Multiple transient point heat sources identification in heat diffusion: application to experimental 2D problems

This paper deals with an inverse problem, which consists of the experimental identification of line heat sources in a homogeneous solid in transient heat conduction. The location and strength of the line heat sources are both unknown. For a single source we examine the case of a source which moves in the system during the experiment. The identification procedure is based on a boundary integral formulation using transient fundamental solutions. The discretized problem is non-linear if the location of the line heat sources is unknown. In order to solve the problem we use an iterative procedure to minimize a cost function comparing the modelled heat source term and the measurements. The proposed numerical approach is applied to experimental 2D examples using measurements provided by an infrared scanner for surface temperatures and heat fluxes. In some particular examples, internal thermocouples can be used. A time regularization procedure associated to future time-steps is used to correctly solve the ill-posed problem.     

The BEM for point heat source estimation: application to multiple static sources and moving sources

This paper deals with an inverse problem, which consists of the identification of point heat sources in a homogeneous solid in transient heat conduction. The location and strength of the line heat sources are both unknown. For a single source we examine the case of a source which moves in the system during the experiment. The two-dimensional and three-dimensional linear heat conduction problems are considered here. The identification procedure is based on a boundary integral formulation using transient fundamental solutions. The discretized problem is non-linear if the location of the line heat sources is unknown. In order to solve the problem we use an iterative procedure to minimize a quadratic norm. The proposed numerical approach is applied to experimental 2D examples using measurements provided by an infrared scanner for surface temperatures and heat fluxes. A numerical example is presented for the 3D application.     

A parameter estimation approach to solve the inverse problem of point heat sources identification

This paper deals with an inverse problem that consists of the identification of multiple line heat sources placed in a homogeneous domain. In the inverse problem under investigation the location and strength of the line heat sources are unknown. The estimation procedure is based on the boundary element method. As the discrete problem is non-linear if the location of the line heat sources is unknown, an iterative procedure has to be applied to find out the location of the sources. The proposed approach has been tested for steady and transient experiments. In the present study we propose an original approach to solve the steady problem. As in the steady heat conduction case we have a limited number of unknown for each source, a “parameter estimation” approach can be applied to estimate the sources. Using the techniques of parameter estimation, we can also estimate the confidence interval of the estimated locations, which permits to design an optimal experiment. We intend to present some numerical and experimental 2D results.     

MIXED CONVECTION IN A RECTANGULAR ENCLOSURE WITH DISCRETE HEAT SOURCES

A numerical study is carried out on mixed convective heat transfer in an enclosure. The discrete heat sources are embedded on a vertical board, which is situated on the bottom wall of an enclosure. An external airflow enters the enclosure through an opening in one vertical wall and exits from another opening in the opposite wall. This study simulates a practical system, such as air-cooled electronic devices with heated elements. Emphasis is placed on the influence of the governing parameters, such as Reynolds number, Re, buoyancy parameter, Gr/Re², location of the heat sources, and the conductivity ratio, rk, on the thermal phenomenon in the enclosure. The coupled equations of the simulated model are solved numerically using the cubic spline collocation method. The computational results indicate that both the thermal field and the average Nusselt number (Nu) depend strongly on the governing parameters, position of the heat sources, as well as the property of the heat-source-embedded board.     

TRANSIENT HEAT-TRANSFER PHENOMENON OF TWO-DIMENSIONAL HYPERBOLIC HEAT CONDUCTION PROBLEM

This article presents a numerical analysis of the two-dimensional hyperbolic heat conduction problem in an anisotropic medium under a point heat source with different boundary conditions. A simple model has been developed to solve the anisotropic problem. In this analysis, the second-order total variation diminishing (TVD) scheme is employed to solve this problem. The effects of boundary conditions and anisotropy on the thermal wave induced by different types of heat sources in the medium are examined in detail. The results show that the transient behavior of the propagation of the two-dimensional thermal wave is muck more complicated than that of the one-dimensional thermal wave due to a circular wave formed to propagate uniformly in alt directions, reflections by boundaries, interaction with each other, and serious discontinuity on the wavefront.     

A NONIDEAL CONTACT PROBLEM OF THERMOELASTICITY FOR TWO SOLIDS WITH A HEAT SOURCE

A generalized Hertz problem of thermoelastic solids in pressure, contacting over convex surfaces, has been discussed. At a certain moment of time a concentrated source of heat starts acting. Therefore the heat flux flowing through the region of contact is nonstationary. The problem considered is axially symmetric. The purpose is twofold: first the problem of thermoelasticity with time variation of temperature is taken into account; second the “paradox of a cooled sphere” has been investigated under time-dependent conditions. There is a possibility that the character of the boundary conditions can change in time. To obtain the solution we have applied the Laplace and Hankel integral transforms. The main point is to discuss the cases when the boundary conditions are such that the problem can be considered in terms of classical thermoelasticity and when the Barber-type boundary conditions have to be used. The solution has been obtained by means of a devised numerical algorithm such that the procedure is simplified. The results have been presented in diagram form suitable for discussion.     

THE METHOD OF CONCENTRATED SOURCES IN HEAT CONDUCTION

In this article a method of an approximate solution of a parabolic initial boundary value problem is presented. As an example the heat conduction equation is considered. This method consists of introducing some fictitious heat sources outside the region under consideration and choosing some points at the boundary of the region where the boundary conditions are strictly satisfied. The intensities of the sources are assumed to vary stepwise in time. The method may also be applied to the solution of inverse problems of heat conduction. Numerical results are given for some illustrative problems.     

NUMERICAL MODELING OF CONJUGATE HEAT TRANSFER DURING IMPINGEMENT OF FREE LIQUID JET ISSUING FROM A SLOT NOZZLE

The conjugate heat transfer from discrete heat sources to a two-dimensional jet of a high Prandtl number fluid discharging from a slot nozzle is considered. The variation of solid and fluid properties with temperature was taken into account in the numerical simulation. The geometry of the free surface was determined iteratively. The influence of different operating parameters such as jet velocity, heat flux, plate thickness, plate material, and the location of the heat generating electronics were investigated. It was found that in addition to jet Reynolds number (Re) plate thickness and its thermal conductivity have significant influence on temperature distribution and average Nusselt number (Nu).     

INVERSE FORCED CONVECTION PROBLEM OF SIMULTANEOUS ESTIMATION OF TWO BOUNDARY HEAT FLUXES IN IRREGULARLY SHAPED CHANNELS

This article deals with the use of the conjugate gradient method of function estimation for the simultaneous identification of two unknown boundary heat fluxes in channels with laminar flows. The irregularly shaped channel in the physical domain is transformed into a parallel plate channel in the computational domain by using an elliptic scheme of numerical grid generation. The direct problem, as well as the auxiliary problems and the gradient equations, required for the solution of the inverse problem with the conjugate gradient method are formulated in terms of generalized boundary-fitted coordinates. Therefore, the solution approach presented here can be readily applied to forced convection boundary inverse problems in channels of any shape. Direct and auxiliary problems are solved with finite volumes. The numerical solution for the direct problem is validated by comparing the results obtained here with benchmark solutions for smoothly expanding channels. Simulated temperature measurements containing random errors are used in the inverse analysis for strict cases involving functional forms with discontinuities and sharp corners for the unknown functions. The estimation of three different types of inverse problems are addressed in the paper: (i) time-dependent heat fluxes; (ii) spatially dependent heat fluxes; and (iii) time and spatially dependent heat fluxes.     

ANALYSIS/FINITE-ELEMENT COMBINED METHODOLOGY ON TEMPERATURE DISTRIBUTION OF A FINITE DOMAIN WITH VARIOUS HEAT SOURCES

Abstract

A new method, involving the combined use of analysis and the finite-element method, is applicable to the heat conduction problem with isolated heat sources. Unlike the finite-element method, the analysis/finite-element combined method is able to discretize the distributed sources with discontinuities into course elements, and the solution is still calculated accurately. The results are compared in tables with exact solutions and other numerical data, and the agreement is found to be good.     

ANALYSIS/FINITE-ELEMENT COMBINED METHODOLOGY ON TRANSIENT HEAT CONDUCTION OF A FINITE DOMAIN WITH PLANAR-DISCRETE SOURCES

Abstract

A useful method, involving the combined use of the analysis and the finite-element methods, is successfully extended to the transient heat conduction problem with isolated heat sources. The results are compared in tables with exact solutions and other numerical data, and the agreement is found to be good. Previously reported analysis /finite-element combined method has been confined to the slow convergence in series solution of analytical method. By using the third Aitken's delta-squared process for accelerating the convergence of infinite series, this restriction is removed, and the new method provides a more powerful solution to transient problems with heat sources     

BEM SOLUTION TO TRANSIENT FREE CONVECTIVE HEAT TRANSFER IN A VISCOUS,ELECTRICALLY CONDUCTING,AND HEAT GENERATING FLUID

The nonlinear partial differential equations for the transient free convective heat transfer in a viscous, electrically conducting, and heat-generating fluid past a vertical porous plate in the presence of free stream oscillations are solved by the boundary element method (BEM). Time-dependent fundamental solutions are employed in a time marching scheme to resolve the field variables. Numerical results are compared with previously reported analytical solutions in order to validate the developed BEM algorithm. These previous studies reported results for simpler versions of our problem, in which the convective effects in the momentum and energy equations were neglected in order to obtain analytical numerical solutions. Our BEM results are shown to be in close agreement with the reported data. The effects of convection currents, the temperature-dependent heat sources (or sinks), the magnetic currents, and the viscous dissipation on the flow and heat transfer characteristics are assessed in a parametric study, which considers a variety of the dimensionless parameters Gr, Ec, Pr, M, and γ. It is observed that γ plays an important role in delaying the fluid flow reversal, present in the case of air, and acts to enhance the effect of Gr in augmenting the rate of heat transfer at the wall. The skin friction is observed to be an increasing function of Gr, Ec, and γ and a decreasing function of M and Pr. However, the rate of heat transfer (in an absolute sense) is an increasing function of M, γ, Gr, and Ec and a decreasing function of Pr, Of all the parameters, the Prandtl number has the strongest effect on the flow and heat transfer characteristics.     

WELL-CONDITIONED NUMERICAL METHOD FOR THE NONLINEAR INVERSE HEAT CONDUCTION PROBLEM

A numerical procedure recently presented by the authors for linear inverse heat conduction problems is extended to nonlinear cases. The method is well conditioned in the sense that it always generates bounded solutions and never generates heat fluxes oscillating with increasing amplitude. The convergence of the algorithm is studied, and a set of examples shows convergence for a wide range of nonlinear problems. In the case of perfect data the accuracy of the estimated heat fluxes increases as the time step is decreased without need for additional stabilization or regularization. Filters allow perturbed data to be treated and facilitate attaining a balance between accuracy and resolving power.     

COMPUTER-AIDED CALCULATION OF APPROXIMATE CLOSED-FORM SOLUTIONS FOR LINEAR TRANSIENT HEAT CONDUCTION PROBLEMS IN MULTILAYERED BODIES

A solution procedure, based on the Kantorovich method, is presented for the derivation of approximate closed-form solutions for linear heat conduction problems in multilayered plane and cylindrical bodies using computers. Constant or space dependent initial conditions; linear time dependent boundary conditions of the first, second, or third kind; contact resistances between the layers; and a homogeneous distributed, time dependent volumetric heat source can be considered. The solution procedure is shown suitable for programming. In order to assess the approximate solution obtained, an error criterion is stated. The accuracy of the method is investigated through a numerical and an analytical example.     

NUMERICAL SIMULATION OF HEAT TRANSFER IN MIST FLOW

A numerical simulation of heat transfer over a row of tubes, in the presence of mist flow, is described. Computations include the solution of the flow field around the tubes, the prediction of the motion of water droplets, and the evaluation of the cooling effect of the water film on the tube surface. The entire analysis is carried out using FENSAP-ICE (Finite Element Navier-Stokes Analysis Package for In-flight icing), a simulation system developed by Newmerical Technologies for icing applications. The numerical model is described, including the Navier-Stokes solution, the water thin film computation, the droplet impingement prediction, and the conjugate heat transfer procedure. The predictions are verified against experimental data for different droplet mass flow rates, showing satisfactory agreement and allowing a useful insight in the physical characteristics of the problem.     

THERMOELASTIC PROBLEM OF ARC CRACK WITH HEAT SOURCE OR DOUBLET

The problem of a circular arc-shaped crack in an infinite plate under the action of a heat source or doublet is solved by using the complex variable function method. The original problem can be considered a superposition of three particular problems. The first problem relates to a heat source applied at the origin point (z = 0) of an infinite continuous medium. In the second problem, the heat flow passes through the arc crack face. In the third problem, tractions are applied on the arc crack face. All three problems can be solved in a closed form. When heat sources are present, the behavior of the complex potentials is studied in detail. Finally, the solution for the original problem is obtained.     

FINITE ANALYTIC NUMERICAL METHOD FOR TRANSIENT HEAT DIFFUSION IN LAYERED COMPOSITE MATERIALS

The finite analytic (FA) method, which has been recently developed and used in fluid flow and heat transfer problems, is presented and extended to the solution of the problem of transient heat conduction in a one-dimensional muMlayered composite slab. The basic idea of the FA method is to incorporate a local analytic solution of the governing equation in the numerical solution of the boundary-value problem. In thii: study, the local analytic solutions are obtained by the standard integral-transformation technique and the associated eigenvalue problem is solved by the Galerkin method. Some numerical examples are computed to demonstrate the applicability of the FA method in dealing with heat diffusion in a composite material. It is shown that the FA method is accurate and provides reductions in computational costs.     

有限热源的热机供热率与效率关系的有限时间热力学分析

讨论了有限热源热机的供热率与效率关系。引人一种完善内可逆模型的方法，对高温热源侧与低温热源侧的传热优化问题进行了分析，得到一些新结论，它对太阳能热机、核动力装置和地热发电装置等热机设计均有一定指导意义。     

太阳能热动力发电系统吸热器换热管试验及数值模拟

吸热器换热管地面试验是空间太阳能热动力发电系统吸热／蓄热器研制的重要阶段，是为了验证相变材料的蓄放热性能。对以共晶盐LiF—CaF2为蓄热介质的换热管进行了27个周期共2511分钟的地面试验，包括变参数试验和稳态试验，获得了容器表面温度和工质出口温度等试验结果。利用焓法建立相变蓄热换热管试验的传热模型，采用试验参数对地面试验进行了数值模拟，得到的结果与试验结果进行了比较，两者比较接近。证明了地面单管试验的成功性，也验证了换热管传热分析软件的可靠性。     

STATE-SPACE FORMULATION TO GENERALIZED THERMOVISCOELASTICITY WITH THERMAL RELAXATION

The model of the equation of generalized thermoviscoelasticity with one relaxation time is established. The state-space formulation for these equations is introduced. The formulation is valid for problems with or without heat sources. The resulting formulation together with the Laplace transform technique is applied to a variety of problems. The solutions to a thermal shock problem and to a problem of layer media, both without heat sources, are obtained. Also, the effects of a plane distribution of heat sources on the whole and semi-space are studied. A numerical method is employed for the inversion of the Laplace transforms. Numerical results are given and illustrated graphically for each problem. Comparisons are made with the results predicted by the coupled theory.