Numerical analysis of solute redistribution during solidification accompanying .DELTA./.GAMMA. transformation.

Kobayashi, Sumio; Nagamichi, Tsuneaki; Gunji, Kôki

doi:10.2355/isijinternational1966.28.543

Cited by 42 publications

(26 citation statements)

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“…For the 0.01P and the 0.10P, the assumption that the rapid growth of g grain starts when the f g reaches more than 0.999 is reasonable. This is almost consistent with the f g at the rapid growth starting temperature, which has already been determined in carbon steels, for example, 0.99 by Kobayashi et al 26) and Yasumoto et al 3) However, for the 0.20P, the result suggests that the pinning effect by the d phase disappears at a relatively low f g of 0.92. There is a possibility that the disappearance is caused by an insufficient dispersion of retained d phase at the P-rich spots.…”

Section: Effect Of Phosphorus On Grain Growthsupporting

confidence: 78%

Analysis on Refinement of Columnar .GAMMA. Grain by Phosphorus in Continuously Cast 0.1 mass% Carbon Steel

2004

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Based on a grain growth model, this paper discusses the g grain refinement by phosphorus (P) in as cast 0.1 mass% C slabs. Two important factors, the starting temperature of the rapid grain growth (T rg ) and the grain growth rate, are especially focused in the model. Local Equilibrium Mapping (LEM), an analytical method with local equilibrium calculations in the micro-segregation maps determined by EPMA, is carried out to evaluate the local transformation temperature and the g phase fraction. The LEM analysis shows that the micro-segregation of P extends the d/g transformation temperature range to a lower temperature and retains the d phase at a lower temperature. The Classical Grain Growth Model (CGM) based on a theory by Burke (1949) and Turnbull (1951) is derived, which assumes a normal grain growth with a parabolic law. The CGM successfully evaluates the g grain growth curve in the as cast 0.1 mass% carbon steels considering the rapid growth after the d/g transformation and the change of the growth rate during cooling. The CGM analysis predicts the decrease of both T rg and growth rate by the P addition. The decrease of T rg was determined to be 160-170 K and agrees with the extension of the d/g transformation temperature range evaluated by the LEM analysis. Hence, the grain growth curves can be predicted by considering the extension of the d/g transformation temperature range due to the P addition and its micro-segregation.KEY WORDS: carbon steel; continuous casting; solidification; grain growth; phosphorous; segregation; phase transformation; austenite; delta ferrite. should emphasize two significant factors when determining the g grain size. The first factor is the temperature at which the rapid grain growth starts, and the second is the grain growth rate within the range of temperature for the rapid growth. The first factor, the starting temperature of rapid growth, is closely related to the d/g transformation temperature range. Since the micro-segregation changes the transformation temperature locally, a new analytical method, i.e. the local equilibrium mapping (LEM), is developed, in which the local equilibrium is determined based on the micro-segregation mapping data by EPMA, using ThermoCalc 7) with a multi-component database. This method can evaluate the local transformation temperature and the phase fraction as a function of temperature.The second factor, the grain growth rate during rapid growth, is evaluated from a grain growth model, i.e. the classical grain growth model (CGM). The basic grain growth equation in the CGM is based on the classical theory proposed by The analytical model is derived by integrating 3,[11][12][13][14][15][16] the basic equation over the duration of the rapid grain growth, considering the temperature dependency of the grain growth rate.9,10) The grain growth rate and the grain size will be predicted by the CGM.Consequently, we will discuss the mechanism of the g grain refinement by phosphorus by analyzing the results of the previous paper by coupling the LEM with the...

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Section: Effect Of Phosphorus On Grain Growthsupporting

confidence: 78%

Analysis on Refinement of Columnar .GAMMA. Grain by Phosphorus in Continuously Cast 0.1 mass% Carbon Steel

2004

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“…(4) The equilibrium partition ratio k the interface and is constant throughout so process. Therefore, (2) applies at (8) and by replacing t and x byf and z, respectively, the equations of diffusion (Eq. (1)) and mass conservation (Eq.…”

Section: Mathematical Formulation Of the Brodyflemings Modelmentioning

confidence: 99%

A mathematical model for solute redistribution during dendritic solidification.

Kobayashi

1988

ISIJ Int.

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SynopsisTo quantify the solute redistribution during solidification, an approximate solution of the Brody-Flemings model of dendritic solidification has been derived on the assumption of parabolic solidification and a constant partition ratio, and compared with an exact solution of the model. It has been shown that the approximate solution has good accuracy.Further, on the basis of the derived equation, an extended mathematical model is proposed, incorporating a thermal model of solidification into the analysis without assumption on the solidification rate. The extended model includes the treatment of multi-component alloy and of 1 transformation, the temperature dependence of diffusion coefficients and the estimation of the diffusion path. The extended model provides the results consistent with the available data.Key words: mathematical modeling; solidification; segregation; sion; carbon steel. dif f u- I. IntroductionIn the earliest description with the Scheil equation for solute redistribution during solidification, the diffusion of solute into solid phase (back diffusion) was neglected.1'2~ Brody and Flemings3~ have extended the theory to include the back diffusion effects based on their solidification model; the model assumed 1) complete diffusion in liquid phase and incomplete back diffusion, 2) a plate-like dendrite geometry, 3) a constant diffusion coefficient, and 4) parabolic or linear solidification rate. Both approximate equations derived by Brody and Flemings for linear and parabolic solidification rate are, however, invalid for rapid diffusion in the solid phase.Clyne and Kruz,4~ and Ohnaka51 have modified the Brody and Flemings equation for parabolic solidification rate; both the modifications are valid for infinite and infinitestimal diffusion coefficient. Further, the present author has derived an exact solution of the solidification model of Brody and Flemings for parabolic solidification rate, and has shown that the accuracy of all the approximate solutions mentioned above are often bad.6~ Although the solidification model of Brody and Flemings with parabolic growth has been rigorously solved, several problems still remain in applying the model to solidification processes : 1) calculation of the exact solution is uneasy because the solution has an infinite series of the confluent hypergeometric functions, 2) no realistic method has been found to estimate the solidification rate or the local solidification time, of which value is necessary for the calculation,

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“…10,13) (6) Retardation of grain growth is partly due to the dendritic micro-segregation that also lowers T g . [12][13][14] Based on the above conditions, the fine-grained g structure in low carbon steels can be achieved by optimal alloying, which stabilizes bcc effectively and causes dendritic micro-segregation to some extent. Figure 1 shows a schematic phase diagram in a Fe-M (bcc-stabilizing element) binary system.…”

Section: Introductionmentioning

confidence: 99%

Influence of Phosphorus on Solidification Structure in Continuously Cast 0.1 mass% Carbon Steel.

2003

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In 0.1 mass% carbon steels with phosphorus contents ranging from 0.01 to 0.2 mass%, slabs were continuously cast to a thickness of 100 mm with a laboratory scale caster. Their macro-and micro-structures were characterized, focusing on the effects of phosphorus addition on the structural evolution during solidification and subsequent cooling. Cast slabs of high phosphorus steels have a fine columnar-g-grain structure. The mean width of the columnar grain was approximately half of that in the cast slabs without the phosphorus addition. Dispersed globular a grains were observed in the a grain structure of high phosphorus steels. The globular grains evolved at the phosphorus-rich spots in the prior-g grain. The micro-segregation of phosphorus results in these structural evolutions. Since the phosphorus enrichment stabilizes bcc (d or a) phase locally at the inter-dendritic region, the phosphorus-rich spot makes d phase retained to lower temperature for the d/g transformation, and provides a predominant site for the g/a transformation. In the high phosphorus casts, therefore, the dispersed d phase are thought to pin the g grain growth more effectively in the dϩg region when the completion of the d/g transformation is remarkably suppressed by the phosphorus segregation.

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Numerical analysis of solute redistribution during solidification accompanying .DELTA./.GAMMA. transformation.

Cited by 42 publications

References 2 publications

Analysis on Refinement of Columnar .GAMMA. Grain by Phosphorus in Continuously Cast 0.1 mass% Carbon Steel

Analysis on Refinement of Columnar .GAMMA. Grain by Phosphorus in Continuously Cast 0.1 mass% Carbon Steel

A mathematical model for solute redistribution during dendritic solidification.

Influence of Phosphorus on Solidification Structure in Continuously Cast 0.1 mass% Carbon Steel.

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