FLOW vp/V e=> Vm ' Vp/V o\u2022-> (b) fLllIPr"~Ul'~~Us_rEc INPUT MIXEDFLOW o e--~ Vm c PLUGFLOW o e\u2022-~ 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)\u2022 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.