ANC CONTROL
Design Calculations for the Sloped Column Lime Feeder
In order to verify if the design of the sloped column lime feeder was backed by theoretical calculations, a few 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, a fractal dimension of 3 essentially implies that the density of lime particles does not change with the size of the lime particles.
2) The 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 2.4 cm corresponds to an upflow velocity of 2.95 mm/s.
Known Dimensions and values
The length of the sloping glass column was 1.3 m while its inner diameter was 2.4 cm. The column was to be inclined at an angle of 22.5 degrees (from the vertical axis).
Calculations
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 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:
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$$
{{V_
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The equation relating the terminal velocity to the particle diameter is:
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$$
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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.
Figure 1: Relationship between tube length, capture velocity, and the smallest particle diameter the tube can capture.
A 1.5 m vertical tube is connected to a 1.3 m slanting tube (at a 22.5 degrees inclination). For the above given conditions, using the above equationa, it was found that a length of 1.3 m has a capture velocity of 0.12 mm/s, and the smallest particle that can be captured would have a diameter of 1.35μm. The slanting tube could have been cut to save space but then the capture velocity the tube would provide and the smallest particle size it could capture would both decrease. This relationship can be observed in figure 1 above.
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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$$
u\left( {d_
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} \over {\sin \alpha }}
$$
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.
Figure 2: Critical velocity vs Capture velocity