DIFFUSION OF NEUTRONS

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DIFFUSION OF NEUTRONS

• OVERVIEW
• Basic Physical Assumptions
• Generic Transport Equation
• Diffusion Equation
• Fermis Age Equation
• Solutions to Reactor Equation

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Basic Physical Assumptions

• Neutrons are dimensionless points
• Neutron neutron interactions are neglected
• Neutrons travel in straight lines
• Collisions are instantaneous
• Background material properties are isotropic
• Properties of background material are known and
  time-independent

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Initial Definitions
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Angular Flux and Current Density
J
dS
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Generic Transport Equation
Arbitrary volume V
We ignore macroscopic forces
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Generic Transport Equation
Gauss Theorem
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Substantial Derivative
z
r
y
x
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Transport (Boltzmann) Equation
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Collision Term
z
r
y
x
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Neutron Transport Equation
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Boundary Condition
Outgoing direction
Outward normal
O
Incoming direction
ns
r
Volume V
O
Rs
Surface S
V
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Delayed Neutrons
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Difficulties

• Mathematical structure is too involved
• Mixed type equation (integro-differential), no
  way to reduce it to a differential equation
• Boundary conditions are given only for a halve of
  the values
• Too many variables (7 in general)
• Angular variable

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Angular Measures
180 Solar disks
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Plane Angles
R
f
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Solid Angles
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One-Group Diffusion Model

• Infinite homogeneous and isotropic medium
• Neutron scattering is isotropic in Lab-system
• Weak absorption Sa ltlt Ss
• All neutrons have the same velosity v. (One-Speed
  Approximation)
• The neutron flux is slowly varying function of
  position

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Derivation
Number of collisions in dV
Neutrons scattered towards dA
Neutrons through dA per 1 second
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Taylors series at the origin
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Ficks Law
CM-System ? Lab-System
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Transport Mean Free Path
Y
Y
Y
lscosY
lscosY2
ls
ltr
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Diffusion Equation
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Leakage Rate
z
Jz
dz
dx
(x,y,z)
dy
y
x
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Diffusion Equation
Time-dependent
Time-independent
Time-independent from a steady source
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Laplaces Operator
Cartesian geometry
Cylindrical geometry
Spherical geometry
n 0 for slab n 1 for cylindrical n 2 for
spherical
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CONDITIONS 1. The neutron flux finite and
non-negative. 2. F - symmetric if there is any
symmetry in the system 3. Boundary conditions
for interface between two different media
neutron flux and neutron current density are
continuous 4. No return from a free surface -
the flux becoming zero at extrapolated length.
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Boundary Condition
Transport eq.
Free surface
Diffusion eq.
Straight line extrapolation from x 0 towards
vacuum
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Plane Infinite Source in Infinite Medium
Transport equation
Q0
x 0
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Point Source in Infinite Medium
r
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Plane Infinite Source in Slab Medium
Infinite
Q0
Slab
x 0
x a/2
x -a/2
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Plane Infinite Source with Reflector
Q0
1
2
2
1
Reflector
Reflector
a
Bare slab
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Age of Neutrons
Energy

• q(E) - number of neutrons, which per
  cubic-centimeter and second pass energy E.
• q(E) ncm-3 s-1
• X-sections depend on E D(E),Ss(E),...

Q
E0
q(E)
E
Slowing down medium
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Fermis Age Equation
EdE
q(EdE)
E
q(E)
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Fermis Age Equation (cont)
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Solutions to the Age Equation
No absorption
x 0
r
No absorption
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Slowing Down Density for Different Fermis Ages
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Migration Area (Length)
Thermal neutron absorbed
Fast neutron borne
r
rs
rth
Fast neutron thermalized
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Diffusion and Slowing Down Parameters for Various
Moderators
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Neutrons in Multiplying Medium
Assumption
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Principles of a Nuclear Reactor
Leakage
E
N2
2 MeV
N1
Fast fission
n n/fission
Energy
Slowing down
Resonance abs.
? 2.5
Non-fuel abs.
Non-fissile abs.
1 eV
Fission
200 MeV/fission
Leakage
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Criticality Condition
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Solution of a Reactor Equation
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Rectangular
Cylinder
Sphere
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Critical Size of a Reactor
We assume bare homogenous reactor For thermal
neutrons we get
Slowing down neutrons
Assumption Reactor is sufficiently big to treat
neutron spectrum independently of space variables
At the beginning slowing down density is t0
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For t gt 0 one has to take into account resonance
capture through p resonance passage factor.
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Volume of an cylindrical reactor with buckling
derived from a critical equation the smallest
critical size
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CRITICALITY EQUATION - physical interpretation
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Thermal leakage
Thermal non - leakage factor