Stochastic model of liquid fuel spraying at high pressures and high
Reynolds numbers
АSKAROVA А.S., BOLEGENOVA S.А., MAXIMOV V.YU., BEKETAYEVA М.Т.
Physics and Technology Faculty
Al-Farabi Kazakh national university
Almaty, Al-Farabi av., 71
REPUBLIC OF KAZAKHSTAN
Abstract: - The paper describes the main features of the combustion of liquid fuel injections, developed a
stochastic model for the atomization of liquid fuels injected into the combustion chamber at high pressures and
high Reynolds numbers. A mathematical model for the combustion of liquid injections at high pressures and
high Reynolds numbers is presented, which includes: the equations of continuity, motion, internal energy, the
K-ε model of turbulence, a system of equations describing the processes of evaporation, mixing, rupture and
coalescence of liquid fuel droplets. A stochastic model of atomization of liquid fuels injected into a combustion
chamber at high pressures and high Reynolds numbers has been developed. On the basis of the proposed
model, computational experiments were carried out to study the combustion of liquid fuel depending on the
injected mass in the combustion chamber under given initial conditions in full. When studying the effect of the
mass of liquid fuel on the processes of ignition and combustion at high pressures and high Reynolds numbers,
the mass values for octane 6 mg and for dodecane 7 mg were taken as the most optimal. A further increase in
the injection mass, both for octane and dodecane at optimal pressures, worsens the combustion process. The
results obtained are of fundamental and practical importance and can be used to develop the theory of
combustion of gaseous and liquid fuels.
Key-Words: - Atomization, high pressure, ignition, injection, numerical simulation, simulation, Reynolds
number, turbulent flows, two-phase media, 3D visualization
Received: July 21, 2021. Revised: March 11, 2022. Accepted: April 13, 2022. Published: May 5, 2022.
1 Introduction
Currently, the main source of generated energy
(about 80%) is the energy of various types of fuels.
Combustion will continue to be the main source of
energy for many years to come, even as the use of
nuclear energy in industrialized countries expands,
and methods of using solar, wind and tidal energy
are intensively developed. The problem of the
formation of harmful substances and the limited
resources of fuel leads to the need to organize more
economical methods of its combustion [1-4].
The combustion of liquid fuels is distinguished by a
number of specific features due to the occurrence of
chemical reactions under conditions of dynamic and
thermal interaction of reagents, intensive mass
transfer during phase transformations, as well as the
dependence of the process parameters on both the
thermodynamic state of the system and its structural
characteristics.
Since the study of combustion is impossible without
its detailed study, the problem of fundamental
research into the regularities of heat and mass
transfer processes during the combustion of various
types of fuels comes to the fore.
Numerical study of the combustion of liquid fuels is
a complex task of thermal physics, since it requires
taking into account a large number of complex
interrelated processes and phenomena. Therefore,
the computational experiment is becoming an
increasingly important element in the study of
combustion processes and the design of various
devices that use the combustion process [5-8].
2 The main features of the
combustion of liquid fuel
2.1 Mathematical model for the formation of
combustion of liquid injections at high
pressures and high Reynolds numbers
Here, the main features of the combustion of liquid
fuel injections are described, a mathematical model
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for the formation of combustion of liquid injections
at high pressures and high Reynolds numbers is
presented: a system of equations describing the
combustion process of atomized liquid fuels [9-12].
The initial and boundary conditions of the problem
under study on the combustion of liquid fuels in a
combustion chamber are given. A stochastic model
of atomization of liquid fuels injected into a
combustion chamber at high pressures and high
Reynolds numbers has been developed.
The continuity equation for the m-th component is
written as follows:
,)( 1m
sc
m
m
m
mDu
t
(1)
where m – is the mass density of the component m,
- total mass density,
u – fluid velocity.
After summing Eq (l) over all phases, the continuity
equation for the liquid is obtained:
s
u
t
)(
(2)
The momentum transfer equation for the liquid
phase:
0
2
( ) 1 2
( ) p-A ( )
3
,
s
uuu k
ta
Fg
(3)
p – fluid pressure.
The value of A0 is equal to zero for laminar flows
and unity in the case of turbulent flow.
The viscous stress tensor has the form:
.)( Iuuu T
(4)
Internal energy equation:
0
0
() ( ) (1 )
cs
IuI u A u
t
J A Q Q
(5)
c
Q
– source term due to heat release as a result of a
chemical reaction,
s
Q
– the heat that the injected
fuel brings.
The equations k - ε of the model for the turbulent
kinetic energy k and its dissipation rate ε have the
form:
2
()3
,
Pr
s
k
kuk k u
t
u k W
(6)
12
12
2
( ) ( )
3
Pr
.
s
s
u c c u
t
c u c c W
k
(7)
The equation of state for a mixture of gases can be
written as:
m
mm WTRp )/(
0
(8)
Specific internal energy:
m
mm TITI )()/()(
(9)
Specific heat capacity at constant pressure has the
form:
m
pmmp TcTc )(/)(
(10)
Enthalpy:
,/)()( 0mmm WTRTITh
(11)
The change in droplet temperature is determined by
the energy balance equation:
,4)(4
3
4223
ddddld QrTRLrTcr
(12)
where сl – the specific heat of the liquid, L(Td) is the
specific heat of vaporization, and Qd is the thermal
conductivity at the droplet surface in a unit volume.
2.2 Stochastic fuel spray model
The task of modeling liquid fuel atomization for
each fragment from the injector at high pressures in
the combustion chamber, when the Weber and
Reynolds numbers are high, is very difficult. To
represent the features of this problem, it is necessary
to consider the physical parameters of the decay. To
do this, consider the main assumptions in our model.
1) At each moment of time, the liquid clot has its
own specific geometric configuration. Each
geometric configuration is defined by the spatial
trajectory of a stochastic particle (FC).
2) At various times, these configurations are
represented by an ensemble of independent
implementations. Stochastic particles are sprayed
through the injector one after another, in this case
each particle has its own path, the duration of which
will be determined below. We consider that the
distribution f(x, t, r) is defined so that f(x,t,r)d3r is
the probability of finding the radial position of the
surface r, in the direction of the x axis, at time t in
the volume d3r. Then this distribution takes the
following form:
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FC
rtxrrtxf ,);,(
(13)
Where rFC – radial position of a stochastic particle.
The spray near the liquid center of the jet or the
region of drops has an insignificant volume, but a
significant mass compared to the gas in the
combustion chamber. Thus, the droplet clusters that
will be located near the injector can be determined
by the distribution f(x, t, r).
3) Suppose that the radial position of a given
stochastic particle
xFC
r,
to different positions along
the axis changes step by step, forming a process in
which the value
xxFC
r,
is obtained by multiplying
by an independent random variable. Thus, it has the
following form:
a
xFCxxFC err ln
,,
(14)
Here the random factor
10
a
determined by
the probability density distribution
q
at
1
1
0
dq
. This coefficient determines the
fragmentation by scale symmetry
rr
and
depends on the probability density distribution
q
, which is basically unknown. However, such
fragmentation with a constant frequency T-1, balance
equation, particle size distribution over time can be
neglected, with increasing spray time, the Fokker-
Planck equation is used [9-11]:
rf
r
r
rT
rf
rT
a
t
trf
2
ln
ln
,2
(15)
trf ,
- normal distribution function,
1
0
drrf
and only the first two logarithmic
moments
can change
trf ,
and in the
same way:
22 lnln
ln
ln
ln
rr
r
(16)
The stochastic equation for the radial position of a
stochastic particle takes the form:
2
, , ,
2
,
ln
ln
2
ln ()
2
FC x x FC x FC x
FC x
r r r dt
TT
r dW t
T
(17)
where
)(
0, tRr eff
injxFC
is determined using the
effective radius of the injector and
)(tdW
-
stochastic Winnner process with
0)( tdW
,
dttdW 2)( 2
[12].
4) Downward in the direction of the x-axis, each
stochastic particle moves with an axial velocity
equal to the effective velocity
)(tUu eff
inj
at the
inlet of the liquid jet. The effective injector radius
and effective injection velocity are calculated using
the jet narrowing factor
C
C
[13], which is based on
the cavitation number
CN
:
3;1,
1
162.0: CN
CN
CCN C
(18)
There:
C
inj
eff
inj C
tU
tU )(
)(
exp
,
geom
injC
eff
inj RCtR )(
,
where
)(
exp tUinj
- given injection speed.
5) The three parameters
ln
,
2
ln
and T must
be determined from (9) according to the decay
mechanism. In the stochastic simulation of
secondary spray in [14], the solution looked like:
0
2
ln
ln
ln
l
rcr
(19)
0
l
- is the size of the "mother" drop formed by
fragmentation along scalar symmetry and
cr
r
- the
typical droplet size obtained. Currently, it is
proposed that the interaction between the
accelerated liquid jet and the gas gives an increase
in the Rayleigh-Taylor instability and then drops are
formed from the free surface kink. In [15], the most
unstable wavelength
RT
of the Rayleigh-Taylor
instability was derived, which has the form:
RT
crit
r
lnln
(20)
RT
crit
r
constconst
lnlnlnln 2
(21)
Where constant is chosen equal to 0.1..
The typical size
cr
r
is included in the Weber
number equation in the form:
1/
2
cr
eff
injgascr rUWe
(22)
The lifetime of each stochastic particle is expressed
as:
bup
V
tT 1
)(
)(
)(
)( tR
tU
teff
inj
eff
inj
fuel
gas
bup
(23)
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If the value of the Weber number is greater than the
critical value, the mathematical model of the
secondary spray is activated.
We propose a new modification of this condition.
When the fuel injection rate decreases:
0
dt
dU eff
inj
, the mass of liquid fuel injected earlier
has a greater velocity than the mass of fuel injected
at the next moment. Thus, we present a new
expression for the Weber number, which is
compared with the critical:
r
uu
r
UfWe pgg
eff
injtuel
2
2
max
(24)
eff
injtU
- incremental fuel injection rate over time.
The relaxation equation for the droplet radius r takes
the following form:
)(
)(
*
r
rr
dt
dr
(25)
Radius
)(
*
r
can be determined after turbulent
expansion of droplets:
3/1
1
*
r
(26)
Parameter
)(
r
can be determined using
dimensional analysis. Turbulent expansion of a
droplet is affected by three main physical quantities:
viscous dissipation in a turbulent gas flow
, liquid
density
1
, and surface tension tensor
. The
quantity is dimensionless and has the form:
5/1
2
1
3
2
)(
(27)
We define the intermittency of turbulence using the
log-normal distribution of the Obukov viscous
dissipation:
x
, where
- viscous
dissipation of the standard κ-ε turbulence model:
2
2
1
22
ln
exp
2
1
)( m
mx
x
dx
m
dxxP
(28)
There
xm ln
1
and
2
12 ln mxm
, at
4
2
12 Reln4.02mm
For a given drop after time, we choose for it the
value of ε from the lognormal distribution and
obtain a new radius value.
3 Problem formulation and results of
3D visualization
3.1 Problem under study
In this work, two types of liquid fuels were used:
octane (C8H18) and dodecane (C12H26). The chemical
reactions for these two fuels are shown below:
2С8Н18 + 50О2 = 16СО2 + 18Н2О
2С12Н26 + 37О2 = 24СО2 + 26Н2О
Based on the created stochastic model of
atomization of liquid fuels injected into the
combustion chamber at high pressures and high
Reynolds numbers, under given initial and boundary
conditions of the problem under study, the
combustion of liquid fuels is considered in a model
combustion chamber with a nozzle located in the
center of the lower part of the chamber, through
which the main part of the liquid fuel consumption
is supplied to the oxidizer flow (heated air). The
combustion process of liquid fuels is considered in a
model combustion chamber with a nozzle located in
the center of the lower part of the chamber, through
which the main part of the liquid fuel flow is
supplied to the oxidizer flow (heated air). The
chamber has a cylinder structure 15 cm high and 2
cm in radius. The initial temperature in the
combustion chamber is 800 K. The number of
control cells is 600. The temperature of the
combustion chamber walls is 353 K. The area of the
injector nozzle is 2×10-4 cm2 [16].
3.2 Results of computer experiments
The results of computer experiments on the effect of
the mass of injected fuel (octane and dodecane) into
the combustion chamber on the combustion process,
which were carried out by us at optimal pressure
values in the combustion chamber.
For octane, the optimal pressure is 100 bar and 80
bar for dodecane, while the mass of injected fuel
varied from 4 to 20 mg.
Further, it will be expedient to present graphs only
for the optimal mass values for octane 6 mg and
dodecane 7 mg.
At high pressure, visualization of the distribution of
droplets of two fuels by size at different points in
time (Fig. 1-2) and temperature profiles in the space
of the combustion chamber at the moment of
ignition of liquid fuel (Fig. 3) are plotted.
Analyzing the data obtained, we can say that high
pressure leads to a decrease in the spray area, which
is quite natural, since the injected liquid experiences
greater resistance. At a pressure of 100 bar, the
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maximum height of the octane spray area is 0.24
cm, and the width is 0.02 cm, and its droplet sizes
range from 6 to 40 µm at the initial spray moment
t=10-4 (Fig. 1a). A further increase in the octane
injection time (t=3.9∙10-4s) leads to an increase in
the area of droplet propagation up to 0.45 cm in
height and up to 0.04 cm in width (Fig. 1b). At the
final moment of spraying time (t=1.1∙10-3s), the area
of distribution of droplets decreases, which indicates
intensive evaporation of liquid fuel (Fig. 1c).
a)
b)
c)
Figure 1. Distribution of octane droplets by size
(rad, mm) in the space of the combustion chamber at
a pressure of 100 bar at different times
а) t=10-4 s, b) t=3.9∙10-4 s, c) t=1.1∙10-3 s
Figure 2 shows similar studies for another type of
fuel - dodecane. For dodecane at a pressure of 80
bar at the time t = 10-4s (Fig. 2a), the maximum
height of the spray area is 0.38 cm and a width of
0.06 cm, and the droplet sizes range from 6 to 40
microns. When the spraying process is not yet
completed t=5∙10-4s (Fig. 2b), the droplets rise to a
great height, in this case the spray area of dodecane
droplets is 1.1 cm in height and 0.15 cm in width of
the chamber. A further increase in the injection time
leads to a significant decrease in the droplet
propagation area (Fig. 2c).
a)
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b)
c)
Figure 2. Size distribution of dodecane droplets
(rad, mm) in the space of the combustion chamber at
a pressure of 80 bar at different times
а) t=10-4 с, b) t=5∙10-4 с, c) t=1∙10-3 с
The combustion mechanism of liquid fuels is
characterized by the fact that the boiling point of
liquid fuels is always lower than the self-ignition
temperature, therefore, the combustion of liquid
fuels occurs in the vapor phase. The combustion
mechanism of liquid fuels includes several stages: a
spark (or other foreign source), ignition of the
vapor-air mixture, combustion of the vapor-air
mixture at the surface of the liquid, an increase in
the evaporation rate due to heat transfer from the
flame (until equilibrium is reached). The ignition
temperature of liquid fuel is the temperature of
heating the liquid base of the fuel, at which the fuel
droplets ignite and the fuel burns continuously.
As can be seen from Fig. 3, ignition for different
fuels occurs at different times. For octane, the
ignition moment is t=7∙10-4 s, while for dodecane
t=8.1∙10-4 s.
a)
b)
Figure 3. Temperature distribution (T, K) in the
space of the combustion chamber at the time of
ignition
а) octane Р=100 bar and t=7∙10-4 s, b) dodecane
Р=80 bar and t=8.1∙10-4 s
а)
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b)
Figure 4. Distribution of the maximum temperature
(T, K) in the space of the combustion chamber at the
moment of time t=1.5 ∙10-3 s
а) octane at P=100 bar, b) dodecane at Р=80 bar
When liquid fuel is atomized in a stationary or
moving gas, a two-phase reacting jet is formed,
which burns, forming a torch of liquid fuel. As
shown in Fig. 4 (a, b), for both octane and dodecane,
at high pressure in the combustion chamber, the
high temperature region increases.
The gas temperature distributions in the space of the
combustion chamber make it possible to see small
differences in the combustion of two types of fuels
with the optimal liquid fuel injection mass: the
highest temperature is observed when burning
octane 1948.79 K (Fig. 4a). When the mixture of
fuel vapors with an oxidizer is ignited, the fuel
begins to burn very quickly, almost the entire area
of the chamber is covered in width by a torch. At the
time t=1.5∙10-3s, the region of high temperatures for
octane (Fig. 4a) is the smallest. Octane vapor rises
to a height of 1.5 cm, while dodecane vapor reaches
1.79 cm. For optimal injection masses (for 6 mg
octane and 7 mg dodecane), at the final time point
(4 ms), the fuel has reacted with the oxidizer
completely, without a trace.
A graphical dependence of the size distribution of
octane and dodecane droplets (Fig. 5,9), temperature
of droplets of both fuels (Fig. 6,10), carbon dioxide
concentration (Fig. 7,11) and soot concentration
(Fig. 8,12) is plotted depending on the mass of
liquid fuel in the combustion chamber.
The dependence of the maximum size of octane
droplets on its injection mass is shown in Figure 5.
Analysis of the graph shows that an increase in the
mass of octane injection to 6 mg leads to a decrease
in the size of its droplets and is 93.61 microns. A
further increase in mass leads to a slight increase in
the radius of fuel droplets. Figure 6 shows the
distribution of the maximum temperature of droplets
over the volume of the combustion chamber from
the injected mass of octane. As we can see,
increasing the mass of the injection leads to a slight
increase in temperature to 561 K for octane at 6 mg.
A further increase in mass leads to a subsequent
decrease in temperature. This result agrees with the
previous Fig. 5, thus, with a decrease in the radius
of the fuel droplets, the temperature of the octane
droplets increases due to the intensive evaporation
of the fuel.
Figure 5. Size distribution of droplets (R, µm) depending
on the mass of liquid fuel (m, g) in the combustion
chamber
Figure 6. Temperature distribution of drops (T, K) in the
combustion chamber depending on the mass of liquid fuel
(m, g)
Thus, we can make a preliminary conclusion that
the optimal mass of octane can be called equal to 6
mg, at which the droplet sizes are minimal and the
droplet temperature reaches its maximum. A further
increase in the mass of injection of octane worsens
the combustion process.
Figure 7 shows the effect of octane mass on
the distribution of carbon dioxide concentration. As
the mass of liquid fuel increases, the amount of
carbon dioxide increases for obvious reasons: the
more fuel, the more CO2 is formed. The minimum
concentration of carbon dioxide equal to 0.0989 g/g
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is formed by injecting octane with a mass equal to 6
mg. This result once again confirms that the optimal
mass values for two fuels selected above are correct.
Figure 8 shows the effect of octane mass on the soot
concentration distribution in the combustion
chamber space. With an increase in the mass of
injected fuel, the soot concentration decreases
monotonically, which is consistent with the previous
Fig. 7. With an injection mass of 6 mg for octane, a
small amount of soot is released equal to 58.4 g/m3.
Thus, the optimal value of the injected mass of
octane was obtained equal to 6 mg, at which the size
of fuel droplets is minimal 93.61 microns, and the
temperature of its drops is maximum 561 K. It is at
this value of the mass of octane that a small amount
of soot is released 58.4 g/m3 and the minimum
amount carbon dioxide 0.0989 g/g.
Figure 7. Distribution of carbon dioxide
concentration (g/g) in the combustion chamber
depending on the mass of liquid fuel (m, g)
Fihure 8. Distribution of soot concentration (g/m3)
in the combustion chamber depending on the mass
of liquid fuel (m, g)
The dependence of the maximum size of dodecane
droplets on its injection mass was obtained (Fig. 9),
which shows that an increase in the injection mass
from 4 mg to 7 mg leads to a decrease in the size of
fuel droplets. With a fuel mass of 7 mg, the droplet
size reaches its minimum and is 93.754 µm, and the
maximum droplet temperature is 644 K (Fig. 10). A
further increase in the mass of dodecane leads to a
slight increase in the radius of its droplets (Fig. 9)
and a subsequent decrease in temperature (Fig. 10).
Figure 9. Size distribution of droplets (R, µm)
depending on the mass of liquid fuel (m, g) in the
combustion chamber
Figure 10. Temperature distribution of drops (T,
K) in the combustion chamber depending on the
mass of liquid fuel (m, g)
Figure 11 shows the distribution of carbon dioxide
concentration depending on the mass of dodecane
injected. As can be seen from the figure, for obvious
reasons, with an increase in the mass of fuel, the
concentration of carbon dioxide increases. With a
mass of fuel equal to 7 mg, the minimum amount of
CO2 is formed, which is 0.101 g/g. We can make a
preliminary conclusion that the optimal value of the
mass of dodecane is 7 mg, at which the
concentration of carbon dioxide is minimal and the
temperature of the fuel droplets is maximum.
An analysis of the dependence of the mass of liquid
fuel on the soot concentration in the space of the
combustion chamber (Fig. 12) shows the opposite
picture than in the case of CO2, with an increase in
the injection mass, the soot concentration decreases.
For dodecane, the minimum amount of soot 25.3
g/m3 is formed by injecting fuel with a mass of 7
mg.
Soot, g/m3
m, g
R, µm
m, g
T, K
m, g
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As a result of the foregoing, the optimal value of the
mass of dodecane was obtained equal to 7 mg, at
which the size of the fuel droplets is minimal and
amounts to 93.754 microns, the maximum
temperature of its drops reaches 644 K, at this value
of the injection mass a small amount of soot 25.3
g/m3 and carbon dioxide are released 0.101 g/g.
Figure 11. Distribution of carbon dioxide
concentration (g/g) in the combustion chamber
depending on the mass of liquid fuel (m, g)
Figure 12. Distribution of soot concentration (g/m3)
in the combustion chamber depending on the mass
of liquid fuel (m, g)
For a more profitable organization of the
combustion process of liquid fuel (octane and
dodecane) in relation to its injected mass under the
conditions of this task, it was established:
1) For octane, the optimal mass is 6 mg,
because it is at this mass value that the size of its
droplets is minimal 93.61 microns, which leads to
an increase in the spray area and intensive
evaporation of the fuel. The evaporation
temperature of octane droplets is maximum 561 K,
which increases the efficiency of fuel combustion.
To minimize emissions of harmful substances, the
optimal mass of octane was chosen equal to 6 mg, at
which a small amount of soot 58.4 g/m3 and a
minimum amount of carbon dioxide 0.0989 g/g are
released.
2) The optimal value of the mass of
dodecane is 7 mg, in this case the spray area of
dodecane droplets is maximum, and the size of fuel
droplets is minimum and is 93.754 microns.
Dodecane drops are heated to a maximum
temperature of 644 K. At this dodecane injection
mass, the following reaction products are formed:
soot, the concentration of which is 25.3 g/m3 and a
small amount of carbon dioxide 0.101 g/g.
4 Conclusion
The paper describes the main features of the
combustion of liquid fuel injections, developed a
stochastic model for the atomization of liquid fuels
injected into the combustion chamber at high
pressures and high Reynolds numbers.
A stochastic model of atomization of liquid fuels
injected into a combustion chamber at high
pressures and high Reynolds numbers has been
developed. On the basis of the proposed model,
computational experiments were carried out to study
the combustion of liquid fuel depending on the
injected mass in the combustion chamber under
given initial conditions in full. A graphical
dependence of all these characteristics has been
constructed, namely, the distribution of octane
droplets by size, the temperature of octane droplets,
carbon dioxide concentration and soot concentration
depending on the mass of liquid fuel in the
combustion chamber.
When studying the effect of the mass of liquid fuel
on the processes of ignition and combustion at high
pressures and high Reynolds numbers, the mass
values for octane 6 mg and for dodecane 7 mg were
taken as the most optimal. With an octane mass of 6
mg, the droplet sizes are minimal and equal to 93.61
μm, and the temperature reaches its maximum of
561 K. In this case, a small amount of CO2 is
released, equal to 98.89 and soot, 58.38 g/m3. For
the same reasons, the optimal weight for dodecane
was 7 mg. The maximum droplet size is 93.75 µm,
the temperature of dodecane droplets is 644.6 K. In
this case, CO2 is formed equal to 10.1 and soot –
25.51 g/m3. A further increase in the injection mass,
both for octane and dodecane at optimal pressures,
worsens the combustion process. The results
obtained are of fundamental and practical
importance and can be used to develop the theory of
combustion of gaseous and liquid fuels.
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m, g
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DOI: 10.37394/232012.2022.17.12
Аskarova А. S., Bolegenova S. А.,
Maximov V. Yu., Beketayeva М. Т.
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Contribution of individual authors to
the creation of a scientific article
(ghostwriting policy)
Aliya Askarova, Saltanat Bolegenova has organized
and executed the experiments.
Valeriy Maximov and Meruyert Beketayeva carried
out the simulation results, interpretation (discussion)
and verification of results.
Follow: www.wseas.org/multimedia/contributor-
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Sources of funding for research
presented in a scientific article or
scientific article itself
Research funded by the Ministry of Education
and Science of the Republic of Kazakhstan
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License 4.0 (Attribution 4.0
International , CC BY 4.0)
This article is published under the terms of the
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WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2022.17.12
Аskarova А. S., Bolegenova S. А.,
Maximov V. Yu., Beketayeva М. Т.
E-ISSN: 2224-3461
123
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