Three-dimensional Heat Conduction Research Articles

Inversion of Infrared Nondestructive Testing with Evolution Algorithm

The technology of infrared nondestructive testing is an important method to detect where the defect is, and its order of severity. In pulse heating stage, obviously there is also temperature difference between defect area and no defect area. So this paper uses temperature difference between defect area and no defect area in the pulse heating stage, according to 1-D transient heat conduction model of finite thickness defect slab under Neumann boundary condition and heat conduction equation, builds a least-square formula to inverse the depth of defect area in the defect slab with evolution algorithm. The results show that this evolutionary algorithm is reliable, and it has only small errors, and is especially suitable for detecting large defect area. To do the more accurate detection, it still needs to find a new data processing method for infrared nondestructive testing from starting with three-dimensional heat conduction inverse problem.

Particle Swarm Optimization-based algorithms for solving inverse heat conduction problems of estimating surface heat flux

An inverse analysis of estimating a time-dependent surface heat flux for a three-dimensional heat conduction problem is presented. A global optimization method known as Particle Swarm Optimization (PSO) is employed to estimate the unknown heat flux at the inner surface of a crystal tube from the knowledge of temperature measurements obtained at the external surface. Three modifications of the PSO-based algorithm, PSO with constriction factor, PSO with time-varying acceleration of the cognitive and social coefficients, and PSO with mutation are carried out to implement the optimization process of the inverse analysis. The results show that the PSO with mutation algorithm is significantly better than other PSO-based algorithms because it can overcome the drawback of trapping in the local optimum points and obtain better inverse solutions. The effects of measurement errors, number of dimensionalities, and number of generations on the inverse solutions are also investigated.

A meshless analysis of three-dimensional transient heat conduction problems

In this paper, we consider a numerical modeling of a three-dimensional transient heat conduction problem. The modeling is carried out using a meshless reproducing kernel particle (RKPM) method. In the mathematical formulation, a variational method is employed to derive the discrete equations. The essential boundary conditions of the formulated problems are enforced by the penalty method. Compared with numerical methods based on meshes, the RKPM needs only scattered nodes, rather than having to mesh the domain of the problem. An error analysis of the RKPM for three-dimensional transient heat conduction problem is also presented in this paper. In order to demonstrate the applicability of the proposed solution procedures, numerical experiments are carried out for a few selected three-dimensional transient heat conduction problems.

Prediction of HAZ Width of Submerged Arc Welded Plates

This paper is focused on deriving an analytical solution to predict the transient temperature distribution on the plate during the process of Submerged Arc Welding (SAW). An analytical solution is derived from the transient three-dimensional heat conduction equation. The energy input that is applied on the plate is taken as the volume of heat lost from the electric arc. The electric arc is assumed to be a moving double-ellipsoidal heat source with a close proximity to a Gaussian distribution. It is observed that the predicted values are in good agreement with the experimental results. HAZ width calculation is also done with the help of the analytical solution of the transient-three dimensional heat conduction equation. Very good agreement between predicted and experimental values has been archived.

Effect of accumulated shear on the microstructure and morphology of severely deformed Cu 60Zr 30Ti 10 metallic glass

Cu 60Zr 30Ti 10 metallic glass sample was subjected to different amount of continuous rotational straining by high pressure torsion. Deformation dependent morphology, microstructural and thermal properties were characterized by scanning electron microscopy, synchrotron radiation, X-ray diffraction, and calorimetry, respectively. In order to estimate the temperature rise in the metallic glass during high pressure torsion, quasi three-dimensional heat conduction equation with a source term was considered. Solutions indicate that the final temperature strongly depends on the total amount of plastic deformation while details of the microstructure are mainly determined by the balance between the local shear rate and temperature.

Thermal Stability of Conduction-Cooled YBCO Pancake Coil

When a high-temperature superconducting (HTS) coil is operated using a conduction-cooling technique, the coil should be impregnated with epoxy resin because the coil is not sufficiently cooled without it. On the other hand, it is important that the coil should have no damaged area in the winding because the damaged area would generate heat locally and the coil would eventually quench. However, whenever we fabricated an impregnated HTS coil wound with YBCO tapes and evaluated its V-I characteristics, the n-value of the coil was much lower than we expected, indicating that the critical currents of some areas in the winding drastically decreased. Therefore, we have started to improve the performance of an impregnated HTS coil wound with YBCO tapes. In this paper, we investigated the cause of the degradation and found that the degradation did not occur when we decreased the radial tensile stress in the windings. Then we fabricated four single-pancake coils, stacked them, and tested them in a conduction-cooled condition. The measured V-I curves were in good agreement with the calculated ones, suggesting that we successfully developed a technique of fabricating an impregnated HTS coil wound with YBCO tapes with no degradation. We also measured thermal runaway currents of a conduction-cooled HTS coil composed of two single-pancake coils wound with YBCO tapes and numerically simulated the thermal properties by using a three-dimensional heat conduction equation in order to study the thermal stability of the YBCO coil. The measured thermal runaway currents were in good agreement with the calculated ones.

Weighted Least-Squares Collocation Method (WLSCM) for 2-D and 3-D Heat Conduction Problems in Irregular Domains

In this article, the weighted least-squares collocation method (WLSCM) is adopted to deal with two- and three-dimensional heat conduction problems in irregular domains. A radial basis function (RBF) is selected to construct the approximation function. To improve the accuracy and stability, some auxiliary points are increased within the domain of interest. Only inner nodes are used to construct the approximation function, and the equilibrium equations are satisfied not only at collocation points but also at auxiliary points, so the equations should be solved in a least-square sense. A 2-D case that has an analytical solution is simulated by the proposed method and the outcome verifies that the present method can obtain desired accuracy and efficiency. Then the current method is adopted to compute one 2-D and two 3-D cases of engineering heat conduction problems in irregular complex domains. The results show that the present method can deal effectively with the heat conduction problems of both 2-D and 3-D irregular domains.

Estimation of front surface temperature and heat flux of a locally heated plate from distributed sensor data on the back surface

We present a new method of solving the three-dimensional inverse heat conduction (3D IHC) problem with the special geometry of a thin sheet. The 3D heat equation is first simplified to a 1D equation through modal expansions. Through a Laplace transform, algebraic relationships are obtained that express the front surface temperature and heat flux in terms of those same thermal quantities on the back surface. We expand the transfer functions as infinite products of simple polynomials using the Hadamard Factorization Theorem. The straightforward inverse Laplace transforms of these simple polynomials lead to relationships for each mode in the time domain. The time domain operations are implemented through iterative procedures to calculate the front surface quantities from the data on the back surface. The iterative procedures require numerical differentiation of noisy sensor data, which is accomplished by the Savitzky–Golay method. To handle the case when part of the back surface is not accessible to sensors, we used the least squares fit to obtain the modal temperature from the sensor data. The results from the proposed method are compared with an analytical solution and with the numerical solution of a 3D heat conduction problem with a constant net heat flux distribution on the front surface.

Variational-asymptotic modeling of the thermoelastic behavior of composite beams

This paper deals with steady state thermoelastic problems in composite beam structure by using variational-asymptotic method. First, the original three-dimensional heat conduction problem is reduced to be a two-dimensional thermal cross-sectional analysis along with an optional one-dimensional heat conduction analysis. The one-dimensional heat conduction analysis exists only if the temperature is not prescribed at any point of the cross-section along the span except the end surfaces. Then we reduce the one-way coupled, three-dimensional thermoelasticity problem into a two-dimensional, one-way coupled thermoelastic cross-sectional analysis and a one-dimensional, one-way coupled, thermoelastic beam analysis. The present theory is implemented into the computer program, variational-asymptotic beam sectional analysis (VABS). Several examples are studied using the present theory and the results from VABS are compared with available analytical solutions and the three-dimensional analysis using the commercial finite element package ANSYS.

Inverse estimation of the inner wall temperature fluctuations in a pipe elbow

Thermal stratification occurs in flows through elbows used for mixing hot and cold fluids. The movement of the interface between the hot and cold fluids over time causes fluctuations of the inner wall temperature, which can lead to the stress variations and structural thermal fatigue. Therefore, accurate inner wall temperature data detailing the fluctuations is essential for analysis and predictions of thermal fatigue of the piping. In the present study, the conjugate gradient method (CGM) is applied to solve the three-dimensional inverse heat conduction problem (IHCP) to estimate the unknown temperature fluctuations on the inner wall of a pipe elbow from simulated outer temperature measurements. First, the direct heat conduction problem is solved using the finite element method (FEM) to produce the outer and inner wall temperatures. Then, the inverse heat conduction problem is solved to estimate the inner wall temperatures based on the outer wall temperature data. The accuracy of the inverse algorithm is then examined by comparing the estimated inner wall temperatures with the exact temperatures from the direct solution. The numerical results show that the inner wall temperature of the elbow can be accurately estimated by using the present algorithm for the test case considered.

Numerical solution of three-dimensional backward heat conduction problems by the time evolution method of fundamental solutions

The time evolution method of fundamental solutions (MFS) is proposed to solve three-dimensional backward heat conduction problems (BHCPs). The time evolution MFS is obtained through the linear superposition of diffusion fundamental solutions. Through a correct treatment of temporal evolution, the MFS can be implemented to solve strongly ill-posed problems. The numerical results demonstrate the accuracy and stability of the MFS for three-dimensional BHCPs with high levels of noise. This represents the first implementation of MFS to solve three-dimensional BHCPs, and demonstrates that time evolution MFS is a stable and powerful numerical scheme which has the potential to significantly improve the solution of three-dimensional backward heat conduction problems.

Transient three-dimensional heat conduction problems with partial heating

An exact solution method is given for 3D heating over a rectangular region on a homogeneous plate. It has application to laser heating and cooling of electronic equipment. Based on Green’s functions and new summation identities, significant improvements are given over the separation of variables method. Machine accuracy is possible, which can be used to both provide verification solutions and intrinsic verification. Basic building blocks for arbitrary space/time heat flux distributions and simplified expressions are provided. The method can also be used for many other boundary conditions using tabulated algebraic identities for summations.

Discrete and continuous adjoint approaches to estimate boundary heat fluxes in falling films

A wavy falling film simulation is considered in which a liquid travels along one side of a thin metal foil that is heated electrically from the opposite side. The direct problem consists of a three-dimensional heat conduction equation on a cuboid domain representing the foil with suitable initial and boundary conditions. The inverse problem consists of determining the heat flux on the film side of the foil from a given distribution of the temperature on the heating side. Two different adjoint approaches for the solution of this inverse problem are compared. In the continuous adjoint approach, the adjoint problem is analytically derived from the direct problem and then discretized. In the discrete adjoint approach, the direct problem is discretized, from which an adjoint code is generated by means of the reverse mode of automatic differentiation. Numerical experiments are reported, demonstrating the advantages and disadvantages of the two approaches.

РЕшЕниЕ трехмерноЙ нестационарноЙ задачИ теплопроводностИ В вариационноЙ постановкЕ

An approach to solving three-dimensional unsteady heat conduction problem in the variational formulation, based on the method of separation of variables and the representation of the coordinate functions in the form of segments of the series in ultraspherical polynomials

Analytical Study on Relaxation Behavior of Thermal Stresses in Nose Cones Composed of Functionally Graded Material Subjected to Three-Dimensional Transient Temperatures

In this study, analytical solutions for the three-dimensional transient temperature fields and the resulting thermal stress fields in a space plane's nose cone composed of a functionally graded material (FGM) are derived by treating the nose cone as a multi-layered hollow hemisphere with distinct thermal and mechanical properties in each layer. The three-dimensional transient heat conduction problem can be solved by applying Fourier-cosine transform, Legendre transform, and Vodicka's method which is a type of integral transform for the one-dimensional multi-layer regions. The associated thermal stress problem can be analyzed by using thermoelastic displacement potential and Papkovich-Neuber's displacement functions. Numerical calculations for the thermal deformations and the thermal stresses are carried out for the FGM hollow hemisphere composed of SiC and Al-Alloy, which has a graded material composition of a power function expression. It is clarified that there is a specific thermal stress behavior in the FGM hollow hemisphere subjected to the three-dimensional transient temperature field, and the effects of the graded composition and number of layers on relaxation characteristics of thermal stress are quantitatively discussed.

Meshless Local Petrov-Galerkin Method for 3D Steady-State Heat Conduction Problems

The Meshless Local Petrov-Galerkin (MLPG) method is applied for solving the three-dimensional steady state heat conduction problems. This method is a truly meshless approach; also neither the nodal connectivity nor the background mesh is required for solving the initial boundary-value problems. The penalty method is adopted to enforce the essential boundary conditions. The moving least squares (MLS) approximation is used for interpolation schemes and the Heviside step function is chosen for representing the test function. The numerical results are compared with the exact solutions of the problem and Finite Difference Method (FDM). This comparison illustrates the accuracy as well as the capability of this method.

Numerical study of transient three-dimensional heat conduction problem with a moving heat source

A numerical study of transient three-dimensional heat conduction problem with a moving source is presented. For numerical solution Douglas-Gunn alternating direction implicit method is applied and for the moving heat source flux distribution Gaussian function is used. An influence on numerical solution of input parameters figuring in flux boundary conditions is examined. This include parameters appearing in Gaussian function and heat transfer coefficient from free convection boundaries. Sensitivity of cooling time from 800 to 500?C with respect to input parameters is also tested.

Channel Temperature Measurement of AlGaN/GaN HEMTs by Forward Schottky Characteristics

Channel temperature measurements of multi-finger AlGaN/GaN HEMTs by forward Schottky characteristics are presented. The temperature dependence of the forward gate-source Schottky junction voltage is investigated and it is used as the temperature sensitive parameter (TSP) by pulsed switching technique. The channel-to-mounting thermal resistance of the tested AlGaN/GaN HEMT sample is 19.6°C/W. Compared with both the measured results by micro-Raman method and simulated results of a three-dimensional heat conduction model, the physical meaning of the channel temperature for AlGaN/GaN HEMT tested by pulsed switching electrical TSP method is investigated quantitatively for the first time.

Three-Dimensional Steady Thermal Stress Analysis of Functionally Graded Hollow Hemisphere

Three-dimensional steady temperature field and the resulting thermal stress field are analyzed in a functionally graded hollow hemisphere (FG hollow hemisphere) subjected to unaxisymmetric heating at the inner and outer hemispherical surfaces. The FG hollow hemisphere whose thermal and mechanical properties vary arbitrarily in the radial direction is approximated as a multi-layered hollow hemisphere with distinct thermal and mechanical constants in each layer. An analytical solution to the three-dimensional heat conduction problem is obtained by introducing Fourier's cosine transform and Legendre transform. The associated three-dimensional thermal stress problem is estimated by the use of thermoelastic displacement potential and Papkovich-Neuber's displacement functions. Numerical calculations of the temperature and thermal stresses are carried out for SiC/Al-Alloy FG hollow hemisphere with three types of graded material composition expressed in the form of a power function. The effects of the graded composition, inner/outer radius ratio, and number of layers on relaxation characteristics of thermal stress are quantitatively discussed.

Modelling of heat processes during the surface treatment of inhomogeneous metal bodies by a high-temperature moving pulse source

A mathematical model capable of describing the thermal action of concentrated energy fluxes on inhomogeneous metal bodies subjected to electromechanical treatment (EMT) is developed. The peculiarities of formulation and solution of a three-dimensional heat conduction problem under high-speed heating and cooling conditions, including the description of the spatial and time configurations of a moving pulse heat source and the account of the latent heat of phase transformations and the temperature dependence of the thermophysical characteristics of a body in specific structural areas, are considered. The obtained numerical results allowed us to study the temperature distribution mechanisms and the rate of temperature change caused by the action of the moving pulse source on the steel sample with a surface layer or covering subjected to EMT. It is shown that a consideration of the temperature dependence of thermophysical coefficients leads to quantitative and qualitative changes in the distribution of temperature and its gradients. Application of a hyperbolic heat conduction equation to the analysis of the EMT processes provides only an inessential refinement of the problem solution.