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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