Positive segregation as a function of buoyancy force during steel ingot solidification

Abstract

We analyze theoretically and experimentally solute redistribution in the dendritic solidification process and positive segregation during solidification of steel ingots. Positive segregation is mainly caused by liquid flow in the mushy zone. Changes in the liquid steel velocity are caused by the temperature gradient and by the increase in the solid fraction during solidification. The effects of buoyancy and of the change in the solid fraction on segregation intensity are analyzed. The relationships between the density change, liquid fraction and the steel composition are considered. Such elements as W, Ni, Mo and Cr decrease the effect of the density variations, i.e. they show smaller tendency to segregate. Based on the modeling and experimental results, coefficients are provided controlling the effects of chemical composition, secondary dendrite arm spacing and the solid fraction.

Introduction

Solute redistribution during solidification of steel ingots determines the growth morphology, phase distribution and the amount of macro-segregation. Macro-segregation in steel ingots is caused by the different solubility of alloying elements in the liquid and solid metal, i.e. by the generation of concentration gradients in both zones, by the impossibility of these two gradients to equalize due to diffusion during solidification, and by the transport of liquid regions enriched by a solute.

One of the reasons for the transport of the segregated liquids may be density differences in the metal due to changes in temperature and composition. Hot liquid metal cools down near the cold surfaces, and the concomitant density increase causes it to sink. In the upper part of an ingot, regions with a higher concentration of alloying elements are formed, i.e. positive segregation occurs. On the other hand, negative segregation takes place in the lower part of the ingot that can be explained by the sedimentation of equiaxed crystals formed in the bulk liquid. Several segregation models, as well as experimental results on segregation intensity, were presented [1–3], and macro-segregation modeling was reviewed [4, 5].

Many variables affect the macroscopic distribution of a solute in a cast ingot, the most important of which are the size of the ingot, solidification rate, heat extraction, chemical composition and superheating. Furthermore, the solid framework of the mushy zone and the presence of equiaxed crystals are very important. The average intensity of the segregation in the central part of the ingot should be small if equiaxed crystals can not be transported [6]. High flux density in the two-phase region results in the early immobilization of equiaxed crystals and reduced segregation in the central part.

Based on theoretical analysis of the positive segregation and on the numerical simulation of the solidification of steel ingot [7], the effect of the temperature gradient on the velocity of interdendritic liquid flow in the two-phase region was analyzed.

Fundamental studies of positive segregation

During the solidification of steel ingots, a heterogeneous solid–liquid zone is usually formed between the solidified outer region and the liquid melt. Liquid melt is in contact with the mold wall, which acts as a heat sink. Therefore melt becomes denser and moves downward (figure 1(a), zone L). Unidirectional solidification is the simplest geometry. Fluid flow during solidification originates from natural convection caused by the density change. Such a density gradient mostly arises from the temperature gradient in the liquid and leads to thermal convection. Because of the rejection of solute elements in the interdendritic liquid, the concentration and density gradient increase. This phenomenon is known as solutal buoyancy. The convection pattern in the melt and the mushy zone is shown schematically in figure 1.

Figure 1.

Schematic diagram of liquid melt flow (L-liquid, S-solid, S+L- solid+liquid).

Cooling starts from the bottom of the ingot. Thus the solid-liquid or mushy zone (S+L in figure 1) is formed near the mold wall in the beginning of solidification. Because the temperature increases in the upward direction, the liquid is denser toward the bottom of the ingot. In the mushy zone, liquid flows upward due to the decrease in density because of the enrichment of solute impurities and alloying elements. Hence, thermal buoyancy opposes the upward motion of the liquid. In this way, the counter-flow of the liquid melt occurs in the boundary region (S+L)/L because the liquid melt flows into the liquid phase due to the temperature difference. In this case, the solution buoyancy forces are opposite to those of the thermal buoyancy, and the former has a dominant effect in the mushy zone. The liquid melt moves through the boundary interface (S+L)/L to form zone L′, where the liquid has a different temperature, density and composition compared with the original liquid in zone L.

Liquid metal flows from the mushy zone into the liquid zone, and the region L′ gradually expands at the expense of the zone L. This mechanism of liquid flow through the boundary interface is limited by the solid fraction f_s. Figure 1 reveals that the increase in the velocity of the interdendritic liquid flow in the mushy zone enables the formation and expansion of region L′. Natural convection into the mushy zone can be described by the Darcy's law [8–13].

The velocity component V_y is

where K and η are the permeability and viscosity of the liquid, g is acceleration due to gravity, ρ_l^o and ρ_l are the liquid densities in the liquid and mushy zones, respectively, and f_l is the liquid fraction.

Changes in η are very small, and thus V_y depends mostly on the changes in K and η. Permeability K can be determined empirically. In the direction perpendicular to the primary dendrite arms, it can be expressed [14] as

Here λ₁ and λ₂ are the primary and secondary dendrite arm spacings, respectively.

The density change dρ_l/dT_l is difficult to determine, therefore, the average value dρ_l/df_l in the mushy zone is usually examined instead. Permeability decreases with the decrease in the liquid fraction. Therefore, the liquid velocity in the mushy zone decreases during solidification. When f_l is smaller than its critical value f_l^c, there is no possibility of interdendritic liquid flow, and V_y becomes

If dρ_l/df_l<0 then the liquid in the mushy zone flows upward, and positive segregation occurs at the top of the ingot. The interdendritic liquid density depends on its temperature and chemical composition and it is a function of f_l

Using the values of the partition coefficient for carbon and other elements, the following expression for V_y can obtained

Here A is a constant, and parameters Z_λ, Z_fl and Z_c control the dendrite arm spacing, liquid fraction and chemical composition, respectively.

Substituting the values of table 1 into equation (4) yields [15] the parameters in equation (5), which are listed in table 2.

Table 1.

Values of parameters used in equations (4) and (5).

	Partition coefficient	∂ρl/∂Cl
Element	k	(g cm−3 wt%−1)
C	0.34	−78
Si	0.59	−17.1
Mn	0.75	−3.32
S	0.024	−30.4
P	0.09	−27.1
Cu	0.96	−1.7
Cr	0.76	−2.61
Ni	0.94	−1.6
Mo	0.56	−3.25
V	0.93	−2.65
W	0.40	−0.5

∂ρ_l/∂T=−0.881×10⁻³ (g cm⁻³ °C⁻¹), .

T_Fe=1809 K (pure Fe).

Casting temperature is 1693 K.

Liquidus temperature of the studied steel is 1693 K.

Table 2.

Values of the parameters in equation (5).

Parameter	Values
A (m−1 s−1)	29.985
Zλ (m2)	λ11.09⋅λ20.91
Zfl (–)	fl3(1−fl)−0.749
	0.01347CC+0.0662CMn+0.2411CS+0.2028CP−
Zc (–)	0.000272CCu+0.0131CCr−0.00015CNi−
	0.01285CMo−0.00544CV−0.0353CW

Equation (4) reveals that temperature variation in the mold during solidification is important in calculating dρ_l/df_l. Using the above relations, the density change and flow velocity can only be determined if temperature is continuously monitored.

Effect of dispersion of dendrite structure

The coefficients Z_λ, Z_fl and Z_c in equation (5) depend on various factors, but all of them are related to the temperature gradient. If the primary dendrites are defined as axially symmetric ellipsoids then their arm spacing can be expressed as [16]

Here ΔT_m is the temperature difference between the liquidus and solidus lines, D is the carbon diffusion coefficient, Γ is the Gibbs–Thomson parameter, k is the partition coefficient of carbon, R is the rate of solidification and G is the temperature gradient.

Ostwald mechanism is mostly used in the theoretical analysis of the formation and growth of the secondary dendrite. Based on this mechanism, Imagumbai [17] gives a general relation for the secondary arm spacing as

Here Θ is the local solidification time, which can be calculated as

The above equations reveal that the dominant factors affecting Z_λ in equation (5) are the change in temperature during the ingot solidification and the properties of the initial material. The solidification time depends on the dimensions and shape of the ingot, as well as its chemical composition and the casting temperature. Despite numerous studies, it is not clear whether the parameter n in equation (7) is a constant or it changes across the surface of the solid–liquid boundary. Steels are multicomponent alloys where variations of n are difficult to determine. Therefore, n is usually taken as 1/3. Upon substituting this value in equation (7), the following expression is obtained:

Effect of solid fractions during solidification

The effect of the solidification dynamics on the liquid flow rate in the two-phase region (solid–liquid) can be obtained from Z_fl in equation (5):

or

The critical value of f_l′ for liquid metal is ∼0.2, and thus, the changes in liquid density can be calculated in the range f_l′<f_l<1. Solid fraction f_s increases during solidification. Therefore, it is necessary to analyze the effect of temperature changes on f_s. For the more detailed analysis of the temperature field of an ingot cross section, a numerical model of the solidification process was developed [7] based on Fourier's differential equation:

Here ρ is the steel density, c_p is the heat capacity, λ is the thermal conductivity and q is latent heat. The change in the solid fraction is given by

The heat flux conducted across the ingot surface can be defined as the sum of convection and radiation heat until the formation of an ‘air gap’ (separation between the ingot surface and mold wall):

After formation of the air gap, the heat is only transferred by radiation and can be expressed as

Here T_s is the surface temperature of the ingot, T_m is the mold temperature, σ is the Stefan-Boltzmann constant, ε_i and ε_m are the emissivities of the ingot and mold, respectively, and T_l is the liquid metal temperature.

Experimental procedure and results

The proposed model is applied to the solidification of an ingot of high-alloy tool steel. The chemical composition of the studied steel is given in table 3.

Table 3.

Chemical composition of the studied steel.

Element	C	Si	Mn	Cr	Ni	V	W	Mo	P	S	Fe
wt%	1.65	0.35	0.32	12.5	0.31	0.22	0.45	0.64	0.03	0.03	83.7

Temperature during ingot solidification was measured with Pt-Rh18-Pt thermocouples fixed at different positions at two-thirds of the ingot height. Ingot has square cross-section. There, we can assume x=y in equation (12) and analyze only one quadrant of the cross-section. The temperature distributions during the ingot solidification, calculated as a function of time and the distance from the surface, are presented in figures 2–4.

Figure 2.

Temperature distribution across the ingot cross section as a function of time (Casting temperature 1693 K).

Figure 4.

Temperature field across the ingot cross section during solidification (Casting temperature 1770 K).

Figure 3.

Temperature distribution across the ingot cross section as a function of distance from the ingot surface (Casting temperature 1770 K).

The temperature of the liquid steel during solidification was measured by three thermocouples at different positions shown in figure 5. The agreement between the measured and calculated values is good, particularly at the ingot surface.

Figure 5.

Calculated and measured temperatures in the ingot during solidification.

The values of temperature gradient and solidification rate obtained from the model can be used to calculate the spacing of the primary and the secondary dendrite arms using equations (6) and (9), respectively. The theoretical values of λ₁ and λ₂ were compared with the experimental values for different casting temperatures (figures 6 and 7).

Figure 6.

Change in primary arm spacing with distance from surface.

Figure 7.

Change in secondary arm spacing with distance from surface.

Interdendritic arm spacing increases from the surface to the center of the ingot. The differences between calculated and experimental values are caused by a number of solidification parameters; thus, it is difficult to derive analytical relations for λ₁ and λ₂. The obtained values of λ₁ and λ₂ can be used to calculate the controlling parameter Z_λ in equation (5), as shown in figure 8.

Figure 8.

Dependence of the Z_λ on temperature gradient.

In figure 8, Z_λ is plotted versus the temperature gradient. Decrease in the temperature gradient increases the effect of Z_λ. This effect is more pronounced at the beginning of the solidification process while at the end it tends to be reduced (figure 9). Explanation of those changes can be found in the fact that the end of the solidification process is followed by the formation of zone equiaxed crystals in the center of the ingot without a pronounced dendrite structure.

Figure 9.

Micrographs of dendrite microstructure of investigated steel for different casting temperatures (×50).

The liquid velocity in the mushy zone depends on the fraction f_l, i.e. on the increment of the solid phase f_s. With increasing solid phase fraction the space available for the liquid metal to flow decreases due to the buoyancy effect. Using the numerical model we obtained df_s/dt during ingot solidification, as shown in figure 10.

Figure 10.

Changes in solid fraction during ingot solidification at different distances from surface (x).

Based on the data of table 2, we can determine changes in the correction factor Z_fl. Its dependence on the distance from the surface is shown in figure 11.

Figure 11.

Change in Z_fl during solidification for different distances from surface (x).

It is clear that near the ingot surface Z_fl has little effect on V_y. The reason for that is the dendrite structure does not form near the surface but undergoes almost instantaneous bulk solidification.

In the central region, there also is no change in Z_fl. The latter has small value because solidification is completed and liquid fraction is small (f_l=0.1) in this region. Considering the effect of the dendrite structure dispersion and the liquid fraction, i.e. the amount of solid phase in the solidification equation (5), it is possible to derive the overall dependence of liquid flow velocity on these parameters. For steels with a fixed chemical composition Z_c, equation (5) is constant; it does not affect the variation of V_y, although it has some effect on its absolute value.

The dependence V_y=f(Z_fl,Z_λ) is shown in figure 12. The velocity of liquid flow (V_y) decreases with decreasing Z_fl, but increases with increasing Z_λ. During solidification, these two parameters behave differently: f_l decreases, but λ₁ and λ₂ increase. In the model of solidification of high alloy tool steel, the dominant effect on V_y is the increase in interdendrite arm spacing (figure 13).

Figure 12.

Changes in V_y as function of Z_fl and Z_λ.

Figure 13.

Variation of V_y for different values of casting temperature (I: zone of equiaxed crystals, II: columnar zone, III: central zone).

In zone II, the increase in velocity V_y is approximately linear versus distance from the ingot surface, whereas V_y in zones I and III exhibits a different trend. In the area with a developed dendrite structure V_y increases almost linearly, while V_y decreases in the central zone. The V_y value is higher for a higher casting temperature due to the increased formation of the dendrite structure and the effect of parameter Z_λ.

Effect of V_y on the formation of positive segregation profile

An equation for global segregation intensity can be derived by integrating the local segregation intensity [5]:

Here C_i is the concentration of a certain element at a certain point and C_o is the average concentration of the element.

Using the experimentally obtained values of local segregation intensity across the ingot cross section, we derived the dependence of the macro-segregation intensity on the liquid metal velocity (figure 14). The dependence of S_i^g on V_y is approximately linear, and it is most pronounced for elements having small partition coefficient (sulfur, phosphorus) and thus prone to segregation. These elements have the greatest effect on the value of the correction factor Z_c (table 1); thus, they indirectly increase the flow rate of the liquid and S_i^g. In this way, the experimental values of M-shaped segregation profiles can be explained, particularly in the upper part of the ingot.

Figure 14.

Effect of V_y on segregation intensity S^g_i.

Comparing the liquid flow rate and profile segregation contour with the images of the microstructure across the cross section of the ingot reveals similar changes (figure 15). The similarity is particularly pronounced in the columnar dendrite zone. However, there are differences at the surface and in the central zone containing equiaxed crystals. The liquid flow rate and the overall segregation intensity increase linearly in the zone of dendrite crystals. In the surface zone, these differences can be neglected, while in the central zone the difference is due to the very small amount of liquid flow because of the high value of the solid fraction f_s.

Figure 15.

Comparison of V_y, S^g_i and the microstructure in high-alloy steel (the top panel shows a micrograph of an axial cross section of an ingot).

The maximum values of V_y and S_i^g coincide not at the ingot center but at the border between the columnar dendrite zone and the central zone. This fact can verify the conclusion that the overall segregation intensity depends greatly on the liquid velocity in the interdendritic region, but only in the range of f_s=0.15–0.70.

Conclusions

Positive macro-segregation in the upper part of a steel ingot during solidification is directly affected by the liquid flow in the mushy zone due to the buoyancy effect and the density decrease. The dispersion of dendrite structures and the chemical composition also contribute to the rate of liquid transfer.

The effects of these three factors can be expressed by the parameters Z_λ, Z_c and Z_fl. Based on the developed here numerical model of solidification, the areas of predominance of each factor have been determined.

In the zone of equiaxed crystals, the prevailing factor is Z_c, i.e. the chemical composition, while in the zone of columnar dendrites, the Z_fl and Z_λ dominate. Global macro-segregation intensity (S_i^g) is depends almost linearly on the liquid velocity. The greatest value of S_i^g is between the columnar dendrite zone and the central zone, in agreement with the experimental data.

Using the measured temperature changes (i.e. the temperature gradient) and the derived mathematical equations, the relationships were obtained between the buoyancy effect, the dispersion of dendrite structures and the presence of segregated elements. The segregation intensity increases with increasing V_y, particularly in for elements with small partition coefficients such as S and P.

Using the steel solidification model and the parameters Z_λ, Z_c and Z_fl, it is possible to determine the liquid velocity and its relation with the segregation intensity.