#### Volume 92, №4

**DIFFUSION MODEL WITH ACCOUNT FOR RELAXATION SUBSTANCE TRANSFER TO DESCRIBE STATIONARY OPERATION OF A REACTOR IN THE PRESENCE OF THREE REGIMES**

A nonlinear boundary-value problem describing the operation of an isothermal fl ow reactor with account for the substance fl ow relaxation is considered. The examined substance source is described by a kinetic law that corresponds to a defi nite type of autocatalytic reaction. Possible solutions under the Dankwerts standard boundary conditions are determined. The limiting form of equations of the relaxation model has been taken as a basis for the mathematical
model. The change in the regimes of operation of reactors in the presence of three solutions of the process equations
is analyzed.

**Author:**
A. I. Moshinskii

**Keywords:**
reactor, stability, autocatalytic law, Peclet number, convective transfer, relaxation transfer

**Page:**
835

**ON THE POSSIBILITY OF IMPROVING THE THERMODYNAMIC CHARACTERISTICS OF MULTIFLOW HEAT TRANSFER SYSTEMS**

The infl uence of the change in the phase state of part of fl ows on optimal organization and minimum possible entropy production in a multifl ow heat-exchanging system is considered. For the Newtonian kinetics of heat transfer, computational formulas for optimal distribution of the total surface between heat exchangers, the
algorithm of selection of contacting fl ows, and the ratio of absolute temperatures of heated and heating fl ows have been obtained

**Author:**
A. M. Tsirlin and A. A. Akhremenkov

**Keywords:**
multifl ow heat transfer, condensation/evaporation heat transfer

**Page:**
844

**NONSTATIONARY TRANSFER OF COMPONENTS OF AN ISOTOPIC MIXTURE AS A RESULT OF THE CHANGE IN CASCADE FLOWS**

A nonstationary process initiated by the change in the values of the fl ows of light and heavy fractions of a cascade
has been modeled using the separation of germanium isotopes as an example. The character of change in the
concentration of isotopes in cascade stages and fl ows as a function of the fi nite values of the fl ows has been shown. It
has been established that in the course of the nonstationary process, isotopes of intermediate masses can be obtained
with a concentration much higher than the maximum attainable values for a three-fl ow cascade.

**Author:**
A. A. Orlov, A. A. Ushakov, and V. P. Sovach

**Keywords:**
cascade, isotope, separation of isotopes, germanium, nonstationary process, mathematical model, isotope mixture

**Page:**
853

**SIMULATION OF HARMONIC TEMPERATURE AND MAGNETIC FIELDS BASED ON THE METHOD OF SEPARATION OF VARIABLES IN REGIONS OF COMPLEX TOROIDAL FORMS**

Basic boundary-value problems of calculating harmonic physical fi elds for toroidal forms (including the cases
of complex sectors and nonlinear boundary-value conditions) are considered on the basis of exact analytical
representation of the solution in the form of a series in toroidal harmonics. The results obtained can be used in
calculating harmonic fi elds in problems of heat and mass transfer and of electromagnetism, coeffi cients of
the medium of specifi c technical objects of toroidal form, as well as a test problem when employing other methods.

**Author:**
S. A. Nekrasov

**Keywords:**
boundary-value problem, harmonic physical fi eld, Laplace equation, separation of variables, toroidal coordinate system

**Page:**
861

**EXPERIMENTAL INVESTIGATION OF THE EFFECT OF PRESSURE ON SEPARATION OF A CARBON DIOXIDE-CONTAINING GASEOUS MIXTURE**

Experimental investigations have been carried out to estimate the effect of pressure on the intensity of separation of

mixture components with enrichment of the resultant mixture in a heavy component for the 0.082CO2 + 0.918N2– N2

system. It is shown that the penetrability of carbon dioxide increases with pressure. For the system considered,

the separation coeffi cient has been determined, and the stability map has been constructed within the linear theory

of stability.

**Author:**
M. S. Moldabekova, M. K. Asembaeva, and O. V. Fedorenko

**Keywords:**
multicomponent mixture, diffusion of gases, two-fl ask method, separation coeffi cient

**Page:**
872

**NONSTATIONARY MASS TRANSFER AND CONTROLLING THE PROCESS DURING THE LAMINAR MOTION OF A LIQUID IN PACKED-COLUMN APPARATUSES**

Approximate solutions have been obtained for the equations of nonstationary mass transfer in packed-column
apparatuses, and a method has been proposed to calculate them under the conditions of laminar motion of a liquid
phase and displacement of a gas. On the basis of the nonstationary-mass-transfer equations, a problem of control
of packed-column apparatuses has been solved. The solutions of the equations have been obtained in the form of an
expansion of functions in a trigonometric series; formulas to determine the values of terms in the series have been
presented. Initial conditions for the equations of nonstationary mass transfer are equations of a stationary process.
Exact and approximate solutions for the stationary-mass-transfer equations have been obtained based on which an
algorithm has been proposed for designing packed-column apparatuses. A calculation of the process of absorption
of an acetone vapor from air by water has been carried out. Plots of the control and of the concentration change in
controlling and with different perturbations have been presented.

**Author:**
D. V. Elizarov, R. R. Shavaleev, and V. V. Elizarov

**Keywords:**
mass transfer, mathematical model, designing, control of the process, packed-type column, packing, absorption

**Page:**
877

**STABILITY OF HYDROGEN FLUORIDE UNDER WATER-GAS-REACTION CONDITIONS**

The prime objective of the investigation was to prove the stability of hydrogen fl uoride under water-gas-reaction
conditions. Calculations of the thermodynamically equilibrium composition of substances in the system of elements
C–H–F–O were conducted, and it has been shown that with excess carbon and at a temperature of 1000 to 2000 K,
the basic components of the mixture are carbon mono- and dioxide, hydrogen, methane, and hydrogen fl uoride, and
carbon fl uorides and oxyfl uorides are absent. At a temperature above 1300 K, the thermodynamically equilibrium
mixture consists of carbon monoxide, hydrogen, and hydrogen fl uoride. Calculation results have been confi rmed by
laboratory experiments in which the only fl uorine-containing substance, i.e., hydrogen fl uoride, was found as part
of the products of interaction of water and carbon in the presence of hydrogen fl uoride at a temperature of the order
of 1500 K. The obtained results can become a basis for the technology of obtaining anhydrous hydrogen fl uoride
from its aqueous solutions, including the azeotropic one, by high-temperature interaction of the steam and carbon at
a temperature above 1000 K.

**Author:**
V. V. Kapustin, D. S. Pashkevich, V. A. Talalov,a D. A. Mukhortov,b Yu. I. Alekseev,c V. B. Petrov,b P. S. Kambur, M. P. Kambur, and O. N. Voznyuk

**Keywords:**
hydrogen fl uoride, carbon, water-gas reaction, heterogeneous combustion

**Page:**
889

**INFLUENCE OF THE TOPOLOGY OF A SOLID BODY ON ITS MASS CONDUCTIVITY**

The problem on the mass transfer in a solid body under conditions where the concentration of the substance distributed
in the body remains unchanged at its surface was formulated and solved analytically on the basis of the generalized
differential mass-conduction (diffusion) equation for bodies different in shape (an unbounded plate, an endless
cylinder, and a sphere). The infl uence of the topology of a solid body on the time changes in the concentration of a
substance distributed in it at its center, the concentration of the distributed substance averaged over the volume of
the body, and the gradient of the concentration of this substance at the surface of the body was numerically analyzed.

**Author:**
S. P. Rudobashta, É. M. Kartashov, and G. A. Zueva

**Keywords:**
distributed substance, kinetics, mass conductivity, topology

**Page:**
899

**PROGNOSIS MODEL FOR INVESTIGATING THE EVAPORATION OF WATER DROPLETS**

A model of heat and mass transfer in a homogeneous water droplet and in a water droplet with a solid inclusion
has been developed with the use of the Ansys Fluent package for investigating the high-temperature evaporation
of such droplets. The infl uence of the size of a water droplet, the thickness of its vapor shell, the thermal diffusivity of
the water in the droplet, the size of an inclusion in it, and the temperature of the gas medium surrounding the droplet
on the time of its existence was determined. The mechanisms of heating homogeneous and inhomogeneous water
droplets were investigated with and without regard for the emission of heat by the water-vapor layer. Approximation
expressions for simulation of the evaporation of homogeneous and inhomogeneous water droplets have been obtained

**Author:**
D. V. Antonov, O. V. Vysokomornaya, G. V. Kuznetsov, and M. V. Piskunov

**Keywords:**
high-temperature gases, vapor–gas mixture, homogeneous and inhomogeneous water droplets, inclusion, evaporation, vaporization, heat emission

**Page:**
907

**INVESTIGATION OF THE INFLUENCE OF HYDRATION ON THE HEAT OF EVAPORATION OF WATER FROM SUCROSE SOLUTIONS**

An analysis has been made of the literature data on sucrose hydration. A platform of the most reliable experimental data has been selected. In a differential evaporation microcalorimeter, the specifi c heat of evaporation of water was measured in the process of continuous dehydration of a solution with an initial content of sucrose of 12.5 wt.%. Curves of the reduced specifi c evaporation heat versus the concentration of the solution at 40, 60, and 80^{o}
C were
obtained. The infl uence of the degree of hydration of sucrose on the specifi c evaporation heat was determined. It has
been shown that in the interval from the initial concentration of the solution, with degree of hydration of sucrose
N = 20, to a content of sucrose of ~65 wt.% (N ~ 6), the specifi c evaporation heat grows from a value that is similar
to the evaporation heat of pure water to values 3 to 4% higher than it. The most signifi cant growth in the specifi c
evaporation heat (7–10%) was recorded on removing water bonded to hydroxyl groups of sucrose in the equatorial
position at N ≤ 6. The concept of strongly and weakly bound hydrate water in sucrose solutions is substantiated.

**Author:**
V. A. Mikhailik, N. V. Dmitrenko, and Yu. F. Snezhkin

**Keywords:**
differential scanning calorimetry, evaporation calorimetry, sucrose, hydration, free and hydrate water, evaporation heat

**Page:**
916

**HYDRODYNAMICS OF A CENTRIFUGAL NONLINEAR-VISCOELASTIC LIQUID FILM**

Film fl ow of nonlinear-viscoelastic fl uid over the surface of a rotating curvilinear extension piece has been considered
with account for the lag. The rheological model used is that of White–Metzner. The problem has been solved by the
method of integral relations with the use of the von Karman (κ = 0) and Leibenson (κ = 1) approximations. Two
algebraic equations with two unknowns have been obtained realized by the method of iterations. Experimental data
are given.

**Author:**
N. Kh. Zinnatullin, G. N. Zinnatullina, and E. I. Kul′ment′eva

**Keywords:**
White–Metzner fl uid, liquid fi lm, centrifugal fi eld, integral relations, experiments

**Page:**
923

**INFLUENCE OF EXTERNAL AND INTRAVOLUMETRIC CHARACTERISTICS OF COMBUSTION ON THE TIME OF THE BURNING-OUT OF SOLID FUEL PARTICLES OF POLYMERIC MICROSTRUCTURE**

The infl uence of the characteristics of external and intravolumetric combustion on the time of burning-out of the
particles of solid fuel of polymetric microstructure has been investigated. In the kinetic, diffusional–relaxational, and
transient regimes of intravolumetric combustion, the mutual infl uence of the activation energy, temperature factor,
sizes of particles, and of the relaxation time on the time of burning-out of fuel particles of polymeric microstructure
has been analyzed.

**Author:**
V. I. Popov

**Keywords:**
fuels of polymeric microstructure; combustion models; kinetic, transient, and diffusional–relaxational regimes; time of burning-out of a particle

**Page:**
929

**MODEL OF VORTICAL DISTURBANCES OF A COMBUSTION WAVE**

A model is suggested for describing the interaction of vortical disturbances with a combustion wave. Unlike the
well-known approaches, the disturbances bear a random character, and their spatial structure is determined by
the correlation function with the scale of a vortex (colored noise).

**Author:**
Yu. V. Polezhaev,* O. G. Stonik, V. D. Geshele, and S. A. Kovalev

**Keywords:**
vibrational combustion, combustion wave, vertical disturbances

**Page:**
936

**FUEL FROM OIL SLUDGES**

Based on detailed investigations of thermal treatment of various types of oil sludges, a thermochemical technology
of obtaining liquid and gaseous fuels by means of steam thermolysis of sludges with subsequent gasifi cation of the
solid residues of thermolysis has been developed, the respective test equipment has been created, and the possibility
of effectively obtaining liquid and gaseous fuels by means of steam thermolysis of sludges has been shown

**Author:**
G. I. Zhuravskii

**Keywords:**
oil sludges, thermochemical technology, testing equipment, steam thermolysis, gasifi cation

**Page:**
940

**GEOMETRIC INVERSE PROBLEM OF RADIATIVE HEAT TRANSFER AS APPLIED TO THE DEVELOPMENT OF ALTERNATE SPACECRAFT ORIENTATION SYSTEMS**

At present, the angular position of small spacecraft is determined using solar sensors, magnetometers, and angularvelocity pickups as a rule, whereas star trackers are used for complicated problems requiring high accuracy. Each
individual system has its advantages and disadvantages. Therefore, combined systems are used to avoid them. Also,
to improve the reliability of a craft, recourse to redundancy of the sensors must be had. All these subtleties result
in the increased materials intensity and in the complicated mutual synchronization of the systems. In the present
work, the authors analyze the possibilities of creating and using the main or alternative orientation system for small
spacecraft with radiative-heat-fl ux sensors on the basis of the methodology of inverse heat-transfer problems. This
system will allow checking and correcting the craft's angular position.

**Author:**
A. V. Nenarokomov, E. V. Chebakov, I. V. Krainova A. V. Morzhukhina, D. L. Reviznikov, and D. M. Titov

**Keywords:**
determination of the spacecraft orientation, heat-fl ux sensors, thermal action on a spacecraft, inverse heat-transfer problems

**Page:**
948

**NUMERICAL INVESTIGATION OF HEAT TRANSFER DURING THE COOLING OF A METAL CYLINDER BY A FLOW OF A GAS-LIQUID MEDIUM IN AN ANNULAR CHANNEL**

A mathematical model of conjugate heat transfer in the heterogeneous system "solid–gas–liquid medium" with
account of the description of bubble vaporization in a liquid fl ow has been proposed. Consideration has been given
to a two-dimensional nonstationary formulation taking account of the axisymmetry of cooling-medium fl ow about the
longitudinal axis of a cylinder. Results of numerical calculations of thermophysical parameters during the cooling of
a high-temperature metal cylinder manufactured from structural-grade steel by a fl ow of a gas–liquid medium in an
annular channel have been given. The infl uence of the fl ow velocity on the value of the concentration of the vapor in
the liquid and the regularities of cooling of the metal cylinder have been assessed. An analysis has been made of the
infl uence of the formation, growth, and motion of vapor bubbles in the liquid on the distribution of the heat fl ux over
the surface of the conjugate boundary of the metal cylinder and the gas–liquid medium.

**Author:**
S. S. Makarov A. M. Lipanov and A. I. Karpova

**Keywords:**
gas–liquid medium, high-temperature metal cylinder, vaporization, conjugate heat transfer, numerical modeling

**Page:**
956

**MODELING EXTERNAL HEAT LOADS ON THE SPACECRAFT IN A THERMAL VACUUM CHAMBER**

A method and an algorithm to solve the problem on reproduction, with maximum possible accuracy, of heat loads
on the surface of the spacecraft in a thermal vacuum chamber with the aid of the radiating system incorporating
thermal shields and halogen incandescent lamps have been presented. In simulation experiments conducted for the
characteristic heat loading of the spacecraft surface, the possibility to substantially improve the accuracy of solution
of the problem by combining the thermal shields and the incandescent lamps into a single radiating system has been
demonstrated.

**Author:**
A. V. Kolesnikov, A. V. Paleshkin, and Sh. O. Syzdykov

**Keywords:**
thermal vacuum chamber, thermal shields, halogen incandescent lamps, modeling, heat loads, optimization, spacecraf

**Page:**
965

**ON THE VALIDITY OF PRACTICAL USE OF A DIRECT EXPLICIT METHOD OF SOLVING A COMBUSTION-LOSS EQUATION**

A new method is offered to solve the combustion-loss equation, making it possible to use the results of rigorous
theoretical computation investigations in the part of gas dynamics and convective heat transfer near the surface of
a body in air fl ow.

**Author:**
V. V. Gorskii

**Keywords:**
thermal shield, ablated shape of a body, ablation, carbon material

**Page:**
971

**INFLUENCE OF JOULE HEAT DISSIPATION ON HEAT EXCHANGE AND MAGNETIC HYDRODYNAMICS OF LIQUID IN A SPHERICAL LAYER. PART II**

Results of a numerical simulation of the convective heat exchange in an electrically conducting liquid layer between
two concentric isothermal spheres are presented. The infl uence of the Joule heat dissipation in this layer on the
structure of the liquid fl ow as well as on the temperature and magnetic-induction fi elds and the distribution of Nusselt
numbers in it was investigated. Equations for simulation of the heat exchange in the spherical liquid layer in the case
where the gravitational acceleration is directed to the center of the concentric spheres have been obtained.

**Author:**
S. V. Solov′ev

**Keywords:**
mathematical simulation, convective heat exchange, Joule heat dissipation, magnetic hydrodynamics, spherical layer, simulation equation

**Page:**
976

**PARABOLIC PROFILE IN HEAT-CONDUCTION PROBLEMS. 3. SEMI-BOUNDED SPACE WITH A DEFINITE EXTERNAL HEAT FLOW**

Some approaches to the optimization of the exponent n of the parabolic temperature profi le for a semi-bounded space
with the second-kind boundary condition, implying the prescription of an external heat fl ow, are considered. An
optimization scheme involving the minimization of a new error norm estimating the absolute value of a temperature
deviation, more effi cient than the scheme of T. Myers with minimization of the Langford norm, as well as new hybrid
integral methods are proposed. With the use of these methods, parabolic solutions have been obtained on the basis
of the temperature momentum integral in different modifi cations and the heat-fl ow momentum integral, and these
solutions defi ne the surface temperature with a very high accuracy. As an illustrative example, the problem on the
heat conduction of a semi-bounded space with a defi nite pulsating heat fl ow at its surface, whose time change is
defi ned by a quadratic parabola, was considered. The approximation errors of the parabolic solutions obtained
using the hybrid integral methods proposed comprise hundredths and thousandths of a percent, and they are smaller
by one to two orders of magnitude than the errors of the analogous solutions obtained by the known heat balance
integral method and refi ned integral method, involving the optimization of the exponent of a parabolic temperature
profi le on the basis of minimization of the Langford norm, and by the combined integral method

**Author:**
V. A. Kot

**Keywords:**
temperature-disturbance front, temperature momentum integral, heat-fl ow momentum integral, heat balance integral method, hybrid integral method

**Page:**
986

**INFLUENCE OF THE MANIFOLD CONFIGURATION OF PULSATING HEAT PIPES ON THEIR PERFORMANCE**

The performance characteristics of three air-to-air pulsating heat pipes with R1233zd as a working fl uid for cooling an electronic cabinet are presented. Each of these pipes comprises two coolers staked in one unit of dimension 36 × 19 × 5 cm^{3}
, constructed with extruded multiport tubes and louvered fi ns brazed together. These pipes differ from
each other only in their manifold confi guration achieved by having fl uid distribution plates at the evaporator and
condenser manifolds to provide a serial connection of the fl ow paths. The experiments were performed with PHPs
of only one type, composed of two coolers in stack and turned through 90o
on their narrow side. A new parameter,
representing the rate of pressure pulsations in a PHP, was introduced into consideration, and it is shown that this
parameter correlates with the PHP performance.

**Author:**
B. Agostini, D. Torresin, and M. Bortolato

**Keywords:**
pulsating heat pipe, two-phase fl ow, electronic cabinet, cooling.

**Page:**
1008

**SYNTHESIS OF REINFORCED CERAMIC MATRIX COMPOSITE BASED ON SiC AND NANOCARBON MESH**

Most modern matrix composite materials employ a variety of carbon nanofi llers to improve their mechanical, electrical, and functional properties. Nanofi llers are separately implanted into the initial ceramic matrix, which complicates the composite manufacturing technology and increases the fi nal cost. In this work, the synthesis of nanocarbon fi llers was carried out using high-temperature (1200^{o} C) pyrolysis of phenolic resin directly inside the silicon carbide matrix. This results in the formation of a continuous 3D nanocarbon mesh uniformly binding and reinforcing the fi nal product throughout the volume. The nanocarbon fi ller synthesized in the SiC matrix contains two allotropic carbon forms: nanographite and nanofi bers. The study of the features of the carbon structure and morphology showed that during the pyrolysis the multilayered nanographite structures had been formed on the surface of SiC grains of average crystallite size equal to 20–35 nm. In the matrix pores, carbon nanofi bers a few micrometers in length and 20–40 nanometers in diameter are synthesized. The reiteration of the phenolic resin impregnation–pyrolysis cycle increased the free carbon content from 4 to 7 wt.% for once and twice impregnated and pyrolyzed samples, respectively, and the Young's modulus, from 50.7 to 94.3 GPa. The obtained carbon content and structure are appropriate to produce C/SiC composite for application of ceramics and electrodes

**Author:**
D. V. Solovei, P. S. Grinchuk, H. M. Abuhimd, M. S. Alshahrani,b M. V. Kiyashko, M. O. Stepkin, A. V. Akulich, and A. A. Khort

**Keywords:**
SiC, nanographite, carbon nanofi bers, ceramic matrix composite

**Page:**
1016

**DETERMINING THE HELIUM PERMEABILITY OF MICROSPHERICAL MEMBRANES FROM THE EXPERIMENTAL TIME DEPENDENCE OF THE ABSORPTION OF HELIUM BY THEM**

A mathematical model has been developed and, on its basis, a problem has been solved on determining the parametric
distributions of microscopic membrane elements (microspheres) over the curve of the absorption of helium by them.

**Author:**
A. S. Vereshchagin, I. V. Kazanin, V. N. Zinoviev, A. Yu. Pak, V. A. Lebiga, and V. M. Fomina,

**Keywords:**
permeability, helium, microsphere, inverse problem, Prony method.

**Page:**
1025

**DYNAMICS OF RADIAL SOAKING OF A MATERIAL WITH A DISPERSED MEDIUM**

Processes of capillary absorption of a liquid by a material have been investigated. A design has been proposed for
a device to assess the physical and chemical condition of the material by the dynamics of its soaking. Taking into
account the device′s design and the specifi c features of the liquid′s travel in the capillary system of the investigated
material, differential equations describing the dynamics of its soaking have been obtained. Results have been
presented and analyzed of experimental investigations of a paper′s circular soaking with different dispersed media.
Based on the obtained data, model dependences have been constructed, using which optimization of the developed
device′s design has been implemented.

**Author:**
N. M. Oliferovich, D. A. Grinyuk, I. O. Orobei, and I. G. Sukhorukova

**Keywords:**
wetting, capillary absorption, dispersed medium, porosity.

**Page:**
1031

**SIMULATION OF THE INJECTION OF A LIQUID INTO A WELL IN A PAYOUT BED WITH HYDRAULIC FRACTURING**

Results of a numerical simulation of the injection of a Newtonian liquid into a well in a payout bed with hydraulic
fracturing are considered. The infl uence of the permeability of the bed and the wellbore storage effect on the
development of an anthropogenic fracture in the bed, the pressure in the well, and the fl ow rate of the liquid at the
borehole was investigated.

**Author:**
A. Ya. Davletbaev and Z. S. Mukhametova

**Keywords:**
low-permeability bed, hydraulic fracturing, wellbore storage effect, injection test

**Page:**
1041

**CHARACTERISTICS OF THERMAL ACTION ON POROUS CELLULOSE ACETATE COMPOSITE MATERIAL**

An analysis is given of the thermal action on an MGA-95 composite porous fi lm cellulose acetate industrial prototype
by differential scanning calorimetry. Investigation of an air-dry sample and a water-saturated sample indicates
redistribution of the relation in the quantity of perfect crystal structures in crystallites having different values for the
enthalpy of melting, which decreases for the low-temperature phase from 6.06 to 0.99 kJ/kg, and for the high-temperature
phase, from 1.99 to 1.72 kJ kg. The total melting enthalpy of the endothermic peaks measures from 8.57 to 6.55 kJ/kg,
which points to a reduction in the crystallinity of the water-saturated MGA-95 polymer composite fi lm sample.

**Author:**
S. I. Lazarev, Yu. M. Golovin, S. V. Kovalev, and A. A. Levin

**Keywords:**
porous fi lm, crystallinity, amorphism, bound water, differential-scanning calorimetry

**Page:**
1050

**INFLUENCE OF HEAT AND MASS EXCHANGE IN A LIQUID RING VACUUM PUMP ON ITS WORKING CHARACTERISTICS**

A mathematical model of the compression of gases in a liquid ring vacuum pump has been developed. This model was
realized in the environment of the universal computational ChemCad program in the multicomponent formulation
with no limitation on the number of phases occurring in the working space of the pump. The adequacy of the model
proposed was confi rmed by the processing of numerous experimental data, and it can be directly integrated into the
ChemCad environment or can be used independently for determining the infl uence of the processes of heat and mass
exchange in a liquid ring vacuum pump on its working characteristics.

**Author:**
É. V. Osipov, É. Sh. Telyakov, R. M. Latyipov and D. Bugembe

**Keywords:**
liquid ring vacuum pump, heat and mass exchange, phase transition, mathematical simulation

**Page:**
1055

**NUMERICAL STUDY OF LAMINAR NATURAL CONVECTION IN A FINNED ANNULUS: LOW ISOTHERMAL BLOCKS POSITIONS**

Natural convection in a fi nned concentric annulus fi lled with air is numerically simulated, using the centered fi nite
difference method if the alternating direction implicit scheme is taken into account. Two isothermal blocks are
attached to the inner cylinder and placed symmetrically in the low section of the annulus. The radius ratio, as well
as the dimensionless width and height of the blocks, are kept constant. The effect of the Rayleigh number Ra varying
between 1000 and 10,000 on the fl ow pattern and heat transfer rate is discussed. It is shown that unicellular and
bicellular fl ows are observed, and the blocks increase the overall heat transfer rate for the studied range of Ra.

**Author:**
S. Touzani, A. Idrissi, A. Cheddadi, and M. T. Ouazzani

**Keywords:**
natural convection, isothermal blocks, concentric annulus, fi nite difference, heat transfer enhancement.

**Page:**
1064

**METHOD OF CONSTRUCTION OF GENERALIZED TEMPERATURE DEPENDENCES OF THE THERMAL CONDUCTIVITY OF FREONS IN VARIOUS STATES OF AGGREGATION**

A method of construction of generalized dependences of the reduced thermal conductivity of Freons on their
temperature on saturation lines of a liquid and a vapor and at a pressure of 0.1 MPa has been constructed, which
is based on the principle of minimization of the Helmholtz free energy. Selection of the scale of the sought function
and of its arguments was carried out from the characteristic-function minimum, which has permitted obtaining
calculation formulas to determine the sought properties of the substances under study with an error not exceeding
the experimental error

**Author:**
B. A. Arutyunov and A. V. Kozlov

**Keywords:**
Freons, Helmholtz free energy minimum, thermal conductivity, reduction scale

**Page:**
1072

**ON THE INFLUENCE OF THE PHYSICOMECHANICAL PROPERTIES OF THE ANISOTROPIC MATERIAL OF THERMOELASTIC SHELLS ON THEIR STRESSED STATE**

Consideration is given to the stressed-strained state of circular cylindrical shells with a localized temperature
distribution over their surface and a temperature difference across the thickness. A study is made of the infl uence of
the physicomechanical properties of the orthotropic material of such shells on the value of stresses in them and the
character of their normal displacement.

**Author:**
B. V. Nerubailo

**Keywords:**
circular cylindrical shell, orthotropic material, thermoelastic stresses, differential equation, Fourier series.

**Page:**
1081