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

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\begin{document}
\begin{enumerate}
\item Crystal-free zone in a glass-ceramic dish

  \paragraph{Background} Some dishes, such as the Corelle line by Corning, are
  made of a high-tech glass-ceramic material in which partial crystallization
  of the interior puts the surface in compression, resulting in high bending
  strength for a thin heat- and chemical-resistant dish.  The glassy phase is
  stable above a temperature $T_c$, and the crystalline phase below that
  temperature.

  During cooling by immersion in a fluid at temperature $T_{fl}$, a surface
  layer cools quickly enough to avoid any crystallization.  The criterion for
  this is roughly that the temperature must fall below $T_{nose}$ before time
  reaches $t_{nose}$, as indicated in the T-T-T
  (time-temperature-transformation) diagram below.

  \begin{center}
    \pdfimageresolution 200
    $\ $\pdfximage{corelle.png}\pdfrefximage\pdflastximage$\ $

    Glass-ceramic ``dish'' (not to scale)

    $\ $\pdfximage{t-t-t.png}\pdfrefximage\pdflastximage$\ $

    T-T-T (time-temperature-transformation) diagram
  \end{center}

  Properties of the glassy phase and process parameters:
  \begin{itemize}
  \item Thermal conductivity: $k=0.4\frac{\rm W}{\rm m\cdot K}$
  \item Density: $\rm\rho=2400\frac{kg}{m^3}$
  \item Heat capacity: $c_p=900\frac{\rm J}{\rm kg\cdot K}$
  \item Heat transfer coefficient: $h=3000\frac{\rm W}{\rm m^2\cdot K}$
  \item Dish thickness: $L=0.01$m (a thick dish)
  \item Temperatures: $T_{init}=1000$K, $T_c=900$K, $T_{nose}=720$K,
    $T_{fl}=300$K; time $t_{nose}=4$ seconds
  \end{itemize}

  \begin{enumerate}
  \item Sketch the temperature distribution during cooling, ignoring phase
    transformations.
  \item \label{expression} Give an expression for the temperature as a function
    of distance into the plate which is approximately valid at short time
    scales.
  \item \label{crystalfree} Using your expression from part \ref{expression},
    estimate the distance into the plate where temperature is $T_{nose}$ when
    time is $t_{nose}$.
  \item Determine whether your expression in part \ref{expression} (and thus
    your calculation in part \ref{crystalfree}) is valid at time $t_{nose}$.
  \end{enumerate}
\end{enumerate}
\end{document}