Аналіз ядра атомного реактора Чорнобильської атомної станції за 5 секунд до вибуху у тривимірному сферичному просторі
This research analyzed the water flow and bubble (void) of nuclear reactor core of Chernobyl Power Plant, for 5 seconds before the explosion, using a mathematical model of two-dimensional spherical coordinates. To solve the problem, we considered this phenomenon as an analogy of Newtonian gravity th...
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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 World Data Center for Geoinformatics and Sustainable Development, NTUU \"KPI\", Kyiv, Ukraine"
},
{
"author": "Petro I. Bidyuk",
"institution": "The Institute for Applied System Analysis NTUU \"KPI\", Kyiv, Ukraine"
}
] |
| author_sort | Matsuki, Yoshio |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2018-03-30T15:25:41Z |
| description | This research analyzed the water flow and bubble (void) of nuclear reactor core of Chernobyl Power Plant, for 5 seconds before the explosion, using a mathematical model of two-dimensional spherical coordinates. To solve the problem, we considered this phenomenon as an analogy of Newtonian gravity theory, which had been solved in Schwarzchild Solution. As a result, the calculated radius of the spherical space of water and bubble indicated the maximum limit, at which the reactor core lost the control of reactor power. And, then, a regression analysis, with exponential model, confirmed the influence of bubble and water to the reactor’s power. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2016.4.09 |
| first_indexed | 2025-07-17T10:22:07Z |
| format | Article |
| fulltext |
Y. Matsuki, P.I. Bidyuk, 2016
88 ISSN 1681–6048 System Research & Information Technologies, 2016, № 4
88
УДК 519.004.942
DOI: 10.20535/SRIT.2308-8893.2016.4.09
ANALYSIS OF THE NUCLEAR REACTOR CORE
OF CHERNOBYL POWER PLANT FOR 5 SECONDS BEFORE
EXPLOSION WITH THE THREE-DIMENSIONAL
SPHERICAL SPACE
Y. MATSUKI, P.I. BIDYUK
Abstracts. This research analyzed the water flow and bubble (void) of nuclear reac-
tor core of Chernobyl Power Plant, for 5 seconds before the explosion, using a
mathematical model of two-dimensional spherical coordinates. To solve the prob-
lem, we considered this phenomenon as an analogy of Newtonian gravity theory,
which had been solved in Schwarzchild Solution. As a result, the calculated radius
of the spherical space of water and bubble indicated the maximum limit, at which
the reactor core lost the control of reactor power. And, then, a regression analysis,
with exponential model, confirmed the influence of bubble and water to the reactor’s
power.
Keywords: nuclear power plant, Chornobyl disaster, critical operation mode, re-
gression analysis, void and water environment.
INTRODUCTION
During a few seconds before the explosion of the reactor core of the Chernobyl
Nuclear Plant in April 1986, the flow rate of the Main Circulation Pump (MCP)
was reduced, and void (bubble) increased in the water flow. Our previous re-
search [1] suggested that the water flow and void explain the process of these five
seconds, while reactor’s power immensely increased toward the explosion. From
this previous result [1], we continued the investigation with the same data, in or-
der to find a rule, which may explain how the water and void were related to the
sudden increase of the reactor power, what resulted in an explosion.
For the further analysis, we assumed the followings: 1) the nuclear reactor
core has potential to control its power; and, such a potential has an analogy from
the gravity force in the space; 2) when the condition of water and void are out of
the reach of the hypothetical gravity, the nuclear reactor loses its control over its
power. Fig. 1 shows the scatter plots of water and void in time series. This figure
shows the decreasing MCP flow rate and increasing void, over time.
METHODOLOGY
Two-dimensional spherical model
Schwarzchild solution [2] of two-dimensional rectangular coordinates is:
)(sin)21()21( 22122 drdrVdrVds ,
Analysis of the nuclear reactor core of Chernobyl power plant, for 5 seconds before explosion
Системні дослідження та інформаційні технології, 2016, № 4 89
where 2ds of this equation (1) is a geodesic, which indicates the status of the reac-
tor core in hypothetical coordinates, V is potential of gravity, r is the radius, and
is the angle of the radius in the two-dimensional spherical coordinates.
If it is applied to the Newton’s gravity, V is inversely proportional to the ra-
dius, r . We take the same assumption in this analysis:
rWV / ,
where W is unknown scalar value, which is specific to the two dimensional space
of this analysis.
Then Wr 2 is a singular point, where 2ds .
In this analysis, the coordinates of Fig. 1 are converted to a two-dimensional
spherical coordinate as well as the time coordinate. In this analysis, the gravity is
static; therefore, time is independent from the special coordinates.
Main circulation pump flow rate (x-coordinate in Fig. 1) and Void (y-
coordinate in Fig.1) are converted to the two-dimensional spherical coordinates
with the following equations:
22 yxr , (1)
x
y1tan .
In this analysis, units of MCP flow rate and void are different; therefore, the
coordinates in the two-dimensional spherical space is purely hypothetical. No at-
tempt is made to find theoretical connection to the nuclear reactor theory; but,
Fig. 1. Geodesic trajectory of MCP flow rate and void
Po
w
er
, %
Void, %
MPC flo
w rate, m
3 /sec
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2016, № 4 90
rather, an attempt is made to find the singular point in the hypothetical space,
which could lead to the explosion. Therefore, only equation (1) is considered.
Exponential model for regression analysis
After the above analysis, if a singular point of the hypothetical spherical space is
identified, regression analysis of the data will be carried out, with the exponential
model:
n
i
jiij XcbaY
1
,exp ,
where a , b , ic ( ni ,....3,2,1 ) are positive constants, and n is number of inde-
pendent variables. mj ,....2,1 , the suffices, j , means j -th observation of the
variable, jiX , , and m is number of observations.
These coefficients, a , b , ic ( ni ,....,3,2,1 ) are calculated numerically,
which are to minimize
m
j
jU
m 1
21
, where
n
i
jiij XcbaYU
1
,exp .
In this analysis, one more model is also tested:
n
i
jij XbaY
1
,exp , (2)
where a andb are positive constants, ni ,...,3,2,1 , and mj ,...,3,2,1 .
Method to test the fitting (predictability) of the exponential model with sam-
pled data
a) calculate the predicted value of Y (i.e., Ŷ ) with the following equation:
n
i
jiij XcbaY
1
,expˆ .
b) calculate the value of R2 by the following equation:
m
j
j
j
m
j
YY
YY
R
1
2
2
12
)(
)ˆ(
,
where
m
j
jYmY
1
/1 .
For the model (2), the same rule applies.
The value of 2R represents the fitting and predictability of the given expo-
nential model upon the given data, and when 0,12 R , it is the perfect match,
Analysis of the nuclear reactor core of Chernobyl power plant, for 5 seconds before explosion
Системні дослідження та інформаційні технології, 2016, № 4 91
while the level of the matching is lower when the value of 2R is lower. In prac-
tice, if 6,0~8,02 R , the fitting of the model in the data is significant. However,
the threshold value depends on the topic and the data of the concerned research
question, therefore the values of 2R need to be considered on the comparative
manner.
Data
The data was taken from Martines, et.al 1989 [3], which we used in [1]. The de-
scriptive statistics are shown in Table 1. Reactor power, MCP flow rate and void
during the time period of 5 seconds before the explosion (between 01h 23 min
38 sec and 01 h 23 min 42, 71 sec on 25 April 1986) are analyzed in this research.
T a b l e 1 . Descriptive Statistics of Parameters (taken from Fig. 3 of [3])
Parameter
Power
(% nominal power)
MCP flow rate
m3/sec
Void, %
Mean 67442,4 9,653 31,819
Median 13220 9,575 34,500
Maximum 227186,7 10,200 40,050
Minimum 0 9,3 12,000
Std, Dev. 90122,460 0,269 9,202
Skewness 0,872 0,713 –0,947
Kurtosis 2,039 2,396 2,837
Observations 16 16 16
Note: Max.: maximum value. Min.: minimum value. Std. Dev.: standard deviation.
Skewness: the measure of the probability distribution leaning to one side of the mean.
Kurtosis: “peakedness” of probability distributions. Observation: number of observations.
RESULT
Geodesic trajectory of MCP flow rate and void in two-dimensional spherical
coordinates
Fig. 2 shows the calculated radius of the hypothetical two-dimensional spherical
coordinates of MCP flow rate and void, in comparison with the reactor power.
When the hypothetical radius reached the length of 40, the reactor power
increased immensely1.. We assume that this is the singular point, where Wr 2 ,
and 2ds .
This observation suggests that there is a potential, which is an analogy from
the earth’s gravity, but it is only in a hypothetical two-dimensional spherical co-
ordinates of water and void in the reactor core. That is, when the geodesic is far
from the center of the coordinates, the gravity force becomes weaker, and then,
the reactor core cannot control its power.
1 Here, the radius doesn’t have unit; because, this coordinate is purely hypothetical.
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2016, № 4 92
Exponential models to explain the explosion
The next step is to describe the sudden increase of reactor power by MCP flow
rate and void.
In our previous study [1], we calculated the first-order estimate with a linear
model. However, upon the observation above in this research, we assumed that
the reactor suddenly increased its power immensely, when the size of radius
reached the singular point in the hypothetical two dimensional space of water and
void of the reactor core, as if the reactor core lost its control over its power. And,
then, we thought that we cannot describe the process of increasing reactor power
with the standard model for the first order estimation, but with exponential
models.
Table 2 shows the results of the regression analysis with two exponential
models and a linear model. The fittings of the exponential models are much better
than the linear model’s, as indicated by the values of 2R . Fig. 3 shows the
comparison between the calculated reactor power and the observed reactor power.
The figure also shows that the exponential models fit well with the observed
power, than the linear model.
This result shows that MCP flow rate and void can explain the increase of
reactor power.
Time, sec
P
ow
er
, %
Power, % (L)
Radius in shencal coordinates (R)
R
ad
iu
s
in
s
he
nc
al
c
oo
rd
in
at
es
Fig. 2. Power and the radius in hypothetical spherical coordinates of water and bubble
Analysis of the nuclear reactor core of Chernobyl power plant, for 5 seconds before explosion
Системні дослідження та інформаційні технології, 2016, № 4 93
T a b l e 2 . Calculated reactor power by two exponential models and a linear model
Model Linear models and calculated coefficients 2R
Exponential
model 1
)void1997,1eMCPflowrat2701,2561,14(exp2,6353Power 0,9587
Exponential
model 2 )voideMCPflowrat(exp1024,67,4451Power 17 0,8819
Linear [1] void6836,794eMCPflowrat475560,64875626Power 0,5712
CONCLUSION AND RECOMMENDATION
In case of the explosion of Chernobyl nuclear reactor, both MCP flow rate and
void showed unusual behaviors. And, then the reactor power increased suddenly.
The exponential curve showed the immense increase of the reactor power, what
resulted in an explosion.
If the reactor core had a capability to control its power, the occurrence of the
explosion suggests that there is a point that led to the explosion. The result of this
research also suggests that there was a limit for controlling the reactor power, and
1
2
3
4
Time, sec
R
ea
ct
or
, P
ow
er
, %
Fig. 3. Comparison of observed reactor power with calculated reactor power: 1 — calcu-
lated power by model-1, %; 2 — power, %; 3 — calculated power by model-2; 4 — cal-
culated power by model
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2016, № 4 94
the limit existed at the singular point of the hypothetical three-dimensional spher-
ical space of water and void in the reactor core.
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. — 2016. — P. 33–42.
2. Dirac P.A.M. General Theory of Relativity / P.A.M. Dirac // A Wiley-Interscience
Publication. — New York: John Wiley & Sons,1975. — Available at:
http://amarketplaceofideas.com/wp-
content/uploads/2014/08/P%2520A%2520M%2520Dirac%2520-
%2520General%2520Theory%2520Of%2520Relativity1.pdf
3. Martinez J.M. An Analysis of the Physical Causes of the Chernobyl Accident /Jose
M. Aragonez, Emilio Mingues, Jose M. Peri, Guillermo Velarde // Nuclear Tech-
nology. — 1990. — Vol. 90. — P. 371–399.
Received 12.07.2016
From the Editorial Board: the article corresponds completely to submitted manuscript.
|
| id | journaliasakpiua-article-88013 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:22:07Z |
| publishDate | 2016 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/f5/2a0a7512d182fb7b8a52468ad674dcf5.pdf |
| spelling | journaliasakpiua-article-880132018-03-30T15:25:41Z Analysis of the nuclear reactor core of Chernobyl Power Plant for 5 seconds before explosion with three-dimensional spherical space Анализ ядра атомного реактора Чернобыльськой атомной станции за 5 секунд до взрыва в трехмерном сферическом пространстве Аналіз ядра атомного реактора Чорнобильської атомної станції за 5 секунд до вибуху у тривимірному сферичному просторі 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 функционирования регрессионный анализ водяная и пустотная среда атомна електростанція Чорнобильська катастрофа критичний режим функціонування регресійний аналіз водяне і пустотне середовище This research analyzed the water flow and bubble (void) of nuclear reactor core of Chernobyl Power Plant, for 5 seconds before the explosion, using a mathematical model of two-dimensional spherical coordinates. To solve the problem, we considered this phenomenon as an analogy of Newtonian gravity theory, which had been solved in Schwarzchild Solution. As a result, the calculated radius of the spherical space of water and bubble indicated the maximum limit, at which the reactor core lost the control of reactor power. And, then, a regression analysis, with exponential model, confirmed the influence of bubble and water to the reactor’s power. Проанализированы поток воды и пустоты (пузырьковые включения) ядра атомного реактора Чернобыльской атомной станции за 5 секунд до взрыва с использованием математической модели в трехмерной сферической системе координат. Для решения проблемы этот феномен рассмотрен по аналогии с законом всемирного тяготения и проанализирован с использованием метода Шварцшильда. Рассчитан радиус сферического пространства воды и пустот, указывающий на максимальный предел, при котором ядро реактора утратило способность контролировать мощность. Выполнен анализ регрессионным методом с помощью экспоненциальной модели и подтверждено влияние пустот и воды на мощность реактора. Проаналізовано потік води та пустоти (бульбашкові включення) ядра атомного реактора Чорнобильської атомної станції за 5 секунд до вибуху з використанням математичної моделі у тривимірній сферичній системі координат. Для вирішення задачі цей феномен розглянуто за аналогією із законом всесвітнього тяжіння та проаналізовано з використанням методу Шварцшільда. Розраховано радіус сферичного простору води та пустот, що вказує на граничне значення, за якого ядро реактора втратило здатність контролювати потужність. Виконано аналіз за регресійним методом за допомогою експоненціальної моделі і підтверджено вплив пустот та води на потужність реактора. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2016-12-15 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/88013 10.20535/SRIT.2308-8893.2016.4.09 System research and information technologies; No. 4 (2016); 88-94 Системные исследования и информационные технологии; № 4 (2016); 88-94 Системні дослідження та інформаційні технології; № 4 (2016); 88-94 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/88013/83765 Copyright (c) 2021 System research and information technologies |
| spellingShingle | атомна електростанція Чорнобильська катастрофа критичний режим функціонування регресійний аналіз водяне і пустотне середовище Matsuki, Yoshio Bidyuk, Petro I. Аналіз ядра атомного реактора Чорнобильської атомної станції за 5 секунд до вибуху у тривимірному сферичному просторі |
| title | Аналіз ядра атомного реактора Чорнобильської атомної станції за 5 секунд до вибуху у тривимірному сферичному просторі |
| title_alt | Analysis of the nuclear reactor core of Chernobyl Power Plant for 5 seconds before explosion with three-dimensional spherical space Анализ ядра атомного реактора Чернобыльськой атомной станции за 5 секунд до взрыва в трехмерном сферическом пространстве |
| title_full | Аналіз ядра атомного реактора Чорнобильської атомної станції за 5 секунд до вибуху у тривимірному сферичному просторі |
| title_fullStr | Аналіз ядра атомного реактора Чорнобильської атомної станції за 5 секунд до вибуху у тривимірному сферичному просторі |
| title_full_unstemmed | Аналіз ядра атомного реактора Чорнобильської атомної станції за 5 секунд до вибуху у тривимірному сферичному просторі |
| title_short | Аналіз ядра атомного реактора Чорнобильської атомної станції за 5 секунд до вибуху у тривимірному сферичному просторі |
| title_sort | аналіз ядра атомного реактора чорнобильської атомної станції за 5 секунд до вибуху у тривимірному сферичному просторі |
| topic | атомна електростанція Чорнобильська катастрофа критичний режим функціонування регресійний аналіз водяне і пустотне середовище |
| topic_facet | nuclear power plant Chornobyl disaster critical operation mode regression analysis void and water environment атомная электростанция Чернобыльская катастрофа критический режим функционирования регрессионный анализ водяная и пустотная среда атомна електростанція Чорнобильська катастрофа критичний режим функціонування регресійний аналіз водяне і пустотне середовище |
| url | https://journal.iasa.kpi.ua/article/view/88013 |
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