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. (Briefly explain again how this is estimated.)
Figure 1: Minimum Plate Settler Spacing vs. Capture Velocity
Figure 2 illustrates the minimum particle size that will roll up the plate settler plotted against plate settler spacing. The line at the order of magnitude 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. (Put this in the figure. Comment about what this means for performance in the paragraph above.)
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) (was a link supposed to be put here?) 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 (what do you mean by different theories?).
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:
Unknown macro: {latex}
\large
% MathType!MTEF!2!1!+-
% feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuamaaBa
% aaleaacaWGJbGaamOCaiaadMgacaWG0bGaamyAaiaadogacaWGHbGa
% amiBaaqabaGccqGH9aqpdaWcaaqaaiabec8aWjaadofacaWGwbGaci
% 4CaiaacMgacaGGUbGaeqiUde3aaWbaaSqabeaacaaIYaaaaaGcbaGa
% aG4maiaaikdacaWGKbWaaSbaaSqaaiaaicdaaeqaaOWaaWbaaSqabe
% aacaaIYaaaaOWaamWaaeaadaWcaaqaaiabgkHiTiaaigdacaaI4aGa
% amOvaiabfA6agjabe27aUjabeg8aYnaaBaaaleaacaWGibGaaGOmai
% aad+eaaeqaaaGcbaGaamizamaaDaaaleaacaaIWaaabaGaaGOmaaaa
% kiaadEgacaGGOaGaeqyWdi3aaSbaaSqaaiaadIeacaaIYaGaam4taa
% qabaGccqGHsislcqaHbpGCdaWgaaWcbaGaamOraiaadYgacaWGVbGa
% am4yaaqabaGccaGGPaaaaaGaay5waiaaw2faamaaCaaaleqabaWaaS
% aaaeaacaaIXaaabaGaamiramaaBaaameaacaWGgbGaamOCaiaadgga
% caWGJbGaamiDaiaadggacaWGSbaabeaaliabgkHiTiaaigdaaaaaaa
% aaaaa!732E!
$$
Q_
Unknown macro: {critical}
= {{\pi SV\sin \theta ^2 } \over {32d_0 ^2 \left[ {{{ - 18V\Phi \nu \rho _
Unknown macro: {H2O}
} \over {d_0^2 g(\rho _
- \rho _
Unknown macro: {Floc}
)}}} \right]^{{1 \over {D_
Unknown macro: {Fractal}
- 1}}} }}
$$
Unknown macro: {latex}
\large
% MathType!MTEF!2!1!+-
% feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuamaaBa
% aaleaacaWGJbGaamyyaiaadchacaWG0bGaamyDaiaadkhacaWGLbaa
% beaakiabg2da9maalaaabaGaamitaiGacogacaGGVbGaai4CaiabeI
% 7aXjabgUcaRiaadofaciGGZbGaaiyAaiaac6gacqaH4oqCaeaacaWG
% tbaaamaadmaabaGaeqiWda3aaeWaaeaadaWcaaqaaiaadofaaeaaca
% aIYaaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOGaay5w
% aiaaw2faaiaadAfaaaa!53CA!
$$
Q_
Unknown macro: {capture}
= L\cos \theta + S\sin \theta } \over S}\left[ {\pi \left( {{S \over 2 \right)^2 } \right]V
$$
Where
S = Tube settler diameter (or spacing)
d0 = size of primary particles
V = upflow velocity
Unknown macro: {latex}
\large
% MathType!MTEF!2!1!+-
% feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdyeaaa!3767!
$$
\Phi
$$
= 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. (Be more specific here. What specific parameters are you varying? Are you trying to confirm that the spacing you calculate in theory will predict performance?)
Figure 3 shows the difference between the 6.35 mm tube and the 23.8 mm tube (What difference? In what way?). For the 6.35mm tube, the critical velocity theory should entirely govern the effluent turbidity produced from experiments with this tube size. (Say it more succinctly that roll-up will dominate effluent performance for these ranges in 6.35 mm tube compared to 23.8 mm tube.)
For the 23.8mm tube, the capture velocity theory governs the size of particles that settle out. (Does it govern it? Is this theory still valid?) This goes along with the theory that there should be minimal to no floc roll up for tube settlers with larger diameters. (This sentence is vague. Be more specific about what you are accomplishing here.)
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 are being run to try to match up experimental data to this theory.