• 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
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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Conceptual Model
A+BC
Reaction Rate
• Intrinsic rate
• Transport-- combine reactants, remove products
Processes
• Reaction
• Advection
• Dispersion
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Coupled effects
Reaction Transport
• Reaction changes K , porous media flow
Cave in Bali
karst, diagenesis
• Reaction changes D, q, 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
Clemson Hydro
Coupled effects
Reaction changes K
A+BC
Precipitation-Dissolutionchange porosity
Couple through porosity, f
Kozeny-Carmen equation
Verma Pruess (1988)
Permeable reactor
Membrane
Karst
http://onlinelibrary.wiley.com/doi/10.1002/hyp.9492/pdf
3
 f   1  fo 
k  ko   

 fo   1  f 
 f  fc 
k  ko 

 fo  fc 
2
n
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
Governing
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 )
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
Clemson Hydro
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
Clemson Hydro
Important Scales
  (  D*  Dh  C )    qC 
C
R
t
 2C
C C
 D 2  qx

R
x
x t
xc = L
Cc = Cc
tc = L2/D
vL
D
RL
Damkohler I, DaI :
;
vCo
Peclet number, Pe:
RL2
Damkohler II, DaII :
DCo
Clemson Hydro
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
Clemson Hydro
Idealized Conceptual Models
1. Reaction during 1-D, steady flow
 
C
R
t
  qC 
 
Q
C
A
Q
Q dC
C 
A
A dx
C
0
t
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
kn
dC
  1 C ; C (0)=Co
dx
q
C  Cin e

k1nx
q
 Cin e

L
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

k1L x
v L
e

travel time x
reaction time L
 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
Clemson Hydro
Non-ideal factors
Preferred flowpaths
Incomplete contact of reactants, affect k1
Reactions alter surface
Precipitates, depletion, longer diffusion time, slower reaction rate
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
  qC 
Q
C
A
x
Q
Q C Q (C  Cin )
C 

A
A x V
R  k1C
 
Q  C  Cin  dC

  k1C
V
dt
Clemson Hydro
Reaction in a mixed region
Q  C  Cin  dC

  k1C
V
dt
No reaction, k1=0, conservative tracer
Q
Q
 t 
 t

V
V
C  Cin 1  e   Co e  Cin 1  e  t*   Co e  t*


k1 > 0, reactive compound
Q kV 
Q kV 

 1 1  t 
 1 1  t
1
 1  e V  Q    Co e V  Q 
C  Cin

 k1V  


1



Q


C  Cin


1
 1 Da t *
 1 Da t *
1  e  1   Co e  1 
1  Da1 
Q
1
1


;
V
tr
mean residence time
k1 
*
t =
t
tr
1
reaction time
Vk1
residence time

 Damkohler number Da I
Q
reaction time
Clemson Hydro
Residence Time Distribution
Pulse tracer test
M  Cout (t1 )Qt
increment of mass out over t ,
or mass that has resided for time, t1
M Cout (t1 )Q

t
MT
MT
normalized
Cin
M
 E (t1 )t
MT
E (t ) 
Cout (t )Q
MT
t
Residence Time Distribution

M T   Cout (t )Qdt
Total mass
0
E (t ) 

Cout (t )Q
C
out
Area = 1
(t )Qdt
Cout
0
if Q is constant
E (t ) 

Cout (t )
C
out
0
(t )dt
t
t1
0
b
F (t )   E (t )dt
a
Cumulative RTD
E
A=1
0
t
t1
Clemson Hydro
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
t
F (t )   E ( )d
Cumulative RTD
0
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
Dead zones
How to use moments to diagnose?
Bypass = local fast paths
Dead zone = local slow paths, diffusion
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Diagnostics
Pulse input into CSTR
Fast path with dead zone
E
Some flow bypasses reactor
does not mix
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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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