Convection In Solidification Research Articles

Enhanced numerical modeling of natural heat convective phase change for generalized non-Newtonian fluids at high Rayleigh number

Numerical modeling of convective heat transfer employing non-Newtonian fluids is a key issue and represent an interesting challenge in many engineering applications. For this reason, this work develops an enhanced numerical model to predict fast and accurately the convective phase change for generalized non-Newtonian fluids. The strongly thermally coupled model is solved by an improved pressure-correction algorithm (SIMPLERnP) and the Finite Volume Method with accuracy of the third order for the transient terms and second order for the convective boundary conditions and the velocity gradients that allows to calculate the apparent viscosity. The study examines the natural heat convective solidification of the ternary Al-27 %Cu-5.25 %Si alloy with the Herschel-Bulkley rheology at a high Rayleigh number (>107). In addressing the main problem, four crucial aspects are discussed: the validation of the numerical scheme with numerical and experimental results; the modeling of the non-linear temperature variation of the liquid phase change fraction; the non-Newtonian effects of the power index (0.1 ≤ n ≤ 1.9) and yield stress (0≤τ0≤5 Pa) on the heat transfer mechanisms; and finally, the time of computation with the high-order numerical scheme and the enhanced pressure-correction algorithm. The main finding is that FVM/SIMPLERnP provides a significant reduction in the time of computation compared to reliable pressure-correction schemes (SIMPLER and IDEAL), being a robust, accurate and efficient scheme to solve complex natural heat convective phase change problems involve generalized non-Newtonian fluids at high-Rayleigh numbers.

Solidification of a hollow sessile droplet under forced convection

This study presents a front-tracking-based numerical analysis of the forced convection solidification of a sessile droplet on a cooling surface. The droplet, a hollow (or compound) droplet with an encapsulated gas core, undergoes a liquid-to-solid phase change in its shell. This phase change starts from the surface. Meanwhile, the surrounding gas, which is characterized by its Reynolds number Re and temperature, moves toward the droplet parallel to the axis of symmetry. When the temperature of the forced flow is below the solidification value (i.e., cold-forced convection), increasing the strength of the forced flow shortens the solidification process. In contrast, increasing the Re number of a hot-forced convection system prolongs solidification. In other words, an increase in the forced flow temperature causes the entire liquid shell to solidify more slowly. Thinner shells require more time to solidify completely than thicker ones. The forced flow does not influence the formation of an apex at the top of the outer droplet surface. The aforementioned apex results from volume expansion. The effects of other parameters, for example, the capillary number and the morphologies of the droplet and cooling surface, are also determined.

Convective mitigation of dendrite growth

The growth of metallic dendrites during the electrodeposition and solidification of metal films is a formidable scientific and industrial problem. It is mostly known for hindering the use of high energy-density solid metal electrodes in rechargeable batteries. Over a century ago, it was experimentally shown that metallic dendrites may be mitigated under the action of mass advection in the liquid adjacent to a solid metal surface which grows out of a melt. Similar observations were reported for electrochemical deposition four decades ago and very recently. However, these insights proved inconclusive; dendrites appear to grow to different extents in many advection-based deposition systems, such as flow batteries, augmented finger batteries, and convective solidification. Here, we scrutinize the contribution of high Peclet metal ion convection in a dilute solution to the growth rate of metal dendrites, emphasizing convective effects near the metal surface. We employ an ideal model system for the isothermal flow of an ion solution along a duct where ions desorb from one solid surface of the duct, transport through the solution, and adsorb (undergo deposition) onto the other side. To account for the limit where metal-ion transport through the solution is the bottleneck for the rate of metal deposition (contributions to dendrite mitigation from near equilibrium surface phenomena are assumed small), we employ the perspective of kinetic stability to elucidate the contribution of ion convection to the mitigation of dendrite growth. We show that while the convection of metal ions appears to enhance dendrite growth --- an intuitive insight --- it also enhances the growth of the bulk solid metal on which the dendrites grow --- a well-known phenomenon. We demonstrate that for a fixed growth rate of the bulk solid metal, the rate of growth of dendrites is smaller in the presence of convective flow compared to the case where ion transport is solely by diffusion. The difference in the relative rate of growth in the presence and absence of convective flow may span orders of magnitude, which explains the absence of significant dendrites in corresponding experiments.

An approach to accelerate the convergence of SIMPLER algorithm for convection-diffusion problems of fluid flow with heat transfer and phase change

A new segregated pressure-based algorithm is developed to solve two-dimensional incompressible fluid flows with convective heat transfer and liquid to solid phase change by the finite volume method. The predictive-correction scheme is based on the SIMPLER algorithm with n added iterative cycles to increase the accuracy of pressure in satisfying the discretized continuity equation. The performance of SIMPLERnP is compared with SIMPLER and IDEAL in the solution of the laminar fluid flow in a lid driven cavity, natural heat convection inside a square cavity, convective heat transfer in the annular space between two concentric cylinders and conjugate natural heat convective solidification of a binary alloy that included the transient heat conduction in the thick mold walls. The results indicate that the proposed algorithm is highly stable, even when a value of 0.99 is used for the under-relaxation of the velocity components, and allows significant improvements in the reduction of the number of iterations and time of computation.

A lattice Boltzmann–cellular automaton study on dendrite growth with melt convection in solidification of ternary alloys**Project supported by the National Natural Science Foundation of China (Grant Nos. 51728601 and 51771118).

A lattice Boltzmann (LB)–cellular automaton (CA) model is employed to study the dendrite growth of Al-4.0 wt%Cu–1.0 wt%Mg alloy. The effects of melt convection, solute diffusion, interface curvature, and preferred growth orientation are incorporated into the coupled model by coupling the LB–CA model and the CALPHAD-based phase equilibrium solver, PanEngine. The dendrite growth with single and multiple initial seeds was numerically studied under the conditions of pure diffusion and melt convection. Effects of initial seed number and melt convection strength were characterized by new-defined solidification and concentration entropies. The numerical result shows that the growth behavior of dendrites, the final microstructure, and the micro-segregation are significantly influenced by melt convection during solidification of the ternary alloys. The proposed solidification and concentration entropies are useful characteristics bridging the solidification behavior and the microstructure evolution of alloys.

Axisymmetric forced convection solidification of a liquid drop on a cold plate

In this paper, we present a fully resolved numerical simulation of the solidification process of a liquid drop on a cold plate under axisymmetric forced convection. The numerical method used is an axisymmetric front-tracking method to represent the interface separating two phases. An interpolation technique is also used to impose the non-slip velocity boundary condition at the solid surface. Because of the forcing flow introduced from the top, the solidifying drop initially experiences an oscillation in the vertical direction, then the oscillation is suppressed as time progresses. In addition, the forcing flow slows down the solidification process. To investigate more in detail this solidification problem, we vary many dimensionless parameters including the Reynolds number (in the range of 25–800), Prandtl number Pr (in the range of 0.01–1.0), Stefan number St (in the range of 0.01–1.0), and Capillary number Ca (in the range of 3.16 × 10−5–0.316) to show their effects on the process. The results show that an increase in Re or Pr, or a decrease in St slows down the solidification process. However, the variation of Re or Pr has a minor effect on the solidified drop height and the conical angle at the solidified drop top. The most affecting parameter is Ca whose variation results in two modes: on-axis and off-axis solidification because of the forcing flow. We also investigate the effects of the growth angle at the triple point and the contact angle at the plate on the solidification process under axisymmetric forced convection.

A lattice Boltzmann study on dendritic growth of a binary alloy in the presence of melt convection

A multi-relaxation-time (MRT) lattice Boltzmann (LB) based model is utilized to simulate the dendritic growth with melt convection in solidification of alloys. It models melt convection by the MRT-LB equation and solute transport by a conservation equation with a pseudo-potential function. The D2Q9 lattice vectors are proposed to describe interface advancement in the liquid-solid transition. Effects of undercoolings, interface curvature and preferred growth orientation are incorporated into the model implicitly. After model validation, dendritic growth under several conditions of pure diffusion and melt convection was numerically investigated, and the solidification entropies were proposed to quantitatively characterize the solidification system. The result shows that the growth behavior, microstructure formation and solute segregation are significantly influenced by melt convection. The solidification entropies reflecting complexity of the solidification system are useful to characterize dendritic growth and solute segregation. This work offers a potential solution for studies of microstructure evolution in solidification of alloys.

Halo formation of Zn-Al alloys under conventional solidification and intensive convection solidification

A halo occurred usually as an envelope of one phase around a primary phase in many alloys after solidification. Its formation mechanism was investigated for hypoeutectic, eutectic and hypereutectic compositions of Zn-Al alloys under conventional solidification and under intensive convection solidification. It was found that the Zn-rich halos occurred in the surroundings of the Al-rich primary phase for the hypereutectic Zn-Al alloys at Al>5 wt% and no halos occurred for the hypoeutectic and eutectic Zn-Al alloys at Al≤5 wt% under conventional solidification. However, the Zn-rich halos were completely absent from the Al-rich phase because of the uniform temperature distribution and enhanced mass transport under intensive convection solidification. Once the intensive convection was interrupted during solidification for the solid-liquid co-existing melt, a halo was formed on the surface of the existing Al-rich phase created either during the primary solidification or the eutectic solidification. Therefore, it was concluded that the halo formation should be a growth-dominant phenomenon not a nucleation-dominant phenomenon. And, the interaction among the solid phases and the liquid phase was responsible for the halo formation, in which the difference in the elasticity modulus and the density of the different phases resulted in the variation of strain energy in the individual phase.

Dendrite remelting of petal-shaped iron-rich nanoparticles in Cu-2 wt% Fe alloy at the nanoscale

Cu-2wt% Fe copper alloy was fabricated under convection solidification condition. Morphologies of iron-rich nanoparticles formed in the Cu-2wt% Fe alloy melt were characterized. Results show that some spherical iron-rich nanoparticle converted into petal-shaped nanoparticle when growing up, and then was split into several smaller spherical or diamond nanoparticles. Dendrite remelting phenomena of iron-rich nanoparticle was observed during solidification process, which is caused by solute redistribution of iron-rich particle as well as the melt convection during solidification process. By taking advantage of dendrite remelting in as-cast Cu-2wt% Fe alloy, the size and quantity of precipitated phase can be potentially controlled.

Simulation of Dendritic Growth with Melt Convection in Solidification of Ternary Alloys**Supported by the National Natural Science Foundation of China under Grant Nos 51306037 and 51371051.

A cellular automaton-lattice Boltzmann coupled model is extended to study the dendritic growth with melt convection in the solidification of ternary alloys. With a CALPHAD-based phase equilibrium engine, the effects of melt convection, solutal diffusion, interface curvature and preferred growth orientation are incorporated into the coupled model. After model validation, the multi dendritic growth of the Al-4.0 wt%Cu-1.0 wt%Mg alloy is simulated under the conditions of pure diffusion and melt convection. The result shows that the dendritic growth behavior, the final microstructure and microsegregation are significantly influenced by melt convection in the solidification.

Modelling of dendritic growth under forced convection in solidification of Al–7Si alloy

An integrated microscale model is used to describe the dendritic growth morphology of Al–7Si alloy under forced convection, which combines the two-dimensional finite difference method (FDM) with the cellular automaton (CA) model. The FDM is used to simulate the induced fluid flow and solute transfer. The CA model is used to depict the dendritic morphology. Simulations are performed to investigate the influences of processing variables on morphological evolution under stirring. Calculated results reveal that at the intermediate undercooling, the growth morphology changes from dendritic to rosette as the rotation speed increases. The dendritic growth is promoted by increasing the undercooling. The rotation speed has a minor influence on microstructure formation for the cases of lower and higher undercoolings. A globulitic structure is formed as the nucleation density is increased. Changing the rotation speed is found to have a negligible influence on morphological evolution for the grain refined alloy.

Solidification of a binary alloy: Finite-element, single-domain simulation and new benchmark solutions

A finite-element simulation of binary alloy solidification based on a single-domain formulation is presented and tested. Resolution of phase change is first checked by comparison with the analytical results of Worster [M.G. Worster, Solidification of an alloy from a cooled boundary, J. Fluid Mech. 167 (1986) 481–501] for purely diffusive solidification. Fluid dynamical processes without phase change are then tested by comparison with previous numerical studies of thermal convection in a pure fluid [G. de Vahl Davis, Natural convection of air in a square cavity: a bench mark numerical solution, Int. J. Numer. Meth. Fluids 3 (1983) 249–264; D.A. Mayne, A.S. Usmani, M. Crapper, h-adaptive finite element solution of high Rayleigh number thermally driven cavity problem, Int. J. Numer. Meth. Heat Fluid Flow 10 (2000) 598–615; D.C. Wan, B.S.V. Patnaik, G.W. Wei, A new benchmark quality solution for the buoyancy driven cavity by discrete singular convolution, Numer. Heat Transf. 40 (2001) 199–228], in a porous medium with a constant porosity [G. Lauriat, V. Prasad, Non-darcian effects on natural convection in a vertical porous enclosure, Int. J. Heat Mass Transf. 32 (1989) 2135–2148; P. Nithiarasu, K.N. Seetharamu, T. Sundararajan, Natural convective heat transfer in an enclosure filled with fluid saturated variable porosity medium, Int. J. Heat Mass Transf. 40 (1997) 3955–3967] and in a mixed liquid–porous medium with a spatially variable porosity [P. Nithiarasu, K.N. Seetharamu, T. Sundararajan, Natural convective heat transfer in an enclosure filled with fluid saturated variable porosity medium, Int. J. Heat Mass Transf. 40 (1997) 3955–3967; N. Zabaras, D. Samanta, A stabilized volume-averaging finite element method for flow in porous media and binary alloy solidification processes, Int. J. Numer. Meth. Eng. 60 (2004) 1103–1138]. Finally, new benchmark solutions for simultaneous flow through both fluid and porous domains and for convective solidification processes are presented, based on the similarity solutions in corner-flow geometries recently obtained by Le Bars and Worster [M. Le Bars, M.G. Worster, Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification, J. Fluid Mech. (in press)]. Good agreement is found for all tests, hence validating our physical and numerical methods. More generally, the computations presented here could now be considered as standard and reliable analytical benchmarks for numerical simulations, specifically and independently testing the different processes underlying binary alloy solidification.

Visualization of convective solidification in a vertical layer of Eutectic Ga-In Melt

A vertical layer of an eutectic gallium indium mixture was subjected to a horizontal temperature gradient. The well stirred melt was first solidified and then remelted in a slow, quasi-steady fashion. Real-time visualization of the processing is provided by radioscopy. It was observed that gravitational segregation leads to 1) a break down of convection, and 2) an early solidification of β-indium, followed by an eutectic solidification of most of the remaining melt. The amount of observed concentrational segregation is higher than predicted by theory. The experimental findings have never been predicted in modeling efforts of similar situations including convection; possible reasons are discussed.

Microstructure refinement with forced convection in aluminium and superalloys

The cooling and average local solidification times were determined for slow solidifiation of Al-4.4 wt% Cu alloy under natural convection and under electromagnetically forced axisymmetric rotation during liquid cooling and solidification in graphite moulds. Cooling rates were measured within situ thermocouples. The conditions needed to stabilize the radial temperature gradient with rotation were established. The microstructure size decreased with increasing rotation, as did the local solidification times. The average grain and dendrite size without imposed rotation is coarser near the mould wall compared with the centre of the casting. This trend is reversed with imposed rotation. Rotation also led to a smaller spread of grain and dendrite size at any chosen height of the casting. These results are discussed in relation to existing theories, and several reasons for an improved heat transfer coefficient with rotation are presented. Forced convective solidification was then carried out for various shapes of integral investment cast Nimonic-90 alloy solidifying under modified conditions that prevented columnar grain formation. Similar results to those recorded for the aluminium case were obtained and are presented here. The major conclusion is that observations indicating a reduction of microstructure spacing during forced convection should also consider improved heat extraction at the mould-metal interface.

Mathematical simulation of thermogravitational convection in solidification of liquid steel

The thermal and hydrodynamic phenomena accompanying crystallization of liquid steel are analyzed numerically.

Thermal Evolution of Plutons: A Parameterized Approach

A conservation-of-energy equation has been derived for the spatially averaged magma temperature in a spherical pluton undergoing simultaneous crystallization and both internal (magma) and external (hydrothermal fluid) thermal convection. The model accounts for the dependence of magma viscosity on crystallinity, temperature, and bulk composition; it includes latent heat effects and the effects of different initial water concentrations in the melt and quantitatively considers the role that large volumes of circulatory hydrothermal fluids play in dissipating heat. The nonlinear ordinary differential equation describing these processes has been solved for a variety of magma compositions, initial termperatures, initial crystallinities, volume ratios of hydrothermal fluid to magma, and pluton sizes. These calculations are graphically summarized in plots of the average magma temperature versus time after emplacement. Solidification times, defined as the time necessary for magma to cool from the initial emplacement temperature to the solidus temperature vary as R(1,3), where R is the pluton radius. The solidification time of a pluton with a radius of 1 kilometer is 5 x 10(4) years; for an otherwise identical pluton with a radius of 10 kilometers, the solidification time is approximately 10(6) years. The water content has a marked effect on the solidification time. A granodiorite pluton with a radius of 5 kilometers and either 0.5 or 4 percent (by weight) water cools in 3.3 x 10(5) or 5 x 10(4) years, respectively. Convection solidification times are usually but not always less than conduction cooling times.