Structures of solutions to the problems for multilayer media and their applications to the calculation of fuel–element thermal fields
The technique of constructing approximate solutions to heat conduction problems of media is presented. The calculation is based on the structures of solutions constructed with the help of the left, normalized-to-the-first-order equations of region boundaries and their characteristic parts. The struc...
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2001
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irk-123456789-784502015-03-18T03:01:53Z Structures of solutions to the problems for multilayer media and their applications to the calculation of fuel–element thermal fields Khazhmuradov, M.A. Experimental methods and computations The technique of constructing approximate solutions to heat conduction problems of media is presented. The calculation is based on the structures of solutions constructed with the help of the left, normalized-to-the-first-order equations of region boundaries and their characteristic parts. The structures of solutions contain uncertain components, and whatever their choice may be, the boundary conditions and the conditions of medium conjugation are fulfilled exactly. The arbitrary choice of uncertain components is used to satisfy the basic differential equation. 2001 Article Structures of solutions to the problems for multilayer media and their applications to the calculation of fuel–element thermal fields / M.A. Khazhmuradov // Вопросы атомной науки и техники. — 2001. — № 1. — С. 66-68. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 61.20.-Ja. http://dspace.nbuv.gov.ua/handle/123456789/78450 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Experimental methods and computations Experimental methods and computations |
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Experimental methods and computations Experimental methods and computations Khazhmuradov, M.A. Structures of solutions to the problems for multilayer media and their applications to the calculation of fuel–element thermal fields Вопросы атомной науки и техники |
| description |
The technique of constructing approximate solutions to heat conduction problems of media is presented. The calculation is based on the structures of solutions constructed with the help of the left, normalized-to-the-first-order equations of region boundaries and their characteristic parts. The structures of solutions contain uncertain components, and whatever their choice may be, the boundary conditions and the conditions of medium conjugation are fulfilled exactly. The arbitrary choice of uncertain components is used to satisfy the basic differential equation. |
| format |
Article |
| author |
Khazhmuradov, M.A. |
| author_facet |
Khazhmuradov, M.A. |
| author_sort |
Khazhmuradov, M.A. |
| title |
Structures of solutions to the problems for multilayer media and their applications to the calculation of fuel–element thermal fields |
| title_short |
Structures of solutions to the problems for multilayer media and their applications to the calculation of fuel–element thermal fields |
| title_full |
Structures of solutions to the problems for multilayer media and their applications to the calculation of fuel–element thermal fields |
| title_fullStr |
Structures of solutions to the problems for multilayer media and their applications to the calculation of fuel–element thermal fields |
| title_full_unstemmed |
Structures of solutions to the problems for multilayer media and their applications to the calculation of fuel–element thermal fields |
| title_sort |
structures of solutions to the problems for multilayer media and their applications to the calculation of fuel–element thermal fields |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2001 |
| topic_facet |
Experimental methods and computations |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/78450 |
| citation_txt |
Structures of solutions to the problems for multilayer media and their applications to the calculation of fuel–element thermal fields / M.A. Khazhmuradov // Вопросы атомной науки и техники. — 2001. — № 1. — С. 66-68. — Бібліогр.: 7 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT khazhmuradovma structuresofsolutionstotheproblemsformultilayermediaandtheirapplicationstothecalculationoffuelelementthermalfields |
| first_indexed |
2025-07-06T02:32:43Z |
| last_indexed |
2025-07-06T02:32:43Z |
| _version_ |
1836863102316969984 |
| fulltext |
STRUCTURES OF SOLUTIONS TO THE PROBLEMS FOR MULTI-
LAYER MEDIA AND THEIR APPLICATIONS TO THE CALCULATION
OF FUEL–ELEMENT THERMAL FIELDS
M.A. Khazhmuradov
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
The technique of constructing approximate solutions to heat conduction problems of media is presented. The cal-
culation is based on the structures of solutions constructed with the help of the left, normalized-to-the-first-order
equations of region boundaries and their characteristic parts. The structures of solutions contain uncertain compo-
nents, and whatever their choice may be, the boundary conditions and the conditions of medium conjugation are ful-
filled exactly. The arbitrary choice of uncertain components is used to satisfy the basic differential equation.
PACS: 61.20.-Ja.
The thermal power of the reactor core is limited,
from the thermal viewpoint, by the maximal tempera-
ture of fuel and fuel element cladding. If the fuel ele-
ments have a finned, intricately-shaped surface, a gas
gap between the fuel and the cladding, or a multi-layer
structure, the existing methods of solving boundary-val-
ue problems give no way of taking into account these
limitations [1]. This paper describes the procedure of
solving boundary-value problems with mixed boundary
conditions and a complicated geometrical shape of fuel
elements.
Mathematically, the problem is presented in terms of
a boundary-value problem:
2 2
2 2
0
0,
( , ), ( , )
0 , ( , ) 0, 1 , 2 ,
s s
s
x y
F x y x y
x y s
∂ Τ ∂ Τ
λ + = ∂ ∂
− ∀ ∈ Ω= ∀ ∈ Ω =
(1)
,,
00
10 0
1
1
0
0
010
ΓΓ
ΓΓ ν∂
Τ∂λ−=
ν∂
Τ∂λ−=Τ T (2)
,,
11
11 1
2
2
1
1
121
ΓΓ
ΓΓ ν∂
Τ∂λ−=
ν∂
Τ∂λ−Τ=Τ (3)
),( 22
2
2 c
c
c
Τ−Τα=
ν∂
Τ∂λ−
Γ
(4)
where F(x,y)=Q/V0 - specific power of heat evolution
(wt/m3, Q – heat evolution power), V0=πR0L=πd2L/4 –
volume of fuel element, s – number of areas, cΤ – am-
bient temperature.
The proposed approach of constructing approximate
solutions of boundary-value problems combines the
possibilities of the R–function method [2] to take into
account the complicated character of boundary condi-
tions (2) – (4) for intricately-shaped fuel elements
(Figs. 1, 2) and the properties of exact solutions to allow
for the effect of concentrated fuel elements.
Fig. 1. Model of a fin-shaped source of an elliptic
type
Fig. 2. Model of the technological cartridge with a
fin-shaped source of an elliptic type
According to the variation principle [3], the bound-
ary-value problem (1) – (4) is equivalent to the variation
problem of finding the function θs(x,y) that leads to the
minimum of the following functional on the set D(A):
66 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 1.
Series: Nuclear Physics Investigations (37), p. 66-68.
0
2
0 0 0 0
2 2
2
I( ) ) 2
) ,
s c
s
s s s c c
F d
d d
Ω
Ω Γ
θ = λ ∇ θ − θ Ω +
+ λ ∇ θ Ω + α θ Γ
∫ ∫
∫ ∫ ∫
(5)
where θs =Ts–f0 is a new unknown function; f0 is a cer-
tain function satisfying the boundary conditions (2) –
(4); s is the region of the i – the boundary.
The functions θs(x,y) (s=0,1,2), minimizing the func-
tional (5), can be represented by structural formulas [4]
as
00 ),( Φ=θ yx , (6)
),(1 yxθ = ,1
)(
)(
00
1
0
0
2
21
0
2
21
0 Φ
−
λ
λ
ω+ωω
ωωω+Φ D (7)
.D
)(
)(D
)(
)(
D
)(
)(
D1
)(
)(
)(
)(1),(
02
2
2
10
2
10
2
2
2
10
2
2
10
2
2
02
2
2
10
2
10
2
2
01
2
1
1
2
20
1
2
20
0
2
2
10
2
2
10
2
2
2
Φ
ω+ωω
ωω
ω+ωω
ωωω
α
λ+
+Φ
ω+ωω
ωω
α
λ−
−
Φ
−
λ
λ
ω+ωω
ωωω+
+Φ
ω+ωω
ωωω
λ
α−=θ yx
(8)
The equations ωs(х,у)=0 are the normalized equa-
tions of boundaries sΓ and satisfy the following condi-
tions [2]:
1) ωs(х,у)∈С2(Ωs),
2) ωs(х,у)>0, ∀(х,у)∈Ωs,
3) ωs(х,у)=0, ∀(х,у)∈Гs,
4) 1=
ν∂
ω∂
Γ s
s
,
where νs is the inward normal of the contour ,sΓ bound-
ing the region Ωs.
For the problem under consideration (Fig. 1) the
equations ωs(х,у) (s=0,1,2) have the following form
ωs(х,у) =
s
sss
R
yyxxR
2
)()( 222 −+−−
,
where Rs is the radius of the region Ωs, and хs is its cen-
ter.
For the finned region Ω1 the equation is written as
follows:
ω1(х,у) =ƒ1+ƒ2 – 2
2
2
1 ff + ,
where
,
2
)()(
1
2
1
2
1
2
1
1 R
yyxxRf −−−−=
2 22
2
f
x y
ϕ=
∂ ϕ ∂ ϕ ϕ + + ∂ ∂
r
rn
i 2
2
21
1
−ϕ+ϕ
∏=ϕ
=
,
where
2
1
2 ( 1)( )cos ,i nx R a
n
π − + α ϕ = − +
2
2
2 ( 1)( )sin i ny R
n
π − − α ϕ = − + α
,
n is the number of fins.
At а=0, the center of the circle forming the fin, will
lie on the axis 0Х. If а=0, then the centers of circles will
be on the circle of radius R1, i.e., on the boundary Г1.
The radius r can be used to control the size of fins and
consequently their number.
Ds is the differential operator of the form
Ds= yyxx
ss
∂
∂
∂
ω∂+
∂
∂
∂
ω∂
,
possessing the properties
,
s
s s
s
ssD
Γ
Γ ν∂
θ∂=θ
which follow from conditions 1), 4) of equations ω
s(х,у).
We now show that the structural formulas (6) – (8)
exactly satisfy the boundary conditions (2) – (4):
,),( 00 Φ=θ yx 01
0
),( Φ=θ Γyx ,
000
0
0
0
0
Φλ−=
ν∂
θ∂λ−
Γ
D ,
000
0
1
1
0
Φλ−=
ν∂
θ∂λ−
Γ
D ,
01
1
),( Φ=θ Γyx ,
02
1
),( Φ=θ Γyx ,
011
1
1
1
1
Φλ−=
ν∂
θ∂λ−
Γ
D ,
011
1
2
2
1
Φλ−=
ν∂
θ∂λ−
Γ
D ,
67
02202
2
2
2
2
Φλ−Φα=
ν∂
θ∂λ−
Γ
D ,
0220222
2
Φλ−Φα=Τα Γ D .
The uncertain element Ф0(х,у) of structural formulas
(6) – (8) can be presented by the expansion [2], [4]
Ф0(х,у)= ∑ ϕ
=
n
i
ii yx
1
),(c , (9)
where ϕi(x,y) stands for the functions of a linearly–inde-
pendent system (Chebyshev, Legendre, Hermite polyno-
mials, etc. [5]), ic denotes the unknown constants.
Substituting (9) into (6) – (8), we obtain
θs(x,y) = s
n
i
s
ii yxyx Ω∈∀∑ Ψ
=
),(,),(c
1
)( , (10)
where n is the number of coordinate functions defined
by the formula n = (к+1)×(к+2)/2 (к being the power of
polynomial), ),()( yxs
iΨ are the elements of basis sys-
tems of functions of types:
),()0( yxiΨ = ),( yxiϕ ,
),()1( yxiΨ = iϕ +b(h3–1)D0 iϕ ,
),()2( yxiΨ =(1–h4b1) iϕ +b2(h5–1)D1 iϕ –
– h6 b3D2 iϕ + h6 b1D2(b3D2 iϕ ),
b =
0
2
21
0
2
21
)(
)(
ω+ωω
ωωω
,
b1 =
2
2
10
2
2
10
)(
)(
ω+ωω
ωωω
,
b2 =
1
2
20
1
2
20
)(
)(
ω+ωω
ωωω
,
b3 =
2
2
10
2
10
)(
)(
ω+ωω
ωω
,
h3 =
1
0
λ
λ
,
h4 =
2
2
λ
α
,
h5 =
2
1
λ
λ
,
h6 = .
2
2
α
λ
The unknown constants сi(i=1,…,n) are calculated
from the minimum condition of functional (5)
∂I(θs)/∂cj = 0, i≤ j ≤ n.
This requirement is equivalent to a set of linear alge-
braic equations
∑ =
=
n
i
iij c
1
A ,,,2,1,
1
njB
n
i
j∑ =
=
where ijA and jB look like
2
( ) ( )( ) ( )2
0
(2 (2)
2 2
d , 0, 1, 2,
s
s ss s
j ji i
ij s
s
i j
d
x x y y
s
= Ω
Γ
∂ Ψ ∂ Ψ∂ Ψ ∂ ΨΑ = + Ω + ∂ ∂ ∂ ∂
+ α Ψ Ψ Γ =
∑ ∫ ∫
∫
∫ ∫ ΩΨ−=Β
Ω 0
.0
)0( dF ij
The integration over the regions Ωs and the bound-
ary cΓ was performed using the Gauss quadrature for-
mulas [6].
The approximate analytical solution of the bound-
ary-value problem (1) – (4), obtained with the help of
the proposed procedure, is written down as
),,(),(),( )(
0 yxyxyx s
s
s θ+ϕ=Τ (11)
where )(
0
sϕ are the temperature fields defined by the
boundary conditions.
Thus, the method is proposed for solving the bound-
ary-value problem with complicated boundary condi-
tions for conjugation of nonuniform media, and the con-
ditions of heat exchange between the surface of the sys-
tem and the environment. Structures from (6) to (8), de-
veloped for each constituent area, exactly satisfy the
conjugation conditions (2) – (4) at the boundaries
., 1+ΓΓ ss They have the properties of the passage to the
limit with decreasing distances between the boundaries
of contacting media. This is of importance, because with
fuel swelling Ω0 in nuclear fuel elements the clearance
between the fuel and cladding is reduced, and the
boundaries Г0 and Г1 are coming together almost in line
[7].
REFERENCES
1. L.V. Kantorovich, V.I. Krylov. Approximate meth-
ods of higher analysis. M.: Phizmatgiz, 1972, 600 р.
(in Russian).
2. V.L. Rvachov. The theory of R-functions and some
of its applications. Kiev: «Naukova dumka», 1982,
552 р. (in Russian).
3. S.G. Mikhlin. Numerical realization of variation
methods. M.: Nauka, 1966, 432 р. (in Russian).
4. M.A. Khazhmuradov. Design automation of engi-
neering systems with due account of thermal restric-
tions. Preprint KIPT 88-14, 1988, 12 p. (in Russian).
5. G. Sege. Orthogonal polynomials. M.: IL., 1962,
500 p. (in Russian).
6. V.M. Krylov, A.T. Shuljga. The reference book on
numerical integration. M.: Nauka, 1966, 372 p., (in
Russian).
7. V.A. Tsykanov, Ye.F. Davydov. Radiation resis-
tance of reactor fuel elements. M.: Atomizdat, 1977,
136 p. (in Russian).
68
M.A. Khazhmuradov
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