Lesson 3: Hand calculations

1. Lesson 3: Hand calculations • Review: MAGICMERV • Buckling equivalence method • Areal density concept • Surface density method

2. Review: MAGICMERV. Which ones can stand alone? • M • A • G • I • C • M • E • R • V 2

3. Hand calculation methods • Buckling shape conversion • Surface density method • Analog density method (not studied) • Solid angle method (not studied) • Usefulness: • Analyst: Starting point for your model • Analyst/Reviewer: Approximate check of results

4. Buckling equivalence • From one-speed diffusion theory, we have: which (ultimately) gives us:

5. Buckling equivalence (2) • For different geometric arrangements, the buckling reduces to Table 8-I in text: • Sphere of radius r • Cylinder (r,h) • Cuboid (a,b,c) • Use d=2 cm (bare) or d=5 cm (H2O reflected) if better information not available

6. Buckling equivalence (2) • For different geometric arrangements, the buckling reduces to Table 8-I in text: • Sphere of radius r • Cylinder (r,h) • Cuboid (a,b,c) • Use d=2 cm (bare) or d=5 cm (H2O reflected) if better information not available

7. 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 • Must be “translated” into one of the MAGICMERV

8. Where does areal density fit?

9. Areal density example • Assume: Your limiting areal density is 0.4 g U235 /cm2 • What does this mean if … • You have a solution tank with a floor area of 2m x 2 m? • You have a solution tank filled with 350 g U235 /cm3? • You have to store units with a maximum of 2 kg U235 each?

10. Surface density method • Basic idea: • 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 < 0.73)

11. Surface density method (2) • Equation: • Conservatism applied to optimum SLAB density (found using Fig. 7.2): • Conservatism applied to most reactive single unit (with worst-case shape and reflection):

12. Surface density method (3) Procedure: • Given a unit fissile mass and H/U ratio, find the unreflected spherical critical mass from Fig. 7-1 (25 mm curve). • Using this value, find f=(unit mass)/(uscm) • If f>0.73, you cannot use the method. • 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. • Use the formula to get your limiting s value (or other values from it). • Apply intelligently (floor or one of the walls).

13. Surface density method (4)

14. Surface density method (5)

15. 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).

16. 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, 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.)