Американский Научный Журнал THE NUMERICAL MODELING OF A FLOW IN PLANE AND RADIAL CONTACT UNITS WITH A STILL GRANULAR LAYER

42 American Scientific Journal № ( 27 ) / 20 19
ХИМИЯ

THE NUMERICAL MODELING OF A FLOW IN PLANE AND RADIAL CONTACT UNITS WITH A
STILL GRANULAR LAYER

Shtern Pavel Gennag’evich
Doct. Sc. Techn.
Koleskin Vladimir Nikolaevich
Cand. Sc. Techn.
Lukyanova Antonina Vladimirovna
Cand. Sc. Phys. -Math.
Yaroslavl State Pedagogical University by K.D. Ushinsky
Yaroslavl, Russia

Abstract . Attempts of strict mathematical modeling heterogeneous media that are made in some
investigations are necessar ily limited by some prior specified theoretical schemes and this fact requires many
experimental data for a practic al realization. The analysis of known works shows that a physical essence of
processes arising at a liquid or gas motion in contact units is insufficiently studied. In articles dealt with a
mechanics of disperse systems, issues of constructing computationa l and theoretical models of concrete industrial
devices that would took into account basic experimental facts and sufficiently simple from engineer’s point of
view are poorly reflected.
The experience of an operation of chemical reactors shows that technic al and economic indicators of an
industrial process are as a rule lower than evaluated values that were obtained during a design step. Now it can be
considered proven that one of the reasons that affects a reactor capacity is heterogeneity of a reagent flo w in a
granular catalyst layer.
In some works [1], [2], [3], [4] computational and theoretical models of a filtration mode of a liquid and gas
flow in the still granular layer that is in a stress -strain state under a load from the carrier phase were propos ed.
Results calculated according these models are in a qualitative agreement with experimental data and a large -scale
heterogeneity of a velo city profile corresponds to a scale of granular layer structure heterogeneity. But using
Darcy’s law or Ergun’s equ ation to describe a motion of the carrier phase in an inhomogeneous granular medium
requires a justification because a formation of boundary layers is possible along borders of the areas with a different
permeability.
In radial units with the still granula r layer (SGL) it is considered that the large -scale heterogeneity of a radial
component of the velocity in the granular layer is caused mostl y by reagent flow features in distributing and
collecting manifolds. So a majority of works dealt with such units investigates a flow with a suction or injection
in channels with perforated walls. A distribution along a distributing manifold length of a cross -section average
value of an axial component of the velocity flow was determined by Meshcherskiy’s equation, an energy approach,
models of potential flows in [5], [6], [7] an so on. The main task was to provide the stable radial velocity at entering
the granular layer.
From the above review a conclusion can be made that physical features of the liquid and gas motion in SGL
are poorly explored both theoretically and experimentally. For example, conflicting results of measuring the
velocity profile in axia l units with SGL, a variety of motion equations used for calculation, a lack of information
about a volume structur e of the granular layer that arises during loading the unit and so on testify it.
So there is a necessity to construct computational and theo retical models of concrete industrial units that
would took into account basic properties of a technological proces s and were sufficiently simple engineer’s
solutions for designing new technological systems.
Keywords: chemical reactor, steam -raw mixture, reagent, large -scale heterogeneity, catalyst, perforated
channel, manifold, working area, velocity field, pressure field
.

American Scientific Journal № (27 ) / 201 9 43
Fig. 1. The scheme of the unit with a radial gas input
In [8] the mathematical modeling of an
incompressible liquid flow in pla ne and radial
contact units with the still granular layer was
developed and methods of a numerical realization of
the model were shown.
The flow area in the unit can be conditionally split
into three sub -areas (fig. 1) — distributing I and
collecting II ma nifolds and working area III , that is the
still granular layer placed between two coaxial
perforated cylindrical shells. A pressure drop in radial
reactors is not large and is about tenths of the
atmosphere.
The velocity of the steam -raw mixture is about 1
m/s and Mach number M << 1, therefore the gas
passing the reactor can be considered as
incompressible.
It is known , for instance [9], [7], that near the axial
zone of the channel an impulse flux for the flow with
the powerful suction (the distributing col lector) and the
injection (the collecting manifold) is several orders of
magnitude higher than the viscous flow. Th e liquid near
the axial zone looks like almost the ideal one and for a
core of the flow a stream in I and II areas can be
considered as poten tial one. The motion of the
incompressible liquid in the working area III is
determined by Ergun’s law where a resi stance term that
is linear over the velocity is neglected.


As a result we obtained a system of equations to find a velocity and pressure fields:
for a total flow area:
div υ = 0 (1)
for I and II areas
rot υ = 0 and �������2
2 +�= ����� (2)
for the working area III
grad p = –f·│ υ│· υ (3)
After writing down the solution of eq. (1) through a current funct ion Ψ as
��= 1
�⋅�������
������� and �������= −1
�⋅�������
�� (4)
we can find a second order partial differential equation of the elliptic type for every area: I, II and III . So, for
instance, the equation for III area looks like:
(�2+�������2)�2������
��2+(�2+��2)�2������
�������2−2����������2������
��������� +2�2�������(����������������
������� −����������������
�� )= 0 (5)

44 American Scientific Journal № ( 27 ) / 20 19
At Γ1 and Γ2 boundaries (see fig.1) between I – III and III – II areas correspondingly the continuity conditions
for the normal components of velocity and the pressure are valid:
��1= ��3 ��2= ��3
�(�)= Δ�1+ �(��� ) �(��)= �(��� )− Δ�2 , (6)
where Δp1,2 is the pressure drop at per forated walls
of the distributing and collecting manifolds; it is equal
�������1,2= ������1,2��1,22 and σ1,2 is the resistance coeffic ient
that corresponds to an average over the manifold side
surface velocity ��1,2 which may be in general a
function of z f the normal component of the velocity at
Γ1 and Γ2 boundaries is specified then the full
determination of the flow parameters can be conducted
for all three areas separately and comes down to solving
the above differential equations for the current function
in every area. That is a direct problem . Matching the
obtained solutions is made by determining the normal
components of the velocity at Γ1 and Γ2 boundaries that
is accomplished using the continuity condition for the
pressure (6) at pass ing these boundaries. To fulfill this
condition at specified normal components of the
velocity ��1 and ��2 the target functio n Φ is constructed
that equals to the average square of the pressure drop at
the boundaries:
������(��1,��2)= 1
������∫ (�(�)−�(���)−�������1) �1
2�� +1
������∫ (�(���)−�(��)−�������2) �2
2�� , (7)

where L is the length of Γ1 and Γ2 boundaries (the
unit height). The procedure of the determination of ��1
and ��2 comes down to the target function Φ
minimization. It is an inverse problem . The solution of
the direct problem in I and II areas wa s obtained
analytically with Green’s function means [10] and the
solution of the direct and inverse problems in III area
was carried out with the help of numerical techniques
on a computer. Results of the evaluation of the
suggested hydrodynamic model carr ied out for various
particular types of radial units are in good agreement
with published ones and experimental dat a obtained in
[8].
With the help of alternative calculations of the model and
theoretical analysis of some particular types of radial units we
have investigated the influence of the layer resistance on the
distribution of relative values of the axial compo nent of the
velocity in the distributing manifold, radial components of the
velocity at Γ1 and Γ2 boundaries (see fig. 1) and the stay
time along the unit height.
We determined the dependence of a coefficient for
decreasing consumption:
�= ��������−��������������
�������� (8)

and a degree of the nonuniformity of the radial
velocity profile at Γ1 boundary
�= ������
�1������� �������2��� �������� (9)

of a dimensionless pressure drop �������̃�= ��� ��������� in
the distri buting manifold. In the range of �������̃� = 0 ÷ 2,5
these dependences can be presented by linear equations
�= 1
6�������̃�; �= 1
2�������̃�. (10)

The obtained equations have a high degree of a
generality because they set a relationship between
dimensionless values and do not depend upon
geometrical sizes of units and its consumable
characteristics in a wide range of these magn itudes.
Besides, the pressure drop in the manifold �������̃� for
industrial units (for example, chemical reactors ) lies as
a rule inside the range that was mentioned above. In the
assumption that the total pressure drop that occurs at
inner and outer per forations and at the granular layer
(see fig.1) is much higher than the drop along
manifolds, i.e. ��� ��������� << 1, and at the condition f = const
we have made the theoretical evaluation of the radial
unit with a low degree of the inhomogeneity flow. On
the base of the obtained solution we have explored a
particular case that is a device with an output in the
barometric environment and without the outer
perforated shell at Γ2 boundary. The analytical solution
of the problem results in (10) also.
So, equations (10) that are obtained by a strict
theoretical way are in good agreement with the results
of a numeri cal experiment that in its turn determined
the limits of theoretical model applicabi lity. It should
be noted that numerical calculations conf irm the
validity of equations (10) not only for units with the
flow output into the atmosphere but also for any
rea ctor, i. e. any unit containing the collecting manifold
with a perforated shell.
All results that were derived in IV chapter of [8,
p. 270] allow developing construction principles for an
engineer method of the radial unit with the still granular
layer design. The method permits to carry out an
estimate of optimal options of a constructive
implementation for devices of various technological
purposes and to define its consumption characteristics
upon the specified total pressure drop, the
inhomogeneity flow degree and some technological
and hydraulic parameters. The detailed definition of
velocity and pressure fields in all areas of the selected

American Scientific Journal № (27 ) / 201 9 45
type of the unit is accomplished by means of a
numerical calculation on the computer according to the
proposed mat hematical model.

References
[1] Scheidegger AE (1960) The Physics of liquid
flow through porous media [Fizika techenija zhidkostej
cherez po ristye sredy]. Gostoptehizdat, Moscow (in
Russian).
[2] Aerow ME et al. (1979) Units with a still
granular layer [Apparaty so statsionarnym zernistym
sloem]. Khimija, Lenigrad (in Russian).
[3] Struminskij VV, Zaichko ND, Zimin VM,
Radchenko ED (1978) Th e problems of the modern
aerodynamics dealt with some technological processes
in the chemical and petrochemical ind ustry [Problemy
sovremennoj aerodinamiki, svjazannye s nekotorymi
tehnologicheskimi protsessami v himicheskoj I
neftehimicheskoj promyshlenno sti]: in the collected
volume “Mechanics of multicomponent media in
technological processes”. Nauka, Moscow (in
Rus sian).
[4]. Stanek V, Szekeli J (1972) The Effekt of Non -
Uniform Porosity in Causing Flow Maldistributions in
Isothermal Packed Beds. Canad. J. Chem. Eng. V. 50,
№ 1. P. 9–14.
[5] Petrow GA (1964) Variable mass hydraulics
[Gidravlika peremennoj massy]. Kha rkiv university
edition, Kharkiv (in Russian).
[6] Goldshtik MA (1984) Transfer processes in a
granular layer [process perenosa w zernistoms sloe].
Institute of thermophysics SO AN USSR, Novosibirsk
(in Russian).
[7] Sergeev SP, Dilman VV, Gen kin VS. (19 74)
Flows distribution in channels with porous walls.
Journal Of Engineering Physics And Thermophysics
[A translation of Inzhenerno -Fizichesk ii Zhurnal]. V.
27. No. 4. P. 588 –595.
[8] Shtern PG (1995) Development of calculation
methods for industrial chem ical reactors [Postroenie
metodov rascheta promyshlennyh himicheskih
reaktorov]: thesis for the PhD degree (in technical
sciences). JSC “Karp ov Institute of Physical
Chemistry”, Moscow. (in Russian).
[9] Sergeev SP (1990) Radial catalytic reactors
with a still granular layer [Radial’nye kataliticheskie
reaktora s nepodvizhnym zernistym sloem]: thesis for
the PhD degree (in technical sciences). JSC “Karpov
Institute of Physical Chemistry”, Moscow. (in
Russian).
[10] Vladimirov VS (1981) Mathematical Physic s
Equations [Uravnenija matematicheskoj fiziki]. Nauka ,
Moscow (in Russian ).