On the constitutive modeling of coupled thermomechanical phase-change problems

This paper deals with a thermodynamically consistent numerical formulation for coupled thermoplastic problems including phase-change phenomena and frictional contact. The final goal is to get an accurate, efficient and robust numerical model, able for the numerical simulation of industrial solidification processes. Some of the current issues addressed in the paper are the following. A fractional step method arising from an operator split of the governing differential equations has been used to solve the nonlinear coupled system of equations, leading to a staggered product formula solution algorithm. Nonlinear stability issues are discussed and isentropic and isothermal operator splits are formulated. Within the isentropic split, a strong operator split design constraint is introduced, by requiring that the elastic and plastic entropy, as well as the phase-change induced elastic entropy due to the latent heat, remain fixed in the mechanical problem. The formulation of the model has been consistently derived within a thermodynamic framework. All the material properties have been considered to be temperature dependent. The constitutive behavior has been defined by a thermoviscous/elastoplastic free energy function, including a thermal multiphase change contribution. Plastic response has been modeled by a J2 temperature dependent model, including plastic hardening and thermal softening. The constitutive model proposed accounts for a continuous transition between the initial liquid state, the intermediate mushy state and the final solid state taking place in a solidification process. In particular, a pure viscous deviatoric model has been used at the initial fluid-like state. A thermomecanical contact model, including a frictional hardening and temperature dependent coupled potential, is derived within a fully consistent thermodinamical theory. The numerical model has been implemented into the computational finite element code COMET developed by the authors. Numerical simulations of solidification processes show the good performance of the computational model developed.     

On the formulation of coupled thermoplastic problems with phase-change

This paper deals with a numerical formulation for coupled thermoplastic problems including phase-change phenomena. The final goal is to get an accurate, efficient and robust numerical model, allowing the numerical simulation of solidification processes in the metal casting industry. Some of the current issues addressed in the paper are the following. A fractional step method arising from an operator split of the governing differential equations has been used to solve the nonlinear coupled system of equations, leading to a staggered product formula solution algorithm. Nonlinear stability issues are discussed and isentropic and isothermal operator splits are formulated. Within the isentropic split, a strong operator split design constraint is introduced, by requiring that the elastic and plastic entropy, as well as the phase-change induced elastic entropy due to the latent heat, remain fixed in the mechanical problem. The formulation of the model has been consistently derived within a thermodynamic framework. The constitutive behavior has been defined by a thermoelastoplastic free energy function, including a thermal multiphase change contribution. Plastic response has been modeled by a J2 temperature dependent model, including plastic hardening and thermal softening. A brief summary of the thermomechanical frictional contact model is included. The numerical model has been implemented into the computational Finite Element code COMET developed by the authors. A numerical assessment of the isentropic and isothermal operator splits, regarding the nonlinear stability behavior, has been performed for weakly and strongly coupled thermomechanical problems. Numerical simulations of solidification processes show the performance of the computational model developed.     

A thermomechanical model for the analysis of light alloy solidification in a composite mould

A coupled thermomechanical model to simulate light alloy solidification problems in permanent composite moulds is presented. This model is based on a general isotropic thermoelasto-plasticity theory and considers the different thermomechanical behaviours of each component of the mould as well as those of the solidifying material during its evolution from liquid to solid. To this end, plastic evolution equations, a phase-change variable and a specific free energy function are proposed in order to derive temperature-dependent material constitutive laws.The corresponding finite element formulation and the staggered scheme used to solve the coupled non-linear system of equations are also presented. Finally, the temperature and displacement predictions of the model are validated with laboratory measurements obtained during an experimental trial.     

An implicit mixed enthalpy–temperature method for phase-change problems

A finite element procedure for phase-change problems is presented. Enthalpy and temperature are interpolated separately and subsequently linked via the appropriate relation in the nodes of the mesh during the solution phase. A novel technique is here used where, depending on the characteristics of the problem, either temperature or enthalpy may be considered as primary variable. The resulting algorithm is both efficient and robust and is further easy to implement and generalize to arbitrary finite elements. The capabilities of the method are illustrated by the solution both isothermal and non-isothermal phase-change problems.     

相变温控导热增强的多孔金属梯度优化设计

高孔隙率金属多孔材料比表面积大、导热性能好且掺混能力强,是理想的相变换热导热增强体材料。增材制造能够精准制备几何高度复杂的微结构,为多孔金属任意梯度设计提供可能。为实现更高的相变换热性能,建立了多孔金属相变温控导热增强的梯度优化设计模型。该优化模型以孔隙率分布为设计变量,以多孔金属用量为约束,以关键位置的温度最低为设计目标,基于考虑相变过程的多孔介质两方程模型为分析方法,通过遗传算法对优化模型进行求解。通过与实验结果的对比,验证了分析方法的有效性。两个具体算例证实了梯度设计能够大幅度提高多孔金属介质导热增强的相变温控性能。     

Hybrid numerical scheme for nonlinear two-dimensional phase-change problems with the irregular geometry

A hybrid numerical scheme combining the Laplace transform and control-volume methods is presented to solve nonlinear two-dimensional phase-change problems with the irregular geometry. The Laplace transform method is applied to deal with the time domain, and then the control-volume method is used to discretize the transformed system in the space domain. Nonlinear terms induced by the temperature-dependent thermal properties are linearized by using the Taylor series approximation. Control-volume meshes in the solid and liquid regions during simulations are generated by using the discrete transfinite mapping method. The location of the phase-change interface and the isothermal distributions are determined. Comparison of these results with previous results shows that the present numerical scheme has good accuracy for two-dimensional phase-change problems. Received on 17 October 1996     

相变传热问题的灵敏度分析与优化设计方法

研究了相变传热问题的优化设计及其灵敏度分析方法. 在有限元-时间差分和等效热容 法求解相变温度场的基础上，提出了相变温度场对设计变量一阶灵敏度的计算方法，给出直 接法和伴随法两种计算格式并分析了它们的特点，建立了相变温度场优化的模型和算法，在有限元分析与优化设计软件JIFEX中实现了该方法. 数值算例表明了灵敏度计算的精度和优 化方法的有效性.     

Computation of Liquid Hydrazine Depressurisation with a Mach-Uniform Staggered Scheme

A numerical method (pressure-correction method using a staggered grid) is coupled to a thermodynamic model for compressible liquid hydrazine. The method is applied to the venting of liquid hydrazine into space, during which the fluid undergoes a large pressure drop. Below the saturation pressure vaporisation occurs. This takes place near the outlet and induces variations of temperature, which may cause solidification and pipe clogging. In order to assess the risk of phase changes, numerical simulations of the venting line have been performed using a quasi one-dimensional approach. The numerical method can handle compressible flows of fluids with nonconvex equation of state at the low Mach numbers that occur during hydrazine venting. A numerical study of the liquid behaviour during strong depressurisation is performed. The method is validated using experimental data, and allows prediction of pressure evolution and vaporisation location along the pipe. This revised version was published online in July 2006 with corrections to the Cover Date.     

Numerical investigation of transient natural convection heat transfer of freezing water in a square cavity

In this study, a transient heat transfer process of freezing water inside a two-dimensional square cavity has been investigated numerically. Water was used as a phase-change medium, and the numerical model has been created with control volume approach by using C++ programming language. To be able to accelerate the numerical calculations, CUT (Consistent-Update-Technique) algorithm has been implemented in the numerical code. Span-wise variations of the vertical component of the velocity have been represented in comparison with the experimental measurements from the literature at various vertical positions to examine the accuracy of the numerical scheme. The influence of natural convection has been considered by comparing the conduction and convection dominated solidification under same boundary conditions. Comparative results have been obtained regarding time-wise variations of the cold wall temperature and the dimensionless effectiveness. Moreover, the streamlines and isotherms have been represented to understand the differences between the conduction and convection driven phase change processes.Results indicate that natural convection becomes remarkable and has different forms at the initial periods of the phase change process. Increasing the effect of natural convection in the cavity increases the cooling rate of water. Near the density inversion temperature of water (4°C), temperature variations fluctuate and counter currents observed in the domain.     

Numerical Analysis of Thermally Coupled Flow Problems with Interfaces and Phase-change Effects

In this work a fixed mesh finite element approach is presented to solve thermally coupled flow problems including moving interfaces between immiscible fluids and phase-change effects. The weak form of the full incompressible Navier-Stokes equations is obtained using a generalized streamline operator (GSO) technique that enables the use of equal order interpolation of the primitive variables of the problem: velocity, pressure and temperature. The interfaces are defined with a mesh of marker points whose motion is obtained applying a Lagrangian scheme. Moreover, a temperature-based formulation is considered to describe the phase-change phenomena. The proposed methodology is used in the analysis of a filling of a step mould and a gravity-driven flow of an aluminium alloy in an obstructed vertical channel.     

An inverse finite element method with an application to extrusion with solidification

The flow and solidification of planar jets are analysed by means of an efficient inverse isotherm finite element method. The method is based on a tessellation that is constructed by isotherms as characteristic co-ordinate lines transverse to the flow direction. Thus opposite sides of finite elements lie on isotherms. The method allows the simultaneous determination of the location of the isotherms with the primary unknowns, namely, the velocity, the pressure, the temperature and the location of the free surface. Thus the determination of the location of the solidification front (which is known to pose significant computational difficulties) is automatic. This facilitates the control of the location of the solidification front by controlling macroscopic variables such as the flow rate, the cooling rate and the capillary design. The location of the solidification may then be suitably chosen to influence the frozen-in orientation and structure in extrusion of high-performance materials such as composites and polymers, in continuous casting of metals and in growth of crystals.     

Numerical solution of solidification in a square prism using an algebraic grid generation technique

The solidification of an infinitely long square prism was analyzed numerically. A front fixing technique along with an algebraic grid generation scheme was used, where the finite difference form of the energy equation is solved for the temperature distribution in the solid phase and the solid–liquid interface energy balance is integrated for the new position of the moving solidification front. Results are given for the moving solidification boundary with a circular phase change interface. An algebraic grid generation scheme was developed for two-dimensional domains, which generates grid points separated by equal distances in the physical domain. The current scheme also allows the implementation of a finer grid structure at desired locations in the domain. The method is based on fitting a constant arc length mesh in the two computational directions in the physical domain. The resulting simultaneous, nonlinear algebraic equations for the grid locations are solved using the Newton-Raphson method for a system of equations. The approach is used in a two-dimensional solidification problem, in which the liquid phase is initially at the melting temperature, solved by using a front-fixing approach. The difference of the current study lies in the fact that front fixing is applied to problems, where the solid–liquid interface is curved such that the position of the interface, when expressed in terms of one of the coordinates is a double valued function. This requires a coordinate transformation in both coordinate directions to transform the complex physical solidification domain to a Cartesian, square computational domain. Due to the motion of the solid–liquid interface in time, the computational grid structure is regenerated at every time step.     

冻土活动层相变温度场Chebyshev拟谱分析

在冻土相变温度热传导机理基础上,应用冻土计算中水热输运过程成熟的通用物理模型,提出冻土活动层温度的一种全新的数值分析方法-谱方法.应用Chebyshev多项式作为基函数将温度解展开,在研究域(或单元)内采用伪谱Chebyshev逼近的谱方法.为了与谱方法的高精度相配合及提高时程积分解的稳定性,本文应用四阶Runger-Kutta法进行时程积分,变物性温度泛函-热导系数、热容量等考虑效应滞后的处理方法.本文提出冻土非线性问题数值计算拟谱分析的理论构架,其计算方法在冻土工程应用中具有一定的理论导向作用和较大的实用价值.     

Solidification problem for the wavy mold

The solidification process on the wavy mold is studied. It is assumed that the mold temperature is kept at a constant level below the melting point of material. The boundaries of the solidified layer are folded. The approximate technique, known as the heat-balance integral method, is applied to determine the location of the solidification front and its perturbations. The solution is obtained with neglecting the square of small quantities. The effect of the folded boundary on the solution is shown in diagrams. Received on 3 January 1998     

Experiments and analysis on the melting of a semitransparent material by radiation

The purpose of this study was to obtain an improved understanding of the physical processes which occur during the melting of a semitransparent material for the case where the melting is effected by irradiation of the material. An analytical model was developed for predicting radiative and thermal conditions, as well as the liquid/solid interface displacement, during the radiation induced phase-change process. Parametric calculations were performed to determine the extent of the influence of some fourteen identifiable dimensionless parameters on the temperature distribution and liquid/solid interface motion. Experimental simulations were conducted in the laboratory using a high intensity lamp to melt a phase-change material within a carefully controlled environment. Provisions were made for measurement of the temperature and the liquid-solid interface displacement. Comparison between the model predictions and the experimental data is shown to be in very good agreement.     

An adaptive finite element procedure for solidification problems

An adaptive mesh regeneration procedure for transient solidification problems has been presented in this paper. The method is based on the local minimum and maximum values of second derivatives of velocity and temperature fields. A numerical example of alloy solidification in a square enclosure is presented for demonstration. Extension to more complicated and odd-shaped problems is straight forward. Received on 8 March 1999     

An Enthalpy Control Volume Method for Transient Mass and Heat Transport with Solidification

A single domain enthalpy control volume method is developed for solving the coupled fluid flow and heat transfer with solidification problem arising from the continuous casting process. The governing equations consist of the continuity equation, the Navier–Stokes equations and the convection–diffusion equation. The formulation of the method is cast into the framework of the Petrov–Galerkin finite element method with a step test function across the control volume and locally constant approximation to the fluxes of heat and fluid. The use of the step test function and the constant flux approximation leads to the derivation of the exponential interpolating functions for the velocity and temperature fields within each control volume. The exponential fitting makes it possible to capture the sharp boundary layers around the solidification front. The method is then applied to investigate the effect of various casting parameters on the solidification profile and flow pattern of fluids in the casting process.     

Petrov-Galerkin methods for natural convection in directional solidification of binary alloys

A Petrov-Galerkin finite element method is presented for calculation of the steady, axisymmetric thermosolutal convection and interface morphology in a model for vertical Bridgman crystal growth of nondilute binary alloys. The Petrov-Galerkin method is based on the formulation for biquadratic elements developed by Heinrich and Zienkiewicz and is introduced into the calculation of the velocity, temperature and concentration fields. The algebraic system is solved simultaneously for the field variables and interface shape by Newton's method. The results of the Petrov-Galerkin method are compared critically with those of Galerkin's method using the same finite element grids. Significant improvements in accuracy are found with the Petrov-Galerkin method only when the mesh is refined and when the formulation of the residual equations is modified to account for the mixed boundary conditions that arise at the solidification interface. Calculations for alloys with stable and unstable solute gradients show the occurrence of classical flow transitions and morphological instabilities in the solidification system.     

Green’s function solution for transient heat conduction in annular fin during solidification of phase change material

The Green’s function method is applied for the transient temperature of an annular fin when a phase change material (PCM) solidifies on it. The solidification of the PCMs takes place in a cylindrical shell storage. The thickness of the solid PCM on the fin varies with time and is obtained by the Megerlin method. The models are found with the Bessel equation to form an analytical solution. Three different kinds of boundary conditions are investigated. The comparison between analytical and numerical solutions is given. The results demonstrate that the significant accuracy is obtained for the temperature distribution for the fin in all cases.     

Visualization of temperature fields and double-diffusive convection using liquid crystals in an aqueous solution crystallizing along a vertical wall

This article presents a simple technique for temperature visualization using liquid crystals in an aqueous solution during the process of cooling and solidification. This method provides a clear picture of the role of double-diffusive convection in producing vertical compositional and density stratification in an initially homogeneous liquid during solidification.