Two-phase Stefan Problem Research Articles

An effective algorithm of the numerical solution to the Stefan problem

The method of the frontal interval of the numerical solution to Stefan problems is proposed. The method allows the calculation of the position of the phase transition boundary with high accuracy at any of time. Using the example of a plane two-phase Stefan problem, which has an exact analytical solution, the advantages of the proposed method in comparison with a variable time stepping method are demonstrated with respect to the calculation of the position of the phase transition boundary. An example of the solution to the one-phase Stefan problem using the frontal interval method is also given.

Estimation of the change in the gas flow rate from the surface of a tungsten plate under pulsed heat load

The work is devoted to the numerical implementation of the tungsten evaporation process model. The tungsten evaporation model is based on solving the two-phase Stefan problem for temperature in the sample area and gas dynamics equations over the sample. The calculation results for parameters corresponding to those used on the BETA facility at BINP SB RAS show that it is possible to use the boundary homogeneous conditions of the Neumann for the velocity when setting the gas density on the plate surface.

A parameters selection criterion of the numerical realisation of the continuous method for the Stefan problems

This paper suggests a selection criterion of the continuous method version for a numerical solution of the Stefan problem which would allow to calculate the phase transition boundary position with a required accuracy for a long period of time and would enable generalization to multidimensional problems. Despite a large number of works deal with the solution to the generalized Stefan problem by the continuous method, the choice of the smoothing interval value for numerical feasibility is not fully clear. A comparison of the calculation accuracy of the phase transition boundary position using different versions of the continuous method was carried out on an example of the well-known 1-D plane two-phase Stefan problem which possesses an analytical solution. The dependence of the total error of the numerical calculation of the phase transition boundary position on the value of the smearing interval is determined from the comparison of numerical and analytical solutions. An analysis of the reason for increase of this error with time at any choice of a constant smoothing interval is given. A version of the continuous method with a variable interval of the delta function smoothing, in which the proposed criterion is carried out, is discussed. The position of the phase transition boundary calculated proposed version matches the analytical solution with a required accuracy over a long period of time.

The two-phase Stefan problem for quasilinear parabolic equations

The two-phase Stefan problem for quasilinear parabolic equations

The solution of the heat conduction problem for a rectangular area with mixed boundary conditions and internal source in general by using fast expansions method

The fundamentals of the fast expansions method are outlined, provided that the mixed Dirichlet-Neumann boundary conditions are specified. Then, using the fast expansions method, the solution of the problem in an explicit analytical form with an accuracy of quadrature of Fourier coefficients is obtained. Such a solution allows to explore temperature fields in a rectangle, depending on the given input data of the task, which include the size of the rectangle, the boundary conditions and the internal source. The resulting solution for a rectangle can also be used to construct a solution to a problem in a curvilinear domain, for a two-phase Stefan problem with a curvilinear boundary, and for other applied problems.

An enthalpy-based finite element method for solving two-phase Stefan problem

Stefan problem is a problem involving phase transition from solid to liquid or vice versa where boundary between solid and liquid regions moves as function of time. This paper presents numerical solution of one-dimensional two-phase Stefan problem by using finite element method. The governing equations involved in Stefan problem consist of heat conduction equation for solid and liquid regions, and also transition equation in interface position (moving boundary). The equations are difficult to solve by ordinary numerical method because of the presence of moving boundary. As consequence, the equations is reformulated into the form of internal energy (enthalpy). By the enthalpy formulation, solution of the heat conduction equations is no longer concerning the phase state of material. The advantage of the enthalpy formulation is that, finite element method can be easily implemented to solve Stefan problem. Numerical simulation of interface position, temperature profile, and temperature history has good agreement with the exact solution. The approximation of interface position using finite element method was found that it is more accurate than the approximation by using Godunov method. The simulation results also reveal that the finite element method for solving Stefan problem have smaller mean absolute error than the Godunov method.

Analytical Approximations for the Closure Time of Freezing Processes in Spheres and Cylindrical Tubes

In this study, an analytical approach is used to find expressions for the closure time in freezing processes for spheres and cylindrical tubes. The starting point is the well-known two-phase Stefan problem. A new characteristic solution is established for extending the theory of constant heat flux ratio. Next, temperature profiles are assumed and substituted into the interface equation, which are then solved for the inward freezing process to get the closure time. Plots are generated to compare the new expressions to previously published experimental results of closure time. The new analytical approximations give reasonable outcomes as discussed in this paper. This paper demonstrates a general approach that can be further applied to different types of phase-change problems.

Phase-change mechanism for evaporation in porous media using volume of fluid: Implicit formulation of interfacial temperature

In this study, a widely-utilized mechanism which relates the rate of phase change in two-phase flows to an interfacial superheat is described and modified to improve its physical consistency and overall performance when compared to the porous two-phase Stefan problem. Maintaining interface temperature at saturation temperature, the rate of phase change is taken to be proportional to the degree of superheating at the interface. The present work demonstrates how the superheat error can be made to meet the same convergence requirements already established for the temperature solution at each time step by comparing the interfacial energy balance to the present methods superheat relation at the fluid-fluid interface. The results are in good agreement with the presented analytical solution for the porous media Stefan problem using propane. The Volume of Fluid (VOF) method, incompressible Navier-Stokes and temperature equations are segregated and modified to incorporate the effects of porous media and the source terms for phase change. For low porosities, a limitation of the rate of phase change is observed due to a restriction in the availability of interstitial pore space occupied by the phase change fluid. At the other extreme, when solid content diminishes, the system is reduced to the classical Stefan problem in free flow. Between these two ranges, an optimum porosity exists where the rate of phase change is maximum.

A model for the evolution of highly reversible martensitic transformations

In this work, we introduce a new system of partial differential equations as a simplified model for the evolution of thermally induced reversible martensitic transformations in low hysteresis alloys. The model is developed in the context of nonlinear continuum mechanics, where the developed theory is mostly static, and cannot capture the influence of dynamics on martensitic microstructures. First, we prove existence of weak solutions; second, we study the physically relevant limit when the interface energy density vanishes, and the elastic constants tend to infinity. The limit problem provides a framework for the moving mask approximation recently introduced by the author. In the last section, we study the limit equations in a one-dimensional setting. After closing the equations with a constitutive relation between the phase interface velocity and the temperature of the one-dimensional sample, the equations become a two-phase Stefan problem with a kinetic condition at the free boundary. Under some further assumptions, we show that the phase interface reaches the domain boundary in finite time.

Adjustment of Thermophysical Rock Mass Properties in Modeling Frozen Wall Formation in Mine Shafts under Construction

Modeling of heat exchange processes in water-saturated rock mass during shafting with artificial freezing is performed. The problem of adjusting thermophysical properties of rock layers by the experimental measurements of temperature in the check thermal wells spaced from the freezing perimeter is analyzed. In terms of the abuilding shafts at Nezhinsky Mining and Processing Plant, significance of adjusting the the thermophysial parameters borrowed from the geological engineering survey data is illustrated. The number of independent adjustment parameters is determined from the analysis of the system of equations in two-dimensional two-phase Stefan problem in the dimensionless form. An inverse Stefan problem is formulated for a horizontal layer of rocks. The numerical algorithm is proposed for the inverse Stefan problem solution based on the gradient descent method. The algorithm minimizes functional of discrepancies between the model and measurement temperatures at the locations of the check wells. The functional of discrepancies in the phase space of the thermophysical properties and the algorithm convergence are analyzed.

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

• New models of nanoparticle melting based on the Maxwell–Cattaneo equation are proposed. • Consider temperature profiles that are continuous or have a jump across the solid–liquid interface. • The jump condition for the temperature is derived by taking a sharp-interface limit of a phase-field model. • The jump condition is the most suitable condition to use in phase-change models based on the Maxwell–Cattaneo equation. • Thermal relaxation is shown to substantially slow the melting process. 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 model of high-power transient heating of tungsten with considering of various erosion effects

Surface melting of tungsten under exposure to a pulsed electron beam was simulated numerically, the evaporation process taken into account. The calculation is based on the experimental time dependence of the total beam power. The model of the tungsten heating process is based on solving the two-phase Stefan problem. The position of the phase boundary depends on discontinuous time- and space-nonlinear coefficients and boundary conditions. The aim of the study is to provide a detailed resolution of the heat flow deep into the material with a fine spatial grid step. As compared with the size of the tungsten plate, the heating depth is very small. The problem statement under consideration is multiscale. Further expansion of the model involves taking into account microcracks. Micro-cracks occur during the cooling process after exposure and affect the temperature of the tungsten surface during the subsequent heating process. The article presents a modeling of cracks of different geometries typical for this process. The results of the calculations correlate with the experimental data obtained on the experimental test facility BETA at BINP SB RAS.

Optimal Control of a Stefan Problem Fully Coupled with Incompressible Navier-Stokes Equations and Mesh Movement

Abstract The optimal control of moving boundary problems receives growing attention in science and technology. We consider the so called two-phase Stefan problem that models a solid and a liquid phase separated by a moving interface. The Stefan problem is coupled with incompressible Navier{Stokes equations. We take a sharp interface model approach and define a quadratic tracking-type cost functional that penalizes the deviation of the interface from the desired state and the control costs. With the formal Lagrange approach and an adjoint system we derive the gradient of the cost functional. The derived formulations can be used to achieve a desired interface position. Among others, we address how to handle the weak discontinuity of the temperature along the interface with mesh movement methods in a finite element framework.

Exact solution for a two-phase Stefan problem with variable latent heat and a convective boundary condition at the fixed face

Recently it was obtained in [Tarzia, Thermal Sci. 21A (2017) 1-11] for the classical two-phase Lam\'e-Clapeyron-Stefan problem an equivalence between the temperature and convective boundary conditions at the fixed face under a certain restriction. Motivated by this article we study the two-phase Stefan problem for a semi-infinite material with a latent heat defined as a power function of the position and a convective boundary condition at the fixed face. An exact solution is constructed using Kummer functions in case that an inequality for the convective transfer coefficient is satisfied generalizing recent works for the corresponding one-phase free boundary problem. We also consider the limit to our problem when that coefficient goes to infinity obtaining a new free boundary problem, which has been recently studied in [Zhou-Shi-Zhou, J. Engng. Math. (2017) DOI 10.1007/s10665-017-9921-y].

Existence and Boundary Regularity for Degenerate Phase Transitions

We study the Cauchy--Dirichlet problem associated to a phase transition modeled upon the degenerate two-phase Stefan problem. We prove that weak solutions are continuous up to the parabolic boundary and quantify the continuity by deriving a modulus. As a byproduct, these a priori regularity results are used to prove the existence of a so-called physical solution.

Computational modeling of freezing of supercooled water using phase-field front propagation with immersed points

Computational modeling of phase change due to solidification, besides correctly capturing the heat transfer, requires accurate evaluation of mass transfer at the phase-interface. In the present study a framework is developed for modeling the solidification of supercooled water using a phase-field approach for the propagation of the interface between ice and supercooled water. Energy equations in the solid and liquid phases are decoupled at the interface by using immersed points to impose the melting temperature as a moving boundary condition. The propagation of the interface is determined by the local energy balance across the interface, which is accounted for in the reconstructed interface points. The model is validated using the known theoretical solutions for the two-phase Stefan problem. An extension of the model to incorporate conjugate heat transfer with a neighboring solid wall is presented, establishing a framework that can be used for further modeling of the interaction of solidifying supercooled water on a solid substrate.

Exact solution for a two-phase Stefan problem with power-type latent heat

A two-phase Stefan problem with latent heat a general power function of position is investigated. The background of the problem can be found in the soil-freezing process during the application of the artificial ground-freezing technique. After introducing the specific engineering condition, the governing equations of a two-phase Stefan problem are developed. An exact solution for the problem is established using the similarity transformation technique and the theory of the Kummer functions. It is proved that the coefficients in the solution can be appropriately determined if certain inequality is satisfied. Special cases of the solution are discussed, and several solutions reported in the literature are recovered. A similar two-phase Stefan problem involving first type of boundary condition is also introduced and solved. The coefficients in the solution can always be properly determined with no additional requirement. In the end, computational examples of the solution are presented and discussed. The exact solution provides a useful benchmark that can be used for verifying general numerical algorithms of Stefan problems, and it is also advantageous in the context of the inverse problem analysis.

Phase-field lattice Boltzmann modeling of boiling using a sharp-interface energy solver.

The main objective of this paper is to extend an isothermal incompressible two-phase lattice Boltzmann equation method to model liquid-vapor phase change problems using a sharp-interface energy solver. Two discrete particle distribution functions, one for the continuity equation and the other for the pressure evolution and momentum equations, are considered in the current model. The sharp-interface macroscopic internal energy equation is discretized with an isotropic finite difference method to find temperature distribution in the system. The mass flow generated at liquid-vapor phase interface is embedded in the pressure evolution equation. The sharp-interface treatment of internal energy equation helps to find the interfacial mass flow rate accurately where no free parameter is needed in the calculations. The proposed model is verified against available theoretical solutions of the two-phase Stefan problem and the two-phase sucking interface problem, with which our simulation results are in good agreement. The liquid droplet evaporation in a superheated vapor, the vapor bubble growth in a superheated liquid, and the vapor bubble rising in a superheated liquid are analyzed and underlying physical characteristics are discussed in detail. The model is successfully tested for the liquid-vapor phase change with large density ratio up to 1000.

Stable and unstable manifolds for quasilinear parabolic problems with fully nonlinear dynamical boundary conditions

We develop a wellposedness and regularity theory for a large class of quasilinear parabolic problems with fully nonlinear dynamical boundary conditions. Moreover, we construct and investigate stable and unstable local invariant manifolds near a given equilibrium. In a companion paper, we treat center, center-stable and center-unstable manifolds for such problems and investigate their stability properties. This theory applies e.g. to reaction-diffusion systems with dynamical boundary conditions and to the two-phase Stefan problem with surface tension.

Ice Layer Spreading along a Solid Substrate during Solidification of Supercooled Water: Experiments and Modeling.

The thermal influence of a solid wall on the solidification of a sessile supercooled water drop is experimentally investigated. The velocity of the initial ice layer propagating along the solid substrate prior to dendritic solidification is determined from videos captured using a high-speed video system. Experiments are performed for varying substrate materials and liquid supercooling. In contrast to recent studies at moderate supercooling, in the case of metallic substrates only a weak influence of the substrate's thermal properties on the ice layer velocity is observed. Using the analytical solution of the two-phase Stefan problem, a semiempirical model for the ice layer velocity is developed. The experimental data are well described for all supercooling levels in the entire diffusion limited solidification regime. For higher supercooling, the model overestimates the freezing velocity due to kinetic effects during molecular attachment at the solid-liquid interface, which are not accounted for in the model. The experimental findings of the present work offer a new perspective on the design of anti-icing systems.