CC BY
13
Колесников // Проблемы анализа риска. -2013. - Т. 10. - № 3. - С. 8-31.
2. Лисанов М.В. Методическое обеспечение и проблемы анализа риска аварий на опасных производственных объектах нефтегазового комплекса / М.В. Лисанов, А.С. Печеркин, С.И. Сумской и др. // Вести газовой науки: Повышение надежности и безопасности объектов газовой промышленности. - М.: Газпром ВНИИГАЗ, 2017. - № 1 (29). -С. 179-186
© Игнатьева Л.И., Шарафутдинова Г.М., Барахнина В.Б., 2021
УДК 536.212
Канарейкин А.И.
кан. техн. наук, доцент МГРИ, г. Москва, РФ
ТЕМПЕРАТУРНЫЕ ПОЛЯ В ТВЭЛАХ ПРИ РАЗЛИЧНЫХ ГЕОМЕТРИЧЕСКИХ СЕЧЕНИЯХ
Аннотация
В статье производится анализ расчета температурного поля в твэле сферического, цилиндрического и эллиптического сечений с внутренним тепловыделением. При этом граничные условия являются граничными условиями первого рода. Решение приведены в соответствующих системах координат. При сопоставлении результатов, автором установлено, что перепад температур между центром твэла и поверхностью не зависит от граничных условий, а полностью определяется внутренним тепловыделением и коэффициентом теплопроводности материала. Что связано с тем, что граничные условия характеризуют лишь абсолютное значение температуры.
Ключевые слова
Температурное поле, твэл, теплопроводность, граничные условия, оператор Лапласа.
The unified approach to the thermal calculation of fuel elements of various nuclear systems is explained by the presence of internal heat release in them. The problem of determining the temperature fields in fuel rods is of independent interest and is also a preliminary stage of calculating the heat transfer process. The most typical geometry of fuel rods is a ball, cylinder, and ellipse with internal heat dissipation. A special place is occupied by long fuel rods with an elliptical cross-section. Their peculiarity is that by manipulating the change in the length of the semi-axes of the ellipse, it is possible to obtain accurate analytical solutions to stationary problems of thermal conductivity for a very wide range of shape changes: from a cylinder (the semi-axes of the ellipse are equal) to a thin plate (one of the semi-axes is significantly larger than the other).
The problems of calculating temperature fields in fuel rods of spherical and cylindrical cross-sections are well understood by the author [1, p. 56] in the presence of internal heat sources, and for the case of an elliptical cross-section, several works are devoted [2, p.74, 3, p. 230].
The determination of the fuel element temperature field is reduced to the solution of the stationary equation of thermal conductivity
AT + q = 0 (1)
Л
where: qv is the specific power of the internal source, X is the coefficient of thermal conductivity.
In the case of a ball-shaped fuel rod, the Laplace operator in a spherical coordinate system has the form
^T+2—(2)
dr r dr Л
for r=R T=Tr, the solution has the form
ISSN 2410-6070_ИННОВАЦИОННАЯ НАУКА_№4 / 2021
T = т +
R 6i
2 f f \Л
l-( R ) ,
(3)
a temperature difference between the center of the ball and its surface
r0 - TR = a^L (4)
0 R 6X where To is the temperature in the center of the ball.
The obtained result is interesting because the temperature difference between the center and the surface does not depend on the boundary conditions, but is completely determined by the internal heat release and the thermal conductivity coefficient of the material.
In the case of a cylindrical fuel rod, the Laplace operator is conveniently viewed in the polar coordinate system
^T + i^T = -3r (5)
dr r dr X
f r
for r=R T=Tr, the solution has the form
T = T +
R 4i
(6)
a temperature difference between the center of the ball and its surface
To - Tr = q^ (7)
0 R 4X
In this case, the temperature difference between the center of the cylinder and its surface does not depend on the boundary conditions.
To obtain a formula describing the temperature field in an elliptical fuel rod, we write down the Poisson equation in elliptic co-ordinates
1
с (ch a - cos ß)
f д2 t д2 тл +
Ii (8)
уда1 dp- j
for a=ao (equation of the body surface) T=Tc, the solution has the form
2
T = Tc + (sh2а0 - sh2а) (9)
4X
a temperature difference between the center of the ellipse and its surface
To - Tc = qC- sh2ao (10)
And in this case, the temperature difference does not depend on the boundary conditions.
In the case of equality of the semi-axes of the ellipse (a=b=R) from the ratio (10), we obtain the temperature difference for a fuel rod with a circular cross-section (7). If one of the semi-axes significantly exceeds the other (a>>b), then the ratio (10) will give the following dependence
a b2
T0 - Tn = av— (11)
0 n 2X
which corresponds to the temperature difference between the center of the heat-generating wall of thickness 2b and its surface. Again, the temperature difference does not depend on the boundary conditions.
In this paper, we present a solution for the distribution of the temperature field in a fuel rod with different cross-sectional cases under boundary conditions of the first kind. Special cases are also given. In all cases, the temperature difference between the fuel element center and the surface does not depend on the boundary conditions, but is completely determined by the internal heat release and the thermal conductivity coefficient of the material. This is due to the fact that the boundary conditions characterize only the absolute value of the
temperature.
As you can see, the obtained relations are quite simple. This is explained by the classical geometry of the configurations under consideration and the stationary nature of the process.
The results presented in the article can be used in further studies and calculations of fuel elements. References:
1. Vlasov N. M., Fedik I. I. Teplovydelyayuschie elements of nuclear rocket engines / M.: TsNII atominform, 2001. 208p.
2. Kanareykin, A. I. Temperature distribution in an elliptical body with an internal heat source under boundary conditions of the first kind // Bulletin of the Kaluga University, series: natural sciences. - 2020, №. 2. - P. 74-76.
3. Kanareykin, A. I. Distribution of the temperature field in a fuel rod with an elliptical cross-section / / Scientific Works of the Kaluga State University named after K. E. Tsiolkovsky, series: natural sciences. - 2016. - P. 230231.
© KaHapeHKHH A.H., 2021
УДК 536.212
Канарейкин А.И.
кан. техн. наук, доцент МГРИ, г. Москва, РФ
СТАЦИОНАРНОЕ РАСПРЕДЕЛЕНИЕ ТЕМПЕРАТУРНЫ В ПРЯМОУГОЛЬНОМ БРУСЕ
Аннотация
В статье производится нахождение распределения температуры в брусе. При этом процесс является стационарным. При этом на каждой грани бруса задана температура. Полученный результат был сопоставлен с решением похожей задачи.
Ключевые слова
Температурное поле, прямоугольный брус, стационарная теплопроводность, граничные условия, метод Фурье.
The problem under consideration is formulated as follows: on two of the side edges of a long rectangular bar (Fig. 1), it is maintained at a given temperature, on the other faces T = 0; find the steady-state temperature at an arbitrary point inside the bar.
z
y
Fig. 1. A bar with the set temperatures on the faces.
