Re: Centrifuge

["John S. Denker" <jsd@MONMOUTH.COM>]

At 08:06 AM 11/3/00 -0500, Ludwik Kowalski wrote:

> 1) Begin by explaining "settling" of mud particles
> suspended in a stationary tube. Two forces acting
> on a single particle can immediately be identified,
> weight and buoyancy. If there were nothing else
> then the particle would be moving vertically down
> with a constant acceleration. Water resistance is
> like friction and it would result in a progressive
> decrease of acceleration (terminal v = constant).

OK so far.

> But there is something else. Thermal agitation creates
> a force which fluctuates randomly and results in
> Brownian motion. The significance of randomness
> depends on the size of a particle; the net random force
> becomes negligible for large particles, such as rocks.

There's a lot more to the story.  Consider a fluid at  near-zero
temperature, so that there are no thermal fluctuations.  The small
particles will fall more slowly.  The downward force goes like the cube of
the radius.  According to Prof. Stokes, the viscous force depends on the
first power of the radius.

> Thus only very large particles travel down along straight
> lines; other particles fall down along zig-zag trajectories.
> This explains why larger particles settle sooner than
> smaller particles. Lower layers are composed mostly of
> larger particles than upper layers.

As discussed above, thermal zig-zagging is not "the" explanation.

For that matter, for really small particles, they don't fall _at all_.  If
you start them out at the bottom, they will diffuse upward.

> 3) The next step is to create a good quantitative problem
> based on the preparatory description.

It depends on what the objective is.  Real sedimentation depends on
 -- particle size and shape
 -- solvent viscosity
 -- solvent density
 -- temperature
 -- centrifugal field
 -- et cetera

If you want to teach all that, fine.  Have at it.  With a little work you
could get the students to understand the relationship between diffusion
constant (in response to concentration gradients) and mobility (in response
to applied force).  That is truly a beautiful result.  The students may be
impressed to learn how recently that was figured out (1905) and who figured
it out.

OTOH for the introductory class, it may be better to hold N-1 of the
independent variables constant, varying only the centrifugal field.  That
results in a very simple functional form.

On the third hand, you can skip thermal effects (i.e. assume not-too-small
particles) and consider only buoyancy, Stokes-type viscous forces, and G
forces.  This leads to the standard centrifugation formulas;  see e.g.
  http://ntri.tamuk.edu/centrifuge/centrifugation.html

Numerous exercises illustrating these ideas already exist in the literature.