In order to verify if the design of the sloped column lime feeder was backed by theoretical calculations, the following assumptions and/or simplifications were made.
Calculations were carried out to estimate the lime particle sizes that the 1.3m sloped tube (acting as a tube settler) could retain and if there was a risk of particle roll-up.
Assumptions
Calculations were made using the following assumptions for simplification:
1)It was assumed that the original lime is solid powder with a fractal dimension of lime particles to be around 3. These solid lime particles continued to dissolve as raw water was continuously added resulting in an effluent solution of saturated lime with a pH of 12.4.
Hence, giving a fractal dimension of 3 essentially implies that the density of lime particles does not change with the size of the lime particles.
2) Density of lime is 2.211 g/m^3 and this remains constant throughout the process if the fractal dimension remains constant.
3) Shape Factor of lime particles = 1 i.e. the lime particles are perfectly spherical.
4) Settling velocity = 10 m/day i.e. 0.12 mm/s. A flow rate of 80 mL/min (as determined by experiment 1)and a tube of inner diameter 2.4cm corresponds to an upflow velocity of 2.95mm/s
The equation of terminal velocity to the particle diameter is:
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$$
V_t = {{gd_0^2 } \over {18\Phi \nu _
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}}{{\rho _
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- \rho _
} \over {\rho _
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}}\left( {{d \over
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}} \right)^{D_
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- 1}
$$
(Perhaps put the design for your tube settler on another page. I would include all relevant equations and summaries in a MathCAD page. I would include the length, angle of inclination, diameter, and predicted capture velocity as well as the critical velocity of floc particles and document these values in the wiki.)
The two constraints are 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 flow; the relevant criteria can be found in the MathCad file
CALCULATIONS
It was assumed that the flow rate of the lime feeder is kept at 80mL/min. The inner diameter of the column is 2.4cm 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:
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$$
{{V_
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} \over {V_c }} = 1 +
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\cos \alpha \sin \alpha
$$
It is also assumed that the smallest particle the tube can capture has the same terminal velocity as the capture velocity, so we get the relationship between the particle size and it's required capture velocity.
Figure 2 shows the change of capture velocity and the particle size it can capture as the function of the slanting tube length. 
Figure 1: Relationship between tube length, capture velocity, and the smallest particle diameter the tube can capture.
A 1.5m vertical tube is connected to a 1.3m slanting tube (at a 22.5degrees inclination). For the above given conditions, a length of 1.3m has a capture velocity of 0.12 mm/s, and the smallest particle that can be captured would have a diameter of 1.35μm. We could cut the slanting tube to save space but then the capture velocity the tube could provide and the smallest particle size it could capture would both decrease. This relationship can be observed in figure 1.
Since the lime particles have varying particle size distribution, there will inevitably be a loss of some extremely fine lime particles that will be washed away with the effluent water during the initial stage of the lime feeder run, however, going by the above calculations, a majority of lime will be retained within the sloped column.
Particle Roll-up Risk
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:
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$$
u\left( {d_
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} \right) \approx {{6d_
} \over S}{{V_
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} \over {\sin \alpha }}
$$
As the particle size increases, terminal velocity becomes much larger than the critical velocity. This is because critical velocity varies linear 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 2.4cm inner diameter could prove the roll up would not happen(see figure 2). (Why? Refer to the figure and explain why? Also, you introduce the concept of critical velocity without any background and very little explanation about what it is.)
Figure 2: Critical velocity vs Capture velocity