An improved lumped analysis for transient heat conduction by using the polynomial approximation method

In this study, unsteady state one-dimensional heat conduction is analyzed using a polynomial approximation method. As a classical lumped model is only applicable for use with Biot numbers of less than 0.1, and additionally, it cannot be used for high-temperature gradients within the region, an improved lumped model is implemented for a typical long slab, long cylinder and sphere. It has been shown that in comparison to a finite difference solution, the improved model is able to calculate average temperature as a function of time for higher value of Biot numbers. The comparison also shows that the presented model has better accuracy when compared with others recently developed models. The simplified relations obtained in this study can be used for engineering calculations in many conditions.     

Approximate analysis for transient heat conduction with radiation in a slab

In an attempt to minimize the numerical computations associated with the solution of transient heat conduction with radiation in a slab, a perturbation type of analysis is being applied to the temperature field and radiation heat flux simultaneously. The resulting partial differential equations for the perturbation functions for the temperature are solved in explicit forms by use of the energy integral methods, while the radiation heat flux is determined by an appropriate scheme of approximating the temperature distribution in the slab. Included in the analysis are the effects of the parameters: the optical thickness, the ratio of conduction transport to radiation and the wall emissivity. It is found that, in a wide range of these governing parameters, the results compare very favorably with those obtained by the numerical solution of the formulated integro-differential equation. With the present analysis, the temperature, conduction and radiation heat fluxes can be predicted without resorting to lengthy numerical analysis.     

An analytic model for transient heat conduction in bi-layered structures with flexible serpentine heaters

Uniform heating of complex surfaces, especially non-developable surfaces, is a crucial problem that traditional rigid heaters cannot solve. Inspired by flexible electronic devices, a novel design for the stretchable heating film is proposed with the flexible serpentine wire embedded in the soft polymer film, which can be attached to non-developable surfaces conformally. It provides a new way for the stretchable heaters to realize uniform heating of complex surfaces. However, the thermal field of flexible serpentine heaters(FSHs) depends on the configurations of the embedded serpentine heating wire, which requires accurate theoretical prediction of real-time temperature distribution. Therefore,the analytical model for the transient heat conduction in FSHs is solved by the separation of variables method and validated by the finite element analysis(FEA) in this paper. Based on this model, the effects of the geometric parameters, such as the radius and the length of the serpentine heaters, on the thermal uniformity are systematically investigated. This study can help to design and fabricate flexible heaters with uniform heating in the future.     

非均质材料等效瞬态热传导时域分析

基于时域自适应算法,结合均匀化技术和有限元方法,从而提出一种瞬态温度场求解模型,用来预测非均质材料等效性能并评估宏观温度场的等效行为.在整个计算中,通过自适应算法保证计算精度和稳定性,避免时间段尺寸变化可能引起的计算误差.在数值算例中,等效分析的结果与利用ANSYS求解的非均质有限元解比较,从计算效率和计算精度综合效果而言,结果是令人满意的.     

Perturbation solution to heat conduction in melting or solidification with heat generation

The Stefan problem involving a source term is considered in this technical note. As an example, planar solidification with time-dependent heat generation in a semi-infinite plane is solved by use of a perturbation technique. The perturbation solution is validated by reducing the problem to the case without heat generation whose exact solution is available. An application to the case with constant heat generation is presented, for which a closed-form solution is obtained. The effects of heat generation and Stefan number on the evolution of solidification are examined using the perturbation solution.     

瞬态热传导问题的加权最小二乘无网格法

加权最小二乘无网格法是一种基于节点信息的纯无网格法,该方法使用最小二乘法建立系统的变分原理,通过移动最小二乘法构造近似函数,控制方程在节点处的残量使用最小二乘法予以消除,边界条件通过罚函数法引入。本文推导了瞬态热传导问题的加权最小二乘无网格计算格式,编制了相应的计算程序,算例结果表明,该方法具有精度高、前后处理简单的优点,是一种高效的的新型无网格法。     

Inverse finite element formulations for transient heat conduction problems

Transient heat conduction problems are normally simulated by the conventional consistent and lumped finite element methods. The discretization error and the physical reality violation in such problems are noticeable and unwanted responses are observed in the results when using the consistent formulations. Although in utilizing the lumped formulations, the violation of physical reality becomes reduced; however, emerging the discretization error would also become obvious to the degree of being quantifiable. In using the inverse finite element method without considering the element shape functions, the element matrices will be obtained by minimizing the governing equation and its generalized discretized corresponding formula. The results obtained by using this method indicates that the reduction in both the discretization error as well as the violation of physical reality would be realized.     

半无限介质瞬态热传导的同伦分析法

In the current work, transient heat conduction in a semi-infinite medium is considered for its many applications in various heat fields. Here, the homotopy analysis method （HAM） is applied to solve this problem and analytical results are compared with those of the exact and integral methods results. The results show that the HAM can give much better approximations than the other approximate methods： Changes in heat fluxes and profiles of temperature are obtained at different times and positions for copper, iron and aluminum.     

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…     

Hybrid graded element model for transient heat conduction in functionally graded materials

This paper presents a hybrid graded element model for the transient heat conduction problem in functionally graded materials (FGMs). First, a Laplace transform approach is used to handle the time variable. Then, a fundamental solution in Laplace space for FGMs is constructed. Next, a hybrid graded element is formulated based on the obtained fundamental solution and a frame field. As a result, the graded properties of FGMs are naturally reflected by using the fundamental solution to interpolate the intra-element field. Further, Stefest’s algorithm is employed to convert the results in Laplace space back into the time-space domain. Finally, the performance of the proposed method is assessed by several benchmark examples. The results demonstrate well the efficiency and accuracy of the proposed method.     

A space marching method for multidimensional transient inverse heat conduction problems

A new method for the solution of multidimensional heat conduction problems is formulated. The developed space marching method allows to determine quickly and exactly unsteady temperature distributions in the construction elements of irregular geometry. The method is especially appropriate for determining transient temperature distribution in thick-wall pressure components based on temperature measurements at the outer surface. Two examples are included to demonstrate the capabilities of the new approach. Received on 28 October 1998     

Approximate solution of non-linear transient heat conduction in cylindrical geometry

The applicability of the Heat Balance Integral method (HBI) using hybrid profiles (HP _n ) for heat diffusion problems in cylindrical geometry is studied. Three test problems are taken to validate the method, namely, TP1 — long hollow cylinder, inner boundary is supplied with constant heat flux and the outer boundary is at infinity, TP2 — the same geometry with the inner boundary at constant temperature, TP3 — the same geometry with convection at the boundary. It is found that HBI in combination with HP _n gives much better solutions as compared to the HBI-polynomial (P _n ) combinations. This method is applied to non-linear cases like temperature dependent thermal conductivity, boiling or condensation at the boundary. For assessment of accuracy Root Mean Square Residual (E _rms) is defined. The residual minimization technique is used to select the parameter in the HP _n profile. It is shown that this procedure yields solutions which are in excellent agreement with the available exact solutions in the case of the three test problems. In the case of nonlinear problems, the value ofE _rms which is comparable to the values in the test cases, shows that the HBI-HP _n is a very good combination for all problems.In dieser Studie wurde die Verwendbarkeit des Wärmebilanz-Integrationsverfahrens (HBI) unter Benutzung von hybriden Profilen (HP _n ) für Wärmeausbreitungsprobleme in zylindrischen Geometrien untersucht. Um das Verfahren zu veranschaulichen wurden drei Testprobleme genommen; wobei für alle drei Fälle die gleiche Geometrie benützt wurde, und zwar ein langer, ausgehöhlter Zylinder. Beim TP-1 ist die innere Grenze mit konstantem Wärmestrom versorgt und die äußere Grenze ist im Unendlichen angenommen worden. Beim TP-2 herrscht an der inneren Grenze eine konstante Temperatur. Beim TP-3 herrscht Konvektion an den Grenzen. Hierbei wurde festgestellt, daß HBI in Verbindung mit HP _n sehr viel bessere Lösungen liefert, als die HBI-Polynom-Kombinationen (P _n ). Dieses Verfahren ist für nicht lineare Fälle, wie temperaturabhängige Wärmeleitfähigkeit, Sieden oder Kondensation and den Grenzen verwendet worden. Für die Abschätzung der Genauigkeit ist das restliche quadratische Mittel definiert worden. Die Residuen-Minimisierungstechnik ist für die Auswahl des Parameters \ benützt worden. Es ist gezeigt worden, daß dieses Verfahren Lösungen liefert, welche mit den benützten genauen Lösungen der drei Testfälle hervorragend übereinstimmen. Im Fall der nicht linearen Probleme zeigt der Wert vonE _rms, der mit den Testfällen vergleichbar ist, daß die HBI-HP _n eine sehr gute Kombination für alle Probleme ist.     

Stationary and transient heat conduction in a non homogeneous material

An apparent thermal conductivity for inhomogeneous materials is widely used. In this paper it is demonstrated that the apparent thermal conductivity for stationary heat conduction is not sufficient to describe the transient heat response of an inhomogeneous medium. In the geometry we used the heat transfer is estimated too high when the stationary thermal conductivity is employed. A numerical solution of the equation of thermal diffusion has been used to check several approximations. For short and for long times a separate approximate analytic expression can be used.

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Stationäre und Übergangswärmeleitung in nichthomogenen Materialien

Zusammenfassung Oft wird eine scheinbare Wärmeleitfähigkeit für inhomogene Materialien verwendet. In diesem Artikel wird gezeigt, daß es im allgemeinen nicht genügt, die instationäre Übergangswärmeleitung mit der stationären Wärmeleitfähigkeit zu beschreiben. In unserer Geometrie gibt dies eine Überschätzung der Wärmeleitung. Ein numerisches Modell für die Wärmediffusions-gleichung ist entwickelt worden, um mehrere Schätzungen der scheinbaren Wärmeleitfähigkeit zu kontrollieren. Für kurze und lange Zeiten stehen unterschiedliche analytische Beziehungen zur Verfügung.

     

Approximate solution of nonlinear transient heat conduction in one dimension

Three approximate methods, viz. the Heat Balance Integral method (HBI), the Modified Heat Balance Integral method (MHBI) and the Double Integral Method (DIM) in combination with hybrid profiles (HP - consisting of an exponential and polynomial) are evaluated for relative merits in solving the onedimensional heat diffusion equation. Applications to two linear test problems (TP1 - semi-infinite solid with constant heat flux at its boundary, TP 2 - semi-infinite solid with step change in its surface temperature) yield the HP-MHBI as the most desirable combination. Applications to nonlinear problems show that these combinations are ideally suited for obtaing reliable and accurate appoximate solutions.

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Näherungslösung von eindimensionaler nicht linearer transienter Wärmeleitung

Zusammenfassung Es wurden drei Näherungsmethoden, die Heat Balance Integral-Methode (HBI), die Modified Heat Balance IntegralMethode (MHBI) und die Double Integral-Methode (DIM) in Verbindung mit hybriden Profilen (HP - bestehend aus Exponentialund Polynomfunktionen) ausgewählt, um die eindimensionale Wärmediffusionsgleichung zu lösen. Anwendungen auf zwei lineare Testprobleme (TP 1 - halb-unendlicher Festkörper mit konstantem Wärmestrom an dessen Grenze, TP2 - halb-unendlicher Festkörper mit sprunghaftem Wechsel der Oberflächentemperatur) ergaben, daß die HP-MHBI-Methode die erwünschte Kombination darstellt. Anwendungen auf nichtlineare Probleme zeigen, daß diese Kombination ideal für das Erzielen zuverlässiger und aussagekräftiger Näherungslösungen ist.

     

Transient heat conduction with uniform heat generation in a slab subjected to convection and radiation cooling

A finite difference scheme with fourth order Runge-Kutta method is employed to determine the unsteady state temperature distribution in a plane slab with uniform heat generation. The plane slab is insulated on one face and subjected to convective and radiative cooling at the other face. The plane slab has a uniform initial temperature and the ambient environment as well as the fluid temperatures are assumed to be constant. Heat conduction is considered to be one dimensional. Results are presented in dimensionless charts over a wide range of parameters.

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Instationäre Wärmeleitung mit gleichförmiger Wärmeerzeugung in einer Platte bei Kühlung durch Konvektion und Abstrahlung

Zusammenfassung Zur Ermittlung der instationären Temperaturverteilung in einer Platte mit gleichförmiger Wärmeerzeugung wird ein Differenzverfahren vierter Ordnung nach Runge-Kutta angewendet. Die ebene Platte ist einseitig isoliert und wird auf der anderen Seite durch Konvektion und Abstrahlung gekühlt. Zu Beginn befindet sich die Platte gleichförmig auf einer bestimmten Anfangstemperatur, die Temperaturen der umgebenden Objekte sowie des Fluids sind ebenfalls konstant. Der Wärmeleitungsvorgang sei eindimensional. Die Ergebnisse sind in dimensionsloser Form für einen weiten Parameterbereich in Diagrammform dargestellt.

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Nomenclature Bi Biot number - c specific heat - h heat transfer coefficient - k thermal conductivity - L half thickness of plate - Nr Radiation number - T absolute temperature - t time - F _o dimensionless time - T _e environment temperature - T _f fluid temperature - T _i initial plate temperature - u internal energy generation number - u internal energy generation - x space coordinate - x* dimensionless space coordinate Greek symbols thermal diffusivity - emissivity - dimensionless temperature - density - Stephan-Boltzman constant     

A high order control volume finite element procedure for transient heat conduction analysis of functionally graded materials

A robust unstructured control-volume finite element method is presented for the solution of two-dimensional transient heat conduction in functionally graded materials (FGMs) with isotropic properties. The material properties at a point in the domain vary exponentially to spatial coordinates. A triangular mesh is chosen for spatial discretization and a fully implicit scheme is adapted for time discretization. Several problems are investigated and the results are successfully validated by using analytical and other numerical solutions available in the literature.     

Fuzzy finite difference method for heat conduction analysis with uncertain parameters

A new numerical technique named as fuzzy finite difference method is proposed to solve the heat conduction problems with fuzzy uncertainties in both the phys- ical parameters and initial/boundary conditions. In virtue of the level-cut method, the difference discrete equations with fuzzy parameters are equivalently transformed into groups of interval equations. New stability analysis theory suited to fuzzy difference schemes is developed. Based on the parameter perturbation method, the interval ranges of the uncertain temperature field can be approximately predicted. Subsequently, fuzzy solutions to the original difference equations are obtained by the fuzzy resolution theorem. Two numerical examples are given to demonstrate the feasibility and efficiency of the presented method for solving both steady-state and transient heat conduction problems.     

二阶非定常热传导反问题的多宗量辨识

引入Bregman距离构造同伦函数,建立了二阶非定常多宗量热传导反问题的一种求解模式,可对导热系数和边界条件等宗量进行识别。时域上采用精细算法,建立了便于敏度分析的有限元正/反演模型,对各宗量进行有效的组合识别。对信息测量误差和初值选取作了初步探讨,数值验证取得了满意的结果。     

Inverse transient heat conduction problems and identification of thermal parameters

This work deals with the estimation of polymers properties. An inverse analysis based on finite element method is applied to identify simultaneously the constants thermal conductivity and heat capacity per unit volume. The inverse method algorithm constructed is validated from simulated transient temperature recording taken at several locations on the surface of the solid. Transient temperature measures taped with infrared camera on polymers were used for identifying the thermal properties. The results show an excellent agreement between manufacturer and identified values.