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ПРОГРАММИРОВАНИЕ
MSC 65P99 DOI: 10.14529/mmp190407
NUMERICAL STUDY OF THE DYNAMICS OF AIR SEPARATION
PROCESS BY PRESSURE SWING ADSORPTION
E.I. Akulinin1, O.O. Golubyatnikov1, D.S. Dvoretsky1, S.I. Dvoretsky1
1Tambov State Technical University, Russian Federation
E-mails: akulinin-2006@yandex.ru, golubyatnikov_ol@mail.ru, dvoretsky@tambov.ru,
sdvoretsky@mail.tstu.ru
Using mathematical modelling and the finite element method, we carry out the
calculation experiments to study the system connections and regularities of pressure
swing adsorption process under the conditions of air separation and oxygen concentration
(production). We study the influence of mode and construction variables on the dynamics
and technological indicators of the effectiveness of this process. Namely, we study the
influence of input variables (composition and temperature of atmospheric air, air pressure at
the compressor outlet) on output variables (extraction degree, oxygen purity, unit capacity,
etc.) of the studied object in a wide range of control variables (adsorption-desorption cycle
time, pressure ratios at adsorption and desorption stages, and oxygen-enriched reverse air
flow coefficient). Also, we study the influence of construction parameters (layer height,
particle diameter and maximum adsorption volume of the adsorbent) on the amount of
adsorption, which is equilibrium with the current concentration of the adsorptive in the
gas mixture flow on the outer surface of the adsorbent granules, the value of the kinetic
adsorption coefficient (the coefficient of external mass transfer of the adsorptive (mainly
nitrogen) from the gas phase into the adsorbent). The results of calculation experiments
allow to establish the most promising mode and construction parameters for the optimal
design of oxygen enrichment systems by pressure swing adsorption with varying pressure.
Keywords: pressure swing adsorption; oxygen; zeolite 13X; mathematical modelling;
numerical study.
Dedicated to Professor T.B. Chistyakova
on the occasion of her anniversary
Introduction
In recent decades, the short-cycle non-heat adsorption method, Pressure Swing
Adsorption (PSA) with molecular sieves, has been widely used to separate gas mixtures and
concentrate gas products on a small scale [1–4]. The performance and quality of the product
gas produced in industrial gas separation units using the adsorption method are achieved
not only by increasing the volume of the adsorbent, but also by optimizing the time
length of the modes and stages that comprise the adsorption-desorption (regeneration)
cycle. To this end, there is need for careful preliminary work is needed, which is based on
mathematical modelling of adsorption processes with different duration of adsorption and
regeneration stages. The mathematical model should take into account the final rate of
the pressure profile relaxation along the adsorbent layer with frequent changes in the cycle
Вестник ЮУрГУ. Серия ≪Математическое моделирование
и программирование≫(Вестник ЮУрГУ ММП). 2019. Т. 12, № 4. С. 95–103 95
E.I. Akulinin, O.O. Golubyatnikov, D.S. Dvoretsky, S.I. Dvoretsky
stages, as well as the effect of continuous changes in the gas mixture filtration conditions
on the dynamics of the adsorption gas separation [5, 6].
Primarily, if the air is enriched with oxygen, then there is adsorption of nitrogen,
which is a more sorbable component, and oxygen, which is a less sorbable component of
air (on the surface and in the micropores of the granules) during PSA cycle including
adsorption and desorption stages. The intensity of the adsorptive mass transfer from the
gas mixture to the adsorbent and back (during desorption of the adsorptive) is determined
by the equilibrium values of component concentrations in the phases and by kinetics (the
adsorptive mass transfer rate from the gas phase to the surface of the granules and into
the adsorbent micropore volume) and back (during desorption). During the adsorption by
N2zeolite adsorbents mainly, the following mass and heat exchange processes take place in
the adsorbers of the PSA unit: 1) diffusion of O2, N2in the gas-air mixture flow; 2) O2, N2
mass transfer and heat exchange between the gas phase and the adsorbent; 3) adsorption of
N2, O2on the surface and in the micropores of zeolite adsorbent granules with the release
of heat, and 4) desorption of N2, O2from the micropores and the surface of the granules
with the heat adsorption [1, 4, 7–9]. The purpose of this work is to use the mathematical
modelling method to study the dynamics, system connections and regularities of the air
separation and oxygen production process in order to increase the efficiency of PSA units
when enriching air with oxygen.
Table 1
Table of notation
a(a∗) component concentration (equilibrium) in the adsorbent, mol/m3
dgr (Ssp) diameter of granules, mm (specific surface, m2/m3)
L(DA) adsorbent layer length (diameter), m
Ggas-air mixture consumption, l/min
kppressure ratio coefficient, kp=Pin
ads/P in
des
P(T) gas mixture pressure, 105Pa (temperature, K)
Qunit capacity, l/min
tads,tdes duration of adsorption and desorption stages, respectively, s
tcadsorption-desorption cycle time, s, tc=tads +tdes
W0limiting adsorption volume, cm3/g
ycomponent concentration in the gas-air mixture, vol.%
β(θ) external mass transfer coefficient, m/s (return flow ratio)
ηrecovery, %
Indices
1, 2, 3 oxygen, nitrogen, argon and impurities
a (g) adsorbent (gas phase)
ads (des) adsorption (desorption)
c adsorption-desorption cycle
in (out) input (output)
min (max) minimum (maximum)
set set value
96 Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming
& Computer Software (Bulletin SUSU MMCS), 2019, vol. 12, no. 4, pp. 95–103
ПРОГРАММИРОВАНИЕ
1. Mathematical Description of the Dynamics of Oxygen
Enrichment Cyclic Adsorption Processes
The technological process of oxygen production by the method of adsorption air
separation with cyclically varying pressure is carried out in a 2-adsorber PSA unit with
a granular zeolite adsorbent 13X [8] designed to produce oxygen with a concentration
of 90...95 vol% of atmospheric air containing oxygen in the amount of 20,8±0,5 vol%,
nitrogen – 78,2 vol% and impurities (argon, carbon dioxide, etc.) – 1±0,5 vol% [7, 8].
Our work [7] demonstrates the mathematical description of the dynamics of the
adsorption air separation and oxygen concentration processes, as well as the algorithm for
solving mathematical model equations, which are a nonlinear partial differential equation
of parabolic type describing: 1) the dynamics of the component-wise material balance
in the gas-air flow taking into account the longitudinal mixing of the gas-air mixture
components in the adsorbent layer; 2) the kinetics of adsorption and desorption processes;
3) the heat distribution in the gas and solid phases taking into account the convective
component and heat conductivity; 4) the dynamics of the flow rate along the height of the
adsorbent; 5) the pressure dynamics of the gas mixture along the length of the adsorbent
(Ergun equation). Herewith, the equilibrium concentration of the sorbed component in
the solid phase was calculated using the Dubinin–Radushkevich equation [9], and the
mass transfer coefficient was calculated using the criterial equation for the case of gas flow
around the layer of spherical particles [10]. To solve the system with the corresponding
initial and boundary conditions, we use the finite element method in the Matlab software
environment.
2. Numerical Study of Oxygen Enrichment Adsorption Process
in Dynamics
Table 2 presents the ranges of variation and the nominal values of the mode
and construction parameters of the PSA unit while studying the dynamics of oxygen
enrichment process. Table 2
The initial data for calculation experiments
Variables Nominal
values
Range of
changes
Variables Nominal
values
Range of
changes
tc, s 8 1–250 L, m 0,3 0,2–1
yin
2, vol% 78,2 – DA, m 0,0334 –
θ1,6 0–3 W0, cm3/g 0,17 0,05–0.5
Pin
ads, 105Pa 3 2–6 B, 10−61/K26,55 –
Pin
des, 105Pa 1 – dgr , mm 2 0,25–8
Gin, l/min 10 – yout
1,set, vol% 90 –
Tin
g, K 293 233–313 Qset, l/min 0,5 –
Fig. 1 shows dependencies of the influence of changes in input variables (concentration
of impurities and atmospheric air temperature ) on output variables (recovery ηand oxygen
concentration yout
1,set ≥90 vol. %) of the PSA unit in a wide range of mode variables (cycle
time tc, pressure ratio at adsorption and desorption stages kpand the coefficient θof
Вестник ЮУрГУ. Серия ≪Математическое моделирование
и программирование≫(Вестник ЮУрГУ ММП). 2019. Т. 12, № 4. С. 95–103 97
E.I. Akulinin, O.O. Golubyatnikov, D.S. Dvoretsky, S.I. Dvoretsky
Fig. 1. Dependences of oxygen recovery ηon: 1 – concentration of impurities in the
initial gas-air mixture yin
3; 2 – atmospheric air temperature Tin
g; 3 – limiting adsorption
volume of the adsorbent W0; 4 – diameter of adsorbent granules dgr
oxygen-enriched reverse air flow) and the adsorbent characteristics (limiting adsorption
volume W0and diameter of the adsorbent particles dgr ). The values yout
3,Tin
g,W0,dgr are
normalized on a scale from 0 to 1.
The analysis of dependencies presented in Fig. 1 shows that increase in the
concentration of weakly sorbed impurities in atmospheric air from 0,5 to 2 vol% and,
consequently, decrease in oxygen concentration from 21,3 to 19,8 vol% lead to decrease
in the extraction degree on average by ∼2 times (from 27,3 to 13,9%, Fig. 1, curve 1)
due to reducing the equilibrium concentration of the adsorptive a∗
2in the adsorbent. It
should be noted that the temperature of the gas-air mixture Tin
g,the limiting adsorption
volume W0and the adsorbent particle diameter dgr significantly affect the efficiency of
the mass transfer process. Therefore, the decrease in the mixture temperature from 313 to
233 K leads to increase in the extraction rate from 16,6 to 99,9% (Fig. 1, curve 2) due to
increase in the equilibrium concentration a∗
2. The increase in W0by an order of magnitude
(from 0,05 to 0,5 cm3/g, Fig. 1, curve 3) leads to increase in the extraction rate also by an
order of magnitude (from 5,2 to 59,7%), which is explained by increase in the equilibrium
concentration of the adsorptive a∗
2in the adsorbent, while decrease in dgr from 8 to 0,25
mm leads to increase in the extraction degree by ∼8 times (from 5,2 to 39,2%, Fig. 1,
curve 4) due to increase in the external surface of particles Ssp and the intensity of the
mass transfer process from gas to solid phase and vice versa.
Fig. 2 presents the study of the influence of atmospheric air characteristics (yin ,Tin
g)
and the adsorbent (W0,dgr ) on the values of mode variables –kp,θ,tc, while ensuring that
specified requirements for oxygen purity are ≥90 vol% and the performance of the PSA
unit (Qset ≥0,5 l/min).
The analysis of the graphs in Fig. 2 (left) shows that with increase in the initial gas-air
mixture (at the same time the oxygen content yin
1decreases), it is necessary to increase the
equilibrium concentration a∗by increase in kpand improving the conditions for adsorbent
regeneration at the desorption stage (increasing the reverse flow rate coefficient θ), and also
reducing the adsorption-desorption cycle time tcdue to more intensive processing of the
adsorbent layer. The analysis of dependencies in Fig. 2 (right) demonstrates that increasing
the temperature Tin
gto ensure specified values of oxygen concentration yout
1,set ≥90 vol% and
the unit capacity Qset ≥0,5 l/min, it is necessary to increase the values of the pressure
ratio coefficient kpand the reverse flow coefficient θwith the simultaneous decrease in the
adsorption-desorption cycle time tc.
98 Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming
& Computer Software (Bulletin SUSU MMCS), 2019, vol. 12, no. 4, pp. 95–103
ПРОГРАММИРОВАНИЕ
Fig. 2. Dependences of the pressure ratio kp(1), reverse flow θ(2), and cycle time tc(3)
coefficients on: concentration of impurities in the initial gas-air mixture yin
3(left),
atmospheric air temperature Tin
g(right)
Fig. 3. Dependences of the pressure ratio kp(1), reverse flow θ(2) and cycle time tc(3)
coefficients on: limiting adsorption volume of the adsorbent W0(left), diameter of the
adsorbent granules dgr (right)
The analysis of the graphs in Fig. 3 (left) shows that when W0increases from 0,1 to 0,5
cm3/g to ensure the specified requirements (yout
1,set ≥90 vol%, Qset ≥0,5 l/min), the value
of the pressure ratio coefficient kpby ∼2.5%, and the value of θwhich characterizes the
fraction of the flow directed to nitrogen desorption from the adsorbent should be reduced
by 2,2%. The latter leads to decrease in the energy consumption of the PSA unit due to
the use of a less efficient compressor. At the same time, the adsorption-desorption cycle
time can be increased by an average of 3 times, which contributes to decrease in the
switching frequency of the PSA unit valves and increase in their service life. Since the
oxygen enrichment process takes place in the external diffusion area, the diameter of the
adsorbent particles dgr increases (Fig. 3 (right)), their specific surface area Ssp decreases
and, consequently, there is decrease in adsorbent mass transfer rate from the gas phase to
the adsorbent and back. Therefore, if the particle diameter of the 13X adsorbent increases
from 2 mm to 4 mm for ensuring the specified values of oxygen concentration and the unit
capacity, then it is necessary to increase kpand θby ∼18%, and 7%, respectively, and the
adsorption-desorption cycle time tcshould be reduced by ∼40%. It is most preferable to
use the adsorbent with the particle diameter less than 0,5 mm (Fig. 3 (right)), however,
this is associated with increase in the adsorbent layer resistance, energy consumption, and
deterioration of equilibrium conditions.
Вестник ЮУрГУ. Серия ≪Математическое моделирование
и программирование≫(Вестник ЮУрГУ ММП). 2019. Т. 12, № 4. С. 95–103 99
E.I. Akulinin, O.O. Golubyatnikov, D.S. Dvoretsky, S.I. Dvoretsky
The analysis of the graphs given in Figs. 2, 3 shows that the temperature of the
initial gas-air mixture Tin
gand the diameter of the adsorbent particles dgr have the most
significant effect on the values of mode parameters, the ratio of adsorption and desorption
pressures kp, the return flow rate coefficient θ, and the cycle time tcensuring the fulfillment
of specified oxygen purity requirements and the PSA unit capacity. Therefore, the use of
additional cooling devices and the adsorbent with a smaller diameter of the adsorbent
granules while reducing the length Lof the adsorbent layer will not only reduce the
dimensions of PSA units, but also increase the degree of oxygen extraction.
Fig. 4 presents the analysis of the influence of changes in the construction parameters of
the PSA unit with oxygen enrichment (the length of the adsorbent layer Land the particle
diameter of the adsorbent dgr ) on the adsorption amount a∗which is equilibrium with the
current concentration of the adsorptive in the gas mixture flow on the outer surface of
the adsorbent granules, the kinetic adsorption coefficient β(the adsorptive external mass
transfer coefficient (nitrogen) from the gas phase to the adsorbent).
Fig. 4. Dependences of equilibrium nitrogen adsorption in the adsorbent (1) and the
nitrogen mass transfer coefficient (2) on the diameter of the granules (left) and the
length of the adsorbent bulk layer (right)
The analysis of the graphs given in Fig. 4 shows that with increase in dgr (Fig. 4 (left))
and L(Fig. 4 (right)), the following pattern is observed: the equilibrium concentration
value a∗
2(Fig. 4 (left), dependence 1) increases (decreases, Fig. 4 (right), dependence 1),
and the value β(Fig. 4 (left), dependence 2) decreases (increases, Fig. 4 (right), dependence
2). In the first case, by decrease in the aerodynamic resistance in the adsorbent layer ∆P
according to the Ergun equation and, accordingly, by increase in the current value of the
gas-air mixture pressure along the length Lof the adsorbent layer. In the second case (with
increase in the height of the adsorbent layer in Fig. 4 (right)), the effect can be explained
by increase in the aerodynamic resistance of the adsorbent layer, decrease in the current
value of the gas-air mixture pressure along the length Lof the layer, and increase in the
specific surface of the adsorbent. The analysis of the graphs given in Fig. 4 also shows
that change in the length of the adsorbent layer leads to increase in equilibrium nitrogen
adsorption and nitrogen mass transfer coefficient by 6 times, which is more significant
compared with decrease in the particle diameter of the adsorbent dgr . In general, the
change of Lor dgr affects the value of the equilibrium concentration a∗
2to a greater extent
and the value of the mass transfer coefficient βto a lesser extent (Fig. 4).
100 Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming
& Computer Software (Bulletin SUSU MMCS), 2019, vol. 12, no. 4, pp. 95–103
ПРОГРАММИРОВАНИЕ
Conclusions
We create prerequisites to formulate and study the problem on optimization (in the
sense of determining the maximum degree of oxygen extraction) of the oxygen enrichment
process in the PSA unit, taking into account the fulfillment of oxygen purity requirements,
the unit capacity, and resource saving of the granular adsorbent. New scientific results
obtained during this work can be used in the development of mathematical and algorithmic
support for designing new automated processes and adsorption process units with cyclically
varying pressure for separating and purifying multicomponent gas mixtures.
Acknowledgements. The work was performed in the project part of the State assignment
no. 10.3533.2017/PCh.
References
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Moscow, KolosS, 2009.
2. Shokroo E., Farsani D., Meymandi H., Yadoliahi N. Comparative Study of Zeolite 5A and
Zeolite 13X in Air Separation by Pressure Swing Adsorption. Korean Journal of Chemical
Engineering, 2016, vol. 33, no. 4, pp. 1391–1401.
3. Wu C., Vermula R., Kothare M., Sircar S. Experimental Study of a Novel Rapid Pressure-
Swing Adsorption Based Medical Oxygen Concentrator: Effect of the Adsorbent Selectivity
of N2 over O2. Industrial and Engineering Chemistry Research, 2016, vol. 55, no. 16,
pp. 4676–4681.
4. Li J.H. The Experimental Study of a New Pressure Equalization Step in the Pressure Swing
Adsorption Cycle of a Portable Oxygen Concentrator. Bio-Medical Materials and Engineering,
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Projects in the Field of Production of Polymeric Sheet Materials. Proceedings of the 2016
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(Science. Education. Innovations)”, St. Petersburg, Saint Petersburg Electrotechnical
University LETI, 2016, pp. 61–64.
6. Chistyakova T.B., Polosin A.N. Computer Modelling System of Industrial Extruders with
Adjustable Configuration for Polymeric Film Quality Control. Proceedings of 2017 IEEE
II International Conference on Control in Technical Systems (CTS), St. Petersburg, Saint
Petersburg Electrotechnical University LETI, 2017, pp. 47–50.
7. Akulinin E.I., Golubyatnikov O.O., Dvoretsky D.S., Dvoretsky S.I. Numerical Study of Cyclic
Adsorption Processes of Air Oxygen Enrichment in Dynamics. Journal of Physics: Conference
Series, 2019, vol. 1278, no. 1, p. 012005.
8. Ruthven D.M., Farooq S., Knaebel K.S. Pressure Swing Adsorption. N.Y., 1993.
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10. Lykov A.V. Teplomassoobmen [Heat and Mass Transfer]. Moscow, Energiya, 1978. (in
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Вестник ЮУрГУ. Серия ≪Математическое моделирование
и программирование≫(Вестник ЮУрГУ ММП). 2019. Т. 12, № 4. С. 95–103 101
E.I. Akulinin, O.O. Golubyatnikov, D.S. Dvoretsky, S.I. Dvoretsky
УДК 661.935+519.633.2 DOI: 10.14529/mmp190407
ЧИСЛЕННОЕ ИССЛЕДОВАНИЕ ДИНАМИКИ ПРОЦЕССА
РАЗДЕЛЕНИЯ ВОЗДУХА МЕТОДОМ КОРОТКОЦИКЛОВОЙ
БЕЗНАГРЕВНОЙ АДСОРБЦИИ
Е.И. Акулинин1, О.О. Голубятников1, Д.С. Дворецкий1, С.И. Дворецкий1
1Тамбовский государственный технический университет, г. Тамбов,
Российская Федерация
С использованием математического моделирования и метода конечных элемен-
тов проведены вычислительные эксперименты по исследованию системных связей и
закономерностей процесса короткоцикловой безнагревной адсорбции при разделении
воздуха и концентрирования (получения) кислорода. Проведено изучение влияния ре-
жимных и конструктивных переменных на динамику и технологические показатели
эффективности данного процесса, а именно: 1) входных переменных (состава и темпе-
ратуры атмосферного воздуха, давления воздуха на выходе компрессора) на выходные
переменные (степень извлечения, чистоту кислорода, производительность установки и
т.п.) объекта исследования в широком диапазоне варьирования управляющих пере-
менных (длительности цикла адсорбция-десорбция, отношения давлений на стадиях
адсорбции и десорбции и коэффициента обратного потока воздуха, обогащенного кис-
лородом); 2) конструктивных параметров (высоты слоя, диаметра частиц и предельно-
го адсорбционного объема адсорбента) на величину адсорбции, равновесной текущей
концентрации адсорбтива в потоке газовой смеси на внешней поверхности гранул ад-
сорбента, значение кинетического коэффициента адсорбции (коэффициента внешней
массоотдачи адсорбтива (преимущественно азота) из газовой фазы в адсорбент). В
ходе анализа результатов вычислительных экспериментов установлены наиболее пер-
спективные режимные и конструктивные параметры для оптимального проектирова-
ния установок обогащения воздуха кислородом методом короткоцикловой адсорбции
с изменяющимся давлением.
Ключевые слова: короткоцикловая безнагревная адсорбция; кислород; цеолит
13X; математическое моделирование; вычислительный эксперимент.
Литература
1. Шумяцкий, Ю.И. Промышленные адсорбционные процессы / Ю.И. Шумяцкий. – М.:
КолосС, 2009.
2. Shokroo, E. Comparative Study of Zeolite 5A and Zeolite 13X in Air Separation by Pressure
Swing Adsorption / E. Shokroo, D. Farsani, H. Meymandi, N. Yadoliahi // Korean Journal
of Chemical Engineering. – 2016. – V. 33, № 4. – P. 1391–1401.
3. Wu, C. Experimental Study of a Novel Rapid Pressure-Swing Adsorption Based Medical
Oxygen Concentrator: Effect of the Adsorbent Selectivity of N2 over O2 / C. Wu, R. Vermula,
M. Kothare, S. Sircar // Industrial and Engineering Chemistry Research. – 2016. – V. 55,
№ 16. – P. 4676–4681.
4. Li, J.H. The Experimental Study of a New Pressure Equalization Step in the Pressure Swing
Adsorption Cycle of a Portable Oxygen Concentrator / J.H. Li // Bio-Medical Materials and
Engineering. – 2014. – V. 24, № 5. – P. 1771–1779.
5. Chistyakova, T.B. Joint Innovative IT Projects in the Field of Production of Polymeric
Sheet Materials / T.B. Chistyakova, A.S. Razygrayev, A.N. Polosin, A.M. Araztaganova
// Proceedings of the 2016 IEEE V Forum ≪Strategic Partnership of Universities and
Enterprises of Hi-Tech Branches (Science. Education. Innovations)≫. – St. Petersburg: Saint
Petersburg Electrotechnical University LETI, 2016. – P. 61–64.
102 Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming
& Computer Software (Bulletin SUSU MMCS), 2019, vol. 12, no. 4, pp. 95–103
ПРОГРАММИРОВАНИЕ
6. Chistyakova, T.B. Computer Modelling System of Industrial Extruders with Adjustable
Configuration for Polymeric Film Quality Control / T.B. Chistyakova, A.N. Polosin
// Proceedings of 2017 IEEE II International Conference on Control in Technical Systems
(CTS). – St. Petersburg: Saint Petersburg Electrotechnical University LETI, 2017. – P. 47–50.
7. Akulinin, E.I. Numerical Study of Cyclic Adsorption Processes of Air Oxygen Enrichment in
Dynamics / E.I. Akulinin, O.O. Golubyatnikov, D.S. Dvoretsky, S.I. Dvoretsky // Journal
of Physics: Conference Series. – 2019. – V. 1278, № 1. – P. 012005.
8. Ruthven, D. M. Pressure Swing Adsorption / D.M. Ruthven, S. Farooq, K.S. Knaebel. – New
York, 1993.
9. Дубинин, М.М. Адсорбция и пористость / М.М. Дубинин. – М.: ВАХЗ, 1972.
10. Лыков, А.В. Тепломассообмен / А.В. Лыков. – М.: Энергия, 1978.
Евгений Игоревич Акулинин, кандидат технических наук, доцент кафедры ≪Тех-
нология и оборудование пищевых и химических производств≫, Тамбовский госу-
дарственный технический университет (г. Тамбов, Российская Федерация), akulinin-
2006@yandex.ru.
Олег Олегович Голубятников, кандидат технических наук, старший преподава-
тель кафедры ≪Технология и оборудование пищевых и химических производств≫,
Тамбовский государственный технический университет (г. Тамбов, Российская Феде-
рация), golubyatnikov_ol@mail.ru.
Дмитрий Станиславович Дворецкий, доктор технических наук, профессор, зав.
кафедрой ≪Технология и оборудование пищевых и химических производств≫, Тамбов-
ский государственный технический университет (г. Тамбов, Российская Федерация),
dvoretsky@tambov.ru.
Станислав Иванович Дворецкий, доктор технических наук, профессор, профессор
кафедры ≪Технология и оборудование пищевых и химических производств≫, Тамбов-
ский государственный технический университет (г. Тамбов, Российская Федерация),
sdvoretsky@mail.tstu.ru.
Поступила в редакцию 27 ноября 2018 г.
Вестник ЮУрГУ. Серия ≪Математическое моделирование
и программирование≫(Вестник ЮУрГУ ММП). 2019. Т. 12, № 4. С. 95–103 103
