When terminal velocity is reached, v s is constant ( v term ), and F net = 0:

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 0 min 10 min 20 min 30 min 40 min 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. 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 0 z (cm) 50 Cin 0 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. 30 min 0 min 10 min 20 min 40 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 30 min 20 min 40 min

13. 0 85% removed 70% h 50% 20% 35% 10% H t3or l3 t2or l2 t1or l1 Time or Distance Concentration Isopleths for Type I Settling

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 30 min 20 min 40 min

15. 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% 70% 73% 52% 59% 67% Depth 40% 47% 55% 59% 63% 46% 51% 24% 36% 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. 37% 50% 70% 74% 79% 80% 59% 69% 33% 37% 45% Depth 70% 28% 31% 35% 49% 57% 60% 30% 44% 24% 29% 54% 15% 23% 26% 40% 52% 50% 40% Time

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% 37% 50% 70% 74% 79% 80% 59% 69% 33% 37% 45% Depth 70% 28% 31% 35% 49% 57% 60% 30% 44% 24% 29% 54% 15% 23% 26% 40% 52% 50% 40% Time

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% 50% 40% 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. 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: 30 min 0 min 10 min 20 min 40 min 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 Derived earlier that the average concentration particles of type i in a column after some settling time is: 30 min 0 min 10 min 20 min 40 min

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

27. 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: 1 df Cumulative fraction of Concentration, fv dv 0 Velocity, v

28. Evaluating fv as function of v 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 1 df fv 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. dv h 0 Velocity, v Cinit 0

29. Evaluating h as function of fv 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. 1 df fv dv 0 Velocity, v 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 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 0 0 1 fv

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