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Transport in Permeable Media
TPM
Leo Pel, Henk Huinink, David Smeulders, Thomas Arends, Hans van Duijn
Faculty of Applied Physics Mechanical Engineering
Eindhoven University of Technology The Netherlands
5 ECTS 2018
Examination : Oral
Transport in porous media 3MT130
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Transport in Permeable Media
TPM Surface tensions
Curved surface
Pressure difference
Unsaturated
Saturated
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TPM
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TPM
Surface tensions
Curved surface
Pressure difference
wnc rp γ2
−=
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Transport in Permeable Media
TPM
trxµγ2
=rg
hργ2
max =
Small pores
• slow absorption
• but very high
Large pores
• fast absorption
• but low
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Transport in Permeable Media
TPM
Liquid ‘fast’ Vapour ‘slow’
Same macroscopic pressure: suction
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TPM
( )z
KDt ∂
∂+∇∇=
∂∂ )()( θθθθ
Richards equation
First order in time and second order in space; require 1. initial condition and 2. boundary conditions Outcome: θ as function x and t
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Transport in Permeable Media
TPM
Some Human Activities that Can Contaminate Groundwater
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Transport in Permeable Media
TPM
Radioactive contaminants
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TPM
Movie Eric Doehne www.getty.edu/conservation/science
Madame John’s Legacy 1788
Cultural Heritage
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Transport in Permeable Media
TPM
Movie Eric Doehne www.getty.edu/conservation/science
Madame John’s Legacy 1788
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Transport in Permeable Media
TPM
Debris from salt weathering (6 months)
Movie Eric Doehne www.getty.edu/conservation/science
Madame John’s Legacy 1788
New Orleans
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Transport in Permeable Media
TPM
Transport of components saturated non-saturated
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Transport in Permeable Media
TPM
The laws of Fick First law:
Second law:
cDtc 2∇=∂∂
Concentration peaks are chopped
02 <∇ c
Concentration valleys are filled up
02 >∇ c
1831-1879
1th : Diffusion
∂∂
∂∂
=∂∂
=∂∂
xCD
xxq
tC x
cTDtc 2∇=∂∂
Porous medium
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TPM
cDtc 2∇=∂∂
cTDtc 2∇=∂∂
Porous medium
Path gets longer: tortuosity
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Transport in Permeable Media
TPM
cDtc 2∇=∂∂
cTDtc 2∇=∂∂
Porous medium
Measure the diffusivity by NMR
Observation time:
Time Length 1 10-6 31 nm 1 10-3 1 µm
1 30 µm
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Transport in Permeable Media
TPM
R. W. Mair et all, Phys Rev Let 1999
Example of diffusion measurement by NMR
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TPM
t∂∂θ
qin qout 0. =∇+∂∂ q
tθ
∂∂
∂∂
=∂∂
=∂∂
xCD
xxq
tC
porx
There is more: Ad/desorption on pore wall Cs(c)
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TPM
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TPM
t∂∂θ
qin qout 0. =∇+∂∂ q
tθ
tS
xCD
xxq
tC
porx
∂∂
+
∂∂
∂∂
=∂∂
=∂∂
Sink term due to binding
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Transport in Permeable Media
TPM
Consider changes in the mass of solute by adsorbing onto the solid soil matrix, given by ρbs, where ρb is the soil bulk density and s is the adsorbed concentration in terms of mass of solute per mass of soil
( )
∂∂
∂∂
=∂+∂
xcD
xtsc
effbρ
Binding
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Transport in Permeable Media
TPM
( )
∂∂
∂∂
=∂+∂
xcD
xtsc
effbρ
Binding
( )
∂∂
∂∂
=∂
∂xcD
xtRc
eff csR bρ+=1With
Retardation factor
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TPM
( )
∂∂
∂∂
=∂+∂
xcD
xtsc
effbρ
( )
∂∂
∂∂
=∂
∂xcD
xtRc
eff csR bρ+=1With
Simplest case Kcsb =ρ
∂∂
+∂∂
=∂∂
xc
KD
xtc eff
1
KR +=1
Diffusion gets ‘slower’
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Transport in Permeable Media
TPM Other mechanism?
Water flow : advection
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TPM
ADVECTION • Chemical transport due to bulk movement of the fluid. • The fastest form of chemical transport in porous
media. • Concentration decreases in the direction of fluid
movement.
xCU
tC
∂∂
−=∂∂CUq −=
Darcy law liquid
ww JUthatNoteJU >→=θ
Darcy law liquid
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TPM
−
∂∂
∂∂
=∂∂ CU
xCD
xtC
effθθ
−
∂∂
∂∂
=∂∂ CU
xCD
xtC
eff
Saturated porous medium
Non-saturated porous medium: Ion transport only in the liquid
Transport can only be liquid of component
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TPM
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TPM
29
Reasons for Spreading: mechanical dispersions
Some solute mass travels faster than average, while some solute mass travels slower than average
Completely dependent on flow
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Transport in Permeable Media
TPM
Completely dependent on flow
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TPM
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Transport in Permeable Media
TPM Dispersion
Mechanical dispersion - caused by motion of the fluid
Longitudinal dispersion – along the streamline
Transverse dispersion – perpendicular to flow path
Flow Direction
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TPM
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Transport in Permeable Media
TPM Experiment
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TPM
Advection, Diffusion, Dispersion
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TPM
−
∂∂
∂∂
=∂∂ CU
xCD
xtC
effθθ
−
∂∂
∂∂
=∂∂ CU
xCD
xtC
eff
Saturated porous medium
Non-saturated porous medium: Ion transport only in the liquid
Deff= diffusion + tortuosity + dispersion
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Transport in Permeable Media
TPM
NaCl
Wind
damage
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TPM
Damage to rising damp in city of Venice
2004 2007
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TPM
How are ions moving?
Characterize the transport?
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Transport in Permeable Media
TPM drying
surface
airflow
Wick action conceptual model
supply
Drying front at surface (drying externally limited)
= Liquid velocity is function of drying rate
= position drying front
moisture flow
u=constant
See also sharp front model C. Hall
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Transport in Permeable Media
TPM drying
surface
airflow
Wick action conceptual model
supply
moisture flow
advection
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Transport in Permeable Media
TPM drying
surface
airflow
Wick action conceptual model
supply
moisture flow
accumulation > crystallization
For NaCl max concentration = 6M
advection
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Transport in Permeable Media
TPM drying
surface
airflow
Wick action conceptual model
supply
moisture flow
advection
(neglect adsorption)
diffusion
accumulation leveling off
competition
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TPM
Peclet number
DLUPe =
liquid velocity
length of the sample
diffusion coefficient of Na in porous medium
:U:L:D
competition advection diffusion
1>Pe1<Pe
accumulation uniform distribution
see e.g.H.P. Huinink et al, Phys. Fluids 14, 1389 (2002)
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TPM
effC cD uCt x x
∂ ∂ ∂ = − ∂ ∂ ∂
Diffusion Advection + = flux
Initial profiles
Advection diffusion equation for transport
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TPM
effC cD uCt x x
∂ ∂ ∂ = − ∂ ∂ ∂
Initial profiles
airflow supply
moisture flow
advection
q=0 Ions can not leave
q= uCo continuous supply
Boundary conditions
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TPM
effC cD uCt x x
∂ ∂ ∂ = − ∂ ∂ ∂ Diffusion Advection
Boundary conditions: Top : q=0 Bottom : q= uCo
+ = flux
Simple solution:
Initial profiles
• Exponential decay
• Width peak =4D/U (e-4~0)
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TPM
effC cD uCt x x
∂ ∂ ∂ = − ∂ ∂ ∂ Diffusion Advection
Boundary conditions: Top : q=0 Bottom : q= uCo
After reaching the solubility limit> crystallization
C*=6 for NaCl
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TPM
Na lower sensitivity longer
measurement time
Signal proportional
to
moisture content
or
Na content
Pulsed NMR signal (spin-echo experiment)
Information on
water and ion
in pores Amplitude spin-echo S~Gρ [1-exp(-TR/T1)] exp(-TE/T2) G = relative sensitivity (for 1H G=1, 23Na=0.1) ρ = density of nuclei
T1 = spin lattice relaxation
TR = repetition time experiment
T2 = spin-spin relaxation time
TE = spin-echo time
see,e.g.,E.L. Hahn,Phys. Rev., 80, 580-594 (1950)
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Transport in Permeable Media
TPM Experimental setup
pump
NMR measurement
Measurements
- NMR moisture profile
- NMR Na profile
(only free ions: no crystals)
1m NaCl reservoir
electrical level control NaCl
Step motor
0% RH air flow
epoxy coating
evaporation shield top
bottom
100
mm
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TPM
( )0
,1
2 .eff
C x t xerfC D t
= −
C CDt x x
∂ ∂ ∂ = ∂ ∂ ∂
9 20.8 10 /D m s−= ×
0 1 2 3 4 5 6 7 8
x 10-5
0
0.5
1
1.5
2
2.5
3
3.5
4
x/sqrt(t) [m s-0.5]
Na
conc
entra
tion
[m]
No airflow > only diffusion
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TPM Results with airflow
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TPM Results
No 6M ????
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TPM
Stone sample saturated with 1 M NaCL solution
position
Con
cent
ratio
n 1
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TPM
Stone sample saturated with 1 M NaCL solution
position
Con
cent
ratio
n
1
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TPM
Stone sample saturated with 1 M NaCL solution
position
Con
cent
ratio
n
1
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TPM
Stone sample saturated with 1 M NaCL solution
position
Con
cent
ratio
n
1
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TPM
Stone sample saturated with 1 M NaCL solution
position
Con
cent
ratio
n
1
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TPM
Limestone sample saturated with 1 M NaCl solution
position
Con
cent
ratio
n
1
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TPM
Stone sample saturated with 1 M NaCL solution
position
Con
cent
ratio
n
1
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TPM
Stone sample saturated with 1 M NaCL solution
position
Con
cent
ratio
n
1
resolution
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Transport in Permeable Media
TPM Results
1D resolution: average over slice
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TPM Results: model fit data
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TPM Results: model fit data
Max concentration = 6 m
Decay width ~ 80 mm
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TPM Integral of concentration
No crystallization
crystallization at 6 m
linear increase ucot
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TPM
Movie Eric Doehne www.getty.edu/conservation/science
Madame John’s Legacy 1788
How to clear a wall (painting) of the salt?
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Transport in Permeable Media
TPM Conservators: poulticing
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TPM
General idea of poulticing
poultice substrate
transport Water absorption
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TPM
poultice substrate What are the mechanisms ???? - time scales?
- efficiency?
- poresize dependence?
General idea of poulticing
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Transport in Permeable Media
TPM Working principle of poulticing
Diffusion
Diffusion of ink in glass of water
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TPM
General idea of poulticing by diffusion
poultice substrate
diffusion Water absorption
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TPM
General idea of poulticing by diffusion
poultice substrate
diffusion
TO KEEP THE DIFFUSION GOING
(maintain sink)
RENEW POULTICE VERY OFTEN
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TPM
General idea of poulticing by diffusion
poultice substrate
diffusion
TO KEEP THE DIFFUSION GOING
(maintain sink)
RENEW POULTICE VERY OFTEN
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Transport in Permeable Media
TPM
2
2
xD
tC
∂∂
=∂∂ C
Desalination by diffusion process:
D (m2s-1) water Bentheimer fired clay brick
NaCl 1.1 10-9 0.4 10-9 0.8 10-9
Na2SO4 1.1 10-9 0.4 10-9 0.85 10-9
In the order of 1 10-9 m2s-1
TIME SCALE?
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TPM
Poulticing side
(where salt comes out)
Time in days
Salt concentration
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TPM
Diffusion Pros • If enough time, can have 100 % efficiency • No pore size dependency
Cons • Slow ( 80% in 10 days for first 25 mm) • Renew poultice very often • Sample wet (long time, bio degradation)) • At end, dry sample (salt damage)
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Transport in Permeable Media
TPM
Equations of moisture and ion transport
∂∂
∂∂
+
∂∂
∂∂
=∂∂
xCD
xxD
xt cθθ
θ
−
∂∂
∂∂
=∂∂ CU
xCD
xtC
effθθ
moisture
salt
So two couple non-linear partial differential equations
+ boundary conditions
We do not learn anything !!!!
The competition
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Transport in Permeable Media
TPM
drying
surface
airflow
How do we get salt effloresence???
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TPM
drying
surface
airflow
moisture flow
Saline drying conceptual model
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TPM
Saline drying conceptual model drying
surface
advection airflow
moisture flow
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TPM
Saline drying conceptual model drying
surface
airflow
moisture flow
accumulation > crystallization
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TPM
Saline drying conceptual model drying
surface
advection
(neglect adsorption)
airflow
moisture flow
accumulation
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TPM
Saline drying conceptual model drying
surface
advection
(neglect adsorption)
airflow
moisture flow
diffusion
accumulation leveling off
competition
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TPM
Peclet number
DLUPe =
liquid velocity
length of the sample
diffusion coefficient of Na in the pores
:U:L:D
competition advection diffusion
1>Pe1<Pe
accumulation uniform distribution
see e.g.H.P. Huinink et al, Phys. Fluids 14, 1389 (2002)
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TPM
Drying experiment
The initially saturated sample is sealed at all sides, except for the top
The sample is moved by step motor
The one-dimensional resolution ~ 1 mm
The measurement of a profile takes ~ 3 hours
Webcam for visual inspection
NMR only free Na ions are measured: no crystals
Pel et al, Applied Physics Letters 81, 2893-2895 (2002)
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TPM
NaCl
Na concentration
profiles at various
drying times
drying surface
0 days
Pel et al, Applied Physics Letters 81, 2893-2895 (2002)
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TPM
drying surface
0 days1
Pe~3
NaCl
Na concentration
profiles at various
drying times
Pel et al, Applied Physics Letters 81, 2893-2895 (2002)
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TPM
drying surface
NaCl max concentration 6 M
crystallization
0 days13
NaCl
Na concentration
profiles at various
drying times
Pel et al, Applied Physics Letters 81, 2893-2895 (2002)
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drying surface
0 days13
6
Pe~0.7
NaCl
Na concentration
profiles at various
drying times
Pel et al, Applied Physics Letters 81, 2893-2895 (2002)
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drying surface
0 days13
6
9
NaCl
Na concentration
profiles at various
drying times
Pel et al, Applied Physics Letters 81, 2893-2895 (2002)
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drying surface
0 days13
6
9
12,15
NaCl
Na concentration
profiles at various
drying times
Pel et al, Applied Physics Letters 81, 2893-2895 (2002)
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0 days
13
6
9
12,15
Pe>1
Pe<1
accumulation
leveling off
Pe number is simple indication
Pel et al, Applied Physics Letters 81, 2893-2895 (2002)
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0 5 10 150.0
0.2
0.4
0.6
0.8
1.0
0
1
2
3
4
5
SavgCavg
Savg
Pe < 1Pe > 1
S
avgC
avg (
mol
l-1)
S avg (-
)
time (days)
Savg is the average (water) saturation of the sample (drying curve)
Savg Cavg represents the total amount of dissolved NaCl
I II III
I: Pe ~ 3 accumulation
II: Pe ~ 0.7 leveling off
III: homogeneous at 6 M
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Transport in Permeable Media
TPM
poultice substrate
advection
General idea of poulticing by advection
Water absorption airflow
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Transport in Permeable Media
TPM
poultice substrate
General idea of poulticing by advection
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Transport in Permeable Media
TPM
Demands on poultice Step 1) Water is absorbed from poulice into substrate
Step 2) Reverse of water flow, i.e., from substrate into poultice
poultice substrate
Absorption
poultice substrate
Advection
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Transport in Permeable Media
TPM
Maximum height > capillary pressure
wnc rp γ2
=
Capillary pressure
Conclusions
1) A porous material will absorb water
2) Small porous will absorb water from larger pores
=
Water wants to stay
in small pores
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Transport in Permeable Media
TPM Demands on poultice
Step 1) Water is absorbed from poulice into substrate
poultice substrate
Absorption
Poulice : Reservoir pores larger than largest pores in substrate
reservoir pores
substrate substrate poultice
pore size
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Transport in Permeable Media
TPM
drying
surface
airflow
Widest pores first rPc
φγ cos2≈
Capillary pressure
Desalination phase
PORES POULTICE SMALLER THAN SUBSTRATE
poultice substrate
Advection
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Transport in Permeable Media
TPM
Calcium-silicate brick
r ∼ 12 nm
Bentheimer sandstone r ∼ 30 µm
Plaster (lime:cement:sand = 4:1:10 (v/v)) r ∼ 0.5 µm
rcalcium-silicate< rplaster< rBentheimer
Ph.D thesis J. Petković TU-Eindhoven (2005)
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Transport in Permeable Media
TPM
0 50 100 150 2000
500
1000
1500
V (m
m3 )
t (h)
Plaster Bentheimer sandstone
0 10 20 30 40 500.0
0.1
0.2
0.3a
150 h
Plaster Bentheimer sandstone
10 h
25 h
75 h
25 h0 h
0 h
θ (m
3 /m3 )
x (mm)
rplaster< rBentheimer
moisture
Ph.D thesis J. Petković TU-Eindhoven (2005)
drying surface
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Transport in Permeable Media
TPM
0 10 20 30 40 500.0
0.1
0.2
0.3a
150 h
Plaster Bentheimer sandstone
10 h
25 h
75 h
25 h0 h
0 h
θ (m
3 /m3 )
x (mm)0 50 100 150 200
0
500
1000
1500
V (m
m3 )
t (h)
Plaster Bentheimer sandstone
rplaster< rBentheimer
moisture
Ph.D thesis J. Petković TU-Eindhoven (2005)
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Transport in Permeable Media
TPM
0 10 20 30 40 500.0
0.1
0.2
0.3a
150 h
Plaster Bentheimer sandstone
10 h
25 h
75 h
25 h0 h
0 h
θ (m
3 /m3 )
x (mm)0 50 100 150 200
0
500
1000
1500
V (m
m3 )
t (h)
Plaster Bentheimer sandstone
rplaster< rBentheimer
moisture
Ph.D thesis J. Petković TU-Eindhoven (2005)
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Transport in Permeable Media
TPM
0 10 20 30 40 500.0
0.1
0.2
0.3a
150 h
Plaster Bentheimer sandstone
10 h
25 h
75 h
25 h0 h
0 h
θ (m
3 /m3 )
x (mm)0 50 100 150 200
0
500
1000
1500
V (m
m3 )
t (h)
Plaster Bentheimer sandstone
rplaster< rBentheimer
moisture
Ph.D thesis J. Petković TU-Eindhoven (2005)
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Transport in Permeable Media
TPM
0 50 100 150 2000
500
1000
1500
V (m
m3 )
t (h)
Plaster Bentheimer sandstone
0 10 20 30 40 500.0
0.1
0.2
0.3a
150 h
Plaster Bentheimer sandstone
10 h
25 h
75 h
25 h0 h
0 h
θ (m
3 /m3 )
x (mm)
rplaster< rBentheimer
moisture
Ph.D thesis J. Petković TU-Eindhoven (2005)
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Transport in Permeable Media
TPM
0 10 20 30 40 500.0
0.2
0.4
0.6
0.8a Plaster Bentheimer sandstone
25 h
10 h
0 h
150 h
75 h
0 h
10 h
25 h
Na c
onte
nt x
103 (m
ol/m
3 )
x (mm)0 50 100 150
0
1000
2000
3000
4000
Na c
onte
nt (m
mol
)t (h)
Plaster Bentheimer sandstone
rplaster< rBentheimer
Na-content
Ph.D thesis J. Petković TU-Eindhoven (2005)
drying surface
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Transport in Permeable Media
TPM
0 10 20 30 40 500.0
0.2
0.4
0.6
0.8a Plaster Bentheimer sandstone
25 h
10 h
0 h
150 h
75 h
0 h
10 h
25 h
Na c
onte
nt x
103 (m
ol/m
3 )
x (mm)0 50 100 150
0
1000
2000
3000
4000
Na c
onte
nt (m
mol
)t (h)
Plaster Bentheimer sandstone
rplaster< rBentheimer
Na-content
Ph.D thesis J. Petković TU-Eindhoven (2005)
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Transport in Permeable Media
TPM
0 10 20 30 40 500.0
0.2
0.4
0.6
0.8a Plaster Bentheimer sandstone
25 h
10 h
0 h
150 h
75 h
0 h
10 h
25 h
Na c
onte
nt x
103 (m
ol/m
3 )
x (mm)0 50 100 150
0
1000
2000
3000
4000
Na c
onte
nt (m
mol
)t (h)
Plaster Bentheimer sandstone
rplaster< rBentheimer
Na-content
Ph.D thesis J. Petković TU-Eindhoven (2005)
Start of crystallization
at surface
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Transport in Permeable Media
TPM
0 10 20 30 40 500.0
0.2
0.4
0.6
0.8a Plaster Bentheimer sandstone
25 h
10 h
0 h
150 h
75 h
0 h
10 h
25 h
Na c
onte
nt x
103 (m
ol/m
3 )
x (mm)0 50 100 150
0
1000
2000
3000
4000
Na c
onte
nt (m
mol
)t (h)
Plaster Bentheimer sandstone
rplaster< rBentheimer
Na-content
Ph.D thesis J. Petković TU-Eindhoven (2005)
Efficiency high
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Transport in Permeable Media
TPM
0 10 20 30 40 500.0
0.2
0.4
0.6
0.8a Plaster Bentheimer sandstone
25 h
10 h
0 h
150 h
75 h
0 h
10 h
25 h
Na c
onte
nt x
103 (m
ol/m
3 )
x (mm)0 50 100 150
0
1000
2000
3000
4000
Na c
onte
nt (m
mol
)t (h)
Plaster Bentheimer sandstone
rplaster< rBentheimer
Na-content
Ph.D thesis J. Petković TU-Eindhoven (2005)
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Transport in Permeable Media
TPM
DLUPe =
Peclet number
Water velocity ???
Mass conservation
0. =∇+∂∂ q
tθ 0)( =∇+
∂∂ U
tθθ
``)()(
1)( dxxtx
xUl
x∫∂
∂= θθ
From measured moisture profiles
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Transport in Permeable Media
TPM
0 10 20 30 40 500
2
4
6
8
10
12
|U| L
x 1
0-9 (
m2 /s)
D = 1 x 10-9 (m2/s)
t (h) 0 10 25 75
Bentheimer sandstoneplaster
75 h
25 h
10 h
0 h
x (mm)
rplaster< rBentheimer
Peclet number as function position
Ph.D thesis J. Petković TU-Eindhoven (2005)
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Transport in Permeable Media
TPM
0 10 20 30 40 500.0
0.1
0.2
0.3a
120 h
Plaster Calcium-silicate brick
120 h
60 h
30 h
60 h
30 h
0 h0 h
θ (m
3 /m3 )
x (mm)0 50 100 150 200
0
500
1000
1500
V (m
m3 )
t (h)
Plaster Calcium-silicate brick
moisture
rcalcium-silicate< rplaster
Ph.D thesis J. Petković TU-Eindhoven (2005)
drying surface
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Transport in Permeable Media
TPM
0 10 20 30 40 500.0
0.1
0.2
0.3a
120 h
Plaster Calcium-silicate brick
120 h
60 h
30 h
60 h
30 h
0 h0 h
θ (m
3 /m3 )
x (mm)0 50 100 150 200
0
500
1000
1500
V (m
m3 )
t (h)
Plaster Calcium-silicate brick
moisture
rcalcium-silicate< rplaster
Ph.D thesis J. Petković TU-Eindhoven (2005)
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Transport in Permeable Media
TPM
0 10 20 30 40 500.0
0.1
0.2
0.3a
120 h
Plaster Calcium-silicate brick
120 h
60 h
30 h
60 h
30 h
0 h0 h
θ (m
3 /m3 )
x (mm)0 50 100 150 200
0
500
1000
1500
V (m
m3 )
t (h)
Plaster Calcium-silicate brick
moisture
rcalcium-silicate< rplaster
Ph.D thesis J. Petković TU-Eindhoven (2005)
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Transport in Permeable Media
TPM
0 10 20 30 40 500.0
0.1
0.2
0.3a
120 h
Plaster Calcium-silicate brick
120 h
60 h
30 h
60 h
30 h
0 h0 h
θ (m
3 /m3 )
x (mm)0 50 100 150 200
0
500
1000
1500
V (m
m3 )
t (h)
Plaster Calcium-silicate brick
moisture
rcalcium-silicate< rplaster
Ph.D thesis J. Petković TU-Eindhoven (2005)
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Transport in Permeable Media
TPM
0 10 20 30 40 500.0
0.2
0.4
0.6a Plaster Calcium-silicate brick
120 h
30 h
60 h
0 h
60 h
30 h
0 h
Na c
onte
nt x
103 (m
ol/m
3 )
x (mm)0 50 100 150 200
0
1000
2000
3000
Na c
onte
nt (µ
mol
)t (h)
Plaster Calcium-silicate brick
Na-content
rcalcium-silicate< rplaster
Ph.D thesis J. Petković TU-Eindhoven (2005)
drying surface
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Transport in Permeable Media
TPM
0 10 20 30 40 500.0
0.2
0.4
0.6a Plaster Calcium-silicate brick
120 h
30 h
60 h
0 h
60 h
30 h
0 h
Na c
onte
nt x
103 (m
ol/m
3 )
x (mm)0 50 100 150 200
0
1000
2000
3000
Na c
onte
nt (µ
mol
)t (h)
Plaster Calcium-silicate brick
Na-content
rcalcium-silicate< rplaster
Ph.D thesis J. Petković TU-Eindhoven (2005)
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Transport in Permeable Media
TPM
0 10 20 30 40 500.0
0.2
0.4
0.6a Plaster Calcium-silicate brick
120 h
30 h
60 h
0 h
60 h
30 h
0 h
Na c
onte
nt x
103 (m
ol/m
3 )
x (mm)0 50 100 150 200
0
1000
2000
3000
Na c
onte
nt (µ
mol
)t (h)
Plaster Calcium-silicate brick
Na-content
rcalcium-silicate< rplaster
Ph.D thesis J. Petković TU-Eindhoven (2005)
Backward flow of salt from plaster into brick
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Transport in Permeable Media
TPM
0 10 20 30 40 500
2
4
6
|U| L
x 1
0-9 (
m2 /s
)
D = 1 x 10-9 (m2/s)
t (h) 0 10 20 30 60
calcium-silicate brickplaster
x (mm)
rcalcium-silicate< rplaster Ph.D thesis J. Petković TU-Eindhoven (2005)
Peclet number as function position
leveling off
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Transport in Permeable Media
TPM
0 10 20 30 40 500.0
0.2
0.4
0.6a Plaster Calcium-silicate brick
120 h
30 h
60 h
0 h
60 h
30 h
0 h
Na c
onte
nt x
103 (m
ol/m
3 )
x (mm)0 50 100 150 200
0
1000
2000
3000
Na c
onte
nt (µ
mol
)t (h)
Plaster Calcium-silicate brick
Na-content
rcalcium-silicate< rplaster
Ph.D thesis J. Petković TU-Eindhoven (2005)
Efficiency low
salt remains in substrate
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Transport in Permeable Media
TPM
ion chromatography
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Transport in Permeable Media
TPM
ion chromatography
MAX DESALINATION
IF
PORES POULTICE SMALLER THEN SUBSTRATE
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Transport in Permeable Media
TPM
Conclusion
Performance =
Poultice property
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Transport in Permeable Media
TPM
Advection Pros • Fast • Object is dry at the end
Cons • Pore size dependent
- adapt poultice to substrate • Renew poultice in time (back diffusion) • Not all salt removed
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Transport in Permeable Media
TPM
Be aware
wetting = advection for ions
Accumulation of ions
drying = advection for ions
Accumulation of ions
= not moved
Diffusion dominant
Advection dominant
So salts are moved in
and
can not be moved out again
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Transport in Permeable Media
TPM
‘Gedanken Experiment’
sample
water
1 cm
Fired clay brick
Permeability ~ 10-8 ms-1
Advection domimant
Concrete
Permeability ~ 10-13 ms-1
Diffusion domimant
Limitations of advection based poulticing ???
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Transport in Permeable Media
TPM
Influence osmotic pressure
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Transport in Permeable Media
TPM
Macroscopic pressure = capillary pressure + osmotic pressure
Water activity (pure water aw=1)
Effective pore size changes
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Transport in Permeable Media
TPM Extreme example
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Transport in Permeable Media
TPM