16593 07 tủ tài liệu bách khoa

CHAPTER 7

SEDIMENTATION AND
FLOTATION
Ross Gregory
WRc Swindon
Frankland Road, Blagrove
Swindon, Wiltshire
England

Thomas F. Zabel
WRc Medmenham
Medmenham, Oxfordshire
England

James K. Edzwald
Professor, Department of Civil and
Environmental Engineering
University of Massachusetts
Amherst, Massachusetts
U.S.A.

Sedimentation and flotation are solid-liquid separation processes used in water
treatment mostly to lower the solids concentration, or load, on granular filters. As a
result, filters can be operated more easily and cost effectively to produce acceptablequality filtered water. Many sedimentation and flotation processes and variants of
them exist, and each has advantages and disadvantages. The most appropriate process for a particular application will depend on the water to be treated as well as
local circumstances and requirements.

HISTORY OF SEDIMENTATION
Early History
Sedimentation for the improvement of water quality has been practiced, if unwittingly, since the day humans collected and stored water in jars and other containers.

7.1


7.2

CHAPTER SEVEN

Water stored undisturbed and then poured or ladled out with little agitation will
improve in quality, and this technique is used to this day.
As societies developed, reservoirs and storage tanks were constructed. Although
constructed for strategic purposes, reservoirs and storage tanks did improve water
quality. Various examples are known that predate the Christian era. Ancient surface
water impounding tanks of Aden were possibly constructed as early as 600 B.C. and
rainwater cisterns of ancient Carthage about 150 B.C. (Ellms, 1928).The castellae and
piscinae of the Roman aqueduct system performed the function of settling tanks,
even though they were not originally intended for that purpose.

Modern Sedimentation
The art of sedimentation progressed little until the industrial age and its increased
need for water. Storage reservoirs developed into settling reservoirs. Perhaps the
largest reservoirs constructed for this purpose were in the United States at Cincinnati, Ohio, where two excavated reservoirs held approximately 1480 ML (392 million
gallons) and were designed to be operated by a fill-and-draw method, though they
never were used in this way (Ellms, 1928). The development of settling basins led to
the construction of rectangular masonry settling tanks that assured more even flow
distribution and easier sludge removal. With the introduction of coagulation and its
production of voluminous sludge, mechanical sludge removal was introduced.
Attempts to make rectangular tanks more cost-effective led to the construction
of multilayer tanks. Very large diameter [60-m (200-ft)] circular tanks also were constructed at an early stage in the development of modern water treatment. Other
industries, such as wastewater treatment, mineral processing, sugar refining, and
water softening, required forms of sedimentation with specific characteristics, and

various designs of settling tanks particular to certain industries were developed.
Subsequently, wider application of successful industrial designs were sought. From
this, circular radial-flow tanks emerged, as well as a variety of proprietary designs of
solids-contact units with mechanical equipment for premixing and recirculation.
The inclined plate settler also has industrial origins (Barham, Matherne, and
Keller, 1956) (Figure 7.1), although the theory of inclined settling dates back to
experiments using blood in the 1920s and 1930s (Nakamura and Kuroda, 1937;
Kinosita, 1949). Closely spaced inclined plate systems for water treatment have their
origins in Sweden in the 1950s, resulting from a search for high-rate treatment processes compact enough to be economically housed against winter weather. Inclined

FIGURE 7.1 Early patent for inclined settling. (Source:
Barham et al., 1956.)


SEDIMENTATION AND FLOTATION

7.3

FIGURE 7.2 The pyramidal Candy floc-blanket
tank. (Source: by PWT.)

tube systems were spawned in the United States in the 1960s. The most recent developments have involved combining inclined settling with ballasting of floc to reduce
plant footprint further (de Dianous, Pujol, and Druoton, 1990).
Floc-Blanket Sedimentation and Other Innovations
The floc-blanket process for water treatment emerged from India about 1932 as the
pyramidal Candy sedimentation tank (Figure 7.2). A tank of similar shape was used
by Imhoff in 1906 for wastewater treatment (Kalbskopf, 1970). The Spaulding Precipitator soon followed in 1935 (Figure 7.3) (Hartung, 1951). Other designs that
were mainly solids contact clarifiers rather than true floc-blanket tanks were also
introduced.
The Candy tank can be expensive to construct because of its large sloping sides,

so less costly structures for accommodating floc blankets were conceived. The aim
was to decrease the hopper component of tanks as much as possible, yet to provide
good flow distribution to produce a stable floc blanket. Development from 1945 progressed from tanks with multiple hoppers or troughs to the present flat-bottom
tanks. Efficient flow distribution in flat-bottom tanks is achieved with either candelabra or lateral inlet distribution systems (Figures 7.4 and 7.5).
An innovation in the 1970s was the inclusion of widely spaced inclined plates in
the floc-blanket region (Figure 7.5). Other developments that also have led to
increased surface loadings include the use of polyelectrolytes, ballasting of floc with
disposable or recycled solids, and improvements in blanket-level control. The principal centers for these developments have been in the United Kingdom, France, and
Hungary.

SEDIMENTATION THEORY
The particle-fluid separation processes of interest to water engineers and scientists
are difficult to describe by a theoretical analysis, mainly because the particles


7.4

CHAPTER SEVEN

FIGURE 7.3 The Spaulding Precipitator solids contact clarifier. (Source: Hartung, 1951.)

involved are not regular in shape, density, or size. Consideration of the theory of
ideal systems is, however, a useful guide to interpreting observed behavior in more
complex cases.
The various regimes in settling of particles are commonly referred to as Types 1
to 4. The general term settling is used to describe all types of particles falling through

FIGURE 7.4 The flat-bottom clarifier with candelabra flow distribution. (Source: by PWT.)

SEDIMENTATION AND FLOTATION

7.5

FIGURE 7.5 The Superpulsator flat-bottom clarifier with lateral-flow distribution. (Source: Courtesy of Infilco Degremont, Inc., Richmond, VA.)

a liquid under the force of gravity and settling phenomena in which the particles or
aggregates are suspended by hydrodynamic forces only. When particles or aggregates rest on one another, the term subsidence applies. The following definitions of
the settling regimes are commonly used in the United States and are compatible
with a comprehensive analysis of hindered settling and flux theory:
Type 1. Settling of discrete particles in low concentration, with flocculation and
other interparticle effects being negligible.
Type 2. Settling of particles in low concentration but with coalescence or flocculation. As coalescence occurs, particle masses increase and particles settle more
rapidly.
Type 3. Hindered, or zone, settling in which particle concentration causes interparticle effects, which might include flocculation, to the extent that the rate of settling is a function of solids concentration. Zones of different concentrations may
develop from segregation of particles with different settling velocities. Two
regimes exist—a and b—with the concentration being less and greater than that
at maximum flux, respectively. In the latter case, the concentration has reached
the point that most particles make regular physical contact with adjacent particles and effectively form a loose structure. As the height of this zone develops,
this structure tends to form layers of different concentration, with the lower layers establishing permanent physical contact, until a state of compression is
reached in the bottom layer.
Type 4. Compression settling or subsidence develops under the layers of zone
settling. The rate of compression is dependent on time and the force caused by
the weight of solids above.


7.6

CHAPTER SEVEN

Settling of Discrete Particles (Type 1)
Terminal Settling Velocity. When the concentration of particles is small, each particle settles discretely, as if it were alone, unhindered by the presence of other particles. Starting from rest, the velocity of a single particle settling under gravity in a
liquid will increase, where the density of the particle is greater than the density of the
liquid.
Acceleration continues until the resistance to flow through the liquid, or drag,
equals the effective weight of the particle. Thereafter, the settling velocity remains
essentially constant. This velocity is called the terminal settling velocity, vt. The terminal settling velocity depends on various factors relating to the particle and the liquid.
For most theoretical and practical computations of settling velocities, the shape
of particles is assumed to be spherical. The size of particles that are not spherical can
be expressed in terms of a sphere of equivalent volume.
The general equation for the terminal settling velocity of a single particle is
derived by equating the forces upon the particle. These forces are the drag fd, buoyancy fb, and an external source such as gravity fg. Hence,
fd = fg − fb

(7.1)

The drag force on a particle traveling in a resistant fluid is (Prandtl and Tietjens, 1957):
CDv2ρA
fd = ᎏ
2

(7.2)

where CD = drag coefficient
v = settling velocity
ρ = mass density of liquid
A = projected area of particle in direction of flow
Any consistent, dimensionally homogeneous units may be used in Eq. 7.2 and all
subsequent rational equations.
At constant (i.e., terminal settling velocity) vt

fg − fb = Vg(ρp − ρ)

(7.3)

where V is the effective volume of the particle, g is the gravitational constant of
acceleration, and ρp is the density of the particle. When Eqs. 7.2 and 7.3 are substituted in Eq. 7.1
CDv2t ρA
ᎏ = Vg(ρp − ρ)
2

(7.4a)

2g(ρp − ρ)V
vt = ᎏᎏ
CDρA

΅

(7.4b)

΅

(7.5)

rearranging:

΄

1/2

When the particle is solid and spherical,
4g(ρp − ρ)d
vt = ᎏᎏ
3CDρ

΄

where d is the diameter of the sphere.

1/2


SEDIMENTATION AND FLOTATION

7.7

FIGURE 7.6 Variation of drag coefficient, CD, with Reynolds number, Re, for
single-particle sedimentation.

The value of vt is the difference in velocity between the particle and the liquid
and is essentially independent of horizontal or vertical movement of the liquid,
although in real situations there are secondary forces caused by velocity gradients,
and so on. Therefore, the relationship also applies to a dense stationary particle with
liquid flowing upward past it or a buoyant particle with liquid flowing downward.
Calculation of vt for a given system is difficult because the drag coefficient, CD,
depends on the nature of the flow around the particle. This relationship can be
described using the Reynolds number, Re (based on particle diameter), as illustrated schematically in Figure 7.6, where
ρvd
Re = ᎏ
µ

(7.6)

and µ is the absolute (dynamic) liquid viscosity, and v is the velocity of the particle
relative to the liquid.
The value of CD decreases as the value of Re increases, but at a rate depending on
the value of Re, such that for spheres only:
Region (a): 10−4 < Re < 0.2. In this region of small Re value, the laminar flow
region, the equation of the relationship approximates to
CD = 24/Re

(7.7)

This, substituted in Eq. 7.1, gives Stokes’ equation for laminar flow conditions:
g(ρp − ρ)d2
vt = ᎏᎏ
18µ

(7.8)

Region (b): 0.2 < Re < 500 to 1000. This transition zone is the most difficult to
represent, and various proposals have been made. Perhaps the most recognized


7.8

CHAPTER SEVEN

representation of this zone for spheres is that promoted by Fair, Geyer, and
Okun (1971):

3
24
CD = ᎏ + ᎏ
+ 0.34
Re Re1/2

(7.9)

For many particles found in natural waters, the density and diameter yield Re values within this region.
Region (c): 500 to 1000 < Re < 2 × 105. In this region of turbulent flow, the value of
CD is almost constant at 0.44. Substitution in Eq. 7.5 results in Newton’s equation:
(ρp − ρ)gd
vt = 1.74 ᎏᎏ
ρ

΄

΅

1/2

(7.10)

Region (d): Re > 2 × 105. The drag force decreases considerably with the development of turbulent flow at the surface of the particle called boundary-layer turbulence, such that the value of CD becomes equal to 0.10. This region is unlikely
to be encountered in sedimentation in water treatment.
Effect of Particle Shape. Equation 7.4b shows how particle shape affects velocity.
The effect of a nonspherical shape is to increase the value of CD at a given value of Re.
As a result, the settling velocity of a nonspherical particle is less than that of a
sphere having the same volume and density. Sometimes, a simple shape factor, Θ, is
determined, for example, in Eq. 7.7:

24Θ
CD = ᎏ
Re

(7.11)

Typical values found for Θ for rigid particles are (Degremont, 1991):
Sand
Coal
Gypsum
Graphite flakes

2.0
2.25
4.0
22

Details on the settling behavior of spheres and nonspherical particles can be found
in standard texts (e.g., Coulson and Richardson, 1978).
Flocculation. A shape factor value is difficult to determine for floc particles
because their size and shape are interlinked with the mechanics of their formation
and disruption in any set of flow conditions. When particles flocculate, a loose and
irregular structure is formed, which is likely to have a relatively large value shape
factor. Additionally, while the effective particle size increases in flocculation, the
effective particle density decreases in accordance with a fractal dimension (Lagvankar and Gemmel, 1968; Tambo and Watanabe, 1979) (see Chapter 6).
Flocculation is a process of aggregation and attrition. Aggregation can occur by
Brownian diffusion, differential settling, and velocity gradients caused by fluid
shear, namely flocculation. Attrition is caused mainly by excessive velocity gradients
(see Chapter 6).
The theory of flocculation detailed in Chapter 6 recognizes the role of velocity

gradient (G) and time (t) as well as particle volumetric concentration Φ. For dilute


SEDIMENTATION AND FLOTATION

7.9

suspensions, optimum flocculation conditions are generally considered only in terms
of G and t:
Gt = constant

(7.12)

Camp (1955) identified optimum Gt values between 104 and 105 for flocculation
prior to horizontal flow settlers. In the case of floc-blanket clarifiers (there being no
prior flocculators), the value of G is usually less than in flocculators, and the value of
Gt is about 20,000 (Gregory, 1979). This tends to be less than that usually considered
necessary for flocculation prior to inclined settling or dissolved air flotation.
In concentrated suspensions, such as with hindered settling, the greater particle
concentration (e.g., volumetric concentration, Φ) contributes to flocculation by
enhancing the probability of particle collisions, and increasing the velocity gradient
that can be expressed in terms of the head loss across the suspension. Consequently,
optimum flocculation conditions for concentrated suspensions may be better represented by Fair, Geyer, and Okun (1971); Ives (1968); and Vostrcil (1971):
GtΦ = constant

(7.13)

The value of the constant at maximum flux is likely to be about 4000, when Θ is measured as the fractional volume occupied by floc, with little benefit to be gained from
a larger value (Gregory, 1979; Vostrcil, 1971).
Measurement of the volumetric concentration of floc particle suspensions is a

problem because of variations in particle size, shape, and other factors. A simple settlement test is the easiest method of producing a measurement (for concentrations
encountered in floc blanket settling) in a standard way (e.g., half-hour settlement in
a graduated cylinder) (Gregory, 1979). A graduated cylinder (e.g., 100 mL or 1 L) is
filled to the top mark with the suspension to be measured: the half-hour settledsolids volume is the volume occupied by the settled suspension measured after 30
min, and it is expressed as a fraction of the total volume of the whole sample.
The process of flocculation continues during conditions intended to allow settlement. Assuming collision between flocculant particles takes place only between particles settling at different velocities at Stokes’ velocities, then the collision
frequencies Νij between particles of size di and dj of concentrations ni and nj is given
by Amirtharajah and O’Melia (1990):
πg(s − 1)
(Nij)d = ᎏᎏ (di − dj)3(di − dj)ninj
72ν

(7.14)

where s is the specific gravity of particles and ν is the kinematic viscosity.
Settlement in Tanks. In an ideal upflow settling tank, the particles retained are
those whose terminal settling velocity exceeds the liquid upflow velocity:
Q
vt ≥ ᎏ
A

(7.15)

where Q is the inlet flow rate to the tank, and A is the cross-sectional area of the tank.
In a horizontal-flow rectangular tank, the settling of a particle has both vertical
and horizontal components, as shown in Figure 7.7:
tQ
L=ᎏ
HW

(7.16)


7.10

CHAPTER SEVEN

FIGURE 7.7 Horizontal and vertical components of settling velocity. (Source: Fair,
Geyer, and Okun, 1971.)

where L = horizontal distance traveled
t = time of travel
H = depth of water
W = width of tank
and for the vertical distance traveled, h:
h = vt

(7.17a)

Hence, the settling time for a particle that has entered the tank at a given level, h, is
h
t=ᎏ
v

(7.17b)

Substitution of this in Eq. 7.16 gives the length of tank required for settlement to
occur under ideal flow conditions:
hQ
L=ᎏ

vHW

(7.18a)

hQ
v=ᎏ
HLW

(7.18b)

or

If all particles with a settling velocity of v are allowed to settle, then h equals H, and,
consequently, this special case then defines the surface-loading or overflow rate of
the ideal tank, v*:
Q
v* = ᎏ
L*W

(7.19a)

Q
v* = ᎏ
A*

(7.19b)


SEDIMENTATION AND FLOTATION

7.11

where L* is the length of tank over which settlement ideally takes place, and A* is the
plan area of tank, with horizontal flow, over which the settlement ideally takes place.
All particles with a settling velocity greater than v* are removed. Particles with a
settling velocity less than v* are removed in proportion to the ratio v:v*.
Particles with a settling velocity v′ less than v* need a tank of length, L′, greater
than L* for total settlement, such that
v′ L*
ᎏ=ᎏ
v* L′

(7.20)

This ratio defines the proportion of particles with a settling velocity of v′ that settle
in a length L*. Equation 7.19a states that settling efficiency depends on the area
available for settling. The same result applies to circular tanks.
Equation 7.19a shows that settling efficiency for the ideal condition is independent of depth H and dependent on only the tank plan area. This principle is sometimes referred to as Hazen’s law. In contrast, retention time, t, is dependent on water
depth, H, as given by
AH
t=ᎏ
Q

(7.21)

In reality, depth is important because it can affect flow stability if it is large and
scouring if it is small.
Predicting Settling Efficiency (Types 1 and 2). For discrete particle settling (Type
1), variation in particle size and density produce a distribution of settling velocities.
The settling velocity distribution may be determined from data collected from a column settling test. The settling column test produces information on x1 (fraction of

particles with settling velocities less than or equal to v1) and v1. This data is used to
produce a settling-velocity analysis curve (Figure 7.8) (Metcalf and Eddy, 1991).
Equation 7.20 defines the proportion of particles with settling velocity v smaller
than v*, which will be removed in a given time. If x* is the proportion of particles
having settling velocities less than or equal to v*, the total proportion of particles
that could be removed in settling is defined by Thirumurthi (1969):
Ft = (1 − x*) +

v
͵ᎏ
dx
v*
x

(7.22)

0

This can be solved using a version of Figure 7.8.
In flocculant settling (Type 2), flocculation occurs as the particles settle. To evaluate the effect of flocculation as a function of basin depth requires a column test with
sampling ports at various depths (Zanoni and Blomquist, 1975). The settling column
should be as deep as the basin being designed. A set of samples is taken every 20 min
or so for at least 2 h. The suspended-solids concentration is determined in each sample and expressed as a percentage difference, removal R, of the original concentration. These results are plotted against time and depth, and curves of equal percentage
removal are drawn. Figure 7.9 is an example for flocculated-particle Type 2 settling,
with increase in settling velocity as settlement progresses. An effective settling rate
for the quiescent conditions of the column can be defined as the ratio of the effective
depth divided by the time required to obtain a given percentage of removal.
For Figure 7.9, any combination of depth hn and time tn on one of the isopercentage lines will establish a settlement velocity vn:
hn
vn = ᎏ

tn

(7.23)


7.12

CHAPTER SEVEN

FIGURE 7.8 Settling-velocity analysis curve for discrete particles. (Source: Camp, 1936; Metcalf and Eddy, Engineers, 1991.
Wastewater Engineering, 3rd ed. New York: McGraw-Hill.
Reproduced by permission of the McGraw-Hill Companies.)

Thus, all particles with a settling velocity equal to or greater than vn will be removed.
Particles with a velocity v less than vn are assumed to be removed in the proportion v/vn.
For the same time tn, this would be the same as taking the depth ratios, h/hn, for
the same reason as in Eq. 7.20. Then the overall removal of particles to a depth hn is
given by

FIGURE 7.9 Settling column and isopercentage settling curves for flocculant particles. (Source: Metcalf and Eddy, Engineers, 1991. Wastewater Engineering, 3rd ed. New York: McGraw-Hill. Reproduced by permission of the
McGraw-Hill Companies.)


7.13

SEDIMENTATION AND FLOTATION

1
∆R = ᎏ
hn

͵

hn

R ⋅ dh

(7.24)

0

This might be solved as [Metcalf and Eddy (1991)]:
n
∆hn Rn + Rn + 1
∆R = Α ᎏ × ᎏᎏ
2
h5
0

΄

΅

(7.25a)

Then with respect to Fig 7.9, this can be written as
∆h1 R1 + R2
∆h2 R2 + R3
∆h3 R3 + R4
∆h4 R4 + R5

∆R = ᎏ × ᎏ + ᎏ × ᎏ + ᎏ × ᎏ + ᎏ × ᎏ
2
2
2
2
h5
h5
h5
h5

΄

΅ ΄

΅ ΄

΅ ΄

΅

(7.25b)
Although of limited practical value, Agrawal and Bewtra (1985) and Ali San (1989)
have suggested improvements for this computation that overcome weaknesses in
the assumptions.
In practice, to design a full-scale settling tank to achieve comparable removal,
the settling rate from the column test should be multiplied by a factor of 0.65 to
0.85, and the detention time should be multiplied by a factor of 1.25 to 1.5 (Metcalf
and Eddy, 1991).

Hindered Settling (Types 3a and 3b)

The following addresses Type 3 settling relevant to clarification. Types 3 and 4 settling as relevant to thickening are addressed in Chapter 16.
Particle Interaction. At high particle concentrations, individual particle behavior
is influenced, or hindered, by the presence of other particles, and the flow characteristics of the bulk suspension can be affected. With increased particle concentration,
the free area between particles is reduced causing greater interparticle fluid velocities and alteration of flow patterns around particles. Consequently, the average settling velocity of the particles in a concentrated suspension is generally less than that
of a discrete particle of similar size.
When particles in a suspension are not uniform in size, shape, or density, individual particles will have different settling velocities. Particles with a settling velocity
less than the suspension increase the effective viscosity. Smaller particles tend to be
dragged down by the motion of larger particles. Flocculation may increase the effective particle size when particles are close together (i.e., flocculation due to differential settling, Eq. 7.14).
Solids Flux. The settling velocity of the suspension, vs, depends on particle concentration in the suspension. The product of velocity and mass concentration, C, is
solids mass flux FM, the mass of solids passing a unit area per unit of time:
FM = vsC

(7.26a)

The equivalent relationship holds for solids volumetric concentration, Φ, to define
solids volume flux, FV:
FV = vsΦ

(7.26b)


7.14

CHAPTER SEVEN

FIGURE 7.10 Typical relationship between flux and concentration
for batch settlement.

The relationship between FM and C is shown in Figure 7.10 and is complex because
vs is affected by concentration. The relationship can be divided into four regions.

Region (a): Type 1 and 2 settling. Unhindered settling occurs such that the flux
increases in proportion to the concentration. A suspension of particles with different settling velocities has a diffuse interface with the clear liquid above.
Region (b): Type 3a settling. With increase in concentration, hindered-flow settling increasingly takes effect, and ultimately a maximum value of flux is reached.
At about maximum flux, the diffuse interface of the suspension becomes distinct
with the clear liquid above when all particles become part of the suspension and
settle with the same velocity.
Region (c): Type 3b settling. Further increase in concentration reduces flux
because of the reduction in settling velocity. In this region, the suspension settles
homogeneously.
Region (d): Type 4 settling. Associated with the point of inflection in the fluxconcentration curve, the concentration reaches the point where thickening can be
regarded to start leading ultimately to compression settling.
Equations for Hindered Settling. The behavior of suspensions in regions (b) and
(c) has attracted considerable theoretical and empirical analysis and is most important in understanding floc-blanket clarification. The simplest and most convenient
relationship is represented by the general equation (Gregory, 1979)
vs = v0 exp (−q Φ)

(7.27)

where q = constant representative of the suspension
v0 = settling velocity of suspension for concentration extrapolated to zero
Φ = volume concentration of the suspension


SEDIMENTATION AND FLOTATION

7.15

Other empirical relationships have been proposed. The most widely accepted and
tested relationship was initially developed for particles larger than 0.1 mm diameter
in rigid particle-fluidized systems. This relationship has been shown to be applicable

to settling and is known as the Richardson and Zaki equation (Coulson and Richardson, 1978):
vs = vtΕn

(7.28)

where Ε = porosity of the suspension (i.e., volume of fluid per volume of suspension, Ε = 1 − Φ)
n = power value dependent on the Reynolds number of the particle
vt = terminal settling velocity of particles in unhindered flow (i.e., absence
of effect by presence of other particles)
For rigid particles, this equation is valid for porosity from about 0.6 (occurring at
around minimum fluidization velocity) to about 0.95. The Reynolds number determines the value of n (Coulson and Richardson, 1978). For a suspension with uniformsize spherical particles, n = 4.8 when Re is less than 0.2. As the value of Re increases,
n decreases until Re is greater than 500 when n equals 2.4.
When Eq. 7.28 is used for flocculent suspensions (Gregory, 1979) correction factors
must be included to adjust for effective volume to account for particle distortion and
compression. If particle volume concentration is measured, for example, by the halfhour settlement test, then because such a test as this is only a relative measurement providing a measure of the apparent concentration, then such adjustments are necessary.
vs = vtk1(1 − k2Φ*)r

(7.29)

where k1, k2 = constants representative of the system
Φ* = apparent solids volumetric concentration
r = power value dependent on the system
Equation 7.28 can be substituted in Eq. 7.26 for flux with 1 − Ε substituted for Φ
(Coulson and Richardson, 1978):
FM = vtΕn(1 − Ε)

(7.30)

Differentiating this equation with respect to Ε gives:
dFM

ᎏ = vtnΕn − 1 − vt(n + 1)Εn
dΕ

(7.31)

The flux FM has a maximum value when dFM/dΕ equals zero and Ε equals Ε+ (the
porosity at maximum flux). Hence, dividing Eq. 7.31 by vt Εn − 1 and equating to zero
produces
0 = n − (n + 1) Ε+

(7.32a)

Ε+
n = ᎏ+
1−Ε

(7.32b)

or

This means that the porosity Ε+, or the volume concentration, at maximum flux, Φ+,
is an important parameter in describing the settling rates of suspensions. In the case
of rigid uniform spheres, if n ranges from 2.4 to 4.6, the maximum flux should occur
at a volumetric concentration between 0.29 and 0.18. In practice, the range of values


7.16

CHAPTER SEVEN

generally found for suspensions of aluminum and iron flocs for optimal coagulant
dose and coagulation pH, when concentration is measured as the half-hour settled
volume, is 0.16 to 0.20 (Gregory, 1979; Gregory, Head, and Graham, 1996), in which
case n ranges from 4.0 to 5.26.
If Eq. 7.31 is differentiated also, then
d2FM
ᎏ
= vt[n(n − 1)Εn − 2 − (n + 1)nΕn − 1]
dΕ2

(7.33)

and when d2FM/dΕ2 = 0 for real values of Ε, a point of inflection will exist, given by
0 = n − 1 − (n + 1) Ε

(7.34a)

n−1
Ε=ᎏ
n+1

(7.34b)

such that

For rigid uniform spheres, if n ranges from 2.4 to 4.6, the point of inflection occurs at
a concentration between 0.59 and 0.35. For nonrigid, irregular-shaped, and multisized particles, the situation is more complex.
The point of inflection is associated with the transition from Type 3 to Type 4 settling. Type 3 and 4 settling in the context of thickening are considered in Chapter 16.
Prediction of Settling Rate. The hindered settling rate can be predicted for suspensions of rigid and uniform spheres using Eqs. 7.5 and 7.28. For suspensions of
nonuniform and flocculent particles, however, settling rate has to be measured. This

is most simply done using a settling column; a 1-L measuring cylinder is usually adequate. The procedure is to fill the cylinder to the top measuring mark with the sample and record at frequent intervals the level of the interface between the suspension
and the clear-water zone. The interface is only likely to be distinct enough for this
purpose if the concentration of the sample is greater than that at maximum flux. The
results are plotted to produce the typical settling curve (Figure 7.11). The slope of
the curve over the constant-settling-rate period is the estimate of the Type 3 settling
rate for quiescent conditions. If the concentration of the sample was greater than
that at the inflection point in the mass flux curve, the transition from region (c) to (d)
in Figure 7.10, then a period of constant settling rate and the compression point (CP)
will not be found as represented by line A.
The compression point signifies the point at which all the suspension has passed
into the Type 4 settling or compression regime. Up to that time, a zone of solids in the
compression regime has been accumulating at the bottom of the suspension with its
upper interface moving upward. The compression point, thus, is where that interface
reaches the top of the settling suspension.

Fluidization
When liquid is moving up through a uniform stationary bed of particles at a low flow
rate, the flow behavior is similar to when the flow is down through the bed.When the
upward flow of liquid is great enough to cause a drag force on particles equal to the
apparent weight (actual weight less buoyancy) of the particles, the particles rearrange to offer less resistance to flow and bed expansion occurs. This process continues as the liquid velocity is increased until the bed has assumed the least stable form
of packing. If the upward liquid velocity is increased further, individual particles sep-


SEDIMENTATION AND FLOTATION

7.17

FIGURE 7.11 Typical batch-settling curves. (Source: Pearse, 1977.)

arate from one another and become freely supported in the liquid. The bed is then

said to be fluidized.
For rigid and generally uniform particles, such as with filter sand, about 10 percent bed expansion occurs before fluidization commences. The less uniform the size
and density of the particles, the less distinct is the point of fluidization. A fluidized
bed is characterized by regular expansion of the bed as liquid velocity increases from
the minimum fluidization velocity until particles are in unhindered suspension (i.e.,
Type 1 settling).
Fluidization is hydrodynamically similar to hindered, or zone Type 3, settling. In
a fluidized bed, particles undergo no net movement and are maintained in suspension by the upward flow of the liquid. In hindered settling, particles move downward, and in the simple case of batch settling, no net flow of liquid occurs. The
Richardson and Zaki equation, Eq. 7.28, has been found to be applicable to both fluidization and hindered settling (Coulson and Richardson, 1978) as have other relationships.
In water treatment, floc-blanket clarification is more a fluidized bed rather than
a hindered settling process. Extensive floc-blanket data (Gregory, 1979) with Φ*
determined as the half-hour settled-solids volume, such that Φ+ tended to be in the
range 0.16 to 0.20, allowed Eq. 7.29 to be simplified to
vs = v0 (1 − 2.5Φ*)

(7.35a)

vs
v0 = ᎏᎏ
(1 − 2.5Φ*)

(7.35b)

or


7.18

CHAPTER SEVEN

The data that allowed this simplification was obtained with alum coagulation (optimal coagulant dose and coagulation pH) of a high-alkalinity, organic-rich river water,
but the value for k2 (in Eq. 7.29) of 2.5 should hold for other types of water producing similar quality floc. The value of k2 can be estimated as the ratio of the concentration at the compression point to the half-hour settled concentration. Values
predicted for v0 by Eq. 7.35b are less, about one-half to one-third, than those likely to
be estimated by Stokes’ equation for vt, assuming spherical particles (Gregory, 1979).
The theory of hindered settling and fluidization of particles of mixed sizes and
different densities is more complex and is still being developed. In some situations,
two or more phases can occur at a given velocity, each phase with a different concentration. This has been observed with floc blankets to the extent that an early but
temporary deterioration in performance occurs with increase in upflow (Gregory
and Hyde, 1975; Setterfield, 1983). An increase in upflow leads to intermixing of the
phases, with further increase in upflow limited by the characteristics of the combined
phase. The theory has been used to explain and predict the occurrence of intermixing and segregation in multimedia filter beds during and after backwash (Patwardan
and Tien, 1985; Epstein and LeClair, 1985).
EXAMPLE PROBLEM 7.1

Predict the maximum volume flux conditions for floc-

blanket sedimentation.
SOLUTION For a floc blanket that can be operated over a range of upflow rates, collect samples of blanket at different upflow rates. For these samples, measure the
half-hour settled volume. Example results are listed below:

Upflow rate (m/h)
Half-hour floc volume (%)
Blanket flux = upflow rate ×
half-hour floc volume (%m/h)

1.6
31

1.95
29

2.5
25

3.05
22

3.65
19

4.2
16

4.7
13

5.15
10

49.6

56.6

62.5

67.1

69.4

67.2

61.1

51.5

These results predict that maximum flux occurs at an upflow rate of 3.7 m/h. If
flux is plotted against upflow rate and against half-hour floc volume, then the maximum flux is located at 3.4 m/h for a half-hour floc volume of 20 percent, as shown in
Figure 7.12.
The above results can be fitted to Eq. 7.35a:
vs = v0 (1 − 2.5Φ*)
3.44 = v0 (1 − 2.5 × 0.2)
v0 = 3.44/0.5 = 6.9 m/h
This means that at the maximum flux, the theoretical terminal settling velocity of
the blanket is 6.9 m/h.The maximum operating rate for a floc blanket in a stable tank
is about 70 percent of this rate, or 4.8 m/h.
Inclined (Tube and Plate) Settling
The efficiency of discrete particle settling in horizontal liquid flow depends on the
area available for settling. Hence, efficiency can be improved by increasing the area.
Some tanks have multiple floors to achieve this. A successful alternative is to use
lightweight structures with closely spaced inclined surfaces.


SEDIMENTATION AND FLOTATION

7.19

FIGURE 7.12 Relationship between blanket flux, blanket concentration, and upflow
rate for Example Problem 7.1.

Inclined settling systems (Figure 7.13) are constructed for use in one of three

ways with respect to the direction of liquid flow relative to the direction of particle
settlement: countercurrent, cocurrent, and cross-flow. Comprehensive theoretical
analyses of the various flow geometries have been made by Yao (1970). Yao’s analysis is based on flow conditions in the channels between the inclined surfaces being
laminar. In practice, the Reynolds number must be less than 800 when calculated
using the mean velocity vθ between and parallel to the inclined surfaces and
hydraulic diameter of the channel dH:
4AH
dH = ᎏ
P

(7.36)


7.20

CHAPTER SEVEN

FIGURE 7.13

Basic flow geometries for inclined settling systems.

where AH is the cross-sectional area of channel-to-liquid flow, and P is the perimeter
of AH, such that Eq. 7.6 becomes:
ρvθdH
Re = ᎏ
µ

(7.37)

Countercurrent Settling. The time, t, for a particle to settle the vertical distance

between two parallel inclined surfaces is:
w
t=ᎏ
v cos θ

(7.38)

where w is the perpendicular distance between surfaces, and θ is the angle of surface
inclination from the horizontal. The length of surface, Lp, needed to provide this
time, if the liquid velocity between the surfaces is vθ, is
w(vθ − v sin θ)
Lp = ᎏᎏ
v cos θ

(7.39a)


SEDIMENTATION AND FLOTATION

7.21

By rearranging this equation, all particles with a settling velocity, v, and greater are
removed if
vθw
v ≥ ᎏᎏ
Lp cos θ + w sin θ

(7.39b)

When many plates or tubes are used

Q
vθ = ᎏ
Nwb

(7.40)

where N is the number of channels made by N + 1 plates or tubes, and b is the dimension of the surface at right angles to w and Q.
Cocurrent Settling. In cocurrent settling, the time for a particle to settle the vertical distance between two surfaces is the same as for countercurrent settling. The
length of surface needed, however, has to be based on downward, and not upward,
liquid flow:
(vθ + v sin θ)
Lp = wᎏᎏ
v cos θ

(7.41a)

Consequently, the condition for removal of particles is given by
vθw
v ≥ ᎏᎏ
Lp cos θ − w sin θ

(7.41b)

Cross-Flow Settling. The time for a particle to settle the vertical distance between
two surfaces is again given by Eq. 7.39. The liquid flow is horizontal and does not
interact with the vertical settling velocity of a particle. Hence
vθw
Lp = ᎏ
v cos θ

(7.42a)

vθw
v≥ᎏ
Lp cos θ

(7.42b)

and

Other Flow Geometries. The above three analyses apply only for parallel surface
systems.To simplify the analysis for other geometries,Yao (1970) suggested a parameter, Sc, defined as
(sin θ + Lr cos θ)
Sc = v ᎏᎏ
vθ

(7.43)

where Lr = L/w is the relative length of the settler.
When v* is the special case that all particles with velocity, v, or greater are
removed, then for parallel surfaces, Sc is equal to 1.0. However, the value for circular
tubes is 4/3 and for square conduits is 11/8 (Yao, 1973). Identical values of Sc for different systems may not mean identical behavior.


7.22

CHAPTER SEVEN

The design overflow rate is also defined by v* in Eq. 7.43, and Yao has shown by
integration of the differential equation for a particle trajectory that the overflow

rate for an inclined settler is given by
k3 K vθ
v* = ᎏ
Lr

(7.44)

where k3 is a constant equal to 8.64 × 102 m3/day ⋅ m−2 and
SC Lr
K = ᎏᎏ
sin θ + Lr cos θ

(7.45a)

For given values of overflow rate and surface spacing and when θ = 0, Eq. 7.45a
becomes
SC
ᎏ = constant
L

(7.45b)

Equation 7.45b indicates that the larger the value of SC the longer the surface length
must be to achieve the required theoretical performance. In practice, compromises
must be made between theory and the hydrodynamic problems of flow distribution
and stability that each different geometry poses.
A tank has been fitted with 2.0 m (6.6 ft) square inclined
plates spaced 50 mm (2.0 in) apart. The angle of inclination of the plates can be
altered from 5° to 85°. The inlet to and outlet from the tank can be fitted in any way
so that the tank can be used for either countercurrent, cocurrent, or cross-flow sedimentation. If no allowances need to be made for hydraulic problems due to flow distribution and so on, then which is the best arrangement to use?

EXAMPLE PROBLEM 7.2

SOLUTION Equation 7.39b for countercurrent flow, Eq. 7.41b for cocurrent flow,
and Eq. 7.42b for cross-flow sedimentation are compared. As an example, the calculation for countercurrent flow at 85° is

v
50
50
ᎏ = ᎏᎏᎏ = ᎏᎏ = 0.223
vθ 2000 cos 85 + 50 sin 85 174.3 + 49.8
The smallest value of v is required. Thus, for the range:
Angle (θ)
Countercurrent (v/vθ)
Cocurrent (v/vθ)
Cross-flow (v/vθ)

5
0.025
0.025
0.025

15
0.026
0.026
0.026

30
0.028
0.029

0.029

45
0.035
0.036
0.035

60
0.048
0.052
0.050

75
0.088
0.106
0.096

85
0.223
0.402
0.287

From the above, little difference exists between the three settling arrangements for
an angle of less than 60°. For angles greater than 60°, countercurrent flow allows settlement of particles with the smallest settling velocity.

Floc-Blanket Clarification
A simple floc-blanket tank has a vertical parallel-walled upper section with a flat or
hopper-shaped base. Water that has been dosed with an appropriate quantity of a

SEDIMENTATION AND FLOTATION

7.23

suitable coagulant, and pH adjusted if needed, is fed downward into the base. The
resultant expanding upward flow allows flocculation to occur, and large floc particles remain in suspension within the tank. Particles in suspension accumulate slowly
at first, but then at an increasing rate due to enhanced flocculation and other effects,
eventually reaching a maximum accumulation rate limited by the particle characteristics and the upflow velocity of the water.When this maximum rate is reached, a floc
blanket can be said to exist.
As floc particles accumulate, the volume occupied by the suspension in the floc
blanket increases and its upper surface rises. The level of the floc blanket surface is
controlled by removing solids from the blanket to keep a zone of clear water or
supernatant liquid between the blanket and the decanting troughs, launders, or
weirs.
A floc blanket is thus a fluidized bed of floc particles even though the process can
be regarded as a form of hindered settling. However, true hindered settling exists
only in the upper section of sludge hoppers used for removing accumulated floc for
blanket-level control. Thickening takes place in the lower section of the sludge hoppers. Excess floc removed from the floc blanket becomes a residue stream and may
be thickened to form sludge (see Chapter 16).
Mechanism of Clarification. Settling, entrainment, and particle elutriation occur
above and at the surface of a blanket. The mechanism of clarification within a floc
blanket is more complex, however, and involves flocculation, entrapment, and sedimentation. In practice, the mean retention time of the water within a blanket is in
excess of the requirements for floc growth to control the efficiency of the process
(i.e., the opportunity for the small particles to become parts of larger and more easily retained floc is substantial, so other factors cause particles to pass through a
blanket).
Physical removal by interception and agglomeration, similar to surface capture in
deep-bed filtration, occurs throughout a floc blanket. Probably the most important
process is mechanical entrapment and straining, in which rising small particles cannot pass through the voids between larger particles that comprise the bulk of the
blanket. (The mechanisms are not the same as in filtration through a fixed bed of
sand, because all the particles are in fluid suspension.) The efficiency of entrapment

is affected by the spacing of the larger suspended floc particles, which, in turn, is
related to floc quality (shape, density, and so on) and water velocity. When suspension destabilization, coagulation, is not optimal, then flocculation will be poor and
will result in a greater number of smaller particles that can pass through the floc
blanket. (See Chapter 6 for material on coagulation and particle destabilization.)
Performance Prediction. Within a floc blanket, the relationship between floc concentration and upflow velocity of the water is represented by Eqs. 7.26 through 7.29
for hindered settling and fluidization. Unsuccessful attempts have been made to
establish a simple theory for predicting solids removal (Cretu, 1968; Shogo, 1971).
The relationships between settled-water quality and floc concentration in the blanket, upflow velocity, and flux (Figure 7.14) are of practical importance for understanding and controlling plant performance (Gregory, 1979). Recently, however,
floc-blanket clarification has been modeled successfully (Hart, 1996; Gregory, Head,
and Graham, 1996; Head, Hart, and Graham, 1997).
The modeling by Head and associates has been successfully tested in dynamic
simulations of pilot and full-scale plants. The modeling is based on the theories and
work of various researchers, including Gould (1967) and Gregory (1979). Although
the model accommodates the principle that the removal rate of primary particles is


7.24

CHAPTER SEVEN

FIGURE 7.14 Typical relationships between settled-water quality and blanket
concentration, upflow velocity and blanket flux. (Source: Gregory, 1979.)

dependent on blanket concentration, removal is simulated on the basis that the blanket region of a clarifier is a continuous stirred-tank reactor (CSTR). To take into
account the possibility of poor coagulation, the model can assume a nonremovable
fraction of solids.
The relationships in Figure 7.14 show that settled-water quality deteriorates
rapidly (point A) as the floc concentration (point B) is decreased below the concentration at maximum flux (point C). Conversely, little improvement in settled-water
quality is likely to be gained by increasing floc concentration to be greater than that
at maximum flux (to the left of points A and B). This is because for concentrations

greater than that at maximum flux, interparticle distances are small enough for
entrapment to dominate the clarification process.
As the concentration decreases below that at maximum flux, interparticle distances increase, especially between the larger particles, and their motion becomes
more intense. Some of the larger particles might not survive the higher shear rates
that develop. Thus, smaller particles may avoid entrapment and escape from the floc
blanket. Consequently, the maximum flux condition represents possible optimum
operating and design conditions. Maximum flux conditions and performance depend
on various factors, which account for differences between waters and particle surface and coagulation chemistry, and are described later.


SEDIMENTATION AND FLOTATION

7.25

Effects of Upflow Velocity. The surface loading for floc-blanket clarification is
expressed as the upflow velocity, or overflow rate. For some floc-blanket systems, the
performance curve is quartic, reflecting an early or premature deterioration in water
quality of limited magnitude with increase in upflow velocity (Figure 7.15). This
deterioration is associated with segregation of particles, or zoning, in the blanket at
low surface loading (Gregory, 1979) because of the wide range in particle settling
velocities. This has been observed not only in the treatment waters with a high silt
content but also with the use of powdered carbon (Setterfield, 1983) and in precipitation softening using iron coagulation (Gregory and Hyde, 1975). As surface loading is increased, remixing occurs at the peak of the “temporary” deterioration as the
lower-lying particles are brought into greater expansion.

FIGURE 7.15 Floc-blanket performance curves showing “temporary” deterioration in settled-water quality. (Source: Setterfield, 1983.)