Code_Aster
®
Version
6.2
Titrate:
WTNL102 - Dimensional mono problem of forced convection
Date:
13/10/04
Author (S):
C. CHAVANT, R. FERNANDES
Key
:
V7.30.102-A
Page:
1/4
Manual of Validation
V7.30 booklet: Thermo hydro-mechanical in porous environment of linear structures
HT-66/04/005/A
Organization (S):
EDF-R & D/AMA
Manual of Validation
V7.30 booklet: Thermo hydro-mechanical in porous environment of linear structures
Document: V7.30.102
WTNL102 - Dimensional mono problem of
forced convection
Summary:
It is about the dimensional mono transport of heat by a flow constant speed. The water resource
is characterized by a linear pressure in space. The reference solution is analytical.
Code_Aster
®
Version
6.2
Titrate:
WTNL102 - Dimensional mono problem of forced convection
Date:
13/10/04
Author (S):
C. CHAVANT, R. FERNANDES
Key
:
V7.30.102-A
Page:
2/4
Manual of Validation
V7.30 booklet: Thermo hydro-mechanical in porous environment of linear structures
HT-66/04/005/A
1
Problem of reference
1.1 Geometry
One places oneself within the framework of a dimensional mono problem in Cartesian co-ordinates.
“structure” considered, is finally a segment length 1
1.2
Boundary conditions and loadings
One imposes a pressure varying P in X linearly = 0 to 0 in X = 1:
() ()
X
P
X
p
-
=
1
In X = 0: the temperature is imposed null
In X = 1: the temperature is imposed on 1.
1.3 Conditions
initial
()
0
=
X
T
everywhere
One is interested in the steady state
=
=
=
0
0 T P
p
X
=
=
=
1
0
1
T
p
X
Code_Aster
®
Version
6.2
Titrate:
WTNL102 - Dimensional mono problem of forced convection
Date:
13/10/04
Author (S):
C. CHAVANT, R. FERNANDES
Key
:
V7.30.102-A
Page:
3/4
Manual of Validation
V7.30 booklet: Thermo hydro-mechanical in porous environment of linear structures
HT-66/04/005/A
2
Reference solution
2.1
Method of calculation
One leaves the equation of the energy [éq 3.1.3-1] of the document [R7.01.11], which in this case
give:
()
()
0
=
+
+
+
Q
M
Div
H
Div
Q
m
H
&
&
éq
2.1-1
In which
H
indicate the enthalpy of water,
M
its mass flow,
m
the mass water contribution and
Q
heat transfer rate.
Taking into account the made assumptions, one sees easily that:
P
M
H
W
X
=
=
M
éq 2.1-2
T
C
H
p
W
=
éq 2.1-3
X
T
Q
T
X
-
=
=
Q
éq 2.1-4
T
C
Q
p
W
W
&
&
=
éq 2.1-5
T
is the thermal coefficient of diffusion process,
W
H
K
µ
int
=
is the hydraulic coefficient of dissemination,
int
K
the intrinsic permeability,
W
,
W
µ
,
p
W
C
are respectively the density, viscosity and
calorific heat with constant pressure of water.
While deferring [éq 2.1-2], [éq 2.1-3], [éq 2.1-4] and [éq 2.1-5] in [éq 2.1-1] one finds:
0
2
2
=
-
+
X
T
X
T
P
C
T
C
T
H
p
W
W
T
p
W
W
&
éq
2.1-6
One poses:
P
C
R
T
H
p
W
W
=
and
T
p
W
W
C
S
=
One obtains
0
2
2
=
-
+
X
T
X
T
R
T
S
&
éq
2.1-7
2.2
Results of reference
In order to obtain the steady state more quickly, one chooses coefficients such as:
1
1
<<
=
P
R
S
H
The solution of [éq 2.1-7] is then:
1
1
-
-
=
R
X-ray
E
E
T
Code_Aster
®
Version
6.2
Titrate:
WTNL102 - Dimensional mono problem of forced convection
Date:
13/10/04
Author (S):
C. CHAVANT, R. FERNANDES
Key
:
V7.30.102-A
Page:
4/4
Manual of Validation
V7.30 booklet: Thermo hydro-mechanical in porous environment of linear structures
HT-66/04/005/A
3 Modeling
With
3.1
Characteristics of modeling A
One makes a modeling with 500 elements, each element thus has a length
500
1
=
H
.
The coefficients are chosen:
1
10
100
1
1
1
int
P
K
C
T
W
p
W
W
µ
These values lead to a Peclet number
10
=
R
and with a Peclet number local
50
1
=
Rh
.
3.2 Functionalities
tested
Order
Option
AFFE_MODELE
D_PLAN_THMD
DEFI_MATERIAU
THM_LIQU
THM_DIFFU
THM_INIT
ELAS
AFFE_CHAR_MECA DDL_IMPO
PRE1
TEMP
STAT_NON_LINE COMP_INCR
RELATION KIT_THM
RELATION_KIT
ELAS
LIQU_SATU
HYDR_UTIL
Discretization in time: 10 pitches of times of 40 S each one
3.3 Results
X
Temperature of reference
Temperature Aster Error
relative
6,00E-01 0,0182710686
0,0182567
0,079%
7,00E-01 0,0497439270
0,0497269
0,034%
8,00E-01 0,1352960260
0,1352760
0,015%
9,00E-01 0,3678507400
0,3678309
0,005%
1,00E+00 1,0000000000
1,0000000
0,0%
4
Summary of the results
A good agreement is obtained between the temperatures calculated by Code_Aster and the values of
reference.