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7/28/2019 1.8 Diffusion
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1.8 TRANSPORT OF IONS IN
FLUIDS 1.8.1 Diffusion in a homogeneous non-
adsorbing system
1.8.2 Adsorption in a porous solid
1.8.3 Diffusion with Adsorption
1.8.4 Electromigration
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Salt dissolving into a solution
Flux
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Diffusion Coefficient
The diffusion coefficient is defined from the
equation:
F = D dC kg/m2/s
dxwhere: F is the flux in kg/m2/s
D is the diffusion coefficient in m2/s
C is the concentration in kg/m3
x is the position.
Thus dC/dx is the concentration gradient.
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Equations for diffusion
Considering a small element of the system the rate atwhich the concentration changes with time will be
proportional to the difference between the flux into itand the flux out of it: dC = A F
dt
where is the volume of the element, A is the crosssectional area and L = /A is the length
F = L dF thus: dC = dF
dx dt dxand: dC = D d2C
dt dx2
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Diffusion
through a
barrier
Increasing time
Position
Concentration
ConcreteStrong
SolutionWeakSolution
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1.8 TRANSPORT OF IONS IN
FLUIDS 1.8.1 Diffusion in a homogeneous non-
adsorbing system
1.8.2 Adsorption in a porous solid
1.8.3 Diffusion with Adsorption
1.8.4 Electromigration
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Definitions
Absorbed water is drawn in to the pores of asolid by capillary suction or osmosis.
Adsorbed ions are bound into the matrix ofa solid and cannot move.
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Concentrations
The concentration of ions in a porous solid (inwhich the pores are filled with fluid) may be
measured in two different ways:Cl kg/m
3 is the concentration of ions per unitvolume of liquid in the pores
andCs kg/m
3 is the total concentration (includingadsorbed ions) per unit volume of the solid.
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Acid and water soluble
If the "acid soluble" concentration ismeasured Cs is obtained
If the "water soluble" concentration ismeasured Cl is obtained.
Ions in solution are the only ones that will
cause corrosion.The ratio of the two concentrations is thecapacity factor:
lC
Cs=
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Distribution ratio
The distribution ratio Kd is defined as the number of
adsorbed ions per unit mass of the solid divided by
the number of dissolved ions per unit volume of thepore fluid thus
Kd = (Cs - Cl) m3/kg
Cl
where is the porosity and is the density thus:
= + Kd.
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Linear isotherm
A simple approximation of the amount of
material which is adsorbed onto the matrix
may be obtained by assuming that at allconcentrations it is proportional to the
concentration of ions in the pore fluid (note
that this implies that the adsorption isreversible). Thus and kd are constant for all
concentrations.
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1.8 TRANSPORT OF IONS IN
FLUIDS 1.8.1 Diffusion in a homogeneous non-
adsorbing system
1.8.2 Adsorption in a porous solid
1.8.3 Diffusion with Adsorption
1.8.4 Electromigration
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Effect of Capacity Factor
The equation for the rate of change of concentrationwill be for the total concentration:
dCs = dFdt dx
thus: dCl = dF
dt dx
From this it may be seen that a high value of willmake the concentration change much more slowly -i.e. if chlorides are penetration into a wall it willdelay the start of corrosion of the steel
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Apparent Diffusion Coefficient
The apparent diffusion coefficient Da (which
is what can be measured by testing the solid
using measurements of total concentration Cs)is defined from:
F = Da dCs kg/m2/s
dx
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Intrinsic Diffusion Coefficient
The intrinsic diffusion coefficient (which is thediffusion coefficient for the pore solution) is definedfrom:
F ' = Di dCl kg/m2/s
dx
where F ' is the flux per unit cross sectional area of
the liquid in the pores. Thus:F = Di dCl kg/m
2/s
dx
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Diffusion Coefficients
It may be seen that:
= Di
Da
for a typical concrete Di = 5 10-12 m2/s
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Diffusion Cells
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Detail of Diffusion Cells
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The NIREX concept
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Spent Fuel Repository
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Transport from
spent fuel
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Diffusion
Data
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Typical Landfill Arrangement
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Permeability Control
Time years
-2
02
4
6
8
1012
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Depth m
Co
ncentration
kg
/m3ofsolid
40 80 400
k = 10-9 m/s D = 5 10-12 m2/s
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Diffusion Control
Time years
-2
0
2
4
6
8
10
12
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Depth m
C
oncentration
kg/m3ofsolid
100 500 5000
k = 10-12
m/s
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Output of contaminants from
base of barrier
0
2
4
6
8
1012
14
16
18
0 2000 4000 6000
Time years
Cumulativeoutp
utkg/m2
Permeabilitycontrol
(base case)
Diffusioncontrol
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Effect of Capacity Factor
1
10
100
1000
10000
1 10 100
Capacity Factor
B
reakthrough
time
Years
Base
case
Diffusion
Control
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1.8 TRANSPORT OF IONS IN
FLUIDS 1.8.1 Diffusion in a homogeneous non-
adsorbing system
1.8.2 Adsorption in a porous solid
1.8.3 Diffusion with Adsorption
1.8.4 Electromigration
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Electromigration
V
Solution
with supply
of negativeions
Solution with
supply of
positive ions
+-
Negative ions, e.g. chloride
Positive ions
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Electromigration
The flux due to electromigration is given by the
following equation:
Where: J is the flux in kg/m2/s, D is the diffusion coefficient in m2/s, z isthe valency of the ion, F is the Faraday constant, E is the electric field in
volts/m, C is the concentration of ions in mol/m3,R = 8.31 J/mol/oK, T is
the temperature in oK
Rearranging this gives the Nernst-Einstein equation:
Where is the conductivity
RT
DzECF=J
ii
i
i
cFz
RTD
22
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Combining Diffusion and
ElectromigrationThe general law governing the ionic movements in concrete
due to the chemical and electrical potential is known as the
Nernst-Planck equation:
Ji
= flux of species i [mol/m2/s]
Di= diffusion coefficient of species i [m2/s]
R= gas constant [8.31 J/mol/K]
T= absolute temperature [K]
V = Voltage, thus V/x = Electric field
x
VcD
RT
Fz
x
cDJ ii
iiii
S l i l i i d diff i
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Solving electromigration and diffusion
the hard wayassuming E is constant
)]
4
erfc(
2
1e
2a[FADc=I
)16
2
2
2
2(
o
= ax
= 2a Dt
where
a =zFE
RT
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Salt bridge measurements
samplecellcell
Reference
electrode
(connected
to data
logger)
small bore pipe4 mm drilledhole
Potassium
chloride
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Electromigration /
diffusion cell
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Section through sample during test
Voltage
Chloride zone Sodium zoneLow resistance (high D) High resistance (low D)
Electrostatic field E is
gradient
?
?
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Modelling a thin slice of the sample for a short time step
Apply Kirchoffs law : current in = current out
Electromigration into element
- set by field E which was
calculated for the last element
Diffusion in and outfixed by
concentration gradient
Electromigration out of
elementwe can set this for
charge neutrality by adjusting
the field E
Final adjustments are needed to get the correct total voltage across
the sample.
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Model output for
current and voltage
Current vs time with no
voltage correction (average)
0.000E+00
5.000E+00
1.000E+01
1.500E+01
2.000E+01
2.500E+01
3.000E+01
0 5 10 15 20
Time hours
TotalCurrentmAmps
Voltage adjustments at different times
-5
0
5
10
15
20
25
30
35
40
45
0.0 10.0 20.0 30.0 40.0 50.0 60.0
Distance from negative side mm
V
oltage 0.000
6.802
17.013
0.000E+00
5.000E+01
1.000E+02
1.500E+02
2.000E+02
2.500E+02
3.000E+02
3.500E+02
4.000E+02
0 5 10 15 20
Time hours
TotalCurrentmAm
ps
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Concentrations in cells at distances from negative side
in mol/m3 vs time in hours
Hydroxyl
050
100
150
200
250
300350
400
0 5 10 15 20
negative
2.5
7.5
17.5
47.5
positive
Potassium
-50
0
50
100
150
200
0 5 10 15 20
negativ
e2.5
7.5
17.5
47.5
Sodium
0
100
200
300
400
500
600
0 5 10 15 20
negative
2.5
7.5
17.5
47.5
positive
Chloride
-100
0
100
200
300
400
500
600
0 5 10 15 20
negative
2.5
7.5
17.5
47.5
positive
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Current in amps at different times in hours vs position
in mm from the negative sideTime = 0
0.00E+00
5.00E-03
1.00E-02
1.50E-02
2.00E-02
2.50E-02
3.00E-02
2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5
potassium
sodium
chloride
hydroxyl
Time = 7
0.00E+00
2.00E-03
4.00E-03
6.00E-03
8.00E-03
1.00E-02
1.20E-02
1.40E-02
2.5 7. 5 12.5 1 7.5 22 .5 27.5 3 2.5 37. 5 4 2.5 47 .5
potassium
sodium
chloride
hydroxyl
Time = 14
0.00E+00
2.00E-03
4.00E-03
6.00E-03
8.00E-03
1.00E-02
1.20E-02
1.40E-02
2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5
potassium
sodium
chloride
hydroxyl
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Current from elements in cells at distances from
negative side in amps vs time in hours
Hydroxyl
-0.001
00.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0 5 10 15 20
2.5
7.5
17.5
47.5
Potassium
-0.005
0
0.005
0.01
0.015
0.02
0 5 10 15 20
2.5
7.5
17.5
47.5
Sodium
0
0.002
0.004
0.006
0.008
0.01
0 5 10 15 20
2.5
7.5
17.5
47.5
Chloride
-0.002
0
0.002
0.004
0.006
0.008
0.01
0 5 10 15 20
2.57.5
17.5
47.5