Modeling and simulation of agglomeration in turbulent

Powder Technology 294 (2016) 373–402

Modeling and simulation of agglomeration in turbulent particle-laden flows: A comparison between energy-based and momentum-based agglomeration models N. Almohammed, M. Breuer∗ Professur f¨ ur Str¨ omungsmechanik, Helmut–Schmidt–Universit¨ at Hamburg, D–22043 Hamburg, Germany

Abstract In the present work, the particle agglomeration in shear flows is investigated in the framework of a hard-sphere model with deterministic collision detection employing two different modeling approaches: the energy-based and the momentum-based agglomeration models. The former is further improved concerning the agglomeration conditions. Moreover, the application area of both models is extended towards fully three-dimensional turbulent flows applying the largeeddy simulation technique. Here, the particles are assumed to be rigid, dry and electrostatically neutral and hence only the cohesion due to the van-der-Waals forces is considered. First, the energy-based and the momentum-based agglomeration model are successfully validated based on theoretical results using an existing laminar and two-dimensional test case. The numerical results are found to be in close agreement with the theory. Then, both agglomeration approaches are used to investigate the dynamic process of the particle agglomeration in a vertical fully developed turbulent channel flow. A detailed comparison of the results obtained using both agglomeration models is reported. Additionally, the performance of both techniques is examined under the influence of varying normal restitution coefficients of the inter-particle collisions. Furthermore, the influence of the inclusion of three sub-models (the feedback effect of the particles on the fluid, the lift forces and the subgrid-scale model for the particles) on the agglomeration process is studied. The results show that a significantly lower agglomeration rate is observed if the sub-models are considered. Next, the agglomeration models are applied to evaluate the effect of the diameter of the primary particles and the wall roughness on the agglomeration behavior. The reduction of the diameter of the particles leads to a stronger cohesive impulse and hence to a higher agglomeration rate. The wall roughness enhances the particle-particle collisions and slightly increases the number of agglomeration processes leading to higher agglomeration rates. The comparative study clearly indicates that both models predict similar trends of the physical behavior of the agglomeration process, but their results deviate slightly from each other. The most important reasons for the differences observed between the numerical results of both models are discussed. Based on the advantages and drawbacks of both models highlighted in this study, it can be concluded that owing to the reduced necessity of empirical parameters and the slightly more accurate results the momentum-based agglomeration model is superior to the energy-based model. Keywords: Agglomeration, Turbulent particle-laden flows, Van-der-Waals force, Energy-based and momentum-based models, Hard-sphere model, Large-eddy simulation

Preprint submitted to Elsevier

March 14, 2016


1 INTRODUCTION

2

1. Introduction Particle-laden turbulent flows are of substantial interest for academic and industrial research due to their significant role in diverse technologies and natural phenomena. The principal difficulty in the prediction of particle-laden turbulent flows can be attributed to their complex flow behavior, which is challenging in many industrial processes. In recent years, the advances in computational fluid dynamics (CFD) combined with the rapid development of high-performance computers and numerical algorithms have provided the basis for further insights into the complex physics of disperse multiphase flows. It is well known that the complexity of these flows strongly increases with the mass loading of the disperse phase, since the particles may influence the turbulence intensity significantly when the particle-fluid interaction (two-way coupling) is taken into account. For a further increase of the volume fraction, the particle-particle interactions play an increasing role. In this case, the particles move by kinetic transport due to the interaction with the turbulent flow and are additionally transported due to the particle-particle collisions. Thus, the inter-particle collisions have to be taken into account within a simulation environment leading to the necessity of four-way coupling. Particle agglomeration is an important mechanism, especially for microscopic particles with diameters in the range of 1 to 20 µm, since it influences the flow dynamics of cohesive powders in process apparatuses, including pneumatic conveying systems, fluidized bed combustors or gas cyclone separators. This phenomenon can have beneficial or negative effects on the performance of powder processing plants. For example, Obermair et al. (2005) found that the particle enlargement due to the agglomeration of fine particles enhances the separation efficiency of gas cyclone separators. Tomas (2007), on the other hand, stated that the particle agglomeration has avalanching effects (i.e., sudden onset of rapid motion of powders) and oscillating mass flow rates in conveyors lead to feeding and dosing problems. Thus, understanding the fundamentals of this phenomenon in turbulent flows is an issue of primary importance. The present paper aims at the modeling and simulation of particle agglomeration in turbulent flows. The first prerequisite for agglomeration is the particle-particle collision. Assuming dry and electrostatically neutral particles, the second prerequisite for the particle agglomeration is an attractive force between the particles known as the molecular van-der-Waals force. Thus, if a collision between particles occurs (first prerequisite fulfilled), it has to be proved whether the cohesive force between the collision partners (second prerequisite) is strong enough for their agglomeration. This can be achieved either by energy-based or momentum-based models and on different levels of complexity allowing strongly simplified considerations or physically more detailed models. In the following a brief literature review on energy-based (EAM) and momentum-based (MAM) agglomeration models is given. Löffler and Muhr (1972) and Hiller (1981) first proposed an energy-based agglomeration condition and applied this model to investigate the particle deposition in fiber filters. Based on an energy balance of the kinetic energy, the van-der-Waals energy (London, 1937; Hamaker, 1937) of the collision partners before and after the impact and the dissipated energy during the collision, a critical relative velocity between the approaching particles is determined. If the actual relative velocity between the collision partners is smaller than this critical value, it is assumed that they agglomerate building up a larger particle (agglomerate). However, this agglomeration model is restricted to a frictionless head-on collision describing a central impact ∗

Corresponding author Email address: [email protected] (M. Breuer)


1 INTRODUCTION

3

without rotation. This agglomeration model was extensively applied by Ho and Sommerfeld (2002), Ho (2004), Sommerfeld (2010) and St¨ ubing and Sommerfeld (2010) in the framework of one-way coupled Euler-Lagrange simulations based on RANS and a stochastic collision model by Sommerfeld (2001). Note that the stochastic model to detect particle-particle collisions relies on reasonable assumptions for the required collision probability. On the other hand, the advantage of the stochastic approach is that it can be applied to large disperse multiphase flow configurations with relatively low computational effort. For example, Ho and Sommerfeld (2003, 2005) used this technique including the simple agglomeration model to predict the separation efficiency of a gas cyclone separator with a very low number of particles (Np = 1000). In their study, they concluded that the inclusion of the agglomeration model enhances the separation efficiency of fine particles (smaller than 2 µm). However, in the above mentioned investigations the assumption of frictionless head-on collisions and the neglect of the particle rotation are rather crude constrictions not justified by real conditions. Therefore, this model was first extended by J¨ urgens (2012) towards oblique collisions allowing relative tangential velocities at the contact point. Later on, Alletto (2014) generalized the model taking the friction at the contact point into account. For this purpose, both the translational kinetic energy and the rotational kinetic energy of the particles are incorporated into the energy balance and the resulting agglomeration conditions. Furthermore, two collision types, namely sticking and sliding collisions, were distinguished within this model. Alletto (2014) first performed an a-priori analysis and then an a-posteriori evaluation with application to a downward directed pipe flow at a low Reynolds number yielding reasonable results but missing a profound validation procedure. The model still has some drawbacks regarding the rebound of the colliding particles not leading to agglomeration. Thus, in the present study it will be improved by revisiting the agglomeration conditions (see Section 2.2.5) and used as an energy-based agglomeration model (EAM). To the best of the authors’ knowledge, only J¨ urgens (2012) and Alletto (2014) carried out LES predictions coupled with a deterministic collision model and an energy-based agglomeration model to predict the agglomeration process in particle-laden turbulent flows. Nevertheless, in addition to the issues mentioned before both studies estimated the required values of the material properties of the particles based on nondistinctive assumptions, especially the maximum contact pressure for the onset of the plastic deformation of the collision partners. To avoid these shortcomings, in the present investigation real mechanical properties of the primary particles (fused quartz) are used (Momentive, 2014) and the maximum contact pressure is accurately estimated (see Appendix A) based on known models (see, e.g., Johnson, 1985). Besides the energy-based models, momentum-based agglomeration conditions were derived in several studies. For example, Weber et al. (2004) incorporated attractive inter-particle forces within the context of rapid granular flows into hard-sphere simulations using a square-well potential. In this model, the particles experience an instantaneous attractive force of diracdelta type at a pre-defined inter-particle distance. An agglomeration process is characterized by a normal relative velocity, which is too small to overcome the cohesive well. Kosinski and Hoffmann (2010) derived a momentum-based agglomeration model overtaking general ideas of Weber et al. (2004) and Weber and Hrenya (2006). They extended the classical hard-sphere model by including a cohesive impulse. Similar to the standard collision model, two cases have to be distinguished: (i) the particles stop sliding during the collision and (ii) the particles continue to slide throughout the entire collision. Without attractive forces the colliding particles bounce back after collision losing mechanical energy due to dissipative processes. However, the inclusion of attractive forces changes the situation completely. The


2 AGGLOMERATION MODELS

4

resulting impulse may not be strong enough to push the collision partners from each other leading to agglomeration. To determine the cohesive impulse, Kosinski and Hoffmann (2010) assumed that the cohesive forces due to the Hamaker interaction are constant during the collision and estimated the entire period of contact. They carried out first tests for their model in a laminar two-dimensional channel flow showing the formation of agglomerates, but a detailed validation is missing. Balakin et al. (2012) employed this model and investigated the agglomeration efficiency of particles in a turbulent but nevertheless two-dimensional shear flow. However, the agreement of the predictions with theoretical investigations is undifferentiated, since the theories also broadly scatter. Kosinski and Hoffmann (2011) extended their original model by combining dissipative forces based on the soft-sphere approach to the JohnsonKendall-Roberts (JKR) analysis of collision dynamics (Johnson et al., 1971) and determined the cohesive impulse based on a dimensional analysis. Unfortunately, they did not compare the new model with their previous model (Kosinski and Hoffmann, 2010). Note that the momentum-based agglomeration model by Kosinski and Hoffmann (2010) does not take the influence of the restitution coefficient on the collision time into account, since a frictionless, fully elastic collision is assumed for the determination of the impact time required for the calculation of the cohesive impulse. Thus, the influence of the restitution coefficient on the duration of the compression and restitution phase is not considered. Furthermore, no distinction is made for the collision times required for the determination of the cohesive force in the normal and tangential direction. To overcome these shortcomings, Breuer and Almohammed (2015) improved this model by taking the dissipative force during the collision into account. Furthermore, they presented a detailed parameter study on the influence of various model parameters on the global agglomeration process in a vertical turbulent particle-laden channel flow. In the present study, this extended agglomeration model denoted momentum-based agglomeration model (MAM) is used for comparison purposes with the energy-based agglomeration model (EAM). The objectives of the present study are as follows: 1. Enhancement of the energy-based agglomeration model by Alletto (2014). 2. Extension of the application area of this extended agglomeration model towards fully three-dimensional turbulent flows simulated by the large-eddy simulation technique combined with a deterministic hard-sphere collision approach. 3. Comparison between the improved energy-based agglomeration model (EAM) and the momentum-based model (MAM) by Breuer and Almohammed (2015). 4. Analysis and final evaluation of both models including their advantages and drawbacks. This paper is organized in the following manner. In Section 2 the energy-based agglomeration model is described in detail. The momentum-based model is only briefly summarized. An overview on the structure models of the arising agglomerate is also presented. Then, a brief description of the applied Euler-Lagrange framework is given in Section 3. The investigated test cases and the used numerical setups are reported in Section 4. A detailed comparison of the results obtained for both agglomeration models is presented and discussed in Section 5. Finally, conclusions concerning the accuracy and efficiency of these models are drawn. 2. Agglomeration models In this section, an enhanced version of the energy-based agglomeration model is derived. The momentum-based agglomeration model by Breuer and Almohammed (2015) is only briefly



5

sketched for the purpose of a direct comparison. The objective of the study is to compare both models with respect to the employed modeling assumptions, the required empirical input parameters and the accuracy of the predicted results. Both agglomeration models are implemented in the CFD code LESOCC (Breuer, 1998, 2000, 2002) briefly described in Section 3. The description starts with the background on inter-particle collisions. 2.1. Particle-particle collisions without cohesion It is well known that in the framework of a hard-sphere model only binary collisions between particles are considered. This assumption is justified, since the simultaneous contact of three or even more particles becomes significant only for very high mass loadings of disperse multiphase flows. In addition, it is assumed that the particle deformation during the collision is neglected (Crowe et al., 1998) implying that the distance between the centers of mass is constant during the collision process and equal to the sum of the particle radii. Furthermore, it is accepted that the friction between the sliding particles obeys Coulomb’s law of kinetic friction. Assuming linear displacements of the centers of two particles during a time step, the collision detection can be carried out purely based on kinematic conditions (Breuer and Alletto, 2012a). Fig. 1 shows a schematic representation of a system of two particles during a particle-particle collision with friction. ω−1 u−2

m1

Particle 1

u−p,r u−1

S1 S1

c Particle 1

c n

fˆt

fˆn

− fˆ

c fˆ

−

u2

−

x1

x−2

−u−c,t,r

r1

− fˆt

S2

− fˆn

r2 S2

u−c,t,r

Particle 2

Particle 2

ω−2

m2

Collision is possible (u−2 - u−1) n

Figure 1: Schematic representation of a particle-particle collision with friction.

Based on the conservation of the translational and angular momentum of the collision partners, the post-collision translational and angular velocities of both particles can be expressed as: m ˆ ˆ f m1 m ˆ ˆ − u+ f 2 = u2 + m2 5 m ˆ ω1+ = ω1− − n × fˆ d1 m1 5 m ˆ ω2+ = ω2− − n × fˆ d2 m2 − u+ 1 = u1 −

(1a) (1b) (1c) (1d)

Here, the subscripts 1 and 2 refer to the collision partners. The symbols u− , u+ and ω − , ω + stand for the translational and the angular velocities of the colliding particles before (−) and



6

after (+) the collision, respectively1 . n denotes the collision-normal unit vector directed from the center of mass of the first particle S1 to the center of mass of the second particle S2 as depicted in Fig. 1. fˆ is the specific impulse vector defined as the integral of the forces acting on the particle during the entire collision normalized by the effective mass m: ˆ Z 1 m1 m2 ˆ . (2) f= f dt with m ˆ = m ˆ m1 + m2 The specific impulse vector fˆ, which indicates a mutual repulsion of the collision partners due to their elastic deformation, pushes the particles apart from each other. The friction between the particles is taken into account. For this purpose, the total impulse vector is divided into a normal and a tangential component as follows (see Fig. 1): fˆ = fˆn + fˆt , where the normal component of the total impulse vector fˆn is given by: Z 1 fˆn = fn,a dt . m ˆ

(3)

(4)

Here, the second index a is introduced to distinguish it from other components discussed later. According to Crowe et al. (1998), the normal impulsive force due to the mechanical deformation can be expressed as: − fˆn,a = fˆn,a n with fˆn,a = −(1 + en,p ) u− (5) 2 − u1 · n ,

where fˆn,a is the magnitude of the normal impulsive force due to mechanical deformation and en,p stands for the normal restitution coefficient for a particle-particle collision. Based on − ˆ Fig. 1, a successful inter-particle collision can only occur if (u− 2 − u1 ) · n < 0, and hence fn,a is always positive in the above equation. The calculation of the tangential component of the total impulse fˆt depends on the collision type (sticking or sliding). 2.1.1. Determination of the collision type To distinguish the collision type, the no-slip condition relying on Coulomb’s law of static friction is used to define a sticking collision. Hence, the condition for the magnitude of the tangential component of the sticking impulse vector fˆst,t can be written as: ˆ fst,t ≤ µst,p fˆn ,

(6)

where µst,p is the coefficient of static friction and the magnitude of the resulting force has to be defined. By substituting Eq. (5) into the above equation, the magnitude of the tangential component given by Eq. (6) can be written as: ˆ (7) fst,t ≤ µst,p fˆn,a . 1

Note that we use standard SI units if not otherwise stated.



7

To determine the total impulse vector for a sticking collision, the tangential restitution coefficient is defined as: + uc,t,r − |et,p | = − and u+ (8) c,t,r = −et,p uc,t,r . uc,t,r

Note that the indices have the following meaning: c = contact point, t = tangential and r = relative. Using et,p , the tangential component of the post-collision velocity vector u+ c,t,r is no − longer zero after the collision, but directed opposite to uc,t,r . Its magnitude is reduced by the factor of |et,p | due to the inelastic deformation of the contact surfaces of the collision partners (Breuer et al., 2012). The tangential velocity of the relative motion at the contact point before (−) and after (+) the impact is given by: − d1 − d2 − − − − − ω + ω2 × n uc,t,r = u2 − u1 − (u2 − u1 ) · n n − 2 1 2 (9) + d1 + d2 + + + + + uc,t,r = u2 − u1 − (u2 − u1 ) · n n − ω + ω2 × n 2 1 2

Inserting Eq. (9) and its corresponding terms given by Eq. (1) into Eq. (8) yields the total impulse vector for a sticking collision: 2 − − fˆst = −(1 + en,p ) u− (10) 2 − u1 · n n − (1 + et,p )uc,t,r . 7 In the above equation, the first term on the right-hand side is the normal impulse vector given by Eq. (5), and the second term represents the tangential impulse vector for a sticking collision: 2 fˆst,t = − (1 + et,p ) u− c,t,r . 7 By substituting Eq. (11) into Eq. (7), the no-slip condition can be written as: − 7 µst,p uc,t,r ≤ fˆn,a , 2 (1 + et,p )

(11)

(12)

If this condition is satisfied, a sticking collision occurs and the colliding particles stop sliding during the collision. Otherwise, the collision partners slide throughout the entire impact. According to Coulomb’s law of kinetic friction, in case of a sliding collision the tangential force deviates from the sticking case. It can be expressed by the product of the coefficient of kinetic friction of the particle µkin,p and the magnitude of the normal force: fˆt = fˆsl,t = −µkin,p fˆn t , (13)

where t is the tangential unit vector defined as: u− c,t,r t = − . uc,t,r

(14)

The minus sign on the right-hand side of Eq. (13) means that the tangential sliding impulse vector points in the direction opposite to the tangential relative velocity at the contact point ˆ u− c,t,r . Thus, the tangential impulse vector for the sliding collision fsl,t reads: −

uc,t,r fˆt = fˆsl,t = −µkin,p fˆn,a − . uc,t,r

(15)



8

2.1.2. Kinetic of the collision partners after the impact To determine the translational and angular velocities of the collision partners after a successful particle-particle impact with friction, the total impulse vector fˆ given by Eq. (3) has to be determined. The normal component of the total impulse acting on a particle is given by Eq. (5), while the tangential component of the total impulse vector is determined based on the impact type as explained before. Thus, two collision types (sticking or sliding) have to be distinguished: • Sticking collision: By inserting both components of the total impulse vector of a sticking collision (fˆn,a and fˆst,t ) given by Eq. (5) and Eq. (11) into Eq. (1), the translational and angular velocities of the particles after a sticking collision can be written as: − 2 m ˆ ˆ + − fn,a n − (1 + et,p ) uc,t,r t u1 = u1 − m1 7 − 2 m ˆ ˆ + − fn,a n − (1 + et,p ) uc,t,r t u2 = u2 + m2 7 (16) m ˆ 10 ω1+ = ω1− + (1 + et,p ) u− c,t,r (n × t) 7 d1 m1 ˆ 10 m ω2+ = ω2− + (1 + et,p ) u− c,t,r (n × t) 7 d2 m2

• Sliding collision: By inserting both components of a sliding collision (fˆn,a and fˆsl,t ) given by Eq. (5) and Eq. (15) into Eq. (1), the new velocities of the particles after a sliding collision read: i m ˆ hˆ + − ˆ fn,a n − µkin,p fn,a t u1 = u1 − m1 i m ˆ hˆ − ˆ u+ = u + f n − µ f t n,a kin,p n,a 2 2 m2 (17) 5 m ˆ + − ˆ µkin,p fn,a (n × t) ω1 = ω1 + d1 m1 5 m ˆ ω2+ = ω2− + µkin,p fˆn,a (n × t) d2 m2 To consider agglomeration, the cohesion between the colliding particles has to be additionally taken into account. In the framework of a hard-sphere model two different approaches for modeling the agglomeration of particles (EAM and MAM) are presented next. 2.2. Energy-based agglomeration model (EAM) Based on the energy balance during the collision process, this model was first introduced by Hiller (1981) assuming a frictionless head-on collision and neglecting the rotation of the colliding partners. Since the assumptions made by Hiller (1981) are not justified by real interparticle collisions, J¨ urgens (2012) extended this model towards oblique collisions. Based on this study, Alletto (2014) generalized the model taking the friction between the collision partners into account. Thus, the rotational energy of the colliding particles and the agglomerate is considered leading to the following energy balance before and after the collision: − − + + − + E˜kin,r + E˜rot + EvdW = E˜kin,r + E˜rot + EvdW + Edis ,

(18)



9

where E˜kin,r and E˜rot denote the relative translational and rotational kinetic energies before (−) and after (+) the impact taking the cohesion into account. EvdW stands for the vander-Waals energy, and Edis is the energy dissipation during the collision process. According to Hiller (1981) and Alletto (2014), it is assumed that the dissipated energy is equal to the difference of the total kinetic energies (i.e., the sum of translational and rotational energies) before and after the collision without considering the effect of the attractive force between the collision partners: − + + − . (19) Edis = Ekin,r − Erot − Ekin,r + Erot

+ + In the above equation, Ekin,r and Erot stand for the relative translational and the rotational kinetic energies after the impact without taking the cohesion into account. This also means that no agglomeration occurs and the particles bounce off after the collision. Note that the translational and rotational energies before the collision are assumed to be identical for both − − − − cases with and without cohesion (i.e., Ekin,r = E˜kin,r and Erot = E˜rot ). This assumption is valid, since the particles before the collision are only marginally influenced by the cohesive − − force. By inserting Eq. (19) into Eq. (18) and eliminating E˜kin,r and E˜rot , the difference of the van-der-Waals energy after and before the collision can be written as: + − + + + + ˜ ˜ ∆EvdW = EvdW − EvdW = Ekin,r − Ekin,r + Erot − Erot . (20)

2.2.1. Agglomeration condition In case of agglomeration, the post-collision relative translational kinetic energy is set equal + to zero in the above relation (E˜kin,r = 0), since the particles stick together (i.e., there is no + relative translational motion between them) and the rotational kinetic energy E˜rot is replaced by the rotational kinetic energy of the arising agglomerate Eag,rot defined later. Hence, the agglomeration condition is expressed as: + + ∆EvdW ≥ Ekin,r + Erot − Erot,ag ,

(21)

with + Ekin,r =

1 + 2 m ˆ (u+ 2 − u1 ) 2

+ and Erot =

2 1 2 1 I1 ω1+ + I2 ω2+ . 2 2

(22)

Physically, the condition (21) means that the difference of the van-der-Waals energy resulting from the cohesive force between the colliding particles is sufficient for an effective agglomeration + of both collision partners. Note that for the calculation of the translational (u+ 1 , u2 ) and angular (ω1+ , ω2+ ) velocities the van-der-Waals force does not have to be taken into account. Thus, depending on the collision type these quantities can be determined based on Eqs. (16) and (17) for a sticking and a sliding collision, respectively. To check the agglomeration condition given by Eq. (21), two unknowns are still required, namely the difference of the van-der-Waals energy ∆EvdW and the rotational energy of the agglomerate Erot,ag . The calculation of these energies is discussed next. 2.2.2. Difference of the van-der-Waals energy According to Israelachvili (2011), the van-der-Waals force for the two geometries most commonly encountered (i.e., two flat surfaces and two spheres) is given by: FvdW =

H fg , 12 δ02

(23)



10

where H stands for the Hamaker constant depending on the material, δ0 denotes the free distance between the surfaces, and fg is a factor depending on the geometry of the bodies. Assume that we have two spheres of radii r1 and r2 in contact at a small distance δ apart. If r1 δ and r2 δ, then the geometry factor is given by: fg =

2 r1 r2 . r1 + r2

(24)

For two flat surfaces in contact at a distance δ, the geometry factor is given by: fg =

2A , πδ

(25)

where A is the contact (deformation) area. The van-der-Waals energy can be determined as follows: EvdW =

Z∞

FvdW dδ ,

(26)

δ0

where δ0 is the distance between the spheres or flat surfaces during the collision. To determine the difference of the van-der-Waals energy, the shape of the colliding bodies has to be taken into account. Before the impact, it is assumed that the collision partners are spherical and hence the van-der-Waals force is determined by substituting Eq. (24) into Eq. (23). After the impact, a flat contact area between the collision partners is assumed as shown in Fig. 2. Therefore, the van-der-Waals force is determined by combining Eqs. (23) and (25). The difference of the van-der-Waals energy ∆EvdW can be expressed as: ∆EvdW =

Z∞

+ FvdW

dδ −

δ0

Z∞

δ0

− FvdW

H H A − dδ = 2 12π δ0 6 δ0

r1 r2 r1 + r2

.

(27)

A glance at Eq. (27) shows that the second term on the right-hand side (the van-der-Waals energy between spheres) is negligible compared to the first one (the van-der-Waals energy between flat surfaces), since δ0 is of the order O(nm). Thus, the above equation is reduced to: ∆EvdW =

H A . 12π δ02

(28)

As shown in Fig. 2, the contact area can be calculated using the dimensions of particle 1 as follows: " 2 # 2 d d 1 1 = π d1 h1 − h21 , (29) A = π a2 = π − − h1 2 2 where a stands for the contact radius, h1 is the depth of the plastic deformation of the particle of diameter d1 . Analogously, the contact area can be expressed based on particle 2 as A = π (d2 h2 − h22 ). If d1 h1 and d2 h2 , the contact area can be approximately calculated as: A = π d1 h1 = π d2 h2 .

(30)



11 h2

u−1 S1

u−2

d1 h − 2 − 1

S2

n d

a

−1 2

Particle 1

Particle 2

h1

Figure 2: Contact surface of two colliding particles.

By inserting Eq. (30) into Eq. (28), the difference of the van-der-Waals energy reads: ∆EvdW =

H d1 h1 . 12 δ02

(31)

The depths of the plastic deformations h1 and h2 are determined by calculating the work required to deform a ductile material (see e.g., Antonyuk, 2006): Epl =

Zh1

p (π d1 h1 ) dh1 +

0

Zh2 0

p (π d2 h2 ) dh2 =

1 π p d1 h21 + d2 h22 , 2

(32)

where p denotes the maximum contact pressure at which the plastic deformation occurs as explained in Appendix A. By inserting Eq. (30) into the above equation, the plastic deformation energy can be written as: d1 1 2 Epl = π p d1 h1 1 + . (33) 2 d2 According to Hiller (1981), the plastic deformation energy is defined as: − Epl = (1 − e2n,p ) Ekin,r ,

(34)

− where Ekin,r stands for the relative kinetic energy before the impact and is given by: − Ekin,r =

1 − 2 m ˆ u− − u . 2 1 2

(35)

By substituting Eq. (35) into Eq. (34) and the resulting equation into Eq. (33), the depth of the plastic deformation h1 can be written as: − h1 = d1 d22 u− 2 − u1

"

# 21 ρ1 ρ2 1 − e2n,p . 6 p¯ (d1 + d2 ) (ρ1 d31 + ρ2 d32 )

(36)



12

Note that h1 depends on the density, the maximum contact pressure and the normal restitution coefficient of the material. Inserting the above equation into Eq. (31) yields the difference of the van-der-Waals energy: ∆EvdW

" # 21 2 ρ ρ 1 − e H 2 2 − 1 2 n,p = d d u − u− . 1 12 δ02 1 2 2 6 p¯ (d1 + d2 ) (ρ1 d31 + ρ2 d32 )

(37)

Note that in this model the difference of the van-der-Waals energies vanishes for a fully elastic collision (en,p = 1), since no kinetic energy is dissipated during the collision. This is consistent with the extended momentum-based agglomeration model described in Section 2.3.1. 2.2.3. Kinetic energy of the agglomerate The rotational kinetic energy of the agglomerate is defined as: Erot,ag =

1 1 1 2 [Iag ] · ωag = Lag · ωag = (Lag,x ωag,x + Lag,y ωag,y + Lag,z ωag,z ) , 2 2 2

(38)

where ωag stands for the angular velocity of the agglomerate, Lag denotes the angular momentum of a two-particle agglomerate about its center of mass and [Iag ] is the moment of inertia tensor of the agglomerate about its center of mass given by:   Ixx Ixy Ixz [Iag ] = Iyx Iyy Iyz  . (39) Izx Izy Izz To calculate the angular velocity of the agglomerate, the following system of equations has to be solved: Lag = [Iag ] ωag .

(40)

As described in Breuer and Almohammed (2015) the angular momentum of a two-particle agglomerate about its center of mass can be written as: Lag = I1 ω1− + I2 ω2− +

m ˆ − (d1 + d2 ) n × (u− 2 − u1 ). 2

(41)

In order to determine the rotational kinetic energy of the agglomerate defined by Eq. (38), the angular velocity of the agglomerate ωag needs to be predicted. For this purpose, the components of the inertia tensor of the two-particle agglomerate have to be calculated first. They can be written as: Ixx = I1 + I2 + m1 a2y,1 + a2z,1 + m2 a2y,2 + a2z,2 Iyy = I1 + I2 + m1 a2x,1 + a2z,1 + m2 a2x,2 + a2z,2 Izz = I1 + I2 + m1 a2x,1 + a2y,1 + m2 a2x,2 + a2y,2 (42) Ixy = Iyx = −m1 (ax,1 ay,1 ) − m2 (ax,2 ay,2 ) Ixz = Izx = −m1 (ax,1 az,1 ) − m2 (ax,2 az,2 ) Iyz = Izy = −m1 (ay,1 az,1 ) − m2 (ay,2 az,2 )



13

with a1 and a2 describing the distance between the center of mass of the agglomerate and the individual centers of mass of both particles: ax,1 = −a1 nx ay,1 = −a1 ny az,1 = −a1 nz ax,2 = +a2 nx ay,2 = +a2 ny az,2 = +a2 nz

(43)

By solving the system of equations (40), now the components of the angular velocity of the center of mass of the two-particle agglomerate are determined as follows: 1 [A11 Lx + A12 Ly + A13 Lz ] D 1 = [A21 Lx + A22 Ly + A23 Lz ] D 1 = [A31 Lx + A32 Ly + A33 Lz ] , D

ωag,x = ωag,y ωag,z

(44)

with 2 2 2 D = Ixx Iyy Izz − Ixx Iyz − Iyy Ixz − Izz Ixy + 2Ixy Ixz Iyz

(45)

The coefficients of Eq. (44) are given by: 2 A11 = Iyy Izz − Iyz

A12 = Ixz Iyz − Izz Ixy A13 = Ixy Iyz − Iyy Ixz

2 A21 = Ixz Iyz − Izz Ixy A22 = Ixx Izz − Ixz A23 = Ixy Ixz − Ixx Iyz 2 A31 = Ixy Iyz − Iyy Ixz A32 = Ixy Ixz − Ixx Iyz A33 = Ixx Iyy − Ixy

(46)

Hence, the rotational kinetic energy of the agglomerate can be determined based on Eq. (38). Thus, the missing quantities in the agglomeration condition (21) are now available. 2.2.4. Kinetics of the collision partners without agglomeration As explained in Section 2.1.1, the colliding particles rebound after the collision if the agglomeration condition is not satisfied. However, also in this case the attractive force changes the resulting impulse balance significantly and this effect should not be neglected. For this purpose, the calculation of the translational and angular velocities of the colliding particles including the cohesion is achieved in two steps: 1. First, the intermediate quantities are determined without considering the cohesive force based on Eq. (16) for a sticking collision and based on Eq. (17) for a sliding collision. 2. Then, the final quantities after an impact taking the cohesion into account are deter˜ 2+ . They are calculated based on the first ˜+ ˜+ ˜ 1+ and ω mined, which are denoted u 2, ω 1, u step by introducing a cohesive impulse force resulting from the van-der-Waals interaction denoted fâg . Here, the conservation of the translational momentum reads: m ˆ ˆ fag , m1 m ˆ ˆ + ˜+ fag . u 2 = u2 + m2 + ˜+ u 1 = u1 −

(47)

Furthermore, it is assumed that the angular velocities are calculated as follows: ˜ 1+ = kω1 ω1+ , ω ˜ 2+ = kω2 ω2+ , ω

(48)



14

Subtracting the relations of Eq. (47) from each other yields the vector of the cohesive impulse fâg . + + ˜+ ˜+ fâg = fâg,n + fâg,t = u 1 − u2 − u1 2 −u

(49)

Note that the tangential component of the cohesive impulse fâg,t can be interpreted as a tangential force resulting from the increase of the friction at the contact point when the cohesion is considered during the collision. Based on the above equation, the magnitudes of the vector of the cohesive impulse in normal and tangential directions are given by: + + ˜+ ˜+ fâg,n = fâg · n = u 2 −u 1 · n − u2 − u1 · n , (50) ˜+ − u ˜ + · t − u+ − u+ · t , fâg,t = fâg · t = u 2

1

2

1

where the tangential unit vector t is given by Eq. (14). Since the relative velocities of the collision partners in both normal and tangential directions are reduced if the van-der-Waals forces are taken into account, new dimensionless factors kn,p and kt,p are introduced as follows:

kn,p

˜+ ˜+ u 2 −u 1 ·n = + u+ − u 2 1 ·n

and kt,p

˜+ ˜+ u 2 −u 1 ·t . = + u+ − u 2 1 ·t

(51)

Substituting the definition of kn,p and kt,p into Eq. (50), the components of the cohesive force in Eq. (49) can be written as: + fâg,n = fâg,n n = −(1 − kn,p ) u+ − u 2 1 ·n n (52) + fâg,t = fâg,t t = −(1 − kt,p ) u − u+ · t t 2

1

Since Eqs. (47) and (48) have four unknowns (kn,p , kt,p , kω1 and kω2 ), Alletto (2014) made some assumptions to determine these factors. The first is that in the energy balance before and after the impact the translational kinetic energies in Eq. (20) are divided into normal and tangential components: + + + + + + ∆EvdW = Ekin,r,n − E˜kin,r,n + Ekin,r,t − E˜kin,r,t + Erot − E˜rot .

(53)

Then, Alletto (2014) distinguished three different cases: (I) The collision partners rebound in the normal direction: In this case, it is assumed that the work done by the van-der-Waals force influences only the normal component of + the relative kinetic energy. Thus, the other contributions remain unchanged: Ekin,r,t = + + + ˜ ˜ Ekin,r,t and Erot = Erot . Then, Eq. (53) can be written as: + + ∆EvdW = Ekin,r,n − E˜kin,r,n

(54)

+ + By rearranging the above equation with respect to Ekin,r,n , dividing the result by E˜kin,r,n and using the new factor kn,p , it can be written as:

2 kn,p

=

+ E˜kin,r,n + Ekin,r,n

2 ˜+ ˜+ u ∆EvdW 2 −u 1 ·n = + , 2 = 1 − + + Ekin,r,n u2 − u1 · n

(55)



15

with + Ekin,r,n =

2 1 + m ˆ (u2 − u+ 1)·n . 2

(56)

2 If kn,p > 0, the coefficients in Eqs. (48) and (52) are kω1 = kω2 = kt,p = 1 according to the assumptions made for this case, and kn,p is calculated based on Eq. (55) . Since kt,p is set to unity, the tangential component of the cohesive impulse fâg,t given by Eq. (52) vanishes. This means that the increase of the friction due to the cohesive force is neglected. Setting kω1 = kω2 = 1 means, on the other hand, that the angular velocities are calculated based on the standard hard-sphere model, since the influence of the tangential component is neglected. Then, the model is equivalent to that of Hiller (1981). 2 < 0, it is assumed (II) The collision partners rebound only in the tangential direction: If kn,p + + + that E˜kin,r,n = 0 and Erot = E˜rot . Therefore, Eq. (53) is reduced to the following form: + + + ∆EvdW = Ekin,r,n + Ekin,r,t − E˜kin,r,t .

(57)

+ + By rearranging the above equation with respect to E˜kin,r,t , dividing the result by Ekin,r,t and using the new factor kt,p , it can be written as:

2 kt,p =

+ E˜kin,r,t + Ekin,r,t

with + Ekin,r,t =

2 + ∆EvdW − Ekin,r,n ˜+ ˜+ u 2 −u 1 ·t = + 2 = 1 − + Ekin,r,t u2 − u+ 1 ·t

2 1 + m ˆ (u2 − u+ 1)·t 2

(58)

(59)

Based on the above assumptions, the coefficients in Eqs. (48) and (52) are kω1 = kω2 = 1, kn,p = 0 and kt,p is calculated based on Eq. (58). Setting kn,p equal to zero implies that the cohesive force in the normal direction is strong enough to hold the colliding particles in contact, but they separate in the tangential direction while fâg,t 6= 0. 2 2 (III) This case is considered if both kn,p < 0 and kt,p < 0. Here, it is assumed that the + = 0 and relative kinetic energy in normal and tangential direction vanishes (i.e., E˜kin,r,n + ˜ Ekin,r,t = 0 ). Thus, the energy balance can be written as: + + + + + Ekin,r,t + Erot − E˜rot . ∆EvdW = Ekin,r,n

(60)

Physically, this case means that the colliding particles rebound neither in the normal nor in the tangential direction, but rather roll or slide over each other during the impact depending on the collision type. By substituting Eq. (22) into the above equation, it can be written as follows: o 1 n + 2 o 1 n + 2 + + + 2 + 2 ˜1 ˜2 I1 ω1 − ω + I2 ω2 − ω = ∆EvdW − Ekin,r,n − Ekin,r,t . (61) 2 2 2 2 ˜ 1+ and ω ˜ 2+ , another assumption Since the above equation still has two unknowns ω was made by Alletto (2014) introducing the dimensionless factor β describing the ratio



16

of the rotational energy of particle 1 in relation to the sum of the rotational energies of both particles: 2 1 I ω1+ 2 1 β= (62) . + 2 + 2 1 1 I ω + I ω 1 2 1 2 2 2 Hence, the difference of the rotational kinetic energy of each particle with and without considering the cohesion between the colliding particles can be written as: o 1 n + 2 + + + 2 ˜1 I1 ω1 − ω = β ∆EvdW − Ekin,r,n − Ekin,r,t , 2 (63) n o 1 + + + 2 + 2 ˜2 I2 ω2 − ω = (1 − β) ∆EvdW − Ekin,r,n − Ekin,r,t . 2

Of course, the summation of both relations of the above equation yields Eq. (61). Based on the definitions of kω2 and kω2 according to Eq. (48), these factors can be expressed as: 2β + + ∆E − E − E kω2 1 = 1 − vdW kin,r,n kin,r,t , 2 I1 ω1+ (64) 2(1 − β) + + 2 kω2 = 1 − 2 ∆EvdW − Ekin,r,n − Ekin,r,t . I2 ω2+

With respect to the above assumptions, the coefficients in Eq. (52) are kn,p = kt,p = 0 and kω1 and kω2 are calculated based on Eq. (64).

2.2.5. Present model extension Regarding the treatment of the particle rebound including the cohesion, the model by Alletto (2014) has the following drawbacks: 2 2 • The third case with kn,p < 0 and kt,p < 0 means that the cohesive force in both directions are strong enough to hold the colliding particles in contact, and hence they do not separate and rather roll (sticking case) or slide (sliding case) over each other. Thus, this condition has to be considered as an agglomeration process of the collision partners for both impact types, since they do not rebound after the collision, but build up an agglomerate. Hence, the agglomeration condition given by Eq. (21) has to be modified as follows: + + 2 2 ∆EvdW ≥ Ekin,r + Erot − Erot,ag or kn,p < 0 & kt,p fˆn,a + fˆn,c

where fˆt stands for the magnitude of the tangential impulse vector explained before. The agglomeration conditions depend on the collision type, and hence two cases are distinguished: (i) in case of a sticking collision, only the first criterion in Eq. (75) is sufficient for the agglomeration. The second criterion is not considered, because it is already assumed that the colliding particles stop sliding during the collision, and hence the tangential component of the relative velocity at the contact point is equal to zero (Kosinski and Hoffmann, 2010). (ii) for a sliding collision, the agglomeration takes place, if both criteria in Eq. (75) are satisfied. Further details can be found in Breuer and Almohammed (2015). 2.3.3. Kinetics of the collision partners without agglomeration If the cohesive forces are not strong enough for a successful agglomeration, the collision partners are separated due to the resulting impulse during the impact. However, also in this case the effect of the cohesive forces has to be taken into account when determining the translational and angular velocities of both particles after the impact. To calculate these quantities, the total impulse vector fˆ given by Eq. (3) has to be determined. The normal component of the total impulse acting on a particle is given by Eq. (69), whereas the tangential component depends on the collision type. In order to determine the collision type, the no-slip condition relying on Coulomb’s law explained in Section 2.1 has to be modified to include the cohesive forces. That is achieved by substituting Eqs. (11) and (66) into Eq. (6) leading to the modified no-slip condition: − 7 µst,p ˆ ˆ uc,t,r ≤ fn,a + fn,c , (76) 2 (1 + et,p )

where fˆn,c denotes the magnitude of the cohesive impulse during the entire collision time (compression and restitution periods). The above equation implies that attractive forces between the particles enhances the friction between the collision partners for any normal restitution coefficient of particle-particle collisions. That indicates that the ratio of sticking to sliding collisions increases when the cohesive force is incorporated into the no-slip condition as observed by Breuer and Almohammed (2015). Since in the case of a sliding collision the normal impulse vector is required for the determination of the tangential force given by Eq. (13), it has to be modified when the cohesive force is taken into account. By inserting Eq. (66) into Eq. (13), the tangential component of the total impulse for a sliding collision now reads: u− c,t,r fˆsl,t = fˆt = −µkin,p fˆn,a + fˆn,c − . uc,t,r

(77)



20

Note that in the case of a sticking collision the tangential component of the total impulse is not influenced by the cohesive force and thus still defined by Eq. (11). Consequently, the following two cases have to be distinguished: • Sticking collision: By inserting both components of the total impulse vector of a sticking collision (fˆn∗ and fˆst,t ) given by Eq. (69) and Eq. (11) into Eq. (1), the translational and angular velocities of the particles after a sticking collision without agglomeration can be written as: − 2 m ˆ ˆ ∗ + − ˆ fn,a + fn,c n − (1 + et,p ) uc,t,r t u1 = u1 − m1 7 − m ˆ 2 + − ∗ ˆ ˆ u2 = u2 + fn,a + fn,c n − (1 + et,p ) uc,t,r t m2 7 (78) − 10 m ˆ − + ω1 = ω1 + (1 + et,p ) uc,t,r (n × t) 7 d1 m1 ˆ 10 m (1 + et,p ) u− ω2+ = ω2− + c,t,r (n × t) 7 d2 m2

By comparison with Eq. (16) it is obvious that solely the normal component is influenced by the cohesive force.

• Sliding collision: By inserting both components of the total impulse vector of a sliding collision (fˆn∗ and fˆsl,t ) given by Eq. (69) and Eq. (77) into Eq. (1), the translational and angular velocities of the particles after a sliding collision without agglomeration read: i m ˆ h ˆ − ∗ ˆ ˆ ˆ u+ = u − f + f n − µ f + f n,a kin,p n,a n,c t 1 1 n,c m1 i m ˆ h ˆ − ˆ∗ n − µkin,p fˆn,a + fˆn,c t f + f u+ = u + n,a n,c 2 2 m2 (79) 5 m ˆ + − ˆ ˆ ω1 = ω1 + µkin,p fn,a + fn,c (n × t) d1 m1 5 m ˆ ω2+ = ω2− + µkin,p fˆn,a + fˆn,c (n × t) d2 m2 Here, the cohesive force has an impact on both the normal and the tangential direction, ∗ but the effect in both directions is different accounted for by fˆn,c and fˆn,c , respectively.

2.4. Kinetics of the agglomerate Independent on the agglomeration model (EAM or MAM) the new properties of the agglomerate have to be determined if an agglomeration process occurs. The translational velocity of the center of mass of the arising agglomerate is predicted based on the conservation of the translational momentum of the collision partners: uag

− m1 u− 1 + m2 u2 . = m1 + m2

(80)

The position of the center of mass of the agglomerate reads: xag =

− m1 x − 1 + m2 x 2 . m1 + m2

(81)



21

To calculate the angular velocity of the agglomerate ωag , the system of equations (40) has to be solved. For this purpose, the moment of inertia tensor of the agglomerate has to be determined which depends on the structure of the arising agglomerate. Kosinski and Hoffmann (2010, 2011), Balakin et al. (2012) and Ho and Sommerfeld (2002) among others, assumed a volumeequivalent agglomerate structure. In this standard model denoted volume-equivalent sphere model (VSM), the density of the agglomerate is equal to that of the primary particles satisfying the conservation of mass (i.e., mag = m1 +m2 ). Thus, the diameter of volume-equivalent sphere of the agglomerate dag can be expressed as: q (82) dag = 3 d31 + d32 .

However, the interstitial space between the agglomerated particles is not taken into account. In order to consider the effect of the porosity of the arising agglomerate structure, the model is extended by Breuer and Almohammed (2015) towards a more general description of the structure of the arising agglomerate denoted closely-packed sphere model (CSM). In this model, it is assumed that an agglomerate is built up from spherical particles including an interstitial space between its primary particles. Consequently, the global density of the agglomerate is reduced compared to that of the primary particles. To account for this effect, a packing fraction of the spheres is assumed by packing the particles into a cube. The packing fraction fpack is defined as: fpack =

Vsphere , Vcube

(83)

where Vsphere stands for the total volume of the spheres packed into a cube and Vcube is the volume of the cube. Hence, the density of the agglomerate is given by ρag = fpack ρo with 0 < fpack < 1. Here, ρo is the density of the agglomerating primary particles. Appropriate values of the packing fraction fpack can be found, for example, in Packomania (2013) and the references cited there. The corresponding values for two, three and four particles are 0.26683, 0.31574 and 0.42099, respectively. The mass conservation of the packed particles yields the diameter of the spherical agglomerate (packing sphere): 1/3 ρ1 d31 + ρ2 d32 . (84) dag = ρag Since both structure models (VSM and CSM) assume spherical agglomerates, the off-diagonal elements of the inertia tensor vanish. Thus, the moment of inertia of the resulting agglomerate is given by Iag = 0.1 mag d2ag . Based on Eq. (40), the angular velocity of the agglomerate can be written as: ωag =

Lag , Iag

(85)

where the angular momentum of the agglomerate Lag is given by Eq. (41). In summary, both structure models satisfy the mass and angular momentum conservation. The main difference between both approaches is that CSM takes the interstitial space between the particles into account by reducing the density of the arising agglomerate mimicking the porosity of the agglomerate. Subsequently, CSM predicts larger diameters than VSM. In the present study, the closely-packed sphere model is applied to the simulations using both agglomeration models due to its advantages against VSM.


3 EULER-LAGRANGE SIMULATION FRAMEWORK

22

3. Euler-Lagrange simulation framework Since the emphasis of this study is on particle agglomeration, the general simulation framework based on a four-way coupled Euler-Lagrange approach is only briefly described concentrating on the specific features, which are of relevance for the agglomeration process. For more details about the research studies using the code LESOCC employed in the present work the interested reader is referred to Breuer (1998, 2000, 2002) for the continuous phase and to Breuer et al. (2006, 2012), Breuer and Alletto (2012a), Alletto and Breuer (2012, 2013) and Breuer and Almohammed (2015) for the disperse phase. 3.1. Continuous phase Complex turbulent flows can be reasonably predicted using the large-eddy simulation technique. In the present study the conservation equations of the filtered quantities are solved based on a classical finite-volume method for arbitrary non-orthogonal and block-structured grids. The entire discretization is second-order accurate in space (midpoint rule with linear interpolations) and time (predictor-corrector scheme based on the low-storage Runge-Kutta method). This eddy-resolving simulation strategy requires a model for the non-resolvable subgrid scales. Here, the well-known Smagorinsky model (Smagorinsky, 1963) with a constant of Cs = 0.065 and the van Driest damping near solid walls is applied. Since the near-wall region is properly resolved (see Section 4), the application of the no-slip condition at solid walls is justified. Furthermore, periodic boundary conditions are applied in streamwise and spanwise direction. A constant mass flux is assured by a procedure dynamically adapting the pressure gradient (Benocci and Pinelli, 1990; Breuer, 2002). For the feedback effect of the particles on the fluid (two-way coupling) the particle-source-in-cell method (Crowe et al., 1977) is used. 3.2. Disperse phase In the last decade, several discrete particle models (DPMs) have been developed. They are employed to predict the dynamic behavior of the particulate phase in complex multiphase flows, such as the trajectories of individual particles and the forces acting on the particles and between them. For this purpose, two approaches of DPMs are commonly used: the hard-sphere model and the soft-sphere model often referred to in the literature as Discrete Element Method (DEM). In the present study, the hard-sphere approach (see, e.g., Hoomans et al., 1996; Crowe et al., 1998) is employed. A detailed review on the models for the calculation of the forces required for DEM simulations and their associated theoretical developments can be found in Zhu et al. (2007). For the prediction of the dispersed phase in the Lagrangian frame of reference a huge number of individual point particles is tracked through the continuous flow field (see, e.g., Elghobashi, 1994; Bini and Jones, 2008; Balachandar and Eaton, 2010; Lee and Balachandar, 2010). Based on Newton’s second law and the assumption of heavy particles in a light fluid (i.e., ρp /ρf 1), only the drag, lift, gravity and buoyancy forces have to be considered (Sommerfeld et al., 2008). The fluid forces are derived from the displacement of a small rigid sphere in a non-uniform flow (Gatignol, 1983; Maxey and Riley, 1983). To consider the effect of the subgrid scales on the particle motion, a relatively simple stochastic model proposed by Pozorski and Apte (2009) for homogeneous isotropic turbulence is applied. To account for the rotation of the particles, Newton’s second law for the angular momentum is considered. The torque acting on a rotating particle is determined analytically taking the rotation of the surrounding fluid into account. For wall-bounded flows the change of the translational and angular particle velocity during the wall impact relies on an elastic hard-sphere model (see, e.g., Hoomans et al., 1996; Crowe


4 TEST CASE AND NUMERICAL SETUP

23

et al., 1977, 1998) including friction based on Coulomb’s law (see, e.g., Crowe et al., 1998; Sommerfeld et al., 2008). The model takes into account the momentum loss of the particle during the wall impact by a wall-normal restitution coefficient en,w , a tangential restitution coefficient et,w , a static coefficient of friction µst,w and a kinetic coefficient of friction µkin,w . This model is adequate for smooth walls. For rough walls a recently developed sandgrain roughness model (Breuer et al., 2012) is applied for the dispersed phase purely relying on generally used roughness parameters such as the mean roughness Rz . The ordinary differential equation for the particle motion is integrated by a fourth-order RungeKutta scheme in physical space. To avoid time-consuming search algorithms, the second integration to determine the particle position on the grid is done in the computational space. Here, an explicit relation between the position of the particle and the cell index containing the particle exists (Breuer et al., 2006), which is required to calculate the fluid forces on the particle. Thus, the result is a highly efficient particle tracking scheme allowing to predict the paths of millions of particles, i.e., high mass loadings. The collisions between particles within the four-way coupled simulation are predicted deterministically by a recently developed collision algorithm described in detail in Breuer and Alletto (2012a). Based on the technique of uncoupling the calculation of particle trajectories is split into two stages. In the first stage particles are moved based on the equation of motion without inter-particle interactions. In the second stage the occurrence of collisions during the first stage is examined for all particles. If a collision is found, the velocities of the collision pair are replaced by the post-collision velocities without changing their position. The post-collision velocities are calculated based on a hard-sphere collision model involving friction between the colliding spheres based on Coulomb’s law as described in Section 2.1. Note that the present simulation methodology has been already validated against experiments in a horizontal channel (Breuer et al., 2012), a vertical channel (Breuer and Alletto, 2012a), a vertical pipe (Breuer and Alletto, 2012b), a horizontal pipe (Alletto and Breuer, 2013) and a cold–flow configuration in a model combustion chamber (Alletto and Breuer, 2012; Breuer and Alletto, 2012a). Prior to the main test case described in the next section, the agglomeration models are validated based on a theoretical model. For a simple laminar shear flow theoretical results including the agglomeration rate and the rate of change of the particle number concentration can be obtained under certain simplifications. These are used to validate the predictions of both agglomeration models presented in Appendix B. The motivation for this simplified test case is given by the lack of experimental data for the agglomeration process under realistic conditions such as the turbulent channel flow. Although a variety of measurements can be found in the literature regarding fluid and particle statistics of particle-laden flows including preferential concentration, to the best of the authors’ knowledge no such data are available for the agglomeration process of particles. 4. Test case and numerical setup The test case is the same as used for the recent study on the momentum-based agglomeration model in Breuer and Almohammed (2015), i.e., a particle–laden vertical plane channel flow at a Reynolds number Re = 11,900 (Reτ = 644) based on the bulk velocity UB and the channel half–width δ. The gravitational acceleration gx points in the main flow direction (gx δ/UB = 2.4 × 10−3 ). Both smooth and rough channel walls are considered, whereas the latter are modeled according to Breuer et al. (2012). Np = 6 × 106 fused quartz particles are


4 TEST CASE AND NUMERICAL SETUP

24

injected into the channel. Two different diameters are considered, i.e., dp = 4 µm and 12 µm leading to dp /δ = 2 × 10−4 and 6 × 10−4 , respectively. The ratio of particle to fluid density is ρp /ρf = 1814 yielding mass loadings of η = 0.122% and η = 3.32%, respectively. The particle relaxation time is τp = 1.06 × 10−4 s for the small particles and τp = 9.54 × 10−4 s for the large particles. For the small particles, that corresponds to Stokes numbers of St+ = τp /τf+ = 1.67 and St = τp /τf = 0.048 depending on the chosen fluid time scale, respectively. The viscous time scale is defined by the kinematic viscosity and the wall shear stress velocity as τf+ = νf /u2τ and the convective time scale is given by the ratio of the channel half-width and the bulk velocity as τf = δ/UB . The Stokes numbers for the large particles are St+ = τp /τf+ = 15.04 and St = τp /τf = 0.431. For the agglomeration model the dimensionless Hamaker constant and Young’s modulus are set to H/(ρf UB2 δ 3 ) = 7.92 × 10−17 and E/(ρf UB2 ) = 7.24 × 108 , respectively. The dimensionless maximum contact pressure p/(ρf UB2 ) = 1.622 × 107 . The Poisson’s ratio is ν = 0.17 and the dimensionless separation between two particles during the contact is δ0 /δ = 1 × 10−8 (see Appendix C). The walls are assumed to be either smooth walls made of the same material as the particles (main case) or rough steel walls with a dimensionless mean roughness of Rz /δ = 5 × 10−4 (Rz = 10 µm) applying the sandgrain roughness model (Breuer et al., 2012) with the constant Csurface = 3 modeling the surface finishing. The smooth wall case is the standard setup. The parameters of the particle-particle and particle-wall collision models are listed in Table 1. For two colliding glass beads, the coefficients en,p , et,p and µkin,p were taken from the measurements of Foerster et al. (1994). An often used value of µst,p = 0.94 (Serway and Vuille, 2007) is applied for the static friction coefficient of the material pairing glass-glass. The values of the corresponding coefficients required for the particle-wall collision involving glass beads in a channel confined by rough walls are also listed in Table 1. The selection of these coefficients is motivated in Breuer and Almohammed (2015). Coefficient

Symbol

Unit Particle

Normal restitution Tangential restitution Static friction Kinetic friction Mean roughness

en,x et,x µst,x µkin,x Rz

− − − − µm

0.97 0.44 0.94 0.092 0.0

Smooth wall

Rough wall

0.97 0.44 0.94 0.092 0.0

0.9 0.3 0.5 0.4 10

Table 1: Restitution and friction coefficients for particle-particle collisions (x = p) and particle-wall collisions (x = w) with smooth and rough walls.

Details of the spatial and temporal resolution are given in Table 2. The grid resolution used lies in the range of the requirements for a wall-resolved LES (Piomelli and Chasnov, 1996). Note that prior to the averaging interval provided in Table 2 different time intervals are considered, in which first of all the unladen flow statistically develops (∆T ∗ = 240), then the particles are released (∆T ∗ = 100), and finally the particles disperse in the channel to assure a proper mixing (∆T ∗ = 520). More details are provided in Breuer and Almohammed (2015).


4 TEST CASE AND NUMERICAL SETUP Domain size Grid points Resolution Dimensionless time step Time averaging interval

25

2πδ × 2δ × πδ 128 × 128 × 128 + + ∆x+ = 61.76, ∆yfirst cell center = 0.63, ∆z = 30.88 ∗ ∆t = ∆t UB /δ = 0.005 ∆T ∗ = ∆T UB /δ = 200

Table 2: Numerical details of the test case.

To analyze the simulation data, different statistics are evaluated. For example, for the local distribution of the number of collision and agglomeration events these quantities are accumulated during the simulation time while considering the agglomeration model (i.e., ∆T ∗ = 200) and then averaged in streamwise and spanwise directions: N

N

N

N

i X k X 1 Ncol (x, y, z) , hNcol (y)i = Ni Nk ∆V (y) i=1 k=1

i X k X 1 Nagg (x, y, z) . hNagg (y)i = Ni Nk ∆V (y) i=1 k=1

(86)

Here, hNcol (y)i and hNagg (y)i denote the mean concentration of the number of particle-particle collisions as a function of the wall-normal coordinate and those leading to agglomeration, respectively. Ni and Nk denote the number of control volumes in streamwise and spanwise directions, and ∆V (y) is the volume of the computational cell depending on the wall distance y. This time interval of ∆T ∗ = 200 (about 33 flow-through times) is motivated as follows: Presently no mechanism for break-up of agglomerates is included. Thus, an equilibrium between agglomeration and break-up processes leading to a statistically steady state can not be reached. On the other hand, it is not expected that agglomeration continues until only a few big agglomerates exist. Since the topic of the present study is to study the influence of different models and different model parameters on the agglomeration process, an appropriate alternative is required. As will be discussed in Section 5 after an initial settlement phase the agglomeration rate shows a nearly linear decrease in the interval t∗ = t UB /δ > 100. Owing to the decreasing number of primary particles by agglomerations and the missing break-up model, this behavior has to be expected and describes a continual (settled) process which is stopped after the dimensionless time interval of ∆T ∗ = 200. As will be shown below the accumulated number of collisions Ncol and agglomeration processes Nagg vary almost linearly after ∆T ∗ = 100. Thus, the temporal behavior of these characteristic quantities can be characterized in terms of its dimensionless frequency fφ (Breuer and Almohammed, 2015): fφ =

Nφ (T ∗ = 200) − Nφ (T ∗ = 100) . (T ∗ = 200) − (T ∗ = 100)

(87)

Furthermore, other particle statistics such as the mean velocity or the volume fraction are calculated on-the-fly based on Alletto and Breuer (2013). For this purpose, the following series is updated if a particle is found in a cell whose center is located at the point (x, y, z) denoted as event: 1 1 Nk Nk hφp (x, y, z)i = φp (x, y, z) + 1− hφp (x, y, z)i(Nk −1) , (88) Nk (x, y, z) Nk (x, y, z)


5 RESULTS: COMPARISON OF THE AGGLOMERATION MODELS

26

where φp can be any characteristic value of interest (e.g., mean velocity or velocity fluctuations). The superscript Nk = Nk (x, y, z) denotes the event number defining the accumulated number of particles found in a specific control volume located at the position (x, y, z). The brackets h· · · i denote the mean value of a particle quantity. 5. Results: Comparison of the agglomeration models The objective of this study is to compare the predictions of both agglomeration models against each other for a particulate turbulent flow. The results are organized as follows. In Section 5.1 the numerical results are analyzed without taking into account the feedback effect of the particles on the continuous flow, the lift forces and the subgrid-scale model for the particles. Next, the effect of the normal restitution coefficient of the particle-particle collisions is presented in Section 5.2 for both agglomeration models. In Section 5.3 the performance of both models is examined under the influence of the three sub-models using the standard value of the normal restitution coefficient and smooth channel walls. The performance of both agglomeration models for different particle sizes is studied in Section 5.4 and the effect of rough channel walls in Section 5.5. Due to their importance the three sub-models are accordingly taken into account in the last three subsections. For all steps the most important reasons for the deviations observed between the results of both models are discussed. 5.1. Effect of the agglomeration model In this section, the performance of both the energy-based and the momentum-based agglomeration model is investigated. Here, the standard computational setup for the large particles (dp = 12 µm) explained in Section 4 is applied assuming smooth channel walls and neglecting the sub-models (the feedback effect, the lift forces and the subgrid-scale model for the particles). Fig. 3(a) shows the number of the agglomerated primary particles Npp (i.e., the total number of the primary particles included in all agglomerates independent of their size) as a function of the dimensionless time. It is clearly visible that the energy-based model predicts a higher number of agglomerated primary particles than the momentum-based agglomeration model. The simulation data also reveal that the energy-based agglomeration model predicts a higher number of agglomerates Na than the momentum-based model (not shown for the sake of brevity). The number of agglomerates of the same type are different for both models. Fig. 3(b) depicts the total number of agglomerates Na at the end of the simulation (i.e., ∆T ∗ = 200) as a function of the number of primary particles included in an agglomerate. This diagram shows that the number of two-particle agglomerates is significantly higher for EAM than for MAM, but a lower number of larger agglomerates is observed for EAM. In other words, for the energy-based agglomeration model the tendency that an existing agglomerate agglomerates with another primary particle or another agglomerate building up a larger agglomerate decreases in comparison with the momentum-based model. However, for primary particles the tendency to agglomerate is slightly higher for EAM compared with MAM. The reason for these observations will become clear in the next paragraph. Since the agglomeration conditions of both models given by Eqs. (21) and (75) cannot be directly compared to each other, some assumptions are made to simplify the relations in order to explain the observations. Breuer and Almohammed (2015) found that for the present test case the dominant particle-particle collisions leading to agglomeration are sticking events. Thus, only the first condition in Eq. (75) has to be satisfied. On the other hand, this implies that in a first guess the rotation of the



27

collision partners and the arising agglomerates can be neglected. Hence, for MAM the cohesive force responsible for a successful agglomeration can be written as: 1

∗ fˆn,c ∝

3/5

ρ0 d0

,

(89)

where ρ0 and d0 stand for the density and the diameter of the agglomerating primary particles. Applying the same assumption to the energy-based agglomeration model, the last two terms on the right-hand side of Eq. (21) cancel out, while the remaining terms can be expressed as a function of ρ0 and d0 as follows: 1/2

+ and Ekin,r ∝ ρ0 d30 .

∆EvdW ∝ ρ0 d20

(90)

300000

100000

EAM MAM

250000

EAM MAM

10000

Na

Npp

200000 150000

1000

100000

100

50000

10

0 0

50

100

(2014)

150

1

200

2

(2014)

t UB /δ

3

4

(a)

6

7

8

9

10

(b)

1.6e+07

0.02

EAM MAM

1.4e+07 1.2e+07

EAM MAM Alletto (2014)

0.015

Nagg /Ncol

Nagg × 20, Ncol

5

Npp in an agglomerate

1e+07 8e+06 6e+06

Collision

4e+06

0.005

Agglomeration

2e+06

0.01

0

0 0

50

100

(2014)

t UB /δ (c)

150

200

0

50

100

150

200

t UB /δ (d)

Figure 3: Results of the energy-based (EAM) and the momentum-based (MAM) agglomeration models for dp = 12 µm: (a) time history of the total number of the agglomerated primary particles Npp , (b) number of agglomerates of the same type (two particles, three particles, ..., etc.) at a dimensionless time of 200, (c) time history of the total number of the accumulated particle-particle collisions Ncol and total number of the accumulated particle-particle collisions leading to agglomeration Nagg and (d) time history of the agglomeration rate. (Note that for the purpose of comparison also the result of the defective model by Alletto (2014) is included here.).



28

Eqs. (89) and (90) show that both agglomeration conditions lead to a lower agglomeration probability when the diameter of the agglomerated particles increases, i.e., for further agglomeration processes between agglomerates and primary particles. This indicates that both techniques reproduce the physical behavior of the agglomeration process in a similar manner, but with different rates. The reason for the higher number of two-particle agglomerates predicted by EAM as shown in Fig. 3(b) can be attributed to the different formulations of the agglomeration conditions. Although the trends regarding a variation of the diameter are qualitatively reproduced by both models in a similar manner, that does not automatically mean that the agglomeration rates are identical. Obviously, the condition regrading the difference + of the van-der-Waals energy ∆EvdW in relation to the relative kinetic energy Ekin,r is more often satisfied for two-particle agglomerates than the corresponding agglomeration condition relying on the cohesive force. If the diameter of the agglomerate reaches a certain size, the trend is reversed. Thus, the energy-based agglomeration model predicts lower numbers of large agglomerates than MAM which is the case when a primary particle collides with an existing agglomerate. A more detailed analysis suggests that the increase or reduction of the number of agglomerates of the same size is similar along the channel width. That means that the number of two-particle agglomerates predicted by EAM is higher than for MAM along the entire channel width, whereas the number of larger agglomerates is lower. Fig. 3(c) depicts the total number of accumulated particle-particle collisions Ncol and the total number of accumulated particle-particle collisions leading to agglomeration Nagg computed by both agglomeration models as a function of time. It shows that the energy-based agglomeration model predicts a higher number of inter-particle collisions than the momentum-based model. To further analyze this behavior, the local distribution of the number of collision events and agglomeration processes using both agglomeration models are considered. Fig. 4(a) depicts the distribution of the concentration of the averaged number of inter-particle collisions and agglomeration processes along the channel width at the end of the simulation. It is worth noting that the total values are not of interest, but solely the distribution over the channel width is of relevance. Obviously the highest numbers of particle-particle collisions and agglomeration processes occur in the near-wall region. Breuer and Almohammed (2015) found that this behavior can be attributed to: (i) the high level of turbulence (particle velocity fluctuations are higher than at other locations as shown in Fig. 5 leading to more inter-particle collisions) and (ii) the high particle volume fraction in the region closest to the channel wall depicted in Fig. 4(b) driven by turbophoresis. Furthermore, it is obvious that both agglomeration models predict almost the same number of inter-particle collisions and agglomeration processes in the direct vicinity of the walls, whereas a significant difference is observed in other regions, especially in the central area. To explore this observation, the distribution of the dimensionless mean diameter hdp /δi of all active particles in the channel (i.e., the remaining primary particles and the existing agglomerates) is calculated during the simulation time. As visible in Fig. 4(c), the energy-based agglomeration model results in a larger mean diameter of the particles along the channel width than MAM. Thus, the higher number of particle-particle collisions predicted by EAM is directly related to the larger mean diameter (i.e., the overall higher number of agglomerates), since the probability of inter-particle collisions between primary particles and agglomerates as well as the agglomerates themselves increases. Fig. 4(b) displays the distribution of the mean particle volume fraction as a function of the dimensionless wall coordinate y + = yuτ /ν using both agglomeration models. Here, the distance y to the wall is made dimensionless with the friction velocity uτ /UB = 0.053 of the one-way coupled case and the kinematic viscosity of the fluid. The highest volume fraction is observed



29

in the region close to the channel walls. As expected, the energy-based agglomeration model predicts a marginally higher volume fraction due to the larger mean diameters of the active particles (Fig. 4(c)) along the channel width compared with MAM. 1e+08

0.0009

EAM MAM

1e+07

EAM MAM

0.0006 0.0003

1e+06

hαp i

hNcol i

100000 10000 1000

hNagg i

100

-1 -0.8 -0.6 -0.4 -0.2

0

1.8e-05

0.2 0.4 0.6 0.8

y/δ

1.2e-05

1

1

(a)

100

1000

(b)

6.09

0.4 0.2 0.1

hNagg i / hNcol i

6.08

hdp /δi × 10−4

10

y+

/δ

6.07 6.06 6.05

0.01 0.005

6.04 -1 -0.8 -0.6 -0.4 -0.2

0

0.2 0.4 0.6 0.8

1

-1 -0.8 -0.6 -0.4 -0.2

y/δ (c)

0

0.2 0.4 0.6 0.8

1

y/δ (d)

Figure 4: Results of the energy-based (EAM) and the momentum-based (MAM) agglomeration models for dp = 12 µm: (a) averaged concentration of the number of collision events hNcol i and the agglomeration processes hNagg i, (b) mean particle volume fraction hαp i along the dimensionless wall coordinate y + (the dashed line denotes the global volume fraction of αp = 1.8 × 10−5 ), (c) mean diameter of the active particles along the dimensionless channel width y/δ and (d) averaged agglomeration rate along the dimensionless channel width y/δ. Dimensionless averaging time ∆T ∗ = 200.

Fig. 3(c) shows that the energy-based model predicts a higher number of agglomeration processes than observed for the momentum-based agglomeration model. This observation can be attributed to the stronger tendency of EAM to predict two-particle agglomerates as explained before (see also Table 3). Since the higher number of agglomerates leads to a larger mean diameter of the particles (Fig. 4(c)), further inter-particle collisions take place. Hence, the global agglomeration rate defined as the total number of particle-particle collisions leading to agglomeration to the total number of collisions (i.e., Nagg /Ncol ) is higher for EAM than for MAM depicted as the time history in Fig. 3(d). At ∆T ∗ = 200 the agglomeration rate is about 0.87% for EAM and about 0.73% for MAM. As mentioned before, the momentum-based agglomeration model leads to a higher number of larger agglomerates including more than two



30

particles. However, the number of these large agglomerates is very small compared to the two-particle agglomerates. Fig. 4(a) shows that both models predict almost the same number of agglomeration processes in the near-wall region, while EAM leads to a higher number of agglomeration processes in the range −0.9 < y/δ < 0.9. For the purpose of direct comparison Fig. 3(d) also includes the time history of the agglomeration rate predicted by the original energy-based agglomeration model by Alletto (2014). At ∆T ∗ = 200 the agglomeration rate is about 0.49% and thus about a factor of 2 smaller than the value for EAM. The reason for this deviation (defective third case in the agglomeration model) is already discussed in Section 2.2.5 and has led to the corrections of this approach by modifying the agglomeration conditions resulting in the EAM employed in the present study. Fig. 4(d) depicts the distribution of the averaged agglomeration rate along the channel width at the end of the simulation time. It is obvious that the agglomeration rate predicted by EAM is slightly higher than for MAM in the region near the walls due to more agglomeration processes occurring there with almost the same number of inter-particle collisions as shown in Fig. 4(a). On the other hand, this behavior is reversed in the central area of the channel within the range −0.4 < y/δ < 0.4. Note also that the highest agglomeration rate of both models is located in the center of the channel although the largest number of collisions and agglomeration processes are found close to the walls. This observation can be attributed to the fact that the velocity gradient as well as the velocity fluctuations and thus the velocity difference between two particles moving in a similar distance to the wall is smallest in the center of the channel leading to a weaker impulsive force between the collision partners. Thus, this force can be more often overcome by the cohesive force between two colliding particles leading to agglomeration. The different values of the agglomeration rates in the central region for both models can be explained in a similar manner as before based on the different agglomeration conditions of EAM and MAM. These rely on either the difference of the van-der-Waals energy ∆EvdW given ∗ (Eq. (70)), respectively. The smaller the relative velocity by Eq. (37) or the cohesive force fˆn,c − − of the collision partners urel = u2 − u− 1 , the lower the difference of the van-der-Waals energy − + 2 (∆EvdW ∼ |urel |). However, the relative kinetic energy Ekin,r scales with |u− rel | . On the ∗ other hand, the cohesive force fˆn,c scales with 1/|u− rel | and thus increases with smaller relative velocities as observed in the channel center. Thus, the tendency of an increased agglomeration rate for smaller differences between the velocities of the collision partners in the central region is correctly reproduced by both models. Nevertheless, due to the complex relationships in both agglomeration models a slightly higher probability of satisfying the agglomeration conditions is found for MAM compared with for EAM. Figs. 3(c) and (d) show that the accumulated number of collisions Ncol and agglomeration processes Nagg vary almost linearly after ∆T UB /δ = 100 allowing the definition of the dimensionless frequencies (Eq. (87)). Table 3 shows the frequencies of the particle-particle collisions fcol and the agglomeration processes fagg . Here, fcol and fagg express how many inter-particle collisions and agglomeration processes occur within a dimensionless time unit, respectively. Thus, these frequencies allow a direct evaluation of the influence of different sub-models and are therefore also provided in Table D.6 for all cases investigated in the following subsections. The comparison of both agglomeration models yields a difference of about 4.3% for the collision frequency and about 19.2% for the agglomeration frequency. That is an evidence that the increase of the collision frequency for EAM is a secondary effect due to the increased mean diameter of the particles (Fig. 4(c)).


5 RESULTS: COMPARISON OF THE AGGLOMERATION MODELS Agglomeration model

fcol

fagg

EAM MAM Difference

8.14 × 104 7.79 × 104 4.3%

6.67 × 102 5.39 × 102 19.2%

31

Table 3: Dimensionless frequencies of the particle-particle collisions and the agglomeration processes using different agglomeration models after a dimensionless time ∆T ∗ = 100.

1.2

0.035

1

0.03

u′p u′p /UB2

0.8 0.6 0.4

0.025 0.02 0.015

hup i /UB

Fig. 5 shows the mean particle velocity and the streamwise and wall-normal fluctuations as well as the Reynolds shear stress determined based on Eq. (88) during the dimensionless time interval of ∆T ∗ = 200 and additionally averaged in both homogeneous directions. Obviously, the particle fluctuations predicted by the energy-based agglomeration model are marginally higher than those computed by the momentum-based model.

0.01

0.2

EAM MAM

0.005

0

0 -1

-0.8 -0.6 -0.4 -0.2

0

0.2 0.4 0.6 0.8

1

-1 -0.8 -0.6 -0.4 -0.2

y/δ

0

0.2 0.4 0.6 0.8

1

0.2 0.4 0.6 0.8

1

y/δ

(a)

(b)

0.0025

0.004 0.003

u′p vp′ /UB2

0.002

0.0015 0.001

vp′ vp′ /UB2

0.002

0.001 0 -0.001 -0.002

0.0005

-0.003 0

-0.004 -1 -0.8 -0.6 -0.4 -0.2

0

0.2 0.4 0.6 0.8

1

-1 -0.8 -0.6 -0.4 -0.2

y/δ (c)

0

y/δ (d)

Figure 5: Results of the energy-based (EAM) and the momentum-based (MAM) agglomeration models for dp = 12 µm: (a) mean particle velocity in streamwise direction, (b) averaged streamwise fluctuations, (c) averaged wall-normal fluctuations and (d) averaged Reynolds shear stress of the particles. Dimensionless averaging time ∆T ∗ = 200.

Furthermore, for both agglomeration models the statistics are nearly identical to the case



32

without agglomeration (not shown here). The main reason why these particle statistics are not noticeably influenced by the agglomeration process is the fact that in the present work the agglomeration rates are quite low and the overall number of arising agglomerates is still below 2.2% of the total number of particles for EAM and below 2% for MAM at the end of the simulation. Finally, a note concerning the fluid statistics is given. As already shown in Breuer and Almohammed (2015), the mean velocities and Reynolds stresses of the continuous phase are hardly influenced by the disperse phase. Solely the wall-normal and spanwise stresses are slightly attenuated by the particles. However, the question whether an agglomeration model (either EAM or MAM) is taken into account or not does not play a role. The reason is the same as mentioned for the particle statistics, i.e., the low number of arising agglomerates. 5.2. Effect of the normal restitution coefficient Since the values of the normal restitution coefficient for the particle-particle collisions en,p often strongly scatter in the literature, this section aims at studying the effect of varying en,p on the agglomeration process. Here, both agglomeration models are applied with different values of en,p . The standard computational setup for the large particles (dp = 12 µm) explained in Section 4 is applied using a normal restitution coefficient of en,p = 0.97 (standard case), en,p = 0.80 and en,p = 0.60, respectively. Smooth channel walls are assumed and the value of the normal restitution coefficient for the particle-wall collisions en,w is set to the same value as en,p , since these coefficients are material-dependent. The three sub-models are again not included. Fig. 6 depicts the influence of the normal restitution coefficient on the number of accumulated inter-particle collisions Ncol and agglomeration processes Nagg as well as the corresponding agglomeration rate as a function of time using the energy-based and the momentum-based agglomeration models. At a first glance, Figs. 6(a), (c) and (e) show that for both agglomeration models the reduction of the normal restitution coefficient leads to a higher number of inter-particle collisions and agglomeration processes. However, the rate of the increase of these quantities with decreasing en,p depends not only on the value of the normal restitution coefficient, but also on the agglomeration model due to the different formulations. It is visible that for the chosen values of en,p the energy-based agglomeration model predicts a higher number of particle-particle collisions than MAM. The reason for these observations will become clear in the last paragraph. Figs. 6(b), (d) and (f) show that the lower the normal restitution coefficient, the higher the agglomeration rate using both models. This finding can be clarified based on the impulsive force and the cohesion (i.e., the difference of the van-der Waals energy for EAM and the cohesive force for MAM) between the collision partners. Eq. (5) indicates that the lower the normal restitution coefficient of the particles, the weaker the impulsive force fˆn,a separating the particles. On the other hand, the reduction of en,p leads to a higher difference of the van-der-Waals ∗ energy (Eq. (37)) and a stronger cohesive force fˆn,c due to a larger difference ∆tˆdif between the restitution and the compression intervals (Eq. (70)). Consequently, the reduction of the impulsive force and the simultaneous increase of the difference of the van-der-Waals energy or the cohesive force with decreasing en,p raises the probability of satisfying the agglomeration conditions, and hence yields more agglomeration processes. Thus, the agglomeration rate increases with decreasing en,p as clearly visible in Figs. 6(b), (d) and (f).


EAM MAM

1.8e+07 1.5e+07 1.2e+07

Collision

6e+06

EAM MAM

0.12

9e+06

0.1 0.08 0.06 0.04

Agglomeration

3e+06

0.02

0

0 0

50

100

150

200

t UB /δ

i

0

50

150

200

150

200

150

200

(b) 0.14

2.1e+07

EAM MAM

1.8e+07 1.5e+07 1.2e+07 9e+06

Collision

6e+06

EAM MAM

0.12

Nagg /Ncol

Nagg × 5, Ncol

100

t UB /δ

i

(a)

0.1 0.08 0.06 0.04

Agglomeration

3e+06

0.02

0

0 0

50

100

150

200

t UB /δ

i

0

50

100

t UB /δ

i

(c)

(d) 0.14

2.1e+07

EAM MAM

1.8e+07 1.5e+07 1.2e+07 9e+06

Collision

Agglomeration

6e+06

EAM MAM

0.12

Nagg /Ncol

Nagg × 5, Ncol

33

0.14

2.1e+07

Nagg /Ncol

Nagg × 5, Ncol


0.1 0.08 0.06 0.04 0.02

3e+06 0

0 0

50

100

t UB /δ

i (e)

150

200

0

i

50

100

t UB /δ (f )

Figure 6: Effect of the normal restitution coefficient en,p on the time history of the accumulated particleparticle collisions Ncol , the total number of the accumulated particle-particle collisions leading to agglomeration Nagg and the agglomeration rate using the energy-based (EAM) and the momentum-based (MAM) agglomeration models for dp = 12 µm: (a) & (b) en,p = 0.97, (c) & (d) en,p = 0.80 and (e) & (f) en,p = 0.60.

As observed in Section 5.1, the energy-based model predicts higher agglomeration rates than the momentum-based model for en,p = 0.97 (Fig. 6(b)). Astonishingly, this trend is reversed when reducing the normal restitution coefficient to en,p = 0.80 and en,p = 0.60. This behavior



34

can be explained by the lower rate of change of the difference of the van-der-Waals energy ∗ ∆EvdW ∼ (1 − e2n,p )1/2 compared with the cohesive force fˆn,c ∼ ∆tˆdif as shown in Table 4. Here, the values of these proportionality factors are listed for the normal restitution coefficients chosen in the present study. Furthermore, their rates of change in relation to the standard case (en,p = 0.97) are given. Based on these values it is clear that the reduction of en,p significantly increases the cohesive force compared to the difference of the van-der-Waals energy leading to a noticeably higher number of agglomeration processes predicted by the momentum-based model with regard to the energy-based model. This issue is also clearly visible based on the agglomeration frequencies fagg given for both models in Table 4, too. en,p

(1 − e2n,p )1/2

∆tˆdif

0.97 0.80 0.60

0.24 0.60 0.80

0.0215 0.1617 0.3880

0.80/0.97 0.60/0.97

2.5 3.3

7.52 18.1

1 − (1 −

p

1 − e2n,p )1/2

0.13 0.37 0.55

2.85 4.23

fagg (EAM) fagg (MAM) 6.67 × 102 21.5 × 102 39.8 × 102 3.22 5.97

5.39 × 102 26.2 × 102 52.9 × 102 4.86 9.81

∗ Table 4: Rates of change of the proportionality factors of ∆EvdW , fˆn,c , fâg,n and the agglomeration frequency for both models as a function of the normal restitution coefficient en,p . (All quantities are dimensionless).

In addition to the findings mentioned above, the simulation data reveal that for en,p = 0.60 a similar tendency of the energy-based model as found for en,p = 0.97 (see Section 5.1) to predict a higher number of two-particle agglomerates and a lower number of larger agglomerates than MAM (not shown for the sake of brevity) is observed. For en,p = 0.80 the number of the two-particle agglomerate is nearly identical for both models. Furthermore, for en,p = 0.80 and en,p = 0.60 the higher number of agglomeration processes predicted by MAM (Figs. 6(d) and (f)) indicates that for this model the probability of a successful agglomeration between an existing agglomerate and a primary particle or another agglomerate building up a larger agglomerate increases in comparison with EAM. In summary, the momentum-based model predicts a higher number and larger sizes of the agglomerates than EAM. Looking again at Figs. 6 (c) and (e), the increase of the number of inter-particle collisions in comparison with the standard case (en,p = 0.97) is attributed to the higher number and larger sizes of the agglomerates predicted by both models for a decreasing restitution coefficient. Note that the number of inter-particle collisions computed by MAM is lower than for EAM despite the higher number and larger sizes of the agglomerates predicted by MAM. To explain this behavior, the magnitudes of the cohesive forces between the particles in the ∗ normal direction fâg,n (EAM: Eq. (52)) and fˆn,c (MAM: Eq. (70)) are compared in terms of their proportionality factors. To simplify the relationship of fâg,n , a constant relative kinetic + energy after the collision without taking the van-der-Waals forces into account Ekin,r,n is asp 1/2 and hence sumed. Thus, based on Eq. (55) the factor kn,p scales with (1 − 1 − e2n,p ) p ∗ scales with ∆tˆdif . The correspondfâg,n ∼ 1 − (1 − 1 − e2n,p )1/2 . As mentioned before, fˆn,c ing values of these proportionality factors and their rates of change in relation to the standard case are given in Table 4. It is obvious that the reduction of the restitution coefficient leads to a stronger increase of the cohesive force in the framework of MAM than found for EAM. That means that the momentum loss during the collision increases with decreasing en,p more



35

strongly for MAM than for EAM leading to a lower probability of further collisions with neighboring particles or agglomerates, since the collision partners have a smaller impulsive force separating them from each other. Consequently, for en,p = 0.80 and en,p = 0.60 lower numbers of particle-particle collisions are predicted by the momentum-based model than by the energy-based model as clearly visible in Figs. 6 (c) and (e). 5.3. Effect of the inclusion of the sub-models In order to reduce the number of influencing parameters, the investigations in the previous sections were carried out without taking the feedback effect of the particles on the fluid (twoway coupling), the lift forces and the subgrid-scale model for the particles (i.e., the effect of the unresolved fluid scales within the framework of LES) into account. In the context of the momentum-based model Breuer and Almohammed (2015) found that the cumulative effect of these sub-models significantly reduces the agglomeration rate. In their study, they concluded: • The feedback of the particles on the fluid leads to a lower number of inter-particle collisions, while the number of agglomeration processes is hardly affected yielding a slightly higher agglomeration rate. • Due to the subgrid-scale model for the particles the particle velocity fluctuations increase leading to a lower number of agglomeration processes. Furthermore, a lower particle concentration is predicted in the direct vicinity of the wall resulting in a lower number of inter-particle collisions. The global agglomeration rate is perceptibly reduced. • The inclusion of the lift forces reduces the global agglomeration rate noticeably, since they enhance the migration of primary particles and agglomerates away from the region close to the walls, where the highest number of inter-particle collisions and agglomerations occurs. Thus, this step aims at extending these investigations towards the energy-based model and the comparison of both agglomeration models when these sub-models are considered. Again, the standard computational setup for the large particles (dp = 12 µm) explained in Section 4 is applied assuming smooth channel walls. At a first glance Fig. 7 shows that both techniques predict similar trends of the agglomeration process as observed in Section 5.1, but with different rates when the sub-models are included (Note the different scaling of the axes in Figs. 3 and 7). Fig. 7(a) reveals that the cumulative effect of the sub-models reduces the number of accumulated particle-particle collisions and agglomeration possesses compared with the case without the sub-models (Fig. 3(c)). This behavior can be explained based on Fig. 7(c). If the sub-models are considered, the volume fraction is significantly reduced in the direct vicinity of the channel walls and slightly increased in the central region due to the migration of the particles out of the near-wall region, where the highest numbers of inter-particle collisions and agglomeration processes occur. Fig. 7(d) depicts the distribution of the concentration of the averaged number of inter-particle collisions and agglomeration processes along the channel width at the end of the simulation. It is obvious that the number of inter-particle collisions and hence agglomeration processes is noticeably reduced when the three sub-models are taken into account. That holds for both agglomeration models. Besides the explanations mentioned above, a further reason for this observation is the reduction of the particle fluctuations, especially in the wall-normal and the spanwise directions, compared with the case without the sub-models (not shown here for the sake of brevity). Note



36

that the level of reduction of the particle fluctuations is not constant along the channel width, but rather a higher level of reduction is observed in the near-wall region than in the central region. In summary, if the three sub-models are taken into account, both agglomeration models predict lower numbers of inter-particle collisions and agglomeration processes, but with different rates. Nevertheless, Fig. 7(d) shows in agreement with Fig. 4(d) that the energy-based model predicts a lower concentration of agglomeration processes in the near-wall region than MAM. Furthermore, a similar tendency as found in Section 5.1 is observed for the central region. Here, EAM predicts a higher concentration of agglomeration processes than MAM. A possible reason for this behavior is the different level of reduction of the particle fluctuations observed along the channel width as will be explained more deeply in the next paragraph. 1.2e+07

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Figure 7: Effect of the inclusion of the feedback of the particles on the fluid (two-way coupling), the subgridscale model for the particles and the lift forces on the results of the energy-based (EAM) and the momentumbased (MAM) agglomeration models for dp = 12 µm: (a) time history of the total number of the accumulated particle-particle collisions Ncol and total number of the accumulated particle-particle collisions leading to agglomeration Nagg , (b) time history of the agglomeration rate, (c) mean particle volume fraction hαp i along the dimensionless wall coordinate y + (the dashed blue line denotes the global volume fraction of αp = 1.8 × 10−5 ), and (d) averaged concentration of the number of collision events hNcol i and agglomeration processes hNagg i. The dashed lines stand for the case without sub-models. (Dimensionless averaging time ∆T ∗ = 200).



37

The lower number of agglomeration processes predicted by both models leads to a lower number of agglomerated primary particles Npp as depicted in Fig. 8(a). Consequently, also the number of agglomerates Na predicted by both models decreases when the sub-models are taken into account. In contrast to the results presented in the preceding sections, for the predictions including the sub-models the energy-based model delivers a higher number not only of the two-particle agglomerates but also of the three-particle agglomerates (Fig. 8(b)) than observed for MAM. This can be clarified based on the discussion presented in Section 5.1 regarding the relationships between the magnitude of the relative velocity between the collision part+ ners |u− rel | and the relative kinetic energy Ekin,r , the difference of the van-der-Waals energy ∗ . Lower particle fluctuations reduce the relative velocities ∆EvdW or the cohesive force fˆn,c + between the collision partners. Based on Eqs. (35) and (37) the energies Ekin,r and ∆EvdW + also decrease, where Ekin,r is more quickly reduced than ∆EvdW . Furthermore, according to ∗ increases. Thus, for both models the reduction of the particle fluctuations raises Eq. (70) fˆn,c the probability of satisfying the agglomeration conditions. However, a general statement why the changes are different for both agglomeration models cannot be made due to the complex formulation of the agglomeration conditions. As mentioned before, the level of reduction of the particle fluctuations due to the sub-models is different along the channel width and hence two regions have to be distinguished. This is visible in Fig. 7(d) showing a lower number of agglomeration processes in the near-wall region predicted by the energy-based model than for MAM. Due to the strong reduction of the particle fluctuations in the near-wall region, a lower probability of satisfying the agglomeration conditions is observed for EAM than for MAM. Thus, near the channel walls the momentum-based model predicts a higher concentration of agglomeration processes than the energy-based model (Fig. 7(d)). In the central region, the weaker reduction of the particle fluctuations raises the probability of satisfying the agglomeration conditions for EAM in a stronger manner than for MAM. Thus, the energy-based model predicts a higher concentration of agglomeration processes in the central region than the momentum-based model. 80000

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38

Overall, the simulation data reveal that along the channel width the number of two-particle agglomerates predicted by the energy-based model is higher than for the momentum-based model (not shown here). Furthermore, the number of three-particle agglomerates computed by EAM is higher in the central region and slightly lower near the walls than for MAM, so that the total number of three-particle agglomerates is higher for EAM than for MAM. The reason for this observation is the higher number of two-particle agglomerates predicted by EAM accompanied by a weaker reduction of the particle fluctuations leading to a higher probability of satisfying the agglomeration conditions of EAM in the central region, if a twoparticle agglomerate collides with a primary particle building up a three-particle agglomerate. On the other hand, the high number of agglomeration processes predicted by MAM in the nearwall region and the lower number of three-particle agglomerates compared with EAM indicate that due to the higher cohesive force a higher probability of satisfying the agglomeration conditions exists for MAM when two-particle agglomerates collide with each other or threeparticle agglomerates collide with primary particles leading to a higher number of four-particle agglomerates (Fig. 8(b)). At the end of the simulation including the three sub-models, the agglomeration rates are about 0.31% for the energy-based model and about 0.23% for the momentum-based model (Fig. 7(b)). The corresponding dimensionless frequency of the agglomeration process fagg decreases significantly from about 6.67 × 102 to about 1.72 × 102 for EAM and from about 5.39 × 102 to about 1.29 × 102 for MAM (see Table D.6). Thus, it can be concluded that the cumulative effect of the three sub-models is not negligible and should be considered for reliable predictions of the agglomeration process. Hence, it will be taken into account in the next simulations. 5.4. Effect of the diameter of the primary particles This step is motivated by the fact that in practical applications of turbulent particle-laden flows the primary particles generally have different sizes. Thus, it is meaningful to investigate the performance of the agglomeration models for different diameters of the primary particles. The results presented in the preceding sections show a noticeable difference between the agglomeration rates predicted by both agglomeration models for dp = 12 µm. To investigate the effect of the size of the primary particles and hence the mass loading, the same number of particles is used, but the diameter is reduced from 12 µm to 4 µm. Thus, the corresponding mass loading decreases from 3.322% to 0.123%, respectively. Here, the standard computational setup explained in Section 4 is applied assuming smooth channel walls. Furthermore, the three sub-models (i.e., the feedback effect of the particles on the continuous phase, the subgrid-scale model and the lift forces) are taken into account due to their significant effect as concluded in Section 5.3. In comparison to Figs. 7 and 8 (dp = 12 µm) Fig. 9 shows a similar tendency of both models regarding the abatement of the agglomeration processes when the diameter of the particles is reduced from 12 µm to 4 µm. However, different rates of attenuation are found. Fig. 9(a) displays the number of the agglomerated primary particles Npp as a function of time. This diagram shows that the energy-based model predicts a higher number of agglomerated primary particles than the momentum-based model, and hence a higher number of agglomerates Na (Fig. 9(b)). Obviously, at the end of the simulation the number of two-particle agglomerates predicted by EAM is noticeably higher than for MAM. On the other hand, the energy-based model predicts a lower number of larger agglomerates than the momentum-based model (Here, the number of three particle-agglomerate is one for EAM and five for MAM). Note that these



39

observations are consistent with those presented in Sections 5.1 and 5.3. In summary, this means that due to the different formulations of the agglomeration conditions of both models a successful agglomeration process using the energy-based model more often occurs for twoparticle agglomerates than for the momentum-based model. If the diameter of the agglomerate exceeds a certain size (e.g., a primary particle collides with an existing agglomerate), the trend is reversed. 10000

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Figure 9: Effect of the diameter of the primary particles on the results of the energy-based (EAM) and the momentum-based (MAM) agglomeration models for dp = 4 µm: (a) time history of the total number of the agglomerated primary particles Npp , (b) number of agglomerates of the same type (two particles, three particles, ..., etc.) at a dimensionless time of 200, (c) time history of the total number of the accumulated particle-particle collisions Ncol and total number of the accumulated particle-particle collisions leading to agglomeration Nagg and (d) time history of the agglomeration rate.

Fig. 9(c) depicts the total number of accumulated particle-particle collisions Ncol and the total number of accumulated particle-particle collisions leading to agglomeration Nagg computed by both agglomeration models as a function of time. It shows that the energy-based agglomeration model predicts a slightly higher number of inter-particle collisions than the momentum-based model. This behavior is due to the higher number of two-particle agglomerates computed by EAM (Fig. 9(b)). It is worth noting that the reduction of the diameter of the primary particles implies larger inter-particle distances between the particles. Consequently, when comparing



40

Fig. 3(a) and Fig. 9(a), the total number of the inter-particle collisions is about two orders of magnitude lower for the small particles in relation to the large particles. Correspondingly, the total number of agglomeration processes is significantly reduced. Fig. 9(c) also shows that although the number of accumulated inter-particle collisions only slightly deviates, the number of agglomeration processes predicted by EAM is noticeably higher than for MAM. This leads to a higher agglomeration rate computed by the energy-based model than observed for the momentum-based model as depicted in Fig. 9(d). At the end of the simulation, the agglomeration rate is about 1.74% for the energy-based model and about 1.1% for the momentum-based model. Note that the corresponding values for the large particles (dp = 12 µm) are 0.31% and 0.23% for EAM and MAM, respectively. Hence, as expected, the agglomeration rates predicted by both agglomeration models increase when the diameter is reduced. In other words, the reduction of the diameter of the primary particles leads to a higher probability of satisfying the agglomeration conditions of both models. This observation can be explained based on Eq. (90) for the energy-based model and based + on Eq. (89) for the momentum-based model. Note that for EAM the kinetic energy Ekin,r decreases more quickly with a decreasing diameter of the particle than the difference of the van-der-Waals energy ∆EvdW . Furthermore, in the momentum-based model the cohesive force ∗ fˆn,c is inversely proportional to the diameter of the primary particles and thus increases for dp = 4 µm in comparison with dp = 12 µm. The corresponding dimensionless frequency of the agglomeration process fagg for the small particles (dp = 4 µm) is about 2.1 × 101 for EAM and about 1.36 × 101 for MAM, whereas for the large particle (dp = 12 µm) it is about 1.72 × 102 for EAM and about 1.29 × 102 for MAM (see Table D.6). Finally, it is worth noting that the increase of the agglomeration rate and the decrease of the agglomeration frequency observed for the reduction of the particle size from dp = 12 µm to dp = 4 µm is not a contradiction, but instead complementary. 5.5. Effect of the wall roughness model Since the walls of industrial apparatuses are seldom ideally smooth as assumed in the previous sections, the effect of the wall roughness on the agglomeration process is investigated using both agglomeration models. Therefore, rough walls with a dimensionless mean roughness Rz = 10 µm as appearing in practice are assumed (see Section 4). The corresponding restitution and friction coefficients are listed in Table 1. The diameter of the primary particles is again set to dp = 12 µm. Note that in comparison to Section 5.3 the properties of the particles are not changed, and hence solely the wall model leads to differences in the results. In order to allow the particles and the fluid phase to adjust to the new situation before considering the agglomeration model, the coefficients listed in Table 1 are used during the dispersion period of the particles. Additionally, the three sub-models are taken into account due to their significant effect as concluded in Section 5.3. According to the wall roughness model by Breuer et al. (2012), the wall-normal unit vector at the contact point of a particle colliding with the wall strongly varies, since its orientation is adjusted leading to a Gaussian distributed random wall-normal unit vector. Thus, after the collision the particle trajectories spread largely. In comparison to the smooth wall depicted in Fig. 7(a), Fig. 10(a) shows that the accumulated number of particle-particle collisions predicted by both models is significantly reduced when the wall roughness is considered. However, the accumulated number of successful agglomeration processes is only slightly affected by the wall roughness model. In order to explain the reasons for the reduction of the number of inter-particle collisions, it is worth noting that the simulation data reveal that for both models the inclusion of the wall roughness enhances the particle



41

fluctuations near the walls in the streamwise, wall-normal and spanwise directions (not shown for the sake of brevity). Furthermore, the particle fluctuations predicted by the energy basedmodel are slightly higher than for the momentum-based model, especially close to the walls. For a detailed explanation for the reasons of the increase of the particle fluctuations the reader is referred to Breuer et al. (2012). 8e+06

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Figure 10: Effect of the wall roughness on the results of the energy-based (EAM) and the momentum-based (MAM) agglomeration models for dp = 12 µm: (a) time history of the total number of the accumulated particleparticle collisions Ncol and total number of the accumulated particle-particle collisions leading to agglomeration Nagg , (b) time history of the agglomeration rate, (c) mean particle volume fraction hαp i along the dimensionless wall coordinate y + , and (d) averaged concentration of the number of collision events hNcol i and agglomeration processes hNagg i close to the walls. The dashed lines stand for the case with smooth walls (Dimensionless averaging time ∆T ∗ = 200).

Since the highest number of inter-particle collisions and those leading to agglomeration occur in the region close to the wall (y + < 10), it is expected that higher particle fluctuations due to the wall roughness enhance the total number of inter-particle collisions in this region. However, a lower number of inter-particle collisions is observed in the near-wall region as visible in Fig. 10(d). This behavior is a direct consequence of the wall roughness, since it considerably alters the rebound behavior of the particles at the wall. Thus, the particles migrate away from the wall leading to a lower volume fraction in the direct vicinity of the wall as clearly visible in



42

Fig. 10(c). Hence, a lower number of inter-particle collisions is found in the near-wall region. A second but less important reason for the reduction of the inter-particle collisions is given by the different values of the restitution and friction coefficients for the particle-wall collisions for smooth and rough walls as given in Table 1. The corresponding dimensionless frequency of the particle-particle collisions fcol decreases appreciably from about 5.85 × 104 (smooth wall) to about 3.91 × 104 (rough wall) for EAM and from about 5.83 × 104 (smooth wall) to about 3.88 × 104 (rough wall) for MAM. This again confirms the significant reduction of the number of the inter-particle collisions when the wall roughness is taken into account. However, a very slight deviation between the dimensionless frequencies of the particle-particle collisions predicted by both models is observed implying almost the same number of the accumulated inter-particle collisions visible in Fig. 10(a). On the other hand, if the wall roughness model is considered, the dimensionless frequency of the agglomeration processes fagg increases from about 1.72 × 102 to about 1.92 × 102 for EAM and from about 1.29 × 102 to about 1.39 × 102 for MAM (see Table D.6). Accordingly, Fig. 10(a) shows that the number of agglomeration processes predicted by the energy-based model is noticeably higher than for MAM. In summary, the inclusion of the wall roughness results in a significant reduction of the number of particle-particle collisions and a slight increase of the agglomeration processes leading to higher agglomeration rates predicted by both models. Fig. 10(b) depicts that for the case with rough walls the energy-based model predicts a higher agglomeration rate than the momentum-based model. At the end of the simulation, the agglomeration rate is about 0.51% for the energy-based model and about 0.36% for the momentum-based model. Note that the corresponding values for the smooth wall are 0.31% and 0.23%, respectively. Fig. 10(d) depicts the local distribution of the concentration of inter-particle collisions and agglomeration processes along the channel width in the near-wall region averaged over the time when the agglomeration model is taken account. Obviously, in comparison to the case with smooth walls the concentrations of the inter-particle collisions and hence also the agglomeration processes are reduced near the walls due to the migration of the particles away from the walls as mentioned before. On the other hand, the migration of the particles increases the volume fraction in the central region as shown in Fig. 10(c). This leads to a slightly higher number of particle-particle collisions and agglomeration processes outside the near-wall region for the rough wall compared to the smooth wall (Fig. 10(d)). However, as explained before a lower number of accumulated inter-particle collisions is observed for the rough wall in comparison to the smooth wall. That implies that the reduction of the number of inter-particle collisions in the region near the walls overwhelms the slight increase in the central region of the channel. Fig. 10(d) also shows that in the near-wall region the momentum-based model predicts a sightly higher concentration of the agglomeration processes than the energy-based model, while this behavior is reversed outside this area (not shown here). To explain this difference, it should be again recalled that the energy-based model predicts slightly higher particle fluctuations near the walls than MAM. Thus, the observation concerning the different distributions of the agglomeration processes for both models can be explained based on the discussion presented in Section 5.3. There, a similar behavior was observed for the case with smooth walls (Fig. 7(d)), but of course with different rates owing to the roughness effect. On average, the number of agglomeration processes predicted by both models for rough walls is higher than for smooth walls. For both types of walls the energy-based model predicts a higher number of agglomeration processes than MAM as stated before. The higher number of agglomeration processes predicted by the energy-based model leads to


6 CONCLUSIONS

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a higher number of agglomerated primary particles Npp as depicted in Fig. 11(a). Hence, the energy-based model predicts a higher number of agglomerates Na than MAM as displayed in Fig. 11(b). In contrast to the case with smooth walls (Fig. 8(b)), both models predict almost the same number of three-particle agglomerates. Furthermore, the energy-based model delivers a higher number of four-particle agglomerates than observed for the momentum-based model (Fig. 11(b)). Looking again at Fig. 10(c), it is obvious that although the total number and the sizes of the agglomerates predicted by EAM are larger than for MAM, the volume fraction using EAM is marginally lower near the walls than for MAM. In the near-wall region this observation is attributed to the lower number of agglomeration processes and hence agglomerates predicted by EAM in comparison with MAM, while the opposite behavior is observed outside this area. On average, the probability of satisfying the agglomeration conditions for EAM is higher than for MAM if three-particle agglomerates collide with primary particles building up four-particle agglomerates (Fig. 11(b)). Thus, a slightly higher number of four-particle agglomerates is predicted by the energy-based model in comparison with the momentum-based model. In other words, for the energy-based model the higher number of inter-particle collisions in the central region of the channel due to the roughness effect enhances the agglomeration rate more significantly than the reduction of the number of agglomeration processes in the near-wall region. For MAM the trend is similar but less pronounced. 80000

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6. Conclusions The present work focuses on the comparison of two different strategies employed for modeling the agglomeration of cohesive particles in the framework of a hard-sphere model with a deterministic collision detection. Here, the agglomeration of rigid, dry and electrostatically neutral particles in a turbulent flow is investigated using energy-based and momentum-based agglomeration models. In the present study, the energy-based model by Alletto (2014) is corrected concerning the agglomeration conditions leading to realistic agglomeration rates fitting quite well to the results of the momentum-based model (Breuer and Almohammed, 2015).


6 CONCLUSIONS

44

Furthermore, the application area is extended towards fully three-dimensional turbulent flows applying the large-eddy simulation. To explore the effects of the different concepts (i.e., the energy-based versus the momentumbased agglomeration model) on the dynamic process of agglomeration, a comparative study of both models with application to a turbulent particle-laden flow in a downward directed vertical channel flow is carried out. The influence of various simulation parameters is investigated. The simulation data reveal that both agglomeration models predict similar trends of the physical behavior of the agglomeration process. However, their results deviate slightly from each other. Possible reasons for the differences between the results of both models can be mainly attributed to the different formulation of the agglomeration conditions. The predictions provide evidence of the following findings: • The energy-based model predicts a higher number of inter-particle collisions, agglomeration processes, agglomerated particles and two-particle agglomerates than the momentumbased model. However, the number and the size of larger agglomerates depend on the sub-models considered in the simulation. For example, in the standard case (i.e., without the three sub-models and smooth walls) the energy-based model predicts a lower number of agglomerates including more than two particles. Furthermore, the energy-based model predicts a higher agglomeration rate than the momentum-based model. At the end of the simulation the total number of arising agglomerates in relation to the total number of particles is still below 2.2% for EAM and below 2% for MAM. Thus, the particle statistics of both models are hardly influenced by the agglomeration process due to relatively low agglomeration rates, and hence they are nearly identical to the case without agglomeration. • For both agglomeration models the reduction of the normal restitution coefficient of the particles en,p leads to a higher number of inter-particle collisions and agglomeration processes. Here, it is worth noting that the rate of increase of these quantities depends not only on the value of the normal restitution coefficient, but also on the agglomeration model. The reason is the different formulation of the agglomeration conditions. The increase of the inter-particle collisions observed for a decreasing restitution coefficient is due to the higher number and larger sizes of the agglomerates predicted by both models. For the chosen values of the normal restitution coefficients (i.e., en,p = 0.97, 0.80 and 0.60) the energy-based model predicts a higher number of particle-particle collisions than MAM. The increase of the number of agglomeration processes is attributed to the fact that the reduction of en,p yields a weaker impulsive force fˆn,a separating the collision partners. Furthermore, for a decreasing value of en,p a larger difference of the van-der∗ Waals energy ∆EvdW and a stronger cohesive force fˆn,c between the colliding particles occur. Hence, the agglomeration rate increases for both models. However, the reduction ∗ of en,p increases fˆn,c more significantly than ∆EvdW . Consequently, different behaviors are observed for both models, where the increase of the agglomeration rate with decreasing en,p is more pronounced for MAM than for EAM. As a direct consequence, for en,p = 0.97 the energy-based model predicts a higher agglomeration rate than the momentum-based model, while the opposite behavior is found for en,p = 0.80 and 0.60. • If the feedback effect of the particles on the continuous phase, the lift forces and the subgrid-scale model for the particles are taken into account, both techniques predict similar trends in comparison with the case without the three sub-models, but with different


6 CONCLUSIONS

45

rates. Using both models, the cumulative effect of the sub-models leads to a significantly lower number of inter-particle collisions and agglomeration processes. The reduction of the number of collisions and hence the agglomeration processes is attributed to the lower volume fraction in the direct vicinity of the wall due to the enhanced migration of the particles from the near-wall region towards the channel center. Thus, the inclusion of the three sub-models leads to lower agglomeration rates predicted by both models. The energy-based model still yields a higher agglomeration rate than the momentum-based model. • Using the same number of primary particles with a diameter of dp = 4 µm, a similar tendency of both models regarding the prediction of the agglomeration processes is found as observed for the case dp = 12 µm, but with different rates of attenuation. The reduction of the diameter of the primary particles by a factor of three significantly reduces the number of inter-particle collisions and hence those leading to agglomeration. The lower number of collisions is due to larger inter-particle distances between the particles when decreasing the diameter of the particles. In the present study, the number of particle-particle collisions is about two orders of magnitude lower for dp = 4 µm than for dp = 12 µm. Nevertheless, the agglomeration rates are larger than for dp = 12 µm. This can be attributed to the fact that the cohesive force inversely scales with the diameter of the particle (i.e., fˆn,c ∼ 1/dp ), and hence it is appropriately higher for the small particles than for the large particles. Hence, despite the lower number of inter-particle collisions for the small particles in comparison with the large particles the probability of satisfying the agglomeration conditions is higher for dp = 4 µm than for dp = 12 µm. As a results of the different formulations of the agglomeration conditions, the energy-based model predicts a higher number of agglomeration processes than MAM and thus a higher agglomeration rate. • For both agglomeration models the inclusion of the wall roughness significantly reduces the number of particle-particle collisions and slightly increases the number of agglomeration processes leading to higher agglomeration rates. The significant reduction of the inter-particle collisions due to the roughness effect is attributed to the lower volume fraction in the near-wall region, since particles migrate away from the wall. For the case with rough walls the energy-based model again predicts a higher agglomeration rate than the momentum-based model. • The simulation results show that for the present test case almost the same computational time is required for both agglomeration models. Furthermore, the agglomeration routine including the collision handling requires about 6% of the total computational time. Advantages and drawbacks of the agglomeration models To determine the difference of the van-der-Waals energy and the impact time required for the agglomeration conditions of the models, a head-on collision is assumed. However, the formulation of the agglomeration conditions and the treatment of the collision partners after an impact taking the cohesion into account are different. Therefore, slightly different results are observed. A common advantage of both agglomeration model is that they take the influence of the normal restitution coefficient into account. In summary, the energy-based model and the momentum-based model have the following advantages and drawbacks.


6 CONCLUSIONS

46

• Advantage of EAM: – Independent of the collision type, the rotation of the colliding particles is taken into account. • Drawbacks of EAM: – In the calculation of the energy dissipation during the collision process the effect of the van-der-Waals force between the collision partners is not taken into account. – In the first step of the calculation, the attractive force is not considered for the determination of the collision type. – The maximum contact pressure p has to be determined based on the compressive strength of the material of the colliding particle, which may not be available in the literature. – An exact calculation of the kinetic energy of the agglomerate is required for the agglomeration condition. For this purpose, a system of three equations with three unknowns has to be solved. – Some rather crude assumptions are made for the kinetics of the collision partners without agglomeration, since it is assumed that they separate either in the normal or the tangential direction. Furthermore, the influence of the van-der-Waals force on the angular velocities of the collision partners after the impact is neglected although it increases the friction at the contact point during the collision. • Advantages of MAM: – The cohesive force is taken into account for the determination of the collision type. – The material parameters E and ν of the particles required for the calculation of the cohesive force are often available in the literature. – The treatment of the collision partners after impact without a successful agglomeration considers the effect of the cohesive force, and hence both components of the impulse (normal and tangential) are realistically calculated. • Drawback of MAM: – For the determination of the impact time the effect of the cohesive force is not taken into account leading to an underestimation of the collision duration and hence the cohesive force. Based on the above mentioned advantages and drawbacks of both agglomeration models and the validation study presented in Appendix B, it can be concluded that owing to the reduced necessity of empirical parameters and the slightly more accurate results, the momentum-based agglomeration model is superior to the energy-based model. As an outlook for future studies employing both agglomeration models, a break-up model of the agglomerates has to be taken into account to allow the application of the agglomeration models to disperse multiphase flow systems of industrial interest. Furthermore, detailed experiments are required to validate the predictions of both techniques.


APPENDIX A ONSET OF PLASTIC YIELD

47

Acknowledgments The project is financially supported by the Deutsche Forschungsgemeinschaft under the contract numbers BR 1847/13-1. All kinds of support are gratefully acknowledged. Appendix A. Onset of plastic yield Any solid material has an upper limit of elastic deformation under normal or tangential stresses. Once the stresses exceed this limit, plastic deformation occurs. It is known that the Hertzian pressure distribution has the following form (Johnson, 1985; Popov, 2010): 1/2 r2 p(r) = p 1 − 2 , (A.1) a where p stands for the maximum contact pressure and a is the radius of the contact area. For this pressure distribution Popov (2010) stated that the components of the shear stress vanish and the principal axes coincide with the coordinate axes for all points along the z-axis (direction of movement). The analytical solution for the components of the stress tensors reads: −1 h a i 1 σxx σyy z z2 = = −(1 + ν) 1 − arctan + 1+ 2 p p a z 2 a 2 −1 z σzz =− 1+ 2 p a

(A.2)

where σxx = σ1 , σyy = σ2 and σzz = σ3 denote the principal stresses, ν is the Poisson’s ratio and z stands for the depth inside the sphere along the axis of symmetry. According to the criterion of Tresca (1869), the maximum shear stress at which the yield occurs is given by: τy =

1 |σ1 − σ3 | 2

(A.3)

For ν = 0.33 the maximum shear stress occurs at a depth of z/a ≈ 0.49 (see, e.g., Johnson, 1985). However, substituting Eq. (A.2) into Eq. (A.3) shows that this depth depends on the Poisson’s ratio. In the present study, fused quartz with ν = 0.17 is used, and hence the corresponding maximum shear stress occurs at a depth of z/a ≈ 0.44 as depicted in Fig. A.12. 1

−σ3 /p −σ1 /p τy /p

0.8 0.6

τmax

0.4 0.2 0 0

0.5

1

1.5

2

2.5

z/a Figure A.12: Distribution of the principal stresses and the maximum shear stress along the z-axis (x = y = 0) for fused quartz with ν = 0.17.


APPENDIX B VALIDATION OF THE AGGLOMERATION MODELS

48

In order to estimate the onset yield stress of the solid material, the criterion of Mises (1913) is applied: r 1 (A.4) σy = (σ1 − σ2 )2 + (σ1 − σ3 )2 + (σ3 − σ2 )2 = σ1 − σ3 , 2 where σy is the uni-axial compression yield stress of the material. Above the yield point, solid materials undergo plastic deformation beneath the contact. In relation to the compressive strength σy of the material of the colliding particles, the onset of the plastic deformation is determined by inserting Eq. (A.2) into Eq. (A.4): −1 h a i 3 σy z z2 = −(1 + ν) 1 − arctan + 1+ 2 (A.5) p a z 2 a

Thus, the maximum contact pressure for fused quartz (ν = 0.17) at z/a ≈ 0.44 is p = 1.466 σy . Appendix B. Validation of the agglomeration models In this section the energy-based and the momentum-based agglomeration models are validated in a simple shear flow based on a theoretical model available in the literature. It is well known that for a monodisperse system consisting of Np primary particles in a volume V , the specific collision rate per unit volume rcol is given by: rcol =

1 N˙ col = K n2 , V 2

(B.1)

where K stands for the size-independent collision kernel and n = Np /V is the particle number concentration (the total number of particles per unit volume of the suspension). Since not all inter-particle collisions lead to agglomeration, the agglomeration rate2 α (dimensionless quantity) is defined as the number of collisions leading to agglomeration Nagg divided by the total number of particle-particle collisions Ncol (i.e., α = Nagg /Ncol ). Thus, the specific rate of agglomeration per unit volume can be expressed as: ragg =

N˙ agg 1 = α K n2 , V 2

(B.2)

According to Hounslow et al. (1988) the rate of decrease of the particle number concentration for a size-independent agglomeration kernel β0 = α K is given by: dn(t) 1 1 = − β0 n2 (t) = − α K n2 (t) , dt 2 2

(B.3)

The integration of Eq. (B.3) yields: Np (t) n(t) 1 ≡ = . Np (0) n(0) 1 + 0.5 α K n(0) t

(B.4)

This relation implies that the number of primary particles reduces with time based on the occurrence of agglomeration processes. For a suspension of neutrally buoyant spheres with a 2

It is also denoted collision efficiency in the literature.



49

diameter dp subjected to a simple shear flow with a constant shear rate γ˙ van de Ven and Mason (1977) proposed the following relation for the theoretical agglomeration rate αth : αth = f (λ)

8H 36 π µf γ˙ d3p

0.18

,

(B.5)

where µf stands for the dynamic viscosity of the fluid. The parameter λ is equal to λ/πdp with the assumption that the characteristic wavelength of the dispersion interaction is λ = 10−7 m. According to van de Ven and Mason (1977) the dimensionless value of f (λ) is equal to 0.95, 0.87 and 0.79 for primary particles with a diameter of 1, 2 and 4 µm, respectively. Balakin et al. (2012) carried out two-dimensional simulations to compare the performance of the agglomeration model proposed by Kosinski and Hoffmann (2010) with the above mentioned theoretical model. To determine the time history of the particle number concentration given by Eq. (B.4), they used the agglomeration rate from their simulation and applied the collision kernel proposed by Saffman and Turner (1956): K=

3 γ˙ 1/3 1/3 V1 + V2 , π

(B.6)

where V1 and V2 are the volumes of the collision partners. However, their predictions do not agree with the theoretical results neither qualitatively nor quantitatively. Possible reasons for these deviations are the assumption of a two-dimensional simulation and the application of an inaccurate agglomeration model leading to a much higher agglomeration rate than given by Eq. (B.5). Furthermore, they employed the size-dependent collision kernel defined by Eq. (B.6) for the model given by Eq. (B.4), which is only valid for a size-independent collision kernel (see, e.g., Kumar, 2013). It is also important to note that Balakin et al. (2012) did not use common material properties, but estimated the values required for the agglomeration model. Therefore, the value of the compressive strength can not be found in the literature prohibiting the application of the energy-based agglomeration model. Thus, owing to the above mentioned drawbacks of the test case by Balakin et al. (2012) in the present study the agglomeration process of polystyrene particles dispersed in a threedimensional shear flow (water) using the energy-based and the momentum-based agglomeration model is validated against the theory. Note that the comparison between both agglomeration models carried out in Section 5 for a particle-laden turbulent channel flow is based on fused quartz particles in air. In addition to the full availability of the material properties of polystyrene particles, when dispersed in water flow, they satisfy the main assumption of neutrally buoyant spheres made by van de Ven and Mason (1977). The size of the rectangular computational domain in the streamwise, wall-normal and spanwise direction is 2δ × 2δ × 0.2 δ, respectively, where δ is the half-width of the channel δ = 0.02 m. An equidistant mesh consisting of 64 × 64 × 10 grid points is used. It is assumed that a constant shear rate γ˙ = 71 s−1 is established by moving the upper and lower walls in opposite directions with a constant velocity uw = γ˙ δ. The Reynolds number based on the wall velocity and δ is Re = uw δ/νf = 28, 400. In this numerical experiment periodic boundary conditions are used in the streamwise and spanwise directions and no-slip conditions hold on the walls. Furthermore, since the flow is steady and laminar, no turbulence model is applied. Note that the required values are made dimensionless using the wall velocity uw = γ˙ δ, the half-width δ and the water density ρf analog to the main test case presented in Section 4. The mechanical properties of polystyrene particles dispersed in water are listed in Table C.5. In the present



50

study, 75,000 primary particles with a diameter of dp = 25 µm are randomly released into the domain with zero velocity after the established flow is frozen. The start of the particle release will be denoted t = 0. The volume-equivalent sphere model is applied for modeling the structure of the agglomerate. The first step in the present validation study is to evaluate the agglomeration rate predicted by both models against the theoretical model by van de Ven and Mason (1977). Fig. B.13(a) shows the time history of the agglomeration rate predicted by the energy-based (EAM) and the momentum-based (MAM) agglomeration model in comparison with the theoretical value given by Eq. (B.5) within a dimensionless time interval of ∆T ∗ = 1400 (corresponding to 20 s). Using a quadratic fitting of the values of f (λ) provided by van de Ven and Mason (1977), the corresponding value of f (λ) for the diameter used in this study is about 0.59 yielding a theoretical agglomeration rate of αth = 3.4%. 0.1

EAM MAM

500000

Nagg × 20, Ncol

0.08

Nagg /Ncol

600000

αth (d = dpp ) αth (d = dp ) EAM MAM

0.06 0.04 0.02

400000 300000

Collision 200000

Agglomeration

100000

0

0 0

200

400

600

800

1000

1200

1400

0

200

400

600

(a)

800

1000 1200 1400

t γ˙

t γ˙

gglomeration

(b)

Figure B.13: Time history of (a) the agglomeration rate predicted by the energy-based (EAM) and the momentum-based (MAM) agglomeration model against the theoretical value given by Eq. (B.5) based on the diameter of either the primary particle dp or mean diameter of the particles dp and (b) the total number of the accumulated particle-particle collisions Ncol and the total number of the accumulated agglomeration processes Nagg .

It is clear that the momentum-based agglomeration model yields slightly more accurate predictions than EAM in comparison with the theoretical model. To analyze this behavior, the time history of the accumulated number of collisions and agglomeration processes predicted by both models is depicted in Fig. B.13(b). It is visible that both models predict almost the same number of agglomeration processes, while the energy-based model results in a higher number of particle-particle collisions than MAM. The main reason for this observation can be attributed to the assumptions made in EAM for the treatment of the kinetics of the collision partners without agglomeration. As explained in Section 2.2.4, it is assumed that the collision partners separate only in the normal or the tangential direction and the van-der-Waals force is neglected when calculating the angular velocities after the collision. It is important to note that the assumption of a constant agglomeration rate as used in the theoretical model (Fig. B.13(a)) is not realistic due to the fact that the agglomeration process leads to an enlargement of the mean particle size and hence to a weaker cohesive force reducing the probability of agglomeration. This behavior is reasonably reproduced by the present agglomeration models. Furthermore, when setting dp in the theoretical model (Eq. (B.5)) equal



51

to the mean diameter of the particles dp , an excellent agreement with the momentum-based model is observed. It is worth mentioning that the agglomeration rate predicted by the improved momentum-based agglomeration model by Breuer and Almohammed (2015) is in much closer agreement with the theory than the original model of Kosinski and Hoffmann (2010) applied by Balakin et al. (2012). In a second step, the decrease of the particle number concentration predicted by EAM and MAM as a function of time is validated against the theory. Since the particle agglomeration process leads to larger particles, a size-dependent collision kernel has to be used. Wang et al. (1998) show that the post-collision treatment scheme leads to different results of the numerical collision kernel. They distinguished three schemes and concluded that corrections to Saffman and Turner (1956) have to be made if one applies the theory to the actual agglomeration process. In the present study, if the collision partners collide and agglomerate, they form a particle of larger size. That means that the agglomerating primary particles are removed immediately from the computational domain and are replaced by an agglomerate. This implies that if per unit volume an agglomeration process occurs due to a successful particle-particle collision, the total number of particles decreases by one. Hence, the total number of particles participating in the collision detection decreases with time. Analogy to Eq. (B.1) the collision kernel at the time tn is defined by Wang et al. (1998) as: K(tn ) =

2 V Ncol (tn → tn+1 ) , Np2 (tn ) ∆t

(B.7)

where ∆t is the simulation time step, Ncol (tn → tn+1 ) stands for the total collision count in the time step tn < t ≤ tn+1 and Np (tn ) denotes the total number of particles (i.e., remaining primary particles and agglomerates of different types) participating in the collision detection. Thus, the reduction of the total number concentration for each time step reads (Wang et al., 1998): Np (tn+1 ) = Np (tn )

1 Np (tn ) 1 + α K(tn ) ∆t V

(B.8)

Fig. B.14 depicts the time history of the dimensionless particle number concentration predicted by both agglomeration models and the theoretical model using α = αth and either K(tn ) = K(tn )EAM or K(tn ) = K(tn )MAM . Obviously, during the first stage both agglomeration models predict almost the same results and agree very well with the theory, while they differ during the remaining time. The deviation between the results in comparison with the theory can be attributed to the slightly higher agglomeration rate calculated by the theoretical approach. Thus, the theoretical curve drops faster than the other two curves predicted by EAM and MAM. Although a reasonable agreement between the theoretical model and the predictions based on both agglomeration models is observed, Fig. B.14 clearly shows that the MAM results are in better agreement with the theoretical model than the energy-based model. The reason for this behavior is the higher number of collisions and hence the lower agglomeration rate predicted by EAM in comparison with the momentum-based model.


APPENDIX C MECHANICAL PROPERTIES OF THE MATERIALS USED IN THE SIMULATIONS52

Np (t)/Np (0)

1

0.9

EAM

0.8

MAM Theory (αth & K(tn )EAM ) Theory (αth & K(tn )MAM )

0.7 0

200

400

600

800

1000

1200

1400

t γ˙ Figure B.14: Time history of the dimensionless particle number concentration Np (t)/Np (0) predicted by EAM and MAM against the theoretical approach using α = αth and either K(tn ) = K(tn )EAM or K(tn ) = K(tn )MAM .

Appendix C. Mechanical properties of the materials used in the simulations Property

Symbol

Hamaker constant Young’s modulus Compressive strength Density Poisson’s ratio

H E σy ρp ν

Unit

Fused quartz

Polystyrene

Joule N/m2 N/m2 kg/m3 −

6.3 × 10−20 7.2 × 1010 1.1 × 109 2200 0.17

2.28 × 10−21 0.3 × 1010 1.0 × 108 1050 0.34

Table C.5: Mechanical properties of fused quartz particles dispersed in air (Momentive, 2014) and polystyrene particles dispersed in water (Chern, 2008; ENGINEERING.com, 2015).

Appendix D. Influence of the sub-models on the dimensionless frequencies

Case

Wall

dp

fcol MAM

EAM

MAM

Without sub-models smooth 12 µm 8.14 × 104 With sub-models smooth 12 µm 5.85 × 104 With sub-models rough 12 µm 3.91 × 104

7.79 × 104 5.83 × 104 3.88 × 104

6.67 × 102 1.72 × 102 1.93 × 102

5.39 × 102 1.29 × 102 1.39 × 102

1.24 × 103

1.23 × 103

2.1 × 101

1.36 × 101

With sub-models

smooth

4 µm

EAM

fagg

Table D.6: Influence of the sub-models on the dimensionless frequencies of the particle-particle collisions and the agglomeration processes predicted with both agglomeration models after a dimensionless time ∆T ∗ = 100.


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