The metallurgy of zinc-coated steel

Validation of corrosion rates measured by the Tafel extrapolation method

The major methods of mathematical modelling of solidification and melting problems are reviewed in this paper. Different analytical methods, nowadays still used as standard references to validate numerical models, are presented. Various mathematical formulations to numerically solve solidification and melting problems are categorized. Relative merits and disadvantages of each formulation are analysed. Recent advances in modelling solidification and melting problems associated with convective motion of liquid phase are discussed. Based on this comprehensive survey, basic guidelines are outlined to choose a correct mathematical formulation for solving solidification and melting problems.

https://iopscience.iop.org/article/10.1088/0965-0393/4/4/004/pdf

Mathematical modelling of solidification and melting: a review

A multiphase/multiscale model is used to predict the columnar-to-equiaxed transition (CET) during solidification of binary alloys. The model consists of averaged energy and species conservation equations, coupled with nucleation and growth laws for dendritic structures. A new mechanism for the CET is proposed based on solutal interactions between the equiaxed grains and the advancing columnar front—as opposed to the commonly used mechanical blocking criterion. The resulting differences in the CET prediction are demonstrated for cases where a steady state can be assumed, and a revised isotherm velocity (VT) vs temperature gradient (G) map for the CET is presented. The model is validated by predicting the CET in previously performed unsteady, unidirectional solidification experiments involving Al-Si alloys of three different compositions. Good agreement is obtained between measured and predicted cooling curves. A parametric study is performed to investigate the dependence of the CET position on the nucleation undercooling and the density of nuclei in the equiaxed zone. Nucleation undercoolings are determined that provide the best agreement between measured and calculated CET positions. It is found that for all three alloy compositions, the nucleation undercoolings are very close to the maximum columnar dendrite tip undercoolings, indicating that the origin of the equiaxed grains may not be heterogeneous nucleation, but rather a breakdown or fragmentation of the columnar dendrites.

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A solutal interaction mechanism for the columnar-to-equiaxed transition in alloy solidification

A solute diffusion model, aimed at predicting microstructure formation in metal castings, is proposed for dendritic solidification of alloys. The model accounts for the different length scales existing in a dendritic structure. This is accomplished by utilizing a multiphase approach, in which not only the various physical phases but also phases associated with different length scales are considered separately. The macroscopic conservation equations are derived for each phase using the volume averaging technique, with constitutive relations developed for the interfacial transfer terms. It is shown that the multiphase model can rigorously incorporate the growth of dendrite tips and coarsening of dendrite arms. In addition, the distinction of different length scales enables the inclusion of realistic descriptions of the dendrite topology and relations to key metallurgical parameters. Another novel aspect of the model is that a single set of conservation equations for solute diffusion is developed for both equiaxed and columnar dendritic solidification. Finally, illustrative calculations for equiaxed, columnar, and mixed columnar-equiaxed solidification are carried out to provide quantitative comparisons with previous studies, and a variety of fundamental phenomena such as recalescence, dendrite tip undercooling, and columnar-to-equiaxed transition (CET) are predicted.

/pdf/a-multiphase-solute-diffusion-model-for-dendritic-alloy-16m40h88y8.pdf

A Multiphase Solute Diffusion Model for Dendritic Alloy Solidification

A columnar-to-equiaxed transition is observed in Al 3 pct Cu solidified directionally from a chill face. The transition occurs when the temperature gradient in the melt ahead of the columnar dendrites decreases to 0.6 ‡C/cm at dendrite growth rates of about 5 x 10-3 cm/s. Increasing the nuclei density by adding 171 ppm of TiB2 to the melt produces a fine-grained structure without columnar growth. Adding 100 ppm TiB2 has no effect on the cast structure or columnar-to-equiaxed transition. The results are considered in relation to the model for the columnar-to-equiaxed transition proposed by Hunt.[2]

The columnar-to-equiaxed transition in Al 3 Pct Cu

Tin-lead alloys were solidified directionally and the position of the columnar to equiaxed transition determined on vertical sections of the ingots. The columnar length was found to increase with decreasing lead concentration and increasing heat transfer coefficient. A mathematical model of the heat flow in the system was used to determine local temperatures, temperature gradients, and velocities in the solidifying alloy. Comparing the position of the columnar to equiaxed transition and local thermal conditions, it was found that the transition occurred when the temperature gradient in the melt at the liquidus temperature was 0.11°C/mm for Sn 10 wt pct Pb, 0.10°C/mm for Sn 5 wt pct Pb, and 0.13°C/mm for Sn 15 wt pct Pb. The position of the transformation was found to be independent of melt superheat for the conditions considered.

/pdf/the-columnar-to-equiaxed-transition-in-tin-lead-alloys-1uhn0evlmc.pdf

The columnar to equiaxed transition in tin-lead alloys

Centerline porosity in horizontally cast steel plates has been examined analytically. The presence and extent of porosity has been calculated on the basis of heat flow in the system and interdendritic fluid flow. The calculated distribution of porosity was found to agree closely with measured values reported in the literature. Several solidification parameters have been proposed to indicate the onset of porosity. The present investigation shows that the parameter G/R1/2 is the best of the parameters suggested, where G andR are the centerline temperature gradient and growth velocity, respectively.

Centerline porosity in plate castings

Very large grains, termed “spangles,” are produced on galvanized sheet steel coatings when lead is added to the zinc bath. The spangles have been attributed to melt undercooling prior to solidification. The present results indicate this is not the case, undercooling being less than 1 °C. The spangle diameter is shown to be dependent on the alloy addition to the bath, large spangles being obtained with Bi and Sb as well as Pb. The spangle size is related to the surface tension of the alloying addition, the size decreasing as the melt vapor surface tension of the alloying element increases. It is proposed that spangles form dendritically from a nucleus in the melt. Alloy additions with low interfacial energies and very limited solid solubility are highly concentrated ahead of the dendrite tip. This decreases the tip radius and increases the dendrite velocity, producing large grains. The basal plane orientation of the samples varies between 17 and 80 deg with respect to the steel sheet surface, which is inconsistent with basal plane dendritic growth in Zn along (1010) directions. It is proposed that solute additions to the melt and growth in a thin liquid layer can modify the dendrite growth direction, accounting for the spangle orientation.

F. Weinberg

Papers

The columnar-to-equiaxed transition in Al 3 Pct Cu

The columnar to equiaxed transition in tin-lead alloys

Centerline porosity in plate castings

Spangle formation in galvanized sheet steel coatings

The pitting corrosion of p-type GaAs single crystals