A tunable aggregation model incorporated in Monte Carlo simulations of spray fluidized bed agglomeration

Journal Pre-proof A tunable aggregation model incorporated in Monte Carlo simulations of spray fluidized bed agglomeration

Abhinandan Kumar Singh, Evangelos Tsotsas PII:

S0032-5910(20)30122-4

DOI:

https://doi.org/10.1016/j.powtec.2020.02.016

Reference:

PTEC 15175

To appear in:

Powder Technology

Received date:

26 October 2019

Revised date:

14 January 2020

Accepted date:

6 February 2020

Please cite this article as: A.K. Singh and E. Tsotsas, A tunable aggregation model incorporated in Monte Carlo simulations of spray fluidized bed agglomeration, Powder Technology(2019), https://doi.org/10.1016/j.powtec.2020.02.016

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier.

Journal Pre-proof A tunable aggregation model incorporated in Monte Carlo simulations of spray fluidized bed agglomeration

Abhinandan Kumar Singh* and Evangelos Tsotsas

Thermal Process Engineering, Otto von Guericke University, Universitätsplatz 2, 39106, Magdeburg, Germany

oo

f

*[email protected] Tel: +49-391-67-54993

Jo u

rn

al

Pr

e-

pr

Fax: +49-391-67-11160

Abstract: Constant volume Monte Carlo (CVMC) model is a discrete and stochastic approach to enhance the understanding of spray fluidized bed (SFB) agglomeration. Previous CVMC model used a simplified approach, considering the formed agglomerates as characteristic spheres with porosity as a function of their fractal properties. This means that other morphological descriptors such as coordination number could not be assessed. In the present work, a tunable aggregation model is developed to reconstruct the SFB agglomerates consisting of monodisperse spherical primary particles by a particle-cluster algorithm using a 1

Journal Pre-proof novel approach of tuning the fractal dimension at a given prefactor (= 1). Various morphological descriptors obtained

from this aggregation model are validated

using

experimental data. Furthermore, this aggregation model is incorporated into the CVMC framework to evaluate both, the morphological descriptors and formation kinetics of SFB agglomerates produced under different process conditions (inlet fluidized gas temperature and binder content).

f

Highlights: A tunable sequential aggregation model was developed to generate agglomerates



Morphological descriptors are needed to understand breakage and other properties



Stochastic model of agglomeration was modified by implementing the aggregation model



Process kinetics were evaluated at different operating conditions



Morphology of different agglomerates was analyzed and compared

Pr

Agglomeration,

Morphological

al

Keywords:

e-

pr

oo



descriptors,

Fractal

dimension,

Aggregation

Jo u

1. Introduction

rn

models, Tunable aggregation model.

Agglomeration is a size enlargement process consisting of many micro-scale mechanisms occurring in either parallel or series [1,2]. Spray fluidized bed (SFB) technique is common in food, chemical and pharmaceutical industries as it produces dust free granules by mixing, drying and binder addition in a single equipment [3,4]. Depending on the process conditions and the materials used, two mechanisms of particle growth can be observed: coating or agglomeration [5]. The border between them, which also makes the boundary of validity of agglomeration models,

has recently been evaluated and discussed in detail in [6].

Agglomeration is the area of interest for the present study. In SFB agglomeration, liquid binder is sprayed onto the primary particles and wets their surface. During fluidization, the 2

Journal Pre-proof particles collide and adhere to form a liquid bridge. Agglomerates are formed when the liquid bridges are dried to solid bonds using fluidizing gas as the drying agent. Microscopic features of the respective agglomerates like porosity, fractal dimension, prefactor and coordination number influence their macroscopic properties like strength and density [7,8]. Morphological descriptors of the SFB agglomerates formed are imperative in understanding product characteristics that affect its behavior. They play a vital role in determining the

f

breakage and droplet (binder) deposition mechanisms during agglomerate formation. The

oo

fractal properties (prefactor and fractal dimension) which are key characteristics of an

pr

agglomerate, usually represent compactness and arrangement of primary particles. It had been seen in the past [9,10] that both prefactor and fractal dimension must be properly ascertained

e-

in order to determine the structure of an agglomerate. Porosity and coordination number (CN)

Pr

are other prominent morphological properties of agglomerates. Porosity is a measure of the internal pores and cavities that affects the strength and density of an agglomerate [11].

al

Coordination number also affects the strength of an agglomerate as it defines the connectivity

rn

of the primary particles with each other.

Jo u

Classically, SFB agglomeration is modeled using population balance equations on a macroscopic scale [12]. In order to solve these equations, initial conditions and at least one aggregation kernel are required [13]. Due to complications in evaluating the kernels, Terrazas et al. introduced a discrete constant volume Monte Carlo (CVMC) model which was able to describe the process on the basis of micro-scale phenomena [1]. However, very simple method was used to predict the kinetics of SFB agglomeration by considering the emerging agglomerates as spheres of constant porosity (= 0.6) [14]. This constant porosity was replaced by variable porosity, which was the function of agglomerate fractal properties, by Singh and Tsotsas [15]. Nevertheless, the emerging agglomerates were still considered as spheres in [15]

3

Journal Pre-proof and other morphological descriptors like the coordination number of primary particles in the agglomerates were not recorded. In recent years, experiments have been conducted to study the morphology of SFB agglomerates formed at different operating conditions using either rigid [16] or deformable [17] primary particles. Numerically, the interest in structure formation by aggregation of small particles has motivated the search for simulation algorithms that might give comparable

f

results to experimental observations. For instance, Dernedde et al. [18] proposed a novel

oo

approach to simulate the structure of SFB agglomerates using a cluster-cluster ballistic

pr

aggregation model, which stores and preserves a complete description of the spatial arrangement of primary particles. However, the formed agglomerates were predicted too

e-

porous and the model was computationally very expensive and slow. In addition to the models

Pr

that are based on the probabilistic approach of sequential addition of primary particles, there are other models based on deterministic approach, such as discrete element method that can

rn

also computationally expensive.

al

take the cohesive force into account [19,20]. However, due to the nature of this method, it is

Jo u

In the present work, we develop a fast and tunable structure formation model for estimating the agglomerate morphology corresponding to its fractal properties. Further, we modified the CVMC model from [15] by implementing this structure formation model to reconstruct emerging SFB agglomerates computationally. Implementation of the structure formation model in the CVMC

simulation can be either in-situ or post-processing. Various

morphological descriptors of agglomerates formed under different operating conditions, such as inlet air temperature and binder content, are predicted and compared with experimental results from [16]. One of the goals of the present work is to provide the foundation for the development

of a

breakage

model by incorporating realistic morphology of SFB

agglomerates. 4

Journal Pre-proof The rest of this paper is organized as follows. In Section 2, we present the morphological descriptors and the properties of an agglomerate associated with it. Section 3 lists various structure-based algorithms and their limitations with regard to SFB agglomerates. The aggregation model used in this study and implementation details are given in Section 4. In Section 5, we describe in detail the CVMC model and the modifications that have been made to implement the aggregation model of Section 4. Then, numerical results and conclusions are presented in Sections 6 and 7, respectively. It should be noted that aggregate size is described

oo

f

by the number of primary particles until Section 6, but time appears explicitly in this section,

pr

so that agglomeration kinetics in terms of size-over-time are also obtained and discussed.

e-

2. Morphological descriptors

Pr

The morphology of SFB agglomerates strongly affects their behavior and properties. The morphological descriptors considered in this work are scalar descriptors, mainly fractal ) describes

al

dimension and prefactor, porosity and coordination number. Fractal dimension (

rn

the arrangement of primary particles in physical space within an agglomerate. A higher

Jo u

specifies that the structure is compact and has more interconnections between the particles, which means that the agglomerate is strong. Whereas, a lower

indicates that the

agglomerate is open and has a tenuous structure. The value of fractal dimension can vary from one (chain-like structure) to three (spherical agglomerate). This can be perceived from Fig. 1 where the morphology of synthetic agglomerates (with 200 primary particles and prefactor equal to 1) is changed at different fractal dimensions. With fixed prefactor, the mean coordination number (MCN) of the agglomerates increases and porosity decreases with an increase in fractal dimension. Agglomerates in this section have been generated using the tunable sequential aggregation (TSA) model, which is explained in detail in Sections 3.2 and 4. Porosity is measured using the gyration method.

5

Journal Pre-proof (Figure 1) Prefactor

describes the packing of primary particles of an agglomerate in

dimensional

space [9]. As indicated in Fig. 2, it has an influence on the degree of compactness of agglomerates. With a fixed fractal dimension (of 3), if the prefactor is increased, the mean coordination number (MCN) of the agglomerate (with 200 primary particles) increases and the porosity decreases, making the agglomerates compact and strong.

oo

f

(Figure 2)

pr

Porosity measures the volume fraction of internal pores as cavities or channels connected with the body’s surface [7]. Depending on the methods of calculating this volume fraction, porosity

e-

is generally classified as porosity by gyration and porosity by convex hull method. Detailed

Pr

descriptions of these methods can be found in [15]. Porosity correlates with the effective density and can influence the mechanical strength and flowability [15,21]. Coordination

al

number (CN) of a primary particle in an agglomerate is defined by the sum of all contact

rn

points that the primary particle has with surrounding primary particles. For two primary

√(

)

Jo u

particles (i, j) with center coordinates (x, y, z) and radius (R), if the condition (

)

(

)

|

|

(1)

is satisfied, then they are in contact. CN is a very popular microscopic parameter associated with packing structure [22]. It influences the elasticity [23], plastic strength [24] and densification rate [25]. Agglomerate strength depends on the porosity and the number of particle-particle contacts measured as a coordination number. Strength (

of the

agglomerate is calculated using the Rumpf model [26] to (

)

.

(2)

6

Journal Pre-proof Here,

and

are mean coordination number and porosity of an agglomerate,

mean cohesive force between the primary particles of diameter

is the

. From Eq. (2) we see that,

as the coordination number increases the strength of agglomerate increases, keeping other parameters (

,

,

) constant. The reason behind this is the increasing number of

contacts with a certain cohesion force between the primary particles.

and

are

necessary to calculate the strength of an agglomerate. Simplifications can be made by or

[4,26,27]. This reduces the number of

f

assuming specific values of either

oo

parameters that need to be known in order to determine the strength of an agglomerate, but is

pr

of uncertain validity, so that Eq. (2) should be preferred for describing breakage behavior.

e-

This urges the requirement for an aggregation model which should be able to reconstruct SFB

Pr

agglomerates and provide various morphological descriptors, including MCN. 3. Aggregation models

al

Aggregation of particles is the process of combining individual primary particles or small

rn

clusters to form large clusters. Model aggregates generally consist of monodisperse spherical

Jo u

particles. Respective aggregation schemes can be easily realized in computer simulations. They are important tools to understand the growth of clusters [28] and provide ways to study the underlying physical phenomena [29]. In general, any aggregate formation begins with a single seed particle. Aggregate growth is then achieved by adding other entities like primary particles or small clusters to the seed particle [30]. Basically, parameters that can be varied to achieve different types of aggregation models can be assigned to four groups [31], as they refer to aggregate formation, particle trajectories, the simulation lattice or algorithm tuning. Aggregate formation occurs, as illustrated in Fig. 3, either through particle-cluster aggregation, where only one primary particle is added each time to an existing aggregate, or

7

Journal Pre-proof through cluster-cluster aggregation, which means that even two clusters may be added to a new cluster. (Figure 3) Particle trajectories are composed by the movements that entities (primary particles or small clusters) undergo in the working lattice until they can stick to the main cluster and form an aggregate [32]. The entities may take a path that consists of many linear pieces or is linear

oo

f

over the whole distance. Erratic, multi-step trajectories are usually obtained by a random walk in all six directions, while the considered entity gets closer to the main cluster in order to form

pr

an aggregate. Aggregates are formed by such trajectories in the diffusion-limited aggregation

e-

(DLA) model [33], which is described in detail in [34]. Alternatively, entities may follow an

Pr

uninterrupted linear trajectory over the entire distance to the main cluster in order to form an aggregate. This type of model is termed as ballistic aggregation model and is described in

al

detail in [35].

rn

Simulation lattice is the network (grid) of equally spaced horizontal and vertical lines that

Jo u

cover the simulated space. Simulation may be on-lattice, when entities may take only specific positions in the simulated space (Fig. 4, left) or off-lattice, when entities may take every position, as seen in Fig. 4 (right).

(Figure 4) Tuning parameters that can have a governing influence on the aggregation algorithm are sticking probability and fractal properties. Sticking probability is the probability of entities to stick to the main cluster to form an aggregate [36]. If an entity sticks to the main cluster after every single collision, then the sticking probability is one [28]. The formed aggregates are tenuous and fractal in structure. The fractal dimension is close to 2.2 and the porosity is high, 8

Journal Pre-proof leading to the DLA model [28]. Sticking probability decreases with increasing number of collisions needed for successful aggregation between the entities and the main cluster. The lowest value of sticking probability is almost zero. The formed aggregates are then closely packed like a sphere with a fractal dimension close to three. The process of formation is termed as reaction-limited aggregation (RLA) or EDEN model and is described in detail in [37]. Figure 5 illustrates exemplarily on-lattice PC aggregates generated using DLA and RLA

oo

(Figure 5)

f

models.

and

. Provided a sufficiently large set of agglomerates of different sizes, the power-

e-

prefactor

pr

Fractal properties of any aggregate are defined by its fractal dimension (

(3)

rn

( ) ,

),

al

with their radii of gyration,

,

Pr

law relationship (which correlates the number of primary particles of the agglomerates,

is supposed to be fulfilled. Assuming validity of the scaling law according to Eq. (3), addition

Jo u

of primary particles to a cluster can be performed with the goal of establishing specific fractal properties. This method of aggregation is termed as a tunable aggregation model and is described in detail in Section 3.2. Meakin [28] gave an overview of six commonly used combinations of the discussed parameters (aggregation formation,

particle trajectories,

simulation lattice

and sticking

probability). The fractal dimensions of clusters generated by simple particle-cluster and cluster-cluster aggregation models are given in Table 1. (Table 1)

9

Journal Pre-proof From Table 1 it is evident that the fractal dimension of cluster-cluster (CC) aggregation models is lower than that of particle-cluster (PC) aggregation models. Consequently, the porosity of CC aggregates is always higher than the porosity of PC aggregates. According to the existing experimental findings, the fractal dimension of spray fluidized bed agglomerates is between 2 and 3, whereas porosity ranges (depending on agglomerate size) from 0.4 to 0.75 [16,38]. Hence, CC algorithms can be excluded due to too low aggregation

(Table 1). Ballistic PC

has been reported to asymptotically (for large particles) result in a constant

oo

f

porosity of about 0.83 by Sutherland [39]. This means that the aggregates formed using the ballistic model are too porous in comparison to real SFB agglomerates. Similar behavior has

pr

been observed in [18]. Therefore, CC and ballistic algorithms are excluded from further

aggregation model.

al

3.1. Sticking probability model

Pr

e-

consideration, which will focus on tuning by the sticking probability and on the tunable

rn

The sticking probability aggregation model of this work is an on-lattice PC aggregation model

Jo u

where the primary particles take a random walk like in DLA [33] to stick to the main cluster. The input parameters are the number of identical spherical primary particles (

) and the

sticking probability ( ). The algorithm of this model (given in Fig. 6) is as following: 1. A seed particle (the first one from the cluster) is placed at the center of a 3D lattice. 2. The next primary particle of unitary diameter is released from a launching radius centered at the seed,

larger than the cluster radius

. (

, where

is the distance

of the center of mass of the cluster at the origin to the farthest particle of the growing cluster and ∆R is a small distance). 3. The released particle performs a random walk on the 3D lattice until it reaches the seed/cluster.

10

Journal Pre-proof 4. If a random number drawn between 0 and 1 (rand) is lower than

(sticking probability),

then the particle is attached to the seed/cluster. Otherwise, the random walk process continues until the seed/cluster is reached again. 5. When the current particle has been attached to the cluster, the next primary particle is generated and released. 6. If the particle is reached at a position that is more than

(killing radius) then it is

eliminated and launched again.

oo

f

7. Steps 2-6 are repeated until the desired number of primary particles in the aggregate is

pr

achieved.

e-

(Figure 6)

Pr

A series of agglomerates made of monodisperse spherical primary particles ranging from 5 to 100 has been generated using this model at various values of the sticking probability and the

al

fractal dimension of each series has been evaluated. Then, 10 realizations of this procedure

rn

have been conducted each time and the results are plotted in Fig. 7. It can be seen that the

Jo u

fractal dimension increases with a decrease in sticking probability. Morphology of the biggest agglomerates (with 100 primary particles) generated synthetically at different sticking probabilities and computation time (on a usual computer with 4-core processor and 4 GB RAM) required for each case after one realization are noted in Table 2. The porosity of the agglomerates in Table 2 is measured using the radius of gyration method. (Figure 7) As the sticking probability decreases, porosity decreases and reaches an asymptotic value. The lowest porosity achieved using this model for

is comparable to the highest

porosities measured for the largest agglomerates in [16], but still higher than the average

11

Journal Pre-proof porosity of SFB agglomerates of around 0.6. At lower sticking probability, the computation time is higher for only marginal change in the structure of an agglomerate, as can be expected. (Table 2) 3.2. Tunable aggregation model The trend of the sticking probability model to produce too high agglomerate porosities

f

motivated the use of a model that can better tune morphological and fractal properties. A

oo

review of the developments in the tunable aggregation model over the years is presented in

pr

Table 3.

e-

(Table 3)

Pr

In the present work, we use the sequential algorithm (SA) proposed by Filippov et al. [40] for the PC aggregation model. This model is off-lattice and the input parameters are the number and

al

and radius of primary particles (

) and the fractal properties (

and

) of the

rn

agglomerates. In this algorithm monodisperse spherical primary particles are added one by

Jo u

one using the following steps:

1. Insert a particle (seed) at the center of a 3D simulation lattice. 2. Place the second particle close to it, by selecting a point on the spherical surface of the seed. 3. For the third and next particles, the correlation of Filippov et al. [40] is used. The center of the next particle is on the sphere of radius ( )

(

,

) ,

such that there is no overlapping as shown in Fig. 8. Here, N is from 3 to

(4) .

4. Randomly a point is selected on the sphere (of radius ) and the overlapping condition is checked. 12

Journal Pre-proof 5. If there is no overlapping then the new particle is attached to the growing cluster (initially a dimer). 6. Steps 3-5 are repeated until the targeted number of primary particles is achieved. (Figure 8)

The morphology of an agglomerate (

) formed using SA with

and

is

shown in Fig. 9. Using this model, the limitation of the sticking probability model is

oo

f

overcome and agglomerates with porosity less than 0.7 can easily be created. The range of morphological descriptors reported by Dadkhah and Tsotsas in their experiments [16] (Table

pr

5) is achieved using this model. However, the limitation in the prefactor (greater than 1.66 not

e-

possible) still remains, whereas prefactors greater than 1.66 are common in practice

Pr

[8,16,41,42] and it would be desirable to use the model without restriction.

rn

al

(Figure 9)

Jo u

4. A novel approach to overcome the limitation of SA (

not possible)

As stated in Table 3, the only limitation of SA is that the prefactor values of possible. This limitation is overcome by tuning

are not

with respect to the porosity correlation

obtained by Singh and Tsotsas [15], ( ) . Keeping the porosity (

(5)

) and the number of primary particles (

fractal dimension by setting the prefactor to ( )

.

) same, we change the

: (6)

13

Journal Pre-proof Equating Eqs. (5) and (6), we get the correlation for tuned fractal dimension at specific prefactor (

) (

).

(7)

Using the power-law relationship (Eq. (3)) to compute the change in the fractal properties with the same radius of gyration, the same correlation as in Eq. (7) is obtained. The logarithmic version of the power law, log ,

(8)

f

log ( )

oo

log

pr

is illustrated in Fig. 10 with straight lines that represent original and tuned combinations of

e-

and .

Pr

(Figure 10)

Table 4 shows the comparison in terms of porosity and mean coordination number of an ) from Exp. B of Dadkhah [38] and agglomerates with the same

al

agglomerate (

rn

number of primary particles synthesized using the SA model. The latter have been obtained and tuned

. Experiment

Jo u

with the genuine fractal properties, as well as with the modified

B has been chosen for comparison because the size of the primary particles measured by Dadkhah for Exp. B is close to monodisperse. Consequently, this experiment replicates in the best way the assumptions behind the monodisperse tunable aggregation model used in this work and can act as a reference to validate the tunable

model, as seen in Fig. 11.

(Table 4) (Figure 11) By tuning the fractal dimension with different prefactor, whereas that can be obtained from keeping the porosity Eq. (5) constant, the formed agglomerates are almost the same (Fig. 11), 14

Journal Pre-proof which can also be seen in nearly constant porosities obtained by the gyration and convex hull methods as well as nearly constant average coordination number (Table 4). Therefore, we can overcome the limitation of SA regarding the value of

and construct all the agglomerates

with this model. However, there is another limitation as depicted in Fig. 10. As the slope is physically not feasible, which

increases the fractal dimension also increases, but

means that slopes of the dashed lines can only be increased until 71.5o . This limits the

)

.

oo

(

f

formation of aggregates with primary particles less than

2 ) at

and

. The reason for using these particular values of

e-

value of

(for Exp. B until the

pr

This restriction is overcome by constructing the aggregates until

(9)

Pr

and k is the compactness of small SFB agglomerates. Using this novel approach of tuned fractal dimension into SA we are able to construct any SFB agglomerates. The algorithm, to

in

the

following

as

tunable

sequential aggregation

(TSA)

model,

is

al

referred

rn

computationally fast. It takes less than 3 seconds to generate an agglomerate with 100

RAM).

Jo u

monodisperse spherical primary particles on the mentioned personal computer (4 cores, 4 GB

5. Constant volume Monte Carlo model and its modification The Monte Carlo method [43] is a stochastic approach based on the use of random numbers and probability statistics [1,2,13,15]. The process of SFB agglomeration (here without breakage and for non-porous primary particles) is simulated with the help of the CVMC algorithm

introduced by Terrazas et al. [1]. The connection between CVMC and the particulate system is provided by the event-driven nature of the method, with a collision between particles in the fluidized bed as the considered event. A brief modeling scheme is presented in Fig. 12. Micro-scale events and processes such as droplet deposition on the particles, collisions of 15

Journal Pre-proof particles, drying of the deposited droplets, Stokes criterion [44] for particles to stick or rebound are incorporated in the algorithm [1,15]. The detailed description of this algorithm is given in [1,15]. For simplicity, all the formed agglomerates in the CVMC model are considered as spheres with diameter [

].

(10)

Terrazas at al. [14] used a constant porosity of

in Eq. (10) to determine the

oo

f

agglomerate diameter. This means that formed agglomerates are assumed to have the same porosity at different operating conditions. This was modified by Singh and Tsotsas [15] by

and

are taken from the empirical correlations obtained by Singh

e-

Eq. (5), the values of

pr

implementing variable values of porosity using Eq. (5). In order to calculate the porosity from

) 2

,

,

is the gas inlet temperature in ℃ and

(11) (12) is the mass fraction of binder in the spray.

rn

where

2

2

al

(

Pr

and Tsotsas [15],

Jo u

The structural change was implemented into the CVMC model to determine the kinetics of the agglomeration process at different operating conditions. However, other morphological features like MCN of the agglomerates were not accessible, though they are very important for the characterization of the product. (Figure 12) The CVMC model of Singh and Tsotsas [15] is modified by introducing the TSA model. Using the TSA model various morphological descriptors are determined. Figure 13 shows two ways of implementing the TSA model into the CVMC simulation: postprocessing and in-situ. In postprocessing, the TSA model is called at the end of the CVMC simulation. The simulation scheme is same as in [15] and after the simulation ends the formed agglomerates 16

Journal Pre-proof are reconstructed using the TSA model with input parameters (

,

,

) from the CVMC

model. Whereas in the in-situ version the TSA model is called at every time step of the CVMC simulation. Unlike in [15], where the porosity is calculated using Eq. (5), here the porosity is calculated using the radius of gyration method by reconstructing the formed agglomerates with the TSA model. Then, this porosity is inserted in Eq. (10) to obtain agglomerate diameter,

.

oo

f

(Figure 13)

, , average

, average MCN) obtained for different

e-

Morphological descriptors (

pr

6. Results and discussions

experiments in [16], namely for Exp. A to Exp. E with glass beads as the primary particles,

Pr

are presented in Table 5. Experiments B, A and C represent the increasing of inlet fluidized

al

gas temperature from 30℃ to 60℃ and 90℃. Similarly, experiments A, D, and E represent increasing binder (Hydroxypropylmethylcellulose, HPMC) mass fraction of 2 wt. %, 6 wt. %

rn

and 10 wt. %. Glass beads with a narrow range of diameters between 450 µm and 631 µm

Jo u

with an average diameter of 520 µm were used for the experiments [16]. Agglomerate growth is measured in terms of the number of primary particles contained in an agglomerate, which in turn increases over time. Moreover, time is an important factor and temporal agglomerate growth will be analyzed later in Section 6.2. 6.1. TSA model Agglomerates with the same number of primary particles as examined by Dadkhah [38] are generated using the TSA model. For each trial, the TSA model takes the values of

and

from the empirical correlations obtained by Singh and Tsotsas [15], i.e. from Eqs. (11) and (12). These values of

and

are used as the input parameters for the TSA model to generate 17

Journal Pre-proof the agglomerates at tuned

(calculated using Eq. (7)) and

. Monodisperse primary

particles with diameter of 520 µm are used in the TSA model. The morphological descriptors of the generated agglomerates from the TSA model are obtained and averaged over five realizations. In each realization a series agglomerates are produced with exactly the same number of primary particles as in the experimental morphological evaluation.

and

of the

agglomerates generated using the TSA model (Table 5) are calculated using the power-law

by the radius of gyration method and of MCN are

oo

evaluation method [7]. Values of

f

relationship on a log scale (Eq. (8)) for the entire sample, similar to the Dadkhah et al.

pr

computed for each agglomerate of a series, then averaged arithmetically for the series, then averaged among five realizations and, finally, given in Table 5. This is also reproducing the

Pr

e-

procedure that Dadkhah has used in her evaluation of tomography data. (Table 5)

al

Morphological descriptors obtained from the TSA model agree very well with the

rn

experimental values [16]. A slight deviation in the results of the TSA model is due to the size

Jo u

dispersity of primary particles used in the experiments while in simulations the primary particles are fully monodisperse with diameter of 520 µm. However, all trends of changes in morphology by changing operating conditions are similar between reconstructed and real agglomerates from [16]. Furthermore, the porosity obtained by the gyration method using the TSA model is compared with Exp. B (in which the primary particles have been close to monodisperse) for specific agglomerates as well as theoretically, by the porosity correlation (Eq. (5)) with fractal properties (

,

) obtained from experiment and simulation (Table 5) in Fig. 14.

Agglomerates simulated using the TSA model are lying close to the curve of Eq. (5) with fractal properties (

,

) obtained from simulation (Table 5). There is a difference to the 18

Journal Pre-proof curve with fractal properties (

,

) obtained from Exp. B (Table 5) but this difference is

small. Figure 14 validates the porosity correlation (Eq. (5)) for the simulated agglomerates in similar way as it was validated previously by Singh and Tsotsas [15] for real agglomerates obtained from experiments [38]. One important aspect of Fig. 14 is that the agglomerates formed using the TSA model, regardless of tuned fractal properties (

, ) at different

, are

in close affinity to the agglomerates examined by Dadkhah in her experiments [38].

oo

f

(Figure 14) The mean coordination numbers for all the agglomerates simulated using the TSA model are

pr

compared to the experimental values in Fig. 15. MCN using the TSA model for the small are nearly similar for all the experiments. The reason behind this

e-

agglomerates (

.

Pr

is the limitation of the TSA model that the small agglomerates are reconstructed at

The spread for experimental data (Fig. 15 (left)) is higher because of polydisperse primary

al

particles used for experiments [16]. The trends of the results from the TSA model are similar

rn

to the experiments [16]. With a high fractal dimension, the porosity is low and MCN is high,

Jo u

leading to a compact structure. On contrary at low fractal dimension, the porosity is high and MCN is low, leading to tenuous structure. The coordination number of compact agglomerates (Exp. C) is the highest and that of the tenuous agglomerates (Exp. E) is the lowest. Consequently, the strength of agglomerates (reconstructed using the TSA model) is calculated using the Rumpf model [26] for the five experiments and compared in Fig. 16. On the one hand, the strength is calculated from Eq. (2) with with

2

and

and

from the TSA model

assumed as 20 N. On the other hand, the strength is calculated

using a simplification in determining the MCN ( and same

and

⁄

[4,26],

from the TSA model,

. From the plots of strength calculated using different ways of obtaining

MCN in Fig. 16 it is clear that agglomerate morphology should be considered as 19

Journal Pre-proof comprehensively as possible (i.e. with MCN and

considered separately) during the

process of breakage. It is also evident from Fig. 16 that the strength of compact agglomerates (Exp. C) is highest, while it is lowest for the tenuous agglomerates (Exp. E). (Figure 15) (Figure 16)

oo

f

6.2. Modified CVMC model In postprocessing, the CVMC modeling scheme is same as the CVMC model of Singh and

pr

Tsotsas [15] (old model) with the TSA model called at the end of the CVMC simulation. SFB

e-

agglomerates with different numbers of primary particles for the different experiments, as

Pr

reconstructed using the TSA model, are tabulated in Fig. 17. Using the in-situ way of implementing the TSA model in the CVMC simulation (new model),

the formed

al

agglomerates are reconstructed by means of the TSA model to calculate the porosity at every

rn

time step of the CVMC simulation. However, the breakage of solid agglomerates is not

Jo u

considered and wetting is similar to [1,15] in the new model. Simulations for the different experiments (Table 5) have been carried out for 400 s, using the same simulation parameters as in [15].

Both ways of implementation of the TSA model in the CVMC simulation provide the same morphological results as in Fig 17. This means that for the same number of primary particles, the agglomerates formed are identical. Increasing the inlet fluidization gas temperature (Exp. B, A and C), the fractal dimension of the formed agglomerates increases and the prefactor decreases, which in turn decreases the porosity and increases the mean coordination number. On contrary, increasing the binder concentration (Exp. A, D and E) leads to a decrease in the

20

Journal Pre-proof fractal dimension at increasing prefactor, resulting in an increase in porosity and a decrease in the mean coordination number. (Figure 17) In Fig. 18 (left) diameters of agglomerates (using Eq. (10)) are calculated by taking the porosity from the TSA model and plotted against the number of primary particles to depict the morphological change for different experiments. The diameter of agglomerates with a similar

oo

f

number of primary particles is lowest for Exp. C and highest for Exp. E. The reason behind this is that at higher inlet gas temperature (Exp. C) the formed agglomerates are compact with

pr

high fractal dimension and coordination number and low porosity (Fig. 17). Contrary, for high

e-

binder content (Exp. E) the agglomerates are tenuous with low fractal dimension and

Pr

coordination number and high porosity (Fig. 17). Therefore, the diameter of a volume equivalent sphere is highest for Exp. E and lowest for Exp. C. Similar conclusions were made

al

by Dadkhah and Tsotsas for the experimentally produced agglomerates [16].

rn

Moreover, in Fig. 18 (right) temporal change of the diameter for different experiments depicts

Jo u

the kinetics of the process. As already mentioned, in contrast to the old model, the agglomerate diameter in the new model is calculated using Eq. (10) with porosity evaluated from the TSA model. This has only little influence on the kinetics of the process (Fig. 18 (right)). The relative diameter of the agglomerates is the ratio of Sauter mean diameter of the population and the initial diameter (

2

) at a given instant of time [15]. As the

temperature of the inlet fluidization gas increases (Exp. B, A and C) the size of deposited droplets decreases and the probability of successful collision becomes smaller [14,15]. Consequently, the growth rate decreases. On the contrary, as the binder concentration increases (Exp. A, D, and E) the growth rate increases. Binder droplets become more viscous when the binder concentration increases and the fulfillment of the Stokes criterion [44] upon 21

Journal Pre-proof wet collision becomes easier. A more viscous liquid can rather dissipate the kinetic energy of colliding particles, leading to a higher growth rate [14,15]. Very similar behavior was seen in [1,15]. However, as can be expected, the new model is computationally slower than the previous model in [15]. (Figure 18) Using the new model (in-situ) the morphology of the agglomerates formed at every time step

oo

f

can provide the foundation for incorporating breakage in the simulation by calculating the strength of agglomerates by means of Eq. (2). Both porosity and coordination number of an

pr

agglomerate, that can be predicted using the new model, are required to fully assess its

e-

strength. Further upon calculation of the surface area of the formed agglomerates, the

Pr

description of the droplet (binder) deposition mechanism in the model could be enhanced.

al

7. Conclusion and outlook

rn

Morphological descriptors are important in order to capture the structure of spray fluidized bed agglomerates and macroscopic properties, like strength, associated with them. From the

Jo u

plethora of aggregation models in literatures, the PC aggregation model offers a good prognosis for SFB agglomerates. Tuning of the aggregation model by means of the sticking probability was not favorable for spray fluidized bed agglomeration. Fractal dimensions obtained in this way were in a reasonable range of 2 to 3, and porosity reached an asymptotic minimum with decreasing sticking probability, but this minimum was still too high compared to the porosity of real SFB agglomerates determined by Dadkhah and Tsotsas in [16]. The tunable PC aggregation model developed by Filippov et al. [40] was therefore used for reconstructing the SFB agglomerates. The fractal dimension and porosity obtained with this model were comparable to those of real SFB agglomerates, but the algorithm was limited to prefactors less than 1.66 according to [40]. 22

Journal Pre-proof A novel approach of tuning the fractal dimension with a given prefactor (equal to one) has been introduced to overcome the limitation of the prefactor. This approach was validated by the experimental data, as the structural features (porosity and coordination number) remained essentially the same when tuned at different prefactor. Irrespectively of the variable fractal dimension for each agglomerate in the TSA model, reconstructed agglomerates were in close affinity to the experimental agglomerates [38]. We have studied in detail the structural properties of agglomerates evaluated by Dadkhah using the TSA model. The results of the

oo

f

TSA model strongly agreed with the experiments [16]. Knowing the coordination number

pr

and porosity of the agglomerates, the strength of the agglomerates was calculated, which could be used further to model breakage. The TSA model generated the agglomerates with

e-

monodisperse primary particles, so that it is a future goal to enhance the model by introducing

Pr

size dispersity.

al

Earlier only the experimental agglomerates validated the porosity correlation (Eq. (5))

rn

developed in [15]. Now, it was found to strongly coincide with the porosity of synthetic agglomerates generated using the TSA model. This validates the porosity correlation and the

Jo u

TSA model conjointly. Further, a new (modified) CVMC model has been developed by incorporating the TSA model to mimic the morphology of agglomerates formed. Two ways of implementing the TSA model in CVMC simulations were presented, and the in-situ implementation was very promising. Though this way of implementation is computationally slower, it has the advantage of breaking path for the future goal of implementing real structural parameters to compute micro-scale mechanisms like droplet deposition and breakage. The new model predicts reasonably the kinetics as well as determines the morphological descriptors of the agglomerates formed at different operating conditions of SFB agglomeration. The prediction is expected to be improved by introducing breakage and

23

Journal Pre-proof enhancing the description of droplet deposition in the CVMC model, which would be a future goal. Acknowledgment The present work was conducted in the frame of the graduate school “Micro–MacroInteractions in Structured Media and Particle Systems” (GRK 1554), funded by the Deutsche

Jo u

rn

al

Pr

e-

pr

oo

f

Forschungsgemeinschaft (DFG).

24

Journal Pre-proof Notation D

m

fractal dimension

-

cohesive force

N

k

prefactor

-

N

number of particles

-

sticking probability

-

radius of particle

oo

R

f

diameter

pr

radius of an aggregate

killing radius

m m

al

z

m

m

temperature of the fluidization gas

℃

position on X-axis

m

mass fraction of the binder

-

position on Y-axis

m

position on Z-axis

m

porosity

-

strength

Pa

rn

x

m

radius of generating sphere

Jo u

T

Pr

launching radius

e-

radius of gyration

m

Greek letters

Subscripts agg

agglomerate

i

particle index 25

Journal Pre-proof j

particle index

lim

limiting value

p

primary particle

cluster-cluster

CH

convex hull

CN

coordination number

CVMC

constant volume Monte Carlo

DLA

diffusion- limited aggregation

HPMC

hydroxypropylmethylcellulose

MCN

mean coordination number

PC

particle-cluster

RLA

reaction-limited aggregation

SA

sequential algorithm

SFB

spray fluidized bed

TSA

tunable sequential aggregation

Jo u

rn

al

Pr

e-

pr

oo

CC

f

Abbreviations

26

Journal Pre-proof References [1]

K. Terrazas-Velarde, M. Peglow, E. Tsotsas, Stochastic simulation of agglomerate formation in fluidized bed spray drying: A micro-scale approach, Chem. Eng. Sci. 64 (2009) 2631–2643. doi:10.1016/j.ces.2009.02.041.

[2]

M. Dernedde, M. Peglow, E. Tsotsas, Stochastic modeling of fluidized bed agglomeration: Determination of particle moisture content, Dry. Technol. 31 (2013) 1764–1771. doi:10.1080/07373937.2013.810638.

f

L. Mörl, S. Heinrich, M. Peglow, Fluidized bed spray granulation, in: Handb. Powder

oo

[3]

Technol., 2007: pp. 21–188. doi:10.1016/S0167-3785(07)80037-6. M. Peglow, S. Antonyuk, M. Jacob, S. Palzer, S. Heinrich, E. Tsotsas, Particle

pr

[4]

&

Co.

KGaA,

Weinheim,

Germany,

2014:

pp.

295–378.

Pr

GmbH

e-

formulation in spray fluidized beds, in: Mod. Dry. Technol., Wiley-VCH Verlag

doi:10.1002/9783527631728.ch20.

A. Bück, E. Tsotsas, K. Sommer, Size Enlargement, in: Ullmann’s Encycl. Ind. Chem.,

al

[5]

rn

Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany, 2014: pp. 1–47.

[6]

Jo u

doi:10.1002/14356007.b02_07.pub2. C. Rieck, A. Bück, E. Tsotsas, Estimation of the dominant size enlargement mechanism in spray fluidized bed processes, AIChE J. (n.d.). [7]

M. Dadkhah, M. Peglow, E. Tsotsas, Characterization of the internal morphology of agglomerates produced in a spray fluidized bed by X-ray tomography, Powder Technol. (2012). doi:10.1016/j.powtec.2012.05.051.

[8]

R. Pashminehazar, A. Kharaghani, E. Tsotsas, Determination of fractal dimension and prefactor of agglomerates with irregular structure, Powder Technol. 343 (2019) 765– 774. doi:10.1016/j.powtec.2018.10.046.

[9]

C.M. Sorensen, G.C. Roberts, The prefactor of fractal aggregates, J. Colloid Interface Sci. 186 (1997) 447–452. doi:10.1006/jcis.1996.4664. 27

Journal Pre-proof [10]

A.M. Brasil, T.L. Farias, M.G. Carvalho, U.O. Koylu, Numerical characterization of the morphology of aggregated particles, J. Aerosol Sci. 32 (2001) 489–508. doi:10.1016/S0021-8502(00)00097-5.

[11]

P. Klobes, K. Meyer, R.G. Munro, Porosity and specific surface area measurements for solid materials, Special pu, NIST, 2006.

[12]

M. Peglow, J. Kumar, R. Hampel, E. Tsotsas, S. Heinrich, Towards a complete population balance model for fluidized-bed spray agglomeration, Dry. Technol. (2007).

oo

[13]

f

doi:10.1080/07373930701438899.

M. Hussain, J. Kumar, M. Peglow, E. Tsotsas, Modeling spray fluidized bed

pr

aggregation kinetics on the basis of Monte-Carlo simulation results, Chem. Eng. Sci.

K. Terrazas-Velarde, M. Peglow, E. Tsotsas, Kinetics of fluidized bed spray

Pr

[14]

e-

101 (2013) 35–45. doi:10.1016/j.ces.2013.06.004.

agglomeration for compact and porous particles, Chem. Eng. Sci. 66 (2011) 1866–

A.K.

Singh,

E.

Tsotsas,

Stochastic

model to

simulate spray fluidized

bed

rn

[15]

al

1878. doi:10.1016/j.ces.2011.01.037.

Jo u

agglomeration: A morphological approach, Powder Technol. 355 (2019) 449–460. doi:10.1016/j.powtec.2019.07.075. [16]

M. Dadkhah, E. Tsotsas, Influence of process variables on internal particle structure in spray

fluidized

bed

agglomeration,

Powder

Technol.

258

(2014)

165–173.

doi:10.1016/j.powtec.2014.03.005. [17]

R. Pashminehazar, A. Kharaghani, E. Tsotsas, Three dimensional characterization of morphology and internal structure of soft material agglomerates produced in spray fluidized

bed

by

X-ray

tomography,

Powder

Technol.

300

(2016)

46–60.

doi:10.1016/j.powtec.2016.03.053. [18]

M. Dernedde, M. Peglow, E. Tsotsas, A novel, structure-tracking monte carlo algorithm for spray fluidized bed agglomeration, AIChE J. 58 (2012) 3016–3029. 28

Journal Pre-proof doi:10.1002/aic.13709. [19]

M. Sommerfeld, S. Stübing, A novel Lagrangian agglomerate structure model, Powder Technol. 319 (2017) 34–52. doi:10.1016/j.powtec.2017.06.016.

[20]

M. Dosta, S. Antonyuk, S. Heinrich, Multiscale simulation of agglomerate breakage in fluidized beds, Ind. Eng. Chem. Res. 52 (2013) 11275–11281. doi:10.1021/ie400244x.

[21]

X. Deng, Z. Huang, W. Wang, R.N. Davé, Investigation of nanoparticle agglomerates properties using Monte Carlo simulations, Adv. Powder Technol. 27 (2016) 1971–

oo

[22]

f

1979. doi:10.1016/j.apt.2016.06.029.

C. Li, Z. Zhou, R. Zou, K. Dong, D. Pinson, A. Yu, Equivalent packing size of

J. Gaume, H. Löwe, S. Tan, L. Tsang, Scaling laws for the mechanics of loose and

Pr

[23]

e-

doi:10.1016/j.powtec.2018.04.035.

pr

spheroidal particles: A microscopic test, Powder Technol. 333 (2018) 286–292.

cohesive granular materials based on Baxter’s sticky hard spheres, Phys. Rev. E. 96

T.-T. Vo, P. Mutabaruka, J.-Y. Delenne, S. Nezamabadi, F. Radjai, Strength of wet

rn

[24]

al

(2017) 1–12. doi:10.1103/PhysRevE.96.032914.

Jo u

agglomerates of spherical particles: Effects of friction and size distribution, EPJ Web Conf. 140 (2017) 08021. doi:10.1051/epjconf/201714008021. [25]

M. Kamyabi, K. Saleh, R. Sotudeh-Gharebagh, R. Zarghami, Effects of the number of particles

and

coordination

number

on

viscous-flow

agglomerate

sintering,

Particuology. 43 (2019) 76–83. doi:10.1016/j.partic.2018.01.015. [26]

H. Rumpf, Grundlagen und Methoden des Granulierens, Chemie Ing. Tech. 30 (1958) 144–158. doi:10.1002/cite.330300307.

[27]

C. Weiler, M. Wolkenhauer, M. Trunk, P. Langguth, New model describing the total dispersion of dry powder agglomerates, Powder Technol. 203 (2010) 248–253. doi:10.1016/j.powtec.2010.05.015.

[28]

P. Meakin, Historical introduction to computer models for fractal aggregates, J. Sol-Gel 29

Journal Pre-proof Sci. Technol. 15 (1999) 97–117. doi:10.1023/A:1008731904082. [29]

S.G. Alves, S.C. Ferreira Jr., M.L. Martins, Strategies for optimize off-lattice aggregate simulations,

Brazilian

J.

Phys.

38

(2008)

81–86.

doi:10.1590/S0103-

97332008000100016. [30]

R.C. Ball, T.A. Witten, Particle aggregation versus cluster aggregation in high dimensions, J. Stat. Phys. 36 (1984) 873–879. doi:10.1007/BF01012946.

[31]

U. Kätzel, R. Bedrich, M. Stintz, R. Ketzmerick, T. Gottschalk-Gaudig, H. Barthel,

oo

f

Dynamic light scattering for the characterization of polydisperse fractal systems: I. Simulation of the diffusional behavior, Part. Part. Syst. Charact. 25 (2008) 9–18.

J.F. Dirkse, J.D. Cawley, A modified ballistic aggregation model, J. Colloid Interface

e-

[32]

pr

doi:10.1002/ppsc.200700004.

[33]

T.A.

Witten,

Sander,

Phys.

Diffusion-limited

Rev.

Lett.

aggregation, 47

a

(1981)

kinetic

critical

1400–1403.

al

phenomenon,

L.M.

Pr

Sci. 170 (1995) 466–476. doi:10.1006/jcis.1995.1126.

P. Meakin, The Vold-Sutherland and Eden models of cluster formation, J. Colloid

Jo u

[34]

rn

doi:10.1103/PhysRevLett.47.1400.

Interface Sci. 96 (1983) 415–424. doi:10.1016/0021-9797(83)90044-9. [35]

M.J. Vold, Computer simulation of floc formation in a colloidal suspension, J. Colloid Sci. 18 (1963) 684–695. doi:10.1016/0095-8522(63)90061-8.

[36]

L.A. Turkevich, H. Scher, Sticking probability scaling in diffusion-limited aggregation, in: Fractals Phys., Elsevier, 1986: pp. 223–229. doi:10.1016/B978-0-444-869951.50043-3.

[37]

C.Y. Wang, P.L. Liu, J.B. Bassingthwaighte, Off-lattice Eden-C cluster growth model, J. Phys. A. Math. Gen. (1995). doi:10.1088/0305-4470/28/8/008.

[38]

M. Dadkhah, Morphological characterization of agglomerates produced in a spray fluidized bed by X-ray tomography, Otto-von-Guericke-University Magdeburg, 2014. 30

Journal Pre-proof [39]

D.N. Sutherland, A theoretical model of floc structure, J. Colloid Interface Sci. 25 (1967) 373–380. doi:10.1016/0021-9797(67)90043-4.

[40]

A.V. Filippov, M. Zurita, D.E. Rosner, Fractal-like aggregates: Relation between morphology and physical properties, J. Colloid Interface Sci. 229 (2000) 261–273. doi:10.1006/jcis.2000.7027.

[41]

C.M. Sorensen, The mobility of fractal aggregates: A review, Aerosol Sci. Technol. 45 (2011) 765–779. doi:10.1080/02786826.2011.560909.

f

M. Lapuerta, R. Ballesteros, F.J. Martos, A method to determine the fractal dimension

oo

[42]

of diesel soot agglomerates, J. Colloid Interface Sci. 303 (2006) 149–158.

N. Metropolis, S. Ulam, The Monte Carlo method., J. Am. Stat. Assoc. 44 (1949) 335–

e-

[43]

pr

doi:10.1016/j.jcis.2006.07.066.

A.A. Adetayo, B.J. Ennis, Unifying approach to modeling granule coalescence

rn

al

mechanisms, AIChE J. 43 (1997) 927–934. doi:10.1002/aic.690430408.

Jo u

[44]

Pr

341. doi:10.1080/01621459.1949.10483310.

31

Journal Pre-proof List of figures Fig. 1. Morphological change of agglomerates (

2

) with the same prefactor (

2

) with varying prefactor and same

)

and varying fractal dimension. Fig. 2. Morphological change of agglomerates ( ).

fractal dimension (

Fig. 3. Particle-cluster aggregation (left) and cluster-cluster aggregation (right).

f

Fig. 4. On-lattice aggregation (left) and off-lattice aggregation (right).

oo

Fig. 5. PC aggregates of 100 primary particles generated on-lattice using DLA (left) and RLA

pr

at sticking probability 0.0001 (right) models.

e-

Fig. 6. Flowchart and pictorial representation of the sticking probability model. Fig. 7. The fractal dimension of agglomerates obtained at different sticking probabilities ( )

Pr

using the sticking probability model.

al

Fig. 8. A representation of aggregation of monodisperse primary particles using the sequential algorithm.

rn

Fig. 9. Porosity using the convex hull (CH) method (top left) and the radius of gyration

Jo u

method (top right) and coordination number distribution (below) of an agglomerate ( ) generated using SA at

and

.

Fig. 10. Logarithmic plots of power law from the origin (dashed lines) with same prefactor (

) and different slopes (

with prefactor (

2

Fig. 11. Agglomerates (

, 2.81, 2.66) leading to the desired plot (solid line)

) and fractal dimension (

2

).

) generated using SA with tuned

and

according to Table

4. Fig. 12. The CVMC modeling scheme for SFB agglomeration. Fig. 13. Two ways of implementing the TSA model in the CVMC simulation: postprocessing (left) and in-situ (right). 32

Journal Pre-proof Fig. 14. Comparison of porosity obtained from Eq. (5) with

,

from Table 5 (from

experiments or simulations) with the porosity of real agglomerates from [38] and agglomerates reconstructed by the means of the TSA model (both evaluation using the gyration method). Fig. 15. Mean coordination number (MCN) of agglomerates for a set of experiments [16] (left) compared with synthetic agglomerates using the TSA model (right). Fig. 16. Strength calculated using Eq. (2) for the agglomerates generated by means of the

f

⁄

(right) for a set of

oo

TSA model with MCN from TSA model (left) and with

pr

experiments [16].

Fig. 17. SFB agglomerates of different sizes reconstructed using the TSA model for different

e-

experimental trials.

agglomerates for

Pr

Fig. 18. Agglomerate diameter with respect to the number of primary particles, along with (left); relative average diameter with respect to time, using

Jo u

rn

al

different model versions for the different experiments (right).

33

Journal Pre-proof List of Tables Table 1. Fractal dimension of aggregates generated by different models according to Meakin [28]. Table 2. Agglomerates (

) generated using the sticking probability model at

different sticking probabilities. Table 3. Tunable methods used by different researchers in the past.

) from Exp. B [38].

model and compared

f

with an agglomerate (

) generated using SA with tuned

oo

Table 4. Agglomerates (

Jo u

rn

al

Pr

e-

pr

Table 5. Experimental and simulation results for each trial.

34

Journal Pre-proof

Jo u

rn

al

Pr

e-

pr

oo

f

Table 1. Fractal dimension of aggregates generated by different models according to Meakin [24]. Model Particle-cluster (PC) Cluster-cluster (CC) Diffusion- limited (off-lattice) 2.50 1.80 Ballistic (off-lattice) 3.00 1.95 Reaction-limited (on-lattice) 3.00 2.09

35

Journal Pre-proof

0.01

0.001

0.922

0.813

0.765

0.745

6

14

523

al

Pr

e-

pr

oo

60

rn

Computation time (s)

0.1

Jo u

Porosity (

) generated using the sticking probability model at

f

Table 2. Agglomerates ( different sticking probabilities. Sticking 1 probability

36

Journal Pre-proof Table 3. Tunable methods used by different researchers in the past. Year Researcher

Method

Comments

1994

Thouy and Jullien [36] CC tuned with

1995

Mackowski [37]

2 not possible No comment on Tuned with the power law and First to use as a tuning factor

2000

Filippov et al. [38]

2009

Chakrabarty et al. [39]

2014

Skorupski et al. [40]

not possible

Jo u

rn

al

Pr

e-

pr

oo

f

Tuned with the power law and not possible for ; made new correlation for PC; using CC 2 particle position (sequential not possible algorithm) FracMAP: PC tuned with the 2 not possible power law not possible FLAGE: Using Filippov model not possible for PC; using CC 2

37

Journal Pre-proof Table 4. Agglomerates ( with an agglomerate (

) generated using SA with tuned ) from Exp. B [34].

model and compared MCN

0.64 0.64 0.64 0.64

0.66 0.61 0.61 0.61

0.64 0.63 0.62 0.63

3.06 3.10 3.08 3.02

al

Pr

e-

pr

oo

f

2.01 1.2 1.0 0.8

rn

SA

2.31 2.68 2.81 2.97

Jo u

Experiment B

38

Journal Pre-proof

E 60 10 2.09 2.24 0.63 2.87 2.05 2.53 0.64 3.09

Jo u

rn

al

Pr

e-

pr

oo

f

Table 5. Experimental and simulation results for each trial. Experiment set A B C D Temperature [℃] 60 30 90 60 Binder [wt. %] 2 2 2 6 Experimental results for each trial [14] [-] 2.45 2.31 2.94 2.24 k [-] 1.76 2.01 0.98 1.96 Average [-] 0.57 0.62 0.53 0.58 Average [-] 3.32 3.10 4.02 2.92 TSA model results for each trial (averaged over 5 realizations) [-] 2.37 2.24 2.72 2.17 k [-] 2.12 2.37 1.48 2.32 Average [-] 0.54 0.59 0.53 0.55 Average [-] 3.61 3.52 3.90 3.31

39

Journal Pre-proof 1.1

Highlights: A tunable sequential aggregation model was developed to generate agglomerates



Morphological descriptors are needed to understand breakage and other properties



Stochastic model of agglomeration was modified by implementing the aggregation model



Process kinetics were evaluated at different operating conditions



Morphology of different agglomerates was analyzed and compared

Jo u

rn

al

Pr

e-

pr

oo

f



40

Journal Pre-proof

e-

pr

oo

f

Graphical abstract

Pr

A tunable sequential aggregation model is incorpareted in Monte Carlo simulations to capture

Jo u

rn

al

the morphological descriptors and process kinectics at various operating conditions.

41

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

Figure 11

Figure 12

Figure 13

Figure 14

Figure 15

Figure 16

Figure 17

Figure 18