Non-axisymmetric Temperature Research Articles

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.

A non-axisymmetric temperature field reconstruction method based on the compressive sensing background oriented Schlieren method

Measurement of the combustion temperature field is an extremely important issue in industrial production. Temperature is one of the key parameters in combustion studies. With the temperature field distribution of the combustion field obtained, heat transfer, heat convection, and heat radiation can be calculated directly and efficiently. Traditional background oriented Schlieren (BOS) is an effective method for non-axisymmetric temperature field measurements, but it requires simultaneous Schlieren imaging at multiple angles for tomographic reconstruction, which will greatly limit its application. In this paper, the compressive sensing algorithm is introduced into the temperature field reconstruction, which establishes the system of equations between the deflection angle and the refractive index gradient. Then, the reconstruction of the non-axisymmetric temperature field is realized by solving the underdetermined system of equations by the method of solving the sparse solution through the compressive sensing. First, light offsets across the non-axisymmetric temperature field are measured by the under-angled BOS system and image processing method. Second, the spatial refractive index field is reconstructed by the compressive sensing BOS method proposed in this paper. Finally, the spatial temperature field is obtained. The experimental results show that by comparing the iRadon reconstruction algorithm and the compressive sensing reconstruction algorithm, the temperature field reconstructed by the compressive sensing under the condition of the under-angled sampling of projection data had a higher accuracy than that reconstructed by the tomographic reconstruction algorithm under the same condition. The average error of the temperature field was reduced from 34.6 to 29.7 K under the same measurement conditions; however, the accuracy is better maintained by using the compressive sensing algorithm under the condition of undersampling projection.

Gravitational radiation from thermal mountains on accreting neutron stars: sources of temperature non-axisymmetry

ABSTRACT The spin distribution of accreting neutron stars in low-mass X-ray binary systems shows a concentration of pulsars well below the Keplerian break-up limit. It has been suggested that their spin frequencies may be limited by the emission of gravitational waves, due to the presence of large-scale asymmetries in the internal temperature profile of the star. These temperature asymmetries have been demonstrated to lead to a non-axisymmetric mass distribution, or ‘mountain’, that generates gravitational waves at twice the spin frequency. The presence of a toroidal magnetic field in the interior of accreting neutron stars has been shown to introduce such anisotropies in the star’s thermal conductivity, by restricting the flow of heat orthogonal to the magnetic field and establishing a non-axisymmetric temperature distribution within the star. We revisit this mechanism, extending the computational domain from (only) the crust to the entire star, incorporating more realistic microphysics, and exploring different choices of outer boundary condition. By allowing a magnetic field to permeate the core of the neutron star, we find that the likely level of temperature asymmetry in the inner crust (ρ ∼ 1013 g cm−3) can be up to 3 orders of magnitude greater than the previous estimate, improving prospects for one day detecting continuous gravitational radiation. We also show that temperature asymmetries sufficiently large to be interesting for gravitational wave emission can be generated in strongly accreting neutron stars if crustal magnetic fields can reach ∼1012 G.

A non-axisymmetric temperature field reconstruction method based on the interferometric fringe schlieren method

The typical non-axisymmetric temperature field distribution in the tail nozzle outlet area of the combustion chamber of an aircraft engine can reflect the operating state of the engine. As an effective non-contact method for measuring a non-axisymmetric temperature field, the traditional background-oriented schlieren (BOS) method has the problem that the extraction accuracy of light deflection is not high. In this paper, interference fringes are introduced into the schlieren measurement, and the highly sensitive fringe offset replaces the traditional background speckle offset for measurement, which realizes accurate reconstruction of the non-axisymmetric temperature field. Firstly, through the interferometric fringe schlieren (IFS) method comprising multiple sets of double-slit interferometric systems and imaging sensors, the offset of the interference fringe after passing through the non-axisymmetric temperature field is measured by image processing. The spatial refractive index field is then reconstructed by the filtered back-projection algorithm under the sector model proposed in this paper and the spatial temperature field is obtained. The experimental results show that the temperature field reconstruction error is smaller for the IFS method than the BOS method, the measurement area is accurate from 0.5 mm for the BOS method to 0.2 mm for the IFS method and the temperature error range is reduced from 50 K for the BOS method to 25 K for the IFS method.

Near-infrared measurement of temperature fields formed by mixed convection from a small heating sphere in water

This study proposed a method for reconstructing three-dimensional (3D) non-axisymmetric temperature fields in water formed by mixed convection surrounding a small heating sphere. The proposed method is based on the temperature dependence of the absorption coefficient of water at a wavelength of 1150 nm in the near-infrared (NIR) region. The absorbance images of a 10-mm-thick water cell containing a 1-mm diameter steel sphere heated via electromagnetic induction were acquired using a two-orthogonal-direction NIR imaging system. The 3D reconstruction was implemented by applying a non-axisymmetric inverse Abel transform to the absorbance images. In addition, particle tracking velocimetry (PTV) was applied to the same images to visualize and quantify the flow field. The results indicated that the temperature distribution within the plume was distorted compared to an axisymmetric one, and the plume shape, length, and angle were dependent on the forced flow and heating power. Further, employing a thermal balance model with temperature distributions, the heat production rates were determined and discussed based on the plume characteristics. The PTV results revealed a mixed convection field inconsistent with the plume shape, which was also verified by comparison with numerical simulations.

3D non-axisymmetric temperature field measurement using rotating tomographic mechanism schlieren method

The tomography schlieren method is useful for calculating the non-axisymmetric temperature field, which is relevant to many industrial applications. However, the traditional tomography schlieren method requires bulky imaging devices, which limits its application. Thus, this paper proposes a non-contact three-dimensional temperature field measurement method based on the rotating tomography schlieren method. Projections from different directions of the temperature field are obtained at different angles using a transmission schlieren system, which is controlled by a designed rotating tomographic mechanism. The calibration schlieren method and the projection reconstruction algorithm are used to calculate the 3D temperature field. Verification experiments of single flame and double flame showed that the relative error of the reconstructed temperature was approximately 3.7% compared with a precision thermocouple.

Stress-deformed state of vertical cylindrical metal shell under temperature climate impact

There are continued the previous executed researches of the tense-deformed condition of a vertical metal thin cylindrical shell under temperature climatic influences in this work. So, in [1] the temperature climatic influences on cylindrical metal shells are investigated, in [2] the parameters of the elastic base simulating the wheat repulse are investigated, in [3] the tense-deformed condition of a shell on the elastic base with piecewise-constant is investigated, and in [4] with a piecewise-linear Winkler coefficient for axisymmetric cooling of the shell. The tense-deformed condition of a vertical cylindrical thin-wall shell lies on a piecewise-constant elastic base under non-axisymmetric temperature climatic influences has been studied in the present study on the basis of membrane theory of shells and a simple edge effect. The decision is received in the Fourier series, containing in the general case of heating-cooling the shell, ten members of the Fourier series.

The influence of the length of heat sources on the external border on the temperature distribution in profiled polar-orthotropic ring plates taking into account there heat exchange with the external environment

We study the influence of N extended heat sources at external boundaries on the nonaxisymmetric temperature distribution on profiled polar-orthotropic ring plates and take into account heat exchange with the external environment. The solution of the stationary heat conduction problem for anisotropic annular plates of a random profile is resolved through the solution of the corresponding Volterra integral equation of the second kind. The formula of a temperature calculations in anisotropic annular plates of an random profile is given. The exact solution of stationary heat conductivity problem for a reverse conical polar-orthotropic ring plate is recorded. The temperature distribution in such anisotropic plate from N extended heat sources at its outer border is more complex than in the case of temperature distribution from N point heat sources at their external border.

Soot temperature measurement within 3D flame by light-field imaging based on wave optics theory

Light-field imaging technique provides a new approach to non-intrusive temperature measurement, which can achieve 3D measurement via single detector equipment. We consider wave characteristics of radiation rays in the light-field imaging model in this work. We propose an improved light-field imaging model of an emitting and absorbing medium by combining wave optics theory and radiation transfer process in semitransparent media. The combination of Fresnel diffraction theory and phase conversion principles of the lens provides a theoretical basis for a description of radiation ray propagation in space and across the lens. The point spread function of the light-field camera can be calculated and used to attain a light-field image in accordance with the convolution imaging model. The light-field image of the flame is regarded as input to reconstruct the soot temperature field of a 3D flame. Inverse problem is formulated as a linear optimization problem and solved by Landweber algorithm. Simulated reconstruction results show that axisymmetric and non-axisymmetric temperature distribution can be reconstructed successfully, thereby verifying the applicability of the presented method. The effect of the measurement noise and the Tikhonov regularization on the reconstructed quality of the temperature are investigated as well. The results show that the temperature can be reconstructed with acceptable accuracy under different noise levels. Tikhonov regularization method is proven available in this problem.

Propagation of heat wave in composite cylinder using Cattaneo- Vernotte theory

Using Cattaneo-Vernotte theory, a general analytical solution for the distribution of non-axisymmetric temperature in a long multi-layer composite cylinder is presented. The temperature distribution due to an initial asymmetric excitation in the cylinder is extracted by solving the hyperbolic heat conduction equation using the separation of variables method. The analytical solution is verified with the results of the finite difference method. In addition, the effect of imperfect interface on the temperature distribution is studied. According to results, the temperature in a multi-layer composite cylinder is dominantly affected by heat waves, and propagate through layers due to the discontinuity between initial and boundary conditions, and their reflections from the interfaces due to the difference in thermal properties of layers. In addition, the heatwaves generated on the imperfect interfaces, due to their thermal resistance, disturb the temperature. Also, the effect of fiber angle on the temperature distribution in radial and circumferential directions is studied. Finally, the temperature distribution is examined in terms of time and it is shown that the temperature distribution in terms of time at a specific point generally has an oscillatory behavior.

Near-infrared measurement of non-axisymmetric temperature and flow fields formed by free convection around a small heating sphere in water

Near-infrared measurement of non-axisymmetric temperature and flow fields formed by free convection around a small heating sphere in water

Evaluation of anisotropic tangential conduction in printed-circuit-board heated-thin-foil heat flux sensors

The effect of the tangential conduction contribution in a thermally anisotropic heated-thin-foil heat flux sensor is examined. A parameter to assess the degree of importance of tangential conduction in the sensor is defined and evaluated in order to justify the need of tangential conduction corrections. Printed circuit boards (PCBs) are typical examples of sensors with anisotropic thermal conduction properties, due to the different conductance values in the directions either parallel or orthogonal to the copper tracks which are placed onto a fiberglass substrate. A parametric study on PCBs with different tracks coverage fraction and copper-to-fiberglass heat conductance ratio is carried out. A revised heated-thin-foil formulation, including a correction for anisotropic thermal properties of the PCB, is experimentally tested. The selected thermo-fluid-dynamic test case is the convective heat transfer of a normally impinging round jet for which axisymmetric maps of the Nusselt number are expected. The anisotropic tangential conduction results in non-axisymmetric temperature distributions. Consequently, if anisotropy is not properly accounted for, non-axisymmetric Nusselt number maps are obtained. The anisotropic conduction effects are shown to be weakly sensitive to the copper tracks coverage fraction while strongly dependent on a parameter called degree of anisotropy, which accounts for the copper-to-fiberglass heat-conductance ratio. Anisotropic conduction effects are found to be almost negligible in PCBs with low values of the degree of anisotropy and/or low values of the tangential conduction degree of importance. Accounting for the anisotropic tangential conduction in the heated-thin-foil formulation allows minimizing the differences between the Nusselt number profiles measured in the directions parallel and orthogonal to the copper tracks. For all the considered PCBs, differences after correction are found to be below the measurement uncertainty expected for an equivalent thermally-isotropic sensor. The proposed approach also allows reducing below the measurement uncertainty the spread between the Nusselt number values measured with all the considered printed circuit boards, providing measurements practically independent of the sensor characteristics.

Numerical Simulation of Temperature Fields in a Three-Dimensional SiC Crystal Growth Furnace with Axisymmetric and Spiral Coils

Three-dimensional numerical simulation platform for silicon carbide crystal growth furnace was established using C programing language, where a physical model of the furnace was built based on cylindrical coordinates; governing equations for electromagnetic and temperature fields were discretized by finite volume method; radiation characteristics were studied with the help of S2S model (surface to surface radiation model); and the least distance method was proposed to check radiation faces visibility efficiently. LU decomposition algorithm based on graphic processing unit (GPU) technology was developed to accelerate the solving process of surface to surface radiation. Then the radiation heat transfer in silicon carbide crystal (SiC) growth chamber and temperature field of silicon carbide growth furnace were studied quantificationally at I = 1250 A and F = 16 kHz. The effects of coil structures (axisymmetric and spiral) on temperature field and its gradient distributions were investigated by standard deviation method. The simulation results demonstrate that spiral electromagnetic coil generates non-axisymmetric temperature field easily; the radiation heat flux is 102~103 times more than conduction heat flux, radiation heat transfer is helpful to increase temperature evenness; the spiral temperature field on the SiC crystal cross-section reduces the poor homogeneity of temperature gradient, which will cause crystal to generate large defects.

Efficient and robust method for simultaneous reconstruction of the temperature distribution and radiative properties in absorbing, emitting, and scattering media

A rapid computational method called generalized sourced multi-flux method (GSMFM) was developed to simulate outgoing radiative intensities in arbitrary directions at the boundary surfaces of absorbing, emitting, and scattering media which were served as input for the inverse analysis. A hybrid least-square QR decomposition–stochastic particle swarm optimization (LSQR–SPSO) algorithm based on the forward GSMFM solution was developed to simultaneously reconstruct multi-dimensional temperature distribution and absorption and scattering coefficients of the cylindrical participating media. The retrieval results for axisymmetric temperature distribution and non-axisymmetric temperature distribution indicated that the temperature distribution and scattering and absorption coefficients could be retrieved accurately using the LSQR–SPSO algorithm even with noisy data. Moreover, the influences of extinction coefficient and scattering albedo on the accuracy of the estimation were investigated, and the results suggested that the reconstruction accuracy decreased with the increase of extinction coefficient and the scattering albedo. Finally, a non-contact measurement platform of flame temperature field based on the light field imaging was set up to validate the reconstruction model experimentally.

Control of Interface Shape by Non-Axisymmetric Solution Convection in Top-Seeded Solution Growth of SiC Crystal

We achieved the convex growth interface shape in top-seeded solution growth of SiC applying non-axisymmetric solution convection induced by non-axisymmetric temperature distribution. The detailed solution flow, temperature distribution and carbon concentration distribution were calculated by 3-dimensional numerical analysis. In the present case, the solution flow below the crystal was unidirectional and the supersaturation was increased along the solution flow direction. By the rotation of the crystal in the unidirectional flow and the temperature distribution, we successfully obtained the crystal with the convex growth interface shape.

Algebraic tomographic reconstruction of two-dimensional gas temperature based on tunable diode laser absorption spectroscopy

Based on tunable diode laser absorption spectroscopy, the combustion gas temperature distribution was reconstructed using the algebraic iterative reconstruction technique. The effects of the projected angle and number of rays on the temperature reconstruction results for both fan beam and parallel beam were investigated. The flow field information was shown in detail. The reconstruction quality increased with increased ray number for both parallel- and fan-beam projections. The trend tended to be smooth when the total number of beams exceeded 100. A virtual ray method combined with the reconstruction algorithms was proposed. The virtual ray method is effective to improve the accuracy of reconstruction results for the symmetric and non-axisymmetric temperature distributions. Moreover, the temperature reconstruction of an infrared gas-field furnace is demonstrated, showing the virtual ray method can be applied to the combustion diagnostics.

Convective heat transfer for viscoelastic fluid in a curved pipe

In this paper, fully developed convective heat transfer of viscoelastic flow in a curved pipe under the constant heat flux at the wall is investigated analytically using a perturbation method. Here, the curvature ratio is used as the perturbation parameter and the Oldroyd-B model is applied as the constitutive equation. In the previous studies, the Dirichlet boundary condition for the temperature at the wall has been used to simplify the solution, but here exactly the non-homogenous Neumann boundary condition is considered to solve the problem. Based on this solution, the non-axisymmetric temperature distribution of Dean flow is obtained analytically and the effect of flow parameters on the flow field is investigated in detail. The current analytical results indicate that increasing the Weissenberg number, viscosity ratio, curvature ratio, and Prandtl number lead to the increase of the heat transfer in the Oldroyd-B fluid flow.

Extension of Stoney’s Formula to Arbitrary Temperature Distributions in Thin Film/Substrate Systems

Abstract Current methodologies used for the inference of thin film stress through curvature measurements are strictly restricted to stress and curvature states that are assumed to remain uniform over the entire film/substrate system. By considering a circular thin film/substrate system subject to nonuniform and nonaxisymmetric temperature distributions, we derive relations between the film stresses and temperature, and between the plate system’s curvatures and the temperature. These relations featured a “local” part that involves a direct dependence of the stress or curvature components on the temperature at the same point, and a “nonlocal” part that reflects the effect of temperature of other points on the location of scrutiny. Most notably, we also derive relations between the polar components of the film stress and those of system curvatures which allow for the experimental inference of such stresses from full-field curvature measurements in the presence of arbitrary nonuniformities. These relations also feature a “nonlocal” dependence on curvatures making full-field measurements of curvature a necessity for the correct inference of stress. Finally, it is shown that the interfacial shear tractions between the film and the substrate are related to the gradients of the first curvature invariant and can also be inferred experimentally.

Nonaxisymmetric thermal stress state of laminated rotational bodies of orthotropic materials under nonisothermic loading

A procedure for numerical investigation of nonaxisymmetric temperature fields and the elastic stress-strain state of laminated rotational bodies of cylindrically and rectilinearly orthotropic materials under nonisothermal loading is proposed. The deformation of orthotropic materials is described by the equations of anisotropic elasticity theory. The equations of state are written in the form of Hooke’s law for homogeneous materials, with additional terms which take into account the thermal deformation, changes in the mechanical properties of materials in the circumferential direction, and their dependence on temperature. A semianalytic finite-element method in combination with the method of successive approximations is used. An algorithm for numerical solution of the corresponding nonlinear boundary problem is elaborated, which is realized as a package of applied FORTRAN programs. Some numerical results are presented.

Deformations of accreting neutron star crusts and gravitational wave emission

Motivated by the remarkably narrow range of measured spin frequencies of ∼20 accreting (and weakly magnetic) neutron stars in the Galaxy, Bildsten conjectured that their spin-up had been halted by the emission of gravitational waves. If so, then the brightest persistent X-ray source on the sky, Scorpius X-1, should be detected by gravitational wave interferometers within 10 years. Bildsten pointed out that small non-axisymmetric temperature variations in the accreted crust will lead to ‘wavy’ electron capture layers, and the resulting horizontal density variations near e− capture layers create a mass quadrupole moment. Neglecting the elastic response of the crust, Bildsten estimated that even e− capture layers in the thin outer crust can develop the quadrupole necessary to balance accretion torque with gravitational waves, for accretion rates We present a full calculation of the crust's elastic adjustment to the density perturbations induced by the temperature-sensitive e− capture reactions. We find that, due to the tendency of the denser material to sink rather than spread sideways, neglecting the elastic response of the crust overestimates, by a factor of 20–50, the Q22 that results from a wavy capture layer in the thin outer crust. However, we find that this basic picture, when applied to capture layers in the deep inner crust, can still generate Q22 in the necessary range, as long as there are ≲5 per cent lateral temperature variations at densities in excess of 1012 g cm−3, and as long as the crustal breaking strain is high enough. By calculating the thermal flow throughout the core and the crust, we find that temperature gradients this large are easily maintained by asymmetric heat sources or lateral composition gradients in the crust. If the composition or heating asymmetries are independent of the accretion rate, then for the induced quadrupole moments have approximately the same scaling, ∝M1/2, as that necessary to balance the accretion torque at the same spin frequency for all M. Temperature gradients in the deep crust lead to a modulation in the thermal emission from the surface of the star that is correlated with Q22. In addition, a ∼0.5 per cent lateral variation in the nuclear charge-to-mass ratio in the crust will also result in a Q22 sufficient to halt spin-up from accretion even in the absence of a lateral temperature gradient. We also derive a general relation between the stresses and strains in the crust and the maximum quadrupole moment they can generate. We show, under quite general conditions, that maintaining a Q22 of the magnitude necessary to balance the accretion torque requires a dimensionless strain at near-Eddington accretion rates, of order the breaking strain of conventional materials. This leads us to speculate that accreting neutron stars reach the same equilibrium spin because they all are driven to the maximum Q22 that the crust can sustain.