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

Abstract. Attempts of strict mathematical modeling heterogeneous media that are made in some
investigations are necessarily limited by some prior specified theoretical schemes and this fact requires many
experimental data for a practical 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 computational 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 technical 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 flow 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 proposed.
Results calculated according these models are in a qualitative agreement with experimental data and a large-scale
heterogeneity of a velocity profile corresponds to a scale of granular layer structure heterogeneity. But using
Darcy’s law or Ergun’s equation to describe a motion of the carrier phase in an inhomogeneous granular medium
requires a justification because a formation of boundary layersis possible along borders of the areas with a different
permeability.
In radial units with the still granular layer (SGL) it is considered that the large-scale heterogeneity of a radial
component of the velocity in the granular layer is caused mostly 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 axial units with SGL, a variety of motion equations used for calculation, a lack of information
about a volume structure of the granular layer that arises during loading the unit and so on testify it.
So there is a necessity to construct computational and theoretical models of concrete industrial units that
would took into account basic properties of a technological process and were sufficiently simple engineer’s
solutions for designing new technological systems Скачать в формате PDF

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 ).

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 ).