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Department of Engineering Physics, Faculty of Engineering, Gadjah Mada University (Study Programs of Engineering Physics & Nuclear Engineering)
Jl. Grafika 2, Yogyakarta 55281, (+62 274) 580882, http://www.tf.ugm.ac.id/
Nuclear Reactor Theory
- Nuclear Reactor Analysis -
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FRM-II
2
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The Neutron Flux
The Neutron Flux is the Main Variable in Nuclear Reactor Theory
o To design and analyze a nuclear reactor it is necessary to predict:• How the neutrons are spatially distributed.
• How the neutron population evolves with time.
o An exact calculation would need to track the neutrons as they move in the system.
This is not possible with current computer capabilities
o We need to use Approximations:• Monte Carlo Methods.
• Analytic Neutron Transport Methods.
• Neutron Diffusion Approximation.
3
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The Neutron Flux
In a reactor neutrons have many different energies:
The neutrons move with velocities
The neutrons interact with a probability per unit length
The reaction rate can be written as
The neutron FLUX is
4
3
( ): neutron density per unit energy
( ) : is the the number of neutrons per cm
with energies in the interval [ , ]
n E
n E dE
E dE E dE
( ) cm/sv E1( ) cmE
3 1
0 0( ) ( ) number of neutron i( ) ( ) nteractions cm s( )n E v E EF E dE E dE
( ) = ( ) ( )E n E v E
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The Neutron Flux
Typical Neutron Energy Spectrum
5Ref.: http://www.tpub.com/content/doe/h1019v1/css/h1019v1_138.htm
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FRM-II
6
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The Diffusion Approximation: Fick´s Law
Neutrons DIFFUSE in the medium as Chemical Species do in solution.
o A net flow of neutrons exists from HIGH Flux to LOW Flux regions.
o For a one-dimemsional, one-energy system:
o Jx is the NET number of neutrons that pass per unit time through an area perpendicular to the direction x.
o D is the Diffusion Coefficent (cm)
7
x
dJ D
dx
x
(x)Jx
Gradient of Flux
( , , )x y zJ D D J J J
Neutron Current Density Vector
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The Equation of Continuity
It is the conservation equation of neutrons in a medium
The Total number of neutrons in a Volume V is
The Rate of Change is:
8
Rate of Change in Rate of of
Number of Neutrons in Neutr
Production
Abso
ons in
Rate of of Rate of of
Neutrons in Neu
rption Leakag
trons f o
e
r m
V V
V V
V
( , )V
n t dV r
r( , )V
dn t dV
dt r( , )
V
n tdV
t
r
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The Equation of Continuity
The Rate of Production in the Volume V is
The Rate of Absorption in the Volume V is
Rate of Leakage through the surface A
9
Production Rate = ( , )V
s t dV r
Absorption Rate = ( , ) ( , )a
V
t t dV r r
J
Surface
Jx
Jz
Jy
Leakage Rate =DivergenceTheorem
A V
J n dA J dV
x y zJ n J J J
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The Diffusion Equation
The Continuity Equation
The Diffusion Eq.
10
( , )( , ) ( , ) ( , )a
V V V V
n ts t dV t t dV J dV
t
rr r r
( , )( , ) ( , ) ( , )a
n ts t t t J
t
rr r r
21( , ) ( ,
( , )( ,) ( , ))) ( ,as t t D t
t
t
vt t
r r
rr rr
Fick´s Lawnv
2
a
LD
Diffusion Length
The integrands must satisfy
The integration is over the same volume V
trtrLtrD
trs
t
tr
trDv,,
1
,
,,
,
11 22
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The Diffusion Equation
Boundary Conditionso The neutron flux can be found by solving the diffusion equation.o It requires the specification of BOUNDARY conditions for the FLUX.
• Physical: The flux must be always POSITIVE.The flux must be FINITE.
• Geometry:Vacuum Boundary Condition: Unreflected Core
Interface Boundary Conditions for two adjacent regions
11
( ) 0d is the extrapolation distance 2.13d Dd
Continuity of the Flux: ( ) ( )
Continuity of the Current: ( ) ( )A n B n
A n B nJ J
A Bn
d
x)
d)=0
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The Diffusion Equation
The Diffusion Variables
Validity of the Fick´s Law Approximation:o Fick´s Law IS NOT an exact relation, but an approximation.o It is not valid:
• In a medium that strongly absorbs neutrons (e.g. near control rods).• Within about three mean free paths of either a neutron source or the surface of a medium.• For strongly anisotropic neutron scattering.
12
2Average cosine of scattering:
3Macroscopic Transport Cross-section: (1 )
1Transport Mean Free Path:
Diffusion Coefficient3
tr s
trtr
tr
A
D
Low A
High A
Reflectors have LOW A
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One-group Reactor Equation
The design of a reactor requireso The calculations of the conditions necessary for criticality.o The calculation of the distribution of neutrons to determine the power distribution in the
system:• Establish the thermal conditions.• Determine the needs for heat removal during operation and abnormal conditions.
The simplest equation is for a “bare” Fast Reactoro One-group Flux and Neutronic Parameters
o One-group Reactor Equation:
13
2 1aD s
v t
0
0
( ) ( )One Energy Flux: ( ) One Energy Equivalent X-Section
x
x
E EE E
dEd
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One-group Reactor Equation
The source of neutronso In a steady state reactor the source of neutrons is mainly the fissions in the fuel.o The average number of neutrons per fission is
o The source can be expressed in terms of the rate of absorptions as
o And in Terms of k∞
14
with
a
Fuela
Fuel NonFue
Fu
la
elf a
a
s f
f
.Fuelf
Fuela
For a one-group reactor k f and as k
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One-group Reactor Equationo In Steady-state, if the fission source does not balance the Leakage and the
absorption, the equation is not satisfied.o The source term is multiplied by 1/keff.
o The Buckling is defined as
o And the keff is
15
2 1( ) 0
1
efff akD
2 2gB
2 1 1m f a
eff
BD k
2 1( ) 0
1
efg f a
fkB
D
2 2
neutrons generated
neutrons lostf f
effg a g a
kDB DB
Material Buckling
Geometric Buckling
2 1aD s
v t
and as k
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Criticality of a Bare ReactorThe necessary condition for the reactor to be critical is
In terms of the Buckling for a critical reactor:
o The equation determines the conditions under which a bare reactor is critical.
• For a given geometry, which determines the buckling, the composition can be calculated.
• For a given composition, the “CRITICAL” buckling can be computed and the geometric dimensions obtained.
16
1effk keff accounts also for the leakage
21ff
c
fe
a
kDB
2
1
c
f
f a a
a
DDB
Material Properties
2 22g m cB B B
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One-group Critical Reactor Equation
Examples of Solutions
17
r
R
2 2
2
1
2
1( ) sin( ) with
4 R f
g
Pr A B r A
B B
r E R
R
Power
Energy per fission
The Solution for the Flux is:
Buckling
2 22
1 ( )( ) 0g
d d rr B r
r dr dr
Spherical Critical Reactor The flux is a function only of the radius r
Homogeneous Reactor
The power is given by the integral
2
04 ( )
R
R fP E r r dr There are many possible values of B that will satisfy the boundary conditions, but the geometrical buckling is the FIRST eigenvalue B1
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One-group Critical Reactor Equation
Examples of Solutions
18
z
r
( , )r z
H/2
H/2
R
22
2
1 ( , ) ( , )( , ) 0m
r z r zr B r z
r r r z
22 2
( , ) ( ) ( )
rg z
r z r z
B
R
BB
Z
22
2
0
2
( , )
3.63
2.4
2
cos( )
05
r
r
R f
z
z
r z A r z
PA
V E
R d
B
B
J B
BH d
Finite Cylindrical Critical ReactorThe flux is a function of the radius r and z
Two Functions:
The solution:
Homogeneous Reactor
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One-group Critical Reactor Equation
Maximum-to-Average Flux and Powero The maximum value of the flux max in a uniform bare reactor is always found at the
center.o The power density is also highest at the center.o The maximun-to-average flux ratio is a measure of the overall variation of the flux
in the system.
For a spherical bare reactor
19
max 2 30
sin( / )lim
4 4rR f R f
P r R P
E R r E R
1avg V
avgR f
R f V
dV PVE VP E dV
Bare Spherical React
3max
or 3.293avg
Too large for a real reactor.Real reactors have FLATTER Flux distributions by using reflectors and distributing the fuel.
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Multi-group Reactor EquationFor thermal reactors and for accurate
solutions it is necessary to solve the diffusion equation the energy dependency to obtain
o More precise description of Cross Section energy dependency.
o More accurate reaction rates (fission, absorption, scattering, etc.)
o The process of moderation and resonance absorption.
o The thermal and fast fission rates.
The energy spectrum is divided into “ENERGY GROUPS”: g1, g2,…,gN
20
g1g2gN ……..
Discretization of the Neutron Energy for Multi-group Calculations
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Multi-group Reactor EquationThe “transfer” of neutrons between groups is accounted for by:
o Scattering Cross-sections (Transfer X_sections)o Fission Spectrum
The Multi-group diffusion equations are:
For Fluxes and X-sections defined as:
21
Group 1 (g1)
Group 2 (g1)
Group n-1 (gn-1)
Group N (gN)
….
Ene
rgy
Incr
easi
ng
Energy Groups for a N-Group Diffusion Calculation
12
1 1
0
1,...,
gN
g g gg ag hh g h
h gg h g s
N
D
g
Transfer out of gTransfer into g
( )g gE dE
1( ) ( )g g
g
dE E E
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Multigroup Diffusion Core Analysis Codes
Modern Core Analysis Codeso They use the Multi-group Diffusion Equations in two or several groups
• Two to six Groups: Fast and Thermal + additional resonance region groups.
• Cross sections are obtained form Advanced Transport based Lattice-Codes (e.g. CASMO-4, WIMS, NEWT):
Energy Averages maintaining Reaction Rates.Spatial-Material Averages: Heterogeneous Cores.
• Corrections to the Diffusion Approximation:Neutronic Information from Lattice-Codes with Neutron Transport Corrections.Advanced formulations of NET NEUTRON CURRENTS across interfaces and in
highly absorbing regions.Algorithms to “reconstruct” the local flux at the fuel rod level.
o Examples: SIMULATE, DIF3D, PARCS.
22
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State-of-the-Art Nodal Methods
23
Ref.: www.fz-juelich.de/ ief/ief-6/2/htr2-flu.html
12
1 1 , ,
0
1,...,
gN
g g g hh g gg ag g hh g h i j k
sD
g N
Node i,j,k
, ,
,
i k
g z
jJ
, ,
,
i k
g z
jJ
, ,
,
i k
g y
jJ , ,k
gi j
Ref.: http://www.polymtl.ca/nucleaire/en/GAN/GAN.php
The Multi-group equations are solved for each node i,j,k in which the reactor is divided.The nodes “homogenize” the heterogeneous reactor.
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Neutron Transport
Transport theory is based on the Boltzmann Equation developed for the kinetic theory of gases.
o The development of nuclear reactors in the 1940 applied the equation to the transport of neutrons in • Reactor design and• Radiation shielding.
o Analytical solutions are very difficult for real 3D-configurations.
o Today, the Transport Equation is solved numerically by discretizing the• Angular, • Energy, and • Time Dependence of the neutron flux and the cross sections.
It is a more accurate description of the neutron
field than the diffusion equation.
24
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Neutron TransportNeutron Transport Methods account for the angular direction
25
( , , ) ( , , )r E vN r E
1( , ) ( , , , )
( , , , )
( , , ) ( , , , )
( ) ( , ) ( , , , )
total
ext
s
f
r E r E tv t
q r E t
dE d r E E r E t
E dE r E d r E t
Fission Source
Scattering
External Neutron Source
Time variation and removal of neutrons
Angular Flux and Neutron Density
z
x
y
dV
dA
is a Solid Angle
d
r
