Theoretical Analysis of the Velocity Gradient
This analysis will be done for both a tube and plate settler. Although tube settlers are used in the lab, plate settlers are used in the Honduras plants. Therefore, a model that takes into account the differences between the two apparatuses must be developed.
Calculation of Ratio of Settling Velocity to Particle Velocity
To begin the calculations for the critical velocity, it is important to note that the maximum velocity at the center of a pipe is two times the average velocity and the maximum velocity between two parallel plates is 1.5 times the average velocity.
In order to determine the critical velocity at which floc particles will begin to roll up the tube and into the effluent, we compare the settling velocity with the particle velocity experienced from the velocity gradient.
The settling velocity can be expressed as follows:
insert vt here
The particle velocity expereince as a result of the velocity gradient can be expressed as follows:
insert vparticle here
where phi is the shape factor, equivalent to 45/24
Therefore, the ratio can be expressed as
insert ratio here
When this ratio is greater than one (ie the settling velocity is greater than the velocity experienced by the floc particles in the tube), the flocs will fall back into floc blanket and fail to travel to the effluent. When this ratio is equal to one, the particles will remain stationary in the tube settler. And when the ratio is less than one, the velocity of the particles will exceed the settling velocity and the floc particles will roll up into the effluent, creating a highter turbidity.
Figure 1: The ratio of Sedimentation Velocity to Fluid Velocity vs. Floc Diameter
Figure 1 shows what size floc could be captured by different tube settler diameters. The lines cross the y value of 1 when the sedimentation velocity matches the upflow velocity at the floc diameter.
Calculation of the Minimum Diameter of the Flocs that Settle from the Sedimentation Velocity Equation
Assuming an upward flow velocity of 100 m/day, the diameter of floc that will roll-up was determined by using a root finding algorithm, and the plate settling or tube diameter was plotted versus the minimum floc diameter.
Figure 2: Plate Spacing or Tube Diameter vs. Minimum Floc Diameter
To ensure that flocs in the plate settler don't roll up, we can calculate the minimum floc diameter that settle from the sedimentation velocity equation. Solving for the floc diameter,
insert floc diameter eqtns
insert S equations and explanations?
Figure 3: Floc Diameter vs. Spacing
Figure 3 shows how the linearized equations provide only tiny divergence for big flocs in small tubes.
Figure 4: Floc Spacing vs. Floc Diameter
Figure 4 graphically displays the linear velocity gradient solutions. The curves are not quite straight on a log log plot. This is due to the quadratic in the velocity profile. We can obtain a very good approximation by using the velocity gradient at the wall and assuming a linear velocity gradient. That assumption makes an analytical solution possible.
insert S eqtn for plate settlers
Figure 5: Minimum spacing vs. Floc Sedimentation
Figure 6: Minimum Plate Settler Spacing vs. Capture Velocity
The Reynold's Number was checked to ensure that the flow in the settler tube is in fact laminar. Next, the entrance region was checked to ensure that the parabolic velocity profile was fully established.
Figure 7: Ratio of Entrance Length to Plate Length vs. Plate Spacing
Figure 7 shows that the entrance regions are always less than the length of the plate setltlers. Thus, based on this analysis, the best designs would include plates that are more closely spaced. All of the plate settlers will have a significant entrance region. It is likely that the only way flocs can successfully pass thorugh the entrance region is to grow into larger flocs so they have a higher sedimentation velocity.
*More detail on the calculation process outlined above can be found in the Math CAD File.