Code_Aster ®
Version
6.4
Titrate:
TPNA01 - Stationary axisymmetric Problème with Date radiation
:

02/06/03
Author (S):
Key J.P. LEFEBVRE
:
V4.41.001-C Page:
1/8

Organization (S): EDF-R & D/AMA
Handbook of Validation
V4.41 booklet: Stationary thermics with radiation
Document: V4.41.001

TPNA01 - Stationary axisymmetric Problème
with radiation

Summary:

This elementary test makes it possible to deal with axisymmetric problem in stationary thermics with a condition
with the limits of the radiation type. The solution is analytical. The problem is dealt with into axisymmetric and in
voluminal.

For modelings presented here, the variations of the results obtained by Code_Aster range between 1 and
2% of the analytically calculated reference.

Handbook of Validation
V4.41 booklet: Stationary thermics with radiation
HT-66/03/008/A

Code_Aster ®
Version
6.4
Titrate:
TPNA01 - Stationary axisymmetric Problème with Date radiation
:

02/06/03
Author (S):
Key J.P. LEFEBVRE
:
V4.41.001-C Page:
2/8

1
Problem of reference

1.1 Geometry
Z
Re = 0.391 m
Re = 0.300 m
R


The hollow roll is supposed infinitely long.

1.2
Material properties

Only the coefficient of conductivity intervenes. Code_Aster makes compulsory the supply of a function
representing the given voluminal enthalpy starting from the voluminal coefficient of heat.

voluminal heat
C
1.00 J/M3 ° C
p =

thermal conductivity
K = 40 W/m° C

1.3
Boundary conditions and loadings

Condition of the radiation type on surface interns cylinder, condition of the convection type
(exchange with the external medium) on external surface.

No the boundary condition on the ends of the cylinder (what amounts imposing a null flow).

T
4

I

intern

surface
K
= (T+ 273.15) 4

- (T + 273.15
ext.
)
N


with = 0.6, =
10
5.73
- 8 W/2 4
m K
and iext
T
= 500.0°
C, T

in

expressed

Centigrade
°
T

external

surface
K
= He [eext
T - T]
N
with H E = 142.0
2
W/m C
°
and eext
T
= 20.0 C
°
Handbook of Validation
V4.41 booklet: Stationary thermics with radiation
HT-66/03/008/A

Code_Aster ®
Version
6.4
Titrate:
TPNA01 - Stationary axisymmetric Problème with Date radiation
:

02/06/03
Author (S):
Key J.P. LEFEBVRE
:
V4.41.001-C Page:
3/8

2
Reference solution

2.1
Method of calculation used for the reference solution

One in the case of lays out of an analytical solution a cylinder infinite length:

T - T
T Log (Re) - T Log (IH)
T (R)
E
I
=
Log (R
I
E
) +

Re
Log (Re) - Log (IH)
Log

IH

2.2
Results of reference

R (m)
T (°C)
.30000
105.55
.32275
99.21
.34550
93.30
.36825
87.76
.39100
82.56

Value of the temperature according to R

R (m)
(W/m2)
.300
11577.49
.391
8822.98

Value of flow according to R

2.3
Uncertainty on the solution

Exact solution.

2.4 References
bibliographical

[1]
Guide validation of the software packages of structural analysis. French company of Mécaniciens
AFNOR 1990 ISBN 2-12-486611-7
Handbook of Validation
V4.41 booklet: Stationary thermics with radiation
HT-66/03/008/A

Code_Aster ®
Version
6.4
Titrate:
TPNA01 - Stationary axisymmetric Problème with Date radiation
:

02/06/03
Author (S):
Key J.P. LEFEBVRE
:
V4.41.001-C Page:
4/8

3 Modeling
With

3.1
Characteristics of modeling

Modeling 2D:

N1
N3
N6
N11
N9

3.2
Characteristics of the grid

2 QUAD8

3.3
Functionalities tested


Order Mot-clé Key word
Argument
factor
simple
DEFI_MATERIAU THER_NL
LAMBDA



BETA

THER_NON_LINE TEMP_INIT
STATIONARY “YES”

CONVERGENCE
RESI_GLOB_RELA
1.E-2


ITER_GLOB_MAXI
9


CRIT_LAGR_RELA
1.E-3


OPTION
“FLUX_ELNO_TEMP”

Handbook of Validation
V4.41 booklet: Stationary thermics with radiation
HT-66/03/008/A

Code_Aster ®
Version
6.4
Titrate:
TPNA01 - Stationary axisymmetric Problème with Date radiation
:

02/06/03
Author (S):
Key J.P. LEFEBVRE
:
V4.41.001-C Page:
5/8

4
Results of modeling A

4.1 Values
tested

The nodes observed have as a co-ordinate Z = 0.0

Identification
Reference
Aster %
difference
temperature
N1 (r=.30000)
105.55
105.52
­ 0.03
N3 (r=.32275)
99.21
99.17
­ 0.04
N6 (r=.34550)
93.30
93.26
­ 0.04
N11 (r=.36825)
87.76
87.73
­ 0.04
N9 (r=.39100)
82.56
82.53
­ 0.04

Identification
Reference
Aster %
difference
flow
net M1 N1 node
11577.49
11533.37
­ 0.38
net m2 N9 node
8822.98
8855.97
­ 0.37

4.2 Remarks



The boundary condition of the radiation type is provided in the form of a function of
temperature interpolated linearly between each point (one discretized the curve using 101 here
points).
Handbook of Validation
V4.41 booklet: Stationary thermics with radiation
HT-66/03/008/A

Code_Aster ®
Version
6.4
Titrate:
TPNA01 - Stationary axisymmetric Problème with Date radiation
:

02/06/03
Author (S):
Key J.P. LEFEBVRE
:
V4.41.001-C Page:
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5 Modeling
B

5.1
Characteristics of modeling

Modeling 3D:



5.2
Characteristics of the grid

32 HEXA20

5.3
Functionalities tested

Order Mot-clé
Key word
Argument
factor
simple
DEFI_MATERIAU THER_NL
LAMBDA



BETA


THER_NON_LINE TEMP_INIT
STATIONNAIRE
“OUI”

CONVERGENCE
RESI_GLOB_RELA
1.E-2


ITER_GLOB_MAXI
10


CRIT_LAGR_RELA
1.E-3


OPTION
“FLUX_ELNO_TEMP”
Handbook of Validation
V4.41 booklet: Stationary thermics with radiation
HT-66/03/008/A

Code_Aster ®
Version
6.4
Titrate:
TPNA01 - Stationary axisymmetric Problème with Date radiation
:

02/06/03
Author (S):
Key J.P. LEFEBVRE
:
V4.41.001-C Page:
7/8

6
Results of modeling B

6.1 Values
tested

The nodes observed have as co-ordinates: y = Z = 0.0

Identification
Reference
Aster
% difference
temperature
NO106 (x=.30000)
105.55
105.51
­ 0.03
NO105 (x=.32275)
99.21
99.17
­ 0.04
NO115 (x=.34550)
93.30
93.26
­ 0.04
NO125 (x=.36825)
87.76
87.73
­ 0.04
NO123 (x=.39100)
82.56
82.52
­ 0.04

Identification
Reference
Aster
% difference
flow
net MA17 node NO106
11577.49
11680.50
+0.89
net MA16 node NO123
8822.98
8968.85
+1.65

6.2 Remarks

The function of flow used in modeling A is also used here.

Handbook of Validation
V4.41 booklet: Stationary thermics with radiation
HT-66/03/008/A

Code_Aster ®
Version
6.4
Titrate:
TPNA01 - Stationary axisymmetric Problème with Date radiation
:

02/06/03
Author (S):
Key J.P. LEFEBVRE
:
V4.41.001-C Page:
8/8

7
Summaries of the results

The taking into account of the conditions of radiation is completely correct in this stationary case.
Let us note that this test utilized a coefficient of thermal conductivity constant, only
non-linearity thus relates to the boundary conditions.

The errors are higher on the calculation of flow, which one can explain by the use of
smoothings with the nodes, carried out starting from the computed values at the points of integration (points of
GAUSS).

Handbook of Validation
V4.41 booklet: Stationary thermics with radiation
HT-66/03/008/A

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