2D Transient Heat Conduction Research Articles

Step-wise transient method for analysis of thermal properties of materials Part 2. Experimental investigations

The subject of the paper which is a continuation of part 1 are issues related to the implementation of the stepwise transient plane source method. Both the measuring device and the testing methodology were developed. Measurements of thermophysical properties k, a and cp of selected materials from the group of thermal insulating foams (polyurethanes, polystyrenes), organic glass polymers and metals, including high thermal conductivity ones, i.e. duralumin at one measurement, were carried out. The measurement time was on the order of a few to several hundred seconds. Procedures for calculating thermal effusivity b and thermal diffusivity a were illustrated. Parameters of the thin-layer heater used in the measurement system were studied, i.e. measurements of the resistance change of the heater in the function of the voltage of the power supply and examination of the heater homogeneity were performed using a thermal imaging camera. The method requires that the thermocouples should be mounted on the faces of the samples. According to authors, this is an advantage of the method, because it allows to limit the applied pressure of the measuring stack on the accuracy of materials thermophysical properties measurements, especially metals. The measurements confirmed the suitability of the method for testing the thermophysical parameters of different groups of materials. The obtained experimentally accuracy of thermophysical parameters of testing materials in relation to reference values did not exceed 10%. The usefulness of the analytical solution of 2D transient heat conduction problem in the considered measuring stack has been confirmed by the results of numerical simulation, which includes heat transfer in a sample and in a flexible thin layer Kapton-nichrome heater.

Simplification of 3D transient heat conduction by reducing it to an axisymmetric heat conduction problem and a new inverse method of the problem solution

Most IHCP-solving methods available in literature are formulated for simple-shaped bodies and cannot be applied to complex geometries. In this work, the 3-D problem is simplified to an axisymmetric analysis. If the nozzle diameter is not much smaller than the diameter of the cylindrical component, a corrected axisymmetric model is proposed. Next, a method is presented for solving the axisymmetric IHCP in a complex domain. It is based on the control volume finite element method. Based on temperature transients measured on the outer surface, the temperature distribution is reconstructed by marching from the known to the unknown boundary. The developed method is applied for temperature identification in a cylindrical component with a nozzle. The presented algorithm is tested using measured temperatures generated from a direct solution. The transient temperature distribution obtained from the method presented in the paper is compared with the values obtained from the direct solution. The proposed method is also used to estimate the unknown boundary condition. The information about the heat transfer coefficient value makes it possible to describe the heat transfer phenomena occurring inside the component.The presented method makes it possible to optimize the power unit start-up and shutdown, contributes to a reduction in heat losses arising during the operations and enables extension of the power unit life. The method can be used in monitoring systems of both conventional and nuclear power plants.

Three-dimensional transient heat conduction analysis by boundary knot method

This paper makes the first attempt to apply the boundary knot method (BKM), in conjunction with dual reciprocity technique, for the solution of three-dimensional transient heat conduction problems. The BKM is a meshless, integration-free, easy-to-program boundary-only numerical technique for high-dimensional problems. The first step of our strategy is to use the finite difference method for temporal derivative to convert the transient heat conduction equation into a nonhomogeneous modified Helmholtz equation. And then the corresponding nonhomogeneous problem is solved using the proposed BKM strategy in conjunction with dual reciprocity technique. Four benchmark numerical examples are investigated in detail, and the numerical results show that the present scheme has the merits of high accuracy, wide applicability, good stability, and rapid convergence and is appealing to solve 3D transient heat conduction problems.

On the sources placement in the method of fundamental solutions for time-dependent heat conduction problems

The method of fundamental solutions is a more and more popular meshless method for solving boundary or initial–boundary value problems. The most important issue in this method is the determination of the positions of the source points. The accuracy of the method depends strongly on the distribution of the source points. In this paper placement of these points for transient heat conduction problems is studied. The problems are initial–boundary value problems and they are considered in a time-space domain. Because of that, the placement of the source points differs from the classical distribution of the source points for boundary values problems. In the paper, four different possible sources distributions are considered for 1D, 2D and 3D transient heat conduction problems. The results show very good accuracy in case of the source points placed in a space much bigger than the considered region, additionally with the negative time coordinate.

The dimension splitting element-free Galerkin method for 3D transient heat conduction problems

By transforming a 3D problem into some related 2D problems, the dimension splitting element-free Galerkin (DSEFG) method is proposed to solve 3D transient heat conduction problems. The improved element-free Galerkin (IEFG) method is used for 2D transient heat conduction problems, and the finite difference method is applied in the splitting direction. The discretized system equation is obtained based on the Galerkin weak form of 2D problem; the essential boundary conditions are imposed with the penalty method; and the finite difference method is employed in the time domain. Four exemplary problems are chosen to verify the efficiency of the DSEFG method. The numerical solutions show that the efficiency and precision of the DSEFG method are greater than ones of the IEFG method for 3D problems.

On code verification of 2D transient heat conduction in composite wall

The objective of the present research paper is to verify and validate the C++ code to solve a 2D transient heat conduction problem. A 2-D wall comprised of four different materials was chosen as a domain to analyse numerically in a commercial software FLUENT and C++. The results were compared and found that the non-uniformity in conductivity arises when there exists non-homogeneity in the material. It is also found that C++ results varies on ‘f’ value in finite volume method’s (FCM’s) discretization scheme which determines the solution scheme as explicit or implicit. The code was executed for explicit and implicit scheme and made sure that it follows Courant–Friedrichs–Lewy (CFL) conditions.

Transient 3D simulation for heating and melting process of PWR core after SBO

In the heat transfer between the core and coolant in a reactor pressure vessel (RPV), a bounding structure such as a shroud has an important influence on the heating and melting process of the core. To study the 3D transient heat transfer and melting process of the Hualong 1 PWR using an IVR strategy, a numerical analysis model has been established based on the 2D transient heat conduction of each rod, which is coupled considering the residual heat of the core, the convective and radiation heat transfer inside the core, and natural convection heat transfer of the outer water chamber. A transient 3D numerical simulation code, 3DTMCOR, has been developed to analyze the melting process of an IVR reactor core with high precision. The effects of heat generation from a zirconium–water reaction, radiation heat transfer, and the adoption of an IVR strategy on the heating and melting process of the reactor core have been investigated. It was found that the first melting time for the same 100 MW PWR from the 3DTMCOR code is about 2500 s later than that of MAAP. The core temperature will first decrease with a decrease in core power within 100 s, then increase as the water level drops down with a zirconium–water reaction of 100 s to 1000 s, then decrease again with a decrease in core power from 1000 s to 4300 s, and increase a second time when water in the RPV is evaporated after 4300 s. At 4860 s, a continuous melting area first appears in the core top. Radiation heat exchange is very important for transferring heat from the fuel rods to structures such as the control rods and bounding structures, which is also the main cooling mode in the core after water is evaporated. The first melting time with the precise radiation exchange model will occur 200 s later than in the results with the empirical model. The water chamber outside the RPV does not have a remarkable effect on the heating and melting process of the core.

The dimension splitting and improved complex variable element‐free Galerkin method for 3‐dimensional transient heat conduction problems

SummaryIn this paper, by combining the dimension splitting method and the improved complex variable element‐free Galerkin method, the dimension splitting and improved complex variable element‐free Galerkin (DS‐ICVEFG) method is presented for 3‐dimensional (3D) transient heat conduction problems. Using the dimension splitting method, a 3D transient heat conduction problem is translated into a series of 2‐dimensional ones, which can be solved with the improved complex variable element‐free Galerkin (ICVEFG) method. In the ICVEFG method for each 2‐dimensional problem, the improved complex variable moving least‐square approximation is used to obtain the shape functions, and the penalty method is used to apply the essential boundary conditions. Finite difference method is used in the 1‐dimensional direction, and the Galerkin weak form of 3D transient heat conduction problem is used to obtain the final discretized equations. Then, the DS‐ICVEFG method for 3D transient heat conduction problems is presented. Four numerical examples are given to show that the new method has higher computational precision and efficiency.

The numerical manifold method for 2D transient heat conduction problems in functionally graded materials

Benefiting from the use of two cover systems, that is, the mathematical cover and the physical cover, the numerical manifold method (NMM) is capable of solving both continuous and discontinuous problems in the same platform. Presently, the NMM is further developed to tackle two-dimensional transient heat conduction problems in the functionally graded materials (FGMs). Firstly, the governing equation, the associated boundary conditions and the initial condition are presented. Then, the fundamentals of the NMM are briefly reviewed. Following, the NMM discrete formulations are derived based on the Galerkin-form weighted residual method and then solved with the backward difference scheme. Finally, for verification, three numerical examples with increasing complexity are tested on uniform mathematical covers composed of square mathematical elements, and our results well demonstrate the advantages of the proposed method in discretization and accuracy; besides, the effects of material gradient on the thermal behavior of FGMs are also examined.

Modeling 2D transient heat conduction problems by the numerical manifold method on Wachspress polygonal elements

• The numerical manifold method is further developed to solve 2D transient heat conduction problems. • The Wachspress polygonal element is extended to thermal analysis. • Regular hexagonal mathematical elements are adopted in numerical examples. • Good agreement between our results and existing solutions is achieved. • The advantages of the NMM in discretization and accuracy are demonstrated. Due to the use of dual cover systems, i.e., the mathematical cover and the physical cover, the numerical manifold method (NMM) is able to solve physical problems with boundary-inconsistent meshes. Meanwhile, n -gons ( n >4) are very impressive, due to their greater flexibility in discretization, less sensitivity to volumetric and shear locking, and better suitability for complex microstructures simulation. In this paper, the NMM, combined with Wachspress-type hexagonal elements, is developed to solve 2D transient heat conduction problems. Based on the governing equations, the NMM temperature approximation and the modified variational principle, the NMM discrete formulations are deduced. The solution strategy to time-dependent global equations and the spatial integration scheme are presented. The advantages of the proposed approach in both discretization and accuracy are demonstrated through several typical examples with increasing complexity. The extension of polygonal elements in unsteady thermal analysis within the NMM is realized.

Simulation of three-dimensional transient heat conduction problem with variable coefficients based on the improved parallel smoothed particle hydrodynamics method

In this work, an improved parallel SPH method is proposed to accurately solve the three-dimensional (3D) transient heat conduction equation with variable coefficients. The improvements are described as follows. Firstly, the first-order symmetric smoothed particle hydrodynamics (SPH) method is extended to the simulating of the 3D problem based on Taylor expansion. Secondly, the concept of stabilized up-wind technique is introduced into the convection term. Thirdly, the MPI parallel technique based on the neighboring particle mark method is introduced into the above improved SPH method, and named the corrected parallel SPH method for 3D problems (CPSPH-3D). Subsequently, the accuracy, convergence and the computational efficiency of the proposed CPSPH-3D method are tested by simulating the 3D transient heat conduction problems with constant/variable coefficient, and compared with the analytical solution. Meanwhile, the capacity of the proposed CPSPH-3D for solving the 3D heat conduction problems with the Dirichlet and Newmann boundaries is illustrated, in which the change of temperature with time under the complex cylindrical area is also considered. The numerical results show that:1) the proposed CPSPH-3D method has the better stability, higher accuracy and computational efficiency than the conventional SPH method no matter whether the particle distribution is uniform; 2) the calculating time can be well reduced by increasing the number of CPUs when the particle number is refined in the simulations of CPSPH-3D. Finally, the temperature variation in the 3D functionally gradient material is predicted by the corrected parallel SPH method, and compared with the other numerical results. The process of temperature variation in the functionally gradient material can be shown accurately.

Numerical prediction of hole profile in laser drilling process and experimental validation

In the present paper, a simple numerical method is proposed to predict the hole contour in a workpiece due to the action of a multi-pulsed laser. A 3D transient heat conduction equation has been used to predict the temperature at various points in the workpiece subjected to multi-pulsed (Gaussian beam (TEM00) mode) laser beam. Subsequently, the hole contour is predicted from the melt isotherm contour in the workpiece. A finite difference implicit splitting scheme has been chosen to solve the above 3D transient heat conduction equation because of several advantages of this techniques over other techniques. The predicted hole contour has been verified with the experimentally obtained hole contour under similar condition of laser and process parameters. It is observed that the numerically predicted hole contour is matching fairly well with the experimental hole contour. The second objective of this paper is to exploit this numerical model to obtain an optimum combination of laser and process parameters for getting a good quality hole. Hence, the proposed numerical model can be used to set optimum process parameters for laser-drilling operation in industrial applications.

Numerical Solution of Unsteady Conduction Heat Transfer in Anisotropic Cylinders

This paper investigates a numerical solution of 2D transient heat conduction in an anisotropic cylinder, subjected to a prescribed temperature over the two end sections and a convective boundary condition over the whole lateral surface. The analysis of this anisotropic heat conduction problem is tedious because the corresponding partial differential equation contains a mixed-derivative. In order to overcome this difficulty, a linear coordinate transformation is used to reduce the anisotropic cylinder heat conduction problem to an equivalent isotropic one, without complicating the boundary conditions but with a more complicated geometry. The alternating-direction implicit finite-difference method (ADI) is used to integrate the isotropic equation combined with boundary conditions. Inverse transformation provides profile temperature in the anisotropic cylinder for full-field configuration. The numerical code is validated by the analytical heat conduction solutions available in the literature such as transient isotropic solution and steady-state orthotropic solution. The aim of this paper is to study the effect of cross-conductivity on the temperature profile inside an axisymmetrical anisotropic cylinder versus time, radial Biot number (Bir), and principal conductivities. The results show that cross-conductivity promotes the effect of Bir according to the principal conductivities. Furthermore, the anisotropy increases the time required to achieve the steady-state heat conduction.

Analytical transformation of volume integral for the time-stepping BEM analysis of 2D transient heat conduction in anisotropic media

Despite the extensive study of the transient heat conduction analysis for isotropic media by the boundary element method (BEM), its relevant researches in transient heat conduction in anisotropic media still remain relatively scarce indeed. In the time-stepping BEM scheme for the transient heat conduction, the transient effect reveals itself as an additional volume integral that conventionally requires domain discretization for direct integration. However, such domain discretization will destroy the BEM׳s most distinctive nature that only the boundary needs to be discretized. In this paper, the domain integral is analytically transformed to the boundary so that the BEM׳s nature of boundary discretization is completely restored. Moreover, the domain mapping technique is combined with this transformed time-stepping scheme to treat the 2D transient heat conduction in anisotropic media. Without any internal treatments or special approximations like in other schemes, the transformed time-stepping scheme can be applied to effectively solve the problem of transient heat conduction in anisotropic media.

An Effective Inverse Heat Transfer Procedure Based on Evolutionary Algorithms to Determine Cooling Conditions of a Steel Continuous Casting Machine

The inverse modeling of heat transfer is a useful tool in analyzing contact heat transfer at the ingot surfaces during the continuous casting process. The determination of the boundary conditions involves an experimental work consisting in the evaluation of the thermal history, generally at the casting surface, experimentally provided by infrared pyrometers. Additionally, numerical simulations, based on the solution of the 2D transient heat conduction equation, are performed in order to be inversely solved in response to the measured thermal data furnished by the sensor. Due to computational time consumption during simulations in searching cooling conditions, this work proposes an interaction between natural inspired algorithms, called evolutionary algorithms, and the numerical model in order to speed up the searching process. The present work aims to compare three algorithms, namely genetic algorithm, improved stochastic ranking evolutionary strategy, and evolutionary strategy with Cauchy distribution. The latter develops a metaheuristic version of an evolutionary strategy workflow, using a Cauchy random number function to generate each individual, instead of the usual uniform distribution function available in almost all programming languages. The surface temperature, solid shell, and molten pool profiles from the determined cooling conditions are analyzed in terms of casting quality.

Thermal–numerical model and computational simulation of pulsed thermography inspection of carbon fiber-reinforced composites

This paper describes a methodology for the modeling and simulation of a pulsed thermography inspection of carbon fiber-reinforced composites. A thermal–numerical model based on the 3D transient heat conduction equation for heterogeneous media is proposed to model the thermal response of the composite medium when a thermal pulse is applied to its surface. The solution of the model is developed using the finite volume method, in which the approximated equations are obtained by performing energy balances for each elementary volume. The experimental validation of the thermal–numerical model consisted of a comparison of the thermal decay curves for defective areas obtained through numerical simulation and with a pulsed thermography inspection of a laminated composite specimen with simulated defects (Teflon inserts). A parametric study was carried out in order to analyze the influence of the irradiation power density, non-uniform heating and characteristics associated with the defects on quantitative variables such as the onset time and thermal contrast. Numerical results showed that the variable thermal contrast is highly sensitivity to changes in the defect thickness, aspect ratio and parameters related to the external stimulation. Furthermore, an increase in the detectability of defects – especially those with lower aspect ratios – can be achieved by increasing the irradiation power.

Parametric integral equations systems in 2D transient heat conduction analysis

Currently the most popular numerical methods used for solving transient heat conduction problems, finite element method (FEM) and boundary element method (BEM), have one fundamental defect – the necessity of discretizing the boundary or the domain. This problem escalates even more, when using an iterative process. An alternative to avoid the mentioned problem are parametric integral equations systems (PIES), which do not require classical discretization of the boundary and the domain while being numerically solved. PIES method was previously used with success to solve steady-state problems. The purpose of this paper is to present PIES method for 2D transient heat conduction problems and present results obtained by solving few numerical examples. Accuracy and effectiveness of the method are shown in comparison with analytical solutions and FEM.

The Interpolating Element-Free Galerkin Method for 2D Transient Heat Conduction Problems

An interpolating element-free Galerkin (IEFG) method is presented for transient heat conduction problems. The shape function in the moving least-squares (MLS) approximation does not satisfy the property of Kronecker delta function, so an interpolating moving least-squares (IMLS) method is discussed; then combining the shape function constructed by the IMLS method and Galerkin weak form of the 2D transient heat conduction problems, the interpolating element-free Galerkin (IEFG) method for transient heat conduction problems is presented, and the corresponding formulae are obtained. The main advantage of this approach over the conventional meshless method is that essential boundary conditions can be applied directly. Numerical results show that the IEFG method has high computational accuracy.

Sequentially Coupled Thermal-Mechanical Analysis of the Brake Disc Used in Air Disc Brake

The basic theory involved in the coupled thermal mechanical analysis is presented briefly. Base on the contact theory, friction heat generation theory and heat transfer theory, the heat distribution ratio between the disc and brake pad is determined, a 3D transient heat conduction differential equation and corresponding boundary conditions are established. Finally, according to the analysis requirement, the business analysis software ABAQUS is chosen as sequentially coupled thermal mechanical analysis solver. In order to know the temperature of discs friction surface during the cycle braking and continuous braking processes, a sequentially coupled thermal mechanical analysis model of the disc is established according to the energy dissipation theory. The temperature field and stress field of the disc during emergency braking, repeat braking and continuous braking processes are carried out based on this model.

Heat Transfer Evaluation on Curved Boundaries in Thermal Lattice Boltzmann Equation Method

An efficient and accurate approach for heat transfer evaluation on curved boundaries is proposed in the thermal lattice Boltzmann equation (TLBE) method. The boundary heat fluxes in the discrete velocity directions of the TLBE model are obtained using the given thermal boundary condition and the temperature distribution functions at the lattice nodes close to the boundary. Integration of the discrete boundary heat fluxes with effective surface areas gives the heat flow rate across the boundary. For lattice models with square or cubic structures and uniform lattice spacing the effective surface area is constant for each discrete heat flux, thus the heat flux integration becomes a summation of all the discrete heat fluxes with constant effective surface area. The proposed heat transfer evaluation scheme does not require a determination of the normal heat flux component or a surface area approximation on the boundary; thus, it is very efficient in curved-boundary simulations. Several numerical tests are conducted to validate the applicability and accuracy of the proposed heat transfer evaluation scheme, including: (i) two-dimensional (2D) steady-state thermal flow in a channel, (ii) one-dimensional (1D) transient heat conduction in an inclined semi-infinite solid, (iii) 2D transient heat conduction inside a circle, (iv) three-dimensional (3D) steady-state thermal flow in a circular pipe, and (v) 2D steady-state natural convection in a square enclosure with a circular cylinder at the center. Comparison between numerical results and analytical solutions in tests (i)–(iv) shows that the heat transfer is second-order accurate for straight boundaries perpendicular to one of the discrete lattice velocity vectors, and first-order accurate for curved boundaries due to the irregularly distributed lattice fractions intersected by the curved boundary. For test (v), the computed surface-averaged Nusselt numbers agree well with published results.