Coupled Upscaling Approaches For Conduction, Convection, and Radiation in Porous Media: Theoretical Developments

This study deals with macroscopic modeling of heat transfer in porous media subjected to high temperature. The derivation of the macroscopic model, based on thermal non-equilibrium, includes coupling of radiation with the other heat transfer modes. In order to account for non-Beerian homogenized phases, the radiation model is based on the generalized radiation transfer equation and, under some conditions, on the radiative Fourier law. The originality of the present upscaling procedure lies in the application of the volume averaging method to local energy conservation equations in which radiation transfer is included. This coupled homogenization mainly raises three challenges. First, the physical natures of the coupled heat transfer modes are different. We have to deal with the coexistence of both the material system (where heat conduction and/or convection take place) and the non-material radiation field composed of photons. This radiation field is homogenized using a statistical approach leading to the definition of radiation properties characterized by statistical functions continuously defined in the whole volume of the porous medium. The second difficulty concerns the different scales involved in the upscaling procedure. Scale separation, required by the volume averaging method, must be compatible with the characteristic length scale of the statistical approach. The third challenge lies in radiation emission modeling, which depends on the temperature of the material system. For a semi-transparent phase, this temperature is obtained by averaging the local-scale temperature using a radiation intrinsic average while a radiation interface average is used for an opaque phase. This coupled upscaling procedure is applied to different combinations of opaque, transparent, or semi-transparent phases. The resulting macroscopic models involve several effective transport properties which are obtained by solving closure problems derived from the local-scale physics.     

Heat propagation in a nonuniform rod of variable cross section

An integral formula is used to average a coupled problem of thermoelasticity for a nonuniform rod of variable cross section. Effective characteristics are found. It is shown that, in addition to the expected effective coefficients, there appear five independent coefficients characterizing the temperature change rate effect on the stresses in the rod, on the longitudinal heat flux, and on the entropy distribution along the length of the rod. A feature of these new coefficients is that they become equal to zero in the case of a uniform rod. The homogenization of the thermoelasticity equations for nonuniform rods allows one to propose a new theory of heat conduction in rods. This new theory differs from the classical one by the fact that some new terms are added to the Duhamel–Neumann law, to the Fourier heat conduction law, and to the entropy expression. These new terms are proportional to the temperature change rate with time. It is also shown that, in the new theory of heat conduction, the propagation velocity of harmonic heat perturbations is dependent on the oscillation frequency and is finite when the frequency tends to infinity.     

A method for calculating intense injection from surfaces of blunt bodies entering planetary atmospheres at hypersonic velocities

Flow past blunt bodies entering planetary atmospheres at hypersonic velocities is studied. A method for calculating the flowfield near the body nose is developed which allows for radiative heat transfer in the P ₁ approximation of the spherical harmonics method but does not take gas viscosity and heat conduction into account. The solution is constructed on the basis of a two-layer flow model, with account for intense injection of ablation products from the body nose due to radiative heat fluxes from the shock layer. The advantages of the method are that the multi-dimensional character of the radiation field is taken into account and the general problem of radiation gasdynamics is solved on the basis of a unified algorithm. The flow past a spherical segment and a spherically-blunted cone re-entering the Earth’s atmosphere at a velocity of 20 km/s and an entry angle of ?10° is calculated.     

Geometric nonlinear formulation for thermal-rigid-flexible coupling system

This paper develops geometric nonlinear hybrid formulation for flexible multibody system with large deformation considering thermal efect. Diferent from the conventional formulation, the heat flux is the function of the rotational angle and the elastic deformation, therefore, the coupling among the temperature, the large overall motion and the elastic deformation should be taken into account. Firstly,based on nonlinear strain–displacement relationship, variational dynamic equations and heat conduction equations for a flexible beam are derived by using virtual work approach,and then, Lagrange dynamics equations and heat conduction equations of the first kind of the flexible multibody system are obtained by leading into the vectors of Lagrange multiplier associated with kinematic and temperature constraint equations. This formulation is used to simulate the thermal included hub-beam system. Comparison of the response between the coupled system and the uncoupled system has revealed the thermal chattering phenomenon. Then, the key parameters for stability, including the moment of inertia of the central body, the incident angle, the damping ratio and the response time ratio, are analyzed. This formulation is also used to simulate a three-link system applied with heat flux. Comparison of the results obtained by the proposed formulation with those obtained by the approximate nonlinear model and the linear model shows the significance of considering all the nonlinear terms in the strain in case of large deformation. At last, applicability of the approximate nonlinear model and the linear model are clarified in detail.     

On applications of the microlocal parameter method in modelling of temperature distributions in composite cylinders

Summary The problems of transient heat conduction in a periodically stratified medium consisting of a large number of alternating concentric cylinders of two homogeneous isotropic rigid materials and in a rotationally periodic cylinder consisting of a large number of circular homogeneous isotropic rigid sectors are considered. The equations of the homogenized models with microlocal parameters are derived by using the homogenization procedure given in [17]. The obtained models take into account certain microlocal effects connected with the microperiodic structure of the considered composites. Some examples of the application of the presented models to the problems of temperature distributions in composite cylinders are detailed. Received 10 March 1997; accepted for publication 23 October 1997     

Control of axisymmetric resonant vibrations and self-heating of shells of revolution with piezoelectric sensors and actuators

The coupled problem of forced vibrations and self-heating of thermoviscoelectroelastic shells of revolution with piezoceramic sensor and actuator under monoharmonic loading is solved. The temperature dependence of the complex characteristics of the passive and piezoactive materials is taken into account. The coupled nonlinear problem of thermoelectroelasticity is solved by time-marching integration, using discrete orhogonalization to integrate the equations of elasticity and explicit finite differencing to solve the heat conduction equation. The effect of the dimensions of the sensor and actuator and self-heating on the sensor voltage and on the active damping of forced vibrations of a circular plate under uniform monoharmonic transverse pressure is studied     

Three-dimensional chemically nonequilibrium viscous shock layer on a catalytic surface with allowance for associated heat transfer

A three-dimensional flow of dissociating air past blunt bodies is investigated in the framework of the thin viscous shock layer theory. Multicomponent diffusion and homogeneous chemical reactions, including dissociation, recombination, and exchange reactions, are taken into account. The generalized Rankine-Kugoniot conditions are specified on the shock wave and the conditions which take into account the heterogeneous catalytic reactions, on the surface of the body. The viscous shock layer equations are solved together with the heat equations inside the coating, which is carbon with a deposited thin film of SiO₂, or quartz. The case of a thermally insulated surface is also considered. The problem for the case of the motion of a body along the re-entry trajectory into Earth's atmosphere is investigated numerically. The temperature of the surface and the heat flux toward it are given as a dependence on the height (tine) of the flight for different cases of the specification of the catalytic reactions. It is shown that the difference between the heat fluxes towards the thermally insulated surface and the fluxes toward the heat-conducting surface in the neighborhood of the stagnation point is of the order of 6–12% for all the cases considered. This makes it possible to decouple the solution of the problem of heat conduction in the body.Translated fron Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 140–146, November–December, 1985.deceased     

A continuum theory for viscoelastic composite beams

Summary A dynamical continuum theory is developed for laminated composite beams. Starting with an assumed displacement- and temperature field, the one-dimensional approximate theory is consistently constructed within the frame of the three-dimensional theory of linear, nonisothermal, anisotropic, coupled viscoelasticity. Each constituent of the beam may possess different constant thickness and mechanical properties. All dynamic interactions between the adjacent constituents are included. Further, the effects of transverse shear and normal strains and rotatory inertia as well as those of cross-sectional distortion are all taken into account. The resulting equations consist of the macroscopic beam equations of motion and heat conduction, the kinematical relations, the initial and boundary conditions and the constitutive equations, and they govern the extensional, flexural and torsional motions of laminated composite beams. The special cases of constituents which made of either isotropic thermoviscoelastic or anisotropic thermoelastic materials are discussed briefly.Supported by the Office of Naval Research.With 1 figure     

The fundamental equations in finite element method of coupled thermo-elastic plane problem

The fundamental equations in finite element method for unsteady temperature field elastic plane problem are derived on the bases of variational principle of coupled thermoelastic problems. In these derivations, elastic plane is divided into three nodes triangular elements, and time interval is divided into linear time elements, in which all the variables, including displacements and temperatures at various nodal points, are varied linearly with time. Two coupled sets of linear algebraic equations of all the unknown displacements and temperatures at every nodal point in every instant (i.e. the terminal values of time elements) are obtained. They are the fundamental equations of the said problem.The total energy in elastic body not only contains the potential energy and heat energy but also contains the kinetic energy, if the rate of change of temperature field with respect to the time in thermoelastic problem is large enough. And the change of displacement is included in the equations of heat conduction. For this reason the variational principle of coupled thermoelastic problems is employed. [1] In this paper, expressions of this principle for plane problems are given. The discretization is carried on then, and Hamilton's action and the potential action of heat flow of elements are derived. Finally they are assembled, so as to get the polar value of the action. And thus the groups of linear algebraic equations in matrix form are obtained.     

Effects of boundary reflection on radiative heat transfer in participating media

Radiative heat transfer through a non-isothermal grey participating medium between two parallel surfaces kept at fixed temperature has been investigated. The integro-differential transfer equations for surface reflection were solved in semi-analytical form by projectional methods; conduction and convection were neglected. It was assumed that reflection from the cold wall was diffuse, while that from the hot wall was either diffuse or specular. The heat flux and the temperature distribution in the participating medium were calculated in each physical condition, in order to compare the effects of different reflection modes on heat transfer. The results show that temperature distributions and heat fluxes are only slightly affected by the particular reflection law, the relative difference being less than 1%. This suggests that diffuse reflection only could be considered for practical applications, since it requires a much simpler computational procedure.     

中心舱体与薄壁梁刚-柔耦合系统的热弹性-结构动力学分析

空间柔性结构受太阳热流冲击而诱发的振动是导致航天器失效的典型模式之一,准确预测结构热致振动的响应及稳定性是卫星设计的基础.针对常见的中心舱体与附属薄壁杆件组成的空间结构,提出了考虑刚-柔耦合、耦合热弹性和耦合热-结构三重耦合效应的热致振动分析理论模型.其中,刚-柔耦合是指舱体姿态角、顶端集中质量转动与柔性附件运动的耦合...     

考虑几何非线性和热效应的刚-柔耦合动力学

温度增高和温度梯度会引起梁的纵向、横向变形位移,在一定程度上对刚-柔耦合规律产生影响.该文考虑热应变,从平面梁的非线性的应变与位移关系式出发,建立了刚体运动、弹性变形和温度相互耦合的有限元离散的热传导方程和动力学方程.研究热流作用下的中心刚体-简支梁系统的刚-柔耦合动力学性质,揭示了几何非线性项和热应变对弹性变形和刚体运动影响.     

Conjugate model for heat and mass transfer of porous wall in the high temperature gas flow

Heat and mass transfer of a porous permeable wall in a high temperature gas dynamical flow is considered. Numerical simulation is conducted on the ground of the conjugate mathematical model which includes filtration and heat transfer equations in a porous body and boundary layer equations on its surface. Such an approach enables one to take into account complex interaction between heat and mass transfer in the gasdynamical flow and in the structure subjected to this flow. The main attention is given to the impact of the intraporous heat transfer intensity on the transpiration cooling efficiency. The project supported by the National Natural Science Foundation of China (19889209) and Russian Foundation for Basic Research (97-02-16943)     

AN ALGEBRAIC MULTIGRID METHOD FOR COUPLED THERMO-HYDRO-MECHANICAL PROBLEMS

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｒｅｃｅｎｔｙｅａｒｓ,ａｇｒｅａｔｄｅａｌｏｆａｔｔｅｎｔｉｏｎｈａｓｂｅｅｎｄｉｒｅｃｔｅｄｔｏｗａｒｄｓｆｕｌｌｃｏｕｐｌｅｄｔｈｅｒｍｏ＿ｈｙｄｒｏ＿ｍｅｃｈａｎｉｃａｌａｎａｌｙｓｉｓｉｎｄｅｆｏｒｍｉｎｇｐｏｒｏｕｓｍｅｄｉａｄｕｅｔｏｅｘｔｒａｃｔｉｏｎｏｆｕｎｄｅｒｇｒｏｕｎｄｆｌｕｉｄｓ(ｗａｔｅｒ,ｐｅｔｒｏｌｅｕｍ ,ｎａｔｕｒａｌｇａｓ)ｉｎｒｅｓｅｒｖｏｉｒ,ｅｎｖｉｒｏｎｍｅｎｔａｎｄｃｏｎｓｔｒｕｃｔｉｏｎｅｎｇｉｎｅｅｒｉｎｇ (Ｌｅｗｉｓａｎ…     

Ideal nonsymmetric 4D-medium as a model of invertible dynamic thermoelasticity

The space-time continuum (4D-medium) is considered, and a generalized model of reversible dynamic thermoelasticity is constructed as a theory of elasticity of an ideal (defect-free) nonsymmetric 4D-medium that is transversally-isotropic with respect to the time coordinate. The definitions of stresses and strains for the space-time continuum are introduced. The constitutive equations of the medium model relating the components of nonsymmetric stress and distortion 4D-tensors are stated. Physical interpretations of all tensor components of the thermomechanical properties are given. The Lagrangian of the generalized model of coupled dynamic thermoelasticity is presented, and the Euler equations are analyzed. It is shown that the three Euler equations are generalized equations of motion of the dynamic classical thermoelasticity, and the last, fourth, equation is a generalized heat equation which allows one to predict the wave properties of heat. An energy-consistent version of thermoelasticity is constructed where the Duhamel-Neumann and Maxwell-Cattaneo laws (a nonclassical generalization of the Fourier law for the heat flow) are direct consequences of the constitutive equations.     

Governing equations in non-isothermal diffusion

Generalizations of Fick's law for the diffusion flux are often considered in the literature by analogy with those for the heat flux. The paper reviews the balance equations for a fluid mixture and provides the equations for the diffusion fluxes. As a consequence, the mass densities are shown to satisfy a system of hyperbolic equations. Moreover, for a binary mixture of ideal gases in stationary conditions, Fick's law is recovered. Next, diffusion fluxes are regarded as constitutive functions and a whole set of thermodynamic restrictions are determined which account for diffusion, heat conduction, viscosity and inhomogeneities. Hyperbolic models for diffusion and heat fluxes are established which involve the co-rotational derivative. The driving term of diffusion turns out to be the gradient of chemical potential rescaled by the temperature.     

Darcy-Forchheimer flow of variable conductivity Jeffrey liquid with Cattaneo-Christov heat flux theory

The role of the Cattaneo-Christov heat flux theory in the two-dimensional laminar flow of the Jeffrey liquid is discussed with a vertical sheet. The salient feature in the energy equation is accounted due to the implementation of the Cattaneo-Christov heat flux. A liquid with variable thermal conductivity is considered in the Darcy-Forchheimer porous space. The mathematical expressions of momentum and energy are coupled due to the presence of mixed convection. A highly nonlinear coupled system of equations is tackled with the homotopic algorithm. The convergence of the homotopy expressions is calculated graphically and numerically. The solutions of the velocity and temperature are expressed for various values of the Deborah number, the ratio of the relaxation time to the retardation time, the porosity parameter, the mixed convective parameter, the Darcy-Forchheimer parameter, and the conductivity parameter. The results show that the velocity and temperature are higher in Fourier's law of heat conduction cases in comparison with the Cattaneo-Christov heat flux model.     

轴对称自由振动功能梯度材料微圆板中的热弹性阻尼

基于经典弹性薄板理论和单向耦合热传导理论,研究了材料性质沿厚度连续变化的功能梯度微圆板的热弹性阻尼特性．首先,考虑热力耦合效应,建立了功能梯度微圆板轴对称横向自由振动微分方程．然后,忽略温度梯度在面内的变化,建立了单向耦合变系数一维热传导方程．采用分层均匀化近似方法,将变系数热传导方程转化为一系列常系数的微分方程,利用上下表面的热边界条件和层间连续性条件获得了微圆板温度场解析解．将所得温度场代入微圆板的自由振动微分方程,得到了包含热弹性阻尼的复频率,从而获得了反映热弹性阻尼水平的逆品质因子．最后,针对材料性质沿板厚按幂函数变化的陶瓷-金属功能梯度微圆板,定量地分析材料梯度指数、几何尺寸、边界条件、温度环境等对微圆板热弹性阻尼的影响．     

Uniqueness theorems in the theory of thermoelasticity for solids with double porosity

In this paper the coupled linear theory of thermoelasticity for solids with double porosity is considered. The governing system of field equations of this theory is based on motion equations, conservation of fluid mass, constitutive equations, extended Darcy’s law for materials with double porosity and Fourier’s law for heat conduction. A wide class of the basic internal and external boundary value problems (BVPs) of steady vibrations is formulated and uniqueness theorems for regular (classical) solutions of these BVPs are proved.