Two-dimensional Heat Conduction Problem Research Articles

Transient Two-Dimensional Heat Conduction Problem With Partial Heating Near Corners

This paper provides a solution for two-dimensional (2D) heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, most notably used in thermal protection systems for re-entry vehicles. The solutions used in this work are generated using Green's functions (GFs). Two approaches are used, which provide solutions for either semi-infinite plates or finite plates with isothermal conditions, which are located a long distance from the heating. The methods are both efficient numerically and have extreme accuracy, which can be used to provide additional solution verification. The solutions have components that are shown to have physical significance. The procedure involves the derivation of previously unknown simpler forms for the summations, in some cases by virtue of the use of algebraic components. Also, a mathematical identity given in this paper can be used for a variety of related problems.

Efficient Numerical Evaluation of Exact Solutions for One-Dimensional and Two-Dimensional Infinite Cylindrical Heat Conduction Problems

Estimation of thermal properties or diffusion properties from transient data requires that a model is available that is physically meaningful and suitably precise. The model must also produce numerical values rapidly enough to accommodate iterative regression, inverse methods, or other estimation procedures during which the model is evaluated again and again. Bodies of infinite extent are a particular challenge from this perspective. Even for exact analytical solutions, because the solution often has the form of an improper integral that must be evaluated numerically, lengthy computer-evaluation time is a challenge. The subject of this paper is improving the computer evaluation time for exact solutions for infinite and semi-infinite bodies in the cylindrical coordinate system. The motivating applications for the present work include the line-source method for obtaining thermal properties, the estimation of thermal properties by the laser-flash method, and the estimation of aquifer properties or petroleum-field properties from well-test measurements. In this paper, the computer evaluation time is improved by replacing the integral-containing solution by a suitable finite-body series solution. The precision of the series solution may be controlled to a high level and the required computer time may be minimized, by a suitable choice of the extent of the finite body. The key finding of this paper is that the resulting series may be accurately evaluated with a fixed number of terms at any value of time, which removes a long-standing difficulty with series solution in general. The method is demonstrated for the one-dimensional case of a large body with a cylindrical hole and is extended to two-dimensional geometries of practical interest. The computer-evaluation time for the finite-body solutions are shown to be hundreds or thousands of time faster than the infinite-body solutions, depending on the geometry.

A NURBS-based scaled boundary finite element method for the analysis of heat conduction problems with heat fluxes and temperatures on side-faces

The scaled boundary finite element method (SBFEM) combined with isogeometric analysis (IGA) is proposed to solve the two-dimensional steady-state heat conduction problems in complex geometries. The main benefit of SBFEM is that the spatial dimension of analyzed domain is reduced by one and the solution is analytical in the radial direction. In this method, only the boundary of the computational domain requires discretization with finite elements leading to the reduction of computational efforts. However, SBFEM suffers from the finite element method related drawbacks. In the case of the complex geometric shapes, a large number of elements are necessary to obtain the exact representation of geometry in finite element method. Isogeometric analysis is a novel numerical technique based on the non-uniform rational B-splines (NURBS), where the geometry can be exactly represented. Moreover, this technique yields superior numerical accuracy, efficiency and convergence property in comparison to finite element method. In the proposed method, the segments of domain boundary with complex geometries are described with NURBS basis functions in IGA, while the straight segments of boundary are represented with polynomial basis functions as in the conventional SBFEM. Thus, the present approach combines the advantages of both SBFEM and IGA. The heat conduction problems of complex geometry can be more effectively handled with the proposed method considering the prescribed heat fluxes and temperatures on side-faces. The accuracy and efficiency of the proposed formulation are demonstrated by modeling five numerical examples involving the complicated geometry.

Meshless Local B-Spline Collocation Method for Two-Dimensional Heat Conduction Problems With Nonhomogenous and Time-Dependent Heat Sources

This paper presents a new approach of meshless local B-spline based finite difference (FD) method for transient 2D heat conduction problems with nonhomogenous and time-dependent heat sources. In this method, any governing equations are discretized by B-spline approximation which is implemented as a generalized FD technique using local B-spline collocation scheme. The key aspect of the method is that any derivative is stated as neighboring nodal values based on B-spline interpolants. The set of neighboring nodes is allowed to be randomly distributed. This allows enhanced flexibility to be obtained in the simulation. The method is truly meshless as no mesh connectivity is required for field variable approximation or integration. Galerkin implicit scheme is employed for time integration. Several transient 2D heat conduction problems with nonuniform heat sources in arbitrary complex geometries are examined to show the efficacy of the method. Comparison of the simulation results with solutions from other numerical methods in the literature is given. Good agreement with reference numerical methods is obtained. The method is shown to be simple and accurate for the time-dependent problems.

Steady heat transfer analysis of 2D plates using the numerical manifold method

In the numerical manifold method (NMM), the mathematical cover system can be inconsistent with the physical boundary, which can alleviate the meshing cost to some extent. In the present paper, the NMM is developed to solve two-dimensional steady heat conduction problems with the square mathematical elements. The corresponding formulations are derived and the proposed approach is validated through a typical example.

Inverse heat conduction estimation of inner wall temperature fluctuations under turbulent penetration

Turbulent penetration can occur when hot and cold fluids mix in a horizontal T-junction pipe at nuclear plants. Caused by the unstable turbulent penetration, temperature fluctuations with large amplitude and high frequency can lead to time-varying wall thermal stress and even thermal fatigue on the inner wall. Numerous cases, however, exist where inner wall temperatures cannot be measured and only outer wall temperature measurements are feasible. Therefore, it is one of the popular research areas in nuclear science and engineering to estimate temperature fluctuations on the inner wall from measurements of outer wall temperatures without damaging the structure of the pipe. In this study, both the one-dimensional (1D) and the two-dimensional (2D) inverse heat conduction problem (IHCP) were solved to estimate the temperature fluctuations on the inner wall. First, numerical models of both the 1D and the 2D direct heat conduction problem (DHCP) were structured in MATLAB, based on the finite difference method with an implicit scheme. Second, both the 1D IHCP and the 2D IHCP were solved by the steepest descent method (SDM), and the DHCP results of temperatures on the outer wall were used to estimate the temperature fluctuations on the inner wall. Third, we compared the temperature fluctuations on the inner wall estimated by the 1D IHCP with those estimated by the 2D IHCP in four cases: (1) when the maximum disturbance of temperature of fluid inside the pipe was 3°C, (2) when the maximum disturbance of temperature of fluid inside the pipe was 30°C, (3) when the maximum disturbance of temperature of fluid inside the pipe was 160°C, and (4) when the fluid temperatures inside the pipe were random from 50°C to 210°C.

Firefly algorithm combined with Newton method to identify boundary conditions for transient heat conduction problems

ABSTRACTFirefly algorithm combined with Newton method (FA–NM) is proposed for identifying time-dependent boundary conditions of two-dimensional transient heat conduction problems with a heat source. The dual reciprocity boundary element method (DRBEM) is applied to solve the direct problem. The improved firefly algorithm has not only a good global search ability, but also a good local search ability. FA–NM can acquire accurate results with much less iterations. Furthermore, different measurement points and noises are also considered. A small number of measurement points cannot obtain precise results. With the increase in the measurement points to a certain value, the results are more accurate. When the measurement noises are not very large, FA–NM can get desirable results. With the decrease in measurement noises, the results are in better agreement with the exact solutions. Numerical results also indicate that the more front the time substep is, the more accurate the estimated boundary conditions are.

A Raviart-Thomas mixed finite element scheme for the two-dimensional three-temperature heat conduction problems

Summary For the two-dimensional three-temperature radiative heat conduction problem appearing in the inertial confinement numerical stimulations, we choose the Freezing coefficient method to linearize the nonlinear equations, and initially apply the well-known mixed finite element scheme with the lowest order Raviart–Thomas element associated with the triangulation to the linearized equations, and obtain the convergence with one order with respect to the space direction for the temperature and flux function approximations, and design a simple but efficient algorithm for the discrete system. Three numerical examples are displayed. The former two verify theoretical results and show the super-convergence for temperature and flux functions at the barycenter of the element, which is helpful for solving the radiative heat conduction problems. The third validates the robustness of this scheme with small energy conservative error and one order convergence for the time discretization. Copyright © 2016 John Wiley & Sons, Ltd.

The application of the Beck`s method combined with the FEM and Trefftz functions to determine the heat transfer coefficient in minichannel

The aim of this study is to determine the heat transfer coefficient between the heated surface and the boiling fluid flowing in a minichannel on the basis of experimental data. The calculation model is based on Beck’s method coupled with the FEM and Trefftz functions. The Trefftz functions used in the Hermite interpolation are employed to construct the shape functions in the FEM. The unknown local values of the heat transfer coefficient at the foil- -fluid contact surface are calculated from Newton’s law. The temperature of the heated foil and the heat flux on the foil surface are determined by solving a two-dimensional inverse heat conduction problem. The study is focused on the identification of the heat transfer coefficients in the subcooled boiling region and the saturated nucleate boiling region. The results are compared with the data obtained through the one-dimensional method. The investigations also reveal how the smoothing of measurement data affects calculation results.

Critical Depth of Buried Isothermal Circular Pipes

ABSTRACTWe address the two-dimensional heat conduction problem due to a periodic array of isothermal pipes buried in a conductive medium. The upper surface of the medium is subjected to convection with a uniform heat transfer coefficient, and the lower surface is insulated. Similar to the concept of critical thickness associated with a slab embedded with isothermal strips, we show that there exists a critical depth such that the heat transfer rate is maximized. As the Biot number tends to infinity, the critical depth approaches zero for a single pipe buried in a semi-infinite medium. For a periodic array of isothermal pipes, there is also a critical Biot number beyond which the critical depth is zero. Furthermore, insulating the pipes reduces the critical depth, and the heat transfer rate does not vary significantly with respect to the depth.

SOLUTION OF MULTIPHASE HEAT CONDUCTION PROBLEMS VIA THE GENERALIZED INTEGRAL TRANSFORM TECHNIQUE WITH DOMAIN CHARACTERIZATION THROUGH THE INDICATOR FUNCTION

The Generalized Integral Transform Technique (GITT) has appeared in the literature as an alternative to conventional discrete numerical methods for partial differential equations in heat transfer and fluid flow. This method permits the automatic control of the error and is easy to program, since there is no need for a discretization. The method has being constantly improved, but there still a vast number of practical problems that has not being solved satisfactory. In several brands of engineering, the transport equations have to be solved for a combination of different phases or materials or inside irregular domains. In this case, the mathematical resource of the Indicator Function can be employed. This function is a representation of the phases or parts of the domain with the numbers 0 and 1 for each phase. According to the method, the Indicator Function is defined by Poisson’s equation, which is added to the system of the transport equations. An integral is done along the curve that defines the interface that will generate the source term in Poisson’ equation used to calculate the Indicator Function distribution. The solution of the system of equations is done using the common GITT approach. Then, an analytical expression for each transformed potential of the indicator function and the other variables are available. Once the transformed potentials are known, the Indicator Function can be analytically operated, and the interface can be represented by an analytical continuous function. In this work, the use of the GITT in conjunction with the Indicator Function is proposed. The methodology is described and some previous results are presented. GITT is applied to a two-dimensional heat conduction problem in a multiphase domain with an irregular geometry, inside a square domain. The methodology presented here can be extended to all brands of convection-diffusion problems already solved via GITT.

Coupling of Adaptive Element Free Galerkin Method with Variational Multiscale Method for Two-Dimensional Transient Heat Conduction Problem

AbstractIn this paper, the coupling of adaptive element free Galerkin method (EFGM) with variational multiscale method is developed to solve two-dimensional transient heat conduction problem. The EFGM is chosen to combine with the adaptive analysis since this method only use a set of particles to generate the discrete equations instead of mesh construction as in finite element method (FEM). This property makes additional nodes can be easily inserted at the high gradient regions to improve the numerical results. The refinement criteria and refinement strategy will be clearly shown for the adaptive analysis procedure. The performance of this coupling method is first validated by solving two-dimensional large localized gradient problem with available theoretical solutions. Next, this method will also be used to solve two-dimensional transient heat conduction problem. The accuracy and efficiency of the method is demonstrated by comparing the results between conventional EFGM and the proposed method. The results obtained show that this method is able to produce higher accuracy results compared with the standard element free Galerkin method.

Analysis of reverse heat transfer for conventional and optimized lubri-cooling methods during tangential surface grinding of ABNT 1020 steel

A numerical thermal model was developed to evaluate the heat flux which is conducted to a rectangular workpiece of steel plate ABNT 1020, thus making it possible to compute the maximum temperature in the grinding surface, taking into account the rectangular distribution of heat flux, the thermal properties of the grinding wheel conventional Al2O3, the piece to be machined and the lubri-refrigerating fluid. The finite volume method was employed for the discretization of the direct thermal problem from the heat diffusion equation associated with the two-dimensional problem of heat conduction in transient regime. The inverse thermal problem was solved by the Golden Section technique. The thermal flux, when compared to the conventional technique of method of application fluid, was reduced by 84.0% in the practices performed with cutting depth of 30µm, at 74.0% in practices with cutting depth of 45µm and 61.2% in the aggressive practices of 60µm, thus demonstrating the applicability of the optimized method for fluid application.

Thermoelastic State of a Half Plane with Curvilinear Crack Under the Conditions of Local Heating

We consider a two-dimensional problem of stationary heat conduction and thermoelasticity for a half plane containing an internal curvilinear crack and heated by a heat flow in a local region. We reduce the problem to two singular integral equations on the crack contour. The numerical solution of the integral equations is obtained by the method of mechanical quadratures. The stress intensity factors at the tips of a parabolic crack are determined.

The Influence of Subsurface Temperature Measurements on the Determination of Transient Wall-Side Boundary Conditions: An Analytical Tool

An analytical approximation for the two-dimensional heat conduction problem with temporally and spatially oscillating thermal boundary conditions is presented. In addition, simplified expressions for the damping and phase shift of the temperature signal between the surface and a “measurement” position inside the wall are derived and their validity in dependency on the inverse Fourier number is shown. The solution algorithm is implemented in a MATLAB program and made available for other researchers. In two case studies, the analytical solution is applied in order to evaluate the ability of experimental systems to measure instantaneous and local heat transfer coefficients as well as temperatures on a boundary.

Parameter estimation in heat conduction using a two-dimensional inverse analysis

ABSTRACTThis article is concerned with a two-dimensional inverse steady-state heat conduction problem. The aim of this study is to estimate the thermal conductivity, the heat transfer coefficient, and the heat flux in irregular bodies (both separately and simultaneously) using a two-dimensional inverse analysis. The numerical procedure consists of an elliptic grid generation technique to generate a mesh over the irregular body and solve for the heat conduction equation. This article describes a novel sensitivity analysis scheme to compute the sensitivity of the temperatures to variation of the thermal conductivity, the heat transfer coefficient, and the heat flux. This sensitivity analysis scheme allows for the solution of inverse problem without requiring solution of adjoint equation even for a large number of unknown variables. The conjugate gradient method (CGM) is used to minimize the difference between the computed temperature on part of the boundary and the simulated measured temperature distribution. The obtained results reveal that the proposed algorithm is very accurate and efficient.

Effect of Laser Pulse Duration on the Error in Determining the Thermal Characteristics of Construction Materials

The errors in determining the thermal characteristics of materials with a finite duration laser pulse method are estimated. Handbook values of the characteristics of materials are compared with numerical solutions of the two-dimensional problem of heat conduction in a sample during exposure to a heat pulse of short duration under experimental conditions.

An approach to consider lateral heat conduction effects in the evaluation process of transient heat transfer measurements using TLC

The standard transient evaluation technique for the experimental determination of heat transfer coefficients with Thermochromic Liquid Crystals is based on a one-dimensional heat conduction model inside a semi-infinite wall. However, complex flow situations induced by obstacles like cylinders, skewed ribs or vortex generators lead to spatially varying heat transfer coefficients and therefore to lateral conduction effects. Kingsley-Rowe et al. [1] developed an analytical empirical correction method considering these effects for flat two-dimensional heat conduction problems and an ideal step change of the fluid reference temperature. This method is expanded to capture three-dimensional heat conduction situations and slow transient fluid reference temperature changes. Transient simulations for twelve different heat transfer coefficient distributions and three different reference temperature histories constitute the pool of data for the determination of the empirical constants and the validation of the method. The results showed an appropriate correction of the heat transfer coefficients received by the standard evaluation method. Therefore the presented method offers the opportunity to consider three-dimensional heat conduction effects in a fast and efficient way in combination with the standard evaluation process.

Heat transfer enhancement of a periodic array of isothermal pipes

We address the problem of two-dimensional heat conduction in a solid slab whose upper and lower surfaces are subjected to uniform convection. In the midsection of the slab there is a periodic array of isothermal pipes of general cross section. The main objective of this work is to find the optimum shapes of the pipes that maximize the Shape Factor (heat transport rate). The Shape Factor is obtained by transforming the periodic array of pipes into a periodic array of strips, using the generalized Schwarz–Christoffel transformation, and applying the collocation boundary element method on the transformed domain. Subsequently we pose the inverse problem, i.e. finding the shape that maximizes the Shape factor given the perimeter of the pipes. For large Biot number the optimum shapes are in agreement with the isothermal case, i.e. circular for sufficiently small perimeters/heat transfer, and elongated towards the surfaces of the slab for larger perimeters/heat transfer. Furthermore, for the isothermal case, we were able to discover a new family of optimum shapes for large thickness of the slab and large perimeters, which do not have their maximum width on the horizontal axis of symmetry. For small Biot number the optimum pipes are flatter than the isothermal ones for a given perimeter. The flatness becomes more apparent for larger perimeters. Most important, for large perimeters there exists a critical thickness which is characterized by maximum heat transfer rate. This is further investigated using the finite element method to obtain the critical thickness of a slab and the critical depth of the periodic array of circular pipes.

Prediction of time-varying heat flux along a hollow cylindrical tube wall using recursive input estimation algorithm and thermal resistance network method

In this study, a model using the thermal resistance network method and the recursive input estimation algorithm was developed to analyze two-dimensional inverse heat conduction problems. The recursive input estimation algorithm consisted of the Kalman filter and the real-time least squares. A two-layered hollow cylindrical tube was assumed with the convection heat transfer boundary condition at the outer wall of the tube. Using the model developed in this study, an unknown time-varying heat flux at the inner wall of the tube was predicted from the known temperatures at some positions on the outer wall of the tube in the longitudinal direction. In the process of predicting the unknown heat flux, the temperature distribution in the two-layered hollow cylindrical tube wall was also obtained. The present model was able to accurately predict the unknown time-varying heat flux profiles of square waves or sinusoidal-like waves with and without spatial variations in the longitudinal direction.