Notes on Colloid transport and filtration in saturated porous media by gR0S34y

VIEWS: 0 PAGES: 53

									 Notes on Colloid transport and
  filtration in saturated porous
                media



Tim Ginn, Patricia Culligan, Kirk Nelson
Purdue Summerschool in Geophysics 2006
       But first, we start with
 Brief review of general reactive transport
  formalism
Outline
   General reactive transport intro
       Multicomponent/two-phase/multireaction
       colloid filtration “Miller lite”
       Stop and smell the characteristic plane - mcad
   Colloid Filtration “Guiness”
       Overview
       Processes catwalk
       Classical approach
       Blocking
   Issues
   Return to macroscale: multisite/population
            Gone to mathcad
   Some analytical solutions - hope it runs
   Just transport
   Irreversible filtration no dispersion
   Reversible filtration no dispersion
   (Dispersion included by superposition.)
Outline
   General reactive transport intro
       Multicomponent/two-phase/multireaction
       colloid filtration “Miller lite”
       Stop and smell the characteristic plane - mcad
   Colloid Filtration “Guiness”
       Overview of colloids in hydrogeology
       Processes catwalk
       Classical approach
       Blocking
   Issues
   Return to macroscale: multisite/population
                                                                               1. Introduction - Background


                                 Particle Sizes
(diameter, m) 10
                 -10      10-9       10-8       10-7          10-6    10-5        10-4         10-3        10-2

                1Å     1 nm                                   1 mm                             1 mm        1 cm

   Soils                                               Clay           Silt              Sand          Gravel


 Microorganisms
                                         Viruses          Bacteria           Protozoa

                                                                                Red blood cell
  Blood cells
                                                                                        White blood cell
  Atoms,
                  Atoms    Molecules Macromolecules
  molecules

                                             Colloids                    Suspended particles

                                 Depth-filtration range

                                 Electron                      Light microscope                Human eye
                                 microscope
       Problems Involving Particle Transport
      through Porous Media in Environmental
               and Health Systems
 Water treatment system
         Deep Bed Filtration (DBF)
         Membrane-based filtration
 Transport of pollutants in aquifers
         Colloidal particle transport1
         Colloid-facilitated contaminant transport2
 Transport of microorganisms
         Pathogen transport in groundwater
         Bioremediation of aquifers
         …
 1.   Ryan, J.N., and M. Elimelech. 1996. Colloids Surf. A, 107:1–56.
 2.   de Jonge, Kjaergaard, Moldrup. 2004. Vadose Zone Journal, 3:321–325
                       …and some more
 In situ bioremediation
        transport of bacteria to contaminants1
        excessive attachment to aquifer grains – biofouling
 Bacteria-facilitated contaminant transport
  (e.g.,DDT2)
 Clinical settings
        Blood cell filtration
        Bacteria and viruses filtration

1.   Ginn et al., Advances in Water Resources, 2002, 25, 1017-1042.
2.   Lindqvist & Enfield. 1992. Appl. Environ. Microbiol, 58: 2211-2218.
Outline
   General reactive transport intro
       Multicomponent/two-phase/multireaction
       colloid filtration “Miller lite”
       Stop and smell the characteristic plane - mcad
   Colloid Filtration “Guiness”
       Overview
       Processes catwalk
       Classical approach
       Blocking
   Issues
   Return to macroscale: multisite/population
        Processes in colloid-surface
                interaction
   Actual colloid,
   Inertia in (arbitrary) velocity field
   Torque, drag due to nonuniform flow
   Diffusion,
   hydrodynamic retardation/lubrication
       Effective increase in viscosity near surface
 Electrostatic (dynamic) interaction
       DLVO (=LvdW + doublelayer model
        electrostatics)
 Buoyancy/gravitational force
Overview
  General reactive transport intro
      Multicomponent/two-phase/multireaction
      colloid filtration “Miller lite”
      Stop and smell the characteristic plane - mcad
  Colloid Filtration “Guiness”
      Overview
      Processes catwalk
      Classical approach – “Colloid filtration theory”
       and some Details
      Blocking
  Issues
  Return to macroscale: multisite/population
Classical take on Processes in
  colloid-surface interaction
 Inert, Spherical colloid to Sphere (flat)
 Inertia in (Stokes) velocity field
 Torque, drag due to nonuniform flow
      approximated
 Diffusion (superposed)
      hydrodynamic retardation/lubrication
 Electrostatic (dynamic) interaction
      DLVO (=LvdW + doublelayer model
       electrostatics )
 Buoyancy/gravitational force added
      So flow must be downward
   Forces And Torques – RT model
 Trajectory Analysis                          Smoluchowski-Levich Solution

(particle has finite                              (particle diameter = 0)
diameter)                   TD
                                                                   TD
                       FG




h      =                                  +
                       FB                     FI = inertial force due to Stokes flow
                                              FD = drag force due to Stokes flow
FI = inertial force due to Stokes flow*       TD = drag torque due to Stokes flow
FD = drag force due to Stokes flow*           FBR = random Brownian force
TD = drag torque due to Stokes flow*
FG = gravitational force
FB = buoyancy force
FvdW = van der Waals force
*with corrections near surface
   Classical CFT :Happel sphere-
               in-cell
 Clean-bed “Filtration Theory”
• Single “collector” represents a solid
phase grain. A fraction h of the particles
are brought to surface of the collector
by the mechanisms of Brownian
diffusion, Interception and/or                                             h0  h D  h I  hG
Gravitational sedimentation.
•A fraction  of the particles that reach                                  Single collector efficiency
the collector surface attach to the

                                                                                     h
surface (electrostatic and ionic strength)

• The single collector efficiency is then
“scaled up” to a macroscopic filtration                                         3(1 n)
                                              Filtration coefficient                  h
coefficient, which can be related to                                              2dc
first-order attachment rate of the
particles to the solid phase of the                                
                                             First-order deposition rate     katt  u
medium.
                                                              
 Bulk “kf” by classical filtration theory
          nC
                f c  kC                  First-order removal
           t
               3   n U
                   1
               h
          k                              Rate = filter coefficient * porewater velocity
                2 dc  n                         => two-step process


      n porosity
      C aqueous phase concentration of colloid suspension
      fc flux of C
      U groundwater (Darcy) specific flux
       fraction of colloids encountering solid surface that stick (empirical2,3)
      h fraction of aqueous colloids that encounter solid surface (modeled1,3-6)
1. Rajagoplan & Tien. 1976. AIChE J. 22: 523-533.      2. Harvey & Garabedian. 1991. ES&T 25: 178-185.
3. Logan et al. 1995. J. Environ. Eng. 121: 869-873.   3. Nelson & Ginn. 2001 Langmuir 17: 5636-5645
4. Tufenkji & Elimelech. 2004 ES&T 38: 529-536.        5. Nelson & Ginn. 2005 Langmuir 21: 2173-2184
     Details1:Happel sphere-in-cell model2
                                          A1   A2
 Happel sphere-in-cell is
  porous medium
 Stokes’ flow field
 h calculated via trajectory
  analysis1
 Additive decomposition
         h=hI+hG+hD
 Initial point of limiting
  trajectory
 h = A1/A2 = sin2qs

1.   Rajagoplan & Tien. 1976. AIChE J.
     22: 523-533.
2.   Happel. 1958. AIChE J. 4: 197-201.
 Detail: Basic solution (analytical)
 due to Rajagopalan & Tien (1976)
 Hydrodynamic retardation effect = the increased drag
force a particle experiences as it approaches a surface.
 a deviation from Stokes’ law             Interception by boundary
                                           condition
 Hydrodynamic correction factors
 Particle velocity expressions gives:     Sedimentation group


                   1 
                                                    U        London van der
     uq r, q  
                                       
                         B s2  D 1   s3  N G sin q          Waals group
                   s1                                   r
                                                                   
                                                         rtd N LO 
                           
                      1  
        ur r, q   t  A 1     
                                    2 m
                                       fr  NG cosq  2               U
                      fr  
                          
                                                       
                                                          
                                                       2     
                                                                  2 




                                                               
where frt, frm, s1, s2, and s3 are the drag correction factors.
               Detail: h vs. 

 irreversible adsorption constant, kirr = f(,h)
 h = fraction of colloids contacting solid phase,
      calculated a priori from RT model
  = fraction of colloids contacting solid phase
      that stick, treated as a calibration parameter
      accounting for all forces and mechanisms
      not considered in calculation of h

Role of electrostatic forces : aside
Detail: Surface Forces in CFT –
DLVO
 RT model uses DLVO theory for surface
  interaction forces:
                    attractive   repulsive for like charges


  potential = van der Waals + double layer

 Theory predicts negligible collection when
  repulsive surface interaction exists  RT
  model neglects double layer force.
Detail: Surface Forces in CFT –
DLVO
 RT model uses DLVO theory for surface
  interaction forces:
                    attractive   repulsive for like charges


  potential = van der Waals + double layer

 Theory predicts negligible collection when
  repulsive surface interaction exists  RT
  model neglects double layer force.
 Thus, double layer force implicit in .
Highlights of Formulae for h
 Yao (1971)
      hydrodynamic retardation and van der Waals force not included

 Rajagopalan and Tien (1976)
      deterministic trajectory analysis
      torque correction factors
      Brownian h added on separately from Eulerian analysis

 Tufenkji and Elimelech (2004)
      convective-diffusion equation solution
      influence of van der Waals force and hydrodynamic retardation on
       diffusion
                                                      D f 
                 fc   UC   D C                  C
                                                        kT
      Diffusion, interception, & sedimentation considered additive

 Nelson and Ginn (2005)
      Particle tracking in Happel cell – all forces together
Outline
   General reactive transport intro
       Multicomponent/two-phase/multireaction
       colloid filtration “Miller lite”
       Stop and smell the characteristic plane - mcad
   Colloid Filtration “Guiness”
       Overview
       Processes catwalk
       Classical approach – “Colloid filtration theory”
        and some Details
       Blocking
   Issues
   Return to macroscale: multisite/population
Dynamic surface blocking (ME)
  initial deposition rate (kinetics)


        rate  a kc      2
                           p

  later, when deposition rate drops due to
   surface coverage (dynamics)

       rate  a kB(s)c
                       2
                       p

  retained particles block sites, B is the
   dynamic blocking function (misnomer).
B's
 B = fraction of particle-surface collisions that
  involve open seats (cake walk).
 Random Sequential Adsorption

                           6 3   40 176 
  Bs  1  4ss    ss    2 s s
                                     2          3
                                3 3 
      Power series in S, for spherical geometry


 Langmuirian Dynamic Blocking
          Bs  1   s
           1/ s
Outline
   General reactive transport intro
       Multicomponent/two-phase/multireaction
       colloid filtration “Miller lite”
       Stop and smell the characteristic plane - mcad
   Colloid Filtration “Guiness”
       Overview
       Processes catwalk
       Classical approach – “Colloid filtration theory”
        and some Details
       Blocking
   Issues
   Return to macroscale: multisite/population
                       Issues
 CFT coarse idealized model
     Chem/env. Engineering, not natural p.m.
     Biofilms, organic matter, asperities,
      heterogeneity (gsd, psd, surface area,
      electrostatic (dynamic), transience, flow
      reversal, temperature, etc.
     Reversibility ???
 CFT good for trend prediction
     Attachment goes up with colloid size, gw
      velocity, ionic strength, etc.
 Ultimately need equs for bulk media
     Lab
     field
Outline
   General reactive transport intro
       Multicomponent/two-phase/multireaction
       colloid filtration “Miller lite”
       Stop and smell the characteristic plane - mcad
   Colloid Filtration “Guiness”
       Overview
       Processes catwalk
       Classical approach – “Colloid filtration theory”
        and some Details
       Blocking
   Issues
   Return to macroscale: See the data !
                  Field/Lab observations
         Microbes 1,2,3 and viruses 4,5 first showed
          apparent multipopulation rates due to
          decreased attachment with scale
                 Sticky bugs leave early
                 Readily explained by subpopulations
                 Some suggest geochemical “heterogeneity”
         Recent surprize is that inert monotype,
          monosize and polysize colloids exhibit same6
1.   Albinger et al., FEMS Microbio Ltr., 124:321 (1994)
2.   Ginn et al., Advances in Water Resources, 25:1017 (2002).
3.   DeFlaun et al., FEMS Microbio Ltr., 20:473 (1997)
4.   Redman et al., EST 35:1798 (2001); Schijven et al., WRR 35:1101 (1999)
5.   Bales et al., WRR 33:639 (1997)
6.   Li et al., EST 38:5616 (2004); Tufenkji and Elim. Langmuir 21:841 (2005)Yoon et al., WRR June 2006
      Ability-based modeling (because
                   we can)
        BTCs (first) exhibit long flat tails
                Two-site, multisite model1 (google “patchwise”)
                Two-population, multipop’n model2 (UAz,
                 Arnold/Baygents)
                Can’t tell the difference
        Profiles (recently) are steeper than expected
                Multipopulation works, not multisite (Li et al in 2), 3
                This is the location of the front in practice
                Upscaling
                Alternative explanations
1.   E.g., Sun et al., WRR 37:209 (2001); “patchwise heterogeneity”, CXTFIT ease of use (sorta)
2.   E.g., Redman et al., EST 35:1798 (2001); Li et al. EST 38:5616 (2004)
3.   Johnson and Li, Langmuir 21:10895 (2005); Comment/Reply
          Research Needs (at least)
          Formal upscaling
                  Forces complex but well understood
                  Approximations tested
                  Analytical results (Smoluchowski-Levitch1)
          Alternative explanations
                  C<-> S -> S’ surface transformations 2
                       Mainly bacteria; need RTD for attachment events
                  Physical straining of larger sizes (a pop’n model)3
                  Reentrainment4
                  Contact (CFT) and surface (multipopn) filtration5
1.   For CFT/Happel cell without interception or sfc forces (LvdW =-hyd. Retardation)
2.   Davros & van de Ven JCIS 93:576 (1983); Meinders et al. JCIS 152:265 (1992); Johnson et al. WRR
     31: 2649 (1995); Ginn WRR 36:2895 (2000)
3.   Bradford et al WRR 38:1327 (2002); Bradford et al. EST 37:2242 (2003)
4.   Grolimund et al WRR 37:571 (2001)          5. Yoon et al. WRR June 2006
Appendix: DNS Approach

 Langevin equation of motion
     Happel sphere-in-cell
     Contemporaneous accounting of all forces
 Solution per colloid
 Calculating h
     Monte carlo colloidal release per qs =>
     P(qs) frequency of attachment per qs
     h as an expectation over P(qs)
Langevin Equation
 Deterministic and Brownian displacements are
  combined per time step:
             du
          mp     Fh  Fe  Fb
             dt
 mp is the particle mass, u is the particle velocity
  vector, Fh is the hydrodynamic force vector, Fe is
  the external force vector, and Fb is the random
  Brownian force vector.


 All three components of random displacement
  must be modeled in the axisymmetric (3D  2D)
  flow field.
Solution
                   R  udet Dt nsR

 R = 3D displacement,
 udet = deterministic velocity vector
 n =3 N(0,1),
sR = standard deviations of Brownian displacements.
 negligible particle inertia assumed
   Dt >> tB (Kanaoka et al., 1983)
   tB particle’s momentum relaxation time (=mp/6map).
Thus, tB << Dt < tu
   tu is the time increment at which udet is considered con
Highlights of numerical solution

  Stokes’ flow in two-dimensions
  R&T (1976) hydrodynamic drag correction
   factors1
  Brownian diffusion algorithm of Kanaoka et
   al. (1983)2 for diffusive aerosols
  Coordinate transformation to 2D model

 1. Brenner, H., Chem. Eng. Sci. 1961, 16, 242-251; Dahneke, B.E., J. Colloid Interface
    Sci., 1974, 48, 520-522.
 2. Kanaoka, C.; Emi, H.; Tanthapanichakoon, W., AIChE J., 1983, 29, 895-902.
Coordinates for diffusion
 The Happel model: 3-D -> 2-D polar coordinates
 convert 3-D Brownian Cartesian displacement to spherical, to
polar ˜                 ˜                   ˜
      Rx  nx 2DBM Dt   Ry  ny 2DBM Dt     Rz  nz 2DBM Dt



y,z, contribute to angular displacements
                              Ry 
                                 ˜                 R 
                                                      ˜
                   ˜
                   Rq  arcsin      ˜
                                        R  arcsin z 
                               r 
                                                r 
And thus to r
                  ˜  R  r 1  sin 2 R  1  sin 2 R  2r
                  Rr ˜ x              ˜             ˜ 
                                          q            
                                                       
Calculating h
                  / 2


            h2
                  0
                         Pcollect qS sin qS cosqS dqS


 qS starting angle of a colloid
 Pc(qS) frequency of contact with the collector.
 reduces to classical equation when
  deterministic (e.g., when Pc(qS) equals one for
  all qS < qLT and zero for all qS > qLT).

 task of stochastic trajectory analysis for h is to
  find Pc(qS).
   Colloid transport and Colloid Filtration Theory
   Classical approach
   Issues
   Direct numerical simulation:
       Approach
       Examples, Convergence, Testing
 Results
 Blocking - pages from Elimelech's site
 Conclusions
Example Brownian Trajectory
         1.6429E-04


         1.6428E-04


         1.6427E-04


         1.6426E-04


         1.6425E-04
 r [m]




         1.6424E-04


         1.6423E-04


         1.6422E-04


         1.6421E-04


         1.6420E-04


         1.6419E-04
                  1.1838   1.184   1.1842   1.1844   1.1846         1.1848   1.185   1.1852   1.1854   1.1856

                                                          q [rad]
           164.44




           164.39




           164.34


r [ m m]
           164.29




           164.24




           164.19
                1.132   1.134   1.136   1.138     1.14   1.142   1.144   1.146

                                            q [rad]
P(qs)

                              Num ber of bacteria collected w ith (Brow nian m otion included) as function of theta-start

                      0.025
                                           ran1 300 rlzns      ran1 1000 rlzns      MT 12K rlzns       ran1 12K rlzns


                      0.020
 Bacteria collected




                      0.015


                      0.010


                      0.005


                      0.000
                              0     0.01    0.02    0.03    0.04    0.05    0.06    0.07    0.08     0.09    0.1        0.11   0.12

                                                                       theta-start
Convergence of a trajectory - 50K
realizations
                                    Convergence of Collection Freq from ts=.0418 (Case 1, ap = .695 microns, dt = 0.5 msec)


                           1


                          0.9


                          0.8


                          0.7
Frequency of collection




                          0.6


                          0.5


                          0.4


                          0.3


                          0.2


                          0.1


                           0
                                1    10                          100                            1000                     10000   100000
                                                                       number of realizations
Convergence to deterministic trajectory analysis of
 Rajagopalan and Tien (when diffusion is neglected),
Parameters: e = 0.2, as = 50 mm, ap = 0.1 mm, and U = 3.4375 * 10-4 m/s.
The approximate analytical solution is h = 1.5 NR2g2AS (Rajagopalan and Tien, 1976).
Convergence of stochastic simulations for
         Smoluchowski-Levich approximation.
Parameters: ap = 0.1 mm, as = 163.5 mm, e = 0.372,
U = 3.4375*10-4 m/sec, m = 8.9*10-4 kg*m/sec, T = 298 K.

              8.4E-03


              8.3E-03

                                            Dt = 10 ms
              8.2E-03                                                  analytical result

                                            D t = 1 ms
              8.1E-03

                                        Dt = 100
              8.0E-03

            h 7.9E-03
              7.8E-03


              7.7E-03


              7.6E-03


              7.5E-03


              7.4E-03
                        0   2000     4000                6000   8000                10000   12000

                                       number_of_realizations
   Colloid transport and Colloid Filtration Theory
   Classical approach
   Issues
   Direct numerical simulation:
       Approach
       Convergence
 Results
       Smoluchowski-Levitch approximation
       General case
 Blocking - pages from Elimelech's site
 Conclusions
Testing comparison to the Smoluchowski-Levich
 approximation (external forces, interception neglected).
Parameters: as = 163.5 mm, e = 0.372, U = 3.4375*10-4 m/sec, m = 8.9*10-4 kg*m/sec,
                T = 298 K, Dt = 1 ms, N = 6000.

     1.E-02
                                       NG04     analytical




     h
 




     1.E-03
              0     0.2         0.4           0.6            0.8   1          1.2
                                               m)
                                          ap (m
            Comparison of h calculations
                    R&T (1976) X N&G         - - - T&E (2004)            o N&G Additive
                  RT_76     NG_04     TE_04       NG_04 additive      RT_76 deterministic NG_04 deterministic
                    R&T (1976) deterministic                     N&G deterministic
    1E-02




    h


   1E-03




    1E-04
            0.2   0.3       0.4        0.5        0.6        0.7       0.8        0.9        1         1.1
                                                           ap (mm)
Conclusions
 Lagrangean analysis is viable tool with modern
  computers

 Stochastic trajectory analysis suggests diffusion
  and sedimentation may not be additive

 More realistic “unit cell” models could be used

 Lagrangean approach allows for arbitrary
  interaction potentials
     Chemical (mineralogical, patchwise) heterogeneity
     Exocellular polymeric substances in bacteria
     Polymer bridging, hysteretic force potentials
                      Parameter                   Value
Parameters
used in
                  Collector radius, as          163.5 mm
stochastic
trajectory            Porosity, e                 0.372
simulations.
                 Approach velocity, U       3.4375 * 10-4 sec


                   Fluid viscosity, m      8.9 * 10-4 kg·m / sec


                 Hamaker constant, H              10-20 J


                 Bacterial density, rp        1070 kg / m3


                   Fluid density, rf           997 kg / m3


               Absolute temperature, T            298 K


                     Time step, Dt                 1 ms


               Number of realizations, N          6000
Modification of CFT to Account for EPS

 Distribution of polymer lengths
  on the cell surface
                                                      Hypothetical cell (drawn to scale)
 Repulsion modeled by steric                                                                  C
                                                                                               O
  force, Fst(h)1,2
                                                                                               L
    depends on polymer density                                      KT2442                    L
    and brush length                                                                           E
                                                                           0.695 mm        h
                                                                                               C
 If sufficient polymers contact
                                                                                               T
  collector, cell attaches                                                                     O
    depends on polymer density,                                                               R
    length, and adhesion forces                      mean polymer length = 160 nm




1. de Gennes. 1987. Adv. Colloid Interface Sci. 27: 189-209.
2. Camesano & Logan. 2000. Environ. Sci. Technol. 34: 3354-3362.
       Theoretical Sticking Efficiency
               Numerical Calculation of Trajectories

 Steric repulsive force
 Polymer bridging
 Interception
 Sedimentation
 Brownian motion
 London van der Waals
  attractive force
 Hydrodynamic retardation
  effect


    Incorporation of Brownian motion and polymer interactions into
    trajectory analysis allows for computation of a theoretical sticking
    efficiency.
      Theoretical Sticking Coefficient

 Incorporation of polymer interactions and Brownian
motion + assumption that polymers control adhesion 
  Trajectory analysis yields the product [h]theo= A1/A2 =sin2qs

 Then we can define a theoretical value for the sticking
efficiency :

                 theo= [h]theo / h

where h is the model result without polymer interactions.
 Comparison of theo with experimental  can serve as a
validation tool for the polymer interaction modeling.
              Pseudomonas putida KT2442
                                                                   KT2442 cells with Congo Red
     Considered for
      bioremediation use1,2
     Congo Red stain
      image  heavy EPS
      coverage on cells                                            White areas

     EPS characteristics                                          indicate EPS


      being studied by
      Camesano et al.
      (WPI)3
1. Nublein et al. 1992. Appl. Environ. Microbiol. 58: 3380-3386.
2. Dobler et al. 1992. Appl. Environ. Microbiol. 58: 1249-1258.    Photo credit: Stephanie Smith
                                                                                 Dept. of Land, Air, & Water Resources
3. Camesano & Abu-Lail. 2002. Biomacromolecules. 3: 661-667.
                  Summary

 CFT trajectory analysis modified for explicit
  inclusion of Brownian motion and bacterial EPS
  interactions
 Brownian trajectory analysis results suggest that
  sedimentation and diffusion may not be additive
  as previously assumed
 Future work
   comparison of h calculations with
  experimental data in the literature
   more realistic modeling of EPS interactions
  (e.g., hysteresis)

								
To top