Mathematical modeling of diffusion transfer for charged particles in the layered composite medium

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 nonstationary 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 

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 

t

t

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  Rj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 D  (t, r) : t  0, r (Rj1, Rj ), R0  0, Rn1   a solution of the system of partial n  j1  differential equations: C j (t, r) a j (t, r)  ( 1 ) 

 2jC j  Drj B j C j   f j (t, r) ;  with following initial conditions: a j   j (C j  j a j ) t

;  С j (t, r)t0  C0j (r); aj (t, r)t0  a0j (r);   ( 2 )  ( 3 )  and such boundary conditions and a system of interface (conjugation) conditions along the geometric coordinate r: [( 102   102 t ) r  (102   102 t )]C1 (t, r ) rR0  1 (t); [( 2n21   2n21 t ) r  ( 2n21   2n21 t )]Cn1 (t, r ) rRn1   n1 (t)   [(i1j i1j t ) r  (i1j  i1j t )]Cj (t,r) [(ij2 i2j t ) r  (i2j  ij2 t )]Cj1(t,r)   rRj  0;   j 1, n;i 1, 2

d 2 Here B j  dr 2  1r (2 j  1) ddr  ( 2j  2j )r 2 is the Bessel operator for n-interface environment, 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 In  r : r (Rj1, Rj ), R0  0, Rn1   for the modified  j1  functions: by boundary conditions B j  q 2j ( p) C *j ( p, r)  j ( p, r)   and interface conditions along the coordinate r: [102 r  102 ]C1* ( p, r) rR0   R*0 ( p) ;  [ 2n21 r   2n21 ]Cn*1 ( p, r) rRn1   R*n1 ( p)   [(i1j ddz  i1j ddt )]C*j( p, z)  (ik2 ddz  ik2 ddt )]C*j1( p, r)   rRj ij ; j 1, n;i 1, 2 .

 

Here 1

Drj j ( p, r)  [ f j ( p, r)  Coj (r)   

j j p    j j a0 j (r)] .   R*0 ( p)  1* ( p)   102 ddr   102  C01 (R0 )  1* ( p)  1,1;    *

Rn1 ( p)   n*1( p)   2n21 ddr   2n21  C0n1 (Rn1)  n*1 ( p)  n1,1; .  ij   i1j ddr   i1j  C0i (r)   i2j ddr   ij2  C0 j1 (r) rRj  ;  q 2j ( p) 

1  p2  p  j (1   j )   j j  2j   

Drj  p   j j    imj   imj   imj  p; i mj   i mj   imj  p; j  1, n;i, m  1, 2 .  ( 4 )  ( 5 )  ( 6 )  ( 7 )  ( 8 )  ( 9 )  ( 10 )  (11)  (12) 

Wherein we have

a0j(r)

 j(r) 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 Cj(p,r)Aj Ij(qjr)Bj Kj(qjr)  j(p,r,)j(p,)2j1d; j1,n1.

Rj1   where j (p,r,), j 1,n 1 is the Cauchy which satisfy following conditions: j (p,r,) r0 j (p,r,) r0  0;     d   .

dr j (p,r,) r0  ddr j (p,r,) r0   2j1.   The Cauchy functions j (p,r,), j 1,n 1 are found in the following form:

 j  D1j Ijj (qjr)  E1j Kjj (qjr);Rj1  r    Rj j (p,r,)   .

  D2j I jj (qjr)  E2j K jj (qjr);Rj1    r  Rj   j at the same time, they must satisfy additional homogeneous conditions for the left and right interfaces 1j21ddr 12j1j 0,

rRj1   1j1 ddr 11jj 0, 

 rlj Let us consider the following functions:

Uj1im(qsRj)(imj ddz imj)Ijj(qsr)rRj (imjRjj imj)Ijj(qsRj)imjRjqs2Ij1,j1(qsRj);  j Uj2im(qsRj)(imjddrimj)Kjj(qsr)rRj (imjRjj imj)Kjj(qsRj)imjRjqs2Kj1,j1(qsRj) 

j jim(qsRj,qsr)Uj1im(qsRj)Kjj(qsr)Uj2im(qsRj)Ijj(qsr). 

To determine the constants D1k ,E1k ,D2k ,E2k and Cauchy functions k(p,r,),k 1,n as a 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;  (D2k  D1k )k k Ikk (qk)  Rkqk2Ik1,k1(qk)    (E2k  E1k )k k Kkk (qk)  Rkqk2Kk1,k1(qk)   qk  2k1 ; 1 D1k Uk112,1(qkRk1)  E1k Uk112,2(qkRk1)  0 ;   

D2k Uk111(qkRk )  E2k Uk,121(qkRk )  0 .  From the algebraic system of equations (21) we find:

D2k  D1k  qk2k Kkk (qk) ,  E2k  E1k  qk2k Ikk (qk) .  (14)  (15)  (16)  (17)  (18)  (19)  (20)  (21)  (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: (23)  (24)  k ( p,r, ) where

qk2k k11 qkRk ,qk  k112 qkRk1,qkr ,Rk1  r   Rk . 11 qkRk1,qkRk  k11 qkRk ,qkr k1 qkRk1,qk  ,Rk1   r  Rk  

 12 im (qk Rk1, qk Rk )  Ukk1i2,1(qk Rk1)Uk,2km1 (qk Rk )  Ukk1i2,2 (qk Rk1)Uk,1km1 (qk Rk )   k  2, n;i, m  1, 2 11 (qn1Rn , qn1Rn1)  Un,1n1i2 (qn1Rn )Unn1,1222 (qn1Rn1) Un,2n1i2 (qn1Rn )Unn1,1122 (qn1Rn1); 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 Aj ,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: Un,111 qnRn  qnln  An Un,121  qnRn  Bn Un,112 qn1Rn  An1 Un,122  qn1Rn  Bn1  1n Un,121 qnRn  An Un,221 qnRn  Bn Un,122  qn1Rn  An1 Un,222  qn1Rn  Bn1  2n  Gn   (25)  Un212,1  qn1Rn1  An1 Un212,2  qn1Rn1  Bn1  Rn1  p Here, the expressions containing the integrals of the Cauchy functions  j ( p,r , ) in (25) and are calculated by the formula:

G*j  Rc2j1jj1 RRjj1 1j111(2q(qjRjRj1j,1q,qjRjj)1) j(p,)2j1d  R2jc2jj11 RRjj1 j11111((qqjj11RRjj,q1,jq1jR1j1)) j1(p,)2j11d .  (26)  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)  Wk1 ( p, r) R0 ( p)  Wkn1 ( p, r)  Rn1 ( p)  j1 1k,j ( p, r) 1j  2k,j ( p, r) 2j    n1 Rj k,j ( p, r, )  j ( p, ) 2 j 1d ; k  1, n  1.   j1 Rj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 (28)  medium Wk1 ( p, r) : k1 1 n c

 1s Фn1(qn1r,qn1Rn1)  2s1 11  (p) s1 qs2s Rs  1

n   2s

2s1R2s11  (p) s1 qs1 s c

Ф012 (q1R0,q1r)       1 W(p,r)   k1        W  ( p,r) : n1k      W (p, z)  

kn1 as follows: 11 ( p,r, )  j  2,n 1j ( p,r, )  j  2,n  j1 q2 j  j c2

s s1 qs21s1 R2s11

s  ( p) 1,n1 ( p, r ,  )  n

c qn21n1 

2s s1 qs21s1 Rs2 s1 1  ( p ) k  n 1

;k 1 ;k  n 1         (31)  (32)  (33)  (34)  012 (q1R0,q1r) j11 (qj Rj ,qj ) 1,2 j  j21 (qj Rj ,qj ) 

1,2 j   Ф012 (q1R0 , q1r )Фn111 (qn1 , qn1Rn1 ) k1 k  Ф (q R ,q r) Ф (q R ,q r)  (p) s1 qs2s Rs2s1  21 k k k 1,2k 11 k k k  ;k 1,n 1,2k    (29)  Functions of the impact of the right boundary condition on the kth segment of the adsorption medium 1 n c 2s

k1  Ф (q R ,q r)

2s1R2s11  22 k k1 k   (p) sk qs1 s  1     (p) Фn12 (qn1r,qn1Rn)1,2n Ф2n2(qn1r,qn1Rn)  1,2k2

 1,2n  k1 Ф (q R ,q r)  ;k 1,n 12 k k1 k 1,2k2    (30)  Functions of the influence of the jth source on the kth segment of the adsorption medium are obtained q21 012 (q1R0,q1 )  111 (q1r,q1R1,) 1  ( p) 012 (q1R0,q1r)  111 (q1,q1R1,) 1,2  21 (q1r,q1R1)1,2 , R0    r  R1

1 1,2  21 (q1,q1R1)1,2 , R0  r    R1 1 impact of the j -th source ( j  2, k  1) on the kth segment ( k  2, n ) of the adsorption medium: q2 j k1 j s j q2s R2s 1 s s c

1s  ( p) 1,2 j2  kj ( p, r, )   k21 (qk Rk , qk r)1,2k  k11 (qk Rk , qk r)

  1,2k   j212 (q j Rj1, q j )    j112 (q j Rj1, q j )    ; j  2, k 1; k  2, n 1,2 j2  Then, in the general case, for the action functions it turns out:

 kk ( p, r, ) 

 q2k [Фk212 (qk Rk1, qk )  

k  ( p)  [Фk212 (qk Rk1, qk r)    1,2k2 1,2k2  Фk112 (qk Rk1, qk )   1,2k2

]   Фk112 (qk Rk1, qk r)   1,2k2 ]  k k [Ф21 (qk Rk , qk r)  1,2k  Ф11 (qk Rk , qk r)  

1,2k ], Rk1    r  R k k [Ф21 (qk Rk , qk )  1,2k  Ф11 (qk Rk , qk )    qn21n1 Ф1n11(qn1,qn1Rn1)1n2(qn1Rn,qn1r)1,2n 2n2(qn1Rn,qn1r)1,2n,Rn  r  Rn1 n1,n1(p,r,)   (р) Ф1n11(qn1r,qn1Rn1)1n2(qn1Rn,qn1)1,2n 2n2(qn1Rn,qn1)1,2n,Rn r   Rn1.   The functions 1k, 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  1, n on the kth segment of the adsorption medium are as follows: j   0 Ф12 (q1R0 , q1r)1,2; 1  j1 c2s 11 j ( p, r)    ( p)  s1 qs21s1 Rs2s11 Ф012 (q1R0 , q1r)1,2 j ;  n1 c2s  s1 qs21s1 Rs2s11 Ф012 (q1R0 , q1r)11 (qn1Rn , qn1Rn1); j  n 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 ) W1 ( p, r),Wn1k ( p, r), 1kj ( p, r), 2kj ( p, r), k,k1 ( p, r, ) are the branch points p   and k

1 p1,2   S  S2   0;  

2  1

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

W1 (t, r)  L1 W1 ( p, r) 

k k  1 

 W1 (is, r)  eist ds  2  k 1  i

 W1 ( p, r)  e pt dp  2 i  i k 1 

 Re W1 (is, r)  eist  ds;  0 k Wn1k (t, r)  L1 Wn1k ( p, r)  R1kj (t, r)  L1 1kj ( p, r) 

 R2kj (t, r)  L1 2kj ( p, r)  1 

 Re Wn1k (is, r)  eist  ds;  0 1 

 Re 1kj (is, r)  eist  ds;  0 1 

 Re 2kj (is, r)  eist  ds;  0 1 i

 W1 ( p, r)  e pt dp  2 i i k k,k1 (t, r, )  L1 k,k1 ( p, r, )  1 

 Re k,k1 (is, r, )  eist  ds.

 0 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 ): (36)  (37)    (38)  ln1 (t)  L ln1 ( p)  n1(t)  ( 2n21 ddr   2n21) C0n1 (r) rR0 mj  ( mj1 ddr   mj1)  C0 j (r)  ( mj 2 ddr   mj 2 )C0 j1 (r) rRj 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  r (r, z,t)  2 (r, z,t)   Q(r, z,t) ,  (41)  r r z2 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  mtm , k,l, m  Z  .  Taking into account the boundary interface conditions for the potential  (r, z, t) and components of the electric field:   (t); . 

(40)    (t); m  1, 2; j  1, n.

(39)  (44)  (45)  (46)   ( p) (r, z,t) rr( p) 0   ( p1) (r, z,t) rr( p) 0

;  ( p)  ( p) (r, z, t) r   ( p1)  ( p1) (r, z,t) r .

.   (r, z, t) z0   (r, z,t) z  0. . 

rr( p) 0 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: 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  2 k,l,m  k,l1,m  rk

zl  k k 1,l,m  ( k   k 1) k,l,m   k 1 k 1,l,m  0;  0,l,m   N 1,l,m  0; Q0,l,m  QN 1,l,m  0.

Qk,l,m 4 0 k  0; .  (47) 

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. 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 diffusionadsorption 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 

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 bucklinginduced 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 AtomicLayer-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 .