Filtration Theory
Monroe L. Weber-Shirk
School of Civil and
Environmental Engineering
Field Trip To CUWTP
Monday at
2:20 pm at
loading dock
Public Health reports
 The decline happened over time and not rapidly as if it were associated with a
centralized intervention
 Chlorine was not responsible for the decline
 Filtration was not responsible for the decline
 The relatively high dose required for an infection would require gross
contamination of the water supply
 Therefore typhoid was generally not waterborne
 There is some evidence that typhoid was greater in the summer. This suggests
multiplication in the environment, most likely in food.
 Improved personal hygiene was likely the dominant factor
 Jakarta and Army evidence that the sources are local: (not centrally distributed like
milk, water, or meat, but food preparation with contaminated hands)
 Improved hygiene reduced contamination of food
 Refrigeration would have reduced the summertime typhoid by reducing
multiplication in food. Home refrigeration happened after the decline began, but
commercial refrigeration
Filtration Outline
 Particle Capture theory
 Transport
 Short range forces
 Grain contact points
 Dimensional Analysis
 Trajectory Models
 Filters
 Rapid
 Slow (lots of detail here…)
References
 Tufenkji, N. and M. Elimelech (2004). "Correlation equation for
predicting single-collector efficiency in physicochemical filtration in
saturated porous media." Environmental-Science-and-Technology
38(2): 529-536.
 Cushing, R. S. and D. F. Lawler (1998). "Depth Filtration:
Fundamental Investigation through Three-Dimensional Trajectory
Analysis." Environmental Science and Technology 32(23): 3793 3801.
 Tobiason, J. E. and C. R. O'Melia (1988). "Physicochemical Aspects of
Particle Removal in Depth Filtration." Journal American Water Works
Association 80(12): 54-64.
 Yao, K.-M., M. T. Habibian, et al. (1971). "Water and Waste Water
Filtration: Concepts and Applications." Environmental Science and
Technology 5(11): 1105.
Overall Filter Performance
Iwasaki (1937) developed relationships
describing the performance of deep bed
filters.
dC
=  0C
dz
dC
=  0 dz
C
C
z
dC
C C =  0 0 dz
0
C 
 ln   =0 z
 C0 
C is the particle concentration [number/L3]
0 is the initial filter coefficient [1/L]  log  C   pC*  1  z
 
0
C
ln
10
z is the media depth [L]
 
 0
The particle’s chances of being caught are the same at all
depths in the filter; pC* is proportional to depth
Particle Removal Mechanisms in
Filters
collector
Transport to a surface
Molecular diffusion
Inertia
Gravity
Interception
Attachment
Straining
London van der Waals
Filtration Performance: Dimensional
Analysis
What is the parameter we are interested in
Effluent concentration
measuring? _________________
How could we make performance
C/C0 or pC*
dimensionless? ____________
What are the important forces?
Inertia
Viscous
London van der Waals
Gravitational
Electrostatic
Thermal
Need to create dimensionless force ratios!
Dimensionless Force Ratios
Reynolds Number
Froude Number
r Vl
Re =
m
V
Fr =
gl
V 2 l
V2
fi = r
l
V
fu = m 2
l
fg = r g
s
fs = 2
W
Weber Number
l

r c2
f Ev =
V
Mach Number
l
M
c (Dp + r g Dz )
2Drag
2
D
p
(
)
Cd 
Pressure/Drag Coefficients C p =
V 2 A
rV2
 (dependent parameters that we measure experimentally)
What is the Reynolds number for
filtration flow?
 What are the possible length scales?
 Void size (collector size) max of 0.7 mm in RSF
 Particle size
 Velocities
 V0 varies between 0.1 m/hr (SSF) and 10 m/hr (RSF)
 Take the largest length scale and highest velocity to find max
Re
 Thus viscosity is generally much more significant than inertia
Vl
Re 

kg  m hr 

3
1000
10
0.7

10
m


3 
m  hr 3600s 
Re  
2
kg 

 0.001

ms 

Choose viscosity!
In Fluid Mechanics inertia is a significant
“force” for most problems
In porous media filtration viscosity is more
Inertia
important that inertia.
We will use viscosity as the repeating
parameter and get a different set of
dimensionless force ratios
Gravitational
Viscous
London
Viscous
Thermal
Viscous
Electrostatic
Viscous
Gravity
velocities
vpore
vg =
(  p   w ) gd p2
18
g =
vg
V0
V
fu   2
l
fg = r g
forces
g =
fg
f
 g
g =
V0
 2
dp
Gravity only helps when
the streamline has a
2
2
(



)
gd
(



)
gd
_________
horizontal component.  g = p w p  g = p w p
18V0
Use this equation
V0
Diffusion (Brownian Motion)
vpore
D
vd 
dc
kT
D B
3 d p
Pe 
Diffusion velocity is
high when the particle
diameter is ________.
small
 L2 
T 
 
V0 d c
D
kB=1.38 x 10-23 J/°K
T = absolute temperature
dc is diameter of the collector
2/3
2 / 3


k BT
-2/3  V0 d c 
 Br = Pe  




 D 
 3 d pV0 d c 
The exponent was obtained from an analytical model
London van der Waals
The London Group is a measure of the
attractive force
H is the Hamaker’s constant
 H = 0.75 1020 J
 Lo
4H
=
9 d 2pV0
Van der Waals force
Viscous force
What about Electrostatic?
 Modelers have not succeeded in describing filter
performance when electrostatic repulsion is
significant
 Models tend to predict no particle removal if
electrostatic repulsion is significant.
 So until we get a better model we will neglect this
force with the understanding that filter
performance is poor if electrostatic repulsion is
significant
Geometric Parameters
What are the length scales that are related to
particle capture by a filter?
Filter depth (z)
______________
Collector diameter (media size) (dc)
__________________________
Particle diameter (dp)
______________
Create dimensionless groups
(dc)
Choose the repeating length ________
R 
dp
dc
z
z 
dc
Number of collectors!
Write the functional relationship
pC*  f   R ,  z ,  g ,  Br ,  Lo 
doubles
If we double depth of filter what does pfz do? ___________
pC*   z f   R ,  g ,  Br ,  Lo 
How do we get more detail on this functional relationship?
Empirical measurements
Numerical models
Numerical Models
Trajectory analysis (similar to the analysis
of flocculation)
A series of modeling attempts with
refinements
Began with a “single collector” model that
modeled London and electrostatic forces
with an attachment efficiency term (a)
pC*   z f   R ,  g ,  Br ,  Lo 
pC* 
z
ln 10 
Addition
assumption
  R   g   Br a
Array of Spheres Model (AOS)
Includes simplified geometry describing the
contact between collectors
Used trajectory analysis to determine which
particles would be captured
Used the numerical model results to
determine the form of the equation based on
dimensional analysis
AOS: The Media Trap
Isolated collectors
Array of spheres model
Collector
Contacts
Contacts Matter!
Two Particle Traps

pC* 
0.029 
 0.48

ln 10 
z
Particles that enter
centered above a
collector are trapped
in the stagnation
point.
This trajectory
analysis ignores
Brownian Motion
0.012
Lo
0.023
R
1.8
g
 R0.38 
Particles that enter on a
streamline that passes
through a contact point
between collectors get
trapped between two
collectors
Collector
contact straining
Array of Spheres Model Results and
Critique
Brownian wasn’t modeled
z
0.012 0.023
1.8 0.38
pC* 
0.029



0.48

 13.6 Br 

Lo
R
g R
ln 10 
 The transport to the media surface by either the
fluid (interception, R), gravity (g), or diffusion
(Br) is followed by an attachment step controlled
by van der Waals (Lo)
 The transport and attachment steps occur in series
and thus removal should be described by the
product of these groups
 More work to be done!
13.6=4.04*As1/3
AOS model deficiencies
z
0.023
1.8 0.38
0.029 0.012


0.48

 13.6 Br 

Lo
R
g R
ln 10 
=1!
z
1.8 0.38
pC* 
0.029

0.48

 13.6 Br 

g R
ln 10 
pC* 
This suggests a third transport mechanisms that is
constant and doesn’t require Brownian motion or
sedimentation! Could be interception, but interception
increases with particle size.
Given this error (and the likelihood that the numerical
model contained errors) the model results from the AOS
model should probably not be used!
Tufenkji and Elimelech with
Analysis by Weber-Shirk
 0   D   I  G
0.715
0.052
 D  2.4 As1/ 3 N R0.081 N Pe
NvdW
N Lo
N vdW
H

=
N Pe
3 d pV0 d c
0.715
0.052
0.052
 D  2.4 As1/ 3 N R0.081 N Pe
N Pe
N Lo
2/3
0.052
 D  2.4 As1/ 3 N Pe
N R0.081 N Lo
NR 
dc
4H
H
N Lo =
N vdW =
2
9

d
pV0
kT
d p2 (  p   w ) g
NG =
18V0

 D  
k BT
N Pe  

  
V
d
3

d
V
d
p 0 c 
 0 c 
As 
Note that my NPe is the inverse of T&E
dp
2 1   5 
2  3  3 5  2 6
  1   
1/ 3
Interception
0.125
0.125
 I  0.55 AS N R1.55 N Pe
N vdW
0.125
 I  0.55 AS N R1.55 N Lo
N Lo  N Pe N vdW
A
=
3 d pV0 d c
Gravity
0.053
G  0.22 N R0.24 NG1.11 NvdW
0.053
0.053
G  0.22 N R0.24 NG1.11 N Lo
N Pe
N Lo
H
 N Pe N vdW =
3 d pV0 d c
N
N vdW  Lo
N Pe
Total removal
 0   D   I  G
2/3
0.052
 D  2.4 As1/ 3 N Pe
N R0.081 N Lo
0.125
 I  0.55 AS N R1.55 N Lo
0.053
0.053
G  0.22 N R0.24 NG1.11 N Lo
N Pe
2/3
0.052
0.125
0.053
0.053
0  2.4 As1/ 3 N Pe
N R0.081 N Lo
 0.55 AS N R1.55 N Lo
 0.22 N R0.24 NG1.11 N Lo
N Pe
2/3
0.072
0.053
0.053
0   2.4 As1/ 3 N Pe
N R0.081  0.55 AS N R1.55 N Lo
 0.22 N R0.24 NG1.11 N Pe
N
 Lo
2/3
0.072
0.053
0   2.4 As1/ 3 N Pe
N R0.081  0.55 AS N R1.55 N Lo
 0.22 N R0.24 NG1.11 N Pe
 N Lo0.053
C 
 ln   = z
 C0 

3 1 
a0
2 dc
3 1     z 
pC 
  a0
2ln 10   d c 
C 
1
*
 log    pC 
z
ln 10 
 C0 
*
pC *  N za0
3 1     z 
Nz 
 
2ln 10   d c 
For particles less than 1 m
2/3
0.053
 D  2.4 As1/ 3 N Pe
N R0.081 N Lo
2/ 3 0.081 0.053
pC *  2.4 As1/ 3 N Pe
N R N Lo N za
1
0
nD
nI
0.1
ng
ntotal
0.01
0.01
0.1
1
particle diameter (m)
10
100
Brownian Motion

pC* 
0.029 
 0.48 

ln 10 
z
0.012
Lo
0.023
R
1.8
g
0.38
R
 13.6 Br 
 Brownian motion dominates the transport and collection of
particles on the order of 1 m and smaller
 Brownian transport (diffusion) leads to nondeterministic
behavior and results in trajectories defined by stochastic
differential equations
 The problem is traditionally decoupled using the
assumption that the Brownian and deterministic transport
mechanisms are additive
 Sedimentation is less important for small particles because
the R group is small and the Br group is large
Filter Performance as function of
particle size
z
0.023
1.8 0.38
pC* 
0.029 0.012


0.48

 13.6 Br 

Lo
R
g R
ln 10 
The exact location of the
minimum varies, but is
generally around 1 m.
For small particles diffusion
dominates and we have
z
pC* 
13.6 Bra
ln 10 
attachment
Estimate Dimensionless Brownian
Transport for a Bacteria Cell
13.6 Br
13.6 Br

k BT
= 13.6 
 3 d V d
p 0 c




2/3

viscosity
1.00E-03 Ns/m2
dp
Particle diameter
1.00E-06 m
kB
Boltzman constant
1.38E-23 J/°K
dc
Collector diameter
T
Absolute temperature
293 °K
V0
Filter approach
velocity
0.1
0.2E-03
m
m/hr



23 J 
1.38

10
293

K






K



= 13.6 
 3 1103 N  s  1106 m   0.10 m hr   0.2 103 m  


2 
 

m
hr
3600
s



 

13.6 Br = 0.025
Advection is 40x greater than diffusion
2/3
The Diffusion Surprise
13.6 Br

k BT
= 13.6 
 3 d V d
p 0 c




2/3
10
 As particle size
1
decreases Brownian
motion becomes more 13.6 Br 0.1
effective
0.01
 Viruses should be
removed efficiently by
0.001
1.E-09
filters (if attachment is
effective)
1.E-08
1.E-07
1.E-06
Particle diameter (m)
1.E-05
How deep must a filter (SSF) be for
diffusion to remove 99% of bacteria?
 Assume a is 1 and dc
is 0.2 mm
1
 a is ____
2
 pfz is ____
 z is _____
3.7 cm
 What does this mean?
If the attachment efficiency
were 1, then we could get great
particle capture in a 1 m deep
filter!
z
pC* 
13.6 Bra
ln 10 
z ln 10  pC *
z 

dc
13.6 Bra
z
z
ln 10  pC * d c
13.6 Bra
ln 10  2   0.2 103 m 
 0.0251
Filtration Technologies
 Slow (Filters→English→Slow sand→Biosand)
 First filters used for municipal water treatment
 Were unable to treat the turbid waters of the Ohio and
Mississippi Rivers
 Rapid (Mechanical→American→Rapid sand)
 Used in Conventional Water Treatment Facilities
 Used after coagulation/flocculation/sedimentation
 High flow rates→clog daily→hydraulic cleaning
 Ceramic
Rapid Sand Filter
(Conventional US Treatment)
Size
(mm)
Anthracite
Influent
Drain
Effluent
Sand
Gravel
Specific Depth
Gravity (cm)
0.70
1.6
30
0.45 - 0.55
2.65
45
5 - 60
2.65
45
Wash water
Filter Design
 Filter media
 silica sand and anthracite coal
smaller particles
 non-uniform media will stratify with _______
at the top
 Flow rates
 2.5 - 10 m/hr
 Backwash rates
 set to obtain a bed porosity of 0.65 to 0.70
 typically 50 m/hr
Backwash
Anthracite
Influent
Drain
Effluent
Sand
Wash water is
treated water!
WHY?
Only clean water
should ever be on
bottom of filter!
Gravel
Wash water
Slow Sand Filtration
First filters to be used on a widespread basis
Fine sand with an effective size of 0.2 mm
Low flow rates (10 - 40 cm/hr)
Schmutzdecke (_____
____) forms on top
filter cake
of the filter
causes high head loss
must be removed periodically
Used without coagulation/flocculation!
Fraction of influent E. coli
remaining in the effluent
Typical Performance of SSF Fed
Cayuga Lake Water
1
0.1
0.05
0
1
2
3
Time (days)
4
5
(Daily samples)
Filter performance doesn’t improve if the filter
only receives distilled water
How do Slow Sand Filters
Remove Particles?
 How do slow sand filters remove particles
including bacteria, Giardia cysts, and
Cryptosporidium oocysts from water?
 Why does filter performance improve with time?
 Why don’t SSF always remove Cryptosporidium
oocysts?
 Is it a biological or a physical/chemical
mechanism?
 Would it be possible to improve the performance
of slow sand filters if we understood the
mechanism?
Slow Sand Filtration Research
Apparatus
Cayuga Lake water
(99% or 99.5% of the
flow)
Manometer/surge tube
Peristaltic
pumps
Manifold/valve block
Sampling Chamber
Auxiliary feeds
(each 0.5% of
the flow)
Sampling tube
Lower to collect sample
To waste
1 liter E.
coli feed
1 liter
sodiu
m
Filter cell with
18 cm of glass beads
Biological and Physical/Chemical
Filter Ripening
Fraction of influent E. coli
remaining in the effluent
Continuously mixed
Cayuga Lake water
1
Quiescent Cayuga Lake
water
1
Physical/chemical
Sodium azide
(3 mM)
Control
0.1
0.05
Gradual growth of
biofilm or ________
predator
_______
0
1
2
3
Time (days)
4
0.1
0.05
5
0
2
4
6
Time (days)
8
What would happen with a short pulse of poison?
10
Biological Poison
1
Fraction of influent E. coli
remaining in the effluent
Control
Sodium azide pulse
Biofilms?
Abiotic?
Sodium chloride pulse
q
predator
0.1
0.08
0
1
2
3
Time—h
4
5
6
Conclusion? _________
predator is removing bacteria
Chrysophyte
long flagellum used for
locomotion and to provide
feeding current
short flagellum
1 µm
stalk used to attach to
substrate (not actually
seen in present study)
Particle Removal by Size
1
Fraction of influent particles
remaining in the effluent
control
3 mM azide
0.1
Effect of
the Chrysophyte
Recall quiescent
vs. mixed?
0.01
What is the physicalchemical mechanism?
0.001
0.8
1
Particle diameter (µm)
10
Role of Natural Particles in SSF
Could be removal by straining
But SSF are removing particles 1 m in
diameter!
To remove such small particles by straining
the pores would have to be close to 1 m
and the head loss would be excessive
Removal must be by attachment to the
sticky particles!
Particle Capture Efficiency
Sand filters are inefficient capturers of
particles
Particles come into contact with filter media
surfaces many times, yet it is common for
filters to only remove 90% - 99% of the
particles.
Failure to capture more particles is due to
attachment
ineffective __________
Remember the diffusion surprise?
Techniques to Increase Particle
Attachment Efficiency
Make the particles stickier
The technique used in conventional water
treatment plants
Control coagulant dose and other coagulant aids
(cationic polymers)
Make the filter media stickier
Potato starch in rapid sand filters?
Biofilms in slow sand filters?
Mystery sticky agent present in surface waters
that is imported into slow sand filters?
Mystery Sticky Agent
Serendipity!
Head loss through a clogged filter decreases
if you add acid
Maybe the sticky agent is acid soluble
Maybe the sticky agent will become sticky
again if the acid is neutralized
Eureka!
Cayuga Lake Seston Extract
Concentrate particles from Cayuga Lake
Acidify with 1 N HCl
Centrifuge
Centrate contains polymer
Neutralize to form flocs
AMP Characterization
volatile solids
Al
13%
Na
Fe
11%
P
S
Si
17%
Ca
carbon
other metals
16%
other nonvolatile solids
56%
Hypothesis:
The organic
fraction is
most
important
How much AMP should be added to a filter?
G (gcarbon/gglass beads)
Organic Carbon Accumulation in
Filters Fed Cayuga Lake Water
day 1
0.001
day 3
0.0001
day 7
0.00001
day 70
0.000001
0.0000001
0.0001
0.0010
0.0100
x (m)
0.1000
Filters fed Cayuga Lake Water
1.0000
Organic Carbon Accumulation
Rate
 Approximately 100 ppb (g/L) of carbon from
Cayuga Lake water is removed in SSF
Total organic carbon
 230 mg TOC /m2/day accumulated in filters fed
Cayuga Lake Water Attachment Mediating Polymer
 100 mg to 2,500 mg AMP as TOC /m2/day fed to
filters (CAMP*V0)
 Calculate application rate of AMP when fed
Cayuga Lake water
C AMPV0 
mg AMPTOC
100 g 1000 L 10cm 1m 24hr
mg






240
L
m3
hr 100cm day 1000 g
m2  day
E. coli Removal as a Function of
Time and AMP Application Rate
7
control
6
100
mgTOC
500 m2  day
2500
pC*
5
4
3
end azide
2
1
0
0
2
4
6
8
Horizontal bars
indicate when AMP
feed was operational
10
for each filter.
time (days)
pC* is proportional to accumulated mass of polymer in filter
Head Loss Produced by AMP
control
100
mg carbon
500
m 2  day
2500
end azide
head loss (m)
1.2
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
time (days)
How much AMP does it take to get 1 m of head loss?
500 mgcarbon
gcarbon
6 days 
3
2
m  day
m2
What do we know about this
Polymer?
Soluble at very low (<1) and at very high
(>13) pH
Forms flocs readily at neutral pH
Contains protein (amino acids)
In acid solution amino acids are protonated and
exist as cations
In basic solution amino acids are deprotonated
and exist as anions
Could be irrelevant!
Dipolar Structure of Amino Acids
R
O
Carboxyl group
..
H—N —CH—C—O—H
Amino group
H
In base solution
In acid solution
R
O
H
+
H—N —CH—C—O—H
H
cation
R
O
..
H—N —CH—C—O
H
anion
Sticky Media vs. Sticky Particles
 Sticky Media
 Potentially treat filter
media at the beginning
of each filter run
 No need to add
coagulants to water for
low turbidity waters
 Filter will capture
particles much more
efficiently
 Sticky Particles
 Easier to add coagulant
to water than to coat
the filter media
Current and Future Research
 Produce the polymer in the lab with an algae culture
 Develop methods to quantify the polymer
 Develop application techniques to optimize filter
performance
 How can we coat all of the media?
 Will the media remain sticky through a backwash?
 Will it be possible to remove particles from the media with a
normal backwash?
 What are the best ways to use this new coagulant?
 Why does the filter performance deteriorate when the AMP
feed is discontinued?
 Characterize the polymer
Conclusions
Filters could remove particles more
efficiently if the attachment
_________ efficiency
increased
SSF remove particles by two mechanisms
Predation
____________
_____________________________
Sticky polymer that coats the sand
pC* is proportional to accumulated mass of
AMP in the filter
pC*   z f   R ,  g ,  Br ,  Lo  a
Contact Points
Polymer Accumulation in a Pore