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

\documentclass{article}
\usepackage{fullpage,pstricks}
\begin{document}
\begin{enumerate}
\item Diffusion of water vapor into a flowing adhesive sheet

  Water vapor in air adsorbs and diffuses into an exposed liquid adhesive
  monomer, such as cyanoacrylate, up to its solubility, possibly initating
  polymerization.  A 5 mm thick sheet of high-viscosity adhesive flows down an
  inclined plane with an exposed section 5 cm long at a 45$^\circ$ angle.
  Estimating the rate of water incorporation into the adhesive is the
  motivation for this problem.
  % Cyanoacrylate data: http://ajh-knives.com/adhesive-pa1.html;
  % 1000 cps (centipoise) = 1000*.01 cgs units = 1000 * 0.001 MKS units = 1.

  \begin{center}
    \input{adhesiflow}
  \end{center}

  Data:
  \begin{itemize}
  \item Adhesive density: $\rm\rho=1060kg/m^3$.
  \item Adhesive viscosity: $\rm\mu=1\frac{kg}{m\cdot s}$.
  \item Water diffusivity in liquid adhesive: $D=10^{-9}\frac{\rm m^2}{\rm s}$.
  \item Water adsorption rate coefficient (based on the Langmuir equation):
    $k''=5\frac{\rm m}{\rm s}$.
    % J_m=\frac{Mp_v}{\sqrt{2\pi MRT}=k''\Delta C,
    % k'' = \frac{Mp_v}{\Delta C\sqrt{2\pi MRT} = ...
    % Assume 5 kPa gives 0.1%=1 kg/m^3 water conc.
    % \frac{.018kg/mol5000kg/ms^2}{1kg/m3\sqrt{2\pi.018kg/mol8.314J/mol-K300K}}
    % = 5.36 m/s
  \end{itemize}

  \begin{enumerate}
  \item \label{exptime} How long does it take for fluid on the top surface to
    flow down the exposed length of the incline?  Is flow laminar?  (Assume
    dissolved water does not affect the adhesive properties.)
  \item Estimate the diffusion length at the bottom of the incline, that is,
    the depth to which the water has diffused during the exposure time
    calculated in part \ref{exptime}.  (If you didn't get an answer to part
    \ref{exptime}, use one second.)
  \item Is transport of water into the adhesive limited by diffusion in the
    liquid, or adsorption of water vapor at the liquid surface?  Explain your
    answer.
  \item Sketch the flow velocity profile and concentration boundary layer
    formed by water diffusion into the liquid adhesive.  Your sketch need not
    be to scale.
  \item Write an equation for the the concentration of dissolved water vs.
    location in the film $C(x,y)$ assuming transport into the liquid is limited
    by diffusion.
  \end{enumerate}
\end{enumerate}
\end{document}