1 NUEN 301 Study Problems, Second Exam, 2013 Material: Notes through page 171. 1. a) Write down the time-dependent energy-dependent neutron conservation equation. Include prompt and delayed neutrons, and include an extraneous source. b) Integrate each term in this equation over an arbitrary volume ΔxΔyΔz and over all energies. What are the units of each integrated term? Give a physical interpretation of each integrated term – say in words what each term means. Pay special attention to the leakage term. 2. Consider Eqs. (6.108) and (6.109) in your notes. These equations are an essentially exact description of neutron and precursor behavior in reactors. Now suppose the reactor is operating in steady state, which means the neutron and precursor populations are steady. Eliminate the precursors from the system of equations and show that the resulting equation for the scalar flux is the same equation you would get if all of the neutrons from fission came out promptly. That is, show that in steady state we can ignore the fact that some fission neutrons are born with a delay. 3. Write down the steady-state two-group diffusion problem for a bare homogeneous shoebox reactor of dimensions a x b x c. Let group 1 be fast neutrons and group 2 be thermal. Assume no upscattering from the thermal group to the fast group, and make reasonable assumptions about where the fission neutrons are born. This should be a complete, well-posed mathematical problem (which means it must include appropriate boundary and initial conditions). Let the center of the reactor have coordinates x=y=z=0. Allow for the presence of extraneous sources. Don’t try to solve the equation – just write it down. 4. a) Write down the k-eigenvalue problem for the thermal scalar flux in a bare homogeneous cylindrical reactor of radius R and height H. Let the center of the reactor have coordinates r=z=0. In this equation, the source of thermal neutrons is downscattering of faster neutrons. b) Describe approximations you can make to express the downscattering source as some factors times the rate at which thermal neutrons cause fission, and make those approximations. This should produce a complete, well-posed mathematical problem for the thermal flux and the k-eigenvalue. c) Guess a separable solution – a product of a function of r and a function of z. Insert this into the equation and obtain two separate eigenvalue problems for the radial and axial functions (with eigenvalues called Br2 and Bz2, respectively). d) Each of the two problems is equivalent to one we solved in class. Write down the solutions, then combine them to obtain the complete solution to the original k-eigenvalue problem. e) What is the geometric buckling in a bare homogeneous cylinder of radius R and height H? What is the fundamental-mode scalar flux? 5. Define the following parameters, which appear in the Point Reactor Kinetics Equations: ρ (give name and give definition in terms of k) β (give name and physical definition [“the fraction of …”])2 Λ (give name, math definition, and the name and physical definition of any terms that appear in the math definition) 6. A large commercial power reactor is critical and operating at a steady-state power level of 1 kW (which is very low). The operators extract some control rods part of the way out of the reactor, introducing positive reactivity. Describe what happens: a) in the first second or two (is there a rapid change in power?) b) in the next few minutes (does the power increase according to some basic functional form?) c) after the power gets high enough to start increasing the reactor temperature 7. A nuclear reactor is operating in steady state with a source present. a) Is the reactor critical, subcritical, or supercritical? b) If the multiplication factor is k0, what would you need to change it to in order to make the neutron population twice as high? 8. a) How quickly can we cause the power in a reactor to decrease from 1 GW to 1 kW? Your discussion should contain both qualitative and quantitative features. [If I ask this on an exam, I expect you to quantify the prompt drop in terms of how much negative reactivity was inserted, quantify the rate of decrease after the prompt drop that you would have if there were no feedback, and then discuss the effects of feedback.] b) How quickly can we cause the power in a reactor to increase from 1 kW to 1 GW? Your discussion should contain both qualitative and quantitative features. [If I ask this on an exam, I expect you to consider the case of ρ>β, which is called “prompt supercritical,” and describe how quickly the power could increase if there were no feedback. Then discuss the effects of feedback.] 9. If there were no delayed neutrons and no feedback, the basic time constant governing how rapidly the neutron population could change would be the prompt neutron lifetime, lp, which is shorter than a millisecond. How do delayed neutrons change this – what is the effective neutron lifetime that governs population changes in the presence of delayed neutrons? [Hint: See problem 3 of HW5.] 10. Someone solved a two-region slab-geometry diffusion problem with the following characteristics: for 0<x<a: region “F”, with DF, Σa,F, νΣf,F, for a<x<b: region “M”, with DM, Σa,M, νΣf,M, She found the following solution: φ(x) = A cosh x / LF( ), 0 ≤ x ≤ a,C exp x / LM( )+ E exp − x / LM( )+ G , a ≤ x ≤ b.⎧ ⎨ ⎩ (1) a) At what net rate per unit area are neutrons crossing from region “M” to region “F” across the surface at x=a? b) At what net rate per unit area are neutrons crossing from region “F” to region “M” across the surface at x=a? c) Suppose a plot of the solution near the interface at x=a looked something like this:3 Judging by this picture, which diffusion coefficient is larger, DF or DM? If scattering is isotropic in the lab frame in both regions, which region has the shorter mean-free path? d) Given the solution described by Eq. (1) above, can you deduce what the