B. Rouben
McMaster University
4D03/6D03 Nuclear Reactor Analysis
2015 Sept.-Dec.
2015-11-30
Additional Exercises 1
1. 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 (ignore the
extrapolation distance). The average one-group properties are D = 1.1 cm,  = 2.4,
and f = 0.0025 cm-1.
(a) Calculate the material buckling.
(b) What is the value of a?
(c) If one fission releases 200 MeV on the average, what is the average value of the
flux?
(d) At what rate is 235U fissioning in the entire reactor?
(e) At what rate is 235U being consumed in the entire reactor?
(f) What fraction of neutrons leaks out of the reactor? What fraction is absorbed
within the reactor?
2. An experiment is set up to measure the reactivity worth of a control rod by the source
multiplication method. The reactor used is a cube of side 4 m, with the bottom
parallel to the ground (ignore the extrapolation length). An external neutron source is
introduced in the subcritical reactor, and a detector at the centre of the reactor
indicates 100. Next, a control rod with a reactivity worth of -10 mk is inserted in the
core, and the new reading is 50. Subsequently, the first control rod is removed and a
second rod, of unknown worth, is inserted. The new detector reading is also 50.
However, it is later determined that the operator making the measurements
inadvertently moved the detector before the last reading, so that it was 1 m above the
centre of the reactor when the last reading was taken. What is the reactivity worth of
the second rod?
3. Consider nuclear material with the following properties in 2 energy groups:
a1 = 0.0012 cm-1, 12 = 0.007 cm-1, a2 = 0.0045 cm-1, f2 = 0.0056 cm-1
f1 = 0 cm-1, D1 = 1.08 cm, D2 = 0.90 cm
(a) Calculate the reactivity of an infinite lattice made of this material. Calculate also
the resonance-escape probability in this lattice.
(b) If a homogeneous cylindrical reactor is made with this material, of 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.
(c) How big would a critical cylindrical reactor be, if D=1.1H?
(d) Which group flux is higher? By how much?
4. A homogeneous solution of 235U and H2O contains 10 g of 235U per litre of solution.
Compute:
(a) the atomic density of 235U and the molecular density of H2O
(b) the thermal utilization
(c) the thermal diffusion area and length
(d) the infinite multiplication factor.
Use: a,U-235= 687 b, f,U-235= 587 b, a,H2O= 0.664 b, D = 0.16 cm.
Use235 and 18 for the atomic masses of U-235 and H2O.
5.
Suppose a homogeneous reactor has 2-group lattice properties as follows:
D1
D2
a1
 a2
12
f2
21
f1
1.269 0.937 0.766E-3 0.406E-2 0.740E-2 0.474E-2
0
0
cm
cm
cm-1
cm-1
cm-1
cm-1
(a) What is the infinite-lattice multiplication constant of the lattice?
(b) If no reactivity devices are in the reactor, what must the leakage then be (in
mk) for criticality?
(c) Use the relationship between the finite-reactor and infinite-lattice
multiplication constants to determine the buckling in the critical reactor.
(d) What is the ratio of the thermal flux to the fast flux in the reactor (neglect
terms in B4)?
(e) If this critical reactor is in the shape of a sphere, what is its diameter?
6.
A reactor is in an approach to critical. Boron (which has a reactivity worth of -8
mk/ppm) is being removed at the rate of 0.015 ppm/minute, and at t = 0 an in-core
detector shows a reading of 80 (arbitrary units). At t = 10 minutes, the detector reads
95.
(a) What do you predict the detector reading will be at t = 30 min?
(b) What is the appropriate time to declare the reactor critical?
7.
A point source of neutrons, of strength S = 108 s-1, is located at point (x1, y1, z1) = (10 cm, 20 cm, 0). Calculate the neutron flux and current (magnitude and direction) at
the point (x, y, z) = (50 cm, 40 cm, 0). Take D = 1 cm and L = 5 cm.
8. We want to make a homogeneous critical reactor with nuclear material with the
following properties: f = 0.0043 cm-1, a = 0.0095 cm-1,  = 2.43, and D = 1.10 cm.
Neglect the extrapolation distance in the following.
a) If we make a cubical reactor, what will be the dimension of the side?
b) If we make a spherical reactor, what will be its diameter?
c) Which of these two reactors would have a smaller value of leakage-to-production,
and what will this value be?
9. A homogeneous critical cylindrical reactor has a height-to-diameter ratio of 1.5:1. Its
nuclear properties in 1 energy group are: a = 0.0078 cm-1, f = 0.0085 cm-1, D
=1.08 cm. What are the dimensions of the reactor? [Neglect the extrapolation
distance.]
10. A research reactor consists of a cubical array of natural-uranium rods in a graphite
moderator. Each side of the cube is 6.2 m long. The reactor operates at a steady
power of 17 MW. The average value of f is 2.4*10-3 cm-1. Energy per fission Ef =
200 MeV, 1 MeV =1.6*10-19 J. [Neglect the extrapolation distance.]
(a) Calculate the buckling
(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 U-235 fissioning in the entire reactor (in nuclides/s) and on the
average per cm3 per s?
11. A CANDU fuel bundle contains 18.8 kg(U). It is introduced in a reactor at a location
where its fission power is 450 kW, and it resides 4 months there. It is then moved to
a new position, where its fission power is 830 kW. When it is removed from the
reactor, its exit burnup is found to be 7,500 MW.d/Mg(U). How long did the bundle
reside in the 2nd position?
12. A parallelepiped reactor with sides in the ratio 1:1.36:1.55 was built with material of
properties: =2.47, f= 0.0027 cm-1, a= 0.0066 cm-1, D=1.12 cm
On testing, it was found to be 1 mk supercritical.
What were the reactor’s dimensions?
13. A point source of neutrons, of strength S = 108 s-1, is located at point (x1, y1, z1) = (-10
cm, 20 cm, 0). Calculate the neutron flux and current (magnitude and direction) at the
point (x, y, z) = (50 cm, 40 cm, 0). Take D = 1 cm and L = 5 cm.
14. An infinite plane source of neutrons is in the y-z plane, of source strength S = 106
n.cm-2.s-1. There is also a point source at (x, y, z) = (20 cm, 0, 0), of strength S’= 10 8
n.s-1. Calcutate the total flux and current (magnitude and direction) at (x, y, z) = (10
cm, 0, 0).
15. A cylindrical reactor is to be made with material of the following properties:
a = 0.0822 cm-1, f = 0.034 cm-1,  = 2.44, D = 1.2 cm.
The designer has the choice between the following proportions for the reactor:
Design 1: H = 1.2d or Design 2: H = 0.8d (where d is the cylinder’s diameter)
Which design will require less fuel, and by what ratio (of volumes)?