4.1. Flow Visualization
We start by replicating the work of Won et al. [
24]. In the figure below, we present cavity stream plots at Reynolds numbers 600 and 1600 and illustrate the relative volumes of the mobile and immobile zones of the membrane surface cavity. As defined by Young and Kabala [
37], the mobile zone refers to the volume of cavity space conducive to
through-flow, and the immobile zone corresponds to the recirculatory volume in the cavity space. Here, through-flow refers to the portion of feed flow that becomes permeate flow. As demonstrated by Won et al. [
24], an increase in Reynolds number results in an increase in the size of the immobile zone. These results are also in agreement with our previous studies on impermeable cavities; see Young and Kabala [
37,
43]. We note that an increase in Reynolds number results in the generation of a secondary, counter-rotating vortex large enough to be observed. It is reasonable to postulate that, under steady flow conditions, particulate captured in this vortex will remain in perpetuity due to flow separation between it and the primary cavity vortex caused by the permeate stream (see the stream plot corresponding to a Reynolds number of 1600). For the reader’s edification, we replicate this simulation at a Reynolds number of 600 for a flow domain containing five sequential cavities (see the supplemental material) and compare the stream plots of the first and last cavities. We see no significant difference in the stream plots, thus supporting the use of a single idealized cavity space for flow-field imaging and particle tracking.
Figure 3.
Vortex formation under steady flow (left), comparison of stream plot lines in first (blue) and last (yellow) cavities of sequential cavity geometry (right).
Figure 3.
Vortex formation under steady flow (left), comparison of stream plot lines in first (blue) and last (yellow) cavities of sequential cavity geometry (right).
In
Figure 4 and
Figure 5 below, we illustrate the deep sweep and vortex ejection mechanisms in the cavity space that result from a sinusoidal feed flow with an average Reynolds number of 600 and 1600 and a non-dimensional period of 4 and amplitude of 0.75. The deep sweep and vortex ejection manifest themselves somewhat differently over the patterned membrane surface than they do in the cavity spaces of impermeable media. The high Reynolds numbers of crossflows over membrane surfaces induce additional mixing in the cavity space, and therefore the generation of secondary vortices comparable in size to the primary cavity vortex. As the through-channel flow volume abruptly increases and decreases, these vortices interact, clouding our ability to easily recognize the deep sweep and the vortex ejection as they occur at lower Reynolds numbers. Despite the presence of these additional vortices, the deep sweep and vortex ejection are still present in the high Reynolds number flows we study in this work, as evidenced by
Figure 4 and
Figure 5. Readers are again directed to the supplemental material to visualize these mechanisms in an impermeable, square cavity space at an average Reynolds number of 10. This sequence clearly demonstrates the deep sweep and vortex ejection streamline patterns.
Similar to steady flow conditions, we observe that an increase in Reynolds number results in an increase in immobile zone volume; this is true over the entire period of the inlet waveform. Notably, during the vortex ejection mechanism, we observe the formation of a large, counter-rotating vortex along the upper channel-wall. In the supplemental material, we verify that existence of this vortex is not strictly an artifact of the selected channel height. For a channel height 10 times that of the pattern depth, the presence of this vortex persists. Therefore, we conclude that the vortex ejection mechanism is not only capable of flushing contaminants from the cavity space, but also of inducing large-scale mixing and shear reversals along boundaries within the immediate vicinity of the cavity geometry.
Finally, we confirm the presence of the deep sweep and vortex ejection mechanisms when the magnitude of the permeate flux is increased by a factor of 10. Because these mechanisms have been previously shown to exist in impermeable cavity spaces (e.g., Kahler and Kabala [
40] and Young and Kabala [
37,
43]), we do not decrease the magnitude of the permeate flux condition. Comparison of the stream plot sequences that result from the two flux conditions result in a decrease in immobile zone volume with an increase in permeate flow volume. It is likely the case that this decrease in immobile zone volume will result in more particle accumulation at the membrane surface, even in the presence of the deep sweep and vortex ejection mechanisms. Support for this conclusion is provided by results of steady flow simulations. Won et al. [
24] found that an increase in the permeation stream area, relative to the vortex stream area, resulted in a higher degree of surface fouling. Jung and Ahn [
35] confirm that an increase in the volume of the “inaccessible” zone results in a decrease in probability that particles will access the membrane surface. After a review of the past two decades of experimental and numerical data on patterned membrane flows, Wang et al. [
36] add further support of this conclusion.
4.2. Particle Tracking
To confirm the ability of the deep sweep and vortex ejection mechanisms to remove particles from the cavity space, we visualize the trajectory of massless particles via integration of the flow field over time. To start, we demonstrate the fate of particles seeded at the geometric boundary between the feed channel and cavity space under steady flow conditions. The results presented below are for a non-dimensional simulation time of 100.
For a Reynolds number of 600, we observe all but 2 of the 9 particles are captured in the recirculation zone; of the remainder, one particle rejoins the bulk flow, and the other settles at the membrane surface. For a Reynolds number of 1600, the only particle that escapes recirculation also settles at the membrane surface. As expected, if we populate the cavity space with a grid of particles, we observe that the particles remain almost entirely in the cavity space. For a Reynolds number of 600 and 1600, 1.5% and 1.9% of the particles leave the cavity space and enter the bulk flow, respectively. In general, the final location of the particles is the same for each of the tested Reynolds numbers. The results presented below are for a non-dimensional simulation time of 200.
To explain the void where there are no particles in the lower-third of the cavity space, we refer back to the stream plots presented in
Figure 3. If we overlay the two plots, we see that the permeation volume remains clear of particles. We also observe a secondary immobile zone on the downstream cavity wall responsible for a small grouping of particles for Re = 1600.
For a rapidly pulsed inlet flow, we observe that, for the boundary particles, all but one is swept away into the bulk flow. Although the particles may enter the cavity space, they are not subject to remain there indefinitely, as in the case of steady flow (see
Figure 6). To reduce computational demand, the non-dimensional simulation time is restricted to 20.
Figure 6.
Trajectories for particles seeded along the boundary of the membrane surface cavity and feed channel, subject to a steady flow with a Reynolds number of 600 (left) and 1600 (right).
Figure 6.
Trajectories for particles seeded along the boundary of the membrane surface cavity and feed channel, subject to a steady flow with a Reynolds number of 600 (left) and 1600 (right).
Figure 7.
Particle positions for a steady feed flow condition; initial position (left), final position (right) for a feed Reynolds number of 600 (red) and 1600 (blue).
Figure 7.
Particle positions for a steady feed flow condition; initial position (left), final position (right) for a feed Reynolds number of 600 (red) and 1600 (blue).
Figure 8.
Final particle positions for a steady feed flow condition (see
Figure 7) are driven by the locations of the recirculation zones in the membrane surface pattern.
Figure 8.
Final particle positions for a steady feed flow condition (see
Figure 7) are driven by the locations of the recirculation zones in the membrane surface pattern.
Figure 9.
Trajectories for particles seeded along the boundary of the membrane surface cavity and feed channel, subject to a rapidly pulsed flow with an average Reynolds number of 600 (top) and 1600 (bottom).
Figure 9.
Trajectories for particles seeded along the boundary of the membrane surface cavity and feed channel, subject to a rapidly pulsed flow with an average Reynolds number of 600 (top) and 1600 (bottom).
If we then seed the cavity space with particles, albeit over a coarser grid to minimize computational demand (180 particles versus the 668 used for steady flow simulation), we observe that approximately half of the particles move into the bulk flow; 56.7% for an average Reynolds number of 600 and 58.9% for 1600. Final particle locations are provided in
Figure 10 below.
Figure 10.
Final location of particles seeded in the cavity space for a rapidly pulsed inlet flow with an average Reynolds number of 600 (red) and 1600 (blue).
Figure 10.
Final location of particles seeded in the cavity space for a rapidly pulsed inlet flow with an average Reynolds number of 600 (red) and 1600 (blue).
Although these plots clearly demonstrate the ability of a rapidly pulsed flow to remove particles from the cavity space via the vortex ejection and the deep sweep, it is necessary to optimize the shape of the waveform, relative to the pattern morphology, to maximize removal. To that end, even with the arbitrary waveform selected for this study, a rapidly pulsed flow is still able to remove approximately fifty times more particles from the cavity space in one-tenth the time – supporting our earlier statement that this type of inlet flow should reduce the need and frequency for traditional membrane cleaning methods.
4.3. Study Limitations
Aside from the numerical error associated with the simulation parameters (e.g., mesh size, precision and accuracy goals, time-step, solver method, etc.), the accuracy of our results is limited by a few simplifying assumptions. Namely, we do not account for the surface roughness of the membrane, which would induce local turbulence. We also enforce a uniform permeate flux in the vertical direction. Realistically, this condition should be imposed normal to the membrane surface (to properly enforce the no-slip condition) and be non-uniform along the membrane length, especially as the membrane fouls. Finally, and most significantly, we simulate only the flow of massless particles – we neglect interactions between the particles and the physical and chemical properties of the membrane itself. Nonetheless, the results provided in this study reveal an accurate first-order representation of the flow field induced over a patterned membrane surface subject to a rapidly pulsed flow.