Effects of the number of particles and coordination number on viscous-flow agglomerate sintering

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Effects of the number of particles and coordination number on viscous-flow agglomerate sintering Mohammadmahdi Kamyabi a,b , Khashayar Saleh b,∗ , Rahmat Sotudeh-Gharebagh a,∗ , Reza Zarghami a a b

Multiphase Systems Research Laboratory, School of Chemical Engineering, College of Engineering, University of Tehran, Tehran 11155/4563, Iran Sorbonne University/Compiègne University of Technology, EA 4297 Transformations Intégrées de la Matière renouvelable, France

a r t i c l e

i n f o

Article history: Received 26 August 2017 Received in revised form 26 December 2017 Accepted 28 January 2018 Available online xxx Keywords: Caking Agglomeration Sintering Viscous flow mechanism Volume of fluid Coordination number

a b s t r a c t The process of sintering of several particles in contact via a viscous flow mechanism was studied numerically using computational fluid dynamics. The volume of fluid technique within a finite volume method was used to simulate bridge formation between particles, as well as densification at different configurational states of the particles. The method was validated by comparing results for two-particle coalescence with the literature. The effect of the number of particles on agglomeration kinetics was studied by comparing bridge growth rate for systems having different numbers of particles in a chain. Although increasing the number of particles led to a decrease in the local bridge growth rate and to slower equilibration, there were no marked differences, when the overall volume of the system was considered. The effect of coordination number on the densification rate was directly studied by changing the number of particles in contact with a central particle. Increasing the coordination number caused the overall rate of densification to increase, but delayed equilibration, analogous to steric effects. These findings describe the configurational state of agglomerates, typical of mesoscale caking. In a multi-scale study, they can be used to characterize caking at a bulk scale to partly address the lack of experimental data in this field. © 2018 Published by Elsevier B.V. on behalf of Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences.

Introduction Coalescence occurs in many physical systems, for example rain formation (Kovetz & Olund, 1969), emulsification and 3D printers (Derby, 2010; Singh, Haverinen, Dhagat, & Jabbour, 2010). If coalescence occurs among particles, then it will probably lead to agglomeration. In particular, undesired agglomeration of industrial powders, which prevents flow at the bulk scale, is referred to as caking (Christakis et al., 2006; Langlet et al., 2013). Caking leads to particle shrinkage and densification and occurs in sequential steps (Hartmann & Palzer, 2011), with two-particle coalescence as the starting point (Jagota & Dawson, 1990; Kamyabi, SotudehGharebagh, Zarghami, & Saleh, 2017). Predicting caking at small scales is very important, because caking at larger scales is influenced by the initial stages of sintering at the small scale. There are different mechanisms leading to caking, based on material properties and operating conditions. Some of these mechanisms

∗ Corresponding authors. E-mail addresses: [email protected] (K. Saleh), [email protected] (R. Sotudeh-Gharebagh).

are dissolution/recrystallization (Cleaver, Karatzas, Louis, & Hayati, 2004; Langlet et al., 2011), volume, surface and grain diffusions (Kirchhof, Schmid, & Peukert, 2009; Seville, Willett, & Knight, 2000), evaporation/re-condensation (Kirchhof et al., 2009), and viscous flow sintering (Kirchhof et al., 2009; Seville et al., 2000). Viscous flow sintering or viscous sintering was introduced as the main mechanism for caking of amorphous particles (Garabedian & Helble, 2001; Katsura, Shinoda, Akatsu, & Wakai, 2015). Amorphous materials become soft and sticky above their glass transition temperatures, slightly decreasing their viscosity to allow flow under the effect of external forces (Martínez-Herrera & Derby, 1994). At temperatures well above the glass transition (T > Tg ), when two particles come into contact, a bridge is formed related to material flow, as shown in Fig. 1. This bridge is the outcome of the natural tendency of the system to reduce its total energy by reducing the relevant total free surface. This occurs as the act of surface tension, excited by the large curvature between a pair of particles, causes dynamic flow and formation of a bonding bridge. This viscous flow mechanism belongs to the capillary driven flow as the general mechanism underlying coalescence. Conventionally, two flow regimes (Eulerian and Stokesian) are considered in capillary driven

https://doi.org/10.1016/j.partic.2018.01.015 1674-2001/© 2018 Published by Elsevier B.V. on behalf of Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences.

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Nomenclature Latin C F I n n N ¯ N L L0 P pavg P0 Pf R0 Rf S S0 Sf Su t teq tvisc V x x0 xf Greek ˛1 ˛2 ı 

П  

Number of contacts External force in the Navier–Stokes equation (N) Identity matrix Total number of particles Normal vector of the surface Coordination number Average coordination number Distance of the farthest particles in a chain at each time step of sintering (m) Initial distance of the farthest particles in a chain (m) Total free perimeter of the agglomerate (m) Average pressure on the contact plane (Pa) Initial total free perimeter of the system of particles in contact (m) Final total free perimeter of the agglomerate (m) Initial radius of the particles (m) Final radius of the agglomerate (m) Total free surface of the agglomerate (m2 ) Initial total free surface of the agglomerate (m2 ) Final total free surface of the agglomerate (m2 ) Suratman number Time (s) Final time of sintering (s) Characteristic time of viscous sintering (s) Velocity of the fluid (m/s) Radius of the bridge (m) Initial radius of the bridge (m) Final radius of the bridge (m)

Volume fraction of phase 1 Volume fraction of phase 2 Dirac delta function (1/m) Viscosity of the fluid (Pa s) Surface stress tensor (N/m2 ) Density of the fluid (kg/m3 ) Surface tension coefficient (N/m)

coalescence, although a third regime has recently been described (Paulsen et al., 2012; Kamyabi et al., 2017). These two regimes are distinguished by Suratman dimensionless number, Su = R0 /2 , where low Su numbers represent Stokesian and high Su numbers represent Eulerian regimes. Comparison of coalescence behavior of different materials containing different particle sizes is possible, only if they belong to the same regime. For example, in the sintering of amorphous particles, although viscosity is decreased, it is still very high (106 –109 Pa s); hence, viscous forces are dominant and sintering occurs mostly within the Stokesian regime. Therefore, its

Fig. 1. Two particles in contact before (left) and after (right) bridge formation. R0 is the initial radius of the particles, L0 is the initial distance of the particles, L is their distance and x is the bridge radius during sintering.

parameters can be only compared with other results dealing with the Stokesian regime of coalescence. Because of extensive costs associated with performing experiments at the particle scale, modeling and numerical simulations of bridge formation are of much interest to researchers (Sprittles & Shikhmurzaev, 2012). Frenkel’s model (Frenkel, 1945) was formulated as the first model describing the kinetics of bridge formation between pairs of particles induced by viscous sintering. This model was based on the energy conservation law, in which the energy released because of free surface changes during coalescence is equal to the viscous dissipation energy. Although Frenkel’s model has several limitations (Hartmann & Palzer, 2011; Kamyabi et al., 2017; Tarafdar & Bergman, 2002), his formulation of this phenomenon introduced surface tension as the driving force of coalescence, with the viscous force as its counterpoise; this framework has been widely accepted. According to this framework, the characteristic time of viscous flow coalescence is tvisc = R0 /, which is used to convert time to a dimensionless parameter, t/tvisc = t/R0 . In contrast to two-particle systems, multi-particle systems are of interest, because these systems reflect a scale intermediate to microscale (two-particle) and macroscale (many-particle) descriptions of caking. In a two-particle system, Jagota and Dawson (1990) observed the formation of localized flow, only within the small area proximal to the forming bridge. They concluded that during viscous sintering, bridges have no significant effect on each other, especially when they are small, i.e., in the early stages of sintering. This conclusion was accepted warily by other researchers. Recently, it has been demonstrated that the kinetics of sintering is influenced by both the number and arrangement of particles in contact, even during the early stages of sintering (Kirchhof et al., 2009; Rasp, Kraft, & Riedel, 2013). Thus, bridges are dependent on each other. To describe multi-particle agglomeration, it is more convenient to consider the sintering process quantitatively. Fig. 2 shows a schematic of the sintering process for three particles in a chain, where the left panel shows the initial shape of the particles, and the right panel shows their state at a given time after sintering begins. Both shrinkage and densification are observed in this process. Clearly, the total free perimeter of the particles (P) (total free surface (S) for 3D views) and distances between particles are reduced during sintering. Only a few studies have investigated the sintering process induced by viscous flow, when more than two primary particles are involved. To our knowledge, experimental investigation of this problem has not yet been carried out. Typically, computational fluid dynamics simulations are used to analyze this phenomenon. Martínez-Herrera and Derby (1995) compared sintering rates of two- and three-particle linear chains using the finite element method. They have found that the overall shrinkage, defined as (L0 − L)/(nR0 ), was faster in the three-particle system, while local shrinkage, defined as (L0 − L)/L0 , behaved conversely. Overall shrinkage considers the total mass/volume of particles, while local shrinkage does not. They also reported the results for sintering of an infinite line of particles in contact, by applying a periodic boundary condition to a two-particle system (MartínezHerrera & Derby, 1994). Surprisingly, densification, defined as (P0 − P)/ (P0 − Pf ), reached its equilibrium state (a final value of 1) faster in the infinite line of particles than within the two-particle system. Using the finite element method, Zhou and Derby (1998) found that asymmetric bridge growth occurred between three Lshaped particles in contact, which led to inward rotation of the outer particles. This asymmetry in the sintering dynamic was not only related to viscous flow, it also occurred in a system having four particles in contact, arranged in a rhombus, sintered via diffusion (Wakai, Chihara, & Yoshida, 2007). Kirchhof et al. (2009) simulated the agglomeration of different numbers and morphologies of particles using a fractional volume

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Fig. 2. Schematic showing the shape evolution during sintering of three particles in a chain at the contact moment (left) and afterwards (right). L0 is the initial distance of the farthest particles in the chain, while L is the distance of the farthest particles in a chain at each time step of sintering.

of fluid method. They found that the rate of bridge growth during the early stages of sintering in multi-particle systems was slightly lower than in a two-particle system. They found that increasing the number of particles in a chain, caused the system to take longer to reach a final (equilibrium) state, although the densification rate (|dS/dt|) was higher. They found that equilibrium was reached faster, and the densification rate was higher in systems having more compact morphologies than widespread morphologies for the same number of particles. Attempting to describe multi-particle coalescence behavior with morphology correction factors, Kirchhof et al. (2009) revealed that using constant morphology correction factors, such as average coordination number, did not adequately describe the overall kinetics of densification. This affirmation seems logical given that several different morphologies of particles may have the same average coordination number, but have different densification kinetics. Clearly, a more detailed study is needed to investigate the exact role of the number of particles and their coordination number on the agglomeration process. In particular, we need to determine how bridge–bridge interactions affect the kinetics of sintering, dependent on whether they are in a linear chain or a compact configuration. Such interactions can be evaluated using changes in bridge formation or densification rates within multi-particle systems. This study presents simulations of the viscous flow mechanism of sintering at the multi-particle scale to investigate the effects of the number of particles and their configurations (using coordination number) on sintering kinetics. We also evaluate whether multi-particle studies are essential, or whether two-particle systems can be used to extrapolate behavior to many-particle systems. Model description

∇ · V = 0.

(1)

Conservation of momentum:



∂V + V · ∇V ∂t







= −∇ P + ∇ ·  ∇ V + (∇ V )T + F.

(2)

Particle-scale viscous flow sintering is specified for two phases: a material phase (particles) and the surrounding phase (air). Density () and viscosity () are determined as weighted averages of the volume fractions of these phases:  = ˛1 1 + (1 − ˛1 ) 2 ,

(3)

and  = ˛1 1 + (1 − ˛1 ) 2 .

Solution method

Scheme

Source

Gradient Pressure interpolation scheme Momentum Volume fraction Pressure-velocity coupling Transient formulation

Green–Gauss cell based PRESTO QUICK Compressive SIMPLE Second order implicit

Coirier (1994) Patankar (1980) Leonard (1979)

(4)

According to these equations, the physical properties are the same as those of each material in its corresponding occupied region, only varying across the interface between phases.

Patankar (1980)

In Eq. (2), F represents all external forces. Neglecting gravity, surface tension (Fs ) becomes the only external force responsible for sintering, acting as the driving force to the flow that causes coalescence. This force can be calculated and introduced into the Navier–Stokes equation, using a continuous surface force (Brackbill, Kothe, & Zemach, 1992) or a continuous surface stress model (Gueyffier, Li, Nadim, Scardovelli, & Zaleski, 1999). In this study, surface tension was estimated using a continuous surface stress model, which has a number of advantages for highly curved surfaces (Lafaurie, Nardone, Scardovelli, Zaleski, & Zanetti, 1994). According to this model, surface tension is given by: F s = ∇ ·˘,

(5)

where П is surface stress tensor, defined by: ˘ = (I − nn)ı,

(6)

where ı is a Dirac delta function, with a value equal to unity only at a phase interface and zero elsewhere; n is the normal vector of the surrounding surface; and  is the surface tension coefficient. Here, n is given by: n=

Coalescence of two or several particles caused by the viscous flow mechanism can be described using the continuity equation incorporating with Navier–Stokes equations as conservation of momentum: Continuity equation:



Table 1 Solution methods implemented in the model.

∇ ˛1 . |∇ ˛1 |

(7)

The phase interface is indicated by solving the continuity equa´ van tion for the volume fraction for one of the phases (Berberovic, ´ Roisman, & Tropea, 2009), as follows: Hinsberg, Jakirlic,

∂˛1 + ∇ · (V˛1 ) = 0. ∂t

(8)

The coalescence process is simulated using a finite volume method, which solves this set of equations numerically, as outlined in the next section. Numerical method Two-particle coalescence was simulated in two and three (2D with an axisymmetric assumption) dimensional coordinate systems. Multi-particle cases were simulated in 2D to reduce computational costs. Eggers, Lister, and Stone (1999) argue that considering this problem in 2D is beneficial, because the leading-order 3D system is asymptotically equivalent to its twodimensional counterpart. The finite volume method was used for solving Eqs. (1), (2), and (8). This method was implemented in Ansys ® Fluent software. The methods associated with both spatial and temporal discretization are listed in Table 1. For different combinations of , , and the initial radius of particles (R0 ), by decreasing

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Fig. 3. Dynamic evolution of coalescence between two particles caused by viscous flow with dimensionless time t/tvisc = t/R0 .

time step (t), a value of t/R0 = 2 × 10−3 was found to be sufficient for obtaining valid results. Calculation of each time step of the simulations took approximately 7 min of real time. The most important issue in our simulations was the procedure for capturing interface deformations. The volume of fluid technique was developed based on discretization of the volume fraction of each phases (e.g., ˛1 for phase 1, and ˛2 = 1 − ˛1 for phase 2) (Hirt & Nichols, 1981). It is a powerful front-capturing method, suitable for numerical simulations of moving/deforming interfaces (Cristini & Tan, 2004). Using this technique, the volume fraction in each control volume (cell) was determined based on the percentage of the phases inside the cell, and the interface position was captured by solving Eq. (8). Results and discussion Simulations were carried out for different values of , , R0 , with results reported in a dimensionless format. Any combination of these variables led to the same result in this format, as long as the Stokesian flow regime prevailed. This reflects that tvisc = R0 / is the characteristic time for viscous sintering. Two-particle system Simulations of sintering were carried out in two-particle systems, with the same R0 . The dynamic evolution of such systems is shown in Fig. 3. Gradual growth of a bridge occurs as sintering progresses, until the two particles merge completely to form a well-shaped single particle. There have been several reports in the literature of coalescence of two-particle systems in the Stokesian regime (Bellehumeur, Bisaria, & Vlachopoulos, 1996; Kirchhof et al., 2009; Paulsen et al., 2012; Rosenzweig & Narkis, 1981). Fig. 4 compares bridge growth from both experimental and numerical studies. These studies were selected for comparison because their Suratman number was low, ensuring that coalescence took place in the Stokesian regime. Good agreement among these results demonstrates that this study reliably simulated the viscous sintering process. However, some

inconsistencies (especially in the early stages of coalescence) were also seen among results, reflecting their different initial conditions or the presence of third coalescence regime (Paulsen et al., 2012), as discussed elsewhere (Kamyabi et al., 2017). Effects of initial conditions on the results are illustrated by comparing results of our 3D simulation for x0 /R0 = 0.1 with those of Kirchhof et al. (2009) for x0 /R0 = 0.125. In contrast, the outcome of our 3D simulation for x0 /R0 = 0.125 is in very close agreement with their results. However, for x0 /R0 = 0.125, only 4900 meshes per primary particle were required, while 31,400 mesh elements per primary particle (106 elements for both phases) were needed for x0 /R0 = 0.1. By increasing the number of the elements, simulations captured the sintering process from an initial condition closer to the real starting point of coalescence (i.e., for smaller values of x0 /R0 ), but at extremely large computational cost. All other simulations were carried out on a network with 31,400 mesh elements per primary particle, i.e., with x0 /R0 = 0.1. Effect of the number of particles Sintering in linear chains containing three and four particles was studied and compared to results for two-particle systems to explore the effect of the number of particles on sintering kinetics. The initial radius of all particles was set to R0 . Fig. 5 (top panel) shows the dimensionless growth of the bridge radius against dimensionless time. Since the results for different bridges in a chain showed no obvious differences, the central bridge was considered to represent the growth dynamic. Clearly, increasing the number of particles caused the bridge to form more slowly. Thus, its growth rate (dx/dt) decreased, as the number of particles increased, during early stages of sintering. In fact, as the number of particles in the chain increased, the ratio of the number of contacts per number of particles (C/n) also increased. This suggests that each particle contributed to more number of bridges, leading to slower overall bridge growth. However, such differences are small (a maximum 10% of x/R0 ), and given that they occur only during the last stages of sintering, they can be neglected, if x/R0 < 0.5. Our results agree well with those reported by Kirchhof et al. (2009), despite the fact that different final values of x for different

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Fig. 4. Comparison of results from this study with experiments and simulations from the literature showing variations of the dimensionless bridge radius (x/R0 ) between two particles as a function of dimensionless time (t/R0 ).

Fig. 6. Dimensionless bridge radius based on initial radius of particles (x/R0 ) versus the modified dimensionless time (based on final radius of agglomerate) for multiple particles in a chain (t/Rf ).

Fig. 5. Dimensionless bridge radius versus dimensionless time (t/R0 ) for multiple particles in a chain, with the dimensionless bridge radius being derived from: (top panel) the initial radius of the particles (x/R0 ) and (bottom panel) the final radius of the agglomerate (x/Rf ).

numbers of particles makes comparison difficult. Fig. 5 (bottom panel) shows bridge radius changes as a dimensionless parameter based on the final radius, Rf . Using this approach, all curves converged to the same value (because xf /Rf = 1), with curve slopes reflecting R1 dx . Clearly, as the number of particles increased, it took dt f

longer to reach the equilibrium. The effect of the number of particles on local shrinkage can also be seen in this figure, with the maximum difference among simulations being about 40% of x/Rf . Although our results agree with those of Kirchhof et al. (2009), Fig. 5 does not cover all total mass/volume values of agglomerates in dimensionless time. Although the two-particle system shows faster local shape evolution than three- and four-particle systems, the larger size of the multi-particle systems needs to be considered. It is useful to compare results against a modified dimensionless time parameter to explore the overall rate of bridge growth. This latter framework takes into account Rf , rather than R0 , providing a better indication of the overall evolution of the system. Fig. 6 shows bridge growth versus this modified dimensionless time parameter. Comparison of Figs. 5 and 6 shows that the order of the curves has been reversed, but the effect of number of particles have become less pronounced. Clearly, increasing the number of particles leads to an increase in overall bridge growth rate, which qualitatively agrees with the results of Martínez-Herrera and Derby (1995), while the term Rf dx/dt remains approximately the same for all cases during initial phases of sintering (as long as t/Rf < 2).

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Fig. 7. Progress of agglomeration for systems having different coordination numbers, with coordination number increasing in panels from left to right (N = 1, 2, 3, 4, 5).

Despite faster local sintering in smaller chains, sintering occurred globally, with the same kinetics for all chain lengths (at least during earlier stages of sintering). Thus, we conclude that the different bridges in a chain create spatial interference with each other, which reduces their growth rate; this reduction is inversely proportional to the total volume of the system. Effect of the coordination number Simulations were carried out on systems, in which a particle with radius R0 was fixed to the center, while different numbers of particles (e.g., N particles) with equal size were placed in contact around it, yielding different coordination numbers for the central particle. This ensured a range of coordination numbers for the central particle. The particles were arranged equally distant from each other to eliminate stress exerted because of asymmetry. Using this approach, we evaluated the effect of exact coordination number, rather than its average value. Fig. 7 shows the progress of agglomeration for systems having coordination numbers from N = 1 to 5, with dimensionless times based on both R0 and Rf . For dimensionless time calculated as t/R0 , the evolution of the system occurred more slowly, with increasing coordination number. In fact, for higher coordination numbers, the steric effects of particles became important, reducing the rate of evolution. However, when t/Rf was used as the dimensionless time, the reverse appeared to occur, with more accelerated overall shape evolution occurring for higher coordination numbers. The evolution of the total free perimeter of the agglomerates was studied to evaluate changes in densification of these systems and to quantify the effect of coordination number on sintering kinetics. Fig. 8 shows the perimeter evolution in the terms of P/P0 as a function of dimensionless time t/R0 for systems having different coordination numbers. Clearly, the evolution rate (|dP/dt|) is constant but has a different value for each system over the early stages of sintering. Increasing the coordination number caused the rate of evolution of the system (|dP/dt|) to increase, although the final equilibrium state was reached later. Using a curve fitting method, Eq. (9) describes these results, defining the perimeter change as a function of coordination number during early stages of sintering.



d P/P0 dt



= −0.1N 0.45

 . R0

(9)

In Fig. 9, the normalized perimeter evolution of the reduced form of (P − Pf )/(P0 − Pf ) is shown versus the dimensionless time, t/Rf . Comparing our results with the outcomes of Kirchhof et al. (2009)

Fig. 8. Evolutions of total free perimeter of the agglomerates in terms of P/P0 for multi-particle systems with different coordination numbers from N = 1 to N = 5, with dimensionless time based on the initial radius of the particle (t/R0 ).

Fig. 9. Evolutions of normalized free perimeter of the agglomerates in terms of (P − Pf ) / (P0 − Pf ) for multi-particle systems with different coordination numbers from N = 1 to N = 5, with the modified dimensionless time based on the final radius of the agglomerate (t/Rf ).

reveals marked differences. Indeed, their results show a unique evolution rate for their given morphologies, while our results do not follow this trend. This is because the coordination number was the same (a constant average value) for all cases considered by Kirchhof et al. (2009), while we varied this number in our study. They obtained the following expression (Eq. (10)) for the densifi-

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Fig. 10. Dimensionless average pressure on the contact plane of particles versus dimensionless time for multi-particle systems with different coordination numbers: (left panel) results for dimensionless parameters based on the initial radius of the particles; (right panel) results for dimensionless parameters based on the final radius of the agglomerates.

Table 2 Equilibrium times of sintering for multi-particle system with different coordination numbers N, normalized with respect to initial particle radius R0 and final agglomerate radius Rf . N

teq /R0

teq /Rf

1 2 3 4 5

4.551724 4.896552 5.241379 5.586207 5.931034

3.218555 2.827025 2.62069 2.498228 2.421335

cation rate for different morphologies during the initial stages of sintering: d



 

S − Sf / S0 − Sf



= −0.67

dt

 Rf

(10)

Given that the particles involved in their configurations did not all have the same coordination number, they used an average coordination number (Kirchhof et al., 2009). Moreover, the total number ¯ were the same of particles (n) and average coordination number (N) ¯ = 1.8) for all their cases. In fact, the coordination (n = 10 and N number was not the parameter directly studied in their work, nor was it an effective parameter in their simulations. Thus, it does not appear in Eq. (10). However, the effect of coordination number on the agglomeration behavior was evident in our study. Using a curve fitting method, we derived the following equation (Eq. (11)) for perimeter evolution, as a function of coordination number:



d (P − P0 ) / (P0 − Pf ) dt



= −0.57N 0.3

 . Rf

(11)

Clearly, the coordination number is a main variable in this equation, highlighting the effect of coordination number on the densification rate. In industrial sintering processes, the completion time of sintering is of economic interest. Table 2 shows the final time of sintering (equilibrium time, teq ) calculated for different coordination numbers. Eq. (12) describes teq as the function of coordination number, as derived from our data: teq  = 0.345N + 4.21. (12) R0 √ Based on Eq. (12), with Rf = R0 N + 1 (from conservation of mass), the following equation can be obtained: teq  0.345N 4.21 = √ +√ . Rf N+1 N+1

(13)

The pressure distribution inside the particles changes over time. Fig. 10 shows the average pressure on the contact plane between particles versus time, where pressure and time were reported based on dimensionless parameters. In this case, the average pressure (p) on the contact plane is negative during the initial stage of sintering, . but increases to the equilibrium value of pavg = R or pavg = √ f

R0 N+1

Conclusions The process of agglomeration caused by viscous flow mechanism was simulated. The simulation method was validated for two-particle sintering systems and applied to multi-particle systems. The effect of the number of particles involved in linear chains of connected particles on the rate of bridge growth was investigated. For the first time, the influence of coordination number on the densification rate was also directly evaluated. The main conclusions of these investigations were that: • Increasing the number of particles in the chain led to a decrease in the bridge growth rate locally, but had very little effect on the overall growth rate. • Increasing the coordination number caused the rate of change of the perimeter of agglomerate to increase but delayed equilibration. Two equations were proposed to describe the effect of coordination number on the densification rate, while two other equations were derived for predicting the equilibrium time of sintering, as a function of the coordination number. By highlighting effective parameters, we observed that bridges are dependent on each other in a multi-particle sintering system. Therefore, two-particle systems (microscale processes) are not sufficient to predict many-particle (larger scale) sintering processes. Clearly, the multi-particle system (mesoscale processes) is an important stage, which needs to be taken into account in simulating the multi-scale phenomenon of caking.

Acknowledgements This work has been supported by the Center for International Scientific Studies & Collaboration (CISSC) and French Embassy in Iran. Iran’s National Elites Foundation (INEF) is also acknowledged for their valuable support (INEF-Grant No. BN096).

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References Bellehumeur, C., Bisaria, M., & Vlachopoulos, J. (1996). An experimental study and model assessment of polymer sintering. Polymer Engineering & Science, 36, 2198–2207. ´ E., van Hinsberg, N. P., Jakirlic, ´ S., Roisman, I. V., & Tropea, C. (2009). Drop Berberovic, impact onto a liquid layer of finite thickness: Dynamics of the cavity evolution. Physical Review E, 79, 036306. Brackbill, J., Kothe, D. B., & Zemach, C. (1992). A continuum method for modeling surface tension. Journal of Computational Physics, 100, 335–354. Christakis, N., Wang, J., Patel, M., Bradley, M., Leaper, M. C., & Cross, M. (2006). Aggregation and caking processes of granular materials: Continuum model and numerical simulation with application to sugar. Advanced Powder Technology, 17, 543–565. Cleaver, J., Karatzas, G., Louis, S., & Hayati, I. (2004). Moisture-induced caking of boric acid powder. Powder Technology, 146, 93–101. Coirier, W. J. (1994). . An adaptively-refined, Cartesian cell-based scheme for the Euler and Navier–Stokes equations. PhD thesis (Vol. 106754) University of Michigan. Cristini, V., & Tan, Y.-C. (2004). Theory and numerical simulation of droplet dynamics in complex flows—A review. Lab on a Chip, 4, 257–264. Derby, B. (2010). Inkjet printing of functional and structural materials: Fluid property requirements, feature stability, and resolution. Annual Review of Materials Research, 40, 395–414. Eggers, J., Lister, J. R., & Stone, H. A. (1999). Coalescence of liquid drops. Journal of Fluid Mechanics, 401, 293–310. Frenkel, J. (1945). Viscous flow of crystalline bodies under the action of surface tension. Journal of Physics (USS R), 9(5), 385. Garabedian, R. S., & Helble, J. J. (2001). A model for the viscous coalescence of amorphous particles. Journal of Colloid and Interface Science, 234, 248–260. Gueyffier, D., Li, J., Nadim, A., Scardovelli, R., & Zaleski, S. (1999). Volume-of-fluid interface tracking with smoothed surface stress methods for three-dimensional flows. Journal of Computational Physics, 152, 423–456. Hartmann, M., & Palzer, S. (2011). Caking of amorphous powders—Material aspects, modelling and applications. Powder Technology, 206, 112–121. Hirt, C. W., & Nichols, B. D. (1981). Volume of fluid (VOF) method for the dynamics of free boundaries. Journal of Computational Physics, 39, 201–225. Jagota, A., & Dawson, P. R. (1990). Simulation of the viscous sintering of two particles. Journal of the American Ceramic Society, 73, 173–177. Kamyabi, M., Sotudeh-Gharebagh, R., Zarghami, R., & Saleh, K. (2017). Principles of viscous sintering in amorphous powders: A critical review. Chemical Engineering Research and Design, 125, 328–347. Katsura, K., Shinoda, Y., Akatsu, T., & Wakai, F. (2015). Sintering force behind shape evolution by viscous flow. Journal of the European Ceramic Society, 35, 1119–1122.

Kirchhof, M., Schmid, H.-J., & Peukert, W. (2009). Three-dimensional simulation of viscous-flow agglomerate sintering. Physical Review E, 80, 026319. Kovetz, A., & Olund, B. (1969). The effect of coalescence and condensation on rain formation in a cloud of finite vertical extent. Journal of the Atmospheric Sciences, 26, 1060–1065. Lafaurie, B., Nardone, C., Scardovelli, R., Zaleski, S., & Zanetti, G. (1994). Modelling merging and fragmentation in multiphase flows with SURFER. Journal of Computational Physics, 113, 134–147. Langlet, M., Benali, M., Pezron, I., Saleh, K., Guigon, P., & Metlas-Komunjer, L. (2013). Caking of sodium chloride: Role of ambient relative humidity in dissolution and recrystallization process. Chemical Engineering Science, 86, 78–86. Langlet, M., Nadaud, F., Benali, M., Pezron, I., Saleh, K., Guigon, P., et al. (2011). Kinetics of dissolution and recrystallization of sodium chloride at controlled relative humidity. KONA Powder and Particle Journal, 29, 168–179. Leonard, B. P. (1979). A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Computer Methods in Applied Mechanics and Engineering, 19, 59–98. Martínez-Herrera, J. I., & Derby, J. J. (1994). Analysis of capillary-driven viscous flows during the sintering of ceramic powders. AIChE Journal, 40, 1794–1803. Martínez-Herrera, J. I., & Derby, J. J. (1995). Viscous sintering of spherical particles via finite element analysis. Journal of the American Ceramic Society, 78, 645–649. Patankar, S. (1980). Numerical heat transfer and fluid flow. CRC Press. Paulsen, J. D., Burton, J. C., Nagel, S. R., Appathurai, S., Harris, M. T., & Basaran, O. A. (2012). The inexorable resistance of inertia determines the initial regime of drop coalescence. Proceedings of the National Academy of Sciences, 109, 6857–6861. Rasp, T., Kraft, T., & Riedel, H. (2013). Discrete element study on the influence of initial coordination numbers on sintering behaviour. Scripta Materialia, 69, 805–808. Rosenzweig, N., & Narkis, M. (1981). Sintering rheology of amorphous polymers. Polymer Engineering & Science, 21, 1167–1170. Seville, J. P. K., Willett, C. D., & Knight, P. C. (2000). Interparticle forces in fluidisation: A review. Powder Technology, 113(3), 261–268. Singh, M., Haverinen, H. M., Dhagat, P., & Jabbour, G. E. (2010). Inkjet printing-process and its applications. Advanced Materials, 22, 673–685. Sprittles, J. E., & Shikhmurzaev, Y. D. (2012). Coalescence of liquid drops: Different models versus experiment. Physics of Fluids, 24(12), 122105. Tarafdar, R. M., & Bergman, T. L. (2002). Detailed numerical and experimental investigation of non-isothermal sintering of amorphous polymer material. Journal of Heat Transfer, 124(3), 553–563. Wakai, F., Chihara, K., & Yoshida, M. (2007). Anisotropic shrinkage induced by particle rearrangement in sintering. Acta Materialia, 55, 4553–4566. Zhou, H., & Derby, J. J. (1998). Three-dimensional finite-element analysis of viscous sintering. Journal of the American Ceramic Society, 81, 533–540.

Please cite this article in press as: Kamyabi, M., et al. Effects of the number of particles and coordination number on viscous-flow agglomerate sintering. Particuology (2018), https://doi.org/10.1016/j.partic.2018.01.015