Fourier Heat Conduction Equation Research Articles

Development of a Solidification Model for TEXAS-VI Code and Application to FARO L14 Analysis

A molten fuel breakup model that considers solidification effects is proposed in this paper. Both the effect of a solid crust layer and the effect of thermal stresses on the fuel particle fragmentation are taken into account in this model. This solidification model predicts the transient temperature profile and crust layer thickness of the fuel particle by numerically solving the Fourier heat conduction equation under specific initial and boundary conditions. This fuel particle breakup model and transient temperature profile model were incorporated into the TEXAS fuel-coolant interaction (FCI) model; this revised TEXAS FCI model is called TEXAS-VI. This paper compares TEXAS-VI to the FARO L14 experiment (FARO L14), for which fuel-coolant mixing and quench data have been published. The FARO L14 pressure history, liquid water pool temperature, and vapor temperature were found to be in good agreement with the revised model predictions. This mixing behavior will also have an impact on FCI explosion energetics. The solidification effect is under investigation for energetics.

Hygrothermal stresses in one-dimensional functionally graded piezoelectric media in constant magnetic field

This paper presents analytical solutions for hygrothermal stresses in one-dimensional functionally graded piezoelectric media. The media are subjected to an external constant magnetic field and could be rested on a Winkler-type elastic foundation. The material properties are assumed to vary through the thickness according to a power-law with different non-homogeneity indices. The governing equations including the Lorentz force effect are written in a unified form that can be used for one-dimensional functionally graded piezoelectric media in Cartesian, cylindrical, and spherical coordinates. The moisture concentration and temperature distributions along the thickness of one-dimensional media are achieved by solving the steady state Fickian moisture diffusion and Fourier heat conduction equations. The two coupled ordinary differential equations in terms of displacement and electric potential are solved analytically. The numerical results illustrate the effects of hygrothermal loading, medium geometry, non-homogeneity indices, external magnetic field, and elastic foundation on the behavior of one-dimensional piezoelectric media. For a uniform temperature rise, the influence of temperature-dependency of material properties on the electromechanical responses is investigated as well. At last, the results are verified with those reported in the literature.

Temperatures and Thermal Diffusivity within a Rangeland Soil Near Oracle, Arizona

Soil temperatures are typically measured at shallow depths near the soil surface where large diurnal variations occur. For depths deeper than about 50 cm analytical and numerical models are often used to simulate the annual variations in daily temperature, which require information about the soil thermal diffusivity (D). An analytic solution to the onedimensional Fourier heat conduction equation was used by Matthias and Warrick (1987) to simulate daily soil temperatures at 20 and 100 cm depths at Safford, Arizona, and at 20 cm depth at Yuma, Arizona, using observed daily soil temperatures at 10 cm depth at each location and an estimate of the average soil thermal diffusivity as inputs. The solution was the sum of a deterministic (mean annual wave) component and a stochastic (daily fluctuations about the mean wave) component. The purpose of the research described in this paper is to use the model described by Matthias and Warrick (1987) to calculate the average thermal diffusivity for a semi-arid rangeland soil at the Page Ranch near Oracle, Arizona. The data used in this analysis were the daily soil temperatures at 50, 100, 300, and 500 cm depths observed during a one-year period from July 1, 1983 to June 30,1984 at the site. To obtain the average D, the daily temperatures at 50 cm depth and an initial estimate of D were input to the model to simulate temperatures at the 100, 300, and 500 cm depths. Observed and simulated temperatures at each depth were then compared. If the root-mean-square-deviations (RMSD) between observed and simulated values were large then an updated value of D was input to the model and temperature values recalculated. This trial-and-error approach continued until RMSD values were minimized. With D∼500 cm2 d-1 good agreement between observed and simulated temperatures was achieved with RMSD values ranging from 0.38 to 1.35 °C for the three depths.

THERMAL MODEL FOR LASER-INDUCED DISCHARGE SURFACE STRENGTHENING

The temperature field due to laser-induced discharge surface strengthening (LIDSS) has significant influence on the microstructure transformation and also the formation quality of discharge pit. A transient axisymmetric thermal model is developed to estimate the temperature distribution during LIDSS based on Fourier heat conduction equation. In the model, a Gaussian heat input distribution is assumed; temperature-dependent material properties are applied and the latent heat of fusion and vaporization is calculated on an enthalpy method. As an application, we use this model to compute the temperature field during the process of tungsten tool electrode machining 1045 steel workpiece and find that the computational results are well consistent with the experimental data.

On the Use of Glycosylated Single-Walled Carbon Nanotubes as a Coolant Additive

A dispersion of single-walled carbon nanotubes (SWCNTs) in an aqueous medium was prepared by functionalizing the SWCNT with D-glucosamine, as confirmed using FT-IR and NMR techniques. The SWCNT and sugar functionalized SWCNT were studied using scanning and transmission electron microscopy. The grafting resulted in a good dispersion of the SWCNTs in water of an amount less than 1 mg ml −1 . The thermal conductivity was measured using high sensitivity thermistor, whose resistance was measured using a 61/2 digit multimeter. Assuming the Fourier heat conduction equation to be valid for an analysis of the present situation (to determine the overall heat transfer coefficient), a 24% increase in the rate of heat transfer was arrived at for dispersions of less than 0.1% sugar functionalized SWCNTs in 1 ml of water. These results are compared with results on SWCNT and MWCNT dispersions in various media in order to demonstrate the superior heat transfer properties of the present medium.

Principle of Virtual Power for thermomechanics of fluids and solids with dissipation

A fundamental variational principle, called the Principle of Virtual Power, is proposed in this paper to govern the irreversible thermomechanical processes for a material body. The principle and the corresponding variational equation of virtual velocity are formulated by integrating the classical field theories of continuum mechanics and thermodynamics. From the variational equation, we establish simultaneously the balance equations of linear momentum, of angular momentum, and of virtual internal power by applying the procedure of relaxation in Lagrangian mechanics. By assuming different types of system functions for internal power consisting of internal energy and newly introduced working dissipation, we are able to derive the constitutive equations for gas, liquid, or solid and the Fourier equation of heat conduction from the balance of virtual internal power. The combination of the law of conservation of mass, the first law of thermodynamics, and the proposed principle forms a complete set of fundamental principles for the thermomechanics of deformable bodies.

Heat conduction and thermal analysis in multilayered plates and shells

Abstract This work solves the Fourier heat conduction equation in the case of layered structures embedding orthotropic layers in order to obtain a calculated temperature profile in the thickness direction. The calculated temperature profile θ ( z ) is obtained for both plate and shell geometries. Several comparisons between the calculated temperature profile and the usual assumption of linear form of θ ( z ) in the thickness direction are given. A wrong temperature profile gives an erroneous thermal load for the static analysis of layered structures. Proposed results show the effect of a wrong use of θ ( z ) on the static response of multilayered plates and shells. It is concluded that in the analysis of multilayered structures the correct solution of the heat conduction equation could be mandatory with respect to any further evaluation.

Non-invasive measurement of the human core temperature

The direct measurement of the temperature of the human body core, while being of great significance for the evaluation of patient health, is inconvenient. This realization has motivated attempts to measure the core temperature indirectly, for the most part at locations which do not require invasion of the body. In the investigation described here, a new approach to noninvasive measurement of the core temperature is set forth. The site of the proposed measurement is on the forehead where the anatomy is relatively uniform amongst individuals and also well established to be modeled accurately. The approach described here consists of simple hardware and a logic-based and customized control algorithm. The connection between the core temperature, the sensor temperature, and the temperature of a sensor-system heater was determined by numerical simulations of the heat transfer in the tissue bed fronted by the forehead skin and backed by the frontal bone. For heat transfer through each tissue layer in the tissue bed, the bioheat equation was employed, while for a resistance element in the sensing system, the Fourier heat conduction equation was used. These were solved in the transient mode. The controller read the temperatures of the core, the sensor, and the ambient, and the algorithm instructed the sensor-system heater to obtain a specific temperature value. The iterative use of the control algorithm resulted in near-equality of the sensor and core temperatures. An optimization routine performed operations needed to find the best performing sensor system. After optimization, that system is capable of yielding core temperature results that are approximately in error by 0.2 °C. A model problem was formulated to assess the capabilities of the sensing system to follow a time-varying core temperature. It was found that for core temperatures that varied by 0.05 °C per minute, the sensor followed the core within 0.2 °C or better.

Thermo-Mechanical Analysis Of Functionally Graded Shells

The thermo-mechanical analysis of a simply supported, functionally graded shell is considered in this work. Refined shell theories are considered to account for grading material variation in the thickness direction. The governing thermodynamical equations are derived from the Principle of Virtual Displacements. The distribution of the temperature field T(z) is not assumed linear in the thickness direction of the layered shells and Fourier's heat conduction equation is solved to provide T(z). Classical and higher order shell theories are implemented in cases of both an equivalent single layer and layer-wise variable description by referring to Carrera's Unified Formulation. The numerical results show temperature, displacement and stress distributions along the thickness direction. Different volume fractions of the metallic and ceramic constituents as well as different shell thickness ratios and orders of expansion are analyzed. These are in good agreement with the quasi-3D solution obtained considering mathematical layers with constant properties in the FGM layer and using high orders of expansion.

Thermal testing and numerical simulation of gypsum wallboards incorporated with different PCMs content

A mathematical model based on the Fourier heat conduction equation for one dimension was developed. The complexity of the mathematical solution of this stiff set of differential equations that use boundary conditions that move with the solid–liquid interface was simplified by using an apparent heat capacity ( c p ap ) dependent on temperature and obtained by Modulated Differential Scanning Calorimetry (MDSC). The performance of this model was confirmed by using a home-made experimental installation for the thermal characterization of solid materials. Theoretical curves obtained for gypsum blocks with three different contents of phase change materials (PCMs) were in agreement with experimental ones, indicating that this thermal process can be reproduced theoretically by using the c p ap of each block and a unique thermal conductivity of the pure gypsum. The other physical and thermal properties were taken from literature or supplied by the manufacturers. Results also indicated that the higher the PCM content, the higher the energy storage capacity of the wallboard and the lower the wall temperature variation. Furthermore, it was found that a block containing a 5 wt.% of microcapsule allows the reduction of gypsum thickness by 8.5%, maintaining the same insulating effect. Thus, these kind of material can be used to improve comfort, save energy in buildings and even reduce the weight of wallboards.

Non-Fourier 3-Omega Method for Thermal Conductivity Measurement

The 3-omega method for thermal conductivity measurement using the hyperbolic heat conduction equation is presented. Mathematical expressions representing the conditions when non-Fourier effects cannot be neglected are formulated. Results obtained from Fourier and hyperbolic heat conduction equations based on parameterized dielectric thermophysical parameters are compared. The analysis is applied to a thin Chemical vapor deposited (CVD) diamond film and bologna meat and it is shown in these cases that by neglecting the non-Fourier effects one can underestimate the thermal conductivity by 50%.

Numerical simulation of non-Fourier effects in combined heat transfer

The non-Fourier effects on transient and steady temperature distribution in combined heat transfer are studied. The processes of coupled conduction and radiation heat transfer in grey, absorbing, emitting, scattering, one-dimensional medium with black boundary surfaces are analysed numerically. The hyperbolic heat conduction equation is solved by flux splitting method, and the radiative transfer equation is solved by P1 approximate method. The transient thermal responses obtained from non-Fourier heat conduction equation are compared with those obtained from the Fourier heat conduction equation. The results show that the non-Fourier effect can be important when the conduction to radiation parameter and the thermal relaxation time are larger. Further, the radiation effect is more pronounced at small values of single scattering albedo and conduction to radiation parameters. Analysis results indicate that the internal radiation in the medium significantly influences the wave nature.

Coupled thermo-mechanical analysis of one-layered and multilayered plates

The paper considers a fully coupled thermo-mechanical analysis of one-layered and multilayered isotropic and composite plates. In the proposed analysis, the temperature is considered a primary variable as the displacement; it is therefore directly obtained from the model and this feature permits the temperature field to be evaluated through the thickness direction in three different cases: – static analysis with imposed temperature on the external surfaces; – static analysis of structures subjected to a mechanical load, with the possibility of considering the temperature field effects; – a free vibration problem, with the evaluation of the temperature field effects. In the first case, imposing a temperature at the top and bottom of the plate, the static response is given in term of displacements, stresses and temperature field; the proposed method is very promising if compared to a partially coupled thermo-mechanical analysis, where the temperature is only considered as an external load, and the temperature profile must be a priori defined: considering it linear through the thickness direction or calculating it by solving the Fourier heat conduction equation. In the second case, a mechanical load is applied. The fully coupled thermo-mechanical analysis gives smaller displacement values than those obtained with the pure mechanical analysis; the temperature effect is not considered in this latter approach. The third case is the free vibration problem. The fully coupled thermo-mechanical analysis permits the effect of the temperature field to be evaluated: larger frequencies are obtained with respect to the pure mechanical analysis. Carrera’s Unified Formulation is applied to obtain several refined models with orders of expansion in the thickness direction, from linear to fourth-order, for displacements and temperature. Both equivalent single layer and layer wise approaches are considered for the multilayered plates. At present, no benchmarks are available within the framework of a fully coupled theory. This work aims to fill this gap.

Coupled thermo-mechanical analysis of one-layered and multilayered isotropic and composite shells

This work considers the fully coupled thermo-mechanical analysis of one-layered and multilayered isotropic and composite shells. The temperature is assumed a primary variable as the displacement; it is therefore directly obtained from the model and this feature permits the temperature field to be evaluated through the thickness direction. Three problems are analyzed: - static analysis of shells with imposed temperature on the external surfaces; - static analysis of shells subjected to a mechanical load, with the possibility of considering the temperature field effects; - a free vibration problem, with the evaluation of the temperature field effects. In the first problem, imposing a temperature at the top and bottom of the shells, the static response is given in terms of displacements, stresses and temperature field; the proposed method is very promising if compared to a partially coupled thermo-mechanical analysis, where the temperature is only considered as an external load, and the temperature profile must be a priori defined (considering it linear through the thickness direction or calculating it by solving the Fourier heat conduction equation). A mechanical load is applied in the second problem. The fully coupled thermo-mechanical analysis gives smaller displacement values than those obtained with the pure mechanical analysis; the temperature effect is not considered in this latter approach. The third problem is the free vibration analysis of shells. The fully coupled thermo-mechanical models permit the effect of the temperature field to be evaluated: larger frequencies are obtained with respect to the pure mechanical models. Several refined theories with orders of expansion in the thickness direction, for displacements and temperature, from linear to fourth-order are obtained in the framework of Carrera's Unified Formulation. Both equivalent single layer and layer wise approaches are considered for the multilayered shells.

Hamilton–Jacobi and quantum theory formulations of thermal-wave propagation under the dual-phase lagging model of heat conduction

Dual-phase lagging model is one of the most promising approaches to generalize the Fourier heat conduction equation, and it can be reduced in the appropriate limits to the hyperbolic Cattaneo–Vernotte and to the parabolic equations. In this paper it is shown that the Hamilton–Jacobi and quantum theory formulations that have been developed to study the thermal-wave propagation in the Fourier framework can be extended to include the more general approach based on dual-phase lagging. It is shown that the problem of solving the heat conduction equation can be treated as a thermal harmonic oscillator. In the classical approach a formulation in canonical variables is presented. This formalism is used to introduce a quantum mechanical approach from which the expectation values of observables such as the temperature and heat flux are obtained. These formalisms permit to use a methodology that could provide a deeper insight into the phenomena of heat transport at different time scales in media with inhomogeneous thermophysical properties.

EFFECT OF THE THROUGH-THE-THICKNESS TEMPERATURE DISTRIBUTION ON THE RESPONSE OF LAYERED AND COMPOSITE SHELLS

This paper considers the thermal stress problem of thick and thin multilayered cylindrical and spherical shells including carbon fiber reinforced layers and/or a central soft core. The following two cases are considered: (i) the temperature distribution in thickness direction is assumed linear; (ii) the temperature distribution in thickness direction is calculated via Fourier's heat conduction equation. Carrera's Unified Formulation and the Principle of Virtual Displacements are used to obtain the governing equations in the case of shells with constant radii of curvature subjected to established temperature conditions on their upper and lower surfaces. Both Equivalent Single Layer and Layer Wise models with an order of expansion in the thickness direction from linear to fourth order are considered. The importance of refined models for a correct evaluation of displacement and stress fields in multilayered shells can be noted. Furthermore, it has been shown that results obtained assuming a linear temperature profile in the thickness direction can be meaningless.

Thermoelasticity that uses fractional heat conduction equation

A survey of nonlocal generalizations of the Fourier law and heat conduction equation is presented. More attention is focused on the heat conduction with time and space fractional derivatives and on the theory of thermal stresses based on this equation.

Computational aspects of temperature-based finite element technique for the phase-change heat conduction problem

Abstract Liquid–solid phase transition is accompanied by a latent heat release, both in isothermal and non-isothermal phase transformations. The latent heat, the discontinuous phase-change function, and the local exchange of field variables during phase-changes increase the difficulties for obtaining a solution for the Fourier heat conduction equation. Celentano et al. [3] (COO) proposed a temperature-based finite element model for solving the multidimensional transient heat conduction problem involving phase-change. The present work addresses the computational aspects of the COO model. The importance of a line-search algorithm for improving the convergence of the Newton–Raphson iterations are explained in detail. While performing the iterations in this kind of fixed domain methods, the phase front moves back and forth fictitiously. The introduced phase-change matrix handles the latent effect efficiently. Three numerical examples including Direct Chill casting are presented, and the benefits and difficulties of the solution technique are elaborately discussed.

Non-Fourier heat conduction in a hollow sphere with periodic surface heat flux

Presented is the analysis of non-Fourier effect in a hollow sphere exposed to a periodic boundary heat flux. The problem is studied by deriving an analytical solution of the hyperbolic heat conduction equation. Using the obtained analytical expression, the temperature profiles at outer and inner surfaces of the sphere are evaluated for various thermal relaxation times. By comparing the results of non-Fourier model with those obtained from Fourier heat conduction equation, the transition process from parabolic model to hyperbolic one is shown. The phase difference and amplitude ratio of boundary surfaces are calculated as functions of the thermal relaxation time and the results are depicted graphically.

Dual-phase lag effects on thermal damage to biological tissues caused by laser irradiation

A dual-phase lag (DPL) bioheat conduction model, together with the broad beam irradiation method and the rate process equation, is proposed to investigate thermal damage in laser-irradiated biological tissues. It is shown that the DPL bioheat conduction model could predict significantly different temperature and thermal damage in tissues from the hyperbolic thermal wave and Fourier's heat conduction models. It is also found that the DPL bioheat conduction equations can be reduced to the Fourier heat conduction equations only if both phase lag times of the temperature gradient ( τ T ) and the heat flux ( τ q ) are zero. This is different from the DPL model for pure conduction materials, for which it can be reduced to the Fourier's heat conduction model provided that τ q = τ T . Effects of laser parameters and blood perfusion on the thermal damage simulated in tissues are also studied. The result shows that the overall effects of the blood flow on the thermal response and damage are similar to those of the time delay τ T .