One-dimensional Temperature Field Research Articles

Vibration Study of Functionally Graded Microcantilever Beams in Fluids Based on Modified Couple Stress Theory by Considering the Physical Neutral Plane

Theoretical studies on the vibration of microcantilever beams in fluids, which are commonly used in micro- and nanoelectromechanical systems (MEMS/NEMS). When microcantilever beams are subjected to photo-thermal excitation, they show that the material properties such as the dynamic response and the one-dimensional temperature field will show significant differences from the macroscopic properties when the size appearance of the microbeam decreases to the scale below a dozen micrometers. In this paper, by correcting the scale constants of the beams, the photothermal vibration model of the microbeam is established using the physical neutral plane theory. The one-dimensional heat transfer equations and scale-corrected temperature field of the microcantilever beam under laser excitation are derived, which are solved by Galerkin’s method, based on the theory of thermoelasticity, the hydrodynamic model of the beams vibrating in incompressible liquids proposed by Sader et al. and the theory of Euler–Bernoulli beams. The equations governing the vibration of micro-cantilever beams corrected for scale effects in different fluids when subjected to photothermal excitation are obtained. The results show that the temperature field, resonant frequency, and quality factor of the microbeam will have a significant upward drift when the size of the microbeam is close to the scale parameter, and the scale effect has a non-negligible influence on the macroscopic performance parameters; the upward drift is gradually weakening when the thickness to scale ratio gradually increases. Finally, the property of the beam is almost the same as that of the theory when the thickness of the beam is 10 times the scale constant. The correction of the theory by the scale effect is insignificant.

Vibration Analysis of Porous Cu-Si Microcantilever Beams in Fluids Based on Modified Couple Stress Theory.

The vibrations in functionally graded porous Cu-Si microcantilever beams are investigated based on physical neutral plane theory, modified coupled stress theory, and scale distribution theory (MCST&SDT). Porous microcantilever beams define four pore distributions. Considering the physical neutral plane theory, the material properties of the beams are computed through four different power-law distributions. The material properties of microcantilever beams are corrected by scale effects based on modified coupled stress theory. Considering the fluid driving force, the amplitude-frequency response spectra and resonant frequencies of the porous microcantilever beam in three different fluids are obtained based on the Euler-Bernoulli beam theory. The quality factors of porous microcantilever beams in three different fluids are derived by estimating the equation. The computational analysis shows that the presence of pores in microcantilever beams leads to a decrease in Young's modulus. Different pore distributions affect the material properties to different degrees. The gain effect of the scale effect is weakened, but the one-dimensional temperature field and amplitude-frequency response spectra show an increasing trend. The quality factor is decreased by porosity, and the degree of influence of porosity increases as the beam thickness increases. The gradient factor n has a greater effect on the resonant frequency. The effect of porosity on the resonant frequency is negatively correlated when the gradient factor is small (n<1) but positively correlated when the gradient factor is large (n>1).

The Influence of the Thermal Effect on the Stress-Strain State of the Soil

Abstract The present paper provides the dependence of the temperature change on time and the depth of the soil massif based on numerical calculations. Mathematical modelling of the one-dimensional soil temperature field via an algorithm and a PC program is carried out without taking into account the influence of the phase transition of moisture in the soil pores during seasonal freezing and thawing using the finite difference method. The amplitude of fluctuations in the temperature regime is obtained as that decreases with depth from the soil surface. It is established that over time heat spreads from the pipeline to the surface of the soil, and over time more intense heating occurs both near the pipeline and in the body of the soil massif.

Dynamic Heat Dissipation Model of Distributed Parameters for Oil-Directed and Air-Forced Traction Transformers and Its Experimental Validation

A traction transformer with narrow oil channels is usually cooled with the ODAF or "Oil Directed Air Forced" method, where its temperature greatly depends on the Joule heat of windings, the conjugate heat transfer in the transformer, and the secondary heat release via oil cooler, together with the oil flowrate generated by oil pump. Neither the thermal-electric analogy nor the CFD simulation approach is qualified to predict the temporal and spatial temperature variations in this type of transformer. In the current work, the distributed parameter models are built for traction transformers and oil coolers with the assumption of a one-dimensional temperature field in the oil flow direction, respectively. Then, the two models are combined with the lumped parameter ones of oil pumps and pipes via the flow rate, temperature and pressure continuities at their interfaces, resulting in the derivation of the dynamic heat dissipation model of oil-directed and air-forced traction transformers. Additionally, an efficient algorithm is proposed for its numerical solution, and the temperature rise experiment is performed for model validation. Finally, the fundamental of dynamic heat dissipation in traction transformers is investigated with the current numerical model and the effects of ambient temperature are studied.

Analysis of 3D crack patterns in a free plate caused by thermal shock using FEM-bifurcation

Damage to components made of brittle material due to thermal shock represents a high safety risk. Predicting the degree of damage is therefore very important to avoid catastrophic failure. An energy-based linear elastic fracture mechanics bifurcation analysis using a three-dimensional finite element model is presented here, which allows the determination of crack length and crack spacing for a defined thermal load in a free plate. It is assumed that a hierarchical crack pattern is formed due to cooling penetration. The constant growth of the ideal regular pattern of hexagons can change into a pattern with a different symmetry by slightly changing the cooling conditions. This bifurcation point is determined by the second derivative of the mechanical potential with respect to the geometry of the crack front. The very high computational effort for the second derivative is reduced by describing the three-dimensional crack front with a limited number of Fourier coefficients. A one-dimensional transient temperature field at a sufficient distance from the plate edge is assumed. For alumina, the crack length and crack spacing curves are computed for different quenching temperatures and heat transfer coefficients. The corresponding final crack lengths are also calculated as a measure of damage. Comparison with a two-dimensional model confirms the expected 1/2 difference in crack spacing. Data from thermal shock experiments are also presented. However, due to the cracks caused by the strong cooling at the edge, these correspond to the results of the two-dimensional model.

Analytical approach for inverse problems theory applications towards determination of thermophysical characteristics of soil

Current paper presents analytical expressions received for investigation of determination of thermophysical characteristics of soil applying the theory of inverse problems. There was considered experimental design with exact measurements and constructed mathematical model for considered case. The analytical expression for transient one-dimensional temperature field was received by Laplace transform. Additional data, such as the heat flux at inlet domain received by conducting numerical simulation of the heat source via computational model. Presented analytical expression for heat transfer parameter allows to determine the soil thermal property without loss of precision, which is crucial in agricultural field. Paper discusses posed peculiarities considered for the inverse problem methodology along with derivation stages of analytical expression. The analytical expression for proposed model is presented both in the frequency and real time domain by applied direct and inverse Laplace transform. The measured outlet input data is interpolated further by the 8-th order polynomial and presented with approximation residuals.

THERMOPHYSICAL CHARACTERISTICS OF THE HVAF COATING FROM QUASICRYSTALLINE ALLOY OF THE Al-Cu-Fe SYSTEM

The article deals with the results of determining the thermal conductivity coefficient from quasicrystalline coatings of the Al-Cu-Fe system in the temperature range up to 900 °C. The Al-Cu-Fe quasicrystalline alloy acquires a great interest for practical use as a material for protective coatings. The Al-Cu-Fe alloy is used to produce thermal barrier coatings in internal combustion engines, non-stick coatings on chemical synthesis equipment and in the food industry and to prevent the icing in aviation. The Al-Cu-Fe quasicrystals have low density, high hardness, high elasticity modulus, high values of corrosion resistance and wear resistance, low coefficient of friction, lowered adhesion, low thermal conductivity in combination with the coefficient of thermal expansion, which is close by its value to some metals. The water atomized Al63Cu25Fe12 powder with a dispersion of +40/-63 μm that has a content of the quasicrystalline phase of about 60 wt. % was used for spraying. The coating was sprayed to the butt of a cylindrical substrate from steel 45 (diameter – 25 mm, height – 10 mm), which before spraying was subjected to jet-abrasive treatment by corundum powder with a determining particle size of 1 mm at air pressure of spraying of 0.5 MPa. The Al-Cu-Fe coating with a thickness of more than 0.8 mm was made by high-speed air-fuel (HVAF) spraying using a burner GVO-RV12 with the following spraying mode: the pressure in the combustion chamber of the burner is 1.0 MPa; the oxidant excess coefficient a ≈ 1.2, the spraying distance is 270 mm. The samples were installed on the side surface of the drum (diameter 120 mm), which rotates at a speed of 2.0 rev/s (the speed of movement of the sputtering spot is 0.8 m/s). Spraying was done in three steps of 10 seconds each with a 30 second cooling time between them. Determination of the temperature dependence of the thermal conductivity of the coating was carried out by solving the inverse problem of thermal conductivity by one-dimensional temperature fields in samples obtained by single-sided jet heating with an industrial hot air torch (at surface temperatures up to 450 °C) and an oxygen-propane welding torch (at temperatures above 450 °C). It is shown that the values of the thermal conductivity coefficient of Al-Cu-Fe quasicrystalline coatings in the range of 20 °С…900 °С vary within λ = 1.9 – 2.31 W/(m×K).

THERMOPHYSICAL CHARACTERISTICS OF THE HVAF COATING FROM QUASICRYSTALLINE ALLOY OF THE Al-Cu-Fe SYSTEM

The article deals with the results of determining the thermal conductivity coefficient from quasicrystalline coatings of the Al-Cu-Fe system in the temperature range up to 900 °C. The Al-Cu-Fe quasicrystalline alloy acquires a great interest for practical use as a material for protective coatings. The Al-Cu-Fe alloy is used to produce thermal barrier coatings in internal combustion engines, non-stick coatings on chemical synthesis equipment and in the food industry and to prevent the icing in aviation. The Al-Cu-Fe quasicrystals have low density, high hardness, high elasticity modulus, high values of corrosion resistance and wear resistance, low coefficient of friction, lowered adhesion, low thermal conductivity in combination with the coefficient of thermal expansion, which is close by its value to some metals. The water atomized Al63Cu25Fe12 powder with a dispersion of +40/-63 μm that has a content of the quasicrystalline phase of about 60 wt. % was used for spraying. The coating was sprayed to the butt of a cylindrical substrate from steel 45 (diameter – 25 mm, height – 10 mm), which before spraying was subjected to jet-abrasive treatment by corundum powder with a determining particle size of 1 mm at air pressure of spraying of 0.5 MPa. The Al-Cu-Fe coating with a thickness of more than 0.8 mm was made by high-speed air-fuel (HVAF) spraying using a burner GVO-RV12 with the following spraying mode: the pressure in the combustion chamber of the burner is 1.0 MPa; the oxidant excess coefficient a ≈ 1.2, the spraying distance is 270 mm. The samples were installed on the side surface of the drum (diameter 120 mm), which rotates at a speed of 2.0 rev/s (the speed of movement of the sputtering spot is 0.8 m/s). Spraying was done in three steps of 10 seconds each with a 30 second cooling time between them. Determination of the temperature dependence of the thermal conductivity of the coating was carried out by solving the inverse problem of thermal conductivity by one-dimensional temperature fields in samples obtained by single-sided jet heating with an industrial hot air torch (at surface temperatures up to 450 °C) and an oxygen-propane welding torch (at temperatures above 450 °C). It is shown that the values of the thermal conductivity coefficient of Al-Cu-Fe quasicrystalline coatings in the range of 20 °С…900 °С vary within λ = 1.9 – 2.31 W/(m×K).

A new predicting model study on U-shaped stamping springback behavior subjected to steady-state temperature field

Springback is the main defect in the U-shaped stamping process, which affects the forming accuracy and shape of parts, and one of the main reasons is that the temperature of materials affects the stress distribution during deformation. This paper establishes a U-shaped predicting model considering one-dimensional temperature field, deformation of bending and anti-bending as well as anisotropy of material to study the influence of temperature field on stress distribution. This theoretical model studies thermodynamic behavior of aluminum material during sheet formability and springback process, such as the variation of sheet thickness, the relative offset of neutral layer and so on. The theoretical calculating model and finite element analysis model of U-shaped stamping parts are established by using Mathematica software and ABAQUS software, and the calculating results are compared by the case 1 so as to verify the accuracy of present work.

Determination of Heat Exchange Law Using Mean Isotherm

Determination of heat exchange between a solid and the environment is a significant inverse thermal physical problem. This heat exchange law irrespective of its nature can be defined easier if mean temperature of such solid is known. When mean temperature of the solid and speed of its change are known, it becomes possible to determine heat flow on the boundary of the solid. In its turn, when heat flow on the boundary, temperature on the boundary and ambient temperature are defined, heat exchange coefficient can be established. Therefore, the main attention is paid to a manner how mean temperature of a solid can be determined in the article. Examining inverse heat transfer problems, input data consist of measurements taken in temperature field as simple as possible mathematically in laboratory conditions. Hence, a symmetrical one-dimensional temperature field is discussed in the article

Grad's Second Problem and Its Solution Within the Framework of Burnett Hydrodynamics

Abstract In his seminal work, Grad not only derived 13 moment equations but also suggested two problems to check his derived equations. These problems are highly instructive as they bring out the character of the equations by examining their solutions to these problems. In this work, we propose Grad's second problem as the potential benchmark problem for checking the accuracy of different sets of higher-order transport equations. The problem definition can be stated as: examination of steady-state solution for a gas at rest in infinite domain upon application of a one-dimensional heat flux. With gas at rest (no bulk velocity), the interest lies in obtaining the solution for pressure and temperature. The problem is particularly interesting with respect to the solution for pressure when Maxwell and hard-sphere molecules are considered. For Maxwell molecules, it is well known that the exact normal solution of Boltzmann equation gives uniform pressure with no stresses in the flow domain. In the case of hard-sphere molecules, direct simulation Monte Carlo (DSMC) results predict nonuniform pressure field giving rise to stresses in the flow domain. The simplistic nature of the problem and interesting results for pressure for different interaction potentials makes it an ideal test problem for examining the accuracy of higher-order transport equations. The proposed problem is solved within the framework of Burnett hydrodynamics for hard-sphere and Maxwell molecules. For hard-sphere molecules, it is observed that the Burnett order stresses do not become zero; they rather give rise to a pressure gradient in a direction opposite to that of temperature gradient, consistent with the DSMC results. For Maxwell molecules, the numerical solution of Burnett equations predicts uniform pressure field and one-dimensional temperature field, consistent with the exact normal solution of the Boltzmann equation.

Determination of thermal diffusivity for rigid body at non-stationary thermal conditions

In this paper we present a new method for calculation of the thermal diffusivity of a rigid body. This method is based on contactless thermal treatment of the examined construction material by an infrared radiation source. Further we analytically find one-dimensional non-stationary temperature field of a rigid body during its heating using data from thermal elements. The calculation of the thermal diffusivity of a rigid body is made using differential heat equation.

Method of Heat Balances for Calculating Heat Transfer in Flat Multilayer Nanostructures

We suggest method of calculation of one-dimensional temperature field in multilayer nanostructures. Our method allows obtaining non-stationary temperature distribution in the periodical and non-periodical spatial structures with a different degree of periodicity. Comparison temperature distributions in the multilayer nanostructures and equivalent continuous samples are performed in this study. In addition we suggest experimental application of our method for estimation an average value of thermal Kapitza resistance.

Non-Fourier thermoelastic behavior of a hollow cylinder with an embedded or edge circumferential crack

In this paper, the non-Fourier heat conduction theory is used to investigate the facture behavior of a hollow cylinder containing an embedded or edge circumferential crack under convective heat transfer boundary conditions. By neglecting the thermo-elastic coupling and inertial effects, the one-dimensional temperature field and the axial thermal stress for an un-cracked hollow cylinder are obtained in the Laplace domain. Then a mode I crack problem is formulated in the cylindrical system by using the superposition method. Integral transform technique is employed to reduce this mixed boundary value problem to a singular integral equation, which is solved numerically with the Gauss–Jacobi quadrature formulas. Finally, the effects of phase lag of heat flux, Biot’s number, and crack geometry on the transient temperature field, axial stress, and stress intensity factors are analyzed. It is found that the C–V heat conduction model gives more conservative results than the Fourier model for the structure safe design against fracture under thermal loading.

An On-line Finite Element Temperature Field Model for Plate Ultra Fast Cooling Process

Taking the element specific-heat interpolation function into account, a one-dimensional (1-D) finite element temperature field model for the on-line control of the ultra fast cooling process was developed based on the heat transfer theory. This 1-D model was successfully implemented in one 4300 mm plate production line. To improve the calculation accuracy of this model, the temperature-dependent material properties inside an element were considered during the modeling process. Furthermore, in order to satisfy the real-time requirements of the on-line model, the variable bandwidth storage method and the Cholesky decomposition method were used in the programming to storage the data and carry out the numerical solution. The on-line application of the proposed model indicated that the deviation between the calculated cooling stop temperature and the measured one was less than ±15 °C.

反射率的数值反演

By analyzing the one-dimensional heat transfer equation under the heat flux boundary condition, a numerical method for inverse computation from the back surface temperature data to the front surface reflectivity data is proposed. The inversely computing program is verified by the positive and inverse computation of one-dimensional heat transfer. The inverse computation results are compared with the reflectivity measured in the experiments, showing that the variation trend of two reflectivity curves are in good agreement. Thus the inverse computation is feasible in one-dimensional temperature field.

Application of Differential Transform Method to Thermoelastic Problem for Annular Disks of Variable Thickness with Temperature-Dependent Parameters

This article analyzes the one-dimensional steady temperature field and related thermal stresses in an annular disk of variable thickness that has a temperature-dependent heat transfer coefficient and is capable of temperature-dependent internal heat generation. The temperature dependencies of the thermal conductivity, Young’s modulus, and the coefficient of linear thermal expansion of the disk are considered, whereas Poisson’s ratio is assumed to be constant. The differential transform method (DTM) is employed to analyze not only the nonlinear heat conduction but also the resulting thermal stresses. Analytical solutions are developed for the temperature and thermal stresses in the form of simple power series. Numerical calculations are performed for an annular cooling/heating fin of variable thickness. Numerical results show that the sufficiently converged analytical solutions are in good agreement with the solutions obtained by the Adomian decomposition method and give the effects of the temperature-dependent parameters on the temperature and thermal stress profiles in the disk. The DTM is useful as a new analytical method for solving thermoelastic problems for a body with temperature-dependent parameters including material properties.

Optical Gaussian beam interaction with one-dimensional thermal wave in the Raman-Nath configuration

Optical Gaussian beam interaction with a one-dimensional temperature field in the form of a thermal wave in the Raman-Nath configuration is analyzed. For the description of the Gaussian beam propagation through the nonstationary temperature field the complex geometric optics method was used. The influence of the refractive coefficient modulation by thermal wave on the complex ray phase, path, and amplitude was taken into account. It was assumed that for detection of the modulated Gaussian beam parameters two types of detector can be used: quadrant photodiodes or centroidal photodiodes. The influence of such parameters as the size and position of the Gaussian beam waist, the laser-screen (detector) distance, the thermal wave beam position and width, as well as thermal wave frequency and the distance between the probing optical beam axis and source of thermal waves on the so-called normal signal was taken into account.

A New Panel Test Facility for Effective Thermal Conductivity Measurements up to 1,650°C

A new steady-state panel test facility is presented which has been designed and constructed for effective thermal-conductivity measurements of insulations in the temperature range between 300 and 1,650°C following ASTM C201-93 and DIN V ENV-1094. Square-shaped samples (length of 400 mm) are used, heated from above and settled on a water-cooled calorimeter system to obtain a one-dimensional steady-state temperature field. The heat is supplied by electrical heating elements freely hanging inside a furnace which is completely constructed from ceramic components to withstand temperatures up to about 1,800°C. The calorimeter system consists of a square central measuring zone (length of 100 mm) surrounded by guard loops to avoid heat losses in all directions. The samples, e.g., a number of fiber mats, one on top of the other up to a maximum height of 110 mm, are open to ambient pressure and atmosphere (air). Measurements include the heat flow rate (taken in the central calorimeter), temperature differences across individual layers of the sample (measured by a series of thermocouples which regularly have to be calibrated), and the thickness of the respective layers (before and after the experiment). The thermal conductivities range from 0.025 to 2 W · m−1 · K−1, and both isotropic and non-isotropic materials can be investigated due to the one-dimensional characteristic of the temperature field. Measurements for alumina fiber mats are presented, and good agreement is found with respective results from other methods and test facilities.

Transient temperature fields in functionally graded materials with different shapes under convective boundary conditions

This paper presents analyses of the transient temperature fields in an infinite plate, an infinite solid cylinder and a solid sphere made of functionally graded materials (FGMs) under convective boundary conditions. The composition and the thermo-physical properties of the infinite FGM plate, the infinite FGM solid cylinder and the FGM solid sphere are of planar symmetric, axially symmetric and spherically symmetric distributions, respectively. The analytical formulae of the one-dimensional transient temperature fields for the three FGM solids are obtained respectively by using the separation-of-variables method and the variable substitution method. Numerical results reveal that the transient temperature fields of the FGM components exhibit similar shape effect to that of homogeneous components. The present work provides valuable basis for the investigation of the thermal shock resistance of FGMs with various shapes.