Lecture 22
Problem Set-5
Objectives
In this lecture you will learn the following
To illustrate the application of material studied in the last few classes.
First we shall illustrate application of diffusion theory with source in the system.
Then we illustrate application of concepts for reactor problems.
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Lecture 22
Problem Set-5
Question-1
1. An infinite bare slab of moderator of thickness ‘2a’ contains uniformly distributed
sources emitting S’’’ neutrons/cm 3-s. Show that the flux in the slab is given by,
where ‘x’ is measured from the centre of the slab and ‘d’ is the extrapolated length.
The governing equation for this source problems
The solution for the homogeneous part can be
written as
Since the source term is constant, the particular integral shall also be a constant.
Hence the solution will be of the form
First we shall obtain C.
Substitution of the solution into the governing equation gives
The boundary conditions for the problem are
At x = 0, J = dΦ/dx = 0, x = a+d, Φ = 0
The symmetric Boundary Condition implies
Thus
Using the flux vanishing at extrapolated boundary, we get
As
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Lecture 22
Problem Set-5
Question-2
Calculate the fuel utilization factor for a fast reactor consisting of a mixture of liquid
sodium and plutonium, in which the plutonium is present to 3.0 w/o. The density of the
mixture is approximately 1.0 g/cm3 . The following properties can be used
Consider 1 cm 3 of the solution.
The mass of the solution = 1.0 g.
Mass of Pu = 0.03 g.
Mass of Na = 0.97 g.
Number density of Pu = 0.03 x 6.023 x 1023 / 239
= 7.56 x 1019 cm-3
Number density of Na = 0.97 x 6.023 x 1023 / 23
= 2.54 x 1022 cm-3
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Lecture 22
Problem Set-5
Question-3
A bare cylindrical reactor of height 200 cm and diameter 100 cm is operating at a steady
power of 200 MW. If the origin is taken at the centre of the reactor, what is the power
density at the point r = 7 cm, z = - 22.7 cm?
The flux variation in a cylindrical reactor is given by
For the problem, r = 7 cm, R = 50 cm, z = -22.7 cm, H = 200 cm
Since power density is proportional to flux
Further, Peak Power to average power = peaking factor.
For cylindrical reactor this is equal to 3.65.
Average power density = 200 MW/ Volume of reactor.
Volume = π x 50 x 50 x 200 = 1.57 x 106 cm3.
Hence, Average Power Density = 127.3 W/cm 3.
Hence Peak Power Density = 464.7 W/cm 3.
Hence Power density at (7, -22.6) = 464.7x0.909=422W/cm 3.
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Lecture 22
Problem Set-5
Question 4
A bare thermal reactor in the shape of a cube consists of a homogeneous mixture of
U235 and graphite. The ratio of atom densities of NFuel /N moderator = 1.0 x 10-5. Using
the two group theory, calculate the critical dimensions. Use the following properties,
Take pε = 1, η = 2.05, L 2 3500 cm2 , τ = 0 cm2 Material
U235 640
Graphite
0.0034
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