3.1. Mesh Quality Assessment
The mesh created for this study is one of the factors influencing the quality of numerical results, which should be independent of the element density, the ratio between the size of each element and the reference element (expansion factor), the quantity of tetrahedral and/or hexahedral elements, mesh refinement, among others. An inadequate mesh for a study negatively impacts the solution’s accuracy, the required simulation time, and the convergence (or divergence) rate of results, among other numerical aspects.
In this research, studies were conducted with the three meshes listed in
Table 3 to evaluate the influence of mesh resolution on the results of the filtering cylindrical hydrocyclone. Fluid velocity results were collected along two lines: one at
(cylindrical part) and the other at
(conical part), as illustrated in
Figure 6.
Figure 7 illustrates the fluid velocity along the
-position at y values of (a)
and (b)
, respectively, for the three developed meshes. It can be observed that the fluid velocity exhibits the same behavior for all three meshes, indicating that the constructed meshes do not influence the results. Furthermore, the similar behavior of the velocity profiles suggests that the results are not affected by mesh refinement at the levels used in this study. Therefore, due to the independence of the results from the analyzed meshes and having an intermediate level of refinement, mesh 02 was used in the subsequent simulations.
3.2. Flow Dynamics in the Hydrocyclone with Cylindrical Filtering Medium
Due to the limited number of studies in the literature on CFD applied to hydrocyclones containing the cylindrical part as a filter and the high complexity of the flow inside these devices, this research aims to contribute to this area, seeking to better understand the behavior of phases during the separation process in this type of equipment.
For the validation of this research, the predicted results of the hydrodynamic parameters obtained herein were compared with the experimental results reported by Façanha [
28]. From the numerical results, a pressure drops of 88 kPa was obtained in the filtering hydrocyclone, resulting in a predicted filtrate flow rate of 0.053 cm
3/s, a Euler number of 1049, a liquid ratio of 32.4, and a separation efficiency of 73.1%. The experimental results for these same parameters reported by Façanha [
28] were 88 kPa for the pressure drop, resulting in an error of 0.0%; 0.052 cm
3/s for the filtrate flow rate, corresponding to an error of 1.9%; 1012 for the Euler number, resulting in an error of 3.6%; and 32.4 for the liquid ratio, representing an error of 0.0%, and 74.9% for the separation efficiency, resulting in an error of 2.4%, which can be considered small given the complexity of the studied phenomenon.
Subsequently, we present an in-depth discussion about different hydrodynamic parameters. In
Figure 8, the pressure field inside the hydrocyclone is represented in the longitudinal
and transversal
planes at different positions:
,
,
,
, and
. It is observed that the pressure in the cylindrical part increases radially from the center to the wall, i.e., there are regions of low pressure near the central axis of the hydrocyclone and regions with higher pressures near the walls due to the centrifugal force caused by tangential velocity. This behavior is reaffirmed in
Figure 9b, where the tangential velocity field inside the HciF is represented in the XY plane with a feed flow rate of
. These results are similar to those observed in conventional and conical filtering hydrocyclones, as reported in the literature [
22,
35,
36,
37,
38].
In
Figure 9a–c, the axial, tangential, and radial velocity fields on the XY plane, respectively, are represented for the filtering cylindrical hydrocyclone (HciF) with volumetric feed flow
. The combination of tangential and radial velocities generates regions of low pressure inside the hydrocyclone, reaching values close to atmospheric pressure in the central region.
A comparison between
Figure 9a,c reveals that the filtration process influenced the characteristics of the axial and radial fluid velocity, confirming the behavior reported in the literature that the radial velocity of the fluid increases from the wall to the axis of the hydrocyclone [
14,
22,
45]. These velocity components induce particles to remain in the external vortex, leading them to exit the hydrocyclone through the lower outlet or “underflow”.
Upon analyzing
Figure 10, which illustrates the distribution of axial velocity components inside the hydrocyclone, a noticeable decrease in velocity is observed from the wall towards the center of the hydrocyclone, consistent with findings reported by Oliveira et al. [
38]. Furthermore, it is noted that the velocities have negative values, indicating recirculation zones. Throughout both the cylindrical and conical parts, it is observed that the recirculation region is more pronounced when there is an increase in radial velocity and a decrease in tangential velocity. This behavior is more accentuated in the conical part of the hydrocyclone, where the radial velocity is higher.
In
Figure 9b, it can be observed that the tangential velocity decreases inside the cylindrical part until it reaches the transition region with the conical part, where a significant decrease in tangential velocity occurs, being overcome by radial and axial velocities upon reaching the conical part, due to the progressive decrease in radius towards the ‘underflow’ outlet. As the fluid moves away from the hydrocyclones’ inlet section, the tangential velocity decreases. This may be a possible explanation for the fluid behavior shown in
Figure 9b, caused by the filtrating cylindrical wall, due to the additional outflow, i.e., the filtrate. The corresponding filtrate flow rate was 0.050 cm³/s, which is much smaller than that observed in the overflow and underflow outlets, but sufficient to make the flow inside the hydrocyclone quite complex.
In
Figure 10, the velocity vector field of the fluid is reported on the longitudinal XY plane for the HciF with a volumetric feed flow rate of 295.7 cm³/s. A chaotic flow pattern is observed in the upper part of the hydrocyclone, known as short circuit or by-pass according to Silva [
40], Façanha [
28], and Salvador et al. [
42]. This flow formation arises due to friction in the upper part of the hydrocyclone, which locally decelerates the flow. As the fluid collides with the wall, it creates a region of chaotic flow near the external wall of the vortex finder, considerably decreasing the material flow. Consequently, a portion of the fluid deviates from the preferred path and, thus, bypasses the separation process, regardless of particle size or density. This phenomenon has been widely observed in various types of hydrocyclones, as reported in the literature [
22,
28,
35,
40,
42,
44,
46].
Figure 11a,b illustrate the volume fraction field on the longitudinal XY plane for the phases (a) fluid and (b) solid particles. It is observed that there is a tendency for a higher concentration of particles in the conical part of the hydrocyclone (
Figure 11b) and water in the cylindrical part (
Figure 11a). This occurs due to the difference in density between the liquid and solid phases. Additionally, in
Figure 11b, the penetration of solid particles into the interior of the cylindrical membrane is observed, which, depending on the intensity of this effect, can increase the flow resistance inside the membrane, leading to a possible partial clogging of the pores, due to the size of the particles being smaller than the pore size. This behavior is better observed in
Figure 12, where the volume fraction fields of the fluid and particles in the filtering medium are represented.
3.3. A Comparison between the Conventional and Filtering Hydrocyclones
In this section, the flow behavior inside a filtering cylindrical hydrocyclone is evaluated and compared with a conventional hydrocyclone, both with the same geometry and volumetric flow rate of 295.7 cm³/s.
Figure 13 shows the pressure field in the longitudinal XY position for both the filtering hydrocyclone (HciF) and the conventional hydrocyclone (Hcon). Upon analysis, it is evident that the pressure field in the filtering hydrocyclone has a higher value near the walls, compared to the conventional hydrocyclone. This can be explained by the presence of the filtering medium, which provides a flow of filtrate and, thus, reduces pressure gradients near the cylindrical wall of the hydrocyclone. The conventional hydrocyclone exhibited a higher pressure drop than that obtained with the filtering hydrocyclone, resulting in a higher Euler number (Eu = 1092) compared to that obtained for the filtering hydrocyclone (Eu = 1012), indicating a higher energy expenditure for processing.
Figure 14,
Figure 15 and
Figure 16 present the axial, tangential, and radial velocity fields, respectively, in the longitudinal XY plane for the filtering (HciF) and conventional (Hcon) hydrocyclones. Upon evaluating these figures, it can be observed that the presence of the filtrate flow in the filtering cylindrical hydrocyclone significantly influences the velocity profiles compared to those observed for the conventional hydrocyclone. Regarding axial velocity (
Figure 14), several recirculation zones emerge in the absence of the filtering medium. The tangential velocity field (
Figure 15) shows a significant decrease in velocity across both the cylindrical and conical regions. Additionally, in the radial velocity field (
Figure 16), higher velocities are clearly observed near the walls of the conventional hydrocyclone, as expected.
The centrifugal field is directly proportional to the tangential velocity and separation efficiency. Therefore, to analyze the tangential velocity, it is necessary to consider the filtrate flow rate, axial, and radial velocities. Examining the behavior of radial velocity, as illustrated in
Figure 16, an increase in this hydrodynamic parameter is observed in certain regions in the cylindrical membrane. This phenomenon causes the transport and elevation of the number of particles near the wall of the hydrocyclone, facilitating their discharge through the lower orifice of the hydrocyclone (underflow). This alteration in behavior affected the overall efficiency, with the conventional hydrocyclone achieving a 12% higher efficiency than the filtering hydrocyclone.
Figure 17 illustrates the water volumetric fraction field on the XY plane for the filtering cylindrical hydrocyclone (HciF) and the conventional hydrocyclone (Hcon). Upon analyzing this figure, an increase in the volumetric fraction of water near the central axis is observed near the “underflow” outlet when comparing HciF with Hcon. According to
Figure 18, when comparing the volumetric fraction fields of particles on the XY plane for (a) HciF and (b) Hcon, a decrease in the particle fraction near the overflow outlet is observed, reducing the short-circuiting phenomenon (see
Figure 19), providing the conventional hydrocyclone with greater efficiency than the filtering cylindrical hydrocyclone.