Sahai&Emi

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ISIJ International. Vol. 36 (1996), No. 6, pp. 667-672
Melt Flow Characterization in Continuous Casting Tundishes
YogeshwarSAHAIand Toshihiko EM11)
Departmentof Materials Science and Engineering, TheOhio State University, 2041 College Rd.. Columbus,OH,4321OUSA.
1)Base Metal Research Station, Institute for AdvancedMaterials Processing, TohokuUniversity. Katahira. Aoba-ku, Sendai,
980-77 Japan.
(Received on December5, l995, accepted in final form on January 18. 1996)
Melt flow in continuous casting tundishes is normal[y characterized by a combinedmodel. The model is
used to analyze the residence time disttibution of fluid in a tundish, In this model, the fluid volume in tundish
is considered to be consisting of the plug flow, well-mixed flow, and dead volumes. Although this model
was proposed over 20 years ago, most researchers have either used it incorrectly or madean assumption
in analyzing melt flow in tundishes. Both approaches may lead to incorrect and misleading calculations
of the dead volume. In this paper, the combinedmodel has been discussed and its correct application to
tundish melt flow has been outlined,
KEYWORDS:continuous casting; tundish; modeling; combinedmodel; plug flow; mixedflow, deadvolume;
clean steel; inclusions.
1. Introduction
The tundish in a continuous casting operation is an
important link between the ladle, a batch vessel, and
the casting mold with a continuous operation. It has
traditionally served as a reservoir and distributor of
molten steel but now, its role has considerably expanded
to deliver metal of desired cleanness and composition.
With the increasing emphasis on the stringent quality
control and reduced cost of production, in future, it may
be required to take up on someadditional responsibilities,
such as refining andprocessing. Thus, the inclusion flota-
tion and separation, and the composition adjustment
have nowbecomeimportant functions of a tundish. The
efficiency and optimization of these processes require a
close control of the molten steel flow characteristics
within the tundish. If the flow of metal in the tundish
is not properly controlled, it mayeven deteriorate the
'quality' of steel produced in the ladle. Thus, tundishes,
in terms of their shapeand use of the flow control devices
(dams, weirs, baffles, pour pads, etc.), are designed to
provide optimumflow characteristics.
To assess the effectiveness of a given tundish design,
researchers have simulated the metal flow either mathe-
matically or physically, before actually using the design
in actual industrial production. Mathematical modeling
has been used by manyresearchers for fiow predictions.
There are several commercial software packages which
can be used for predicting melt flow and residence time
distribution in tundishes. In spite of all claims for their
user friendliness, these codes are not very simple to use,
and can only be used by highly trained professionals.
The results obtained by these computer programs, Iike
any other computer program, are dependent on the as-
667
surnptions and the boundary conditions used in solving
them. Thus, any inappropriate boundary condition may
lead to erroneous and misleading results. Evennow, the
use of these codes to treat free surface or multi-phase
flow which mayexist in a tundish is relatively difficult
and cumbersometask. It is therefore recommendedthat
the results of a mathematical model must be verified by
actual experiments, such as water modeling.
Watermodeling, on the other hand, is relatively straight
forward experimentation and can be madeby relatively
less experienced personnel. A carefully planned experi-
mentmaygive very useful results, and the correct inter-
pretation of these results mayprovide an insight into the
tundish design. Water modeling in full or reduced scale
model of a tundish has been a very popular meansof
physical simulation of melt flow in tundishes. It is a rel-
atively quick and less expensive method of qualitative
and semi-quantitative study of the melt flow and hence,
the design assessment of a tundish. In such studies, a
tracer (e.g, dye, acid, or salt) is injected in the incoming
water stream and its concentration at the exit is record-
ed as a function of time. The plot of the exit concen-
tration against time is knownas the Residence Time
Distribution (RTD) curve. The RTDof the fluid in
a tundish is analyzed to characterize the flow which,
normally, includes the determination of the extent of
mixing (plug and mixed volumes) and dead volume in
the tundish. Thechoice of dye as a tracer also provides
fiow visualization which mayput the results obtained by
the RTDanalysis in proper perspective,
Oneof the models which has been extensively used
for the analysis of the RTDcurve is a combined or
mixed model for the calculation of the plug fiow, mixed
flow, and dead volumes in a tundish. In the plug fiow
@1996 ISIJ
ISIJ International. Vol.
region, the longitudinal mixing is non-existent, however,
there maybe transverse mixing to any extent. In the
plug flow, all fluid elements have equa] residence times
in the tundish. The mixed flow is the other extreme
where the mixing in a tundish is maximumpossible.
The dead volumel) is the fluid that movesslowly in the
tundish and stays for longer than two times the mean
residence time (see definition* below). A review of the
tundish modeling literature shows that the combined
mode]has beenmisinterpreted and used incorrectly. The
assumption used in the calculation of the dead volume
maylead to a significant error in the dead volume. The
authors are not aware of any publication in the open
literature where this model has been correctly applied
(without the assumption) to the tundish flow. The pur-
pose of this paper is to discuss the combinedmodel and
outline its application to the tundish melt flow character-
ization. Basedon the authors' experience of dealing with
the researchers in steel and related (ceramic flow control
device manufactures) industries, especially in the North
America, the purpose is clearly worthwhile. Most of the
theoretical framework and definitions presented in this
paper are taken from Levenspiel's book.1)
2. CombinedModel
The simplest type of a combinedmodel and the one
most frequently used for the flow characterization in
tundishes assumesthat the following three kinds of flow
regions are present in the total volume of the fluid in a
tundish.
Plug fiow region,
Mixed region, and
36 (1996), No. 6
Deadregion.
3 Actrve Volume
Anycombination of the plug flow and well-mixed flow
volumes maybe termed as an active volume. Consider
that fluid fiowing in a vessel maybe represented by a
combination of the plug flow and well mixed regions as
shownin Figs. l(a) and 1(b). Theorder of the two regions
is reversed in the two models. The two models give anidentical tracer response to a pulse or any other type of
input for a linear system. A Iinear system is one in which
any change in the input or stimulus signal results in
a corresponding proportional change in the output or
response signal. The residence time distribution curve is
shownin Fig. 2. As shownin this figure, the minimum
residence time (O*i.) corresponds to the plug volume
fraction (Vp/V), and maximumconcentration (C~..) is
equal to the inverse of the well mixed volume fraction
(V/V*). WhereVp, V*, and Vare the plug, mixed, and
total volumes, respectively. The equation of the expo-
nential decay curve is given in the figure.
4. DeadRegion
For the simplicity of discussion, the deadvolumemaybe divided into two types. In the first type, the liquid in
the dead region is considered to be completely stagnant
such that the incoming fluid does not even enter this
region. Figure 3schematically represents a system with
this type of dead volume. In the second type, the fiuid
in this region movevery slowly, and as a result sorne
fluid stays muchlonger in the vessel. In fact, the fluid
in the dead region continually exchangeswith the fluid
(a)
fl
~.Y_~{:.~
c INPUT C
PLUGFLOW
o e--> VP
MIXEDFLOW
vp/V e=>
Vm
'
Vp/V o•->
(b)
fLllIPr"~Ul'~~Us_rEc INPUT MIXEDFLOW
o e--~
Vm
c
PLUGFLOW
o e•-~
VP
c
v !V e'-~'
Fig, l.
Acombinedmodel representing plug volume and
mixed volume.
C ::-Lmax Vm
1,
C
[- )J
v"'p vl~
(e-l~v_'vl Q=Q ACTIVEREGION
Va
Omin
=
=y Time
,
eV
Fig. 2. C-curve for a combinedmodel presented in Fig, l.
DEAD
REGION Qd= O
Vd
Frg. 3. Flow through active and dead (stagnant) regions of a
combinedmodel.
* Meanresidence time is the ratio of the volume of liquid to the volumetric flow rate.
@1996 ISIJ 668
ISIJ International, Vol. 36 (1996), No. 6
Q
Qa
ACTIVE
REGION
V,
Qd DEAD
REGION
Vd
Q
ACTIVE QREGION
Va
Qd
DEAD
REGION
Vd
Flow through active and dead (slow moving) regions
of a combinedmodel.
C
Q
1.2
0.8
0.6
0.4
0.2
O
Total Area up to e= 2
Q*
Q
Area Q-dQ
Fig. 4.
in the active region. Thus, the fluid which stays in the
vessel for a period longer than two times the meanresi-
dence time is considered as the dead volume. Twoalter-
native waysof schematically representing a system with
slow movingdead volume are shownin Fig. 4. Thedead
volurne in most of the normally operating tundishes falls
in the second type, and is characterized by a long tail
extending beyondthe two times the meanresidence time.
The average residence time of the fluid for any given
tundish at a constant volumetric flow rate remains con-
stant. Thus, the slower movingfluid or deadvolumestays
10nger in the tundish at the expense of other fluid. In
other words, if somefluid assumesmuchlonger residence
time in the tundish, an equivalent amountof other fluid
has, accordingly, a shorter residence time in the tundish.
This faster moving melt maynot spend sufficient time
to separate and fioat out the non-metallic inclusions.
Also, molten metal in the dead (slow moving) regions
mayloose sufficient heat, andmaystart to solidify metal.
Thus, tundishes are designed to have dead volume as
small as possible.
Consider a combined model consisting of an active
(plug fiow and well-mixed fiow) and a dead regions. As
depicted in Figs. 3and 4, Iet the total volume of the
system be Vwhich is divided into an active volume of
V* and a dead volume of Vd. Let the total volumetric
flow rate through the system be Qwhich is also divided
in Q* through the active region and Qdthrough the dead
region. For completely stagnant deadvolume (represent-
ed in Fig. 3). QdWill be zero and Qwill be equal to Q*.
For a dead region with slowly moving fluid, a typical
experimentally obtained RTDis shownin Fig. 5. ARTD
curve corresponding to a pulse input is knownas the
C-curve. Let the dimensionless meantime of the C-curve
upto the cutoff point of dimensionless time, O=2be ~c,then
e~c
=
measuredmeantime upto e=2 measuredt~c
meanresidence time ~t
.(1)
O 1 2 43
~.=Jy
.~ e
v Q.
Fig. 5. A typical residence time distribution curve for flow in
a tundish
e~c
tc V./Qa
_
V.
.
Q ,.......,.(2)
~ ~ V/Q ~ V Qa """~
Va
_
Qa
. ~
..........(3)V~Q c
Thus, the dead volume fraction
Vd I Qa . O~c """""(4)V Q
The term Q./Q is the area under the C-curve from e=0
to 2, and represents the fractional volumetric flow rate
through the active region. With the presence of dead
region(s), the measuredaverage dimensionless residence
time,
ec
......
..........(5)
If the dead region is completely stagnant (as in Fig.
3) so that the fiowing fluid does not enter or leave the
region, the volumeof the system through which the fluid
flows in the system is effectively reduced to Va (or QalQ
is one in Eq. (4)). Thus, the deadvolume fraction will be
V
= I -ec """""(6)
The dead volume fraction with stagnant volume is
given by Eq. (6), which is a special case of Eq. (4). The
dead volume with the slowly moving fluid is given by
Eq. (4).
5. Application of the CombinedModel to Melt Flow in
Tundishes
As stated earlier, a typical experimental C-curve ob-
tained in water model studies or in an actual tundish
showsan extended tail beyond the time, 0=2. This in-
dicates the existence of a slow moving flow through the
dead regions. As shownschematically in Fig. 6, there
mayexist dead regions on the downstreamside of the
damsand weirs, or near the end wall. Figures 7and 8
are taken from literatures where detailed flow patterns
in different tundishes are predicted by the solution of
the Navier Stokes' equation. Figure 7is taken from the
669 C 1996 Is[J
ISIJ International, Vol. 36 (1996), No. 6
DeadVolume
.\~..x\",r*
"'//.
'//..
Table l. Estimated error in dead volume calculation by
ignoring the QdlQ.
Qd/Q Correct Vd/V (Eq. (4)) Elrol In V IV
O
0.01
0.05
O. 1
O, l
O, I09
O, 145
0,19
O
90/0
450/0
900/0
Frg. 6. Representation of melt flow in a tundish having dead
volumeand exchangeof liquid with the active volume.
Inlet End
O.OS M/S
=-~>
~
\
-- L
'/'L'~~._*.
~'
li
L 1"'I-ff'-!T
-
"
' : i:~
.,~~
l\
,,('
Continuous temperature measl
l
/ ~ / /////
~/ ///~/ / / 't
~//// /~ /ll/
, ,~
~
); (/// \~~
~// ~/~/'7/__
_ _
-"~~:P(///_
___'~~/// '
_
-
/ - //' ~_
\
~
~/ f// ,Channel /;;
Strand
\\\- - /
Fig. 7. Predicted low pattern in a selected plane ofthe Armco
KansasCity bloom caster tundish (Ref. 2)).
measurement
Fig. 8. Predicted velocity profile in an induction heated tun-
dish at the Muroran works of the Nippon Steel Cor-
poration (Ref. 3)).
work of Lowry and Sahai2) and showsflow pattern in a
selected plane of the ArmcoKansasCity bloom caster
tundish. The figure clearly depicts the slow movingdead
volumes behind the damand weir. Figure 8shows ve-
locity profile predicted by Suzuki et al.3) in an induc-
tion heated tundish at the Muroranworks of the Nippon
Steel Corporation. In these figures, the velocity vectors
are proportional to the magnitude of velocity at that
Iocation. These figures show regions of very high ve-
locity and thus, high turbulence and regions of good
mixing, and regions with very slow moving fluid causing
dead volumes.
The fiuid in these dead regions is constantly inter-
changedwith the main flow (in active volume) of the tun-
dish. Theseregions should not be considered as stagnant
deadvolume regions. Thus, the deadvolume fraction in
tundishes is given by Eq. (4). There are two approaches
used by researchers in the analysis and modeling of the
melt flow in tundishes. The first one, which has been
most widely used (e.g. Refs. 4)-6)), is the use of Eq. (6).
Thus, the model assumesthat the area under the curve
@1996 ISIJ 670
from the time, 0=2 to co in Fig. 5is zero. This area rep-
resents the fraction of the volumetric flow rate through
the dead regions (QdlQ)• Onecan visualize in any water
modeling experiments of a typical tundish design that
there is always an exchangeof liquid between the main
flow (active volume) and the so called dead voiume re-
gions. This assumption maylead to someerror in the
ca]culation.
In the secondapproach (e.g. Ref. 7)), the deadvolume
fraction has been considered to be equal to the area
under the curve from the time 0=2 to co. Both ap-
proaches normally lead to incorrect determination of the
dead volume. Table I gives someestimates of error in
the dead volume calculation by using Eq. (6). In these
calculations, it is assumedthat the deadvolumewithout
any flow through the dead region is lO"/o of the total
volume. With the fractional flow through the dead re-
gions (Qd/Q) of l, 5, and 10 */o, the correct dead volume
calculated from Eq. (4) and the error by using Eq. (6)
are given in the table. It can be seen that the error may
be as high as 90 o/o with the 10 o/. exchangeor cross flow
between the dead and active regions.
6. Calculation of the Plug and Mixed Volumes
After calculation of the dead volume, it remains to
evaluate the plug flow and well-mixed flow volumes in
the tundish. For this, two approaches, based on elther
the use of the combinedmodel or the use of dlspersion
model,1) are suggested here. Thechoice of the approach
should depend upon the shape of the experimental C-
curve. Ideally, a plug flow and a mixed flow in series give
a C-curve as shownin Fig.2. In such a RTDcurve, the
concentration rises vertically which is followed by an
exponential decay of concentration. If the experimental
C-curve has a general shape of this type, a combined
modelmaybe used. Thecombinedmodel approach has
normally been used by manyresearchers, in which the
dimensionless time of the first appearance of dye at the
tundish exit, e~i* is equal to the fractional plug flow
volume. The difference provides the well-mixed volume
fraction. Thus, the following equations maybe used:
Vp
=0 ..........(7)V ~i* '
V~
_1- Vp V ..........(8)V V V ""
The second approach may be used when there is
significant deviation in the shape of the experimental
C-curve from that shown in Fig. 2. In this approach,
ISIJ International, Vol.
the dispersion model for the closed system should be
employed. It is suggested that the variance of the RTD
curve, (T2, should be calculated using Eqs. (9) through
(12). Fromthe C-curve, the dimensionless meanof the
RTDcan be calculated directly by
~=
ocdo
Cde
and the dimensionless variance is given by
(o-e)2cdo
a2=
Cde
.
(9)
. (IO)
If the concentration measurementsare taken at equal
time intervals
~
Ciei
~=
i
..........(ll)
~
Ci """"
and
~, e~Ci
a2= -~2
.....
..........(12)
~
Ci
Equation (13) gives the relationship between the vari-
ance and the dispersion number(DC/UL) for a closed
system. De is the effective longitudinal dispersion, U
is the longitudinal velocity, and L is the length of the
tundish. Thus, the dispersion numberfor a given con-
figuration can be calculated. Thevalue of the dispersion
numberprovides the deviation from the plug flow. Dis-
persion numberis zero for the plug flow and infinity for
the well-mixed flow system. Thus, a higher dispersion
numberindicates moremixed region and a smaller plug
fiow region.
)J
-
)[
-
2
a2
=2
De De ULl -exp (1 3)2UL UL De
7. AnExampleof the DeadVolumeCalculation
Consider a typical RTDexperiment for melt flow in a
tundish, whose result is plotted in Fig. 5. It shows a
typical RTDcurve which extends well beyond the di-
mensionless time of e=2.
This C-curve has been analyzed by the combined
model for the dead volume in the system.
Area under the entire curve,
"
~
CiAe I ..........(14)
0=0
Meanresidence time for the entire curve,
671
36 (1996), No. 6
*
~
Ciei
~=
0=0
=1
.....,
,.........(15)
"
~
Ci
0=0
Meanresidence time upto e=2,
Ciei
e~c= e=0 ..........(16)=0.857
....,.
Ci
e=0
Area under the curve upto e=2,
Q•
=
CiA0=0.9134
.......
..........(17)Q 0=0
FromEq. (4), the dead volume fraction,
Vd I Q' • ec=0217 ..........(18)V Q
In this example, the flow rate through the dead region
is estimated to be about 9o/o, and the calculated dead
volume by neglecting this volumetric flow rate through
the dead region (i.e, from Eq. (6)) is 14.3 o/o. Thecorrect
dead volume is 21.70/0 which is about 520/0 more than
that predicted by Eq. (6). It should be mentioned here
that any changein the tundish configuration such as, use
of different flow control devices, change of total flow
rate, depth of liquid in the tundish, could change both
the value of ~c and the Q.lQ-
8. Conclusions
Acombinedmodel which is commonlyused to char-
acterize the fluid flow in tundishes is discussed. Most
researchers havemadean assumption in the deadvolume
calculation that the volumetric fiow rate through the
dead volume is negligible. This may result in a sig-
nificant error in the calculated value. Others have con-
sidered the dead volume as the area under the C-curve
after two times the meanresidence time. Both approaches
are incorrect and maylead to misleading results. The
correct procedure for the dead volume calculation is
outlined, and an example illustrating the use of the cor-
rect formulation is presented.
Nomenclature
C: Dimensionless concentration
C~** : Maximumconcentration in a C-curve
D. : Effective longitudinal diffusivity in a tundish
L : Length of a tundishQ: Total volumetric flow rate in through a tundish
Q* : Volumetric flow rate through the active region
of a tundish
Qd: Volumetric flow rate through the dead region of
a tundish
t : Meanresidence time (= VIQ)
t~c
: Measuredmeanresidence time upto two times
the f
U: Longitudinal velocity of liquid in a tundish
V: Volumeof liquid in a tundish
C 1996 ISIJ
ISIJ International, Vol. 36 (1 996), No. 6
V.:
Vd:
V*:
Vp:
a2 :
e:
O:
e~c:
e :min
Volumeof active region in a tundish
Volumeof dead region in a tundish
Volumeof mixed flow region in a tundish
Volumeof plug flow region in a tundish
Dimensionless variance of the RTDcurve
Dimensionless time (= time tlT)Dimensionless average meanresidence time
Dimensionless average meanresidence time up
to 0=2
Dimensionless time of first appearanceof tracer
at the tundish exit
l)
2)
3)
4)
5)
6)
7)
REFERENCES
O. Levenspiel: Chemical Reaction Engineering, John Wiley &
Sons, Inc., NewYork, (1972).
M. L. Lowry and Y. Sahai: Steelmaking Conf. Proc.. ISS
Publication, (1989), 71.
I. Suzuki, S. Noguchi, Y. Kashiwakura, T. Horie and M. Saito:
Tundish Metallurgy, Vol, 1, ISS Publication, (1990), 201.
L. J. Heaslip and A. McLean:Continuous Casting, Vol. 1, ISS
Publication, (1983), 93.
J. Knoepkeand J. Mastervich: Steelmaking Conf. Proc., ISS
Publication, (1986), 777.
H. Chen and R. D. Phelke: Steelmaking Conf. Proc., ISS
Publication, (1 994), 695,
L. K. Chiang: Steelmaking Conf. Proc., ISS Publication, (1992),
437.
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