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\title{An Investigation of Dynamics of Oxygen Burning Out
of Lift-reactors and Pecular Features of Gas
Flowing Through a Layer of Immovable Catalyzer}
\author{S.~F.~Urmancheyev\thanks{The Bashkir State University, Ufa, RUSSIA},
C.~I.~Mikhaylenko\thanks{The Institute of Mechanics of R.A.S., Ufa, RUSSIA},
R.~R.~Vezirov and E.~G.~Telyashev\thanks{The Ufa State Oil Technology
University, Ufa, RUSSIA}}
\begin{document}
\maketitle
\begin{abstract}
The work presents a mathematical model of the flow of dispersed particles
accelerated with a stream of reacting gas in the two-dimensional domain.
The model provides to take into account interactions of gas streams and
the dispersed phase. Computerized experiments have allowed to find the areas
for fuel gas to be injected in order to maximize the rate of oxygen burning
out of catalyzer pores. Special features of gas flowing through a layer of
immovable built-in porous medium are studied, these being determined by
surface deformation of the layer. From the practical point of view the
solution of the problem is closely related to theoretical calculations
and optimization of the processes in lift-reactors and reactors with
a layer of immovable catalyzer.
\end{abstract}
\section{\large\bf Introduction}
Investigations into dynamics of the flow of dispersed particles are of much
interest for calculating catalytic reactors with a layer of moving catalyzer.
One of the designs of such units is a lift-reactor, with a fine-dispersed
catalyzer fed into the reaction zone and accelerated with a gas stream
(Fig.~1). The absence of symmetry in the distribution pattern of the stream
sets up prerequisites for the problem to arise in connection with proportional
intermixing at the bottom of the reactor. This problem is of special
significance if one takes the reagent as an accelerating gas to prepare a
gas/particles mixture for the principal reaction; for example, when in order
to suppress phenol formed during the catalytic cracking process, it is
necessary to burn oxygen out of gas mixture fed into the reaction zone
together with the catalyzer, by means of propane in the stream of
accelerating gas. Efficient intermixing can be realized at the expense of
rational arrangement of the areas of gas injection.
Considering that gas injection is performed through the walls of the reactor
reservoir, the control over the adjacent flow is extremely important in
optimizing the design features of the most different chemical reactors.
As a design modification it is possible to apply surface deformation of a
layer of porous medium to favour intermixing among gas components to the best.
The present paper is aimed at mathematically simulating a dispersed catalyzer
flowing in the riser of a lift-reactor and at determining the areas of
combustible gas to be injected so that to ensure the greatest possible
oxygen burn-out in the reactor operating zone. Among the aims is also the
investigation of some peculiar features of gas flowing through a layer of
immovable catalyzer.
The mathematical model of the flow of the reacting gas mixed with dispersed
catalyzer is based upon equations of dynamics of continuous media
(Nigmatulin,~1990). Since the flow of the dispersed phase at the initial
stage occurs with the interaction of particles with each other, a necessity
has arisen to take this interaction into account in the model. It seems
worthwhile to draw on the concepts presented by Savage (Savage,~1979).
\begin{figure}[hbt]
\centerline{\epsfig{file=fonea.eps, height=7cm}
\hspace{.5cm}\epsfig{file=foneb.eps, height=7cm}}
\vspace{.3cm}
\centerline{\sf (a)\hspace{3.3cm}(b)\ }
\caption{Scheme of operating domain of the lift-reactor (a);
catalyser concentration field at ordinary gas injection (b).}
\end{figure}
\section{\large\bf A Mathematical Modelling of a Fine-Dispersed Catalyzer
Flowing along the Channel}
Let us consider a flow of the two-phase gas/fine-dispersed catalyzer mixture
at the bottom of a lift-reactor, with acceleration of the catalyzer up to
the transportation regime at the expense of gas injection (Fig.~1).
As the catalyzer enters the operating zone on leaving the regeneration
chamber, its temperature will be high enough ($\sim 650\,{}^\circ$C). Due to
high heat capacity of the catalyzer and its considerable content by mass,
the temperature of the injected gas becomes equal rapidly to that of the
catalyzer. Thus, there is a reason to think that the conditions of the process
are isothermic. Filtration heating and heat realease in the course of
subsidiary chemical reactions are neglected.
Let us assume that a gas stream containing solid particles corresponds to
the hypotheses of dynamics of multiphase media (Nigmatulin,~1990) with
regard both to interphase forces and interaction of the particles.
In this case the behavior of interphase forces is supposed to be
conditioned by the state of velocity nonequilibrium at sufficiently slow
flows determined by Stokes' frictional force.
Interaction among particles in a dense flow of the dispersed phase suggests
such medium to be a collision one. Pursuing a goal of practical results and
ignoring the deformational peculiar features of granular media, let us take
up a simplified formulation of the problem on the flow of dispersed medium,
when components of the given stress tensor, that is responsible for contact
pulse transmission among particles, are determined only by the deviator of
deformational velocities, with the dependence between viscosity coefficient
and solid phase content by volume taken into account (Savage,~1979).
The solution of the problem in the two-dimensional domain requires boundary
conditions to be laid down properly. It is necessary to note in this connection
that the assumption accepted in dynamics of multiphase media on the dominating
role of dispersed phase viscosity in the forces of phase interaction as
compared to their action under shear deformations should not lead to ignoring
tangential stresses in the same phase. This conclusion is based on the
necessity to interpret the boundary conditions and their influence upon
the method of solving the problem in physical terms.
Relying on the given arguments, let us write a system of equations for
the process under investigation:
\dmn{\fd{\rho_i}{t}+\nabla\rho_i\vec v_i=0\,,}
\dmn{\begin{array}{l}
\ds \fd{\rho_iv_i}{t}+\nabla\rho_iv_i|\vec v_i|= \\
\qquad = -\alpha_i\,{\rm grad}\,p +
\mu_i \Delta\vec v_i + \vec F_{ij} - \rho_i\vec g\,,
\end{array}}
where
\begin{description}
\item[$\rho_i$] is the density of the $i$-th phase averaged by space,
kg$\cdot$m${}^{-3}$;
\item[$a_i$] is the content of the $i$-th phase by volume;
\item[$\vec v_i$] is the velocity vector of the $i$-th phase, m$\cdot$s${}^{-1}$;
\item[$p$] is the dispersive phase pressure,
kg$\cdot$m${}^{-1}\cdot$s${}^{-2}$;
\item[$\mu_i$] is the phase dynamical viscosities,
kg$\cdot$m${}^{-1}\cdot$s${}^{-1}$;
\item[$\vec F_{ij}$] is the vector of phase interaction forces,
kg$\cdot$m${}^{-3}$;
\item[$\vec g$] is the free fall acceleration, kg/m${}^{-2}\cdot$s${}^{-2}$.
\end{description}
The equation of state for the dispersive phase is as follows:
\dmn{p=\rho_1^0RT_c\qquad(T_c=\rm const)\,,}
where
\begin{description}
\item[$\rho_1^0$] is the true density of the 1st phase, kg$\cdot$m${}^{-3}$;
\item[$R$] is the gas constant for the given gas mixture, \\
m${}^2\cdot$s${}^{-2}\cdot$K${}^{-1}$;
\item[$T_c$] is the temperature in the reactor, K.
\end{description}
The following relationships are added:
\dmn{\rho_i=\alpha_i\rho_i^0\,;\qquad a_1 + a_2 = 1\,.}
Expressions for phase interaction forces are taken in the form:
\dmn{\vec F_{ij}=-\vec F_{ji}=\eta_\mu\alpha_i\alpha_ja^{-2}
(\vec v_i-\vec v_j)\,;}
where
\begin{description}
\item[$\eta_\mu$] is the structural ratio;
\item[$a$] is the diameter of particles in the dispersed phase, m.
\end{description}
Besides, let us treat the law of variations in viscosity of a dispersed phase
in terms of the formula put forward by Savage (Savage,~1979) and on the basis
of Bagnold's experiments:
\dm{\mu_2=\beta\left(\frac{\alpha_{2\ast}-\alpha_{20}}
{\alpha_{2\ast}-\alpha_{2}}\right)\,,
\quad\alpha_{20}<\alpha_2<\alpha_{2\ast}\,;}
where
\begin{description}
\item[$\beta$] is the empirical ratio;
\item[$a_{2\ast}$] is the greatest possible concentration of the dispersed phase;
\item[$a_{20}$] is the least concentration required for ``fluidity'' of the
dispersed phase.
\end{description}
In combination with formulae (5) and (6) the system of equations (1)--(4)
makes up a closed model of fine-grained catalyzer flowing. This system has
been numerically realized by means of a computer code based on the modified
Finite Volume Method, with the SIMPLE algorithm applied.
Boundary conditions correspond to the design illustrated in Fig.~1(a).
\dm{\vec v_i\Big|_{(inp)}=\vec v_{(inp)}\,,
\qquad \vec v_i\Big|_{(inj)}=\vec v_{(inj)}\,,}
\dm{\qquad \vec v_1\Big|_{(wall)}=0\,,
\qquad \fd{\vec v_2}{y}\Big|_{(wall)}=0\,,}
\dm{p\Big|_{(out)}=p_{(out)}\,,}
\dm{ \qquad\alpha_1\Big|_{(inp)}=\alpha_{(inp)}\,,
\qquad\alpha_1\Big|_{(inj)}=\alpha_{(inj)}\,.}
Figure~1(b) shows the result of calculating the field of the dispersed
phase concentration and attests to irregular distribution of the catalyzer
over the zone of the reactor chamber under discussion.
\section{\large\bf Calculations of Oxygen Concentration}
Catalyzer is supplied into the reactor chamber on leaving the regeneration
zone, and, as special investigations show, there is a gas mixture with
remaining oxygen among other components. Since a lift-reactor may be used
in catalytic cracking process, this oxygen may result in forming
phenol and subsequently dioxins unwanted from the environmental standpoint.
For the remaining oxygen to be removed off the zone of principal chemical
reactions most efficiently, it has been suggested that oxygen should be
burnt out at the stage of accelerating the catalyzer at the bottom the
lift-reactor chamber by injecting combustible and accelerating gases in
combination. It has been also taken into account that this combustible gas
and its products are neutral both for the efficiency of catalytic cracking
process and catalyzer activity.
In view of oxygen considerably small initial concentration and heat moderate
total input because of oxidation reaction, let us consider the process of
burning out oxygen in the context of the mathematical model described above
with additional equations to determine concentrations
of the given components of the gas mixture:
\dm{\fd{c_m\rho_1}{t}+\nabla c_m\rho_1\vec v_1 = {\cal J}_m\,,}
where
\begin{description}
\item[$m=1$] corresponds to oxygen, and $m=2$ corresponds to combustible gas;
\item[${\cal J}_m = {\cal J}(c_1,c_2,T)$] is the intensity of the chemical reaction.
\end{description}
Figure~2(a) presents the calculating data for oxygen concentrations at
normal gas injection. It is evident that in this case oxygen does not
completely burn out of the zone in question, that being associated with
catalyzer irregular distribution. Numerical experiments have allowed to find
the best mode of injecting combustible gas at fixed consumption. The
associated field of oxygen concentrations is shown in Fig.~2(b).
\begin{figure}[htb]
\centerline{\epsfig{file=ftwo.eps, height=7cm}
}
\vspace{.3cm}
\centerline{\sf (a)\hspace{2.1cm}(b)\ \ \ \ }
\caption{Oxygen concentration field at ordinary gas injection (a) and
after optimization (b).}
\end{figure}
\section{\large\bf The Investigation into Peculiar Features of Gas Flowing
through a Layer of Immovable Catalyzer}
It is known from the literature about the effect related to the passage of gas
through a layer of porous or granulated medium, which is called the effect of
``hare ears''. This effect makes itself evident in the formation of a
macroscopic ear-like heterogeneity of the velocity field behind the
porous layer. Opinions differed on how to explain this phenomenon, for
instance, by variations in porosity just near the wall or by the deformational
pattern of a porous medium. Based on the experimental investigations,
Goldshtik~(1984) found out that the ``hare ears'' effect owed its
origin to surface deformations of a porous medium.
\begin{figure*}[htb]
\centerline{\epsfig{file=fthreea.eps, height=7cm}\hspace{.6cm}
\epsfig{file=fthreeb.eps, height=7.1cm}}
\vspace{.3cm}
\centerline{\sf (a)\hspace{6.5cm}(b)\ \ \ \ \ }
\caption{Scheme of the channel with a porous layer (a); velocity
distribution of the gas flow on cross-sections 1 and 2 (b); line 3
corresponds to plane layer surface.}
\end{figure*}
The above given model has suggested to make some numerical investigations of
this phenomenon. In doing so, the velocity of particles of the dispersed
medium is assumed to be zero ($\vec v_2=0$), and the medium occupies a
portion of the channel as displayed in Fig.~3(a). Besides,
it is assumed that $\alpha_2 = \rm const$.
Figure~3(b) shows the calculated gas horizontal velocity in sections (1)
and (2), where the ``hare ears'' effect is clearly displayed. For comparison
purposes line (3) is given in Fig.~3(b), that corresponds to the plane s
urface of the porous layer. Thus, one should conclude that it is precisely
the outer surface deformation of the laeyr that is responsible for the
effect to be revealed. It should also be mentioned that the shape of
deformation makes no special difference. The only important thing that the
surface of the porous medium must form an acute angle ($\varphi<\pi/2$) with
the wall of the channel.
\begin{figure}[h]
\centerline{\epsfig{file=ffive.eps, width=5.8cm}}
\caption{Fields of velocities vectors (arrows) and that of gas pressure in
channel (correspond to the scheme in Fig.~3(a).}
\end{figure}
A comparison of the pressure field (Fig.~4) with the vector field of
velocities shows it most clearly that the direction in velocity under
flowing is perpendicular to pressure isolines.
Thus, the ``hare ears'' effect arises from cumulation of the flow with
a jet stream formed near the wall. This is also evidenced by calculations
made for porous layers with the reverse deformation (Fig.~5(a)). In Fig.~5(b)
a jump in horizontal velocity is clearly seen in sections (1) and (2) in the
central portion of the channel with the formation of a single jet.
\begin{figure*}[htb]
\centerline{\epsfig{file=ffoura.eps, height=7cm}\hspace{.6cm}
\epsfig{file=ffourb.eps, height=7.1cm}}
\vspace{.3cm}
\centerline{\sf (a)\hspace{6.5cm}(b)\ \ \ \ \ }
\caption{Scheme of the channel with a V-shaped porous layer (a);
distribution of gas flow velocity in cross-sections 1 and 2 (b).}
\end{figure*}
\section{\large\bf Conclusion}
The present paper is an attempt to describe a fine-grained catalyzer
flowing along the lift-reactor channel within the region of acceleration and
to determine the pattern in the course of chemical reaction depending on
reagent supplying conditions. A possibility is shown on the basis of the
given mathematical model to optimize the arrangement of combustible gas
injection zones and to efficiently remove a gas component unwanted in the
process, namely, oxygen.
The investigation into peculiar features of a gas flow behind the catalyzer
immovable layer has resulted in a clear-cut interpretation of the ``hare ears''
effect. This effect is determined to correlate with gas flow cumulation near
the wall and to depend on the value of the angle formed between the outer
surface of a catalyzer layer (or a porous medium in general) and the
wall of the channel.
\section*{\large\bf References}
\begin{enumerate}
\item
Nigmatulin~R.~I. (1990) Dynamics of Multiphase Media. v.~1.~---
Hemisphere.~Washington.
\item
Savage~S.~B. (1979) Gravity flow of cohesionless granular materials in
chutes and channels.~--- J.~Fluid~Mech., v.~92., pt.~1., pp.~53--96.
\item
Patankar~S.~V. (1980) Numerical Heat Transfer and Fluid Flow.~---
Hemisphere.~Washington.
\item
Goldshtik~M.~A. (1984) Transfer Processes in Granular Layer.~---
Novosibirsk. (In Russian)
\end{enumerate}
\end{document}