Equations still to be made pretty and alpha or 'a' written as symbol
A common parameter used to describe sedimentation is the capture velocity. Capture velocity is the velocity of the floc that is only barely settled-out, or "captured", during the sedimentation process. To gain a more profound understanding of this crucial parameter, we will walk through the derivation of the capture velocity equation.
The figure below illustrates the longest path that a floc is able to travel in a sedimentation tube and still be captured.
Figure 1. Longest possible path of a floc through a sedimentation tube.
The capture velocity is the velocity that the particle must travel to fall a distance 'x' (see figure above) in the same amount of time that a particle traveling at
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\large[V_\alpha]
will travel L
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\large[+ b\tan (\alpha )]
.
Now, we know that one way to determine the travel time of a particle is to divide the distance travelled by the particle by the particle velocity. Therefore, we can equate the following travel times:
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\large[\frac
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V_c \cos (\alpha ) = \frac{{L + d\tan(\alpha )}}{{V_\alpha }} = \frac{{L\sin (\alpha ) + \frac
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\cos (\alpha )}}{{V_
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}}]
Solving for the capture velocity by equating the first and third expressions, we find that:
Where L is the length of the tube, d is the inner diameter of the tube, Vup is the vertical velocity through one tube, and a is equal to sixty degrees.
From the capture velocity equation just derived, it is evident that capture velocity is dependent on the diameter, length, and angle of the tube settlers, as well as the flow rate through the settlers.