Code_Aster
®
Version
6.4
Titrate:
TTNL02 - Thermal transient with phase shift
Date:
11/06/03
Author (S):
J.P. LEFEBVRE
Key
:
V4.22.002-B
Page:
1/8
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/03/008/A
Organization (S):
EDF-R & D/AMA
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
Document
:
:
V4.22.002
TTNL02 - Thermal transient with change
of phase
Summary:
This elementary test makes it possible to deal with one-way problem in non-linear transitory thermics and of
to check the taking into account of a liquid/solid phase shift by Code_Aster while introducing by
the intermediate of the voluminal enthalpy latent heat of fusion. The solution is analytical and utilizes
functions of erf error and erfc. The problem is dealt with in the plane and voluminal cases.
For modelings presented here, the variations of the results obtained by Code_Aster range between 1 and
4% of the analytically calculated reference.
Code_Aster
®
Version
6.4
Titrate:
TTNL02 - Thermal transient with phase shift
Date:
11/06/03
Author (S):
J.P. LEFEBVRE
Key
:
V4.22.002-B
Page:
2/8
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/03/008/A
1
Problem of reference
1.1 Geometry
T
I
= 740 °C
T
m
= 660 °C
T
0
= 580 °C
0
L
Solid phase
Liquid phase
L = 0.1m
1.2
Material properties
They are the subscripted characteristics of aluminum by S for the solid phase and L for the phase
fluid. They are supposed to be constant within each phase.
3
9
3
3
6
3
3
J/m
H
C
T
J/kg
L
C
W/m
C
W/m
K
C
J/m
C
J/m
kg/m
m
1
S
1
6
1
S
S
10
1.08048
voluminal
enthalpy
of
variation
660.0
fusion
of
E
températur
10
437.44
fusion
of
latent
heat
95
210
thermics
t-piece
conductivi
10
2.58
10
3.00
voluminal
heat
2390
kg/m
2550
voluminal
mass
3
=
°
=
=
°
=
°
=
°
=
°
=
=
=
K
C
C
1.3
Boundary conditions and loadings
Temperature imposed at the ends.
L
=
=
in
C
740.0
in
C
580.0
X
T
0
X
T
I
0
°
°
=
=
1.4 Conditions
initial
Uniform initial temperature
C
740.0
°
=
=
I
init
T
T
Code_Aster
®
Version
6.4
Titrate:
TTNL02 - Thermal transient with phase shift
Date:
11/06/03
Author (S):
J.P. LEFEBVRE
Key
:
V4.22.002-B
Page:
3/8
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/03/008/A
2
Reference solution
2.1
Method of calculation used for the reference solution
One has an semi-analytical solution utilizing the functions of errors:
erf
E
erfc
E
()
()
and
X
dt
X
dt
T
T
X
X
=
=
-
-
+
2
2
2
2
0
This solution is valid for a semi-infinite medium, it could thus be used only in one
field of variation limited of the variable of time.
That is to say
X
T
the position of the solid interface/fluid. Are
S
L
T
S
D
T
total
T
S
=
=
and
2
where
D
D
S
L
and
the solid diffusivity of the mediums and fluid indicate
D
K
C
D
K
C
S
S
S
L
L
L
=
=
,
. The solution of the equation of
heat is form:
T X T
T
T
T
X
D T
X X
T X T
T
T
T
D
D
X
D T
X X
S
m
S
T
L
I
m
I
S
L
L
T
(,)
erf () erf
(,)
erfc
erfc
=
+
-
= +
-
0
0
2
2
if
if
The data of
T
total
is enough to define the solution, one thus fixes
420.0
=
total
T
2.2
Results of reference
TIME:
X-coordinate
0.5.1.0.1.5 2.0.2.5.3.0
.000
580.
580.
580.
580.
580.
580.
.005
682.43
661.33
647.50
638.74
632.69
628.20
.010
726.05
705.75
692.06
682.43
675.24
669.63
.015
738.11
728.70
718.44
709.60
702.23
696.06
.020
739.86
737.22
731.99
726.05
720.27
714.94
.025
740.
739.50
737.56
734.47
730.81
727.00
.030
740.
739.93
739.39
738.11
736.20
733.88
.035
740.
739.99
739.88
739.45
738.61
737.40
.040
740.
740.
739.98
739.86
739.55
739.00
.045
740.
740.
740.
739.97
739.87
739.65
.050
740.
740.
740.
740.
739.97
739.89
.055
740.
740.
740.
740.
740.
739.97
.060
740.
740.
740.
740.
740. 740.
.065
740.
740.
740.
740.
740.
740.
.070
740. 740. 740. 740. 740. 740.
.075
740. 740. 740. 740. 740. 740.
.080
740. 740. 740. 740. 740. 740.
.085
740. 740. 740. 740. 740. 740.
.090
740. 740. 740. 740. 740. 740.
.095
740. 740. 740. 740. 740. 740.
.100
740. 740. 740. 740. 740. 740.
Code_Aster
®
Version
6.4
Titrate:
TTNL02 - Thermal transient with phase shift
Date:
11/06/03
Author (S):
J.P. LEFEBVRE
Key
:
V4.22.002-B
Page:
4/8
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/03/008/A
TIME:
X-coordinate
3.5.4.0.4.5 5.0.5.5.6.0
.000 580.
580. 580.
580. 580.
580.
.005 624.68 621.84 619.48 617.48 615.25 614.25
.010 665.09 661.33 657.43 653.65 650.37 647.49
.015 690.83 686.33 682.43 678.99 675.95 673.22
.020 710.11 705.75 701.81 698.25 709.92 692.06
.025 723.23 719.60 716.17 712.95 720.89 707.09
.030 731.34 728.70 726.05 723.43 728.48 718.44
.035 735.89 734.18 732.34 730.43 733.42 726.53
.040 738.21 737.22 736.07 734.79 736.44 731.99
.045 739.29 738.77 738.11 737.33 738.18 735.47
.050 739.74 739.50 739.15 738.71 739.12 737.56
.055 739.91 739.81 739.65 739.42 739.60 738.75
.060 739.97 739.93 739.86 739.75 739.83 739.39
.065 739.99 739.98 739.95 739.90 739.93 739.72
.070 740.
739.99 739.98 739.96 739.97 739.88
.075 740.
740. 740.
739.99 739.99 739.95
.080 740.
740. 740.
740. 740.
739.98
.085 740.
740. 740.
740. 740.
739.99
.090 740.
740. 740.
740. 740.
740.
.095 740.
740. 740.
740. 740.
740.
.100 740.
740. 740.
740. 740.
740.
(In °C, according to the X-coordinate in meter and of time in seconds).
Note:
One limits oneself to the variations during the 6 first second, beyond 10 seconds the condition with
limit at the end
L
X
=
is not assured any more.
2.3
Uncertainty on the solution
Unknown factor, due to the evaluation of the functions of error.
2.4 References
bibliographical
[1]
Mr. Necati Özisik - Heat Conduction - Chapter 10: Phase-change problems example 10-3 -
John Wiley & Sounds.
Code_Aster
®
Version
6.4
Titrate:
TTNL02 - Thermal transient with phase shift
Date:
11/06/03
Author (S):
J.P. LEFEBVRE
Key
:
V4.22.002-B
Page:
5/8
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/03/008/A
3 Modeling
With
3.1
Characteristics of modeling
Modeling 2D:
N6
N11
N16
N21
3.2
Characteristics of the mesh
20 QUAD8
3.3
Functionalities tested
Order
Key word factor
Simple key word
Argument
DEFI_MATERIAU THER_NL
LAMBDA
BETA
THER_NON_LINE TEMP_INIT
CHAM_NO
INCREMENT
CONVERGENCE
RESI_GLOB_RELA
1.E-2
ITER_GLOB_MAXI
25
CRIT_LAGR_RELA
1.E-3
FILING
PARM_THETA
0.8
LAGRANGIAN
RHO
4
R 4
FILING
3.4 Notice
The latent heat of fusion is provided via the enthalpy on an interval of 0.01 °C.
H
=1.08 10
T
=0.01
Temperature
Voluminal enthalpy
9
Code_Aster
®
Version
6.4
Titrate:
TTNL02 - Thermal transient with phase shift
Date:
11/06/03
Author (S):
J.P. LEFEBVRE
Key
:
V4.22.002-B
Page:
6/8
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/03/008/A
4
Results of modeling A
4.1 Values
tested
The nodes observed have as a co-ordinate y = 0.0
Identification
temperature
Reference
Aster %
difference
T = 0.5 S N6 (X = 0.005)
682.43
676.24
0.908
T = 1.0 S N6 (X = 0.005)
661.33
640.58
3.137
T = 3.0 S N6 (X = 0.005)
628.20
615.72
1.987
T = 6.0 S N6 (X = 0.005)
614.25
605.37
1.446
T = 0.5 S N11 (X = 0.010)
726.05
736.92
+1.497
T = 1.0 S N11 (X = 0.010)
705.75
710.18
+0.628
T = 3.0 S N11 (X = 0.010)
669.63
648.30
3.185
T = 6.0 S N11 (X = 0.010)
647.49
629.97
2.707
T = 0.5 S N16 (X = 0.015)
738.11
741.19
+0.418
T = 1.0 S N16 (X = 0.015)
728.70
737.23
+1.170
T = 3.0 S N16 (X = 0.015)
696.06
692.52
0.508
T = 6.0 S N16 (X = 0.015)
673.22
652.62
3.059
T = 0.5 S N21 (X = 0.020)
739.86
741.60
+0.235
T = 1.0 S N21 (X = 0.020)
737.22
741.16
+0.534
T = 3.0 S N21 (X = 0.020)
714.94
723.46
+1.192
T = 6.0 S N21 (X = 0.020)
692.06
685.31
0.976
Calculation by finite elements requires a discretization in times of
S
10
5
T
4
-
=
at least for
first pitches. The boundary condition imposed at the origin making pass the temperature abruptly of
740.°C with 580.°C. One observes on the level of the first pitches of time some oscillations which
stabilize then rather quickly, despite everything the maximum temperature is exceeded, it does not have there
respect of the discrete maximum. This phenomenon is observed at the time of the thermal shocks, only one
particular digital processing on the level of the matrix of mass can cure this last.
Code_Aster
®
Version
6.4
Titrate:
TTNL02 - Thermal transient with phase shift
Date:
11/06/03
Author (S):
J.P. LEFEBVRE
Key
:
V4.22.002-B
Page:
7/8
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/03/008/A
5 Modeling
B
5.1
Characteristics of modeling
Modeling 3D:
No324
No312 No300 No277
5.2
Characteristics of the mesh
20 HEXA20
5.3
Functionalities tested
Order
Key word factor
Simple key word
Argument
DEFI_MATERIAU THER_NL
LAMBDA
BETA
THER_NON_LINE
TEMP_INIT VALE
740.0
INCREMENT
CONVERGENCE
RESI_GLOB_RELA
1.E-2
ITER_GLOB_MAXI
25
CRIT_LAGR_RELA
1.E-3
PARM_THETA
0.8
LAGRANGIAN
RHO
4
R
4
Code_Aster
®
Version
6.4
Titrate:
TTNL02 - Thermal transient with phase shift
Date:
11/06/03
Author (S):
J.P. LEFEBVRE
Key
:
V4.22.002-B
Page:
8/8
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/03/008/A
6
Results of modeling B
6.1 Values
tested
The nodes observed have as co-ordinates: X = y = 0.005
Identification
Temperature
Reference
Aster %
difference
T = 0.5 S No324 (Z = 0.005)
682.43
671.47
1.59
T = 1.0 S No324 (Z = 0.005)
661.33
698.97
3.38
T = 3.0 S No324 (Z = 0.005)
628.20
615.54
2.02
T = 6.0 S No324 (Z = 0.005)
614.25
612.41
0.30
T = 0.5 S No312 (Z = 0.010)
726.05
733.77
+1.06
T = 1.0 S No312 (Z = 0.010)
705.75
707.08
+0.19
T = 3.0 S No312 (Z = 0.010)
669.63
647.77
3.26
T = 6.0 S No312 (Z = 0.010)
647.49
645.01
0.38
T = 0.5 S No300 (Z = 0.015)
738.11
740.86
+0.373
T = 1.0 S No300 (Z = 0.015)
728.70
736.05
+1.01
T = 3.0 S No300 (Z = 0.015)
696.06
691.42
0.67
T = 6.0 S No300 (Z = 0.015)
673.22
670.77
0.36
T = 0.5 S No277 (Z = 0.020)
739.86
741.34
+0.20
T = 1.0 S No277 (Z = 0.020)
737.22
740.36
+0.48
T = 3.0 S No277 (Z = 0.020)
714.94
722.46
+1.05
T = 6.0 S No277 (Z = 0.020)
692.06
696.09
0.58
7
Summaries of the results
The error obtained compared to the analytical solution remains reasonable for the points of observation
listed in the tables. Let us announce however that the thermal shock imposed on the beginning of the transient
cause oscillations (when one observes the variation in the temperature in a point during
times) which diminish quickly and which disappeared at time T = 0.5 S.