Аналіз вибуху ядерного реактора Чорнобильської АЕС у квітні 1986 р. за допомогою тензорних рівнянь
This research analyzes the process of the explosion of the reactor core of Chernobyl nuclear plant in April 1986, using the tensor equations. These tensor equations show a movement of a vector in the three dimensional curvature coordinates of time, water flow, and void. The equations shows that this...
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The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
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System research and information technologies| _version_ | 1867334316646727680 |
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| author | Matsuki, Yoshio Bidyuk, Petro I. |
| author_facet | Matsuki, Yoshio Bidyuk, Petro I. |
| author_institution_txt_mv | [
{
"author": "Yoshio Matsuki",
"institution": "The Laboratory for Econometrics and Forecasting at the World Data Center for Geoinformatics and Sustainable Development, National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
},
{
"author": "Petro I. Bidyuk",
"institution": "The Department of the Mathematical Methods of System Analysis of the Educational and Scientific Complex \"Institute for Applied System Analysis\" of National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
}
] |
| author_sort | Matsuki, Yoshio |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2018-03-30T15:35:30Z |
| description | This research analyzes the process of the explosion of the reactor core of Chernobyl nuclear plant in April 1986, using the tensor equations. These tensor equations show a movement of a vector in the three dimensional curvature coordinates of time, water flow, and void. The equations shows that this vector moves along the geodesic in the curvature coordinates, which is described by fundamental tensor (gµν), Christoffel symbol (Γαµνσ) and Ricci tensor (Rµν), where µ, ν, σ, α are suffixes that indicate the coordinates. The solution of the tensor equations shows that the geodesic of the vector has a singular point, which describes a turning point of the reactor core from the normal operation to the explosion, which we reported in our previous articles [1, 2]. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2017.2.06 |
| first_indexed | 2025-07-17T10:22:54Z |
| format | Article |
| fulltext |
Y. Matsuki, P.I. Bidyuk, 2017
62 ISSN 1681–6048 System Research & Information Technologies, 2017, № 2
TIДC
ПРОБЛЕМИ ПРИЙНЯТТЯ РІШЕНЬ
І УПРАВЛІННЯ В ЕКОНОМІЧНИХ, ТЕХНІЧНИХ,
ЕКОЛОГІЧНИХ ТА СОЦІАЛЬНИХ СИСТЕМАХ
UDC 519.004.942
DOI: 10.20535/SRIT.2308-8893.2017.2.06
CURVATURE COORDINATES TO DESCRIBE THE EXPLOSION
OF CHERNOBYL’S REACTOR CORE IN APRIL 1986,
USING THE TENSOR EQUATIONS
Y. MATSUKI, P.I. BIDYUK
Abstracts. This research analyzes the process of the explosion of the reactor core of
Chernobyl nuclear plant in April 1986, using the tensor equations. These tensor
equations show a movement of a vector in the three dimensional curvature coordi-
nates of time, water flow, and void. The equations shows that this vector moves
along the geodesic in the curvature coordinates, which is described by fundamental
tensor ( g ), Christoffel symbol (
) and Ricci tensor ( R ), where , , ,
are suffixes that indicate the coordinates. The solution of the tensor equations
shows that the geodesic of the vector has a singular point, which describes a turning
point of the reactor core from the normal operation to the explosion, which we re-
ported in our previous articles [1, 2].
Keywords: nuclear power plant, Chornobyl disaster, critical operation mode, re-
gression analysis, void and water environment.
INTRODUCTION
In the previous research [1, 2], we analyzed the physical parameters [3] of nuclear
reactor core, which were observed during the time period of five seconds before
the explosion of Chernobyl nuclear plant, which happened in April 1986. With
the first-order empirical analysis [1], we found that two parameters, water flow
and void of the nuclear core, were strongly related to the sudden increase of the
reactor power that led to the explosion. And, then, we found that the exponential
model with these two parameters can explain the process of the immense power
increase [2]. We also found that there was a turning point in the process between
the normal operation mode and the explosion.
In this research we analyzed the explosion conditions, using the tensor equa-
tions, which show evolution of the three dimensional curvature coordinates of
time, water flow, and void.
METHODOLOGY
At first, we assume that there is a scalar field that manages the water flow and
void in the reactor core; and then, the derivatives of the scalar field produce a ten-
Curvature coordinates to describe the explosion of Chernobyl’s reactor core in april 1986, …
Системні дослідження та інформаційні технології, 2017, № 2 63
sor field that is to describe the coordinates evolution in the three dimensional
space.
Let S be a scalar field. It can be considered as a function of the three coor-
dinates x ( 2,1,0 ). Here 0 corresponds to time (t), and 2,1 for coor-
dinates of two parameters (water flow and void). Then
xSS .
Here S
dx
d
S , S
dx
d
S , and
x
dx
d
x ; and, the derivative of a
scalar field ( S ) is a covariant vector field by its definition [4].
And then, the movement of the vector in the curvature coordinates of time,
water flow and void is described by tensor equations:
AAA ,: . (1)
Here A is a vector field. And
is the Christoffel Symbol of the second
kind, while:
g ,
A
dx
d
A , ,
,g ,
g
dx
d
g , : and, g are1 fundamental tensors, which determine coordinate
system and curvature of the space. The g also lower and raise suffixes of vec-
tors (contra-variant vector, A , and covariant vector, A ), and which vary from a
point to a point in a curvature space [4]. Also, it is noted that “:” denotes the co-
variant differentiation. The covariant derivative of A is :A , and its relation to
,A is shown in the equation (1).
If we assume that there is no scalar field in the three dimensional space of
time, water flow and void, the variables evolve straight in the space. In such a
case, the geodesic of the variables is a straight line. On the other hand, if there is a
scalar field, the variables move along curvature coordinates. So, in this case, the
«curvature» gives the information about the evolvement of the variables.
For a scalar field, S , its derivative is described with the covariant differen-
tiation. And now we consider two differentiations in succession:
`
ASASS ,:,: .
1 Here “are” is appropriate to use, because g are plural because and vary for time
and spatial coordinates.
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2017, № 2 64
To deal with the second covariant derivative of a tensor, we use the Rie-
mann-Christoffel tensor (or, covariant tensor):
,,R
. With the symmetries of the first and the last suffixes, we get
RR , which is called as Ricci tensor:
,,R . (2)
These
and R are indicators of the curvature of the physical coordi-
nates.
In the curved space (Riemann space), we can also calculate the distance ds
between a point x and a neighboring point dxx , by:
dxdxgds2 .
This gives the geodesic of the movement of a vector along the curvature co-
ordinates.
RESULTS
Tensor equations of three dimensional curvature coordinates
According to the theory of the curvature tensors [4], if there is only one scalar
field with no other physical field, the curvature space is empty; and, it is de-
scribed by:
0R . (3)
Then we will calculate the geodesic of the movement of the vector by:
22
22
21
11
20
00
2 )()()( dxgdxgdxgds . (4)
Here time ( t ) and two coordinates ( x and y ) form a space, i.e., 0xt ,
1xx , 2xy , where 0x , 1x and 2x are contra-variant vectors, and 00g , 11g ,
22g are fundamental tensors that lower the suffixes of the contra-variant vectors,
to convert them to covariant vectors2.
And then, we assume a static coordinate system, where g are constant in
time. Therefore, 00, g , where and are 0, 1, 2 (while, 0 is for time, and 1
and 2 are coordinates for water flow and void).
Where space is not flat, a vector of the variables moves along the curvature
coordinates. This curvature coordinate system is nonlinear; so it is difficult to find
solutions. In order to overcome this difficulty, we use spherically symmetric field
to get solutions fairly easily. It is Schwarzchild’ special solution [4]:
22222 dWrVdrUdtds . (5)
2 In order to describe a physical space, we need to use covariant vectors.
Curvature coordinates to describe the explosion of Chernobyl’s reactor core in april 1986, …
Системні дослідження та інформаційні технології, 2017, № 2 65
Here U , V , W are functions of r only. For replacing the equation (5), we
use the following equation:
2222222 drdredteds . (6)
From the equation (5), we determine the values of g as follows:
2
00 eg , 2
11 eg , 2
22 rg , and 0g for .
And, then, 200 eg , 211 eg , 222 rg , and 0g for .
Then, we calculate the Christoffel symbols as follows:
22
,
22
1,00
11
001001
111
00 ')(22)( eveeggg r ;
22
,
22
1,00
00
001001
000
10 )(22)( eeeggg r ;
22
,
22
1,11
11
111111
110
11 )(22)( eeeggg r ;
1
,
22
1,22
22
221221
222
12 ))((22)( rrrggg r ;
2
,
22
1,22
11
221221
111
22 )(22)( rereggg r .
With these Christoffel symbols rewrite the equation (2) as follows:
222
00
2
e
r
R ; (7)
r
R
'2
''''' 2
11
; (8)
1)''1( 2
22 errR . (9)
Here drd , 22 drd , and drd' .
From the equation (3),
0221100 RRR . (10)
From the equations (7), (8) and (10):
0 .
For large values of r , the space must become approximately flat; so, when
r , 0 and 0 .
From the equations (9) and (10): 1)21( 2 er , and then:
1)( 2 re . (11)
By integrating the equation (11), we get:
mrre 22 .
Here m is a constant of integration.
Because 2
00 eg , we get:
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2017, № 2 66
r
m
g
2
100 . (12)
From the equations (4), (6), and (12), we get the equation to calculate the geo-
desic of the movement of a vector along the curvature coordinates:
222
1
22 2
1
2
1
drdr
r
m
dt
r
m
ds . (13)
The equation (13) becomes singular at mr 2 , because then 000 g and
11g .
Estimation of the magnitude of the scalar field
Thus, mr 2 is the critical radius of the geodesic equation (13). From our previ-
ous research [2], we assume that the value of 40r . Now, we neglect the angu-
lar component, 2x , therefore the critical length of r occurs when the void
(%) becomes the maximum, while the water flow (m3/sec) becomes minimum.
The critical value of the void is 40 (%).3
From our previous research [2], the water flow and void influenced the reac-
tor power. With the total output energy (Mega Joules) as the dependent variable,
the relations are shown in the following equation:
)7552.02934.3809.11(exp8.1903 voideMCPflowratyTotalEnerg . (14)
As far as we neglected the angular component, the critical length of r takes
place when the void acquires its critical (maximum) value, and the water flow
approaches its minimum level. With keeping this in mind, equation (14) allows to
compute the total energy that is to be released in critical cases using the values of
MCP flow rate and of the void. From the equation developed in [2], it was shown
that the 40 percent of void is leading to approximately 4100.5 Mega Joules of
the total output kinetic energy of the reactor.
CONCLUSIONS AND RECOMMENDATIONS
Thus, it was shown that the curvature coordinates, fundamental tensor ( g ),
Christoffel symbol (
) and Ricci tensor ( R ) appear as indicators of devel-
opment the processes in time, which describe the curvature of the coordinates of
time, water flow and void, along which a vector of the variables considered
evolves. The track of the movement of a hypothetical particle was described by
the tensor equation (4), 22
22
21
11
20
00
2 )()()( dxgdxgdxgds , given ds is a
distance of the vector’s movement, 0x is the coordinate of time, 1x and 2x are
coordinates for water flow and void. And then, the coordinates are converted into
the symmetrical two dimensional curvature coordinates; after that the fundamen-
tal tensors were calculated.
3 Here, 2x does not mean “squared x ”. It is the suffix for the third coordinate, .
Curvature coordinates to describe the explosion of Chernobyl’s reactor core in april 1986, …
Системні дослідження та інформаційні технології, 2017, № 2 67
The tensor equations considered showed qualitatively and quantitatively that
there is a minimum distance to describe evolution of the process of possible ex-
plosion, as in the minimum radius of three-dimensional curvature coordinates.
The order of magnitude of the reactor power is approximately in the order of
810 (Mega Joules-1), at the singular point in the curvature space.
REFERENCE
1. Matsuki Y. Empirical Analysis of Chernobyl Nuclear Reactor Core for 5 seconds be-
fore the Explosion / Y. Matsuki, P.I. Bidyuk // System Research & Information
Technology, 3/2016, October 2016. — P. 33–41.
2. Matsuki Y. Analysis of the nuclear reactor core of Chernobyl Power Plant, for 5 sec-
onds before explosion, with the three-dimensional spherical space / Y. Matsuki,
P.I. Bidyuk // System Research & Information Technology, 4/2016, December
2016. — P. 88–94.
3. Jose M. Martinez. An Analysis of the Physical Causes of the Chernobyl Accident /
Jose M. Martinez, Jose M. Aragonez, Emilio Mingues, Jose M. Peri and Gui-
llermo Velarde // Nuclear Technology. —Vol. 90. — June, 1990. — P. 371–399.
Madrid Polytechnic University, Institute of Nuclear Fusion, J. Gutlierez Abascol,
2, 28006 Madrid, Spain http://life-upgrade.com/DATA/Artikel%20zu%
20Tschernobyl%20in%20Nuclear%20Technology%20Vol%2090.pdf (last ac-
cess, 22 November, 2015).
4. Dirac P.A.M. General Theory of Relativity / P.A.M. Dirac // Florida University,
A Wiley-Interscience Publication. — John Wiley & Sons, New York, 1975. — P. 69.
— Available at: http://amarketplaceofideas.com/wp-content/uploads/2014/08/
P%2520A%2520M%2520Dirac%2520-%2520General%2520Theory%2520Of%
2520Relativity1.pdf (last access, 5 July 2016).
Received 30.09.2016
From the Editorial Board: the article corresponds completely to submitted manuscript.
|
| id | journaliasakpiua-article-108779 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:22:54Z |
| publishDate | 2017 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/5a/4594c272336cb5a288a66fcb3f47265a.pdf |
| spelling | journaliasakpiua-article-1087792018-03-30T15:35:30Z Curvature coordinates to describe the explosion of Chernobyl’s reactor core in April 1986, using the tensor equations Анализ взрыва ядерного реактора Чернобыльской АЭС в апреле 1986 г. при помощи тензорных уравнений Аналіз вибуху ядерного реактора Чорнобильської АЕС у квітні 1986 р. за допомогою тензорних рівнянь Matsuki, Yoshio Bidyuk, Petro I. nuclear power plant Chornobyl disaster critical operation mode regression analysis void and water environment атомная электростанция Чернобыльская катастрофа критический режим работы регрессивный анализ пустота и водная среда атомна електростанція Чорнобильська катастрофа критичний режим роботи регресійний аналіз порожнеча та вода This research analyzes the process of the explosion of the reactor core of Chernobyl nuclear plant in April 1986, using the tensor equations. These tensor equations show a movement of a vector in the three dimensional curvature coordinates of time, water flow, and void. The equations shows that this vector moves along the geodesic in the curvature coordinates, which is described by fundamental tensor (gµν), Christoffel symbol (Γαµνσ) and Ricci tensor (Rµν), where µ, ν, σ, α are suffixes that indicate the coordinates. The solution of the tensor equations shows that the geodesic of the vector has a singular point, which describes a turning point of the reactor core from the normal operation to the explosion, which we reported in our previous articles [1, 2]. Проанализирован процесс взрыва активной зоны ядерного реактора на Чернобыльской атомной электростанции в апреле 1986 г. при помощи тензорных уравнений, которые демонстрируют движение вектора в трехмерных координатах кривой времени, потока воды и пустоты. Уравнения показывают, что этот вектор движется вдоль геодезической прямой в координатах кривой, которая описывается фундаментальным тензором (gµν), символом Кристоффеля (Γαµνσ) и тензором Риччи (Rµν), где µ, ν, σ, α — индексы, обозначающие координаты. Решение тензорных уравнений показывает, что геодезическая прямая вектора имеет сингулярную точку, которая описывает точку вращения активной зоны ядерного реактора от нормального функционирования до взрыва. Проаналізовано процес вибуху активної зони ядерного реактора на Чорнобильській атомній електростанції у квітні 1986 р. за допомогою тензорних рівнянь, які демонструють рух вектора в тривимірних координатах кривої часу, потоку води та пустоти. Рівняння показують, що цей вектор рухається вздовж геодезичної прямої у координатах кривої, яка описується фундаментальним тензором (gµν), символом Крістофеля (Γαµνσ) і тензором Річчі (Rµν), де µ, ν, σ, α — індекси, які позначають координати. Розв’язання тензорних рівнянь показує, що геодезична пряма вектора має сингулярну точку, яка описує точку обертання активної зони ядерного реактора від нормального функціонування до вибуху. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2017-06-26 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/108779 10.20535/SRIT.2308-8893.2017.2.06 System research and information technologies; No. 2 (2017); 62-67 Системные исследования и информационные технологии; № 2 (2017); 62-67 Системні дослідження та інформаційні технології; № 2 (2017); 62-67 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/108779/103722 Copyright (c) 2021 System research and information technologies |
| spellingShingle | атомна електростанція Чорнобильська катастрофа критичний режим роботи регресійний аналіз порожнеча та вода Matsuki, Yoshio Bidyuk, Petro I. Аналіз вибуху ядерного реактора Чорнобильської АЕС у квітні 1986 р. за допомогою тензорних рівнянь |
| title | Аналіз вибуху ядерного реактора Чорнобильської АЕС у квітні 1986 р. за допомогою тензорних рівнянь |
| title_alt | Curvature coordinates to describe the explosion of Chernobyl’s reactor core in April 1986, using the tensor equations Анализ взрыва ядерного реактора Чернобыльской АЭС в апреле 1986 г. при помощи тензорных уравнений |
| title_full | Аналіз вибуху ядерного реактора Чорнобильської АЕС у квітні 1986 р. за допомогою тензорних рівнянь |
| title_fullStr | Аналіз вибуху ядерного реактора Чорнобильської АЕС у квітні 1986 р. за допомогою тензорних рівнянь |
| title_full_unstemmed | Аналіз вибуху ядерного реактора Чорнобильської АЕС у квітні 1986 р. за допомогою тензорних рівнянь |
| title_short | Аналіз вибуху ядерного реактора Чорнобильської АЕС у квітні 1986 р. за допомогою тензорних рівнянь |
| title_sort | аналіз вибуху ядерного реактора чорнобильської аес у квітні 1986 р. за допомогою тензорних рівнянь |
| topic | атомна електростанція Чорнобильська катастрофа критичний режим роботи регресійний аналіз порожнеча та вода |
| topic_facet | nuclear power plant Chornobyl disaster critical operation mode regression analysis void and water environment атомная электростанция Чернобыльская катастрофа критический режим работы регрессивный анализ пустота и водная среда атомна електростанція Чорнобильська катастрофа критичний режим роботи регресійний аналіз порожнеча та вода |
| url | https://journal.iasa.kpi.ua/article/view/108779 |
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