Intellectual information technologies for the study of filtration
in multidimensional nanoporous particles media
Dmytro Mykhalyka, Mykhaylo Petryka, Igor Boykoa, Yuriy Drohobytskiya, Vasyl
Kovbashyna
a
    Ternopil Ivan Puluj National Technical University, 56 Ruska str., Ternopil 46001, Ukraine

                Abstract
                High-performance intellectual information technologies for the nanoporous filtration systems
                research based on the mathematical model of the two-level transport "filtration-
                consolidation" in the system of nanopores in intraparticle spaces, which includes two
                subspaces of particles of different sizes has been considered.
                The high-speed analytical solution of the model, which allows calculations parallelization on
                multi-core computers has been found using the operational Heaviside’s method, Laplace
                integral, and Fourier integral transformations.
                The high-performance software complex was built on top of the mode, with a modern
                approach to software design and keeping in mind software engineering best practices.
                Numerical modeling of filtration kinetics process research has been done using developed
                software.

                Keywords 1
                Filtration processes, numerical modeling, parallel computing, science-intensive technologies,
                multidimensional nanoporous particles media


1. Introduction
    Complex systems and processes design in the field of environmental protection, emission
reduction, medicine, liquids or gases filtration requires a new high-performance information systems
creation for their research based on scientific mathematical models with high-quality physical
substantiation of the composition of their elements, connections between them and parameters that
determine efficiency their progress and work.
    The proposed information research technology of nanoporous filtration systems is based on the
phenomenological model of a solid-liquid liquid that we developed, containing various-sized
nanoporous moisture-containing particles as a multi-level porous system with interparticle and
intraparticle networks for fluid express flows. Mathematical models of the two-level transport
"filtration-consolidation" in the system "interparticle space - nanoporous particles" are considered,
which take into account the internal flow of liquid from particles, along with the flow of liquid in the
skeleton [1, 2].
    We consider the nanoporous particles containing liquid as a porous layer subjected to
unidimensional pressing (Fig. 1). The liquid flowing occurs inside the particles, outside the
nanoporous particles and between these two spaces. The nanoporous particles are separated by the
porous network. The layer of particles is considered a double-porosity media. Fig. 1 illustrates two
levels of the considered elementary volume: level 1(a) for the system of macropores in interparticle

ITTAP’2022: 2nd International Workshop on Information Technologies: Theoretical and Applied Problems, November 22–24, 2022,
Ternopil, Ukraine
EMAIL: dmykhalyk@gmail.com (A. 1); petrykmr@gmail.com (A. 2); boyko.i.v.theory@gmail.com (A. 3); daodrg@gmail.com (A. 4);
kovbashyn_v@tntu.edu.ua (A. 5)
ORCID 0000-0001-9032-695X (A. 1); 0000-0001-6612-7213 (A. 2); 0000-0003-2787-1845 (A. 3); 0000-0002-3333-1573 (A.4); 0000-
0002-5504-1606 (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)
spaces and level 2 (b and c) for the system of nanopores in intraparticle spaces, which includes two
subspaces of particles of different sizes: intraparticle spaces 1 – subspace of nanoporous particles with
a radius of at least R1 and intraparticle spaces 2 – a subspace of nanoporous particles with a radius of
at least R2 (R1> R2). The model assumed, that in nanoporous media, the first porosity level is formed
by the interparticle network with low storage capacity, while two-second levels of porosities are
formed by the intraparticle network with high storage capacity.




                                                                                   h
                                                                                                              R1              R2



                                                                                                b) particle 1          c) particle 2

                                                                                                                   R1 > R2

                       a) layer

Figure 1: Example figure Schematization of mass transfer in a two‐level system of pores


2. Mathematical model
   The mathematical model of the considered transfer, taking into account the specified physical
factors, can be described in the form of such a system of boundary value problems for equations in
partial derivatives, formulated both for the interparticle space and for two intraparticle networks
versus the pressure in the liquid phase.:

    2.1.          Consolidation equation for a layer

  Problem A is to find a limited solution of the consolidation equation for a layer of
multidimensional nanoporous particles media in the domain D1   t , z  : t  0, 0  z  h :

             P1  t , z            2 P1         1
                                                          R
                                                                                      1   2
                                                                                               R


                                                 R1 t 0                              R2 t 0
                              b1            1          P2 ( t , x1 , z ) dx1   2           P3 (t , x2 , z ) dx2                   (1)
                 t                 z 2

   with the initial condition:

                                        P1  t , z  t  0  PE ,                                                                      (2)

   the boundary conditions (for variable z)

                                                                    P1
                                    P1  t , z  z 0  0 ;             z h  0             (impermeability condition);               (3)
                                                                    z
     2.2.               Consolidation equations for particles
    Problems В1,2: to find the limited solutions of the consolidation equations for the nanoporous
particles (radius Ri) in the domain D2                     t , x , x , z  : t  0, x <R , x <R , 0  z  h :
                                                                     1    2                          1   1        2       2


                                            Pi     2 P
                                                 bi 2i , i =2,3, j =1, 2                                                                 (4)
                                            t      x j

    with the initial conditions:

                                           Pi t 0  PE ( z ), i  2,3                                                                    (5)

    the boundary conditions (for radial variable xj) are :

                                            Pi
                                                 x 0  0
                                                          ;              Pi (t , x j , z ) x  R  P1  t , z  .                         (6)
                                            x j j                                           j   j




     2.3.               Nomenclature

    P1 - liquid pressure in interparticle space, P2 , P3 - liquid pressure in intraparticle space 1 and
intraparticle space 2 (interior of spherical particles 1 and 2) in accordance, b1 - is a consolidation
coefficient in interparticle space, b2 , b3 - consolidation coefficients in intraparticle space 1 and
intraparticle space 2, 1, , 2 - elasticity factor of the particles 1 and 2 in accordance, h - is layer
thickness, R1, R2 - radius of particles 1 and 2.


3. The analytical solution of the model
    A pressure profiles in interparticle spaces and intraparticle spaces 1 and intraparticle spaces 2.
The analytical solution of the problem is found using the operational Heaviside’s method, Laplace
integral and Fourier integral transformations.
    Applying the finite integral Fourier transform (cos) [3, 4]:
                                Rj                                     Rj
                                                                                                                                    2, j  1
                            
                                0
                                                            
    Fc  Pi  t , x j , z     Pi (t , x j , z ) x j , m j dx j   Pi (t , x j , z ) cos m j x j dx j  Pim j  t , z , i  
       
                                                                       0                                                            3, j  2
                                                                                                                                              ;

                                
     Fc1  Pim j  t , z     Pim j  t , z 
                                                          2 P (t, z) cos x  P  t, x , z , i  2, j  1 ;
                                                     x j ,m                  

                                                                 
                                                                 j

                            m 0                                                                   
                                                      x , 
                                                                                      im j               mj   j       i       j
                                                                                                     3, j  2
                                                                     2
                                j
                                                               R           1 m1  0
                                                        j   mj


         P  R  2 Pi                                                                                   2, j  1
    Fc  i2                               
                            x j , m j dx j   m2 j Pim j (t , z )   1 j  m j P1  t , z  , i  
                                                                             m

                                                                                                          3, j  2
        x j  0 x j
                         2


                                                2m j  1
                          
    were  x j , m j  cos m j x j , m j 
                                                   2R j
                                                               , m j  0,  - are the spectral functions and spectral

numbers of integral Fourier transform (cos), we obtain the solutions of the problems В1, B2 :
                            2   1 b2m21t
                                         1   m
                                                         2 
                                                                          t
                                                                             b2m21  t  
   P2  t, x, z   PE  z               e   cosm1 x    1 1 b2m1  e
                                                                  m
                                                                                              P1  , z  dz cosm1 x, x  R1
                            R1 m1 0 m1                 R1 m1 0         0




                                    2   1 b3m22 t
                                                 2   m
                                                                   2 
                                                                                           t
                                                                                              b3m22  t  
        P3  t, x, z   PE  z                  e    cos   x           1
                                                                                  m2
                                                                                     b3 m2 
                                                                                            e                 P1  , z  dz cosm2 x, x  R2
                                    R2 m2 0 m2
                                                            m2
                                                                   R3 m2 0                0

                                                                                                    (7)
   Substituting the expressions (7) into the consolidation equation (1), after a series of
transformations and successive application to the problem (1)-(3) of the integral Laplace transform [3]
and the finite integral Fourier transform (sin):

                                                 h                                      h
                      Fs  P1*  s, z     P1*  s, z   V  z , n  dz   P1*  s, z   sin n zdz  P1,*n ( s ) ,
                                                 0                                      0
                                                  
                                                                     V ( z , n )
                                                                              2  *
                      Fs1  P1,*n  s     P1,*n  s               2
                                                                                   P1,n  s  sin n z  P1*  s, z  ,
                                               n 0          V ( z , n )     h n 0

                          d 2 P1*  s  
                                           n P1,n  s, z  ,
                                               2 *
                      Fs         2
                              dz        

                       2n  1                                             2n  1
   were V ( z, n )  sin      – are the spectral functions and n              – are the spectral
                         2h                                                2h
numbers of integral Fourier transformation (Sin-Fourier).
   Applying the integral operator of the inverse integral Laplace transformation to expression (8) we
obtain [5]:

               n  s                        1            2 1                1        *
                    1  2 12                        2                                    P1 n ( s ) 
                  b         R           s b     2
                                                                 R 2
                                                                                s b     2   
                   1          1 m 1  0     2     m 1            2    m 2  0     3     m 2  


                      1   1           s / b2         1             s / b3     PE
                        2 2                       2 2 2                               
                      b1  R 1 m 1  0 s b 2   m1
                                                  2
                                                           R2                       2  
                                                                  m 2  0 s b3   m 2   n


                1   1   1   2 (1   )     1
                                           PE
                   b1            b1             n
   Using seris [3, 5]
                                                       R b  s
                                                             1       
                                                       1 2 th  R1 
                                                     2   2      
                                                           s  b2    
                                     m1  0 s b2   m               
                                                                 1


                        
                                                               2    
                                                                   1
                                         s                   R1 R1 b2  s         1
                        2                                       th  R1 
                                     2
                      m  0 m s  b2m
                                                  2
                                         m  0  m
                                                      b2m
                                                            
                                                      s 2  2
                                                            
                                                                    2 s  b2 

   as result we obtain
                                                                                                            1
                                 b2              s                        b        s       
     P ( s )   b1 n  s  1
       *
                                         s  th     R1    2 1    3 s  th        R2   
      1n
                                R1                                                   b       
                                                 b2                     R2         3       
                                                                                                                  (8)
                            b2  s                1    b3  s                 1
                    2  1        th        R1   2                th    R2   PE
                           R1   s  b2                    R2              
                                                                      s  b3           n
                                                                                

   Introducing the notation

                                             b2             s                     b           s 
             s,  n   s  b1 n  1          s  th    R1    2 1    3   s  th   R
                                                                                                b 2 
                                            R1              b2                   R2           3 

  and applying the integral operator of the inverse Laplace transformation, we obtain the formula for
making the transition to the original in equation (8):
                                                                                s        
                                                                       sh     R1     
                      2 1       1          1 1       1     1          b2        
     P1, n (t )  PE    L                P      L                * L
                                      
                                                               
                                                                           
                                            E                                            
                     n              n                        n           s         s
                             s ,          n      s ,            R1ch      R1 
                                                                        b2        b2 
                                                                                       s        
                                                                               sh      R2     
                                             2 (1   ) 1       1     1          b3        
                                        PE             L              *L  s
                                                n
                                                                     n
                                                                        
                                                              s ,           R2 ch
                                                                                            s    
                                                                                              R2 
                                                                              b3         b3                   (9)
   where L1 ... - integral operator of inverse L laplce transformation, " * " – is an operator of
convolution of both functions.
   Now we can consider the next equation:

                                            b2             s                     b            s 
                       s  b1 n  1            s  th    R1    2 1    3    s  th   R 0
                                            R1                                                  b 2 
                                                           b2                   R2            3     .

   Replacing i s   or s   2 , we obtain:

                                                  b2   R1                  b     R 
                      2  b1n2  1             tg        2 (1   ) 3 tg  2   0 .                   (10)
                                                 R1  b2                  R2  b3 

   According to Heviside theorem one can obtain the expression of transfer to original [4]:
                                                                                                         
                                                                                                         
         1                                                1                                             
      L                                                                                                 
             s  b  n    b2                       s                     b                 s     
                  1       1            s  th          R1    2 1    3     s  th        R 2  
                                                       b2                                       b3     
                            R1                                            R2                        

           1                                              e st ds
      
          2 i
                                  b2                                        b                       
                                                                                                            
                                                        s                                        s
                s  b n   
                     1       1                sth       R1    2 1    3      sth        R 2 
                                  R1                  b2                    R2               b3      


                                                              e st
                                                                                                                                ,
          j0 d                        b2                   s                     b                 s    
                   s  b1 n  1              s  th       R1    2 1    3     s  th        R2  
               ds                                           b2                                       b3    
                                   R1                                            R2                      
                                                                                                                 s   2    jn       (11)
                     ____
   where  jn , j= 1, ; n  0,  – the roots of transcendental equation (10).

   Calculating of the denominator in (11):



               d                  b                   s                   b         s    
                    s  b1 n  1 2           s  th    R1   2 1    3 s  th    R2  
               ds                                      b                             b    
                                  R1                   2                  R2         3  
                                                                                                                s   2jn


                                                                                                               
                                                                                                               
           b2  1          R1  R1          1                       b3  1          R2  R2          1          
 1  1         tg  jn                          2 (1   )         tg  jn                           
          2R1  jn        b2  b2    2     R                  2R2  jn        b3  b3    2     R2  
                                      cos  jn                                               cos  jn      
              
              
                                           
                                               b2
                                                   
                                                                       
                                                                                                      
                                                                                                           b3  


   Then, as result, the expression (11) will have the vie:
                                                                                                   
                                                                                                             2 t
                                                    1                                                     e jn
   L 1                                                                                                          .
                 n       b2                    s                     b
                                                   R1    2 1    3
                                                                                           s      j 1 1    jn    
          s  b1  1          s  th 
                                                b2
                                                                               s  th       R 
                                                                                           b3 2  
                       R1                                          R2                         
   were
                                                           
                                                           
                b2  1        R1  R1                      
   
  jn 1 1      tg  jn
               2R1  jn 
                                   
                                   
                                b2  b2 2 
                                                 1
                                                       R 
                                                            
                                          cos  jn      
                                                      b  
                                                      2 
                                                                      .
                                                                  
                                                                  
                     b3  1         R2  R2            1          
          2(1 )      tg  jn                               
                    2R2  jn           
                                      b3  b3 2            R2  
                                              cos  jn         
                        
                                                     
                                                            b3  

We calculate the Laplace originals of expressions:
                   s        
             sh      R1                                2
                   b2                      i (1)k e b2k t                                  2             2
    L 1  
           s
                             
                               d  s
                                                                                      b2
                                                                                                2 
                                                                                                    eb2k t .
              R1  ch
                        s
                          R1  k  0                  s                                     R1 k  0
                                            R1  ch     R1 
            b
          2          b 2        ds  b2           b2 
                                                                          s  k2
                       s        
                  sh     R2                              2
                       b3                      i (1)k eb3k t                                   2             2
          L1 
               s
                                 
                                   d  s
                                                                                        b3
                                                                                                   2 
                                                                                                       eb2 k t     (12)
                            s                            s                                       R2 k  0
                   R2ch      R2  k  0        R2ch      R2 
                 b
               3         b3         ds  b3        b3 
                                                                             s k2
In result of this transforms, we obtain the original of function P1
                                                                        2                         
                                                                   2 jn                            
                                                             b2 k           t                   
                                                                            b2 
                                                                                                     
                                                     1e                      
                                                                                                      
                                                                                                    
                                                           2  2jn                                 
                                        
                                                            k                                     
                                2jnt 
                                                                    b                              
                 2
P1  t , z   PE 
                     
                              e 1    2       
                                                                       2                             s in  z
                                              2 
                                                                                                               n ,   (13)
                                                                                                      
                 h n  0 j 1  ( jn )                                                                   
                                           1
                                             R1 k  0                                        2           n
                                                                                        2 jn  
                                                                                    b3  k       t
                                        
                                                                                              b3  
                                                                                                   
                                                                     2  1 e
                                                                                        
                                                                                                     
                                                       2 (1   ) 2                               
                                                                    R2 k  0
                                                                                 2  2jn  
                                        
                                                                                 k            
                                                                                          b3  
                                                                                                   


which describe the pressure distributions in the interparticle space.
                 ____
Here  jn , j= 1, ; n  0,  – the roots of transcendental equation (10).


k 
        2k  1  , k  0,  – are the roots of equation ch               s    
                                                                              R1   0 , (s=iɳ, i- imaginary unit),
                                                                               
          2 R1                                                            b2 
   k 
           2k  1  , k  0,  – are the roots of equation ch   s    
                                                                     R2   0 , (s=iµ)
                                                                      
            2 R2                                                 b3 
          2n  1
   n            – are the spectral numbers of integral Fourier transformation (Sin-Fourier).
           2h
    Substituting into formulas (7) the analytical expression of pressure distributions in the interparticle
space P1(t,z), calculated according to (13), we obtain the final expressions for determining the time-
space distributions of pressures P2(t,x,z) and P3(t,x,z) in the spaces of nanoporous particles:
intraparticle spase 2 and intraparticle spase 3 in accordance.

4. Numerical modelisation and discussion.
   As a part of the simulation stage, a special software complex was developed to study the internal
kinetics processes of filtration in multidimensional nanoporous particle media. Such software was
built with a modern approach to software design and keeping in mind software engineering best
practices.
   The main goal is to allow the quick study of filtration processes for scientists and staff by reducing
the time from inputting parameters of nanoporous media to the graphical visualization of key
performance metrics of the filtration process.
   Results of the filtration kinetics process study are presented below. The process parameters used
for simulations are: h=0.01m, R1=0.008 m, R2=0.004 m, b1 = 10-7 m2/s, b2 = 2 10-7 m2/s, b3 = 10-8
m2/s, β1 = 0.1, β2 = 0.15, ε = 0.5. The media consists of two types of multidimensional nanoporous
particles of different kinetic properties.




Figure 2: Distribution of dimensionless pressure in the intraparticles space1 P2(t,x,z) versus time t, [s]
in different sections of dimensionless layer: а) Z=0.05; b) Z=0.25; c) Z=0.5; d) Z=1 (Z=z/h);
1 – X=1.0 2 – X=0.8; 3 ‐ X=0.6; 4 – X=0.4; 5 – X=0.05 (X=x1/R1)
   Figure 2 shows the dimensionless liquid pressure profiles inside the porous particles of first type
P2(t,x,z) in time t[s]. The temporal pressure profiles were simulated for different layer sections: Z=1
(top of layer), Z=0.5 and Z=0.25 (a middle sections of layer), and Z=0 (surface of the filter medium.
In the proposed images are clear to observe that liquid pressure is higher in the center of particles
(X=0.05) and decline in direction of liquid expulsion on a particles surface X=1 ( xm  Rm ). At the
particles edge the pressure in micropores tends to the pressure in macropores P1(t,z). Also, it is worth
to note, that liquid pressure declines rapidly on the particles surface (X=1) than in the middle sections
(0.4, 0.6, 0.8) or particles center (X=0.05).
   The difference between the temporal pressure profiles becomes more significant for particles
located on top of the later (Z=0). However, even in sections close to the central axe of particles (X=0),
the liquid pressure drops rather rapidly.




Figure 3. Distribution of dimensionless pressure in the intraparticles space2 P3(t,x,z) versus time t, [s]
in different sections of dimensionless layer: а) Z=0.05; b) Z=0.25; c) Z=0.5; d) Z=1 (Z=z/h);
1 – X=1.0 2 – X=0.8; 3 ‐ X=0.6; 4 – X=0.4; 5 – X=0.05 (X=x2/R2)

   Figure 3 shows the temporal profiles of dimensionless liquid pressure inside of porous particles of
second type (small) in time t[s]. Same as before, the temporal pressure profiles were simulated for
four different sections of media layer: Z=1, 0.25, 0.5 and 0.05. The consolidation coefficient for these
types of particles characterizes less destroyed cellular tissue compared to the particles of the first type.
Like in a previous example, the presented profiles show liquid pressure drops on the surface of
particles (X=1) are more rapid than for sections close to the particle’s center (X=0.05), and overall
decline is more significant when Z leads to 0. However, appreciable retardation of liquid pressure
drop can be detected in micropores of particles.
     Figure 4. Distribution of dimensionless pressure in the interparticles space P1(t,z): 1 ‐ Z = 0.05,
                           2 ‐ Z = 0.3, 3 ‐ Z = 0.5, 4 ‐ Z = 0.7; 5 ‐ Z=1.0 (Z=z/h)

   Figure 4 shows distributions of pressure profiles in interparticles space P1(t,z) at different sections
of nanoporous media.

5. Conclusions
    During this research, was developed a foundation of scientific information technologies for
nanoporous filtration systems with multidimensional nanoporous particles. A phenomenological
model of the solid-liquid expression of a liquid containing various-sized nanoporous moisture-
containing particles as a multi-level porous system with interparticle and intraparticle networks for
liquid flows is formulated.
    The filtration-consolidation equations were formulated for both intrerparticle and two intraparticle
networks considering the pressure profiles. It was assumed, that for nanoporous media, the
interparticle network forms the first porosity with a low storage capacity, while the intraparticle
network forms two-second porosities with a high storage capacity. High-speed analytical solutions
describe the spatiotemporal pressure distributions in the interparticle space, intraparticle space
(particles of radius R1), and intraparticle space (particles of radius R2, R1>R2) are obtained for real
nanoporous geomedia with two high characteristics of stability and permeability. Numerical
simulation results showed a joint pressure drop in the intraparticle network and an increase in the
consolidation kinetics for the two types of differently sized nanoporous particles.
    In the framework of scientific information technologies, specialized software has been created for
the study of nanoporous filtration systems in media with multidimensional nanoporous particles based
on the described mathematical model. The main goals pursued in software design were to allow the
quick and detailed study of filtration processes in nanoporous for scientists, the ability to run on any
modern platforms, high-performance numerical modeling, and friendly UI/UX. The use of software
engineering best practices made it possible to create a software design that could easily be expanded
or evolved by adding new classes of scientific and special services, as well as future improvements, to
meet new requirements.

6. References

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        Kluwer, 1990.
    [2] M. Petryk, E. Vorobiev Numerical and Analytical Modelling of Solid-Liquid Expression from
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    [3] G. Doetsch Handbuch der Laplace-Transformation: Band I: Theorie der Laplace-
        Transformation. Springer Basel AG, 2013.
[4] M. Lenyuk, M. Petryk Integral Fourier, Bessel transforms with spectral parameters in
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    Dumka, 2000.
[5] M. Petryk, T. Gancarczyk, O. Khimich Methods of Mathematical Modeling and Identification
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