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 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
.
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 (
= 251.5
m) in comparison to platelet graphite agglomerates (
= 87.0
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
= 250.3
m. Conversely, the agglomerates of spherical graphite particles exhibit a reduction in size between 30 s and 90 s, reaching a size of
= 178.2
m. Afterwards, the agglomerates undergo a second increase until an agglomeration time of 600 s, at which point the size reaches
= 415.9
m. The final agglomerate size was determined to be 358.7
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
= 1.3
m and
= 27.1
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
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
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,
, 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 (
= 1000 kg/m
3,
= 1
Pas) are assumed in the stirred tank used. The power input per kilogram of liquid in the stirred tank is
= 5.74 W/kg. This value was calculated using the following parameters:
n = 1200 1/min,
= 700 g,
M = 0.0320 Nm. This results in a smallest vortex size of
= 21.8
m, which is much smaller than the agglomerates at 30 s (
= 251.6
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
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
= 128.0
m to
= 185.7
m, while the agglomerates themselves remain highly branched structures with a sphericity value of
0.46 and a packing density value
0.31. In torque phase III, the agglomerates are densified (
= 0.44) and more spherical (
= 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
= 0.54 to
= 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
= 0.70 (torque phase IV). During the plateau of
= 0.60, the agglomerates densify, as evidenced by the increase in packing density from
= 0.40 to
= 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
= 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
[
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
= 2.20 prior to the application of an external shear force. After the force was applied, the fractal dimension was found to be
= 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 (
= 2.23) than the fractal dimension of agglomerates consisting of spherical graphite (
= 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
= 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
= 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
= 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
exhibits two distinct peak regions, in contrast to the joint distribution of
, 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
= 88.0
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
.
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
= 87.0
m and
= 94.8
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
= 415.9
m, the agglomerate size of agglomerates obtained from experiment PG80 is
= 405.1
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
= 649.5
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
= 0.55 to
= 0.46, before it increases again to a value of
= 0.63 at the end of torque phase III. The same trend can be seen in the packing density (
Figure 10B), which decreases from
= 0.41 to
= 0.31 at the end of torque phase I and increases to the end of torque phase III to a value of
= 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
= 2.26 to
= 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
= 250.7
m, whereas the agglomerate size in experiment PG80 reaches a value of
= 649.5
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
= 2.27 compared to
= 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
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.