PROBLEM OF PROCESS INGENEERING

J. K. Brimacombe

Design of continuous casting machines based on a heat-f low analysis: state-of -the-art review

Resume

The design of the mould and spray sections of a continuous casting machine for steel has been examined in detail with the aid of a heat-flow mathematical model. The thermal requirements t h a t the design must meet have been estab- lished using experimental data obtained from commercial continuous-casting machines. For the mould, the requirement IS the solidification of a sufficiently thick shell; for the sprays, the requirements are the minimization of surface reheating and the maintenance of a reasonably high solidifi- cation r'Hte. Correlations a r e developed which >elate the working mould length, casting speed and shell thickness a t , the mould exit. Design curves are also presented of spray

'length, spray heat transfer coefficient and water flux dis- tributions for two sizes of billets. The method by which the theoretical spray water flux distribution can be related to

. <practical spray parameters - spray pressure, nozzle type and arrangement - is also discussed.

On a Btudik en detail, B l'aide d'un modele mathematique de 1'6coulement de chaleur, la conception du moule et des sec- tions de refroidissement d'une machine de coulee continue pour l'acier. Les exigences thermiques B rencontrer ont BtC! fixdes B l'aide de donn6es experimentales provenant de ma- chines industrielles. Pour le moule, l'exigence est la forma- tion d'une carapace solide suffisamment Bpaisse alors que pour les jets l'exigence consiste en un rechauffement mini- mal de la surface et le maintien d'un taux de solidification assez 61evB. On a developpd des correlations qui relient la longueur effective du moule, le taux de coulee, et 1'6paisseur de la carapace B l a sortie du moule. L'article inclut des courbes permettant d16tablir la longueur de l a zone des jets, leur coefficient de transmission de chaleur e t les distributions de debit d'eau pour deux dimensions de billettes. Finalement, on discute de la nlethode pour relier la distribution du debit theorique d'eau aux parametres pratiques des jets (pression

' - . - . .. , ,. . ' d'eau aux jets, type et configuration des gicleurs). 1 '*

.I- 4 - L r L . .h

Introduction

T h e mould a n d s p r a y chamber of a continuous c a s t i n g machine m u s t fulfill i m p o r t a n t thermal requirements if steel i s t o be cast efficiently w i t h a minimum of i n t e r n a l o r external defects. T h e mould must be capable of ex- t r a c t i n g sufficient heat f r o m t h e incoming metal t o solidify a shell which can support t h e liquid pool a t t h e mould exit, a n d thereby minimize breakouts. I t i s also

_r _ important t h a t t h e mould be able t o remove t h e heat uniformly t o avoid t h e formation of locally t h i n regions in t h e shell which could r u p t u r e a n d lead also t o break- outs o r s u r f a c e cracks. T o achieve a uniform shell w i t h s u f f i c i e n t thickness f o r a given s e t of c a s t i n g conditions, several mould p a r a m e t e r s h a r e t o be regulated. T h e most i m p o r t a n t of these a r e t h e working length, t h e t a p e r a n d t h e corner radius.

Continuing t h e h e a t extraction process begun in t h e mould, t h e s p r a y s m u s t remove sufficient h e a t f r o m t h e steel t o virtually complete solidification of t h e cast sec- tion. T h e r a t e a t which t h e h e a t extraction proceeds i s critical t o t h e smooth operation of t h e process, because undercooling can result i n excessively long liquid pools a n d overcooling can lead t o t h e formation of cracks a t the bending o r s t r a i g h t e n i n g rolls. T h e h e a t extraction

J.K. Brimacombe is Associate Professor in the Department of Metallurgy, University of British Columbia, Vancouver, r,"

in t h e s p r a y s m u s t also be a r r a n g e d s o t h a t t h e r e i s a smooth transition of t h e s u r f a c e temperature, w i t h a minimum of reheating, a s t h e steel passes f r o m t h e mould t o the s p r a y s a n d f r o m t h e s p r a y s t o t h e radiation cooling zones. Excessive r e h e a t i n g h a s been shown"' t o be a p r i m a r y cause of halfway cracks in strand-cast steel. T h i s aspect of t h e s p r a y design will be discussed in more detail in t h e paper. T h e r e a r e two m a j o r vari- ables which can be adjusted t o optimize t h e performance of t h e s p r a y s : w a t e r f l u x distribution a n d length of t h e s p r a y zone. F r o m a n operational standpoint, t h e optimum h e a t extraction can be achieved by a suitable combination of s p r a y nozzle types, w a t e r pressures a n d nozzle a r - rangements.

I t i s impo~utant t o realize t h a t , in addition t o t h e thermal requirements outlined above, a continuous cast- i n g machine h a s vital mechanical functions. T h e oscilla- tion of t h e mol~ld, t h e s u p p o r t a n d guidance of t h e s t r a n d w i t h pinch rolls, bending a n d s t r a i g h t e n i n g rolls and, i n some cases, c u t t i n g of t h e s t r a n d w i t h mechanical s h e a r s all m u s t be carefully designed a n d controlled if trouble- f r e e casting is t o be achieved.

T h e need f o r a continuous casting machine t o meet both thermal a n d mechanical requirements h a s been recognized f o r many y e a r s by designers a n d operators alike. T h e design of continuous c a s t i n g machines, a s a result, h a s improved, b u t largely due t o changes arrived a t by empirical methods. T h a t a theoretical approach h a s not played a more i m p o r t a n t role, particularly i n t h e . thermal aspects of design, h a s not been d u e t o a lack

uf mathematics or difficulty in solving differential equa- tions. Rather i t has been due t o a lack of knowledge of, and the inability to express in quantitative terms, t h e conditions which lead to such problems a s cracks and breakouts. Without this information, i t is exceedingly difficult to define quantitatively the criteria which t h e machine design must meet.

The work described in t h i s paper i s a n attempt t o put some of the thermal aspects of the design of con- tinuous casting machines on a more rational basis. The criteria required f o r t h i s purpose have been established through a series of studies relating to heat flow in t h e mould and sprays, and t h e causes of internal cracks, which have been conducted a t the University of British C ~ l u m b i a . " ~ ' The theoretical design of continuous cast- ing machines t o meet these criteria for a range of cast- ing conditions has been developed with the aid of a mathematical model. This model embodies a quantitative description of t h e heat flow in t h e solidifying steel and can reliably predict the three-dimensional temperature distribution and shell profile in the strand. A brief description of t h e mathematical model follows.

Mathematical Description of Heat Flow In the Continuous Casting of Steel I n the continuous casting of steel, heat flows through the solid shell in the transverse (x,y) and axial (z) di- rections by conduction; heat also flows in t h e axial di- rection a s a result of t h e bulk motion of the descending strand. This latter mechanism t u r n s o u t to be consider- ably more important than axial conduction f o r steel cast- ing, due to t h e relatively high withdrawal rates and low thermal conductivity of steel; consequently, heat con- duction in the axial direction can be neglected. Under these circumstances, steady-state heat flow in the solid ahell can be described by

where the symbols a r e listed a t t h e end of the paper. Heat flow in the liquid pool is more complex t h a n in

the solid shell due to t h e turbulent mixing imparted by the input metal stream. Thus, in addition to normal thermal conduction, heat, or more correctly superheat, can also be transferred from t h e liquid pool to the solid shell by forced convection. The r a t e a t which the heat flows from the liquid pool can be calculated roughly by defining an effective liquid thermal conductivity, kt, which may be five- to ten-fold larger than the normal value t o account f o r convective heat flow, and substi- tuting i t f o r k, in Eq. (1). The magnitude of the effective conductivity has some physical significance in that, f o r the range mentioned above, the superheat is almost fully extracted in the mould. Fortunately, the almost a r b i t r a r y assignment of a value f o r k, has a minor effect on t h e thickness of and temperatures in the solid shell. This i s due to the fact t h a t the superheat is normally a small fraction of t h e latent heat of solidification.

The most, important process in continuous casting, of course, is solidification of t h e steel and t h e release of latent heat. The precise manner in which the latent heat is released between t h e solidus and liquidus tempera- tures, however, is difficult to describe mathematically due to the complexity of solidification involving several components. Rather than embarking on a program to

mathematically simulate t h e solidification process - segregation, dendrite growth, interdendritic fluid flow and so f o r t h - which in itself requires several question- able assumptions, a relationship between enthalpy and temperature is normally assumed. I n t h e present model, this relationship has been obtained f r o m t h e iron-carbon phase diagram; i.e., equilibrium freezing has been as- '- sumed. Again i t i s fortunate t h a t t h e exact f o r m of t h e enthalpy-temperature function employed does not appear to affect the calculated temperature field significantly."'

The extraction of heat from the surface of the s t r a n d proceeds by different mechanisms in each of the three cooling zones. Of these, the heat extraction processes a r e most complex in the mould and spray regions. I n t h e mould, heat t r a n s f e r from t h e steel surface is influenced markedly by t h e formation of a g a p between t h e solid shell and the mould face. As t h e width of t h e gap appears to be very small, considerable heat can flow from t h e shell surface to the mould by conduction through gas in the m t 1 ) a s well a s by radiation. A complication in describing the rate of heat flow arises, however, because t h e gap width does not appear to be constant, b u t varies in both vertical and horizontal directions over the mould face. Unfortunately, the magnitude of t h i s variation and t h e resultant effect on the r a t e of heat extraction a r e un- known. I n the absence of mechanistic data of t h i s kind, e f f o r t s have been made to obtain useful heat-flow d a t a in the mould from n heat balance on the mould cooling watel.. By this method, the average mould heat flux, q,, can be obtained f o r a given s e t of casting conditions, the most important parameter of which appears to be the mould dwell time, t.,. Lait, Brimacombe and Weinberg"' have shown t h a t q,, and t, a r e related by the following Ca expression C. Z

q,, = 64 - 5.3&, (cal cm-' sec.-l. sec.) (2) 4 f o r a wide range of casting conditions, a s shown in Fig- 4 ure 1. The solid line in the f i g u r e is a plot of Eq. ( 2 ) , which can be seen to pass through the d a t a obtained from several different casting machines. The instantaneous heat flux, q.,, which varies with time [equivalent to dis- tance (= z i u ) I below the meniscus, is then given by

q,,, = 64 - 8 4 t (cal cm-= sec.-1, sec.) (3)

It is this expression, which f i r s t was measured by Savage and Pritchard"', t h a t can be used to describe mathematically the rate of heat extraction in t h e mould. I t is important to note that, according to Eq. ( 3 ) , q, is :I function only of t or distance below t h e meniscus, whereas, a s suggested above, i t probably also should vary with horizontal position relative to the corners. This refinement will require measurements to be made of t h e heat-flow distribution over the entire mould face using an a r r a y of thermoco~lples inserted in the mould wall. An experimental program for this purpose is currently being -planned by o u r group.

In t h e s p r a y chamber, heat is removed from t h e strand by fast-moving water droplets, which, having been ejected by the spray nozzle, penetrate the steam film ad- jacent to the steel surface and evaporate. Like the mould, this process is too complex to be treated theoretically. Instead, the rate of heat extraction, q., can best be characterized mathematically by a spray heat t r a n s f e r coefficient, h,, a s follows :

q. = h. (To - T.) ( 4 , where To and T, a r e the steel surface temperature and spray water temperature respectively. Values of h., of 4

0 * o MRM 101

DWELL TIME, s

Figure 1. Average mould heat flux as a function of dwell time [from Lait. Brimacombe and Weinberg(4'].

course, must be determined experimentally f o r the spray conditions - water flux, spray pressure, nozzle type and steel surface temperature - common t o continuous casting. Some experiments have been undertaken t o obtain these coefficients by Mizikar"' and Miiller and J e s ~ h a r . ' ~ ' However, the range of spray conditions over which d a t a a r e available is not complete enough t o cover all t h e types of sprays used in continuous casting.

The simplest of t h e heat flow zones to characterize mathematically is the radiation cooling zone below t h e sprays. Here the r a t e of heat extraction from the surface,

a q , is given by t h e well-established Stefan-Boltzmann equation:

where T, is the ambient temperature of the immediate environment around the radiation cooling zone. The value of emissivity normally adopted f o r t h e oxidized s t r a n d surface is 0.8.

The expressions f o r q,, q, and q, [Eqs. (3) to ( S ) ] a r e a n integral p a r t of t h e mathematical model, because they a r e required f o r the formulation of the boundary condition a t t h e surface ( x = 0, y = 0). I n mathematical terms, this boundary condition can be expressed a s fol- lows :

where q. = qi ( i - m,s,r) for each of the three cooling zones. Equations (6) and (7) a r e a simple expression of t h e continuity of heat flow a t the surface.

Another boundary condition can be specified if the heat flow can be assumed to be symmetrical about t h e centerplanes ( x = '/2, y = '/2) of t h e cast section of width X and thickness Y. Under this condition, heat does not flow across the centerplanes; t h a t i s

This assunlption results in considerable simplification of the solution, because the temperature distribution need only be calculated over one quarter of the cast sec- tion, the other quarters being identical.

The boundary condition f o r the withdrawal (z) di- rection is a statement of t h e temperature of t h e steel a t the meniscus (z = 0 ) , which can be s e t equal to the pouring temperature, T, :

Equation ( 1 ) and t h e boundary conditions listed above a r e a complete mathematical description of heat flow in continuous casting within t h e framework of the iissumptions t h a t have been made. The solution t o Eq. (1) is the essence of the mathematical model. This solution callnot be achieved by analytical means, however, due t o the non-linearity of t h e boundary conditions, t h e release of latent heat and, in addition, t h e need to allow the thermal conductivity of the steel to vary with tem- perature. For these reasons, a numerical approach i s required, for which the finite-difference method i s ideal- ly suited. I n t h i s method, t h e steel is divided into a l i ~ ~ m b e r of discrete, rectangular elements of width Ax and length Ay, a s shown in F i g u r e 2. The material with- in each element, o r node, then is assumed to have both ~ ~ n i f o r m temperature and thermophysical properties.

For the explicit method of finite differences which has been employed in the model, i t i s most convenient to 1 transform Eq. (1) into a n equivalent two-dimensional, I

unsteady-state equation by substituting u t f o r z.

where 11 is the withdaawal r a t e and H. = p,C.T, is t h e enthalpy of the solid steel. Then the enthalpy of the nodes in a two-dimensional slice of steel of unit thickness can .. be calculated over successive intervals of time a s t h e . slice descends through the three cooling zones, a s seen in Figure 3.

I The numerical calculation is performed in the fol-

lowing way. The enthalpy and temperature of all nodes ' i 1

are initially assigned values corresponding to the enthal- Figure 3. Slice of steel moving through cooling zones. py and temperature of the incoming steel. Then over the next time step, At, the enthalpies of the nodes are recal- culated using an explicit form of the finite-difference a,ld for the corner node: approximation of Eq. (11). For interior nodes (i, j ) , this equation, expressed as a central difference, is

l i t ( l , l ) = H (1.1) +2nt { ( - - T I , I ) k 1

H ' ( i , j ) = H ti, j) + A t [ { 'T.."l -2Tt.1 + ~ . . . I - I ) (AY)~ q~ AY - T1.1 )} (16) , k z

( T ~ + I . J - 2 Ttrl + T , - I . ~ ) 1

where the temperature-dependent thermal conductivity is a linear function of Ti,]

For surface nodes, the equation is a s follows for the face, x = 0:

where and kz are the average thermal conductivities 1 between Tnrj and TI,j, and between TI,* and Ti., respec- tively. For the centerliqe nodes, the finite-difference equations are identical to . (12), except that T1+,,l is set

plane and T i , j + ~ is set equal t I I

The newly calculated enthalpies, H' ( i , j ) , can be con- verted to temperatures from a knowledge of the enthalpy- 1 temperature relationship, and the calculation can be re- peated over successive time intervals. Because, a t each level, the temperature distribution is fixed, the three-

I dimensional temperature field is obtained by this proce- dure. The solid shell is then given by the locus of nodes

i 1

a t the solidus temperature, and the pool depth by the A (T1.j712 - 2 Tlsj+~ TI,,-I + TI-j-1') istance below the meniscus (= ut) a t which the center-

+ T (AyP ine node has reached the solidus temperature. -

In order to achieve a convergent solution using the + 2 (TlSi - q s - . . , . . . ( Ax)' k 1 licit method, the magnitude of the time interval be-

uccessive calculations must be constrained, de- and for the face, y = 0 : on the size of Ax and Ay used for the node

ns. The "stability criterion" which governs the ql (i, 1 ) = H (i, I ) + A t {ki,, (Ti+l*l - 2 Ti.1 Ti-1.1) nce has been thoroughly discussed by Sargent

( AxIP A ( T ~ + I , I ' - 2 T ~ + I ~ I Ti-ltl + T ~ - I ~ I ' ) .k $73- +4 (ax)2 , ' ; In order to run the mathematical model, the follow- , .

ing input data must be supplied: la] section size; [b] pouring temperature ;

~ . % ' . .. ,

[d] working mould length ; [el casting speed; [f] details of spray chamber.

The output from the model, as mentioned above, is d the three-dimensional temperature distribution of the strand.

Verification of the Mathematical Model Before the mathematical model could be employed with confidence, its ability to reliably predict the temperature distribution and pool profile, using data from industrial machines, had to be tested carefully. The process of verification was accomplished by comparing the model predictions of pool profile to measured profiles which had been obtained from radioactive tracer tests. In these

radioactive gold ( A U " ~ ) alloyed with lead was added to the liquid pool of a commercial continuous cast- ing machine; later, when the cast section had solidified and cooled sufficiently, it was sectioned, surface-ma- chined and autoradiographs of each section were made to determine the distribution of the radioactive gold. A composite of autoradiographs from a 14-cm-square steel billet cast a t Western Canada Steel Ltd., Vancouver, is displayed in Figure 4. The dark regions in the autoradio- graphs are the gold-rich areas of the liquid pool; the light strips running down either side of the longitudinal sections indicate the solid shell. Measurements of the shell thickness have been made from the autoradiographs and these are presented in Figure 5. Also plotted in Fig- ure 5 is the pool profile calculated using the mathematical

Meniscus

Bottom of Mold-

Figure 4. Composite of autoradiographs of a 14-cm-square steel billet cast at Western Canada Steel.

spray 4 I I

Figure 5. Comparison of model-predicted shell thickness and surface temperature with values measured from the 14-cni steel billet.

model for the particular set of casting conditions. The agreement between the measured and calculated profiles can be seen to be reasonably good over the upper regicns of the pool, whereas near the pool bottom the calculated shell is marginally thinner than the measured shell. This result would be expected if, due to stagnant conditions in the lower regions of the pool, the radioactive tracer did not penetrate fully to the solidus isotherm. A comparison of measured and calculated surface temperatures is also given in Figure 5, which again shows that agreement is acceptably good. The favourable results of this compari- son, and others like it, are a clear demonstration of the reliability of the mathematical model in its present form.

Mould Design - Working Mould Length In order to design a mould using the mathematical model for a given set of casting conditions, the relationships among the important mould parameters - corner radius, taper and working length - and the space-dependent heat flux in the mould must be known. These relation- ships are needed to define the surface boundary condi- tion [Eq. (6)]. Unfortunately, only the effect of casting time (or distance, z ) on the mould heat flux [Eq. ( 3 ) l has been established with reasonable certainty: thus, predictions of an optimum corner radius or taper cannot be undertaken a t this time. Nevertheless, the fact that Eq. (3) does appear to be accurate for a wide variety of casting conditions means that a reliable prediction of the working mould length can be made.

From a design standpoint, one is normally interested in the working mould length, ZM, required to solidify a shell of a given thickness, Yu, for a given casting rate, section size and steel grade. A plot of mould length against casting rate, with Yu as parameter, has been obtained using the mathematical model and is shown in Figure 6. Here i t can be seen that the mould length is a

Cor!mnq Speed kmtsec)

Cast~nq Speed (m/mmn)

Figure 6. Worki~ig mould length as a function of casting speed.

linear function of casting r a t e ; and i t should be noted f u r t h e r t h a t the slope of each line i s simply t h e casting time required to solidify a shell of thickness YM. Given a casting speed and shell thickness, F i g u r e 6 can be used to determine the working mould length required f o r t h e casting of both billets and slabs, because Eq. ( 3 ) holds reasonably well f o r both cases. Some caution should be exercised, however, a s there i s now evidence t h a t t h e mould heat flux i s influenced slightly by the carbon con- tent of t h e steel being ~ a s t . " ' ~ ' ~ ' I t appears certain t h a t for t h e casting of steel with carbon contents less than about 0.2% or g r e a t e r than 0.6%, t h e mould heat flux can be 15 to 20% less than f o r steels with intermediate carbon levels. Correspondingly, the solid shell will be thinner a t the mould exit.

F o r purposes of comparison, values of working mould lengths employed in industrial casting machines have also been plotted in F i g u r e 6. The size and shape of t h e points plotted correspond roughly to the size and shape of the sections cast. I t is interesting to note t h a t t h e lengths of t h e working mould used have a constant value of 60 to 70 cm, independent of the size of t h e section being cast. Thus, slabs cast a t 1 cm sec.' have a shell thickness a t t h e mould bottom in excess of 2 cm (0.8 in.), whereas billets cast a t 5 to 6 em set.-' have a shell which is half a s thick, o r 0.9 cm (0.35 in.). The thicker shell in t h e case of t h e slab is clearly required to cope with its g r e a t e r tendency t o bulge along t h e wide face, a s rompared to a billet.

Although moulds currently in use tend to have t h e same fixed length, i t seems likely that, i n the f u t u r e , the length will have to be increased if higher casting rates a r e to be achieved. F i g u r e 6 can be used to predict the increase required, if i t is assumed t h a t the shell thickness now obtained routinely a t t h e mould bottom f o r a given section size i s to be maintained. These "optimum" values of shell thickness a r e given in Table I. Thus, f o r

Table 1. Shell thickness at mould exit for different cast sections

Section Cast Shell Thickness at

Mould Exit (cm). Yu Small billets

(10 X 10 cm) Medium billets

(15 X 15 cm) Blooms and

Small Slabs Large Slabs

example, if one wishes t o determine t h e length of mould required t o cast a 10- by 10-cm billet a t 8 cm set.-', a value of Ym of 0.9 cm is taken from Table I ; a working mould length of 110 cm is then read from Figure 6. Finally, to calculate t h e total length of mould required, about 10 cm is added to t h e working mould length to account f o r the freeboard above t h e meniscus.

Spray Design

The approach taken in the design of t h e spray system i s different from t h a t adopted f o r t h e design of t h e mould f o r two reasons. I n the f i r s t place, relatively more i s known about t h e relationship between heat extraction r a t e s and spray parameters; secondly, t h e spray system is more flexible, from t h e standpoint of controlling t h e r a t e of heat removal from t h e strand, than t h e mould. Thus, a t the present time, i t is relatively easier t o de- sign a spray system t h a t will realistically achieve a desired temperature distribution in t h e strand.

I n order to determine t h e best spray design for a given s e t of casting parameters, an important question must be answered a t the outset: what a r e t h e optimum thermal conditions t h a t the sprays must meet? Two criteria have been set in this work t o define such con- ditions: minimization of halfway crack formation and maintenance of a reasonably high solidification rate. As will be shown i n the next section, the f i r s t criterion can be met by minimizing reheating of t h e surface of t h e strand either in or below t h e sprays. The second criterion, on t h e other hand, can be met by maintaining t h e sur- face temperature of the s t r a n d in t h e sprays between 1000 and 1100°C.

Once optimum thermal conditions have been defined, the mathematical model can be employed to calculate t h e heat flux distribution, expressed in terms of heat trans- f e r coefficients, required to meet these conditions. The final step in t h e design i s t h e translation of t h e distribu- dj tion of spray heat t r a n s f e r coefficients into a spray water flux distribution and thence into workable s p r a y nozzle variables. The accuracy with which t h e latter stage of t h e design can be undertaken depends strongly on our knowledge of spray characteristics and heat removal rates.

[a) Halfway Cracks

Halfway cracks, such a s t.hose seen i n t h e sulphur p r i n t of t h e billet in F i g u r e 7, have been shown to be caused by

Figure 7. Sulphur print of a transverse section from a 10- by 15-cm billet showing the presence of halfway cracks (from Grill, Erimacombe and Weinberg''"').

reheating of the surface of the strand.") The reheating is the result of a sharp reduction in the rate of heat ex- traction from the surface which can occur as the strand passes from the mould to the sprays, from the sprays to the radiation cooling zone or from one spray nozzle to the next spray nozzle. The reheating forces the surface to expand and, in so doing, imposes a tensile strain on the interior, hotter regions of the solid shell, which, being weaker, can crack.

The stress distribution that results from reheating can be calculated approximately using finite-element analysis."" An example of such a distribution, calcu- lated for a reheating of 118°C of the surface of the 10- by 15-cm billet, seen in Figure 7, is shown in Figure 8. The length of the lines in Figure 8 i s proportional t o calculated stress in the x- and y-directions in the trans- verse plane. Lines ending with a perpendicular bar denote compressive stresses; those without bars, tensile stresses. The numbers next to the stress lines a r e the calculated temperatures in degrees Centigrade. Relatively large tensile stresses can be seen parallel to the wide face near the solidification front; the location of the high- stress region further can be seen to correspond close- ly to that of the actual halfway cracks visible in Figure 7.

1000 kglcm2 - tenston - compression

Figure 8. Stress distribution, calculated using finite-element analysis, in the transverse section of a 10- by 15-cm billet after 118°C surface reheat below the sprays (from Grill. Brimacombe and Weinberg"").

That steel close to the solidus temperature i s suscel tible to crack formation is a direct result of its low strength and d ~ c t i l i t y . " ~ * ' ~ ' There is strong evidence,"" for example, that above approximately 1350°C steel has a ductility-to-fracture of 0.2 to 0.3%. The reason for these mechanical properties can be seen in Figures 9 and 10. Shown in Figure 9 is a scanning electron micro- graph of an open halfway crack from the billet in Fig- ure 7. A smooth dendrite branch, showing no signs of deformation, is visible inside the crack. This is a clear indication that the cracks formed in a temperature zone where residual liquid still remained between dendrite branches. Further evidence of the smoothness of the crack surface can be obtained by pulling open a halfway crack with an Instron machine, and examining the ex- posed surface. A photograph of such a surface taken on the scanning electron microscope is shown in Figure 10.

The surface, again, is very smooth and undulating, but also is covered with manganese sulphides. Presumably the high concentration of sulphur arises from the resi- dual liquid into which sulphur would have positively segregated. With this type of hot tearing mechanism, i t would be more correct to call these cracks "halfway tears".

Figure 9. Scanning electron micrograph of an open halfway crack in an as-cast section of the 10- by 15-cm billet (from Vandrunen, Brimacombe and Weinberg"').

Figure 10. Scarlning electron microgra )f the exposed surface of a halfway crack. Note both the s~.,~,thness of the surface and the high concentration of manganese sulphides (from Vandrunen. Bri~nacombe and Weinberg"').

From the design standpoint, the maximum allowable reheat, below which halfway cracks will not form, must be defined because, under industrial conditions, i t is

factors, of which the two most important a r e cast struc- 2 I 1 o o MOI FIOW 5nolyse of Spmys ture and steel composition. I t i s now known,"7*" t h a t " Zone tor o 1715 cm

Square Steel Bnllel steel which has a predominantly equiaxed structure, such as is obtained by casting a t low temperatures, can resist 3 the formation of cracks even with large reheats of 200°C. 2 On the other hand, steel cast with a large columnar zone is very susceptible to cracking with much lower surface reheating. In the same way, steel composition has been shown by Vom Ende and V ~ g t " ~ ' to influence strongly the susceptibility of steel to crack formation. For exam- ple, steel with carbon contents between .17 and .24% or greater than 0.6% is more prone to cracking than other carbon grades. Also, steel with sulphur levels above 0.025% phosphorus contents above 0.03% or manganese concentrations in excess of 1.0% is reported to be more subject to cracking. Although not studied by them, i t is certain t h a t copper and tin concentrations will exert a similar effect on crack susceptibility.

With these factors to consider, i t is difficult to define a maximum allowable surface reheat for all grades of steel. Nevertheless, if the common grades have a ductility- to-fracture of 0.2 to 0.3% and a thermal expansion co- efficient of 0.2% per 10O0C, the maximum reheat should be about 100°C. However. i t mav be wiser to adopt a

- Mold dwell time 28 7 sec

~ d t o c e Surloce Temp IT) 1100 1150 1200

. ---- \ - \

92

r . I ' \ 1 -

\\ ' \ I -

46, ; I i

Shell Thnchners (cm)

safety factor and take SO'C as t i e maximum value.-

(b) Spray Heat Transfer Coefficient Profiles

Although difficult to obtain under industrial condi- tions, i t is a simple exercise to control both the reheating of the surface and the absolute value of the surface tem- perature using the mathematical model. This can be ac- complished by changing the surface boundary condition Selow the mould from a "heat flux" to a "known tempera-

Dtslonce From Mtdtoce (cm)

E 5 4 3 2 1 0 1 2 3 4 5 l " " " " ' 1 Heat Flow Analys~s of SOr0yS Zone far a 10 16 crn Square Steel Billel

f 1 , , , , , , , , , ] ; 9 0 0 V) Mtdtoce Shell

Surloce Temp (2) ThnckneSs k m )

Mold dwell llme 15 sec

Figure 12. Heat-flow analysis of spray zone for a 17.15-cm- square steel billet.

ture" boundary condition. Thus, instead of using Eqs. (6) and (7), a sub-mould surface temperature profile which meets the thermal requirements is assigned.

For our design purposes, a colistant surface tempera- ture profile can be taken over most of the theoretical spray zone. The form of the temperature profile was ob- tained in a separate computer run using the mathematical model, with the assumption t h a t heat was extracted be- low the mould by radiation only. The surface temperature profile was then taken from the resulting calculated temperature field a t a point where the mid-face surface temperature equalled a pre-set value. Normally, a value of llOO°C was used, a s this resulted in adequate solidifi- cation rates. This artificial procedure was adopted to ensure that the surface temperature profile near the bottom of the sprays was reasonably close t o the dis- tribution due to radiation alone. The surface temperature profiles calculated for 10.16- and l7.lbcm-square billets a r e shown in the uppermost graphs of Figures 11 and 12 respectively.

In the upper region of the sprays, the surface tem- perature was allowed to change over an arbitrary dis- tance of 60 cm from the profile existing a t the mould exit t o the constant surface temperature profile. This transition period can be seen for the mid-face surface temperature of 10.16- and 17.15-cm-square billets in the center graphs of Figures 11 and 12 respectively. I t is interesting to note t h a t the surface temperature of the smaller billet must be reduced by 40°C during the transi- tion period, whereas no reduction is required for the larger billet. The higher mid-face temperature found a t the mould exit for the 10.16-cm billet is a result of a shorter mould dwell time (15 sec.) as compared to the 17.15-cm billet (28.7 sec.) .

Figure 11. Heat-flow analysis of spray zone for a 10.16-crn- square steel billet.

Once a surface temperature profile has been assigned, the spray heat flux profile can be calculated using re-

arranged forms of Eqs. (14), (15) and (16). For the face, x = 0 . the expression employed is

and for the face, y = 0, i t is

and for the corner

q. (1. 1) = AX Ay

~ ( A Y + Ax)

Finally the spray heat flux profile can be converted into a heat transfer coefficient profile by dividing the

- q, by the surface temperature driving force a t each nodal point. Heat transfer coefficient profiles calculated in this manner can be seen for 10.16- and 17.15-cm-square billets in the left-hand graphs of Figures 11 and 12 re- spectively. In these graphs, the heat transfer coefficients a r e plotted a s contours, with distance from the mid-face as abscissa and time below the mould a s ordinate. The most interesting feature of the plots in both cases is the high value of heat transfer coefficients calculated for the central regions of the face in the upper spray zone. The coefficients then slope downward toward the corners and the lower regions of the sprays, as one would expect. The corners require less cooling, because two-dimensional heat flow exerts its full effect there; farther down in the sprays, less cooling is necessary because the rate of conduction of heat to the surface is lower due to the in- creased thickness of the solid shell (see right-hand graphs in Figures 11 and 12). A comparison of the contour maps of h, reveals further that the smaller billet requires more intense cooling near the top of the spray zone (Fig. 11) than the larger billet. This is a result of the need to lower the surface temperature by 40°C from the mould exit in the former case.

In practice, i t is very difficult to continuously alter spray conditions down the strand to obtain the smooth contours seen in Figures 11 and 12. However, the con- tinuously varying values of h, can be approximated rea- sonably by a series of zones over which the h, profiles are constant. A possible combination of zones for each of the two sizes of billets is presented to the right of the h, contour maps in Figures 11 and 12. The numbers in brackets are the values of h, in cal cm-' set.-I "C-I a t the mid-face for each zone. The selection of the length of

each zone and the number involved i s determined by t h e need to prevent excessive reheating of the surface a s the strand passes from zone to zone.

[c) Over-all Length of Spray Zone

The over-all length of the spray zone required for given conditions is a function of the maximum reheating of the surface that can be allowed below the sprays. If no reheating is to be tolerated, the sprays should be turned off theoretically a t a point where the calculated h, profile i s equal to the profile t h a t would be obtained by radiation cooling alone.

A s discussed earlier, i t i s probably unnecessary to avoid reheating altogether in industrial practice, but rather i t would be sufficient to prevent reheating above a set limit. The effect of time in the sprays on reheating was examined using the mathematical model, where the sprays were "turned off" simply by changing the surface boundary condition from a "known temperature" to a radiative heat flux boundary condition. The results ob- tained for the 10.16- and 17.15-cm billets can be seen in the center graph of Figures 11 and 12 respectively. For the 10.16-cm billet, a spray time of 57 sec. results in an unacceptably high reheat of 104°C a t the mid-face, whereas 135 sec. of controlled spraying yields a reheat of only 17°C. Similarly, for the 17.15-cm billet a spray time of 95 sec. produces a reheat of 92°C. and a longer spray time of 252 sec. results in a reheat of 15°C.

These data have been replotted in a more useful form in Figures 13 and 14, respectively, for the two billet sizes.

steel b ~ l l e t Assumed mold dwell

C o s t ~ n g Speed (cm set') Figure 13. Over-all length of spray zone as a function of casting speed for a 10.16-cm-square billet.

Here, t h e over-all spray length is presented a s a function of casting speed ( t h e slope of t h e lines is t h e spraying time), with extent of surface reheating a t t h e mid-face a s t h e parameter. The maximum surface reheat given in each g r a p h i s t h e reheat obtained if no s p r a y cooling existed below t h e mould. Thus, f o r t h e 10.16-cm billet, which normally would be cast a t 5 to 6 cm set.-', a spray chamber with a length of 650 cm can be seen to be re- quired to keep surface reheating below 40°C. Similarly, for t h e 17.15-cm billet, which usually would be cast a t about 3 cm set.-I, a 600-cm-long spray zone would be needed t o reduce reheating below 40°C. I t i s interesting t h a t despite t h e difference in size and casting speed, the spray zones required have roughly t h e same over-all lengths.

c 5 0 0 - N O Z Z I ~ = N ao ro 3 - Spray Prbaaurr - 90 PSI# H t 000- r-- Spray Prwsur* - 40 psig w 8 F

' 6 2 Ib l o 4 8 :2 36 i WATER FLUX (#al./min., fV )

steel blllet Assumed mold dwell

800 - f

Moxlmum Reheat = 2 0 3 "C

Cost~ng Speed (cm se;')

Figure 14. Over-all length of spray zone a s a function of casting speed for a 17.15-cm-square billet.

[dl Spray Water Flux Profiles

T o carry t h e spray design a s t a g e f u r t h e r and t r a n s - form h, profiles into spray water flux profiles, experi- mental data relating t h e two variables must be available. Mizikar"' has reported data of this kind f o r a narrow range of spray conditions, and a graph from his paper is reproduced in F i g u r e 15. The information in this plot pertains specifically t o the '/4 GGlO ( S p r a y Systems) nozzle.

Taking the 10.16-cm billet a s an example, t h e h, pro- files f o r each of the four suggested spray zones have been converted into equivalent water flux profiles using

Figure 15. Influence of spray water flux from a '/4 GGlO nozzle on the heat transfer coefficient [from Mi~ikar'~']].

F i g u r e 15. Both spray heat t r a n s f e r coefficient and wa- t e r flux profiles can be seen in F i g u r e 16. I t is important to note t h a t t h e flux of spray water required in the mid- face region decreases markedly from about 0.85 cm:' cm-% sec:' in zone I to about 0.1 em:' cm-% set.-I in zone IV. Near the corners, on the other hand, very little spray water cooling is needed in any of the zones.

Clearly, if there is to be any degree of flexibility in the choice of spray nozzles, which becomes important a t this stage of the design, many more plots like F i g u r e 1 5 will have to be generated f o r a wide variety of nozzles and spray conditions. A study f o r t h i s purpose is cur- rently in progress in our group.

(e) Spray Pressure and Nozzle Position I , ' The final stage in the spray design, in which water I

flux profiles a r e linked to nozzle position and spray pres- s u r e for each zone, again requires sound experimental data. MizikarC7' has provided such information f o r the 14 GGlO nozzle, and plots of spray water flux against position relative to the spray center, taken f r o m his study, a r e presented in F i g u r e 1'7. These profiles, es- pecially a t the closest nozzle distance of 4 in. (10 cm), can be seen to have a peak a t the center of t h e spray, which is quite unlike the calculated water flux profiles in F i g u r e 16. The dissimilarity of t h e profiles may in- dicate t h a t t h e 5 / j . GGlO nozzle a t close spray distances is not ideally suited to this size of billet.

Assuming t h a t t h e '/4 GGlO nozzle is the only type available, the nozzle-to-billet distance and spray pressure for each zone can best be obtained f r o m F i g u r e 18, also reproduced from the work %y Mizikar. Each g r a p h in F i g u r e 18 gives water flux vs spray distance f o r a dif- ferent annulus within the spray pattern. Thus, f o r zone 1 i n F i g u r e 16, where t h e spray water flux required over the mid-face region is 0.85 cm>m-' sec." or 12.5 gal min-' ft", the spray distance should be 5.5 in. o r 14 cm, assuming a spray pressure of 40 psig.

Because the measured spray profile f o r t h e 34 GGlO nozzle does not match t h e shape of t h e required profile, i t i s probably wise to compare the required water flux to the measured value a t some point other than t h e mid- face. Taking the corner f o r t h i s purpose ( 2 in.), the & water flux delivered by t h e nozzle there will be 5 gal min-' ft-a o r 0.34 cm3 cm" set.-' from F i g u r e 18. T h i s i s ;1 higher value t h a n is needed, according t o the mathe-

" l k d !:kk.q 0

:;-J ;; 0 Ol

0 "X-J ! ; L 7 - ] 0 01

0 S p a y Heat Transter , C o e f f ~ c ~ e n l s Spray Water Flux Dlshlbutlm

(col cm-'set- OC-') (cm' cm-'rec-') 1/4 GGK) Nozzles a1 4 0 ~ s ~

Figure 16. Profiles of spray heat transfer coefficient and water flux for the four spray zones in a theoretical 10.16-cm billet machine.

&-* S p r a y Dlstance - 4 In. 2 4 - Nozzle - I , GG 10

r--r S p r a y Distance - 8 In. 2 2 - S p r a y Pressure - 40 psig r n . - - ~ ~ r n Spray Distance - 12 In.

18-

- nLI 16- .L

- 0

-M 12- X 2 2

Y 1G- > a E

8- VI

NOZZLE DISTANCE (In I

Figure 18. Spray water flux as a function of spray distance for the different annuli in the spray pattern from a 1/4 GGlO nozzle (from Mizikar"').

the h, profile. A spray nozzle with a smaller spray angle 5 4 3 2 1 0 1 2 3 4 5 may be another solution.

R A D I U S (in.) The distance between nozzles in zone I is determined , by the need to maintain a uniform water flux over the -

Figure 17. Spray water flux as a function of radial position zone. The calculation of this distance requires knowledge %om the spray center for a 1/4 GGlO nozzle (from Mizikar"'). of the spray water flux resulting from overlapping spray patterns. Mizikar has shown, f o r example, t h a t the water

matical model (Figure 16), and may result in overcooling flux between two sprays is about 25-30% greater than J o f the corners. Lowering the spray pressure to 30 psig would be expected from the algebraic sum of contribu-

may alleviate this problem without significantly altering tions t o the flux from the individual nozzles. Taking this

I

I. The values for spray distance and nozzle spacing have een placed in brackets f o r zone I11 and IV, a s i t may e possible that a different combination of lower pres-

sure, closer spray distance and wider nozzle spacing ould be used to a greater a2vantage. Clearly, experience

I is an essential ingredient in arriving a t the best com- bination of variables. C Table ii. Proposed spray zone cqnfiguration for a 10.16-cm- square billet cast at 5.5 cm set.-

Distance of Time in Length of Nozzle to Nozzle

S ~ r a v Zone Zone Billet Face S~acine

. . . . Nozzles: Spray Systems Type M GGlO at 40 psig

Liquid Pool Depths Knowledge of t h e depth of the liquid pool for the ideal- ized conditions treated here is important because, under any circumstances, the pool depth normally controls the position of the cutting stand. Calculated values of the pool depth are presented in Figure 19 a s a function of

Sections

Costing Speed (in mi;')

40.0' 4 0 8 0 120 160

I I I I , ,

36.0 - - 120

32.0 -

C a s t ~ n g Speed (cm set-I)

Figure 19. Liquid pool depths as a function of casting speed for a range of sectinn sizes.

casting speed f o r a range of section sizes, These graphs pertain t o medium-carbon steels specifically, although the carbon grade does not appeer to have a large in- fluence on pool depth.

Summary The design of the mould and spray zones of a continuous casting machine has been developed with the aid of a mathematical model. For the mould, heat flow data from commercial continuous casting machines have been em- ployed t o predict the relationships among mould length, shell thickness a t the mould exit and casting speed. This information, combined with a knowledge of the exit shell thickness obtained in commercial machines, makes i t possible to predict the mould length required if higher casting speeds a r e to be adopted in future for a range of section sizes.

For the sprays, the design is based on the need to satisfy two thermal requirements: minimization of half- way crack formation and maintenance of an acceptably high solidification rate. The f i r s t requirement is met by minimizing any reheating of the surface of the strand; the second by maintaining the surface temperature pro- file constant, with a mid-face temperature between 1000 and 1100°C. The method of design by which these con- ditions can be achieved in the sprays has been demon- strated for 10.lr;- and 17.15-em-square billets. The design involves the calculation of the spray heat transfer co- efficient distributionremploying the mathematical model and, from this, the determination of the spray water flux distribution, the nozzle displacement and the spray pressure using empirical spray correlations.

Acknowledgments The author is most grateful to Dr. G. Vandrunen f o r u important ideas and helpful discussions in the early stages of this work. Thanks a r e also due to B. Prabhakar for assistance with the computing. The combined gen- erosity of the federal Department of Energy, Mines and Resources and the National Research Council, who s u p ported this study, is sincerely appreciated.

Symbols

A. B = constants in thermal conductivity equation (13) C. = specific heat of the solid steel, cal g-I "C-1 h. = spray heat transfer coefficient, cal cm-2 set.-1 "C-1 H., H , 11' = enthalpy of solid steel, at time t and at time t + At.

cal cm-3 i, j k.. k l

kivi- - k ~ . k r

q,,,. qn,

q.9 q r . q o

t. t, To. T.

T,. T I Tp, Tisj

U

X S

f; YM

= integers denoting nodal position = thermal conductivity of solid steel and liquid steel.

cal cm-I see.-' "C-1 = thermal conductivity of the node (i, j) = thermal conductivity averaged between nodes (1, j)

and (2, j) and between (i, 1) and (i, 2) = instantaneous and average heat flux in the mould,

cal cm-2 sec.-I = instantaneous heat flux in the spray zone, the ra

diation cooling zone and from the surfce = time and dwell time, sec. = temperature of the surface of the steel, and of the

ambient environment, "C = temperatureof the solid steel and of the liquid steel = pouring temperature, and temperature of the node

(i. i) = caiting speed, cm sec.-1 = transverse direction. cm = length of billet, cm = transverse direction, cm = width of billet, cm = shell thickness at mould exit, cm

tee1 E n g . 47: 53 (1970). = working mould length, c n ( 8 ) H. Miiller and R. Jeschar. A r c h Eisenhuttenwes. 4 4 :

€ = missivity of strand surface ( Y 0.8) 589 (1973). h = density of solid steel ( 9 ) R.J. S a r j a n t and M.R. Slack. J. I r o n Steel Inst. 1 7 4 :

= Stefan Boltzmann constant 428 (1964).

I Q ' References (10) G. Vandrunen, J.K. Brimacornbe and F. Weinberg. Un- published work. ( 1 ) G, Vandrunen, J.K. Brimacombe and F. Weinberg. (11) S.N. Singh and K.E. Blazek. P ~ ' o c e e d i n ~ 8 of the NOH Ironmaking Steelmaking. 2 : 125 (1975). B O S Confel-ence, Atlantic C i t y . 57: 16 (1974). (2) J.K. Brimacornbe and F. Weinberg. J. Iron Steel Inst. (12) A. Grill and J.K. Brimacombe, 8 : 76 (1976). l r o n ~ n a k

211: 24 (1973). ing Steelmaking. ( 3 ) J.K. Brimacombe, J.E. L a i t and F. Weinberg. Pro- (1:i) A. Grill, J.K. Brimacombe and F. Weinberg. Ironmak-

ceedings of t h e Second I S I Conference on Mathematical ing Steelmaking. 8 : 38 (1976). Pvocess MotEsle Applied in Iron- and Steelmaking, (14) C.J. Adams. Proceedings of the N O H - B O S Conference, A m s t e ~ . d a m : 174 (1975). Pittsburgh. 5 4 : 290 (1971).

( 4 ) J.E. Lait, J.K. Brimacombe and F. Weinberg. Iron- ( 1 6 ) C.W. Briggs. T h e Metallurgy of Steel Castings. Mc- ?nuking Steelmaking. 1 : 90 (1974). Graw-Hill, New York: 328 (1946).

( 5 ) J.E. Lait, J.K. Brimacornbe and F. Weinberg. Iron- (16) H. Vom Ende and G. Vogt. J. Iron Steel Inst. 210: making Steelmaking. 1 : 35 (1974). 889 (1972).

(6) J. Savage and W.H. Pritchard, J. Iron Steel Inst. 178: ( 1 7 ) W. Poppmeier. P ~ o c e e d i n g s of the I S 1 Conference o n 269 (1954). the Solidificatiow of Metals: 393 (1967).

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Reprintea from CANADIAN METALLURGICAL QUARTERLY Volume 15 Number 2 (1976)

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