Page History

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The two constraints were the tube's length and the terminal velocity of the particle. This terminal velocity should be larger than the capture velocity. The length should be large enough to let the flow in the slanting tube to become a fully developed laminar flow.
It was assumed that the flow rate of the lime feeder is kept at 80 mL/min. The inner diameter of the column is 2.4 cm giving an upflow velocity of 2.95mm/s.
Capture velocity is a function of the geometry of the tube and the equation relating the capture velocity to the geometry of lime feeder is:

{latex}
\large
$$
{{V_{ \uparrow Plate} } \over {V_c }} = 1 + {L \over S}\cos \alpha \sin \alpha
$$
{latex}

The equation relating the terminal velocity to the particle diameter is:

{latex}
\large
$$
V_t = {{gd_0^2 } \over {18\Phi \nu _{H_2 O} }}{{\rho _{Floc_0 } - \rho _{H_2 O} } \over {\rho _{H_2 O} }}\left( {{d \over {d_0 }}} \right)^{D_{Fractal} - 1}
$$
{latex}

In order to obtain the relationship between the particle size and its required capture velocity, it was also assumed that the smallest particle the tube can capture has the same terminal velocity as the capture velocity.

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To measure if the particle has a risk of roll-up, the relationship between the critical velocity and terminal velocity was also calculated. Since the particle diameter is small compared to the diameter of the tube, and the flow is fully developed, the linearized equation for critical velocity (floc roll up velocity) that is used for tube settler designing can be used here:

{latex}
\large
$$
u\left( {d_{Floc} } \right) \approx {{6d_{Floc} } \over S}{{V_{ \uparrow Plate} } \over {\sin \alpha }}
$$
{latex}

As the particle size increases, terminal velocity becomes much larger than the critical velocity. This is because critical velocity varies linearly with respect to the particle diameter but terminal velocity is proportional to the square of the particle diameter. However, if the slanting tube's diameter decreases, the critical velocity will increase and there will be a higher risk of particle roll up, but with the present apparatus, the 2.4 cm inner diameter could ensure that roll up will not occur. Figure 2 shows the change of capture velocity and the particle size it can capture as the function of the slanting tube length.

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