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

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\newcommand{\PSbox}[3]{\mbox{\rule{0in}{#3}\special{psfile=#1}\hspace{#2}}}
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\begin{enumerate}
\item Dimensional analysis: catalytic combustion of carbon monoxide

  \begin{enumerate}
  \item\label{pi1} The uniformity can be described by the ratio of oxygen
    concentration at the center of a sphere to its concentration just inside
    the surface.  It should depend on the reaction rate coefficient $k$, sphere
    radius $R$, and diffusivity $D$:
    $$\frac{C_{\rm center}}{C_{\rm surf}} = f(k, R, D)$$

  \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. (10)

    There are four parameters, with units as follows:
    \begin{center}
      \begin{tabular}[h]{l|cccc|}
        Parameter & $\frac{C_{\rm center}}{C_{\rm surf}}$ & $k$ & $R$ & $D$ \\
        \hline
        Units & (dimensionless) & s$^{-1}$ & cm & $\rm\frac{cm^2}{sec}$ \\
        \hline
      \end{tabular}
    \end{center}
    The base units are cm and sec, so there are two independent units.  The
    Buckingham pi theorem says that these four parameters with two independent
    units gives two dimensionless parameters.

  \item Let's choose $R$ and $D$ to eliminate in order to make the others
    dimensionless.  The concentration ratio which we'll call $\pi_C$ is already
    dimensionless, so we just need to make the dimensionless $\pi_k$ by
    finding $a$ and $b$ such that $kD^aR^b$ is dimensionless:
    \begin{center}
      \begin{tabular}[h]{l|rrr|}
        Base units             & sec & cm \\ \hline
        $k$                    & -1  &  0 \\
        $D^{-1}$               &  1  & -2 \\
        $R^2$                  &  0  &  2 \\ \hline
        Total                  &  0  &  0 \\ \hline
      \end{tabular}
    \end{center}
    So the dimensionless $\pi_k$ is $\frac{kR^2}{D}$.  This is the same
    dimensionlesss number as we derived in class for 1-D reaction/diffusion in
    carbon-carbon composite fabrication.

    Therefore, $\pi_C = f(\pi_k)$, $\frac{C_{\rm center}}{C_{\rm surf}} =
    f\left(\frac{kR^2}{D}\right)$.

  \item Since uniformity is a function of $\pi_k$, we want to keep that
    constant through this change.  We're told that $D$ doesn't change much, so
    if $R$ doubles, $k$ must fall by a factor of four in order to give the same
    $\pi_k$, and the same $\pi_C$.

  \item We know that for slow reaction, small radius and fast diffusion, the
    concentration should be uniform, so when $\pi_k$ is zero, $\pi_C$ should
    be one.

    As $\pi_k$ increases, the reaction consumes more oxygen, and it diffuses
    out more slowly, so there is less oxygen in the center, until $\pi_C$
    eventually falls to zero.

    An analytical solution to this would use Bessel functions, and is beyond
    the scope of 3.185.  But qualitatively, it should behave something like
    the equation derived for uniformity due to reaction/diffusion in a flat
    plate, which went like:
    $$\frac{C_{\rm center}}{C_{\rm surf}} =
    \frac{\cosh(0)}{\cosh\left(\frac{L}{2}\sqrt{\frac{k}{D}}\right)} =
    \frac{1}{\cosh\left(\frac{\sqrt{\pi_k}}{2}\right)}$$
    For small $x$, $\cosh(x)$ behaves like $1+\frac{1}{2}x^2$, so for small
    $\pi_k$, this looks like:
    $$\pi_C \simeq \frac{1}{1+\frac{1}{2}\left(\frac{1}{2}\sqrt{\pi_k}\right)}
    \simeq \frac{1}{1+a\pi_k}$$
    where $a$ is a constant.  This means that there will be a non-zero slope
    at $\pi_k=0$, and the curve will look something like:

    \begin{center}
      \PSbox{picpik.ps}{260pt}{135pt}
    \end{center}

    \item For small and large $\pi_k$, these graphs look more uniform and less
      uniform, something like:

    \hspace{1.3in}Small $\pi_k$\hspace{2in}Large $\pi_k$

    \hspace{0.3in}
    \PSbox{COcombust1.ps}{190pt}{135pt}
    \PSbox{COcombust2.ps}{190pt}{135pt}
  \end{enumerate}
\end{enumerate}
\end{document}