dope exponentialz [Re: India intercepted ...]

Dave Long dl at silcom.com
Fri Sep 12 15:56:56 PDT 2003



> Measuring nearly eight feet in length and nine inches in diameter,
> the tubes were made of a special alloy, 6061-T6, known to be both
> light and exceptionally strong. Similar tubes are used in a wide range
> of commercial products, from bicycle frames to aircraft parts. But
> they also are useful in the construction of machines known as gas
> centrifuges, which enrich uranium into the key material for nuclear
> weapons.

I've been wondering what the deal is
with unexportable tubes -- why would
anyone going for nuke tech be held up
by a matter of metallurgy, especially
when the US managed it* so long ago?

After reading a bit too much Herbert
Kornfeld, it seemed obvious that th'
numbaz be needin' crunchin':

The basic problem is to concentrate
the U-235 several times over usual.
(very roughly three doublings, or
exp(2) times.  I have no idea what
weapons grade might require)

The ideal concentration of a gas in 
a gravity gradient goes as:
	exp(-Mgh/kT)
so if we wish to boost a fraction
by exp(2), we need to solve for:
	(M1-M0)gh/kT = 2

The mass difference between 238 and
235 isn't very large:
	M(U-238) - M(U-235) =
	3 amu = 5e-24 g
and if we just use the gravitational
field of the earth, and solve for h,
we get:
	h = 2kT/(M1-M0)g =
	2 (1.38e-16 erg/K)(300K)/(5e-24g)(980 cm/s^2) =
	170 km
so if we had a space elevator handy,
and enough uranium to fill up a very
long tube, we might be able to enrich
it fairly easily.

(of course, using the space elevator,
we'd be looking at 230K instead of 300,
but even so we're still talking 130km,
while neglecting the dropoff in g; it
doesn't seem that refrigeration buys
us very much in this application)

How about multi-stage processes?  It's
probably a good thing that they're on
the wrong side of the exponential here:
	exp(a)exp(b) = exp(a+b)
so running twice as many processes in a
cascade only buys us a length reduction
of half, and it would take thousands of
stages to bring the tube length down to
around 100m.

k and M we're stuck with, and altering
h and T aren't so helpful, so what can
we do with g?  In ancient Egypt, it was
possible to increase metal flow into a
mold by spin casting instead of pouring;
the modern equivalent is the centrifuge.

According to a brief communication in a
recent Nature, one can pull 1.4e6 g with
no moving parts in a microfluidic system;
which would bring h down to 12 cm, which,
although much smaller than before, still
is safely bigger than the 30 micrometer
radius of the chamber.

But what about a macroscopic centrifuge?
The same calculation says that if one is
willing to cascade for a dozen steps, and
can generate average gradients of 1e5 g,
then a tube of radius 10cm could make a
decent gas centrifuge component.

I guess it's the need for high tensile
strength at low mass that makes this a
problem amenable to export control.  If
the tube isn't strong enough for those
accelerations, cascade length gets out
of hand quickly: wit'out the g's, G's,
GC's be wack.

-Dave

:: :: ::

* the US still uses a similar process to
the 1940's one; it's much closer to gas
chromatography than gas centrifuge, and
has the redeeming feature that it needs
large facilities.


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