MATHEMATICAL MODELLING and RESEARCH for DIFFUSION
PROCESSES in MULTILAYER and NANOPOROUS MEDIA
M. PETRYK, O. SHABLIY, M. LENIUK, P. VASYLUK
Ivan Puluy Technical State University
Laboratory of Mathematical Modelling of Technological Processes
Rue Ruska, 56, 46001 Ternopil, Ukraine
Abstract
The problem of mass transfer for diffusion and adsorption in non-regular
disperse and porous multilayer media with no stationary regimes of mass
exchange processes on the mass exchanged surfaces, which is described by
systems of differential equations with boundary conditions and contact
conditions, is introduced. The exact analytical solution of the problem by the
application of Laplace, Fourier and Bessel integral transforms and the
fundamental function method of Cauchy is established. Models were selected in
accordance with component distribution in the multilayer media Me-MeO area.
The solution of equation systems for different models as well as the
investigation of the functions made it possible to find the effective meaning of
the diffusion coefficient. The investigations will allow to correct multilayer
chemical composition, to understand better the high-temperature oxidation
mechanism and adsorption mechanism in nanoporous n-component structures.
1. Introduction
The development of modern technologies in chemical technology, engineering
ecology and biotechnology raises new problems in the kinetics, mechanism and
acceleration of mass transfer processes for diffusion and adsorption in
multilayer media, which one considers as well as technological materials. To
study such processes it is necessary to develop new simulation methods and
mathematical models to describe the internal kinetics considering different
regimes and no stationary modes of change of mass on the surfaces of mass for
homogeneous media. Multilayer oxide media (Fe-Cr, Fe-Tb, Fe-Dy, etc.) are
used in the fields of semi-conductor engineering, nuclear energy, building
technology, thin films and covering, mineral filament production.
1
2. Analysis of scientific content of modelling methodology
The mathematical modeling of mass transfer processes for different
homogeneous media and the creation of analytical solutions are described in the
work of Lykov and Mykhaylov [1], Ufiland [2], Sergienko et a.l [3], Kärger and
Ruthven [9], Fraissard, et al. [6,7,10,11], Mathieu-Balster and Sicard [12],
Zapolsky et al. [13], de Boor [14], Mongiovi [15]. These methods use integral
transformations for the simulation of these processes. The methods of Fourier,
Laplace, Fourier-Bessel, Weber, Hankel and Hilbert integral transformations
will be used intensively for the solution of mass transfer problems, especially
for diffusion and adsorption processes in regular media.
The theory of integral hybrid transformations and application for
simulation of mass transfer problems has been developed in the work of Leniuk
and Petryk [4]. Considering that the framework of Fourier, Bessel, Weber and
Hankel differential operators, boundary conditions and contact conditions offer
a more general view, we have created unified Fourier, Fourier-Bessel, Weber
and Hankel transformations for unlimited, half-limited and limited non-regular
media. This work is devoted to the simulation of mass transfer for diffusion and
adsorption processes.
3. The Mathematical Model of Diffusion Process in Multilayer Media
The definition of the diffusion kinetics characteristics enables one to update an
elemental multilayer chemical composition. By selecting a model one considers
the construction of media, modes of mass transfer and experimental
concentration profiles on each unit in multilayer media. For example,
experimental concentration distributions of Al, Cr and Si in oxide media - fig. 1.
The Fe-Cr multilayer systems are applied as a structural material in
nuclear engineering and in mineral fiber production. Oxides, including
multilayered ones, find wide application in semi-conductor engineering. The
creation of multilayered oxides on a surface provides high heat resistance in the
interval of 1273-1623 K. The oxide is based on components such as: Cr, Al, Si.
Amongst others, alloys (mass %) such as Fe-35Cr-(0.5, 3)Al are of great
interest. When the content of Al is 0.5% oxides on the alloy surface are formed
according to the scheme - Me-Al2O3-FeCr2O4. If the Al content is raised to 3%,
then the scheme changes to Me-FeCr2O4-Al2O3. This is accompanied by a sharp
rise in the heat resistance, because external Al oxide formation prevents oxygen
penetration into the internal layers.
3.1. DETERMINATION of KINETIC PROCESS PARAMETER
Determination of kinetic diffusion characteristics allows to correct the chemical
composition of the alloy as well as to predict its service life. The alloys 1020
2
mm specimens were held at 1200С for 20 h and weighed every 5 h [21, 22].
When choosing a mode, the working conditions as well as the element
distribution pattern in the Me–MeO zone were taken into account. The Al, Cr,
and Si distribution curves include peaks at the places where the corresponding
oxides are formed (fig. 1).
80
C,%
70
60
S1
50
40
30
20
10
0
0
50
100
Al x 10
Cr
150
200
Si x 10
z ,мкм
Figure 1. Experimental concentration distributions of Al, Cr and Si
expérimentales d s concentrations des Al, Cr, Si en multicouche d’oxyde
~
The effective diffusion coefficients D for the elements Al, Cr and Si were
determined by means of inverse problem solution using mathematical modelling
methods, and the parameters obtained were used to determine new
concentration profiles for the diffusion media investigated.
3.2. METHODOLOGY of ANALYTICAL SOLUTION CONSTRUCTION of
MODEL
Based on the results of experimental work as well as numerical investigations
on kinetic coefficient determination (effective diffusion coefficients Dk, k=1,n)
a mathematical model of the mass transfer diffusion process for a multilayer
oxide medium was built according (fig. 2) to the scheme Me-Al2O3-FeCr2O4 in
the form of an equation system of second-order partial derivatives:
Ck (t , z )
 2 Ck
  k2Ck  Dk
t
z 2
(1)
According to the initial conditions:
3
Ck ( t ,z )t 0   k ( z ), k  1,n
(2)
y
C1
C3
C2
D1
D2
Z
l0
l1
l2
lnl
Figure 2. Schematic representation of the multilayer structure of an oxide
medium
the boundary conditions:
 0
0  d
0
0 
(



)
 11
  11
11
11

C1 (t , z )z  l 0  g10 (t )
t dz
Cn 1
z    0
z
(3)
and contact conditions:
 k
k  
k
k 
 (  j1   j1 t ) z   j1   j1 t  Ck


 



 (  kj2   kj2 )   kj2   kj1  Ck  1 
 0 (4)
t z
t 

 z  lk
j  12,k  1,n
The exact analytical solution of the problem (1)-(4) can be constructed using
Fourier integral transformations, determined by the following integral operators
[4, 21]:
- the integral operator of direct action:
lk
~
F, n Ck (t , z )   Ck (t , z )Vk ( z,  ) k  C (t )
(5)
l k 1
4
-
the integral operator of inverse action:

~
2 ~
F,1n C (t ,  )   C (t ,  )Vk ( z )2 ( )d  Ck (t ) k  1, n
 0
- the identity principal of integral Fourier transformation of Laplace
differential operator:


(6)
 
 n 2

1
~
F, n   2 Ck (t , z )( z  lk 1 )(lk  z )  2C (t ,  )   1D1 110 V1 (l0 ) g10 (t ) 
k 1 z

(7)
n 1
li
i 1
l i 1
  ki  Ci (t , z )Vi ( z ) i dz.
The following designations will be used: Vk ( z,  )
of the Fourier integral transformation:
k  1,2,3 - the components
n
Vk ( z ,  )   C 21, k 1bk1 1 ( k 1, 2 ( ) cosbk z   k 1,1 ( ) sin bk z );
k1 1
Vn 1 ( z ,  )   n 2 ( ) cosbn 1 z   n1 sin bn 1 z;
n
k  
j k
 n ( ) 
1 C11, j
1 C11,n 1
1
; n 
;  n1 
;
Dk C 21, j
Dn C 21,n Dn
Dn1

bn1

n 2 ( )
2
  n1 ( ) 2

1
; bk2 
2  k k
Dk
; k k2   12   k2 , k  1, n;
k
c j1,k       ; m, j  1,2; k  1,2,3 ;  mj
,  mjk , - the experimental
k
2j
k
1j
k
1j
k
2j
constants, which determine conditions of mass transfer on the mass exchanging
surfaces lk, k=0,n (boundary conditions and contact conditions);
 01 (  )  v11011 (b, l 0 );  02 (  )  v1102 (b, l 0 ) ;
 jm (  )   j 1, 2 (  )1mj (bi l j ; bi 1l j )   j 1,1 (  )2jm (b j l j ; bi 1l j );
km
k
 jm
(bk , l k , bk 1l k )  v11j (bk l k )v 22
(bk 1l k )  v 21j (bk l k )v12km (bk 1l k );
k
k
vk1jm ( bs lk )   kjm qs sh qs lk   jmk ch qs lk ; v kjm2 ( bs lk )   kjm qs ch qs lk   jmk shqs lk
s k,k  1 j,m  1,2 k  1,n
As a result of the integral operators (5)-(7), applied to the equations (1)-(4), the
Cauchy problem can be obtained:
5
~(t ,  )
dC
~(t ,  )  F (t )
 (k1  2 )C
dt
~
C (t ,  )t 0  ~( )
(8)
(9)
The solution to the Cauchy problem (8)-(9) is the function [5]:
t
~(t ,  )  e ( k1 )t ~( )  e ( k1 2 )(t  ) F ( )d
C

(10)
0
The determination of the main solutions of the boundary problem (1)-(4):
- Grin function [4]:
Wk ,1 ( z, l 0 , t ) 
2

e

( k1  2 ) t
Vk ( z,  )V1 (l0 ,  ) 2 ( )d
(11)
0
-
fundamental function (Cauchy function ) [4]:
 k ,k1 ( z,  , t , ) 
2

e

( k1  2 )( t  )
Vk ( z,  )Vk1 ( ,  ) 2 ( )d
(12)
0
As a result the exact analytical solution of the boundary problem (1)-(4), which
describes diffusion process in the multilayered medium, has the following form:
 1D1  3 k
C (t , z )  Wk ,1 ( z, l0 , t )k (t )  11     k , k ( z,  , t , ) g10 (t   )dd (13)
 0 0 k 1 l 1
1
k
3.3. MODEL TESTING, SIMULATION and ANALYSES of the DIFFUSIVE
MULTILAYERS.
The experimental concentration curves, which were used in the model as
boundary conditions r and initial conditions, are presented in fig. 3 [21].Using
the obtained analytical model solution (13) as well as the first experimental Al
concentration profiles (fig. 3.), the inverse problem of the effective diffusion
~
coefficient D determination for Al as a function of a specimen thickness can be
solved. The results of computer simulation – numerical values of the effective
~
diffusion coefficient D by the experimental Al concentration distribution data
are presented in fig. 4. This is the first stage of model testing. As it can be seen,
the diffusion coefficient dependence on specimen thickness coordinates is
determined by the experimental concentration profile.
6
0.8
0.6
C Al_exp. k
C Cr_exp.k
0.4
C Si_exp. k
0.2
0.0031
0
0
0.2
0.4
0.6
0.8
1
0
1.2
1.4
1.6
1.8
2
2.0
lk
Figure 3. Experimental concentration curves of Al, Cr and Si
5.0 10  54 10 5
D Al
10 5
0
0
0.4
0.6
0.8
1
0
1.2
1.4
1.6
1.8
2
2.0
lk
Figure 4. The results of numeric modeling on the effective diffusion coefficient
~
D for Al determination, [mkm2/s]
0.4
0.3
C Al_model ( 1000  z)
0.2
C Al_experm
k
0.1
0
0
0
0
0.2
0.4
0.6
0.8
1
z  lk
1.2
1.4
1.6
1.8
2
2.0
Figure 5. Model and experimental concentration distributions of Al vs.
specimen thickness
7
The following stage of the investigation implies that the diffusion coefficient
constituent values obtained are placed into the analytical solution (13) – model
distribution. With the resulting, more precise, model distribution, numerical
modelling of the Al concentration profiles is performed. The results of the
modelling are presented in fig. 4. Along with the numerical model concentration
distribution obtained, the experimental distribution is placed in the graph. This
constitutes the second stage in testing the proposed model.
As it can be seen in fig. 5, the mathematical modelling results,
possessing sufficient accuracy, are close to their corresponding experimental
parameters. The maximum error value does not exceed 2-3%.
By the same scheme (using model (13) and experimental concentration
profiles) the investigations on effective Cr and Si diffusion coefficient
determination as well as their simulation profiles were carried out. Fig. 6-9
present the results of simulation on effective Cr and Si diffusion coefficient
calculation, as well as simulation results given together with experimental data
for Cr and Si concentration distribution. The Cr distribution curve has three
~
peaks. The effective diffusion coefficient D was determined for each of them.
~
For D calculation the Cr concentration distribution curve was divided into
three areas (first area – from 88 to 116 mkm, second area – from 116 to 138
mkm, third area – from 138 to 166 mkm). The experimental data and the
simulation data correspond quite well.
~
~
~
The obtained values of DCr , D Al , DSi confirm the fact that the component
mobility decreases at transition from internal to external oxides, which in the
best way secures their protective properties. The layered phase oxide content
was determined using methods.
1. 10
D Crk1.
-4
5 10
5
0
0
0
0.2
0.4
0.6
0.8
1
lk
1.2
1.4
1.6
1.8
2
2.0
~
Figure 6. Numerical modelling results on the effective diffusion coefficient D
determination for Cr, [mkm2/s]
8
0.8
0.8
0.6
C Cr_model ( 1000  z)
0.4
C Cr_experm
k
0.2
0
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
z  lk
0
2
2.0
Figure 7. Model and experimental concentration distributions of Cr
vs. multilayer structure thickness


D Si 2 10 5
k
i
0
0
0.2
0.4
0.6
0.8
1
0
1.2
1.4
1.6
1.8
2
2.0
lk
~
Figure 8. Numerical modelling results on the effective diffusion coefficient D
determination for Si, [mkm2/s]
0.5
0.4
C Si_model ( 1000  z) 0.3
C Si_experm
k
0.2
0.1
0
0
0
0
0.2
0.4
0.6
0.8
1
z  lk
1.2
1.4
1.6
1.8
2
2.0
Figure 9. Model and experimental concentration distributions of Si
vs. specimen thickness
9
4. Mathematical Modelling of Mass Transfers in Multilayer Porous
Media.
Definition of the process.
The processes of mass transfer by adsorption in multilayered porous media
are introduced (fig. 10). The liquid phase with an adsorption substance
concentration C (t , r , z ) supplies the porous media of the "solid-liquid" system
in a working chamber in directions z and r. As a result of this process, a
substance in a liquid phase is transformed into a solid phase of adsorption media
with concentration a (t , r , z ) [16].
z
ln
l1
l0
r
Figure 10. The working chamber of process
The mathematical model, which describes the kinetics of this process, is
considered as the boundary problem by system: to create in the area
D1  t  0,r  ( 0, ),z : z ( l0 ,l1 )  ( l1 , );l0  0
the limited solution of the system of partial differential equations:
10
 2 1  
Ck ak
 2 Ck

 k2 Ck  Drk  2 
C

D
 f k ( t,r,z ) (14)
zk
 k
t
t
r r 
z 2
 r
a k
  k C k   k a k 
t
(15)
with initial conditions:
Ck t , r , z  t  0  C0 k r , z , ak t , r , z  t  0  a0 k r , z 
(16)
the boundary condition for variable z:
 0
0   
0
0 
(  11
 11
  11   11 )C1  t,r,z 
t  z
t

z 0
 w1  t,r  ;
C 2
z
0
(17)
z 
the contact conditions for variable z:
  1 1    1 1  

 1 1    1 1  
  j1   j1    j1   j1  C1 (t,r,z )    j 2   j 2    j 2   j 2  C2 (t,r,z )
t  z
t 
t  z
t 

 

  j1(t,r ) (18)
z  l1
the boundary condition for variable r:

Ck  t,r,z 
r
r 0
 0;
Ck
z
(19)
0
r 
Here: Dz , Dr  the components of effective diffusion coefficients for
k
k
coordinate z and r,
k
- the adsorption constants,
k
- the mass transfer
coefficients, k - section number of porous media.
Using in a problem (14)-(19) the Laplace integral transformations for variable
t
and the Bessel integral transformation for variable r [4, 16], we obtain the
following boundary problem: to create in the area I1*  z : z  (l0 , l1 )  (l1 , ); l0  0
the limited solution of the equation system:
~
d 2Ck*
~
~
 qk2Ck2 ( p,  , z )   Dzk1 Fk* ( p,  , z )   g~k* ( p,  , z )
2
dz
(20)
11
with boundary conditions:
 0 d
0  *
 1* ( p,  ),
 11  11  C1 ( p,  ,z )
dz


z  l0
dC*2 ( p,  ,z )
dz
 0 (21)
z 
and contact conditions:
 1 d

 1 d
1  *
1  *
 j1 dz   j1  C1 ( p,  ,z )   j 2 dz   j 2  C2 ( p,  ,z )





  j1    , j  1,2 (22)
z  l1
Isi
Fk*  fk* ( p,r,z )  Cok ( r,z ) 
k k
 

a ok ( r,z );1* ( p,z )  *0 ( p,r )   110   110 Co1 ( r,z )
p  k k
 z

z l
0
 *o (
p,r )  o1 ( r ) ;


 j1 (r )    1j1
qk2 







  1j1 C01 (r , z )    1j 2
  1j 2 Co1 (r , z )
z
z





z l1


1  p 2  p  k (1   k )  k2   k  kk2
1
 D rk  2  


Dzk 
p   k k
 Dzk ( p   k  k )


 p 2  p  k (1   k )  k2  Dzk  2   k  k (k*  Drk  2 )

Fixing a direction, where Re qk  0 , creates a solution of a problem (20)-(22)
using the fundamental function method of Cauchy [4]. As a result, we obtain a
unique exact solution of a problem (20)-(22):
2
~
~
~
~ * ( p,  ) 
C k* ( p,  , z )  W1*k ( p,  2 , z )
 R *j1,k ( p,  2 , z )~ j1 (  ) 
1
j 1

l1

~
~*
2
*
~ * ( p ,  ,  ) d  H
H k*1 ( p,  2 , z ,  ) g
1
 k 2 ( p,  , z,  )g~2 ( p,  ,  )d
(23)
l1
l0
The Cauchy functions of system (20) are:
12
~
H11* ( p,  2 , z ,  ) 
0
2
1 11 (q1l0 , q1 z ) S1 ( p,  ,  ), l0  z    l1
;
 0
q1( p) 11
(q1l0 , q1 ) S1 ( p,  2 , z ), l0    z  l1
C
~
H12* ( p,  2 , z,  )   21 110 (q1l0 , q1 z )eq2 ( l1 ) ,
( p )
C
~
H 21* ( p,  2 , z,  )   11 110 (q1l0 , q1 )e q2 ( zl1 ) ,
( p)
~*
H 22
( p,  2 , z ,  ) 
 q ( l )
2
1 e 2 1 S 2 ( p,  , z ), l1  z    
;

q2 ( p) e q2 ( z l1 ) S 2 ( p,  2 ,  ), l1    z  
The Grin functions of boundary conditions are (21):
S ( p,  2 , z )
C q
~
~
W11* ( p,  2 , z )   1
, W12* ( p,  2 , z )  11 1 e q2 ( z l1 ) ,
( p )
( p )
The Grin function contact conditions (22) are:
1
1
22
q2  22
1 q   1 0
0
11
( q,l0 ,q1z ), R*211, ( p,  2 ,z )  12 2 12 11
( q1l0 ,q1z; )
( p )
( p )
 ( q l ,q z )
21( p,  2 ,z ) q2 ( z  l1 ) *
*
2
R11
(
p,

,z
)


e
,R21,2 ( p,  2 ,z )  11 1 0 1 eq2 ( z  l1 )
,2
( p )
( p )
*
2
R111
, ( p,  ,z )  
Here :
m2
m1
 mjk ( qi lm, qi z )  V jk
( qi lm )chqi z  V jk
( qi lm )shqi z,
01
02
11
j1( q1l0 ,q1l1 )  V11
( q1l0 )V 12
j 2 ( q1l1 )  V11 ( q1l0 )V j 2 ( q1l1 ),
1
1
1
1
1
S1 ( p,  2 , z )  ( 12
q2  12
) 121 ( q1l1 ,q1 z )  (  22
q2   22
)11
( q1l1 ,q, z )
1
S2 ( p,  2 ,z )  11( q1l0 ,q1l1 )122 ( q2l1 ,q2 z )   21( q1l0 ,q1l1 )12
( q2l1 ,q2 z )
Special points of the problem (19)-(21) solution are:
p1, 2  


1
S1 (  2 )  S 2 (  2 )  0 and p   .
2
Here:
S1   k (1   k )  k2  Drk  2 ;



S 2  (  k  k  k2  Drk  2 ) 2   k  k 1  2 k   2 k2  Drk  2  0.
13
Thus, in the area Re p  0 we can determine the following original analytical
functions [4]:
1 i *
1  *
2
pt
2
ist
W
(
p,

,z
)e
dp

 1k
 Re W ( is,  ,z )e  ds;
2 i i
 0  1k
W1k ( t,  2 ,z ) 



1
~
~
R j1,k (t ,  2 , z )   Re R *j1,k (is ,  2 , z )eist ds;

0



1
~
~
H kj (t ,  2 , z )   Re H kj (is ,  2 , z,  )eist ds;

Z kj ( t,  2 ,z, ) 
0


1
 j j t
2
ist
 Re H kj ( is,  ,z, ) e  e

0


  j j  is 
 j j Dzj1
2
 12 12
ds,
Returning to the original solutions of the problem (20)-(22) and after
mathematical transformations, we obtain the exact solution of the problem (14)(19):
t 

0 0
0
Ck (t , r , z )    W1k (t   , r ,  , z ) 0 ( ,  ) dd   W1k (t , r ,  , z ) 01 (  ) d 
2 
t  l1
   R j1,k (t , r ,  , z ) j1 (  ) d     (t   , r ,  , z,  ) Dz11[ f1 ( ,  ,  ) 
j 1 0
0 0 l0
t 
 C01 (  ,  ) t ( )] ddd     H k 2 (t   , r ,  , z,  ) Dz21[ f 2 ( ,  ,  ) 
0 0 l1
 l1
 C02 (  ,  ) t ( ) ddd    Z k1 (t , r ,  , z,  )a01 (  ,  )dd 
0 l0

   Z k 2 (t , r ,  , z,  )a02 (  ,  ) dd ; k  1,2
(24)
0 l1
ak (t , r , z )  e
  k k t
t
a0k (r , z )   k  Ck (t   , r , z )e k k d
(25)
0
Here, the main solutions of the boundary problem (14)-(19) are:

W1k ( t ,r,  ,z, )   W1k ( t ,  2 ,z )J 0 (  ) d  , k  1, 2 ;
(26)
0
14

R j1,k ( t,r,  , )   R j1,k ( t,  2 ,z )J 0 (  r )J 0 (  ) d  ;
j,k  1, 2
(27)
0

H kj ( t,r,  ,z, )   H kj ( t,  2 ,z, )J 0 (  r )J 0 (  ) d  ;
k , j  1, 2 (28)
0

Z kj ( t,r,  ,z, )   Z kj ( t,  2 ,z, )J 0 (  r )J 0 (  ) d  ;
k , j  1, 2 (29)
0
5. Conclusions
The models developed and exact solutions obtained allow a more general
view, which enables one to analyze and investigate different variants of
the schemes and regimes of diffusion and adsorption process.
Using the obtained analytical solutions of the proposed models as well as
experimental concentration profiles and solving the inverse problems, the
kinetic parameters of internal processes (the effective diffusion coefficients, the
adsorption constants, and the mass transfer coefficients as functions of
geometrical coordinates of layer media) are defined.
The methodology of the process model solution can be further developed
and applied to:
- Simulation of diffusion processes in multilayered media and adsorption
processes in n-component media ( n  2 ) and media of complicated structure;
- Models which consider mass transfer by diffusion and
convection/adsorption processes as well as non-linear models.
15
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