Numerical solutions for general inverse heat conduction problem

A family of numerical methods for determining the space-and time-variable heat transfer coefficient, based on experimentally acquired interior temperature-time data, is presented. Newton-type methods are utilized to compute simultaneously the unknown heat transfer coefficient components. To reduce the influence of random errors in the measurement data on the estimated heat transfer coefficients, the noisy data are smoothed using least squares approximation by cubic splines. Three test examples using experimental and random simulated data are used to illustrate the computation efficiency and generality of the present methods.     

Whole time domain solution of inverse heat conduction problem in multi-layer media

The unknown surface conditions in composite media is estimated by minimizing the nonlinear least squares error between the computed and measured temperatures over the whole time domain. This approach shown to be stable, efficient and accurate. The unknown surface conditions are assumed to have an abrupt change at unknown time. A sensitivity analysis is conducted to learn more insight into the nature of difficulties that can be encountered in the estimation of the parameters associated with the inverse problem. The stability and accuracy of the method is demonstrated by several numerical examples which provide very strict test conditions. Received on 7 March 1997     

变几何域传热的表面热流反演方法

变几何域的表面热流反演是一类特殊的热传导逆问题,在再入飞行器烧蚀型防热材料的表面热流反演中具有工程实用价值.本文首先对变几何域传热的正问题计算方法进行了校核验证,然后建立了求解变几何域表面热流反演问题的顺序函数法和共轭梯度法;给出了这两种反演方法的基本思想和算法推导,并针对典型算例进行了仿真.结果表明:两种反演方法都能计算出较好的反演结果,并且算法受测量噪声的影响较小,具有较好的鲁棒性;反演算法能适应不同的几何域变化函数,但几何域变化量的测量误差在表面热流的反演结果中会有较为直接的反映.     

Analytical solution for laminar flow through leaky tube

The laminar flow through a leaky tube is investigated, and the momentum and conservation of energy equations are solved analytically. By using the Hagen-Poiseuille velocity profile and defining unknown functions for the axial and radial velocity components, the pressure and mass transfer equations are obtained, and their profiles are plotted according to different parameters. The results indicate that the axial velocity, the radial velocity, the mass transfer parameter, and the pressure in the tube decrease as the fluid moves along the tube.     

CHAOS-REGULARIZATION HYBRID ALGORITHM FOR NONLINEAR TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEM

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Fundamental solution method for inverse source problem of plate equation

The elastic plate vibration model is studied under the external force. The size of the source term by the given mode of the source and some observations from the body of the plate is determined over a time interval, which is referred to be an inverse source problem of a plate equation. The uniqueness theorem for this problem is stated, and the fundamental solution to the plate equation is derived. In the case that the plate is driven by the harmonic load, the fundamental solution method (FSM) and the Tikhonov regularization technique are used to calculate the source term. Numerical experiments of the Euler-Bernoulli beam and the Kirchhoff-Love plate show that the FSM can work well for practical use, no matter the source term is smooth or piecewise.     

An exact mathematical solution for entrance-region laminar heat transfer with axial conduction

Summary A relatively simple mathematical scheme is proposed by which the entrance-region temperature solution for laminar flow heat transfer with axial conduction can be rigorously obtained.For Poiseuille pipe flow (parabolic velocity profile) with uniform wall heat flux, the accurate first twelve eigenvalues, eigenfunctions, and the coefficients of the series expansion which are required in the temperature solution have been determined for Peclet numbers of 5, 10, 20, 30, 50, and 100. In addition, asymptotic expressions for the eigenvalues and the eigenfunction,R _n (1), are derived. It is possible to use the asymptotic equation to predict, with satisfactory accuracy, even the first few eigenvalues for all the Peclet numbers considered.By employing the computed eigenvalues and the relevant constants, the effect of axial conduction on the entrance-region temperature profile and local Nusselt numbers has been examined and reported.This work was performed under the auspices of the U.S. Atomic Energy Commission.     

Analytical solution of the overdetermined inverse heat conduction problem with an application to monitoring thermal stresses

The paper presents a technique for determining the transient temperature distribution, when input data as thermocouple responses are known at several interior locations. If the temperature field is known, then the thermal stresses can be calculated. The problem is overdetermined and is solved using a least squares method that minimizes the error between the computed and measured thermocouple temperatures. The present method incorporates the advantages of simplicity and accuracy of analytical solutions. Several numerical examples and measurements are presented as an indication of the accuracy of the presented method. Received on 10 January 1997     

Determination of nonuniform heat flux density on boundary in inverse heat conduction problem

This paper develops reducing deviation method to solve inverse heat conduction problem with nonuniform heat flux density on boundary. When the thermocouple is placed a bit far from the boundary, there is no problem on non-convergence in this method. A calculation example is given.     

PRECISE INTEGRAL ALGORITHM BASED SOLUTION FOR TRANSIENT INVERSE HEAT CONDUCTION PROBLEMS WITH MULTI-VARIABLES

IntroductionIHCPs (InverseHeatConductionProblems)arecloselyassociatedwithmanyengineeringaspects,andwelldocumentedintheliteratures,coveringtheidentificationsofthermalparameters[1,2 ],boundaryshapes[3],boundaryconditions[4 ]andsource_relatedterms[5 ,6 ]etc .Howeveritseemsthatonlylittleworkisdirectlyconcernedwithmulti_variablesidentificationsbyauthors’knowledge.Tsengetal.presentedanapproachtodeterminingtwokindsofvariables[7],butonlygavefewnumericalexamplestodeterminethemsimultaneously .Thesol…     

基于数据驱动模型求解热传导反问题

热传导反问题求解在工程领域具有重要的应用价值.本文发展数据驱动模型识别了管道内壁几何形状和皮肤肿瘤生长参数等热传导反问题.在管道内壁几何形状识别问题中,首先采用随机生成模型结合有限元法求解热传导正问题,并采用有效导热系数转化的思想,建立机器学习模型,求解了测点温度与有效导热系数之间的抽象映射关系,进而实现管道内壁几何形...     

A boundary identification method for an inverse heat conduction problem with an application in ironmaking

A boundary identification problem in inverse heat conduction is studied, based on data from internal measurement of temperature and heat flux. Formulated as a sideways heat conduction equation, a spatial continuation technique is applied to extend the solution to a known boundary condition at the desired boundary position. Recording the positions traversed in the continuation for each time instant yields the boundary position trajectory and hence the solution of the identification problem. A prospective application of the method can be found in the ironmaking blast furnace, where it is desired to monitor the thickness of the accreted refractory wall based on measurement of its internal state. Simulations featuring noisy measurement data demonstrate the feasibility of the identification method for blast furnace wall thickness estimation.     

Optimization of heat transfer through finned dissipators cooled by laminar flow

In the present work, the problem of optimizing the shape and the spacing of the fins of a thermal dissipator cooled by a fluid in laminar flow is studied. For a particular finned conduit, the velocity and temperature distributions on the transversal section are determined with the help of a finite element model and a global heat transfer coefficient is calculated. A polynomial lateral profile is proposed for the fins and the geometry is optimized in order to make the heat transfer coefficient as high as possible with the smallest dimensions or the lowest hydraulic resistance to the flow. The optimum fin profile and spacing, obtained by means of a genetic algorithm, are finally shown for different situations. Increases of 45% are obtained in the heat transfer coefficient referring to the maximum values which can be obtained with rectangular fin profiles.     

Pertubation solution of non-newtonian fluid axial laminar flow through eccentric annuli

Using the perturbation method, the axial laminar flow of Non-Newtonian fluid through an eccentric annulus is studied in the present paper. The relative eccentricity ε is taken as a perturbation parameter, and the first order perturbation solutions of the problem, such as velocity field, limit velocity and pressure gradient, are all obtained.     

Solution of the laminar duct flow problem by a boundary integral equation method

Summary A boundary integral equation method is proposed for approximate numerical and exact analytical solutions to fully developed incompressible laminar flow in straight ducts of multiply or simply connected cross-section. It is based on a direct reduction of the problem to the solution of a singular integral equation for the vorticity field in the cross section of the duct. For the numerical solution of the singular integral equation, a simple discretization of it along the cross-section boundary is used. It leads to satisfactory rapid convergency and to accurate results. The concept of hydrodynamic moment of inertia is introduced in order to easily calculate the flow rate, the main velocity, and the fRe-factor. As an example, the exact analytical and, comparatively, the approximate numerical solution of the problem of a circular pipe with two circular rods are presented. In the literature, this is the first non-trivial exact analytical solution of the problem for triply connected cross section domains. The solution to the Saint-Venant torsion problem, as a special case of the laminar duct-flow problem, is herein entirely incorporated.     

Formulation of Gyarmati's principle for heat conduction equation

Gyarmati's principle is formulated in various pictures for the heat conduction phenomenon in solid. Since the heat current density and the internal energy function can be given in three different pictures for heat conduction phenomena, we get the nine forms of the principle from which the heat conduction equation can be derived. This formulation has been shown using the generalized picture. In the subsequent section the principle is formulated in proper picture from which three proper pictures namely Fourier, entropy and energy follow.

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Formulierung des Prinzips von Gyarmati für Wärmeleitprobleme

Zusammenfassung Das Prinzip von Gyarmati wird in verschiedenen Arten für Wärmeleitphänomene formuliert. Da die Wärmestromdichte und die innere Energie in drei verschiedenen Arten für Wärmeleitphänomene angegeben werden können, erhalten wir die neun Formen des Prinzips, von denen die Wärmeleitgleichung abgeleitet werden kann. In dieser Formulierung wird ein verallgemeinertes -Bild verwendet. Im folgenden Teil wird das Prinzip in einem geeigneten -Bild formuliert, von dem drei geeignete Bilder folgen, nämlich das Fourier-, das Entropie- und das Energie-Bild.

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Nomenclature rate of entropy production - dissipation potential function of thermodynamic forces only - dissipation potential function of fluxes only - v volume of the system - x _i thermodynamic forces - J _i thermodynamic currents - f number of irreversible processes taking place in the system - L_iK phenomenological coefficients representing conductivity of the material - R_iK phenomenological coefficients representing resistances - density of the material - a specific value of the extensive transport quantity - _i state parameters, the gradients of which give rise to the thermodynamic forces - _i source density of a_i - s specific entropy - T absolute temperature - J _q heat current density vector - heat conductivity coefficient - L_qq phenomenological coefficient corresponding to heat conductivity coefficient - x _q thermal dissipative force - _q entropy production due to heat transfer - u specific internal energy - L phenomenological coefficient in picture - c_V specific heat at constant volume     

A note on heat transfer in laminar flow through a gap

Summary The problem of heat transfer in laminar flow through a gap between two semi-infinite parallel plates at constant temperature was recently studied by Agrawal¹⁾. He solved this problem with the use of infinite Fourier sine series and derived an expression for the local temperature profile and the local Nusselt number as a function of the distance along the gap. A detailed solution for Péclèt number Pe=1 is given. Far enough from the entrance of the gap the local temperature profile of the fluid is almost independent of it's initial temperature. In this paper this limit temperature profile is expressed with the confluent hypergeometric function and the corresponding Nusselt number as a function of Pe is calculated.     

A semi-numerical method for solving inverse heat conduction problems

A semi-numerical method is presented for solving the inverse heat conduction problems in homogeneous and composite bodies. The presented solution does not require both the initial temperature distribution in the body and the whole temperature-time history at the temperature sensor locations. Sample calculations confirm that this approach produces stable and accurate results for both exact and noisy data. The extension of the method presented to two or three dimensions is straightforward.