NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. V : Flux, Current, etc.
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V. Flux, Current, Etc.
I.
Introduction
We are well aware that a reactor’s behavior depends on the production and loss rates of neutrons
in the reactor. We also know that some production and loss is due to neutron-nucleus reactions,
and that some is due to neutron leakage.
It is also true that reaction rates depend on the neutron scalar flux and that leakage rates
depend on the neutron current. In this chapter we study scalar flux, current, and what we can do
with them.
II. Scalar Flux: one-speed (mono-energetic) case
Consider some volume, V, in which the density of neutrons is n neutrons/cm3, with each moving
at speed v but in many different directions:
What is the rate at which neutrons make tracks in this volume? In time Δt,
each neutron travels:
and there are nV of them. Thus,
path-length in Δt =
(1)
or, if we divide by VΔt,
(2)
This is a general result:
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NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. V : Flux, Current, etc.
III. Scalar Flux: distribution of speeds (energies)
Neutrons do not bounce around with a single energy in nuclear reactors –– they have a
distribution of energies.
Question: How many neutrons in a commercial reactor have exactly 1 MeV of kinetic
energy?
Answer:
Note:
Every density is a limit. Mathematically, the density of things at some point x is
where x is a point in phase space and V is a phase-space volume that contains the point x.
If we had asked how many neutrons have energies between 0.99 and 1.01 MeV, the answer
would have been different.
We define the energy-dependent neutron density n(r,E) such that:
n(r,E) d3r dE
= number of neutrons in
(6)
Likewise, we define the energy-dependent scalar flux:
φ(r,E)
≡
(7)
Example
In a certain (imaginary) reactor, the energy-dependent scalar flux in the moderator is given by:
Question: What is the density of neutrons that have energies between 1 and 10 keV?
Solution: Since v(E) = (2E/m)1/2, where m is neutron mass, we have [see Eq. (7)]:
Thus,
answer =
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NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. V : Flux, Current, etc.
36
=
=
IV. Reaction Rates
We already know that
provided that
This holds for the energy-dependent case as well:
energy-dependent reaction-rate
density at position r
=
(8)
reminder –– units:
Important!
1)
φ(r,E) is a density in space and in energy. It is path length rate
It does not make sense to ask what the path-length rate is at r and/or at E. It does
make sense to ask what the path-length rate density is.
2)
Σ(r,E) is not a density in space or in energy. It does make sense to ask what Σ is at
r and at E.
3)
Σs(r,Ei→Ef) is not a density in space or in incident energy, Ei. It is a density in
How many neutrons have outgoing energy of exactly 100 keV after a scattering
event?
Another definition:
φ(r) ≡
(9)
Thus, the total scalar flux is the
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NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. V : Flux, Current, etc.
37
Examples
You have been analyzing a reactor and you have computed the energy-dependent scalar flux,
φ(r,E), everywhere in the reactor and for all neutron energies. You also know all the
macroscopic cross sections everywhere in the reactor.
Q1: How many neutrons are in the reactor?
ntot [neutrons] =
n(r) [n/cm3] =
n(r,E) [n/(cm3-MeV)] =
Thus,
ntot [neutrons] =
Q2: Assuming each fission produces 200 MeV, what is the reactor power?
power [MeV/s] =
energy per fission = 200 [MeV/fission].
fission rate =
fission rate density =
Thus,
power [MeV/s] =
Q3: What is the rate at which neutrons are produced by fission in the reactor?
rate [n/s] =
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NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. V : Flux, Current, etc.
rate density
⇒
38
=
rate [n/s] =
Q4: What is the rate at which neutrons scatter from energies above 1 MeV to energies below 1
MeV in the reactor?
rate [n/s] =
rate density
=
=
⇒
rate [n/s] =
V. Neutron Current and Leakage Rates: one-speed (monoenergetic) case
In this section we will find that the net leakage of neutrons across a surface can be written in
terms of a vector quantity called
We will find that the net leakage rate, across a flat surface whose unit normal is en, is just
where A is the area and J is the neutron net current density [n/cm2-s]. Sometimes we will simply
call this the “neutron current density” or the “net current density” or the “net current” or even
just the “current.” All of these terms refer to the same thing, which most precisely is called the
“neutron net current density.”
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NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. V : Flux, Current, etc.
39
Consider a beam of mono-energetic, mono-directional neutrons crossing a surface:
(Ω is just a unit vector in the direction of the beam.) The number of neutrons crossing the
surface in time Δt is:
number crossing
area A in time Δt =
(10)
where θ is the angle between Ω and ex. Now, because
and because
we have
number crossing
area A in time Δt =
(11)
Thus, the crossing rate per unit area is:
crossing rate
per unit area
=
(12)
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NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. V : Flux, Current, etc.
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Now consider the same beam again, but with a different surface:
The only difference between this case and the last case is the orientation of the surface. In
particular, the unit vector normal to the surface is now ey instead of ex; thus,
Everything else is the same, so
crossing rate
per unit area
=
(13)
In fact, given a surface with any old orientation
we have
crossing rate
per unit area
=
(14)
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NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. V : Flux, Current, etc.
Notice that, no matter what the orientation of the surface was, we needed the quantity
in order to compute the crossing rate. In our simple beam example, this is the net current
density:
J = net current density =
[only for a beam!!!] (15)
This formula is valid only for mono-directional beams.
Note: The net current density has the same units as the scalar flux. It does not have the
same physical meaning. This may seem confusing at first. I’m sorry. That’s just the way it is.
Consider next a control volume in which several different mono-directional beams are
crossing:
In this case, the neutron current density is the sum of that from each beam:
J =
[for 3 beams] (16)
This generalizes to the many-beam case:
J =
[for many beams] (17)
In general, when neutrons are not moving around in beams, the exact formula for J is an
integral over all directions. This exact treatment of J leads to equations that are difficult to
solve, so we often approximate J in terms of the scalar flux using the diffusion approximation.
We will discuss this in detail in a later chapter.
Note that in the many-beam case, the current density from some beams may cancel the current
density from others. For example, given two beams of the same intensity but opposite directions,
we would have:
which means
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NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. V : Flux, Current, etc.
J
=
=
=
[opposing beams of same I] (18)
Thus, J is related to the net rate at which neutrons cross surfaces, which is why we call it the
We need often need to use the net current density to calculate
across surfaces. Here’s how it goes. Consider a point rs on some surface, with outward unit
normal en(rs):
Then
net leakage rate density
outward through surface =
at point rs
(19)
Note that this is an
To get the net leakage rate through some surface, we integrate the net leakage rate density. If
we denote an infinitesimal surface area as d2r, then
net leakage rate
outward through
surface S
=
(20)
It really is that simple!
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NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. V : Flux, Current, etc.
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Examples
You have been analyzing a parallelepiped reactor, which is given by:
–a/2 ≤ x ≤ +a/2 ,
–b/2 ≤ y ≤ +b/2 ,
–c/2 ≤ z ≤ +c/2 ,
and you have somehow figured out that the net current density is:
⎡
πx
πy
πz ⎤
J r = J x, y, z = J 0 ⎢ e x sin
+ e y sin
+ e z sin ⎥ [n/cm2-s]
a
b
c ⎦
⎣
()
(
)
Note: This means that the three components of the net current density are:
Jx
=
Jy
=
Jz
=
J0 sin(πz/c).
Q1: At what net rate do neutrons leak outward through the top (z=+c/2) surface of the reactor?
net leakage rate
outward through
=
top surface
=
=
Q2: What is the total net rate at which neutrons leak out of the reactor?
We will compute leakage out of the bottom (z=–c/2) surface, then write the others by inspection.
net leakage rate
outward through
=
bottom surface
=
=
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NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. V : Flux, Current, etc.
The other surfaces are similar; the final answer is
net leakage rate
out of reactor
= [top + bottom + back + front + right + left] leakage rates
=
=
Q3: At what net rate do neutrons leak from back to front across the surface x=–a/4?
Since we said back to front, the unit normal we need is +ex. Thus,
net leakage rate
to front through
=
surface at x=–a/4
=
=
Note that this leakage rate
That just means more neutrons are flowing to the back than to the front across this surface.
If the question had asked for the net rate of leakage from front to back across the surface
x=–a/4, we would get
of the same magnitude, because we would have used the unit normal –ex instead of ex.
Neutron Current & Leakage Rates: distribution of energies
We know that neutrons don’t travel about monoenergetically –– neutrons in reactors are
distributed in energy. As we did with the scalar flux, we define an energy-dependent net current
density:
J(r,E) dE
= net current density of neutrons whose energy is
in the interval dE about the energy E.
(21)
We see that the product J(r,E)dE has units of net current density, which are:
n/cm2-s.
Thus, the energy-dependent net current density, J(r,E), must have units of
n/(cm2-s-MeV).
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NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. V : Flux, Current, etc.
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We define the total net current density as:
J(r) ≡ “total net current density” ≡ .
(22)
Example
You have been analyzing a parallelepiped reactor, which is given by:
–a/2 ≤ x ≤ +a/2 ,
–b/2 ≤ y ≤ +b/2 ,
–c/2 ≤ z ≤ +c/2 ,
and you have found that the energy-dependent net current density is:
( )
(
)
J r, E = J x, y, z, E = J 0
Question 1:
Answer:
E ⎡
πx
πy
πz ⎤
e x sin
+ e y sin
+ e z sin ⎥ .
⎢
a
b
c ⎦
E02 ⎣
What is the net neutron loss rate due to leakage from the reactor?
First let’s compute net loss rate out of the back face (x=–a/2) of the reactor:
net loss back =
=
=
=
=
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NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. V : Flux, Current, etc.
=
=
Similarly,
net loss front =
The other faces are handled the same way; the final answer is
net loss from reactor =
Question 2:
What are the units of J0? How about E0?
Answer:
1) We know from our answer above that (area)(J0) gives neutrons/time; thus,
2) We know that exponents are dimensionless; thus
Summary
Our basic goal in this course is to gain an understanding of how reactors behave and why they
behave that way. This requires, at the very least, an understanding of neutron production and
loss rates in reactors.
In this chapter we discussed neutron-nucleus reaction rates. We found that
•
reaction rate =
,
•
reaction rate density at r =
•
energy-dependent RRD at (r,E) = Σ(r,E)φ(r,E) , where
•
φ(r,E) = energy-dependent scalar flux [n/(cm2-s-MeV)] ,
•
φ(r,E) = n(r,E)v(E) ,
•
n(r,E) = energy-dependent neutron density [n/(cm3-MeV)] , and
•
v(E) = speed of neutron whose kinetic energy is E.
,
Production of neutrons in reactors is mainly by fission. The production rate due to fission, by
the above equations, is
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NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. V : Flux, Current, etc.
prod rate =
, where ν is # of neutrons/fission.
Loss of neutrons in reactors is by absorption and leakage. By the above equations,
loss rate, absorption =
.
The other neutron-loss mechanism in reactors is leakage, which we also discussed in this chapter.
We found that
•
net outleakage rate =
, where en = outward unit normal,
•
J(r) = (total) net current density =
•
J(r,E) = energy-dependent net current density [n/(cm2-s-MeV)] .
,
The loss rate due to leakage is just the net outleakage rate.
Thus, if we know the energy-dependent scalar flux, the energy-dependent net current density,
and all macroscopic cross sections, we can compute neutron production and loss rates in a
reactor.
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