TO: Clifford K. Beck
FROM: Raymond L. Murray
SUBJECT: Reactor Size and Mass

It has been suggested that changes in the volume of the reactor core
from the original proposal be made to accomodate possible larger necessary masses.
Consideration should be given to the following facts, however:

• 1. The critical mass is a rather insensitive function of the water content
  in the region around NH/NU= 400. An estimate of the variation from
  the minimum, Mo, from bare-reactor theory
  
  is plotted in Figure 1. Taking the minimum critical mass to occur at
  NH/NU = 360, there is no more than a 10% increase in going as low as
  250 or as high as 470.
•  
• 2. The solution volume for a given mass of uranium varies inversely with
  the NH/NU ratio. Neglecting the presence of other elements and ignoring
  displacement, the relation is
  
  The variation of volume is plotted on the same graph, using the relation
  

The conclusion is that if the critical mass of the N. C. State reactor
is actually higher than the expected 915 grams, the excess material may still be
easily accomodated by the container, providing a slightly lower NH/NU be used.
On the other hand, if the mass turns out to be lower, the mixture can be diluted
to fit the vessel.

Calculations leading to the formulas appear in the Appendix.

Assumptions:

• 1. Bare spherical reactor of critical radius R, with pure U-235 in water.
•  
• 2. Critical equation ---(1)
  where k is the infinite multiplication constant, [eta]f, where [eta] = 2.1,
  , m² is the migration area ~ 34 cm² for
  systems in the range of
•  
• 3. Ratio of mass of U-235 to mass (and volume) of water
  ---(2)
•  
• 4. Volume and mass of water
  ---(3)

Analysis:

• 1. Assume cross sections [sigma]U = 640 barns , [sigma]H = 0.31 barns.
•  
• 2. Solve equations (1), (2) and (3) for MU, using the approximation that
  , where the latter is 2065. The result is
  ---(4)
•  
• 3. The minimum of this function occurs when the two terms in the denominator
  are equal, ie at NH/NU = 360 , which checks experiment fairly well.
  At that point MU = 3.2 kg = Mo which is a reasonable bare reactor mass.
•  
• 4. The ratio of M in general to the minimum mass, as quoted in the body
  of this note, is computed to be ---(5)
•  
• 5. The volume ratio is obtained from equations (2), (3), (5) as