background image
Code_Aster
®
Version
3
Titrate:
HSNL100 - Heating of a cable per Joule effect
Date:
03/12/96
Author (S):
Mr. AUFAURE
Key:
V7.21.100-A
Page:
1/6
Manual of Validation
V7.21 booklet: Thermomechanical nonlinear statics of the linear structures
HI-75/96/067/A
Organization (S):
EDF/IMA/MN
Manual of Validation
V7.21 booklet: Thermomechanical nonlinear statics of the linear structures
V7.21.100 document
HSNL100 - Heating of a cable per Joule effect
Summary:
This test relates to to it thermal transient independent of the space of the electric cables subjected to the Joule effect
and the thermo dynamic elasticity of these cables.
Interest:
·
to test the functions of evolution of the heating of a conducting cable per Joule effect, then of sound
cooling with the ambient air (operator
DEFI_THER_JOULE
[U4.21.09]),
·
to test the influence of the variation in temperature of a cable on the evolution of the arrow (operator
DYNA_NON_LINE
[U4.32.02]).
background image
Code_Aster
®
Version
3
Titrate:
HSNL100 - Heating of a cable per Joule effect
Date:
03/12/96
Author (S):
Mr. AUFAURE
Key:
V7.21.100-A
Page:
2/6
Manual of Validation
V7.21 booklet: Thermomechanical nonlinear statics of the linear structures
HI-75/96/067/A
1
Problem of reference
1.1 Geometry
A cable of section 1.71 10
­ 3
and of range 100 m hang in the field of gravity, with
temperature of 0°C. It is the seat, during 25 seconds, of a current of short-circuit which carries its
temperature with 1636°C. One follows the evolution of the position of balance.
T = 0°
T = 1636°
1.2
Properties of materials
Properties of conducting metal:
E = 5.4 10
10
AP
= 2761.4 kg/m
3
= 23. 10
­ 6
°C
­ 1
resistivity (T: temperature):
(T) = 3.25 10
­ 8
(1. + 3.6 10
­ 3
(T20.))
m
C
p
= 2.457.646 J m
­ 3
°C
­ 1
Convection coefficient for the losses of heat by the side wall of the cable:
0.5 J m
­ 2
°C
­ 1
1.3
Boundary conditions and loadings
The cable has its fixed ends. It is subjected to gravity and dilates by the Joule effect due to one
current of 70.000 A during 25 seconds.
1.4 Conditions
initial
The cable forms a chain, at the temperature of 0°C.
background image
Code_Aster
®
Version
3
Titrate:
HSNL100 - Heating of a cable per Joule effect
Date:
03/12/96
Author (S):
Mr. AUFAURE
Key:
V7.21.100-A
Page:
3/6
Manual of Validation
V7.21 booklet: Thermomechanical nonlinear statics of the linear structures
HI-75/96/067/A
2
Reference solution
2.1
Method of calculation used for the reference solution
·
Of first part, the test consists in checking if Aster tabule well two function-temperature
of heating and cooling. They are analytical functions, of exponential nature
compared to time, resulting from the exact integration of the equation of heat independent of
space but comprising a term of exchange of the Fourier type. This equation and its solution are
in [bib1] [R3.08.02].
The numerical value of the coefficients is in [bib2] [U4.21.09].
·
In second part, one compares the arrow of the problem of reference, at a given moment, with
theoretical arrow of the chain balance static of an inextensible cable, same length,
range and temperature. One can make it because the extensionnelle rigidity of the cable (produced EA)
is large and that the variation in temperature is slow.
As there is not analytical solution with the problem of reference, it is admitted that one
regular heating of 1600° in 25 seconds causes a quasi-static evolution. Speed
of descent of the medium of the cable is indeed about 0,3 m/s, whereas the speed of one
pendular motion with 0° reaches a value at least 30 times higher.
The static curve of balance of an inextensible cable whose ends are of level [2.1-a],
of range
S
, of linear weight
W
(
gA) and of horizontal voltage
H
has as an equation [bib3]:
Z
H
W
W
H
S
X
ws
H
=
-




-




cosh
cosh
.
2
2
éq 2.1-1
From where the length L is deduced:
W L
H
ws
H
2
2
=
sinh
.
éq 2.1-2
Z
X
F
P (S, 0)
O (0, 0)
Appear 2.1-a: Curves of balance of a cable
H
, which is constant along the cable since there is no horizontal force external, is given,
according to [éq 2.1-2], by the transcendent equation:
sinh X
L
S X
=
éq 2.1-3
where:
X
ws
H
=
2.
background image
Code_Aster
®
Version
3
Titrate:
HSNL100 - Heating of a cable per Joule effect
Date:
03/12/96
Author (S):
Mr. AUFAURE
Key:
V7.21.100-A
Page:
4/6
Manual of Validation
V7.21 booklet: Thermomechanical nonlinear statics of the linear structures
HI-75/96/067/A
The equation [éq 2.1-3] has a positive root
X
O
provided that:
L
S
>
1.
X
0
X
L
S
sinh X
1
The arrow
F
results then from [éq 2.1-1]:
(
)
F
cosh
.
=
-
S
X
X
O
O
2
1
The length of the cable with
T
O
, which intervenes in coefficient in [éq 2.1-3], rises the length
data with 0° by the equation of dilation:
()
()
(
)
L T
L
T
=
+
0
1
.
2.2 References
bibliographical
[1]
Mr. AUFAURE, G. DEVESA: Modeling of the cables in Code_Aster. Document
[R3.08.02] (1996).
[2]
Mr. AUFAURE: Operator DEFI_THER_JOULE. Document [U4.21.09] (1994).
[3]
H. MAX IRVINE: Cable structures. The MIT Close (1981).
background image
Code_Aster
®
Version
3
Titrate:
HSNL100 - Heating of a cable per Joule effect
Date:
03/12/96
Author (S):
Mr. AUFAURE
Key:
V7.21.100-A
Page:
5/6
Manual of Validation
V7.21 booklet: Thermomechanical nonlinear statics of the linear structures
HI-75/96/067/A
3 Modeling
With
3.1
Characteristics of modeling
Cable of 100 m range modelized by 20 elements of cable of the 1
Er
command. No the time of the analysis
dynamics: 0.25 seconds.
3.2 Functionalities
tested
Order
Key word
factor
Key word
Keys
DEFI_THER_JOULE
PARA_COND_1D
[U4.21.09]
AFFE_CHAM_NO
AFFE
SIZE
GROUP_NO
NOM_CMP
FUNCTION
[U4.26.01]
CREA_RESU
CHAM_GD
TYPE_RESU
NOM_CHAM
LIST_INST
CHAM_NO
[U4.26.02]
AFFE_CHAR_MECA
TEMP_CALCULEE
[U4.25.01]
STAT_NON_LINE
EXCIT
CHARGE
[U4.32.01]
DYNA_NON_LINE
EXCIT
CHARGE
[U4.32.02]
background image
Code_Aster
®
Version
3
Titrate:
HSNL100 - Heating of a cable per Joule effect
Date:
03/12/96
Author (S):
Mr. AUFAURE
Key:
V7.21.100-A
Page:
6/6
Manual of Validation
V7.21 booklet: Thermomechanical nonlinear statics of the linear structures
HI-75/96/067/A
4
Results of modeling A
4.1 Values
tested
Identification
Reference
Aster
% difference
Tabulation of 6 values of a function f01 of heating-cooling
moment ­ 0.1
12.0°C
12.0°C
1E6
moment 0.
1.0°C
1.0°C
1E6
moment 10.
44051.93°C
44051.93°C
1E6
moment 20.
2.9999°C
2.9999°C
1E6
moment 30.
88102.86°C
88102.86°C
1E6
moment 40.
4.9998°C
4.9998°C
1E6
Tabulation of 3 values of a function f02 of heating-cooling
moment ­ 0.1
15.0°C
15.0°C
1E6
moment 0.
15.0°C
15.0°C
1E6
moment 40.
15.0°C
15.0°C
1E6
Cable subjected to one 3
ème
function of heating f1
arrow at the moment 6.25s (T = 167°C)
1.614397 m
1.583216
1.9
arrow at the moment 12.50s (T = 441°C)
3.682028 m
3.640127
1.1
arrow at the moment 18.75s (T = 892°C)
6.157222 m
6.092494
1.1
arrow at the moment 25.00s (T = 1636°C)
9.244288 m
9.121316
1.3
4.2 Remarks
The dynamic arrow calculated by Aster is lower, from approximately 1%, with the static arrow with same
temperature. This difference results from mechanical inertia.
4.3 Parameters
of execution
Version: 3.06.11
Machine: CRAY C90
Overall dimension memory:
8 MW
Time CPU To use:
280 seconds
5
Summary of the results
The passage of this test guarantees that there no was regression of Code_Aster for the analysis of
evolution of the sag of the cables subjected to the Joule effect.