# Lesson3 ```Lesson 3: Hand calculations
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Review: MAGICMERV
Buckling equivalence method
Areal density concept
Surface density method
Review: MAGICMERV.
Which ones can stand alone?
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M
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2
Hand calculation methods
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Buckling shape conversion
Surface density method
Analog density method (not studied)
Solid angle method (not studied)
Usefulness:
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Analyst: Starting point for your model
Analyst/Reviewer: Approximate check of
results
Buckling equivalence
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From one-speed diffusion theory, we have:
  J  r    D  r    r 
which (ultimately) gives us:
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 D  r  B 2   a  r     r    f  r    r 

2

 r 
2
B 
 constant
 r 
Buckling equivalence (2)
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For different geometric arrangements, the
buckling reduces to Table 8-I in text:
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  
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
 r d 
2
 2.405    
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 
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 r  d   h  2d 
2
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Cylinder (r,h)
        
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 
 

 a  2d   b  2d   c  2d 
2
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Cuboid (a,b,c)
2
2
Use d=2 cm (bare) or d=5 cm (H2O reflected)
if better information not available
2
Buckling equivalence (2)
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For different geometric arrangements, the
buckling reduces to Table 8-I in text:

  


 r d 
2
 2.405    

 

 r  d   h  2d 
2

Cylinder (r,h)
        

 
 

 a  2d   b  2d   c  2d 
2
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Cuboid (a,b,c)
2
2
Use d=2 cm (bare) or d=5 cm (H2O reflected)
if better information not available
2
Areal density
Alternate definition of fissile mass density
based (almost always) on floor area coverage
Problem: Assumes uniform coverage
Cannot be used directly in posted controls
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Must be “translated” into one of the MAGICMERV
Where does areal density fit?
Areal density example
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Assume: Your limiting areal density is
0.4 g U235 /cm2
What does this mean if …
1. You have a solution tank with a floor
area of 2m x 2 m?
2. You have a solution tank filled with 350
g U235 /cm3?
3. You have to store units with a
maximum of 2 kg U235 each?
Surface density method
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Basic idea:
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What is the minimum spacing of a 2D array
of unstacked units for criticality?
Express as an “areal density”
An infinite array of basic units with
spacing d can be limited by a
compressed, water reflected
arrangement, corrected for unit
reactivity
s, s0 = Areal densities (g/cm2)
f = “fraction critical” (MUST be &lt; 0.73)
Surface density method (2)
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Equation:
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Conservatism applied to optimum SLAB
density (found using Fig. 7.2):
  0.54 0 1  1.37 f 
0.54 0
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Conservatism applied to most reactive
single unit (with worst-case shape and
reflection):
11.37 f 
Surface density method (3)
Procedure:
1. Given a unit fissile mass and H/U ratio, find
the unreflected spherical critical mass from
Fig. 7-1 (25 mm curve).
2. Using this value, find f=(unit mass)/(uscm)
3. If f&gt;0.73, you cannot use the method.
4. For H/U ratio, use Fig. 7-2 to get BOTH the
critical thickness (300 mm curve) AND the
concentration. The product of these is s0.
5. Use the formula to get your limiting s value
(or other values from it).
6. Apply intelligently (floor or one of the walls).
Surface density method (4)
Surface density method (5)
Homework
Homework 3-1
You fill a shoebox (15 cm x 20 cm x 30 cm) with a
fissile solution and find that it is exactly critical. What
would be the approximate radius of a critical sphere
of the same material? (Ans. = 11.645 cm)
Homework 3-2
Work problem 8.7 from the text (but just for the
surface density method).
Homework
Homework 3-3
It is common practice to assume that a cylinder with
a height/diameter ratio of 1.00 has the highest
reactivity (lowest buckling). Use your calculus to
show that based on the formula (ignoring the
extrapolation distance), the actual value for the most
reactive H/D (for fixed volume) is 0.924.
(You may use a spreadsheet to show that this is
optimum.)
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