Chemical analyses with computed values for the liquidus and solidus temperatures
1. Introduction
The quest for billets and blooms production in the continuous casting of carbon steels with more stringent quality demands in internal soundness, free from surface defects, and internal porosity has increased the need for more insight in the appraisal of the phenomena associated with solidification under industrial conditions. Medium and high carbon steels cover a broad range of manufacturing products; nevertheless, the production of this type of material with a constant high percentage of primechoice products remains a tough subject to analyze, understand, and more difficult to attain in practice. These carbon steels exhibit relatively low liquidus and solidus temperatures, with these values decreasing even further as carbon levels increase. It is understood that the blooms produced must be 100% crystallized (solid) at least before cutting to the delivered lengths is accomplished. It is realized that the larger the crosssection of the produced cast product the more time it requires in order to solidify completely. Bigsized cast products (or blooms) are more desirable for mainly two reasons: (a) large crosssections are associated with large values of mass per unit length, hence, productivity is favored, and more important (b) large crosssections are subject to larger values of area reduction once rolled, giving products with smaller possibility to quality degradation. On the other hand, the proper control of the cooling intensity in the secondary cooling zones (airmist spray zones) upon the solidifying product in order to avoid surface/subsurface defects in the unbending regions of the caster, places limitations on the casting speeds, and therefore productivities are not always at the desired levels. In practice, fundamental operation parameters like casting speed, casting temperature, and coolingwater consumption at the secondary (spray) zones per produced mass of steel are among the most critical ones that the operator should keep in mind, once these have been analyzed, and their impact upon quality has been realized. Furthermore, from the early stages of medium and highcarbon steels casting, the operator has appreciated that it has been impossible to attain the high levels of productivity as with low and mediumlow carbon steels without loss of internal quality. This is so because the low solidus temperatures that these grades exhibit become even lower in a dynamic way due to the local cooling rates that affect the solidification mechanism. Indeed, microsegregation phenomena become more pronounced for these types of steels reducing even further the temperature at which the product becomes 100% solid, or in other terms, the solid fraction becomes one. There are some correlations for the solidus temperatures that unfortunately do not hold appreciably well, under varying cooling conditions, as it normally happens in the industrial continuous casting process. Nevertheless, the liquidus temperature is computed with great precision based upon the liquidsteel chemical analysis, and in this way, the industrial parameter known as superheat, which is the difference between the casting and liquidus temperatures, is calculated correctly; it is known that superheat is of paramount importance in the casting process. In this study, an attempt was carried out in order to shed some light into the effect of the various casting parameters upon the internal quality of the produced blooms in the continuous caster of Stomana, Pernik, Bulgaria. The biggest size of blooms produced currently in this caster is 300 x 250 mm x mm, and most medium and highcarbon grades are produced in this size. A heat transfer and a microsegregation model were coupled and put into effect in order to facilitate the analysis of solidification along the caster length. Consequently, the solid fraction in the mushy zone, which is actually the intermediate zone between liquid and solid, was computed across a bloom section at any point along the caster, or in other words, from the meniscus level in the mold till the point of analysis. In addition to this, temperature and localcooling rate distributions were also computed in a similar manner. Different operating conditions were fed into the simulation model in order to compute the required metallurgical lengths, or in other words, the effective cutting lengths that obeyed the unity solid fractions in the mushy zone along the centerline of a bloom.
2. Quality problems associated with internal soundness/central porosity
From the early times of the continuous casting for medium and high carbon steels, it has been realized that central porosity as a quality problem seems inevitable. Figure 1 shows a picture from a macroetched high carbon bloom crosssection.
It is pointed out that for high carbon steels the tendency for central porosity generation is very large no matter how well liquid steel is treated, and how successful the casting process is performed. With time, the continuous casters manufacturers realized that this problem can be abided if electromagnetic stirring (EMS) was applied not only in the mold but in specific positions along the strand, mostly known as strand (SEMS) and final (FEMS) positions. FEMS position is considered the position at which the mushy zone along the centerline becomes solid, or the position around the final solidification of the product. This is an effective technical solution to lessen the problem and is currently applied in some caster installations worldwide. Another promising technical solution is the dynamic soft reduction, in which the part of the bloom which approaches final solidification is subject to a compressive force that slightly reduces its size in one direction but mechanically eliminates the central porosity problem. It is interesting to note that there are installations worldwide that currently apply both technical solutions for internal porosity minimization. However, no matter whether an installation applies one technical solution or another, it is really intriguing to try to figure out why and how this happens, and most importantly up to what extent, depending upon the various casting operating conditions. Without some knowledge upon this problem for a specific installation, one may not appreciate all the phenomena involved in, and maybe a definitive solution may not be successfully attained even after the installation of the discussed technical solutions. That is why it was decided to get some extra information on the subject before the installation of any technical solution might be applied at the Stomana bloom caster.
3. Literature review mostly oriented to quality problems for medium and high carbon steels
Superheat was one of the most fundamental factors recognized from the early years of continuous casting especially for medium and high carbon steels. In an early report [1], pilot plant tests were performed casting 150 x 150 mm x mm billets of high carbon steels. It was proven that at low superheats or even subliquidus casting temperatures, the centerline segregation was minimized. The electromagnetic stirring at the mold (MEMS) exhibited some benefits, and the application of EMS at the strand (S) and final (F) stages of solidification started being installed in some casters worldwide. In a study [2], it was found that the combination of EMS, that is, (S+F)EMS for blooms and (M+S+F)EMS for billets, is the most effective method for reducing macrosegregation among various EMS conditions, causing them to solidify more rapidly during the final stages of solidification, providing more finely distributed porosities and segregation spots along the central region. The optimum liquid pool thickness was found to decrease as the carbon content increased, which may be attributed to longer solidification times in the solid fraction range from f_{s}= 0.3 to 0.7. By gaining experience [3] in an actual caster installation, they concluded that a mold tube with a parabolic taper was proven good enough qualitywise for the continuous casting of medium and high carbon steels for carbon contents up to 0.55%. In another study [4], a coupled model was developed consisting of a cellular automaton scheme (CA) simulating the grain structure formation during solidification, and a finite difference scheme simulating the macroscopic heat transfer and solute transport in the continuous casting process. Columnar to equiaxed dendritic transition was effectively reproduced. The effect of superheat on the solidification structure was analyzed, verifying the empirical fact that increasing superheat the columnar dendritic growth increases against the equiaxed one. Under industrial conditions [5], SEMS applied in the continuous casting of 150 x 150 mm x mm billets reduced centerline segregation up to a degree. However, increasing field intensity, deterioration upon the attained quality improvement was recorded. Some interesting fundamental research, as well as industrial achievements regarding medium and high carbon steels were presented in the recent European continuous casting conference [618], revealing the broad research and practice that may be developed in the field during the coming years. In Ref. [6], they used the liquid–solid interface energy as the main property in order to study the microsegregation during solidification. They concluded that convection effects influenced microsegregation behavior of the studied high carbon (C ≤ 0.7%), and high manganese steels. In another work [7], a 3D mathematical model was used to analyze the characteristics of magnetic field, flow field, and solidification of molten steel in the mold with electromagnetic stirring for a 260 x 300 mm x mm bloom. A dominant swirling motion at the transverse direction described the flow in the mold; the electromagnetic field was computed with similar values to those measured. They took under consideration the airgap formation in the corners of the bloom adjacent to the mold. A summary of the actions taken to increase the productivity of Tenaris casters and to ensure highquality standards in the produced round blooms for low, medium, and high carbon steels was presented in Ref. [8]. An inhouse heat transfer model was developed to simulate the temperature distribution at various casting conditions. In addition to this, a rigidviscoplastic model for simulating the thermal strain effects was developed to assess the potential risk of internal and surface cracks. Industrial practice was improved [9] in the field of highcarbon steel casting by the introduction of EMS not only in the mold, but in specific positions in the strand, and sometimes in the position of the final solidification front. Typical values for solid fraction along the central axis where the FEMS is effective were found to be in the range of 0.1–0.4. Just for the sake of sense, comparing the findings between the published works [2] and [9] for the valid solidfraction range for a successful FEMS application, it is derived that depending upon a specific caster installation different approximations may yield to optimum solutions. In a recent monumental industrial installation [10], the excellent quality results in the production of –among others – medium and high carbon steels were successfully attained by a 2strand vertical caster (in order to avoid extra strains from the bending/unbending of the strand, which are inevitable to customary curved casters) for the production of big sections of blooms (up to 420 x 530 mm x mm). The caster managed to attain high quality results with the simultaneous application of a soft reduction system plus a moveable FEMS system per strand. Chaotic phenomena seem to take place [11] in the continuous casting of steel billets, and specifically porosity chains seem to follow chaotic behavior along a cast billet. In other words, the chain of voids that are formed in the central zone along a billet and are generally coupled with segregation exhibit a spacing fluctuation in an erratic, random manner; stochastic modelling was applied on the basis of empirical time series in order to capture much of the dynamics. The lowering of the solidus temperature with respect to carboncontent increase for steels seems to be magnified by the boron effect [12]. Indeed, high Mn medium carbon steels with B higher than 40 ppm exhibit a very low solidification temperature at about 1140°C. In general, it has been verified that for high carbon steels (C≤1%), the addition of B introduces the possibility of a retrograde melting phenomenon retaining liquid at temperatures around 1100°C. In actual practice, a proper secondary cooling scheme should be applied in order to minimize remelting behavior. In a similar study [13], low and medium carbon boronsteels (B≈30–40 ppm) exhibit a sharp decrease of hot ductility at about 1100°C for a 0.7%Mn content. It was explained on the basis of BN formation after the initial MnS formation; castability is therefore reduced for these types of steels. The implementation of soft reduction at the Voestalpine Stahl Donawitz bloom caster for the continuous casting of high carbon (0.80–1.05%C) rounds exhibited positive quality results [14]. On the other hand, the EMS underlined the proper positioning limitations that made the implementation very difficult to attain reliable and reproducible quality results. Proper design can be successful in as far as quality results are concerned even for small radius casters. In a revamping case [15], good quality results were obtained in a relatively small radius (~5m) caster through multiradius unbending. Furthermore, the addition of MEMS, SEMS, and FEMS gave rise to the successful casting of medium and high carbon steels with billets crosssections of 110 x 110 up to 160 x 160 mm x mm. Proper design by reducing roll pitch in the areas where soft reduction was applied and implementing EMS and proper secondary cooling led to the required quality improvements in the revamping of another caster [16] for the production of high carbon steels. The need for fundamental research is illustrated in the following two published works. In the first one [17], an in situ material characterization by bending (IMCB) 3pointbending test was developed to simulate crack formation that takes place during continuous casting for most carbon grades; after the test, the strains were calculated using a simulation model in ABAQUS. In the next work [18], it was explicitly verified that microsegregation phenomena are of paramount importance in the calculation of the final solidification front in order to apply soft reduction efficiently; specifically, an error of about 40°C in the estimation of the solidus temperature may result in an uncertainty of about 1.2 m in the determination of the correct soft reduction point.
4. Microsegregation effects
The liquidus temperature for a specific steel chemical analysis is calculated with very good accuracy. However, the correct calculation of the solidus temperature is not always that easy. Nevertheless, there are some formulas for the computation of the solidus temperature based upon the specific chemical analysis. One is given by equation (1) as presented in reference [19]:
The Simple model [20] for microsegregation gives a thorough fundamental analysis for the computation of both the liquidus and solidus temperatures. According to this, the computation of the liquidus temperature is given by equation (2) based on chemical analysis only:
However, for the computation of the solidus temperature the same model [20] requires more computational effort, as the solid fraction in the mushy zone (in which solid and liquid coexist) depends not only upon temperature but local coolingrates as well. This can be described as:
The function












1466  1366  1325  0.77  0.28  0.76  0.01  0.02  0.32  0.00 

1493  1435  1404  0.44  0.23  0.63  0.01  0.02  0.08  0.00 
Equation (2) was used for the calculation of the liquidus temperatures, and equation (1) for the solidus temperature
5. Heattransfer mathematical approach
The general 3D heattransfer equation that describes the temperature distribution inside the solidifying body is given by the following equation according to Refs. [21, 22]:
The source term
Furthermore,
The boundary conditions applied in order to solve (6) are as follows.
The heat flux in the mold was computed based on a recent treatment [7] that takes under consideration the air gap formation at the corners of the bloom, in conjunction with another older analysis [24] that came up with more precise heattransfer coefficients in the mold. In the latter, the heat transfer coefficient at any position inside the mold is given by:
The heat flux at any position in the mold is given by:
Integrating (8), the average value for heat flux that is transferred through the walls of the mold is:
Finally, the mold heat transfer coefficient was adjusted for the airgap effect at the corners of the bloom [7], according to the following sets of equations:
This formulation reasonably neglects the effect of contact resistance between the solidified shell of steel and the copper mold; this is a valid approach for blooms and big sections generally, as the soft shell bulges a bit and stays in contact with the copper mold in the central areas of the mold, retracting somewhat at the corners. Furthermore, this analysis was performed on similar sized sections (300 x 260 mm x mm) [7]. The heat fluxes due to water spraying and strand radiation in the secondary cooling zones were calculated using the following expressions:
where
where
where
In mathematical terms, considering one quartersection of the bloom assuming symmetry is valid, the aforementioned boundary conditions below mold can be written in compact form:
where
Finally, the initial temperature of the pouring liquid steel is supposed to be the temperature of liquid steel in the tundish:
The thermophysical properties of carbon steels were obtained from an older published work [27], in which the properties were given as functions of carbon content for the liquid, mushy, solid, and transformationtemperature domain values; this also gave rise to the advantage of eliminating the source factor (
6. Numerical solution
This specific article is part of a series of published works with respect to the numerical solution of the heat transfer equation in 2D and 3D domains [2830]. The strongly implicit method as practiced by Patankar [23] was applied. Although a grid of 200 x 200 nodal points was sufficient to stabilize results with a maximum error of 10^{2}°C for each nodal point in the bloom quartersection, a final grid of 400 x 400 nodal points was selected for the final computations. The selected time interval (
Calculation of the temperature distribution inside a bloom crosssection at a specific location at the caster.
Calculation of the local coolingrates distribution at the same crosssection and position.
Calculation of the solid fraction in the mushy zone at the same crosssection and position.
The dynamic computation of the solidus temperature (at the solid fraction value equal to 1,
In this way, the computed results were stored in the disk and the system proceeded for the next time step; stepbystep the solidification front reached the center of the bloom. In average, about 30–40 iterations per time step sufficed for convergence. The local cooling rates were computed at every nodal point, as follows:
7. Results and discussion
The Stomana bloom caster has a casting radius of 12 m, with MEMS and no extra EMS or soft reduction along its four strands. Nevertheless, it is interesting to know the effect of critical casting parameters in order to maximize casting speed and hence, productivity, keeping the most attainable good internal porosity as much as possible. Consequently, the effect of parameters like
The adopted values for the casting parameters and the results for the required solidification lengths, for the MC and HC grades selected, are presented in Table 2. Initially it was designed for 12 cases; some more runs were added as of existing preliminary data (included in the statistical analysis).
The effect of airgap formation upon the temperature distribution in the bloom corners was also verified in this study in accordance with a recent research work published in [7]. Specifically, Figure 7 depicts typical results revealing this important effect. Increasing the percentage of contact loss in the bloom corners due to airgap formation inside the mold the exitmold temperatures in the corners increase as well. On the other hand, a statistical analysis that was performed for the effect of the casting conditions upon the required solidification length for the two selected grades as presented in Table 2 showed that the airgap formation inside the mold has no effect upon






(K) 




(K) 


1  33.37  0.70  45  0.536  12  36.21  0.55  45  0.480  12 
2  32.43  0.70  30  0.466  12  35.29  0.55  30  0.480  12 
3  33.13  0.70  30  0.200  12  35.93  0.55  30  0.200  12 
4  26.12  0.55  30  0.200  12  26.40  0.40  30  0.200  12 
5  26.03  0.55  30  0.200  0  26.40  0.40  30  0.200  0 
6  33.37  0.70  45  0.536  0  36.02  0.55  45  0.561  0 
7  26.12  0.55  45  0.561  0  26.27  0.40  45  0.606  0 
8  27.04  0.55  45  0.200  0  26.53  0.40  45  0.450  0 
9  33.37  0.70  45  0.536  6  27.13  0.40  45  0.200  0 
10  28.85  0.625  30  0.469  6  36.12  0.55  45  0.480  6 
11  27.04  0.55  45  0.200  6  28.95  0.45  30  0.479  6 
12  29.58  0.625  30  0.200  6  27.13  0.40  45  0.200  6 
13  30.01  0.65  30  0.468  0  29.63  0.45  30  0.200  6 
14  32.43  0.70  30  0.466  0  —  —  —  —  — 









47.589  ***  64.064  *** 

0.0716  ***  0.052  *** 

–2.636  ***  –2.209  *** 
Intercept  –1.782  ***  –0.383  *** 
Correlation coefficient  0.9997  0.9999  
Standard error of estimate  0.049  0.0545  
Fvalue  16770, on 3 and 10 DF  26920, on 3 and 9 DF 
In this way, a regression formula of the form
is derived, where
More than 99% statistical importance is signified by 3stars (***) on Table 3, according to R. Figure 8 summarizes the results presented on Table 3 in graphical form. For practical purposes the maximum allowable caster length was considered to be 33 m (instead of 34 m that the Stomana caster actually is), introducing a small safety factor.
Figures 9 and 10 illustrate results for the localcooling rates and temperatures inside a bloom for the MC grade selected, under the same casting conditions at a specific location from the liquidsteel meniscus level in the mold. Figure 9 shows that there is a distribution of local coolingrate values at any instance inside the bloom. For this case, a simple statistical analysis gave an average value
Figure 11 depicts the local cooling rates inside a bloom section at about 24.98 m from the liquid steel meniscus level. In this case, the statistical analysis gave an average value for the local cooling rates of μ = 0.204 °C/s, with a standard cooling rate of 0.074, and min and max values of 0 and 1.682, respectively. The local microsegregation analysis gave a solidus temperature of 1307°C which is shown in Figure 12, and which is very close to the solidus temperature of 1306°C computed from the average coolingrate of a section at the instance under discussion. One may wonder about the large error upon the estimate of the shell thickness if the whole analysis relied only upon the a priori calculated value (
Some more results of this nature are presented in Figures 13 and 14 at different casting conditions than those presented in the previous set of data shown in Figures 9 and 10 for the MC steel grade selected. Figure 13 exhibits local cooling rates at an approximate distance of 25 m from the liquid steel meniscus level in the mold. In this case, a short statistical analysis for the local cooling rates gave an average value of μ = 0.264 °C/s, with a standard deviation of 0.087, and min and max values of 0.001 and 1.112, respectively.
Figure 14 shows the temperature distribution inside the bloom section at the conditions presented in Figure 13. In this case, the computed solidus temperature at the solidification front was found to be 1389°C, a temperature that also defines the shell thicknesses (Sx and Sy) along the x and y axes.
Similar results are also presented in Figures 15 and 16 for the HC steel grade selected. At different continuous casting conditions than the ones presented in Figures 11 and 12, these results show data just a couple of meters apart from the final stage of solidification. Figure 15 illustrates in graphical form that the local cooling rates remain as a distribution of values with an average value of μ = 0.230 °C/sec, with a standard error of σ = 0.073, and min and max values of 0 and 1.729, respectively. Figure 16 depicts the temperature distribution inside the bloom section, defining the shell thicknesses along the x and y axes (Sx and Sy), at the computed solidus temperature of 1306°C.
One last comment about the computed values of local cooling rates: once the shell formation has been created inside the mold, the average values of the local cooling rates are more or less stabilized to specific values. For example, for the cases presented in the Figures 9 through 16, the overall average values for the local cooling rates are about 0.234°C/s for the MC and 0.217°C/s for the HC, respectively. Supplying these values to the microsegregation model, the values of 1389°C and 1306°C for the solidus temperatures of the MC and HC grades can be deduced, respectively. In this way, a priori calculated values for the solidus temperatures may be used with better precision in heat transfer applications that rely upon preselected values only.
The prediction of the grain size of the solidified metal structure as described in Ref. [27], together with the percentage of the equiaxed zone that may be formed depending on the prevailing heattransfer conditions may be one part for future work; another part could be the analysis of thermal stressstrain phenomena that act upon the solidifying shell.
8. Conclusions
The installation of strand EMS has been decided for the Stomana caster. The target is to come up with blooms having better internal soundness than the one presented in Figure 1. However, some fundamental points are covered with the help of this study. Summarizing:
The casting conditions with respect to the required casting length for complete solidification have been analyzed, and the effect upon expected productivity per grade is known.
The appreciation of the solidification phenomena is impossible without coupling the microsegregation analysis, especially for medium and high carbon grades. Once the SEMS is installed, the specific range of critical solid fractions required for the proper performance of the stirrers can be linked with the operating conditions of the caster.
The difficulty in predicting the solidus temperature at varying cooling conditions especially for high carbon steels has been illustrated. Nevertheless, a shortcut that can generate solidus temperatures to be used as a priori values in heat transfer models has been presented; it only takes some runs for the appropriate heattransfer models to compute average values for the local cooling rates, which then may be fed to standalone microsegregation models to calculate the corresponding solidus temperatures. Consequently, simple heat transfer models can simulate casting with less error in the involved solidification phenomena for medium and high carbon steels, by taking as input the solidus temperatures derived in the aforementioned manner.
The great importance of heat transfer analysis on the domain of medium and high carbon solidification is proven once more. It is of paramount importance that mankind has this kind of tool for shedding light into similar type of complex industrial conditions.
Nomenclature
Δx, Δy; Distance between adjacent nodal points along the x and yaxis, respectively, in m
ε; Emissivity
ρ; Density, in kg m^{3}
σ; The Stefan–Boltzmann constant, which is equal to 5.67 10^{8} W m^{2} K^{4}
ω; Overrelaxation factor
Subscripts
0; Referring to an initial value
F, f; Fluid (WF referring to water as cooling fluid)
S, s; Surface, or solid
r, c; Radiation, convection
m; Average value, or referring to the mold
ag; Air gap is considered
Acknowledgments
The author is grateful to the topmanagement of Sidenor SA for the continuous support upon these types of studies, as well as for the permission of publishing this piece of work. My continuous gratitude and respect to Professor Rabi Baliga from the Mechanical Engineering Department of McGill University, Montreal, Canada, who introduced me to the field of computational fluidflow and heat transfer should also be acknowledged.
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