Calculation of temperature fields in ampoules under the radiation treatment by electrons with energy 10 MeV
To obtain the temperature fields in ampoules under the radiation treatment we use the one-dimensional equations of the thermal conductivity on the coordinates along the axis which is congruent with the direction of the beam. The results of the calculations of temperature fields within the ampoule ar...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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Bakai, A.S. Tanatarov, L.V. Gonchar, V.Yu. Sergiyeva, G.G. 2015-04-18T18:55:41Z 2015-04-18T18:55:41Z 2005 Calculation of temperature fields in ampoules under the radiation treatment by electrons with energy 10 MeV / A.S. Bakai, L.V. Tanatarov, V.Yu. Gonchar, G.G. Sergiyeva // Вопросы атомной науки и техники. — 2005. — № 4. — С. 32-39. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 81.65.K; 28.41.T https://nasplib.isofts.kiev.ua/handle/123456789/80540 To obtain the temperature fields in ampoules under the radiation treatment we use the one-dimensional equations of the thermal conductivity on the coordinates along the axis which is congruent with the direction of the beam. The results of the calculations of temperature fields within the ampoule are discussed for two cases: for the ampoule with the molten fluorides mix between three tested Hastelloy specimens (in section 1); and for the ampoule with exhalations of ftuorides salts (in section 2). These two situations are differed as in the energy losses of electron beam, as well in the mechanism of heat transport, that leads to essentially different temperature fields inside of the ampoule. It is shown that for ampoule with fluoride salts exhalations the stationary temperatures strongly depend from heat transport mechanisms and from layers thickness, that leads to conclusion about necessity to take into account both mechanisms of heat transport (thermal radiation and thermal conductivity) simultaneously. При розрахунках температурних полей в ампулі під випромінюванням електронами нами використовувались одновимірні рівняння теплопровідності із вісью х, яка збігається із напрямком потоку електронів. Одержані коордінатні залежності температури для двох випадків: 1) для ампули із с розплавом солей флюориду між випробуваними зразками Хастелоя; 2) для ампули із парами солей флюориду між шарами ампули.. Показано, що у другому випадку сталий розподіл температури істотно залежить від механизму переносу тепла і товщини шарів, що дозволяє зробити висновок про необхідність одночасно брати до уваги обидва механізми переносу тепла (теплове випромінювання і теплопровідніть шарів). Для вычисления температурных полей в ампуле под облучением электронами нами использовались одномерные уравнения теплопроводности с осью х, совпадающей с направлением потока электронов. Получены координатные зависимости температуры для двух случаев: 1) для ампулы с расплавом солей флюорида между испытуемыми образцами Хастеллоя; 2) для ампулы с парами солей флюорида между образцами Хастеллоя. Показано, что во втором случае установившееся распределение температуры существенно зависит от механизма переноса тепла и от толщины слоев, что позволяет сделать вывод о необходимости принимать во внимание оба механизма переноса тепла одновременно( тепловое излучение и теплопроводность слоев). This work was supported in part by STCU, Project # 294. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Calculation of temperature fields in ampoules under the radiation treatment by electrons with energy 10 MeV Розрахунки температурних полей у ампулах при радіаційних випробуваннях електронами з енергією 10 МеВ Расчеты температурных полей в ампулах при радиационных испытаниях электронами с энергией 10 МэВ Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Calculation of temperature fields in ampoules under the radiation treatment by electrons with energy 10 MeV |
| spellingShingle |
Calculation of temperature fields in ampoules under the radiation treatment by electrons with energy 10 MeV Bakai, A.S. Tanatarov, L.V. Gonchar, V.Yu. Sergiyeva, G.G. |
| title_short |
Calculation of temperature fields in ampoules under the radiation treatment by electrons with energy 10 MeV |
| title_full |
Calculation of temperature fields in ampoules under the radiation treatment by electrons with energy 10 MeV |
| title_fullStr |
Calculation of temperature fields in ampoules under the radiation treatment by electrons with energy 10 MeV |
| title_full_unstemmed |
Calculation of temperature fields in ampoules under the radiation treatment by electrons with energy 10 MeV |
| title_sort |
calculation of temperature fields in ampoules under the radiation treatment by electrons with energy 10 mev |
| author |
Bakai, A.S. Tanatarov, L.V. Gonchar, V.Yu. Sergiyeva, G.G. |
| author_facet |
Bakai, A.S. Tanatarov, L.V. Gonchar, V.Yu. Sergiyeva, G.G. |
| publishDate |
2005 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Розрахунки температурних полей у ампулах при радіаційних випробуваннях електронами з енергією 10 МеВ Расчеты температурных полей в ампулах при радиационных испытаниях электронами с энергией 10 МэВ |
| description |
To obtain the temperature fields in ampoules under the radiation treatment we use the one-dimensional equations of the thermal conductivity on the coordinates along the axis which is congruent with the direction of the beam. The results of the calculations of temperature fields within the ampoule are discussed for two cases: for the ampoule with the molten fluorides mix between three tested Hastelloy specimens (in section 1); and for the ampoule with exhalations of ftuorides salts (in section 2). These two situations are differed as in the energy losses of electron beam, as well in the mechanism of heat transport, that leads to essentially different temperature fields inside of the ampoule. It is shown that for ampoule with fluoride salts exhalations the stationary temperatures strongly depend from heat transport mechanisms and from layers thickness, that leads to conclusion about necessity to take into account both mechanisms of heat transport (thermal radiation and thermal conductivity) simultaneously.
При розрахунках температурних полей в ампулі під випромінюванням електронами нами використовувались одновимірні рівняння теплопровідності із вісью х, яка збігається із напрямком потоку електронів. Одержані коордінатні залежності температури для двох випадків: 1) для ампули із с розплавом солей флюориду між випробуваними зразками Хастелоя; 2) для ампули із парами солей флюориду між шарами ампули.. Показано, що у другому випадку сталий розподіл температури істотно залежить від механизму переносу тепла і товщини шарів, що дозволяє зробити висновок про необхідність одночасно брати до уваги обидва механізми переносу тепла (теплове випромінювання і теплопровідніть шарів).
Для вычисления температурных полей в ампуле под облучением электронами нами использовались одномерные уравнения теплопроводности с осью х, совпадающей с направлением потока электронов. Получены координатные зависимости температуры для двух случаев: 1) для ампулы с расплавом солей флюорида между испытуемыми образцами Хастеллоя; 2) для ампулы с парами солей флюорида между образцами Хастеллоя. Показано, что во втором случае установившееся распределение температуры существенно зависит от механизма переноса тепла и от толщины слоев, что позволяет сделать вывод о необходимости принимать во внимание оба механизма переноса тепла одновременно( тепловое излучение и теплопроводность слоев).
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/80540 |
| citation_txt |
Calculation of temperature fields in ampoules under the radiation treatment by electrons with energy 10 MeV / A.S. Bakai, L.V. Tanatarov, V.Yu. Gonchar, G.G. Sergiyeva // Вопросы атомной науки и техники. — 2005. — № 4. — С. 32-39. — Бібліогр.: 8 назв. — англ. |
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2025-11-25T22:19:42Z |
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2025-11-25T22:19:42Z |
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1850562663123255296 |
| fulltext |
PACS: 81.65.K; 28.41.T
CALCULATION OF TEMPERATURE FIELDS IN AMPOULES
UNDER THE RADIATION TREATMENT BY ELECTRONS WITH
ENERGY 10 MeV
O.S. Bakai, L.V. Tanatarov, V.Yu. Gonchar and G.G. Sergiyeva
National Science Center “Kharkiv Institute for Physics and Technology”
Kharkiv, Ukraine, gsergeeva@kipt.kharkov.ua
To obtain the temperature fields in ampoules under the radiation treatment we use the one-dimensional equations
of the thermal conductivity on the coordinates along the axis which is congruent with the direction of the beam. The
results of the calculations of temperature fields within the ampoule are discussed for two cases: for the ampoule with
the molten fluorides mix between three tested Hastelloy specimens (in section 1); and for the ampoule with
exhalations of ftuorides salts (in section 2). These two situations are differed as in the energy losses of electron
beam, as well in the mechanism of heat transport, that leads to essentially different temperature fields inside of the
ampoule. It is shown that for ampoule with fluoride salts exhalations the stationary temperatures strongly depend
from heat transport mechanisms and from layers thickness, that leads to conclusion about necessity to take into
account both mechanisms of heat transport (thermal radiation and thermal conductivity) simultaneously.
INTRODUCTION Here the results of the calculations of temperature
fields within the ampoule are discussed for two cases:
for the ampoule with the molten fluorides mix between
three tested metallic spacimens (in section 1); and for
the ampoule with exhalations of ftuorides salts (in
section 2). The later situation can take place in the case
when the temperature of the molten fluorides mix is
compared with the sublimation temperature, and the
tested metallic specimens are located in exhalations of
fluorides salts. These two situations are differed in the
energy losses of electron beam, as well in the
mechanism of heat transport, that leads to essentially
different temperature fields inside of the ampoule.
When the projecting the EITF test bench the choice
of the geometry of the irradiated ampoules was made so
that the energy losses of electron beam were equal to the
amount of thermal energy irradiated by target. During
the treatment mean current was regulated so that the
temperature of the surfaces of ampoules was 650ºС.
temperature fields within the ampoule is heterogeneous
because of the intrinsic heterogeneity of its construction
[1], and the finite values of the ampoule contents’
thermal conductivity coefficient. It is known that the
corrosion processes are controlled by the velocity of
chemical reactions and the diffusion transport which are
essentially temperature dependent. That’s why the
knowledge of the temperature distribution inside
ampoules is very crucial when one analyzing the tests
results. The distribution of energy losses of the electron
beam within the ampoule was received by means of
Monte-Carlo calculations in Ref. [2]. The ampoules and
the assemblies are such that with sufficient accuracy we
can consider the distribution of the energy losses and of
the temperature as one-dimensional and essentially
dependent only on the coordinates along the axis which
is congruent with the direction of the beam. This means
that we can use the one-dimensional equation of the
thermal conductivity to obtain the temperature fields.
We have not precition values of thermal conductivity
coefficients of the molten fluorides mix, and of tested
metallic specimens at the temperature ºC, and
the data from Refs.[3-4] was used at the numerical
estimation.
1. THE CALCULATION OF TEMPERATURE
FIELDS WITHIN THE AMPOULE
WITH THE MOLTEN FLUORIDES MIX
Taken into account that the sizes of intrinsic
heterogeneity (≤ 0.2 cm) along axis x, which is
congruent with the direction of the beam (see fig. 1), are
much less than vertical dimensions of ampoule (5 cm
along axis y) [1], we can use the one-dimensional
equation of the thermal conductivity for calculation of
the temperature fields )(xT
QdivJ
t
Tc Ev =+
∂
∂ .
650≥T Here and κEE divJQTdivJ −=∇−= ,κ is thermal
conductivity coefficient. Energy flux ,
transported at point x by electron beam with initial
energy is equal to
)(xJ E
According to our estimations the temperature in the
middle of the ampoule is nearly ten degrees greater than
on the surface of the ampoule. This result allow us to
consider the evalution data from Refs.[3-4] satisfactory
if the analysis of experimantal data for corrosion of
tested metallic spacemens does not require calculations
of greater accuracy.
0E
,)(
,'
'
)'()()(
0
S
WbJ
dx
x
xE
SE
WbJxJ
E
x
b
EE
=−
∂
∂
−−= ∫
− (1)
_______________________________________________________________________________
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2005. №.4.
Серия: Физика радиационных повреждений и радиационное материаловедение (87), с. 32-39. 32
33
where is flux energy on irradiated surface of
the ampoule, which is equal to beam power
S
'
'
)'()( 0 dx
x
xEExE
x
b
∫
− ∂
∂
−=)( bJ E −
W on
1cm , is energy losses of electron
at point x. Here , (2)
2 )(/)( xxxE ϕ≡∂∂ is the energy of an electron at point x.
Fig. 1. Temperature fields within the ampoule with molten fluoride salts under the radiation treatment
Energy losses due to the processes of ionization and
atomic excitation, are well discribed by Bloch-Bethe
formula for a target with thickness less than the track
lengh for electron[5]. The target ampoule is made from
the Carbon-Carbon (C-C) composite material. But for
C-C composite material lengh of electron track is
~2сm, and this formula cannot be used for thick layer
(here thickness of C-C composite layer is 1.5 сm (see
fig. 1)). The region
Hastelloy Nickel-Molybdenum-Chromium alloy), the
thickness of layer is equal cmhhh 06.0321 =Δ=Δ=Δ ,
(see fig. 1). According Ref. [2] the mean values of
energy losses in these layers are equal to
( )3,65.0,7,8
3
333231 cm
MeV
cm
MeV
cm
MeV
hhh ≈≈≈ ϕϕϕ .
bxaaxb <<−<<− ,
are the first (g1)and the second (g2) C-C composite
layers ( ) with thermal
conductivity coefficient
.09.0,6.0
,5,7
3433
3231
cm
MeV
cm
MeV
cm
MeV
cm
MeV
ff
ff
==
==
ϕϕ
ϕϕ
For the thermal conductivity coefficients fκ of molten
fluoride salts and tested metallic specimens
cmacmb 5.0,2 ==
hκ at
we used evalution data from
Refs. [3-4]:
cmKsek
erg710 KTK 9231000 ≥≥7.0| 900 ≈= KTgκ × [3]. For a thick target
the main problem in energy losses calculations consists
in taking into account multiple scattering of electrons,
and in the calculation of the energy losses profile for
each layer we used the results of the Monte Carlo
computer modeling method from Ref. [2]. For the first
C-C composite layer (g1) the mean value is
sekKcm
erg7102.0≈≈ hf κκ × .
At thermal balance we can find the stationary
distrbution of temperature fields within the ampoule
from thermal conductivity equations with the source of
heat (energy flux ) and sink of heat due to thermal
radiaton by C-C composite layers in camera where
EJ
3
1 /3 cmMeVg ≈ϕ=)(1 xgϕ , and for second C-C
composite layer (g2) it is
4Tσ
σ is Stephan-Boltzman constant. Let
be the temperature fields for C-C composite sides of the
ampoule, is the temperature field for i-th
molten fluorides liquid mix layer (i=1,2,3.4), and
is the temperature field for i-th tested metallic
specimen hi (i=1,2,3).
)(),( 21 xTxT gg
3
22 /03.0)( cmMeVx gg == ϕϕ .
)(xTfi
5.05.0 ≤≤− xThe inside of the ampoule has four the
layers of molten fluorides liquid mix (the thickness of
each layer is equal
fi
)(xThi
), cmffff 205.04321 =Δ=Δ=Δ=Δ
and three layers of tested metallic specimens (species of
1.1. THE TEMPERATURE FIELD FOR and the first integral of thermal conductivity is equal to
THE C-C COMPOSITE FIRST LAYER
'.)'(
|
)(
5.0
1
0
5.0
11
dxx
SE
W
x
T
x
xT
x
f
x
f
f
f
f
∫
−
−=
−=
=
∂
∂
−
∂
∂
ϕ
κκThermal conductivity equation for the C-C
composite first layer
(9)
5.02 −≤≤− x is
)(
)(1 xJ
xx
xT
x E
g
g ∂
∂
=
∂
∂
∂
∂ κ ,
After integration of (9) with the use second boundary
2
14
1 |)2( −=∂
∂
=− x
g
gg x
T
T κσ
condition (6) we receive the equation for the temperature , (4) fields
')')('(
)5.0]()(
)2([)]5.0()([
5.0
1
0
5.0
2
1
0
4
111
dxxxx
SE
W
xdxx
SE
W
TTxT
x
f
g
gfff
−−
−+−
−−=−−
∫
∫
−
−
−
ϕ
ϕ
σκwhere ~ 650ºC is the temperature of the )2(1 −gT
irradiated surface of the first C-C composite layer. ,
Second boundary condition is
5.015.01 |)(|)( −=−= ∂
∂
=
∂
∂
xffxgg xT
x
xT
x
κκ , (5)
and then after integrating (4) from to x we get 2−
and the temperature field with mean values of energy
losses is equal )2()(|
)(
2
11 −−=
∂
∂
−
∂
∂
−= EEx
g
g
g
g JxJ
x
T
x
xT
κκ , (6)
)10(].)5.0(
2
5.1
)2()[5.0()]5.0()([
0
1
0
1
4
111
+−−
−−+=−−
x
SE
W
SE
W
TxTxT
fg
gfff
ϕϕ
σκ
and taking into account the first boundary condition at
x= (4), we receive
2−
'
'
)'()2()2(
)()2(
)(
20
4
1
4
1
1
dx
dx
xdE
SE
WTJ
xJT
x
xT
x
gE
Eg
g
g
∫
−
−−=−−
−+−=
∂
∂
σ
σκ
34
. (7)
The substitution of x=-0.295 into (10) gives us the
equation for the temperature at the right
hand boundary of the first layer of molten salts
)295.0(1 −fT
]2/205.05.1()2(~[205.0
)]5.0()295.0([
11
0
4
2
11
fg
fff
SE
WT
TT
ϕϕσ
κ
−−−=
=−−−
So after integrating (7) from to x, we receive
the temperature field for the first C-C composite layer
:
2− .
)(1 xTg
In the first layer of Hastelloy 1h
'')''('
)2()2()]2()([
2
'
2
1
0
4
111
dxxdx
SE
W
TxTxT
x x
g
gggg
∫ ∫
− −
−
−−+=−−
ϕ
σκ 235.0295.0 −≤≤− x the energy flux is
,
where .
∫
∫ ∫∫ ∫
−
−− −
−=
==
x
g
x x
x
g
x x
g
xxxdx
dxxdxdxxdx
2
1
2 '
1
2
'
2
1
)')('('
')''('''')''('
ϕ
ϕϕ
110
235.0
1
0
205.05.1)295.0(
],')'()295.0([)(
fg
x
fE
EE
dxxE
SE
WxJ
ϕϕ
ϕ
−−=−
−−= ∫
− ,
' and after integration we receive )(1 xTh
]}.2/)295.0(
205.05.1[)2({
)295.0()]295.0()([
1
11
0
4
1
11
h
fgg
fhf
x
SE
WT
xTxT
ϕ
ϕϕσ
κ
++
++−−×
×+=−−
(11)
The temperature difference on the boundaries of first C-
C composite layer is
].
2
5.1)2([5.1
)]2()5.0([
1
0
4
1
11
gg
ggg
SE
WT
TT
ϕσ
κ
−−=
=−−−
(8) The substitution of x=-0.235 into (11) gives us the
equation for the temperature at the )235.0(1 −hT
right hand boundary of the first layer of Hastelloy
).11(]}2/06.0205.05.1[
)2({06.0)295.0(|
111
0
4
11235.01
a
SE
W
TTT
hfg
gffhf
ϕϕϕ
σκκ
++−
−−+−=−1.2. THE TEMPERATURE FIELDS
IN THE INSIDE OF THE AMPOULE
Taking into account that all seven layers in the
inside of the ampoule have essentially different values
of energy losses, we must receive thermal conductivity
equations separately.
In the second layer of molten salts 2f
03.0235.0 −≤≤− x the energy flux is
In the first layer of molten salts 1f
,06.0205.05.1)235.0(
],')'()235.0([)(
1110
235.0
1
0
hfg
x
fE
EE
dxxE
SE
WxJ
ϕϕϕ
ϕ
−−−=−
−−= ∫
−
295.05.0 −≤≤− x the energy flux is equal
dxxEE
dxxE
SE
WxJ
g
x
fE
)()5.0(
],')'()5.0([)(
5.0
2
10
5.0
1
0
∫
∫
−
−
−
−=−
−−=
ϕ
ϕ
,
that leads to the equation for the temperature )(2 xTf
1.3. THE TEMPERATURE FIELD FOR SECOND
THE C-C COMPOSITE LAYER
]}.2/)235.0(06.0
205.05.1[)2({
)235.0()]235.0()([
21
11
0
4
1
12
fh
fgg
hff
x
SE
WT
xTxT
ϕϕ
ϕϕσ
κ
+++
++−−×
×+=−−
(12) 25.0 ≤≤ xIn second the C-C composite layer the
energy flux is equal
;205.006.0205.006.0205.0
06.0205.05.1)5.0(
],')'()5.0([)(
43322
1110
5.0
2
0
fhfhf
hfg
x
gE
EE
dxxE
SE
WxJ
ϕϕϕϕϕ
ϕϕϕ
ϕ
−−−−−
−−−−=
−= ∫
Taking x=-0.03 we find the temperature )03.0(2 −fT at
the right hand boundary of the second layer of molten
salts:
and the first integral of thermal conductivity equation is
]}2/205.006.0
205.05.1[)2({
205.0)235.0()03.0(
21
11
0
4
1
12
fh
fgg
hfff
SE
WT
TT
ϕϕ
ϕϕσ
κκ
++
++−−×
×+−=−
'.)'(|
5.0
1
0
5.0
22 dxx
SE
W
x
T
x
T x
fx
g
g
g
g ∫
−
−= −=
∂
∂
−
∂
∂
ϕκκ (18) . (13)
Using second boundary condition
)5.0()5.0( 42 fg TT =
after integration of (18) we receive second integral of
thermal conductivity equation For the rest temperature fields
in the inside of
the ampoule the similar calculations are consequtively
repeated, and we determine the temperature fields by
means of the system of equations:
)(),(),(),( 4332 xTxTxTxT fhfh
]}.2/)5.0(205.006,0
205.006,0205.006.0
205.05.1[)2({
)5.0()]5.0()([
243
3221
11
0
4
2
42
gfh
fhfh
fgg
fgg
x
SE
WT
xTxT
ϕϕϕ
ϕϕϕϕ
ϕϕσ
κ
−+++
+++++
++−−×
×−=−
. (19)
Second Hastelloy layer Taking x=2 in (19) we can find the temperature
at the right hand boundary of the second layer
of C-C composite.
.03.003.0
]},2/)03.0(06.0205.0
5.1[)2(){03.0
()]03.0()([
211
1
0
4
1
22
≤≤−
++++
+−−+
+=−−
x
x
SE
WT
xTxT
fhf
gg
fhf
ϕϕϕ
ϕσ
κ )2(2gT
(14) Thus, in sections 1.1-1.3 we received the equations
for the calculations of the temperature fields in each
layer of ampoule under the radiation treatment by
electrons with energy 10 MeV. These equations ought
to be completed by the equation for determination of the
temperature
)2(1 −gTThird molten salts layer , for which above we used the
result of the experimental measurements[2]. We get this
equation from summarizing all the equations for the
temperature difference on the boundaries of each layer:
( )
.235.003.0
]},2/03.0235.006.0
205.05.1[)2({
)03.0()]03.0()([
221
11
0
4
1
23
≤≤
−+++
++−−×
×−=−
x
x
SE
WT
xTxT
hfh
fgg
hff
ϕϕϕ
ϕϕσ
κ
(15)
].
)()(
)([)]2()2([
2
5.0
2
5.0
295.0
4
295.0
235.0
3
03.0
03.0
235.0
03.0
32
03.0
235.0
2
235.0
295.0
1
295.0
5.0
1
5.0
2
1
0
4
2
4
1
dxdx
dxdxdx
dxxdxdxx
dxx
SE
WTT
gf
hfh
fhf
ggg
∫∫
∫∫ ∫
∫∫∫
∫
++
++++
++++
+=+−
−
−
−
−
−
−
−
−
−
ϕϕ
ϕϕϕ
ϕϕϕ
ϕσ
(20)
Third Hastelloy layer
.295.0235.0.
]},2/)235.0(06.0
235.006.0205.0
5.1[)2(){235.0
()]235.0()([
33
221
1
0
4
1
33
≤≤
−++
++++
+−−−
−=−
x
x
SE
WT
xTxT
ff
fhf
gg
fhf
ϕϕ
ϕϕϕ
ϕσ
κ
(16) This estimation comes to agreement with the results of
measurements [1] and of our calculations
650)2()2( 21 ≈≈− gg TT ºC.
1.4. THE RESULTS OF CALCULATIONS
OF THE TEMPERATURE FIELDS Fourth molten salts layer
35
.5.0295.0
]},2/)295.0(06.0
205.006.0205.0
5.1[)2(){295.0
()]295.0()([
33
221
1
0
4
1
34
≤≤
−++
++++
+−−−
−=−
x
x
SE
WT
xTxT
fh
fhf
gg
hff
ϕϕ
ϕϕϕ
ϕσ
κ
(17)
We caried out calculations of the temperature
fieldsfor each layer of the ampoule under radiation
treatment by electrons with energy 10 MeV. The area of
an am¬poules assemblage м . The
parameters of electron beam are: initial energy
power beam /сm 2 . For all
layers of the ampoule under radiation treatment we
2cS 3200 =
;100 MeVE = kwW 5=
The third layer h3 of Hastelloy received nine equations with 0/ SEW kiki ϕϕ =′ , where
k is label of layer, and i is number of this layer:
.)(01.92739.0)235.0()295.0(
];2/)235.0(4.0317.1[
2.0
235.0)235.0()(
:/104.0,295.0235.0
33
33
7
3
KTT
x
xTxT
cmergx
fh
fh
h
≈−≅
−−−×
×
−
+=
×≈′≤≤ ϕ
The C-C composite first layer
:/1098.1,5.02 7
1 cmergx g ×≈′−≤≤− ϕ
.)67.5923()5.0(
],2/)2(137.4[
7.0
2923)(
1
11
KT
xxxT
g
gg
+≈−
′+−
+
+≅ ϕ
.
The fourth layer f4 of molten salts
.)(96.92535.1)295.0()5.0(
];2/)295.0(09.0341.1[
2.0
295.0)295.0()(
:/1009.0,5.0295.0
34
34
7
4
KTT
x
xTxT
cmergx
hf
hf
f
≈−≅
−−−×
×
−
−=
×≈′≤≤ ϕ
The first layer of molten salts
:/1095.4,295.05.0 7
1 cmergx f ×≈′−≤≤− ϕ
.66.0)5.0()295.0(
)];295.0(95.4
167.1[
2.0
295,0)5.0()(
11
11
+−≅−
+−
−
+
+−≅
gf
gf
TT
x
xTxT
, The C-C composite second layer
:/1002,0,25.0 7
2 cmergx g ×≈′≤≤ ϕ
.02.9237.2)5.0()2(
];2/)5.0(02.0359.1[
7.0
)5.0()5.0()(
42
42
KTT
x
xTxT
fg
fg
≈−≅
−−−×
×
−
+=
The first layer of Hastelloy 1h
./1026.7,235.0295.0 7
1 cmergx h ×≈′−≤≤− ϕ
]2/)295.0(26.7153.0[
2.0
295.0)295.0()( 11 +−
+
+−= xxTxT fh
, The results of calculations of temperature fields in
ampoules under the radiation treatment by the electrons
with energy 10 MeV show that typical for the energy
losses of electron beam maximum is situated within the
first half of the middle of an ampoule and leads to the
highest temperatures in the first and second layers of
molten salts and the first Hastelloy (nearly 7ºC
relatively of the temperature of the surfaces of an
ampoule, see fig.1). We can suppose that such small
(~1%) temperature heterogeneities weakly become
apparent in corrosion process and we may does not take
account these heterogeneities at the analysis of the
testing metallic specimens if the analysis of
experimantal data for corrosion of tested metallic
specimens does not require calculations of greater
accuracy.
02.0)295.0()235.0( 11 −−=− fh TT ;
.36.929]36.6)2([| 1275.0max,1 KKTT gxh ≅+−≈−=
The second layer of molten salts 2f
:/103.3,03.0235.0 7
2 cmergx f ×≈′−≤≤− ϕ
.]65.0)235.0([)03.0(
];2/)235.0(3.328.0[
2.0
235.0)235.0()(
12
12
KTT
x
xTxT
hf
hf
−−≅−
+−−×
×
+
+−=
The second layer h2 of Hastelloy But these temperature heterogeneities it is need to
take into account in the case when the temperature in
the molten fluoride salts (~660ºC ) is compared with
their sublimation temperature and over long period (~
700 hours) of testing time metallic specimens were
located in exhalations of fluoride salts under
atmospheric pressure. This situation considerably
distinguishes from above studied and requires to be
separately studied because the energy losses of electron
beam in the ampoule without molten salts essentially
distinguish from the losses in the ampoule with molten
salts as well as the mechanisms of heat transport.
.67.92833.0)03.0()03.0(
];2/)03.0(62.496.0[
2.0
03.0)03.0()(
:/1062.4,03.003.0
22
22
7
2
KTT
x
xTxT
cmergx
fh
fh
h
≈−−≅
+−−×
×
+
+−=
×≈′≤≤− ϕ
The third layer f3 of molten salts
.)(40.92714.1)03.0()235.0(
];2/)03.0(4.0
1.1[
2.0
03.0)03.0()(
:/104.0,235.003.0
23
23
7
3
KTT
x
xTxT
cmergx
hf
hf
f
≈−≅
−−
−−
−
+=
×≈′≤≤ ϕ
2. THE CALCULATION OF TEMPERATURE
FIELDS WITH IN THE AMPOULE
WITH EXHALATIONS OF FLUORIDE
SALTS UNDER THE RADIATION
In these section we calculate the temperature fields
for an ampoule which is located in camera with
36
temperature ºC of walls, and is irradiated by
electrons with energy 10 MeV . The inside of the
ampoule contains three layers of tested Hastelloy
spacemens, and four spaces between spacemens which
are filled up by exhalations of fluoride salts under
atmospheric pressure. This means that thermal
conductivity equations between spacemens are
homogenius
37
300 ≅T
0/ 22 =∂∂ xTnexκ , (2.1)
where exκ is thermal conductivity coefficient of the
exhalations of fluoride salts. At thermal balance we can
find the stationary distribution of temperature fields
within the ampoule from thermal conductivity equations
with the source of heat (energy flux ) and sink of
heat due to thermal radiation by C-C composite layers
in camera, and to thermal radiation by inner layers of
the ampoule. The thermal conductivity equations for
five solid state layers ( two are C-C composite ones with
n=1 and n=5 and three layers are tested Hastelloy
specimens) are
EJ
5...2,1,/2 =−=∂∂ nxT nnn ϕκ , (2.2)
where nκ is tne thermal conductivity coefficient of n-
layer. For energy losses of electron
)(/)( xxxE nn ϕ=∂∂
in each layer we use the results of Monte-Carlo
calculations [8] (see fig.2, where empty squares are
energy losses of secondary electrons, and filled squares
are energy losses of primary electrons).With taking into
account both mechanisms of heat exchange (thermal
radiation and thermal conductivity) the boundary
conditions for left-hand and right-hand boundaries of n-
th layer are
,)()(|
,)()(|
21,1
4
2,
4
1,12
2,11
4
2,1
4
1,1
lTTTT
x
T
lTTTT
x
T
nnexnn
n
n
nnexnn
n
n
−+−=
∂
∂
−+−=
∂
∂
++=
−−=
κσκ
κσκ
α
α (2.3)
Fig. 2. The results of Monte-Carlo calculations [8] of
the energy losses of electron ( ) / ( )n nE x x xϕ∂ ∂ = for
three Hastelloy layers within the ampoule with
exhalations of fluoride salts between layers
where l is thickness of the streak of fluorides salts
exhalations between solid state layers, is the
temperature on left boundary n-th layer at
α,nT
,1=α and
on right one at .2=α Integration of (2.2) from left
boundary n-th layer to x leads to 1nx
dxx
x
T
x
T x
x
n
n
n
n
n
n
)(|
1
1 ∫−∂
∂
=
∂
∂
= ϕκκ α
, (2.4)
where is coordinate of αnx −α boundary of n-th
layer. Later on we would suppose that nn x ϕϕ ≈)( ,
and after second integration of (2.4) we have
.')'(
)(|])([
1
111
∫ −−
−−
∂
∂
=− =
x
x
n
n
n
nnnn
n
dxxx
xx
x
TTxT
ϕ
κκ α
(2.5)
2nxx = At
lTTll
lTTl
l
x
TTT
nnnexnn
nnnnn
n
n
nnnn
/)(2/
)(2/
|][
2,11
2
4
2,1
4
1
2
212
−
−
=
−+−
−−=−
−
∂
∂
=−
κϕ
σϕ
κκ α
,
where is thickness of n-th layer. At we
receive
nl 2nxx =
nnnnnn
ex
nnnn
lTTTT
l
TTTT
ϕκ
σ
=−+−+
+−+−
−+
−+
)(
)(
2,111,12
4
2,1
4
1,
4
1,1
4
2
.
Let us to normalize the temperature on boundariesof n-
th layer:
0201 /,/ TTVTTu nnnn ≡≡ ,
and from two last equations we receive the system of
equations of fourth order for : nn Vu ,
2
)( 1
4
1
4 n
nnnnnnn VVVbubu μνν +++=++ −− ,(2.6)
nnnn
nnnnn
VVV
Vuuuu
μν
ν
−−+−
−=−+−
−−
++
)(
)(
1
4
1
4
1
44
1
. (2.7)
3
0
3
0
4
0
,,
TllT
b
T
l ex
n
n
n
nn
n σ
κν
σ
κ
σ
ϕμ === . (2.8)
Thus we received the system of the equations with
boundary conditions for dimensionless temperatures on
camera walls: Summarizing
(2.7) from n=1 to n with taking into account boundary
conditions leads to
.1,1 01 ==+ VuN
38
1n -
4 4
2 21 1
4
1 1 1
4 4
3 2 3 2
4 4
2 2 21 1
4 4
1 1
4 4
2 21 1
4
( )
1 ( 1)
( )
( )
....................................................
( )
( )
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
n nn n
n nn n
n
u u u u
V V
u u u u
V V V V
u u u u
V V V V
u
n
n m
n
n m
n
n m
n
- -
- -- -
- + - =
= - + - -
- + - =
= - + - -
- + - =
= - + - -
+ 4 4
1 1 1 1
1
1
(1 )
n n n
n
i
i
u V V u un n
n m
- -
-
=
= + + + -
- + - е
, (2.9)
and to the balance equation
∑
=
++=+++
N
i
iNN VVuu
1
4
1
4
1 )1(2 μννν . (2.10)
Rewriting (2.6) as
2
)( 1
4
1
4 n
nnnnnnn VVuVbuu μνν +++−=+ −− ,
and equating right-hand part of (2.10) and (2.9), we
receive the equation
]
2
1[ 1
4
1
1
1 uub
Vu
n
n
i
inn
nnn
νμμνλ
λ
−−−++=
+=
∑
=
−
(2.11)
With taken into account that nex κκ p we can consider
0≈ν , that let us express all :
,
and so on to , that let us to receive the
equation for determination of the temperature on left-
hand boundary of first C-C composite layer,
)( 1uVV nn =
∑∑
−
==
− +−+−++=
1
1
4
1
44/14
1
1
4
1 1])2([
N
i
i
N
i
inN uuV μμλ
)( 111 uVV =
:1u
[ ]{ }44/14
321
4
121
4
1
4
1
4
1
1
1
1
1)]2/1(1[
Λ+++−+=−
−−+−−+
μμλμ
μ
u
uu
b
u
. (2.12)
Here
4/14
4321
4
133 ]1[ Λ++++−+=Λ μμμλ u ,
4/14
54321
4
144 ]1[ Λ+++++−+=Λ μμμμλ u ,
.]1[ 4/1
54321
4
155 μμμμμλ +++++−+=Λ u
Thus, determination of the temperature 0111 TuT =
on the left-hand boundary of C-C composite first layer
for stationary regime demands of solution (2.12), which
contain with whole and fractional degrees.
Analitical solution of this equation is difficult to find,
and stationary distribution of the temperature for all
layers of an ampoule was found by numerical method of
solution of ten equations of fourth order (2.6-2.8) with
boundary conditions .1,1 01 ==+ VuN
2.1. NUMERICAL SOLUTION OF SYSTEM
EQUATIONS (2.6-2.7)
For each layer with n=1,2,3,4,5 from (2.6-2.7) we
have two reccurent formula
]2/)
([),,(
1
4
1
41
1
nnnn
nnnnnnn
VVu
bubVVufV
μνν −−−+
++==
−−
−
− (2.13)
nnnnnn
nnnnnn
VVVuu
VVVufuu
μνν
ν
−−+−++
+===+
−−
−++
)(
),,(
1
4
1
4
4
121
4
1 (2.14)
Setted value from (2.13) we find with taking
into account
1u 1V
10 =V :
( )2/)( 10
4
011
4
1
1
11 μνν −−−++= − VVububV
For given and we find the solution of
equations (2.13-2.14) for then and so
on. Last step we do for . Value is need to
come an agreement with second boundary condition
1u 1V
32 ,uV 43, uV
65, uV 6u
161 ==+ uuN , that we can do using variation
method Neuton-Raphson for initial value . At
numerical solution we used the results of Monte-
Carlo calculations [8] (see fig. 2) and the mean
values energy losses of electron are
,
1u
,/1066.0
,/109.9
/1086.13
,/1052.14
,/1098.1
7
5
7
4
7
3
7
2
7
1
cmerg
cmerg
cmerg
cmerg
cmerg
×≈
×≈
×≈
×≈
×≈
ϕ
ϕ
ϕ
ϕ
ϕ
sekKcmerg /103.0 4×≈exκand value [3]. On
fig. 3-4 we see the results of solution for two situations:
i)on fig.3 we see temperature fields for ampoule without
molten salts with taken into account only thermal
radiation as between ampoule and camera wall, and so
between camera layers (i.e. exa κκ ≈ =0). As it is seen,
the temperature on right-hand of first C-C composite
layer increases on ~5K from ºC on left-hand
boundary.
64011 ≈T
,...16
1u
creases to to 60042 ≈T ºC. For second C-C composite
layer layer 5205251 ≈≈ TT ºC.
Thus we see that for the ampoule with fluorides salts
exhalations the stationary temperatures depend from
heat transport mechanisms and from layers thickness.
This leads to conclusion about necessity to take into
account both mechanisms of heat transport (thermal
radiation and thermal conductivity) simultaneously.
Carried out calculations show that all sinks of heat
(thermal radiation by C-C composite layers in camera
and heat exchange between solid state layers and streaks
with fluorides salts exhalations inside ampoule) don’t
lead to overheating of layers of irradiated ampoule.
This work was supported in part by STCU, Project #
294.
Fig. 3. Temperature fields within the ampoule without
molten fluoride salts under the radiation treatment (with
taken into account only thermal radiation between REFERENCES
layers)
In Hastelloy layers we have
ºC, and then the
temperature decreases to ºC for third
Hastelloy layer, and to second C-C composite layer
from values ºC to ºC.
1. V.M. Аzhazhа, O.S. Bakai, I.V. Gurin, А.N.
Dovbnya, N.V. Demidov, А.I. Zykov, E.S. Zlunin, S.D.
Lavrinenko, L.K. Myakushko, О.А. Repihov, А.V.
Tоrgovkin, B.M. Shirokov, B.I. Shramenko. Stend dlya
radiatsionnyh ispytanij konstruktsionnyh materialov v
uslovyah solevogo reaktora //Procceding XVI
international conference: physics of construction matter
and radiation materials science in condition of salts
reactor”, 6-11september 2004 г., Аlushtа, Crimea,
Ucraine (ННЦ ХФТИ, Харьков), c. 270
68032312221 ≈≈≈≈ TTTT
6404241 ≈≈ TT
55052 ≈T55551 ≈T
ii) For determination of the contributions of each the
mechanism of heat transport on fig. 4 we see the results
of numerical solution of system equations (2.7-2.8) with
taking into account as heat exchange between solid
state layers and streaks with fluoride salts exhalations
and with small [3] and so thermal
radiation between solid state layers. At this case for the
2. А.С. Бакай, М.И. Братченко, С.В. Дюльдя.
Моделирование профилей поглощения энергии
электронных пучков в имитационных
экспериментах по изучению радиационной
стойкости хастеллоя в среде расплавленных
фторидов //Труды XVI международной конференции
по физике радиационных явлений и радиационному
материаловедению, 6-11 сентября 2004г., Алушта,
Крым, Украина. ННЦ ХФТИ, Харьков, c. 274.
gexa κκκ 310~~ −
temperature for first C-C composite layer we have
ºC. For Hastelloy layers the
temperature has maxima for l first and second Hastelloy
layers
5901211 ≈≈ TT
3. В.С. Чиркин. Теплофизические свойства
материалов ядерной техники. М.: “Атомиздат”,
1968,
484 с.
4. А.С. Бакай, А.В. Чечкин, В.В. Жук.
Коррозионные свойства сплавов хастеллоя в
расплавах фторидных солей. Препринт ННЦ ХФТИ
(в печати, 2004)
5. L. Katz, A.S. Penfold //Rev.Mod.Phys. 1952, v. 24,
p. 28–35.
6. Handbuch der Physic, Springer Verlag, Berlin, b. 54,
1958.
7. И.М. Неклюдов, Б.В. Борц, В.В. Ганн, Г.Д.
Толсто¬ луцкая. Методология имитации
радиационных повреждений с помощью
ускорителей электронов и протонов //Вопросы
атомной науки и техники. Серия ФРП и РМ. 2003,
#3, с. 62.
Fig. 4. Temperature fields within the ampoule without
molten salt under the radiation treatment (with taken
into account thermal radiation between layers as well
as the heat exchange between layers and streaks with
fluoride salts exhalations)
8. O.S. Bakaj, M.I. Bratchenko, S.V. Dyul’dya (to be
published).
64532312221 ≈≈≈≈ TTTT ºC, and then slowly de-
РАСЧЕТЫ ТЕМПЕРАТУРНЫХ ПОЛЕЙ В АМПУЛАХ ПРИ РАДИАЦИОННЫХ ИСПЫТАНИЯХ
ЭЛЕКТРОНАМИ С ЭНЕРГИЕЙ 10МЭВ
А.С. Бакай, Л.В. Танатаров, В.Ю. Гончар и Г.Г. Сергеева
Для вычисления температурных полей в ампуле под облучением электронами нами использовались одномерные уравнения
теплопроводности с осью х, совпадающей с направлением потока электронов. Получены координатные зависимости температуры для
двух случаев: 1) для ампулы с расплавом солей флюорида между испытуемыми образцами Хастеллоя; 2) для ампулы с парами солей
флюорида между образцами Хастеллоя. Показано, что во втором случае установившееся распределение температуры существенно
39
зависит от механизма переноса тепла и от толщины слоев, что позволяет сделать вывод о необходимости принимать во внимание оба
механизма переноса тепла одновременно( тепловое излучение и теплопроводность слоев).
РОЗРАХУНКИ ТЕМПЕРАТУРНИХ ПОЛЕЙ У АМПУЛАХ ПРИ РАДІАЦІЙНИХ ВИПРОБУВАННЯХ
ЕЛЕКТРОНАМИ З ЕНЕРГІЄЮ 10 МЕВ
О.С. Бакай, Л.В. Танатаров, В.Ю. Гончар, і Г.Г. Сергієва
При розрахунках температурних полей в ампулі під випромінюванням електронами нами використовувались одновимірні
рівняння теплопровідності із вісью х, яка збігається із напрямком потоку електронів. Одержані коордінатні залежності температури
для двох випадків: 1) для ампули із с розплавом солей флюориду між випробуваними зразками Хастелоя; 2) для ампули із парами
солей флюориду між шарами ампули.. Показано, що у другому випадку сталий розподіл температури істотно залежить від механизму
переносу тепла і товщини шарів, що дозволяє зробити висновок про необхідність одночасно брати до уваги обидва механізми переносу
тепла (теплове випромінювання і теплопровідніть шарів).
CALCULATION OF TEMPERATURE FIELDS IN AMPOULES
UNDER THE RADIATION TREATMENT BY ELECTRONS WITH
ENERGY 10 MeV
INTRODUCTION
When the projecting the EITF test bench the choice of the geometry of the irradiated ampoules was made so that the energy losses of electron beam were equal to the amount of thermal energy irradiated by target. During the treatment mean current was regulated so that the temperature of the surfaces of ampoules was 650ºС. temperature fields within the ampoule is heterogeneous because of the intrinsic heterogeneity of its construction [1], and the finite values of the ampoule contents’ thermal conductivity coefficient. It is known that the corrosion processes are controlled by the velocity of chemical reactions and the diffusion transport which are essentially temperature dependent. That’s why the knowledge of the temperature distribution inside ampoules is very crucial when one analyzing the tests results. The distribution of energy losses of the electron beam within the ampoule was received by means of Monte-Carlo calculations in Ref. [2]. The ampoules and the assemblies are such that with sufficient accuracy we can consider the distribution of the energy losses and of the temperature as one-dimensional and essentially dependent only on the coordinates along the axis which is congruent with the direction of the beam. This means that we can use the one-dimensional equation of the thermal conductivity to obtain the temperature fields. We have not precition values of thermal conductivity coefficients of the molten fluorides mix, and of tested metallic specimens at the temperature ºC, and the data from Refs.[3-4] was used at the numerical estimation.
Taken into account that the sizes of intrinsic heterogeneity ( 0.2 cm) along axis x, which is congruent with the direction of the beam (see fig. 1), are much less than vertical dimensions of ampoule (5 cm along axis y) [1], we can use the one-dimensional equation of the thermal conductivity for calculation of the temperature fields
.
Energy losses due to the processes of ionization and atomic excitation, are well discribed by Bloch-Bethe formula for a target with thickness less than the track lengh for electron[5]. The target ampoule is made from the Carbon-Carbon (C-C) composite material. But for C-C composite material lengh of electron track is ~2сm, and this formula cannot be used for thick layer (here thickness of C-C composite layer is 1.5 сm (see fig. 1)). The region
are the first (g1)and the second (g2) C-C composite layers ( ) with thermal conductivity coefficient × [3]. For a thick target the main problem in energy losses calculations consists in taking into account multiple scattering of electrons, and in the calculation of the energy losses profile for each layer we used the results of the Monte Carlo computer modeling method from Ref. [2]. For the first C-C composite layer (g1) the mean value is , and for second C-C composite layer (g2) it is
.
For the thermal conductivity coefficients of molten fluoride salts and tested metallic specimens at we used evalution data from Refs. [3-4]: × .
At thermal balance we can find the stationary distrbution of temperature fields within the ampoule from thermal conductivity equations with the source of heat (energy flux ) and sink of heat due to thermal radiaton by C-C composite layers in camera where is Stephan-Boltzman constant. Let be the temperature fields for C-C composite sides of the ampoule, is the temperature field for i-th molten fluorides liquid mix layer (i=1,2,3.4), and is the temperature field for i-th tested metallic specimen (i=1,2,3).
. (19)
Taking x=2 in (19) we can find the temperature at the right hand boundary of the second layer of C-C composite.
(20)
2. THE CALCULATION OF TEMPERATURE FIELDS WITH IN THE AMPOULE
WITH EXHALATIONS OF FLUORIDE SALTS UNDER THE RADIATION
Fig. 4. Temperature fields within the ampoule without molten salt under the radiation treatment (with taken into account thermal radiation between layers as well as the heat exchange between layers and streaks with fluoride salts exhalations)
REFERENCES
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