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

\documentclass{article}
\usepackage{fullpage,lmodern}
\usepackage[T1]{fontenc}
\begin{document}
\begin{enumerate}
\item Freezing Lake

  A shallow (10 cm deep) body of water initially at 10$^\circ$C is exposed to
  air at -10$^\circ$C, and begins to freeze from the top.  Here we will use the
  finite difference method to calculate the temperature profile across the
  liquid water and ice, and model the growth of the ice as well as the heat
  conduction.  We will neglect buoyancy instabilities in the water and treat it
  as stagnant, and also neglect the small change in overall thickness due to
  the lower density of ice.

  \noindent Properties:
  \begin{center}
    \begin{tabular}{l|c|c|}
      Material                             &  Liq. water & Sol. water \\ \hline
      Conductivity $k$, $\rm\frac{W}{m\cdot K}$     & 0.56 & 2.3 \\
      Heat capacity $c_p$, $\rm\frac{J}{kg\cdot K}$ & 4200 & 2100 \\
      Density $\rho$, $\rm\frac{kg}{m^3}$           & 1000 & 920 \\
      Heat of fusion $\Delta H_f$, $\rm\frac{J}{kg}$ &
      \multicolumn{2}{|c|}{334,000} \\ \hline
    \end{tabular}
  \end{center}
  % Water k=~0.00135 from my CRC 67th Ed. pp. E-5, dense ice k at
  % http://naca.larc.nasa.gov/reports/1954/naca-tn-3143/index.cgi?thumbnail4#start
  % given as ~16 BTU-in/(hr/ft^2-F) = 2.3 W/m-K, approx. ice cp at
  % http://www.cgd.ucar.edu/csm/models/cpl/cpl4.0/doc9.html

  \begin{enumerate}
  \item Write the difference equation corresponding to the following
    differential equation at an internal node using the forward Euler
    algorithm:
    $$\frac{\partial H}{\partial t} -
    \frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right) =0$$
    You can ``close'' this equation in your finite difference calculation by
    creating separate arrays to hold enthalpy and temperature, so $T_{i,n}$ is
    calculated from $H_{i,n}$, and $H_{i,n+1}$ is calculated from $T_{i,n}$ and
    its spatial neighbors.

  \item At the bottom of the lake ($x=10$cm), we will assume that the lake bed
    is a perfect insulator, so the boundary condition is equivalent to a {\em
      symmetry plane}.  At the top surface ($x=0$), the heat flux is given by a
    heat transfer coefficient: $q_x = h(T_{\rm air}-T_{\rm water})$.

    Write the difference equations for the top and bottom finite difference
    nodes in your system.

  \item Calculate the thermal diffusivities of liquid and solid water.  Which
    one will determine the largest stable timestep size using the forward Euler
    ({\em i.e.} explicit) algorithm?

  \item Estimate the following three timescales:
    \begin{enumerate}
    \item Time required to reach steady-state by conduction through 10 cm of
      liquid water.
    \item \label{condlimit} Time required for {\em conduction}-limited freezing
      of the whole lake, assuming ice is at the air temperature at the top
      surface and temperature profile across the ice is linear.
    \item \label{convlimit} Time required for {\em convection}-limited freezing
      of the whole lake, assuming the ice is always at its melting point.
    \end{enumerate}

    For \ref{condlimit} and \ref{convlimit}, you can model the freezing process
    by setting the heat flux required to freeze at a certain rate equal to the
    heat flux due to conduction or convection:
    $$\rho\Delta H_f\frac{dX}{dt} = k\frac{\Delta T}{X} {\rm or} = h\Delta T$$
    and solve the resulting differential equation to determine thickness frozen
    $X$ vs. time $t$.

  \item Using at least ten nodes in the $x$-direction, perform a finite
    difference calculation to predict the temperature profiles throughout the
    freezing process with a top heat transfer coefficient of
    10$\rm\frac{W}{m^2\cdot K}$.  Provide plots at the times when the top of
    the lake just starts to freeze, when the lake is half frozen, and when the
    lake has just finished freezing.\footnote{This calculation can be performed
      in a spreadsheet, but due to the large number of timesteps required and
      the computational inefficiency of spreadsheets, it may take a long time
      to run; a C, FORTRAN or Matlab implementation will run several times
      faster.}
  \end{enumerate}
\end{enumerate}
\end{document}
%%% Local Variables:
%%% TeX-PDF-mode: t
%%% End: