Uniform Heat Source Research Articles

An exact thermodynamical model of power-law temperature time scaling

In this paper a physical model for the anomalous temperature time evolution (decay) observed in complex thermodynamical system in presence of uniform heat source is provided. Measures involving temperatures T with power-law variation in time as T(t)∝tβ with β∈R shows a different evolution of the temperature time rate Ṫ(t) with respect to the temperature time-dependence T(t). Indeed the temperature evolution is a power-law increasing function whereas the temperature time rate is a power-law decreasing function of time.Such a behavior may be captured by a physical model that allows for a fast thermal energy diffusion close to the insulated location but must offer more resistance to the thermal energy flux as soon as the distance increases. In this paper this idea has been exploited showing that such thermodynamical system is represented by an heterogeneous one-dimensional distributed mass one with power-law spatial scaling of its physical properties. The model yields, exactly a power-law evolution (decay) of the temperature field in terms of a real exponent as T∝tβ (or T∝t−β) that is related to the power-law spatial scaling of the thermodynamical property of the system. The obtained relation yields a physical ground to the formulation of fractional-order generalization of the Fourier diffusion equation.

Numerical simulation of large spot laser + MIG arc brazing–fusion welding of Al alloy to galvanized steel

A finite element model was developed to investigate the thermal process of large spot laser+MIG arc brazing–fusion welding. The laser was treated as a Gaussian plane heat source, the MIG arc was performed as a modified double ellipse Gaussian plane heat source, in which the arc distortion was taken into consideration and the overheated droplet was treated as a uniform body heat source. The calculated weld bead geometry and heat-affected width of zinc coating had good agreement with experimental results. The temperature field, especially for the brazed interface, showed non-uniform and asymmetric distribution. The thermal cycles at brazed interface had obvious bimodal characteristic at arc center and laser spot center and the high-temperature zone at the brazed interface was widened duo to the introduction of laser beam compared with conventional MIG brazing–fusion welding. A fundamental processing window was determined based on the founded model to satisfy a certain energy condition, in which the Al alloy was fully penetrated and steel plate was not melted.

Numerical simulation on MIG arc brazing-fusion welding of aluminum alloy to galvanized steel plate

Based on the difference in melting point between aluminum alloy and steel, metal inert gas (MIG) arc brazing-fusion welding process was developed to join thin 6013 aluminum alloy plate to galvanized steel plate. A finite element model for lap joint was established to study its thermal process. In modeling, MIG arc was treated as the double ellipse Gaussian plane heat source and the overheated metal droplet was considered as the uniform body heat source. The effects of the zinc coating and the overlap gap were taken into consideration. The brazed seam width and the width of heat-affected zinc coating on the back side of galvanized steel plate were used to validate the calculated results. The results show that the calculated results are in good agreement with experimental results. The calculation results indicate that the temperature is not uniformly distributed in the brazing-fusion welded joint. There is a great difference in the reaction temperature and time at different positions on the brazed interface. In order to satisfy the energy condition in which aluminum plate is fully penetrated and steel plate is not melted, a corresponding relationship that the welding current and welding speed are synchronized to be increased must be followed.

Power generation characteristics of a sandwich-type self-heating thermoelectric generator with spatially varying embedded heat source

Summary Power generation characteristics of a sandwich-type thermoelectric generator in which the heat source is embedded into thermoelectric elements are investigated. Our previous work on a similar concept only considered a uniform heat source distribution inside thermoelectric elements. In this work, the effect of the spatial distribution of a heat source is examined. In particular, the effect of the concentration of heat source near the one end, that is, the hot end, is intensively studied as a potential means of improving the efficiency of the device. Although the effects of heat source concentration in impractical cases without heat transfer limitations on the cold side remain ambiguous, it become clear that heat source concentration indeed has positive effects in more realistic cases with finite heat transfer coefficients imposed on the cold side. Because of the relatively low efficiency of typical thermoelectric generation, a significant amount of heat must be dissipated from the cold end of the thermoelectric element. Greater heat source concentration near the hot end leads to more effective utilization of available heat source, reduces the amount of heat rejected at the cold end, and lowers the hot end temperature of the thermoelectric element. Overall, it is suggested that heat source concentration can be used as a method to achieve more efficient operation and better structural integrity of the system. Copyright © 2015 John Wiley & Sons, Ltd.

УСОВЕРШЕНСТВОВАНИЕ АНАЛИТИЧЕСКИХ МЕТОДОВ РАСЧЕТОВ ТЕМПЕРАТУРНЫХ ПОЛЕЙ В СИСТЕМЕ РЕЗАНИЯ

This paper presents the detailed analytical method of calculation of temperature fields for the blade tool cutting process. The author considers in details the heat sources forms during cutting, the possibility and necessity of schematization both of the heat sources and the forms of interacting bodies. The calculation of temperature in the zone of primary plastic deformation is carried out taking into account the mutual influence of temperature on the intensity of shear stresses within this area. The author determined that the distribution of intensity of friction source along the back surface is stipulated by the nature of changes in friction induced shear stresses and is little different from the uniform one. The author offered the accurate solution of the issue of calculating temperature within the half-space from the action of a rectangular constant rate heat source. Thus, it became possible to determine the rate of constant heat flows on the front and back surfaces. The author tested a hypothesis about the constancy of heat flows rate. For this purpose, each final heat source was presented as a combination of elementary uniform heat sources. The research proved the large non-uniformity of rate distribution of heat sources that act on the cutting tool surfaces. This made it possible to identify new patterns of temperature distribution on the front and back surfaces of tool within various combinations of heat conductivity of the processed and tool materials, as well as the tool wear. The author identified the essential influence of body deferred on the front surface on the amount and direction of heat flows within the front and back surfaces – to the tool or to the blank part, and eventually on the temperature of cutting tool contact surfaces. It allowed development of guidelines for selecting optimal tool material for processing of blank parts with different heat conductivity.

Free-boundary models of a meltwater conduit

We analyse the cross-sectional evolution of an englacial meltwater conduit that contracts due to inward creep of the surrounding ice and expands due to melting. Making use of theoretical methods from free-boundary problems in Stokes flow and Hele–Shaw squeeze flow we construct an exact solution to the coupled problem of external viscous creep and internal heating, in which we adopt a Newtonian approximation for ice flow and an idealized uniform heat source in the conduit. This problem provides an interesting variant on standard free-boundary problems, coupling different internal and external problems through the kinematic condition at the interface. The boundary in the exact solution takes the form of an ellipse that may contract or expand (depending on the magnitudes of effective pressure and heating rate) around fixed focal points. Linear stability analysis reveals that without the melting this solution is unstable to perturbations in the shape. Melting can stabilize the interface unless the aspect ratio is too small; in that case, instabilities grow largest at the thin ends of the ellipse. The predictions are corroborated with numerical solutions using boundary integral techniques. Finally, a number of extensions to the idealized model are considered, showing that a contracting circular conduit is unstable to all modes of perturbation if melting occurs at a uniform rate around the boundary, or if the ice is modelled as a shear-thinning fluid.

Adiabatic eigenflows in a vertical porous channel

AbstractThe existence of an infinite class of buoyant flows in a vertical porous channel with adiabatic and impermeable boundary walls, called adiabatic eigenflows, is discussed. A uniform heat source within the saturated medium is assumed, so that a stationary state is possible with a net vertical through-flow convecting away the excess heat. The simple isothermal flow with uniform velocity profile is a special adiabatic eigenflow if the power supplied by the heat source is zero. The linear stability analysis of the adiabatic eigenflows is carried out analytically. It is shown that these basic flows are unstable. The only exception, when the power supplied by the heat source is zero, is the uniform isothermal flow, which is stable. The existence of adiabatic eigenflows and their stability analysis is extended to the case of spanwise lateral confinement, viz. in the case of a vertical rectangular channel. A generalisation of this study to a vertical channel with an arbitrary cross-sectional shape is also presented.

Buoyant-MHD Flows in HCLL Blankets Caused by Spatially Varying Thermal Loads

In the helium-cooled lead lithium (HCLL) blanket concept convective phenomena caused by nonuniform thermal conditions due to bulk neutron volumetric heating can occur. Buoyancy can become very important and modify the velocity distribution and related heat transfer performance of the blanket. A numerical study has been performed to investigate liquid metal flows driven by buoyant forces in a breeder unit (BU) of a HCLL test blanket module (TBM) under the influence of intense uniform magnetic fields. According to the last design review, two internal cooling plates subdivide the fluid domain into three slender flow regions, which are thermally and electrically coupled through common walls. First, a uniform volumetric heat source is considered to identify the basic convective patterns that establish in the liquid metal. Results are then compared with those obtained by applying a realistic radial distribution of the power density as obtained from a neutronic analysis. This paper summarizes the main effects of spatial gradients of a neutron thermal load on velocity and temperature distribution in magnetohydrodynamic flows in a BU of a HCLL TBM.

Closed-form expressions of the thermo-mechanical fields induced by a uniform heat source acting over an isotropic half-space

We provide closed-form expressions of displacements, stress and temperature fields produced in an isotropic linearly elastic half-space by a uniform heat flow acting over an arbitrarily shaped region of its boundary.The result is obtained by applying recent results of potential theory which allow one to replace the domain integrals expressing the thermo-mechanical fields by one-dimensional integrals extended to the boundary of the heated region.Algebraic expressions for evaluating the thermo-mechanical fields at an arbitrary point of the half-space are derived solely as a function of the heat flux and of the position vectors which define the boundary of the heated region assumed to be polygonal.Representative numerical examples show the effectiveness of the proposed approach to typical problems in thermo-mechanical contact, with particular reference to the evaluation of the thermal spreading resistance.

Thermal quadrupole approaches applied to improve heat transfer computations in multilayered materials with internal heat sources

An extension to the classical quadrupole method is proposed which allows computing temperature and heat fluxes anywhere inside a multilayer material containing localized and/or distributed heat sources. The distributed heat sources are not limited to being uniform in each layer. By using the superposition principle and through a treatment which depends on the relative position of the heat sources and the observation point, we get a closed-form analytical expression for the temperature/flux vector which yields stable results over an arbitrary time scale. This source-sampled quadrupole method is based on the transfer formulation related to a T-scheme two-port network representation. We propose another approach based on the impedance formulation related to the same T-scheme two-port network representation; it leads to a global impedance matrix formulation which provides first the heat flux vector and then the temperature vector. Alternatively we also propose an approach based on the admittance formulation related to a Π-scheme two-port network representation with admittances; it leads to a global admittance matrix formulation which provides first the temperature vector and then the heat flux vector. Both impedance and admittance matrix formulations are easier to program than the source-sampled quadrupole method but their computing time is slightly higher. The three proposed methods are illustrated on two four-layer slabs, one with a uniform heat source distribution in each layer and the other one with exponential heat source profiles.

Three dimensional simulations of penetrative convection in a porous medium with internal heat sources

The problem of penetrative convection in a fluid saturated porous medium heated internally is analysed. The linear instability theory and nonlinear energy theory are derived and then tested using three dimensions simulation. Critical Rayleigh numbers are obtained numerically for the case of a uniform heat source in a layer with two fixed surfaces. The validity of both the linear instability and global nonlinear energy stability thresholds are tested using a three dimensional simulation. Our results show that the linear threshold accurately predicts the onset of instability in the basic steady state. However, the required time to arrive at the basic steady state increases significantly as the Rayleigh number tends to the linear threshold.

EFFECT OF CHEMICAL REACTION ON MHD FLOW OF A VISCO-ELASTIC FLUID THROUGH POROUS MEDIUM

The effect of heat and mass transfer on free convective flow of a visco-elastic incompressible electrically conducting fluid past a vertical porous plate through a porous medium with time dependant oscillatory permeability and suction in the presence of a uniform transverse magnetic field, heat source and chemical reaction has been studied in this paper. The novelty of the present study is to analyze the effect of chemical reaction, time dependant fluctuative suction and permeability of the medium on a visco-elastic fluid flow. It is interesting to note that presence of sink contributes to oscillatory motion leading to flow instability. Further it is remarked that presence of heat source and low rate of thermal diffusion counteract each other in the presence of reacting species.

Numerical simulation of cooling electronic components mounted in a vertical wall by natural convection

This paper is a numerical work to investigate thermal interactions between heat fluxes generated by electronic devices mounted on a vertical printed circuit board (PCB). In fact, the effect of some parameters such as Grashof number (Gr), the distance (S) between the heat sources and the upper exit distance (Le) was studied. The results show that a regular and uniform heat sources distribution in the inlet is very important in order to take advantage from the leading edge of the boundary layer and to obtain the necessary heat dissipation. The impact of parameters on the heat dissipation, characterized by the Nusselt number, has a different importance. For a Prandtl number Pr = 0.71, the (Gr) increases the heat exchange that is reflected by the increase of Nusselt number, and also participates to the formation of recirculation zones. The total Nusselt number Nu is increased when (Gr) is multiplied by 102. With regards to the distance (S) between the heat sources, the results show that Nu increases about 7% if the distance (S) doubles, and becomes approximately 17.8% when (S) quadruples. At the end, the heat transfer increases when increasing the distance (Le) of the upper exit length, especially on the second component. The total Nusselt number increases by 1.6% when (Le) increases by about 67%.

Characteristics of Natural Convection Heat Transfer in an Array of Discrete Heat Sources

This study aims to investigate experimentally the natural convection flow over an array of discrete surfaces mounted on a flat vertical substrate. The investigation covers the case where the discrete surfaces/substrate unit forms a wall of a closed cavity. Experimental measurements of the flow and thermal fields were conducted. The experimental study is aimed at investigating the influence of the Rayleigh number on the velocity and thermal fields inside the cavity. Two cases of heat source distribution were considered: uniform heat source distribution and non-uniform heat source distribution. Measurements of vertical and horizontal velocity components were obtained for three heating values: 10, 15, and 20 W corresponding to Rayleigh number values of Ra = 9.06 × 109, Ra 1.19 × 1010, and Ra 1.41 × 1010. For each case, the velocity measurements were conducted at three different vertical locations, located at the middle of each of the three heaters. Velocity measurements were also obtained for non-uniform heater distribution at Ra 9.53 × 109. The energy loss was found to be about 10% of the power input in each case. Empirical correlations were obtained for the data for uniformly distributed heaters.

A linear stability analysis of thermal convection in spherical shells with variable radial gravity based on the Tau-Chebyshev method

The onset of thermal convection in a non-rotating spherical shell is investigated using linear theory. The Tau-Chebyshev spectral method is used to integrate the linearized equations. We investigate the onset of thermal convection by considering two cases of the radial gravitational field (i) a local acceleration, acting radially inward, that is proportional to the distance from the center r, and (ii) a radial gravitational central force that is proportional to r−n. The former case has been widely analyzed in the literature, because it constitutes a simplified model that is usually used, in astrophysics and geophysics, and is studied here to validate the numerical method. The latter case was analyzed since the case n=5 has been experimentally realized (by means of the dielectrophoretic effect) under microgravity condition, in the experimental container called GeoFlow, inside the International Space Station. Our study is aimed to clarify the role of (i) a radially inward central force (either proportional to r or to r−n), (ii) a base conductive temperature distribution provided by either a uniform heat source or an imposed temperature difference between outer and inner spheres, and (iii) the aspect ratio η (ratio of the radii of the inner and outer spheres), on the critical Rayleigh number. In all cases the surface of the spheres has been assumed to be rigid. The results obtained with the linear theory based on the Tau-Chebyshev spectral method are compared with those of the integration of the full non-linear equations solved by using the spectral element method. By using the Tau-Chebyshev method, we were able to explore new cases that have not been previously reported in the literature.

Temperature analysis of one-dimensional NiTi shape memory alloys under different loading rates and boundary conditions

Superelastic polycrystalline NiTi shape memory alloys under tensile loading accompany the strain localization and propagation phenomena. Experiments showed that the number of moving phase fronts and the mechanical behavior are very sensitive to the loading rate due to the release/absorption of latent heat and the material’s inherent temperature sensitivity of the transformation stress. In this paper, the moving heat source method based on the heat diffusion equation is used to study the temperature evolution of one-dimensional superelastic NiTi specimen under different loading rates and boundary conditions with moving heat sources or a uniform heat source. Comparisons of temperature variations with different boundary conditions show that the heat exchange at the boundaries plays a major role in the nonuniform temperature profile that directly relates to the localized deformation. Analytical relation between the front temperature of a single phase front, the inherent Clausius–Clapeyron relation (sensitivity of the material’s transformation stress with temperature), heat transfer boundary conditions and the loading rate is established to analyze the nucleation of new phase fronts. Finally, the rate-dependent stress hysteresis is also simply discussed by using the results of temperature analyses.

A canonical model for stratified flow in estuaries and rivers

We present both fully resolved direct numerical simulation and large eddy simulation results for turbulent stratified flow in an open channel with periodic boundaries in the stream-wise direction and no-slip vertical sidewalls in the span-wise direction. A uniform heat source term is applied in the top 20% of the domain, approximating the solar heat flux into a river system. After each time step the total scalar flux input into the domain is uniformly removed allowing a fully developed flow field to be evolved and statistics collected. This approach allows fully developed stratified flow to be simulated with a density profile which includes a lower non-stratified region, steep thermocline and an upper laminar mixed layer region. The Reynolds number for the flows is in the range 5400-7300 and the bulk Richardson number in the range 0-0.4. References R. P. Garg, J. H. Ferziger, S. G. Monismith, and J. R. Koseff. Stably stratified turbulent channel flows. I. Stratification regimes and turbulence suppression mechanism. Phys. Fluids , 12(10):2569--2594, 2000. doi:10.1063/1.1288608 . S. Komori, H. Ueda, F. Ogino, and T. Mizushina. Turbulence structure in stably stratified open-channel flow. J. Fluid Mech. , 130:13--26, 1983. doi:10.1017/S0022112083000944 . Lei Wang and Xi-Yun Lu. Large eddy simulation of stably stratified turbulent open channel flows with low- to high-Prandtl number. Int. J. Heat and Mass Transfer , 48(10):1883 -- 1897, 2005. doi:10.1016/j.ijheatmasstransfer.2004.12.017 . J. R. Taylor, S. Sarkar, and V. Armenio. Large eddy simulation of stably stratified open channel flow. Phys. Fluids , 17(11):116602, 2005. doi:10.1063/1.2130747 . J. S. Turner. A note on wind mixing at the seasonal thermocline. Deep-Sea Res. , 16(Suppl.):297--300, 1969. P. E. Holloway. A criterion for thermal stratification in a wind-mixed System. J. Phys. Oceanogr. , 10:861--869, 1980. 2.0.CO;2>doi:10.1175/1520-0485(1980)010 2.0.CO;2 J. H. Simpson and J. R. Hunter. Fronts in the Irish sea. Nature , 250:404--406, 1974. doi:10.1038/250404a0 . M. Bormans and I. T. Webster. A mixing criterion for turbid rivers. Environmental Modelling and Software with Environment Data News , 12(4):329--333, 1997. doi:10.1016/S1364-8152(97)00032-7 . M. Germano, U. Piomelli, P. Moin, and W. H. Cabot. A dynamic subgrid-scale stress model. Phys. Fluids A , 3(7):1760--1765, 1991. doi:10.1063/1.857955 . S. B. Pope. Turbulent Flows . Cambridge University Press, 2000. S. W. Armfield and R. Street. Fractional step methods for the Navier--Stokes equations on non-staggered grids. ANZIAM Journal , 42:C134--C156, 2000. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/593 . S. W. Armfield, S. E. Norris, P. Morgan, and R. Street. A parallel non-staggered Navier--Stokes solver implemented on a workstation cluster. In S. Armfield, P. Morgan, and K. Srinvas, editors, Proceedings of the Second International Conference on Computational Fluid Dynamics , pages 30--45, Sydney, July 15-19 2002. J. H. Ferziger and M. Peric. Computational Methods for Fluid Dynamics . Springer, third edition, 2002.

Development of Latent Fingerprints on Thermal Paper by the Controlled Application of Heat

Apparatus to produce a spatially and temporally uniform heat source is described and this is used to visualize latent fingerprints deposited onto thermal paper by raising the temperature of the paper. Results show an improvement over previous research when fingerprint deposits are aged or the developed fingerprints faint; visualization being enhanced by the use of a blue LED light source of 465nm peak wavelength. An investigation of the components in fingerprint sweat likely to affect the solubility and hence color change of the dye present in the thermal paper has shown that polar protic solvents able to donate a proton are favored and a polar amino acid found commonly in eccrine fingerprint sweat (lysine) has been shown able to produce the desired color change. Aged fingerprint deposits on thermal paper from a variety of sources up to 4years old have been visualized with this technique.

Convective Motion and Heat Transfer in a Slowly Rotating Fluid Quasi-Sphere With Uniform Heat Source and Axial Gravity

The laminar natural convection of a rotating fluid quasi-sphere in the presence of an axial gravity field and uniform heat source is presented. The influence of the Rayleigh number Ra and the Taylor number Ta on the flow pattern and heat transfer rate from the fluid quasi-sphere is discussed. The governing nonsteady, three-dimensional Navier–Stokes equations for an incompressible fluid, formulated in a Cartesian coordinate system, have been numerically solved by using the h/p spectral element method. It is shown that for a given Ta number, as the Ra number is increased, the heat transfer on the northern hemisphere is enhanced whereas the average Nusselt number on the southern hemisphere is reduced. On the other hand for a given Ra number, as the Ta number is increased, the heat transfer is a function of the convective motion intensity. It has been found that for low and high Ra numbers the heat transfer rate slightly depends on the rotation rate. However at intermediate Ra numbers, the net effect of an increased rotation rate is a reduction of the heat transfer through the wall, hence an increase of the maximum temperature of the fluid sphere is observed. We show that the net effect of the Coriolis force is to damp the convective motion and to allow a redistribution of the vorticity field.

Linear stability of natural convection in a multilayer system of fluid and porous layers with internal heat sources

The onset of thermal natural convection in a horizontal multilayer system consisting of a homogeneous porous layer sandwiched between two fluid layers has been simulated by an accurate numerical method. The porous and fluid layers include uniform heat sources. Flow in the porous medium has been governed by Darcy–Brinkman’s law. On the other hand, the Navier–Stokes equations with the Boussinesq approximation have ruled over the clear fluid layers. The lower and upper rigid surfaces are assumed to be fixed at the equal temperatures TL and TU. The eigenvalues and eigenfunctions of the linear stability analysis have been solved by utilizing the compound matrix method (CMM). The CMM reaches accurate results in a very efficient manner. Moreover, the method removes the stiffness from the equations of the stability system. The results indicate that the onset of convection and the nature of convection cells depend on the relative depths of layers. It has been observed that the thickness of the lower fluid layer increases the critical Rayleigh number of the upper fluid layer and stabilizes it.