DOI: 10.17277/amt.2019.01.pp.055-072

The Study of Cyclic Adsorption Air Separation and Oxygen Concentration Processes

V.G. Matveykin1, E.I. Akulinin2*, N.V. Posternak1, S.A. Skvortsov2, S.I. Dvoretsky2

1 OJSC "Corporation "Roskhimzashchita", 19, Morshanskoye shosse, Tambov, 392000, Russia 2 Tambov State Technical University, 106, Sovetskaya St., Tambov, 392000, Russia

* Corresponding author: Tel .: +7 (909) 231 40 61. E-mail: akulinin-2006@yandex.ru

Abstract

The paper presents the dynamics of the developed mathematical model of pressure swing adsorption (PSA) process, which is carried out in a dual-adsorber unit with a 13X zeolite adsorbent used for air separation with the aim of oxygen concentration. The authors formulate and solve the regularized problem of identifying the kinetic parameters for the mathematical model - the mass and oxygen transfer coefficients for oxygen and nitrogen. Numerical studies of the effect of raw materials load (air composition, environment temperature and pressure) and control variables ("adsorption -desorption" cycle time, pressure at the compressor outlet, laws of changing the opening degree of the inlet and discharge valves of the PSA unit) on the dynamics and performance indicators of cyclic adsorption process of air oxygen enrichment have been carried out. The mathematical and algorithmic support for the creation of automated processes and PSA units for gas mixtures separation and purification has been developed.

Keywords

Pressure swing adsorption, zeolite adsorbent, adsorption isotherm, kinetics, mathematical model, parametric identification, calculation experiment.

© V.G. Matveykin, E.I. Akulinin, N.V. Posternak, S.A. Skvortsov, S.I. Dvoretsky, 2019

Introduction

In recent decades, the use of cyclic adsorption processes for separating gas mixtures and concentrating target products has become increasingly common. Short-cycle processes for adsorptive separation of gas mixtures are widely used in industry for air oxygen enrichment, drying gases without heating, separating hydrocarbons, concentrating carbon dioxide, extracting hydrogen, methane, etc. One of the urgent tasks in the field of adsorption separation is air oxygen enrichment. Typical substances that accompany oxygen are nitrogen, argon, carbon dioxide. A feature of the adsorptive oxygen concentration is the fact that in gas-air mixtures the components associated with oxygen have higher values of adsorption selectivity [1].

The analysis of numerous works by foreign and Russian scientists in the field of adsorption separation of multicomponent gas mixtures and concentration of the target product (hydrogen, oxygen, carbon dioxide,

etc.) made it possible to determine the place of this article among other works, its relevance and perspectivity [2-11].

Thus, the works [2-8] present the results of numerical studies of the effect of mode variables (pressure, temperature, flow rate of the initial mixture) on the dynamics and efficiency of the adsorption separation of two (H2-CO2), three (H2-CO2-CO), four (H2-CO2-CO-CH4), five (H2-CO2-CO-CH4-N2) and six (H2-CO2-CO-H2O-Ar-N2) component mixtures and hydrogen concentration using active carbon and metal-organic compounds as adsorbents. In the works [2, 3], when calculating the equilibrium conditions of a multicomponent mixture, sorption isotherms of individual substances are used. The calculation experiment allowed to study the features of the ten-adsorber unit with vacuum regeneration (in the English literature - VPSA) and the four-adsorber PSA unit with a metal-organic adsorbent of a new type. The possibility of obtaining hydrogen with the purity

of 99.981 vol. % at the extraction degree of 81.6 % [2] and 99.9 vol. % at the extraction degree of 48.05 % [4], respectively, was established. The results of numerical studies of the effect of the number of pressure equalization stages, their sequence and the use of combinations of different adsorbent layers on the purity and degree of hydrogen extraction from a two-component mixture (H2-CH4) in the PSA unit using the Langmuir - Freundlis equilibrium isotherm are presented in [6, 7]. It has been established that the use of a six-sorbent unit with two pressure equalization operations provides the best combination of the hydrogen purity (~ 99 vol. %) while achieving the extraction degree of ~ 83 %.

In [9], the calculation experiment investigated the dependences of the purity of extracted carbon dioxide from a nine-component gas mixture using active carbon and found that using the seven-adsorber PSA unit (instead of three or four adsorbers) allows increasing the purity of the produced carbon dioxide from 95.1 vol. % up to 98.9 vol. % while reducing the extraction degree from 90.2 % to 86.1 %. In [10, 11], the mathematical models of the dynamics of the cyclic adsorption process for producing CO2 from a two-component gas mixture (CO2-N2) on zeolite 13X using the Langmuir isotherm were studied, and the problem of the optimal design of PSA units (vacuum-pressure VPSA and fractional vacuum-pressure FVPSA types) by the complex criterion - the ratio of energy consumption of the PSA unit to the purity of the produced carbon dioxide, was formulated and investigated. It has been established that the units implemented according to the FVPSA and VPSA schemes provide the production of carbon dioxide with the purity of ~ 90 vol. % and ~ 72 vol. %, respectively, and the specific power of the unit according to the FVPSA scheme is on average 2.5 times higher.

Over the last decade, the number and range of consumers of air separation products have significantly increased, and the annual increase in oxygen demand is on average ~ 4-5 % due to the increased demand in the steel and chemical industry, aluminum production, aviation and other industries and social spheres.

A significant proportion of oxygen consumers uses in their activities not so much pure oxygen as air enriched with oxygen from 30 to 90 vol. %. For these reasons, in recent years, the adsorption method of separating air is becoming more common as the most profitable method for consumers who use oxygen and nitrogen unevenly in time.

The units separating the air mixture by adsorption using the PSA method differ in the way of creating the

driving force (the difference in equilibrium concentrations at adsorption and desorption stages) and use synthetic zeolites and activated carbons as adsorbents. Pressure-type units operate from an overpressure source, and production gas can be directly discharged to the consumer. The given costs of electricity for oxygen production with the concentration of 90 vol. % in PSA units range from 1.5 to 1.8 kWh/m , which is several times higher than the costs of obtaining oxygen by the method of low-temperature rectification. Therefore, pressure-type units are distinguished by low productivity and are used in industries where the problem of oxygen delivery and storage is acute. The main advantages of PSA units are their autonomy, mobility, reliability, and quick access to the stationary periodic mode. Energy costs in units, where oxygen is obtained at almost atmospheric pressure, and vacuuming is used for nitrogen de sorption, are significantly lower and amount to ~ 0.5-0.7 kWh/m .

The highest values of the oxygen extraction degree and productivity are achieved in Vacuum PSA units, in which the adsorption stage is carried out at an overpressure and the desorption stage - under vacuum. Increasing the level of the unit automation for separating components of the air mixture and concentrating oxygen is associated both with the difficulties of mathematical modeling and optimization of mass and heat transfer processes within the adsorber, and with the complexity of considering the mutual connections of all included devices. As a rule, the flow chart of the PSA process includes two - four apparatus-adsorbers filled with granular adsorbent, flow boosters (air compressor, vacuum pump, etc.), receivers, and valves designed to increase and decrease the pressure in adsorbers (desorbers) and air flow control [12-19].

Impurities of water and carbon dioxide contained in the separated air are traced in the frontal layers of the adsorbent and have practically no impact on the efficiency of nitrogen adsorption. The limiting purity of oxygen produced in adsorption units is 95.7 % (4.3 % is accounted for by argon, which is adsorbed on zeolites as well as oxygen). In industry, an oxygen-argon mixture is produced in adsorption units with the purity of 90-95 % [20].

The aim of this work is to study the effectiveness of cyclic adsorption processes for air separation and oxygen concentration, mathematical and algorithmic support for creating automated PSA units for air oxygen enrichment.

The current state analysis of the PSA technology and equipment

The analysis of the current state of the PSA technology for purifying and separating gas mixtures allowed to identify a generalized flow chart of the PSA process [21-31] (Fig. 1).

The PSA process of a gas mixture is implemented in the environment with the following parameters: the air composition (vector yenv of oxygen, nitrogen, argon and other impurities concentrations), temperature Tenv and barometric pressure Benv of the environment [32]. The pressure in the system is created by the flow rate boosters FB (compressor, blower, vacuum pump, etc.). The initial gas mixture with concentration, flow rate,

temperature and pressure yin, Gin, Tin,Pin, respectively enters the unit inlet. Through the inlet valves K1,i (i = 1, n), the gas mixture or atmospheric air enters the adsorbers A1,i (i = 1, n), where the process of selective adsorption of one or several gas mixture components is carried out. At the unit outlet, using check valves K3,i, a stream of concentrated production gas mixture is formed with concentration, flow rate, temperature and pressure y out, Gout, Tout, P out,

respectively. Part of the production flow through the respective heat exchanger Tk and the throttle Thk is sent to the adsorbers (valves K2,i are open) to carry out the process of the adsorbate desorption. The desorbed gas mixture is discharged by the flow booster FB out,1

with the composition yout,\ flow rate Go

temperature Tout,t and pressure Pout,\ respectively, into the atmosphere.

When implementing the adsorption schemes for air separation and purification, the following process organization schemes can be used: pressure (PSA - the adsorption pressure is excessive relative to atmospheric, while the desorption pressure is atmospheric), vacuum - pressure (VPSA - adsorption pressure is excessive relative to atmospheric and the desorption pressure is below atmospheric), vacuum pressure (VSA - the adsorption pressure is atmospheric, while the desorption pressure is below atmospheric) [33-35].

The main advantage of PSA units is the simplicity of their organization, and the disadvantage is the low extraction degree of the target product compared to other classes of units [1]. The main advantage of VPSA units is high efficiency in extracting target components, and the disadvantage is the complexity of instrumentation. VSA units reach a compromise between the efficiency and complexity of instrumentation, which led to their wide distribution in portable gas concentrators [36].

The adsorbers used in adsorption units can have different constructive designs that affect the structure of the flows in the adsorption layer (Fig. 2).

At the axial direction (Fig. 2a), the gas flow moves along the axis of the adsorber. The main advantage of this type of adsorbers is the simplicity of the design, and the disadvantages are the high aerodynamic resistance of the layer. At the radial direction (Fig. 2b), the flow is directed to the central cavity and moves through the adsorption layer to the periphery. This provides low aerodynamic resistance, the ability to provide high flow rates through the adsorber, the disadvantages are: the complexity of the design, the possibility of the stream leakage due to the relatively small size of the adsorbent layer.

->out,1

Fig.1. Generalized flow chart of the PSA process

Fig. 2. The direction of the gas flow in the adsorber:

a - axial; b -radial; c - variable

The advantage of the adsorber with a conical insert (Fig. 2c) is the ability to obtain a variable cross-section, providing a uniform flow rate over the entire height of the adsorber.

To increase the efficiency of the gas mixture adsorption separation process, a multilayer structure of adsorbents in the adsorber can be applied, where each layer is focused on the selective absorption of certain components of the gas mixture. An example can be the use in the frontal layer of adsorbents with high activity on water vapor, which protects the subsequent layers of the adsorbent from loss of sorption activity on the target components of the gas mixture.

The technological scheme of the PSA process (Fig. 1) can have from one to several adsorbers. The increase in the number of adsorbers allows the increase in the extraction degree of the target component, but at the same time capital costs get higher, the complexity of the control system increases, and the reliability of the unit decreases [37]. By performance, PSA units are distinguished by low productivity - up to 2 Nm /h; average productivity -2-20 Nm /h and high productivity - more than 20 Nm3/h.

Activated carbons, zeolites, silica gels, and active alumina are widely used as adsorbents in cyclic adsorption processes [38, 39].

In the adsorption technique, zeolites of types A, X, M are used with a low value of the silica module, which determines the structure of the crystal lattice of the zeolite and its adsorption properties. Silica gels are mainly used for drying gases, purifying mineral oils and as a carrier of catalysts. Activated (active) carbon has a very large specific surface per unit mass, which accounts for its high adsorption properties with respect to the sorption of high-molecular compounds.

The current state analysis of mathematical modeling of cyclic adsorption processes

The current state analysis of mathematical modeling of cyclic adsorption processes has shown that, to date, mathematical models constructed by an experimental-analytical method [40-55] are the most widely used.

The analysis of works in the field of mathematical modeling of cyclic adsorption separation of gas mixtures made it possible to establish that, in general, the mathematical model includes a system of equations of general material balance; component-wise material balance in the gas (taking into account diffusion, convection in the gas phase, as well as the internal source/drain of the substance as a result of adsorption

or desorption) and solid phases (taking into account diffusion, as well as the internal source/drain of the substance as a result of adsorption or desorption); thermal balance in the gas phase (taking into account thermal conductivity, convection, as well as thermal effect as a result of adsorption or desorption) and the adsorbent (taking into account thermal conductivity, as well as thermal effect as a result of adsorption or desorption); conservation of momentum (a variation of the Navier - Stokes equation); adsorption kinetics -desorption (taking into account the rate of mass transfer from the gas to the solid phase and back during adsorption - desorption); equilibrium in the gas-adsorbent system (adsorption isotherms of the components) [56, 57]; other relationships between model variables, initial and boundary conditions.

The equations of component-wise material balance are written in the form of a system of partial differential equations of a parabolic type [58]:

d (Vg Ck) d ck 6 -+—- +

(1 -e \da

d x

d t

dt

= D

d 2ck

x,k dx2

(1)

where vg is gas flow rate (m/s); ck is molar

concentration of the k component of the gas mixture (mol/m ); e is porosity of the adsorbent layer, taking

33

into account the porosity of the particles (m /m ); ak is amount of sorption (adsorbate concentration in

the adsorbent) (mol/m ); Dx,k is effective coefficient

of longitudinal mixing of the k component of the gas mixture (m /s); x is spatial coordinate of the adsorbent layer (m); t is time (s).

In equation (1), the first term describes the convective transfer of the substance in the adsorbent layer; the second term is the accumulation rate of the component in the mixture in the gas phase; the third and fourth terms are the sorption rate and the longitudinal mixing of the k component in the adsorbent layer, respectively.

The effective coefficient of longitudinal mixing Dx in early works on the adsorption separation of gas mixtures was identified with the molecular diffusion coefficient. At present, two main components are distinguished in longitudinal mixing (diffusion): molecular diffusion and turbulent mixing, which arises as a result of recombination of flows around the particles of the adsorbent. In practical calculations, the formula [1] is most often used to estimate the coefficient of longitudinal mixing:

Dx = 0.7Dm + 0.5dgrvg ,

e

where Dm is molecular diffusion coefficient; dgr is particle diameter of the adsorbent; vg is gas velocity.

Equation (1) for an unambiguous solution should be supplemented with initial and boundary conditions:

- initial conditions

ck(x,0) = c0(x); k = 1,nk ; 0 < x < L ;

- boundary conditions at the stage of adsorption

ck (0, t ) = cf(t ); ^^ = 0, k = Vn k ; dx

- boundary conditions at the stage of desorption

ck ( L, t ) = Ckout(t ); ^^ = 0, k = Vn k.

dx

To describe the sorption kinetics in the external diffusion region of the process, the equation [57] is used:

a dt

= Pk (ck - ck ),

where ck is the equilibrium molar concentration of the

3 1

k component of the gas mixture (mol/m ); pk is the mass transfer coefficient related to the concentration of the adsorptive in the gas phase (1/s).

For the internal diffusion adsorption process, the driving force is written as the difference between the values of the equilibrium sorption and the current sorption in the adsorbent (the Glukaf formula) [57]:

dk = P 2(ak- ak), dt

where ak is the equilibrium value of sorption of the k

3 2

component of the gas mixture (mol/m ); pk is the internal diffusion kinetic coefficient in the adsorbent granules (1/s).

To describe the kinetics of adsorption in the mixed-diffusion region, the mass transfer equation for the adsorptive from the gas phase to the solid phase of the adsorbent (through the phase boundary) is applied in the following form [59, 60]:

dak — F -F-(tgh(e(Vg -v*))+1)+Fk1,

dt

2

k = 1,2,3.

(2)

where Fk - the right part of the kinetics equation for nonstationary convective (external) mass transfer,

11 * 1 Fk =Pk (ck - ck); Pk is the mass transfer coefficient

related to the concentration of the adsorptive in the gas

phase; c* is the concentration of the adsorptive at the

interface or the equilibrium current value of adsorption

ak ; Fk is the right part of the kinetics equation the

internal diffusion adsorption process,

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Fk =P2h(a*k - ak); Pit is the kinetic coefficient; a* is the amount of adsorption equilibrium to the current concentration of the adsorptive c k in the gas mixture flow on the outer surface of the granules; e is the formal coefficient setting the dimensions of the mixed-diffusion region; vg* is the velocity of the gas mixture

which determines the transition from the diffusion region to the kinetic region of the adsorptive transfer; with initial conditions

ak (x,0) - a°(x), 0 < x < L , k — 1, nk .

Equation (2) is a description of the adsorption kinetics for the mixed-diffusion transfer region of the adsorptive across the phase boundary: when the velocity of the gas mixture is below the transition

velocity vg* , the adsorption process is limited by the

external mass transfer process with the coefficient Pk, otherwise - by the internal diffusion process in the granules of the adsorbent with the kinetic coefficient

Pk. The hyperbolic tangent and the formal coefficient

e along with v* describe a continuous transition from

the external mass transfer region to the internal diffusion adsorption process with zeolite adsorbents CaA, 13X.

The isotherms described by the equations of Dubinin-Radushkevich and Langmuir-Freindlich [56, 61] are most often used as equations of sorption isotherms in multicomponent gas mixtures.

To describe the processes of heat propagation in the gas mixture flow and the adsorbent along its length, partial equations of parabolic type are most often used [60, 62]:

g dTg ( x, t) g dTg( x, t )

cp p+ p pg vg-

dx

--^ud[Ta(x,t) -Tg(x,t)] -^^[v -Tg(x,t)] 8 8 d a

= x

d% g ar2

0 < x < L ,

(3)

c pp

pv a

^^ + a ^d[[(x,t) -Tg(x,t)]-

dt

2 ads

dak(x,t) d Ta(x,t)

a-T2-,

-Z hk

k

dt

dx2

(4)

where cg, pg is the specific heat and molar density of the gas mixture, J/(mol-K), mol/m3, respectively; Tg is the temperature of the gas mixture, K; A,g is the

coefficient of thermal conductivity of the gas mixture, W/(m-K); a is the heat transfer coefficient from the surface of the adsorbent granules to the gas mixture

3

flow, W/(K-m2); Sud = (1 -e)—is the specific surface

r gr

coefficient of the adsorbent granules, m2/m3; Kenv is heat transfer coefficient from the gas mixture flow to the environment, W/(K-m2); dA is the adsorber diameter, m; Tenv is the environment temperature, K, where c P is specific heat capacity of the adsorbent,

,3. ,„ads

is

J/(kg K); pa is the adsorbent density, kg/m ; hk the adsorption heat of the k component of the gas mixture, J/mol; Xa is the coefficient of the adsorbent

thermal conductivity, W/(m-K); with initial and boundary conditions at the adsorption and desorption stages similar to the conditions written above for equation (1).

In equation (3), the first term describes the accumulation of heat in the gas phase; the second term - the convective component of heat transfer; the third term - the heat transfer from the gas phase to the solid phase (the adsorbent); the fourth term - the heat transfer from the gas phase to the environment through the wall of the adsorber; the fifth term - the longitudinal thermal conductivity of the gas phase along the height of the adsorbent layer. In equation (4), the first term describes the enthalpy of the solid phase (the adsorbent); the second term - the heat transfer from the solid phase (the adsorbent) to the gas phase; the third term - the release of the heat of the gas mixture components sorption; the fourth term - the thermal conductivity in the adsorbent along the vertical axis of the adsorber.

The dynamics of changes in pressure and velocity of the gas mixture in the adsorbent layer is most often described by the Ergun equation [63]:

dP dx

150 (1 -so )2 (2rgrV)s0

^gvg + 1.75M gpg

(i -

>)vg2

2rgr Vs 0

where s o is the porosity of the adsorbent layer without taking into account the porosity of the particles, m3/m3; y is the sphericity coefficient of the adsorbent

granules; is the dynamic viscosity of the gas mixture, Pa s; Mg is the molar mass of the gas mixture, kg/mol; pg is the gas mixture density (mol/m3); r is the adsorbent granule radius, m.

The ideal gas state equation has the following

form:

P(x, t) = R Tg (x, t)Z Ck (x, t),

(5)

R is the universal gas constant, J/(mol-K).

For the numerical solution of the system of nonlinear partial differential equations (1) - (5) with the corresponding initial and boundary conditions, the method of straight lines was used according to which the derivatives of the spatial variable x were approximated by finite difference formulas. In this case, the time derivative remains in continuous form. It results in the system of ordinary differential equations along a given family of straight lines with initial and boundary conditions, which can be solved by some numerical method, for example, the fourth-order Runge-Kutta with automatic step selection. The method of straight lines has quite acceptable accuracy and speed of convergence for practice.

The experimental study of air oxygen enrichment process

The flow chart of the experimental dual-adsorber PSA unit for oxygen concentration is shown in Fig. 3: A1, A2 is adsorbers with granulated zeolite adsorbent 13X; K2, K3, K4, K8 is control valves; K5, K7 is check valves; K6 is pass-through valve; R is receiver of oxygen enriched production air. Further, the concentration of oxygen and nitrogen will be denoted

by y = ^2,) vol. %.

The PSA unit while enriching air with oxygen operates as follows. The flow of atmospheric air is formed by compressor C and inlet valves K1, K3 with

flow rate Gin, initial composition yk", where

k = {1 - 02, 2 - N2}, pressure P^ and temperature

Tgin . At the initial moment of time, the valves K1, K4,

K6, K8 are open. The air flow through the valve K1 enters the adsorber A1, in which the pressure rises to

the value Pj^. for a certain length of time [0, tads ] and

k

Fig. 3. Dual-adsorber PSA unit

the adsorption process of predominantly nitrogen and to a lesser extent oxygen and argon takes place over

the time tads (impurities are not adsorbed). Oxygen

enriched air in the adsorber A1 enters the receiver R through the valve K5 and then it is removed to the consumer through the valve K8 with the flow rate

Gout, composition yout

Pc

and

/^»out,1 out,1

G , composition y

^ out ,1

pressure

temperature Tgout. At the same time, part of oxygen

enriched air flow in the adsorber A1, through the valve K6, enters the adsorber A2, where desorption of nitrogen, oxygen and argon takes place under pressure

Pdœ . The air flow saturated with nitrogen (waste) enters the outlet of the adsorption concentrator from the desorber A2 through the valve K4 with the flow rate

pressure Pou1,1 and

temperature T o

At the moment of time t = tc /2 = tads, where tc is the duration of the "adsorption-desorption" cycle, the valves are switched: valves K1, K4, K5 close, valves K2, K3 open and valves K6, K8 still remain open. The atmospheric air through the valve K3 is fed by the compressor C to the adsorber A2, in which pressure is raised (pressure up-pu) for a certain length of time

[0, tads] to the value Pad and the adsorption process of mainly nitrogen and to a lesser extent oxygen and argon (impurities are not adsorbed) during the segment of time tc / 2 < t < tc . Oxygen enriched air in the

adsorber A2 enters the receiver through the valve K7 and then it is removed to the consumer through the

valve K8 with the flow rate G

out

out

out

composition y At the same

Tout

pressure Pout and temperature Tg

time, part of oxygen enriched air flow in the adsorber A1 enters the adsorber A1 through valve K6, where pressure is first released (pressure down-pd), and then nitrogen, oxygen and argon are desorbed under

pressure Pdiens . The air flow saturated with nitrogen (waste) enters the outlet of the adsorption concentrator from the desorber A1 through the valve K2 with flow

rate GouU, composition youU, pressure PouU and temperature 7gou1,1.

Upon expiration of time tc, one complete cycle of the adsorption concentration process is completed, after which the cycles are repeated during the entire operation period [0, tf] of the PSA unit.

The implementation of cyclic operation modes of the PSA unit is carried out by an automated control system using a software setpoint control device and control valves K1, K2, K3, K4, K8 in accordance with the periodic switching cyclogram.

A schematic diagram of an automated experimental unit for the oxygen adsorption concentration, which implements the described PSA scheme (Fig. 3) and the cyclogram (Fig. 4) is shown in Fig. 5.

R

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ads

Ki K; Ki K, K< Ki K: Ki

1

; /6 \ / / N / \ r S "s ^J \ ,'A, V t \ ' J \ t

!

!

i

i

pu ds ¿pd ^des

tjl tc/2

k

Fig. 4. Dynamics of pressure changes in adsorbers and cyclogram of control valves switching

Fig. 5. Schematic diagram of automated experimental unit of oxygen adsorption concentration

The atmospheric air under pressure of (2.0 - 6.0)-105 Pa is fed to the inlet of the oxygen concentrator (point "a") through the filtering unit 2, which traps water and oils. The pressure regulator 3 maintains the pressure (0.5 - 0.8)-105 Pa on the pneumorel 9, 12, 13, 14, 15, 16, 17, which are set at points "c" of the scheme. Under the pressure, the membrane blocks of elements 9, 12, 13, 14, 15, 16, 17 move upwards. For the sensor 4, the pressure gauge 5

sets the adsorption pressure Pallid (15 - 2.4)-105 Pa in the receiver 6. The oxygen concentrator is started by switching the pneumotumbler 8. The generator of rectangular pneumatic impulses (includes pneumorel 9, variable pneumatic resistance 10 and pneumatic

capacitance 11) periodically sets pressure P^ or Pdns at the point "d". At the initial time of the generator operation, the membrane blocks of the elements 12, 13, 14, 15, 16, 17 move downwards. Through the upper chambers of the elements 15, 16, 17, the atmospheric air begins to flow into the adsorber 18. In the adsorber 18, the pressure increases to a certain

value Paidns and the adsorption process of predominantly nitrogen and to a lesser extent oxygen and argon is carried out (impurities are not adsorbed). At the outlet of the adsorber 18, oxygen enriched air is formed. Part of the air flow enters the receiver 23

through the check valve 21, and the other part is fed into the adsorber 20 through variable pneumatic

resistance under pressure Pd^. The process of nitrogen desorption is carried out in the adsorber 20, and at its exit a gas mixture is formed with a high concentration of nitrogen, which through the upper chambers of the elements 12, 13, 14 enters the atmosphere. After a

half-cycle time tc / 2 , the generator sets pressure Pifs, and the membrane blocks of the elements 12, 13, 14, 15, 16, 17 move upwards. While the adsorber 20 goes into adsorption mode, and the adsorber 18 in desorption mode. The continuity of the process is achieved by cyclically switching the adsorbers at regular intervals. The flow rate of the production gas mixture is controlled by a variable resistance 25 on the rotameter 26. The main tuning parameters of the concentrator are the half-cycle time determined by the resistance 10 and the return flow value determined by the resistance of the element 19. The oxygen concentration is measured by the gas analyzer 25.

The results of experimental studies of the oxygen

concentration y1out in the production gas-air mixture (at the outlet of the PSA unit in the steady state) depending on the half cycle time tc / 2 for different

values of pressure Pads presented in Fig. 6.

at the adsorption stage are

..out

y. ,vol.%

60 50 40 30 20

37

52

67

a)

\ , vol.% 60

82 tJ2, S

50

•10

30

20

A A

□ □ A □ A

[ A o ' D.......... A

n

37

52

67

b)

Fig. 6. Results of experimental studies, Pa:

Pdns = 2-2 -105; b - A - Pds

82 LI2, s

a - A - -Cr 5-2T05, □ - Pd = 2.2 -105; b - A - Pd = 3.7 -105, □ - Pd = 2.7 -105

ads

Modeling and algorithmization of the dynamics of the PSA unit operation

The PSA unit as a system, in which the process of air separation and oxygen concentration is carried out, can be represented as a set of interacting subsystems: environment, flow booster (compressor, blower, etc.), "adsorber-desorber", receiver, subsystem valves, and control system.

When developing a mathematical model of the technological process, we will adhere to the principle of an autonomous mathematical description of the processes carried out in each subsystem and the matching of the subsystem models among themselves into a single mathematical system model.

Here we also give the equations of the mathematical model of the central system-forming element of the PSA unit - the "adsorber - desorber" subsystem.

During the adsorption of O2, N2, granulated zeolitic adsorbent 13X in the adsorbers Ai, A2 of the PSA unit, the following mass and heat exchange processes take place:

a) the mass transfer of O2, N2, and heat exchange between the air mass and the adsorbent;

b) the distribution of air components in the gas phase due to the convection;

c) the distribution of heat in the air flow and the adsorbent due to the convection and thermal conductivity;

d) the adsorption of O2, N2 on the surface and in the micropores of the zeolite adsorbent granules with the heat release, leaching of O2 from the adsorbent at the adsorption stage and desorption of N2 from micropores and from the surface of the granules with the heat absorption.

The mathematical description of the processes in the adsorber include the following assumptions:

i) the atmospheric air is predominantly a two-component air mixture and is considered as an ideal

gas, which is quite acceptable at pressures in the adsorber up to 200-105 Pa;

2) the granular zeolite NaX of spherical shape with a diameter of 2 mm is used as an adsorbent;

3) longitudinal mixing of O2, N2 components in the air flow in the axial direction and thermal losses to the environment are absent.

The mathematical description of the "adsorber-desorber" subsystem in the PSA unit includes the following equations:

- total material balance in the adsorber

d (vg Pg)

(

d x

+ Pa

d a

1 + d a2 ^

d t d t

+ -

ÔPg d t

= 0,

(6)

where vg is linear velocity of the gas mixture, m/s; pg

is molar density of the gas mixture, mol/m ; pa is bulk

3

density of the adsorbent, kg/m ; a1, a2 is adsorbate concentrations (oxygen and nitrogen), respectively, mol/kg;

- component-based material balance

d (vg ck ) d ck d ak n

• + 8-- + pa-- = 0,

d t d t

g

d x

k = {1 - O2,2 -N2}, thermal balance for the gas phase

d Tg d Tg

(7)

vgc„gp ——

g pg g dx

ÔVg

d x

■ 8cpgpg■

d t

+ P^ + Kt SUd(?g -Ta) = 0:

(8)

where cpg is specific heat of the gas mixture, J/(mol-K); s is adsorption layer porosity; Sud is coefficient of the specific sUrface of the adsorbent particles, m2/m3; Ta is temperature of the adsorbent,

K; KT is heat transfer coefficient, W/(m2-K);

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- heat balance for the adsorbent

2 g T 2 Г

Pa(cpa +YjCpkai^^ + PaZ| Ш1

k_1

- KT Sud(Tg -Ta)

k=14

STL

&

dak dt

—

= 0,

where cpa,cpk is specific heat capacities of the

adsorbent and adsorbate, respectively, k = 1, 2, J/(mol-K); AHk is thermal effect of the k air component sorption тепловой, J/mol; A,a is thermal conductivity coefficient of the adsorbent, W/(m-K); - adsorption kinetics

dak dt

_ Pk (a* - ak ), k = {1 - О 2,2 - N2}, (10)

- equilibrium conditions calculated by the Langmuir-Freindlich adsorption isotherm formal for zeolites [64]:

* b1,kck exp(b2,k /Ta)

a* =—2-,-+

1 +Êb3,}c] exp(b4,k /Ta) j=1

b5,kck exp(b6,k / Ta)

+ -

, k = 1,2 , (11)

1 + Z 63,jCj exP(64,k / Ta) j=1

- momentum conservation

*(Pg Tg) _ , (1 -e)2

dt

-_- Л

^(dgr9)2 e3 RgVg

- B(C1 + C2)M,

gr

1 - e

g ^dgr9e3 g

(12)

where A, B are known constants.

Let us formulate the problem of identifying kinetic parameters Pk , k — {1 -02,2-N2} by the

output experimental signals y^ (Pi,p2),

j — 1, m, i — 1, d, where m - the number of output measured coordinates of the control object; d - the number of experimental points for a separate output coordinate of the control object depending on the half-cycle of adsorption tads — tc / 2 .

Then a non-negative function is constructed

m d r -i2

F (P1, P2) — Hk j - yj (Pidns, tads, i, P1, P2)] ,

j—1 i—1

where yj (P^, tads,,-,P1,P2) is solution of the mathematical model equations (6) - (12) (with the

corresponding initial and boundary conditions) of the process of air oxygen enrichment in the PSA unit for

fixed values Pans, tads,i and kinetic parameters P1, P2 .

In a finite-dimensional numerical Euclidean space

(9) Emd , the value of F is equal to the square of the

distance between the vectors

y

and

y(Cs,tads,P1,P2) •

Let us rewrite this function in another form:

F (P1, P 2) _ y e - y ( Pads, tads, P1, P 2)

The function F is defined on the set V c E , where l < md; l is the given natural number.

The task of determining parameters P1, P2 is to

find P* eV c El such that

F (p*)— min F (P)

in case of constraints in the form of the mathematical model equations (6) - (12) of the air oxygen enrichment process in the PSA unit.

Despite the presence of constraint equations (6) -(12), we obtained the problem of the unconditional minimum of the function F(P), since P1, P2 are included in F (P) through the solution y (P1, P2), which takes into account all the mathematical properties of constraint equations (6) - (12).

Based on the physical meaning of the problem, it

*

would be necessary to find P so that the solution y(Pails, tads, P*, P2) was as close as possible to the true value of the vector y of the control object state variables measured at the output of the PSA unit in magnitude

F (P1, P2) — | |y - y (P2, tads, P1, P*2)|| Ei-

However, the vector y is unknown, therefore we have to work with the "perturbed" function F (p).

The regularization of a problem is the process of transforming it into a correctly set one. Regularization of the extremal problem formulated above consists in transforming a convex function F (p) into a strictly or uniformly convex one, which ensures the uniqueness of the solution P . Let us construct a continuous nonnegative parametric function

0(p ) — F (p ) + aQ(P),

2

E

where the parameter a > 0; Q is non-negative continuous function such that ®(p) is an uniformly convex function. An uniformly convex function Q(Pi, P2 ) = Pi + P2 can be taken as Q.

If F (P) is convex, and Q is a uniformly convex

function, then for any a > 0 function 0(P) will be

*

uniformly convex and the problem of finding Pa e V such that

o(pa)=mpn 0(Pi, P2) (13)

p1,p2

is set correctly and its solution pa is unique for each

fixed a. For determining pa , high-speed quasiNewton methods can be used [65].

As a result of solving the regularized problem

* *

(13), the values of kinetic parameters P1 <i,P2 <a of the

adsorption oxygen concentration process were

determined: P*a = 5.776 s-1, P2a = 1.925 s-1.

The adequacy of the mathematical model was tested on a set of experimental data obtained under conditions different from those under which they were obtained, according to which parametric identification

was carried out. The function y°ut 0^, tadv, P1, P2)

(solving the mathematical model equations (6) - (12)

for given values P^ = 2.7-105; 3.7-105 Pa, fads (from

**

7 to 82 s), kinetic parameters P1 = P1P2 = P2 a and

values of ordinates y^f'6, I = 1, 2, ..., 10 are shown in Fig. 7.

The mismatch of calculated by the model (6) -(12) and experimental data (Fig. 7) was estimated by the following formula:

..out

•V, ,vo\.% 60

50

40

30--

20

..........*............1....................................

n_8 ' /---—-__ □

IT 2 I

i! 1 1 ■ ' 1 ■ 1 1 1 -'— .111 . ..

37

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52

67

82 L/2s

Fig. 7. Verification of mathematical model adequacy of the process of oxygen adsorption concentration at the inlet pressure: 1 - 2.7-105 Pa; 2 - 3.7-105 Pa; A, □ - experiment, — calculation by model

„ out/nin , n o \ out,e

max y (.Pads, Íad8>í,ß! a, ß2 a)- yu-

i=1,d' ' 1

The mathematical model (6) - (12) with found

* -1 * -1 values P1ai = 5.776 s , P2a = 1.925 s was

considered adequate to the technological process of the adsorption oxygen concentration, if

Smax = max

i=1,d

yOUt(PaÍdns,tads,i,ßl,a,ß2,a) - y^

out,e

,out,e

y1,i

<S,

where 5 is the measurement error of oxygen concentration y1out at the production outlet of the PSA unit, which is 15 %. The verification of the adequacy of the mathematical model of adsorption oxygen concentration in the PSA unit showed that the maximum relative error of the mathematical model of the adsorption oxygen concentration process 5max

was 13.2 %, which allows using this model with found

**

values P1 =P1ai, P2 =P2a for the purposes of

analyzing the oxygen adsorption concentration process, optimizing and controlling this process.

The numerical analysis of the dynamics of the dual-adsorber PSA unit operation

In order to examine the systemic links, patterns, and increase the efficiency of the PSA unit, calculation experiments were conducted to study the dynamics and "statics" during the adsorption oxygen concentration in the air-gas mixture for a dual-adsorber technological scheme with 13X granular zeolite adsorbent (see Fig. 3). The main parameters of the pilot PSA dual-adsorber unit are presented in Table 1.

In Table 1: dA is the inner diameter of the adsorber shell (the adsorbent bulk layer); L is the height of the adsorbent bulk layer; dK1 = dK2 is the

bore section of cut-off valves; VR is the receiver volume.

Table 1

The characteristics of the pilot PSA dual-absorber unit

Parameter Value Parameter Value

dA, m 0.050 d K1 = d K 2, m 0.0014

L, m 0.500 Vr, m3 0.002

dgr, m 0.002 ßi; ß2, s-1 5.776; 1.925

Table 2

Values of the process parameters at the nominal point and ranges of their variation

Parameter The value in the working (nominal) point Range of variation

?ads = ?c/2, S 40 10 - 90

Pm x105, Pa 3 2.0 - 5.2

DOUt 5 P x105, Pa 1 0.9 - 1.1

P °uu X105, Pa 0.75 0.25 - 1.0

y|n % vol. 20.8 20.3 - 21.3

y3n% vol. 1.0 0.5 - 1.5

Tm,T ,K g T oc 298 273 - 323

dK6, mm 0.5 0.31 - 0.80

Variables and ranges of their changes are presented in Table 2.

A series of calculation experiments was conducted to study the effect of the half-cycle time tc / 2 (the duration of the adsorption stage tads), the

pressure Pin and temperature of the gas-air mixture Tgin at the compressor outlet, the diameter of the K6

throttle dK6 on the concentration >'1out and the degree

of oxygen extraction n.

Fig. 8 shows graphs of the oxygen concentration

dependencies >'1out in the production flow from the half-cycle duration tads = tc / 2 at various pressures

P in and, therefore, pressure Paidns at the adsorption stage.

The increase in P in leads to the increase in the oxygen concentration >'1out in the production flow and, accordingly, its sensitivity to changes in the half-cycle time tads = tc / 2 . All graphs have a pronounced

extreme character, so it is possible to choose the

*

optimal half-cycle time value tads, which provides the maximum concentration >'1out at various input pressure

values Pin . In this case, the range of values tads ,

including the optimal value (in case of a given optimality criterion), it is advisable to limit the interval [27 - 67] s. The analysis of the graphs in Fig. 9 shows that the time of the transient process (the transition of the unit to a periodic stationary mode) corresponds on average to 20-40 "adsorption - desorption" cycles,

i.e. tst ~ (20-40)-tc.

The analysis of the graphs in Fig. 10 shows that at

Tgin= 273 and 298 K the curves monotonously increase

over the entire range of pressure Pin variation. At the same time, up to the value of Pin « 3.7-105 Pa the sensitivity of the concentration >'1)ut to Pin is noticeably higher than in the next section. At Tgin = 323 K, the graph acquires an extreme character, and the maximum is reached at P in= 3.9-105 Pa and amounts to ~ 41 % vol. The further increase in Tgin leads to the

decrease in the oxygen content y1out in the production air flow. This is because the increase in Tgin leads to

the heating of the adsorption layer and the decrease in the equilibrium adsorptive concentration in the adsorbent. The greatest deviation of the curves (corresponding to 273 and 323 K) is observed at the

end segment of the variation in Pm and is ~ 1.2 % vol.

»» ^

y, , vol.%

40

20

2

3

1

out

Fig. 8. Dependencies y1 on /ads = tc / 2 at P , Pa:

1 - 2.2-105; 2 - 2.7-105; 3 - 3.7-105; 4 - 5.2-105

Fig. 9. Dependencies y on the time

of the unit operation t for iads, s:

1 - 10; 2 - 40; 3 - 65

-.OUT

>; ,voi.% 45 -

r|. %

JO

i 1 ! i .................1.................

| --- ............i..................

> ■—-- 2

/j/ | I

/ / / : T 1 : i ; ;

18.50

17.50

J

^ V-'.......

¿A ................

2 7

3.2 3,7 4.2

out

4.7 /"n,X105,Pa

Fig. 10. Dependencies y1out on Pln at temperature of initial mixture Tgin, K: 1 - 273; 2 - 298; 5 - 323

This phenomenon is explained by the increase in the adsorbent temperature, which leads, on the one hand, to the decrease in the equilibrium adsorptive concentration in the adsorbent, and, on the other hand, increases the desorption rate at the stage of the adsorbent regeneration.

The analysis of the graphs in Fig. 11 demonstrates that the oxygen extraction degree n is affected both

Pin and Tgin. At Tgin = 273 K the dependence has an

extreme nature and the extremum is reached at

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Pin = 4-105 Pa, and at Tgin = 298 K and Tgin = 323 K

the graphs monotonously increase proportionally Pin . At Tgin = 323 K a sharp decrease in the sensitivity of

the extraction degree n to the change in Pin is observed starting from Pin = 3.9-105 Pa, and at Tgin = 298 K the dependence n on P in becomes almost linear.

The dependencies of oxygen concentration y1out on the half-cycle time tads for various values Tgin are presented in Fig. 12. The graphs are of extreme nature at temperatures Tgin = 298 and 323 K. The maximum

yout is reached at tads «40 s. At Tgin = 273 K the

graph takes the form of a saturation curve, the "plateau" of the graph corresponds to the value

yout= 42,7 % vol.

It should be noted that the more tads , the greater deviations are observed between the curves. The maximum mismatch of the curves corresponding

to Tgin = 273 and 323 K is reached at tads = 60 s and

Pm,X10.Pa

Fig. 11. Dependencies of oxygen extraction degree n on Pln at 7gin, K: 1 - 273; 2 - 298; 3 -323

amounts ~ 5 % vol. These changes in the graphs can be explained as follows. At small values of tads < 30 s, the heat exchange between the adsorbent and the gas phase is not sufficiently intensive (due to the inertia of the heating or cooling process of the adsorbent),

therefore the effect of Tgin is insignificant, and at tads - 30 the effect of the inertia of the thermal processes is reduced.

The graphs analysis presented in Fig. 13 shows that the extraction degree is proportional tads at all

Tgin. At tads > 40 s, the maximum n is reached at the

lowest Tgn (273 K). The type of the considered curves as a whole correlates with the graphs of oxygen concentrations y1out in the production flow from the half cycle time yf* at Tgin = 273; 298; 323 K.

Fig. 12. Dependencies y™ on tads at Tgin , K: 1 - 273; 2 - 298; 3 - 323

— AM&T

34 ■

Fig. 13. Dependencies n on tads at Tgn, K:

1 - 273; 2 - 298; 3 - 323

.......1......\......i............-......i......i...... ::::::: i i i i i i i iii ......1......!......1.....-b" : Jr i Jr\ : i i S iii: i i i 1 iii!

i i i i i i i ¡¡¡iii: jf i i : : : i i Jf

i --i—-¿¿L \ \ \ i iii! i i i i ill!

15 20 25 30 35 40 45 50 55 60 6 5 70 75 taAsS

Fig. 15. Dependencies of n from tads at dKe , mm:

1 - 0.31; 2 - 0.62; 3 - 0.8

Dependencies of oxygen concentration >iout in the production air flow from tads for different values of the nominal diameter dK(6 of the purge valve K6 are

shown in Fig. 14. The graphs analysis indicates that with the increase of dK6 in the duration of the

adsorption stage tads ensuring the achievement of the maximum oxygen concentration at the unit outlet should decrease from 62 s (at dK6 = 0.345 mm) to 27 s

(at dK6 = 0.715 mm). Fig. 15 presents dependency

graphs of n from tads for various values of dK6 .

At dK6 = 0.31, the dependency is directly proportional

and close to linear, since the flow directed to the desorption is too small and it takes a long time to regenerate the adsorbent; at dK6 = 0.8, the dependency

is also close to linear, but inversely proportional,

>T! voi.%

...........|........... ...........

i

> \i

V 5

: : ___________I___________!___________ 4

87/ads.S

Fig. 14. Dependencies of _y1out from tads at dK6 , mm:

1 - 0,345; 2 - 0.5; 3 - 0.62; 4 - 0.715

which is explained by the excess flow directed to the nitrogen desorption, as a result of which nitrogen is desorbed before the desorption stage ends; at dK6 = 0.62 mm, the dependency acquires an extreme

character, since the intermediate state is reached.

The graphs analysis presented in Fig. 16 shows

that the maximum value Pin corresponds to the maximum peak of the air velocity vg = 0.18 m/s, since

the velocity rate is proportional to the pressure difference P in at the compressor outlet and the pressure P^ at the inlet to the adsorber. When these pressures are equalized, the air velocity gradually decreases to some steady-state value, approximately

equal to 0.03 m/s for all considered values Pin . Similar dependences were obtained at the desorption stage (Fig. 16, t > 1270 s).

v„ ,m/s

0.18 -

0.13

0.08 -

0.03

0,07

0.12

IV

!/\2Yv II r\, |

j

............1............

; 1

1240

1250

1260

1270

1280

1290

A-icls, S

Fig. 16. Dependencies of vg (x = 0) in adsorber from t at P*n, Pa:

1 - 2-105; 2 - 3-105; 3 - 4-105

Fig. 17. Dependencies of Vg in adsorber from t at dK6 , mm:

1 - 0.345; 2 - 0.5; 3 - 0.715

The graphs analysis in Fig. 17 specifies that varying the diameter of the throttle dK6 in the

investigated range does not affect the appearance of the curves obtained. Reducing the diameter of the purge throttle leads to the slight increase in the gas velocity at the beginning of the adsorption stage and the decrease at its end. These changes in the graphs are explained by the fact that the increase in the throttle diameter leads to the increase in the flow rate of the purge mixture for the regeneration of the second adsorber, which is reflected in the value of the difference in the air pressure directly proportional to its speed.

The velocity value of the flow entering the adsorber plays an important role, since it determines the adsorbent abrasion under alternating loads (in the cycles of lifting and pressure relief in the adsorbers). The interaction of the moving gas stream with the adsorbent layer leads to the effect of limited "fluidization" of the layer when the granules of the adsorbent begin to shift relative to each other, which leads to abrasion of the adsorbent and the appearance of a significant amount of dust in the product stream.

Even at the velocity of the filtered air flow at a much lower rate of the beginning of fluidization, the abrasion of the adsorbent granules can be quite strong [26]. This is due to the impact on the granules of changing "side" forces, called the forces of Karman, causing oscillating displacement of the granules relative to each other. Both destructive effects are likely to occur when the stages change, when large pressure gradients occur. Therefore, it is extremely important to control the gas flow rates in the frontal layer of the adsorbent during the transition periods of adsorption oxygen concentration process.

Fig. 18. Dynamics of opening degree of inlet 1 and discharge 2 valves of the PSA unit

v„,m/s

0.05

Fig. 19. Dependencies of Vg (x = 0)

in the adsorber from the time at Pin , Pa:

1 - 3105; 2 - 4-105; 3 - 5105

The analysis of operating experience of PSA units shows that the gas flow rate in the adsorber (depending on the size of the adsorber, the diameter of the adsorbent particles in it, the values of adsorption and desorption pressures) should not exceed 0.05-0.3 m/s.

Fig. 18 displays the graphs of the step change in the opening degree of the inlet and discharge valves from the time with the control frequency of 4 s. As a result, the air velocity in the front layer of the adsorbent is not higher than 0.08 m/s (Fig. 19). Thus, controlling the opening degree of the valves is an effective means of ensuring the absence of abrasion of the expensive adsorbent.

Conclusion

The calculation experiments conducted with the developed mathematical model established that it is

advisable to use pressure P in at the compressor outlet, a temporary program ) for opening control valves

Ki (K2), the duration of the adsorption stage tads (as half cycle time) and the diameter of the purge throttle dK6 as control actions allowing to effectively control

the modes implemented in the PSA unit. The range of values tads, including the optimal value (maximum

concentration value y°ut ), is advisable to limit the interval to 27-67 s, and the time for the unit to reach a periodic stationary mode on average corresponds to 20-40 adsorption-desorption cycles.

It has been established that by finding the law of opening the inlet and discharge valves of the PSA unit, it is possible to ensure the air flow rate that does not lead to abrasion of the adsorbent during the implementation of cyclic adsorption-desorption processes. At the same time, the influence of the gas flow rate limitation on the purity of the production oxygen, the extraction degree and the capacity of the PSA unit requires further research.

The results of numerical analysis, mathematical and algorithmic support for the operation of the PSA dual-adsorption unit, presented in this paper, can be used to design new automated processes and adsorption process units with cyclically varying pressure to separate and purify multicomponent gas mixtures.

The research was financially supported by the Russian Ministry of Education and Science within the framework of project No. 10.3533.2017.

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