Analysis of metal mould heat transfer coefficients during continuous casting of steel

Transcription

1 Analysis of metal mould heat transfer coefficients during continuous casting of steel V. K. de Barcellos* 1,2, C. R. F. Ferreira 2, C. A. dos Santos 3 and J. A. Spim 1,2 Heat transfer coefficients in the mould were determined by the inverse method and they are related to both process conditions and carbon equivalents of steels. Experimental data were obtained from an industrial plant by temperatures measured in moulds of the continuous casting machine by thermocouples placed in the mould wall in known positions. The temperatures are compared to profiles of simulations by the numerical model of both solidification and heat transfer processes previously examined. As a result, the numerical model calculates the heat transfer coefficients in the metal/mould interface for heats cast with different process parameters. The results make possible the determination of expressions for the calculation of the interfacial metal/ mould heat transfer coefficients that include the effects of steels grades, mould faces, casting speed, mould taper, mould section and pouring temperatures for a range of process parameters. Keywords: Continuous casting, Heat transfer coefficient, Numerical model, Steel solidification Introduction In order to understand mould heat transfer it is important to know how the heat transfers from the metal to the cooling fluid occurs. For a global analysis heat transfer the relative influence of every active heat transfer mechanism must be initially analysed. In this stage, both liquid and solid metal, lubricant, air gap formed in the metal/mould interface, as well as the mould wall and the cooling fluid are considered as sources of thermal change. The solid metal layer formed along the mould works as an increasing barrier, which stems from the heat flow, that is, it constitutes a thermal resistance that relatively increases the influence of the solid layer thermal conduction on the heat global fluid. The mould walls, however, do not represent an important thermal resistance to the heat conduction in the system because, besides the order of magnitude of the walls thickness, mould materials, usually copper and its alloys, have a great capacity of transferring heat due to their high thermal conductivity. For the thermal contact between both metal and mould, three regions can be observed: one from the meniscus up to the initial point of solidification, where the liquid metal is separated from the mould by a layer of mould powder; the second one has a good thermal contact between the solid metal and third the mould wall, due to the ferrostatic pressure on the solidified layer by the liquid metal. Furthermore, its behaviour has 1 Post-graduation Programme in Mining, Metallurgy and Materials Engineering of the Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil 2 Foundry Laboratory, Technology Center, the Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil 3 Pontifical Catholic University of Rio Grande do Sul, Porto Alegre, RS, Brazil *Correspondence author, vinicius.karlinski@ufrgs.br an important influence on superficial quality of the strand. In the third one, there is either a physical separation or a well defined air gap formation between both metal and mould, making difficult the heat flow. This separation can vary according to the steel chemical composition and its ferrite potential (FP), which represents a sticking or depression tendency. 1 6 Solidification of steels starts with either d-ferrite or c- austenite nucleation. In the case of carbon steels, the peritectic reaction LzdRc occurs with a C concentration range between 0?09 and 0?53 wt-%. Its effectiveness can be obtained by adding elements such as C, N, Ni, Mn, Cu, responsible for austenite formation. Owing to the difference in thermophysical properties between c- austenite and d-ferrite, this reaction originates undesirable phenomena such as stress, volumetric variation, segregation and inclusions precipitation, according to Matsuura et al. 7 Figure 1 schematically shows the peritectic reaction, where d-ferrite, firstly formed, is encapsulated by c-austenite from the Lzd reaction. Some studies have been carried out to understand changes in the heat flux by the peritectic reaction. Singh and Blazek 8 analysed the heat flux in the mould region by the nominal carbon contents expressed in weight per cent. For concentrations y0?10% carbon, the heat flux is minimal. Over this concentration range, the heat flux increases and it is kept constant (concentrations over 0?25%). This behaviour is explained by an excessive shrinkage upon solidification and cooling in steels containing 0?10%C that influences directly in the shell growth and the thermal resistance. Grill and Brimacombe 9 proposed a mechanism based on phases drc reaction to explain the heat flux rate. They compared 0?10%C steel to others with larger carbon concentrations and observed that the former suffers a more intense reaction in the solid state than the latter, with a volume contraction of 0?38%, which reduces the ß 2010 Institute of Materials, Minerals and Mining Published by Maney on behalf of the Institute Received 29 June 2009; accepted 10 September 2009 DOI / X Ironmaking and Steelmaking 2010 VOL 37 NO 1 47

2 1 Schematic drawing of peritectic reaction, showing d- ferrite encapsulation by c-austenite heat flux through the mould wall. Similar results were found by Suzuki et al. 10 During the steel s solidification, the intensity of the volumetric variation caused by peritectic transformation LzdRc makes the strand during continuous casting tend to depressions and stickers on the mould wall. For better comprehension of this behaviour, Wolf 11 developed a simple technique to estimate the potential for cracking or surface defects in cast carbon and alloy steels. The approach involves the FP of carbon and low alloy steels, where FP represents the proportion of d- ferrite present during solidification. This concept outlines a steel s tendency to either shrink or expand during solidification inside the mould. For either carbon or low alloy steels, the FP is defined by equation (1) FP~2 : 5(0 : 5{%C eq ) (1) where C eq is the carbon equivalent (wt-%), which classifies the influence of alloying elements present in the steel into those that stabilise either ferrite or austenite. The carbon equivalent is calculated using equation (2) C eq ~%Cz0 : 02%Mn{0 : 1%Siz0 : 04%Ni{ 0 : 04%Cr{0 : 1%Mo{0 : 7%S (2) where FP.1?0 stands for hypoperitectic steels (indicative of a fully ferritic structure), and FP,0?0 stands for hyperperitectic steels (indicating a fully austenitic structure), whereas FP between 0?0 and 1?0 is defined as the peritectic range, with mixture of the d and c phases. For steels with a tendency to form depressions (type A steels), FP values range between 0?85 and 1?05, whereas for those with a tendency to stick to the mould wall (type B steels), these values are FP.1?05 or FP,0?85. In this work a proposed method based on the solution of an inverse heat transfer problem is employed to determine the heat transfer coefficients h i along the metal/mould interface. This method involves the measurement of experimental temperatures obtained by thermocouples placed in the mould wall in known positions. Those temperatures are compared to profiles of simulated ones by a numerical model of both solidification and heat transfer, previously examined by Spim and Garcia. 12 Specifically this work describes the following activities: (i) development of a method to determine the metal/mould heat transfer coefficients based on the solution of the inverse heat conduction problem (IHCP) (ii) development of a computer simulation of steel solidification (iii) mathematical analysis of mould temperatures were made to calculate the interfacial heat transfer coefficients between the strand and the mould (iv) analysis of the influence of process parameters [pouring temperature, casting speed (CS)], steel grades and mould geometry (taper, section, mould faces) on the mould heat transfer (v) attainment of empirical equations for the calculation of the heat transfer coefficients as a function of the process parameters and chemical composition of steel, for the analysed moulds. Mathematical modelling of heat transfer In the present work, the adopted numerical model uses the finite difference method for simulating solidification behaviour. The mathematical formulation of heat transfer is based on the general equation of heat conduction, which is given for one-dimensional heat flux by equation (3) rc LT Lt ~ L Lx k LT Lx z : q (3) where r is density (kg m 23 ); c is specific heat (J kg 21 K 21 ); k is thermal conductivity (W m 21 K 21 ); LT/Lt is cooling rate (K s 21 ), T is temperature (K) and : q represents the term associated to the internal heat generation due to the phase change. Approximating equation (3) by finite difference method, we obtain rc 0 T nz1 i Dt {T n i ~k T n iz1 {2T n i zt n i{1 Dx 2 (4) where i is the element position according to x axis, n and nz1 refer to temperatures before and after the incremental time interval Dt respectively, and the stability criteria are given by Dt,(Dx 2 /2a) where a5k/ rc is the thermal diffusivity (m 2 s 21 ). The term c represents the apparent specific heat of the material and includes the effect of the release of latent heat L (J kg 21 ) and changes of solid fraction with the temperature (Lf S / LT), given by c95[c2l(lf S /LT)]. The solid fraction depends on a number of parameters involved in the system. However, it is quite reasonable to assume the solid fraction varying only with temperature. For carbon steels, f S is appropriately described by the lever rule, given by equation (5) f S ~ 1 TL {T (5) 1{K o T F {T where k o is the partition coefficient, T L is the liquidus temperature and T F is the solvent fusion temperature. In the present work, k o is adopted as 0? Ironmaking and Steelmaking 2010 VOL 37 NO 1

3 The model also permits the insertion of physical properties as a function of temperature, considering the amount of solid as liquid fractions. At the range of temperatures, where solidification occurs for metallic alloys, the physical properties will be evaluated taking into account the amount of liquid and solid that coexists in equilibrium at each temperature given by k SL ~ ðk S {k L Þf S zk L (6) r SL ~ ðr S {r L Þf S zr L (7) c SL ~ ðc S {c L Þf S zc L {L Lf S (8) LT where subindices S and L indicate respectively solid and liquid states. Using an analogy between a thermal system and the passive elements of an electrical circuit, the final equation used by the solidification mathematical model is given by T nz1 i ~ C ti where Dt ðr ti{1 zr ti Þ T i{1 n z " # Dt Dt 1{ { Ti n C ti R tiz1 zr ti C ti ðr ti{1 zr ti Þ z C ti Dt T n (9) iz1 R tiz1 zr ti C ti ~DxDyDzr i c i (10) where nz1 is the index associated to the future time, n is the index corresponding to the actual time, Dt is the increment in time (s), i is the position, DxDyDz is the nodal element volume (m 3 ) and C ti is the thermal capacitance (J K 21 ) which represents the energy accumulated in a volume element i. The thermal resistance at the heat flux line from point iz1 ori21 to the point i is given by R ti{1 ~ Dx i{1 k i{1 DyDz R ti ~ Dx i k i DyDz Dx iz1 (11a) (11b) R tiz1 ~ (11c) k iz1 DyDz For the mesh elements in contact with the mould, the thermal resistance is given by R t0 ~ 1 (12) h i DyDz where h i is the metal/mould heat transfer coefficient (W m 22 K 21 ). More details of the overall mathematical model can be seen in previous articles The thermophysical properties of steel and mould used in the mathematical model are listed in Table 1. Boundary conditions Figure 2 shows the finite difference mesh of the onedimensional horizontal section of the strand and mould. The number of elements or nodes consists of 72 in the 2 Schematic of grid distribution in vertical and transversal plane of strand and mould strand, and 12 in the mould for the 150 mm square section mould, where Dx and Dy are the dimensions of the volume elements. For the 180 and 240 mm section moulds, the mesh contains 90 and 120 nodes in the strand, and 15 and 22 in the mould respectively. As the grid mesh is one-dimensional for the strand and mould, a line until the strand centre in the middle of the face has been considered, as showed in Fig. 2. The mesh consists of a thin horizontal slice of the mould and strand section. Initially, the slice is located at the meniscus and at greater times is allowed to move downwards at a rate equal to the withdrawal CS. Thus, after each movement, the line moves a vertical distance equal to Dt6CS. The time interval Dt is calculated by the mathematical model and corresponds to a stability criterion for the numerical programme. The following assumptions were adopted in the heat transfer model: (i) the temperature at the meniscus (t50) is uniform and assumed to be equal to the tundish metal temperature (ii) the heat transfer coefficient of the water is considered constant in the mould (iii) (iv) (v) the heat flux in the casting direction is negligible variation in heat flux due to mould oscillation, metal level fluctuations at the meniscus, mould deformation and segregation are ignored the top and bottom surface of the copper mould are considered isolated, without thermal loss Table 1 Thermophysical properties of steel and mould used in heat transfer model Property Value 28 Metal Mould Liquidus thermal conductivity k l,wm 21 K Solidus thermal conductivity k s,wm 21 K Liquidus heat capacity c l,jkg 21 K Solidus heat capacity c s,jkg 21 K Liquidus density r l,kgm Solidus density r s,kgm Latent heat L, Jkg Ironmaking and Steelmaking 2010 VOL 37 NO 1 49

4 (vi) the effect of forced convection due to the liquid metal turbulence is neglected (vii) the thermophysical metal properties (specific heat, thermal conductivity, density) are constants in the liquid and solid phases. they only change with the temperature in the mushy zone, being the solid fraction [equation (5)] calculated with the lever rule (viii) the thermophysical properties are constant for the mould material (ix) the transformation temperatures (solidus and liquidus) are parameters that depend on chemical composition. They were calculated with empirical equations used by Thomas et al., 17 shown respectively below T L ~1537{88(%C){25(%S){5(%Cu){ 8(%Si){5(%Mn){2(%Mo){4(%Ni){ 1 : 5(%Cr){18(%Ti){30(%P){2(%V) (13) T S ~1535{200(%C){183 : 9(%S){ 12 : 3(%Si){6 : 8(%Mn){4 : 3(%Ni){ 1 : 4(%Cr){4 : 1(%Al){124 : 5(%P) (14) (x) the CS and the pouring temperature are constant and correspond to the average values in the time (xi) the heat transfer coefficients in the interface mould/cooling water are determined and used constantly throughout the mould. Methodology for determination of metal/mould interfacial heat transfer coefficients The method used to determine the metal/mould heat transfer coefficients is based on the solution of the IHCP. In this work, the method consists of using the thermocouple temperature measurements connected at known locations inside the mould wall. Applications of the IHCP are described in previous articles. 3,18 22 In this work the model is employed to determine heat transfer coefficients along the metal/mould interface. This method involves the measure of experimental temperatures obtained by thermocouples placed in the metal and/or mould in known positions. Those temperatures are compared to profiles of simulated ones by a numerical model of both solidification and heat transfer, previously examined. The procedure used to determine h i is performed by adopting an initial value of h (h i ), by which temperatures of each position in a numerical mesh are calculated by a mathematical model for both mould and metal at different time intervals. In every interaction, the h i correction is carried out with either increasing or decreasing of a Dh i value and, therefore, new temperatures are estimated. A one-dimensional model was used to determine the metal/mold heat transfer coefficients by IHCP in the middle of each face of the mould because in this region the flux is essentially unidirectional. 21 According to Thomas, in models of steel continuous casting axial heat conduction (z direction) can be ignored because it is small relative to axial heat conduction (x and y directions), as indicated by the small Peclet number (casting speed6shell thickness/thermal diffusivity) Flowchart of optimisation algorithm for determining heat transfer coefficients in metal/mould interface Optimisation strategy for searching heat transfer coefficient along mould Search routines for determining heat transfer coefficients in the metal/mould interface, along the mould length, were developed by optimisation strategies for increasing both the processing speed and the results accuracy obtained from the mathematical model of solidification. Figure 3 shows the flow chart of the optimisation algorithm for determining heat transfer coefficients in the metal/mould interface. The routine acts interactively. Initially, the mesh determines a temperature profile of both metal and mould in the meniscus region, and it also compares the simulated temperature with the experimental one in the monitored point from a convenient initial value of h i W m 22 K 21. Compared temperatures within the range of 1uC are accepted. The test is carried out when the numerical mesh moves from the meniscus region to the next monitored point. At this time, the h i representativeness is verified by analysing whether or not the simulated temperature is within the tolerable range. If necessary, the system can either increase or decrease the h i value and also repeat calculations from the previous monitored point with a new h i value. In each stage, the processing follows up to the distance of the next thermocouple, and successively, along the mould. The heat transfer coefficient in the mould/cooling water interface is only determined for the 240 mm section mould in the distance of 120 mm of the meniscus and it was considered constant along the mould. Comparisons of experimental and simulated temperatures of the corresponding points to the Tp4 and Tp5 50 Ironmaking and Steelmaking 2010 VOL 37 NO 1

5 4 Thermocouples arrangement along copper mould of continuous casting machine thermocouples were carried out to determine the coefficients. In order to start the calculations an initial value of W m 22 K 21 is attributed, if necessary an increment or decrement in the value of 5000 W m 22 K 21 is made. Experimental procedure Heat transfer coefficients both along the mould length and in three different mould faces were determined from temperature data obtained by thermocouples strategically positioned in the mould walls. In all monitored heats, moulds were monitored with type K thermocouples inserted along the mould height. Heat transfer coefficients were correlated with the steels FP, which tend to stick or depress from the wall. Experimental data were obtained from an industrial plant by temperatures measured in moulds of the continuous casting machine. Relevant specifications of the machine are shown in Table 2. The moulds were monitored with 30 thermocouples inserted in the three faces of the central position, arranged in the following way: 10 thermocouples in the centre of the outer face, 10 thermocouples in the centre of the inner face, and 10 thermocouples in the centre of the side face. The thermocouples arrangements along the moulds are shown in Fig. 4. For the 240 mm square section mould the thermocouples were placed from the meniscus Table 2 Machine specifications Mould dimensions, mm ; ; Type of machine Bow type caster with curved mould Type of mould Parabolic with taper inside Electromagnetic stirrer Mould stirrer and movable final stirrer Lubrication Powder Length of mould, mm 801 Average mould level (meniscus), mm 170 Mould wall thickness ( mm), mm 22 Mould wall thickness ( mm), mm Mould wall thickness ( mm), mm Machine linear length, m 24 Machine radius, m 9. 0 Number of strands 3 Ladle size, t 65 Tundish capacity, t 12 Average temperature of the mould cooling water at the inlet, uc 27 Average temperature of the mould cooling water at the outlet, uc 32 Ironmaking and Steelmaking 2010 VOL 37 NO 1 51

6 region, 170 mm from the mould top, up to 31 mm above the mould exit, which results in a total of 10 strategic points of faces monitoring. The thermocouples Tp1, Tp2, Tp3, Tp4, Tp6, Tp7, Tp8 and Tp10 were inserted at 5?4 mm from the mould hot face (metal/mould interface) and the Tp5 and Tp9 at 15?4 mm in the same height from Tp4 and Tp8 respectively. For the square section moulds with 180 and 150 mm, the thermocouples were inserted at 5?5 and 6?0 mm from the mould hot face respectively. For these moulds, Tp1 was placed above meniscus at 15 mm. The Tp2 stands for the reference of meniscus temperature A total of 13 heats were cast and mould wall temperatures were measured (one heat with parabolic square section mould with 150 mm, one heat with section mould 180 mm, nine heats with the 240 mm section mould and three heats with the 240 mm linear section mould). The compositions and casting conditions of these heats are listed in Table 3. During casts, temperatures were continuously monitored at the specific positions as a function of time, the average heat time was y50 min per heat cycle. Results The analysed heats were simulated and measured the heat transfer coefficients in the mould gap using the solidification model. The obtained results were analysed and correlated among them to find the influence of the boundary conditions and process parameters. It was found that the heat transfer coefficients in the mould gap are influenced by different process parameters, such as: steel grade, CS, pouring temperature, mould section, mould faces and mould taper. Because of this the influence of different process parameters was examined. Influence of steel grade Figure 5 shows the influence of steel grade on the heat transfer coefficients in the mould gap. The main difference in the casting conditions between the heats is the carbon content, being two low carbon steels and two high carbon. Heats 3 and 4 are of high carbon steel with ferritic potential that define them as steel type B, with high index tendency to stick to the mould wall. Heats 1 and 2 are of low carbon steel of type A with high index tendency to contraction during solidification. For all graphs, it is observed that coefficients achieve their maximum values in the meniscus region due to the best 5 Influence of steel grade on heat transfer during solidification in mould thermal contact between the liquid metal and the mould wall. As these coefficients move along the mould, they drastically decrease, keeping both a constant feature for type A steel, and a small increase for type B steel. In Table 3 Casting conditions and chemical compositions for heats investigated Heat no. Section mould, mm Steel composition of heats, wt-% Ferritic potential Casting conditions C Mn Ni Si Cr S Mo C eq FP Type CS, m min 21 Tp, uc P A P A P B P B L B L B L B P B P B P B P B P B P B Ironmaking and Steelmaking 2010 VOL 37 NO 1

7 6 Influence of mould faces on heat flux general, the results show that the heat transfer coefficients are higher for high carbon steels. This effect is greatest in the meniscus region (170 mm). On the results, it can be seen that the heat transfer coefficients are directly influenced by the shrinkage of the solidified shell and the expansion of the gap during the peritectic reaction. This behaviour in the mould leads to changes on the heat flux over the length of the mould. The results behaviour is very similar with to findings of other workers. 3,4,8,24 Influence of mould faces All heats were cast with a continuous caster of the curved type. Therefore, the bending of the mould makes the gap thickness different for each face and consequently, the heat transfer in each face is also changed. Figure 6 shows the heat transfer coefficients calculated for each face, showing small variations of heat transfer coefficients along the mould. It can be observed that the heat flow is bigger in the outer face. This can be explained by the better contact of the strand to the mould wall in function of the bending. Influence of CS Figure 7 shows the influence of a CS on the heat flux. It can be seen that the heat transfer coefficients are affected by CS changes between heats of similar steel grade. According to Chow et al., 24 the increased heat transfer coefficients at higher CSs mainly next to meniscus can be explained by three reasons. First, the shorter residence time of the steel at high CS results in thinner shells that deform easily under the ferrostatic pressure, ultimately reducing the mould/strand gap. Second, these shorter 8 Influence of mould taper on heat transfer during solidification in mould residence times result in hotter billet surface temperatures, which increase the thermal gradient for heat flow. Third, there is less thermal contracting at these hotter strand shell temperatures, which improves mould/strand contact. This behaviour can also be seen in Fig. 5. Influence of mould taper Figure 8 shows the profiles of heat flux when comparing heats cast under similar process conditions but in different moulds. One with a conventional single taper and the other with a continuously changing taper (parabolic taper) toward the bottom of the mould. It is clear from the profiles of heat transfer coefficients that the heat extracted by the parabolic mould is higher, except for the upper part of the mould. In this region, the size of the air gap between the mould and the strand surface is smaller and hence the heat transfer is similar. On the other hand, the heat transfer at the bottom half of the mould is most sensitive to the taper, owing to the thermal contraction and steel shrinkage which for single taper result in a gradually increasing air gap between mould strand interface and hence a loss of heat transfer. The study about the mould dimensions and taper has been discussed by several authors, 3,25,26 and all of them agree that the heat transfer is affected by the mould taper changes. Influence of mould sections Figure 9 shows the heat transfer coefficients for similar steel grades with different mould sections. The curves are the average heat transfer coefficients between the faces. In the results, it can be observed that the heat 7 Influence of CS on heat transfer during solidification in mould 9 Influence of mould sections on heat transfer during solidification in mould Ironmaking and Steelmaking 2010 VOL 37 NO 1 53

8 10 Influence of pouring temperature on heat transfer during solidification in mould transfer coefficients are higher for smaller mould sections. This behaviour could be explained by the fact that the moulds with smaller sections have thinner mould wall thickness, thus providing less resistance to the heat extraction promoted by the cooling water. Moreover, the metal volume that must be solidified is minor for smaller mould sections; therefore, the solidified shell thickness will be formed quickly. This situation makes possible the increase in the CS that consequently increases the heat flux, as previously discussed. Pouring temperature Pouring temperature (Tp) was found to have no significant effect on the mould heat transfer inside the temperature range analysed in the heats of Fig. 10. It shows the heat transfer coefficients from heats cast under similar steel grade and CS but with different superheats. Other authors have also reported that an increase in casting temperature has a negligible effect on the heat transfer profiles in the mould. 24,27 Determination of empirical equation to measure heat transfer coefficients in mould In several studies of literature, average mould heat transfer coefficients are commonly used in solidification modelling. The focus of this study is to develop an equation that can predict the heat transfer coefficients along the length of the mould from the simulation of the heats with different process parameters and steel grades. From the results of y90 heats simulated and analysed, a general equation to estimate the heat transfer coefficients was obtained. For this case, the heat transfer coefficients of each heat were defined as average values of the heat transfer coefficients measured in the three mould faces. The resultant expression is h~{(section 0 : 57z91 : 43) ½6 : 6zln(Dist{169 : 7) Šz CS C eq 300z3500 (15) a comparison between heat transfer coefficients obtained by equation and by numerical model; b comparison between experimental temperatures inside mould wall and simulated temperatures obtained with heat transfer coefficients determined by equation (15) 11 Simulation results of heats with parabolic section mould 240 mm where Section is the mould section, Dist is the distance from the meniscus, CS is the casting speed and C eq is the carbon equivalent. Equation (15) is valid for square sections of 150, 180 and 240 mm with parabolic moulds and process parameters in the following ranges showed in Table 4. For validation, this equation was inserted in the numerical model to measure the temperatures inside the mould walls in the same position where the thermocouple was placed. The results obtained with the simulation using the equation were compared with experimental temperatures of the heats. Additionally, the equation results were compared to the heat transfer coefficients generated by the numerical model. Figures show the comparison of the results for square section moulds of 240, 180 and 150 mm. Although the results show that the graphs are slightly different, it is apparent that the curve behaviours are similar. Using the approach with equation (15), the heat transfer coefficients can be reasonably predicted and enable the simulate of the mould heat transfer with more accuracy. Thus, it allows greater use of computer modelling in evaluation and subsequent optimisation of the simulation performance. Table 4 Range of operating conditions Casting speed, m min 21 Pouring temperature, uc C eq, wt-% Mould section, mm Min. Max. Min. Max. Min. Max Ironmaking and Steelmaking 2010 VOL 37 NO 1

9 2. The mathematical model was efficient to determine both the FP and its influence on the heat transfer regime during solidification of steels along the parabolic mould with square sections of 150, 180 and 240 mm and 801 mm in height. Calculated heat transfer coefficients were in accordance with the heat transfer tendency, presenting both high values for the region under the meniscus, where the thermal contact is more effective, and low ones as steel solidification develops. These coefficients also highlight the difference existing between types A and B steels heat transfer. 3. The results make possible the determination of expressions for the calculation of the interfacial metal/ mould heat transfer coefficients of the analysed moulds that include the effects of steel grades, CS and mould sections for determined range of process parameters. 12 Simulation results of heats with parabolic section mould 180 mm Conclusions The main conclusions drawn from the study are as follows. 1. A system of numerical models of the continuous casting process has been developed and validated with plant measurements. It has been applied to simulate the heat transfer throughout the solidifying steel shell as it moves through the continuous casting mould. 13 Simulation results of heats with parabolic section mould 150 mm Acknowledgements The authors would like to thank the Gerdau Company (Gerdau Aços Especiais Piratini, state of Rio Grande do Sul, Brazil) for cooperation, support and help. The authors would also like to thank agency National Council for Scientific and Technological Development of Brazil (CNPq) for the financial support. References 1. J. K. Brimacombe: Can. Metall. Q., 1973, 15, J. K. Brimacombe, I. V Samarasekera and J. E. Lait: Iron Steel Soc. AIME, 1984, 2, S. Chandra, J. K. Brimacombe and I. V. Samarasekera: Ironmaking Steelmaking, 1993, 20, (2), R. B. Mahapatra, J. K. Brimacombe and I. V. Samarasekera: Metall. Trans. B, 1991, 22B, (6), K. C. Mills, T. J. H. Billany, A. S. Normanton, B. Walker and P. Grieveson: Ironmaking Steelmaking, 1991, 18, (4), M. M. Wolf and W. Kurz: Solidification and casting of metals, ; 1979, Sheffield, The Metals Society. 7. K. Matsuura, H. Maruyama, M. Kudoh and Y. Itoh: ISIJ Int., 1995, 35, (12), S. N. Singh and K. E. Blazek: J. Met., 1974, 26, A. Grill and J. K. Brimacombe: Ironmaking Steelmaking, 1976, 3 (2), M. Suzuki, C. H. Yu, H. Sato, Y. Tsui, H. Shibata and T. Emi: ISIJ Int., 1996, 36, M. M. Wolf: Proc. 1st Eur. Conf. on Continuous casting, Vol. 2, 23 25; 1991, Florence, AIME. 12. J. A. Spim and A. Garcia: Materials Science & Engineering, 2000, 277, N. Cheung and A. Garcia: Eng. Appl. Artif. Intell., 2001, 14, (2), C. A. Santos, J. M. V. Quaresma and A. Garcia: J.Alloys Compd, 2001, 319, C. A. Santos, E. L. Fortaleza, C. R. F. Ferreira, J. A. Spim and A. Garcia: Model. Simul. Materi. Sci. Eng., 2005, 13, J. E. Spinelli, J. P. Tosetti, C. A. Santos, J. A. Spim and A. Garcia: J. Mater. Process. Technol., 2004, 150, (3), B. G. Thomas, I. V. Samarasekera and J. K. Brimacombe: Metall. Trans. B, 1987, 18B, H. B. Yin and M. Yao: J. Mater. Process. Technol., 2007, 183, C. A. M. Pinheiro, I. V. Samarasekera, J. K. Brimacombe and B. N. Walker: Ironmaking Steelmaking, 2000, 27, (1), K. Ho and R. D. Pehlke: AFS Trans,., 1984, 92, C. A. Santos, A. Garcia, C. R. Frick and J. A. Spim: Inverse Prob. Sci. Eng., 2006, 14, C. A. Santos, J. A. Spim and A. Garcia: Eng. Appl. Artif. Intell., 2003, 16, B. G. Thomas: Modeling of continuous casting, in making, shaping and treating of steel: continuous casting, Vol. 5, Chapter 5, 1 24; 2003, Pittsburgh, PA, AISE Steel Foundation. Ironmaking and Steelmaking 2010 VOL 37 NO 1 55

10 24. C. Chow, I. V. Samarasekera, B. N. Walker and G. Lockhart: Ironmaking Steelmaking, 2002, 29, (1), C. Chow and I. V. Samarasekera: Ironmaking Steelmaking, 2002, 29, (1), N. Fukada, Y. Marukawa, K. Abe and T. Ando: Can. Metall. Q., 1999, 38, (5), I. V. Samarasekera and J. K. Brimacombe: Int. Met. Rev., 1978, 23, (6), R. D. Pehlke, A. Jeyarajan and H. Wada: Summary of thermal properties for casting alloys and mold materials, University of Michigan, Ann Arbor, MI, USA, Ironmaking and Steelmaking 2010 VOL 37 NO 1