since deposition efficiency showed dependency on the Reynolds number, a new set of Adhesion parameter experiments were carried out at a different Reynolds number to check if the decrease in the deposition efficiency could still be observed. In the new experiments, the Reynolds number was set to Re=1733 and the threshold velocity was m/s. Then the deposition efficiencies were calculated for each experiment. The results are shown in Figure 4.9 and they indicated that the deposition efficiency did not decrease from Ad3 to Ad DE Adhesion parameter (x10 4 ) Figure 4.9: The effect of Adhesion parameter on deposition efficiency (Re=1733) In addition to the analysis based on the particle velocity, the deposition efficiency was analysed based on the particle slip velocity(v s ). The slip velocity of the particle is generally defined as the difference in the velocity of the particle and the continuous phase [42]. Since, the new Adhesion parameter experiments were all assigned an inlet fluid velocity of m/s, ideally the particle slip velocity would be close to the fluid velocity if the particles are deposited (i.e since V s = υ 0 V then a near zero value of V would indicate deposition of a particle). However, particle slip velocity is known to vary from one section of the flow to another[55]. For example, the particles that were deposited on the obstruction of the flow cell generally showed high slip velocity. This was expected because of the fact that the fluid attained high velocity as it passed through the narrow section around the 69

86 Chapter 4. Results and discussion obstruction. On the other hand, in the section after the obstruction (Figure 4.15 section C) there were eddy formations and the fluid velocity was much lower than the section containing the obstruction. Visual inspection of the particles deposited in Section C showed that the slip velocity was generally around 1.1 m/s. Hence, it was decided to use a slip velocity value of 1.1 m/s or above as a basis to determine deposition efficiency (the reasoning here is that, if the slip velocity of a particle deposited on a section where the fluid velocity is the lowest is 1.1m/s, then it is safe to assume that the particles deposited on other sections would have higher slip velocities than 1.1m/s). A new threshold derived part was created using STAR-CCM+ and the deposition efficiencies were calculated DE Adhesion parameter (x10 4 ) Figure 4.10: The effect of Adhesion parameter on deposition efficiency (particle slip velocity) The result showed that as the Adhesion parameter was increased the particle deposition efficiency also increased. The increase was observed from Ad=1722 to Ad=8613 but a slight decrease was observed from Ad=8613 to Ad= In addition to the investigation of the amount of deposited particles, the location of deposition of particles in the flow cell was also investigated visually for each 70

87 Chapter 4. Results and discussion experiment. These investigations revealed an increasing particle deposition with increasing Adhesion parameter in most sections of the flow cell. In particular, Section A showed linear increase in particle deposition with Adhesion parameter. Once again particle deposition in Section B was minimal in all experiments apart from the few deposited particles at the front edge of the section. Most of the deposited particles were observed in Section C. A review of scientific literature was carried out to assess the validity of these experimental observations. The review revealed that particle deposition/adhesion increases with increasing surface energy [56] [57] [58]. Since the only Adhesion parameter variable in the experiments was W coh and since it is the sum of surface energies of particle-particle or particle-wall interactions then it is imperative to say that the experimental observations were in agreement with prior studies. Flow direction Ad1 Ad2 Ad3 Ad4 Figure 4.11: Front view of particle deposition (Adhesion parameter) Tabor parameter results In order to understand the effect of the Tabor parameter on the deposition of particles, four numerical simulations with different Tabor parameter values were carried out. In these experiments Young s modulus and Poisson ratio were independently varied. Simulations with high values of Young s modulus and Poisson s ratio values were compared with simulations with low values of these two parameters. The 71

88 Chapter 4. Results and discussion Flow direction Ad1 Ad2 Ad3 Ad4 Figure 4.12: Bottom view of particle deposition (Adhesion parameter) values of these parameters for each simulation are presented in Table 3.3. The Tabor parameter experiments revealed lower deposition efficiency for experiments with high Young s modulus values than for those with lower Young s modulus. Experiments Tab1 and Tab4 both were assigned a Young s modulus of Pa while experiments Tab2 and Tab3 were assigned Pa. Tab1 and Tab4 had deposition efficiencies of 13.7% and 13.2%, respectively which contrasted with the deposition efficiencies of Tab2 and Tab3, which were 69.1% and 67%. In addition to the variation of deposition efficiency with Young s modulus, variation of deposition efficiency with Poisson s ratio was also investigated. Experiments Tab1 and Tab4 were assigned identical parameters except Poisson s ratio values. In Tab1 the value was 0.35 while in Tab4 it was 0.1. The simulation results revealed that although there was no significant difference in the deposition efficiency between the two experiments, it was apparent that a small increase can be observed in the deposition efficiency as Poisson s ratio increased. The same trend was observed in experiments Tab2 and Tab3. Since Young s modulus is a measure of stiffness of a particle, large value of Young s moduli (rigid particles) lead to increase in tear-off force. Hence, it is expected that large Young s modulus values lead to lower deposition. Literature review on the subject matter revealed a similar trend with the observations found 72

89 Chapter 4. Results and discussion Name Tabor parameter Threshold velocity(m/s) DE(%) Tab Tab Tab Tab Table 4.4: Tabor parameter results in this study. A pioneer research done by Muller et al.[33] showed that the tearoff force is entirely dependent on the Tabor parameter and it decreased when the Tabor parameter was increased. This suggested that the deposition of particles is facilitated by low tear-off force and hence high Tabor parameter. Similar conclusions were made by papers written by Cheng et al.[59], Maguis[60], Greenwald[61] and Feng[62]. Flow direction Tab1 Tab2 Tab3 Tab4 Figure 4.13: Front view of particle deposition (Tabor parameter) 73

90 Chapter 4. Results and discussion Flow direction Tab1 Tab2 Tab3 Tab4 Figure 4.14: Bottom view of particle deposition (Tabor parameter) 4.2 Laboratory experiments Laboratory experiments were carried out for various Reynold s number values and influence of the Reynolds number on the particle deposition behaviour in a flow cell was investigated. Polystyrene particles with the work of cohesion value of J/m 2 were used during the experiments. Hence, comparison of results from laboratory experiments and numerical experiments (Low cohesion simulations) were carried out. The results from the laboratory experiments were images taken by a microscope during different stages of the experiment. Due to limited capacity, the microscope can only take images at specified locations in the flow cell. Three separate locations were chosen. These were, Section A, Section B and Section C. During each laboratory experiment images of the particle depositions at the above three specified locations were taken using a microscope. Each experiment was carried out for 15 minutes and images of deposition of the particles were taken at 5, 10 and 15 minute marks. However, it was observed that there was no appreciable difference both in the number of particles deposited and the location of deposition between images taken for each experiment. Due to this, it was decided that only 74

91 Chapter 4. Results and discussion Inlet Outlet front edge back edge Section A Section B Section C Figure 4.15: Flow cell locations of interest a single representative image for each experiment at each specified location to be presented. Due to lack of appropriate measurement technique, the deposition efficiency of particles in the laboratory experiments was not possible to estimate. But it was possible to visually observe the increase in deposition of particles with the Reynolds number. Analysis of images from Section A showed that the number of particles deposited increased when the Reynolds number was increased. This trend was also observed in the simulation results from low cohesion experiments although the increase was less prominent in Section A than Section C. Section B did not show appreciable particle deposition in the laboratory experiments. The few particles that were deposited did so usually at the back edge of the section and this could well be considered as part of Section A. The results from the simulations suggested that particles would adhere at the front edge of Section B but it was not observed appreciably during the laboratory experiments. Several factors might have contributed to this difference. The possible explanations are 75

92 Chapter 4. Results and discussion (a) Re-333 (b) Re-800 (d) Re-1733 (c) Re-1266 (e) Re-2200 Figure 4.16: Particle deposition in Section A. the presence of recurring bubbles near this section and the susceptibility of experimental set-up to flow and mechanical disturbances. Analysis of Section C showed that particle deposition increased with increased Reynolds number. Generally, most particles deposited on the left and right corners of this section. This was in agreement with simulation results. One interesting thing that emerged when experimental results were compared with simulation results was that, the simulations predicted that majority of the particles would deposit in Section C. However, the laboratory experiments revealed that Polystyrene particles deposited on Section A just as much as in Section C. In addition, Section A seemed to show a more linear increase in the deposition of particles with increasing Reynolds number than Section C or B. In both the simulations and the laboratory experiments, Section B experienced the least deposition of all sections. 76

93 Chapter 4. Results and discussion (a) Re-333 (b) Re-800 (d) Re-1733 (c) Re-1266 (e) Re-2200 Figure 4.17: Particle deposition in Section B. (a) Re-333 (b) Re-800 (d) Re-1733 (c) Re-1266 (e) Re-2200 Figure 4.18: Particle deposition in Section C. 77