Mathematical modeling of diffusion transfer for charged
particles in the layered composite medium
Oksana Petryk, Igor Boyko, Yurii Stoianov, Stepan Balaban and Julia Nestor
Ternopil Ivan Puluj National Technical University, 56, Ruska Street, Ternopil, 46001, Ukraine
Abstract
Using the methods of the integral Laplace transform and fundamental Cauchy functions, for
the first time, an exact analytical solution of the mathematical model of the adsorption mass
transfer of charged particles for an inhomogeneous cylindrical limited composite medium
with a symmetrical cavity and a system of n interface boundaries with specified 2n+2 non-
stationary mass transfer modes was constructed for the first time. New recurrent algorithms
and computational procedures have been developed for constructing influence functions
generated by system inhomogeneities, boundary conditions, and a system of interface
conditions.
Keywords 1
Micro- and nanoporous systems, inhomogeneous cylindrical media, mass transfer,
inhomogeneous boundary conditions, interface conditions, Laplace transformations,
fundamental functions.
1. Introduction
The development of modern nanotechnologies and the aging of the latest nanostructures and
materials pose new challenges for studying the mechanisms of kinetics and intensification of
diffusion-adsorption mass transfer in heterogeneous multilayer medias of various configurations. The
study of the processes of adsorption mass transfer in inhomogeneous media with a cellular structure
today requires the development of new qualitative modeling methods that make it possible to describe
the complex mechanisms of the system of interface interactions between all components of the
transfer and unsteady modes of mass transfer on mass transfer surfaces. Problems of mathematical
modeling of diffusion-adsorption mass transfer in homogeneous and inhomogeneous porous media
and methods for constructing mathematical solutions of such models were considered in papers [1-3].
For homogeneous media of adsorption mass transfer, the methods of integral transformations of
Fourier, Laplace, Weber, Hankel and Hilbert were used. For inhomogeneous media, numerous
methods were mainly used [4-6]. Mathematical theory of integral transformations and their
applications for mass transfer problems for inhomogeneous and porous media, taking into account the
system of mechanisms of interface interactions between transfer elements and non-stationary modes
of mass transfer on mass transfer surfaces (taking into account the spectral parameter in the boundary
conditions and the system of interface conditions (conjugation)]. assumptions on the structures of
differential operators Fourier, Bessel, Weber, Hankel, boundary conditions and interface conditions,
integral Fourier, Bessel, Weber and Hankel transforms with a spectral parameter for inhomogeneous
n+1-complex bounded, semi-bounded and unbounded are constructed and the main solutions of
transport models are constructed (fundamental Cauchy and Green functions) are the functions of the
influence of the inhomogeneities of the problem, the boundary conditions and the system of interface
ITTAP’2022: 2nd International Workshop on Information Technologies: Theoretical and Applied Problems, November 22–24, 2022,
Ternopil, Ukraine
EMAIL: oopp3@ukr.net (A. 1); boyko.i.v.theory@gmail.com (A. 2); yuriy556s@gmail.com (A. 3); Balabantep57@gmal.com (A. 4);
nazarko.julia26@gmail.com (A. 5)
ORCID: 0000-0001-8622-4344 (A. 1); 0000-0003-2787-1845 (A. 2); 0000-0003-1848-2258 (A. 3); 0000-0003-4829-0353 (A. 4); 0000-
0003-0737-8965 (A. 5)
© 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
conditions. In paper [2] we considered a mathematical model of adsorption mass transfer in an
inhomogeneous limited n-interface nanoporous medium, the exact analytical solution of the model is
constructed and the components of the influence matrices of (principal) solutions of the system are
written out.
Figure 1: Geometric scheme of a layered microstructure with marked coordinates of the boundaries
of the separation of media
2. Mathematical description of the problem
Adsorption mass transfer in an inhomogeneous limited cylindrical n-interface adsorption medium
in coordinate r filled with n adsorbents with different physicochemical characteristics is considered.
The geometric scheme of the studied layered microstructure is as shown in Fig.1. The mathematical
model of such a transfer, taking into account the non-stationarity of mass transfer on mass transfer
surfaces (edge surfaces and contact surfaces r R j 1 , j 1, n ) and the physical assumptions given in
[2-4], can be described in the form of such a mixed boundary value problem: to construct limited in
n 1
the region Dn (t , r ) : t 0, r ( R j 1 , R j ), R0 0, Rn1 a solution of the system of partial
j 1
differential equations:
C j (t , r ) a j (t , r )
2j C j Drj B j C j f j (t , r ) ; (1)
t t
a j
j (C j j a j ) (2)
t ;
with following initial conditions:
С j (t , r )t 0 C0 j (r ); a j (t , r )t 0 a0 j (r ); (3)
and such boundary conditions and a system of interface (conjugation) conditions along the geometric
coordinate r:
[(120 120 ) ( 120 120 )] C1 (t , r ) r R 1 (t );
t r t 0
(4)
n 1
[( 22
n 1
22n 1 ) ( 22n 1 22 )] C n 1 (t , r ) r R n 1 (t )
t r t n 1
j j j j j
[(i1 i1 t ) r (i1 i1 t )]C j (t, r ) [(i 2 i 2 t ) r (i 2 i 2 t )]C j 1 (t, r ) 0;
j j j
r Rj (5)
j 1, n; i 1,2
d2 1 d
Here B j 2
(2 j 1) ( 2j 2j )r 2 is the Bessel operator for n-interface environment,
dr r dr
C j , a j - mass concentrations of the adsorbent, respectively, in the liquid phase (interparticle space)
and the solid phase (in micro- and nanopores of adsorbent grains) for the jth layer of the adsorption
medium, j 1, n 1 .
2.1. Methodology for constructing the analytical solution of the model and
recurrent algorithms for calculating the matrices of the impact functions
(main solutions)
Assuming that the desired vector functions C (t , r ), a(t , r ) are Laplace originals, we apply the
Laplace integral transformation with respect to the time variable t to the boundary value problem (1)-
(5). As a result, we obtain a boundary value problem: to construct a solution of the system of Bessel
n 1
differential equations bounded on the set I n r : r ( R j 1 , R j ), R0 0, Rn 1 for the modified
j 1
functions:
B q 2j ( p) C *j ( p, r ) j ( p, r ) (6)
j
by boundary conditions
[120 120 ] C1* ( p, r ) R*0 ( p ) ; [ 22n 1 22n 1 ] Cn*1 ( p , r ) R* n1 ( p ) (7)
r r R0 r r Rn 1
and interface conditions along the coordinate r:
j d j d k d k d
[(i1 dz i1 dt )]C j ( p, z) (i 2 dz i 2 dt )]C j 1 ( p, r ) ij ; j 1, n; i 1,2 .
* *
(8)
rRj
Here
1 j j
j ( p, r ) [ f j ( p, r ) Co j (r ) a0 (r )] . (9)
Drj p j j j
d
R* ( p ) 1* ( p ) 120 120 C01 ( R0 ) 1* ( p ) 1,1 ;
0
dr
(10)
d n 1
*
Rn 1 ( p ) ( p ) 22n 1 22
*
n 1 C0n1 ( Rn 1 ) n 1 ( p ) n 1,1 ; .
*
dr
d d
ij i1j i1j C0i (r ) i 2j i j2 C0 j 1 (r ) ; (11)
dr dr r Rj
1
q 2j ( p) p 2 p j (1 j ) j j 2j
Drj p j j
(12)
p; p; j 1, n; i, m 1, 2 .
j
im
j
im
j
im
j
im
j
im
j
im
Wherein we have
j (r )
a0 j ( r )
a j ( p, r )
C j ( p, r ); j 1, n 1 . (13)
p j j p j j
If we fix a branch of the two-sheeted function, on which Re q j ( p ) 0 , due to the properties of the
functions that form the fundamental system of solutions of the Bessel equation (6), we construct the
solution of the inhomogeneous boundary value problem (6)-(8) by the method of Cauchy functions
[3]:
Rj
2 1
Cj ( p, r) Aj Ij (qj r) Bj K j (qj r) j ( p, r, )j ( p, ) j d; j 1, n 1. (14)
Rj1
where j ( p, r , ), j 1, n 1 is the Cauchy which satisfy following conditions:
( p, r , ) j ( p, r , ) 0;
j
r 0 r 0
d d
. (15)
( p, r , ) ( p , r , )
2 j 1
.
dr j r 0 dr j r 0
The Cauchy functions j ( p, r , ), j 1, n 1 are found in the following form:
D1 I (q j r ) E1 K (q j r ); R j 1 r R j
( p, r , )
j j j j j j j
. (16)
j
D2 I (q j r ) E2 K (q j r ); R j 1 r R j
j j j j j j j
at the same time, they must satisfy additional homogeneous conditions for the left and right interfaces
j1 d
12 12j1 j 0, (17)
dr rRj1
j d j
11 11 j 0, (18)
dr r l j
Let us consider the following functions:
d
Uj1 (qs Rj ) (imj imj ) I (qsr) (imj imj )I (qs Rj ) imj Rj qs2I 1, 1(qs Rj ) ;
j j
im
dz j j
rRj Rj j j j j
(19)
d
U (qs Rj ) ( imj ) K (qsr)
j2 j
(imj imj )K (qs Rj ) imj Rj qs2K 1, 1(qs Rj )
j j
im
dr im
rRjj j
Rj j j j j
j (qs Rj , qsr) Uj1 (qs Rj ) K (qsr) Uj2 (qs Rj ) I (qsr) .
im im j j im j j
(20)
To determine the constants D1k , E1k , D2k , E2k and Cauchy functions ( p, r , ), k 1, n as a
k
result of their properties determined by conditions (19), (20), we obtain the algebraic system of
equations:
( D2k D1k ) I k k (qk ) ( E2k E1k ) K k k (qk ) 0 ;
k
( D2k D1k ) k I k k (qk ) Rk qk2 I k 1, k 1 (qk )
k 1 (21)
( E2k E1k ) k K k k (qk ) Rk qk2 K k 1, k 1 (qk ) ;
qk 2 k 1
k 1,1 k 1,2
D1k U 12
( qk Rk 1 ) E1k U 12
( qk Rk 1 ) 0 ;
D2k U
k1
11
( qk Rk ) E2k U
k ,2
11
( qk Rk ) 0 .
From the algebraic system of equations (21) we find:
D2k D1k qk 2 k K k k ( qk ) , E2k E1k qk 2 k I k k ( qk ) . (22)
As a result of the single-valued openness of the algebraic system (21), the Cauchy functions
k ( p, r , ); k 1, n 1 are defined and, as a result of symmetry with respect to the diagonal r ,
have the following structure:
qk
2k
k
11
qk Rk ,qk
k 1 q R ,q r ,R r R
k k 1 k
k 1 k
12
( p,r, ) . (23)
k
11
k k k
qk Rk 1,qk Rk k q R ,q r k 1 q R ,q ,R r R
k k 1 k
k 1 k
11 12
where
im (qk Rk 1 , qk Rk ) Uk k1,1 (qk Rk 1 )Uk ,2k (qk Rk ) Uk k1,2 (qk Rk 1 )Uk ,1k (qk Rk )
i2 m1 i2 m1
k 2, n; i, m 1, 2
(24)
11 (qn1 Rn , qn1 Rn1 ) Un,1n1 (qn1 Rn )Unn1,2
1
(qn1 Rn1 ) Un,2n1 (qn1 Rn )Unn1,1
1
(qn1 Rn1 );
i2 22 i2 22
i 1, 2.
With known Cauchy functions j
( p,r, ) , the boundary conditions at the points r R0 and
r Rn 1 the interface conditions (7) for determining the unknown coefficients A j ,B j , j 1,n 2
participating in the structures (14) of the general solution of the boundary value problem (6)-(8)
C j ( p,r ) give an algebraic system with 2n+ 2 equations:
Un,111 qn Rn qnln An Un,211 qn Rn Bn Un,112 qn1Rn An1 Un,212 qn1Rn Bn1 1n
n,1
U21 qn Rn An U21 qn Rn Bn U22 qn1Rn An1 U22 qn1Rn Bn1 2n Gn
n ,2 n ,1 n ,2
(25)
n1,1
U22 qn1Rn1 An1 U22 qn1Rn1 Bn1 Rn1 p
n 1,2
Here, the expressions containing the integrals of the Cauchy functions j
( p,r, ) in (25) and are
calculated by the formula:
c1j Rj
j1 (qj Rj1, qj ) 2 j 1
c2j Rj1
j1 (qj1Rj1, qj1) 2 1
(q R ,q R ) ( p, ) d 2j11 j1( p, ) j1 d .
Gj
*
2 j 1
12
j
11
(26)
R j Rj1 j j 1 j j 1 Rj
Rj 11
(q R , q R
j1 j j1 j 1 )
11
Next we suppose that the fulfilled condition for the unique solvability of the boundary value problem
(6)-(8), that is, the determinant of the algebraic system (25) is non-zero:
( p) det A ( p) 0 . (27)
As a result of the unique solvability of the algebraic system (25) and the substitution of the obtained
values Ak , Bk , D1k , D2k E1k , E2k , k 1, n 1 into (14), the components of the solution of the boundary
value problem (6)-(8) is can be obtained. After a series of transformations (expanding the
determinants Ak I k k (qk r ) Bk K k k (qk r ), k 1, n 1 ) we obtain recursive expressions for calculating
the components Ck ( p, r ) of the vector function C ( p, r ) - the solution of the boundary value
problem (6)-(8) in the form:
n
Ck ( p, r ) Wk1 ( p, r ) R0 ( p) Wkn1 ( p, r ) Rn1 ( p) 1k , j ( p, r ) 1 j 2k , j ( p, r ) 2 j
j 1
n 1
Rj (28)
k , j ( p, r , ) j ( p, )
2 j 1
d ; k 1, n 1.
j 1 R j 1
Here the main solutions of the boundary value problem (6)-(8) have are obtained as follows.
Functions of the influence of the left boundary condition R0 ( p ) on the kth segment of the adsorption
medium Wk1 ( p, r ) :
1
Ф
21 (q1R1, q1r)1,2 Ф11 (q1R1, q1r)1,2 ; k 1
1 1
( p)
1 k1 c1s
2s 2s 1
Ф21 (q Rk ,qk r)1,2k Ф11 (q Rk ,qk r)1,2k ; k 1, n
k 1 k
Wk1 ( p, r)
( p) s1 qs Rs
k k (29)
1 n c1s
q 2s
R
Фn1 (q r, qn1Rn1)
2s 1 11 n1
k n 1
( p) s1 s s
Functions of the impact of the right boundary condition on the kth segment of the adsorption medium
Wn1k ( p, r ) :
1 n c2s
Ф0 (q R , q r)
2s1 2s1 1 12 1 0 1
( p) s1 qs1 Rs
; k 1
1 n c2s
2s1 2s1 1
Ф22 (q Rk1,qk r)1,2k2 Ф12 (q Rk1,qk r)1,2k2 ; k 1, n
k 1 k 1
Wkn1 ( p, z)
( p) sk qs1 Rs
k k (30)
1
Фn (qn1r, qn1Rn ) 1,2 n Ф22n (qn1r, qn1Rn ) 1,2n ; k n 1
( p ) 12
Functions of the influence of the jth source on the kth segment of the adsorption medium are obtained
as follows:
0 1 , R0 r R1
q121 12 (q1R0 , q1 ) 11 (q1r, q1R1 ,)1,2 21 (q1r, q1R1 )1,2
1
11 ( p, r, )
0
( p) (q R , q r) (q , q R ,) (q , q R ) , R r R
1 1
(31)
12 1 0 1 11 1 1 1 1,2 21 1 1 1 1,2 0 1
j 2, n
j 1
c2
q j j 2s1 2ss1 1
2
s 1 qs 1 Rs
1 ( p, r, )
0
(q1R0 , q1r) j 11 (q j Rj , q j )1,2 j j 21 (q j Rj , q j )1,2
j
(32)
j
( p) 12
j 2, n
n c2s
q n21n 1 2 s 1
q R s2 s 1 1 (33)
1,n 1 ( p , r , ) s 1
s 1
Ф
0 n 1
( q1 R 0 , q1 r )Ф ( q n 1 , q n 1 R n 1 )
( p ) 12 11
impact of the j -th source ( j 2, k 1 ) on the kth segment ( k 2, n ) of the adsorption medium:
2
k 1 c1
q j j 2 s 2s s 1
s j q
s Rs
kj ( p, r , )
k
k
(qk Rk , qk r ) 1,2 k
(qk Rk , qk r ) 1,2 k
( p) 21 11
(34)
j 221 ( q j R j 1 , q j ) 1,2 j 2 j 121 (q j R j 1 , q j ) 1,2
j 2 ; j 2, k 1; k 2, n
Then, in the general case, for the action functions it turns out:
q 2 k [Ф 22 (qk Rk 1 , qk ) 1,2 k 2 Ф12 (qk Rk 1 , qk ) 1,2
k 2 ]
k 1 k 1
kk ( p, r , ) k k 1
k 2 ]
( p) [Ф 22 (qk Rk 1 , qk r ) 1,2 k 2 Ф12 (qk Rk 1 , qk r ) 1,2
k 1
(35)
k Фk 11 (qk Rk , qk r ) 1,2 k ], Rk 1 r Rk
[Фk 21 (qk Rk , qk r ) 1,2
k Фk 11 (qk Rk , qk ) 1,2 k ], Rk 1 r Rk
[Фk 21 (qk Rk , qk ) 1,2
q2n1 n c2s
k,n1 ( p, r, ) n1
p 11 sk qs1 Rs
Фn1 (q ;qn1Rn1) Фk221 (qk Rk1, qk r)1,2k2 Ф12k1(qk Rk1, qk r)1,2
2s1 2s11 11 n1
k2
n1 n n 22 (qn1Rn , qn1r)1,2n , Rn r Rn1
qn21n1 Ф11 (qn1, qn1Rn1) 12 (qn1Rn , qn1r)1,2
n
n1,n1 ( p, r, ) n1
( р) Ф (q r, q R ) n (q R , q ) n (q R , q ) , R r R .
11 n1 n1 n1 12 n1 n n1 1,2n 22 n1 n n1 1,2n n n1
The functions 1k , j ( p, z ); k 1, n 1; j 1, n of the influence of inhomogeneities of the first condition
of the j-th interface 1 j , j 1, n on the kth segment of the adsorption medium are as follows:
Ф0 (q R , q r ) ; j 1
12 1 0 1 1,2
1 j 1 c2s
11 j ( p, r ) 2 s1 2 s1 1 Ф12 (q1 R0 , q1r ) 1,2 j ; j 2, n 1
0
(36)
( p) s 1 qs 1 Rs
n 1 c
2 22s 1 Ф 0
(q1 R0 , q1r )11 (qn 1 Rn , qn 1 Rn 1 ); j n
s 1 qs 1 Rs
s 1 s 1 12
To obtain final analytical solutions that define a detailed mathematical model, it is necessary to
perform the transition from images to originals. This is done in the following way. The singular points
of the main solutions of the boundary value problem (6) - (8)
W1k ( p, r ),Wn 1k ( p, r ), 1kj ( p, r ), 2kj ( p, r ), k ,k ( p, r , ) are the branch points p and
1
1
p1,2 S1 S 2 0;
2 (37)
S1 k (1 k ) k2 ; S2 ( k k k ) 2 k k (1 2 k ) 2 k2 0 .
So, when passing to the Laplace originals, the integral over the Bromwich contour can be replaced by
an integral over the imaginary axis
i i
1 1
1
W1k (t , r ) L W1k ( p, r ) W1k ( p, r ) e dp
pt
W1k ( p, r ) e pt dp
2 i i 2 i i
1 1
W1k (is, r ) e ds 0 Re W1k (is, r ) e ds;
ist ist
2
1
0
Wn 1k (t , r ) L1 Wn1k ( p, r ) Re Wn1k (is, r ) eist ds;
(38)
1
R1kj (t , r ) L 1kj ( p, r ) Re 1kj (is, r ) eist ds;
1
0
1
R2kj (t , r ) L1 2kj ( p, r ) Re 2kj (is, r ) eist ds;
0
1
0
k,k1 (t , r , ) L1 k, k1 ( p, r , ) Re k, k1 (is, r , ) eist ds.
As a result of the unique solution of the algebraic system (25), taking into account the obtained main
solutions of problem (6)-(8) and formulas (38), we obtain a unique solution to the original boundary
value problem (1)-(5):
t t
Ck (t , r ) W1k (t , r ) R0 (t ) d Wn 1k (t , r ) Rn1 (t )d
0 0
n t
1kj (t , r ) 1 j ( ) 2kj (t , r ) 2 j ( ) d
j 1
0
t n 1 Rk1
k , k1 (t ; r , ) f k1 ( , ) C0k ( ) ( )
2 ak1
d d (39)
1
0 k1 1 Rk11
t n 1 Rk1
k k
k1 k1 2 ak1
1 1
k ,k1 (t ; r , ) e a0k ( ) d d ;
0 k1 1 Rk11
Dzk 1
1
t
ak (t , z ) k e k k ( t ) Ck ( , z )d e k k t a0k ( z ).
0
In the expressions (39) the following designations are made:
d
1 (t ) L 1 ( p ) 0 (t ) (110 110 ) C01 (r ) (t );
dr r R0
d
ln1 (t ) L ln1 ( p) n 1 (t ) ( 22n 1 22n 1 ) C0n1 (r ) (t ); . (40)
dr r R0
d d
mj ( mj1 mj 1 ) C0 j (r ) ( mj 2 mj 2 )C0 j1 ( r ) (t ); m 1, 2; j 1, n.
dr dr rRj
In order to describe the distribution of the electric field created by charged particles, which is formed
inside the layered structure under study, we use the Poisson equation, which has the form:
1 2 (r , z , t ) Q(r , z, t )
r (r , z, t ) , (41)
r r z 2
4 0 (r )
where the charge distribution in the structure is:
Q (r , z ) e Ck (t , r ) ak (t , z ) , (42)
and the permittivity of the system is now defined as follows:
N
(r ) (i ) (r r(i ) ) (r r(i 1) ) , r( N 1) . (43)
i 0
The solution of equation (41) is sought on a multidimensional grid, which is given in the following
form [7, 8]:
klm r , t , z : rk k rk , zl l zl , tm mtm , k , l , m Z . (44)
Taking into account the boundary interface conditions for the potential ( r , z , t ) and components of
the electric field:
( p ) (r , z , t ) ( p 1) (r , z , t ) ;
r r( p ) 0 r r( p ) 0
( p ) (r , z , t ) ( p 1) (r , z , t ) . (45)
( p) ( p 1) .
r r r( p ) 0
r r r( p ) 0
In addition, it is taken into account that there is no electric field outside the microstructure, which
gives the following condition:
(r , z , t ) z 0 ( r , z , t ) z 0. . (46)
Thus, the finite difference scheme for finding solutions to the Poisson equation acquires the following
form:
k ,l ,m k 1,l ,m 0;
2 k ,l , m k 1,l , m k ,l 1,m 2k ,l ,m k ,l 1,m Qk ,l , m
0;
rk zl 4 0 k
(47)
k k 1,l ,m ( k k 1 ) k ,l ,m k 1k 1,l , m 0; .
0,l ,m N 1,l ,m 0;
Q0,l ,m QN 1,l ,m 0.
3. Discussion of the results
In order to implement the developed mathematical model, the concentration distributions were
calculated using relations (39). For this, the technological parameters of the microstructure
experimentally studied in the papers [9, 10] were used. The microstructure under study is created
from films of materials Al203/SiO2. In our calculations, it was assumed that the film thickness Al203 is
0.1 µm and for thickness SiO2 is 0.09 µm.
In Figure 2 are shown the results of calculations of the concentration С (t , r ) for charged particles
in the sample under study, as well as the cross section of the spatial dependence of the plane, which
more demonstrates the radial distribution of particles in the layers of the microstructure.
Figure 2: Spatial dependence of the concentration of charged particles in the intercrystalite space
and the cross section of this dependence
Dependencies shown in Figure 2 demonstrate the concentration distribution of charged particles in the
microstructure, which is formed by 15 layers of Al203 and 16 layers of SiO2. As can be seen from the
figure, during a given time interval (up to 30 minutes), charged particles accumulate in the layers of
the microsystem in such a way that they are contained mainly in the layers of the Al203 material (light
stripes), in contrast to the layers of SiO2 where particles do not actually accumulate.
Further in Figure 3 are shown the results of calculating the concentration in the micropores of the
layers of the studied microscopic sample. As can be seen from the calculated dependences, the
concentration of charged frequencies in micropores is less than in the intercrystalite space, which is
evidence that further filling of pores with particles is impossible. This is a sign of the achievement of
phase equilibrium between the substance in the intercrystalite space of the microstructure layers and
the substance in the pores. It should be noted that the cross section of the spatial dependence by a
plane for this dependence is actually similar to the dependence shown in Figure 2 with the difference
that the concentration increase does not occur immediately, but approximately 10 minutes after the
adiabatic start of the process.
Figure 3: Spatial dependence of the concentration of charged particles in the porous space and the
cross section of this dependence
Further, Figure 4 shows the result of calculating the spatial electrostatic potential created inside the
microstructure by the charges accumulated in it. As can be seen from the calculated dependence, the
electric field actually exists only inside the layers, which are mainly filled with charged particles. At
the boundaries of the layers formed Al203 and SiO2, the potential of the electric field drops
significantly and rapidly.
Figure 4: Spatial dependence of the electric field potential inside the studied microstructure
4. Concussions
In the proposed paper, a mathematical model of adsorption mass transfer in a limited cylindrical
layered microporous medium is developed and analytical solutions are obtained for the first time,
which generally describe the influence of factors of the internal kinetics of charged particle transfer,
the main among which is the influence of non-stationary conditions of the system of n-interface
relationships. . This makes it possible to model and build graphical concentration profiles of the
adsorbate in macro- and micropores, to calculate the distribution of the electric potential inside the
studied sample, to carry out a comprehensive analysis of the internal kinetics of mass transfer both at
the macro level and at the level of micro- and nanopores of adsorbent particles, design optimal
technological schemes and investigate for optimality various non-stationary modes of diffusion-
adsorption mass transfer for multi-complex adsorption media with different physic-chemical
conditions. The obtained model solutions and effective recurrent matrix algorithms for constructing
matrices of influence functions of the boundary mass transfer problem are important in formulating
and solving inverse mass transfer problems - to determine the kinetic parameters of the process, which
makes it possible to check the adequacy of the modeling parameters and physical experiment.
5. References
[1] R. Krishna, J. M. van Baten, Investigating the non-idealities in adsorption of CO2-bearing
mixtures in cation-exchanged zeolites, Sep. Purif. Technol. 206 (2018) 208–217.
[2] M. R. Petryk, I. V. Boyko, O. M. Khimich, M. M. Petryk, High-Performance Supercomputer
Technologies of Simulation of Nanoporous Feedback Systems for Adsorption Gas Purification,
Cybern. Syst. Anal. 56 (2020) 835–847.
[3] M. R. Petryk, A. Khimich, M. M. Petryk, J. Fraissard. Experimental and computer simulation
studies of dehydration on microporous adsorbent of natural gas used as motor fuel, Fuel. 239
(2019) 1324–1330.
[4] B. Puértolas.; M. V. Navarro, J. M. Lopez, R. Murillo, A. M. Mastral, T. Garcia. Modelling the
heat and mass transfers of propane onto a ZSM-5 zeolite, Sep. Purif. Technol. 86 (2012) 127–
136.
[5] B. Puértolas, M. Comesaña-Hermo, L.V. Besteir, M. Vázquez-González, M.A. Correa-Duarte.
Challenges and Opportunities for Renewable Ammonia Production via Plasmon-Assisted
Photocatalysis, Adv. Energy Mater. 12 (2022) 2103909-1– 2103909-23.
[6] L .F. De Freitas, B. Puértolas, J. Zhang, J. W. Medlin, E. Nikolla. Tunable Catalytic Performance
of Palladium Nanoparticles for H2O2 Direct Synthesis via Surface-Bound Ligands, ACS Catal.
10 (2020) 5202–5207.
[7] H. Deng. A Heaviside function-based density representation algorithm for truss-like buckling-
induced mechanism design, Int J Numer. Methods Eng. 119 (2019) 1069–1097.
[8] A. Z. Khurshudyan. An identity for the Heaviside function and its application in representation
of nonlinear Green’s function, Comput. Appl. Math. 12 (2019) 32-1–32-12.
[9] M. Broas, O. Kanninen, V. Vuorinen, M Tilli, M. Paulasto-Kröckel. Chemically Stable Atomic-
Layer-Deposited Al2O3 Films for Processability, ACS Omega. 7 (2017) 3390–3398.
[10] D. Arl, V. Rogé, N. Adjeroud, B. R. Pistillo, M. Sarr, N. Bahlawane, D. Lenoble. SiO2 thin film
growth through a pure atomic layer deposition technique at room temperature, RSC Adv. 31
(2020) 18073–18081.