Stefan Problem Research Articles

Computational Thermochemistry for Modelling Oxidation During the Conveyance Tube Manufacturing Process

Conveyance tube manufacturing via a hot-finished, welded route is an energy-intensive process which promotes rapid surface oxidation. During normalisation at approximately 950 °C to homogenise the post-weld microstructure, an oxide mill scale layer grows on tube outer surfaces. Following further thermomechanical processing, there is significant yield loss of up to 3% of total feedstock due to scale products, and surface degradation due to inconsistent scale delamination. Delaminated scale is also liable to contaminate and damage plant tooling. The computational thermochemistry software, Thermo-Calc 2023b, with its diffusion module, DICTRA, was explored for its potential to investigate oxidation kinetics on curved geometries representative of those in conveyance tube applications. A suitable model was developed using the Stefan problem, bespoke thermochemical databases, and a numerical solution to the diffusion equation. Oxide thickness predictions for representative curved surfaces revealed the significance of the radial term in the diffusion equation for tubes of less than a 200 mm inner radius. This critical value places the conveyance tubes’ dimensions well within the range where the effects of a cylindrical coordinate system on oxidation, owing to continuous surface area changes and superimposed diffusion pathways, cannot be neglected if oxidation on curved surfaces is to be fully understood.

An enthalpy-based model for the physics of ice-crystal icing

Ice-crystal icing (ICI) in aircraft engines is a major threat to flight safety. Due to the complex thermodynamic and phase-change conditions involved in ICI, rigorous modelling of the accretion process remains limited. The present study proposes a novel modelling approach based on the physically observed mixed-phase nature of the accretion layers. The mathematical model, which is derived from the enthalpy change after accretion (the enthalpy model), is compared with an existing pure-phase layer model (the three-layer model). Scaling laws and asymptotic solutions are developed for both models. The onset of ice accretion, the icing layer thickness and solid ice fraction within the layer are determined by a set of non-dimensional parameters including the Péclet number, the Stefan number, the Biot number, the melt ratio and the evaporative rate. Thresholds for freezing and non-freezing conditions are developed. The asymptotic solutions present good agreement with numerical solutions at low Péclet numbers. Both the asymptotic and numerical solutions show that, when compared with the three-layer model, the enthalpy model presents a thicker icing layer and a thicker water layer above the substrate due to mixed-phased features and modified Stefan conditions. Modelling in terms of the enthalpy poses significant advantages in the development of numerical methods to complex three-dimensional geometrical and flow configurations. These results improve understanding of the accretion process and provide a novel, rigorous mathematical framework for accurate modelling of ICI.

A class of free boundary problems describing the local-nonlocal cross-diffusion models with two species in ecology

We study a class of free boundary problems describing the local-nonlocal cross-diffusion models with two species in ecology. The first is a native species with nonlocal cross-diffusion, and the second is an invasion species with local cross-diffusion. Since the local and nonlocal cross-diffusions are taken into account, the models are described by the strongly coupled systems of integral and differential equations. In addition, the position of each free boundary is determined by the Stefan condition. By using the transformations of function and variable, various estimates and contraction mapping theorem, we show the global existence, uniqueness and regularity of solutions for the free boundary problems. Besides, the long time behavior of the solution for a special type of local-nonlocal cross-diffusion competition model is also obtained.

A collocation heat polynomials method for one-dimensional inverse Stefan problems

The inverse one-phase Stefan problem in one dimension, aimed at identifying the unknown time-dependent heat flux P(t) with a known moving boundary position s(t), is investigated. A previous study (Kassabek et al., 2021) attempted to reconstruct the unknown heat flux P(t) using the Variational Heat Polynomials Method (VHPM). In this paper, we develop the Collocation Heat Polynomials Method (CHPM) for the reconstruction of the time-dependent heat flux P(t). This method constructs an approximate solution as a linear combination of heat polynomials, which satisfies the heat equation, with the coefficients determined using the collocation method. To address the resulting ill-posed problem, Tikhonov regularization is applied. As an application, we demonstrate the effectiveness of the method on benchmark problems. Numerical results show that the proposed method accurately reconstructs the time-dependent heat flux P(t), even in the presence of significant noise. The results are also compared with those obtained in Kassabek et al. (2021) using the VHPM.

A consistent, volume preserving, and adaptive mesh refinement-based framework for modeling non-isothermal gas–liquid–solid flows with phase change

This work expands on our recently introduced low Mach enthalpy method (Thirumalaisamy and Bhalla 2023) for simulating the melting and solidification of a phase change material (PCM) alongside (or without) an ambient gas phase. The method captures PCM’s volume change (shrinkage or expansion) by accounting for density change-induced flows. We present several improvements to the original work. First, we introduce consistent time integration schemes for the mass, momentum, and enthalpy equations, which enhance the method stability. Demonstrating the effectiveness of this scheme, we show that a system free of external forces and heat sources can conserve its initial mass, momentum, enthalpy, and phase composition. This allows the system to transition from a non-isothermal, non-equilibrium, phase-changing state to an isothermal, equilibrium state without exhibiting unrealistic behavior. Furthermore, we show that the low Mach enthalpy method accurately simulates thermocapillary flows without introducing spurious phase changes. To reduce computational costs, we solve the governing equations on adaptively refined grids. We investigate two cell tagging/untagging criteria and find that a gradient-based approach is more effective. This approach ensures that the moving thin mushy region is always captured at fine grid levels, even when it temporarily falls within a subgrid level. We propose an analytical model to validate advanced computational fluid dynamics (CFD) codes used to simulate metal manufacturing processes (welding, 3D printing). These processes involve a heat source (like a laser) melting metal or its alloy in the presence of an ambient (inert) gas. Traditionally, studies relied on artificially manipulating material properties to match complex experiments for validation purposes. Leveraging the analytical solution to the Stefan problem with a density jump, this model offers a straightforward approach to validating multiphysics simulations involving heat sources and phase change phenomena in three-phase flows. Lastly, we demonstrate the practical utility of the method in modeling porosity defects (gas bubble trapping) during metal solidification. A field extension technique is used to accurately apply surface tension forces in a three-phase flow situation. This is where part of the bubble surface is trapped within the (moving) solidification front.

Studying a cylindrical moving boundary problem in ductal carcinoma in situ

In this paper, a mathematical model of ductal carcinoma in situ (DCIS) is considered as a moving boundary problem in cylindrical coordinates. Using the front-fixing technique, first, we transform the problem into a nonlinear parabolic problem on a fixed domain, and then some essential properties of the solution, such as the uniqueness of the solution, are shown. A numerical approach based on the finite difference method is established for solving the fixed domain problem. The solvability of the proposed finite difference method is proved, and the stability and convergence of the numerical solution are shown. For a test case problem, analytical and numerical results are compared to demonstrate the ability and effectiveness of the numerical method.

Triple junction benchmark for multiphase-field models combining capillary and bulk driving forces

Abstract A benchmark problem is formulated which is well suited for the validation of mesoscopic phase-field models for grain-boundary migration in polycrystals. First, an analytical steady-state solution of the sharp moving boundary problem is derived for a symmetric lamellar structure, which is valid for arbitrary bulk driving forces and triple junction angles. Characteristic quantities are identified to reduce the parameter space which in turn allows a systematic comparison of simulations and analytical results. Various multiphase-field (MPF) formulations are compared which approximate the sharp interface problem in terms of a diffuse regularization. An interfacial thickness convergence study reveals that the model error is largely dependent on the ratio of bulk to interfacial stabilizing force as well as the underlying model formulation. An additional grid convergence study highlights the efficiency of a more advanced discretization scheme. The results can be used to guide the selection of appropriate models and to estimate the interface thickness and spatial resolution required to achieve a given accuracy target. The post-processing framework consists of a fully automated determination of well-defined metrics from the phase field simulation data, eliminating human bias and facilitating reproducibility. The corresponding code is made openly available to assist the materials science and engineering community in validating MPF, multi-order parameter and similar model developments. We believe that this work provides a reliable benchmark procedure to better understand the potentials and limitations of current multiphase-field models as well as alternative approaches.

Mathematical model for heat transfer during underground coal gasification process

Purpose. To develop a mathematical model of the “coal-gas” medium heat transfer during underground coal gasification to predict the combustion face advance velocity and the duration of gasification column mining. Methodology. To detect the temperature fields in the coal seam and gas, depending on the displacement length of the phase transition boundary, a boundary-value problem of mathematical physics has been developed. To solve this boundary-value problem, Boltzmann transformations, as well as methods for solving ordinary differential equations, are used. Newton-Raphson method, which has quadratic convergence, is used to find the roots of the transcendental equation. Findings. Tendencies in the formation of mathematical models when studying temperature fields around an underground gasifier have been analyzed, with highlighting their disadvantages. A mathematical model of heat transfer during underground gasification has been developed, taking into account the phase transition boundaries of the “coal-gas” medium. A computational experiment was conducted to determine the temperature of the phase transition boundary based on the length of the gasification column and the duration of the process. Originality. A mathematical model for heat transfer during coal gasification in the form of a boundary-value problem of mathematical physics, which consists of parabolic heat-transfer equations, the Stefan condition at the phase transition boundary, and the Dirichlet boundary conditions, has been constructed. As a result of solving the boundary-value problem, a self-similar solution has been obtained for the distribution of the coal seam and gas temperature fields, as well as the position of the phase transition boundary depending on the gasification duration and on the medium density, thermal conductivity coefficients, specific heat capacity of gas and coal, specific calorific value and temperature of coal combustion, initial coal temperature and constant temperature of the gasification process. The conducted analysis of numerical calculations provides a deeper understanding of the dynamics of underground coal gasification process and makes necessary corrections to achieve the maximum process efficiency. Practical value. A methodology for determining the displacement length of the phase transition boundary of the “coal-gas” medium has been developed, taking into account the change in the combustion face temperature along the gasification zone length depending on the duration of this process. Application of the methodology makes it possible to predict the time of mining the gasified coal column for drawing up a calendar plan for mining operations.

Well-posedness and stability for the two-phase periodic quasistationary Stokes flow

The two-phase horizontally periodic quasistationary Stokes flow in \mathbb{R}^{2} , describing the motion of two immiscible fluids with equal viscosities that are separated by a sharp interface, parameterized as the graph of a function f=f(t) , is considered in the general case when both gravity and surface tension effects are included. Using potential theory, the moving boundary problem is formulated as a fully nonlinear and nonlocal parabolic problem for the function f . Based on abstract parabolic theory, it is shown that the problem is well-posed in all subcritical spaces \mathrm{H}^{r}(\mathbb{S}) , with r\in(3/2,2) . Moreover, the stability properties of the flat equilibria are analyzed in dependence on the physical properties of the fluids.

On the factors and the degree of their effect on subglacial melt and changes in the state of Antarctic subglacial lakes

ABSTRACT This study presents the results of mathematical modeling of the degree of effects of various characteristics on basal conditions in Antarctica. The model is based on the numerical solution of the one-dimensional Stefan problem. Five factors determining the nature of subglacial processes have been studied: ice thickness, snow-firn thickness, surface mass balance, air temperature, and geothermal heat flux. It was found that on the Antarctic plateau, slope, and coast, the geothermal heat flux has the greatest influence on the basal conditions. It is the heat flux that is responsible for the transition of most Antarctic lakes from a stable state to an active one (except for the ice stream regions). In areas where the ice sheet is thinner than 1,500 m, air temperature is the second most important factor affecting subglacial conditions. In areas where the glacier is thicker, the ice thickness begins to have a greater effect. The snow-firn thickness and snow accumulation have little effect on subglacial melt in most parts of Antarctica. Calculations over a 100-year period show that if meltwater accumulates on the bedrock rather than being completely channeled, it leads to a decrease in the average rate of subglacial melting by approximately 10 times.

Analysis and optimization of dynamic and static thermal rectification in shell-and-tube phase change thermal rectifiers

Thermal rectification devices based on solid-liquid phase change material (PCM) can realize efficient and reversible control of forward and reverse heat flux. In this study, an analytical solution for a shell-and-tube phase change thermal rectifier is derived based on Fourier's law of heat conduction. The validity of the analytical model is confirmed by the numerical model verified by the exact solution of the single-phase Stefan problem. The influence of thermophysical properties, structural parameters, and initial temperature on the dynamic and static thermal rectification effects of the shell-and-tube heterojunction composed of polydimethylsiloxane (PDMS) and hexadecane (C16) is investigated and optimized. Results indicate that the proposed analytical model for the shell-and-tube phase change thermal rectifier enables rapid and precise prediction of thermal rectification effects. The presence of solid-liquid phases of C16 under forward and reverse heat transfer modes reduced the thermal resistance difference, resulting in poor initial rectification performance (only about 1.17). The optimized thermal rectification effect is significantly increased to 1.56, which is significantly improved by 33.56 %. Moreover, the dynamic rectification effect of the shell-and-tube thermal rectifier is notably higher than the static rectification effect. The average dynamic thermal rectification ratio within the first 50 s before and after optimization is 2.19 and 7.80, respectively, which is several times higher than the static thermal rectification ratio. This paper further analyzes the impact of initial temperature and finds that an increase in initial temperature can further enhance the dynamic thermal rectification effect. Raising the initial temperature to 5 K above the phase change temperature can increase the improvement of the dynamic rectification effect to 256.36 % in the first 50 s. The significance of this study lies in its potential to advance the understanding and practical application of thermal rectification based on PCM.

Existence of maximal solutions for the financial stochastic Stefan problem of a volatile asset with spread

In this work, we consider the outer Stefan problem for the short-time prediction of the spread of a volatile asset traded in a financial market. The stochastic equation for the evolution of the density of sell and buy orders is the Heat Equation with a space–time white noise, posed in a moving boundary domain with velocity given by the Stefan condition. This condition determines the dynamics of the spread, and the solid phase [s−(t),s+(t)] defines the bid–ask spread area wherein the transactions vanish. We introduce a reflection measure and prove existence and uniqueness of maximal solutions up to stopping times in which the spread s+(t)−s−(t) stays a.s. non-negative and bounded. For this, we define an approximation scheme, and use some of the estimates of Hambly et al. (2020) for the Green’s function and the associated to the reflection measure obstacle problem. Analogous results are obtained for the equation without reflection corresponding to a signed density.

Boiling heat transfer by phase-field method

AbstractIn this work, we propose and test the validity of a phase-field method tailored specifically for modeling boiling phenomena. The method relies on numerical solutions of the Navier–Stokes equations coupled with a phase-field method and the energy equation. The continuity and Navier–Stokes equations have been modified introducing a source term that accounts for phase change. Likewise, in the conservative Allen–Cahn equation (phase-field method) a source term that accounts for the volume is introduced. The system of governing equations is solved using a projection-correction method and equations are discretized using a second-order finite difference approach. Thanks to the numerical discretization employed, a constant coefficient Poisson equation for pressure is obtained, which can be efficiently solved using FFT-based direct solvers. The proposed method is validated against several benchmarks: an interface undergoing vaporization at a constant rate, the Stefan problem, the adsorption problem, and the growth of a 2D vapor bubble. For all the benchmarks, the present method well matches with analytical and archival literature results for a wide range of vapor-to-liquid density ratios, from $$\rho _v/\rho _l = 1$$ ρ v / ρ l = 1 down to $$\rho _v/\rho _l \simeq 5 \times 10^{-4}$$ ρ v / ρ l ≃ 5 × 10 - 4 (where $$\rho _v$$ ρ v identifies the vapor density and $$\rho _l$$ ρ l the liquid density).

Approximate Solution of Fuzzy Sedimentary Ocean Basin Boundary Value Problem Using Variational Iteration Method

The problem that is considered in this paper is developed from the migration of the coastline over time in sedimentary ocean basins that may be considered as a moving boundary problem with a variable latent heat transfer equation as a governing equation if one of the boundaries is unknown, with boundary and initial conditions are given. This model discusses the shoreline movement within sedimentary basins. The main objective of the paper is to use fuzzy logic to formulate this problem when modelling sand particles migration in saltwater by utilizing fuzzy numbers whenever replacing some of the problem's parameters. In reality, the proposed model is more general than the nonfuzzy or crisp models. By employing the variational iteration method, the approximate numerical solution of the proposed problem has been found, and computer software written in Mathematica 11 have been used to obtain the results.

Development of an improved mass transfer model for condensation with dynamic feedback regulation

In the field of condensation research, accurate and efficient numerical simulation models are highly desired. One of the widely used phase change mass transfer models is known as the Lee model. However, this model has limitations due to uncertainties in the mass transfer intensity factor, which can lead to simulation failures and impede large-scale engineering applications. This study presents an improved mass transfer model which employs a tuning function to dynamically and accurately modify the mass transfer intensity factor in each two-phase cell via feedback adjustment. This approach enables immediate and customized adjustment of the factor for each grid within the limit of the tuning function, significantly reducing the errors caused by repeated artificial adjustments and related uncertainties. The adjustment capabilities of the hyperbolic tangent and softsign functions are evaluated. The softsign function is superior in both regulation efficiency and accuracy. The model's accuracy is verified by comparing with previous experiments and simulations, as well as by the one-dimensional Stefan problem. The proposed model achieves high prediction accuracy, reduces the risk of simulation dispersion, and significantly enhances computational efficiency with the iteration number reduced by nearly half under certain conditions compared to the Lee model. Furthermore, the new model shows robustness as the prediction accuracy is insensitive to variations in critical parameters. The improved model is expected to provide valuable assistance for future research and applications in flow condensation with outstanding computational accuracy, efficiency, and stability.

A singular two-phase Stefan problem and particles interacting through their hitting times

A singular two-phase Stefan problem and particles interacting through their hitting times

A front-fixing numerical method for a fluvio-deltaic sedimentation process with thespace fractional derivative and variable sediment flux

This paper centers around a space-fractional mathematical model for a fluvio-deltaic sedimentation process which involves a space-fractional derivative (Caputo derivative) and time dependent variable sediment flux to investigates the movement of shoreline in a sedimentary ocean basin. This model is a specific case of a basic shoreline model and analogous to a Stefan problem. The numerical solution to the problem is acquired by employing a front-fixing explicit finite difference method. The consistency, stability and convergence of the numerical scheme are theoretically analyzed. The effects of variable sediment flux on the movement of shoreline position and the height of sediments are also assessed for different cases.

Phase-field formulated meshless simulation of axisymmetric Rayleigh-Taylor instability problem

A formulation of the immiscible Newtonian two-liquid system with different densities and influenced by gravity is based on the Phase-Field Method (PFM) approach. The solution of the related governing coupled Navier-Stokes (NS) and Cahn-Hillard (CH) equations is structured by the meshless Diffuse Approximate Method (DAM) and Pressure Implicit with Splitting of Operators (PISO). The variable density is involved in all the terms. The related moving boundary problem is handled through single-domain, irregular, fixed node arrangement in Cartesian and axisymmetric coordinates. The meshless DAM uses weighted least squares approximation on overlapping subdomains, polynomial shape functions of second-order and Gaussian weights. This solution procedure has improved stability compared to Chorin's pressure-velocity coupling, previously used in meshless solutions of related problems. The Rayleigh-Taylor instability problem simulations are performed for an Atwood number of 0.76. The DAM parameters (shape parameter of the Gaussian weight function and number of nodes in a local subdomain) are the same as in the authors’ previous studies on single-phase flows. The simulations did not need any upwinding in the range of the simulations. The results compare well with the mesh-based finite volume method studies performed with the open-source code Gerris, Open-source Field Operation and Manipulation (OpenFOAM®) code and previously existing results.

Simulating film boiling with sharp interface and direct calculation of mass transfer to investigate the hydrodynamic and thermal transport phenomena near the interface

The current study presents a computational fluid dynamics (CFD) model for film boiling using two fluids, including liquid nitrogen. Ansys Fluent was customized using user-defined functions (UDFs) to accurately model thermal and fluid dynamic interactions. The UDFs accommodate mass transfer at the liquid-vapor interface, maintaining a sharp interface by accounting for the saturation temperature. Direct heat and mass transfer calculations are implemented rather than relying on empirical correlations. Mesh analysis was performed with grid sizes of 615, 410, 307, and 123 μm, with the 307 μm grid selected for its agreement with average Nusselt number values and accurate bubble shape and height. Verification using the Stefan problem showed a maximum error of 0.45 % for interface displacement and 0.1 % for temperature distribution. Validation against the Berenson correlation demonstrated good agreement in Nusselt number values. Qualitative validation of the developed method is performed using the shapes of the growing bubble. The results obtained show that the sharp interface is preserved throughout the simulation, with temperature contours showing significant variations during bubble growth and departure. Local heat transfer coefficient analysis identified peak values of 2170 W/m²-K at locations of minimal film thickness. Velocity vector analysis revealed velocities of 2 m/s in the vapor phase, influencing bubble shape during growth and departure. The study highlights the impact of thermal film thickness on heat transfer, with the highest Nusselt number of 5.69 observed at a film thickness of 0.5 mm. The results demonstrate the application of VOF sharp interface tracking and direct calculation of interfacial conditions utilizing customized Ansys Fluent in the modeling of film boiling as a way to understand the fundamental aspects of the fluid and thermal behavior.

Phase-field-based lattice Boltzmann method for containerless freezing.

In this paper we first propose a phase-field model for the containerless freezing problems, in which the volume expansion or shrinkage of the liquid caused by the density change during the phase change process is considered by adding a mass source term to the continuum equation. Then a phase-field-based lattice Boltzmann (LB) method is further developed to simulate solid-liquid phase change phenomena in multiphase systems. We test the developed LB method by the problem of conduction-induced freezing in a semi-infinite space, the three-phase Stefan problem, and the droplet solidification on a cold surface, and the numerical results are in agreement with the analytical and experimental solutions. In addition, the LB method is also used to study the rising bubbles with solidification. The results of the present method not only accurately capture the effect of bubbles on the solidification process, but also are in agreement with the previous work. Finally, a parametric study is carried out to examine the influences of some physical parameters on the sessile droplet solidification, and it is found that the time of droplet solidification increases with the increase of droplet volume and contact angle.