Semibounded Bodies Research Articles

Heat treatment of ground surface using an electric arc plasma generator

The article presents a generalization of the results of experimental and theoretical studies on the intensive thermal effect on the surface of foundations from cohesive soils (as opposed to deep heat treatment) using an electric arc generator of low-temperature plasma (plasmatron). A plasmatron with massive consumable electrodes made of borosilicated graphite and a wide "blurred" plasma torch was used for heat treatment. Physical and mathematical modeling of the thermal effect on the surface of clay soil by a moving high-temperature source has been performed. For the first time, a special installation was manufactured and experimental studies were carried out on the heat treatment of ground surfaces on the site with an electric arc plasma torch. It is established that the effective heat flux density reaches 2.5-.3.5 kW/cm2. The main part of the thermal energy for the used plasmatron enters the material due to radiant heat transfer. An analytical mathematical model of the temperature field of a three-dimensional semi-bounded body in the technological process under consideration is given by an essentially nonlinear boundary value problem for a nonlinear heat equation with nonlinear boundary conditions of the second kind. When solving thermophysical problems, the ground semi-bounded space was considered as a quasi-homogeneous medium having a constant initial temperature and initial thermophysical parameters that change in the process of temperature increase. The effective values of the thermophysical characteristics were determined on the basis of experimental data. Calculations and experiments have shown that the temperature boundary leading to significant changes in the structural properties of soils drops to a depth of 4-6 cm from the surface, despite the boiling of the ground melt with a temperature of 2500 ...2800 K on the ground surface. Therefore, a new technological principle of heat treatment is proposed, which consists in building up the ground melt in layers of 4-5 cm up from the initial surface. The impossibility of obtaining a positive effect even with the use of a powerful high-temperature source of exposure in the case of traditional surface heat treatment technology has been confirmed. The new plasma technology of surface heat treatment of ground surfaces to the stage of silicate melt allows to obtain a common layer of the required thickness. At the same time, the efficiency of surface heat treatment is significantly increased. The new technology of layer-by-layer surface deposition reduces cracking in the layer when the melt cools, but does not eliminate this negative process.

Решение краевой задачи тепломассопереноса методом источников для условий лучистого обогрева почвы

The purpose of the research is to find, predict and regulate the temperature and humidity regime of the soil for the conditions of radiant heating of cultivation facilities using dark-type ceiling infrared emitters. A system of differential equations (in dimensional and dimensionless forms) is presented, reflecting the relationship of thermal and mass transfer processes in colloidal capillary-porous bodies in the case of surface radiant heating. A partial analytical solution of this system of differential equations for a semi-bounded body is considered with the exclusion of the phenomenon of thermal and thermal conductivity and period diffusion processes. On the example of milling peat, taking into account the initial data, the solution of the boundary value problem of heat and mass transfer by the method of sources is obtained, which is one-dimensional non-stationary fields of moisture content and temperature. It is established that under the given initial and boundary conditions, as well as taking into account the thermophysical properties of milling peat, the required moisture content will be achieved in six hours, the temperature in three hours. By adjusting the thermal power of the infrared radiation source, and, therefore, the intensity of moisture evaporation from the soil surface into the environment, it is possible to control the rate of heating and drying of the layer (for example, to determine the time or frequency of watering the soil). To clarify the values of moisture content and soil temperature by coordinate and over time, it seems expedient to consider analytical solutions that took into account not only basic, but also conjugate processes of heat and mass diffusion

Method for determining the thermal diffusivity and thermal conductivity coefficient by temperatures of plate surface as a semi-bounded body

The studied numerical and analytical model of a semi-bounded body is used to simultaneously determine the thermophysical characteristics (TFC): thermal diffusivity at and thermal conductivity coefficient λt of the material which make it easy to determine the volumetric heat capacity сt . Temperature distribution over the plate cross-section at the end of the calculated time interval τ is described by a power function, its exponent n depends on the Fourier number Fo. The values of TFC were calculated from the dynamics of changes in surface temperatures T(xp = Rp , τ) and T(xp = 0, τ) of the plate with a thickness Rp heated under boundary conditions of the second kind q = const. The temperature T(xp = 0, τ) was used to determine the time moment τe , at which the temperature perturbation reached the adiabatic surface xp = 0 (T(Rp , τe ) – Tb (0, τe = 0) = 0.1 K). Calculations of TFC (at and λt ) were performed using formulas whose parameters were found by solving a nonlinear system of three algebraic equations by selecting the Fourier number corresponding to τe . The author studied the complexity and accuracy of TFC calculation using the test (initial) temperature fields of a plate made of refractory material by the finite difference method. Dependences of TFC on the temperature ai (T ), λi (T ) and ci (T ) were set by polynomials. Temperatures of the plate with a thickness of Rp = 0.04 m with initial conditions Tb = T(xp , τ = 0) = 300, 900, 1200, 1800 K (0 ≤ xp ≤ Rp ) were calculated for a specific heat flow q = 5000 W/m2. The heating time to τe was 105 – 150 s. The average mass temperature Tm, pl of the plate during the τe increased by 5 – 11 K. The TFC values were restored by solving the inverse thermal diffusivity problem for 10 time points τi + 1 = τi + Δτ. The arithmetic mean deviations of TFC (Tm, pl ) from the initial values for calculations at Tb = 300, 900, 1200, 1800 K were less than 2.5 %. It was established that the values of at and λt obtained for the time moments ti are practically constant, therefore, a simplified calculation of at, o and λt, o is possible only from the values of temperatures T(Rp , τe ) and T(0, τe ) at the end of heating. The values of at, o and λt, o , which were calculated immediately for the entire heating time, differed from the initial values of the accepted heat exchange conditions by about 2 %. The parameters of simple algebraic formulas for calculating at, o and λt, o were found by solving a system of three nonlinear equations n = n( Fo), at, o = a(Tb , T(Rp , τe ), Rp , n, τe ), Fo = Fo(at, o , Rp , τe ) and expressions for λt, o = λ(Rp , q, n, Tb , T(Rp , τe )). The proposed method significantly simplifies the solution of the inverse problem of thermal conductivity.

Investigation of the Fracture of a Semibounded Body Compressed Along a Near-Surface Interface Crack

We consider the problem of compression of a piecewise homogeneous half plane by forces directed along a near-surface crack located on the interface of two materials. This problem belongs to the class of nonclassical problems of fracture mechanics because, for the analyzed loading scheme, the stressstrain state realized in the body is homogeneous and the corresponding expressions for stresses and displacements in the vicinity of the crack do not contain singular components. Due to the fact that the stress intensity factors are equal to zero, for the analyzed problem, the classical Griffith–Irwin fracture criteria are inapplicable. In the indicated situation, the onset of crack propagation is associated with the local loss of stability of the equilibrium state of a part of the material located in the region adjacent to the crack. By using the approaches of the three-dimensional linearized theory of stability of deformable bodies, we propose a mathematical statement of the problem and an approach to its investigation. By using the representations of stresses and displacements in terms of complex potentials, we analyze the case where the roots of the corresponding characteristic equation are identical for each material.

Analytical study of temperature field during microwave drying of the material

Mathematical models of heating wet materials in a microwave field are analyzed. A thermal conductivity model is presented that considers two internal sources of heat, positive and negative. A negative heat source considers moisture output during drying of the material. A formula is obtained for calculating the temperature of a semi-bounded body under the action of internal heat sources for boundary conditions of the first kind. The results of calculations of the temperature of water and a dense layer of wheat grain are presented, depending on the duration of the microwave field and the specific power for the heating period. It is shown that in order to obtain reliable results, it is necessary to correctly set the efficiency of the microwave chamber. The results of calculations of moisture content and temperature of a layer of wheat grain for a period of constant drying speed are presented.

Determination of thermal diffusivity of the material by numerical-analytical model of a semi-bounded body

A mathematical description of the material thermal diffusivity aт in a semi-bounded body is proposed with a relatively simple algorithm for its numerical and analytical by solving the inverse problem of thermal conductivity. To solve the problem, it is necessary to obtain the temperature values of the unbounded plate as a result of a thermophysical experiment. A plate can be conditionally considered as a semi-bounded body as long as the Fourier number Fo ≤ Foк (Foк ≈ 0.04–0.06). It is assumed that the temperature distribution over cross-section of the heated layer of the plate R is sufficiently described by a power function whose exponent depends linearly on the Fourier number. A simple algebraic expression is obtained for calculating aт in the time interval ∆τ from the dynamics of temperature change T(Rп , τ) of a plate surface with thickness Rп heated under boundary conditions of the second kind. Temperature of the second surface of the plate T(0, τ) is used only to determine the time of the end of experiment τк. The moment of time τк, in which the temperature perturbation reaches the adiabatic surface x = 0, can be set by the condition T(Rп , τк) – T(0, τ = 0) = 0,1 K. The method of approximate calculation of dynamics of changes in depth of the heated layer R by the values of Rп , τк , and τ is proposed. Calculation of a т for the time interval ∆τ is reduced to an iterative solution of a system of three algebraic equations by matching the Fourier number, for example, using a standard Microsoft Excel procedure. Estimation of the accuracy of a т calculation was made by the test (initial) temperature field of the refractory plate with the thickness Rп = 0.05 m, calculated by the finite difference method under the initial condition T(x, τ = 0) = 300 (0 ≤ x ≤ Rп) at radiation-convective heating. The heating time was 260 s. Calculation of aт, i was performed for 10 time moments τi + 1 = τi + Δτ, τ = 26 s. Average mass temperature of the heated layer for the whole time was T = 302 K. The arithmetic-mean absolute deviation of aт(T = 302 K) from the initial value at the same temperature was 2.8 %. Application of the method will simplify the conduct and processing of experiments to determine the thermal diffusivity of materials.

Determination of Thermal Diffusivity of Material by the Numerical-Analytical Model of a Semi-Bounded Body

A mathematical description of a material’s thermal diffusivity ae in a semi-bounded body is proposed with a relatively simple algorithm for its numerical and analytical calculation by solving the inverse problem of thermal conductivity. To solve the problem, it is necessary to obtain the temperature values of the unbounded plate as a result of a thermophysical experiment. A plate can be conditionally considered as a semi-bounded body as long as Fourier number Fo ≤ Foe (Foe = 0.04–0.06). It is assumed that the temperature distribution over a cross-section of the heated layer of the plate with thickness R is sufficiently described by a power-like function whose exponent depends linearly on the Fourier number. A simple algebraic expression is obtained for calculating ahc in time interval Δτ from the dynamics of temperature change T(Rp, τ) of a plate surface with thickness Rp heated at boundary conditions of the second kind. Temperature T(0, τ) of the second surface of the plate is used only to determine end time τe of the experiment. Moment of time τe, at which the temperature perturbation reaches adiabatic surface x = 0, can be set by the condition T(Rp, τe) – T(0, τ = 0) = 0.1 K. An approximate method of calculating the dynamics of changes in depth of heated layer R by values of Rp, τe, and τ is proposed. The calculation of ahc for time interval Δτ is reduced to an iterative solution of a system of three algebraic equations by matching the Fourier number, for example, using a standard Microsoft Excel procedure. Estimation of the accuracy of calculation of ahc at radiation-convective heating was performed using the initial temperature field of the refractory plate with thickness Rp = 0.05 m, calculated by the finite difference method under initial condition T(x, τ = 0) = 300 (0 ≤ x ≤ Rp). The heating time was 260 s. Calculation of ahc, i has been performed for ten time moments τi + 1 = τi + Δτ, Δτ = 26 s. Average mass temperature of the heated layer for the entire time was τe T = 302 K. The arithmetic-mean absolute deviation of ae (T = 302 K) from the initial value at the same temperature was 2.8%. Application of the method will simplify conducting and processing experiments to determine thermal diffusivity of materials.

Integral method of solving heat-conduction problems with the second-kind boundary condition. 3. Pulsed laser heating

Exact and approximate solutions to the non-stationary problem on the heat conduction in a semi-bounded body exposed to a pulsed laser radiation flow have been obtained. The action of rectangular, triangular and parabolic laser radiation pulses on this body was investigated. Polynomial relations have been constructed on the basis of the boundary-characteristic method with introduction into consideration of the temperature-disturbance front, and they made it possible to obtain practically exact solutions for the temperature function and its time derivative at both the stages of heating and cooling of the body. It is shown by some examples that the success in solving problems on the pulsed plasma heating of bodies is associated in many respects with the necessity of definition of the time law of movement of the temperature-disturbance front with the use of the Pade diagonal approximation, which excludes, practically completely, the divergence of the power series defining the law of movement of this front, in particular, in small time intervals. The approach proposed for solving heat-conduction problems with the second-kind boundary condition allows one to simply and effectively find solutions for isotherms and lines of equal heating and cooling. Analysis of the results obtained allows the conclusion that the effectiveness of solving various technological problems, based on the use of pulsed laser radiation, is determined by the success in solving the problems on control of the time shape of a laser pulse and determination of the temperature fields in the body on the basis of polynomial representations.

HEAT AND MASS TRANSFER IN SOLID – LIQUID SYSTEM WITH SUPERFICIAL HEAT SOURCE

The interaction between a solid (zinc) and a liquid reagent (a solution of hydrochloric acid) was studied theoretically and experi-mentally. These processes are used in the dimensional processing of metals in order to give those appropriate shapes. A cylindrical zinc billet was used as the object of the study. The billet lateral surface is protected by an acid-resistant coating and the dissolution process occurred only along the plane of the work piece section. The experimental installation scheme with the description of ex-perimental researches technique is resulted too. The zinc billet dissolution proceeds with the release of the gaseous phase during the nucleation of the bubbles, growth and separation from the solid surface where they nucleate. It promotes intensive mixing of the liq-uid and turbulizes the boundary diffuse layer. As a result, the process of maso-visualization is intensified. This process is controlled by diffusion and its intensity is determined by the mass-transfer coefficient. The value of the mass-transfer coefficient is presented in this work. The interaction is accompanied by significant thermal effects. This enhances the process of dissolving zinc with hydro-chloric acid. The heat source is located on the interaction surface and spreads by convection in a liquid reagent and thermal conduc-tivity in a solid, respectively. The temperature at the interaction surface is important to know, since it determines the values of the physical quantities that are used to mass-exchange processes calculation. The process of thermal conductivity in a semibounded body with a continuously operating surface heat source is considered in the condition of constant convective heat flow into liquid medium. A differential heat equation with a constantly acting heat source on the solid surface with the corresponding initial and boundary conditions is used to describe the dissolution process. The results of experiments are shown in the graphs form. The temperature of the interaction surface is determined as a function of time. It is also found that the largest surface temperature gradients are observed at the initial periods. The temperature field in a bar-shaped metal blank was determined theoretically and experimentally and it ade-quacy has been confirmed.

МАТЕМАТИЧЕСКАЯ МОДЕЛЬ РАСПРОСТРАНЕНИЯ ТЕПЛА В СИСТЕМЕ ДВУХ ПОЛУОГРАНИЧЕННЫХ ТЕЛ НА СТАДИИ ОСТЫВАНИЯ

МАТЕМАТИЧЕСКАЯ МОДЕЛЬ РАСПРОСТРАНЕНИЯ ТЕПЛА В СИСТЕМЕ ДВУХ ПОЛУОГРАНИЧЕННЫХ ТЕЛ НА СТАДИИ ОСТЫВАНИЯ

Modeling of Temperature Conditions for a Braking System with Regard for the Heat Sensitivity of Materials

We obtain a numerical-analytic solution of the thermal problem of friction for two semibounded bodies with regard for the heat sensitivity of materials and changes in the relative sliding velocity with time (braking with constant deceleration). The linearization of the corresponding boundary-value problem of heat conduction is performed by using the Kirchhoff substitution and the method of linearizing parameters. We study the changes in the temperatures of the working surfaces of the pad and the disk in the process of braking for two frictional couples with and without taking into account the temperature dependences of their thermal properties.

A programmed realization of methods for the diagnostics of the thermal properties of materials

A solution of the direct problem of thermal conduction when a linear pulsed heat source acts in the contact plane of two semibounded bodies, obtained by the method of finite differences, is presented. The results of a programmed realization and experimental investigations are given.

The detection of defects in materials using thermography

A three-dimensional solution of the direct problem of heat conduction for a semi-bounded isotropic body with a defect when it is heated by pulsed thermal radiation and cools due to convective heat transfer to the surroundings is presented. The results of modeling and experiment are given.

The principles of data processing based on the numerical solution of the nonlinear heat-conduction problem Pt. 2. The coefficient functional optimization problem in a variational formulation

The solution of the coefficient heat-conduction problem when a linear pulsed heat source acts in the plane of contact of two semibounded bodies, obtained using variational-iterational calculus methods, is presented. The results of tests on a number of heat-insulating materials are given.

The principles of information processing based on the numerical solution of the nonlinear heat-conduction problem. Pt. 1. The three-dimensional discrete mathematical model

A three-dimensional solution of the direct problem of heat conduction when a linear pulsed heat source acts in the plane of contact of two semibounded bodies is presented. The solution is obtained by the method of finite differences. An example of the modeling is considered.

Identification of the thermal properties of materials using methods of the calculus of variations

A solution of the heat-conduction problem when a linear pulsed source of heat acts in the plane of contact of two semibounded bodies, obtained using methods of the variational-iterational calculus, is presented. The results of tests of a number of heat-insulating materials are given.

Features of the transient processes in a ground in which solar energy is accumulated

Algorithms of the transient processes arising in a ground in which solar energy is accumulated with the use of individual vertical heat exchangers when the work of solar collectors is stopped or resumed are proposed. The results of solving problems on the nonstationary heat conduction in semibounded bodies by the method proposed were compared with classical results.

Quasistatic Thermoelastic Fields in a Half-Space Heated by a Circular Surface Heat Source

An accurate analytical solution of a classical problem, related to unbound thermoelasticity in a homogeneous and isotropic linearly-elastic half-space has been obtained in a closed integral form. Important partial cases, when the expressions for temperature and stresses take a simpler form, have been analyzed. The asymptotics of the found solutions have been investigated at small and large values of dimensionless time near the heat source and at a distance from it. The fields of isotherms and isobars have been studied. Unapparent properties of spatial inversion of both temperature and thermal stresses have been found. Maximum values of all components of the thermoelastic field at different values of the Poisson ratio have been found. For steady-state heating, a general solution, expressed in terms of a set of elliptical integrals, has been constructed in an explicit form. It has been shown that only compression stresses are possible in a steady-state mode in a semibounded body. The deformation profile of a free boundary has been determined. Possibility of mechanical destructions in the heated half-space has been analyzed. The graphs illustrating the spatial distribution and the time evolution of thermoelastic fields are given. The constructed solution can serve as a Green function for a set of other problems, having similar geometrical and boundary conditions.

Analytical Solution of the Problem of Heat Conduction of a Semi-Bounded Body with an Envelope and Its Application to Control and Identification of Heat Transfer Processes

Analytical Solution of the Problem of Heat Conduction of a Semi-Bounded Body with an Envelope and Its Application to Control and Identification of Heat Transfer Processes

Thermal Nondestructive Testing of the Thermal Properties of Materials Using Multifactor Transformation Functions

A solution of the direct problem of heat conduction when a linear pulsed heat source acts in the plane of contact of two semibounded bodies, obtained by the method of finite differences, is presented. The results of modeling are given. Using a discrete model, a method of determining (identifying) the thermal properties of materials using multifactor transformation functions is considered.