State Temperature Distribution Research Articles

An analytical model of three-hot-wire acoustic particle velocity sensors

A thermal-based acoustic vector sensor consisting of three heated wires can detect acoustic particle velocity instead of pressure. This paper proposes an analytical model for calculating the temperature distribution and acoustic-caused temperature perturbation of the sensor with a three-hot-wire configuration. In this model, the nonlinear effect caused by non-uniform thermal conductivity is considered. The nonlinear effects will not only affect the temperature of the three hot wires, but also the frequency response of the sensor. Based on the stationary temperature distribution calculated by the nonlinear model, we correct the heat diffusion coefficient in the acoustic perturbation model. The nonlinear stationary model and the corrected perturbation model are found to be in good agreement with the numerical results. Furthermore, we design a chip to measure the temperature distribution of the three heated wires and compare the sensitivity of the sensor with the corrected model. The experimental results can verify the proposed model. This study provides a theoretical basis for sensor performance optimization.

On the application of the Fourier method to solve the problem of correction of thermographic images

The work is devoted to the construction of computational algorithms implementing the method of correction of thermographic images. The correction is carried out on the basis of solving some ill-posed mixed problem for the Laplace equation in a cylindrical region of rectangular cross-section. This problem corresponds to the problem of the analytical continuation of the stationary temperature distribution as a harmonic function from the surface of the object under study towards the heat sources. The cylindrical region is bounded by an arbitrary surface and plane. On an arbitrary surface, a temperature distribution is measured (and thus is known). It is called a thermogram and reproduces an image of the internal heat-generating structure. On this surface, which is the boundary of the object under study, convective heat exchange with the external environment of a given temperature takes place, which is described by Newton’s law. This is the third boundary condition, which together with the first boundary condition corresponds to the Cauchy conditions - the boundary values of the desired function and its normal derivative. The problem is ill-posed. In this paper, using the Tikhonov regularization method, an approximate solution of the problem was obtained, stable with respect to the error in the Cauchy data, and which can be used to build effective computational algorithms. The paper considers algorithms that can significantly reduce the amount of calculations.

Correction of thermographic images based on the minimization method of Tikhonov functional

The paper considers the method of correction of thermographic images (thermograms) obtained by recording in the infrared range of radiation from the surface of the object under study using a thermal imager. A thermogram with a certain degree of reliability transmits an image of the heat-generating structure inside the body. In this paper, the mathematical correction of images on a thermogram is performed based on an analytical continuation of the stationary temperature distribution as a harmonic function from the surface of the object under study towards the heat sources. The continuation is carried out by solving an ill-posed mixed problem for the Laplace equation in a cylindrical region of rectangular cross-section. To construct a stable solution to the problem, the principle of the minimum of the Tikhonov smoothing functional we used.

Transient Surface Charge Characteristics of DC-GIL Insulator Under Thermal-Electric Coupled Fields

The insulator in direct current gas-insulated transmission lines (DC-GIL) would suffer discharge risk due to surface charge accumulation under thermal-electric coupled fields. In this paper, the transient surface charge accumulation characteristics of a basin-type DC-GIL insulator is investigated via finite element method based on a three-dimension horizontally installed GIL model. The stationary temperature distribution of the model is obtained and then applied to the transient simulation of charge. Weak form partial differential equation is employed to deal with the ion transportation equation. Equations and parameters in the simulation are optimized to reduce the computing memory and time. Results indicate that the charge accumulation is accelerated due to the promotion of conduction through the insulator under thermal gradient. Higher charge density is obtained under thermal gradient. And the surface charge density of the convex surface is higher due to the promoted conduction. The highest field strength increases and the corresponding location moves along the convex surface during the transient process. This could attribute to the influence of transient charge behavior under thermal gradient on the electric field distribution. This study indicates that the thermal gradient and transient charge accumulation should be considered when dealing with the insulation characteristics of DC-GIL with insulators.

The heating of a stream of particles by thermal counter radiation

Introduction. It is accepted that the depth of heating of the dust/gas/air mixture by the radiation of combustion products SR is equal to the length LR of the free path of radiation in the mixture. Numerical simulation of combustion of a gas-air mixture that has inert particles, taking into account the re-radiation of heat by heated particles of the fresh mixture, led to ratio SR >> LR. In this work, the analytical assessment of ratio χS = SR/LR is performed.One-dimensional problem model. The co-authors determined stationary temperature distribution over the flow of initially cold monodisperse particles suspended in vacuum. Particle velocity V is directed toward a heat-radiating, absolutely black surface that is permeable by particles. Simplifying assumptions are used: radiation consists of two oppositely-directed flows of electromagnetic energy; interaction between particles and radiation is described in the approximation of geometric optics; the temperature inside the particle is the same. Problem solving. It is shown that χS is determined by V=Vcp / (εT 0,5, σTb)3 , where cp, εT, σ, Tb are, respectively, heat capacity per unit volume of the suspended matter, integral emissivity of the particle material, the Stefan-Boltzmann constant, and the surface temperature. For ≤ 2.8, re-emission can be neglected: χS ≈ 1. At ≤ 1.2, temperature distribution regulates re-emission: χS ≈ 5 –1/(2 – εT) >> 1.Solution discussion. The analytical solution satisfactorily describes the available numerical solutions and experimental data for the case of combustion of a dust/gas/air mixture after specifying the parameters of a simplified model: the radiating surface should be understood as the flame front, Tb is the combustion temperature, and cp is the overall heat capacity of the mixture. The estimate ≤ 1.2 indicates the final high temperature of the gas suspension, the possibility of its autoignition far from the flame, and the need to change initial assumptions when simulating re-emission.Conclusions. Analytical evaluations make it possible to employ ratios SR >> LR and SR ≈ LR for the suspension over a thermal radiation source in vacuum. Conditions for the application of the results of simplified simulation of re-emission to the combustion of a dust/gas/air mixture are formulated.

Application of the Minimum Principle of a Tikhonov Smoothing Functional in the Problem of Processing Thermographic Data

The paper considers a method for correcting thermographic images. Mathematical processing of thermograms is based on the analytical continuation of the stationary temperature distribution as a harmonic function from the surface of the object under study to the heat sources. The continuation is performed by solving an ill posed mixed problem for the Laplace equation in a cylindrical region of rectangular cross-section. The cylindrical area is bounded by an arbitrary surface and plane. The Cauchy conditions are set on the surface-the boundary values of the desired function and its normal derivative. Inhomogeneous conditions of the first kind are set on the side faces of the cylinder. The problem is the inverse of the corresponding mixed problem for the Poisson equation. In this paper, an approximate solution of the problem is obtained that is stable with respect to the error in the Cauchy data and inhomogeneity in the boundary conditions. In the course of constructing an approximate solution, the problem is reduced to the Fredholm integral equation of the first kind, which is solved using the minimum smoothing functional principle. The convergence of the approximate solution of the problem is proved when the regularization parameter is matched to the error in the data.

Effect of diffuse phonon boundary scattering on heat flow

We propose a model for the thermal conductivity of 2D-samples when the mean free paths due to the phonon–phonon interactions exceed the sample dimensions. The physical mechanisms that ensure the stationary heat flux and the stationary nonequilibrium temperature distribution are examined. A recursive equation is derived to quantify the contribution of phonon scattering to the net heat flux in the pure elastic isotropic diffusive regime, as a function of the number of scattering at the boundaries. As the length to width ratio L/W increases above unity, our model shows an increase in the phonon mean free path compared to that predicted by the Casimir model in which the lateral walls are assumed to be black bodies. The present model also reveals that the multiple phonon interactions with the lateral walls lead to a stationary, but non-linear temperature distribution. The dependence of the heat flux and the thermal conductivity on the sample dimensions is then shown to be non-monotonic. This infers that the location of the thermometers on the sample influences experimental measurements of the thermal conductivity.

The Theory of Diffusion Zone Formation during Surface Modification of Materials with Allowance for the Thermodiffusion Effects

Using asymptotic methods in the theory of differential equations, an analytical approximate solution to the problem of diffusion of interstitial atoms in a restricted medium under the condition of a constant temperature gradient is obtained. As follows from the analysis, this solution is applicable to plate samples undergoing thermal and ionic modification of one of the surfaces, whose thickness does not exceed the diffusion penetration depth of impurities, provided that a stationary temperature distribution over the thickness is established. For the certification of the proposed method, a comparison of the approximate solution with the numerical solution of the thermal diffusion problem is carried out. Conditions are obtained under which the contribution of thermal diffusion effects to the total diffusion mass transfer of the impurity is significant.

A Variation Model of Thermal Breakdown of a High-Voltage DC Cable Electrical Insulation

On the basis of the presented differential form of the mathematical model describing the steady-state thermal conductivity in the layer of electrical insulation of a single-core high-voltage dc cable, the variation form of this model is elaborated. The analysis of stationary points of the functional obtained is carried out and a critical combination of the main parameters preceding the thermal breakdown of this layer has been determined. To find the best trial functions allowable for the functional, which approximate the stationary temperature distribution in the polymer layer of electrical insulation, residuals are compared that arise when substituting these functions into an original differential equation.

3D Simulation of Anisotropic Heat Transfer in Outer Layers of Magnetized Neutron Stars

The results of three-dimensional numerical simulation of heat transfer in the outer layers of magnetized neutron stars are presented. Determining the structure of the magnetic field on a neutron star surface is an important task of modern astrophysics. In the presence of strong magnetic fields, the medium becomes anisotropic, and the laws of heat conduction change. The tensor coefficient of thermal conductivity for magnetized degenerate plasma was obtained by Bisnovatyi-Kogan and Glushikhina by solving the Boltzmann equation with the Chapman–Enskog method. In this paper, the temperature distribution on the surface of a magnetized neutron star is obtained for magnetic fields of the dipolar and quadrupolar type, as well as for their superposition. To numerically solve the stationary temperature distribution problem in the outer layers of a magnetized neutron star, the basic (support) operator numerical method was extended to a three-dimensional case. The problem was solved on the grid which consists of tetrahedra.

Thermal flux-flow regime in long Josephson tunnel junctions

We study thermal transport induced by soliton dynamics in a long Josephson tunnel junction operating in the flux-flow regime. A thermal bias across the junction is established by imposing the superconducting electrodes to reside at different temperatures, when solitons flow along the junction. Here, we consider the effect of both a bias current and an external magnetic field on the thermal evolution of the device. In the flux-flow regime, a chain of magnetically-excited solitons rapidly moves along the junction driven by the bias current. We explore the range of bias current triggering the flux-flow regime at fixed values of magnetic field, and the stationary temperature distribution in this operation mode. We evidence a steady multi-peaked temperature profile which reflects on the average soliton distribution along the junction. Finally, we analyse also how the friction affecting the soliton dynamics influences the thermal evolution of the system.

Improving the Methods for Estimating Radiation Swelling and Progressive Dimensional Changes of the Elements of VVER-1000 Internals

The problem in the computer prediction of radiation swelling in the elements of reactor internals when justifying VVER-1000 safe operation is quite relevant. In this regard, the approaches of the current Standard Program (PM-T.0.03.333-15) for assessing the technical condition and lifetime extension of VVER-1000 internals were considered. The paper presents various models of radiation swelling for 08Х18Н10Т steel of which elements of internals are made. A comparative analysis of the prediction results for various models has been performed. It is shown that the approaches of the current Standard Program, which take into account factors such as the stress-strain state and radiation creep of the material, are the most relevant. The influence of various fuel campaigns and their sequence on the nature of distribution and the maximum value of radiation swelling, as well as on the stress state in the baffle after long-term operation has been investigated. It is determined that the calculated stationary temperature distribution in the baffle may differ significantly depending on the input data on heat release and accumulated fluence, therefore, it must be calculated for each fuel campaign, and the calculation of the input data averaged for all fuel campaigns may lead to significant errors in the results. From the point of view of the reliability, it is also important to take into account the sequence of fuel campaigns, since, depending on the radiation swelling kinetics, the resulting baffle form can significantly change. The consequences of a possible decrease in cooling efficiency on the external surface of the baffle in the case of reducing the gap between the inner cavity and the baffle are considered, which may cause a local temperature increase in the baffle, but does not lead to a noticeable change in the radiation swelling. However, the radial displacement of the baffle due to thermal expansion increases significantly, which can cause an increase in stresses in the contact area, both in the wall of the cavity and in the baffle. According to the results of the computer analysis, recommendations were formulated to improve the approaches to estimating radiation swelling and progressive dimensional changes in the elements of VVER-1000 internals.

Thermal radiation and conduction in functionally graded thermal barrier coatings. Part II: Experimental thermal conductivities and heat transfer modeling

To further study the heat transfer process inside the functionally graded thermal barrier coatings (FGTBCs) beyond the companion part of the present paper concerning their radiative properties, this paper takes both conductive and radiative heat transfer into account and establishes a comprehensive heat transfer model considering both conductive and radiative heat transfer to estimate the thermal insulation performance of FGTBCs. Thermal conductivities of more than thirty yttria-stabilized zirconia (YSZ)/NiCoCrAlY FGTBC samples were determined based on their thermal diffusivities determined by the laser flash analysis. To predict the overall thermal insulation performance of 8YSZ/NiCoCrAlY FGTBCs, a comprehensive heat transfer model based on the two-flux method and multi-band model was built to solve the stationary temperature distribution and heat flux inside thermal barrier coatings (TBCs) and FGTBCs. Results indicate that FGTBCs show worse thermal insulation performance compared to conventional YSZ TBCs; we also investigated the influence of the number of layers. The temperature of engine blade is increased by 30–50K if coated with an FGTBC instead of a conventional TBC. Conduction heat flux is significantly greater than radiative flux for both TBCs and FGTBCs. This paper gives a detailed analysis by simultaneously considering the heat conduction and thermal radiation in FGTBCs, which is helpful for achieving better thermal design of YSZ/NiCoCrAlY FGTBCs. Along with the Part I, this two-part article presents a deep and thorough study on the thermal insulation performance of YSZ/NiCoCrAlY FGTBCs.

Stationary Temperature Distribution in a Rotating Ring-Shaped Target

For a rotating ring-shaped target irradiated by a heavy-ion beam, a differential equation for computing the stationary distribution of the temperature averaged over the cross section is derived. The ion-beam diameter is assumed to be equal to the ring width. Solving this equation allows one to obtain the stationary temperature distribution along the ring-shaped target depending on the ion-beam, target, and cooling-gas parameters. Predictions are obtained for the rotating target to be installed at the DC-280 cyclotron. For an existing rotating target irradiated by an ion beam, our predictions are compared with the measured temperature distribution.

Some mathematical problems related to the second order optimal shape of a crystallisation interface

We consider the problem to optimise the stationary temperature distribution and the equilibrium shape of the solid-liquid interface in a two-phase system subject to a temperature gradient. The interface satisfies the minimisation principle of the free energy while the temperature is solving the heat equation with radiation boundary conditions at the outer wall. Under the condition that the temperature gradient is uniformly negative in the direction of crystallisation, we can expect that the interface has a global representation as a graph. We reformulate this condition as a pointwise constraint on the gradient of the state, and we derive the first order optimality system for a class of objective functionals that account for the second derivatives of the surface and for the surface temperature gradient.

Gelfand-type problem for two-phase porous media

We consider a generalization of the Gelfand problem arising in Frank-Kamenetskii theory of thermal explosion. This generalization is a natural extension of the Gelfand problem to two-phase materials, where, in contrast to the classical Gelfand problem which uses a single temperature approach, the state of the system is described by two different temperatures. We show that similar to the classical Gelfand problem the thermal explosion occurs exclusively owing to the absence of stationary temperature distribution. We also show that the presence of interphase heat exchange delays a thermal explosion. Moreover, we prove that in the limit of infinite heat exchange between phases the problem of thermal explosion in two-phase porous media reduces to the classical Gelfand problem with renormalized constants.

Large Deflection of a Heated Elliptical Plate Under Stationary Temperature

The large deflection of a heated elliptic plate with clamped edges using Berger's method has been investigated under stationary temperature distribution. The deflection is obtained in terms of Mathieu function Of the first kind of order 2m.

Temperature distributions in a periodically stratified layer with slant lamination

The paper deals with the two-dimensional stationary temperature distribution in a composite layer. The nonhomogenous body is assumed to be composed of periodically repeated two-layered laminae. The layering is inclined with an arbitrary angle to the boundary planes. The lower and upper boundary planes are assumed to be kept at given temperatures. The considered problem is solved within the framework of the homogenized model with microlocal parameters, where the continuity thermal conditions on interfaces are satisfied.

Determination of Stationary Temperature Distribution in Shielded Cables of Finite Length

Thermal energy management is one of the most important aspects for the interior automobile design and often represents a bottleneck concerning the integration of electrical components in modern cars. Thus, the size of current bars and cables has to be dimensioned correctly. On the one hand, these connecting structures must not be too small for thermal reasons. Inadequate materials and component dimensions result in hotspot generation and overheating which could entail fatal damages. On the other hand, oversizing has to be avoided for reasons of space and weight reduction. In this paper, we present a new approach to quickly and accurately compute the heat distribution in electric shielded cables of finite length. The derivation of a nonlinear system of ordinary differential equations allows computing the cable temperatures for the stationary case by a fixed point method. The heat generated inside the cable by the Joule effect is taken into account as well as the thermal energy emitted via the surface by convection and radiation and to adjacent components by conduction. To compare the simulation results to realistic settings, an experimental study was performed. Apart from the good accordance of computations and measurements, we show further advantages concerning calculation times and industrial practicability.

Spontaneous Mirror Symmetry Breaking in the Limited Enantioselective Autocatalysis Model: Abyssal Hydrothermal Vents as Scenario for the Emergence of Chirality in Prebiotic Chemistry

The emergence of chirality in enantioselective autocatalysis for compounds unable to transform according to the Frank-like reaction network is discussed with respect to the controversial limited enantioselectivity (LES) model composed of coupled enantioselective and non-enantioselective autocatalyses. The LES model cannot lead to spontaneous mirror symmetry breaking (SMSB) either in closed systems with a homogeneous temperature distribution or in closed systems with a stationary non-uniform temperature distribution. However, simulations of chemical kinetics in a two-compartment model demonstrate that SMSB may occur if both autocatalytic reactions are spatially separated at different temperatures in different compartments but coupled under the action of a continuous internal flow. In such conditions, the system can evolve, for certain reaction and system parameters, toward a chiral stationary state; that is, the system is able to reach a bifurcation point leading to SMSB. Numerical simulations in which reasonable chemical parameters have been used suggest that an adequate scenario for such a SMSB would be that of abyssal hydrothermal vents, by virtue of the typical temperature gradients found there and the role of inorganic solids mediating chemical reactions in an enzyme-like role.