Mathematical model for the unidirectional solidification of metals: I. cooled molds

This paper presents a new mathematical model that can be used to predict the solidification rate and the temperature distribution during the unidirectional solidification of metals n molds cooled by fluids such as air and water. The model differs from other analytical methods presented in the literature in the sense that it is more general in application and easier to manipulate, while retaining the advantage of convenience over numerical techniques. The proposed model permits the measurement of the Newtonian heat transfer coefficient at the metal/mold interface. In order to verify the applicability to air cooled molds, the model is compared with experimental results in the literature for the cases of lead, tin and lead-tin eutectic. Finally, the case of water cooled molds is examined, the model being compared with experimental results obtained in this work for lead and aluminum.     

Mathematical model for the unidirectional solidification of metals: I. cooled molds

This paper presents a new mathematical model that can be used to predict the solidification rate and the temperature distribution during the unidirectional solidification of metals n molds cooled by fluids such as air and water. The model differs from other analytical methods presented in the literature in the sense that it is more general in application and easier to manipulate, while retaining the advantage of convenience over numerical techniques. The proposed model permits the measurement of the Newtonian heat transfer coefficient at the metal/mold interface. In order to verify the applicability to air cooled molds, the model is compared with experimental results in the literature for the cases of lead, tin and lead-tin eutectic. Finally, the case of water cooled molds is examined, the model being compared with experimental results obtained in this work for lead and aluminum.     

Heat flow parameters affecting the unidirectional solidification of pure metals

Unidirectional solidification of pure metals has been studied as a function of liquid superheat, heat transfer coefficient of the metal/mold interface(h _i ), and mold material. Experimental results show that the position of the solid/liquid interface (X) varies with time (l) following the relationshipX(t) = A . t ^1/2 -B, where, for a fixed metal,B is a constant only dependent on the superheat through a parabolic law andA is another constant that is independent of the superheat but dependent on the mold heat sink capacity,i.e., mold material and coefficienth _i At the same time, it has been calculated that the constantA varies withh _i through an error-function type law tending asymptotically to Lyubov’s exact analytic solution.     

A model of the interfacial heat-transfer coefficient during unidirectional solidification of an aluminum alloy

A model is presented for the prediction of the interfacial heat-transfer coefficient during the unidirectional solidification vertically upward of an Al-7 wt pct Si alloy cast onto a water cooled copper chill. It has been experimentally determined that the casting surfaces were convex toward the chill, probably due to the deformation of the initial solidified skin of the casting. The model was, therefore, based upon a determination of the (macroscopic) nominal contact area between the respective rough surfaces and, within this region, the actual (microscopic) contact between the casting and the chill surfaces. The model produced approximate agreement with both experimentally determined values of the heat-transfer coefficient and the measured curvature of the casting surface and showed a reasonable agreement with measured temperatures in the casting and the chill also. A common experimental technique for the experimental determination of the heat-transfer coeffcient involves the assumption of one-dimensional heat transfer only. An implication of the approach adopted in this model is that the heat transfer in the region of the casting-chill interface may be two-dimensional, and the subsequent error in the experimentally determined values is discussed.     

Mathematical model of COREX melter gasifier: Part II. Dynamic model

A dynamic model of the COREX melter gasifier is developed to study the transient behavior of the furnace. The effect of pulse disturbance and step disturbance on the process performance has been studied. This study shows that the effect of pulse disturbance decays asymptotically. The step change brings the system to a new steady state after a delay of about 5 hours. The dynamic behavior of the melter gasifier with respect to a shutdown/blow-on condition and the effect of tapping are also studied. The results show that the time response of the melter gasifier is much less than that of a blast furnace.     

Mathematical model of the thermal processing of steel ingots: Part II. Stress model

A mathematical model has been developed to predict the internal stresses generated in a steel ingot during thermal processing. The thermal history of the ingot has been predicted by a finite-element, heat-flow model, the subject of the first part of this two-part paper, which serves as input to the stress model. The stress model has been formulated for a two-dimensional transverse plane at mid-height of the ingot and is a transient, elasto-viscoplastic, finite-element analysis of the thermal stress field. Salient features of the model include the incorporation of time-temperature and temperature-dependent mechanical properties, and volume changes associated with nonequilibrium phase transformation. Model predictions demonstrate that the development of internal stresses in the ingot during thermal processing can be directly linked to the progress of the phase transformation front. Moreover, the low strain levels calculated indicate that metallurgical embrittlement must be very important to the formation of cracks in addition to the development of high tensile stresses. B. G. THOMAS, formerly a Graduate Student at the University of British Columbia     

Numerical models for casting solidification: Part II. Application of the boundary element method to solidification problems

A new numerical model, which is based on the boundary element method, was proposed for the simulation of solidification problems, and its application was demonstrated for solidification of metals in metal and sand molds. Comparisons were made between results from this model and those from the explicit finite difference method. Temperature recovery method was successfully adopted to estimate the liberation of latent heat of freezing in the boundary element method. A coupling method was proposed for problems in which the boundary condition of the interface consisting of inhomogeneous bodies is governed by Newton’s law of cooling in the boundary element method. It was concluded that the boundary element method which has several advantages, such as the wide variety of element shapes, simplicity of data preparation, and small CPU times, will find wide application as an alternative for finite difference or finite element methods, in the fields of solidification problems, especially for complex, three-dimensional geometries. On leave from Kyung-Pook National University, Taegu, Korea     

Chlorine fluxing for removal of magnesium from molten aluminum: Part II. Mathematical model

This second part of a two-part article presents a mathematical model for the “chlorine fluxing” of aluminum alloys, in particular, for the “demagging” of Al-Mg alloys such as those resulting from the recycling of used beverage cans. The model is based on the experimental results described in Part I and, in conformity with those results, assumes that neither the reaction kinetics at the melt-bubble interface, nor mass transfer on the gas side of that interface, are rate determining. With the introduction of one correction factor (applied to the surface renewal model for mass transfer on the melt side of the melt-gas interface), the model fitted the experimental data well, once measured values for the bubble size and rise velocity were introduced. The model was then used to predict the progress of demagging operations on an industrial scale. Computed results for these larger melts suggest that gross emissions of chlorine/chlorides are avoidable in bringing the magnesium content down to a critical value (which depends on operating characteristics such as bubble size). A multiple-step strategy is suggested when a batch of alloy is to be brought to yet-lower magnesium levels. In that strtegy, the chlorine content of the injected gas is reduced as the processing of the batch proceeds. The predicted effects of other operating changes (deeper nozzle submergence, broad bubble size distribution, etc.) are reported.     

The heat-transfer coefficient during the unidirectional solidification of an Al-Si alloy casting

The heat-transfer coefficient was measured during the unidirectional solidification of Al-7 wt pct Si alloy castings against a water-cooled Cu chill. Heat-transfer coefficients in the range of 2.5 to 9 kW m⁻² K⁻¹ were obtained with solidification vertically upward associated with higher values than solidification vertically downward. Horizontal solidification was associated with intermediate values. Profiles taken across the diameters of the casting surfaces at the interface with the chill showed them to be convex toward the chill by an amount which would produce a mean gap between the casting and the chill that would account for a significant proportion of the measured heat-transfer coefficient. The convex casting surfaces were attributed to the deformation of the initial casting skin by the thermal stress produced during solidification. Heat transfer during casting solidification is shown to be a complex mechanism controlled by the microscale surface roughness of the respective surfaces, mesoscale deformation of the casting skin by thermal stress, and macroscale movements of the casting and the chill due to their relative thermal expansion and contraction.     

Numerical simulation of solidification of molten aluminum alloys in cylindrical molds

A finite element model for the solidification of molten metals and alloys in cylindrical molds is developed using the energy equation in terms of temperature and enthalpy. TheNewton-Raphson technique was used to solve the resulting nonlinear algebraic equations. A computer program is developed to calculate the enthalpy, temperature, and fraction solid per the classical Lever rule, Scheile equation, and Brody-Flemings models. Cooling curves are calculated for pure metal (aluminum), two eutectic alloys (Al-33.2 pct Cu and Al-12.6 pct Si), and three hypoeutectic alloys (Al-2.2 pct Cu, Al-4.5 pct Cu and Al-7 pct Si) and are compared with the experimental curves.     

The solidification of pure metals (and eutectics) under uni-directional heat flow conditions: II. Solidification in the presence of superheat

The generalized integral-profile method previously developed² to predict the solidification rates of pure metals and eutectic alloys under a range of cooling conditions has been extended to treat solidification rates in the presence of super heat. Heat transfer within the liquid metal has been characterized in terms of a single parameter for which a heat balance on the liquid yields an ordinary nonlinear equation. This equation is additional to the two equations derived previously for the growth of the solid metal, and the three equations have been solved simultaneously using a Runge-Kutta technique. Solutions have been obtained for heat transfer conditions that can be reproduced accurately in laboratory experiments. The results of a series of such experiments are also presented and shown to agree very well with the theoretical predictions. In carrying out the analysis, it has been shown that a number of different cooling and solidification modes can occur during the solidification of superheated liquid metal, the solidification process following one of two possible routes through these modes. This presents a useful approach to the analysis of this solidification problem and, indeed, of more complicated problems since the logic of the computer program used in the analysis is closely related to the logic of the solidification process. The work described in this paper was carried out while the authors were with the John Percy Research Group in Process Metallurgy at Imperial College, London.     

A suggested effective method for unidirectional solidification under rotating magnetic field in the space experiments

In order to save costs and time during the unidirectional solidification experiments, it is necessary to obtain more and more information by solidifying one sample in a rotating magnetic field (RMF) to be performed on the space station. The macro- and microstructure developing during solidification is significantly influenced by the intensity and frequency of magnetic field, the solid/liquid interface velocity as well as the temperature gradient. A method by which one of the aforementioned four parameters can repeatedly be changed during the solidification is described. By this method, the transient phenomena occurring during parameters modification can also be observed. The results of earth experiments carried out by this method with Al7Si1Fe alloy are presented.     

Mathematical modeling of porosity formation in solidification

Shrinkage porosity and gas porosity occur simultaneously and at the same location when conditions are such that both may exist in a solidifying casting. Porosity formation in a solidifying alloy is described numerically, including the possible evolution of dissolved gases. The calculated amount and size of the porosity formed in Al-4.5 pct Cu plate castings compares favorably with measured values. The calculated distribution of porosity in sand cast Al-4.5 pct Cu plates of 1.5 cm thickness matches experimental measurements. The decrease of the hydrogen content by strong degassing and the increase of mold chilling power are recommended to produce sound aluminum alloy castings. The calculated results for steel plate castings are in agreement with the experimental work of Pellini. The present modeling has clarified the basis of empirical rules for soundness and suggests that the simultaneous occurrence of shrinkage and gas evolution is an essential mechanism in the formation of porosity defects.     

Solidification of Nb-bearing superalloys: Part II. Pseudoternary solidification surfaces

Equilibrium distribution coefficients and pseudoternary solidification surfaces for experimental superalloys containing systematic variations in Fe, Nb, Si, and C were determined using quenching experiments and microstructural characterization techniques. In agreement with previous results, the distribution coefficient, k, for Nb and Si was less than unity, while the “solvent” elements (Fe, Ni, and Cr) exhibited little tendency for segregation (k ≈ 1). The current data were combined with previous results to show that an interactive effect between k _Nb and nominal Fe content exists, where the value of k _Nb decreases from 0.54 to 0.25 as the Fe content is increased from ≈2 wt pct to ≈47 wt pct. This behavior is the major factor contributing to formation of relatively high amounts of eutectic-type constituents observed in Fe-rich alloys. Pseudoternary γ-Nb-C solidification surfaces, modeled after the liquidus projection in the Ni-Nb-C ternary system, were proposed. The Nb compositions, which partially define the diagrams, were verified by comparison of calculated amounts of eutectic-type constituents (via the Scheil equation) and those measured experimentally, and good agreement was found. The corresponding C contents needed to fully define the diagrams were estimated from knowledge of the primary solidification path and k values for Nb and C.     

A nonmetal interaction model for the segregation of trace metals during solidification of Fe-Ni-S,Fe-Ni-P,and Fe-Ni-S-P alloys

As nonmetals are added to the Fe-Ni system, segregation coefficients(k) of trace constituents change dramatically. For example, as the S content of the metallic liquid increases from 0 to ≈31 wt pct, the molar k(Ge) between solid and liquid metal increases from 0.6 to ~120. Little of this change can be ascribed to temperature. Also, these changes are not linear. In the case of the Fe-Ni-S system, the largest changes are seen between 20 and 30 wt pct S. Here we present a model for the interaction of nonmetals with other species in metallic systems. The model assumes that the nonmetal either repels or attracts tracers (E) in the metallic liquid, causing those tracers to either segregate strongly into the solid metal or to have enhanced solubility in the liquid. Examples of both types of behavior are given. The model predicts that the activity coefficient of the tracer (γ_E) should correlate linearly with (1 - αnX_N @#@), where X_N is the mole fraction of the nonmetal in the metallic liquid,n is a stoichiometry factor related to the speciation of N in the liquid, and a is a constant. Indeed, good linear correlations of n(k) vs n (1 -αnX_N @#@) are found for all elements where there is a measurable effect. Thus, if the composition of the metallic liquid is known, a segregation coefficient of a trace constituent may be pre-dicted— even if the temperature, exact Fe/Ni ratio, and information about the activity coef-ficient in the solid phase are unknown. The nonmetal interaction model presented here can be related to more traditional methods of modeling activity coefficients(i.e., power law expan-sions) and can be shown to be a special case of this type of parameterization. Comparison of model predictions of first-order (sulfur-E) interaction coefficients (ε) to measured values yields acceptable agreement for some elements, such as P and Ge, and all elements except Ni agree to within a factor of 3. The predictive model described above, based on equilibrium experi-ments, may be used to evaluate the segregation coefficients extracted from the dynamic (“plane front solidification”) experiments of Sellamuthu and Goldstein.^[10,11] Contrary to claims, reliable segregation coefficients are not extractable from dynamic experiments.