XII.1 XII. NEUTRON TRANSPORT THEORY Our goal is to determine the distribution of neutrons as a function of time, their position in space, and their velocities, i.e., we want to know . If we know the distribution of neutrons in a nuclear reactor, we can determine the space/time distribution of various nuclear reactions. For example, the power production and its space/time distribution in the reactor core is determined by the space/time distribution of fission reactions. Based on the concepts of the neutron number density, the neutron cross sections, and the neutron reaction rates that were introduced in Chapter II, we can derive the so called “neutron transport equation” in order to describe the behavior of neutrons (“neutron transport”) mathematically. We will find out that the equation that governs the behavior of neutrons has the form of a linearized Boltzmann equation. First, we will introduce the basic assumptions, and then give the basic defi- nitions of various quantities and their physical interpretation. We will then derive the most gen- eral time-dependent form of neutron transport equation and discuss the applicable boundary conditions. Several simplified examples will be discussed. In the case of isotropic scattering and neutron sources we will derive the integral form of the neutron transport equation for angular and scalar flux, as well as for infinte and finite geometries. Finaly, we will present the relationship between the adjoint and forward space in a unified, yet simple formulation that might be helpful for understanding the physical meaning of adjoint functions, deducing the equation satisfied by these functions and finding the adjoint space formulation for a variety of applications. In the chapters that follow we will discuss various approximations of the general neutron trans- port equations, their validity, and their usefulness in numerical solution methods of neutron trans- port problems. CHAPTER TWELVE nrvt , , ( )
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
XII. NEUTRON TRANSPORT THEORY
Our goal is to determine the distribution of neutrons as a function of time, their position in space,
and their velocities, i.e., we want to know . If we know the distribution of neutrons in a
nuclear reactor, we can determine the space/time distribution of various nuclear reactions. For
example, the power production and its space/time distribution in the reactor core is determined by
the space/time distribution of fission reactions.
Based on the concepts of the neutron number density, the neutron cross sections, and the neutron
reaction rates that were introduced in Chapter II, we can derive the so called “neutron transport
equation” in order to describe the behavior of neutrons (“neutron transport”) mathematically. We
will find out that the equation that governs the behavior of neutrons has the form of a linearized
Boltzmann equation. First, we will introduce the basic assumptions, and then give the basic defi-
nitions of various quantities and their physical interpretation. We will then derive the most gen-
eral time-dependent form of neutron transport equation and discuss the applicable boundary
conditions. Several simplified examples will be discussed. In the case of isotropic scattering and
neutron sources we will derive the integral form of the neutron transport equation for angular and
scalar flux, as well as for infinte and finite geometries. Finaly, we will present the relationship
between the adjoint and forward space in a unified, yet simple formulation that might be helpful
for understanding the physical meaning of adjoint functions, deducing the equation satisfied by
these functions and finding the adjoint space formulation for a variety of applications.
In the chapters that follow we will discuss various approximations of the general neutron trans-
port equations, their validity, and their usefulness in numerical solution methods of neutron trans-
port problems.
CHAPTERTWELVE
n r v t, ,( )
XII.1
XII.1. Basic Assumptions, Definitions, and Physical Interpretations
XII.1.1 Basic Assumptions
In order to simplify our derivation of the neutron transport equation, we introduce the following
assumptions and simplifications:
• The Boltzmann equation that is used in the kinetic theory of gases is non-linear. However, in the case of neutron transport, the neutron density is so much smaller than that of the scattering atoms, that the neutron-neutron interactions can be neglected. In addition, we can assume that the equilibrium distribution of the scatter-ing centers (atoms) is unaffected by the presence of the neutrons. These assumptions enable the Boltzmann equation to be linearized.
• Neutrons may be considered as “mathematical” points, that travel in straight lines between the collision points. In addition, the collisions may be considered instanta-neous.
• The gravitational force (and any other force) acting on “free” neutrons is neglected in comparison with the nuclear forces.
• The decay of a “free” neutron is neglected (T1/2 ~ 11 minutes).
• Thermal motion of the target nuclei is neglected.
• The material properties are assumed to be isotropic (thus, the differential scattering cross section depends only on the angle between incoming direction and outgoing neutron direction).
• The cross sections for all materials are assumed to be know and independent of time.
• We are solving for the statistically “expected” or “mean” values of neutron density distribution.
XII.1.2 Neutron Number Density
The total “expected” number of neutrons in the six-dimensional “volume element” (dr dv) at time
t is given by
(XII. 1)
which defines the expected number of neutrons in dr around r with velocities in dv around v at time
t. The graphical representation of this six-dimensional phase-space volume element is given in
Figure XII.1.
However, the total number of neutrons in the six-dimensional “volume element” must be the
same, regardless of our definition of the phase-space volume, namely
n r v t, ,( )drdv
XII.2
. (XII. 2)
FIGURE XII.1. Graphical representation of the six-dimensional phase-space volume element (dr dv)
In our derivation of the neutron transport equation, we will be using the energy-dependent angu-
lar neutron density as defined in Chapter II:
(XII. 3)
as the number of neutrons per unit volume around r, per unit energy around E, per unit solid angle
around Ω, at time t. The graphical representation of this six-dimensional phase-space volume ele-
ment (dr dE dΩ)is presented in Figure XII.2.
n r v t, ,( )drdv n r v Ω t, , ,( )drdvdΩ n r E Ω t, , ,( )drdEdΩ= =
FIGURE XII.2. Graphical representation of the six-dimensional phase-space volume element (dr dE dΩ)
XII.1.3 Angular neutron flux
We have already introduced the angular neutron flux as
ψ(r,E,Ω,t) = v(E) n(r,E,Ω,t) [neutrons / (cm2 s MeV ster)]
There are two common definitions for the angular neutron flux
ψ(r,E,Ω,t) dΩ dE is the total number of neutrons passing at r through an area of 1 cm2, perpendicular to Ω, per second with energy in dE around E with directions in dΩ round Ω at time t, or is the path length per unit volume about r, passed by neutrons with energies in dE about E and direction dΩ about Ω, per second at time t.
FIGURE XII.3. Illustration of the physical interpretation of angular flux
XII.1.4 Scalar neutron flux
In most cases for the calculation of the reaction rates we are not interested in angular dependence
of the flux, but in the combined effect of all neutrons coming from all directions. In this case, we
define the scalar neutron flux, or simply the neutron flux:
φ(r,E,t) = scalar flux [neutrons/(cm2 s MeV)]
(XII. 4)
Physically,
φ(r,E,t) dE = path length per unit volume about r, traveled by neutrons per second with energies in dE about E, at time t.
Ω
dΩr
1 cm2
φ r E t, ,( ) ψ r E Ω t, , ,( ) Ωd4π∫=
XII.4
We can also interpret the neutron flux as the number of neutrons that penetrates a sphere of a 1-
cm2 cross sectional area, located at r, per second with energies in dE about E, at time t.
FIGURE XII.4. Illustration of the physical interpretation of the scalar flux
The total neutron flux is obtained when the energy-dependent scalar flux is integrated over
energy:
in [neutrons/(cm2 s)] (XII. 5)
We can also define the thermal or the fast neutron flux by integrating over thermal or fast energy
range:
(XII. 6)
(XII. 7)
The scalar flux can be multiplied by macroscopic cross sections to obtain the reaction-rate densi-
ties. (Recall from Chapter II that the macroscopic cross sections are reaction probabilities per unit
path length.) Thus, we can define the energy dependent fission rate density at time t as:
[reactions/(cm3 s MeV)] (XII. 8)
The total fission rate in the volume V at time t is defined as:
[reactions/s] (XII. 9)
1 cm2
φ r t,( ) φ r E t, ,( ) Ed
0
∞
∫=
φth r t,( ) φ r E t, ,( ) Ed
0
Eth
∫=
φf r t,( ) φ r E t, ,( ) Ed
Eth
∞
∫=
F f r E t, ,( ) Σ f r E,( )φ r E t, ,( )=
F f t( ) r EΣ f r E,( )φ r E t, ,( )d
0
∞
∫dV∫=
XII.5
XII.1.5 Angular current density
Now, let us consider an arbitrary volume V surrounded by surface S (Figure XII.5).
FIGURE XII.5. An arbitrary volume V with surface S
At each point let es be the outer surface normal (a unit vector). Thus, if > 0, the neu-
trons are streaming out of the volume V through the surface element dS, and for < 0, the
neutrons are streaming into the volume V through the surface element dS. We can define the
angular neutron current (a vector) as
j(r,E,Ω,t) = Ω ψ(r,E,Ω,t) [neutrons/(cm2 s MeV sterad)]
Physically, the following dot product (a scalar)
es•j(r,E,Ω,t) dS dΩ dE
can be interpreted as the number of neutrons crossing the surface element dS about r, with ener-
gies in dE about E and direction dΩ about Ω, per unit second at time t. Positive value means the
neutron flow is across dS in direction of es; negative value means the flow is in direction of –es.
We can also define the angular current density (a scalar) as:
The angular current density can be used to determine the neutron leakage across surfaces.
XII.1.6 Net current density
The "net motion" of neutrons through some surface element is described by the net neutron cur-
rent density (a vector), which is defined as the integral over all directions of the angular current
density (a vector):
Ω
es
VS
dS
r S∈ es Ω•
es Ω•
XII.6
(XII. 10)
Physically, the following dot product [ es•J(r,E,t) dS dE ] can be interpreted as the net number of
neutrons crossing the surface element dS about r, with energies in dE about E and direction dΩabout Ω, per unit second at time t. Positive value means flow is across dS in direction of es; neg-
ative value means flow is in direction of –es.
The net current density (a scalar) can be defined as
(XII. 11)
The net current density can be used to determine a net leakage across surfaces. The detailed
energy dependence can be reduced by integrating over all energies, in order to obtain the energy-
independent net current density (a vector):
[neutrons/(cm2 s)] (XII. 12)
or the energy-independent net current density (a scalar):
(XII. 13)
XII.1.7 Partial current density (a scalar)
We can also define the neutron flow through the surface element dS in positive or negative direc-
tion with respect to the surface normal es
J±(r,E,t) = partial current density in direction ± es
Thus,
J±(r,E,t)dS dE
can be interpreted as the number of neutrons crossing the surface element dS in the direction ±es,
where es is normal to dS about r, with energies in dE about E, per second at time t. Partial cur-
rents are always positive scalars.
J r E t, ,( ) j r E Ω t, , ,( ) Ωd4π∫ Ωψ r E Ω t, , ,( ) Ωd
4π∫= =
J r E t, ,( ) es J r E t, ,( )• es Ω• ψ r E Ω t, , ,( ) Ωd4π∫= =
J r t,( ) J r E t, ,( ) Ed
0
∞
∫ E ΩΩψ r E Ω t, , ,( )d4π∫d
0
∞
∫= =
J r t,( ) es J r E t, ,( )• Ed
0
∞
∫ E Ω es Ω•( )ψ r E Ω t, , ,( )d4π∫d
0
∞
∫= =
XII.7
(XII. 14)
(XII. 15)
Then, the net current density (a scalar) can be calculated from the partial current densities as:
(XII. 16)
or
(XII. 17)
where we integrated over all energy range. The partial current density can be used to determine a
leakage across surfaces in a particular direction (in or out).
XII.1.8 Net neutron current (a scalar)
In the reactor physics we often use the "net current" for the quantity that is actually the net current
density as defined above. The "net current" is defined as
(XII. 18)
which is the total number of neutrons that are crossing the surface S per unit time, or the net rate
with which the neutrons are crossing the surface S. The dimension of the "net current" is [neu-
trons/s].
J+
r E t, ,( ) es Ω• ψ r E Ω t, , ,( ) Ωdes Ω 0>•
∫=
J–
r E t, ,( ) es Ω• ψ r E Ω t, , ,( ) Ωdes Ω 0<•
∫=
J r E t, ,( ) es J r E t, ,( )• J+
r E t, ,( ) J–
r E t, ,( )–= =
J r t,( ) es J r t,( )• J+
r t,( ) J–
r t,( )–= =
J t( ) E J r E t, ,( ) dS•S∫d
0
∞
∫ E J r E t, ,( ) es
• SdS∫d
0
∞
∫= =
XII.8
XII.2. Integro-Differential Form of Neutron Transport Equation
XII.2.1 Derivation
We would like to write a balance equation that describes the time rate of change of the neutron
population in the volume V (but only those with energies in dE around E and with directions in
dΩ round Ω), as the difference between the production rate and the loss rate:
(1) = = Production rate - Loss rate = (3) - (2) (XII. 19)
There are several contributions to the loss (2) and production (3) rates:
(2) Loss Rate = (4) Rate at which neutrons collide and leave V by absorption
or by scattering out of dE dΩ+ (5) Net leakage rate of neutrons with energies in dE around E and
with directions in dΩ round Ω through S.
(3) Production Rate = (6) Rate at which neutrons in V scatter into dE dΩ+ (7) Rate at which prompt fission or (8) delayed neutrons are born in V
with energies in dE around E and with directions in dΩ round Ω+ (9) Rate at which neutrons from “external” source [source that does
not depend on the n(r,E,Ω,t] are born in V with energies in dE around E
and with directions in dΩ round Ω.
Let’s determine each of these contributions:
(4) Loss rate due to neutron collisions in V for neutrons with energies in dE around E and with
directions in dΩ round Ω at time t:
(XII. 20)
(5) Net leakage rate of neutrons through S:
We need an expression for the net rate at which neutrons stream across a surface. Consider neu-
trons impinging on a surface element dS at time t, and count how many have crossed at time t+∆t
(Figure XII.6). The number of neutrons that pass through the surface element dS in the direction
of Ω (with speed v) during ∆t is equal to the number of neutrons contained in the volume element
t∂∂
n r E Ω t, , ,( ) rdV∫ dEdΩ
Σt r E,( )vn r E Ω t, , ,( ) rdV∫ dEdΩ
XII.9
(∆d dS). The “effective” distance ∆d that neutrons traveled in the direction of the normal to the
surface is
(XII. 21)
Thus, the number of neutrons contained in the volume element (∆d dS) at any time t with energy
in dE around E and direction in dΩ around Ω is:
(XII. 22)
Then, the number of neutrons at r with energy in dE around E and direction in dΩ around Ω that
pass through a surface element dS with outer normal es per unit time at time t is
(XII. 23)
FIGURE XII.6. Neutron streaming through the surface element dS
The net leakage rate of neutrons with energies in dE around E and with directions in dΩ around Ωthrough S is then:
(XII. 24)
Let’s recall the Divergence Theorem:
If V is a simple solid region whose boundary surface S has positive (outward) ori-
entation, and F(r) is a vector function whose (x,y,z) components have continuous
partial derivatives on an open region that contains V, then
∆d v∆t αcos( ) v∆t Ω es•( )= =
n r E Ω t, , ,( ) v∆t Ω es•( )( )dS
vn r E Ω t, , ,( ) Ω es•( )dS[ ] dEdΩ
α
time ttime t + ∆t
es
Ω
distancetravelled = v∆t
∆d = v∆t cosα= v∆t Ω•es
dS
∆d
“effective”distance
vn r E Ω t, , ,( ) Ω es•( )dSS∫ dEdΩ
XII.10
(XII. 25)
where the vector differential operator "del"is defined as
(XII. 26)
Also, the divergence of F(r) is a scalar function given by
.
Now we can write Eq. XII.24 in terms of a volumetric integral:
or, having in mind that "del" operator is the spatial derivative that does not operate on Ω or v, we
can rewrite it as
(XII. 27)
where is the gradient vector of the neutron nnumber density.
(6) Production rate in V due to neutrons scattering into dE around E, dΩ around Ω from other
energies E’ and directions Ω’ at time t:
(XII. 28)
(7) Production rate at which prompt fission neutrons are born in V with energies in dE around E
and with directions in dΩ round Ω at time t:
(XII. 29)
where: is the total number of neutrons produced in a fission event that is caused by
a neutron with energy E’
es F r( )• SdS∫ ∇ F r( )• rd
V∫=
∇
∇ ix∂
∂j
y∂∂
kz∂
∂++=
divF r( ) ∇ F r( )•x∂
∂Fx
y∂∂Fy
z∂∂Fz+ += =
vn r E Ω t, , ,( ) Ω es•( )dSS∫ dEdΩ ∇ Ω vn r E Ω t, , ,( )( )• rd
V∫ dEdΩ=
vn r E Ω t, , ,( ) Ω es•( )dSS∫ dEdΩ vΩ ∇• n r E Ω t, , ,( ) rd
V∫ dEdΩ=
∇ n r E Ω t, , ,( )
dE' Σs r E, ' E Ω' Ω•,→( )v'n r E' Ω' t, , ,( ) Ω'd4π∫
0
∞
∫ rdV∫
dEdΩ
dE' 1 β E'( )–( )ν E'( )Σ f r E, ' E→( )v'n r E' Ω' t, , ,( ) Ω'd4π∫
0
∞
∫ rdV∫
χ p E( )
4π---------------dEdΩ
ν E'( )
XII.11
is the total fraction of delayed neutrons in a fission event caused by a neu-
tron with energy E’,
Is the total fraction of all fission neutrons that are prompt
The spectrum (energy distribution) of the prompt fission neutrons. It is nor-
malized to 1:
(XII. 30)
We also assumed that the fission neutrons are emitted isotropically (the 1/4π term).
(8) Production rate at which delayed fission neutrons are born in V with energies in dE around E
and with directions in dΩ round Ω at time t:
(XII. 31)
where is the probable number of fission products in precursor group j, in dr
around r, at time t.
is the decay constant of the delayed neutron precursor in group j,
the fraction of delayed neutrons in a fission event, caused by a neutron with
energy E’, that are emitted from the j-th precursor group, with j = 1, 2, ..., 6,
is the total fraction of delayed neutrons in a fission event,
caused by a neutron with energy E’,
The spectrum (energy distribution) of the delayed neutrons in a fission
event emitted from precursor group j. It is normalized to 1:
(XII. 32)
β E'( )
1 β E'( )–
χ p E( )
χ p E( ) Ed
0
∞
∫ 1=
χd j, E( )4π
------------------ λ jC j r t,( ) rdV∫ dEdΩ
C j r t,( ) rd
λ j
β j E'( )
β E'( ) β j E'( )j 1=
6
∑=
χd j, E( )
χp,j E( ) Ed
0
∞
∫ 1=
XII.12
(9) “External” source rate at which neutrons are born in V with energies in dE around E and with
directions in dΩ round Ω at time t:
(XII. 33)
Often we can assume that the external source is isotropic:
such as the presence of Cf source. Another posibility is a monodirectional beam of neutrons com-
ing from some other source (for example, an accelerator):
.
Now, collecting the terms (1) through (9), we have: