Nonstationary Temperature Research Articles

Application of an Ensemble Stationary-Based Category-Based Scoring Support Vector Regression to Improve Drought Prediction in the Upper Colorado River Basin

Recent above-normal temperatures, which exacerbated the impacts of precipitation deficits, are recognized as the primary driver of droughts in the Upper Colorado River Basin (UCRB), USA. This research aims to enhance drought prediction models by addressing structural changes in non-stationary temperature time series and minimizing drought misclassification through the ES-CBS-SVR model, which integrates ESSVR and CBS-SVR. The research investigates whether this coupling improves prediction accuracy. Furthermore, the model’s performance will be tested in a region distinct from those originally used to evaluate its generalizability and effectiveness in forecasting drought conditions. We used a change point detection technique to divide the non-stationary time series into stationary subsets. To minimize the chances of drought mis-categorization, category-based scoring was used in ES-CBS-SVR. In this study, we tested and compared the ES-CBS-SVR and SVR models in the Upper Colorado River Basin (UCRB) using data from the Global Land Data Assimilation System (GLDAS), where the periods 1950–2004 and 2005–2014 were used for training and testing, respectively. The results indicated that ES-CBS-SVR outperformed SVR consistently across of the drought indices used in this study in a higher portion of the UCRB. This is mainly attributed to variable hyperparameters (regularization constant and tube size) used in ES-CBS-SVR to deal with structural changes in the data. Overall, our analysis demonstrated that the ES-CBS-SVR can predict drought more accurately than traditional SVR in a warming climate.

SULFIDE CAPACITY AND PROPERTIES OF MODIFIED BUCKET SLAGS

Ladle slag during the entire period of iron refining must have a sufficient degree of ability to hold sulfides while simultaneously ensuring the minimization of iron losses. In ladles with recycled cast iron entering the desulphurization department, slags are formed already during the release of cast iron from the blast furnace or pouring from the mixer, which differ significantly from blast furnace slag in terms of physical and chemical properties. The basicity and sulphide capacity of slags decreases, which due to their active mixing with cast iron is accompanied by an increase in sulfur in the metal.The composition of slags, formed from the remains of blast furnace or mixer slags and products of chemical reactions of sulfur removal by injection of mixtures based on lime and magnesium, needs adjustment to increase the sulfide capacity while simultaneously thinning them to prevent excessive losses of iron with the slag. The use of an additional amount of fluidized lime for adjustment leads to an increase in the processing cost. It is proposed to use a slag modifier based on CaO- and Al2O3-containing steel processing waste to reduce the costs of fluidized lime. The rational specific consumption of modifiers was determined depending on the initial composition of ladle slags and the change in their sulfide capacity was calculated using models based on the concept of optical basicity.The change in melting point, viscosity, and melting point of modified ladle slags was determined, and the concentration path of CaO- and Al2O3-containing waste modified with additives was investigated. It is shown that for the typical composition of the ladle slag of cast iron desulfurization plant of PrJSC «Kamet-Stal», the sulfide capacity values ​​range from 1.0.10-4 to 20.0.10-4, which corresponds to blast furnace slags of the CaO-SiO2-MnO-Al2O3-MgO system with increased desulfurizing ability. The change in the amount of the solid fraction in the final slag in the course of refining and the effect of the basicity of the modified slag on the sulfur distribution coefficient in non-stationary temperature conditions were determined. With the application of the linear programming method, the range of rational chemical composition of ladle slag for the conditions of cast iron desulfurization plant of PrJSC «Kamet-Stal» was determined. Modified ladle slag with a basicity of 1.11-1.41 and a content of 40-51% CaO; 36-51 %SiO2 and 6-21 %Al2O3 are proposed to be considered as satisfying the given range of selected metallurgical properties.

Non-axisymmetric coupled non-stationary problem of thermoelectroelasticity for a long piezoceramic cylinder

A new closed solution to the non-axisymmetric coupled non-stationary problem of thermoelectroelasticity was constructed for a long piezoceramic cylinder for the case of satisfaction of the first and the third kind boundary conditions. Cylindrical surfaces were made as electrodes and connected to a measurement device with large input resistance. Limitation of a temperature change “load” rate made it possible to include equations of statics, electrostatics and thermal conductivity in the initial formula. The finite biorthogonal transforms are applying to explore a non-selfadjoint system of differential equations and to develop a closed solution. The obtained relations made it possible to determine the temperature and electric fields, and the stress-strain state in the piezoceramic cylinder, as well as the potential difference between cylindrical surfaces (electrodes) under non-stationary non-axisymmetric temperature impact.

Analysis of the thermomechanical behavior of a three-layer rectangular plate of the ceramic-metal-ceramic structure due to non-stationary convective heating

A rectangular isotropic layered plate is considered, which is convectively heated by the external environment. The thermal stress state of the plate is determined in two stages. At the first stage, the nonstationary temperature field is determined from the ratios of the non-stationary thermal conductivity problem. At the second stage, using the five-modal mathematical model of the shear theory of thermoelasticity, the parameters of the stress-strain state are determined.

RESIDUAL - STRESS AND TEMPERATURE FIELDS IN SURFACE TESTING: FINITE-ELEMENT ANALYSIS

Numerous advancements have been made to treat part surfacing, however, the presence of surface imperfections resulting from finishing processes has raised concerns about their potential to serve as stress concentrators. To address this concern, the present study utilized the power of the finite element method and cutting-edge software applications. The main aim of the study is to evaluate the stress-strain condition of material surfaces post-finishing, leveraging the comprehensive capabilities offered by these software systems. A significant component of the investigation centered on the influence of non-stationary temperature fields, as monitored by dynamic thermoelements, on the stress-strain dynamics within material surfaces. Visualization techniques were employed to depict the specified field functions, revealing notable variations in temperature distribution. The findings demonstrated that the standard function identified the highest temperature area within the 3-7-4 node segment, while the alternative function pinpointed it within the 4-node segment, indicated by the red area. These outcomes highlight the valuable role of non-stationary temperature fields in balancing the mechanical and physical aspects of the finishing process. This study contributes significant insights into the post-finishing stress-strain state of material surfaces, emphasizing the potential advantages of using modern software systems for in-depth exploration.

MATHEMATICAL MODELING OF THE NON-STATIONARY THERMAL STATE OF A COMPOSITE COATING OF CLOSE TO EQUIMOLAR COMPOSITION DURING LASER REMELTING

Based on assumptions and limitations, a mathematical model of the non-stationary thermal state of a composite coating during laser remelting on a metal substrate to obtain an alloy of equimolar or close to equimolar composition has been developed. The model includes the boundary value problem of calculating the nonstationary thermal state of the temperature distribution in the solid, two-phase and liquid regions, considering the heat release of crystallization according to the theory of a quasi-equilibrium two-phase zone in the presence of porosity of a mixture of metal powders. Based on the mathematical model, the computer program “Composite coating thermal state” was created, which allows to determine the non-stationary temperature distribution in the formed coating and substrate, the depth of penetration of the high-entropy coating layer, to evaluate the shrinkage of the molten coating under different technological conditions: laser power, the diameter of its focal spot, as well as the speed of movement along the surface of the processed material. The created program also allows to determine the dynamics of the position of two-dimensional liquidus and solidus lines for each metal included in the mixture. Using the created program, a computer simulation of the process of unsteady heating of a mixture of three metal powders in the same mass fractions: nickel, chromium and iron was carried out. Non-stationary temperature fields in the composite coating are obtained, considering its penetration and heating of the substrate. The depth of penetration of all metals, including the most low-melting one, was determined during laser remelting on a metal substrate in the production of a high-entropy alloy. The practical significance of the developed computer model lies in its use to predict the penetrationdepth of a composite coating under specified technological modes of laser remelting. At the same time, its adaptation is necessary, based on experimental data on the porosity and thickness of the coating before and after melting of simple metal systems with similar thermophysical properties.

Mathematical Modeling of the Heat Transfer Process in Spherical Objects with Flat, Cylindrical and Spherical Defects

This paper proposes a method for determining the optimal parameters for the thermal testing of plant tissues of fruits and vegetables containing surface and subsurface defects in the form of areas of plant tissues with different thermophysical characteristics. Based on well-known mathematical models for objects of predominantly flat, cylindrical and spherical shapes containing flat, spherical and cylindrical regions of defects, numerical solutions of three-dimensional, non-stationary temperature fields were found, making it possible to measure the power and time of the thermal exposure of the sample surface to the radiation from infrared lamps using the finite element method. This made it possible to ensure the reliable detection of a temperature contrast of up to 4 °C between the defect and defect-free regions of the test object using modern thermal imaging cameras. In this case, subsurface defects can be detected at a depth of up to 3 mm from the surface. To determine the parameters of mathematical models of temperature fields, such as thermal conductivity and a coefficient of the thermal diffusivity of plant tissues, a new method of a pulsed heat flux from a flat heater is proposed; this differs in the method of processing experimental data and makes it possible to determine the required characteristics with high accuracy during the active stage of the experiment in a period not exceeding 1–3 min.

Improving the technology for Inflow Profile Evaluation in horizontal Wells by temperature changes Due to calorimetric Mixing

The paper is devoted to the problem of increasing the effectiveness of temperature logging quantitative interpretation for the inflow profile evaluation in horizontal production wells draining heterogeneous reservoirs with low permeability. Such wells are characterized by an extremely uneven distribution of inflow along the length of the wellbore. One of the ways of quantitative interpretation of temperature logging is based on the effect of calorimetric mixing. It’s widely used to quantify the share of local producing intervals in the total flow rate.The low accuracy of interpretation is associated with the lack of reliable information about the temperature of the fluid flowing into the wellbore. The authors propose an estimate of this parameter based on the similarity of the behavior of the temperature field vs time in the near-wellbore zone of a reservoir during periods of stable production and the periods of the well shut-in. This relationship is confirmed by the results of modeling the temperature field of the “well – reservoir” system, taking into account changes in a wide range of reservoir permeability and thermal properties of the reservoir, the geometry of hydraulic fractures in the reservoir, the flowing fluids, as well as the parameters of the well production targets. The logging technology recommended by the authors involves the registration of several temperature profiles along the length of the wellbore at the beginning of the well production with the maximum rate and drawdown and during the well shut-in. Their integrated analysis based on the behavior of the temperature field features in time identified on the basis of modeling makes it possible to evaluate with a high degree of certainty the dynamics of the temperature of the gas-liquid mixture coming from reservoirs to the wellbore during production periods. This provides the required accuracy in the quantitative assessment of the inflow profile from mixing anomalies.The proposed approaches to the interpretation of thermograms are applicable in the analysis of the results of non-stationary temperature logging results in horizontal and vertical wells during the production from heterogeneous reservoirs both through perforation and multi-stage hydraulic fracturing.

RESEARCH OF HEAT RESISTANCE OF CARBON-CARBON COMPOSITE MATERIALS WITH PROTECTIVE COATINGS OBTAINED USING FUNCTIONALLY ACTIVE CHARGES

This study focuses on the development of protective coatings for carbon-carbon composite materials (CCСM) used in high-temperature processes in aerospace engineering. CСCM have limitations due to their susceptibility to oxidation, erosion, and burnout in gas streams. Our research is aimed at creating protective coatings using functionally active charges (FAC) obtained under non-stationary temperature conditions and improving the performance of composites. The aim of our study is to identify the optimal compositions of powders for chromium-alloyed protective coatings using functionally active charges to increase the heat resistance of the working surface. We analyzed various methods of obtaining protective coatings, including chemical-thermal and liquid-phase saturation methods, to find out their peculiarities in interaction with the CCСM matrix and changes in their mechanical properties. In addition to the traditional methods, we have investigated the method of saturating the surface with a solid phase in an active gas environment using FAС, which are obtained under non-stationary temperature conditions. This method provides high quality coatings, reduces processing time and makes it possible to work at high temperatures depending on the composition of the spray coatings. Much attention was paid to the problem associated with chemical interaction and formation of carbide phases, which are important for ensuring the stability of coatings under high temperature conditions. Experimental studies include a factorial experiment to determine the compositions of powder mixtures that provide high heat resistance. Various independent variables, such as the content of chromium, silicon, titanium, and aluminum, were considered, taking into account their influence on the physical and mechanical properties of coatings. Particular attention was paid to the optimization of the parameters of thermal autoinitiation of the functionally active charge under process conditions. Regression equations are presented to evaluate the dependence of the wear resistance of coatings on the autoinitiation parameters and the content of alloying elements. The analysis of the study results includes the construction of three-dimensional graphical dependencies for optimizing the composition of powdered CBA in the Cr-Al-Ti and Cr-Al-Si systems. In practice, chromium-aluminum-siliconized coatings obtained under isothermal conditions, when tested for heat resistance, have a more porous surface through which oxygen penetrates to the surface of the HCCM. Compared to the coatings obtained using FAS, the heat resistance is 1.5-1.7 times higher, which can be explained by the higher concentration of chromium, aluminum, silicon, and titanium, which contribute to the formation of protective oxide membraness. The low-porosity surface of the coatings obtained using functionally active layers prevents oxygen from entering the material, contributing to the formation of oxide protective membranes such as SiO2, TiO2, Cr2O3, Al2O3.

Synthesis and sensory properties of tungsten (VI) oxide-based nanomaterials

The purpose of this work was to develop a methodology for the synthesis of WO3-based nano-scale materials, to provide their characterization, and to study their sensory properties. The nanopowder was made by slowly adding nitric acid to an aqueous solution of ammonium paratungstate,(NH4)10W21O41·xH2O, followed by centrifugation, drying, and calcination. The size of tungsten trioxide grains, which was 10-20 nm, was determined by transmission electron microscopy. According to X-ray phase analysis, the powder, which was calcined at a temperature of 500 °C, mainly consisted of a triclinic phase. Subsequently, diammine palladium (II) nitrate and terpeniol were added to the WO3 nanopowder to form a paste. The resulting paste was applied to a special dielectric substrate and calcined to a temperature of 750 °C. As a result, a fragile tungsten trioxide-based gel formed. The mass fraction of palladium in the fragile gel was 3%. The sensory properties of the obtained gas-sensitive material were studied under stationary (300 °C) and non-stationary temperature conditions (quick heating to 450 °C and slow cooling to 100 °C). A sharp increase in the sensitivity of a tungsten trioxide-based sensor was observed under non-stationary temperature conditions which depended on the composition of the gas-sensitive layer

Qualitative Properties of the Solution of a Conjugate Problem of Thermal Convection

The joint convection of two viscous heat-conducting liquids in a three-dimensional layer bounded by flat solid walls was studied. The upper wall is thermally insulated, and the lower wall has a non-stationary temperature field. The liquids are immiscible and separated by a flat interface with complex conjugation conditions set on it. The evolution of this system in each liquid was described by the Oberbeck–Boussinesq equations. The solution of the problem was sought for velocities that are linear in two coordinates and temperature fields that are quadratic functions of the same coordinates. Thus, the problem was reduced to a system of 10 nonlinear integro-differential equations. Its conjugate and inverse nature is determined by the four functions of time. Integral redefinition conditions were set to find them. The physical meaning of the integral conditions is the closeness of the flow. The inverse initial-boundary value problem describes convection near the temperature extremum point on the lower solid wall in a two-layer system. For small Marangoni numbers, the problem was approximated linearly (the Marangoni number is analogous to the Reynolds number in the Navier–Stokes equations). Using the obtained a priori estimates, sufficient conditions were identified for the non-stationary solution to become a stationary one over time.

Full-scale strain gauge studies of structures under conditions of high temperatures and elastoplastic deformations

The features of full-scale strain gauge studies of structures operated at temperatures up to 550°C, in which local elastic-plastic deformations occur, are considered. A method for determining the creep of strain gauges based on measuring the output signals during cyclically varying deformation is considered. Algorithms for accounting for errors associated with the creep of strain gauges at nonstationary temperatures are proposed. An algorithm for calculating stresses based on measured deformations at variable elastic and elastic-plastic loading of the structure is determined.

Constructing solutions for two-dimensional quasi-static problems of thermomechanics in terms of stresses for bodies with plane-parallel boundaries

A methodology to construct solutions for two-dimensional quasi-static thermomechanical problems for bodies with plane-parallel boundaries (2D-QS thermomechanical problems) is proposed. This approach begins with selecting equations for the plane quasi-static thermoelasticity problem in terms of stresses. The methodology approximates the distribution of non-zero stress tensor components through the body's thickness using cubic polynomials, with coefficients expressed in terms of integral characteristics of the stress tensor components over the thickness variable and their specified boundary values on the body’s lower and upper surfaces. Consequently, the original two-dimensional boundary problem is simplified to a one-dimensional boundary problem for the integral characteristics. For an infinite layer, solutions are found using the Fourier transform along the longitudinal coordinate, while for a strip plate, a finite integral transformation is applied along the transverse coordinate. General solutions for 2D-QS thermomechanical problems are formulated for the selected bodies under non-stationary volume forces and temperature fields. The resulting expressions for the stress tensor components are presented as convolutions of functions representing the boundary values on the bases (and end cross-sections for strip-plates) and functions describing homogeneous solutions to the one-dimensional boundary problems for the integral characteristics of the stress tensor components.

Численное моделирование нагрева защитной стенки модели резервуара типа "стакан в стакане" в условиях пожара в основном резервуаре

Introduction. Within the framework of this article, the need to solve a thermal engineering problem aimed at ensuring the fire safety of tanks with a protective wall is substantiated. Computational modeling was also carried out using the developed mathematical model to determine non-stationary temperature fields and stress-strain states that arise in the bodies of a model tank with a protective wall under fire conditions, with a nominal capacity of 700 and 5000 m3. Purpose and objectives. Experimental and theoretical assessment of the non-stationary temperature field of the protective wall of a "glass in a glass" type tank with oil and petroleum products to predict its stability in case of fire. The task is to develop and test a mathematical model in order to carry out numerical simulation of the interaction of a fire flame with the protective wall of a tank to determine its non-stationary temperature fields. Research methods. Numerical modeling was carried out on the basis of a modern computing platform – COMSOL Multiphysics, which allows you to conveniently and accurately simulate complex and complex physical processes, including the interaction of an oil fire flame with the tank body and its protective wall. Results and its discussion. As a result of numerical modeling, the values of the distribution of thermal fields and the stress-strain state on the main tank and the protective wall of the "glass in a glass" type tank were obtained. The calculation results are visualized in the form of isofields, which also allows you to clearly understand the distribution of temperature fields and stresses on each belt of the tank. Conclusions: The results of numerical modeling were obtained to determine the non-stationary temperature fields and stress-strain state that arise in the body of a mathematical model of a tank with a protective wall under fire conditions. The developed and justified numerical modeling method was tested on a reservoir with the most common nominal capacity in operational practice, 5000 m3, with its corresponding geometric parameters. The results of a complex numerical calculation showed that the temperature value of the protective wall body in its upper zone can significantly exceed its critical heating temperature and amount to about 720 °C, at which a stress-strain state occurs. Keywords: reservoir; protective wall; oil; mathematical modeling; mathematical model; fire.

THE EQUATION OF THE NON-STATIONARY THERMAL CONDUCTIVITY PROBLEM OF THE THERMOMETRICAL DEVICE CONSTRUCTIVE ELEMENT OF WEAPON SYSTEMS AND MILITARY EQUIPMENT

Means of contact thermometry are widely used in various complexes of devices of weapons systems and military equipment. Such means basically involve the use of mechanical contact, in particular, bimetallic temperature transmitters, which are the most widely used thermometer variants in practice. To ensure the proper level of metrological and operational characteristics of bimetallic thermometers, reliable and adequate methods of calculating thermodynamic parameters (distributions of temperature, deformations, stresses caused by force and temperature loads) of the bimetallic package are necessary. The presented work describes an approach to the calculation of non-stationary temperature distribution in a solid-state element of a complex shape, which simulates the operation of a bimetallic thermometer sensor using the contact method. The calculated dependences obtained are the basis for modeling bimetallic thermometers of various structural forms and standard sizes. In order to constructively improve the thermometric system, it is proposed to make bimetallic plates non-one-pieces, but with additional structural inclusions, in the presence of which the temperature distribution and, accordingly, the stress state will approach to uniaxial one. Keywords: differential equations, integration constants, displays of primary heat exchangers, calculation dependencies, information transfer, metrological, operational characteristics, temperature, density, thermal conductivity of the medium.

Analysis of non-stationary temperature field generated by a shaftless screw conveyor heated by Joule–Lenz effect

Analysis of non-stationary temperature field generated by a shaftless screw conveyor heated by Joule–Lenz effect

Constructing a physical-mathematical model of grain mass self-heating by a rod site of rectangular cross-section

This paper considers the issue related to improving the energy and resource efficiency of the process of storing raw materials of plant origin, namely, preventing self-heating of grain masses in elevator silos. It was noted that the effectiveness of the analysis of self-heating of grain mass increases with the use of mathematical models of temperature fields of grain mass during storage together with data obtained experimentally. A physical-mathematical model has been built that describes a two-dimensional localized non-stationary temperature field of seed material generated by a homogeneous rod cell with a rectangular cross-section. A technique for accelerating the convergence of the series is proposed for the constructed analytical solution, which is based on the selection and analytical calculation of the sum of the component of slow convergence. The adequacy of the physical-mathematical model has been proven by calculations and by comparing the temperature of the self-heating site, obtained theoretically, and the temperature obtained under industrial setting. The established temperature kinetics of grain mass volumes during storage, obtained experimentally and theoretically, correlate with each other in the duration range from 0 to 30 days with a correlation coefficient of at least 0.98. This proves the possibility of applying forecasts of the temperature of self-heating sites in the volume of grain mass, obtained by using the physical-mathematical model built, under industrial setting. A limitation of the study is that the model is not universal. It is another stage on the way to a universal model. A limitation of the study is that for storage periods of more than 30 days, a new excess temperature forecast must be made

Нестаціонарна математична модель розподілу температур в шарах соняч-ної панелі

This paper presents the results of mathematical modeling of non-stationary temperature fields in a typical solar panel under real environmental conditions. The mathematical model is based on a system of nonlinear ordinary differential equations with corresponding initial and boundary conditions. The model takes into account radiation losses from the surface of the panel, which are determined by the Stefan–Boltzmann law, and convective losses due to free and forced convection. The solar flux density was considered constant, but its value depended on the solar panel setting angle. The temperature dependence of the solar cell efficiency was calculated using a standard method. A computational algorithm was developed in C++ using standard mathematical libraries with a linearization of the system of ordinary differential equations. The results were visualized using the gnuplot graphing utility. The temperature distribution in each of the solar panel layers was obtained as a function of the ambient temperature. It was found that an increase in the ambient temperature leads to a significant decrease, up to 40%, in the solar panel efficiency. With increasing ambient temperature, the time of transition to steady operation increases. The solar panel temperature was related to the blackness degree of the protective glass. It was shown that in the Kirchhoff approximation it is necessary that the blackness degree of the selective coating of the protective glass be a maximum, which reduces the temperature of the system and increases its efficiency. The solar panel temperature was related to the wind speed. It was shown that the convective losses increase with the wind speed, which has a favorable effect on the solar panel temperature regime. The results of the study showed the effect of various external environmental factors on the temperature regime of a solar panel and a way to maximize its efficiency by optimizing its parameters. The results may be used in the development and production of improved solar panels with minimum temperature effects on their efficiency.

DETERMINING HEAT LOSSES FROM THE BUILDING ENVELOPE USING THE NON-STATIONARY METHOD

One of the main causes of the climate change is accumulation of huge amount of carbon dioxide (CO2) in the atmosphere emitted from the combusting of organic fuels (coal, oil products and natural gas), consequently, to slow down the progress of the global warming is directly related to the limitation of CO2 emission which could be achieved through the rational use of fuel and energy in every sector (industrial, household, transport and building sectors), introduction of energy-saving measures including highly efficient technologies and innovative methods. The building sector accounts for about 40% of the energy saving potential, therefore reduction of energy losses is the best way to reduce energy consumption of buildings. To calculate the heat loss from the building envelope, it is necessary to know the thermal conductivity coefficient (λ) of each construction element. Currently developed methods of λ determination are entirely based on the laboratory test using the stationary regime. For more realistic results, it is necessary to take into account the daily variability of temperature and non-stationary thermal conductivity processes. Solving the non-stationary thermal conductivity tasks are associated with the significant difficulties due to the application of the relatively complex mathematical equations. Usually, the theory of non-stationary thermal conductivity refers to the method of separation of variables or the so-called Laplace Transform, which requires the use of operational counting methods. The article presents an innovative method for determining the coefficient of thermal conductivity (λ) of each construction element in the non-stationary temperature regime, which enables determination of heat losses from the building envelope in real environment using the precise definition of thermal flow velocity.

METHOD OF CALCULATING THE UNSTATIONARY TEMPERATURE FIELD PARTS OF THE SPHERICAL FORM OF THE STRUCTURE FLIGHT APPARATUS OF UNMANNED AVIATION COMPLEXES

The design of unmanned aircraft vehicles during operation is exposed to significant temperature effects, which causes changes in the dielectric characteristics of radio technical elements and can lead to failures of on-board radio electronic equipment, as a result of which the task of determining the amount of temperature change in the compartments where the radio electronic equipment is located is performed. At the same time, some external elements of the aircraft vehicle can have complex structures with shapes close to spherical and different thicknesses in each cross-sectional structure.
 There is a need to develop a method for calculating the non-stationary temperature field with the purpose of finding out the real conditions in which all the elements of the aircraft structure work, which would help to determine how the temperature of the structure changes during any flight. At the same time, it is necessary to take into account the three dimensions of temperature gradients on spherical parts of structures of various shapes, namely nose part, rounded wing and others. In them, as a rule, each cross-section of the wall has its own thickness and, according to each cross-section, the temperature distribution along the thickness of the wall will correspond, that is, it is necessary to ensure the redistribution of temperature along the structural element, the calculation of which month. In addition, it is necessary to note that the wall of the aircraft can be multi-layered and the properties of its material can change along with the change in temperature. Considering the fact that it is impossible to take into account all the listed features by analytical methods, when choosing a calculation method, it will be advisable to stop at numerical methods.
 One of the most well-known numerical methods is the finite difference method with an implicit calculation scheme, which has an indisputable advantage for modeling - absolute stability with sufficient changes in time and coordinate steps. We will use the level of thermal conductivity in spherical coordinates in order to increase the accuracy of the calculation, for which solution, after finding out the boundary conditions, we will perform a one-dimensional calculation of the temperature by thickness in each mesh section of the structure, and also perform a calculation of the two-dimensional "heat flow" along the solid between the points, that are located on the same thickness, but have a temperature difference. Thus, when calculating the temperature field, the sphericity and multi-lay redness of the structural elements is taken into account, and the property of the material of the aircraft structure changes depending on the temperature.