Conventional Conventional FiltrationFiltration

CE 547CE 547

Filtration is a unit operation of separating Filtration is a unit operation of separating solids from liquids.solids from liquids.

Types of FiltersTypes of Filters (to create pressure differential to (to create pressure differential to force the water through the filter):force the water through the filter):

Gravity filtersGravity filters Pressure filtersPressure filters Vacuum filtersVacuum filters

In terms of mediaIn terms of media Perforated platesPerforated plates Septum of woven materialsSeptum of woven materials Granular materials (such as sand)Granular materials (such as sand)

Sand FiltersSand Filters Slow sand filters (1.0 – 10 mSlow sand filters (1.0 – 10 m33/m/m22.d).d) Rapid sand filter (100 – 200 mRapid sand filter (100 – 200 m33/m/m22.d).d)

Medium Specification for Granular FiltersMedium Specification for Granular Filters

1.1. Medium is the most important Medium is the most important component of granular filterscomponent of granular filters

Small grain sizes tend to have higher head Small grain sizes tend to have higher head losseslosses

Large grain sizes may not be effective in Large grain sizes may not be effective in filteringfiltering

The most effective grain sizes are found from The most effective grain sizes are found from previous experienceprevious experience

2.2. Effective size of medium is specified in Effective size of medium is specified in terms of :terms of :

Effective sizeEffective sizeUniformity coefficientUniformity coefficient

What is Effective Size?What is Effective Size?It is the size of sieve opening that passes the 10% It is the size of sieve opening that passes the 10% finer of the medium sample (the 10finer of the medium sample (the 10thth percentile size, percentile size, PP1010))

What is Uniformity Coefficient?What is Uniformity Coefficient?It is the ratio of the size of the sieve opening that It is the ratio of the size of the sieve opening that passes the 60% finer of the medium sample (Ppasses the 60% finer of the medium sample (P6060) to the ) to the size of the sieve opening that passes the 10% finer of size of the sieve opening that passes the 10% finer of the medium sample (Pthe medium sample (P1010). It is P). It is P6060 to P to P1010..

For Slow Sand FiltersFor Slow Sand Filters PP1010 = 0.25 – 0.35 mm = 0.25 – 0.35 mm PP6060/P/P1010 = 2 – 3 = 2 – 3

For Rapid Sand FiltersFor Rapid Sand Filters PP1010 = 0.45 mm and higher = 0.45 mm and higher PP6060/P/P1010 = 1.5 and higher = 1.5 and higher

Example 7.1Example 7.1

Linear Momentum Equation Linear Momentum Equation Applied to Filters (Fig 7.10)Applied to Filters (Fig 7.10)

Movement of water through a filter bed is Movement of water through a filter bed is similar to the moment of water in parallel similar to the moment of water in parallel pipes, except that the motion is not straight pipes, except that the motion is not straight but tortuous. If momentum equation applied but tortuous. If momentum equation applied on water flow in the downward direction of the on water flow in the downward direction of the element, then:element, then:

dt

ddlkAFFdpAF

dt

ddlkA

dt

dVdaVdF

FFAdpppAF

sshg

s

shg

2

22

2

FF22 = unbalanced force in downward (z) direction (inertia = unbalanced force in downward (z) direction (inertia force)force)p = hydrostatic pressurep = hydrostatic pressureA = cross-sectional area of the cylindrical element of the A = cross-sectional area of the cylindrical element of the fluidfluidFFgg = weight of water in the element = weight of water in the elementFFshsh = shear force acting on the fluid along the surface = shear force acting on the fluid along the surface areas of the grainsareas of the grainsddV = volume of elementV = volume of elementdl = differential length of the elementdl = differential length of the elementAAss = surface area of the grains = surface area of the grainsk = factor which converts As into an area such that k = factor which converts As into an area such that ( kA( kAssdl = dl = ddV )V ) = porosity of the bed= porosity of the bedaa22 = acceleration of fluid element in downward (z) = acceleration of fluid element in downward (z) directiondirection = fluid mass density= fluid mass density = fluid element velocity in the (z) direction= fluid element velocity in the (z) directiont = timet = time

SinceSince

KKii = proportionality constant = proportionality constant

V = average water velocityV = average water velocity

22

.

VAKF

or

dkAdl

ddlkA

dl

ddlkA

thus

dl

d

dt

dl

dl

d

dt

d

si

sss

SinceSince

PPss = drop in pressure due to shear = drop in pressure due to shear forceforce

= liquid viscosity= liquid viscosityl = length of pipel = length of pipeD = diameter of pipeD = diameter of pipe

)(32

sin

2

2

2

equationPoiseuilleHagenD

lVP

ce

FFVAKpA

then

FFdpAF

s

gshsi

shg

KKss = proportionality constant = proportionality constant

rrHH = hydraulic radius = (area of flow / wetted perimeter) = hydraulic radius = (area of flow / wetted perimeter)

thenthen

Since FSince Fgg is constant, it can be included in K is constant, it can be included in Kii and K and Kss and and removed from the equation. Then:removed from the equation. Then:

This is a good linear momentum equation which can be This is a good linear momentum equation which can be applied to any filterapplied to any filter

H

sssh r

VAKF

gH

sssi F

r

VAKVAKpA

2

Hsis

H

sssi r

VKVKA

r

VAKVAKpA

22

If particles are sphericalIf particles are spherical

d = diameter of particled = diameter of particle

dv

S

dS

dv

p

p

p

p

6

62

3

ThusThus

rrH H = (area of flow / wetted perimeter) = = (area of flow / wetted perimeter) = (volume of flow / wetted area)(volume of flow / wetted area)if N = number of grainsif N = number of grainsvvpp = volume of each grain = volume of each grain

ThusThus

Since V can be expressed as = (vSince V can be expressed as = (vss / / ))vvss = superficial velocity = superficial velocity

1.6

lSd

As

6111 d

S

v

NS

Nv

rp

p

p

p

H

Head Loss in Grain Head Loss in Grain FiltersFilters

There are two categoriesThere are two categories head loss in clean filtershead loss in clean filters head loss due to the deposited materialshead loss due to the deposited materials

A. Clean-filter Head LossA. Clean-filter Head Loss

SSpp = surface area of a particle = surface area of a particle N = number of grains in the bedN = number of grains in the bedVolume of bed grains = SVolume of bed grains = S00l(1-l(1-))SS00 = empty bed or surficial area of the bed = empty bed or surficial area of the bed = porosity of bed= porosity of bedl = length of bedl = length of bed

ps NSA

if vif vpp is volume of a grain, then: is volume of a grain, then:

1

1

0

0

lSv

SA

thus

v

lSN

p

ps

p

Now we haveNow we have

)4(

)3(61

)2(16

)1(

0

2

s

H

s

Hsis

VV

dr

lSd

A

r

VKVKApA

Substitute (2), (3), and (4) in (1)Substitute (2), (3), and (4) in (1)

Re

115075.1

1

Re

115075.1

1

Re

sin

1366

1

2

20

2

20

2

20

p

pss

s

s

si

s

f

where

fd

VlS

d

VlSpA

then

dV

ce

dV

KK

d

VlSpA

ffpp is a form of friction factor. Since: is a form of friction factor. Since:

g

V

d

lfh

then

filtertheacrossheadh

hp

ce

g

V

d

lfp

then

SA

spL

L

L

sp

2

12

sin

2

12

2

3

2

2

0

Since particles are not spherical and Since particles are not spherical and

= shape factor= shape factor

ddpp = sieve diameter = sieve diameter

After backwash, the grain particles are After backwash, the grain particles are allowed to settle. That means, the particles allowed to settle. That means, the particles will deposit layer by layer, and hence will deposit layer by layer, and hence particles will be of different sizes. particles will be of different sizes.

pdd 3131

6

The bed is said to be stratified, and the head loss The bed is said to be stratified, and the head loss will be the sum of head losses of each layer. Then:will be the sum of head losses of each layer. Then:

If xIf xii is the fraction of the d is the fraction of the dii particles in the i particles in the ithth layer, layer, thenthen

Assume Assume is the same throughout the bed, then is the same throughout the bed, then

i

sipiL dg

Vlfh

3

2

2

12

lxl ii

i

ipi

s

i

sipiL d

xf

g

Vl

dg

Vlxfh

2

12

2

12 2

33

2

Example 7.2Example 7.2

Head Losses Due to Deposited Head Losses Due to Deposited MaterialsMaterials

If q = deposited materials per unit volume of the If q = deposited materials per unit volume of the bed, then:bed, then:

If hIf hL0L0 = clean-bed head loss, then = clean-bed head loss, then

C = concentration of solids introduced into bedC = concentration of solids introduced into bed

l = length of the filter bedl = length of the filter bed

qbah

tsconsarebanda

qah

a

bd

lnlnln

tan

l

CtVq

qahhhhhaedtotal

s

biLdLL

00,

How to determine (a) and (b) in:How to determine (a) and (b) in:

It is clear that the equation represents a It is clear that the equation represents a straight line if ln (hstraight line if ln (hdd) was plotted against ln ) was plotted against ln (q). (q).

qbah

from

qah

d

bd

lnlnln

This means that only two data points This means that only two data points are needed in order to determine (a) are needed in order to determine (a) and (b). If we have:and (b). If we have:

1

2

1

2 lnln1

1

1

2

1

2

22

11

ln

ln

q

q

h

h

d

d

d

bd

bd

d

d

q

ha

and

q

q

h

h

b

then

qah

qah

Examples 7.3 - 7.5Examples 7.3 - 7.5

Backwashing Head Loss in Backwashing Head Loss in Granular FiltersGranular Filters

During backwashing, the filter bed expands. During backwashing, the filter bed expands. For this to happen, a force must be applied: For this to happen, a force must be applied:

llee = expanded depth of the bed = expanded depth of the bed l’ = the difference between the level at the l’ = the difference between the level at the

trough and the limit of bed expansiontrough and the limit of bed expansion hhLbLb = backwashing head loss = backwashing head loss ww = specific weight of water = specific weight of water

eLbw llhforce '

weight of suspended solids is:weight of suspended solids is:

ee = expanded porosity of bed = expanded porosity of bed

pp = specific weight of particles = specific weight of particles

weight of column waterweight of column water

peess lW 1

weecw llW '

Weights of suspended grains and Weights of suspended grains and column water are acting downward column water are acting downward against the backwashing force. Thus:against the backwashing force. Thus:

Solving for backwashing head loss:Solving for backwashing head loss:

0'1' weepeeweLb lllllh

w

wpeeLb

lh

1

IfIf vvbb = backwashing velocity = backwashing velocity v = settling velocity of grainsv = settling velocity of grains

Then, and according to the following Then, and according to the following empirical equation:empirical equation:

22.0

v

vbe

for stratified bedfor stratified bed

if xi = fraction of particles in layer I, if xi = fraction of particles in layer I, then:then:

So,So,

w

wpieieiLbLb

lhh

,,

,

1

eiie lxl ,

ieiw

wpeLb x

lh ,1

Assuming expanded mass = Assuming expanded mass = unexpanded mass; then:unexpanded mass; then:

Then, the fraction bed expression is:Then, the fraction bed expression is:

ie

iie

peep

pieiepi

ll

then

ll

or

ll

,,

,,

1

1

11

11

eie

ie x

l

l

1

1

11

,

Study Example 7.6Study Example 7.6

Cake FiltrationCake Filtration

This is happening when solids This is happening when solids precipitate on the surface of the filter precipitate on the surface of the filter medium. In cake filtration, the medium. In cake filtration, the general momentum equation can be general momentum equation can be applied:applied:

s

si

s

dV

KK

d

VlSpA

1366

12

20

Because solids are tightly packed in Because solids are tightly packed in cake filtration, the first term is equal cake filtration, the first term is equal to zero due to the fact that the inertia to zero due to the fact that the inertia force must disappear upon entrance of force must disappear upon entrance of flow into the cake and the flow is not flow into the cake and the flow is not accelerated. So:accelerated. So:

s

ss

dV

K

d

VlSpA

13612

20

because A = Sbecause A = S00; and 36K; and 36Kss = 150; = 150; then:then:

dlV

dddp

or

dVd

V

dl

dp

then

dVd

V

l

p

dVd

Vlp

s

s

s

s

s

s

s

11501

11501

11501

11501

3

3

2

3

2

3

2

if if pp is the density of solids, the mass is the density of solids, the mass (dm) in the differential thickness of (dm) in the differential thickness of cake (dl) is:cake (dl) is:

= specific cake resistance= specific cake resistance

dmS

Vdp

simplyor

S

dmV

dddlV

dddp

then

dlSdm

s

pss

p

0

033

0

1

1150111501

1

integrating from Pintegrating from P22 to P to P11

mmc c = total mass of solids collected on the filter bed= total mass of solids collected on the filter bed

If the resistance of the filter medium to be If the resistance of the filter medium to be included, then:included, then:

RRmm = resistance of medium = resistance of medium

cs m

S

VPPP

021

m

cs R

S

mVP

0

Determination of Determination of V = volume of filtrate collected at any time, tV = volume of filtrate collected at any time, t

if mif mcc = cV = cVc = mass of cake collected per unit volume of the filtrate, c = mass of cake collected per unit volume of the filtrate, thenthen

= average specific cake resistance (since = average specific cake resistance (since changes with time) changes with time)

0S

dt

dV

Vs

0202 SP

RV

SP

c

V

t m

the equation represents a straight line with the equation represents a straight line with slope m of :slope m of :

by determining the slope of the line from the by determining the slope of the line from the experimental data, experimental data, can be determined as: can be determined as:

202 SP

cm

c

SP

202

WhereWhere

n1 and n2 are the number of elements n1 and n2 are the number of elements in the respective groupin the respective group

2

1

1

1

2

1

1

1

1

1

2

11

1

2

1

n n

ii

n n

Vn

Vn

Vt

nVt

nm

Example 7.7Example 7.7

Design of Cake Filtration Design of Cake Filtration EquationEquation

Practically, the resistance of the filter medium is negligible. Practically, the resistance of the filter medium is negligible. Then:Then:

Introducing a new parameter called the filter yield, Lf, Introducing a new parameter called the filter yield, Lf, defined as:defined as:

Lf = the amount of cake formed per unit area of the filter per Lf = the amount of cake formed per unit area of the filter per unit timeunit time

VSP

c

V

t202

tS

cVL f

0

If t was called the formation time (tIf t was called the formation time (tff) that is ) that is when cake starts to form. Also some filters when cake starts to form. Also some filters operate on a cycle. Call cycling time as toperate on a cycle. Call cycling time as tcc. In . In such cases tsuch cases tff may be expressed as a fraction of may be expressed as a fraction of ttcc. So:. So:

ThenThen

In vacuum filtration, f is equal to the fraction of submergence of In vacuum filtration, f is equal to the fraction of submergence of the drum.the drum.

)( cf toffractiontt

c

f ft

cPL

2

For compressible cakes, For compressible cakes, is not constant and is not constant and the above equation must be modified. In such the above equation must be modified. In such cases:cases:

s = measure of cake compressibilitys = measure of cake compressibility

If s = zero, then the cake is incompressible If s = zero, then the cake is incompressible and and = = 00. For compressible cakes:. For compressible cakes:

sP 0

c

s

f ft

cPL

0

12

Determination of Cake Filtration Determination of Cake Filtration ParametersParameters

What design parameters?What design parameters? LLff

--PP

ff ttcc

cc , s, s

Ps

P s

lnlnln 0

0

This is a straight line equation with This is a straight line equation with slope = sslope = s

1

1

1

1

ln1

ln1

1

0

2

1

1

1

2

1

1

1

ln1

1ln

2

1

ln1

1ln

2

1

n n

ii Pn

s

n

n n

ii

n n

ii

e

and

Pn

Pn

nns

Examples 7.8 and 7.9Examples 7.8 and 7.9