Stefan Problem Research Articles

Impact of Geometry on a Thermal-Energy Storage Finned Tube during the Discharging Process

This work focused on the modelling of latent heat thermal energy storage systems. The mathematical modelling of a melting and solidification process has time-dependent boundary conditions because the interface between solid and liquid phases is a moving boundary. The heat transfer analysis needs the interface position over time to predict the temperature inside the liquid and the solid regions. This work started by solving the classical two-phase (one-dimensional) Stefan problem through a Matlab implementation of the analytical model. The same physical problem was numerically simulated using ANSYS FLUENT, and the good match of analytical and numerical results validated the numerical model, which was used for a more interesting problem: comparing three different latent heat TES configurations during the discharging process to evaluate the most efficient in terms of maximum average discharging power. The three axial heat conduction structures changed only for the fin shape (rectangular, trapezoidal and fractal), keeping constant the volume fractions of steel, aluminium and PCM to perform a proper comparison. Results showed that the trapezoidal fin profile performs better than the rectangular one, and the fractal fin profile geometry was revealed as the best for faster thermal exchange when the solidifying frontier moves away from the steel ring. In conclusion, the average discharging power for the three configurations was evaluated for a time corresponding to a reference value (10%) of the liquid fraction: the rectangular fin profile provided 950.8 W, the trapezoidal fin profile 979.4 W and the fractal fin profile 1136.6 W, confirming its higher performance compared with the other two geometries.

Legendre wavelet residual approach for moving boundary problem with variable thermal physical properties

Abstract The main aim of the current article is to describe an uni-dimensional moving boundary problem with conduction and convection effect when thermal conductivity and specific heat varying linearly with temperature and time. The Mathematical model has nonlinearity due to presence of variable thermal conductivity and specific heat. A Legendre wavelet residual approach is introduced to get the solution of the problem with high accuracy. The surface heat flux is taken as an exponent function of time while latent heat is presented as an exponent function of position. Galerkin technique is used to obtain the numerical solution in case of constant physical properties while collocation technique is used for variable thermal physical properties. When it is considered that thermal physical properties are constant then obtained numerical solution was compared with exact solution and found in good acceptance. The effect of convection and variable thermal conductivity with time and temperature on the location of the moving layer thickness is analyzed. Further the effect of Peclet number and other physical parameters on the location of moving layer thickness are discussed in detail.

A novel hybrid method based cubic B-spline for one-dimensional Stefan problem with moving PCM, size-dependent thermal conductivity and periodic boundary condition

A novel hybrid method based cubic B-spline for one-dimensional Stefan problem with moving PCM, size-dependent thermal conductivity and periodic boundary condition

An efficient explicit immersed boundary-reconstructed lattice Boltzmann flux solver for isothermal fluid-structure interaction problems with large deformations and complex geometries

In this paper, a novel approach for isothermal fluid-structure interaction (FSI) problems is proposed that not only inherits the advantages of a reconstructed lattice Boltzmann flux solver (RLBFS) in solving the fluid field, but also retains the efficiency found in the explicit boundary condition-enforced immersed boundary method (EIB) for implementation of boundary conditions on the solid wall. Hence, the new approach (EIB-RLBFS) is highly suitable for complex FSI problems such as large deformations and complex geometries. Furthermore, the arbitrary Lagrangian-Eulerian (ALE) approach is integrated with EIB-RLBFS to solve moving boundary problems efficiently. The structural dynamics are solved by a finite difference method for moving objects and flexible structures. The overall framework of EIB-RLBFS is simple yet robust, where a fractional method is applied to split the FSI process into a predictor step and a corrector step. The RLBFS is applied in the predictor step to predict the intermediate flow field, while the EIB is invoked in the corrector step to impose the no-slip boundary conditions within the flow field, thereby, correcting the predicted flow field to satisfy the no-slip boundary conditions. The proposed approach (EIB-RLBFS) has been tested and evaluated extensively in terms of accuracy and efficiency on several benchmark cases such as, a 2D self-propelled unconstrained flapping/undulatory foil, a 2D filament, a 3D streamwise rotating sphere, and a 3D flapping flag. Results obtained using the proposed approach are in good agreement with previous studies, substantiating the capability and flexibility of EIB-RLBFS for solving FSI problems.

Spalling rate of concretes subject to combined leaching and external sulfate attack

A spalling rate model is established to estimate the damage of concretes subject to the combined action of leaching and external sulfate attack (ESA). Two sub-models are developed to consider the different damage patterns. The profiles of sulfate and hydroxide ions are obtained from the solution of a dual moving boundary problem (MBP). The supersaturation of ettringite crystal is evaluated from these profiles, the effective stress is calculated through poromechanics for pore crystallization, and the spalling depth is determined using the fracture criterion. The experimental validation confirms the proportionality between the spalling depth and the square root of exposure time, and the models give conservative results but on the correct magnitude order for spalling depth. The parametric analysis shows the foremost impact of aluminate content and sulfate diffusivity on damage rate. The main assumptions in models lead to conservative predictions, but yet the model results are meaningful for engineering use.

Immersed Interface and Diffuse-Domain Approach for Current-Potential Distributions and Electrodeposition Problems

Several numerical approaches are being developed to solve the moving boundary models which predict the morphological evolution of the electrode surface during electrodeposition. Despite these significant research efforts over a broad range of materials, scales, and purposes, the prediction of the shape evolution of the electrodes over time is still a challenging problem. One needs to solve these moving boundary problems, as the potential and current distributions within the cell get affected by the geometry.1-2 In this work, we will demonstrate a recently developed numerical framework based on the diffused domain approach using phase-field models for two classical example, Hull cell and copper deposition in trenches.3 The original model equation was carefully reformulated as a diffuse-domain/immerse domain approach for steady state current-potential distribution problems.4-5 Then the shape change problem is cast as a transient phase-field problem to predict the shape changes of the electrode during electrodeposition. A comparative analysis has also been performed for the efficiency, accuracy, speed, and convergence of the present solutions for the Hull cell with other available numerical and even analytical methods. It can be stated that the present phase-field model is robust and efficient for predicting the shape changes of the electrode over time without oscillations. Acknowledgments This research was supported by the Assistant Secretary for Energy Efficiency and Renewable Energy, Office of Vehicle Technologies of the US Department of Energy (DoE) through the Advanced Battery Materials Research Program (Battery500 Consortium) References V. R. Subramanian, and R. E. White, J. Electrochem. Soc, 149(10) C498 (2002).R. Alkire, T. Bergh, and R. L. Sani, J. Electrochem. Soc., 125, 1981 (1978).T. Jang, L. Mishra, S. A. Roberts, A. Subramaniam, M. Uppaluri, M. P. Gururajan, J. Zhang, and V. R. Subramanian, “BattPhase – A convergent, non-oscillatory, efficient algorithm and code for predicting shape changes in lithium metal batteries using phase-field models – 1. Secondary Current Distribution,” ECSarXiv (2022), doi:10.1149/osf.io/k2vu6C. S. Peskin, Acta Numer., 11, 479 (2003).X. Li, J. Lowengrub, A. Ratz and A. Voigt, Commun. Math. Sci., 7, 81 (2009).

A Stefan problem with nonlinear thermal conductivity

The solution is obtained and validated by an existence and uniqueness theorem for the following nonlinear boundary value problem which was proposed in 1974 by Cho and Sunderland to represent a Stefan problem with a nonlinear temperature‐dependent thermal conductivity on the semi‐infinite line . The modified error function of two parameters is introduced to represent the solution of the problem above, and some properties of the function are established. This generalizes the results obtained in earlier studies.

Inverse two-phase nonlinear Stefan and Cauchy-Stefan problems: A phase-wise approach

We develop a novel phase-wise sequential numerical approach based on the method of fundamental solutions (MFS) for inverse two-phase nonlinear Stefan and Cauchy-Stefan problems in one dimension (1D). By treating each phase independently, the inverse two-phase nonlinear Stefan problem splits into two single-phase inverse problems: an inverse nonlinear boundary identification problem and an inverse linear one-phase Stefan problem. Along with the reconstruction of boundary data, the simultaneous reconstruction of the boundary and initial data is also considered. Numerical investigations show the robustness and efficiency of the proposed method in reconstructing the data.

Remarks on the Control of Two-Phase Stefan Free-Boundary Problems

This paper concerns the null controllability of the two-phase 1D Stefan problem with distributed controls. This is a free-boundary problem that models solidification or melting processes. In each phase, a parabolic equation, completed with initial and boundary conditions, must be satisfied; the phases are separated by a phase change interface, where an additional free-boundary condition is imposed (the so-called Stefan condition). We assume that two localized sources of heating/cooling controls act on the system (one in each phase). We prove the following local null controllability result: the temperatures can be steered to zero and, simultaneously, the interface can be steered to a prescribed location provided the initial data and the interface position are sufficiently close to the targets. The ingredients of the proofs are a compactness-uniqueness argument (which gives appropriate observability estimates adapted to constraints) and a fixed-point formulation and resolution of the controllability problem (which gives the result for the nonlinear system). We also prove a negative result corresponding to the case where only one control acts on the system and the interface does not collapse to the boundary.

A One-Phase Space-Fractional Stefan Problem With No Liquid Initial Domain

Taking into account the recent works \cite{RoTaVe:2020} and \cite{Rys:2020}, we consider a phase-change problem for a one dimensional material with a non-local flux, expressed in terms of the Caputo derivative, which derives in a space-fractional Stefan problem. We prove existence of a unique solution to a phase-change problem with the fractional Neumann boundary condition at the fixed face $x=0$, where the domain, at the initial time, consists of liquid and solid. Then we use this result to prove the existence of a limit solution to an analogous problem with solid initial domain, when it is not possible to transform the domain into a cylinder.

Two-phase inverse Stefan problems solved by heat polynomials method

In this paper, two-phase inverse Stefan problem are investigated using the method of heat polynomials. The approximate solution of the problem is presented in the form of linear combination of heat polynomials. We present numerical results illustrating an application of the heat polynomials method for typical benchmark test examples. Due to ill-posedness of the problem, the regularization will be taken into account. Numerical results are discussed and compared with results obtained by another method.

A numerical solution of a non-classical Stefan problem with space-dependent thermal conductivity, variable latent heat and Robin boundary condition

A numerical solution of a non-classical Stefan problem with space-dependent thermal conductivity, variable latent heat and Robin boundary condition

Quasi-equilibrium approximation of the solidification process for micro phase change materials taking into account curvature and surface tension

Phase change materials (PCMs) have superior energy storage properties, which can be used to solve the problem of energy absorption and release at various times, intensities and locations. Energy storage using PCMs is a promising way to make the rational use of energy, which could help improving our environment. Owing to their significant applications, we investigate the solidification process of microparticles, which is also known as the Stefan problem. Due to the tiny size of microparticles, quasi-equilibrium approximation of such Stefan’s problem is considered, where the effect of the curvature and the surface tension are incorporated. Here, we determine a simple but yet effective iterative model for the moving boundaries of micro PCMs (MPCMs) by assuming the homogeneous temperature distribution in molten regimes in order to avoid solving more complicated diffusion equations. We purposely investigate the solidification process of spherical and ellipsoidal MPCMs and find that both the curvature and the surface tension promote the solidification process. Due to these outcomes, peanut-shaped MPCMs are proposed to demonstrate the usage of both effects for fabricating smart MPCMs. Finally, the validity of the model is confirmed by a Computational Fluid Dynamics (CFD) method.

A sharp numerical method for the simulation of Stefan problems with convective effects

We present a numerical method for the solution of interfacial growth governed by the Stefan model coupled with incompressible fluid flow. An algorithm is presented which takes special care to enforce sharp interfacial conditions on the temperature, the flow velocity and pressure, and the interfacial velocity. The approach utilizes level-set methods for sharp and implicit interface tracking, hybrid finite-difference/finite-volume discretizations on adaptive quadtree grids, and a pressure-free projection method for the solution of the incompressible Navier-Stokes equations. The method is first verified with numerical convergence tests using a synthetic solution. Then, computational studies of ice formation on a cylinder in crossflow are performed and provide good quantitative agreement with existing experimental results, reproducing qualitative phenomena that have been observed in past experiments. Finally, we investigate the role of varying Reynolds and Stefan numbers on the emerging interface morphologies and provide new insights around the time evolution of local and average heat transfer at the interface.

Accelerating phase-change heat conduction simulations on GPUs

Numerical simulation of the phase-change heat conduction process plays an important role in numerous engineering applications such as metal and alloy production and casting. Because of the increasing demands on the efficiency of the computational models utilized in advanced process control and optimization, high-performance computational models are becoming a necessary building block of practical applications. Thus, in this paper, we focus on accelerating the phase-change heat conduction simulations on GPUs. In addition to taking the advantage of thousands of GPU cores, the current paper presents a set of optimization methods to improve the resource usage of GPUs. The proposed methods include thread rescheduling, global function design, and a data caching mechanism, which helps establish an appropriate mapping from the numerical algorithm to the GPU hardware feature, thus remarkably enhancing the solving efficiency. We have validated the accuracy of the accelerated model with the Stefan problem and evaluated its performance with a three-dimensional steel solidification problem. A performance boost has been demonstrated, where the optimized GPU solver achieves a 333.6x speed-up compared to the non-parallel CPU solver. The performance characteristic of the accelerated solver makes it a powerful and promising tool for practical industrial usage.

Unstructured, Pressure-Based Algorithm Using Implicit Runge–Kutta for Moving Boundary Problems

An unstructured, pressure-based algorithm, IRK-SIMPLER, is developed for simulating two-dimensional incompressible flows over moving bodies. The Navier–Stokes equations in the arbitrary Lagrangian Eulerian form are discretized using collocated, median-dual control volumes with a flux correction method. The semidiscrete form is integrated in time using a diagonally implicit Runge–Kutta method. The unstructured discretization of the spatial computational domain is composed of a circular body-fitted grid, a rectangular patch grid, and a background grid. The motion of the bodies is decomposed into a series of rotations and translations. Rotation is achieved by rotating the body-fitted grid along with the body and flipping the edges of the appropriate cells. Translation is achieved by morphing the patch grid as the body moves. For large translational displacements, the rectangular patch grid is moved with the body and rematched with the background grid to obtain a new conforming grid. The algorithm is validated for multiple incompressible flow cases.

Stress-strain state of coal wells Karaganda basin

The article considers the problem of determining the thermoelastic stresses that arise when drilling a coal seam with a depth of at least 800 m. The stress-strain state is a term that has penetrated from the mechanics of a deformable solid body into various areas, both natural and technological. Usually, three types of stress-strain state are distinguished: the first type of elastic work - up to 35% of the breaking load; the second type of elastic-plastic work - up to 75% of the breaking load, while cracks appear and their number increases with increasing load; the third type of destruction - cracks appear like an avalanche and every detail is destroyed. For the first time, we succeeded in solving analytically the Stefan problem for a cylinder of finite dimensions, with a moving interface, by selecting an integral transformation. A similar solution for a well in coal seams (as well as for other applications) gives an analytical solution to the deformation-wave process in a rock mass, discovered experimentally in the late 70s of the last century (scientific discovery of 1985, priority of 1978). Our experimental and theoretical results fit into the model of macroscopic localization of plastic flow. This model shows that the localization of plastic flow in solids (and in a coal seam) has a pronounced wave character. At the same time, at the stages of easy slip, linear and parabolic strain hardening, as well as at the stage of preliminary destruction, the observed patterns of localization are different types of wave processes.

A Fundamental Moving Boundary Problem of 1D Commingled Preferential Darcian Flow and Non-Darcian Flow Through Dual-Layered Porous Media

In consideration of vertical formation heterogeneity, a basic nonlinear model of 1D commingled preferential Darcian flow and non-Darcian flow with the threshold pressure gradient (TPG) in a dual-layered formation is presented. Non-Darcian flow in consideration of the TPG happens in the low-permeability tight layer, and the Darcian kinematic equation holds in the other high-permeability layer. The similarity transformation method is applied to analytically solve the model. Moreover, the existence and uniqueness of the analytical solution are proved strictly. Through analytical solution results, some significant conclusions are obtained. The existence of the TPG in the low-permeability tight layer can intensify the preferential Darcian flow in the high-permeability layer, and the intensity of the preferential Darcian flow is very sensitive to the dimensionless layer thickness ratio. The effect of the layer permeability ratio and layer elastic storage ratio on the production sub-rate is more sensitive than that of the layer thickness ratio. In addition, it is strictly demonstrated that moving boundary conditions caused by the TPG should be incorporated into the model. When the moving boundary is neglected, the preferential Darcian flow in the high-permeability layer will be exaggerated. Eventually, solid theoretical foundations are provided here, which are very significant for solving non-Darcian seepage flow problems in engineering by numerical simulation validation and physical experiment design. Furthermore, they are very helpful for better understanding the preferential flow behavior through the high-permeability paths (such as fractures) in the water flooding development of heterogeneous low-permeability reservoirs; then, the efficient profile control technology can be further developed to improve oil recovery.

Arbitrary Lagrangian-Eulerian-type conserved discrete unified gas kinetic scheme for the simulations of transonic continuum and rarefied gas flows with moving boundaries

In this paper, an arbitrary Lagrangian-Eulerian (ALE) framework is incorporated into the conserved discrete unified gas-kinetic scheme (CDUGKS) to solve the transonic continuum and rarefied gas flows with moving boundary. This is a continuation of our earlier work [Y. Wang et al., Phys. Rev. E, 100(6), 063310 (2019)]. Compared to the original low-speed ALE-DUGKS, in which only the governing equation of the distribution function is solved, the mesh motion velocity is introduced in the proposed ALE-CDUGKS for updating both the distribution function and the conservative flow variables. For a flow in the continuum regime, the potential energy double-distribution-functions framework and the circle equilibrium distribution function model are incorporated for inviscid and viscous flows. In the rarefied flow regime, the technique of unstructured velocity-space mesh is introduced to decrease the total number of discrete particle-velocity points and reduce the computational load. In addition, a loosely-coupled algorithm for simulating the fluid-structure interaction problem (airfoil flutter) is also presented. As a result, under this unified framework based on the distribution function, the numerical simulations have relatively high computational efficiency for flows in both continuum and rarefied regimes. A series of flows around a stationary or moving airfoil in the continuum regime is simulated, and a plunging airfoil in rarefied gas flow is also studied. The consistent and good results obtained from the above test cases demonstrate the capability of the proposed ALE-CDUGKS for solving the compressible moving boundary problems with the rarefied gas effect.

A multi-resolution particle method with high order accuracy for solid-liquid phase change represented by sharp moving interface

This paper develops a highly accurate, multi-resolution particle method to simulate solid-liquid phase change coupled with the thermal flow. Instead of including the latent heat in the governing equation, the heat equations for solid and liquid phases are solved separately. A sharp interface model is proposed to represent the solid-liquid interface explicitly. The sharp interface, represented by discrete nodes, provides the Neumann boundary condition for pressure and the Dirichlet boundary condition for velocity/temperature, respectively. Based on temperature gradients in the solid and liquid phases, the positions of these interface nodes are updated every time step. The Eulerian-based formulation, rather than the conventional Lagrangian-based one, is utilized to minimize time step-dependent error. Up to 4th order spatial discretization scheme is adopted based on the Least Square Moving Particle Semi-implicit (LSMPS) scheme. Moreover, a geometry-based multi-resolution scheme is introduced to dynamically refine the spatial resolution near the interface for saving computational cost. The 1-D Stefan problem is firstly simulated to verify the accuracy of the proposed sharp interface model. Then, the consistency of the multi-resolution scheme is investigated by a convergence study of the Taylor-Green vortex problem. After that, numerical simulations of natural convection in a cavity are performed with different spatial resolutions and high order schemes. Resulted computational costs are compared and discussed. Finally, the problems of melting by natural convection with different Rayleigh numbers are investigated. The results achieved so far indicate that the multi-resolution and high order schemes have great potential to save computational cost.