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In this paper, the simulation of freezing within a cold storage unit is undertaken, featuring a container equipped with distinctive branch-shaped fins attached to the lower cold surface. The primary mode of freezing is conduction, causing the simplification of governing equations and resulting in two key equations. The Galerkin method is employed for numerical modeling, accompanied by an adaptive grid for enhanced accuracy. Unsteady terms are discretized using implicit formulation, and the resulting numerical procedure is rigorously validated against benchmarks, revealing commendable accuracy. To enhance cold storage efficiency, a dual approach is introduced, extending beyond conventional fin applications to include nanoparticles dispersed within the water. This approach significantly amplifies the system’s performance by enhancing the conduction mode of heat transfer. Two pivotal variables, the volume fraction ($\varphi$) of the nanofluid and its shape factor (m), are central to the investigation. Notably, the presence of nanoparticles results in a minimum freezing period of 8.08s, while the longest process takes 11.6s. Further exploration reveals that an increase in both m and $\varphi$ correlates with a notable decrease in the freezing period, reducing by 9.97% and 30.33%, respectively. This study advances understanding of cold storage dynamics and introduces innovative methods for optimizing efficiency. The strategic use of branch-shaped fins and the incorporation of nanoparticles represent crucial breakthroughs in heat transfer. The findings underscore the importance of considering these factors for optimal performance, making this study a pivotal contribution to cold storage technology.

The main aim of this paper is to examine the discharging process with insertion of wavy surface and changing shape of nanoparticles. Contours were presented in the form of contours and profiles of energy and temperatures. To get the acceptable accuracy, adaptive grid is employed and time steps for each iteration are variable. The outputs indicate that augmenting A and selection of platelet shape lead to a faster solidification. With augment of $A$, 14% reduction has been reported for ($m=3$). Such percentage augments with the rise of $m$ and 14.03% reduction were reported for ($m=5.7$). At $m=5.7$, augmenting $A$ from 0.1 to 0.3 makes the time to reduce from 44.16s to 37.96s. Lower level of energy was reported for platelet shapes, which means higher liquid fraction of domain. Temperature declines with augment of $A$ and $A=0.1$ prolongs process of about 14.03% in the existence of platelet shape.

We review the use of phase field methods in solidification modeling, describing their fundamental connection to the physics of phase transformations. The inherent challenges associated with simulating phase field models across multiple length and time scales are discussed, as well as how these challenges have been addressed in recent years. Specifically, we discuss new asymptotic analysis methods that enable phase field equations to emulate the sharp interface limit even in the case of quite diffuse phase-field interfaces, an aspect that greatly reduces computation times. We then review recent dynamic adaptive mesh refinement algorithms that have enabled a dramatic increase in the scale of microstructures that can be simulated using phase-field models, at significantly reduced simulation times. Combined with new methods of asymptotic analysis, the adaptive mesh approach provides a truly multi-scale capability for simulating solidification microstructures from nanometers up to centimeters. Finally, we present recent results on 2D and 3D dendritic growth and dendritic spacing selection, which have been made using phase-field models solved with adaptive mesh refinement.

A finite element method with a semi-implicit time update and an adaptive mesh refinement is used to numerically simulate characteristic growth morphologies in binary eutectic alloys for varying process conditions. The evolution equations are based on a recently developed phase-field model.¹ Microstructure formations in typical temperature-composition regions of the eutectic phase diagram are computed showing single cellular primary phase growth, melting of eutectics, eutectic and off-eutectic solidification. We consider 2D and 3D lamellar two-phase growths, analyze the angle conditions at the eutectic triple junctions for different surface entropy data and discuss the occurrence of wetting along the solid-solid interface.

Experimental and microstructure simulation approaches were taken to investigate the morphological evolutions of primary particles in an Al-20wt% pct Cu alloy under LSPSF (low superheat pouring with a shear field) rheocasting conditions. The results indicate that crystals are globular and present in non-entrapped eutectic, after 3s of solidification. The morphology of these crystals during the subsequent free growth is determined by both the number of free crystals and the cooling intensity of melt. Analyzed results from microstructure simulation and two stability models suggest that the primary globular particles formed in the earlier stage of solidification can attain growth stability up to a larger size scale.

The non-isothermal polycrystalline solidification of a binary alloy is simulated by employing a phase field model which takes into account the heat transition and the random crystallographic orientation. The stochastic nucleation is taken into account in the simulation through the Poisson seeding algorithm and a kinetic calculation for binary melts based on the classical nucleation theory. Different microstructures are obtained under various cooling conditions. It is found that the grain structure becomes finer with increasing the cooling rate, which agrees with experimental result.

The conditions of local Lorentz invariance (LLI) breakdown, obtained during neutron emission from a sonicated cylindrical bar of AISI 304 steel, were reproduced in a system made of a mole of mercury. After 3 min, a part of the liquid transformed into solid state material, in which isotopes were found with both higher and lower atomic mass with respect to the starting material. Changes in the atomic weight without production of gamma radiation and radionuclides are made possible by deformed space–time reactions.

The dependence of tensile properties on the length scale of the dendritic morphology of Al–Cu, Al–Ag and Al–Ag–Cu alloys is experimentally investigated. These alloys were directionally solidified (DS) under a wide range of cooling rates $(Ṫ)$, permitting extensive microstructural scales to be examined. Experimental growth laws are proposed relating the primary dendritic arm spacing, ${\lambda }_1$ to $Ṫ$ and tensile properties to ${\lambda }_1$. It is shown that the most significant effect of the scale of ${\lambda }_1$ on the tensile properties is that of the ternary alloy, which is attributed to the more homogeneous distribution of the eutectic mixture for smaller ${\lambda }_1$ and by the combined reinforcement roles of the intermetallics present in the ternary eutectic: Al₂Cu and nonequilibrium Ag₃Al.

On the basis of Xu’s interfacial wave theory, the stability of dendritic growth in a convective binary alloy melt with buoyancy effect is studied using the asymptotic method. The resulting asymptotic solution of equations reveals that the stability mechanism of dendritic growth in the binary alloy melt with buoyancy-driven convection is similar to that in a pure melt. Dendritic growth is stable above and unstable below a critical stability number ${𝜀}_{\ast }$, which is determined by the quantization condition. In particular, there is a critical morphological number in the binary alloy melt. When the morphological number is less than the critical morphological number, the tip growth velocity increases, the tip curvature radius and oscillation frequency decrease, and the interface becomes thinner and smooth. When the morphological number is larger than the critical morphological number, the tip growth velocity decreases, the tip curvature radius and oscillation frequency increase, and the interface becomes fatter and rough. The result demonstrates that in a microgravity environment, there is a critical initial concentration such that below it thermal diffusion dominates, the tip growth velocity increases, the tip curvature radius and oscillation frequency decrease, and the interface becomes thinner and smooth; above it, solute diffusion dominates, the tip growth velocity decreases, the tip curvature radius and oscillation frequency increase, and the interface becomes fatter and rough.

Development of numerical code for evaluating the solidification of water has been scrutinized in this work. The container has two circular and sinusoidal cold walls at bottom and top surfaces. Galerkin-based code has been employed to model this phenomenon. To elevate the conductivity of phase change material (PCM), alumina particles with nanosized were utilized with incorporating different shapes. The conductivity of nanoencapsulated phase change material (NEPCM) is a function of concentration and shapes of nanoparticles. The freezing process is mainly dominated by conduction and selecting curved shaped and adding nanoparticles can affect this mechanism. Verification test reveals the good accommodation and applying adaptive grids leads to higher accuracy. As shape coefficient increases, the period of process declines around 10.65% owing to stronger conduction. Also, mixing water with alumina nanopowders with blade shape causes decrement in needed time around 32.51%. Besides, outputs reveal that utilizing blade shape of powders has more effect on required time than that of cylindrical shape.

To discover the efficacy of loading CuO nanoparticle on freezing within a container with a narrow fin, a numerical procedure was implemented in this paper. The pure phase change material (PCM) is H₂O and various sizes of nanopowders have been mixed with this material. The ignoring of velocity terms in equations leads to a mathematical model involving nanoparticle enhanced PCM (NEPCM) properties and an associated source term. Testing the procedure with the prediction of the previously published data shows good implementation of the numerical method. The used grid in this study is viable to become finer in special regions and this option can increase the accuracy of the model. The maximum impacts of dp and $\varphi$ on freezing time are 20% and 41.28%, respectively, and both factors make the process faster. The freezing period changes from 500.12 s to 293.58 s involving the nanomaterial with a fraction of 0.04 and radius of 20 nm.

Numerical simulation was offered for scrutinizing the freezing of water within the complex container. The container has elliptic left adiabatic wall while the right wall is sinusoidal wall and maintained at cold temperature. The drawback of water has been removed by adding alumina nanoparticles. For this modeling, different ranges of volume and shape factor of nanoparticles have been scrutinized by incorporating FEM. The configuration of grid alters with change of time and verification test has been presented which proved good accuracy. As bigger shape factor has been selected, the time of process decline less than 4% for cylinder shape and this percentage augments around 78.22% for blade shape. As nanoparticle fraction increases, the required time declines around 26.84%. The impact of blade shape in view of adding nanoparticles is 25.74% greater than that of cylinder shape.

Alumina nanoparticles with various shapes have been loaded into water to accelerate the freezing phenomena within closed cavity. The mathematical model contains unsteady conduction equation in the existence of unsteady source terms for phase changing. In this model, three properties of NEPCM are involved and for calculating those parameters, single phase approximation has been utilized. By using FEM, outputs for impact of shape and volume of nanoparticles have been reported. Also, the method was verified by comparing the outputs with the previous data. Increasing the volume of nanoparticles to twice the range makes the needed time decrease by about 13.82%. When $\varphi =0.02$, replacing nanoparticles with $m=8.6$ instead of $m=4.8$ makes freezing time decrease by about 3.92%.

Numerical unsteady approach based on Galerkin method with the inclusion of adaptive mesh was executed in this paper to simulate the freezing of water within a container. The domain has two triangular and trapezoidal cold surfaces and two adiabatic walls. The governing equations show the domination of conduction and transient source term has been involved. Galerkin method for modeling was applied and good accommodation has been reported based on comparison with previous data. Adding nano-powders makes the amount of heat to be released with stronger conduction and freezing time decreases about 26.76% when $m=8.6$. Augmenting the fraction of powders can make the rate of process increase. For the greatest fraction of powders, an augmenting shape factor can enhance the speed of process about 6.95%.

This work includes the finite element modeling of solidification process within the container with curved top wall. The container has three insulated wall and one cold wall which is situated on top wall and it has sinusoidal configuration. Nanoparticles with low fraction were loaded inside the water to improve the thermal features. In deriving the mathematical model, two main assumptions were incorporated: (1) neglecting the impact of velocity; (2) considering homogeneous mixture for NEPCM. To improve the accuracy of numerical approach, the variable style of grid with time has been employed and also, implicit method was considered for time-dependent two terms. Increasing fraction of nano-powders has its maximum effect when the second level of dp has been selected in which required time reduces around 41.33%. Also, in existence of maximum size of particles, changing $\varphi$ causes reduction of time around 12.41%. With altering dp from 30 to 40nm, the freezing period reduces around 19.99% while augmenting up to 50nm can lead to reduction of time around 49.29%.

The morphological stability of lamellar eutectic growth with the anisotropic effect of surface tension is studied by means of the interfacial wave (IFW) theory developed by Xu in the 1990s. We solve the related linear eigenvalue problem for the case that the Peclet number is small and the segregation coefficient parameter is close to the unit. The stability criterion of lamellar eutectic growth with the anisotropic surface tension is obtained. The linear stability analysis reveals that the stability of lamellar eutectic growth depends on a stability critical number ${𝜀}_{\ast }$. Similar to the case of isotropic surface tension, the system involves two types of global instability mechanisms: the “exchange of stability” invoked by the non-oscillatory, unstable modes and the “global wave instabilities” invoked by four types of oscillatory unstable modes, namely antisymmetric–antisymmetric (AA-), symmetric–symmetric (SS-), antisymmetric–symmetric (AS-) and symmetric–antisymmetric (SA-) modes. The anisotropic surface tension, by decreasing the corresponding stability critical number ${𝜀}_{\ast }$, stabilizes the “exchange of stability” mechanism and “global wave instability” mechanism invoked by AA-, SA- and SS-modes. However, by increasing the corresponding stability critical number ${𝜀}_{\ast }$, the anisotropic surface tension destabilizes the “global wave instability” mechanism invoked by AS-mode.

When a rod is vertically withdrawn from a granular layer, oblique force chains can be developed by effective shearing. In this study, the force-chain structure in a rod-withdrawn granular layer was experimentally investigated using a photoelastic technique. The rod is vertically withdrawn from a two-dimensional granular layer consisting of bidisperse photoelastic disks. During the withdrawal, the development process of force chains is visualized by the photoelastic effect. By systematic analysis of photoelastic images, force chain structures newly developed by the rod withdrawing are identified and analyzed. In particular, the relation between the rod-withdrawing force $F_{\mathrm{\text{w}}}$, total force-chains force $F_{\mathrm{\text{t}}}$, and their average orientation $𝜃$ are discussed. We find that the oblique force chains are newly developed by withdrawing. The force-chain angle $𝜃$ is almost constant (approximately $20^{\circ }$ from the horizontal), and the total force $F_{\mathrm{\text{t}}}$ gradually increases by the withdrawal. In addition, $F_{\mathrm{\text{t}}}sin𝜃$ shows a clear correlation with $F_{\mathrm{\text{w}}}$.

Galerkin method for scrutinizing the treatment of system during solidification of water has been implemented in existence of alumina nanoparticle. Cylinder and blade shapes of nanoparticles have been dispersed in to water in considering concentration less than 0.04. With neglecting the gravity force impact during freezing process, there exist only energy equations which should involve phase change term as mathematical model. Such equation has been coupled with equation for fraction of solid phase and final equations have been solved numerically. The configuration of grid has been considered variable according to position of ice front. As amount of $m$ increases, the freezing period declines around 10.7% owing to greater conductivity of Nanoparticle Enhanced Phase Change Material (NEPCM). With dispersing nanoparticles, the required time for solidification declines around 47.7%.

The Sierpinski fractal is introduced to construct the porous metal foam. Based on this fractal description, an unsteady heat transfer model accompanied with solidification phase change in fractal porous metal foam embedded with phase change material (PCM) is developed and numerically analyzed. The heat transfer processes associated with solidification of PCM embedded in fractal structure is investigated and compared with that in single-pore structure. The results indicate that, for the solidification of phase change material in fractal porous metal foam, the PCM is dispersedly distributed in metal foam and the existence of porous metal matrix provides a fast heat flow channel both horizontally and vertically, which induces the enhancement of interstitial heat transfer between the solid matrix and PCM. The solidification performance of the PCM, which is represented by liquid fraction and solidification time, in fractal structure is superior to that in single-pore structure.

Inspired by the snowflake structure, an innovative Koch-fractal fin is proposed to optimize the fin geometry of the latent heat thermal energy storage (LHTES) units. A model of unsteady heat transfer accompanied with phase change is developed and numerically analyzed to investigate the effect of fin structure on the discharging process of a LHTES unit. The dynamic response of heat release, the dynamic temperature response and solidification front evolution of a LHTES unit with Koch-fractal fins are discussed and compared with the corresponding radial fins. Furthermore, a comprehensive evaluation of thermal performance of LHTES units is conducted in terms of the TES capacity, TES rate and solidification time. The results indicate that the heat release rate of a LHTES unit with Koch-fractal fins is faster than that with radial fins. Moreover, because the Koch-fractal fins have advantages of higher specific surface area, faster heat flow path from point to surface and smaller thermal resistance arising from the reasonable spatial layout, the evolution of solidification front is faster and the temperature distribution is more uniform. The results of quantitative evaluation show that although the TES capacity is identical, the TES rate of a LHTES unit with Koch-fractal fins is six times that with radial fins.