• Project
• Describing Methods
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Reactive Transport
Silver dichromate forming Leisegang rings in a test
tube experiment
http://www.insilico.hu/liesegang/index.html
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Reactive Transport
Conceptual
A+BC
A
A
C
B
B
Reactions require transport
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Conceptual Model
A+BC
Reaction Rate
• Intrinsic rate
• Transport-- combine reactants, remove products
• Dispersion
• Diffusion
Scale
Processes
• Advection
Time scale for reaction, tr
Time scale for transport, tt
Damkohler Numbers: tt/tr
DaI: tadvect/treact
DaII: tdisperse/treact
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Reaction Locations and Mixing
Bulk Material
Fluid— Multi-scale mixing, dispersion, length scale
Solid– Diffusion dominant
Interfaces
Fluid-Solid
Liquid-Gas
No flow at interfacediffusion
Mass transfer over stagnant layer
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Important Scales
Time scales
advection: tcf 
L
v
L2
dispersion: tcd 
D
C
reaction:
tcr  o
R
tcd L2 v vL
Peclet number, Pe (advection/diffusion rate):


tca D L D
L
t
RL
Damkohler I, DaI (reaction/advection rate): ca  v 
;
C
tcr
vCo
o
R
L2
tcd
RL2
D
Damkohler II, DaII (reaction/diffusion rate):


tcr Co DCo
R
1st order, k1  R / Co
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Coupled effects
Reaction Transport
• Reaction changes K , porous media flow
Cave in Bali
karst, diagenesis
• Reaction changes D, subsequent reaction rate
Biofouling, reactor performance
• Heat affects reaction rate
Biofouling in pipe
Geothermal, remediation
• Other chemicals, competing/synergistic reaction
Bioprocesses, waste water treatment
• Precipitates affect density, flow
Flocculation, mixtures
• Stress affects reaction, reduces K, flow
Geyser in Yellowstone
Diagenesis, sintering
Styolites in limestone
http://www.seatrekbali.com/seasons-greetings-seatrek/
http://www.merusonline.com/biology
http://onlinelibrary.wiley.com/doi/10.1002/hyp.9492/pdf
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Coupled effects
Reaction changes K
A+BC
Precipitation-Dissolution-change porosity
Couple through porosity, f
 
Kozeny-Carmen equation k  k  ff   11ff 
Verma Pruess (1988) k  k  f  f 


f f 
Permeable reactors
Karst
3
2
o
o
o
n
c
o
o
http://onlinelibrary.wiley.com/doi/10.1002/hyp.9492/pdf
c
Clemson Hydro
Governing Equation
Advection-Dispersion-Reaction
c
S
t
= D +A
Ms
c 3
Lc
 
c=C
Storage
c C

t t
Advective Flux
A = qC
Diffusive Flux (Fick’s Law)
D =- D * C
Dispersive Flux
Dh = - DhC
Source
S=R
C
   ( DC )     qC   R
t
homogeneous, flux divergence free
Governing
C
  (  D  Dh  C )    qC 
R
t
*
C
 D 2C  q C  R
t
Clemson Hydro
Simulation of Advection-Dispersion-Reaction
C
 D 2C  q  C  R
t
C  N1
n J  0
n  J  N2
n  qC  0
n  qC  N 3
n  J   k (C  N 4 )
Governing
Boundary
Dirichlet, Specify Conc
Neuman, Specify diffusive flux
Specify advective flux
Cauchy, flux proportional to
gradient
Initial Conditions
C(x,y,z,0) = Ci
Parameters
D:
hydrodynamic dispersion
R:
reaction ratekinetics
m, r: fluid properties
k:
permeability
q, P
T

Ci
Flow, pressure
Temperature
Stress
other conc
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Idealized Conceptual Models
Case 1. 1-D flow, Steady
Steady state, C changes with x, not t
Plug flow reactor
Along streamtube/flowpath
Reactive wall
Case 2. Thoroughly mixed, transient
Transient, C changes with t, not x
Tank reactor, CSTR
Pore, Pond, Lake, Atmosphere
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Permeable
reactive barrier
Permeable material that sorbs or breaks down
contaminants on contact.
• Metallic iron reduce chlorinated solvents
• Limestone, phosphate precipitate metals
• Activated carbon, zeolites sorb
contaminants
• Compost, mulch, sawdustbiodegradation
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Idealized Conceptual Models
1. Reaction during 1-D, steady flow
 
C
R
t
  qC 
Q
C
A
Q
Q dC
    C 
A
A dx
dC
0
dt
R   k1Cn
3
1 M s Lf
T L3f L3c
 
C
R
t
Q dC
  k1Cn
A dx
Clemson Hydro
Idealized Conceptual Models
Case 1. Reaction during 1-D flow
C  Cin e

k1nx
q
 Cin e

k1 x
v
Time scale for reaction
1
k1
Time scale for advection (travel time):
L Ln

v
q

C
e
Cin
k1 x
v
e

k1nL x
q L
e

travel time x
reaction time L
dC
Ak1n

C ; C (0)=Co
dx
Q
 e  DaI x '
Clemson Hydro
More general case
other reactions, non-uniform v
First-order Decay Chain
x
x
1
t    dx take integral along flowpath
v 0v
Distance along flowpath
Temporal changes map out to spatial zones
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Non-ideal factors
Preferred flowpaths
Incomplete contact of reactants, affect k1
Non-ideal interface,
Water, precipitates, longer diffusion time
Storage along tube without reaction
Matrix diffusion
Reactions alter flow
Precipitation, dissolution, biofilm
Result:
q, k1, D change with time/space
Clemson Hydro
Idealized Conceptual Model
Case 2. Reaction in mixed region
CSTR, Lake, Ocean, pore
A
Q
  qC  C
A
Q
Q C Q (C  Cin )
    C 

A
A x V
R   k1C
x
Clemson Hydro
Reaction in a mixed region
Q kV 
Q kV 

 1 1  t 
 1 1  t
1
 1  e V  Q    Co e V  Q 
C  Cin

 k1V  


1  Q 


Q
1
1


V
tr
residence time
k1 
1
reaction time
Vk1
residence time

 Damkohler number Da I
Q
reaction time
No reaction, k1=0, conservative tracer
Q
Q
 t
 t

C  Cin 1  e V   Coe V


Clemson Hydro
Residence Time Distribution
Pulse tracer test
M  Cout (t )Qt
increment of mass out
M Cout (t )Q

t
MT
MT
normalized
M
 E ( t ) t
MT
E (t ) 
Cout (t )Q
MT
Cin
define RTD
t
Residence Time Distribution

M T   Cout (t )Qdt
Cumulative mass out
0
E (t ) 

Cout (t )Q
C
out
Area = 1
Cout
(t )Qdt
t
0
if Q is constant
E (t ) 

Cout (t )
C
out
E
(t )dt
0
b
F (t )   E (t )dt
a
Cumulative RTD
A=1
0 a
t
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Residence Time Distribution
Step tracer test
Co
F (t ) 
C (t )
Cumulative RTD
Co
Cin
t
Co
Cout
t
1.0
F
t
Clemson Hydro
Residence Time Distribution
b
F (t )   E (t )dt
Cumulative RTD
a
E (t ) =
dF (t )
dt
RTD
E
tm
Moments of RTD
mean residence time (first moment)
t
1.0
F

tm   tE (t )dt
0
Variance of the residence time (second moment)
t

    t  tm  E (t )dt
2
2
0
Skewness of the residence time (third moment)
s 
1
3


t  tm 
3/2  
3
E (t )dt
0
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Diagnostics
Plug Flow Column
Ideal
E
Bypass
Low K zones
How to use moments to diagnose?
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Diagnostics
Pulse input into CSTR
Dead zone
E
Bypass
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Residence time and Reaction
•
•
•
•
•
Residence time = time molecule in reactor
First-order rxn only depends on residence time
Other rxn also depend on mixing
Macro-mixingflow paths
Micro-mixingmechanical dispersion, diffusion
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