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

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\begin{enumerate}
\item Dimensional analysis: catalytic combustion of carbon monoxide

  CuO is a good catalyst for the reaction
  $$\rm 2CO(g)+O_2(g)\Rightarrow 2CO_2(g)$$
  CuO is coated on the internal surfaces of porous Al$_2$O$_3$ spheres of
  radius $R$, so the gases containing CO and O$_2$ can diffuse in through the
  pores with diffusivity $D$ and react on the internal CuO surfaces.  (The
  Al$_2$O$_3$ is used as a substrate because of its high strength and stability
  at the high temperatures reached due to combustion; porous CuO would sinter
  into dense solid spheres at such temperatures.)

  \begin{center}
    \input{convandsphere}
  \end{center}

  Because the pores are a lot smaller than the sphere, you can treat this as a
  homogeneous chemical reaction throughout the porous solid.  Assume for this
  problem that O$_2$ is the limiting reagent, and the rate of reaction is
  proportional to its concentration with ``homogeneous'' rate constant $k$, so
  $G=-kC_{\rm O_2}$.  The concentration of oxygen in the pores just inside the
  surface of the sphere is approximately equal to that in the gas outside, so
  that is our outer surface boundary condition.

  If the concentration of oxygen throughout each sphere is very uniform, then
  the reaction will be occurring on all of the inner pore surfaces within each
  sphere, so the CuO will be well-utilized.  On the other hand, if we are
  flowing CO gas through this bunch of spheres, the larger the spheres, the
  less resistance to flow there will be.  So we want the spheres to be as large
  as possible without being so large that there is no reaction at their center.

  \begin{enumerate}
  \item\label{pi1} Write an expression for a parameter which describes the
    uniformity of oxygen concentration, in terms of the other problem
    parameters.

  \item Identify the number of parameters, and the number of independent units,
    and use the Buckingham pi theorem to determine the number of independent
    dimensionless parameters in the problem.

  \item Write a new expression for the dimensionless uniformity parameter in
    part \ref{pi1} in terms of the other dimensionless parameter(s).

  \item If you double the sphere radius, how much does $k$ have to change in
    order to yield the same uniformity?  (The reaction rate coefficient $k$
    tends to be a much stronger function of temperature than diffusivity, so
    changing the temperature will change $D$ a small amount and $k$ a large
    amount.)

  \item Draw a sketch (or sketches) showing how you think that uniformity
    parameter will vary as a function of the other parameter(s), making sure
    your sketch captures the transition from strongly non-uniform to roughly
    uniform concentration in a sphere.

  \item Sketch the concentration of oxygen in a sphere as a function of radius
    for large and small values of the other dimensionless parameter(s) of which
    uniformity is a function.  If there is more than one such parameter, make
    sketches for all combinations of large and small values of each
    parameter.
  \end{enumerate}
\end{enumerate}
\end{document}