materials

Article Investigations of Liquid Steel Viscosity and Its Impact as the Initial Parameter on Modeling of the Steel Flow through the Tundish

Marta Sl˛ezak´ 1,* and Marek Warzecha 2

1 Department of Ferrous Metallurgy, Faculty of Metals Engineering and Industrial Computer Science, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Kraków, Poland 2 Department of Metallurgy and Metal Technology, Faculty of Production Engineering and Materials Technology, Cz˛estochowaUniversity of Technology, Al. Armii Krajowej 19, 42-201 Cz˛estochowa,Poland; [email protected] * Correspondence: [email protected]

Received: 14 September 2020; Accepted: 5 November 2020; Published: 7 November 2020

Abstract: The paper presents research carried out to experimentally determine the dynamic viscosity of selected iron solutions. A high temperature rheometer with an air bearing was used for the tests, and ANSYS Fluent commercial software was used for numerical simulations. The experimental results obtained are, on average, lower by half than the values of the dynamic viscosity coeﬃcient of liquid steel adopted during ﬂuid ﬂow modeling. Numerical simulations were carried out, taking into account the viscosity standard adopted for most numerical calculations and the average value of the obtained experimental dynamic viscosity of the analyzed iron solutions. Both qualitative and quantitative analysis showed diﬀerences in the ﬂow structure of liquid steel in the tundish, in particular in the predicted values and the velocity proﬁle distribution. However, these diﬀerences are not signiﬁcant. In addition, the work analyzed two diﬀerent rheological models—including one of our own—to describe the dynamic viscosity of liquid steel, so that in the future, the experimental stage could be replaced by calculating the value of the dynamic viscosity coeﬃcient of liquid steel using one equation. The results obtained support the use of the author’s rheological model for the above; however, this model still needs to be reﬁned and extended to a wide range of alloying elements, mainly the extension of the carbon range.

Keywords: steel; viscosity; tundish; numerical modelling

1. Introduction Today, the continuous casting process is the dominant method of steel casting. The basic aggregates of the steel continuous casting machine are tundish and mold. The multitude of functions fulﬁlled by the tundish is the reason for the undertaking of many studies by scientists and steelmakers from all over the world of science and industry. The positive aspects of the ﬂow of liquid steel through the tundish are widely used in practice. Research undertaken by scientists is still necessary to identify speciﬁc parameters of technological processes and to further optimize the process. They can be partly carried out directly on devices operating in industrial conditions, using speciﬁc control and measuring devices. The results of such research (called industrial) are of great practical importance, as they enable their direct application in real conditions. Unfortunately, due to the speciﬁcity of the processes carried out in metallurgical devices, conducting industrial research is limited in scope and usually concerns individual measurements. In a situation where it is impossible, diﬃcult and/or expensive to carry out relevant tests directly on a working device, model tests using physical [1,2] or numerical [3–5]

Materials 2020, 13, 5025; doi:10.3390/ma13215025 www.mdpi.com/journal/materials Materials 2020, 13, 5025 2 of 20 modeling techniques are conducted. Due to the simpliﬁcations used, such research also has some limitations, but they have nevertheless become equal research tools for scientists. This applies to both laboratory tests carried out on physical models and numerical tests carried out using computational ﬂuid dynamics (CFD). Simulating the process is cheaper and requires much less input, material or human. When conducting numerical simulations, one should be aware not only that the mathematical models used are simpliﬁed, but also how the simpliﬁcation aﬀects the reliability of the results obtained using them. When modeling the ﬂow of liquid steel in steel aggregates, including the tundish, parameters such as density and viscosity of liquid steel are often taken from the literature data (the most commonly adopted density of liquid steel is about 7000 kg/m3 for the ladle process and 7010–7040 kg/m3 for ﬂows in tundish, while the viscosity is about 0.006–0.007 Pas). How big a mistake do we make in this way? The calculation results presented in this article are an attempt to at least partially answer the above questions. Rheological tests were carried out to experimentally determine the viscosity of the tested steel grades. Rheology is interested in the behavior of bodies under the inﬂuence of force applied to them. Reverse issues, i.e., the potential response of the system to the given stress, are also of interest to this ﬁeld of science [6,7]. In rheological terms, we are interested in bodies in various states of aggregation: solid, liquid, solid–liquid [8–10]. In this work, rheology has been included in relation to material problems, and the area of interest has become iron solutions and their viscosity. In the literature on the subject, there are numerical data showing that the physical quantity, which is viscosity, is low in the case of iron solutions, similar to water viscosity [11,12]. There are also theories that completely liquid steel is a Newtonian ideal. Thus, it is assumed that the viscosity, i.e., the ratio of shear stress to shear rate [6–9], depends only on temperature changes, while independent of the force with which we interact with the system. The conclusion is that the viscosity curve of Newton’s perfectly viscous ﬂuid is always a straight line, regardless of changes in the shear rate value, i.e., the force that we give to the system under test [6–8].However, the development of today’s technology has also resulted in the development of measuring apparatus that can allow the veriﬁcation and testing within much wider limits and using diﬀerent values of measurement parameters. What is more, the sensitivity of the devices can capture the diﬀerences, often very small, between individual species of iron solutions that diﬀer in their chemical composition. At present, the subject of rheology of iron alloys has been mainly dealt with in relation to metal ﬂow and the possibility of its formation in liquid states, in relation to the so-called thixoforming [13–15]. Often, these studies were accompanied by experiments conducted with light metals, mainly aluminum and magnesium [16–22]. The authors have previously conducted rheological studies in relation to both liquid and solid–liquid iron solutions, from the point of view of observing the ﬂow of material used in various metallurgical processes [23–25]. The authors of this work made an eﬀort to experimentally determine the viscosity of the steel grades they were investigating as part of a larger project. In the next stage, they checked—through model tests—how the introduction of measured values into the numerical model would aﬀect the numerically obtained steel ﬂow forecasts through the tundish, in particular on the ﬂow structure and velocity proﬁle.

2. Experimental and Numerical Investigations Data from real rheological measurements can be used to model the process of metal behavior in the tundish of a continuous steel casting device. In software used for modeling the ﬂuid ﬂow process, there are usually built-in thermodynamic bases that contain the values of basic physical quantities [12]. However, with this approach to the problem and issues, we return to the use of unitary, single data, while the problem can be much more complex. In rheological terms, the value of dynamic viscosity coeﬃcient is not a single point, but the curve is drawn for diﬀerent temperatures, for diﬀerent chemical compositions, using variable values of rheological parameters (shear stress, shear rate) [26]. Only with this look can you fully understand Materials 2020, 13, 5025 3 of 20 the process and try to reproduce it by computer simulation. In the real process, the system is not aﬀected—most often—by one force of the same size, but we deal with a system of forces and a complexMaterials problem 2020, 13,and x FOR phenomenon. PEER REVIEW Thanks to modern research equipment and modern software,3 of 20 it is possibleis possible to carry to out carry high-temperature, out high-temperature, advanced advanced measurements measurements and reproduce and reproduce the entire the complexity entire of encounteredcomplexity processes.of encountered Thanks processes. to the Thanks use of experimentalto the use of experimental and numerical and studies, numerical the studies, paper presentsthe whatpaper impact presents some what assumptions impact some (simpliﬁcations) assumptions (simplifications) may have when may applied have when to the applied initial to andthe initial boundary conditionsand boundary of the numericalconditions of model the numerical of steel model ﬂow throughof steel flow the through tundish, the in tundish, particular in particular the liquid the steel parameter—whichliquid steel parameter is viscosity—on—which is viscosity changing—on the changing ﬂow structure the flow instructure the working in the working space of space a tundish. of a tundish. 2.1. Calibration Measurements 2.1. Calibration Measurements High-temperature rheological tests were performed on the FRS1600 rheometer equipped with a standardHigh DSR301-temperature head (Figure rheological1). This tests device were is performed a prototype on the machine FRS1600 used rheometer to study equipped high-temperature with a standard DSR301 head (Figure 1). This device is a prototype machine used to study high-temperature metallurgical systems, and was developed in cooperation with employees of Anton Paar GmbH (Graz, metallurgical systems, and was developed in cooperation with employees of Anton Paar GmbH Austria) and the AGH Department of Ferrous Metallurgy (Kraków, Poland). The rheometer is equipped (Graz, Austria) and the AGH Department of Ferrous Metallurgy (Kraków, Poland). The rheometer is withequipped a resistance with tube a resistance furnace, tube enabling furnace, the enabling temperature the temperature in the sample in the to besample reached to be to reached about 1530to ◦C. In addition,about 1530 during °C. In the addition measurement,, during the the measurement, sample resides the insample a protective resides in atmosphere a protective of atmosphere argon, fed into the furnaceof argon, from fed into below. the furnace from below.

FigureFigure 1. 1.Diagram Diagram ofof FRS1600FRS1600 high high temperature temperature rheometer. rheometer.

TheThe high-temperature high-temperature rheometer rheometer is a a very very sensitive sensitive device device (torque (torque accuracy accuracy of 0.001 ofmNm), 0.001 and mNm), and thethe head is is equipped equipped with with anan air airbearing. bearing. In this In thisrheometer rheometer,, the torque the torque values valuesare measured are measured by the by the headhead andand thenthen the the software software calculate calculatess the the values values of shear of shear stress, stress, viscosity viscosity etc. The etc. parameters The parameters of the of geometry of using measurement system are implemented in the Rheoplus software (Anton Paar the geometry of using measurement system are implemented in the Rheoplus software (Anton Paar GmbH, Graz, Austria ) before starting the experiments. GmbH, Graz,The control Austria of )the before furnace starting is possible the experiments. from the rheometer software, which allows us to program Thethe experiments control of thewith furnace changes is of possible temperature. from the rheometer software, which allows us to program the experimentsTemperature with is changes measured of in temperature. the furnace (thermocouple in the furnace) and then the temperature Temperaturein the sample is is calculated measured automatically in the furnace by (thermocouplethe software. In the in theRheoplus furnace) software and then, the temperature the temperature in thecalibration sample is equation calculated wasautomatically implemented. The by theaccuracy software. of temperature In the Rheoplus measure software,ment is verif theied temperature during calibrationhigh-temperature equation was measurement implemented. of the Theviscosity accuracy pattern. of temperature measurement is veriﬁed during high-temperatureThe apparatus measurement can carry out of the rotational viscosity and pattern. oscillation tests as well as material softening tests. The measurement apparatus can is made carry in out a stationary rotational crucible, and oscillation in which a sample tests as of well the test ased material material softening is placed. tests. Inside the crucible, the spindle is immersed in the tested material. The crucible is placed inside the The measurement is made in a stationary crucible, in which a sample of the tested material is placed. ceramic protective tube, which is part of the heating furnace. The furnace is made of four SiC heating Insideelements the crucible, heated theby electric spindleity. is The immersed whole is protected in the tested from material. the outside The with crucible insulating is placedmaterial. inside The the ceramictemperature protective inside tube, the which furnace ispart is controlled of the heating by changing furnace. theThe power furnace supply is in made the measuring of four SiC and heating elementscontrol heated system. by The electricity. rate of heating The and whole maintaining is protected a constant from temperature the outside are with set insulatingin the Rheoplus material. The temperatureRheometer software inside the control furnace panel. is controlled The measuring by changing head driven the bypower an electric supply motor in the controls measuring the and controlrotational system. movements The rate ofof heatingthe spindle. and The maintaining spindle is suspended a constant on temperature a ceramic tube are located set in thein the Rheoplus air Rheometerbearing. software To ensure control low temperatures, panel. The the measuring head is cooled head by driven water byand an air. electric The rheometer motor controls uses a the rotational movements of the spindle. The spindle is suspended on a ceramic tube located in the

Materials 2020, 13, 5025 4 of 20

Materials 2020, 13, x FOR PEER REVIEW 4 of 20 air bearing. To ensure low temperatures, the head is cooled by water and air. The rheometer uses aconcentric concentric cylinder cylinder system system with with a rotary a rotary internal internal cylinder. cylinder. This is This called is calledthe Searle the measuring Searle measuring system system[27]. [27]. For these tests, a measuring spindle with a perforated side surface, 26.6 mm in diameter, and a cruciblecrucible with smooth smooth internal internal walls walls with with an an internal internal diameter diameter of of30 30mm mm were were used. used. Therefore, Therefore, the themeasurements measurements were were carried carried out out in in a normalized a normalized system system with with a a narrow narrow shear shear gap, gap, described in ISO3219ISO3219 and and DIN53019 DIN53019 [27 [27].]. The The diagram diagram below below (Figure (Figure2) presents 2) presents the measuring the measu rangering of range the device, of the usingdevice, a systemusing a withsystem an with internal an internal cylinder cylinder with a diameter with a diameter of 26.6 mm of 26.6 (for mm simplicity (for simplicity marked withmarked the numberwith the 27)number and the 27) symbol and the CC symbol (concentric CC (concentric cylinder). cylinder).

Figure 2. An envelope of the rheometer measuring range using a 26.6 mm spindle. Figure 2. An envelope of the rheometer measuring range using a 26.6 mm spindle. The measuring range of the device is very large. Even when using the “large” spindle and The measuring range of the device is very large. Even when using the “large” spindle and conducting research in so-called narrow shear gap, we can test a variety of substances, from water to conducting research in so-called narrow shear gap, we can test a variety of substances, from water to liquid steel and even liquid glass. liquid steel and even liquid glass. Figure3 schematically presents the measuring system: internal (bob—Figure3a) and external Figure 3 schematically presents the measuring system: internal (bob – Figure 3a) and external (cup—Figure3b) used. Materials for measuring tools for testing were selected in such a way that (cup – Figure 3b) used. Materials for measuring tools for testing were selected in such a way that the the surface of the tool did not react with the tested sample. The tools are made of zirconia-stabilized surface of the tool did not react with the tested sample. The tools are made of zirconia-stabilized aluminum oxide (96% Al2O3 + 1% ZrO2). aluminum oxide (96% Al2O3 + 1% ZrO2).

(a) (b)

Figure 3. Measuring tools used for research, (a) bob; (b) cup.

Materials 2020, 13, x FOR PEER REVIEW 4 of 20 concentric cylinder system with a rotary internal cylinder. This is called the Searle measuring system [27]. For these tests, a measuring spindle with a perforated side surface, 26.6 mm in diameter, and a crucible with smooth internal walls with an internal diameter of 30 mm were used. Therefore, the measurements were carried out in a normalized system with a narrow shear gap, described in ISO3219 and DIN53019 [27]. The diagram below (Figure 2) presents the measuring range of the device, using a system with an internal cylinder with a diameter of 26.6 mm (for simplicity marked with the number 27) and the symbol CC (concentric cylinder).

Figure 2. An envelope of the rheometer measuring range using a 26.6 mm spindle.

The measuring range of the device is very large. Even when using the “large” spindle and conducting research in so-called narrow shear gap, we can test a variety of substances, from water to liquid steel and even liquid glass. Figure 3 schematically presents the measuring system: internal (bob – Figure 3a) and external (cup – Figure 3b) used. Materials for measuring tools for testing were selected in such a way that the surface of the tool did not react with the tested sample. The tools are made of zirconia-stabilized aluminum oxide (96% Al2O3 + 1% ZrO2). Materials 2020, 13, 5025 5 of 20

(a) (b) Materials 2020, 13, x FOR PEER REVIEW 5 of 20 FigureFigure 3. 3. MeasuringMeasuring tools tools used used for for research research,, (a (a) )bob bob;; (b (b) )cup cup.. Before performing the correct rheological measurements, a motor adjustment test and measurementBefore performing in the air the (air correct check) rheological were performed measurements, (Figure 4), a motor to verify adjustment the technical test and condition measurement of the indevice. the air During (air check) the measurement, were performed the (Figure torque4), on to the verify measuring the technical head condition was measured of the device.along with During the thechange measurement, in the yaw the angle. torque The on value the measuringof the measured head was torque measured (blue curve) along withshou theld be change within in the the error yaw angle.limits of The ±0. value05 µN of (red the dashed measured lines). torque (blue curve) should be within the error limits of 0.05 µN ± (red dashed lines).

Figure 4. Air test chart.

The rheometer works correctly and the air bearing is functional if the measured torque is within the error range with the increasing angleangle ofof deﬂection.deflection. The resultin resultingg chart indicates that the device is in veryvery goodgood condition.condition. Before the measurements were started, the measuring spindle inertia was calibrated, thanksthanks toto which which the the correction correction was was made made for for a givena given tool tool used used for for measurements. measurements. Before Before the measurementsthe measurements were were taken, taken, low (atlow room (at room temperature) temperature) and high and temperaturehigh temperature measurements measurements on water on andwater so-called and so-called viscosity viscosity standards. standards. First, First, the dynamic the dynamic viscosity viscosity coeﬃ coefficientcient of water of water and and reference reference oil wereoil were measured. measured. The The following following measuring measuring system system was was used used for for the the following following tests,tests, whichwhich waswas later used for measuring liquid steel. The results will be presented as ﬂow curves (dependence between used. for measuring liquid steel. The results will be presented as flow curves (dependence between γ η 훾̇——shearshear rate and η—viscosity—viscosity coecoefficientﬃcient values).values). The referencereference oiloil usedused waswas characterized characterized by by low low values values of of dynamic dynamic viscosity viscosity coe coefficient,ﬃcient, equal equal to 9to mPas. 9 mPas. The The authors authors obtained obtained the the viscosity viscosity coe ﬃcoefficientcient value value about about 8.9 mPas8.9 mPas in changeable in changeable shear shear rate 1 conditionsrate conditions (between (between 30 and 30 90and s− 90). s−1). The viscosity coecoefficientﬃcient values obtained during the measurement of the reference oil are within the limits of the measurementmeasurement errorerror andand conﬁrmconfirm thatthat thethe devicedevice hashas beenbeen properlyproperly calibrated.calibrated. Then, water viscosity measurements were carried out at room temperature, which in the above conditions is a Newtonian liquid withwith aa viscosity ofof 1 mPas [[7].7]. During the tests, a linear dependence of water viscosity over time was obtained, regardless of changes in shear rate values—the viscosity coefficient value was about 1 mPa s. Next, at the elevated temperatures, the standard glass (germ. DGG Standardglas I) was used. The chemical composition of the standard glass is presented in a Table 1.

Table 1. Chemical composition of standard glass.

Chemical composition SiO2 Al2O3 Fe2O3 TiO2 SO3 CaO MgO Na2O K2O % 71.72 1.23 0.191 0.137 0.436 6.73 4.18 14.95 0.338

The viscosity value of standard glass at 1400 °C (the highest measured temperature for the referenced glass) should be 14.8 Pas. The measurement of standard glass was performed in 1400 °C. The obtained results were sufficient. Authors obtained the viscosity value coefficient equal to the 14.8 Pas. During rheometric measurements of low viscosity liquids, the problem of laminar flow disturbance in the shear gap is a big problem. Then the equations enabling the calculation of the dynamic viscosity coefficient lose their validity [6]. Research on the existence of critical rotational

Materials 2020, 13, 5025 6 of 20 of water viscosity over time was obtained, regardless of changes in shear rate values—the viscosity coeﬃcient value was about 1 mPa s. Next, at the elevated temperatures, the standard glass (germ. DGG Standardglas I) was used. The chemical composition of the standard glass is presented in a Table1.

Table 1. Chemical composition of standard glass.

Chemical Composition SiO2 Al2O3 Fe2O3 TiO2 SO3 CaO MgO Na2OK2O % 71.72 1.23 0.191 0.137 0.436 6.73 4.18 14.95 0.338

The viscosity value of standard glass at 1400 ◦C (the highest measured temperature for the referenced glass) should be 14.8 Pas. The measurement of standard glass was performed in 1400 ◦C. The obtained results were suﬃcient. Authors obtained the viscosity value coeﬃcient equal to the 14.8 Pas. During rheometric measurements of low viscosity liquids, the problem of laminar ﬂow disturbance in the shear gap is a big problem. Then the equations enabling the calculation of the dynamic viscosity coeﬃcient lose their validity [6]. Research on the existence of critical rotational speed for a system with a movable internal cylinder was carried out by Taylor. He ﬁrst predicted, on the basis of theoretical considerations, and later proved the existence of vortices named after him, which signal the transition of the laminar ﬂow (of the examined medium) into a transient state, and consequently the complex ﬂow-turbulent. In the case of laminar ﬂow, there is no case of layer mixing. Taylor’s vortices testify to the ﬂow linearity disturbance [6]. In order to carry out the measurements correctly, taking into account the fact that there is a critical/limit value of shear rate, for which the ﬂow of liquid steel, characterized by relatively small values of dynamic viscosity, becomes non-laminar—a maximum 1 value of the applied shear rate of 10 s− was adopted. This value was calculated from the formulas, taking into account the geometry of the concentric cylinder system used and the estimated value of the dynamic viscosity coeﬃcient of the test substance [27].

2.2. Analysis of the Chemical Composition of the Samples Four steel samples were subjected to chemical analysis. The samples for analysis were prepared by cutting and grinding. The analysis was performed using a WAS Foundry-Master optical emission spectrometer. The device is calibrated for analysis of iron alloys. The wavelength measurement range is from 160 to 800 nm. It is controlled by means of a PC, has a vacuum chamber in which a set of CCD photosensitive elements is placed. During the analysis, the spark chamber is rinsed with argon, pre-burning and measurement of the intensity of individual components. The result of the analysis is the percentage chemical composition of the sample. The following tests were carried out for the iron solutions given in Table2.

Table 2. Chemical composition of investigated alloys.

Steel Grade C Mn Si Cr Ni Mo B 30MnB4 0.289 0.9 0.11 0.19 0.56 0.01 0.002 27MnB4 0.293 0.98 0.25 0.21 0.07 0.008 0.002 C45 0.41 0.68 0.24 0.08 0.07 0.01

Liquidus temperature values (Table3) were calculated for each of the above species (Table2). The calculated temperature values are necessary to determine the measurement scheme, assuming that the steel under test is to be in a completely liquid state. Materials 2020, 13, 5025 7 of 20

Table 3. Liquidus temperature values for the analyzed steel grades [◦C].

Steel Grade Liquidus Temperature (Scheil Algorithm) 30MnB4 1511 27MnB4 1509 C45 1503

For each of grades, the Scheil algorithm (representing the non-equilibrium crystallization) was used to calculate the values of their liquidus temperatures [24]: Thermodynamic databases—CompuTherm LLC—supplied together with the ProCAST software package were used for the calculations.

2.3. Experimental Research on Viscosity Measurement Rheological tests were carried out for three steel grades that diﬀer in their chemical composition (Table2). The tests were carried out from room temperature up to 1530 ◦C. From the liquidus temperature (appropriate for each of the iron solutions tested), the measurements were performed every 10 degrees. Tests for each steel grade were carried out twice to verify the results obtained. Figure5a shows a steel sample before measurement. The sample has a rectangular shape with a height of 46 mm and a base side equal to 20 mm. Figure5b shows a steel sample after correctly carrying out measurement. The sample surface did not oxidize. The change in the dimensions of the Materials 2020, 13, x FOR PEER REVIEW 7 of 20 sample indicates that it completely melted and assumed the diameter of the crucible (30 mm).

(a) (b)

FigureFigure 5. 5.(a ()a Steel) Steel sample sample before before measurement; measurement; (b ()b steel) steel sample sample after after measurement. measurement.

. SteelSteel tests tests were were conducted conducted in conditionsin conditions of variable of variable values values of rheological of rheological parameters parameters (γ—shear (훾̇— rateshear 1 −1 valuerate invalue the rangein the of range 1–10 sof− 1),– the10 s purpose), the purpose of which of was which to determine was to determine the impact the of theimpact above of variables the above onvariables the value on of the dynamic value of viscositythe dynamic coeﬃ viscositycient of liquid coefficient steel, and of liquid thus to steel, attempt and tothus determine to attempt their to rheologicaldetermine nature. their rheological The tests werenature. carried The tests out from were the carr liquidusied out temperaturefrom the liquidus value temperature (appropriate value for each(appropriate of the iron for solutions each of tested)the iron up solutions to a temperature tested) up of to1530 a temperature◦C with 10 of degree 1530 °C steps. with For10 degree each value steps. ofFor shear each rate, value at a given of shear temperature, rate, at for a given a given temperature, chemical composition, for a given the chemicalmeasurement composition, was carried the outmeasurement for a minimum was of carried three minutes, out for witha minimum a frequency of three of data minutes, reading with every a frequency minute. The of resultsdata reading were presentedevery minute. in the The form results of viscosity were presented curves and in ﬂow the form curves. of viscosity curves and flow curves. Figures 6–8 present curves for three steel grades. The results are presented for the highest temperature at which the measurements were carried out, i.e., 1530 °C. The results are presented only for this temperature, because the viscosity/flow curves were the same in the case of the tests carried out above the liquidus temperature for each steel grade. Thus, the viscosity results of overheated iron solutions were the same as for these liquids in their liquidus temperature. Due to the fact that the results of the experiments were to be used to model the continuous casting process in the tundish (where temperatures are around 1560 °C), viscosity and flow curves are presented for the maximum temperature achievable in the rheometer—1530 °C.

(a) (b)

Figure 6. (a) Viscosity curve for 30MnB4 steel at 1530 °C; (b) flow curve for 30MnB4 steel at 1530 °C.

Materials 2020, 13, x FOR PEER REVIEW 7 of 20

Materials 2020, 13, 5025 8 of 20

(a) (b) Figures6–8 present curves for three steel grades. The results are presented for the highest Figure 5. (a) Steel sample before measurement; (b) steel sample after measurement. temperature at which the measurements were carried out, i.e., 1530 ◦C. The results are presented only for thisSteel temperature, tests were conducted because the in viscosity conditions/ﬂow of curvesvariable were values the of same rheological in the case parameters of the tests (훾̇— carriedshear rateout abovevalue thein the liquidus range temperatureof 1–10 s−1), forthe eachpurpose steel of grade. which Thus, was theto determine viscosity results the impact of overheated of the above iron variablessolutions onwere the the value same of as the for dynamic these liquids viscosity in their coefficient liquidus of temperature. liquid steel, Dueand thus to the to fact attempt that the to determineresults of thetheir experiments rheological were nature. to beThe used tests to were model carr theied continuous out from the casting liquidus process temperature in the tundish value (appropriate(where temperatures for each of are the around iron solutions 1560 ◦C), tested) viscosity up andto a temperature ﬂow curves areof 1530 presented °C with for 10 the degree maximum steps. temperature achievable in the rheometer—1530 ◦C. For each value of shear rate, at a given temperature,. for a given chemical composition, the measurementViscosity curveswas carried are the out dependence for a minimum between of γthree—shear minutes, rate and withη—dynamic a frequency viscosity of data coe readingﬃcient. everyFlow minute. curves The are results the dependence were presented between in theτ—shear form of stress viscosity and ηcurves—dynamic and flow viscosity curves. coe ﬃcient [9]. BlackFigures lines on 6 the–8 present graphs indicate curves for the three trend. steel grades. The results are presented for the highest temperatureIn addition, at which during the measurements measurements, were it was carried observed out, i.e. that, 1530 superheated °C. The results steels are exhibit presented the same only fordynamic this tem viscosityperature, coe becauseﬃcient as the steels viscosity/ at liquidusflow temperaturecurves were (forthe analyzedsame in the steel case grades), of the sotests it has carried been outassumed above that the liquidus steel at a temperature ladle temperature for each of steel 1560 grade.◦C will Thus have, the viscosity same dynamic results viscosity of overheated coeﬃcient iron solutionsas at 30 ◦C were lower, the but same still as above for these liquidus liquids temperature in their liquidus of the given temperature. alloy. Due to the fact that the resultsThe of coursethe experiments of the curves were is similarto be used to the to curve model of the a perfectly continuous viscous casting ﬂuid. process The lowest in the values tundish of (wherethe dynamic temperatures viscosity are coe aroundﬃcient 1560 were °C), obtained viscosity for and C45 flow steel—about curves are 0.0255 presented Pas—while for the the maximum highest temperaturevalue of the dynamic achievable viscosity in the coerheometerﬃcient— was1530 for °C 30MnB4. grade—0.0298 Pas. A value of 0.0271 Pas for the dynamic viscosity coeﬃcient was obtained for grade 27MnB4.

(a) (b) Materials 2020, 13, x FOR PEER REVIEW 8 of 20

Figure 6. ((aa)) Viscosity Viscosity curve for 30MnB4 steel at 1530 °C;◦C; ( b) flow ﬂow curve for 30MnB4 steel at 1530 ◦°C.C.

(a) (b)

FigureFigure 7. 7. ((aa)) Viscosity Viscosity curve curve for for 27MnB4 27MnB4 steel steel at at 1530 1530 °C;◦C; ( (bb)) flow ﬂow curve curve for for 27MnB4 27MnB4 steel steel at at 1530 1530 °C.◦C.

(a) (b)

Figure 8. (a) Viscosity curve for C45 steel at 1530 °C; (b) flow curve for C45 steel at 1530 °C.

Viscosity curves are the dependence between γ̇ —shear rate and η—dynamic viscosity coefficient. Flow curves are the dependence between τ—shear stress and η—dynamic viscosity coefficient [9]. Black lines on the graphs indicate the trend. In addition, during measurements, it was observed that superheated steels exhibit the same dynamic viscosity coefficient as steels at liquidus temperature (for analyzed steel grades), so it has been assumed that steel at a ladle temperature of 1560 °C will have the same dynamic viscosity coefficient as at 30 °C lower, but still above liquidus temperature of the given alloy. The course of the curves is similar to the curve of a perfectly viscous fluid. The lowest values of the dynamic viscosity coefficient were obtained for C45 steel—about 0.0255 Pas—while the highest value of the dynamic viscosity coefficient was for 30MnB4 grade—0.0298 Pas. A value of 0.0271 Pas for the dynamic viscosity coefficient was obtained for grade 27MnB4. Figures 6b, 7b and 8b present flow curve graphs—i.e., the relationship between shear rate and shear stress. This presentation of results is typical for rheological analysis of fluid behavior. In order to verify the influence of the chemical composition on the values of dynamic viscosity coefficient, it is necessary to conduct detailed tests in two-component systems containing iron with an alloying additive, so that it is possible to determine the influence of a given alloying component on the value of dynamic viscosity coefficient. With complex chemical composition, it is difficult to make unequivocal conclusions about the impact of selected elements on the value of the dynamic viscosity coefficient of iron solutions. At this stage, one can only suspect that the increase in carbon content may affect the reduction in the dynamic viscosity coefficient of the iron solution. However, the increase in manganese content seems to cause a slight increase in the value of the dynamic viscosity coefficient of such a solution.

2.4. Rheological Models In the next part of the work, the obtained data were verified using rheological models, namely the Newton model [9] and the author’s empirical MKH1 model [23,27], using the chemical

Materials 2020, 13, x FOR PEER REVIEW 8 of 20

(a) (b)

MaterialsFigure2020 ,7.13 (,a 5025) Viscosity curve for 27MnB4 steel at 1530 °C; (b) flow curve for 27MnB4 steel at 1530 °C. 9 of 20

(a) (b)

Figure 8. (a) Viscosity curve for C45 steel at 1530 ◦°C;C; ( b) ﬂowflow curve for C45 steel at 1530 ◦°CC..

FiguresViscosity6b, 7 curvesb and 8b are present the dependence ﬂow curve graphs—i.e., between γ̇ — theshear relationship rate and between η—dynamic shear rate viscosity and shearcoefficient. stress. This presentation of results is typical for rheological analysis of ﬂuid behavior. In order to verifyFlow the curves inﬂuence are the of dependence the chemical between composition τ—shear on thestress values and η of— dynamicdynamic viscosityviscosity coecoefficientﬃcient, it[9]. is Black necessary lines toon conduct the graphs detailed indicate tests the in trend. two-component systems containing iron with an alloying additive,In addition, so that it during is possible measurements to determine, it the was inﬂuence observed ofa that given superheated alloying component steels exhibit on the the value same of dynamic viscosityviscosity coecoefficientﬃcient. Withas steels complex at liquidus chemical temperature composition, (for it analy is diﬃzedcult steel to make grades), unequivocal so it has conclusionsbeen assumed about that the steel impact at a of ladle selected temperature elements of on 1560 the value °C will of the have dynamic the same viscosity dynam coeic ﬃ viscositycient of ironcoefficient solutions. as at At30 this°C lower, stage, but one still can above only suspect liquidus that temperature the increase of inthe carbon given alloy. content may aﬀect the reductionThe course in the dynamicof the curves viscosity is similar coeﬃ cientto the of curve the iron of a solution. perfectly However, viscous fluid. the increase The lowest in manganese values of contentthe dynamic seems viscosity to cause coefficient a slight increasewere obtained in the for value C45 of steel the— dynamicabout 0.0255 viscosity Pas— coewhileﬃcient the ofhighest such avalue solution. of the dynamic viscosity coefficient was for 30MnB4 grade—0.0298 Pas. A value of 0.0271 Pas for the dynamic viscosity coefficient was obtained for grade 27MnB4. 2.4. RheologicalFigures 6b Models, 7b and 8b present flow curve graphs—i.e., the relationship between shear rate and shearIn stress. the next This part presentation of the work, of theresults obtained is typical data for were rheological veriﬁed using analysis rheological of fluid models, behavior. namely In order the Newtonto verify modelthe influence [9] and of the the author’s chemical empirical composition MKH1 on model the values [23,27 of], usingdynamic the viscosity chemical coefficient, composition it ofis necessary the tested to samples conduct fordetailed calculation, tests in astwo well-component as the rheologicalsystems containing parameter iron (shear with stress)an alloying and temperatureadditive, so that value. it is possible to determine the influence of a given alloying component on the value of dynamicThe idea viscosity and types coefficient. of diﬀerent With models complex which are chemical used to composition, calculate the viscosity it is difficult of liquids to make were presentedunequivocal in conclusions a previous paper about [ 23the,24 impact]. In paper of selected [23], the elements author on described the value in of detail the dynamic the way toviscosity obtain thecoefficient empirical of modelsiron solutions. dedicated At forthis the stage, calculation one can of only the viscositysuspect that of liquid the increase iron solutions. in carbon content may The affect purpose the reduction of this analysisin the dynamic was to check viscosity whether coefficient the data of obtained the iron through solution. the However, experiment the canincrease be easily in manganese calculated content using available,seems to cause selected a slight rheological increase models. in the value Thus, of whether the dynamic based viscosity on data fromcoefficient thermodynamic of such a solution. databases or not, during process modeling, the values are assumed to be similar or signiﬁcantly diﬀerent from the values of the dynamic viscosity coeﬃcient obtained through the2.4. experiment.Rheological Models FiguresIn the next9–11 part show of viscosity the work, curves the obtained for three data (selected) were ofverified the four using iron rheological solutions analyzed. models, Eachnamely of the graphs Newton presented model [9] has and three the viscosity author curves:’s empirical derived MKH1 from modelreal measurements, [23,27], using calculated the chemical from Newton’s model (signed Newton-model), and calculated from the author’s model MKH1 [23,24]. The original MKH1 model was described in detail in [24], and only its mathematical form is presented below:

Newton model [9] • τ η = . (1) γ Materials 2020, 13, x FOR PEER REVIEW 9 of 20 composition of the tested samples for calculation, as well as the rheological parameter (shear stress) and temperature value. The idea and types of different models which are used to calculate the viscosity of liquids were presented in a previous paper [23,24]. In paper [23], the author described in detail the way to obtain the empirical models dedicated for the calculation of the viscosity of liquid iron solutions. The purpose of this analysis was to check whether the data obtained through the experiment can be easily calculated using available, selected rheological models. Thus, whether based on data from thermodynamic databases or not, during process modeling, the values are assumed to be similar or significantly different from the values of the dynamic viscosity coefficient obtained through the experiment. Figures 9–11 show viscosity curves for three (selected) of the four iron solutions analyzed. Each of the graphs presented has three viscosity curves: derived from real measurements, calculated from Newton’s model (signed Newton-model), and calculated from the author’s model MKH1 [23,24]. The original MKH1 model was described in detail in [24], and only its mathematical form is presented below:

Newton model [9] Materials 2020, 13, 5025 10 of 20 τ η = γ̇ (1) MKH1 model [23,24] • MKH1 model [23,24] 3p p3 η = −η(0=.0086260.008626× MnMn3) −3 (0(.003720.00372× √3 Ni) + (0.10360.1036 ×3 S√)+S ) + − 3 p × − × × (2) (0.02244(0.02244 × √Cu3 )Cu− ()0.00933(0.00933× MoMo) )++(0(0.007434.007434 ×CC))++ (2) (0.3179 × logT×) + (0.0187− × τ) −×0.129 × (0.3179 logT) + (0.0187 τ) 0.129 × × −

Materials 2020, 13, x FOR PEER REVIEW 10 of 20 Figure 9. Viscosity curve for 30MnB4 steel at 1530 C. Figure 9. Viscosity curve for 30MnB4 steel at 1530 ◦°C.

Figure 10. Viscosity curve for C45 steel at 1530 C. Figure 10. Viscosity curve for C45 steel at 1530 °C.◦

(a)

Materials 2020, 13, x FOR PEER REVIEW 10 of 20

Materials 2020, 13, 5025 Figure 10. Viscosity curve for C45 steel at 1530 °C. 11 of 20

Materials 2020, 13, x FOR PEER REVIEW 11 of 20

(a)

(b)

FigureFigure 11. 11.Investigated Investigated tundish tundish with with ((aa)) boundary conditions conditions and and (b) ( bmesh) mesh used used for the for simulations. the simulations.

Newton’sNewton model’s model is is the most most well well-known,-known, basic basicmodel modelused to usedcalculate to the calculate value of thethe viscosity value of the viscositycoefficient coeﬃ ofcient liquids, of liquids, both ideal both—showing ideal—showing the invariability the invariability of the value of of the dynamicvalue of viscosity the dynamic coefficient under varying shear stress conditions—and non-Newtonian—showing non-linear viscosity coeﬃcient under varying shear stress conditions—and non-Newtonian—showing non-linear dependence between shear rate and shear stress. dependence between shear rate and shear stress. In Newton’s model, only values of τ—shear stress and γ̇—. shear rate are taken into account for calculatingIn Newton’s the model,η—viscosity only value values [27]. of τ—shear stress and γ—shear rate are taken into account for calculatingThe the MKH1η—viscosity model is an value empirical [27]. model based on rheological measurements [23,28]. This model hasThe been MKH1 developed model to is calculate an empirical viscosity model of totally based liquid on rheological and in the semi measurements-solid state iron [23, solutions.28]. This model has beenMultiple developed linear to calculate regression viscosity analysis of implementedtotally liquid inand the in the STATISTICA semi-solid version state iron 10.0 solutions. and ClementineMultiple linear SPSS regressionversion 12.0 analysissoftware implementedwere used for developing in the STATISTICA the model. version To construct 10.0 andthe models Clementine SPSSabout version 2000 12.0 records software were were used used contained for developing the following the data: model. chemical Toconstruct composition the— modelselements about 2000conce recordsntration were in used % (C, contained Mn, Si, P, S, the Cu, following Cr, Ni, Mo, data: V); n chemical—rotational composition—elements speed value; temperature concentration value; γ̇—shear rate value; τ—shear stress value; η—dynamic viscosity coefficient value. After correlation. in % (C, Mn, Si, P, S, Cu, Cr, Ni, Mo, V); n—rotational speed value; temperature value; γ—shear analysis, only one rheological parameter was chosen as the independent variable. The other τ η rateindependent value; —shear variables stress we value;re chemical—dynamic composition viscosity elements coe andﬃcient temperature value. Aftervalue. correlation analysis, The calculated coefficient of multiple determination for the above equation is R2 = 0.842. The Fisher–Snedecor coefficient value is 1259.098; and the critical (read from tables for n1 = 7, n2 = 1983) F is 2.01. The hypothesis test conducted with the p value confirmed the relationship between the variables analyzed. The standard estimation error is 0.12 [24]. The dynamic viscosity coefficient was considered the output variable, and the other data were entered as input data. Only one rheological parameter—shear stress—was put into the formula. In the case of the MKH1 model for calculating the viscosity values using the chemical composition of alloy elements (Mn—manganese, Ni—nickel, S—sulphur, Mo—molybdenum, C— carbon), T—temperature, and the rheological parameter is τ—shear stress value and an absolute term.

Materials 2020, 13, 5025 12 of 20 only one rheological parameter was chosen as the independent variable. The other independent variables were chemical composition elements and temperature value. The calculated coeﬃcient of multiple determination for the above equation is R2 = 0.842. The Fisher–Snedecor coeﬃcient value is 1259.098; and the critical (read from tables for n1 = 7, n2 = 1983) F is 2.01. The hypothesis test conducted with the p value conﬁrmed the relationship between the variables analyzed. The standard estimation error is 0.12 [24]. The dynamic viscosity coeﬃcient was considered the output variable, and the other data were entered as input data. Only one rheological parameter—shear stress—was put into the formula. In the case of the MKH1 model for calculating the viscosity values using the chemical composition of alloy elements (Mn—manganese, Ni—nickel, S—sulphur, Mo—molybdenum, C—carbon), T—temperature, and the rheological parameter is τ—shear stress value and an absolute term. In the MKH 1 model, the viscosity values of liquid iron solutions are calculated by using its chemical composition, temperature value and rheological parameter (such as shear stress). This point of view is completely diﬀerent from the viscosity models existing in the literature. Discussions about other types of formulas allowing one to calculate viscosity values (including viscosity of liquid metals) are widely analyzed in another authors’ paper [23,24]. In this paper, the authors would like to show and emphasis that during the modelling of the metal ﬂow (i.e., in the tundish) taking only into account the simple value of the viscosity coeﬃcient of liquid steel or calculating it by using the most well-known model can give improper/faulty values of the viscosity of the liquid iron solution. This, in turn, may cause errors/simpliﬁcations of the simulation. In Figures9–11, the three di ﬀerent viscosity curves presented:

With the rhombus-measured values of the viscosity coeﬃcient of investigated liquid iron solutions • (Table2); With the square-calculated values of the viscosity coeﬃcient by using Newton’s model. In this • case, the values of shear rate (set during measurement) and obtained shear stress values were put into Newton’s model (1), and the viscosity values were calculated. With the triangular-calculated values of the viscosity coeﬃcient by using empirical MKH1 model. • In this case, the values of obtained shear stress, temperature value, and chemical composition of iron solutions were put into the empirical model MKH1 (2), and the viscosity values were calculated. The analyses show that the nature of the curve derived from the measurements and the viscosity curve calculated from the MKH1 model are similar. However, the course of the viscosity curve calculated from Newton’s model is diﬀerent. The authors would like to show that existing and well-known formulas, such as Newton’s law, might not be suﬃcient when calculating the viscosity values (during modeling process), and it is necessary to use real data from the experiment or investigate (improve) the empirical model dedicated to calculating the value of the viscosity of liquid high-temperature solutions such as liquid iron alloys. The nature of the curve derived from the measurements and the viscosity curve calculated from the MKH1 model are similar. However, the course of the viscosity curve calculated from the Newton’s model is completely diﬀerent. The model MKH1 rather underestimates the values of the dynamic viscosity coeﬃcient of steel. This is probably due to the fact that the MKH1 model was developed using data obtained for iron solutions with a carbon content of about and above 0.50%. This model should be revised, taking into account a wider range of chemical compositions, including coal. However, graph analysis shows that the Newton model, which is often used in ﬂow modeling programs, incorrectly describes real data obtained during high-temperature rheological studies of iron solutions. Therefore, in order to carry out computer simulations of the real process in which the tested medium is liquid steel, it would be most appropriate to use data from real measurements, and then reﬁne the MKH1 model so that it can be able to calculate the values of the dynamic viscosity coeﬃcient of liquid iron in a wide range of chemical compositions. Materials 2020, 13, 5025 13 of 20

In the next part, during numerical simulations of steel ﬂow in the tundish, the constant and experimentally obtained viscosity values of liquid steel were used.

2.5. Numerical Simulations of Steel Flow in the Tundish Tundish model and computational mesh were set with Gambit 2.4 computer program (Ansys, Inc., Canonsburg, PA, USA). Due to the symmetry tundish criteria, the half of the tundish was calculated. Thanks to that, a ﬁner grid could be used with similar computational time. A computational grid was generated for the investigated object (tetra type mesh) with reﬁnement in the area of the wall and submerged entry nozzles (SENs) (mesh was locally reﬁned in areas with high velocity gradients, using the gradient adaption function to obtain a more accurate solution). Mesh quality has been checked with the skew angle criterion. This coeﬃcient, for the computational grid used in the current investigation, is 0.61 and is in the required range. During the adaptation of the computational grid, a dimensionless parameter y+ was taken into account. According to ANSYS Fluent manual, this coeﬃcient should be within 30 to 60, and in the current mesh its value was 54. Thanks to that, the suﬃcient mesh quality is maintained in this study. The computational domain included 720291 cells (with an average size of 15 mm; in the regions of inlet, the outlet and wall mesh were reﬁned and were about 5–10 mm), this number of elements is suﬃcient to predict ﬂow structure in the tundish. In a previous investigation done by the author, a similar object was tested with similar initial and boundary conditions, and the results were validated with industrial and water model experiments. Similar mesh size was also proven to be enough for grid independence by other authors [29]. CFD simulations of liquid steel ﬂow through the tundish were carried out with the use of ANSYS Fluent commercial code ver. 16.0. For turbulence modeling, Standard k-ε model has been used. Detailed description of the models can be found in ANSYS Fluent Theory Guide [30]. Based on the literature and our own data, a simulation model describing the studied phenomenon was formulated. For non-isothermal calculations, the following boundary conditions were formulated: the bottom and side walls of the ladle meet the conditions of a stationary wall with heat loss of 5 kW/m2, − and the metal free surface is a wall with zero tangential stresses with heat loss of 12.5 kW/m2. Figure 11 − presents schematically the boundary conditions used in numerical simulations and mesh used for numerical simulations. At the tundish inlet (shroud), liquid steel ﬂows with a velocity of 1.1 m/s with a turbulence intensity of 5% and the initial temperature of 1823 K. For the current study, convergence criteria were set to 1e-6 for continuity equation and 1e-3 for other equations. In addition, monitoring points were used in the tundish to test changes in liquid steel velocity between iterations. Monitoring points were set in regions of outlets, inlet and inside the tundish. The solution was found to be converged once the change in liquid steel velocity in monitoring points reached a steady solution and did not exceed 1%. Numerical calculations were carried out for two variants. In the ﬁrst variant, the viscosity value used in most calculations, the results of which are widely published (adopted from the literature), in a range about 0.006–0.007 Pas [2,31–33], was introduced as the viscosity parameter of liquid steel. The experimentally determined viscosity values (obtained for the steel grades tested in this study) were about four times higher than the literature values. Therefore, in order to perform numerical simulation for the counter-variant, a viscosity value of 0.025 Pas was assumed, which was considered representative for the steel grades tested. Figure 12 shows the distribution of the numerically predicted liquid steel velocity ﬁeld on transverse tundish planes, successively passing through the shroud and then between each submerged entry nozzle (SEN). Even this cursory qualitative analysis reveals that for both variants, the liquid ﬂow structure in the ladle is very similar, although of course some slight diﬀerences can be observed in the ﬂow structure and in the distribution of velocity values. Materials 2020, 13, x FOR PEER REVIEW 13 of 20 quality has been checked with the skew angle criterion. This coefficient, for the computational grid used in the current investigation, is 0.61 and is in the required range. During the adaptation of the computational grid, a dimensionless parameter y+ was taken into account. According to ANSYS Fluent manual, this coefficient should be within 30 to 60, and in the current mesh its value was 54. Thanks to that, the sufficient mesh quality is maintained in this study. The computational domain included 720291 cells (with an average size of 15 mm; in the regions of inlet, the outlet and wall mesh were refined and were about 5–10 mm), this number of elements is sufficient to predict flow structure in the tundish. In a previous investigation done by the author, a similar object was tested with similar initial and boundary conditions, and the results were validated with industrial and water model experiments. Similar mesh size was also proven to be enough for grid independence by other authors [29]. CFD simulations of liquid steel flow through the tundish were carried out with the use of ANSYS Fluent commercial code ver. 16.0. For turbulence modeling, Standard k-ɛ model has been used. Detailed description of the models can be found in ANSYS Fluent Theory Guide [30]. Based on the literature and our own data, a simulation model describing the studied phenomenon was formulated. For non-isothermal calculations, the following boundary conditions were formulated: the bottom and side walls of the ladle meet the conditions of a stationary wall with heat loss of −5 kW/m2, and the metal free surface is a wall with zero tangential stresses with heat loss of −12.5 kW/m2. Figure 11 presents schematically the boundary conditions used in numerical simulations and mesh used for numerical simulations. At the tundish inlet (shroud), liquid steel flows with a velocity of 1.1 m/s with a turbulence intensity of 5% and the initial temperature of 1823 K. For the current study, convergence criteria were set to 1e-6 for continuity equation and 1e-3 for other equations. In addition, monitoring points were used in the tundish to test changes in liquid steel velocity between iterations. Monitoring points were set in regions of outlets, inlet and inside the tundish. The solution was found to be converged once the change in liquid steel velocity in monitoring points reached a steady solution and did not exceed 1%. Numerical calculations were carried out for two variants. In the first variant, the viscosity value used in most calculations, the results of which are widely published (adopted from the literature), in a range about 0.006–0.007 Pas [2,31–33], was introduced as the viscosity parameter of liquid steel. The experimentally determined viscosity values (obtained for the steel grades tested in this study) were about four times higher than the literature values. Therefore, in order to perform numerical simulation for the counter-variant, a viscosity value of 0.025 Pas was assumed, which was considered representative for the steel grades tested. Figure 12 shows the distribution of the numerically predicted liquid steel velocity field on transverse tundish planes, successively passing through the shroud and then between each submerged entry nozzle (SEN). Even this cursory qualitative analysis reveals that for both variants, the liquid flow structure in the ladle is very similar, although of course some slight differences can be observed in the flow structure and in the distribution of velocity values. Materials 2020, 13, 5025 14 of 20

Materials 2020, 13, x FOR PEER REVIEW 14 of 20 (a)

(b)

FigureFigure 12.12. NumericallyNumerically predictedpredicted velocityvelocity ﬁeldfield ofof liquidliquid steelsteel onon transversetransverse tundishtundish planes:planes: ((aa)) basebase viscosity,viscosity, (b(b)) experimentally experimentally obtained obtained viscosity. viscosity.

FiguresFigures 1313 and and 14 14 show show the the numerically numerically predicted predicted liquid liquid steel steel velocity velocity ﬁeld field on on longitudinal longitudinal tundishtundish planes.planes. The ﬁrstfirst passes through the shroud andand isis characterizedcharacterized by higherhigher velocityvelocity values, andand thethe secondsecond passespasses throughthrough outﬂowsoutflows (SENs)(SENs) ofof thethe tundish.tundish. SimilarSimilar toto thethe transverse planes,planes, somesome didifferencesﬀerences inin thethe distributiondistribution ofof velocityvelocity ﬁeldsfields cancan bebe observed.observed.

(a)

(b)

Figure 13. Numerically predicted velocity field of liquid steel in the longitudinal tundish plane passing through the shroud: (a) base viscosity, (b) experimentally obtained viscosity.

Materials 2020, 13, x FOR PEER REVIEW 14 of 20

(b)

Figure 12. Numerically predicted velocity field of liquid steel on transverse tundish planes: (a) base viscosity, (b) experimentally obtained viscosity.

Figures 13 and 14 show the numerically predicted liquid steel velocity field on longitudinal tundish planes. The first passes through the shroud and is characterized by higher velocity values, and the second passes through outflows (SENs) of the tundish. Similar to the transverse planes, some differences in the distribution of velocity fields can be observed. Materials 2020, 13, 5025 15 of 20

(a)

(b)

Figure 13. 13. NumericallyNumerically predicted predicted velocity velocity ﬁeld field of liquid of liquid steel steelin the in longitudinal the longitudinal tundish tundish plane passing plane throughpassing through the shroud: the shroud: (a) base viscosity,(a) base viscosity, (b) experimentally (b) experimentally obtained obtained viscosity. viscosity.

The tundish working space can be divided into two characteristic regions. The ﬁrst of them is located in the area of the shroud (steel inlet from the casting ladle). This tundish area is characterized by a high level of ﬂow turbulence. Liquid steel ﬂows with impetus into the tundish and is directed to the turbulence inhibitor, installed at the tundish bottom, whose main task is to protect the refractory lining of the tundish, reduce the speed of steel and stabilize its ﬂow. This tundish region is characterized by much higher liquid velocity values than the rest of the working area. Dynamic viscosity is negligible in this part of the ladle because eddy viscosity dominates. The remaining part of the ladle is characterized by signiﬁcantly lower liquid ﬂow velocities (except for small areas at SEN-s neighborhood) and signiﬁcantly lower turbulence intensity. In this region, diﬀerences in the ﬂow structure calculated for individual viscosities can already be observed (Figures 13 and 14). Because treatments such as the ﬂotation of non-metallic inclusions from steel into the slag layer occur mainly in this tundish region, changes in the ﬂow structure, even small and subtle, will aﬀect the prediction of the simulation of non-metallic inclusions’ removal into the slag in this tundish zone. Therefore, it might inﬂuence results, especially in death zones, which then might inﬂuence, for instance, the prediction of non-metallic inclusion removal. It is important to build up a simulation model as close to reality as possible. Of course, mesh density and other numerical model parameters have an inﬂuence on the computational results, but with the very high mesh density (which with todays computing power is easy to reach) and with the same model settings for both investigated cases, the results can be properly compared with each other. MaterialsMaterials 20202020, 13,,13 x ,FOR 5025 PEER REVIEW 15 16of 20 of 20

(a)

(b)

MaterialsFigureFigure 20 20 14., 14.1 3Numerically, xNumerically FOR PEER REVIEW predicted predicted velocity ﬁeld field of of liquid liquid steel steel in the in thelongitudinal longitudinal tundish tundish plane plane passing 16 of 20 passingthrough through submerged submerged entry nozzles entry nozzles (SENs): ( (SENsa) base): ( viscosity,a) base viscosity, (b) experimentally (b) experimentally obtained viscosity.obtained viscosity.The quantitative analysis of differences in the forecasted liquid steel speed profiles was carried out byThe comparing quantitative the values analysis obtained of diﬀerences for individual in the forecasted lines passing liquid through steel speedthe tundish proﬁles working was carried area. TheoutThe byarrangement comparingtundish working of the the values linesspace obtainedis can shown be divided forin individualFigure into 15 twoand lines characteristicthe passing results through obtained regions. the in tundishThe Figure first working16. of themThe lines area.is locatedareThe loc arrangement inated the atarea the of following of the the shroud lines heights: is(steel shown inletline in inletfrom Figure (shroud) the 15 casting and at the 0.ladle).4 resultsm, lineThis obtained outlet tundish at inarea0.4 Figure and is characterized 0. 167 m. The (ladle lines is by0.are a8 highm located high). level at of the flow following turbulence heights:. Liquid line steel inlet flows (shroud) with at impetus 0.4 m, line into outlet the tundish at 0.4 and and 0.7 is mdirected (ladle is to 0.8 the m turbulence high). inhibitor, installed at the tundish bottom, whose main task is to protect the refractory lining of the tundish, reduce the speed of steel and stabilize its flow. This tundish region is characterized by much higher liquid velocity values than the rest of the working area. Dynamic viscosity is negligible in this part of the ladle because eddy viscosity dominates. The remaining part of the ladle is characterized by significantly lower liquid flow velocities (except for small areas at SEN-s neighborhood) and significantly lower turbulence intensity. In this region, differences in the flow structure calculated for individual viscosities can already be observed (Figures 13 and 14). Because treatments such as the flotation of non- metallic inclusions from steel into the slag layer occur mainly in this tundish region, changes in the flow structure, even small and subtle, will affect the prediction of the simulation(a) line of noninlet- metallic inclusions’ removal into (theb) line slag outlets in this tundish zone. Therefore, it might influence results, especially in death zones, which then might influence, for FigureFigure 15. 15. LocationLocation of of measuring measuring lines: lines: (a (a) )passing passing through through the the shroud, shroud, ( (bb)) passing passing through through the the SENs. SENs. instance, the prediction of non-metallic inclusion removal. It is important to build up a simulation model as close to reality as possible. Of course, mesh density and other numerical model parameters have an influence on the computational results, but with the very high mesh density (which with todays computing power is easy to reach) and with the same model settings for both investigated cases, the results can be properly compared with each other.

(a)

(b)

Materials 2020, 13, x FOR PEER REVIEW 16 of 20

The quantitative analysis of differences in the forecasted liquid steel speed profiles was carried out by comparing the values obtained for individual lines passing through the tundish working area. The arrangement of the lines is shown in Figure 15 and the results obtained in Figure 16. The lines are located at the following heights: line inlet (shroud) at 0.4 m, line outlet at 0.4 and 0.7 m (ladle is 0.8 m high).

(a) line inlet (b) line outlets

Materials 2020Figure, 13, 502515. Location of measuring lines: (a) passing through the shroud, (b) passing through the SENs. 17 of 20

(a)

Materials 2020, 13, x FOR PEER REVIEW 17 of 20

(b)

(c)

FigureFigure 16. Liquid 16. Liquid steel steel velocity velocity value value on lines on linespassing passing through through shroud shroud and outﬂows and outflows (SENs); (SENs) (a) shroud;; (a) (b) SENs,shroud h ;= (b0.4;) SENs, (c) SENs, h=0.4; h(c=) SENs,0.7. h=0.7.

The flow structure and its turbulence changes according to the viscosity value discussed, according to Figures 12–14, are confirmed by the results shown in Figure 16. These show velocity magnitude at lines passing through different tundish working zones. Velocity values are given for numerical calculations performed with the standard (base) viscosity value and for viscosity values obtained experimentally. For lines passing through the shroud, and so the region characterized by higher liquid velocity, the values are very close to each other and there are almost no differences. The other situation can be seen in other regions of the tundish, where liquid steel velocities are lower. Velocity lines obtained for lines passing through the submerged entry nozzles, for both highs (0.4 and 0.7), slightly differ from each other. Again, in regions closer to the liquid steel inlet (shroud), the values are very similar, but in the more distant zones, the velocity profile changes and the differences even reach about 20%. This was observed at both fluid levels and can be very important in the liquid surface area, were non-metallic inclusions can react with covering slag. Both qualitative and quantitative analysis showed differences in the flow structure of liquid steel in the tundish, in particular in the predicted values and the velocity profile distribution. However, these differences are not significant, and in most of the analyzed working areas of the tundish, they are negligible. On the other hand, one should not uncritically take values from the literature data, especially if the case deviates from others. Although there are other effects that have more influence on the simulation results than the presented difference in viscosity, the current investigation is focused on basic research and the purpose is show that the choice of correct viscosity values might influence the results, especially in death zones, which then might influence, for instance, the prediction of non-metallic inclusion removal or when one investigates the slag re-entrainment process into the liquid steel. It is important to build a simulation model as close to reality as possible. To sum up, if there is such a possibility, the value of steel viscosity should be determined by means of experimental tests, while in the absence of such a possibility by entering a value from publicly available results of experimental tests in this area, it should be ensured that they are as close as possible to the analyzed case—then, when numerically forecasting the flow found in the ladle, one will not make a significant mistake. However, this further emphasizes the importance of conducting experimental research to determine the viscosity of various steel grades.

3. Conclusions

Materials 2020, 13, 5025 18 of 20

The ﬂow structure and its turbulence changes according to the viscosity value discussed, according to Figures 12–14, are conﬁrmed by the results shown in Figure 16. These show velocity magnitude at lines passing through diﬀerent tundish working zones. Velocity values are given for numerical calculations performed with the standard (base) viscosity value and for viscosity values obtained experimentally. For lines passing through the shroud, and so the region characterized by higher liquid velocity, the values are very close to each other and there are almost no diﬀerences. The other situation can be seen in other regions of the tundish, where liquid steel velocities are lower. Velocity lines obtained for lines passing through the submerged entry nozzles, for both highs (0.4 and 0.7), slightly diﬀer from each other. Again, in regions closer to the liquid steel inlet (shroud), the values are very similar, but in the more distant zones, the velocity proﬁle changes and the diﬀerences even reach about 20%. This was observed at both ﬂuid levels and can be very important in the liquid surface area, were non-metallic inclusions can react with covering slag. Both qualitative and quantitative analysis showed diﬀerences in the ﬂow structure of liquid steel in the tundish, in particular in the predicted values and the velocity proﬁle distribution. However, these diﬀerences are not signiﬁcant, and in most of the analyzed working areas of the tundish, they are negligible. On the other hand, one should not uncritically take values from the literature data, especially if the case deviates from others. Although there are other eﬀects that have more inﬂuence on the simulation results than the presented diﬀerence in viscosity, the current investigation is focused on basic research and the purpose is show that the choice of correct viscosity values might inﬂuence the results, especially in death zones, which then might inﬂuence, for instance, the prediction of non-metallic inclusion removal or when one investigates the slag re-entrainment process into the liquid steel. It is important to build a simulation model as close to reality as possible. To sum up, if there is such a possibility, the value of steel viscosity should be determined by means of experimental tests, while in the absence of such a possibility by entering a value from publicly available results of experimental tests in this area, it should be ensured that they are as close as possible to the analyzed case—then, when numerically forecasting the ﬂow found in the ladle, one will not make a signiﬁcant mistake. However, this further emphasizes the importance of conducting experimental research to determine the viscosity of various steel grades.

3. Conclusions Based on the results of the experimental measurements and numerical simulations carried out, the following conclusions were drawn:

A higher amount of carbon in the chemical composition of steel causes a lower liquidus temperature. • At the investigated temperature, this iron solution probably does not have solid particles which could aﬀect its rheological behavior. Steel C45, with the higher content of carbon, has the lowest value of the viscosity coeﬃcient. With complex chemical composition, it is diﬃcult to make unequivocal conclusions about the • impact of selected elements on the value of the dynamic viscosity coeﬃcient of iron solutions The character of viscosity and ﬂow curves, obtained from rheological measurements, is quite • similar to the course of a perfectly viscous ﬂuid curve. The nature of the curve derived from the measurements and that of the viscosity curve calculated • from the MKH1 model are similar. The character of the viscosity curve calculated from Newton’s model is completely diﬀerent. • Existing and well-known formulas, such as Newton’s law, might not be suﬃcient when calculating • the viscosity values—during the modeling process—of liquid iron solutions. The author’s rheological model still needs to be reﬁned and extended to a wide range of • alloying elements. Based on numerical predictions, qualitative and quantitative analysis showed diﬀerences in the • ﬂow structure of liquid steel in the tundish. Materials 2020, 13, 5025 19 of 20

Diﬀerences received for calculations with diﬀerent steel viscosities are not signiﬁcant, and in most • of the analyzed working areas of the tundish, they are negligible. Nevertheless, investigations are focused on basic research and it was shown that the choice of • correct viscosity values might inﬂuence the results, especially in death zones, which then might inﬂuence, for instance, the prediction of non-metallic inclusion removal or dragging slag into the metal.

Author Contributions: Conceptualization, M.W.; data curation, M.S.´ and M.W.; formal analysis, M.S.´ and M.W.; investigation, M.S.´ and M.W.; methodology, M.S.´ and M.W.; writing–original draft, M.S.´ and M.W. All authors have read and agreed to the published version of the manuscript. Funding: Acknowledgements to the National Centre for Research and Development for ﬁnancial support (project No PBS2/A5/32/2013). Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1. Chang, S.; Cao, X.; Zou, Z.; Isac, M.; Guthrie, R.I.L. Microbubble Swarms in a Full-Scale Water Model Tundish. Metall. Mater. Trans. B 2016, 47, 2732–2743. [CrossRef] 2. Cwudzinski, A. Numerical and physical modeling of liquid steel behaviour in one strand tundish with gas permeable barrier. Arch. Metall. Mater. 2018, 63, 589–596. [CrossRef] 3. Zepeda-Diaz, F.A.; Garcia-Hernandez, S.; Jesús Barreto, J.; Gutierrez, E. Mathematical Modelling of the Eﬀects of Transient Phenomena on Steel Cleanness during Tundish Transfer Practices. ISIJ Int. 2019, 59, 51–59. [CrossRef] 4. Mabentsela, A.; Akdogan, G.; Bradshaw, S. Numerical and physical modelling of tundish slag entrainment in the steelmaking process. J. S. Afr. Inst. Min. Metall. 2017, 117, 469–483. [CrossRef] 5. Merder, T.; Pieprzyca, J. Optimization of Two-Strand Industrial Tundish Work with Use of Turbulence Inhibitors: Physical and Numerical Modeling. Steel Res. Int. 2012, 83, 1029–1038. [CrossRef] 6. Ferguson, J.; Kembłowski, Z. Fluid Rheology; Marcus: Łód´z,Poland, 1995. 7. Kembłowski, Z. Non-Newtonian Fluid Rheometry; WN-T: Warszawa, Poland, 1973. 8. Reiner, M. Theoretical Rheology; PWN: Warszawa, Poland, 1958. 9. Newton, I. Philosophiae Naturalis Principia Mathematica; S. Pepys Reg. Sor. Preses: London, UK, 1686. 10. Izaak, P. Rheology of Ceramic Suspensions; AGH: Kraków, Poland, 2012. 11. Malkin, A.; Isayev, A. Rheology Concepts, Methods and Applications; ChemTec Publishing: Toronto, ON, Canada, 2006. 12. Mazumdar, D.; Evans, J. Modeling of Steelmaking Processes; CRC Press Taylor and Francis Group: Boca Raton, FL, USA, 2010. 13. Rogal, Ł.; Dutkiewicz, J. Development of technology and microstructure characterization of X210CrW12 and HS6-5-2 tool steels after thixotropic forming. Pol. Metall. 2010, 2010, 846–853. 14. Rogal, Ł.; Dutkiewicz, J. Efect of annealing on microstructure, phase composition and mechanical properties of thixo-cast 100Cr6 steel. Mater. Character. 2012, 68, 123–130. [CrossRef] 15. Rogal, Ł.; Dutkiewicz, J.; Czeppe, T.; Bonarski, J.; Olszowska-Sobieraj, B. Characteristics of 100Cr6 bearing steel after thixoforming process performed with prototype device. Trans. Nonferrous Met. Soc. China 2010, 20, 1033–1037. [CrossRef] 16. Koke, J.; Modigell, M. Flow behaviour of semi-solid metal alloys. J. Non-Newton. Fluid Mech. 2003, 112, 141–160. [CrossRef] 17. Modigell, M.; Koke, J. Rheological modelling on semi-solid metal alloys and simulation of thixocasting processes. J. Mater. Process. Technol. 2001, 111, 53–58. [CrossRef] 18. Rogal, Ł.; Dutkiewicz, J.; Atkinson, H.V.; Lity´nska-Dobrzy´nska,L.; Modigell, M. Characterization of semi-solid processing of aluminium alloy 7075 with Sc and Zr additions. J. Mater. Sci. Eng. A 2013, 580, 362–373. [CrossRef] 19. Thadela, S.; Mandal, B.; Das, P.; Roy, H.; Lohar, A.K.; Samanta, S.K. Rheological behavior of semi-solid TiB2 reinforced Al composite. Trans. Nonferrous Met. Soc. China 2015, 25, 2827–2832. [CrossRef] Materials 2020, 13, 5025 20 of 20

20. Behnamfard, S.; Taherzadeh Mousavian, R.; Azari Khosroshahi, R.; Brabazon, D. A comparison between hot-rolling process and twin-screw rheo-extrusion process for fabrication of aluminum matrix nanocomposite. J. Mater. Sci. Eng. A 2019, 760, 152–157. [CrossRef] 21. Sl˛ezak,M.;´ Bobrowski, P.; Rogal, Ł. Microstructure analysis and rheological behavior of magnesium alloys at semi-solid temperature range. J. Mater. Eng. Perform. 2018, 27, 4593–4605. [CrossRef] 22. Sl˛ezak,M.´ Study of semi-solid magnesium alloys (with RE elements) as a non-Newtonian ﬂuid described by rheological models. Metals 2018, 8, 222. [CrossRef] 23. Korolczuk-Hejnak, M. Empiric formulas for dynamic viscosity of liquid steel based on rheometric measurements. High Temp. 2014, 52, 667–674. [CrossRef] 24. Korolczuk-Hejnak, M. Determination of the Dynamic Viscosity Coeﬃcient Value of Steel Based on the Rheological Measurements; Wydawnictwa AGH: Krakow, Poland, 2014. 25. Miłkowska-Piszczek, K.; Korolczuk-Hejnak, M. An analysis of the inﬂuence of viscosity on the numerical simulation of temperature distribution, as demonstrated by the CC proces. Arch. Metall. Mater. 2013, 58, 1267–1274. [CrossRef] 26. Terzieﬀ, P. The viscosity of liquid alloys. J. Alloys Compd. 2008, 453, 233–240. [CrossRef] 27. Mezger, T. The Rheology Handbook. For Users of Rotational and Oscillatory Rheometers, 2nd Revision; Vincentz Network GmbH&Co: Hannover, Germany, 2006. 28. Sl˛ezak,M.´ Mathematical models for calculating the value of dynamic viscosity of a liquid. Arch. Metall. Mater. 2015, 60, 581–589. [CrossRef] 29. Cloete, J.H.; Akdogan, G.; Bradshaw, S.M.; Chibwe, D.K. Physical and numerical modelling of a four-strand steelmaking tundish using ﬂow analysis of diﬀerent conﬁgurations. J. S. Afr. Inst. Min. Metall. 2015, 115, 355–362. [CrossRef] 30. ANSYS Fluent Ver. 16.0—Theory Guide; ANSYS Inc.: Canonsburg, PA, USA, 2015. 31. Chang, S.; Zhong, L.; Zou, Z. Simulation of ﬂow and heat ﬁelds in a seven-strand tundish with gas curtain for molten steel continuous-casting. ISIJ Int. 2015, 55, 837–844. [CrossRef] 32. Khana, M.F.; Hussaina, A.; Usmanib, A.Y.; Yadavc, R.; Jafria, S.A.H. Multiphase ﬂow modeling of molten steel and slag ﬂow for diﬀerent tundish conﬁgurations. Mat. Tod. Proc. 2018, 5, 24915–24923. [CrossRef] 33. Qin, X.; Cheng, C.; Li, Y.; Zhang, C.; Zhang, J.; Jin, Y. A Simulation study on the ﬂow behavior of liquid steel in tundish with annular argon blowing in the upper nozzle. Metals 2019, 9, 225. [CrossRef]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).