EVGENY AKULININ OPTIMIZATION AND ANALYSIS OF OLEG GOLUBYATNIKOV PRESSURE SWING ADSORPTION …...

Chemical Industry & Chemical Engineering Quarterly

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Chem. Ind. Chem. Eng. Q. 26 (1) 89−104 (2020) CI&CEQ

89

EVGENY AKULININ

OLEG GOLUBYATNIKOV DMITRY DVORETSKY

STANISLAV DVORETSKY

Tambov State Technical University, Tambov, Russia

SCIENTIFIC PAPER

UDC 66.074.5.081.3:66.074.35:66.083

OPTIMIZATION AND ANALYSIS OF PRESSURE SWING ADSORPTION PROCESS FOR OXYGEN PRODUCTION FROM AIR UNDER UNCERTAINTY

Article Highlights • PSA oxygen production unit is optimized under uncertainty • Optimal modes of the PSA oxygen production unit are studied • A constraint on gas flow velocity in the frontal layer of adsorbent of the PSA unit is

introduced • Limiting gas flow velocity provides for resource saving of granulated adsorbent • The fulfilment of gas flow velocity constraint is ensured by controlling the PSA unit's

valves Abstract

Pressure swing adsorption (PSA) units are widely used for atmospheric air separation and oxygen concentration. However, the efficiency of such install-ations is reduced due to accidental changes in the characteristics of the atmo-spheric air to be separated. The article formulates and solves the problem of optimizing the regimes of operation of PSA units with zeolite adsorbent 13X, according to the criterion of oxygen recovery rate in the conditions of interval uncertainty of composition, temperature and pressure of atmospheric air. The optimization problem also takes into account the fulfillment of the requirements on purity of oxygen, productivity of the unit and resource saving of granulated adsorbent from granule abrasion. It is proposed to provide adsorbent saving by limiting the speed of incoming flow in the frontal layer of the adsorbent by means of "soft" stepwise change of the degree of opening of control inlet and outlet valves of the unit. The problem (including the search for time change programs for the degree of opening of control valves) was solved with the use of the developed mathematical model of cyclic heat- and mass exchange processes of adsorption-desorption in a PSA unit and a heuristic iterative algo-rithm. The comparative analysis of the results of the optimization problem sol-ution, with and without taking into account the constraint on the gas flow velo-city in the frontal layer of the adsorbent, is carried out. The influence of the specified requirements for the performance of the PSA unit and the purity of oxygen on the degree of its recovery has been studied.

Keywords: pressure swing adsorption, zeolite, mathematical modeling, optimization, numerical simulation, uncertainty.

The most common and effective method of ind-ustrial air separation on solid adsorbents is the method of pressure swing adsorption (PSA), devel-

Correspondence: E.I. Akulinin, Tambov State Technical Univer-sity, Sovetskaya str., 106, 392000 Tambov, Russia. E-mail: [email protected] Paper received: 14 April, 2019 Paper revised: 10 September, 2019 Paper accepted: 18 September, 2019

https://doi.org/10.2298/CICEQ190414028A

oped in 1960 in the USA by Charles W. Skarstrom. Concentration of oxygen from the atmospheric air by the PSA method is carried out practically in iso-thermal conditions and allows to obtain a gas mixture containing up to 95.5% of oxygen at the outlet.

The Skarstrom method (PSA) is widely used in chemical engineering [1–20]. In its implementation, adsorbents (mainly zeolites of types 13X, 5A, LiX) with a low adsorption heat in relation to the separated components of gas mixtures are used (typical values

E. AKULININ et al.: OPTIMIZATION AND ANALYSIS OF PRESSURE SWING... Chem. Ind. Chem. Eng. Q. 26 (1) 89−104 (2020)

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of h/RT for PSA fall within the range of 1-10). In sys-tems where the target component is poorly sorbed (e.g., nitrogen and oxygen in comparison with water vapor in humid air on silica gel or hydrogen in syn-thesis gas on microporous sorbents), almost pure gases can be produced while ensuring their recovery rate up to 85-90%. When differences in adsorption heat for components are insignificant (e.g., nitrogen, oxygen and argon in the air on microporous sorbents), the purity of the product can be achieved only through a significant reduction in its recovery [1,21–24].

Annual growth in oxygen demand averages 4– -5% due to increased demand in ferrous metallurgy, chemical industry, aluminum production and other industries and social sphere. Thus, the essential share of consumers uses not so much pure oxygen, as the air enriched with oxygen from 40 to 90%. [25–28].

In modern adsorption plants for the separation of gas mixtures, Rapid PSA (RPSA) and ultra rapid PSA (URPSA) processes are implemented with the use of multi-adsorber flow diagrams [29–33]. The main idea behind the RPSA and URPSA processes is to reduce cycle times compared to conventional PSA. In RPSA and URPSA units, in comparison with tradit-ional PSA units, the pressure drop increases on the average by an order of magnitude and, accordingly, the length of the adsorbent layer is reduced, which makes it possible to reduce the size of the install-ations by several times and reduce the cycle time up to ten seconds. They are effective when using a fairly narrow range of adsorbent granule sizes (0.25–2 mm).

Increase of gas flow velocity in PSA processes inevitably leads to increase of aerodynamic resist-ance of an adsorbent layer, adsorbent granule abras-ion rate and reduction of its service life. Experimental and computational studies, as well as the analysis of the literature sources [18,24] showed that at the full jump opening of the inlet and outlet valves of the PSA unit, the velocity of the gas flow in the frontal layer of the adsorbent can reach the values of fluidization, when the granules of the sorbent begin to move rel-ative to each other, which leads to abrasion of the adsorbent and the appearance of significant amounts of dust in the product flow.

It is worth noting that even at gas flow rates lower than the rate of fluidization, the abrasion of ads-orbent granules can be quite significant. This is due to the impact on the granules of changing "lateral" forces or the so-called lateral von Karman forces, causing the oscillating displacement of the granules relative to each other. Both destructive effects are likely to occur when the technological stages of ads-orption and desorption change, when large pressure

gradients of the adsorbent layer appear. The pre-sence of dust in the product flow can lead to failure of solenoid valves, with the help of which the gas dis-tribution inside the unit is carried out, which substan-tially increases the operating costs of PSA units [24].

There are a number of design solutions to red-uce adsorbent abrasion in the layer: hardening and reduction of adsorbent granule roughness, direction of filtered flow from top to bottom, immobilization of adsorbent granules in the layer.

One of the effective ways of solving this prob-lem, as suggested in this work, is limiting the gas flow velocity in the frontal layer of the adsorbent, which virtually eliminates the abrasion of adsorbent gra-nules. For this purpose, it is necessary to use control solenoid inlet and outlet valves, the programs of open-ing of which, along with other regime variables, can be determined by solving an optimization problem.

Optimization problems of operation regimes of PSA units for oxygen enrichment were formulated and solved in the works of the authors: Jiang et al. [34], Cruz et al. [35], Santos et al. [21], Liu et al. [36], Hossein-zadeh Hejazi et al. [19] and Ding et al. [37]. These studies considered 2-bed RPSAs [21,34-36], VSA [19,34,35] and PVSA [37] units with a capacity of up to 4 l/min at NTP. Both industrial zeolites 13X [35], 5A [35,36], LiLSX [37] were used as adsorbents, which allow to obtain oxygen with concentration up to 95 vol.%, and promising experimental sorbents AgLiLSX [21] and Ag-ETS-10 [19], which have high argon selectivity and allow to obtain oxygen with con-centration up to 99.5 vol.%, but with a sufficiently low degree of recovery ∼27.3% and lower [19,21].

The works [21,34-36] used oxygen recovery rate as the goal function determining the efficiency of the PSA unit; the works [19,37] used energy consump-tion; the works [35,36] used profit from the use of the PSA unit; and in [37] the productivity of the unit was used. As optimized variables, the following were con-sidered: duration of adsorption and desorption cycle steps [19,21,34-37], valve capacity [21,34,35,37], pressure in the receiver [34], pressure at the steps of adsorption and desorption [19,35,36], rate of dis-charge [36] and inlet flow rate [19], backflow rate [36], air flow rate at the compressor and vacuum pump outlet [37], as well as the length of the adsorbent bulk layer [19]. As constraints in the optimization problem, the operating procedure requirements for oxygen purity [19,21,34,35,37] and productivity of the unit [21] were used.

The work [34] has applied a simultaneous tail-ored methods approach for the solution of the opti-mization problem, which is more economical in the


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sense of reducing the time spent on obtaining a sol-ution. However, in the majority of other works [19,21,35-37] optimization problems were solved with the use of the so-called black-box approach, which is widely used both for solving practical and scientific problems [34,38].

An important aspect of the study of the effi-ciency of the PSA air separation units is to take into account the influence of uncertain parameters - the composition, temperature and pressure of the atmo-spheric air. During the operation of the PSA unit, uncertain parameters may take on random values in some intervals, depending on the operating condit-ions of the units and, in particular, on the climatic and geographical characteristics. For instance, when operating PSA medical concentrators indoors, the ambient temperature may vary from 293 to 303 K, the pressure may vary from 0.75×105 to 1.0×105 Pa at altitudes up to 2 km above sea level, and the concen-tration of oxygen in the air may vary from 18 to 21%.

A new approach is proposed in this paper to account for the interval uncertainty in the design of PSA air separation units. The idea of the approach is to define such values of the regime parameters, at which the optimal value of the efficiency criterion (for example, the rate of oxygen recovery) is achieved and the requirements for the PSA unit (for the purity of oxygen and the productivity of the unit) are met, regardless of the values of uncertain parameters from

the given intervals of their possible changes. The purpose of this work is to set and solve the

problem of optimization of the regime parameters of the two-adsorber PSA unit for air oxygenation. As an optimization criterion the rate of oxygen recovery, cal-culated in the cyclic steady state (CSS) regime, will be used, and the operating procedure requirements on the purity of oxygen, productivity of the unit and resource saving of the adsorbent will be taken into account.

Mathematical description of pressure swing adsorption processes of oxygen concentration

The technological process of separation of atmospheric air containing oxygen in the amount of 18-21 vol.%, nitrogen 78-80 vol.%, argon and other impurities 1-2 vol.% is carried out in the two-adsorber PSA unit with granulated synthetic zeolite adsorbent 13X [13,14] (Figure 1).

Atmospheric air is supplied to the unit after pre-drying with the overpressure inP from 2×105 to 6×105 Pa. Pressure rise in adsorbers is performed by “soft” opening of control inlet valves v1 (v2), through which air is supplied to the layer of granulated bulk ads-orbent. The bulk of the oxygen-enriched product air stream from the adsorber A1 (A2) is directed to the consumer through the check valve v5 (v7) and product storage-tank STP, while the other part of the stream is directed through the throttle v6 to the adsorber A2 (А1)

Figure 1. Structural diagram of the PSA unit as an object of optimization.


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for washing the adsorbent layer and desorption of mainly nitrogen, after which through the control outlet valve v4 (v3) it is directed for further processing. The v8 valve is required for manual control of the amount of gas extracted from the product storage-tank STP. The pressure cyclogram in the adsorber A1 is shown in Figure 2a. As the adsorbers operate alternately, the pressure cyclogram for the adsorber A2 is similar to that of the adsorber A1, but with a shift of half a cycle of "adsorption-desorption". Figure 2b shows a cyclo-gram of operation of the valves in the unit. Compared to the standard Skarstrom cycle, which includes 4 steps, the pressure feed and adsorption steps, as well as blowdown and desorption, are combined in this unit as in RPSA units.

The analysis of the technological process of air separation and oxygen concentration by the PSA method [13,14,21-28,34-44] allowed to determine:

1) regime parameters u - duration adst ( = =ads des c / 2t t t ) of adsorption step (half cycle); pressure inP at compressor outlet; backflow coef-ficient θ (for adsorbent regeneration); programs ψ ψ

1 4v v( ), ( )t t , ∈ ads[0, ],t t ψ ψ2 3v v( ), ( ),t t ∈ ads c[ , ]t t t

of opening of control inlet v1, v2 and outlet v3, v4 valves; with ψ ψ ψ= =

1 21 v v( ) ( ),t t ψ ψ ψ= =3 42 v v( ) ( )t t ,

as adsorbers work alternately; 2) design parameters d - diameter of DA of ads-

orber, length L of the layer of granulated adsorbent 13X, granule diameter dgr, throughput capacity vK of inlet and outlet valves of the unit;

3) uncertain parameters ξ = env env env{ , , }y T P - concentration of yenv gases in the air, temperature of Tenv and ambient pressure of Penv;

4) output variables z, which determine the effi-ciency of the PSA unit - the rate of oxygen recovery η , the concentration outy of product flow compo-nents at the outlet of the unit and productivity outG of the PSA unit.

When air (adsorbate) components are ads-orbed, the following mass and heat exchange pro-

cesses take place in the adsorbent layer: 1) diffusion of adsorbate in the gas-air mixture flow; 2) mass and heat exchange between the gas phase and adsor-bent; 3) adsorption of predominantly N2 on the sur-face and in micropores of zeolite adsorbent granules with heat generation and desorption of predominantly N2 from micropores and from the surface of granules with heat absorption.

The analysis of the results of physical modeling of these processes has shown that diffusion of ads-orbate (N2, O2, Ar + impurities) and distribution of heat in gas and solid phases are carried out mainly in the axial direction relative to the movement of the gas mixture along the length of the adsorbent layer. At the same time, the process of enrichment of the gas-air mixture at adsorption (N2, O2, Ar + impurities) by gra-nulated zeolite adsorbent with granule diameter 0.25– –2 mm takes place in the external-diffusion region (determined by the coefficient of external mass trans-fer), as well as equilibrium ratios of adsorbate con-centrations in the phases, which is in line with [21,34,35].

The following assumptions were made in the mathematical description of the air oxygenation pro-cess:

1) initial gas-air mixture is 3-component (con-tains 1 – oxygen O2 with concentration of 18-21 vol.%, 2 – nitrogen N2 with concentration of 78-80 vol.%; 3 – argon Ar and impurities with concentration of 1-2 vol.%) and is considered as ideal gas, which is quite acceptable at pressure in the adsorber up to 200×105 Pa [45];

2) diffusion of adsorbate and heat propagation in gas and solid phases are carried out only in the axial direction of the gas mixture flow in the adsorber (along the length of the adsorbent layer) [1,21–28,46];

3) granulated zeolite 13X with a granule dia-meter of 1.6 mm is used as an adsorbent [14];

4) adsorption equilibrium (adsorption isotherm) is described by the Dubinin-Raduskevich equation [47];

Figure 2. Cyclograms of pressure in adsorber A1 (а) and operation of PSA unit’s valves (b).


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5) desorption branches of adsorption isotherms (N2, O2, Ar + impurities) for zeolite 13X coincide with adsorption branches [14];

6) gas temperature in the receiver R is equal to the gas temperature at the adsorber outlet, heat losses to the environment are negligibly small.

According to the accepted assumptions, the mathematical model of the PSA process of atmo-spheric air separation and oxygen concentration has the following form (Table 1).

The mathematical model, Eqs. (1)–(12), represents a system of differential equations of the parabolic type in partial derivatives and is supple-mented by the corresponding initial (Table 2) and boundary conditions (Table 3) for the adsorption and desorption steps.

Formulas for calculating model coefficients and parameter values are given in the nomenclature sec-tion.

To solve Eqs. (1)–(12) with initial and boundary conditions we used the method of straight lines [48] in the MatLab software environment, and the solution of the system (1)–(12) was carried out before the onset of the CSS regime [21,34] of the PSA unit. Reaching the CSS regime was determined when the condition was met:

ε−− ≤out out1, 1, 1n ny y (13)

where ε – a small positive number, ε = 1.0×10-3 [21,35]; the time of the CSS onset tst was determined by = *

st ct n t , where n* is the number of the cycle at which the condition (13) is valid. Average comput-ational time (CPU time) of the CSS regime startup was ∼210 s (Win7x64, Intel Core i7-7700, DDRIV 16 Gb).

The analysis of the adequacy of the mathem-atical model, Eqs. (1)–(12), was carried out with the use of the factual RMS error:

Table 1. Mathematical model of the process of air separation and oxygen concentration in the PSA unit

Parameter Expression Eq. No.

Component mass balance ε ν

ε∂ ∂ ∂− ∂ ∂ + + = = ∂ ∂ ∂ ∂ ∂

g g( , ) ( , ) ( , )(1 )

( ) ( , ) , 1,2,3k k kk

y x t a x t y x t D x y x t kt t x x x

(1)

Linear driving force model (kinetics) ( )β∂= −

∂*

sp( , )

( , )kk k k

a x t S a a x tt

(2)

Dubinin-Radushkevich isotherm ρσ −

= − ×

22g S,* 0

a * 2 5exp lg1.013 10 0.0224

kk

k k k

T PWa B

V P y (3)

Heat propagation in the gas-air mixture ( )αρ ρ ν λε

∂ ∂ ∂+ − − =

∂ ∂ ∂

2g g g

g g g g g sp a g g 2

( , ) ( , )( , ) ( , )p p

T x t T x t Tc с S T x t T x t

t x x (4)

Heat propagation in the adsorbent ρ α λ∂ ∂∂ + − − = ∂ ∂ ∂a

2a a

a sp a g a 2

( , ) ( , )( , )( , ) ( , ) k

p kk

T x t T x ta x tс S T x t T x t ht t x

(5)

Flow continuity equation νν

∂ ∂ − =

∂ ∂

g

g 0k

kk

k

yy

x x

(6)

Gas phase momentum balance (Ergun equation)

( )( )

( )ε εμ ν ρ ν

ςες ε

− −∂ = − + ∂

2

2g g g g g2 3

grgr

150 1 11.75

P Mx dd

(7)

Valves equations ( )λψ λ= − = =in in out

v v v v v , 1,4, 1,i i i i i

G K P P i m

( )= −out out outads ads SG G P P

(8)

Throttler equation θ=out out

in ads adsdes in

des

( ) ( )( )

G t P tG

P t(9)

Set pressure equation in the adsorber ( )ε∂

= −∂ ×

in inads/des des v inA

A3 inA60 10

iK P GP P P

t P V (10)

Set pressure equation in the product storage-tank

( )∂= −

∂ ×

out outinS S ads

S3 inS60 10

P K P G P Pt P V (11)

Product storage-tank purging equation ( )∂= − −

∂

outS, out

S,S

( )( )k

k ky G t y y t

t V (12)


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Table 2. Initial conditions of the mathematical model, Eqs. (1)–(12), for the steps of adsorption and desorption

Adsorption Adsorption Desorption

= = ≤ ≤0, 1, 0t n x L : = × = ≤ ≤ads , 2,3,..., 0 :t n t n x L = × = ≤ ≤des , 1,2,..., 0 :t n t n x L

( ) = 0,0 ( )k ky x y x ( ) =ads desdes, ( , )k ky x t y x t ( ) =des ads

ads, ( , )k ky x t y x t

( ) =,0 0ka x

( ) =ads desdes, ( , )k ka x t a x t ( ) =des ads

ads, ( , )k ka x t a x t

( ) ( )= 0g g,0T x T x ( ) ( )=ads des

g g des, ,T x t T x t ( ) ( )=des adsg g ads, ,T x t T x t

( ) ( )= 0a a,0T x T x ( ) ( )=ads des

a a des, ,T x t T x t ( ) ( )=des adsa a ads, ,T x t T x t

( ) ( )ν ν= 0g g,0x x

( ) ( )ν ν=ads des

g g des, ,x t x t ( ) ( )ν ν=des adsg g ads, ,x t x t

( ) ( )= 0,0P x P x =ads desdes( , ) ( , )P x t P x t =des ads

ads( , ) ( , )P x t P x t

( ) = 0A 0P P ( )=ads des

A A des( )P t P t ( )=des adsA A ads( )P t P t

( ) = 0S 0P P ( )=S S ads( )P t P t

( ) = 0S, S,0k ky y ( ) ( )= out

S, adsk ky t y t

Table 3. Boundary conditions of the mathematical model (1)–(12) for the steps of adsorption and desorption

Adsorption Desorption

x = 0: x = L: x = 0: x = L:

= in(0, ) ( )k ky t y t ∂

=∂

( , )0ky L t

x

∂=

∂(0, )

0ky tx

= ads( , ) ( , )k ky L t y L t

= ing g(0, ) ( )T t T t

∂

=∂g( , )

0T L t

x ∂

=∂

(0, )0gT t

x = ads

g g( , ) ( , )T L t T L t

α∂= −

∂ina

sp a g(0, )

( (0, ) ( ))T t S T t T t

x

∂=

∂a( , )

0T L t

x

∂=

∂a(0, )

0T t

x α∂

= −∂a

sp a g( , )

( ( , ) ( , ))T L t

S T L t T L tx

νε

=in

gA

( )(0, )

G ttS

ν∂

=∂g( , )

0L tx

ν∂

=∂g(0, )

0t

x ν

ε=

indes

gA

( )( , )

G tL t

S

= A(0, ) ( )P t P t = A(0, ) ( )P t P t

( )( )δ=

= −2

out out,e out,eММ 1, 1, 1,

1

1100 ( ) ( ) / ( )

N

i i ii

y t y t y tN

where i is the measurement number, N is the total number of measurements. RMS error δММ at N = 24 was 5.2%, which allows using the mathematical model for technological calculation, optimization of

cyclic regimes and design of PSA units for air separ-ation and oxygen concentration (Figure 3).

Optimization of cyclic processes of the adsorption separation of atmospheric air under uncertainty of parameters

The purpose of the PSA air separation unit is to produce oxygen with a given concentration of out

1,defy

Figure 3. Dependence of concentration out

1y of product oxygen on: a) the duration adst of adsorption step; b) the length of adsorbent layer L of the granulated zeolite adsorbent 13X: 1– grd =1.6 mm, 2– grd = 0.5 mm (, – experimental data, calculation by the model).


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vol.% and in a given amount (the efficiency of the PSA unit - the flow rate of product air outG with a given concentration of oxygen out

1y , must be not lower than the specified one out

defG ). As a criterion of optimality that determines the

efficiency of oxygen PSA units with productivity up to 4 l/min at NTP, it is advisable to use the rate of oxygen recovery η in the CSS regime, which is noted in [21,34-36]. Oxygen recovery rate η characterizes the share of product flow in relation to the flow sup-plied for separation:

η =out out1

in in1

100y Gy G

We shall introduce constraints that take into account the requirements of the operating procedure of the PSA unit in the CSS regime:

- for the purity of oxygen out1y at the outlet of the

unit, - for the productivity outG of the unit and - gas flow rate gv in the frontal layer of ads-

orbent, which should not be lower than the specified values

out1,defy , out

defG and higher than +gv , respectively.

In addition, we introduce a limit on the pressure drop in the adsorbent layer, calculated using the Ergun Equation (7). The pressure in the closing (final) adsorbent layer at the absorption out

adsP and desorption out

desP steps should be greater than or equal to the atmospheric pressure Penv.

Problem statement. Let vector:

ξ = env env env{ , , }y T P

of uncertainty parameters belong to an area Ξ , ξ ξ ξ− +Ξ = ≤ ≤{ } , i.e., ξ ∈ Ξ . We shall introduce a

set of approximation points ξ i , ∈ 1i J , ξ ∈ 1i S , evenly

covering the uncertainty area ξ ξ ξ− +Ξ = ≤ ≤{ } , and a set of critical points ξ ξ= ∈ Ξ ∈2 2{ : , }l lS l J , in which the above introduced constraints of the optimization problem can be violated.

The problem of optimizing regime parameters of the PSA unit under the interval uncertainty of the parameters ξ = env env env{ , , }y T P is formulated as fol-lows: at fixed values of the vector of design para-meters = A gr v{ , , , }d D L d K it is necessary to deter-mine the regime variables:

λ λθ ψ ψ λ∗ ∗ ∗ ∗ ∗ ∗= =inads 1 2{ , , , , , 1,20}u t P

such that the average value of the rate of oxygen recovery in the CSS regime, Eq. (13), reached the maximum value, i.e.

ω η ξ∈

= 1

*( ) max ( , , )iiu

i J

I u d u (14)

in the form of mathematical model (1)–(12) equations and constraints:

- purity of product oxygen, out1y :

( )( ) ( )ξ ξ= − ≤out out1 st 1,def 1, , , , 0g z d u y y d u (15)

- PSA unit productivity, outG :

( )( ) ( )ξ ξ= − ≤out out2 st def, , , , 0g z d u G G d u (16)

- velocity of gas mixture gv in the frontal layer of adsorbent:

( )( ) ( )ξ ξ +

∈= − ≤

с3 st g g[0, ]

, , max ( , , ) 0t t

g z d u v d u v (17)

- pressure gradient in the adsorbent layer at the adsorption step:

( )( )ξ ξ= − ≤out4 st env ads, , ( , , ) 0g z d u P P d u (18)

- pressure gradient in the adsorbent layer at the desorption step:

( )( )ξ ξ= − ≤out5 st env des, , ( , , ) 0g z d u P P d u (19)

- ranges of change of optimized variables, − +≤ ≤u u u :

≤ ≤ads0.5 20t s, × ≤ ≤ ×5 in 52 10 6 10P Pа,

θ≤ ≤0 6, λψ≤ ≤10 1, λψ≤ ≤20 1, λ = 1,20 (20)

- ranges of change in uncertain variables ξ ξ ξ− +≤ ≤ :

≤ ≤env,118 21y vol.%, ≤ ≤env,278 80y vol.%,

≤ ≤env,31 2y vol.%, ≤ ≤env293 303T K,

× ≤ ≤ ×5 5env0.75 10 1 10P Pа (21)

Here, η= out outst { , , }z y G is the vector of output vari-

ables in the CSS regime, which is determined by solving Eqs. (1)-(12) of the mathematical model before the onset of the CSS regime (13); ω −i weight coefficients satisfying the conditions of ω ≥ 0i ,

ω∈ = 1ii J . Since it is not known with what proba-bility the uncertain parameters can take some values from the given ranges ξ ξ ξ− +≤ ≤ , it is assumed that they will be distributed in accordance with the equi-probable law. Then the coefficients ωi will be the same for all approximation points ξ ∈ 1,i i J , i.e., ω =

11/i JK , = ∈

1 11, ,Ji K i J . Changes in the timing of the opening of inlet

λλ λψ ∈1 ads( ), [0, ]t t t and outlet λ

λ λψ ∈2 ads c( ), [ , ]t t t t


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valves were approximated by step-functions with a number of steps equal to 20.

Taking into account the constraint on the max-imum air velocity gv in the frontal layer of the ads-orbent ensures the protection of granulated adsorbent from destruction due to aerodynamic impact with the full jump-opening of control valves, which helps to increase the service life of the adsorbent.

In accordance with the calculations (Figure 4), as well as the data from literature [18,24], it can be concluded that the oscillating shifts of granules relat-ive to each other, causing the adsorbent abrasion during the operation of the PSA unit, arise at speeds of ∼0.15-0.25 m/s, which are 2-2.5 times less than the rate of fluidization of the layer. Calculation of the fluid-ization rate νf was carried out by the Todes formula [49], which allows to calculate with high accuracy the critical velocity of gas flow at homogeneous fluidiz-ation of the particle layer (the formula is given in the nomenclature).

Figure 4. Dependence of fluidization rate ν f and flow rate in the

adsorbent frontal layer ν g (0) on the pressure in the adsorber during the adsorption step in

adsP at the adsorbent particle diameter dgr: 1– 0.5 mm; 2– 1 mm; 3– 1.5 mm (∆ – experimental

data, full line – calculation by the model).

The formulated optimization problem, Eqs. (14)- –(21), belongs to the class of nonlinear programming problems, the solution of which was carried out by the method of sequential quadratic programming in the MatLab software environment (solver fmincon) [50] with the use of the optimized black-box approach [19,21,35-37] and the developed heuristic iterative algorithm.

The global maximum in the problem (14)-(21) can be guaranteed if the goal function and constraints

( )( )ξ =st , , , 1,5jg z d u j are convex functions with respect to the optimization variables, which is a necessary and sufficient Kuhn-Tucker condition [50].

It should be mentioned that the increase in pres-sure at the compressor outlet Pin simultaneously con-tributes to the increase in oxygen recovery (goal func-

tion) and energy consumption of the PSA unit. There-fore, it is expedient to search for the minimum value of the pressure Pin, at which: all the constraints of the problem ( )( )ξ =st , , , 1,5jg z d u j would be fulfilled; the maximum degree of oxygen recovery and the minimum energy consumption would be provided. Therefore, it is proposed to determine the pressure at the compressor outlet Pin by an iterative procedure.

Algorithm of solving the optimization problem, Eqs. (14)–(21)

Step 1. Set the initial numbers of iterations k = 1, ν = 1 and the initial value of pressure = ×in( ) 52 10kP Pa at the compressor outlet.

Step 2. Set the approximation points ξ ξ∈ ∈1 1, ,i ii J S , the initial set of critical points

ν νξ ξ− −= ∈ Ξ ∈( 1) ( 1)2 2{ : , }l lS l J and the initial approx-

imations of the regime variables (0)u . The initial set of critical points (0)

2S will be formed from the angular points ξ − , ξ + of uncertainty area Ξ in the assumption of convexity of functions describing the constraints ( )( )ξ =st , , , 1,5jg z d u j .

Step 3. Using the sequential quadratic program-ming method, we find a solution to the auxiliary problem:

ν ω η ξ∈

= 1

( , )( ) max ( , , )k iiu

i J

I u d u (А)

at links (1)-(12), ranges of change of optimized (20) and uncertain (21) variables, constraints (15)-(19), which are calculated in approximation points ξ ∈ ∈1 1,i S i J and critical points ξ ∈ ∈2 2,l S l J , and we define values of ν( , )( )kI u and of vector

ν ν ν ν λ ν λ νθ ψ ψ λ= =( , ) ( ) in( ) ( ) ( ) ( )ads 1 2{ , , , , , 1,20}k k, k, k, k, k,u t P .

If the solution of the auxiliary problem (A) is obtained, i.e.,

ν ν ν ν λ ν λ νθ ψ ψ λ= =( , ) ( ) in( ) ( ) ( ) ( )ads 1 2{ , , , , , 1,20}k k, k, k, k, k,u t P ,

we move to the next step. If the solution of the auxiliary task (A) has not

been obtained, i.e., at least one constraint ( )( )ξ ≤st , , 0i

jg z d u , ( )( )ξ ≤st , , 0ljg z d u , = 1,5j ,

∈ ∈1 1ξ ,i S i J , ξ ∈ ∈2 2,l S l J has not been fulfilled, then we check the fulfillment of the conditions:

1) if < ×in( ) 56 10kP Pa, we accept that = + Δin( ) in(k)kP P P ( ΔP =0.1×105 Pа), k = k+1 and

pass to step 3; 2) if = ×in( ) 56 10kP Pa, the algorithm finishes its

work and the solution of the problem (14)-(21) cannot be obtained for the specified requirements to the unit

out1,defy , out

defG , ν +g .

Step 4. To determine the new critical points where the constraints (15)-(19) ( )( )ν ξ ≤( , )

st , , 0kjg z d u ,


97

= 1,5j , ξ ∈ Ξ are violated, we solve five extremum problems:

( )( )ν

ξξ( , )

stmax , ,kjg z d u , ξ ξ ξ− +≤ ≤ , = 1,5j

and define five points νξ ( )q , = 1,5q which deliver the

maximum of functions ( )( )ν ξ =( , )st , , , 1,5k

jg z d u j , respectively. In case the constraint functions

( )( )ν ξ =( , )st , , , 1,5k

jg z d u j are non-convex with res-pect of uncertain parameters ξ, it is necessary to use global search methods, for example, the Global-Search in MatLab method [50], to solve extremum problems.

Step 5. Check the fulfillment of j×q constraints (15)-(19) ( )( )ν νξ ≤( , ) ( )

st , , 0kj qg z d u , = 1,5j , = 1,5q ,

and form a new set of critical points: ν ν ν−= ( ) ( 1) ( )2 2S S R ,

( )( ){ }ν ν ν νξ= > = =( ) ( ) ( , ) ( )stξ : , , 0, 1,5, 1,5k

q j qR g z d u j q . If the set ν( )R is empty, then the solution of the

problem at ν-th iteration is obtained ν∗ = ( , )ku u and the algorithm finishes its work; otherwise, we accept ν ν= +1 and go to step 3.

If the solution of the auxiliary problem (A) at step 3 of the algorithm was not obtained, the following variants are possible:

1) to “soften” the requirements to the unit, i.e., to reduce the required oxygen purity out

1,defy or unit pro-ductivity out

defG , to increase the value of the velocity constraint

+νg in the frontal layer of the adsorbent,

and to solve the problem (14)-(21) again with the help of the given algorithm;

2) to reduce the range of changes in the values of uncertain parameters ξ ξ ξ− +≤ ≤ and solve the problem (14)-(21) anew, using the above algorithm.

It should be noted that the number of iterations necessary to obtain the optimal solution of the prob-lem (14)-(21) using the algorithm described above, as a rule, does not exceed 2-3. A certain disadvantage of the developed algorithm is an increase in the number of critical points at each iteration and, accordingly, the number of constraints to be considered.

Results of the optimization problem solution under uncertainty for cyclic processes of atmospheric air adsorption separation

The optimization problem (14)-(21) was solved for the different values of oxygen purity out

1,defy (15) and productivity of the PSA unit out

defG (16) taking into account the constraint (17) on the gas flow velocity ν g and without the constraint (17) for the following cases:

1) outdefG = 2 l/min, out

1,defy = 40, 50, 60, 70, 80, 90 vol.% without the constraint (17) on velocity ν g (Fig-ure 5a and b) and taking into account the constraint (17) on velocity ν g (Figure 6a and b);

2) out1,defy = 90 vol.%, out

defG = 0.5, 1, 1.5, 2 l/min without constraint (17) on velocity ν g (Figure 5c and

Figure 5. Results of the optimization problem (14)–(21) solution without constraint (17) on the velocity of gas flow gv : at different values

of product oxygen purity out1,defy and out

defG = 2 l/min (a, b); and at different values of unit productivity outdefG and out

1,defy = 90 vol.% (c, d).


98

d) and taking into account constraint (17) on velocity ν g (Figure 6c and d);

3) out1,defy = 90 vol.%, out

defG =2 l/min, ν g = 0.15, 0.2, 0.3, 0.4, 0.5 m/s (Figure 7).

The initial data for the solution of the problem (14)-(21) are given in Table 4.

Tables 5 and 6 show the obtained programs of time changes in valve opening and CPU time of the optimization problem solution.

Analysis of graphs in Figure 8 testifies to the fact that at full jump opening of the inlet valve (Figure 8a, curve 1) the flow rate in the frontal layer of the ads-orbent (Figure 8b, curve 1) exceeds the rate of fluid-ization (Figure 4, curve 3). Oscillating motion of the particles relative to each other and abrasion of the

adsorbent (ν g = 0.65> fv ≈ 0.4 m/s) is observed. Opti-mal programs of “soft” stepwise opening of valves allow to provide the velocity in frontal adsorbent layer, not exceeding maximum admissible value of ν g .

The comparative analysis of the obtained opti-mal regimes without the constraint (17) on the gas flow velocity ν g (Figure 5a) and taking into account the constraint (17) (Figure 6a) allows to draw a con-clusion that the fulfillment of the requirements on resource saving of adsorbent leads to the increase of energy consumption (pressure inP increases), but at the same time the increased degree of oxygen rec-overy η is achieved. For example, at out

1,defy = 90 vol.% the account of constraint (17) provides the higher rec-overy rate η on ∼17 %, and the pressure increases

Figure 6. Results of the optimization problem (14)–(21) solution under constraint (17) on the velocity of gas flow +

gv = 0.2 m/s at different values of product oxygen purity out

1,defy and outdefG = 2 l/min (a, b); and at different values of unit productivity out

defG and out1,defy = 90 vol.% (c, d).

Figure 7. Results of the optimization problem (14)–(21) solution at different values of +

gv , out1,defy = 90 vol.% и out

defG = 2 l/min.


99

Table 4. Initial data for solving the optimization problem

Initial data Value

Composition of gas-air mixture, k 1- O2, 2- N2, 3- Ar

Zeolite adsorbent 13X

Isotherm parameters

Maximum adsorption volume W0, cm3/g 0.17

Parameter B of Dubinin-Radushkevich equation, ×10-6 1/К2 6.55

Affinity ratio, σ1 1.1

σ2 1

σ3 1.1

Design parameters

Number of adsorbers in the PSA unit 2

Internal diameter of adsorber DA, m 0.04

Height (length) of adsorbent layer L, m 0.2

Diameter of adsorbent granules dgr, mm 1.6

Valve throughput capacity Kv, l/min 15

Volume of product storage-tank VS , l 2

Set values of constraints

Oxygen concentration out1,defy , vol. % 40, 50, 60, 70, 80, 90

Productivity of the unit outdefG , l/min at NTP 0.5, 1, 1.5, 2

Velocity at the frontal layer of the adsorbent ν+g , m/s 0.08, 0.15, 0.2, 0.3, 0.4, 0.5

Regime variable

Duration of adsorption step (half-cycle) adst , s 0.5–30

Pressure at the compressor outlet inP , 510× Pa 2–6

Backflow coefficient θ , rel. units 0–6

Degree of valve opening λψ j , rel. units 0–1

Uncertain variable

Concentrations of components in the initial mixture, = −in 2env 4y y b ac , vol. %

oxygen in1y 18–21

nitrogen in2y 78–80

argon and impurities in3y 1–2

Temperature of the initial mixture =ing envT T , К 293–303

Pressure at the unit outlet =indes envP P , × 510 Pa 0.75–1

Table 5. Valve opening programs obtained as a result of solving the optimization problem (14)–(21) without constraint on ν+g in (17)

Requirements ψ1 ( )t , rel. unit ψ 2 ( )t , rel. unit CPU time,min out1,defy =40 vol. %, out

defG =2 l/min 1 1 145 out1,defy =50 vol. %, out





defG =2 l/min 1 1 158 outdefG =0.5 l/min, out

1,defy =90 vol. % 1 1 115 outdefG =1.0 l/min, out

1,defy =90 vol. % 1 1 142 outdefG =1.5 l/min, out

1,defy =90 vol. % 1 1 141

Table 6. Valve opening programs obtained as a result of solving the optimization problem (14)–(21) with constraint on ν+g = 0.2 m/s


defG =2 l/min 0.26, 0.83, 1 0.45, 1 318 out1,defy =50 vol. %, out

defG =2 l/min 0.24, 0.71, 1 0.41, 1 327


100

Table 6. Continued


defG =2 l/min 0.21, 0.54, 1 0.35, 0.95, 1 331 out1,defy =70 vol. %, out

defG =2 l/min 0.19, 0.47, 1 0.32, 0.91, 1 352 out1,defy =80 vol. %, out

defG =2 l/min 0.17, 0.39, 0.95, 1 0.29, 0.88, 1 345 out1,defy =90 vol. %, out

defG =2 l/min 0.11, 0.19,0.33, 0.56, 1 0.25, 0. 34, 0.72, 0.97, 1 404 outdefG =0.5 l/min, out

1,defy =90 vol. % 0.13, 0.27, 0.56, 1 0.29, 0.51, 0.91, 1 377 outdefG =1.0 l/min, out

1,defy =90 vol. % 0.13, 0.26, 0.52, 1 0.28, 0.51, 0.89, 1 372 outdefG =1.5 l/min, out

1,defy =90 vol. % 0.11, 0.22, 0.42, 0.81, 1 0.25, 0.41, 0.82, 1 388

Figure 8. Programs (degrees) of opening of inlet valves (a) and velocity of gas-air flow in the frontal layer of the adsorbent (b): 1 – no constraint (17) on the maximum permissible velocity ν+

g ; and under constraint (17): 2– ν+g = 0.15 m/s; 3– ν+

g = 0.2 m/s; 4– ν +g = 0.3 m/s.

inP ≈1×105 Pa. Accounting for the constraint (17) on the velocity of the gas flow gv leads to an increase in the duration of the absorption and desorption steps, which reduces the number of operations of the control solenoid valves of the unit and, accordingly, prolongs their service life. It is especially evident at out

1,defy = 90 vol.%, when the difference in duration of tads of the adsorption step is 40% (Figures 5a and b).

The analysis of graphs in Figures 5c–d and Figures 6c–d shows that with the increase in the unit productivity out

defG , there is also an increase in the oxygen recovery rate of η. This regularity can be exp-lained by the increase of the inlet gas flow rate due to the increase of the compressor outlet pressure inP and the decrease of the flow tapped for adsorbent regeneration. For example, at the increase of out

defG by 4 times (from 0.5 to 2 l/min) a significant increase of oxygen recovery rate η (taking into account the cons-traint (17) - by 3 times (Figure 6c), without taking into account the constraint (17) - by 4 times (Figure5c)) is observed. This is several times higher than the inc-rease in energy consumption of the compressor due to pressure rise inP (taking into account the constraint (17) - by 1.3 times (Figure 6d), without taking into account the constraint (17) - by 1.1 times (Figure 5d)).

Decrease in maximum admissible value of velo-city +

gv from 0.5 to 0.2 m/s allows to raise oxygen recovery rate η on average by 15%, simultaneously increasing pressure inP at the compressor outlet by 12% (Figure 9), and decrease of +

gv from 0.2 to 0.15

m/s is inexpedient, as at pressure inP increase by

33% the oxygen recovery rate η grows only by ∼20% (Figure 9).

Figure 9. Dependence of PSA unit productivity outG on the set

values of product purity out1,defy and out

defG = 2 l/min: 1 – without constraint (17), 2 – under constraint (17).

Comparative analysis of the graphs in Figure 9 shows that, at the required purity of oxygen out

1,defy in the range from 40 to 70 vol.%, the productivity of the PSA unit can be increased by 13% when the cons-traint (17) on the velocity gv of gas flow is taken into account.

CONCLUSION

The problem of optimization of regime variables of PSA process of air separation and oxygen concen-


101

tration under interval uncertainty of initial composition of atmospheric air, temperature and ambient pressure was formulated and solved. A heuristic iterative algo-rithm for solving the optimization problem using the criterion of the oxygen recovery rate and taking into account operating procedure requirements on oxygen purity, unit productivity and gas flow velocity in the frontal layer of the adsorbent was developed. Gas flow velocity constraint in the adsorbent layer was provided by a stepwise change in the degree of open-ing of inlet and outlet valves in time, which made it possible to increase the adsorbent service life along with the achievement of the maximum degree of oxy-gen recovery and purity.

Statement of the optimization problem under partial uncertainty of the initial data and the dev-eloped algorithm of its solution can be used in the modernization of existing and design of new res-ource-saving PSA units for air oxygenation and other adsorption plants of separation and purification of multi-component gas mixtures, in which it is neces-sary to use expensive or unique granular adsorbents (silver-containing types of zeolites, sodium forms, per-ovskite adsorbents, etc.).

Funding. This work was supported by the Min-istry of Education and Science of the Russian Feder-ation (grant number 10.3533.2017).

Nomenclature

English symbols А1, А2 – adsorbers Ar – Archimedes number a – concentration in the adsorbent, mole/m3

a* – equilibrium adsorption value, mole/m3 B – parameter of the Dubinin thermal equation, К-2 b – molar volume constant, l/mole С – compressor

apс – specific heat capacity of adsorbent, J/(kg×K) pgс – specific heat capacity of gas-air mixture, J/(mol K)

AD – internal diameter of the adsorber, m gD – diffusion coefficient in gas phase, m2/s

d – vector of design parameters grd – diameter of adsorbent granules, m

E – characteristic adsorption energy, J/mole G – flow rate, l/min

outG – unit productivity, l/min out

1G – gas-air mixture flow rate at the unit outlet, l/min g – constant of free fall acceleration, m/s2 h – sorption heat, J/mole K – kinetic coefficient of the rate of pressure rise and fall in the adsorber, s-1

vK – throughput capacity of inlet and outlet valves, l/min

L – height (length) of the adsorbent layer, m M – mathematical expectation

gM – molar mass of gas-air mixture, kg/mole NTP – normal temperature and pressure n – number of adsorption-desorption cycles P – pressure, Pa

inP – pressure at the compressor outlet, Pa out

1P – pressure at the unit outlet, Pа Pr – Prandtl number Ps – saturation pressure, Pa PSА – pressure swing adsorption R – universal gas constant, J/(mole×К) Re – Reynolds number Recr – critical Reynolds number S – adsorber cross-section area, m2

spS – area of specific surface of pores (meso- and macropores) of the adsorbent, m2/m3 STС – compressor storage-tank STP – product storage-tank T – temperature, К

= inenv gT T – temperature of the initial gas-air mixture at

the unit inlet, К out

1T – temperature of the gas-air mixture at the unit outlet, К t – time, s

= =ads des c / 2t t t – half-cycle time (adsorption, desorp-tion), s u – vector of regime variables V – volume, m3

*V – molar volume, sm3/mmole v1, v2, v3, v4 – control valves v6 – throttle v5, v7 – backflow valves v8 – manual valve W0 – adsorption capacity, cm3/g x – spatial coordinate, m y – concentration in gas phase, mole/m3

= inenvy y – composition of atmospheric air, vol.% inky – inlet concentration of k-th component in gas-air

mixture, vol.% outky – outlet concentration of k-th component in gas-

air mixture, vol.% out,1y – concentration of gas-air mixture components

at the outlet of the unit, vol.%

Greek symbols α – heat transfer coefficient from adsorbent to gas-air mixture, Wt/(K×m2) ρ – mass transfer coefficient attributable to the concentration of adsorbate, m/s ε – adsorbent porosity coefficient, m3/m3 φ – thermal coefficient of limiting adsorption, rel. unit η – recovery rate, rel. unit


102

θ – backflow coefficient; defines the amount of outlet flow returning for adsorbent regeneration, rel. unit λa – heat transfer coefficient of adsorbent, Wt/(m×K)

gλ – heat transfer coefficient of gas mixture,

Wt/(m×K) μ g – dynamic viscosity of gas-air mixture, Pa×s ν – velocity of gas-air mixture, m/s νf – fluidization rate, m/s χ – latent condensation heat for volumetric liquid phase, J/mole ξ – vector of uncertain variables ρ – density, kg/m3

ρa – adsorbent density, kg/m3 ρ g

– molar density of gas, mole/m3 ρ * – molar density, g/sm3 ρ*

cr – adsorbate density at critical temperature, g/sm3 σ – affinity ratio ς – sphericity coefficient of adsorbent granules τ – index of the Dubinin thermal equation υ * – ratio of the components of atomic diffusion volumes ψ – degree of opening of inlet and outlet valves, rel. unit ω – weight coefficient

Subscripts + – maximum admissible value of a parameter – – minimum admissible value of a parameter A – adsorber a – adsorbent ads – adsorption step b – boiling с – cycle cr – critical C – compressor def – given des – desorption step e – experimental value env – environment, ambient g – gas phase i – countable index in – inlet j – countable index k – number of a component of the gas-air mixture nc – under normal conditions out – outlet P – product S – product storage-tank st – steady state v –valve λ – step number for a valve opening step-function

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[55] S.A. Skvortsov, E.I. Akulinin, O.O. Golubyatnikov, D.S. Dvoretsky, S.I. Dvoretsky, Journal of Physics: Confe-rence Series, 1015(3) (2018) 032002

[56] Y.I. Sumyatsky, Industrial adsorption processes, Moscow, 2009, p. 182

[57] D.N. Zhedyaevskiy, V.D. Kosmin, V.A. Lukyanov, S.S. Kruglov, Hydro-mechanical processes. Guidelines for practical training in the laboratory of processes and apparatus of oil and gas processing, Moscow, 2012, p. 76.

APPENDIX

( )υ

−

=

∏

3g g, g,

g 21/3*

10 /k kk k

kk

T M MD

P, calculated by the Fuller, Schettler, and Gidding's method [51];

*1υ =16.6,

*2υ =17.9,

*3υ =16.1 [51]; k = 1,2,3: 1 – oxygen, 2 – nitrogen, 3 – argon and impurities;

τ τϕχτ

− = + +

1 1 1* *

aads, ln lnk k

k kk k

Ta ah Ea a

[47]; τ τϕχ

τ

− = + + +

1 1 1* *

ades, 1 ln lnk k

k kk k

Ta ah Ea a

[47];

= × 14.187 4.754E

B [47]; χ1=6824.5 J/mole, χ2=5581 J/mole, χ3=5694 J/mole [51]; τ=2 [47];


104

μν

ρ= cr g

fg gr

Red

[49]; ( )( )ε ε ε ε

=− + ×

cr3 3 3

ArRe

150 1 / 1.75 / Ar / 1.75 [49];

ρμ

=3gr g

2g

gAr

d [49];

ρρ = g,nc g,nc

gg nc

T PT P

;g = 9.81

m/s2;

( ) = − −

Sg

exp / 760kk

k

FP AT C

, Ak, Fk, Ck – constants of the Antoine equation for the k-th component of gas-air

mixture [51];

α =0,83

g

gr

0.24Re λd

[49]; βε

⋅=

0.64 1/3g7.9 Re Pr

4kD

[54]; μ

ρ= g

g g

PrD

[54]; ρ

μ ε ε−= g eq g

g

2Re

3 (1 )w d

; ε=eqsp

4S

d – equivalent

diameter of adsorbent pore ducts, m [54]; =sp gr6 /S d [49];

ρ= g,*

*k

kk

MV [47];

ϕρ ρ − −= g b,0.434 ( )*b, 10 k kT T

k k [47]; ( )

ρρ

ϕ

=

−

b,*cr,

cr, b,

lg

0.434

k

kk

k kT T [47]; ρ =*

cr, 1000k

kk

Mb [47];

= cr,

cr,

18

k

k

RTb

P [47];

θ = ads des

des ads

( ) ( )( ) ( )

P t G tP t G t

, rel. unit, θ≤ ≤ ads

des

( )0

( )P tP t

[56];

B = 6.55×10-6 K-2 [14, 55]; apс = 1000 J/(kg K); pgс = 830 J/(mol K); =ads 1K s-1, =des 4K s-1 [52, 53]; W0 = 0.17

sm3/g [14, 55]; ε = 0.39 [14, 55]; λ =a 0.139 Wt/(m K) [14];

λg =0.0259 Wt/(m K) [49]; μ g =1.81×10-5 Pa s [49]; ρa =2140 kg/m3; σ1 = 1.1, σ 2 = 1, σ 3 = 1.1 [14]; ς =1

for round granules [57]; λψ≤ ≤0 1j .

EVGENY AKULININ

OLEG GOLUBYATNIKOV DMITRY DVORETSKY

STANISLAV DVORETSKY

Tambov State Technical University, Tambov, Russia

NAUČNI RAD

OPTIMIZACIJA I ANALIZA PROCESA ADSORPCIJE SA PROMENLJIVIM PRITISKOM ZA PROIZVODNJU KISEONIKA IZ VAZDUHA

Postrojenja za adsorpciju sa promenljivim pritiskom (PSA) se široko koriste za razdva-janje atmosferskog vazduha i koncentrisanje kiseonika. Međutim, efikasnost takvih pos-trojenja je smanjena zbog slučajnih promena u karakteristikama atmosferskog vazduha koji se razdvaja. U ovom radu je formulisan i rešen problem optimizacije režima rada PSA postrojenja sa zeolitnim adsorbentom 13X prema kriterijumu brzine izdvajanja kiseonika u uslovima periodične nesigurnosti sastava, temperature i pritiska atmosfer-skog vazduha. Problem optimizacije uzima u obzir, takođe, zahtev za čistoćom kiseo-nika, produktivnošću postrojenja i uštedom granulisanog adsorbenta zbog abrazije gra-nula. Predlaže se obezbeđivanje uštede adsorbenta ograničavanjem ulaznog protoka u frontalnom sloju adsorbenta pomoću „meke“ stepenaste promene otvaranja kontrolnih ulaznih i izlaznih ventila postrojenja. Problem (uključujući traženje programa za pro-menu vremena za stepen otvaranja upravljačkih ventila) rešen je primenom razvijenog matematičkog modela cikličnih procesa toplotne i masene razmene adsorpcije-desorp-cije u PSA postrojenja i heurističkog iterativnog algoritma. Izvršena je uporedna analiza rezultata rešenja problema optimizacije sa i bez uzimanja u obzir ograničenja brzine protoka gasa u frontalnom sloju adsorbenta. Izučavan je uticaj određenih zahteva na performanse PSA postrojenja i čistoće kiseonika na stepen njegovog izdvajanja.

Ključne reči: adsorpcija sa promenljivim pritiskom, zeolit, matematičko modelo-vanje, optimizacija, numerička simulacija, nesigurnost.

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