root/trunk/matml/transport/problems/ladle2/ladle2-solution.tex

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\begin{enumerate}
\item Ladle Metallurgy II: Natural Convection

  \begin{enumerate}
  \item The Prandtl number calculation should have preceeded any boundary layer
    sketching, as it would have shown that $\rm Pr<1$ so the velocity and
    thermal boundary layers are about the same size.  In any case, your sketch
    should have looked something like:
    \begin{center}
      \input{/home/hazelsct/3.185/figs/ladleflow}
    \end{center}

  \item First the Grashof number, using the liquid metal height as the length
    scale:
    $${\rm Gr}_x = \frac{g\beta\Delta Tx^3}{\nu^2} =
    \frac{\rm9.8\frac{m}{s^2}\cdot 10^{-5}K^{-1}\cdot 100K\cdot (2m)^3\cdot
      (7000\frac{kg}{m^3})^2}{\rm(0.005\frac{kg}{m\cdot s})^2} =
    1.54\times10^{11}.$$
    The Prandtl number is:
    $${\rm Pr} = \frac{\nu}{\alpha} = \frac{\mu c_p}{k} =
    \frac{\rm0.005\frac{kg}{m\cdot s}\cdot 700\frac{J}{kg\cdot K}}
    {\rm15\frac{W}{m\cdot K}} = 0.23.$$
    The Grashof-Prandtl product is then $3.5\times10^{10}$, which is above
    $10^9$ so flow is turbulent.

  \item The GrPr product puts this solidly in the $10^9-10^{12}$ r\'{e}gime
    where we can use the turbulent flow Nusselt number correlation:
    $${\rm Nu}_L = 0.245{\rm Gr}_L^{2/5}
    {\rm Pr^{7/15}(1+0.494Pr^{2/3})^{-2/5}} =
    0.245\cdot 3.0\times10^4\cdot 0.443 = 3237,$$
    $$h_L = \frac{{\rm Nu}_Lk}{L} =
    \frac{\rm3237\cdot 15\frac{W}{m\cdot K}}{\rm2m} =
    24300\frac{\rm W}{\rm m^2\cdot K}.$$
    This is a factor of sixty larger than a typical correlation for air, and
    significantly larger than one would calculate for water or other
    non-conducting fluids.  Liquid metals are like that.

  \item With the boundary layer flowing down the sides of the ladle and to the
    bottom, when the gate is opened, the cooler metal from the boundary layers
    will exit first, followed by the hotter metal from the interior.  That the
    flow rate down the boundary layers is similar to the flow rate through the
    nozzle means that this effect is quite significant, and as a result,
    thermal management in the downstream processes (tundish and continuous
    caster) can be very difficult.
  \end{enumerate}
\end{enumerate}
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