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Lesson 3: Hand calculations Review: MAGICMERV Buckling equivalence method Areal density concept Surface density method Review: MAGICMERV. Which ones can stand alone? M A G I C M E R V 2 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 Buckling equivalence From one-speed diffusion theory, we have: J r D r r which (ultimately) gives us: D r B 2 a r r f r r 2 r 2 B constant r Buckling equivalence (2) For different geometric arrangements, the buckling reduces to Table 8-I in text: r d Sphere of radius r 2 2.405 r d h 2d 2 Cylinder (r,h) a 2d b 2d c 2d 2 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) For different geometric arrangements, the buckling reduces to Table 8-I in text: r d Sphere of radius r 2 2.405 r d h 2d 2 Cylinder (r,h) a 2d b 2d c 2d 2 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 Must be “translated” into one of the MAGICMERV Where does areal density fit? Areal density example 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 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) Surface density method (2) Equation: Conservatism applied to optimum SLAB density (found using Fig. 7.2): 0.54 0 1 1.37 f 0.54 0 Conservatism applied to most reactive single unit (with worst-case shape and reflection): 11.37 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>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.)