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

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\begin{enumerate}
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\item Electron beam centrifugal atomization of metal

  There are several methods for producing a spray from a liquid whose resulting
  size distribution is quite broad, that is, there is a wide range of spray
  droplet sizes that result from that process, and when they solidify, the
  resulting powder is a mixture of spheres of various sizes.  For some
  materials applications, it's much better to have a powder with a narrow
  distribution, that is, with most of the droplets having the same size.  One
  way to achieve this is by centrifugal atomization, in which we rotate a solid
  cylinder and melt it at a controlled rate so droplets break off at a size
  determined by the balance between centrifugal and surface tension forces: $$R
  \sim \sqrt{\frac{\gamma}{\rho\omega^2 r}}$$
  where $R$ is the droplet size, $\gamma$ is the surface tension, $\rho$ the
  liquid density, $\omega$ the rotation rate and $r$ the distance from the
  rotation axis where the liquid droplet breaks free.  An arrangement which
  produces this result is pictured below.  The cylinder, called the ingot, is
  held vertically and rotated quickly while melting slowly from the top such
  that a thin film of liquid is accelerated out to the edges, where the liquid
  breaks into droplets.
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  For this vertical atomization arrangement, we would like to calculate two
  things:
  \begin{itemize}
  \item The amount of heat needed to melt and atomize at a certain rate.
  \item The temperature distribution in the rotating ingot at steady-state.
  \end{itemize}
  We will use titanium as the atomized material here, which has the following
  properties:
  \begin{itemize}
  \item Melting point: 1667$^\circ$C = 1940 K
  \item Radiative emissivity: 0.55
  \item Thermal conductivity: 20$\rm\frac{W}{m\cdot K}$
  \item Density: 4700$\rm\frac{kg}{m^3}$
  \item Molar mass: 0.0479$\rm\frac{kg}{mol}$
  \item Heat capacity: 700$\rm\frac{J}{kg\cdot K}$
  \item Heat of fusion: 300$\rm\frac{kJ}{kg}$
  \item Vapor pressure constants: $A=23200{\rm K}$, $B=11.74$, $C=-0.66$, $D=0$
    $$\log_{10}p_v({\rm torr}) = -\frac{A}{T} + B + C\log_{10}T + DT$$
  \item Heat of vaporization: $\Delta H_e=9.2\frac{\rm MJ}{\rm kg}$
  \end{itemize}

  Source: E. Brandes, ed., {\em Smithells Metals Handbook} (6th edition),
  Boston: Butterworth \& Co., 1983.

  \begin{enumerate}
  \item \label{radloss} Assuming the chamber is cold and black, and that the
    liquid film is all at the melting point, estimate the radiative heat loss
    from the top surface of the ingot.

  \item \label{evaploss} Assuming ideal Langmuir evaporation into a vacuum,
    calculate the evaporation rate and heat loss due to evaporation.

  \item \label{totalflux} If we would like to melt and atomize at a rate of 1
    cm of ingot per minute, what is the required power density of the heat
    source?  You may neglect losses from the sides of the ingot, but include
    energy required to heat the titanium from 300 K to its melting point, and
    to melt it, and the losses in parts \ref{radloss} and \ref{evaploss}.

  \item Is the process more energy-efficient if it goes faster or slower?

  \item \label{tempdist} The ingot bottom temperature and initial temperature
    are both 300 K.  If the ingot is 1 m long, can it be considered
    semi-infinite?  When the process reaches steady-state, what is the
    relationship between temperature and distance from the top of the ingot?

  \item Use your answer from part \ref{tempdist} to calculate the heat flux
    into the top of the solid ingot.  Which of the energy components from part
    \ref{totalflux} does this relate to?  (Radiative/evaporative losses, heat
    of fusion, heat of raising the titanium to its melting point)

  \item Suppose the ingot were turned on its side, and hit on the top by a
    fixed (not scanning) electron beam while spun like a rolling pin.  Give at
    least one advantage or disadvantage this different form of the process
    would have vs. that pictured above.
  \end{enumerate}
\end{enumerate}
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