1. Introduction
Macrosegregations are very typical defects in metal materials now, which can cause performance failures due to nonuniform composition distribution and structure [
1,
2,
3]. Regarding macrosegregation, the fundamental formation theory was pioneered by Fleimings and co-workers about 50 years ago and is now extensively accepted [
4]. In the classical theory, microsegregation and interdendritic fluid flow are two essential aspects for macrosegregation formation, and the latter usually plays a decisive role during the process [
5,
6]. It can safely be said that all types of macrosegregation form within the liquid–solid zone, and it is the result of slow interdendritic flow in most cases, but there is much left to study to fully understand and control macrosegregation in practice [
4,
6]. Hence, numerous studies have attempted to measure, simulate, and control different characteristics of interdendritic liquid flow for specific processes [
7]. For example, because the velocity magnitude can probably be calculated by Darcy’s Law if the permeability is obtained, many researchers are interested in the experimental measurement of permeability and mathematical models of how permeability varies with fraction solid, columnar or equiaxed structures, different processing, etc. [
8,
9,
10,
11,
12,
13,
14]. However, because the structures are somewhat conceptual and the pressure gradients are assumed or approximate, the calculated velocities always deviate from the real. At the same time, flow velocity in the pure liquid zone is always simulated by many researchers, but it is still very difficult to correctly simulate the flow velocity in the mushy zone, especially for complex fluid flow or large-size casting or with different actual dendritic grains of complex morphology [
2,
6,
7,
15,
16]. Additionally, if based on the actual result (like the segregation content ratio) and local solute redistribution equation [
5,
6], the velocity magnitude can be calculated by at least two contents, but it is only a relative value with respect to a reference content that may not be available to compare (e.g., original liquid content that ignores element diffusion or liquid transfer) and is always a cubic average value with a large scale (e.g., centimeter) for the shortcomings of content measuring methods. Therefore, as to the complex solidifying process, it is always difficult to investigate precisely the characteristics of interdendritic liquid flow for the formation mechanism of different actual segregation morphology.
Solute redistribution is the result of different kinds of segregation, especially for some elements with low distribution coefficients, such as carbon in steel. So it should be useful to obtain firstly the actual element distribution in order to study the specific segregation formation mechanism. However, because some complicated phenomena like eutectic reaction occur in the final solidification, it is always not easy to measure directly the element change from the casting dendrite grain center to its margin (almost always at least 0.5 mm long) by existing technologies (e.g., wet chemistry, optical emission spectrometry, X-ray spectroscopy or fluorescence) [
17], not to mention many casting grains. Therefore, on the basis of the nonuniform distribution, hot pickling is always used to etch the samples in order to reveal the structure and related element distribution [
18,
19,
20,
21]. By different grayscales, different structures (e.g., columnar, equiaxed, macrosegregation) can be distinguished from the etched structure picture [
15,
20,
21,
22]. Meanwhile, the height of the sample surface becomes uneven after being etched because the electrochemical reaction of hot pickling appears at different levels or kinds in different zones of smooth surface. For example, different surface heights can induce different grayscales, and different grayscales can be indicative of segregated carbon element content in steel alloy [
15,
22,
23], so the height on the sample surface after etching can also probably be indicative of carbon element content. Moreover, the grayscale value can be more easily influenced by measuring processes such as light source features, and the large area and multiscale distribution of element content is very difficult to measure directly by existing methods, so the etched surface height could be a possible method to study the element redistribution phenomenon in a large area. Although an entirely accurate quantitative correlation equation between the etched surface height and the element content at a certain position is now difficult to establish, plenty of etched surface heights (abbreviated as
in this article) may be partly suitable for investigating some segregation formation processes by simultaneously using some statistical methods.
In this study, the
values at different surface places are measured firstly in GCr15 bearing steel bloom of continuous casting for the reason that its high-content carbon element (mass fraction, 1.02%) is easy to segregate during solidifying and the fluid flow is strong enough during the continuous casting process [
15,
19]. Secondly, the magnitude and the direction of velocity in interdendritic fluid flow (abbreviated as
in this article) are calculated by establishing corresponding models on the basis of the
. Finally, the results and the verification are discussed. In all, this study demonstrates a new approach for measuring the velocity of interdendritic fluid flow during metal solidification, which may provide a unique alternative method for understanding the interdendritic fluid flow, macrosegregation formation, and hence, material properties.
3. Model Description
Figure 9a shows the macrostructure of the red rectangular area from
Figure 4. The white part displays the dendrite grain morphology, and the black part displays the interdendritic area, which is characteristic of high element content. The horizontal dashed arrow corresponds to the vertical dashed arrow in
Figure 3.
Figure 9b is the etched surface height distribution along the horizontal dashed arrow in
Figure 9a. Taking F position in the inner curve side under 20 °C as an example, by comparing
Figure 9a with
Figure 9b, we found that the heights in the dendrite centers always had a relative maximum value of height (e.g., (1), (2) and (5) points marked by the intersection between the red dashed line and horizontal dashed arrow), and the heights in the black interdendritic areas always had a relative minimum value of height (e.g., (3), (4) and (6) points marked by the intersection between the green dashed line and horizontal dashed arrow).
In addition, the heights may seem not to be larger or smaller when the point is closer to or farther from the grain center. This is because the surface macrostructure is photographed by a SELP1650 digital camera, which cannot display three-dimensional morphology (i.e., the height difference) and some dendrites are cut off by cross section. Notwithstanding, the above discussion indicates that there is some correlation between the etched surface height and segregating element content. Carbon is the element segregated most in GCr15 bearing steel solidification [
17,
19]. Therefore, based on the characteristics of
distribution, a quantitative correlation equation between
and the dimensionless quantity of carbon content is put forward firstly, given by Equation (1).
is the dimensionless quantity of carbon content. At a certain position, 50,001 effective carbon contents are rearranged in descending order. Then, if the 12.5 mm measuring line at each position is treated as a solidification unit for the reason that every point on a measuring line has almost the same solidifying condition, the rearranged contents can be displayed by liquid fraction as a horizontal axis from a statistical average angle, as shown in
Figure 10.
That is to say, the highest
can be regarded the final solidifying point, and the lowest is the original solidifying point. The content in solidus increases when liquid fraction is decreasing.
where
is the dimensionless quantity of carbon content at a point.
is the etched surface height at a point.
is the average height of the etched surface at each position.
In order to analyze the interdendritic fluid flow of macrosegregation that mainly happens in the middle and late period, the period with 0 to 0.7 liquid fraction in
Figure 10 will be the focus of research (e.g.,
Figure 11). According to the near-equilibrium solidification model (NESM) where no diffusion occurs in solids and finite diffusion occurs in liquids [
7,
26,
27], it is assumed that the distribution in
Figure 11 should be expressed well by Equation (3) at all positions. Equation (2) is obtained by using the Scheil equation form to effectively describe element distribution in the near-equilibrium solidification model [
28,
29].
As
can be considered as solid content
on the basis of Equation (1), Equation (3) can be obtained on the basis of Equation (2) and be used to fit the content distribution like that in
Figure 10 using the Levenberg–Marquardt algorithm [
30]. These fitting calculations were carried out using Microsoft Excel software.
where
is the solute content at the solid phase of the solidifying interface.
is the effective solute redistribution coefficient at the solidifying interface in the NESM.
is the original solute content before solidification.
is the liquid fraction.
where
,
, and
is the original dimensional quantity of carbon content. By Equation (1),
should be equal to 1 under ideal conditions and is the calculated result of
.
The difference of the distribution characteristics between the 0 to 0.7 liquid fraction and the 0.7 to 1.0 liquid fraction may be attributed to rare interdendritic flow in the initial period with nucleation and original growth. In fact, it is not easy to obtain the whole element distribution of a dendrite grain, especially for the distribution in the final stage of solidification, because some more complicate reactions (e.g., eutectic reaction) will occur for multicomponent alloys such as steel. Moreover, only in the two-dimensional surface of the actual ingot, the morphology appearing in the surface may not be representative of the three-dimensional grains structure. Owing to the above reasons, 12.5 mm long is measured to obtain enough data, which is more than 10 times a dendrite’s width. Thus, even though the surface morphology information displays incompletely for the whole grain structures at some points, the statistically average features of a rearranged distribution of a great many heights could represent effectively the segregating element content distribution of the position by eliminating the deviation of original data as far as possible.
All fitted results at each positions are shown in
Table 2,
Table 3,
Table 4 and
Table 5. The determination coefficient
is calculated as Equation (4) [
30]. When
is closer to 1, the fitted formula can express the original data more effectively. Because all R
2 are almost bigger than 0.9 (more than half of them bigger than 0.95), it means that Equation (3) is a good formula to quantitatively describe the content distribution with 0 to 0.7 liquid fraction at all positions.
where
is the original value.
is the average of the original value.
is the corresponding regression value.
Based on fitted parameters and Equation (3),
is obtained on the basis of
value. Meanwhile, the fitted
is almost equal to 1 in
Table 2,
Table 3,
Table 4 and
Table 5, which indicates that Equation (2) and the NESM are suitable to describe the distribution because the ideal
is equal to 1, further indicating that the above method is suitable to calculate the
value.
Meanwhile, according to the theory suggested by Burton and his co-workers [
27,
31],
at the solidifying interface in the NESM can also be calculated theoretically by Equation (5). Then, the average thickness of concentration boundary layer
can be obtained by Equation (6).
Then,
where
is the average thickness of the concentration boundary layer.
is the mass diffusivity of the carbon element, 0.005 m
2/s [
32].
is the ideal solute redistribution coefficient and is chosen as 0.1 for the carbon element [
33].
is the corresponding growth velocity magnitude in the solidification unit along the measuring line at each position, which can be calculated by Equation (7).
where
is the measuring length at each position, 12.5 mm.
and
are the liquidus temperature (1466 °C) and solidus temperature (1328 °C) of GCr15.
is the cooling rate (°C/s), which is calculated by thermal simulation of the finite element model, and the governing equations are Equations (8) and (9) [
33]. Axial heat conduction in the casting direction was ignored.
where
is the temperature.
is time.
is the thermal conductivity.
is the density.
is the specific heat. The evolution of the latent heat during the solidification was incorporated into the calculation by using the effective specific heat method, as shown in Equation (9) [
34].
where
is the effective specific heat.
is the latent heat and
is the solid fraction.
Combining measured
in
Table 2,
Table 3,
Table 4 and
Table 5 with the aforementioned equations,
can be calculated as shown in
Table 2,
Table 3,
Table 4 and
Table 5. Regardless of whether it is laminar flow or turbulent flow, there is a boundary layer during the initial stage that can be assumed to feature laminar flow characteristics, especially in the limited length on a flat plate [
35]. Thus, in this research, the interdendritic flow on dendritic grain structure in each position can be considered as the flat plate flow model because there is only enough space to flow during the solidifying process. According to the dimensionless equation in the transport phenomena of the boundary layer theory [
32,
35], the average thickness of the velocity boundary layer
can be calculated by Equation (10) after getting
.
where
is Schmidt number.
is the dynamic viscosity, 0.0067 Pa·s, and
is the mass diffusivity of the carbon element, 0.005 m
2/s, and
is the density, 7600 kg/m
3 [
32].
Based on the boundary layer theory [
34,
35], the change of thickness of the velocity boundary layer
with velocity
and flow length
can be described by Equation (11), the average thickness
in
long zone can be expressed by Equation (12).
From Equation (12), the equivalent flow velocity
can be calculated by Equation (13):
where
is obtained by Equation (10).
is chosen as 1 mm based on the measuring size.
Meanwhile, there is also the original thickness of the concentration boundary layer without fluid flow, which can be calculated by Equation (14), only considering molecular diffusion [
36].
Based on Equation (14), when
= 1.01, the original thickness of the concentration boundary layer
can be obtained by Equation (15). Then the original thickness of the velocity boundary layer
can also be calculated by Equation (10).
On the basis of
and
,
and
can be obtained. Then, the corresponding velocity can be calculated. Finally, the difference between
and
is the magnitude of interdendritic fluid flow velocity
, as shown in Equation (16). The corresponding flowchart is also shown in
Figure 12.
Following is the calculation method for the direction of interdendritic flow velocity with respect to the above
. In
Figure 4, the measuring heights of 1 mm length α and β should have the same features if there is no fluid flow because the left and right sides are almost under the same solidifying conditions for their limited distance. Then the standard deviation
as shown in Equation (17) [
30], is firstly introduced to measure the fluctuation extent of left or right side heights.
where
is the etched surface height at
point.
is the average value of the etched surface height.
is the number of the point.
where
is the difference value.
is the standard deviation of the left side, and
is the standard deviation of the right side.
As shown in Equation (18), the standard deviation difference between left side and right side is introduced to measure the velocity direction. As a matter of fact, because interdendritic fluid flow can induce more stochastic solidification behavior [
7,
36], it was assumed in this research that the main fluid flow direction is from right side to left side if
is positive, and the main fluid flow direction is from left side to right side if
is negative. The absolute value of
should indicate the extent along the growth direction. Actually, interdendritic fluid flow exists in three-dimensional space, and the calculated velocity magnitude above is also a three-dimensional value. Considering the surface from the inner curve side to outer curve side as
x-positive axis, the grain growth direction from bloom left edge to center as
y-positive axis, and the casting direction as
z-positive axis, as shown in
Figure 13, only the velocity direction component
on
xz plane can be obtained because only left and right side heights on the cross section are measured here.
As to the degree
, its value can be calculated by Equation (19), assuming the position with the largest absolute value of difference has the same or symmetric degree (0° or 180°) to grain growth direction. The degree calculation flowchart is also shown in
Figure 14.
where
is the degree of fluid flow direction on the
xz plane with respect to the
x-positive axis in
Figure 13, and arccos is the anticosine sign.
is the standard deviation difference, which has the maximum absolute value among all points. The denominator in Equation (19) is the negative value of the absolute of
.