Liquidus Front Research Articles

Heat transfer analysis for the solidification of a binary eutectic system under imposed movement of the material

The current article deals with a moving boundary problem describing solidification of a eutectic alloy in a semi-infinite medium. The process of solidification of a eutectic alloy is considered under imposed movement of material in the mushy zone in place of liquidus zone due to imposing an insulated boundary condition at liquidus front. It is assumed that solid fraction $$f_{\text {s}}$$ has a linear, quadratic and cubic relationship with distance within the mushy zone between the solidus and liquidus. An exact solution of the problem is obtained with the help of similarity transformation. A numerical example of the solidification of Al–Cu alloy is presented to demonstrate the application of the current analysis. Solidification of eutectic system is discussed in the absence of material movement and in the presence of material movement in each case of solid fraction distribution. Thus, the temperature profile and moving interfaces in each region are shown for different Peclet numbers Pe. In addition, heat extraction Q from the surface $$x=0$$ is shown with respect to the time for different Pe. The novelty of the current study is transition process becomes fast when material moves in the direction of freeze and hence time for complete freezing of the alloy reduces. Moreover, mushy zone becomes thinner when material moves in the direction of freeze. A comparative study and error analysis between the present work and Tien and Geiger (ASME J Heat Transf 89:230–233, 1967)[9] in linear case of solid fraction are presented in figure and tables. The application of the present analysis is useful for both eutectic and solid solution alloys.

A numerical simulation of columnar solidification: influence of inertia on channel segregation

We investigate the role of the inertia of the flow through the dendritic mushy zone in the numerical prediction of channel segregations during columnar solidification. The contribution of inertia is included in the momentum transport equation through the quadratic Forchheimer correction term. The study reveals a significant influence of the Forchheimer term in the vicinity of the liquidus front, i.e. at high liquid fractions. The natural convective flow field in this region is modified due to the additional inertial drag. This strongly influences the convective transport of solute and thereby incurs a modification of the dynamics of the advancement of the mushy zone. The most notable consequence is a significant decrease in the predicted channel segregation.

Alignment and Permeability of Al-7Si Alloy Directional Solidification with the Application of a Pulsed Magnetic Field

An experiment on the directional solidification of Al-7Si alloys under the influence of a pulsed magnetic field (PMF) is performed. Maximal peak value of PMF intensity up to 0.226 T between two iron cores was generated when a capacitor bank of 120 µF capacitance with initial voltage of 1000 V was triggered to a pair of solenoids. The PMF decreases the temperature gradient ahead of the liquidus front and increases the width of the mushy zone. The convection of symmetrical vortices seems to charge the intersectional growth of primary dendrite stems and the decrease of dendrite arms spacing under a PMF with higher magnetic intensities. Using Darcy's permeability law and Lehmann's model, the order of magnitude of the interdendritic liquid flow velocities with the application of PMF is evaluated. A strong damped effect of the mushy zone in the bulk flow of the melt is revealed.

Microstructural Development in a Ternary Al-Cu-Si Alloy during Transient Solidification

Macrosegregation and microporosity formation numerical models are dependent on microstructural parameters, such as primary and secondary arm spacings to provide the permeability coefficient in the mushy zone. As can be observed in the literature, growth models for ternary alloys for unsteady solidification are rarely found. In this paper, the primary (λ1) dendrite arm spacing was measured along the length of an Al-Cu-Si alloy casting and correlated with transient solidification thermal variables. A combined theoretical and experimental approach has been carried out to quantitatively determine such thermal variables, i.e., transient metal/mold heat transfer coefficient, liquidus isotherm velocity and cooling rate ahead the liquidus front.

Influence of Solidification Thermal Parameters on the Columnar-to-Equiaxed Transition of Aluminum-Zinc and Zinc-Aluminum Alloys

Understanding the interaction between the parameters involved in the columnar-to-equiaxed transition (CET) has gained considerable attention over the last two decades in the study of the structure of ingot castings. The present investigation was undertaken to investigate experimentally the directional solidification of Al-Zn and Zn-Al (ZA) alloys under different conditions of superheat and heat-transfer efficiencies at the metal/mold interface. The CET is observed; grain sizes are measured and the observations are related to the solidification thermal parameters: cooling rates, growth rates, thermal gradients, and recalescence determined from the temperature vs time curves. The temperature gradient in the melt, measured during the transition, is between –0.338 °C/mm and 0.167 °C/mm. In addition, there is an increase in the velocity of the liquidus front faster than the solidus front, which increases the size of the mushy zone. The size of the equiaxed grains increases with distance from the transition, an observation that was independent of alloy composition. The observations indicate that the transition is the result of a competition between coarse columnar dendrites and finer equiaxed dendrites. The results are compared with those previously obtained in lead-tin alloys.

Modelling of the thermosolutal convection and macrosegregation in the solidification of an Fe‐C binary alloy

PurposeThe motivation for this work is to establish a model that not only includes the main factors resulting in macrosegregation but also retains simplicity and consistency for the sake of potential application in casting practice.Design/methodology/approachA mathematical model for the numerical simulation of thermosolutal convection and macrosegregation in the solidification of multicomponent alloys is developed, in which the coupled macroscopic mass, momentum, energy and species conservation equations are solved. The conservation equations are discretized by using the control volume‐based finite difference method, in which an up‐wind scheme is adopted to deal with the convection term. The alternative direction implicit procedure and a line‐by‐line solver, based on the tri‐diagonal matrix algorithm, are employed to iteratively solve the algebraic equations. The velocity‐pressure coupling is handled by using the SIMPLE algorithm.FindingsBased on the present study, the liquid flow near the dendritic front is believed to play an important role in large‐scale transport of the solute species. The numerical or experimental results in the literatures on the formation of channel segregation, especially those about the location of the initial flow as well as the morphology of the liquidus front, are well supported by the present investigation.Research limitations/implicationsThe modelling is limited to dealing with the thermosolutal convection of two‐dimensional cases. More complicated phenomena (e.g. crystal movement) and 3D geometry should be considered in future research.Practical implicationsThe present model can be used to analyze the effects of process parameters on macrosegregation and, with further development, could be applied as a useful tool in casting practice.Originality/valueThe numerical simulation demonstrates the capability of the model to simulate the thermosolutal convection and macrosegregation in alloy solidification. It also shows that the present model has good application potential in the prediction and control of channel segregation.

Analysis of solidification parameters during solidification of lead and aluminum base alloys

Experiments were carried out in which the conditions of columnar to equiaxed transition (CET) in directional solidification of dendritic alloys are known, in alloy systems such as Pb–Sn, Al–Cu, Al–Si, Al–Mg and Al–Zn. These experiments allow determining that the transition does not occur in an abrupt form in the samples and it is presented when the gradient in the liquid ahead of the columnar dendrites reaches critical and minimum values, being negative in most of the cases. The transition occurs when the temperature gradients in the melt ahead of the columnar dendrites are in the range of −0.80 to 1.0 °C/cm for Pb–Sn, −11.41 to 2.80 °C/cm for Al–Cu, −4.20 to 0.67 °C/cm for Al–Si, −1.67 to 0.91 °C/cm for Al–Mg, −11.38 to 0.91 °C/cm for Al–Zn. Two interfaces are defined, assumed to be macroscopically flat, the liquidus and solidus interfaces. When the CET occurs, the speed of the liquidus front accelerates much faster than the speed of the solidus front, the values were 0.004 to 0.01, 0.02 to 0.48, 0.12 to 0.89, 0.10 to 0.18 and 0.09 to 0.18 cm/s, respectively. Also, the average supercooling of 0.63 to 2.75 °C for Pb–Sn, 0.59 to 1.15 °C for Al–Cu, 0.67 to 1.25 °C for Al–Si, 0.69 to 1.15 °C for Al–Mg and 0.85 to 1.40 °C for the Al–Zn alloys were measured, which provides the driving force to surmount the energy barrier required to create a viable solid–liquid interface. The results obtained in all alloys systems are compared and analyzed.

Grain refinement induced by electromagnetic stirring: A dendrite fragmentation criterion

The influence of electromagnetic stirring (EMS) on grain refinement has been studied for two copper-base alloys (Cu-1 wt pct Ni-1 wt pct Pb-0.2 wt pct P and Cu-4 wt pct Zn-4 wt pct Sn-4 wt pct Pb) solidified in a Bridgman furnace. Metallographic inspection of the specimens, temperature measurements during solidification, and numerical simulations performed with CALCOSOFT revealed that the efficiency of EMS is strongly dependent upon the penetration of the liquid in the mushy zone and therefore upon the position of the convection vortices with respect to the liquidus front. In particular, the low-concentration alloy could be grain refined only at high power and when the coil was moved close to the liquidus front. These results were analyzed on the basis of a dendrite fragmentation criterion similar to Flemings’ criterion for local remelting of the mushy zone. Considering that the component of the fluid flow velocity along the thermal gradient, \(u_{l,G} = \frac{{u_e \cdot \nabla T}}{{\left\| {\nabla T} \right\|}}\), must be larger than the casting speed, Vc, dendrite fragmentation occurs if $$C_R \approx \frac{1}{{V_c }}\frac{K}{{g_l \cdot \mu }}\frac{{B_0^2 }}{{\mu _0 d_{ind} }} > 1$$ at some depth within the mushy zone where dendrite arms are sufficiently developed, typically 8 λ2, where λ2 is the final secondary dendrite arm spacing, K is the permeability of the mushy zone, gl is the volume fraction of liquid, μ is the dynamic viscosity, B0 is the magnetic field, μ0 is the permeability of vacuum, and dind is the distance between the inductor and the liquidus front.

Directional and single-crystal solidification of Ni-base superalloys: Part I. The role of curved isotherms on grain selection

The development of crystallographic texture during directional solidification has been quantitatively analyzed in columnar castings of the Ni-base superalloys, CMSX4 and CM186LC, produced with a range of cooling rates and liquidus front curvatures. It is proposed that the more diffuse crystallographic texture developed in CMSX4 relative to CM186LC results from a combination of the differing local orientation stability condition and the alloys’ solidification characteristics. The implications of these additional factors on the evolution of the axial grain texture, the grain orientations produced in singlecrystal processing, and the stability of spurious grains in processing CMSX4 are discussed. An experimental method is presented to quantitatively analyze the grain selection process in the case of curved liquidus isotherms by retaining the stereology of the primary 〈001〉 dendrite growth direction and the local thermal gradient vector. This can account for the stability of spuriously nucleated edge grains in a single-crystal matrix.

Solidification parameters during the columnar-to-equiaxed transition in lead-tin alloys

The columnar-to-equiaxed transition (CET) was studied in lead-tin alloys, which were solidified directionally from a chill face. The main parameters analyzed include the temperature gradients, solidification velocities of the liquidus and solidus fronts, and grain size. The transition was observed to occur when the temperature gradient in the melt decreased to values between −0.8 °C/cm and 1 °C/cm. In addition, there is an increase in the velocity of the liquidus front faster than the solidus front, which increases the size of the mushy zone. The size of the equiaxed grains increases with distance from the transition, an observation that was independent of alloy composition. The comparison with available analytical models and the observations indicate that the transition is the result of a competition between coarse columnar dendrites and finer equiaxed dendrites.

The effect of mold rotation on solidification process of an Al-Cu alloy

The effect of mold rotation on the transport process and resultant macrosegregation pattern during solidification of an Al-Cu alloy contained in a vertical axisymmetric annular mold cooled from the inner wall is numerically investigated. The mold initially at rest starts to rotate at a prescribed angular velocity simultaneously with the beginning of cooling. Computed results for a representative case show that the mold rotation essentially suppresses the development of both thermal and solutal convections in the melt, creating distinct characteristics such as the liquidus front, flow pattern and temperature distribution from those for the stationary mold. Thermal convection which develops at the early stages of cooling is soon extinguished by the rotating flow induced during spin-up, and thus does not effectively remove the initial superheat from the melt. On the other hand, solutal convection, though it weakens considerably and is confined within the mushy zone, still predominates over the solute redistribution process. With increasing the angular velocity, the solute transport in the axial direction is enhanced, whereas that in the radial direction is reduced. The final macrosegregation formed in the mold rotating at moderate angular velocities appears to be favorable in comparison with the stationary casting, in that not only relatively homogenized composition is achieved, but also a severely positive-segregated channel is restrained.

Effect of rotation on fluid motion and channel formation during unidirectional solidification of a binary alloy

When off-eutectic binary alloys are cast by means of unidirectional solidification (UDS), perturbations in density and permeability along the liquidus front may result in the nucleation and subsequent growth of vertical flow channels in the two-phase (solid/liquid) mushy region. The channels subsequently become locations of severe compositional nonhomogeneity which are commonly termed freckles. In this study UDS of hypereuteclic NH 4Cl-H 2O has been numerically simulated to determine the effects of both steady and intermittent mold rotation. For slow, steady rotation up to 120 rpm, it was found that related inerlial effects were insufficient to significantly alter buoyancy induced mechanisms responsible for channel formation. However, for intermittent rotation corresponding to successive spin-up and spin-down of the mold, channel nuclealion was confined to the centerline and outer radius of the casting. The elimination of channels from the core of the casting is attributed to the impulsive change in angular frequency associated with spin-up and its effect on establishing an Ekman layer along the liquidus front. The front is washed by flow within the layer, thereby eliminating the perturbations responsible for channel nucleation.

Unidirectional solidification of a binary alloy and the effects of induced fluid motion

Solidification of a binary substance in a bottom-chilled geometry is numerically simulated under conditions for which large-scale convection results from a compositionally induced density inversion in the mushy region. Model results successfully predict characteristics of mushy region channel formation which have previously been observed but have heretofore eluded prediction. Channel growth is predicted to begin at the liquidus front and, due to localized freezing point depression (melting), to propagate downward toward the chill plate. The channels act as paths of least resistance for the transport of interdendritic fluid into the overlying bulk fluid. Fluid escaping from the mushy zone forms buoyant plumes at the mouth of each channel, and the plumes are sustained by penetration of bulk liquid across the liquidus front and subsequent advection toward the channels. Predictions for channel development and flow in the overlying melt agree qualitatively with previous experimental observations.

NUMERICAL ANALYSIS OF BINARY SOLID-LIQUID PHASE CHANGE WITH BUOYANCY AND SURFACE TENSION DRIVEN CONVECTION

The effects of thermo/diffusocapillary convection on the solidification of aqueous NH4Cl in a rectangular cavity have been simulated numerically using a newly developed continuum model. Diffusocaptttary convection is negligible relative to thermocapillary convection, and for a 20 × 20 mm cavity in a one-gravity environment, thermocapillary effects are most pronounced during the early stages of solidification, when flow conditions are characterized by three major cells. One cell, driven by solutal buoyancy forces, extends from the mushy region to the melt and separates top and bottom melt region cells driven primarily by surface tension and buoyancy forces, respectively. With increasing time, however, the top cell strengthens and eventually envelops the entire melt. In terms of the strength of the flow, the liquidus front morphology, and the amount of solid formed, final conditions differ only slightly from those predicted for pure thermal/solutal convection.

Solidification of a binary mixture in a square cavity with a free surface

Solidification of an NH4Cl-H2O solution in a square cavity with a free surface has been studied numerically and experimentally. Solidification is characterized by the coexistence of solid, mushy (solid plus liquid), and liquid regions, and flow within the liquid and mushy regions is driven by surface tension gradients, as well as by opposing thermal/solutal buoyancy forces. Theoretical and experimental results indicate that (i) surface tension forces spawn a cellular flow at the top of the cavity and enhance its development with increasing time, (ii) water enrichment of the top cell induces remelting of adjoining solid, and (iii) final steady-state conditions are characterized by a liquidus front whose thickness increases with increasing depth. However, predicted and observed results differ with respect to the chronological pattern of multicellular flow development and the time required to achieve steady-state conditions. Differences are attributed to selected model assumptions and to uncertainties concerning mushy region structure.

An Experimental Investigation of Binary Solidification in a Vertical Channel With Thermal and Solutal Mixed Convection

Experimental measurements and photographic observations are presented for the solidification of a binary, aqueous ammonium chloride solution in a vertical channel. Transient liquidus front progressions and temperature measurements are used to characterize the influences of flow rate, superheat, composition, and chill wall temperature on binary phase-change behavior. Experimental results are compared with previously reported model predictions and discrepancies are used to critically assess model assumptions and limitations.

Numerical simulation of binary solidification in a vertical channel with thermal and solutal mixed convection

Abstract A continuum model is used to investigate the solidification of an aqueous ammonium chloride solution in the presence of an imposed forced flow. Calculations are performed to determine the effect of flow rate, chill wall temperature, and entry liquid composition on momentum, energy and species transfer within both the multiphase and bulk liquid regions. Small mass flow rates and small solidification rates are shown to enhance the free flow of chilled and water-enriched interdendritic fluids across the permeable liquidus front and hence to enhance species redistribution.

Selective Freezing of a Dilute Salt Solution on a Cold Ice Surface

The growth of a solid–liquid, two-phase region during selective freezing of a dilute, eutectic-forming, salt solution over a subcooled ice slab is investigated experimentally and theoretically. The morphology of the two-phase region and the kinetics of the solid–liquid interface observed for a NaCl-H2O system are described photographically. The motion of the two-phase, liquidus front, recorded by a telescopic device that amplifies the local phenomena of the two-phase region, is presented along with the measured transient temperature distribution of the system. Based on the assumption that the solution element of the two-phase region is in local thermodynamic equilibrium with the solid phase, a similarity model is developed to predict the dependence of the freezing rate on various controlling parameters of the system. Transient heat conduction in the ice slab is also included in the model to study the effect of the wall. Comparison is made between the analytical and the experimental results and found to be good.

Two-Dimensional Flow of Liquid During the Late Stage of Solidification of Alloys

This is a mathematical analysis of the two-dimensional interdendritic liquid flow occurring during late stages of solidification, i.e., when the liquidus front reaches the center of the slab. The liquid flow considered is that caused by density change accompanying freezing. The liquid-solid region is considered a porous medium where the liquid permeability is a function of the solid fraction. Analytical solution for the two-dimensional velocity and pressure fields in solidifying Al-4.5 percent Cu alloy is obtained from a mathematical model which includes the concept of conservation of mass and Darcy’s law to correlate the pressure and velocity. Assuming that there is no gas-bubble formation and that the solidified metal in the mushy zone remains rigid, the liquid moving with a creeping velocity in the interdendritic regions is under tension (when no external pressure is applied). The magnitude of this tension increases with increasing depth of the solidifying ingot and is a strong function of the cooling rate. For example, a negative pressure of the order of 100 atm in the interdendritic liquid of solidifying Al-4.5 percent Cu alloy is estimated at ∼90 percent solidification. On the basis of the present analysis, an estimate can be made of pressures required to suppress blow-hole formation during the later stages of freezing arising from the solidification shrinkage.