2. Figure: 06-04 2

4. When terminal velocity is reached, vs is constant (vterm), and Fnet = 0:
If Re is less than ~100, CD  24/Re, and:

6. Discrete Settling of a Suspension of Particles that Reach Their Terminal Settling Velocity Rapidly
z =0
h(t)
z =Z
Note: z is a generic coordinate for depth (0 at top of column and Z at bottom); h(t) is distance fallen by particles in time t, computed as vtermt.

7. 0 min
10 min
20 min
30 min
40 min h
Inferences
At any t >0, the particles are present in only part of the column.
The particles’ settling velocity can be computed as h/t after any t.
If the entire water column were mixed after time t, the concentration in the mixture (Cmixed) would be a weighted average of the concentrations in the two layers. So, if h < Z:
If the computed h is ≥ Z, Cmixed will be zero.

8. Distance Fallen, h (cm)
Conc. (mg/L)
Velocity (cm/min)
10 min
20 min
40 min
1.5
0.12
1.2
2.4
4.8
5.3
0.70
7.0
14.0
28.0
1.8 24 48 96
1.4
5.0 50
100
200
If particles with different terminal velocities are present, they might behave independently
0 min
10 min
20 min
40 min
0 cm
50 cm

9. The total particle concentration at any depth is the sum of the concentrations of the particles at that depth
0 min
10 min
20 min
40 min
z (cm) 50
Cin
C (mg/L)

10. 0 min
10 min
20 min
30 min
If Several Groups of Independent Particles are Present (Type I Settling)
Interfaces develop instantly between layers of water with various groups of particles.
Each group of particles is present in only part of the column; the faster vterm, the smaller the portion of the column in which those particles are present.
The top layer has no particles; the next layer down has only the slowest-settling particles, at their original concentration; the next layer has the two slowest-settling groups of particles, each at their original concentrations; etc.
Each interface sinks at a constant rate equal to vterm of the next faster-settling group of particles.

11. In reality, suspensions contain particles of virtually all velocities, so the concentration distribution is almost continuous rather than stair-step.
10 min
20 min
40 min
30 min
0 min

12. The particles in suspension can be treated as an infinite number of groups with different settling velocities, each with a differential concentration, dC. As in the example with four types of particles, the interface between any two particle types falls at a steady velocity:
0 min
10 min
20 min
30 min
40 min

13. Concentration Isopleths for Type I Settling
t1 or l1
10%
20%
35%
50%
70%
85% removed
t2 or l2
t3 or l3
Time or Distance H

14. In a settling tank operating as a PFR, the suspension behaves like a test column on a conveyor belt
Influent contains all particles at Ci,0
Particles settle as the column moves through the PFR
Effluent contains each type of particle at Ci,mixed
0 min
10 min
20 min
30 min
40 min

15. If Particles Do Not Behave Independently…
0 min
10 min
20 min
30 min
If Particles Do Not Behave Independently…
Often, particle growth continues during settling. This is referred to as Type II settling.
Particles settle faster the longer they are in the system, so they cannot be assigned a single vterm.
Effluent can still be viewed as mixture of different groups of particles, each present in part of the column and absent from other parts.
A generic design approach exists that can be used for both Type I and Type II settling. A simpler approach exists that is applicable only to Type I settling. Both are described next.

16. Generic Design Approach for Both Types I and II Settling
Fill a column at least as long as the anticipated settling depth with water of interest. Allow suspension to settle, taking samples from several ports at various times. Plot % removal for each sample as shown below.
70%
77%
83%
84%
85%
Depth
52%
59%
67%
70%
73%
40%
47%
55%
59%
63%
24%
36%
46%
51%
57%
15%
23%
36%
40%
51%
Time

17. Generic Design for Both Types I and II Settling
Sketch isopleths of constant removal on the plot.
Time
37%
28%
33%
45%
59%
69%
79%
74%
70%
50%
54%
44%
30%
29%
24%
31%
35%
49%
57%
52%
40%
26%
23%
15%
60%
80%
Depth

18. Generic Design for Both Types I and II Settling
Make preliminary choices for residence time (t) and depth. In this case, choose the time and depth indicated by the red dot. Estimate the % removal at the bottom of the column at that time (here, 42%); assume 100% removal at the top.
100%
Time
37%
28%
33%
45%
59%
69%
79%
74%
70%
50%
54%
44%
30%
29%
24%
31%
35%
49%
57%
52%
40%
26%
23%
15%
60%
80%
Depth

19. Generic Design for Both Types I and II Settling
Divide the column at time t into hypothetical layers between the points of known % removal. Approximate the % removal in those layers and the fraction of the column height that they occupy.
100%
Dz1
90%
Dz2
75%
80%
Depth
Dz3
65%
70%
60%
Dz4
55%
40%
50%
Time

20. Generic Design for Both Types I and II Settling
Overall removal efficiency is weighted average of removal efficiencies in the different layers
If settling tank operates as a PFR, conditions in effluent will be same as in test column at t=t.
Each term in summation is removal efficiency in a layer times fraction of the column that the layer occupies.

21. Generic Design for Both Types I and II Settling
In example, Dzi/Z values from top to bottom are 0.12, 0.17, 0.18, and 0.53, respectively, so:

22. Generic Design for Both Types I and II Settling
Predicted removal depends on both depth (Z) and residence time (t)
t depends on Z; to make the design parameters independent of one another, overflow rate, Z/t, is used instead of t
Increasing Z at constant O/F increases h; increasing O/F at constant Z decreases h.

23. Simplified Design for Type I Settling
Approach based on groups of particles with similar vterm values, rather than groups in a particular layer of water; analogous to census based on age groups vs. geographic regions
Dfv,j is the fraction of the particle concentration that has a certain terminal settling velocity, and hj is the removal efficiency for particles with that settling velocity; S(Dfi) = 1.0.
Each term in summation is removal efficiency for a group of particles times the fraction of the total concentration that the particles represent.

24. 10 min
20 min
40 min
30 min
0 min
Treat a continuous distribution as an infinite number of discrete groups with different settling velocities; each group comprises a differential fraction of the total concentration:
Now, dfv is the fraction of the particle concentration that has terminal settling velocities in a small range (v to v + dv), and h is the average removal efficiency for particles in that group.
To evaluate integral, express both h and fv as functions of vterm.

25. Evaluating hi as Function of v
10 min
20 min
40 min
30 min
0 min
Evaluating hi as Function of v
Derived earlier that the average concentration particles of type i in a column after some settling time is:

26. 10 min
20 min
40 min
30 min
0 min
Recall that the velocity required to fall Z in time t is vcrit. So, for particles with constant vi,term:

27. Cumulative fraction of Concentration, fv
Evaluating fv as function of v
dfv is the fraction of the particle concentration with terminal settling velocities between v and v + dv . fv can therefore be defined as the cumulative fraction of the particle concentration with settling velocity <v:
Velocity, v
Cumulative fraction of Concentration, fv 1 dv df

28. Evaluating fv as function of v
Velocity, v fv 1 dv df
In a settling test, the fraction of the particle concentration remaining at depth h after time t is the fraction with terminal vterm < h/t. is
Take sample at, say, h = 30 cm and t = 60 min. If C is 4 mg/L and Cinit was 10 mg/L, plot a point at v = 0.5 cm/min, fv = 0.4.
Repeat for other h and/or t to develop entire f vs. v curve. Note that this can be done with any length column, and after any settling times; does not require experiment with dimensions or duration comparable to conditions in full-scale system.
Cinit h

29. Evaluating h as function of fv
Velocity, v fv 1 dv df
Now, we have data for fv vs. v and an expression for h vs. v. From those, compute h vs. fv for a given choice of vcrit.
vcrit (cm/min)
0.2
Col. 1
Col. 2
Col. 3
Col. 4
Col. 5
v (cm/min) fv
dfv h
h dfv :
Choose a preliminary vcrit. Col. 1 is the range of v values. Col.2 is the corresponding fv, and Col.3 is the difference in successive fv’s. Col.4 is either v/vcrit or 1.0, whichever is smaller. Col.5 is the product of Col.3 and Col.4.

30. Evaluating hoverall
vcrit (cm/min)
0.2
Col. 1
Col. 2
Col. 3
Col. 4
Col. 5
v (cm/min) fv
dfv h
h dfv : 1
hoverall is area under curve h
hoverall can be evaluated as the sum of the values in Col.5 or as the area under a plot of h vs. fv.
fv for vcrit fv 1

31. Design for Type I Settling
Predicted removal depends only on properties of suspension (fv vs. v) and vcrit (which determines h). Recall that vcrit is usually reported as O/F. Changing Z while holding O/F constant (same Q and A) has no effect on hoverall .
Increasing Z at constant O/F increases distance particles have to fall and time available for them to fall
For Type I settling, settling velocity remains constant throughout, so benefit of additional time exactly offsets ‘cost’ of additional distance
For Type II settling, benefit of additional time exceeds cost of additional distance to fall, because particles accelerate during the extra time they spend in the basin

32. Enhancing Sedimentation with “Tube” or “Lamellar” Settlers