Solution Of Heat Conduction Problem Research Articles

Singular boundary method for heat conduction problems with certain spatially varying conductivity

The singular boundary method (SBM) is a recent boundary-type meshless collocation method, in which the solution of a given problem is expanded as a linear combination of the fundamental solutions in terms of the source points. The method circumvents the fictitious boundary long perplexing the method of fundamental solution (MFS) by the introduction of the concept of origin intensity factors (OIFs). This paper documents the first attempt to extend the method to heat conduction problems in nonhomogeneous materials. We derive the fundamental solutions of heat conduction problems with the thermal conductivity of the quadratic, exponential and trigonometric material variations in three directions. Furthermore, we firstly theoretically derive the value of the OIF for the natural logarithm function, which is later extended to the OIFs for a group of fundamental solutions. The feasibility, accuracy and stability of the presented SBM formulation are confirmed for both two- (2D) and three-dimensional (3D) examples.

A Series Solution for Heat Conduction Problem with Phase Change in a Finite Slab

A two-dimensional differential transform method is applied to solve one-dimensional phase change problems in a slab of finite thickness, which is subjected to convective thermal loading at one surface and a constant prescribed temperature at the other. In the problems, the initial temperature of the slab does not necessarily have to be the same as the fusion temperature. A series solution is derived for the temperature profile in the melting or solidifying slab with temperature-dependent thermal conductivity and volumetric heat capacity. The latent heat effect of the phase change is incorporated into the temperature-dependent heat capacity. Numerical results demonstrate the effects of the temperature-dependent parameters on the transient temperature profile of the slab.

Comprehesive Solution of Heat Conduction Problem Using Mathematical Model of Heat and Mass Transfer in Permafrost Soils

Technical solutions are proposed for the beds and foundations of buildings and structures on permafrost soils, which are designed with use of a software package for modeling heat and moisture transfer in soils by the method of thermal and hydraulic resistances.

Assessment of the convergence of solutions of integro-differential equations of heat conduction in the conditions of the relaxation system

Modeling of relaxation processes is possible in the presence of relaxation components of a system, leading to consideration of integro-differential equations (IDE) of heat conduction taking into account relaxation functions and definition of the estimates of the convergence of their solutions. In this paper, we build solutions in times of system relaxation by the method of successive approximations. This method allows explicitly specify the time interval on which there is the solution, unlike the existing methods, involving the introduction of relative time. The function, which defines the scope of application of IDE of heat conduction only at intervals of system relaxation, is given. As appears from the definition of boundedness of input functions and their derivatives, the equation has bounded solutions at all critical points of relaxation. The conducted analysis determines the boundedness, uniformity and existence of solutions under the influence of relaxation, allowed to carry out the evaluation of the integral terms of the heat conduction equation. In this paper, the boundaries of existence and uniformity of solutions of heat conduction problems for IDE taking into account the thermal memory are defined. Theorems on the existence of uniformity and convergence of solutions of the integrodifferential equation on the interval 0<FO≤FOr and the boundedness of functions |Q(X, FO), |FO Q (X, FO)| at FO, which are responsible for thermophysical properties of material, are proved. This allows to perform calculations at times of relaxation of the system of locally-nonequilibrium state of material in problems of heat and mass transfer

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The meshless local radial basis function-based differential quadrature (RBF-DQ) method is applied on two-dimensional heat conduction for different irregular geometries. This method is the combination of differential quadrature approximation of derivatives and function approximation of radial basis function. Four different geometries with regular and irregular boundaries are considered, and numerical results are compared with those gained by finite element (FE) solution achieved by COMSOL commercial code. Outcomes prove that current technique is in very good agreement with FEM and this fact that RBF-DQ method is an accurate and flexible method in solution of heat conduction problems.

Exact Solution for Heat Conduction Problem of a Sector of a Hollow Cylinder

In this article, the heat conduction problem of a sector of a finite hollow cylinder is studied as an exact solution approach. The governing equations are in the form of non-homogeneous partial differential equation (PDE) with non-homogeneous boundary conditions. In order to solve the PDE equation, generalized finite Hankel, periodic Fourier, Fourier and Laplace transforms are applied. Three different boundary conditions as case studies for simulations are presented and verified with the result which extracted from finite element method. The results are shown that this approach is suitable and systematic for solving heat conduction problems in cylindrical coordinate.

Mathematical simulation of three-dimensional temperature fields by projection-structural method using the S-functions

The article suggests a new approximate analytical approach to the mathematical simulation of three-dimensional temperature fields in bodies, limited by cylindrical surface of complex shape and two parallel planes, whose generators are perpendicular to the planes. The application of the final integral transformations on the coordinate, parallel to the generators of the cylindrical surface, allowed reducing the solution of three-dimensional heat conduction problems to the solution of the corresponding two-dimensional problems with a parameter of the Sturm-Liouville problem. Built analytical structures of approximate solutions of the heat conduction problems in the image area exactly satisfy the corresponding boundary conditions. Geometrical information is taken into account by the S-functions. The basic functions, contained in the approximate analytical structures of solutions of the heat conduction problems in the images area, are continuously differentiable. These qualities of the basic functions are possible to obtain due to the fact that the S-functions permitted to build the equations of smooth surfaces of bodies and the equations of smooth boundaries of areas of complex shape. According to the theorem of Kantorovich-Krylov, in these cases the basic functions, built using complete systems of functions in a particular coordinate system, also form complete systems of continuously differentiable functions. Undetermined coefficients of analytical structures of solutions in the image area are discovered from the corresponding algebraic systems with a parameter of the Sturm-Liouville problem, obtained by the Bubnov-Galerkin method.

Analytical Method for Heat Conduction Problem with Internal Heat Source in Irregular Domains

In engineering field, analytical methods can only be applied to classic heat transfer problems with regular geometric boundaries. It is difficult to apply analytical methods in solving the mathematical and physical equations in nonorthogonal boundary of irregular domains. In this paper, we presented a method by conformal mapping from the solution for heat conduction problems in regular domains to solve problems in irregular domains.

A Comparative Study of Gradient-Type Methods for Recovering Time-Dependent Heat Sources

Gradient-based methods belong to offline techniques which use all the available data reading to set the unknown thermal coefficients. General application and rigorous mathematical investigation of these methods make them robust tools for analyzing inverse heat problems. In the present study, four search techniques in gradient-type methods through the use of adjoint problem are employed for the evaluation of heat sources in arbitrary geometries. The methods include Fletcher–Reeves version of the conjugate gradient method (CGM), Hestenes–Stiefel version of the CGM, Broydon–Fletcher–Goldfarb–Shanno version of quasi-Newton method and a novel form of the gradient-type method. The search direction in the novel method appears to have originated as a hybridization of a conjugate gradient and a quasi-Newton method. A conservative cell-based version of finite volume scheme is applied for the solution of heat-conduction problem in irregular domains. To highlight the flexibility and advantages of these algorithms, their performance and accuracy are presented and compared for two-dimensional heat-conduction problems.

Mesh-free modeling of two-dimensional heat conduction between eccentric circular cylinders

In this paper, a numerical investigation of heat conduction problem for an irregular geometry using mesh-free local radial basis function-based differential quadrature (RBF-DQ) method has been performed. This method is the combination of differential quadrature approximation of derivatives and function approximation of radial basis function. The method can be used to directly approximate the derivatives of dependent variables on a scattered set of knots. In this study, knots were distributed irregularly in the solution domain using the Halton sequences. The method is applied to a two-dimensional geometry consisting of two eccentric cylinders. The inner and outer walls are maintained at different temperatures and . The obtained results from numerical simulations are compared with those gained by finite volume (FV) method. Outcomes prove that current technique is in very good agreement with finite volume method and this is due to the fact that RBF-DQ method is an accurate and flexible method in solution of heat conduction problems. Key words: Mesh-free method, conduction, radial basis function, eccentric circular cylinders

New derivations of the fundamental solution for heat conduction problems in three-dimensional general anisotropic media

This work presents two new methods to derive the fundamental solution for three-dimensional heat transfer problems in the general anisotropic media. Initially, the basic integral equations used in the definition of the general anisotropic fundamental solution are revisited. We show the relationship between three, two and one-dimensional integral definitions, either by purely algebraic manipulation as well as through Fourier and Radon transforms. Two of these forms are used to derive the fundamental solutions for the general anisotropic media. The first method gives the solution analytically for which the solution for the orthotropic case agrees with the well known result obtained by the domain mapping, while the fundamental solution for the general anisotropic media is new. The second method expresses the solution by a line integral over a semi-circle. The advantages and disadvantages of the two methods are discussed with numerical examples.

Local RBF-DQ method for two-dimensional transient heat conduction problems

The meshless local radial basis function-based differential quadrature (RBF-DQ) method is applied on two-dimensional heat conduction for different irregular geometries. This method is the combination of differential quadrature approximation of derivatives and function approximation of radial basis function. Four different geometries with regular and irregular boundaries are considered, and numerical results are compared with those gained by finite element (FE) solution achieved by COMSOL commercial code. Outcomes prove that current technique is in very good agreement with FEM and this fact that RBF-DQ method is an accurate and flexible method in solution of heat conduction problems.

Approximate analytic solution of heat conduction problems with a mismatch between initial and boundary conditions

We consider a heat conduction problem for an infinite plate with a mismatch between initial and boundary conditions. Using the method of integral relations, we obtain an approximate analytic solution to this problem by determining the temperature perturbation front. The solution has a simple form of an algebraic polynomial without special functions. It allows us to determine the temperature state of the plate in the full range of the Fourier numbers (0≤F<∞) and is especially effective for very small time intervals.

Explicit full field analytic solutions for two-dimensional heat conduction problems with finite dimensions

This study presents explicit analytical solutions of heat conduction problems for isotropic media with finite dimensions. The geometry configurations considered in this study include composite layer, wedge and circular media. The boundary conditions are assumed to be either thermal isolation or isothermal. The full field analytical solutions of temperature and heat fluxes for the composite layered media subjected to an embedded heat source are derived first by Fourier transform technique in conjunction with the image method. The corresponding problems of composite wedge and circular media are constructed by conformal mapping and the solutions of composite layer media. The explicit full field solutions are expressed in simple closed-forms which can be easily used in engineering applications. The numerical calculations of the temperature and heat fluxes distributions are provided in full field configuration base on the available analytic solutions.

Inverse Heat Conduction Using Measured Back Surface Temperature and Heat Flux

In the high-energy laser heating of a target, the temperature and heat flux at the heated surface are not directly measurable, but they can be estimated by solving an inverse heat conduction problem based on the measured temperature and/or the heat flux at the accessible (back) surface. In this study, the one-dimensional inverse heat conduction problem in a finite slab is solved by the conjugate gradient method, using measured temperature and heat flux at the accessible (back) surface. Simulated measurement data are generated by solving a direct problem, in which the front surface of the slab is subjected to high-intensity periodic heating. Two cases are simulated and compared, with the temperature or heat flux at the heated front surface chosen as the unknown function to be recovered. The results show that the latter choice (i.e., choosing back surface heat flux as the unknown function) can give better estimation accuracy in the inverse heat conduction problem solution. The front surface temperature can be computed with high precision as a by-product of the inverse heat conduction problem algorithm. The robustness of this inverse heat conduction problem formulation is tested by different measurement errors and frequencies of the input periodic heating flux.

Analytical Exact Solutions of Heat Conduction Problems for a Three-Phase Elliptical Composite

Analytical Exact Solutions of Heat Conduction Problems for a Three-Phase Elliptical Composite

Additional boundary conditions in nonstationary problems of heat conduction

The integral method of thermal balance, with determining the front of temperature disturbance and introducing additional boundary conditions, is used to obtain analytical solutions of heat conduction problems for plate, cylinder, and sphere at the boundary conditions of the third kind. The distribution of isotherms and the velocities of their motion are analyzed.

On the Truly Meshless Solution of Heat Conduction Problems in Heterogeneous Media

A truly meshless method based on the weighted least-squares (WLS) approximation and the method of point collocation is proposed to solve heat conduction problems in heterogeneous media. It is shown that, in the case of strong heterogeneity, accurate and smooth solutions for temperature and heat flux can be obtained by applying the WLS approximation in each homogeneous domain and using a double-stage WLS approximation technique together with a proper neighbor selection criterion at each interface.

Analytical solution of heat-conduction problems with a variable initial condition based on determining a temperature perturbation front

With the aid of the integral heat-balance method an analytical solution of the problem of unsteady-state heat conduction has been obtained for an infinite plate with a variable initial condition. To increase the accuracy of solution by the integral method additional boundary conditions are introduced which are determined from the initial differential equation and basic boundary conditions, including those prescribed at the temperature perturbation front.

Analytical solution for heat conduction problem in composite slab and its implementation in constructal solution for cooling of electronics

Abstract The constructal solution for cooling of electronics requires solution of a fundamental heat conduction problem in a composite slab composed of a heat generating slab and a thin strip of high conductivity material that is responsible for discharging the generated heat to a heat sink located at one end of the strip. The fundamental 2D heat conduction problem is solved analytically by applying an integral transform method. The analytical solution is then employed in a constructal solution, following Bejan, for cooling of electronics. The temperature and heat flux distributions of the elemental heat generating slabs are assumed to be the same as those of the analytical solution in all the elemental volumes and the high conductivity strips distributed in the different constructs. Although the analytical solution of the fundamental 2D heat conduction problem improves the accuracy of the distributions in the elemental slabs, the results following Bejan’s strategy do not affirm the accuracy of Bejan’s constructal solution itself as applied to this problem of cooling of electronics. Several different strategies are possible for developing a constructal solution to this problem as is indicated.