Mushy Layer Research Articles

Effective transport rates and transport-induced melting and solidification in mushy layers

Mushy layers are known to occur in magma chambers, sea-ice, and metal castings. They are often modeled as a porous layer in which a fluid and solid matrix exist in thermal and compositional equilibria. In nonreactive porous media, both advective and diffusive transport rates for heat and solute differ. In mushy layers, however, the temperature and composition of the fluid phase are constrained by the liquidus relationship giving rise to effective transport rates that are intermediate between those for heat and solute in passive porous layers. The transport of heat and solute, even if the invading fluid is itself in equilibrium, is also accompanied by a degree of solidification or melting due to the difference in the transport rates for these two quantities. In this paper, analytical expressions for the effective velocity and diffusivity in a mushy layer and for the degree of melting or solidification accompanying the passage of a front with a different temperature and composition are derived and compared with the predictions of a numerical model and found to be in very good agreement. Characteristic parameters for a few mushy systems are calculated and appropriate transport and melting/freezing regimes are indicated.

10.2478/s11814-009-0156-0

In directional solidification, compositional convection can be driven by an unstable density gradient in the melt. In this paper convective instabilities in liquid and mushy layers during solidification of a horizontal binary melt are analyzed by using the propagation theory. The self-similar stability equations are used to examine the boundary-mode and mushy-layer-mode of convection. The effects of velocity conditions at the liquid-mush interface on the onset of convection are discussed. The critical Darcy-Rayleigh number for the convection in the mushy layer decreases with increasing the temperature of a cooled boundary.

Maximal Potential Energy Transport: A Variational Principle for Solidification Problems

We analyze numerically the mechanisms controlling the spacing of chimneys--channels devoid of solid--in two-dimensional mushy layers formed by solidifying a binary alloy. Chimneys are the principal conduits through which buoyancy effects material transport out of the mushy layer and into the liquid from which it formed. Experiments show a coarsening of chimney spacing; we pursue the hypothesis that the spacing adjusts to optimize material transport and hence maximize the rate of removal of potential energy stored in the mushy layer. The optimal solute flux increases approximately linearly with the mush Rayleigh number. However, for spacings below a critical value, the chimneys collapse and solute fluxes cease, revealing a hysteresis between chimney convection and no flow. The results are consistent with a variational principle controlling the dynamics of this dissipative system.

Convective instability of directional crystallization in a forced flow: The role of brine channels in a mushy layer on nonlinear dynamics of binary systems

The linear analysis of morphological instability of the crystallization process with a mushy layer is carried out. The instability is caused by the influence of a forced oceanic flow leading to the formation of brine channels in the anisotropic and heterogeneous phase transition domain. Two instability criteria are found for the cases of non-turbulent and turbulent boundary conditions at the mushy layer–ocean interface. The theory under consideration shows that the instability domain will depend highly on the physical and operating parameters of the system.

Linear stability of solutal convection in a mushy layer subjected to gravity modulation

This study concentrates on instability problem due to solutal convection in a mushy layer of solidifying binary alloys, under the effect of gravity modulation. The gravitational field consists of two parts: a constant part and an externally imposed time-dependent periodic part. The time-dependent part of the gravity is expressed in Fourier series. Using Hill’s equation and the Floquet theory, the convective threshold is obtained and effect of gravity modulation is examined. We compare our result with some known results. Further with the help of different graphs we discuss the effect of various physical parameters on the stability of the mushy region.

Shear flow, phase change and matched asymptotic expansions: Pattern formation in mushy layers

In many scientific and engineering problems the solidification of an alloy leads to a highly convoluted crystalline matrix modeled as a thermodynamically controlled reactive porous medium called a mushy layer. We analyze the interaction of an external shear flow with a solidifying mushy layer through a corrugated mush–liquid interface. We find that the external flow can drive forced convective motions within the mushy layer resulting in the formation of a pattern of dissolution and solidification features transverse to the overall flow. Here we seek to lay bare the underlying processes through a systematic comparison of matched asymptotic expansions and numerical solutions. The success of our modeling effort draws substantially upon understanding gleaned from the fluid mechanics of boundary layers and the theory of multi-component solidification. The results have a broad range of implications in geophysics and materials science.

Investigating the effect of slag bath conditions on the existence of multiphase emulsion zones in PGM smelting furnaces using computation fluid dynamics

The presence of chrome in electric arc furnaces smelting platinum group metals (PGM’s) has a number of potentially negative consequences. In cases where the slag chrome content is above the saturation limit the existence of near-stagnant conditions near the slag/matte interface increases the risk of chromite spinels settling and consolidating into a “mushy layer”, a three-phase suspension of slag, chromite and matte. The hold-up of matte above the elevation of the slag/matte interface can lead to the attack of freeze linings and copper cooling elements potentially causing failures of the furnace lining and significant downtimes as well as major safety risks. This paper investigates the relationship between typical furnace operating parameters and the behavior of the slag bath with respect to the formation of the “mushy” layer at the slag/matte interface using computational fluid dynamics (CFD). The extent of the potential “mushy” layer is seen to increase with decreasing electrode immersion and furnace power. Electrode immersion is, however, a considerably stronger driver is this regard. The CFD modelling results have aided in selecting appropriate furnace electrode immersion/power combinations intended to minimize “mushy” layer formation.

Geochemical evolution of historical lavas from Askja Volcano, Iceland: Implications for mechanisms and timescales of magmatic differentiation

The mechanisms and the timescales of magmatic evolution were investigated for historical lavas from the Askja central volcano in the Dyngjufjöll volcanic massif, Iceland, using major and trace element and Sr, Nd, and Pb isotopic data, as well as 238U– 230Th– 226Ra systematics. Lavas from the volcano show marked compositional variation from magnesian basalt through ferrobasalt to rhyolite. In the magnesian basalt–ferrobasalt suite (5–10 wt% MgO), consisting of lavas older than 1875 A.D., 87Sr/ 86Sr increases systematically with increasing SiO 2 content; this suite is suggested to have evolved in a magma chamber located at ∼600 MPa through assimilation and fractional crystallization. On the other hand, in the ferrobasalt–rhyolite suite (1–5 wt% MgO), including 1875 A.D. basalt and rhyolite and 20th century lavas, 87Sr/ 86Sr tends to decrease slightly with increasing SiO 2 content. It is suggested that a relatively large magma chamber occupied by ferrobasalt magma was present at ∼100 MPa beneath the Öskjuvatn caldera, and that icelandite and rhyolite magmas were produced by extraction of the less and more evolved interstitial melt, respectively, from the mushy boundary layer along the margin of the ferrobasalt magma chamber, followed by accumulation of the melt to form separate magma bodies. Ferrobasalt and icelandite lavas in the ferrobasalt–rhyolite suite have a significant radioactive disequilibrium in terms of ( 226Ra/ 230Th), and its systematic decrease with magmatic evolution is considered to reflect aging, along with assimilation and fractional crystallization processes. Using a mass-balance model in which simultaneous fractional crystallization, crustal assimilation, and radioactive decay are taken into account, the timescale for the generation of icelandite magma from ferrobasalt was constrained to be <∼3 kyr which is largely dependent on Ra crystal–melt partition coefficients we used.

Melt segregation and matrix compaction: the mush continuity equation, compaction/segregation time, implications

SUMMARY Two-phase aggregates, a special case of which is partially molten systems, are commonly modelled as an interpenetrating flow of two viscous liquids and are therefore described in terms of the fluid mechanics. The governing equation set of fluid mechanics describing a two-phase aggregate is incomplete as it consists of nine equations (momentum and mass conservation equations for each phase, and an energy conservation equation) in ten unknowns (six velocity components, two pressures, temperature and melt fraction). In this paper, the general equation closing the governing equation set is derived. It explicitly expresses the condition of the coupling between the liquid percolation and porosity volume change due to which the medium continuity is maintained. Therefore, the equation may be referred to as the mush continuity equation. The character of the closed set solutions is demonstrated using 1-D isothermal models by examples of melt segregation inside a bounded low-melting-degree zone and sediment compaction at the bottom of a reservoir. The compaction/segregation style and time are controlled by the compaction/segregation parameter γc= (L/δc)2 and compaction/segregation length depending on the mushy layer thickness, L, matrix permeability, k, characteristic porosity, φ0, and matrix, η, and fluid, μ, viscosities. At low γc, which approximately corresponds to low-viscosity liquids (kimberlites, carbonatites), the compaction/segregation time scales as L−1, which effectively constrains the molten region thickness at the moment of the segregation, and appears to explain the small volume and clustering of low viscosity magma eruptions characteristic of kimberlites. At high γc the porosity evolves through a formation of a series of isolated waves, which could provide an explanation for the rhythmicity observed in large layered plutons.

Perturbation and stability analyses of nonlinear buoyant flow in mushy layers

We consider the problem of weakly nonlinear buoyant flow and its stability in horizontal dendrite layers, often referred to as mushy layers, during solidification of binary alloys. A mushy layer is a layer that is formed adjacent to the solidification front and is composed of both solid dendrites and melt. Understanding the convective flow driven by the buoyancy force within the mushy layer is known to be important for proper production of high-quality solidified alloy. In the present study, no restriction is imposed on the thickness of the mushy layer, and a number of simplifying assumptions made in previous related nonlinear analyses are lifted here in order to determine the results on the basis of a more realistic model. Using perturbation and stability analyses and some numerical approaches, we determine the steady stable solutions to the weakly nonlinear problem under a certain range of the parameter values where such analyses are valid. The relevant parameters for the present problem are the Rayleigh number representing the buoyancy, the Stefan number representing the latent heat of solidification, a concentration ratio parameter representing the composition of the solid dendrites in the mushy layer, and the far-field temperature parameter. We found, in particular, that depending on the range of parameter values, stable convective flow is possible in the form of subcritical or supercritical down-hexagons, with down-flow at the cells’ centers and up-flow at the cells’ boundaries, and subcritical or supercritical up-hexagons, with up-flow at the cells’ centers and down-flow at the cells’ boundaries, supercritical rectangles, supercritical squares and supercritical rolls.

On weakly nonlinear evolution of convective flow in a passive mushy layer

The problem of weakly nonlinear convective flow in a mushy layer, with a permeable mush–liquid interface and constant permeability, is studied under operating conditions for an experiment. A Landau type nonlinear evolution equation for the amplitude of the secondary solutions, which is based on the Landau theory and formulation for the Rayleigh, R , number close to its critical value, R c , is developed. Using numerical and analytical methods, the solutions to the evolution equation are calculated for both supercritical and subcritical conditions. We found, in particular, that for R < R c , the amplitude of the secondary solutions decays with time. For R > R c , the tendency for chimney formation in the mushy layer increases with time. In addition, in such a supercritical regime, the basic flow is linearly unstable and we see the presence of steady flow for large values of time. These results suggest a possible slow transition to turbulence in such a flow system.

Convective instabilities during the solidification of an ideal ternary alloy in a mushy layer

We consider a model for the solidification of an ideal ternary alloy in a mushy layer that incorporates the effects of thermal and solutal diffusion, convection and solidification. Our results reveal that although the temperature and solute fields are constrained to the liquidus surface of the phase diagram, the system still admits double-diffusive modes of instability. Additionally, modes of instability exist even in situations in which the thermal and solute fields are each individually stable from a static point of view. We identify these instabilities for a general model in which the base-state solution and its linear stability are computed numerically. We then highlight these instabilities in a much simpler model that admits an analytical solution.

Interactions between steady and oscillatory convection in mushy layers

We study nonlinear, two-dimensional convection in a mushy layer during solidification of a binary mixture. We consider a particular limit in which the onset of oscillatory convection just precedes the onset of steady overturning convection, at a prescribed aspect ratio of convection patterns. This asymptotic limit allows us to determine nonlinear solutions analytically. The results provide a complete description of the stability of and transitions between steady and oscillatory convection as functions of the Rayleigh number and the compositional ratio. Of particular focus are the effects of the basic-state asymmetries and non-uniformity in the permeability of the mushy layer, which give rise to abrupt (hysteretic) transitions in the system. We find that the transition between travelling and standing waves, as well as that between standing waves and steady convection, can be hysteretic. The relevance of our theoretical predictions to recent experiments on directionally solidifying mushy layers is also discussed.

On Nonlinear Evolution Approach for Convective Flow during Alloy Solidification

We consider the problem of nonlinear buoyant flow in a horizontal mushy layer during alloy solidification. We study the nonlinear evolution of such flow based on a recently developed realistic model for the mushy layer. The evolution approach is based on a Landau type equation for the amplitude of the secondary nonlinear solution, which is derived in this article. Using both analytical and computational methods, we calculate the solution to the evolution equation for both subcritical and supercritical regimes. We find, in particular, that for a passive mush, where the permeability is constant, and supercritical regime, the primary solution is linearly unstable to the secondary solution which becomes a steady stable solution for sufficiently large time, while the secondary solution decays to zero for the subcritical regime. On the other hand, for a realistic reactive mush, where the permeability is variable, the secondary flow can break down in a finite time for either supercritical or subcritical regime, which indicates existence of some kind of bursting behavior. These results are then compared to the corresponding ones based on the weakly nonlinear theory.

Solidification from a Cooled Boundary with a Mushy Layer Under Conditions of Nonturbulent and Turbulent Heat and Mass Transfer in the Ocean

Solidification from a Cooled Boundary with a Mushy Layer Under Conditions of Nonturbulent and Turbulent Heat and Mass Transfer in the Ocean

Solidification of binary aqueous solution cooled from above

AbstractFreezing and melting phenomena are important in many different fields, including crystal growth, casting, metallurgy, geophysics, and oceanography. Solidification of a multi‐component solution is the one often observed in nature. In order to investigate basic features of the freezing processes of binary systems, we conducted a series of laboratory experiments in a rectangular box cooled from above using aqueous NaNO3 solution. During the freezing, the solid phase always grows into many needle‐like crystals called the mushy layer. We measured the growth of the mushy layer thickness, the solid fraction, the temperature, and the concentration distributions. The average solid fraction is found to increase with time in the mushy layer. This causes a slow descent of the released solute in the mushy layer and its eventual fall into the liquid region below because of gravity. We propose a one‐dimensional model to explain the horizontally‐averaged mushy layer growth. In the model, the estimate of a heat flux at the mushy‐liquid interface due to natural convection is found essential for a correct prediction. The proposed theory predicts well the growth of the mushy‐layer and the average solid fraction, once the convective heat flux is properly given. © 2009 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/htj.20278

Nonlinear dynamics of directional solidification of ternary solutions with mushy layers

We present new analytic results relating to the nonstationary dynamics of Stefan-type problems for the unidirectional solidification of ternary solutions in the presence of two mushy layers. Explicit analytical solutions (thicknesses of mushy layers, growth rates of their boundaries, temperature and impurity concentration fields, solid fractions) of the nonlinear model under consideration are found. Model predictions are in good agreement with existing experimental data.

Solidification of a ternary melt from a cooled boundary, or nonlinear dynamics of mushy layers

We present a mathematical model and its analytical solution describing directional solidification of a ternary (three-component) system cooled from below. We focus on the solidification theory in the presence of two distinct mushy layers: (1) solidification along a liquidus surface is characterized by a primary mushy layer, and (2) solidification along a cotectic line is characterized by a secondary (cotectic) mushy layer. We consider the case when the phase transition temperatures in two mushy layers represent arbitrary functions of the compositions. We obtain an exact analytical solution of the nonlinear set of equations and boundary conditions in the case of a self-similar solidification scenario. Model predictions are in good agreement with existing experimental data.

On Perturbation and Marginal Stability Analysis of Magneto-Convection in Active Mushy Layer

This present study considers the problem of steady magneto-convection in a horizontal mushy layer with variable permeability and an impermeable mush–liquid interface during directional solidification of binary alloys. We model the flow by introducing a uniform magnetic field in the mushy layer which is considered as a porous medium where Darcy’s law holds and the permeability is a function of the local solid volume fraction. Basic-state solutions are obtained analytically using the no-flow condition. With the help of multiple shooting techniques, we obtain numerical solutions to the linear perturbation system for non-magnetic and magnetic cases. Numerical results are presented showing the effects of the magnetic field and the permeability of the layer. These results demonstrate that the application of an external magnetic field has stabilizing effects on the convection and can reduce the tendency for chimney formation in the mushy layer. In addition, variable permeability, which corresponds to an active mushy layer, indicates more stable and realizable flow system as compared to the case of constant permeability.

Stability of buoyancy-driven convection in directional solidification of a binary melt

In directional solidification, compositional convection can be driven by an unstable density gradient in the melt. In this paper convective instabilities in liquid and mushy layers during solidification of a horizontal binary melt are analyzed by using the propagation theory. The self-similar stability equations are used to examine the boundary-mode and mushy-layer-mode of convection. The effects of velocity conditions at the liquid-mush interface on the onset of convection are discussed. The critical Darcy-Rayleigh number for the convection in the mushy layer decreases with increasing the temperature of a cooled boundary.