# Volume 23 № 4 2021

**1. Albina F. Gizzatullina, Elena S. Baymetova, Mikhail E. Hval’ko, Aleksej Yu. Armyanin**

**Parametric Study of Convective Heat Exchange in Cooling System**P. 393-402.

**Download.**

The paper is devoted to the parametric studies of forced convective heat transfer in the cooling system of a closed-type power plant. The study is based on the numerical modeling of the problem of conjugate convective heat transfer between the cooled air flow and the coolant (ethylene glycol) moving along the working elements of the cooling system; the elements are in the form of ribbed aluminum tubes. The simulation was performed with the open integrated platform OpenFOAM using the chtMultiRegionFoam solver designed for solving conjugate heat transfer problems. A detailed study of the heating of the tube and coolant at various speed of the air inflow in the heat exchanger was carried out. In the power plant, the initial temperature of the air flow was equal to 253 K, the ethylene glycol temperature was 213 K, and the flow speed varied in the range of 2 m/s to 10 m/s. The main results of numerical simulation are presented in the form of the continuous distribution of temperature over the tubes, gas, and coolant medium and, also, of heat fluxes at the interface between the media. The investigation of the character of the heat flux distribution over the tube surface shows that the maximum wall heat flux is on the lateral side of the tube and reaches 1600 W/m^{2}. The analysis of the obtained data shows that an increase in the air flow rate increases the heating of the tube and its ribs. For a tube of more than 25 cm in length, it is necessary to increase the speed of the coolant movement to maintain the cooling process along the entire length of the ribbed tube. The coolant heating is the same for all the speed modes and it is invariant to the velocity of the oncoming stream of the cooled gas in the considered range of speeds and temperatures of the gas and liquids. At the same time, for the gas velocity of 10 m/s the heating rate of the coolant is almost twice as much as that at the gas flow velocity of 2 m/s. After the interaction of air and the cooling element, the total temperature difference depends significantly on the cooled air flow rate. It is shown that the maximum cooling of the gas is reached at the average air flow velocity of 6 m/s.

**DOI:** https://doi.org/10.15350/17270529.2021.4.35

**2. Larisa S. Goruleva, Evgenii Yu. Prosviryakov**

**The Couette-Poiseuille Inhomogeneous Shear Flow at the Motion of the Lower Boundary of the Horizontal Layer**P. 403-411. Download.

In the present paper, an exact solution of the Navier-Stokes equations and incompressibility equation is obtained. The solution describes the steady-state isobaric and gradient inhomogeneous shear flow of a viscous incompressible fluid. The fluid motion under constant pressure is induced by the motion of the lower boundary of an infinite horizontal fluid layer. The Poiseuille-type inhomogeneous fluid flow is considered under the condition that the horizontal gradients of pressure and velocities are specified as constant. The isothermal shear flow of a viscous incompressible fluid is described by an overdetermined system of partial equations. The quadratically nonlinear system consists of four equations, in which two components of velocity and pressure are to be computed by integration. In order to find a nontrivial exact solution (different from the zero), the differential constraint method is applied, which allows to find a redundant equation in the overdetermined system. A family of exact solutions to the Navier-Stokes equations is used as the differential constraints of overdetermined nonlinear equations describing shear flows of fluids. The exact integration of the Navier-Stokes equations is performed in the Lin-Sidorov-Aristov class. The velocity field and the pressure field are linear forms with respect to two ((horizontal or longitudinal) coordinates. The linear form coefficients depend on the third (vertical or transverse) coordinate. Owing to the structure of the exact solution, the incompressibility equation is automatically satisfied. Thus, it is the incompressibility equation that acts as the redundant equation. The obtained polynomial exact solution of the Navier-Stokes equations for incompressible fluids is analyzed. The Couette-type inhomogeneous flow is characterized by a fourth-degree polynomial. A fifth-degree polynomial is used to describe the modified Poiseuille flow. The study of the localization of the polynomial roots has shown the existence of the nonmonotonic profile of specific kinetic energy with two zero values. In other words, there are counterflows in the fluid.

**DOI:**https://doi.org/10.15350/17270529.2021.4.36

**3. Boris B. Benderskiy, Alena A. Chernova**

**Investigation of Heat Transfer in the Combustion Chamber of a Solid-Propellant Rocket Motor Within the Framework of the Homogeneous Gas Model**P. 412-419. Download.

The article analyzes the possibility of applying the homogeneous gas model for the correct numerical simulation of coupled heat transfer processes in the flow paths of the combustion chamber of a solid-propellant rocket engine. The estimation of the density value of heat flows assumes the solution of a conjugate problem of heat exchange in a three-dimensional statement taking into account the flow characteristics such as compressibility, anisotropy, heterogeneity and turbulence. To simplify the mathematical description of the intra-chamber processes it is necessary to make certain physically-based assumptions. The goal of the work is to evaluate the correctness of considering only convective heat flows at the transition from heterogeneous to homogeneous combustion products in mathematical modeling. In this case, the gas thermophysical parameters correspond to the real parameters of combustion products characteristic of subsonic flows occurring in the pre-nozzle volume of the combustion chamber of a solid-propellant rocket engine. In this work, a numerical simulation of the problem of conjugate heat transfer in the pre-nozzle volume of the combustion chamber of a solid-propellant rocket engine with a recessed nozzle was performed within the framework of a viscous thermally conductive compressible gas model. The simulation was carried out on the basis of the finite volume method. In the mathematical description of the intra-chamber processes occurring in the flow paths of the combustion chamber of a solid-propellant rocket engine system under consideration, an assumption was made about the properties of solid propellant combustion products. Namely, the amount of k-phase was assumed small and was not considered. The gas mixture was taken to be homogeneous and isotropic, the chemical kinetics of solid fuel was not considered. The combustion process of the solid fuel charge was not considered either and was replaced by distributed blowing. To describe the heat exchange between the outer wall of the combustion chamber and the environment, a model of natural convection was used that meets the conditions of the model blowdown. The correctness of the assumption about the homogeneity of combustion products and of the neglect of the radiant heat transfer in calculating the heat transfer coefficient on the impermeable surfaces of the combustion chamber was investigated. As a result of the calculations, the distributions of the heat transfer coefficient along the generatrix of a recessed nozzle were obtained; the comparison of the obtained results with the experimental data confirmed the correctness of the assumption made. The analysis of the data obtained shows that in the study of internal gas dynamics and heat transfer in the subsonic regions of a solid-propellant rocket engine the application of the assumption of the homogeneous composition of combustion products is correct and justified.

**DOI:**https://doi.org/10.15350/17270529.2021.4.37

**4. Nikolay G. Galkin, Dmitry T. Yan, Konstantin N. Galkin, Alexander V. Nepomnyashchiy**

**Establishment of Relationship between the Reflection and Refraction Coefficients and the Porosity Index for Mesoporous Silicon Layers with Different Surface Morphology**P. 420-430. Download.

A method based on the interference method is proposed allowing the evaluation of the porosity of mesoporous silicon (meso-PS) layers. The meso-PS layer porosity was evaluated using the visible and near-infrared regions of the reflection spectra and the dispersionless refractive index diagram of porous silicon known in the literature; the meso-PS film thickness was specified. It is shown that for calculating the dispersionless refractive index it is necessary to use only the peaks of maxima and minima in the interference pattern in the wavelength range of (450-900 nm). The interference found in the reflection spectra in the mid- and far-infrared range of (9-30 μm) is not associated with the interference in the meso-PS layer, but refers to a thicker (tens of microns) layer in the silicon substrate, in which after anodic etching the dispersionless refractive index decreases slightly. The refractive index values were estimated based on the reflection spectra of meso-PS samples with interference features in the visible and near-IR ranges and on the thickness values according to the data obtained from the SEM images of the sample transverse cleavages. It is found that in the meso-PS layers the patterns of interference are observed for the limited range of thicknesses and only in the layers with a poorly developed morphology. In the range of the PS layer thicknesses up to 6.5 μm, the porosity increases with an increase in the anodizing current density and at keeping the time of anodizing. When the porosity of the PS layer is below 20 % or in the case of large values of the PS layer thickness (14.2 μm and more), this method cannot be used due to an increase in the absorption coefficient in meso-PS or a sharp development of the relief of the upper layer of meso-PS and the breakdown of interference in its layer in the visible and near-infrared range.

**DOI:**https://doi.org/10.15350/17270529.2021.4.38

**5.Anatoly A. Shakov,**

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**Svetlana F. Lomayeva, Vjacheslav A. Pozdeev, Andrey A. Shushkov, Alexander V. Vakhrushev**

**Organic Films Synthesized on the Iron Surface in Low-Temperature Plasma of Adamantane**P. 431-440. Download.

A method based on the interference method is proposed allowing the evaluation of the porosity of mesoporous silicon (meso-PS) layers. The meso-PS layer porosity was evaluated using the visible and near-infrared regions of the reflection spectra and the dispersionless refractive index diagram of porous silicon known in the literature; the meso-PS film thickness was specified. It is shown that for calculating the dispersionless refractive index it is necessary to use only the peaks of maxima and minima in the interference pattern in the wavelength range of (450-900 nm). The interference found in the reflection spectra in the mid- and far-infrared range of (9-30 μm) is not associated with the interference in the meso-PS layer, but refers to a thicker (tens of microns) layer in the silicon substrate, in which after anodic etching the dispersionless refractive index decreases slightly. The refractive index values were estimated based on the reflection spectra of meso-PS samples with interference features in the visible and near-IR ranges and on the thickness values according to the data obtained from the SEM images of the sample transverse cleavages. It is found that in the meso-PS layers the patterns of interference are observed for the limited range of thicknesses and only in the layers with a poorly developed morphology. In the range of the PS layer thicknesses up to 6.5 μm, the porosity increases with an increase in the anodizing current density and at keeping the time of anodizing. When the porosity of the PS layer is below 20 % or in the case of large values of the PS layer thickness (14.2 μm and more), this method cannot be used due to an increase in the absorption coefficient in meso-PS or a sharp development of the relief of the upper layer of meso-PS and the breakdown of interference in its layer in the visible and near-infrared range.

**DOI:**https://doi.org/10.15350/17270529.2021.4.39

**6. Anatoly L. Beltyukov, Natalia V. Olyanina, Vladimir I. Ladyanov**

**Investigations of the Viscosity of the Ternary Co**

_{81}

**(B,Si)**

_{19}

**and Co**

_{75}

**(B,Si)**

_{25}

**Melts**P.441-452. Download.

The temperature (polytherms) and concentration (isotherms) dependences of the kinematic viscosity of melts of the ternary Co_{81}(B,Si)_{19}
and Co_{75}(B,Si)_{25} systems have been experimentally studied. The polytherms of liquid alloys obtained in the heating and cooling regimes coincide and are well described by the exponential Arrhenius equation. The concentration dependences of the viscosity of ternary systems for the quasi-binary Co_{81}B_{19}-Co_{81}Si_{19} and Co_{75}B_{25}-Co_{75}Si_{25}
sections at fixed temperatures were plotted using polytherms. For both systems, the extreme points on the viscosity isotherms corresponding to binary alloys have similar values at the same temperatures. The viscosity isotherms of the quasi-binary sections are close to the additive sum of the viscosity values of binary alloys with a small positive deviation. When the temperature of the melt changes the character of the viscosity isotherms does not change significantly. The calculations of the concentration dependences of the dynamic viscosity of the melts of the Сo_{81}B_{19}-Co_{81}Si_{19} and Сo_{75}B_{25}-Co_{75}Si_{25}
systems were carried out according to the equations based on the data on the thermodynamic properties of the melt and/or the physical properties of its components. The calculations were carried out according to the following equations: Kozlov-Romanov-Petrov; Kaptay; Chhabra; Sato; Hirai; Budai-Benkö-Kaptay and Živković-(Budai-Benkö-Kaptay). For both ternary systems, the best agreement with the experimental data is shown by the Kozlov-Romanov-Petrov equation both in the nature of the isotherm and in the viscosity values. The maximum relative deviation of the viscosity isotherm calculated by the Kozlov-Romanov-Petrov equation from the isotherm obtained in the experiment does not exceed 15 %.

**DOI:**https://doi.org/10.15350/17270529.2021.4.40

**7. Ivan A. Elkin, Konstantin S. Stolbov, Alevtina A. Chulkina, Artemy N. Beltyukov**

**Layer-by-layer Consolidation of Cu- and Ti-based Metal Powders by Spot Pulsed Electric Current Sintering**P.453-465. Download.

Additive manufacturing (AM) has enormous prospects in the modern industry. Thanks to these technologies, it is possible to produce products with a complex shape, as well as to obtain materials with unique complex properties. The range of objects that can be produced by additive manufacturing is extremely wide, so there is a constant search for new methods. The method proposed in this paper can be considered, to some extent, as a development of the pulsed electric current sintering (PECS) method. In the conventional PECS method, the entire volume of powder is poured into a dielectric matrix, compressed by electrodes, and a powerful electric current pulse is passed through it. PECS allows obtaining bulk material with a different degree of porosity from virtually any electrically conductive powder. The disadvantages of PECS include limits on shapes of the materials obtained and the requirement for a pressing system and a high-power electric energy source. Additionally, selecting the modes of electro pulse sintering is necessary for each type of powder, since under unstable operating modes, the electric explosion of the powder or inhomogeneous sintering of the powder is possible. In the proposed method, powder is sintered layer-by-layer and each layer is sintered point-by-point. This makes it possible to use a low-power pressing system and a low-power electric energy source. In addition, during the sintering of powders, we can change the level of mechanical and electrical effects at each sintering point, which ensures the creation of the required porous structure in the volume of a sample. Since this additive manufacturing method is under development, the aim of this work was to study the influence of production modes, materials of used powders, their dispersion and morphology on the quality of the materials formed. Using the proposed method, bulk samples were obtained from powders having different chemical compositions (Cu, Ti, mechanically alloyed tin bronze), and particles of various shapes (dendritic, dumbbell, stone-shaped). Using X-ray diffraction, electron microscopy, and micro hardness measurements, the structural phase states and porosity of the obtained bulk consolidated materials were studied. It was found that the porosity of the sintered samples mainly depends on the powder morphology. Dispersed powders with compact particles provide denser samples. The powders with high electrical resistivity are consolidated with higher density because the heat generation along the boundaries of the powder particles depends on the specific resistivity of the material, as well as on the area of the contact and the transient resistance between the particles. Our method is characterized by little thermal influence on the sintered powders due to the rapid process of sintering and the relatively low heating temperature of the powder particles. When the mechanically alloyed powders are sintered, the nanostructured state is retained in consolidated materials. The decrease of the electrode scanning step leads to the increase in the uniformity of the sintered compacts and the improvement of their surface quality.

**DOI:**https://doi.org/10.15350/17270529.2021.4.41

**8. Anna G. Knyazeva, Natalia V. Bukrina**

**Numerical Study of the Effect of the Type of Refractory Particles on Product Formation Under Bulk Synthesis Conditions**P.466-477. Download.

Methods of combustion for the synthesis of composites are widely used in modern chemical and metallurgical technologies. According to experimental data, the most homogeneous product could be obtained by bulk synthesis. Such modes are often used to produce composite materials. However the high reaction rate of bulk synthesis significantly complicates the control of the formation of the product. One way to reduce the synthesis temperature consists in the inert particles introducing into the reaction mixture. Additionally the heating rate can be changed by changing the reactor wall thickness. However, the specifics of the conjugate heat exchange of the reaction mixture with the reactor walls under various synthesis conditions are rarely analyzed in publications. To demonstrate the role of these factors, a two-dimensional model of bulk synthesis of a composite from pure elements with the addition of inert particles is proposed in this work. The heating of the reactor walls by thermal radiation from a device whose temperature varies according to a given law is taken into account. The heat contact between reactor walls and powder mixture is assumed as ideal. The chemical reactions are described by a summary reaction scheme that corresponds to the synthesis of Ni_{3}Al from elements 3Ni+Al. The kinetic law takes into account the retardation of the reaction by the product layer and contains corresponding parameter. The effective thermal-physical properties of the mixture in the reactor depend on the properties of the initial components and the volume fraction of inert particles. The proposed model is implemented numerically. Process dynamics for WC, TiC, and Al_{2}O_{3} - particles are analyzed. Steel is used as the material of the reactor walls. It is established that the synthesis process in the volume is inhomogeneous, that is directly related to the conjugate heat exchange conditions. In spite of this, for the chosen set of reactor parameters and particle fraction, the result is the same: there are almost no unspent reagents in the final products. The dynamics of temperature change in different parts of the reactor may not correspond to the classical thermal explosion. This indicates that the transformation of the reagents into the product occurs in two stages. The first stage ends with an incomplete transformation. Then, due to the heat stored in the compact and in the reactor walls, the reactions are accelerated again, which leads to the formation of a final composite. Changing the fraction of inert particles in the mixture has an ambiguous effect on the dynamics of synthesis, That is due to the thermophysical properties changing with the fraction of particles and a decrease in the total heat release in the reaction.

**DOI:**https://doi.org/10.15350/17270529.2021.4.42

**9. Svetlana G. Menshikova, Vadim V. Brazhkin, Anastasia S. Danilova**

**Study of the Influence of High Pressures on the Solidification of the Al**

_{86}

**Ni**

_{6}

**Co**

_{4}

**Gd**

_{2}

**Tb**

_{2}

**Liquid Alloy**P.478-485. Download.

The purpose of the present paper is to study the possibility of the formation of new phases in the Al_{86}Ni_{6}Co_{4}Gd_{2}Tb_{2}
alloy upon rapid solidification of its high-temperature melt under high pressure. Samples for research were obtained under high pressure of 3, 5 and 7 GPa in a high-pressure chamber of the "toroid" type. The heating and melting of the sample were carried out by passing an alternating current through the sample placed in a hexagonal boron nitride crucible. High pressure punches served as current leads. The melts were cooled at a rate of 1000 deg/sec, the temperature of the melt before quenching was 1500 °C. The experimental scheme is as follows: pressure setting → pulse heating → holding at a set pressure and temperature → cooling without depressurization to room temperature → reduction of high pressure to atmospheric. The microstructure of the samples of the alloy of the eutectic composition Al_{86}Ni_{6}Co_{4}Gd_{2}Tb_{2}, obtained depending on the quenching temperature (1500 °C) and high pressure (3, 5 and 7 GPa), was investigated by the methods of X-ray diffraction analysis, optical and electron microscopy. The cooling rate was 1000 deg/sec. Due to the combination of a high solidification rate and mechanical compaction under high pressure, samples of the alloy of the composition Al_{86}Ni_{6}Co_{4}Gd_{2}Tb_{2}
with a fine structure and high density were obtained. At pressures of 5-7 GPa, the formation of new phases was observed in the alloy. The studies show that the average microhardness of a sample obtained, in particular, under pressure of 7 GPa, is high (~1700 MPa) due to the solid solution and precipitation hardening. It is almost twice as high as the average microhardness in the original sample. The results obtained show the fundamental possibility of using the method of solidification of the melt under high pressure to change the level of properties of aluminum alloys used in industry without changing their chemical composition by modifying the structure and changing the composition of the structural components of a sample.

**DOI:**https://doi.org/10.15350/17270529.2021.4.43

**10. Evgeny I. Salamatov, Elena B. Dolgusheva**

**Methods for Calculating the Lattice Thermal Conductivity of Metals at High and Low Temperatures**P.486-496. Download.

The Molecular Dynamics (MD) method seems to be the most promising for determining the lattice contribution to the overall thermal conductivity of metals and metal alloys. In the paper, the method is used for studying the lattice thermal conductivity of aluminum at high and low temperatures with a proven potential. It is shown that at high temperatures, standard algorithms are more convenient for calculating the lattice thermal conductivity coefficient. In this case, the thermal conductivity coefficient is calculated using the Fourier equation, and the MD calculation is used to simulate a stationary non-equilibrium state with a linear temperature gradient at a length comparable to the size of the calculated cell. This approach is shown to give the values of the lattice thermal conductivity coefficient in good agreement with the results of the first-principles calculations. With a decrease in the size of the base crystallite, the thermal conductivity coefficient decreases due to the depletion of the low-frequency section of the phonon spectrum, the contribution of which to thermal conductivity becomes insignificant with increasing temperature. At high temperatures, the thermal conductivity coefficient does not depend on the size of the crystallite and coincides with that obtained from the first-principles calculations. To calculate the thermal conductivity at low temperatures, a new method is proposed, based on the homogeneous heat equation given on an infinite line. In this case, the task of the MD method is to obtain a stationary non-equilibrium temperature distribution in the system in the form of a Gaussian curve, which is the fundamental solution of the equation. The approximation of the system relaxation from the non-equilibrium to equilibrium state allows finding the thermal diffusivity coefficient related to the thermal conductivity coefficient. The test calculations carried out for a thin aluminum film at low temperatures with different initial conditions showed that the obtained thermal diffusivity coefficient does not depend on the parameters of the initial Gaussian distribution, which suggests the suitability of the proposed method for studying the lattice thermal conductivity.

**DOI:**https://doi.org/10.15350/17270529.2021.4.44