Three-dimensional Heat Conduction Research Articles

Effect of Heat input on 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 and the kinetic energy of filler droplets specifically driven by gravity, electromagnetic force, arc drag force, carring mass, momentum and thermal energy. These driving forces periodically impinge onto the base metal, leading to a liquid weld puddle. The electric arc is assumed to be a moving double central conicoidal 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. Analyses of micro-structural changes are critically investigated to comprehend the HAZ softening and phenomena.

Efficient reconstruction of local heat fluxes in pool boiling experiments by goal-oriented adaptive mesh refinement

In this paper, we consider the efficient estimation of local boiling heat fluxes from transient temperature measurements in the heater close to the heater surface. For accurate prediction, heat flux estimation is formulated as a transient three-dimensional (3D) inverse heat conduction problem (IHCP). This inverse problem is ill-posed and cannot be treated straightforwardly by established numerical methods. In order to obtain a regularized stable solution, a large-scale time-dependent PDE-constrained optimization problem has to be solved and an appropriate stopping criterion for the termination of the iterative solution process has to be chosen. Since the boiling heat flux is non-uniformly distributed on the heater surface due to the strong local activity of the boiling process, the use of a fixed uniform spatial discretization is not efficient. Instead, an adaptive mesh refinement strategy can be used to obtain an appropriate discretization which significantly reduces the total computational effort. In this work, we present an automatic algorithm incorporating an adaptive mesh refinement via a heat flux-based a-posteriori error estimation technique. The suggested algorithm can cope with both spatially point-wise or highly resolved temperature observations efficiently. It is applied to real measurement data obtained from two different types of pool boiling experiments. The numerical results show that the computational effort can be reduced significantly for given estimation quality. This adaptive IHCP solution technique can be also viewed as an efficient soft sensor to deduce unmeasurable local boiling heat fluxes.

Meshless analysis of three-dimensional steady-state heat conduction problems

Steady-state heat conduction problems arisen in connection with various physical and engineering problems where the functions satisfy a given partial differential equation and particular boundary conditions, have attracted much attention and research recently. These problems are independent of time and involve only space coordinates, as in Poisson's equation or the Laplace equation with Dirichlet, Neuman, or mixed conditions. When the problems are too complex, it is difficult to find an analytical solution, the only choice left is an approximate numerical solution. This paper deals with the numerical solution of three-dimensional steady-state heat conduction problems using the meshless reproducing kernel particle method (RKPM). A variational method is used to obtain the discrete equations. The essential boundary conditions are enforced by the penalty method. The effectiveness of RKPM for three-dimensional steady-state heat conduction problems is investigated by two numerical examples.

High-fidelity real-time simulation on deployed platforms

We present a certified reduced basis method for high-fidelity real-time solution of parametrized partial differential equations on deployed platforms. Applications include in situ parameter estimation, adaptive design and control, interactive synthesis and visualization, and individuated product specification. We emphasize a new hierarchical architecture particularly well suited to the reduced basis computational paradigm: the expensive Offline stage is conducted pre-deployment on a parallel supercomputer (in our examples, the TeraGrid machine Ranger); the inexpensive Online stage is conducted “in the field” on ubiquitous thin/inexpensive platforms such as laptops, tablets, smartphones (in our examples, the Nexus One Android-based phone), or embedded chips. We illustrate our approach with three examples: a two-dimensional Helmholtz acoustics “horn” problem; a three-dimensional transient heat conduction “Swiss Cheese” problem; and a three-dimensional unsteady incompressible Navier–Stokes low-Reynolds-number “eddy-promoter” problem.

High pressure torsion of Cu-based metallic glasses

Cu-Zr-Ti metallic glass was subjected to high pressure torsion applying different revolution times (180s, 120s, 60s). Both deformation and deformation rate dependent microstructural and thermal properties were characterized by scanning electron microscopy, 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 saturation temperature strongly depends on the revolution time, i. e. on the deformation rate.

Application of optimization techniques and the enthalpy method to solve a 3D-inverse problem during a TIG welding process

Tungsten inert Gas (TIG) welding takes place in an atmosphere of inert gas and uses a tungsten electrode. In this process heat input identification is a complex task and represents an important role in the optimization of the welding process. The technique used to estimate the heat flux is based on solution of an inverse three-dimensional transient heat conduction model with moving heat sources. The thermal fields at any region of the plate or at any instant are determined from the estimation of the heat rate delivered to the workpiece. The direct problem is solved by an implicit finite difference method. The system of linear algebraic equations is solved by Successive Over Relaxation method (SOR) and the inverse problem is solved using the Golden Section technique. The golden section technique minimizes an error square function based on the difference of theoretical and experimental temperature. The temperature measurements are obtained using thermocouples at accessible regions of the workpiece surface while the theoretical temperatures are calculated from the 3D transient thermal model.

On stationary heat conduction in 3D symmetric domains: an application of extended thermodynamics

In this paper, we study some three-dimensional stationary heat conduction problems in rarefied gases at rest, using the linearized 13-moment equations of extended thermodynamics. It follows that, in this linear theory, the temperature is still described by the Fourier law of heat conduction, while in contrast with the Navier–Stokes law, the stress tensor does not vanish. Furthermore, a non-equilibrium temperature is introduced and differences between this temperature and the kinetic one are predicted.

An Inverse Problem in Estimating the Volumetric Heat Generation for a Three-Dimensional Encapsulated Chip

A three-dimensional inverse heat conduction problem is solved in the present study by using the conjugate gradient method (CGM) and the general-purpose commercial code CFD−ACE+ to estimate the strength of the unknown heat generation for an encapsulated chip in a three-dimensional irregular domain. The advantage of calling CFD−ACE+ code as a subroutine in the present inverse calculation lies in that many difficult but practical 3D inverse problem can be solved under this construction since the general-purpose commercial code has the ability to solve the direct problem easily. The results obtained by using the CGM to solve this 3D inverse problem are justified based on the numerical experiments using the simulated exact and inexact measurements. It is concluded that reliable heat generation can be estimated by the present inverse algorithm.

過渡応答法によるディンプル面熱伝達率計測における固体壁内三次元熱伝導補正(熱工学,内燃機関,動力など)

The transient technique using infrared-thermography or liquid-crystal has widely been used for measuring the distribution of local heat transfer coefficients. In this technique, wall surface temperature is measured, and the heat transfer coefficient is calculated so as to accord the measured temperature with the theoretical solution of one-dimensional heat conduction problem. In actual cases of complicated surface geometry, however, three-dimensional heat conduction, caused by the three-dimensionality of the wall surface and the distribution of heat transfer coefficient, occurs in the wall. In this study, the heat transfer coefficient on the hemispherically dimpled surface was measured with an infrared camera, while the three-dimensional heat conduction in the wall was numerically calculated. In the compensation process, modification of the heat transfer coefficient was repeated till the numerical result agreed with the measured surface temperature. The present results showed that the heat transfer coefficient near the dimple edge was overrated, while that within the cavity was underrated. The maximum error induced by the three-dimensional heat conduction was 60% on the leading edge of the dimple, and the error in the other area was about 20% at most. At the dimple edge, the convex geometry increased the surface area where the heat flew into the wall, and consequently temperature rise became larger than the flat part. On the other hand, within the dimple, the concave geometry formed the radially-expanding heat conduction area, and the temperature became lower. The principal factor contributing to the error of the measurement is the three-dimensionality of the surface.

Estimation of local nucleate boiling heat flux using a three-dimensional transient heat conduction model

Boiling heat transfer is still hard to model and to predict. Among all kinds of studies on boiling processes, the estimation of the local heat fluxes at the boiling surface is fundamental. In this article, we solve a three-dimensional transient inverse heat conduction problem (IHCP) arising from a single-bubble nucleate boiling experiment to reconstruct the local boiling heat flux from a high-resolution temperature field measured at the back side of a thin heating foil. From the mathematical point of view, this problem belongs to the class of ill-posed inverse problems and cannot be solved straightforwardly by classical numerical methods. Thus, an iterative regularization strategy based on the conjugate gradient method is applied to the solution of the considered IHCP. Ring-shaped local boiling heat fluxes which undergo a significant change during a single-bubble cycle are observed.

Analytical Method for Determining Heat Flux from Temperature-Sensitive-Paint Measurements in Hypersonic Tunnels

This paper describes the analytical method used for temperature-sensitive-paint measurements to obtain quantitative global heat flux diagnostics in hypersonic tunnels. An exact inverse solution of the one-dimensional unsteady heat conduction equation for a polymer layer (temperature-sensitive paint or temperature-sensitive paint and an insulator layer) on a base of any material is obtained to calculate heat flux into the polymer surface as an explicit function of a time-dependent surface temperature change measured by temperature-sensitive paint. In addition, an analytical solution of the three-dimensional unsteady heat conduction equation is given to consider the effect of the lateral heat conduction. Simulations and experiments are conducted to examine the analytical method and assess the relevant factors regarding the measurement uncertainty.

Mould cooling simulation for injection moulding using a fast boundary element method approach

The existing cooling simulations for injection moulding are mostly based on the boundary element method (BEM). In this paper, a fast BEM approach for mould cooling analysis is developed. The actual problem is decoupled into a one-dimensional transient heat conduction problem within the thin part and a cycle-averaged steady state three-dimensional heat conduction problem of the mould. The BEM is formulated for the solution of the mould heat transfer problem. A dynamic allocation strategy of integral points is proposed when using the Gaussian integral formula to generate the BEM matrix. Considering that the full and unsymmetrical influence matrix of the BEM may lead to great storage space and solution time, this matrix is transformed into a sparse matrix by two methods: the direct rounding method or the combination method. This approximated sparsification approach can reduce the storage memory and solution time significantly. For validation, six typical cases with different element numbers are presented. The results show that the error of the direct rounding method is too large while that of the combination method is acceptable.

Development of a Three-Dimensional Time-Dependent Calculation Scheme for Molten Salt Reactors and Validation of the Measurement Data of the Molten Salt Reactor Experiment

This paper presents the development, validation, and results of a three-dimensional, time-dependent, coupled-neutronics–thermal-hydraulic calculational scheme for channel-type molten salt reactors (MSRs). The reactor physics part is based on diffusion theory, extended by a term representing the flow of the fuel through the core. The calculation of the temperature field is done by modeling all fuel channels, which are coupled to each other by a three-dimensional heat conduction equation. For the purpose of validation, the results of the MSR Experiment (MSRE) natural-circulation experiment and the thermal feedback coefficients of the reactor have been calculated and compared.With the aid of a code system developed to implement this scheme, calculations were carried out for the normal operating state of the MSRE and some debris-induced channel-blocking-incident transients. In the case of the MSRE, it is shown that the severity of such an incident strongly depends on the degree of channel blocking and that high-temperature gradients in the moderator can connect thermally the adjacent fuel channels. Results are included for an unblocking transient (i.e., the debris suddenly exits the core, and the fuel flow reverts to the normal operating pattern), and it was demonstrated that during the unblocking large power peaks can be induced.

Efficient solution of a three-dimensional inverse heat conduction problem in pool boiling

In this paper, we consider a three-dimensional transient inverse heat conduction problem arising in pool boiling experiments, i.e., the reconstruction of the surface heat flux from pointwise temperature observations inside a heater. We show that the inverse problem is ill-posed and utilize Tikhonov regularization and conjugate gradient methods together with a discrepancy stopping rule for a stable solution. We investigate the proper choice of regularization terms, which not only affects stability of the reconstructions but also greatly influences the quality of reconstructions in the case of limited observations. For the numerical solution of the governing partial differential equation, a space-time finite element method is used. This allows us to compute exact gradients for the discretized Tikhonov functional, and enables the use of conjugate gradient methods for the solution of the regularized inverse problem. We discuss further aspects of an efficient implementation, including a multilevel optimization strategy, together with an implementable stopping criterion. Finally, the proposed algorithms are applied to the reconstruction of local boiling heat fluxes from experimental data.

(156) Serologic profile of HBS antigen carriers

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 and the kinetic energy of filler droplets specifically driven by gravity, electromagnetic force, arc drag force, carring mass, momentum and thermal energy. These driving forces periodically impinge onto the base metal, leading to a liquid weld puddle. The electric arc is assumed to be a moving double central conicoidal 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. Analyses of micro-structural changes are critically investigated to comprehend the HAZ softening and phenomena.

Transient thermal effects in Alpine permafrost

Abstract. In high mountain areas, permafrost is important because it influences the occurrence of natural hazards, because it has to be considered in construction practices, and because it is sensitive to climate change. The assessment of its distribution and evolution is challenging because of highly variable conditions at and below the surface, steep topography and varying climatic conditions. This paper presents a systematic investigation of effects of topography and climate variability that are important for subsurface temperatures in Alpine bedrock permafrost. We studied the effects of both, past and projected future ground surface temperature variations on the basis of numerical experimentation with simplified mountain topography in order to demonstrate the principal effects. The modeling approach applied combines a distributed surface energy balance model and a three-dimensional subsurface heat conduction scheme. Results show that the past climate variations that essentially influence present-day permafrost temperatures at depth of the idealized mountains are the last glacial period and the major fluctuations in the past millennium. Transient effects from projected future warming, however, are likely larger than those from past climate conditions because larger temperature changes at the surface occur in shorter time periods. We further demonstrate the accelerating influence of multi-lateral warming in steep and complex topography for a temperature signal entering the subsurface as compared to the situation in flat areas. The effects of varying and uncertain material properties (i.e., thermal properties, porosity, and freezing characteristics) on the subsurface temperature field were examined in sensitivity studies. A considerable influence of latent heat due to water in low-porosity bedrock was only shown for simulations over time periods of decades to centuries. At the end, the model was applied to the topographic setting of the Matterhorn (Switzerland). Results from idealized geometries are compared to this first example of real topography, and possibilities as well as limitations of the model application are discussed.

Determination of the convective heat transfer coefficient in three-dimensional inverse heat conduction problems

A thin steel plate probe (24 mm wide×24 mm long×3 mm thick) surrounded by insulation except for the exposed surface was constructed. Using the temperature measurements of the steel plate and the heat flux measured by a Gardon gauge and with the assistance of numerical calculations, two methods based on three-dimensional inverse conduction problems are developed to determine the convective heat transfer coefficient in fire experiments. The first one is based on optimisation of the predictions of the temperature of the steel plate using different heat transfer coefficients, while the second adopts a “predictor–corrector” method to determine the instantaneous heat transfer coefficient. Validation of both methods is accomplished by performing experiments in the cone calorimeter at a known constant heat flux. Subsequently, the two methods are applied to experiments in enclosures to examine the sensitivities of the two methods.

Effect of ellipse orientation on the thermoelastic behaviour of skew laminated composite plate with elliptical cutout

An effort is made to study the thermoelastic behaviour of a cross-ply laminated composite skew plate with elliptical cutout subjected to pressure and non-linearly varying temperature loading in the present analysis. Orientation of the elliptical cut out is varied from 0° to 180° with respect to horizontal at an interval of 30° in the anti clockwise direction is considered for the present analysis. A three-dimensional heat conduction analysis in fibre reinforced composite laminates has been simulated by finite element method to get realistic temperature in the laminate under different thermal boundary conditions. A finite element method, which works on the basis of three-dimensional theory of elasticity, is employed to evaluate the stresses and deformations. The effect of orientation due to pressure loading on the stresses and transverse deflection is observed to be insignificant. The magnitudes of the in-plane normal stresses, σ x and σ y, for temperature loading are greatly affected by ellipse orientation and are observed to be minimum at the ellipse orientation of 0° and 90°, respectively. The in-plane and inter-laminar shear stresses are observed to be minimum at the ellipse orientation of 90°.

Three-dimensional unsteady heat conduction analysis by triple-reciprocity boundary element method

The conventional boundary element method (BEM) requires a domain integral in heat conduction analysis with heat generation or an initial temperature distribution. In this paper it is shown that the three-dimensional heat conduction problem can be solved effectively using the triple-reciprocity BEM without internal cells. In this method, the distributions of heat generation and initial temperature are interpolated using integral equations and time-dependent fundamental solutions are used. A new computer program was developed and applied to solving several problems.

Three-dimensional inverse transient heat transfer analysis of thick functionally graded plates

In this paper, a three-dimensional transient inverse heat conduction (IHC) procedure is presented to estimate the unknown boundary heat flux of thick functionally graded (FG) plates. For this purpose, the conjugate gradient method (CGM) in conjunction with adjoint problem is used. A recently developed three-dimensional efficient hybrid method is employed to solve variable-coefficient initial-boundary-value differential equations of direct problem as a part of the inverse solution. The accuracy of the inverse analysis is examined by simulating the exact and noisy data for problems with different types of boundary conditions and material properties. In addition to rectangular domain, skew plates are considered. The results obtained show good accuracy for the estimation of boundary heat fluxes.