1 NUEN 301 Final Exam, Section 501, 2008 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 cylindrical container of radius 1 meter and height 2 meters holds a mixture of H2O and U. The interaction of the container walls with neutrons is negligible. The container is surrounded by vacuum. Other info: - 0.9 g of H2O per cm3 of mixture, - 5 atoms of 235U per 10,000 molecules of H2O, - 95 atoms of 238U per 10,000 molecules of H2O, - mixture temperature ≈ 100°C - neutron temperature ≈ 400°C [10] a) Give an expression for the resonance-escape probability that I could punch into a calculator. [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 thermal diffusion coefficient, Dth. [10] e) Do the same for the fast nonleakage probability. (If your answer depends on the answer to previous parts of this problem, you should just use a symbol to represent that previous part, instead of writing out an expression containing numbers.) [10] f) Do the same for the thermal nonleakage probability. (If your answer depends on the answer to previous parts of this problem, you should just use a symbol to represent that previous part, instead of writing out an expression containing numbers.) [15] g) Use the symbol a to represent the correct answer to part (a), b for part (b), c for part (c), d for part (d), e for part (e), and f for part (f). In terms of these symbols, what is the multiplication factor of the reactor? [30] 2. In an example in your notes, we estimated that k∞ = 1.08 for a mixture of D2O and natural U, with 1 U atom per 500 D2O molecules. 7/1000 of the U atoms are 235U, and the other 993/1000 U atoms are 238U. Design a critical reactor using this material. That is, find a reactor size and shape such that the multiplication factor would be approximately 1 if your reactor were surrounded by vacuum. Assume a density of 1 g D2O per cm3 of mixture, a mixture temperature of 20 degrees C, and a neutron temperature of 100 degrees C. If you make other assumptions, be sure to state them.2 [15] 3. a) A large commercial power reactor is critical and operating at a steady-state power level of 1 kW (which is very low). The operators move some control rods partway out of the reactor, introducing a small amount of positive reactivity. They do not change anything else. Sketch the reactor power as a function of time, putting labels and marks on your graph. You may need two graphs – one for early time and one for later – to show what happens. 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] b) If a reactor, operating in steady state at low power, is made slightly supercritical, what is the effect of delayed neutrons? That is, what is the difference between what would happen in the real world (with delayed neutrons) and what would happen if all neutrons were prompt? Be specific about functional forms for the neutron population as a function of time and about the differences or similarities in the two cases. A high-school student should understand most of your answer. [10] 4. a) Write down the one-speed time-dependent diffusion equation for a bare homogeneous spherical reactor of radius R. (The unknown is the scalar flux as a function of position and time.) Include appropriate conditions so that equation plus conditions = well-posed mathematical problem. Allow for fission and extraneous sources. Spell out any derivative operators that appear (i.e., don’t just use a “∇” symbol). Do not try to solve the equation. [10] b) Write down the energy-dependent k-eigenvalue diffusion problem for a bare homogeneous spherical reactor of radius R. (The unknown eigenfunction is the scalar flux as a function of position and energy.) Include appropriate conditions so that equation plus conditions = well-posed mathematical problem. Spell out any derivative operators that appear (i.e., don’t just use a “∇” symbol). Do not try to solve the equation. 5. Consider a critical cube of extrapolated width 2 meters, operating at steady state and producing a power of 2 GW [which is 1.25 x 1022 MeV/s]. Suppose that each fission produces 195 MeV of energy, and suppose that there are 1.05 total fissions per thermally-induced fission (which means the fast-fission factor is roughly 1.05). [15] a) If the thermal fission cross section, Σf,th, is 0.75 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. [15] b) Let the correct answer to (a) be called “φ0”. Let x = 0 be at the center of the reactor, so that x=1 meter is on an extrapolated boundary. If the thermal diffusion coefficient, Dth, is 1.5 cm, then at what net rate are neutrons flowing outward (toward the surface) across the plane at x=0.5 m? Express your answer in terms of Dth and φ0. [5] 6. a) A reactor is operating in steady state at a low power level. What can you say about reactor criticality (critical, subcritical, or supercritical) and whether an extraneous source is present? [5] b) A reactor at a low power level has its power increasing linearly with time. What can you say about reactor criticality and whether an extraneous source is present? [5] c) A reactor at a low power level has its power increasing exponentially with time. What can you say about reactor criticality and whether an extraneous source is present? [5] d) If we integrate the scalar flux over a volume, a time interval, and all neutron energies, what is the physical meaning of the result?3 Extra A certain mixture has k∞>1. You have figured out that a bare cube of this material with extrapolated width 2 meters would be critical. [5] a)