A Numerical Method for Multiphase Incompressible Thermal Flows with Solid–Liquid and Liquid–Vapor Phase Transformations

ABSTRACT

A numerical method for multiphase incompressible thermal flows with solid–liquid and liquid–vapor phase transformations is presented. The flow is mainly driven by thermocapillary force and vaporization. Based on the level set method and mixture continuum model, a set of governing equations valid for solid, liquid, and vapor phases is derived, considering phase boundary conditions as source terms in the transport equations. The vaporization process is treated as a source term in the continuity equation. The model developed is applied to the laser welding process, where the flow is coupled with optical phenomena. Formation and collapse of a laser-created hole is simulated.     

Numerical simulation of the quenching process in liquid jet impingement

Direct numerical simulation is performed for quenching of a hot plate in liquid jet impingement. The flow and thermal characteristics associated with the quenching process, which includes film boiling in the fluid region as well as transient conduction in the solid region, are investigated by solving the conservation equations of mass, momentum and energy in the liquid, gas and solid phases. The liquid–vapor and liquid–air interfaces are tracked by the sharp-interface level-set method modified to treat the effect of phase change. The computations demonstrate that the boiling curve of wall heat flux versus temperature does not depend on the transient or steady-state heating conditions. The effects of initial solid temperature and solid properties on the quenching characteristics are quantified.     

A NUMERICAL STUDY OF SOLIDIFICATION IN THE PRESENCE OF A FREE SURFACE UNDER MICROGRAVITY CONDITIONS

A numerical study of the relative importance of Marangoni effects under microgravity conditions is presented. The mathematical formulation adopted is based on the enthalpy porosity method. One of the advantages of the fixed grid method is that a unique set of equations and boundary conditions is used for the whole domain, including both solid and liquid phases. The governing equations written in a vorticity-velocity formulation are discretized using a finite volume technique on a staggered grid. A fully implicit method has been adopted for the mass and momentum equations, while the temperature field is solved separately in order to evaluate the variation in the local liquid mass fraction. The resulting algebraic system of equations is solved using a preconditioned BI-CGStab method. Numerical results modelling the free surface, including the effects on it of Marangoni convection, are presented. The influence of the presence of argon in the gap above the free surface is investigated. During the numerical simulations presented in this paper 161 2 41 and 641 2 161 uniform meshes on the whole computational domain for values of Marangoni number ( Ma ) up to 16,120 and Rayleigh number ( Ra ) of 5 have been used.     

Computational modelling of heat and mass transfer during the high-temperature heat treatment of wood

During the heat treatment process, the heat and mass transfer takes place between the solid and the drying medium, and the moisture evaporation occurs within the solid due to the capillarity action and diffusion. In the current work, the three-dimensional Navier–Stokes equations along with the energy and concentration equations for the fluid coupled with the energy and mass conservation equations for the solid (wood) are solved to study the conjugate heat treatment behaviour. Whitaker’s continuum approach has been used to obtain the equations for the liquid and vapour migration within the solid. Three moisture phases are accounted for: free water, bound water, and water vapour. The model equations are solved numerically by the commercial package FEMLAB for the temperature and moisture content histories under different treatment conditions. The model validation is carried out via a comparison between the predicted values with those obtained experimentally. The comparison of the numerical and experimental results shows good agreement, implying that the proposed numerical algorithm can be used as a useful tool in designing high-temperature wood treatment processes.     

Three-dimensional simulation of saturated film boiling on a horizontal cylinder

Three-dimensional simulations of film boiling on a horizontal cylinder have been performed. A finite difference method is used to solve the equations governing the conservation of mass, momentum and energy in vapor and liquid phases. A level set formulation for tracking the liquid–vapor interface is modified to include the effect of phase change at the liquid–vapor interface and to treat the no-slip condition at the fluid–solid interface. From the numerical simulations, the effects of cylinder diameter and gravity on the interfacial motion and heat transfer in film boiling are quantified. The heat transfer coefficients obtained from numerical analysis are found to compare well with those predicted from empirical correlations reported in the literature.     

Diffusion mass transport in liquid phase epitaxial growth of semiconductors

A numerical simulation model for the mass transport occurring during the liquid phase epitaxial growth of AlGaAs is presented. The mass transport equations in the liquid and solid phases, and the relationships between concentrations and temperature obtained from the phase diagram constitute the governing equations. These equations together with appropriate interface and boundary conditions were solved numerically by the Finite Element Method. Numerical results show the importance of diffusion into the solid phase, affecting the composition of grown layers. Simulation results agree with experiments.     

Thermal Dissolution of a Spherical Particle with a Moving Boundary

A numerical problem for mass and energy transport considering thermodynamic equilibrium is solved around a spherical particle in the absence of hydrodynamic effects in a binary solution. The analysis includes the radial convective term generated due to the differences on density between the solid and liquid phases. Because the transport of mass and energy compete in the process, how the rate of dissolution is affected by compositional diffusivity or thermal diffusivity is distinguished. The partial differential equations are discretized with the finite difference method in space, and the resulting set of ordinary differential equations in time is solved by the method of lines. The numerical solution for the thermal dissolution of a spherical particle in a binary melt is compared with heat-balance integral method for small times of the process. The solutions are found to agree for conditions close to the dissolution regime.     

A study on transient ice formation of laminar flow inside externally supercooled rectangular duct

The transient process of solidification of laminar liquid flow (water) submitted to super-cooling was investigated both theoretically and experimentally. In this study an alternative analytical formulation and numerical approach were adopted resulting in the unsteady model with temperature dependent thermophysical properties in the solid region. The proposed model is based upon the fundamental equations of energy balance in the solid and liquid regions as well as across the solidification front. The basic equations and the associated boundary and initial conditions were made dimensionless by using the Landau transformation to immobilize the moving front and render the problem to a fixed plane type problem. A laminar velocity profile is admitted in the liquid domain and the resulting equations were discretized using the finite difference approach. The numerical predictions obtained were compared with the available results based on other models and concepts such as Neumann analytical model, the apparent thermal capacity model due to Bonacina and the conventional fixed grid energy model due to Goodrich. To obtain further comparisons and more validation of the model and the numerical solution, an experimental rig was constructed and instrumented permitting very well controlled experimental measurements. The numerical predictions were compared with the experimental results and the agreement was found satisfactory.     

Numerical Simulation of Evaporation-Induced Particle Line Formation on a Moving Substrate

Numerical simulations are performed for evaporation-induced particle line formation on a moving substrate by solving the conservation equations of mass, momentum, energy, vapor concentration, and particle concentration in the liquid–gas phases. The liquid–gas interface and the liquid–gas–solid contact line are tracked by using a level-set method, which is modified to include the effects of contact line, phase change, and particle concentration. The numerical results for liquid evaporation and particle deposition in confined convective coating between two parallel plates showed that the substrate velocity is a key parameter determining the particle deposition pattern and the particle line formation can be controlled by varying the substrate velocity.     

A LEVEL SET METHOD FOR INCOMPRESSIBLE TWO-FLUID FLOWS WITH IMMERSED SOLID BOUNDARIES

ABSTRACT

A numerical method is presented for computing incompressible gas–liquid (or two-fluid) flows with immersed solid boundaries on fixed Cartesian meshes. A level set technique for tracking the gas–liquid interface is modified to treat the contact angle condition at the gas–liquid–solid interline as well as the no-slip condition at the fluid–solid interface. The no-slip condition is imposed by introducing another level set for fluid–solid phases and an effective viscosity formulation. In the immersed solid region where the level set function for gas–liquid phases is not well defined, its zero level set is calculated so that the contact angle condition should be satisfied where the three phases meet. The numerical method is validated through computations of interfacial motion subject to Taylor instability, single-fluid flow past a circular cylinder, and bubbles adhering to a cylindrical solid.     

ANALYSIS OF SOLIDIFICATION AND VISCOPLASTIC STRESSES INCORPORATING A MOVING BOUNDARY: AN APPLICATION TO SIMULATION OF THE CENTRIFUGAL CASTING PROCESS

Coupled equations governing temperature and stress fields incorporating solidification in a growing body with a moving boundary are developed based on the thermomechanics of a mixture of liquid and solid, and a finite-element method for stress analysis as well as temperature distribution is formulated, where a unified inelastic constitutive relationship capable of describing both elastic-viscoplastic solids and viscous fluids is employed. Examples of numerical calculation of temperature affected by solidification and stresses are presented for some practical cases of centrifugal casting processes. In order to verify the results of simulation, calculated temperatures were compared with the measured data.     

Mathematical model for turbulent flow,heat transfer,and solidification

A steady‐state, two‐dimensional numerical model has been used to describe coupled liquid steel's turbulent flow and heat transfer with solidification for Fe‐C binary alloy in a crystallizer of inverse casting. The solid‐liquid phase change phenomena have been modeled by using continuum formulations and considering the mushy zone as porous media. The turbulence flow in the crystallizer has been accounted for using a modified version of the low‐Reynolds‐number κ?ε turbulence model. The flow pattern in the liquid zone and the temperature distribution in the solid, mushy, and liquid regions have been predicted. The numerical analysis indicates that the residence time of the mother sheet in the crystallizer is one of the key parameters. The effects of some other main parameters on the solidification behavior have also been studied, such as the thickness and the initial temperature of the mother sheet, and the superheat degree of liquid steel. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(7): 582–592, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10112     

NUMERICAL SIMULATION OF TRANSIENT COOLING OF A HOT SOLID BY AN IMPINGING FREE SURFACE JET

This paper treats transient cooling of a hot solid by an impinging circular free surface liquid jet. The flow and thermal fields in the liquid as well as the temperature distributions in the hot solid have been predicted numerically. The Navier-Stokes equations for incompressible fluid flow in an axisymmetric coordinate system and the transient heat conduction equation for a solid have been solved by a finite difference method. The hydrodynamics of the liquid film and the heat transfer processes have been investigated to understand the physics of the phenomena.     

A Level-Set Method for the Direct Numerical Simulation of Particle Motion in Droplet Evaporation

A sharp-interface level-set (LS) method is presented for direct numerical simulation (DNS) of particle motion in droplet evaporation. The LS formulation for liquid–gas flows is extended to liquid–gas–solid flows by treating the moving solid region as a high-viscosity fluid phase. The evaporation effect is accurately implemented by imposing the coupled temperature and vapor fraction conditions at the interface. The LS method is tested through computations of particle sedimentation in single-phase and two-phase fluids. The DNS of particle motion in droplet evaporation demonstrates the pinning phenomena of the liquid–gas–solid contact line.     

3-D modeling of the dynamics and heat transfer characteristics of subcooled droplet impact on a surface with film boiling

The impact of a subcooled water and n-heptane droplet on a superheated flat surface is examined in this study based on a three-dimensional model and numerical simulation. The fluid dynamic behavior of the droplet is accounted for by a fixed-grid, finite-volume solution of the incompressible governing equations coupled with the 3-D level-set method. The heat transfer inside each phase and at the solid–vapor/liquid–vapor interface is considered in this model. The vapor flow dynamics and the heat flux across the vapor layer are solved with consideration of the kinetic discontinuity at the liquid–vapor and solid–vapor boundaries in the slip flow regime. The simulated droplet dynamics and the cooling effects of the solid surface are compared with the experimental findings reported in the literatures. The comparisons show a good agreement. Compared to the water droplet, it is found that the impact of the n-heptane droplet yields much less surface temperature drop, and the surface temperature drop mainly occurs during the droplet-spreading stage. The effects of the droplet’s initial temperature are also analyzed using the present model. It shows that the droplet subcooling degree is related closely to the thickness of the vapor layer and the heat flux at the solid surface.     

A three-dimensional model of droplet impact and solidification

A three-dimensional model has been developed to simulate the fluid dynamics, heat transfer and phase-change that occur when a molten metal droplet falls onto a flat substrate. The model is an extension of one developed by Bussmann et al. [Phys. Fluids 11 (1999) 1406] and combines a fixed-grid control volume discretization of the fluid flow and energy equations with a volume-tracking algorithm to track the droplet free surface, and an improved fixed velocity method to track the solidification front. Surface tension is modeled as a volume force acting on fluid near the free surface. Contact angles are applied as a boundary condition at liquid-solid contact lines. The energy equations in both the liquid and solid portions of the droplet are solved using the Enthalpy method. Heat transfer within the substrate is by conduction alone. Thermal contact resistance at the droplet-substrate interface is included in the model. We studied the deposition of tin droplets on a stainless steel surface using both experiments and numerical simulations. The results of two different scenarios are presented: the normal impact of a 2.7 mm tin droplet at 1 m/s, and of the oblique impact of a 2.2 mm tin droplet at 2.35 m/s onto a surface inclined at 45° to the horizontal. Images obtained from numerical model were compared with experimental photographs and found to agree well.     

Direct numerical simulation of flow boiling in a finned microchannel

Direct numerical simulations of bubble growth and heat transfer associated with flow boiling in a finned microchannel are performed by solving the conservation equations of mass, momentum and energy in the liquid and vapor phases. The phase interfaces are determined by a sharp-interface level-set method which is modified to include the effect of phase change at the liquid–vapor interface and to treat the no-slip and contact-angle conditions on the immersed solid surface of fins. The effects of fin height, spacing, and length on the flow boiling in a microchannel are investigated to find the better conditions for heat transfer enhancement.     

Numerical Simulation of Bubble Growth and Heat Transfer During Flow Boiling in a Surface-Modified Microchannel

Flow boiling in a microchannel without or with surface modifications, such as fins, grooves, and cavities, has received significant attention as an effective cooling method for high-power microelectronic devices. However, a general predictive approach for the boiling process has not yet been developed because of its complexity involving the bubble dynamics coupled with boiling heat transfer in a microscale channel. In this study, direct numerical simulations for flow boiling in a surface-modified microchannel are performed by solving the conservation equations of mass, momentum, and energy in the liquid and vapor phases. The bubble surfaces are determined by a sharp-interface level-set method, which is modified to include the effect of phase change at the liquid–vapor interface and to treat the no-slip and contact-angle conditions on immersed solid surface of microstructures. This computation demonstrates that the surface-modified microchannel enhances boiling heat transfer significantly compared to a plain microchannel. The effects of various surface modifications on the bubble growth and heat transfer are investigated to find better conditions for boiling enhancement.     

Lattice Boltzmann simulation of non-Fourier heat conduction with phase change

The lattice Boltzmann method (LBM) combined with the enthalpy method is a very effective method to solve the solid–liquid phase transition problem. However, when the heat flux is very high or the time of the process is in the same order of magnitude as the relaxation time, it is necessary to consider the non-Fourier effect in heat conduction. At this time, whether the LBM-BGK format based on Bhatnagar-Gross-Krook (BGK) approximation is still valid remains to be discussed. In this paper, the hyperbolic lattice Boltzmann method (HLBM) is combined with the enthalpy method to solve the non-Fourier solid–liquid phase change problem. By solving the non-Fourier heat conduction problem and the Fourier solid–liquid phase change problem, the numerical solution is compared with the analytical solution to verify the accuracy of the algorithm. The effect of different relaxation times on the solid–liquid phase transition is analyzed. In addition, the effect of changes in thermal diffusivity due to state changes on the non-Fourier solid–liquid phase transition is discussed.     

A Level Set Method for Analysis of Film Boiling on an Immersed Solid Surface

A numerical method is presented for simulating film boiling on an immersed (or irregularly shaped) solid surface. The level set formulation for tracking the phase interfaces is modified to include the effect of phase change at the liquid–vapor interface and to treat the no-slip condition at the fluid–solid interface. The boundary or matching conditions at the phase interfaces are accurately imposed by incorporating the ghost fluid approach based on a sharp-interface representation. The numerical method is tested through computations of bubble rise in a stationary liquid, single-phase fluid flow past a circular cylinder, and film boiling on a horizontal cylinder.