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h2. Recommendations

Based on the calculations associated with the critical velocity theory, the best way to avoid floc roll up is to maximize the plate settler spacing. Figure 1 shows the minimum plate settler spacing that will produce acceptable results. From the graph, it can be estimated that this diameter is approximately 5 mm.

!Minimum plate settler spacing v. Capture velocity.png|width=600px,align=centre!
_Figure 1: Minimum Plate Settler Spacing vs. Capture Velocity_

Figure 2 shows the minimum particle size that will roll up the plate settler plotted against plate settler spacing. The line at the order of magnitue of colloidal particle size shows that at a plate settler spacing of approximately 17 mm and a tube diameter of 23 mm there should theoretically be no floc roll up

!Plate spacing vs floc diameter.png|width=600px,align=centre!

_Figure 2: Plate Settler Spacing vs. Floc Diameter_

Although the critical velocity theory suggests that larger plate settler spacing will produce the best results, the capture velocity theory (link) suggests that failure will occurs with a larger plate settler spacing. Theoretically, at different terminal velocities (which can be converted to a particle diameter) different theories will govern the behavior of the floc particles.

By plotting the plant flow rates against the terminal settling velocity for both the critical and capture velocity theories (which can be converted into a particle size), you can see which theory should govern the plate settler behavior. The equations relating the critical and capture velocity are as follows:

{latex}
\large
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$$
Q_{critical}  = {{\pi SV\sin \theta ^2 } \over {32d_0 ^2 \left[ {{{ - 18V\Phi \nu \rho _{H2O} } \over {d_0^2 g(\rho _{H2O}  - \rho _{Floc} )}}} \right]^{{1 \over {D_{Fractal}  - 1}}} }}
$$
{latex}



{latex}
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$$
Q_{capture}  = {{L\cos \theta  + S\sin \theta } \over S}\left[ {\pi \left( {{S \over 2}} \right)^2 } \right]V
$$

{latex}

Where

S = Tube settler diameter (or spacing)
d0 = size of primary particles
V = upflow velocity
{latex}
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% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdyeaaa!3767!
$$
\Phi 
$$
 {latex}= Shape Factor
V = Predicted Terminal Settling Velocity





Since, with our experiments, all of these variables will be held constant except for the spacing, we can analyze these relationships between critical and capture velocity theories for different tube diameters.

Figure 3 shows the difference between the 6.35 mm tube and the 23.8 mm tube. For the 6.35mm tube, the critical velocity theory should entirely govern the effluent turbidity produced from experiments with this tube size.

For the 23.8mm tube, the capture velocity theory governs the size of particles that settle out. This goes along with the theory that there should be minimal to no floc roll up for tube settlers with larger diameters

!6.35mm Terminal Velocity vs Plant Flow Rate.png|width=400px,align=left! !23.8mm Terminal Velocity vs plant Flow rate.png|width=400px,align=right!











_Figure 3: Plant Flow Rate vs. Terminal Velocity (Particle Size) for 6.35 mm tube and 23.8 mm tube_

Based on this analysis, a larger tube would be more effective because the minimum size of particles that are settled out is larger. However, this theory needs to be tested, so the Ramp State Experiments (link) are being run to try to match up experimental data to this theory.