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Session 3 : Reactor Physics
• Nuclear reactor theory is used to predict the behavior of neutrons in
nuclear reactors.
• The concepts in nuclear reactor theory such as nuclear reactions,
fission process, neutron cross sections, and the moderation and
diffusion of neutrons will be introduced.
• The neutron diffusion equation can be used to predict the nuclear
power distribution in a nuclear reactor and the neutron multiplication
factor.
• The nuclear reactor theory treats the static and transient behavior of
nuclear reactors as well as nuclear reactor design and operations.
Reactor Physics
• In this session, we will discuss the followings :
– Nuclear Reaction
• Compound Nucleus Model
• Classification of Nuclear Reaction
• Types of Nuclear Reaction
• Mass-Energy Balance in Nuclear Reaction
– Neutron Cross Section
– Reaction Rates
– Neutron Diffusion
– Neutron Life Cycle
– Critical Equations
– Reactor Dynamics
– Reactor Operation
Nuclear Reaction-Introduction
• The design of all nuclear systems – reactors, radiation shields,
isotopic generators, and so on – depends fundamentally on the way
in which nuclear radiation (neutrons, gamma rays, and charged
particles) interacts with matter.
• This section will present the mechanism of neutron interaction with
matter and various types of them.
Nuclear Reaction
• Nuclear Reaction
– Consider the incident particle a of mass m0 and velocity v0 strikes the
stationary (v=0) target nucleus X of mass M.
– As the result of the reaction, the target nucleus transforms, a new product
nucleus Y and a particle b is emitted.
a + X Y + b or X(a, b)Y
where a = incident particle (p, n, d, α, γ)
X = target nucleus
b = emitted particle (p, n, d, t, α, γ)
Y = product nucleus
mass m0
velocity v0
mass M
velocity v=0
Example: Nuclear Reactions
■ Fission
■ Radiative capture
■ Charged particle ejection
*
1 235 236 92 141 1
0 92 92 02.4n U U Kr Ba n E
*
1 238 239 239
0 92 92 92n U U U
*
1 10 11 4 7
0 5 5 2 3n B B Li
Example : Nuclear Reactions
• The First Nuclear Reaction (Rutherford, 1919)
• Neutron Discovery (Chadwick, 1932)
• Nuclear Reaction with Accelerator (Cockroft-Walton, 1932)
14 4 17 1 14 17
7 2 8 1 ( , ) ( , )ReactionN O p N p O p
9 4 12 1 9 12
4 2 6 0 ( , ) ( , )ReactionBe C n Be n C n
7 1 4 4 7 4
3 1 2 2 ( , ) ( , )ReactionLi P He Li p He p
Example: Nuclear Reactions
pSiAl 30
14
27
13 SipAl 30
14
27
13 ),(
pAlpAl *27
13
27
13*27
13
27
13 ),( AlppAl
nHeB 4
2
9
4HenB 4
2
9
4 ) ,(
2
2
21
1
1
235
92 ZZnU A
Z
A
Z
ThnTh 233
90
232
90 ThnTh 233
90
232
90 ),(
Compound Nucleus Model
• Compound Nucleus Model (N. Bohr,1936):
1. Formation of compound nucleus in excited state.
• Its energy equal to kinetic energy of incident neutron + binding energy of the neutron in the compound nucleus.
• It stays in a quasi-stationary state relatively long (10-14 sec), compared to transit time of neutron thru the nucleus.
1. Excited nucleus decay by emitting particles from the nucleus
• Mostly radiative capture often with beta ray accompanied
• Sometime emits least bound nucleons.
n + X C* Y + y (= n, n’, γ, p, α, 2n, 3n, pn)
Neutron Interactions
• Two-step interaction
– Neutron and target coalesce to form a compound nucleus
– The compound decays
• Elastic scattering : kinetic energy conserved
• Inelastic scattering : kinetic energy is not conserved
• Radiative capture
• Fission
• Charged particle (proton, alpha) ejections
• Multiple neutrons
• Potential scattering
– No compound formation
– Billiard ball like collision
1 2
1 2
1
0
*1
0
1
1
0*
1 11
01 1
4 3
2 2
1 1
0
1 1
0
2
3
...
A
Z
A
Z
A
Z
A A
Z Z
A AA
Z ZZ
A
Z
A
Z
A
Z
n X
n X
X
X X n
n X X p X
X
n X
n X
1 1
0 0
A A
Z Zn X n X
Classification of Nuclear Reaction
• Nuclear Reaction
• Classification of reactions by the emitted Particles:
– Elastic scattering: (n,n)
– Inelastic scattering: (n,n')
– Capture reaction or Radiative capture reaction: (n,γ)
– Spallation reaction or fission: (n,f)
– Charged-particle reaction: (n,p) or (n,α)
– Neutron generation reaction: (n,2n) or (n,3n)
n + X Y + y or X(n, y)Y
where n = incident neutron
X = target nucleus
Y = product nucleus
y = emitted particle( n, n’,γ, p, α, 2n, 3n)
Nuclear Reaction Classification
• Classification of reaction by the incident energy:
– Thermal neutron energies ≈ (1/40) eV
– Epithermal neutron energies ≈ 1 eV
– Slow-neutron energies ≈ 1 keV
– Fast-neutron energies ≈ 0.1∼10 MeV
– Low-energy charged particles ≈ 0.1∼10 MeV
– High energies ≈ 10∼100 MeV
• Classification of reaction by the target nucleus:
– Light nuclei : A ≤ 25
– Medium-weight nuclei : 25< A <150
– Heavy nuclei : A ≥ 150
Types of Nuclear Reactions
Scattering
Reaction
Absorption
Reaction
① Elastic Scattering
② Inelastic Scattering
③ Capture Reaction
④ Charged Particle Emission
⑤ Fission
⑥ (n,2n), (n,3n) etc
X(a, a)X, Preservation of Kinetic energy
X(a, a')X*, excitation of the target nucleus
γ ray emitted
59Co(n, r)60Co
X(a, b)Y
BAnUA
Z
A
Z2
2
1
1 235
92
Nuclear
Reaction
pSiAl 30
14
27
13
Elastic Scattering Reaction
• Elastic Scattering Reaction
– No energy transferred into nucleus excitation
– Preservation of momentum and kinetic energy
– The target nucleus gains the amount of kinetic energy that
the neutron loses
xXXx
Absorption Reaction
• Radiative Capture : (n, γ)
• Charged Particle Emission
– (n, p), (n, d), (n, α), ….
– Nuclei with low atomic number
• Neutron Emission
– (n,2n), (n,3n)…..
– Fast neutron
• Nuclear Fission : (n, f)
– Nuclei with large atomic number
– 92U235 92U
238
*1 1
1
1
1
4
2
1
2
1 2
1 2
2
3
A A A
Z Z Z
A
Z
A
Z
A
Z
A
Z
A
Z
A A
Z Z
X n X X
X p
X d
X
X n
X n
X X n
Mass-Energy Balance
in Nuclear Reaction
• Q = Difference in the rest mass energies
(before reaction – after reaction)
• Threshold Energy
2
, , = x X y YQ E E mc
Q < 0 : Endothermic Reaction – Required Threshold Energy
Q > 0 : Exothermic Reaction
yYXx )()()()( ymYmxmXmm
yYQXx |||| QyYXx
Momentum Balance vmM
mVVmMmv
)(
Internal Energy Change EmM
MEVmMEE
2)(
2
1
Required Energy for Nuclear Reaction ? || QE
|| QM
mME
m
mM v
V
E
Neutron Cross Section
• The extent to which neutrons interact with nuclei is described in
terms of quantities known as cross sections.
• In order to describe the neutron balance, cross sections are crucial
information together with neutron flux.
• The multiplication of cross sections and fluxes are the reaction rate
per unit volume and per second.
• The measured cross section data have been evaluated and
collected in ENDF, JEF, JENDL, etc.
• In this section, we will discuss the followings:
– Microscopic Cross Section
– Microscopic Scattering Cross Section
– Microscopic Absorption Cross Section
– Resonance Absorption
– Macroscopic Cross Section
Microscopic Neutron Cross Section
• Microscopic cross section σ:
• Physical Meaning of Microscopic Cross section
– The probability of a particular reaction occurring between a neutron and a nuclide
– a measure of the relative occurrence probability of a given reaction
– the effective area of the particular nuclear reaction
The larger effective area, the greater the probability for reaction
• Measurement of Cross Section
Incident Neutron Target Nucleus
Incident Neutron
Detector
Target
Nucleus
Cross Section= Nuclear Reaction Rate
Intensity of Incident Neutron
Microscopic Neutron Cross Section
• Microscopic cross section units:
– cm2 (units of area)
– barns (1 barn = 10-24 cm2)
• Components of total microscopic cross section :
– Summation of Microscopic Scattering Cross section and
Microscopic Absorption Cross section
ast σσσ
αpfγiet σσσσσσσ
Microscopic Cross Section
• Microscopic Cross section :
1barn = 10-24 cm2
Target Nucleus
Incident Neutron
Total
fission
Scattering Scattering
Total
fission
U-235 U-238
Pu-239 Total
fission
Scattering
Why do we need to
slowing down neutron velocity?
• Fission Cross section of Fissionable Materials
(U235, Pu239)
– Fission cross section in thermal energy region is much larger than
Fission cross Section in fast energy region
Fast Neutron
Thermal Neutron
Microscopic Scattering Cross Section: σs
• Light nuclei:
– constant from low energy to MeV region
– fairly wide resonances in MeV region
– smooth function of energy above
MeV region
• Heavy nuclei:
– constant at low energy region
– sharp resonance in resonance region
– smooth function of energy above resonance region
• Heavy nuclei in slowing down neutrons at high energy:
– more effective than light nuclei because of inelastic scattering
– Threshold energy Eth > the energy of the first excited state ε1
H-1
Be-9 C-12
H-2
Na-23
1th εA
1AE
H-1
Be-9 C-12
H-2
Na-23
Microscopic Cross Section of Ag and Cd
Total XS ~ Absorption XS
.
Cd-48
Total, Absorption XS Ag-109
Total, Absorption XS
.
Microscopic
Elastic Scattering Cross Section: 𝜎𝑠
• Light nuclei:
– constant from low energy to MeV region
– fairly wide resonances in MeV region
– smooth function of energy above MeV region
• Heavy nuclei:
– constant at low energy region
– sharp resonance in resonance region
– constant between resonances
– smooth function of energy above resonance region
ies σσσ
Microscopic Scattering Cross section of H
Microscopic
Absorption Cross Section: 𝜎𝑎
• Microscopic absorption cross section
• Microscopic neutron producing cross section
Capture
Fission
Particle Emission
Neutron Alpha
γ
a γ f p α
f
p
α
σ σ σ σ σ
σ : n,γ cross section
σ : n,f cross section
σ : n,p cross section
σ : n,α cross section
2
3
(n, 2n) reaction
(n, 3n) reaction
( , n) reaction
n
n
n
σ
σ
σ
Typical Absorption Cross Section vs.
Neutron Energy
Resonance Absorption
• Resonance Absorption
– Very large resonance absorption at 1eV ~ 100 keV energy region
– Depends on nuclei and neutron energy
Resonance Absorption Cross section of U235
Resonace Region
Macroscopic Cross Section
• The chance for a reaction is given by the sum of all microscopic
cross sections in the material
Neutron
Macroscopic Cross Section
• Macroscopic cross section: Σ [cm-1]
≡ the probability of neutron interaction per unit track length with a
nucleus X
• Neutron mean free path: λ [cm] = 1/Σ
≡ the average distance that a neutron travels in the substance
before it experiences the nuclear reaction under consideration
≡ the reciprocal of the macroscopic cross section
X X X
X
X
1 1 2 2 i i i i
i
1 2 i i
i
Σ N σ for the individualisotope
N : the number of a nucleus X per cubic centimeter
σ : the microscopic cross section of a nucleus X
Σ N σ N σ N σ N σ for the mixture
Σ Σ Σ Σ
Example
• Find macroscopic thermal absorption cross section got iron, which
has a density of 7.86 g/cm3.
The microscopic cross section for absorption of iron is 2.56 barns
and the atomic weight is 55.847 g.
• Solution Calculate the atomic density of iron, Calculate the macroscopic cross section
A
23
3
22
3
ρNN =
M
g atoms7.86 6.022 10
cm mole =
g55.847
mole
atoms = 8.48 10
cm
-24 222
3
a= N
atoms 1 10 cm = 8.48 10 (2.56 barns)
cm 1 barn
= 0.217 cm
σa
Reaction Rates
• The neutron balance equation needs to be solved to predict the
neutron population throughout a reactor.
• The neutron balance equation requires expressions for the rates at
which various nuclear events will occur at any given location and
involving neutrons of any given energy.
• In this section, we will discuss;
– Neutron Flux
– Reaction Rates
– Neutron Moderation
Neutron Flux
• Neutron flux
– The number of neutrons of energy E, in the interval dE,
that penetrates a sphere of a 1-cm2 cross section, located at r,
per second.
Illustation of the flux, ( ,E), at r r
Neutron Flux
• Neutron flux
– The total path length covered by all neutrons in
one cubic centimeter (1-cm3) during one second (1-sec)
– Scalar sum of the contributions from all neutrons which will be moving in
all directions
– Or, can be considered to be comprised of many neutron beams
traveling in various directions
• In each beam, all neutrons move in a same direction
• No neutron – neutron interaction is assumed
2
3
= nv
where = neutron flux (neutrons/cm -sec)
n = neutron density (neutrons/cm )
v = neutron velocity (cm/sec)
1 2 3
2
2
= I + I + I +
where = neutron flux (neutrons/cm -sec)
I = neutron beam intensity (neutrons/cm -sec) = n vn
K
Reaction Rates
• Reaction rate
– The number of neutron-nucleus interactions taking place in a cubic
centimeter in one second
– Note
• The flux is the total path length of all the neutrons in
a cubic centimeter (1-cm3) in a second (1-sec)
• The macroscopic cross section is the probability of having an interaction per
centimeter path length of a neutron
• Therefore, the multiplication of those two is the total number of interactions
in that cubic centimeter in a second – reaction rate!!
3
2
1
where R = reaction rates (reactions/cm -sec)
= neutron flux (neutrons/cm -sec)
= macroscopic cross section (cm )
R
Reaction Rates
• Example
– A one cubic centimeter of a reactor with fission cross section of 0.1 cm-1.
– Thermal neutron flux is 1013 neutrons/cm2-sec.
– What is the fission reaction rate?
13 -1
2
12
3
neutrons = (1 x 10 ) x (0.1 cm )
cm -sec
fissions = 1 10
cm -sec
f fR
Reaction Rates
• Reactor power calculation
– The total number of fissions in a reactor core per second is the
multiplication of the average fission reaction rate per unit volume by the
total volume of the core.
– 3.12x1010 fissions release 1 watt-second of energy
– From flux to power
-6 -71 fission = 200 MeV, 1 MeV = 1.602 x 10 ergs, 1 erg = 1 x 10 watt-sec
10
-7 -6
1 erg 1 MeV 1 fission fissions1 watt =3.12x10
1 x 10 watt-sec 1.602x10 erg 200 MeV second
th f
10
2
th
-1
f
3
Σ VP =
fissions3.12x10
watt-sec
where P = power (watts)
thermal flux (neutrons/cm -sec)
Σ = fission cross section (cm )
V = volume of core (cm )
Neutron Moderation
rf : crow-flight distance from fast neutron birth to thermalization
rth : crow-flight distance from thermalization to absorption
Fast neutron
Fission
Thermal neutron
Thermal neutron absorption
2 2 2migration area : thM L
2r = 6f
r = 6th thL
Collision
Neutron Moderation
• Moderation, thermalization, or slowing down
– The process of reducing the energy of a neutron to
the thermal region by (in)elastic scattering
• Fission neutrons are born at an average energy level of 2 MeV
• Fission cross section is low at high energy (~MeV) and high at low energy
(< 1 eV)
• After a number of collisions with nuclei, the speed of a neutron is reduced to
have approximately the same average kinetic energy as the atoms of the
medium in which the neutron is undergoing elastic scattering => thermal
energy (0.025 eV at 20 °C)
• Thermal neutron: Maxwellian distribution
– Ideal moderating material (moderator)
• Large scattering cross section
• Small absorption cross section
• Large energy loss per collision
Neutron Moderation
• Energy loss in scattering
– Transfer of kinetic energy of neutron to a target nucleus
energy loss of neutron
E
Incident
neutron
Scattered neutron
Target
nucleus Recoiling nucleus
E
min
2
Grazing collision E = E
Head-on collision E =E = α E
A-1where α = , A = atomic mass number
A+1
1Average energy E = (1+ )E
2
1Average fractional energy loss (1- )
2
E
E
Neutron Moderation
• Average logarithmic energy decrement (ξ)
– Energy loss per scattering collision in logarithmic scale
• Definition
• A constant for each type of material
• Independent of the initial energy
– Average number of scattering collisions to thermalize
(from 2 MeV to 0.025 eV)
ii f
f
i
f
Eξ=ln(E )-ln(E )=ln
E
where E = average initial neutron energy before scattering collision
E = average final neutron energy after scattering collision
2( 1) 1 2ξ=1- ln( ) (for A > 10)
22 1
3
A A
A AA
62 10ln
0.025 18.2= (collisions)
ξ ξn
Neutron Moderation
• Macroscopic slowing down power (ξΣs)
– The measure of how rapidly a neutron will slow down in the material
– Still not sufficient to represent the effectiveness of moderator
– e.g. Boron has a good LED (ξ) and a good MSDP(ξΣs), but boron is not
a good moderator due to its high neutron absorptions
• Moderating ratio (ξΣs/Σa)
– The most complete measure of the effectiveness of a moderator
Neutron Moderation
• Summary of the parameters
Nucleus A α ΔE/E ξ n MSDP MR
Hydrogen 1 0 0.5 1.000 18
H2O 0.920 20 1.425 62
Deuterium 2 0.111 0.725
D2O 0.590 31 0.177 4830
Helium 4 0.360 0.32 0.427 42 9x10-6 51
Beryllium 9 0.640 0.209 86 0.154 126
Boron 10 0.669 0.165 0.171 105 0.092 0.00086
Carbon 12 0.716 0.14 0.158 114 0.083 216
Oxygen 16 0.779 0.120
Sodium 23 0.840 0.0825
Iron 56 0.931 0.0357
Uranium 238 0.983 0.008 0.00838 21718
Neutron Diffusion
• In order to design a nuclear reactor properly, we need to know how
the neutrons are distributed throughout the system.
• It is a reasonably good approximation that the neutrons undergo
diffusion in the reactor medium much like the diffusion of one gas in
another.
• This section will introduce the neutron diffusion equation and its
solution.
– Fick’s Law
– Neutron Diffusion Equation
– Boundary Conditions
– Diffusion Length
Fick’s Law
• Diffusion of Neutron
– Solute diffuse from higher concentration to lower concentration
• Rate of solute flow is proportional to negative of the gradient
• Can be used to approximate the behavior of neutron in reactor
– Neutron diffuses from higher flux region to lower flux region
– Limitations
• Near the boundary
• Near the material interface (strong variation of material properties)
• Near the isolated source
• Too strong absorbing medium
• Too strong anisotropic scattering medium
x
2
x
2
dJ = -D
dx
where J = the net number of neutrons passing per a second per cm
perpendicular to the x-direction (neutrons/cm sec)
D = diffusion coefficient (cm)
Neutron Diffusion Equation
• Neutron diffusion equation – Equation of continuity or neutron balance
4 3 2 1 rate of change in
rate of neutron rate of neutron rate of neutron number of neutrons = - -
production in V absorption in V leakage from Vin V
1V V
d nndV dV
dt t
2 fV V
sdV dV
3 aV
dV
4A V
dA dV J n J
f a
n
t
J
vn
Neutron Diffusion Equation
21
vf a D
t
D J Fick’s law
Divergence theorem
Neutron Diffusion Equation
• Steady state neutron diffusion equation
– Balance equation
• LHS : neutron loss (leakage and absorption)
• RHS : neutron production
– Multiplication factor
• k has been introduced to adjust ν factor in order to make
the loss and source balanced
• Rearranging the equation for k
• Shows the physical meaning of k as a neutron multiplication factor
2 1a fD
k
2
production
leakage + absorption
f
a
dVk
D dV
Boundary Conditions
• Boundary conditions
– Must be physical (flux must be real, non-negative, and finite)
– Mathematically an appropriate set of BCs must be set
– ϕ or dϕ/dn or a linear combination of those two be specified
• At the surface
– Flux zero at the extrapolated distance
• From transport theory
• Interface conditions
– Continuity of flux and current
1 1where d = extrapolation distance
n d
d
d
0.71 2.13 ( )3
trtrd D D
A B A Bn nJ J
Neutron Diffusion Equation
• Example: Solution of diffusion equation with a point source
– Isotropic point source with strength, S (neutrons/sec) at r=0
– Diffusion equation
– General solution
– Boundary conditions
• B.C. #1: finite flux as r→ ∞, then B=0
• B.C. #2: source condition
2
2
2 2
10 for 0
where diffusion area (cm )a
rL
DL
/ /
( )r L r Le e
r A Br r
/2 2
0 0lim 4 ( ) lim 4 4 , ( )
4 4
r L
x x
d S SeS r J r r D DA A r
dr D Dr
Diffusion Length
• Diffusion length
– Consider a mono-energetic neutron in an infinite homogeneous
moderator
• The neutron moves in complicated, zig-zag paths
• Will eventually be absorbed
• The probability that the neutron is absorbed between r and r+dr
• Average of the square of the “crow-flight” distance from the source to the
point the neutron is absorbed
• Diffusion length
2( )4( ) a r r drdn
p r drS S
2 2 2
0( ) 6r r p r dr L
22
6a
D rL
Neutron Life Cycle
• In this section, we will discuss the followings;
– Multiplication Factor
– Six Factor Formula
– Neutron Life Cycle
Multiplication Factor
• Multiplication factor (k)
– Measure of change in fission neutron population from any one
generation to subsequent generation
– Effective multiplication factor in a finite reactor (keff)
– relationship of multiplication factor and reactor power
• keff < 1 : sub-criticality, power decrease
• keff = 1 : criticality, constant power
• keff > 1 : super-criticality, power increase
number of fissions in any one generation
number of fissions in the immediately preceding generationk
: six factor formula
where : infinite multiplication factor
eff f t f tk k L L pfP P
k pf
Six Factor Formula
• Six factors
– Thermal utilization factor f
– Neutron production factor η
• The average number of neutrons produced per thermal neutron absorbed in
the fuel
the probability that a neutron will be absorbed in fuel
the probability that the neutron will be absorbed in the core
(fuel)
(fuel) (other material)
a
a a
f
for all the fuel material(fuel)
i fi
i
ai
i
Six Factor Formula
• Six factors
– Fast fission factor ε
– Resonance escape probability p
• Neutron slowdown probability to thermal energy (< 1eV) without resonance
capture
0
0
the number of neutrons produced by fissions at all energies
the number of neutrons produced by thermal fission
( ) ( ) ( )
( ) ( ) ( )thermal
f
E
f
E E E dE
E E E dE
Six Factor Formula
• Six factors
– Fast non-leakage factor Pf
• Fraction of non-leaked neutrons from the system during the slowing-down
from fission energy to thermal energy
– Thermal non-leakage factor Pt
• Fraction of the thermal neutrons that do not leak out of the system during
thermal diffusion
2
2
2
2
1exp ,
1
where = Fermi age(cm )
B = geometrical buckling of the system
f fP B or PB
2 2
2 2
1
1
where L = thermal diffusion area (cm )
tPL B
Neutron Life Cycle
107
106
105
104
103
102
101
100
10-1
10-2
10-3
10-4
U235 Fast Fission
No
Fast Neutron Leakage
(1-Pf)No
U238 Resonance Absorption
(1-p)PfNo
Absorption other materials
(Not in Fuel)
(1-f)pPtPfNo
Thermal Neutron Leakage p(1-Pt)PfNo
PfNo
pPfNo
fpPtPfNo
pPtPfNo
U235 Thermal Fission
fpPtPfNo
Fast Neutron
No
Neutron Energy (eV)
Reactor Multiplication Factor k = No
fpPtPfNo
Neutron Life Cycle
• Thermal-hydraulic feedback in terms of four factors
– If a moderator temperature increases (Mod. Den. decreases)
• f increase
• p, Pf, Pt decreases
• keff decreases (in under-moderated region)
– If a fuel temperature increases
• p decreases ⇒ keff decreases
• Heterogeneous reactor in terms of four factors
– Thermal non-leakage factor Lt • Fraction of the thermal neutrons that do not leak out of the system during thermal
diffusion
homo
homo
homo
homo (dominant effect)
hetero
hetero
hetero
hetero
f f
p p
, ,homoheterok k
Critical Equation
• A critical equation determines the condition under which a given
bare reactor is critical.
• The critical equation can be used to find the critical reactor size
when its composition and the amount of fuel are given and/or to find
critical mass when its size is given.
• This is a simplified approach using analytic expression of critical
equation. In real world design, the search for critical mass / critical
size can be done using reactor core design codes.
Critical Equations
• One-group critical equation
– One-group steady state diffusion equation for an infinite homogeneous
reactor
– Rearranging
– Define buckling (material buckling)
– Then one-group reactor equation
– One-group critical equation
– Non-leakage probability
2
a aD k
2
2
10
k
L
2
2
1kB
L
2 2 0B
2 21
1
k
L B
2 2
1
1P
L B
Critical Equations
• Critical reactor size
– Space dependent equation
• Solution for spherical geometry
• Boundary condition then
• Fundamental solution and
– The relationship between material buckling and geometrical buckling
• B2 (material) < B2 (geometrical) : sub-criticality
• B2 (material) = B2 (geometrical) : criticality
• B2 (material) > B2 (geometrical) : super-criticality
– When the composition in a reactor is given, the critical size of the
reactor can be determined
( ) 0r R
2 2 0B sin( )
( )Br
r Ar
2
2 , 1,2,3,n
B nR
K
2
2BR
sin( )
( )
r
Rr Ar
Geometrical
buckling
Reactor Dynamics
• Nuclear reactors are not always in critical condition at constant
power.
• It is necessary for a reactor to be supercritical to start it up or raise
its power level, whereas it must be subcritical to shut it down or
reduce power.
• This section will study the behavior of the neutron population in a
noncritical reactor;
– Reactivity
– Reactivity Coefficients
– Doppler Feedback
– Prompt and Delayed Neutrons
– Reactor Kinetics
– Neutron Poisons
– Reactor Reactivity Control
Reactivity
• The number of neutrons in the core after n generations
• Reactivity
– Fractional change in neutron population per generation
– Reactivity vs. core power
– Units
0
n
n effN N k
1eff
eff
k
k
0 1.0 reactor: critical
0 1.0 reactor power : supercritical
0 1.0 reactor power subcritical
= reactor: prompt critical
k
k
k
eff
eff
eff
:
[ ] 100 [ ]
Δ Δ
dollars or centsk k
ρ ρk k
Δk Δk Δk Δk1 pcm = 0.00001 , 1 % = 0.01 = 1000 pcm, 1 mk = 0.001 = 100 pcm
k k k k
Reactivity Coefficients
• Multiplication factor
– Many parameters ~ function(T)
– Change in T ⇒ change in k ⇒ reactivity change
– Important bearing on the operation and safety of reactors
• Temperature coefficients
– The extent of reactivity change due to temperature change
– “temperature coefficients of reactivity”
f tk fp P P
TT
d
d
1 11
k
k k
T 2
1 1ln
T T
dk dk dk
k d k d dT
Reactivity Coefficients
• Effects of temperature reactivity coefficients
– When αT is positive (>0)
• Increase in T increase in k increase in power level increase in T
…. power keep increasing
• Decrease in T …. reactor shutdown
• Inherently unstable reactor
– When αT is negative (<0)
• Increase in T decrease in k decrease in power level decrease in T
…. back to original state
• Inherently stable reactor
• Types of temperature coefficients
– Fuel temperature coefficients
– Moderator temperature coefficients
T T TM F
Reactivity Coefficients
• Fuel Temperature Coefficient
– Provides prompt feedback through Doppler broadening
– Negative coefficient inherent reactor safety
• P increase fuel temp increase Doppler broadening (more neutron capture by U-238) negative reactivity feedback P decrease : stable
• Moderator Temperature Coefficient
– Positive(+) in an over-moderated core, negative(-) in a under-moderated core
– Provides delayed feedback due to the time for heat to be transferred to moderator
– Negative coefficient inherent reactor safety
• P increase Tm increase mod. Density decrease less moderation, p decrease (more resonance capture in U-238), f increase negative feedback P decrease : stable
TTf
f
TTm
m
Reactivity Coefficients
• Pressure Coefficient
– Change in reactivity per unit change in pressure
– Pressure increase mod. Den. Increase more moderation
positive effect P increase
– The magnitude of the coefficient is small in PWR
– More important in BWR due to larger density change associated with
boiling of coolant or moderator
• Void Coefficient
– Change in reactivity per percent change in void volume
– BWR: P increase more void formation in moderator replace the
volume of moderator less moderation negative feedback in under-
moderated core P decrease
– positive in sodium cooled fast reactor
• Power Coefficient of reactivity
)())(( ,P
T
P
T
TdP
d j
j
jT
j
j j
P
Doppler Feedback
• Doppler broadening
– Mechanism
• Stationary nuclei absorb only neutron of energy E0
• If the nucleus is moving away from the neutron, the velocity (and energy) of
the neutron must be greater than E0 to undergo resonance absorption
• Likewise, if the nucleus is moving toward the neutron, the neutron needs
less energy than E0 to be absorbed
• Raising the temperature causes the nuclei to vibrate more rapidly within their
lattice structures, effectively broadening the energy range of neutrons that
may be resonantly absorbed in the fuel
Ca
ptu
re C
ross S
ection
0oK
20oC
1000oC
E0
Doppler Feedback
• Doppler Effect to the reactivity
– As temperature rises
• Resonance broadens (Doppler broadening)
• Absorption cross section goes down
• Neutron flux in resonances increase with temperature (less energy self-
shielding)
• More neutron capture in the resonances
• Negative reactivity effect
– The most important mechanism of inherent reactor safety
– The most prompt effect of the power level change
– In LWRs, U-238 is principal contributor over the core life,
Pu-240 becomes important later in core life.
Contribution is small for U-235 and Pu-239
Prompt and Delayed Neutrons
• Prompt neutrons
– The great majority (> 99%) of the neutrons produced in fission are
released within about 10-13 seconds of the actual fission event
– Prompt neutron generation time
• LWR ~ 10-4 seconds
– Thermalization time ~10-6 seconds
– Thermal diffusion time ~ 10-4 seconds
– Fission to prompt neutron production ~ 10-13 seconds
• Fast reactor ~ 10-6 seconds
– Prompt neutron spectrum
1.036
0
( ) 0.453 sinh 2.29
0.73
( ) 1.98
E
p
E e E
E MeV
E E E dE MeV
Prompt and Delayed Neutrons
• Delayed neutrons
– Are emitted immediately following the first beta decay of a neutron-rich
fission fragment (delayed neutron precursor)
– Characteristic half-life determined by that of the precursor of the actual
neutron emitter
– Average delayed neutron generation time (U-235) ~12.5 seconds
87 87 86
35 36 3655.9sec instantaneous
n
stableBr Kr Kr
group Half-life (sec) Decay
constant(Sec-1)
Energy
KeV yield fraction
1 55.72 0.0124 250 0.00052 0.000215
2 22.72 0.0305 560 0.00346 0.001424
3 6.22 0.111 405 0.00310 0.001274
4 2.30 0.301 450 0.00624 0.002568
5 0.610 1.14 - 0.00182 0.000748
6 0.230 3.01 - 0.00066 0.000273
total 0.0158 0.0065
Delayed neutron data for thermal fission of U-235
Nucleus fraction
U-233 0.0026
U-235 0.0065
U-238 0.0148
Pu239 0.0021
Delayed neutron fraction β
Prompt and Delayed Neutrons
• Average neutron generation time (Λ)
– Example
• Given that a prompt neutron generation time is 5x10-5 seconds and a
delayed neutron generation time is 12.5 seconds. Calculate the average
generation time (β = 0.0065)
– With delayed neutrons, the reactor power level control becomes easier
(1 )average prompt delayed
55 10 (1-0.0065)+ 12.5 (0.0065)
= 0.0813 seconds
average
Reactor Kinetics
• Prompt neutron only
– Number of fissions
• The absorption of a neutron from one generation leads to, ℓp sec later, k∞
neutrons in the next generation
– Number of fission after time t when k∞ ≠ 1
( ) ( )
( )( ) ( )
( ) 1( )
F p F
FF p F p
FF
p
N t l k N t
dN tN t l N t l
dt
dN t kN t
dt l
/( ) (0)
; Reactor Period1
t T
F F
p
N t N e
lT
k
4
F F0
for 1.001
1 100.1sec
0.001
in 1 sec, N /N 22,000
k
T
Reactor Kinetics
• With delayed neutrons in an infinite homo medium
– Thermal flux and delayed neutron precursor concentration concentration
– Trial solution
– Graphical solution
– Reactor period
(1 ) aT Tp T
a
d k p Cl
dt
a TkdCC
dt p
0
tC C etAe
1 1
p
p p
l
l l
1 2 1
1 2
t t t
T Ae A e e
1
1T
t
TT e
Reactor Kinetics
• With delayed neutrons
– Time delay due to delayed neutrons
– For large negative reactivity,
only longest-lived precursors remain
shortly after insertion
1
180secondsT
Neutron Poisons
• Fission product poisoning
– FP absorbs neutrons to some extent
– FP accumulates
– Absorption cross section1/v behavior : important for thermal reactor
– FP poison removes neutrons from the reactor and therefore it will affect
the thermal utilization factor and keff.
• Xe-135
– Strong neutron absorber, ~2.6x106 b
– Formed directly as a fission product
– Radioactive decay of tellurium-135
Ba(stable) Cs Xe I Te 135β
yr102.3
135β
9.2hr
135β
6.57hr
135β
19sec
135
6
Idt
dIITfI
TaXXITfX XXIdt
dX
Neutron Poisons
• Xe-135
– Reaches equilibrium state after ~40 hours of power operation
– Equilibrium concentrations
• Increases as flux level increases but there is a limit (0.052dk/k for U-235)
– Xenon after shutdown
• Removal: no absorption, radioactive decay
• Production: no fission yield, radioactive decay of I Xe poisoning effect will change!!
• Peak after ~11 hours
– Reactor dead time
• Reactivity worth of Xe > control rods & SB
• Cannot restart the reactor !!
– Step change of power
• Cause Xe concentration change
• Xe instability
I
TfII
TaXX
ITfX IX
Neutron Poisons
• Sm-149: Strong neutron absorber ~40,000 barns
– Equilibrium concentration
• Reactivity worth ~0.005 dk/k • Independent of flux level
– Samarium after shutdown • No absorption, no radioactive decay
• but production from Pm decay
• The amount of Sm will increase to
~0.04 dk/k for high flux
• In the beginning of next cycle, Sm will burn down back to equilibrium level
Sm(stable) Pm Nd 149β
2.212d
149β
1.73hr
149
Pdt
dPPTfP TaPP PP
dt
dS
P
TfPP
aS
fPS
Reactor Reactivity Control
• Control rods
– Rods made of neutron-absorbing materials ( Ag, In, Cd, B, Hf ) which can be moved into or out of the reactor core
– Types
• Regulating rod ⇒ power level / power distribution control
• Shutdown rod (safety rod) ⇒ Reactor shutdown (scram, trip)
– (integral/differential) Rod worth ρ = ρout - ρin
• varies depending on the location in the core
• Highest worth when inserted at the highest flux location
– Can compensate for rapid reactivity change
– Increase peak-to-average power density
– Usually the control rods alone is not enough to compensate for the
excess reactivity at the beginning of cycle
– Must be able to satisfy shutdown margin with N-1 control rods
Reactor Reactivity Control
• Burnable poison rods
– High neutron absorption cross section
– Converted into a material of relatively low absorption cross section as a result of neutron absorption
– Compensate for the excess reactivity of the fuel in the beginning of cycle
– Excess reactivity control with soluble boron alone requires too high boron concentration positive MTC
– No adverse effect to moderator temperature coefficients
– Can be used for shape flux profiles (local and global)
– Residuals of burnable absorbers can degrade neutron economy
– Gd, Er, B
• Non-burnable poison
– Relatively constant neutron absorption characteristics over core life
– The absorption of a neutron by one isotope in the material produces another isotope also with high absorption cross section
– Power shaping, power peaking reduction near moderator region
– Hf
Reactor Reactivity Control
• Chemical shim
– Soluble boron absorber, H2BO3, in the moderator (ppm)
– Compensate for the fuel burnup, poison buildup, temperature defects
– Spatially uniform effect
– Possible to increase or decrease amount of poison in the core during
reactor operation
– Adverse effect to moderator temperature coefficients when too high
concentration in the moderator (more absorption than moderate)
less negative MTC
– Soluble boron concentration must be
adjusted to compensate for the reduced
excess reactivity as the core burns
Reactor Operation
• Basic concepts related to the nuclear reactor operations will be
introduced.
• Startup of reactor
– Source neutrons from irradiated fuels and/or installed source
– Subcritical multiplication is used to increase power level in the source
range
• Estimated critical position
– The position of control rods that can result in criticality of reactor
– Take into accounts all of the changes in conditions: time since shutdown,
temperature, pressure, fuel burnup, samarium and xenon poisoning
Reactor Operation
• Core power distribution
– To achieve larger power output while satisfying minimum DNBR
• Need to flatten the power across the assemblies
• Use reflectors, enrichment zoning, burnable poisons
– To lower the radiation damage to reactor vessel
• Low leakage loading pattern (L3P)
• Low-low leakage loading pattern(L4P)
– Power tilt
• A core power distribution problem
• Non-symmetrical variation of core power in one quadrant of the core relative
to the others
• Shutdown margin
– The reactivity required to make a reactor subcritical from its present
condition assuming all control rods fully inserted except for the single
rod with the highest integral worth, which is assumed fully withdrawn
Reactor Operation
• Temperature variations
– Temperature change of the reactor has a significant effect to the
reactivity of the core
– Power (temperature) defect at startup
• Pressure
– Pressure affects the density of moderator and thus reactivity
– The effects are more noticeable at BWR
• Power level
– Once the power level is increased over the point of adding heat, then it
affects the reactivity through temperature variations
• Flow
– For BWR, increasing the flow rate decreases the fraction of steam voids
in the coolant and results in a positive reactivity
Reactor Operation
• Core burnup
– As a reactor is operated, fissile atoms of fuel are consumed
– For PWR, chemical shim concentration must be reduced to compensate
for the negative reactivity effect
– For BWR, control rods must be withdrawn
– As burnup increases, the delayed neutron fraction decreases
• Shutdown
– A reactor is subcritical and sufficient shutdown reactivity exists, no
gaining of criticality
– Following a large negative reactivity insertion, power level undergoes a
rapid drop (prompt drop), then the final rate of decrease will be
determined by the decay of the delayed neutron precursors
Reactor Operation
• Decay heat
– About 7 % of the 200 MeV produced by an average fission is released
at some time after the instant fission, from decay of fission products
– After a reactor shutdown from full power operation, the initial decay heat
is 5 ~ 6 % of the thermal rating of the reactor
– The decay heat generation rate diminishes to less than 1 %,
1 hour after shutdown
– Continued removal of heat is required for an appreciable time after
shutdown
Thank you
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