Code_Aster ®
Version
7.2
Titrate:
WTNP103 - Diffusion of air dissolved in water (plan)


Date:
20/10/04
Author (S):
S. GRANET, C. CHAVANT Key
:
V7.32.103-A Page:
1/8

Organization (S): EDF-R & D/AMA

Handbook of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
Document: V7.32.103

WTNP103 - Diffusion of air dissolved in water (plan)

Summary:

Here a problem at temperature and constant saturation are considered. By boundary conditions suitable
one imposes a water pressure and a steam pressure constants. A gas pressure is imposed on one
edge of the field (null flows of the other with dimensions). Only pressures of dry air and dissolved air connected by the law of
Henry evolve/move. This problem brings back in an equation for the pressure of dry air of type “equation of
heat “. The reference solution will be then a thermal calculation ASTER.
Handbook of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
WTNP103 - Diffusion of air dissolved in water (plan)


Date:
20/10/04
Author (S):
S. GRANET, C. CHAVANT Key
:
V7.32.103-A Page:
2/8

1
Problem of reference

1.1 Geometry

y
C

D

With
X
B

Co-ordinates of the points (m):

To 0 0
C
1 0,5
B
1
0
D 0 0,5

1.2
Properties of material

One gives here only the properties whose solution depends, knowing that the command file
contains other data of material (thermal conductivity, moduli of elasticity…) who finally
do not play any part in the solution of the dealt with problem.

Liquid water
Density (kg.m-3)
103
Specific heat with constant pressure (J.K-1)
0.
Dynamic viscosity of liquid water (Pa.s)
0.001
thermal dilation coefficient of the liquid (K-1)
0.
Permeability relating to water
Kr

W (S) = 0 5
.
Vapor
Specific heat (J.K-1)
0.
Mass molar (kg.mol-1)
0,01
Gas
Specific heat (J.K-1)
0.
Mass molar (kg.mol-1)
0,01
Permeability relating to gas
Kr

gz (S) = 0 5
.
Viscosity of the gas (kg.m-1.s-1)
0.001
Dissolved air
Specific heat (J.K-1)
0.
Constant of Henry (Pa.m3.mol-1)
50000
Initial State
Porosity
1
Temperature (K)
300
Gas pressure (Pa)
1.01E5
Steam pressure (Pa)
1000
Capillary pressure (Pa)
1.E6
Initial saturation in liquid
0,4
Constants
Constant of perfect gases
8,32
Coefficients
Homogenized density (kg.m-3)
2200
homogenized
Isotherm of sorption
S (P

c) =
.
0 4
Coefficient of Biot
0
Fick Vapor (m2.s-1)
FV=0
Fick dissolved air (m2.s-1)
FA=6. E-10
Intrinsic permeability (m2)
Kint = 1.E-19
Handbook of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
WTNP103 - Diffusion of air dissolved in water (plan)


Date:
20/10/04
Author (S):
S. GRANET, C. CHAVANT Key
:
V7.32.103-A Page:
3/8

1.3
Boundary conditions and loadings

On the whole of the field, one wants:

0
p = cte = p
W
W
1 = 0
0
= cte =
K
W
W
W
0
p = cte = p
vp
vp
F = 0

vp
S (p) = cte = S
C
0
0
T = cte = T
= 1
ol
ol
ol
M
= M = M
have
vp
AD

On all the edges: Hydraulic flows and null thermics.

One now will linearize pvp according to pw.

Linear writing of pvp function of pw:

Section 4.2.3 of the reference document Aster [R7.01.11] gives us the relation
:
ol
dp
M
vp
vp dpw
=
. If this expression is linearized one obtains
:
p
RT
vp
W
0
ol
p M

0
ol
p M

p = vp
vp p + 0
p - vp
vp
0
p which one can write in the form:
vp
0
W
vp
0
W
RT
RT
W

W


p = Ap + B








éq 1.3-1
vp
W

0
ol
p M
0
ol
p M
with
vp
vp
With =
and
0
vp
vp
0
B = p -
p
0
RT
vp
0
W
RT
W
W

On
:

AB
edge



p
= 115000
gz

PC = 10e6

Handbook of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
WTNP103 - Diffusion of air dissolved in water (plan)


Date:
20/10/04
Author (S):
S. GRANET, C. CHAVANT Key
:
V7.32.103-A Page:
4/8

2
Reference solution

2.1
Method of calculation

2.1.1 Calculation of the conservation of the mass of air

The conservation of the gas mass is written:

to dmair + div (M + M) = 0 éq
2.1.1-1
have
AD
dt

It is written that the total water mass and the total mass of air are preserved (because there is no flow
from gas water nor at the edge) and one obtains:

m
= m + m = S (
0
-) + 1
(- S) (
0
-)
air
have
AD
0
AD
AD
0
have
have
thus
D (m + m) = S D + 1
(- S) D éq
2.1.1-2
have
AD
0
AD
0
have
ol
M
ol
have
D
=
dP
M AD

have
have and D
=
dP
RT
AD
have
K H
DM

M ol
M ol dP
air = S
have + 1
(- S
have
have
)

dt

0. K
0
RT dt

H


Calculation speeds:
Farmhouse = (- P)
éq
2.1.1-3
gz
have
have
since F = 0 and P = 0
vp
vp

and

M
= (- P) - F C with C =
AD
AD
lq
lq
AD
AD
AD
AD

RT
Like P = P + P = P =
P
lq
W
AD
AD
have
K H
ol
RT
M AD
M
=
(- P) -.
.F P
AD
AD
lq
have
AD
have
K
K
H
H

[éq 2.1.1-1] can then be simplified in the following form:
dPas
C
= Ldiv (P)
have
dt

ol
ol
M
M
C =
have
S
+ 1
(-
have
S)

.
0 K
0
RT
with
H


ol
0
RT
0
M have
L = +
+
F
.

have
gz
AD
lq
AD
K
K
H
H

Equation of the heat whose one knows the result.
Handbook of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
WTNP103 - Diffusion of air dissolved in water (plan)


Date:
20/10/04
Author (S):
S. GRANET, C. CHAVANT Key
:
V7.32.103-A Page:
5/8

2.2
Results of reference

With the preceding numerical values, one finds:

P =
RT
105
0
0
P =
P = 4992
have
AD
have
K H
ol
M
ol
M
0
have
0
=
P = 4
.
0 and 0
AD
0
=
P = 02
.
0

have
have
RT
AD
AD
RT

0
3
=
4.10-
=

vp
vp

The constants of the equation of heat are then:

- 6
C =,
2 4810
16
L =,
1 4.10-

2.3 Uncertainties

Uncertainties are rather large because the analytical solution is an approximate solution of
fact of the linearization of the equations.
Handbook of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
WTNP103 - Diffusion of air dissolved in water (plan)


Date:
20/10/04
Author (S):
S. GRANET, C. CHAVANT Key
:
V7.32.103-A Page:
6/8

3 Modeling
With

3.1
Characteristics of modeling A

Modeling in plane deformations. 20 elements QUAD8.

3.2 Functionalities
tested

Order Option



AFFE_MODELE
D_PLAN_THH 2D


DEFI_MATERIAU
THM_LIQU


THM_GAZ
THM_VAPE_GAZ
THM_AIR_DISS
THM_DIFFU
THM_INIT
ELAS
AFFE_CHAR_MECA DDL_IMPO PRE1


PRE2
TEMP
STAT_NON_LINE COMP_INCR
RELATION
KIT_THH


RELATION_KIT
ELAS

LIQU_AD_GAZ_VAPE
HYDR_UTIL
Discretization in time: 100 steps of time of 5e7 S each one

3.3 Results
X (m)
Time (S)
PRE2 Aster
Thermal PRE2 calculation
Relative error
0,2 3e9s 1.128E4 1.120E4 0.73%
0,2 5e9s 1.127E4 1.224E4 0.24%

Comparison near
Sion of dry air, thermal calculation

1,14E+05

1,12E+05

Not; t=5E8s

1,10E+05

Not; t=3E9s

1,08E+05

S
Not; t=5E9s

1,06E+05
Pa

TEMP; t=5E8s
1,04E+05

TEMP; t=3E9s

1,02E+05

TEMP; t=5E9s

1,00E+05

9,80E+04

0
0,5
1

X
Handbook of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
WTNP103 - Diffusion of air dissolved in water (plan)


Date:
20/10/04
Author (S):
S. GRANET, C. CHAVANT Key
:
V7.32.103-A Page:
7/8

4
Summary of the results

The Aster results are in very good agreement with the semi-analytical solution.
Handbook of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
WTNP103 - Diffusion of air dissolved in water (plan)


Date:
20/10/04
Author (S):
S. GRANET, C. CHAVANT Key
:
V7.32.103-A Page:
8/8

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Handbook of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
HT-66/04/005/A

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