when terminal velocity is reached, v s is constant (v term ), and f net = 0: if re is less than...

3

6G p pF d g

3

6B p LF d g

2

2p

D D p L

vF C A

0G B DF F F

2

3 06 2

termp p L D p L

vd g C A

2

18p p L

term

d gv

When terminal velocity is reached, vs is constant (vterm), and Fnet = 0:

If Re is less than ~100, CD 24/Re, and:

0 min 10 min 20 min 40 min

Discrete Settling of a Suspension of Particles that Reach Their Terminal Settling Velocity Rapidly

30 minz =0

z =Z

h(t)

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.

0 min 10 min 20 min 40 min30 min

1. At any t >0, the particles are present in only part of the column.

2. The particles’ settling velocity can be computed as h/t after any t.

3. 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:

Inferences

h

0 00mixed

Z h h Z hC C C

Z Z Z

If the computed h is ≥ Z, Cmixed will be zero.

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 2.4 24 48 96

1.4 5.0 50 100 200

0 min 10 min 20 min 40 min

If particles with different terminal velocities are

present, they might behave independently

50 cm

0 cm

0 min 10 min 20 min 40 min

The total particle concentration at any depth is the sum of the concentrations of the particles at that depth

C (mg/L)Cin0

z (c

m)

0

50

0 min 10 min 20 min

1. Interfaces develop instantly between layers of water with various groups of particles.

2. 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.

3. 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.

4. Each interface sinks at a constant rate equal to vterm of the next faster-settling group of particles.

30 min

If Several Groups of Independent Particles are Present (Type I Settling)

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 min30 min0 min

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

h

t1 or l1

10%20% 35%

50%

70%

85% removed

t2 or l2 t3 or l3Time or Distance

0

H

Concentration Isopleths for Type I Settling

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

20 min 30 min 40 min

0 min 10 min 20 min

1. Often, particle growth continues during settling. This is referred to as Type II settling.

2. Particles settle faster the longer they are in the system, so they cannot be assigned a single vterm.

3. Effluent can still be viewed as mixture of different groups of particles, each present in part of the column and absent from other parts.

30 min

If Particles Do Not Behave Independently…

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.

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.

Generic Design Approach for Both Types I and II Settling

Time

Dep

th

Sketch isopleths of constant removal on the plot.

Generic Design for Both Types I and II Settling

Time50%

60%

70%

80%

40%

Dep

th

Generic Design for Both Types I and II Settling

Time50%

60%

70%

80%

40%

Dep

th

Make preliminary choices for residence time () 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%

Generic Design for Both Types I and II Settling

Time50%

70%

80%

40%

Dep

th100%

Divide the column at time 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.

90%

75%

65%

55%

z1

z2

z3

z4

• 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

1 21 2 ... jn

overall n jj

zzz z

Z Z Z Z

Each term in summation is removal efficiency in a layer times fraction of the column that the layer occupies.

Generic Design for Both Types I and II Settling

joverall j

j

z

Z

In example, zi/Z values from top to bottom are 0.12, 0.17, 0.18, and 0.53, respectively, so:

Generic Design for Both Types I and II Settling

0.90 0.12 0.75 0.17

0.65 0.18 0.55 0.53

0.64

overall

• Predicted removal depends on both depth (Z) and residence time ()

• depends on Z; to make the design parameters independent of one another, overflow rate, Z/, is used instead of

Increasing Z at constant O/F increases ; increasing O/F at constant Z decreases .

Generic Design for Both Types I and II Settling

O/F/crit

Z Z Q Qv

LWZ Q LW A

• 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

Simplified Design for Type I Settling

1 ,1 2 ,2 , ,...overall v v n v n j v jj

f f f f fv,j is the fraction of the particle concentration that has a certain terminal settling velocity, and j is the removal efficiency for particles with that settling velocity; (fi) = 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.

10 min 20 min 40 min30 min0 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:

1

,

0

overall j v j vj

f df Now, dfv is the fraction of the particle concentration that has terminal settling velocities in a small range (v to v dv), and is the average removal efficiency for particles in that group.

To evaluate integral, express both and fv as functions of vterm.

10 min 20 min 40 min30 min0 min

Derived earlier that the average concentration particles of type i in a column after some settling time is:

,0,

1 if

0 if

ii i

i mixed

i

hC h Z

C Z

h Z

if

0 if i i

ii

h Z h Z

h Z

, ,01 , so:i i mixed iC C

Evaluating i as Function of v

10 min 20 min 40 min30 min0 min

, if

0 if i term crit i

i

i

v v h Z

h Z

Recall that the velocity required to fall Z in time is vcrit. So, for particles with constant vi,term:

, ,i term i termi

crit crit

v vh

Z v 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:

Evaluating fv as function of v

Velocity, v

Cu

mu

lati

ve f

ract

ion

of

Co

nce

ntr

atio

n,

f v

0

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

Evaluating fv as function of v

Velocity, v

f v

0

1

dv

df

0 Cinit

h

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.

Now, we have data for fv vs. v and an expression for vs. v. From those, compute vs. fv for a given choice of vcrit.

Evaluating as function of fv

Velocity, v

f v

0

1

dv

df

vcrit (cm/min) 0.2

Col. 1 Col. 2 Col. 3 Col. 4 Col. 5

v (cm/min) fv dfv 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.

Evaluating overall

vcrit (cm/min) 0.2

Col. 1 Col. 2 Col. 3 Col. 4 Col. 5

v (cm/min) fv dfv dfv

: : : : :

overall can be evaluated as the sum of the values in Col.5 or as the area under a plot of vs. fv.

1

0

overall vdf

0

1

fv0 1

fv for vcrit

overall is area under curve

• Predicted removal depends only on properties of suspension (fv vs. v) and vcrit (which determines ). Recall that vcrit is usually reported as O/F. Changing Z while holding O/F constant (same Q and A) has no effect on overall .

• 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

Design for Type I Settling

Enhancing Sedimentation with “Tube” or “Lamellar” Settlers