1 Nuke 301 Study Problems, First Exam, 2013 Material: Notes through Chapter VI “Case 3.” Lamarsh Chapter 1. Lewis Chapters 1, 2, and 4 1. You must compute the rate at which neutrons are produced in various reactors. You also must compute the rate at which neutrons are lost in each reactor. The only production mechanisms of significance are 1) neutron-induced fission in the reactor fuel, 2) spontaneous fission in Cf-252 (only in some of the reactors). The only loss mechanisms of significance are 1) absorption, 2) leakage. Compute each of the production rates and also the absorption rate for the cases listed below. Be very precise with your notation and answers. Include units. Make sure someone else could easily follow your logic and calculations. In parts (a)-(d) the cross sections you are given have been appropriately averaged, not only over all nucleus velocities, but also over all neutron energies. This means that a reaction-rate density is one of these macroscopic cross sections times the total scalar flux – you don’t have to worry about energy integrals in (a)-(d). a) Brick-shaped homogeneous reactor: –a/2 < x <a/2 , –b/2 < y < b/2 , –c/2 < z <c/2 , operating in steady state. You are given the total scalar flux as a function of position: , Σt = 5 cm–1, Σa = 0.2 cm–1, Σf = 0.06 cm–1, νfuel= 2.5 n/fission. where A = 2 E9 n/cm2-s, C = 1 E10 n/cm2-s. In this part you must also calculate the net leakage rate out of the reactor. (Note that the reactor is operating in steady state – this gives you enough information to figure out the leakage rate, even though you don’t know the net current density, once you calculate the other rates!) You are given that the Cf-252 is distributed uniformly throughout the reactor, and that: N252 = Cf-252 density = 1E18 atoms/cm3, λf,252 = decay constant for spontaneous fission of Cf-252 ≈ 0.01 year–1. ν252 = neutrons emitted per spontaneous fission of Cf-252 = 3.8 n/fission φ(x, y, z) = A + Ccosπxa⎛ ⎝ ⎞ ⎠ cosπyb⎛ ⎝ ⎞ ⎠2 b) Another shoebox reactor, same dimensions and material properties as in part (a). The same Cf-252 is distributed with the same density. This time, though, you are told that: φ(x,y,z) = A, for all x,y,z in the reactor, and there is no leakage at all. If the reactor is operating in steady state, what is the value of the constant A? c) A spherical reactor, one meter in diameter, critical and operating in steady state. Material properties are as in part (a), except that Σf = 0.09 cm–1, and there is no Cf. If the average scalar flux in the reactor is: = 1E9 n/cm2-s, then what is the rate per cm2 at which neutrons are leaking out of the outer surface (r=50cm)? d) A commercial pressurized-water reactor. The reactor has 193 fuel assemblies arranged in a roughly circular pattern (if you are looking down on it from the top). Each fuel assembly is 20 cm × 20 cm × 300 cm. A nuclear engineer divided each assembly into 20 axial segments (each 15 cm tall), and she did some calculations that produced the following results: . That is, she has this volume-averaged flux, φij, stored away for i=1,...,193 and j=1,...,20. She also has very good estimates for the flux-weighted average cross sections in each axial layer of each assembly: , along with the same kind of average for the total and fission cross sections. For this reactor, write down expressions, in terms of her estimates and known volumes, for the production rate from fission and the loss rate from absorption. (Ignore other production & loss mechanisms.) e) A slab reactor, infinite in the y and z directions, thickness a in the x direction, with uniform material properties. For this reactor, the production and loss rates are infinite, because the volume is infinite. Since that’s not very interesting, you should find expressions for the production & absorption rates per unit y-z area. That is, consider a Δy × Δz × a chunk of the slab, compute the rates, and divide by ΔyΔz. Your expressions should contain the energy-dependent scalar flux (which is a function of x and E) and the energy-dependent cross sections (which you can assume have been averaged over nucleus motion and don’t depend on position). φ ≡1VoldVVo l∫φφij=1Vijdx dydzjth axial regionof ith assembly∫∫∫dE0∞∫φ(x,y,z, E)Σa,ij≡1Vijφijdx dy dzjth axial regi onof ith assembly∫∫∫dE0∞∫Σa(x, y, z,E)φ(x, y, z, E)3 f) Back to a brick-shaped reactor of dimensions a × b × c, as in part (a). You don’t know the cross sections this time, but the scalar flux is as given in part (a). Now suppose the total net current density is given by the following expression: J x, y, z( )= −D∇φx, y, z( ) , where D is a known constant with units of length. Carry out the gradient operation to obtain an expression for the net current density. (This means taking derivatives and being careful with unit vectors.) Then use the result to obtain an expression for the net rate at which neutrons leak from the reactor. (This will involve integrals over surfaces.) In every step be clear about what is a vector and what is a scalar. g) If the net outleakage rate from part (f) agrees with the net outleakage rate from part (a), what value must the constant D have? 2. Microscopic cross sections depend fundamentally on a certain speed. What speed? 3. What is the physical interpretation of a macroscopic cross section? 4. What is the physical interpretation of the total scalar flux? If you integrate the total scalar flux over a volume and over a time interval, what is the physical interpretation of the answer? In general, if you integrate the total scalar flux over a surface area, what is the physical significance of the answer? 5. What is the physical interpretation of the net current density? If you integrate the net current density over a volume what is the physical significance of the answer? 6. What is the macroscopic total cross section, at a neutron speed of 2200 m/s, in a mixture of 10 grams of He-4 and 10 grams of hydrogen? Data: σtHe = 0.8 barns, σtH = 38