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

\documentclass{article}
\usepackage{pstricks}
\usepackage{fullpage}
\begin{document}
\begin{enumerate}
\item Freezing by radiation and convection

  Castings with large open tops often ``freeze off'' by radiation and
  convection, forming a solid shell on top and trapping the liquid beneath it.
  (Because the liquid shrinks during solidification, this results in a large
  shrinkage cavity.)  Here you will analyze the rate of solidification downward
  from the top surface in an low-carbon steel ingot casting due to these
  factors.

  \begin{center}
    \input{ingotcast}
  \end{center}

  Assume that the temperature is uniform, and for part \ref{qtotal}, the
  environment around the casting is gray along with the solid shell.

  Iron data:
  \begin{itemize}
  \item Electrical conductivity near melting point: $\rm\sigma=5\times10^{5}
    (\Omega\cdot m)^{-1}$.
  \item Wiedmann-Franz constant: $L=2.45\times10^{-8} \frac{\rm W\Omega}{\rm
      K^2}$.
  \item Density: $\rm\rho=7500\frac{kg}{m^3}$.
  \item Heat capacity: $c_p=500\frac{\rm J}{kg\cdot K}$
  \item Melting point: $T_m=1800{\rm K}$.
  \item Heat of fusion: $\Delta H_f=2.67\times10^5\frac{J}{kg}$
  \item Radiative emissivity: $\epsilon=0.6$
  \item Radiation constant: $\rm\sigma=5.67\times10^{-8}\frac{W}{m^2K^4}$.
  \item Heat transfer coefficient to air: $h=100\frac{\rm W}{\rm m^2\cdot K}$.
  \end{itemize}

  \begin{enumerate}
  \item Estimate the thermal conductivity of iron near its melting point.
  \item \label{qtotal} Write an expression for the total radiative and
    convective heat flux from the top surface of the solidifying metal shell to
    the surrounding environment.
  \item \label{htotal} Assuming the environment is much colder than the shell
    and absorbs all radiation ({\em i.e.} is ``black''), calculate a total
    ``heat transfer coefficient'' which is the ratio between heat flux and
    absolute temperature.
  \item \label{unifthick} Use your heat transfer coefficient from part
    \ref{htotal} to estimate the thickness of solid metal $Y$ at which
    temperature can no longer be considered uniform (where the Biot number
    reaches 0.1).
  \item Estimate the rate of growth of the solid while solidification rate is
    limited by radiation/convection from the top surface (that is, while solid
    temperature can be considered uniform).  How long does it take to reach the
    thickness calculated in part \ref{unifthick}?
  \item Set up the equation for solidification rate limited by both
    radiation/convection from the top and also (quasi-steady-state) conduction
    through the solid.
  \end{enumerate}
\end{enumerate}
\end{document}