Correlation for Mold Heat Flux Measured in a Thin-Slab Casting Mold

Transcription

1 orrelation for Mold Heat Flux Measured in a Thin-Slab asting Mold Prathiba Duvvuri, ryan Petrus, and rian G. Thomas Department of Mehanial Siene and Engineering University of Illinois at Urbana-hampaign 6 W. Green Street Urbana, IL 68 US Phone - (7) -699 Fax - (7) bgthomas@uiu.edu Keywords: ontinuous asting, Thin slab aster, Mold heat flux, Non-linear multiple regression INTRODUTION Maintaining appropriate heat transfer in the steel ontinuous asting mold is ritial to strand surfae quality [-], mold life [4], asting produtivity [5] and operating safety in preventing breakouts [6,7]. onsiderable researh on mold heat transfer has been based on omputational modeling and numerial simulations. However, with advanes in mold instrumentation and data olletion, plant measurements an be analyzed to investigate these phenomena in atual operation. Most researh on this topi investigated the effet of mold heat transfer on quality and breakouts onsidering a partiular asting variable, suh as mold powder properties, asting speed, or steel omposition [6-9]. Xia et al [7] investigated the dependene of integral heat flux on these asting variables, and used the results to analyze breakouts. iutti et al [8] developed an equation to predit mold heat flux (averaged over the hot fae) as a funtion of these asting variables by performing multiple regression using data olleted from a onventional slab aster produing low and medium arbon steels. Heth et al [9] studied the effet of super heat, osillation mark depth and mold powder onsumption in addition to the effet of steel omposition on mold heat removal, and investigated surfae quality of rak sensitive steel grades as a funtion of mold heat flux. These papers all studied onventional slab asters. Santillana [] et al studied the effet of asting powder and mold plate thikness on mold heat transfer in a thin slab aster, using plant measurements and the OND model [] to simulate temperature in the strand and mold. Data analysis tehniques are extensively used to identify relationships between proess variables and in developing models for prediting proess outomes [7-9]. In ontinuous asting of steel, there are signifiant opportunities for omprehensive study of plant measurements of the wide range of asting variables and their relationship to safety, quality, and prodution goals. This paper applies data fundamental analysis methods to predit mold heat flux in a thin slab aster as a funtion of asting onditions, based on extensive plant measurements of over years at the Nuor steel mill in Deatur, L. Different empirial equations to predit mold heat flux are developed with nonlinear multiple regression analysis. ased on statistial analysis, the models are evaluated and ompared with a previous equation in the literature. The best models are presented, and an be applied to predit mold heat transfer in future work to understand and improve the ontinuous asting proess. METHODOLOGY The Nuor Deatur steel mill has two ompat Strip Prodution (SP) slab asters with a slab thikness of 9 mm, respetively alled the North and South asters. The mill maintains a large data base of measurements of various onditions in the asters, most reorded every 5 seonds. The goal of the present work is to utilize this database to learn about and model the relationships between heat flux in the mold and these measurements. First, the effet of individual asting onditions on mold heat flux is studied by seletively filtering the data so that all other pertinent asting onditions are as onstant as possible. Then, a model to predit mold heat flux as a funtion of asting variables is developed using nonlinear multiple ISTeh 4 Proeedings. 4 by IST. 88

2 regression. This is aomplished in four steps: data extration, data pre-proessing, model development, and evaluating the models. Data extration Sine the molds of ontinuous asters are onstantly ooled by water, the spatially-averaged heat flux extrated from a mold hot fae (hereafter simply alled mold heat flux) at a given time an be alulated from measurements of the volumetri flow rate and temperature rise of the ooling water as 5 GwwT GT 6 Q 6W Z W Z () where Q is the heat flux (MW/m ), G is the flow rate (l/min), w is the density (kg/l), w is the speifi heat apaity (J/kg ), and T is the temperature rise ( ) of the ooling water for a mold fae with working (i.e. in ontat with steel) length Z (m) and strand width W (m). Measurement data is stored in the Level II database of the Nuor mill. Strutured Query Language (SQL) is employed to extrat the required data, whih is loated in different tables of the database. The query utilizes appropriate andidate keys (primary olumns suh as heat number) to ross-referene the data from different tables and to redue the run time. This work analyzed 6, heats on eah aster over years, overing a wide range of operating onditions. heat at Nuor Deatur typially lasts around 5 minutes. To haraterize the measurements of asting onditions during eah heat, the measured data were averaged over a minute interval, starting minutes after ladle open so as to exlude transient effets during ladle hanges. Heats that did not last this long, for example due to a breakout ourring, were not inluded. Data at every 5 seonds were extrated using SQL ode and the average, maximum, and minimum of the measurements over the minute interval were omputed and saved. In addition, the standard deviation was alulated for the mold level. The mold powder properties are reported by plant metallurgists, inluding the breakpoint temperature (whih is onsidered a measure of the rystallization or melting temperature), and the visosity. The final version of the query ontains 5 lines of SQL ode and takes less than 5 seonds to ollet years of data. Data preproessing Pre-proessing is a very ruial step in data modeling, in order to remove inomplete, noisy and inonsistent data that would distort the final model. To restrit the study to heats with steady asting onditions and reliable measurements, primary filters were applied, whih required eah seleted heat to satisfy the onditions summarized in Table I. In the table, variation is the differene between the maximum and minimum measured value during the minute interval seleted for eah heat. Table I. North aster: Primary filters for asting variables asting variable Filter riterion Remaining heats (of 45 total) onstant asting speed variation in V mm/s 7 Mold powder type exluding trial powders 5 onstant mold width variation in W mm 7 Realisti super heat s Realisti mold level standard deviation l mm 556 ox plots and density plots for the olleted data are shown in Figure, to visualize the distribution of values for eah variable. To investigate the influene of eah asting ondition individually, by keeping the other onditions as onstant as possible, a seondary filter is applied to every variable exept for the one under investigation. good seondary filter for a given variable (when not under individual investigation) should leave a large number of heats, with all measurements falling within a narrow range. The seondary filters used in this work are given in Table II. The properties of the asting powders are tabulated in Table III. 88 ISTeh 4 Proeedings. 4 by IST.

3 asting speed (m/min) arbon (%)..4.6 Visosity (Pa.s) Melting temperature ( o ) asting speed (m/min)..4.6 arbon (%) Visosity (Pa.s) Melting temperature ( o ) Mold width (mm) Super heat ( o ) Mold plate thk. (mm) Mold lvl. st. dev. (mm) Mold width (mm) 4 5 Super heat ( o ) Mold plate thk. (mm) Mold lvl. st. dev. (mm) Figure. ox plots and density plots presenting data distribution of asting variables for fixed fae of North aster at Nuor mill for two years Table II. North aster: Operating onditions and seondary filters for asting variables Measurement range (after primary filters) Range for seondary filter asting variable asting speed (m/min) arbon perent (%) Mold powder see Table III P4 Mold width (mm) Super heat ( ) Mold plate thikness (mm) Mold level standard deviation (mm) Table III. Mold powder properties (alulated at ) Powder asiity Visosity (Pa-s) reak point temperature ( ) P P P P P P P Model development The model development step starts by assuming a general struture for equations to predit mold heat flux as a funtion of asting variables. In general, numerial tools an determine parameters for the model that best fit the measurements, but not ISTeh 4 Proeedings. 4 by IST. 88

4 the struture of the model. For this work, the struture was partly based on an equation developed by iutti [8], stated in the literature: Flow % Q 4.6 T V.5exp.7 () where Q is the mold heat flux (kw/m ), is the mold slag visosity (Pa-s), T Flow is the melting temperature of the powder ( ), V is the asting speed (m/min), and % is the arbon amount (weight %). The expression inside parentheses aounts for the known drop in mold heat flux for periteti steels. The predited drop in heat flux is a bell-shaped (Gaussian) urve over arbon ontent. The bell urve has a minimum at.7 weight % arbon, where it subtrats a fration of.5 (5. %) of the heat flux relative to a non-periteti steel under the same onditions. The value.7 ontrols the width of the heat flux drop. The urrent work extends the iutti equation to inlude other parameters aording to the following struture: Q x V T W s t l.5 exp x x x x4 x5 x6 x7 x8 break 9 % () where Q is the predited mold heat flux (MW/m ). The fitting parameters x i, i =,,.., 9 are hosen to best math the measurements. The variables, or measurements used as a basis for the predition, are asting speed V (m/min), mold slag visosity (Pa-s), break point temperature of the powder T break ( ), width of the slab W (mm), temperature superheat s ( ), thikness of the mold plate t (mm), standard deviation of the mold level l (mm), and arbon amount % (weight %). Rather than using iutti s expression to inorporate the heat flux drop for periteti steels, Eq. gives this drop as a funtion of %, inluding the effets of other alloys from the relation of lazek et al []. Speifially, and determine the range of periteti steels depending upon the measured omposition of eah heat as follows l.5mn.77si.l.9ni.6mo V r r W (4).967.6l.6Mn.Si.4l.5l Si.4Ni Mo V r r r Ni W (5) where l, Mn, Si, et., are the element weight perentages of the steel omposition in eah heat. Then, non-linear regression analysis is performed in MTL [] to find the best fit values for the parameters that minimize the error (speifially the sum of squares of the differenes between eah measured heat flux and the predited value from Eq. ) using the 556 heats after applying primary filters. The minimization is performed using fminsearh funtion in MTL whih employs the Nelder-Mead Simplex algorithm [4]. This funtion is sensitive to the initial guess. Therefore, to determine a good initial guess, a linear regression is first performed in Mirosoft Exel, by taking the logarithm to make Eq. linear. Then, nonlinear regression is performed in MTL for initial guesses randomly distributed in the neighborhood of the linear best fit. Initially, the signifiane of eah of the eight asting variables inluded in Eq. on mold heat flux is not lear. So, to find the best model, this analysis is performed with different ombinations of asting variables. With eight variables, there are ISTeh 4 Proeedings. 4 by IST.

5 different potential models, onsidering all possible ombinations. Rather than test every ombination, stepwise forward seletion is used. Models are developed in a sequene, beginning with a model with only one asting variable. In eah subsequent model, the one new asting variable that results in least RSS error is added into the equation [5]. Model evaluation mong the different models from this stepwise forward seletion, the best model is seleted, aounting both for auray and simpliity, based on statistial measures, namely the residual sum of squares (RSS) and kaike Information riterion (I). The residual sum of squares (RSS) is a measure of disrepany between the data and estimation model. It is alulated as the sum of squares of the differenes between the observed and predited values. n ( i ( i)) (6) i RSS y f x where y i is the observed value and f ( x i ) is the model predited value, and n is the total number of observations. smaller RSS indiates a better fit. The oeffiient of determination (R ) is the ratio of explained variation to the total variation of the data and expresses the goodness of fit of a regression as follows. n ( yi f ( xi)) n i R, y y n i n i (y y) i i (7) R always lies between and, with indiating a perfet predition of all data, and indiating extreme variability. In this work, it is only employed to examine the relationship between mold heat flux and individual asting variables. It is always possible to derease RSS by inreasing the number of variables in the model, even if the variable does not really ontribute to the auray of the predition. Other statistial measures, suh as I, whih penalizes adding insignifiant variables, are better for omparing different model strutures. The kaike Information riterion (I) [6] is a measure of the relative quality of a statistial model. The I for a model is, I L k (8) where L is the maximum likelihood of the measured data ourring given the best possible set of parameters, and k is the number of parameters. Using RSS as a measurement of the total error, the maximum likelihood orresponds to the smallest possible RSS, i.e. with the best fit parameters. Using this minimum RSS, the maximum log-likelihood value an be alulated as n RSS L ln (9) n I measures the trade-off between the goodness of fit and omplexity of the model. smaller I indiates a better model. The models are developed using fixed fae of Nuor North aster, whih is treated as training data. The models are then tested with data from the loose fae of the North aster and both faes of the South aster. The performane of the optimum model to predit mold heat flux is then ompared to the iutti predition (Eq. ) for the fixed faes of North and South asters. ISTeh 4 Proeedings. 4 by IST. 885

6 MODEL VERIFITION efore developing a pratial model with the omplete database, the proedure for hoosing best fit parameters desribed in the previous setion is verified with a known equation, alulating the average heat flux for heats assuming a mold heat flux model form: Q x G T W x x x4 () where G is the water flow rate (l/min), T is the rise in water temperature ( ), and W is the mold width (m). The influene of eah of the above hosen variables on mold heat flux is shown in Figures -4. Regressions using only one ( G ), two ( G and T ), and all three variables are shown in Figures 5-7 to illustrate the use of I in judging the model. With the addition of eah variable, I drops indiating an improved model. There is only a moderate drop when the seond variable is inluded, but a large drop when all three variables are inluded. This illustrates a problem in modelling nonlinear relationships, that it is diffiult to separate the effets of the parameters. In addition, trends may appear weak and preditions poor if even one important and ross-orrelated variable is missing from the model. heats Figure. Inrease in heat flux with water flow rate Figure. Variation in heat flux with rise in water temperature Figure 4. Variation in heat flux with mold width Predited heat flux (MW/m ).5 heats.5 heats.5 heats tual heat flux (MW/m ) Figure 5. Regression with only water flow rate as a variable. I = tual heat flux (MW/m ) Figure 6. Regression with water flow rate and rise in temperature as variables. I = tual heat flux (MW/m ) Figure 7. Regression with all variables. I = -47. Perfet math of predited heat flux with atual heat flux The best fit for x i are ( /.85 ),, and - respetively whih exatly math those of the original equation (Eq. ), given the fixed working mold length of 85 mm. Thus, the proedure is reliable. 886 ISTeh 4 Proeedings. 4 by IST.

7 RESULTS ND DISUSSION Heat extration from the four mold faes is ompared in Figures 8 and 9. Figure 8 shows larger heat flux from the wide faes than the narrow faes, in agreement with previous studies [8]. Figure 9 shows that there is no signifiant differene in the heat flux extrated between fixed and loose wide faes. Therefore, model development was performed using only the fixed fae data. The loose fae was used as testing data for omparing the developed models. Figure 8.Total of average heat flux on two narrow faes versus total of average heat flux on two wide faes Figure 9. omparison of heat flux between loose and fixed sides of wide fae Influene of individual variables In order to investigate the influene of eah of the eight asting variables on mold heat flux, the effet of other asting parameters must be made as insignifiant as possible. In the urrent work, this was done by applying respetive seondary filters to make all variables, exept the one under study, as onstant as possible, as desribed in the methodology setion. Figure. verage heat flux plotted against mold width after applying primary filters Figure. Variation of asting speed with mold width after applying primary filters Figure. Variation of heat flux with mold with after applying seondary filters n illustration of why this is important is provided in Figures -. In Figure, the heat flux is plotted against mold width for every heat in the entire data set, with only primary filters applied to ensure reliable, steady measurements. The linear regression shows that heat flux tends to derease with mold width. However, intuitively, with inreasing mold width, low heat flux at the orners will beome less important as high heat flux at the region of good ontat over the rest of the wide fae beomes larger. Thus mold heat flux is expeted to inrease with inrease in mold width, whih is opposite of what is seen in the data. Figure offers an explanation for this, showing ross-orrelation between asting speed and mold width. The ommon pratie is to derease asting speed as mold width inreases to maintain onstant throughput for quality reasons. Sine heat flux has a stronger dependene on asting speed, this leads to a net derease in heat flux with inreasing mold width as seen in Figure. However, Figure shows that, when seondary filters are applied to make asting speed onstant, there is a small positive orrelation observed between heat flux and mold width, as expeted. ISTeh 4 Proeedings. 4 by IST. 887

8 There are other ross-orrelations in the data, related to mold powders. Speifi mold powders are seleted for partiular grades, and some mold powders are related in omposition, leading to related properties (visosity and break point). lso, there are only a small number of powders in the data set after primary filtering, so sparseness of data may also lead to false orrelations. These are aounted for by studying measurements for only a single powder, using the seondary filters. asting speed asting speed has the most signifiant influene on mold heat flux, as observed in Figure, whih is well known from many previous studies [7,8,7]. s asting speed is inreased, the residene time of the steel in the mold dereases. This makes the solidifying steel shell thinner, ausing steeper temperature gradients, leading to higher heat flux. The resistane to heat flow aross the gap between the shell and mold also dereases, owing to the drop in mold powder onsumption (kg/m ) []. Figure. Inrease in mold average heat flux with inrease in asting speed Figure 4. Effet of arbon on average heat flux arbon ontent It is well known [7-9,8] that less heat is removed for periteti steels ompared to lower and higher arbon steels. The larger ontration of the steel during the periteti phase hange inreases the gap between the shell and mold fae resulting in lower heat flux. However, the Nuor Deatur mill does not ast periteti steels. s shown in Figure 4, this effet annot be observed from the data. For general appliation of the model to other asters, the term and variable (%) aounting for this effet of steel omposition was retained. Mold powder properties Mold powder properties are known to have a onsiderable influene on mold heat flux [7,8,9]. In this work, the different mold slag properties are haraterized by the visosity and break point temperature. Figure 5 shows that, as reported in the literature [9], higher break point temperature orrelates with a lower heat flux. This heat flux drop ours beause a thiker solidified slag layer forms between the liquid slag layer and the mold, whih inreases the gap resistane. Figure 5. Mold heat flux dereases as break point temperature of mold powder inreases Figure 6. Variation of mold heat flux with visosity of mold powder 888 ISTeh 4 Proeedings. 4 by IST.

9 Figure 6 shows a very slight effet of mold heat flux inreasing with inreasing slag visosity. This disagrees with previously reported results [7,8,9] that higher visosity is related to lower heat flux. This is likely due to ross-orrelations and the small data set for mold powders, as desribed previously. Mold width Figure 7 shows a small inrease in mold heat flux with inreasing mold width. s disussed at the beginning of this setion, this is likely due to the drop in heat flux at the mold orners beoming less important as the length of the region of good ontat inreases. Figure 7. Influene of mold width on mold heat flux Figure 8. Effet of super heat on mold heat flux Superheat Figure 8 shows that mold heat flux inreases very slightly with inreasing superheat of the inoming liquid steel. This is expeted beause the higher orresponding liquid temperatures at the top surfae should lessen menisus freezing and hook formation, leading to shallower osillation marks and less gap resistane. However, superheat temperature is measured in the tundish, leading to satter, so the observed effet is very small, even after making other asting parameters as onstant as possible to isolate the effet. This is in agreement with a previous study [9] whih showed little effet of superheat on mold heat flux. Mold level standard deviation Figure 9 shows that inreasing level flutuations (as indiated by the standard deviation in mold level), leads to slightly lower heat flux. This is expeted beause higher mold level flutuations result in deeper osillation marks, inreasing gap resistane, and thus reduing mold heat flux. s the standard deviation is less than mm, the deviation measurements appear to be filtered for the benefit of the mold level ontrol system, before being stored in the database. This may be why this trend is not observed in the data. Figure 9. Variation of heat flux with mold level standard deviations Figure. Little effet of mold plate thikness on mold heat flux ISTeh 4 Proeedings. 4 by IST. 889

10 Mold plate thikness It is expeted that as mold plate thikness dereases, the resistane to heat flow dereases slightly, resulting in higher heat flux. Santillana et al [] measured even higher than expeted inrease in mold heat flux as mold plate thikness dereases, owing to the inreased hot fae temperature of the mold dereasing slag layer thikness and further dereasing resistane to heat flow. ut as shown in Figure, there appears to be little orrelation between heat flux and measured mold plate thikness in this data. The effet is onfounded by possible ross orrelations beause water flow rate may be adjusted with mold plate thikness. In addition, plate thikness is measured only when the mold is hanged, and therefore does not aount for wear over the ourse of a ampaign. s a proxy for this wear, Figures -4 show mold heat flux over the four longest ampaigns in the data set plotted against the number of heats ast using that mold, using seondary filters for speed, omposition and mold powder to aount for hanging asting onditions. There is a weak negative relationship, whih may be due to the wear inreasing surfae roughness, and thus inreasing gap resistane. Figure. Slight derease in heat flux with heats on mold for ampaign Figure. Effet not learly seen for ampaign Figure. Slight derease in heat flux with heats on mold for ampaign Figure 4. Slight derease in heat flux with heats on mold for ampaign 4 Model development and evaluation The proedure desribed in the methodology setion is performed to find best fit parameters for different ombinations of asting variables. The results are summarized in Table IV. The suessive models are developed through stepwise forward seletion, meaning that the single variable that most dereases RSS is added to make the next model. The exeption to this is the arbon term, as disussed below. RSS Error and I at eah step illustrate the signifiane of the asting variable added. s disussed in the individual effets setion, some of these variables have strong ross-orrelations with eah other. These make it diffiult for numerial tehniques to onverge to a unique set of best-fit parameters. There may be more than one set of best-fit parameters that give similar preditions. Therefore, inluding ross-orrelated variables together, suh as width and asting speed, may improve preditions of heat flux. However, it is diffiult to extrat onlusions about the underlying physis. s seen above and in the literature, asting speed has a lear, strong influene on the average mold heat flux. Therefore, the first, simplest model is fitted with asting speed alone. lthough the effet of arbon is not detetable with the urrent data set as peritetis are not ast at Nuor Deatur, for general appliation of the model to other asters, the arbon term is important. Therefore, the seond model inluded asting speed and the expression aounting for drop in heat flux for periteti grades. RSS did not derease and I inreased from model to model, due to the lak of data, but the periteti expression is inluded in the remaining models for generality to other asters. The remaining variables were added in stepwise order. derease in RSS and I with the addition of break point temperature, width, visosity and mold level deviation in the respetive order indiate their inlusion is signifiant. ut the ontribution of visosity and mold level deviation is very small. In fat, though heat flux appeared to have little orrelation with visosity, as shown in Figure 6, model 5 shows that there is a weak derease in heat flux as visosity inreases. ased on I, model 8 appears to be the best preditive model. 89 ISTeh 4 Proeedings. 4 by IST.

11 Table IV. Different equations with statistial estimates Equation RSS Error I.97 Q.544 V Q V + -% -.5 exp -. ( - ) %.78.5 exp. ( ) break Q V -.44 T % exp. ( ) break Q V T.7 W %.8.5 exp. ( ) break Q V T W -.7 μ % exp. ( ) break Q V T W.8 l % exp. ( ) break Q V T W l.5 s % exp. ( ) break Q V T W l s -.7 t nother measure of performane of the new model equations, originally fit to data from the fixed fae of the North aster, is to test their auray in prediting the loose fae of the North aster and both faes of the South aster. Figure 5 ompares the RSS errors for these models and iutti s equation (Eq. ) on these four mold faes. The models developed in the urrent work give better preditions than the iutti equation. Due to symmetry between the faes, these models predit well for the North aster loose fae. There is an inrease in RSS for both faes of South aster on addition of mold level deviation, superheat, and mold plate thikness. The respetive variables may not have signifiant effet on heat flux, the effet may be different in magnitude between the two asters, the way they are inluded in the equation ould need modifiation, or the measurements may be too inaurate to disern the effet. For example, mold plate thikness is measured only at the start of a ampaign, and whatever wear ours during the ampaign is not measured. 89 ISTeh 4 Proeedings. 4 by IST.

12 Figure 5. Performane of prediting equation on other asters; white markers are for models in Table IV, grey markers are for iutti equation (Eq. ) Thus, model 5, whih is the best model before the jump in RSS, is onsidered to be superior to the other equations with respet to simpliity and auray shown again as follows. % Q.8 V.5 exp. Tmelt W ( ) () This model is ompared with the iutti equation graphially for the fixed faes of the North and South asters in Figure 6. The developed equation (Eq. ) is learly prediting better than the iutti equation. Predited heatflux (MW/m ).5 North aster-fixed fae 556 heats.5 North aster-fixed fae 556 heats Developed equation iutti equation Developed equation iutti equation tual heatflux (MW/m ) tual heatflux (MW/m ) tual heatflux (MW/m ) tual heatflux (MW/m ) Figure 6.omparison of new model 5 to iutti equation for fixed faes of North and South asters The satter still shows that this equation has room for improvement. One possibility is that the model needs to aount for other asting variables, like mold osillation mark depth and frequeny, whih are not onsidered in the present work. etter seletion of asting variables, the struture of the model, and the handling of odependent variables ould also lead to better results..5 South aster-fixed fae 67 heats.5 South aster-fixed fae 67 heats ONLUSIONS verage mold heat flux over the wide fae of a thin slab asting mold is investigated using measurements of eight asting variables. Of the tested variables, asting speed has a lear, strong influene on heat flux. reak point temperature of mold powder, mold width and powder visosity have a weaker effet that was not seen individually, but was found through nonlinear multiple regression of the data. The influene of arbon ontent is not observed beause of non-availability of data for periteti steels. The effet of superheat, mold plate thikness, and mold level standard deviation are not evident in the 89 ISTeh 4 Proeedings. 4 by IST.

13 data, although no onlusion an be drawn about whether this is due to an atual lak of relationship, or a weakness in the data or methodology. n equation for prediting mold heat flux as a funtion of asting variables is developed. The developed model mathes Nuor Deatur s asters better than the iutti equation, whih was originally developed for a thik slab aster. Though harateristi to the plant, the equation is expeted to behave well on other asters, due to its good performane on test data that was not used for model development. The results of this paper, being based on plant measurements, provide a greater understanding of mold thermal behavior, and the methodology an be extended easily to other asters and phenomena. KNOWLEDGEMENTS We would like to thank Ron O Malley, ob Williams, and others at Nuor Steel Deatur for providing aess and valuable guidane for this projet. We would also like to thank the members of the ontinuous asting onsortium at the University of Illinois at Urbana-hampaign for support of this researh. REFERENES. R.. Mahapatra, J. K. rimaombe, I. V. Samarasekera, N. Walker, E.. Paterson, and J. D. Young, "Mold behavior and its influene on quality in the ontinuous asting of steel slabs: Part i. Industrial trials, mold temperature measurements, and mathematial modeling", Metallurgial Transations, Vol., No. 6, 99, pp T.J.H. illany,.s. Normanton, K.. Mills, and P. Grieveson, Surfae raking in ontinuously ast produts, Ironmaking and Steelmaking, Vol.8, No.6,99, pp Hiraki, Sei, K. Nakajima, T. Murakami, and T. Kanazawa, "Influene of mold heat fluxes on longitudinal surfae raks during high speed ontinuous asting of steel slab", In Steelmaking onferene, Vol. 77, 994, pp Tada, Kihio, S. Kasai, fira Ihihara, and H. Onishi, "Improvement in Servie Life of ontinuous asting Mold", Kawasaki Steel Tehinal Report, No.7, 987, pp J.K. Park,.G. Thomas, I.V. Samarasekera, and U. Sok Yoon, Thermal and mehanial behavior of opper molds during thin-slab asting (I): Plant trial and mathematial modeling, Metallurgial and Materials Transations : Proess Metallurgy and Materials Proessing Siene, Vol., No., June, pp W.H. Emling and S.Dawson, Mold instrumentation for breakout detetion and ontrol, Steelmaking onferene Proeedings, Vol. 74, 99, pp G. Xia, H.P. Narzt and. Furst, Investigation of mould thermal behaviour by means of mould instrumentation, Iron making and steelmaking, Vol., No.5, 4, pp iutti, M. Valdez, T. Perez, G. DiGresia, W. alante, and J. Petroni, Mould Thermal Evaluation in a Slab ontinuous asting Mahine, Steelmaking onferene Proeedings, Vol. 85,, pp Heht, Mihael, Zhiyuan Zhu, Helmut Lahmund, and K. H. Take, "Mould investigations on a thik slab aster." Revue de Métallurgie, Vol., No., 5, pp Santillana, L.. Hibbeler,.G. Thomas,. Hamoen,. Kamperman, and W.V.D. Knoop, Heat transfer in funnelmould asting: effet of plate thikness, ISIJ international, Vol. 48, No., 8, pp Y.. Meng, and. G. Thomas, "Heat-transfer and solidifiation model of ontinuous slab asting: OND," Metallurgial and Materials Transations, Vol. 4, No. 5,, pp K.E. lazek, O. Lanzi III, P.L. Gano, and, D.L. Kellogg, "alulation of the periteti range for steel alloys", Iron & steel tehnology, Vol. 5, No. 7, 8, pp J.. Nelder and R. Mead, simplex method for funtion minimization,omputer Journal,Vol. 7, No. 4, 965, pp Optimization Toolbox, User s Guide R b, MTL", The MathWorks,, pp J.. Morgan, J.F. Tatar, alulation of the residual sum of squares for all possible regressions, Tehnometris, Vol.4, No., May 97, pp John I. Marden, "Multivariate Statistis Old Shool", Department of statistis, University of Illinois at Urbana- hampaign,, pp M. Sadat,. Honarvar Gheysari, and S. Sadat, The effets of asting speed on steel ontinuous asting proess, Heat and Mass Transfer, Vol. 47, No.,, pp S.N. Singh and K.E. lazek, Heat transfer and skin formation in a ontinuous-asting mold as a funtion of steel arbon ontent, Journal Of Metallurgy, Vol. 6, No., Otober 974, pp.7-,6. 9. K. Watanabe,M. Suzuki, K. Murakami, H. Kondo,. Miyamoto, and T. Shiomi. "The effet of rystallization of mold powder on the heat transfer in ontinuous asting mold", Journal of the Iron and Steel Institute of Japan, Vol. 8, No., 997, pp. 5-. ISTeh 4 Proeedings. 4 by IST. 89

14 894 ISTeh 4 Proeedings. 4 by IST.