Code_Aster
®
Version
7.2
Titrate:
WTNP106 - Heating of a porous environment désaturé with dissolved air
Date:
13/10/04
Author (S):
S. GRANET, C. CHAVANT
Key
:
V7.32.106-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: V7.32.106
WTNP106 - Heating of a désaturé porous environment
with dissolved air
Summary:
One heats a porous environment of which the pores are filled with a mixture of water (fluid and vapor) and of air (dry and
dissolved in water). Initial saturation in fluid is 50%, the loading is a uniform heat flux
on the edges of the field. The modeling made by only one element corresponds to the modeling of one
homogeneous problem in space.
The reference solution is an approximate analytical solution.
Code_Aster
®
Version
7.2
Titrate:
WTNP106 - Heating of a porous environment désaturé with dissolved air
Date:
13/10/04
Author (S):
S. GRANET, C. CHAVANT
Key
:
V7.32.106-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
h=
1
m
1m
X
y
Z
B
With
C
D
E
F
Co-ordinates of the points (m):
To - 0,5 - 0,5
C
0,5 0,5
B
0,5 - 0,5
D - 0,5 0,5
Code_Aster
®
Version
7.2
Titrate:
WTNP106 - Heating of a porous environment désaturé with dissolved air
Date:
13/10/04
Author (S):
S. GRANET, C. CHAVANT
Key
:
V7.32.106-A
Page:
3/8
Manual of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
HT-66/04/005/A
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
)
Heat with constant pressure (J.K
- 1
)
thermal expansion factor of the fluid (K
- 1
)
Dynamic viscosity of liquid water (Pa.s)
Permeability relating to water
10
3
4180
0.
0.001
()
1
=
S
Kr
W
Vapor
Specific heat (J.K
- 1
)
Initial enthalpy (latent heat of vaporization)
J/kg
Mass molar (kg.mol
- 1
)
1900
2,5E6.
0,018
Gas
Specific heat (J.K
- 1
)
Mass molar (kg.mol
- 1
)
Permeability relating to gas
Viscosity of the gas (kg.m
- 1
.s
- 1
)
1900
0,018
()
1
=
S
Kr
gz
1,8
E
- 5
Dissolved air
Specific heat (J.K
- 1
)
Constant of Henry (Pa.m
3
.mol
- 1
)
1900
50000
Skeleton
Heat-storage capacity with constant stress (J.K
- 1
) 1050
Initial State
Porosity
Temperature (K)
Gas pressure (AP)
Steam pressure (AP)
Initial saturation in fluid (AP)
0,3
300
1E5
3700
0,5
Constants
Constant of perfect gases
8,315
Coefficients
homogenized
Homogenized density (kg.m
- 3
)
Isotherm of sorption
2200
()
(
)
0
0
12
10
5
.
0
C
vp
C
C
P
P
P
P
S
-
-
-
=
-
With
3700
0
=
vp
P
0
0
=
C
P
1.3
Boundary conditions and loadings
On all the edges:
Heat flux
6
10
.
=
N
Q
ext.
Hydraulic flow no one
Code_Aster
®
Version
7.2
Titrate:
WTNP106 - Heating of a porous environment désaturé with dissolved air
Date:
13/10/04
Author (S):
S. GRANET, C. CHAVANT
Key
:
V7.32.106-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 steam pressure starting from the temperature
We suppose the linear curve of saturation. It is thus written:
C
P
S
S
S
+
=
0
éq
2.1.1-1
The equation [éq 2.2.3.3-2] of the reference document [R7.01.11] gives then:
(
)
(
)
(
)
(
)
(
)
C
have
have
have
have
AD
AD
AD
AD
C
vp
vp
vp
vp
C
W
W
P
S
S
m
P
S
S
m
P
S
S
m
P
S
m
-
-
-
=
+
-
=
-
-
-
=
=
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
éq
2.1.1-2
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:
=
+
0
vp
W
m
m
(
)
(
)
(
)
0
1
0
0
=
-
-
+
-
S
P
S
vp
vp
C
vp
W
éq
2.1.1-3
=
+
0
have
AD
m
m
(
)
(
)
(
)
(
)
0
1
0
0
0
0
=
-
+
-
-
+
-
S
S
P
S
AD
AD
have
have
C
have
AD
éq 2.1.1-4
[R7.01.11] [éq 4.1.4-1] gives in addition:
(
)
(
)
(
)
(
)
(
)
-
+
-
+
-
-
-
+
-
-
=
+
T
T
m
W
m
vp
ol
vp
gz
vp
H
W
ol
vp
C
C
W
ol
vp
vp
vp
H
W
ol
vp
gz
gz
H
W
ol
vp
vp
vp
T
dT
H
H
R
M
T
T
p
p
K
R
M
p
p
RT
M
p
p
K
M
p
p
K
RT
M
p
p
0
2
0
0
0
0
0
0
0
0
0
ln
)
1
1
(
ln
éq
2.1.1-5
Coupling of the equations [éq 2.1.1-3], [éq 2.1.1-4] and [éq 2.1.1-5], for which it is necessary to add the equation of
perfect gases for the vapor, the dry air and the dissolved air as well as the law of Henry are strongly a system
nonlinear that we will solve in small disturbances, which makes it possible to linearize it.
Code_Aster
®
Version
7.2
Titrate:
WTNP106 - Heating of a porous environment désaturé with dissolved air
Date:
13/10/04
Author (S):
S. GRANET, C. CHAVANT
Key
:
V7.32.106-A
Page:
5/8
Manual of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
HT-66/04/005/A
All made calculations, one obtains:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
) (
)
(
)
(
)
(
)
(
)
(
)
=
-
+
-
+
+
-
=
-
-
-
-
+
-
+
+
-
-
+
-
-
-
=
-
-
-
-
+
-
-
+
-
-
-
+
-
0
1
1
0
1
.
1
.
1
.
0
1
.
1
.
1
2
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
T
T
H
H
R
M
R
T
K
P
M
P
RT
M
P
P
T
S
RT
P
M
K
RP
S
RT
S
K
S
M
K
RT
S
P
P
S
S
P
T
S
RT
P
M
K
RP
S
K
RT
S
P
P
S
RT
M
S
S
P
m
W
m
vp
ol
vp
H
have
W
ol
vp
W
W
ol
vp
vp
vp
have
ol
vp
H
have
have
AD
H
ol
vp
H
have
AD
have
W
have
AD
have
AD
vp
vp
ol
vp
H
have
vp
W
H
vp
W
have
W
vp
W
ol
vp
vp
W
vp
éq 2.1.1-6
2.1.2 Calculation of the temperature
The equation [éq 3.2.4.3-1] of the reference document [R7.01.11] gives:
T
C
p
T
Q
gz
m
gz
+
-
=
0
3
éq
2.1.2-1
(since the other expansion factors are null).
The equation [éq 3.2.4.3-2] gives:
(
)
T
S
lq
m
gz
3
1
-
=
éq
2.1.2-2
One thus obtains:
(
) (
)
T
C
p
p
S
Q
have
vp
lq
+
+
-
-
=
0
1
éq
2.1.2-3
In this problem,
Q
is anything else only the heat brought per unit of volume.
While calling
Flight
the total volume of the part and
Surfing
its side surface and
T
the time of application
flows:
N
Q.
ext.
Flight
Surfing
T
Q
=
éq
2.1.2-4
Code_Aster
®
Version
7.2
Titrate:
WTNP106 - Heating of a porous environment désaturé with dissolved air
Date:
13/10/04
Author (S):
S. GRANET, C. CHAVANT
Key
:
V7.32.106-A
Page:
6/8
Manual of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
HT-66/04/005/A
2.1.3 System to be solved
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
) (
)
(
)
(
)
(
)
()
()
=
×
-
-
-
-
-
+
-
-
-
+
+
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
+
-
0
.
0
0
1
0
1
0
)
1
(
1
1
.
1
.
1
.
)
1
.(
.
1
1
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
N
Q
ext.
have
W
vp
lq
lq
m
W
m
vp
ol
vp
H
W
have
ol
vp
W
ol
vp
vp
H
ol
vp
H
have
AD
have
ol
vp
H
have
have
AD
have
AD
have
AD
H
vp
W
H
have
vp
W
ol
vp
vp
vp
W
ol
vp
vp
W
Flight
Surfing
T
P
T
P
P
S
C
S
T
H
H
R
M
R
T
K
P
M
RT
M
P
RT
S
K
S
M
K
RT
S
S
RT
P
M
K
RP
S
S
S
K
RT
S
K
RP
S
RT
M
p
S
S
RT
M
S
S
éq
2.1.2-5
0
S
S
0
T
0
vp
p
0
vp
H
0
vp
(calculated)
lq
5,00E-01 - 1,00E-12 3,00E+02 3,70E+03 2,50E+06 2,67E-02 1,00E+03
0
R
0
S
(calculated)
S
C
p
lq
C
L
p
vp
C
0
C
(calculated)
2,20E+03 3,00E-01 2,93E+03 1,05E+03 4,18E+03 1,90E+03 2,78E+06
N
Q
.
ext.
T
Surfing
Flight
1,00E+06
10
400
1,00E+04
The following results are obtained:
After resolution of this system, one obtains:
-
=
7
.
45
144
.
0
99500
4
.
29
have
W
vp
P
T
P
P
What gives in term of result Aster (increment):
PRE1 PRE2
DT PVP
(V3)
9.95
E
4.7.5E1 1.44E-1 2.94E1
2.2 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:
WTNP106 - Heating of a porous environment désaturé with dissolved air
Date:
13/10/04
Author (S):
S. GRANET, C. CHAVANT
Key
:
V7.32.106-A
Page:
7/8
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. A Q8 element.
Discretization in time: only one pitch of time: 10
S.
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 FLUX_THM_REP
FLUN_HYDR1
STAT_NON_LINE COMP_INCR RELATION
KIT_THH
RELATION_KIT
ELAS
LIQU_AD_GAZ_VAPE
HYDR_UTIL
3.3 Values
tested
Node Urgent Field Component
(S)
Reference
(analytical)
Aster Difference
(%)
NO1
DEPL
TEMP 10
S
0.1440 0.1439
0.08%
NO1
DEPL
PRE1
10 S
9.95 10
4
9.95
10
4
0.02%
NO1
DEPL
PRE2
10 S
75
73.3
2.21%
NO1
VARI_ELNO_ELGA
V3
10 S
29.4
29.5
0.2%
Code_Aster
®
Version
7.2
Titrate:
WTNP106 - Heating of a porous environment désaturé with dissolved air
Date:
13/10/04
Author (S):
S. GRANET, C. CHAVANT
Key
:
V7.32.106-A
Page:
8/8
Manual of Validation
V7.32 booklet: Thermo hydro-mechanical in porous environment unsaturated
HT-66/04/005/A
4
Summary of the results
The solution ASTER is in very good agreement with the analytical solution except for the gas pressure.
The weak differences are due to the linearization.