root/trunk/matml/transport/problems/tundish/tundish.tex

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\begin{enumerate}
\item New tundish design\footnote{This problem was inspired by Robert Hyers
    (now on the faculty at the University of Massachusetts at Amherst), and the
    design shown here is informally known as the ``Hyers Tundish''.}

  A tundish is a bathtub-shaped vessel which holds liquid steel between the
  ladle and continuous caster.  It is used to float ceramic inclusion particles
  out of liquid steel, and maintain a constant liquid level and head of
  pressure above the continuous caster, providing a steady flow rate and
  velocity of metal into the caster.

  A new design for tundishes has been proposed which uses vertical plates of
  refractory ceramics to separate the flow into several channels, as shown
  below.
  \begin{center}
    \PSbox{newtun.ps}{5in}{2.5in}
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  Data:
  \begin{itemize}
  \item Steel $\rm\rho=7000\frac{kg}{m^3}$, $\rm\eta=5.2\times10^{-3}\frac{kg}
    {m\cdot s}$; typical ceramic particle $\rm\rho=3700\frac{g}{cm^3}$
  \item Tundish dimensions: $H=1$m, $L=1.5$m, $W_c=0.15$m, $W=1$m
  \item Flow rate: $Q=2\frac{\rm liters}{\rm sec}$ (=0.002$\rm\frac{m^3}{s}$)
  \end{itemize}

  \begin{enumerate}
  \item \label{uav} Using the flow rate $Q$ given above, and assuming that flow
    is distributed equally to each of the four vertical channels, calculate the
    average velocity of molten steel through those channels.

  \item Assuming flow is fully-developed, what is the Reynolds number of the
    metal flow in the channels?  What would the Reynolds
    number be without the vertical plates between the channels (that is, one
    big open box)?  Will the flow likely be laminar in each situation (with and
    without the plates)?

  \item \label{maxu} Treating a channel as a pair of parallel plates with
    fully-developed steady-state laminar flow between them, calculate the
    maximum velocity of molten metal in a channel.

  \item Consider a ceramic particle which enters a channel at the bottom center
    and rises through the molten steel as it is being carried forward by the
    flow of metal in the center of the channel.  Its $y$-direction velocity is
    the maximum velocity of the fluid (since it is in the center of the
    channel), and its $z$-direction velocity is its upward terminal velocity.
    What is the minimum upward terminal velocity this particle can have and
    still reach the top before it gets to the end of the channel?  (If you
    didn't get an answer to part \ref{maxu}, use your value from part \ref{uav}
    as the molten steel flow velocity in the center.)

  \item What is the spherical particle size corresponding to this terminal
    velocity?  (If you assume Stokes flow around the particle, be sure to check
    it!)

  \item Calculate the entrance length for flow in these channels.  What does
    this tell you about the validity of our assumptions above?

  \item Considering only the faces of the plates (not the ends), and assuming
    uniform velocity at the entrances of the channels, estimate the drag force
    on a single plate.
  \end{enumerate}
\end{enumerate}
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