METHOD FOR DETERMINING AVERAGE VOLUME PARAMETERS OF THE DYNAMICS OF THE ELECTROTHERMAL STATE OF MEDIA USING THE GAUSS – OSTROGRADSKY THEOREM
In this paper, using the methodology of spatial averaging of partial differential equations, based on the application of the divergence theorem, the average volume parameters of the dynamics of the electrothermal state of media associated with the geometric shape of the medium limited in space are d...
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| author | Matiakh, S. Surzhyk Т., Т. Sheiko , I. |
| author_facet | Matiakh, S. Surzhyk Т., Т. Sheiko , I. |
| author_institution_txt_mv | [
{
"author": " S. Matiakh",
"institution": "Institute of Renewable Energy of NAS of Ukraine, Kyiv, Ukraine , National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute», Kyiv, Ukraine"
},
{
"author": "Т. Surzhyk Т.",
"institution": "Institute of Renewable Energy of NAS of Ukraine, Kyiv, Ukraine "
},
{
"author": "I. Sheiko ",
"institution": "Institute of Renewable Energy of NAS of Ukraine, Kyiv, Ukraine "
}
] |
| author_sort | Matiakh, S. |
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| collection | OJS |
| datestamp_date | 2026-07-18T06:32:20Z |
| description | In this paper, using the methodology of spatial averaging of partial differential equations, based on the application of the divergence theorem, the average volume parameters of the dynamics of the electrothermal state of media associated with the geometric shape of the medium limited in space are determined. It is shown that the volume-averaged parameters of the dynamics of the electrothermal state of media can be represented by the parameters averaged over the surface surrounding the volume. This allows us to significantly simplify the analysis of the final results of research into the processes of interaction of solar radiation with the active surfaces of elements of solar energy systems, in particular, the influence of the surrounding medium.           |
| doi_str_mv | 10.36296/1819-8058.2024.4(79).64-67 |
| first_indexed | 2025-07-17T11:39:43Z |
| format | Article |
| fulltext |
64
Відновлювана енергетика. №4/2024 | Сонячна енергетика
UDK 621 https://doi.org/10.36296/1819-8058.2024.4(79)64-67
METHOD FOR DETERMINING AVERAGE VOLUME PARAMETERS OF THE DYNAMICS OF THE
ELECTROTHERMAL STATE OF MEDIA USING THE GAUSS – OSTROGRADSKY THEOREM
Received Nov. 14, 2024; accepted Nov. 27, 2024
Available online Dec. 11, 2024
Matiakh S.1, Surzhyk Т.2, Sheiko I.3
Author for correspondence: Sheiko Iryna,
e-mail: irina_sheiko@ukr.net
Abstract. In this paper, using the methodology of spatial aver-
aging of partial differential equations, based on the application
of the divergence theorem, the average volume parameters of
the dynamics of the electrothermal state of media associated
with the geometric shape of the medium limited in space are de-
termined. It is shown that the volume-averaged parameters of
the dynamics of the electrothermal state of media can be repre-
sented by the parameters averaged over the surface surround-
ing the volume. This allows us to significantly simplify the analysis of the final results of research into the
processes of interaction of solar radiation with the active surfaces of elements of solar energy systems, in
particular, the influence of the surrounding medium.
Keywords: electrothermal state of the medium, heat and mass transfer, vector analysis, partial differential
equations, divergence theorem.
МЕТОД ВИЗНАЧЕННЯ СЕРЕДНЬООБ’ЄМНИХ ПАРАМЕТРІВ ДИНАМІКИ ЕЛЕКТРОТЕПЛОВОГО
СТАНУ СЕРЕДОВИЩ З ВИКОРИСТАННЯМ ТЕОРЕМИ ГАУСА – ОСТРОГРАДСЬКОГО
Отримано 14 лист. 2024 р.; рекомендовано до публікації 27 лист. 2024 р.
Доступно онлайн 11 груд. 2024 р.
Матях С. В.1, Суржик Т. В.2, Шейко І. О.3
Автор для кореспонденції: Шейко Ірина,
e-mail: irina_sheiko@ukr.net
Анотація. Анотація. В роботі з використанням методології
просторового усереднення рівнянь в частинних похідних на
основі застосування теореми про дивергенцію визначено се-
редньооб’ємні параметри динаміки електротеплового
стану середовищ, пов’язаних з геометричною формою обме-
женого в просторі середовища. Показано, що усереднені за
об’ємом параметри динаміки електротеплового стану се-
редовищ можливо представити через усереднені за поверх-
нею, що оточує об’єм, параметри. Це дозволяє значно спростити аналіз кінцевих результатів дослі-
джень процесів взаємодії сонячного випромінювання з активними поверхнями елементів сонячних
енергетичних систем, зокрема, впливу оточуючого середовища.
Ключові слова: електротепловий стан середовища, тепломасоперенос, векторний аналіз, рівняння в
частинних похідних, теорема про дивергенцію.
1 PhD
https://orcid.org/0000-0002-1707-3519
2 Dr. of Science (Engin.)
https://orcid.org/0000-0002-1418-7748
3 Junior Researcher
https://orcid.org/0000-0002-5770-3677
1, 2, 3 Institute of Renewable Energy of NAS
of Ukraine, Kyiv, Ukraine
1 National Technical University of Ukraine
«Igor Sikorsky Kyiv Polytechnic Institute»,
Kyiv, Ukraine
1 канд. техн. наук
https://orcid.org/0000-0002-1707-3519
2 д-р техн. наук
https://orcid.org/0000-0002-1418-7748
3 мол. наук. співроб.
https://orcid.org/0000-0002-5770-3677
1, 2, 3 Інститут відновлюваної енергетики НАН
України, Київ, Україна
1 Національний технічний університет Ук-
раїни «Київський політехнічний інститут
імені Ігоря Сікорського», Київ, Україна
65
Відновлювана енергетика. №4/2024 | Сонячна енергетика
Introduction. In the 60s of the 20th century, at the Institute
of Electrodynamics of the National Academy of Sciences of
Ukraine, on the initiative of Doctor of Technical Sciences,
Professor Yury Petrovych Yemets, research was proposed
into the methods for calculating and analyzing the aver-
aged electrophysical characteristics of media with different
spatial structures of inhomogeneities and taking into ac-
count complicating factors, in particular, anisotropy caused
by the Hall effect. The results of research, carried out over
the next 50 years by his students and colleagues, made it
possible to obtain original data on the anomalous nature of
the effect of electrical conductivity anisotropy on the aver-
aged electrical and electrophysical characteristics of heter-
ogeneous low-temperature plasma, semiconductors and
other media used in various electrical engineering and
power generation devices. What is important in these
works is that they were based on non-trivial methods of
mathematical physics, and the dependencies for the aver-
aged characteristics were compared with data on the local
distribution of fields in the region of inhomogeneities and
their surrounding media.
It is known that during the interaction of solar radiation
with any media, one of the basic processes is the conver-
sion of solar radiation energy into thermal energy, which
leads to an increase in their temperature. For solar thermal
energy systems, such a process is positive and necessary
from a practical point of view. For photovoltaic systems de-
signed to produce electrical energy, heating of active ele-
ments has a negative effect, since for all known photocon-
verters, an increase in their temperature leads to a
decrease in the coefficient of conversion of solar radiation
energy.
Today, models and methods for calculating the processes
of interaction of solar radiation with active surfaces of ele-
ments of solar thermal and photovoltaic systems have been
developed in sufficient detail at the level of Maxwell's
equations and the partial differential equation of thermal
conductivity, which describe the distribution of the corre-
sponding parameters (solar radiation intensity, heat re-
lease density, temperature and temperature gradients) in
space and time. In particular, using the synergetic method-
ology of I. Prigozhin [1, 2], the main causes of the emer-
gence of self-oscillating modes as a result of the emergence
of instabilities in nonlinearly related processes of energy
conversion of renewable sources have been formulated [3],
an analysis of the stability of nonlinear thermal processes
in media with different spatial structure of disturbances
was performed [4], which, under conditions of instability,
can lead to the formation of spatially inhomogeneous
structures with the possible realization of fluctuations in
time and a corresponding decrease in the reliability of func-
tioning and resource due to the processes of degradation
of the structure of functional materials. It should be noted
that existing experimental methods of electron microscopy
and X-ray phase analysis allow determining the morphology
of only the surface, and do not provide the opportunity to
determine the volumetric distributions of the above pa-
rameters.
Since in solar thermal and photovoltaic energy, mainly in-
tegral parameters are experimentally determined, namely:
the intensity of solar radiation, the average temperature of
the active surfaces of solar collectors and photoconverters,
and in some cases the initial temperature of the coolants
when selecting heat in solar collectors and photoconvert-
ers, there is a practical need to develop a method for tran-
sitioning from models described by partial differential
equations to models described by ordinary differential
equations for volume-averaged parameters. With this ap-
proach, the question of how to determine the volume-av-
eraged stress fluxes associated with the geometric shape of
a spatially confined medium and the influence on this me-
dium through the corresponding boundary conditions is
open.
To solve this issue, it is proposed to use the methodology
of transition from a local description of the change of pa-
rameters in time and space to a description of the change
of volume-averaged parameters only in time. This method-
ology, the essence of which lies in the theory of spatial av-
eraging of partial differential equations in space, was devel-
oped in the works of academicians of the National Academy
of Sciences of Ukraine V.O. Marchenko and E.Ya. Khruslov,
authors of the theory of averaging boundary value prob-
lems in mathematical physics [5]. The basis of this method-
ology is the use of integral theorems of vector analysis, in
particular the fundamental theorem of divergence (Gauss-
Ostrogradsky theorem) [6].
Setting the problem. The basic initial and simplest model
of the electrothermal state of the medium in the form of a
parabolic equation is considered
,v
w
q q
t
+ =
(1)
where: pw c T= – specific heat, а , ,pc T – den-
sity, specific heat capacity and temperature, respectively;
q – heat flow density;
vq – volume density of heat generation due to absorption
of electromagnetic radiation.
The magnitude of the heat flow vector q is determined by
the Fourier law [6]
,q T= − (2)
where – coefficient of thermal conductivity.
The volume density of heat release
vq is determined by
the relation
,vq П= − (3)
66
Відновлювана енергетика. №4/2024 | Сонячна енергетика
where П – Umov-Poynting vector (П Е Н= ), а
,Е Н – intensity of the electric and magnetic fields in
the medium, respectively.
The methodology of reducing the volume heat release
density
vq to a divergent form. Because
vq is a scalar
quantity with dimension W/m3, then it is naturally to ana-
lyze the function П , which is also scalar with dimen-
sion W/m3. For this purpose, according to [5], we use the
operation
( ) ( ) ( ).n Е Н Н Е Е Н = = − (4)
Using the system of Maxwell's equations, it is possible to show that the relation holds
,
В D
n Н Е Е
t t
= − − +
(5)
where: ,В Н D Е = = – induction of magnetic
and electric fields, respectively;
, , – electrical conductivity, magnetic and
dielectric permeability.
At , const − from (5) the relation follows
( ) ( ) 2 2
2 , , .
2 2
w w Н Е
П Е w w
t t
= − − − = =
(6)
In terms of physical meaning and dimensionality, the func-
tions ,w w correspond, in sinusoidal current circuits,
to the energies stored in inductances and capacitancesand
do not contribute to thermal energy.
Therefore, volumetric heat release value 2 ,vq Е=
i.e. .vq П= −
From a mathematical point of view, this is explained by the
fact that functions ,
w w
t t
are sinusoidal, which
when averaged over time T for a period, that significantly
exceeds the period of electromagnetic oscillations, are
equal to zero.
Thus, the use of the divergence theorem [5] taking into ac-
count the fact that vq П= − , given
, pc const − , leads to the following expression for
the average temperature /ср
V
Т TdV V=
( )
,
, .
vср sсрср
p
ss
vср sср
Q Q SdТ
dt c V
П dS q dS
Q Q
S S
−
=
= − =
(7)
Here V – volume bounded by a closed surface S ;
dS П dS= – vector of the elementary plane to the
surface S with unit normal П ;
vсрQ – average over surface S amount of thermal energy
of the electromagnetic field falling on the surface S ;
sсрQ – average over surface S amount of thermal energy
that is transferred to the external environment.
Equation (7) allows us to determine the growth dynamics
of the average temperature over time (at vср sсрQ Q ),
temperature value срТ in saturation mode (at
0
срdТ
dt
= ) and its reduction (at vср sсрQ Q ) after the
electromagnetic field is removed due to the dissipation of
thermal energy into the surrounding environment.
It is worth noting that the nature of change срТ over time
depends essentially on the relationship between V і S ,
which demonstrates different patterns when the character-
istic dimensions of the environment and its shape change.
It was established that dependence (7) is quantitatively or
qualitatively confirmed by experimental data for a limited
range of power values vсрQ , sсрQ , geometric parameters
V , S and physical characteristics , рс .
Note that this provides a basis for applying the Marchenko-
Khruslov methodology to other media, power values, and
equipment sizes. A crucial aspect here is that the paramet-
67
Відновлювана енергетика. №4/2024 | Сонячна енергетика
ers vсрQ , sсрQ are significantly easier to determine exper-
imentally or calculate on the surface S rather than deter-
mining their local distribution within the volume V . This
conclusion, in particular, is confirmed in our work on the
dynamics of photovoltaic cell heating under solar radiation
[8 - 10], heating of various composite materials under mi-
crowave radiation [11] and air heating in localized areas us-
ing locally distributed electric heaters, as well as in electro-
mass transfer processes [12].
Conclusions. It has been established that it is advisable to
use the Gauss-Ostrogradsky theorem to average the pa-
rameters of the dynamics of the electrothermal state over
the volume of the media. For this, a necessary condition is
the possibility of representing differential operators in spa-
tial coordinates in the original models in a divergent form.
It has been demonstrated that the volume-averaged pa-
rameters of the dynamics of the electrothermal state of the
media can be expressed through parameters averaged over
the surface surrounding the volume. This greatly simplifies
the analysis of final results and practical conclusions, in par-
ticular, the influence of the surrounding medium.
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37-40 (in Ukrainian). https://doi.org/10.36296/1819-
8058.2023.1(72)37-40.
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http://nbuv.gov.ua/UJRN/vien_2014_4_9.
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nation of the Averaged Temperature of Photovoltaic
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0000683870.
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http://nbuv.gov.ua/UJRN/vien_2015_1_9
http://nbuv.gov.ua/UJRN/vien_2014_4_9
http://jnas.nbuv.gov.ua/article/UJRN-0000698849
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http://jnas.nbuv.gov.ua/article/UJRN-0000683870
http://jnas.nbuv.gov.ua/article/UJRN-0000683870
|
| id | veorgua-article-488 |
| institution | Vidnovluvana energetika |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-19T01:14:22Z |
| publishDate | 2024 |
| publisher | Institute of Renewable Energy National Academy of Sciences of Ukraine |
| record_format | ojs |
| resource_txt_mv | veorgua/c0/31af6d4e8ac8206446a63b54a32956c0.pdf |
| spelling | veorgua-article-4882026-07-18T06:32:20Z METHOD FOR DETERMINING AVERAGE VOLUME PARAMETERS OF THE DYNAMICS OF THE ELECTROTHERMAL STATE OF MEDIA USING THE GAUSS – OSTROGRADSKY THEOREM МЕТОД ВИЗНАЧЕННЯ СЕРЕДНЬООБ’ЄМНИХ ПАРАМЕТРІВ ДИНАМІКИ ЕЛЕКТРОТЕПЛОВОГО СТАНУ СЕРЕДОВИЩ З ВИКОРИСТАННЯМ ТЕОРЕМИ ГАУСА – ОСТРОГРАДСЬКОГО Matiakh, S. Surzhyk Т., Т. Sheiko , I. electrothermal state of the medium, heat and mass transfer, vector analysis, partial differential equations, divergence theorem. електротепловий стан середовища, тепломасоперенос, векторний аналіз, рівняння в частинних похідних, теорема про дивергенцію. In this paper, using the methodology of spatial averaging of partial differential equations, based on the application of the divergence theorem, the average volume parameters of the dynamics of the electrothermal state of media associated with the geometric shape of the medium limited in space are determined. It is shown that the volume-averaged parameters of the dynamics of the electrothermal state of media can be represented by the parameters averaged over the surface surrounding the volume. This allows us to significantly simplify the analysis of the final results of research into the processes of interaction of solar radiation with the active surfaces of elements of solar energy systems, in particular, the influence of the surrounding medium.           Анотація. В роботі з використанням методології просторового усереднення рівнянь в частинних похідних на основі застосування теореми про дивергенцію визначено середньооб’ємні параметри динаміки електротеплового стану середовищ, пов’язаних з геометричною формою обмеженого в просторі середовища.  Показано, що усереднені за об’ємом параметри динаміки електротеплового стану середовищ можливо представити через усереднені за поверхнею, що оточує об’єм, параметри. Це дозволяє  значно  спростити аналіз кінцевих результатів досліджень процесів взаємодії сонячного випромінювання з активними поверхнями елементів  сонячних енергетичних систем, зокрема, впливу оточуючого середовища. Institute of Renewable Energy National Academy of Sciences of Ukraine 2024-12-10 Article Article application/pdf https://ve.org.ua/index.php/journal/article/view/488 10.36296/1819-8058.2024.4(79).64-67 Vidnovluvana energetika ; No. 4(79) (2024): Scientific and applied Journal renewable energy ; 64-67 Возобновляемая энергетика; ##issue.no## 4(79) (2024): Scientific and applied Journal renewable energy ; 64-67 Відновлювана енергетика; № 4(79) (2024): Науково-прикладний журнал Відновлювана енергетика; 64-67 2664-8172 1819-8058 10.36296/1819-8058.2024.4(79) en https://ve.org.ua/index.php/journal/article/view/488/397 Copyright (c) 2024 S. Matiakh, Т. Surzhyk Т., I. Sheiko https://creativecommons.org/licenses/by-nc-nd/4.0 |
| spellingShingle | electrothermal state of the medium heat and mass transfer vector analysis partial differential equations divergence theorem. Matiakh, S. Surzhyk Т., Т. Sheiko , I. METHOD FOR DETERMINING AVERAGE VOLUME PARAMETERS OF THE DYNAMICS OF THE ELECTROTHERMAL STATE OF MEDIA USING THE GAUSS – OSTROGRADSKY THEOREM |
| title | METHOD FOR DETERMINING AVERAGE VOLUME PARAMETERS OF THE DYNAMICS OF THE ELECTROTHERMAL STATE OF MEDIA USING THE GAUSS – OSTROGRADSKY THEOREM |
| title_alt | МЕТОД ВИЗНАЧЕННЯ СЕРЕДНЬООБ’ЄМНИХ ПАРАМЕТРІВ ДИНАМІКИ ЕЛЕКТРОТЕПЛОВОГО СТАНУ СЕРЕДОВИЩ З ВИКОРИСТАННЯМ ТЕОРЕМИ ГАУСА – ОСТРОГРАДСЬКОГО |
| title_full | METHOD FOR DETERMINING AVERAGE VOLUME PARAMETERS OF THE DYNAMICS OF THE ELECTROTHERMAL STATE OF MEDIA USING THE GAUSS – OSTROGRADSKY THEOREM |
| title_fullStr | METHOD FOR DETERMINING AVERAGE VOLUME PARAMETERS OF THE DYNAMICS OF THE ELECTROTHERMAL STATE OF MEDIA USING THE GAUSS – OSTROGRADSKY THEOREM |
| title_full_unstemmed | METHOD FOR DETERMINING AVERAGE VOLUME PARAMETERS OF THE DYNAMICS OF THE ELECTROTHERMAL STATE OF MEDIA USING THE GAUSS – OSTROGRADSKY THEOREM |
| title_short | METHOD FOR DETERMINING AVERAGE VOLUME PARAMETERS OF THE DYNAMICS OF THE ELECTROTHERMAL STATE OF MEDIA USING THE GAUSS – OSTROGRADSKY THEOREM |
| title_sort | method for determining average volume parameters of the dynamics of the electrothermal state of media using the gauss – ostrogradsky theorem |
| topic | electrothermal state of the medium heat and mass transfer vector analysis partial differential equations divergence theorem. |
| topic_facet | electrothermal state of the medium heat and mass transfer vector analysis partial differential equations divergence theorem. електротепловий стан середовища тепломасоперенос векторний аналіз рівняння в частинних похідних теорема про дивергенцію. |
| url | https://ve.org.ua/index.php/journal/article/view/488 |
| work_keys_str_mv | AT matiakhs methodfordeterminingaveragevolumeparametersofthedynamicsoftheelectrothermalstateofmediausingthegaussostrogradskytheorem AT surzhyktt methodfordeterminingaveragevolumeparametersofthedynamicsoftheelectrothermalstateofmediausingthegaussostrogradskytheorem AT sheikoi methodfordeterminingaveragevolumeparametersofthedynamicsoftheelectrothermalstateofmediausingthegaussostrogradskytheorem AT matiakhs metodviznačennâserednʹoobêmnihparametrívdinamíkielektroteplovogostanuseredoviŝzvikoristannâmteoremigausaostrogradsʹkogo AT surzhyktt metodviznačennâserednʹoobêmnihparametrívdinamíkielektroteplovogostanuseredoviŝzvikoristannâmteoremigausaostrogradsʹkogo AT sheikoi metodviznačennâserednʹoobêmnihparametrívdinamíkielektroteplovogostanuseredoviŝzvikoristannâmteoremigausaostrogradsʹkogo |