On the approximate solution of the third boundary-value problem of heat-conduction theory for a circle

An efficient analytical approximate representation of the solution of the third boundary-value problem of heatconduction theory for a circle is obtained. A uniform evaluation of the error of the approximate formula ensures the convergence of the numerical algorithm. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 82, No. 2, pp. 403–408, March–April, 2009.     

The Validity of Using the Microscopic Hyperbolic Heat Conduction Model Under a Harmonic Fluctuating Boundary Heating Source

The validity of using the microscopic hyperbolic heat conduction model under a harmonic fluctuating boundary heating source is investigated. It is found that using the microscopic hyperbolic heat conduction model is essential when . The phase shift between the electron-gas and solid-lattice temperatures is found to be . This phase shift reaches a fixed value of 1.5708 rad at very large values of . It is found that the use of the microscopic hyperbolic heat conduction model is essential when ¯>1×10⁹ rads^–1 for most metallic layers independent of their thickness.     

Overshooting Phenomenon in the Hyperbolic Microscopic Heat Conduction Model

The overshooting phenomenon under the effect of the microscopic hyperbolic heat conduction model is investigated. A map tracing the region within which the overshooting phenomenon occurs is presented. The two most important parameters which control the overshooting phenomenon are found to be the first and second time-derivatives of the temperature at t=0. However, in order for the overshooting to appear, a higher initial value of the second time-derivative of the temperature change is required than the initial value of the first time-derivative of the temperature. Overshooting is more likely to appear in the parabolic, rather than in the hyperbolic, microscopic heat conduction model.     

The Hyperbolic Heat Conduction Equation in an Anisotropic Material

In this work, the phase-lag concept in the wave theory of heat conduction is extended to describe the thermal behavior of an anisotropic material. This is achieved by assuming that there are phase lags of different magnitudes between each component of the heat flux vector and the summation of temperature gradients in all directions of the orthogonal coordinate system. Also, expressions are provided to specify the locations of the principal coordinate axes, the principal thermal conductivities, and the principal thermal relaxation times.     

Use of the Laplace transformation for data reduction in the flash method of measuring thermal diffusivity

This paper presents a new way to reduce data in the laser flash method of measuring thermal diffusivity. Experimental temperature vs time data are first transformed by using the Laplace transformation, and then they are fitted with an appropriate theoretical formula. The data reduction procedure is more efficient and enables the use of more realistic models of heat conduction in the sample, because the theoretical formulae for transformed temperatures have a simpler form than those for nontransformed ones. Some examples of the theoretical formulae of transformed temperatures are included here for one- and two-dimensional heat transfer, respectively. The models described take into account a finite pulse time and heat losses from the sample. Two fitting algorithms are proposed. Experimentally, the data reduction procedure has been tested for a correction of the finite pulse time effect in the flash method. The results show that the accuracy of our procedure is comparable with other data reduction methods. Provided that the shape and duration of the pulse are known, this procedure allows elimination of the finite pulse time effect on calculation of the thermal diffusivity for any transformable heat pulse time function, even in cases where the other specialized data reduction procedures have failed.     

Gradient method for inverse heat conduction problem in nanoscale

An inverse heat conduction problem for nanoscale structures was studied. The conduction phenomenon is modelled using the Boltzmann transport equation. Phonon‐mediated heat conduction in one dimension is considered. One boundary, where temperature observation takes place, is subject to a known boundary condition and the other boundary is exposed to an unknown temperature. The gradient method is employed to solve the described inverse problem. The sensitivity, adjoint and gradient equations are derived. Sample results are presented and discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

双曲型积分微分方程混合元法的误差估计

基于Raviart-Thomas空间Vh×Wh,本文研究了双曲型积分微分方程初边值问题混合元方法的L2,L∞误差估计。给出了未知函数u,ut和乱utt伴随速度P,散度divP逼近解的最优阶L2误差估计。还得到了逼近u及P的拟最优阶L∞误差估计。     

Direct error in constitutive equation formulation for inverse heat conduction problem

We present a mixed numerical formulation that handles discontinuities well for scalar hyperbolic partial differential equations. The formulation is based on a least‐square error in the constitutive equation. It is motivated by scalar inverse diffusion problems with interior data and applies to convection of a passive scalar in a discontinuous compressible flow field. We motivate the need for a mixed formulation by discretizing using an irreducible finite element method and discuss some of the limitations of that approach. We then develop and prove that the mixed formulation is well posed and verify that it works for problems with continuous and discontinuous thermal conductivity distributions.     

Analytical Prediction of Quench Energies of Cooled Superconductors Based on the Hyperbolic Heat Conduction Model

The critical energy characteristics of cooled composite superconductors is analytically predicted based on the one-dimensional hyperbolic heat conduction model. The temperature dependence of the Ohmic heat generation, the finite speed of heat transfer, and the finite duration and finite length of the thermal disturbances are taken into account in the present model. The critical energies are calculated using a model based on the analytical solution of the hyperbolic heat conduction equation by the Laplace transformation method. The computational model results show that the critical energy depends on the relaxation time and disturbance duration. It is found that the hyperbolic conduction model predicts a lower-critical energy as compared to the predictions of the parabolic heat conduction model.     

受移动热源作用的两端固定杆的广义热-弹耦合问题

基于带有一个热松弛时间的Lord-Shulman广义热弹性理论,研究了受移动热源作用的两端固定的均质各向同性杆的热-弹动态响应。该文给出了杆的广义热-弹耦合的控制方程,借助拉普拉斯积分变换及其数值反变换对控制方程进行了求解。计算得到了杆内温度、应力及位移的分布规律,从其分布图上可以看出,温度、应力及位移随移动热源速度的增大而减小。     

Determination of the limit of applicability of the parabolic equation of heat conduction

A criterion is proposed which enables breakdown of the parabolic model of heat conduction for isotropic materials, in which there are no phase transitions, to be revealed, and which defines the conventional boundary between the linear and nonlinear regimes of nonequilibrium thermodynamics in heat problems. __________ Translated from Izmeritel’naya Tekhnika, No. 6, pp. 38–42, June, 2008.     

Transient Behavior of a Thermoelectric Device Under the Hyperbolic Heat Conduction Model

The major objective of this work is to describe the dynamic thermal behavior of thermoelectric generators and refrigerators under the effect of the hyperbolic heat conduction model. In practical situations, these devices work under transient operating conditions due to the time change in the imposed current, voltage, and hot or cold temperatures. Results for transient temperature distributions were obtained for different parameters. The coefficient of performance was obtained as a function of time for increasing current flow.     

Thermal Behavior of Perfect and Imperfect Contact Composite Slabs Under the Effect of the Hyperbolic Heat Conduction Model

This work studies the heat transfer mechanisms during rapid heating of two-layer composite thin slabs from a macroscopic point of view using the hyperbolic heat conduction model. The composite slabs consist of two thin metal layers which may be in perfect or imperfect thermal contact. The effects of parameters such as the two films' thickness ratio, thermal conductivity ratio, heat capacity ratio, thermal relaxation time, and interfacial heat transfer coefficient on the thermal behavior of the composite slabs are investigated.     

On a three-dimensional axisymmetric boundary-value problem of nonlinear elastic deformation: Asymptotic solution and exponentially small error

In this paper, we study a three-dimensional axisymmetric boundary-value problem of a slender cylinder composed of a nonlinearly elastic material subjected to an axial force. Starting from the field equations, after a transformation and proper scalings, we identify a small variable and two small parameters, which characterize the present problem. Then, by an approach involving compound series-asymptotic expansions, a nonlinear ODE is derived, which governs the axial strain (the first-term in the series expansion). By imposing the zero radial displacement conditions at two ends, we manage to get the analytical solution of the axial strain, from which all other physical quantities can be deduced and thus the three-dimensional displacement field can be determined. Graphical results are presented, which show that there are two boundary layers near the two ends while the middle part is in a state of almost uniform extension. The asymptotic structure of the analytical solution is derived, which offers clear explanations to the structure of the deformed configuration and shows that the thickness of both boundary layers is of the order of the radius. We also point out the relevance of the present results to the St. Venant’s problem. In particular, we obtain the explicit uniformly-valid exponentially small error term, when the obtained deformed configuration is compared to the configuration of a uniform extension.     

Transient Free Convection Fluid Flow in a Vertical Microchannel as Described by the Hyperbolic Heat Conduction Model

The transient hydrodynamics and thermal behaviors of fluid flow in an open-ended vertical parallel-plate microchannel are investigated analytically under the effect of the hyperbolic heat conduction model. The model that combines both the continuum approach and the possibility of slip at the boundary is adopted in this study. The effects of Knudsen number Kn and thermal relaxation time τ on the microchannel hydrodynamics and thermal behaviors are investigated using the hyperbolic and the parabolic heat conduction models. It is found that as Kn increases, the slip in the hydrodynamic and thermal boundary condition increases. Also, this slip increases as τ decreases.     

Model of the interaction of electromagnetic and thermal fields in delay media

We have solved the problem on the propagation of a high-frequency electromagnetic field in a half-space filled with a medium with delayed electric and magnetic polarizations and conduction currents under the action on the half-space of an arbitrary combination of plane fields, as well as of surface currents and charges distributed over the half-space surface. A model of the heating of the half-space medium as a result of the electromagnetic-to-thermal and vice versa energy conversion has been considered. A numerical study of the heating depending on the delay time has been made. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 82, No. 1, pp. 177–183, January–February, 2009.     

Multivariate numerical derivative by solving an inverse heat source problem

A method for approximating multivariate numerical derivatives is presented from multidimensional noise data in this paper. Starting from solving a direct heat conduction problem using the multidimensional noise data as an initial condition, we conclude estimations of the partial derivatives by solving an inverse heat source problem with an over-specified condition, which is the difference of the solution to the direct problem and the given noise data. Then, solvability and conditional stability of the proposed method are discussed for multivariate numerical derivatives, and a regularized optimization is adopted for overcoming instability of the inverse heat source problem. For achieving partial derivatives successfully and saving amount of computation, we reduce the multidimensional problem to a one-dimensional case, and give a corresponding algorithm with a posterior strategy for choosing regularization parameters. Finally, numerical examples show that the proposed method is feasible and stable to noise data.     

On the numerical solution of the Dirichlet initial boundary-value problem for the heat equation in the case of a torus

The initial-boundary-value problem for the heat equation in the case of a toroidal surface with Dirichlet boundary conditions is considered. This problem is reduced to a sequence of elleptic boundary-value problems by a Laguerre transformation. The special integral representation leads to boundary-integral equations of the first kind and the toroidal surface gives one-dimensional integral equations with a logarithmic singularity. The numerical solution is realized by a trigonometric quadrature method in cases of open or closed smooth boundaries. The results of some numerical experiments are presented.     

Three-dimensional thermo-viscoelastic material

A three-dimensional model of the generalized thermo-viscoelasticity is established. The formulation is applied to the generalized thermo-viscoelasticity theories: Lord–Shulman and Green–Lindsay as well as to the dynamic coupled theory. The resulting nondimensional coupled equations together with the Laplace and double Fourier transforms techniques are applied to a specific problem of a half space subjected to thermal shock and traction-free surface. The inverses of Fourier transforms and Laplace transforms are obtained numerically by using the complex inversion formula of the transform together with Fourier expansion techniques. Numerical results for the temperature, thermal stress, strain, and displacement distributions are represented graphically.     

Heat conduction in a solid slab embedded with a pipe of general cross-section: Shape Factor and Shape Optimization

We address the problem of two-dimensional heat conduction in a solid slab embedded with an isothermal, symmetric pipe of general cross-section. Similar formulations have applications in continuum mechanics and electricity. The main objective of this work is to develop a Shape Optimization algorithm that will reveal the optimal shapes of the pipe such that the conduction rate is maximized or minimized. This is achieved by optimizing the Shape Factor. To obtain the Shape Factor we transform the pipe into a strip using the generalized Schwarz-Christoffel transformation, and develop an integral equation of the first kind for the temperature gradient using Fourier transform techniques. The integral equation is solved both numerically and analytically/asymptotically. The fact that the Shape Factor is a monotonic function of the length of the strip suggests a Shape Optimization formulation where the objective function is the length of the strip and the variables of the optimization are the parameters of the generalized Schwarz-Christoffel transformation. Optimal shapes for the problem of minimizing the conduction rate are computed numerically and validated with an analytical solution. Numerical results for maximizing the transport rate are also obtained. The versatility and the robustness of the numerical optimization algorithm offers opportunities for improving the design of similar processes with non-linear equality and inequality constraints.