Fourier Heat Conduction Equation Research Articles

Improvement and Validation of the System Analysis Model and Code for Heat-Pipe-Cooled Microreactor

Heat-pipe-cooled microreactors (HPMR) use a passive high-temperature alkali metal heat pipe to directly transfer the heat of solid core to the hot end of the intermediate heat exchanger or thermoelectric conversion device, thus avoiding a single point failure. To analyze and evaluate the transient safety characteristics of an HPMR system under accident conditions, such as heat pipe failure in the core or a loss of system heat sink and other accidents, a previously developed model for transient analysis of a heat-pipe-cooled space nuclear reactor power system (HPSR) was improved and validated in this study. The models improved mainly comprise: (1) An entire 2-D solid-core heat transfer model is established to analyze the accident conditions of core heat pipe failure and system heat sink loss. In this model, radial and axial Fourier heat conduction equations are used to divide the core into r-θ direction control volumes. The physical parameters of the material in the control volume are calculated according to the volume-weighted average. (2) By coupling the heat transfer limit model and the two-dimensional thermal resistance network model, the transient model of a heat pipe for HPMR system analysis is improved. (3) Conversion system models are established to simulate the system characteristics of the advanced HPMR concept, such as thermoelectric conversion, Stirling conversion, and the open Brayton conversion analysis model. Based on the improved models, the HPMR system analysis program TAPIRSD was developed, which was verified by experimental data of the separated conversion components and the ground nuclear test device KRUSTY. The maximum deviation of the power output predicted by the energy conversion model is less than 8%. The accident conditions of the KRUSTY tests, such as load change, core heat pipe failure, and heat sink loss accident, were studied by using TAPIRSD. The results show that the simulation results of the TAPIRSD code agree well with the experimental data of the KRUSTY prototype reactor. The maximum error between the TAPIRSD code prediction and the measured value of the core temperature under accident conditions is less than 10 K, and the maximum deviation is less than 2%. The results show that the developed code can predict the transient response process of the HPMR system well. At the same time, the accuracy and reliability of the improved model are proved. The TAPIRSD is suitable for system transient analysis of different types of HPMRs and provides an optional tool for the system safety characteristics analysis of HPMR.

An alternative for the determination of thermal diffusivity using 1D Fourier solution: Talbot’s method

Thermal diffusivity is a parameter of fundamental importance in heat transfer studies of solids. This parameter can be determined from the transient Fourier heat conduction equation using different techniques. In this research, a new algorithm based on Talbot’s method was proposed to determine the thermal diffusivity using a one-dimensional Fourier heat conduction equation. Furthermore, a free and open-source program was developed to perform the calculations. Experimental data in heating and cooling conditions of ham were used to test the programming. It was found that n and ω are strongly related and affect the results. The developed program in Scilab using Talbot’s method, the Levenberg-Marquardt algorithm of optimization (lsqrolve routine), n = 100 and ω = 1.8 obtained r2 of 0.999 and 0.998; thermal diffusivities of 0.98 × 10−7 m2/s and 1.13 × 10−7 m2/s with the time required to run the programming from 0.6 to 1.1 s.

Experimental Evaluation of Fire Resistance Limits for Steel Constructions with Fire-Retardant Coatings at Various Fire Conditions

The experimental evaluation of fire resistance limits for steel constructions with fire-retardant coatings consists of a lot of experiments on the heating of steel structures of buildings by solving a heat engineering problem at various fire conditions. Building design implies the assessment of compliance of actual fire resistance limits for steel constructions with the required limits. Fire resistance limits for steel constructions are determined for “standard” temperature mode, and this can lead to overestimated fire resistance and underestimated heat influence for a real fire. Estimation of the convergence for “standard” temperature mode and possible “real” fire mode, as well as of the compliance of actual fire resistance limits with real fire conditions, was realized in the following stages: mathematical modeling of real fire development by the field model in software package Fire Dynamics Simulation (FDS) with various fire loads and mathematical modeling of steel construction heating for the standard temperature mode obtained by modeling “real” fire modes (the finite difference method of solving the Fourier heat conduction equation at external and internal nonlinearities was used for modeling the process of steel structure heating with the implementation in the ANSYS mechanical software package). Experiments of the assessment of fire-protective paint’s effectiveness were carried out for standard temperature mode and obtained by modeling “real” fire modes. The equivalent fire duration dependence on fire load type was determined. This dependence can be taken into account in determination of fire resistance limits for steel constructions in warehouse building roofing. Fire-protective paint effectiveness was estimated for “standard” temperature mode and various other temperature modes.

Analysis of Temperature, Displacement, and Stress in Shafts Made of Functionally Graded Materials with Various Grading Laws

An exact solution for thermo‐elastic analyses of radially graded circular hollow shafts made of functionally graded materials (FGMs) is studied. The material properties of functionally graded (FG) shafts are assumed to be temperature‐dependent (TD) and graded in the radial direction according to various grading patterns such as linear law of material gradation (LLMG), power law of material gradation (PLMG), and exponential law of material gradation (ELMG). Temperature fields of radially graded FG shafts are obtained analytically for the different grading patterns, using a steady‐state Fourier equation of heat conduction. Subsequently, based on linear strain‐displacement relations, thermo‐elastic equations are solved analytically to obtain displacement and stress fields as functions of radial distances, material gradient indices, and temperature gradients. The present mathematical derivations for thermo‐elastic analyses and solution methods are successfully validated by comparing with the results of published literature. Numerical results are presented to highlight the importance of material parameters, temperature distributions, radial distances, and aspect ratios on the temperature profiles, displacement, and stress fields of the FG shafts under thermal/mechanical loads. The numerical results have explicitly shown that the thermo‐elastic analyses of FG shafts are significantly influenced by material gradient indices, temperature gradients, and temperature‐dependent properties in high‐temperature environmental applications.

On the Determination of Thermal Conductivity From Frequency Domain Thermoreflectance Experiments

Abstract The Fourier and the hyperbolic heat conduction equations were solved numerically to simulate a frequency-domain thermoreflectance (FDTR) experiment. Numerical solutions enable isolation of pump and probe laser spot size effects and use of realistic boundary conditions. The equations were solved in time domain and the phase lag between the temperature of the transducer (averaged over the probe laser spot) and the modulated pump laser signal was computed for a modulation frequency range of 200 kHz–200 MHz. Numerical calculations showed that extracted values of the thermal conductivity are sensitive to both the pump and probe laser spot sizes, while analytical solutions (based on Hankel transform) cannot isolate the two effects. However, for the same effective (combined) spot size, the two solutions are found to be in excellent agreement. If the substrate (computational domain) is sufficiently large, the far-field boundary conditions were found to have no effect on the computed phase lag. The interface conductance between the transducer and the substrate was found to have some effect on the extracted thermal conductivity. The hyperbolic heat conduction equation yielded almost the same results as the Fourier heat conduction equation for the particular case studied. The numerically extracted thermal conductivity value (best fit) for the silicon substrate considered in this study was found to be about 82–108 W/m/K, depending on the pump and probe laser spot sizes used.

Mathematical modelling of thermoelasticity problems for thin biperiodic cylindrical shells

The objects of consideration are thin linearly thermoelastic Kirchhoff-Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential and axial directions (biperiodic shells). The aim of this contribution is to formulate and discuss two new averaged mathematical models for the analysis of selected dynamic thermoelasticity problems for the shells under consideration: the non-asymptotictolerance and the consistent asymptotic models. The starting equations are the well-known governing equations of linear Kirchhoff-Love theory of thin elastic cylindrical shells combined with Duhamel–Neumann thermoelastic constitutive relations and coupled with the known linearized Fourier heat conduction equation in which the heat sources are neglected. For the microperiodic shells under consideration, the starting equations mentioned above have highly oscillating, non-continuous and periodic coefficients. The tolerance model is derived applying the tolerance averaging technique and a certain extension of the known stationary action principle. It has constant coefficients depending also on a cell size. Hence, this model makes it possible to study the effect of a microstructure size on the global shell thermoelasticity (the length-scale effect). The consistent asymptotic model is obtained using the consistent asymptotic approach. It has constant coefficients being independent of the period lengths. Moreover, the comparison between the tolerance model for biperiodic shells proposed here and the known tolerance model for cylindrical shells with a periodic structure in the circumferential direction only (uniperiodic shells) is presented.

Development and validation of a method to predict the soil thermal conductivity using thermal piezocone penetration testing (T-CPTU)

A new in situ thermal piezocone penetration test (T-CPTU) system is developed to determine the thermal properties of soil. It is expected to overcome most of the shortcomings observed in existing in situ test techniques. Based on Fourier's heat conduction equation and pore pressure dissipation theoretical equation, a method for calculating the thermal conductivity, namely the predicted temperature method, was proposed. The accuracy of the T-CPTU probe testing process and thermal conductivity calculation results were verified by numerical simulation, laboratory large-scale model tank tests, and thermal needle tests. Finally, the field data of T-CPTUs at three sites in Nanjing, China, were collected and compared with the laboratory thermal needle tests. The results indicated that the thermal conductivities obtained using T-CPTU were accurate and closer to those of laboratory thermal needle tests for most soils. The thermal conductivities of the undisturbed soil samples measured in the laboratory were lower than those obtained by T-CPTU.

Non-linear free vibrations of the column loaded with a mass element and a local heat source

In this paper, the problem of non-linear vibrations of an articulated column subjected to two loads was considered. The first load is axial compression associated with the action of the mass element. The second is the heat load in the form of a local heat source. In such a system, the problem analysis was divided into two parts. The first illustrates the influence of temperature on the physical and mechanical properties of the structure. The second - concerning non-linear vibrations of the system. The problem of heat flow in the column was formulated on the basis of the classical Fourier heat conduction equation and solved with the use of the Finite Element Method. Distribution of material properties (Young’s modulus) was obtained in subsequent moments of exposure of the column to the heat source. The obtained results were used in the problem of system vibrations, which was formulated on the basis of Hamilton’s principle. Due to the non-linearity, the perturbative small parameter method was used to finally derive the differential equations describing the behaviour of the system during vibrations. Numerical calculations were carried out on the basis of which the characteristic curves of the system (on the load - natural frequency plane) for various stages of heating the column were determined. Moreover, the influence of the heating time of the system on its non-linear natural frequency was presented. The significant influence of the longitudinal inertia of the mass element causing the load on the vibrations of the structure was shown. The mathematical model and the correctness of the performed numerical simulations were verified by experimental research.

Three-dimensional analysis model of electric heating fabrics considering the skin metabolism

In low temperature environment, electric heating clothing can provide extra heat for human body through built-in heat source, so it has better thermal insulation effect. The thermal analysis is the initial step for electric heating clothing design. The current thermal analysis of electric heating textiles focuses on the fabric itself instead of the effect of skin tissue metabolism and heat production. In order to improve the accuracy of skin surface temperature prediction, the biological heat transfer need be modeled to analyze the internal temperature distribution of the heating suit system. In this paper, a three-dimensional (3D) thermal analysis model of electric heating clothing combined with human skin tissue is established. Firstly, the coupling analysis of Fourier heat conduction and Pennes biological heat transfer equation is carried out. Then the reliability of the 3D thermal analysis model is verified by finite element analysis (FEA). The results show that the fitting error between the three-dimensional model analysis data and FEA simulation data is 5°C, which proves that the model can accurately predict the system temperature. Finally, we make further research about the effects of ambient temperature, clothing layer thickness, and input power on the maximum skin surface temperature. This study provides theoretical foundation for the design of wearable thermal management fabric.

Thermo-mechanical behaviour of multi-layered media based on the Lord-Shulman model

The relaxation time model is introduced to analyse the thermo-mechanical problem for multi-layered media in this paper. The Fourier heat conduction equation is extended by introducing a relaxation time based on the Lord-Shulman model. For multi-layered media, the analytical layer-element method is employed to establish the global stiffness matrix equations by assembling the correlative layer-elements in the Laplace-Hankel domain. Then, solutions for displacement and temperature in the time domain are obtained by solving the global matrix equations and implementing a numerical inversion procedure. For numerical implementations, computational overflows are avoided by eliminations of positive exponentials in the stiffness matrix, and numerical instabilities in special cases due to singularity are removed by replacing singular layer-elements with well-conditioned ones. Comparisons with the existing solutions and the FEM analysis demonstrate the accuracy and efficiency of the proposed method. Finally, a series of numerical examples are presented to discuss the effects of transient load type, stratification, relaxation time and the surface heat transfer coefficient.

Exact Axisymmetric Thermal Analysis of Functionally Graded Disks with Continuously Hyperbolically Varying Thickness

An exact thermal analysis of radially functionally graded (FG) disks with continuously varying thickness is performed by steady-state 1-D Fourier heat conduction equation at specific surface temperatures. By employing a simple-power material grading pattern together with the convergent/divergent hyperbolic disk profiles, the differential equation is obtained in the form of Euler-Cauchy type. Analytical solution of the differential equation gives the temperature field and heat flux distributions in the radial direction in a closed form. A numerical study is conducted to visualize both the temperature and heat flux variations with respect to the disk profile parameter for hyperbolic disks made of SUS-304 /ZrO2 (Stainless steel/Zirconium oxide) metal-ceramic pairs. Those exact expressions are also used to study parametrically the effects of both the inhomogeneity and profile parameters on the temperature field of the disks made of hypothetic FG metal-ceramic pairs. It is revealed that heat conduction behavior of such disks is strictly affected from the variation of both inhomogeneity and disk profile parameters.

Assessment on jet direction effect on drag and heat reduction efficiency for hypersonic vehicles

Introducing jet for drag reduction and thermal protection in hypersonic flows has attained worldwide attentions due to its giant potential, and this paper focused on the drag and heat reduction efficiency of unit mass flow rate for different jet directions in hypersonic flows. The developed code employed conjugate heat transfer approach to solve unsteady Reynolds-averaged Navier-Stokes equations and Fourier's heat conduction equation simultaneously, and then it was thoroughly validated by two classical experiments. Based on that, the flow field characteristics induced by different jet directions under the constant mass flow rate of jet were numerically investigated. Besides, two parameters were introduced to quantitatively evaluate the drag and heat reduction efficiency of different jet directions. The obtained results show that though the total mass flow rate of jet is constant, jet direction still has a decisive influence in manipulating the flow structures, and the rear jet would result in more complicated shock waves due to the interaction between the rear jet flow and spike. The force produced by the jet is the key role in determining the total drag, and the variation of pressure distribution along the blunt body is not the direct presentation of drag reduction performance. The opposing jet has the highest drag reduction efficiency, and lateral jet has the highest heat flux reduction. However, compared with the lateral and rear jets, the opposing jet has better thermal protection performance from the overall perspective as it can also protect the aerodome head from severe aerodynamic heating.

Nonlinear axisymmetric thermomechanical response of FGM circular plates

Based on von Karman plate theory, axisymmetric geometrically nonlinear governing equations of simple power-law type functionally graded material (FGM) circular plates are derived. Material properties are assumed to be temperature-independent, and thermal or/and mechanical loads are considered in the derivation. Considering clamped boundary conditions, the dimensionless critical (lowest) buckling temperature (temperature difference) 14.684 of the systems is obtained by analyzing linear-eigenvalue problem. With this constant and an analytical formula proposed in this paper, accurate dimensional critical (lowest) buckling temperature (temperature difference) of any specified clamped FGM circular plate can be easily calculated. Moreover, two-point boundary value problem posed by the governing equations and the clamped boundary conditions is solved using the shooting method. Thermal buckling response under thermal load and geometrically nonlinear mechanical behaviors under thermomechanical load of the system are discussed. When the temperature difference of the upper and lower surfaces of FGM plates exists, the solution of one-dimension Fourier heat conduction equation is an infinite series, which depicts a nonlinear temperature field (NLTF) along the thickness direction of plates. The effects of the initial terms of the series on the accuracy of solutions are examined. The results reveal it is necessary to take enough the initial terms of the series even for thinner FGM plates. Typical load–deflection curves of the clamped FGM circular plates with different temperature fields (along the thickness) are presented. For a given temperature field, the clamped FGM circular plates under transverse uniform load are a hard stiffness nonlinear system. When the values of thermal and mechanical load and their loading sequence meet certain combinations, the clamped circular plate will inevitably undergo secondary buckling (snap-through buckling).

Semi-Analytical Model for Heat and Mass Transfer Evaluation of Vapor Bubbling

Multi-stage refrigeration systems cover a wide range of possibilities and are diffusing more and more. The idea that inspired this work derived from the need to have a tool to model the energy behavior of the intercooler inside a multi-stage refrigeration system. In this work, a semi-analytical model of a single bubble, injected into the liquid of an intercooler of a multi-stage system, has been developed. The developed model is a set of equations derived from the Fourier equation for heat conduction in defined conditions and includes the effects of sensible and latent heat. The vapor bubble is supposed to be injected in the saturated liquid contained in a tank at a defined depth, at an intermediate pressure. The model has been implemented in Matlab and the results show the influence of the liquid surface tension, the injection depth and the thermal diffusivity of the vapor. The model developed here is a useful low-cost tool for evaluating heat transfer optimization of a separator/intercooler of a multi-stage refrigeration system.

Buckling Analysis of Steel H Column with Thermal Gradient Along the Flanges

This paper presents an analytical study about the axial compression buckling of H-section columns with one side of the flanges exposed to fire. The fire-induced thermal gradient along the flanges is obtained by using simplified Fourier heat conduction equation. The pre-buckling stress and bowing effect due to the non-uniform temperature distribution through the cross-section is considered. The analysis is accomplished by using the Rayleigh–Ritz method. The analytical solution is validated by finite element analysis using ANSYS. Finally, the parametric study is performed for different sections including different values of length, thickness ratio and width-to-height ratio.

Finite-volume-based conservative numerical scheme in cylindrical coordinate system to predict material removal during micro-EDM on Inconel 718

Micro-EDM is a thermal energy-based non-contact machining process, able to machine any electrically conductive material, making it one of the preferred methodologies for micro-manufacturing. Before making any improvement in the process, understanding of the process physics is very important. A conservative numerical scheme to model the micro-EDM process using the finite volume method with Fourier heat conduction equation in cylindrical coordinates as the governing equation has been attempted. The model was formulated with Gaussian distribution of input heat flux, temperature-dependent material properties, latent heat and fixed percentage of heat to the workpiece. The proposed numerical scheme was validated with simplified analytical results. Numerical models considering temperature-dependent and temperature-independent thermal conductivity and specific heat were considered. The simulation results showed a near-hemispherical crater. The model was validated experimentally by comparing the numerically simulated craters with experimentally generated ones. Results showed that the numerical model with temperature-dependent material properties was able to predict the crater with the highest accuracy on comparing with the experimental results. The temperature-dependent model predicted the crater dimensions with relative error ranging between 6 and 13% for radius and 3 and 14% for depth.

A multiphase SPH framework for solving the evaporation and combustion process of droplets

PurposeThis paper aims to introduce a two-dimensional smoothed particle hydrodynamics (SPH) framework for simulating the evaporation and combustion process of fuel droplets.Design/methodology/approachTo solve the gas–liquid two-phase flow problem, a multiphase SPH method capable of handling high density-ratio problems is established. Based on the Fourier heat conduction equation and Fick’s law of diffusion, the SPH discrete equations are derived. To effectively characterize the phase transition problem, inspired by volume of fluid method, the concept of liquid phase mass fraction of the SPH particles is proposed. The one-step global reaction model of n-hexane is used for the vapor combustion.FindingsThe evaporation and combustion process of single droplet conforms to the law. The framework works out well when the evaporation of multiple droplets involves coalescence process. Three different kinds of flames are observed in succession in the combustion process of a single droplet at different inflow velocity, which agree well with the results of the experiment.Originality/valueTo the best of the authors’ knowledge, this is the first computational framework that has the capability to simulate evaporation and combustion with SPH method. Based on the particle nature of SPH method, the framework has natural advantages in interface tracking.

Free vibration analysis of smart laminated carbon nanotube-reinforced composite cylindrical shells with various boundary conditions in hygrothermal environments

Abstract Free vibration analysis is carried out for piezoelectric coupled carbon nanotube (CNT)-reinforced composite cylindrical shells with the influences of various boundary conditions and hygrothermal environmental conditions for the first time. A simple and effective non-iterative mathematical method is used to calculate the natural frequencies. The equilibrium equations of motion are obtained based on the first-order shear deformation shell theory with the coupling effects of piezoelectricity, temperature, and moisture, respectively based on the Maxwell equation, the Fourier heat conduction equation, and the Fickian moisture diffusion equation. The Mori-Tanaka micromechanics model is used to estimate the resulting material properties for a composite shell reinforced with CNTs. Presented methodology and attained results are validated with the existing results in the literature. The effects of the boundary conditions, lamination stacking sequence, volume fraction and orientation of CNTs, piezoelectricity, and geometry of the shell on the natural frequencies of the shell are investigated. A moderate effect of temperature/moisture variation on the natural frequencies is also observed. It is found that the influence of structural boundary conditions is more significant at higher CNT volume fractions and for thicker and shorter shells, and the piezoelectricity effect is more obvious at higher circumferential mode. The model and results presented in this study can be utilized to determine vibration characteristics of smart CNT-reinforced composites subjected to hygrothermal loading as well as mechanical loading.

Analytical solution for magneto-thermo-elastic responses of an annular functionally graded sandwich disk by considering internal heat generation and convective boundary condition

In this paper, an exact solution for stress distribution of a rotating sandwich disk subjected to magnetic, thermal fields, and convection heat transfer with the consideration of internal heat generation is presented. The sandwich disk is composed of three different layers in which the central layer is made of functionally graded material, whereas the inner and outer layers are made from pure metal and ceramic materials, respectively. All mechanical and thermal properties of the central layer are assumed to obey a power-law form in the radial direction with different non-homogeneity constants. Axisymmetric temperature distributions in each layer are obtained by solving Fourier heat conduction equation. By substituting the constitutive equations in equilibrium equation, a second-order differential equation in terms of radial displacement for each layer is derived by considering centrifugal force and Lorentz magnetic force obtained from Maxwell’s relations. Thereafter, the governing equations are analytically solved and radial displacement, radial and circumferential stresses for the current disk are obtained. Numerical simulations reveal the effects of internal heat generation, convection heat transfer, magnetic field, material properties, and angular velocity on the magneto-thermo-elastic response of the functionally graded sandwich disk. Also, a comparison study between metal, functionally graded, ceramic, and functionally graded sandwich disks is performed.

The Current Status and Development of Semi-solid Powder Forming (SPF)

Semi-solid powder forming is a promising near-net-shaped forming technology, which has the advantages of powder metallurgy and semi-solid forming, such as fine grains, low forming pressure and short process flow. It was used to prepare wide solidification range alloys and its composites. Until now, there have been many studies on the parameters, microstructure and mechanical properties of this technology, but few on the forming principles. Because deformation, solidification and densification occur simultaneously, the forming mechanism is very complex. The liquid fraction is the key factor influencing the microstructure and mechanical properties. Semi-solid compression of porous materials was carried out to study the deformation mechanism of semi-solid powders. The combination mechanism and densification process for semi-solid powder rolling has been analyzed, and the compaction behavior of powders in the semi-solid state has been studied. Shima porous yield criterion and Doraivelu plastic yield criterion were applied in the simulation of semi-solid powder rolling. Based on the Fourier heat conduction equation and the related parameters of semi-solid powder, the rolling force and relative density were simulated by using the Marc finite element software platform. The simulation results are in agreement with the experimental results. Although some achievements have been made in the theoretical research and numerical simulation, the yield criteria and mathematical models suitable for semi-solid powder forming need to be further established. In addition, further optimization of this technology and its application in commercial applications should be the research direction.