Fluid Flow and Mechanisms of Momentum Transfer in a Six-Strand Tundish

Abstract

Flow in a six-strand billet tundish, using turbulence inhibitors (TIs), was characterized using inputs of a pulsed tracer and mathematical simulations. It was found that to control turbulence attaining high fluid fractions under plug flow patterns, the key parameter for designing TIs is the dissipation rate of kinetic energy. TI designs that induce steep dissipation gradients are less efficient as flow controllers than those designs that yield more prolonged dissipation gradients from the inhibitor bottom to the bulk flow. A direct relationship between the dissipation of kinetic energy and the linear acceleration of the smallest turbulent eddies in the flow was established through dimensional analysis. The inhibitor with the highest linear accelerations of eddies in the viscous sublayer at the Kolmogorov scale, for a given liquid flow rate, yields the better flow control.

Appendix A

From a dimensional analysis of turbulent flows it is possible to find a link between velocities of eddies with length scales λ, assumed to be longer than the length scale of the viscous sublayer of the boundary layer or the Kolmogorov’s scale η, but smaller than the integral scale of turbulence:

$$ \nu_{\lambda } = \left( {{\frac{\varepsilon }{\lambda }}} \right)^{{\frac{1}{3}}} $$

(A1)

The dissipation rate of kinetic energy can be related with integral scales of velocity and length according to:

$$ \varepsilon \approx {\frac{{(\Updelta U)^{3} }}{L}} $$

(A2)

where ΔU is the change in the average velocity over a distance equal to a large size scale in the flow, whose magnitude can be as large as the geometric dimensions, L, of the inhibitor (for instance its height or even the bath height). Using Eq. [A2] in Eq. [A1] leads to:

$$ \nu_{\lambda } = \Updelta U\left( {{\frac{\lambda }{L}}} \right)^{{\frac{1}{3}}} $$

(A3)

Thus, the velocities of eddies with a scale λ are smaller than the velocity of the main flow by a factor of $\left( {{\frac{\lambda }{L}}} \right)^{{\frac{1}{3}}}$. The reductions of velocity and scale are matched by corresponding Reynolds number as follows:

$$ \text{Re}_{\lambda } = {\frac{{\nu_{\lambda }}}{v}} = {\frac{{\Updelta U\left({{\frac{\lambda }{L}}} \right)^{{\frac{1}{3}}} \lambda }}{v}} = {\frac{\Updelta UL}{v}}\,{\frac{{\left( {{\frac{\lambda }{L}}} \right)^{{\frac{1}{3}}} \lambda }}{L}} = \text{Re}_{\text{L}} \left( {{\frac{\lambda }{L}}} \right)^{{\frac{4}{3}}} $$

(A4)

where $\text{Re}_{\text{L}} = \frac{k^{2}}{\nu \varepsilon}\approx{\frac{\Updelta UL}{v}}$ is the Reynolds number corresponding to the integral length scale and can be calculated directly from CFD simulations. On the other hand, Kolmogorov’s scale is the smallest length in the flow, valid in the subviscous layer, and is given by:

$$ \eta = {\frac{L}{{\text{Re}_{L}^{{\frac{4}{3}}} }}} = \left( {{\frac{{v^{3} }}{\varepsilon }}} \right)^{{\frac{1}{4}}} $$

(A5)

Since λ > η, the characteristic period of turbulent-fluctuating motion of eddies with a size λ will be:

$$ T_{\lambda } = \left( {{\frac{{\lambda^{2} }}{\varepsilon }}} \right)^{{\frac{1}{3}}} \sim {\frac{\lambda }{\nu \lambda }} $$

(A6)

The acceleration of these eddies will be:

$$ a_{\lambda } = {\frac{{d\nu_{\lambda } }}{{dT_{\lambda } }}} = {\frac{{\nu_{\lambda } }}{\lambda }}\sim {\frac{\lambda }{{T_{\lambda }^{2} }}}\sim \left( {{\frac{{\varepsilon^{{\frac{2}{3}}} }}{{\lambda^{{\frac{1}{3}}} }}}} \right) $$

(A7)

Equation [A7] is useful to estimate the acceleration of eddies with sizes λ, which can be of the order of Taylor’s scale[20] or even close to the integral scale of the flow, L. However, someone would like to know the acceleration of the smallest scales eddies from well defined flow variables. In other words, if it is possible to express the acceleration of the smallest scales, then we will have a quantitative idea of the motion of the smallest eddies responsible for the dissipation of kinetic energy. In the viscous sublayer, the local Reynolds number Re _η  = 1 and then the velocity scale $\nu_{\eta } = {\frac{\nu }{\eta }}$ and the corresponding period of fluctuating motion for small eddies is:

$$ T_{\eta } = {\frac{\eta }{{\nu_{\eta } }}}\sim {\frac{{\eta^{2} }}{\nu }}\sim \left( {{\frac{\eta }{\varepsilon }}} \right)^{{\frac{1}{2}}} $$

(A8)

Therefore, the accelerations of these eddies will be:

$$ a_{\eta } = {\frac{{\partial \nu_{\eta } }}{\partial t}}\sim {\frac{{\nu_{\eta } }}{{T_{\eta } }}}\sim {\frac{{\nu_{\eta }^{2} }}{\eta }}\sim {\frac{{\nu_{\eta }^{3} }}{\nu }}\sim \frac{\varepsilon^{\frac{3}{4}}}{\nu^{\frac{1}{4}}} $$

(A9)

According to Levich this proportionality becomes into an equality through a constant[21]:

$$ a_{\eta } = \sqrt 3 \frac{\varepsilon^{\frac{3}{4}}}{\nu^{\frac{1}{4}}} $$

(A10)

Equation [A10] has the advantage over Eq. [A7] because it does not depend on any length scale to be defined; instead it depends on flow variables which are directly accessible through CFD simulations like k − ε, and ν is a fluid property.