A consideration of the thermodynamics of continuous media using the entropy flux assumption of muller and the inclusion of more constitutive variables

In this paper we attempt to study the effect of the entropy flux of Muller and of the inclusion of other constitutive variables on the new heat equation, which is capable of predicting a finite velocity of propagation of the heat disturbance, and the form of the stress tensor. We define a new free energy function. The heat equation obtained is shown to be a generalisation of some obtained so far. It turns out that the inclusion of the constitutive variables T, the material time derivative of the temperature and the velocity gradient may be necessary and the adoption of Muller's new entropy flux in thermoelastic materials explored.     

On constitutive equations for thermoelastic dielectric continuum in terms of invariants

Electro-thermomechanical behavior of a thermoelastic dielectric body subject to external loading has been investigated theoretically in the present analysis. The theory is formulated in the context of continuum electrodynamics. The reaction of the body subject to external loads is expressed in symmetric stress, electrical polarization and heat flux. The solid medium is assumed to be linear, dielectric, isotropic, incompressible and dependent on temperature gradient. It has been observed that, as a result of thermodynamic constraints, the stress potential function is dependent on the deformation tensor, the electric field vector and the temperature, while the heat flux vector function is dependent on the deformation tensor, the electric field vector, the temperature and temperature gradient. To determine arguments of the stress potential and the heat flux vector functionals, findings of the theory of invariants have been used as a method because of that isotropy constraint is imposed on the material. As a result, constitutive equations of symmetric stress, polarization field and heat flux vector have been obtained in both material and spatial coordinates and asymmetric stress has been found using the expressions of symmetric stress and polarization field.     

Bandwidth reduction by automatic renumbering

Consider a solid heat conductor with a non-linear constitutive equation for the heat flux. If the material is anisotropic and inhomogeneous, the heat conduction equation to be satisfied by the temperature field θ(x, t) is, Here L (θ, x ) [grad θ] is a vector-valued function of θ, x , grad θ which is linear in grad θ, In the present paper, the application of the finite element method to the solution of this class of problems is demonstrated. General discrete models are developed which enable approximate solutions to be obtained for arbitrary three-dimensional regions and the following boundary and initial conditions: (a) prescribed surface temperature, (b) prescribed heat flux at the surface and (c) linear heat transfer at the surface. Numerical examples involve a homogeneous solid with a dimensionless temperature-diffusivity curve of the form κ = κ₀(l + σT). The resulting system of non-linear differential equations is integrated numerically.     

Temperature distribution and heat flow around a crack of arbitrary orientation in a functionally graded medium

This paper investigates the heat transfer problem of an infinite functionally graded medium containing an arbitrarily oriented crack under uniform remote heat flux. In the mathematical treatment the crack is approximated as a perfectly insulating cut. By using Fourier transformation, the mixed boundary value problem is reduced to a Cauchy-type singular integral equation for an unknown density function. The singular integral equation is then solved by representing the density function with a Chebyshev polynomial-based series and solving the resulting linear equation using a collocation technique. The temperature field in the vicinity of the crack and the crack-tip heat flux intensity factor are presented to quantify the effect of crack orientation and grading inhomogeneity on the heat flow around the crack.     

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

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

Determining heat flux from surface temperature measurements in pulsed gasdynamic processes

The surface temperature of a body of revolution in a pulsed supersonic nitrogen flow has been measured using anisotropic thermoelements based on bismuth single crystals. A method for calculating the heat flux toward the body surface using data on the surface temperature variations is proposed, which is based on solving the heat conduction equation in a semibounded space. The errors of temperature measurements and heat flux calculations are estimated using linear regression analysis.     

Boundary element solution of thermal creep flow in microfluidic devices

Flow in rarefied gases can be caused by a tangential temperature gradient along the contour boundaries (tangential heat flux), without the presence of any other external driven force, inducing a fluid motion from colder to hotter regions. This phenomenon is known as thermal creep and has gained importance in recent years in connection with micro-scale gas flow systems. Prediction of the flow field in micro-systems can be obtained by using continuum based models under appropriate boundary conditions accounting for the slip velocity due to tangential shear rate and heat flux. In this work a boundary integral equation formulation for Stokes slip flow, based on the normal and tangential projection of the Green's integral representational formulae for the velocity field is presented. The tangential heat flux is evaluated in terms of the tangential gradient of the temperature integral representational formulae presenting singularities of the Cauchy type, which are removed by the use of an auxiliary potential field. These formulations are used to evaluate the performance of different microfluidic devices.     

Transient heat transfer model for normal zone propagation. Part 1 — theory of a bare helium-cooled superconductor

The paper presents a physical model, involving no adjustable parameters of transient heat transfer for normal zone propagation in a composite helium-cooled superconductor. Evaluations are given showing that the main contribution to transient heat flux is made by transverse heat conduction in the coolant. When the zone front passes through a fixed point, the temperature at that point increases drastically, a great temperature gradient appears in coolant which results in transient cooling of the superconductor. The zone propagation equation becomes an integro-differential with a convolution integral. The transient heat transfer intensity is characterized by the introduced dimensionless parameter ?.The problem has been solved analytically for the model with constant coefficients and jump-wise heat release. Formulae have been obtained for the propagation and recovery velocities describing the effect of transient heat transfer (via parameter ?).     

Inverse approach for estimating boundary properties in a transient fin problem

A solution methodology is proposed for an inverse estimation of boundary conditions from the knowledge of transient temperature data. A forward model based on prevalent time-dependent heat conduction fin equation is solved using a fully implicit finite volume method. First, the inverse model is formulated and accomplished for time-invariant heat flux at the fin base, and later extended to transient heat flux, base temperature and average heat transfer coefficient. Secondly, the Nusselt number is then replaced with Rayleigh number in the forward model to realistically estimate the base temperature, which varies with respect to time, based on in-house transient fin heat transfer experiments. This scenario further corroborates the validation of the proposed inverse approach. The experimental set-up consists of a mild steel \(250 \times 150 \times 6\, \hbox {mm}^3\) fin mounted centrally on an aluminium base \(250 \times 150 \times 8\, \hbox {mm}^3\) plate. The base is attached to an electrical heater and insulated with glass-wool to prevent heat loss to surroundings. Five calibrated K-type thermocouples are used to measure temperature along the fin. The functional form of the unknown parameters is not known beforehand; sensitivity studies are performed to determine suitability of the estimation and location of sensors for the inverse approach. Conjugate gradient method with adjoint equation is chosen as the inverse technique and the study is performed as a numerical optimization; subsequently, the estimates show satisfactory results.     

Multi-axial thermo-mechanical analysis of power plant components from 9–12% Cr steels at high temperature

The aim of this paper is to analyze local changes of stress and strain states in a power plant component under a transient thermal environment. A robust constitutive model is developed to describe inelastic behavior of advanced 9–12% Cr heat-resistant steels at high temperature and in a multi-axial stress state. The model includes the constitutive equation for the inelastic strain rate tensor, the evolution equation for a tensor-valued state variable to reflect hardening/recovery processes and two evolution equations for two scalar-valued variables that characterize softening and damage states. The model is calibrated against experimental creep curves and verified for inelastic responses under different isothermal and non-isothermal loading paths. Steam temperature and loading profiles that correspond to an idealized start-up, holding and shut-down sequence of a power plant component are assumed. To estimate the thermal fields, transient heat transfer analysis is performed. The results are applied in the subsequent structural analysis using the developed inelastic constitutive model. The outcome is a multi-axial thermo-mechanical fatigue loop which can be used for damage assessment.     

Equivalent constitutive equations of honeycomb material using micro-polar theory to model thermo-mechanical interaction

For describing the properties for micro-polar cellar material, the definitions of representative element as well as the corresponding equivalent physical quantities in cellular are introduced in this paper. It involves the Cauchy stress, couple stress, displacement gradient, strain, torsion tensor, temperature gradient and the heat flux respectively. The general principle and mode of solving the boundary value problem with respect to the construction of the equivalent constitutive equations is investigated, and the thermo-mechanical interaction is modeled for the micro-polar cellar material. In the context, a standard method for formulizing the boundary value problem in accordance with the specified representative element volume (REV) is developed, and thereupon the equivalent constitutive equations are deduced. For honeycomb materials, the analytical formula for the equivalent elastic coefficient tensor, temperature coefficients of equivalent stress, equivalent Fourier coefficients and the temperature gradient coefficients of couple stress of the honeycomb materials would be formed.     

A thermodynamic analysis of rigid heat conductors

A thermodynamic approach to rigid heat conductors is proposed: it introduces the heat flux vector as independent variable while its temporal evolution is governed by a first order differential equation. The form of the second law is that proposed by Müller wherein the entropy flux and the entropy source are not given a priori but determined through constitutive equations. Restrictions on the constitutive equations are placed by the second law. Some properties, valid in the vicinity of equilibrium are established. In particular, it is shown that the present theory leads to a hyperbolic heat conduction equation, allowing for the propagation of heat as a thermal wave with a finite velocity. The concept of thermodynamic forces and fluxes is also introduced. The latter are seen to derive from a potential function plus an additional term. Finally, it is established under which conditions symmetry relations are satisfied.     

Equivalent constitutive equations of honeycomb material using micro-polar theory to model thermo-mechanical interaction

For describing the properties for micro-polar cellar material, the definitions of representative element as well as the corresponding equivalent physical quantities in cellular are introduced in this paper. It involves the Cauchy stress, couple stress, displacement gradient, strain, torsion tensor, temperature gradient and the heat flux respectively. The general principle and mode of solving the boundary value problem with respect to the construction of the equivalent constitutive equations is investigated, and the thermo-mechanical interaction is modeled for the micro-polar cellar material. In the context, a standard method for formulizing the boundary value problem in accordance with the specified representative element volume (REV) is developed, and thereupon the equivalent constitutive equations are deduced. For honeycomb materials, the analytical formula for the equivalent elastic coefficient tensor, temperature coefficients of equivalent stress, equivalent Fourier coefficients and the temperature gradient coefficients of couple stress of the honeycomb materials would be formed.     

薄板在周期热流作用下的热响应(Ⅰ):温度响应

基于具有热流延迟相的双曲型热传导方程,研究了薄板在周期热流作用下的温度响应。首先采用分离变量法,求解了以热流矢量为基本未知量的热传导方程,得到了板内热流场分布,然后再利用能量守恒方程,获得了板内温度响应的解析表达式。通过计算,分析了板内温度响应随不同热流矢量延迟相以及边界热流频率的变化趋势,并与经典的Fourier热传导方程所得到的结果进行了比较。结果表明,在高频热流加热下,双曲型热传导模型所给出的温度响应与经典的Fourier热传导模型具有显著的差别。     

Thermal boundary layers over a shrinking sheet: an analytical solution

In this paper, the heat transfer over a shrinking sheet with mass transfer is studied. The flow is induced by a sheet shrinking with a linear velocity distribution from the slot. The fluid flow solution given by previous researchers is an exact solution of the whole Navier–Stokes equations. By ignoring the viscous dissipation terms, exact analytical solutions of the boundary layer energy equation were obtained for two cases including a prescribed power-law wall temperature case and a prescribed power-law wall heat flux case. The solutions were expressed by Kummer’s function. Closed-form solutions were found and presented for some special parameters. The effects of the Prandtl number, the wall mass transfer parameter, the power index on the wall heat flux, the wall temperature, and the temperature distribution in the fluids were investigated. The heat transfer problem for the algebraically decaying flow over a shrinking sheet was also studied and compared with the exponentially decaying flow profiles. It was found that the heat transfer over a shrinking sheet was significantly different from that of a stretching surface. Interesting and complicated heat transfer characteristics were observed for a positive power index value for both power-law wall temperature and power-law wall heat flux cases. Some solutions involving negative temperature values were observed and these solutions may not physically exist in a real word.     

Effective thermal properties of viscoelastic composites having field-dependent constituent properties

This study introduces a micromechanical model for predicting effective thermal properties (linear coefficient of thermal expansion and thermal conductivity) of viscoelastic composites having solid spherical particle reinforcements. A representative volume element (RVE) of the composites is modeled by a single particle embedded in the cubic matrix. Periodic boundary conditions are imposed to the RVE. The micromechanical model consists of four particle and matrix subcells. Micromechanical relations are formulated in terms of incremental average field quantities, i.e., stress, strain, heat flux and temperature gradient, in the subcells. Perfect bonds are assumed along the subcell’s interfaces. Stress and temperature-dependent viscoelastic constitutive models are used for the isotropic constituents in the micromechanical model. Thermal properties of the particle and matrix constituents are temperature dependent. The effective coefficient of thermal expansion is derived by satisfying displacement and traction continuity at the interfaces during thermo-viscoelastic deformations. This formulation leads to an effective time–temperature–stress-dependent coefficient of thermal expansion. The effective thermal conductivity is formulated by imposing heat flux and temperature continuity at the subcells’ interfaces. The effective thermal properties obtained from the micromechanical model are compared with analytical solutions and experimental data available in the literature. Finally, parametric studies are also performed to investigate the effects of nonlinear thermal and mechanical properties of each constituent on the overall thermal properties of the composite.     

Radiative-conductive transmission of heat through optically dense media

The problem of heat transmission by conduction and radiation through a semiinfinite optically dense medium is analyzed, with incident external radiation and with convective heat transfer taken into account. An expression for the radiant thermal flux is derived from the solution to the equation of radiation flux propagation by the method of associative asymptotic expansions. The effect of the temperature gradient at the surface on the emissivity of the body is established for the medium range of absorptivity values.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 23, No. 3, pp. 459–464, September, 1972.     

Stability of uniform temperature fields in linear heat conductors with memory

In this paper we consider the asymptotic behavior of temperature fields in linear heat conductors with memory. Using the method of energy integrals, we establish stability for the history problem both in the case when the governing heat equation is parabolic in character and when the heat equation is hyperbolic and predicts finite wave speeds.     

Reformulation of the Nosé–Hoover thermostat for heat conduction simulation at nanoscale

This paper investigates the heat conduction phenomena at nanoscale by reformulating the Nosé–Hoover thermostat for nonequilibrium molecular dynamics simulation. Inspired by the Nosé–Hoover mechanics, the feedback temperature force caused by the temperature control is reformulated to a more general level, aiming at (1) controlling the temperature locally at several distinct spots and (2) eliminating the rigid-body translation and rotation which are unexpectedly introduced into the system due to the feedback temperature force, so that accurate trajectories of atoms can be generated and the heat conduction simulation at nanoscale can be performed successfully. Correspondingly, the definition of temperature is modified; the expression of the Hamiltonian is generalized. To demonstrate the capability and feasibility of this newly formulated algorithm, we studied heat conduction phenomenon in a beam-like finite size specimen via our in-house developed computer code. The results, temperature distributions across the specimen, are shown to reach steady state after a period of time which cannot be achieved by the original Nosé–Hoover thermostat. Also, it is concluded that the heat conduction at nanoscale exhibits the same feature of Fourier’s law at macroscopic scale, namely that the heat flux is linearly proportional to the temperature gradient, if the temperature is averaged over a sufficiently large time interval and large spatial region. The thermal conductivity can thereafter be calculated based on the linear relation between the heat flux and the temperature gradient. It is found that the obtained numerical value of thermal conductivity matches the experimental result very well.