Application of Electrodynamic Catalytic Reactors for Intensification of
Heat and Mass Exchange Processes of Heterophase Catalysis
EKATERINA A. SHULAEVA
Department "Automated Technology and Information Systems"
Branch of the Ufa State Petroleum Technological University in Sterlitamak
Oktyabrya Ave., 2.Sterlitamak, 453118 Bashkortostan Republic
RUSSIA
Abstract: - This article discusses a model of an electrodynamic reactor. It is fundamentally different in the way
of supplying energy to the reaction zone from the reactors currently in operation in industry. This significantly
reduces energy consumption, simplifies process control and increases the efficiency of the reactor. The
environmental advantage of the developed reactor is also of great importance. There are no emissions of gases
into the atmosphere, which are formed in large quantities during fuel combustion in superheating furnaces, the
consumption of water is reduced, which is used only in a closed cycle to cool the microwave generator and
circulator as a matching load during its operation. The overall efficiency of the electrodynamic reactor is 1.2
times higher than that of existing industrial ones. A method for calculating thermodynamic processes in
electrodynamic reactors is proposed. It allows you to determine the technological parameters of the process to
ensure a given temperature distribution and provides the maximum yield of the target reaction products with the
minimum possible energy consumption of electromagnetic radiation.
Key-Words: - microwave, electrodynamic reactor, catalyst, temperature distribution, heterophase catalysis, heat
and mass transfer
Received: March 19, 2021. Revised: January 3, 2022. Accepted: February 8, 2022. Published: March 2, 2022.
1 Introduction
The creation of energy-saving and resource-saving
technologies for the rational use of natural resources
in the petrochemical industry is a very urgent
problem. One of the ways to solve which is the use
of various physical methods of influencing
technological environments, in particular,
electromagnetic radiation of the microwave range,
as one of the effective methods of energy transfer.
The use of microwave as a heat carrier for
heating media of various natures is one of the ways
to increase the efficiency of modern chemical
production. It is stimulating research on the use of
microwave in chemical technology. Currently, in
Russia, the USA, UK, Canada, Japan and other
countries of the world, there are chemical
laboratories focused on research in this field of
chemistry and on the creation of special microwave
units for carrying out specific chemical processes.
The use of microwave in the chemical and
petrochemical industry is constrained by the lack of
reliable data on the specifics of chemical
transformations occurring under the action of
microwave taking into account the specific nature of
energy transfer in the reaction zone and due to the
lack of methods for calculating technological units
and reaction devices that use this type of energy
transfer. It is also necessary to take into account that
the efficiency of transformation of the energy of the
electromagnetic field into heat, which is necessary
for carrying out chemical transformations. First of
all, this is determined by the properties of the
technological environment, which is a heat
converter [1]. Therefore, the study of the effect of
electromagnetic radiation in the microwave range on
various technological environments requires the
introduction of additional parameters that determine
the efficiency of conversion of electromagnetic
energy into thermal energy. This is an important
both theoretical and practical task aimed at optimal
design of technological cycles of oil refining and
operation of reaction devices and technological
units.
2 Problem Formulation
The main feature when calculating an
electrodynamic reactor, in comparison with a
traditional catalytic reactor, is the need to take into
account the properties of the electromagnetic field
and the electrophysical properties of the catalyst
substance. These parameters are directly related to
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the determination of the characteristic dimensions of
the reaction unit, i.e. its diameter and height [2].
Another important characteristic determined by
the parameters of the electromagnetic field and
affecting the design parameters of the reactor is the
minimum diameter of the reaction device, it is
determined from the wavelength of electromagnetic
radiation. When using microwave the body of the
reaction device is a resonator. The main condition
for the propagation of electromagnetic waves in the
resonator is the ratio [3]:
cr
, (1)
where λ – wavelength of electromagnetic radiation,
m; λcr – critical wavelength, m.
The critical wavelength for waves of the Emn type
is determined by the following relation:
mn
P
a
2
Ecr
, (2)
where a is the radius of the waveguide, m; Pmn are
the roots of the Bessel function Jm of order m, the
values of which are given in Table 1 [4].
Table 1. Roots of the Bessel function Jm of order
m
m
n
1
2
3
Pmn
0
2,405
5,52
8,654
1
3,832
7,016
10,173
2
5,136
8,417
11,620
Thus, the radius of the reaction units can be
determined:
2mn
P
a
. (3)
Similarly, you can determine the critical wavelength
for waves of the Hmn type:
mn
P
a
2
Hcr
, (4)
where P'mn are the roots of the Bessel function Jm',
the values of which are given in Table 2 [4].
Table 2. Roots of the Bessel function Jm’of order
m
m
n
1
2
3
Pmn’
0
3,832
7,016
10,174
1
1,841
5,332
8,536
2
3,054
6,705
9,965
Thus, the minimum diameter of the reaction
device, in which electromagnetic waves of the E01
and H11 types can propagate without loss at a
wavelength of 0.12 m, corresponding to a frequency
of 2450 Hz, is 0.092 m.
Based on the required capacity of the reactor, the
volume of the catalyst is determined, taking into
account the limitations stated above. Thus:
W
V
VG
K
, (5)
where VK is the catalyst volume, m3; VG – gas
volumetric velocity, m3/h; W – volumetric feed rate,
h-1.
To assess the energy efficiency of the catalytic
process under the action of microwave, the thermal
balance of a microwave unit is calculated by the
formula:
Qm+Qin = Qр+Qout+Qlos, (6)
where Qm is the heat released during the absorption
of electromagnetic radiation, J/h; Qin is the heat
introduced into the reaction plant by the flow of the
gas mixture, J/h, defined as Qin = G • c • Tin, where
G is the amount of substance entering the reactor,
mol/h, c is the molar heat capacity, J / (mol • K) , Tin
is the temperature of the gas mixture at the entrance
to the reactor, K; Qp – heat absorbed / released
during chemical reactions, J/h; Qout is the heat
carried away from the reaction plant, J/h; Qlos – heat
losses in the reaction plant, J/h, as a rule, for the
purposes of technological calculation is taken at 5%,
of the heat entering the reaction plant.
Thus, the thermal efficiency of the reaction plant:
м
p
Q
Q
. (7)
The power of the magnetron required for
carrying out chemical transformations can be
calculated based on the heat balance equation of the
reaction plant (6):
Qm = Qр+Qout – Qin +Qlos. (8)
Thus, expressions (1-7) allow for the
technological calculation of the reaction device. The
peculiarities of this calculation include the need to
determine the "total" depths of microwave
absorption for the catalysts used in each specific
technological process on the basis of experimental
data. Another feature is the need to select the
diameter of the reaction device, based on the
wavelength of the microwave generator.
Reactor design for carrying out processes under
microwave. On the basis of laboratory research, a
special reaction device was developed for the
implementation of endothermic processes using
microwave heating [5]. The schematic diagram of
the reaction device (electrodynamic reactor) is
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shown in Fig. 1. The reactor has an inner diameter
of 100 mm, a maximum height of 1340 mm, and a
wall thickness of 5 mm. The installation includes a
continuous microwave source with a fixed
oscillation frequency of 2450 MHz and an
adjustable output power of 0-5 kW.
Fig. 1: Schematic diagram of a microwave catalytic
reactor for endothermic heterophase reactions: 1 –
reactor vessel; 2 – microwave generator; 3 – top
cover of the reactor; 4 – catalyst; 5 – waveguide; I –
raw material; II – contact gas.
The reactor (resonator for a microwave
generator) is a vertical cylindrical thermally
insulated vessel and consists of three main units
(Figure 2): input of raw materials and microwave
radiation, a reaction zone and a unit for output of
reaction products.
Fig. 2: Model of an electrodynamic reactor in
section: 1 – input unit; 2 – catalyst; 3 – output node;
4 – reaction units.
The upper cover of the reactor plays the role of a
radiating antenna that ensures a uniform distribution
of energy over the cross section of the reactor. Alloy
steel resistant to corrosive media in a wide
temperature range and having sufficient electrical
conductivity, is proposed as a material for the
manufacture of the reactor. The last condition is
necessary to prevent losses of electromagnetic
energy.
The fluoroplastic membrane is installed in the
injection unit, which ensures the tightness of the
reaction zone. The raw material is introduced into
the reaction chamber with tangential input. The
outgoing gas is discharged from the bottom.
Distribution grids for the catalyst are installed in the
reaction chamber they are made of heat-resistant
ceramics permeable to microwaves.
A matching chamber for absorbing residual
radiation is located in the reaction product outlet.
Residual radiation due to its incomplete absorption
by the catalyst substance penetrates through the
membrane of the matching chamber and is absorbed
by water. The inclined surface of the matching
chamber attenuates the direct reflection of
microwave radiation into the magnetron. In the
matching chamber, crushed catalyst particles are
collected.
The matching chamber is equipped with a hatch
by means of the fittings it is possible to regulate the
level of filling the chamber with liquid. Control and
measurement devices are mounted on the hatch
cover [6].
The advantages of the reactor include the ability
to operate in a wide temperature range and
resistance to aggressive media. Alloy steel was
chosen as the structural material of the reactor,
which has proven itself in the petrochemical
industry for the manufacture of various apparatus
and equipment.
Through modular construction of the reaction
zone may change the size of the reactor, while
maintaining the desired ratio of height to diameter.
This ratio can vary from minimum values to 9.75.
At present, the use of electromagnetic radiation
of the microwave range is becoming more and more
widespread for carrying out various physical
transformations as a way of more efficient, from the
standpoint of energy consumption per unit of the
product obtained, and faster heating of substances of
various nature, which is a new direction in the
creation of energy and resource-saving technologies.
In contrast to the traditional convective
mechanism of heat transfer, when using microwave
heating, electromagnetic radiation, being absorbed
by the substance, leads to the volumetric dissipation
of the energy of the electromagnetic field in the
substance and, as a consequence, to the volumetric
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heating of the latter, which significantly reduces the
heating time and gives a greater uniformity of
heating.
The dissipated energy depends both on the
parameters of the electromagnetic field (frequency,
amplitude of the electric and magnetic field strength
vectors) and on the electrophysical properties of the
substance (real and imaginary parts of the dielectric
and magnetic permeability, conductivity of the
medium).
The use of this heating method has a number of
significant advantages, in particular, the use of heat
carriers is excluded, which ensures cleanliness, there
is no need for separation, and the mass of the
process stream is reduced. Also, this type of heating
has low inertia, which facilitates process control and
increases safety. Therefore, the creation of reaction
devices, in which the role of the coolant is played by
electromagnetic radiation of the microwave range, is
a very promising task.
3 Problem Solution
The model of functioning of an electrodynamic
reactor (Fig. 3) is as follows [2]: when microwave is
absorbed by a catalyst substance, volumetric heat
sources appear in it, the power density of which
(W/m3) is determined by the expression:
EE
vx
K
F
P
xq
exp
1
)(
, (9)
where P is the power of electromagnetic radiation
absorbed by the catalyst substance, W; F is the
cross-sectional area of the reactor, m2; δE – "total"
depth of absorption of electromagnetic radiation, m;
ε is the porosity of the catalyst layer; K is a
dimensionless coefficient that depends on the
physical properties of the material and takes into
account the ability of the substance to absorb
electromagnetic energy; x – coordinate directed
along the reactor axis.
Fig. 3: Diagram of an electrodynamic reactor: 1 –
gas phase; 2 – solid phase; 3 – contact gas.
One of the important tasks in the design of a
reactor plant of this class is to find the temperature
distribution in the solid and gas phase when the
gaseous medium is blown through the catalyst bed
with a mass flow rate Q under microwave heating
conditions in a one-dimensional approximation, we
investigate two heat balance equations: the first
equation is the heat balance gas and solid phase, the
second is the heat balance of the solid phase for the
elementary volume of the reactor.
For gas and solid:
v
g
pg
s
s
g
g
sssggpg
q
x
T
Qc
x
T
x
T
x
TcTc
t
1
,
(10)
where cpg is the averaged isobaric heat capacity of
the gas phase, J/(kg•K); cs is the averaged heat
capacity of the solid phase, J/(kg•K); Tg – gas
temperature, Ts – solid phase temperature,
–
porosity,
g – effective thermal conductivity of gas,
W/(m•K);
s – effective thermal conductivity of a
solid, W/(m•K); Q is the mass velocity of the gas
phase, kg/(m2•s); qv – volumetric power of heat
sources, W/m3.
vgs
s
s
sss
qTT
x
T
x
Tc
t
1
,
(11)
where
is the effective volumetric heat transfer
coefficient, W/(m3•K).
Simplifying the last equations, we get:
v
g
pg
s
s
g
g
s
ss
g
gpg
q
x
T
Qc
x
T
x
T
t
T
c
t
T
c
2
2
2
2
1
, (12)
vgs
s
s
s
ss
qTT
x
T
t
T
c
2
2
1
. (13)
Subtracting (12) from (13), we obtain the
following relations connecting the temperatures of
the solid Ts and gas Tg phases:
gs
g
pg
g
g
g
gpg
TT x
T
Qc
x
T
t
T
c
2
2
. (14)
Thus, to find the temperature distribution of the
gas and the solid phase over the height of the
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electrodynamic reactor, it is necessary to solve the
system of equations (12), (14).
Expressions (12) and (13) are the solution to the
system of equations (12), (14).
The temperature of the gas phase in an
electrodynamic reactor can be determined from the
following expression:
x
g
pg
s
g
gg
pg
gg
pg
g
pg
gg
pg
g
pg
g
x
Qc
T
Qc
x
QcQc
C
x
QcQc
CT
0
2
2
2
2
1
2
exp
4
2
2
4
exp
2
4
exp
dx
Qc
gg
pg
2
4
sh
2
, (15)
where C1 and C2 are the integration constants
obtained from the boundary conditions of the
problem.
The temperature of the solid phase in an
electrodynamic reactor can be determined from the
following expression:
x
s
v
g
s
s
ss
s
q
T
xCxCT
0
21
)(
1
shch
dx
s
sh
, (16)
where C1 and C2 are the integration constants
obtained from the boundary conditions of the
problem.
Thus, the solution of the system of equations
(15) and (16) by the method of sequential
approximation makes it possible to find the
temperature distribution in the solid and gas phase
along the height of the microwave reactor during
purging of the chemically neutral gas phase.
Estimation of the coefficients of effective
thermal conductivity and heat transfer. For the
correct solution of the systems of equations
described above, it is necessary to know the
coefficients of effective thermal conductivity,
effective heat transfer, as well as the coefficient of
heat transfer to the side surface of the reactor.
To determine them, you can use the formulas
that summarize the experimental studies that were
carried out by a number of authors. These
dependences have a satisfactory accuracy in finding
the corresponding coefficients.
Effective thermal conductivity coefficient in a
fixed catalyst bed. In [2], the following equations
were experimentally established to find the effective
thermal conductivity of a fixed catalyst bed:
for spherical granules:
Qd
Dk
g
s
pg
s01915,0
15,11
112,0
, (17)
for cylindrical granules:
Qd
Dk
g
s
pg
s0348,0
97,11
112,0
, (18)
where
g
is the coefficient of thermal conductivity
of the gas, W/(m•K); Dp – reactor diameter, m;
s
–
thermal conductivity of the catalyst material,
W/(m•K); Q is the mass velocity of the gas phase,
kg/(m2•s);
– layer porosity,
– dynamic viscosity
coefficient, Pa•s; dk is the diameter of the catalyst
particles, m (in the case of cylindrical granules, dk =
φ dc, where dc is the cylinder diameter).
The sphericity coefficient φ is the ratio of the
surface of the equal-sized sphere Fs to the surface of
the catalyst granule Fg:
2
g
s
F
F
V
d
dF
, (19)
where dF, dV are the diameters of the balls, which
are equivalent to the catalyst granule in terms of
surface and volume.
For spherical granules φ = 1, for cylindrical
granules the values of φ are presented in the table 3.
Heat transfer coefficient from the particle
surface. As a generalization of the experimental data
in [2], an expression is given for determining the
volumetric coefficient of effective heat transfer in
a fixed bed:
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6,1Re
535,0
3,0
3
2
g
p
p
c
Qc
, (20)
where
1
Re QFk
is the Reynolds criterion; Fk
is the surface area of the particle, φ is the coefficient
of sphericity of the particle.
Table 3. Sphericity coefficients for cylindrical
catalyst granules
Body
Cylinder
h=d/2
h=d/6
h=d/20
h=d/30
φ
0,827
0,594
0,323
0,220
Body
Cylinder
h=d
h=1,5d
h=5d
h=10d
φ
0,438
0,860
0,691
0,580
Heat transfer coefficient to the wall. To find the
effective coefficient of heat transfer to the reactor
wall, you can use the dependence proposed in [2], as
follows:
365,0
6,3
Qd
Dk
p
g
. (21)
Thus, finding the effective heat transfer
coefficients, the effective heat capacity in a fixed
catalyst bed, the heat transfer coefficient from the
side surface of the reactor complete the problem of
finding the temperature distribution in the solid
(catalyst) and gas (reaction mixture) phases over the
height of the electrodynamic reactor.
Characteristic time of transition to stationary
state. When simulating physicochemical processes
in electrodynamic reactors according to the
proposed mathematical model, which is described
by the system of equations (12) - (15), only
stationary modes of operation were considered,
therefore, it becomes necessary to assess the time
domain of application of these stationary solutions
to non-stationary processes that can occur,
according to for various reasons in these reaction
devices.
Such an estimate can be made by a parameter
called the characteristic time of transition to a
stationary state, which allows, firstly, to
qualitatively evaluate the influence of various
parameters on the dynamic characteristics of such a
system, and secondly, to determine the time domain
of application of the stationary system of equations
(12) - (15).
Let us estimate the characteristic time of
establishment of a stationary state from the
temperature distribution in the gas phase, which is
decisive for finding the rate of a chemical reaction
and, accordingly, the degree of conversion, for
which we write the energy equation for the gas
phase in the one-dimensional approximation for an
elementary volume
gsef
g
pg
g
gef
g
gpg
TTa x
T
Gc
x
T
t
T
c
2
2
,
. (22)
Representing the last equation in a difference
form, assuming that (Ts-Tg)~Tg. For the
characteristic size, it is convenient to take the value
Е, which determines the depth of penetration of the
electromagnetic wave into the substance, and,
discarding the second-order quantities, we obtain
E
g
pggef
g
gpg
T
GcTa
T
c
, (23)
where – is the characteristic time of the onset of
the stationary regime.
Thus, simplifying the last equation, we obtain
E
pg
ef
gpg
Gc
a
c
. (24)
As can be seen from expression (24), an increase
in the heat capacity of the gas leads to a significant
increase in the time required to establish a stationary
regime. An increase in the gas density also leads to
an increase in the transition time to a stationary
state, also to an increase in the transition time to a
stationary state, and an increase in the porosity of
the solid phase. The mass velocity of the gas phase
also has a significant effect on the time of transition
to the stationary regime. As can be seen from the
obtained expression, with an effective heat transfer
coefficient ef~G0,7, with an increase in the mass
velocity, the denominator of the expression slightly
decreases, which, in general, leads to an increase in
the time of the onset of the stationary regime. An
increase in the penetration depth also leads to an
increase in the time of the onset of the stationary
regime. Thus, we have proposed (24), which makes
it possible to qualitatively assess the influence of
various parameters on the characteristic time of the
onset of a stationary state. For a more accurate
assessment, it is necessary to solve non-stationary
equations describing the physicochemical processes
occurring in electrodynamic reactors, which may be
a topic for further research in this area.
Numerical experiments were carried out
according to the proposed model, the results of
which are shown in the figures. Fig. 4 presents
graphs showing the effect of the mass flow rate of
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the gas flow on the final temperature distribution in
the solid and gas phases.
As can be seen from the graphs presented, at low
values of the mass velocity of the gas phase, higher
values of the steady-state temperature of the solid
phase are observed, which is caused by a decrease in
heat transfer from the heated catalyst to the blown
gas phase. On the other hand, it becomes necessary
to maintain optimal values of the mass rate to
increase the yield of the target products during the
course of the reaction. Thus, it is possible to
determine the optimal values of the mass velocity
when assessing the yield of the target product and
the maximum value of the thermal efficiency of the
electrodynamic reactor.
The influence of the porosity of the solid phase
layer (Fig. 5) has a noticeable effect on the
temperature distribution along the height of the
reactor, primarily due to a decrease in the mass of
the solid phase, while, with a constant penetration
depth of electromagnetic radiation, a significant
increase in the power of volumetric energy sources
occurs, thereby a more intense heating the catalyst
with greater porosity.
Fig. 4: Distribution of temperature in the solid and
gas phase along the height of the electrodynamic
reactor at different mass velocities of the gas flow.
Fig. 5: Distribution of temperatures in the solid and
gas phases along the height of the reactor at
different porosities of the solid phase.
An assessment of the effect of the penetration
depth of electromagnetic radiation into the catalyst
substance on the final temperature distributions in
the solid and gas phase is shown in Fig. 6. As can be
seen from the graphs, an increase in the penetration
depth δ leads to a decrease in the maximum
temperatures in the solid phase and a decrease in the
temperature gradient in the gas phase by at the
initial stage with a shift of the maximum of the
curves of the temperature profile of the solid and
gas phases to the lower part of the reactor and,
accordingly, a higher level of temperatures at the
outlet of the reactor.
Based on these data, it can be concluded that the
more efficient, from the standpoint of thermal
efficiency, is the catalyst, which, other things being
equal, has a smaller penetration depth of
electromagnetic radiation at the same level of
supplied energy, but it is necessary to take into
account the fact that with a decrease in depth
penetration, the temperature at the inlet to the
reactor rises, which can lead to overheating of the
upper layers of the catalyst, and, as a consequence,
to its destruction.
Fig. 6: Temperature distribution in the solid and gas
phases along the height of the reactor depending on
the depth of penetration of electromagnetic radiation
into the catalyst substance.
The influence of the gas thermal conductivity
coefficient on the temperature distribution is
illustrated by the graph shown in Fig. 7.
Fig. 7: Distribution of temperature in the solid and
gas phase along the height of the reactor depending
on the coefficient of thermal conductivity of the gas
phase.
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At lower values of the gas phase thermal
conductivity coefficient, with other parameters
unchanged, a slightly higher temperature level is
observed, especially at the initial stage, in the lower
part of the reactor the temperature distribution is
leveled. Based on this, it can be concluded that the
thermal conductivity of the gas phase does not have
a strong effect on the temperature distribution over
the height of the reactor.
4 Conclusion
By varying various technological, design parameters
and physical properties, both of the catalyst, which
is the main heat transformer in microwave
technology, and the physical properties of the
reaction mixture, it is possible to achieve optimal
conditions for carrying out chemical transformations
in a microwave field, both from the standpoint of
the thermodynamics of a electrodynamic reactor and
positions of the chemistry of specific
transformations.
The presented model makes it possible to
determine the optimal values of the parameters of
electrodynamic reactors at a given value of the yield
of the target products, to evaluate the influence of
various technological parameters of reaction devices
on the thermal efficiency of the processes taking
place in them, taking into account the supply of heat
to the reaction zone by means of microwave.
The electrodynamic reactor is fundamentally
different from the reactors currently operating in
industry in the way of supplying energy to the
reaction zone, which significantly reduces energy
consumption, simplifies the process control and
increases the efficiency of the installation.
References:
[1] Rakhmankulov D.L., Bikbulatov I.Kh., Shulaev
N.S., Shavshukova S.Yu., Microwave
Radiation and Intensification of Chemical
Processes, M.: Chemistry, 2003.
[2] Shulaeva E.A., Improvement of technological
production on the basis of modeling the
processes of chemical technologies:
monograph, Ufa: Publishing House "Oil and
Gas Business", 2018.
[3] Shulaeva E.A., Shulaev N.S., Calculation and
Modeling of the Temperature Conditions of
Electrodynamic Chemical Reactors, Chemical
and Petroleum Engineering, Vol.52, No.1,
2016, pp. 3-9.
[4] Janke E., Emde F., Lesh F., Special functions:
formulas, graphs, tables, M.: Science, 1977.
[5] Bikbulatov I.Kh., Daminev R.R., Shulaev N.S.,
Shulaev S.N., 1998, Pat. of the Russian
Federation No. 2116826, appl. 27.01.1997,
publ. 10.08.1998.
[6] Bikbulatov I.Kh., Daminev R.R., Shulaev N.S.,
Shulaeva E.A., Feoktistov L.R., Modeling the
Process of Dehydrogenation of Butenes in an
Electrodynamic Catalytic Reactor, Butlerov
Communications, Vol.24, No.1, 2011, pp. 99-
104.
[7] N. Golden Stepha, D. Kavin Jacob, Numerical
simulation for convective heat and mass
transfer effect of micropolar nanofluid flow
with Variable Viscosity and radiation, WSEAS
Transactions on Heat and Mass Transfer,
Volume 16, 2021, pp. 29-33.
[8] Yedilkhan Amirgaliyev, Murat Kunelbayev,
Aliya Kalizhanova, Beibut Amirgaliyev, Ainur
Kozbakova, Omirlan Auelbekov, Nazbek
Kataev, The study of thermal and convective
heat transfer in flat solar collectors, WSEAS
Transactions on Heat and Mass Transfer,
Volume 15, 2020, pp. 55-63.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Ekaterina A. Shulaeva developed a mathematical
model and performed numerical modeling of an
electrodynamic reactor.
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
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on POWER SYSTEMS
10.37394/232016.2022.17.4
Ekaterina A. Shulaeva
E-ISSN: 2224-350X
44
Volume 17, 2022
