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Momentum equation:
(rui)1 ­
(rui uj)52­P
Fmeff1­ui­uj 1­uj­xi2Grg [6]
Mass transfer equation:
1 u
1 v
1 w
5 Deff 1­
­z22 [7]
Equation that describes the turbulent kinetic energy:
(rk) 1 ­
­xi 1rui k 2
­xi2 5 G 2 r« [8]
Equation that describes the dissipation rate of turbulence
Fig. 3\u2014Geometric dimensions for the TI&D tundish arrangement (m): (a) ­
(r«) 1 ­
­xi 1rui« 2
­xi2 5
k (C1 G 2 C2 r«
2) [9]baffles and (b) turbulence inhibitor.
G 5 mt
­xi 1
­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\u2014Schematics 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
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
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
[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
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\u2019s 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
[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
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
Fig. 8\u2014Experimental 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\u2014Experimental RTD curves for the tundish arrangements: (a) BT,
(b) BWIP, and (c) TI&D.
Fig. 9\u2014Experimental 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\u2014Schematic representation of the flow behavior using a step in the
upper side of the dam. considerable improvements,