Code_Aster
®
Version
7.2
Titrate:
WTNP103 - Dissemination of air dissolved in water (plan)
Date:
20/10/04
Author (S):
S. GRANET, C. CHAVANT
Key
:
V7.32.103-A
Page:
1/8
Manual of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
HT-66/04/005/A
Organization (S):
EDF-R & D/AMA
Manual of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
Document: V7.32.103
WTNP103 - Dissemination 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.
Code_Aster
®
Version
7.2
Titrate:
WTNP103 - Dissemination of air dissolved in water (plan)
Date:
20/10/04
Author (S):
S. GRANET, C. CHAVANT
Key
:
V7.32.103-A
Page:
2/8
Manual of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
HT-66/04/005/A
1
Problem of reference
1.1 Geometry
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
)
Specific heat with constant pressure (J.K
- 1
)
Dynamic viscosity of liquid water (Pa.s)
thermal expansion factor of the fluid (K
- 1
)
Permeability relating to water
10
3
0.
0.001
0.
()
5
.
0
=
S
Kr
W
Vapor
Specific heat (J.K
- 1
)
Mass molar (kg.mol
- 1
)
0.
0,01
Gas
Specific heat (J.K
- 1
)
Mass molar (kg.mol
- 1
)
Permeability relating to gas
Viscosity of the gas (kg.m
- 1
.s
- 1
)
0.
0,01
()
5
.
0
=
S
Kr
gz
0.001
Dissolved air
Specific heat (J.K
- 1
)
Constant of Henry (Pa.m
3
.mol
- 1
)
0.
50000
Initial State
Porosity
Temperature (K)
Gas pressure (AP)
Steam pressure (AP)
Capillary pressure (AP)
Initial saturation in fluid
1
300
1.01E5
1000
1.
E
6
0,4
Constants
Constant of perfect gases
8,32
Coefficients
homogenized
Homogenized density (kg.m
- 3
)
Isotherm of sorption
Coefficient of Biot
Fick Vapor (m
2.
S
- 1
)
Fick dissolved air (m
2.
S
- 1
)
Intrinsic permeability (m
2
)
2200
()
4
.
0
=
C
P
S
0
FV=0
FA=6. E-10
Kint = 1.E-19
y
X
With
B
C
D
Code_Aster
®
Version
7.2
Titrate:
WTNP103 - Dissemination of air dissolved in water (plan)
Date:
20/10/04
Author (S):
S. GRANET, C. CHAVANT
Key
:
V7.32.103-A
Page:
3/8
Manual of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
HT-66/04/005/A
1.3
Boundary conditions and loadings
On the whole of the field, one wants:
1
)
(
0
0
1
0
0
0
0
0
=
=
=
=
=
=
=
=
=
=
=
=
=
T
cte
T
S
cte
p
S
F
p
cte
p
cte
K
p
cte
p
C
vp
vp
vp
W
W
W
W
W
ol
AD
ol
vp
ol
have
M
M
M
=
=
On all the edges: Hydraulic flows and null thermics.
One now will linearize
vp
p
according to
W
p
.
Writing of p
vp
linear function of p
W
:
Section 4.2.3 of the reference document Aster [R7.01.11] gives us the relation
:
W
W
ol
vp
vp
vp
dp
RT
M
p
dp
=
. If this expression is linearized one obtains
:
-
+
=
0
0
0
0
0
0
W
W
ol
vp
vp
vp
W
W
ol
vp
vp
vp
p
M
RT
p
p
p
M
RT
p
p
that one can write in the form:
B
Ap
p
W
vp
+
=
éq 1.3-1
with
0
0
W
ol
vp
vp
M
RT
p
With
=
and
0
0
0
0
W
W
ol
vp
vp
vp
p
M
RT
p
p
B
-
=
6
10
115000
E
PC
p
gz
=
=
:
AB
edge
On
Code_Aster
®
Version
7.2
Titrate:
WTNP103 - Dissemination of air dissolved in water (plan)
Date:
20/10/04
Author (S):
S. GRANET, C. CHAVANT
Key
:
V7.32.103-A
Page:
4/8
Manual of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
HT-66/04/005/A
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:
0
)
(
=
+
+
AD
have
air
div
dt
DM
M
M
éq
2.1.1-1
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:
)
) (
1
(
)
(
0
0
0
0
have
have
AD
AD
AD
have
air
S
S
m
m
m
-
-
+
-
=
+
=
thus
have
AD
AD
have
D
S
D
S
m
m
D
)
1
(
)
(
0
0
-
+
=
+
éq
2.1.1-2
have
ol
have
have
dP
RT
M
D
=
and
have
H
ol
AD
AD
dP
K
M
D
=
dt
dP
RT
M
S
K
M
S
dt
DM
have
ol
have
H
ol
have
air
-
+
=
)
1
(
0
.
0
Calculation speeds:
)
(
have
gz
have
P
-
=
have
M
éq
2.1.1-3
since
0
=
vp
F
and
0
=
vp
P
and
AD
AD
lq
lq
AD
D
C
F
P
-
-
=
)
(
has
M
with
AD
AD
C
=
Like
have
H
AD
AD
W
lq
P
K
RT
P
P
P
P
=
=
+
=
have
AD
H
ol
AD
have
H
lq
AD
D
P
F
K
M
P
K
RT
-
-
=
.
.
)
(
has
M
[éq 2.1.1-1] can then be simplified in the following form:
)
(
have
have
P
Ldiv
dt
dP
C
=
with
+
+
=
-
+
=
AD
H
ol
have
lq
AD
H
gz
have
ol
have
H
ol
have
F
K
M
K
RT
L
RT
M
S
K
M
S
C
.
)
1
(
0
0
0
.
0
Equation of the heat whose one knows the result.
Code_Aster
®
Version
7.2
Titrate:
WTNP103 - Dissemination of air dissolved in water (plan)
Date:
20/10/04
Author (S):
S. GRANET, C. CHAVANT
Key
:
V7.32.103-A
Page:
5/8
Manual of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
HT-66/04/005/A
2.2
Results of reference
With the preceding numerical values, one finds:
4992
10
0
0
5
=
=
=
have
H
AD
have
P
K
RT
P
P
4
.
0
0
0
=
=
have
ol
have
have
P
RT
M
and
02
.
0
0
0
=
=
AD
ol
AD
AD
P
RT
M
3
0
10
.
4
-
=
=
vp
vp
The constants of the equation of heat are then:
16
6
10
.
4
,
1
4810
,
2
-
-
=
=
L
C
2.3 Uncertainties
Uncertainties are rather large because the analytical solution is an approximate solution of
fact of the linearization of the equations.
Code_Aster
®
Version
7.2
Titrate:
WTNP103 - Dissemination of air dissolved in water (plan)
Date:
20/10/04
Author (S):
S. GRANET, C. CHAVANT
Key
:
V7.32.103-A
Page:
6/8
Manual of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
HT-66/04/005/A
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 pitches 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 pressure of dry air, thermal calculation
9,80E+04
1,00E+05
1,02E+05
1,04E+05
1,06E+05
1,08E+05
1,10E+05
1,12E+05
1,14E+05
0
0,5
1
X
AP
S
Not; t=5E8s
Not; t=3E9s
Not; t=5E9s
TEMP; t=5E8s
TEMP; t=3E9s
TEMP; t=5E9s
Code_Aster
®
Version
7.2
Titrate:
WTNP103 - Dissemination of air dissolved in water (plan)
Date:
20/10/04
Author (S):
S. GRANET, C. CHAVANT
Key
:
V7.32.103-A
Page:
7/8
Manual of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
HT-66/04/005/A
4
Summary of the results
The Aster results are in very good agreement with the semi-analytical solution.
Code_Aster
®
Version
7.2
Titrate:
WTNP103 - Dissemination of air dissolved in water (plan)
Date:
20/10/04
Author (S):
S. GRANET, C. CHAVANT
Key
:
V7.32.103-A
Page:
8/8
Manual of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
HT-66/04/005/A
Intentionally white left page.