Colloidal Transport: Modeling the “Irreversible” attachment of colloids
to porous media
Summer School in Geophysical Porous Media, Purdue UniversityJuly 28, 2006
Son-Young Yi, Natalie Kleinfelter, Feng Yue, Gaurav Saini, Guoping Tang, Jean E Elkhoury, Murat Hamderi, Rishi Parashar
Advisors: Patricia Culligan, Timothy Ginn, Daniel Tartakovsky
Jean E. ElkhouryGrad. Student
Geophysics, UCLA
Son-Young YiPhD
Mathematics, Purdue
Guoping TangPhD
Civil Engg., Northeastern
Gaurav SainiGrad. Student
Env. Engg., OSU
Murat HamderiGrad. Student
Civil Engg., Drexel
Rishi ParasharGrad. Student
Civil Engg., Purdue
Feng YueGrad. Student
Civil Engg., UIUC
Natalie KleinfelterPhD
Mathematics, Purdue
3
• Motivation• Classical Filtration Theory• Experimental Setup• Limitation of Classical Theory
• Single Population Model• Multi Population Model
• 5-population model• 2-population model
• Conclusions• Future Work
Outline
4
Motivation
• Decrease in attachment rate with transport distance indicates deficiencies in classical clean-bed filtration theory.
• Modeling unique experimental dataset to explore alternative approaches to describe irreversible attachment of colloids.
• Distance dependent attachment rate constant• “Multi-population” based modeling approach.
5
Classical Clean-Bed Filtration Theory
ηD: Brownian diffusion
ηI: Interception
ηG: Gravitational Settling
6
Variation of η with velocity & dp
0
0.05
0.1
0.15
0.2
0.25
0.3
0 5 10 15 20 25
particle size (micron)
eta
fast flow
Medium flow
slow flow
7
Macro-scale Approach
First-order deposition rate
kirr = λu
8
Classical Transport Equations2
2hC S C CD ut t x x
Rate of Change of Fore Fluid Concentration
Rate of Change of Adsorbed Concentration
Hydrodynamic dispersion term
Advection term
kirr=attachment rate constant
CktS
irr
9
Classical Theory Inferences
Kirr
Kirr
KirrC
10
Experimental Set-up (Yoon et al. 2006)
• Porous Media: Glass beads (dc= 4mm), surface roughness = 2 μm (rough beads only!)
• Colloids: dp=1-25 μm (d50=7 μm), fluorescent• C0 = 50 ppm • Injection period ≈ 11.5 PV (fast/medium) & 9.5 (slow)• Flushing period ≈ 11 PV (fast/medium) & 7 (slow)• Fast (0.0522 cm/s), Medium (0.0295 cm/s)& slow (0.0124
cm/s) flow• Laser induced fluorescence and digital image processing
11
Schematics
(Yoon et al., 2006)
12
Rough Beads (Contact + Surface Filtration)
13
Smooth Beads (Contact Filtration)
14
Breakthrough Curve
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25Pore volumes
Fast Flow RoughMedium Flow RoughSlow Flow Rough
C/C0
15
Concentration Profile (d~13 cm)
0
0.5
1
1.5
2
2.5
3
3.5
0 5 10 15 20 25
PV
(C+S
)/C0
Fast Flow Rough
Medium Flow Rough
Slow Flow Rough
Sirr
16
Collector Efficiency
0
2
4
6
8
10
12
14
16
18
0.0000 0.0020 0.0040 0.0060 0.0080 0.0100
collector efficiency (alpha*eta)
Dept
h
Fast flow rough
medium flow rough
slow flow rough
17
Irreversible Adsorption
0
2
4
6
8
10
12
14
16
18
0 0.5 1 1.5 2 2.5 3 3.5
Sirr/C0
Dept
h
Fast flow rough
Medium flow rough
Slow flow rough
18
CXTFIT Analysis
• Fast flow, rough beads
• Deterministic Equilibrium Model
• Predictions:• D = 0.2834 cm2/s• kirr = 0.165/s 0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25
PV
C/C0
Experimental values
D = 0.042 cm2/s
Kirr = 1.48 x10-4/s
Classic Transport Model can not explain the observed behavior
19
Variation in irreversible attachment with depth (constant kirr)
0
3
6
9
12
15
18
0 1 2 3 4Sirr/C0
Dept
h (c
m)
Slow Flow Rough-ObservedSlow Flow Rough-SimulatedMedium Flow Rough-ObservedMedium Flow Rough-ObservedFast Flow Rough-ObservedFast Flow Rough-Simulated
20
Approaches Explored
• Distance dependent attachment rate constant• Multi-Population approach
• 5-population• 2-population
21
Distance dependent kirr
0
2
4
6
8
10
12
14
16
0 1 2 3 4Sirr/C0
Dept
h (c
m)
Slow Flow Rough-Observed
Slow Flow Rough-Simulated
Slow Flow Rough-StepFunction like kirr Assumption
Step Function like kirr
kirr
8 cm
Dep
th
3.5 x10-4
1.2 x10-4
22
Summary
• Constant or step function distribution of rate constant (kirr) does not explain the observed behavior.
• Particle sorption can be predicted given the distribution of kirr with depth (which is unlikely!)
23
Multi-Population Modeling
Governing equations
tS
zC
Dz
Cu
tC massimassimassimassi
,2,
2,,
massimassirrimassi Ckt
S,,,
,
i : population
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Analytical Solution
Dtutzerfc
Duz
Dtutzerfc
Duz
CktzfCktzC massoimassirrimassoimassirrimassi
22)1(exp
22)1(exp
21
),;,(),;,( ,,,,,,,,,
Runkel (1996)
H21 2,,2u
DkH massirriwhere
,
(t<τ)
),;,(),;,(),;,( ,,,,,,,,,,,,, massoimassirrimassoimassirrimassoimassirrimassi CkTtzfCktzfCktzC
(t>τ)
25
5-Population Model
26
Predictions with constant kirr
0
2
4
6
8
10
12
14
16
0 5 10 15 20Sirr/Co
Dept
h (c
m)
Slow Flow Rough 1.5 micronSlow Flow Rough 3 micron
Slow Flow Rough 7 micronSlow Flow Rough 12 micron
Slow Flow Rough 18 micron
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Simulation Inputs
Average particle size (μm) kirr (10-4)
Mass fraction (input) weighted Mass fraction
1.5 0 0.04 0.16
3 0.17 0.16 0.32
7 2.5 0.37 0.31
12 8.6 0.33 0.17
18 15.5 0.10 0.04
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Slow Flow: Absolute mass fraction as input
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7
(C+S)final/C0 or Sirr/C0
Dep
th, c
m
ObservedPredicted
Medium flow: Absolute mass fraction as input
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7
(C+S)final/C0 or Sirr/C0
Dep
th, c
m
ObservedPredicted
Fast flow: Absolute mass fraction as input
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7
(C+S)final/C0 or Sirr/C0
Dep
th, c
m
ObservedPredicted
29
Slow Flow: Weighted mass fraction as input
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7
(C+S)final/C0 or Sirr/C0
Dep
th, c
m
ObservedPredicted
Medium flow: Weighted mass fraction as input
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7
(C+S)final/C0 or Sirr/C0
Dep
th, c
m
ObservedPredicted
Fast flow: Weighted mass fraction as input
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7
(C+S)final/C0 or Sirr/C0
Dep
th, c
m
ObservedPredicted
30
Slow Flow: Weighted mass fraction and weighted kirr
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7
(C+S)final/C0 or Sirr/C0
Dep
th, c
m
ObservedPredicted
Medium flow: Weighted mass fraction and weighted kirr
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7
(C+S)final/C0 or Sirr/C0
Dep
th, c
m
ObservedPredicted
Fast flow: Weighted mass fraction and weighted Kirr
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7
(C+S)final/C0 or Sirr/C0
Dep
th, c
m
ObservedPredicted
31
Summary of 5-Population model results• 5-population model provides good fit to the
observed data.• Mass fraction weighting alone does not explain
the observations.• Mass fraction weighted with specific surface
provides good fit, using kirr values from slow flow test.
• Very good fits were obtained when kirr were weighted according to breakthrough data.
32
Motivation for 2-population model
33
2-Population Model
• Population distribution not available.• Assumptions:
• kirr(pop1) > kirr(pop2)
• Approach• Least square fit to observed data (Sirr/C0)• Predicted rate constants and concentrations (C0 (1) &
C0 (2))
34
2-Population model fit
0
3
6
9
12
15
18
0 1 2 3 4
Sirr/C0
Dept
h
Medium flow-observed
Medium flow-fitted
Fast flow-observed
Fast flow-fitted
Slow flow-observed
Slow flow-fitted
35
2-Population model predictions
Slow Medium Fast
Kirr 1(Experimental Kirr
values)
5.34E-03(2.36E-04)
4.44E-03(2.25E-04)
1.75E-03(1.48E-04)
Kirr 21.69E-04 1.70E-04 4.75E-05
C0 (1)/ C0(1+2)3.75E-02(3.75%)
3.34E-02(3.34%)
7.60E-02(7.60%)
36
Summary Results
• 2-population model approach is a potent tool when particle distribution is unknown.
• Optimized k-values found using 2-population model are of the same order as the observed values.
• A small fraction (~5%) of population having high kirr values can explain variations in kirr with transport distance.
37
Conclusions• Multi-population models capture the trend of decreasing
kirr with transport distance.• Multi-population models can be used to obtain
reasonable predictions if particle population distribution is known.
• If particle population distribution is unknown, a 2-population model with optimization can be used to obtain parameters for predictions.
• In homogenized, clean bed-filters, decreases in kirr with transport distance are best explained by distributions in particle populations and not medium properties.
38
Suggestions for future work
• Analysis of Reversible attachment (Srev)• k = f (particle size, media roughness,fluid
velocity)?• Quantitative measurements of C & S (use
of fluorescent bacteria…)• Explore 1-site model with long-tail
distribution functions for residence time.
Questions
Supplementary Slides….
41
Properties of particles
42
43
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