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 pulling particles 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:
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$$
v_z = {1 \over {4\mu }}\left( {{
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$$
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, small r is the distance from the tube wall to its center. 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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$$
Q = {{ - \pi R^4 } \over {8\mu }}\left( {{
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$$
Solving this equation for the pressure differential gives:
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$$
\left( {{
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$$
This can then be plugged into the velocity profile equation to yield, after simplification:
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$$
v_z = {{2Q} \over {\pi R^4 }}\left(
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$$
The differential of this velocity with respect to the position within the tube gives the velocity gradient:
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$$
dv_z } \over {dr = {{ - 4Q} \over {\pi R^4 }}r
$$
It is important to note that the profile only depends on the tube radius, the flow rate, and the distance from the tube wall.