基于有限元的一类stefan问题数值模型研究

该文给出基于有限元方法的一类一维stefan问题的数值求解过程及算法.模型的建立基于已知的相变界面和固定边界处测得的温度和热流.模型的精度通过与Neumann获得的解析解的比较而得到验证.文中所讨论的模型可以用于反Stefan问题中自由边界的实时跟踪或者控制.最后,比较了已有的有限元模型,给出了仿真结果.     

A mathematical model for nanoparticle melting with size-dependent latent heat and melt temperature

In this paper, we study the melting of a spherical nanoparticle. The model differs from previous ones in that a number of features have been incorporated to match experimental observations. These include the size dependence of the latent heat and a cooling condition at the boundary (as opposed to the fixed temperature condition used in previous studies). Melt temperature variation and density change are also included. The density variation drives the flow of the outer fluid layer. The latent heat variation is modelled by a new relation, which matches experimental data better than previous models. A novel form of Stefan condition is used to determine the position of the melt front. This condition takes into account the latent heat variation, the energy required to create new surface and the kinetic energy of the displaced fluid layer. Results show that melting times can be significantly faster than predicted by previous theoretical models; for smaller particles, this can be around a factor 3. This is primarily due to the latent heat variation. The previously used fixed temperature boundary condition had two opposing effects on melt times: the implied infinite heat transfer led to faster melting but also artificially magnified the effect of kinetic energy, which slowed down the process. We conclude that any future models of nanoparticle melting must be based on the new Stefan condition and account for latent heat variation.     

A finite difference method for a Stefan problem

Many physical problems involve initial boundary value problems for parabolic differential equations in which part of the boundary is not given a priori but is found as part of the solution. These problems have been considered under the name of «Stefan problems». Stefan problems occur in such physical processes as the melting of solids and the crystallizing of liquids. Existence and uniqueness for various one dimensional Stefan problems have been shown by several authors ([1], [4], [5], [8], [9]). Some multidimensional Stefan problems were considered by ([6], [11]). The purpose of this paper is to approximate the solution of a Stefan problem using an implicit finite difference analog for the heat equation and an explicit finite difference analog for the differential equation describing the free boundary. Also, we shall show that the finite difference solution we obtain converges uniformly to the actual solution. Numerous numerical schemes for various Stefan problems have been successfully employed by several authors ([2], [3], [10], [12], [14], [16], [17]). In Chapter I we formulate a continuous time Stefan problem, and in Chapter II we describe a finite difference scheme for approximating results. Chapter IV contains the main result which shows that the finite difference solution converges to the actual solution. Finally, in Chapter V we give a numerical example.     

Parameter identification of a multi-stage,multi-load-demand experimental refrigeration plant

Parameter estimation of a multi-stage, multi-load-demand refrigeration plant is addressed. It is shown that dominating system dynamics are those of heat exchangers, and their heat-transfer-related parameters affect system statics but barely influence system dynamics. A novel identification procedure focused on the heat exchangers is presented. It is based on non-measurable refrigerant phase-change zones, considering an overall heat transfer coefficient at each zone. Consistent values of all parameters are obtained considering only steady-state experimental data and some orders of magnitude found in the literature. The identified parameters are validated considering the two-stage, two-load-demand plant configuration.     

Boundary Geometric Control of a Nonlinear Diffusion System with Time‐Dependent Spatial Domain

A Stefan problem represents a distributed parameter system with a time‐dependent spatial domain. This paper addresses the boundary control of the position of the moving liquid–solid interface in the case of nonlinear Stefan problem with Neumann actuation. The main idea consists in deriving an equivalent linear model by means of Cole‐Hopf tangent transformation, i.e. under a certain physical assumption, the original nonlinear Stefan problem is converted to a linear one. Then, the geometric control law is deduced directly from that developed, by the authors of the present paper, for the linear Stefan problem. Based on the fact that the Cole‐Hopf transformation is bijective, it is shown that the developed control law yields a stable closed‐loop system. The performance of the controller is evaluated through numerical simulation in the case of stainless steel melting characterized by a temperature‐dependent thermal conductivity, which is nonlinear. The objective is to control the position of the liquid–solid interface by manipulating a heat flux at the boundary.     

太阳能热发电蓄热系统的混杂建模与仿真

蓄热系统是太阳能热发电站中保证电站可靠性和经济性的重要组成部分,其工作状况直接影响到太阳能热发电站的正常运行。本文针对单罐式蓄热系统,分析了系统在蓄热和放热模式下的动态数学模型,在此基础上,提出了蓄热系统的混杂动态系统模型。该模型将蓄热和放热模式统一在一个框架下,能够对蓄热系统的各种操作模式及其循环交替过程进行动态模拟。混杂模型采用MATLAB软件进行仿真,仿真结果:表明了模型的有效性。     

Effect of heat generation on mixed convection in porous cavity with sinusoidal heated moving lid and uniformly heated or cooled bottom walls

Effect of heat generation and absorption on mixed convection flows in a sinusoidal heated lid-driven square cavity filled with a porous medium is investigated numerically. Both the vertical walls of the enclosure are insulated while the bottom wall is uniformly heated or cooled. The top wall is moving at a constant speed and is heated sinusoidally. The governing equations and boundary conditions are non-dimensionalized and solved numerically by using finite volume method approach along with SIMPLE algorithm together with non-uniform grid system. The effect of Darcy and heat generation parameters are investigated in terms of the flow, heat transfer, and Nusselt number. The results for stream function and isotherm are plotted and it is found that there have significant influence with the presence of heat generation and porous medium.

     

Optimal setpoint control of complex heat exchanger systems

This paper presents an approach for calculating the best way of distributing the streams following through a certain class of complex heat exchanger systems in order to achieve maximum heat recovery within the system. A computer code has been developed by which the described method is demonstrated, off-line, for two real cases. This program can be readily integrated into an over-all, on-line computer control system for any complex process consisting of an exchanger system of this class. Using an accurate and detailed heat exchanger model, the exit temperatures of each exchanger are calculated by a simple mathematical procedure based on Gilmour's design method. This procedure has been included in a general model for the complete scheme of the system. The scheme is made up of a series of heat exchanger groups with parallel paths in each group. The optimal distribution of the streams within a group is found by the direct search method of Hooke and Jeeves, modified to include constraints; while the overall optimization of the system is achieved via dynamic programming.     

The heat equation as grid smoothing in a local optimization procedure

We consider a local optimization technique, where starting from a preliminary version of the grid under consideration, we try to improve the part of the grid that really needs this improvement. When this procedure is performed, the processed grids may result irregular, so a smoothing step must be taken into account. We propose a smoothing approach based on an iterative formula resembling the explicit difference schemes for the heat equation. This is a quite general approach, however to fix the ideas it is described in the context of quadrilateral grid generation and the variational approach is considered as the base method for the solution of planar grid generation. Some numerical experiments are presented to show the efficiency of the proposed method.     

Modeling laser heated thin film media for heat assisted magnetic recording

In the present study, we concern laser induced heat generation and its effect on lubricant depletion on the multilayer disk structure. A mathematical model is applied to explore the local laser heating from the point of view of the interaction between the disk media and the electromagnetic wave. The absorption rate of optical energy in each layer of disk structure is derived from the Maxwell equations, which is incorporated into the energy equation by a source term. We simulate the unsteady heat transfer and lubricant depletion processes on a multilayer disk structure. The effect of multi-cycle heating on the lubricant depletion is investigated. Comparison between the present results and those obtained by applying a heat flux boundary condition is also made. The present simulations provide useful information on the media structure design of hard disk drives in heat assisted magnetic recording.     

Isoparametric axisymmetric shell elements with temperature gradients for heat conduction

This paper presents a finite element formulation for axisymmetric shell heat conduction where temperature gradients through the shell thickness are retained as primary nodal variables. The element geometry is constructed using the coordinates of the nodes lying on the middle surface of the shell and the middle surface nodal point normals. The element temperature field is approximated in terms of element approximation functions, the nodal temperature, and the nodal temperature gradients. The weak formulation of the two-dimensional Fourier heat conduction equation in cylindrical coordinate system is constructed. The finite element properties of the shell element are then derived using the weak formulation and the element temperature field approximation. The formulation permits linear temperature gradients through the shell thickness. Distributed heat flux as well as convective boundaries are permitted on all four faces of the element. Furthermore, the element can also have internal heat generation as well as orthotropic material properties. The superiority of the formulation in terms of efficiency and accuracy is demonstrated. Numerical examples are presented and a comparison is made with the theoretical results.     

Numerical implementation of a discontinuous finite element algorithm for phase-change problems

This paper is devoted to the numerical solution of phase-change problems in two dimensions. The technique of finite elements is employed. The discretization is carried out using linear isoparametric elements and special attention is given to the accurate integration of functions presenting discontinuities at arbitrarily curved interfaces. This type of function arises in a natural way when dealing with phase-change problems because the enthalpy attains a discontinuity at the phase change temperature. To integrate the discontinuous functions in the phase-changing elements a second mapping is performed from the master element onto a new one for which the interface iis a straight line. The integrals are calculated using the Gaussian technique applied to each part of the divided element, which may be triangular or quadrilateral. The discontinuous integration technique improves the behaviour of the numerical method avoiding any possible loss of latent heat due to an inaccurate evaluation of the residual vector. Some important aspects of the solution of the nonlinear system of equations are discussed and several numerical examples are presented together with the details of the computational implementation of the algorithm.     

Determination of the time-dependent reaction coefficient and the heat flux in a nonlinear inverse heat conduction problem

ABSTRACT

Diffusion processes with reaction generated by a nonlinear source are commonly encountered in practical applications related to ignition, pyrolysis and polymerization. In such processes, determining the intensity of reaction in time is of crucial importance for control and monitoring purposes. Therefore, this paper is devoted to such an identification problem of determining the time-dependent coefficient of a nonlinear heat source together with the unknown heat flux at an inaccessible boundary of a one-dimensional slab from temperature measurements at two sensor locations in the context of nonlinear transient heat conduction. Local existence and uniqueness results for the inverse coefficient problem are proved when the first three derivatives of the nonlinear source term are Lipschitz continuous functions. Furthermore, the conjugate gradient method (CGM) for separately reconstructing the reaction coefficient and the heat flux is developed. The ill-posedness is overcome by using the discrepancy principle to stop the iteration procedure of CGM when the input data is contaminated with noise. Numerical results show that the inverse solutions are accurate and stable.     

Evaluation of heat transfer coefficients during upward and downward transient directional solidification of Al–Si alloys

Aluminum alloys with silicon as a major alloying element consist of a class of alloys which provides the most significant part of all shaped castings manufactured. This is mainly due to the outstanding effect of silicon in the improvement of casting characteristics, combined with other physical properties such as mechanical properties and corrosion resistance. In general, an optimum range of silicon content can be assigned to casting processes. For slow cooling rate processes (sand, plaster, investment), the range is 5 to 7 wt%; for permanent molds, 7 to 9%; and for die castings, 8 to 12%. Since most casting parts are produced considering there is no dominant heat flow direction during solidification, it seems to be adequate to examine both upward and downward growth directions to better understand foundry systems. The way the heat flows across the metal/mold interface strongly affects the evaluation of solidification and plays a remarkable role in the structural integrity of castings. Gravity or pressure die casting, continuous casting, and squeeze casting are some of the processes where product quality is more directly affected by the interfacial heat transfer conditions. Once information in this area is accurate, foundrymen can effectively optimize the design of their chilling systems to produce sound castings. The present work focuses on the determination and evaluation of transient heat transfer coefficients from the experimental cooling curves during solidification of Al 5, 7, and 9 wt% Si alloys. The method used is based on comparisons between experimental data and theoretical temperature profiles furnished by a numerical solidification model, which applies finite volume techniques. In other words, the resulting data were compared with a solution for the inverse heat conduction problem. The necessary solidification thermodynamic input data were obtained by coupling the software ThermoCalc Fortran interface with the solidification model. A comparison between upward and downward transient metal/mold heat transfer coefficients is conducted.     

On-line identification of a heat exchanger with a process computer—A case study

For on-line identification and parameter estimation of industrial processes with process computers an identification program package was developed. Three appropriate identification methods can be selected: recursive least squares, recursive instrumental variables and recursive correlation analysis with least squares. The program package also includes: signal generation, determination of model order and time delay, data filtering for the elimination of low frequent disturbances, model verification and plotting of intermediate and final results. Practical results and comparisons with the identification package are shown for an industrial size steam-heated heat exchanger.     

Three dimensional solid-shell transition finite elements for heat conduction

This paper presents a finite element formulation for a special class of finite elements referred to as ‘Solid-Shell Transition Finite Elements’ for three dimensional heat conduction. The solid-shell transition elements are necessary in applications requiring the use of both three dimensional solid elements and the curved shell elements. These elements permit transition from the solid portion of the structure to the shell portion of the structure. A novel feature of the formulation presented here is that nodel temperatures as well as nodal temperature gradients are retained as primary variables. The element geometry is defined in terms of coordinates of the nodes as well as the nodal point normals for the nodes lying on the middle surface of the element. The temperature field with the element is approximated in terms of element approximation functions, nodal temperatures and nodal temperature gradients. The properties of the transition element are then derived using the weak formulation (or the quadratic functional) of the Fourier heat conduction equation in the Cartesian coordinate system and the element temperature approximation. The formulation presented here permits linear temperature distribution in the element thickness direction.

Convective boundaries as well as distributed heat flux is permitted on all six faces of the elements. Furthermore, the element formulation also permits internal heat generation and orthotropic material behavior. Numerical examples are presented firstly to illustrate the accuracy of the formulation and secondly to demonstrate its usefulness in practical application. Numerical results are also compared with the theoretical solutions.     

热泵相变储热器动态特性分析及强化传热研究

热泵相变储热器传输过程具有动态性,提出基于四阶动态响应特性分析的热泵相变储热器动态特性分析及强化传热模型。构建热泵相变储热器的欠阻尼振荡输出特征分析模型,以系统功率突变为控制状态约束对象进行热泵相变储热器动态驱动响应模拟,采用二项单调衰减分量、二项欠阻尼振荡分量作为控制约束变量进行热泵相变储热器动态强化传输控制,建立热泵相变储热器的输出功率调制模型,结合热泵相变储热器动态状态量初始值及变化值进行强化传热过程控制,提高热泵相变储热器的输出性能。仿真结果表明,采用该模型进行热泵相变储热器的输出动态控制能力较好,强化传输的输出增益较高,提高了热泵相变储热器的输出响应能力。     

Using bulk micromachined structures to enhance pool boiling heat transfer

This paper presents the important results of enhancing the boiling heat transfer of the pressurized water reactors (PWRs) by using LIGA or LIGA-like techniques to add microstructures on the surface of heater elements. The heater elements were made of 10 mm × 80 mm silicon strips with different in-line square micro-pin-fin configurations of 200 μm fin width, 35 μm fin height, and different inter-fin spacing values of 200, 400, 600, 800, 1000 μm and infinity. The experiments were conducted in de-ionized water at the atmospheric pressure. The input power, heater temperature, steam generation rate and video images of boiling phenomena were continuously recorded. Their relationships was studied and used to evaluate the total boiling heat transfer performance. The optimized microstructures can then be mass-fabricated on PWR tubes by using LIGA or LIGA-like technology. The experimental results suggest that by adding micro-sized in-line pin-fin arrays on heater surface and modifying heater surface morphology, the boiling process can be greatly enhanced through the improvements of vapor nucleation and vapor evolution processes at heater surface, which yields a low wall superheat and achieves a higher boiling heat transfer efficiency. The video images showed that the bubble nucleation sites are located immediately on top of each micro-pin fins. At current experimental setup, the 200 μm-spacing heater has the highest steam generation efficiency.     

Applications of nonlinear Volterra equations of the first kind to the control problem for heat exchange dynamics

We consider a class of nonlinear integral Volterra equations of the first kind related to the automatic control problem for a nonlinear dynamical system (object) which is a black box with vector input and no output feedback. The characteristic features of the algorithms are illustrated on the example of mathematical modeling for nonlinear heat exchange processes.