monroe l. weber-shirk s chool of civil and environmental engineering flocculation
TRANSCRIPT
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Monroe L. Weber-Shirk School of Civil and
Environmental Engineering
Flocculation
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Overview
Flocculation definition Types of flocculatorsMechanical DesignFractal Flocculation Theory applied to
hydraulic flocculatorsCFD analysis of hydraulic flocculatorsHydraulic Flocculator Design
0.001 0.01 0.1 11 10
4
0.001
0.01
0.1
1
10
Floc diameter (mm)
Ter
min
al v
eloc
ity (
mm
/s)
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What is Flocculation?
A process that transforms a turbid suspension of tiny particles into a turbid suspension of big particles!
RequiresSticky particles (splattered with adhesive nanoglobs)Successful collisions between particles
Flocs are fractals (“the same from near as from far”)The goal of flocculation is to reduce the number of
colloids (that haven’t been flocculated)One goal is to understand why some colloids are
always left behind (turbidity after sedimentation) Another goal is to learn how to design a flocculator
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Mechanical Flocculation
Shear provided by turbulence created by gentle stirring
Turbulence also keeps large flocs from settling so they can grow even larger!
Presumed advantage is that energy dissipation rate can be varied independent of flow rate
Disadvantage is the reactors have potential short circuiting (some fluid moves quickly from inlet to outlet)
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Recommended G and Gq values: Turbidity or Color Removal (Mechanical flocculators)
Type
“Velocity gradient” (G) (1/s)
Energy Dissipation Rate
(e)+ Gqq*
(minute)
Low turbidity, color removal
20-70 0.4 – 4.9 50,000-250,000
11 - 210
High turbidity, solids removal
70-180 4.9 - 32 80,000-190,000
7 - 45
Sincero and Sincero, 1996 Environmental Engineering: A Design Approach
mW
kg
* Calculated based on G and Gq guidelines+ average value assuming viscosity is 1 mm2/s
G is the wrong parameter… (Cleasby, 1984)
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Mechanical Design: mixing with paddles
3 3
2D P paC A a V
GV
“v
elocit
y grad
ient”
Drag co
effici
ent
Projec
ted ar
ea of
padd
les
Ratio o
f rela
tive t
o
abso
lute v
elocit
y of p
addle
s
Reactor volume1.9DC
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Mechanical Flocculators?
Waste (a small amount of) electricityRequire unnecessary mechanical
componentsHave a wide distribution of energy
dissipation rates (highest near the paddles) that may break flocs
Have a wide distribution of particle residence times (completely mixed flow reactors)
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Ten State Standards
The detention time for floc formation should be at least 30 minutes with consideration to using tapered (i.e., diminishing velocity gradient) flocculation. The flow‑through velocity should be not less than 0.5 nor greater than 1.5 feet per minute.
Agitators shall be driven by variable speed drives with the peripheral speed of paddles ranging from 0.5 to 3.0 feet per second. External, non-submerged motors are preferred.
Flocculation and sedimentation basins shall be as close together as possible. The velocity of flocculated water through pipes or conduits to settling basins shall be not less than 0.5 nor greater than 1.5 feet per second. Allowances must be made to minimize turbulence at bends and changes in direction.
Baffling may be used to provide for flocculation in small plants only after consultation with the reviewing authority. The design should be such that the velocities and flows noted above will be maintained.
http://10statesstandards.com/waterrev2012.pdf
Hydraulic flocculators allowed only by special permission!
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Hydraulic Flocculators
Horizontal baffle
Vertical baffle
Pipe flow
Gravel bedVery low flows and pilot plants
A bad idea (cleaning would require a lot of work)
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Flocculator Geometry
2 Channels
Upper Baffles
Port between channels
Entrance
Lower Baffles
Exit
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Why aren’t hydraulic flocculators used more often?
Simple construction means that there aren’t any items that private companies (venders) can sell as specialized components
Consulting firms want to be able to pass the design responsibility off to a vender
The presumed operation flexibility of mechanical flocculators (variable speed motor driving a slow mixing unit)
Poor documentation of design approach for hydraulic flocculators (special permission required to use in the US!)
Using electricity is cool, design innovation is suspect… Prior to AguaClara we didn’t have a design algorithm based on
the fundamental physics
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Schulz and Okun (on Hydraulic flocculators)
Recommend velocity between 0.1 and 0.3 m/sDistance between baffles at least 45 cmWater must be at least 1 m deepQ must be greater than 10,000 m3/day (115 L/s)
These aren’t universal constants!
We need to understand the real constraints so we can scale the designs correctlyWhat length scale could make a dimensionless parameter?
Surface Water Treatment for Communities in Developing Countries, (1984) by Christopher R. Schulz and Daniel A. Okun. Intermediate Technology Publications
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AguaClara Flocculator Design Evolution
We began using conventional guidelines based on “velocity gradient” but we were aware that this system was fundamentally incorrect (it doesn’t capture the physics of turbulence)
We were concerned that by using a defective model we could potentially produce defective designs
If the model doesn’t capture the physics, then it won’t scale correctly (and we are designing plants for scales that hadn’t been tested)
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Edge of Knowledge Alert
Why would we ever think that the baffled flocculators invented over 100 years ago were the optimal design for flocculation?
We have better coagulants, shouldn’t that influence flocculator design?
We are only now beginning to understand the physics of fractal flocculation
We are improving the design of hydraulic flocculators based on our evolving understanding of the physics of flocculation
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The Challenge of Flocculation
We would like to knowHow do particles make contact to aggregate?What determines the time required for two flocs
to collide?How strong are flocs?
The challenge of the large changes in scaleThe Al(OH)3 nanoglobs begin at a scale of 100
nanometersFlocs end at a scale of 100 micrometers
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Hydraulic Flocculation Theory
Turbulence is caused by the expansion that results when the water changes direction as it flows around each baffle
Colloids and flocs are transported to collide with each other by turbulent eddies and over short distances by viscous shear.
Flocs grow in size with each successful collisionColloids have a hard time attaching to large flocs*
(maybe because the surface shear is too high???)Flocculation model and collision potential for reactors with flows characterized by high Peclet numbers Monroe L. Weber-Shirk
, a, and Leonard W. Liona
http://dx.doi.org/10.1016/j.watres.2010.06.026*hypothesis from 2012!
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How do the flocs grow?
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Exponential growth
How many sequential collisions are required to make a 1 mm particle starting from 1 mm particles?
How much larger in volume is the 1 mm diameter particle?
____________1,000,000,000 !
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Doubling Collisions
1 collision 1+1=22 collisions 2+2=43 collisions 4+4=84 collisions 8+8=16Number of original particles in the floc = 2
n
What is n to obtain 1,000,000,000 = 2n?
n=30This assumes volume is conserved!
log(1000000000) log(2)
log(1000000000)
log(2)
n
n
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Fractal Dimensions
What happens to the density of a floc as it grows larger?
1
0FractalDd d i
diam
eter o
f prim
ary
parti
cle
diam
eter o
f flo
c
Number of primary particles in the floc
Frac
tal d
imen
sion
If volume is conserved, what is DFractal? ____3
3
0FractalDV V i
Floc density approaches the density of water because the floc includes water
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Fractal Flocculation
Fractal geometry explains the changes in floc density, floc volume fraction, and, ultimately, sedimentation velocity as a function of floc size
The fractal dimension of flocs is approximately 2.3 (based on floc measurements)
3
0FractalDV V i
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Floc Volume Fraction
The fraction of the reactor volume that is occupied by flocs
For fractal dimensions less than 3 the floc volume fraction increases as floc size increases
Use conservation of volume to estimate initial
Floc FlocFloc
Suspension Floc
V C
V
0 3( )Floc Clay Al OH
3 3
0 3
ClayAl OH Al OHClay
Floc Clay Al OH
C C CC
Floc
Floc
dominates
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Floc Volume Fraction
0.001 0.01 0.1 1 101 10
5
1 104
0.001
0.01
0.1
1
Fractal dimension = 1Fractal dimension = 2.3Fractal dimension = 3
Floc diameter (mm)
Flo
c V
olum
e F
ract
ion
0
3
0
FractalD
Floc Floc
d
d
Primary particle diameter (clay + coagulant)
“super fluffy” flocs
Dense flocs
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Buoyant Density of Flocs
0.001 0.01 0.1 1 100.1
1
10
100
1000
1 104
Fractal dimension = 1Fractal dimension = 2.3Fractal dimension = 3
Floc diameter (mm)
Flo
c de
nsity
- w
ater
den
sity
(kg
/m^3
)
2 0 2
3
0FractalD
Floc H O Floc H O
d
d
Will these flocs settle faster than the primary particles?
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0.001 0.01 0.1 11 10
4
0.001
0.01
0.1
1
10
Floc diameter (mm)
Ter
min
al v
eloc
ity (
mm
/s)
Floc Terminal Velocity
1 mm
Upflow velocity for floc blankets
Capture velocity for AguaClara plate settlers
Why flocculation is necessary!
The model takes into account the changing density of flocs
0 2
2 2
120
018
FractalD
Floc H Ot
H O H O
gd dV
d
DFractal = 2.3 and d0 = 1 mm
2
2 2
2
18
Floc H Ot
H O H O
gdV
shape factor for drag on flocs
Laminar flow
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Analytical Model of the Flocculation Process
The floc porosity increases with floc diameterThe velocity between flocs is a function of
whether the separation distance is less or greater than the Kolmogorov scale
The time required per collision is a function of the relative velocity between flocs, the average separation distance between flocs, and the floc size In the next slides we will explore how to characterize
collision time for flocsWe will assume that collisions occur between similar
sized flocs. That assumption will need to be evaluated, but it is probably a good assumption for the initial growth of flocs.
Are the two flocs in different eddies?
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How much water is cleared (filtered) from a floc’s perspective?
Volume cleared is proportional to a collision area defined by a circle with diameter = sum of the floc diameters
Volume cleared is proportional to timeVolume cleared is proportional to the
relative velocity between flocs
2Cleared Floc rV d v t
2Flocd
trv
rv t 2Flocd
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How much volume must be cleared before a collision occurs?
What is the average volume ofwater “occupied” by a floc?
Need to know floc diameter (dFloc)
And floc volume fraction (fFloc)
3
6 Floc
FlocOccupied
d
V
Floc volumeSuspension volume
3
6 Floc
OccupiedFloc
dV
13
6Separation FlocFloc
L d
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Use dimensional analysis to get a relative velocity given a length scale
2
3
L
T
,rv f L
1
3rv L
Viscous range Inertial range
, ,rv f L
rv L
2L
T
,rv Lf Assume linear
The origin of the G notation
13 4
KL
Re=
1
laminar turbulent
If the flocculator has laminar flow, then this side doesn’t apply and the G, Gq approach applies.
L is separation distance
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Summary for Particle Collisions
1
3r Separationv Lr Separationv L
3
6 Floc
OccupiedFloc
dV
tc is average time per collision
Floc separation distance
13
6Separation FlocFloc
L d
viscous inertia
13 4
KL
2Cleared Floc r cV d v t
113 2
23
1 6 1
6c
Floc
t
11 2 39
89
1 6 1
6Floc
c
Floc
dt
0
3
0
FractalD
Floc Floc
d
d
6Floc
cFloc r
dt
v
Is tc a function of d? Yes!
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Successful Collision Models
113 2
23
1 6 1
6c
Floc
t
11 2 39
89
1 6 1
6Floc
c
Floc
dt
1
2234.8
viscousc FlocN t
13
89
25.6
inertialc FlocFloc
N td
viscous inertia
Time for one collision
Number of successful collisions
G is fractional surface coverage of colloids with adhesive nanoglobsdFloc is perhaps mean floc size of flocs that are capturing colloids
For completeness we should probably include a correction for hydrodynamic effects that make it difficult for non porous particles to approach closely. This may increase the time for the first few collisions when flocs aren’t very porous
234.8
viscousc FlocN Gt
cc
tN
t Number of collisions is equal
to time over collision time
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Relationship between laminar and turbulent
13
823 9
2
23
138
92
2
9
3
2 229 99
2
3
14.8 5.6
4.8
15.6
6
306
Floc FlocFloc
Floc
FlocFloc
Clay
Clay ClayClay
Clay Clay Clay
Gd
G
d
G
d
C CG GN
V Mm
The ratio of Gq to y is the number of clay particles raised to the 2/9th power!
Set the collisions equal to get the relationship
Use the target settled water turbidity as the best guess for a relationship between conventional Gq and y. For a turbidity of 1 NTU the number of clay particles is 150,000,000 per liter.Schulz and Okun suggest 20,000 as a minimum value of Gq equivalent to = y 65 m2/3. But this isn’t calculated correctly. Should take their Gq and energy dissipation rate to make the conversion.
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0.001 0.01 0.1 1 100
10
20
30
1 NTU suspension10 NTU suspension100 NTU suspension
Floc diameter (mm)
Col
lisio
n tim
e (s
)Minimum time to grow from
colloid to large floc
20 collisions to grow from 1 mm to 0.4 mm
How much time is required to produce a 0.4 mm floc?
DFractal = 2.3d0 = 1 mme = 6 mW/kg
0
21 1 22 33 23
0
1 6
6
Fractal
viscous
D
c Floc
dt
d
0
8 311 2 8 99
3 3 9
0
1 6
6
Fractal
inertial
D
c Floc
dt d
d
InertialViscous
20
log nFractal
dn D
d
13 4
KL
400 s
120 s40 s
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Initial Floc Growth: Phase 1(the 50% solution)
Initial growth phase of flocculation can be modeled with the equations on the previous slide (summing the collision times until the maximum floc size is reached)
The end of the initial growth phase is reached when a significant fraction of the flocs reach a size that can be removed by sedimentation.
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Observations: Collision time model
The previous analysis was tracking the time required to make the first big flocs
For a highly turbid suspension it may only take a few seconds to produce visible flocs
This is why successful flocculation can be observed very early in a flocculator
This doesn’t mean that a flocculator with a residence time of 100 s will perform well
We need to track the colloids that are left behind! Performance is based on _______________residual turbidity
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Remember our Goal?
Introduce pC*Sloppy parlance… log removalWhat is the target effluent turbidity for a
water treatment plant?What is pC* for a water treatment plant
treating 300 NTU water?
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Tracking the residual turbidity: Phase 2 of Flocculation
After the initial production of flocs at their maximum size the interaction of the colloids change
Most of the collisions are ineffective because collisions with flocs that are maximum size apparently are useless and most flocs are their maximum size
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Flocculation Model: Integrating and tracking residual turbidity
ColloidColloid
c
dCkC
dN
Colloidc
Colloid
dCkdN
C
0
Colloid
Colloid
CColloid
cCColloid
dCk dN
C
0
ln Colloidc
Colloid
CkN
C
The change in colloid concentration with respect to the potential for a successful collision is proportional to the colloid concentration (the fraction of the colloids swept up is constant for a given number of collisions)
Integrate from initial colloid concentration to current colloid concentration
Separate variables
Classic first order reaction with number of successful collisions replacing time
1
2234.8
viscousc FlocN t
13
89
25.6
inertialc FlocFloc
N td
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Laminar Flow Cases
0
2 32 13 3 2
10
4.8
Fractal
Viscous
D
Colloidc
Colloid
C dN t
d
0
3
0
FractalD
Floc Floc
d
d
Only colloids can collide effectively (big flocs are useless)
Colloids can attach to all flocs
0
2 13 21
ln4.8
ColloidColloid
Colloid Colloid
Ct
k C C
0
0
2 2 3 13 3 2
01ln
4.8
FractalD
ColloidColloid
Colloid Colloid
Cdt
k C d C
Floc volume fraction that matters for successful collisions is the colloid (small floc) fraction
0
ln Colloidc
Colloid
CkN
C
Floc volume fraction is a function of floc size (d) (which is a function of the reaction progression and shear conditions in the reactor
1
2234.8
viscousc FlocN t
1
2234.8
viscousc FlocN t
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10 100 1000 100000.1
1
10
100
colloids aggregateeverything aggregates
Time (s)
Tur
bidi
ty (
NT
U)
Comparison of Flocculation Hypotheses
0
2 13 21
ln4.8
ColloidColloid
Colloid Colloid
Ct
k C C
0
0
2 2 3 13 3 2
01ln
4.8
FractalD
ColloidColloid
Colloid Colloid
Cdt
k C d C
If colloids could aggregate with all of the flocs then colloids would aggregate VERY rapidly
Ten state standards 30 minute flocculation time.The graph is laminar flow case, full scale flocculators are turbulent flow. 100 NTU clay suspension, G = 0.1
Show that pC* is proportional to time for “everything aggregates” model
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Coiled Tube Flocculation Residual Turbidity Analyzer
Dr. Karen Swetland Dissertation research
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Two Phase Floc Model: PACl
0.1 1 10 1000
0.5
1
1.5
2
=1200s, 5 NTU=1200s, 15 NTU=1200s, 50 NTU=1200s, 150 NTU=1200s, 500 NTU=800s, 5 NTU=800s, 15 NTU=800s, 50 NTU=800s, 150 NTU=800s, 500 NTUFitted Model
*G**̂ 2/3
pC*
Initial floc growth
1
2* 3
0
3log( ) 2
2 3Coag
CCapture
epC W Gt N
V
1CN
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Two Phase Floc Model: Alum
0.1 1 10 1000
0.5
1
1.5
2
=1200s, 5 NTU=1200s, 15 NTU=1200s, 50 NTU=1200s, 150 NTU=1200s, 500 NTU=800s, 5 NTU=800s, 15 NTU=800s, 50 NTU=800s, 150 NTU=800s, 500 NTUFitted Model
*G**̂ 2/3
pC*
1
2* 3
0
3log( ) 2
2 3Coag
CCapture
epC W Gt N
V
Initial floc growth1C
N
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Solving for pC*: Lambert W or ProductLog Function (Laminar flow)
Use Wolframalpha to solve for C*
W is the Lambert W function
Log is base 10
2* 3
0
3log( ) 2
2 3Coag
Capture
epC W Gt
V
* 2
302
*3
lnCoag
Capture
CGt
VC
2
30
3 2* 2 3
Coag
Capture
W GtVC e
0 0
2233
0lnColloid Colloid Coag
Colloid Colloid Capture
C CGt
C C V
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Laminar Flow Floc Model
Velocity gradient
Flocculation time
Fractional surface coverage of colloid by
coagulant
Floc volume fraction
Sedimentation tank capture velocity
Sedimentation velocity of ???
1
2
1
2* 3
0
3log( ) 2
2 3Coag
CCapture
epC W Gt N
V
Lambert W Function
Collisions to make first big flocs = 0.4
Empirical
0.4Coag
mm
s
0.12Capture
mmV
s
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Turbulent Flow Case
0
2 38 139 3
20
5.6
Fractal
Inertial
D
Colloidc
Colloid Floc
C dN t
d d
8 139
25.6
Inertial
Colloidc
Colloid Colloid
CN t
d
Only colloids can collide and attach effectively
Colloids can attach to all flocs
0
3
0
FractalD
Floc Floc
d
d
0
8 12 391
ln5.6
ColloidColloid Colloid
Colloid Colloid
Cdt
k C C
0
0
8 2 3 129 33
01ln
5.6
FractalD
ColloidColloid Floc
Colloid Floc Colloid
Cd dt
k C d C
0
ln Colloidc
Colloid
CkN
C
1
38
92
5.6Inertialc Floc
Floc
N td
Why are almost all of these collisions in the inertial range?
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100 1000 100000.1
1
10
100
3 NTU10 NTU30 NTU100 NTU300 NTU
time (s)
Tur
bidi
ty (
NT
U)
Turbulent Flow Flocculator:“Big Flocs are useless” hypothesis
0
8 12 391
ln5.6
ColloidColloid Colloid
Colloid Colloid
Cdt
k C C
The predicted performance is quite similar given a wide range of influent turbiditiesThe predictions seem reasonable
Ten state standards 30 minute flocculation time.
After the big flocs become non reactive, then the average separation distance between the remaining flocs is larger and thus the collisions between active particles is dominated by inertia.
G = 0.1 for these plots e = 2.6 mW/kg
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Turbulent Flow Flocculator: Colloids can attach to all flocs hypothesis
The predicted performance is quite different from what we observe. High turbidity would be very easy to treat if this hypothesis were true!Flocculators would be tiny!
0
0
8 2 3 129 33
01ln
5.6
FractalD
ColloidColloid Floc
Colloid Floc Colloid
Cd dt
k C d C
Ten state standards 30 minute flocculation time.
10 100 1000 100000.1
1
10
100
3 NTU10 NTU30 NTU100 NTU300 NTU
time (s)
Tur
bidi
ty (
NT
U)
Here the separation distance between reactive flocs might be less than the Kolmogorov length scale
For these plots G= 0.1e= 2.6 mW/kgpC* proportional to time
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Solving for pC*: Lambert W or ProductLog (Turbulent Flow)
0 0
8 189 39
0 2
3
lnColloid Colloid Coag
Colloid Colloid CaptureColloid
C C t
C C Vd
1* 8 3
908 2
* 9 3
lnCoag
CaptureColloid
C t
VC d
18
390 2
3
9 8* 8 9
Coag
Capture
Colloid
tW
Vd
C e
18 3
* 90 2
3
9log( ) 8
8 9Coag
CaptureColloid
e tpC W
Vd
Use Wolframalpha to solve for C*
W is the Lambert W function
Log is base 10
0
* Colloid
Colloid
CC
C
0
* log Colloid
Colloid
CpC
C
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Lambert W Function
Performance varies very little over wide range of inputs.
Diminishing returns on investment
18 3
* 90 2
3
9log( ) 8
8 9Coag
CaptureColloid
e tpC W
Vd
18 39
0 2
3
8
9Coag
CaptureColloid
t
Vd
0.1 1 10 100 10000
1
2
3
pC*
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8
3
*2
9
1
3
0
9log( ) 8
8 9Coll
C
C
oapture
id
oag
V
tC W
d
ep
Proposed Turbulent Flocculation Sedimentation Model (missing phase 1)
Energy dissipation rate
Flocculation time
Fractional surface coverage of colloid by coagulant
Initial floc volume fraction
Sedimentation tank capture velocity
Sedimentation velocity of ???
-log(fraction remaining)
Characteristic colloid size
Lambert W Function
What does the plant operator control? ________
What does the engineer control? __________
G1
3t CaptureV
What changes with the raw water? __________𝜙0
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Turbulent Flocculation Sedimentation Model Questions
Must represent a characteristic sedimentation velocity of the floc suspension. It could be the average terminal velocity of the full size flocs in the flocculator effluent
Is a characteristic size of a colloid. Is a function of the characteristic size of the
coagulant precipitate, geometry of the colloids, and loss of coagulant to reactor walls
Coag
2
3Colloidd
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Fractal Flocculation Conclusions
It is difficult to flocculate to a low residual turbidity because the time between effective collisions increases as the number of colloids and nonsettleable flocs decreases
Colloids don’t attach to full size flocsPerhaps because the surface shear on the big
floc is too high for a colloid to attach (surface shear increases with diameter)
If we could routinely break up full size flocs perhaps we could speed the aggregation process
Perhaps related to deformability?
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What is the model missing?
The model doesn’t include any particle aggregation that occurs in the sedimentation tank. Residence time in the sed tank is long and energy dissipation rate is low. Flocculation in the sed tank (especially if there is a floc blanket) could be very important.
Thus real world performance is likely better than model predictions.
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Flocculator Collision Potential
The majority of the collisions in a turbulent hydraulic flocculator are in the inertial range and the collision potential is determined by flocculator residence time, q, and energy dissipation rate, e.
The collision potential is given the symbol y and has the dimension of m2/3 and this length scale will be a property of the reactor geometry
The next set of notes provides guidance for designing the geometry of a flocculator to obtain a target collision potential
1
3
0
t dt
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Required Collision Potential*
0 20 40 60 80 1001
10
100
3 NTU10 NTU30 NTU100 NTU300 NTU
ψ (m^2/3)
Tur
bidi
ty (
NT
U)
*The model has not been validated for turbulent hydraulic flocculator. Thus this is only a rough estimate.
Agalteca, Alauca, Marcala 2 were designed to have y=100 m2/3
We are currently using y=75 m2/3
0 0
2 83 9
8
9
lnColloid ColloidCapture Colloid
Coag Colloid Colloid
C CV d
C C
1
3
0
t dt
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Review
Why is it that doubling the residence time in a flocculator doesn’t double pC* for the flocculator?
Why does increasing floc volume fraction decrease the time between collisions?
Which terms in the model are determined by the flocculator design?
8
3
*2
9
1
3
0
9log( ) 8
8 9Coll
C
C
oapture
id
oag
V
tC W
d
ep
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Surface coverage (G) of clay by coagulant precipitate
In our modeling work we didn’t cover how to calculate what fraction of the clay surfaces are coated with coagulant. The following slides are the equation heavy derivation of that coverage. We assume coagulant precipitate has some characteristic diameter when it sticks to the clay and that the clay can be represented as a cylinder. We also assume that the coagulant sticks to everything including reactor walls. The coagulant approaches the clay surface randomly and thus accumulates in a Poisson distribution. The random bombardment means that some coagulant is wasted in double coverage of previously covered clay.
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Floc Model Equations for G
Clay Total
ClayTotalA A
ClayTotal Wall
SA
SA SA
Wall Tube TubeSA D L
2
4Clay
ClayTotal Tube Tube ClayClay
SASA D L
V
1
1Clay TotalA A
Wall
ClayTotal
SA
SA
3
26
6
ClayClay Clay Clay
ClaySphereClay Clay Clay
DSA SA D
V D V
6Clay ClaySphere
Clay Clay
SA
V D
23
2Tube Tube ClaySphere
ClayTotal ClayClay
D LSA
D
Surface area to volume ratio for clay normalized by surface area to volume ratio for a sphere
Surface area of clay divided by total surface area of clay + reactor walls
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Ratio of clay surface area to total surface area (including reactor walls)
2
1
13
2
Clay TotalA ATube Tube
Tube Tube ClaySphereClay
Clay
D LD L
D
12
13
Clay TotalA AClay
Tube ClaySphere Clay
D
D
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Poisson distribution and coagulant loss to walls combined to get surface coverage
2
1Coag CoagPerClay
A AClay TotalClay
D N
SAe
3
1
6
CoagPerClay Coag Clay Clay
Clay Coag Clay ClayCoag
N C V
SA SA CD
1
21
3
Clay TotalA AClay
Tube ClaySphere Clay
D
D
6Clay ClaySphere
Clay Clay
SA
V D
2
Coag CoagPerClay Coag Clay
Clay Clay Coag ClaySphere
D N D
SA D
1A ACoag Clay Clay Total
Clay Coag ClaySphere
D
De
The surface coverage is reduced due to stacking (which is handled by the Poisson distribution) and by the loss of coagulant to the reactor walls.
loss of coagulant to the reactor walls
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Clay platelet geometry
ClayPlateletHD
ClayPlatelet
H
D
2
4Platelet ClayPlatelet ClayPlateletV D H
3
6Platelet ClayV D
3
4Platelet HD ClayPlateletV D
3 3
4 6Platelet HD ClayPlatelet ClayV D D
1
32
3ClayPlatelet ClayHD
D D
D.Clay is the diameter of a sphere with equal volume as the clay platelet
Model clay as cylinder with height and diameter
D.ClayPlatelet is the diameter of a cylinder given ratio of height to diameter and equal volume spherical diameter
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Clay platelet geometry
2 2
3 32 22 2
24 3 3ClayPlatelet Clay HD Clay
HD HD
SA D D
2
ClayPlateletClaySphere
Clay
SA
D
2
31 2
2 3ClaySphere HDHD
224ClayPlatelet ClayPlatelet ClayPlatelet ClayPlateletSA D D H
1
32
3ClayPlatelet ClayHD
D D
ClayPlateletHD
ClayPlatelet
H
D
2
321 2
2 3ClayPlatelet HD ClayHD
SA D
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Surface coverage of clay
1A ACoag Clay Clay Total
Clay Coag ClaySphere
D
De
2
31 2
2 3ClaySphere HDHD
1 12
31
Coag Clay
ClayClay CoagClaySphere
Tube Clay
D
DD
De
12
13
Clay TotalA AClay
Tube ClaySphere Clay
D
D
12
3
Clay TotalA A
ClayClaySphereClaySphere
Tube Clay
D
D
2
3
1 1
21 2
2 3 3
1Coag Clay
Clay CoagClay
HDHD Tube Clay
D
DD
D
e
2
3
1 1
1 2
2 3
1Coag Clay
Clay Coag
HDHD
D
De
If you neglect wall loses
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Solving for Coagulant Dose
2
3
1 1
21 2
2 3 3
1Coag Clay
Clay CoagClay
HDHD Tube Clay
D
DD
D
e
2
3 21 2ln 1
2 3 3Clay Coag Clay
Coag Coag HDHD Tube Clay Clay
D DC
D D
0 0
2 8
3 9
8
9
lnColloid ColloidCaptureColloid
Coag Colloid Colloid
C CVd
C C
Coagulant dose is inside coagulant volume fractionSolve for dose…
Given a target residual turbidity, solve for required G, then solve for coagulant dose
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Loss of clay to reactor walls
0.001 0.01 0.1 1 100.001
0.01
0.1
1
5 NTU50 NTU500 NTU5000 NTU
Tube diameter (m)
Fra
ctio
n of
coa
gula
nt o
n cl
ay
Loss of coagulant to reactor walls can dominate for low turbidities and small reactors. For the LFSRSF in India treating 5 NTU water and injecting coagulant into 7.5 cm pipes the clay only gets 23% of the applied coagulant!There may be a way to design a larger contact chamber for the coagulant to reduce losses to the walls.
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Effect of stacking and wall loss in a 10 cm diameter flocculator with 20 NTU
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
No Stacking and No Wall LossNo Wall LossStacking and Wall Losses
Coagulant concentration (mg/L)
Fra
ctio
nal
clay
sur
face
cov
erag
e
Stacking due to random bombardment of the clay surface with coagulant nanoglobs. Stacking becomes significant for high surface coverage. Wall losses also cause a significant reduction in clay surface coverage.
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