Solution Of Heat Flow Research Articles

A simple calibration procedure to determine thermal effusivity values measured by using a thermal microscope

A thermal microscope is a useful tool for investigating the spatial distribution of the thermal transport properties of materials. However, for materials with relatively high thermal effusivities, it is well known that the values calculated on the basis of the conventional one-dimensional heat flow solution are higher than the values provided in the literature. In this study, we developed a simple calibration procedure for a thermal microscope to measure the thermal effusivities of materials by using several reference materials whose thermal effusivities are known. It is expected that the temperature response will be influenced by not only the thermal effusivity but also the heat capacity per unit volume. However, reference samples with different heat capacities per unit volume were used. In comparison with the calculated values obtained with the conventional one-dimensional heat flow solution, the values for pure iron obtained with our calibration procedure, without considering the heat capacity per unit volume, were closer to the values provided in the literature. We found that this procedure is useful for calibrating a thermal microscope easily for measuring thermal effusivity.

Similarity Solutions for Hydromagnetic Free Convective Heat and Mass Transfer Flow along a Semi-Infinite Permeable Inclined Flat Plate with Heat Generation and Thermophoresis

The problem of steady, two-dimensional, laminar, hydromagnetic flow with heat and mass transfer over a semi-infinite, permeable inclined flat plate in the presence of thermophoresis and heat generation is studied numerically. A similarity transformation is used to reduce the governing non-linear partial differential equations into ordinary ones. The obtained locally similar equations are then solved numerically by applying Nachtsheim-Swigert shooting iteration technique with sixth-order Runge-Kutta integration scheme. Comparisons with previously published work are performed and the results are found to be in very good agreement. Numerical results for the dimensionless velocity, temperature and concentration profiles as well as for the skin-friction coefficient, wall heat transfer and particle deposition rate are obtained and reported graphically for various values of the parameters entering into the problem.

A numerical model for the thermocapillary flow and heat transfer in a thin liquid film on a microstructured wall

PurposeThis paper aims to study thermocapillarity‐induced flow of thin liquid films covering heated horizontal walls with 2D topography.Design/methodology/approachA numerical model based on the 2D solution of heat and fluid flow within the liquid film, the gas above the film and the structured wall is developed. The full Navier‐Stokes equations are solved and coupled with the energy equation by a finite difference algorithm. The movable gas‐liquid interface is tracked by means of the volume‐of‐fluid method. The model is validated by comparison with theoretical and experimental data showing a good agreement.FindingsIt is demonstrated that convective motion within a film on a structured wall exists at any nonzero Marangoni number. The motion is caused by surface tension gradients induced by temperature differences at the gas‐liquid interface due to the spatial structure of the heated wall. These simulations predict that the maximal flow velocity is practically independent from the film thickness, and increases with increasing temperature difference between the wall and the surrounding gas. It is found that an abrupt change in wall temperature causes rupture of the liquid film. The thermocapillary convection notably enhances heat transfer in liquid films on heated structured walls.Research limitations/implicationsOur solutions are restricted to the case of periodic wall structure, and the flow is enforced to be periodic with a period equal to that of the wall.Practical implicationsThe reported results are useful for design of the heat transfer equipment.Originality/valueNew effects in thermocapillary convection are presented and studied using a developed numerical model.

Nonsimilar Solutions for Heat and Mass Transfer Flow in an Electrically Conducting Viscoelastic Fluid over a Stretching Sheet Saturated in a Porous Medium with Suction/Blowing

Nonsimilar Solutions for Heat and Mass Transfer Flow in an Electrically Conducting Viscoelastic Fluid over a Stretching Sheet Saturated in a Porous Medium with Suction/Blowing

Coupled solution of heat and moisture flow in unsaturated clay barriers in a repository geometry

AbstractAn analytical solution in the Laplace transform domain is obtained for the transient heat and moisture transport in an unsaturated clay buffer with a geometry simulating repository conditions. A numerical inversion scheme based on Crump's method is used to obtain the time‐domain solution. The coupled effect of thermally driven moisture transport is especially investigated because of its importance to alter the flow field in low‐permeability buffers. The practical background is based on the case of an engineering bentonite barrier placed in a drift excavated in rock in the context of underground disposal of high‐level radioactive waste. Parametric study has been performed to assess the effects of dimensionless geometry and material parameters on flow field. Despite the simplified assumptions required in order to obtain analytical expressions, the results incorporate the main mechanisms involved in the coupled thermo‐hydraulic (T–H) problem, and they may be eventually used for validation purposes. Copyright © 2006 John Wiley & Sons, Ltd.

Traveling wave solutions of harmonic heat flow

We prove the existence of a traveling wave solution of the equation \( u_t=\Delta u+|\nabla u|^2u \) in an infinitely long cylinder of radius R, which connects two locally stable and axially symmetric steady states at x3 = ±∞. Here u is a director field with values in \( {\mathbb{S}}^2 \subset {\mathbb{R}}^3 : |u|=1 \) The traveling wave has a singular point on the cylinder axis. Letting R→ ∞ we obtain a traveling wave defined in all space.

Periodic solution of transient heat flow through multilayer walls and flat roofs by complex finite Fourier transform technique

An analytical solution method for the estimation of space heat gain through multilayer walls and flat roofs submitted to periodic boundary conditions is developed. Formulation of the transient heat transfer problem consisting of differential equation and the initial and boundary conditions is converted into dimensionless form and it is solved by the application of complex finite Fourier transform (CFFT) technique. This analytical solution is used to determine hourly variation of heat flow into the space through the walls and the temperature at the inner surface of the walls. A computer program is developed based on the analytical model to perform numerical calculations considering six different wall and two different roof constructions, characteristics to the buildings in Turkey. The CFFT technique provides itself as an alternative to other analytical and numerical methods and it appears to be useful in that it allows the calculations to be performed for various multilayer wall and roof constructions and for various climatological locations with varying ambient air temperatures and solar heat inputs.

A remark on the finite time singularity of the heat flow for harmonic maps

In this paper we study the finite time singularities for the solution of the heat flow for harmonic maps. We derive a gradient estimate for the solution across a finite time singularity. In particular, we find that the solution is asymptotically radial around the isolated singular point in space at a finite singular time. It would be more desirable to understand whether the solution is continuous in space at a finite singular time.

Effect of gap size and spacing on heat conduction between materials with irregularly spaced plane strip contacts

A previous paper developed solutions for conductive heat flow in the x, y plane where there is a uniform flow at infinite values of y and where y = 0 represents the interface between two materials. This interface consists of an arbitrary pattern of perfectly conducting strips and non-conducting gaps. It was assumed that thermoelastic effects are negligible. The solution took the form of an equation with one unknown parameter per gap, which can be found by a simple process involving numerical integration. Using this method, the present paper gives solutions for a range of gap sizes and spacings. These are compared with approximate solutions determined by simplifying the general solutions sufficiently for them to be solved analytically. It is shown that the sizes of the gaps are the dominant parameters. The spacings are relatively unimportant unless two gaps are very close. The extent to which the approximate forms can be used to give solutions for random patterns of gaps, using the statistics of gap sizes, is discussed.

Harmonic and quasi-harmonic spheres, part III. Rectifiablity of the parabolic defect measure and generalized varifold flows

We study weakly convergent sequences of suitable weak solutions of heat flows of harmonic maps or approximated harmonic maps. We prove a dimensional stratification for the space-time concentration measure and verify that the concentration measure, viewed as a generalized varifold, moves according to the generalized varifold flow rule which reduces to the Brakke's flow of varifold provided that the limiting harmonic map flow is suitable. We also establish an energy quantization for the density of the limiting varifold.

Transient 3D heat flow analysis for integrated circuit devices using the transmission line matrix method on a quad tree mesh

This paper presents a 3D transmission line matrix (TLM) implementation for the solution of transient heat flow in integrated semiconductor devices. The implementation uses a rectangular discontinuous mesh to allow for local mesh refinement. This approach is based on a quad tree meshing technique which can have a complex geometry using blocks of varying sizes. Each such block can have a maximum of two adjacent blocks on any vertical side and a maximum of four blocks on the top or bottom.The TLM implementation is based on a physical extraction of a resistance and capacitance network and then the creation of the appropriate TLM matrix. The formulation allows for temperature-dependent material parameters and a non-uniform time stepping.The simulator is first tested using a 2D example of a heat source in a rectangular region. Using this example the numerical error is determined and found to be less than 0.4%. Next, non-linearities are included, and a number of non-uniform time stepping algorithms are tested. Then, a 3D problem is also compared to an analytical solution and again the error is very small. Finally, an example of a full solution of heat flow in a realistic Si trench device is presented.

A 3D thermal simulation tool for integrated devices-Atar

This paper presents a novel three-dimensional (3D) thermal simulation tool for semiconductor integrated devices. The simulator is used to automatically generate an accurate 3D physical model of the device to be simulated from layout information. The simulator produces an appropriate mesh of the device based on a rectangular block structure. The mesh is automatically created such that a fine mesh is produced around heat generation regions, but a moderate number of blocks are used for the entire device. This paper first confirms that the simulator produces an accurate solution to the nonlinear differential equation describing the heat flow. Then model generation from three example technologies (silicon trench, GaAs mesa structures, silicon on insulator) is presented. The potential of the simulator to quickly and easily explore the effect of layout and process variations is illustrated, with the simulation of a two-transistor GaAs power cell as a large example. The program incorporates a transient solver based on a transmission line matrix (TLM) implementation using a physical extraction of a resistance and capacitance network. The formulation allows for temperature dependent material parameters and a nonuniform time stepping. An example of a full transient solution of heat flow in a realistic Si trench device is presented.

Shear Heating in Granular Layers

—Heat-flow measurements imply that the San Andreas Fault operates at lower shear stresses than generally predicted from laboratory friction data. This suggests that a dramatic weakening effect or reduced heat production occur during dynamic slip. Numerical studies intimate that grain rolling or localization may cause weakening or reduced heating, however laboratory evidence for these effects are sparse. We directly measure frictional resistance (μ), shear heating and microstructural evolution with accumulated strain in layers of quartz powder sheared at a range of effective stresses (σ n = 5 - 70 MPa) and sliding velocities (V = 0.01 - 10 mm/s). Tests conducted at σ n ≥ 25 MPa show strong evidence for shear localization due to intense grain fracture. In contrast, tests conducted at low effective stress (σ n = 5 MPa) show no preferential fabric development and minimal grain fracture hence we conclude that non-destructive processes such as grain rolling/sliding, distributed throughout the layer, dominate deformation. Temperature measured close to the fault increases systematically with σ n and V, consistent with a one-dimensional heat-flow solution for frictional heating in a finite width layer. Mechanical results indicate stable sliding $(\mu \sim 0.6)$ for all tests, irrespective of deformation regime, and show no evidence for reduced frictional resistance at rapid slip or high effective stresses. Our measurements verify that the heat production equation $(q =\mu \sigma_n V)$ holds regardless of localization state or fracture regime. Thus, for quasistatic velocities (V≤ 10 mm/s) and effective stresses relevant to earthquake rupture, neither grain rolling/sliding or shear localization appear to be a viable mechanism for the dramatic weakening or reduced heating required to explain the heat flow paradox.

Analytical solutions for heat flow in multiple pass welding

The purpose of this paper is to present analytical solutions for the heat distribution in the base metal during and after serial heating in multiple pass arc welding. Recent literature on welding heat transfer has covered extensively numerical models for different aspects of the process. Most of these aspects are non-linear problems in heat flow, e.g. varying thermal properties, inclusion of radiative transfer in the arc, and/or convective flow in the welding pool. The basic analytical models in use were established long ago in classical papers, but these refer to single heat cycles. The present paper addresses the circumstances of multiple cycles. In these cases, depending on the bead length and the interpass time, as shown by the proposed solution, the thermal effects of each successive pass will accumulate. When modelling the heat transfer in each pass, the effects of previous passes enter into the differential equations as non-homogeneous initial conditions. The complexity introduced by these conditions requires special treatment: an analytical solution employing the Green’s function method was used. The main focus in this work is on welding of thin plates. Simple changes to the models take account of the convective heat loss to the surroundings and enable temperature dependent properties to be calculated. The solutions were checked against experimental results obtained from a specially designed laboratory setup. Thermal cycles provided by the analytical solutions compare well with temperature histories measured at different locations during three pass gas metal arc welding of a 0.5 in (∼1 cm) AISI 304 stainless steel plate. Measured data and model results show very good agreement. The analytical solutions are also extended to the geometries of moderately thick plates.

The Impurity Effect in the Space Shuttle Dendritic Growth Experiments with Succinonitrile

The high purity succinonitrile (SCN) used in the recent space shuttle dendritic growth experiments of Glicksman and co-workers was reported to be 99.999% pure. This material is treated as a very dilute alloy with a composition C o = 0.001%. The dendrite tip temperature ( T t) is determined from both heat flow and interfacial equilibrium considerations. Only the latter is affected by the presence of impurities. For the experimentally measured values of the tip growth rate ( R) and tip radius ( r t), the values of T t estimated on the basis of the Ivantsov heat flow solution are in excess of the equilibrium melting point of succinonitrile. A consideration of the impurity effect suggests that the microgravity experiments correspond to what is considered to be the rapid solidification regime. The chemical Peclet number, p = Rr t/2 D L, where D L is the diffusivity of the impurity, is greater than one-half (p > 0.50) in a large number of experiments and greater than one ( p > 1) in some experiments.

Wall transient heat flow using time-domain analysis

If values of ambient temperature and room temperature are known at hourly intervals, the heat flow to or from the room at some specified time can be computed in terms of the values of recent temperatures and heat flows, weighted by values of a series of N wall transfer coefficients, b k, or c k, and d k, ( k = 0 to N). These coefficients have hitherto been found by frequency domain methods. It is shown here that they can be evaluated using elementary time domain solutions for wall heat flow. A multi-layer wall with surface films representing radiant and convective exchange is discussed. Using suitable transmission matrices, solutions for the steady-progressive and transient temperature profiles through the wall are derived. The value of N and the wall d k values follow immediately from the wall decay times of the transient solution, together with the choice of 1 hour as time interval. Using a development of Fourier analysis, the steady-progressive and transient solutions are combined so as to provide the time-domain solution for the response to an imposed ramp in ambient or room temperature. Combination of three such ramps forms a triangular temperature pulse of excitation and the infinite series of wall response factors—hourly values of heat flow at and after the temperature peak—serve to describe the response of the wall when excited by such a pulse. The values of b k and c k follow from the response factors. The procedure is illustrated numerically.

Bubbling of the heat flows for harmonic maps from surfaces

In this article we prove that any Palais-Smale sequence of the energy functional on surfaces with uniformly L2-bounded tension fields converges pointwise, by taking a subsequence if necessary, to a map from connected (possibly singular) surfaces, which consist of the original surfaces and finitely many bubble trees. We therefore get the corresponding results about how the solutions of heat flow for harmonic maps from surfaces form singularities at infinite time. © 1997 John Wiley & Sons, Inc.

A remark onp-harmonic heat flows

IN this note we prove the existence of the global weak solutions of p-harmonic heat flows, in particular 1 1 (see Lemma 2), and it is in itself a significance.

Solution For Heat Flow in Soil with a Heat Source at a Fixed Depth

AbstractWe developed a computationally efficient model for estimating the transient soil temperature distributions resulting from a heat source at a fixed depth beneath a greenhouse. The model was developed for use as a submodel of the air‐soil systems in greenhouses where the soil is used as a heat storage medium, with a network of buried pipes acting as a heat exchanger. The model assumes soil thermal properties are spatially and temporally constant and that energy transfer between the greenhouse and soil is primarily a result of net radiation and sensible heat transfer. First‐order energy transfer terms are also used to account for lateral energy exchanges with the surrounding soil and the heat source is included as a plane at a fixed depth. This was achieved by obtaining a Green's function solution in the Laplace domain and performing a numerical inversion with a fast Fourier‐transform algorithm, which requires very little computational time. Model parameters are optimized using a Simplex algorithm during a 3‐d simulation to show the ability of the method to describe the greenhouse soil temperature. This method provides a reasonable description of the heat flux under these conditions.

Two-Dimensional Parallel Computing Solution of Coupled Heat and Moisture Flow in Unsaturated Soil

A two-dimensional parallel algorithm for coupled heat and moisture transfer in unsaturated soil is presented. A theoretical formulation, which was recently published in the literature is adopted for the work. The algorithm, based on a numerical finite difference explicit solution scheme, is programmed in the concurrent language Occam and executed on a parallel computing network of transputers. An assessment of the two-dimensional algorithm in terms of its overall computational performance is also performed, together with an investigation of its accuracy. The ability of the numerical algorithm to fully model the complex, interrelated, coupled nature of the two-dimensional transient and steady-state phenomenon was demonstrated, including the gravitational effects. The computational efficiency of the approach was found to be very high. Further, a superior computational performance was achieved when applying the two-dimensional approach to a larger network of processors as well as to a larger size of problem. The accuracy of the model was also established by comparison of results with values produced by an alternative, completely independent numerical approach. Excellent correlation was obtained.