An analytical model for nodular eutectic grain predictions during solidification
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I. INTRODUCTION
NODULAR iron has been increasingly used in replacing other engineering alloys due to its unique properties and reduced production costs.[1] Accordingly, considerable research has been done on this material, which has led to numerous reports on properties,[1–4] and solidification mechanisms.[4,5] Most research efforts on the solidification of nodular iron have been focused on the nucleation and growth aspects of graphite spheroids. In general,[5] it is believed that nucleation of graphite spheroids occurs at substrates such as sulfides or saltlike carbides, while growth is controlled by diffusion of carbon through a surrounding austenite shell. From the solidification point of view, it is known that nodular iron is probably the only engineering material that solidifies as a globular eutectic. Since graphite nucleation is dominant at the onset of solidification, it is a key factor in establishing the final population of eutectic grains. The exhibited grain density, in turn, effects the degree of segregation,[6,7] the nodular iron properties,[8] and the chilling tendency.[9] In addition, the density of potential nuclei can be influenced by the iron chemistry, type and amount of spheroidizing agents, as well as solidification conditions (degree of undercooling and cooling rates[10]). The density of nuclei N, or the nucleation rates due to heterogeneous or homogeneous nucleation, can be described by[11] dN c 5 c1 exp 2 22 dt DT
1
or by empirical relations
[12–16]
2
[1]
T N 5 2nc3 DT n21 t t
[2] [3]
where DT is the degree of undercooling (8C), and ci (i 5 1, 2, 3) and n are nucleation constants. Unfortunately, the c1 and c2 constants are not easily determined. Moreover, the c3 and n constants in the empirical relations are not well defined. ´ EDWARD FRAS, Professor, is with the Foundry Engineering Institute, University of Mining and Metallurgy, 30-059 Cracow, Poland. HUGO F. LOPEZ, Associate Professor, is with the Materials Department, University of Wisconsin–Milwaukee, Milwaukee, WI 53201. Manuscript submitted August 20, 1998. METALLURGICAL AND MATERIALS TRANSACTIONS B
II. ANALYSIS During the solidification of nodular iron of eutectic composition, two stages can be distinguished: stage I, which corresponds to alloy cooling from the pouring temperature T0 to the temperature at the onset of eutectic solidification Tn , or in a timeframe (t) from t 5 0 to tn; and stage II, which is defined by the onset of eutectic solidification time tn until maximum undercooling (DTm 5 Tn 2 Tu) is reached at time tu. In stage II, the melt temperature drops from Tn to the maximum undercooling temperature Tu (Figure 1). During solidification, the temperature of the melt can be determined from the Fourier equation as
l¹2T 1 L
of the type
N 5 c3 DT n
This, in turn, makes it difficult to provide adequate predictions of nucleation events. Hence, in this work, an analytical solution is proposed that relates the expected volumetric nuclei density N to the maximum degree of undercooling, DTm. Predictions of the sol
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