Fourier Heat Conduction Equation Research Articles

Thermal fatigue behavior of an iron-based laser sintered material

Direct metal laser sintering is a rapid manufacturing technique to make intricate and near net-shaped parts. An iron-based laser sintered metal was studied to evaluate its thermal fatigue properties. The test was performed using cylindrical specimens in a high power induction heating system equipped with a laser strain gauge for a contactless surface strain measurement. Initiation of thermal fatigue cracks occurred preferentially at pores and layer interfaces, while propagation of cracks followed along phase boundaries and thin inter-dendritic phases and showed an inter-granular fracture. By using the fundamental Fourier equation for heat conduction, the temperature cycle was modeled and calculated. A thermo elastic ideal plastic model was used to deduce the thermal stress based on surface strain experimentally measured. Finally, the temperature distribution, thermal stresses and mechanical strains were discussed with respect to thermal fatigue damage.

열유속 상지연이 존재하는 열전도 현상에 대한 연구

In most engineering applications related with the heat conduction phenomena, a conventional Fourier heat conduction equation has been successfully applied and it has supplied quite reasonable results. However, it is well known that the Fourier heat conduction equation is failed in the application to the extremely small space and short time, in other words, a nano-scale system and a pico-second time. In this study, non-Fourier effect was evaluated in the heat conduction by considering the concept of a phase lag model. The results show the existence of a heat wave, which means that the heat is transferred with a finite speed while an infinite speed of heat transfer is assumed in the conventional Fourier heat conduction. In addition, the copper and the gold are tested to evaluate the phase lag time between the heat flux and the temperature gradient. The results show that the gold has the heat wave speed faster than that of the copper consistent with the prediction based on an actual experiment.

Thermo-fluid dynamics of Loop Heat Pipe operation

The paper presents a thermo-fluid dynamic model for the transient operations of a Loop Heat Pipe. The Fourier heat conduction equation in a hollow cylinder is solved to determine the temperature distribution of the compensation chamber and the cavity. A one-dimensional transient model is also derived for the vapor temperature in the condenser section of the Loop Heat Pipe. The thermo-fluid dynamic characteristic of a Stainless Steel/Ammonia Loop Heat Pipe is analyzed.

Initial version of an integrated thermal hydraulics and neutron kinetics 3D code X3D

A theoretical concept for the description of 3D nuclear kinetics, heat conduction and the thermal-hydraulic single and two-phase flow phenomena in a 3D light water reactor core simulated by parallel channels is presented. The heat generation within each fuel element is described by a 3D core kinetics code employing a two-group diffusion theory model. Finite differences are used to approximate the diffusion equations for fast and thermal neutron fluxes. The resulting algebraic equations are solved by an iterative procedure. The heat transport out of a fuel element is determined from a finite difference version of the Fourier heat conduction equation applied in the radial direction only. An implicit solution for the temperature in each radial cell is found using a tri-diagonal algorithm. The heat transfer rate into the coolant channel is calculated from the clad and coolant temperatures. The thermal-hydraulic part of the code is based on the generally applicable and self-contained ‘coolant channel module’ (CCM) developed by Hoeld [Hoeld, A., 2004a, A theoretical concept for a thermal-hydraulic 3D parallel channel core model. PHYSOR 2004, April 25–29, Chicago, USA; Hoeld, A., 2004b, Are separate-phase thermal-hydraulic models better than mixture-fluid approaches. It depends. Rather not. International Conference on ‘Nuclear Engineering for New Europe 2004’. September 6–9, Portoroz, Slov; Hoeld, A., 2005, A thermal-hydraulic drift-flux based mixture-fluid model for the description of single- and two-phase flow along a general coolant channel. NURETH-11, October 2–6, Avignon, France], which offers a new approach for the simulation of steady state and transient behaviour of single- or two-phase flow within a general coolant channel. The concept is demonstrated in the experimental code X3D, concentrating in a first stage on the steady state behaviour of a hypothetical BWR or PWR core which consists of a central channel and four quadrants. Initial results are presented to demonstrate the viability of the proposed concept. This code therefore tests a different theoretical approach of how to determine the essential 3D thermal-hydraulic mass flow distribution into parallel channels by maintaining equal pressure decrease terms over all channels.

Heat conduction in microstructured materials

The phonon Boltzmann equation is solved numerically in order to study the phonon thermal conductivity of micro/nanostructured thin films with open holes in a host material. We focused on the size effect of embedded pores and film thickness on the decrease in thermal conductivity of the film. Simulations have revealed that the temperature profiles in the micro/nanostructured materials are very different from those in their bulk counterparts, due to the ballistic nature of the microscale phonon transport. These simulations clearly demonstrate that the conventional Fourier heat conduction equation cannot be applied to study heat conduction in solids at microscale. The effective thermal conductivity of thin films with micro/nanoholes is calculated from the applied temperature difference and the heat flux. In the present paper, the effective thermal conductivity is shown as a function of the size of the micro/nanoholes and the film thickness. For example, when the size of the hole becomes approximately 1/20th the phonon mean free path in a film, the thickness is 1/10th the mean free path of phonons and the effective thermal conductivity decreases to as low as 6% of the bulk value. The distribution of holes also affects the reduction in the effective thermal conductivity. Thin films embedded with staggered-hole arrays have slightly lower effective thermal conductivities than films with aligned-hole arrays. The cross-sectional area in the thermal transport direction is a significant parameter with respect to the reduction of thermal conductivity. The results of the present study may prove useful in the development of artificial micro/nanostructured materials, including thermoelectrics and low-k dielectrics.

Effect of Superheat, Mold, and Casting Materials on the Metal/Mold Interfacial Heat Transfer During Solidification in Graphite-Lined Permanent Molds

Heat transfer during the solidification of an Al-Cu-Si alloy (LM4) and commercial pure tin in single steel, graphite, and graphite-lined metallic (composite) molds was investigated. Experiments were carried out at three different superheats. In the case of composite molds, the effect of the thickness of the graphite lining and the outer wall on heat transfer was studied. Temperatures at known locations inside the mold and casting were used to solve the Fourier heat conduction equation inversely to yield the casting/mold interfacial heat flux transients. Increased melt superheats and higher thermal conductivity of the mold material led to an increase in the peak heat flux at the metal/mold interface. Factorial experiments indicated that the mold material had a significant effect on the peak heat flux at the 5% level of significance. The ratio of graphite lining to outer steel wall and superheat had a significant effect on the peak heat flux in significance range varying between 5 and 25%. A heat flux model was proposed to estimate the maximum heat flux transients at different superheat levels of 25 to 75 °C for any metal/mold combinations having a thermal diffusivity ratio (α R) varying between 0.25 and 6.96. The heat flow models could be used to estimate interfacial heat flux transients from the thermophysical properties of the mold and cast materials and the melt superheat. Metallographic analysis indicated finer microstructures for castings poured at increased melt superheats and cast in high-thermal diffusivity molds.

Mathematical modeling and experimental analysis of the hardened zone in laser treatment of a 1045 AISI steel

The aim of this work is to develop a mathematical model to predict the depth of laser treated zone in the LTH process. The Fourier equation of heat conduction is solved by using the Finite Difference Method in cylindrical coordinates in order to study the temperature distribution produced in a workpiece and hence to obtain the depth to which hardening occurs. The theoretical simulations are compared with results produced experimentally by a CO2 laser operating in continuous wave, showing good agreement.

Effect of chill thickness and superheat on casting/chill interfacial heat transfer during solidification of commercially pure aluminium

Heat transfer at the metal/chill interface during solidification of commercially pure aluminium square bar castings with cast iron chill at one end was investigated. Experiments were carried out for different chill thicknesses and superheats. The inner surface temperature of the chill initially was found to increase at a faster rate for higher superheats. The effect of chill thickness on the inner surface temperature of the chill was observed only after the heat from the solidifying casting had sufficient time to diffuse to the interior of the chill material. Inverse analysis of the non-linear one-dimensional Fourier heat conduction equation indicated the occurrence of peak heat flux at the end of filling of the mould. The effect of superheat on heat flux was minimal after filling. However, the effect of chill thickness had a significant effect on the heat flux after the occurrence of peak heat flux. Higher heat flux transients were estimated for castings poured at higher superheats. The corresponding heat transfer coefficients were also estimated and reported. The heat flux model presented in this work can be used for determination of casting/chill interfacial heat flux as a function of chill thickness and superheat. These heat flux transients could be used as boundary conditions during numerical simulation of solidification of the casting.

A SIMPLE MODEL OF URBAN CANYON ENERGY BUDGET AND ITS VALIDATION

An urban canyon energy budget simulation model is described. The model is simple and computationally inexpensive enough to permit characterization of the spatial and temporal variability of canyon energy exchanges by running multiple cases. It is two-dimensional and dry and avoids the necessity for computational fluid dynamical calculations by using parameterizations for the down-canyon wind profile and cross-canyon vortex flow, and by employing an exchange coefficient formulation for sensible heat loss from the walls. These are combined with radiation calculations using the procedure of Arnfield (1976, 1982) and substrate heat fluxes computed using the Fourier heat conduction equation and wall and floor thermal property information. The model is validated against the measured canyon energy budget data of Nunez and Oke (1977) and reproduces these data quite well, with Willmott indices of agreement in the range 0.91 to 1.00 and root-mean-square errors of 14 to 46 W m-2 for individual facets and for the total canyon system. The poorest correspondence was for the substrate heat flux, the maxima for which were overestimated, suggesting use of excessively large facet thermal properties. These inputs were adjusted based on the measurement-simulation discrepancies, leading to further improvement in model performance. The adjusted thermal properties are consistent both with published data and with the available information on the morphological characteristics of the canyon. [Key words: urban canyon energy budget, urban climate, numerical simulation model.]

Unsteady heat conduction in the soil layers above underground repository for spent nuclear fuel

Due to radioactivity of spent nuclear fuel, a repository is expected to act as a heat source of exponentially decreasing intensity over hundreds of years. In case of underground emplacement of such a heat source, the temperature changes in the soil layers surrounding the heat source may have important implications such as evaporation of the water contained in the soil and its subsequent condensation. Assuming a uniformly distributed power generation over a horizontal, relatively thin, circular zone of several thousand meters diameter located several hundred meters below the ground surface, the temperature variations along the vertical centerline of the heating zone was estimated analytically under simplifying assumptions. Unsteady one-dimensional heat conduction in a semi-infinite solid with an exponentially decreasing heat flux at the proximal end was considered. The corresponding solution of the Fourier equation for heat conduction contains Error Functions of complex arguments. The evaluation of the Error Functions for discrete space and time parameters was performed applying analytical procedures and using standard tables. The results show temperature distributions in the soil at various time points over thousands of years after underground emplacement of spent nuclear fuel.

Correlation and modeling of the occurrence of different crystalline forms of isotactic polypropylene as a function of cooling rate and uniaxial stress in thin and thick parts

AbstractA study of structure development in thin melt spun isotactic polypropylene filaments is described, which is then applied to the prediction of the behavior of thick parts. Conditions under which different crystalline forms of polypropylene are obtained as a function of cooling rate and spinline stress were investigated. Continuous cooling transformation (CCT) curves are developed. This also allows us to develop a map of crystalline form as a function of these variables. We have applied the CCT curves and this map to predict the development of cross‐sectional variation structure in thick filaments and rods. This is applied in particular to the quenching of a cylindrical rod and the structural characterizations observed through the cross section are compared with predictions from the CCT curves and solutions of Fourier's transient heat conduction equation.

Acoustic wave prediction in flowing steam–water two-phase mixture

The transient two-fluid model has been used to develop a general relation for acoustic waves with steam–water two-phase mixture in one-dimensional flowing system. The analysis is valid in principle over the whole void fraction region. Both the mechanical and thermal nonequilibrium are considered. The vapor–liquid interface heat flux is derived from the one dimensional Fourier heat conduction equation to evaluate the interphase evaporation or condensation rate. Calculations are performed for pressures from 0.07 to 16.0 MPa, void fractions from 0.0 to 1.0. Higher frequency limit of sonic velocities are only relied on the pressures and void fractions, and are insensitive to the bubble radius, tube diameter, and the velocity difference between the two phases. The predicted sonic velocities are compared with some experimental data for low pressures. Good agreement has been achieved between the predictions and the experimental data.

Assessment of Metal/Mould Interfacial Heat Transfer during Solidification of Cast Iron

The problem of heat transfer between a ferrous alloy and a sand mould/chill has been investigated. An inverse technique has been used to measure the heat flow at the metal/mould interface during solidification of cast iron in dry sand and graphite moulds. In the case of sand moulds, the Fourier heat conduction equation was modified to take into account the packed bed nature of the sand mould, the effect of convection of the gases and the evolution and absorption of heat due to mould reactions. The interfacial heat transfer coefficient and interfacial heat flux were estimated from measured temperatures inside the mould during the solidification of the cast iron. The temporal variation of the heat flux transients during the early stages of the experiment was explained on the basis of the formation of an initial solid skin and the nucleation of an air gap resulting from the combined action of the movement of the solid shell away from the mould wall and mould dilation. At later times the interfacial heat transfer from the cast iron casting was found to be dominated by radiation which accounted for as much as 80% of the overall heat flux.

Temperature distribution during conventional and microwave wood heating

The paper presents a comparison between the experimental and theoretical distribution of temperature in wood during conventional and microwave heating. Theoretical models of temperature distribution, based on heat conduction, were established and compared with experimental data. The solutions of Fourier's heat conduction equations under adequate initial and boundary conditions properly describe experimental temperature distributions both in conventional and microwave heating.

Fourier's heat conduction equation: History, influence, and connections

The equation describing the conduction of heat in solids has, over the past two centuries, proved to be a powerful tool for analyzing the dynamic motion of heat as well as for solving an enormous array of diffusion‐type problems in physical sciences, biological sciences, earth sciences, and social sciences. This equation was formulated at the beginning of the nineteenth century by one of the most gifted scholars of modern science, Joseph Fourier of France. A study of the historical context in which Fourier made his remarkable contribution and the subsequent impact his work has had on the development of modern science is as fascinating as it is educational. This paper is an attempt to present a picture of how certain ideas initially led to Fourier's development of the heat equation and how, subsequently, Fourier's work directly influenced and inspired others to use the heat diffusion model to describe other dynamic physical systems. Conversely, others concerned with the study of random processes found that the equations governing such random processes reduced, in the limit, to Fourier's equation of heat diffusion. In the process of developing the flow of ideas, the paper also presents, to the extent possible, an account of the history and personalities involved.

Pulsed laser heating of steel surfaces — Fourier and electron kinetic theory approaches

Application of Fourier theory to heat conduction due to high power laser irradiation may give closed form solution to the problem. On the other hand, the heat flux through a given plane depends on the electron energy distribution through the material and at the scale of distance required to examine the problem, the material can no longer be considered as being homogeneous continuum, therefore, errors may occur when considering the Fourier theory in laser heating process. The problem requires to be examined in the quantum field. The present study examines the pulse laser heating process when considering both Fourier conduction and electron-kinetic theory approaches. Analytical solution to Fourier heat conduction equation are obtained for the intensity step input pulse while numerical scheme is introduced to solve the heat transfer equation resulted from kinetic theory approach. It is found that both Fourier and electron kinetic theory approaches result in similar temperature profiles for a step input intensity pulse.

An urban canyon energy budget model and its application to urban storage heat flux modeling

To obtain a local-scale urban energy balance by either measurement or modeling it is necessary to determine storage heat flux (Δ Q s). This flux cannot be measured directly due to the complexity of the urban surface. The Grimmond et al. Objective Hysteresis Model (OHM) [C.S.B. Grimmond, H.A. Cleugh, T.R. Oke, An objective urban heat storage model and its comparison with other schemes, Atmos. Environ., 25B (1991) 311–326] of local-scale ΔQ s combines empirical equations for individual surface types in the proportion that they are present within the urban area. One surface type for which there is very limited field data is the urban canyon, which consists of the walls of adjacent buildings, the horizontal street-level area separating them (roadways, gardens, parking lots, etc.) and the enclosed air volume. Here the storage heat flux of an urban canyon and the resulting OHM parameters are investigated with a numerical model of a dry urban canyon energy budget. Substrate heat fluxes are derived from simulated surface and substrate temperatures; the latter evolve through time according to the finite difference form of the Fourier heat conduction equation. When compared against measured fluxes, the model performed satisfactorily. Numerical experiments show significant effects on the OHM parameters due to changes in the ratio of building height to separation distance and building wall thermal properties. Effects of intermediate significance were attributable to canyon orientation, wind speed and the timing of the between-building air temperature regime. Air temperature and the timing of the wind speed curve showed only minor significance.

MODELING OF LASER HEATING OF SOLID SUBSTANCE INCLUDING ASSISTING GAS IMPINGEMENT

Laser heat treatment finds wide application in industry because of its precision operation, its ability to be used for local heating, and its low cost. In general, an assisting gas jet is introduced coaxially with the laser beam for shielding the region treated from the oxygen. To explore the effects of the gas jet on the heating mechanism, it is essential to perform a simulation of the process. The present study is conducted to simulate three-dimensional laser heating of steel substrate when subjected to impinging gas. The gas jet is considered to impinge to the workpiece surface coaxially with the laser beam. The k-e model with and without corrections and the Reynolds stress model are tested under conditions of constant heat flux introduced from the solid wall. As a result and in accordance with previous studies, the low-Re k-ϵ model is selected to account for the turbulence. However, the transient Fourier heat conduction equation is considered to compute the temperature profiles in the solid substrate. A numerical method using the control volume approach is introduced to solve the governing flow and energy equations. The study is extended to include four gas jet velocities. It is found that the impinging gas jet velocity has a considerable effect on the resulting gas side temperature. Moreover, as the radial distance from the heated spot center increases, the temperature at the surface decreases rapidly. In addition, the temperature profiles inside the solid substrate are not influenced considerably by the assisting gas jet velocity.

Numerical simulation of continuous induction steel bar end heating with material properties depending on temperature and magnetic field

Continuous induction steel bar end heating is investigated by means of numerical calculations. A numerical model is used for the calculation of the three-dimensional eddy current and the heating process. The differential equations describing the electromagnetic field are integrated as an A-/spl phi/ formulation. For the calculation of the temperature fields, Fourier's heat-conduction equation is used. On principle, the finite element method is used for the numerical solution. The effect of material properties depending on temperature and magnetic field are taken into consideration in an iterative manner. The results of simulation correspond well to experimental data and give good transparency of the process. The computing time, however, is too long for an effective realization of optimization processes.

Information-theoretical approach to radiative transfer

The maximum entropy formalism is used to obtain the radiation and matter distribution functions for radiative systems is steady nonequilibrium states, under the gray approximation. The radiation distribution function is expanded in a smallness parameter, which vanishes at equilibrium. In the first near-equilibrium approximation, we derive the results of near-equilibrium diffusion theory. This may be regarded as an analogue to the kinetic-theoretical result, according to which in the first Enskog approximation, the Fourier heat conduction equation is obtained. The theory is also developed up to the second order, leading to results which apply to situations further away from equilibrium than those corresponding to near-equilibrium diffusion theory. A simple application is analyzed.