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

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\begin{enumerate}
\item Chemical Vapor Infiltration for Carbon-Carbon Composites

  Carbon-carbon composites (graphite fibers in a graphite matrix) are
  outstanding heat-resistant materials used, for example, to protect the
  hottest parts of the Space Shuttle from the high temperatures of re-entry.
  These are most often fabricated by a process known as chemical vapor
  infiltration (CVI), in which a fiber preform is heated and bathed in a
  carbon-bearing gas such as acetylene.  The resulting reaction at the surface
  of the fibers deposits the graphite matrix, which slowly closes off the pores
  until a nearly dense composite has been fabricated.

  \begin{center}
    \PSbox{cvifig.ps}{4.7in}{1.9in}
  \end{center}

  \begin{enumerate}
  \item What type of reaction is occurring here, homogeneous or heterogeneous?

  \item {\em Macroscopically}, what type of reaction does this look like?  That
    is, is the reaction occurring only on the outer surfaces of the preform, or
    throughout it?

  \item Write down some possible rate-limiting steps for this problem.

  \item \label{difeq} Consider a spherical preform of radius $R$.  Write a mass
    balance for the gas in a spherical shell of thickness $\Delta r$ which
    incorporates gas consumption by the reaction but ignores accumulation.
    Turn this mass balance into a differential equation for the concentration
    profile throughout the preform.

    You may assume:
    \begin{itemize}
    \item The time required to deposit a significant amount of carbon is much
      shorter than the time scale of diffusion, so the concentration profile
      reaches a quasi-steady-state which changes very slowly as the pores fill
      up with carbon.
    \item The reacting gas is in dilute solution with an inert gas.
    \item Diffusion through the preform is described by Fick's first law with
      uniform effective diffusivity $D_{\rm eff}$.
    \item Gas in the preform is spherically symmetric, so concentration is only
      a function of distance from the center $r$.
    \item The reaction is first order and proportional to the surface area of
      fibers in the preform, {\em I.e.} ``generation'' rate of acetylene per
      unit volume is given by $G_A=-k_A''aC_A$, where $k_A''$ is the
      heterogeneous reaction constant and $a$ is the fiber preform surface area
      per unit volume.
    \end{itemize}

  \item Solve the differential equation from part \ref{difeq}.  (Hint: assume a
    solution of the form $C_A=f(r)/r$ and solve for $f(r)$.)

  \item Fit this solution to the boundary conditions:
    \begin{itemize}
    \item $C_A$ is bounded (non-infinite) at the center, so $f(0)=0$.
    \item $C_A=C_{As}$ at the outer surface of the preform.
    \end{itemize}

  \item Consider the dimensionless quantity:
    $$N=\frac{k_A''aR^2}{D_{\rm eff}}$$
    Will the deposition be more uniform for small or large values of $N$?
    (Note that deposition rate is proportional to $C_A$, and is thus determined
    by the ratio of concentration at the center vs. near the surface
    $C_{A,0}/C_{A,s}$.  To calculate that ratio, note also that
    $\lim_{x\rightarrow 0}\frac{\sinh bx}{x}=b$.)

  \item Based on what you know about temperature dependence of diffusivity and
    reaction rates, will the deposition be more uniform at high or low
    temperatures?  (Your answer might include: in which temperature range is
    the process reaction-limited, or diffusion-limited?)

  \item What kind of problems will occur at low temperatures?  At high
    temperatures?  Is there a way to overcome such problems, or are we stuck
    with an inherently very slow process for making carbon-carbon composites?
\end{enumerate}
\end{enumerate}
\end{document}