Numerical Simulation Of Heat Conduction Research Articles

Numerical Simulation of Heat Conduction in Laser Ablation Based on Optimal Weight Factor

Numerical Simulation of Heat Conduction in Laser Ablation Based on Optimal Weight Factor

Observation-based temperature field simulation at Zhurong landing site, Mars

Modeling the temperature field near the Martian surface is critical for many scientific exploration tasks, such as detecting liquid water and analyzing the existence of saline ice. Meteorological conditions on Mars are highly dramatic, with a daily temperature change of up to 80–100 K. Most previous tasks of surface temperature monitoring on Mars are based on satellite observations, lacking in-situ measured data. Recently, two Martian missions at mid-low latitudes in the northern hemisphere, InSight lander and Zhurong rover, carried out near-surface temperature observations. However, the temperature monitoring of the Zhurong rover obtained data for only some short periods in its working days; thus, the amount of recorded temperature data is inadequate for a whole-day analysis at the landing site. Here we reconstruct the near-surface temperature at the Zhurong landing site by incorporating the continuous temperature data observed at the InSight lander, simultaneously referring to the Martian Climate Database; then, the reconstructed data are used to constrain the numerical simulation of the response of shallow subsurface under the Zhurong landing site. The numerical simulation of heat conduction shows that the daily temperature change under the Zhurong landing site mainly influences the uppermost depth of 0–30 cm, with a daily average temperature of ∼225 K. During the traveling duration of the Zhurong rover (i.e., summer of Mars), the seasonal temperature change within the top 1 m is significant and is related to the thermal properties of possible subsurface media (e.g., soil, ice, and sandstones). Although there might be aqueous activities in Utopia Planitia, our results show that from the perspective of temperature field, there is little possibility of liquid water in the shallow subsurface under the Zhurong landing site. The proposed method in this study provides a new way for the temperature field simulation of the subsurface in areas with insufficient local observations, especially on extraterrestrial objects.

Combined EBSD and Computer-Assisted Quantitative Analysis of the Impact of Spark Plasma Sintering Parameters on the Structure of Porous Materials

The paper presents the experimental, numerical, and theoretical investigation of the microstructure of nickel aluminide samples manufactured by spark plasma sintering using electron backscatter diffraction and computer assisted software. The aim of the work was to reveal the evolution of the microscopic and macroscopic parameters related to the microstructure of the material and its dependence on the applied sintering parameters—temperature and pressure. The studied porous samples with different relative density were extracted from various planes and then tested by electron backscatter diffraction to evaluate the crystallographic orientation in every spot of the investigated area. On this foundation, the grain structure of the samples was determined and carefully described in terms of the grain size, shape and boundary contact features. Several parameters reflecting the grain morphology were introduced. The application of the electric current resulting in high temperature and the additional external loading leads to the significant changes in the structure of the porous sample, such as the occurrence of lattice reorientation resulting in grain growth, increase in the grain neighbours, or the evolution of grain ellipticity, circularity, grain boundary length, and fraction. Furthermore, the numerical simulation of heat conduction via a finite element framework was performed in order to analyse the connectivity of the structures. The numerical results related to the thermal properties at the micro- and macroscopic scale—local heat fluxes, deviation angles, and effective thermal conductivity—were evaluated and studied in the context of the microstructural porosity. Finally, the effective thermal conductivity of two-dimensional EBSD maps was compared with those obtained from finite element simulations of three-dimensional micro-CT structures. The relationship between the 2D and 3D results was derived by using the analytical Landauer model.

Numerical Simulation of Heat Conduction and Convection from Geothermal Zone to Shale Oil Reservoir

Numerical Simulation of Heat Conduction and Convection from Geothermal Zone to Shale Oil Reservoir

Thermal analysis of heat transfer in pipe cooling concrete structure by a meshless RBF-FD method combined with an indirect model

Numerical simulation of heat conduction in the concrete structure with cooling pipes is meaningful and full of challenges. In this paper, the radial basis function finite difference method (RBF-FD) combined with a self-correcting prediction model (SCM) is proposed for thermal analysis of heat transfer in concrete structures with pipe cooling system. Using the Multiquadric radial basis function (MQ-RBF), the SCM is proposed to give full play to advantages of the localized meshless method. This technique allows use of more number of nodes in the whole domain to solve the ill-conditioning problem associated with multiple regions of rapid variation and increases the accuracy of models. A multiple-scale technique is applied to determine the shape parameter in the MQ-RBF and the fictitious nodes method is modified by using just one fictitious node outside the domain to deal with boundary conditions of the third kind in tiny holes. To show the flexibility and efficiency of the proposed scheme, we consider several examples with different number, size, and positions of pipes. Meanwhile, a guidance of determination of effected area near a tiny hole is provided and a simple optimization design of cooling pipe system is performed. The results demonstrate advantages of the proposed model which can effectively handle heat conduction problem with multiple holes subjected to the third kind boundary conditions.

A partition of unity finite element method for nonlinear transient diffusion problems in heterogeneous materials

This work studies for the first time the solution of a nonlinear problem using an enriched finite element approach. Such problems can be highly demanding computationally. Hence, they can significantly benefit from the efficiency of the enriched finite elements. A robust partition of unity finite element method for solving transient nonlinear diffusion problems is presented. The governing equations include nonlinear diffusion coefficients and/or nonlinear source terms in both homogeneous and heterogeneous materials. To integrate the equations in time, we consider a linearly semi-implicit scheme in the finite element framework. As enrichment procedures, we consider a combination of exponential expansions to be injected in the finite element basis functions on coarse meshes. The proposed method shows a large reduction in the number of degrees of freedom required to achieve a fixed accuracy compared to the conventional finite element method. In addition, the proposed partition of unity method shows a stable behavior in treating both internal and external boundary layers in nonlinear diffusion applications. The performance of the proposed method is also used for the numerical simulation of heat conduction in functionally graded materials.

Effect of deformation on the thermal conductivity of granular porous media with rough grain surface

AbstractHeat transfer in granular porous media is an important phenomenon that is relevant to a wide variety of problems, including geothermal reservoirs and enhanced oil recovery by thermal methods. Resistance to flow of heat in the contact area between the grains strongly influences the effective thermal conductivity of such porous media. Extensive experiments have indicated that the roughness of the grains' surface follows self‐affine fractal stochastic functions, and thus, the contact resistance cannot be accounted for by models based on smooth surfaces. Despite the significance of rough contact area, the resistance has been accounted for by a fitting parameter in the models of heat transfer. In this Letter we report on a study of conduction in a packing of particles that contains a fluid of a given conductivity, with each grain having a rough self‐affine surface, and is under an external compressive pressure. The deformation of the contact area depends on the fractal dimension that characterizes the grains' rough surface, as well as their Young's modulus. Excellent qualitative agreement is obtained with experimental data. Deformation of granular porous media with grains that have rough self‐affine fractal surface is simulated. Thermal contact resistance between grains with rough surfaces is incorporated into the numerical simulation of heat conduction under compressive pressure. By increasing compressive pressure, thermal conductivity is enhanced more in the grains with smoother surfaces and lower Young's modulus. Excellent qualitative agreement is obtained with the experimental data.

接点温度法による局所熱伝導率計測の適用条件

The purpose of this study is a development of the local thermal conductivity measurement method by contact temperature method. In this paper, the applicable conditions of this method were examined. First, by the numerical simulation of heat conduction and the measurement test for non-conductive materials coated with gold film, it was found that the proposed method with the platinum wire probe is useful for thermal conductivity of 1 to 100 W/mK. Spatial resolution of this system was also examined. It was noted the resolution for low conductivity sample is approximately 100 μm.

Numerical Simulation of Heat Conduction for Error Analysis on Heat Flux Sensor Embedded in Flat Steel Plate

A numerical simulation model of heat flux sensors embedded in a flat plate is established. Each sensor has four thermal couples and is inserted into the specified hole. The problem is defined as a steady heat conduction problem with specified boundary conditions and solved by the finite element method. The results of the simulation case demonstrate that the maximum heat flux appears near the sensor shell. The average heat flux of the plate is much smaller than the maximum. Due to exiting of the contact heat resistance, the temperature of the sensor is much lower than that of the plate at horizontal surface. The maximum temperature difference appears on the bottom shell of the sensor. The maximum temperature difference between the simulation results and the experimental data at test points is 1.5 K. The model is verified and could be accepted for the further errors analysis.

Numerical simulation of heat conduction in a random ballistic deposited grain aggregate

AbstractA numerical simulation of heat conduction in a grain aggregate is carried out to determine the packing fraction (the spatial fraction occupied by grains or 1‐porosity) dependence of the thermal conductivity of the aggregate. Arrangements of grains are given by random ballistic deposition. It is found that the packing fraction dependence is well approximated by an exponential function. The number of contacts between grains is the crucial quantity in the conduction, and its packing fraction dependence is exponential. Heat conduction is found to be anisotropic, where the conduction along the deposition direction is more efficient than that perpendicular to the deposition direction.

有限要素解析におけるソース項の保存型離散化（六面体要素の場合）

The conventional Galerkin finite element solution is mesh dependent, and its discretization for Poisson's equation can not satisfy the conservation law at a nodal level when unstructured linear meshes are used. This research tries to solve these problems by introducing a new concept of the virtual nodal domain(Vnd) for a linear hexahedral element, and distributing the source term to a nodal algebraic equation in proportion to the volume of the Vnd. The Vnd is evaluated using a second-order flux existing within a linear element. We proved that the total Vnd of the eight nodes equals to the volume of the element, which guarantees that our scheme is also elementally conservative. Numerical simulation of heat conduction with both Dirichlet and Neumann boundary conditions shows that the accuracy has been improved obviously comparing with the conventional Galerkin FEM for unstructured hexahedral meshes, especially for bad quality elements. Our scheme can be introduced into any commercial FEM code quite easily.

Conservative Discretization of the Source Term in Finite Element Analyses (In the Case of Tetrahedral Elements)

The conventional Galerkin finite element solution is mesh-dependent and sometimes can not converge on the original Poisson's equation locally when a unstructured linear mesh is used. We try to solve these two problems by introducing a new concept of the virtual nodal domain(Vnd) for a linear tetrahedral element and distributing the source term to a nodal algebraic equation in proportion to the Vnd. The Vnd is evaluated from the conservation law using a second-order flux. We have proved theoretically that the total Vnd belong to the four nodes equals to the volume of the element. Numerical simulation of heat conduction with both Dirichlet and Neumann boundary conditions shows that the accuracy has been improved obviously comparing with the conventional Galerkin FEM for unstructured tetrahedral meshes, especially for bad quality elements.

有限要素解析におけるソース項の保存型離散化(四辺形要素の場合)

The conventional Galerkin finite element solution is mesh-dependent, and its discretization for Poisson's equation can not satisfy the conservation law over a nodal domain when unstructured linear meshes are used. This research tries to solve these problems by introducing a new concept of the virtual nodal domain(Vnd) for a linear quadrilateral element, and distributing the source term to a nodal algebraic equation in proportion to the area of the Vnd. The Vnd is evaluated using a second-order flux existing within a linear element. We proofed that the total Vnd of the four nodes equals to the area of the element, which guarantees that our scheme is also elementally conservative. Numerical simulation of heat conduction with both Dirichlet and Neumann boundary conditions shows that the accuracy has been improved obviously comparing with the conventional Galerkin FEM for unstructured quadrilateral meshes, especially for bad quality elements. Our scheme can be introduced into any commercial FEM code quite easily.

A neural network based estimation of tumour parameters from a breast thermogram

This paper proposes the estimation of the position and size of a spherical tumour in a human breast using the temperatures obtained on the surface of the breast through a breast thermogram in conjunction with artificial neural networks. The first part of the work involves the numerical simulation of heat conduction in a cancerous breast by employing the Pennes bio-heat transfer equation using a finite element based commercial solver COMSOL. The surface temperatures thus obtained from these simulations are trained against the tumour parameters by using artificial neural networks (ANN). An optimal neural network can then be utilized to retrieve parameters with good accuracy. The forward heat transfer problem was solved for two cases: (i) constant heat generation rate inside the tumour and (ii) variable heat generation rate inside the tumour, thereby generating 447 temperature data vectors in each case. In this study, the “measured” temperatures correspond to those numerically generated for a known size and position of the tumour. For the case of no noise in the “measured” data, accuracies of 90% in position and 95% in the radius for the constant heat generation rate were estimated using ANN, while another network retrieved the same with accuracies of 88% and 98% respectively for the case of varying heat generation rate. In practice, depending upon the resolution of an IR camera, the number of data points at which temperatures are recorded can be conveniently chosen from a breast thermogram. Hence, sensitivity studies were conducted to investigate the effect of the size of the data vector (“measured” temperatures) on the retrievals, and it is seen that an optimum indeed exists.

Numerical simulation of transient phonon heat transfer in silicon nanowires and nanofilms

This work proposes a numerical simulation of heat conduction in silicon nanowires and nanofilms. Boltzmann equation for phonons is solved in the relaxation time approximation. The equation is integrated in an axisymmetric cylindrical two dimensional geometry. Solid angle integration is done by means of Discrete Ordinate Method. Moreover, in contrast to other models published in literature, spectral dependency of relaxation times and acoustic wave dispersion are taken into account in this numerical resolution. Consequently, thermal profiles are obtained for silicon nanowires and nanofilms in steady state allowing computation of thermal conductivity and/or thermal conductance. Besides, we solve the unsteady Boltzmann equation in order to obtain nanosystems temporal evolution. The results obtained with this code match nanofilms and nanowires already predicted thermal profiles in steady state. In unsteady condition, diffusive state (Fourier) is discussed for nanowires and nanofilms. At low temperatures, ballistic phenomenons are seen in nanofilms, whereas, in nanowires, due to boundary scattering, diffusion regime is observed.

Numerical simulation of heat conduction for the growth of anisotropic layered GaSe crystals

AbstractIn this report, we present the usage of a second rank cylindrical conductivity tensor which we derived to simulate the crystal growth processes of a layered compound GaSe in a cylindrical enclosure by directional solidification. Use of such a tensor is inevitable in the simulations of the growth of highly anisotropic crystals having layered structure, since the crystallographic orientation of the grown material is not necessarily aligned with the ampoule symmetry. Using the finite difference control volume approach in 3D, we solved transient heat conduction equation for a highly anisotropic solid in a cylindrical enclosure. We obtained sloped thermal fields and isothermal surfaces and the magnitudes of the slopes are strong functions of both azimuthal angle and growth orientation. The results showed that the orientation of the crystallographic axes of GaSe in the enclosure is quite effective in the steady and the transient fields, isotherms, and axial and radial temperature gradient within the material. Increase of Bi number decreases the magnitude of the slope of isothermal surface. Anisotropy of the conductivity seems to be effective in the orientation of the growth direction of the resulting crystal within the cylindrical ampoule. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerical Simulation of Heat Conduction to Liquids from a Thin Vertical Cylinder

The paper presents numerical simulations of heat conduction around a circular vertical cylinder immersed in liquids. A finite volume formulation is used, and the numerical analysis is performed in unsteady state with an explicit scheme. The numerical predictions are compared with experiments performed on liquids to find the temperature inside the cylinder, where a thermocouple is located, and at the wall of the insulated coaxial container, where the liquid is poured. The cylinder is immersed vertically. The numerical results are in good agreement with the temperature at the wall of the container. The experimental temperature measurement of the thermocouple located inside the probe is intermediate between the numerical temperatures on the axis and on the surface of the probe. The natural convection phenomenon is evidenced in the experiments, after a certain time from the beginning of heating, in some of the liquids used, except glycerol. Natural convection is not considered in the present numerical simulations, which solve only the heat conduction equation.

Constructal Placement of High-Conductivity Inserts in a Slab: Optimal Design of “Roughness”

This paper addresses the fundamental problem of how to facilitate the flow of heat across a conducting slab heated from one side. Available for distribution through the system is a small amount of high-conductivity material. The constructal method consists of optimizing geometrically the distribution of the high-conductivity material through the material of lower conductivity. Two-dimensional distributions (plate inserts) and three-dimensional distributions (pin inserts) are optimized based on the numerical simulation of heat conduction in a large number of possible configurations. Results are presented for the external and internal features of the optimized architectures: spacings between inserts, penetration distances, tapered inserts and constant-thickness inserts. The use of optimized pin inserts leads consistently to lower global thermal resistances than the use of plate inserts. The side of the slab that is connected to the high-conductivity intrusions is in effect a “rough” surface. This paper shows that the architecture of a rough surface can be optimized for minimum global contact resistance. Roughness can be designed.

The numerical simulation of heat conduction in irregularly‐shaped materials of thermally‐dependent properties

AbstractThe numerical simulation of conduction heat transfer in arbitrarily‐shaped regions is relevant to many engineering applications including the casting of plastics and metals, cold region problems and latent heat storage. This class of problem offers its particular challenges. The accurate computational simulation of conduction with thermally‐dependent properties requires a solution technique with good conservation properties. The boundary‐fitted co‐ordinates of an arbitrary solution domain typically yield grids which are non‐orthogonal and which, consequently, make energy conservation difficult to model.In the current article, a technique is described to simulate accurately the conduction heat transfer in materials of thermally‐dependent properties in irregular domains. The method combines boundary‐fitted co‐ordinates with the finite volume method, FVM, to produce a numerical technique which will accurately solve this non‐linear conduction problem using a grid which may be highly skewed. This capability is achieved through a unique treatment of the cross‐derivative terms that arise when the heat conduction equation is transformed to a non‐orthogonal grid. The cross‐derivative terms represent the non‐normal components of the heat fluxes into the skewed control volume. The tangent components of the heat fluxes are interpreted in a special way to produce finite difference expressions which accurately model the cross‐derivative partial.The numerical procedure is validated by comparing it against a purely analytical mathematical method. Although the numerical results have been obtained using a highly skewed grid, they exhibit close agreement with the analytically‐derived solution.

Dynamic thermal behaviour of a wall

A new technique for dealing with the problem of the numerical simulation of heat conduction in solids is presented. This method is based on that of Milne but does not have any limitation in the time step amplitude. It has, therefore, the advantage of being more accurate and requires less computational time than the traditional methods. The mathematics of the method are shown and a comparison with the Crank-Nicolson technique is made.