About possibility to calculate radioactive pollutants taking into account
climatic and meteorological peculiarities.
HIKMAT HASANOV1, ISMAYIL ZEYNALOV2
Caspian Sea Department
Ministry of Science and Education Republic of Azerbaijan
Institute of Geography named after academician H. Aliyev
AZ1143, Baku, H. Javid Ave.115
AZERBAIJAN
Abstract: - It is developed the model of three-zones radioactive pollutions, which are exhausted while an
explosion at Atomic Power Plant. It is formulated how radioactive pollutions behave in each zone. The problem
of the radioactive pollutants distribution from the explosion epicenter for the far zone is solved. The algorithm
for analytical solution is given. It is stated that there is the space limit for pollutions distribution from the
epicenter. Quantitative discussions are provided. Effect of the climatic and meteo peculiarities are considered.
Key-Words: - radioactive pollutions, far limit, radial symmetry, the coefficient of diffusion, climatic
peculiarities
Received: April 25, 2024. Revised: September 17, 2024. Accepted: October 20, 2024. Published: November 7, 2024.
1 Introduction
In case of force-majeure at Atomic Power Plant
(APP), there are essential exhaust of radioactive
pollutions in environment. These pollutions can be
represented as particles, which later settle onto the
Earth’s surface. The area of radioactive precipitation
might be very wide and expand on large distance
from APP through international borders. In forming
radioactive pollutions, it is necessary to take into
account the climatic and geographic factors, which
make a contribution into this propagation. The
matter of this paper is to clear how a) radioactive
pollutions from APP propagate after explosion at
APP, and b) different atmospheric peculiarities
affect the distribution of radioactive pollutions in
space.
Especially for this type of problem, we developed a
model for radioactive pollutants propagation. In this
model it is suggested that an explosion has occurred
at APP, and radioactive pollutants were thrown out
up to certain height and then move radially from
epicenter which is located at the explosion point
(maybe the APP tube). Then we take some
simplifications like:
• The pollutants move from the epicenter
strictly radially without any turbulence and rotations
in propagation plane;
• All the pollutants move radially within tine
plane so that no mixing with pollutants moving at
parallel plane;
• The pollutants are thrown homogeneously
to all the directions from the explosion epicenter
independently on the azimuthal angle, so we can
propose that there is the radial symmetry of the
problem.
In our model, it is proposed that after explosion with
radioactive pollutions at APP three zones of fallout
are valid:
1) Near zone, where intensive mixing
substances take place together with the great speed
of substances motion from the APP center. This
zone is located near the epicenter of explosion;
2) Medium zone, where propagation wave has
a great speed, which is much more in comparison
with the pollutants mixing;
3) Far zone, where radial propagation of
pollutants sharply reduces and can be neglected with
the pollutants diffusion, so in solving the problem
we should take into account just mixing factor.
These three zones gently cross over to each other
but are managed by different models and laws. In
this paper the authors have developed model for
describing, how radioactive pollutions are
distributed from the epicenter in the far zone.
EARTH SCIENCES AND HUMAN CONSTRUCTIONS
DOI: 10.37394/232024.2024.4.20
Hikmat Hasanov, Ismayil Zeynalov
E-ISSN: 2944-9006
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Volume 4, 2024
2 Problem Formulation
To the authors’ stand point, maximal harmful effects
should be expected in far zone since the effect of
radioactivity will be long-term and could be
minimize anyhow. So, it is preferable to modelling
how radioactive pollutions are distributed in space
after passing large distance from the explosion
epicenter.
For the third case in above classification, when we
try to describe how pollutants can affect the
periphery, for solving the problem of pollutions
distributed radially from the APP center we use the
radial equation of diffusion
(1)
In the representation (1), it is proposed that the
coefficient of diffusion is non-linear and depends
upon the concentration of pollutants. The function n
= n (r, t) to be found is the pollutants radial
distribution from the explosion center. The
dependence
is empirical and should be
determined from the field measurements but in this
paper, we accept some law more probably valid
from practice, namely
(2)
where and are constant values and should be
found from experiments. Additionally, the
coefficient should take into account the climatic
and geographic peculiarities of the territory under
consideration to ensure relevant description of
radioactive pollutants.
It is well known, that account of climatic, meteo-
and geographic peculiarities are highly important for
forecasting, how radioactive pollutants are formed
and propagate in atmosphere; as it was recently
described by one of authors in [1]. The coefficient
of diffusion introduced accordingly to the formula
(2) gives an opportunity to correctly evaluate the
effect of these peculiarities on the final spatial
radioactive pollutants distribution.
For modelling the distribution of pollutants, we have
to take the next conditions: one condition by time
n (t=0) = n0 (3)
and two conditions by coordinate
and
(4)
Physically, the conditions (3) and (4) have the next
matter. Solution of the problem is made only after
pollutants reach the point rf from the explosion
center, moreover we can not minimize rf to zero
because in the given statement the problem is
correct just for the far zone described in above
classification. So, we accept that we are looking for
the distribution of pollutants after they come to
point rf. The concentration of pollutions and their
changes by time at the radius rf from the epicenter
are the time-dependent as it follows from the
conditions (4).
The equation (1) can be changed to
(5)
If make formal substitution of required function as
, (6)
then we obtain the next equation
(7)
In the substitution (6), the constants and are
the same as in (2). The equation (7) in non-linear
one and the author has already developed the
algorithm for solving such a type of equations in his
paper [2].
3 Stationary case for pollutions
distribution after explosion
This case describes the distribution of pollutants
after long time passing from the explosion when
process became balanced and time-independent.
Mathematically, this case is obtained if in the
equation (7) one takes
=0. Under such a
condition, we have
, (8)
and after formal transformations finally one gets
(9)
with obvious
= C1, after that we can find the
solution of equation (9) in a view
(10)
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In the equation (10), the values C1 and C2 are
constant which to be found from the conditions (4)
but for asymptotic case t, which physically
corresponds to the stationary case. For this, we have
to come back to the required function n (r).
Using the conditions (4) provides two algebraic
equations for finding C1 and C2, namely
(11)
and
(12)
The values
and
are defined as the
asymptotic case for the functions and
at t, respectively.
By solving the last two equations we can finally
write
(13)
and respectively,
(14)
If put values C1 and C2 in the solution (10) and keep
in mind the substitution (6), one finally gets for the
distribution of radioactive pollutants after explosion
at periphery the next formal law
(15)
Herein, a and b are some constants which are
expressed through C1 and C2. The formula (15) is in
accordance with the result obtained by authors in
their early publication [3].
The key points for understanding the far zone model
are the next:
1) The equations (8) – (15) are valid for balanced
process of radioactive pollutants distribution in
far zone after explosion at APP. As it was
proposed by our classification in the beginning
of this paper, this zone is characterized by poor
convective radial speed of pollutants and
exceeding mixing pollutants in propagation plane
but not between the planes.
2) The law (15) gives an opportunity to formally
calculate the far limit of radioactive pollutants
expansion
. As it is seen from the dependence
(15), this limit can be calculated as
.
As it is can be logically concluded, the ratio
is critical for describing the radioactive
pollutants distribution in the far zone. The
analysis shows that the formula (15) as well as
the ratio
is very sensitive to the climatic and
meteo peculiarities of the territory where the
APP is located and radioactive pollutants are
distributed.
3) If after explosion a few radioactive
components/substances are polluted, then one
needs to solve the same number of the equations
as it has been made in this paper. The algorithm
(2) – (15) should be realized for each component
after that we can conclude that each radioactive
component has own far limit
, (i – the
number of components threw up to atmosphere
after explosion at APP) which is defined by the
physical properties of the component. As the
analyses shows, component with more mass (in
atomic units) will expand less in comparison
with lighter components. So, we finally have
different
, thereto the more is mass of
substance, the less will be corresponding
.
The approach given in this paper allows calculating
the total level of radioactive pollutants moved from
the explosion epicenter to periphery as well.
Accordingly to the diffusion theory, the equation
(16)
describes the intensity of the radioactive pollutants
flow through unit length of covering line from the
explosion epicenter. So, for the closed line
radioactive (equi-polluted line) pollutions one
has
In case of pollutants expansion in form of concentric
circumferences which is corresponded to the radial
symmetry of the problem solved, one has
If the pollutants are distributed homogeneously by
circumference and the physical properties of the
process do not change at the circumference, then the
last integral will be reduced to the final view
(17)
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DOI: 10.37394/232024.2024.4.20
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The last formula describes the total circular level of
radioactive pollutants at the distance R from the
explosion epicenter. Moreover, if one knows how
the coefficient depends upon the climatic,
meteorological and geographic peculiarities of given
territory, it will be possible to find the dependence
of the radioactivity level on indicated physical
factors. The magnitude can be measured by
wireless monitoring system, which has been
proposed in [4].
Of course, it should be taken into account that the
result (17) is relevant just for the stationary case,
when all the physical magnitudes included into the
algorithm (2) – (17) are time-independent.
4 Conclusion
1. In the present paper it is developed the model
of radioactive pollutants propagation in space
after the explosion at APP. The model
introduces three zones of propagation, each of
these zones is characterizes by specific criteria.
2. The problem of the radioactive pollutants
distribution in the far zone proves to be non-
linear. This is explained by the fact, that the
diffusion coefficient in the far zone depends
upon the concentration of radioactive isotopes
exhausted while the explosion. This type of
concentration is expected to be closer to the
reality. The algorithm for solving such a
problem is found.
3. The algorithm for calculating the far limit in
the radioactive pollutants propagation after
explosion at APP is given. It is proposed that
after the explosion radioactive isotopes are
exhausted into atmosphere and move radially
from the explosion epicenter. The calculations
show that this spatial limit for radioactive
pollutants depends upon the explosion
conditions and physical characteristics of the
pollutants.
4. It is obvious that this far limit is different for
various radioactive isotopes, which are
exhausted while explosion and move in
atmosphere from the explosion epicenter. In
this paper, the general formula for calculating
the far limit for arbitrary radioactive isotope
could be throwing up into atmosphere is given.
5. The analytical formula for calculating the total
doze of the radioactive pollutions of territory is
found. This formula allows to determine the
pollutions degree in the simplest case when it is
suggested that after the explosion radioactive
pollutants are distributed from the epicenter in
form of the concentric circumferences.
However, the linear integral for calculating the
radioactive pollutions propagation for any
arbitrary case is also provided. Additionally,
one can calculate the circular density of the
radioactive pollutions degree. Indeed, if one
uses the
(18)
The last formula is much more convenient for
making an experiment because the value
describes the radioactivity degree on the unit length
of the circumference with the radius R.
6. The analysis of the formula (17) shows that the
level of radioactive pollutions is directly
proportional to temperature. This leads to very
important conclusion, the level of radioactive
pollutions are much more intensive and wide in
hot regions in comparison with cold ones. In
other words, the explosion at APP in hot regions
will result in much harmful damages than that in
cold regions under otherwise equal conditions.
References:
[1] Hasanov H., Zeynalov I., The use of Satellite
Data in the Detection of Radioactive Fallout on
the Territory of Azerbaijan. 1st International
Computer Science, Engineering and Information
Technology Congress (ICSITY 2022) At:
WARSHAV/ POLAND pp.57-62
[2] H. G. Hasanov Method of Two-dimensional
Nonlinear Laplace Transformation for Solving
the Navier – Stokes Equation. GSTF Intern.
Journal of Computing (JoC), Vol.3, #1, 2013,
pp. 151-156
[3] H.G. Hasanov, I.M. Zeynalov. The Role of
Meteorological and Geographic Specifications in
Formation of Radioactive Precautions.
ICSSIETCONGRESS3st International Congress
on Social Sciences, Innovation and Educational
Technologies. August 26-28 2022 Tetova/
MAKEDONIA. pp. 173-180
[4] H.G. Hasanov1, I.M. Zeynalov2. Development of
a Radioactive Precipitation Monitoring System
Based on Wireless Technology Training.
ICSSIET CONGRESS 2st International
Congress of Educational Sciences and Linguists
(ICEL 2023) July 20-21, 2023 Warsaw/
POLAND PROCEEDINGS BOOK. pp. 8-14
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Contribution of individual authors to
the creation of a scientific article
(ghostwriting policy)
Hikmat Hasanov carried Physical and mathematical
modelling the problem.
Ismayil Zeynalov has organized and executed the
experiments, literature review.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
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