Verification of a mathematical model for the solution of the Stefan problem using the mushy layer method

The use of solar energy has limitations due to its periodic availability: solar plants do not operate at night and are ineffective in dull weather. The solution of this problem involves the introduction of energy storage and duplication systems into the conversion loop. Among the energy storage systems, solid–liquid phase transition modules have significant energy, ecologic, and cost advantages. Physical processes in modules of this type are described by a system of non-stationary nonlinear partial differential equations with specific boundary conditions at the phase interface. The verification of a method for solving the Stefan problem for a heat-storage material is presented in this paper. The use of the mushy layer method made it possible to simplify the classical mathematical model of the Stefan problem by reducing it to a nonstationary heat conduction problem with an implicit heat source that takes into account the latent heat of transition. The phase transition is considered to occur in an intermediate zone determined by the solidus and liquidus temperatures rather than in in infinite region. To develop a Python code, use was made of an implicit computational scheme in which the solidus and liquidus temperatures remain constant and are determined in the course of numerical experiments. The physical model chosen for computer simulation and algorithm verification is the process of ice layer formation on a water surface at a constant ambient temperature. The numerical results obtained allow one to determine the temperature fields in the solid and the liquid phase and the position of the phase interface and calculate its advance speed. The algorithm developed was verified by analyzing the classical analytical solution of the Stefan problem for the one-dimensional case at a constant advance speed of the phase interface. The value of the verification coefficient was determined from a numerical solution of a nonlinear equation with the use of special built-in Python functions. Substituting the data for the physical model under consideration into the analytical solution and comparing them with the numerical simulation data obtained with the use of the mushy layer method shows that the results are in close agreement, thus demonstrating the correctness of the computer algorithm developed. These studies will allow one to adapt the Python code developed on the basis of the mushy layer method to the calculation of heat storage systems with a solid-liquid phase transition with account for the features of their geometry, the temperature level, and actual boundary conditions.

Mathematical modeling of bulk and directional crystallization with the moving phase transition layer

This paper is devoted to the mathematical modeling of a combined effect of directional and bulk crystallization in a phase transition layer with allowance for nucleation and evolution of newly born particles. We consider two models with and without fluctuations in crystal growth velocities, which are analytically solved using the saddle-point technique. The particle-size distribution function, solid-phase fraction in a supercooled two-phase layer, its thickness and permeability, solidification velocity, and desupercooling kinetics are defined. This solution enables us to characterize the mushy layer composition. We show that the region adjacent to the liquid phase is almost free of crystals and has a constant temperature gradient. Crystals undergo intense growth leading to fast mushy layer desupercooling in the middle of a two-phase region. The mushy region adjacent to the solid material is filled with the growing solid phase structures and is almost desupercooled.

Convection in a mushy layer along a vertical heated wall

Motivated by the mushy zones of sea ice, volcanoes and icy moons of the outer solar system, we perform a theoretical and numerical study of boundary-layer convection along a vertical heated wall in a bounded ideal mushy region. The mush is comprised of a porous and reactive binary alloy with a mixture of saline liquid in a solid matrix, and is studied in the near-eutectic approximation. Here, we demonstrate the existence of four regions and study their behaviour asymptotically. Starting from the bottom of the wall, the four regions are (i) an isotropic corner region; (ii) a buoyancy dominated vertical boundary layer; (iii) an isotropic connection region; and (iv) a horizontal boundary layer at the top boundary with strong gradients of pressure and buoyancy. Scalings from numerical simulations are consistent with the theoretical predictions. Close to the heated wall, the convection in the mushy layer is similar to a rising buoyant plume abruptly stopped at the top, leading to increased pressure and temperature in the upper region, whose impact is discussed as an efficient melting mechanism.

The Stabilization of Waste Funnel Glass of CRT by SiO2 Film Coating Technique

The funnel glass of the CRT monitor contains about 22–28% of lead oxide, of which lead is a highly toxic species and hazardous to the environment. This study proposes a process to form a protective layer of SiO2 film coating on the funnel glass to reduce the hazardous effect of lead leaching to the environment. The film coating benefits from the advantages of the sol–gel method. There are two key procedures of the stabilization technique, including the alkaline treatment and the formation of SiO2 coating from TEOS. The results show that the funnel glass powder treated with 10 M NaOH can produce a mushy layer on the surface. The mushy layer, which comprises OH− and water, can promote the formation of the SiO2 film layer on the surface of funnel glass powder. The conditions of the SiO2 film coating proposed in this study are: alkaline treatment by 10 M NaOH, the addition ratio of TEOS and funnel glass powder 2: 1, reaction temperature 40 °C, and reaction time 1.3 h. The EDS and ESCA results show that the Pb peak intensity on the surface of funnel glass decreases with the film coating. In the TCLP test, the leaching amount for Pb of the SiO2 film coated funnel glass powders is 0.7 mg/L, which is far lower than the standard in Taiwan EPA. Based on the experimental results, the formation mechanism of the SiO2 film layer on the surface of waste funnel glass powder is proposed. This study demonstrates that the SiO2 film coating is a potentially effective method to solve the problem of the waste funnel glass.

МОДЕЛЮВАННЯ ЕНЕРГОПЕРЕНОСУ В ФАЗОПЕРЕХІДНОМУ ТЕПЛОАКУМУЛЯТОРІ СОНЯЧНОЇ ТЕРМОДИНАМІЧНОЇ УСТАНОВКИ

Проведено числове дослідження процесів фазового переходу «тверде тіло – рідина» в тепловому акумуляторі сонячної термодинамічної установки. В основі математичної моделі та числового алгоритму покладено метод “Mushy Layer”, який відображає фізичну суть явища. Комп’ютерне моделювання задачі Стефана дозволило виявити особливості процесу фазового переходу, визначити розподіл температур в рідкій та твердий фазі, товщину твердої та рідкої фаз теплоакумулюючого матеріалу, швидкість руху границі розподілу фаз.

The effect of density changes on crystallization with a mushy layer

A nonlinear problem with two moving boundaries of the phase transition, which describes the process of directional crystallization in the presence of a quasi-equilibrium two-phase layer, is solved analytically for the steady-state process. The exact analytical solution in a two-phase layer is found in a parametric form (the solid phase fraction plays the role of this parameter) with allowance for possible changes in the density of the liquid phase accordingly to a linearized equation of state and arbitrary value of the solid fraction at the boundary between the two-phase and solid layers. Namely, the solute concentration, temperature, solid fraction in the mushy layer, liquid and solid phases, mushy layer thickness and its velocity are found analytically. The theory under consideration is in good agreement with experimental data. The obtained solutions have great potential applications in analysing similar processes with a two-phase layer met in materials science, geophysics, biophysics and medical physics, where the directional crystallization processes with a quasi-equilibrium mushy layer can occur. This article is part of the theme issue ‘Patterns in soft and biological matters’.

Three-dimensional convection, phase change, and solute transport in mushy sea ice

<p>Sea ice is a porous mushy layer composed of ice crystals and interstitial brine. The dense brine tends to sink through the ice, driving convection. Downwelling at the edge of convective cells leads to dissolution of the ice matrix and the development of narrow, entirely liquid brine channels. The channels provide an efficient pathway for drainage of the cold, saline brine into the underlying ocean. This brine rejection provides an important buoyancy forcing for the polar oceans, and causes variation of the internal structure and properties of sea ice on seasonal and shorter timescales. This process is inherently multiscale, with simulations requiring resolution from O(mm) brine-channel scales to O(m) mushy-layer dynamic scales.</p><p> </p><p>We present new, fully 3-dimensional numerical simulations of ice formation and convective brine rejection that model flow through a reactive porous ice matrix with evolving porosity. To accurately resolve the wide range of dynamical scales, our simulations exploit Adaptive Mesh Refinement using the Chombo framework. This allows us to integrate over several months of ice growth, providing insights into mushy-layer dynamics throughout the winter season. The convective desalination of sea ice promotes increased internal solidification, and we find that convective brine drainage is restricted to a narrow porous layer at the ice-ocean interface. This layer evolves as the ice grows thicker over time. Away from this interface, stagnant sea ice consists of a network of previously active brine channels that retain higher solute concentrations than the surrounding ice. We investigate the response of ice growth and brine drainage to varying atmospheric cooling conditions, and consider the potential implications for ice-ocean brine fluxes, nutrient transport, and sea ice ecology.</p><p><br><br></p>