B. Rouben
McMaster University
4D03/6D03 Nuclear Reactor Analysis
2015 Sept.-Dec.
2015-11-30
Additional Exercises 2
1. The figure below shows the neutron cycle for a finite, critical reactor. The notes in the figure refer to an arbitrary unit of time. Calculate the number of epithermal and fast neutrons captured per unit time.
2. Consider a reactor with a mean neutron-generation time L of 0.0014 s and 1 delayed-neutron-precursor group with a delayed-neutron fraction of 0.0055 and a decay constant of 0.012 s-1.
The reactor is running in steady state with a neutron density n = 4*108 cm-3 and at a neutron power of 1 (arbitrary units), when a perturbation suddenly inserts positive reactivity, and the neutron power jumps to a value of 1.15.
(a) What was the reactivity inserted by the perturbation?
(b) If this reactivity is not countered or changed, when will the neutron power reach 1.35?
(c) In the initial steady state, what was the value of the delayed source, in cm-3.ms-1?
3. A research reactor fuelled with 235U is operated in critical steady state at a power of 40 MW. It is in the shape of a sphere with a diameter of 2.2 m. The average one-group properties are n = 2.4, Sf = 0.0025 cm-1, and D = 1.1 cm.
(a) Calculate the material buckling.
(b) What is the value of Sa?
(c) If one fission releases 200 MeV (and 1 MeV = 1.6*10-13 J), what is the average value of the thermal flux?
(d) What fraction of neutrons leaks out of the reactor?
4. A subcritical reactor is on its approach to critical. At some point in time, a detector in the core has a reading of 5 (arbitrary units). The approach to critical is halted, and to determine the reactivity value, a control rod whose reactivity is known to be -0.2 mk is inserted into the core; the detector then reads 4.5 units.
(a) What was the reactivity of the core without the control rod?
(b) If the control rod is removed and the approach to critical is resumed, what will be the reading of the detector when the core reactivity reaches -0.6 mk?
(c) What is the general criterion which must be satisfied for the calculation method which you used in parts (a) and (b) to be reasonably accurate?
5. The decay constants and direct yields in fission of I-135 and Xe-135 are:
o lI = 2.92*10-5 s-1
o lX = 2.10*10-5 s-1
o gI = 0.0638
o gX = 0.0025
The Xe-135 neutron-absorption cross section and the fission cross section are:
o sX = 3.20*10-18 cm2
o Sf = 0.002 cm-1
At a certain point in a transient, the Xe-135 concentration is 2*1013 nuclides.cm-3, and the rates of production of Xe-135 from I-135 decay and directly from fission are 6*109 cm-3.s-1 and 3*108 cm-3.s-1 respectively.
(a) What is the instantaneous I-135 concentration?
(b) What is the instantaneous flux?
(c) What are the instantaneous net rates of change of the I-135 and Xe-135 concentrations?
6. A certain “standard” CANDU reactor has a fuel-temperature reactivity coefficient of -0.009 mk/oC. When the reactor is at full power (FP), the average fuel temperature is 690 oC. At hot shutdown (0% neutron power), the average fuel temperature is 270 oC.
(a) If the reactor power is increased from FP to 105% FP, estimate the change in reactivity (including its sign), assuming that there is no other change besides a fuel-temperature change.
(b) If coolant-density changes are also considered, do you think that the change in reactivity as the power is increased to 105% FP is likely to be algebraically greater or smaller than the value you calculated in (b)? Explain why.
7. Consider nuclear material with the following properties in 2 energy groups:
Sa1 = 0.0012 cm-1, S1®2 = 0.007 cm-1, Sa2 = 0.0045 cm-1, nSf2 = 0.0056 cm-1
D1 = 1.08 cm, D2 = 0.90 cm
(a) Calculate the reactivity of an infinite lattice made of this material.
(b) Now let’s make a homogeneous cylindrical reactor with this material. Its dimensions are diameter 5.2 m and height 5 m [neglect the extrapolation distance]. Calculate the fast and thermal non-leakage probabilities for this reactor, and its reactivity.
8. (a) At the beginning of life of a CANDU reactor, all fuel is fresh. The excess reactivity of the core is compensated for by doing two things. One of these is to add boron poison to the moderator. What is the second?
(b) Sketch how and explain why the boron concentration will have to evolve over the
first 100 Full-Power Days of operation (at which time, approximately, refuelling
starts).
9. The decay constants and direct yields in fission of I-135 and Xe-135 are:
o lI = 2.92*10-5 s-1
o lX = 2.10*10-5 s-1
o gI = 0.0638
o gX = 0.0025
The Xe-135 neutron-absorption cross section and the fission cross section are:
o sX = 3.20*10-18 cm2
o Sf = 0.002 cm-1
At a certain point in a transient, the Xe-135 concentration is 1.8*1013 nuclides.cm-3, and the rates of production of Xe-135 from I-135 decay and directly from fission are 5.7*109 cm-3.s-1 and 2.8*108 cm-3.s-1 respectively.
(a) What is the instantaneous I-135 concentration?
(b) What is the instantaneous flux?
(c) What are the instantaneous net rates of change of the I-135 and Xe-135 concentrations?
(d) What is the rate of change of the xenon load?
10. In the standard CANDU design:
(a) Describe briefly the two functions of the zone controllers.
(b) Describe briefly three functions of the adjusters
11.
(a) Use the data given at the beginning of Problem 1. If the average neutron flux at full power is 7.0*1013 n.cm-2.s-1, what are the Xe-135 and I-135 concentrations if the reactor is operated at 80% full power?
(b) If from that point the reactor is shut down instantaneously, what are the instantaneous rates of change of the Xe-135 and I-135 concentrations?
(c) In the approximation that these rates of change do not decrease appreciably in the first 10 minutes, what is the fractional increase in the xenon reactivity load at 10 minutes (compared to its initial value)?
12.
(a) Why does the lattice reactivity drop in the first few days of operation of a new CANDU reactor, and then rise?
(b) In which two ways is the excess reactivity of the initial core compensated?
13. The unknown reactivity worth of a control rod is to be measured by the source multiplication method. The reactor used is a parallelepiped with a square horizontal base of side 2 m and a height of 4 m. The base is on the x-y plane, and the sides of the base lie along the positive x and y axes (i.e., one corner of the base is at the origin of co-ordinates).
In steady state with the external neutron source inserted, a detector located at position (x, y, z) = (40 cm, 40 cm, 200 cm) indicates a reading of 80 (arbitrary units). Next, a control rod with a reactivity worth of -0.8 mk is inserted in the core, and the detector reads 74. The known control rod is removed and at the same time the detector is moved to a new position, (x, y, z) = (80 cm, 60 cm, 100 cm). The unknown rod is then inserted and the detector now reads 92. Determine the reactivity worth of the unknown rod.
14. A homogeneous critical cylindrical reactor of diameter 5.2 m is made of nuclear material with the following properties in 2 energy groups:
Sa1 = 0.0011 cm-1, S1®2 = 0.007 cm-1, Sa2 = 0.0048 cm-1, nSf2 = 0.0058 cm-1
D1 = 1.05 cm, D2 = 0.95 cm
(a) What is the axial length of this reactor? (Ignore extrapolation lengths and realise that this reactor is large.)
(b) What is the leakage in this reactor, in mk?
(c) Calculate the values of the fast and thermal non-leakage probabilities. How do you understand that the product of these non-leakage probabilities is not exactly equal to [1-(Leakage expressed as an absolute number)]?
15.
(a) Name the routine perturbation in CANDU reactors which the liquid zone controllers must counteract.
(b) Will leakage be higher in a peaked flux shape or a flattened flux shape? Explain very briefly.
(c) (d) Explain clearly why the fuel average exit burnup is higher when a reactor is refuelled daily than when it’s refuelled in batch mode.
16. Three isotropic point sources of neutrons are located in an infinite non-multiplying medium with the following properties: absorption cross section Sa = 0.04 cm-1 and diffusion coefficient D = 1.3 cm. Each source is of strength 108 neutrons.s-1. The sources are located at the vertices of an equilateral triangle in the x-y plane, with the following (x,y) co-ordinates:
· Source 1 at (0,0)
· Source 2 at (100 cm, 0)
· Source 3 at (50 cm, 50Ö3 cm)
Calculate the neutron flux and the neutron current (magnitude and direction) at the centre of the equilateral triangle.
17.A reactor with 480 channels has a total fission power of 3200 MW. It is operated with a top-to-bottom tilt of 5%, i.e., [Power (top half)-Power (bottom half)]/Total Power = 0.05. Every channel has 12 fuel bundles with 19.2 kg(U) each. The average exit burnup of fuel is 195 MW.h/kg(U) in the top half of the reactor and 188 MW.h/kg(U) in the bottom half.
(a) What is the refuelling rate (in bundles/day) in each half of the reactor, and in the entire reactor?
(b) What is the average residence time of a bundle in the top half, and in the bottom half?
18. A research reactor is in the shape of a parallelepiped with a square base of side 5.2 m and a height of 6.8 m. The reactor is filled uniformly with a fuel of one-group properties nSf = 0.0072 cm-1 (and n = 2.45) and Sa = 0.0070 cm-1. The reactor operates steadily at a fission power of 15 MW. The average value of energy per fission Ef = 200 MeV, and 1 eV =1.6*10-19 J. [Neglect the extrapolation distance.]
(a) What is the value of the diffusion coefficient?
(b) What is the average value of the neutron flux?
(c) What is the maximum value of the neutron flux?
(d) At what rate is the fuel consumed in the entire reactor (in nuclides.s-1) and at the centre of the reactor (in nuclides.cm-3.s-1)?
19. Consider a homogeneous nuclear reactor with one delayed-neutron group. The reactor is cubical, with side equal to 4 m. The neutronic parameters of the reactor are:
Assume that the extrapolated size of the reactor is equal to its physical size.
a) Calculate keff.
b) The reactor is maintained sub-critical by the addition of 10B, which has a microscopic cross section of approximately 4000 b (). If , what is the numerical density of boron atoms?
20. Consider a reactor with six delayed-neutron groups, with the following parameters:
The reactor is initially operating at a steady-state power of 1000 MW. A control rod that was initially in the core is accidentally ejected at t=0, yielding a 2 mk reactivity increase. What is the reactor power immediately after rod ejection? Use the prompt jump approximation.
21. Consider a reactor with one delayed-neutron group, with the following
parameters:
The reactor is initially in steady-state operation at a power of 1000 MW. A control rod with a reactivity worth of 2 mk is inserted in the reactor at t=0. At t=2 s, a second, identical, control rod is inserted. What is the reactor power at t=4 s?
Notes:
Use the prompt jump approximation.
The prompt jump approximation is also valid when the reactor is not initially critical.