The fourth-dimention concept in the finite element analysis of transient heat transfer problems

This study presents the finite element analysis for the transient heat transfer problems by introducing the Fourth-dimension concept. Time is treated as an additional dimension in the solution domain thereby Increasing the number of dimensions by one. For instance, a three-dimensional transient problem can be considered as a four-dimensional problem in the x, y, z, t domain and a two-dimensional transient problem can be considered as a three-dimensional steady-state problem in the x, y, t domain, respectively. The variational principle of the finite element method and the techniques existing for steady-state problems can be directly utilized. Numerical calculations were performed for heat conduction problems, laminar-turbulent convective heat transfer problems, and radiative heat transfer problems.     

Simulation of transient heat conduction using one‐dimensional mapped infinite elements

Many engineering problems exist in physical domains that can be said to be infinitely large. A common problem in the simulation of these unbounded domains is that a balance must be met between a practically sized mesh and the accuracy of the solution. In transient applications, developing an appropriate mesh size becomes increasingly difficult as time marches forward. The concept of the infinite element was introduced and implemented for elliptic and for parabolic problems using exponential decay functions. This paper presents a different methodology for modeling transient heat conduction using a simplified mesh consisting of only two‐node, one‐dimensional infinite elements for diffusion into an unbounded domain and is shown to be applicable for multi‐dimensional problems. A brief review of infinite elements applied to static and transient problems is presented. A transient infinite element is presented in which the element length is time‐dependent such that it provides the optimal solution at each time step. The element is validated against the exact solution for constant surface heat flux into an infinite half‐space and then applied to the problem of heat loss in thermal reservoirs. The methodology presented accurately models these phenomena and presents an alternative methodology for modeling heat loss in thermal reservoirs. Copyright © 2010 John Wiley & Sons, Ltd.     

A meshless analysis of three-dimensional transient heat conduction problems

In this paper, we consider a numerical modeling of a three-dimensional transient heat conduction problem. The modeling is carried out using a meshless reproducing kernel particle (RKPM) method. In the mathematical formulation, a variational method is employed to derive the discrete equations. The essential boundary conditions of the formulated problems are enforced by the penalty method. Compared with numerical methods based on meshes, the RKPM needs only scattered nodes, rather than having to mesh the domain of the problem. An error analysis of the RKPM for three-dimensional transient heat conduction problem is also presented in this paper. In order to demonstrate the applicability of the proposed solution procedures, numerical experiments are carried out for a few selected three-dimensional transient heat conduction problems.     

Hybrid finite element analysis of transient thermoelastic fracture problems subjected to general heat transfer conditions

This paper presents the singular characteristics of heat flux in the vicinity of the crack-tip for two dimensional transient thermoelastic fracture problems subjected to general heat transfer conditions at crack surfaces. Based on a restricted variational principle, a rigorous hybrid finite element procedure is then developed to perfectly describe the singularities of heat flux and thermal stress induced at the crack-tip. For verification purposes, the examples of transient thermoelastic problems with insulated crack surfaces are first analyzed. Excellent agreements between the computed results and referenced solutions can be drawn. To evaluate the influence of heat convection and radiation on the computation of temperature distributions and thermal stress intensity factors, several numerical examples are also presented.     

Homogenization of transient heat transfer problems for some composite materials

The article presented is devoted to the homogenization of transient heat transfer problems in some composite materials. The mathematical model used in the FEM computation is based on the effective modules method introduced for periodic composites. The effective heat conductivity is calculated in the closed form; effective heat capacity and mass density for the composite are obtained by simple spatial averaging. Such a homogenization scheme makes it possible to significantly simplify the numerical analysis of transient heat phenomena in various types of composites. Computational experiments performed using symbolic mathematics show the variability of effective heat conductivity for 2D and 3D composites as a function of the reinforcement volume ratio, of composite components conductivity coefficients as well as of the probabilistic moments of material properties versus volume ratio. Finally, using the Finite Element Method program, the comparison of transient heat transfer problem for the real and homogenized composites models is carried out.     

The dimension splitting and improved complex variable element‐free Galerkin method for 3‐dimensional transient heat conduction problems

In this paper, by combining the dimension splitting method and the improved complex variable element‐free Galerkin method, the dimension splitting and improved complex variable element‐free Galerkin (DS‐ICVEFG) method is presented for 3‐dimensional (3D) transient heat conduction problems. Using the dimension splitting method, a 3D transient heat conduction problem is translated into a series of 2‐dimensional ones, which can be solved with the improved complex variable element‐free Galerkin (ICVEFG) method. In the ICVEFG method for each 2‐dimensional problem, the improved complex variable moving least‐square approximation is used to obtain the shape functions, and the penalty method is used to apply the essential boundary conditions. Finite difference method is used in the 1‐dimensional direction, and the Galerkin weak form of 3D transient heat conduction problem is used to obtain the final discretized equations. Then, the DS‐ICVEFG method for 3D transient heat conduction problems is presented. Four numerical examples are given to show that the new method has higher computational precision and efficiency.     

On an approximate method of solution of nonlinear heat-conduction problems

An approximate method is presented for the solution of problems of transient heat conduction in solids with thermal conductivity and specific heat linearly dependent on temperature.     

埋地热含蜡原油管道的非稳态传热问题

我国生产的原油大多为含蜡原油，加热输送是含蜡原油的主要输送方式。埋地热含蜡原油管道的运行中涉及若干复杂的非稳态传热问题。从管内原油传热（内部传热）和管道与外部环境的传热（外部传热）两方面，分析和总结了含蜡原油管道非稳态传热问题的研究现状，介绍了管道在土壤中传热的影响因素，比较了其各种解析和数值解法，阐述了管道停输状态下管内含蜡原油相变传热的规律及研究方法，讨论了非稳定流动状态下管内油流的水力—热力耦合问题及求解方法，指出了埋地热含蜡原油管道的非稳态传热方面需进一步研究的问题。     

Analytical Solutions to Models of Local Nonequilibrium Heat Transfer

High Temperature - A series of boundary-value problems of local nonequilibrium heat transfer is considered in terms of the theory of transient heat conduction for hyperbolic-type equations (wave...     

The effects of dynamical treatment of thermal stresses on a circular region due to continuous point heat source

The investigation referring to the effects of inertia term on the plane transient thermal stress problems are treated for a solid circular cylinder under a continuous point heat source.     

Variationally consistent computational homogenization of transient heat flow

A framework for variationally consistent homogenization, combined with a generalized macro‐homogeneity condition, is exploited for the analysis of non‐linear transient heat conduction. Within this framework the classical approach of (model‐based) first‐order homogenization for stationary problems is extended to transient problems. Homogenization is then carried out in the spatial domain on representative volume elements (RVE), which are (in practice) introduced in quadrature points in standard fashion. Along with the classical averages, a higher order conservation quantity is obtained. An iterative FE²‐algorithm is devised for the case of non‐linear diffusion and storage coefficients, and it is applied to transient heat conduction in a strongly heterogeneous particle composite. Parametric studies are carried out, in particular with respect to the influence of the ‘internal length’ associated with the second‐order conservation quantity. Copyright © 2009 John Wiley & Sons, Ltd.     

A parallel domain decomposition boundary element method approach for the solution of large-scale transient heat conduction problems

This paper develops a parallel domain decomposition Laplace transform BEM algorithm for the solution of large-scale transient heat conduction problems. In order to tackle large problems the original domain is decomposed into a number of sub-domains, and a Laplace transform method is utilized to avoid time marching. A procedure is described which provides a good initial guess for the domain interface temperatures, and an iteration procedure is carried out to satisfy continuity of temperature and heat flux at the domain interfaces. The decomposition procedure significantly reduces the size of any single problem for the BEM, which significantly reduces the overall storage and computational burden and renders the application of the BEM to large transient conduction problems on modest computational platforms. The procedure is readily implemented in parallel and applicable to 3D problems. Moreover, as the approach described herein readily allows adaptation and integration of traditional BEM codes, it is expected that the domain decomposition approach coupled to parallel implementation should prove very competitive to alternatives proposed in the literature such as fast multipole acceleration methods that require a complete re-write of traditional BEM codes.     

A least-squares front-tracking finite element method analysis of phase change with natural convection

This paper focuses on the numerical modelling of phase-change processes with natural convection. In particular, two-dimensional solidification and melting problems are studied for pure metals using an energy preserving deforming finite element model. The transient Navier–Stokes equations for incompressible fluid flow are solved simultaneously with the transient heat flow equations and the Stefan condition. A least-squares variational finite element method formulation is implemented for both the heat flow and fluid flow equations. The Boussinesq approximation is used to generate the bulk fluid motion in the melt. The mesh motion and mesh generation schemes are performed dynamically using a transfinite mapping. The consistent penalty method is used for modelling incompressibility. The effect of natural convection on the solid/liquid interface motion, the solidification rate and the temperature gradients is found to be important. The proposed method does not possess some of the false diffusion problems associated with the standard Galerkin formulations and it is shown to produce accurate numerical solutions for convection dominated phase-change problems.     

Time‐step constraints in transient coupled finite element analysis

In transient finite element analysis, reducing the time‐step size improves the accuracy of the solution. However, a lower bound to the time‐step size exists, below which the solution may exhibit spatial oscillations at the initial stages of the analysis. This numerical ‘shock’ problem may lead to accumulated errors in coupled analyses. To satisfy the non‐oscillatory criterion, a novel analytical approach is presented in this paper to obtain the time‐step constraints using the θ‐method for the transient coupled analysis, including both heat conduction–convection and coupled consolidation analyses. The expressions of the minimum time‐step size for heat conduction–convection problems with both linear and quadratic elements reduce to those applicable to heat conduction problems if the effect of heat convection is not taken into account. For coupled consolidation analysis, time‐step constraints are obtained for three different types of elements, and the one for composite elements matches that in the literature. Finally, recommendations on how to handle the numerical ‘shock’ issues are suggested. Copyright © 2015 John Wiley & Sons, Ltd.     

Thermoelastic analysis of a partially insulated crack in a strip under thermal impact loading using the hyperbolic heat conduction theory

In this paper, the transient temperature and thermal stresses around a partially insulated crack in a thermoelastic strip under a temperature impact are obtained using the hyperbolic heat conduction theory. Fourier and Laplace transforms are applied and the thermal and mechanical problems are reduced to solving singular integral equations. Numerical results show that the hyperbolic heat conduction parameters, the thermal conductivity of crack faces, and the geometric size of the strip have significant influence on the dynamic temperature and stress field. The results based on hyperbolic heat conduction show much higher temperature and much more dynamic thermal stress concentrations in the very early stage of impact loading comparing to the Fourier heat conduction model. It is suggested that to design materials and structures against fracture under transient thermal loading, the hyperbolic model is more appropriate than the Fourier heat conduction model.     

Transient free convection about a vertical flat plate embedded in a porous medium

The problems of transient free convection in a porous medium adjacent to a vertical semi-infinite flat plate with a step increase in wall temperature and surface heat flux are considered in this paper. By assuming a temperature profile in each case, the governing equation for the boundary layer thickness is obtained by an integral method. These governing equations are first-order partial differential equations of the hyperbolic type that can be solved exactly by the method of characteristics and approximately by the method of integral relations. The results based on the method of characteristics clearly indicate that during the initial stage when the leading edge effect is not being felt, heat is transferred as if by transient 1-dimensional heat conduction. At a later time, depending on the vertical location, the heat transfer characteristics change from transient 1-dimensional heat conduction to steady 2-dimensional convection. The thickness of the boundary layer is shown to be increasing with time until it reaches steady state where its value remains constant thereafter. The growth rate of the boundary layer thickness exhibits a discontinuity at the end of the transient period and the beginning of the steady state period. On the other hand, the results based on the method of integral relations show that the boundary layer thickness grows continuously with time and approaches the steady state value asymptotically; the growth rate of the boundary layer thickness decreases from a finite value to zero continuously as the steady state is approached. Except between the end of the transient period and the beginning of the steady state period, the results based on the method of integral relations are in good agreement with those based on the method of characteristics.     

Transient heat conduction analysis of solids with small open-ended tubular cavities by boundary face method

This paper applies the boundary face method (BFM) to solve transient heat conduction problems for the first time. Rather than using a transformation scheme, a direct solution of the boundary integral equation (BIE) with time domain fundamental solution is performed in this application. To avoid the domain integrals, the boundary integral equation is solved by the time stepping convolution method. For problems on structures that contain a large number of open-ended tubular shaped cavities in small diameters, a curvilinear tube element is employed to approximate the variables on the cavity surface. Furthermore, to perform integration and boundary variable approximation on the end faces that are intersected by the tubular cavity, a triangular element with negative part is adopted. With the two types of specified elements, the BFM is implemented to solve transient heat conduction problems on structures with open-ended tubular shaped cavities of small size which are usually inconvenience in finite element implementations. Three numerical examples on different structures are presented to illustrate the validity and efficiency of the method.     

Numerical analysis of two-dimensional lagging thermal behavior under short-pulse-laser heating on surface

In this work, the dual-phase-lag (DPL) model of heat conduction is introduced in treating the transient heat conduction problems in finite rigid mediums under short-pulse-laser heating. Two-dimensional numerical solutions in a rectangular and an axially symmetric system are given by finite difference method. Calculations are performed to exhibit various two-dimensional lagging thermal behavior of conduction heat transfer, such as wavy, wavelike, and diffusive behavior.     

Acoustic resonance in plate heat exchangers: Résonance acoustique des échangeurs de chaleur à plaques

A ‘whistling' sound, which develops under transient conditions in some automobile air conditioning systems equipped with plate type evaporators, was identified as acoustic resonance. The ‘whistle' was reproduced in the laboratory under steady-state conditions. Testing of these heat exchangers was done first with R134a and then with nitrogen. Nitrogen testing proved to be much faster and easier and provided results comparable to the results obtained using R134a. The evidence presented in this work suggests that the acoustic resonance in this type of heat exchanger is similar in nature to the acoustic resonance problems reported for tube array in duct type heat exchangers. This is to the authors' knowledge the first time that acoustic resonance problems have been reported in the literature for plate heat exchangers.