granular filter design
TRANSCRIPT
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PIERO M. ARMENANTENJIT
Depth (or Deep-Bed)
Filtration
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PIERO M. ARMENANTENJIT
Depth (or Deep-Bed) Filtration
Depth filtration consists of passing a liquid,typically containing only a small amount ofsolids, through a porous bed where the solids
become trapped
Solid entrapment occurs within the entire filterbed or a significant part of it
Different bed materials are used in theindustrial practice
Depth filtration is typically a batch process
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PIERO M. ARMENANTENJIT
Slow Sand Filters vs. Rapid Filters
In slow sand filters, water flows downwardstrough a sand bed. This is one of the oldestmethods to remove solids (and other material
as well) from water. The first filters of this type
were built in England in 1829.
Sand filters operate not only because theparticles in the water are trapped in the bed,
but also because the upper layer of the bed(called the Shmutzdecke) becomes colonizedby bacteria after some time, forming agelatinous gel responsible for most of the
particle entrapment and filtration action.
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Slow Sand Filters vs. Rapid Filters
The water throughput in slow sand filters islow. These filters are cleaned infrequently byremoving the top sand layer.
Rapid filters were developed in the U.S. toincrease the water throughput (which alsoincreases the pressure drop across them) andby cleaning them frequently via fluidization.
In rapid filters there is not enough time
between cleaning (backwashing) operations togenerate a Schmutzdecke. The filtering actionoccurs throughout the entire filter bed. Thisproduces a better utilization of the entire filter.
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Example of Slow Sand Filter
After Droste,Theory and Practice of Water and Wastewater Treatment, 1997, pp. 450.
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Example of Rapid Multimedia Filter
After Droste,Theory and Practice of Water and Wastewater Treatment, 1997, pp. 418.
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Direction of Flow in Deep-Bed Filters
Upflow
Downflow (most common)
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Examples of Deep-Bed Filters
Granular-bed filters
Conventional mono-medium downflow filter
Conventional dual-medium downflow filter
Conventional mono-medium deep-beddownflow filter
Deep-bed upflow filter
Pulsed-bed filter
Traveling-bridge filter
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PIERO M. ARMENANTENJIT
Examples of Deep-Bed Filters
(continued)
Granular-bed filters (continued)
Continuous backwash deep-bed upflow
filter Slow sand filter
Fast sand filter
Pressure filters
Cartridge filters
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PIERO M. ARMENANTENJIT
Examples of Deep-Bed Filters
Conventional Monomedium Conventional Dual Medium
Downflow Downflow
After Metcalf and Eddy, Wastewater Engineering, 1991, p. 252
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PIERO M. ARMENANTENJIT
Examples of Deep-Bed Filters
Conventional Monomedium Deep Bed Upflow
Deep-Bed Downflow
After Metcalf and Eddy, Wastewater Engineering, 1991, p. 252
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Examples of Deep-Bed Filters
Pulsed-Bed Traveling Bridge
After Metcalf and Eddy, Wastewater Engineering, 1991, p. 253
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Examples of Deep-Bed Filters
Continuous Backwash Slow Sand
Deep-Bed Upflow
After Metcalf and Eddy, Wastewater Engineering, 1991, p. 253
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PIERO M. ARMENANTENJIT
Physical Characteristics of Commonly
Used Granular-Medium Filters
After Metcalf and Eddy, Wastewater Engineering, 1991, p. 250
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Dynasand Filter
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Example of Pressure Filter
After Metcalf and Eddy, Wastewater Engineering, 1991, p. 256
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Example of Pressure
Filter Operation
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Backwashing of Depth Filters
Because of the solid build-up within or on thefilter medium the resistance offered to filtration
increases with time
Backwashing is an operation conducted to
remove the filtered solids by inverting thedirection of the liquid flow while using clearliquid
In conventional filters in which the slurry
velocity is downward backwashing produces alifting of the filter medium with consequentdislodging of the filtered solids that can be
collected from the top of the filter
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Classification of Solid Medium Particles
During Backwashing
During backwashing the larger medium particles
tend to sediment to the bottom of the filter while
the lighter particles rise to the top
When the filter is put back into operation the
incoming slurry encounters the smaller particles
first. This is clearly undesirable since, as a
result, the filtering action will be provided
primarily by the top layer where the smaller
particles are
Dual- and multi-media systems are designed to
reduce the magnitude of this problem
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Dual and Multimedia Systems
Such systems (working in downward flow)utilize as filter media small heavier particles
(typically sand) at the bottom and lighter butlarger particles (typically coal) on top
During backwashing the lighter, largerparticles will sediment more slowly than thesmaller but heavier particles and will remain atthe top
This will result in a more appropriate soliddistribution in which the slurry will firstencounter the larger particles as it enters the
filter from the top
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PIERO M. ARMENANTENJIT
Common Depth Filter Media
Type of
Medium
Medium
Material
Particle
Size (mm)
Filter
Depth (in)
Monomedia
(a) coarse
(b) fine
Anthracite Coal
Sand
1.3 - 1.7
0.35 - 0.60
36 - 60
10 - 20
Dual Media Anthracite Coal
Sand
1.0 - 1.1
0.45 - 0.6
20 - 30
10 - 12
Multimedia Anthracite Coal
Sand
Garnet,
Metal Oxides
1.0 - 1.1
0.45 - 0.55
0.25 - 0.4
0.25 - 0.4
18 - 24
8 - 12
2 - 4
2 - 4
After Eckenfelder, Industrial Wastewater Pollution Control, p.383
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PIERO M. ARMENANTENJIT
Stratification of Filter Medium Particles in Dual-
and Multimedia Systems After Backwashing
After Metcalf and Eddy, Wastewater Engineering, 1991, p. 255
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Flow Control During Depth Filtration
Flow Rate
Driving Force
Filter Resistance=
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NJIT
Flow Control Strategies for Depth Filtration
Fixed Head (4 filters in parallel) Variable Head (4 filters in parallel)
After Metcalf and Eddy, Wastewater Engineering, 1991, p. 258
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Flow Control Strategies for Depth Filtration
Pulsed-Bed Filter Variable Head and Flow (4 filters in parallel)
After Metcalf and Eddy, Wastewater Engineering, 1991, p. 258
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Analysis of
Depth Filtration
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NJIT
Analysis of Suspended Solids
Removal and Pressure Drop
in Depth Filters
As the suspension moves through the filter
bed some of the particles are captured by thefilter and are removed from the suspension.
Equations can be written to describe:
the removal of particles by the filter, and
the pressure drop (or headloss) of the fluidas it passes through the filter bed
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Removal of Suspended Solids in Depth Filters
The rate of particle removal from suspension will depend
on several parameters such as:
concentration of solids in suspension, X
type of solids in suspension
amount of solids deposited in filter per unit volume, q
vertical location within the filter, z
fluid superficial velocity, us
size of particles, Dp
void fraction, (void volume/total bed volume)
time, t
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Superficial Velocity
The superficial (or approach) velocity is definedas the velocity of the liquid as it flows through across section equal to that of the tank (or filtervessel) in the absence of the medium. It is also
equal to the total flow rate divided by the totalcross-sectional area normal to flow, i.e.:
uQ
As =
where:A = cross sectional area of empty tank
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Void Fraction (Porosity)
The void fraction (also called porosity), , ofa bed is defined as the ratio:
=
void volume
total volume of bed
Because of its definition the void fraction mustbe within the range 0-1.
The void fraction in a depth filter can changewith time as more suspended solids are
removed by the filter.
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Removal of Suspended Solids in
Depth Filters
L
pD
suA
dz
X
Q
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Mass Balance for the Suspended Solids
Moving Through a Section of the Bed
Rate of accumulation
of solids within the layer
Rate of flow of
solids into the layer
Rate of flow of
solids out of the layer
=
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Mass Balance for the Suspended Solids
Moving Through a Section of the Bed
( )
q
tt
X
tAdz Q X Q X
X
zdz+
= +
The term q/t is the rate of deposition ofsolids per unit bed volume in the filter layer of
thickness dz
The term X/t is the rate of change of solid
suspension concentration as a function of time
The term X/z is the rate of change ofconcentration as a function of filter depth z
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Mass Balance for the Suspended Solids
Moving Through a Section of the BedA simplification of the above equation yields:
( )
q
t
tX
t
uX
zs+ =
since
uQ
As =
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Mass Balance for the Suspended Solids
Moving Through a Section of the BedSince the fluid contained in a layer is typicallysmall in comparison with the flow passingthrough it one can safely assume that:
( )
q
tt
X
t>>
i.e., the mass balance becomes:
=u Xz
qt
s
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Mass Balance for the Suspended Solids
Moving Through a Section of the Bed The rate of solids deposition per unit bed
volume q/t is very difficult to estimate
In general, it is reasonable to assume that therate of solid removal is proportional to the
concentration in the solid suspension:
q
tX i.e.,
X
zX
In practice, extensive experimental data arenecessary to predict the removal rate of solids
in depth filters
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Generalized Rate Equation
The experimental data can be analyzed using thefollowing general equation:
( )
dX
dz a z
r Xq
q
n o
u
m
=
+
1
1
1
where:
a, n, m = experimentally determined constants
ro = initial rate of removal constant (length-1)
q = amount of solids deposited in unit filtervolume (mass/volume)
qu= ultimate amount of solids deposited in unitfilter volume (mass/volume)
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Modification of the Generalized Rate
Equation
Initially, when the value of q 0 the term inparenthesis in the previous equation is equal to 1and the rate equation becomes:
( )
dX
dz azr Xn o=
+
1
1
The term in brackets is called the retardation
factor.
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Determination of the Constants in the
Generalized Rate EquationThe value of ro is obtained by plotting theexperimental rate of removal for very shallow filterdepths for which one can assume that:
( )
1
11
+
a z
n
and
dX
dzr Xo ln
X
Xr zo o=
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Determination of the Constants in the
Generalized Rate EquationOnce ro is known a and n are obtained from:
( )
= +
r X
dX dz az
o
n1
1
using a trial-and-error approach (or a non-linearregression algorithm) until the values ofa and nthat produce a straight line when plotting the term
in parenthesis vs. z are obtained
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Limitations of the Applicability of Theoretical
Analysis to Depth Filter Design and Operation Predictive equations to determine the rate of solid
removal and solid buildup in the filter as a function of
time, and size distribution and concentration of
solids in the wastewater are typically quite complex
They typically require numerical integration ofdifferential equation as well as the estimation of
constants from preliminary experiments
In practice, depth filters are sized largely on the basis
of past experience and the use of semiempiricalequations to correlate pilot plant data for scale up
purpose
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Suspended Solids Removal in Filters:
OMelias Approach OMelia (1975) has proposed a theoretical
approach to determine the efficiency of solidsremoval from wastewaters using depth filters.
This approach is based on the consideration ofdifferent mechanisms of particle removal.
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Suspended Solids Removal in Filters:
OMelias ApproachThe fraction, , of particles remaining in thewastewater after passing through a monolayer offilter medium is given by:
= +
+
4 072 0 002413
23
18
158
56
25
Pe LoD
DGr
D
D
p
s
p
s
. .
Then, the ratio of the effluent to influent particle
concentration, f, can be calculated from:
( )ln fL
Dc s=
3
21
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OMelias Approach: Nomenclature
=
+
1
1 15 15
5
5 6. .
( )
=
===
=
113
13
D
k joule K
T
erg
s
s
L
diameter of solid particles in suspension
density of solid particles in suspension
liquid viscosity
= Bolzmann constant = 1.38 10
absolute temperature (K)
Ha = Hamaker constant (typically 10
-23 /
)
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OMelias Approach: Nomenclature
The nondimensional numbers in the precedingequations are defined as:
( )
PeD D u
kT
LoHa
D u
GrD g
u
p s s
L p s
p s L
L s
=
=
=
3
9
2
9
2
2
c = dimensioness collision efficiency (=1 for idealdestabilization)
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Semiempirical Equations to Size and
Operate Depth Filters Many semiempirical equations are used to
interpret and analyze pilot plant data
An example of a semiempirical equation to sizeand operate depth filters is:
tk H
X us=
where: k = empirical constantt = total run time before backwash is carried out
H = available head before backwash is carried out
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Mechanical Energy Losses of a Fluid
Moving in a Conduit As a fluid moves in a conduit mechanical
energy losses occur as a result of friction with
the wall of the conduits and turbulence. This
phenomenon can also be interpreted as aconversion of some of the mechanical energyto thermal energy.
In pressurized pipes this energy loss is
typically reported in terms of pressure drops. In open channels this loss is typically reported
in terms of headloss.
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Mechanical Energy Losses of a Fluid
Moving in a ConduitThe mechanical energy losses must be accountedfor in the mechanical energy balance for the fluid.
1
2
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Mechanical Energy Balance
A mechanical energy balance in a fluid at twodifferent sections (1 and 2) in a conduit gives the
familiar Bernoulli equation:
v
g z
P
g W
v
g z
P
g HL1
1
1 2
2
2
2 2+ + + = + + +
where:
v = fluid velocity
P = fluid pressure
z = fluid height (with respect to a reference height)
W = mechanical energy input (e.g., via a pump)
HL = headloss due to friction
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Pressure Drop vs. Headloss
It is common in the industrial practice to referto the pressure drop encountered as a liquid
passes through a flow resistance (e.g., agranular bed) in terms of headloss and vice
versa The headloss is the energy loss expressed in
terms of an equivalent head of the liquid, i.e.,the liquid height that produces a hydrostatic
pressure equal to the pressure drop
To convert a pressure term into a headlossterm just remember the equation for
hydrostatic pressure
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Pressure Drop vs. Headloss (cont.'d)
P g hL= (SI units)
Pg
ghL
c
= (English units)
Example: P corresponding to a head of 5 ft of water
Pkg
m
m
sft
m
ftPa= =1000 9 8 5
0 3045
114 920 5
3 2.
., .
P lbft
ft
slb ft
lb s
ft lbft
psimm
f
f= = =62 4332 174
32 1745 312 15 2 1653
2
2
2.
.
.. .
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Pressure Drop Across Depth Filters
L
pD
suA
P
Q
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Pressure Drop Across Granular Media
The pressure drop across a depth filter depends
on a number of factors including:
bed depth, L
effective diameter of filter medium particles, Dp
shape factor of filter medium particles, L
void fraction, (void volume/total bed volume)
superficial velocity of fluid, us
fluid density, L
fluid viscosity, L
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Pressure Drop in an Empty Pipe
uD
L
The pressure drop across an empty pipe given by:
P fL
DuL= 2
2
where:
P = pressure drop across length of pipe L
L = length of pipe
u = average fluid velocity
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Friction Factorffor Pressure Drop in
an Empty PipeThe (Fanning) friction factorfin the pressure dropequation for empty pipes is given by:
f = 0 07911 4.Refor turbulent flow
f =24
Refor laminar flow
where:
Re =u DL
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Friction Factorffor Pressure Drop in
an Empty Pipe
After Bird, Steward and Lightfoot,Transport Phenomena, 1960, p. 184
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Pressure Drop Across Granular Media
Similarly to what found for the pressure drop in
empty pipes the pressure drop across granularmedia is given by:
P f
L
D up p L s= 22
where:
P = pressure drop across length of bed L
L = length of bedus = superficial fluid velocity
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Effective Particle DiameterDpThe effective particle diameter is defined as:
DV
Ap
p
p
=6
where:
Vp = volume of filter medium particle
Ap = surface area of filter medium particle
This definition is important to determine the area
of the particles if their volume in known since:
AV
Dp
p
p
=6
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Effective Particle DiameterDp
(continued)Remark: the above definition for Dp was chosenso that for the case of a sphere it is always:
( )D V
ADD
Dpp sphere
p sphere
p sphere
p sphere
p sphere= = =6 66
3
2
,
,
,
,
,
Important: one should be careful in checking
definitions in textbooks since a number ofdefinitions for the effective particle diameterexists
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Particle Reynolds Number
The effective Reynolds number, Rep is defined as:
RepL p sD u=
It has been found experimentally that:
forRep
110
flow is turbulent
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Friction Factorfp for Pressure Drop
Across Granular MediaThe friction factorfp in the pressure drop equationfor granular media is given by:
( )fpp
=
75 1
2
3Re
for laminar flow
fp =
08751
3.
for turbulent flow
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Pressure Drop Across Granular Media
- The Ergun EquationCombining together all the expressions givenabove one obtains the Ergun Equation forpressure drop in granular media:
( )PL
Du
p p
L s= +
1501 175
13
2
Re.
where the first term and the second term in
brackets are the laminar contribution and theturbulent contribution, respectively.
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Pressure Drop Across Granular Media - The
Blake-Kozeny and Burke-Plummer EquationsSometime the laminar and the turbulent contributions
in the Ergun equation are considered separately (this
is actually the way in which each contribution was
originally determined). In such a case one obtains the
Blake-Kozeny and Burke-Plummer equations, i.e.:
( )P
L
Du
p p
L s=
150 12
3
2
Re
Blake-Kozeny equation
P LD
up
L s=
175 1 32.
Burke-Plummer equation
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Average Particle Size of Sieved
Fractions of Medium Typically, sieves are used to determine the
particle size distribution of particulate filtermedia (e.g., sand)
Sieves come in different "mesh" sizes, each
one corresponding to the size of the sieveopening
The larger the mesh size the smaller the
opening
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Average Particle Size of Sieved
Fractions of Medium Table of representative mesh sizes vs. particle
sizes:
Mesh size 10 16 20 28 32
Sieve Opening(mm)
1.68 1 0.841 0.595 0.5
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Shape Factor of Filter Medium
ParticlesThe filter medium particle shape factor, p, isdefined as:
p =Surface area of sphere having same volume as particle
Surface area of particle
i.e.,
p
p
p
p
sph p
D
D
V
A
V
D A
sph
sph
= =6 6
2
3
where Dsph is the diameter of a sphere having thesame volume as the particle.
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Relationship Between Dp, Dsph, and pSince:
DV
Ap
p
p
=6
and:
psph
p
pD
V
A=
1 6
then:
D Dp p sph=
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Approximate Relationship Between
Dp and Sieve OpeningThe assumption is often made that:
D Dsph p
where Dp is the average size of the particles
whose size is between two sieve openings
D D Dp s s= 1 2
and where Ds1 and Ds2 are the sieve openings.Then:
D D Dp p sph p p=
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Shape Factor of Filter Medium
Particles (continued)Values ofp:
spheres p = 1
cylinders (with H = D) p = 0.874
cubes p = 0.806
rounded sand p = 0.82
average sand p = 0.75
crushed coal and angular sand p = 0.73
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Particle Sphericity and Porosity
Description Sphericity, p Typical Porosity,
Spherical 1.00 0.38
Rounded 0.98 0.38Worn 0.94 0.39
Sharp 0.81 0.40
Angular 0.78 0.43
Crushed 0.70 0.48
After Droste,Theory and Practice of Water and Wastewater Treatment, 1997, pp. 420.
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Characterization of Filter Media
Sieve analysis is commonly used tocharacterize the particle size distribution offilter media.
The mean and standard deviation are the
appropriate statistical parameters that can beused to describe the particle population.
A straight line is typically obtained by plottingthe cumulative weight percentage of the solidsvs. the particle size on normal probability-
logarithmic paper.
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Effective Grain Size and Uniformity
Coefficient of a Filter MediumTwo parameters are commonly used tocharacterize filter bed particle sizes. They are:
Effective Grain Size (d10) = the particle size incorrespondence of the 10 percentile by weight,using sieve analysis
Uniformity Coefficient (UC) = d60/d10
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Characteristics of Depth Filter Media
Type ofMedium
Density(g/cm3)
Uniformity Coefficient(UC)
Range Typical
Dual Media
Coal
Sand
1.5
2.65
1.3 - 1.8
1.2 - 1.65
1.5
1.4
Multimedia
CoalSandGarnet
1.52.654.1
1.3 - 1.81.2 - 1.65
--
1.51.4--
After Sundstrom and Klei, Wastewater Treatment, 1979, p. 228
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Characteristics of Filter Bed Material
Material Shape p s/water d10 (mm)
Silica Sand Rounded 0.82 2.65 0.42 0.4-1.0
Silica Sand Angular 0.73 2.65 0.53 0.4-1.0
Ottawa Sand Spherical 0.95 2.65 0.40 0.4-1.0
Silica Gravel Rounded 2.65 0.40 1.0-50
Garnet 3.1-4.3 0.2-0.4
Crushed
Anthracite
Angular 0.72 1.50-1.75 0.55 0.4-1.4
Plastic Any characteristics of choice
After Droste,Theory and Practice of Water and Wastewater Treatment, 1997, pp. 420.
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The Ergun Equation for a Stratified
Bed of the Same Filter MediumThe Ergun equation can also be rewritten for amedium made of the same material (e.g., sand)but made of particles with a given particle size
distribution as:
( )P L uD
L s
pj
j
j
j
j
pji
n
= +
=
2
31
1501 175
1
Re.
where:j = fraction of particles (based on mass) having
a particle size between two sieve openings
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Most Common Equations for the
Calculation of Pressure Drop AcrossGranular Media
Ergun equation
- Blake-Kozeny equation (laminar regime)- Burke-Plummer (turbulent regime)
Fair-Hatch equation
Rose equation
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Pressure Drop Across Granular Media
- The Fair-Hatch EquationThe Fair-Hatch equation can also be used topredict pressure drop in granular material:
( )P k
L
D ups= 36
12
3 2
where:
k = non-dimensional filtration constant (equal
to 5 if based on sieve openings, or 6 ifbased on size of separation)
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The Fair-Hatch Equation for a Stratified
Bed of the Same Filter MediumThe Fair-Hatch equation can also be rewritten fora medium made of the same material (e.g., sand)but made of particles with a given particle size
distribution as
( )P k Lu
Ds
j
j
j
pjj
n
=
=36
12
3 21
where:j= fraction of particles (based on mass) having
a particle size between two sieve openings
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Pressure Drop Across Granular Media
- The Rose EquationPressure drop for beds made or uniform sizeparticles:
P C
L
Du
Dp
L s=1067
1
4
2.
where CD = drag coefficient for spheres givenfrom graph or from:
CDp p
= + +24 3
0 34
Re Re
.
with: RepL s pu D=
and D
V
ADp
p
p
p p= 6
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The Rose Equation for a Stratified
Bed of the Same Filter MediumThe Rose equation can also be rewritten for a
medium made of the same material (e.g., sand)but having a given particle size distribution:
P L u C D
L s Dj
j
j
pjj
n
==
1067 12 41
.
where:
j= fraction of particles (based on mass) havinga particle size between two sieve openings
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Pressure Drop in Clean Multimedia
FiltersThe total pressure drop in multimedia filters isjust the sum of the pressure drops produced byeach layer of medium:
P PC Cjj
n
==
1
where
PC = total pressure drop in clean filter
PCj = pressure drop in the jth layer of medium
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Pressure Drop in Clean Multimedia
Filters (continued)For example, the pressure drop of a dual mediumfilter made of sand and anthracite having each aperfectly homogeneous particle size (UC = 1) is:
P P PC anthracite sand = +If the sizes of, say, the anthracite particles are not
identical stratification will occur with the largerparticles typically on top. In such a case one can
determine the pressure drop of each layer withinthe anthracite medium and sum all the pressure
drop contributions as described above.
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Clean Filter Pressure Drop vs.
Dirty Filter Pressure Drop The equations developed above apply to clean filters
in which the characteristics of the filter medium are
known
As solids from the suspension are filtered andtrapped in the filter medium the pressure drop across
the medium increases
Calculation of the new pressure drop can still be
carried out using the equations for granular media for
clean filters given above provided that the newcombined distribution of all the solids (due to filtered
solids as well as filter media solids) is known
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Pressure Drop in Dirty Filters
Several expressions have been derived to predictthe pressure drop in dirty filters. They require theknowledge of the volume of deposited solids perunit bed volume. For example, the Ivesexpression is:
( ) ( ) PL
P
Lb b
D C
=
+ + + +
1 2 1 1
2
where:
subscripts D orC refer to the dirty and clean filter,( )b = =
=
1 packing constant
volume of deposited particles per unit bed volume
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Pressure Drop in Dirty Filters
Another approach to the calculation of the pressure dropin dirty filters is by summing the contribution of all layers
at each time for each layer containing a known amount of
filtered solids. This implies solving the equation:
( ) P t P p t D C jj
n
= + = ( )1where both PD(t) and pj(t) are functions of time.
PD(t) = Total pressure drop across dirty filter
PC = Total pressure drop across clean filter
pj(t) = Incremental pressure drop across the jth layerin filter
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Relationship Between Pressure Drop
in a Filter Layer and Amount ofMaterial Deposited
The following equation states that the incrementalpressure drop in the jth layer of the filter at time tdue to the amount of solids deposited is afunction of the amount of solids, q, that has beendeposited in that layer
( ) ( )
[ ]p t q tj j=
where qj(t) = amount of deposited solids per unitbed volume in jth layer at time t
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Relationship Between Pressure Drop
in a Filter Layer and the Amount ofMaterial Deposited
After Metcalf and Eddy, Wastewater Engineering, 1991, p. 267
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Design Parameters for Depth Filters
The most important design parameter fordepth filters is the hydraulic loading, definedas the volumetric flow rate per unit crosssectional area.
Typical hydraulic loading values are in therange 1-10 gpm/ft2.
Depth filtration units are typically cylindrical orrectangular in shape.
The surface area of a bed is about 1600 ft2
(150m
2). The typical range is: 400-2100 ft
2(35-190
m2).
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Design Parameters for Depth Filters
In medium to large filter installations (Q > 10mgd) four beds are typically installed.
Wastewater pretreatment with coagulants isoften common prior to depth filtration, in order
to remove colloidal particles.
Backwashing typically results in a 15-30% bedexpansion. Water flow rates per unit areaduring backwashing are in the range 10-20gmp/ft
2(6.8-13.6 L/m
2s). Application times are
in the range 5-15 minutes.
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Design Features of MonomediumFilter Beds for Wastewater Treatment
Characteristic Range TypicalShallow bed (stratified)Sand
Depth, cm (in.) 25-30 (10-12) 28 (11)Effective size, mm 0.35-0.6 0.45Uniformity coefficient 1.2-1.6 1.5Filtration rate, m/h
(gal/ft2/min)
5-15 (2-6) 7 (3)
AnthraciteDepth, cm (in.) 30-50(12-20) 40 (16)Effective size, mm 0.8-1.5 1.3Uniformity coefficient 1.3-1.8 1.6Filtration rate, m/h(gal/ft2/min)
5-15 (2-6) 7 (3)
Conventional (stratified)Sand
Depth, cm (in.) 50-76 (20-30) 60 (24)Effective size, mm 0.4-0.8 0.65Uniformity coefficient 1.2-1.6 1.5Filtration rate, m/h
(gal/ft
2
/min)
5-15 (2-6) 7 (3)
Anthracite
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Design Features of MonomediumFilter Beds for Wastewater Treatment
(Continued)Characteristic Range Typical
Deep bed (unstratified)SandDepth, cm (in.) 90-180 (36-
72)120 (48)
Effective size, mm 2-3 2.5Uniformity coefficient 1.2-1.6 1.5Filtration rate, m/h(gal/ft2/min)
5-24 (2-10) 12 (5)
AnthraciteDepth, cm (in.) 90-215 (36-
84)150 (60)
Effective size, mm 2-4 2.75Uniformity coefficient 1.3-1.8 1.6Filtration rate, m/h(gal/ft2/min)
5-24 (2-10) 12 (5)
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Design Parameters for Pressure
Depth FiltersEffective Size, mm Filtration Rate, m/h
(gal/ft2/h)
0.35 25-35 (615-860)
0.55 40-50 (980-1230)
0.75 55-70 (1350-1720)
0.95 70-90 (1720-2210)
After Droste,Theory and Practice of Water and Wastewater Treatment, 1997, pp. 448and Dregmont (1979).
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Cyclical Operation of Depth Filters
The operation of depth filter is intrinsicallycyclical as a result of solids accumulating inthe filter and the necessity of their removal.
Typically two or more units are used so that
backwashing can be conducted withoutinterrupting the treatment.
Most depth filters are designed so thatbackwashing takes place once per dayoperation.
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Backwashing
During backwashing water is pumped upward,i.e., in the opposite direction of the suspension
during normal operation
The backwashing flow expands the bed to
dislodge all the particles removed duringfiltration
In order for backwashing to be effective the
filter medium must be fluidized
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Backwashing
Regular
Bed
Expanded
Bed
Backwash
Water
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Fluidization of Solids in Depth Filters
Steps in the fluidization of solids in depth filters:
1. At a low upflow velocity of the backwash water the
solids in the bed remain stationary
2. As the upflow velocity is increased the pressure
drops across the bed also increases (Ergunequation)
3. For a critical value of the upflow velocity the
minimum fluidization velocity is achieved, the
particles begin to loosen up, and the bed begins to
expand
4. At higher velocities the porosity of the bed
increases and the bed continues to expand
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Minimum Pressure Drop for Fluidization
to Occur During Backwashing When the incipient (or minimum) fluidization
velocity is achieved the actual weight of the
solid bed is supported by the drag force
generated by the water on the solid particles. The actual weight of the bed is equal to the
weight of the solid less that of the water
displaced by the solids (buoyancy effect).
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Minimum Pressure Drop for Fluidization
to Occur During BackwashingFrom a force balance on a particle at the point ofincipient fluidization it must be that:
drag force gravity force buoyancy force=
( ) ( )P L gmf mf mf s L= 1 where: Pmf= pressure drop at the point of
incipient fluidization
Lmf= height of bed at the point of
incipient fluidizationmf= bed void fraction at the point of
incipient fluidization
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Relationship Between Bed Height and
Bed Porosity (Void Fraction) DuringBackwashing
If the cross section of the bed, A, is constant andno solids are lost with the backwash water the
mass of solids in the bed is constant. Hence:
( ) ( )L A L A1 1 2 21 1 =
where the subscripts 1 and 2 refer to two levels ofbed expansion (depending on the fluid velocities).
L
L1
2
2
1
1
1=
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Relationship Between Bed Height and
Bed Porosity (Void Fraction) DuringBackwashing (continued)
In particular it must be that:
L
Lmf
o
o
mf=
1
1
where:
subscript mf= at incipient fluidization
subscript o = resting bed (before fluidization)
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Porosity at the Incipient Fluidization Point
Particle Size (mm)
0.06 0.10 0.20 0.40
Particle Material Porosity at Incipient Fluidization, mf
Sharp Sand
(s = 0.67)
0.6 0.58 0.53 0.49
Round Sand
(s = 0.86)
0.53 0.48 0.43 (0.42)
Anthracite Coal(s = 0.63)
0.61 0.6 0.56 0.52
After Leva et al., U.S. Bur. Mines Bull., 1951
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Backwashing Water Velocity to Achieve
Fluidization of Filter MediumThe superficial velocity at which fluidizationbegins, us mf, can be obtained by combining theequation for the pressure drop in the bed (using
the Ergun equation) as it begins to fluidize:
( )PL
Dumf
p mf
mfmf
mf p
L s mf = +
1501 175
13
2
Re.
with the equation forP at incipient fluidization:
( ) ( )P L gmf mf mf s L= 1
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Backwashing Water Velocity to Achieve
Fluidization of Filter Medium (cont.d)
By recalling that the flow through small particles
is typically laminar one can re-write the Ergun
equation as:
( )P
L
Dumf
p mf
mf
mf p
L s mf
150 12
3
2
Re
(i.e., the Blake-Kozeny equation)
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Backwashing Water Velocity to Achieve
Fluidization of Filter Medium (cont.d)The resulting equation is:
( )u
D gs mf
p s L mf
mf
=
2 3
150 1
Recalling that:
D D Dp p sph p p=
the above equation can also be written as:
( )u
D gs mf
s p s L mf
mf
=
2 2 3
150 1
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Backwashing Water Velocity to Achieve
Fluidization of Filter Medium (cont.d)The equation for the superficial velocity at whichfluidization begins:
( )u
D gs mf
p s L mf
mf=
2 3
150 1
is valid for:
RepL p s mf D u=