Equivalence between dual-phase-lagging and two-phase-system heat conduction processes

We show the equivalence between the dual-phase-lagging heat conduction and the Fourier heat conduction in two-phase systems subject to lack of local thermal equilibrium. This provides an additional tool for studying the two heat-conduction processes and shows the possibility of thermal oscillation and resonance in two-phase-system heat conduction, a phenomenon observed experimentally. Such thermal waves and possibly resonance come from the macroscale coupled conductive terms.     

Dual-phase-lagging heat conduction based on Boltzmann transport equation

In this work, the dual-phase-lagging model of the microscale heat conduction is re-derived analytically from the Boltzmann transport equation. Based on this model, the delay/advanced partial differential equations governing the microscale heat conduction are established. The method of separation of variables is applied to solve such delay/advanced partial differential equations. Finally, the oscillation of the microscale heat conduction is investigated.     

The modeling of nanoscale heat conduction by Boltzmann transport equation

The discrete Boltzmann transport equation is firstly formulated by accounting for the impact of particle collisions on the distribution function in the discrete Liouville equation. Based on this equation, we re-derive the improved dual-phase-lagging heat conduction model with lagging effect in time and nonlocal effect in space. By taking into account the contribution of the higher order moments of the distribution function to the heat flux, we show that the discrete Boltzmann transport equation can give rise to the well-known Guyer–Krumhansl heat conduction model.     

Integral equation solution for hyperbolic heat conduction with surface radiation

Temperature distributions for the hyperbolic heat conduction in a semi-infinite medium with surface radiation are found from the solutions of a nonlinear Volterra equation for the surface temperature. The integral equation is obtained by the Laplace transform. This method has the advantage that the temperature distributions do not involve numerical oscillations around the thermal wave front.     

Difference scheme for hyperbolic heat conduction equation with pulsed heating boundary

horotctionIt is well known that the basic law of heatconduction is the Fourier law. It has the formq = --k' vT, and one-dimensional heat conduchondifferenhal equation is. The aboveequations are derived from the hypothesis that thevelocity to establish the thermal balance is infinitelygreat. In the modem heat conduchon theory heattransacts in materials in a licited velocity. The factorsto affect the velocity are the thermal propelles of thematerials. In order tO describe this Problem, the sch…     

An approximate analytic method for solving 1D dual-phase-lagging heat transport equations

In this study, we develop an approximate analytic method for solving 1D dual-phase-lagging heat conduction equations, which are derived based on the original dual-phase-lagging model without the first-order Taylor series approximation. The approximate analytic solution is obtained by employing the method of separation of variables. The coefficients of the series solution are then approximated by polynomials. The numerical method is illustrated with two simple examples.     

A new algorithm for direct and backward problems of heat conduction equation

Direct heat conduction problem (DHCP) and backward heat conduction problem (BHCP) are numerically solved by employing a new idea of fictitious time integration method (FTIM). The DHCP needs to consider the stability of numerical integration in the sense that the solution may be divergent for a specific time stepsize and specific spatial stepsize. The BHCP is renowned as strongly ill-posed because the solution does not continuously depend on the given data. In this paper, we transform the original parabolic equation into another parabolic type evolution equation by introducing a fictitious time variable, and adding a fictitious viscous damping coefficient to enhance the stability of numerical integration of the discretized equations by employing a group preserving scheme. When 10 numerical examples are amenable, we find that the FTIM is applicable to both the DHCP and BHCP. Even under seriously noisy initial or final data, the FTIM is also robust against disturbance. More interestingly, when we use the FTIM, we do not need to use different techniques to treat DHCP and BHCP as that usually employed in the conventional numerical methods. It means that the FTIM can unifiedly approach both the DHCP and BHCP, and the gap between direct problems and inverse problems can be smeared out.     

Explicit full field analytic solutions for two-dimensional heat conduction problems with finite dimensions

This study presents explicit analytical solutions of heat conduction problems for isotropic media with finite dimensions. The geometry configurations considered in this study include composite layer, wedge and circular media. The boundary conditions are assumed to be either thermal isolation or isothermal. The full field analytical solutions of temperature and heat fluxes for the composite layered media subjected to an embedded heat source are derived first by Fourier transform technique in conjunction with the image method. The corresponding problems of composite wedge and circular media are constructed by conformal mapping and the solutions of composite layer media. The explicit full field solutions are expressed in simple closed-forms which can be easily used in engineering applications. The numerical calculations of the temperature and heat fluxes distributions are provided in full field configuration base on the available analytic solutions.     

A regularized integral equation method for the inverse geometry heat conduction problem

This paper addresses a new technique for solving the inverse geometry heat conduction problem of the Laplace equation in a two-dimensional rectangle, which, named regularized integral equation method (RIEM), consists of three parts. First of all, the Fourier series expansion technique is used to calculate the temperature field u(x, y). Second, we consider a Lavrentiev regularization by adding a term αg(x) to obtain a second kind Fredholm integral equation. The termwise separable property of the kernel function allows us to transform the inverse geometry heat conduction problem into a two-point boundary value problem and therefore, an analytical regularized solution is derived in the final part by using orthogonality. Principally, the RIEM possesses the following advantages: it does not need any guess of the initial profile, it does not need any iteration and a regularized closed-form solution can be obtained. The uniform convergence and error estimate of the regularized solution u^α(x, y) are proved and a boundary geometry p(x) is solved by half-interval method. Several numerical examples present the effectiveness of our novel approach in providing excellent estimates of unknown boundary shapes from given data.     

Nonlinear inverse heat conduction problem of surface temperature estimation by calibration integral equation method

A calibration integral equation method is proposed for estimating the surface temperature in the context of a nonlinear inverse heat conduction problem. The temperature-dependent thermophysical properties and probe positioning are implicitly accounted in the integral equation formulation through calibration tests. A first kind Chebyshev expansion is applied to represent the temperature-dependent property transform function. The undetermined expansion coefficients associated with the Chebyshev expansion are then estimated through two calibration tests. Regularization of the ill-posed problem is achieved by the future-time method. The optimal regularization parameter is estimated using a phase plane and cross-correlation phase plane analyses. Numerical simulation for stainless steel yields highly favorable surface temperature prediction.     

Relativistic heat conduction

The hyperbolic heat conduction equation (HHCE), which acknowledges the finite speed of heat propagation, is based on microscopic evidence from the kinetic theory and statistical mechanics. However, it was argued that the HHCE could violate the second law of thermodynamics. This paper shows that a HHCE-like equation (RHCE) can be derived directly from the theory of relativity, as a direct consequence of space-time duality, without any consideration of the microstructure of the heat-conducting medium. This approach results in an alternative expression for the heat flux vector that is more compatible with the second law. Therefore, the RHCE brings the classical field theory of heat conduction into agreement with other branches of modern physics.     

He's homotopy perturbation method for two‐dimensional heat conduction equation: Comparison with finite element method

Heat conduction appears in almost all natural and industrial processes. In the current study, a two‐dimensional heat conduction equation with different complex Dirichlet boundary conditions has been studied. An analytical solution for the temperature distribution and gradient is derived using the homotopy perturbation method (HPM). Unlike most of previous studies in the field of analytical solution with homotopy‐based methods which investigate the ODEs, we focus on the partial differential equation (PDE). Employing the Taylor series, the gained series has been converted to an exact expression describing the temperature distribution in the computational domain. Problems were also solved numerically employing the finite element method (FEM). Analytical and numerical results were compared with each other and excellent agreement was obtained. The present investigation shows the effectiveness of the HPM for the solution of PDEs and represents an exact solution for a practical problem. The mathematical procedure proves that the present mathematical method is much simpler than other analytical techniques due to using a combination of homotopy analysis and classic perturbation method. The current mathematical solution can be used in further analytical and numerical surveys as well as related natural and industrial applications even with complex boundary conditions as a simple accurate technique. © 2010 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20292     

Numerical solution for the linear transient heat conduction equation using an Explicit Green’s Approach

This paper presents a novel numerical solution algorithm for the linear transient heat conduction equation using the ‘Explicit Green’s Approach’ (ExGA). The method uses the Green’s matrix that represents the domain of the problem to be solved in terms of the physical properties and geometrical characteristic. The Green’s matrix is the problem discrete Green’s function determined numerically by the Finite Element Method (FEM). The ExGA allows explicit time marching with time step larger than the one required by FEM, without losing precision. The ExGA numerical results are quite accurate when compared to analytical solutions and to numerical solutions obtained by the FEM.     

Solution of the inverse heat conduction problem described by the Poisson equation for a cooled gas-turbine blade

The presented paper describes a method of solving the inverse problems of heat conduction, consisting in solving the Poisson equation for a simply connected region instead of the Laplace equation for a multiply connected one, like a gas-turbine blade provided with cooling channels. The considered method consists in determining unknown values of the source (heat sink) power in the cooling channels for a given external heat transfer situation to achieve as close as possible an isothermal outer surface. Afterwards the temperature and heat flux distributions at the cooling channel walls are determined. Since the unknown source power is sought, the problem is an inverse one. Taking into account the sought values the method is reckoned among the class of the fictitious source methods and presents an optimization scheme. Using an exemplary gas turbine blade cooling configuration, the results of the calculation obtained with this method have been compared to the results achieved with an inverse method using the boundary element method for a multiple connected region.The results obtained with both methods within the optimization scheme approximated each other. Nevertheless, the results for the inverse method shown in the present paper gave nearly no oscillations, which is important in case of the blades with other geometric features of the cooling channels.     

Well-posedness and regularity of weakly coupled wave-plate equation with boundary control and observation

We consider the open-loop system of a weakly coupled linear wave-plate equation with Dirichlet boundary control and collocated observation. It is shown that the system is well-posed in the sense of D. Salamon and regular in the sense of G. Weiss. With the multiplier method, the feedthrough operator is explicitly represented. This work was supported by the National Natural Science Foundation of China (No.60774014), Program for the Top Young Academic Leaders of Higher Learning Institutions of Shanxi and the National Research Foundation of South Africa.     

A self-adaptive LGSM to recover initial condition or heat source of one-dimensional heat conduction equation by using only minimal boundary thermal data

The present study is concerned with the recovery of an unknown initial condition for a one-dimensional heat conduction equation by using only the usual two boundary conditions of the direct problem for heat equation. The algorithm assumes a function for the unknown initial condition and derives an inverse problem for estimating a spatially-dependent heat source F(x) in T_t(x, t) = T_xx(x, t) + F(x). A self-adaptive Lie-group shooting method, namely a Lie-group adaptive method (LGAM), is developed to find F(x), and then by integrations or by solving a linear system we can extract the information for unknown initial condition. The new method possesses twofold advantages in that no a priori information of unknown functions is required and no extra data are needed. The accuracy and efficiency of present method are confirmed by comparing the estimated results with some exact solutions.     

Local effects of longitudinal heat conduction in plate heat exchangers

In a plate heat exchanger, heat transfer from the hot to the cold fluid is a multi-dimensional conjugate problem, in which longitudinal heat conduction (LHC) along the dividing walls often plays some role and can not be neglected. Large-scale, or end-to-end, LHC is always detrimental to the exchanger’s effectiveness. On the contrary, if significant non-uniformities exist in the distribution of either convective heat transfer coefficient, small-scale, or local, LHC may actually enhance the exchanger’s performance by improving the thermal coupling between high heat transfer spots located on the opposite sides of the dividing wall.