Thermal state modeling in thermosensitive elements of microelectronic devices with reach-through foreign inclusions
The nonlinear boundary axially symmetric problem of heat conduction for the thermosensitive piecewise homogeneous layer with reach-through cylindrical inclusion that generates heat has been considered. Using the introduced function, the partial linearization of the original problem has been carri...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2012
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irk-123456789-1183142017-05-30T03:05:45Z Thermal state modeling in thermosensitive elements of microelectronic devices with reach-through foreign inclusions Gavrysh, V.I. The nonlinear boundary axially symmetric problem of heat conduction for the thermosensitive piecewise homogeneous layer with reach-through cylindrical inclusion that generates heat has been considered. Using the introduced function, the partial linearization of the original problem has been carried out. With the proposed piecewiselinear approximation of temperature at the boundary surface of the foreign inclusion and on the contact surface of the homogeneous elements of the layer, the problem has been completely linearized. The analytical solution of this problem of finding the introduced function using Hankel integral transform has been formed. The formulae for calculating the desired temperature have been derived. 2012 Article Thermal state modeling in thermosensitive elements of microelectronic devices with reach-through foreign inclusions / V.I Gavrysh.// Semiconductor Physics Quantum Electronics & Optoelectronics. — 2012. — Т. 15, № 3. — С. 247-251. — Бібліогр.: 13 назв. — англ. 1560-8034 PACS 74.25.fc http://dspace.nbuv.gov.ua/handle/123456789/118314 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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English |
| description |
The nonlinear boundary axially symmetric problem of heat conduction for the
thermosensitive piecewise homogeneous layer with reach-through cylindrical inclusion
that generates heat has been considered. Using the introduced function, the partial
linearization of the original problem has been carried out. With the proposed piecewiselinear
approximation of temperature at the boundary surface of the foreign inclusion and
on the contact surface of the homogeneous elements of the layer, the problem has been
completely linearized. The analytical solution of this problem of finding the introduced
function using Hankel integral transform has been formed. The formulae for calculating
the desired temperature have been derived. |
| format |
Article |
| author |
Gavrysh, V.I. |
| spellingShingle |
Gavrysh, V.I. Thermal state modeling in thermosensitive elements of microelectronic devices with reach-through foreign inclusions Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Gavrysh, V.I. |
| author_sort |
Gavrysh, V.I. |
| title |
Thermal state modeling in thermosensitive elements of microelectronic devices with reach-through foreign inclusions |
| title_short |
Thermal state modeling in thermosensitive elements of microelectronic devices with reach-through foreign inclusions |
| title_full |
Thermal state modeling in thermosensitive elements of microelectronic devices with reach-through foreign inclusions |
| title_fullStr |
Thermal state modeling in thermosensitive elements of microelectronic devices with reach-through foreign inclusions |
| title_full_unstemmed |
Thermal state modeling in thermosensitive elements of microelectronic devices with reach-through foreign inclusions |
| title_sort |
thermal state modeling in thermosensitive elements of microelectronic devices with reach-through foreign inclusions |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2012 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118314 |
| citation_txt |
Thermal state modeling in thermosensitive elements of
microelectronic devices with reach-through foreign inclusions / V.I Gavrysh.// Semiconductor Physics Quantum Electronics & Optoelectronics. — 2012. — Т. 15, № 3. — С. 247-251. — Бібліогр.: 13 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT gavryshvi thermalstatemodelinginthermosensitiveelementsofmicroelectronicdeviceswithreachthroughforeigninclusions |
| first_indexed |
2025-07-08T13:42:33Z |
| last_indexed |
2025-07-08T13:42:33Z |
| _version_ |
1837086439173521408 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 247-251.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
247
PACS 74.25.fc
Thermal state modeling in thermosensitive elements of
microelectronic devices with reach-through foreign inclusions
V.I. Gavrysh
National University “Lviv Polytechnic”, 28-a, S. Bandery str., 79013 Lviv, Ukraine
E-mail: ikni.pz@gmail.com
Abstract. The nonlinear boundary axially symmetric problem of heat conduction for the
thermosensitive piecewise homogeneous layer with reach-through cylindrical inclusion
that generates heat has been considered. Using the introduced function, the partial
linearization of the original problem has been carried out. With the proposed piecewise-
linear approximation of temperature at the boundary surface of the foreign inclusion and
on the contact surface of the homogeneous elements of the layer, the problem has been
completely linearized. The analytical solution of this problem of finding the introduced
function using Hankel integral transform has been formed. The formulae for calculating
the desired temperature have been derived.
Keywords: temperature, thermal conductivity, steady-state, ideal thermal contact,
foreign reach-through inclusion, thermosensitive.
Manuscript received 08.12.11; revised version received 15.05.12; accepted for
publication 14.06.12; published online 25.09.12.
1. Introduction
For microelectronic devices, the light-emitting elements
based on organic materials are commonly developed.
The basis for such a device is an organic microparticle
with electroluminescent properties, i.e. when it is
stimulated by current, it emits light. Due to it, the
mentioned elements based on organic materials
reflecting the random color will be very thin and will be
implemented on flexible substrates. Using two vacuum
units of thermovacuum spraying, from one of them a
thin film structure of organic materials is formed, and
from another – electrodes. In such a combination, ready
LED is obtained. To improve the efficiency of organic
LEDs, as better and stable operation parameters
(brightness, time of operation, reliability, etc.), the effect
of large temperature gradients and absolute values of
temperature should be taken into account.
Some researches of the temperature conditions for
nodes and separate elements of microelectronic devices
have been made previously [ 81 ].
Hereinafter the boundary axially symmetric
problem of heat conduction for a single element or node
of microelectronic devices that is modulated with
thermosensitive piecewise homogeneous layer with heat-
generating reach-through foreign cylindrical inclusion
has been considered.
2. Formulation of the problem
Let us consider an isotropic, in the sense of
thermophysical properties, thermosensitive piecewise-
homogeneous layer, which consists of n homogeneous
elements that differ in geometric and thermophysical
parameters, and which is assigned to a cylindrical
coordinate system ( φ )Or z with the beginning on one of
its edges, and it contains reach-through foreign inclusion
with the radius R . On the contact surfaces
,π2φ0:),φ,( zRSR
( , φ, ) : , 0 φ 2π, 1, 1 ,i iS r z r R i n
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 247-251.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
248
where iz is the thickness of the i-th element layer, the
ideal thermal contact takes place.
In the region
0 ( , φ, ) : ,0 φ 2π, 0 nr z r R z z ,
of inclusion, the uniformly distributed internal heat
sources with the capacity 0q have influence. On the
boundary surfaces of the layer
0 ( ,φ,0) : , 0 φ 2π ,K r r
( ,φ, ) : , 0 φ 2πn nK r z r , the boundary
conditions of the second kind are set (Figure).
3. Construction of partially linear mathematical
model
The distribution of steady state axially symmetric
temperature field ),( zrt in the system under
consideration is obtained by solving the nonlinear heat
equation [9, 10]
)(
),,(λ),,(λ
1
0 rRSq
z
t
tzr
zr
t
tzrr
rr
(1)
with the following boundary conditions
0
0, 0,
n
r r z zz
t
t t t
r z z
(2)
where
n
i
iii zzNrRSttttzr
1
10 ),()}()](λ)(λ[)(λ{),,(λ
is the thermal conductivity coefficient of piecewise-
homogeneous layer; )(λ),(λ 0 tti thermal conductivity
coefficients of materials of i-th layer and inclusion,
respectively;
Thermosensitive isotropic piecewise homogeneous layer with
heat generating reach-through foreign cylindrical inclusion.
0 0z ; 1 1( , ) ( ) ( )i i iN z z S z z S z z ;
1, ζ 0
(ζ) 0, 5 0, 5, ζ 0
0, ζ 0
S
asymmetric unit function
[11].
Let us introduce the function
,)(
),(
0
ζ)ζ(λ
)(ζ)ζ(λ)](ζ))ζ(λ)ζ(λ(
)(ζ))ζ(λ)ζ((ζ))ζ(λ)ζ(λ(
),()[(),(ζ)ζ(λ
1
),(
0
),(
),(
0
1
),(
),(
0
),(
),(
0
1
),(
0
11
1
1
1
izzS
izrt
di
zzSdzzSd
zzSdd
zzNrRSzzNd
i
zrt
ii
zrt
zRt
i
i
zrt
zRt
i
zrt
zRt
i
n
i
zrt
iii
ii
i
i
i
(3)
by differentiation of which with respect to r and z we
obtain
),,(),,(λ
),,(),,(λ
2
1
zrF
zz
t
tzr
zrF
rr
t
tzr
(4)
where
.),(]))(λ)(λ[()(),(
)],())(λ()(
))(λ[(),(
1
102
1
1
1
1
n
i
i
Rr
i
iii
i
n
i
zzN
z
t
ttrRSzrF
zzS
zzr
t
tzzS
zzr
t
tzrF
i
i
Taking into account the expressions (4), the
original equation (1) takes the following form:
).()],([
),(
11
02
12
2
rRSqzrF
z
zrFr
rrzr
r
rr
(5)
Using the equation (3) the boundary conditions can
be written as:
.0,0
0
nzzzrr zzr
(6)
Thus, the introduced function represented by the
expression (3) has allowed to convert the non-linear heat
conduction problem (1), (2) to partially linear equations
with discontinuous coefficients (5) and completely linear
boundary conditions (6).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 247-251.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
249
4. Absolutely linearized mathematical model
Let the approximate functions ),(),,( izrtzRt look as
1( ) ( ) ( ) ( )*
1 1
1
( , ) ( ) ( ),
miR iR iR i
k k k
k
t R z t t t S z z
1( ) ( ) ( )
1 1
1
( , ) ( ) ( ),
li i i
i j j j
j
t r z t t t S r r
(7)
where
( )* ( )* ( )* ( )*
1 2 1є ]0;z [; ... ;i i i i
k i mz z z z
1 2 1*є ] ; [; ... ;j lr R r r r r ml, – number of
fragmentation of intervals *;R r and 0; iz
respectively;
( ) ( )( 1, 1), ( 1, 1)iR i
k jt k m t j l unknown
approximated temperature values; *r value of radial
coordinate, where the temperature practically equals to
zero (is found from the corresponding linear model).
Substituting the expression (7) into the equation
(5), we obtain the linear differential equation with partial
derivatives relative to the introduced function
.)()](
)()(δ)(
1
[
1
1
0
)(
1
1
1
)(
m
k
k
i
n
i
l
j
j
j
i
rRSqzF
rRSrrzF
r
(8)
Here
;)(δ)](λ)(λ)[()(
*)()(
1
)(
10
)()(
1
)( i
k
iR
ki
iR
k
iR
k
iR
k
k
i zzttttzF
( ) ( 1) ( 1) ( 1)
1 1 1[( ) ( ) λ ( ) ( )j i i i
i j j j i j iF z r t t t S z z
( ) ( ) ( )
1 1)( λ ( ) ( )];i i i
j j i j it t t S z z
2
2
1
r
r r r z
Laplace operator in the
cylindrical coordinate system;
(ζ)
δ (ζ)
ζ
d S
d
asymmetric Dirac delta function [11].
5. Construction of the analytical solution for the
boundary value problem (8), (6)
Applying the Hankel integral transform by the
coordinate r to the equation (8) and boundary conditions
(6), we obtain the ordinary differential equations with
constant coefficients
)ξ(
ξ
])()ξ(
ξ
)()ξ(ξ[
1
0
1
1
)(
1
1
1
1
)(
1
2
2
2
RJ
Rq
zFRJ
R
zFrJ
dz
d
m
k
k
i
n
i
l
j
j
ij
(9)
and boundary conditions
0
0,
nz z z
d d
dz dz
(10)
where 0
0
( ξ)r J r dr
is the transformant of the
function ( , )r z ; ξ – central Hankel transform
parameter; (ζ )vJ – Bessel function of the first kind of
thv order.
Let us rewrite the above general view of Eq. (9) in
the following form
)}.ξ(
ξ
)]()(ξ))(λ
)(λ()()ξ(
))(λ)()())(ξ1(
)(λ)()())(ξ1((
)ξ([{
ξ
1
12
0)()()(
1
)(
10
1
1
)()(
11
)(
1
)()(
1
)1(
1
)1()1(
111
1
1
1
1
ξ
2
ξ
1
**
RJ
Rq
zzSzzcht
tttRRJ
tttzzSzzch
tttzzSzzch
rJrecec
i
k
i
k
iR
ki
iR
k
m
k
iR
k
iR
k
i
ji
i
j
i
jii
i
ji
i
j
i
jii
n
i
l
j
jj
zz
Here, 1 2,c c − integration constants.
Using the boundary conditions (10), we obtain a
partial solution to the problem (9), (10):
)}.ξ(
ξ
))](ξ
ξ
ξ
)()(ξ))((λ)(λ(
)()ξ()))(λ)(
))(ξ
ξ
ξ
)())(ξ1((
)(λ)())(ξ
ξ
ξ
)())(ξ1((()ξ([{
ξ
1
12
0)(
)()()(
1
)(
10
1
1
)()(
11
)(
1
)()(
1
)1(
1
)1()1(
11
1
1
1
111
*
**
RJ
Rq
zzsh
zsh
zch
zzSzzchtt
ttRRJttt
zzsh
zsh
zch
zzSzzch
tttzzsh
zsh
zch
zzSzzchrJr
i
kn
n
i
k
i
k
iR
ki
iR
k
m
k
iR
k
iR
k
i
ji
i
j
i
j
in
n
ii
i
ji
i
j
i
jin
n
n
i
l
j
iijj
(11)
Applying the inverse Hankel integral transform to
the equation (11), we find the expression for the
function
0
0
ξ ( ξ) ξ.J r d
(12)
By substitution of the specific dependence of
thermal conductivity coefficients of materials of each
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 247-251.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
250
element layer and inclusion in relations (3), (12) and by
comparison of the obtained expressions for the function
on surfaces ( 1, 1),R i
S S i n , we come to systems
of nonlinear algebraic equations for determination of the
unknown temperature values
( ) ( 1, )iR
kt k m and
( ) ( 1, )i
jt j l .
The desired temperature field for the considered
system is determined from the nonlinear algebraic
equations obtained using the relations (3), (12) after
substituting to them the specific expression of
dependences of the thermal conductivity coefficients for
material elements of the layer and inclusion.
6. Partial illustration and analysis of the results
obtained
In many practical cases, there is such dependence of the
thermal conductivity coefficient on temperature [11, 12]:
0λ λ (1 )k t , (13)
where k,λ0 are pivotal and temperature coefficients of
thermal conductivity.
Then, using Exps (3) and (12) yields a formula for
determining the temperature t for the case of 2-D layer
(n = 2) in the areas
1 1{( , φ, ) : ,0 φ 2π,0 }r z r R z z ,
1
10
1
1 )(211
k
k
t
; (14)
2 1 2{( , φ, ) : ,0 φ 2π, }r z r R z z z ,
2
20
2
2 )(211
k
k
t
; (15)
3 1{( , φ, ) : ,0 φ 2π,0 }r z r R z z ,
0
30
0
0 )(211
k
k
t
; (16)
4 1 2{( , φ, ) : ,0 φ 2π, }r z r R z z z ,
.
)(211
0
40
0
0
k
k
t
(17)
Here, ;)
2
1(λ
0
10
11
z
tt
k
;)
2
λλ
λλ(;
1
2
0
21
0
10
1
0
212
zz
mm tt
kk
;)
2
1(λ;
0
00
000
0
)1()1(
3
zz
vv tt
k
;2,1)
2
λλ
(λλ 0
0
0
0
00
0
)(
itt
kk
Rr
ii
i
i
v
(2) (1)
4 0
0
;v v m
z r R
the value of
temperature ( , 0)t r is such that is equal to the
environment temperature, 1( , ), ( , )t R z t R z are calculated
using the formula (14).
7. Conclusions
The introduced function described by the expression
(3) allowed to partially linearize the original nonlinear
heat equation (1) and completely linearize the boundary
conditions (2). The proposed piecewise-linear
approximation of the temperature expressions (7) on the
boundary surface RS of the foreign inclusion and of
contact surfaces ( 1, 1)iS i n for homogeneous
elements of the layer gave the opportunity to completely
linearize the equation (5). And therefore, it was possible
to apply the Hankel integral transform to the obtained
boundary linear problem to the introduced function
and to construct an analytical solution for finding it. The
2-D layer with the dependence of the thermal
conductivity coefficient of the layers and inclusion
described by the expressions (13) has been considered.
On this basis, the formulae (14)-(17) for the calculation
of the temperature ),( zrt at any point of the considered
system have been shown.
References
1. А.F. Barvinskyy, V.І. Gavrysh, Nonlinear thermal
conductivity problem for nonhomogeneous layer
with internal heat sources // Journal of Mechanical
Engineering, 12(1), p. 47-53 (2009) in Ukrainian.
2. V.І. Gavrysh, D.V. Fedasyuk, The method for
calculation of temperature fields for
thermosensitive piecewise homogeneous strip with
foreign inclusion // Industrial Heat Engineering,
32(5), p. 18-25 (2010) in Ukrainian.
3. V.І. Gavrysh, D.V. Fedasyuk, А.І. Kosach,
Boundary thermal conductivity problem for the
layer with cylindrical inclusion // Physicochemical
Mechanics of Materials, 46(5), p. 115-120 (2010),
in Ukrainian.
4. V.I. Gavrysh, D.V. Fedasyuk, Thermal simulation
of heterogeneous structural components in
microelectronic devices // Semiconductor Physics,
Quantum Electronics & Optoelectronics, 13, No.4,
p. 439-443 (2010).
5. V.I. Gavrysh, A.I. Kosach, Simulation of
temperature conditions in the elements of
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 247-251.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
251
microelectronic devices // Tekhnologiya i
konstruirovanie v elektron. apparature, №1-2 (90),
p. 27-30 (2011) in Russian.
6. V.І. Gavrysh, А.І. Kosach, Boundary thermal
conductivity problem for the layer with cylindrical
inclusion // Physicochemical Mechanics of
Materials 47(6), p. 52-58 (2011) in Ukrainian.
7. V.I. Gavrysh, Thermal simulation of heterogeneous
structural components in microelectronic devices //
Semiconductor Physics, Quantum Electronics &
Optoelectronics, 14(4), p. 478-481 (2011).
8. V.I. Gavrysh, А.І. Kosach, Simulation of
temperature conditions in electronic devices of
piecewise-homogeneous structure // Electronic
Modeling, 33(4), p. 99-113 (2011) in Russian.
9. Ya.S. Podstryhach, V.A. Lomakin, Yu.M. Kolyano,
Thermoelasticity of Heterogeneous Body Structure.
Nauka, Moscow, 1984 (in Russian).
10. Yu.M. Kolyano, Methods of Heat Conductivity and
Thermoelasticity of Heterogeneous Bodies. Kyiv,
Naukova Dumka, 1992 (in Russian).
11. G. Korn, T. Korn, Mathematical Handbook for
Scientists and Engineers. Nauka, Moscow, 1977 (in
Russian).
12. V.A. Lomakin, Theory of Elasticity of
Heterogeneous Bodies. Moscow University
Publishing House, 1976 (in Russian).
13. R. Berman, Thermal Conductivity of Solids. Mir,
Moscow, 1979 (in Russian).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 247-251.
PACS 74.25.fc
Thermal state modeling in thermosensitive elements of microelectronic devices with reach-through foreign inclusions
V.I. Gavrysh
National University “Lviv Polytechnic”, 28-a, S. Bandery str., 79013 Lviv, Ukraine
E-mail: ikni.pz@gmail.com
Abstract. The nonlinear boundary axially symmetric problem of heat conduction for the thermosensitive piecewise homogeneous layer with reach-through cylindrical inclusion that generates heat has been considered. Using the introduced function, the partial linearization of the original problem has been carried out. With the proposed piecewise-linear approximation of temperature at the boundary surface of the foreign inclusion and on the contact surface of the homogeneous elements of the layer, the problem has been completely linearized. The analytical solution of this problem of finding the introduced function using Hankel integral transform has been formed. The formulae for calculating the desired temperature have been derived.
Keywords: temperature, thermal conductivity, steady-state, ideal thermal contact, foreign reach-through inclusion, thermosensitive.
Manuscript received 08.12.11; revised version received 15.05.12; accepted for publication 14.06.12; published online 25.09.12.
1. Introduction
For microelectronic devices, the light-emitting elements based on organic materials are commonly developed. The basis for such a device is an organic microparticle with electroluminescent properties, i.e. when it is stimulated by current, it emits light. Due to it, the mentioned elements based on organic materials reflecting the random color will be very thin and will be implemented on flexible substrates. Using two vacuum units of thermovacuum spraying, from one of them a thin film structure of organic materials is formed, and from another – electrodes. In such a combination, ready LED is obtained. To improve the efficiency of organic LEDs, as better and stable operation parameters (brightness, time of operation, reliability, etc.), the effect of large temperature gradients and absolute values of temperature should be taken into account.
Some researches of the temperature conditions for nodes and separate elements of microelectronic devices have been made previously [
8
1
-
].
Hereinafter the boundary axially symmetric problem of heat conduction for a single element or node of microelectronic devices that is modulated with thermosensitive piecewise homogeneous layer with heat-generating reach-through foreign cylindrical inclusion has been considered.
2. Formulation of the problem
Let us consider an isotropic, in the sense of thermophysical properties, thermosensitive piecewise-homogeneous layer, which consists of n homogeneous elements that differ in geometric and thermophysical parameters, and which is assigned to a cylindrical coordinate system
(
φ)
Orz
with the beginning on one of its edges, and it contains reach-through foreign inclusion with the radius
R
. On the contact surfaces
{
}
,
π
2
φ
0
:
)
,
φ
,
(
£
£
=
z
R
S
R
EMBED Equation.3{
}
(,
φ,):,0φ2π,1,1,
ii
SrzrRin
=>££=-
where
i
z
is the thickness of the i-th element layer, the ideal thermal contact takes place.
In the region
{
}
0
(,
φ,):,0φ2π,0
n
rzrRzz
W=£££££
,
of inclusion, the uniformly distributed internal heat sources with the capacity
0
q
have influence. On the boundary surfaces of the layer
{
}
0
(,
φ,0):,0φ2π,
K
rr
=<¥££
EMBED Equation.3{
}
(,
φ,):,0φ2π
nn
K
rzr
=<¥££
, the boundary conditions of the second kind are set (Figure).
3. Construction of partially linear mathematical model
The distribution of steady state axially symmetric temperature field
)
,
(
z
r
t
in the system under consideration is obtained by solving the nonlinear heat equation [9, 10]
)
(
)
,
,
(
λ
)
,
,
(
λ
1
0
r
R
S
q
z
t
t
z
r
z
r
t
t
z
r
r
r
r
-
-
-
=
=
ú
û
ù
ê
ë
é
¶
¶
¶
¶
+
ú
û
ù
ê
ë
é
¶
¶
¶
¶
(1)
with the following boundary conditions
0
0,0,
n
rr
zz
z
t
ttt
rzz
®¥®¥
=
=
¶¶¶
====
¶¶¶
(2)
where
å
×
-
-
×
-
+
=
=
-
n
i
i
i
i
z
z
N
r
R
S
t
t
t
t
z
r
1
1
0
)
,
(
)}
(
)]
(
λ
)
(
λ
[
)
(
λ
{
)
,
,
(
λ
is the thermal conductivity coefficient of piecewise-homogeneous layer;
-
)
(
λ
),
(
λ
0
t
t
i
thermal conductivity coefficients of materials of i-th layer and inclusion, respectively;
Thermosensitive isotropic piecewise homogeneous layer with heat generating reach-through foreign cylindrical inclusion.
0
0
z
=
;
11
(,)()()
iii
NzzSzzSzz
-+-+
=---
;
1,
ζ0
(
ζ)0,50,5,ζ0
0,
ζ0
S
±
>
==-
<
ì
ï
í
ï
î
m
asymmetric unit function [11].
Let us introduce the function
,
)
(
)
,
(
0
ζ
)
ζ
(
λ
)
(
ζ
)
ζ
(
λ
)]
(
ζ
))
ζ
(
λ
)
ζ
(
λ
(
)
(
ζ
))
ζ
(
λ
)
ζ
(
(
ζ
))
ζ
(
λ
)
ζ
(
λ
(
)
,
(
)[
(
)
,
(
ζ
)
ζ
(
λ
1
)
,
(
0
)
,
(
)
,
(
0
1
)
,
(
)
,
(
0
)
,
(
)
,
(
0
1
)
,
(
0
1
1
1
1
1
ï
þ
ï
ý
ü
-
+
×
ò
+
-
×
-
-
×
-
+
+
-
×
-
-
-
ï
î
ï
í
ì
-
+
×
=
J
-
+
+
-
+
=
-
-
-
ò
ò
ò
ò
å
ò
-
-
-
i
z
z
S
i
z
r
t
d
i
z
z
S
d
z
z
S
d
z
z
S
d
d
z
z
N
r
R
S
z
z
N
d
i
z
r
t
i
i
z
r
t
z
R
t
i
i
z
r
t
z
R
t
i
z
r
t
z
R
t
i
n
i
z
r
t
i
i
i
i
i
i
i
i
l
(3)
by differentiation of which with respect to
r
and
z
we obtain
),
,
(
)
,
,
(
λ
),
,
(
)
,
,
(
λ
2
1
z
r
F
z
z
t
t
z
r
z
r
F
r
r
t
t
z
r
+
¶
J
¶
=
¶
¶
+
¶
J
¶
=
¶
¶
(4)
where
.
)
,
(
]
))
(
λ
)
(
λ
[(
)
(
)
,
(
)],
(
)
)
(
λ
(
)
(
)
)
(
λ
[(
)
,
(
1
1
0
2
1
1
1
1
å
×
¶
¶
-
×
-
=
-
×
=
¶
¶
-
-
´
´
=
¶
¶
å
=
=
-
=
-
+
-
+
=
-
n
i
i
R
r
i
i
i
i
i
n
i
z
z
N
z
t
t
t
r
R
S
z
r
F
z
z
S
z
z
r
t
t
z
z
S
z
z
r
t
t
z
r
F
i
i
Taking into account the expressions (4), the original equation (1) takes the following form:
[
]
).
(
)]
,
(
[
)
,
(
1
1
0
2
1
2
2
r
R
S
q
z
r
F
z
z
r
F
r
r
r
z
r
r
r
r
-
×
-
=
¶
¶
+
+
×
¶
¶
+
¶
J
¶
+
ú
û
ù
ê
ë
é
¶
J
¶
¶
¶
-
(5)
Using the equation (3) the boundary conditions can be written as:
.
0
,
0
0
=
¶
J
¶
=
¶
J
¶
=
¶
J
¶
=
J
=
=
¥
®
¥
®
n
z
z
z
r
r
z
z
r
(6)
Thus, the introduced function
J
represented by the expression (3) has allowed to convert the non-linear heat conduction problem (1), (2) to partially linear equations with discontinuous coefficients (5) and completely linear boundary conditions (6).
4. Absolutely linearized mathematical model
Let the approximate functions
)
,
(
),
,
(
i
z
r
t
z
R
t
look as
1
()()()()*
11
1
(,)()(),
m
iRiRiRi
kkk
k
tRztttSzz
-
+-
=
=+--
å
1
()()()
11
1
(,)()(),
l
iii
ijjj
j
trztttSrr
-
+-
=
=+--
å
(7)
where
()*()*()*()*
121
є ]0;z[;...;
iiii
kim
zzzz
-
£££
121
*
є ]; [; ...;
jl
rRrrrr
-
£££
EMBED Equation.3m
l
,
– number of fragmentation of intervals
*
;
Rr
ùé
ûë
and
]
[
0;
i
z
respectively;
()()
(1,1),(1,1)
iRi
kj
tkmtjl
=-=--
unknown approximated temperature values;
-
*
r
value of radial coordinate, where the temperature practically equals to zero (is found from the corresponding linear model).
Substituting the expression (7) into the equation (5), we obtain the linear differential equation with partial derivatives relative to the introduced function
J
.
)
(
)]
(
)
(
)
(
δ
)
(
1
[
1
1
0
)
(
1
1
1
)
(
å
-
-
´
´
å
å
-
+
-
-
¢
-
=
DJ
-
=
-
=
-
=
-
m
k
k
i
n
i
l
j
j
j
i
r
R
S
q
z
F
r
R
S
r
r
z
F
r
(8)
Here
;
)
(
δ
)]
(
λ
)
(
λ
)[
(
)
(
*
)
(
)
(
1
)
(
1
0
)
(
)
(
1
)
(
i
k
iR
k
i
iR
k
iR
k
iR
k
k
i
z
z
t
t
t
t
z
F
-
¢
×
-
-
=
-
+
+
+
()(1)(1)(1)
111
[
()()
λ()()
jiii
ijjjiji
F
zrtttSzz
---
+++-
=-××--
()()()
11
)
(
λ()()];
iii
jjiji
tttSzz
+++
×
--×-
2
2
1
r
rrrz
¶¶¶
D=+-
¶¶¶
æö
ç÷
èø
Laplace operator in the cylindrical coordinate system;
(
ζ)
δ(ζ)
ζ
dS
d
±
±
=-
asymmetric Dirac delta function [11].
5. Construction of the analytical solution for the boundary value problem (8), (6)
Applying the Hankel integral transform by the coordinate
r
to the equation (8) and boundary conditions (6), we obtain the ordinary differential equations with constant coefficients
)
ξ
(
ξ
]
)
(
)
ξ
(
ξ
)
(
)
ξ
(
ξ
[
1
0
1
1
)
(
1
1
1
1
)
(
1
2
2
2
R
J
R
q
z
F
R
J
R
z
F
r
J
dz
d
m
k
k
i
n
i
l
j
j
i
j
-
å
×
+
å
+
å
×
-
=
J
-
J
-
=
=
-
=
x
(9)
and boundary conditions
0
0,
n
zzz
dd
dzdz
==
JJ
==
(10)
where
0
0
(
ξ)
rJrdr
¥
J=J
ò
is the transformant of the function
(,)
rz
J
;
ξ
– central Hankel transform parameter;
(
ζ)
v
J
– Bessel function of the first kind of
th
-
v
order.
Let us rewrite the above general view of Eq. (9) in the following form
)}.
ξ
(
ξ
)]
(
)
(
ξ
))
(
λ
)
(
λ
(
)
(
)
ξ
(
))
(
λ
)
(
)
(
))
(
ξ
1
(
)
(
λ
)
(
)
(
))
(
ξ
1
((
)
ξ
(
[
{
ξ
1
1
2
0
)
(
)
(
)
(
1
)
(
1
0
1
1
)
(
)
(
1
1
)
(
1
)
(
)
(
1
)
1
(
1
)
1
(
)
1
(
1
1
1
1
1
1
1
ξ
2
ξ
1
*
*
R
J
R
q
z
z
S
z
z
ch
t
t
t
t
R
RJ
t
t
t
z
z
S
z
z
ch
t
t
t
z
z
S
z
z
ch
r
J
r
e
c
e
c
i
k
i
k
iR
k
i
iR
k
m
k
iR
k
iR
k
i
j
i
i
j
i
j
i
i
i
j
i
i
j
i
j
i
i
n
i
l
j
j
j
z
z
+
-
-
×
-
-
å
×
-
×
-
-
×
-
×
-
-
-
-
-
-
×
-
-
-
´
´
å
å
+
+
=
J
-
+
+
-
=
+
+
+
+
-
+
-
-
+
-
+
-
=
-
=
-
Here,
12
,
cc
− integration constants.
Using the boundary conditions (10), we obtain a partial solution to the problem (9), (10):
)}.
ξ
(
ξ
))]
(
ξ
ξ
ξ
)
(
)
(
ξ
))(
(
λ
)
(
λ
(
)
(
)
ξ
(
)))
(
λ
)
(
))
(
ξ
ξ
ξ
)
(
))
(
ξ
1
((
)
(
λ
)
(
))
(
ξ
ξ
ξ
)
(
))
(
ξ
1
(((
)
ξ
(
[
{
ξ
1
1
2
0
)
(
)
(
)
(
)
(
1
)
(
1
0
1
1
)
(
)
(
1
1
)
(
1
)
(
)
(
1
)
1
(
1
)
1
(
)
1
(
1
1
1
1
1
1
1
1
*
*
*
R
J
R
q
z
z
sh
z
sh
z
ch
z
z
S
z
z
ch
t
t
t
t
R
RJ
t
t
t
z
z
sh
z
sh
z
ch
z
z
S
z
z
ch
t
t
t
z
z
sh
z
sh
z
ch
z
z
S
z
z
ch
r
J
r
i
k
n
n
i
k
i
k
iR
k
i
iR
k
m
k
iR
k
iR
k
i
j
i
i
j
i
j
i
n
n
i
i
i
j
i
i
j
i
j
i
n
n
n
i
l
j
i
i
j
j
+
-
-
-
-
×
-
-
´
´
å
-
×
-
-
´
´
-
+
-
-
-
-
-
-
×
-
+
å
å
+
-
-
-
×
=
J
-
+
+
-
=
+
+
+
+
-
+
-
-
+
-
=
-
=
-
+
-
(11)
Applying the inverse Hankel integral transform to the equation (11), we find the expression for the function
J
0
0
ξ(ξ)ξ.
Jrd
¥
J=×J
ò
(12)
By substitution of the specific dependence of thermal conductivity coefficients of materials of each element layer and inclusion in relations (3), (12) and by comparison of the obtained expressions for the function
J
on surfaces
(1,1)
,
R
i
SSin
=-
, we come to systems of nonlinear algebraic equations for determination of the unknown temperature values
()
(1,)
iR
k
tkm
=
and
()
(1,)
i
j
tjl
=
.
The desired temperature field for the considered system is determined from the nonlinear algebraic equations obtained using the relations (3), (12) after substituting to them the specific expression of dependences of the thermal conductivity coefficients for material elements of the layer and inclusion.
6. Partial illustration and analysis of the results obtained
In many practical cases, there is such dependence of the thermal conductivity coefficient on temperature [11, 12]:
0
λλ(1)
kt
=-
,
(13)
where
k
,
λ
0
are pivotal and temperature coefficients of thermal conductivity.
Then, using Exps (3) and (12) yields a formula for determining the temperature
t
for the case of 2-D layer (n = 2) in the areas
11
{(,
φ,):,0φ2π,0}
rzrRzz
W=>£££<
,
1
1
0
1
1
)
(
2
1
1
k
k
t
J
+
J
l
-
-
=
;
(14)
212
{(,
φ,):,0φ2π,}
rzrRzzz
W=>££££
,
2
2
0
2
2
)
(
2
1
1
k
k
t
J
+
J
l
-
-
=
;
(15)
31
{(,
φ,):,0φ2π,0}
rzrRzz
W=££££<
,
0
3
0
0
0
)
(
2
1
1
k
k
t
J
+
J
l
-
-
=
;
(16)
412
{(,
φ,):,0φ2π,}
rzrRzzz
W=£££££
,
.
)
(
2
1
1
0
4
0
0
0
k
k
t
J
+
J
l
-
-
=
(17)
Here,
;
)
2
1
(
λ
0
1
0
1
1
=
ú
û
ù
ê
ë
é
-
=
J
z
t
t
k
EMBED Equation.3;
)
2
λ
λ
λ
λ
(
;
1
2
0
2
1
0
1
0
1
0
2
1
2
z
z
m
m
t
t
k
k
=
ú
û
ù
ê
ë
é
-
+
-
=
J
J
+
J
=
J
EMBED Equation.3;
)
2
1
(
λ
;
0
0
0
0
0
0
0
)
1
(
)
1
(
3
=
=
ú
û
ù
ê
ë
é
-
=
J
J
+
J
-
J
=
J
z
z
v
v
t
t
k
EMBED Equation.3;
2
,
1
)
2
λ
λ
(
λ
λ
0
0
0
0
0
0
0
)
(
=
ú
û
ù
ê
ë
é
-
+
-
=
J
=
i
t
t
k
k
R
r
i
i
i
i
v
(2)(1)
40
0
;
vvm
zrR
==
+
J=J-J+JJ
the value of temperature
(,0)
tr
is such that is equal to the environment temperature,
1
(,),(,)
tRztRz
are calculated using the formula (14).
7. Conclusions
The introduced function
J
described by the expression (3) allowed to partially linearize the original nonlinear heat equation (1) and completely linearize the boundary conditions (2). The proposed piecewise-linear approximation of the temperature expressions (7) on the boundary surface
R
S
of the foreign inclusion and of contact surfaces
(1,1)
i
Sin
=-
for homogeneous elements of the layer gave the opportunity to completely linearize the equation (5). And therefore, it was possible to apply the Hankel integral transform to the obtained boundary linear problem to the introduced function
J
and to construct an analytical solution for finding it. The 2-D layer with the dependence of the thermal conductivity coefficient of the layers and inclusion described by the expressions (13) has been considered. On this basis, the formulae (14)-(17) for the calculation of the temperature
)
,
(
z
r
t
at any point of the considered system have been shown.
References
1. А.F. Barvinskyy, V.І. Gavrysh, Nonlinear thermal conductivity problem for nonhomogeneous layer with internal heat sources // Journal of Mechanical Engineering, 12(1), p. 47-53 (2009) in Ukrainian.
2. V.І. Gavrysh, D.V. Fedasyuk, The method for calculation of temperature fields for thermosensitive piecewise homogeneous strip with foreign inclusion // Industrial Heat Engineering, 32(5), p. 18-25 (2010) in Ukrainian.
3. V.І. Gavrysh, D.V. Fedasyuk, А.І. Kosach, Boundary thermal conductivity problem for the layer with cylindrical inclusion // Physicochemical Mechanics of Materials, 46(5), p. 115-120 (2010), in Ukrainian.
4. V.I. Gavrysh, D.V. Fedasyuk, Thermal simulation of heterogeneous structural components in microelectronic devices // Semiconductor Physics, Quantum Electronics & Optoelectronics, 13, No.4, p. 439-443 (2010).
5. V.I. Gavrysh, A.I. Kosach, Simulation of temperature conditions in the elements of microelectronic devices // Tekhnologiya i konstruirovanie v elektron. apparature, №1-2 (90), p. 27-30 (2011) in Russian.
6. V.І. Gavrysh, А.І. Kosach, Boundary thermal conductivity problem for the layer with cylindrical inclusion // Physicochemical Mechanics of Materials 47(6), p. 52-58 (2011) in Ukrainian.
7. V.I. Gavrysh, Thermal simulation of heterogeneous structural components in microelectronic devices // Semiconductor Physics, Quantum Electronics & Optoelectronics, 14(4), p. 478-481 (2011).
8.
V.I. Gavrysh, А.І. Kosach, Simulation of temperature conditions in electronic devices of piecewise-homogeneous structure // Electronic Modeling, 33(4), p. 99-113 (2011) in Russian.
9.
Ya.S. Podstryhach, V.A. Lomakin, Yu.M. Kolyano, Thermoelasticity of Heterogeneous Body Structure. Nauka, Moscow, 1984 (in Russian).
10.
Yu.M. Kolyano, Methods of Heat Conductivity and Thermoelasticity of Heterogeneous Bodies. Kyiv, Naukova Dumka, 1992 (in Russian).
11.
G. Korn, T. Korn, Mathematical Handbook for Scientists and Engineers. Nauka, Moscow, 1977 (in Russian).
12.
V.A. Lomakin, Theory of Elasticity of Heterogeneous Bodies. Moscow University Publishing House, 1976 (in Russian).
13.
R. Berman, Thermal Conductivity of Solids. Mir, Moscow, 1979 (in Russian).
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
247
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