Theoretical Analysis of the Velocity Gradient
This analysis will be done for both a tube and plate settler. Although tube settlers are used in the lab, plate settlers are used in the Honduras plants. Therefore, a model that takes into account the differences between the two apparatuses must be developed.
Calculation of Ratio of Settling Velocity to Particle Velocity
To begin the calculations for the critical velocity, it is important to note that the maximum velocity at the center of a pipe is two times the average velocity and the maximum velocity between two parallel plates is 1.5 times the average velocity.
In order to determine the critical velocity at which floc particles will begin to roll up the tube and into the effluent, we compare the settling velocity with the particle velocity experienced from the velocity gradient.
The settling velocity of a particle in a tube settler can be expressed as follows:
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$$
V_t = {{gd_0 ^{\left( {3 - D_
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} \right)} d^{\left( {D_
- 1} \right)} } \over {18\Phi \nu }}\left( {{{\rho _
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} \over {\rho _
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}} - 1} \right)
$$
Where:
g = Gravity
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\large $$d_o $$
= size of the primary particles
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\large $$D_
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$$
= fractal dimension of the floc particles
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\large $$\Phi $$
= shape factor for drag on flocs which is equal to
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\large $$\nu $$
= viscosity
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\large $$\rho _
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$$
= density of the floc particle
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\large $$\rho _
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$$
= density of water
The particle velocity expereinced as a result of the velocity gradient can be expressed as follows:
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$$
V_
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= V_
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V_\alpha \left[ {1 - \left( {{{{{d_
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} \over {{{d_
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} \over 2}}}} \right)^2 } \right]
$$
Where
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\large $$V_\alpha $$
= directional velocity in the tube settler
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\large $$d_
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$$
= diameter of the tube settler
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\large $$ d_
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= the diameter of floc particles
Therefore, the ratio between the settling velocity of the particle and the velocity experienced as a result of the velocity gradient can be expressed as by the below equation.
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$$
\Pi _V = {{{{g\sin (\alpha )d_0 ^2 } \over {18\Phi \nu }}{{\rho _
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- \rho _
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} \over {\rho _
}}\left( {{{d_
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} \over
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}} \right)^{D_
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- 1} } \over {V_
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{{V_
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} \over {\sin (\alpha )}}\left[ {1 - \left( {{{{{d_
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} \over 2} - d_
} \over {{{d_
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} \over 2}}}} \right)^2 } \right]}}
$$
This ratio is a function of particle diameter, tube diameter, upflow velocity and the angle of the plate settler. When this ratio is greater than one (ie the settling velocity is greater than the velocity experienced by the floc particles in the tube), the flocs will fall back into floc blanket and fail to travel to the effluent. When this ratio is equal to one, the particles will remain stationary in the tube settler. When the ratio is less than one, the velocity of the particles will exceed the settling velocity and the floc particles will roll up into the effluent, creating a highter turbidity.
Figure 1 shows the floc size that could be captured by different tube settler diameters. When the ratio has a value of 1, the sedimentation velocity matches the upflow velocity of the floc particle, and this is the critical diameter, i.e. the floc diameter in which floc roll up will begin, for each of the given tube diameters and plate settler spacing.
Figure 1: The ratio of Sedimentation Velocity to Fluid Velocity vs. Floc Diameter
The graph is cut off at a particle size of 1x10^3 because this is on the order of magnitude of size of a colloidal particle. Therefore, for plate spacings of 2 and 5 cm and for tube diameters of 5 cm, it is predicted that there will be no floc roll up under AguaClara conditions.
Calculation of the Minimum Diameter of the Flocs that Settle from the Sedimentation Velocity Equation
Assuming an upward flow velocity of 100 m/day, which used in the newer AguaClara plants, the diameter of floc that will roll-up was determined by using a root finding algorithm, and the plate settler spacing or tube diameter was plotted versus the minimum floc diameter.
Figure 2: Plate Spacing or Tube Diameter vs. Minimum Floc Diameter
The minimum floc diameter corresponds to the minimum size of particles that will still settle out of the tube and return to the floc blanket instead of going into the effluent. The line at 10^-3 represents the order of magnitude of the size of colloidal particles. There are no particles that will be smaller than this order of magnitude, therefore with larger plate settler spacing, floc roll-up could theroetically be eliminated.
To ensure that flocs in the plate settler don't roll up, we can calculate the minimum floc diameter that settles from the sedimentation velocity equation.
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$$
d = d_0 \left( {{{18V_t \Phi \nu _
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} \over {gd_0^2 }}{{\rho _
} \over {\rho _
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- \rho _
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}}} \right)^{{1 \over {D_
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- 1}}}
$$
This equation is based on a linearization of the velocity profile at the wall. This linearization was done by assuming that the velocity gradient is linear instead of parabolic. The slope of the velocity gradient at the edge of the tube is used to describe the entire velocity profile. Since this is an approximation, it is not accurate for all spacings and floc diameters. For small floc sizes this linearization is valid and produces an analytical solution. This linearized approach was used because the final results are very similar, and this linearized approach makes it much easier to solve for the critical velocity.
Figure 3 illustrates that linearized equations provide only tiny divergence for big flocs in small tubes.
Figure 3: Floc Diameter vs. Spacing
Figure 4 graphically displays the linear velocity gradient solutions. The curves are not quite straight on a log log plot. This is due to the quadratic in the velocity profile. We can obtain a very good approximation by using the velocity gradient at the wall and assuming a linear velocity gradient. That assumption makes an analytical solution possible.
Figure 4: Floc Spacing vs. Floc Diameter
The equation below represents the spacing of the plate settlers to capture a floc particle of a particular size. This is based on the critical velocity model.
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$$
S = V_
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{{108\Phi \nu _
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d^2 } \over {g\sin ^2 (\alpha )d_0^3 }}\left( {{
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\over d}} \right)^{D_
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} {{\rho _
} \over {\rho _
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- \rho _
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$$
Where:
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\large$$ alpha $$
= The angle of the tube settler (60 degrees)
Figure 5 represents the minimum spacing that will capture a floc particle with a particular settling velocity. The settling velocity can be translated into a floc diameter by the terminal settling velocity equation.
Figure 5: Minimum spacing vs. Floc Sedimentation Velocity
Figure 6 represents the absolute minimum plate settler spacing that will capture floc particles with a settling velocity of 10 m/day in an AguaClara plant. Theoretically, any spacing below the intersection of two lines would produce a worsened effluent turbidity because larger particles are no longer being captured by the plate settler.
Figure 6: Minimum Plate Settler Spacing vs. Capture Velocity
In order to complete the calculations, the Reynold's Number was checked to ensure that the flow in the settler tube is in fact laminar. The reason why we must use laminar flow is because turbulent flow would produce a much larger range of results, and would be more unpredictable. Additionally, we can model laminar flow, whereas with turbulent flow, a statistical method must be employed. Next, the entrance region was checked to ensure that the parabolic velocity profile was fully established. This was done by calculating the distance that it will take laminar flow to become fully developed, and make sure that this entrance region is shorter than the length of the tube settler.
The entrance regions are always less than the length of the plate settlers (Figure 7). Thus, based on this analysis, the best designs would include a plate settler with a larger spacing.
Figure 7: Entrance region vs. Plate Settler Spacing
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*More detail on the calculation process outlined above can be found in the Math CAD File.