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\title{NUMERICAL SIMULATION OF 2D GAS TRANSFER DYNAMICS IN
AN ADSORBING OPEN SYSTEM OF THE INTERACTING PARTICLES}
\author{Kh.T.Kholmurodov$^{*}$, V.K.Fedyanin, I.V.Puzynin\\
Laboratory of Computing Techniques and Automation, \\
Joint Institute for Nuclear Research, Dubna, Russia; \\
$^{*}$ Permanent address: Tajik Technical University, Dushanbe; \\
E-mail: mirzo@lcta6.jinr.dubna.su}
\date{}
\maketitle
We investigate the gas transfer dynamics under the unfluence of an "active
element". In the term of "active element" we imply polymer (D=1), surface
(D=2) and crystall (in particularly, zeolite) (D=3). The active element may
adsorbed (or desorbed) particles from a carrying gas stream. For the
description of gas transfer dynamics in the adsorpbate- adsorbent open system
we proposed (see [1-6]) a model in which, unlike the well- known sorption
dynamics models, the interactions between adatoms and activated complexes
(AC) are explicitly taken into account. The equations of the model for
dimensionless concentrations $\nu$ in gas phase and $\theta$ in sorbent look
like:

\vskip 2truemm

$$\frac{\partial\nu}{\partial\tau}+\frac{\partial\theta}{\partial\tau}+
(\stackrel{\rightarrow}v \stackrel{\rightarrow}\nabla)\nu = d\Delta\nu ,
\eqno(1)$$
$$\frac{\partial\theta}{\partial\tau}= v_{ads}-v_{des} =
\Bigl\{\frac{\nu}{ap_0}-1 \Bigr\}
\Phi(T,\varepsilon,\varepsilon_1,z,\theta),
\eqno(2)$$
$$\frac{1}{ap_0} = \psi(\theta).
\eqno(3)$$

\vskip 2truemm

  All designations can be find elsewhere in [1-6]. In the works [2,3],
within the framework of the proposed model, in three approximations of
particle correlation effects, namely, in the molecular field approximation
(MFA), quasichemical approximation (QCA) and polynomial fission (PF) the
gas isotherms at different values of adatoms interaction parameter
$\beta\varepsilon$ $(\beta=(k_BT)^{-1})$ were obtained. The numerical results
for the fixed values of the particles interaction energy $\varepsilon$, such
that $\beta\varepsilon =1.0,1.5$ and $2.0$, are shown in Figs.1.1-1.3,
respectively. The results of MFA are drawn by the dashed lines, of QCA- by
the solid lines, and of the PF approximation- by the signs of "+". The
analysis of these results shows the essential nonlinear dependence of surface
coverage rate $\theta$ on the gas concentration $\nu$ with the growth of
$\beta\varepsilon$. In [4], on the basis of the numerical solution the system
of eqs.(1)-(3), for 1D physical adsorption a "hiden mechanism" of
concentrational wave formation is founded that is owed to the parameter
$\beta\varepsilon$.

\vskip 2truemm

  In this report we simulate the two-dimensional gas transfer on the active
surface with the help of the numerical solution of eqs.(1)-(3). For the time
integration of these equations we developed an iterative scheme, based on
the continuous analog ot the Newton's method [7]. The numerical results of
the evolution of concentration $\nu(\xi_1,\xi_2,\tau)$ for the five moments
of time in the interval $[0\div 1]$ with the step $\Delta\tau= 0.2$ are shown
in Figs.2.1-2.3. The Figs.2.1.-2.2 correspond, respectively, to the following
values of adatom interaction parameter: $\beta\varepsilon=1.0,1.5,2.0$.
>From the Figs.2.1-2.3 it is easily to see the irregular evolutions of the
concentration $\nu(\xi_1,\xi_2,\tau)$ in the gas stream (also such a picture
exists for the concentration $\theta(\xi_1,\xi_2,\tau)$ in the adsorbent).
Thus, the introducing of the interactions in the adsorptional system change
essentially the gas transfer dynamics and we observe the concentration wave
formations in the system.

\newpage

\centerline{\Large \bf References}

\vskip 5truemm
\noindent
1. Fedyanin V.K., Puzynin I.V., Kholmurodov Kh.T.
 JINR Rapid Comm. 1994. N 1-64. P.26.			 \\
2. Puzynin I.V, Fedyanin V.K., Kholmurodov Kh.T.
 Preprint JINR, P17-94-135, Dubna, 1995;
 Journal of physical chemistry. 1995. V.69. N 3. P.492.  \\
3. Puzynin I.V, Fedyanin V.K., Kholmurodov Kh.T.
 Journal of physical chemistry. 1995. V.69. N 8. P.2062. \\
4. Puzynin I.V, Fedyanin V.K., Kholmurodov Kh.T.
 Preprint JINR, P17-95-405, Dubna, 1995;
 Journal of physical chemistry (to be published).	 \\
5. Puzynin I.V, Fedyanin V.K., Kholmurodov Kh.T.
 Preprint JINR, P17-95-406, Dubna, 1995;
 Journal of physical chemistry (to be published).	 \\
6. Kholmurodov Kh.T., Fedyanin V.K., Puzynin I.V.
 Proseeding of the International Conference PLDS-2,
 Dubna, 1995; Journal "Physics of Low-Dimensional
 Structures",10/11 (1995) pp.181-186.                    \\
7. Puzynin I.V., Vinitsky S.I.
 Muon Catlyzed Fusion, 1988, V.3, P.307.

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