Uncoupling Capture Velocity & Velocity Gradient
Floc roll-up occurs in tube settlers when the torque caused by a differential in the velocity profile exceeds the force of gravity whereby particles fall back out into the sedimentation basin. The Plate Settler Spacing team hypothesized that holding the length to diameter ratio in tubes of different diameters constant at 20 would decrease the capture velocity's sensitivity to flow rate. Two diameters representing the extremes in lamella spacing (23.5 mm and 6.35 mm) were tested in this experiment. Each experiment was run with the tube settler at two different heights (1.3 cm and 2.7 cm) above the floc blanket-clear water interface.
Initially the team tried to visualize velocity gradient failure by slowly incrementing the flow rate through the 15.35 mm diameter tube until what appeared to be roll-up occurred. This happened around a velocity gradient of 12 1/s. The team then used this value as a mean velocity gradient and bracketed it with three equally spaced values lower and higher to generate a reasonably wide range for experimentation (6, 9, 12, 15, and 18 1/s were tested). In order to calculate the flow rates associated with these velocity gradients, the team needed to find a relationship between the volumetric flow rate and the velocity gradient established in the tubes.
This relationship was derived in the following way:
First, begin with the description of the velocity profile for laminar flow in a cylindrical tube set at a vertical angle alpha:
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\large
$$
v_z = {1 \over {4\mu \sin \alpha }}(\partial p} \over {\partial z)(r^2 - R^2 )
$$
Here, mu is the kinematic viscosity of the fluid flowing through the tube (in this case, water), dp/dz represents the pressure difference across the tube, R is the tube radius, and small r represents the distance between the tube center and the position of itnerest. The key is to determine a relationship that eliminates this pressure term in the equation.
We know that the flow rate can be defined as:
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\large
$$
Q = {{ - \pi R^4 } \over {8\mu }}\left( {{
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\over
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}} \right)
$$
Solving this equation for the pressure differential and inserting the term v ratio (equal to 2 for tubes) gives:
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\large
$$
(\partial p} \over {\partial z) = {{ - 4v_ratio \mu Q} \over {\pi R^4 }}
$$
This can then be plugged into the velocity profile equation to yield, after simplification:
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\large
$$
v_z = {{v_ratio Q} \over {\pi R^4 \sin \alpha }}(R^2 - r^2 )
$$
The differential of this velocity with respect to the position within the tube gives the velocity gradient:
\large
$$
dv_z } \over {dr = {{ - 2v_ratio} Q} \over {\pi R^4 \sin \alpha }}r
$$
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This term can be simplified by substituting V alpha, which represents the average velocity through the cylindrical tube.
\large
$$
V_\alpha = {Q \over {\pi R^2 \sin \alpha }}
$$
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\large
$$
dv_z } \over {dr = {{ - 2v_ratio
V_\alpha } \over {R^2 }}r
$$
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It is important to note that the profile only depends on the tube radius, the flow rate, and r, the distance from the tube wall, which corresponds to the tube radius minus the diameter of the floc particle of interest.The velocity gradient can either be analyzed in terms of the specific radial position of a particle in the tube, or in terms of the velocity gradient that all particles experience at the tube wall. This term corresponds to the previous equation evaluated at the tube wall:
\large
$$
dv_z } \over {dr|
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= {{ - 2v_ratio}
V\alpha } \over R}
$$
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Results and Discussion
As mentioned in the experimental methods, five velocity gradients (6, 9, 12, 15, 18 1/s) were tested for two tube diameters - 6.35 mm and 15.3 mm. For each of these tubes, two placements of the tube settler were also tested - 1.3 cm and 2.7 cm above the floc blanket. These placements were designated as "low" and "high" heights, respectively, in the following analysis.
15.3 mm Diameter Tube
For the 15.3 mm tube, the team collected usable data from five out of six runs performed with the tube settler placed at the low height placement and for three runs at the high height placement. Figures 1a. and 1b. below are graphs of the tube settler effluent turbidity against velocity gradient for the low and high height runs, respectively.
Figure 1a., Average Turbidity vs. Velocity Gradient for the 15.35 mm tube diameter, low height
Figure 1b., Average Turbidity vs. Velocity Gradient for the 15.35 mm tube diameter, high height
While the team originally postulated that failure would occur at the 12 1/s velocity gradient and beyond, the results from Figures 1a,b indicate failure for all velocity gradients tested. To shed light on this development, the capture velocities associated with the tested velocity gradients were calculated and the effluent turbidity was plotted against these capture velocities in Figures 2a and 2b below.
Figure 2a., Average Turbidity vs. Capture Velocity for the 15.35 mm tube diameter, low height 
Figure 2b., Average Turbidity vs. Capture Velocity for the 15.35 mm tube diameter, high height
The capture velocities depicted in Figure 2 explain the poor performance measured for the runs. AguaClara plants operate with a capture velocity of 10 m/day, while the capture velocities used in the experiments are about 9-27 times greater than those used in plants. The trend in the effluent turbidity depicted in both Figures 1 and 2 are consistent with the performance expected in this range of capture velocities. The turbidity generally increases before leveling off. From the data, we can postulate that the average settling velocity of the particles fall between 135.5 m/day and 180.2 m/day. Above this average settling velocity, the bulk of the particles would be pulled up into the effluent without settling. This makes it impossible to distinguish the effects of the velocity gradient from capture velocity on poor performance.
Comparing the results from the high height runs to the low height runs, it can be seen that the average effluent turbidity increases as the tube is placed closer to the floc blanket. The significance of this understanding is discussed further in the conclusions section below.
6.35 mm Diameter Tube
For the 6.35 mm tube, multiple runs were performed; however, the collected data suggested a problem with the 6.35 mm tube experiments. Figure 3 below is a graph of the average effluent turbidities against velocity gradients for both the low and high placements of the tube settler.
Figure 3., Average Turbidity vs. Velocity Gradient, 6.35 mm diameter tube; low height, high height
The blue line in Figure 3 represents the low placement of the tube settler and the red line, high placement. The trend holds that the effluent turbidity increases as the tube settler is placed closer to the floc blanket. However, Figure 3 indicates that for all velocity gradients, no failure occurred in the 6.35 mm tube. All effluent turbidities were below 1 NTU. The team's original hypothesis was that failure would occur at the 12 1/s velocity gradient and beyond. As in the 15.3 mm tube, the capture velocities corresponding to the velocity gradients used in the 6.35 mm tube were calculated. Figure 4 below is a plot of the effluent turbidity from the 6.35 mm against the capture velocities used in the experiments.
Figure 4., Average Turbidity vs. Velocity Gradient, 6.35 mm diameter tube; low height, high height
Figure 4 indicates capture velocities that are 3-10 times greater than those used in AguaClara plants. With this understanding, the results for the 6.35 mm tube experiments do not make sense. If failure is not occurring, the only parameter that should be affecting performance is the capture velocity. Figure 5 below is a comparison of performance due to capture velocity between the 6.35 mm and the 15.3 mm tube.
Figure 5., Comparison of Performance Due to Capture Velocity between 6.35 mm and 15.35 mm diameter tubes
Because the two capture velocities circled in Figure 5 are fairly similar, similar performance was expected. However, it can be seen that the measured turbidity between the two capture velocities were significantly different. Due to this observation, the team believes that the flow rates used in the 6.35 mm tube experiments were too low, resulting in a long residence time of the effluent in the turbidimeter. The long residence time might allow floc particles to settle out in the turbidimeter vial creating a clear effluent zone that may result in readings that underestimate the effluent turbidity.
Conclusion
Analysis of the data resulted in three realizations that were pertinent to team's future experimental plans. The first was that the main failure of the experiments was that the capture velocities were too high, resulting in poor performance and making it impossible to test for rollup. For future experiments, the team plans to fix capture velocity at 10 m/day for all tubes tested to ensure adequate performance so that performance deterioration due specifically to floc rollup can be observed. Secondly, the team observed that effluent turbidity readings increased as the tube settler was placed closer to the floc blanket. The team postulated that the effects of floc rollup would be better observed when the tube settler is placed closer to the floc blanket since the magnitude of failure would be greater. Lastly, the team observed inaccurate data collection in the small diameter tube possibly due to flocs settling out in the turbidimeter as a result of very low flow rates. In order to address this issue, the team plans to test the flow rates used in the small tube with unflocculated 100 NTU water. Readings less than 100 NTU in the effluent turbidimeter would be indicative of particle settling in the turbidimeter. Since clay has a slower settling velocity than floc particles, the effects of particle settling would be magnified for floc particles. If settling proves to be a problem, the team plans to bundle small tubes to increase the total flow to the turbidimeter, while still being able to maintain low flow rates in individual tubes.