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Momentum equation: t (rui)1 xj (rui uj)52P xi 1 xj Fmeff1uiuj 1ujxi2Grg [6] Mass transfer equation: C t 1 u C x 1 v C y 1 w C z 5 Deff 1 2C x2 1 2C y2 1 2C z22 [7] Equation that describes the turbulent kinetic energy: t (rk) 1 xi 1rui k 2 meff sk k xi2 5 G 2 r« [8] Equation that describes the dissipation rate of turbulence energy:(b) Fig. 3—Geometric dimensions for the TI&D tundish arrangement (m): (a) t (r«) 1 xi 1rui« 2 meff ss k xi2 5 1 k (C1 G 2 C2 r« 2) [9]baffles and (b) turbulence inhibitor. where G 5 mt ui xi 1 ui xi 1 ui xj2 [10] Effective viscosity is the sum of laminar viscosity and turbu- lent viscosity:(a) meff 5 ml 1 mt [11] Turbulent viscosity is related to the turbulent energy and dissipation rate of turbulent energy by mt 5 CDrk2/« [12] The values for the constants in this k-« model C1, C2, CD ,(b) sk , and s« are 1.43, 1.92, 0.09, 1.00, and 1.30, respectively; Fig. 4—Schematics of tundish arrangements employed in this study (m): these values were taken from Spalding.[23] (a) BWIP arrangement and (b) TI&D arrangement. In the mass transfer equations, Deff 5 Dm 1 Dt is the effective mass transfer diffusivity, which is the summation of molecular and turbulent diffusivities, respectively. The Table I. Basic Parameters for the Water Model turbulent diffusivity Dt is related to the turbulent viscosity mt byParameter Model st 5 mt /(rDt) [13]Water volume for 27 ton 0.1475 m3 Water model depth at 27 ton 0.253 m Since turbulent flow generally carries mass over an equiva- Nozzle penetration 0.065 m lent Prandtl mixing length,[24] this coefficient was assumedWater flow rate 0.02 m3 min21 to equal one. Then, from Eq. [13], we obtain METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 31B, DECEMBER 2000—1507 Dt 5 mt /r [14] explained previously, until reaching the steady state of the fluid flow. The velocity field calculated was then employed to solve Eq. [7] for the tracer concentration under unsteady-B. Boundary Conditions state conditions. Here, it is implicitly assumed that the pres- Nonslipping conditions were applied as boundary condi- ence of the tracer does not affect the water density to an tions to all solid surfaces of the tundish including baffles, appreciable extent. The initial condition to solve Eq. [7] is dams, and interior walls of the tundish. Near any solid sur- stated as follows: face, there is a very thin laminar sublayer. Between the laminar sublayer and the turbulent core, there is also a buffer at t 5 0 and x0, y0, z0 C 5 M Vnozzle [23] sublayer, which is in a state between laminar and turbulent flow. Consequently, for nodes near a solid wall, the so- where M is the total mass of the tracer and V is the volume called wall functions are required to calculate the values of of the water column in the ladle nozzle from the injection a variable, since in those places, very steep gradients occur. point to the nozzle tip assuming a perfect mixing. The termsIf a node in the 3-D domain is in the laminar sublayer, a x0, y0, and z0 are the coordinates of the nozzle tip in the 3-linear relationship between the wall stress and the velocity D domain.gradient is assumed: tw 5 m Dv Dn [15] D. Numerical Solution The continuity equation, the momentum and mass transferIf the node is beyond the laminar sublayer, the logarithmic equations, and the initial and all boundary conditions werelaw is applied in order to calculate the wall shear stress: rewritten in a finite difference scheme using the 3-D mesh shown in Figure 5. A dense mesh was employed near thevp v* 5 1 kv ln (Ey+) [16] tundish bottom and near the tundish walls just to avoid inconsistencies when the wall function is applied as a bound- where ary condition. A dense mesh was also employed in the outlet- longitudinal planes as well as in the proximity of the stopper v* 5 !twr [17] rods. The total number of cells in this 3-D domain counted 80,000, which ensures reliable calculations of fluid flow and and mass transfer. The numerical algorithm used to solve those equations isy+ 5 rv* Dnp /m [18] known as PISO[26,27] (pressure implicit with splitting opera- where kv is the Von Karman’s constant (0.42), E is an empiri- tions). This is a time marching procedure: a predictor is cal constant (9.81) taken from Reference 25, and vp is the followed by one or more corrector steps for each time-step, velocity of the fluid near the wall. using a noniterative splitting of operations of discretized The boundary conditions for k and « in this sublayer were continuity, momentum, kinetic energy, dissipation rate of calculated with the previous knowledge of y+ through kinetic energy, and pressure equations. In this way, the veloc- ity fields at the end of each step are close approximations y+ 5 rk1/2p C1/4m Dnp m [19] of the turbulent change equations. A criterion for convergence was established when the sum where kp is the turbulent kinetic energy at the near-wall grid of all residuals for the variables was less than 1026. Velocity point p and Dnp is the distance of point p to the wall. Equation fields at steady state were first calculated and later they were [19] is an empirical fit of turbulent flow data for y+ between employed to solve the mass transfer equation. values of 10 to 20. In this study, Eq. [15] is used when y+ The mathematical model was run in a workstation Silicon is smaller than 12, and Eq. [19] is used when y+ is larger Graphics (Silicon Graphics S.A. de C.V., Mexico) Model O2 than this value. with R 10000 processors at the Laboratory for Simulation of On the top free surface of the bath and in symmetry Materials Processing of IPN-ESIQIE, Department of Metallurgy planes, the fluxes of momentum and mass, as well as the and Materials Engineering. The computer results were stored gradients of the turbulent kinetic energy and the dissipation in magnetic tapes to be arranged in a special format for further rate of kinetic energy, were set equal to zero. analysis by feeding them into commercial plotting software At the entry jet, the flow profile was assumed to be flat known as Tecplot (Adaptive Research, Alhambra, CA). and calculated by Uin 5 Q/Anozzle [20] IV. RESULTS AND DISCUSSION The inlet values for k and « were calculated with the follow- A. Water Modeling Experimentsing equations: Figures 6(a) through (c) show the experimental RTDkin 5 0.01U 2in [21] curves for the BT and the tundishes with the BWIP and TI&D arrangements, respectively, showing the interior and«in 5 2k3/2in /Dnozzle [22] exterior outlet signals. The BT shows an unequal distribution of the tracer to both outlets and the minimum residence isC. Initial Conditions smaller in the interior outlet than in the exterior one. The concentration peak is higher in the exterior outlet and theEquations [5], [6], [8], and [9] were solved together with their boundary conditions using the auxiliary expressions, difference between maximum concentration time and the 1508—VOLUME 31B, DECEMBER 2000 METALLURGICAL AND MATERIALS TRANSACTIONS B (a) (a) (b) (b) Fig. 8—Experimental RTD curves for two distances between the dam and the entry nozzle using the TI&D arrangement: (a) 0.23 m and (b) 0.24 m. Table II. Flow Characteristics Results from the Experimental Total RTD Curves Arrangement VDead VPlug VMixed tcalc su2 D/UL BT 0.0755 0.2071 0.7174 304.18 0.2171 0.123(c) BWIP 0.1054 0.2761 0.6185 362.06 0.2427 0.141 TI&D 0.0693 0.3461 0.5846 318.44 0.1961 0.110Fig. 6—Experimental RTD curves for the tundish arrangements: (a) BT, (b) BWIP, and (c) TI&D. Fig. 9—Experimental total RTD curves. minimum residence time is smaller for this outlet. It is appar- ent from Figure 6(a) that the interior outlet exhibits larger dispersion data for the concentration of the tracer with some characteristics of bypass flow. Using the BWIP arrangement, the RTD curves exhibitFig. 7—Schematic representation of the flow behavior using a step in the upper side of the dam. considerable improvements,