Code_Aster ®
Version
6.2
Titrate:
WTNL102 - Dimensional mono Problème of forced convection
Date:
13/10/04
Author (S):
C. CHAVANT, R. FERNANDES Key
:
V7.30.102-A Page:
1/4

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

Handbook of Validation
V7.30 booklet: Thermo hydro-mechanical in porous environment of linear structures
Document: V7.30.102

WTNL102 - Dimensional mono Problème 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.
Handbook of Validation
V7.30 booklet: Thermo hydro-mechanical in porous environment of linear structures
HT-66/04/005/A

Code_Aster ®
Version
6.2
Titrate:
WTNL102 - Dimensional mono Problème of forced convection
Date:
13/10/04
Author (S):
C. CHAVANT, R. FERNANDES Key
:
V7.30.102-A Page:
2/4

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

p = 0

p = P
X = 1

X = 0
T =1

T = 0

1.2
Boundary conditions and loadings

One imposes a pressure varying P in X linearly = 0 to 0 in X = 1: p (X) = P (1 - X)

In X = 0: the temperature is imposed null
In X = 1: the temperature is imposed on 1.

1.3 Conditions
initial

T (X) = 0 everywhere
One is interested in the steady state
Handbook of Validation
V7.30 booklet: Thermo hydro-mechanical in porous environment of linear structures
HT-66/04/005/A

Code_Aster ®
Version
6.2
Titrate:
WTNL102 - Dimensional mono Problème of forced convection
Date:
13/10/04
Author (S):
C. CHAVANT, R. FERNANDES Key
:
V7.30.102-A Page:
3/4

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:
m
H +
Q + Div
& &
(M
H) + Div (Q) = 0
éq
2.1-1
In which H indicates the enthalpy of water, M its flow mass, m the mass water contribution and
Q heat flow.

Taking into account the made assumptions, one sees easily that:
M = M = P







éq 2.1-2
X
W H
H = C Pt









éq 2.1-3
W
T

Q = Q = -









éq 2.1-4
X
T
X

Q = C Pt
&










éq 2.1-5
W
W &

K
is the thermal coefficient of diffusion process,
int
=
is the hydraulic coefficient of diffusion, K
T
H
µ
int
W
the intrinsic permeability, µ, p
C are respectively the density, viscosity and
W
W
W
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:
p
2
C

W
W
p
H
T
T + C
P
&

éq
2.1-6
W
W
- T = 0
2


T
T
X X
One poses:

R = C p H P
W
W T
and
p
wCw
S =

T

One obtains
2
T
T
S & + R
- T = 0 éq
2.1-7
2
X X
2.2
Results of reference

In order to obtain the steady state more quickly, one chooses coefficients such as:
S
1
=
<<1
R
P
H
The solution of [éq 2.1-7] is then:
X-ray
E - 1
T =

R
E - 1
Handbook of Validation
V7.30 booklet: Thermo hydro-mechanical in porous environment of linear structures
HT-66/04/005/A

Code_Aster ®
Version
6.2
Titrate:
WTNL102 - Dimensional mono Problème of forced convection
Date:
13/10/04
Author (S):
C. CHAVANT, R. FERNANDES Key
:
V7.30.102-A Page:
4/4

3 Modeling
With

3.1
Characteristics of modeling A

1
One makes a modeling with 500 elements, each element thus has a length H =
.
500
The coefficients are chosen:


1
W
C p
1
W
µ
1
W

K
100
int

10
T
P
1
1
These values lead to a number of Peclet R = 10 and to a Peclet number local Rh =
.
50

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 steps 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.

Handbook of Validation
V7.30 booklet: Thermo hydro-mechanical in porous environment of linear structures
HT-66/04/005/A

Outline document