=Paper=
{{Paper
|id=Vol-3628/short2
|storemode=property
|title=Mathematical model of diffusion in a layered zeolite medium with spherical symmetry
|pdfUrl=https://ceur-ws.org/Vol-3628/short2.pdf
|volume=Vol-3628
|authors=Igor Boyko,Stepan Balaban,Vasyl Kovbashyn
|dblpUrl=https://dblp.org/rec/conf/ittap/BoykoBK23
}}
==Mathematical model of diffusion in a layered zeolite medium with spherical symmetry==
https://ceur-ws.org/Vol-3628/short2.pdf
Mathematical model of diffusion in a layered zeolite medium
with spherical symmetry
Igor Boyko, Stepan Balaban and Vasyl Kovbashyn
Ternopil Ivan Puluj National Technical University, 56, Ruska Street, Ternopil, 46001, Ukraine
Abstract
Based on the methods of integral transformations and in comparison with difference
approaches, a mathematical model is constructed that describes the nonequilibrium diffusion
process of mass transfer of volatile substances in multilayer media containing microporous
inclusions. The effect of various types of interface conditions on the efficiency of adsorption
processes has been studied. It has been established that the results obtained using direct
analytical methods of mathematical modeling correlate well with the results obtained by
using finite difference schemes approximating the diffusion equation boundary conditions
compared to it.
Keywords 1
Mass transfer, Poisson’s equation, finite difference method, layered zeolite medium
1. Introduction
A variety of zeolite materials are widely used in various and relevant areas of modern technology
[1-5]. In particular, materials such as ZSM-5 zeolite or silica gel have found their application in
exhaust gas cleaning systems [6-8]: both in the case of internal combustion engines and in cleaners of
chemical and energy industries. This predetermines one of the key roles of zeolites in the protection
and purification of the environment. The practical utility of these materials is determined by their
physicochemical properties, which consist in the presence of micropores in these materials, which
function in the mode of traps for molecules of various hydrocarbons. This effect is directly explained
by the emergence of Van Eder Waals forces at the boundary between micropores and the external
environment, as a result of which, due to attraction, they are able to capture light gas molecules, such
as methane, propane, water vapor, etc.
The problems associated with the use of zeolites and constantly requiring additional research are
the need to provide conditions for the effective functioning of traps for gaseous hydrocarbons under
various environmental conditions: pressure, temperature, percentage of gases in the adsorbed mixture.
Also a poorly studied case is the simultaneous use of several different zeolite media at the same time.
The mentioned scientific and technical problems lead to the need for additional rethinking of the
existing mathematical models of diffusion in microporous media and their modification. In the
proposed work, we propose the implementation of a mathematical model that describes the diffusion
of hydrocarbons in a layered zeolite sample with spherical symmetry. Based on the proposed model,
the concentration distributions of methane absorption were calculated for a homogeneous zeolite and
a medium consisting of different layers of zeolites. The general case is considered when the trapped
particles can carry a charge.
2. Statement of the problem. Mathematical model of diffusion in the layered
zeolite medium
Proceedings ITTAP’2023: 3rd International Workshop on Information Technologies: Theoretical and Applied Problems, November 22–24,
2023, Ternopil, Ukraine, Opole, Poland
EMAIL: boyko.i.v.theory@mail.com (A. 1); Balabantep57@gmal.com (A. 2); kovbashyn_v@tntu.edu.ua (A. 3);
ORCID: 0000-0003-2787-1845 (A. 1); 0000-0003-4829-0353 (A. 2); 0000-0002-5504-1606 (A. 3);
© 2020 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)
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
The process of diffusion of light hydrocarbons in a layered zeolite medium is considered, the
schematic geometric diagram of which is shown in Fig. 1. It is assumed that a stable adsorption
process occurs in each of the layers of the test sample, which is described at the macrolevel of the
zeolite medium by the following equation:
Ck (t , r , z ) ak (t , r , z ) 1 C 2Ck
k2Ck Dk 2 r 2 k f k (t , r , z ) . (1)
t t r r r z 2
where Dk - is the coefficient of diffusion for k-th layer.
At the macrolevel, which corresponds to the micropore-target boundary and the zeolite medium,
diffusion is described by the following condition [9, 10]:
ak
k (Ck k ak ) . (2)
t
Figure 1: Geometric diagram of cross‐section a multilayer sample of zeolite material
For the absorption process of hydrocarbons, there are also input initial conditions that set the input
concentrations in the zeolite medium and micropores:
Сk (t , z )t 0 C0k ( z ); ak (t , z )t 0 a0k ( z ) . (3)
The interface boundary conditions of the third kind, describing the vertical transport of
hydrocarbons in the diffusion process, are as follows:
0 0 0
[(11 11 t ) z ( 11 11 t )] C1 (t , z ) z l0 0 (t );
0
[( n 1 n 1 ) ( n 1 n 1 )] C (t , z ) n 1 (t );
22 22
t z
22 22
t
n 1 z ln1
(4)
k k k k k
[( j1 j1 t ) z ( j1 j1 t )]Ck (t , z ) [( j 2 j 2 t ) z ( j 2 j 2 t )]Ck 1 (t , z )
k k k
z lk
0; k 1, n; j 1, 2
Accounting for the radial symmetry of the sample also leads to boundary conditions at the
boundaries of each of the layers, as well as at the boundaries of the sample with the environment.
dC da
These boundary conditions recognize the homogeneity of the flow I k Dk k k k k within
dr dr
the sample and look like as follows:
Ck r r 0 Ck 1 r r 0 ;
k k
dCk dak dC da (5)
Dk dr k k dr Dk 1 k 1 k 1 k 1 k 1 .
dr dr r rk
r rk
In the case when particles trapped by micropores have a charge, then in the mean field model, the
potential (r , z , t ) created by them is a continuous function of coordinates, its explicit form is
obtained by finding solutions to the Poisson equation:
1 2
r , (6)
r r r
2
4 0 (r )
where the permittivity of the sample is:
N
(r ) k ( z zk ) ( z zk 1 ) . (7)
k 1
The density of charges distributed over the entire volume of the studied sample is expressed as
follows:
2
1
Ck (r , r cos ) ak (r , r cos ) sin d d
V 0 0
2
(8)
3
3 Ck (r , r cos ) ak ( r , r cos ) sin d .
2R 0 0
For the potential created by distributed charges inside the sample, within each of the zeolite layers, the
following limiting conditions are met:
k (r ) r r k 1 (r ) r r ;
k k
( r ) ( r ) . (9)
k k k 1 k 1 .
r r rk r r rk
3. Implementation of the mathematical model of diffusion in a multilayer
medium
For the direct implementation of the mathematical model given by equations (1)-(5) and additional
conditions (6), (7) and (9), the finite difference method is used. To this end, we will replace the area
n
of the sample Dn (t , r , r cos ) : t 0, r
(r , r ), l 0, l
k 1
k k 1 1 n 1 , 0 with a discrete
spatial grid that looks like as follows:
Dmkl t , r , z : tm mtm , rk k rk , zl l zl ,, k , l , m Z .
(10)
Performing now the approximation of the derivatives of the first and second orders according to the
relations:
df f fk
k 1 ;
dr h
(11)
d2 f f k 1 2 f k f k 1
; h r
dr 2 h2
we obtain a difference scheme, which is the implementation of the proposed mathematical model in a
form accessible to direct modeling:
Ck ,l , m 1 Ck ,l , m ak ,l , m 1 ak ,l , m
k2 Ck ,l , m
tm t m
D Ck 1,l , m 2Ck ,l , m Ck 1,l , m Ck ,l , m Ck 1,l , m Ck ,l 1, m 2Ck ,l , m Ck ,l 1, m f ;
rk zl
k k ,l , m
2
rk rk 2
ak ,l , m 1 ak ,l , m k (Ck ,l , m k ak ,l , m ) tm 0;
0 0 C C C C C
11 11 k ,l ,m 110 k ,l ,mz tk ,l 1,m1 110 k ,l ,m t k ,l ,m1 l
l m m
(12)
Ck ,l , m Ck 1,l , m 0;
Dk Ck 1,l , m Dk Dk 1 Ck ,l , m Dk 1Ck 1,l , m k k ak 1,l , m
k k k 1 k 1 ak ,l , m k 1 k 1 ak 1,l , m 0;
k ,l , m k 1,l , m 0; 0,l , m N 1,l , m 0;
2 k ,l , m k 1,l , m k ,l 1, m 2 k ,l , m k ,l 1, m
k ,l , m 0;
rk zl 4 0 k
2
k k 1,l , m ( k k 1 ) k ,l , m k 1 k 1,l , m 0.
4. Results and discussion
In order to verify the developed mathematical model of diffusion in layered samples, the
concentrations of adsorbed hydrocarbons were calculated. For this, the following geometric
parameters of the sample under study were chosen: the number of layers were taken 100, and the
thickness of an individual layer was taken to be 10 µm. In this case, three different cases were
considered: the sample layers were created only from silica gel; sample layers are made of ZSM-5
zeolite only; the sample layers are formed by alternating these materials. For each of these cases of
the structure of the studied sample, the concentration curves of methane and propane were calculated.
The results of these calculations are shown in Fig. 2a, b. The temperature value was chosen as typical
for technological processes of hydrocarbon adsorption and equal to 380 K The arrow in both figures
marks the moment of time corresponding to the desorption process.
0,0012
0,14 desorption
propane b ZSM-5 /silica gel
metane a desorption
ZSM-5
0,12 ZSM-5 /silica gel silica gel
ZSM-5 0,0009
0,10 silica gel
a(t,z)
0,08
a(t,z)
0,0006
0,06
0,04
0,0003
0,02
0,00 0,0000
0 50 100 150 200 250 300 350 400 450 500 550 600
0 50 100 150 200 250 300 350 400
t, s t, s
Figure 2: The dependence of the concentration of methane (a) and propane (b) in the process of
adsorption‐desorption using different types of materials in the test sample.
As can be seen from the dependencies shown in Fig. 2a, 2b, the adsorption-desorption process of
methane proceeds somewhat faster than the analogous process for propane. This is due to the fact that
micropores in the used zeolite materials have a characteristic size closer to the effective diameter of
propane molecules, which entails large Van der Waals forces. As can be seen from Fig. 2a, in the case
of methane adsorption in the range from 150 to 300 s, a "plateau" is formed with an almost constant
concentration close to the maximum. In the case of propane, this effect also takes place, but for such a
"plateau" the concentration value is almost two times less than the maximum.
As can be seen from Fig. 2a, b, in cases where methane and propane are used, the dependences of
the concentrations of the trapped substance on time have qualitatively similar dependences, the
maximum values of which are actually formed in the same time interval. However, it should be noted
that the use of only silica gel (blue dotted lines) in the sample demonstrates a weak adsorption
efficiency of hydrocarbon molecules, which is very low in the case of methane and slightly higher in
the case of propane. It should be noted that, as can be seen from the calculated dependences, the use
of only ZSM-5 zeolite (red dashed line) in the test sample has advantages over the case of using only
silica gel, since the concentration of adsorbed hydrocarbons is almost four times higher in the cases of
both gases. It should be noted that the highest adsorption efficiency is achieved when using a sample
in which layers of ZSM-5 zeolite and silica gel alternate.
5. Conclusions
A mathematical model is constructed that describes diffusion processes in layered zeolite samples
with spherical symmetry. It has been established by direct calculations that the highest adsorption
efficiency is achieved for a sample formed by alternating layers of ZSM-5 zeolite and silica gel. The
results obtained can be applied in further studies of the processes of hydrocarbon adsorption by
microporous materials.
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