Granular Filter Design

7/29/2019 Granular Filter Design

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Depth (or Deep-Bed)

Filtration


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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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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



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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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(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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Conventional Monomedium Conventional Dual Medium

Downflow Downflow

After Metcalf and Eddy, Wastewater Engineering, 1991, p. 252


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Conventional Monomedium Deep Bed Upflow

Deep-Bed Downflow



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Pulsed-Bed Traveling Bridge



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Continuous Backwash Slow Sand

Deep-Bed Upflow



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Physical Characteristics of Commonly

Used Granular-Medium Filters



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Dynasand Filter


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Example of Pressure Filter



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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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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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Stratification of Filter Medium Particles in Dual-

and Multimedia Systems After Backwashing



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Flow Control During Depth Filtration

Flow Rate

Driving Force

Filter Resistance=


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PIERO M. ARMENANTE

NJIT

Flow Control Strategies for Depth Filtration

Fixed Head (4 filters in parallel) Variable Head (4 filters in parallel)



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NJIT

Flow Control Strategies for Depth Filtration

Pulsed-Bed Filter Variable Head and Flow (4 filters in parallel)



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Analysis of

Depth Filtration


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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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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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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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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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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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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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- 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



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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



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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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- 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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- 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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PIERO M. ARMENANTE

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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



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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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NJIT

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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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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NJIT


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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PIERO M. ARMENANTE

NJIT


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=

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