Sahai&Emi
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Sahai&Emi

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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.

C1996 ISIJ 672