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

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\begin{enumerate}
\item Joule heating of a titanium rod

  \begin{center}
    $\ $\pdfximage{ticurrent.png}\pdfrefximage\pdflastximage$\ $
  \end{center}

  At time $t=0$, the temperature in a titanium rod 0.25 cm ($2.5\times10^{-3}$
  m) in diameter is uniform and equal to its surface temperature.  A switch is
  turned on so that a current passes lengthwise through the rod, heating it
  through a mechanism known as ``Joule heating''.  Assume the current is
  uniformly distributed throughout the rod, so it is heated uniformly, with a
  uniform heat generation rate per unit volume $\dot{q}$.

  Titanium data:
  \begin{itemize}
  \item Thermal conductivity: $k=20{\rm\frac{W}{m\cdot K}}$
  \item Density: $\rm\rho=4700\frac{kg}{m^3}$
  \item Heat capacity: $c_p=700{\rm \frac{J}{kg\cdot K}}$
  \end{itemize}

  \begin{enumerate}
  \item At a current of 10 amperes, the heat generation rate per unit volume
    $\dot{q}$ is approximately $\rm5\times10^6\frac{W}{m^3}$.  Use the
    uniform-generation solution to the heat conduction equation to calculate
    the maximum temperature difference between the center of the rod and its
    surface.

    %Current density $J$ is $2\times10^6\rm\frac{amps}{m^2}$,
    %$\rho_e=1.12\times10^{-6}\rm\frac{W\cdot m}{A^2}$ (ps6), so
    %$\dot{q}=J^2\rho_e=4.65\times10^6\frac{\rm W}{\rm m^3}$.

  \item If the current through the rod doubles to 20 amps, the heat generation
    rate $\dot{q}$ quadruples.  How does this affect the difference between the
    maximum temperature in the center and the temperature at the surface?

  \item Assuming the surface is held at a constant temperature, sketch the
    temperature $T$ as a function of $r$ across the rod for several different
    times $t$.  (This should have one curve for $t=0$, one for the steady-state
    at $t=\infty$, and others for several intermediate times.)

  \item Approximately how long will it take for the center of the rod to reach
    its maximum temperature?
  \end{enumerate}
\end{enumerate}
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