Method of preventing formation of segregations during continuous casting

A method for preventing segregations in continuous casting by deforming the continuous strand plactically during the solidification in such a way that the cross-sectional area of the strand is physically reduced on a mount corresponding substantially to the solidification and cooling shrinkage of the material along the solidifying strand length. The method avoids upward or downward transport of melt in the solidifying strand. The reduction in most cases will be 2-6% and can be accomplished with apparatus having a number of pairs of strand reducing rolls or jets along the strand, to reduce it a number of times, each time less than the total desired reduction. The degree of reduction of the strand from casting to the final strand corresponds to the solidification and cooling shrinkage.

BACKGROUND OF THE INVENTION 
This invention relates to a method and an apparatus for preventing the 
formation of macro-segregations in continuous metal casting, and 
particularly carbide segregations. 
In the continuous casting of high-carbon steels, for example ball bearing 
steels, high-speed tool steels and also other steels with high carbon 
content, distinctive carbide segregations appear which render the material 
unsuitable for many fields of application. The same kind of carbide 
segregations also can arise when the aforesaid steels are cast in 
conventional molds and during ESR-remelting at high melting rates. 
Carbide segregations are formed during the solidification of the inner 
parts of an ingot. Due to the large solidification intervals of the 
steels, relatively thick zones of semi-solidified material are formed 
therein. In said zones dendrites form a porous network of solidified metal 
with a lower than average content of impurities in the material. Residual 
molten metal with higher carbon content is located in the intermediate 
spaces between the dendrites. During the solidification, the metal 
shrinks, partly as solidification shrinkage of about 4% and partly as 
cooling shrinkage in metal already solidified. 
The metal or material solidifies from the outer surfaces inward to the 
center of the material. This results in several solidification front 
existing during the solidification process which grow toward the material 
center. In continuous casting, furthermore, a strand with unsolidified 
material in the center moves from a mold downward. Depending on the 
dimensions of the strand and its casting rate, the solidification zone, 
i.e., the zone within which semi-solidified material is present, varies in 
the longitudinal direction of the strand with respect to length and other 
dimensions. When the solidification zone has an unfavorable configuration, 
i.e. when it is long and thick, high stresses arise between the 
solidification fronts of the solid and semi-solidified zones. These 
stresses arise due to a difference in the cooling rate between the shell 
surfaces and the interior solid dendrite phase. As the material solidifies 
the solidification front is moved towards the center of the strand. 
Hereby, the temperature decreases when the solidification front passes. 
This decrease is often higher than the temperature decrease at the outer 
surface. This is the case when the solidification fronts meet in the 
center. 
Thus, the stresses arise due to a temperature difference between the solid 
surface and the solid interior phase. As a result of these stresses, the 
fronts separate. The shrinkage gives rise to a pressure differential, 
which sucks down the melt through the porous semi-solidified material. 
This melt is enriched with impurities and alloying elements and, 
consequently, macro-carbide segregations are formed in the center of the 
strand. Corresponding macro-segregation conditions prevail for all alloys 
with large solidification intervals and give rise to segregations. These 
macro-segregations occur with respect to all alloying and non-metallic 
elements present in molten metals. 
When the solidification zone is long, the metal solidifies and shrinks in 
the central portions of the strand, relatively large amounts of melt must 
be transported to the semi-solified zone. As a result thereof, substantial 
macro-segregations arise which form pores and cracks in the central 
portion. In continuous casting it is also known that carbide segregations 
can be reduced by carrying out the casting very slowly. The casting rate, 
however, in that case must be reduced so much that the process is 
uneconomical. 
A process for controlling continuous casting against the formation of 
center segregation and center porosity is described in U.S. Pat. No. 
3,974,559 to Kawawa et al. According to this process, a continuous casting 
is formed in a mold 11 and is then passed through a secondary cooling zone 
12 in order to form a thick solidification shell on the strand. 
Thereafter, a series of reducing rolls are positioned at the front portion 
of the crater end within the strand in order to check the movement of the 
molten steel which contains impurities concentrated by reasons of the 
previous solidification within the secondary cooling zone. This patent 
discloses and claims that the reduction rate should be less than 1.5% for 
each pair of the reducing rolls in order to avoid center cracks which can 
be formed by too great a reduction in the slab thickness. This patent does 
not recognize the problem of macro-segregations across the ingot 
cross-section and along its length during a continuous casting operation 
and the related problems caused by the concentrated impurities within the 
liquid phase. Also, the disclosure of this patent does not take into the 
account the effects of the cooling shrinkage in the semi-solidified phase 
and in the surrounding solid metal shell but rather focuses its entire 
discussion on the minimum reduction rate on the rate of solidification 
shrinkage in the front end of the crater end within the strand in order to 
present a solution for the problem of center segregation and porosity. The 
fact is that it is not possible to eliminate the macro-segregations by 
only compensating for solidification shrinkage. Elimination of 
macro-segregations can only be effected by taking solidification shrinkage 
as well as cooling shrinkage in the solid and semi-solid phases into 
account. This was apparently not realized by Kawawa et al. 
The method disclosed by Kawawa et al cannot be used on the strand just 
below the mold because the high reduction rate utilized would cause the 
molten metal to be pressed upward into the tundish since the solid shell 
of the strand constitutes only a small portion of the cross-sectional 
area. The Kawawa patent is premised on first forming a substantial solid 
shell on the strand during which time impurities are concentrated in the 
molten phase and to then press the concentrated molten steel backward from 
moving toward the front of the crater end. In this manner, a positive 
forcing of the concentrated molten steel backward along the strand length 
occurs as an integral part of the process. 
SUMMARY OF THE INVENTION 
The applicants of the present invention have discovered macro-segregation 
fluid flow mechanics which can be used as the basis for a casting process 
in which transport of molten steel either upwardly or downwardly in the 
ingot is prevented. This has been done by the enunciation of a theoretical 
model, consisting of a differential material balance equation, which can 
be solved in a manner to determine the necessary deformation rate in order 
to prevent flow of molten metal either upwardly or downwardly along the 
ingot in the case of a vertical casting machine or backward and forward in 
the case of a horizontal casting machine. The process evolved requires the 
successive application of deformation roll pairs from immediately below 
the mold until the point where the ingot strand has completely solidified. 
The result of the practice of this process is the attainment of a goal long 
sought in the continuous casting industry of producing a strand of 
constant composition across the transverse cross-section and 
longitudinally along the strand length as well. 
The present invention relates to a method of preventing the aforesaid 
formation of macro-segregations during continuous casting. The invention 
is characterized in that the cast strand during solidification is 
subjected to plastic deformation, so that the cross-section area of the 
strand is reduced from just below the mold to in a series of steps to 
degree specified by the described theorectical model which has been 
heretofore available.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
The formation of suctions, stresses and cracks in a semi-solidified area 
depends on the configuration of the solidification zone. FIG. 1 shows a 
solidification zone having a favorable configuration with respect to 
suctions, stresses and cracks, because the semi-solidified material 2 has 
a short extension in the vertical direction, i.e. in the longitudinal 
direction of the strand. The semi-solidified material 2 is surrounded by 
molten material 1 and solidified material 3. A mold 4 encloses the strand 
1, 2, and 3. A solidification zone of the configuration shown in FIG. 1 
arises at a very low-rate continuous casting, at normal ESR-recasting and 
at the casting of a thick, short ingot. 
FIG. 2 shows a solidification zone having an unfavorable configuration, 
because the semi-solidified material 2 has an enlarged vertical extension. 
This type of solidification zone is formed during normal and rapid 
continuous casting, high-rate ESR-remelting and the casting of a long, 
narrow ingots. When a metal, which is cast by normal or rapid continuous 
casting, as in FIG. 2, shrinks at the center, relatively large amounts of 
melt 1 are transported downward from above due to the relatively large 
area with semi-solidified material. As a result thereof, substantial 
segregations arise, as mentioned above, over a larger area, in the form of 
so-called macro-segregations, which give rise to pores and cracks in the 
central portion and variations in composition in the transverse and 
longitudinal directions. The process described with reference to FIG. 2 is 
the normal process during continuous casting. The casting rate is so high, 
that the solidification zone is relatively long. The above known technique 
for preventing carbide segregations consists of low-rate casting whereby a 
small solidification zone, according to FIG. 1, is formed. This process, 
however, has an unfavorable economic result. 
According to the present invention, the strand is deformed plastically so 
that the cross-sectional area reduction substantially corresponds to the 
solidification and combined cooling shrinkages of the material as further 
detailed herein. Preferably, the plastic deformation of the strand is 
effected from immediately below the mold as shown in FIG. 3 to the 
position where the strand is fully solidified. This reduction length 
extends from the top position where the strand has all three phases of 
solid, semi-solid, and liquid present to the fully solid position. When 
the central portions solidify and this material shrinks by solidification, 
the strand is subjected to a reducing working so that its cross-section 
area is reduced to a dimension corresponding to the area of the solidified 
and entirely welded-together material over the cross-section area of the 
strand. Due to this process, melt cannot be sucked down into the 
semi-solidified material 2 or pushed back up into the mold. Consequently, 
the formation of macro-segregations in the transverse and longitudinal 
directions and along the length as well as of pores and cracks in the 
central portions are prevented. In order to attain this result, it is 
necessary to carry out the successive reduction steps in a precise manner 
which is determined by a valid theoretical understanding of the 
macro-segregation process. 
In FIG. 3 a device is shown, by which a working operation for deforming the 
strand can be carried out. The molten metal is poured down through the 
mold or chill 4 and solidifies substantially immediately on the surface. 
The solidified strand is passed down and out of the mold 4, and thereafter 
is introduced between a plurality of roll pairs 5--5'. Each of the roll 
pairs 5--5' has a spaced relationship between the rolls which brings about 
an area reduction corresponding to the theoretically derived deformation 
rate required at each roll pair. The strand 6, thus, from the first roll 
pair and downward is subjected to deformation forces so that it is 
entirely welded-together at its center after completion of the process. No 
cooling section is interposed between the mold and the first of the roll 
pairs. After the last roll pair, the strand is entirely solidified. Due to 
this successive working, the molten metal 1 (so-called "melt") will not be 
forced upward or downward through the semi-solidified metal. This process 
is described below by reference to a theoretical model which forms the 
basis for the present invention. 
MOLTEN FLUID FLOW MECHANICS 
Referring now to FIG. 4, the microscopic semi-solid domain D.sub.0 shown in 
FIG. 3 is shown enlarged with solid dendrites 25 forming therein by 
solidification of the liquid metal 26 which is within the semi-solidified 
material 2. The dendrites can be attached to the internal matrix formed by 
the solidified metal 3 or can be relatively free-floating in the 
solidifying matrix within the semi-solidified zone 2 as shown by dendrite 
27. 
FIG. 5 shows a semi-solidified domain volume 28 within the microscopic 
metal domain D.sub.0 ' identified in FIG. 4. This volume 28 has a solid 
portion 29 and a molten portion 30 and thus represents the solidification 
interface. Within volume 28 the solidification portion can be defined as 
A.multidot.dy, where A is the cross-sectional area of the domain volume 
and dy is the incremental growth of the solid portion 29 over a given time 
period. The solidification shrinkage resulting from dy changing phase and 
physical properties from liquid to solid is one of the causes of molten 
metal to be sucked down into the liquid phase 30. This solidification 
shrinkage described in microcosm for volume 28 is repeated for all such 
domain volumes within the semi-solidified zone 2 of FIG. 3. In aggregate, 
this shrinkage process causes liquid metal to flow into volume 28 and also 
to be drawn through each such domain volume. Another cause of this molten 
metal flow is the cooling shrinkage as detailed below. 
As shown by FIG. 6, the molar fraction or concentration of carbon in the 
solid phase 29 of FIG. 5 is denoted as X.sub.s and is lower than the 
fraction in the liquid phase 30 which is denoted as X.sub.L. This 
concentration differential described for carbon is also a phenomenon 
encountered for other elements such as sulphur and phosphorous which 
together with carbon and other elements cause macro-segregation problems 
in continuous cast metal strands. The concentration of these elements 
tends to increase in the liquid phase over the duration of the casting 
operation and to thus produce macro-segregation across the ingot 
cross-section as well as along its length. By casting according to the 
present invention such macro-segregation can be controlled and thus be 
avoided. 
THEORY OF INVENTION 
The theoretical basis on which the invention rests can be described by 
reference to FIGS. 5 and 6 in terms of a generalized material balance 
equation for domain volume 28. The concentration of a given alloying or 
non-metallic element in the liquid metal flowing into the volume 28 is 
denoted as X.sub.L ' and the concentration of that element in the outgoing 
liquid metal which is drawn through the liquid phase portion 30 due to 
solidification and cooling shrinkage in other domain volumes within the 
semi-solidified zone 2 is referred to as X.sub.L ". The molar fraction in 
the solid phase can be generally expressed by the symbol X.sub.s and in 
the liquid phase by X.sub.L. These concentration measures are expressed as 
molar fractions so that the total of all such elements is unity as 
follows: 
EQU 1=X.sub.C +X.sub.S +X.sub.P +X.sub.Fe +X.sub.other elements Eq. (1) 
For carbon, C, the molar fraction is calculated as: 
##EQU1## 
where, M=molar weight of each of the designated elements identified by 
periodic chart subscripts and E=other alloying or non-metallic elements 
which are present in the melt such as Si and Mn. 
Next an expression is formulated for the material balance for the domain 
volume 28 of FIG. 5 considering that an incremental volume dy.multidot.A 
solidifies over a given time period. This balance for any one of the 
alloying or non-metallic elements, E, can be used to construct a general 
expression from the following terms for which reference to FIG. 5 should 
be made for the cross-sectional area, A, and dimensions .lambda. and y of 
the domain element 28: 
##EQU2## 
Where: X.sub.s =molar fraction of E in the solid phase volume y.multidot.A 
V.sub.m.sup.s =molar volume of the solid phase y.multidot.A 
X.sub.L =molar fraction of E in the liquid phase volume 
(.lambda.-y).multidot.A 
V.sub.m.sup.L =molar volume of the liquid phase (.lambda.-y).multidot.A 
V.sub.in =volume of liquid entering the domain volume .lambda..multidot.A 
V.sub.out =volume of liquid leaving the domain volume .lambda..multidot.A 
The material balance can then be formulated as follows using the above 
defined terms: 
EQU I+II+III-IV=V+VI Eq. (9) 
where the symbolized terms are as defined above in equations 3 through 8. 
The resulting material balance is a differential equation and can be 
solved by a computer run for any alloying or non-metallic element, E. The 
solution of the equation describes the content of element, E, in volume 
element 28 when the solidification process has been completed, i.e. 
formation of a solid cast ingot. This is also a description of 
macrosegregation. For avoidance of macrosegregation the moving strand must 
be reduced in thickness at a deformation rate, D.sub.r, over the 
semi-solidified region so that 
EQU V.sub.in =V.sub.out =0 Eq. (10) 
whereby liquid metal flow through the liquid phase 30 of domain volume 28 
in FIG. 4 is prevented. 
The required deformation force must be applied immediately below the mold 
and throughout the cast ingot length to the position where the ingot has a 
solid cross-section since macro-segregation phenomenon occurs during the 
entire casting time period at economical casting rates. It is insufficient 
to permit a substantial ingot wall thickness to form first and to then 
apply outside deformation forces since considerable macrosegregation will 
have already occurred in the metal forming such an ingot wall; thus 
resulting in variation in concentration of all of the elements, E, both 
across the ingot and along its length. 
The above material balance differential equation (9) can be solved in a 
manner to obtain the necessary deformation rate, D.sub.r, in order to 
satisfy the condition of equation (10) above. The necessary deformation 
rate, D.sub.r, is determined by considering the physical properties of the 
strand at each deformation position along the length of the ingot. The 
force between opposing rollers in each roller pair 5 is set accordingly. 
The solution of equation (9) thus produces a model by which the 
deformation necessary to fully compensate gainst flow through each domain 
volume and thus to compensate for solidification shrinkage in the entire 
strand as well as to compensate for cooling shrinkage. 
From equation (9) it can be determined that the solidification shrinkage 
S.sub.sh when incremental volume dy.multidot.A solidifies is described by: 
##EQU3## 
where the terms are as above defined. This equation can be solved for the 
deformation rate necessary to compensate for solidification shrinkage in 
terms of dy. The term, dy, can then be calculated from cooling shrinkage 
data. 
Two additional factors also enter into the determination of the necessary 
deformation rate, D.sub.r. Compensation must be made for the cooling 
shrinkage of the solid material in the semi-solidified zone, represented 
by molten volume 30 in FIG. 5. This shrinkage otherwise causes liquid 
metal to be sucked down into the zone. Such shrinkage equals 
.alpha..multidot..DELTA.T.multidot.1 where .alpha. is the coefficient of 
thermal expansion (or contraction), .DELTA.T is the decrease in 
temperature and 1 is a unit length. 
Further, one must compensate for cooling shrinkage of the solid material 
surrounding the semi-solidified area. This material will, when cooling, 
deform the soft semi-solidified material, whereby this cooling shrinkage 
will be a positive contribution to the deformation of the strand. 
Equation (10) gives the condition for avoiding macrosegregations. This 
condition is expressed as in equation (12) below, which means that the 
deformation rate (D.sub.r) is equal to the algebraic sum of all different 
shrinkages. 
To sum up, equation (12) gives the deformation rate of the strand by 
external force on its surface over a given time period, for avoiding 
marco-segregations as follows: 
##STR1## 
In the above theoretical model it is assumed that no plastic deformation 
occurs due to thermal stresses only, i.e., that the thermal stresses do 
not reach such a magnitude. 
The described theoretical model embodied in equations (9) to (12) and the 
supporting equations permits calculation of the extent to which the ingot 
strand must be deformed in order to fully avoid macro-segregations. When 
the necessary deformation rate, D.sub.r, is attained at multiple positions 
along the length of the solidifying ingot the concentration of carbon, 
sulfur and other non-metallic elements as well as alloying elements can be 
controlled so as to be uniform across the ingot cross-section and along 
its length. 
Control against macrosegregation over the length of the continuous cast 
strand has not been heretofore possible since the molten phase fluid flow 
mechanics which form the basis for the above described theoretical model 
had not been known until enunciated by the inventors of the present 
invention. The continuous casting of ingots having uniform properties and 
composition in both transverse and longitudinal directions is now possible 
due to the formulation of the described theoretical model which represents 
the general solution to the problem of controlling against 
macrosegregations in continuous cast metal strands and is thus a 
commercial development of broad implications. The compensation for the 
solidification and cooling shrinkages must start immediately below the 
mold and this is practical due to the large static pressure of the molten 
steel against which the reducing rollers work. More specific operational 
aspects of using this theoretical model in practice are described below. 
FIG. 7 shows the apparatus of the present invention applied to a horizontal 
casting machine in which the solidified ingot 6 is disposed in a 
horizontal direction. As in FIG. 3 molten metal 1 is surrounded in the 
lower parts of mold 4 by the semi-solid phase 2 which is further 
solidified to form the solid phase 3 in the horizontal casting machine. A 
series of roller pairs 5--5' are utilized as reduction rolls from a 
position immediately below the mold 4 until the point where the ingot 
strand has completely solidified. The number of rolls, the bending radius, 
R, depend upon the metal being cast, the cooling rate, and the dimensions 
of the strand, etc. 
In both FIGS. 3 and 7 the first reduction roller pair 5--5' is used to 
plastically deform the ingot strand at the position where the strand 
consists of a solid metal skin of only a few centimeters in thickness and 
this position usually occurs about 1 meter below the top of the liquid 
phase surface within mold 4. 
In the continuous casting of workpieces with rectangular cross-section, 
so-called slabs, the corners and portions adjacent thereto are cooled much 
more rapidly than the remaining parts of the strand. As a result thereof, 
the solidification shrinkage, which causes the sucking down of melt 1 into 
the semi-solidified material 2, takes place in the central strand 
portions, which solidify at a later time. This implies that only the broad 
sides of a strand with rectangular cross-section shall be worked. It is 
accentuated thereby that a strand, due to the stronger cooling at the 
corners, tends to assume a greater thickness at the center of the broad 
sides where the material is hotter. 
FIG. 8 shows in a schematic way a device according to an embodiment of the 
invention, at which only a portion of the broad sides of a strand is 
intended to be worked. A strand 6 with convex broad sides 7 is cast in a 
mold 4 (see FIG. 3) and worked between two plane rolls 8, 9. Thereby only 
that portion of the convex broad sides is worked which has contact with 
the plane rolls. After the working, the strand has a reduced cross-section 
area, because the strand has assumed a less convex configuration while the 
areas at the corners of the strand are substantially unworked. The 
convexity of the strand can be adjusted at casting so that, as a result of 
the necessary reduction of the cross-section by working with rolls, the 
strand after the working has a rectangular cross-section. 
The reduction of the strand according to the embodiments described above 
and in the following must be so great, that it slightly exceeds the 
reduction in area which corresponds to the solidification shrinkage 
occurring. The reduction must be carried out in multiple steps, as 
indicated in FIG. 3, so that a substantially continuous area reduction is 
obtained which is adjusted to and corresponds to the deformation rate 
specified by the above theoretical model. Thensile stresses in the 
solidifying material are hereby avoided and only moderate compressive 
stresses are obtained. The number of reduction steps is determined by 
practical factors, especially by the casting rate and, thereby, the length 
of the solidification zone. In high-speed continuous casting machines, 
with a solidification zone length of up to 20 meters, the working can take 
place in 20 to 40 steps, i.e. roller pairs, while in slower operating 
machines, for example an ESR-machine, the working is carried out in a few 
steps. 
A suitable total reduction of the cross-sectional area of the strand 
generally is 1-10%, preferably 2-6%. For steel, a suitable reduction 
generally is 4%. 
The rolls 8, 9 are arranged to rotate at the same circumferential speed as 
the rate of the cast strand at said roll pair. A plurality of roll pairs 
similar to the roll pair 8, 9 can be positioned with different spaced 
relationship to the mold, as shown in FIG. 3. 
Another embodiment is shown in FIG. 9, according to which the strand 6 is 
cast with rectangular cross-section and plane broad sides, and the working 
is carried out with rolls 10, 11, which are cambered, i.e. so designed as 
to have a diameter decreasing from the center to both ends. According to 
this embodiment, a strand is obtained after the working which has the 
smallest thickness at its center and increasing thickness to the short 
sides of the substantially rectangular cross-section of the strand. In 
general, the above information with respect to the plane rolls 8, 9 
according to FIG. 8 and the roll pairs in FIG. 9 applies also to this 
embodiment. A corresponding working of strands with square cross-section, 
octagonal cross-section round cross-section or a cross-section of another 
shape can be carried out by means of tools, which enclose the strand as 
completely as possible, because the cooling of the strand at such 
cross-sections is more symmetrical than for strands with rectangular 
cross-section. 
In order to illustrate this, FIG. 10 shows schematically a device for 
working a strand with substantially square cross-section. The strand 6 is 
worked by means of two rolls 12, 13, which are provided with grooves 14, 
the configuration of which corresponds to the shape of the strand at two 
diagonally opposite corners. The grooves 14 are given such a depth, that 
they together substantially enclose the strand, which is being worked, 
also along its sides. When several roll pairs similar to the rolls 12, 13 
are arranged one after the other, the axles of such roll pairs can form an 
angle of 90.degree. with each other in order to work the strand 
symmetrically. 
A further embodiment of the invention is shown schematically in FIG. 11. A 
strand 6 is worked here by means of two opposed reciprocating forging 
tools 15, 16 with working surfaces facing toward each other, which 
surfaces between themselves form a space adjusted to the shape of the 
strand and to the type of working, to which the strand is to be subjected. 
Said space tapers to wedge shape in the direction of strand movement in 
order to subject the strand to the desired reduction with respect to its 
cross-section area. The arrows 17, 18 in FIG. 11 indicate the direction of 
movement of the tools 15, 16. At this device, the strand 6 is advanced one 
step when the forging tools 15, 16 move away from each other, and is 
deformed when said tools move toward each other. By working the strand by 
means of forging tools 16, 17 conical in the longitudinal direction of the 
strand 6, an almost continuous reduction of the cross-section of the 
strand is obtained. 
The working surfaces of the forging tools 15, 16 can, perpendicularly to 
the longitudinal direction of the strand 6, be formed plane, convex or 
concave, depending on the corss-sectional shape of the strand 6. 
According to a further embodiment of the invention, the reduction of the 
cross-section of the strand 6 is effected by controlled cooling of the 
strand 6 as illustrated in FIG. 14. Immediately after its leaving the mold 
4, (FIG. 14), the strand 6 has a cross-section corresponding to the inner 
form of the mold 4. In FIG. 12 a rectangular cross-section of a strand is 
shown as an example. The corners 19 and the areas immediately adjacent 
thereto are colder than the center on the broad sides 20 of the strand 6 
and the material inside thereof. The solidification process is shown by 
way of example in FIG. 12, with solidified material 3 at the colder 
portions and semi-solidified material 2 in the interior of the strand. Due 
to this temperature difference, the strand is thinner adjacent the corners 
19 than at its center, because solidification shrinkage and cooling 
shrinkage have occurred adjacent the corners 19, whereby the strand 
assumes a convex appearance as shown in FIG. 13. According to this 
embodiment, a reduction of the cross-section area of the strand 6 is 
obtained thereby, that the broad sides of the strand 6 are subjected to 
forced cooling, whereby the surface layer of the convex portions and 
solidified material 21 inside thereof are contracted and deform the 
centrally located semi-solidified material. Thereby the necessary 
deformation of the strand is obtained. The cooling, thus, is started 
during the final solidification phase of the strand, as appears from 
above. 
This embodiment can be applied also to strands with other cross-sections. 
In the case of square, octagonal, round or like shape of the strand, the 
forced cooling is carried out so that all sides or outer surfaces of the 
strand are cooled. This implies, that the entire outer shell of the strand 
shrinks as a result of the cooling shrinkage, whereby the necessary 
reduction of the cross-section takes place and the inner semi-solidified 
strand material is deformed. This is possible by controlling the cooling 
shrinkage terms in equation (13) in a positive manner according to the 
above theoretical model. 
The forced cooling in the apparatus of FIG. 14 is effected by a plurality 
of nozzles 22 which eject coolant 23 against the strand 6 in the above 
indicated places. The coolant may be water, water-air mixture or steam. 
The invention may be embodied in other specific forms without departing 
from the spirit or essential characteristics thereof. The present 
embodiment is therefore to be considered in all aspects as illustrative 
and not restrictive, the scope of the invention being indicated by the 
appended claims rather than by the foregoing description, and all changes 
which come within the meaning and range of equivalency of the claims are 
therefore intended to be embraced therein. The mechanic plastic working, 
for example, can be varied in different ways, and also the cooling device, 
if cooling is used for bringing about the cross-section reduction, can be 
modified in a suitable way within the scope of the invention.