• Project
•
•
•
•
1. Proposal. Describe the problem you propose to analyze. Include
Background: describe the problem you intend to analyze, give motivation for doing the
analysis, cite literature.
Objective: Describe what you intend to achieve as a result of doing the simulation
Method: Describe the analysis you will conduct. This should include a description of the
mathematical problem, including governing equation, boundary and initial conditions,
parameters, and geometry.
–
–
–
•
•
Verification: Describe existing analyses you would use to verify your results. This could be an analytical solution or
existing numerical solution.
Validation or calibration: Describe data that you would use to calibrate your simulation. Identify the approach you
would use for calibration.
Approach: Outline 3-5 analyses of increasing complexity that ends with the final goal.
Results: Describe the results you expect to get from the analyses.
References: Identify and cite at least 3 papers
Clemson Hydro
Transport with Fluid-Solid
Reactions
Ion exchange resin
Gas chromatograph
Mineralized vein
http://www.insilico.hu/liesegang/index.html
Clemson Hydro
Reaction Locations
Bulk Material (homogeneous rxn)
Fluid—Multi-scale mixing
Solid– Diffusion dominant
Interfaces (heterogeneous rxn)
Fluid-Solid—No flow at interfacediffusion
Liquid-Gas
Clemson Hydro
Conceptual Model
Cf
fluid
Cf
Cs
Cs
solid
Clemson Hydro
Concentrations
• In water:
• In soil:
Mass in Water
mg
Cw 

Volume of Water Lwater
Mass in Water
mg
Cs 

Mass of Soil
kg soil
• On surfaces: C  Mass on Surface  mg
ss
2
Area
m
Clemson Hydro
Processes
Sorption: bonding, but similar species as aqueous
Precipitation: change species
Clogging: significant thickness
Dissolution: remove solid
Biofilm: growth of filmreactions, clogging
Matrix diffusion: species into matrix, store/react
Clemson Hydro
Reaction Rates
• Fast relative to transport
– Equilibrium
– Partitioning between fluid/solid
– Cs = f(Cf)
• Similar or slower than transport
– Disequilibrium, kinetics important
dCs
 f (C f , Cs )
dt
– Reaction time scale: 1/k1
– Diffusion time scale: L2/D
– Advection time scale: L/v
k1 L
v
k1 L2
DaII 
D
DaI 
Clemson Hydro
Sorption Isotherms
https://www.soils.org/publications/sssaj/articles/67/4/1140
Clemson Hydro
Equilibrium Sorption
Mass Contaminant
Mass Soil Solids
Mass Contaminant
Conc. Water=
Vol. Water
Concentration
sorbed (mass/mass)
Concentration sorbed
(mass/mass)
Conc. Sorbed=
Linear
Isotherm
Slope = Distribution Coefficient, Kd
Good for low concentrations
Concentration in water
Sorption sites fill at
high concentrations
Non- linear Isotherm
Examples
High concentrations
Freundlich: Cs  kC n
Langmuir: Cs 
Concentration in water
 C
1 C
Clemson Hydro
Important Concept, FluidSolid Surface
Porous media, Two overlapping domains
Equilibrium Partitioning
Fluid conc
Fluid concentration, Cf[Ms/Lf3]
Solid surface concentration, Cs[Ms/Mso]
Linear:
Cs  K d C f
Solid conc
Freundlich: Cs  kC nf
Langmuir: Cs 
 C f
1  C f
Langmuir w/ competition: Cs 
 C f 1
1   C f 1  C f 2 
Clemson Hydro
Effects of Equilibrium Sorption
on Transport of a Plume
Langmuir isotherm: Cs 
Breakthrough curves
cP ,max k LC f
1  k LC f
Source as mass flux over a circular area
( x  x1 )2  ( y  y1 )2  R 2
A point is in the circle if
Chromatographic eff
( x  x1 )2  ( y  y1 )2  R 2
Clemson Hydro
Application of pulse test
to determine ne and R
Average linear flow velocity
v=L/tm,w
tm=9215 s (from first moment, conservative tracer)
L=300m (from set up)
v=300m/9215 s =0.032 m/s
L
Effective porosity
Flux = q =0.01 m/s (specified in model)
Effective porosity =q/v = 0.01/0.032 = 0.31
Compare to porosity specified in model=0.3
Retardation factor
vc=L/tm,c
tm=20700 s (from first moment sorbing compound)
v=300m/20700 s =0.014 m/s
Chromatographic effect
R=vw/vc= 0.032/0.014=2.3
Clemson Hydro
Rate of change due to sorption
dM s , f
dM s , s
; rate of mass in fluid opposite to rate on solid

dt
dt
d M s ,s
d M s, f
; divide by control volume

dt Vc
dt Vc
d M s ,s M os
d M s, f V f
;

dt M osVc
dt VcV f
V f  vol. of fluid, M os  mass of solids
dCs  b
dCn
;

dt
dt
concentration in water, C  M s , f / V f
concentration on solids, Cs  M s ,s / M os
also using porosity, n  V f / Vc ; bulk density,  b  M os / Vc
dCs
dCn
;
  b
dt
dt
rearrange to get rate of change of C related to rate of Cs
Clemson Hydro
Governing Equation
Advection-Dispersion w/ surface reaction
c
S
t
= D +A
 
Ms
c 3
Lc
c Cn
c = Cn 
t
t
Storage
Advective Flux
Diffusive Flux (Fick’s Law)
Dispersive Flux
Source
Sn
Governing
dC
dC
  b s
dt
dt
A = qC
D = - nD * C
Dh = - nDhC
 M so M s 1 
 3

L
M
T
 c
so

 M s L3f 1 
 3 3 
 L f Lc T 
 L3f 
n  porosity  3 
 Lc 
 M so 
 b  bulk density  3 
 Lc 
Cn
Cs
  (  n  D  Dh  C )    qC 
  b
t
t
*
Clemson Hydro
Governing Eq.
AD w/Equilibrium Sorption, Linear Isotherm
Cn
 bCs
  (  n  D  Dh  C )    qC 

t
t
q
C
 Cs
  (   D*  Dh  C )  C 
 b
n
t
n t
*
Cs  K d C
Cs
C
 Kd
t
t
Linear Isotherm
q
C

C
  (   D*  Dh  C )  C 
  b Kd
n
t
n
t
C   b

  (  DC )  vC 
1  K d   0
t 
n

D
v
C
  (  C )  C 
0
R
R
t

 
R  1  b K d 
Retardation factor
n


Clemson Hydro
Governing Eq.
AD w/Equilibrium Sorption, Langmuir Isotherm
Cn
Cs
  (  n  D  Dh  C )    qC 
  b
t
t
q
C
 Cs
  (   D*  Dh  C )  C 
 b
n
t
n t
*
Cs 
 C
1  C
Langmuir Isotherm
 C
 1

Cs   C
 C  C
1

  






 1   C 1   C 2  t
 1   C 2  t
t
t 1   C




q
C
 
  (   D*  Dh  C )  C 
 b
n
t
n
 C

1


 1   C 2  t



C   b 
1
1 
  (  DC )  vC 
  0

t 
n  1   C 2  


D
v
C
  (  C )  C 
0
R
R
t
   

1
R  1  b 

Retardation factor

n  1   C 2  


Clemson Hydro
Governing Eq.
AD w/Equilibrium Sorption, Linear Isotherm
Comsol format
Cn
 bCs
*
  (  n  D  Dh  C )  qC 

t
t
Cn  C 
  (  n  D*  Dh  C )  qC 
t


 t  sorption
Cn
 C
  K d b ; expand terms on rhs
t
t
n
C
C

  (  n  D*  Dh  C )  qC  C  n
 bK d
 KdC b
t
t
t
t
 b  (1  n )  P ; from definition,  P is grain density
  (  n  D*  Dh  C )  qC 
n
C

n 
 
  n  b K d 
 KdC  P  n P  P 
t
t
t
t 
 t

n

C


n
  (  n  D*  Dh  C )  qC  C   n   b K d 
  K d C P 1  n   K d C  P
t
t
t
t

n



C


  (  n  D*  Dh  C )  qC  C 1  K d  P   K d C P 1  n     n   b K d 
0
t
t
t


n
C
  (  n  D*  Dh  C )  qC  C 1  K d  P    n   b K d 
 0; if grain density constant
t
t
C
  (  n  D*  Dh  C )  qC   n   b K d 
 0; if porosity constant
t
q
  K  C
  (   D*  Dh  C )  C   1  b d 
 0; divide through by porosity
n
n  t

D
q
C
  (  C ) 
C 
0
R
nR
t

 
R  1  b K d 
n


  (  n  D*  Dh  C )  qC  C
Cs  K d C
Linear Isotherm
Cs
C
 Kd
t
t
For Reference
Retardation factor
Clemson Hydro
Nonequilibrium (Kinetic) Sorption
Macroscopic, Two adjacent domains
Cf
Fluid concentration, Cf ,[mol/m3]
Solid surface concentration, Cs , [mol/m2]
kads: sorption rate constant [1/(m s)]
kdes: desorption rate constant [1/s]
Fluid
Cs
solid
Transport bulk fluid solid = Solid rxn rate
kads
Cf
kdes
Cs
Cf
dCs
 k absC f  k desCs
dt
J boundary 
dCs
dt
1st order sorption kinetics
reversible
Jboundary
Mass flux boundary condition
on fluid
Cs
Clemson Hydro
Nonequilibrium Sorption Kinetics
First order
kth order
Langmuir kinetic
Langmuir with competition
Elovich
Power
dCs 
 kads
dt 
dCs 
 kads
dt 
n
C  kdesCs
  f
n k
C  kdesCs
  f
dCs 
n
  kads  (Co  Cs )C f  kdesCs
dt 

n

dCs ,i 
n 
  kads ,i  Co   Cs , j  C nf  kdes ,iCs ,i
dt
 
j 1


dCs
 kads exp(  PC f )
dt
dCs 
n
  kads  C kf Csm
dt 

Cf: concentration of s1 in fluid [Ms1/L3f]; Cs concentration on solid [Ms1/Msolid]; kads: sorption
rate constant[units vary], kdes: desorption rate constant [1/T], : porosity [L3w/L3T], : bulk
density[Msolid/L3T], P constant [L3f/Ms1]
Limousin et al. (2007) http://www.sciencedirect.com/science/article/pii/S0883292706002629#
http://www.sciencedirect.com/science/article/pii/S0883292706002629#
Clemson Hydro
Non-equilibrium Sorption
Pore-scale
Cf
Specify rate of change of Cs
Fluid
First-order irreversible kinetics
Cs
dCs 
n 
  k ads  C f  k ads _ eC f
dt 
b 
solid
First-order reversible
Solid
dCs 
n 
  k ads  C f  k desCs
dt 
b 
dCs
 k ads _ eC f  k desCs
dt
Fluid
dC f
dt

 b dCs
n dt
Clemson Hydro
Important Concept, FluidSolid Surface
Porous media, Two overlapping domains
Dual Porosity, Dual Permeability
Two domains
(fractures, matrix)
(fluid, solid)
(liquid, gas)
Usually contrasting k
Mass transfer between domains
dCs
 k ads _ eC f  k desCs
dt
dC f
dt

 b dCs
n dt
Clemson Hydro
Example
First-order non-equilib sorption
Reversible and irreversible
Breakthrough curves
water
solid
Cwater
Non-reversible
Left behind on solid
Clemson Hydro
Governing Equation
Advection-Dispersion w/ surface reaction
Dual Porosity Approach, with concentrations in both domains
C An
  (  n  D  Dh  C A )    qC A 
  k1 (C A  CB )
t
*
CB n
  (  nD CB ) 
   k1 (C A  CB )
t
*
Advection
Diffusion only
 L2f 
  specific surface area  3 
 Lc 
Clemson Hydro
Clogging of a flow channel from precipitation on wall
Non-equilibrium sorption
Pipes clogged
with precipitate
Cementation of pore space
biofilm
Plaque clogging artery
Biofilm
Clemson Hydro
Biofilm
Conceptual Model
Growth/decay of biomass
•
•
•
•
•
uptake of nutrients, increase in thickness
decay, decrease in thickness
3D geometry on surface
interaction with flow
fluid sheardetachment
Mass transfer to biofilm
• transport through fluid
• mass transfer through stagnant water layer
• mass transfer within biofilm
Reactions within biofilm
• first-order, monod, growth/death other
• vary within biofilm
http://wyss.harvard.edu/viewmedia/133/bacterial-biofilmhttp://www.bti.umn.edu/bond/bond_lab___university_of_minnesota.html
1;jsessionid=6F46332D65A9586919824B047248B4E0.wyss2
Clemson Hydro
Clogging and Channeling
Fluid concentration, Cf ,[mol/m3]
Solid surface concentration, Cs , [mol/m2]
Solid concentration, Cs , [Ms/Mos]
Cf
Fluid
Macroscopic model
Cs
dCs M w Mol M L3c Lc
 2

dt 
TLc Mol M T
solid
𝐿
[ 𝑇𝑐 ]velocity of interface (moving mesh).
REV Model
dCs b
M s M os L3s
L3s

 3
3
dt  TM os Lc M s LcT
Use to calc rate of change in porosity
3
3
3
dn dLp d  Lc  Ls 
dL3s
dCs  b
 3 

1


1

dt Lc dt
L3c dt
L3c dt
dt 
3
 n   1  no 
k  ko   

 no   1  n 
2
Use porosity change to get k change
Clemson Hydro
Clogging of flow channel from precipitation
Non-equilibrium sorption
Cf: fluid concentration [mol/m3]
Cs: concentration on solid [mol/m2]
kads: sorption rate constant [m/s]
kerode: erosion rate constant[mol/m2]
tw: wall shear rate[1/s]
tw: critical wall shear stress for erosion[1/s]
w: thickness of layer along wall [m]
Mvol: Molar volume [m3/mol]
Reaction
Flux out of
fluid
Movement
of wall
Cf
Fluid
w
solid
dCs
 k adsC f  k shear (t w  t w ,threshold )
dt
J boundary  
Cf
dCs
dt
dw dCs

M vol
dt
dt
Cs
Jboundary
w
Cs
Clemson Hydro
Example
Geometry (mm)
No flow
0.001m/s
P=0
Fluid
No flow
No flux
Cf=1
Outflow
No diffusive flux
Transport
In water
Flux out = -rxn
Physics
• Laminar flow
Surface
Viscosity = f(C_m) reaction
• Transport, rxn
C_substrate
C_microbe population
non-reactive |
reactive
Clemson Hydro
Biofilm growth and clogging
https://vimeo.com/65554224
Baseline, fluid shear has
no effect
https://vimeo.com/65554293
Less sensitive to shear
https://vimeo.com/65554294
More sensitive to shear
Clemson Hydro
Clemson Hydro
Strategy
1.
2.
3.
4.
5.
Geometry, definitions, physics
Flow
Flow+transport
Flow+transport+surface rxn
Flow+transport+surface rxn+deformed mesh
Clemson Hydro
concentration
non-reactive |
reactive
Clemson Hydro
Clemson Hydro
http://ac.els-cdn.com/S0008622304002155/1-s2.0-S0008622304002155-main.pdf?_tid=2c787188-8be711e2-b17d-00000aab0f02&acdnat=1363183759_8cfe4ddd81d00a9db89666573888753f
Clemson Hydro
Clemson Hydro
Clemson Hydro