Page History

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Starting from a capture velocity of 0.1 mm/s, the team chose to test three different up-flow velocities: 0.86, 1.73, and 4.32 mm/s. The first value is that currently used by AguaClara, and the range represents a considerable spectrum of flow rates, and thus velocity gradients, experienced in the tube settlers. The velocity of the influent water increases as it enters the settlers due to hydraulic contraction, and this is a velocity of interest to the team since it also represents the average flow rate through the settler, taking into account also its angle of inclination. These values for this experiment are: 1, 2, and 5 mm/s, and can be calculated from the following relationship:

{latex}
\large
$$
{{V_\alpha }} = {{ V_{up} } \over {\sin \alpha }}
$$
{latex}

Failure is defined as П V, the ratio of the average floc's settling velocity (a function of both the capture velocity and the floc's density, and thus also a function of the flocculation process), to the velocity experienced by the particle at its diameter exposed to the effluent stream. Particles on the bottom wall of the settler that experience a higher velocity at their exposed edge than the velocity at which they settle out of the tube will experience a torque upwards that causes them to exit with the effluent. Thus when П V is less than one under specific circumstances, roll-up is expected to occur. A value of unity suggests an equilibrium point where roll-up may or may not occur.

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1. Choose a range of tube diameter (1" ½" 3/8" ¼") based on available materials and to ensure a good range of spacings.
2. Fix capture velocity at 10 m/day based on the value used for AguaClara plants
3. The team used the following relationship for Vα to find the necessary length for each tube diameter required to achieve average velocities of 1.04 mm/s, 2.31 mm/s, and 5.77 mm/s.

{latex}
\large
$$
{{V_\alpha  \sin (\alpha )} \over {V_c }} = {L \over S}\cos \alpha \sin \alpha  + \sin ^2 \alpha
$$

{latex}

4. From the V _α 's, the team determined the associated velocity gradients by the equation:

{latex}
\large
$$
{{\partial v_z } \over {\partial r}} = {{ - 2v_{ratio} Q} \over {\pi R^4 \sin \alpha }}r
$$
{latex}

where
v _ratio = 2 for tubes and 1.5 for plates
α = 60 degrees
R = radius of the tube

5. The necessary flow rates were calculated from the capture velocities, diameters, and lengths determined above by the following relationship:

{latex}
\large
$$
Q = \left[ {L\cos (\alpha ) + d^2 \sin (\alpha )} \right]{{\pi n_{Tube} V_C } \over 4}
$$
{latex}

where
L = length of the tube
d = diameter of the tube
α = 60 degrees
r = radius of the tube
n _tube = number of tubes

6. П _V ratios were determined for each tube evaluated by the following equation:

{latex}
\large
$$
\Pi _V  = {{{{g\sin (\alpha )d_0 ^2 \rho _{floc.0} (C_{Alum,} C_{Clay} ) - \rho _{H20} } \over {18\varphi \nu \rho _{H20} }}({{d_{floc} (V_C )} \over {d_0 }})^{d_{fractal} - 1} } \over {{{2V_{UP} } \over {\sin \alpha }}(1 - ({{{d \over 2} - d_{floc} (V_C )} \over {{d \over 2}}}))^2 }}
$$
{latex}

where
d ₀ = diameter of clay particle
d _floc (V _C ) - diameter of floc captured based on capture velocity
d _fractal number generally between 2-3 for flocs that describes the volume of dirt in the floc compared to the volume of water. A value of 3 indicates that the floc has no water within it.
ρ _floc.0 (C _Alum,C _Clay) - density of floc based on concentration of alum and clay, respectively
ρ _H20 - density of water
ν - kinematic viscosity of water
Φ - shape factor

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