6.7 Appendix G: The Neutron Diffusion Equation
221
and N > , we should consequently use values corresponding to the average radial
displacement that a neutron will undergo between its last collision and reaching
the surface at r, that is, their average displacement perpendicular to the spherical
surface. This will presumably be less than λ t due to the neutrons’ random motions.
For the moment, let us represent this average radial displacement as λ r ; how this
is determined is taken up following (6.106) below.
Now, reverse the order of the terms in (6.95), and both multiply and divide by
2 λ r :
⎛
⎜
⎝
net e f f usion rate
inside to outside
at radius r
⎞
⎟
⎠ = −
1
4
A v
(N > − N < )
2 λ r
(2 λ r ).
(6.96)
The square bracket in (6.96) is the change in N divided by the distance over which
that change occurs, that is, the derivative of N with respect to radial distance:
⎛
⎜
⎝
net e f f usion rate
inside to outside
at radius r
⎞
⎟
⎠ = −
1
2
A v λ r
∂ N
∂r
r
= −2 π r
2
v λ r
∂ N
∂r
r
,
(6.97)
where we have substituted for the area of the sphere and used partial derivatives as a
reminder that N is a function of both position and time. Be sure to understand why
factors of 2 were included with the factors of λ r in (6.96): the surfaces of density
N < and N > are each distance λ r from the surface at radius r, and so the distance
over which the change (N > − N < ) occurs is 2 λ r .
We come now to the second sub-step of this part of the derivation. We desire an
expression for the net rate of change of N per unit volume due to random neutron
motions. To do this, apply (6.97) to a spherical shell within the core that extends from
inner radius r to outer radius r + dr, as shown in Fig. 6.10. For neutrons arriving
from inside the shell,
rate o f neutrons f rom
within r entering shell
= −2 π r
2
v λ r
∂ N
∂r
r
.
(6.98)
At the same time, neutrons exit the shell by passing through the surface at r + dr:
rate o f neutrons exiting
shell f rom within
= −2 π (r + dr)
2
v λ r
∂ N
∂r
r +dr
.
(6.99)
Notice that in writing these expressions, we evaluate (∂N/∂r) at the inner and
outer surfaces of the shell. It follows that the net rate of neutron flux into the shell is
given by the entry rate, (6.98), minus the exit rate, (6.99); the overall result could in
fact be a loss (and will be so at the outer surface of the core):
221
and N > , we should consequently use values corresponding to the average radial
displacement that a neutron will undergo between its last collision and reaching
the surface at r, that is, their average displacement perpendicular to the spherical
surface. This will presumably be less than λ t due to the neutrons’ random motions.
For the moment, let us represent this average radial displacement as λ r ; how this
is determined is taken up following (6.106) below.
Now, reverse the order of the terms in (6.95), and both multiply and divide by
2 λ r :
⎛
⎜
⎝
net e f f usion rate
inside to outside
at radius r
⎞
⎟
⎠ = −
1
4
A v
(N > − N < )
2 λ r
(2 λ r ).
(6.96)
The square bracket in (6.96) is the change in N divided by the distance over which
that change occurs, that is, the derivative of N with respect to radial distance:
⎛
⎜
⎝
net e f f usion rate
inside to outside
at radius r
⎞
⎟
⎠ = −
1
2
A v λ r
∂ N
∂r
r
= −2 π r
2
v λ r
∂ N
∂r
r
,
(6.97)
where we have substituted for the area of the sphere and used partial derivatives as a
reminder that N is a function of both position and time. Be sure to understand why
factors of 2 were included with the factors of λ r in (6.96): the surfaces of density
N < and N > are each distance λ r from the surface at radius r, and so the distance
over which the change (N > − N < ) occurs is 2 λ r .
We come now to the second sub-step of this part of the derivation. We desire an
expression for the net rate of change of N per unit volume due to random neutron
motions. To do this, apply (6.97) to a spherical shell within the core that extends from
inner radius r to outer radius r + dr, as shown in Fig. 6.10. For neutrons arriving
from inside the shell,
rate o f neutrons f rom
within r entering shell
= −2 π r
2
v λ r
∂ N
∂r
r
.
(6.98)
At the same time, neutrons exit the shell by passing through the surface at r + dr:
rate o f neutrons exiting
shell f rom within
= −2 π (r + dr)
2
v λ r
∂ N
∂r
r +dr
.
(6.99)
Notice that in writing these expressions, we evaluate (∂N/∂r) at the inner and
outer surfaces of the shell. It follows that the net rate of neutron flux into the shell is
given by the entry rate, (6.98), minus the exit rate, (6.99); the overall result could in
fact be a loss (and will be so at the outer surface of the core):