Numerical study of melting inside concentric and eccentric horizontal annulus

This paper presents numerical investigations on melting of phase change material using N-eicosane inside a cylindrical container. Numerical simulations are performed for symmetric melting of phase change material between two cylinders in concentric and eccentric arrays using the FLUENT software which is sub-cooled initially to 1 °C. Inner cylindrical tube is considered hot wall while outer tube is insulated. Predicted result shows that melting rate is the same approximately for concentric and eccentric array before time of 15 min. After this time, melting rate decreases in concentric array. It is due to the pure conduction between hot tube and cold solid phase change material.     

A comparison of numerical models for one-dimensional Stefan problems

In this paper, we present a critical comparison of the suitability of several numerical methods, level set, moving grid and phase field model, to address two well-known Stefan problems in phase transformation studies: melting of a pure phase and diffusional solid-state phase transformations in a binary system. Similarity solutions are applied to verify the numerical results. The comparison shows that the type of phase transformation considered determines the convenience of the numerical techniques. Finally, it is shown both numerically and analytically that the solid-solid phase transformation is a limiting case of the solid–liquid transformation.     

A numerical method to compute solidification and melting processes

This paper deals with thermodynamically consistent numerical predictions of solidification and melting processes of pure materials using moving grids. Till date, enthalpy-porosity-based formulations of numerical codes have been generally the popular choice, although because of an artificial numerical smearing of the interface, it is virtually impossible to reproduce a sharp melting/solidification front that is supposed to exist for phase changes of pure substances. Numerical techniques based on moving grid methods have been relatively less used as they rely on complex and time-consuming adaptive grid generations. Using the moving grid approach, the authors present a method to solve solidification and melting problems. A simple linear interpolation is used to slide grid nodes along the interface to handle the otherwise obtained grid skewness near the interface. The numerical approach employed is validated with standard test cases available in the literature. The capability of capturing very complex flow field structures and the superiority of the present approach over enthalpy-porosity-based formulations is discussed. The authors also demonstrate the ability of the set-up computer code to solve complex thermofluid processes such as occur during crystal growth in Czochralski reactors.     

Phase-Field Modeling and Numerical Simulation for Ice Melting

In this paper, we propose a mathematical model and present numerical simulations for ice melting phenomena. The model is based on the phase-field modeling for the crystal growth. To model ice melting, we ignore anisotropy in the crystal growth model and introduce a new melting term. The numerical solution algorithm is a hybrid method which uses both the analytic and numerical solutions. We perform various computational experiments. The computational results confirm the accuracy and efficiency of the proposed method for ice melting.     

A Stefan problem in combustion

This paper deals with a mathematical model of a condensed two-phase combustion process which describes combustion of solid materials in which melting occurs. The paper shows the existence of a weak solution of the resulting differential equations system and, furthermore, shows that the phase change set (the set where the temperature is equal to the given constant melting temperature) is not a front but a whole mushy region. For this mushy region an estimate in measure is given.This work has been supported by the Deutsche Forschungsgemeinschaft.     

Surface wave propagation in a liquid-saturated porous solid layer lying over a heterogeneous elastic solid half-space

Dispersion equation is derived for the propagation of Rayleigh type surface waves in a liquid saturated porous solid layer lying over an inhomogeneous elastic solid half-space. Effect of heterogeneity on the phase velocity is studied by taking different numerical values of heterogeneity factor for particular models. Dispersion curves have been drawn showing the effect of heterogeneity on the phase velocity.     

Numerical study of melting in an enclosure with discrete protruding heat sources

In order to explore the capability of a solid–liquid phase change material (PCM) for cooling electronic or heat storage applications, melting of a PCM in a vertical rectangular enclosure was studied. Three protruding generating heat sources are attached on one of the vertical walls of the enclosure, and generating heat at a constant and uniform volumetric rate. The horizontal walls are adiabatic. The power generated in heat sources is dissipated in PCM (n-eicosane with the melting temperature, T_m = 36 °C) that filled the rectangular enclosure. The advantage of using PCM is that it is able to absorb high amount of heat generated by heat sources due to its relatively high energy density. To investigate the thermal behaviour and thermal performance of the proposed system, a mathematical model based on the mass, momentum and energy conservation equations was developed. The governing equations are next discretised using a control volume approach in a staggered mesh and a pressure correction equation method is employed for the pressure–velocity coupling. The PCM energy equation is solved using the enthalpy method. The solid regions (wall and heat sources) are treated as fluid regions with infinite viscosity and the thermal coupling between solid and fluid regions is taken into account using the harmonic mean of the thermal conductivity method. The dimensionless independent parameters that govern the thermal behaviour of the system were next identified. After validating the proposed mathematical model against experimental data, a numerical investigation was next conducted in order to examine the thermal behaviour of the system by analyzing the flow structure and the heat transfer during the melting process, for a given values of governing parameters.     

Numerical simulation of immersion quenching process of an engine cylinder head

In this article, we present the numerical simulations of a real cylinder head quench cooling process employing a newly developed boiling phase change model using the commercial CFD code AVL-FIRE v8.5. Separate computational domains constructed for the solid and liquid regions are numerically coupled at the interface of the solid–liquid boundaries using the AVL-Code-Coupling-Interface (ACCI) feature. The boiling phase change process triggered by the dipping hot metal and the ensuing two-phase flow is handled using an Eulerian two-fluid method. Multitude of flow features such as vapor pocket generation, bubble clustering and their disposition, are captured very effectively during the computation, in addition to the variation of the temperature pattern within the solid region. A comparison of the registered temperature readings at different monitoring locations with the numerical results generates an overall very good agreement and indicates the presence of intense non-uniformity in the temperature distribution within the solid. Overall, the predictive capability of the new boiling model is well demonstrated for real-time quenching applications.     

A mathematical boundary integral equation analysis of standard viscoelastic solid polymers

Structural analysis of viscoelastic solid polymers is one of the most important subjects in engineering structures. Several attempts have been so far made for the integral equation approach to viscoelastic problems. From the basic assumptions of viscoelastic constitutive equations and weighted residual techniques, a simple but effective boundary element formulation (BEF) is implemented for the standard linear solid (SLS) viscoelastic models. The SLS model provides an approximate representation of the observed behavior of a real polymer in its viscoelastic range. This formulation needs only Kelvin’s fundamental solution of isotropic elastostatics with material constants prescribed as explicit functions of time. This approach avoids the use of relaxation functions and mathematical transformations, and it is able to solve the quasistatic viscoelastic problems with any load time-dependence and boundary conditions. As an application, a numerical example is provided to validate the proposed formulation. The problem of the pressurization of thick-walled cylindrical viscoelastic tanks made of PMMA polymer is completely analyzed by this approach.     

学英语（英文）

<正>There are 6questions in total,presenting various different question types.While you attempt to resolve the problems,remember to be creative.During accomplishing these flexible mathematical exercises,you can inspire your mathematical thinking.1.The melting point of a solid is the temperature at which the solid becomes a liquid.The melting point of bromine is 1/30of the melting point of the melting point of nitrogen.Write and     

Modelling ultra-fast nanoparticle melting with the Maxwell–Cattaneo equation

The role of thermal relaxation in nanoparticle melting is studied using a mathematical model based on the Maxwell–Cattaneo equation for heat conduction. The model is formulated in terms of a two-phase Stefan problem. We consider the cases of the temperature profile being continuous or having a jump across the solid–liquid interface. The jump conditions are derived from the sharp-interface limit of a phase-field model that accounts for variations in the thermal properties between the solid and liquid. The Stefan problem is solved using asymptotic and numerical methods. The analysis reveals that the Fourier-based solution can be recovered from the classical limit of zero relaxation time when either boundary condition is used. However, only the jump condition avoids the onset of unphysical “supersonic” melting, where the speed of the melt front exceeds the finite speed of heat propagation. These results conclusively demonstrate that the jump condition, not the continuity condition, is the most suitable for use in models of phase change based on the Maxwell–Cattaneo equation. Numerical investigations show that thermal relaxation can increase the time required to melt a nanoparticle by more than a factor of ten. Thus, thermal relaxation is an important process to include in models of nanoparticle melting and is expected to be relevant in other rapid phase-change processes.     

Numerical modelling of EHD effects on heat transfer and bubble shapes of nucleate boiling

In this article, mathematical and numerical models are developed to study pure electrohydrodynamic (EHD) effects on heat transfer and bubble shapes when an initial bubble attached to a superheated horizontal wall in nucleate boiling. In the modelling of EHD effects on heat transfer, an undeformed bubble is considered; the electric body force and Joule heat are added to the momentum and energy equations; governing equations for heat, fluid flow and electric fields are coupled numerically and solved using a non-orthogonal body-fitted mesh system with necessary interfacial treatments at the gas–liquid boundary. While, to study the pure effect of EHD on the deformation of the bubble, the evaluation of a deformable bubble without heat transfer is simulated by volume of fluid (VOF) method based on an axial symmetric Cartesian coordinate system. The simulations indicate that EHD can effectively enhance heat transfer rate of nucleate boiling by influencing the motion of the ring vortex around the bubble and that bubble can be elongated due to the pull in axial direction and push in the negative radial direction by the electric field force.     

A study of the elastic deformations in a thermoelastic inhomogeneous solid of revolution

We construct a mathematical model for studying the elastic deformations in a thermoelastic inhomogeneous solid of revolution applicable to sliding bearings. The method of numerical solution is based on the grid method and the relaxation method. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 91–94.     

Three‐dimensional solidification with two possible crystallization states: Existence of solutions with flow in the melt

In this paper we discuss the existence of solutions of a system of nonlinear and singular partial differential equations constituting a phase field model with convection for non‐isothermal solidification/melting of certain metallic alloys in the case where two different kinds of crystallization are possible. Each one of these crystallization states is described by its own phase field, while the liquid phase is described by another one. The model also allows the occurrence of fluid flow in non‐solid regions, which are a priori unknown, and then we have a free‐boundary value problem. Thus, the model relates the evolutions of these three phase fields, the temperature of the solidification/melting process and the fluid flow in non‐solid regions. Copyright © 2009 John Wiley & Sons, Ltd.     

A finite element method for steady-state conduction-advection phase change problems

A new finite element: technique is developed to solve steady-state conduction-advection problems with a phase change. The energy balance equation at the solid/liquid interface is employed to calculate the velocity of the solid/liquid interface in the Lagrangian frame. The position of the solid/liquid interface in the Eulerian frame is determined based on the composition of the velocity of the solid/liquid interface in the Lagrangian frame and the steady-state velocity of a rigid body. The interface position and the finite element mesh are continuously updated during an incremental process. No artificial diffusion is needed with this new finite element approach. An analytical solution for solidification of a pure material with a radiative boundary condition is also developed in this paper. Numerical experimentation is conducted and the corresponding results are compared with analytical solutions. The numerical results agree well with analytical solutions.     

A Phase-Change Model with a Zone of Coexistence of Phases

A mathematical model for change of phase is presented, accountingfor the presence of regions in which liquid and solid coexist.The basic variables are temperature and solid fraction v. Westart from a relationship of the type =(v), supposed valid inthermodynamical equilibrium. Then for dynamical processes weintroduce a perturbation causing v to be less than its equilibriumvalue in any solidification process. This solid fraction deficiencyis governed by an ordinary differential equation containing_t, in the forcing term. The heat-balance equation is in turncoupled to the ordinary differential equation through the termv_t, ( is latent heat). Some existence and uniqueness resultsare proved and some monotonicity properties are described forpure melting or pure solidification processes.     

Computation of melting with natural convection inside a rectangular enclosure heated by discrete protruding heat sources

This paper presents the results of a numerical investigation of the heat transfer by natural convection during the melting of a phase change material (PCM, n-eicosane with melting point of 36 °C) contained in a rectangular enclosure. This latest is heated by three discrete protruding heat sources (simulating electronic components) placed on one of its vertical walls. The power generated by heat sources is dissipated in PCM. The advantage of using this cooling scheme is that the PCMs are able to absorb high amount of heat generated by the heat sources, without acting the fan during the charging process (melting of the PCM). The thermal behavior and thermal performance of the proposed PCM based-heat sink are numerically investigated by developing a mathematical model based on the mass, momentum and energy conservation equations. The obtained numerical results show the impact of various key parameters on the cooling capacity of the PCM-based heat sink. Correlations encompassing a wide range of parameters were developed in terms of the dimensionless secured operating time (time required by one of the electronic components before reaching its critical temperature, T_cr ∼ 75 °C) and the corresponding liquid fraction, using the asymptotic computational fluid dynamics (ACFD) technique.     

Single phase limit for melting nanoparticles

The melting of a spherical or cylindrical nanoparticle is modelled as a Stefan problem by including the effects of surface tension through the Gibbs–Thomson condition. A one-phase moving boundary problem is derived from the general two-phase formulation in the singular limit of slow conduction in the solid phase, and the resulting equations are studied analytically in the limit of small time and large Stefan number. Further analytical approximations for the temperature distribution and the position of the solid–melt interface are found by applying an integral formulation together with an iterative scheme. All these analytical results are compared with numerical solutions obtained using a front-fixing method, and are shown to provide good approximations in various regimes. The inclusion of surface tension, which acts to decrease the melting temperature as the particle melts, is shown to accelerate the melting process. Unlike the classical one-phase Stefan problem without surface tension, the solid–melt interface exhibits blow-up at some critical radius of the particle (which for metals is of the order of a few nanometres), a phenomenon that has been observed experimentally. An interesting feature of the model is the prediction that surface tension drives superheating in the solid particle before blow-up occurs.     

Numerical simulation study on gas–solid two-phase flow in pre-calciner

A three-dimensional numerical simulation of DD (dual combustion and denitratior process) pre-calciner for cement production was conducted in this paper. In Euler coordinate system, the fluid phase is expressed with RNG k–ε two-equation model and the solid phase is expressed with particle stochastic trajectory model in Lagrange coordinate system. Four mixture fractions are deduced in this article to simulate the gas compositions. The results of numerical simulation predicted the burn-out ratio of coal and the decomposition ratio of limestone particles along with particle trajectories. It also supplied theoretical foundation for industrial analysis of the coupling relation between coal combustion and calcium carbonate decomposition.     

Tribalj dam-break and flood wave propagation

In this paper we present numerical simulations for the dam-break flood wave propagation from Tribalj accumulation to the town of Crikvenica (Croatia). The mathematical models we used were the one-dimensional open channel flow and the two-dimensional shallow water equations. They were solved with the well-balanced finite volume numerical schemes which additionally include special numerical treatment of the wetting/drying front boundary. These schemes were tested on CADAM test problems. The aim of this study was to assess potential damage in the village of Tribalj and the town of Crikvenica. Results of these simulations were used as the basis for urban planning and micro-zoning of the flood-risk areas. Several different dam-break scenarios were considered, ranging from sudden dam disappearance to partial and dynamic breach formation.