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

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\begin{enumerate}
\item Water etching of magnesium oxide ceramic MEMS parts

  It has been proposed that some MEMS (Micro Electro-Mechanical Systems) device
  parts for high temperature applications could be made out of magnesium oxide
  by water etching, due to the slight solubility of magnesium oxide in water.
  The etching reaction is:
  $$\rm MgO + H_2O\Rightarrow Mg^{2+} + 2 OH^-$$

  A magnesium oxide plate 5 cm (0.05 m) long partially covered by a polymer
  etching mask is placed in a water tank near an impeller which generates a
  free stream water velocity of 1 m/s past the plate.

  \begin{center}
    $\ $\pdfximage{mgetch.png}\pdfrefximage\pdflastximage$\ $
  \end{center}

  Data:
  \begin{itemize}
  \item Mg$^{2+}$ diffusivity in water: $D=10^{-5}\frac{\rm cm^2}{\rm s}$
    ($\rm =10^{-9}\frac{m^2}{s}$).
  \item Mg$^{2+}$ solubility in this particular solution:
    $C_s=\rm10^{-2}\frac{mol}{cm^3}$.
  \end{itemize}

  \begin{enumerate}
  \item \label{magsketch} Sketch the shapes of the velocity and diffusion
    boundary layers above the plate on one sketch, clearly indicating which of
    the two boundary layers will be larger.

  \item If the magnesium oxide dissolution is limited by mass transfer of
    Mg$^{2+}$ ions away from the interface, will the dissolution rate be
    uniform across the plate?  If not, indicate on your sketch from part
    \ref{magsketch} where on the plate you expect to see the highest and lowest
    dissolution rates.

  \item \label{magdims} Calculate the dimensionless numbers on which the
    Nusselt number depends using the whole length of the plate as your
    lengthscale.

  \item \label{magnuss} Calculate the {\em local} Nusselt number Nu$_x$ at the
    back end of the plate ($x=L$) based on your answers to part \ref{magdims}.

  \item Calculate the magnesium oxide dissolution rate at the back end of the
    plate.  Is this the largest or smallest dissolution rate you expect to see
    over the plate?

  \item So you do the experiment, and the magnesium oxide dissolution is
    uniform across the whole plate!  Cool!  Furthermore, the dissolution rate
    is just one tenth of the minimum rate which your transport analysis had
    predicted.  Not so cool---but for high-value-added applications such as
    this, quality is much more important than manufacturing cost so that's not
    a problem.

    How could this have happened?  (Hint: what assumption in this analysis
    might have been violated?)
  \end{enumerate}
\end{enumerate}
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