Assumptions made
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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The angle of inclination and laminar flow regime allows certain sized lime particles to settle back into the column and thus prevent unnecessary lime loss. Thus the primary column would be used as a storage vessel for the suspended lime bed while the slanted tube above it would allow more lime particles to settle back to the column below, making the process more economical. (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.