1 NUEN 301 Final Exam, Section 502, 2009 Rules: Make reasonable assumptions when you need to, and state them! Don’t write where the staple will go. Notes: A data/formula sheet is provided. There are 200 possible points, plus an extra-credit problem worth 10 more (5% more). 1. A spherical container of radius 4 meters holds a mixture of D2O and Pu. The interaction of the container walls with neutrons is negligible. The container is surrounded by vacuum. Other info: - heavy-water density is 1.1 g/cm3, - 10 atoms of 239Pu per 10,000 molecules of D2O, - 1 atom of 240Pu per 10,000 molecules of D2O, - mixture temperature ≈ 20°C - neutron temperature ≈ 100°C [10] a) Give an expression for the resonance-escape probability that I could punch into a calculator. In the following, if an answer depends on the answer to a previous part of this problem, use a symbol to represent the correct answer to that previous part. Use the symbol a to represent the correct answer for part (a), b for part (b), c for part (c), d for part (d), e for part (e), f for part (f ), and g for part (g). [10] b) Do the same for the product of the thermal reproduction factor and thermal utilization. [10] c) Do the same for the product of the fast reproduction factor and fast utilization. [5] d) Do the same for the geometric buckling. [5] e) Do the same for the thermal diffusion coefficient, Dth. [10] f) Do the same for the fast nonleakage probability. [10] g) Do the same for the thermal nonleakage probability. [10] h) In terms of the symbols that represent the correct answers to parts (a)-(g), what is the multiplication factor of the reactor? [10] i) Suppose that the concentration of 239Pu and 240Pu in this reactor is chosen so that it is exactly critical, and suppose it is operating in steady state at a low power level. What is the thermal flux as a function of position? [30] 2. A bare cubical reactor of width 3 meters contains a homogeneous mixture of D2O and U. It is critical. If the square of the thermal diffusion length is 150 cm2, approximately what is the mixture k∞?2 [15] 3. A large commercial power reactor is critical and operating at a steady-state power level of 1 GW. The operators move some control rods partly out of the reactor, introducing positive reactivity. They do not change anything else. Eventually a new steady state is established at a power of 3 GW. A few minutes later, at time t0, the operators quickly return the control rods to their initial positions, thus inserting negative reactivity. Sketch the reactor power as a function of time, starting just before t0, putting labels and marks on your graph. [Hint: Think carefully about what happens as the reactor cools back down as power decreases.] Point out and describe interesting features of the graph, as if you were explaining to a bright high-school student how a reactor works. Some of your explanation should be qualitative and some should talk about the actual functional form of P(t). [15] 4. a) Write down the energy-dependent time-dependent diffusion problem for a bare homogeneous cylindrical reactor of radius R and height H. (The unknown is the scalar flux as a function of position and time.) This should be a complete, well-posed mathematical problem (which means it must include appropriate conditions). Allow for fission and extraneous sources, and don’t forget that some neutrons are delayed. Spell out any derivative operators that appear (i.e., don’t just use a “∇” symbol). Let z=0 be at the center of the cylinder. Do not try to solve the equation. [15] b) Write down the k-eigenvalue diffusion problem for the thermal flux in a bare homogeneous cylindrical reactor of radius R and height H. This should be a complete, well-posed mathematical problem (which means it must include appropriate conditions). Spell out any derivative operators that appear (i.e., don’t just use a “∇” symbol). Let z=0 be at the center of the cylinder. Describe how the equation accounts for the reality that neutrons are born fast, even though the fast flux is not in the equation. (The unknowns in your equation should be “k” and the thermal flux.) Do not try to solve the equation. 5. Consider a bare homogeneous critical cube of extrapolated width 2 meters, operating at steady state and producing a power of 3.2 GW [which is 2 x 1022 MeV/s]. Suppose each fission produces 190 MeV of energy, and suppose there are 1.10 total fissions per thermally-induced fission (which means the fast-fission factor is roughly 1.10). [15] a) If the thermal fission cross section, Σf,th, is 0.7 cm-1, what is the thermal flux (n/cm2-s) at the center of the reactor? Give an expression that I could punch into a calculator. [10] b) Let y = 0 be at the center of the reactor, so that y = −1 meter defines an extrapolated boundary. Let the correct answer to part (a) be called “φ0”. If the thermal diffusion coefficient, Dth, is 0.75 cm, then at what rate are neutrons flowing out of the reactor through the surface at y ≈ –1 m? Give an expression that I could punch into a calculator if I knew φ0.3 6. Three monoenergetic and monodirectional neutron beams strike a very cold material. Each beam completely covers the face it strikes. The block is 2 cm deep in the z direction. The following data apply: Σt(v) = Σγ(v) = Σt(v) (i.e., every collision is a capture) n1 = neutron density in first beam = 2×105 n/cm3 ; v1 = neutron speed in first beam = 106 cm/s ; n2 = neutron density in second beam = 3×104 n/cm3 ; v2 = neutron speed in second beam = 5×105 cm/s ; n3 = neutron density in second beam = 4×106 n/cm3 ; v3 = neutron speed in second beam = 105 cm/s ; [10] a) What is φ(x,y,z), for 0≤x≤6 cm and 0≤y≤2 cm and 0≤z≤2 cm ? [10] b) What is J(x,y,z), for 0≤x≤6 cm and 0≤y≤2 cm and 0≤z≤2 cm ? [10] Extra Suppose a given mixture has k∞ = 1.1, Lth2 = 50 cm2, and τth = 60 cm2. a) If the mixture is compressed to twice its original density, what are its new values of k∞, Lth2, and τth? b) Suppose that a bare sphere of the initial mixture