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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adsorbing system




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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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adsorbing system




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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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adsorbing system




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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

1.8 diffusion

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