Circular Heat Source Research Articles

Coherent vortical structures in numerical simulations of buoyant plumes from wildland fires

The structure and dynamics of buoyant plumes arising from surface-based heat sources in a vertically sheared ambient atmospheric flow are examined via simulations of a three-dimensional, compressible numerical model. Simple circular heat sources and asymmetric elliptical ring heat sources that are representative of wildland fires of moderate intensity are considered. Several different coherent vortical structures that dominate the plume structure and evolution are evident in the simulations, and these structures correspond well with those observed in plumes from wildland fires. For the circular source, these structures include: (i) a counter-rotating vortex pair aligned with the plume trajectory that is associated with a bifurcation of the plume, (ii) transverse shear-layer vortices on the upstream face of the plume, and (iii) vertically oriented wake vortices that form periodically with alternating sign on either side of the downstream edge of the plume base. For the elliptical ring source, a streamwise counter-rotating vortex pair is apparent on each flank, and a transverse horizontal vortex is observed above the head of the source. In all simulations the plume cross section is represented poorly by a self-similar Gaussian distribution.

THREE-DIMENSIONAL CALCULATION OF TEMPERATURE IN A ROTATING DISK SUBJECTED TO AN ECCENTRIC CIRCULAR HEAT SOURCE AND SURFACE COOLING

This article derives, for the first time, an appropriate analytical solution for the calculation of three-dimensional temperature distribution inside a rotating disk subjected to an eccentric circular heat source and surface cooling. This problem is encountered in, among others, the pin-on-disk frictional device, thrust ball bearing, or machining in a lathe. In order to obtain this solution under an explicit and simple form, we assume that the cooling occurs over the whole frictional surface of the disk, including the heated region. A numerical study, using the finite-volume method, is also conducted in order to define the limits of validity of the above assumption. It is shown that the analytical solution is accurate for a wide range of heat convection coefficients ( h ≤ 1,000 W/m2K for high Peclet numbers and h ≤ 5,000 W/m2K for low ones).

On the mechanics of the grinding process, Part III—thermal analysis of the abrasive cut-off operation

This is Part III of a 3 part series on the Mechanics of the Grinding Process. Part I deals with the stochastic nature of the grinding process, Part II deals with the thermal analysis of the fine grinding process and this paper (Part III) deals with the thermal analysis of the cut-off operation. Heat generated in the abrasive cut-off operation can affect the life of resin bonded grinding wheels and cause thermal damage to the workpiece. Thermal analysis of the abrasive cut-off operation can, therefore, provide guidelines for proper selection of the grinding conditions and optimization of the process parameters for improved wheel life and minimal thermal damage to the workpiece. In this investigation, a new thermal model of the abrasive cut-off operation is presented based on statistical distribution of the abrasive grains on the surface of the wheel. Both cutting and ploughing/rubbing that take place between the abrasive grains and the work material are considered, depending on the depth of indentation of the abrasives into the work material. In contrast to the previous models, where the apparent contact area between the wheel and the workpiece was taken as the heat source, this model considers the real area of contact, namely, the cumulative area of actual contacting grains present at the interface as the heat source. It may be noted that this is only a small fraction of the total contact area as only a small percentage of the abrasive grains present on the surface of the cut-off wheel are in actual contact with the workpiece at any given time and even a smaller fraction of them are actual cutting grains taking part in the cut-off operation. Since, the Peclet number, N Pe in the case of cut-off grinding is rather high (a few hundred), the heat flow between the work and the contacting abrasive grains can be considered to be nearly one-dimensional. In this paper, we consider the interaction between an abrasive grain and the workpiece at the contact interface. Consequently, the heat source relative to the grain is stationary and relative to the workpiece is fast moving. The interface heat source on the grain side as well as on the workpiece side is equivalent to an infinitely large plane heat source with the same heat liberation intensity as the circular disc heat source. However, it will be shown in the paper that the contacting times are different. For example, the abrasive grain contacts the heat source, as it moves over the wheel-work interface, for a longer period of time (∼ milliseconds) whereas the workpiece contacts the heat source for shorter period of time (∼ a few microseconds). The temperature in the grinding zone is taken as the sum of the background temperature due to the distributed action of the previous active grains operating in the grinding zone (global thermal analysis) and the localized temperature spikes experienced at the current abrasive grain tip-workpiece interfaces (local thermal analysis), similar to the work reported in the literature [Proc Roy Soc (London) A 453 (1997) 1083]. The equivalent thermal model developed in the present investigation is simple and represents the process more realistically, especially the heat partition. The model developed provides a better appreciation of the cut-off operation; a realistic estimation of the heat partition between the wheel, the workpiece, and the chip; thermal gradients in the workpiece due to abrasive cut-off operation, and an insight into the wear of the cut-off wheels.

On the mechanics of the grinding process, Part II—thermal analysis of fine grinding

Thermal analysis of fine grinding is conducted taking into consideration the stochastic nature of the distribution of abrasive grains and its role under fine grinding (dry) conditions to determine the grinding temperatures and the heat partition at the contacting interface. The analysis considers the grain–workpiece interactions at the local level and the wheel–workpiece interactions at the global level. The workpiece temperature in the grinding zone is taken as the sum of the background temperature due to distributed action of all the previous active grains operating in the grinding zone (global thermal analysis) and the localized temperature spikes experienced at the current abrasive grain tip–workpiece interfaces (local thermal analysis), similar to the work reported in the literature. Since the Peclet number, N Pe, in the case of fine grinding is very high (a few hundred), the heat flow between the work and the contacting abrasive grains can be considered to be nearly one-dimensional. In this paper, we consider the interaction between an abrasive grain and the workpiece at the contact interface. Consequently, the heat source relative to the grain is stationary and relative to the workpiece is fast moving. The interface heat source on the grain side as well as on the workpiece side is equivalent to an infinitely large plane heat source (with the same heat liberation intensity as the circular disc heat source). However, it will be shown in the paper that the contacting times are different. For example, the abrasive grain contacts the heat source, as it moves over the interface, for a longer period of time (~milliseconds) whereas the workpiece contacts the heat source for a shorter period of time (~a few microseconds). The equivalent thermal model developed in the present investigation is simple and represents the process more realistically, especially the heat partition. The analytical results reported here are found to be in good agreement with both the analytical and experimental results reported in the literature by other researchers.

An Exact Explicit Analytical Solution of the Steady-State Temperature in a Half Space Subjected to a Moving Circular Heat Source

So far in the literature, the distribution of stationary temperature over the surface of a half space subjected to a moving circular heat source has been reported in an integral or asymptotic form. In this paper, an exact explicit analytical solution is provided, which allows the determination of the temperatures over the contact area with a very short computational time, regardless of the value of the Peclet number. The solution is based on special functions (Bessel and hypergeometric functions) that are pre-programmed under a formal calculation software (e.g., Maple). The results of the proposed solution are in agreement with the asymptotic models available in the literature.

Analytical model for thermal resistance due to multiple moving circular contacts on a coated body

The heat transfer at the interface of two solids in sliding/rolling contact depends on the constriction phenomenon which occurs at the vicinity of asperities. In order to study this problem, the micro-contacts are represented by multiple moving circular heat sources on the surface of a body. The studied body is constituted of a substrate and a surface coating. The thermal constriction resistance due to those contacts is determined analytically in this paper. The solution is developed by using the integral Fourier transforms, and it is valid regardless of the velocity and the relative contact size values. To cite this article: A. Baïri, C. R. Mecanique 331 (2003).

Efficient finite element approach for modelling of actual welded structures

As the basis of the finite element analysis of a welding process, two mathematical models for the welding heat source are presented, based on a Gaussian distribution of power density. One is the prolonged Gaussian heat source model, and the other is the prolonged point heat source model. Finite element simulations of a bead on plate welding process were carried out using the circular disc heat source model, prolonged Gaussian heat source model, and prolonged point heat source model. Comparisons of the temperature, distortion, and stress obtained show that the combination of the new models and a reasonable mesh can greatly increase the computational efficiency while maintaining a relatively high accuracy. In the present paper, both finite element predictions of distortion for huge complex welded beams and workshop experiments were carried out. The numerical results were in reasonable agreement with the experimentally determined distortion values.

Analyse du phénomène de constriction thermique au sein d'un milieu cylindrique avec retournement à [formula omitted] des lignes de flux

We present the analytical solution and a physical analysis of the thermal constriction resistance for a configuration in which the heat flux lines undergo a π reversal. The configuration considered is an adiabatic cylinder except one of its bases, where three boundary conditions are imposed: a circular heat source, three annular surfaces which correspond to two adiabatic surfaces and one isothermal surface (heat sink). We study the size influence of the source, of the heat sink, and of the cylinder thickness on the thermal constriction resistance. To cite this article: V. Gatto et al., C. R. Mecanique 330 (2002) 615–621.

General solutions for stationary/moving plane heat source problems in manufacturing and tribology

General solutions (both transient and steady state) for the temperature rise at any point due to stationary/moving plane heat sources of different shapes ( elliptical, circular, rectangular, and square) and heat intensity distributions ( uniform, parabolic, and normal) are presented using the Jaeger’s classical heat source method (J.C. Jaeger, Moving sources of heat and the temperature at sliding contacts, Proc. Royal Society of NSW 76 (1942) 203–224). Starting from an instantaneous point heat source solution, an elliptical moving heat source with different heat intensity distributions, namely, uniform, parabolic and normal, was used as the basic plane heat source and its solution for the temperature rise at any point was derived. This analysis was then extended to other plane heat sources, such as circular, rectangular, and square heat sources to cover a range of manufacturing processes and tribological problems experienced in engineering practice. In addition, the analysis presented here is valid for both transient and steady state conditions while most analyses to date are strictly for quasi-steady state conditions. The solutions for the stationary heat sources are obtained from the moving heat source solution by simply equating the velocity of sliding to zero. Further, the analysis can be used to determine the temperature distribution not only at the surface but also with respective to the depth which again is a very important consideration in most manufacturing and tribological applications since it effects the subsurface deformation, metallurgical changes, hardness variation, and residual stresses. It can also be used to determine the maximum and average temperatures within the area of the heat source. Thus, the analysis presented here is believed to be comprehensive.

Temperature field in a microperiodic two-layered composite caused by a circular laser heat source

The paper is connected with the determination of␣temperature field in a periodically layered half-space caused by a laser heating. Two types of thermal loadings distributed on a boundary of the body is considered: uniform and normal (Gaussian). The influence of geometrical and thermal parameters of the layered composite on the temperature field is analyzed.

Magnetic Field Assisted Finishing of Ceramics—Part I: Thermal Model

A thermal model for magnetic field assisted polishing of ceramic balls/rollers is presented. The heat source at the area of contact between the balls and the abrasives where material removal takes place is approximated to a disk. The disk heat source is considered as a combination of a series of concentric circular ring heat sources with different radii. Each ring in turn is considered as a combination of a series of infinitely small arc segments and each arc segment as a point heat source. Jaeger’s classical moving heat source theory (Jaeger, 1942; Carslaw and Jaeger, 1959) is used in the development of the model, starting from an instantaneous point heat source, to obtain the general solution (transient and steady-state) of the moving circular ring heat source problem and finally the moving disc heat source problem. Due to the formation of fine scratches during polishing (on the order of a few micrometers long), the conditions are found to be largely transient in nature. Calculation of the minimum flash temperatures and minimum flash times during polishing enables the determination if adequate temperatures can be generated for chemo-mechanical polishing or not. This model is applied in Part II for magnetic float polishing (MFP) of ceramic balls and in Part III for magnetic abrasive finishing (MAF) of ceramic rollers.

Analytical modeling of spreading resistance in flux tubes, half spaces, and compound disks

A review of previously published models and solutions pertinent to the issue of modeling thermal resistances of diamond on copper heat sink systems is presented. The many particular solutions are shown to be special cases of the comprehensive model developed for a circular heat source in perfect thermal contact with the top surface of a compound disk which consists of two isotropic layers in perfect thermal contact. The bottom surface of the compound disk is subjected to a convective or contact cooling condition. Whenever possible simple models and correlation equations are presented for ease of computation. Bounds are presented for estimating the overall thermal resistance of several important cases.

帯板のレーザ割断における温度上昇の抑制

Brittle materials like semiconductor wafers and ceramic substrates are usually cut by diamond tools, where the generation of microcracks cannot be avoided. Recently, thermal stress cleaving of a thin strip using a point heat source is studied to cut brittle materials without leaving any microcracks on the cleaved surfaces. However, this method often causes thermal damages to the workpieces because of the very high temperature reached around the heating point. It is desirable to cut the electronic substrates, on which precise circuits were already formed, with minimizing the temperature rise. In the present paper, based on the plane thermoelasticity, the time dependent thermal stress intensity factor of a line crack has been obtained for two heating ways : by two point heat sources and a circular zone heat source located symmetrically with respect to the crack line. It was found that a zone heating is better than a point heating in the case of a thin strip cleaving under the restriction of the temperature rise. The most successful cleaving is generally achieved when a radius of a heating region is expanded up to the crack tip. The present calculation was compared with the experimental results and good agreement was found.

On moving heat sources

The two-dimensional thermal problem due to relative motion of a medium and a suddenly activated circular heat source is solved for several boundary conditions. The solutions can be interpreted as for a moving heat source in a stationary medium or a medium moving past a stationary heat source. Uniform and non-uniform temperature, and uniform and non-uniform heat flux boundary conditions are considered. The effect of velocity and radial direction on the temperature distribution is examined. Average, steady-state Nusselt numbers are derived. The transient response of a continuous line source is obtained as a limiting case of the prescribed heat flux solution.

A contact method of determination of thermophysical properties of rocks from core samples

The zone of the action of thermal disturbances around a circular heat source on the surface of a semi-infinite body is estimated with the aim of using contact methods of determination of thermophysical properties of materials from core samples.

Quasi-steady-state three-dimensional temperature distribution induced by a moving circular gaussian heat source in a finite depth solid

An analytical solution to the three-dimensional quasi-stationary problem in a finite depth and width solid with a circular Gaussian moving heat source at the body surface is developed and analyzed. The temperature distribution and the axial coordinate at which the maximum midplane temperature is achieved are presented as a function of Peclet number, solid thickness and width. The dependence of the maximum midplane temperature on the process parameters is highlighted. Combinations of process parameters for which the solution to the three-dimensional problem can be approximated by those to simpler models are pointed out.

Expanding heat source model for thermal spalling of TiB2 in electrical discharge machining

A model is presented to explain the recently reported mechanism of thermal spalling for shaping high melting point ceramics by electrical discharge machining. Since previous models fail to explain the experimental observations completely, an expanding circular heat source created by growth of plasma is assumed to act on the surface. Erosion of materials by spalling is caused by thermally induced compressive stresses during heating-up periods and tensile stresses during cooling-down periods. This model explains material removal for anodic erosion in general (wire-cutting machines) and for cathodic erosion (die-sinking machines) whenever long pulse duration is used. Simulation of the model for TiB2 provides a local melt front that penetrates to a depth of submicrometer, then recedes as pulse duration increases. Spalling develops flakes with thickness correlated to pulse duration. The results were verified by the experimental observations which showed that large flakes having the predicted maximum thickness as well as few quenched spherical droplets containing titanium were obtained.

NUMERICAL SIMULATION OF LAMINAR FLOW AND HEAT TRANSFER FOR LIQUID JET IMPINGEMENT COOLING OF A CIRCULAR HEAT SOURCE WITH ANNULAR COLLECTION OF THE SPENT FLUID

Axisymmetric, laminar jet impingement cooling of a circular heat source has been simulated under conditions for which discharge of the spent fluid may influence heat transfer from the source. The simulation specifically addresses an annular configuration for which the jet is discharged from a circular tube and, after impingement, is redirected 180° to exit through an annular passage. The flow is strongly influenced by two recirculation zones located adjacent to the jet and in the corner formed by intersection of the outer wall of the annulus with the heater/substrate surface. For jet velocity profiles that are parabolic or uniform, respectively, the radial distribution of the local Nusselt number is characterized by a local maximum or minimum at the stagnation point. If the outer wall of the annulus intersects the outer edge of the heater, the Nusselt number exhibits a local maximum near the edge, irrespective of the jet velocity profile. Parametric calculations have been performed to assess the effects o...

Thermal properties of the multilayer infinite plate structure

Using the double Fourier integral transform method, the analytical temperature solutions to the single-layer and multilayer structures with infinite lateral dimensions have been derived. For the single-layer structure, temperature dependence of the thermal conductivity has been taken into account using the Kirchhoff transformation. The use of discrete rectangular heat sources rather than circular heat sources on the top surface of the structure makes this model particularly useful for the thermal analysis of common solid-state and integrated circuit devices and packages. For studying the thermal properties of a large class of solid-state device geometries, this model can be utilized to approximate actual devices with reasonable accuracy. The general device geometries for which this model can be employed to represent the actual devices are determined. Compared with the model where the finite extent of the chip lateral dimensions is considered, the present model is much more computationally efficient for cases where the heat source size is much smaller than the substrate or package size.

Local heating of an anisotropic body by a circular heat source

The solution and analysis of the two-dimensional nonstationary heat-conduction problem are presented for an anisotropic (orthotropic) body heated by a circular heat source.