The R-functions method in mathematical modeling of convective heat transfer in fuel cartridge with fuel rods
Conjugate boundary value problems of heat transfer in cases, when viscous incompressible fluid flows via channels of non-canonical sections, bypassing the bundle of rods, is considered. The influence of the packaging pattern on the velocity and temperature distribution is researched. The theory of R...
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Maksimenko-Sheyko, K.V. Uvarov, R.A. Sheyko, T.I. 2017-01-15T11:26:03Z 2017-01-15T11:26:03Z 2013 The R-functions method in mathematical modeling of convective heat transfer in fuel cartridge with fuel rods / K.V. Maksimenko-Sheyko, R.A. Uvarov, T.I. Sheyko // Вопросы атомной науки и техники. — 2013. — № 3. — С. 205-209. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 02.70.-c, 44.05.+e, 67.40.Hf https://nasplib.isofts.kiev.ua/handle/123456789/111846 Conjugate boundary value problems of heat transfer in cases, when viscous incompressible fluid flows via channels of non-canonical sections, bypassing the bundle of rods, is considered. The influence of the packaging pattern on the velocity and temperature distribution is researched. The theory of R-functions in combination with the Ritz variational method were used for the solution. Different packaging of fuel rods are examined. Each package contains 91 rods, and the corresponding equations are constructed with the new design tools of the theory of R-functions. Розглянуто пов’язанi крайовi задачi теплообмiну для випадкiв, коли в’язка нестисла рiдина рухається по каналах неканонiчного перерiзу, обтiкаючи пучок стрижнiв. Дослiджено вплив виду пакування на розподiл швидкостi i температури. Для розв’язування використовувалася теорiя R-функцiй у поєднаннi з варiацiйним методом Рiтца. Розглянуто рiзнi пакування ТВЕлiв. Кожне пакування мiстить 91 стрижень, а вiдповiднi рiвняння побудованi з використанням нових конструктивних засобiв теорiї R-функцiй. Рассмотрены сопряженные краевые задачи теплообмена для случаев, когда вязкая несжимаемая жидкость движется по каналам неканонического сечения, обтекая пучок стержней. Исследовано влияние вида упаковки на распределение скорости и температуры. Для решения использовалась теория R-функций в сочетании с вариационным методом Ритца. Рассмотрены различные упаковки ТВЭлов. Каждая упаковка содержит 91 стержень, а соответствующие уравнения построены с использованием новых конструктивных средств теории R-функций. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Ядернo-физические методы и обработка данных The R-functions method in mathematical modeling of convective heat transfer in fuel cartridge with fuel rods Метод R-функцiй у математичному моделюваннi конвективного теплообмiну у паливнiй касетi твелiв Метод R-функций в математическом моделировании конвективного теплообмена в топливной кассете твэлов Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
The R-functions method in mathematical modeling of convective heat transfer in fuel cartridge with fuel rods |
| spellingShingle |
The R-functions method in mathematical modeling of convective heat transfer in fuel cartridge with fuel rods Maksimenko-Sheyko, K.V. Uvarov, R.A. Sheyko, T.I. Ядернo-физические методы и обработка данных |
| title_short |
The R-functions method in mathematical modeling of convective heat transfer in fuel cartridge with fuel rods |
| title_full |
The R-functions method in mathematical modeling of convective heat transfer in fuel cartridge with fuel rods |
| title_fullStr |
The R-functions method in mathematical modeling of convective heat transfer in fuel cartridge with fuel rods |
| title_full_unstemmed |
The R-functions method in mathematical modeling of convective heat transfer in fuel cartridge with fuel rods |
| title_sort |
r-functions method in mathematical modeling of convective heat transfer in fuel cartridge with fuel rods |
| author |
Maksimenko-Sheyko, K.V. Uvarov, R.A. Sheyko, T.I. |
| author_facet |
Maksimenko-Sheyko, K.V. Uvarov, R.A. Sheyko, T.I. |
| topic |
Ядернo-физические методы и обработка данных |
| topic_facet |
Ядернo-физические методы и обработка данных |
| publishDate |
2013 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Метод R-функцiй у математичному моделюваннi конвективного теплообмiну у паливнiй касетi твелiв Метод R-функций в математическом моделировании конвективного теплообмена в топливной кассете твэлов |
| description |
Conjugate boundary value problems of heat transfer in cases, when viscous incompressible fluid flows via channels of non-canonical sections, bypassing the bundle of rods, is considered. The influence of the packaging pattern on the velocity and temperature distribution is researched. The theory of R-functions in combination with the Ritz variational method were used for the solution. Different packaging of fuel rods are examined. Each package contains 91 rods, and the corresponding equations are constructed with the new design tools of the theory of R-functions.
Розглянуто пов’язанi крайовi задачi теплообмiну для випадкiв, коли в’язка нестисла рiдина рухається по каналах неканонiчного перерiзу, обтiкаючи пучок стрижнiв. Дослiджено вплив виду пакування на розподiл швидкостi i температури. Для розв’язування використовувалася теорiя R-функцiй у поєднаннi з варiацiйним методом Рiтца. Розглянуто рiзнi пакування ТВЕлiв. Кожне пакування мiстить 91 стрижень, а вiдповiднi рiвняння побудованi з використанням нових конструктивних засобiв теорiї R-функцiй.
Рассмотрены сопряженные краевые задачи теплообмена для случаев, когда вязкая несжимаемая жидкость движется по каналам неканонического сечения, обтекая пучок стержней. Исследовано влияние вида упаковки на распределение скорости и температуры. Для решения использовалась теория R-функций в сочетании с вариационным методом Ритца. Рассмотрены различные упаковки ТВЭлов. Каждая упаковка содержит 91 стержень, а соответствующие уравнения построены с использованием новых конструктивных средств теории R-функций.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/111846 |
| citation_txt |
The R-functions method in mathematical modeling of convective heat transfer in fuel cartridge with fuel rods / K.V. Maksimenko-Sheyko, R.A. Uvarov, T.I. Sheyko // Вопросы атомной науки и техники. — 2013. — № 3. — С. 205-209. — Бібліогр.: 6 назв. — англ. |
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| fulltext |
THE R-FUNCTIONS METHOD IN MATHEMATICAL
MODELING OF CONVECTIVE HEAT TRANSFER IN FUEL
CARTRIDGE WITH FUEL RODS
K.V. Maksimenko-Sheyko∗, R.A. Uvarov, T.I. Sheyko
Podgorny Institute for Mechanical Engineering Problems of NAS of Ukraine, Kharkov, Ukraine
(Received March 27, 2013)
Conjugate boundary value problems of heat transfer in cases, when viscous incompressible fluid flows via channels
of non-canonical sections, bypassing the bundle of rods, is considered. The influence of the packaging pattern on
the velocity and temperature distribution is researched. The theory of R-functions in combination with the Ritz
variational method were used for the solution. Different packaging of fuel rods are examined. Each package contains
91 rods, and the corresponding equations are constructed with the new design tools of the theory of R-functions.
PACS: 02.70.-c, 44.05.+e, 67.40.Hf
1. INTRODUCTION
At the core of modern nuclear power plants the nu-
clear fuel is concentrated in the fuel elements (fuel
rods). One of the main problems in the reactor cal-
culation process is a definition of the temperature
field, since the requirements to the reliability of the
fuel rods are highest. The failure of several fuel rods,
while the reactor has thousands of them, can lead to
an emergency situation. In this case, there is a need
to meet the relevant transport problems in the dual
formulation, as these solutions allow to research the
real heat transfer processes, which essentially show
the mutual influence of the moving fluid, the channel
walls, fuel rods, etc.
Consider the conjugate heat transfer boundary
value problems for cases, when viscous incompressible
fluid flows through the channel of the non-canonical
section, bypassing the rods. It is assumed that the
physical properties of the fluid are constant, laminar
flow is hydro-dynamically stable, and heat exchange
process is stationary. Conditions of the first, second
and third kind can be specified on the outer surface
of the channel. It is assumed that the change of the
heat flux along the channel due to the axial thermal
conductivity is negligible compared to the change of
the heat flux due to convection. It is also assumed
that the pipe walls and the inner rods are made of
an isotropic material, and the thermal conductivity
of the latter can be considered as constant in this
temperature range [1-3].
The presence of internal heat sources in the reac-
tor components complicates both the heat equation,
and the methods for its solution. A heat transfer in
the bundle of infinite cylinders bypassed lengthwise is
considered in [1,3]. A symmetry of the temperature
field due to the symmetry of the system is assumed,
and only the region is considered (Fig. 1).
The aim of this work is the improvement of de-
sign tools and algorithms of the R-functions method
for mathematical and computer modeling of the con-
jugate problem of convective heat transfer in arrays
of fuel rods and the study of the packaging influ-
ence on the velocity and temperature distributions.
Fig.1. Location of operating channels in the
reactor core by the pattern: hallway-on the left;
checkerboard-on the right
2. MAIN PART
The basic system of equations describing the process
of heat transfer in a flow of viscous fluid with constant
physical properties of the liquid and the temperature
has the form
DT
Dτ = a4T + qV
ρcp
+ µΦ
ρcp
,
D~V
Dτ = − 1
ρ
~∇p + ν4~V ,
div ~V = 0 ,
where D
Dτ = ∂
∂τ + (~V · ~∇) is a substantial (or total)
derivative,
µΦ is a dissipation function;
a = λ
ρcp
is a thermal diffusivity;
∗Corresponding author E-mail address: sheyko@ipmach.kharkov.ua
ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2013, N3(85).
Series: Nuclear Physics Investigations (60), p.205-209.
205
cp is a heat capacity of the medium;
qV is a power of internal heat sources.
For stationary processes, the temperature of an
object is independent of time, and the heat conduc-
tion equation, while bypassing the fuel rods length-
wise, takes the form
Vz
∂T
∂z
= a4T +
qV
ρcp
,
and a mathematical model of the velocity field for
laminar flow is given by
4Vz = −∇P
µl
= −C ,
where ∇P is a constant pressure drop along the pipe
on the randomly selected portion of length l.
In the thermal stability area, when ∂T
∂z = const,
we will take
−div(λ∇T ) = qV − VzCl .
Thus, the mathematical model of heat transfer in
laminar flow of the fluid in the cartridge with fuel
rods is reduced to the system of equations
{ 4Vz = −C, in Ωb
⋂
Ωtν ;
−div(λi∇Ti) = Fi, in Ωb,
where {
F1 = −Vz, in Ωb
⋂
Ωtν ;
F2 = qV , in Ωtν ,
with boundary conditions
Vz
∣∣∣∂Ωb
T
∂Ωtν
= 0,
∂T
∂n
+ hT |∂Ωb
= 0,
T1 |∂Ωtν = T2 |∂Ωtν ,
λ1
∂T1
∂n1
|∂Ωtν = λ2
∂T2
∂n2
|∂Ωtν . (1)
Consider a typical constructive pattern of the re-
actor core, which is collected of a large number of
fuel cartridges [2, 3]. Cartridges are hexagonal cas-
ings, possessing the fuel rods. Construct an equa-
tion of fuel cartridge with 91 fuel rods and the tri-
angular packaging moved apart (Fig. 2,a), which is
called checkerboard sometimes. Note that ordinary
technique used in the theory of R-functions allows to
obtain 93 R-operations in the equation as a result.
Cumbersome formula will lead not only to increase
of the computation time, but also, perhaps, to some
symmetry breaking due to the non-associativity of R-
operations. Therefore, technique developed in [4, 5, 6]
will be used for the construction of the equation of
hexagonal casing.
Consider the equation of a line
σ ≡ Rν − x ≥ 0
and a periodic function
µν =
4
3π
∑
k
(−1)k+1 sin[(2k − 1)3θ]
(2k − 1)2
.
The result is
ωb ≡ Rν − r cos µν = 0 ,
where r =
√
x2 + y2, θ = arctg y
x .
To construct a triangle packing of fuel rods, define
f1 = R2 − µ2
x − µ2
y ≥ 0,
where
µx =
4hx
π2
∑
k
(−1)k+1
sin
[
(2k − 1)πx
hx
]
(2k − 1)2
,
µy =
4hy
π2
∑
k
(−1)k+1
sin
[
(2k − 1)πx
hy
]
(2k − 1)2
,
f2 = R2 − µ2
x1 − µ2
y1 ≥ 0,
where
µx1 =
4hx
π2
∑
k
(−1)k+1
sin
[
(2k−1)π(x−hx/2)
hx
]
(2k − 1)2
,
µy1 =
4hy
π2
∑
k
(−1)k+1
sin
[
(2k−1)π(y−hy/2)
hx
]
(2k − 1)2
,
Then the equation of fuel cartridge is
ω ≡ ωb ∧0 ωtν ≥ 0, ωtν ≡ (f1 ∨0 f2).
Construction of the ω(x, y) is performed
for the following values of parameters:
R = 0.2; hx = 2.32; hy = 1.35; n0 = 6; rk = 6.7.
It should be noted that R-operations
are used only twice in new method
of the cartridge equation construction.
a
b
Fig.2. Fuel cartridge with 91 fuel rods located by: a
- checkerboard pattern; b - distribution with cyclic
symmetry and central fuel rod
Let us also consider the construction of the func-
tion ω(x, y) ≡ ωb ∧0 ωtν , when fuel rod is distributed
with cyclic symmetry nd times along a circle of radius
R, nb times along a circle of radius R1, and nc times
along a circle of radius R2.
206
To construct the equations of the boundary of fuel
rods distributed with cyclic symmetry nd times along
a circle of radius R, the function
ω0 ≡ 1
2Rtν
(
R2
tν − (x−R)2 − y2
)
and the formula
µd =
8
ndπ
∑
k
(−1)k+1 sin
[
(2k − 1)ndθ
2
]
(2k − 1)2
,
will be used. The result is
ωtν1 ≡ R2
tν − (r cosµd −R)2 − (r sin µd)2
2Rtν
≥ 0 .
To construct the equations of the boundary of fuel
rods distributed with cyclic symmetry nb times along
a circle of radius R1, the function
ω01 ≡ 1
2Rtν
(
R2
tν − (x−R1)2 − y2
)
and the formula
µb =
8
nbπ
∑
k
(−1)k+1 sin
[
(2k − 1)nbθ
2
]
(2k − 1)2
,
will be used. The result is
ωtν2 ≡ R2
tν − (r cos µb −R1)2 − (r sin µb)2
2Rtν
≥ 0 .
To construct the equations of the boundary of fuel
rods distributed with cyclic symmetry nc times along
a circle of radius R2, the function
ω02 ≡ 1
2Rtν
(
R2
tν − (x−R2)2 − y2
)
and the formula
µc =
8
ncπ
∑
k
(−1)k+1 sin
[
(2k − 1)ncθ
2
]
(2k − 1)2
,
will be used. The result is
ωtν3 ≡ R2
tν − (r cosµc −R2)2 − (r sin µc)2
2Rtν
≥ 0 .
Thus, the equation of the boundary of the car-
tridge with 91 fuel rods has the form
ω ≡ (ωb ∧0 ωtν1 ∨0 ωtν2 ∨0 ωtν3) ≥ 0 ,
when nd = 38, R = 6, nb = 32, R1 =
4.5, nc = 21, R2 = 3 and it is a seven-
parametric (nb, nd, nc, R, R1, R2, Rtν) family of
curves (Fig. 2,b). It should be noted that the R-
operations were used only three times. If the central
fuel rod is present, the result is
ω ≡ ωb ∧0 ωtν1 ∨0 ωtν2 ∨0 ωtν3 ∨0
Rtν
2 − x2 − y2
2Rtν
≥ 0 ,
when nc = 20.
When the equation of the cartridge and fuel rods
are known, we can rewrite the problem (1) in the
form { 4Vz = −C,
−div(λ∇T ) = F,
with boundary conditions
Vz
∣∣∣∂Ωb
T
∂Ωtν
= 0,
∂T
∂n
+ hT |Ωb
= 0,
T1 |∂Ωtν = T2 |∂Ωtν ,
λ1
∂T1
∂n1
∣∣∣∣
∂Ωtν
= λ2
∂T2
∂n2
∣∣∣∣
∂Ωtν
,
where
λ = λ1
1− sgn(ωtν)
2
+ λ2
1 + sgn(ωtν)
2
,
F = −Vz
1− sgn(ωtν)
2
+ qV
1 + sgn(ωtν)
2
.
The R-functions method in conjunction with the
Ritz variational method were used for the solving.
The structure of solution for the problem of laminar
longitudinal flow of fuel rods by fluid has the form
Vz = ωp1 ,
where ω(x, y) ≡ ωb ∧0 ωtν ≥ 0 is equation of
the boundary of cartridge’s cross-section,
p1 =
∑N
i=1 cikϕik(x, y) is the undefined component,
which will be found, minimizing the functional
I =
∫
Ω
[
(∇Vz)2 − 2CVz
]
dΩ .
Note that the solution Vz is obtained analytically
and used without any further treatment (approxima-
tion, interpolation). Therefore, the resulting distrib-
ution of the velocity is substituted in the right side
of the heat conductivity equation. The structure of
solution for the problem of determining the temper-
ature field was used both as exactly satisfying the
boundary conditions on ∂Ωb
u = p2 + ωb(−D1p2 + hp2) ,
and as T = p2 where, as before,
p2 =
N∑
i=1
dikϕik(x, y) .
It should be noted that the boundary conditions
∂T
∂n
+ hT |Ωb
= 0 ,
λ1
∂T1
∂n1
∣∣∣∣
∂Ωtν
= λ2
∂T2
∂n2
∣∣∣∣
∂Ωtν
are natural and result from the Ritz functional
I =
∫
Ω
[
λ(∇T )2 − 2FT
]
dΩ +
∫
∂Ωb
hT 2d∂Ωb .
As an approximation of ϕik(x, y) cubic splines of
Schoenberg with N = 6400, 10000 were used. Com-
putational experiments were carried out in POLYE-
RL system developed in the department of applied
mathematics and computational methods of IPMash
207
NASU. The results of researches for different pack-
ages of fuel rods are shown below (Figs. 3-6). Each
package contains 91 fuel rods with same other condi-
tions λ1 = 1, λ2 = 10, h = 1, qV = 4.
Different distributions of the researched fields
were taken by changing the parameter values.
V
T
Fig.3. Picture of the velocity and temperature
distribution in the cartridge with fuel rods arranged
in checkerboard pattern
V
T
Fig.4. Picture of the velocity and temperature
distribution in the cartridge with cyclically spaced
fuel rods and central fuel rod
V
T
Fig.5. Picture of the velocity and temperature
distribution in the cartridge with cyclically spaced
fuel rods without central fuel rod
Fig.6. Graphs of the temperature field for different
packages of fuel rods in the cross section : 1 is
checkerboard pattern; 2 is cyclical symmetry with
central fuel rod; 3 is cyclical symmetry without
central fuel rod
Analyzing the results, we can conclude that the
presence of fuel rods in the central area leads to a
higher temperature. Therefore, changing the packag-
ing pattern and types of symmetry, we can adjust the
nature of the flow and temperature distribution in the
cartridge, achieving the value stated by technical re-
quirements. Analysis of the velocity and temperature
distribution allows to conclude that the consideration
of the velocity field of the cell (see Fig. 1), if it is suffi-
ciently far from the border, is appropriate. However,
the temperature field at the same time is far from
reality, as evidenced by the results obtained for the
whole cartridge.
208
3. CONCLUSIONS
It is shown that the R-functions method is effective
for solving the problems of the physical field calcula-
tion in complex shape structural elements of nuclear
power plants. The developed design tools for con-
structing the equations of domain boundaries with
translational and cyclic symmetry types allowed to
significantly reduce the number of operations with
subsequent automation of the process, and, hence,
to reduce the time for solving the problems. These
experiments allow designers to choose certain types
of packaging according to the specifications. During
this, the essential point is to calculate the tempera-
ture field for the whole cartridge. Mathematical mod-
eling and associated computer experiment are indis-
pensable in cases, when the natural experiment is im-
possible or difficult, for various reasons. In addition,
working with a mathematical model of the process
and computer experiment allows to investigate the
properties and behavior of the process in different
situations painlessly, relatively quickly, and without
significant cost. At the same time, the computational
experiments with object models allow, based on mod-
ern numerical methods, to study them deeply in de-
tails. The reliability of analytical identification of
geometric objects is approved by their visualization,
while the reliability of calculation methods, results
and conclusions is confirmed by comparison with the
information known from the literature and the analy-
sis of the numerical convergence of solutions and the
calculation of the residual.
References
1. A.P. Slesarenko, D.A. Kotulsky. Regional-
analytical and variational methods in solving the
conjugative problems of convective heat transfer
// Heat mass exchange MIF-2000, Proceedings of
IV Minsk international forum (Belarus, Minsk,
May 2000). Minsk: IHME AS of Belarus, 2000,
v. 3, p. 135-142 (in Russian).
2. B.S. Petukhov. Heat transfer and resistance in
laminar flows of liquids in pipes. Moscow: ”En-
ergy”, 1967, p. 412 (in Russian).
3. B.S. Petukhov, L.G.Genin, S.A. Kovalev. Heat
exchange in nuclear power plants. Moscow: ”At-
omizdat”, 1974, p. 367 (in Russian).
4. V.L.Rvachev. The theory of R-functions and its
several applications. Kiev: ”Naukova dumka”,
1982, p. 552 (in Russian).
5. K.V.Maksimenko-Sheyko. The R-functions in
mathematical modelling of geometrical objects
and physical phields, Kharkiv: ”IPMach NASU”,
2009, p. 306 (in Russian).
6. K.V.Maksimenko-Sheyko, A.M. Matsevity,
A.V.Tolok, T.I. Sheyko. The R-functions and
the inverse problem of analytical geometry
in three-dimensional space // Informational
technologies, Moscow, 2007, N10, p. 23-32 (in
Russian).
МЕТОД R-ФУНКЦИЙ В МАТЕМАТИЧЕСКОМ МОДЕЛИРОВАНИИ
КОНВЕКТИВНОГО ТЕПЛООБМЕНА В ТОПЛИВНОЙ КАССЕТЕ ТВЭЛОВ
К.В.Максименко-Шейко, Р.А.Уваров, Т.И.Шейко
Рассмотрены сопряженные краевые задачи теплообмена для случаев, когда вязкая несжимаемая жид-
кость движется по каналам неканонического сечения, обтекая пучок стержней. Исследовано влияние
вида упаковки на распределение скорости и температуры. Для решения использовалась теория R-
функций в сочетании с вариационным методом Ритца. Рассмотрены различные упаковки ТВЭлов.
Каждая упаковка содержит 91 стержень, а соответствующие уравнения построены с использованием
новых конструктивных средств теории R-функций.
МЕТОД R-ФУНКЦIЙ У МАТЕМАТИЧНОМУ МОДЕЛЮВАННI КОНВЕКТИВНОГО
ТЕПЛООБМIНУ У ПАЛИВНIЙ КАСЕТI ТВЕЛIВ
К.В.Максименко-Шейко, Р.О.Уваров, Т.I.Шейко
Розглянуто пов’язанi крайовi задачi теплообмiну для випадкiв, коли в’язка нестисла рiдина рухається
по каналах неканонiчного перерiзу, обтiкаючи пучок стрижнiв. Дослiджено вплив виду пакування на
розподiл швидкостi i температури. Для розв’язування використовувалася теорiя R-функцiй у поєд-
наннi з варiацiйним методом Рiтца. Розглянуто рiзнi пакування ТВЕлiв. Кожне пакування мiстить
91 стрижень, а вiдповiднi рiвняння побудованi з використанням нових конструктивних засобiв теорiї
R-функцiй.
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