Multidimensional Heat Conduction Research Articles

An algorithm for solving multidimensional inverse heat conduction problem

The unknown time-dependent boundary heat flux of a multidimensional body is determined from temperature measurements inside the body or on its boundary. The sequential function specification method with the assumption that future-boundary heat flux varies linearly with time is used to solve the inverse problem. The method of discretization used by the proposed algorithm is the boundary element method. The effectiveness of the algorithm is illustrated by solving a sample two-dimensional problem.

Model reduction for the resolution of multidimensional inverse heat conduction problems

For large linear heat conduction systems, it is proposed here to solve an inverse heat conduction problem (IHCP) that consists in the identification of several time-varying thermal solicitations from simulations of measured temperatures. For this inversion, instead of using a detailed model of large size, this one is first transformed into a reduced model. The latter is built with identified dominant eigenmodes of the system leading to a reduced state representation that links the inputs (unknown solicitations) to the outputs (simulated temperatures). The procedure is sequential and uses future time steps. At first, a numerical 2D IHCP is provided: two time-varying heat flux densities are estimated from various positions of two sensors. A specific study on static and dynamic sensitivities is made. An example of a 3D IHCP is also given. The method is particularly interesting in this last case where, at each time step, the resolution of a system of order 9 (the reduced model) takes the place of a system of order 1331 (the detailed model).

A recursive algorithm for multidimensional inverse heat conduction problems by means of mode reduction

A recursive algorithm is developed to solve the inverse heat conduction problem of estimating time-varying strength of a heat source from the knowledge of temperature readings taken inside the domain. It is based on the Kalman filtering technique. Although the recursive nature of the Kalman filter algorithm allows the possibility of on-line estimation in place of batch form off-line estimation, the straightforward implementation of this algorithm to multidimensional partial differential equations has not been feasible due to the tremendous requirement of computer time and memory. In the present investigation we overcome this difficulty by employing the Karbunen–Loève Galerkin procedure which reduces the governing partial differential equation to a minimum number of ordinary differential equations. The performance of the present technique of inverse analysis is evaluated by several numerical experiments, and is found to be very accurate as well as efficient.

A Simplified Numerical Solution of Unsteady Heat Conduction Problems with an Application to a Short Cylinder

A simple numerical approach is introduced, for solving transient one-dimensional heat conduction problems, in both a plate and a long cylinder with or without internal heat generation, using spreadsheets. Details of developing the nodal equations and writing them in spreadsheet format are presented. Then, the formulated problems are used to solve transient multi-dimensional heat conduction in a short cylinder applying the principle of the separation of variables. This reduces the two-dimensional nature of the problem into a couple of one-dimensional problems that are easily solved using spreadsheets. Temperature at any location and time is obtained by combining both solutions. The method allows the easy change in time increment and Biot number in a way that make it general for solving transient heat conduction in a short cylinder. It is believed that this method is appropriate for teaching transient conduction numerically to undergraduate heat transfer students.

Numerical method for the solution of non-linear two-dimensional inverse heat conduction problem using unstructured meshes

A new method of solving 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 which is based on temperature measurements at the outer surface, is especially appropriate for determining transient temperature distribution in thick-wall pressure components. Two examples are included to demonstrate the capabilities of the new approach. Copyright © 2000 John Wiley & Sons, Ltd.

RELAP5-3D Multidimensional Heat Conduction Enclosure Model for RBMK Reactor Application

A heat conduction enclosure model is conceived and implemented by RELAP5-3D between heat structures. The suggested model uses a lumped parameter model that is generally applicable to multidimensional calculational domain. This new model is applied to calculation of RBMK reactor core graphite blocks and is compared to the commercially available Fluid Dynamics Analysis Package (FIDAP) finite element code. Reasonably good agreement between the results of RELAP5-3D and FIDAP is obtained. The new heat conduction enclosure model gives RELAP5-3D a general multidimensional heat conduction capability. It also provides new routes for temperature cooloff of the RBMK graphite blocks from the ruptured channel to the surrounding ones. This ability to predict graphite temperature cooloff is very important during accidents or for transient simulation, especially concerning long-term coolability of the RBMK reactor core.

An implicit method for reconstructing dynamic three-dimensional phase boundaries under low Peclet number conditions

This article develops an implicit inverse method for reconstructing dynamic multidimensional phase boundaries. The technique is suitable for problems having small liquid phase Peclet numbers, Pe r = (V̂ rL̂/ α ̂ ), where V̂ r is the characteristic liquid phase velocity scale evaluated relative to the solid phase velocity scale, L̂ is a characteristic length scale, and α ̂ is a characteristic thermal diffusivity. Under these conditions, a multidimensional Stefan problem emerges. Explicit front-tracking procedures are eliminated by incorporating the latent heat effect in an effective, temperature dependent specific heat. Time-sequential reconstruction is then performed by solving a multidimensional nonlinear inverse heat conduction problem. As an illustration, evolving phase boundaries are reconstructed within moving and stationary plates subject to concentrated, high energy density heat sources. It is found that boundaries can be accurately reconstructed using either exact or noisy temperature measurements.

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.

Solution of inverse heat conduction problems using control volume approach

A new method for solving multidimensional inverse heat conduction problems is presented. Using control volume methods, the partial heat conduction equation is replaced by a system of ordinary differential equations in time, which are then solved sequentially. The procedure is started at a spatial node where temperature sensor is located and sequentially marches through space to the surface node. The accuracy of the method was demonstrated by comparison of the calculated surface heat flux and temperature with the known exact solution. Two numerical experiments, one- and two-dimensional inverse heat conduction problems, are solved to illustrate the effectiveness, computational efficiency and good accuracy of the presented method. In the third example, the inverse formulation is applied to a set of experimental data. The method can easily be extended to three-dimensional cases.

A guide for the use of the function specification method for 2D inverse heat conduction problems

The solution of the multidimensional inverse heat conduction problem (IHCP) requires much practice and several runs to determine the many parameters involved in the solution. This paper describes a step by step guide to identifying all of the parameters involved in the function specification method with spatial regularization: namely the number of future temperatures, the value of the regularization coefficient and the sensor locations. It also intends to show the new difficulties that arise from solving a 2D IHCP compared to a ID IHCP. The issue of finding the appropriate number of sensors and their locations so as to obtain a desired surface heat flux spatial resolution is addressed. An experimental set-up was built for the validation of the methodology. It also shows that the determination of surface temperature and flux that vary in both time and space from remote temperature measurements is possible when the sensors are properly placed.

An Introduction To a User-Friendly Numerical Fire Endurance Algorithm

This paper adopts the simplest numerical procedure for solving the multi-dimensional nonlinear transient heat conduction model, which "occasionally" is used to simulate fire endurance and related fire engineering problems. Due to its superb simplicity, average fire professionals can easily implement this relatively user-friendly algorithm customized for the rarely published fire (convective or Robin) boundary conditions. Further, the danger of constant incident (Neumann) flux is discussed. This paper also discusses the cogent investigation of converging solutions, numerical difficulties, comparisons of analytical solutions of a simple benchmark case, and nonlinear problems arising from thermophysical properties, temporal fire including variable combined heat transfer coefficient, and a heat source or sink. Language: en

Symbolic Computation to Estimate Two-Sided Boundary Conditions in Two-Dimensional Conduction Problems

A method using symbolic computation is proposed to determine the boundary conditions in two-dimensional inverse heat conduction problems. The method uses symbols to represent the unknown boundaries and then executes the finite difference method to calculate the temperature field. The calculated results are expressed explicitly as functions of the undetermined boundaries. Then, a direct comparison between the output symbolic temperature and the measurement temperature are made to construct a set of linear equations. Thus, the linear least-squares method is adopted to solve the linear equations. Finally, the unknown boundary conditions are determined. Results from the examples confirm that the proposed method is applicable in solving the multidimensional inverse heat conduction problems. In the example problems, three kinds of measuring methods are adopted to estimate the surface temperature.

A finite element analysis for inverse heat conduction problems

This paper presents an efficient inverse analysis technique based on a sensitivity coefficient algorithm to estimate the unknown boundary conditions of multidimensional steady and transient heat conduction problems. Sensitivity coefficients were used to represent the temperature response of a system under unit loading conditions. The proposed method, coupled with the sensitivity analysis in the finite element formulation, is capable of estimating both the unknown temperature and heat flux on the surface provided that temperature data are given at discrete points in the interior of a solid body. Inverse heat conduction problems are referred to as ill-posed because minor inaccuracy or error in temperature measurements cause a drastic effect on the predicted surface temperature and heat flux. To verify the accuracy and validity of the new method, two-dimensional steady and transient problems are considered. Their surface temperature and heat flux are evaluated. From a comparison with the exact solution, the effects of measurement accuracy, number and location of measuring points, a time step, and regularization terms are discussed. © 1998 Scripta Technica. Heat Trans Jpn Res, 26(6): 345–359, 1997

A Finite Element Analysis for Inverse Heat Conduction Problems.

We present an efficient inverse analysis method based on a sensitivity coefficient algorithm to estimate unknown boundary conditions of multidimensional heat conduction problems. Sensitivity coefficients were used to represent the temperature response of a system under unit loading conditions. The proposed method, coupled with sensitivity analysis in the finite element formulation, is capable of estimating unknown both temperature and heat flux on the surface provided that temperature data are given at discrete points in the interior of a solid body. Inverse heat conduction problems are referred to as ill-posed ones because minor inaccuracy or error in temperature measurement causes a drastic effect on the predicted surface temperature and heat flux. To verify the accuracy and validity of the new method, two-dimensional steady-state and transient problems are considered and their surface temperature and heat flux are evaluated. From a comparison with exact solutions, the effects of measurement accuracy, number and location of measuring points, a time step, and regularization terms are discussed.

Modelling of heat and mass transfer processes during core melt discharge from a reactor pressure vessel

The objective of this paper is to study the heat and mass trasnfer processes related to core melt discharge from a reactor vessel in a light water reactor severe accident. The phenomenology modelled includes the convection in, and heat transfer from, the melt pool in contact with the vessel lower head wall, the fluid dynamics and heat transfer of the melt flow in the growing discharge hole and multi-dimensional heat conduction in the ablating lower head wall. A research programme is underway at the Royal Institute of Technology (Kungliga Tekniska Högskolan, KTH) to (1) identify the dominant heat and mass transfer processes determining the characteristics of the lower head ablation process: (2) develop and validate efficient analytical/computational models for these processes; (3) apply models to assess the character of the melt discharge process in a reactor-scale situation; (4) determine the sensitivity of the melt discharge to structural differences and variations in the in-vessel melt progression scenarios. The paper also presents a comparison with recent results of vessel hole ablation experiments conducted at KTH with a melt simulant.

The boundary estimation in two-dimensional inverse heat conduction problems

A direct method is developed to estimate the boundary condition in two-dimensional inverse heat conduction problems. At the beginning of the study, finite-difference methods are employed to discretize the problem domain and then a linear inverse model is constructed to identify the boundary condition. The linear least-squares method is adopted for the linear model and thus iteration times can be limited to one cycle and the uniqueness of the solutions can be identified easily. Results from the examples confirm that the proposed method is effective and applicable to solution of multidimensional inverse heat conduction problems. In the example problems, three kinds of measuring methods are adopted to estimate the surface temperature. The result shows that only a few measuring points is sufficient to estimate the surface temperature when the measurement errors are neglected. When the measurement errors are considered, more measuring points are needed in order to increase the congruence of the estimated results to the exact solutions.

Multidimensional Steady Conduction in Convecting, Radiating, and Convecting-Radiating Fins and Fin Assemblies

This article reviews the literature on multidimensional heat conduction in single fins and fin assemblies. The surface heat dissipation mechanisms include pure convection, pure radiation, and simultaneous convection and radiation. Two-dimensional heat transfer results are presented for longitudinal, radial, and cylindrical fins. The discussion also covers square and polygonal fins on round tubes and composite fins. These heat transfer results encompass a variety of situations, such as a nonconstant base temperature, nonuniform ambient temperature, asymmetric thermal boundary conditions, anisotropy of fin material, variable heat transfer coefficient, and internal heat generation. The article provides a comprehensive collection of important analytical and numerical results for two-dimensional effects in extended surfaces, and, as such, both scientific and practice-oriented heat transfer workers would find the information convenient to use for analysis and design.

DIRECT SENSITIVITY COEFFICIENT METHOD FOR SOLVING TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEMS BY FINITE-ELEMENT SCHEME

Abstract A simple, but versatile, algorithm called the direct sensitivity coefficient (DSC) method has been developed to deal with multidimensional inverse heat conduction (IHC) problems. Different sensitivity coefficients have been used to assess the physical responses of the domain considered under unit loading conditions. The concept of the finite-element discretization has been applied to evaluate the global responses under total loading conditions. The method developed is capable of dealing with unknown surface heat flux, surface temperature, heal transfer coefficient, or any combination of them. In this article the formulation as well as the theory adopted is discussed. The finite-element procedures for solving a two-dimensional steady-state problem art specifically illustrated. Analysis for several selected benchmark problems is then presented, emphasizing accuracy estimations. Both linear and nonlinear problems are considered. Recommendations for further work are also included in this article.

Green's Function Approach to Optimal Control of Unsteady Heat Conduction Field.

This paper presents a new methodology for optimal control of an unsteady heat conduction field with heat sources. The optimal control problem considered is that of determining the time changes of heat-source intensities to produce desired temperatures at several reference points. Using Green's functions, which are pre-calculated numerically under a point heat source approximation, it is shown that the distributed parameter system described by a multi dimensional heat conduction equation can be reduced to a lumped parameter system maintaining the distributed property. Thus the optimal control problem becomes low-dimensional and a fast optimal control scheme can be developed for an arbitrarily shaped region. Some examples demonstrate the capability of the proposed method.

A general optimization method using adjoint equation for solving multidimensional inverse heat conduction

A three-dimensional formulation is presented to solve inverse heat conduction as a general optimization problem by applying the adjoint equation approach coupled to the conjugate gradient algorithm. The formulation consists of the sensitivity problem, the adjoint problem and the gradient equations. A solution algorithm is presented for the estimation of the surface condition (i.e. heat flux or temperature), space dependent thermal conductivity and heat capacity from the knowledge of transient temperature recordings taken within the solid. In this approach, no a priori information is needed about the unknown function to be determined. It is shown that the problems involving a priori information about the unknown function become special cases of this general approach.