Torque Mapping as a Novel Method for In-Line Characterisation of Spherical Agglomeration Process Kinetics

Abstract

This study presents a novel method to determine the kinetics of spherical agglomeration in-line via monitoring the stirrer torque. In order to confirm that the torque can be used to characterise the process phases, samples were taken at different times during the agglomeration process and the morphology of agglomerates is examined using X-ray microtomography. It can be shown that if the stirrer torque is decreasing, the fast growth regime starts. Additionally, the influence of binding-liquid concentration as well as the primary particle shape was studied. Both have an influence on the process kinetics as well as on the different growth mechanisms of agglomerates during spherical agglomeration. These two growth mechanisms, distribution and immersion, can be identified by the morphological parameters sphericity, packing density and fractal dimension. In particular, the concept of fractal dimension was not used so far on agglomerates obtained from spherical agglomeration.

1. Introduction

Agglomeration is a size-enlargment process, where small primary particles interact and form agglomerates. The driving force for agglomeration are the interparticle forces, which were first described by Rumpf [1]. Processes based on capillary forces between particles are high-shear agglomeration [2], fluidised bed agglomeration [3] or spherical agglomeration [4]. The main difference between these agglomeration processes is the surrounding fluid, which is air for high-shear and fluidised bed agglomeration and a liquid for spherical agglomeration.

One of the first publications corresponding to spherical agglomeration is the patent of Cattermole [5] in 1904. The basis for spherical agglomeration is a three-phase system which is agitated by a stirrer. This three-phase system itself consists of a solid phase represented by the primary particles, which are suspended in a suspension liquid (typically water). Then a second liquid phase is added, which is immiscible with the suspension liquid and preferably wets the primary particles. This second liquid is called binding-liquid.

Due to the turbulent mixing of the stirrer the binding-liquid droplets and the particles collide in the beginning of the process. The wetting properties, characterised via the contact angle $\delta$, between the binding-liquid droplets and the particles (in the surrounding suspension liquid) decide if the particles are getting wetted. Hereby the size difference between the droplet and the particles play an important role for the growth mechanism, which can follow an immersion mechanism or a distribution mechanism [6].

Agglomeration growth following an immersion mechanism is caused by big binding-liquid droplets and small particles. A collision of them results in a particle-covered droplet. Dependent on the contact angle $\delta$, the particles stick onto the droplet surface [7] or penetrate it [8]. The distribution mechanism is characterised by a small binding-liquid droplet compared to the primary particles. After the collision and attachment, highly branched agglomerates are produced [9].

One of the first attempts to study the kinetic of this process was by Kawashima and Capes [10]. They took different samples out of the suspension and studied the agglomerate size with a microscope. According to them, the agglomerate growth follows a first order kinetic, which is related to a “restricted in space” growth.

Bemer [4] studied the agglomerate growth with a back-scattered light detector. This signal can be used to estimate the number and size of agglomerates. Based on his experiments, Bemer phrased the three process phases of spherical agglomeration: wetting period (divided into flocculation and zero-growth regime), fast-growth regime and equilibrium phase, which are depicted in Figure 1.

As the name says, in the wetting period the particles first are wetted by the binding-liquid droplets, as described above. Subsequently, a zero-growth regime may ensue if the entire surface area of the binding-liquid is wetted by particles. Due to turbulence and collision between the flocculated structures, the binding-liquid can be exposed on the surface again [11] and then the fast-growth regime starts. It is characterised by a sharp increase in the agglomerate size with the agglomeration time. If the binding forces in the agglomerates are balanced with the shear forces applied through the turbulent mixing, the equilibrium agglomerate size is reached and therefore the equilibrium phase. The wetting phase and the fast growth regime can also be optically distinguished via the turbidity of the suspension. With the sharp increase in agglomerate size, the turbidity also decreases sharply. This sharp increase in the agglomerate size is also reported by Kawashima and Capes [10], Müller and Löffler [9], Blandin et al. [12] and Subero-Couroyer et al. [8].

So far, the process kinetics are determined via measuring the time-dependent agglomerate size, as described above. There exist some measuring techniques in the literature, like the mentioned back-scattered light detector by Bemer [4], but authors also used laser diffraction [7,9], sieve analysis [13] and 2D image analysis of samples taken out of the agglomeration process [9,10] or by online 2D image analysis of images taken through the transparent stirrer vessel wall [8,12]. All these measuring techniques have in common that they cannot be performed in-line.

A recently published paper by Schreier and Bröckel [14] mentioned that torque mapping during spherical agglomeration can be used to detect the end of the wetting phase and the beginning of the fast-growth regime. In this paper, we want to clarify that this assumption is right. To prove it, different samples were taken out of spherical agglomeration experiments, where the torque is measured precisely. The samples were characterised with X-ray microtomography (μCT) and analysed by 3D image analysis [15]. The results are compared with the measured torque to gain insight about what is happening in the different torque gradients. Therefore not only the agglomerate size is determined, but also the morphology represented by sphericity, packing density and fractal dimension. In particular, the concept of fractal dimension has so far not been applied on agglomerates produced during spherical agglomeration. Also, the influence of primary particle shape and binding-liquid concentration is studied in detail.

2. Materials and Methods

2.1. Materials

As Solid phase graphite particles with a spherical shape (SGB 23 L, Graphit Kropfmühl GmbH, Hauzenberg, Germany) and platelet shape (SGA 20, Graphit Kropfmühl GmbH) are used. The size distribution of these particles are determined by laser diffraction (Analysette 22 Micro Tec Plus, Fritsch, Idar-Oberstein, Germany, Figure 2A). The graphite particles show similar size distributions with a range between $x_{10}$ and $x_{90}$ values of $x_{10,\mathrm{Platelet}}$ = 6.8 $\mathsf{\mu }$m and $x_{90,\mathrm{Platelet}}$ = 31.4 $\mathsf{\mu }$m compared to $x_{10,\mathrm{Spherical}}$ = 10.9 $\mathsf{\mu }$m and $x_{90,\mathrm{Spherical}}$ = 35.9 $\mathsf{\mu }$m. The different shapes of the particles can be seen in the SEM images in Figure 2B (spherical graphite) and in Figure 2C (platelet graphite). Spherical graphite is produced out of platelet graphite by a spheronisation process, which is described by Frey [16].

The suspension liquid for spherical agglomeration is water, whereas Highly Liquid Paraffin Oil (Merck KGaA, Darmstadt, Germany, CAS-Nr. 8002-74-2) is used as binding-liquid with an average density of $\rho$ = 850 kg/m³ and an average dynamic viscosity of $\eta$ = 35 mPas.

2.2. Agglomeration Experiment

The agglomeration experiments are carried out in a rheometer (Kinexus Ultra Plus, Netzsch, Selb, Germany) with a torque resolution of 0.05 nNm. In order to install the one litre stirred vessel with four baffels and the inclined, four-bladed stirrer, a self-made fixture is mounted. Graphite particles ($m_{\mathrm{G}}$ = 28 g) are suspended in water ($m_{\mathrm{W}}$ = 560 g) for t = 5 min with n = 700 rpm. Meanwhile, paraffin oil is emulsified in water ($m_{\mathrm{W}}$ = 140 g) externally, using a dispersing unit (Ultra Turrax, IKA, Staufen im Breisgau, Germany) at n = 16,000 rpm for t = 2 min. To study the influence of the binding-liquid concentration, $m_{\mathrm{P}}$ = 8.4 g and $m_{\mathrm{P}}$ = 22.4 g are used. These represent a binding-liquid concentration of 30% and 80%, referring to the amount of graphite. The emulsion is then added to the suspension and the agglomeration process is started with a stirrer speed of n = 1200 rpm. After a total agglomeration time of $t_{\mathrm{Agglomeration}}$ = 3600 s (1 h), the agglomeration is stopped and the suspension is sieved with a mesh size of 100 $\mathsf{\mu }$m to obtain the agglomerates. They are dried at a temperature of T = 80 °C until they are free flowing.

The total agglomeration time of 1 h is chosen due to the fact that afterwards the shock-freezing of the ethanol/dry ice mixture was not fast enough any more, so further sampling was not possible (see Section 2.3.1).

In Section 3, the results from different parameter sets of spherical agglomeration are discussed. To study the influence of the primary particle shape, the same binding-liquid concentration of 30% paraffin oil is used. These experiments are abbreviated with SG30 (spherical graphite with 30% paraffin oil) and PG30 (platelet graphite with 30% paraffin oil). Additionally, another agglomeration experiment is carried out with 80% paraffin oil, which is abbreviated with PG80 (platelet graphite with 80% paraffin oil).

2.3. Agglomerate Morphology

The methodology of studying the agglomerate morphology was published by Schreier and Bröckel in 2024 [15]. All further developments on the method to characterise the morphology of the agglomerates over the whole agglomeration experiment are presented here.

2.3.1. Sample Preparation

The aim of the sample preparation for the following $\mathsf{\mu }$CT scan is to produce a cylindrical sample, in which the agglomerates are fixed during the $\mathsf{\mu }$CT scan. Additionally, the mechanical influence during sampling should be as small as possible. Consequently, taking samples out of the agglomeration experiment with a tube and shock-freezing them in an ethanol/dry ice mixture is a suitable combination.

This kind of sample preparation is suitable for samples taken in the beginning of the agglomeration experiment, which is described in detail by Schreier and Bröckel [15]. With increasing agglomeration time and therefore increasing agglomerate size, the sedimentation velocity additionally increases and at some point becomes higher than the freezing velocity. As a result of the increased agglomerate size, the agglomerates sediment very quickly in the tube and are not separated enough any more. To reduce the sedimentation velocity, the only option is to use a medium with a high viscosity, which can further be shock-frozen with the ethanol/dry ice mixture.

An alginat-water mixture of $\beta$ = 1.7 wt.% with a dynamic viscosity of $\eta$ = 850 mPas reduces the sedimentation sufficiently enough and can also be shock-frozen in the ethanol/dry ice mixture. Consequently, the sample is therefore taken out of the agglomeration experiment and transferred into the alginat-water mixture. With a new tube (inner diameter $d_{\mathrm{i}}$ = 5 mm), the agglomerates are taken again and then shock-frozen. The timing, where the sample preparation technique is changed, was determined via the torque minimum (see Section 3).

2.3.2. $\mathsf{\mu }$CT Scan

A $\mathsf{\mu }$CT (Skyscan 1272, Bruker, Billerica, MA, USA) is used for this study. The power of the X-ray source is set to 4.8 W/40 keV with an exposure time of 3000 ms and rotation steps of 0.3° per proejction image. This results in 1200 projection images per scan. The scan position along the z-axis is chosen to be 0.5 mm above the copper pin up to a scan height of $h\approx$ 3 mm. Additional scans along the z-axis of the sample are omitted to minimise scan time. The chosen settings result in a total scan time of $t\approx$ 3 h and a resolution of 2.4 $\mathsf{\mu }$m/voxel for the tubes with $d_{\mathrm{i}}$ = 4 mm and a resolution of 3.0 $\mathsf{\mu }$m/voxel for tubes with $d_{\mathrm{i}}$ = 5 mm.

During an agglomeration experiment, up to eight samples are analysed with the $\mathsf{\mu }$CT. The sampling times are adapted to the corresponding torque curves which results in the experiment with the spherical graphite (SG30); the timing was 30 s, 90 s, 150 s, 210 s, 270 s, 600 s, 900 s, 1200 s. The same sampling interval is applied to the platelet graphite with 30% paraffin oil (30 s, 90 s, 150 s, 210 s, 600 s, 900 s, 3595 s). Due to the higher binding-liquid concentration of 80% paraffin, the sample timing was adapted to the torque curve. This results in sample timing of 30 s, 60 s, 120 s, 180 s, 360 s, 1680 s and 2880 s.

2.3.3. 3D Image Processing

The image processing and analysis routine and the computation of the morphometric parameters are carried out in a self-written script based on MatLab (R2022a) together with the toolboxes Image Processing, Statistics and Machine Learning and Curve Fitting. The image-processing operations are in italics. The MatLab commands are given in brackets to make the whole implementation easier for the reader.

The aim of the 3D image processing is to obtain a representation of every individual agglomerate within the scanned volume of the $\mathsf{\mu }$CT. For samples taken at the beginning of the agglomeration experiment, the 3D image processing is described elsewhere [15] in detail. Therefore, only the changes for agglomerates taken later in the agglomeration experiment are described here. An overview of all individual steps of the image processing is depicted in Figure 3.

For samples taken after the torque minimum, two main adjustments had to be done in the image processing, the filling up of the agglomerate boundaries and the deletion of big air bubbles caused by the modified sample preparation described in Section 2.3.1.

The main steps of filling up the agglomerate boundaries so far were to add the corresponding binding-liquid amount and then perform a 3D Fill Holes (imfill) to close any remaining holes in the agglomerates. This strategy works well with big paraffin droplets; the surface is covered by particles. During spherical agglomeration, the agglomerates become more compact and the binding-liquid is homogenously distributed within the agglomerates. This leads to the fact that the determination of the binding-liquid is influenced by the partial volume effect [17]. This leads to a mixed grey value for very small binding-liquid droplets and therefore using a global Threshold for binarisation could lead to a missing of these small binding-liquid droplets. Consequently, there are still holes inside the agglomerates because the addition of the binding-liquid does not contain information about the small binding-liquid droplets. If these small binding-liquid droplets are located directly at the surface of agglomerates, the output after the addition is an open pore, which cannot be closed by the 3D Fill Holes. Consequently, a Closing (imclose) [18,19,20] algorithm has to be applied to close the outer contour before the 3D Fill Holes.

Due to the transfer of the sample from the stirred vessel into the alginat-water mixture (see Section 2.3.1), there can exist big air bubbles between agglomerates or inside the frozen matrix. Additionally, air bubbles have a similar grey value compared to the paraffin oil used as binding-liquid. Consequently, they can only be excluded via the size, which is called Despeckle. Therefore, in an extra step a dataset is created, which consists only of the big air bubbles. This is done via setting the Threshold which is also used to determine the binding-liquid. Afterwards, the size of every object is calculated, whereas the value for the Despeckle is determined individually for every $\mathsf{\mu }$CT scan. All objects smaller than this value are deleted. This dataset is then subtracted from the 3D volume before applying the Distance Transform Watershed.

2.3.4. Morphological Parameters

The morphological parameters used in this study are the volume-equivalent sphere diameter $x_{\mathrm{V}}$, the sphericity according to Wadell $\Psi$, the packing density ${\rho }_{\mathrm{pack}}$ and the fractal dimension $D_{\mathrm{f}}$. The definitions and determination methods are described here. The sphericity according to Wadell $\Psi$ [21] represents the ratio between the volume-equivalent sphere diameter $x_{\mathrm{V}}$ and the surface-equivalent sphere diameter $x_{\mathrm{S}}$ raised to the power of 2 (Equation (1))

$$\Psi ={\left({\displaystyle \frac{x_{\mathrm{V}}}{x_{\mathrm{S}}}}\right)}^2\mathrm{with}{x}_{\mathrm{V}}=\sqrt[3]{{\displaystyle \frac{6V}{\pi }}}\mathrm{and}{x}_{\mathrm{S}}=\sqrt{{\displaystyle \frac{S}{\pi }}}$$

(1)

Therefore, the volume V and the surface S of every agglomerate is determined during the 3D image processing (see Section 2.3.3). The volume is calculated via counting the object voxels, whereas the surface is determined via a surface estimation [22].

The packing density ${\rho }_{\mathrm{pack}}$ in general is a measure of the compactness of an object consisting of several small objects [23]. Agglomerates consist of several small objects (primary particles), whereas the space-filling objects are represented by the solid particles and the total volume is the convex hull volume of the corresponding agglomerate (see Equation (2))

$${\rho }_{\mathrm{pack}}={\displaystyle \frac{V_{\mathrm{Particles}}}{V_{\mathrm{Convex}\mathrm{hull}}}}$$

(2)

The fractal dimension $D_{\mathrm{f}}$ in this study is determined by the box-counting method [24]. In general, this method is based on an equally spaced grid, which is placed over an object. Then all cubes (3D) that cover one piece of the object are counted. Then the box size is increased and the counting is repeated. The data points boxsize (L) and number of object boxes ($N\left(\mathrm{L}\right)$) are plotted against each other in a log-log plot. The fractal dimension then represents the slope of the straight line (Equation (3)).

$$D_{\mathrm{f}}=\underset{L\to 0}{lim}{\displaystyle \frac{\log \left(N\right(L\left)\right)}{\log (1/L)}}$$

(3)

3. Results and Discussion

The objective of this study is to identify which phase of the process corresponds to which slope in the torque curve in order to utilise torque mapping for process characterisation in the future. To achieve this goal, samples were extracted from the agglomeration process at different times and analysed using a combination of $\mathsf{\mu }$CT and 3D image processing.

In this section, all data are expressed as the 50% quantile value of the corresponding number weighted distribution.

3.1. Influence of Primary Particle Shape

In order to investigate the influence of primary particle shape on the torque behaviour of the stirrer during spherical agglomeration, a concentration of 30% paraffin oil in accordance with the graphite amount is used. Figure 4 illustrates the torque curves together with the volume-equivalent sphere diameter $x_{\mathrm{V}}$.

As illustrated in Figure 4, three distinct slopes of the torque curve can be identified from the experiment with platelet graphite (Figure 4A) and four different slopes from the experiment with spherical graphite (Figure 4B). In the initial stages of the experiment, the torque decreases in both cases until an agglomeration time of 210 s (PG30) or 150 s (SG30), which is related to torque phase I. In experiment PG30, the torque remains at a minimum value until an agglomeration time of 1900 s (torque phase II), after which it subsequently increases until the maximum agglomeration time of 3600 s (torque phase III). In experiment SG30, torque phase II is concentrated in the minimum at 150 s and directly increases up to 600 s (torque phase III). After 600 s, until 3600 s the torque remains stable in a plateau, which is designated as torque phase IV.

It can be observed that after 30 s of agglomeration, the mean agglomerate size (Figure 4C) is greater for spherical graphite agglomerates ($x_{\mathrm{V},\mathrm{SG}30,30\mathrm{s}}$ = 251.5 $\mathsf{\mu }$m) in comparison to platelet graphite agglomerates ($x_{\mathrm{V},\mathrm{PG}30,30\mathrm{s}}$ = 87.0 $\mathsf{\mu }$m). Subsequently, the agglomerates of platelet graphite continue to grow throughout the entire agglomeration experiment, spanning from 30 s to 3600 s, reaching a final size of $x_{\mathrm{V},\mathrm{PG}30,3595\mathrm{s}}$ = 250.3 $\mathsf{\mu }$m. Conversely, the agglomerates of spherical graphite particles exhibit a reduction in size between 30 s and 90 s, reaching a size of $x_{\mathrm{V},\mathrm{SG}30,90\mathrm{s}}$ = 178.2 $\mathsf{\mu }$m. Afterwards, the agglomerates undergo a second increase until an agglomeration time of 600 s, at which point the size reaches $x_{\mathrm{V},\mathrm{SG}30,600\mathrm{s}}$ = 415.9 $\mathsf{\mu }$m. The final agglomerate size was determined to be 358.7 $\mathsf{\mu }$m.

The initial size difference between agglomerates at the onset of agglomeration and their development over time can be attributed to the varying growth mechanisms, described in Section 1. If the binding-liquid droplets are smaller than the particles, the agglomerate growth is governed by a distribution mechanism. Conversely, the immersion mechanism involves a relatively large binding-liquid droplet and small particles, which results in particle-covered droplets. It has been demonstrated that the platelet graphite particles follow a distribution mechanism (Figure 5A), whereas the spherical graphite particles follow an immersion mechanism (Figure 5C).

An emulsion with one part of the suspension liquid and the binding-liquid is produced externally in a rotor-stator system (see Section 2.2), which leads to a paraffin droplet size distribution between $x_{10}$ = 1.3 $\mathsf{\mu }$m and $x_{90}$ = 27.1 $\mathsf{\mu }$m (characterised with laser diffraction Analysette 22 Micro Tec Plus, Fritsch, Idar-Oberstein, Germany). This size distribution is smaller than the particle size distribution when coalescence does not occur during the agglomeration experiment. In the experiment involving spherical graphite particles, there are larger paraffin droplets observed (Figure 5C). But why do the paraffin droplets coalesce during agglomeration of spherical graphite particles, whereas the coalescence is inhibited during agglomeration of platelet graphite particles?

The platelet structure is characterised by a large planar surface and a small extension in depth. If it is assumed that the laser diffraction mainly determines the largest projection surface of the particles and thus the planar side of the platelets, then the particles have a low intrinsic mass for the same particle size but different particle shape. This results in a significantly greater number of particles for platelet-shaped particles than for spherical ones, given the same total mass of particles. If the total number of particles is higher, the probability of collisions between particles and binding-liquid droplets is increased, while the probability of collision between the binding-liquid droplets is reduced. This prevents the coalescence of the binding-liquid droplets. Consequently, there is a greater coalescence of droplets in the agglomeration of spherical graphite particles than in the agglomeration of platelet graphite particles, which results in the formation of larger droplets and a change in the growth mechanism.

Another characteristic of the immersion mechanism is that the agglomerate size decreases at the initiation of the agglomeration experiment before increasing again. This behaviour was also measured by Blandin et al. [12]. They assumed that the agglomerates are becoming denser at the beginning of the agglomeration process. However, due to the limitations of the 2D image analysis employed, no conclusions can be drawn about the internal structure of the agglomerates. The $\mathsf{\mu }$CT images used in this study demonstrate that the paraffin droplets within the agglomerates are also becoming smaller (Figure 5D). One potential explanation for this observation is that the agglomerates are disrupted by the repeated emulsification of the droplets due to the turbulence within the stirred vessel.

To reinforce this assumption, the micro-length scale according to Kolmogorov ${\eta }_{\mathrm{K}}$ is estimated. Zhou und Kresta [25] have shown that this correlates with the minimum droplet size that can be generated by turbulence in a stirred tank. The Kolmogorov length scale, ${\eta }_{\mathrm{K}}$, is an estimation of the size of the smallest turbulent vortices. All objects larger than the smallest vortices are affected by shear [26]. To estimate the size of these smallest vortices, the material data for pure water ($\rho$ = 1000 kg/m³, $\eta$ = 1$·{10}^{-3}$ Pas) are assumed in the stirred tank used. The power input per kilogram of liquid in the stirred tank is $ϵ$ = 5.74 W/kg. This value was calculated using the following parameters: n = 1200 1/min, $m_{\mathrm{water}}$ = 700 g, M = 0.0320 Nm. This results in a smallest vortex size of ${\eta }_{\mathrm{K}}$ = 21.8 $\mathsf{\mu }$m, which is much smaller than the agglomerates at 30 s ($x_{\mathrm{V}50,\mathrm{SG}30,30\mathrm{s}}$ = 251.6 $\mathsf{\mu }$m). It can therefore be postulated that the agglomerates are subjected to shear forces within the agitated vessel, resulting in the breakup of the agglomerates at a high paraffin content.

In contrast, the agglomerates consisting of platelet graphite particles following the distribution mechanism exhibit a slight increase in the paraffin droplet within the agglomerates (Figure 5B) due to coalescence with free binding-liquid droplets, which can be observed in the $\mathsf{\mu }$CT scans after 30 s agglomeration time. Simultaneously, there can be observed an increase in the number of isolated microagglomerates until 150 s. This isolation of microagglomerates is followed by an increase in sphericity. At the end of torque phase I (210 s), the microagglomerates interconnect again and build highly branched structures. This change in structure can also be detected by the decrease in sphericity and packing density (Figure 6A,B). During torque phase II, the agglomerate size increases from $x_{\mathrm{V},210\mathrm{s}}$ = 128.0 $\mathsf{\mu }$m to $x_{\mathrm{V},1200\mathrm{s}}$ = 185.7 $\mathsf{\mu }$m, while the agglomerates themselves remain highly branched structures with a sphericity value of $\Psi \approx$ 0.46 and a packing density value ${\rho }_{\mathrm{pack}}\approx$ 0.31. In torque phase III, the agglomerates are densified (${\rho }_{\mathrm{pack},3595\mathrm{s}}$ = 0.44) and more spherical (${\Psi }_{3595\mathrm{s}}$ = 0.58).

A comparison of the shape parameters of agglomerates growing via the distribution mechanism with those resulting from the immersion mechanism reveals that the latter leads to the formation of more spherical structures than the former (Figure 6A). During the initial stages of agglomeration, the sphericity increases from ${\Psi }_{\mathrm{SG}30,30\mathrm{s}}$ = 0.54 to ${\Psi }_{\mathrm{SG}30,150\mathrm{s}}$ = 0.60 in torque phase I and then remains stable until 600 s (end of torque phase II). Finally, the sphericity increases once more, reaching a value of ${\Psi }_{\mathrm{SG}30,1200\mathrm{s}}$ = 0.70 (torque phase IV). During the plateau of $\Psi$ = 0.60, the agglomerates densify, as evidenced by the increase in packing density from ${\rho }_{\mathrm{pack},\mathrm{SG}30,30\mathrm{s}}$ = 0.40 to ${\rho }_{\mathrm{pack},\mathrm{SG}30,600\mathrm{s}}$ = 0.53 over the same time interval.

The different growth mechanisms can also be observed in the fractal dimension (Figure 6C). Fractal objects have a rough and fragmented geometrical shape, which can be divided into smaller amounts that are a copy of the original object [27]. They are self-similar on different length scales. If these requirements are fulfilled, then a fractal dimension can be assigned to such objects. Common geometrical objects, such as cuboids and spheres, have a fractal dimension of $D_{\mathrm{f}}$ = 3.0, which is equal to the space dimension. Fractal objects, such as agglomerates, which consist of many small primary particles that adhere to each other, can have a fractal dimension value $D_{\mathrm{f}}\in [1,3]$ [28,29].

In general, the size of agglomeration partners as well as the application of an external shear force has an influence on the resulting fractal dimension. The particle–cluster agglomeration leads to more compact agglomerates and therefore to higher values in the fractal dimension [28]. Clusters are defined as objects which already consist of more primary particles and are therefore bigger than the primary particles. However, there also exist cluster–cluster agglomerations, which result in lower values of the fractal dimension [30], as both agglomeration partners have an approximately equal size. Furthermore, external shear forces have been shown to influence the fractal dimension too. Sonntag and Russel [31] flocculated polystyrene and analysed the fractal dimension of these flocs. The polystyrene flocs were observed to have a fractal dimension of $D_{\mathrm{f}}$ = 2.20 prior to the application of an external shear force. After the force was applied, the fractal dimension was found to be $D_{\mathrm{f}}$ = 2.48. The authors propose that this difference can be attributed to the rearrangement and compaction of the flocs, which occurs when the shear rate exceeds a certain value. This reorganisation and densification of the flocs can be attributed to the rearrangement of the weaker, string-like segments and the redistribution of particles within the flocs. It can be anticipated that this rearrangement and compaction process will occur during the spherical agglomeration process, given that the high turbulence within the stirred vessel facilitates these effects.

If these findings are applied to spherical agglomeration, it can be concluded that the particle–cluster agglomeration is related to the immersion mechanism, whereas the distribution mechanism is related to the cluster–cluster agglomeration. This phenomenon can be identified in Figure 6C, where the fractal dimension of agglomerates consisting of platelet particles is initially lower ($D_{\mathrm{f},\mathrm{PG},30\mathrm{s}}$ = 2.23) than the fractal dimension of agglomerates consisting of spherical graphite ($D_{\mathrm{f},\mathrm{SG},30\mathrm{s}}$ = 2.44). In both experiments, the fractal dimension increases with agglomeration time, in accordance with the volume-equivalent sphere diameter. With regard to the experiment of spherical graphite particles, the fractal dimension reaches a plateau after 150 s (end of torque phase I) at $D_{\mathrm{f}}$ = 2.48 until 600 s (end of torque phase III). This is consistent with the plateau observed in the sphericity (Figure 6A) and the volume-equivalent sphere diameter (Figure 4C). Subsequently, a slight increase is identified up to a fractal dimension of $D_{\mathrm{f}}$ = 2.51 in torque phase IV. In contrast, the fractal dimension of agglomerates consisting of platelet graphite particles exhibits a steady increase until 3600 s, reaching a value of $D_{\mathrm{f}}$ = 2.50 in conjunction with the volume-equivalent sphere diameter.

Until now, the shape parameters have been analysed individually. However, as previously stated, the shape parameters also depend on the agglomerate size. In order to visualise this relationship, the joint number distribution of the shape parameter and the agglomerate size were determined using two-dimensional kernel density estimators [15]. As the number of agglomerates decreases significantly over the entire process, this characterisation is only meaningful at the beginning of agglomeration. Figure 7 shows the joint distribution of sphericity and volume-equivalent sphere diameter as well as the joint distribution of packing density and volume-equivalent sphere diameter for both experiments PG30 and SG30.

It can be seen in Figure 7 that, for both experiments, the smallest agglomerates are more compact and spherical than the larger agglomerates. This correlation was a priori expected, as the smallest agglomerates, regardless of the growth mechanisms discussed previously, contain a smaller amount of binding-liquid and are most affected by the partial volume effect [17]. In contrast, the largest agglomerates produced by the immersion mechanism initially have a high amount of binding-liquid (low packing density) and are less spherical due to the rough surface produced by the primary particles. The largest agglomerates produced by the distribution mechanism are highly branched. This is the reason why they have low values in packing density and sphericity.

The most noticeable difference in the joint distributions can be seen in the volume-equivalent sphere diameter-based fractal dimension (see Figure 8A,B). It is evident that $q_{0,\mathrm{PG}30}(x_{\mathrm{V}},D_{\mathrm{f}})$ exhibits two distinct peak regions, in contrast to the joint distribution of $q_{0,\mathrm{SG}30}(x_{\mathrm{V}},D_{\mathrm{f}})$, which displays a single peak.

Figure 8(A1–A5) illustrate some examples of the three-dimensional structure of agglomerates derived from experiment PG30. A comparison of agglomerates A2 and A3 reveals that these two agglomerates have the same volume-equivalent sphere diameter of $x_{\mathrm{V}}$ = 88.0 $\mathsf{\mu }$m, but different fractal dimensions. Agglomerate A2, with a fractal dimension of 2.37, is more spherical and compact, indicating that it follows the particle–cluster agglomeration mechanism described previously. In contrast, agglomerate A3, with a fractal dimension of 2.06, has a compact core and a side arm of comparable size to the core. This agglomerate follows a cluster–cluster agglomeration mechanism, which is characterised by smaller fractal dimensions in general. Furthermore, when comparing agglomerates A4 and A5 with A3, it can be observed that there are also agglomerates present which have side arms; however, the volume of these side arms is smaller in comparison with the core of the agglomerate. Consequently, A4 has a higher fractal dimension of 2.24 in comparison with A3, and the fractal dimension of A5 is also higher with a value of 2.37.

Comparing the structure of agglomerates from experiment PG30 with agglomerates obtained from the experiment SG30, the fractal dimension shows only one broad peak in the joint distribution $q_{0,\mathrm{SG}30}(x_{\mathrm{V}},D_{\mathrm{f}})$. Figure 8(B1–B5) also shows examples of agglomerate structures obtained from this experiment. It is directly visible that the agglomerates are not highly branched and the core is bigger compared to the agglomerates obtained by the distribution mechanism.

3.2. Influence of Binding-Liquid Concentration

Due to the fact that the platelet graphite with 30% paraffin oil shows no torque phase IV, there was the question if a higher binding-liquid concentration would lead to an appearance of torque phase IV. This was indeed the fact with 80% paraffin oil (Figure 9B), but then torque phase IV did not show a plateau like the spherical graphite, but a steady increase in the torque between 360 s and 3000 s. The minimum was reached directly after 100 s (torque phase I) which is immediately followed by torque phase III between 100 s and 360 s. Between 3000 s and 3600 s, the torque slope increases again in torque phase V, which is not in the focus of this study. Torque phase II is also concentrated in the minimum at 100 s.

In Figure 9C, the volume-equivalent sphere diameter dependent on the agglomeration time is shown. Both agglomeration experiments start with a similar agglomerate size after 30 s with $x_{\mathrm{V},30\mathrm{s},\mathrm{PG}30}$ = 87.0 $\mathsf{\mu }$m and $x_{\mathrm{V},30\mathrm{s},\mathrm{PG}80}$ = 94.8 $\mathsf{\mu }$m. Afterwards, the agglomerate size in experiment PG80 increases faster than the agglomerate size in experiment PG30. At the end of torque phase III, where the agglomerates of experiment SG30 reached almost their final size of $x_{\mathrm{V},\mathrm{SG}30,600\mathrm{s}}$ = 415.9 $\mathsf{\mu }$m, the agglomerate size of agglomerates obtained from experiment PG80 is $x_{\mathrm{V},\mathrm{PG}80,360\mathrm{s}}$ = 405.1 $\mathsf{\mu }$m. Due to the high binding-liquid concentration, the agglomerate size reaches no plateau but still increases until 2880 s up to a value of $x_{\mathrm{V},\mathrm{PG}80,2880\mathrm{s}}$ = 649.5 $\mathsf{\mu }$m. This was expected a priori, because an increase in binding-liquid concentration should lead to a faster agglomeration kinetic as well as to an increase in agglomerate size.

Nevertheless, the increased binding-liquid concentration leads to a faster agglomeration kinetic, but not to a change of the growth mechanism observed. This can be seen in the different shape parameters shown in Figure 10.

In Figure 10A, the sphericity dependent on the agglomeration time is shown. At the end of torque phase I, the sphericity also decreases from ${\Psi }_{60\mathrm{s}}$ = 0.55 to ${\Psi }_{120\mathrm{s}}$ = 0.46, before it increases again to a value of ${\Psi }_{360\mathrm{s}}$ = 0.63 at the end of torque phase III. The same trend can be seen in the packing density (Figure 10B), which decreases from ${\rho }_{\mathrm{pack},60\mathrm{s}}$ = 0.41 to ${\rho }_{\mathrm{pack},120\mathrm{s}}$ = 0.31 at the end of torque phase I and increases to the end of torque phase III to a value of ${\rho }_{\mathrm{pack},360\mathrm{s}}$ = 0.54. The decrease in these two values can only be seen in one sample during experiment PG80, because torque phase II is concentrated in a minimum at 100 s, whereas torque phase II in experiment PG30 lasts between 210 s and 1900 s.

As mentioned before, the volume-equivalent sphere diameter in experiment PG80 does not reach a tableau like in experiment SG30. At the same time, the torque reaches a tableau in torque phase IV in experiment SG30, but steadily increases in experiment PG80 in torque phase IV. This trend can also be seen in the fractal dimension, which is shown in Figure 10C. During the whole agglomeration experiment, the fractal dimension increases from $D_{\mathrm{f},30\mathrm{s}}$ = 2.26 to $D_{\mathrm{f},2880\mathrm{s}}$ = 2.54 and shows therefore higher values than the fractal dimension of agglomerates obtained in experiment PG30. Nevertheless, the agglomerate size in the end differs widely in $x_{\mathrm{V},\mathrm{PG}30,3595\mathrm{s}}$ = 250.7 $\mathsf{\mu }$m, whereas the agglomerate size in experiment PG80 reaches a value of $x_{\mathrm{V},\mathrm{PG}80,2880\mathrm{s}}$ = 649.5 $\mathsf{\mu }$m. The steady increase of agglomerate size and fractal dimension during experiment PG30 is related to the slower agglomeration kinetic, which can be seen in the long lasting torque phase II. In contrast, the steady increase in agglomerate size and fractal dimension during experiment PG80 is related to the high binding-liquid concentration, which leads to a third growth period seen in the blackberry structure of the agglomerates in the end of the experiment. Nevertheless, the fractal dimension also starts at the same value of $D_{\mathrm{f},30\mathrm{s},\mathrm{PG}80}$ = 2.27 compared to $D_{\mathrm{f},30\mathrm{s},\mathrm{PG}30}$ = 2.23. This shows that the agglomerates still follow a distribution mechanism with a higher binding-liquid concentration. This can also be seen in the joint number distribution $q_0(x_{\mathrm{V}},D_{\mathrm{f}})$ in the beginning after 30 s agglomeration time (Figure 11). There are also two peaks visible instead of the broad peak which is related to the immersion mechanism.

4. Conclusions

The aim of this study is to check the assumption that torque mapping during spherical agglomeration can be used as an in-line measurement technique for characterising the three distinct process phases of spherical agglomeration, namely wetting period, fast growth regime and equilibrium phase. The second aim was to introduce the fractal dimension as a structure parameter for agglomerates obtained by spherical agglomeration.

It can be outlined from the results shown in Section 3 that the change between the wetting period and the fast growth regime takes place during torque phase I where the torque is decreasing to a minimum. The end of the fast growth regime takes place at the end of torque phase III, the increasing part of the torque curve. This can be seen in the time-dependent fractal dimension in combination with the volume-equivalent sphere diameter. As long as the agglomerates grow in size (which is the case during torque phases I to III), the fractal dimension is increasing, whereas a stable agglomerate size (torque phase IV) corresponds with a stable fractal dimension.

Additionally, the growth mechanism depends on the primary particle shape. Platelet-shaped particles followed the distribution mechanism, whereas the spherical-shaped particles followed an immersion mechanism. If the same total mass of particles is used (as in this study), the number concentration of primary particles can highly differ, because of the different mass of the primary particles (see Section 3.1). The different growth mechanisms can be seen in the morphological parameters’ sphericity, packing density and also in the fractal dimension. Highly branched structures which are produced during the distribution mechanism show lower sphericity values, packing densities and fractal dimension. During the agglomeration process, in both cases of growth mechanisms, the agglomerates become more compact and spherical. This leads to an increase in all morphological parameters.

A higher concentration of binding-liquid as used in this study increases the speed, in which the agglomerates are formed and also the size of agglomerates, but it did not change the growth mechanism observed. This can clearly be seen by comparing experiments PG30 and PG80. At a higher binding-liquid concentration, the torque phase II is concentrated in the minimum, whereas it lasts between 210 s and 1900 s at a lower binding-liquid concentration.

In the future, this technique can be employed to characterise the process kinetics in-line by monitoring the stirrer torque. This will enable the identification of the optimal parameters and time interval for the two-dimensional selective spherical agglomeration, as described by Schreier and Bröckel [14]. This will facilitate the improvement of the quality of shape selectivity.