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

\documentclass{article}
\usepackage{fullpage}
\newcommand{\PSbox}[3]{\mbox{\rule{0in}{#3}\special{psfile=#1}\hspace{#2}}}
\begin{document}
\begin{enumerate}
\item Diffusion of water vapor into a flowing adhesive sheet

  \begin{enumerate}
  \item Time required to flow down the exposed length is that length divided by
    velocity on top of the film, which is the maximum velocity:
    $t=L_{exp}/u_{max}$.  Taking $x$ as the flow direction and $y$ as the
    distance from the top surface, maximum velocity is given by:
    $$u_{max} = u_x|{y=0} = \frac{\rho g\sin\theta L^2}{2\mu} =
    \frac{\rm1060\frac{kg}{m^3}\cdot9.8\frac{m}{s^2}\cdot0.707\cdot(0.005m)^2}
    {\rm2\cdot1\frac{kg}{m\cdot s}} = 0.0918\frac{\rm m}{\rm s}.$$
    $$t = \frac{L_{exp}}{u_{max}} = \frac{\rm0.05m}{\rm0.0918\frac{m}{s}} =
    0.545{\rm seconds}.$$

    Is flow laminar?  Check the Reynolds number using
    $u_{av}=\frac{2}{3}u_{max}$ as the velocity:
    $$Re = \frac{\rho u_{av}L}{\mu} =
    \frac{\rm1060\frac{kg}{m^3}\cdot0.0612\frac{m}{s}\cdot0.005m}
    {\rm1\frac{kg}{m\cdot s}}=0.324.$$
    Since this is below 20, we're well within the laminar flow r\'{e}gime.

  \item For an order of magnitude, one can just estimate the diffusion length
    using:
    $$L_D \simeq \sqrt{Dt} = \sqrt{\rm10^{-9}\frac{m^2}{s}\cdot0.545s} =
    2.3\times10^{-5}{\rm m}.$$
    About 20 microns, which is quite shallow.  Even if a small amount of water
    diffuses several times more deeply, this is still well within the top
    region with roughly uniform velocity.

  \item To determine the rate-limiting step, check the Biot number, using the
    diffusion depth as the lengthscale:
    $${\rm Bi} = \frac{k''L}{D} =
    \frac{\rm5\frac{m}{s}\cdot2.3\times10^{-5}m}{\rm10^{-9}\frac{m^2}{s}}
    \simeq10^5.$$
    There's no question this is diffusion-limited.

  \item Your sketches should have looked something like:
    \begin{center}
        \PSbox{adhesiflow.ps}{204pt}{230pt}      
    \end{center}

  \item Since this is cleary diffusion-limited, the top surface will rapidly
    reach equilibrium with the air, giving a constant top surface concentration
    of water.  The initial condition has a uniform (and hopefully low)
    concentration of water.

    Since the diffusion depth is tiny compared to the adhesive film thickness,
    and the top surface is a free surface where $\partial u_x/\partial y=0$,
    the velocity is roughly uniform and equal to $u_{max}$ throughout the
    diffusion boundary layer.

    These and the much thinner boundary layer thickness than exposed surface
    length $\delta_C\ll L_{exp}$, give us the conditions for using the error
    function to describe diffusion in the $y$-direction from the top, with time
    replaced by exposure time $t=x/u_{max}$.  Since the surface concentration
    is higher than the inital concentration, it is convenient to use the erfc
    form:
    $$\frac{C-C_{init}}{C_S-C_{init}} = 
    {\rm erfc}\left(\frac{y}{2\sqrt{Dx/u_{max}}}\right).$$
  \end{enumerate}
\end{enumerate}
\end{document}