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

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\begin{enumerate}
\item Heat conduction and diffusion in alloy casting

  In the die-casting of a relatively thick roughly plate-shaped metal alloy
  part, the liquid metal alloy cools and reaches a roughy uniform distribution
  at the melting point at time $t=0$, then solidifies with a plane front from
  the sides.  The rate of solidification is limited by two types of heat
  transfer: conduction through the mold and to the environment can be
  represented by $q_x = h(T_s-T_{env})$ ($T_s$ is the outer surface metal
  temperature), and conduction through the already-solidified metal of
  thickness $Y$.

  \begin{enumerate}
  \item Sketch the temperature profile ($T$ vs. $x$, where $x$ is the distance
    from one side of the mold) across the solid metal shell and liquid metal
    interior for short times (small Bi) and long times (large Bi).

  \item Derive a simple expression for the growth rate $dY/dt$ which is valid
    for short times (small Bi).

  \item Derive a simple expression for the growth rate $dY/dt$ which is valid
    for long times.

  \item Sketch the relationship between solidified shell thickness $Y$ and time
    $t$, showing the transition from convection- to conduction-limited
    growth.
  \end{enumerate}
  
  As the metal solidifies, the lower solubility of the alloying element
  (solute) results in its ``rejection'' into the liquid.  At steady-state, the
  concentration of the solute in the solid is that of the liquid $C_L$, but in
  the liquid at the liquid-solid interface, the concentration is much higher,
  say $5C_L$.

  \begin{enumerate} \setcounter{enumii}{4}
  \item Assuming that solidification front velocity $U$ is constant, that
    diffusion is slow enough that the diffusion boundary layer is much smaller
    than the part thickness, and that the effect of fluid flow is negligible,
    derive the steady-state general equation for concentration as a function of
    distance into the liquid from the moving interface $x'$ (that is, in the
    frame of reference of the moving interface).

  \item Fit this general solution to the boundary conditions in the liquid at
    $x'=0$ and $x'=\infty$ to give the particular concentration profile here.

  \item Approximately how thick is the solute-enriched layer in the liquid (an
    expression, not a number)?
  \end{enumerate}
\end{enumerate}
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